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Speed Limit for Information Acquisition in Stochastic Learning Dynamics
Shuta Kobayashi, Andreas Dechant
TL;DR
The analysis considers how trainable parameters acquire information during stochastic learning, including comparisons between mini-batch SGD and full-batch gradient descent. The paper models SGD as a stochastic modified equation and derives a Fisher-information-flow speed limit with separate drift and noise information budgets. In basis-function linear regression, modal Fisher-information flows peak at characteristic relaxation times, with larger-eigenvalue modes peaking earlier and latent variables combining these modal times differently.
Problem
The analysis considers how trainable parameters acquire information during stochastic learning, including comparisons between mini-batch SGD and full-batch gradient descent.
Method
The paper models SGD as a stochastic modified equation and derives a Fisher-information-flow speed limit with separate drift and noise information budgets.
Results
In basis-function linear regression, modal Fisher-information flows peak at characteristic relaxation times, with larger-eigenvalue modes peaking earlier and latent variables combining these modal times differently.
Takeaways & Limitations
Fisher-information speed limits provide a quantitative framework for diagnosing when and how different aspects of the data-generating mechanism are acquired during stochastic learning.
Takeaways & Limitations
The analytical treatment assumes the basis-function approximation is accurate, so the residual approximation error satisfies δZ(x) ≃ 0.
Abstract
from arXiv · showhide
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.
END MATTER
The supplementary analysis derives the stochastic modified equation and evaluates Fisher-information speed limits in linear regression. Near convergence, modal information flows peak on characteristic relaxation times, with earlier peaks for modes having larger eigenvalues.
- From mini-batch SGD to the SME: The stochastic modified equation approximates mini-batch SGD by Gaussian drift and diffusion, with χ = ε reproducing SGD’s natural covariance scaling while χ = 1 gives conventional diffusion scaling.The Gaussian approximation is obtained for N ≫ m ≫ 1 by matching the mini-batch covariance over one interval.
- Numerical and Analytical evaluation of Fisher information: The analytical evaluation obtains Fisher information from SME moment dynamics, whereas finite-N SGD estimates means and covariances from 5000 independent trajectories.The SME results require no stochastic-trajectory estimation of these moments.
- Analytical results near convergence: The Fisher-information flow initially grows, peaks near convergence, and then decreases toward its stationary value; the speed-limit bound holds throughout and becomes tight at the peak.The peak value agrees with the asymptotic drift information budget in the analytical Ornstein–Uhlenbeck approximation.
- Analytical results near convergence: In the near-convergence regime, the dynamics reduces to a multivariate Ornstein–Uhlenbeck process when the state-dependent diffusion correction is negligible.H, Dst, and the isotropic initial covariance are simultaneously diagonalizable, enabling mode-resolved analysis.
- Basis-function linear regression: Each modal Fisher-information flow reaches its drift information budget at its maximum, and its peak shares the characteristic relaxation time of that budget.For comparable initial modal variances, modes with larger λ_i peak earlier.
- Basis-function linear regression: Different latent variables combine modal contributions through weights e_{uz,i}, producing distinct information-acquisition dynamics from the same modal learning timescales.The modal peak times depend on learning dynamics, whereas latent-variable dependence enters through the modal weights.
A. Single-parameter linear regression for a linear function
The single-parameter linear-regression analysis examines Fisher-information flow during stochastic learning toward a linear target. It shows that the flow peaks near convergence, with peak timing linked to drift relaxation and the speed-limit bound becoming tight in the small-deviation regime.
- Model and dynamics: The near-convergence evolution reduces to an Ornstein–Uhlenbeck process when the second-order diffusion term can be neglected.The analysis defines the parameter deviation from the optimum and derives its mean and variance.
- Fisher-information flow: Under SGD scaling χ = ε, decreasing the learning rate shifts the Fisher-information-flow peak to later times.The corresponding peak value is derived analytically from the single-parameter model.
- Speed-limit saturation: When the parameter deviation is sufficiently small, the speed-limit inequality becomes tight because its right-hand side approaches the peak Fisher-information-flow value.The limiting right-hand side is mH/σ^2 in the stated small-deviation regime.
- Peak and convergence times: The Fisher-information flow reaches its peak when transient drift information has decayed to the scale of the stationary contribution, so peak and drift-convergence times coincide to logarithmic order.This correspondence is established when the stationary variance is much smaller than the initial variance and the analytical consistency condition holds.
B. Basis-function linear regression for a general target function
The paper analyzes Fisher-information flow in basis-function linear regression by resolving learning dynamics into eigenmodes of the data covariance and diffusion matrices. Under small residual mismatch, modal information peaks follow eigenvalue ordering and share characteristic peak and convergence timescales, whereas large mismatch disrupts this simple ordering.
- Model and residual error: Basis-function regression represents a general target as f_θ(x)=θ^⊤Ψ(x), while residual approximation error depends on the chosen basis functions.The model remains linear in trainable parameters, but the fixed basis functions can represent nonlinear target structure.
- Near-convergence dynamics: Near convergence, the dynamics reduce to an Ornstein–Uhlenbeck process, enabling Fisher information and modal flow to be analyzed through mean and covariance evolution.The analysis assumes simultaneous diagonalizability of the relevant matrices and uses the resulting eigenbasis to define mode-resolved flows.
- Modal Fisher-information flow: Each eigenmode’s Fisher-information-flow peak coincides with its drift-information contribution, while the total flow generally peaks differently because modal peak times differ.Equality for the total bound requires all modes to reach their peaks simultaneously.
- Characteristic timescales: Each eigenmode’s Fisher-information-flow peak time and convergence time have the same characteristic timescale, linking information-acquisition timing to the mode’s eigenvalue.The result applies under the analyzed near-convergence conditions.
- Eigenvalue ordering: For appropriately chosen basis functions with δ_Z(x)≈0, eigenmode peaks occur in descending order of the covariance eigenvalues λ_i.When δ_Z(x) is large, this simple eigenvalue ordering does not generally hold.