SGD with memory: fundamental properties and stochastic acceleration

An important open problem is the theoretically feasible acceleration of mini-batch SGD-type algorithms on quadratic problems with power-law spectrum. In the non-stochastic setting, the optimal exponent $\xi$ in the loss convergence $L_t\sim C_Lt^{-\xi}$ is double that in plain GD and is achievable using Heavy Ball (HB) with a suitable schedule; this no longer works in the presence of mini-batch noise. We address this challenge by considering first-order methods with an arbitrary fixed number $M$ of auxiliary velocity vectors (*memory-$M$ algorithms*). We first prove an equivalence between two forms of such algorithms and describe them in terms of suitable characteristic polynomials. Then we develop a general expansion of the loss in terms of *signal and noise propagators*. Using it, we show that losses of stationary stable memory-$M$ algorithms always retain the exponent $\xi$ of plain GD, but can have different constants $C_L$ depending on their *effective learning rate* that generalizes that of HB. We prove that in memory-1 algorithms we can make $C_L$ arbitrarily small while maintaining stability. As a consequence, we propose a memory-1 algorithm with a time-dependent schedule that we show heuristically and experimentally to improve the exponent $\xi$ of plain SGD.