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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.
In this paper we propose a new method to learn the underlying acyclic mixed graph of a linear non-Gaussian structural equation model given observational data. We build on an algorithm proposed by Wang and Drton, and we show that one can augment the h
One popular trend in meta-learning is to learn from many training tasks a common initialization for a gradient-based method that can be used to solve a new task with few samples. The theory of meta-learning is still in its early stages, with several
In recent years, constrained optimization has become increasingly relevant to the machine learning community, with applications including Neyman-Pearson classification, robust optimization, and fair machine learning. A natural approach to constrained
We propose two novel conditional gradient-based methods for solving structured stochastic convex optimization problems with a large number of linear constraints. Instances of this template naturally arise from SDP-relaxations of combinatorial problem
We investigate 1) the rate at which refined properties of the empirical risk---in particular, gradients---converge to their population counterparts in standard non-convex learning tasks, and 2) the consequences of this convergence for optimization. O