Supermodular $\mf$-divergences and bounds on lossy compression and generalization error with mutual $\mf$-information

• URL: http://arxiv.org/abs/2206.11042v1
• Date: Tue, 21 Jun 2022 09:17:06 GMT
• Title: Supermodular $\mf$-divergences and bounds on lossy compression and generalization error with mutual $\mf$-information
• Authors: Saeed Masiha, Amin Gohari, Mohammad Hossein Yassaee
• Abstract summary: We introduce super-modular $mf$-divergences and provide three applications for them. We provide a connection between the generalization error of algorithms with bounded input/output mutual $mf$-information and a generalized rate-distortion problem. Our bound is based on a new lower bound on the rate-distortion function that strictly improves over previously best-known bounds.
• Score: 17.441807469515254
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we introduce super-modular $\mf$-divergences and provide three applications for them: (i) we introduce Sanov's upper bound on the tail probability of sum of independent random variables based on super-modular $\mf$-divergence and show that our generalized Sanov's bound strictly improves over ordinary one, (ii) we consider the lossy compression problem which studies the set of achievable rates for a given distortion and code length. We extend the rate-distortion function using mutual $\mf$-information and provide new and strictly better bounds on achievable rates in the finite blocklength regime using super-modular $\mf$-divergences, and (iii) we provide a connection between the generalization error of algorithms with bounded input/output mutual $\mf$-information and a generalized rate-distortion problem. This connection allows us to bound the generalization error of learning algorithms using lower bounds on the rate-distortion function. Our bound is based on a new lower bound on the rate-distortion function that (for some examples) strictly improves over previously best-known bounds. Moreover, super-modular $\mf$-divergences are utilized to reduce the dimension of the problem and obtain single-letter bounds.

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