Accelerated Bregman proximal gradient methods for relatively smooth convex optimization

We consider the problem of minimizing sum two convex functions: one is differentiable and relatively smooth with respect to a reference function, other can be nondifferentiable but simple optimize. investigate triangle scaling property Bregman distance generated by function present accelerated proximal gradient (ABPG) methods that attain an $$O(k^{-\gamma })$$ convergence rate, where $$\gamma \in (0,2]$$ exponent (TSE) distance. For Euclidean distance, we have =2$$ recover rate Nesterov’s methods. non-Euclidean distances, TSE much smaller (say \le 1$$ ), show relaxed definition intrinsic always equal 2. exploit develop adaptive ABPG converge faster in practice. Although theoretical guarantees on fast seem out reach general, our obtain empirical $$O(k^{-2})$$ rates numerical experiments several applications provide posterior certificates for rates.