We develop two new stochastic Gauss-Newton algorithms for solving a class of non-convex stochastic compositional optimization problems frequently arising in practice. We consider both the expectation and finite-sum settings under standard assumptions, and use both classical stochastic and SARAH estimators for approximating function values and Jacobians. In the expectation case, we establish $\mathcal{O}(\varepsilon^{-2})$ iteration-complexity to achieve a stationary point in expectation and estimate the total number of stochastic oracle calls for both function value and its Jacobian, where $\varepsilon$ is a desired accuracy. In the finite sum case, we also estimate $\mathcal{O}(\varepsilon^{-2})$ iteration-complexity and the total oracle calls with high probability. To our best knowledge, this is the first time such global stochastic oracle complexity is established for stochastic Gauss-Newton methods. Finally, we illustrate our theoretical results via two numerical examples on both synthetic and real datasets.
We aim to find a solution $\bm{x}\in\mathbb{C}^n$ to a system of quadratic equations of the form $b_i=\lvert\bm{a}_i^*\bm{x}\rvert^2$, $i=1,2,\ldots,m$, e.g., the well-known NP-hard phase retrieval problem. As opposed to recently proposed state-of-the-art nonconvex methods, we revert to the semidefinite relaxation (SDR) PhaseLift convex formulation and propose a successive and incremental nonconvex optimization algorithm, termed as \texttt{IncrePR}, to indirectly minimize the resulting convex problem on the cone of positive semidefinite matrices. Our proposed method overcomes the excessive com…
We develop a novel primal-dual algorithm to solve a class of nonsmooth and nonlinear compositional convex minimization problems, which covers many existing and brand-new models as special cases. Our approach relies on a combination of a new nonconvex potential function, Nesterov's accelerated scheme, and an adaptive parameter updating strategy. Our algorithm is single-loop and has low per-iteration complexity. Under only general convexity and mild assumptions, our algorithm achieves $\mathcal{O}(1/k)$ convergence rates through three different criteria: primal objective residual, dual objective…
We consider a general class of regression models with normally distributed covariates, and the associated nonconvex problem of fitting these models from data. We develop a general recipe for analyzing the convergence of iterative algorithms for this task from a random initialization. In particular, provided each iteration can be written as the solution to a convex optimization problem satisfying some natural conditions, we leverage Gaussian comparison theorems to derive a deterministic sequence that provides sharp upper and lower bounds on the error of the algorithm with sample-splitting. Cruc…
The Ohio State University（俄亥俄州立大学）；Duke University（杜克大学）
Two types of zeroth-order stochastic algorithms have recently been designed for nonconvex optimization respectively based on the first-order techniques SVRG and SARAH/SPIDER. This paper addresses several important issues that are still open in these methods. First, all existing SVRG-type zeroth-order algorithms suffer from worse function query complexities than either zeroth-order gradient descent (ZO-GD) or stochastic gradient descent (ZO-SGD). In this paper, we propose a new algorithm ZO-SVRG-Coord-Rand and develop a new analysis for an existing ZO-SVRG-Coord algorithm proposed in Liu et al.…
We investigate the effectiveness of convex relaxation and nonconvex optimization in solving bilinear systems of equations under two different designs (i.e.$~$a sort of random Fourier design and Gaussian design). Despite the wide applicability, the theoretical understanding about these two paradigms remains largely inadequate in the presence of random noise. The current paper makes two contributions by demonstrating that: (1) a two-stage nonconvex algorithm attains minimax-optimal accuracy within a logarithmic number of iterations. (2) convex relaxation also achieves minimax-optimal statistical…
Motivated by the increasing availability of high-performance parallel computing, we design a distributed parallel algorithm for linearly-coupled block-structured nonconvex constrained optimization problems. Our algorithm performs Jacobi-type proximal updates of the augmented Lagrangian function, requiring only local solutions of separable block nonlinear programming (NLP) problems. We provide a cheap and explicitly computable Lyapunov function that allows us to establish global and local sublinear convergence of our algorithm, its iteration complexity, as well as simple, practical and theoreti…
We develop theoretically guaranteed stochastic methods for outlier-robust PCA. Outlier-robust PCA seeks an underlying low-dimensional linear subspace from a dataset that is corrupted with outliers. We are able to show that our methods, which involve stochastic geodesic gradient descent over the Grassmannian manifold, converge and recover an underlying subspace in various regimes through the development of a novel convergence analysis. The main application of this method is an effective differentially private algorithm for outlier-robust PCA that uses a Gaussian noise mechanism within the stoch…
A General Recipe for Parameter-Free Nonconvex Optimization via Higher-Order Regularization
Graduate School of Information Science and Technology, University of Tokyo（东京大学信息理工学研究科）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）
The paper proves convergence to global optima for a class of distributed algorithms for nonconvex optimization in network-based multi-agent settings. Agents are permitted to communicate over a time-varying undirected graph. Each agent is assumed to possess a local objective function (assumed to be smooth, but possibly nonconvex). The paper considers algorithms for optimizing the sum function. A distributed algorithm of the consensus+innovations type is proposed which relies on first-order information at the agent level. Under appropriate conditions on network connectivity and the cost objectiv…
In this paper, we propose and analyze algorithms for zeroth-order optimization of non-convex composite objectives, focusing on reducing the complexity dependence on dimensionality. This is achieved by exploiting the low dimensional structure of the decision set using the stochastic mirror descent method with an entropy alike function, which performs gradient descent in the space equipped with the maximum norm. To improve the gradient estimation, we replace the classic Gaussian smoothing method with a sampling method based on the Rademacher distribution and show that the mini-batch method copes…
University of California, Davis（加州大学戴维斯分校）；Princeton University（普林斯顿大学）
Bandit algorithms have been predominantly analyzed in the convex setting with function-value based stationary regret as the performance measure. In this paper, motivated by online reinforcement learning problems, we propose and analyze bandit algorithms for both general and structured nonconvex problems with nonstationary (or dynamic) regret as the performance measure, in both stochastic and non-stochastic settings. First, for general nonconvex functions, we consider nonstationary versions of first-order and second-order stationary solutions as a regret measure, motivated by similar performanc…
In this paper, we develop a variant of the well-known Gauss-Newton (GN) method to solve a class of nonconvex optimization problems involving low-rank matrix variables. As opposed to the standard GN method, our algorithm allows one to handle general smooth convex objective function. We show, under mild conditions, that the proposed algorithm globally and locally converges to a stationary point of the original problem. We also show empirically that the GN algorithm achieves higher accurate solutions than the alternating minimization algorithm (AMA). Then, we specify our GN scheme to handle the s…
In this paper, we address the trajectory planning problem in uncertain nonconvex static and dynamic environments that contain obstacles with probabilistic location, size, and geometry. To address this problem, we provide a risk bounded trajectory planning method that looks for continuous-time trajectories with guaranteed bounded risk over the planning time horizon. Risk is defined as the probability of collision with uncertain obstacles. Existing approaches to address risk bounded trajectory planning problems either are limited to Gaussian uncertainties and convex obstacles or rely on sampling…
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 t…
We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: $L(x,y)=f(x)+Q(x,y)+g(y)$, where $f:\R^n\rightarrow\R\cup{+\infty}$ and $g:\R^m\rightarrow\R\cup{+\infty}$ are proper lower semicontinuous functions, and $Q:\R^n\times\R^m\rightarrow \R$ is a smooth $C^1$ function which couples the variables $x$ and $y$. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize $L$. We work in a nonconvex setting, just assuming that the function $L$ satisfies the Kurdyka-Łojasiewicz in…
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
Federated Learning is a popular distributed learning paradigm in machine learning. Meanwhile, composition optimization is an effective hierarchical learning model, which appears in many machine learning applications such as meta learning and robust learning. More recently, although a few federated composition optimization algorithms have been proposed, they still suffer from high sample and communication complexities. In the paper, thus, we propose a class of faster federated compositional optimization algorithms (i.e., MFCGD and AdaMFCGD) to solve the nonconvex distributed composition problem…
Department of Statistics, University of California, Davis（加利福尼亚大学戴维斯分校统计系）；Department of Operations Research and Financial Engineering, Princeton University（普林斯顿大学运筹学与金融工程系）
In this paper, we propose and analyze zeroth-order stochastic approximation algorithms for nonconvex and convex optimization, with a focus on addressing constrained optimization, high-dimensional setting and saddle-point avoiding. To handle constrained optimization, we first propose generalizations of the conditional gradient algorithm achieving rates similar to the standard stochastic gradient algorithm using only zeroth-order information. To facilitate zeroth-order optimization in high-dimensions, we explore the advantages of structural sparsity assumptions. Specifically, (i) we highlight an…
Riemannian ascent--descent for nonconvex nonconcave minimax landscapes: convergence to basin saddle points and applications to distributionally robust optimization
We present $\textit{Free-MESSAGE}^{p}$, the first zeroth-order algorithm for (weakly-)convex mean-semideviation-based risk-aware learning, which is also the first three-level zeroth-order compositional stochastic optimization algorithm whatsoever. Using a non-trivial extension of Nesterov's classical results on Gaussian smoothing, we develop the $\textit{Free-MESSAGE}^{p}$ algorithm from first principles, and show that it essentially solves a smoothed surrogate to the original problem, the former being a uniform approximation of the latter, in a useful, convenient sense. We then present a comp…
The paper considers a distributed algorithm for global minimization of a nonconvex function. The algorithm is a first-order consensus + innovations type algorithm that incorporates decaying additive Gaussian noise for annealing, converging to the set of global minima under certain technical assumptions. The paper presents simple methods for verifying that the required technical assumptions hold and illustrates it with a distributed target-localization application.
Min-Max Optimisation for Nonconvex-Nonconcave Functions Using a Random Zeroth-Order Extragradient Algorithm
School of Engineering Australian National University（工程学院澳大利亚国立大学）；CIICADA Lab（CIICADA实验室）；Department of Electrical Engineering Polytechnique Montréal（电气工程系蒙特利尔理工学院）；GERAD & Mila（GERAD与Mila）；Department of Mathematics and Industrial Engineering Polytechnique Montréal（数学与工业工程系蒙特利尔理工学院）；GERAD；Department of Electrical and Electronic Engineering University of Melbourne（电气与电子工程系墨尔本大学）
