Shift-Curvature, SGD, and Generalization

A longstanding debate surrounds the related hypotheses that low-curvature minima generalize better, and that SGD discourages curvature. We offer a more complete and nuanced view in support of both. First, we show that curvature harms test performance through two new mechanisms, the shift-curvature and bias-curvature, in addition to a known parameter-covariance mechanism. The three curvature-mediated contributions to test performance are reparametrization-invariant although curvature is not. The shift in the shift-curvature is the line connecting train and test local minima, which differ due to dataset sampling or distribution shift. Although the shift is unknown at training time, the shift-curvature can still be mitigated by minimizing overall curvature. Second, we derive a new, explicit SGD steady-state distribution showing that SGD optimizes an effective potential related to but different from train loss, and that SGD noise mediates a trade-off between deep versus low-curvature regions of this effective potential. Third, combining our test performance analysis with the SGD steady state shows that for small SGD noise, the shift-curvature may be the most significant of the three mechanisms. Our experiments confirm the impact of shift-curvature on test loss, and further explore the relationship between SGD noise and curvature.