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Zhaosong Lu

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7 papers

2 author rows

NeurIPS Conference 2024 Conference Paper

Single-Loop Stochastic Algorithms for Difference of Max-Structured Weakly Convex Functions

Quanqi Hu
Qi Qi
Zhaosong Lu
Tianbao Yang

In this paper, we study a class of non-smooth non-convex problems in the form of $\min_{x}[\max_{y\in\mathcal Y}\phi(x, y) - \max_{z\in\mathcal Z}\psi(x, z)]$, where both $\Phi(x) = \max_{y\in\mathcal Y}\phi(x, y)$ and $\Psi(x)=\max_{z\in\mathcal Z}\psi(x, z)$ are weakly convex functions, and $\phi(x, y), \psi(x, z)$ are strongly concave functions in terms of $y$ and $z$, respectively. It covers two families of problems that have been studied but are missing single-loop stochastic algorithms, i. e. , difference of weakly convex functions and weakly convex strongly-concave min-max problems. We propose a stochastic Moreau envelope approximate gradient method dubbed SMAG, the first single-loop algorithm for solving these problems, and provide a state-of-the-art non-asymptotic convergence rate. The key idea of the design is to compute an approximate gradient of the Moreau envelopes of $\Phi, \Psi$ using only one step of stochastic gradient update of the primal and dual variables. Empirically, we conduct experiments on positive-unlabeled (PU) learning and partial area under ROC curve (pAUC) optimization with an adversarial fairness regularizer to validate the effectiveness of our proposed algorithms.

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JMLR Journal 2023 Journal Article

A Parameter-Free Conditional Gradient Method for Composite Minimization under Hölder Condition

Masaru Ito
Zhaosong Lu
Chuan He

In this paper we consider a composite optimization problem that minimizes the sum of a weakly smooth function and a convex function with either a bounded domain or a uniformly convex structure. In particular, we first present a parameter-dependent conditional gradient method for this problem, whose step sizes require prior knowledge of the parameters associated with the Hölder continuity of the gradient of the weakly smooth function, and establish its rate of convergence. Given that these parameters could be unknown or known but possibly conservative, such a method may suffer from implementation issue or slow convergence. We therefore propose a parameter-free conditional gradient method whose step size is determined by using a constructive local quadratic upper approximation and an adaptive line search scheme, without using any problem parameter. We show that this method achieves the same rate of convergence as the parameter-dependent conditional gradient method. Preliminary experiments are also conducted and illustrate the superior performance of the parameter-free conditional gradient method over the methods with some other step size rules. [abs] [ pdf ][ bib ] &copy JMLR 2023. ( edit, beta )

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ICML Conference 2023 Conference Paper

Generalized-Smooth Nonconvex Optimization is As Efficient As Smooth Nonconvex Optimization

Ziyi Chen 0002
Yi Zhou 0017
Yingbin Liang
Zhaosong Lu

Various optimal gradient-based algorithms have been developed for smooth nonconvex optimization. However, many nonconvex machine learning problems do not belong to the class of smooth functions and therefore the existing algorithms are sub-optimal. Instead, these problems have been shown to satisfy certain generalized-smooth conditions, which have not been well understood in the existing literature. In this paper, we propose a notion of $\alpha$-symmetric generalized-smoothness that substantially extends the existing notions and covers many important functions such as high-order polynomials and exponential functions. We study the fundamental properties and establish descent lemmas for the functions in this class. Then, to solve such a large class of nonconvex problems, we design a special deterministic normalized gradient descent algorithm that achieves the optimal iteration complexity $\mathcal{O}(\epsilon^{-2})$, and also prove that the popular SPIDER variance reduction algorithm achieves the optimal sample complexity $\mathcal{O}(\epsilon^{-3})$. Our results show that solving generalized-smooth nonconvex problems is as efficient as solving smooth nonconvex problems.

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NeurIPS Conference 2014 Conference Paper

An Accelerated Proximal Coordinate Gradient Method

Qihang Lin
Zhaosong Lu
Lin Xiao

We develop an accelerated randomized proximal coordinate gradient (APCG) method, for solving a broad class of composite convex optimization problems. In particular, our method achieves faster linear convergence rates for minimizing strongly convex functions than existing randomized proximal coordinate gradient methods. We show how to apply the APCG method to solve the dual of the regularized empirical risk minimization (ERM) problem, and devise efficient implementations that can avoid full-dimensional vector operations. For ill-conditioned ERM problems, our method obtains improved convergence rates than the state-of-the-art stochastic dual coordinate ascent (SDCA) method.

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ICML Conference 2014 Conference Paper

Rank-One Matrix Pursuit for Matrix Completion

Zheng Wang 0011
Ming-Jun Lai
Zhaosong Lu
Wei Fan 0001
Hasan Davulcu
Jieping Ye

Low rank matrix completion has been applied successfully in a wide range of machine learning applications, such as collaborative filtering, image inpainting and Microarray data imputation. However, many existing algorithms are not scalable to large-scale problems, as they involve computing singular value decomposition. In this paper, we present an efficient and scalable algorithm for matrix completion. The key idea is to extend the well-known orthogonal matching pursuit from the vector case to the matrix case. In each iteration, we pursue a rank-one matrix basis generated by the top singular vector pair of the current approximation residual and update the weights for all rank-one matrices obtained up to the current iteration. We further propose a novel weight updating rule to reduce the time and storage complexity, making the proposed algorithm scalable to large matrices. We establish the linear convergence of the proposed algorithm. The fast convergence is achieved due to the proposed construction of matrix bases and the estimation of the weights. We empirically evaluate the proposed algorithm on many real-world large scale datasets. Results show that our algorithm is much more efficient than state-of-the-art matrix completion algorithms while achieving similar or better prediction performance.

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ICML Conference 2013 Conference Paper

A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems

Pinghua Gong
Changshui Zhang
Zhaosong Lu
Jianhua Huang
Jieping Ye

Non-convex sparsity-inducing penalties have recently received considerable attentions in sparse learning. Recent theoretical investigations have demonstrated their superiority over the convex counterparts in several sparse learning settings. However, solving the non-convex optimization problems associated with non-convex penalties remains a big challenge. A commonly used approach is the Multi-Stage (MS) convex relaxation (or DC programming), which relaxes the original non-convex problem to a sequence of convex problems. This approach is usually not very practical for large-scale problems because its computational cost is a multiple of solving a single convex problem. In this paper, we propose a General Iterative Shrinkage and Thresholding (GIST) algorithm to solve the nonconvex optimization problem for a large class of non-convex penalties. The GIST algorithm iteratively solves a proximal operator problem, which in turn has a closed-form solution for many commonly used penalties. At each outer iteration of the algorithm, we use a line search initialized by the Barzilai-Borwein (BB) rule that allows finding an appropriate step size quickly. The paper also presents a detailed convergence analysis of the GIST algorithm. The efficiency of the proposed algorithm is demonstrated by extensive experiments on large-scale data sets.

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NeurIPS Conference 2011 Conference Paper

Penalty Decomposition Methods for Rank Minimization

Yong Zhang
Zhaosong Lu

In this paper we consider general rank minimization problems with rank appearing in either objective function or constraint. We first show that a class of matrix optimization problems can be solved as lower dimensional vector optimization problems. As a consequence, we establish that a class of rank minimization problems have closed form solutions. Using this result, we then propose penalty decomposition methods for general rank minimization problems. The convergence results of the PD methods have been shown in the longer version of the paper. Finally, we test the performance of our methods by applying them to matrix completion and nearest low-rank correlation matrix problems. The computational results demonstrate that our methods generally outperform the existing methods in terms of solution quality and/or speed.

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