Information-Theoretic Bounds on Transfer Generalization Gap Based on Jensen-Shannon Divergence

TL;DR

This paper derives novel information-theoretic bounds on the transfer generalization gap using generalized Jensen-Shannon divergences, capturing domain shift and sensitivity of the transfer learner, applicable even with unbounded loss functions.

Contribution

It introduces new bounds based on generalized Jensen-Shannon divergence that handle domain shift and unbounded losses, improving upon previous KL divergence-based bounds.

Findings

Bounds are tighter when domain shift is significant.

Applicable to unbounded loss functions with bounded cumulant generating functions.

Numerical example demonstrates the bounds' effectiveness.

Abstract

In transfer learning, training and testing data sets are drawn from different data distributions. The transfer generalization gap is the difference between the population loss on the target data distribution and the training loss. The training data set generally includes data drawn from both source and target distributions. This work presents novel information-theoretic upper bounds on the average transfer generalization gap that capture $(i)$ the domain shift between the target data distribution $P'_Z$ and the source distribution $P_Z$ through a two-parameter family of generalized $(\alpha_1,\alpha_2)$-Jensen-Shannon (JS) divergences; and $(ii)$ the sensitivity of the transfer learner output $W$ to each individual sample of the data set $Z_i$ via the mutual information $I(W;Z_i)$. For $\alpha_1 \in (0,1)$, the $(\alpha_1,\alpha_2)$-JS divergence can be bounded even when the support of $P_Z$ is not included in that of $P'_Z$. This contrasts the Kullback-Leibler (KL) divergence $D_{KL}(P_Z||P'_Z)$-based bounds of Wu et al. [1], which are vacuous under this assumption. Moreover, the obtained bounds hold for unbounded loss functions with bounded cumulant generating functions, unlike the $\phi$-divergence based bound of Wu et al. [1]. We also obtain new upper bounds on the average transfer excess risk in terms of the $(\alpha_1,\alpha_2)$-JS divergence for empirical weighted risk minimization (EWRM), which minimizes the weighted average training losses over source and target data sets. Finally, we provide a numerical example to illustrate the merits of the introduced bounds.