Line Search for Convex Minimization
Abstract
Golden-section search and bisection search are the two main principled algorithms for 1d minimization of quasiconvex (unimodal) functions. The first one only uses function queries, while the second one also uses gradient queries. Other algorithms exist under much stronger assumptions, such as Newton's method for twice-differentiable, strongly convex functions with Lipschitz gradients. However, to the best of our knowledge, there is no principled exact line search algorithm that for general convex functions — including piecewise-linear and max-compositions of convex functions — that takes advantage of convexity.
We propose two such algorithms: $\Delta$-Bisection is a variant of bisection search that uses (sub)gradient information and convexity to speed up convergence, while $\Delta$-Secant is a variant of golden-section search and uses only function queries.
Both algorithms are based on a refined definition of the optimality region $\Delta$ containing the minimum point, for general convex functions.
While bisection search reduces the $x$ interval by a factor 2 at every iteration, $\Delta$-Bisection reduces the (sometimes much) smaller $x^*$-gap $\Delta^x$ (the $x$ coordinates of $\Delta$) by at least a factor 2 at every iteration. Similarly, $\Delta$-Secant also reduces the $x^*$-gap by at least a factor 2 every second function query.
Moreover, and possibly more importantly, the $y^*$-gap $\Delta^y$ (the $y$ coordinates of $\Delta$) also provides a refined stopping criterion, which can also be used with other algorithms.
Experiments on a few convex functions confirm that our algorithms are always faster than their quasiconvex counterparts, often by more than a factor 2.
We further design a quasi-exact line search algorithm based on $\Delta$-Secant. It can be used with gradient descent as a replacement for backtracking line search, for which some parameters can be finicky to tune — and we provide examples to this effect, on strongly-convex and smooth functions. We provide convergence guarantees, and confirm the efficiency of quasi-exact line search on a few single- and multivariate convex functions.
Authors
Laurent Orseau, Marcus Hutter
Venue
arXiv