Fisher information sets speed limits on learning in neural networks
Speed Limit for Information Acquisition in Stochastic Learning Dynamics
Machine Learning
Summary
Learning in neural networks happens step-by-step and involves some randomness. The authors describe how fast these networks can gather information about hidden features in their training data by thinking of learning as a random process. They found a mathematical rule that limits the speed of information gain, which depends on the balance between steady learning forces and the noise from randomness. They checked their rule with simple models and found it correctly predicts how and when networks learn different hidden details.
What this means in practice
- •For machine learning engineers: Design training schedules that respect inherent speed limits on how fast neural networks can learn different data features.
- •For signal processing developers: Evaluate when noisy adaptive algorithms reach information limits in changing environments by relating learning speed to noise levels.
Tested on simulated data.
Authors
Shuta Kobayashi, Andreas Dechant
Abstract
Neural networks acquire internal representations through learning. In this work, we formulate stochastic gradient descent (SGD) as a Markovian stochastic process and derive a Fisher-information flow speed limit that bounds the rate at which trainable parameters can acquire information about latent variables in the data-generating process. The resulting inequality decomposes the information flow into drift and noise contributions, thereby quantifying the roles of deterministic learning forces and SGD-induced fluctuations from an information-theoretic perspective. We verify the bound in analytically tractable basis-function linear regression, where the information budget predicted by the bound reproduces the ordering and characteristic time scales with which different latent variables are encoded in the learned parameters. These results establish Fisher-information speed limits as a quantitative framework for diagnosing when and how different aspects of the data-generating mechanism are acquired during stochastic learning.