  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.
