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
H\"older 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.

通过对 Shor 子梯度分析的推广，我们将子梯度方法的经典收敛速度理论扩展到可适用于非 Lipschitz 函数。我们证明了在任何具有局部 Lipschitz 性的凸函数中，确定性投影子梯度算法的全局 O（1/√T）收敛速度。我们还表明，对于具有最多二次增长的凸函数，随机投影子梯度方法的收敛速度为 O（1/√T），在强凸性或较弱的二次下限条件下，该速度可进一步提高至 O (1/T)。