Reinforcement learning (RL) is an important field of research in machine learning that is increasingly being applied to complex optimization problems in physics. In parallel, concepts from physics have contributed to important advances in RL with developments such as entropy-regularized RL. While these developments have led to advances in both fields, obtaining analytical solutions for optimization in entropy-regularized RL is currently an open problem. In this paper, we establish a mapping between entropy-regularized RL and research in non-equilibrium statistical mechanics focusing on Markovian processes conditioned on rare events. In the long-time limit, we apply approaches from large deviation theory to derive exact analytical results for the optimal policy and optimal dynamics in Markov Decision Process (MDP) models of reinforcement learning. The results obtained lead to a novel analytical and computational framework for entropy-regularized RL which is validated by simulations. The mapping established in this work connects current research in reinforcement learning and non-equilibrium statistical mechanics, thereby opening new avenues for the application of analytical and computational approaches from one field to cutting-edge problems in the other.
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Existing online learning algorithms for adversarial Markov Decision Processes achieve ${O}(\sqrt{T})$ regret after $T$ rounds of interactions even if the loss functions are chosen arbitrarily by an adversary, with the caveat that the transition function has to be fixed. This is because it has been shown that adversarial transition functions make no-regret learning impossible. Despite such impossibility results, in this work, we develop algorithms that can handle both adversarial losses and adversarial transitions, with regret increasing smoothly in the degree of maliciousness of the adversary. More concretely, we first propose an algorithm that enjoys $\widetilde{{O}}(\sqrt{T} + C^{\textsf{P}})$ regret where $C^{\textsf{P}}$ measures how adversarial the transition functions are and can be at most ${O}(T)$. While this algorithm itself requires knowledge of $C^{\textsf{P}}$, we further develop a black-box reduction approach that removes this requirement. Moreover, we also show that further refinements of the algorithm not only maintains the same regret bound, but also simultaneously adapts to easier environments (where losses are generated in a certain stochastically constrained manner as in Jin et al.[2021]) and achieves $\widetilde{{O}}(U + \sqrt{UC^{\textsf{L}}} + C^{\textsf{P}})$ regret, where $U$ is some standard gap-dependent coefficient and $C^{\textsf{L}}$ is the amount of corruption on losses.
Everything else being equal, simpler models should be preferred over more complex ones. In reinforcement learning (RL), simplicity is typically quantified on an action-by-action basis -- but this timescale ignores temporal regularities, like repetitions, often present in sequential strategies. We therefore propose an RL algorithm that learns to solve tasks with sequences of actions that are compressible. We explore two possible sources of simple action sequences: Sequences that can be learned by autoregressive models, and sequences that are compressible with off-the-shelf data compression algorithms. Distilling these preferences into sequence priors, we derive a novel information-theoretic objective that incentivizes agents to learn policies that maximize rewards while conforming to these priors. We show that the resulting RL algorithm leads to faster learning, and attains higher returns than state-of-the-art model-free approaches in a series of continuous control tasks from the DeepMind Control Suite. These priors also produce a powerful information-regularized agent that is robust to noisy observations and can perform open-loop control.
We study the problem of conservative off-policy evaluation (COPE) where given an offline dataset of environment interactions, collected by other agents, we seek to obtain a (tight) lower bound on a policy's performance. This is crucial when deciding whether a given policy satisfies certain minimal performance/safety criteria before it can be deployed in the real world. To this end, we introduce HAMBO, which builds on an uncertainty-aware learned model of the transition dynamics. To form a conservative estimate of the policy's performance, HAMBO hallucinates worst-case trajectories that the policy may take, within the margin of the models' epistemic confidence regions. We prove that the resulting COPE estimates are valid lower bounds, and, under regularity conditions, show their convergence to the true expected return. Finally, we discuss scalable variants of our approach based on Bayesian Neural Networks and empirically demonstrate that they yield reliable and tight lower bounds in various continuous control environments.
Mobility systems often suffer from a high price of anarchy due to the uncontrolled behavior of selfish users. This may result in societal costs that are significantly higher compared to what could be achieved by a centralized system-optimal controller. Monetary tolling schemes can effectively align the behavior of selfish users with the system-optimum. Yet, they inevitably discriminate the population in terms of income. Artificial currencies were recently presented as an effective alternative that can achieve the same performance, whilst guaranteeing fairness among the population. However, those studies were based on behavioral models that may differ from practical implementations. This paper presents a data-driven approach to automatically adapt artificial-currency tolls within repetitive-game settings. We first consider a parallel-arc setting whereby users commute on a daily basis from an individual origin to an individual destination, choosing a route in exchange of an artificial-currency price or reward, while accounting for the impact of the choices of the other users on travel discomfort. Second, we devise a model-based reinforcement learning controller that autonomously learns the optimal pricing policy by interacting with the proposed framework considering the closeness of the observed aggregate flows to a desired system-optimal distribution as a reward function. Our numerical results show that the proposed data-driven pricing scheme can effectively align the users' flows with the system optimum, significantly reducing the societal costs with respect to the uncontrolled flows (by about 15% and 25% depending on the scenario), and respond to environmental changes in a robust and efficient manner.
In this paper, we study risk-sensitive Reinforcement Learning (RL), focusing on the objective of Conditional Value at Risk (CVaR) with risk tolerance $\tau$. Starting with multi-arm bandits (MABs), we show the minimax CVaR regret rate is $\Omega(\sqrt{\tau^{-1}AK})$, where $A$ is the number of actions and $K$ is the number of episodes, and that it is achieved by an Upper Confidence Bound algorithm with a novel Bernstein bonus. For online RL in tabular Markov Decision Processes (MDPs), we show a minimax regret lower bound of $\Omega(\sqrt{\tau^{-1}SAK})$ (with normalized cumulative rewards), where $S$ is the number of states, and we propose a novel bonus-driven Value Iteration procedure. We show that our algorithm achieves the optimal regret of $\widetilde O(\sqrt{\tau^{-1}SAK})$ under a continuity assumption and in general attains a near-optimal regret of $\widetilde O(\tau^{-1}\sqrt{SAK})$, which is minimax-optimal for constant $\tau$. This improves on the best available bounds. By discretizing rewards appropriately, our algorithms are computationally efficient.
A crucial problem in reinforcement learning is learning the optimal policy. We study this in tabular infinite-horizon discounted Markov decision processes under the online setting. The existing algorithms either fail to achieve regret optimality or have to incur a high memory and computational cost. In addition, existing optimal algorithms all require a long burn-in time in order to achieve optimal sample efficiency, i.e., their optimality is not guaranteed unless sample size surpasses a high threshold. We address both open problems by introducing a model-free algorithm that employs variance reduction and a novel technique that switches the execution policy in a slow-yet-adaptive manner. This is the first regret-optimal model-free algorithm in the discounted setting, with the additional benefit of a low burn-in time.
While deep reinforcement learning (RL) has fueled multiple high-profile successes in machine learning, it is held back from more widespread adoption by its often poor data efficiency and the limited generality of the policies it produces. A promising approach for alleviating these limitations is to cast the development of better RL algorithms as a machine learning problem itself in a process called meta-RL. Meta-RL is most commonly studied in a problem setting where, given a distribution of tasks, the goal is to learn a policy that is capable of adapting to any new task from the task distribution with as little data as possible. In this survey, we describe the meta-RL problem setting in detail as well as its major variations. We discuss how, at a high level, meta-RL research can be clustered based on the presence of a task distribution and the learning budget available for each individual task. Using these clusters, we then survey meta-RL algorithms and applications. We conclude by presenting the open problems on the path to making meta-RL part of the standard toolbox for a deep RL practitioner.
A mainstream type of current self-supervised learning methods pursues a general-purpose representation that can be well transferred to downstream tasks, typically by optimizing on a given pretext task such as instance discrimination. In this work, we argue that existing pretext tasks inevitably introduce biases into the learned representation, which in turn leads to biased transfer performance on various downstream tasks. To cope with this issue, we propose Maximum Entropy Coding (MEC), a more principled objective that explicitly optimizes on the structure of the representation, so that the learned representation is less biased and thus generalizes better to unseen downstream tasks. Inspired by the principle of maximum entropy in information theory, we hypothesize that a generalizable representation should be the one that admits the maximum entropy among all plausible representations. To make the objective end-to-end trainable, we propose to leverage the minimal coding length in lossy data coding as a computationally tractable surrogate for the entropy, and further derive a scalable reformulation of the objective that allows fast computation. Extensive experiments demonstrate that MEC learns a more generalizable representation than previous methods based on specific pretext tasks. It achieves state-of-the-art performance consistently on various downstream tasks, including not only ImageNet linear probe, but also semi-supervised classification, object detection, instance segmentation, and object tracking. Interestingly, we show that existing batch-wise and feature-wise self-supervised objectives could be seen equivalent to low-order approximations of MEC. Code and pre-trained models are available at //github.com/xinliu20/MEC.
The rapid changes in the finance industry due to the increasing amount of data have revolutionized the techniques on data processing and data analysis and brought new theoretical and computational challenges. In contrast to classical stochastic control theory and other analytical approaches for solving financial decision-making problems that heavily reply on model assumptions, new developments from reinforcement learning (RL) are able to make full use of the large amount of financial data with fewer model assumptions and to improve decisions in complex financial environments. This survey paper aims to review the recent developments and use of RL approaches in finance. We give an introduction to Markov decision processes, which is the setting for many of the commonly used RL approaches. Various algorithms are then introduced with a focus on value and policy based methods that do not require any model assumptions. Connections are made with neural networks to extend the framework to encompass deep RL algorithms. Our survey concludes by discussing the application of these RL algorithms in a variety of decision-making problems in finance, including optimal execution, portfolio optimization, option pricing and hedging, market making, smart order routing, and robo-advising.
This paper serves as a survey of recent advances in large margin training and its theoretical foundations, mostly for (nonlinear) deep neural networks (DNNs) that are probably the most prominent machine learning models for large-scale data in the community over the past decade. We generalize the formulation of classification margins from classical research to latest DNNs, summarize theoretical connections between the margin, network generalization, and robustness, and introduce recent efforts in enlarging the margins for DNNs comprehensively. Since the viewpoint of different methods is discrepant, we categorize them into groups for ease of comparison and discussion in the paper. Hopefully, our discussions and overview inspire new research work in the community that aim to improve the performance of DNNs, and we also point to directions where the large margin principle can be verified to provide theoretical evidence why certain regularizations for DNNs function well in practice. We managed to shorten the paper such that the crucial spirit of large margin learning and related methods are better emphasized.
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