Acceleration by Stepsize Hedging I: Multi-Step Descent and the Silver Stepsize Schedule
Abstract¶
Can we accelerate convergence of gradient descent without changing the algorithm - just by carefully choosing stepsizes? Surprisingly, we show that the answer is yes. Our proposed Silver Stepsize Schedule optimizes strongly convex functions in k^log_rho 2 approx k^(0).7864 iterations, where rho = 1+sqrt(2) is the silver ratio and k is the condition number. This is intermediate between the textbook unaccelerated rate k and the accelerated rate sqrt(k) due to Nesterov in 1983. The non-strongly convex setting is conceptually identical, and standard black-box reductions imply an analogous accelerated rate epsilon^-log_rho 2 approx epsilon^(-0).7864. We conjecture and provide partial evidence that these rates are optimal among all possible stepsize schedules. The Silver Stepsize Schedule is constructed recursively in a fully explicit way. It is non-monotonic, fractal-like, and approximately periodic of period k^log_rho 2. This leads to a phase transition in the convergence rate: initially super-exponential (acceleration regime), then exponential (saturation regime).