Research map / Learning theory and algorithms / Kernels and regression theory
Tight Bounds on the Smallest Eigenvalue of the Neural Tangent Kernel for Deep ReLU Networks
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a single hidden layer of roughly order N neurons rather than requiring all hidden layers to be wide | Deep ReLU networks with Gaussian weight initialization · Finite-width empirical NTK | smallest eigenvalue: \evmin{\bar{K}^{(L)}} \geq\sum_{k=2}^{L} \xi_{k-1}\; \mu_r(\sigma)^2\; \bigOmg{d\prod_{l=1}^{L-1}n_l \prod_{\substack{l= | Helped |
| Use a deep ReLU network in the limiting-width setting under the stated data assumptions | Deep ReLU networks · Limiting NTK | smallest eigenvalue: \mu_r(\sigma)^2\; \Omega(d) \leq \evmin{K^{(L)}} \leq L\mathcal{O}(d) | Helped |
| Use activation-pattern analysis to bound feature-map Lipschitzness instead of multiplying layer operator-norm bounds | Deep ReLU feature maps · Hidden-layer feature maps | squared Lipschitz constant: \norm{g_k}_{\mathrm{Lip}}^2 =\bigO{ \frac{\prod_{l=0}^k n_l}{\mathop{\rm min}\nolimits_{l\in[0,k]}n_l}\, \prod_{l=1}^{k- | Helped |
| Require all hidden layers to be wide to obtain a positive smallest-eigenvalue bound | Deep ReLU networks · Finite-width empirical NTK | smallest eigenvalue | No clear effect |
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