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1. The Projected Bellman Equation in Reinforcement Learning

   https://doi.org/10.1109/TAC.2024.3409647  

   Meyn, Sean (January 2024, IEEE Transactions on Automatic Control)

   Astolfi, Alessandro (Ed.)

   Q-learning has become an important part of the reinforcement learning toolkit since its introduction in the dissertation of Chris Watkins in the 1980s. In the original tabular formulation, the goal is to compute exactly a solution to the discounted-cost optimality equation, and thereby obtain the optimal policy for a Markov Decision Process. The goal today is more modest: obtain an approximate solution within a prescribed function class. The standard algorithms are based on the same architecture as formulated in the 1980s, with the goal of finding a value function approximation that solves the so-called projected Bellman equation. While reinforcement learning has been an active research area for over four decades, there is little theory providing conditions for convergence of these Q-learning algorithms, or even existence of a solution to this equation. The purpose of this paper is to show that a solution to the projected Bellman equation does exist, provided the function class is linear and the input used for training is a form of epsilon-greedy policy with sufficiently small epsilon. Moreover, under these conditions it is shown that the Q-learning algorithm is stable, in terms of bounded parameter estimates. Convergence remains one of many open topics for research. 

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2. Convex Q Learning in a Stochastic Environment

   Lu, Fan and (December 2023, Proceedings of the IEEE Conference on Decision Control)

   The paper introduces the first formulation of convex Q-learning for Markov decision processes with function approximation. The algorithms and theory rest on a relaxation of a dual of Manne's celebrated linear programming characterization of optimal control. The main contributions firstly concern properties of the relaxation, described as a deterministic convex program: we identify conditions for a bounded solution, a significant connection between the solution to the new convex program, and the solution to standard Q-learning with linear function approximation. The second set of contributions concern algorithm design and analysis: (i) A direct model-free method for approximating the convex program for Q-learning shares properties with its ideal. In particular, a bounded solution is ensured subject to a simple property of the basis functions; (ii) The proposed algorithms are convergent and new techniques are introduced to obtain the rate of convergence in a mean-square sense; (iii) The approach can be generalized to a range of performance criteria, and it is found that variance can be reduced by considering ``relative'' dynamic programming equations; (iv) The theory is illustrated with an application to a classical inventory control problem. 

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3. Learning Bellman Complete Representations for Offline Policy Evaluation

   Jonathan Chang, Kaiwen Wang (January 2022, Proceedings of the 39th International Conference on Machine Learning)

   We study representation learning for Offline Reinforcement Learning (RL), focusing on the important task of Offline Policy Evaluation (OPE). Recent work shows that, in contrast to supervised learning, realizability of the Q-function is not enough for learning it. Two sufficient conditions for sample-efficient OPE are Bellman completeness and coverage. Prior work often assumes that representations satisfying these conditions are given, with results being mostly theoretical in nature. In this work, we propose BCRL, which directly learns from data an approximately linear Bellman complete representation with good coverage. With this learned representation, we perform OPE using Least Square Policy Evaluation (LSPE) with linear functions in our learned representation. We present an end-to-end theoretical analysis, showing that our two-stage algorithm enjoys polynomial sample complexity provided some representation in the rich class considered is linear Bellman complete. Empirically, we extensively evaluate our algorithm on challenging, image-based continuous control tasks from the Deepmind Control Suite. We show our representation enables better OPE compared to previous representation learning methods developed for off-policy RL (e.g., CURL, SPR). BCRL achieve competitive OPE error with the state-of-the-art method Fitted Q-Evaluation (FQE), and beats FQE when evaluating beyond the initial state distribution. Our ablations show that both linear Bellman complete and coverage components of our method are crucial. 

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4. Neural Network Approximation for Pessimistic Offline Reinforcement Learning

   https://doi.org/10.1609/aaai.v38i14.29517  

   Wu, Di; Jiao, Yuling; Shen, Li; Yang, Haizhao; Lu, Xiliang (March 2024, Proceedings of the AAAI Conference on Artificial Intelligence)

   Deep reinforcement learning (RL) has shown remarkable success in specific offline decision-making scenarios, yet its theoretical guarantees are still under development. Existing works on offline RL theory primarily emphasize a few trivial settings, such as linear MDP or general function approximation with strong assumptions and independent data, which lack guidance for practical use. The coupling of deep learning and Bellman residuals makes this problem challenging, in addition to the difficulty of data dependence. In this paper, we establish a non-asymptotic estimation error of pessimistic offline RL using general neural network approximation with C-mixing data regarding the structure of networks, the dimension of datasets, and the concentrability of data coverage, under mild assumptions. Our result shows that the estimation error consists of two parts: the first converges to zero at a desired rate on the sample size with partially controllable concentrability, and the second becomes negligible if the residual constraint is tight. This result demonstrates the explicit efficiency of deep adversarial offline RL frameworks. We utilize the empirical process tool for C-mixing sequences and the neural network approximation theory for the Holder class to achieve this. We also develop methods to bound the Bellman estimation error caused by function approximation with empirical Bellman constraint perturbations. Additionally, we present a result that lessens the curse of dimensionality using data with low intrinsic dimensionality and function classes with low complexity. Our estimation provides valuable insights into the development of deep offline RL and guidance for algorithm model design. 

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5. Mean-Field Multiagent Reinforcement Learning: A Decentralized Network Approach

   https://doi.org/10.1287/moor.2022.0055  

   Gu, Haotian; Guo, Xin; Wei, Xiaoli; Xu, Renyuan (March 2024, Mathematics of Operations Research)

   One of the challenges for multiagent reinforcement learning (MARL) is designing efficient learning algorithms for a large system in which each agent has only limited or partial information of the entire system. Whereas exciting progress has been made to analyze decentralized MARL with the network of agents for social networks and team video games, little is known theoretically for decentralized MARL with the network of states for modeling self-driving vehicles, ride-sharing, and data and traffic routing. This paper proposes a framework of localized training and decentralized execution to study MARL with the network of states. Localized training means that agents only need to collect local information in their neighboring states during the training phase; decentralized execution implies that agents can execute afterward the learned decentralized policies, which depend only on agents’ current states. The theoretical analysis consists of three key components: the first is the reformulation of the MARL system as a networked Markov decision process with teams of agents, enabling updating the associated team Q-function in a localized fashion; the second is the Bellman equation for the value function and the appropriate Q-function on the probability measure space; and the third is the exponential decay property of the team Q-function, facilitating its approximation with efficient sample efficiency and controllable error. The theoretical analysis paves the way for a new algorithm LTDE-Neural-AC, in which the actor–critic approach with overparameterized neural networks is proposed. The convergence and sample complexity are established and shown to be scalable with respect to the sizes of both agents and states. To the best of our knowledge, this is the first neural network–based MARL algorithm with network structure and provable convergence guarantee. 

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