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This paper gives a unified information-theoretic accounting of instance-dependent lower bounds for bandit regret, sequential testing, best-arm identification, and Markov decision processes. It recasts the standard derivation as an exact ledger that tracks where each approximation loses information, and it identifies which losses an algorithm can measure from its own runs. For ML researchers, the practical value is diagnostic: it shows which gap between standard algorithms and classical limits comes from the uniform-quantification step, which matters most for model identification and Wald's test. The evaluation is on seven published sources, so it remains a theoretical framework rather than an empirically validated method.
Instance-dependent sequential limits as an exact information ledger
arXiv:2610.42133v1 Announce Type: new Abstract: Instance-dependent limits for bandit regret, sequential tests, best-arm identification and Markov decision processes are proved by one chain. It runs from a divergence decomposition through data processing to a statistic, a testing step that turns a class guarantee into a requirement on the history, and optimization over alternatives and allocations. Only the first step is an identity. This work writes the chain as an exact ledger. By the chain rule, data processing loses exactly a conditional divergence, which vanishes for arms in natural exponential families at the per-arm counts and reward sums. A one-parameter family of martingale identities extends the decomposition to every tilt between two instances. The classical limits follow by relaxations with named defects. All defects but the asymptotic step's can be computed from an algorithm's own runs, the uniform quantification's only when the class guarantee is a number. On seven published sources, from movie ratings to engine replacement, the ledger locates the difference between standard algorithms and the classical limits. For identification among trained models and for Wald's test the largest term is usually the uniform quantification's.
Abstract:Instance-dependent limits for bandit regret, sequential tests, best-arm identification and Markov decision processes are proved by one chain. It runs from a divergence decomposition through data processing to a statistic, a testing step that turns a class guarantee into a requirement on the history, and optimization over alternatives and allocations. Only the first step is an identity. This work writes the chain as an exact ledger. By the chain rule, data processing loses exactly a conditional divergence, which vanishes for arms in natural exponential families at the per-arm counts and reward sums. A one-parameter family of martingale identities extends the decomposition to every tilt between two instances. The classical limits follow by relaxations with named defects. All defects but the asymptotic step's can be computed from an algorithm's own runs, the uniform quantification's only when the class guarantee is a number. On seven published sources, from movie ratings to engine replacement, the ledger locates the difference between standard algorithms and the classical limits. For identification among trained models and for Wald's test the largest term is usually the uniform quantification's.
Machine Learning (cs.LG); Information Theory (cs.IT); Statistics Theory (math.ST); Machine Learning (stat.ML)
arXiv:2610.42133 [cs.LG]
(or arXiv:2610.42133v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.42133 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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