Title: Convergence Rates for Deterministic and Stochastic Subgradient Methods Without Lipschitz Continuity,
Abstract: We extend the classic convergence rate theory for subgradient methods to
apply to non-Lipschitz functions. For the deterministic projected subgradient
method, we present a global $O(1/\sqrt{T})$ convergence rate for any convex
function which is locally Lipschitz around its minimizers. This approach is
based on Shor's classic subgradient analysis and implies generalizations of the
standard convergence rates for gradient descent on functions with Lipschitz or
Hölder continuous gradients. Further, we show a $O(1/\sqrt{T})$ convergence
rate for the stochastic projected subgradient method on convex functions with
at most quadratic growth, which improves to $O(1/T)$ under either strong
convexity or a weaker quadratic lower bound condition.