A Descent Lemma Beyond Lipschitz Gradient Continuity: First-Order Methods Revisited and Applications

Abstract

The proximal gradient and its variants is one of the most attractive first-order algorithm for minimizing the sum of two convex functions, with one being nonsmooth. However, it requires the differentiable part of the objective to have a Lipschitz continuous gradient, thus precluding its use in many applications. In this paper we introduce a framework which allows to circumvent the intricate question of Lipschitz continuity of gradients by using an elegant and easy to check convexity condition which captures the geometry of the constraints. This condition translates into a new descent lemma which in turn leads to a natural derivation of the proximal-gradient scheme with Bregman distances. We then identify a new notion of asymmetry measure for Bregman distances, which is central in determining the relevant step-size. These novelties allow to prove a global sublinear rate of convergence, and as a by-product, global pointwise convergence is obtained. This provides a new path to a broad spectrum of problems arising in key applications which were, until now, considered as out of reach via proximal gradient methods. We illustrate this potential by showing how our results can be applied to build new and simple schemes for Poisson inverse problems.

Bregman Proximal Mappings and Bregman–Moreau Envelopes Under Relative Prox-Regularity

“…For relatively prox-regular functions, our results guarantee stationarity with respect to the equivalent inf-projected problem that involves the Bregman-Moreau envelope. Based on the gradient formulas for the Bregman-Moreau envelope, we conclude this section by highlighting a local equivalence between alternating minimization and recent Bregman proximal gradient algorithms [6,18,5].…”

“…Moreover, in analogy to the Euclidean setting, for which strongly amenable functions provide a source of examples for prox-regular functions, we introduce relatively amenable functions. Their definition is based on a recent generalization of functions that have a Lipschitz continuous gradient to L-smooth adaptable (or relatively smooth) functions [6,18], i.e., functions that are convex relative to a function that generates the Bregman distance.…”

“…In the following we generalize this concept to the Bregman distance case. To this end the recently introduced generalization of L-smooth functions to L-smooth adaptable (L-smad) functions [6,18] (called relatively smooth functions in [40]) is used. We state a slightly modified version, where we introduce an additional open subset V ⊆ int(dom φ) of int(dom φ) and require the property to hold only on V instead of int(dom φ).…”

Scalable preprocessing for sparse scRNA-seq data exploiting prior knowledge

“…This is not true for the Poisson Log-Likelihood. However, (Bauschke et al, 2016) utilized a different definition of smoothness and derived a generalized algorithm (NoLips) that is capable of achieving a sublinear rate of convergence for a class of non-Lipschitz continuous functions including the Poisson Log-Likelihood. Here we use a custom alternating minimization approach using the NoLips algorithm to optimize the Poisson Log-Likelihood with additional modifications to allow for faster computation for sparse matrices and ability to parallelize the computation (see supplementary materials).…”

“…UNCURL is capable of running on larger datasets comprising of up to millions of cells. UNCURL uses a fast public NMF package (Pedregosa et al, 2011) for the log-normal and Gaussian distributions, while using SPNoLips (Sparse-Parallel-NoLips, described in supplementary methods), which is a custom implementation based on the NoLips algorithm (Bauschke et al (2016)) for the Poisson distribution. Both implementations are capable of using sparse matrices as input for memory and runtime advantages, and are parallelizable.…”