Box Thirding (B3) Algorithm

• Box Thirding (B3) is an anytime best arm identification method that employs hierarchical ternary comparisons to screen arms effectively in fixed-budget stochastic bandit settings.
• The algorithm organizes arms into boxes where the best arm is lifted, the median is deferred, and the weakest is discarded, ensuring efficient sample allocation and robust elimination.
• Beyond bandit problems, the term ‘B3’ also appears in file-system crash testing and tricriteria optimization, reflecting its broader cross-domain usage.

The explicit title "Box Thirding" is used for a 2026 algorithm for Best Arm Identification (BAI) under fixed-budget constraints, particularly in anytime settings and regimes with large $N$, where exhaustive evaluation is impossible within a limited budget $T$ (Hwang et al., 20 Feb 2026). In that work, B3 performs iterative ternary comparisons inside hierarchical boxes: among three arms, the best-performing arm is explored further, the median is deferred, and the weakest is discarded. The same three-letter label $B^3$ also appears in file-system research as "Bounded Black-Box Crash Testing" (Mohan et al., 2018), and "box thirding" also describes a tricriteria box-decomposition mechanism in discrete multi-objective optimization (Daechert et al., 2013). The primary modern usage of the exact phrase "Box Thirding" is therefore bandit-theoretic, but the term has a broader cross-domain history.

1. Terminological scope and disambiguation

The arXiv record associates closely related labels with three distinct technical constructions.

Usage	Domain	Core idea
Box Thirding (B3)	Stochastic bandits	Hierarchical ternary comparison for anytime BAI
$B^3$	File systems	Bounded black-box crash testing
Box thirding	Tricriteria optimization	Splitting a search box into up to three subboxes

In the bandit setting, B3 is an anytime BAI algorithm designed for "insufficient sampling" regimes, especially when $T < N$ or, more generally, when the candidate set retained by an algorithm is too small to cover all $\epsilon$-best arms (Hwang et al., 20 Feb 2026). In the file-system setting, $B^3$ denotes a practical crash-consistency methodology that exhaustively tests a bounded space of workloads and crash points without inspecting or modifying file-system code (Mohan et al., 2018). In tricriteria optimization, the phrase refers to a geometric update rule in which each newly discovered nondominated point cuts a current box along up to three coordinate directions, with linear bounds on the number of boxes and scalarizations (Daechert et al., 2013).

This terminological overlap matters because the three constructions share only a superficial naming similarity. A plausible implication is that "B3" is best treated as a context-dependent abbreviation rather than a single established concept across the literature.

2. Bandit formulation and the data-poor regime

In "Box Thirding: Anytime Best Arm Identification under Insufficient Sampling," the underlying model is a stochastic multi-armed bandit with $N$ arms, unknown reward distributions $\nu_i$, and 1-sub-Gaussian rewards $X$ with means $T$0, ordered as $T$1 for analysis (Hwang et al., 20 Feb 2026). The goal is BAI rather than cumulative-reward maximization: after $T$2 pulls, the algorithm outputs $T$3, and performance is measured by simple regret

$T$4

or by the fixed-budget misidentification probability $T$5.

The paper emphasizes the distinction among fixed-budget BAI, fixed-confidence BAI, and anytime BAI. The B3 algorithm belongs to the anytime class: it does not assume prior knowledge of $T$6, maintains a recommendation at every time $T$7, and can be stopped arbitrarily. This design is motivated by settings in which resources are uncertain.

A central construct is the candidate set $T$8, the set of arms that can still plausibly be returned as output under some reward instance. Its size is $T$9. The paper defines a "data-poor condition" for $B^3$0 by

$B^3$1

where $B^3$2 is the number of $B^3$3-best arms. In this regime, the dominant difficulty decomposes into two parts: screening, meaning inclusion of a near-best arm in $B^3$4, and estimation, meaning discrimination among the arms already retained. That decomposition is the organizing principle for the B3 analysis.

3. Hierarchical ternary mechanism

B3 organizes arms into boxes $B^3$5, indexed by a level $B^3$6 and a deferment index $B^3$7, with each box holding up to three arms and their empirical means (Hwang et al., 20 Feb 2026). When a box becomes full, B3 applies a local ternary rule:

• the largest arm is LIFTed,
• the median arm is SHIFTed,
• the smallest arm is DISCARDed.

LIFT promotes the arm to level $B^3$8, allocates an additional $B^3$9 pulls, updates its empirical mean, and places it in $B^3$0. SHIFT keeps the arm at the same level but increases its deferment index to $B^3$1 without extra pulls. DISCARD removes the arm permanently. The paper denotes the local update as ARRANGE_BOX.

The full anytime algorithm repeatedly sweeps through boxes from higher levels to lower ones, and from larger deferment indices down to smaller ones. A box is arranged only when it is full and the destination boxes for LIFT and SHIFT are not full. If $B^3$2 has room, the algorithm introduces a new arm uniformly from those not already discarded and not present in any box. At stopping time, it returns the arm in $B^3$3 with the largest empirical mean, where $B^3$4 is the current highest nonempty level.

The per-level budget factor is inherited from a local Sequential Halving analysis. The paper uses $B^3$5, the solution of

$B^3$6

This geometric budget schedule lets B3 allocate more samples to arms that survive to higher levels while remaining anytime. The crucial design choice is that B3 does not preselect a fixed subset and run a fixed-budget routine on it; instead, it maintains a global candidate pool with incremental admission, promotion, deferment, and elimination.

4. Guarantees, rates, and relation to other BAI procedures

The analysis of B3 separates total error into non-inclusion error and within-set misidentification (Hwang et al., 20 Feb 2026). Under the data-poor condition, the paper gives a general non-inclusion bound of the form

$B^3$7

where $B^3$8 is the best mean inside the candidate set. This makes the size of the candidate set a first-order quantity.

The paper then compares candidate-set growth across algorithms. Uniform Sampling (US) has $B^3$9, Bracketing Sequential Halving (BSH) has $T < N$0, and B3 also has $T < N$1. Thus B3 matches US in screening capacity while preserving a stronger elimination structure than uniform allocation.

For within-set discrimination, the paper reanalyzes Sequential Halving (SH) with per-level budgets $T < N$2, showing that for $T < N$3,

$T < N$4

B3 is then shown to inherit SH-like within-set performance: $T < N$5

The main theorem combines the two terms and yields a total misidentification exponent proportional to

$T < N$6

Under a polynomial gap structure,

$T < N$7

the resulting simple-regret bound is

$T < N$8

The paper also states an $T < N$9-style sufficient budget condition: $\epsilon$0

Relative to baselines, US maximizes screening but has weaker within-set discrimination, SH is not anytime and requires $\epsilon$1, and bracketing methods such as BUCB and BSH are anytime but lose a $\epsilon$2 factor in screening capacity. The stated contribution of B3 is to combine $\epsilon$3 with SH-like discrimination in a single anytime procedure.

5. Empirical behavior, implementation, and scope conditions

The experimental section evaluates B3 on synthetic settings and on the New Yorker Cartoon Caption Contest dataset (Hwang et al., 20 Feb 2026). In contest 893, each caption is an arm and $\epsilon$4. The paper defines $\epsilon$5 as the empirical proportion of Funny plus Somewhat Funny responses after preprocessing, and simulates three reward models: high noise with $\epsilon$6, moderate noise with $\epsilon$7, and a deterministic case with $\epsilon$8. The baselines are B3, US, BUCB, and BSH; the budget is $\epsilon$9, repeated over 1,000 runs.

The reported behavior follows the screening-versus-estimation decomposition. In the deterministic case, US achieves very low simple regret because once the best arm enters the candidate set there is no estimation error; B3 is close. Under moderate and high noise, within-set misidentification becomes material, US plateaus, BUCB and BSH improve more slowly because they sample fewer arms, and B3 achieves the best or near-best simple regret. The appendix further reports Bernoulli and heavy-tailed Kumaraswamy rewards, along with data-rich regimes such as $B^3$0 and $B^3$1, where B3 is described as comparable to SH with tuned $B^3$2 and better than UCB-E.

For implementation, the paper specifies a table or dictionary of boxes $B^3$3, state variables $B^3$4, $B^3$5, and the discard set $B^3$6, together with incremental sample counts and top-down scheduling. Memory is stated as $B^3$7, and the hierarchy depth as $B^3$8. The paper also describes a modified B3 for non-data-poor regimes, in which, once all arms have been initially examined, the algorithm raises a base level $B^3$9 and reintroduces previously discarded arms with more samples.

The paper is explicit about limitations. The theory assumes 1-sub-Gaussian rewards. The constants and rates depend on $N$0 and on the gap structure. In very low-noise regimes, uniform sampling may be marginally better because screening dominates. The authors therefore recommend B3 as a robust default when the operative regime is not known in advance.

6. Other meanings of B3 and box thirding

In file-system research, $N$1 denotes "Bounded Black-Box Crash Testing," a crash-consistency methodology that treats the file system as a black box accessible only through the POSIX system-call interface and the resulting block-level I/O (Mohan et al., 2018). The method bounds workload length, operation types, arguments, initial state, and crash points, then exhaustively generates workloads within that bounded space. Crash points are restricted to persistence-related calls such as fsync(), fdatasync(), sync, and msync(). The implementation uses CrashMonkey and ACE. The paper reports that the tools rediscovered 24 of 26 known crash-consistency bugs, found 10 new crash-consistency bugs in Linux file systems, and found one new crash-consistency bug in FSCQ. Here, the three $N$2's stand for bounded, black-box, and crash testing, not for box decomposition.

In discrete tricriteria optimization, the 2013 paper does not explicitly use the label "Box Thirding," but it presents a three-dimensional box-decomposition update that fits that description (Daechert et al., 2013). Each newly discovered nondominated point splits a current search box into up to three subboxes,

$N$3

and the paper develops a $N$4-split rule, based on individual subsets and neighbors, to avoid redundant boxes. The resulting tricriteria algorithm requires at most $N$5 scalarized subproblems to generate the entire nondominated set, and this improves to $N$6 with an $N$7-constraint scalarization and a specific box-selection rule.

These alternative uses show that "B3" and "box thirding" are not field-invariant terms. In the bandit literature, Box Thirding is an anytime arm-elimination algorithm built around local ternary comparisons. In file systems, $N$8 is a bounded crash-testing methodology. In tricriteria optimization, box thirding is a geometric decomposition rule for objective-space search. The shared vocabulary reflects a recurrent pattern of structured triadic reduction, but the technical objects, guarantees, and use cases are distinct.