Adaptive Bregman Alternating Projections for Feasible Gromov-Wasserstein Learning

2026-10-03

AI Summary

This study addresses the computational challenges arising from the non-convexity and coupled constraints of the Gromov-Wasserstein problem, as well as the persistent feasibility gap inherent in fixed-penalty methods. To overcome these limitations, we propose an adaptive KL-BAPG algorithm that integrates fixed-penalty warm-starting with an increasing penalty strategy, combining Bregman alternating proximal gradient updates, Sinkhorn-inspired projection diameter protection, and delayed power backtracking line search. Theoretically, we establish the finite termination of the algorithm, the existence of stationary points for the original problem, and convergence bounds superior to those of relaxed formulations. Empirically, while maintaining comparable accuracy, the proposed method reduces marginal feasibility gaps by 62–99% and stationarity residuals by 28–98% relative to fixed-penalty baselines.

Abstract

The Gromov-Wasserstein (GW) problem compares structured distributions without requiring a shared feature space or known correspondences, but its nonconvex objective and coupled marginal constraints make computation challenging. Bregman alternating projected gradient (BAPG) uses inexpensive alternating row and column updates, yet its fixed-penalty relaxation leaves a persistent feasibility gap. We propose Adaptive KL-BAPG (A-KL-BAPG), which combines a finite fixed-penalty burn-in with a guarded increasing-penalty phase. At each tail iteration, the method reuses BAPG's alternating updates and backtracks a delayed-power step until a Sinkhorn-inspired projective-diameter safeguard is satisfied. We prove finite termination of the backtracking at each iteration and show that the feasibility gap vanishes asymptotically. We further establish a best-iterate $O(1/\log N)$ bound for the weighted squared corrected residual and, under a support regularity condition, the existence of a stationary accumulation point for the original GW problem. This distinguishes A-KL-BAPG from fixed-penalty BAPG, whose stationarity guarantees are given for the relaxed problem. Experiments show that A-KL-BAPG achieves a favorable balance of accuracy, objective value, feasibility, and stationarity relative to BAPG variants, projection-based methods, and task-specific baselines. For synthetic and real graph alignment problems, it closely matches the accuracy and objective value of fixed-penalty KL-BAPG while reducing the marginal feasibility gap by 62-99% and the projected stationarity residual by 28-98%. Heterogeneous domain adaptation experiments show a similar pattern: A-KL-BAPG maintains comparable target accuracy and objective values while achieving better feasibility and stationarity than fixed-penalty KL-BAPG.

Problem

• Gromov-Wasserstein
• feasibility gap
• Bregman alternating projections
• nonconvex optimization
• marginal constraints

Innovation

• Gromov-Wasserstein
• Adaptive Bregman Alternating Projections
• Feasibility Gap
• Backtracking Safeguard
• Convergence Guarantee