# Instance-Log-Optimality of Portfolio-Based E-Processes and their Sequential Hypothesis Tests

arXiv: https://arxiv.org/abs/2504.02818 · HTML: https://mlsanity.com/paper/2504.02818/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Apr 2025.

## Built on by (most cited first, 12 of 12)

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- [Sequential Structure-Sensitive Residual Diagnostics for PDE Inverse Problems](https://mlsanity.com/paper/2607.02101/) (2026)
- [Global Sequential Testing for Multi-Stream Auditing](https://mlsanity.com/paper/2602.21479/) (2026)
- [Multi-Armed Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2603.17925/) (2026)
- [Semi-Supervised Hypothesis Testing by Betting on Predictions](https://mlsanity.com/paper/2605.28533/) (2026)
- [Optimizing Social Utility in Sequential Experiments](https://mlsanity.com/paper/2605.06520/) (2026)
- [Confidence Horizons](https://mlsanity.com/paper/2608.03889/) (2026)

## Abstract (arXiv)

We consider the problem of sequential hypothesis testing using e-processes. For a rich class of composite testing problems---which include bounded mean testing, equal mean testing for bounded random tuples, and some key ingredients of two-sample and independence testing as special cases---we show that any e-process satisfying a certain sublinear regret bound is asymptotically and almost surely instance-log-optimal for a composite alternative. This is a strong notion of optimality that has not previously been established for the aforementioned problems, and we provide explicit test supermartingales and e-processes satisfying this notion in a more general case. Furthermore, we derive matching lower and upper bounds on the expected rejection time in the high-confidence regime for the resulting sequential tests in all of these cases. The proofs of these results make weak, algorithm-agnostic moment assumptions and rely on a proof technique involving the aforementioned regret and a family of numeraire portfolios. Finally, we discuss how all of these theorems hold in a distribution-uniform sense, a notion of log-optimality that is stronger still and seems to be new to the literature.