We propose a projection-free conditional gradient-type algorithm for smooth stochastic multi-level composition optimization, where the objective function is a nested composition of $T$ functions and the constraint set is a closed convex set. Our algorithm assumes access to noisy evaluations of the functions and their gradients, through a stochastic first-order oracle satisfying certain standard unbiasedness and second moment assumptions. We show that the number of calls to the stochastic first-order oracle and the linear-minimization oracle required by the proposed algorithm, to obtain an $ε$-stationary solution, are of order $\mathcal{O}_T(ε^{-2})$ and $\mathcal{O}_T(ε^{-3})$ respectively, where $\mathcal{O}_T$ hides constants in $T$. Notably, the dependence of these complexity bounds on $ε$ and $T$ are separate in the sense that changing one does not impact the dependence of the bounds on the other. Moreover, our algorithm is parameter-free and does not require any (increasing) order of mini-batches to converge unlike the common practice in the analysis of stochastic conditional gradient-type algorithms.
Projection operators are fundamental algorithmic operators in Analysis and Optimization. It is well known that these operators are firmly nonexpansive; however, their composition is generally only averaged and no longer firmly nonexpansive. In this note, we introduce the modulus of averagedness and provide an exact result for the composition of two linear projection operators. As a consequence, we deduce that the Ogura-Yamada bound for the modulus of the composition is sharp.
In this work, we study a novel class of projection-based algorithms for linearly constrained problems (LCPs) which have a lot of applications in statistics, optimization, and machine learning. Conventional primal gradient-based methods for LCPs call a projection after each (stochastic) gradient descent, resulting in that the required number of projections equals that of gradient descents (or total iterations). Motivated by the recent progress in distributed optimization, we propose the delayed projection technique that calls a projection once for a while, lowering the projection frequency and …
In this paper, we propose a new inexact version of the projected subgradient method to solve nondifferentiable constrained convex optimization problems. The method combine $ε$-subgradient method with a procedure to obtain a feasible inexact projection onto the constraint set. Asymptotic convergence results and iteration-complexity bounds for the sequence generated by the method employing the well known exogenous stepsizes, Polyak's stepsizes, and dynamic stepsizes are established.
In this paper, we consider the convex, finite-sum minimization problem with explicit convex constraints over strongly connected directed graphs. The constraint is an intersection of several convex sets each being known to only one node. To solve this problem, we propose a novel decentralized projected gradient scheme based on local averaging and prove its convergence using only local functions' smoothness.
Robustness studies of black-box models is recognized as a necessary task for numerical models based on structural equations and predictive models learned from data. These studies must assess the model's robustness to possible misspecification of regarding its inputs (e.g., covariate shift). The study of black-box models, through the prism of uncertainty quantification (UQ), is often based on sensitivity analysis involving a probabilistic structure imposed on the inputs, while ML models are solely constructed from observed data. Our work aim at unifying the UQ and ML interpretability approaches…
This paper proposes a distributed stochastic projection-free algorithm for large-scale constrained finite-sum optimization whose constraint set is complicated such that the projection onto the constraint set can be expensive. The global cost function is allocated to multiple agents, each of which computes its local stochastic gradients and communicates with its neighbors to solve the global problem. Stochastic gradient methods enable low computational cost, while they are hard and slow to converge due to the variance caused by random sampling. To construct a convergent distributed stochastic p…
Hard-thresholding-based algorithms have seen various advantages for sparse optimization in controlling the sparsity and allowing for fast computation. Recent research shows that when techniques of the Newton-type methods are integrated, their numerical performance can be improved surprisingly. This paper develops a gradient projection Newton pursuit algorithm that mainly adopts the hard-thresholding operator and employs the Newton pursuit only when certain conditions are satisfied. The proposed algorithm is capable of converging globally and quadratically under the standard assumptions. When i…
In this paper, we focus on solving a distributed convex aggregative optimization problem in a network, where each agent has its own cost function which depends not only on its own decision variables but also on the aggregated function of all agents' decision variables. The decision variable is constrained within a feasible set. In order to minimize the sum of the cost functions when each agent only knows its local cost function, we propose a distributed Frank-Wolfe algorithm based on gradient tracking for the aggregative optimization problem where each node maintains two estimates, namely an e…
We consider convex constrained optimization problems that also include a cardinality constraint. In general, optimization problems with cardinality constraints are difficult mathematical programs which are usually solved by global techniques from discrete optimization. We assume that the region defined by the convex constraints can be written as the intersection of a finite collection of convex sets, such that it is easy and inexpensive to project onto each one of them (e.g., boxes, hyper-planes, or half-spaces). Taking advantage of a recently developed continuous reformulation that relaxes th…
It is well-known that the sequence of iterations of the composition of projections onto closed affine subspaces converges linearly to the projection onto the intersection of the affine subspaces when the sum of the corresponding linear subspaces is closed. Inspired by this, in this work, we systematically study the relation between the projection onto intersection of halfspaces and hyperplanes, and the composition of projections onto halfspaces and hyperplanes. In addition, as by-products, we provide the Karush-Kuhn-Tucker conditions for characterizing the optimal solution of convex optimizati…
Halıcıoğlu Data Science Institute, University of California, San Diego（加州大学圣地亚哥分校哈利哲奥卢数据科学研究所）；University of California, Davis（加州大学戴维斯分校）；University of Waterloo（滑铁卢大学）
We study stochastic optimization algorithms for constrained nonconvex stochastic optimization problems with Markovian data. In particular, we focus on the case when the transition kernel of the Markov chain is state-dependent. Such stochastic optimization problems arise in various machine learning problems including strategic classification and reinforcement learning. For this problem, we study both projection-based and projection-free algorithms. In both cases, we establish that the number of calls to the stochastic first-order oracle to obtain an appropriately defined $ε$-stationary point is…
We herein propose a variant of the projected inexact Levenberg--Marquardt method (ILMM) for solving constrained nonsmooth equations. Since the orthogonal projection onto the feasible set may be computationally expensive, we propose a local ILMM with feasible inexact projections. By using assumption of semi-smoothness and an error bound condition which is weaker than the standard full rank assumption, we establish its convergence results, including results on its rate. Finally, some computacional results are reported in order to illustrate the advantages of the new schemes.
Stochastic bilevel optimization, which captures the inherent nested structure of machine learning problems, is gaining popularity in many recent applications. Existing works on bilevel optimization mostly consider either unconstrained problems or constrained upper-level problems. This paper considers the stochastic bilevel optimization problems with equality constraints both in the upper and lower levels. By leveraging the special structure of the equality constraints problem, the paper first presents an alternating implicit projected SGD approach and establishes the $\tilde{\cal O}(ε^{-2})$ s…
We consider a linear iterative solver for large scale linearly constrained quadratic minimization problems that arise, for example, in optimization with PDEs. By a primal-dual projection (PDP) iteration, which can be interpreted and analysed as a gradient method on a quotient space, the given problem can be solved by computing sulutions for a sequence of constrained surrogate problems, projections onto the feasible subspaces, and Lagrange multiplier updates. As a major application we consider a class of optimization problems with PDEs, where PDP can be applied together with a projected cg meth…
Projection-Free Functional Constrained Optimization for Risk Aversion and Sparsity Control
H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（赫尔曼·米利顿·斯图尔特工业与系统工程学院，佐治亚理工学院）；College of Management of Technology, EPFL（技术管理学院，苏黎世联邦理工学院）
Inexact Primal-Dual Gradient Projection Methods for Nonlinear Optimization on Convex Set
School of Information Science and Technology, ShanghaiTech University（上海科技大学信息科学与技术学院）；Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences（中国科学院上海微系统与信息技术研究所）；University of Washington（华盛顿大学）；Tongji University（同济大学）；University of Chinese Academy of Sciences（中国科学院大学）
In this paper, we consider a distributed constrained optimization problem with delayed subgradient information over the time-varying communication network, where each agent can only communicate with its neighbors and the communication channel has a limited data rate. We propose an adaptive quantization method to address this problem. A mirror descent algorithm with delayed subgradient information is established based on the theory of Bregman divergence. With non-Euclidean Bregman projection-based scheme, the proposed method essentially generalizes many previous classical Euclidean projection-b…
Indian Institute of Technology Kanpur（印度理工学院坎普尔分校）；University of Maryland, College Park（马里兰大学帕克分校）
This work presents the first projection-free algorithm to solve stochastic bi-level optimization problems, where the objective function depends on the solution of another stochastic optimization problem. The proposed $\textbf{S}$tochastic $\textbf{Bi}$-level $\textbf{F}$rank-$\textbf{W}$olfe ($\textbf{SBFW}$) algorithm can be applied to streaming settings and does not make use of large batches or checkpoints. The sample complexity of SBFW is shown to be $\mathcal{O}(ε^{-3})$ for convex objectives and $\mathcal{O}(ε^{-4})$ for non-convex objectives. Improved rates are derived for the stochastic…
