Benjamin Grimmer We prove the first convergence guarantees for a subgradient method minimizing a generic Lipschitz function over generic Lipschitz inequality constraints. No smoothness or convexity (or weak convexity) assumptions are made. Instead, we utilize a sequence of recent advances in Lipschitz unconstrained minimization, which showed convergence rates of $O(1/\delta\epsilon^3)$ towards reaching a “Goldstein” stationary point, that … Read more
Recently Grimmer [1] showed for smooth convex optimization by utilizing longer steps periodically, gradient descent’s state-of-the-art O(1/T) convergence guarantees can be improved by constant factors, conjecturing an accelerated rate strictly faster than O(1/T) could be possible. Here we prove such a big-O gain, establishing gradient descent’s first accelerated convergence rate in this setting. Namely, we … Read more
This work establishes provably faster convergence rates for gradient descent in smooth convex optimization via a computer-assisted analysis technique. Our theory allows nonconstant stepsize policies with frequent long steps potentially violating descent by analyzing the overall effect of many iterations at once rather than the typical one-iteration inductions used in most first-order method analyses. We … Read more
Constrained optimization problems where both the objective and constraints may be nonsmooth and nonconvex arise across many learning and data science settings. In this paper, we show a simple first-order method finds a feasible, ϵ-stationary point at a convergence rate of O(ϵ−4) without relying on compactness or Constraint Qualification (CQ). When CQ holds, this convergence is measured by … Read more
This work considers minimizing a convex sum of functions, each with potentially different structure ranging from nonsmooth to smooth, Lipschitz to non-Lipschitz. Nesterov’s universal fast gradient method provides an optimal black-box first-order method for minimizing a single function that takes advantage of any continuity structure present without requiring prior knowledge. In this paper, we show … Read more
We study convergence rates of the classic proximal bundle method for a variety of nonsmooth convex optimization problems. We show that, without any modification, this algorithm adapts to converge faster in the presence of smoothness or a Hölder growth condition. Our analysis reveals that with a constant stepsize, the bundle method is adaptive, yet it … Read more
(Renegar, 2016) introduced a novel approach to transforming generic conic optimization problems into unconstrained, uniformly Lipschitz continuous minimization. We introduce radial transformations generalizing these ideas, equipped with an entirely new motivation and development that avoids any reliance on convex cones or functions. Perhaps of greatest practical importance, this facilitates the development of new families of … Read more
The first part of this work established the foundations of a radial duality between nonnegative optimization problems, inspired by the work of (Renegar, 2016). Here we utilize our radial duality theory to design and analyze projection-free optimization algorithms that operate by solving a radially dual problem. In particular, we consider radial subgradient, smoothing, and accelerated … Read more
Minimax optimization has become a central tool for modern machine learning with applications in generative adversarial networks, robust optimization, reinforcement learning, etc. These applications are often nonconvex-nonconcave, but the existing theory is unable to identify and deal with the fundamental difficulties posed by nonconvex-nonconcave structures. In this paper, we study the classic proximal point method … Read more
Often in the analysis of first-order methods, assuming the existence of a quadratic growth bound (a generalization of strong convexity) facilitates much stronger convergence analysis. Hence the analysis is done twice, once for the general case and once for the growth bounded case. We give a meta-theorem for deriving general convergence rates from those assuming … Read more