arXiv subjects

Mengdi Wang

Publications and source records attributed to Mengdi Wang.

At least 271 records · Page 15Linked to original sources

arXiv2019

A Single Time-Scale Stochastic Approximation Method for Nested Stochastic Optimization

We study constrained nested stochastic optimization problems in which the objective function is a composition of two smooth functions whose exact values and derivatives are not available. We propose a single time-scale stochastic approximation algorithm, which we call the Nested Averaged Stochastic Approximation (NASA), to find an approximate stationary point of the problem. The algorithm has two auxiliary averaged sequences (filters) which estimate the gradient of the composite objective function and the inner function value. By using a special Lyapunov function, we show that NASA achieves the sample complexity of ${\cal O}(1/ε^{2})$ for finding an $ε$-approximate stationary point, thus outperforming all extant methods for nested stochastic approximation. Our method and its analysis are the same for both unconstrained and constrained problems, without any need of batch samples for constrained nonconvex stochastic optimization. We also present a simplified variant of the NASA method for solving constrained single level stochastic optimization problems, and we prove the same complexity result for both unconstrained and constrained problems.

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arXiv2019

Solving Discounted Stochastic Two-Player Games with Near-Optimal Time and Sample Complexity

In this paper, we settle the sampling complexity of solving discounted two-player turn-based zero-sum stochastic games up to polylogarithmic factors. Given a stochastic game with discount factor $γ\in(0,1)$ we provide an algorithm that computes an $ε$-optimal strategy with high-probability given $\tilde{O}((1 - γ)^{-3} ε^{-2})$ samples from the transition function for each state-action-pair. Our algorithm runs in time nearly linear in the number of samples and uses space nearly linear in the number of state-action pairs. As stochastic games generalize Markov decision processes (MDPs) our runtime and sample complexities are optimal due to Azar et al (2013). We achieve our results by showing how to generalize a near-optimal Q-learning based algorithms for MDP, in particular Sidford et al (2018), to two-player strategy computation algorithms. This overcomes limitations of standard Q-learning and strategy iteration or alternating minimization based approaches and we hope will pave the way for future reinforcement learning results by facilitating the extension of MDP results to multi-agent settings with little loss.

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arXiv2019

Improved Oracle Complexity of Variance Reduced Methods for Nonsmooth Convex Stochastic Composition Optimization

We consider the nonsmooth convex composition optimization problem where the objective is a composition of two finite-sum functions and analyze stochastic compositional variance reduced gradient (SCVRG) methods for them. SCVRG and its variants have recently drawn much attention given their edge over stochastic compositional gradient descent (SCGD); but the theoretical analysis exclusively assumes strong convexity of the objective, which excludes several important examples such as Lasso, logistic regression, principle component analysis and deep neural nets. In contrast, we prove non-asymptotic incremental first-order oracle (IFO) complexity of SCVRG or its novel variants for nonsmooth convex composition optimization and show that they are provably faster than SCGD and gradient descent. More specifically, our method achieves the total IFO complexity of $O\left((m+n)\log\left(1/ε\right)+1/ε^3\right)$ which improves that of $O\left(1/ε^{3.5}\right)$ and $O\left((m+n)/\sqrtε\right)$ obtained by SCGD and accelerated gradient descent (AGD) respectively. Experimental results confirm that our methods outperform several existing methods, e.g., SCGD and AGD, on sparse mean-variance optimization problem.

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arXiv2019

Reinforcement Learning in Feature Space: Matrix Bandit, Kernels, and Regret Bound

Exploration in reinforcement learning (RL) suffers from the curse of dimensionality when the state-action space is large. A common practice is to parameterize the high-dimensional value and policy functions using given features. However existing methods either have no theoretical guarantee or suffer a regret that is exponential in the planning horizon $H$. In this paper, we propose an online RL algorithm, namely the MatrixRL, that leverages ideas from linear bandit to learn a low-dimensional representation of the probability transition model while carefully balancing the exploitation-exploration tradeoff. We show that MatrixRL achieves a regret bound ${O}\big(H^2d\log T\sqrt{T}\big)$ where $d$ is the number of features. MatrixRL has an equivalent kernelized version, which is able to work with an arbitrary kernel Hilbert space without using explicit features. In this case, the kernelized MatrixRL satisfies a regret bound ${O}\big(H^2\widetilde{d}\log T\sqrt{T}\big)$, where $\widetilde{d}$ is the effective dimension of the kernel space. To our best knowledge, for RL using features or kernels, our results are the first regret bounds that are near-optimal in time $T$ and dimension $d$ (or $\widetilde{d}$) and polynomial in the planning horizon $H$.

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arXiv2019

Near-Optimal Time and Sample Complexities for Solving Discounted Markov Decision Process with a Generative Model

In this paper we consider the problem of computing an $ε$-optimal policy of a discounted Markov Decision Process (DMDP) provided we can only access its transition function through a generative sampling model that given any state-action pair samples from the transition function in $O(1)$ time. Given such a DMDP with states $S$, actions $A$, discount factor $γ\in(0,1)$, and rewards in range $[0, 1]$ we provide an algorithm which computes an $ε$-optimal policy with probability $1 - δ$ where \emph{both} the time spent and number of sample taken are upper bounded by \[ O\left[\frac{|S||A|}{(1-γ)^3 ε^2} \log \left(\frac{|S||A|}{(1-γ)δε} \right) \log\left(\frac{1}{(1-γ)ε}\right)\right] ~. \] For fixed values of $ε$, this improves upon the previous best known bounds by a factor of $(1 - γ)^{-1}$ and matches the sample complexity lower bounds proved in Azar et al. (2013) up to logarithmic factors. We also extend our method to computing $ε$-optimal policies for finite-horizon MDP with a generative model and provide a nearly matching sample complexity lower bound.

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arXiv2019

Sample-Optimal Parametric Q-Learning Using Linearly Additive Features

Consider a Markov decision process (MDP) that admits a set of state-action features, which can linearly express the process's probabilistic transition model. We propose a parametric Q-learning algorithm that finds an approximate-optimal policy using a sample size proportional to the feature dimension $K$ and invariant with respect to the size of the state space. To further improve its sample efficiency, we exploit the monotonicity property and intrinsic noise structure of the Bellman operator, provided the existence of anchor state-actions that imply implicit non-negativity in the feature space. We augment the algorithm using techniques of variance reduction, monotonicity preservation, and confidence bounds. It is proved to find a policy which is $ε$-optimal from any initial state with high probability using $\widetilde{O}(K/ε^2(1-γ)^3)$ sample transitions for arbitrarily large-scale MDP with a discount factor $γ\in(0,1)$. A matching information-theoretical lower bound is proved, confirming the sample optimality of the proposed method with respect to all parameters (up to polylog factors).

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arXiv2019

Feature-Based Q-Learning for Two-Player Stochastic Games

Consider a two-player zero-sum stochastic game where the transition function can be embedded in a given feature space. We propose a two-player Q-learning algorithm for approximating the Nash equilibrium strategy via sampling. The algorithm is shown to find an $ε$-optimal strategy using sample size linear to the number of features. To further improve its sample efficiency, we develop an accelerated algorithm by adopting techniques such as variance reduction, monotonicity preservation and two-sided strategy approximation. We prove that the algorithm is guaranteed to find an $ε$-optimal strategy using no more than $\tilde{\mathcal{O}}(K/(ε^{2}(1-γ)^{4}))$ samples with high probability, where $K$ is the number of features and $γ$ is a discount factor. The sample, time and space complexities of the algorithm are independent of original dimensions of the game.

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arXiv2019

Towards Coherent and Cohesive Long-form Text Generation

Generating coherent and cohesive long-form texts is a challenging task. Previous works relied on large amounts of human-generated texts to train neural language models. However, few attempted to explicitly improve neural language models from the perspectives of coherence and cohesion. In this work, we propose a new neural language model that is equipped with two neural discriminators which provide feedback signals at the levels of sentence (cohesion) and paragraph (coherence). Our model is trained using a simple yet efficient variant of policy gradient, called negative-critical sequence training, which is proposed to eliminate the need of training a separate critic for estimating baseline. Results demonstrate the effectiveness of our approach, showing improvements over the strong baseline -- recurrent attention-based bidirectional MLE-trained neural language model.

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arXiv2019

RL4health: Crowdsourcing Reinforcement Learning for Knee Replacement Pathway Optimization

Joint replacement is the most common inpatient surgical treatment in the US. We investigate the clinical pathway optimization for knee replacement, which is a sequential decision process from onset to recovery. Based on episodic claims from previous cases, we view the pathway optimization as an intelligence crowdsourcing problem and learn the optimal decision policy from data by imitating the best expert at every intermediate state. We develop a reinforcement learning-based pipeline that uses value iteration, state compression and aggregation learning, kernel representation and cross validation to predict the best treatment policy. It also provides forecast of the clinical pathway under the optimized policy. Empirical validation shows that the optimized policy reduces the overall cost by 7 percent and reduces the excessive cost premium by 33 percent.

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arXiv2019

Learning to Control in Metric Space with Optimal Regret

We study online reinforcement learning for finite-horizon deterministic control systems with {\it arbitrary} state and action spaces. Suppose that the transition dynamics and reward function is unknown, but the state and action space is endowed with a metric that characterizes the proximity between different states and actions. We provide a surprisingly simple upper-confidence reinforcement learning algorithm that uses a function approximation oracle to estimate optimistic Q functions from experiences. We show that the regret of the algorithm after $K$ episodes is $O(HL(KH)^{\frac{d-1}{d}}) $ where $L$ is a smoothness parameter, and $d$ is the doubling dimension of the state-action space with respect to the given metric. We also establish a near-matching regret lower bound. The proposed method can be adapted to work for more structured transition systems, including the finite-state case and the case where value functions are linear combinations of features, where the method also achieve the optimal regret.

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arXiv2018

Graph-Adaptive Pruning for Efficient Inference of Convolutional Neural Networks

In this work, we propose a graph-adaptive pruning (GAP) method for efficient inference of convolutional neural networks (CNNs). In this method, the network is viewed as a computational graph, in which the vertices denote the computation nodes and edges represent the information flow. Through topology analysis, GAP is capable of adapting to different network structures, especially the widely used cross connections and multi-path data flow in recent novel convolutional models. The models can be adaptively pruned at vertex-level as well as edge-level without any post-processing, thus GAP can directly get practical model compression and inference speed-up. Moreover, it does not need any customized computation library or hardware support. Finetuning is conducted after pruning to restore the model performance. In the finetuning step, we adopt a self-taught knowledge distillation (KD) strategy by utilizing information from the original model, through which, the performance of the optimized model can be sufficiently improved, without introduction of any other teacher model. Experimental results show the proposed GAP can achieve promising result to make inference more efficient, e.g., for ResNeXt-29 on CIFAR10, it can get 13X model compression and 4.3X practical speed-up with marginal loss of accuracy.

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arXiv2018

Adaptive Low-Nonnegative-Rank Approximation for State Aggregation of Markov Chains

This paper develops a low-nonnegative-rank approximation method to identify the state aggregation structure of a finite-state Markov chain under an assumption that the state space can be mapped into a handful of meta-states. The number of meta-states is characterized by the nonnegative rank of the Markov transition matrix. Motivated by the success of the nuclear norm relaxation in low rank minimization problems, we propose an atomic regularizer as a convex surrogate for the nonnegative rank and formulate a convex optimization problem. Because the atomic regularizer itself is not computationally tractable, we instead solve a sequence of problems involving a nonnegative factorization of the Markov transition matrices by using the proximal alternating linearized minimization method. Two methods for adjusting the rank of factorization are developed so that local minima are escaped. One is to append an additional column to the factorized matrices, which can be interpreted as an approximation of a negative subgradient step. The other is to reduce redundant dimensions by means of linear combinations. Overall, the proposed algorithm very likely converges to the global solution. The efficiency and statistical properties of our approach are illustrated on synthetic data. We also apply our state aggregation algorithm on a Manhattan transportation data set and make extensive comparisons with an existing method.

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arXiv2018

Dimensionality Reduction for Stationary Time Series via Stochastic Nonconvex Optimization

Stochastic optimization naturally arises in machine learning. Efficient algorithms with provable guarantees, however, are still largely missing, when the objective function is nonconvex and the data points are dependent. This paper studies this fundamental challenge through a streaming PCA problem for stationary time series data. Specifically, our goal is to estimate the principle component of time series data with respect to the covariance matrix of the stationary distribution. Computationally, we propose a variant of Oja's algorithm combined with downsampling to control the bias of the stochastic gradient caused by the data dependency. Theoretically, we quantify the uncertainty of our proposed stochastic algorithm based on diffusion approximations. This allows us to prove the asymptotic rate of convergence and further implies near optimal asymptotic sample complexity. Numerical experiments are provided to support our analysis.

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arXiv2018

Diffusion Approximations for Online Principal Component Estimation and Global Convergence

In this paper, we propose to adopt the diffusion approximation tools to study the dynamics of Oja's iteration which is an online stochastic gradient descent method for the principal component analysis. Oja's iteration maintains a running estimate of the true principal component from streaming data and enjoys less temporal and spatial complexities. We show that the Oja's iteration for the top eigenvector generates a continuous-state discrete-time Markov chain over the unit sphere. We characterize the Oja's iteration in three phases using diffusion approximation and weak convergence tools. Our three-phase analysis further provides a finite-sample error bound for the running estimate, which matches the minimax information lower bound for principal component analysis under the additional assumption of bounded samples.

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arXiv2018

Estimation of Markov Chain via Rank-Constrained Likelihood

This paper studies the estimation of low-rank Markov chains from empirical trajectories. We propose a non-convex estimator based on rank-constrained likelihood maximization. Statistical upper bounds are provided for the Kullback-Leiber divergence and the $\ell_2$ risk between the estimator and the true transition matrix. The estimator reveals a compressed state space of the Markov chain. We also develop a novel DC (difference of convex function) programming algorithm to tackle the rank-constrained non-smooth optimization problem. Convergence results are established. Experiments show that the proposed estimator achieves better empirical performance than other popular approaches.

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arXiv2018

Scalable Bilinear $π$ Learning Using State and Action Features

Approximate linear programming (ALP) represents one of the major algorithmic families to solve large-scale Markov decision processes (MDP). In this work, we study a primal-dual formulation of the ALP, and develop a scalable, model-free algorithm called bilinear $π$ learning for reinforcement learning when a sampling oracle is provided. This algorithm enjoys a number of advantages. First, it adopts (bi)linear models to represent the high-dimensional value function and state-action distributions, using given state and action features. Its run-time complexity depends on the number of features, not the size of the underlying MDPs. Second, it operates in a fully online fashion without having to store any sample, thus having minimal memory footprint. Third, we prove that it is sample-efficient, solving for the optimal policy to high precision with a sample complexity linear in the dimension of the parameter space.

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arXiv2018

Approximation Methods for Bilevel Programming

In this paper, we study a class of bilevel programming problem where the inner objective function is strongly convex. More specifically, under some mile assumptions on the partial derivatives of both inner and outer objective functions, we present an approximation algorithm for solving this class of problem and provide its finite-time convergence analysis under different convexity assumption on the outer objective function. We also present an accelerated variant of this method which improves the rate of convergence under convexity assumption. Furthermore, we generalize our results under stochastic setting where only noisy information of both objective functions is available. To the best of our knowledge, this is the first time that such (stochastic) approximation algorithms with established iteration complexity (sample complexity) are provided for bilevel programming.

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arXiv2018

Multi-Level Stochastic Gradient Methods for Nested Composition Optimization

Stochastic gradient methods are scalable for solving large-scale optimization problems that involve empirical expectations of loss functions. Existing results mainly apply to optimization problems where the objectives are one- or two-level expectations. In this paper, we consider the multi-level compositional optimization problem that involves compositions of multi-level component functions and nested expectations over a random path. It finds applications in risk-averse optimization and sequential planning. We propose a class of multi-level stochastic gradient methods that are motivated from the method of multi-timescale stochastic approximation. First we propose a basic $T$-level stochastic compositional gradient algorithm, establish its almost sure convergence and obtain an $n$-iteration error bound $O (n^{-1/2^T})$. Then we develop accelerated multi-level stochastic gradient methods by using an extrapolation-interpolation scheme to take advantage of the smoothness of individual component functions. When all component functions are smooth, we show that the convergence rate improves to $O(n^{-4/(7+T)})$ for general objectives and $O (n^{-4/(3+T)})$ for strongly convex objectives. We also provide almost sure convergence and rate of convergence results for nonconvex problems. The proposed methods and theoretical results are validated using numerical experiments.

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