The cost of useful natural gradient updates

Kakade's natural policy gradient method has been studied extensively in recent years, showing linear convergence with and without regularization. We study another natural gradient method based on the Fisher information matrix of the state-action distributions which has received little attention from the theoretical side. Here, the state-action distributions follow the Fisher-Rao gradient flow inside the state-action polytope with respect to a linear potential. Therefore, we study Fisher-Rao gradient flows of linear programs more generally and show linear convergence with a rate that depends on the geometry of the linear program. Equivalently, this yields an estimate on the error induced by entropic regularization of the linear program which improves existing results. We extend these results and show sublinear convergence for perturbed Fisher-Rao gradient flows and natural gradient flows up to an approximation error. In particular, these general results cover the case of state-action natural policy gradients.

We consider the problem of approximating a function by an element of a nonlinear manifold which admits a differentiable parametrization, typical examples being neural networks with differentiable activation functions or tensor networks. Natural gradient descent (NGD) for the optimization of a loss function can be seen as a preconditioned gradient descent where updates in the parameter space are driven by a functional perspective. In a spirit similar to Newton's method, a NGD step uses, instead of the Hessian, the Gram matrix of the generating system of the tangent space to the approximation manifold at the current iterate, with respect to a suitable metric. This corresponds to a locally optimal update in function space, following a projected gradient onto the tangent space to the manifold. Still, both gradient and natural gradient descent methods get stuck in local minima. Furthermore, when the model class is a nonlinear manifold or the loss function is not ideally conditioned (e.g., the KL-divergence for density estimation, or a norm of the residual of a partial differential equation in physics informed learning), even the natural gradient might yield non-optimal directions at each step. This work introduces a natural version of classical inertial dynamic methods like Heavy-Ball or Nesterov and show how it can improve the learning process when working with nonlinear model classes.

We study the query complexity of sampling from high-dimensional Gaussian distributions using gradient information. In the standard oracle model, exact gradients expose only matrix-vector products with the precision matrix, leading to polynomial approximation barriers and a characteristic $\sqrtκ$ dependence on the condition number. We show that this barrier disappears when the sampler is allowed to query \emph{smoothed scores}, namely gradients of the logarithms of the Gaussian-convolved densities. For a Gaussian target with precision matrix $Λ$, a smoothed-score query at noise level $τ$ gives access to the resolvent $(Λ+τ^{-1}I)^{-1}$. Combining geometrically spaced noise levels with sinc-quadrature rational approximation, we obtain a sampler with $q=O\!\left(\bigl(\logκ+\log(e\sqrt d/δ_{\rm TV})\bigr)\log(e\sqrt d/δ_{\rm TV})\right)$ smoothed-score queries for total variation error $δ_{\rm TV}$, improving the condition-number dependence from $\sqrtκ$ to logarithmic. We also study finite-bit gradient oracles. Using coordinatewise quantization of the transformed smoothed-score answers and a final dithering step, we obtain a sampling scheme whose total communicated gradient information is polylogarithmic in $κ$; in particular, for fixed dimension and accuracy, the bit complexity is $O(\log^2κ)$. To complement these upper bounds, we introduce a channel-synthesis, or reverse-Shannon, converse technique for sampling lower bounds. This converts total-variation simulation guarantees into communication requirements and yields an $Ω(\logκ)$ lower bound on the required gradient information. Together, these results identify smoothed scores as a provably more informative oracle for sampling and give nearly matching upper and lower bounds for its finite-bit complexity.