Exploring Weight Importance and Hessian Bias in Model Pruning

TL;DR

This paper investigates the mechanics of model pruning, revealing how importance measures relate to Hessian structures and highlighting the limitations of magnitude-based pruning, especially in neural networks.

Contribution

It provides a theoretical framework connecting importance measures to Hessian structures and compares different pruning methods with precise asymptotic formulas.

Findings

Hessian-based importance captures weight significance in linear models.

Magnitude-based pruning can fail when important weights decrease in magnitude.

Implicit regularization via Hessian structure aids in identifying important weights.

Abstract

Model pruning is an essential procedure for building compact and computationally-efficient machine learning models. A key feature of a good pruning algorithm is that it accurately quantifies the relative importance of the model weights. While model pruning has a rich history, we still don't have a full grasp of the pruning mechanics even for relatively simple problems involving linear models or shallow neural nets. In this work, we provide a principled exploration of pruning by building on a natural notion of importance. For linear models, we show that this notion of importance is captured by covariance scaling which connects to the well-known Hessian-based pruning. We then derive asymptotic formulas that allow us to precisely compare the performance of different pruning methods. For neural networks, we demonstrate that the importance can be at odds with larger magnitudes and proper initialization is critical for magnitude-based pruning. Specifically, we identify settings in which weights become more important despite becoming smaller, which in turn leads to a catastrophic failure of magnitude-based pruning. Our results also elucidate that implicit regularization in the form of Hessian structure has a catalytic role in identifying the important weights, which dictate the pruning performance.