Rethinking learning rate schedules for stochastic optimization

There is a stark disparity between the learning rate schedules used in the practice of large scale machine learning and what are considered admissible learning rate schedules prescribed in the theory of stochastic approximation. Recent results, such as in the 'super-convergence' methods which use oscillating learning rates, serve to emphasize this point even more. One plausible explanation is that non-convex neural network training procedures are better suited to the use of fundamentally different learning rate schedules, such as the ``cut the learning rate every constant number of epochs'' method (which more closely resembles an exponentially decaying learning rate schedule); note that this widely used schedule is in stark contrast to the polynomial decay schemes prescribed in the stochastic approximation literature, which are indeed shown to be (worst case) optimal for classes of convex optimization problems. The main contribution of this work shows that the picture is far more nuanced, where we do not even need to move to non-convex optimization to show other learning rate schemes can be far more effective. In fact, even for the simple case of stochastic linear regression with a fixed time horizon, the rate achieved by any polynomial decay scheme is sub-optimal compared to the statistical minimax rate (by a factor of condition number); in contrast the ```''cut the learning rate every constant number of epochs'' provides an exponential improvement (depending only logarithmically on the condition number) compared to any polynomial decay scheme. Finally, it is important to ask if our theoretical insights are somehow fundamentally tied to quadratic loss minimization (where we have circumvented minimax lower bounds for more general convex optimization problems)? Here, we conjecture that recent results which make the gradient norm small at a near optimal rate, for both convex and non-convex optimization, may also provide more insights into learning rate schedules used in practice.