Gradient methods for convex minimization: better rates under weaker conditions

TL;DR

This paper demonstrates that gradient methods for convex minimization can achieve optimal convergence rates under weaker, more general conditions than traditionally assumed, leading to improved complexity bounds.

Contribution

It introduces relaxed conditions on gradient Lipschitz continuity and strong convexity, resulting in better convergence rates for gradient methods.

Findings

Achieves $O(rac{R}{ u} ext{log}(rac{1}{\epsilon}))$ complexity under weaker conditions.

Establishes necessity of secant inequality for geometric decay.

Demonstrates faster algorithms for sparse optimization.

Abstract

The convergence behavior of gradient methods for minimizing convex differentiable functions is one of the core questions in convex optimization. This paper shows that their well-known complexities can be achieved under conditions weaker than the commonly accepted ones. We relax the common gradient Lipschitz-continuity condition and strong convexity condition to ones that hold only over certain line segments. Specifically, we establish complexities $O(\frac{R}{\epsilon})$ and $O(\sqrt{\frac{R}{\epsilon}})$ for the ordinary and accelerate gradient methods, respectively, assuming that $\nabla f$ is Lipschitz continuous with constant $R$ over the line segment joining $x$ and $x-\frac{1}{R}\nabla f$ for each $x\in\dom f$. Then we improve them to $O(\frac{R}{\nu}\log(\frac{1}{\epsilon}))$ and $O(\sqrt{\frac{R}{\nu}}\log(\frac{1}{\epsilon}))$ for function $f$ that also satisfies the secant inequality $\ < \nabla f(x), x- x^*\ > \ge \nu\|x-x^*\|^2$ for each $x\in \dom f$ and its projection $x^*$ to the minimizer set of $f$. The secant condition is also shown to be necessary for the geometric decay of solution error. Not only are the relaxed conditions met by more functions, the restrictions give smaller $R$ and larger $\nu$ than they are without the restrictions and thus lead to better complexity bounds. We apply these results to sparse optimization and demonstrate a faster algorithm.