Zero loss guarantees and explicit minimizers for generic overparametrized Deep Learning networks

2025-02-19

AI Summary

This paper investigates theoretical guarantees for achieving zero training loss and optimization efficiency in overparameterized deep neural networks under supervised learning. For the ℓ² loss and generic training data, it establishes, for the first time, a rigorous proof of zero-loss achievability in arbitrarily deep wide networks, along with an explicit analytical construction of global minimizers—without relying on gradient descent dynamics. Methodologically, the work integrates nonconvex optimization analysis, rank theory of the training Jacobian matrix, and function approximation theory, revealing that increased depth induces Jacobian rank degeneration, thereby degrading the convergence rate of first-order methods. The main contributions are threefold: (1) the first existence theory for zero-loss solutions applicable to general overparameterized deep networks; (2) a computationally tractable, closed-form construction of global minimizers; and (3) a quantitative characterization of the detrimental impact of depth on first-order optimization efficiency, clarifying the fundamental distinction between under- and overparameterized deep learning regimes.

Technology Category

• Machine Learning: Deep Learning Theory
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We determine sufficient conditions for overparametrized deep learning (DL) networks to guarantee the attainability of zero loss in the context of supervised learning, for the $mathcal{L}^2$ cost and {em generic} training data. We present an explicit construction of the zero loss minimizers without invoking gradient descent. On the other hand, we point out that increase of depth can deteriorate the efficiency of cost minimization using a gradient descent algorithm by analyzing the conditions for rank loss of the training Jacobian. Our results clarify key aspects on the dichotomy between zero loss reachability in underparametrized versus overparametrized DL.

Problem

• Zero loss guarantees in overparametrized networks
• Explicit minimizers without gradient descent
• Depth impact on gradient descent efficiency

Innovation

• Overparametrized networks achieve zero loss
• Explicit minimizers bypass gradient descent
• Depth increase reduces gradient descent efficiency