A novel analysis of gradient descent under directional smoothness

We develop new sub-optimality bounds for gradient descent that depend on the conditioning of the objective along the path of optimization, rather than on global, worst-case constants. Key to our proofs is directional smoothness, a measure of gradient variation that we use to develop upper-bounds on the objective. Minimizing these upper-bounds requires solving an implicit equation to obtain an adapted step-size; we show that this equation is straightforward to solve for convex quadratics and leads to new guarantees for a classical step-size sequence. For general functions, we prove that exponential search can be used to obtain a path-dependent convergence guarantee with only a log-log dependency on the global smoothness constant. Experiments on quadratic functions showcase the utility of our theory and connections to the edge-of-stability phenomenon.