Memorization Capacity for Additive Fine-Tuning with Small ReLU Networks
Jy-yong Sohn, Dohyun Kwon, Seoyeon An, Kangwook Lee
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.
BibTeX
@InProceedings{pmlr-v244-sohn24a,
title = {Memorization Capacity for Additive Fine-Tuning with Small ReLU Networks},
author = {Sohn, Jy-yong and Kwon, Dohyun and An, Seoyeon and Lee, Kangwook},
booktitle = {Proceedings of the Fortieth Conference on Uncertainty in Artificial Intelligence},
pages = {3264--3278},
year = {2024},
editor = {Kiyavash, Negar and Mooij, Joris M.},
volume = {244},
series = {Proceedings of Machine Learning Research},
month = {15--19 Jul},
publisher = {PMLR},
pdf = {https://raw.githubusercontent.com/mlresearch/v244/main/assets/sohn24a/sohn24a.pdf},
url = {https://proceedings.mlr.press/v244/sohn24a.html},
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.}
}