Yuanhao Wang

Yuanhao Wang, Qinghua Liu, Chi Jin

Reinforcement learning from Human Feedback (RLHF) learns from preference signals, while standard Reinforcement Learning (RL) directly learns from reward signals. Preferences arguably contain less information than rewards, which makes preference-based RL seemingly more difficult. This paper theoretically proves that, for a wide range of preference models, we can solve preference-based RL directly using existing algorithms and techniques for reward-based RL, with small or no extra costs. Specifically, (1) for preferences that are drawn from reward-based probabilistic models, we reduce the problem to robust reward-based RL that can tolerate small errors in rewards; (2) for general arbitrary preferences where the objective is to find the von Neumann winner, we reduce the problem to multiagent reward-based RL which finds Nash equilibria for factored Markov games with a restricted set of policies. The latter case can be further reduced to adversarial MDP when preferences only depend on the final state. We instantiate all reward-based RL subroutines by concrete provable algorithms, and apply our theory to a large class of models including tabular MDPs and MDPs with generic function approximation. We further provide guarantees when K-wise comparisons are available.

Yuanhao Wang, Qinghua Liu, Yu Bai et al.

A unique challenge in Multi-Agent Reinforcement Learning (MARL) is the curse of multiagency, where the description length of the game as well as the complexity of many existing learning algorithms scale exponentially with the number of agents. While recent works successfully address this challenge under the model of tabular Markov Games, their mechanisms critically rely on the number of states being finite and small, and do not extend to practical scenarios with enormous state spaces where function approximation must be used to approximate value functions or policies. This paper presents the first line of MARL algorithms that provably resolve the curse of multiagency under function approximation. We design a new decentralized algorithm -- V-Learning with Policy Replay, which gives the first polynomial sample complexity results for learning approximate Coarse Correlated Equilibria (CCEs) of Markov Games under decentralized linear function approximation. Our algorithm always outputs Markov CCEs, and achieves an optimal rate of $\widetilde{\mathcal{O}}(ε^{-2})$ for finding $ε$-optimal solutions. Also, when restricted to the tabular case, our result improves over the current best decentralized result $\widetilde{\mathcal{O}}(ε^{-3})$ for finding Markov CCEs. We further present an alternative algorithm -- Decentralized Optimistic Policy Mirror Descent, which finds policy-class-restricted CCEs using a polynomial number of samples. In exchange for learning a weaker version of CCEs, this algorithm applies to a wider range of problems under generic function approximation, such as linear quadratic games and MARL problems with low ''marginal'' Eluder dimension.

Qinghua Liu, Yuanhao Wang, Chi Jin

An ideal strategy in zero-sum games should not only grant the player an average reward no less than the value of Nash equilibrium, but also exploit the (adaptive) opponents when they are suboptimal. While most existing works in Markov games focus exclusively on the former objective, it remains open whether we can achieve both objectives simultaneously. To address this problem, this work studies no-regret learning in Markov games with adversarial opponents when competing against the best fixed policy in hindsight. Along this direction, we present a new complete set of positive and negative results: When the policies of the opponents are revealed at the end of each episode, we propose new efficient algorithms achieving $\sqrt{K}$-regret bounds when either (1) the baseline policy class is small or (2) the opponent's policy class is small. This is complemented with an exponential lower bound when neither conditions are true. When the policies of the opponents are not revealed, we prove a statistical hardness result even in the most favorable scenario when both above conditions are true. Our hardness result is much stronger than the existing hardness results which either only involve computational hardness, or require further restrictions on the algorithms.

Yuanhao Wang, Dingwen Kong, Yu Bai et al.

A natural goal in multiagent learning besides finding equilibria is to learn rationalizable behavior, where players learn to avoid iteratively dominated actions. However, even in the basic setting of multiplayer general-sum games, existing algorithms require a number of samples exponential in the number of players to learn rationalizable equilibria under bandit feedback. This paper develops the first line of efficient algorithms for learning rationalizable Coarse Correlated Equilibria (CCE) and Correlated Equilibria (CE) whose sample complexities are polynomial in all problem parameters including the number of players. To achieve this result, we also develop a new efficient algorithm for the simpler task of finding one rationalizable action profile (not necessarily an equilibrium), whose sample complexity substantially improves over the best existing results of Wu et al. (2021). Our algorithms incorporate several novel techniques to guarantee rationalizability and no (swap-)regret simultaneously, including a correlated exploration scheme and adaptive learning rates, which may be of independent interest. We complement our results with a sample complexity lower bound showing the sharpness of our guarantees.

Guocheng Qian, Yuanhao Wang, Jinjin Gu et al.

Imaging is usually a mixture problem of incomplete color sampling, noise degradation, and limited resolution. This mixture problem is typically solved by a sequential solution that applies demosaicing (DM), denoising (DN), and super-resolution (SR) sequentially in a fixed and predefined pipeline (execution order of tasks), DM$\to$DN$\to$SR. The most recent work on image processing focuses on developing more sophisticated architectures to achieve higher image quality. Little attention has been paid to the design of the pipeline, and it is still not clear how significant the pipeline is to image quality. In this work, we comprehensively study the effects of pipelines on the mixture problem of learning-based DN, DM, and SR, in both sequential and joint solutions. On the one hand, in sequential solutions, we find that the pipeline has a non-trivial effect on the resulted image quality. Our suggested pipeline DN$\to$SR$\to$DM yields consistently better performance than other sequential pipelines in various experimental settings and benchmarks. On the other hand, in joint solutions, we propose an end-to-end Trinity Pixel Enhancement NETwork (TENet) that achieves state-of-the-art performance for the mixture problem. We further present a novel and simple method that can integrate a certain pipeline into a given end-to-end network by providing intermediate supervision using a detachable head. Extensive experiments show that an end-to-end network with the proposed pipeline can attain only a consistent but insignificant improvement. Our work indicates that the investigation of pipelines is applicable in sequential solutions, but is not very necessary in end-to-end networks. \RR{Code, models, and our contributed PixelShift200 dataset are available at \url{https://github.com/guochengqian/TENet}

Ding-Jiun Huang, Yuanhao Wang, Shao-Ji Yuan et al.

Creating high-fidelity, animatable 3D talking heads is crucial for immersive applications, yet often hindered by the prevalence of low-quality image or video sources, which yield poor 3D reconstructions. In this paper, we introduce SuperHead, a novel framework for enhancing low-resolution, animatable 3D head avatars. The core challenge lies in synthesizing high-quality geometry and textures, while ensuring both 3D and temporal consistency during animation and preserving subject identity. Despite recent progress in image, video and 3D-based super-resolution (SR), existing SR techniques are ill-equipped to handle dynamic 3D inputs. To address this, SuperHead leverages the rich priors from pre-trained 3D generative models via a novel dynamics-aware 3D inversion scheme. This process optimizes the latent representation of the generative model to produce a super-resolved 3D Gaussian Splatting (3DGS) head model, which is subsequently rigged to an underlying parametric head model (e.g., FLAME) for animation. The inversion is jointly supervised using a sparse collection of upscaled 2D face renderings and corresponding depth maps, captured from diverse facial expressions and camera viewpoints, to ensure realism under dynamic facial motions. Experiments demonstrate that SuperHead generates avatars with fine-grained facial details under dynamic motions, significantly outperforming baseline methods in visual quality.

Aaron Mishkin, Ahmed Khaled, Yuanhao Wang et al. · princeton

We develop new sub-optimality bounds for gradient descent (GD) that depend on the conditioning of the objective along the path of optimization rather than on global, worst-case constants. Key to our proofs is directional smoothness, a measure of gradient variation that we use to develop upper-bounds on the objective. Minimizing these upper-bounds requires solving implicit equations to obtain a sequence of strongly adapted step-sizes; we show that these equations are straightforward to solve for convex quadratics and lead to new guarantees for two classical step-sizes. For general functions, we prove that the Polyak step-size and normalized GD obtain fast, path-dependent rates despite using no knowledge of the directional smoothness. Experiments on logistic regression show our convergence guarantees are tighter than the classical theory based on $L$-smoothness.

Junfeng Wu, Dongliang Luo, Weizhi Zhao et al.

In this work, we reveal the limitations of visual tokenizers and VAEs in preserving fine-grained features, and propose a benchmark to evaluate reconstruction performance for two challenging visual contents: text and face. Visual tokenizers and VAEs have significantly advanced visual generation and multimodal modeling by providing more efficient compressed or quantized image representations. However, while helping production models reduce computational burdens, the information loss from image compression fundamentally limits the upper bound of visual generation quality. To evaluate this upper bound, we focus on assessing reconstructed text and facial features since they typically: 1) exist at smaller scales, 2) contain dense and rich textures, 3) are prone to collapse, and 4) are highly sensitive to human vision. We first collect and curate a diverse set of clear text and face images from existing datasets. Unlike approaches using VLM models, we employ established OCR and face recognition models for evaluation, ensuring accuracy while maintaining an exceptionally lightweight assessment process <span style="font-weight: bold; color: rgb(214, 21, 21);">requiring just 2GB memory and 4 minutes</span> to complete. Using our benchmark, we analyze text and face reconstruction quality across various scales for different image tokenizers and VAEs. Our results show modern visual tokenizers still struggle to preserve fine-grained features, especially at smaller scales. We further extend this evaluation framework to video, conducting comprehensive analysis of video tokenizers. Additionally, we demonstrate that traditional metrics fail to accurately reflect reconstruction performance for faces and text, while our proposed metrics serve as an effective complement.

Shange Tang, Yuanhao Wang, Chi Jin

Elo rating, widely used for skill assessment across diverse domains ranging from competitive games to large language models, is often understood as an incremental update algorithm for estimating a stationary Bradley-Terry (BT) model. However, our empirical analysis of practical matching datasets reveals two surprising findings: (1) Most games deviate significantly from the assumptions of the BT model and stationarity, raising questions on the reliability of Elo. (2) Despite these deviations, Elo frequently outperforms more complex rating systems, such as mElo and pairwise models, which are specifically designed to account for non-BT components in the data, particularly in terms of win rate prediction. This paper explains this unexpected phenomenon through three key perspectives: (a) We reinterpret Elo as an instance of online gradient descent, which provides no-regret guarantees even in misspecified and non-stationary settings. (b) Through extensive synthetic experiments on data generated from transitive but non-BT models, such as strongly or weakly stochastic transitive models, we show that the ''sparsity'' of practical matching data is a critical factor behind Elo's superior performance in prediction compared to more complex rating systems. (c) We observe a strong correlation between Elo's predictive accuracy and its ranking performance, further supporting its effectiveness in ranking.

Johanna Karras, Yuanhao Wang, Yingwei Li et al.

Given a person and a garment image, virtual try-on (VTO) aims to synthesize a realistic image of the person wearing the garment, while preserving their original pose and identity. Although recent VTO methods excel at visualizing garment appearance, they largely overlook a crucial aspect of the try-on experience: the accuracy of garment fit -- for example, depicting how an extra-large shirt looks on an extra-small person. A key obstacle is the absence of datasets that provide precise garment and body size information, particularly for "ill-fit" cases, where garments are significantly too large or too small. Consequently, current VTO methods default to generating well-fitted results regardless of the garment or person size. In this paper, we take the first steps towards solving this open problem. We introduce FIT (Fit-Inclusive Try-on), a large-scale VTO dataset comprising over 1.13M try-on image triplets accompanied by precise body and garment measurements. We overcome the challenges of data collection via a scalable synthetic strategy: (1) We programmatically generate 3D garments using GarmentCode and drape them via physics simulation to capture realistic garment fit. (2) We employ a novel re-texturing framework to transform synthetic renderings into photorealistic images while strictly preserving geometry. (3) We introduce person identity preservation into our re-texturing model to generate paired person images (same person, different garments) for supervised training. Finally, we leverage our FIT dataset to train a baseline fit-aware virtual try-on model. Our data and results set the new state-of-the-art for fit-aware virtual try-on, as well as offer a robust benchmark for future research. We will make all data and code publicly available on our project page: https://johannakarras.github.io/FIT.

Yuanhao Wang, Cheng Zhang, Gonçalo Frazão et al.

We introduce GarmentCrafter, a new approach that enables non-professional users to create and modify 3D garments from a single-view image. While recent advances in image generation have facilitated 2D garment design, creating and editing 3D garments remains challenging for non-professional users. Existing methods for single-view 3D reconstruction often rely on pre-trained generative models to synthesize novel views conditioning on the reference image and camera pose, yet they lack cross-view consistency, failing to capture the internal relationships across different views. In this paper, we tackle this challenge through progressive depth prediction and image warping to approximate novel views. Subsequently, we train a multi-view diffusion model to complete occluded and unknown clothing regions, informed by the evolving camera pose. By jointly inferring RGB and depth, GarmentCrafter enforces inter-view coherence and reconstructs precise geometries and fine details. Extensive experiments demonstrate that our method achieves superior visual fidelity and inter-view coherence compared to state-of-the-art single-view 3D garment reconstruction methods.

Jiawei Ge, Yuanhao Wang, Wenzhe Li et al.

This paper examines multiplayer symmetric constant-sum games with more than two players in a competitive setting, including examples like Mahjong, Poker, and various board and video games. In contrast to two-player zero-sum games, equilibria in multiplayer games are neither unique nor non-exploitable, failing to provide meaningful guarantees when competing against opponents who play different equilibria or non-equilibrium strategies. This gives rise to a series of long-lasting fundamental questions in multiplayer games regarding suitable objectives, solution concepts, and principled algorithms. This paper takes an initial step towards addressing these challenges by focusing on the natural objective of equal share -- securing an expected payoff of C/n in an n-player symmetric game with a total payoff of C. We rigorously identify the theoretical conditions under which achieving an equal share is tractable and design a series of efficient algorithms, inspired by no-regret learning, that provably attain approximate equal share across various settings. Furthermore, we provide complementary lower bounds that justify the sharpness of our theoretical results. Our experimental results highlight worst-case scenarios where meta-algorithms from prior state-of-the-art systems for multiplayer games fail to secure an equal share, while our algorithm succeeds, demonstrating the effectiveness of our approach.

Rui Li, Darius Rückert, Yuanhao Wang et al.

Neural rendering with implicit neural networks has recently emerged as an attractive proposition for scene reconstruction, achieving excellent quality albeit at high computational cost. While the most recent generation of such methods has made progress on the rendering (inference) times, very little progress has been made on improving the reconstruction (training) times. In this work, we present Neural Adaptive Scene Tracing (NAScenT), the first neural rendering method based on directly training a hybrid explicit-implicit neural representation. NAScenT uses a hierarchical octree representation with one neural network per leaf node and combines this representation with a two-stage sampling process that concentrates ray samples where they matter most near object surfaces. As a result, NAScenT is capable of reconstructing challenging scenes including both large, sparsely populated volumes like UAV captured outdoor environments, as well as small scenes with high geometric complexity. NAScenT outperforms existing neural rendering approaches in terms of both quality and training time.

Darius Rückert, Yuanhao Wang, Rui Li et al.

In this paper, we present Neural Adaptive Tomography (NeAT), the first adaptive, hierarchical neural rendering pipeline for multi-view inverse rendering. Through a combination of neural features with an adaptive explicit representation, we achieve reconstruction times far superior to existing neural inverse rendering methods. The adaptive explicit representation improves efficiency by facilitating empty space culling and concentrating samples in complex regions, while the neural features act as a neural regularizer for the 3D reconstruction. The NeAT framework is designed specifically for the tomographic setting, which consists only of semi-transparent volumetric scenes instead of opaque objects. In this setting, NeAT outperforms the quality of existing optimization-based tomography solvers while being substantially faster.

Chi Jin, Qinghua Liu, Yuanhao Wang et al.

A major challenge of multiagent reinforcement learning (MARL) is the curse of multiagents, where the size of the joint action space scales exponentially with the number of agents. This remains to be a bottleneck for designing efficient MARL algorithms even in a basic scenario with finitely many states and actions. This paper resolves this challenge for the model of episodic Markov games. We design a new class of fully decentralized algorithms -- V-learning, which provably learns Nash equilibria (in the two-player zero-sum setting), correlated equilibria and coarse correlated equilibria (in the multiplayer general-sum setting) in a number of samples that only scales with $\max_{i\in[m]} A_i$, where $A_i$ is the number of actions for the $i^{\rm th}$ player. This is in sharp contrast to the size of the joint action space which is $\prod_{i=1}^m A_i$. V-learning (in its basic form) is a new class of single-agent RL algorithms that convert any adversarial bandit algorithm with suitable regret guarantees into a RL algorithm. Similar to the classical Q-learning algorithm, it performs incremental updates to the value functions. Different from Q-learning, it only maintains the estimates of V-values instead of Q-values. This key difference allows V-learning to achieve the claimed guarantees in the MARL setting by simply letting all agents run V-learning independently.

Yuanhao Wang, Ruosong Wang, Sham M. Kakade

A fundamental question in the theory of reinforcement learning is: suppose the optimal $Q$-function lies in the linear span of a given $d$ dimensional feature mapping, is sample-efficient reinforcement learning (RL) possible? The recent and remarkable result of Weisz et al. (2020) resolved this question in the negative, providing an exponential (in $d$) sample size lower bound, which holds even if the agent has access to a generative model of the environment. One may hope that this information theoretic barrier for RL can be circumvented by further supposing an even more favorable assumption: there exists a \emph{constant suboptimality gap} between the optimal $Q$-value of the best action and that of the second-best action (for all states). The hope is that having a large suboptimality gap would permit easier identification of optimal actions themselves, thus making the problem tractable; indeed, provided the agent has access to a generative model, sample-efficient RL is in fact possible with the addition of this more favorable assumption. This work focuses on this question in the standard online reinforcement learning setting, where our main result resolves this question in the negative: our hardness result shows that an exponential sample complexity lower bound still holds even if a constant suboptimality gap is assumed in addition to having a linearly realizable optimal $Q$-function. Perhaps surprisingly, this implies an exponential separation between the online RL setting and the generative model setting. Complementing our negative hardness result, we give two positive results showing that provably sample-efficient RL is possible either under an additional low-variance assumption or under a novel hypercontractivity assumption (both implicitly place stronger conditions on the underlying dynamics model).

Guodong Zhang, Yuanhao Wang, Laurent Lessard et al.

Smooth minimax games often proceed by simultaneous or alternating gradient updates. Although algorithms with alternating updates are commonly used in practice, the majority of existing theoretical analyses focus on simultaneous algorithms for convenience of analysis. In this paper, we study alternating gradient descent-ascent (Alt-GDA) in minimax games and show that Alt-GDA is superior to its simultaneous counterpart~(Sim-GDA) in many settings. We prove that Alt-GDA achieves a near-optimal local convergence rate for strongly convex-strongly concave (SCSC) problems while Sim-GDA converges at a much slower rate. To our knowledge, this is the \emph{first} result of any setting showing that Alt-GDA converges faster than Sim-GDA by more than a constant. We further adapt the theory of integral quadratic constraints (IQC) and show that Alt-GDA attains the same rate \emph{globally} for a subclass of SCSC minimax problems. Empirically, we demonstrate that alternating updates speed up GAN training significantly and the use of optimism only helps for simultaneous algorithms.

Yi Tian, Yuanhao Wang, Tiancheng Yu et al.

We study online learning in unknown Markov games, a problem that arises in episodic multi-agent reinforcement learning where the actions of the opponents are unobservable. We show that in this challenging setting, achieving sublinear regret against the best response in hindsight is statistically hard. We then consider a weaker notion of regret by competing with the \emph{minimax value} of the game, and present an algorithm that achieves a sublinear $\tilde{\mathcal{O}}(K^{2/3})$ regret after $K$ episodes. This is the first sublinear regret bound (to our knowledge) for online learning in unknown Markov games. Importantly, our regret bound is independent of the size of the opponents' action spaces. As a result, even when the opponents' actions are fully observable, our regret bound improves upon existing analysis (e.g., (Xie et al., 2020)) by an exponential factor in the number of opponents.

Yuanhao Wang, Kefan Dong

We consider the adversarial Markov Decision Process (MDP) problem, where the rewards for the MDP can be adversarially chosen, and the transition function can be either known or unknown. In both settings, Follow-the-PerturbedLeader (FPL) based algorithms have been proposed in previous literature. However, the established regret bounds for FPL based algorithms are worse than algorithms based on mirrordescent. We improve the analysis of FPL based algorithms in both settings, matching the current best regret bounds using faster and simpler algorithms.

Guodong Zhang, Yuanhao Wang

Smooth game optimization has recently attracted great interest in machine learning as it generalizes the single-objective optimization paradigm. However, game dynamics is more complex due to the interaction between different players and is therefore fundamentally different from minimization, posing new challenges for algorithm design. Notably, it has been shown that negative momentum is preferred due to its ability to reduce oscillation in game dynamics. Nevertheless, the convergence rate of negative momentum was only established in simple bilinear games. In this paper, we extend the analysis to smooth and strongly-convex strongly-concave minimax games by taking the variational inequality formulation. By connecting momentum method with Chebyshev polynomials, we show that negative momentum accelerates convergence of game dynamics locally, though with a suboptimal rate. To the best of our knowledge, this is the \emph{first work} that provides an explicit convergence rate for negative momentum in this setting.

Yuanhao Wang, Jian Li

This paper studies minimax optimization problems $\min_x \max_y f(x,y)$, where $f(x,y)$ is $m_x$-strongly convex with respect to $x$, $m_y$-strongly concave with respect to $y$ and $(L_x,L_{xy},L_y)$-smooth. Zhang et al. provided the following lower bound of the gradient complexity for any first-order method: $Ω\Bigl(\sqrt{\frac{L_x}{m_x}+\frac{L_{xy}^2}{m_x m_y}+\frac{L_y}{m_y}}\ln(1/ε)\Bigr).$ This paper proposes a new algorithm with gradient complexity upper bound $\tilde{O}\Bigl(\sqrt{\frac{L_x}{m_x}+\frac{L\cdot L_{xy}}{m_x m_y}+\frac{L_y}{m_y}}\ln\left(1/ε\right)\Bigr),$ where $L=\max\{L_x,L_{xy},L_y\}$. This improves over the best known upper bound $\tilde{O}\left(\sqrt{\frac{L^2}{m_x m_y}} \ln^3\left(1/ε\right)\right)$ by Lin et al. Our bound achieves linear convergence rate and tighter dependency on condition numbers, especially when $L_{xy}\ll L$ (i.e., when the interaction between $x$ and $y$ is weak). Via reduction, our new bound also implies improved bounds for strongly convex-concave and convex-concave minimax optimization problems. When $f$ is quadratic, we can further improve the upper bound, which matches the lower bound up to a small sub-polynomial factor.

Yuanhao Wang, Guodong Zhang, Jimmy Ba

Many tasks in modern machine learning can be formulated as finding equilibria in \emph{sequential} games. In particular, two-player zero-sum sequential games, also known as minimax optimization, have received growing interest. It is tempting to apply gradient descent to solve minimax optimization given its popularity and success in supervised learning. However, it has been noted that naive application of gradient descent fails to find some local minimax and can converge to non-local-minimax points. In this paper, we propose \emph{Follow-the-Ridge} (FR), a novel algorithm that provably converges to and only converges to local minimax. We show theoretically that the algorithm addresses the notorious rotational behaviour of gradient dynamics, and is compatible with preconditioning and \emph{positive} momentum. Empirically, FR solves toy minimax problems and improves the convergence of GAN training compared to the recent minimax optimization algorithms.

Yuanhao Wang, Jiachen Hu, Xiaoyu Chen et al.

We study the problem of regret minimization for distributed bandits learning, in which $M$ agents work collaboratively to minimize their total regret under the coordination of a central server. Our goal is to design communication protocols with near-optimal regret and little communication cost, which is measured by the total amount of transmitted data. For distributed multi-armed bandits, we propose a protocol with near-optimal regret and only $O(M\log(MK))$ communication cost, where $K$ is the number of arms. The communication cost is independent of the time horizon $T$, has only logarithmic dependence on the number of arms, and matches the lower bound except for a logarithmic factor. For distributed $d$-dimensional linear bandits, we propose a protocol that achieves near-optimal regret and has communication cost of order $\tilde{O}(Md)$, which has only logarithmic dependence on $T$.

Kefan Dong, Yuanhao Wang, Xiaoyu Chen et al.

A fundamental question in reinforcement learning is whether model-free algorithms are sample efficient. Recently, Jin et al. \cite{jin2018q} proposed a Q-learning algorithm with UCB exploration policy, and proved it has nearly optimal regret bound for finite-horizon episodic MDP. In this paper, we adapt Q-learning with UCB-exploration bonus to infinite-horizon MDP with discounted rewards \emph{without} accessing a generative model. We show that the \textit{sample complexity of exploration} of our algorithm is bounded by $\tilde{O}({\frac{SA}{ε^2(1-γ)^7}})$. This improves the previously best known result of $\tilde{O}({\frac{SA}{ε^4(1-γ)^8}})$ in this setting achieved by delayed Q-learning \cite{strehl2006pac}, and matches the lower bound in terms of $ε$ as well as $S$ and $A$ except for logarithmic factors.