Individually Conditional Individual Mutual Information Bound on Generalization Error

We propose an information-theoretic bound on the generalization error based on a combination of the error decomposition technique of Bu et al. and the conditional mutual information (CMI) construction of Steinke and Zakynthinou. In a previous work, Haghifam et al. proposed a different bound combining the two aforementioned techniques, which we refer to as the conditional individual mutual information (CIMI) bound. However, in a simple Gaussian setting, both the CMI and the CIMI bounds are order-wise worse than that by Bu et al. This observation motivated us to propose the bound, which overcomes this issue by reducing the conditioning terms in the conditional mutual information. In the process of establishing this bound, a conditional decoupling lemma is established, which also leads to a meaningful dichotomy and comparison among these information-theoretic bounds. As an application of the proposed bound, we analyze the noisy and iterative stochastic gradient Langevin dynamics and provide an upper bound on its generalization error.