planning_horizon: Planning horizon for a distance to the decision threshold

In seqbench: Anytime-Valid Sequential Benchmarking of Algorithms

Planning horizon for a distance to the decision threshold

Description

Number of instances after which the confidence sequence is expected to have declared the true hypothesis, when the true mean paired difference is at distance distance from the nearest threshold -margin or margin. Two modes:

Usage

planning_horizon(design, distance, sd = NULL, max_t = 1e+06)

Arguments

design	A comparison_design().
distance	Positive distance ⁠|mu| - margin⁠ (superiority) or ⁠margin - |mu|⁠ (equivalence), in loss units.
sd	Optional standard deviation of the paired difference, loss units.
max_t	Search limit.

Details

• sd = NULL (default): the guaranteed horizon ⁠t*(Delta) = min{t : 2 w_t < |Delta|}⁠ of the predictable-plug-in Hoeffding boundary, whose half-width w_t is deterministic. On the coverage event (probability at least 1 - alpha) the Hoeffding comparison has stopped with the correct declaration by then (theory note, Prop. 2). It is distribution-free and, for small margins, very conservative: it can exceed what the default betting boundary needs by two or three orders of magnitude.

• sd given (standard deviation of the paired difference, loss units): an approximation that plugs sd into the empirical-Bernstein half-width in place of the running variance estimate. It is not a bound; it is the order of magnitude a variance-adaptive boundary needs, and the betting boundary is usually faster still. Use a pilot or a conservative guess for sd.

Both charge the full half-width twice (worst-case position of the centre); typical stopping times are about half of the returned value, as measured in the package's pilot study. Both are computable before any data are collected and serve to size budget and n_max.

Value

An integer number of instances, or Inf if not reached by max_t.

Examples

design <- comparison_design(margin = 0.02, bounds = c(0, 1))
planning_horizon(design, distance = 0.05)             # guaranteed, Hoeffding
planning_horizon(design, distance = 0.05, sd = 0.1)   # variance-based approximation

seqbench documentation built on Oct. 8, 2026, 5:07 p.m.