Deep Learning Through A Telescoping Lens: A Simple Model Provides Empirical Insights On Grokking, Gradient Boosting & Beyond

TL;DR

This model presents a pedagogical formalism allowing us to isolate components of the training process even in complex contemporary settings, providing a lens to reason about the effects of design choices such as architecture&optimization strategy, and reveals surprising parallels between neural network learning and gradient boosting.

Abstract

Deep learning sometimes appears to work in unexpected ways. In pursuit of a deeper understanding of its surprising behaviors, we investigate the utility of a simple yet accurate model of a trained neural network consisting of a sequence of first-order approximations telescoping out into a single empirically operational tool for practical analysis. Across three case studies, we illustrate how it can be applied to derive new empirical insights on a diverse range of prominent phenomena in the literature -- including double descent, grokking, linear mode connectivity, and the challenges of applying deep learning on tabular data -- highlighting that this model allows us to construct and extract metrics that help predict and understand the a priori unexpected performance of neural networks. We also demonstrate that this model presents a pedagogical formalism allowing us to isolate components of the training process even in complex contemporary settings, providing a lens to reason about the effects of design choices such as architecture & optimization strategy, and reveals surprising parallels between neural network learning and gradient boosting.

Figures (14)

• Figure 1: Illustration of the telescoping model of a trained neural network.Unlike the more standard framing of a neural network in terms of an iteratively learned set of parameters, the telescoping model takes a functional perspective on training a neural network in which an arbitrary test example's initially random prediction, $f_{\bm{\theta}_0}(\mathbf{x})$, is additively updated by a linearized adjustment $\Delta \tilde{f}_t(\mathbf{x})$ at each step $t$ as in \ref{['eq:modeldef']}.
• Figure 2: Approximation error of the telescoping ($\tilde{f}_{\bm{\theta}_t}(\mathbf{x})$, red) and the linear model (${f}^{lin}_{\bm{\theta}_t}(\mathbf{x})$, gray).
• Figure 3: Double descent in MSE (top) and effective parameters $p^{0}_{\hat{\mathbf{s}}}$ (bottom) on CIFAR-10.
• Figure 4: Grokking in mean squared error on a polynomial regression task (1, replicated from kumargrokking) and in misclassification error on MNIST using a network with large initialization (2, replicated from liu2022omnigrok) (top), against effective parameters (bottom). Column (3) shows test results on MNIST with standard initialization (with and without sigmoid activation) where time to generalization is quick and grokking does not occur.
• Figure 5: Neural Networks vs GBTs: Relative performance (top) and behavior of kernels (bottom) with increasing test data irregularity using the houses dataset.