Speed limits for information acquisition in stochastic learning dynamics
Shuta Kobayashi, Andreas Dechant
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 data-generating parameters. The resulting inequality decomposes the upper bound on 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 data-generating parameters are encoded in the learned parameters. These results establish Fisher-information speed limits as a quantitative framework for understanding which properties of the data-generating process are acquired, in what order, and how rapidly they can be encoded in the trainable parameters.
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