In this paper, we study the efficiency of a {\bf R}estarted {\bf S}ub{\bf G}radient (RSG) method that periodically restarts the standard subgradient method (SG). We show that, when applied to a broad class of convex optimization problems, RSG method can find an $\epsilon$-optimal solution with a low complexity than SG method. In particular, we first … Read more
The paper studies the solution of stochastic optimization problems in which approximations to the gradient and Hessian are obtained through subsampling. We first consider Newton-like methods that employ these approximations and discuss how to coordinate the accuracy in the gradient and Hessian to yield a superlinear rate of convergence in expectation. The second part of … Read more
This paper studies the matrix completion problem under arbitrary sampling schemes. We propose a new estimator incorporating both max-norm and nuclear-norm regularization, based on which we can conduct efficient low-rank matrix recovery using a random subset of entries observed with additive noise under general non-uniform and unknown sampling distributions. This method significantly relaxes the uniform … Read more
Machine learning theory typically assumes that training data is unbiased and not adversarially generated. When real training data deviates from these assumptions, trained models make erroneous predictions, sometimes with disastrous effects. Robust losses, such as the huber norm are designed to mitigate the effects of such contaminated data, but they are limited to the regression … Read more
High-order optimality conditions for convexly-constrained nonlinear optimization problems are analyzed. A corresponding (expensive) measure of criticality for arbitrary order is proposed and extended to define high-order $\epsilon$-approximate critical points. This new measure is then used within a conceptual trust-region algorithm to show that, if derivatives of the objective function up to order $q \geq 1$ … Read more
Abstract—In this article, we introduce new techniques to solve the nonlinear regression problem and the nonlinear classification problem. Our benchmarks suggest that our method for regression is significantly more effective when compared to classical methods and our method for classification is competitive. Our list of classical methods includes least squares, random forests, decision trees, boosted … Read more
This paper provides a review and commentary on the past, present, and future of numerical optimization algorithms in the context of machine learning applications. Through case studies on text classification and the training of deep neural networks, we discuss how optimization problems arise in machine learning and what makes them challenging. A major theme of … Read more
Stochastic approximation techniques play an important role in solving many problems encountered in machine learning or adaptive signal processing. In these contexts, the statistics of the data are often unknown a priori or their direct computation is too intensive, and they have thus to be estimated online from the observed signals. For batch optimization of … Read more
The paper answers several open questions of the alternating direction method of multipliers (ADMM) and the block coordinate descent (BCD) method that are now wildly used to solve large scale convex optimization problems in many fields. For ADMM, it is still lack of theoretical understanding of the algorithm when the objective function is not separable … Read more
In this paper, we combine ideas from machine learning (ML) and operations research and management science (OR/MS) in developing a framework, along with specific methods, for using data to prescribe optimal decisions in OR/MS problems. In a departure from other work on data-driven optimization and reflecting our practical experience with the data available in applications … Read more