2022/12/04 by Andrea Della Vecchia, Della Vecchia, Andrea, Ernesto De Vito +3
Computer Science · Engineering · Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Sparse and Compressive Sensing Techniques #Statistical Methods and Inference #Statistics Theory (math.ST) #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2212.01866
openalex publication_date 2022/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a natural extension of classical empirical risk minimization, where the hypothesis space is a random subspace of a given space. In particular, we consider possibly data dependent subspaces spanned by a random subset of the data, recovering as a special case Nystrom approaches for kernel methods. Considering random subspaces naturally leads to computational savings, but the question is whether the corresponding learning accuracy is degraded. These statistical-computational tradeoffs have been recently explored for the least squares loss and self-concordant loss functions, such as the logistic loss. Here, we work to extend these results to convex Lipschitz loss functions, that might not be smooth, such as the hinge loss used in support vector machines. This unified analysis requires developing new proofs, that use different technical tools, such as sub-gaussian inputs, to achieve fast rates. Our main results show the existence of different settings, depending on how hard the learning problem is, for which computational efficiency can be improved with no loss in performance.