Adaptive Bregman Proximal Stochastic Gradient with a Stabilized Barzilai--Borwein Step Size

Bregman proximal stochastic gradient (BPSG) methods bring variance-reduced composite optimization to objectives whose geometry is poorly captured by Euclidean smoothness. Their performance, however, remains sensitive to the step size: raw stochastic curvature estimates can fluctuate sharply, whereas line searches add repeated proximal evaluations. We introduce Ada-BPSG, a line-search-free BPSG method that couples the SAGA gradient table with a stabilized Barzilai--Borwein (BB) candidate. A mediant aggregates incremental secant information so that nearly singular local ratios receive little weight, and an explicit safeguard translates the resulting curvature estimate into the bounded step-size sequence required for convergence. This design yields a direct analytical chain from relative smoothness and component-wise variance control to convergence in finite-dimensional normed spaces. We prove an O(n/K) ergodic rate for convex objectives, a restarted linear rate under relative quadratic growth, and an O(1/K) bound for a Bregman proximal residual in the nonconvex setting. On logistic regression and sparse nonnegative matrix factorization, Ada-BPSG combines low objective values with substantially less sensitivity to the initial step size than standard variance-reduced baselines, while avoiding line search.