arXiv subjects

Pan Xu

Publications and source records attributed to Pan Xu.

84 records · Page 5Linked to original sources

arXiv2019

An Improved Convergence Analysis of Stochastic Variance-Reduced Policy Gradient

We revisit the stochastic variance-reduced policy gradient (SVRPG) method proposed by Papini et al. (2018) for reinforcement learning. We provide an improved convergence analysis of SVRPG and show that it can find an $ε$-approximate stationary point of the performance function within $O(1/ε^{5/3})$ trajectories. This sample complexity improves upon the best known result $O(1/ε^2)$ by a factor of $O(1/ε^{1/3})$. At the core of our analysis is (i) a tighter upper bound for the variance of importance sampling weights, where we prove that the variance can be controlled by the parameter distance between different policies; and (ii) a fine-grained analysis of the epoch length and batch size parameters such that we can significantly reduce the number of trajectories required in each iteration of SVRPG. We also empirically demonstrate the effectiveness of our theoretical claims of batch sizes on reinforcement learning benchmark tasks.

cs.LG↗

arXiv2018

Sample Efficient Stochastic Variance-Reduced Cubic Regularization Method

We propose a sample efficient stochastic variance-reduced cubic regularization (Lite-SVRC) algorithm for finding the local minimum efficiently in nonconvex optimization. The proposed algorithm achieves a lower sample complexity of Hessian matrix computation than existing cubic regularization based methods. At the heart of our analysis is the choice of a constant batch size of Hessian matrix computation at each iteration and the stochastic variance reduction techniques. In detail, for a nonconvex function with $n$ component functions, Lite-SVRC converges to the local minimum within $\tilde{O}(n+n^{2/3}/ε^{3/2})$ Hessian sample complexity, which is faster than all existing cubic regularization based methods. Numerical experiments with different nonconvex optimization problems conducted on real datasets validate our theoretical results.

math.OC↗

arXiv2018

Balancing Relevance and Diversity in Online Bipartite Matching via Submodularity

In bipartite matching problems, vertices on one side of a bipartite graph are paired with those on the other. In its online variant, one side of the graph is available offline, while the vertices on the other side arrive online. When a vertex arrives, an irrevocable and immediate decision should be made by the algorithm; either match it to an available vertex or drop it. Examples of such problems include matching workers to firms, advertisers to keywords, organs to patients, and so on. Much of the literature focuses on maximizing the total relevance---modeled via total weight---of the matching. However, in many real-world problems, it is also important to consider contributions of diversity: hiring a diverse pool of candidates, displaying a relevant but diverse set of ads, and so on. In this paper, we propose the Online Submodular Bipartite Matching (\osbm) problem, where the goal is to maximize a submodular function $f$ over the set of matched edges. This objective is general enough to capture the notion of both diversity (\emph{e.g.,} a weighted coverage function) and relevance (\emph{e.g.,} the traditional linear function)---as well as many other natural objective functions occurring in practice (\emph{e.g.,} limited total budget in advertising settings). We propose novel algorithms that have provable guarantees and are essentially optimal when restricted to various special cases. We also run experiments on real-world and synthetic datasets to validate our algorithms.

cs.DS↗

arXiv2018

Finding Local Minima via Stochastic Nested Variance Reduction

We propose two algorithms that can find local minima faster than the state-of-the-art algorithms in both finite-sum and general stochastic nonconvex optimization. At the core of the proposed algorithms is $\text{One-epoch-SNVRG}^+$ using stochastic nested variance reduction (Zhou et al., 2018a), which outperforms the state-of-the-art variance reduction algorithms such as SCSG (Lei et al., 2017). In particular, for finite-sum optimization problems, the proposed $\text{SNVRG}^{+}+\text{Neon2}^{\text{finite}}$ algorithm achieves $\tilde{O}(n^{1/2}ε^{-2}+nε_H^{-3}+n^{3/4}ε_H^{-7/2})$ gradient complexity to converge to an $(ε, ε_H)$-second-order stationary point, which outperforms $\text{SVRG}+\text{Neon2}^{\text{finite}}$ (Allen-Zhu and Li, 2017) , the best existing algorithm, in a wide regime. For general stochastic optimization problems, the proposed $\text{SNVRG}^{+}+\text{Neon2}^{\text{online}}$ achieves $\tilde{O}(ε^{-3}+ε_H^{-5}+ε^{-2}ε_H^{-3})$ gradient complexity, which is better than both $\text{SVRG}+\text{Neon2}^{\text{online}}$ (Allen-Zhu and Li, 2017) and Natasha2 (Allen-Zhu, 2017) in certain regimes. Furthermore, we explore the acceleration brought by third-order smoothness of the objective function.

cs.LG↗

arXiv2018

Onion Curve: A Space Filling Curve with Near-Optimal Clustering

Space filling curves (SFCs) are widely used in the design of indexes for spatial and temporal data. Clustering is a key metric for an SFC, that measures how well the curve preserves locality in moving from higher dimensions to a single dimension. We present the {\em onion curve}, an SFC whose clustering performance is provably close to optimal for the cube and near-cube shaped query sets, irrespective of the side length of the query. We show that in contrast, the clustering performance of the widely used Hilbert curve can be far from optimal, even for cube-shaped queries. Since the clustering performance of an SFC is critical to the efficiency of multi-dimensional indexes based on the SFC, the onion curve can deliver improved performance for data structures involving multi-dimensional data.

cs.CG↗

arXiv2018

A PTAS for a Class of Stochastic Dynamic Programs

We develop a framework for obtaining polynomial time approximation schemes (PTAS) for a class of stochastic dynamic programs. Using our framework, we obtain the first PTAS for the following stochastic combinatorial optimization problems: \probemax: We are given a set of $n$ items, each item $i\in [n]$ has a value $X_i$ which is an independent random variable with a known (discrete) distribution $π_i$. We can {\em probe} a subset $P\subseteq [n]$ of items sequentially. Each time after {probing} an item $i$, we observe its value realization, which follows the distribution $π_i$. We can {\em adaptively} probe at most $m$ items and each item can be probed at most once. The reward is the maximum among the $m$ realized values. Our goal is to design an adaptive probing policy such that the expected value of the reward is maximized. To the best of our knowledge, the best known approximation ratio is $1-1/e$, due to Asadpour \etal~\cite{asadpour2015maximizing}. We also obtain PTAS for some generalizations and variants of the problem and some other problems.

cs.DS↗

arXiv2018

Stochastic Variance-Reduced Cubic Regularized Newton Method

We propose a stochastic variance-reduced cubic regularized Newton method for non-convex optimization. At the core of our algorithm is a novel semi-stochastic gradient along with a semi-stochastic Hessian, which are specifically designed for cubic regularization method. We show that our algorithm is guaranteed to converge to an $(ε,\sqrtε)$-approximately local minimum within $\tilde{O}(n^{4/5}/ε^{3/2})$ second-order oracle calls, which outperforms the state-of-the-art cubic regularization algorithms including subsampled cubic regularization. Our work also sheds light on the application of variance reduction technique to high-order non-convex optimization methods. Thorough experiments on various non-convex optimization problems support our theory.

cs.LG↗

arXiv2017

Third-order Smoothness Helps: Even Faster Stochastic Optimization Algorithms for Finding Local Minima

We propose stochastic optimization algorithms that can find local minima faster than existing algorithms for nonconvex optimization problems, by exploiting the third-order smoothness to escape non-degenerate saddle points more efficiently. More specifically, the proposed algorithm only needs $\tilde{O}(ε^{-10/3})$ stochastic gradient evaluations to converge to an approximate local minimum $\mathbf{x}$, which satisfies $\|\nabla f(\mathbf{x})\|_2\leqε$ and $λ_{\min}(\nabla^2 f(\mathbf{x}))\geq -\sqrtε$ in the general stochastic optimization setting, where $\tilde{O}(\cdot)$ hides logarithm polynomial terms and constants. This improves upon the $\tilde{O}(ε^{-7/2})$ gradient complexity achieved by the state-of-the-art stochastic local minima finding algorithms by a factor of $\tilde{O}(ε^{-1/6})$. For nonconvex finite-sum optimization, our algorithm also outperforms the best known algorithms in a certain regime.

math.OC↗

arXiv2017

Allocation Problems in Ride-Sharing Platforms: Online Matching with Offline Reusable Resources

Bipartite matching markets pair agents on one side of a market with agents, items, or contracts on the opposing side. Prior work addresses online bipartite matching markets, where agents arrive over time and are dynamically matched to a known set of disposable resources. In this paper, we propose a new model, Online Matching with (offline) Reusable Resources under Known Adversarial Distributions (OM-RR-KAD), in which resources on the offline side are reusable instead of disposable; that is, once matched, resources become available again at some point in the future. We show that our model is tractable by presenting an LP-based adaptive algorithm that achieves an online competitive ratio of 1/2 - eps for any given eps greater than 0. We also show that no non-adaptive algorithm can achieve a ratio of 1/2 + o(1) based on the same benchmark LP. Through a data-driven analysis on a massive openly-available dataset, we show our model is robust enough to capture the application of taxi dispatching services and ride-sharing systems. We also present heuristics that perform well in practice.

cs.AI↗

arXiv2017

Speeding Up Latent Variable Gaussian Graphical Model Estimation via Nonconvex Optimizations

We study the estimation of the latent variable Gaussian graphical model (LVGGM), where the precision matrix is the superposition of a sparse matrix and a low-rank matrix. In order to speed up the estimation of the sparse plus low-rank components, we propose a sparsity constrained maximum likelihood estimator based on matrix factorization, and an efficient alternating gradient descent algorithm with hard thresholding to solve it. Our algorithm is orders of magnitude faster than the convex relaxation based methods for LVGGM. In addition, we prove that our algorithm is guaranteed to linearly converge to the unknown sparse and low-rank components up to the optimal statistical precision. Experiments on both synthetic and genomic data demonstrate the superiority of our algorithm over the state-of-the-art algorithms and corroborate our theory.

stat.ML↗

arXiv2016

Communication-efficient Distributed Estimation and Inference for Transelliptical Graphical Models

We propose communication-efficient distributed estimation and inference methods for the transelliptical graphical model, a semiparametric extension of the elliptical distribution in the high dimensional regime. In detail, the proposed method distributes the $d$-dimensional data of size $N$ generated from a transelliptical graphical model into $m$ worker machines, and estimates the latent precision matrix on each worker machine based on the data of size $n=N/m$. It then debiases the local estimators on the worker machines and send them back to the master machine. Finally, on the master machine, it aggregates the debiased local estimators by averaging and hard thresholding. We show that the aggregated estimator attains the same statistical rate as the centralized estimator based on all the data, provided that the number of machines satisfies $m \lesssim \min\{N\log d/d,\sqrt{N/(s^2\log d)}\}$, where $s$ is the maximum number of nonzero entries in each column of the latent precision matrix. It is worth noting that our algorithm and theory can be directly applied to Gaussian graphical models, Gaussian copula graphical models and elliptical graphical models, since they are all special cases of transelliptical graphical models. Thorough experiments on synthetic data back up our theory.

stat.ML↗

arXiv2014

Mining Maximal Cliques from an Uncertain Graph

We consider mining dense substructures (maximal cliques) from an uncertain graph, which is a probability distribution on a set of deterministic graphs. For parameter 0 < α < 1, we present a precise definition of an α-maximal clique in an uncertain graph. We present matching upper and lower bounds on the number of α-maximal cliques possible within an uncertain graph. We present an algorithm to enumerate α-maximal cliques in an uncertain graph whose worst-case runtime is near-optimal, and an experimental evaluation showing the practical utility of the algorithm.

cs.DS↗