Bottleneck Structure in Learned Features: Low-Dimension vs Regularity Tradeoff
This work provides theoretical insights into feature learning in deep networks, which is foundational for improving model interpretability and design in machine learning.
The paper tackles the problem of understanding the structure of learned features in deep neural networks with large depth and L2 regularization, proving that as depth increases, hidden representations become approximately low-dimensional and weight matrices exhibit a bottleneck structure with most singular values near 1 and others decaying rapidly.
Previous work has shown that DNNs with large depth $L$ and $L_{2}$-regularization are biased towards learning low-dimensional representations of the inputs, which can be interpreted as minimizing a notion of rank $R^{(0)}(f)$ of the learned function $f$, conjectured to be the Bottleneck rank. We compute finite depth corrections to this result, revealing a measure $R^{(1)}$ of regularity which bounds the pseudo-determinant of the Jacobian $\left|Jf(x)\right|_{+}$ and is subadditive under composition and addition. This formalizes a balance between learning low-dimensional representations and minimizing complexity/irregularity in the feature maps, allowing the network to learn the `right' inner dimension. Finally, we prove the conjectured bottleneck structure in the learned features as $L\to\infty$: for large depths, almost all hidden representations are approximately $R^{(0)}(f)$-dimensional, and almost all weight matrices $W_{\ell}$ have $R^{(0)}(f)$ singular values close to 1 while the others are $O(L^{-\frac{1}{2}})$. Interestingly, the use of large learning rates is required to guarantee an order $O(L)$ NTK which in turns guarantees infinite depth convergence of the representations of almost all layers.