Acceleration via Fractal Learning Rate Schedules

When balancing the practical tradeoffs of iterative methods for large-scale optimization, the learning rate schedule remains notoriously difficult to understand and expensive to tune. We demonstrate the presence of these subtleties even in the innocuous case when the objective is a convex quadratic. We reinterpret an iterative algorithm from the numerical analysis literature as what we call the Chebyshev learning rate schedule for accelerating vanilla gradient descent, and show that the problem of mitigating instability leads to a fractal ordering of step sizes. We provide some experiments and discussion to challenge current understandings of the "edge of stability" in deep learning: even in simple settings, provable acceleration can be obtained by making negative local progress on the objective.