Memorization Capacity for Additive Fine-Tuning with Small ReLU Networks
This provides theoretical insights into fine-tuning for machine learning practitioners, but it is incremental as it extends existing memorization capacity concepts to fine-tuning scenarios.
The paper tackles the problem of mathematically analyzing fine-tuning by introducing a new measure called Fine-Tuning Capacity (FTC), which quantifies how many samples a neural network can fine-tune, showing that for ReLU networks, 2-layer networks require Θ(N) neurons and 3-layer networks require Θ(√N) neurons to fine-tune N samples.
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 $m=Θ(N)$ neurons for 2-layer networks, and with $m=Θ(\sqrt{N})$ 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.