Stochastic Model-Based Minimization of Weakly Convex Functions

Abstract

We consider a family of algorithms that successively sample and minimize simple stochastic models of the objective function. We show that under reasonable conditions on approximation quality and regularity of the models, any such algorithm drives a natural stationarity measure to zero at the rate O(k −1/4 ). As a consequence, we obtain the first complexity guarantees for the stochastic proximal point, proximal subgradient, and regularized Gauss-Newton methods for minimizing compositions of convex functions with smooth maps. The guiding principle, underlying the complexity guarantees, is that all algorithms under consideration can be interpreted as approximate descent methods on an implicit smoothing of the problem, given by the Moreau envelope. Specializing to classical circumstances, we obtain the long-sought convergence rate of the stochastic projected gradient method, without batching, for minimizing a smooth function on a closed convex set.

Convergence of a stochastic subgradient method with averaging for nonsmooth nonconvex constrained optimization

“…An essential step in the analysis of stochastic recursive algorithms by the differential inclusion method is the chain rule on a path (see [9] and the references therein). For an absolutely continuous function p : [0, ∞) → Ê n we denote by • p(·) its weak derivative: a measurable function such that…”

The importance of better models in stochastic optimization

“…By providing a "relative" noise condition on f , Assumption A4 allows for a broader class of functions without global Lipschitz properties (as are typically assumed [8]), such as the phase retrieval and matrix completion objectives (Examples 1 and 2). It can allow exponential growth, addressing the challenges in Ex.…”

“…To describe convergence and stability guarantees in non-convex (even non-smooth) settings, we require appropriate definitions. Finding global minima of non-convex functions is computationally infeasible [26], so we follow established practice and consider convergence to stationary points, specifically using the convergence of the Moreau envelope [8,13]. To formalize, for x ∈ R n and λ ≥ 0, the Moreau envelope and associated proximal map are…”

“…For large enough λ, the minimizer x λ := prox F/λ (x) is unique whenever F is weakly convex. Adopting the techniques pioneered by Davis and Drusvyatskiy [8] for convergence of stochastic methods on weakly convex problems, our convergence machinery relies on the Moreau envelope's connections to (near) stationarity:…”

“…The three properties (8) imply that any nearly stationary point x of F λ (x)-when ∇F λ (x) 2 is small-is close to a nearly stationary point x λ of the original function F (·). To prove convergence for weakly convex methods, then, it is sufficient to show that ∇F λ (x k ) → 0.…”