Conjecture: eliminating the logarithmic factor in last-iterate SGD bounds

Prove that the expected last-iterate convergence rate of Stochastic Gradient Descent for convex and L-smooth stochastic optimization can be achieved at O(1/√T) without the logarithmic ln(T) factor.

Background

Building on analogies with convex Lipschitz settings where specialized step-size schedules can remove logarithmic factors, the authors conjecture that a similar elimination is possible in the convex smooth stochastic setting. Establishing this would improve the known O(ln(T)/√T) rate to O(1/√T) and refine our understanding of SGD’s last-iterate behavior without variance assumptions.