Memorization Capacity for Additive Fine-Tuning with Small ReLU Networks

TL;DR

This paper introduces the Fine-Tuning Capacity (FTC), a new measure to analyze how many samples a small ReLU network can memorize during additive fine-tuning of large pre-trained models, revealing different scaling laws for 2- and 3-layer networks.

Contribution

It defines FTC as a novel metric for memorization in fine-tuning and provides tight bounds for 2- and 3-layer ReLU networks, extending understanding of memorization capacity in this context.

Findings

FTC scales linearly with N for 2-layer networks

FTC scales with the square root of N for 3-layer networks

Results recover known memorization capacities when N equals K

Abstract

Fine-tuning large pre-trained models is a common practice in machine learning applications, yet its mathematical analysis remains largely unexplored. In this paper, we study fine-tuning through the lens of memorization capacity. Our new measure, the Fine-Tuning Capacity (FTC), is defined as the maximum number of samples a neural network can fine-tune, or equivalently, as the minimum number of neurons ($m$) needed to arbitrarily change $N$ labels among $K$ samples considered in the fine-tuning process. In essence, FTC extends the memorization capacity concept to the fine-tuning scenario. We analyze FTC for the additive fine-tuning scenario where the fine-tuned network is defined as the summation of the frozen pre-trained network $f$ and a neural network $g$ (with $m$ neurons) designed for fine-tuning. When $g$ is a ReLU network with either 2 or 3 layers, we obtain tight upper and lower bounds on FTC; we show that $N$ samples can be fine-tuned with $m=\Theta(N)$ neurons for 2-layer networks, and with $m=\Theta(\sqrt{N})$ neurons for 3-layer networks, no matter how large $K$ is. Our results recover the known memorization capacity results when $N = K$ as a special case.