Recent advances in neural network pruning have shown how it is possible to reduce the computational costs and memory demands of deep learning models before training.
We focus on this framework and propose a new pruning at initialization algorithm that leverages the Neural Tangent Kernel (NTK) theory to align the training dynamics
of the sparse network with that of the dense one. Specifically, we show how the usually neglected data-dependent component in the NTK's spectrum can be taken into
account by providing an analytical upper bound to the NTK's trace obtained by decomposing neural networks into individual paths. This leads to **our Path eXclusion (PX),
a foresight pruning method designed to preserve the parameters that mostly influence the NTK's trace**. PX is able to find lottery tickets (i.e. good paths) even at high
sparsity levels and largely reduces the need for additional training. When applied to pre-trained models it extracts subnetworks directly usable for several downstream
tasks, resulting in performance comparable to those of the dense counterpart but with substantial cost and computational savings.

Our paper introduces a novel pruning algorithm, Path eXclusion (PX), designed to enhance the efficiency of neural networks by pruning at initialization. Leveraging Neural Tangent Kernel (NTK) theory, PX focuses on identifying and retaining the most critical network paths, ensuring minimal performance loss even when the network is transferred to new tasks.

The PX algorithm iteratively prunes network weights that have minimal impact on the NTK trace, thus preserving
essential paths and maintaining the training dynamics of the resulting subnetwork aligned with its dense counterpart.
Improving on current pruning methods focusing on the NTK theory, our approach leverages a new upper bound for the NTK
trace, which takes into account the network's input-output paths based on **architecture and weight values**
(captured by the Path Kernel \(J_\theta^v\)) and **how data maps onto such paths** (captured by the Path Activation
Matrix \(J_v^f(X)\)). Formally, our upper bound is defined as
\[\text{Tr}[\Theta(X,X)] = \|\nabla_\theta f(X,\theta)\|_F^2 = \|J_v^f(X) J_\theta^v \|_F^2 \leq \|J_v^f(X)\|_F^2 \cdot \|J_\theta^v \|_F^2, \]
which can be efficiently computed using automatic differentiation.

**NTK Theory and Network Paths.**Utilizing NTK theory to express its trace via network paths, providing a robust theoretical foundation for pruning.**Path eXclusion (PX).**Pruning weights based on their impact on the NTK trace, ensuring that crucial network paths are retained. The saliency function derived from the NTK trace ensures positive scores for parameters, preventing layer collapse.**Iterative Pruning Process.**Gradually refining the mask to focus on the most significant connections, enhancing efficiency and performance.

PX demonstrates robust performance across various neural network architectures and tasks, including large pre-trained vision models. It maintains the transferability and effectiveness of pruned networks. The algorithm's theoretical rigor and practical efficiency make it a versatile tool for modern neural network pruning.

```
@inproceedings{iurada2024finding,
author = {Iurada, Leonardo and Ciccone, Marco and Tommasi, Tatiana},
title = {Finding Lottery Tickets in Vision Models via Data-driven Spectral Foresight Pruning},
booktitle = {CVPR},
year = {2024},
}
```