  Variable metric proximal gradient (VM-PG) is a widely used class of convex
optimization method. Lately, there has been a lot of research on the
theoretical guarantees of VM-PG with different metric selections. However, most
such metric selections are dependent on (an expensive) Hessian, or limited to
scalar stepsizes like the Barzilai-Borwein (BB) stepsize with lots of
safeguarding. Instead, in this paper we propose an adaptive metric selection
strategy called the diagonal Barzilai-Borwein (BB) stepsize. The proposed
diagonal selection better captures the local geometry of the problem while
keeping per-step computation cost similar to the scalar BB stepsize i.e.
$O(n)$. Under this metric selection for VM-PG, the theoretical convergence is
analyzed. Our empirical studies illustrate the improved convergence results
under the proposed diagonal BB stepsize, specifically for ill-conditioned
machine learning problems for both synthetic and real-world datasets.
