GradMetaNet: An Equivariant Architecture for Learning on Gradients

Michael Bronstein (USI) · Yoav Gelberg (University of Oxford, Sakana AI) · Yam Eitan (Technion - Israel Institute of Technology, Technion) · Aviv Navon (Bar-Ilan University) · Aviv Shamsian (Bar Ilan University) · Theo Putterman (University of California, Berkeley) · Haggai Maron (Technion / NVIDIA Research)

curvature informationequivariant blocksequivariant designgradient representationgradient statisticsgradientsgradmetanetinr editinglearned optimizationloss landscape curvaturenatural gradient optimization pruningrank-1 decompositionuniversality results

Gradients of neural networks encode valuable information for optimization, editing, and analysis of models. Therefore, practitioners often treat gradients as inputs to task-specific algorithms, e.g., using gradient statistics for pruning or optimization. Recent works explore *learning* algorithms that operate directly on gradients but use architectures that are not specifically designed for gradient processing, hindering their applicability. In this paper, we present a principled approach for designing architectures that process gradients. Our approach is guided by three principles: (1) equivariant design that preserves neuron permutation symmetries, (2) processing sets of gradients across multiple data points to capture curvature information, and (3) efficient gradient representation through rank-1 decomposition. Based on these principles, we introduce GradMetaNet, a novel architecture for learning on gradients, constructed from simple equivariant blocks. We prove universality results for GradMetaNet, and show that previous approaches cannot approximate natural gradient-based functions that GradMetaNet can. We then demonstrate GradMetaNet's effectiveness on a diverse set of gradient-based tasks for *MLPs* and *transformers*, such as learned optimization, INR editing, and loss landscape curvature estimation.