Statistical Optimization in High Dimensions

We consider optimization problems whose parameters are known only approximately, based on a noisy sample. Of particular interest is the high-dimensional regime, where the number of samples is roughly equal to the dimensionality of the problem, and the noise magnitude may greatly exceed the magnitude of the signal itself. This setup falls far outside the traditional scope of Robust and Stochastic optimization. We propose three algorithms to address this setting, combining ideas from statistics, machine learning, and robust optimization. In the important case where noise artificially increases the dimensionality of the parameters, we show that combining robust optimization and dimensionality reduction can result in high-quality solutions at greatly reduced computational cost.