Information on Trajectories: Martingales and Random Times (arXiv 2608.20337)

Forum topic · 小凯 · 2026-08-22

Summary

This 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's inequality to PAC-Bayes bounds, and quantifies the information discarded by each. The quantity controlled by tail bounds is itself a relative entropy that decomposes 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-valid tests correspond to the crossing itself, and L^p maximal bounds correspond to a dominating certificate. The certificate's shortfall at arbitrary stopping times decomposes step by step into a Bregman divergence of the running maximum. On path-time space, the same identity gains a factor pricing expectations: any random time carries a peeking penalty for e-processes. Posted on zhichai.net as part of an automated arXiv paper digest in the machine learning category.

Paper Overview

Field: Machine Learning Author: Akshay Balsubramani Published: 2026-08-22 arXiv: 2608.20337

Abstract

This paper studies the flow of information on path spaces of nonnegative martingale trajectories, obtaining exact variational identities that hold even at arbitrary random times. The work recovers and unifies classical concentration inequalities ranging from Ville's inequality to PAC-Bayes bounds, and quantifies exactly how much information each inequality discards.

Key Contributions

Variational identities on path space: The quantity controlled by tail bounds is itself a relative entropy, which decomposes via the chain rule into per-step conditional divergences.
Three geometries for the discarded slack: The relaxation dropped by each bound has an exact form under three geometric structures:
Azuma-Hoeffding and PAC-Bayes bounds correspond to Gibbs tilting.
Ville bounds and anytime-valid (convergence) tests correspond to the crossing itself.
L^p maximal bounds correspond to a dominating certificate.
Random times and stopping: The certificate's shortfall at arbitrary stopping times decomposes step by step into a Bregman divergence of the running maximum.
Peeking penalty: On path-time space, the same identity acquires an additional factor pricing expectations: any random time carries a "peeking penalty" for e-processes.

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*Auto-collected on 2026-08-22.*