Bridging Neural Tangent Kernel Analysis and Reinforcement Learning for Deep Function Approximation

The empirical state-space Neural Tangent Kernel (NTK) describes the local learning geometry of a finite-width neural network, but computing it explicitly is almost always impractical in terms of computation and memory costs. Here, we show that many useful NTK statistics that characterize, for example, the dimensionality of learned updates or how two models or learning rules relate, can instead be efficiently approximated to very high accuracy via matrix-free products using randomized trace estimation. Namely, we use Hutch++ to estimate the NTK trace, Frobenius norm, effective rank, and alignment. Furthermore, we show that the positive-semidefinite structure of the NTK yields one-sided estimators that require only forward- or reverse-mode automatic differentiation. We validate these estimators across MLPs, recurrent GRUs, and a natural-language Transformer with up to 410 million parameters, in which the state-space contains high-dimensional four-tensors. We demonstrate orders-of-magnitude speedups, with the fastest estimator in a given application depending on the ratio of parameter and state dimensions. Equipped with these estimators, we examine rich and lazy RNN training using hidden-state NTK alignment and use NTK alignment as a regularizer for data-scarce knowledge distillation. We find that this regularization can modestly improve generalization, especially in very data-scarce settings. Together, these results suggest state-space NTK diagnostics are practical even at large scales.

In training a neural network with gradient descent (GD), each iteration induces a linear operator that governs the first-order updates to a model's internal state variables. We call this operator the Global Empirical Neural Tangent Kernel (GeNTK). In finite-width networks, GeNTK is typically intractable to form, leading prior work to focus on restrictive settings, such as tracking only outputs or infinite-width limits. Here, we bridge this gap by studying the structure of GeNTK across many models. Formulating the model state as the solution to a global implicit constraint, we derive GeNTK as a product of two operators: K, accounting for immediate parameter-to-state interactions, and P, describing internal state-to-state dependencies. For a broad class of "weight-based models", such as RNNs and transformers, we prove a universal Kronecker-core theorem showing that K admits an exact, forward-pass-computable form given by the Gram matrix of weight-site variables. This core gives a measurable state-dependent restriction on Euclidean GD, termed "self-referential bias": current activity filters which adjoint directions can drive a first-order update. For recurrent models (GRUs and RNNs), we examine the spectrum of GeNTK and formalize notions of low-rank, biased learning in terms of reduced spatial or temporal ranks of the full operator. Practically, this explains why some RNN initializations yield better learning, while some plateau, by biasing the GeNTK and thus restricting learning to a small domain of learnable tasks. To demonstrate computational viability, we validate the main theorems in a self-attention transformer and a pretrained Pythia transformer language model with 410 million parameters. Overall, our results expose tractable local rank and mode-selective biases in the finite-width GD geometry and open the door to principled study of the GeNTK, a bridge between theory and practice.

The Neural Tangent Kernel (NTK) framework explains optimization in over-parameterized neural networks via approximately linearized dynamics, yielding exponential convergence guarantees. However, existing results are often overly pessimistic and do not match the fast training in practice, as they depend on the smallest NTK eigenvalue, which is typically extremely small in practice. In this work, we develop sharper convergence guarantees by characterizing the interaction between data labels and the NTK eigen-spectrum. We identify two key phenomena, Label-NTK alignment and Residual-NTK alignment, showing that projections of labels and residuals onto NTK eigenvectors scale with the corresponding eigenvalues. We provide empirical evidence and theoretical justification under mild data assumptions. Exploiting these alignment properties, we derive a refined convergence bound that depends on the full spectrum and closely matches practical training dynamics, significantly improving over classical worst-case results. We further obtain improved generalization bounds. Experiments on MLPs and CNNs across multiple datasets validate our theory.