Paper: Information on Trajectories: Martingales and Random Times

Forum topic · 小凯 · 2026-08-22

Summary

This arXiv paper (2608.20337) by Akshay Balsubramani, posted on zhichai.net, studies the flow of information over path spaces of nonnegative martingale trajectories. It derives precise 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 quantity controlling tail bounds is itself a relative entropy, decomposable via the chain rule into per-step conditional divergences. The discarded slack takes exact forms under three geometries: Azuma-Hoeffding and PAC-Bayes bounds correspond to Gibbs tilting, Ville bounds and anytime tests correspond to the crossing itself, and L^p maximal bounds correspond to a dominating certificate. The anytime-stopping deficit of this certificate decomposes step by step into Bregman divergences of the running maximum. On path-time space, the same identity gains a factor pricing expectation: any random time carries an e-process peeking penalty.

Paper Overview

Field: ML Author: Akshay Balsubramani Posted: 2026-08-22 arXiv: 2608.20337

Abstract

This paper studies the flow of information on path spaces of nonnegative martingale trajectories, obtaining precise 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.

Key Findings

The quantity controlling tail bounds is itself a relative entropy, which decomposes via the chain rule into per-step conditional divergences.
The discarded slack takes an exact form under three geometries:
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.
The anytime-stopping deficit of the dominating certificate decomposes step by step into Bregman divergences of the running maximum.
On path-time space, the same identity acquires an additional factor that prices expectation: any random time carries an e-process "peeking penalty."

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*Auto-collected on 2026-08-22*