Optimal last-iterate rate for SGD in convex smooth stochastic optimization

Determine the optimal worst-case last-iterate convergence rate in expectation for Stochastic Gradient Descent applied to convex and L-smooth stochastic optimization problems without uniform gradient variance assumptions, specifically by establishing whether the best possible rate is O(1/√T) or necessarily involves an O(ln(T)/√T) factor.

Background

The paper proves an expected last-iterate bound of order O(ln(T)/√T) for SGD in convex and smooth stochastic problems without uniform variance assumptions, closing a long-standing gap in the literature. However, the authors note that the presence of the logarithmic factor may not be inherent and the exact optimal worst-case rate remains unknown. Clarifying this would settle the tightness of last-iterate bounds in the smooth convex stochastic setting and guide algorithmic design.