Tradeoff between accuracy and information in loss difference learning
An Accuracy--Information Tradeoff for Loss-Difference Conditional Mutual Information
Machine LearningInformation Theory
Summary
This paper looks at how much information about training choices is hidden in a model’s differences in loss, which are used to judge how well it's learned. The authors show that if a model is accurate, it must also contain a substantial amount of this loss-difference information. They analyze this for certain types of models and conditions, finding that the information cannot be made very small even when models generalize well. They also explain when this relationship holds and when it does not.
What this means in practice
- •For machine learning engineers: Understand fundamental limits on information leakage in loss differences to better interpret model training and generalization tradeoffs.
- •For data privacy officers: Use theoretical insights about loss-difference information to assess risks of data leakage through model outputs under different noise conditions.
A theory result. No direct application yet.
Authors
Hazar Yueksel
Abstract
Loss-difference conditional mutual information (ld-CMI) uses the smallest of the standard observations in the supersample hierarchy of generalization bounds: it measures what a learner's loss differences reveal about which candidate of each pair it was trained on. Accuracy is known to force information into the model; data processing does not carry such lower bounds to losses. We show, by bounding three moments of the loss differences, that accuracy also forces ld-CMI. For linear predictors with a smooth convex loss of nonzero slope at zero, such as the logistic loss, plus a regularizer whose curvature and growth are both of power $r\ge2$, on product distributions over a scaled sign cube in dimension at least linear in $n$, every proper learner with expected excess risk at most $\varepsilon$ on these distributions at the optimal sample size $n\asymp\varepsilon^{-2+2/r}$ has worst-case ld-CMI of order $n$ bits, and $Θ(n/(1+(τ/\varepsilon)^2))$ bits under Gaussian noise of standard deviation $τ$ on the loss differences. The same holds without a regularizer, at $n\asymp\varepsilon^{-2}$. Consequently, range-scaled ld-CMI bounds cannot vanish on these distributions, although every proper learner's generalization gap is $O(n^{-1/2})$. We also show that model-level information does not determine noisy loss-difference information, and that the growth, slope and dimension conditions are needed, the last up to a logarithm.