Autonomous agents operating in domains such as robotics or video game simulations must adapt to changing tasks without forgetting about the previous ones. This process called Continual Reinforcement Learning poses non-trivial difficulties, from preventing catastrophic forgetting to ensuring the scalability of the approaches considered. Building on recent advances, we introduce a benchmark providing a suite of video-game navigation scenarios, thus filling a gap in the literature and capturing key challenges: catastrophic forgetting, task adaptation, and memory efficiency. We define a set of various tasks and datasets, evaluation protocols, and metrics to assess the performance of algorithms, including state-of-the-art baselines. Our benchmark is designed not only to foster reproducible research and to accelerate progress in continual reinforcement learning for gaming, but also to provide a reproducible framework for production pipelines helping practitioners to identify and to apply effective approaches. https://sites.google.com/view/continual-nav-bench

This paper introduces a novel backup strategy for Monte-Carlo Tree Search (MCTS) tailored for highly stochastic and partially observable Markov decision processes. We adopt a probabilistic approach, modeling both value and action-value nodes as Gaussian distributions, to introduce a novel backup operator that computes value nodes as the Wasserstein barycenter of their action-value children nodes; thus, propagating the uncertainty of the estimate across the tree to the root node. We study our novel backup operator when using a novel combination of $L^1$-Wasserstein barycenter with $\alpha$-divergence, by drawing a crucial connection to the generalized mean backup operator. We complement our probabilistic backup operator with two sampling strategies, based on optimistic selection and Thompson sampling, obtaining our Wasserstein MCTS algorithm. We provide theoretical guarantees of asymptotic convergence of $\mathcal{O}(n^{-1/2})$, with $n$ as the number of visited trajectories, to the optimal policy and an empirical evaluation on several stochastic and partially observable environments, where our approach outperforms well-known related baselines.

We investigate the regret-minimisation problem in a multi-armed bandit setting with arbitrary corruptions. Similar to the classical setup, the agent receives rewards generated independently from the distribution of the arm chosen at each time. However, these rewards are not directly observed. Instead, with a fixed $\varepsilon\in (0,\frac{1}{2})$, the agent observes a sample from the chosen arm’s distribution with probability $1-\varepsilon$, or from an arbitrary corruption distribution with probability $\varepsilon$. Importantly, we impose no assumptions on these corruption distributions, which can be unbounded. In this setting, accommodating potentially unbounded corruptions, we establish a problem-dependent lower bound on regret for a given family of arm distributions. We introduce CRIMED, an asymptotically-optimal algorithm that achieves the exact lower bound on regret for bandits with Gaussian distributions with known variance. Additionally, we provide a finite-sample analysis of CRIMED’s regret performance. Notably, CRIMED can effectively handle corruptions with $\varepsilon$ values as high as $\frac{1}{2}$. Furthermore, we develop a tight concentration result for medians in the presence of arbitrary corruptions, even with $\varepsilon$ values up to $\frac{1}{2}$, which may be of independent interest. We also discuss an extension of the algorithm for handling misspecification in Gaussian model.

Monte-Carlo Tree Search (MCTS) is a widely-used online planning strategy that effectively combines Monte-Carlo sampling with forward tree search to make optimal decisions in various real-world scenarios. Its success hinges on the application of the Upper Confidence bound for Trees (UCT) algorithm, an extension of the Upper Confidence bound (UCB) method used in multi-arm bandits. However, theoretical investigations of UCT have been found incomplete due to an error in the estimated "logarithmic" bonus term used for action selection in the tree. This issue was addressed by introducing a "polynomial" exploration bonus, which effectively balances the exploration-exploitation trade-off. However, most theoretical studies have focused on deterministic Markov Decision Processes (MDPs), neglecting stochastic settings. In this paper, we introduce Stochastic-Power-UCT, an MCTS algorithm explicitly designed for stochastic MDPs using power mean as the value estimator. We conduct a comprehensive study of the polynomial convergence assurance for selecting the optimal action and estimating the value at the root node within our methodology. Our findings demonstrate that Stochastic-Power-UCT shares the same convergence rate as UCT, with UCT being a special case of Stochastic-Power-UCT. Furthermore, we validate our theoretical results through empirical assessments across diverse stochastic MDP environments, providing empirical evidence to support our method’s theoretical claims

We study the problem of episodic reinforcement learning in continuous state-action spaces with unknown rewards and transitions. Specifically, we consider the setting where the rewards and transitions are modeled using parametric bilinear exponential families. We propose an algorithm, that a) uses penalized maximum likelihood estimators to learn the unknown parameters, b) injects a calibrated Gaussian noise in the parameter of rewards to ensure exploration, and c) leverages linearity of the bilinear exponential family transitions with respect to an underlying RKHS to perform tractable planning. We provide a frequentist regret upper-bound for our algorithm which, in the case of tabular MDPs, is order-optimal with respect to H and K, where H is the episode length and K is the number of episodes. Our analysis improves the existing bounds for the bilinear exponential family of MDPs by square root of H and removes the handcrafted clipping deployed in existing RLSVI-type algorithms.

We study reinforcement learning (RL) for decision processes with non-Markovian reward, in which high-level knowledge in the form of reward machines is available to the learner. Specifically, we investigate the efficiency of RL under the average-reward criterion, in the regret minimization setting. We propose two model-based RL algorithms that each exploits the structure of the reward machines, and show that our algorithms achieve regret bounds that improve over those of baselines by a multiplicative factor proportional to the number of states in the underlying reward machine. To the best of our knowledge, the proposed algorithms and associated regret bounds are the first to tailor the analysis specifically to reward machines, either in the episodic or average-reward settings. We also present a regret lower bound for the studied setting, which indicates that the proposed algorithms achieve a near-optimal regret. Finally, we report numerical experiments that demonstrate the superiority of the proposed algorithms over existing baselines in practice.

In decision-making problems such as the multi-armed bandit, an agent learns sequentially by optimizing a certain feedback. While the mean reward criterion has been extensively studied, other measures that reflect an aversion to adverse outcomes, such as mean-variance or conditional value-at-risk (CVaR), can be of interest for critical applications (healthcare, agriculture). Algorithms have been proposed for such risk-aware measures under bandit feedback without contextual information. In this work, we study contextual bandits where such risk measures can be elicited as linear functions of the contexts through the minimization of a convex loss. A typical example that fits within this framework is the expectile measure, which is obtained as the solution of an asymmetric least-square problem. Using the method of mixtures for supermartingales, we derive confidence sequences for the estimation of such risk measures. We then propose an optimistic UCB algorithm to learn optimal risk-aware actions, with regret guarantees similar to those of generalized linear bandits. This approach requires solving a convex problem at each round of the algorithm, which we can relax by allowing only approximated solution obtained by online gradient descent, at the cost of slightly higher regret. We conclude by evaluating the resulting algorithms on numerical experiments.

We revisit the method of mixtures, or Laplace method, to study the concentration phenomenon in generic (possibly multidimensional) exponential families. Using the duality properties of the Bregman divergence associated with the log-partition function of the family to construct nonnegative martingales, we establish a generic bound controlling the deviation between the parameter of the family and a finite sample estimate, expressed in the local geometry induced by the Bregman pseudo-metric. Our bound is time-uniform and involves a quantity extending the classical information gain to exponential families, which we call the Bregman information gain.For the practitioner, we instantiate this novel bound to several classical families, e.g., Gaussian (including with unknown variance or multivariate), Bernoulli, Exponential, Weibull, Pareto, Poisson and Chi-square, yielding explicit forms of the confidence sets and the Bregman information gain. We further compare the resulting confidence bounds to state-of-the-art time-uniform alternatives and show this novel method yields competitive results. Finally, we apply our result to the design of generalized likelihood ratio tests for change detection, capturing new settings such as variance change in Gaussian families.

We consider the problem of regret minimization in non-parametric stochastic bandits. When the rewards are known to be bounded from above, there exists asymptotically optimal algorithms, with asymptotic regret depending on an infimum of Kullback-Leibler divergences (KL). These algorithms are computationally expensive and require storing all past rewards, thus simpler but non-optimal algorithms are often used instead. We introduce several methods to approximate the infimum KL which reduce drastically the computational and memory costs of existing optimal algorithms, while keeping their regret guaranties. We apply our findings to design new variants of the MED and IMED algorithms, and demonstrate their interest with extensive numerical simulations.

We consider reinforcement learning in a discrete, undiscounted, infinite-horizon Markov decision problem (MDP) under the average reward criterion, and focus on the minimization of the regret with respect to an optimal policy, when the learner does not know the rewards nor transitions of the MDP. In light of their success at regret minimization in multi-armed bandits, popular bandit strategies, such as the optimistic \texttt{UCB}, \texttt{KL-UCB} or the Bayesian Thompson sampling strategy, have been extended to the MDP setup. Despite some key successes, existing strategies for solving this problem either fail to be provably asymptotically optimal, or suffer from prohibitive burn-in phase and computational complexity when implemented in practice. In this work, we shed a novel light on regret minimization strategies, by extending to reinforcement learning the computationally appealing Indexed Minimum Empirical Divergence (\texttt{IMED}) bandit algorithm. Traditional asymptotic problem-dependent lower bounds on the regret are known under the assumption that the MDP is \emph{ergodic}. Under this assumption, we introduce \texttt{IMED-RL} and prove that its regret upper bound asymptotically matches the regret lower bound. We discuss both the case when the supports of transitions are unknown, and the more informative but a priori harder-to-exploit-optimally case when they are known. Rewards are assumed light-tailed, semi-bounded from above. Last, we provide numerical illustrations on classical tabular MDPs, \textit{ergodic} and \textit{communicative} only, showing the competitiveness of \texttt{IMED-RL} in finite-time against state-of-the-art algorithms. \texttt{IMED-RL} also benefits from a lighter complexity.

We introduce GLRklUCB, a novel algorithm for the piecewise iid non-stationary bandit problem with bounded rewards. This algorithm combines an efficient bandit algorithm, klUCB, with an efficient, parameter-free, change-point detector, the Bernoulli Generalized Likelihood Ratio Test, for which we provide new theoretical guarantees of independent interest. Unlike previous non-stationary bandit algorithms using a change-point detector, GLRklUCB does not need to be calibrated based on prior knowledge on the arms' means. We prove that this algorithm can attain a $O(\sqrt{TA\Upsilon_T\log(T)})$ regret in $T$ rounds on some “easy” instances in which there is sufficient delay between two change-points, where $A$ is the number of arms and $\Upsilon_T$ the number of change-points, without prior knowledge of $\Upsilon_T$. In contrast with recently proposed algorithms that are agnostic to $\Upsilon_T$, we perform a numerical study showing that GLRklUCB is also very efficient in practice, beyond easy instances.

Extending model-based regret minimization strategies for Markov decision processes (MDPs) beyond discrete state-action spaces requires structural assumptions on the reward and transition models. Existing parametric approaches establish regret guarantees by making strong assumptions about either the state transition distribution or the value function as a function of state-action features, and often do not satisfactorily capture classical problems like linear dynamical systems or factored MDPs. This paper introduces a new MDP transition model defined by a collection of linearly parameterized exponential families with $d$ unknown parameters. For finite-horizon episodic RL with horizon $H$ in this MDP model, we propose a model-based upper confidence RL algorithm (Exp-UCRL) that solves a penalized maximum likelihood estimation problem to learn the $d$-dimensional representation of the transition distribution, balancing the exploitation-exploration tradeoff using confidence sets in the exponential family space. We demonstrate the efficiency of our algorithm by proving a frequentist (worst-case) regret bound that is of order $\tilde O(d\sqrt{H^3 N})$, sub-linear in total time $N$, linear in dimension $d$, and polynomial in the planning horizon $H$. This is achieved by deriving a novel concentration inequality for conditional exponential families that might be of independent interest. The exponential family MDP model also admits an efficient posterior sampling-style algorithm for which a similar guarantee on the Bayesian regret is shown.

We consider a regret minimization task under the average-reward criterion in an unknown Factored Markov Decision Process (FMDP). More specifically, we consider an FMDP where the state-action space $\mathcal X$ and the state-space $\mathcal S$ admit the respective factored forms of $\mathcal X = \otimes_{i=1}^n \mathcal X_i$ and $\mathcal S=\otimes_{i=1}^m \mathcal S_i$, and the transition and reward functions are factored over $\mathcal X$ and $\mathcal S$. Assuming a known a factorization structure, we introduce a novel regret minimization strategy inspired by the popular UCRL strategy, called DBN-UCRL, which relies on Bernstein-type confidence sets defined for individual elements of the transition function. We show that for a generic factorization structure, DBN-UCRL achieves a regret bound, whose leading term strictly improves over existing regret bounds in terms of the dependencies on the size of $\cS_i$’s and the diameter. We further show that when the factorization structure corresponds to the Cartesian product of some base MDPs, the regret of DBN-UCRL is upper bounded by the sum of regret of the base MDPs. We demonstrate, through numerical experiments on standard environments, that DBN-UCRL enjoys a substantially improved regret empirically over existing algorithms that have frequentist regret guarantees.

In this paper we study a multi-arm bandit problem in which the quality of each arm is measured by the Conditional Value at Risk (CVaR) at some level alpha of the reward distribution. While existing works in this setting mainly focus on Upper Confidence Bound algorithms, we introduce a new Thompson Sampling approach for CVaR bandits on bounded rewards that is flexible enough to solve a variety of problems grounded on physical resources. Building on a recent work by Riou & Honda (2020), we introduce B-CVTS for continuous bounded rewards and M-CVTS for multinomial distributions. On the theoretical side, we provide a non-trivial extension of their analysis that enables to theoretically bound their CVaR regret minimization performance. Strikingly, our results show that these strategies are the first to provably achieve asymptotic optimality in CVaR bandits, matching the corresponding asymptotic lower bounds for this setting. Further, we illustrate empirically the benefit of Thompson Sampling approaches both in a realistic environment simulating a use-case in agriculture and on various synthetic examples.

The stochastic multi-arm bandit problem has been extensively studied under standard assumptions on the arm's distribution (e.g bounded with known support, exponential family, etc). These assumptions are suitable for many real-world problems but sometimes they require knowledge (on tails for instance) that may not be precisely accessible to the practitioner, raising the question of the robustness of bandit algorithms to model misspecification. In this paper we study a generic \emph{Dirichlet Sampling} (DS) algorithm, based on pairwise comparisons of empirical indices computed with \textit{re-sampling} of the arms' observations and a data-dependent \textit{exploration bonus}. We show that different variants of this strategy achieve provably optimal regret guarantees when the distributions are bounded and logarithmic regret for semi-bounded distributions with a mild quantile condition. We also show that a simple tuning achieve robustness with respect to a large class of unbounded distributions, at the cost of slightly worse than logarithmic asymptotic regret. We finally provide numerical experiments showing the merits of DS in a decision-making problem on synthetic agriculture data.

We consider the problem of online linear regression in the stochastic setting. We derive high probability regret bounds for online $\textit{ridge}$ regression and the $\textit{forward}$ algorithm. This enables us to compare online regression algorithms more accurately and eliminate assumptions of bounded observations and predictions. Our study advocates for the use of the forward algorithm in lieu of ridge due to its enhanced bounds and robustness to the regularization parameter. Moreover, we explain how to integrate it in algorithms involving linear function approximation to remove a boundedness assumption without deteriorating theoretical bounds. We showcase this modification in linear bandit settings where it yields improved regret bounds. Last, we provide numerical experiments to illustrate our results and endorse our intuitions.

We consider a variant of the stochastic multi-armed bandit problem where arms are known to be organized into different groups having the same mean. The groups are unknown but a lower bound $q$ on their size is known. This situation typically appears when each arm can be described with a list of categorical attributes, and the (unknown) mean reward function only depends on a subset of them, the others being redundant. In this case, $q$ is linked naturally to the number of attributes considered redundant, and the number of categories of each attribute. For this structured problem of practical relevance, we first derive the asymptotic regret lower bound and corresponding constrained optimization problem. They reveal the achievable regret can be substantially reduced when compared to the unstructured setup, possibly by a factor $q$. However, solving exactly the exact constrained optimization problem involves a combinatorial problem. We introduce a lower-bound inspired strategy involving a computationally efficient relaxation that is based on a sorting mechanism. We further prove it achieves a lower bound close to the optimal one up to a controlled factor, and achieves an asymptotic regret $q$ times smaller than the unstructured one. We believe this shows it is a valuable strategy for the practitioner. Last, we illustrate the performance of the considered strategy on numerical experiments involving a large number of arms.

We consider a stochastic multi-armed bandit problem specified by a set of one-dimensional family exponential distributions endowed with a unimodal structure. The unimodal structure is of practical relevance for several applications. We introduce IMED-UB, an algorithm that exploits provably optimally the unimodal-structure, by adapting to this setting the Indexed Minimum Empirical Divergence (IMED) algorithm introduced by Honda and Takemura (2015). Owing to our proof technique, we are able to provide a concise finite-time analysis of the IMED-UB algorithm, that is simple and yet yields asymptotic optimality. We finally provide numerical experiments showing that IMED-UB competes favorably with the recently introduced state-of-the-art algorithms.

We consider the problem of planning in a Markov Decision Process (MDP) with a generative model and limited computational budget. Despite the underlying MDP transitions having a graph structure, the popular Monte-Carlo Tree Search algorithms such as UCT rely on a tree structure to represent their value estimates. That is, they do not identify together two similar states reached via different trajectories and represented in separate branches of the tree. In this work, we propose a graph-based planning algorithm, which takes into account this state similarity. In our analysis, we provide a regret bound that depends on a novel problem-dependent measure of difficulty, which improves on the original tree-based bound in MDPs where the trajectories overlap, and recovers it otherwise. Then, we show that this methodology can be adapted to existing planning algorithms that deal with stochastic systems. Finally, numerical simulations illustrate the benefits of our approach.

we consider the problem of sequential change-point detection where both the change-points and the distributions before and after the change are assumed to be unknown. For this problem of primary importance in statistical and sequential learning theory, we derive a variant of the Bayesian Online Change Point Detector proposed by \cite{fearnhead2007line} which is easier to analyze than the original version while keeping its powerful message-passing algorithm. We provide a non-asymptotic analysis of the false-alarm rate and the detection delay that matches the existing lower-bound. We further provide the first explicit high-probability control of the detection delay for such approach. Experiments on synthetic and real-world data show that this proposal outperforms the state-of-art change-point detection strategy, namely the Improved Generalized Likelihood Ratio (Improved GLR) while compares favorably with the original Bayesian Online Change Point Detection strategy.

The upper confidence reinforcement learning (UCRL2) algorithm introduced in \citep{jaksch2010near} is a popular method to perform regret minimization in unknown discrete Markov Decision Processes under the average-reward criterion. Despite its nice and generic theoretical regret guarantees, this algorithm and its variants have remained until now mostly theoretical as numerical experiments in simple environments exhibit long burn-in phases before the learning takes place. In pursuit of practical efficiency, we present UCRL3, following the lines of UCRL2, but with two key modifications: First, it uses state-of-the-art time-uniform concentration inequalities to compute confidence sets on the reward and (component-wise) transition distributions for each state-action pair. Furthermore, to tighten exploration, it uses an adaptive computation of the support of each transition distribution, which in turn enables us to revisit the extended value iteration procedure of UCRL2 to optimize over distributions with reduced support by disregarding low probability transitions, while still ensuring near-optimism. We demonstrate, through numerical experiments in standard environments, that reducing exploration this way yields a substantial numerical improvement compared to UCRL2 and its variants. On the theoretical side, these key modifications enable us to derive a regret bound for UCRL3 improving on UCRL2, that for the first time makes appear notions of local diameter and local effective support, thanks to variance-aware concentration bounds.

In this paper we propose the first multi-armed bandit algorithm based on re-sampling that achieves asymptotically optimal regret simultaneously for different families of arms (namely Bernoulli, Gaussian and Poisson distributions). Unlike Thompson Sampling which requires to specify a different prior to be optimal in each case, our proposal RB-SDA does not need any distribution-dependent tuning. RB-SDA belongs to the family of Sub-sampling Duelling Algorithms (SDA) which combines the sub-sampling idea first used by the BESA and SSMC algorithms with different sub-sampling schemes. In particular, RB-SDA uses Random Block sampling. We perform an experimental study assessing the flexibility and robustness of this promising novel approach for exploration in bandit models.

We consider the problem of robust and adaptive model predictive control (MPC) of a linear system, with unknown parameters that are learned along the way (adaptive), in a critical setting where failures must be prevented (robust). This problem has been studied from different perspectives by different communities. However, the existing theory deals only with the case of quadratic costs (the LQ problem), which limits applications to stabilisation and tracking tasks only. In order to handle more general (non-convex) costs that naturally arise in many practical problems, we carefully select and bring together several tools from different communities, namely non-asymptotic linear regression, recent results in interval prediction, and tree-based planning. Combining and adapting the theoretical guarantees at each layer is non trivial, and we provide the first end-to-end suboptimality analysis for this setting. Interestingly, our analysis naturally adapts to handle many models and combines with a data-driven robust model selection strategy, which enables to relax the modelling assumptions. Last, we strive to preserve tractability at any stage of the method, that we illustrate on two challenging simulated environments.

Leveraging an equivalence property in the state-space of a Markov Decision Process (MDP) has been investigated in several studies. This paper studies equivalence structure in the reinforcement learning (RL) setup, where transition distributions are no longer assumed to be known. We present a notion of similarity between transition probabilities of various state-action pairs of an MDP, which naturally defines an equivalence structure in the state-action space. We present equivalence-aware confidence sets for the case where the learner knows the underlying structure in advance. These sets are provably smaller than their corresponding equivalence-oblivious counterparts. In the more challenging case of an unknown equivalence structure, we present an algorithm called ApproxEquivalence that seeks to find an (approximate) equivalence structure, and define confidence sets using the approximate equivalence. To illustrate the efficacy of the presented confidence sets, we present C-UCRL, as a natural modification of UCRL2 for RL in undiscounted MDPs. In the case of a known equivalence structure, we show that C-UCRL improves over UCRL2 in terms of \emph{regret} by a factor of $\sqrt{SA/C}$, in any communicating MDP with $S$ states, $A$ actions, and $C$ classes, which corresponds to a massive improvement when $C\ll SA$. To the best of our knowledge, this is the first work providing regret bounds for RL when an equivalence structure in the MDP is efficiently exploited. In the case of an unknown equivalence structure, we show through numerical experiments that C-UCRL combined with ApproxEquivalence outperforms UCRL2 in ergodic MDPs.

We consider change-point detection in a fully sequential setup, when observations are received one by one and one must raise an alarm as early as possible after any change. We assume that both the change points and the distributions before and after the change are unknown. We consider the class of piecewise-constant mean processes with sub-Gaussian noise, and we target a detection strategy that is uniformly good on this class (this constrains the false alarm rate and detection delay). We introduce a novel tuning of the GLR test that takes here a simple form involving scan statistics, based on a novel sharp concentration inequality using an extension of the Laplace method for scan-statistics that holds doubly-uniformly in time. This also considerably simplifies the implementation of the test and analysis. We provide (perhaps surprisingly) the first fully non-asymptotic analysis of the detection delay of this test that matches the known existing asymptotic orders, with fully explicit numerical constants. Then, we extend this analysis to allow some changes that are not-detectable by any uniformly-good strategy (the number of observations before and after the change are too small for it to be detected by any such algorithm), and provide the first robust, finite-time analysis of the detection delay.

A Budgeted Markov Decision Process (BMDP) is an extension of a Markov Decision Process to critical applications requiring safety constraints. It relies on a notion of risk implemented in the shape of an upper bound on a constrains violation signal that -- importantly -- can be modified in real-time. So far, BMDPs could only be solved in the case of finite state spaces with known dynamics. This work extends the state-of-the-art to continuous spaces environments and unknown dynamics. We show that the solution to a BMDP is the fixed point of a novel Budgeted Bellman Optimality operator. This observation allows us to introduce natural extensions of Deep Reinforcement Learning algorithms to address large-scale BMDPs. We validate our approach on two simulated applications: spoken dialogue and autonomous driving.

We consider the problem of online reinforcement learning when several state representations (mapping histories to a discrete state space) are available to the learning agent. At least one of these representations is assumed to induce a Markov decision process (MDP), and the performance of the agent is measured in terms of cumulative regret against the optimal policy giving the highest average reward in this MDP representation. We propose an algorithm (UCB-MS) with O(sqrt(T)) regret in any communicating Markov decision process. The regret bound shows that UCB-MS automatically adapts to the Markov model. This improves over the currently known best results in the literature that gave regret bounds of order O(T^(2/3)).

We study the problem of learning the transition matrices of a set of Markov chains from a single stream of observations on each chain. We assume that the Markov chains are ergodic but otherwise unknown. The learner can sample Markov chains sequentially to observe their states. The goal of the learner is to sequentially select various chains to learn transition matrices uniformly well with respect to some loss function. We introduce a notion of loss that naturally extends the squared loss for learning distributions to the case of Markov chains, and further characterize the notion of being \emph{uniformly good} in all problem instances. We present a novel learning algorithm that efficiently balances \emph{exploration} and \emph{exploitation} intrinsic to this problem, without any prior knowledge of the chains. We provide finite-sample PAC-type guarantees on the performance of the algorithm. Further, we show that our algorithm asymptotically attains an optimal loss.

The problem of reinforcement learning in an unknown and discrete Markov Decision Process (MDP) under the average-reward criterion is considered, when the learner interacts with the system in a single stream of observations, starting from an initial state without any reset. We revisit the minimax lower bound for that problem by making appear the local variance of the bias function in place of the diameter of the MDP. Furthermore, we provide a novel analysis of the \texttt{\textsc{KL-Ucrl}} algorithm establishing a high-probability regret bound scaling as $\widetilde {\mathcal O}\Bigl({\textstyle \sqrt{S\sum_{s,a}{\bf V}^\star_{s,a}T}}\Big)$ for this algorithm for ergodic MDPs, where $S$ denotes the number of states and where ${\bf V}^\star_{s,a}$ is the variance of the bias function with respect to the next-state distribution following action $a$ in state $s$. The resulting bound improves upon the best previously known regret bound $\widetilde {\Ocal}(DS\sqrt{AT})$ for that algorithm, where $A$ and $D$ respectively denote the maximum number of actions (per state) and the diameter of MDP. We finally compare the leading terms of the two bounds in some benchmark MDPs indicating that the derived bound can provide an order of magnitude improvement in some cases. Our analysis leverages novel variations of the transportation lemma combined with Kullback-Leibler concentration inequalities, that we believe to be of independent interest.

We consider the problem of streaming kernel regression, when the observations arrive sequentially and the goal is to recover the underlying mean function, assumed to belong to an RKHS. The variance of the noise is not assumed to be known. In this context, we tackle the problem of tuning the regularization parameter adaptively at each time step, while maintaining tight confidence bounds estimates on the value of the mean function at each point. To this end, we first generalize existing results for finite-dimensional linear regression with fixed regularization and known variance to the kernel setup with a regularization parameter allowed to be a measurable function of past observations. Then, using appropriate self-normalized inequalities we build upper and lower bound estimates for the variance, leading to Bernstein-like concentration bounds. The latter is used in order to define the adaptive regularization. The bounds resulting from our technique are valid uniformly over all observation points and all time steps, and are compared against the literature with numerical experiments. Finally, the potential of these tools is illustrated by an application to kernelized bandits, where we revisit the Kernel UCB and Kernel Thompson Sampling procedures, and show the benefits of the novel adaptive kernel tuning strategy.

We consider parametric exponential families of dimension $K$ on the real line. We study a variant of boundary crossing probabilities coming from the multi-armed bandit literature, in the case when the real-valued distributions form an exponential family of dimension $K$. Formally, our result is a concentration inequality that bounds the probability that $\mathcal{B}^\psi(\hat \theta_n,\theta^\star)\geq f(t/n)/n$, where $\theta^\star$ is the parameter of an unknown target distribution, $\hat \theta_n$ is the empirical parameter estimate built from $n$ observations, $\psi$ is the log-partition function of the exponential family and $\mathcal{B}^\psi$ is the corresponding Bregman divergence. From the perspective of stochastic multi-armed bandits, we pay special attention to the case when the boundary function $f$ is logarithmic, as it enables to analyze the regret of the state-of-the-art KL-ucb and KL-ucb+ strategies, whose analysis was left open in such generality. Indeed, previous results only hold for the case when $K=1$, while we provide results for arbitrary finite dimension $K$, thus considerably extending the existing results. Perhaps surprisingly, we highlight that the proof techniques to achieve these strong results already existed three decades ago in the work of T.L. Lai, and were apparently forgotten in the bandit community. We provide a modern rewriting of these beautiful techniques that we believe are useful beyond the application to stochastic multi-armed bandits.

We consider a variation on the problem of prediction with expert advice, where new forecasters that were unknown until then may appear at each round. As often in prediction with expert advice, designing an algorithm that achieves near-optimal regret guarantees is straightforward, using aggregation of experts. However, when the comparison class is sufficiently rich, for instance when the best expert and the set of experts itself changes over time, such strategies naively require to maintain a prohibitive number of weights (typically exponential with the time horizon). By contrast, designing strategies that both achieve a near-optimal regret and maintain a reasonable number of weights is highly non-trivial. We consider three increasingly challenging objectives (simple regret, shifting regret and sparse shifting regret) that extend existing notions defined for a fixed expert ensemble; in each case, we design strategies that achieve tight regret bounds, adaptive to the parameters of the comparison class, while being computationally inexpensive. Moreover, our algorithms are anytime, agnostic to the number of incoming experts and completely parameter-free. Such remarkable results are made possible thanks to two simple but highly effective recipes: first the "abstention trick" that comes from the specialist framework and enables to handle the least challenging notions of regret, but is limited when addressing more sophisticated objectives. Second, the "muting trick" that we introduce to give more flexibility. We show how to combine these two tricks in order to handle the most challenging class of comparison strategies.

We present spectral methods of moments for learning sequential models from a single trajectory, in stark contrast with the classical literature that assumes the availability of multiple i.i.d. trajectories. Our approach leverages an efficient SVD-based learning algorithm for weighted automata and provides the first rigorous analysis for learning many important models using dependent data. We state and analyze the algorithm under three increasingly difficult scenarios: probabilistic automata, stochastic weighted automata, and reactive predictive state representations controlled by a finite-state policy. Our proofs include novel tools for studying mixing properties of stochastic weighted automata.

Rejection sampling is a technique for sampling from difficult distributions. However, its use is limited due to a high rejection rate. Common adaptive rejection sampling methods either work only for very specific distributions or without performance guarantees. In this paper, we present pliable rejection sampling (PRS), a new approach to rejection sampling, where we learn the sampling proposal using a kernel estimator. Since our method builds on rejection sampling, the samples obtained are with high probability i.i.d. and distributed according to f. Moreover, PRS comes with a guarantee on the number of accepted samples.

We consider a multi-armed bandit problem where the reward distributions are indexed by two sets –one for arms, one for type– and can be partitioned into a small number of clusters according to the type. First, we consider the setting where all reward distributions are known and all types have the same underlying cluster, the type’s identity is, however, unknown. Second, we study the case where types may come from different classes, which is significantly more challenging. Finally, we tackle the case where the reward distributions are completely unknown. In each setting, we introduce specific algorithms and derive non-trivial regret performance. Numerical experiments show that, in the most challenging agnostic case, the proposed algorithm achieves excellent performance in several difficult scenarios.

We consider a reinforcement learning setting where the learner also has to deal with the problem of finding a suitable state-representation function from a given set of models. This has to be done while interacting with the environment in an online fashion (no resets), and the goal is to have small regret with respect to any Markov model in the set. For this setting, recently the BLB algorithm has been proposed, which achieves regret of order T^2/3, provided that the given set of models is finite. Our first contribution is to extend this result to a countably infinite set of models. Moreover, the BLB regret bound suffers from an additive term that can be exponential in the diameter of the MDP involved, since the diameter has to be guessed. The algorithm we propose avoids guessing the diameter, thus improving the regret bound.

We consider an agent interacting with an environment in a single stream of actions, observations, and rewards, with no reset. This process is not assumed to be a Markov Decision Process (MDP). Rather, the agent has several representations (mapping histories of past interactions to a discrete state space) of the environment with unknown dynamics, only some of which result in an MDP. The goal is to minimize the average regret criterion against an agent who knows an MDP representation giving the highest optimal reward, and acts optimally in it. Recent regret bounds for this setting are of order O(T^2/3) with an additive term constant yet exponential in some characteristics of the optimal MDP. We propose an algorithm whose regret after T time steps is O(\sqrtT), with all constants reasonably small. This is optimal in T since O(\sqrtT) is the optimal regret in the setting of learning in a (single discrete) MDP.

In the setting of active learning for the multi-armed bandit, where the goal of a learner is to estimate with equal precision the mean of a finite number of arms, recent results show that it is possible to derive strategies based on finite-time confidence bounds that are competitive with the best possible strategy. We here consider an extension of this problem to the case when the arms are the cells of a finite partition P of a continuous sampling space X \subset \Real^d. Our goal is now to build a piecewise constant approximation of a noisy function (where each piece is one region of P and P is fixed beforehand) in order to maintain the local quadratic error of approximation on each cell equally low. Although this extension is not trivial, we show that a simple algorithm based on upper confidence bounds can be proved to be adaptive to the function itself in a near-optimal way, when |P| is chosen to be of minimax-optimal order on the class of \alpha-Hölder functions.

This paper aims to take a step forwards making the term ``intrinsic motivation'' from reinforcement learning theoretically well founded, focusing on curiosity-driven learning. To that end, we consider the setting where, a fixed partition P of a continuous space X being given, and a process \nu defined on X being unknown, we are asked to sequentially decide which cell of the partition to select as well as where to sample \nu in that cell, in order to minimize a loss function that is inspired from previous work on curiosity-driven learning. The loss on each cell consists of one term measuring a simple worst case quadratic sampling error, and a penalty term proportional to the range of the variance in that cell. The corresponding problem formulation extends the setting known as active learning for multi-armed bandits to the case when each arm is a continuous region, and we show how an adaptation of recent algorithms for that problem and of hierarchical optimistic sampling algorithms for optimization can be used in order to solve this problem. The resulting procedure, called Hierarchical Optimistic Region SElection driven by Curiosity (HORSE.C) is provided together with a finite-time regret analysis.

We consider the problem of recovering the parameter alpha in R^K of a sparse function f, i.e. the number of non-zero entries of alpha is small compared to the number K of features, given noisy evaluations of f at a set of well-chosen sampling points. We introduce an additional randomisation process, called Brownian sensing, based on the computation of stochastic integrals, which produces a Gaussian sensing matrix, for which good recovery properties are proven independently on the number of sampling points N, even when the features are arbitrarily non-orthogonal. Under the assumption that f is Hölder continuous with exponent at least 1/2, we provide an estimate a of the parameter such that ||\alpha - a||_2 = O(||eta||_2\sqrt{N}), where eta is the observation noise. The method uses a set of sampling points uniformly distributed along a one-dimensional curve selected according to the features. We report numerical experiments illustrating our method.

The problem of selecting the right state-representation in a reinforcement learning problem is considered. Several models (functions mapping past observations to a finite set) of the observations are given, and it is known that for at least one of these models the resulting state dynamics are indeed Markovian. Without knowing neither which of the models is the correct one, nor what are the probabilistic characteristics of the resulting MDP, it is required to obtain as much reward as the optimal policy for the correct model (or for the best of the correct models, if there are several). We propose an algorithm that achieves that, with a regret of order T^{2/3} where T is the horizon time.

We consider the problem of reinforcement learning in high-dimensional spaces when the number of features is bigger than the number of samples. In particular, we study the least-squares temporal difference (LSTD) learning algorithm when a space of low dimension is generated with a random projection from a high-dimensional space. We provide a thorough theoretical analysis of the LSTD with random projections and derive performance bounds for the resulting algorithm. We also show how the error of LSTD with random projections is propagated through the iterations of a policy iteration algorithm and provide a performance bound for the resulting least-squares policy iteration (LSPI) algorithm.

We consider least-squares regression using a randomly generated subspace GP\subset F of finite dimension P, where F is a function space of infinite dimension, e.g.~L2([0,1]^d). GP is defined as the span of P random features that are linear combinations of the basis functions of F weighted by random Gaussian i.i.d.~coefficients. In particular, we consider multi-resolution random combinations at all scales of a given mother function, such as a hat function or a wavelet. In this latter case, the resulting Gaussian objects are called {\em scrambled wavelets} and we show that they enable to approximate functions in Sobolev spaces H^s([0,1]^d). As a result, given N data, the least-squares estimate \hat g built from P scrambled wavelets has excess risk ||f^* - \hat g||\P^2 = O(||f^||^2_{H^s([0,1]^d)}(\log N)/P + P(\log N )/N) for target functions f^\in H^s([0,1]^d) of smoothness order s>d/2. An interesting aspect of the resulting bounds is that they do not depend on the distribution \P from which the data are generated, which is important in a statistical regression setting considered here. Randomization enables to adapt to any possible distribution. We conclude by describing an efficient numerical implementation using lazy expansions with numerical complexity \tilde O(2^d N^{3/2}\log N + N^2), where d is the dimension of the input space.

We consider the problem of learning, from K input data, a regression function in a function space of high dimension N using projections onto a random subspace of lower dimension M. From any linear approximation algorithm using empirical risk minimization (possibly penalized), we provide bounds on the excess risk of the estimate computed in the projected subspace (compressed domain) in terms of the excess risk of the estimate built in the high-dimensional space (initial domain). We apply the analysis to the ordinary Least-Squares regression and show that by choosing M=O(\sqrt{K}), the estimation error (for the quadratic loss) of the ``Compressed Least Squares Regression is O(1/\sqrt{K}) up to logarithmic factors. We also discuss the numerical complexity of several algorithms (both in initial and compressed domains) as a function of N, K, and M.