  Stochastic Gradient Descent (SGD) is a widely deployed optimization procedure
throughout data-driven and simulation-driven disciplines, which has drawn a
substantial interest in understanding its global behavior across a broad class
of nonconvex problems and noise models. Recent analyses of SGD have made
noteworthy progress in this direction, and these analyses have innovated
important and insightful new strategies for understanding SGD. However, these
analyses often have imposed certain restrictions (e.g., convexity, global
Lipschitz continuity, uniform Holder continuity, expected smoothness, etc.)
that leave room for innovation. In this work, we address this gap by proving
that, for a rather general class of nonconvex functions and noise models, SGD's
iterates either diverge to infinity or converge to a stationary point with
probability one. By further restricting to globally Holder continuous functions
and the expected smoothness noise model, we prove that -- regardless of whether
the iterates diverge or remain finite -- the norm of the gradient function
evaluated at SGD's iterates converges to zero with probability one and in
expectation. As a result of our work, we broaden the scope of nonconvex
problems and noise models to which SGD can be applied with rigorous guarantees
of its global behavior.
