Summary
A 2026 arXiv paper (2608.20337) by Akshay Balsubramani studies information flow on path spaces of nonnegative martingale trajectories, deriving exact variational identities that hold even at arbitrary random times. The work recovers and unifies classical concentration inequalities from Ville to PAC-Bayes and quantifies the information discarded by each. The tail-bound-controlled quantity is itself a relative entropy, decomposable via the chain rule into per-step conditional divergences. Under three geometries, the discarded slack takes exact forms: Azuma-Hoeffding and PAC-Bayes bounds correspond to Gibbs tilting; Ville bounds and anytime tests correspond to the crossing itself; L^p maximal bounds correspond to a dominating certificate, whose arbitrary-stopping deficit decomposes stepwise into Bregman divergences of running maxima. On path-time space, the identity gains a pricing-expectation factor: any random time carries an e-process peeking penalty.
Paper Overview
Field: Machine Learning
Author: Akshay Balsubramani
Published: 2026-08-22
arXiv: 2608.20337
Summary
This paper studies the flow of information on the path space of nonnegative martingale trajectories, obtaining exact variational identities that hold even at arbitrary random times. The work recovers and unifies classical concentration inequalities spanning Ville to PAC-Bayes, and quantifies the information discarded by each inequality.
The quantity controlled by tail bounds is itself a relative entropy, which decomposes via the chain rule into per-step conditional divergences. Under three geometric structures, the discarded slack takes exact forms:
- Azuma-Hoeffding and PAC-Bayes bounds correspond to Gibbs tilting.
- Ville bounds and anytime tests correspond to the crossing itself.
- L^p maximal bounds correspond to a dominating certificate, whose arbitrary-stopping deficit decomposes stepwise into Bregman divergences of the running maximum.
On path-time space, the same identity acquires an additional pricing-expectation factor: any random time carries a "peeking penalty" in the form of an e-process.
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*Auto-collected on 2026-08-22*
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