On The Concurrence of Layer-wise Preconditioning Methods and Provable Feature Learning

• URL: http://arxiv.org/abs/2502.01763v1
• Date: Mon, 03 Feb 2025 19:08:32 GMT
• Title: On The Concurrence of Layer-wise Preconditioning Methods and Provable Feature Learning
• Authors: Thomas T. Zhang, Behrad Moniri, Ansh Nagwekar, Faraz Rahman, Anton Xue, Hamed Hassani, Nikolai Matni,
• Abstract summary: We show that layer-wise preconditioning methods are provably necessary from a statistical perspective. We show that SGD is a suboptimal feature when extending beyond ideal isotropic inputs. We show that standard tools like Adam preconditioning and batch-norm only mildly mitigate these issues.
• Score: 22.486361028522374
• License:
• Abstract: Layer-wise preconditioning methods are a family of memory-efficient optimization algorithms that introduce preconditioners per axis of each layer's weight tensors. These methods have seen a recent resurgence, demonstrating impressive performance relative to entry-wise ("diagonal") preconditioning methods such as Adam(W) on a wide range of neural network optimization tasks. Complementary to their practical performance, we demonstrate that layer-wise preconditioning methods are provably necessary from a statistical perspective. To showcase this, we consider two prototypical models, linear representation learning and single-index learning, which are widely used to study how typical algorithms efficiently learn useful features to enable generalization. In these problems, we show SGD is a suboptimal feature learner when extending beyond ideal isotropic inputs $\mathbf{x} \sim \mathsf{N}(\mathbf{0}, \mathbf{I})$ and well-conditioned settings typically assumed in prior work. We demonstrate theoretically and numerically that this suboptimality is fundamental, and that layer-wise preconditioning emerges naturally as the solution. We further show that standard tools like Adam preconditioning and batch-norm only mildly mitigate these issues, supporting the unique benefits of layer-wise preconditioning.

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