---
title: Adjacent-BAI in Non-Stationary Linear Bandits
url: https://www.emergentmind.com/topics/adjacent-bai
type: topic
---

# Adjacent-BAI in Non-Stationary Linear Bandits

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" [2603.10346], 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 [2603.10346].

## 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 $t$, the learner selects $x_t\in\mathcal X$ and observes
\[
r_t=x_t^\top\theta_t+\epsilon_t,
\]
where $\epsilon_t$ is zero-mean $1$-sub-Gaussian noise. The hindsight-best arm is defined through the time-average parameter
\[
\bar\theta=\tfrac1T\sum_{t=1}^T\theta_t,\qquad
x_*=\arg\max_{x\in\mathcal X}x^\top\bar\theta,
\]
and uniqueness of $x_*$ is assumed. The fixed-budget objective is to minimize the identification error probability $\Pr(\widehat x\neq x_*)$ [2603.10346].

A further quantity used throughout the analysis is the minimum gap among extreme points. Let $V\subset\mathcal X$ be the set of vertices of $\mathrm{conv}(\mathcal X)$. Then
\[
\Delta_{(1)}=\min_{x\in V\setminus\{x_*\}}(x_*-x)^\top\bar\theta.
\]
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 $G$-optimal design yields a minimax-optimal error probability of $\exp(-\Theta(T/H_G))$, where $H_G$ scales proportionally with the dimension $d$; the motivation for Adjacent-BAI is that this complexity can be overly pessimistic for arm sets with richer geometric structure [2603.10346].

## 2. Adjacency as the operative geometry

The defining structural notion is adjacency in the edge graph of the arm polytope. Two distinct vertices $x,x'\in V$ are called adjacent if the line segment $\mathrm{conv}\{x,x'\}$ is an edge of $\mathrm{conv}(\mathcal X)$. The set of ordered adjacent pairs is
\[
I=\{(x,x')\in V\times V:\ x\neq x',\ x,x'\text{ adjacent}\}.
\]

The key structural statement is the Adjacency Lemma: a vertex $x$ can be beaten by some $y\in V$, meaning $(y-x)^\top\theta>0$, if and only if it can be beaten by an adjacent vertex $z$ 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 $\mathrm{conv}(\mathcal X)$ rather than by arbitrary global comparisons among vertices [2603.10346].

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 $V^2$.

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

For a design $\lambda\in\triangle_{\mathcal X}$, define
\[
A(\lambda)=\sum_{x\in\mathcal X}\lambda_x\,xx^\top.
\]
Fixing the target minimum gap $\Delta_{(1)}$, the complexity introduced in the paper is
\[
H_{\mathrm{Adjacent}}(\mathcal X,\Delta_{(1)})
=
\min_{\lambda\in\triangle_{\mathcal X}}
\max_{(x,x')\in I}
\frac{\|x-x'\|^2_{A(\lambda)^{-1}}}{\Delta_{(1)}^2}.
\]
This quantity depends explicitly on the arm set through its adjacency structure, in contrast to the classical benchmark $H_G$ [2603.10346].

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 $\Delta_{(1)}$ but with different best arms, such that
\[
\max\bigl\{\Pr_{\theta}(\widehat x\neq x_*),\ \Pr_{\theta'}(\widehat x\neq x_*')\bigr\}
\ge
\tfrac14\exp\!\Bigl(-\tfrac{4T}{H_{\mathrm{Adjacent}}(\mathcal X,\Delta_{(1)})}\Bigr).
\]
Equivalently, the achievable error exponent is at best of order $\exp(-\Theta(T/H_{\mathrm{Adjacent}}))$.

The proof strategy uses a two-phase KL argument. The horizon is split into $\tfrac T2$ exploration rounds and $\tfrac T2$ verification rounds. In the exploration phase, one instance uses parameter $0$ and the other a small shift $v$, forcing the KL divergence to scale as $\tfrac T2\,v^\top A(\lambda)v$. The feasibility constraints induced by the minimum-gap condition reduce to linear inequalities for a chosen adjacent pair $(x,x')$. Minimizing edge-wise yields the closed-form inner value
\[
\min_{\theta,v} v^\top A(\lambda)v
\Longrightarrow
\frac{4\,\Delta_{(1)}^2}{\|x-x'\|^2_{A(\lambda)^{-1}}},
\]
which is exactly the quantity that leads to $H_{\mathrm{Adjacent}}$ [2603.10346].

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

The algorithm is built from an adjacent-restricted variant of the classical $\mathcal{XY}$-design. The unrestricted design is
\[
\lambda^{XY}
=
\arg\min_{\lambda}
\max_{(x,x')\in V^2}
\|x-x'\|^2_{A(\lambda)^{-1}},
\]
whereas the Adjacent-optimal design uses only edges:
\[
\lambda^{\mathrm{Adjacent}}
=
\arg\min_{\lambda\in\triangle_{\mathcal X}}
\max_{(x,x')\in I}
\|x-x'\|^2_{A(\lambda)^{-1}}.
\]
Once $\lambda^*$ is obtained, one sets
\[
H_{\mathrm{Adjacent}}
=
\frac1{\Delta_{(1)}^2}
\max_{(x,x')\in I}\|x-x'\|^2_{A(\lambda^*)^{-1}}.
\]

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

1. Compute the edge set $I$ of $\mathrm{conv}(\mathcal X)$, for example via convex-hull computation plus adjacency or via linear programs.
2. Solve
   \[
   \lambda^*\leftarrow \arg\min_{\lambda\in\triangle_{\mathcal X}}
   \max_{(x,x')\in I}\|x-x'\|^2_{A(\lambda)^{-1}}.
   \]
3. Round $\lambda^*$ into an explicit static allocation $x_1,\dots,x_T$ such that
   \[
   \frac1T\sum_{t=1}^T x_t x_t^\top \approx A(\lambda^*)
   \]
   up to a small constant factor.
4. Draw a uniform random permutation $\pi$ of $[T]$.
5. Play $x_{\pi(t)}$ at each round and observe $r_t$.
6. Form the least-squares estimator
   \[
   \widehat\theta
   =
   \Bigl(\sum_{t=1}^T x_t x_t^\top\Bigr)^{-1}
   \sum_{t=1}^T x_{\pi(t)}r_t.
   \]
7. Output
   \[
   \widehat x=\arg\max_{x\in\mathcal X}x^\top\widehat\theta.
   \]

The budget condition stated in the pseudocode is $T\ge d^2$. 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 [2603.10346].

## 5. Matching upper bound and comparison with classical complexity

Theorem 4 establishes a matching upper bound for Adjacent-BAI under the assumptions $\|x\|_2,\|\theta_t\|_2\le1$. Writing
\[
I^{x_*}=\{x':(x_*,x')\in I\},
\]
the algorithm satisfies
\[
\Pr(\widehat x\neq x_*)
\le
|I^{x_*}|
\exp\!\Bigl(-\frac{T}{36\,H_{\mathrm{Adjacent}}(\mathcal X,\Delta_{(1)})}\Bigr).
\]
In particular,
\[
\Pr(\widehat x\neq x_*)\le \exp(-\Omega(T/H_{\mathrm{Adjacent}})).
\]
Since the lower bound has the same $T/H_{\mathrm{Adjacent}}$ exponent up to constants, the paper concludes that the arm-set-dependent complexity is tight [2603.10346].

The proof uses a sub-Gaussian concentration property of the least-squares estimator: for any direction $z$,
\[
z^\top(\widehat\theta-\bar\theta)
\]
is $O\!\bigl(\|z\|_{(\sum x_t x_t^\top)^{-1}}\bigr)$-sub-Gaussian. The Adjacency Lemma is then applied again, reducing the failure event to the existence of an adjacent neighbor $x'$ that beats $x_*$. A union bound over $|I^{x_*}|$ edges yields an edge-wise error term of the form
\[
\exp\bigl(-c\,T\,\Delta_{(1)}^2/\|x'-x_*\|^2_{A^{-1}}\bigr),
\]
and the rounding guarantee together with the definition of $\lambda^*$ converts this into the stated bound.

The comparison with the traditional complexity measure is explicit. The minimax-optimal stationary-style complexity is
\[
H_G(\Delta_{(1)})=\frac{d}{\Delta_{(1)}^2},
\qquad
\text{since }\min_{\lambda}\max_x \|x\|^2_{A(\lambda)^{-1}}=d.
\]
One always has
\[
H_{\mathrm{Adjacent}}\le 4\,H_G,
\]
but the gap can be strict. For $\mathcal X$ equal to the set of $K$ equally spaced points on the unit circle in $\mathbb R^2$, adjacent pairs become arbitrarily close as $K\to\infty$, so
\[
\max_{(x,x')\in I}\|x-x'\|^2_{A(\lambda)^{-1}}\to0,
\qquad
H_{\mathrm{Adjacent}}\ll H_G.
\]
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 $\mathcal{XY}$-allocations. If a convex-hull routine is available, the adjacency graph $I$ can be found in
\[
O(K\log K+K^{\lfloor d/2\rfloor}),
\]
or, alternatively, in polynomial time in $(K,d)$ by solving $O(K^2)$ small linear programs. The optimization
\[
\min_{\lambda\in\triangle_{\mathcal X}}
\max_{(x,x')\in I}\|x-x'\|^2_{A(\lambda)^{-1}}
\]
is described as a modest-sized semidefinite program, or it can be approximated with off-the-shelf first-order methods in $O(d^3)$ per iteration. Rounding $\lambda^*$ into a fixed allocation of length $T$ uses a standard deterministic-rounding procedure and again requires $T\ge d^2$ [2603.10346].

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 $H_{\mathrm{Adjacent}}$ 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 [2603.10346].

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 $\mu^\top L\mu\le R$ [2005.09841]. 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 [1111.5972]. 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 $H_{\mathrm{Adjacent}}$, whereas in other domains the same shorthand can refer to distinct constructions.

Source: https://www.emergentmind.com/topics/adjacent-bai