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Optimal sequential tests yield log-optimal e-processes

NeurIPS 2026 · Learning theory and algorithms › Sample complexity · approach: Optimal mean estimation and subsampling · 0 citations (OpenAlex)

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Shows that asymptotically optimal sequential tests can be aggregated, via weighted indicators of stopping times (WAIT e-processes), into e-processes achieving asymptotically optimal log-growth, and characterizes which notions of test optimality suffice.

Problem

Existing results show e-processes are complete for sequential testing and that log-optimal e-processes yield optimal tests, but not the converse: whether asymptotically optimal sequential tests can be converted into asymptotically log-optimal e-processes.

Approach

Aggregate a level-indexed family of sequential tests into M_t = sum_k w_k 1{tau_k <= t}; this is a valid e-process whose log-growth under Q equals rho*I, where rho is the exponential growth rate of the cumulative weight profile W(x) = sum_{b_k<=x} w_k, provided tests are first-order optimal at rate I.

Key results

• WAIT aggregate under Q satisfies (1/t) log M_t -> rho*I almost surely whenever tests are almost-surely first-order optimal and log W(x)/x -> rho.
• Unweighted schedules alpha_k ~ 1/[k(log k)^p] with p>1 achieve full log-rate I, while alpha_k = 2^{-k} gives log-rate 0.
• Weighted schedule M^dy with weights (6/pi^2) 2^k/k^2 at levels 2^{-k} recovers full rate I; expectation optimality alone is shown insufficient via a counterexample.

Surprising findings

• Unweighted counting aggregate with the natural schedule 2^{-k} yields zero log-growth rate despite valid levels.
• Expectation-optimal tests alone do not guarantee log-optimal aggregates; a random rate Y/I can arise.

Limitations

• Results are pointwise for each fixed alternative Q, not uniform over a class of alternatives.
• Assumes the input tests are optimal; existence of such tests for a given model is not addressed.
• Gives only first-order limits, no second-order terms or nonasymptotic lower bounds on M_t.

Experimental setup

Metrics: log-growth rate (1/t) log M_t threshold time T_alpha / log(1/alpha)

Builds on: Ville's inequality e-processes universal portfolio theory

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