Han Zhong

Binghui Li, Jikai Jin, Han Zhong et al. · stanford

It is well-known that modern neural networks are vulnerable to adversarial examples. To mitigate this problem, a series of robust learning algorithms have been proposed. However, although the robust training error can be near zero via some methods, all existing algorithms lead to a high robust generalization error. In this paper, we provide a theoretical understanding of this puzzling phenomenon from the perspective of expressive power for deep neural networks. Specifically, for binary classification problems with well-separated data, we show that, for ReLU networks, while mild over-parameterization is sufficient for high robust training accuracy, there exists a constant robust generalization gap unless the size of the neural network is exponential in the data dimension $d$. This result holds even if the data is linear separable (which means achieving standard generalization is easy), and more generally for any parameterized function classes as long as their VC dimension is at most polynomial in the number of parameters. Moreover, we establish an improved upper bound of $\exp({\mathcal{O}}(k))$ for the network size to achieve low robust generalization error when the data lies on a manifold with intrinsic dimension $k$ ($k \ll d$). Nonetheless, we also have a lower bound that grows exponentially with respect to $k$ -- the curse of dimensionality is inevitable. By demonstrating an exponential separation between the network size for achieving low robust training and generalization error, our results reveal that the hardness of robust generalization may stem from the expressive power of practical models.

Wei Xiong, Han Zhong, Chengshuai Shi et al.

Offline reinforcement learning (RL) aims at learning an optimal strategy using a pre-collected dataset without further interactions with the environment. While various algorithms have been proposed for offline RL in the previous literature, the minimax optimality has only been (nearly) established for tabular Markov decision processes (MDPs). In this paper, we focus on offline RL with linear function approximation and propose a new pessimism-based algorithm for offline linear MDP. At the core of our algorithm is the uncertainty decomposition via a reference function, which is new in the literature of offline RL under linear function approximation. Theoretical analysis demonstrates that our algorithm can match the performance lower bound up to logarithmic factors. We also extend our techniques to the two-player zero-sum Markov games (MGs), and establish a new performance lower bound for MGs, which tightens the existing result, and verifies the nearly minimax optimality of the proposed algorithm. To the best of our knowledge, these are the first computationally efficient and nearly minimax optimal algorithms for offline single-agent MDPs and MGs with linear function approximation.

Xiaoyu Chen, Han Zhong, Zhuoran Yang et al.

We study human-in-the-loop reinforcement learning (RL) with trajectory preferences, where instead of receiving a numeric reward at each step, the agent only receives preferences over trajectory pairs from a human overseer. The goal of the agent is to learn the optimal policy which is most preferred by the human overseer. Despite the empirical successes, 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. Our algorithm achieves the regret 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.

Han Zhong, Wei Xiong, Sirui Zheng et al.

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

Wei Xiong, Han Zhong, Chengshuai Shi et al.

Existing studies on provably efficient algorithms for Markov games (MGs) almost exclusively build on the "optimism in the face of uncertainty" (OFU) principle. This work focuses on a different approach of posterior sampling, which is celebrated in many bandits and reinforcement learning settings but remains under-explored for MGs. Specifically, for episodic two-player zero-sum MGs, a novel posterior sampling algorithm is developed with general function approximation. Theoretical analysis demonstrates that the posterior sampling algorithm admits a $\sqrt{T}$-regret bound for problems with a low multi-agent decoupling coefficient, which is a new complexity measure for MGs, where $T$ denotes the number of episodes. When specialized to linear MGs, the obtained regret bound matches the state-of-the-art results. To the best of our knowledge, this is the first provably efficient posterior sampling algorithm for MGs with frequentist regret guarantees, which enriches the toolbox for MGs and promotes the broad applicability of posterior sampling.

Yunchang Yang, Han Zhong, Tianhao Wu et al.

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

Jiachen Hu, Han Zhong, Chi Jin et al.

Sim-to-real transfer trains RL agents in the simulated environments and then deploys them in the real world. Sim-to-real transfer has been widely used in practice because it is often cheaper, safer and much faster to collect samples in simulation than in the real world. Despite the empirical success of the sim-to-real transfer, its theoretical foundation is much less understood. In this paper, we study the sim-to-real transfer in continuous domain with partial observations, where the simulated environments and real-world environments are modeled by linear quadratic Gaussian (LQG) systems. We show that a popular robust adversarial training algorithm is capable of learning a policy from the simulated environment that is competitive to the optimal policy in the real-world environment. To achieve our results, we design a new algorithm for infinite-horizon average-cost LQGs and establish a regret bound that depends on the intrinsic complexity of the model class. Our algorithm crucially relies on a novel history clipping scheme, which might be of independent interest.

Shuang Qiu, Ziyu Dai, Han Zhong et al.

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.

Han Zhong, Jiachen Hu, Yecheng Xue et al.

While quantum reinforcement learning (RL) has attracted a surge of attention recently, its theoretical understanding is limited. In particular, it remains elusive how to design provably efficient quantum RL algorithms that can address the exploration-exploitation trade-off. To this end, we propose a novel UCRL-style algorithm that takes advantage of quantum computing for tabular Markov decision processes (MDPs) with $S$ states, $A$ actions, and horizon $H$, and establish an $\mathcal{O}(\mathrm{poly}(S, A, H, \log T))$ worst-case regret for it, where $T$ is the number of episodes. Furthermore, we extend our results to quantum RL with linear function approximation, which is capable of handling problems with large state spaces. Specifically, we develop a quantum algorithm based on value target regression (VTR) for linear mixture MDPs with $d$-dimensional linear representation and prove that it enjoys $\mathcal{O}(\mathrm{poly}(d, H, \log T))$ regret. Our algorithms are variants of UCRL/UCRL-VTR algorithms in classical RL, which also leverage a novel combination of lazy updating mechanisms and quantum estimation subroutines. This is the key to breaking the $Ω(\sqrt{T})$-regret barrier in classical RL. To the best of our knowledge, this is the first work studying the online exploration in quantum RL with provable logarithmic worst-case regret.

Wei Xiong, Hanze Dong, Chenlu Ye et al.

This paper studies the alignment process of generative models with Reinforcement Learning from Human Feedback (RLHF). We first identify the primary challenges of existing popular methods like offline PPO and offline DPO as lacking in strategical exploration of the environment. Then, to understand the mathematical principle of RLHF, we consider a standard mathematical formulation, the reverse-KL regularized contextual bandit for RLHF. Despite its widespread practical application, a rigorous theoretical analysis of this formulation remains open. We investigate its behavior in three distinct settings -- offline, online, and hybrid -- and propose efficient algorithms with finite-sample theoretical guarantees. Moving towards practical applications, our framework, with a robust approximation of the information-theoretical policy improvement oracle, naturally gives rise to several novel RLHF algorithms. This includes an iterative version of the Direct Preference Optimization (DPO) algorithm for online settings, and a multi-step rejection sampling strategy for offline scenarios. Our empirical evaluations on real-world alignment experiment of large language model demonstrate that these proposed methods significantly surpass existing strong baselines, such as DPO and Rejection Sampling Optimization (RSO), showcasing the connections between solid theoretical foundations and their potent practical implementations.

Miao Lu, Han Zhong, Tong Zhang et al.

The sim-to-real gap, which represents the disparity between training and testing environments, poses a significant challenge in reinforcement learning (RL). A promising approach to addressing this challenge is distributionally robust RL, often framed as a robust Markov decision process (RMDP). In this framework, the objective is to find a robust policy that achieves good performance under the worst-case scenario among all environments within a pre-specified uncertainty set centered around the training environment. Unlike previous work, which relies on a generative model or a pre-collected offline dataset enjoying good coverage of the deployment environment, we tackle robust RL via interactive data collection, where the learner interacts with the training environment only and refines the policy through trial and error. In this robust RL paradigm, two main challenges emerge: managing distributional robustness while striking a balance between exploration and exploitation during data collection. Initially, we establish that sample-efficient learning without additional assumptions is unattainable owing to the curse of support shift; i.e., the potential disjointedness of the distributional supports between the training and testing environments. To circumvent such a hardness result, we introduce the vanishing minimal value assumption to RMDPs with a total-variation (TV) distance robust set, postulating that the minimal value of the optimal robust value function is zero. We prove that such an assumption effectively eliminates the support shift issue for RMDPs with a TV distance robust set, and present an algorithm with a provable sample complexity guarantee. Our work makes the initial step to uncovering the inherent difficulty of robust RL via interactive data collection and sufficient conditions for designing a sample-efficient algorithm accompanied by sharp sample complexity analysis.

Han Zhong, Yutong Yin, Shenao Zhang et al.

Large Language Models (LLMs) have demonstrated remarkable capabilities in complex reasoning tasks, yet generating reliable reasoning processes remains a significant challenge. We present a unified probabilistic framework that formalizes LLM reasoning through a novel graphical model incorporating latent thinking processes and evaluation signals. Within this framework, we introduce the Bootstrapping Reinforced Thinking Process (BRiTE) algorithm, which works in two steps. First, it generates high-quality rationales by approximating the optimal thinking process through reinforcement learning, using a novel reward shaping mechanism. Second, it enhances the base LLM by maximizing the joint probability of rationale generation with respect to the model's parameters. Theoretically, we demonstrate BRiTE's convergence at a rate of $1/T$ with $T$ representing the number of iterations. Empirical evaluations on math and coding benchmarks demonstrate that our approach consistently improves performance across different base models without requiring human-annotated thinking processes. In addition, BRiTE demonstrates superior performance compared to existing algorithms that bootstrap thinking processes use alternative methods such as rejection sampling, and can even match or exceed the results achieved through supervised fine-tuning with human-annotated data.

Jianliang He, Han Zhong, Zhuoran Yang

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 $\tilde{\mathcal{O}}(\mathrm{poly}(d, \mathrm{sp}(V^*)) \sqrt{Tβ} )$ regret, where $d$ and $β$ 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.

Guhao Feng, Han Zhong · pku

Reinforcement Learning (RL) encompasses diverse paradigms, including model-based RL, policy-based RL, and value-based RL, each tailored to approximate the model, optimal policy, and optimal value function, respectively. This work investigates the potential hierarchy of representation complexity -- the complexity of functions to be represented -- among these RL paradigms. We first demonstrate that, for a broad class of Markov decision processes (MDPs), the model can be represented by constant-depth circuits with polynomial size or Multi-Layer Perceptrons (MLPs) with constant layers and polynomial hidden dimension. However, the representation of the optimal policy and optimal value proves to be $\mathsf{NP}$-complete and unattainable by constant-layer MLPs with polynomial size. This demonstrates a significant representation complexity gap between model-based RL and model-free RL, which includes policy-based RL and value-based RL. To further explore the representation complexity hierarchy between policy-based RL and value-based RL, we introduce another general class of MDPs where both the model and optimal policy can be represented by constant-depth circuits with polynomial size or constant-layer MLPs with polynomial size. In contrast, representing the optimal value is $\mathsf{P}$-complete and intractable via a constant-layer MLP with polynomial hidden dimension. This accentuates the intricate representation complexity associated with value-based RL compared to policy-based RL. In summary, we unveil a potential representation complexity hierarchy within RL -- representing the model emerges as the easiest task, followed by the optimal policy, while representing the optimal value function presents the most intricate challenge.

Xutong Liu, Siwei Wang, Jinhang Zuo et al.

We introduce a novel framework of combinatorial multi-armed bandits (CMAB) with multivariant and probabilistically triggering arms (CMAB-MT), where the outcome of each arm is a $d$-dimensional multivariant random variable and the feedback follows a general arm triggering process. Compared with existing CMAB works, CMAB-MT not only enhances the modeling power but also allows improved results by leveraging distinct statistical properties for multivariant random variables. For CMAB-MT, we propose a general 1-norm multivariant and triggering probability-modulated smoothness condition, and an optimistic CUCB-MT algorithm built upon this condition. Our framework can include many important problems as applications, such as episodic reinforcement learning (RL) and probabilistic maximum coverage for goods distribution, all of which meet the above smoothness condition and achieve matching or improved regret bounds compared to existing works. Through our new framework, we build the first connection between the episodic RL and CMAB literature, by offering a new angle to solve the episodic RL through the lens of CMAB, which may encourage more interactions between these two important directions.

Zhihan Liu, Miao Lu, Wei Xiong et al.

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

Jose Blanchet, Miao Lu, Tong Zhang et al.

In this paper, we study distributionally robust offline reinforcement learning (robust offline RL), which seeks to find an optimal policy purely from an offline dataset that can perform well in perturbed environments. In specific, we propose a generic algorithm framework called Doubly Pessimistic Model-based Policy Optimization ($P^2MPO$), which features a novel combination of a flexible model estimation subroutine and a doubly pessimistic policy optimization step. Notably, the double pessimism principle is crucial to overcome the distributional shifts incurred by (i) the mismatch between the behavior policy and the target policies; and (ii) the perturbation of the nominal model. Under certain accuracy conditions on the model estimation subroutine, we prove that $P^2MPO$ is sample-efficient with robust partial coverage data, which only requires the offline data to have good coverage of the distributions induced by the optimal robust policy and the perturbed models around the nominal model. By tailoring specific model estimation subroutines for concrete examples of RMDPs, including tabular RMDPs, factored RMDPs, kernel and neural RMDPs, we prove that $P^2MPO$ enjoys a $\tilde{\mathcal{O}}(n^{-1/2})$ convergence rate, where $n$ is the dataset size. We highlight that all these examples, except tabular RMDPs, are first identified and proven tractable by this work. Furthermore, we continue our study of robust offline RL in the robust Markov games (RMGs). By extending the double pessimism principle identified for single-agent RMDPs, we propose another algorithm framework that can efficiently find the robust Nash equilibria among players using only robust unilateral (partial) coverage data. To our best knowledge, this work proposes the first general learning principle -- double pessimism -- for robust offline RL and shows that it is provably efficient with general function approximation.

Han Zhong, Tong Zhang

The proximal policy optimization (PPO) algorithm stands as one of the most prosperous methods in the field of reinforcement learning (RL). Despite its success, the theoretical understanding of PPO remains deficient. Specifically, it is unclear whether PPO or its optimistic variants can effectively solve linear Markov decision processes (MDPs), which are arguably the simplest models in RL with function approximation. To bridge this gap, we propose an optimistic variant of PPO for episodic adversarial linear MDPs with full-information feedback, and establish a $\tilde{\mathcal{O}}(d^{3/4}H^2K^{3/4})$ regret for it. Here $d$ is the ambient dimension of linear MDPs, $H$ is the length of each episode, and $K$ is the number of episodes. Compared with existing policy-based algorithms, we achieve the state-of-the-art regret bound in both stochastic linear MDPs and adversarial linear MDPs with full information. Additionally, our algorithm design features a novel multi-batched updating mechanism and the theoretical analysis utilizes a new covering number argument of value and policy classes, which might be of independent interest.

Han Zhong, Wei Xiong, Jiyuan Tan et al.

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 NEs 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 suggests that the data-dependent term in the upper bound 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.

Tianhao Wu, Yunchang Yang, Han Zhong et al.

Policy optimization methods are one of the most widely used classes of Reinforcement Learning (RL) algorithms. However, theoretical understanding of these methods remains insufficient. Even in the episodic (time-inhomogeneous) tabular setting, the state-of-the-art theoretical result of policy-based method in \citet{shani2020optimistic} is only $\tilde{O}(\sqrt{S^2AH^4K})$ where $S$ is the number of states, $A$ is the number of actions, $H$ is the horizon, and $K$ is the number of episodes, and there is a $\sqrt{SH}$ gap compared with the information theoretic lower bound $\tildeΩ(\sqrt{SAH^3K})$. To bridge such a gap, we propose a novel algorithm Reference-based Policy Optimization with Stable at Any Time guarantee (\algnameacro), which features the property "Stable at Any Time". We prove that our algorithm achieves $\tilde{O}(\sqrt{SAH^3K} + \sqrt{AH^4K})$ regret. When $S > H$, our algorithm is minimax optimal when ignoring logarithmic factors. To our best knowledge, RPO-SAT is the first computationally efficient, nearly minimax optimal policy-based algorithm for tabular RL.

Yunchang Yang, Tianhao Wu, Han Zhong et al.

In this paper, we present a reduction-based framework for conservative bandits and RL, in which our core technique is to calculate the necessary and sufficient budget obtained from running the baseline policy. For lower bounds, we improve the existing lower bound for conservative multi-armed bandits and obtain new lower bounds for conservative linear bandits, tabular RL and low-rank MDP, through a black-box reduction that turns a certain lower bound in the nonconservative setting into a new lower bound in the conservative setting. For upper bounds, in multi-armed bandits, linear bandits and tabular RL, our new upper bounds tighten or match existing ones with significantly simpler analyses. We also obtain a new upper bound for conservative low-rank MDP.