We consider minimization problems with the well-known Polya-Lojasievich condition and Lipshitz-continuous gradient. Such problem occurs in different places in machine learning and related fields. Furthermore, we assume that a gradient is available with some relative inexactness. We propose some adaptive gradient-type algorithm, where the adaptivity took place with respect to the smoothness parameter and the level of the gradient inexactness. The theoretical estimate of the the quality of the output point is obtained and backed up by experimental results.
Due to its applications in many different places in machine learning and other connected engineering applications, the problem of minimization of a smooth function that satisfies the Polyak-Łojasiewicz condition receives much attention from researchers. Recently, for this problem, the authors of recent work proposed an adaptive gradient-type method using an inexact gradient. The adaptivity took place only with respect to the Lipschitz constant of the gradient. In this paper, for problems with the Polyak-Łojasiewicz condition, we propose a full adaptive algorithm, which means that the adaptivit…
Communication delays and synchronization are major bottlenecks for parallel computing, and tolerating asynchrony is therefore crucial for accelerating parallel computation. Motivated by optimization problems that do not satisfy convexity assumptions, we present an asynchronous block coordinate descent algorithm for nonconvex optimization problems whose objective functions satisfy the Polyak-Lojasiewicz condition. This condition is a generalization of strong convexity to nonconvex problems and requires neither convexity nor uniqueness of minimizers. Under only assumptions of mild smoothness of …
Convergence of the Safeguarded Augmented Lagrangian Method under the Polyak-Lojasiewicz constraint qualification for Constrained Composite Optimization
We introduce a notion of inexact model of a convex objective function, which allows for errors both in the function and in its gradient. For this situation, a gradient method with an adaptive adjustment of some parameters of the model is proposed and an estimate for the convergence rate is found. This estimate is optimal on a class of sufficiently smooth problems in the presence of errors. We consider a special class of convex nonsmooth optimization problems. In order to apply the proposed technique to this class, an artificial error should be introduced. We show that the method can be modifie…
A Local Polyak-Lojasiewicz and Descent Lemma of Gradient Descent For Overparametrized Linear Models
Center for Innovation in Data Engineering and Science (IDEAS) Department of Statistics and Data Science, the Wharton School University of Pennsylvania（创新数据工程与科学中心（IDEAS）统计与数据科学系，沃顿商学院，宾夕法尼亚大学）；Center for Innovation in Data Engineering and Science (IDEAS) University of Pennsylvania（创新数据工程与科学中心（IDEAS）宾夕法尼亚大学）；Center for Innovation in Data Engineering and Science (IDEAS) Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania（创新数据工程与科学中心（IDEAS）应用数学与计算科学研究生组，宾夕法尼亚大学）；Mathematical Institute for Data Science (MINDS) Department of Electrical and Computer Engineering Johns Hopkins University（数据科学数学研究所（MINDS）电气与计算机工程系，约翰霍普金斯大学）；Department of Electrical and Systems Engineering（电气与系统工程系；放射学系，宾夕法尼亚大学）；Department of Radiology University of Pennsylvania
In this paper, we derive a new linear convergence rate for the gradient method with fixed step lengths for non-convex smooth optimization problems satisfying the Polyak-Lojasiewicz (PL) inequality. We establish that the PL inequality is a necessary and sufficient condition for linear convergence to the optimal value for this class of problems. We list some related classes of functions for which the gradient method may enjoy linear convergence rate. Moreover, we investigate their relationship with the PL inequality.
Minimization of a smooth function on a sphere or, more generally, on a smooth manifold, is the simplest non-convex optimization problem. It has a lot of applications. Our goal is to propose a version of the gradient projection algorithm for its solution and to obtain results that guarantee convergence of the algorithm under some minimal natural assumptions. We use the Lezanski-Polyak-Lojasiewicz condition on a manifold to prove the global linear convergence of the algorithm. Another method well fitted for the problem is the conditional gradient (Frank-Wolfe) algorithm. We examine some conditio…
Norm-Constrained Flows and Sign-Based Optimization: Theory and Algorithms
Institute of Data Science and Artificial Intelligence, Innopolis University（数据科学与人工智能研究所，因诺波利斯大学）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Université de Mons（蒙斯大学）
We present a novel communication-efficient Newton-type algorithm for finite-sum optimization over a distributed computing environment. Our method, named DINO, overcomes both theoretical and practical shortcomings of similar existing methods. Under minimal assumptions, we guarantee global sub-linear convergence of DINO to a first-order stationary point for general non-convex functions and arbitrary data distribution over the network. Furthermore, for functions satisfying Polyak-Lojasiewicz (PL) inequality, we show that DINO enjoys a linear convergence rate. Our proposed algorithm is practically…
{We consider alternating minimization procedures for convex optimization problems with variable divided in many block, each block being amenable for minimization with respect to its variable with freezed other variables blocks. In the case of two blocks, we prove a linear convergence rate for alternating minimization procedure under Polyak-Lojasiewicz condition, which can be seen as a relaxation of the strong convexity assumption. Under strong convexity assumption in many-blocks setting we provide an accelerated alternating minimization procedure with linear rate depending on the square root o…
We study the convergence rate of gradient-based local search methods for solving low-rank matrix recovery problems with general objectives in both symmetric and asymmetric cases, under the assumption of the restricted isometry property. First, we develop a new technique to verify the Polyak-Lojasiewicz inequality in a neighborhood of the global minimizers, which leads to a local linear convergence region for the gradient descent method. Second, based on the local convergence result and a sharp strict saddle property proven in this paper, we present two new conditions that guarantee the global …
Conditional Value-at-Risk ($\mathrm{CV@R}$) is one of the most popular measures of risk, which has been recently considered as a performance criterion in supervised statistical learning, as it is related to desirable operational features in modern applications, such as safety, fairness, distributional robustness, and prediction error stability. However, due to its variational definition, $\mathrm{CV@R}$ is commonly believed to result in difficult optimization problems, even for smooth and strongly convex loss functions. We disprove this statement by establishing noisy (i.e., fixed-accuracy) li…
We propose a practical inexact augmented Lagrangian method (iALM) for nonconvex problems with nonlinear constraints. We characterize the total computational complexity of our method subject to a verifiable geometric condition, which is closely related to the Polyak-Lojasiewicz and Mangasarian-Fromowitz conditions. In particular, when a first-order solver is used for the inner iterates, we prove that iALM finds a first-order stationary point with $\tilde{\mathcal{O}}(1/ε^4)$ calls to the first-order oracle. If, in addition, the problem is smooth and a second-order solver is used for the inner i…
We develop a class of algorithms, as variants of the stochastically controlled stochastic gradient (SCSG) methods (Lei and Jordan, 2016), for the smooth non-convex finite-sum optimization problem. Assuming the smoothness of each component, the complexity of SCSG to reach a stationary point with $\mathbb{E} \|\nabla f(x)\|^{2}\le ε$ is $O\left (\min\{ε^{-5/3}, ε^{-1}n^{2/3}\}\right)$, which strictly outperforms the stochastic gradient descent. Moreover, SCSG is never worse than the state-of-the-art methods based on variance reduction and it significantly outperforms them when the target accurac…
Oden Institute, UT Austin（德克萨斯大学奥斯汀分校奥登研究所）；Department of Mathematics, UT Austin（德克萨斯大学奥斯汀分校数学系）
We prove that the norm version of the adaptive stochastic gradient method (AdaGrad-Norm) achieves a linear convergence rate for a subset of either strongly convex functions or non-convex functions that satisfy the Polyak Lojasiewicz (PL) inequality. The paper introduces the notion of Restricted Uniform Inequality of Gradients (RUIG)---which is a measure of the balanced-ness of the stochastic gradient norms---to depict the landscape of a function. RUIG plays a key role in proving the robustness of AdaGrad-Norm to its hyper-parameter tuning in the stochastic setting. On top of RUIG, we develop a…
A Continuous-Time Analysis of Smoothed Matrix-Polar Spectral Gradient Flows for Muon-Type Optimization
Zhejiang University（浙江大学）；National University of Singapore（新加坡国立大学）；Huzhou Normal University（湖州师范学院）；State Key Laboratory of Industrial Control Technology, the Institute of Cyber-Systems and Control, College of Control Science and Engineering, Zhejiang University（浙江大学控制科学与工程学院网络控制系统研究所工业控制技术国家重点实验室）；Huzhou Institute of Zhejiang University（浙江大学湖州研究院）
Quasar convexity is a condition that allows some first-order methods to efficiently minimize a function even when the optimization landscape is non-convex. Previous works develop near-optimal accelerated algorithms for minimizing this class of functions, however, they require a subroutine of binary search which results in multiple calls to gradient evaluations in each iteration, and consequently the total number of gradient evaluations does not match a known lower bound. In this work, we show that a recently proposed continuized Nesterov acceleration can be applied to minimizing quasar convex …
We derive bounds on the path length $ζ$ of gradient descent (GD) and gradient flow (GF) curves for various classes of smooth convex and nonconvex functions. Among other results, we prove that: (a) if the iterates are linearly convergent with factor $(1-c)$, then $ζ$ is at most $\mathcal{O}(1/c)$; (b) under the Polyak-Kurdyka-Lojasiewicz (PKL) condition, $ζ$ is at most $\mathcal{O}(\sqrtκ)$, where $κ$ is the condition number, and at least $\widetildeΩ(\sqrt{d} \wedge κ^{1/4})$; (c) for quadratics, $ζ$ is $Θ(\min\{\sqrt{d},\sqrt{\log κ}\})$ and in some cases can be independent of $κ$; (d) assumi…
Optimization Insights into Deep Diagonal Linear Networks
MaLGa, DIBRIS, Università degli Studi di Genova（基因瓦大学MaLGa、DIBRIS分校）；MaLGa, DIMA, Università degli Studi di Genova（基因瓦大学MaLGa、DIMA分校）；Center for Brains, Minds and Machines, MIT（MIT大脑、心智与机器中心）；Istituto Italiano di Tecnologia（意大利技术研究院）；Instituto de Alta investigaciòn (IAI), Universidad de Tarapacà（塔拉帕卡大学高级研究机构）
Benign Nonconvex Landscape for Policy Optimization: Infinite-Horizon Discounted MDPs with General State and Action Spaces
Naveen Jindal School of Management, University of Texas at Dallas（德克萨斯大学达拉斯分校纳文·金达尔管理学院）；H. Milton Stewart School of Industrial and Systems Engineering, Georgia Tech（佐治亚理工学院H.米尔顿·斯图尔特工业与系统工程学院）
