Researchers analyze how much predetermined stepsizes can speed up gradient descent in smooth convex optimization. They establish a lower bound matching the so-called silver rate, up to a doubly logarithmic correction factor. The result indicates that this rate is essentially the best achievable acceleration for the setting studied.
A new arXiv preprint studies how many oracle queries are needed to find a point where the residual of a nonexpansive self-map on a compact convex set falls below a tolerance, measured in a general norm. The analysis covers stochastic fixed-point equations, quantifying the query cost of computing an approximate fixed point in this setting. Although cross-listed under machine learning, the contribution is primarily optimization-theoretic.
A newly posted paper builds explicit counterexamples to Rockafellar's sum conjecture, showing pairs of maximally monotone operators that meet the interior-domain qualification yet whose sum fails maximal monotonicity. One example is constructed on the Banach space c0 and another on l1. The results settle a long-standing open question in convex analysis and optimization theory.