arXiv subjects

Yingbin Liang

Publications and source records attributed to Yingbin Liang.

At least 109 records · Page 6Linked to original sources

arXiv2022

Lower Bounds and Accelerated Algorithms for Bilevel Optimization

Bilevel optimization has recently attracted growing interests due to its wide applications in modern machine learning problems. Although recent studies have characterized the convergence rate for several such popular algorithms, it is still unclear how much further these convergence rates can be improved. In this paper, we address this fundamental question from two perspectives. First, we provide the first-known lower complexity bounds of $\widetildeΩ(\frac{1}{\sqrt{μ_x}μ_y})$ and $\widetilde Ω\big(\frac{1}{\sqrtε}\min\{\frac{1}{μ_y},\frac{1}{\sqrt{ε^{3}}}\}\big)$ respectively for strongly-convex-strongly-convex and convex-strongly-convex bilevel optimizations. Second, we propose an accelerated bilevel optimizer named AccBiO, for which we provide the first-known complexity bounds without the gradient boundedness assumption (which was made in existing analyses) under the two aforementioned geometries. We also provide significantly tighter upper bounds than the existing complexity when the bounded gradient assumption does hold. We show that AccBiO achieves the optimal results (i.e., the upper and lower bounds match up to logarithmic factors) when the inner-level problem takes a quadratic form with a constant-level condition number. Interestingly, our lower bounds under both geometries are larger than the corresponding optimal complexities of minimax optimization, establishing that bilevel optimization is provably more challenging than minimax optimization.

cs.LG↗

arXiv2021

Provably Faster Algorithms for Bilevel Optimization

Bilevel optimization has been widely applied in many important machine learning applications such as hyperparameter optimization and meta-learning. Recently, several momentum-based algorithms have been proposed to solve bilevel optimization problems faster. However, those momentum-based algorithms do not achieve provably better computational complexity than $\mathcal{\widetilde O}(ε^{-2})$ of the SGD-based algorithm. In this paper, we propose two new algorithms for bilevel optimization, where the first algorithm adopts momentum-based recursive iterations, and the second algorithm adopts recursive gradient estimations in nested loops to decrease the variance. We show that both algorithms achieve the complexity of $\mathcal{\widetilde O}(ε^{-1.5})$, which outperforms all existing algorithms by the order of magnitude. Our experiments validate our theoretical results and demonstrate the superior empirical performance of our algorithms in hyperparameter applications.

cs.LG↗

arXiv2021

Self-Secure Capacity-Achieving Feedback Schemes of Gaussian Multiple-Access Wiretap Channels with Degraded Message Sets

The Schalkwijk-Kailath (SK) scheme, which achieves the capacity of the point-to-point white Gaussian channel with feedback, is secure by itself and also achieves the secrecy capacity of the Gaussian wiretap channel with feedback, i.e., the SK scheme is a self-secure capacity-achieving (SSCA) feedback scheme for the Gaussian wiretap channel. For the multi-user wiretap channels, recently, it has been shown that Ozarow's capacity-achieving feedback scheme for the two-user Gaussian multiple-access channel (GMAC) is the SSCA feedback scheme for the two-user Gaussian multiple-access wiretap channel (GMAC-WT). In this paper, first, we propose a capacity-achieving feedback scheme for the two-user GMAC with degraded message sets (GMAC-DMS), and show that this scheme is the SSCA feedback scheme for the two-user GMAC-WT with degraded message sets (GMAC-WT-DMS). Next, we extend the above scheme to the two-user GMAC-DMS with noncausal channel state information at the transmitters (NCSIT), and show that the extended scheme is capacity-achieving and also a SSCA feedback scheme for the two-user GMAC-WT-DMS with NCSIT. Finally, we derive outer bounds on the secrecy capacity regions of the two-user GMAC-WT-DMS with or without NCSIT, and numerical results show the rate gains by the feedback.

cs.IT↗

arXiv2021

Faster Algorithm and Sharper Analysis for Constrained Markov Decision Process

The problem of constrained Markov decision process (CMDP) is investigated, where an agent aims to maximize the expected accumulated discounted reward subject to multiple constraints on its utilities/costs. A new primal-dual approach is proposed with a novel integration of three ingredients: entropy regularized policy optimizer, dual variable regularizer, and Nesterov's accelerated gradient descent dual optimizer, all of which are critical to achieve a faster convergence. The finite-time error bound of the proposed approach is characterized. Despite the challenge of the nonconcave objective subject to nonconcave constraints, the proposed approach is shown to converge to the global optimum with a complexity of $\tilde{\mathcal O}(1/ε)$ in terms of the optimality gap and the constraint violation, which improves the complexity of the existing primal-dual approach by a factor of $\mathcal O(1/ε)$ \citep{ding2020natural,paternain2019constrained}. This is the first demonstration that nonconcave CMDP problems can attain the complexity lower bound of $\mathcal O(1/ε)$ for convex optimization subject to convex constraints. Our primal-dual approach and non-asymptotic analysis are agnostic to the RL optimizer used, and thus are more flexible for practical applications. More generally, our approach also serves as the first algorithm that provably accelerates constrained nonconvex optimization with zero duality gap by exploiting the geometries such as the gradient dominance condition, for which the existing acceleration methods for constrained convex optimization are not applicable.

math.OC↗

arXiv2021

Bilevel Optimization: Convergence Analysis and Enhanced Design

Bilevel optimization has arisen as a powerful tool for many machine learning problems such as meta-learning, hyperparameter optimization, and reinforcement learning. In this paper, we investigate the nonconvex-strongly-convex bilevel optimization problem. For deterministic bilevel optimization, we provide a comprehensive convergence rate analysis for two popular algorithms respectively based on approximate implicit differentiation (AID) and iterative differentiation (ITD). For the AID-based method, we orderwisely improve the previous convergence rate analysis due to a more practical parameter selection as well as a warm start strategy, and for the ITD-based method we establish the first theoretical convergence rate. Our analysis also provides a quantitative comparison between ITD and AID based approaches. For stochastic bilevel optimization, we propose a novel algorithm named stocBiO, which features a sample-efficient hypergradient estimator using efficient Jacobian- and Hessian-vector product computations. We provide the convergence rate guarantee for stocBiO, and show that stocBiO outperforms the best known computational complexities orderwisely with respect to the condition number $κ$ and the target accuracy $ε$. We further validate our theoretical results and demonstrate the efficiency of bilevel optimization algorithms by the experiments on meta-learning and hyperparameter optimization.

cs.LG↗

arXiv2021

Doubly Robust Off-Policy Actor-Critic: Convergence and Optimality

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

cs.LG↗

arXiv2021

A Unified Off-Policy Evaluation Approach for General Value Function

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

cs.LG↗

arXiv2021

CRPO: A New Approach for Safe Reinforcement Learning with Convergence Guarantee

In safe reinforcement learning (SRL) problems, an agent explores the environment to maximize an expected total reward and meanwhile avoids violation of certain constraints on a number of expected total costs. In general, such SRL problems have nonconvex objective functions subject to multiple nonconvex constraints, and hence are very challenging to solve, particularly to provide a globally optimal policy. Many popular SRL algorithms adopt a primal-dual structure which utilizes the updating of dual variables for satisfying the constraints. In contrast, we propose a primal approach, called constraint-rectified policy optimization (CRPO), which updates the policy alternatingly between objective improvement and constraint satisfaction. CRPO provides a primal-type algorithmic framework to solve SRL problems, where each policy update can take any variant of policy optimization step. To demonstrate the theoretical performance of CRPO, we adopt natural policy gradient (NPG) for each policy update step and show that CRPO achieves an $\mathcal{O}(1/\sqrt{T})$ convergence rate to the global optimal policy in the constrained policy set and an $\mathcal{O}(1/\sqrt{T})$ error bound on constraint satisfaction. This is the first finite-time analysis of primal SRL algorithms with global optimality guarantee. Our empirical results demonstrate that CRPO can outperform the existing primal-dual baseline algorithms significantly.

cs.LG↗

arXiv2021

Gradient Free Minimax Optimization: Variance Reduction and Faster Convergence

Many important machine learning applications amount to solving minimax optimization problems, and in many cases there is no access to the gradient information, but only the function values. In this paper, we focus on such a gradient-free setting, and consider the nonconvex-strongly-concave minimax stochastic optimization problem. In the literature, various zeroth-order (i.e., gradient-free) minimax methods have been proposed, but none of them achieve the potentially feasible computational complexity of $\mathcal{O}(ε^{-3})$ suggested by the stochastic nonconvex minimization theorem. In this paper, we adopt the variance reduction technique to design a novel zeroth-order variance reduced gradient descent ascent (ZO-VRGDA) algorithm. We show that the ZO-VRGDA algorithm achieves the best known query complexity of $\mathcal{O}(κ(d_1 + d_2)ε^{-3})$, which outperforms all previous complexity bound by orders of magnitude, where $d_1$ and $d_2$ denote the dimensions of the optimization variables and $κ$ denotes the condition number. In particular, with a new analysis technique that we develop, our result does not rely on a diminishing or accuracy-dependent stepsize usually required in the existing methods. To our best knowledge, this is the first study of zeroth-order minimax optimization with variance reduction. Experimental results on the black-box distributional robust optimization problem demonstrates the advantageous performance of our new algorithm.

cs.LG↗

arXiv2021

Proximal Gradient Descent-Ascent: Variable Convergence under KŁ Geometry

The gradient descent-ascent (GDA) algorithm has been widely applied to solve minimax optimization problems. In order to achieve convergent policy parameters for minimax optimization, it is important that GDA generates convergent variable sequences rather than convergent sequences of function values or gradient norms. However, the variable convergence of GDA has been proved only under convexity geometries, and there lacks understanding for general nonconvex minimax optimization. This paper fills such a gap by studying the convergence of a more general proximal-GDA for regularized nonconvex-strongly-concave minimax optimization. Specifically, we show that proximal-GDA admits a novel Lyapunov function, which monotonically decreases in the minimax optimization process and drives the variable sequence to a critical point. By leveraging this Lyapunov function and the KŁ geometry that parameterizes the local geometries of general nonconvex functions, we formally establish the variable convergence of proximal-GDA to a critical point $x^*$, i.e., $x_t\to x^*, y_t\to y^*(x^*)$. Furthermore, over the full spectrum of the KŁ-parameterized geometry, we show that proximal-GDA achieves different types of convergence rates ranging from sublinear convergence up to finite-step convergence, depending on the geometry associated with the KŁ parameter. This is the first theoretical result on the variable convergence for nonconvex minimax optimization.

math.OC↗

arXiv2021

Improving Sample Complexity Bounds for (Natural) Actor-Critic Algorithms

The actor-critic (AC) algorithm is a popular method to find an optimal policy in reinforcement learning. In the infinite horizon scenario, the finite-sample convergence rate for the AC and natural actor-critic (NAC) algorithms has been established recently, but under independent and identically distributed (i.i.d.) sampling and single-sample update at each iteration. In contrast, this paper characterizes the convergence rate and sample complexity of AC and NAC under Markovian sampling, with mini-batch data for each iteration, and with actor having general policy class approximation. We show that the overall sample complexity for a mini-batch AC to attain an $ε$-accurate stationary point improves the best known sample complexity of AC by an order of $\mathcal{O}(ε^{-1}\log(1/ε))$, and the overall sample complexity for a mini-batch NAC to attain an $ε$-accurate globally optimal point improves the existing sample complexity of NAC by an order of $\mathcal{O}(ε^{-1}/\log(1/ε))$. Moreover, the sample complexity of AC and NAC characterized in this work outperforms that of policy gradient (PG) and natural policy gradient (NPG) by a factor of $\mathcal{O}((1-γ)^{-3})$ and $\mathcal{O}((1-γ)^{-4}ε^{-1}/\log(1/ε))$, respectively. This is the first theoretical study establishing that AC and NAC attain orderwise performance improvement over PG and NPG under infinite horizon due to the incorporation of critic.

cs.LG↗

arXiv2020

Sample Complexity Bounds for Two Timescale Value-based Reinforcement Learning Algorithms

Two timescale stochastic approximation (SA) has been widely used in value-based reinforcement learning algorithms. In the policy evaluation setting, it can model the linear and nonlinear temporal difference learning with gradient correction (TDC) algorithms as linear SA and nonlinear SA, respectively. In the policy optimization setting, two timescale nonlinear SA can also model the greedy gradient-Q (Greedy-GQ) algorithm. In previous studies, the non-asymptotic analysis of linear TDC and Greedy-GQ has been studied in the Markovian setting, with diminishing or accuracy-dependent stepsize. For the nonlinear TDC algorithm, only the asymptotic convergence has been established. In this paper, we study the non-asymptotic convergence rate of two timescale linear and nonlinear TDC and Greedy-GQ under Markovian sampling and with accuracy-independent constant stepsize. For linear TDC, we provide a novel non-asymptotic analysis and show that it attains an $ε$-accurate solution with the optimal sample complexity of $\mathcal{O}(ε^{-1}\log(1/ε))$ under a constant stepsize. For nonlinear TDC and Greedy-GQ, we show that both algorithms attain $ε$-accurate stationary solution with sample complexity $\mathcal{O}(ε^{-2})$. It is the first non-asymptotic convergence result established for nonlinear TDC under Markovian sampling and our result for Greedy-GQ outperforms the previous result orderwisely by a factor of $\mathcal{O}(ε^{-1}\log(1/ε))$.

cs.LG↗

arXiv2020

Convergence of Meta-Learning with Task-Specific Adaptation over Partial Parameters

Although model-agnostic meta-learning (MAML) is a very successful algorithm in meta-learning practice, it can have high computational cost because it updates all model parameters over both the inner loop of task-specific adaptation and the outer-loop of meta initialization training. A more efficient algorithm ANIL (which refers to almost no inner loop) was proposed recently by Raghu et al. 2019, which adapts only a small subset of parameters in the inner loop and thus has substantially less computational cost than MAML as demonstrated by extensive experiments. However, the theoretical convergence of ANIL has not been studied yet. In this paper, we characterize the convergence rate and the computational complexity for ANIL under two representative inner-loop loss geometries, i.e., strongly-convexity and nonconvexity. Our results show that such a geometric property can significantly affect the overall convergence performance of ANIL. For example, ANIL achieves a faster convergence rate for a strongly-convex inner-loop loss as the number $N$ of inner-loop gradient descent steps increases, but a slower convergence rate for a nonconvex inner-loop loss as $N$ increases. Moreover, our complexity analysis provides a theoretical quantification on the improved efficiency of ANIL over MAML. The experiments on standard few-shot meta-learning benchmarks validate our theoretical findings.

cs.LG↗

arXiv2020

Finite-Time Analysis for Double Q-learning

Although Q-learning is one of the most successful algorithms for finding the best action-value function (and thus the optimal policy) in reinforcement learning, its implementation often suffers from large overestimation of Q-function values incurred by random sampling. The double Q-learning algorithm proposed in~\citet{hasselt2010double} overcomes such an overestimation issue by randomly switching the update between two Q-estimators, and has thus gained significant popularity in practice. However, the theoretical understanding of double Q-learning is rather limited. So far only the asymptotic convergence has been established, which does not characterize how fast the algorithm converges. In this paper, we provide the first non-asymptotic (i.e., finite-time) analysis for double Q-learning. We show that both synchronous and asynchronous double Q-learning are guaranteed to converge to an $ε$-accurate neighborhood of the global optimum by taking $\tildeΩ\left(\left( \frac{1}{(1-γ)^6ε^2}\right)^{\frac{1}ω} +\left(\frac{1}{1-γ}\right)^{\frac{1}{1-ω}}\right)$ iterations, where $ω\in(0,1)$ is the decay parameter of the learning rate, and $γ$ is the discount factor. Our analysis develops novel techniques to derive finite-time bounds on the difference between two inter-connected stochastic processes, which is new to the literature of stochastic approximation.

cs.LG↗

arXiv2020

Non-asymptotic Convergence of Adam-type Reinforcement Learning Algorithms under Markovian Sampling

Despite the wide applications of Adam in reinforcement learning (RL), the theoretical convergence of Adam-type RL algorithms has not been established. This paper provides the first such convergence analysis for two fundamental RL algorithms of policy gradient (PG) and temporal difference (TD) learning that incorporate AMSGrad updates (a standard alternative of Adam in theoretical analysis), referred to as PG-AMSGrad and TD-AMSGrad, respectively. Moreover, our analysis focuses on Markovian sampling for both algorithms. We show that under general nonlinear function approximation, PG-AMSGrad with a constant stepsize converges to a neighborhood of a stationary point at the rate of $\mathcal{O}(1/T)$ (where $T$ denotes the number of iterations), and with a diminishing stepsize converges exactly to a stationary point at the rate of $\mathcal{O}(\log^2 T/\sqrt{T})$. Furthermore, under linear function approximation, TD-AMSGrad with a constant stepsize converges to a neighborhood of the global optimum at the rate of $\mathcal{O}(1/T)$, and with a diminishing stepsize converges exactly to the global optimum at the rate of $\mathcal{O}(\log T/\sqrt{T})$. Our study develops new techniques for analyzing the Adam-type RL algorithms under Markovian sampling.

cs.LG↗

arXiv2020

Spectral Algorithms for Community Detection in Directed Networks

Community detection in large social networks is affected by degree heterogeneity of nodes. The D-SCORE algorithm for directed networks was introduced to reduce this effect by taking the element-wise ratios of the singular vectors of the adjacency matrix before clustering. Meaningful results were obtained for the statistician citation network, but rigorous analysis on its performance was missing. First, this paper establishes theoretical guarantee for this algorithm and its variants for the directed degree-corrected block model (Directed-DCBM). Second, this paper provides significant improvements for the original D-SCORE algorithms by attaching the nodes outside of the community cores using the information of the original network instead of the singular vectors.

stat.ML↗

arXiv2020

Momentum Q-learning with Finite-Sample Convergence Guarantee

Existing studies indicate that momentum ideas in conventional optimization can be used to improve the performance of Q-learning algorithms. However, the finite-sample analysis for momentum-based Q-learning algorithms is only available for the tabular case without function approximations. This paper analyzes a class of momentum-based Q-learning algorithms with finite-sample guarantee. Specifically, we propose the MomentumQ algorithm, which integrates the Nesterov's and Polyak's momentum schemes, and generalizes the existing momentum-based Q-learning algorithms. For the infinite state-action space case, we establish the convergence guarantee for MomentumQ with linear function approximations and Markovian sampling. In particular, we characterize the finite-sample convergence rate which is provably faster than the vanilla Q-learning. This is the first finite-sample analysis for momentum-based Q-learning algorithms with function approximations. For the tabular case under synchronous sampling, we also obtain a finite-sample convergence rate that is slightly better than the SpeedyQ \citep{azar2011speedy} when choosing a special family of step sizes. Finally, we demonstrate through various experiments that the proposed MomentumQ outperforms other momentum-based Q-learning algorithms.

cs.LG↗

arXiv2020

History-Gradient Aided Batch Size Adaptation for Variance Reduced Algorithms

Variance-reduced algorithms, although achieve great theoretical performance, can run slowly in practice due to the periodic gradient estimation with a large batch of data. Batch-size adaptation thus arises as a promising approach to accelerate such algorithms. However, existing schemes either apply prescribed batch-size adaption rule or exploit the information along optimization path via additional backtracking and condition verification steps. In this paper, we propose a novel scheme, which eliminates backtracking line search but still exploits the information along optimization path by adapting the batch size via history stochastic gradients. We further theoretically show that such a scheme substantially reduces the overall complexity for popular variance-reduced algorithms SVRG and SARAH/SPIDER for both conventional nonconvex optimization and reinforcement learning problems. To this end, we develop a new convergence analysis framework to handle the dependence of the batch size on history stochastic gradients. Extensive experiments validate the effectiveness of the proposed batch-size adaptation scheme.

math.OC↗