Anisotropic Proximal Gradient

This paper studies a novel algorithm for nonconvex composite minimization which can be interpreted in terms of dual space nonlinear preconditioning for the classical proximal gradient method. The proposed scheme can be applied to composite minimization problems whose smooth part exhibits an anisotropic descent inequality relative to a reference function. In the convex case this is a dual characterization of relative strong convexity in the Bregman sense. It is proved that the anisotropic descent property is closed under pointwise average if the dual Bregman distance is jointly convex and, more specifically, closed under pointwise conic combinations for the KL-divergence. We analyze the method's asymptotic convergence and prove its linear convergence under an anisotropic proximal gradient dominance condition. This is implied by anisotropic strong convexity, a recent dual characterization of relative smoothness in the Bregman sense. Applications are discussed including exponentially regularized LPs and logistic regression with nonsmooth regularization. In the LP case the method can be specialized to the Sinkhorn algorithm for regularized optimal transport and a classical parallel update algorithm for AdaBoost. Complementary to their existing primal interpretations in terms of entropic subspace projections this provides a new dual interpretation in terms of forward-backward splitting with entropic preconditioning

Sublabel-Accurate Relaxation of Nonconvex Energies

We propose a novel spatially continuous framework for convex relaxations based on functional lifting. Our method can be interpreted as a sublabel-accurate solution to multilabel problems. We show that previously proposed functional lifting methods optimize an energy which is linear between two labels and hence require (often infinitely) many labels for a faithful approximation. In contrast, the proposed formulation is based on a piecewise convex approximation and therefore needs far fewer labels. In comparison to recent MRF-based approaches, our method is formulated in a spatially continuous setting and shows less grid bias. Moreover, in a local sense, our formulation is the tightest possible convex relaxation. It is easy to implement and allows an efficient primal-dual optimization on GPUs. We show the effectiveness of our approach on several computer vision problems

Convex relaxations for large-scale graphically structured nonconvex problems with spherical constraints: An optimal transport approach

In this paper we derive a moment relaxation for large-scale nonsmooth optimization problems with graphical structure and spherical constraints. In contrast to classical moment relaxations for global polynomial optimization that suffer from the curse of dimensionality we exploit the partially separable structure of the optimization problem to reduce the dimensionality of the search space. Leveraging optimal transport and Kantorovich--Rubinstein duality we decouple the problem and derive a tractable dual subspace approximation of the infinite-dimensional problem using spherical harmonics. This allows us to tackle possibly nonpolynomial optimization problems with spherical constraints and geodesic coupling terms. We show that the duality gap vanishes in the limit by proving that a Lipschitz continuous dual multiplier on a unit sphere can be approximated as closely as desired in terms of a Lipschitz continuous polynomial. The formulation is applied to sphere-valued imaging problems with total variation regularization and graph-based simultaneous localization and mapping (SLAM). In imaging tasks our approach achieves small duality gaps for a moderate degree. In graph-based SLAM our approach often finds solutions which after refinement with a local method are near the ground truth solution

Discrete-Continuous ADMM for Transductive Inference in Higher-Order MRFs

This paper introduces a novel algorithm for transductive inference in higher-order MRFs, where the unary energies are parameterized by a variable classifier. The considered task is posed as a joint optimization problem in the continuous classifier parameters and the discrete label variables. In contrast to prior approaches such as convex relaxations, we propose an advantageous decoupling of the objective function into discrete and continuous subproblems and a novel, efficient optimization method related to ADMM. This approach preserves integrality of the discrete label variables and guarantees global convergence to a critical point. We demonstrate the advantages of our approach in several experiments including video object segmentation on the DAVIS data set and interactive image segmentation

Dualities for Non-Euclidean Smoothness and Strong Convexity under the Light of Generalized Conjugacy

Relative smoothness and strong convexity have recently gained considerable attention in optimization. These notions are generalizations of the classical Euclidean notions of smoothness and strong convexity that are known to be dual to each other. However, conjugate dualities for non-Euclidean relative smoothness and strong convexity remain an open problem, as noted earlier by Lu, Freund, and Nesterov [SIAM J. Optim., 28 (2018), pp. 333–354]. In this paper, we address this question by introducing the notions of anisotropic strong convexity and smoothness as the respective dual counterparts. The dualities are developed under the light of generalized conjugacy, which leads us to embed the anticipated dual notions within the superclasses of certain upper and lower envelopes. In contrast to the Euclidean case, these inclusions are proper in general, as showcased by means of counterexamples