New computational guarantees for solving convex optimization problems with first order methods, via a function growth condition measure

Abstract

Motivated by recent work of Renegar, we present new computational methods and associated computational guarantees for solving convex optimization problems using first-order methods. Our problem of interest is the general convex optimization problem f[superscript ∗] = min[subscript x∈Q] f(x),where we presume knowledge of a strict lower bound f[subscript slb] < f[superscript ∗]. [Indeed, f[subscript slb] is naturally known when optimizing many loss functions in statistics and machine learning (least-squares, logistic loss, exponential loss, total variation loss, etc.) as well as in Renegar’s transformed version of the standard conic optimization problem; in all these cases one has f[subscript slb] = 0 < f[superscript ∗] .] We introduce a new functional measure called the growth constant G for f(·), that measures how quickly the level sets of f(·) grow relative to the function value, and that plays a fundamental role in the complexity analysis. When f(·) is non-smooth, we present new computational guarantees for the Subgradient Descent Method and for smoothing methods, that can improve existing computational guarantees in several ways, most notably when the initial iterate x[superscript 0] is far from the optimal solution set. When f(·) is smooth, we present a scheme for periodically restarting the Accelerated Gradient Method that can also improve existing computational guarantees when x[superscript 0] is far from the optimal solution set, and in the presence of added structure we present a scheme using parametrically increased smoothing that further improves the associated computational guarantees