Weighted Risk Minimization & Deep Learning

Importance weighting is a key ingredient in many algorithms for causal inference and related problems, such as off-policy evaluation for reinforcement learning. Recently, theorists proved that on separable data, unregularized linear networks, trained with cross-entropy loss and optimized by stochastic gradient descent converge in direction to the max margin solution. This solution depends on the location of data points but not their weights, nullifying the effect of importance weighting. This paper asks, for realistic deep networks, for which all datasets are separable, what is the effect of importance weighting? Lacking theoretical tools for analyzing modern deep (nonlinear, unregularized) networks, we investigate the question empirically on both realistic and synthetic data. Our results demonstrate that while importance weighting alters the learned model early in training, its effect diminishes to negligible with indefinite training. However, this diminishing effect does not occur in the presence of L2-regularization. These results (i) support the broader applicability of theoretical findings by Soudry et al (2018), who analyze linear networks; (ii) call into question the practice of importance weighting; and (iii) suggest that its usefulness interacts strongly with the early stopping criteria and regularization methods that interact with the loss function.