Search

In-context learning (ICL) is a cornerstone of large language model (LLM) functionality, yet its theoretical foundations remain elusive due to the complexity of transformer architectures. In particular, most existing work only theoretically explains how the attention mechanism facilitates ICL under certain data models. It remains unclear how the other building blocks of the transformer contribute to ICL. To address this question, we study how a two-attention-layer transformer is trained to perform ICL on $n$-gram Markov chain data, where each token in the Markov chain statistically depends on the previous $n$ tokens. We analyze a sophisticated transformer model featuring relative positional embedding, multi-head softmax attention, and a feed-forward layer with normalization. We prove that the gradient flow with respect to a cross-entropy ICL loss converges to a limiting model that performs a generalized version of the induction head mechanism with a learned feature, resulting from the congruous contribution of all the building blocks. In the limiting model, the first attention layer acts as a $\mathit{copier}$, copying past tokens within a given window to each position, and the feed-forward network with normalization acts as a $\mathit{selector}$ that generates a feature vector by only looking at informationally relevant parents from the window. Finally, the second attention layer is a $\mathit{classifier}$ that compares these features with the feature at the output position, and uses the resulting similarity scores to generate the desired output. Our theory is further validated by experiments., Comment: 100 pages, 10 figures

Chain-of-Thought (CoT) prompting and its variants have gained popularity as effective methods for solving multi-step reasoning problems using pretrained large language models (LLMs). In this work, we analyze CoT prompting from a statistical estimation perspective, providing a comprehensive characterization of its sample complexity. To this end, we introduce a multi-step latent variable model that encapsulates the reasoning process, where the latent variable encodes the task information. Under this framework, we demonstrate that when the pretraining dataset is sufficiently large, the estimator formed by CoT prompting is equivalent to a Bayesian estimator. This estimator effectively solves the multi-step reasoning problem by aggregating a posterior distribution inferred from the demonstration examples in the prompt. Moreover, we prove that the statistical error of the CoT estimator can be decomposed into two main components: (i) a prompting error, which arises from inferring the true task using CoT prompts, and (ii) the statistical error of the pretrained LLM. We establish that, under appropriate assumptions, the prompting error decays exponentially to zero as the number of demonstrations increases. Additionally, we explicitly characterize the approximation and generalization errors of the pretrained LLM. Notably, we construct a transformer model that approximates the target distribution of the multi-step reasoning problem with an error that decreases exponentially in the number of transformer blocks. Our analysis extends to other variants of CoT, including Self-Consistent CoT, Tree-of-Thought, and Selection-Inference, offering a broad perspective on the efficacy of these methods. We also provide numerical experiments to validate the theoretical findings., Comment: 150 pages, 18 figures, 3 tables

Diffusion models have revolutionized various application domains, including computer vision and audio generation. Despite the state-of-the-art performance, diffusion models are known for their slow sample generation due to the extensive number of steps involved. In response, consistency models have been developed to merge multiple steps in the sampling process, thereby significantly boosting the speed of sample generation without compromising quality. This paper contributes towards the first statistical theory for consistency models, formulating their training as a distribution discrepancy minimization problem. Our analysis yields statistical estimation rates based on the Wasserstein distance for consistency models, matching those of vanilla diffusion models. Additionally, our results encompass the training of consistency models through both distillation and isolation methods, demystifying their underlying advantage., Comment: 28 pages, 2 figures

In this work, from a theoretical lens, we aim to understand why large language model (LLM) empowered agents are able to solve decision-making problems in the physical world. To this end, consider a hierarchical reinforcement learning (RL) model where the LLM Planner and the Actor perform high-level task planning and low-level execution, respectively. Under this model, the LLM Planner navigates a partially observable Markov decision process (POMDP) by iteratively generating language-based subgoals via prompting. Under proper assumptions on the pretraining data, we prove that the pretrained LLM Planner effectively performs Bayesian aggregated imitation learning (BAIL) through in-context learning. Additionally, we highlight the necessity for exploration beyond the subgoals derived from BAIL by proving that naively executing the subgoals returned by LLM leads to a linear regret. As a remedy, we introduce an $\epsilon$-greedy exploration strategy to BAIL, which is proven to incur sublinear regret when the pretraining error is small. Finally, we extend our theoretical framework to include scenarios where the LLM Planner serves as a world model for inferring the transition model of the environment and to multi-agent settings, enabling coordination among multiple Actors., Comment: 47 pages, accepted by ICML 2024

Large Language Models (LLMs) like GPT-4 have revolutionized natural language processing, showing remarkable linguistic proficiency and reasoning capabilities. However, their application in strategic multi-agent decision-making environments is hampered by significant limitations including poor mathematical reasoning, difficulty in following instructions, and a tendency to generate incorrect information. These deficiencies hinder their performance in strategic and interactive tasks that demand adherence to nuanced game rules, long-term planning, exploration in unknown environments, and anticipation of opponents' moves. To overcome these obstacles, this paper presents a novel LLM agent framework equipped with memory and specialized tools to enhance their strategic decision-making capabilities. We deploy the tools in a number of economically important environments, in particular bilateral bargaining and multi-agent and dynamic mechanism design. We employ quantitative metrics to assess the framework's performance in various strategic decision-making problems. Our findings establish that our enhanced framework significantly improves the strategic decision-making capability of LLMs. While we highlight the inherent limitations of current LLM models, we demonstrate the improvements through targeted enhancements, suggesting a promising direction for future developments in LLM applications for interactive environments., Comment: 39 pages, 4 figures

Offline Reinforcement Learning (RL) has shown promising results in learning a task-specific policy from a fixed dataset. However, successful offline RL often relies heavily on the coverage and quality of the given dataset. In scenarios where the dataset for a specific task is limited, a natural approach is to improve offline RL with datasets from other tasks, namely, to conduct Multi-Task Data Sharing (MTDS). Nevertheless, directly sharing datasets from other tasks exacerbates the distribution shift in offline RL. In this paper, we propose an uncertainty-based MTDS approach that shares the entire dataset without data selection. Given ensemble-based uncertainty quantification, we perform pessimistic value iteration on the shared offline dataset, which provides a unified framework for single- and multi-task offline RL. We further provide theoretical analysis, which shows that the optimality gap of our method is only related to the expected data coverage of the shared dataset, thus resolving the distribution shift issue in data sharing. Empirically, we release an MTDS benchmark and collect datasets from three challenging domains. The experimental results show our algorithm outperforms the previous state-of-the-art methods in challenging MTDS problems. See https://github.com/Baichenjia/UTDS for the datasets and code., Comment: Accepted by Artificial Intelligence (AIJ)

We study infinite-horizon average-reward Markov decision processes (AMDPs) in the context of general function approximation. Specifically, we propose a novel algorithmic framework named Local-fitted Optimization with OPtimism (LOOP), which incorporates both model-based and value-based incarnations. In particular, LOOP features a novel construction of confidence sets and a low-switching policy updating scheme, which are tailored to the average-reward and function approximation setting. Moreover, for AMDPs, we propose a novel complexity measure -- average-reward generalized eluder coefficient (AGEC) -- which captures the challenge of exploration in AMDPs with general function approximation. Such a complexity measure encompasses almost all previously known tractable AMDP models, such as linear AMDPs and linear mixture AMDPs, and also includes newly identified cases such as kernel AMDPs and AMDPs with Bellman eluder dimensions. Using AGEC, we prove that LOOP achieves a sublinear $\tilde{\mathcal{O}}(\mathrm{poly}(d, \mathrm{sp}(V^*)) \sqrt{T\beta} )$ regret, where $d$ and $\beta$ correspond to AGEC and log-covering number of the hypothesis class respectively, $\mathrm{sp}(V^*)$ is the span of the optimal state bias function, $T$ denotes the number of steps, and $\tilde{\mathcal{O}} (\cdot) $ omits logarithmic factors. When specialized to concrete AMDP models, our regret bounds are comparable to those established by the existing algorithms designed specifically for these special cases. To the best of our knowledge, this paper presents the first comprehensive theoretical framework capable of handling nearly all AMDPs., Comment: ICLR 2024

This paper studies minimax optimization problems defined over infinite-dimensional function classes of overparameterized two-layer neural networks. In particular, we consider the minimax optimization problem stemming from estimating linear functional equations defined by conditional expectations, where the objective functions are quadratic in the functional spaces. We address (i) the convergence of the stochastic gradient descent-ascent algorithm and (ii) the representation learning of the neural networks. We establish convergence under the mean-field regime by considering the continuous-time and infinite-width limit of the optimization dynamics. Under this regime, the stochastic gradient descent-ascent corresponds to a Wasserstein gradient flow over the space of probability measures defined over the space of neural network parameters. We prove that the Wasserstein gradient flow converges globally to a stationary point of the minimax objective at a $O(T^{-1} + \alpha^{-1})$ sublinear rate, and additionally finds the solution to the functional equation when the regularizer of the minimax objective is strongly convex. Here $T$ denotes the time and $\alpha$ is a scaling parameter of the neural networks. In terms of representation learning, our results show that the feature representation induced by the neural networks is allowed to deviate from the initial one by the magnitude of $O(\alpha^{-1})$, measured in terms of the Wasserstein distance. Finally, we apply our general results to concrete examples including policy evaluation, nonparametric instrumental variable regression, asset pricing, and adversarial Riesz representer estimation., Comment: Submitted

Conditional diffusion models serve as the foundation of modern image synthesis and find extensive application in fields like computational biology and reinforcement learning. In these applications, conditional diffusion models incorporate various conditional information, such as prompt input, to guide the sample generation towards desired properties. Despite the empirical success, theory of conditional diffusion models is largely missing. This paper bridges this gap by presenting a sharp statistical theory of distribution estimation using conditional diffusion models. Our analysis yields a sample complexity bound that adapts to the smoothness of the data distribution and matches the minimax lower bound. The key to our theoretical development lies in an approximation result for the conditional score function, which relies on a novel diffused Taylor approximation technique. Moreover, we demonstrate the utility of our statistical theory in elucidating the performance of conditional diffusion models across diverse applications, including model-based transition kernel estimation in reinforcement learning, solving inverse problems, and reward conditioned sample generation., Comment: 92 pages, 5 figures

In a sequential decision-making problem, the information structure is the description of how events in the system occurring at different points in time affect each other. Classical models of reinforcement learning (e.g., MDPs, POMDPs) assume a simple and highly regular information structure, while more general models like predictive state representations do not explicitly model the information structure. By contrast, real-world sequential decision-making problems typically involve a complex and time-varying interdependence of system variables, requiring a rich and flexible representation of information structure. In this paper, we formalize a novel reinforcement learning model which explicitly represents the information structure. We then use this model to carry out an information-structural analysis of the statistical hardness of general sequential decision-making problems, obtaining a characterization via a graph-theoretic quantity of the DAG representation of the information structure. We prove an upper bound on the sample complexity of learning a general sequential decision-making problem in terms of its information structure by exhibiting an algorithm achieving the upper bound. This recovers known tractability results and gives a novel perspective on reinforcement learning in general sequential decision-making problems, providing a systematic way of identifying new tractable classes of problems., Comment: 59 pages, 5 figures

We study the dynamics of gradient flow for training a multi-head softmax attention model for in-context learning of multi-task linear regression. We establish the global convergence of gradient flow under suitable choices of initialization. In addition, we prove that an interesting "task allocation" phenomenon emerges during the gradient flow dynamics, where each attention head focuses on solving a single task of the multi-task model. Specifically, we prove that the gradient flow dynamics can be split into three phases -- a warm-up phase where the loss decreases rather slowly and the attention heads gradually build up their inclination towards individual tasks, an emergence phase where each head selects a single task and the loss rapidly decreases, and a convergence phase where the attention parameters converge to a limit. Furthermore, we prove the optimality of gradient flow in the sense that the limiting model learned by gradient flow is on par with the best possible multi-head softmax attention model up to a constant factor. Our analysis also delineates a strict separation in terms of the prediction accuracy of ICL between single-head and multi-head attention models. The key technique for our convergence analysis is to map the gradient flow dynamics in the parameter space to a set of ordinary differential equations in the spectral domain, where the relative magnitudes of the semi-singular values of the attention weights determines task allocation. To our best knowledge, our work provides the first convergence result for the multi-head softmax attention model., Comment: 141 pages, 7 figures

We study the Constrained Convex Markov Decision Process (MDP), where the goal is to minimize a convex functional of the visitation measure, subject to a convex constraint. Designing algorithms for a constrained convex MDP faces several challenges, including (1) handling the large state space, (2) managing the exploration/exploitation tradeoff, and (3) solving the constrained optimization where the objective and the constraint are both nonlinear functions of the visitation measure. In this work, we present a model-based algorithm, Variational Primal-Dual Policy Optimization (VPDPO), in which Lagrangian and Fenchel duality are implemented to reformulate the original constrained problem into an unconstrained primal-dual optimization. Moreover, the primal variables are updated by model-based value iteration following the principle of Optimism in the Face of Uncertainty (OFU), while the dual variables are updated by gradient ascent. Moreover, by embedding the visitation measure into a finite-dimensional space, we can handle large state spaces by incorporating function approximation. Two notable examples are (1) Kernelized Nonlinear Regulators and (2) Low-rank MDPs. We prove that with an optimistic planning oracle, our algorithm achieves sublinear regret and constraint violation in both cases and can attain the globally optimal policy of the original constrained problem.

Bilevel optimization has been recently applied to many machine learning tasks. However, their applications have been restricted to the supervised learning setting, where static objective functions with benign structures are considered. But bilevel problems such as incentive design, inverse reinforcement learning (RL), and RL from human feedback (RLHF) are often modeled as dynamic objective functions that go beyond the simple static objective structures, which pose significant challenges of using existing bilevel solutions. To tackle this new class of bilevel problems, we introduce the first principled algorithmic framework for solving bilevel RL problems through the lens of penalty formulation. We provide theoretical studies of the problem landscape and its penalty-based (policy) gradient algorithms. We demonstrate the effectiveness of our algorithms via simulations in the Stackelberg Markov game, RL from human feedback and incentive design., Comment: Shorter version accepted to ICML 2024

In this paper, we extend mean-field Langevin dynamics to minimax optimization over probability distributions for the first time with symmetric and provably convergent updates. We propose mean-field Langevin averaged gradient (MFL-AG), a single-loop algorithm that implements gradient descent ascent in the distribution spaces with a novel weighted averaging, and establish average-iterate convergence to the mixed Nash equilibrium. We also study both time and particle discretization regimes and prove a new uniform-in-time propagation of chaos result which accounts for the dependency of the particle interactions on all previous distributions. Furthermore, we propose mean-field Langevin anchored best response (MFL-ABR), a symmetric double-loop algorithm based on best response dynamics with linear last-iterate convergence. Finally, we study applications to zero-sum Markov games and conduct simulations demonstrating long-term optimality., Comment: ICLR 2024 spotlight

Autonomous Driving (AD) encounters significant safety hurdles in long-tail unforeseen driving scenarios, largely stemming from the non-interpretability and poor generalization of the deep neural networks within the AD system, particularly in out-of-distribution and uncertain data. To this end, this paper explores the integration of Large Language Models (LLMs) into AD systems, leveraging their robust common-sense knowledge and reasoning abilities. The proposed methodologies employ LLMs as intelligent decision-makers in behavioral planning, augmented with a safety verifier shield for contextual safety learning, for enhancing driving performance and safety. We present two key studies in a simulated environment: an adaptive LLM-conditioned Model Predictive Control (MPC) and an LLM-enabled interactive behavior planning scheme with a state machine. Demonstrating superior performance and safety metrics compared to state-of-the-art approaches, our approach shows the promising potential for using LLMs for autonomous vehicles., Comment: Accepted to LLMAgent workshop @ICLR2024

We study high-dimensional multi-armed contextual bandits with batched feedback where the $T$ steps of online interactions are divided into $L$ batches. In specific, each batch collects data according to a policy that depends on previous batches and the rewards are revealed only at the end of the batch. Such a feedback structure is popular in applications such as personalized medicine and online advertisement, where the online data often do not arrive in a fully serial manner. We consider high-dimensional and linear settings where the reward function of the bandit model admits either a sparse or low-rank structure and ask how small a number of batches are needed for a comparable performance with fully dynamic data in which $L = T$. For these settings, we design a provably sample-efficient algorithm which achieves a $ \mathcal{\tilde O}(s_0^2 \log^2 T)$ regret in the sparse case and $ \mathcal{\tilde O} ( r ^2 \log^2 T)$ regret in the low-rank case, using only $L = \mathcal{O}( \log T)$ batches. Here $s_0$ and $r$ are the sparsity and rank of the reward parameter in sparse and low-rank cases, respectively, and $ \mathcal{\tilde O}(\cdot)$ omits logarithmic factors involving the feature dimensions. In other words, our algorithm achieves regret bounds comparable to those in fully sequential setting with only $\mathcal{O}( \log T)$ batches. Our algorithm features a novel batch allocation method that adjusts the batch sizes according to the estimation accuracy within each batch and cumulative regret. Furthermore, we also conduct experiments with synthetic and real-world data to validate our theory.

This paper investigates posterior sampling algorithms for competitive reinforcement learning (RL) in the context of general function approximations. Focusing on zero-sum Markov games (MGs) under two critical settings, namely self-play and adversarial learning, we first propose the self-play and adversarial generalized eluder coefficient (GEC) as complexity measures for function approximation, capturing the exploration-exploitation trade-off in MGs. Based on self-play GEC, we propose a model-based self-play posterior sampling method to control both players to learn Nash equilibrium, which can successfully handle the partial observability of states. Furthermore, we identify a set of partially observable MG models fitting MG learning with the adversarial policies of the opponent. Incorporating the adversarial GEC, we propose a model-based posterior sampling method for learning adversarial MG with potential partial observability. We further provide low regret bounds for proposed algorithms that can scale sublinearly with the proposed GEC and the number of episodes $T$. To the best of our knowledge, we for the first time develop generic model-based posterior sampling algorithms for competitive RL that can be applied to a majority of tractable zero-sum MG classes in both fully observable and partially observable MGs with self-play and adversarial learning., Comment: NeurIPS 2023

We design and analyze reinforcement learning algorithms for Graphon Mean-Field Games (GMFGs). In contrast to previous works that require the precise values of the graphons, we aim to learn the Nash Equilibrium (NE) of the regularized GMFGs when the graphons are unknown. Our contributions are threefold. First, we propose the Proximal Policy Optimization for GMFG (GMFG-PPO) algorithm and show that it converges at a rate of $O(T^{-1/3})$ after $T$ iterations with an estimation oracle, improving on a previous work by Xie et al. (ICML, 2021). Second, using kernel embedding of distributions, we design efficient algorithms to estimate the transition kernels, reward functions, and graphons from sampled agents. Convergence rates are then derived when the positions of the agents are either known or unknown. Results for the combination of the optimization algorithm GMFG-PPO and the estimation algorithm are then provided. These algorithms are the first specifically designed for learning graphons from sampled agents. Finally, the efficacy of the proposed algorithms are corroborated through simulations. These simulations demonstrate that learning the unknown graphons reduces the exploitability effectively.

This paper studies two fundamental problems in regularized Graphon Mean-Field Games (GMFGs). First, we establish the existence of a Nash Equilibrium (NE) of any $\lambda$-regularized GMFG (for $\lambda\geq 0$). This result relies on weaker conditions than those in previous works for analyzing both unregularized GMFGs ($\lambda=0$) and $\lambda$-regularized MFGs, which are special cases of GMFGs. Second, we propose provably efficient algorithms to learn the NE in weakly monotone GMFGs, motivated by Lasry and Lions [2007]. Previous literature either only analyzed continuous-time algorithms or required extra conditions to analyze discrete-time algorithms. In contrast, we design a discrete-time algorithm and derive its convergence rate solely under weakly monotone conditions. Furthermore, we develop and analyze the action-value function estimation procedure during the online learning process, which is absent from algorithms for monotone GMFGs. This serves as a sub-module in our optimization algorithm. The efficiency of the designed algorithm is corroborated by empirical evaluations.

We study multi-agent reinforcement learning (MARL) for the general-sum Markov Games (MGs) under the general function approximation. In order to find the minimum assumption for sample-efficient learning, we introduce a novel complexity measure called the Multi-Agent Decoupling Coefficient (MADC) for general-sum MGs. Using this measure, we propose the first unified algorithmic framework that ensures sample efficiency in learning Nash Equilibrium, Coarse Correlated Equilibrium, and Correlated Equilibrium for both model-based and model-free MARL problems with low MADC. We also show that our algorithm provides comparable sublinear regret to the existing works. Moreover, our algorithm combines an equilibrium-solving oracle with a single objective optimization subprocedure that solves for the regularized payoff of each deterministic joint policy, which avoids solving constrained optimization problems within data-dependent constraints (Jin et al. 2020; Wang et al. 2023) or executing sampling procedures with complex multi-objective optimization problems (Foster et al. 2023), thus being more amenable to empirical implementation.

We study reinforcement learning (RL) for learning a Quantal Stackelberg Equilibrium (QSE) in an episodic Markov game with a leader-follower structure. In specific, at the outset of the game, the leader announces her policy to the follower and commits to it. The follower observes the leader's policy and, in turn, adopts a quantal response policy by solving an entropy-regularized policy optimization problem induced by leader's policy. The goal of the leader is to find her optimal policy, which yields the optimal expected total return, by interacting with the follower and learning from data. A key challenge of this problem is that the leader cannot observe the follower's reward, and needs to infer the follower's quantal response model from his actions against leader's policies. We propose sample-efficient algorithms for both the online and offline settings, in the context of function approximation. Our algorithms are based on (i) learning the quantal response model via maximum likelihood estimation and (ii) model-free or model-based RL for solving the leader's decision making problem, and we show that they achieve sublinear regret upper bounds. Moreover, we quantify the uncertainty of these estimators and leverage the uncertainty to implement optimistic and pessimistic algorithms for online and offline settings. Besides, when specialized to the linear and myopic setting, our algorithms are also computationally efficient. Our theoretical analysis features a novel performance-difference lemma which incorporates the error of quantal response model, which might be of independent interest., Comment: 129 pages, 1 figure

Personalized pricing, which involves tailoring prices based on individual characteristics, is commonly used by firms to implement a consumer-specific pricing policy. In this process, buyers can also strategically manipulate their feature data to obtain a lower price, incurring certain manipulation costs. Such strategic behavior can hinder firms from maximizing their profits. In this paper, we study the contextual dynamic pricing problem with strategic buyers. The seller does not observe the buyer's true feature, but a manipulated feature according to buyers' strategic behavior. In addition, the seller does not observe the buyers' valuation of the product, but only a binary response indicating whether a sale happens or not. Recognizing these challenges, we propose a strategic dynamic pricing policy that incorporates the buyers' strategic behavior into the online learning to maximize the seller's cumulative revenue. We first prove that existing non-strategic pricing policies that neglect the buyers' strategic behavior result in a linear $\Omega(T)$ regret with $T$ the total time horizon, indicating that these policies are not better than a random pricing policy. We then establish that our proposed policy achieves a sublinear regret upper bound of $O(\sqrt{T})$. Importantly, our policy is not a mere amalgamation of existing dynamic pricing policies and strategic behavior handling algorithms. Our policy can also accommodate the scenario when the marginal cost of manipulation is unknown in advance. To account for it, we simultaneously estimate the valuation parameter and the cost parameter in the online pricing policy, which is shown to also achieve an $O(\sqrt{T})$ regret bound. Extensive experiments support our theoretical developments and demonstrate the superior performance of our policy compared to other pricing policies that are unaware of the strategic behaviors., Comment: The paper has been accepted by JASA

We take the first step in studying general sequential decision-making under two adaptivity constraints: rare policy switch and batch learning. First, we provide a general class called the Eluder Condition class, which includes a wide range of reinforcement learning classes. Then, for the rare policy switch constraint, we provide a generic algorithm to achieve a $\widetilde{\mathcal{O}}(\log K) $ switching cost with a $\widetilde{\mathcal{O}}(\sqrt{K})$ regret on the EC class. For the batch learning constraint, we provide an algorithm that provides a $\widetilde{\mathcal{O}}(\sqrt{K}+K/B)$ regret with the number of batches $B.$ This paper is the first work considering rare policy switch and batch learning under general function classes, which covers nearly all the models studied in the previous works such as tabular MDP (Bai et al. 2019; Zhang et al. 2020), linear MDP (Wang et al. 2021; Gao et al. 2021), low eluder dimension MDP (Kong et al. 2021; Gao et al. 2021), generalized linear function approximation (Qiao et al. 2023), and also some new classes such as the low $D_\Delta$-type Bellman eluder dimension problem, linear mixture MDP, kernelized nonlinear regulator and undercomplete partially observed Markov decision process (POMDP)., Comment: 48 pages

In this paper, we study representation learning in partially observable Markov Decision Processes (POMDPs), where the agent learns a decoder function that maps a series of high-dimensional raw observations to a compact representation and uses it for more efficient exploration and planning. We focus our attention on the sub-classes of \textit{$\gamma$-observable} and \textit{decodable POMDPs}, for which it has been shown that statistically tractable learning is possible, but there has not been any computationally efficient algorithm. We first present an algorithm for decodable POMDPs that combines maximum likelihood estimation (MLE) and optimism in the face of uncertainty (OFU) to perform representation learning and achieve efficient sample complexity, while only calling supervised learning computational oracles. We then show how to adapt this algorithm to also work in the broader class of $\gamma$-observable POMDPs.

We examine online safe multi-agent reinforcement learning using constrained Markov games in which agents compete by maximizing their expected total rewards under a constraint on expected total utilities. Our focus is confined to an episodic two-player zero-sum constrained Markov game with independent transition functions that are unknown to agents, adversarial reward functions, and stochastic utility functions. For such a Markov game, we employ an approach based on the occupancy measure to formulate it as an online constrained saddle-point problem with an explicit constraint. We extend the Lagrange multiplier method in constrained optimization to handle the constraint by creating a generalized Lagrangian with minimax decision primal variables and a dual variable. Next, we develop an upper confidence reinforcement learning algorithm to solve this Lagrangian problem while balancing exploration and exploitation. Our algorithm updates the minimax decision primal variables via online mirror descent and the dual variable via projected gradient step and we prove that it enjoys sublinear rate $ O((|X|+|Y|) L \sqrt{T(|A|+|B|)}))$ for both regret and constraint violation after playing $T$ episodes of the game. Here, $L$ is the horizon of each episode, $(|X|,|A|)$ and $(|Y|,|B|)$ are the state/action space sizes of the min-player and the max-player, respectively. To the best of our knowledge, we provide the first provably efficient online safe reinforcement learning algorithm in constrained Markov games., Comment: 59 pages, a full version of the main paper in the 5th Annual Conference on Learning for Dynamics and Control

In this paper, we conduct a comprehensive study of In-Context Learning (ICL) by addressing several open questions: (a) What type of ICL estimator is learned by large language models? (b) What is a proper performance metric for ICL and what is the error rate? (c) How does the transformer architecture enable ICL? To answer these questions, we adopt a Bayesian view and formulate ICL as a problem of predicting the response corresponding to the current covariate, given a number of examples drawn from a latent variable model. To answer (a), we show that, without updating the neural network parameters, ICL implicitly implements the Bayesian model averaging algorithm, which is proven to be approximately parameterized by the attention mechanism. For (b), we analyze the ICL performance from an online learning perspective and establish a $\mathcal{O}(1/T)$ regret bound for perfectly pretrained ICL, where $T$ is the number of examples in the prompt. To answer (c), we show that, in addition to encoding Bayesian model averaging via attention, the transformer architecture also enables a fine-grained statistical analysis of pretraining under realistic assumptions. In particular, we prove that the error of pretrained model is bounded by a sum of an approximation error and a generalization error, where the former decays to zero exponentially as the depth grows, and the latter decays to zero sublinearly with the number of tokens in the pretraining dataset. Our results provide a unified understanding of the transformer and its ICL ability with bounds on ICL regret, approximation, and generalization, which deepens our knowledge of these essential aspects of modern language models.

Diffusion models have demonstrated highly-expressive generative capabilities in vision and NLP. Recent studies in reinforcement learning (RL) have shown that diffusion models are also powerful in modeling complex policies or trajectories in offline datasets. However, these works have been limited to single-task settings where a generalist agent capable of addressing multi-task predicaments is absent. In this paper, we aim to investigate the effectiveness of a single diffusion model in modeling large-scale multi-task offline data, which can be challenging due to diverse and multimodal data distribution. Specifically, we propose Multi-Task Diffusion Model (\textsc{MTDiff}), a diffusion-based method that incorporates Transformer backbones and prompt learning for generative planning and data synthesis in multi-task offline settings. \textsc{MTDiff} leverages vast amounts of knowledge available in multi-task data and performs implicit knowledge sharing among tasks. For generative planning, we find \textsc{MTDiff} outperforms state-of-the-art algorithms across 50 tasks on Meta-World and 8 maps on Maze2D. For data synthesis, \textsc{MTDiff} generates high-quality data for testing tasks given a single demonstration as a prompt, which enhances the low-quality datasets for even unseen tasks., Comment: Accepted by NeurIPS 2023. 22 pages

In online reinforcement learning (online RL), balancing exploration and exploitation is crucial for finding an optimal policy in a sample-efficient way. To achieve this, existing sample-efficient online RL algorithms typically consist of three components: estimation, planning, and exploration. However, in order to cope with general function approximators, most of them involve impractical algorithmic components to incentivize exploration, such as optimization within data-dependent level-sets or complicated sampling procedures. To address this challenge, we propose an easy-to-implement RL framework called \textit{Maximize to Explore} (\texttt{MEX}), which only needs to optimize \emph{unconstrainedly} a single objective that integrates the estimation and planning components while balancing exploration and exploitation automatically. Theoretically, we prove that \texttt{MEX} achieves a sublinear regret with general function approximations for Markov decision processes (MDP) and is further extendable to two-player zero-sum Markov games (MG). Meanwhile, we adapt deep RL baselines to design practical versions of \texttt{MEX}, in both model-free and model-based manners, which can outperform baselines by a stable margin in various MuJoCo environments with sparse rewards. Compared with existing sample-efficient online RL algorithms with general function approximations, \texttt{MEX} achieves similar sample efficiency while enjoying a lower computational cost and is more compatible with modern deep RL methods.

In this paper, we study offline Reinforcement Learning with Human Feedback (RLHF) where we aim to learn the human's underlying reward and the MDP's optimal policy from a set of trajectories induced by human choices. RLHF is challenging for multiple reasons: large state space but limited human feedback, the bounded rationality of human decisions, and the off-policy distribution shift. In this paper, we focus on the Dynamic Discrete Choice (DDC) model for modeling and understanding human choices. DCC, rooted in econometrics and decision theory, is widely used to model a human decision-making process with forward-looking and bounded rationality. We propose a \underline{D}ynamic-\underline{C}hoice-\underline{P}essimistic-\underline{P}olicy-\underline{O}ptimization (DCPPO) method. \ The method involves a three-stage process: The first step is to estimate the human behavior policy and the state-action value function via maximum likelihood estimation (MLE); the second step recovers the human reward function via minimizing Bellman mean squared error using the learned value functions; the third step is to plug in the learned reward and invoke pessimistic value iteration for finding a near-optimal policy. With only single-policy coverage (i.e., optimal policy) of the dataset, we prove that the suboptimality of DCPPO almost matches the classical pessimistic offline RL algorithm in terms of suboptimality's dependency on distribution shift and dimension. To the best of our knowledge, this paper presents the first theoretical guarantees for off-policy offline RLHF with dynamic discrete choice model.

Policy optimization methods with function approximation are widely used in multi-agent reinforcement learning. However, it remains elusive how to design such algorithms with statistical guarantees. Leveraging a multi-agent performance difference lemma that characterizes the landscape of multi-agent policy optimization, we find that the localized action value function serves as an ideal descent direction for each local policy. Motivated by the observation, we present a multi-agent PPO algorithm in which the local policy of each agent is updated similarly to vanilla PPO. We prove that with standard regularity conditions on the Markov game and problem-dependent quantities, our algorithm converges to the globally optimal policy at a sublinear rate. We extend our algorithm to the off-policy setting and introduce pessimism to policy evaluation, which aligns with experiments. To our knowledge, this is the first provably convergent multi-agent PPO algorithm in cooperative Markov games., Comment: ICML 2023

Most offline reinforcement learning (RL) methods suffer from the trade-off between improving the policy to surpass the behavior policy and constraining the policy to limit the deviation from the behavior policy as computing $Q$-values using out-of-distribution (OOD) actions will suffer from errors due to distributional shift. The recently proposed \textit{In-sample Learning} paradigm (i.e., IQL), which improves the policy by quantile regression using only data samples, shows great promise because it learns an optimal policy without querying the value function of any unseen actions. However, it remains unclear how this type of method handles the distributional shift in learning the value function. In this work, we make a key finding that the in-sample learning paradigm arises under the \textit{Implicit Value Regularization} (IVR) framework. This gives a deeper understanding of why the in-sample learning paradigm works, i.e., it applies implicit value regularization to the policy. Based on the IVR framework, we further propose two practical algorithms, Sparse $Q$-learning (SQL) and Exponential $Q$-learning (EQL), which adopt the same value regularization used in existing works, but in a complete in-sample manner. Compared with IQL, we find that our algorithms introduce sparsity in learning the value function, making them more robust in noisy data regimes. We also verify the effectiveness of SQL and EQL on D4RL benchmark datasets and show the benefits of in-sample learning by comparing them with CQL in small data regimes., Comment: ICLR 2023 notable top 5%

We study the offline contextual bandit problem, where we aim to acquire an optimal policy using observational data. However, this data usually contains two deficiencies: (i) some variables that confound actions are not observed, and (ii) missing observations exist in the collected data. Unobserved confounders lead to a confounding bias and missing observations cause bias and inefficiency problems. To overcome these challenges and learn the optimal policy from the observed dataset, we present a new algorithm called Causal-Adjusted Pessimistic (CAP) policy learning, which forms the reward function as the solution of an integral equation system, builds a confidence set, and greedily takes action with pessimism. With mild assumptions on the data, we develop an upper bound to the suboptimality of CAP for the offline contextual bandit problem., Comment: 76 page, 5 figures

We study the incentivized information acquisition problem, where a principal hires an agent to gather information on her behalf. Such a problem is modeled as a Stackelberg game between the principal and the agent, where the principal announces a scoring rule that specifies the payment, and then the agent then chooses an effort level that maximizes her own profit and reports the information. We study the online setting of such a problem from the principal's perspective, i.e., designing the optimal scoring rule by repeatedly interacting with the strategic agent. We design a provably sample efficient algorithm that tailors the UCB algorithm (Auer et al., 2002) to our model, which achieves a sublinear $T^{2/3}$-regret after $T$ iterations. Our algorithm features a delicate estimation procedure for the optimal profit of the principal, and a conservative correction scheme that ensures the desired agent's actions are incentivized. Furthermore, a key feature of our regret bound is that it is independent of the number of states of the environment., Comment: 35 pages, adding an impossible result (Lemma 3.2) with its proof in Section D.1

We study decentralized learning in two-player zero-sum discounted Markov games where the goal is to design a policy optimization algorithm for either agent satisfying two properties. First, the player does not need to know the policy of the opponent to update its policy. Second, when both players adopt the algorithm, their joint policy converges to a Nash equilibrium of the game. To this end, we construct a meta algorithm, dubbed as $\texttt{Homotopy-PO}$, which provably finds a Nash equilibrium at a global linear rate. In particular, $\texttt{Homotopy-PO}$ interweaves two base algorithms $\texttt{Local-Fast}$ and $\texttt{Global-Slow}$ via homotopy continuation. $\texttt{Local-Fast}$ is an algorithm that enjoys local linear convergence while $\texttt{Global-Slow}$ is an algorithm that converges globally but at a slower sublinear rate. By switching between these two base algorithms, $\texttt{Global-Slow}$ essentially serves as a ``guide'' which identifies a benign neighborhood where $\texttt{Local-Fast}$ enjoys fast convergence. However, since the exact size of such a neighborhood is unknown, we apply a doubling trick to switch between these two base algorithms. The switching scheme is delicately designed so that the aggregated performance of the algorithm is driven by $\texttt{Local-Fast}$. Furthermore, we prove that $\texttt{Local-Fast}$ and $\texttt{Global-Slow}$ can both be instantiated by variants of optimistic gradient descent/ascent (OGDA) method, which is of independent interest., Comment: ICLR 2023

We study a heterogeneous agent macroeconomic model with an infinite number of households and firms competing in a labor market. Each household earns income and engages in consumption at each time step while aiming to maximize a concave utility subject to the underlying market conditions. The households aim to find the optimal saving strategy that maximizes their discounted cumulative utility given the market condition, while the firms determine the market conditions through maximizing corporate profit based on the household population behavior. The model captures a wide range of applications in macroeconomic studies, and we propose a data-driven reinforcement learning framework that finds the regularized competitive equilibrium of the model. The proposed algorithm enjoys theoretical guarantees in converging to the equilibrium of the market at a sub-linear rate., Comment: 44 pages

Learning policies from fixed offline datasets is a key challenge to scale up reinforcement learning (RL) algorithms towards practical applications. This is often because off-policy RL algorithms suffer from distributional shift, due to mismatch between dataset and the target policy, leading to high variance and over-estimation of value functions. In this work, we propose variance regularization for offline RL algorithms, using stationary distribution corrections. We show that by using Fenchel duality, we can avoid double sampling issues for computing the gradient of the variance regularizer. The proposed algorithm for offline variance regularization (OVAR) can be used to augment any existing offline policy optimization algorithms. We show that the regularizer leads to a lower bound to the offline policy optimization objective, which can help avoid over-estimation errors, and explains the benefits of our approach across a range of continuous control domains when compared to existing state-of-the-art algorithms., Comment: Old Draft, Offline RL Workshop, NeurIPS'20

Motivated by the human-machine interaction such as training chatbots for improving customer satisfaction, we study human-guided human-machine interaction involving private information. We model this interaction as a two-player turn-based game, where one player (Alice, a human) guides the other player (Bob, a machine) towards a common goal. Specifically, we focus on offline reinforcement learning (RL) in this game, where the goal is to find a policy pair for Alice and Bob that maximizes their expected total rewards based on an offline dataset collected a priori. The offline setting presents two challenges: (i) We cannot collect Bob's private information, leading to a confounding bias when using standard RL methods, and (ii) a distributional mismatch between the behavior policy used to collect data and the desired policy we aim to learn. To tackle the confounding bias, we treat Bob's previous action as an instrumental variable for Alice's current decision making so as to adjust for the unmeasured confounding. We develop a novel identification result and use it to propose a new off-policy evaluation (OPE) method for evaluating policy pairs in this two-player turn-based game. To tackle the distributional mismatch, we leverage the idea of pessimism and use our OPE method to develop an off-policy learning algorithm for finding a desirable policy pair for both Alice and Bob. Finally, we prove that under mild assumptions such as partial coverage of the offline data, the policy pair obtained through our method converges to the optimal one at a satisfactory rate.

This paper studies offline policy learning, which aims at utilizing observations collected a priori (from either fixed or adaptively evolving behavior policies) to learn an optimal individualized decision rule that achieves the best overall outcomes for a given population. Existing policy learning methods rely on a uniform overlap assumption, i.e., the propensities of exploring all actions for all individual characteristics must be lower bounded. As one has no control over the data collection process, this assumption can be unrealistic in many situations, especially when the behavior policies are allowed to evolve over time with diminishing propensities for certain actions. In this paper, we propose Pessimistic Policy Learning (PPL), a new algorithm that optimizes lower confidence bounds (LCBs) -- instead of point estimates -- of the policy values. The LCBs are constructed using knowledge of the behavior policies for collecting the offline data. Without assuming any uniform overlap condition, we establish a data-dependent upper bound for the suboptimality of our algorithm, which only depends on (i) the overlap for the optimal policy, and (ii) the complexity of the policy class we optimize over. As an implication, for adaptively collected data, we ensure efficient policy learning as long as the propensities for optimal actions are lower bounded over time, while those for suboptimal ones are allowed to diminish arbitrarily fast. In our theoretical analysis, we develop a new self-normalized type concentration inequality for inverse-propensity-weighting estimators, generalizing the well-known empirical Bernstein's inequality to unbounded and non-i.i.d. data. We complement our theory with an efficient optimization algorithm via Majorization-Minimization and policy tree search, as well as extensive simulation studies and real-world applications that demonstrate the efficacy of PPL.

We study the hidden-action principal-agent problem in an online setting. In each round, the principal posts a contract that specifies the payment to the agent based on each outcome. The agent then makes a strategic choice of action that maximizes her own utility, but the action is not directly observable by the principal. The principal observes the outcome and receives utility from the agent's choice of action. Based on past observations, the principal dynamically adjusts the contracts with the goal of maximizing her utility. We introduce an online learning algorithm and provide an upper bound on its Stackelberg regret. We show that when the contract space is $[0,1]^m$, the Stackelberg regret is upper bounded by $\widetilde O(\sqrt{m} \cdot T^{1-1/(2m+1)})$, and lower bounded by $\Omega(T^{1-1/(m+2)})$, where $\widetilde O$ omits logarithmic factors. This result shows that exponential-in-$m$ samples are sufficient and necessary to learn a near-optimal contract, resolving an open problem on the hardness of online contract design. Moreover, when contracts are restricted to some subset $\mathcal{F} \subset [0,1]^m$, we define an intrinsic dimension of $\mathcal{F}$ that depends on the covering number of the spherical code in the space and bound the regret in terms of this intrinsic dimension. When $\mathcal{F}$ is the family of linear contracts, we show that the Stackelberg regret grows exactly as $\Theta(T^{2/3})$. The contract design problem is challenging because the utility function is discontinuous. Bounding the discretization error in this setting has been an open problem. In this paper, we identify a limited set of directions in which the utility function is continuous, allowing us to design a new discretization method and bound its error. This approach enables the first upper bound with no restrictions on the contract and action space.

We study sample efficient reinforcement learning (RL) under the general framework of interactive decision making, which includes Markov decision process (MDP), partially observable Markov decision process (POMDP), and predictive state representation (PSR) as special cases. Toward finding the minimum assumption that empowers sample efficient learning, we propose a novel complexity measure, generalized eluder coefficient (GEC), which characterizes the fundamental tradeoff between exploration and exploitation in online interactive decision making. In specific, GEC captures the hardness of exploration by comparing the error of predicting the performance of the updated policy with the in-sample training error evaluated on the historical data. We show that RL problems with low GEC form a remarkably rich class, which subsumes low Bellman eluder dimension problems, bilinear class, low witness rank problems, PO-bilinear class, and generalized regular PSR, where generalized regular PSR, a new tractable PSR class identified by us, includes nearly all known tractable POMDPs and PSRs. Furthermore, in terms of algorithm design, we propose a generic posterior sampling algorithm, which can be implemented in both model-free and model-based fashion, under both fully observable and partially observable settings. The proposed algorithm modifies the standard posterior sampling algorithm in two aspects: (i) we use an optimistic prior distribution that biases towards hypotheses with higher values and (ii) a loglikelihood function is set to be the empirical loss evaluated on the historical data, where the choice of loss function supports both model-free and model-based learning. We prove that the proposed algorithm is sample efficient by establishing a sublinear regret upper bound in terms of GEC. In summary, we provide a new and unified understanding of both fully observable and partially observable RL., Comment: We changed the title from the first version. We fixed a technical issue in the first version regarding the $\ell_2$ eluder technique (Lemma D.2)

We study reserve price optimization in multi-phase second price auctions, where seller's prior actions affect the bidders' later valuations through a Markov Decision Process (MDP). Compared to the bandit setting in existing works, the setting in ours involves three challenges. First, from the seller's perspective, we need to efficiently explore the environment in the presence of potentially nontruthful bidders who aim to manipulates seller's policy. Second, we want to minimize the seller's revenue regret when the market noise distribution is unknown. Third, the seller's per-step revenue is unknown, nonlinear, and cannot even be directly observed from the environment. We propose a mechanism addressing all three challenges. To address the first challenge, we use a combination of a new technique named "buffer periods" and inspirations from Reinforcement Learning (RL) with low switching cost to limit bidders' surplus from untruthful bidding, thereby incentivizing approximately truthful bidding. The second one is tackled by a novel algorithm that removes the need for pure exploration when the market noise distribution is unknown. The third challenge is resolved by an extension of LSVI-UCB, where we use the auction's underlying structure to control the uncertainty of the revenue function. The three techniques culminate in the $\underline{\rm C}$ontextual-$\underline{\rm L}$SVI-$\underline{\rm U}$CB-$\underline{\rm B}$uffer (CLUB) algorithm which achieves $\tilde{ \mathcal{O}}(H^{5/2}\sqrt{K})$ revenue regret when the market noise is known and $\tilde{ \mathcal{O}}(H^{3}\sqrt{K})$ revenue regret when the noise is unknown with no assumptions on bidders' truthfulness.

It is quite challenging to ensure the safety of reinforcement learning (RL) agents in an unknown and stochastic environment under hard constraints that require the system state not to reach certain specified unsafe regions. Many popular safe RL methods such as those based on the Constrained Markov Decision Process (CMDP) paradigm formulate safety violations in a cost function and try to constrain the expectation of cumulative cost under a threshold. However, it is often difficult to effectively capture and enforce hard reachability-based safety constraints indirectly with such constraints on safety violation costs. In this work, we leverage the notion of barrier function to explicitly encode the hard safety constraints, and given that the environment is unknown, relax them to our design of \emph{generative-model-based soft barrier functions}. Based on such soft barriers, we propose a safe RL approach that can jointly learn the environment and optimize the control policy, while effectively avoiding unsafe regions with safety probability optimization. Experiments on a set of examples demonstrate that our approach can effectively enforce hard safety constraints and significantly outperform CMDP-based baseline methods in system safe rate measured via simulations., Comment: Accepted to ICML 2023

The cooperative Multi-A gent R einforcement Learning (MARL) with permutation invariant agents framework has achieved tremendous empirical successes in real-world applications. Unfortunately, the theoretical understanding of this MARL problem is lacking due to the curse of many agents and the limited exploration of the relational reasoning in existing works. In this paper, we verify that the transformer implements complex relational reasoning, and we propose and analyze model-free and model-based offline MARL algorithms with the transformer approximators. We prove that the suboptimality gaps of the model-free and model-based algorithms are independent of and logarithmic in the number of agents respectively, which mitigates the curse of many agents. These results are consequences of a novel generalization error bound of the transformer and a novel analysis of the Maximum Likelihood Estimate (MLE) of the system dynamics with the transformer. Our model-based algorithm is the first provably efficient MARL algorithm that explicitly exploits the permutation invariance of the agents. Our improved generalization bound may be of independent interest and is applicable to other regression problems related to the transformer beyond MARL.

We study the offline reinforcement learning (RL) in the face of unmeasured confounders. Due to the lack of online interaction with the environment, offline RL is facing the following two significant challenges: (i) the agent may be confounded by the unobserved state variables; (ii) the offline data collected a prior does not provide sufficient coverage for the environment. To tackle the above challenges, we study the policy learning in the confounded MDPs with the aid of instrumental variables. Specifically, we first establish value function (VF)-based and marginalized importance sampling (MIS)-based identification results for the expected total reward in the confounded MDPs. Then by leveraging pessimism and our identification results, we propose various policy learning methods with the finite-sample suboptimality guarantee of finding the optimal in-class policy under minimal data coverage and modeling assumptions. Lastly, our extensive theoretical investigations and one numerical study motivated by the kidney transplantation demonstrate the promising performance of the proposed methods.

We study offline reinforcement learning under a novel model called strategic MDP, which characterizes the strategic interactions between a principal and a sequence of myopic agents with private types. Due to the bilevel structure and private types, strategic MDP involves information asymmetry between the principal and the agents. We focus on the offline RL problem, where the goal is to learn the optimal policy of the principal concerning a target population of agents based on a pre-collected dataset that consists of historical interactions. The unobserved private types confound such a dataset as they affect both the rewards and observations received by the principal. We propose a novel algorithm, Pessimistic policy Learning with Algorithmic iNstruments (PLAN), which leverages the ideas of instrumental variable regression and the pessimism principle to learn a near-optimal principal's policy in the context of general function approximation. Our algorithm is based on the critical observation that the principal's actions serve as valid instrumental variables. In particular, under a partial coverage assumption on the offline dataset, we prove that PLAN outputs a $1 / \sqrt{K}$-optimal policy with $K$ being the number of collected trajectories. We further apply our framework to some special cases of strategic MDP, including strategic regression, strategic bandit, and noncompliance in recommendation systems., Comment: 62 pages

In view of its power in extracting feature representation, contrastive self-supervised learning has been successfully integrated into the practice of (deep) reinforcement learning (RL), leading to efficient policy learning in various applications. Despite its tremendous empirical successes, the understanding of contrastive learning for RL remains elusive. To narrow such a gap, we study how RL can be empowered by contrastive learning in a class of Markov decision processes (MDPs) and Markov games (MGs) with low-rank transitions. For both models, we propose to extract the correct feature representations of the low-rank model by minimizing a contrastive loss. Moreover, under the online setting, we propose novel upper confidence bound (UCB)-type algorithms that incorporate such a contrastive loss with online RL algorithms for MDPs or MGs. We further theoretically prove that our algorithm recovers the true representations and simultaneously achieves sample efficiency in learning the optimal policy and Nash equilibrium in MDPs and MGs. We also provide empirical studies to demonstrate the efficacy of the UCB-based contrastive learning method for RL. To the best of our knowledge, we provide the first provably efficient online RL algorithm that incorporates contrastive learning for representation learning. Our codes are available at https://github.com/Baichenjia/Contrastive-UCB., Comment: ICML 2022

While single-agent policy optimization in a fixed environment has attracted a lot of research attention recently in the reinforcement learning community, much less is known theoretically when there are multiple agents playing in a potentially competitive environment. We take steps forward by proposing and analyzing new fictitious play policy optimization algorithms for zero-sum Markov games with structured but unknown transitions. We consider two classes of transition structures: factored independent transition and single-controller transition. For both scenarios, we prove tight $\widetilde{\mathcal{O}}(\sqrt{K})$ regret bounds after $K$ episodes in a two-agent competitive game scenario. The regret of each agent is measured against a potentially adversarial opponent who can choose a single best policy in hindsight after observing the full policy sequence. Our algorithms feature a combination of Upper Confidence Bound (UCB)-type optimism and fictitious play under the scope of simultaneous policy optimization in a non-stationary environment. When both players adopt the proposed algorithms, their overall optimality gap is $\widetilde{\mathcal{O}}(\sqrt{K})$., Comment: ICML 2021

We study decentralized policy learning in Markov games where we control a single agent to play with nonstationary and possibly adversarial opponents. Our goal is to develop a no-regret online learning algorithm that (i) takes actions based on the local information observed by the agent and (ii) is able to find the best policy in hindsight. For such a problem, the nonstationary state transitions due to the varying opponent pose a significant challenge. In light of a recent hardness result \citep{liu2022learning}, we focus on the setting where the opponent's previous policies are revealed to the agent for decision making. With such an information structure, we propose a new algorithm, \underline{D}ecentralized \underline{O}ptimistic hype\underline{R}policy m\underline{I}rror de\underline{S}cent (DORIS), which achieves $\sqrt{K}$-regret in the context of general function approximation, where $K$ is the number of episodes. Moreover, when all the agents adopt DORIS, we prove that their mixture policy constitutes an approximate coarse correlated equilibrium. In particular, DORIS maintains a \textit{hyperpolicy} which is a distribution over the policy space. The hyperpolicy is updated via mirror descent, where the update direction is obtained by an optimistic variant of least-squares policy evaluation. Furthermore, to illustrate the power of our method, we apply DORIS to constrained and vector-valued MDPs, which can be formulated as zero-sum Markov games with a fictitious opponent.