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Theory/Reinforcement Learning and Planning

Yuan Cheng, Jing Yang, Yingbin Liang

Reinforcement learning (RL) under changing environment models many real-world applications via nonstationary Markov Decision Processes (MDPs), and hence gains considerable interest. However, theoretical studies on nonstationary MDPs in the literature have mainly focused on tabular and linear (mixture) MDPs, which do not capture the nature of unknown representation in deep RL. In this paper, we make the first effort to investigate nonstationary RL under episodic low-rank MDPs, where both transition kernels and rewards may vary over time, and the low-rank model contains unknown representation in addition to the linear state embedding function. We first propose a parameter-dependent policy optimization algorithm called PORTAL,and further improve PORTAL to its parameter-free version of Ada-PORTAL, which is able to tune its hyper-parameters adaptively without any prior knowledge of nonstationarity. For both algorithms, we provide upper bounds on the average dynamic suboptimality gap, which show that as long as the nonstationarity is not significantly large, PORTAL and Ada-PORTAL are sample-efficient and can achieve arbitrarily small average dynamic suboptimality gap with polynomial sample complexity.

Theory/Reinforcement Learning and Planning

Udaya Ghai, Arushi Gupta, Wenhan Xia, Karan Singh, Elad Hazan

We investigate robust model-free reinforcement learning algorithms designed for environments that may be dynamic or even adversarial. Traditional state-based policies often struggle to accommodate the challenges imposed by the presence of unmodeled disturbances in such settings. Moreover, optimizing linear state-based policies pose an obstacle for efficient optimization, leading to nonconvex objectives, even in benign environments like linear dynamical systems.Drawing inspiration from recent advancements in model-based control, we intro- duce a novel class of policies centered on disturbance signals. We define several categories of these signals, which we term pseudo-disturbances, and develop corresponding policy classes based on them. We provide efficient and practical algorithms for optimizing these policies.Next, we examine the task of online adaptation of reinforcement learning agents in the face of adversarial disturbances. Our methods seamlessly integrate with any black-box model-free approach, yielding provable regret guarantees when dealing with linear dynamics. These regret guarantees unconditionally improve the best-known results for bandit linear control in having no dependence on the state-space dimension. We evaluate our method over various standard RL benchmarks and demonstrate improved robustness.

Theory/Reinforcement Learning and Planning

Yunchang Yang, Han Zhong, Tianhao Wu, Bin Liu, Liwei Wang, Simon Du

We study stochastic delayed feedback in general single-agent and multi-agent sequential decision making, which includes bandits, single-agent Markov decision processes (MDPs), and Markov games (MGs). We propose a novel reduction-based framework, which turns any multi-batched algorithm for sequential decision making with instantaneous feedback into a sample-efficient algorithm that can handle stochastic delays in sequential decision making. By plugging different multi-batched algorithms into our framework, we provide several examples demonstrating that our framework not only matches or improves existing results for bandits, tabular MDPs, and tabular MGs, but also provides the first line of studies on delays in sequential decision making with function approximation. In summary, we provide a complete set of sharp results for single-agent and multi-agent sequential decision making with delayed feedback.

Theory/Reinforcement Learning and Planning

Xiaoyan Hu, Ho-fung Leung

The options framework yields empirical success in long-horizon planning problems of reinforcement learning (RL). Recent works show that options help improve the sample efficiency in online RL. However, these results are no longer applicable to scenarios where exploring the environment online is risky, e.g., automated driving and healthcare. In this paper, we provide the first analysis of the sample complexity for offline RL with options, where the agent learns from a dataset without further interaction with the environment. We derive a novel information-theoretic lower bound, which generalizes the one for offline learning with actions. We propose the PEssimistic Value Iteration for Learning with Options (PEVIO) algorithm and establish near-optimal suboptimality bounds for two popular data-collection procedures, where the first one collects state-option transitions and the second one collects state-action transitions. We show that compared to offline RL with actions, using options not only enjoys a faster finite-time convergence rate (to the optimal value) but also attains a better performance when either the options are carefully designed or the offline data is limited. Based on these results, we analyze the pros and cons of the data-collection procedures.

Theory/Reinforcement Learning and Planning

Hanlin Zhu, Paria Rashidinejad, Jiantao Jiao

We propose A-Crab (Actor-Critic Regularized by Average Bellman error), a new practical algorithm for offline reinforcement learning (RL) in complex environments with insufficient data coverage. Our algorithm combines the marginalized importance sampling framework with the actor-critic paradigm, where the critic returns evaluations of the actor (policy) that are pessimistic relative to the offline data and have a small average (importance-weighted) Bellman error. Compared to existing methods, our algorithm simultaneously offers a number of advantages:(1) It achieves the optimal statistical rate of $1/\sqrt{N}$---where $N$ is the size of offline dataset---in converging to the best policy covered in the offline dataset, even when combined with general function approximators.(2) It relies on a weaker \textit{average} notion of policy coverage (compared to the $\ell_\infty$ single-policy concentrability) that exploits the structure of policy visitations.(3) It outperforms the data-collection behavior policy over a wide range of specific hyperparameters. We provide both theoretical analysis and experimental results to validate the effectiveness of our proposed algorithm. The code is available at https://github.com/zhuhl98/ACrab.

Theory/Reinforcement Learning and Planning

Eric Eaton, Marcel Hussing, Michael Kearns, Jessica Sorrell

The replicability crisis in the social, behavioral, and data sciences has led to the formulation of algorithm frameworks for replicability --- i.e., a requirement that an algorithm produce identical outputs (with high probability) when run on two different samples from the same underlying distribution. While still in its infancy, provably replicable algorithms have been developed for many fundamental tasks in machine learning and statistics, including statistical query learning, the heavy hitters problem, and distribution testing. In this work we initiate the study of replicable reinforcement learning, providing a provably replicable algorithm for parallel value iteration, and a provably replicable version of R-Max in the episodic setting. These are the first formal replicability results for control problems, which present different challenges for replication than batch learning settings.

Theory/Reinforcement Learning and Planning

Blake Bordelon, Paul Masset, Henry Kuo, Cengiz Pehlevan

Reinforcement learning has been successful across several applications in which agents have to learn to act in environments with sparse feedback. However, despite this empirical success there is still a lack of theoretical understanding of how the parameters of reinforcement learning models and the features used to represent states interact to control the dynamics of learning. In this work, we use concepts from statistical physics, to study the typical case learning curves for temporal difference learning of a value function with linear function approximators. Our theory is derived under a Gaussian equivalence hypothesis where averages over the random trajectories are replaced with temporally correlated Gaussian feature averages and we validate our assumptions on small scale Markov Decision Processes. We find that the stochastic semi-gradient noise due to subsampling the space of possible episodes leads to significant plateaus in the value error, unlike in traditional gradient descent dynamics. We study how learning dynamics and plateaus depend on feature structure, learning rate, discount factor, and reward function. We then analyze how strategies like learning rate annealing and reward shaping can favorably alter learning dynamics and plateaus. To conclude, our work introduces new tools to open a new direction towards developing a theory of learning dynamics in reinforcement learning.

Theory/Reinforcement Learning and Planning

Yigit Efe Erginbas, Thomas Courtade, Kannan Ramchandran, Soham Phade

Online pricing has been the focus of extensive research in recent years, particularly in the context of selling an item to sequentially arriving users. However, what if a provider wants to maximize revenue by selling multiple items to multiple users in each round? This presents a complex problem, as the provider must intelligently offer the items to those users who value them the most without exceeding their highest acceptable prices. In this study, we tackle this challenge by designing online algorithms that can efficiently offer and price items while learning user valuations from accept/reject feedback. We focus on three user valuation models (fixed valuations, random experiences, and random valuations) and provide algorithms with nearly-optimal revenue regret guarantees. In particular, for any market setting with $N$ users, $M$ items, and load $L$ (which roughly corresponds to the maximum number of simultaneous allocations possible), our algorithms achieve regret of order $O(NM\log\log(LT))$ under fixed valuations model, $\widetilde{O}(\sqrt{NMLT})$ under random experiences model and $\widetilde{O}(\sqrt{NMLT})$ under random valuations model in $T$ rounds.

Theory/Reinforcement Learning and Planning

Hyunin Lee, Yuhao Ding, Jongmin Lee, Ming Jin, Javad Lavaei, Somayeh Sojoudi

We first raise and tackle a ``time synchronization'' issue between the agent and the environment in non-stationary reinforcement learning (RL), a crucial factor hindering its real-world applications. In reality, environmental changes occur over wall-clock time ($t$) rather than episode progress ($k$), where wall-clock time signifies the actual elapsed time within the fixed duration $t \in [0, T]$. In existing works, at episode $k$, the agent rolls a trajectory and trains a policy before transitioning to episode $k+1$. In the context of the time-desynchronized environment, however, the agent at time $t_{k}$ allocates $\Delta t$ for trajectory generation and training, subsequently moves to the next episode at $t_{k+1}=t_{k}+\Delta t$. Despite a fixed total number of episodes ($K$), the agent accumulates different trajectories influenced by the choice of interaction times ($t_1,t_2,...,t_K$), significantly impacting the suboptimality gap of the policy. We propose a Proactively Synchronizing Tempo ($\texttt{ProST}$) framework that computes a suboptimal sequence {$t_1,t_2,...,t_K$} (= { $t_{1:K}$}) by minimizing an upper bound on its performance measure, i.e., the dynamic regret. Our main contribution is that we show that a suboptimal {$t_{1:K}$} trades-off between the policy training time (agent tempo) and how fast the environment changes (environment tempo). Theoretically, this work develops a suboptimal {$t_{1:K}$} as a function of the degree of the environment's non-stationarity while also achieving a sublinear dynamic regret. Our experimental evaluation on various high-dimensional non-stationary environments shows that the $\texttt{ProST}$ framework achieves a higher online return at suboptimal {$t_{1:K}$} than the existing methods.

Theory/Reinforcement Learning and Planning

Yashaswini Murthy, Mehrdad Moharrami, R. Srikant

Many policy-based reinforcement learning (RL) algorithms can be viewed as instantiations of approximate policy iteration (PI), i.e., where policy improvement and policy evaluation are both performed approximately. In applications where the average reward objective is the meaningful performance metric, often discounted reward formulations are used with the discount factor being close to $1,$ which is equivalent to making the expected horizon very large. However, the corresponding theoretical bounds for error performance scale with the square of the horizon. Thus, even after dividing the total reward by the length of the horizon, the corresponding performance bounds for average reward problems go to infinity. Therefore, an open problem has been to obtain meaningful performance bounds for approximate PI and RL algorithms for the average-reward setting. In this paper, we solve this open problem by obtaining the first non-trivial finite time error bounds for average-reward MDPs which go to zero in the limit as policy evaluation and policy improvement errors go to zero.

Theory/Reinforcement Learning and Planning

Laixi Shi, Gen Li, Yuting Wei, Yuxin Chen, Matthieu Geist, Yuejie Chi

This paper investigates model robustness in reinforcement learning (RL) via the framework of distributionally robust Markov decision processes (RMDPs). Despite recent efforts, the sample complexity of RMDPs is much less understood regardless of the uncertainty set in use; in particular, there exist large gaps between existing upper and lower bounds, and it is unclear if distributional robustness bears any statistical implications when benchmarked against standard RL. In this paper, assuming access to a generative model, we derive the sample complexity of RMDPs---when the uncertainty set is measured via either total variation or $\chi^2$ divergence over the full range of uncertainty levels---using a model-based algorithm called distributionally robust value iteration, and develop minimax lower bounds to benchmark its tightness. Our results not only strengthen the prior art in both directions of upper and lower bounds, but also deliver surprising messages that learning RMDPs is not necessarily easier or more difficult than standard MDPs. In the case of total variation, we establish the minimax-optimal sample complexity of RMDPs which is always smaller than that of standard MDPs. In the case of $\chi^2$ divergence, we establish the sample complexity of RMDPs that is tight up to polynomial factors of the effective horizon, and grows linearly with respect to the uncertainty level when it approaches infinity.

Theory/Reinforcement Learning and Planning

Jose Blanchet, Miao Lu, Tong Zhang, Han Zhong

We study distributionally robust offline reinforcement learning (RL), which seeks to find an optimal robust policy purely from an offline dataset that can perform well in perturbed environments. We propose a generic algorithm framework Doubly Pessimistic Model-based Policy Optimization ($\texttt{P}^2\texttt{MPO}$) for robust offline RL, which features a novel combination of a flexible model estimation subroutine and a doubly pessimistic policy optimization step. Here the double pessimism principle is crucial to overcome the distribution shift incurred by i) the mismatch between behavior policy and the family of target policies; and ii) the perturbation of the nominal model. Under certain accuracy assumptions on the model estimation subroutine, we show that $\texttt{P}^2\texttt{MPO}$ is provably sample-efficient with robust partial coverage data, which means that the offline dataset has good coverage of the distributions induced by the optimal robust policy and perturbed models around the nominal model. By tailoring specific model estimation subroutines for concrete examples including tabular Robust Markov Decision Process (RMDP), factored RMDP, and RMDP with kernel and neural function approximations, we show that $\texttt{P}^2\texttt{MPO}$ enjoys a $\tilde{\mathcal{O}}(n^{-1/2})$ convergence rate, where $n$ is the number of trajectories in the offline dataset. Notably, these models, except for the tabular case, are first identified and proven tractable by this paper. To the best of our knowledge, we first propose a general learning principle --- double pessimism --- for robust offline RL and show that it is provably efficient in the context of general function approximations.

Theory/Reinforcement Learning and Planning

Bhavya, Lenart Treven, Cansu Sancaktar, Sebastian Blaes, Stelian Coros, Andreas Krause

Reinforcement learning algorithms commonly seek to optimize policies for solving one particular task. How should we explore an unknown dynamical system such that the estimated model allows us to solve multiple downstream tasks in a zero-shot manner? In this paper, we address this challenge, by developing an algorithm -- OPAX -- for active exploration. OPAX uses well-calibrated probabilistic models to quantify the epistemic uncertainty about the unknown dynamics. It optimistically---w.r.t. to plausible dynamics---maximizes the information gain between the unknown dynamics and state observations. We show how the resulting optimization problem can be reduced to an optimal control problem that can be solved at each episode using standard approaches. We analyze our algorithm for general models, and, in the case of Gaussian process dynamics, we give a sample complexity bound andshow that the epistemic uncertainty converges to zero. In our experiments, we compare OPAX with other heuristic active exploration approaches on several environments. Our experiments show that OPAX is not only theoretically sound but also performs well for zero-shot planning on novel downstream tasks.

Theory/Reinforcement Learning and Planning

Lenart Treven, Jonas Hübotter, Bhavya, Florian Dorfler, Andreas Krause

Reinforcement learning algorithms typically consider discrete-time dynamics, even though the underlying systems are often continuous in time. In this paper, we introduce a model-based reinforcement learning algorithm that represents continuous-time dynamics using nonlinear ordinary differential equations (ODEs). We capture epistemic uncertainty using well-calibrated probabilistic models, and use the optimistic principle for exploration. Our regret bounds surface the importance of the measurement selection strategy (MSS), since in continuous time we not only must decide how to explore, but also when to observe the underlying system. Our analysis demonstrates that the regret is sublinear when modeling ODEs with Gaussian Processes (GP) for common choices of MSS, such as equidistant sampling. Additionally, we propose an adaptive, data-dependent, practical MSS that, when combined with GP dynamics, also achieves sublinear regret with significantly fewer samples. We showcase the benefits of continuous-time modeling over its discrete-time counterpart, as well as our proposed adaptive MSS over standard baselines, on several applications.

Theory/Reinforcement Learning and Planning

Gellert Weisz, András György, Csaba Szepesvari

We consider online reinforcement learning (RL) in episodic Markov decision processes (MDPs) under the linear $q^\pi$-realizability assumption, where it is assumed that the action-values of all policies can be expressed as linear functions of state-action features. This class is known to be more general than linear MDPs, where the transition kernel and the reward function are assumed to be linear functions of the feature vectors. As our first contribution, we show that the difference between the two classes is the presence of states in linearly $q^\pi$-realizable MDPs where for any policy, all the actions have approximately equal values, and skipping over these states by following an arbitrarily fixed policy in those states transforms the problem to a linear MDP. Based on this observation, we derive a novel (computationally inefficient) learning algorithm for linearly $q^\pi$-realizable MDPs that simultaneously learns what states should be skipped over and runs another learning algorithm on the linear MDP hidden in the problem. The method returns an $\epsilon$-optimal policy after $\text{polylog}(H, d)/\epsilon^2$ interactions with the MDP, where $H$ is the time horizon and $d$ is the dimension of the feature vectors, giving the first polynomial-sample-complexity online RL algorithm for this setting. The results are proved for the misspecified case, where the sample complexity is shown to degrade gracefully with the misspecification error.

Theory/Reinforcement Learning and Planning

Jiayi Huang, Han Zhong, Liwei Wang, Lin Yang

While numerous works have focused on devising efficient algorithms for reinforcement learning (RL) with uniformly bounded rewards, it remains an open question whether sample or time-efficient algorithms for RL with large state-action space exist when the rewards are \emph{heavy-tailed}, i.e., with only finite $(1+\epsilon)$-th moments for some $\epsilon\in(0,1]$. In this work, we address the challenge of such rewards in RL with linear function approximation. We first design an algorithm, \textsc{Heavy-OFUL}, for heavy-tailed linear bandits, achieving an \emph{instance-dependent} $T$-round regret of $\tilde{O}\big(d T^{\frac{1-\epsilon}{2(1+\epsilon)}} \sqrt{\sum_{t=1}^T \nu_t^2} + d T^{\frac{1-\epsilon}{2(1+\epsilon)}}\big)$, the \emph{first} of this kind. Here, $d$ is the feature dimension, and $\nu_t^{1+\epsilon}$ is the $(1+\epsilon)$-th central moment of the reward at the $t$-th round. We further show the above bound is minimax optimal when applied to the worst-case instances in stochastic and deterministic linear bandits. We then extend this algorithm to the RL settings with linear function approximation. Our algorithm, termed as \textsc{Heavy-LSVI-UCB}, achieves the \emph{first} computationally efficient \emph{instance-dependent} $K$-episode regret of $\tilde{O}(d \sqrt{H \mathcal{U}^*} K^\frac{1}{1+\epsilon} + d \sqrt{H \mathcal{V}^* K})$. Here, $H$ is length of the episode, and $\mathcal{U}^*, \mathcal{V}^*$ are instance-dependent quantities scaling with the central moment of reward and value functions, respectively. We also provide a matching minimax lower bound $\Omega(d H K^{\frac{1}{1+\epsilon}} + d \sqrt{H^3 K})$ to demonstrate the optimality of our algorithm in the worst case. Our result is achieved via a novel robust self-normalized concentration inequality that may be of independent interest in handling heavy-tailed noise in general online regression problems.

Theory/Reinforcement Learning and Planning

Yige Hong, Qiaomin Xie, Yudong Chen, Weina Wang

We study the infinite-horizon restless bandit problem with the average reward criterion, in both discrete-time and continuous-time settings.A fundamental goal is to efficiently compute policies that achieve a diminishing optimality gap as the number of arms, $N$, grows large. Existing results on asymptotic optimality all rely on the uniform global attractor property (UGAP), a complex and challenging-to-verify assumption. In this paper, we propose a general, simulation-based framework, Follow-the-Virtual-Advice, that converts any single-armed policy into a policy for the original $N$-armed problem. This is done by simulating the single-armed policy on each arm and carefully steering the real state towards the simulated state. Our framework can be instantiated to produce a policy with an $O(1/\sqrt{N})$ optimality gap. In the discrete-time setting, our result holds under a simpler synchronization assumption, which covers some problem instances that violate UGAP. More notably, in the continuous-time setting, we do not require \emph{any} additional assumptions beyond the standard unichain condition. In both settings, our work is the first asymptotic optimality result that does not require UGAP.

Theory/Reinforcement Learning and Planning

Thanh Nguyen-Tang, Raman Arora

We seek to understand what facilitates sample-efficient learning from historical datasets for sequential decision-making, a problem that is popularly known as offline reinforcement learning (RL). Further, we are interested in algorithms that enjoy sample efficiency while leveraging (value) function approximation. In this paper, we address these fundamental questions by (i) proposing a notion of data diversity that subsumes the previous notions of coverage measures in offline RL and (ii) using this notion to \emph{unify} three distinct classes of offline RL algorithms based on version spaces (VS), regularized optimization (RO), and posterior sampling (PS). We establish that VS-based, RO-based, and PS-based algorithms, under standard assumptions, achieve \emph{comparable} sample efficiency, which recovers the state-of-the-art sub-optimality bounds for finite and linear model classes with the standard assumptions. This result is surprising, given that the prior work suggested an unfavorable sample complexity of the RO-based algorithm compared to the VS-based algorithm, whereas posterior sampling is rarely considered in offline RL due to its explorative nature. Notably, our proposed model-free PS-based algorithm for offline RL is \emph{novel}, with sub-optimality bounds that are \emph{frequentist} (i.e., worst-case) in nature.

Theory/Reinforcement Learning and Planning

Emmeran Johnson, Ciara Pike-Burke, Patrick Rebeschini

Policy Mirror Descent (PMD) is a general family of algorithms that covers a wide range of novel and fundamental methods in reinforcement learning. Motivated by the instability of policy iteration (PI) with inexact policy evaluation, unregularised PMD algorithmically regularises the policy improvement step of PI without regularising the objective function. With exact policy evaluation, PI is known to converge linearly with a rate given by the discount factor $\gamma$ of a Markov Decision Process. In this work, we bridge the gap between PI and PMD with exact policy evaluation and show that the dimension-free $\gamma$-rate of PI can be achieved by the general family of unregularised PMD algorithms under an adaptive step-size. We show that both the rate and step-size are unimprovable for PMD: we provide matching lower bounds that demonstrate that the $\gamma$-rate is optimal for PMD methods as well as PI and that the adaptive step-size is necessary to achieve it. Our work is the first to relate PMD to rate-optimality and step-size necessity. Our study of the convergence of PMD avoids the use of the performance difference lemma, which leads to a direct analysis of independent interest. We also extend the analysis to the inexact setting and establish the first dimension-optimal sample complexity for unregularised PMD under a generative model, improving upon the best-known result.

Theory/Reinforcement Learning and Planning

Qinghua Liu, Gellert Weisz, András György, Chi Jin, Csaba Szepesvari

While policy optimization algorithms have played an important role in recent empirical success of Reinforcement Learning (RL), the existing theoretical understanding of policy optimization remains rather limited---they are either restricted to tabular MDPs or suffer from highly suboptimal sample complexity, especial in online RL where exploration is necessary. This paper proposes a simple efficient policy optimization framework---Optimistic NPG for online RL. Optimistic NPG can be viewed as simply combining of the classic natural policy gradient (NPG) algorithm [Kakade, 2001] with optimistic policy evaluation subroutines to encourage exploration. For $d$-dimensional linear MDPs, Optimistic NPG is computationally efficient, and learns an $\epsilon$-optimal policy within $\tilde{\mathcal{O}}(d^2/\epsilon^3)$ samples, which is the first computationally efficient algorithm whose sample complexity has the optimal dimension dependence $\tilde{\Theta}(d^2)$. It also improves over state-of-the-art results of policy optimization algorithms [Zanette et al., 2021] by a factor of $d$. For general function approximation that subsumes linear MDPs, Optimistic NPG, to our best knowledge, is also the first policy optimization algorithm that achieves the polynomial sample complexity for learning near-optimal policies.