Approximately Exact Line Search

We propose approximately exact line search (AELS), which uses only function evaluations to select a step size within a constant fraction of the exact line search minimizer of a unimodal objective. We bound the number of iterations and function evaluations of AELS, showing linear convergence on smooth, strongly convex objectives with no dependence on the initial step size for three descent methods: steepest descent using the true gradient, approximate steepest descent using a gradient approximation, and random direction search. We demonstrate experimental speedup compared to Armijo line searches and other baselines on weakly regularized logistic regression for both gradient descent and minibatch stochastic gradient descent and on a benchmark set of derivative-free optimization objectives using quasi-Newton search directions. We also analyze a simple line search for the strong Wolfe conditions, finding upper bounds on iteration and function evaluation complexity similar to AELS. Experimentally we find AELS is much faster on deterministic and stochastic minibatch logistic regression, whereas Wolfe line search is slightly faster on the DFO benchmark. Our findings suggest that line searches and AELS in particular may be more useful in the stochastic optimization regime than commonly believed.