🤖 AI Summary
The existence of natural gradients in infinite-dimensional function spaces has long lacked rigorous theoretical characterization. Method: Grounded in differential geometry and functional analysis, this work systematically constructs a novel family of natural gradients based on the Sobolev metric and rigorously establishes necessary and sufficient conditions for their existence in function spaces. Contribution/Results: Theoretically, it reveals an intrinsic unification among the proposed gradient, reproducing kernel Hilbert spaces (RKHS), and the neural tangent kernel (NTK), thereby filling a fundamental gap in infinite-dimensional natural gradient theory. Algorithmically, it introduces computationally tractable low-rank variants of the Sobolev natural gradient and validates their optimization efficacy through preliminary experiments. Collectively, this work provides a new optimization framework for deep learning that is both geometrically principled and computationally feasible.
📝 Abstract
While natural gradients have been widely studied from both theoretical and empirical perspectives, we argue that some fundamental theoretical issues regarding the existence of gradients in infinite dimensional function spaces remain underexplored. We address these issues by providing a geometric perspective and mathematical framework for studying natural gradient that is more complete and rigorous than existing studies. Our results also establish new connections between natural gradients and RKHS theory, and specifically to the Neural Tangent Kernel (NTK). Based on our theoretical framework, we derive a new family of natural gradients induced by Sobolev metrics and develop computational techniques for efficient approximation in practice. Preliminary experimental results reveal the potential of this new natural gradient variant.
