Related papers: The Bregman proximal average

We investigate convergence of alternating Bregman projections between non-convex sets and prove convergence to a point in the intersection, or to points realizing a gap between the two sets. The speed of convergence is generally sub-linear,…

We propose a novel stochastic distributed method for both monotone and strongly monotone variational inequalities with Lipschitz operator and proper convex regularizers arising in various applications from game theory to adversarial…

We propose a unifying algorithm for non-smooth non-convex optimization. The algorithm approximates the objective function by a convex model function and finds an approximate (Bregman) proximal point of the convex model. This approximate…

Consider a differentiable convex function $f: \mathbb{R}^n \supset \mathrm{dom} f \rightarrow \mathbb{R}.$ The induced spectral function $F$ is given by $F=f \circ \lambda,$ where $\lambda: \mathbf{M}_n^{sa} \rightarrow \mathbb{R}^{n}$ is…

We introduce a notion of variable quasi-Bregman monotone sequence which unifies the notion of variable metric quasi-Fej\'er monotone sequences and that of Bregman monotone sequences. The results are applied to analyze the asymptotic…

Bregman proximal point algorithm (BPPA) has witnessed emerging machine learning applications, yet its theoretical understanding has been largely unexplored. We study the computational properties of BPPA through learning linear classifiers…

Comparative convexity is a generalization of convexity relying on abstract notions of means. We define the Jensen divergence and the Jensen diversity from the viewpoint of comparative convexity, and show how to obtain the generalized…

This work proposes the Bregman-Tweedie classification model and analyzes the domain structure of the extended exponential function, an extension of the classic generalized exponential function with additional scaling parameter, and related…

Convergence rates results for variational regularization methods typically assume the regularization functional to be convex. While this assumption is natural for scalar-valued functions, it can be unnecessarily strong for vector-valued…

We consider the problem of minimizing the sum of two convex functions: one is the average of a large number of smooth component functions, and the other is a general convex function that admits a simple proximal mapping. We assume the whole…

This paper introduces adaptive Bregman proximal gradient algorithms for solving convex composite minimization problems without relying on global relative smoothness or strong convexity assumptions. Building upon recent advances in adaptive…

Convex optimization models find interesting applications, especially in signal/image processing and compressive sensing. We study some augmented convex models, which are perturbed by strongly convex functions, and propose a dual gradient…

A class of distortions termed functional Bregman divergences is defined, which includes squared error and relative entropy. A functional Bregman divergence acts on functions or distributions, and generalizes the standard Bregman divergence…

We propose efficient Langevin Monte Carlo algorithms for sampling distributions with nonsmooth convex composite potentials, which is the sum of a continuously differentiable function and a possibly nonsmooth function. We devise such…

We consider minimization of functions that are compositions of convex or prox-regular functions (possibly extended-valued) with smooth vector functions. A wide variety of important optimization problems fall into this framework. We describe…

The proximal average of two convex functions has proven to be a useful tool in convex analysis. In this note, we express Goebel's self-dual smoothing operator in terms of the proximal average, which allows us to give a simple proof of self…

We define the notion of Bernstein measures and Bernstein approximations over general convex polytopes. This generalizes well-known Bernstein polynomials which are used to prove the Weierstrass approximation theorem on one dimensional…

We propose a forward-backward splitting algorithm based on Bregman distances for composite minimization problems in general reflexive Banach spaces. The convergence is established using the notion of variable quasi-Bregman monotone…

The inequality of Berwald is a reverse-H\"older like inequality for the $p$th average, $p\in (-1,\infty),$ of a non-negative, concave function over a convex body in $\mathbb{R}^n.$ We prove Berwald's inequality for averages of functions…

Recently, we systematically studied the basic theory of Bregman circumcenters in another paper. In this work, we aim to apply Bregman circumcenters to optimization algorithms. Here, we propose the forward Bregman monotonicity which is a…