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.

The success of sequential decision-making approaches, such as reinforcement learning (RL), is closely tied to the availability of a reward feedback. However, designing a reward function that encodes the desired objective is a challenging task. In this work, we address a more realistic scenario: sequential decision making with preference feedback provided, for instance, by a human expert. We aim to build a theoretical basis linking preferences, (non-Markovian) utilities, and (Markovian) rewards, and we study the connections between them. First, we model preference feedback using a partial (pre)order over trajectories, enabling the presence of incomparabilities that are common when preferences are provided by humans but are surprisingly overlooked in existing works. Second, to provide a theoretical justification for a common practice, we investigate how a preference relation can be approximated by a multi-objective utility. We introduce a notion of preference-utility compatibility and analyze the computational complexity of this transformation, showing that constructing the minimum-dimensional utility is NP-hard. Third, we propose a novel concept of preference-based policy dominance that does not rely on utilities or rewards and discuss the computational complexity of assessing it. Fourth, we develop a computationally efficient algorithm to approximate a utility using (Markovian) rewards and quantify the error in terms of the suboptimality of the optimal policy induced by the approximating reward. This work aims to lay the foundation for a principled approach to sequential decision making from preference feedback, with promising potential applications in RL from human feedback.

In the standard Reinforcement Learning (RL) paradigm, the action space is assumed to be fixed and immutable throughout the learning process. However, in many real-world scenarios, not all actions are available at every decision stage. The available action set may depend on the current environment state, domain-specific constraints, or other (potentially stochastic) factors outside the agent’s control. To address these realistic scenarios, we introduce a novel paradigm called Sleeping Reinforcement Learning, where the available action set varies during the interaction with the environment. We start with the simpler scenario in which the available action sets are revealed at the beginning of each episode. We show that a modification of UCBVI achieves regret of order $\widetilde{\mathcal{O}}(H\sqrt{SAT})$, where $H$ is the horizon, $S$ and $A$ are the cardinalities of the state and action spaces, respectively, and $T$ is the learning horizon. Next, we address the more challenging and realistic scenario in which the available actions are disclosed only at each decision stage. By leveraging a novel construction, we establish a minimax lower bound of order $\Omega(\sqrt{T 2^{A/2}})$ when the availability of actions is governed by a Markovian process, establishing a statistical barrier of the problem. Focusing on the statistically tractable case where action availability depends only on the current state and stage, we propose a new optimistic algorithm that achieves regret guarantees of order $\widetilde{\mathcal{O}}(H\sqrt{SAT})$, showing that the problem shares the same complexity of standard RL.

*Policy gradient* (PG) methods are effective *reinforcement learning* (RL) approaches, particularly for continuous problems. While they optimize stochastic (hyper)policies via action- or parameter-space exploration, real-world applications often require deterministic policies. Existing PG convergence guarantees to deterministic policies assume a fixed stochasticity in the (hyper)policy, tuned according to the desired final suboptimality, whereas practitioners commonly use a dynamic stochasticity level. This work provides the theoretical foundations for this practice. We introduce PES, a phase-based method that reduces stochasticity via a deterministic schedule while running PG subroutines with fixed stochasticity in each phase. Under gradient domination assumptions, PES achieves last-iterate convergence to the optimal deterministic policy with a sample complexity of order $\widetilde{\mathcal{O}}(\epsilon^{-5})$. Additionally, we analyze the common practice, termed SL-PG, of jointly learning stochasticity (via an appropriate parameterization) and (hyper)policy parameters. We show that SL-PG also ensures last-iterate convergence with a rate $\widetilde{\mathcal{O}}(\epsilon^{-3})$, but to the optimal stochastic (hyper)policy only, requiring stronger assumptions compared to PES.

*Restless* Multi-Armed Bandits (MABs) are a general framework designed to handle real-world decision-making problems where the expected rewards evolve over time, such as in recommender systems and dynamic pricing. In this work, we investigate from a theoretical standpoint two well-known structured subclasses of restless MABs: the *rising* and the *rising concave* settings, where the expected reward of each arm evolves over time following an unknown *non-decreasing* and a *non-decreasing concave* function, respectively. By providing a novel methodology of independent interest for general restless bandits, we establish new lower bounds on the expected cumulative regret for both settings. In the rising case, we prove a lower bound of order $\Omega(T^{2/3})$, matching known upper bounds for restless bandits; whereas, in the rising concave case, we derive a lower bound of order $\Omega(T^{3/5})$, proving for the first time that this setting is provably more challenging than stationary MABs. Then, we introduce Rising Concave Budgeted Exploration (RC-BE($\alpha$)), a new regret minimization algorithm designed for the rising concave MABs. By devising a novel proof technique, we show that the expected cumulative regret of RC-BE($\alpha$) is in the order of $\widetilde{\mathcal{O}}(T^{7/11})$. These results collectively make a step towards closing the gap in rising concave MABs, positioning them between stationary and general restless bandit settings in terms of statistical complexity.

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

The (expected cumulative) regret is the customary index to judge the performance of online sequential decision-making algorithms. In the traditional form, it is defined as the expected sum over a learning horizon T of the sub-optimality gaps ∆It (i.e., expected instantaneous regret) the agent suffers when playing arm It at round t. In this paper, we propose and investigate a generalization of this notion, named g-(expected cumulative) regret, obtained by applying a transformation function g to the sub-optimality gaps, making the agent suffer g(∆It) instead of just ∆It . Intuitively, function g embeds the “perception” that the agent manifests when performing a sub-optimal decision. We first show that sublinear g-regret is not achievable for a generic transformation function g. Then, we introduce a mild condition on g and provide instance-dependent and worst-case (i.e., minimax) lower bounds for the g-regret. Finally, we show that state-of-the-art stochastic bandit algorithms with no modification surprisingly display optimal performances for the g-regret. Specifically, we prove that UCB1 matches (up to constant factors) the instance-dependent lower bound regardless of function g and that MOSS matches (up to constant factors) the minimax lower bound at least for a wide class of transformation functions.

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.

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.

Policy gradient (PG) methods are successful approaches to deal with continuous reinforcement learning (RL) problems. They learn stochastic parametric (hyper)policies by either exploring in the space of actions or in the space of parameters. Stochastic controllers, however, are often undesirable from a practical perspective because of their lack of robustness, safety, and traceability. In common practice, stochastic (hyper)policies are learned only to deploy their deterministic version. In this paper, we make a step towards the theoretical understanding of this practice. After introducing a novel framework for modeling this scenario, we study the global convergence to the best deterministic policy, under (weak) gradient domination assumptions. Then, we illustrate how to tune the exploration level used for learning to optimize the trade-off between the sample complexity and the performance of the deployed deterministic policy. Finally, we quantitatively compare action-based and parameter-based exploration, giving a formal guise to intuitive results.

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.

*Constrained Reinforcement Learning* (CRL) tackles sequential decision-making problems where agents are required to achieve goals by maximizing the expected return while meeting domain-specific constraints, which are often formulated on expected costs. In this setting, *policy-based* methods are widely used since they come with several advantages when dealing with continuous-control problems. These methods search in the policy space with an *action-based* or *parameter-based* exploration strategy, depending on whether they learn directly the parameters of a stochastic policy or those of a stochastic hyperpolicy. In this paper, we propose a general framework for addressing CRL problems via *gradient-based primal-dual* algorithms, relying on an alternate ascent/descent scheme with dual-variable regularization. We introduce an exploration-agnostic algorithm, called C-PG, which exhibits global last-iterate convergence guarantees under (weak) gradient domination assumptions, improving and generalizing existing results. Then, we design C-PGAE and C-PGPE, the action-based and the parameter-based versions of C-PG, respectively, and we illustrate how they naturally extend to constraints defined in terms of *risk measures* over the costs, as it is often requested in safety-critical scenarios. Finally, we numerically validate our algorithms on constrained control problems, and compare them with state-of-the-art baselines, demonstrating their effectiveness.

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 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.

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.

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.