Maximum a posteriori (MAP) inference in discrete-valued Markov random fields is a fundamental problem in machine learning that involves identifying the most likely configuration of random variables given a distribution. Due to the difficulty of this combinatorial problem, linear programming (LP) relaxations are commonly used to derive specialized message passing algorithms that are often interpreted as coordinate descent on the dual LP. To achieve more desirable computational properties, a number of methods regularize the LP with an entropy term, leading to a class of smooth message passing algorithms with convergence guarantees. In this paper, we present randomized methods for accelerating these algorithms by leveraging techniques that underlie classical accelerated gradient methods. The proposed algorithms incorporate the familiar steps of standard smooth message passing algorithms, which can be viewed as coordinate minimization steps. We show that these accelerated variants achieve faster rates for finding $ε$-optimal points of the unregularized problem, and, when the LP is tight, we prove that the proposed algorithms recover the true MAP solution in fewer iterations than standard message passing algorithms.
Maximum a posteriori (MAP) inference is a fundamental computational paradigm for statistical inference. In the setting of graphical models, MAP inference entails solving a combinatorial optimization problem to find the most likely configuration of the discrete-valued model. Linear programming (LP) relaxations in the Sherali-Adams hierarchy are widely used to attempt to solve this problem, and smooth message passing algorithms have been proposed to solve regularized versions of these LPs with great success. This paper leverages recent work in entropy-regularized LPs to analyze convergence rates…
NTT DATA Mathematical Systems Inc.（NTT DATA 数理系统公司）；The University of Tokyo（东京大学）
In this paper, we develop a new accelerated stochastic gradient method for efficiently solving the convex regularized empirical risk minimization problem in mini-batch settings. The use of mini-batches is becoming a golden standard in the machine learning community, because mini-batch settings stabilize the gradient estimate and can easily make good use of parallel computing. The core of our proposed method is the incorporation of our new "double acceleration" technique and variance reduction technique. We theoretically analyze our proposed method and show that our method much improves the min…
Department of Mathematics, University of California, Los Angeles, CA 90095（加州大学洛杉矶分校数学系）；Department of Mathematics, University of South Carolina, Columbia, SC 29208（南卡罗来纳大学数学系）
In modern large-scale machine learning applications, the training data are often partitioned and stored on multiple machines. It is customary to employ the "data parallelism" approach, where the aggregated training loss is minimized without moving data across machines. In this paper, we introduce a novel distributed dual formulation for regularized loss minimization problems that can directly handle data parallelism in the distributed setting. This formulation allows us to systematically derive dual coordinate optimization procedures, which we refer to as Distributed Alternating Dual Maximizat…
University of Alberta（阿尔伯塔大学）；University of Chicago（芝加哥大学）；Purdue University（普渡大学）
Nesterov's accelerated gradient methods (AGM) have been successfully applied in many machine learning areas. However, their empirical performance on training max-margin models has been inferior to existing specialized solvers. In this paper, we first extend AGM to strongly convex and composite objective functions with Bregman style prox-functions. Our unifying framework covers both the $\infty$-memory and 1-memory styles of AGM, tunes the Lipschiz constant adaptively, and bounds the duality gap. Then we demonstrate various ways to apply this framework of methods to a wide range of machine lear…
Hong Kong University of Science and Technology（香港科技大学）
In regularized risk minimization, the associated optimization problem becomes particularly difficult when both the loss and regularizer are nonsmooth. Existing approaches either have slow or unclear convergence properties, are restricted to limited problem subclasses, or require careful setting of a smoothing parameter. In this paper, we propose a continuation algorithm that is applicable to a large class of nonsmooth regularized risk minimization problems, can be flexibly used with a number of existing solvers for the underlying smoothed subproblem, and with convergence results on the whole a…
Regularized nonlinear acceleration (RNA) estimates the minimum of a function by post-processing iterates from an algorithm such as the gradient method. It can be seen as a regularized version of Anderson acceleration, a classical acceleration scheme from numerical analysis. The new scheme provably improves the rate of convergence of fixed step gradient descent, and its empirical performance is comparable to that of quasi-Newton methods. However, RNA cannot accelerate faster multistep algorithms like Nesterov's method and often diverges in this context. Here, we adapt RNA to overcome these issu…
Stochastic dual coordinate ascent (SDCA) is an effective technique for solving regularized loss minimization problems in machine learning. This paper considers an extension of SDCA under the mini-batch setting that is often used in practice. Our main contribution is to introduce an accelerated mini-batch version of SDCA and prove a fast convergence rate for this method. We discuss an implementation of our method over a parallel computing system, and compare the results to both the vanilla stochastic dual coordinate ascent and to the accelerated deterministic gradient descent method of \cite{ne…
Accelerated Algorithm for Sparse Regularized Partial Optimal Transport
Hanoi University of Science and Technology（河内理工大学）；High School for the Gifted（天才高中）；VinUniversity；Rochambeau French International School（罗尚博法语国际学校）；RMIT University（皇家墨尔本理工大学）；Aalto University（阿尔托大学）；Linköping University（林雪平大学）
Optimization plays a key role in machine learning. Recently, stochastic second-order methods have attracted much attention due to their low computational cost in each iteration. However, these algorithms might perform poorly especially if it is hard to approximate the Hessian well and efficiently. As far as we know, there is no effective way to handle this problem. In this paper, we resort to Nesterov's acceleration technique to improve the convergence performance of a class of second-order methods called approximate Newton. We give a theoretical analysis that Nesterov's acceleration technique…
Optimization plays a key role in machine learning. Recently, stochastic second-order methods have attracted much attention due to their low computational cost in each iteration. However, these algorithms might perform poorly especially if it is hard to approximate the Hessian well and efficiently. As far as we know, there is no effective way to handle this problem. In this paper, we resort to Nestrov's acceleration technique to improve the convergence performance of a class of second-order methods called approximate Newton. We give a theoretical analysis that Nestrov's acceleration technique c…
University of California, Berkeley（加州大学伯克利分校）；University of Texas, Austin（德克萨斯大学奥斯汀分校）
We present several new complexity results for the entropic regularized algorithms that approximately solve the optimal transport (OT) problem between two discrete probability measures with at most $n$ atoms. First, we improve the complexity bound of a greedy variant of Sinkhorn, known as \textit{Greenkhorn}, from $\widetilde{O}(n^2\varepsilon^{-3})$ to $\widetilde{O}(n^2\varepsilon^{-2})$. Notably, our result can match the best known complexity bound of Sinkhorn and help clarify why Greenkhorn significantly outperforms Sinkhorn in practice in terms of row/column updates as observed by~\citet{A…
Georgia Institute of Technology（佐治亚理工学院）；Virginia Tech（弗吉尼亚理工大学）
We study the problem of finding the Nash equilibrium in a two-player zero-sum Markov game. Due to its formulation as a minimax optimization program, a natural approach to solve the problem is to perform gradient descent/ascent with respect to each player in an alternating fashion. However, due to the non-convexity/non-concavity of the underlying objective function, theoretical understandings of this method are limited. In our paper, we consider solving an entropy-regularized variant of the Markov game. The regularization introduces structure into the optimization landscape that make the soluti…
Soochow University（苏州大学）；University of Science and Technology of China（中国科学技术大学）；State Grid Corporation（国家电网公司）
Subspace segmentation assumes that data comes from the union of different subspaces and the purpose of segmentation is to partition the data into the corresponding subspace. Low-rank representation (LRR) is a classic spectral-type method for solving subspace segmentation problems, that is, one first obtains an affinity matrix by solving a LRR model and then performs spectral clustering for segmentation. This paper proposes a group norm regularized factorization model (GNRFM) inspired by the LRR model for subspace segmentation and then designs an Accelerated Augmented Lagrangian Method (AALM) a…
The growing amount of high dimensional data in different machine learning applications requires more efficient and scalable optimization algorithms. In this work, we consider combining two techniques, parallelism and Nesterov's acceleration, to design faster algorithms for L1-regularized loss. We first simplify BOOM, a variant of gradient descent, and study it in a unified framework, which allows us to not only propose a refined measurement of sparsity to improve BOOM, but also show that BOOM is provably slower than FISTA. Moving on to parallel coordinate descent methods, we then propose an ef…
Alternating gradient-descent-ascent (AltGDA) is an optimization algorithm that has been widely used for model training in various machine learning applications, which aims to solve a nonconvex minimax optimization problem. However, the existing studies show that it suffers from a high computation complexity in nonconvex minimax optimization. In this paper, we develop a single-loop and fast AltGDA-type algorithm that leverages proximal gradient updates and momentum acceleration to solve regularized nonconvex minimax optimization problems. By leveraging the momentum acceleration technique, we pr…
The linear primal-dual hybrid gradient (PDHG) method is a first-order method that splits convex optimization problems with saddle-point structure into smaller subproblems. Unlike those obtained in most splitting methods, these subproblems can generally be solved efficiently because they involve simple operations such as matrix-vector multiplications or proximal mappings that are fast to evaluate numerically. This advantage comes at the price that the linear PDHG method requires precise stepsize parameters for the problem at hand to achieve an optimal convergence rate. Unfortunately, these step…
A Momentum Accelerated Algorithm for ReLU-based Nonlinear Matrix Decomposition
School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）；School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）
Understanding efficiency in high dimensional linear models is a longstanding problem of interest. Classical work with smaller dimensional problems dating back to Huber and Bickel has illustrated the benefits of efficient loss functions. When the number of parameters $p$ is of the same order as the sample size $n$, $p \approx n$, an efficiency pattern different from the one of Huber was recently established. In this work, we consider the effects of model selection on the estimation efficiency of penalized methods. In particular, we explore whether sparsity, results in new efficiency patterns wh…
Graph Learning Should Move Beyond Restrictive Views of Spectral and Message-Passing GNNs
RWTH Aachen University（亚琛工业大学）；Massachusetts Institute of Technology（麻省理工学院）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Stanford University（斯坦福大学）；University of California San Diego（加州大学圣地亚哥分校）；Univ. de Montréal（蒙特利尔大学）；Mila（Mila人工智能研究所）；Technion – Israel Institute of Technology（技术学院–以色列理工学院）
The accelerated proximal point algorithm (APPA), also known as "Catalyst", is a well-established reduction from convex optimization to approximate proximal point computation (i.e., regularized minimization). This reduction is conceptually elegant and yields strong convergence rate guarantees. However, these rates feature an extraneous logarithmic term arising from the need to compute each proximal point to high accuracy. In this work, we propose a novel Relaxed Error Criterion for Accelerated Proximal Point (RECAPP) that eliminates the need for high accuracy subproblem solutions. We apply RECA…
Bridging the Gap between Newton-Raphson Method and Regularized Policy Iteration
Laboratory for Information and Decision Systems, Massachusetts Institute of Technology（信息与决策系统实验室，麻省理工学院）；Department of Mechanical Engineering, Tsinghua University（机械工程系，清华大学）；College of Artificial Intelligence and the School of Vehicle and Mobility, Tsinghua University（人工智能学院和车辆与移动系统学院，清华大学）
An Accelerated Alternating Partial Bregman Algorithm for ReLU-based Matrix Decomposition
School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）；School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）
Inverse problems for accelerated MRI typically incorporate domain-specific knowledge about the forward encoding operator in a regularized reconstruction framework. Recently physics-driven deep learning (DL) methods have been proposed to use neural networks for data-driven regularization. These methods unroll iterative optimization algorithms to solve the inverse problem objective function, by alternating between domain-specific data consistency and data-driven regularization via neural networks. The whole unrolled network is then trained end-to-end to learn the parameters of the network. Due t…
