We study an online market-making problem in which a learner sequentially posts bid and ask prices for a single asset while interacting with traders holding private valuations. Unlike existing online learning formulations that assume fully censored feedback, we introduce an action-dependent feedback model inspired by real limit order books: when a trade occurs, the trader’s valuation remains hidden, whereas when no trade occurs, informative feedback about supply and demand is revealed. We show that this additional information fundamentally changes the learnability of the problem. In the stochastic setting with i.i.d. market prices, we propose an elimination-based algorithm that achieves $\widetilde O(\sqrt{T})$ regret with high probability, without requiring any smoothness assumptions on the distribution of trader valuations. We then extend this result to a broad class of mean-reverting price processes by considering both local, autoregressive dynamics and a weaker global drift condition based on cumulative deviations from the mean. Under either assumption, we establish high-probability $\widetilde O(\sqrt{T})$ regret bounds, relying on a new concentration inequality of independent interest. Finally, in the adversarial setting with oblivious prices, we design an explore-then-perturb algorithm that guarantees $\tilde O(T^{2/3})$ regret in expectation. Our results quantify the value of observing the order book in online market making and demonstrate that even limited, action-dependent feedback can substantially improve regret guarantees compared to standard bandit feedback models.

Abstract Many real-world machine learning applications are characterized by a huge number of features, leading to computational and memory issues, as well as the risk of overfitting. Ideally, only relevant and non-redundant features should be considered to preserve the complete information of the data and limit the dimensionality. Dimensionality reduction and feature selection are common preprocessing techniques addressing the challenge of efficiently dealing with high-dimensional data. Dimensionality reduction methods control the number of features in a dataset while minimizing information loss. Feature selection aims to identify the most relevant features for a task, discarding the less informative ones. Previous works have proposed approaches that aggregate features depending on their correlation without discarding any of them and preserving their interpretability through aggregation with the mean. A limitation of these works is the assumption of linearity in the relationship between features and targets. In this paper, we relax this assumption in two ways. First, we propose a bias-variance analysis for general regression models with additive Gaussian noise, leading to a first dimensionality reduction algorithm (NonLinCFA). Then, we extend the approach assuming that a generalized (non-)linear model regulates the data generation process. A deviance analysis leads to a second dimensionality reduction algorithm (GenLinCFA), applicable to larger classes of regression problems and in classification. In both cases, the main focus is to preserve the interpretability of the reduced features through the aggregation with the mean of groups of original features. Finally, we test the algorithms on synthetic and real-world datasets, performing regression and classification and showing competitive performance.

Abstract In reinforcement learning, the performance of an algorithm is typically evaluated along two dimensions: computational and statistical complexity. While theoretical researchers often prioritize statistical efficiency—minimizing the number of samples needed to reach the desired accuracy—practitioners focus mainly on reducing computational costs, such as training time and resource consumption. Bridging these two perspectives requires algorithms able to deliver strong statistical guarantees while remaining computationally efficient in practice. In this paper, we introduce , a meta-algorithm designed to enhance state-of-the-art RL algorithms by improving their computational efficiency while maintaining competitive sample efficiency. is based on the novel notion of W-step Markov decision process (MDP), where, instead of performing a single action and transitioning to the next state, the agent executes a sequence of W actions before observing the resulting state and collecting the discounted W -step cumulative reward. First, we provide a theoretical analysis of the suboptimality introduced in the optimal policy performance when planning in a W -MDP, highlighting the impact of the environment stochasticity. Second, we apply to GPOMDP, a well-known policy gradient method, and theoretically investigate the advantages of learning in the W -MDP in terms of variance reduction and improved sample complexity. Finally, empirical evaluations confirm that reduces computational costs while preserving sample efficiency.

We study goal-conditioned Hierarchical Reinforcement Learning (HRL), where a high-level agent instructs sub-goals to a low-level agent. Under the assumption of a sparse reward function and known hierarchical decomposition, we propose a new algorithm to learn optimal hierarchical policies. Our algorithm takes a low-level policy as input and is flexible enough to work with a wide range of low-level policies. We show that when the algorithm that computes the low-level policy is optimistic and provably efficient, our HRL algorithm enjoys a regret bound which represents a significant improvement compared to previous results for HRL. Importantly, our regret upper bound highlights key characteristics of the hierarchical decomposition that guarantee that our hierarchical algorithm is more efficient than the best monolithic approach. We support our theoretical findings with experiments that underscore that our method consistently outperforms algorithms that ignore the hierarchical structure.

We tackle average-reward infinite-horizon POMDPs with an unknown transition model but a known observation model, a setting that has been previously addressed in two limiting ways: (i) frequentist methods relying on suboptimal stochastic policies having a minimum probability of choosing each action, and (ii) Bayesian approaches employing the optimal policy class but requiring strong assumptions about the consistency of employed estimators. Our work removes these limitations by proving convenient estimation guarantees for the transition model and introducing an optimistic algorithm that leverages the optimal class of deterministic belief-based policies. We introduce modifications to existing estimation techniques providing theoretical guarantees separately for each estimated action transition matrix. Unlike existing estimation methods that are unable to use samples from different policies, we present a novel and simple estimator that overcomes this barrier. This new data-efficient technique, combined with the proposed $\textit{Action-wise OAS-UCRL}$ algorithm and a tighter theoretical analysis, leads to the first approach enjoying a regret guarantee of order $\mathcal{O}(\sqrt{T \log T})$ when compared against the optimal policy, thus improving over state of the art techniques. Finally, theoretical results are validated through numerical simulations showing the efficacy of our method against baseline methods.

Recent advances in Reinforcement Learning (RL) largely benefit from the inclusion of Deep Neural Networks, boosting the number of novel approaches proposed in the field of Deep Reinforcement Learning (DRL). These techniques demonstrate the ability to tackle complex games such as Atari, Go, and other real-world applications, including financial trading. Nevertheless, a significant challenge emerges from the lack of interpretability, particularly when attempting to comprehend the underlying patterns learned, the relative importance of the state features, and how they are integrated to generate the policy’s output. For this reason, in mission-critical and real-world settings, it is often preferred to deploy a simpler and more interpretable algorithm, although at the cost of performance. In this paper, we propose a novel algorithm, supported by theoretical guarantees, that can extract an interpretable policy (e.g., a linear policy) without disregarding the peculiarities of expert behavior. This result is obtained by considering the advantage function, which includes information about why an action is superior to the others. In contrast to previous works, our approach enables the training of an interpretable policy using previously collected experience. The proposed algorithm is empirically evaluated on classic control environments and on a financial trading scenario, demonstrating its ability to extract meaningful information from complex expert policies.

Parallel data collection has redefined Reinforcement Learning (RL), unlocking unprecedented efficiency and powering breakthroughs in large-scale real-world applications. In this paradigm, $N$ identical agents operate in $N$ replicas of an environment simulator, accelerating data collection by a factor of $N$. A critical question arises: Does specializing the policies of the parallel agents hold the key to surpass the $N$ factor acceleration? In this paper, we introduce a novel learning framework that maximizes the entropy of collected data in a parallel setting. Our approach carefully balances the entropy of individual agents with inter-agent diversity, effectively minimizing redundancies. The latter idea is implemented with a centralized policy gradient method, which shows promise when evaluated empirically against systems of identical agents, as well as synergy with batch RL techniques that can exploit data diversity. Finally, we provide an original concentration analysis that shows faster rates for specialized parallel sampling distributions, which supports our methodology and may be of independent interest.

The increasing integration of renewable energy sources (RESs) is transforming traditional power grid networks, which require new approaches for managing decentralized energy production and consumption. Microgrids (MGs) provide a promising solution by enabling localized control over energy generation, storage, and distribution. This paper presents a novel reinforcement learning (RL)-based methodology for optimizing microgrid energy management. Specifically, we propose an RL agent that learns optimal energy trading and storage policies by leveraging historical data on energy production, consumption, and market prices. A digital twin (DT) is used to simulate the energy storage system dynamics, incorporating degradation factors to ensure a realistic emulation of the analysed setting. Our approach is validated through an experimental campaign using real-world data from a power grid located in the Italian territory. The results indicate that the proposed RL-based strategy outperforms rule-based methods and existing RL benchmarks, offering a robust solution for intelligent microgrid management.

We address the learning problem in the context of infinite-horizon average-reward POMDPs. Traditionally, this problem has been approached using $\textit{Spectral Decomposition}$ (SD) methods applied to samples collected under non-adaptive policies, such as uniform or round-robin policies. Recently, SD techniques have been extended to accommodate a restricted class of adaptive policies such as $\textit{memoryless policies}$. However, the use of adaptive policies has introduced challenges related to data inefficiency, as SD methods typically require all samples to be drawn from a single policy. In this work, we propose $\texttt{Mixed Spectral Estimation}$, which generalizes spectral estimation techniques to support a broader class of $\textit{belief-based policies}$. We solve the open question of whether spectral methods can be applied to samples collected from multiple policies, and we provide finite-sample guarantees for our approach under standard observability and ergodicity assumptions. Building on this data-efficient estimation method, we introduce the $\texttt{Mixed Spectral UCRL}$ algorithm. Through a refined theoretical analysis, we demonstrate that it achieves a regret bound of $\widetilde{\mathcal{O}}(\sqrt{T})$ when compared to the optimal policy, without requiring full knowledge of either the transition or the observation model. Finally, we present numerical simulations that validate the theoretical analysis of both the proposed estimation procedure and the $\texttt{Mixed Spectral UCRL}$ algorithm.

In reinforcement learning, we typically refer to *unsupervised* pre-training when we aim to pre-train a policy without a priori access to the task specification, i.e., rewards, to be later employed for efficient learning of downstream tasks. In single-agent settings, the problem has been extensively studied and mostly understood. A popular approach casts the unsupervised objective as maximizing the *entropy* of the state distribution induced by the agent's policy, from which principles and methods follow. In contrast, little is known about state entropy maximization in multi-agent settings, which are ubiquitous in the real world. What are the pros and cons of alternative problem formulations in this setting? How hard is the problem in theory, how can we solve it in practice? In this paper, we address these questions by first characterizing those alternative formulations and highlighting how the problem, even when tractable in theory, is non-trivial in practice. Then, we present a scalable, decentralized, trust-region policy search algorithm to address the problem in practical settings. Finally, we provide numerical validations to both corroborate the theoretical findings and pave the way for unsupervised multi-agent reinforcement learning via state entropy maximization in challenging domains, showing that optimizing for a specific objective, namely *mixture entropy*, provides an excellent trade-off between tractability and performances.

In this work, we study the statistical properties of the off-policy estimation problem, i.e., estimating expectations under a target policy using samples collected from a different policy. We begin by presenting a novel minimax concentration lower bound that highlights the fundamental limits of off-policy estimation. We then analyze two well-known importance weighting (IW) techniques: vanilla IW and self-normalized importance weighting (SN). For both methods, we derive concentration and anti-concentration results, showing that their concentration rates are provably suboptimal compared to our lower bound. Observing that this undesired behavior arises from the heavy-tailed nature of the IW and SN estimators, we propose a new class of parametric estimators based on a transformation using the power mean (PM), which is no longer heavy-tailed. We study the theoretical properties of the PM estimator in terms of bias and variance. We show that, with suitable (possibly data-driven) tuning of its parameters, the PM estimator satisfies two key properties under certain conditions: ( i ) it achieves a subgaussian concentration rate that matches our lower bound and ( ii ) it maintains differentiability with respect to the target policy. Finally, we validate our approach through numerical simulations on both synthetic datasets and contextual bandits, comparing it against standard off-policy evaluation and learning baselines. 1

In several real-world sequential decision problems, at every step, the learner is required to select different actions. Every action affects a specific part of the system and generates an observable intermediate effect. In this paper, we introduce the Factored-Reward Bandits (FRBs), a novel setting able to effectively capture and exploit the structure of this class of scenarios, where the reward is computed as the product of the action intermediate observations. We characterize the statistical complexity of the learning problem in the FRBs, by deriving worst-case and asymptotic instance-dependent regret lower bounds. Then, we devise and analyze two regret minimization algorithms. The former, F-UCB, is an anytime optimistic approach matching the worst-case lower bound (up to logarithmic factors) but fails to perform optimally from the instance-dependent perspective. The latter, F-Track, is a bound-tracking approach, that enjoys optimal asymptotic instance-dependent regret guarantees. Finally, we study the problem of performing best arm identification in this setting. We derive an error probability lower bound, and we develop F-SR, a nearly optimal rejection-based algorithm for identifying the best action vector, given a time budget.2

In this paper, we investigate the optimal online configuration of episodic Markov decision processes when the space of the possible configurations is continuous. Specifically, we study the interaction between a learner (referred to as the configurator) and an agent with a fixed, unknown policy, when the learner aims to minimize her losses by choosing transition functions in online fashion. The losses may be unrelated to the agent's rewards. This problem applies to many real-world scenarios where the learner seeks to manipulate the Markov decision process to her advantage. We study both deterministic and stochastic settings, where the losses are either fixed or sampled from an unknown probability distribution. We design two algorithms whose peculiarity is to rely on occupancy measures to explore with optimism the continuous space of transition functions, achieving constant regret in deterministic settings and sublinear regret in stochastic settings, respectively. Moreover, we prove that the regret bound is tight with respect to any constant factor in deterministic settings. Finally, we compare the empiric performance of our algorithms with a baseline in synthetic experiments.

Approximate value iteration (AVI) is a family of algorithms for reinforcement learning (RL) that aims to obtain an approximation of the optimal value function. Generally, AVI algorithms implement an iterated procedure where each step consists of (i) an application of the Bellman operator and (ii) a projection step into a considered function space. Notoriously, the Bellman operator leverages transition samples, which strongly determine its behavior, as uninformative samples can result in negligible updates or long detours, whose detrimental effects are further exacerbated by the computationally intensive projection step. To address these issues, we propose a novel alternative approach based on learning an approximate version of the Bellman operator rather than estimating it through samples as in AVI approaches. This way, we are able to (i) generalize across transition samples and (ii) avoid the computationally intensive projection step. For this reason, we call our novel operator projected Bellman operator (PBO). We formulate an optimization problem to learn PBO for generic sequential decision-making problems, and we theoretically analyze its properties in two representative classes of RL problems. Furthermore, we theoretically study our approach under the lens of AVI and devise algorithmic implementations to learn PBO in offline and online settings by leveraging neural network parameterizations. Finally, we empirically showcase the benefits of PBO w.r.t. the regular Bellman operator on several RL problems.

Autoregressive processes naturally arise in a large variety of real-world scenarios, including stock markets, sales forecasting, weather prediction, advertising, and pricing. When facing a sequential decision-making problem in such a context, the temporal dependence between consecutive observations should be properly accounted for guaranteeing convergence to the optimal policy. In this work, we propose a novel online learning setting, namely, Autoregressive Bandits (ARBs), in which the observed reward is governed by an autoregressive process of order $k$, whose parameters depend on the chosen action. We show that, under mild assumptions on the reward process, the optimal policy can be conveniently computed. Then, we devise a new optimistic regret minimization algorithm, namely, AutoRegressive Upper Confidence Bound (AR-UCB), that suffers sublinear regret of order $\tilde{O} ( \frac{(k+1)^{3/2}\sqrt{nT}}{(1-\Gamma)^2} )$, where $T$ is the optimization horizon, $n$ is the number of actions, and $\Gamma < 1$ is a stability index of the process. Finally, we empirically validate our algorithm, illustrating its advantages w.r.t. bandit baselines and its robustness to misspecification of key parameters.

We consider the problem of learning an $\varepsilon$-optimal policy in a general class of continuous-space Markov decision processes (MDPs) having smooth Bellman operators. Given access to a generative model, we achieve rate-optimal sample complexity by performing a simple, \emph{perturbed} version of least-squares value iteration with orthogonal trigonometric polynomials as features. Key to our solution is a novel projection technique based on ideas from harmonic analysis. Our $\widetilde{O}(\epsilon^{-2-d/(\nu+1)})$ sample complexity, where $d$ is the dimension of the state-action space and $\nu$ the order of smoothness, recovers the state-of-the-art result of discretization approaches for the special case of Lipschitz MDPs $(\nu=0)$. At the same time, for $\nu\to\infty$, it recovers and greatly generalizes the $O(\epsilon^{-2})$ rate of low-rank MDPs, which are more amenable to regression approaches. In this sense, our result bridges the gap between two popular but conflicting perspectives on continuous-space MDPs.

Posterior sampling allows exploitation of prior knowledge on the environment's transition dynamics to improve the sample efficiency of reinforcement learning. The prior is typically specified as a class of parametric distributions, the design of which can be cumbersome in practice, often resulting in the choice of uninformative priors. In this work, we propose a novel posterior sampling approach in which the prior is given as a (partial) causal graph over the environment's variables. The latter is often more natural to design, such as listing known causal dependencies between biometric features in a medical treatment study. Specifically, we propose a hierarchical Bayesian procedure, called C-PSRL, simultaneously learning the full causal graph at the higher level and the parameters of the resulting factored dynamics at the lower level. We provide an analysis of the Bayesian regret of C-PSRL that explicitly connects the regret rate with the degree of prior knowledge. Our numerical evaluation conducted in illustrative domains confirms that C-PSRL strongly improves the efficiency of posterior sampling with an uninformative prior while performing close to posterior sampling with the full causal graph.

In this paper, we propose a novel generalization of rested and restless bandits where the evolution of the arms’ expected rewards is governed by a graph defined over the arms. An edge connecting a pair of arms $(i,j)$ represents the fact that a pull of arm $i$ triggers the evolution of arm $j$, and vice versa. Interestingly, rested and restless bandits are both special cases of our model for some suitable (degenerate) graphs. Still, the model can represent way more general and interesting scenarios. We first tackle the problem of computing the optimal policy when no specific structure is assumed on the graph, showing that it is NP-hard. Then, we focus on a specific structure forcing the graph to be composed of a set of fully connected subgraphs (i.e., cliques), and we prove that the optimal policy can be easily computed in closed form. Then, we move to the learning problem presenting regret minimization algorithms for deterministic and stochastic cases. Our regret bounds highlight the complexity of the learning problem by incorporating instance-dependent terms that encode specific properties of the underlying graph structure. Moreover, we illustrate how the knowledge of the underlying graph is not necessary for achieving the no-regret property.

Obtaining no-regret guarantees for reinforcement learning (RL) in the case of problems with continuous state and/or action spaces is still one of the major open challenges in the field. Recently, a variety of solutions have been proposed, but besides very specific settings, the general problem remains unsolved. In this paper, we introduce a novel structural assumption on the Markov decision processes (MDPs), namely $\nu-$smoothness, that generalizes most of the settings proposed so far (e.g., linear MDPs and Lipschitz MDPs). To face this challenging scenario, we propose two algorithms for regret minimization in $\nu-$smooth MDPs. Both algorithms build upon the idea of constructing an MDP representation through an orthogonal feature map based on Legendre polynomials. The first algorithm, Legendre-Eleanor, archives the no-regret property under weaker assumptions but is computationally inefficient, whereas the second one, Legendre-LSVI, runs in polynomial time, although for a smaller class of problems. After analyzing their regret properties, we compare our results with state-of-the-art ones from RL theory, showing that our algorithms achieve the best guarantees.

In several real-world sequential decision problems, at every step, the learner is required to select different actions. Every action affects a specific part of the system and generates an observable intermediate effect. In this paper, we introduce the Factored-Reward Bandits (FRBs), a novel setting able to effectively capture and exploit the structure of this class of scenarios, where the reward is computed as the product of the action intermediate observations. We characterize the statistical complexity of the learning problem in the FRBs, by deriving worst-case and asymptotic instance-dependent regret lower bounds. Then, we devise and analyze two regret minimization algorithms. The former, F-UCB, is an anytime optimistic approach matching the worst-case lower bound (up to logarithmic factors) but fails to perform optimally from the instance-dependent perspective. The latter, F-Track, is a bound-tracking approach, that enjoys optimal asymptotic instance-dependent regret guarantees.

Stochastic Rising Bandits (SRBs) model sequential decision-making problems in which the expected reward of the available options increases every time they are selected. This setting captures a wide range of scenarios in which the available options are learning entities whose performance improves (in expectation) over time (e.g., online best model selection). While previous works addressed the regret minimization problem, this paper focuses on the fixed-budget Best Arm Identification (BAI) problem for SRBs. In this scenario, given a fixed budget of rounds, we are asked to provide a recommendation about the best option at the end of the identification process. We propose two algorithms to tackle the above-mentioned setting, namely R-UCBE, which resorts to a UCB-like approach, and R-SR, which employs a successive reject procedure. Then, we prove that, with a sufficiently large budget, they provide guarantees on the probability of properly identifying the optimal option at the end of the learning process and on the simple regret. Furthermore, we derive a lower bound on the error probability, matched by our R-SR (up to constants), and illustrate how the need for a sufficiently large budget is unavoidable in the SRB setting. Finally, we numerically validate the proposed algorithms in both synthetic and realistic environments.

Recent works have studied *state entropy maximization* in reinforcement learning, in which the agent's objective is to learn a policy inducing high entropy over states visitation (Hazan et al., 2019). They typically assume full observability of the state of the system, so that the entropy of the observations is maximized. In practice, the agent may only get *partial* observations, e.g., a robot perceiving the state of a physical space through proximity sensors and cameras. A significant mismatch between the entropy over observations and true states of the system can arise in those settings. In this paper, we address the problem of entropy maximization over the *true states* with a decision policy conditioned on partial observations *only*. The latter is a generalization of POMDPs, which is intractable in general. We develop a memory and computationally efficient *policy gradient* method to address a first-order relaxation of the objective defined on *belief* states, providing various formal characterizations of approximation gaps, the optimization landscape, and the *hallucination* problem. This paper aims to generalize state entropy maximization to more realistic domains that meet the challenges of applications.

Machine learning algorithms are designed to capture complex relationships between features. In this context, the high dimensionality of data often results in poor model performance, with the risk of overfitting. Feature selection, the process of selecting a subset of relevant and non-redundant features, is an essential step to mitigate these issues. However, classical feature selection approaches do not inspect the causal relationship between features and the target variable, which can lead to misleading results in real-world applications. Causal discovery, instead, aims to identify causal relationships between features with observational data. In this paper, we propose a novel methodology at the intersection between feature selection and causal discovery, focusing on time series. We introduce a causal feature selection approach that relies on the forward and backward feature selection procedures and leverages transfer entropy to estimate the causal flow of information. In this context, we provide theoretical guarantees on the regression and classification errors for both the exact and the finite-sample cases. Finally, we present numerical validations on synthetic and real-world regression problems, showing results competitive w.r.t. the considered baselines.

Feeders play a crucial role in efficiently transporting materials, parts, and components to work areas in manufacturing industries. They employ various technologies and mechanisms tailored to the specific requirements of the manufacturing process. With the growing demand for flexibility and technology across diverse applications, feeders are evolving into smarter systems, thanks to the integration of industrial robots into their architecture. The challenges associated with grasping tasks, such as object variability, environmental dynamics, and real-time adaptability, underscore the need for advanced cognitive capabilities. Consequently, the application of Reinforcement Learning (RL) becomes instrumental in enabling robots to learn and adapt their grasping strategies autonomously. When combined with Deep Learning (DL), RL techniques drive robots to refine their grasping actions through iterative experiences, ultimately enhancing overall performance and dexterity. The primary goal of this work is to establish a comprehensive pipeline, starting from the 3D model of the object to be grasped and culminating in a trained algorithm capable of managing the hierarchical nature of agents. These agents detect the object to grasp and make decisions regarding the suggested point of picking. This complete workflow facilitates a seamless transition between graspable objects, introducing new flexible capabilities to a smart feeder equipped with an industrial robot. As a result, the integration of RL and DL not only addresses the inherent challenges in grasping tasks but also empowers the feeder system to adapt to a variety of objects dynamically, thereby optimizing efficiency in a manufacturing setting.

Machine learning methods have a groundbreaking impact in many application domains, but their application on real robotic platforms is still limited.Despite the many challenges associated with combining machine learning technology with robotics, robot learning remains one of the most promising directions for enhancing the capabilities of robots. When deploying learning-based approaches on real robots, extra effort is required to address the challenges posed by various real-world factors. To investigate the key factors influencing real-world deployment and to encourage original solutions from different researchers, we organized the Robot Air Hockey Challenge at the NeurIPS 2023 conference. We selected the air hockey task as a benchmark, encompassing low-level robotics problems and high-level tactics. Different from other machine learning-centric benchmarks, participants need to tackle practical challenges in robotics, such as the sim-to-real gap, low-level control issues, safety problems, real-time requirements, and the limited availability of real-world data. Furthermore, we focus on a dynamic environment, removing the typical assumption of quasi-static motions of other real-world benchmarks.The competition's results show that solutions combining learning-based approaches with prior knowledge outperform those relying solely on data when real-world deployment is challenging.Our ablation study reveals which real-world factors may be overlooked when building a learning-based solution.The successful real-world air hockey deployment of best-performing agents sets the foundation for future competitions and follow-up research directions.

In this paper, we introduce a novel variation of multi-armed bandits called bandits with ranking feedback. Unlike traditional bandits, this variation provides feedback to the learner that allows them to rank the arms based on previous pulls, without quantifying numerically the difference in performance. This type of feedback is well-suited for scenarios where the arms' values cannot be precisely measured using metrics such as monetary scores, probabilities, or occurrences. Common examples include human preferences in matchmaking problems. Furthermore, its investigation answers the theoretical question on how numerical rewards are crucial in bandit settings. In particular, we study the problem of designing no-regret algorithms with ranking feedback both in the stochastic and adversarial settings. We show that, with stochastic rewards, differently from what happens with non-ranking feedback, no algorithm can suffer a logarithmic regret in the time horizon $T$ in the instance-dependent case. Furthermore, we provide two algorithms. The first, namely DREE, guarantees a superlogarithmic regret in $T$ in the instance-dependent case thus matching our lower bound, while the second, namely R-LPE, guarantees a regret of $\mathcal{\widetilde O}(\sqrt{T})$ in the instance-independent case. Remarkably, we show that no algorithm can have an optimal regret bound in both instance-dependent and instance-independent cases. Finally, we prove that no algorithm can achieve a sublinear regret when the rewards are adversarial.

Achieving the no-regret property for Reinforcement Learning (RL) problems in continuous state and action-space environments is one of the major open problems in the field. Existing solutions either work under very specific assumptions or achieve bounds that are vacuous in some regimes. Furthermore, many structural assumptions are known to suffer from a provably unavoidable exponential dependence on the time horizon $H$ in the regret, which makes any possible solution unfeasible in practice. In this paper, we identify _local linearity_ as the feature that makes Markov Decision Processes (MDPs) both _learnable_ (sublinear regret) and _feasible_ (regret that is polynomial in $H$). We define a novel MDP representation class, namely _Locally Linearizable MDPs_, generalizing other representation classes like Linear MDPs and MDPS with low inherent Belmman error. Then, i) we introduce **Cinderella**, a no-regret algorithm for this general representation class, and ii) we show that all known learnable and feasible MDP families are representable in this class. We first show that all known feasible MDPs belong to a family that we call _Mildly Smooth MDPs_. Then, we show how any mildly smooth MDP can be represented as a Locally Linearizable MDP by an appropriate choice of representation. This way, **Cinderella** is shown to achieve state-of-the-art regret bounds for all previously known (and some new) continuous MDPs for which RL is learnable and feasible.

Inverse Reinforcement Learning (IRL) deals with the problem of deducing a reward function that explains the behavior of an expert agent who is assumed to act *optimally* in an underlying unknown task. Recent works have studied the IRL problem from the perspective of recovering the *feasible reward set*, i.e., the class of reward functions that are compatible with a unique optimal expert. However, in several problems of interest it is possible to observe the behavior of multiple experts with different degree of optimality (e.g., racing drivers whose skills ranges from amateurs to professionals). For this reason, in this work, we focus on the reconstruction of the feasible reward set when, in addition to demonstrations from the optimal expert, we observe the behavior of multiple *sub-optimal experts*. Given this problem, we first study the theoretical properties showing that the presence of multiple sub-optimal experts, in addition to the optimal one, can significantly shrink the set of compatible rewards, ultimately mitigating the inherent ambiguity of IRL. Furthermore, we study the statistical complexity of estimating the feasible reward set with a generative model and analyze a uniform sampling algorithm that turns out to be minimax optimal whenever the sub-optimal experts' performance level is sufficiently close to that of the optimal expert.

In bandit best-arm identification, an algorithm is tasked with finding the arm with highest mean reward with a specified accuracy as fast as possible. We study multi-fidelity best-arm identification, in which the algorithm can choose to sample an arm at a lower fidelity (less accurate mean estimate) for a lower cost. Several methods have been proposed for tackling this problem, but their optimality remain elusive, notably due to loose lower bounds on the total cost needed to identify the best arm. Our first contribution is a tight, instance-dependent lower bound on the cost complexity. The study of the optimization problem featured in the lower bound provides new insights to devise computationally efficient algorithms, and leads us to propose a gradient-based approach with asymptotically optimal cost complexity. We demonstrate the benefits of the new algorithm compared to existing methods in experiments. Our theoretical and empirical findings also shed light on an intriguing concept of optimal fidelity for each arm.

Abstract One of the central issues of several machine learning applications on real data is the choice of the input features. Ideally, the designer should select a small number of the relevant, nonredundant features to preserve the complete information contained in the original dataset, with little collinearity among features. This procedure helps mitigate problems like overfitting and the curse of dimensionality, which arise when dealing with high-dimensional problems. On the other hand, it is not desirable to simply discard some features, since they may still contain information that can be exploited to improve results. Instead, dimensionality reduction techniques are designed to limit the number of features in a dataset by projecting them into a lower dimensional space, possibly considering all the original features. However, the projected features resulting from the application of dimensionality reduction techniques are usually difficult to interpret. In this paper, we seek to design a principled dimensionality reduction approach that maintains the interpretability of the resulting features. Specifically, we propose a bias-variance analysis for linear models and we leverage these theoretical results to design an algorithm, Linear Correlated Features Aggregation (LinCFA), which aggregates groups of continuous features with their average if their correlation is “sufficiently large”. In this way, all features are considered, the dimensionality is reduced and the interpretability is preserved. Finally, we provide numerical validations of the proposed algorithm both on synthetic datasets to confirm the theoretical results and on real datasets to show some promising applications.

This work addresses the mediator feedback problem, a bandit game where the decision set consists of a number of policies, each associated with a probability distribution over a common space of outcomes. Upon choosing a policy, the learner observes an outcome sampled from its distribution and incurs the loss assigned to this outcome in the present round. We introduce the policy set capacity as an information-theoretic measure for the complexity of the policy set. Adopting the classical EXP4 algorithm, we provide new regret bounds depending on the policy set capacity in both the adversarial and the stochastic settings. For a selection of policy set families, we prove nearly-matching lower bounds, scaling similarly with the capacity. We also consider the case when the policies' distributions can vary between rounds, thus addressing the related bandits with expert advice problem, which we improve upon its prior results. Additionally, we prove a lower bound showing that exploiting the similarity between the policies is not possible in general under linear bandit feedback. Finally, for a full-information variant, we provide a regret bound scaling with the information radius of the policy set.

With the continuous growth of the global economy and markets, resource imbalance has risen to be one of the central issues in real logistic scenarios. In marine transportation, this trade imbalance leads to Empty Container Repositioning (ECR) problems. Once the freight has been delivered from an exporting country to an importing one, the laden will turn into empty containers that need to be repositioned to satisfy new goods requests in exporting countries. In such problems, the performance that any cooperative repositioning policy can achieve strictly depends on the routes that vessels will follow (i.e., fleet deployment). Historically, Operation Research (OR) approaches were proposed to jointly optimize the repositioning policy along with the fleet of vessels. However, the stochasticity of future supply and demand of containers, together with black-box and non-linear constraints that are present within the environment, make these approaches unsuitable for these scenarios. In this paper, we introduce a novel framework, Configurable Semi-POMDPs, to model this type of problems. Furthermore, we provide a two-stage learning algorithm, “Configure & Conquer” (CC), that first configures the environment by finding an approximation of the optimal fleet deployment strategy, and then “conquers” it by learning an ECR policy in this tuned environmental setting. We validate our approach in large and real-world instances of the problem. Our experiments highlight that CC avoids the pitfalls of OR methods and that it is successful at optimizing both the ECR policy and the fleet of vessels, leading to superior performance in world trade environments.

Uncertainty quantification has been extensively used as a means to achieve efficient directed exploration in Reinforcement Learning (RL). However, state-of-the-art methods for continuous actions still suffer from high sample complexity requirements. Indeed, they either completely lack strategies for propagating the epistemic uncertainty throughout the updates, or they mix it with aleatoric uncertainty while learning the full return distribution (e.g., distributional RL). In this paper, we propose Wasserstein Actor-Critic (WAC), an actor-critic architecture inspired by the recent Wasserstein Q-Learning (WQL), that employs approximate Q-posteriors to represent the epistemic uncertainty and Wasserstein barycenters for uncertainty propagation across the state-action space. WAC enforces exploration in a principled way by guiding the policy learning process with the optimization of an upper bound of the Q-value estimates. Furthermore, we study some peculiar issues that arise when using function approximation, coupled with the uncertainty estimation, and propose a regularized loss for the uncertainty estimation. Finally, we evaluate our algorithm on standard MujoCo tasks as well as suite of continuous-actions domains, where exploration is crucial, in comparison with state-of-the-art baselines. Additional details and results can be found in the supplementary material with our Arxiv preprint.

Behavioral Cloning (BC) aims at learning a policy that mimics the behavior demonstrated by an expert. The current theoretical understanding of BC is limited to the case of finite actions. In this paper, we study BC with the goal of providing theoretical guarantees on the performance of the imitator policy in the case of continuous actions. We start by deriving a novel bound on the performance gap based on Wasserstein distance, applicable for continuous-action experts, holding under the assumption that the value function is Lipschitz continuous. Since this latter condition is hardy fulfilled in practice, even for Lipschitz Markov Decision Processes and policies, we propose a relaxed setting, proving that value function is always H\"older continuous. This result is of independent interest and allows obtaining in BC a general bound for the performance of the imitator policy. Finally, we analyze noise injection, a common practice in which the expert's action is executed in the environment after the application of a noise kernel. We show that this practice allows deriving stronger performance guarantees, at the price of a bias due to the noise addition.

According to the main international reports, more pervasive industrial and business-process automation, thanks to machine learning and advanced analytic tools, will unlock more than 14 trillion USD worldwide annually by 2030. In the specific case of pricing problems, which constitute the class of problems we investigate in this paper, the estimated unlocked value will be about 0.5 trillion USD per year. In particular, this paper focuses on pricing in e-commerce when the objective function is profit maximization and only transaction data are available. This setting is one of the most common in real-world applications. Our work aims to find a pricing strategy that allows defining optimal prices at different volume thresholds to serve different classes of users. Furthermore, we face the major challenge, common in real-world settings, of dealing with limited data available. We design a two-phase online learning algorithm, namely PVD-B, capable of exploiting the data incrementally in an online fashion. The algorithm first estimates the demand curve and retrieves the optimal average price, and subsequently it offers discounts to differentiate the prices for each volume threshold. We ran a real-world 4-month-long A/B testing experiment in collaboration with an Italian e-commerce company, in which our algorithm PVD-B - corresponding to A configuration - has been compared with human pricing specialists - corresponding to B configuration. At the end of the experiment, our algorithm produced a total turnover of about 300 KEuros, outperforming the B configuration performance by about 55%. The Italian company we collaborated with decided to adopt our algorithm for more than 1,200 products since January 2022.

In the sequential decision making setting, an agent aims to achieve systematic generalization over a large, possibly infinite, set of environments. Such environments are modeled as discrete Markov decision processes with both states and actions represented through a feature vector. The underlying structure of the environments allows the transition dynamics to be factored into two components: one that is environment-specific and another that is shared. Consider a set of environments that share the laws of motion as an example. In this setting, the agent can take a finite amount of reward-free interactions from a subset of these environments. The agent then must be able to approximately solve any planning task defined over any environment in the original set, relying on the above interactions only. Can we design a provably efficient algorithm that achieves this ambitious goal of systematic generalization? In this paper, we give a partially positive answer to this question. First, we provide a tractable formulation of systematic generalization by employing a causal viewpoint. Then, under specific structural assumptions, we provide a simple learning algorithm that guarantees any desired planning error up to an unavoidable sub-optimality term, while showcasing a polynomial sample complexity.

In Reinforcement Learning, the performance of learning agents is highly sensitive to the choice of time discretization. Agents acting at high frequencies have the best control opportunities, along with some drawbacks, such as possible inefficient exploration and vanishing of the action advantages. The repetition of the actions, i.e., action persistence, comes into help, as it allows the agent to visit wider regions of the state space and improve the estimation of the action effects. In this work, we derive a novel operator, the All-Persistence Bellman Operator, which allows an effective use of both the low-persistence experience, by decomposition into sub-transition, and the high-persistence experience, thanks to the introduction of a suitable bootstrap procedure. In this way, we employ transitions collected at any time scale to update simultaneously the action values of the considered persistence set. We prove the contraction property of the All-Persistence Bellman Operator and, based on it, we extend classic Q-learning and DQN. After providing a study on the effects of persistence, we experimentally evaluate our approach in both tabular contexts and more challenging frameworks, including some Atari games.

The most relevant problems in discounted reinforcement learning involve estimating the mean of a function under the stationary distribution of a Markov reward process, such as the expected return in policy evaluation, or the policy gradient in policy optimization. In practice, these estimates are produced through a finite-horizon episodic sampling, which neglects the mixing properties of the Markov process. It is mostly unclear how this mismatch between the practical and the ideal setting affects the estimation, and the literature lacks a formal study on the pitfalls of episodic sampling, and how to do it optimally. In this paper, we present a minimax lower bound on the discounted mean estimation problem that explicitly connects the estimation error with the mixing properties of the Markov process and the discount factor. Then, we provide a statistical analysis on a set of notable estimators and the corresponding sampling procedures, which includes the finite-horizon estimators often used in practice. Crucially, we show that estimating the mean by directly sampling from the discounted kernel of the Markov process brings compelling statistical properties w.r.t. the alternative estimators, as it matches the lower bound without requiring a careful tuning of the episode horizon.

Inverse reinforcement learning (IRL) denotes a powerful family of algorithms for recovering a reward function justifying the behavior demonstrated by an expert agent. A well-known limitation of IRL is the ambiguity in the choice of the reward function, due to the existence of multiple rewards that explain the observed behavior. This limitation has been recently circumvented by formulating IRL as the problem of estimating the feasible reward set, i.e., the region of the rewards compatible with the expert's behavior. In this paper, we make a step towards closing the theory gap of IRL in the case of finite-horizon problems with a generative model. We start by formally introducing the problem of estimating the feasible reward set, the corresponding PAC requirement, and discussing the properties of particular classes of rewards. Then, we provide the first minimax lower bound on the sample complexity for the problem of estimating the feasible reward set of order ${\Omega}\left( \frac{H^3SA}{\epsilon^2} \left( \log \left(\frac{1}{\delta}\right) + S \right)\right)$, being $S$ and $A$ the number of states and actions respectively, $H$ the horizon, $\epsilon$ the desired accuracy, and $\delta$ the confidence. We analyze the sample complexity of a uniform sampling strategy (US-IRL), proving a matching upper bound up to logarithmic factors. Finally, we outline several open questions in IRL and propose future research directions.

In many real-world sequential decision-making problems, an action does not immediately reflect on the feedback and spreads its effects over a long time frame. For instance, in online advertising, investing in a platform produces an instantaneous increase of awareness, but the actual reward, i.e., a conversion, might occur far in the future. Furthermore, whether a conversion takes place depends on: how fast the awareness grows, its vanishing effects, and the synergy or interference with other advertising platforms. Previous work has investigated the Multi-Armed Bandit framework with the possibility of delayed and aggregated feedback, without a particular structure on how an action propagates in the future, disregarding possible dynamical effects. In this paper, we introduce a novel setting, the Dynamical Linear Bandits (DLB), an extension of the linear bandits characterized by a hidden state. When an action is performed, the learner observes a noisy reward whose mean is a linear function of the hidden state and of the action. Then, the hidden state evolves according to linear dynamics, affected by the performed action too. We start by introducing the setting, discussing the notion of optimal policy, and deriving an expected regret lower bound. Then, we provide an optimistic regret minimization algorithm, Dynamical Linear Upper Confidence Bound (DynLin-UCB), that suffers an expected regret of order $\widetilde{\mathcal{O}} \Big( \frac{d \sqrt{T}}{(1-\overline{\rho})^{3/2}} \Big)$, where $\overline{\rho}$ is a measure of the stability of the system, and $d$ is the dimension of the action vector. Finally, we conduct a numerical validation on a synthetic environment and on real-world data to show the effectiveness of DynLin-UCB in comparison with several baselines.

In Reinforcement Learning (RL), an agent acts in an unknown environment to maximize the expected cumulative discounted sum of an external reward signal, i.e., the expected return. In practice, in many tasks of interest, such as policy optimization, the agent usually spends its interaction budget by collecting episodes of *fixed length* within a simulator (i.e., Monte Carlo simulation). However, given the discounted nature of the RL objective, this data collection strategy might not be the best option. Indeed, the rewards taken in early simulation steps weigh exponentially more than future rewards. Taking a cue from this intuition, in this paper, we design an a-priori budget allocation strategy that leads to the collection of trajectories of different lengths, i.e., *truncated*. The proposed approach provably minimizes the width of the confidence intervals around the empirical estimates of the expected return of a policy. After discussing the theoretical properties of our method, we make use of our trajectory truncation mechanism to extend Policy Optimization via Importance Sampling (POIS, Metelli et al., 2018) algorithm. Finally, we conduct a numerical comparison between our algorithm and POIS: the results are consistent with our theory and show that an appropriate truncation of the trajectories can succeed in improving performance.

We investigate the problem of bandits with expert advice when the experts are fixed and known distributions over the actions. Improving on previous analyses, we show that the regret in this setting is controlled by information-theoretic quantities that measure the similarity between experts. In some natural special cases, this allows us to obtain the first regret bound for EXP4 that can get arbitrarily close to zero if the experts are similar enough. While for a different algorithm, we provide another bound that describes the similarity between the experts in terms of the KL-divergence, and we show that this bound can be smaller than the one of EXP4 in some cases. Additionally, we provide lower bounds for certain classes of experts showing that the algorithms we analyzed are nearly optimal in some cases.

Policy evaluation via Monte Carlo (MC) simulation is at the core of many MC Reinforcement Learning (RL) algorithms (e.g., policy gradient methods). In this context, the designer of the learning system specifies an interaction budget that the agent usually spends by collecting trajectories of *fixed length* within a simulator. However, is this data collection strategy the best option? To answer this question, in this paper, we consider as quality index the variance of an unbiased policy return estimator that uses trajectories of different lengths, i.e., *truncated*. We first derive a closed-form expression of this variance that clearly shows the sub-optimality of the fixed-length trajectory schedule. Furthermore, it suggests that adaptive data collection strategies that spend the available budget sequentially might be able to allocate a larger portion of transitions in timesteps in which more accurate sampling is required to reduce the variance of the final estimate. Building on these findings, we present an *adaptive* algorithm called **R**obust and **I**terative **D**ata collection strategy **O**ptimization (RIDO). The main intuition behind RIDO is to split the available interaction budget into mini-batches. At each round, the agent determines the most convenient schedule of trajectories that minimizes an empirical and robust estimate of the estimator's variance. After discussing the theoretical properties of our method, we conclude by assessing its performance across multiple domains. Our results show that RIDO can adapt its trajectory schedule toward timesteps where more sampling is required to increase the quality of the final estimation.

The Maximum Entropy (Max-Ent) framework has been effectively employed in a variety of Reinforcement Learning (RL) tasks. In this paper, we first propose a novel Max-Ent framework for policy evaluation in a distributional RL setting, named *Distributional Maximum Entropy Policy Evaluation* (D-Max-Ent PE). We derive a generalization-error bound that depends on the complexity of the representation employed, showing that this framework can explicitly take into account the features used to represent the state space while evaluating a policy. Then, we exploit these favorable properties to drive the representation learning of the state space in a Structural Risk Minimization fashion. We employ state-aggregation functions as feature functions and we specialize the D-Max-Ent approach into an algorithm, named *D-Max-Ent Progressive Factorization*, which constructs a progressively finer-grained representation of the state space by balancing the trade-off between preserving information (bias) and reducing the effective number of states, i.e., the complexity of the representation space (variance). Finally, we report the results of some illustrative numerical simulations, showing that the proposed algorithm matches the expected theoretical behavior and highlighting the relationship between aggregations and sample regimes.

Off-policy methods are the basis of a large number of effective Policy Optimization (PO) algorithms. In this setting, Importance Sampling (IS) is typically employed for off-policy evaluation, with the goal of estimating the performance of a target policy, given samples collected with a different behavioral policy. However, in Monte Carlo simulation, IS represents a variance minimization approach. In this field, a suitable behavioral distribution is employed for sampling, allowing diminishing the variance of the estimator below the one achievable when sampling from the target distribution. In this paper, we analyze IS in these two guises in the context of PO. We provide a novel view of off-policy PO, showing a connection between the policy improvement and variance minimization objectives. Then, we illustrate how minimizing the off-policy variance can, in some circumstances, lead to a policy improvement, with the advantage, compared with direct off-policy learning, of implicitly enforcing a trust region. Finally, we present numerical simulations on continuous RL benchmarks, with a particular focus on the robustness to small batch sizes.

Automated Reinforcement Learning (AutoRL) is a relatively new area of research that is gaining increasing attention. The objective of AutoRL consists in easing the employment of Reinforcement Learning (RL) techniques for the broader public by alleviating some of its main challenges, including data collection, algorithm selection, and hyper-parameter tuning. In this work, we propose a general and flexible framework, namely ARLO: Automated Reinforcement Learning Optimizer, to construct automated pipelines for AutoRL. Based on this, we propose a pipeline for offline and one for online RL, discussing the components, interaction, and highlighting the difference between the two settings. Furthermore, we provide a Python implementation of such pipelines, released as an open-source library. Our implementation is tested on an illustrative LQG domain and on classic MuJoCo environments, showing the ability to reach competitive performances requiring limited human intervention. We also showcase the full pipeline on a realistic dam environment, automatically performing the feature selection and the model generation tasks.

Convex Reinforcement Learning (RL) is a recently introduced framework that generalizes the standard RL objective to any convex (or concave) function of the state distribution induced by the agent's policy. This framework subsumes several applications of practical interest, such as pure exploration, imitation learning, and risk-averse RL, among others. However, the previous convex RL literature implicitly evaluates the agent's performance over infinite realizations (or trials), while most of the applications require excellent performance over a handful, or even just one, trials. To meet this practical demand, we formulate convex RL in finite trials, where the objective is any convex function of the empirical state distribution computed over a finite number of realizations. In this paper, we provide a comprehensive theoretical study of the setting, which includes an analysis of the importance of non-Markovian policies to achieve optimality, as well as a characterization of the computational and statistical complexity of the problem in various configurations.

Distribution drift is an important issue for practical applications of machine learning (ML). In particular, in streaming ML, the data distribution may change over time, yielding the problem of concept drift, which affects the performance of learners trained with outdated data. In this article, we focus on supervised problems in an online nonstationary setting, introducing a novel learner-agnostic algorithm for drift adaptation, namely importance weighting for drift adaptation (IWDA), with the goal of performing efficient retraining of the learner when drift is detected. IWDA incrementally estimates the joint probability density of input and target for the incoming data and, as soon as drift is detected, retrains the learner using importance-weighted empirical risk minimization. The importance weights are computed for all the samples observed so far, employing the estimated densities, thus, using all available information efficiently. After presenting our approach, we provide a theoretical analysis in the abrupt drift setting. Finally, we present numerical simulations that illustrate how IWDA competes and often outperforms state-of-the-art stream learning techniques, including adaptive ensemble methods, on both synthetic and real-world data benchmarks.

Learning in a lifelong setting, where the dynamics continually evolve, is a hard challenge for current reinforcement learning algorithms. Yet this would be a much needed feature for practical applications. In this paper, we propose an approach which learns a hyper-policy, whose input is time, that outputs the parameters of the policy to be queried at that time. This hyper-policy is trained to maximize the estimated future performance, efficiently reusing past data by means of importance sampling, at the cost of introducing a controlled bias. We combine the future performance estimate with the past performance to mitigate catastrophic forgetting. To avoid overfitting the collected data, we derive a differentiable variance bound that we embed as a penalization term. Finally, we empirically validate our approach, in comparison with state-of-the-art algorithms, on realistic environments, including water resource management and trading.

Several recent works have been dedicated to unsupervised reinforcement learning in a single environment, in which a policy is first pre-trained with unsupervised interactions, and then fine-tuned towards the optimal policy for several downstream supervised tasks defined over the same environment. Along this line, we address the problem of unsupervised reinforcement learning in a class of multiple environments, in which the policy is pre-trained with interactions from the whole class, and then fine-tuned for several tasks in any environment of the class. Notably, the problem is inherently multi-objective as we can trade off the pre-training objective between environments in many ways. In this work, we foster an exploration strategy that is sensitive to the most adverse cases within the class. Hence, we cast the exploration problem as the maximization of the mean of a critical percentile of the state visitation entropy induced by the exploration strategy over the class of environments. Then, we present a policy gradient algorithm, alphaMEPOL, to optimize the introduced objective through mediated interactions with the class. Finally, we empirically demonstrate the ability of the algorithm in learning to explore challenging classes of continuous environments and we show that reinforcement learning greatly benefits from the pre-trained exploration strategy w.r.t. learning from scratch.

The goal in the standard reinforcement learning problem is to find a policy that optimizes the expected return. However, such an objective is not adequate in a lot of real-life applications, like finance, where controlling the uncertainty of the outcome is imperative. The mean-volatility objective penalizes, through a tunable parameter, policies with high variance of the per-step reward. An interesting property of this objective is that it admits simple linear Bellman equations that resemble, up to a reward transformation, those of the risk-neutral case. However, the required reward transformation is policy-dependent, and requires the (usually unknown) expected return of the used policy. In this work, we propose two general methods for policy evaluation under the mean-volatility objective: the direct method and the factored method. We then extend recent results for finite sample analysis in the risk-neutral actor-critic setting to the mean-volatility case. Our analysis shows that the sample complexity to attain an $\epsilon$-accurate stationary point is the same as that of the risk-neutral version, using either policy evaluation method for training the critic. Finally, we carry out experiments to test the proposed methods in a simple environment that exhibits some trade-off between optimality, in expectation, and uncertainty of outcome.

In reinforcement learning, we encode the potential behaviors of an agent interacting with an environment into an infinite set of policies, called policy space, typically represented by a family of parametric functions. Dealing with such a policy space is a hefty challenge, which often causes sample and computational inefficiencies. However, we argue that a limited number of policies are actually relevant when we also account for the structure of the environment and of the policy parameterization, as many of them would induce very similar interactions, i.e., state-action distributions. In this paper, we seek for a reward-free compression of the policy space into a finite set of representative policies, such that, given any policy $\pi$, the minimum Rényi divergence between the state-action distributions of the representative policies and the state-action distribution of $\pi$ is bounded. We show that this compression of the policy space can be formulated as a set cover problem, and it is inherently NP-hard. Nonetheless, we propose a game-theoretic reformulation for which a locally optimal solution can be efficiently found by iteratively stretching the compressed space to cover the most challenging policy. Finally, we provide an empirical evaluation to illustrate the compression procedure in simple domains, and its ripple effects in reinforcement learning.

We propose a novel formulation for the Inverse Reinforcement Learning (IRL) problem, which jointly accounts for the compatibility with the expert behavior of the identified reward and its effectiveness for the subsequent forward learning phase. Albeit quite natural, especially when the final goal is apprenticeship learning (learning policies from an expert), this aspect has been completely overlooked by IRL approaches so far. We propose a new model-free IRL method that is remarkably able to autonomously find a trade-off between the error induced on the learned policy when potentially choosing a sub-optimal reward, and the estimation error caused by using finite samples in the forward learning phase, which can be controlled by explicitly optimizing also the discount factor of the related learning problem. The approach is based on a min-max formulation for the robust selection of the reward parameters and the discount factor so that the distance between the expert’s policy and the learned policy is minimized in the successive forward learning task when a finite and possibly small number of samples is available. Differently from the majority of other IRL techniques, our approach does not involve any planning or forward Reinforcement Learning problems to be solved. After presenting the formulation, we provide a numerical scheme for the optimization, and we show its effectiveness on an illustrative numerical case.

When the agent’s observations or interactions are delayed, classic reinforcement learning tools usually fail. In this paper, we propose a simple yet new and efficient solution to this problem. We assume that, in the undelayed environment, an efficient policy is known or can be easily learnt, but the task may suffer from delays in practice and we thus want to take them into account. We present a novel algorithm, Delayed Imitation with Dataset Aggregation (DIDA), which builds upon imitation learning methods to learn how to act in a delayed environment from undelayed demonstrations. We provide a theoretical analysis of the approach that will guide the practical design of DIDA. These results are also of general interest in the delayed reinforcement learning literature by providing bounds on the performance between delayed and undelayed tasks, under smoothness conditions. We show empirically that DIDA obtains high performances with a remarkable sample efficiency on a variety of tasks, including robotic locomotion, classic control, and trading.

This paper is in the field of stochastic Multi-Armed Bandits (MABs), i.e., those sequential selection techniques able to learn online using only the feedback given by the chosen option (a.k.a. arm). We study a particular case of the rested and restless bandits in which the arms’ expected payoff is monotonically non-decreasing. This characteristic allows designing specifically crafted algorithms that exploit the regularity of the payoffs to provide tight regret bounds. We design an algorithm for the rested case (R-ed-UCB) and one for the restless case (R-less-UCB), providing a regret bound depending on the properties of the instance and, under certain circumstances, of $\widetilde{\mathcal{O}}(T^{\frac{2}{3}})$. We empirically compare our algorithms with state-of-the-art methods for non-stationary MABs over several synthetically generated tasks and an online model selection problem for a real-world dataset. Finally, using synthetic and real-world data, we illustrate the effectiveness of the proposed approaches compared with state-of-the-art algorithms for the non-stationary bandits.

In the maximum state entropy exploration framework, an agent interacts with a reward-free environment to learn a policy that maximizes the entropy of the expected state visitations it is inducing. Hazan et al. (2019) noted that the class of Markovian stochastic policies is sufficient for the maximum state entropy objective, and exploiting non-Markovianity is generally considered pointless in this setting. In this paper, we argue that non-Markovianity is instead paramount for maximum state entropy exploration in a finite-sample regime. Especially, we recast the objective to target the expected entropy of the induced state visitations in a single trial. Then, we show that the class of non-Markovian deterministic policies is sufficient for the introduced objective, while Markovian policies suffer non-zero regret in general. However, we prove that the problem of finding an optimal non-Markovian policy is NP-hard. Despite this negative result, we discuss avenues to address the problem in a tractable way and how non-Markovian exploration could benefit the sample efficiency of online reinforcement learning in future works.

There is a rising interest in industrial online applications where data becomes available sequentially. Inspired by the recommendation of playlists to users where their preferences can be collected during the listening of the entire playlist, we study a novel bandit setting, namely Multi-Armed Bandit with Temporally-Partitioned Rewards (TP-MAB), in which the stochastic reward associated with the pull of an arm is partitioned over a finite number of consecutive rounds following the pull. This setting, unexplored so far to the best of our knowledge, is a natural extension of delayed-feedback bandits to the case in which rewards may be dilated over a finite-time span after the pull instead of being fully disclosed in a single, potentially delayed round. We provide two algorithms to address TP-MAB problems, namely, TP-UCB-FR and TP-UCB-EW, which exploit the partial information disclosed by the reward collected over time. We show that our algorithms provide better asymptotical regret upper bounds than delayed-feedback bandit algorithms when a property characterizing a broad set of reward structures of practical interest, namely α-smoothness, holds. We also empirically evaluate their performance across a wide range of settings, both synthetically generated and from a real-world media recommendation problem.

Warehouse Management Systems have been evolving and improving thanks to new Data Intelligence techniques. However, many current optimizations have been applied to specific cases or are in great need of manual interaction. Here is where Reinforcement Learning techniques come into play, providing automatization and adaptability to current optimization policies. In this paper, we present Storehouse, a customizable environment that generalizes the definition of warehouse simulations for Reinforcement Learning. We also validate this environment against state-of-the-art reinforcement learning algorithms and compare these results to human and random policies.

In several e-commerce scenarios, pricing long-tail products effectively is a central task for the companies, and there is broad agreement that Artificial Intelligence (AI) will play a prominent role in doing that in the next future. Nevertheless, dealing with long-tail products raises major open technical issues due to data scarcity which preclude the adoption of the mainstream approaches requiring usually a huge amount of data, such as, e.g., deep learning. In this paper, we provide a novel online learning algorithm for dynamic pricing that deals with non-stationary settings due to, e.g., the seasonality or adaptive competitors, and is very efficient in terms of the need for data thanks to assumptions such as, e.g., the monotonicity of the demand curve in the price that are customarily satisfied in long-tail markets. Furthermore, our dynamic pricing algorithm is paired with a clustering algorithm for the long-tail products which aggregates similar products such that the data of all the products of the same cluster are merged and used to choose their best price. We first evaluate our algorithms in an offline synthetic setting, comparing their performance with the state of the art and showing that our algorithms are more robust and data-efficient in long-tail settings. Subsequently, we evaluate our algorithms in an online setting with more than 8,000 products, including popular and long-tail, in an A/B test with humans for about two months. The increase of revenue thanks to our algorithms is about 18% for the popular products and about 90% for the long-tail products.

The Off-Policy Evaluation (OPE) problem consists in evaluating the performance of new policies from the data collected by another one. OPE is crucial when evaluating a new policy online is too expensive or risky. Many of the state-of-the-art OPE estimators are based on the Inverse Propensity Scoring (IPS) technique, which provides an unbiased estimator when the full support assumption holds, i.e., when the logging policy assigns a non-zero probability to each action. However, there are several scenarios where this assumption does not hold in practice, i.e., there is deficient support, and the IPS estimator is biased in the general case.In this paper, we consider two alternative estimators for the deficient support OPE problem. We first show how to adapt an estimator that was originally proposed for a different domain to the deficient support setting.Then, we propose another estimator, which is a novel contribution of this paper.These estimators exploit additional information about the actions, which we call side information, in order to make reliable estimates on the unsupported actions. Under alternative assumptions that do not require full support, we show that the considered estimators are unbiased.We also provide a theoretical analysis of the concentration when relaxing all the assumptions. Finally, we provide an experimental evaluation showing how the considered estimators are better suited for the deficient support setting compared to the baselines.

The classic Reinforcement Learning (RL) formulation concerns the maximization of a scalar reward function. More recently, convex RL has been introduced to extend the RL formulation to all the objectives that are convex functions of the state distribution induced by a policy. Notably, convex RL covers several relevant applications that do not fall into the scalar formulation, including imitation learning, risk-averse RL, and pure exploration. In classic RL, it is common to optimize an infinite trials objective, which accounts for the state distribution instead of the empirical state visitation frequencies, even though the actual number of trajectories is always finite in practice. This is theoretically sound since the infinite trials and finite trials objectives are equivalent and thus lead to the same optimal policy. In this paper, we show that this hidden assumption does not hold in convex RL. In particular, we prove that erroneously optimizing the infinite trials objective in place of the actual finite trials one, as it is usually done, can lead to a significant approximation error. Since the finite trials setting is the default in both simulated and real-world RL, we believe shedding light on this issue will lead to better approaches and methodologies for convex RL, impacting relevant research areas such as imitation learning, risk-averse RL, and pure exploration among others.

In several real-world applications, a learner has access to multiple environment simulators, each with a different precision (e.g., simulation accuracy) and cost (e.g., computational time). In such a scenario, the learner faces the trade-off between selecting expensive accurate simulators or preferring cheap imprecise ones. We formalize this setting as a multi-fidelity variant of the stochastic best-arm identification problem, where querying the original arm is expensive, but multiple and biased approximations (i.e., fidelities) are available at lower costs. The learner's goal, in this setting, is to sequentially choose which simulator to query in order to minimize the total cost, while guaranteeing to identify the optimal arm with high probability. We first derive a lower bound on the identification cost, assuming that the maximum bias of each fidelity is known to the learner. Then, we propose a novel algorithm, Iterative Imprecise Successive Elimination (IISE), which provably reduces the total cost w.r.t. algorithms that ignore the multi-fidelity structure and whose cost complexity upper bound mimics the structure of the lower bound. Furthermore, we show that the cost complexity of IISE can be further reduced when the agent has access to a more fine-grained knowledge of the error introduced by the approximators.Finally, we numerically validate IISE, showing the benefits of our method in simulated domains.

In this paper, we study the learning problem in two-player general-sum Markov Games. We consider the online setting where we control a single player, playing against an arbitrary opponent to minimize the regret. Previous works only consider the zero-sum Markov Games setting, in which the two agents are completely adversarial. However, in some cases, the two agents may have different reward functions without having conflicting objectives. This involves a stronger notion of regret than the one used in previous works. This class of games, called general-sum Markov Games is far to be well understood and studied. We show that the new regret minimization problem is significantly harder than in standard Markov Decision Processes and zero-sum Markov Games. To do this, we derive a lower bound on the expected regret of any “good” learning strategy which shows the constant dependencies with the number of deterministic policies, which is not present in zerosum Markov Games and Markov Decision Processes. Then we propose a novel optimistic algorithm that nearly matches the proposed lower bound. Proving these results requires overcoming several new challenges that are not present in Markov Decision Processes or zero-sum Markov Games.

Abstract Real-world sequential decision-making tasks are generally complex, requiring trade-offs between multiple, often conflicting, objectives. Despite this, the majority of research in reinforcement learning and decision-theoretic planning either assumes only a single objective, or that multiple objectives can be adequately handled via a simple linear combination. Such approaches may oversimplify the underlying problem and hence produce suboptimal results. This paper serves as a guide to the application of multi-objective methods to difficult problems, and is aimed at researchers who are already familiar with single-objective reinforcement learning and planning methods who wish to adopt a multi-objective perspective on their research, as well as practitioners who encounter multi-objective decision problems in practice. It identifies the factors that may influence the nature of the desired solution, and illustrates by example how these influence the design of multi-objective decision-making systems for complex problems.

Abstract We study the problem of identifying the policy space available to an agent in a learning process, having access to a set of demonstrations generated by the agent playing the optimal policy in the considered space. We introduce an approach based on frequentist statistical testing to identify the set of policy parameters that the agent can control, within a larger parametric policy space. After presenting two identification rules (combinatorial and simplified), applicable under different assumptions on the policy space, we provide a probabilistic analysis of the simplified one in the case of linear policies belonging to the exponential family. To improve the performance of our identification rules, we make use of the recently introduced framework of the Configurable Markov Decision Processes, exploiting the opportunity of configuring the environment to induce the agent to reveal which parameters it can control. Finally, we provide an empirical evaluation, on both discrete and continuous domains, to prove the effectiveness of our identification rules.

Abstract Policy gradient (PG) algorithms are among the best candidates for the much-anticipated applications of reinforcement learning to real-world control tasks, such as robotics. However, the trial-and-error nature of these methods poses safety issues whenever the learning process itself must be performed on a physical system or involves any form of human-computer interaction. In this paper, we address a specific safety formulation, where both goals and dangers are encoded in a scalar reward signal and the learning agent is constrained to never worsen its performance, measured as the expected sum of rewards. By studying actor-only PG from a stochastic optimization perspective, we establish improvement guarantees for a wide class of parametric policies, generalizing existing results on Gaussian policies. This, together with novel upper bounds on the variance of PG estimators, allows us to identify meta-parameter schedules that guarantee monotonic improvement with high probability. The two key meta-parameters are the step size of the parameter updates and the batch size of the gradient estimates. Through a joint, adaptive selection of these meta-parameters, we obtain a PG algorithm with monotonic improvement guarantees.

Policy Optimization (PO) is a widely used approach to address continuous control tasks. In this paper, we introduce the notion of mediator feedback that frames PO as an online learning problem over the policy space. The additional available information, compared to the standard bandit feedback, allows reusing samples generated by one policy to estimate the performance of other policies. Based on this observation, we propose an algorithm, RANDomized-exploration policy Optimization via Multiple Importance Sampling with Truncation (RANDOMIST), for regret minimization in PO, that employs a randomized exploration strategy, differently from the existing optimistic approaches. When the policy space is finite, we show that under certain circumstances, it is possible to achieve constant regret, while always enjoying logarithmic regret. We also derive problem-dependent regret lower bounds. Then, we extend RANDOMIST to compact policy spaces. Finally, we provide numerical simulations on finite and compact policy spaces, in comparison with PO and bandit baselines.

In a reward-free environment, what is a suitable intrinsic objective for an agent to pursue so that it can learn an optimal task-agnostic exploration policy? In this paper, we argue that the entropy of the state distribution induced by finite-horizon trajectories is a sensible target. Especially, we present a novel and practical policy-search algorithm, Maximum Entropy POLicy optimization (MEPOL), to learn a policy that maximizes a non-parametric, $k$-nearest neighbors estimate of the state distribution entropy. In contrast to known methods, MEPOL is completely model-free as it requires neither to estimate the state distribution of any policy nor to model transition dynamics. Then, we empirically show that MEPOL allows learning a maximum-entropy exploration policy in high-dimensional, continuous-control domains, and how this policy facilitates learning meaningful reward-based tasks downstream.

The reward function is widely accepted as a succinct, robust, and transferable representation of a task. Typical approaches, at the basis of Inverse Reinforcement Learning (IRL), leverage on expert demonstrations to recover a reward function. In this paper, we study the theoretical properties of the class of reward functions that are compatible with the expert’s behavior. We analyze how the limited knowledge of the expert’s policy and of the environment affects the reward reconstruction phase. Then, we examine how the error propagates to the learned policy’s performance when transferring the reward function to a different environment. We employ these findings to devise a provably efficient active sampling approach, aware of the need for transferring the reward function, that can be paired with a large variety of IRL algorithms. Finally, we provide numerical simulations on benchmark environments.

The linear contextual bandit literature is mostly focused on the design of efficient learning algorithms for a given representation. However, a contextual bandit problem may admit multiple linear representations, each one with different characteristics that directly impact the regret of the learning algorithm. In particular, recent works showed that there exist “good” representations for which constant problem-dependent regret can be achieved. In this paper, we first provide a systematic analysis of the different definitions of “good” representations proposed in the literature. We then propose a novel selection algorithm able to adapt to the best representation in a set of $M$ candidates. We show that the regret is indeed never worse than the regret obtained by running \textsc{LinUCB} on best representation (up to a $\ln M$ factor). As a result, our algorithm achieves constant regret if a “good” representation is available in the set. Furthermore, we show the algorithm may still achieve constant regret by implicitly constructing a “good” representation, even when none of the initial representations is “good”. Finally, we validate our theoretical findings in a number of standard contextual bandit problems.

Many real-world domains are subject to a structured non-stationarity which affects the agent's goals and the environmental dynamics. Meta-reinforcement learning (RL) has been shown successful for training agents that quickly adapt to related tasks. However, most of the existing meta-RL algorithms for non-stationary domains either make strong assumptions on the task generation process or require sampling from it at training time. In this paper, we propose a novel algorithm (TRIO) that optimizes for the future by explicitly tracking the task evolution through time. At training time, TRIO learns a variational module to quickly identify latent parameters from experience samples. This module is learned jointly with an optimal exploration policy that takes task uncertainty into account. At test time, TRIO tracks the evolution of the latent parameters online, hence reducing the uncertainty over future tasks and obtaining fast adaptation through the meta-learned policy. Unlike most existing methods, TRIO does not assume Markovian task-evolution processes, it does not require information about the non-stationarity at training time, and it captures complex changes undergoing in the environment. We evaluate our algorithm on different simulated problems and show it outperforms competitive baselines.

This paper considers sequential decision-making problems where the interactions between an agent and its environment are affected by delays. Delays may be present in the state observation, in the action execution, or in the reward collection. We consider the delayed Markov Decision Process (MDP) framework both in the case of deterministic and stochastic delays. Given the hardness of the delayed MDP problem, we use a heuristic approach to design an algorithm that uses the belief over the current unobserved state to select its action. We design a self-attention prediction module which, given the last observed state and the following sequence of actions, estimates the beliefs over the following states. This algorithm is able to deal with deterministic delays and could potentially be extended to stochastic delays. We empirically evaluate the effectiveness of the proposed approach in both deterministic and stochastic control problems affected by deterministic delays.

Importance Sampling (IS) is a widely used building block for a large variety of off-policy estimation and learning algorithms. However, empirical and theoretical studies have progressively shown that vanilla IS leads to poor estimations whenever the behavioral and target policies are too dissimilar. In this paper, we analyze the theoretical properties of the IS estimator by deriving a novel anticoncentration bound that formalizes the intuition behind its undesired behavior. Then, we propose a new class of IS transformations, based on the notion of power mean. To the best of our knowledge, the resulting estimator is the first to achieve, under certain conditions, two key properties: (i) it displays a subgaussian concentration rate; (ii) it preserves the differentiability in the target distribution. Finally, we provide numerical simulations on both synthetic examples and contextual bandits, in comparison with off-policy evaluation and learning baselines.

We study the role of the representation of state-action value functions in regret minimization in finite-horizon Markov Decision Processes (MDPs) with linear structure. We first derive a necessary condition on the representation, called universally spanning optimal features (UNISOFT), to achieve constant regret in any MDP with linear reward function. This result encompasses the well-known settings of low-rank MDPs and, more generally, zero inherent Bellman error (also known as the Bellman closure assumption). We then demonstrate that this condition is also sufficient for these classes of problems by deriving a constant regret bound for two optimistic algorithms (LSVI-UCB and ELEANOR). Finally, we propose an algorithm for representation selection and we prove that it achieves constant regret when one of the given representations, or a suitable combination of them, satisfies the UNISOFT condition.

The Configurable Markov Decision Process framework includes two entities: a Reinforcement Learning agent and a configurator that can modify some environmental parameters to improve the agent's performance. This presupposes that the two actors have the same reward functions. What if the configurator does not have the same intentions as the agent? This paper introduces the Non-Cooperative Configurable Markov Decision Process, a setting that allows having two (possibly different) reward functions for the configurator and the agent. Then, we consider an online learning problem, where the configurator has to find the best among a finite set of possible configurations. We propose two learning algorithms to minimize the configurator's expected regret, which exploits the problem's structure, depending on the agent's feedback. While a naive application of the UCB algorithm yields a regret that grows indefinitely over time, we show that our approach suffers only bounded regret. Furthermore, we empirically show the performance of our algorithm in simulated domains.

In most of the transfer learning approaches to reinforcement learning (RL) the distribution over the tasks is assumed to be stationary. Therefore, the target and source tasks are i.i.d. samples of the same distribution. Unfortunately, this assumption rarely holds in real-world conditions, e.g., due to seasonality or periodicity, evolution in the environment or faults in the sensors/actuators. In the context of this work, we consider the problem of transferring value functions through a variational method when the distribution that generates the tasks is time-variant, proposing a solution that leverages this temporal structure inherent in the task generating process. Furthermore, by means of a finite-sample analysis, the previously mentioned solution is theoretically compared to its time-invariant version. Finally, the experimental evaluation of the proposed technique is carried out on the lake Como water system representing a real-world scenario and on three different RL environments with three distinct temporal dynamics.

Temporal-Difference off-policy algorithms are among the building blocks of reinforcement learning (RL). Within this family, Q-Learning is arguably the most famous one, which has been widely studied and extended. The update rule of Q-learning involves the use of the maximum operator to estimate the maximum expected value of the return. However, this estimate is positively biased, and may hinder the learning process, especially in stochastic environments and when function approximation is used. We introduce the Weighted Estimator as an effective solution to mitigate the negative effects of overestimation in Q-Learning. The Weighted Estimator estimates the maximum expected value as a weighted sum of the action values, with the weights being the probabilities that each action value is the maximum. In this work, we study the problem from the statistical perspective of estimating the maximum expected value of a set of random variables and provide bounds to the bias and the variance of the Weighted Estimator, showing its advantages over other estimators present in literature. Then, we derive algorithms to enable the use of the Weighted Estimator, in place of the Maximum Estimator, in online and batch RL, and we introduce a novel algorithm for deep RL. Finally, we empirically evaluate our algorithms in a large set of heterogeneous problems, encompassing discrete and continuous, low and high dimensional, deterministic and stochastic environments. Experimental results show the effectiveness of the Weighted Estimator in controlling the bias of the estimate, resulting in better performance than representative baselines and robust learning w.r.t. a large set of diverse environments.

MushroomRL is an open-source Python library developed to simplify the process of implementing and running Reinforcement Learning (RL) experiments. Compared to other available libraries, MushroomRL has been created with the purpose of providing a comprehensive and flexible framework to minimize the effort in implementing and testing novel RL methodologies. The architecture of MushroomRL is built in such a way that every component of a typical RL experiment is already provided, and most of the time users can only focus on the implementation of their own algorithms. MushroomRL is accompanied by a benchmarking suite collecting experimental results of state-of-the-art deep RL algorithms, and allowing to benchmark new ones. The result is a library from which RL researchers can significantly benefit in the critical phase of the empirical analysis of their works. MushroomRL stable code, tutorials, and documentation can be found at https://github.com/MushroomRL/mushroom-rl.

This paper presents a study of the policy improvement step that can be usefully exploited by approximate policy-iteration algorithms. When either the policy evaluation step or the policy improvement step returns an approximated result, the sequence of policies produced by policy iteration may not be monotonically increasing, and oscillations may occur. To address this issue, we consider safe policy improvements, i.e., at each iteration, we search for a policy that maximizes a lower bound to the policy improvement w.r.t. the current policy, until no improving policy can be found. We propose three safe policy-iteration schemas that differ in the way the next policy is chosen w.r.t. the estimated greedy policy. Besides being theoretically derived and discussed, the proposed algorithms are empirically evaluated and compared on some chain-walk domains, the prison domain, and on the Blackjack card game.

Abstract In real-world applications, inferring the intentions of expert agents (e.g., human operators) can be fundamental to understand how possibly conflicting objectives are managed, helping to interpret the demonstrated behavior. In this paper, we discuss how inverse reinforcement learning (IRL) can be employed to retrieve the reward function implicitly optimized by expert agents acting in real applications. Scaling IRL to real-world cases has proved challenging as typically only a fixed dataset of demonstrations is available and further interactions with the environment are not allowed. For this reason, we resort to a class of truly batch model-free IRL algorithms and we present three application scenarios: (1) the high-level decision-making problem in the highway driving scenario, and (2) inferring the user preferences in a social network (Twitter), and (3) the management of the water release in the Como Lake. For each of these scenarios, we provide formalization, experiments and a discussion to interpret the obtained results.

Traditional model-based reinforcement learning approaches learn a model of the environment dynamics without explicitly considering how it will be used by the agent. In the presence of misspecified model classes, this can lead to poor estimates, as some relevant available information is ignored. In this paper, we introduce a novel model-based policy search approach that exploits the knowledge of the current agent policy to learn an approximate transition model, focusing on the portions of the environment that are most relevant for policy improvement. We leverage a weighting scheme, derived from the minimization of the error on the model-based policy gradient estimator, in order to define a suitable objective function that is optimized for learning the approximate transition model. Then, we integrate this procedure into a batch policy improvement algorithm, named Gradient-Aware Model-based Policy Search (GAMPS), which iteratively learns a transition model and uses it, together with the collected trajectories, to compute the new policy parameters. Finally, we empirically validate GAMPS on benchmark domains analyzing and discussing its properties.

What is a good exploration strategy for an agent that interacts with an environment in the absence of external rewards? Ideally, we would like to get a policy driving towards a uniform state-action visitation (highly exploring) in a minimum number of steps (fast mixing), in order to ease efficient learning of any goal-conditioned policy later on. Unfortunately, it is remarkably arduous to directly learn an optimal policy of this nature. In this paper, we propose a novel surrogate objective for learning highly exploring and fast mixing policies, which focuses on maximizing a lower bound to the entropy of the steady-state distribution induced by the policy. In particular, we introduce three novel lower bounds, that lead to as many optimization problems, that tradeoff the theoretical guarantees with computational complexity. Then, we present a model-based reinforcement learning algorithm, IDE3AL, to learn an optimal policy according to the introduced objective. Finally, we provide an empirical evaluation of this algorithm on a set of hard-exploration tasks.

In many Reinforcement Learning (RL) applications, the goal is to find an optimal deterministic policy. However, most RL algorithms require the policy to be stochastic in order to avoid instabilities and perform a sufficient amount of exploration. Adjusting the level of stochasticity during the learning process is non-trivial, as it is difficult to assess whether the costs of random exploration will be repaid in the long run, and to contain the risk of instability.We study this problem in the context of policy gradients (PG) with Gaussian policies. Using tools from the safe PG literature, we design a surrogate objective for the policy variance that captures the effects this parameter has on the learning speed and on the quality of the final solution. Furthermore, we provide a way to optimize this objective that guarantees stable improvement of the original performance measure. We evaluate the proposed methods on simulated continuous control tasks.

We consider Inverse Reinforcement Learning (IRL) about multiple intentions, \ie the problem of estimating the unknown reward functions optimized by a group of experts that demonstrate optimal behaviors. Most of the existing algorithms either require access to a model of the environment or need to repeatedly compute the optimal policies for the hypothesized rewards. However, these requirements are rarely met in real-world applications, in which interacting with the environment can be expensive or even dangerous. In this paper, we address the IRL about multiple intentions in a fully model-free and batch setting. We first cast the single IRL problem as a constrained likelihood maximization and then we use this formulation to cluster agents based on the likelihood of the assignment. In this way, we can efficiently solve, without interactions with the environment, both the IRL and the clustering problem. Finally, we evaluate the proposed methodology on simulated domains and on a real-world social-network application.

We study finite-armed stochastic bandits where the rewards of each arm might be correlated to those of other arms. We introduce a novel phased algorithm that exploits the given structure to build confidence sets over the parameters of the true bandit problem and rapidly discard all sub-optimal arms. In particular, unlike standard bandit algorithms with no structure, we show that the number of times a suboptimal arm is selected may actually be reduced thanks to the information collected by pulling other arms. Furthermore, we show that, in some structures, the regret of an anytime extension of our algorithm is uniformly bounded over time. For these constant-regret structures, we also derive a matching lower bound. Finally, we demonstrate numerically that our approach better exploits certain structures than existing methods.

In most Reinforcement Learning (RL) studies, the considered task is assumed to be stationary, i.e., it does not change its behavior or its characteristics over time, as this allows to generate all the convergence properties of RL techniques. Unfortunately, this assumption does not hold in real-world scenarios where systems and environments typically evolve over time. For instance, in robotic applications, sensor or actuator faults would induce a sudden change in the RL settings, while in financial applications the evolution of the market can cause a more gradual variation over time. In this paper, we present an adaptive RL algorithm able to detect changes in the environment or in the reward function and react to these changes by adapting to the new conditions of the task. At first, we develop a figure of merit onto which a hypothesis test can be applied to detect changes between two different learning iterations. Then, we extended this test to sequentially operate over time by means of the CUmulative SUM (CUSUM) approach. Finally, the proposed change-detection mechanism is combined (following an adaptive-active approach) with a well known RL algorithm to make it able to deal with non-stationary tasks. We tested the proposed algorithm on two well-known continuous-control tasks to check its effectiveness in terms of non-stationarity detection and adaptation over a vanilla RL algorithm.

The choice of the control frequency of a system has a relevant impact on the ability of reinforcement learning algorithms to learn a highly performing policy. In this paper, we introduce the notion of action persistence that consists in the repetition of an action for a fixed number of decision steps, having the effect of modifying the control frequency. We start analyzing how action persistence affects the performance of the optimal policy, and then we present a novel algorithm, Persistent Fitted Q-Iteration (PFQI), that extends FQI, with the goal of learning the optimal value function at a given persistence. After having provided a theoretical study of PFQI and a heuristic approach to identify the optimal persistence, we present an experimental campaign on benchmark domains to show the advantages of action persistence and proving the effectiveness of our persistence selection method.

We are interested in how to design reinforcement learning agents that provably reduce the sample complexity for learning new tasks by transferring knowledge from previously-solved ones. The availability of solutions to related problems poses a fundamental trade-off: whether to seek policies that are expected to immediately achieve high (yet sub-optimal) performance in the new task or whether to seek information to quickly identify an optimal solution, potentially at the cost of poor initial behaviour. In this work, we focus on the second objective when the agent has access to a generative model of state-action pairs. First, given a set of solved tasks containing an approximation of the target one, we design an algorithm that quickly identifies an accurate solution by seeking the state-action pairs that are most informative for this purpose. We derive PAC bounds on its sample complexity which clearly demonstrate the benefits of using this kind of prior knowledge. Then, we show how to learn these approximate tasks sequentially by reducing our transfer setting to a hidden Markov model and employing spectral methods to recover its parameters. Finally, we empirically verify our theoretical findings in simple simulated domains.

In a wide range of applied problems involving fluid flows, Computational Fluid Dynamics (CFD) provides detailed quantitative information on the flow field, at variable level of fidelity and computational cost. However, CFD alone cannot predict high-level functional properties that are not easily obtained from the equations of fluid motion. In this work, we present a data-driven framework to extract these additional information, such as medical diagnostic output, from CFD solutions. This is a challenging task because of the huge data dimensionality of CFD, and the limited training data that can be typically gathered due to the large computational cost of CFD. By pursuing a traditional Machine Learning (ML) pipeline of pre-processing, feature extraction, and model training, we demonstrate that informative features can be extracted from CFD data. Two experiments, pertaining to different application domains, support our claim that the convective properties implicit into a CFD solution can be leveraged to retrieve functional information that does not admit an analytical definition. Despite the preliminary nature of our study and the relative simplicity of both the geometrical and CFD models, for the first time we demonstrate that the combination of ML and CFD can diagnose a complex system in terms of high-level functional properties.

The use of reinforcement learning in algorithmic trading is of growing interest, since it offers the opportunity of making profit through the development of autonomous artificial traders, that do not depend on hard-coded rules. In such a framework, keeping uncertainty under control is as important as maximizing expected returns. Risk aversion has been addressed in reinforcement learning through measures related to the distribution of returns. However, in trading it is essential to keep under control the risk of portfolio positions in the intermediate steps. In this paper, we define a novel measure of risk, which we call reward volatility, consisting of the variance of the rewards under the state-occupancy measure. This new risk measure is shown to bound the return variance so that reducing the former also constrains the latter. We derive a policy gradient theorem with a new objective function that exploits the mean-volatility relationship. Furthermore, we adapt TRPO, the well-known policy gradient algorithm with monotonic improvement guarantees, in a risk-averse manner. Finally, we test the proposed approach in two financial environments using real market data.

Inverse Reinforcement Learning addresses the problem of inferring an expert's reward function from demonstrations. However, in many applications, we not only have access to the expert's near-optimal behaviour, but we also observe part of her learning process. In this paper, we propose a new algorithm for this setting, in which the goal is to recover the reward function being optimized by an agent, given a sequence of policies produced during learning. Our approach is based on the assumption that the observed agent is updating her policy parameters along the gradient direction. Then we extend our method to deal with the more realistic scenario where we only have access to a dataset of learning trajectories. For both settings, we provide theoretical insights into our algorithms' performance. Finally, we evaluate the approach in a simulated GridWorld environment and on the MuJoCo environments, comparing it with the state-of-the-art baseline.

In the contextual linear bandit setting, algorithms built on the optimism principle fail to exploit the structure of the problem and have been shown to be asymptotically suboptimal. In this paper, we follow recent approaches of deriving asymptotically optimal algorithms from problem-dependent regret lower bounds and we introduce a novel algorithm improving over the state-of-the-art along multiple dimensions. We build on a reformulation of the lower bound, where context distribution and exploration policy are decoupled, and we obtain an algorithm robust to unbalanced context distributions. Then, using an incremental primal-dual approach to solve the Lagrangian relaxation of the lower bound, we obtain a scalable and computationally efficient algorithm. Finally, we remove forced exploration and build on confidence intervals of the optimization problem to encourage a minimum level of exploration that is better adapted to the problem structure. We demonstrate the asymptotic optimality of our algorithm, while providing both problem-dependent and worst-case finite-time regret guarantees. Our bounds scale with the logarithm of the number of arms, thus avoiding the linear dependence common in all related prior works. Notably, we establish minimax optimality for any learning horizon in the special case of non-contextual linear bandits. Finally, we verify that our algorithm obtains better empirical performance than state-of-the-art baselines.

Multi-Armed Bandit (MAB) techniques have been successfully applied to many classes of sequential decision problems in the past decades. However, non-stationary settings -- very common in real-world applications -- received little attention so far, and theoretical guarantees on the regret are known only for some frequentist algorithms. In this paper, we propose an algorithm, namely Sliding-Window Thompson Sampling (SW-TS), for nonstationary stochastic MAB settings. Our algorithm is based on Thompson Sampling and exploits a sliding-window approach to tackle, in a unified fashion, two different forms of non-stationarity studied separately so far: abruptly changing and smoothly changing. In the former, the reward distributions are constant during sequences of rounds, and their change may be arbitrary and happen at unknown rounds, while, in the latter, the reward distributions smoothly evolve over rounds according to unknown dynamics. Under mild assumptions, we provide regret upper bounds on the dynamic pseudo-regret of SW-TS for the abruptly changing environment, for the smoothly changing one, and for the setting in which both the non-stationarity forms are present. Furthermore, we empirically show that SW-TS dramatically outperforms state-of-the-art algorithms even when the forms of non-stationarity are taken separately, as previously studied in the literature.

How can we effectively exploit the collected samples when solving a continuous control task with Reinforcement Learning? Recent results have empirically demonstrated that multiple policy optimization steps can be performed with the same batch by using off-distribution techniques based on importance sampling. However, when dealing with off-distribution optimization, it is essential to take into account the uncertainty introduced by the importance sampling process. In this paper, we propose and analyze a class of model-free, policy search algorithms that extend the recent Policy Optimization via Importance Sampling (Metelli et al., 2018) by incorporating two advanced variance reduction techniques: per-decision and multiple importance sampling. For both of them, we derive a high-probability bound, of independent interest, and then we show how to employ it to define a suitable surrogate objective function that can be used for both action-based and parameter-based settings. The resulting algorithms are finally evaluated on a set of continuous control tasks, using both linear and deep policies, and compared with modern policy optimization methods.

Configurable Markov Decision Processes (Conf-MDPs) have been recently introduced as an extension of the usual MDP model to account for the possibility of configuring the environment to improve the agent’s performance. Currently, there is still no suitable algorithm to solve the learning problem for real-world Conf-MDPs. In this paper, we fill this gap by proposing a trust-region method, Relative Entropy Model Policy Search (REMPS), able to learn both the policy and the MDP configuration in continuous domains without requiring the knowledge of the true model of the environment. After introducing our approach and providing a finite-sample analysis, we empirically evaluate REMPS on both benchmark and realistic environments by comparing our results with those of the gradient methods.

Policy Search (PS) is an effective approach to Reinforcement Learning (RL) for solving control tasks with continuous state-action spaces. In this paper, we address the exploration-exploitation trade-off in PS by proposing an approach based on Optimism in the Face of Uncertainty. We cast the PS problem as a suitable Multi Armed Bandit (MAB) problem, defined over the policy parameter space, and we propose a class of algorithms that effectively exploit the problem structure, by leveraging Multiple Importance Sampling to perform an off-policy estimation of the expected return. We show that the regret of the proposed approach is bounded by $\widetilde{\mathcal{O}}(\sqrt{T})$ for both discrete and continuous parameter spaces. Finally, we evaluate our algorithms on tasks of varying difficulty, comparing them with existing MAB and RL algorithms.

We consider the transfer of experience samples in reinforcement learning. Most of the previous works in this context focused on value-based settings, where transferring instances conveniently reduces to the transfer of (s,a,s’,r) tuples. In this paper, we consider the more complex case of reusing samples in policy search methods, in which the agent is required to transfer entire trajectories between environments with different transition models. By leveraging ideas from multiple importance sampling, we propose robust gradient estimators that effectively achieve this goal, along with several techniques to reduce their variance. In the case where the transition models are known, we theoretically establish the robustness to the negative transfer for our estimators. In the case of unknown models, we propose a method to efficiently estimate them when the target task belongs to a finite set of possible tasks and when it belongs to some reproducing kernel Hilbert space. We provide empirical results to show the effectiveness of our estimators.

Mutual information has been successfully adopted in filter feature-selection methods to assess both the relevancy of a subset of features in predicting the target variable and the redundancy with respect to other variables. However, existing algorithms are mostly heuristic and do not offer any guarantee on the proposed solution. In this paper, we provide novel theoretical results showing that conditional mutual information naturally arises when bounding the ideal regression/classification errors achieved by different subsets of features. Leveraging on these insights, we propose a novel stopping condition for backward and forward greedy methods which ensures that the ideal prediction error using the selected feature subset remains bounded by a user-specified threshold. We provide numerical simulations to support our theoretical claims and compare to common heuristic methods.

Most of the research in Reinforcement Learning (RL) focuses on balancing exploration and exploitation. Indeed, the reasons for the success or failure of an RL algorithm often deal with the choice between the execution of exploratory actions and the exploitation of actions that are known to be good. In the context of Multi-Armed Bandits (MABs), many algorithms have addressed this dilemma. In particular, Thompson Sampling (TS) is a solution that, besides having good theoretical properties, usually works very well in practice. Unfortunately, the success of TS in MAB problems has not been replicated in RL, where it has shown to scale very poorly w.r.t. the dimensionality of the problem. Nevertheless, the application of TS in RL, instead of more myopic strategies such as ε-greedy, remains a promising solution. This paper addresses such issue proposing several algorithms to use TS in RL and deep RL in a feasible way. We present these algorithms explaining the intuitions and theoretical considerations behind them and discussing their advantages and drawbacks. Furthermore, we provide an empirical evaluation on an increasingly complex set of RL problems, showing the benefit of TS w.r.t. other sampling strategies available in classical and more recent RL literature.

Exploring high-dimensional state spaces and finding sparse rewards are central problems in reinforcement learning. Exploration strategies are frequently either naïve (e.g., simplistic-greedy or Boltzmann policies), intractable (i.e., full Bayesian treatment of reinforcement learning) or rely heavily on heuristics. The lack of a tractable but principled exploration approach unnecessarily complicates the application of reinforcement learning to a broader range of problems. Efficient exploration can be accomplished by relying on the uncertainty of the state-action value function. To obtain the uncertainty, we maintain an ensemble of value function estimates and present an optimistic Bellman equation (OBE) for such ensembles. This OBE is derived from a relative entropy maximization principle and yields an implicit exploration bonus resulting in improved exploration during action selection. The implied exploration bonus can be seen as a well-principled type of intrinsic motivation and exhibits favorable theoretical properties. OBE can be applied to a wide range of algorithms. We propose two algorithms as an application of the principle: Optimistic Q-learning and Optimistic DQN which outperform comparison methods on standard benchmarks.

How does the uncertainty of the value function propagate when performing temporal difference learning? In this paper, we address this question by proposing a Bayesian framework in which we employ approximate posterior distributions to model the uncertainty of the value function and Wasserstein barycenters to propagate it across state-action pairs. Leveraging on these tools, we present an algorithm, Wasserstein Q-Learning (WQL), starting in the tabular case and then, we show how it can be extended to deal with continuous domains. Furthermore, we prove that, under mild assumptions, a slight variation of WQL enjoys desirable theoretical properties in the tabular setting. Finally, we present an experimental campaign to show the effectiveness of WQL on finite problems, compared to several RL algorithms, some of which are specifically designed for exploration, along with some preliminary results on Atari games.

In 2017, Internet ad spending reached 209 billion USD worldwide, while, e.g., TV ads brought in 178 billion USD. An Internet advertising campaign includes up to thousands of sub-campaigns on multiple channels, e.g., search, social, display, whose parameters (bid and daily budget) need to be optimized every day, subject to a (cumulative) budget constraint. Such a process is often unaffordable for humans and its automation is crucial. As also shown by marketing funnel models, the sub-campaigns are usually interdependent, e.g., display ads induce awareness, increasing the number of impressions-and, thus, also the number of conversions-of search ads. This interdependence is widely exploited by humans in the optimization process, whereas, to the best of our knowledge, no algorithm takes it into account. In this paper, we provide the first model capturing the sub-campaigns interdependence. We also provide the IDIL algorithm, which, employing Granger Causality and Gaussian Processes, learns from past data, and returns an optimal stationary bid/daily budget allocation. We prove theoretical guarantees on the loss of IDIL w.r.t. the clairvoyant solution, and we show empirical evidence of its superiority in both realistic and real-world settings when compared with existing approaches.

Pay-per-click advertising includes various formats (e.g., search, contextual, and social) with a total investment of more than 140 billion USD per year. An advertising campaign is composed of some subcampaigns-each with a different ad-and a cumulative daily budget. The allocation of the ads is ruled exploiting auction mechanisms. In this paper, we propose, for the first time to the best of our knowledge, an algorithm for the online joint bid/budget optimization of pay-per-click multi-channel advertising campaigns. We formulate the optimization problem as a combinatorial bandit problem, in which we use Gaussian Processes to estimate stochastic functions, Bayesian bandit techniques to address the exploration/exploitation problem, and a dynamic programming technique to solve a variation of the Multiple-Choice Knapsack problem. We experimentally evaluate our algorithm both in simulation-using a synthetic setting generated from real data from Yahoo!-and in a real-world application over an advertising period of two months.

In many real-world problems, there is the possibility to configure, to a limited extent, some environmental parameters to improve the performance of a learning agent. In this paper, we propose a novel framework, Configurable Markov Decision Processes (Conf-MDPs), to model this new type of interaction with the environment. Furthermore, we provide a new learning algorithm, Safe Policy-Model Iteration (SPMI), to jointly and adaptively optimize the policy and the environment configuration. After having introduced our approach and derived some theoretical results, we present the experimental evaluation in two explicative problems to show the benefits of the environment configurability on the performance of the learned policy.

In this paper, we propose a novel reinforcement-learning algorithm consisting in a stochastic variance-reduced version of policy gradient for solving Markov Decision Processes (MDPs). Stochastic variance-reduced gradient (SVRG) methods have proven to be very successful in supervised learning. However, their adaptation to policy gradient is not straightforward and needs to account for I) a non-concave objective function; II) approximations in the full gradient computation; and III) a non-stationary sampling process. The result is SVRPG, a stochastic variance-reduced policy gradient algorithm that leverages on importance weights to preserve the unbiasedness of the gradient estimate. Under standard assumptions on the MDP, we provide convergence guarantees for SVRPG with a convergence rate that is linear under increasing batch sizes. Finally, we suggest practical variants of SVRPG, and we empirically evaluate them on continuous MDPs.

We consider the transfer of experience samples (i.e., tuples < s, a, s’, r >) in reinforcement learning (RL), collected from a set of source tasks to improve the learning process in a given target task. Most of the related approaches focus on selecting the most relevant source samples for solving the target task, but then all the transferred samples are used without considering anymore the discrepancies between the task models. In this paper, we propose a model-based technique that automatically estimates the relevance (importance weight) of each source sample for solving the target task. In the proposed approach, all the samples are transferred and used by a batch RL algorithm to solve the target task, but their contribution to the learning process is proportional to their importance weight. By extending the results for importance weighting provided in supervised learning literature, we develop a finite-sample analysis of the proposed batch RL algorithm. Furthermore, we empirically compare the proposed algorithm to state-of-the-art approaches, showing that it achieves better learning performance and is very robust to negative transfer, even when some source tasks are significantly different from the target task.

In the last two decades, online advertising has become the most effective way to sponsor a product or an event. The success of this advertising format is mainly due to the capability of the Internet channels to reach a broad audience and to target different groups of users with specific sponsored announces. This is of paramount importance for media agencies, companies whose primary goal is to design ad campaigns that target only those users who are interested in the sponsored product, thus avoiding unnecessary costs due to the display of ads to uninterested users. In the present work, we develop an automatic method to find the best user targets (a.k.a. contexts) that a media agency can use in a given Internet advertising campaign. More specifically, we formulate the problem of target optimization as a Learning from Logged Bandit Feedback (LLBF) problem, and we propose the TargOpt algorithm, which uses a tree expansion of the target space to learn the partition that efficiently maximizes the campaign revenue. Furthermore, since the problem of finding the optimal target is intrinsically exponential in the number of the features, we propose a tree-search method, called A-TargOpt, and two heuristics to drive the tree expansion, aiming at providing an anytime solution. Finally, we present empirical evidence, on both synthetically generated and real-world data, that our algorithms provide a practical solution to find effective targets for Internet advertising.

Policy optimization is an effective reinforcement learning approach to solve continuous control tasks. Recent achievements have shown that alternating online and offline optimization is a successful choice for efficient trajectory reuse. However, deciding when to stop optimizing and collect new trajectories is non-trivial, as it requires to account for the variance of the objective function estimate. In this paper, we propose a novel, model-free, policy search algorithm, POIS, applicable in both action-based and parameter-based settings. We first derive a high-confidence bound for importance sampling estimation; then we define a surrogate objective function, which is optimized offline whenever a new batch of trajectories is collected. Finally, the algorithm is tested on a selection of continuous control tasks, with both linear and deep policies, and compared with state-of-the-art policy optimization methods.

We consider the problem of transferring value functions in reinforcement learning. We propose an approach that uses the given source tasks to learn a prior distribution over optimal value functions and provide an efficient variational approximation of the corresponding posterior in a new target task. We show our approach to be general, in the sense that it can be combined with complex parametric function approximators and distribution models, while providing two practical algorithms based on Gaussians and Gaussian mixtures. We theoretically analyze them by deriving a finite-sample analysis and provide a comprehensive empirical evaluation in four different domains.

This paper is about the estimation of the maximum expected value of an infinite set of random variables.This estimation problem is relevant in many fields, like the Reinforcement Learning (RL) one.In RL it is well known that, in some stochastic environments, a bias in the estimation error can increase step-by-step the approximation error leading to large overestimates of the true action values. Recently, some approaches have been proposed to reduce such bias in order to get better action-value estimates, but are limited to finite problems.In this paper, we leverage on the recently proposed weighted estimator and on Gaussian process regression to derive a new method that is able to natively handle infinitely many random variables.We show how these techniques can be used to face both continuous state and continuous actions RL problems.To evaluate the effectiveness of the proposed approach we perform empirical comparisons with related approaches.

We study, to the best of our knowledge, the first Bayesian algorithm for unimodal Multi-Armed Bandit (MAB) problems with graph structure. In this setting, each arm corresponds to a node of a graph and each edge provides a relationship, unknown to the learner, between two nodes in terms of expected reward. Furthermore, for any node of the graph there is a path leading to the unique node providing the maximum expected reward, along which the expected reward is monotonically increasing. Previous results on this setting describe the behavior of frequentist MAB algorithms. In our paper, we design a Thompson Sampling-based algorithm whose asymptotic pseudo-regret matches the lower bound for the considered setting. We show that -as it happens in a wide number of scenarios- Bayesian MAB algorithms dramatically outperform frequentist ones. In particular, we provide a thorough experimental evaluation of the performance of our and state-of-the-art algorithms as the properties of the graph vary.

This paper is about the study of B-FQI, an Approximated Value Iteration (AVI) algorithm that exploits a boosting procedure to estimate the action-value function in reinforcement learning problems. B-FQI is an iterative off-line algorithm that, given a dataset of transitions, builds an approximation of the optimal action-value function by summing the approximations of the Bellman residuals across all iterations. The advantage of such approach w.r.t. to other AVI methods is twofold: (1) while keeping the same function space at each iteration, B-FQI can represent more complex functions by considering an additive model; (2) since the Bellman residual decreases as the optimal value function is approached, regression problems become easier as iterations proceed. We study B-FQI both theoretically, providing also a finite-sample error upper bound for it, and empirically, by comparing its performance to the one of FQI in different domains and using different regression techniques.

Logged data is one of the most widespread form of recorded information, since it can be acquired by almost any system and stored at a little cost. Customarily, the interaction logs between the system and a user (or environment) present the structure of a sequential decision process: given a context, the system performs an action and the user provides a feedback about it. This structure is common to a wide range of real-world micro-economic applications, e.g., e-commerce websites and advertisement campaigns. The problem of learning a policy from such logged interactions to take more profitable decisions in the future is known as the Learning from Logged Bandit Feedback (LLBF) problem. In this paper, we propose RADT, an algorithm specifically shaped for the LLBF setting and based on a risk-averse learning method which exploits the joint use of regression trees and statistical confidence bounds. Differently from existing techniques developed for this setting, RADT generates policies aiming to maximize a lower bound on the expected reward and provides a clear characterization of those features in the context that influence the process the most. Finally, we provide a wide experimental campaign over both synthetic and real-world datasets showing empirical evidence that RADT outperforms both state-of-the-art machine learning classification and regression techniques and existing methods addressing the LLBF setting.

Inverse Reinforcement Learning (IRL) is an effective approach to recover a reward function that explains the behavior of an expert by observing a set of demonstrations. This paper is about a novel model-free IRL approach that, differently from most of the existing IRL algorithms, does not require to specify a function space where to search for the expert's reward function. Leveraging on the fact that the policy gradient needs to be zero for any optimal policy, the algorithm generates a set of basis functions that span the subspace of reward functions that make the policy gradient vanish. Within this subspace, using a second-order criterion, we search for the reward function that penalizes the most a deviation from the expert's policy. After introducing our approach for finite domains, we extend it to continuous ones. The proposed approach is empirically compared to other IRL methods both in the (finite) Taxi domain and in the (continuous) Linear Quadratic Gaussian (LQG) and Car on the Hill environments.

Policy gradient methods are among the best Reinforcement Learning (RL) techniques to solve complex control problems. In real-world RL applications, it is common to have a good initial policy whose performance needs to be improved and it may not be acceptable to try bad policies during the learning process. Although several methods for choosing the step size exist, research paid less attention to determine the batch size, that is the number of samples used to estimate the gradient direction for each update of the policy parameters. In this paper, we propose a set of methods to jointly optimize the step and the batch sizes that guarantee (with high probability) to improve the policy performance after each update. Besides providing theoretical guarantees, we show numerical simulations to analyse the behaviour of our methods.

Evolutionary game theory provides the principal tools to model the dynamics of multi-agent learning algorithms. While there is a long-standing literature on evolutionary game theory in strategic-form games, in the case of extensive-form games few results are known and the exponential size of the representations currently adopted makes the evolutionary analysis of such games unaffordable. In this paper, we focus on dynamics for the sequence form of extensive-form games, providing three dynamics: one realization equivalent to the normal-form logit dynamic, one realization equivalent to the agent-form replicator dynamic, and one realization equivalent to the agent-form logit dynamic. All the considered dynamics require polynomial time and space, providing an exponential compression w.r.t. the dynamics currently known and providing thus tools that can be effectively employed in practice. Moreover, we use our tools to compare the agent-form and normal-form dynamics and to provide new "hybrid" dynamics.

Inverse Reinforcement Learning (IRL) deals with the problem of recovering the reward function optimized by an expert given a set of demonstrations of the expert's policy.Most IRL algorithms need to repeatedly compute the optimal policy for different reward functions.This paper proposes a new IRL approach that allows to recover the reward function without the need of solving any "direct" RL problem.The idea is to find the reward function that minimizes the gradient of a parameterized representation of the expert's policy.In particular, when the reward function can be represented as a linear combination of some basis functions, we will show that the aforementioned optimization problem can be efficiently solved.We present an empirical evaluation of the proposed approach on a multidimensional version of the Linear-Quadratic Regulator (LQR) both in the case where the parameters of the expert's policy are known and in the (more realistic) case where the parameters of the expert's policy need to be inferred from the expert's demonstrations.Finally, the algorithm is compared against the state-of-the-art on the mountain car domain, where the expert's policy is unknown.

Multi–Armed Bandits (MABs) have been widely considered in the last decade to model settings in which an agent wants to learn the action providing the highest expected reward among a fixed set of available actions during the operational life of a system. Classical techniques provide solutions that minimize the regret due to learning in settings where selecting an arm has no cost. Though, in many real world applications the learner has to pay some cost for pulling each arm and the learning process is constrained by a fixed budget B. This problem is addressed in the literature as the Budgeted MAB (BMAB). In this paper, for the first time, we study the problem of Budgeted Continuous–Armed Bandit (BCAB), where the set of the possible actions consists in a continuous set (e.g., a range of prices) and the learner suffers from a random reward and cost at each round. We provide a novel algorithm, named B–Zoom, which suffers a regret of, where d is the Zooming dimension of the problem. Finally, we provide an empirical analysis showing that, despite a lower average performance, the proposed approach is more robust to adverse settings as compared to existing algorithms designed for BMAB.

This paper is about the estimation of the maximum expected value of a set of independent random variables. The performance of several learning algorithms (e.g., Q-learning) is affected by the accuracy of such estimation. Unfortunately, no unbiased estimator exists. The usual approach of taking the maximum of the sample means leads to large overestimates that may significantly harm the performance of the learning algorithm. Recent works have shown that the cross validation estimator—which is negatively biased—outperforms the maximum estimator in many sequential decision-making scenarios. On the other hand, the relative performance of the two estimators is highly problem-dependent. In this paper, we propose a new estimator for the maximum expected value, based on a weighted average of the sample means, where the weights are computed using Gaussian approximations for the distributions of the sample means. We compare the proposed estimator with the other state-of-the-art methods both theoretically, by deriving upper bounds to the bias and the variance of the estimator, and empirically, by testing the performance on different sequential learning problems.

Many real-world control applications, from economics to robotics, are characterized by the presence of multiple conflicting objectives. In these problems, the standard concept of optimality is replaced by Pareto-optimality and the goal is to find the Pareto frontier, a set of solutions representing different compromises among the objectives. Despite recent advances in multi-objective optimization, achieving an accurate representation of the Pareto frontier is still an important challenge. In this paper, we propose a reinforcement learning policy gradient approach to learn a continuous approximation of the Pareto frontier in multi-objective Markov Decision Problems (MOMDPs). Differently from previous policy gradient algorithms, where n optimization routines are executed to have n solutions, our approach performs a single gradient ascent run, generating at each step an improved continuous approximation of the Pareto frontier. The idea is to optimize the parameters of a function defining a manifold in the policy parameters space, so that the corresponding image in the objectives space gets as close as possible to the true Pareto frontier. Besides deriving how to compute and estimate such gradient, we will also discuss the non-trivial issue of defining a metric to assess the quality of the candidate Pareto frontiers. Finally, the properties of the proposed approach are empirically evaluated on two problems, a linear-quadratic Gaussian regulator and a water reservoir control task.

Classical approximate dynamic programming techniques based on state-space gridding become computationally impracticable for high-dimensional problems. Policy search techniques cope with this curse of dimensionality issue by searching for the optimal control policy in a restricted parameterized policy space. We here focus on the case of discrete action space and introduce a novel policy parametrization that adopts particles to describe the map from the state space to the action space, each particle representing a region of the state space that is mapped into a certain action. The locations and actions associated with the particles describing a policy can be tuned by means of a recently introduced policy gradient method with parameter-based exploration. The task of selecting an appropriately sized set of particles is here solved through an iterative policy building scheme that adds new particles to improve the policy performance and is also capable of removing redundant particles. Experiments demonstrate the scalability of the proposed approach as the dimensionality of the state-space grows.

This paper is about learning a continuous approximation of the Pareto frontier in Multi-Objective Markov Decision Problems (MOMDPs).We propose a policy-based approach that exploits gradient information to generate solutions close to the Pareto ones.Differently from previous policy-gradient multi-objective algorithms, where n optimization routines are used to have n solutions, our approach performs a single gradient-ascent run that at each step generates an improved continuous approximation of the Pareto frontier.The idea is to exploit a gradient-based approach to optimize the parameters of a function that defines a manifold in the policy parameter space so that the corresponding image in the objective space gets as close as possible to the Pareto frontier.Besides deriving how to compute and estimate such gradient, we will also discuss the non-trivial issue of defining a metric to assess the quality of the candidate Pareto frontiers.Finally, the properties of the proposed approach are empirically evaluated on two interesting MOMDPs.

To perform many common industrial robotic tasks, e.g. deburring a work-piece, in small and medium size companies where a model of the work-piece may not be available, building a geometrical model of how to perform the task from a data set of human demonstrations is highly demanded. In many cases, however, the human demonstrations may be sub-optimal and noisy solutions to the problem of performing a task. For example, an expert may not completely remove the burrs that result in deburring residuals on the work-piece. Hence, we present an iterative algorithm to estimate a noise-free geometrical model of a work-piece from a given dataset of profiles with deburring residuals. In a case study, we compare the profiles obtained with the proposed method, nonlinear principal component analysis and Gaussian mixture model/Gaussian mixture regression. The comparison illustrates the effectiveness of the proposed method, in terms of accuracy, to compute a noise-free profile model of a task.

This paper investigates the use of second-order methods to solve Markov Decision Processes (MDPs). Despite the popularity of second-order methods in optimization literature, so far little attention has been paid to the extension of such techniques to face sequential decision problems. Here we provide a model-free Reinforcement Learning method that estimates the Newton direction by sampling directly in the parameter space. In order to compute the Newton direction we provide the formulation of the Hessian of the expected return, a technique for variance reduction in the sample-based estimation and a finite sample analysis in the case of Normal distribution. Beside discussing the theoretical properties, we empirically evaluate the method on an instructional linear-quadratic regulator and on a complex dynamical quadrotor system.

Multi-agent learning is a challenging open task in artificial intelligence. It is known an interesting connection between multi-agent learning algorithms and evolutionary game theory, showing that the learning dynamics of some algorithms can be modeled as replicator dynamics with a mutation term. Inspired by the recent sequence-form replicator dynamics, we develop a new version of the Q-learning algorithm working on the sequence form of an extensive-form game allowing thus an exponential reduction of the dynamics length w.r.t. those of the normal form. The dynamics of the proposed algorithm can be modeled by using the sequence-form replicator dynamics with a mutation term. We show that, although sequence-form and normal-form replicator dynamics are realization equivalent, the Q-learning algorithm applied to the two forms have non-realization equivalent dynamics. Originally from the previous works on evolutionary game theory models form multi-agent learning, we produce an experimental evaluation to show the accuracy of the model.

This paper investigates the use of policy gradient techniques to approximate the Pareto frontier in Multi-Objective Markov Decision Processes (MOMDPs). Despite the popularity of policy-gradient algorithms and the fact that gradient-ascent algorithms have been already proposed to numerically solve multi-objective optimization problems, especially in combination with multi-objective evolutionary algorithms, so far little attention has been paid to the use of gradient information to face multi-objective sequential decision problems. Three different Multi-Objective Reinforcement-Learning (MORL) approaches are here presented. The first two, called radial and Pareto following, start from an initial policy and perform gradient-based policy-search procedures aimed at finding a set of non-dominated policies. Differently, the third approach performs a single gradient-ascent run that, at each step, generates an improved continuous approximation of the Pareto frontier. The parameters of a function that defines a manifold in the policy parameter space are updated following the gradient of some performance criterion so that the sequence of candidate solutions gets as close as possible to the Pareto front. Besides reviewing the three different approaches and discussing their main properties, we empirically compare them with other MORL algorithms on two interesting MOMDPs.

This paper investigates the use of policy gradient techniques to approximate the Pareto frontier in Multi-Objective Markov Decision Processes (MOMDPs). Despite the popularity of policy gradient algorithms and the fact that gradient ascent algorithms have been already proposed to numerically solve multi-objective optimization problems, especially in combination with multi-objective evolutionary algorithms, so far little attention has been paid to the use of gradient information to face multi-objective sequential decision problems. Two different Multi-Objective Reinforcement-Learning (MORL) approaches, called radial and Pareto following, that, starting from an initial policy, perform gradient-based policy-search procedures aimed at finding a set of non-dominated policies are here presented. Both algorithms are empirically evaluated and compared to state-of-the-art MORL algorithms on three MORL benchmark problems.

Evolutionary game theory combines game theory and dynamical systems and is customarily adopted to describe evolutionary dynamics in multi-agent systems. In particular, it has been proven to be a successful tool to describe multi-agent learning dynamics. To the best of our knowledge, we provide in this paper the first replicator dynamics applicable to the sequence form of an extensive-form game, allowing an exponential reduction of time and space w.r.t. the currently adopted replicator dynamics for normal form. Furthermore, our replicator dynamics is realization equivalent to the standard replicator dynamics for normal form. We prove our results for both discrete-time and continuous-time cases. Finally, we extend standard tools to study the stability of a strategy profile to our replicator dynamics.

This paper presents a study of the policy improvement step that can be usefully exploited by approximate policy-iteration algorithms. When either the policy evaluation step or the policy improvement step returns an approximated result, the sequence of policies produced by policy iteration may not be monotonically increasing, and oscillations may occur. To address this issue, we consider safe policy improvements, i.e., at each iteration we search for a policy that maximizes a lower bound to the policy improvement w.r.t. the current policy. When no improving policy can be found the algorithm stops. We propose two safe policy-iteration algorithms that differ in the way the next policy is chosen w.r.t. the estimated greedy policy. Besides being theoretically derived and discussed, the proposed algorithms are empirically evaluated and compared with state-of-the-art approaches on some chain-walk domains and on the Blackjack card game.

In the last decade, policy gradient methods have significantly grown in popularity in the reinforcement--learning field. In particular, they have been largely employed in motor control and robotic applications, thanks to their ability to cope with continuous state and action domains and partial observable problems. Policy gradient researches have been mainly focused on the identification of effective gradient directions and the proposal of efficient estimation algorithms. Nonetheless, the performance of policy gradient methods is determined not only by the gradient direction, since convergence properties are strongly influenced by the choice of the step size: small values imply slow convergence rate, while large values may lead to oscillations or even divergence of the policy parameters. Step--size value is usually chosen by hand tuning and still little attention has been paid to its automatic selection. In this paper, we propose to determine the learning rate by maximizing a lower bound to the expected performance gain. Focusing on Gaussian policies, we derive a lower bound that is second--order polynomial of the step size, and we show how a simplified version of such lower bound can be maximized when the gradient is estimated from trajectory samples. The properties of the proposed approach are empirically evaluated in a linear--quadratic regulator problem.

This paper is about solving multi-objective control problems using a model-free batch-mode reinforcement-learning approach. Although many real-world applications have several conflicting objectives, reinforcement-learning (RL) literature has mainly focused on single-objective control problems. As a consequence, in the presence of multiple objectives, the usual approach is to consider many single-objective control problems (resulting from different combinations of the original problem objectives), each one solved using standard RL techniques. The algorithm proposed in this paper is an extension of Fitted Q-iteration (FQI) that enables to learn the control policies for all the linear combinations of preferences (weights) assigned to the objectives in a single training process. The key idea of multi-objective FQI (MOFQI) is to enlarge the continuous approximation of the action-value function, which is performed by single-objective FQI over the state-action space, also to the weight space. The approach is demonstrated on an interesting real-world application for multi-objective RL algorithms: the optimal operation of a multi-purpose water reservoir.

This paper is about dimensionality reduction by variable selection in high-dimensional real-world control problems, where designing controllers by conventional means is either impractical or results in poor performance.

In this paper we address the combination of batch reinforcement-learning (BRL) techniques with direct policy search (DPS) algorithms in the context of robot learning. Batch value-based algorithms (such as fitted Q-iteration) have been proved to outperform online ones in many complex applications, but they share the same difficulties in solving problems with continuous action spaces, such as robotic ones. In these cases, actor-critic and DPS methods are preferable, since the optimization process is limited to a family of parameterized (usually smooth) policies. On the other hand, these methods (e.g., policy gradient and evolutionary methods) are generally very expensive, since finding the optimal parameterization may require to evaluate the performance of several policies, which in many real robotic applications is unfeasible or even dangerous. To overcome such problems, we exploit the fitted policy search (FPS) approach, in which the expected return of any policy considered during the optimization process is evaluated offline (without resorting to the robot) by reusing the data collected in the initial exploration phase. In this way, it is possible to take the advantages of both BRL and DPS algorithms, thus achieving an effective learning approach to solve robotic problems. A balancing task on a real two-wheeled robotic pendulum is used to analyze the properties and evaluate the effectiveness of the FPS approach.

Transfer reinforcement learning (RL) methods leverage on the experience collected on a set of source tasks to speed-up RL algorithms. A simple and effective approach is to transfer samples from source tasks and include them in the training set used to solve a target task. In this paper, we investigate the theoretical properties of this transfer method and we introduce novel algorithms adapting the transfer process on the basis of the similarity between source and target tasks. Finally, we report illustrative experimental results in a continuous chain problem.

The main objective of transfer in reinforcement learning is to reduce the complexity of learning the solution of a target task by effectively reusing the knowledge retained from solving a set of source tasks. In this paper, we introduce a novel algorithm that transfers samples (i.e., tuples 〈s, a, s', r〉) from source to target tasks. Under the assumption that tasks have similar transition models and reward functions, we propose a method to select samples from the source tasks that are mostly similar to the target task, and, then, to use them as input for batch reinforcement-learning algorithms. As a result, the number of samples an agent needs to collect from the target task to learn its solution is reduced. We empirically show that, following the proposed approach, the transfer of samples is effective in reducing the learning complexity, even when some source tasks are significantly different from the target task.

Learning in real-world domains often requires to deal with continuous state and action spaces. Although many solutions have been proposed to apply Reinforce- ment Learning algorithms to continuous state problems, the same techniques can be hardly extended to continuous action spaces, where, besides the computation of a good approximation of the value function, a fast method for the identification of the highest-valued action is needed. In this paper, we propose a novel actor-critic approach in which the policy of the actor is estimated through sequential Monte Carlo methods. The importance sampling step is performed on the basis of the values learned by the critic, while the resampling step modifies the actor’s policy. The proposed approach has been empirically compared to other learning algo- rithms into several domains; in this paper, we report results obtained in a control problem consisting of steering a boat across a river.

In this paper, we present a dead reckoning sensor to support reliable odometry on mobile robots. This sensor is based on a pair of optical mice rigidly connected to the robot body and its main advantages are 1) this localization system is independent from the kinematics of the robot, 2) the measurement given by the mice is not subject to slipping, since they are independent from the traction wheels, nor to crawling, since they measure displacements in any direction 3) it is a low-cost solution with a precision comparable to classical shaft encoders. Since we have redundant measures it is possible to detect non-systematic errors; in this paper, an automatic procedure to reduce non-systematic errors of the sensor is presented and validated with experimental results on a real mobile robot.

In this paper, we present a dead reckoning sensor to support reliable odometry on mobile robots. This sensor is based on a pair of optical mice rigidly connected to the robot body and its main advantages are 1) this localization system is independent from the kinematics of the robot, 2) the measurement given by the mice is not subject to slipping, since they are independent from the traction wheels, nor to crawling, since they measure displacements in any direction 3) it is a low-cost solution with a precision comparable to classical shaft encoders. We present the mathematical model of the sensor, its implementation, and some experimental evaluations using the standard UMBmark benchmark for odometry.

We present a mobile robot localization method for known 2D environments, which is an evidence accumulation method where the complexity is reduced by means of a multi-resolution scheme. The method is called multi-resolution evidence accumulation (MUREA). The added values of the work include: the capability of the system to accept both raw sensor data as well as independently generated localization estimates; the capability of the system to be both a global or local localization system, depending on the availability of a global estimate; and the capability of the system to give out an accurate estimate when required (e.g. before its regular completion), which could be called any-time localization. We elaborate and evaluate a strategy for travelling the search-space, which expands the pose current estimate alternating between subspaces. Real experiments, based on omnidirectional sensing in an indoor environment, are presented.