Theoretical Breakthrough: ReLU DNNs Achieve Optimal Convergence Rates in High-Dimensional Classif...
By Zihan Zhang, Lei Shi, Ding-Xuan Zhou
Published on November 24, 2025| Vol. 1, Issue No. 1
Content Source
This is a curated briefing. The original article was published on stat.ML updates on arXiv.org.
Summary
This paper investigates binary classification on high-dimensional data, specifically under the Tsybakov noise condition and a "compositional assumption" where the conditional class probability function is a composition of simpler functions. A key finding is the derivation of an optimal convergence rate for classifiers, which surprisingly depends on a lower intrinsic dimension d* rather than the full input dimension d. Crucially, the research demonstrates that ReLU deep neural networks (DNNs) trained with hinge loss can achieve this optimal rate, up to a logarithmic factor, providing strong theoretical backing for their effectiveness in real-world, high-dimensional classification tasks.
Why It Matters
This research delivers a significant theoretical underpinning for the practical success of Deep Neural Networks, particularly those leveraging ReLU activations, in tackling complex classification problems. For AI professionals, this isn't just an abstract mathematical proof; it directly addresses the "curse of dimensionality," a pervasive challenge in fields like computer vision, natural language processing, and bioinformatics where input data often has thousands or millions of features. The finding that optimal convergence rates can be independent of the overall input dimension d, instead relying on a smaller "effective" dimension d*, provides a profound theoretical justification for why DNNs generalize well even with immense input sizes.
This work suggests that the inherent compositional structure often present in real-world data - which DNNs are exceptionally good at learning hierarchically - is key to their efficiency. Understanding these fundamental theoretical limits and knowing that ReLU networks can reach them boosts confidence in designing and deploying deep learning models. It can guide future architectural innovations, prompt research into identifying and exploiting d* in diverse datasets, and potentially lead to more robust, interpretable, and theoretically sound AI systems. Ultimately, this strengthens the bridge between theoretical understanding and practical application, helping practitioners make more informed choices about model selection and training strategies in high-dimensional scenarios.