  Recent works have shown that high probability metrics with stochastic
gradient descent (SGD) exhibit informativeness and in some cases advantage over
the commonly adopted mean-square error-based ones. In this work we provide a
formal framework for the study of general high probability bounds with SGD,
based on the theory of large deviations. The framework allows for a generic
(not-necessarily bounded) gradient noise satisfying mild technical assumptions,
allowing for the dependence of the noise distribution on the current iterate.
Under the preceding assumptions, we find an upper large deviations bound for
SGD with strongly convex functions. The corresponding rate function captures
analytical dependence on the noise distribution and other problem parameters.
This is in contrast with conventional mean-square error analysis that captures
only the noise dependence through the variance and does not capture the effect
of higher order moments nor interplay between the noise geometry and the shape
of the cost function. We also derive exact large deviation rates for the case
when the objective function is quadratic and show that the obtained function
matches the one from the general upper bound hence showing the tightness of the
general upper bound. Numerical examples illustrate and corroborate theoretical
findings.
