What is the long-run distribution of stochastic gradient descent? A large deviations analysis
This provides foundational insights into SGD's behavior for researchers in optimization and machine learning, though it is theoretical and incremental in analyzing existing methods.
The paper tackles the problem of understanding the long-run distribution of stochastic gradient descent (SGD) in non-convex settings, showing that it resembles a Boltzmann-Gibbs distribution where critical regions are visited exponentially more often than non-critical ones, with iterates concentrated around the minimum energy state.
In this paper, we examine the long-run distribution of stochastic gradient descent (SGD) in general, non-convex problems. Specifically, we seek to understand which regions of the problem's state space are more likely to be visited by SGD, and by how much. Using an approach based on the theory of large deviations and randomly perturbed dynamical systems, we show that the long-run distribution of SGD resembles the Boltzmann-Gibbs distribution of equilibrium thermodynamics with temperature equal to the method's step-size and energy levels determined by the problem's objective and the statistics of the noise. In particular, we show that, in the long run, (a) the problem's critical region is visited exponentially more often than any non-critical region; (b) the iterates of SGD are exponentially concentrated around the problem's minimum energy state (which does not always coincide with the global minimum of the objective); (c) all other connected components of critical points are visited with frequency that is exponentially proportional to their energy level; and, finally (d) any component of local maximizers or saddle points is "dominated" by a component of local minimizers which is visited exponentially more often.