[Paper Review] Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning

Charles H. Martin, Michael W. Mahoney|arXiv (Cornell University)|Oct 2, 2018

Statistical Mechanics and Entropy67 references74 citations

TL;DR

The paper uses Random Matrix Theory to analyze DNN weight matrices, showing training induces implicit self-regularization and identifies a 5+1 phase taxonomy (including heavy-tailed regimes) that explains generalization gaps and batch-size effects.

ABSTRACT

Random Matrix Theory (RMT) is applied to analyze weight matrices of Deep Neural Networks (DNNs), including both production quality, pre-trained models such as AlexNet and Inception, and smaller models trained from scratch, such as LeNet5 and a miniature-AlexNet. Empirical and theoretical results clearly indicate that the DNN training process itself implicitly implements a form of Self-Regularization. The empirical spectral density (ESD) of DNN layer matrices displays signatures of traditionally-regularized statistical models, even in the absence of exogenously specifying traditional forms of explicit regularization. Building on relatively recent results in RMT, most notably its extension to Universality classes of Heavy-Tailed matrices, we develop a theory to identify 5+1 Phases of Training, corresponding to increasing amounts of Implicit Self-Regularization. These phases can be observed during the training process as well as in the final learned DNNs. For smaller and/or older DNNs, this Implicit Self-Regularization is like traditional Tikhonov regularization, in that there is a "size scale" separating signal from noise. For state-of-the-art DNNs, however, we identify a novel form of Heavy-Tailed Self-Regularization, similar to the self-organization seen in the statistical physics of disordered systems. This results from correlations arising at all size scales, which arises implicitly due to the training process itself. This implicit Self-Regularization can depend strongly on the many knobs of the training process. By exploiting the generalization gap phenomena, we demonstrate that we can cause a small model to exhibit all 5+1 phases of training simply by changing the batch size. This demonstrates that---all else being equal---DNN optimization with larger batch sizes leads to less-well implicitly-regularized models, and it provides an explanation for the generalization gap phenomena.

Motivation & Objective

• Motivate a practical theory of regularization for deep learning beyond explicit techniques like dropout or weight norms.
• Characterize the energy landscape of DNNs by analyzing layer weight matrices with RMT-derived metrics.
• Introduce operationally defined phases of training that map to increasing self-regularization.
• Demonstrate how training knobs (e.g., batch size) influence phase transitions and generalization.

Proposed method

• Model each DNN layer weight matrix W as W = W_rand + Δsig to separate random and signal components.
• Analyze the empirical spectral density (ESD) of X = (1/N) W^T W and fit it with Marchenko-Pastur (MP) theory and heavy-tailed universality classes.
• Define and compute capacity metrics from spectra: Hard Rank, Matrix Entropy, Stable Rank, and MP Soft Rank.
• Develop and validate a 5+1 phases taxonomy (Random-like, Bleeding-out, Bulk+Spikes, Bulk-decay, Heavy-Tailed, Rank-collapse) corresponding to implicit regularization levels.
• Exhibit phase transitions by manipulating training knobs (e.g., batch size) on smaller models and compare with pre-trained larger models.

Experimental results

Research questions

• RQ1 Can Random Matrix Theory explain how DNN training induces regularization without explicit penalties?
• RQ2 What are the spectral signatures (ESD) of weight matrices that reflect different levels of implicit self-regularization?
• RQ3 How do training parameters, especially batch size, drive transitions between the identified phases and affect generalization?

Key findings

• Older/smaller models show weak, Tikhonov-like implicit regularization with a signal-noise separation in MP terms.
• Modern large models exhibit heavy-tailed self-regularization with no clean signal-noise separation and finite spectral support.
• Phases can be observed during training and in final models, with MP Soft Rank decreasing and Stable Rank also declining as implicit regularization increases.
• Batch size reduction can drive a small model through all 5+1 phases, linking generalization gaps to implicit regularization.
• Explicit regularization can induce the Rank-collapse phase, illustrating how regularization strength shapes spectrum and capacity.