Gradient methods for convex minimization: better rates under weaker conditions

H. Zhang, W. Yin

Submitted for publication

Overview

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 xinmathrm{dom} f.

Then we improve the rates above to O(frac{R}{nu}mathrm{log}(frac{1}{epsilon})) and O(sqrt{frac{R}{nu}}mathrm{log}(frac{1}{epsilon})) for function f that also satisfies the secant inequality langle nabla f(x), x- x^*ranglege nu|x-x^*|^2 for each xin mathrm{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.

Citation

H. Zhang and W. Yin, Gradient methods for convex minimization: better rates under weaker conditions, Rice CAAM technical report 13-04, 2013.


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