International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking Methods

The convex feasibility problem (CFP) is at the core of the modeling of many problems in various areas of science. Subgradient projection methods are important tools for solving the CFP because they enable the use of subgradient calculations instead of orthogonal projections onto the individual sets of the problem. Working in a real Hilbert space, we show that the sequential subgradient projection method is perturbation resilient. By this we mean that under appropriate conditions the sequence generated by the method converges weakly, and sometimes also strongly, to a point in the intersection of the given subsets of the feasibility problem, despite certain perturbations which are allowed in each iterative step. Unlike previous works on solving the convex feasibility problem, the involved functions, which induce the feasibility problem's subsets, need not be convex. Instead, we allow them to belong to a wider and richer class of functions satisfying a weaker condition, that we call "zero-convexity". This class, which is introduced and discussed here, holds a promise to solve optimization problems in various areas, especially in non-smooth and non-convex optimization. The relevance of this study to approximate minimization and to the recent superiorization methodology for constrained optimization is explained.Comment: Mathematical Programming Series A, accepted for publicatio

Nonconvex proximal splitting: batch and incremental algorithms

Within the unmanageably large class of nonconvex optimization, we consider the rich subclass of nonsmooth problems that have composite objectives---this already includes the extensively studied convex, composite objective problems as a special case. For this subclass, we introduce a powerful, new framework that permits asymptotically non-vanishing perturbations. In particular, we develop perturbation-based batch and incremental (online like) nonconvex proximal splitting algorithms. To our knowledge, this is the first time that such perturbation-based nonconvex splitting algorithms are being proposed and analyzed. While the main contribution of the paper is the theoretical framework, we complement our results by presenting some empirical results on matrix factorization.Comment: revised version 12 pages, 2 figures; superset of shorter counterpart in NIPS 201

Distributionally Robust Learning with Weakly Convex Losses: Convergence Rates and Finite-Sample Guarantees

We consider a distributionally robust stochastic optimization problem and formulate it as a stochastic two-level composition optimization problem with the use of the mean--semideviation risk measure. In this setting, we consider a single time-scale algorithm, involving two versions of the inner function value tracking: linearized tracking of a continuously differentiable loss function, and SPIDER tracking of a weakly convex loss function. We adopt the norm of the gradient of the Moreau envelope as our measure of stationarity and show that the sample complexity of $\mathcal{O}(\varepsilon^{-3})$ is possible in both cases, with only the constant larger in the second case. Finally, we demonstrate the performance of our algorithm with a robust learning example and a weakly convex, non-smooth regression example

A proximal method for composite minimization

Abstract. We consider minimization of functions that are compositions of prox-regular functions with smooth vector functions. A wide variety of important optimization problems can be formulated in this way. We describe a subproblem constructed from a linearized approximation to the objective and a regularization term, investigating the properties of local solutions of this subproblem and showing that they eventually identify a manifold containing the solution of the original problem. We propose an algorithmic framework based on this subproblem and prove a global convergence result

Towards Optimization and Robustification of Data-Driven Models

In the past two decades, data-driven models have experienced a renaissance, with notable success achieved through the use of models such as deep neural networks (DNNs) in various applications. However, complete reliance on intelligent machine learning systems is still a distant dream. Nevertheless, the initial success of data-driven approaches presents a promising path for building trustworthy data-oriented models. This thesis aims to take a few steps toward improving the performance of existing data-driven frameworks in both the training and testing phases. Specifically, we focus on several key questions: 1) How to efficiently design optimization methods for learning algorithms that can be used in parallel settings and also when first-order information is unavailable? 2) How to revise existing adversarial attacks on DNNs to structured attacks with minimal distortion of benign samples? 3) How to integrate attention models such as Transformers into data-driven inertial navigation systems? 4) How to address the lack of data problem for existing data-driven models and enhance the performance of existing semi-supervised learning (SSL) methods? In terms of parallel optimization methods, our research focuses on investigating a delay-aware asynchronous variance-reduced coordinate descent approach. Additionally, we explore the development of a proximal zeroth-order algorithm for nonsmooth nonconvex problems when first-order information is unavailable. We also extend our study to zeroth-order stochastic gradient descent problems. As for robustness, we develop a structured white-box adversarial attack to enhance research on robust machine learning schemes. Furthermore, our research investigates a group threat model in which adversaries can only perturb image segments rather than the entire image to generate adversarial examples. We also explore the use of attention models, specifically Transformer models, for deep inertial navigation systems based on the Inertial Measurement Unit (IMU). In addressing the problem of data scarcity during the training process, we propose a solution that involves quantizing the uncertainty from the unlabeled data and corresponding pseudo-labels, and incorporating it into the loss term to compensate for noisy pseudo-labeling. We also extend the generic semi-supervised method for data-driven noise suppression frameworks by utilizing a reinforcement learning (RL) model to learn contrastive features in an SSL fashion. Each chapter of the thesis presents the problem and our solutions using concrete algorithms. We verify our approach through comparisons with existing methods on different benchmarks and discuss future research directions