An Accuracy--Information Tradeoff for Loss-Difference Conditional Mutual Information

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 $\Theta(n/(1+(\tau/\varepsilon)^2))$ bits under Gaussian noise of standard deviation $\tau$ 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.