  Generalization and optimization guarantees on the population loss in machine
learning often rely on uniform convergence based analysis, typically based on
the Rademacher complexity of the predictors. The rich representation power of
modern models has led to concerns about this approach. In this paper, we
present generalization and optimization guarantees in terms of the complexity
of the gradients, as measured by the Loss Gradient Gaussian Width (LGGW).
First, we introduce generalization guarantees directly in terms of the LGGW
under a flexible gradient domination condition, which we demonstrate to hold
empirically for deep models. Second, we show that sample reuse in finite sum
(stochastic) optimization does not make the empirical gradient deviate from the
population gradient as long as the LGGW is small. Third, focusing on deep
networks, we present results showing how to bound their LGGW under mild
assumptions. In particular, we show that their LGGW can be bounded (a) by the
$L_2$-norm of the loss Hessian eigenvalues, which has been empirically shown to
be $\tilde{O}(1)$ for commonly used deep models; and (b) in terms of the
Gaussian width of the featurizer, i.e., the output of the last-but-one layer.
To our knowledge, our generalization and optimization guarantees in terms of
LGGW are the first results of its kind, avoid the pitfalls of predictor
Rademacher complexity based analysis, and hold considerable promise towards
quantitatively tight bounds for deep models.
