Inverse design of implicit bias by symmetry, stochasticity and holomorphic parametrizations

Can we design a model such that its stochastic training favours a desired class of solutions without enforcing an explicit penalty? Under suitable conditions, the interplay between symmetries of a model's weight parametrization and stochastic training favours particular solutions, inducing an implicit bias. Building on this mechanism, we develop a framework for inverse-designing such biases by constructing novel parametrizations and their associated symmetries. We show how holomorphic functions make this construction and calculation simple and explicit. Specifically, we introduce a new parametrization that biases learned weights toward the binary values $\{-1,+1\}$. Numerical experiments confirm the theoretical predictions. They also show that our parametrization reproduces the same preference induced by an explicitly regularized model without adding a penalty to the training loss.