"Deep lifts": Neural networks can generalize even with the dumbest optimizer

Sunil Madhow, Yu-Xiang Wang

Given a labeled dataset $\{(x_i, y_i)\}_{i = 1}^n$, consider drawing a parameter vector $\hatθ$ uniformly at random from those that interpolate the data within a width-$W$ depth-$D$ ReLU architecture with continuous weights bounded within $[-M, M]$. We study the generalization properties of $\hatθ$ in the case where the dataset can, in fact, be explained by a smaller $w$-by-$d$ ReLU network. Despite the obliviousness of the selection rule for $\hatθ$, it adaptively generalizes at the rate $\tilde O\left(\frac{wdWD}{n}\right)$, rather than being governed by the VC dimension of the full $W$-by-$D$ hypothesis class. The reason is due to a particular inductive bias in ReLU networks: the $w$-by-$d$ parameter *witnesses* an exponential volume of $W$-by-$D$ interpolating parameters, which dwarfs the volume of poorly generalizing parameters (conditioned on interpolation). We explicitly construct this set of near-equivalent $W$-by-$D$ parameters by a proof technique called *deep lifting*, and then tie the achievable volume to stability properties of the $w$-by-$d$ witness. Examples show that these techniques allow us to convert approximation-theoretic results into generalization…