Constrained Learning with Non-Convex Losses

Abstract

Though learning has become a core technology of modern information processing, there is now ample evidence that it can lead to biased, unsafe, and prejudiced solutions. The need to impose requirements on learning is therefore paramount, especially as it reaches critical applications in social, industrial, and medical domains. However, the non-convexity of most modern learning problems is only exacerbated by the introduction of constraints. Whereas good unconstrained solutions can often be learned using empirical risk minimization (ERM), even obtaining a model that satisfies statistical constraints can be challenging, all the more so a good one. In this paper, we overcome this issue by learning in the empirical dual domain, where constrained statistical learning problems become unconstrained, finite dimensional, and deterministic. We analyze the generalization properties of this approach by bounding the empirical duality gap, i.e., the difference between our approximate, tractable solution and the solution of the original (non-convex) statistical problem, and provide a practical constrained learning algorithm. These results establish a constrained counterpart of classical learning theory and enable the explicit use of constraints in learning. We illustrate this algorithm and theory in rate-constrained learning applications.

Iterative Implicit Gradients for Nonconvex Optimization with Variational Inequality Constraints

“…In the existing literature, classical meta-learning, hyperparameter optimization problems do not consider constraints. There is an imperative need for constrained learning to incorporate safety, fairness, and other high-level specifications [24], [5]. Main motivation arises from the fact that meta-learning can be formed as a bilevel optimization problem where the "inner" optimization deals with the adaptation for a particular task and the "outer" optimization can be modified for meta-training with the safety constraints in order to restrict biased and risky scenarios.…”