When you quantize a neural network to low-precision representations, you introduce new mathematical constraints that effectively regularize the model. Quantization restricts the set of possible parameter values by forcing weights and activations to be mapped onto a discrete set of levels. Mathematically, if you denote the original parameter vector as $w$ and the quantized version as $Q(w)$, then quantization can be described as enforcing the constraint:

where each $q_k$ is a quantization level. This means that after quantization, every parameter must satisfy $w_i ∈ Q$ for all i. The effect is similar to adding an implicit regularizer to the loss function, since the optimization is now performed over a restricted subset of the parameter space. In practice, this can be viewed as a projection of the full-precision solution onto the quantized set, which often leads to a reduction in overfitting and increased generalization. The regularization effect is implicit: rather than adding a penalty term to the loss, quantization directly limits the model's expressiveness by reducing the set of representable solutions.

However, this regularization comes with a trade-off between numerical stability and model expressiveness. Under reduced precision, the model's parameters and activations can no longer represent subtle variations, which may limit the network's ability to fit complex functions. At the same time, quantization can improve numerical stability by reducing the sensitivity of the model to small perturbations, since the quantized values are less likely to change dramatically in response to small updates. The balance between these two effects is crucial: too little precision may overly constrain the model, harming its ability to learn, while too much precision may negate the benefits of regularization and stability. Choosing the right quantization level is therefore a matter of balancing the need for stable, generalizable models against the desire for maximum expressiveness.

Fine-tuning is effective for improving the performance of quantized models because it allows the network to adapt its parameters within the constraints of the quantized space. Immediately after quantization, the model may suffer a drop in accuracy due to the abrupt change in parameter values. By continuing to train the model, you enable the optimization process to find new parameter configurations that are both compatible with the quantization constraints and better aligned with the training objective. Theoretically, fine-tuning can be seen as a way to optimize the loss function on the quantized manifold, seeking local minima that were inaccessible to the original full-precision solution but are optimal within the quantized setting. This process can often recover a significant portion of the performance lost during quantization, especially when the quantization levels are not excessively coarse.