Zhaoran Wang

Xiao-Yang Liu, Ziyi Xia, Hongyang Yang et al.

The financial market is a particularly challenging playground for deep reinforcement learning due to its unique feature of dynamic datasets. Building high-quality market environments for training financial reinforcement learning (FinRL) agents is difficult due to major factors such as the low signal-to-noise ratio of financial data, survivorship bias of historical data, and model overfitting. In this paper, we present FinRL-Meta, a data-centric and openly accessible library that processes dynamic datasets from real-world markets into gym-style market environments and has been actively maintained by the AI4Finance community. First, following a DataOps paradigm, we provide hundreds of market environments through an automatic data curation pipeline. Second, we provide homegrown examples and reproduce popular research papers as stepping stones for users to design new trading strategies. We also deploy the library on cloud platforms so that users can visualize their own results and assess the relative performance via community-wise competitions. Third, we provide dozens of Jupyter/Python demos organized into a curriculum and a documentation website to serve the rapidly growing community. The open-source codes for the data curation pipeline are available at https://github.com/AI4Finance-Foundation/FinRL-Meta

Yifei Min, Tianhao Wang, Ruitu Xu et al.

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.

Jiayang Li, Jing Yu, Qianni Wang et al.

In a Stackelberg congestion game (SCG), a leader aims to maximize their own gain by anticipating and manipulating the equilibrium state at which the followers settle by playing a congestion game. Often formulated as bilevel programs, large-scale SCGs are well known for their intractability and complexity. Here, we attempt to tackle this computational challenge by marrying traditional methodologies with the latest differentiable programming techniques in machine learning. The core idea centers on replacing the lower-level equilibrium problem with a smooth evolution trajectory defined by the imitative logit dynamic (ILD), which we prove converges to the equilibrium of the congestion game under mild conditions. Building upon this theoretical foundation, we propose two new local search algorithms for SCGs. The first is a gradient descent algorithm that obtains the derivatives by unrolling ILD via differentiable programming. Thanks to the smoothness of ILD, the algorithm promises both efficiency and scalability. The second algorithm adds a heuristic twist by cutting short the followers' evolution trajectory. Behaviorally, this means that, instead of anticipating the followers' best response at equilibrium, the leader seeks to approximate that response by only looking ahead a limited number of steps. Our numerical experiments are carried out over various instances of classic SCG applications, ranging from toy benchmarks to large-scale real-world examples. The results show the proposed algorithms are reliable and scalable local solvers that deliver high-quality solutions with greater regularity and significantly less computational effort compared to the many incumbents included in our study.

Boxiang Lyu, Zhaoran Wang, Mladen Kolar et al.

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.

Pangpang Liu, Zhuoran Yang, Zhaoran Wang et al.

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

Ruitu Xu, Yifei Min, Tianhao Wang et al.

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.

Nuoya Xiong, Zhaoran Wang, Zhuoran Yang

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

Rui Ai, Boxiang Lyu, Zhaoran Wang et al.

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 Contextual-LSVI-UCB-Buffer (CLUB) algorithm which achieves $\tilde{O}(H^{5/2}\sqrt{K})$ revenue regret, where $K$ is the number of episodes and $H$ is the length of each episode, when the market noise is known and $\tilde{O}(H^{3}\sqrt{K})$ revenue regret when the noise is unknown with no assumptions on bidders' truthfulness.

Jianqing Fan, Zhaoran Wang, Zhuoran Yang et al.

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

Siyu Chen, Yitan Wang, Zhaoran Wang et al.

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

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.

Yifei Zuo, Yutong Yin, Zhichen Zeng et al.

Transformer architectures have achieved remarkable success in various domains. While efficient alternatives to Softmax Attention have been widely studied, the search for more expressive mechanisms grounded in theoretical insight-even at greater computational cost-has been relatively underexplored. In this work, we bridge this gap by proposing Local Linear Attention (LLA), a novel attention mechanism derived from nonparametric statistics through the lens of test-time regression. First, we show that LLA offers theoretical advantages over Linear and Softmax Attention for associative memory via a bias-variance trade-off analysis. Next, we address its computational challenges and propose two memory-efficient primitives to tackle the $Θ(n^2 d)$ and $Θ(n d^2)$ complexity. We then introduce FlashLLA, a hardware-efficient, blockwise algorithm that enables scalable and parallel computation on modern accelerators. In addition, we implement and profile a customized inference kernel that significantly reduces memory overheads. Finally, we empirically validate the advantages and limitations of LLA on test-time regression, in-context regression, associative recall and state tracking tasks. Experiment results demonstrate that LLA effectively adapts to non-stationarity, outperforming strong baselines in test-time training and in-context learning, and exhibiting promising evidence for its scalability and applicability in large-scale models. Code is available at https://github.com/Yifei-Zuo/Flash-LLA.

Quanquan Gu, Zhaoran Wang, Han Liu

In this paper, we study the estimation of the $k$-dimensional sparse principal subspace of covariance matrix $Σ$ in the high-dimensional setting. We aim to recover the oracle principal subspace solution, i.e., the principal subspace estimator obtained assuming the true support is known a priori. To this end, we propose a family of estimators based on the semidefinite relaxation of sparse PCA with novel regularizations. In particular, under a weak assumption on the magnitude of the population projection matrix, one estimator within this family exactly recovers the true support with high probability, has exact rank-$k$, and attains a $\sqrt{s/n}$ statistical rate of convergence with $s$ being the subspace sparsity level and $n$ the sample size. Compared to existing support recovery results for sparse PCA, our approach does not hinge on the spiked covariance model or the limited correlation condition. As a complement to the first estimator that enjoys the oracle property, we prove that, another estimator within the family achieves a sharper statistical rate of convergence than the standard semidefinite relaxation of sparse PCA, even when the previous assumption on the magnitude of the projection matrix is violated. We validate the theoretical results by numerical experiments on synthetic datasets.

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.

Harsh Mahesheka, Zhixian Xie, Zhaoran Wang et al.

Learning from Demonstrations, particularly from biological experts like humans and animals, often encounters significant data acquisition challenges. While recent approaches leverage internet videos for learning, they require complex, task-specific pipelines to extract and retarget motion data for the agent. In this work, we introduce a language-model-assisted bi-level programming framework that enables a reinforcement learning agent to directly learn its reward from internet videos, bypassing dedicated data preparation. The framework includes two levels: an upper level where a vision-language model (VLM) provides feedback by comparing the learner's behavior with expert videos, and a lower level where a large language model (LLM) translates this feedback into reward updates. The VLM and LLM collaborate within this bi-level framework, using a "chain rule" approach to derive a valid search direction for reward learning. We validate the method for reward learning from YouTube videos, and the results have shown that the proposed method enables efficient reward design from expert videos of biological agents for complex behavior synthesis.

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.

Yixuan Wang, Simon Zhan, Zhilu Wang et al.

In model-based reinforcement learning for safety-critical control systems, it is important to formally certify system properties (e.g., safety, stability) under the learned controller. However, as existing methods typically apply formal verification \emph{after} the controller has been learned, it is sometimes difficult to obtain any certificate, even after many iterations between learning and verification. To address this challenge, we propose a framework that jointly conducts reinforcement learning and formal verification by formulating and solving a novel bilevel optimization problem, which is differentiable by the gradients from the value function and certificates. Experiments on a variety of examples demonstrate the significant advantages of our framework over the model-based stochastic value gradient (SVG) method and the model-free proximal policy optimization (PPO) method in finding feasible controllers with barrier functions and Lyapunov functions that ensure system safety and stability.

Gene Li, Junbo Li, Anmol Kabra et al.

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).

Han Zhong, Zhuoran Yang, Zhaoran Wang et al.

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 myopic, 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 $(π^*, ν^*)$ such that (i) $π^*$ is the optimal policy for the leader when the followers always play their best response, and (ii) $ν^*$ is the best response policy of the followers, which is a Nash equilibrium of the followers' game induced by $π^*$. 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 myopic followers.

Yingjie Fei, Zhuoran Yang, Yudong Chen et al.

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.

Haque Ishfaq, Qiwen Cui, Viet Nguyen et al.

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 $\widetilde{O}(\mathrm{poly}(d_EH)\sqrt{T})$ where $T$ is the time elapsed, $H$ is the planning horizon and $d_E$ is the $\textit{eluder dimension}$ of $\mathcal{F}$. In the linear setting, our algorithm reduces to LSVI-PHE, a variant of RLSVI, that enjoys an $\widetilde{\mathcal{O}}(\sqrt{d^3H^3T})$ regret. We complement the theory with an empirical evaluation across known difficult exploration tasks.

Prashant Khanduri, Siliang Zeng, Mingyi Hong et al.

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 {\aname}~requires $\mathcal{O}(ε^{-3/2})$ iterations (each using ${\cal O}(1)$ samples) to find an $ε$-stationary solution. The $ε$-stationary solution is defined as the point whose squared norm of the gradient of the outer function is less than or equal to $ε$. 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.

Zhuoran Yang, Yufeng Zhang, Yongxin Chen et al.

We consider the optimization problem of minimizing a functional defined over a family of probability distributions, where the objective functional is assumed to possess a variational form. Such a distributional optimization problem arises widely in machine learning and statistics, with Monte-Carlo sampling, variational inference, policy optimization, and generative adversarial network as examples. For this problem, we propose a novel particle-based algorithm, dubbed as variational transport, which approximately performs Wasserstein gradient descent over the manifold of probability distributions via iteratively pushing a set of particles. Specifically, we prove that moving along the geodesic in the direction of functional gradient with respect to the second-order Wasserstein distance is equivalent to applying a pushforward mapping to a probability distribution, which can be approximated accurately by pushing a set of particles. Specifically, in each iteration of variational transport, we first solve the variational problem associated with the objective functional using the particles, whose solution yields the Wasserstein gradient direction. Then we update the current distribution by pushing each particle along the direction specified by such a solution. By characterizing both the statistical error incurred in estimating the Wasserstein gradient and the progress of the optimization algorithm, we prove that when the objective function satisfies a functional version of the Polyak-Łojasiewicz (PL) (Polyak, 1963) and smoothness conditions, variational transport converges linearly to the global minimum of the objective functional up to a certain statistical error, which decays to zero sublinearly as the number of particles goes to infinity.

Zhuoran Yang, Chi Jin, Zhaoran Wang et al.

The classical theory of reinforcement learning (RL) has focused on tabular and linear representations of value functions. Further progress hinges on combining RL with modern function approximators such as kernel functions and deep neural networks, and indeed there have been many empirical successes that have exploited such combinations in large-scale applications. There are profound challenges, however, in developing a theory to support this enterprise, most notably the need to take into consideration the exploration-exploitation tradeoff at the core of RL in conjunction with the computational and statistical tradeoffs that arise in modern function-approximation-based learning systems. We approach these challenges by studying an optimistic modification of the least-squares value iteration algorithm, in the context of the action-value function represented by a kernel function or an overparameterized neural network. We establish both polynomial runtime complexity and polynomial sample complexity for this algorithm, without additional assumptions on the data-generating model. In particular, we prove that the algorithm incurs an $\tilde{\mathcal{O}}(δ_{\mathcal{F}} H^2 \sqrt{T})$ regret, where $δ_{\mathcal{F}}$ characterizes the intrinsic complexity of the function class $\mathcal{F}$, $H$ is the length of each episode, and $T$ is the total number of episodes. Our regret bounds are independent of the number of states, a result which exhibits clearly the benefit of function approximation in RL.

Jiayang Li, Jing Yu, Yu Marco Nie et al.

In a social system, the self-interest of agents can be detrimental to the collective good, sometimes leading to social dilemmas. To resolve such a conflict, a central designer may intervene by either redesigning the system or incentivizing the agents to change their behaviors. To be effective, the designer must anticipate how the agents react to the intervention, which is dictated by their often unknown payoff functions. Therefore, learning about the agents is a prerequisite for intervention. In this paper, we provide a unified framework for learning and intervention in games. We cast the equilibria of games as individual layers and integrate them into an end-to-end optimization framework. To enable the backward propagation through the equilibria of games, we propose two approaches, respectively based on explicit and implicit differentiation. Specifically, we cast the equilibria as the solutions to variational inequalities (VIs). The explicit approach unrolls the projection method for solving VIs, while the implicit approach exploits the sensitivity of the solutions to VIs. At the core of both approaches is the differentiation through a projection operator. Moreover, we establish the correctness of both approaches and identify the conditions under which one approach is more desirable than the other. The analytical results are validated using several real-world problems.

Yining Wang, Yi Chen, Ethan X. Fang et al.

We consider the stochastic contextual bandit problem under the high dimensional linear model. We focus on the case where the action space is finite and random, with each action associated with a randomly generated contextual covariate. This setting finds essential applications such as personalized recommendation, online advertisement, and personalized medicine. However, it is very challenging as we need to balance exploration and exploitation. We propose doubly growing epochs and estimating the parameter using the best subset selection method, which is easy to implement in practice. This approach achieves $ \tilde{\mathcal{O}}(s\sqrt{T})$ regret with high probability, which is nearly independent in the ``ambient'' regression model dimension $d$. We further attain a sharper $\tilde{\mathcal{O}}(\sqrt{sT})$ regret by using the \textsc{SupLinUCB} framework and match the minimax lower bound of low-dimensional linear stochastic bandit problems. Finally, we conduct extensive numerical experiments to demonstrate the applicability and robustness of our algorithms empirically.

Yingjie Fei, Zhuoran Yang, Zhaoran Wang et al.

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.

Lingxiao Wang, Zhuoran Yang, Zhaoran Wang

Empowered by expressive function approximators such as neural networks, deep reinforcement learning (DRL) achieves tremendous empirical successes. However, learning expressive function approximators requires collecting a large dataset (interventional data) by interacting with the environment. Such a lack of sample efficiency prohibits the application of DRL to critical scenarios, e.g., autonomous driving and personalized medicine, since trial and error in the online setting is often unsafe and even unethical. In this paper, we study how to incorporate the dataset (observational data) collected offline, which is often abundantly available in practice, to improve the sample efficiency in the online setting. To incorporate the possibly confounded observational data, we propose the deconfounded optimistic value iteration (DOVI) algorithm, which incorporates the confounded observational data in a provably efficient manner. More specifically, 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 by a multiplicative factor, which decreases towards zero when the confounded observational data are more informative upon the adjustments. Our algorithm and analysis serve as a step towards causal reinforcement learning.

Shuang Qiu, Xiaohan Wei, Zhuoran Yang et al.

We consider online learning for episodic stochastically constrained Markov decision processes (CMDPs), which plays a central role in ensuring the safety of reinforcement learning. Here the loss function can vary arbitrarily across the episodes, and 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 Markov decision processes (MDPs) 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.

Qiaomin Xie, Yudong Chen, Zhaoran Wang et al.

We develop provably efficient reinforcement learning algorithms for two-player zero-sum finite-horizon Markov games with simultaneous moves. To incorporate function approximation, we consider a family of Markov games where the reward function and transition kernel possess a linear structure. Both the offline and online settings of the problems are considered. In the offline setting, we control both players and aim to find the Nash Equilibrium by minimizing the duality gap. In the online setting, we control a single player playing against an arbitrary opponent and aim 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, where $d$ is the linear dimension, $H$ the horizon and $T$ the total number of timesteps. Our results do not require additional assumptions on the sampling model. Our setting requires overcoming several new challenges that are absent in Markov decision processes or turn-based Markov games. In particular, to achieve optimism with simultaneous moves, we construct both upper and lower confidence bounds of the value function, and then compute the optimistic policy by solving a general-sum matrix game with these bounds as the payoff matrices. As finding the Nash Equilibrium of a general-sum game is computationally hard, our algorithm instead solves for a Coarse Correlated Equilibrium (CCE), which can be obtained efficiently. To our best knowledge, such a CCE-based scheme for optimism has not appeared in the literature and might be of interest in its own right.

Lingxiao Wang, Qi Cai, Zhuoran Yang et al.

Policy gradient methods with actor-critic schemes demonstrate tremendous empirical successes, especially when the actors and critics are parameterized by neural networks. However, it remains less clear whether such "neural" policy gradient methods converge to globally optimal policies and whether they even converge at all. We answer both the questions affirmatively in the overparameterized regime. In detail, we prove that neural natural policy gradient converges to a globally optimal policy at a sublinear rate. Also, we show that neural vanilla policy gradient converges sublinearly to a stationary point. Meanwhile, by relating the suboptimality of the stationary points to the representation power of neural actor and critic classes, we prove the global optimality of all stationary points under mild regularity conditions. Particularly, we show that a key to the global optimality and convergence is the "compatibility" between the actor and critic, which is ensured by sharing neural architectures and random initializations across the actor and critic. To the best of our knowledge, our analysis establishes the first global optimality and convergence guarantees for neural policy gradient methods.

Chi Jin, Zhuoran Yang, Zhaoran Wang et al.

Modern Reinforcement Learning (RL) is commonly applied to practical problems with an enormous number of states, where 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.

Yixuan Lin, Kaiqing Zhang, Zhuoran Yang et al.

This paper considers a distributed reinforcement learning problem in which a network of multiple agents aim to cooperatively maximize the globally averaged return through communication with only local neighbors. A randomized communication-efficient multi-agent actor-critic algorithm is proposed for possibly unidirectional communication relationships depicted by a directed graph. It is shown that the algorithm can solve the problem for strongly connected graphs by allowing each agent to transmit only two scalar-valued variables at one time.

Sen Na, Zhuoran Yang, Zhaoran Wang et al.

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.

Kean Ming Tan, Zhaoran Wang, Tong Zhang et al.

Sliced inverse regression is a popular tool for sufficient dimension reduction, which replaces covariates with a minimal set of their linear combinations without loss of information on the conditional distribution of the response given the covariates. The estimated linear combinations include all covariates, making results difficult to interpret and perhaps unnecessarily variable, particularly when the number of covariates is large. In this paper, we propose a convex formulation for fitting sparse sliced inverse regression in high dimensions. Our proposal estimates the subspace of the linear combinations of the covariates directly and performs variable selection simultaneously. We solve the resulting convex optimization problem via the linearized alternating direction methods of multiplier algorithm, and establish an upper bound on the subspace distance between the estimated and the true subspaces. Through numerical studies, we show that our proposal is able to identify the correct covariates in the high-dimensional setting.

Chris Junchi Li, Zhaoran Wang, Han Liu

Solving statistical learning problems often involves nonconvex optimization. Despite the empirical success of nonconvex statistical optimization methods, their global dynamics, especially convergence to the desirable local minima, remain less well understood in theory. In this paper, we propose a new analytic paradigm based on diffusion processes to characterize the global dynamics of nonconvex statistical optimization. As a concrete example, we study stochastic gradient descent (SGD) for the tensor decomposition formulation of independent component analysis. In particular, we cast different phases of SGD into diffusion processes, i.e., solutions to stochastic differential equations. Initialized from an unstable equilibrium, the global dynamics of SGD transit over three consecutive phases: (i) an unstable Ornstein-Uhlenbeck process slowly departing from the initialization, (ii) the solution to an ordinary differential equation, which quickly evolves towards the desirable local minimum, and (iii) a stable Ornstein-Uhlenbeck process oscillating around the desirable local minimum. Our proof techniques are based upon Stroock and Varadhan's weak convergence of Markov chains to diffusion processes, which are of independent interest.

Yuan Xie, Boyi Liu, Qiang Liu et al.

When learning from a batch of logged bandit feedback, the discrepancy between the policy to be learned and the off-policy training data imposes statistical and computational challenges. Unlike classical supervised learning and online learning settings, in batch contextual bandit learning, one only has access to a collection of logged feedback from the actions taken by a historical policy, and expect to learn a policy that takes good actions in possibly unseen contexts. Such a batch learning setting is ubiquitous in online and interactive systems, such as ad platforms and recommendation systems. Existing approaches based on inverse propensity weights, such as Inverse Propensity Scoring (IPS) and Policy Optimizer for Exponential Models (POEM), enjoy unbiasedness but often suffer from large mean squared error. In this work, we introduce a new approach named Maximum Likelihood Inverse Propensity Scoring (MLIPS) for batch learning from logged bandit feedback. Instead of using the given historical policy as the proposal in inverse propensity weights, we estimate a maximum likelihood surrogate policy based on the logged action-context pairs, and then use this surrogate policy as the proposal. We prove that MLIPS is asymptotically unbiased, and moreover, has a smaller nonasymptotic mean squared error than IPS. Such an error reduction phenomenon is somewhat surprising as the estimated surrogate policy is less accurate than the given historical policy. Results on multi-label classification problems and a large- scale ad placement dataset demonstrate the empirical effectiveness of MLIPS. Furthermore, the proposed surrogate policy technique is complementary to existing error reduction techniques, and when combined, is able to consistently boost the performance of several widely used approaches.

Yingxiang Yang, Adams Wei Yu, Zhaoran Wang et al.

We propose a nonparametric method for detecting nonlinear causal relationship within a set of multidimensional discrete time series, by using sparse additive models (SpAMs). We show that, when the input to the SpAM is a $β$-mixing time series, the model can be fitted by first approximating each unknown function with a linear combination of a set of B-spline bases, and then solving a group-lasso-type optimization problem with nonconvex regularization. Theoretically, we characterize the oracle statistical properties of the proposed sparse estimator in function estimation and model selection. Numerically, we propose an efficient pathwise iterative shrinkage thresholding algorithm (PISTA), which tames the nonconvexity and guarantees linear convergence towards the desired sparse estimator with high probability.

Zhuoran Yang, Lin F. Yang, Ethan X. Fang et al.

Existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying "true" statistical models. To address this issue, we take a first step towards taming model misspecification by studying the high-dimensional sparse phase retrieval problem with misspecified link functions. In particular, we propose a simple variant of the thresholded Wirtinger flow algorithm that, given a proper initialization, linearly converges to an estimator with optimal statistical accuracy for a broad family of unknown link functions. We further provide extensive numerical experiments to support our theoretical findings.

Xingguo Li, Junwei Lu, Raman Arora et al.

We propose a general theory for studying the \xl{landscape} of nonconvex \xl{optimization} with underlying symmetric structures \tz{for a class of machine learning problems (e.g., low-rank matrix factorization, phase retrieval, and deep linear neural networks)}. In specific, we characterize the locations of stationary points and the null space of Hessian matrices \xl{of the objective function} via the lens of invariant groups\removed{for associated optimization problems, including low-rank matrix factorization, phase retrieval, and deep linear neural networks}. As a major motivating example, we apply the proposed general theory to characterize the global \xl{landscape} of the \xl{nonconvex optimization in} low-rank matrix factorization problem. In particular, we illustrate how the rotational symmetry group gives rise to infinitely many nonisolated strict saddle points and equivalent global minima of the objective function. By explicitly identifying all stationary points, we divide the entire parameter space into three regions: ($\cR_1$) the region containing the neighborhoods of all strict saddle points, where the objective has negative curvatures; ($\cR_2$) the region containing neighborhoods of all global minima, where the objective enjoys strong convexity along certain directions; and ($\cR_3$) the complement of the above regions, where the gradient has sufficiently large magnitudes. We further extend our result to the matrix sensing problem. Such global landscape implies strong global convergence guarantees for popular iterative algorithms with arbitrary initial solutions.

Xiang Lyu, Will Wei Sun, Zhaoran Wang et al.

We consider the estimation and inference of graphical models that characterize the dependency structure of high-dimensional tensor-valued data. To facilitate the estimation of the precision matrix corresponding to each way of the tensor, we assume the data follow a tensor normal distribution whose covariance has a Kronecker product structure. A critical challenge in the estimation and inference of this model is the fact that its penalized maximum likelihood estimation involves minimizing a non-convex objective function. To address it, this paper makes two contributions: (i) In spite of the non-convexity of this estimation problem, we prove that an alternating minimization algorithm, which iteratively estimates each sparse precision matrix while fixing the others, attains an estimator with an optimal statistical rate of convergence. (ii) We propose a de-biased statistical inference procedure for testing hypotheses on the true support of the sparse precision matrices, and employ it for testing a growing number of hypothesis with false discovery rate (FDR) control. The asymptotic normality of our test statistic and the consistency of FDR control procedure are established. Our theoretical results are backed up by thorough numerical studies and our real applications on neuroimaging studies of Autism spectrum disorder and users' advertising click analysis bring new scientific findings and business insights. The proposed methods are encoded into a publicly available R package Tlasso.

Kean Ming Tan, Zhaoran Wang, Han Liu et al.

Sparse generalized eigenvalue problem (GEP) plays a pivotal role in a large family of high-dimensional statistical models, including sparse Fisher's discriminant analysis, canonical correlation analysis, and sufficient dimension reduction. Sparse GEP involves solving a non-convex optimization problem. Most existing methods and theory in the context of specific statistical models that are special cases of the sparse GEP require restrictive structural assumptions on the input matrices. In this paper, we propose a two-stage computational framework to solve the sparse GEP. At the first stage, we solve a convex relaxation of the sparse GEP. Taking the solution as an initial value, we then exploit a nonconvex optimization perspective and propose the truncated Rayleigh flow method (Rifle) to estimate the leading generalized eigenvector. We show that Rifle converges linearly to a solution with the optimal statistical rate of convergence for many statistical models. Theoretically, our method significantly improves upon the existing literature by eliminating structural assumptions on the input matrices for both stages. To achieve this, our analysis involves two key ingredients: (i) a new analysis of the gradient based method on nonconvex objective functions, and (ii) a fine-grained characterization of the evolution of sparsity patterns along the solution path. Thorough numerical studies are provided to validate the theoretical results.

Zhaoran Wang, Quanquan Gu, Han Liu

We study the fundamental tradeoffs between computational tractability and statistical accuracy for a general family of hypothesis testing problems with combinatorial structures. Based upon an oracle model of computation, which captures the interactions between algorithms and data, we establish a general lower bound that explicitly connects the minimum testing risk under computational budget constraints with the intrinsic probabilistic and combinatorial structures of statistical problems. This lower bound mirrors the classical statistical lower bound by Le Cam (1986) and allows us to quantify the optimal statistical performance achievable given limited computational budgets in a systematic fashion. Under this unified framework, we sharply characterize the statistical-computational phase transition for two testing problems, namely, normal mean detection and sparse principal component detection. For normal mean detection, we consider two combinatorial structures, namely, sparse set and perfect matching. For these problems we identify significant gaps between the optimal statistical accuracy that is achievable under computational tractability constraints and the classical statistical lower bounds. Compared with existing works on computational lower bounds for statistical problems, which consider general polynomial-time algorithms on Turing machines, and rely on computational hardness hypotheses on problems like planted clique detection, we focus on the oracle computational model, which covers a broad range of popular algorithms, and do not rely on unproven hypotheses. Moreover, our result provides an intuitive and concrete interpretation for the intrinsic computational intractability of high-dimensional statistical problems. One byproduct of our result is a lower bound for a strict generalization of the matrix permanent problem, which is of independent interest.

Zhuoran Yang, Zhaoran Wang, Han Liu et al.

We study parameter estimation and asymptotic inference for sparse nonlinear regression. More specifically, we assume the data are given by $y = f( x^\top β^* ) + ε$, where $f$ is nonlinear. To recover $β^*$, 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. In addition, we provide an efficient algorithm that provably converges to a stationary point. We also access the uncertainty of the obtained estimator. Specifically, based on any stationary point of the objective, we construct valid hypothesis tests and confidence intervals for the low dimensional components of the high-dimensional parameter $β^*$. Detailed numerical results are provided to back up our theory.

Xinyang Yi, Zhaoran Wang, Constantine Caramanis et al.

Linear regression studies the problem of estimating a model parameter $β^* \in \mathbb{R}^p$, from $n$ observations $\{(y_i,\mathbf{x}_i)\}_{i=1}^n$ from linear model $y_i = \langle \mathbf{x}_i,β^* \rangle + ε_i$. We consider a significant generalization in which the relationship between $\langle \mathbf{x}_i,β^* \rangle$ and $y_i$ is noisy, quantized to a single bit, potentially nonlinear, noninvertible, as well as unknown. This model is known as the single-index model in statistics, and, among other things, it represents a significant generalization of one-bit compressed sensing. We propose a novel spectral-based estimation procedure and show that we can recover $β^*$ in settings (i.e., classes of link function $f$) where previous algorithms fail. In general, our algorithm requires only very mild restrictions on the (unknown) functional relationship between $y_i$ and $\langle \mathbf{x}_i,β^* \rangle$. We also consider the high dimensional setting where $β^*$ is sparse ,and introduce a two-stage nonconvex framework that addresses estimation challenges in high dimensional regimes where $p \gg n$. For a broad class of link functions between $\langle \mathbf{x}_i,β^* \rangle$ and $y_i$, we establish minimax lower bounds that demonstrate the optimality of our estimators in both the classical and high dimensional regimes.

Zhaoran Wang, Quanquan Gu, Han Liu

Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this paper, we study the statistical limits of convex relaxations. Particularly, we consider two problems: Mean estimation for sparse principal submatrix and edge probability estimation for stochastic block model. We exploit the sum-of-squares relaxation hierarchy to sharply characterize the limits of a broad class of convex relaxations. Our result shows statistical optimality needs to be compromised for achieving computational tractability using convex relaxations. Compared with existing results on computational lower bounds for statistical problems, which consider general polynomial-time algorithms and rely on computational hardness hypotheses on problems like planted clique detection, our theory focuses on a broad class of convex relaxations and does not rely on unproven hypotheses.

Zhaoran Wang, Quanquan Gu, Yang Ning et al.

We provide a general theory of the expectation-maximization (EM) algorithm for inferring high dimensional latent variable models. In particular, we make two contributions: (i) For parameter estimation, we propose a novel high dimensional EM algorithm which naturally incorporates sparsity structure into parameter estimation. With an appropriate initialization, this algorithm converges at a geometric rate and attains an estimator with the (near-)optimal statistical rate of convergence. (ii) Based on the obtained estimator, we propose new inferential procedures for testing hypotheses and constructing confidence intervals for low dimensional components of high dimensional parameters. For a broad family of statistical models, our framework establishes the first computationally feasible approach for optimal estimation and asymptotic inference in high dimensions. Our theory is supported by thorough numerical results.

Zhaoran Wang, Huanran Lu, Han Liu

Sparse principal component analysis (PCA) involves nonconvex optimization for which the global solution is hard to obtain. To address this issue, one popular approach is convex relaxation. However, such an approach may produce suboptimal estimators due to the relaxation effect. To optimally estimate sparse principal subspaces, we propose a two-stage computational framework named "tighten after relax": Within the 'relax' stage, we approximately solve a convex relaxation of sparse PCA with early stopping to obtain a desired initial estimator; For the 'tighten' stage, we propose a novel algorithm called sparse orthogonal iteration pursuit (SOAP), which iteratively refines the initial estimator by directly solving the underlying nonconvex problem. A key concept of this two-stage framework is the basin of attraction. It represents a local region within which the `tighten' stage has desired computational and statistical guarantees. We prove that, the initial estimator obtained from the 'relax' stage falls into such a region, and hence SOAP geometrically converges to a principal subspace estimator which is minimax-optimal within a certain model class. Unlike most existing sparse PCA estimators, our approach applies to the non-spiked covariance models, and adapts to non-Gaussianity as well as dependent data settings. Moreover, through analyzing the computational complexity of the two stages, we illustrate an interesting phenomenon that larger sample size can reduce the total iteration complexity. Our framework motivates a general paradigm for solving many complex statistical problems which involve nonconvex optimization with provable guarantees.