Variance-based regularization with convex objectives
This work addresses variance regularization for machine learning practitioners, offering an incremental improvement over existing methods.
The paper tackles the problem of balancing approximation and estimation error in risk minimization by developing a convex surrogate for variance that provides certificates of optimality, achieving faster convergence rates than empirical risk minimization in some scenarios and improving test performance on classification problems.
We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximation and estimation error. Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator. In particular, we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance. We give corroborating empirical evidence showing that in practice, the estimator indeed trades between variance and absolute performance on a training sample, improving out-of-sample (test) performance over standard empirical risk minimization for a number of classification problems.