Adjacent-BAI in Non-Stationary Linear Bandits

• Adjacent-BAI is a fixed-budget algorithm that identifies the hindsight-best arm in non-stationary linear bandits by exploiting the geometry of adjacent vertices of the arm set.
• The method reduces global comparisons to local swaps along edges defined by the convex hull, thereby yielding an arm-set-dependent complexity measure that can be significantly smaller than the classical dimension-based benchmark.
• A matching upper bound is established via a two-phase KL approach and least-squares estimation, ensuring error probabilities decay as exp(-Θ(T/HAdjacent)) up to constant factors.

Searching arXiv for the target paper and closely related best-arm identification work in bandits. Adjacent-BAI is a fixed-budget best-arm-identification procedure for non-stationary linear bandits in which the statistical difficulty is characterized by the geometry of adjacent vertices of $\mathrm{conv}(\mathcal X)$ rather than by a dimension-driven minimax surrogate. In the formulation studied in "On The Complexity of Best-Arm Identification in Non-Stationary Linear Bandits" (Maynard-Zhang et al., 11 Mar 2026), a learner interacts for $T$ rounds with a finite arm set $\mathcal X\subset\mathbb R^d$ under an obliviously chosen sequence of parameters $\{\theta_t\}_{t=1}^T$, and seeks to identify the hindsight-best arm $x_*=\arg\max_{x\in\mathcal X}x^\top\bar\theta$, where $\bar\theta=\tfrac1T\sum_{t=1}^T\theta_t$. The central contribution is an arm-set-dependent complexity $H_{\mathrm{Adjacent}}$ and a static allocation rule, Adjacent-BAI, whose error exponent matches a corresponding lower bound up to constants, thereby refining the classical $G$-optimal benchmark (Maynard-Zhang et al., 11 Mar 2026).

1. Formal setting and objective

The problem is posed with a finite arm set

$$\mathcal X\subset\mathbb R^d,\qquad |\mathcal X|=K,\qquad \mathrm{span}(\mathcal X)=\mathbb R^d,$$

and an unknown parameter sequence $\{\theta_t\}_{t=1}^T\subset\mathbb R^d$ chosen by an oblivious adversary. At round $T$0, the learner selects $T$1 and observes

$T$2

where $T$3 is zero-mean $T$4-sub-Gaussian noise. The hindsight-best arm is defined through the time-average parameter

$T$5

and uniqueness of $T$6 is assumed. The fixed-budget objective is to minimize the identification error probability $T$7 (Maynard-Zhang et al., 11 Mar 2026).

A further quantity used throughout the analysis is the minimum gap among extreme points. Let $T$8 be the set of vertices of $T$9. Then

$\mathcal X\subset\mathbb R^d$0

This gap parameter enters both the lower and upper bounds through the eventual complexity measure. In the same setting, it is also recalled that uniformly sampling arms from the $\mathcal X\subset\mathbb R^d$1-optimal design yields a minimax-optimal error probability of $\mathcal X\subset\mathbb R^d$2, where $\mathcal X\subset\mathbb R^d$3 scales proportionally with the dimension $\mathcal X\subset\mathbb R^d$4; the motivation for Adjacent-BAI is that this complexity can be overly pessimistic for arm sets with richer geometric structure (Maynard-Zhang et al., 11 Mar 2026).

2. Adjacency as the operative geometry

The defining structural notion is adjacency in the edge graph of the arm polytope. Two distinct vertices $\mathcal X\subset\mathbb R^d$5 are called adjacent if the line segment $\mathcal X\subset\mathbb R^d$6 is an edge of $\mathcal X\subset\mathbb R^d$7. The set of ordered adjacent pairs is

$\mathcal X\subset\mathbb R^d$8

The key structural statement is the Adjacency Lemma: a vertex $\mathcal X\subset\mathbb R^d$9 can be beaten by some $\{\theta_t\}_{t=1}^T$0, meaning $\{\theta_t\}_{t=1}^T$1, if and only if it can be beaten by an adjacent vertex $\{\theta_t\}_{t=1}^T$2 in the edge graph. This reduces the hard alternatives from all pairs of vertices to edge-wise competitors. In consequence, the worst-case identification difficulty is driven by local swaps along edges of $\{\theta_t\}_{t=1}^T$3 rather than by arbitrary global comparisons among vertices (Maynard-Zhang et al., 11 Mar 2026).

This restriction is not merely combinatorial. It determines which alternative instances can feasibly reverse the ranking of the best arm while respecting the gap constraints used in the lower-bound construction. A plausible implication is that Adjacent-BAI should be most beneficial when the edge geometry is substantially easier than the full pairwise geometry of $\{\theta_t\}_{t=1}^T$4.

3. Arm-set-dependent complexity and the lower bound

For a design $\{\theta_t\}_{t=1}^T$5, define

$\{\theta_t\}_{t=1}^T$6

Fixing the target minimum gap $\{\theta_t\}_{t=1}^T$7, the complexity introduced in the paper is

$\{\theta_t\}_{t=1}^T$8

This quantity depends explicitly on the arm set through its adjacency structure, in contrast to the classical benchmark $\{\theta_t\}_{t=1}^T$9 (Maynard-Zhang et al., 11 Mar 2026).

Theorem 2 gives the corresponding arm-set-dependent lower bound. For any possibly adaptive algorithm, there exist two non-stationary instances, both with minimum gap at least $x_*=\arg\max_{x\in\mathcal X}x^\top\bar\theta$0 but with different best arms, such that

$x_*=\arg\max_{x\in\mathcal X}x^\top\bar\theta$1

Equivalently, the achievable error exponent is at best of order $x_*=\arg\max_{x\in\mathcal X}x^\top\bar\theta$2.

The proof strategy uses a two-phase KL argument. The horizon is split into $x_*=\arg\max_{x\in\mathcal X}x^\top\bar\theta$3 exploration rounds and $x_*=\arg\max_{x\in\mathcal X}x^\top\bar\theta$4 verification rounds. In the exploration phase, one instance uses parameter $x_*=\arg\max_{x\in\mathcal X}x^\top\bar\theta$5 and the other a small shift $x_*=\arg\max_{x\in\mathcal X}x^\top\bar\theta$6, forcing the KL divergence to scale as $x_*=\arg\max_{x\in\mathcal X}x^\top\bar\theta$7. The feasibility constraints induced by the minimum-gap condition reduce to linear inequalities for a chosen adjacent pair $x_*=\arg\max_{x\in\mathcal X}x^\top\bar\theta$8. Minimizing edge-wise yields the closed-form inner value

$x_*=\arg\max_{x\in\mathcal X}x^\top\bar\theta$9

which is exactly the quantity that leads to $\bar\theta=\tfrac1T\sum_{t=1}^T\theta_t$0 (Maynard-Zhang et al., 11 Mar 2026).

4. Adjacent-optimal design and the Adjacent-BAI procedure

The algorithm is built from an adjacent-restricted variant of the classical $\bar\theta=\tfrac1T\sum_{t=1}^T\theta_t$1-design. The unrestricted design is

$\bar\theta=\tfrac1T\sum_{t=1}^T\theta_t$2

whereas the Adjacent-optimal design uses only edges: $\bar\theta=\tfrac1T\sum_{t=1}^T\theta_t$3 Once $\bar\theta=\tfrac1T\sum_{t=1}^T\theta_t$4 is obtained, one sets

$\bar\theta=\tfrac1T\sum_{t=1}^T\theta_t$5

Adjacent-BAI is then a static-allocation algorithm with random permutation. Its prescribed workflow is:

1. Compute the edge set $\bar\theta=\tfrac1T\sum_{t=1}^T\theta_t$6 of $\bar\theta=\tfrac1T\sum_{t=1}^T\theta_t$7, for example via convex-hull computation plus adjacency or via linear programs.
2. Solve

$\bar\theta=\tfrac1T\sum_{t=1}^T\theta_t$8

1. Round $\bar\theta=\tfrac1T\sum_{t=1}^T\theta_t$9 into an explicit static allocation $H_{\mathrm{Adjacent}}$0 such that

$H_{\mathrm{Adjacent}}$1

up to a small constant factor.

1. Draw a uniform random permutation $H_{\mathrm{Adjacent}}$2 of $H_{\mathrm{Adjacent}}$3.
2. Play $H_{\mathrm{Adjacent}}$4 at each round and observe $H_{\mathrm{Adjacent}}$5.
3. Form the least-squares estimator

$H_{\mathrm{Adjacent}}$6

1. Output

$H_{\mathrm{Adjacent}}$7

The budget condition stated in the pseudocode is $H_{\mathrm{Adjacent}}$8. The use of a randomized permutation is part of the stated procedure, and the final decision rule is purely empirical maximization under the least-squares estimate (Maynard-Zhang et al., 11 Mar 2026).

5. Matching upper bound and comparison with classical complexity

Theorem 4 establishes a matching upper bound for Adjacent-BAI under the assumptions $H_{\mathrm{Adjacent}}$9. Writing

$G$0

the algorithm satisfies

$G$1

In particular,

$G$2

Since the lower bound has the same $G$3 exponent up to constants, the paper concludes that the arm-set-dependent complexity is tight (Maynard-Zhang et al., 11 Mar 2026).

The proof uses a sub-Gaussian concentration property of the least-squares estimator: for any direction $G$4,

$G$5

is $G$6-sub-Gaussian. The Adjacency Lemma is then applied again, reducing the failure event to the existence of an adjacent neighbor $G$7 that beats $G$8. A union bound over $G$9 edges yields an edge-wise error term of the form

$$\mathcal X\subset\mathbb R^d,\qquad |\mathcal X|=K,\qquad \mathrm{span}(\mathcal X)=\mathbb R^d,$$0

and the rounding guarantee together with the definition of $$\mathcal X\subset\mathbb R^d,\qquad |\mathcal X|=K,\qquad \mathrm{span}(\mathcal X)=\mathbb R^d,$$1 converts this into the stated bound.

The comparison with the traditional complexity measure is explicit. The minimax-optimal stationary-style complexity is

$$\mathcal X\subset\mathbb R^d,\qquad |\mathcal X|=K,\qquad \mathrm{span}(\mathcal X)=\mathbb R^d,$$2

One always has

$$\mathcal X\subset\mathbb R^d,\qquad |\mathcal X|=K,\qquad \mathrm{span}(\mathcal X)=\mathbb R^d,$$3

but the gap can be strict. For $$\mathcal X\subset\mathbb R^d,\qquad |\mathcal X|=K,\qquad \mathrm{span}(\mathcal X)=\mathbb R^d,$$4 equal to the set of $$\mathcal X\subset\mathbb R^d,\qquad |\mathcal X|=K,\qquad \mathrm{span}(\mathcal X)=\mathbb R^d,$$5 equally spaced points on the unit circle in $$\mathcal X\subset\mathbb R^d,\qquad |\mathcal X|=K,\qquad \mathrm{span}(\mathcal X)=\mathbb R^d,$$6, adjacent pairs become arbitrarily close as $$\mathcal X\subset\mathbb R^d,\qquad |\mathcal X|=K,\qquad \mathrm{span}(\mathcal X)=\mathbb R^d,$$7, so

$$\mathcal X\subset\mathbb R^d,\qquad |\mathcal X|=K,\qquad \mathrm{span}(\mathcal X)=\mathbb R^d,$$8

This example indicates that the relevant difficulty may be governed by local edge geometry rather than ambient dimension.

6. Computational profile and terminological scope

The computational overhead is described as comparable in spirit to stationary $$\mathcal X\subset\mathbb R^d,\qquad |\mathcal X|=K,\qquad \mathrm{span}(\mathcal X)=\mathbb R^d,$$9-allocations. If a convex-hull routine is available, the adjacency graph $\{\theta_t\}_{t=1}^T\subset\mathbb R^d$0 can be found in

$\{\theta_t\}_{t=1}^T\subset\mathbb R^d$1

or, alternatively, in polynomial time in $\{\theta_t\}_{t=1}^T\subset\mathbb R^d$2 by solving $\{\theta_t\}_{t=1}^T\subset\mathbb R^d$3 small linear programs. The optimization

$\{\theta_t\}_{t=1}^T\subset\mathbb R^d$4

is described as a modest-sized semidefinite program, or it can be approximated with off-the-shelf first-order methods in $\{\theta_t\}_{t=1}^T\subset\mathbb R^d$5 per iteration. Rounding $\{\theta_t\}_{t=1}^T\subset\mathbb R^d$6 into a fixed allocation of length $\{\theta_t\}_{t=1}^T\subset\mathbb R^d$7 uses a standard deterministic-rounding procedure and again requires $\{\theta_t\}_{t=1}^T\subset\mathbb R^d$8 (Maynard-Zhang et al., 11 Mar 2026).

The paper’s summary positions Adjacent-BAI as the first fixed-budget, non-stationary linear-bandit algorithm whose error probability both depends tightly on the arm-set-dependent complexity $\{\theta_t\}_{t=1}^T\subset\mathbb R^d$9 and is minimax-optimal up to constants. Within the scope of the provided materials, this identifies Adjacent-BAI as a geometry-sensitive refinement of static design methods for non-stationary linear-bandit identification (Maynard-Zhang et al., 11 Mar 2026).

The label is, however, not globally unique across literatures. In the supplied materials, closely related terminology also appears in a graph-smoothness-constrained best-arm-identification setting for spectral bandits, where fixed-confidence sample complexity is governed by a max-min game over alternatives satisfying $T$00 (Kocák et al., 2020). It also appears in statistical genetics as an “adjacent-marker Bayesian association and interaction” model for partitioning SNPs into LD-blocks and selecting marginal or epistatic disease-associated markers under a block-based Bayesian framework (Zhang et al., 2011). This suggests that, in technical usage, “Adjacent-BAI” should be interpreted from context: in non-stationary linear bandits it denotes the edge-restricted optimal-design algorithm built around $T$01, whereas in other domains the same shorthand can refer to distinct constructions.