On the fast convergence of minibatch heavy ball momentum
This work addresses a gap between theory and practice for machine learning optimization, offering theoretical justification for widely used stochastic momentum methods, though it is incremental as it focuses on quadratic problems with large batches.
The paper tackles the lack of theoretical guarantees for stochastic momentum methods in optimization by showing that minibatch heavy ball momentum achieves fast linear convergence on quadratic problems with sufficiently large batches, matching deterministic rates, and provides numerical evidence that the bounds are sharp.
Simple stochastic momentum methods are widely used in machine learning optimization, but their good practical performance is at odds with an absence of theoretical guarantees of acceleration in the literature. In this work, we aim to close the gap between theory and practice by showing that stochastic heavy ball momentum retains the fast linear rate of (deterministic) heavy ball momentum on quadratic optimization problems, at least when minibatching with a sufficiently large batch size. The algorithm we study can be interpreted as an accelerated randomized Kaczmarz algorithm with minibatching and heavy ball momentum. The analysis relies on carefully decomposing the momentum transition matrix, and using new spectral norm concentration bounds for products of independent random matrices. We provide numerical illustrations demonstrating that our bounds are reasonably sharp.