Max Staats

Matthias Thamm, Max Staats, Bernd Rosenow

Neural networks have been used successfully in a variety of fields, which has led to a great deal of interest in developing a theoretical understanding of how they store the information needed to perform a particular task. We study the weight matrices of trained deep neural networks using methods from random matrix theory (RMT) and show that the statistics of most of the singular values follow universal RMT predictions. This suggests that they are random and do not contain system specific information, which we investigate further by comparing the statistics of eigenvector entries to the universal Porter-Thomas distribution. We find that for most eigenvectors the hypothesis of randomness cannot be rejected, and that only eigenvectors belonging to the largest singular values deviate from the RMT prediction, indicating that they may encode learned information. In addition, a comparison with RMT predictions also allows to distinguish networks trained in different learning regimes - from lazy to rich learning. We analyze the spectral distribution of the large singular values using the Hill estimator and find that the distribution cannot in general be characterized by a tail index, i.e. is not of power law type.

Max Staats, Matthias Thamm, Bernd Rosenow

Deep neural networks have been successfully applied to a broad range of problems where overparametrization yields weight matrices which are partially random. A comparison of weight matrix singular vectors to the Porter-Thomas distribution suggests that there is a boundary between randomness and learned information in the singular value spectrum. Inspired by this finding, we introduce an algorithm for noise filtering, which both removes small singular values and reduces the magnitude of large singular values to counteract the effect of level repulsion between the noise and the information part of the spectrum. For networks trained in the presence of label noise, we indeed find that the generalization performance improves significantly due to noise filtering.

Max Staats, Matthias Thamm, Bernd Rosenow

Large annotated datasets inevitably contain noisy labels, which poses a major challenge for training deep neural networks as they easily memorize the labels. Noise-robust loss functions have emerged as a notable strategy to counteract this issue, but it remains challenging to create a robust loss function which is not susceptible to underfitting. Through a quantitative approach, this paper explores the limited overlap between the network output at initialization and regions of non-vanishing gradients of bounded loss functions in the initial learning phase. Using these insights, we address underfitting of several noise robust losses with a novel method denoted as logit bias, which adds a real number $ε$ to the logit at the position of the correct class. The logit bias enables these losses to achieve state-of-the-art results, even on datasets like WebVision, consisting of over a million images from 1000 classes. In addition, we demonstrate that our method can be used to determine optimal parameters for several loss functions -- without having to train networks. Remarkably, our method determines the hyperparameters based on the number of classes, resulting in loss functions which require zero dataset or noise-dependent parameters.

Max Staats, Matthias Thamm, Bernd Rosenow

This work analyzes singular-value spectra of weight matrices in pretrained transformer models to understand how information is stored at both ends of the spectrum. Using Random Matrix Theory (RMT) as a zero information hypothesis, we associate agreement with RMT as evidence of randomness and deviations as evidence for learning. Surprisingly, we observe pronounced departures from RMT not only among the largest singular values -- the usual outliers -- but also among the smallest ones. A comparison of the associated singular vectors with the eigenvectors of the activation covariance matrices shows that there is considerable overlap wherever RMT is violated. Thus, significant directions in the data are captured by small singular values and their vectors as well as by the large ones. We confirm this empirically: zeroing out the singular values that deviate from RMT raises language-model perplexity far more than removing values from the bulk, and after fine-tuning the smallest decile can be the third most influential part of the spectrum. To explain how vectors linked to small singular values can carry more information than those linked to larger values, we propose a linear random-matrix model. Our findings highlight the overlooked importance of the low end of the spectrum and provide theoretical and practical guidance for SVD-based pruning and compression of large language models.