Sparse + Group-Sparse Dirty Models: Statistical Guarantees without Unreasonable Conditions and a Case for Non-Convexity

Imposing sparse + group-sparse superposition structures in high-dimensional parameter estimation is known to provide flexible regularization that is more realistic for many real-world problems. For example, such a superposition enables partially-shared support sets in multi-task learning, thereby striking the right balance between parameter overlap across tasks and task specificity. Existing theoretical results on estimation consistency, however, are problematic as they require too stringent an assumption: the incoherence between sparse and group-sparse superposed components. In this paper, we fill the gap between the practical success and suboptimal analysis of sparse + group-sparse models, by providing the first consistency results that do not require unrealistic assumptions. We also study non-convex counterparts of sparse + groupsparse models. Interestingly, we show that these are guaranteed to recover the true support set under much milder conditions and with smaller sample size than convex models, which might be critical in practical applications as illustrated by our experiments.