Kevin Han Huang

Kevin Han Huang, Ni Zhan, Elif Ertekin et al.

Incorporating group symmetries into neural networks has been a cornerstone of success in many AI-for-science applications. Diagonal groups of isometries, which describe the invariance under a simultaneous movement of multiple objects, arise naturally in many-body quantum problems. Despite their importance, diagonal groups have received relatively little attention, as they lack a natural choice of invariant maps except in special cases. We study different ways of incorporating diagonal invariance in neural network ansätze trained via variational Monte Carlo methods, and consider specifically data augmentation, group averaging and canonicalization. We show that, contrary to standard ML setups, in-training symmetrization destabilizes training and can lead to worse performance. Our theoretical and numerical results indicate that this unexpected behavior may arise from a unique computational-statistical tradeoff not found in standard ML analyses of symmetrization. Meanwhile, we demonstrate that post hoc averaging is less sensitive to such tradeoffs and emerges as a simple, flexible and effective method for improving neural network solvers.

Kevin Han Huang, Peter Orbanz, Morgane Austern

We provide universality results that quantify how data augmentation affects the variance and limiting distribution of estimates through simple surrogates, and analyze several specific models in detail. The results confirm some observations made in machine learning practice, but also lead to unexpected findings: Data augmentation may increase rather than decrease the uncertainty of estimates, such as the empirical prediction risk. It can act as a regularizer, but fails to do so in certain high-dimensional problems, and it may shift the double-descent peak of an empirical risk. Overall, the analysis shows that several properties data augmentation has been attributed with are not either true or false, but rather depend on a combination of factors -- notably the data distribution, the properties of the estimator, and the interplay of sample size, number of augmentations, and dimension. As our main theoretical tool, we develop an adaptation of Lindeberg's technique for block dependence. The resulting universality regime may be Gaussian or non-Gaussian.