Stochastic Gradient Descent-like relaxation is equivalent to Metropolis dynamics in discrete optimization and inference problems
This provides a theoretical link between two fundamental algorithms in machine learning, potentially improving training efficiency for researchers and practitioners.
The paper tackles the question of whether Stochastic Gradient Descent (SGD) differs from Metropolis Monte Carlo dynamics, showing that in discrete optimization and inference problems, an SGD-like algorithm closely matches Metropolis dynamics with a temperature dependent on mini-batch size, enabling optimization of mini-batch size for efficiency in hard inference problems.
Is Stochastic Gradient Descent (SGD) substantially different from Metropolis Monte Carlo dynamics? This is a fundamental question at the time of understanding the most used training algorithm in the field of Machine Learning, but it received no answer until now. Here we show that in discrete optimization and inference problems, the dynamics of an SGD-like algorithm resemble very closely that of Metropolis Monte Carlo with a properly chosen temperature, which depends on the mini-batch size. This quantitative matching holds both at equilibrium and in the out-of-equilibrium regime, despite the two algorithms having fundamental differences (e.g.\ SGD does not satisfy detailed balance). Such equivalence allows us to use results about performances and limits of Monte Carlo algorithms to optimize the mini-batch size in the SGD-like algorithm and make it efficient at recovering the signal in hard inference problems.