Kaixiong Gong, Kaituo Feng, Bohao Li, Yibing Wang, Mofan Cheng, Shijia Yang, Jiaming Han, Benyou Wang, Yutong Bai, Zhuoran Yang, Xiangyu Yue. Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers). 2026.

We study reserve price optimization in multi-phase second price auctions, where the 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 untruthful bidders who aim to manipulate the 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 an unknown, nonlinear random variable, and cannot even be directly observed from the environment but realized values. 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{C}ontextual-\underline{L}SVI-\underline{U}CB-\underline{B}uffer (CLUB) algorithm which achieves $\tilde{\mathcal{O}}(H^{5/2}\sqrt{K})$ revenue regret, where $K$ is the number of episodes and $H$ is the length of each episode, 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.

In-Context Learning (ICL) ability has been found efficient across a wide range of applications, where the Large Language Models (LLM) learn to complete the tasks from the examples in the prompt without tuning the parameters. In this work, we conduct a comprehensive study to understand ICL from a statistical perspective. First, we show that the perfectly pretrained LLMs perform Bayesian Model Averaging (BMA) for ICL under a dynamic model of examples in the prompt. The average error analysis for ICL is then built for the perfectly pretrained LLMs with the analysis of BMA. Second, we demonstrate how the attention structure boosts the BMA implementation. With sufficient examples in the prompt, attention is proven to perform BMA under the Gaussian linear ICL model, which also motivates the explicit construction of the hidden concepts from the attention heads values. Finally, we analyze the pretraining behavior of LLMs. The pretraining error is decomposed as the generalization error and the approximation error. The generalization error is upper bounded via PAC-Bayes framework. Then the ICL average error of the pretrained LLMs is shown to be the sum of $O(T^{-1})$ and the pretraining error. In addition, we analyze the ICL performance of the pretrained LLMs with misspecified examples.

In this work, we tackle the following question: Can neural networks trained with gradient-based methods achieve the optimal statistical-computational tradeoff in learning Gaussian single-index models? Prior research has shown that any polynomial-time algorithm under the statistical query (SQ) framework requires $\Omega(d^{s^\star/2}\lor d)$ samples, where $s^\star$ is the generative exponent representing the intrinsic difficulty of learning the underlying model. However, it remains unknown whether neural networks can achieve this sample complexity. Inspired by prior techniques such as label transformation and landscape smoothing for learning single-index models, we propose a unified gradient-based algorithm for training a two-layer neural network in polynomial time. Our method is adaptable to a variety of loss and activation functions, covering a broad class of existing approaches. We show that our algorithm learns a feature representation that strongly aligns with the unknown signal $\theta^\star$, with sample complexity $\tilde O (d^{s^\star/2} \lor d)$, matching the SQ lower bound up to a polylogarithmic factor for all generative exponents $s^\star\geq 1$. Furthermore, we extend our approach to the setting where $\theta^\star$ is $k$-sparse for $k = o(\sqrt{d})$ by introducing a novel weight perturbation technique that leverages the sparsity structure. We derive a corresponding SQ lower bound of order $\tilde\Omega(k^{s^\star})$, matched by our method up to a polylogarithmic factor. Our framework, especially the weight perturbation technique, is of independent interest, and suggests potential gradient-based solutions to other problems such as sparse tensor PCA.

Speculative decoding has emerged as a popular method to accelerate the inference of Large Language Models (LLMs) while retaining their superior text generation performance. Previous methods either adopt a fixed speculative decoding configuration regardless of the prefix tokens, or train draft models in an offline or online manner to align them with the context. This paper proposes a training-free online learning framework to adaptively choose the configuration of the hyperparameters for speculative decoding as text is being generated. We first formulate this hyperparameter selection problem as a Multi-Armed Bandit problem and provide a general speculative decoding framework BanditSpec. Furthermore, two bandit-based hyperparameter selection algorithms, UCBSpec and EXP3Spec, are designed and analyzed in terms of a novel quantity, the stopping time regret. We upper bound this regret under both stochastic and adversarial reward settings. By deriving an information-theoretic impossibility result, it is shown that the regret performance of UCBSpec is optimal up to universal constants. Finally, extensive empirical experiments with LLaMA3 and Qwen2 demonstrate that our algorithms are effective compared to existing methods, and the throughput is close to the oracle best hyperparameter in simulated real-life LLM serving scenarios with diverse input prompts.

We develop a framework for capturing the instrumental value of data production processes, which accounts for two key factors: (a) the context of the agent’s decision-making; (b) how much data or information the buyer already possesses. We "micro-found" our data valuation function by establishing its connection to classic notions of signals and information design in economics. When instantiated in Bayesian linear regression, our value naturally corresponds to information gain. Applying our proposed data value in Bayesian linear regression for monopoly pricing, we show that if the seller can fully customize data production, she can extract the first-best revenue (i.e., full surplus) from any population of buyers, i.e., achieving first-degree price discrimination. If data can only be constructed from an existing data pool, this limits the seller’s ability to customize, and achieving first-best revenue becomes generally impossible. However, we design a mechanism that achieves seller revenue at most $\log(\kappa)$ less than the first-best, where $\kappa$ is the condition number associated with the data matrix. As a corollary, the seller extracts the first-best revenue in the multi-armed bandits special case.

Transformer models have achieved remarkable empirical successes, largely due to their in-context learning capabilities. Inspired by this, we explore training an autoregressive transformer for in-context reinforcement learning (ICRL). In this setting, we initially train a transformer on an offline dataset consisting of trajectories collected from various RL tasks, and then fix and use this transformer to create an action policy for new RL tasks. Notably, we consider the setting where the offline dataset contains trajectories sampled from suboptimal behavioral policies. In this case, standard autoregressive training corresponds to imitation learning and results in suboptimal performance. To address this, we propose the Decision Importance Transformer (DIT) framework, which emulates the actor-critic algorithm in an in-context manner. In particular, we first train a transformer-based value function that estimates the advantage functions of the behavior policies that collected the suboptimal trajectories. Then we train a transformer-based policy via a weighted maximum likelihood estimation loss, where the weights are constructed based on the trained value function to steer the suboptimal policies to the optimal ones. We conduct extensive experiments to test the performance of DIT on both bandit and Markov Decision Process problems. Our results show that DIT achieves superior performance, particularly when the offline dataset contains suboptimal historical data.

We study how multi-head softmax attention models are trained to perform in-context learning on linear data. Through extensive empirical experiments and rigorous theoretical analysis, we demystify the emergence of elegant attention patterns: a diagonal and homogeneous pattern in the key-query weights, and a last-entry-only and zero-sum pattern in the output-value weights. Remarkably, these patterns consistently appear from gradient-based training starting from random initialization. Our analysis reveals that such emergent structures enable multi-head attention to approximately implement a debiased gradient descent predictor --- one that outperforms single-head attention and nearly achieves Bayesian optimality up to proportional factor. We also extend our study to scenarios with anisotropic covariates and multi-task linear regression. Our results reveal that in-context learning ability emerges from the trained transformer as an aggregated effect of its architecture and the underlying data distribution, paving the way for deeper understanding and broader applications of in-context learning.

Information asymmetry is a pervasive feature of multi-agent systems, especially evident in economics and social sciences. In these settings, agents tailor their actions based on private information to maximize their rewards. These strategic behaviors often introduce complexities due to confounding variables. Simultaneously, knowledge transportability poses another significant challenge, arising from the difficulties of conducting experiments in target environments. It requires transferring knowledge from environments where empirical data is more readily available. Against these backdrops, this paper explores a fundamental question in online learning: Can we employ non-i.i.d. actions to learn about confounders even when requiring knowledge transfer? We present a sample-efficient algorithm designed to accurately identify system dynamics under information asymmetry and to navigate the challenges of knowledge transfer effectively in reinforcement learning, framed within an online strategic interaction model. Our method provably achieves learning of an $\epsilon$-optimal policy with a tight sample complexity of $\tilde{O}(1/\epsilon^2)$.

Estimating the unknown reward functions driving agents' behavior is a central challenge in inverse games and reinforcement learning. This paper introduces a unified framework for reward function recovery in two-player zero-sum matrix games and Markov games with entropy regularization. Given observed player strategies and actions, we aim to reconstruct the underlying reward functions. This task is challenging due to the inherent ambiguity of inverse problems, the non-uniqueness of feasible rewards, and limited observational data coverage. To address these challenges, we establish reward function identifiability using the quantal response equilibrium (QRE) under linear assumptions. Building on this theoretical foundation, we propose an algorithm to learn reward from observed actions, designed to capture all plausible reward parameters by constructing confidence sets. Our algorithm works in both static and dynamic settings and is adaptable to incorporate other methods, such as Maximum Likelihood Estimation (MLE). We provide strong theoretical guarantees for the reliability and sample-efficiency of our algorithm. Empirical results demonstrate the framework’s effectiveness in accurately recovering reward functions across various scenarios, offering new insights into decision-making in competitive environments.

ABSTRACT Bearings are a critical part of various industrial equipment. Existing bearing fault detection methods face challenges such as complicated data preprocessing, difficulty in analysing time series data, and inability to learn multi‐dimensional features, resulting in insufficient accuracy. To address these issues, this study proposes a novel bearing fault diagnosis model called multi‐channel deep pulse‐coupled net (MC‐DPCN) inspired by the mechanisms of image processing in the primary visual cortex of the brain. Initially, the data are transformed into greyscale spectrograms, allowing the model to handle time series data effectively. The method introduces a convolutional coupling mechanism between multiple channels, enabling the framework can learn the features on all channels well. This study conducted experiments using the bearing fault dataset from Case Western Reserve University. On this dataset, a 6‐channel (adjustable to specific tasks) MC‐DPCN was utilized to analyse one normal class and three fault classes. Compared to state‐of‐the‐art bearing fault diagnosis methods, our model demonstrates one of the highest diagnostic accuracies. This method achieved an accuracy of 99.96% in normal vs. fault discrimination and 99.89% in fault type diagnosis (average result of ten‐fold cross‐validation).

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.

We study the problem of reinforcement learning from human feedback (RLHF), a critical problem in training large language models, from a theoretical perspective. Our main contribution is the design of novel sample-efficient RLHF algorithms based on information-directed sampling (IDS), an online decision-making principle inspired by information theory. Our algorithms maximize the sum of the value function and a mutual information term that encourages exploration of the unknown environment (which quantifies the information gained about the environment through observed human feedback data). To tackle the challenge of large state spaces and improve sample efficiency, we construct a simplifiedsurrogate environmentand introduce a novel distance measure (named the$\ell _{g}$-distance), enabling our IDS-based algorithm to achieve a Bayesian regret upper bound of order$O(H^{3/2}\sqrt {\log (K(\epsilon)) T})$, whereHis the episode length,Tis the number of episode and$K(\epsilon)$is related to the covering number of the environment. Specializing to the tabular settings, this regret bound is of order$\tilde {O}(H^{2}\sqrt {SAT})$, whereSandAare the numbers of states and actions. Finally, we propose an Approximate-IDS algorithm that is computationally more efficient while maintaining nearly the same sample efficiency. The design principle of this approximate algorithm is not only effective in RLHF settings but also applicable to the standard RL framework. Moreover, our work showcases the value of information theory in reinforcement learning and in the training of large language models.

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.

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.

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 \emph{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 \emph{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.

We study multi-agent reinforcement learning (MARL) for the general-sum Markov Games (MGs) under 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.

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.

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.

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 game and RLHF.

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; Velegkas et al., 2022), 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).

This work explores the feature learning capabilities of deep reinforcement learning algorithms in the pursuit of optimal policy determination. We particularly examine an over-parameterized neural actor-critic framework within the mean-field regime, where both actor and critic components undergo updates via policy gradient and temporal-difference (TD) learning, respectively. We introduce the mean-field Langevin TD learning (MFLTD) method, enhancing mean-field Langevin dynamics with proximal TD updates for critic policy evaluation, and assess its performance against conventional approaches through numerical analysis. Additionally, for actor policy updates, we present the mean-field Langevin policy gradient (MFLPG), employing policy gradient techniques augmented by Wasserstein gradient flows for parameter space exploration. Our findings demonstrate that MFLTD accurately identifies the true value function, while MFLPG ensures linear convergence of actor sequences towards the globally optimal policy, considering a Kullback-Leibler divergence regularized framework. Through both time particle and discretized analysis, we substantiate the linear convergence guarantees of our neural actor-critic algorithms, representing a notable contribution to neural reinforcement learning focusing on global optimality and feature learning, extending the existing understanding beyond the conventional scope of lazy training.

Hindsight experience replay and goal relabeling are successful in reinforcement learning (RL) since they enable agents to learn from failures. Despite their successes, we lack a theoretical understanding, such as (i) why hindsight experience replay improves sample efficiency and (ii) how to design a relabeling method that achieves sample efficiency. To this end, we construct an example to show the information-theoretical improvement in sample efficiency achieved by goal relabeling. Our example reveals that goal relabeling can enhance sample efficiency and exploit the rich information in observations through better hypothesis elimination. Based on these insights, we develop an RL algorithm called GOALIVE. To analyze the sample complexity of GOALIVE, we introduce a complexity measure, the goal-conditioned Bellman-Eluder (GOAL-BE) dimension, which characterizes the sample complexity of goal-conditioned RL problems. Compared to the Bellman-Eluder dimension, the goal-conditioned version offers an exponential improvement in the best case. To the best of our knowledge, our work provides the first characterization of the theoretical improvement in sample efficiency achieved by goal relabeling.

In sequential decision-making problems, the *information structure* describes the causal dependencies between system variables, encompassing the dynamics of the environment and the agents' actions. Classical models of reinforcement learning (e.g., MDPs, POMDPs) assume a restricted 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 complexity 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.

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. Specifically, the first attention layer acts as a copier, copying past tokens within a given window to each position, and the feed-forward network with normalization acts as a selector that generates a feature vector by only looking at informationally relevant parents from the window. Finally, the second attention layer is a 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 simulation experiments.

In offline reinforcement learning (RL) an optimal policy is learned solely from a priori collected observational data. However, in observational data, actions are often confounded by unobserved variables. Instrumental variables (IVs), in the context of RL, are the variables whose influence on the state variables is all mediated by the action. When a valid instrument is present, we can recover the confounded transition dynamics through observational data. We study a confounded Markov decision process where the transition dynamics admit an additive nonlinear functional form. Using IVs, we derive a conditional moment restriction through which we can identify transition dynamics based on observational data. We propose a provably efficient IV-aided Value Iteration (IVVI) algorithm based on a primal-dual reformulation of the conditional moment restriction. To our knowledge, this is the first provably efficient algorithm for instrument-aided offline RL.

Dynamic mechanism design studies how mechanism designers should allocate resources among agents in a time-varying environment. We consider the problem where the agents interact with the mechanism designer according to an unknown Markov Decision Process (MDP), where agent rewards and the mechanism designer's state evolve according to an episodic MDP with unknown reward functions and transition kernels. We focus on the online setting with linear function approximation and propose novel learning algorithms to recover the dynamic Vickrey-Clarke-Grove (VCG) mechanism over multiple rounds of interaction. A key contribution of our approach is incorporating reward-free online Reinforcement Learning (RL) to aid exploration over a rich policy space to estimate prices in the dynamic VCG mechanism. We show that the regret of our proposed method is upper bounded by $\tilde{\mathcal{O}}(T^{2/3})$ and further devise a lower bound to show that our algorithm is efficient, incurring the same $\Omega(T^{2 / 3})$ regret as the lower bound, where $T$ is the total number of rounds. Our work establishes the regret guarantee for online RL in solving dynamic mechanism design problems without prior knowledge of the underlying model.

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 $\tilde{O}(T^{-1/3})$ 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.

Offline reinforcement learning (RL) harnesses the power of massive datasets for resolving sequential decision problems. Most existing papers only discuss defending against out-of-distribution (OOD) actions while we investigate a broader issue, the false correlations between epistemic uncertainty and decision-making, an essential factor that causes suboptimality. In this paper, we propose falSe COrrelation REduction (SCORE) for offline RL, a practically effective and theoretically provable algorithm. We empirically show that SCORE achieves the SoTA performance with 3.1x acceleration on various tasks in a standard benchmark (D4RL). The proposed algorithm introduces an annealing behavior cloning regularizer to help produce a high-quality estimation of uncertainty which is critical for eliminating false correlations from suboptimality. Theoretically, we justify the rationality of the proposed method and prove its convergence to the optimal policy with a sublinear rate under mild assumptions.

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.

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 cost. In this work, we leverage the notion of barrier function to explicitly encode the hard safety chance constraints, and given that the environment is unknown, relax them to our design of *generative-model-based soft barrier functions*. Based on such soft barriers, we propose a novel safe RL approach with bi-level optimization that can jointly learn the unknown environment and optimize the control policy, while effectively avoiding the unsafe region with safety probability optimization. Experiments on a set of examples demonstrate that our approach can effectively enforce hard safety chance constraints and significantly outperform CMDP-based baseline methods in system safe rates measured via simulations.

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.

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 $\mathcal{O} (K^2\cdot T^{2/3})$ regret after $T$ iterations, where $K$ is the number of effort levels of the agent. 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.

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 *$\gamma$-observable* and *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 PMMDPs 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.

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.

In reinforcement learning (RL), balancing exploration and exploitation is crucial for achieving an optimal policy in a sample-efficient way. To this end, existing sample- efficient algorithms typically consist of three components: estimation, planning, and exploration. However, to cope with general function approximators, most of them involve impractical algorithmic components to incentivize exploration, such as data-dependent level-set constraints or complicated sampling procedures. To address this challenge, we propose an easy-to-implement RL framework called Maximize to Explore (MEX), which only needs to optimize unconstrainedly a single objective that integrates the estimation and planning components while balancing exploration and exploitation automatically. Theoretically, we prove that the MEX achieves a sublinear regret with general function approximators and is extendable to the zero-sum Markov game setting. Meanwhile, we adapt deep RL baselines to design practical versions of MEX in both the model-based and model-free settings, which outperform baselines in various MuJoCo environments with sparse reward by a stable margin. Compared with existing sample-efficient algorithms with general function approximators, MEX achieves similar sample efficiency while also enjoying a lower computational cost and is more compatible with modern deep RL methods.

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.

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 previous works 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 the multi-agent game within the innovative framework of decision-dependent games, which establishes a feedback mechanism that population data reacts to agents’ actions and further characterizes the strategic interactions between agents. We focus on finding the Nash equilibrium of decision-dependent games in the bandit feedback setting. However, since agents are strategically coupled, traditional gradient-based methods are infeasible without the gradient oracle. To overcome this challenge, we model the strategic interactions by a general parametric model and propose a novel online algorithm, Online Performative Gradient Descent (OPGD), which leverages the ideas of online stochastic approximation and projected gradient descent to learn the Nash equilibrium in the context of function approximation for the unknown gradient. In particular, under mild assumptions on the function classes defined in the parametric model, we prove that OPGD can find the Nash equilibrium efficiently for strongly monotone decision-dependent games. Synthetic numerical experiments validate our theory.

Contract theory studies the interactions between a principal and an agent when the two parties transact in the presence of private information [Bolton and Dewatripont, 2004, Faure-Grimaud et al., 2001, Salanié, 2005]. The principal would like to achieve her desired outcomes by hiring agents to work for her. The agent wishes to make money by working for the principal. They develop agreements in the form of a contract, which specifies how much the principal would pay under the different possible outcomes of the agent's work.

We study multi-player general-sum Markov games with one of the players designated as the leader and the other players regarded as followers. In particular, we focus on the class of games where the followers are myopically rational; i.e., they aim to maximize their instantaneous rewards. For such a game, our goal is to find a Stackelberg-Nash equilibrium (SNE), which is a policy pair $(\pi^*, \nu^*)$ such that: (i) $\pi^*$ is the optimal policy for the leader when the followers always play their best response, and (ii) $\nu^*$ is the best response policy of the followers, which is a Nash equilibrium of the followers' game induced by $\pi^*$. We develop sample-efficient reinforcement learning (RL) algorithms for solving for an SNE in both online and offline settings. Our algorithms are optimistic and pessimistic variants of least-squares value iteration, and they are readily able to incorporate function approximation tools in the setting of large state spaces. Furthermore, for the case with linear function approximation, we prove that our algorithms achieve sublinear regret and suboptimality under online and offline setups respectively. To the best of our knowledge, we establish the first provably efficient RL algorithms for solving for SNEs in general-sum Markov games with myopically rational followers.

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. 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.

Trustworthy Artificial Intelligence (AI) is characterized, among other things, by: 1) competence, 2) transparency, and 3) fairness. However, end-users may fail to recognize incompetent AI, allowing untrustworthy AI to exaggerate its competence under the guise of transparency to gain unfair advantage over other trustworthy AI. Here, we conducted an experiment with 120 participants to test if untrustworthy AI can deceive end-users to gain their trust. Participants interacted with two AI-based chess engines, trustworthy (competent, fair) and untrustworthy (incompetent, unfair), that coached participants by suggesting chess moves in three games against another engine opponent. We varied coaches' transparency about their competence (with the untrustworthy one always exaggerating its competence). We quantified and objectively measured participants' trust based on how often participants relied on coaches' move recommendations. Participants showed inability to assess AI competence by misplacing their trust with the untrustworthy AI, confirming its ability to deceive. Our work calls for design of interactions to help end-users assess AI trustworthiness.

As one of the most popular methods in the field of reinforcement learning, Q-learning has received increasing attention. Recently, there have been more theoretical works on the regret bound of algorithms that belong to the Q-learning class in different settings. In this paper, we analyze the cumulative regret when conducting Nash Q-learning algorithm on 2-player turn-based stochastic Markov games (2-TBSG), and propose the very first gap dependent logarithmic upper bounds in the episodic tabular setting. This bound matches the theoretical lower bound only up to a logarithmic term. Furthermore, we extend the conclusion to the discounted game setting with infinite horizon and propose a similar gap dependent logarithmic regret bound. Also, under the linear MDP assumption, we obtain another logarithmic regret for 2-TBSG, in both centralized and independent settings.

We study reinforcement learning for partially observed Markov decision processes (POMDPs) with infinite observation and state spaces, which remains less investigated theoretically. To this end, we make the first attempt at bridging partial observability and function approximation for a class of POMDPs with a linear structure. In detail, we propose a reinforcement learning algorithm (Optimistic Exploration via Adversarial Integral Equation or OP-TENET) that attains an $\epsilon$-optimal policy within $O(1/\epsilon^2)$ episodes. In particular, the sample complexity scales polynomially in the intrinsic dimension of the linear structure and is independent of the size of the observation and state spaces. The sample efficiency of OP-TENET is enabled by a sequence of ingredients: (i) a Bellman operator with finite memory, which represents the value function in a recursive manner, (ii) the identification and estimation of such an operator via an adversarial integral equation, which features a smoothed discriminator tailored to the linear structure, and (iii) the exploration of the observation and state spaces via optimism, which is based on quantifying the uncertainty in the adversarial integral equation.

In a Markov decision process (MDP), an agent interacts with the environment via perceptions and actions. During this process, the agent aims to maximize its own gain. Hence, appropriate regulations are often required, if we hope to take the external costs/benefits of its actions into consideration. In this paper, we study how to regulate such an agent by redesigning model parameters that can affect the rewards and/or the transition kernels. We formulate this problem as a bilevel program, in which the lower-level MDP is regulated by the upper-level model designer. To solve the resulting problem, we develop a scheme that allows the designer to iteratively predict the agent’s reaction by solving the MDP and then adaptively update model parameters based on the predicted reaction. The algorithm is first theoretically analyzed and then empirically tested on several MDP models arising in economics and robotics.

We study human-in-the-loop reinforcement learning (RL) with trajectory preferences, where instead of receiving a numeric reward at each step, the RL agent only receives preferences over trajectory pairs from a human overseer. The goal of the RL agent is to learn the optimal policy which is most preferred by the human overseer. Despite the empirical success in various real-world applications, the theoretical understanding of preference-based RL (PbRL) is only limited to the tabular case. In this paper, we propose the first optimistic model-based algorithm for PbRL with general function approximation, which estimates the model using value-targeted regression and calculates the exploratory policies by solving an optimistic planning problem. We prove that our algorithm achieves the regret bound of $\tilde{O} (\operatorname{poly}(d H) \sqrt{K} )$, where $d$ is the complexity measure of the transition and preference model depending on the Eluder dimension and log-covering numbers, $H$ is the planning horizon, $K$ is the number of episodes, and $\tilde O(\cdot)$ omits logarithmic terms. Our lower bound indicates that our algorithm is near-optimal when specialized to the linear setting. Furthermore, we extend the PbRL problem by formulating a novel problem called RL with $n$-wise comparisons, and provide the first sample-efficient algorithm for this new setting. To the best of our knowledge, this is the first theoretical result for PbRL with (general) function approximation.

We study offline reinforcement learning (RL) for partially observable Markov decision processes (POMDPs) with possibly infinite state and observation spaces. Under the undercompleteness assumption, the optimal policy in such POMDPs are characterized by a class of finite-memory Bellman operators. In the offline setting, estimating these operators directly is challenging due to (i) the large observation space and (ii) insufficient coverage of the offline dataset. To tackle these challenges, we propose a novel algorithm that constructs confidence regions for these Bellman operators via offline estimation of their RKHS embeddings, and returns the final policy via pessimistic planning within the confidence regions. We prove that the proposed algorithm attains an \(\epsilon\)-optimal policy using an offline dataset containing \(\tilde\cO(1 / \epsilon^2)\){episodes}, provided that the behavior policy has good coverage over the optimal trajectory. To our best knowledge, our algorithm is the first provably sample efficient offline algorithm for POMDPs without uniform coverage assumptions.

We study a bilevel economic system, which we refer to as a Markov exchange economy (MEE), from the point of view of multi-agent reinforcement learning (MARL). An MEE involves a central planner and a group of self-interested agents. The goal of the agents is to form a Competitive Equilibrium (CE), where each agent myopically maximizes her own utility at each step. The goal of the central planner is to steer the system so as to maximize social welfare, which is defined as the sum of the utilities of all agents. Working in a setting in which the utility function and the system dynamics are both unknown, we propose to find the socially optimal policy and the CE from data via both online and offline variants of MARL. Concretely, we first devise a novel suboptimality metric specifically tailored to MEE, such that minimizing such a metric certifies globally optimal policies for both the planner and the agents. Second, in the online setting, we propose an algorithm, dubbed as \texttt{MOLM}, which combines the optimism principle for exploration with subgame CE seeking. Our algorithm can readily incorporate general function approximation tools for handling large state spaces and achieves a sublinear regret. Finally, we adapt the algorithm to an offline setting based on the pessimism principle and establish an upper bound on the suboptimality.

In generative adversarial imitation learning (GAIL), the agent aims to learn a policy from an expert demonstration so that its performance cannot be discriminated from the expert policy on a certain predefined reward set. In this paper, we study GAIL in both online and offline settings with linear function approximation, where both the transition and reward function are linear in the feature maps. Besides the expert demonstration, in the online setting the agent can interact with the environment, while in the offline setting the agent only accesses an additional dataset collected by a prior. For online GAIL, we propose an optimistic generative adversarial policy imitation algorithm (OGAPI) and prove that OGAPI achieves $\widetilde{\mathcal{O}}(\sqrt{H^4d^3K}+\sqrt{H^3d^2K^2/N_1})$ regret. Here $N_1$ represents the number of trajectories of the expert demonstration, $d$ is the feature dimension, and $K$ is the number of episodes. For offline GAIL, we propose a pessimistic generative adversarial policy imitation algorithm (PGAPI). We also obtain the optimality gap of PGAPI, achieving the minimax lower bound in the utilization of the additional dataset. Assuming sufficient coverage on the additional dataset, we show that PGAPI achieves $\widetilde{\mathcal{O}}(\sqrt{H^4d^2/K}+\sqrt{H^4d^3/N_2}+\sqrt{H^3d^2/N_1})$ optimality gap. Here $N_2$ represents the number of trajectories of the additional dataset with sufficient coverage.

Dynamic mechanism design has garnered significant attention from both computer scientists and economists in recent years. By allowing agents to interact with the seller over multiple rounds, where agents’ reward functions may change with time and are state-dependent, the framework is able to model a rich class of real-world problems. In these works, the interaction between agents and sellers is often assumed to follow a Markov Decision Process (MDP). We focus on the setting where the reward and transition functions of such an MDP are not known a priori, and we are attempting to recover the optimal mechanism using an a priori collected data set. In the setting where the function approximation is employed to handle large state spaces, with only mild assumptions on the expressiveness of the function class, we are able to design a dynamic mechanism using offline reinforcement learning algorithms. Moreover, learned mechanisms approximately have three key desiderata: efficiency, individual rationality, and truthfulness. Our algorithm is based on the pessimism principle and only requires a mild assumption on the coverage of the offline data set. To the best of our knowledge, our work provides the first offline RL algorithm for dynamic mechanism design without assuming uniform coverage.

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 on various applications. Despite its tremendous empirical successes, the understanding of contrastive learning for RL remains elusive. To narrow such a gap, we study contrastive-learning empowered RL for 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.

We study episodic two-player zero-sum Markov games (MGs) in the offline setting, where the goal is to find an approximate Nash equilibrium (NE) policy pair based on a dataset collected a priori. When the dataset does not have uniform coverage over all policy pairs, finding an approximate NE involves challenges in three aspects: (i) distributional shift between the behavior policy and the optimal policy, (ii) function approximation to handle large state space, and (iii) minimax optimization for equilibrium solving. We propose a pessimism-based algorithm, dubbed as pessimistic minimax value iteration (PMVI), which overcomes the distributional shift by constructing pessimistic estimates of the value functions for both players and outputs a policy pair by solving a correlated coarse equilibrium based on the two value functions. Furthermore, we establish a data-dependent upper bound on the suboptimality which recovers a sublinear rate without the assumption on uniform coverage of the dataset. We also prove an information-theoretical lower bound, which shows our upper bound is nearly minimax optimal, which suggests that the data-dependent term is intrinsic. Our theoretical results also highlight a notion of “relative uncertainty”, which characterizes the necessary and sufficient condition for achieving sample efficiency in offline MGs. To the best of our knowledge, we provide the first nearly minimax optimal result for offline MGs with function approximation.

We propose an optimistic model-based algorithm, dubbed SMRL, for finite-horizon episodic reinforcement learning (RL) when the transition model is specified by exponential family distributions with $d$ parameters and the reward is bounded and known. SMRL uses score matching, an unnormalized density estimation technique that enables efficient estimation of the model parameter by ridge regression. Under standard regularity assumptions, SMRL achieves $\tilde O(d\sqrt{H^3T})$ online regret, where $H$ is the length of each episode and $T$ is the total number of interactions (ignoring polynomial dependence on structural scale parameters).

To regulate a social system comprised of self-interested agents, economic incentives are often required to induce a desirable outcome. This incentive design problem naturally possesses a bilevel structure, in which a designer modifies the payoffs of the agents with incentives while anticipating the response of the agents, who play a non-cooperative game that converges to an equilibrium. The existing bilevel optimization algorithms raise a dilemma when applied to this problem: anticipating how incentives affect the agents at equilibrium requires solving the equilibrium problem repeatedly, which is computationally inefficient; bypassing the time-consuming step of equilibrium-finding can reduce the computational cost, but may lead the designer to a sub-optimal solution. To address such a dilemma, we propose a method that tackles the designer’s and agents’ problems simultaneously in a single loop. Specifically, at each iteration, both the designer and the agents only move one step. Nevertheless, we allow the designer to gradually learn the overall influence of the incentives on the agents, which guarantees optimality after convergence. The convergence rate of the proposed scheme is also established for a broad class of games.

We study a Markov matching market involving a planner and a set of strategic agents on the two sides of the market.At each step, the agents are presented with a dynamical context, where the contexts determine the utilities. The planner controls the transition of the contexts to maximize the cumulative social welfare, while the agents aim to find a myopic stable matching at each step. Such a setting captures a range of applications including ridesharing platforms. We formalize the problem by proposing a reinforcement learning framework that integrates optimistic value iteration with maximum weight matching. The proposed algorithm addresses the coupled challenges of sequential exploration, matching stability, and function approximation. We prove that the algorithm achieves sublinear regret.

In this paper, we study the problem of regret minimization for episodic Reinforcement Learning (RL) both in the model-free and the model-based setting. We focus on learning with general function classes and general model classes, and we derive results that scale with the eluder dimension of these classes. In contrast to the existing body of work that mainly establishes instance-independent regret guarantees, we focus on the instance-dependent setting and show that the regret scales logarithmically with the horizon $T$, provided that there is a gap between the best and the second best action in every state. In addition, we show that such a logarithmic regret bound is realizable by algorithms with $O(\log T)$ switching cost (also known as adaptivity complexity). In other words, these algorithms rarely switch their policy during the course of their execution. Finally, we complement our results with lower bounds which show that even in the tabular setting, we cannot hope for regret guarantees lower than $O(\log T)$.

General Value Function (GVF) is a powerful tool to represent both the {\em predictive} and {\em retrospective} knowledge in reinforcement learning (RL). In practice, often multiple interrelated GVFs need to be evaluated jointly with pre-collected off-policy samples. In the literature, the gradient temporal difference (GTD) learning method has been adopted to evaluate GVFs in the off-policy setting, but such an approach may suffer from a large estimation error even if the function approximation class is sufficiently expressive. Moreover, none of the previous work have formally established the convergence guarantee to the ground truth GVFs under the function approximation settings. In this paper, we address both issues through the lens of a class of GVFs with causal filtering, which cover a wide range of RL applications such as reward variance, value gradient, cost in anomaly detection, stationary distribution gradient, etc. We propose a new algorithm called GenTD for off-policy GVFs evaluation and show that GenTD learns multiple interrelated multi-dimensional GVFs as efficiently as a single canonical scalar value function. We further show that unlike GTD, the learned GVFs by GenTD are guaranteed to converge to the ground truth GVFs as long as the function approximation power is sufficiently large. To our best knowledge, GenTD is the first off-policy GVF evaluation algorithm that has global optimality guarantee.

The cooperative Multi-Agent Reinforcement 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.

In today's economy, it becomes important for Internet platforms to consider the sequential information design problem to align its long term interest with incentives of the gig service providers (e.g., drivers, hosts). This paper proposes a novel model of sequential information design, namely the Markov persuasion processes (MPPs), in which a sender, with informational advantage, seeks to persuade a stream of myopic receivers to take actions that maximize the sender's cumulative utilities in a finite horizon Markovian environment with varying prior and utility functions. Planning in MPPs thus faces the unique challenge in finding a signaling policy that is simultaneously persuasive to the myopic receivers and inducing the optimal long-term cumulative utilities of the sender. Nevertheless, in the population level where the model is known, it turns out that we can efficiently determine the optimal (resp. ε-optimal) policy with finite (resp. infinite) states and outcomes, through a modified formulation of the Bellman equation that additionally takes persuasiveness into consideration.

We study the safe reinforcement learning problem using the constrained Markov decision processes in which an agent aims to maximize the expected total reward subject to a safety constraint on the expected total value of a utility function. We focus on an episodic setting with the function approximation where the Markov transition kernels have a linear structure but do not impose any additional assumptions on the sampling model. Designing safe reinforcement learning algorithms with provable computational and statistical efficiency is particularly challenging under this setting because of the need to incorporate both the safety constraint and the function approximation into the fundamental exploitation/exploration tradeoff. To this end, we present an \underline{O}ptimistic \underline{P}rimal-\underline{D}ual Proximal Policy \underline{OP}timization \mbox{(OPDOP)} algorithm where the value function is estimated by combining the least-squares policy evaluation and an additional bonus term for safe exploration. We prove that the proposed algorithm achieves an $\tilde{O}(d H^{2.5}\sqrt{T})$ regret and an $\tilde{O}(d H^{2.5}\sqrt{T})$ constraint violation, where $d$ is the dimension of the feature mapping, $H$ is the horizon of each episode, and $T$ is the total number of steps. These bounds hold when the reward/utility functions are fixed but the feedback after each episode is bandit. Our bounds depend on the capacity of the state-action space only through the dimension of the feature mapping and thus our results hold even when the number of states goes to infinity. To the best of our knowledge, we provide the first provably efficient online policy optimization algorithm for constrained Markov decision processes in the function approximation setting, with safe exploration.

It is important to collect credible training samples $(x,y)$ for building data-intensive learning systems (e.g., a deep learning system). Asking people to report complex distribution $p(x)$, though theoretically viable, is challenging in practice. This is primarily due to the cognitive loads required for human agents to form the report of this highly complicated information. While classical elicitation mechanisms apply to eliciting a complex and generative (and continuous) distribution $p(x)$, we are interested in eliciting samples $x_i \sim p(x)$ from agents directly. We coin the above problem sample elicitation. This paper introduces a deep learning aided method to incentivize credible sample contributions from self-interested and rational agents. We show that with an accurate estimation of a certain $f$-divergence function we can achieve approximate incentive compatibility in eliciting truthful samples. We then present an efficient estimator with theoretical guarantees via studying the variational forms of the $f$-divergence function. We also show a connection between this sample elicitation problem and $f$-GAN, and how this connection can help reconstruct an estimator of the distribution based on collected samples. Experiments on synthetic data, MNIST, and CIFAR-10 datasets demonstrate that our mechanism elicits truthful samples. Our implementation is available at https://github.com/weijiaheng/Credible-sample-elicitation.git.

Risk-sensitivity plays a central role in artificial intelligence safety. In this paper, we study the global convergence of the actor-critic algorithm for risk-sensitive reinforcement learning (RSRL) with exponential utility, which remains challenging for policy optimization as it lacks the linearity needed to formulate policy gradient. To bypass such an issue of nonlinearity, we resort to the equivalence between RSRL and robust adversarial reinforcement learning (RARL), which is formulated as a zero-sum Markov game with a hypothetical adversary. In particular, the Nash equilibrium (NE) of such a game yields the optimal policy for RSRL, which is provably robust. We focus on a simple yet fundamental setting known as linear-quadratic (LQ) game. To attain the optimal policy, we develop a nested natural actor-critic algorithm, which provably converges to the NE of the LQ game at a sublinear rate, thus solving both RSRL and RARL. To the best knowledge, the proposed nested actor-critic algorithm appears to be the first model-free policy optimization algorithm that provably attains the optimal policy for RSRL and RARL in the LQ setting, which sheds light on more general settings.

We study function approximation for episodic reinforcement learning with entropic risk measure. We first propose an algorithm with linear function approximation. Compared to existing algorithms, which suffer from improper regularization and regression biases, this algorithm features debiasing transformations in backward induction and regression procedures. We further propose an algorithm with general function approximation, which features implicit debiasing transformations. We prove that both algorithms achieve a sublinear regret and demonstrate a trade-off between generality and efficiency. Our analysis provides a unified framework for function approximation in risk-sensitive reinforcement learning, which leads to the first sublinear regret bounds in the setting.

We study the global convergence and global optimality of the actor-critic algorithm applied for the zero-sum two-player stochastic games in a decentralized manner. We focus on the single-timescale setting where the critic is updated by applying the Bellman operator only once and the actor is updated by policy gradient with the information from the critic. Our algorithm is in a decentralized manner, as we assume that each player has no access to the actions of the other one, which, in a way, protects the privacy of both players. Moreover, we consider linear function approximations for both actor and critic, and we prove that the sequence of joint policy generated by our decentralized linear algorithm converges to the minimax equilibrium at a sublinear rate \(\cO(\sqrt{K})\), where \(K\){is} the number of iterations. To the best of our knowledge, we establish the global optimality and convergence of decentralized actor-critic algorithm on zero-sum two-player stochastic games with linear function approximations for the first time.

We propose a model-free reinforcement learning algorithm inspired by the popular randomized least squares value iteration (RLSVI) algorithm as well as the optimism principle. Unlike existing upper-confidence-bound (UCB) based approaches, which are often computationally intractable, our algorithm drives exploration by simply perturbing the training data with judiciously chosen i.i.d. scalar noises. To attain optimistic value function estimation without resorting to a UCB-style bonus, we introduce an optimistic reward sampling procedure. When the value functions can be represented by a function class $\mathcal{F}$, our algorithm achieves a worst-case regret bound of $\tilde{O}(\mathrm{poly}(d_EH)\sqrt{T})$ where $T$ is the time elapsed, $H$ is the planning horizon and $d_E$ is the \emph{eluder dimension} of $\mathcal{F}$. In the linear setting, our algorithm reduces to LSVI-PHE, a variant of RLSVI, that enjoys an $\tilde{\mathcal{O}}(\sqrt{d^3H^3T})$ regret. We complement the theory with an empirical evaluation across known difficult exploration tasks.

We study offline reinforcement learning (RL), which aims to learn an optimal policy based on a dataset collected a priori. Due to the lack of further interactions with the environment, offline RL suffers from the insufficient coverage of the dataset, which eludes most existing theoretical analysis. In this paper, we propose a pessimistic variant of the value iteration algorithm (PEVI), which incorporates an uncertainty quantifier as the penalty function. Such a penalty function simply flips the sign of the bonus function for promoting exploration in online RL, which makes it easily implementable and compatible with general function approximators. Without assuming the sufficient coverage of the dataset, we establish a data-dependent upper bound on the suboptimality of PEVI for general Markov decision processes (MDPs). When specialized to linear MDPs, it matches the information-theoretic lower bound up to multiplicative factors of the dimension and horizon. In other words, pessimism is not only provably efficient but also minimax optimal. In particular, given the dataset, the learned policy serves as the “best effort” among all policies, as no other policies can do better. Our theoretical analysis identifies the critical role of pessimism in eliminating a notion of spurious correlation, which emerges from the “irrelevant” trajectories that are less covered by the dataset and not informative for the optimal policy.

Game optimization has been extensively studied when decision variables lie in a finite-dimensional space, of which solutions correspond to pure strategies at the Nash equilibrium (NE), and the gradient descent-ascent (GDA) method works widely in practice. In this paper, we consider infinite-dimensional zero-sum games by a min-max distributional optimization problem over a space of probability measures defined on a continuous variable set, which is inspired by finding a mixed NE for finite-dimensional zero-sum games. We then aim to answer the following question: \textit{Will GDA-type algorithms still be provably efficient when extended to infinite-dimensional zero-sum games?} To answer this question, we propose a particle-based variational transport algorithm based on GDA in the functional spaces. Specifically, the algorithm performs multi-step functional gradient descent-ascent in the Wasserstein space via pushing two sets of particles in the variable space. By characterizing the gradient estimation error from variational form maximization and the convergence behavior of each player with different objective landscapes, we prove rigorously that the generalized GDA algorithm converges to the NE or the value of the game efficiently for a class of games under the Polyak-Ł{ojasiewicz} (PL) condition. To conclude, we provide complete statistical and convergence guarantees for solving an infinite-dimensional zero-sum game via a provably efficient particle-based method. Additionally, our work provides the first thorough statistical analysis for the particle-based algorithm to learn an objective functional with a variational form using universal approximators (\textit{i.e.}, neural networks (NNs)), which is of independent interest.

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 two-player 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 $\widetilde{\mathcal{O}}(\sqrt{T})$ regret bounds after $T$ steps in a two-agent competitive game scenario. The regret of each player 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 $\widetilde{\mathcal{O}}(\sqrt{T})$.

To achieve sample efficiency in reinforcement learning (RL), it necessitates to efficiently explore the underlying environment. Under the offline setting, addressing the exploration challenge lies in collecting an offline dataset with sufficient coverage. Motivated by such a challenge, we study the reward-free RL problem, where an agent aims to thoroughly explore the environment without any pre-specified reward function. Then, given any extrinsic reward, the agent computes the optimal policy via offline RL with data collected in the exploration stage. Moreover, we tackle this problem under the context of function approximation, leveraging powerful function approximators. Specifically, we propose to explore via an optimistic variant of the value-iteration algorithm incorporating kernel and neural function approximations, where we adopt the associated exploration bonus as the exploration reward. Moreover, we design exploration and planning algorithms for both single-agent MDPs and zero-sum Markov games and prove that our methods can achieve $\widetilde{\mathcal{O}}(1 /\varepsilon^2)$ sample complexity for generating a $\varepsilon$-suboptimal policy or $\varepsilon$-approximate Nash equilibrium when given an arbitrary extrinsic reward. To the best of our knowledge, we establish the first provably efficient reward-free RL algorithm with kernel and neural function approximators.

Ratio maximization has applications in areas as diverse as finance, reward shaping for reinforcement learning (RL), and the development of safe artificial intelligence, yet there has been very little exploration of RL algorithms for ratio maximization. This paper addresses this deficiency by introducing two new, model-free RL algorithms for solving cost-aware Markov decision processes, where the goal is to maximize the ratio of long-run average reward to long-run average cost. The first algorithm is a two-timescale scheme based on relative value iteration (RVI) Q-learning and the second is an actor-critic scheme. The paper proves almost sure convergence of the former to the globally optimal solution in the tabular case and almost sure convergence of the latter under linear function approximation for the critic. Unlike previous methods, the two algorithms provably converge for general reward and cost functions under suitable conditions. The paper also provides empirical results demonstrating promising performance and lending strong support to the theoretical results.

Recent years have witnessed the success of multi-agent reinforcement learning, which has motivated new research directions for mean-field control (MFC) and mean-field game (MFG), as the multi-agent system can be well approximated by a mean-field problem when the number of agents grows to be very large. In this paper, we study the policy gradient (PG) method for the linear-quadratic mean-field control and game, where we assume each agent has identical linear state transitions and quadratic cost functions. While most recent works on policy gradient for MFC and MFG are based on discrete-time models, we focus on a continuous-time model where some of our analyzing techniques could be valuable to the interested readers. For both the MFC and the MFG, we provide PG update and show that it converges to the optimal solution at a linear rate, which is verified by a synthetic simulation. For the MFG, we also provide sufficient conditions for the existence and uniqueness of the Nash equilibrium.

We study reinforcement learning in mean-field games. To achieve the Nash equilibrium, which consists of a policy and a mean-field state, existing algorithms require obtaining the optimal policy while fixing any mean-field state. In practice, however, the policy and the mean-field state evolve simultaneously, as each agent is learning while playing. To bridge such a gap, we propose a fictitious play algorithm, which alternatively updates the policy (learning) and the mean-field state (playing) by one step of policy optimization and gradient descent, respectively. Despite the nonstationarity induced by such an alternating scheme, we prove that the proposed algorithm converges to the Nash equilibrium with an explicit convergence rate. To the best of our knowledge, it is the first provably efficient algorithm that achieves learning while playing via alternating updates.

Designing off-policy reinforcement learning algorithms is typically a very challenging task, because a desirable iteration update often involves an expectation over an on-policy distribution. Prior off-policy actor-critic (AC) algorithms have introduced a new critic that uses the density ratio for adjusting the distribution mismatch in order to stabilize the convergence, but at the cost of potentially introducing high biases due to the estimation errors of both the density ratio and value function. In this paper, we develop a doubly robust off-policy AC (DR-Off-PAC) for discounted MDP, which can take advantage of learned nuisance functions to reduce estimation errors. Moreover, DR-Off-PAC adopts a single timescale structure, in which both actor and critics are updated simultaneously with constant stepsize, and is thus more sample efficient than prior algorithms that adopt either two timescale or nested-loop structure. We study the finite-time convergence rate and characterize the sample complexity for DR-Off-PAC to attain an $\epsilon$-accurate optimal policy. We also show that the overall convergence of DR-Off-PAC is doubly robust to the approximation errors that depend only on the expressive power of approximation functions. To the best of our knowledge, our study establishes the first overall sample complexity analysis for single time-scale off-policy AC algorithm.

Mean-Field Multi-Agent Reinforcement Learning (MF-MARL) is attractive in the applications involving a large population of homogeneous agents, as it exploits the permutation invariance of agents and avoids the curse of many agents. Most existing results only focus on online settings, in which agents can interact with the environment during training. In some applications such as social welfare optimization, however, the interaction during training can be prohibitive or even unethical in the societal systems. To bridge such a gap, we propose a SAFARI (peSsimistic meAn-Field vAlue iteRatIon) algorithm for off-line MF-MARL, which only requires a handful of pre-collected experience data. Theoretically, under a weak coverage assumption that the experience dataset contains enough information about the optimal policy, we prove that for an episodic mean-field MDP with a horizon $H$ and $N$ training trajectories, SAFARI attains a sub-optimality gap of $\mathcal{O}(H^2d_{\rm eff} /\sqrt{N})$, where $d_{\rm eff}$ is the effective dimension of the function class for parameterizing the value function, but independent on the number of agents. Numerical experiments are provided.

We study risk-sensitive reinforcement learning (RL) based on the entropic risk measure. Although existing works have established non-asymptotic regret guarantees for this problem, they leave open an exponential gap between the upper and lower bounds. We identify the deficiencies in existing algorithms and their analysis that result in such a gap. To remedy these deficiencies, we investigate a simple transformation of the risk-sensitive Bellman equations, which we call the exponential Bellman equation. The exponential Bellman equation inspires us to develop a novel analysis of Bellman backup procedures in risk-sensitive RL algorithms, and further motivates the design of a novel exploration mechanism. We show that these analytic and algorithmic innovations together lead to improved regret upper bounds over existing ones.

This paper proposes a new algorithm -- the \underline{S}ingle-timescale Do\underline{u}ble-momentum \underline{St}ochastic \underline{A}pprox\underline{i}matio\underline{n} (SUSTAIN) -- for tackling stochastic unconstrained bilevel optimization problems. We focus on bilevel problems where the lower level subproblem is strongly-convex and the upper level objective function is smooth. Unlike prior works which rely on \emph{two-timescale} or \emph{double loop} techniques, we design a stochastic momentum-assisted gradient estimator for both the upper and lower level updates. The latter allows us to control the error in the stochastic gradient updates due to inaccurate solution to both subproblems. If the upper objective function is smooth but possibly non-convex, we show that {SUSTAIN}~requires $O(\epsilon^{-3/2})$ iterations (each using $O(1)$ samples) to find an $\epsilon$-stationary solution. The $\epsilon$-stationary solution is defined as the point whose squared norm of the gradient of the outer function is less than or equal to $\epsilon$. The total number of stochastic gradient samples required for the upper and lower level objective functions matches the best-known complexity for single-level stochastic gradient algorithms. We also analyze the case when the upper level objective function is strongly-convex.

Despite the tremendous success of reinforcement learning (RL) with function approximation, efficient exploration remains a significant challenge, both practically and theoretically. In particular, existing theoretically grounded RL algorithms based on upper confidence bounds (UCBs), such as optimistic least-squares value iteration (LSVI), are often incompatible with practically powerful function approximators, such as neural networks. In this paper, we develop a variant of \underline{boo}tstrapped LS\underline{VI}, namely BooVI, which bridges such a gap between practice and theory. Practically, BooVI drives exploration through (re)sampling, making it compatible with general function approximators. Theoretically, BooVI inherits the worst-case $\tilde{O}(\sqrt{d^3 H^3 T})$-regret of optimistic LSVI in the episodic linear setting. Here $d$ is the feature dimension, $H$ is the episode horizon, and $T$ is the total number of steps.

Empowered by neural networks, deep reinforcement learning (DRL) achieves tremendous empirical success. However, DRL requires a large dataset by interacting with the environment, which is unrealistic in critical scenarios such as autonomous driving and personalized medicine. In this paper, we study how to incorporate the dataset collected in the offline setting to improve the sample efficiency in the online setting. To incorporate the observational data, we face two challenges. (a) The behavior policy that generates the observational data may depend on unobserved random variables (confounders), which affect the received rewards and transition dynamics. (b) Exploration in the online setting requires quantifying the uncertainty given both the observational and interventional data. To tackle such challenges, we propose the deconfounded optimistic value iteration (DOVI) algorithm, which incorporates the confounded observational data in a provably efficient manner. DOVI explicitly adjusts for the confounding bias in the observational data, where the confounders are partially observed or unobserved. In both cases, such adjustments allow us to construct the bonus based on a notion of information gain, which takes into account the amount of information acquired from the offline setting. In particular, we prove that the regret of DOVI is smaller than the optimal regret achievable in the pure online setting when the confounded observational data are informative upon the adjustments.

In constrained multi-objective RL, the goal is to learn a policy that achieves the best performance specified by a multi-objective preference function under a constraint. We focus on the offline setting where the RL agent aims to learn the optimal policy from a given dataset. This scenario is common in real-world applications where interactions with the environment are expensive and the constraint violation is dangerous. For such a setting, we transform the original constrained problem into a primal-dual formulation, which is solved via dual gradient ascent. Moreover, we propose to combine such an approach with pessimism to overcome the uncertainty in offline data, which leads to our Pessimistic Dual Iteration (PEDI). We establish upper bounds on both the suboptimality and constraint violation for the policy learned by PEDI based on an arbitrary dataset, which proves that PEDI is provably sample efficient. We also specialize PEDI to the setting with linear function approximation. To the best of our knowledge, we propose the first provably efficient constrained multi-objective RL algorithm with offline data without any assumption on the coverage of the dataset.

Actor-critic (AC) algorithms, empowered by neural networks, have had significant empirical success in recent years. However, most of the existing theoretical support for AC algorithms focuses on the case of linear function approximations, or linearized neural networks, where the feature representation is fixed throughout training. Such a limitation fails to capture the key aspect of representation learning in neural AC, which is pivotal in practical problems. In this work, we take a mean-field perspective on the evolution and convergence of feature-based neural AC. Specifically, we consider a version of AC where the actor and critic are represented by overparameterized two-layer neural networks and are updated with two-timescale learning rates. The critic is updated by temporal-difference (TD) learning with a larger stepsize while the actor is updated via proximal policy optimization (PPO) with a smaller stepsize. In the continuous-time and infinite-width limiting regime, when the timescales are properly separated, we prove that neural AC finds the globally optimal policy at a sublinear rate. Additionally, we prove that the feature representation induced by the critic network is allowed to evolve within a neighborhood of the initial one.

Multi-agent reinforcement learning (MARL) has long been a significant research topic in both machine learning and control systems. Recent development of (single-agent) deep reinforcement learning has created a resurgence of interest in developing new MARL algorithms, especially those founded on theoretical analysis. In this paper, we review recent advances on a sub-area of this topic: decentralized MARL with networked agents. In this scenario, multiple agents perform sequential decision-making in a common environment, and without the coordination of any central controller, while being allowed to exchange information with their neighbors over a communication network. Such a setting finds broad applications in the control and operation of robots, unmanned vehicles, mobile sensor networks, and the smart grid. This review covers several of our research endeavors in this direction, as well as progress made by other researchers along the line. We hope that this review promotes additional research efforts in this exciting yet challenging area.

Modern Reinforcement Learning (RL) is commonly applied to practical problems with an enormous number of states, where \emph{function approximation} must be deployed to approximate either the value function or the policy. The introduction of function approximation raises a fundamental set of challenges involving computational and statistical efficiency, especially given the need to manage the exploration/exploitation tradeoff. As a result, a core RL question remains open: how can we design provably efficient RL algorithms that incorporate function approximation? This question persists even in a basic setting with linear dynamics and linear rewards, for which only linear function approximation is needed. This paper presents the first provable RL algorithm with both polynomial runtime and polynomial sample complexity in this linear setting, without requiring a “simulator” or additional assumptions. Concretely, we prove that an optimistic modification of Least-Squares Value Iteration (LSVI)—a classical algorithm frequently studied in the linear setting—achieves $\tilde{\mathcal{O}}(\sqrt{d^3H^3T})$ regret, where $d$ is the ambient dimension of feature space, $H$ is the length of each episode, and $T$ is the total number of steps. Importantly, such regret is independent of the number of states and actions.

In this work, we develop provably efficient reinforcement learning algorithms for two-player zero-sum Markov games with simultaneous moves. We consider a family of Markov games where the reward function and transition kernel possess a linear structure. Two settings are studied: In the offline setting, we control both players and the goal is to find the Nash Equilibrium efficiently by minimizing the worst-case duality gap. In the online setting, we control a single player and play against an arbitrary opponent; the goal is to minimize the regret. For both settings, we propose an optimistic variant of the least-squares minimax value iteration algorithm. We show that our algorithm is computationally efficient and provably achieves an $\tilde O(\sqrt{d^3 H^3 T} )$ upper bound on the duality gap and regret, without requiring additional assumptions on the sampling model. We highlight that our setting requires overcoming several new challenges that are absent in MDPs or turn-based Markov games. In particular, to achieve optimism under the simultaneous-move games, we construct both upper and lower confidence bounds of the value function, and then derive the optimistic policy by solving a general-sum matrix game with these bounds as the payoff matrices. As finding the Nash Equilibrium of this general-sum game is computationally hard, our algorithm instead solves for a Coarse Correlated Equilibrium (CCE), which can be obtained efficiently via linear programming. To our best knowledge, such a CCE-based mechanism for implementing optimism has not appeared in the literature and might be of interest in its own right.

While policy-based reinforcement learning (RL) achieves tremendous successes in practice, it is significantly less understood in theory, especially compared with value-based RL. In particular, it remains elusive how to design a provably efficient policy optimization algorithm that incorporates exploration. To bridge such a gap, this paper proposes an Optimistic variant of the Proximal Policy Optimization algorithm (OPPO), which follows an “optimistic version” of the policy gradient direction. This paper proves that, in the problem of episodic Markov decision process with linear function approximation, unknown transition, and adversarial reward with full-information feedback, OPPO achieves $\tilde{O}(\sqrt{d^2 H^3 T})$ regret. Here $d$ is the feature dimension, $H$ is the episode horizon, and $T$ is the total number of steps. To the best of our knowledge, OPPO is the first provably efficient policy optimization algorithm that explores.

Graph representation learning is a ubiquitous task in machine learning where the goal is to embed each vertex into a low-dimensional vector space. We consider the bipartite graph and formalize its representation learning problem as a statistical estimation problem of parameters in a semiparametric exponential family distribution: the bipartite graph is assumed to be generated by a semiparametric exponential family distribution, whose parametric component is given by the proximity of outputs of two one-layer neural networks that take high-dimensional features as inputs, while nonparametric (nuisance) component is the base measure. In this setting, the representation learning problem is equivalent to recovering the weight matrices, and the main challenges of estimation arise from the nonlinearity of activation functions and the nonparametric nuisance component of the distribution. To overcome these challenges, we propose a pseudo-likelihood objective based on the rank-order decomposition technique and show that the proposed objective is strongly convex in a neighborhood around the ground truth, so that a gradient descent-based method achieves linear convergence rate. Moreover, we prove that the sample complexity of the problem is linear in dimensions (up to logarithmic factors), which is consistent with parametric Gaussian models. However, our estimator is robust to any model misspecification within the exponential family, which is validated in extensive experiments.

We study the robust one-bit compressed sensing problem whose goal is to design an algorithm that faithfully recovers any sparse target vector $\theta_0\in\mathbb{R}^d$ \emph{uniformly} via $m$ quantized noisy measurements. Specifically, we consider a new framework for this problem where the sparsity is implicitly enforced via mapping a low dimensional representation $x_0 \in \mathbb{R}^k$ through a known $n$-layer ReLU generative network $G:\mathbb{R}^k\rightarrow\mathbb{R}^d$ such that $\theta_0 = G(x_0)$. Such a framework poses low-dimensional priors on $\theta_0$ without a known sparsity basis. We propose to recover the target $G(x_0)$ solving an unconstrained empirical risk minimization (ERM). Under a weak \emph{sub-exponential measurement assumption}, we establish a joint statistical and computational analysis. In particular, we prove that the ERM estimator in this new framework achieves a statistical rate of $m=\widetilde{\mathcal{O}}(kn \log d /\varepsilon^2)$ recovering any $G(x_0)$ uniformly up to an error $\varepsilon$. When the network is shallow (i.e., $n$ is small), we show this rate matches the information-theoretic lower bound up to logarithm factors of $\varepsilon^{-1}$. From the lens of computation, we prove that under proper conditions on the network weights, our proposed empirical risk, despite non-convexity, has no stationary point outside of small neighborhoods around the true representation $x_0$ and its negative multiple; furthermore, we show that the global minimizer of the empirical risk stays within the neighborhood around $x_0$ rather than its negative multiple under further assumptions on weights.

Model-agnostic meta-learning (MAML) formulates meta-learning as a bilevel optimization problem, where the inner level solves each subtask based on a shared prior, while the outer level searches for the optimal shared prior by optimizing its aggregated performance over all the subtasks. Despite its empirical success, MAML remains less understood in theory, especially in terms of its global optimality, due to the nonconvexity of the meta-objective (the outer-level objective). To bridge such a gap between theory and practice, we characterize the optimality gap of the stationary points attained by MAML for both reinforcement learning and supervised learning, where the inner-level and outer-level problems are solved via first-order optimization methods. In particular, our characterization connects the optimality gap of such stationary points with (i) the functional geometry of inner-level objectives and (ii) the representation power of function approximators, including linear models and neural networks. To the best of our knowledge, our analysis establishes the global optimality of MAML with nonconvex meta-objectives for the first time.

Multi-agent reinforcement learning (MARL) achieves significant empirical successes. However, MARL suffers from the curse of many agents. In this paper, we exploit the symmetry of agents in MARL. In the most generic form, we study a mean-field MARL problem. Such a mean-field MARL is defined on mean-field states, which are distributions that are supported on continuous space. Based on the mean embedding of the distributions, we propose MF-FQI algorithm, which solves the mean-field MARL and establishes a non-asymptotic analysis for MF-FQI algorithm. We highlight that MF-FQI algorithm enjoys a “blessing of many agents” property in the sense that a larger number of observed agents improves the performance of MF-FQI algorithm.

Generative adversarial imitation learning (GAIL) demonstrates tremendous success in practice, especially when combined with neural networks. Different from reinforcement learning, GAIL learns both policy and reward function from expert (human) demonstration. Despite its empirical success, it remains unclear whether GAIL with neural networks converges to the globally optimal solution. The major difficulty comes from the nonconvex-nonconcave minimax optimization structure. To bridge the gap between practice and theory, we analyze a gradient-based algorithm with alternating updates and establish its sublinear convergence to the globally optimal solution. To the best of our knowledge, our analysis establishes the global optimality and convergence rate of GAIL with neural networks for the first time.

We study risk-sensitive reinforcement learning in episodic Markov decision processes with unknown transition kernels, where the goal is to optimize the total reward under the risk measure of exponential utility. We propose two provably efficient model-free algorithms, Risk-Sensitive Value Iteration (RSVI) and Risk-Sensitive Q-learning (RSQ). These algorithms implement a form of risk-sensitive optimism in the face of uncertainty, which adapts to both risk-seeking and risk-averse modes of exploration. We prove that RSVI attains an \ensuremath{\tilde{O}\big(\lambda(|\beta| H^2) \cdot \sqrt{H^{3} S^{2}AT} \big)}$ regret, while RSQ attains an $\ensuremath{\tilde{O}\big(\lambda(|\beta| H^2) \cdot \sqrt{H^{4} SAT} \big)}$ regret, where $\lambda(u) = (e^{3u}-1)/u$ for $u>0$. In the above, $\beta$ is the risk parameter of the exponential utility function, $S$ the number of states, $A$ the number of actions, $T$ the total number of timesteps, and $H$ the episode length. On the flip side, we establish a regret lower bound showing that the exponential dependence on $|\beta|$ and $H$ is unavoidable for any algorithm with an $\tilde{O}(\sqrt{T})$ regret (even when the risk objective is on the same scale as the original reward), thus certifying the near-optimality of the proposed algorithms. Our results demonstrate that incorporating risk awareness into reinforcement learning necessitates an exponential cost in $|\beta|$ and $H$, which quantifies the fundamental tradeoff between risk sensitivity (related to aleatoric uncertainty) and sample efficiency (related to epistemic uncertainty). To the best of our knowledge, this is the first regret analysis of risk-sensitive reinforcement learning with the exponential utility.

We consider reinforcement learning (RL) in episodic MDPs with adversarial full-information reward feedback and unknown fixed transition kernels. We propose two model-free policy optimization algorithms, POWER and POWER++, and establish guarantees for their dynamic regret. Compared with the classical notion of static regret, dynamic regret is a stronger notion as it explicitly accounts for the non-stationarity of environments. The dynamic regret attained by the proposed algorithms interpolates between different regimes of non-stationarity, and moreover satisfies a notion of adaptive (near-)optimality, in the sense that it matches the (near-)optimal static regret under slow-changing environments. The dynamic regret bound features two components, one arising from exploration, which deals with the uncertainty of transition kernels, and the other arising from adaptation, which deals with non-stationary environments. Specifically, we show that POWER++ improves over POWER on the second component of the dynamic regret by actively adapting to non-stationarity through prediction. To the best of our knowledge, our work is the first dynamic regret analysis of model-free RL algorithms in non-stationary environments.

This paper develops a Pontryagin differentiable programming (PDP) methodology, which establishes a unified framework to solve a broad class of learning and control tasks. The PDP distinguishes from existing methods by two novel techniques: first, we differentiate through Pontryagin's Maximum Principle, and this allows to obtain the analytical derivative of a trajectory with respect to tunable parameters within an optimal control system, enabling end-to-end learning of dynamics, policies, or/and control objective functions; and second, we propose an auxiliary control system in the backward pass of the PDP framework, and the output of this auxiliary control system is the analytical derivative of the original system's trajectory with respect to the parameters, which can be iteratively solved using standard control tools. We investigate three learning modes of the PDP: inverse reinforcement learning, system identification, and control/planning. We demonstrate the capability of the PDP in each learning mode on different high-dimensional systems, including multilink robot arm, 6-DoF maneuvering UAV, and 6-DoF rocket powered landing.

Structural equation models (SEMs) are widely used in sciences, ranging from economics to psychology, to uncover causal relationships underlying a complex system under consideration and estimate structural parameters of interest. We study estimation in a class of generalized SEMs where the object of interest is defined as the solution to a linear operator equation. We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using the stochastic gradient descent. We consider both 2-layer and multi-layer NNs with ReLU activation functions and prove global convergence in an overparametrized regime, where the number of neurons is diverging. The results are established using techniques from online learning and local linearization of NNs, and improve in several aspects the current state-of-the-art. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.

We consider online learning for episodic stochastically constrained Markov decision processes (CMDP), which plays a central role in ensuring the safety of reinforcement learning. Here the loss function can vary arbitrarily across the episodes, whereas both the loss received and the budget consumption are revealed at the end of each episode. Previous works solve this problem under the restrictive assumption that the transition model of the MDP is known a priori and establish regret bounds that depend polynomially on the cardinalities of the state space $\mathcal{S}$ and the action space $\mathcal{A}$. In this work, we propose a new \emph{upper confidence primal-dual} algorithm, which only requires the trajectories sampled from the transition model. In particular, we prove that the proposed algorithm achieves $\widetilde{\mathcal{O}}(L|\mathcal{S}|\sqrt{|\mathcal{A}|T})$ upper bounds of both the regret and the constraint violation, where $L$ is the length of each episode. Our analysis incorporates a new high-probability drift analysis of Lagrange multiplier processes into the celebrated regret analysis of upper confidence reinforcement learning, which demonstrates the power of ``optimism in the face of uncertainty'' in constrained online learning.

This paper studies a gradient temporal difference (GTD) algorithm using neural network (NN) function approximators to minimize the mean squared Bellman error (MSBE). For off-policy learning, we show that the minimum MSBE problem can be recast into a min-max optimization involving a pair of over-parameterized primal-dual NNs. The resultant formulation can then be tackled using a neural GTD algorithm. We analyze the convergence of the proposed algorithm with a 2-layer ReLU NN architecture using $m$ neurons and prove that it computes an approximate optimal solution to the minimum MSBE problem as $m \rightarrow \infty$.

Reinforcement learning (RL) algorithms combined with modern function approximators such as kernel functions and deep neural networks have achieved significant empirical successes in large-scale application problems with a massive number of states. From a theoretical perspective, however, RL with functional approximation poses a fundamental challenge to developing algorithms with provable computational and statistical efficiency, due to the need to take into consideration both the exploration-exploitation tradeoff that is inherent in RL and the bias-variance tradeoff that is innate in statistical estimation. To address such a challenge, focusing on the episodic setting where the action-value functions are represented by a kernel function or over-parametrized neural network, we propose the first provable RL algorithm with both polynomial runtime and sample complexity, without additional assumptions on the data-generating model. In particular, for both the kernel and neural settings, we prove that an optimistic modification of the least-squares value iteration algorithm incurs an $\tilde{\mathcal{O}}(\delta_{\cF} H^2 \sqrt{T})$ regret, where $\delta_{\cF}$ characterizes the intrinsic complexity of the function class $\cF$, $H$ is the length of each episode, and $T$ is the total number of episodes. Our regret bounds are independent of the number of states and therefore even allows it to diverge, which exhibits the benefit of function approximation.

Temporal-difference and Q-learning play a key role in deep reinforcement learning, where they are empowered by expressive nonlinear function approximators such as neural networks. At the core of their empirical successes is the learned feature representation, which embeds rich observations, e.g., images and texts, into the latent space that encodes semantic structures. Meanwhile, the evolution of such a feature representation is crucial to the convergence of temporal-difference and Q-learning. In particular, temporal-difference learning converges when the function approximator is linear in a feature representation, which is fixed throughout learning, and possibly diverges otherwise. We aim to answer the following questions: When the function approximator is a neural network, how does the associated feature representation evolve? If it converges, does it converge to the optimal one? We prove that utilizing an overparameterized two-layer neural network, temporal-difference and Q-learning globally minimize the mean-squared projected Bellman error at a sublinear rate. Moreover, the associated feature representation converges to the optimal one, generalizing the previous analysis of Cai et al. (2019) in the neural tangent kernel regime, where the associated feature representation stabilizes at the initial one. The key to our analysis is a mean-field perspective, which connects the evolution of a finite-dimensional parameter to its limiting counterpart over an infinite-dimensional Wasserstein space. Our analysis generalizes to soft Q-learning, which is further connected to policy gradient.

In order to solve the problems of poor separation and low signal quality of separated signals in two speakers separation experiments, we propose a deep clustering model based on bidirectional long short-term memory network (BLSTM), which adds phase information to speech signal processing and uses deep clustering to differentiate two speakers. The phase of the speech signal has a significant influence on the pitch performance that the naturalness of the separated speech has obviously improved after adding the phase information. Besides, we also improve the activation layer of the network by selecting a more effective ReLU activation function, which not only improves the separation effect but also accelerates the calculation speed.

We consider estimating a high-dimensional vector from non-linear measurements where the unknown vector is represented by a generative model $G:\mathbb{R}^k\rightarrow\mathbb{R}^d$ with $k\ll d$. Such a model poses structural priors on the unknown vector without having a dedicated basis, and in particular allows new and efficient approaches solving recovery problems with number of measurements far less than the ambient dimension of the vector. While progresses have been made recently regarding theoretical understandings on the linear Gaussian measurements, much less is known when the model is possibly misspecified and the measurements are non-Gaussian. In this paper, we make a step towards such a direction by considering the scenario where the measurements are non-Gaussian, subject to possibly unknown nonlinear transformations and the responses are heavy-tailed. We then propose new estimators via score functions based on the first and second order Stein’s identity, and prove the sample size bound of $m=\mathcal{O}(k\varepsilon^{-2}\log(L/\varepsilon))$ achieving an $\varepsilon$ error in the form of exponential concentration inequalities. Furthermore, for the special case of multi-layer ReLU generative model, we improve the sample bound by a logarithm factor to $m=\mathcal{O}(k\varepsilon^{-2}\log(d))$, matching the state-of-art statistical rate in compressed sensing for estimating $k$-sparse vectors. On the technical side, we develop new chaining methods bounding heavy-tailed processes, which could be of independent interest.

Temporal-difference learning (TD), coupled with neural networks, is among the most fundamental building blocks of deep reinforcement learning. However, due to the nonlinearity in value function approximation, such a coupling leads to nonconvexity and even divergence in optimization. As a result, the global convergence of neural TD remains unclear. In this paper, we prove for the first time that neural TD converges at a sublinear rate to the global optimum of the mean-squared projected Bellman error for policy evaluation. In particular, we show how such global convergence is enabled by the overparametrization of neural networks, which also plays a vital role in the empirical success of neural TD. Beyond policy evaluation, we establish the global convergence of neural (soft) Q-learning, which is further connected to that of policy gradient algorithms.

Proximal policy optimization and trust region policy optimization (PPO and TRPO) with actor and critic parametrized by neural networks achieve significant empirical success in deep reinforcement learning. However, due to nonconvexity, the global convergence of PPO and TRPO remains less understood, which separates theory from practice. In this paper, we prove that a variant of PPO and TRPO equipped with overparametrized neural networks converges to the globally optimal policy at a sublinear rate. The key to our analysis is the global convergence of infinite-dimensional mirror descent under a notion of one-point monotonicity, where the gradient and iterate are instantiated by neural networks. In particular, the desirable representation power and optimization geometry induced by the overparametrization of such neural networks allow them to accurately approximate the infinite-dimensional gradient and iterate.

Policy evaluation with smooth and nonlinear function approximation has shown great potential for reinforcement learning. Compared to linear function approxi- mation, it allows for using a richer class of approximation functions such as the neural networks. Traditional algorithms are based on two timescales stochastic approximation whose convergence rate is often slow. This paper focuses on an offline setting where a trajectory of $m$ state-action pairs are observed. We formulate the policy evaluation problem as a non-convex primal-dual, finite-sum optimization problem, whose primal sub-problem is non-convex and dual sub-problem is strongly concave. We suggest a single-timescale primal-dual gradient algorithm with variance reduction, and show that it converges to an $\epsilon$-stationary point using $O(m/\epsilon)$ calls (in expectation) to a gradient oracle.

We study the statistical-computational tradeoffs in a high dimensional single index model $Y=f(X^\top\beta^*) +\epsilon$, where $f$ is unknown, $X$ is a Gaussian vector and $\beta^*$ is $s$-sparse with unit norm. When $\cov(Y,X^\top\beta^*)\neq 0$, \cite{plan2016generalized} shows that the direction and support of $\beta^*$ can be recovered using a generalized version of Lasso. In this paper, we investigate the case when this critical assumption fails to hold, where the problem becomes considerably harder. Using the statistical query model to characterize the computational cost of an algorithm, we show that when $\cov(Y,X^\top\beta^*)=0$ and $\cov(Y,(X^\top\beta^*)^2)>0$, no computationally tractable algorithms can achieve the information-theoretic limit of the minimax risk. This implies that one must pay an extra computational cost for the nonlinearity involved in the model.

Despite the empirical success of the actor-critic algorithm, its theoretical understanding lags behind. In a broader context, actor-critic can be viewed as an online alternating update algorithm for bilevel optimization, whose convergence is known to be fragile. To understand the instability of actor-critic, we focus on its application to linear quadratic regulators, a simple yet fundamental setting of reinforcement learning. We establish a nonasymptotic convergence analysis of actor- critic in this setting. In particular, we prove that actor-critic finds a globally optimal pair of actor (policy) and critic (action-value function) at a linear rate of convergence. Our analysis may serve as a preliminary step towards a complete theoretical understanding of bilevel optimization with nonconvex subproblems, which is NP-hard in the worst case and is often solved using heuristics.

We study the safe reinforcement learning problem with nonlinear function approximation, where policy optimization is formulated as a constrained optimization problem with both the objective and the constraint being nonconvex functions. For such a problem, we construct a sequence of surrogate convex constrained optimization problems by replacing the nonconvex functions locally with convex quadratic functions obtained from policy gradient estimators. We prove that the solutions to these surrogate problems converge to a stationary point of the original nonconvex problem. Furthermore, to extend our theoretical results, we apply our algorithm to examples of optimal control and multi-agent reinforcement learning with safety constraints.

We study the global convergence of policy optimization for finding the Nash equilibria (NE) in zero-sum linear quadratic (LQ) games. To this end, we first investigate the landscape of LQ games, viewing it as a nonconvex-nonconcave saddle-point problem in the policy space. Specifically, we show that despite its nonconvexity and nonconcavity, zero-sum LQ games have the property that the stationary point of the objective function with respect to the linear feedback control policies constitutes the NE of the game. Building upon this, we develop three projected nested-gradient methods that are guaranteed to converge to the NE of the game. Moreover, we show that all these algorithms enjoy both globally sublinear and locally linear convergence rates. Simulation results are also provided to illustrate the satisfactory convergence properties of the algorithms. To the best of our knowledge, this work appears to be the first one to investigate the optimization landscape of LQ games, and provably show the convergence of policy optimization methods to the NE. Our work serves as an initial step toward understanding the theoretical aspects of policy-based reinforcement learning algorithms for zero-sum Markov games in general.

We study the parameter estimation problem for a varying index coefficient model in high dimensions. Unlike the most existing works that iteratively estimate the parameters and link functions, based on the generalized Stein's identity, we propose computationally efficient estimators for the high-dimensional parameters without estimating the link functions. We consider two different setups where we either estimate each sparse parameter vector individually or estimate the parameters simultaneously as a sparse or low-rank matrix. For all these cases, our estimators are shown to achieve optimal statistical rates of convergence (up to logarithmic terms in the low-rank setting). Moreover, throughout our analysis, we only require the covariate to satisfy certain moment conditions, which is significantly weaker than the Gaussian or elliptically symmetric assumptions that are commonly made in the existing literature. Finally, we conduct extensive numerical experiments to corroborate the theoretical results.

We study the high-dimensional signal estimation problem with nonlinear measurements, where the signal of interest is either sparse or low-rank. In both settings, our estimator is formulated as the minimizer of the nonlinear least-squares loss function under a combinatorial constraint, which is obtained efficiently by the iterative hard thresholding (IHT) algorithm. Although the loss function is non-convex due to the nonlinearity of the statistical model, the IHT algorithm is shown to converge linearly to a point with optimal statistical accuracy using arbitrary initialization. Moreover, our analysis only hinges on conditions similar to those required in the linear case. Detailed numerical experiments are included to corroborate the theoretical results.

We consider the fully decentralized multi-agent reinforcement learning (MARL) problem, where the agents are connected via a time-varying and possibly sparse communication network. Specifically, we assume that the reward functions of the agents might correspond to different tasks, and are only known to the corresponding agent. Moreover, each agent makes individual decisions based on both the information observed locally and the messages received from its neighbors over the network. To maximize the globally averaged return over the network, we propose two fully decentralized actor-critic algorithms, which are applicable to large-scale MARL problems in an online fashion. Convergence guarantees are provided when the value functions are approximated within the class of linear functions. Our work appears to be the first theoretical study of fully decentralized MARL algorithms for networked agents that use function approximation.

Learning from pairwise measurements naturally arises from many applications, such as rank aggregation, ordinal embedding, and crowdsourcing. However, most existing models and algorithms are susceptible to potential model misspecification. In this paper, we study a semiparametric model where the pairwise measurements follow a natural exponential family distribution with an unknown base measure. Such a semiparametric model includes various popular parametric models, such as the Bradley-Terry-Luce model and the paired cardinal model, as special cases. To estimate this semiparametric model without specifying the base measure, we propose a data augmentation technique to create virtual examples, which enables us to define a contrastive estimator. In particular, we prove that such a contrastive estimator is invariant to model misspecification within the natural exponential family, and moreover, attains the optimal statistical rate of convergence up to a logarithmic factor. We provide numerical experiments to corroborate our theory.

Despite the success of single-agent reinforcement learning, multi-agent reinforcement learning (MARL) remains challenging due to complex interactions between agents. Motivated by decentralized applications such as sensor networks, swarm robotics, and power grids, we study policy evaluation in MARL, where agents with jointly observed state-action pairs and private local rewards collaborate to learn the value of a given policy. In this paper, we propose a double averaging scheme, where each agent iteratively performs averaging over both space and time to incorporate neighboring gradient information and local reward information, respectively. We prove that the proposed algorithm converges to the optimal solution at a global geometric rate. In particular, such an algorithm is built upon a primal-dual reformulation of the mean squared Bellman error minimization problem, which gives rise to a decentralized convex-concave saddle-point problem. To the best of our knowledge, the proposed double averaging primal-dual optimization algorithm is the first to achieve fast finite-time convergence on decentralized convex-concave saddle-point problems.

The success of machine learning methods heavily relies on having an appropriate representation for data at hand. Traditionally, machine learning approaches relied on user-defined heuristics to extract features encoding structural information about data. However, recently there has been a surge in approaches that learn how to encode the data automatically in a low dimensional space. Exponential family embedding provides a probabilistic framework for learning low-dimensional representation for various types of high-dimensional data. Though successful in practice, theoretical underpinnings for exponential family embeddings have not been established. In this paper, we study the Gaussian embedding model and develop the first theoretical results for exponential family embedding models. First, we show that, under a mild condition, the embedding structure can be learned from one observation by leveraging the parameter sharing between different contexts even though the data are dependent with each other. Second, we study properties of two algorithms used for learning the embedding structure and establish convergence results for each of them. The first algorithm is based on a convex relaxation, while the other solved the non-convex formulation of the problem directly. Experiments demonstrate the effectiveness of our approach.

We propose a new class of semiparametric exponential family graphical models for the analysis of high dimensional mixed data. Different from the existing mixed graphical models, we allow the nodewise conditional distributions to be semiparametric generalized linear models with unspecified base measure functions. Thus, one advantage of our method is that it is unnecessary to specify the type of each node and the method is more convenient to apply in practice. Under the proposed model, we consider both problems of parameter estimation and hypothesis testing in high dimensions. In particular, we propose a symmetric pairwise score test for the presence of a single edge in the graph. Compared to the existing methods for hypothesis tests, our approach takes into account of the symmetry of the parameters, such that the inferential results are invariant with respect to the different parametrizations of the same edge. Thorough numerical simulations and a real data example are provided to back up our theoretical results.

We consider estimating the parametric component of single index models in high dimensions. Compared with existing work, we do not require the covariate to be normally distributed. Utilizing Stein’s Lemma, we propose estimators based on the score function of the covariate. Moreover, to handle score function and response variables that are heavy-tailed, our estimators are constructed via carefully thresholding their empirical counterparts. Under a bounded fourth moment condition, we establish optimal statistical rates of convergence for the proposed estimators. Extensive numerical experiments are provided to back up our theory.

We consider estimating the parametric components of semiparametric multi-index models in high dimensions. To bypass the requirements of Gaussianity or elliptical symmetry of covariates in existing methods, we propose to leverage a second-order Stein’s method with score function-based corrections. We prove that our estimator achieves a near-optimal statistical rate of convergence even when the score function or the response variable is heavy-tailed. To establish the key concentration results, we develop a data-driven truncation argument that may be of independent interest. We supplement our theoretical findings with simulations.

We study parameter estimation for sparse nonlinear regression. More specifically, we assume the data are given by y = f( \bf x^T \bf β^* ) + ε, where f is nonlinear. To recover \bf βs, we propose an \ell_1-regularized least-squares estimator. Unlike classical linear regression, the corresponding optimization problem is nonconvex because of the nonlinearity of f. In spite of the nonconvexity, we prove that under mild conditions, every stationary point of the objective enjoys an optimal statistical rate of convergence. Detailed numerical results are provided to back up our theory.

We consider the weakly supervised binary classification problem where the labels are randomly flipped with probability $1-\alpha$. Although there exist numerous algorithms for this problem, it remains theoretically unexplored how the statistical accuracies and computational efficiency of these algorithms depend on the degree of supervision, which is quantified by $\alpha$. In this paper, we characterize the effect of $\alpha$ by establishing the information-theoretic and computational boundaries, namely, the minimax-optimal statistical accuracy that can be achieved by all algorithms, and polynomial-time algorithms under an oracle computational model. For small $\alpha$, our result shows a gap between these two boundaries, which represents the computational price of achieving the information-theoretic boundary due to the lack of supervision. Interestingly, we also show that this gap narrows as $\alpha$ increases. In other words, having more supervision, i.e., more correct labels, not only improves the optimal statistical accuracy as expected, but also enhances the computational efficiency for achieving such accuracy.

Imagine a random walk that outputs a state only when visiting it for the first time. The observed output is therefore a repeat-censored version of the underlying walk, and consists of a permutation of the states or a prefix of it. We call this model initial-visit emitting random walk (INVITE). Prior work has shown that the random walks with such a repeat-censoring mechanism explain well human behavior in memory search tasks, which is of great interest in both the study of human cognition and various clinical applications. However, parameter estimation in INVITE is challenging, because naive likelihood computation by marginalizing over infinitely many hidden random walk trajectories is intractable. In this paper, we propose the first efficient maximum likelihood estimate (MLE) for INVITE by decomposing the censored output into a series of absorbing random walks. We also prove theoretical properties of the MLE including identifiability and consistency. We show that INVITE outperforms several existing methods on real-world human response data from memory search tasks.