Intrinsic training dynamics of deep neural networks
This work addresses a fundamental theoretical challenge in deep learning, providing incremental insights into implicit bias and dimensionality reduction in training dynamics.
The paper tackles the problem of understanding whether gradient-based training in deep neural networks can be described by lower-dimensional dynamics, showing that for ReLU networks, the flow can be rewritten as an intrinsic dynamic in a lower dimension for any initialization, and generalizing balanced initializations to relaxed ones for linear networks.
A fundamental challenge in the theory of deep learning is to understand whether gradient-based training in high-dimensional parameter spaces can be captured by simpler, lower-dimensional structures, leading to so-called implicit bias. As a stepping stone, we study when a gradient flow on a high-dimensional variable $θ$ implies an intrinsic gradient flow on a lower-dimensional variable $z = φ(θ)$, for an architecture-related function $φ$. We express a so-called intrinsic dynamic property and show how it is related to the study of conservation laws associated with the factorization $φ$. This leads to a simple criterion based on the inclusion of kernels of linear maps which yields a necessary condition for this property to hold. We then apply our theory to general ReLU networks of arbitrary depth and show that, for any initialization, it is possible to rewrite the flow as an intrinsic dynamic in a lower dimension that depends only on $z$ and the initialization, when $φ$ is the so-called path-lifting. In the case of linear networks with $φ$ the product of weight matrices, so-called balanced initializations are also known to enable such a dimensionality reduction; we generalize this result to a broader class of {\em relaxed balanced} initializations, showing that, in certain configurations, these are the \emph{only} initializations that ensure the intrinsic dynamic property. Finally, for the linear neural ODE associated with the limit of infinitely deep linear networks, with relaxed balanced initialization, we explicitly express the corresponding intrinsic dynamics.