Bandits with Single-Peaked Preferences and Limited Resources

## Abstract

Abstract We study an online stochastic matching problem in which an algorithm sequentially matches U U users to K K arms, aiming to maximize cumulative reward over T T rounds under budget constraints. Without structural assumptions, computing the optimal matching is NP-hard, making online learning computationally infeasible. To overcome this barrier, we focus on single-peaked preferences —a well-established structure in social choice theory, where users’ preferences are unimodal with respect to a common order over arms. We devise an efficient algorithm for the offline budgeted matching problem, and leverage it into an efficient online algorithm with a regret of O ~ ​ ( U ​ K ​ T 2 / 3 ) \tilde{O}(UKT^{2/3}) . Our approach relies on a novel PQ tree-based order approximation method. If the single-peaked structure is known, we develop an efficient UCB-like algorithm that achieves a regret bound of O ~ ​ ( U ​ T ​ K ) \tilde{O}(U\sqrt{TK}) .

## 1 Introduction

Modern recommendation systems often face the challenge of personalization at scale—learning individual user preferences while simultaneously satisfying global resource allocation constraints. To illustrate, consider a content platform that must decide which content creators to commission daily, where each creator has a different cost and produces ephemeral content on specific topics. Each user has preferences over all creators’ content styles and topics. After commissioning a subset of creators that fit the platform’s budget, it matches each user to content from one of these creators, where the same creator’s content can be recommended to multiple users. The challenge lies in learning individual user preferences for each creator’s content while selecting which creators to commission and how to assign their content to maximize user satisfaction.

This problem fits the combinatorial multi-armed bandit framework, where the decision-maker must choose structured action sets [8], such as assigning each user to an item. The goal is to maximize cumulative reward, or equivalently, minimize regret by balancing exploration and exploitation. Unfortunately, combinatorial problems like the one in the example above are NP-complete even for offline settings. Therefore, traditional approaches settle for weaker notions of $\alpha$-regret [8], competing against the best efficient computable solution rather than the optimal one. This compromise can be unsatisfying, especially when the solution space is highly structured or the stakes of poor decisions are high.

In this paper, we circumvent this computational barrier by focusing on *single-peaked* (hereinafter SP) preferences—a structure that has been extensively studied in social choice theory since the seminal work of Black [5]. Specifically, it implies that there is an order of the arms such that each user’s utility is unimodal. SP preferences appear naturally in numerous domains: voters’ preferences over political candidates along an ideological spectrum, consumers’ preferences for products varying in a single attribute, and users’ preferences for content recommendations based on genre similarity. The single-peaked property has proven helpful in circumventing impossibility results in voting theory [1] and transforming NP-hard problems into polynomial-time solvable ones [15, 12].

### 1.1 Our Contributions

We study a stochastic combinatorial bandit problem over $T$ rounds with $K$ arms and $U$ users. Each arm $k$ has a cost $c_{k}$, and the learner can select any subset of arms within a budget $B$, and then assign users to the selected arms. Each user derives a stochastic satisfaction from each arm, and the goal is to maximize total user satisfaction. As the general case is NP-complete, we focus on SP preferences (see LABEL:{def:single-peaked-def}), which enable efficient solutions. Our main contributions are as follows.

#### Efficient offline algorithm.

We develop SP-Matching, an efficient algorithm that optimally solves the budget-constrained matching problem in $O(K^{2}B+K^{2}U)$ time when preferences are SP.

#### Learning with known SP structure

We consider the case of known SP order and user peaks (but not the cardinal preferences). Such an assumption is justified if, for example, preferences rely on a known ideological spectrum where user peaks correspond to stated or inferred inclinations. We design an efficient UCB-based algorithm, termed MvM, that achieves $\tilde{O}(U\sqrt{TK})$ regret. The key property used in our approach is the existence of a *maximal preferences matrix* in the confidence set of all plausible SP matrices—that is, a matrix that element-wise dominates all others in the set. We use this maximal matrix as the basis for optimistic matching using our offline algorithm.

#### Learning with unknown SP structure

We also consider the more challenging case, where the SP structure is unknown. We introduce an explore-then-commit algorithm, named EMC, that devotes an initial exploration phase to collecting reward estimates. Then, a novel PQ tree-based procedure [6], which we refer to as Extract-Order, recovers an approximate SP order from these estimates. Given this order, the reward estimates are projected onto a nearby SP matrix, and the corresponding offline matching is solved using SP-Matching; the resulting matching is committed to for the remainder of the rounds. A careful approximation and concentration analysis yields a regret bound of $\tilde{O}(UKT^{2/3})$, while all computational steps run in $\mathrm{poly}(U,K,B)$ time.

#### Lower bounds

To demonstrate that SP preferences do not trivialize the learning problem, we establish several lower bounds on the worst-case regret. We show that even if the SP order is known, a regret of $\Omega(U\sqrt{TK})$ is unavoidable. Furthermore, even if both the SP order and user peaks are known, every algorithm incurs a regret of $\Omega(U\sqrt{T}+\sqrt{TK})$.

### 1.2 Related Work

Our research intersects the combinatorial and matching threads of the multi-armed bandits literature, as well as single-peaked preferences from social choice theory. We provide a review of each of these strands of literature below.

#### Multi-armed bandits.

The multi-armed bandit framework provides the foundation for sequential decision-making under uncertainty [7, 31, 23]. This setting has been extended to the combinatorial domain, where actions are solutions of a combinatorial optimization problem [9, 22]. While efficient learning algorithms exist for certain combinatorial structures [26], many practical instances involve computationally intractable optimization problems. To address this computational barrier, prior work introduced the notion of $\alpha$-regret [8], which compares the learner’s performance against the best efficiently computable solution rather than the true optimum. This approach has led to the development of general frameworks for achieving $\alpha$-regret guarantees in computationally hard settings [30, 28]. Notably, the single-peaked structure we impose enables us to circumvent computational intractability entirely, allowing us to achieve standard regret bounds rather than settle for $\alpha$-regret. Our setting further incorporates per-round budget constraints, which align with the cardinality and knapsack-type restrictions studied in prior works [27, 28]. A closely related line of work considers bandit learning for matching problems [10, 24, 20, 21]. Another perspective arises in content recommendation, which can be naturally viewed as a matching problem between users and creators. This has motivated a line of research on strategic content providers [25, 18, 19]. Particularly, Ben-Porat and Torkan [4] study a variant where arms become unavailable if they are not selected sufficiently often, and their proposed algorithm has runtime exponential in $K$. In contrast, in our setting, arms remain continuously available (subject to budget constraints), and we devise efficient algorithms.

#### Single-peaked preferences.

The study of single-peaked preferences dates back to seminal work in social choice theory [5, 1]. From a computational perspective, several problems have been investigated, including recognizing whether a given preferences profile admits a single-peaked order and constructing such an order when it exists [3, 13]. While recognition and construction of single-peaked orders can be solved efficiently, the problem becomes computationally intractable when preferences are nearly but not perfectly single-peaked. Specifically, finding an order that minimizes various distance measures from single-peakedness has been shown to be NP-hard [16, 11, 14]. In [Section 4](https://arxiv.org/html/2510.09425v1#S4), we contribute to this literature by developing a technique to extract single-peaked orders from approximately single-peaked cardinal preferences.

The single-peaked assumption has profound implications across different domains. In social choice theory, it circumvents classical impossibility results and enables positive outcomes that fail under general preferences. Most notably, Black’s median voter theorem [5] guarantees the existence of a Condorcet winner under single-peaked preferences, resolving a fundamental challenge in voting theory. From an algorithmic perspective, single-peakedness has been shown to simplify computational challenges: numerous voting problems that are NP-hard under general preferences become polynomial-time solvable when restricted to single-peaked structures [15, 12]. Our work reveals a similar phenomenon in an entirely distinct setting. We demonstrate that in our cardinal matching model with budget constraints, the single-peaked structure enables efficient optimization for problems that are computationally intractable under general preferences.

## 2 Problem Definition

In this section, we formally introduce our model and notation ([Section 2.1](https://arxiv.org/html/2510.09425v1#S2.SS1)), address the case of general preferences ([Section 2.2](https://arxiv.org/html/2510.09425v1#S2.SS2)), and focus on single-peaked preferences ([Section 2.3](https://arxiv.org/html/2510.09425v1#S2.SS3)).

### 2.1 Model

We formalize the Constrained Bandit Recommendation problem ($\mathsf{CBR}$ for brevity) as follows. An instance of $\mathsf{CBR}$ consists of the tuple $(T,U,K,(D_{u,k})_{u,k},(c_{k})_{k},B)$, whose components are described below.
The problem involves $U$ users indexed by $[U]=\{1,\ldots,U\}$ and $K$ items (arms) indexed by $[K]=\{1,\ldots,K\}$. The satisfaction of each user $u$ with an item $k$ is stochastic and stationary over time, drawn from a distribution $D_{u,k}$ supported on $[0,1]$ with expected value $\theta_{u,k}$. We denote the expected reward matrix $\Theta\in[0,1]^{U\times K}$ by setting its $(u,k)$-th entry to $\theta_{u,k}$. The distributions are assumed to be independent across users and arms.

The goal of the learner is to repeatedly recommend arms to users so as to maximize their overall satisfaction. Interaction proceeds over $T$ rounds. In each round $t\in[T]$, the learner chooses a *matching* $\pi_{t}:[U]\to[K]$, namely a function assigning each user a single item. The expected value of the matching is the total expected satisfaction of the users with their assigned arms,

$$ $V(\pi_{t};\Theta)=\sum_{u\in[U]}\theta_{u,\pi_{t}(u)}.$ $$

However, not all matchings are feasible. The algorithm has a budget constraint on the selected arms: each selected arm (i.e., there is at least one user $u$ such that $\pi_{t}(u)=k$) has a cost of $c_{k}$, and the total cost of the selected arms in every round should remain below a known budget $B\in\mathbb{N}$. Formally, for a matching $\pi$, let $\text{Im}(\pi)=\{k\in[K]:\exists u\in[U]\text{ s.t. }\pi(u)=k\}$. Hence, a matching is feasible only if $\sum_{k\in\text{Im}(\pi_{t})}c_{k}\leq B$. We denote the set of all feasible matchings as $\Pi$. Without loss of generality, we assume that $c_{k}\leq B$ for all $k\in[K]$, since otherwise such arms could never be selected. As in the motivating example of [Section 1](https://arxiv.org/html/2510.09425v1#S1), the platform pays the content creators (arms) per creation, so multiple users can be matched to the same arm without incurring additional cost.

At the beginning of each round $t$, the learner has access to the history of past actions and observed rewards, denoted by $H_{t}$. Upon selecting a matching $\pi_{t}$, a random reward $r^{t}_{u,\pi_{t}(u)}\sim D_{u,\pi_{t}(u)}$ is generated for each pair $(u,\pi_{t}(u))$. We assume semi-bandit feedback, meaning the learner observes $r^{t}_{u,\pi_{t}(u)}$ for every matched pair $(u,\pi_{t}(u))$. The performance of the learner $\mathcal{A}$ on a $\mathsf{CBR}$ instance $\mathcal{I}$ is measured using the standard notion of regret,

$$ $R_{T}(\mathcal{A},\mathcal{I})=T\cdot\max_{\pi\in\Pi}V(\pi;\Theta)-\mathbb{E}\left[\sum_{t=1}^{T}\sum_{u\in[U]}r^{t}_{u,\pi_{t}(u)}\right],$ $$

where the expectation is over the algorithm’s randomness and the realized rewards. As is standard in bandit settings, we assume $T\geq K,U$. As a useful intermediate step, we also consider the offline version of the problem, where the algorithm is given the expected reward matrix $\Theta$ and must compute the optimal feasible matching, i.e., $\operatorname*{arg\,max}_{\pi\in\Pi}V(\pi;\Theta)$.

### 2.2 Warm Up: General Preferences

Without structural assumptions on $\Theta$, achieving low regret with polynomial time algorithms is hopeless. In particular, computing $\operatorname*{arg\,max}_{\pi\in\Pi}V(\pi;\Theta)$ is computationally intractable. Furthermore, due to a reduction from the maximum coverage problem [17], it is NP-hard to approximate the offline problem within any factor better than $(1-1/e)$. Since an efficient online algorithm with sublinear regret would imply an efficient approximation algorithm to the offline problem, such an online algorithm cannot exist. Due to space limitations, we defer the full analysis to [Appendix E](https://arxiv.org/html/2510.09425v1#A5).

The literature typically addresses such cases, where the offline problem is NP-complete and thus cannot serve as a comparable benchmark, using the notion of $\alpha$-regret [8, 28]. Although computationally feasible, achieving sublinear $\alpha$-regret still implies linear regret.

In the online setting, the primary obstacle is computational rather than statistical. Indeed, optimal regret bounds can be achieved by applying the CUCB algorithm [8], which maintains element-wise UCB estimates for the reward matrix and repeatedly computes the optimal matching on these estimates. However, this optimization is NP-hard under general preferences, making CUCB computationally impractical. This motivates our assumption of structural properties for the expected reward matrix $\Theta$, which enables efficient optimization without sacrificing regret guarantees.

### 2.3 Single-Peaked Preferences

Single-peaked preferences arise naturally in numerous applications and, as we show, this structure enables polynomial-time algorithms to achieve sublinear regret. We define it formally as follows.

###### Definition 1 (PSP matrix) .

We say that a matrix $\Theta$ is *perfectly single-peaked* (PSP for brevity) if for every user $u\in[U]$, there exists an index $p(u;\Theta)\in[K]$ such that

$$ $\theta_{u,1}\leq\cdots\leq\theta_{u,p(u;\Theta)}\geq\cdots\geq\theta_{u,K}.$ (1) $$

In other words, the entries of every row are unimodal. The index $p(u;\Theta)$ is called the *peak* of user $u$, or simply $p(u)$ when $\Theta$ is clear from context; if multiple indices attain the maximum, we fix an arbitrary choice. More generally,

###### Definition 2 (SP matrix and SP order) .

We say that a matrix $\Theta$ is *single-peaked* (or SP) if there exists a (total) order $\prec$ on the arms such that
the matrix obtained by permuting the columns of $\Theta$ according to $\prec$ is PSP. In such a case, we say that $\prec$ is an *SP order* of $\Theta$.

Note that any PSP matrix is also SP, with the identity order serving as its SP order. In contexts where the relevant order matters, we shall write that $\Theta$ is SP w.r.t. $\prec$. Furthermore, we say that a $\mathsf{CBR}$ instance is (P)SP if its expected reward matrix $\Theta$ is (P)SP.

Figure: (a) An SP instance

To illustrate the PSP and SP properties, consider [Figure 1](https://arxiv.org/html/2510.09425v1#S2.F1). The instance comprises three users and five arms, and each subfigure illustrates the expected rewards for each user. The instance in [Figure 1(a)](https://arxiv.org/html/2510.09425v1#S2.F1.sf1) is not PSP, since there are “valleys” that contradict [(1)](https://arxiv.org/html/2510.09425v1#S2.Ex3). If we reorder the arms according to the order $\prec$ such that $1\prec 2\prec 5\prec 4\prec 3$, we obtain [Figure 1(b)](https://arxiv.org/html/2510.09425v1#S2.F1.sf2). Indeed, we observe a unimodal preference shape for each user, indicating that the matrix is PSP. Thus, the instance in [Figure 1(a)](https://arxiv.org/html/2510.09425v1#S2.F1.sf1) is SP and $\prec$ is its SP order.

We note that the SP order need not be unique; multiple SP orders may exist for the same matrix. For example, the reverse order of any SP order is also an SP order.

###### Definition 1 (PSP matrix) .

###### Definition 2 (SP matrix and SP order) .

## 3 Offline Algorithm for SP instances

In this section, we present an efficient and optimal algorithm for computing $\operatorname*{arg\,max}_{\pi\in\Pi}V(\pi;\Theta)$ for SP instances. W.l.o.g., we shall assume that the SP order is the identity order, i.e., the instance is PSP. Indeed, an SP order can be extracted using our Extract-Order algorithm (see [Algorithm 1](https://arxiv.org/html/2510.09425v1#alg1)) with $\varepsilon=0$ in $O(UK^{2})$ time, after which the columns can be reordered accordingly.

The main observation used by our algorithm relates to the optimal matching conditioned on a given subset of arms. Namely, we show that if the set of selected arms is $S$, the optimal matching assigns each user to the arm in $S$ that is closest to their peak. We formalize this in [Lemma 1](https://arxiv.org/html/2510.09425v1#Thmtheorem1).

###### Lemma 1 .

Fix any PSP matrix $\Theta$ with peaks $p(\cdot)$, and any arm subset $S=\{k_{1},\ldots,k_{m}\}\subseteq K$ with $k_{1}<\cdots<k_{m}$. Let $\pi^{\star}\in\operatorname*{arg\,max}_{\pi,\,\mathrm{Im}(\pi)\subseteq S}V(\pi;\Theta)$. For any user $u$, if $k_{j}\leq p(u)\leq k_{j+1}$ for some $j$ with $1\leq j<m$, then $\pi^{\star}(u)\in\{k_{j},k_{j+1}\}$; otherwise, $\pi^{\star}(u)\in\{k_{1},k_{m}\}$.

[Lemma 1](https://arxiv.org/html/2510.09425v1#Thmtheorem1) guides the design of a dynamic programming-based approach for finding the optimal matching for a given PSP matrix $P$, costs $c(\cdot)$, and budget $B$. We call this algorithm SP-Matching and explain its essence here, deferring full details to [Appendix A](https://arxiv.org/html/2510.09425v1#A1). The algorithm maintains a table $F(k,b)$ representing the maximum achievable reward when arm $k$ is the rightmost *selected* arm, the total budget is at most $b$, and we consider only users $u$ such that $p(u)\leq k$. We add an auxiliary degenerate arm 0 such that $c_{0}=0$ and $P_{u,0}=0$ for every user $u$, and initialize $F(0,b)=0$ for every $b\in\{0\}\cup[B]$.

The recurrence relation for $F(k,b)$ follows from [Lemma 1](https://arxiv.org/html/2510.09425v1#Thmtheorem1).
When $k$ is the arm with the highest index among the selected arms and $i$ is the second highest, users with peaks before $i$ are unaffected by the inclusion of arm $k$, while those with peaks between $i$ and $k$ are assigned to whichever of the two arms offers a higher reward. Consequently, $F(k,b)$ is given by

$$ $\max_{\begin{subarray}{c}i:0\leq i<k,\\ b\geq c_{i}+c_{k}\end{subarray}}\left[F(i,b-c_{k})+\sum_{u:i<p(u)\leq k}\max\{P_{u,i},P_{u,k}\}\right].$ $$

This maximum is always well-defined since arm 0 always satisfies its constraint. The optimal value is then

$$ $V^{\star}=\max_{k\in[K]}\left[F(k,B)+\sum_{u:p(u)>k}P_{u,k}\right],$ $$

where users with peaks beyond all selected arms are assigned to the last selected arm $k$. To enable efficient computation, we precompute for all pairs $(i,k)$ the sums $\sum_{u:i<p(u)\leq k}\max\{P_{u,i},P_{u,k}\}$ in $O(K^{2}U)$ time. This allows each entry in $F$ to be filled in $O(K)$ time, and since $F$ has $KB$ entries, the total runtime is $O(K^{2}U+K^{2}B)$.

###### Theorem 2 .

For any PSP matrix $P$, SP-Matching computes an optimal matching in time $O(K^{2}U+K^{2}B)$.

###### Lemma 1 .

###### Theorem 2 .

## 4 Online Algorithm for SP instances

In this section, we consider the more challenging online setup. Although the underlying expected reward matrix is SP, our reward estimates are generally not guaranteed to preserve this structure. For motivation, suppose the true reward matrix is $\Theta$, but we only have access to a noisy estimate $\bar{\Theta}$. If $\bar{\Theta}$ is itself SP, we could apply known methods to extract its SP order [3, 13] and use the SP-Matching algorithm to obtain an approximately optimal matching w.r.t. $\Theta$. However, since $\bar{\Theta}$ only approximates $\Theta$, $\bar{\Theta}$ may not be SP. Moreover, the SP order of $\Theta$, which is crucial for our offline algorithm, remains unknown and may not be recoverable directly from $\bar{\Theta}$. Therefore, leveraging the fact that the instance is SP is non-trivial.

To tackle this challenge, we take a three-step approach. First, we define the concept of approximate single-peaked matrices that relaxes the strict SP condition while remaining amenable to analysis. Second, we present an efficient procedure to extract a plausible SP order from empirical data. Finally, we combine these tools in an explore-then-commit algorithm that achieves sublinear regret.

### 4.1 Approximate Single-Peaked Matrices

We begin our analysis by defining the concept of approximately single-peaked (ASP) matrices.

###### Definition 3 .

We say a matrix $P$ is *$\delta$-approximately single-peaked* (or $\delta$-ASP) w.r.t. an order $\prec$ if for every $i,j,l\in[K]$ such that $i\prec j\prec l$ and user $u\in U$, it holds that

$$ $P_{u,j}\geq\min\{P_{u,i},P_{u,l}\}-\delta.$ (2) $$

If $P$ is $\delta$-ASP w.r.t. $\prec$, we say $\prec$ is its *$\delta$-ASP order*, analogously to SP order. The parameter $\delta$ quantifies the tolerable violation of the single-peaked property. When $\delta=0$, this condition is equivalent to the matrix being SP w.r.t. $\prec$, as we prove in the following proposition.

###### Proposition 3 .

A matrix is SP w.r.t. an order $\prec$ if and only if it is 0-ASP w.r.t. $\prec$.

The $\delta$-ASP condition proves useful beyond the strict $\delta=0$ case. In fact, it follows directly from [(2)](https://arxiv.org/html/2510.09425v1#S4.Ex6) that if $P$ is SP w.r.t. $\prec$ and $\tilde{P}$ satisfies $\lVert P-\tilde{P}\rVert_{\infty}\leq\delta$, then $\tilde{P}$ is $2\delta$-ASP w.r.t. $\prec$. This implies that a noisy estimate of an SP matrix remains ASP. Conversely, given any $\delta$-ASP matrix and its ASP order, we can construct a matrix that is (exact) SP with respect to the same order, with each entry differing from the original by at most $\delta$.

###### Lemma 4 .

Let $\tilde{P}$ be a $\delta$-ASP matrix w.r.t. $\prec$. There exists a matrix $P$ which is SP w.r.t. $\prec$, and satisfies $\lVert P-\tilde{P}\rVert_{\infty}\leq\delta$.

We prove [Lemma 4](https://arxiv.org/html/2510.09425v1#Thmtheorem4) by construction, which can be implemented in $O(UK)$ time. The detailed construction is given in [Appendix B](https://arxiv.org/html/2510.09425v1#A2). [Lemma 4](https://arxiv.org/html/2510.09425v1#Thmtheorem4) enables us to work with SP matrices even when the estimate matrix is not SP, while not sacrificing the approximation quality. However, [Lemma 4](https://arxiv.org/html/2510.09425v1#Thmtheorem4) requires the ASP order $\prec$, which we do not have.

###### Definition 3 .

###### Proposition 3 .

###### Lemma 4 .

### 4.2 Extract-Order Procedure

Figure: Algorithm 1 Extract-Order

Relating to the previous subsection, even if we know that the estimate matrix $\bar{\Theta}$ is ASP w.r.t. the SP order of $\Theta$, this order is still unknown. In what follows, we develop the Extract-Order algorithm, which extracts *an* ASP order from $\bar{\Theta}$ (not necessarily the SP order of $\Theta$).

We start with some intuition. Finding an order boils down to solving a set of contiguity constraints—subsets of arms that must be contiguous in any valid order. To illustrate, fix an SP matrix $P$ and suppose there exists a user $u$ and a partition of $[K]$ into two sets $A_{1}$ and $A_{2}$. Let $\beta=\min_{k\in A_{1}}P_{u,k}$ and $\gamma=\max_{k\in A_{2}}P_{u,k}$. If $\beta-\gamma>0$, then every arm in $A_{1}$ is preferred by user $u$ over every arm in $A_{2}$. In this case, any order $\prec$ that places an arm $j\in A_{2}$ between two arms $i,l\in A_{1}$ violates the 0-ASP condition ([(2)](https://arxiv.org/html/2510.09425v1#S4.Ex6)). Hence, according to [Proposition 3](https://arxiv.org/html/2510.09425v1#Thmtheorem3), $\prec$ cannot be an SP order of $P$. Thus, any SP order of $P$ places the arms in $A_{1}$ in a contiguous block. As a result, any order that satisfies all such contiguity constraints is an SP order of $P$.

Next, assume we have a matrix $\tilde{P}$ such that $\lVert P-\tilde{P}\rVert_{\infty}\leq\varepsilon$ for the SP matrix $P$. Since every entry in $\tilde{P}$ only approximates $P$, we cannot hope to discover all the contiguity constraints of $P$. For instance, if $\min_{k\in A_{1}}\tilde{P}_{u,k}=\beta-\varepsilon$ and $\max_{k\in A_{2}}\tilde{P}_{u,k}=\gamma+\varepsilon$ for the user $u$ and partition $A_{1},A_{2}$, then $\min_{k\in A_{1}}\tilde{P}_{u,k}-\max_{k\in A_{2}}\tilde{P}_{u,k}=\beta-\varepsilon-\gamma-\varepsilon$. In the case that this expression is negative, we would not discover the contiguity constraint of $A_{1}$. Similarly, we might enforce incorrect constraints. Thus, we should aim to discover as many contiguity constraints as possible from $\tilde{P}$, while remaining consistent with the contiguity constraints of $P$. To that end, if a triplet $u,A_{1}$ and $A_{2}$ satisfies

$$ $\min_{k\in A_{1}}\tilde{P}_{u,k}-\max_{k\in A_{2}}\tilde{P}_{u,k}>2\varepsilon,$ (3) $$

we can safely add a contiguity constraint on $A_{1}$ since the proximity between $\tilde{P}$ and $P$ guarantees $\min_{k\in A_{1}}P_{u,k}-\max_{k\in A_{2}}P_{u,k}>0$; hence, this constraint also applies to the matrix $P$.

The Extract-Order algorithm, described in [Algorithm 1](https://arxiv.org/html/2510.09425v1#alg1), formalizes this approach using a PQ tree [6], an efficient data structure that specializes in encoding and resolving contiguity constraints. PQ trees maintain sets of items that must appear consecutively in any valid order, and can be used both to determine whether a consistent order exists and to efficiently construct such an order. Extract-Order initializes a PQ tree (Line [3](https://arxiv.org/html/2510.09425v1#alg1.l3)), and for each user $u$, it sorts the arms by preference values (Lines [4](https://arxiv.org/html/2510.09425v1#alg1.l4)-[5](https://arxiv.org/html/2510.09425v1#alg1.l5)). Whenever there is a significant gap (exceeding $2\varepsilon$) between consecutive preference values, which exactly corresponds to the case described in [(3)](https://arxiv.org/html/2510.09425v1#S4.Ex7), it adds a contiguity constraint requiring all higher-valued arms to appear consecutively (Line [7](https://arxiv.org/html/2510.09425v1#alg1.l7)). Finally, the algorithm returns an order on the columns of $\tilde{P}$ that satisfies all the imposed contiguity constraints, if such an order exists (Line [9](https://arxiv.org/html/2510.09425v1#alg1.l9)). Otherwise, it fails and returns nothing (Line [11](https://arxiv.org/html/2510.09425v1#alg1.l11))

Notably, the constraints enforced by this procedure are not only consistent with the SP order of $P$, but also ensure bounded valley depth (in the sense of [(2)](https://arxiv.org/html/2510.09425v1#S4.Ex6)). We formalize this in the following lemma.

###### Lemma 5 .

Let $\tilde{P}$ be a matrix such that $\|\tilde{P}-P\|_{\infty}\leq\varepsilon$ for some SP matrix $P$. Extract-Order($\tilde{P},\varepsilon$) returns an order $\prec$ such that $\tilde{P}$ is $\left(2K\varepsilon\right)$-ASP w.r.t. $\prec$.

The runtime complexity of Extract-Order is dominated by sorting user preferences and PQ tree operations. Sorting preferences for all users requires $O(UK\log K)$ time. Subsequently, for each user, we may add up to $K-1$ contiguity constraints to the PQ tree, where each addition takes $O(K)$ time, yielding $O(UK^{2})$ time for all constraints. The final feasibility check and order extraction from the PQ tree requires $O(K)$ time. Thus, the overall runtime is $O(UK^{2})$.

###### Lemma 5 .

### 4.3 Explore-then-Commit Algorithm

Figure: Algorithm 2 Explore-then-Match-and-Commit (EMC)

We now present the EMC algorithm, which we implement in [Algorithm 2](https://arxiv.org/html/2510.09425v1#alg2), that combines the tools developed in the previous subsections to achieve sublinear regret. It operates in five phases. First, given an input parameter $N$, the algorithm explores by pulling each arm $k\in[K]$ for $N$ rounds and collects empirical means $\bar{\Theta}$. Second, it applies Extract-Order and extracts an order $\prec$ from $\bar{\Theta}$. The parameter $\varepsilon$ is chosen so that $\lVert\Theta-\bar{\Theta}\rVert_{\infty}\leq\varepsilon$ holds with high probability. In the third phase, it uses $\bar{\Theta}$ and the order $\prec$ to construct the matrix $\tilde{\Theta}$, which is SP w.r.t. $\prec$ and is element-wise close to $\bar{\Theta}$. Fourth, it executes SP-Matching on $\tilde{\Theta}$ to find an optimal matching $\tilde{\pi}$. And in the fifth and last phase, it exploits—it plays $\tilde{\pi}$ for the remaining $T-NK$ rounds. Importantly, the computational effort in Lines [3](https://arxiv.org/html/2510.09425v1#alg2.l3)–[5](https://arxiv.org/html/2510.09425v1#alg2.l5) is only $O(K^{2}U+K^{2}B)$, ensuring that the algorithm is computationally efficient. This is in sharp contrast to the exponential runtime we obtain for general instances in [Section 2.2](https://arxiv.org/html/2510.09425v1#S2.SS2).

The regret analysis for the commitment phase hinges on the approximation quality between the matrix $\tilde{\Theta}$ used for the matching and the true expected reward matrix $\Theta$. Since $\tilde{\Theta}$ is constructed to be close to the empirical estimates $\bar{\Theta}$, and $\bar{\Theta}$ concentrates around $\Theta$ with sufficient exploration, the value gap $V(\tilde{\pi};\tilde{\Theta})-V(\tilde{\pi};\Theta)$ is bounded by the estimation errors and the approximation errors from [Lemma 4](https://arxiv.org/html/2510.09425v1#Thmtheorem4) and Extract-Order. By optimizing the choice of exploration rounds $N$, we balance the exploration cost against the quality of the resulting approximation.

###### Theorem 6 .

EMC$\left(N=\left\lceil T^{2/3}(\ln T)^{1/3}\right\rceil\right)$ yields an expected cumulative regret of $O(UKT^{2/3}(\ln T)^{1/3})$.

Moreover, the computational phases in Lines [3](https://arxiv.org/html/2510.09425v1#alg2.l3)-[5](https://arxiv.org/html/2510.09425v1#alg2.l5) take $O(K^{2}U+K^{2}B)$, which makes it feasible in practical scenarios. This is in sharp contrast to the exponential runtime we obtain for general instances in [Section 2.2](https://arxiv.org/html/2510.09425v1#S2.SS2).

###### Theorem 6 .

## 5 Known SP Structure

In this section, we leverage additional structural information to design an efficient algorithm with improved regret guarantees. Specifically, we consider the setting in which both the SP order and user peaks are known. Although in standard stochastic bandit problems knowledge of each user’s peak (i.e., the optimal arm) is sufficient to eliminate regret, here, matching users to their peaks can violate budget constraints. Consequently, identifying the optimal matching requires exploration of the arms’ cardinal utilities.

These structural assumptions enable an optimistic, UCB-style algorithm that achieves a regret bound of $\tilde{O}(U\sqrt{TK})$—a rate that is comparable to our lower bound in [Section 6](https://arxiv.org/html/2510.09425v1#S6). Our approach involves the construction of optimistic reward estimates using *confidence sets* [31, Chapter 8.4], as detailed in [Section 5.1](https://arxiv.org/html/2510.09425v1#S5.SS1), while preserving the single-peaked structure. Then, we use those estimates to design the algorithm described in [Section 5.2](https://arxiv.org/html/2510.09425v1#S5.SS2).

### 5.1 Confidence Sets and Maximal Matrix

Given the known SP order $\prec$ and user peaks $p(\cdot)$, we construct confidence sets for the expected reward matrix that contain the true parameters with high probability. Intuitively, these sets capture all matrices that are statistically plausible given the data observed so far, and naturally shrink as more data is collected. We begin by establishing notation for standard confidence bounds. Given a history $H_{t}$ at round $t$, we denote by $\bar{\theta}_{u,k}(t)$ and $n_{u,k}(t)$ the empirical mean reward and the number of pulls of the pair $(u,k)$ observed up to round $t$, respectively. The upper confidence bound is defined as $UCB_{u,k}(t)=\bar{\theta}_{u,k}(t)+\sqrt{\frac{2\ln T}{n_{u,k}(t)}}$, with the lower confidence bound $LCB_{u,k}(t)$ defined analogously using subtraction. We formally define the confidence set as follows.

###### Definition 4 (Confidence set) .

The *confidence set* $\mathcal{C}^{t}=\mathcal{C}^{t}(\prec,p,H_{t})$
contains a matrix $P$ if and only if it is consistent with $\prec$ and $p(\cdot)$, and for any $u,k$ it holds that $LCB_{u,k}(t)\leq P_{u,k}\leq UCB_{u,k}(t)$.

The standard combinatorial UCB approach requires picking the optimistic matching in every time $t$, namely

$$ $\pi_{t}\in\operatorname*{arg\,max}_{\pi\in\Pi}\max_{P\in\mathcal{C}^{t}}V(\pi;P).$ (4) $$

This discrete optimization problem over a convex set may be tricky, as different matrices yield different optimal matchings. However, a structural property of $\mathcal{C}^{t}$ simplifies this optimization problem. As we demonstrate shortly in [Lemma 7](https://arxiv.org/html/2510.09425v1#Thmtheorem7), $\mathcal{C}^{t}$ includes a unique *maximal* matrix $P^{t}$ in an element-wise sense; thus, to find the optimistic matching $\pi_{t}$, we should only find the optimal matching w.r.t. $P^{t}$.

###### Lemma 7 .

For any confidence set $\mathcal{C}^{t}(\prec,p,H_{t})$, there exists a unique element-wise maximal matrix $P^{t}\in\mathcal{C}^{t}$ such that $P^{t}_{u,k}\geq P_{u,k}$ for all $P\in\mathcal{C}^{t}$ and all $u\in[U],k\in[K]$. Furthermore, this matrix is given by

$$ $P^{t}_{u,k}=\begin{cases}\min_{i:k\preceq i\preceq p(u)}UCB_{u,i}(t),&k\preceq p(u)\\ \min_{i:p(u)\preceq i\preceq k}UCB_{u,i}(t),&p(u)\preceq k\end{cases}.$ $$

We refer to the above matrix as the *maximal matrix* of $\mathcal{C}^{t}$. This maximal matrix could not be exploited in the previous section, as its characterization fundamentally relies on knowledge of both the order $\prec$ and the peaks $p(\cdot)$.

###### Definition 4 (Confidence set) .

###### Lemma 7 .

### 5.2 The Match-via-Maximal Algorithm

Figure: Algorithm 3 Match-via-Maximal (MvM)

Next, we present a combinatorial UCB-based algorithm, which we refer to as MvM ([Algorithm 3](https://arxiv.org/html/2510.09425v1#alg3)). MvM acts optimistically—in each round, it selects the optimal matching w.r.t. the corresponding maximal matrix. The regret analysis of MvM involves standard UCB-like arguments. By the construction of the confidence sets and applying Hoeffding’s inequality with a union bound over all $(u,k)$ pairs and time steps, the actual expected reward matrix $\Theta$ is in $\mathcal{C}^{t}$ for all $t\in[T]$ with probability at least $1-\frac{2KU}{T^{3}}$. Employing the clean event technique, we ignore the complementary (bad) event, which occurs with low probability. In every round $t$, we have

$$ $\displaystyle\max_{\pi\in\Pi}V(\pi;\Theta)-V(\pi_{t};\Theta)\leq V(\pi_{t};P^{t})-V(\pi_{t};\Theta)\leq\sum_{u\in[U]}2\sqrt{\frac{2\ln T}{n_{u,\pi_{t}(u)}}}.$ (5) $$

Summing Inequality ([5](https://arxiv.org/html/2510.09425v1#S5.E5)) over all rounds, we obtain the following theorem:

###### Theorem 8 .

For any SP instance with known SP order $\prec$ and peaks $p(\cdot)$, MvM achieves regret at most $O(U\sqrt{TK\ln T})$ with per-round runtime of $O(K^{2}U+K^{2}B)$.

###### Theorem 8 .

#### Unknown separated peaks.

The assumption of known peaks can be relaxed when each user’s peak is separated from their second-best option by a positive gap $\Delta_{\text{peak}}$. We stress that $\Delta_{\text{peak}}$ is unknown to the algorithm. Under this condition, we can utilize a mixed exploration-UCB approach to obtain a regret of
$O\left(\frac{UK\ln T}{\Delta_{\text{peak}}^{2}}+U\sqrt{TK\ln T}\right)$, where the first term represents the peak-identification cost and the second term corresponds to applying [Algorithm 3](https://arxiv.org/html/2510.09425v1#alg3) for the remaining rounds. We defer the full details to [Appendix C](https://arxiv.org/html/2510.09425v1#A3).

## 6 Lower Bounds for Known Order

In this section, we show that the class of SP instances still poses a significant challenge for the learning algorithm. We demonstrate this by establishing lower bounds on the regret for the most lenient information knowledge: when the SP order is known. W.l.o.g., we assume the identity SP order and formulate our results in terms of PSP instances.

We present our results in the following theorem, which provides a lower bound on the worst-case expected regret for both known and unknown peaks scenarios.

###### Theorem 9 .

For any algorithm, the worst-case regret over PSP instances is $\Omega(\max\{U\sqrt{T},\sqrt{TK}\})$ when user peaks are known, and $\Omega(U\sqrt{TK})$ when user peaks are unknown.

These lower bounds naturally apply to the unknown order case as well, since algorithms operate with strictly less information in that setting. Notably, the lower bound for unknown user peaks matches that of the general preferences case, as detailed in [Appendix E](https://arxiv.org/html/2510.09425v1#A5).

###### Theorem 9 .

## 7 Discussion

We studied a budgeted matching problem under single-peaked preferences. While the general offline problem is NP-hard to approximate within a constant factor, imposing a single-peaked structure enables polynomial-time optimization. When the SP structure is unknown, we introduce an explore-then-commit approach that first extracts an approximate order, then commits to a near-optimal policy, yielding $\tilde{O}(UKT^{2/3})$ regret. For the more lenient case of known SP order and user peaks, we developed an optimistic algorithm based on maximal matrices that achieves $\tilde{O}(U\sqrt{TK})$ regret. Furthermore, we developed lower bounds for the case of known and unknown user peaks. We also conducted an experimental validation, which is deferred to [Appendix F](https://arxiv.org/html/2510.09425v1#A6) due to space constraints.

We identify several interesting directions for future work. First, comparing our algorithmic results and the lower bounds, we observe several gaps. For the unknown SP order, EMC achieves a regret of $\tilde{O}(UKT^{2/3})$, while our lower bound is $\Omega(U\sqrt{TK})$. Future work could either propose an efficient algorithm with better regret or introduce a stronger lower bound under efficient implementation. For instance, solving the optimization problem in [Equation 4](https://arxiv.org/html/2510.09425v1#S5.E4) for confidence sets that do not require the SP order (and are thus non-convex) would lead to the same $\tilde{O}(U\sqrt{TK})$ regret. For known order and known peaks, our MvM algorithm achieves $\tilde{O}(U\sqrt{TK})$ regret (Theorem [8](https://arxiv.org/html/2510.09425v1#Thmtheorem8)), which exceeds the lower bound by a factor of $\sqrt{\ln T}\cdot\max\{\sqrt{K},U\}$. Future work could also resolve this gap.

Second, our results suggest that single-peaked preferences could potentially simplify other computationally challenging problems in online learning, suggesting an alternative route to a solution rather than $\alpha$-regret. Finally, previous work has explored generalizations of single-peaked preferences to more complex structures, including single-peaked preferences on trees [32] and cycles [29]. Extending our algorithmic framework and theoretical guarantees to these richer preference structures presents an intriguing challenge.

## References

- Arrow [1950]
Kenneth J Arrow.
A difficulty in the concept of social welfare.
*Journal of political economy*, 58(4):328–346, 1950.
- Auer et al. [2002]
Peter Auer, Nicolò Cesa-Bianchi, Yoav Freund, and Robert E. Schapire.
The nonstochastic multiarmed bandit problem.
*SIAM Journal on Computing*, 32(1):48–77, 2002.
doi: 10.1137/S0097539701398375.
URL [https://doi.org/10.1137/S0097539701398375](https://doi.org/10.1137/S0097539701398375).
- Bartholdi III and Trick [1986]
John Bartholdi III and Michael A Trick.
Stable matching with preferences derived from a psychological model.
*Operations Research Letters*, 5(4):165–169, 1986.
- Ben-Porat and Torkan [2023]
Omer Ben-Porat and Rotem Torkan.
Learning with exposure constraints in recommendation systems.
In *Proceedings of the ACM Web Conference 2023*, WWW ’23, page 3456–3466, New York, NY, USA, 2023. Association for Computing Machinery.
ISBN 9781450394161.
doi: 10.1145/3543507.3583320.
URL [https://doi.org/10.1145/3543507.3583320](https://doi.org/10.1145/3543507.3583320).
- Black [1948]
Duncan Black.
On the rationale of group decision-making.
*Journal of political economy*, 56(1):23–34, 1948.
- Booth and Lueker [1976]
Kellogg S. Booth and George S. Lueker.
Testing for the consecutive ones property, interval graphs, and graph planarity using pq-tree algorithms.
*Journal of Computer and System Sciences*, 13(3):335–379, 1976.
ISSN 0022-0000.
doi: https://doi.org/10.1016/S0022-0000(76)80045-1.
URL [https://www.sciencedirect.com/science/article/pii/S0022000076800451](https://www.sciencedirect.com/science/article/pii/S0022000076800451).
- Bubeck et al. [2012]
Sébastien Bubeck, Nicolo Cesa-Bianchi, et al.
Regret analysis of stochastic and nonstochastic multi-armed bandit problems.
*Foundations and Trends® in Machine Learning*, 5(1):1–122, 2012.
- Chen et al. [2013]
Wei Chen, Yajun Wang, and Yang Yuan.
Combinatorial multi-armed bandit: General framework and applications.
In Sanjoy Dasgupta and David McAllester, editors, *Proceedings of the 30th International Conference on Machine Learning*, volume 28 of *Proceedings of Machine Learning Research*, pages 151–159, Atlanta, Georgia, USA, 17–19 Jun 2013. PMLR.
URL [https://proceedings.mlr.press/v28/chen13a.html](https://proceedings.mlr.press/v28/chen13a.html).
- Combes et al. [2015]
Richard Combes, Mohammad Sadegh Talebi Mazraeh Shahi, Alexandre Proutiere, and marc lelarge.
Combinatorial bandits revisited.
In C. Cortes, N. Lawrence, D. Lee, M. Sugiyama, and R. Garnett, editors, *Advances in Neural Information Processing Systems*, volume 28. Curran Associates, Inc., 2015.
URL [https://proceedings.neurips.cc/paper_files/paper/2015/file/0ce2ffd21fc958d9ef0ee9ba5336e357-Paper.pdf](https://proceedings.neurips.cc/paper_files/paper/2015/file/0ce2ffd21fc958d9ef0ee9ba5336e357-Paper.pdf).
- Das and Kamenica [2005]
Sanmay Das and Emir Kamenica.
Two-sided bandits and the dating market.
In *IJCAI*, volume 5, page 19, 2005.
- Elkind and Lackner [2014]
Edith Elkind and Martin Lackner.
On detecting nearly structured preference profiles.
In *Proceedings of the AAAI Conference on Artificial Intelligence*, volume 28, 2014.
- Elkind and Lackner [2016]
Edith Elkind and Martin Lackner.
Structure in dichotomous preferences, 2016.
URL [https://arxiv.org/abs/1505.00341](https://arxiv.org/abs/1505.00341).
- Escoffier et al. [2008]
Bruno Escoffier, Jérôme Lang, and Meltem Öztürk.
Single-peaked consistency and its complexity.
In *ECAI*, volume 8, pages 366–370, 2008.
- Escoffier et al. [2021]
Bruno Escoffier, Olivier Spanjaard, and Magdaléna Tydrichová.
Measuring nearly single-peakedness of an electorate: Some new insights.
In *International Conference on Algorithmic Decision Theory*, pages 19–34. Springer, 2021.
- Faliszewski et al. [2009]
Piotr Faliszewski, Edith Hemaspaandra, Lane A Hemaspaandra, and Jörg Rothe.
The shield that never was: Societies with single-peaked preferences are more open to manipulation and control.
In *Proceedings of the 12th Conference on Theoretical Aspects of Rationality and Knowledge*, pages 118–127, 2009.
- Faliszewski et al. [2011]
Piotr Faliszewski, Edith Hemaspaandra, and Lane A Hemaspaandra.
The complexity of manipulative attacks in nearly single-peaked electorates.
In *Proceedings of the 13th conference on theoretical aspects of rationality and knowledge*, pages 228–237, 2011.
- Feige [1998]
Uriel Feige.
A threshold of ln n for approximating set cover.
*J. ACM*, 45(4):634–652, July 1998.
ISSN 0004-5411.
doi: 10.1145/285055.285059.
URL [https://doi.org/10.1145/285055.285059](https://doi.org/10.1145/285055.285059).
- Hron et al. [2023]
Jiri Hron, Karl Krauth, Michael Jordan, Niki Kilbertus, and Sarah Dean.
Modeling content creator incentives on algorithm-curated platforms.
In *The Eleventh International Conference on Learning Representations*, 2023.
- Immorlica et al. [2024]
Nicole Immorlica, Meena Jagadeesan, and Brendan Lucier.
Clickbait vs. quality: How engagement-based optimization shapes the content landscape in online platforms.
In *Proceedings of the ACM Web Conference 2024*, WWW ’24, page 36–45, New York, NY, USA, 2024. Association for Computing Machinery.
ISBN 9798400701719.
doi: 10.1145/3589334.3645353.
URL [https://doi.org/10.1145/3589334.3645353](https://doi.org/10.1145/3589334.3645353).
- Kong et al. [2022]
Fang Kong, Junming Yin, and Shuai Li.
Thompson sampling for bandit learning in matching markets, 2022.
URL [https://arxiv.org/abs/2204.12048](https://arxiv.org/abs/2204.12048).
- Kong et al. [2024]
Fang Kong, Zilong Wang, and Shuai Li.
Improved analysis for bandit learning in matching markets.
In *The Thirty-eighth Annual Conference on Neural Information Processing Systems*, 2024.
URL [https://openreview.net/forum?id=07N0qoaZ2L](https://openreview.net/forum?id=07N0qoaZ2L).
- Kveton et al. [2015]
Branislav Kveton, Zheng Wen, Azin Ashkan, and Csaba Szepesvari.
Tight Regret Bounds for Stochastic Combinatorial Semi-Bandits.
In Guy Lebanon and S. V. N. Vishwanathan, editors, *Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics*, volume 38 of *Proceedings of Machine Learning Research*, pages 535–543, San Diego, California, USA, 09–12 May 2015. PMLR.
URL [https://proceedings.mlr.press/v38/kveton15.html](https://proceedings.mlr.press/v38/kveton15.html).
- Lattimore and Szepesvári [2020]
Tor Lattimore and Csaba Szepesvári.
*Bandit algorithms*.
Cambridge University Press, 2020.
- Liu et al. [2020]
Lydia T Liu, Horia Mania, and Michael Jordan.
Competing bandits in matching markets.
In *International Conference on Artificial Intelligence and Statistics*, pages 1618–1628. PMLR, 2020.
- Mladenov et al. [2020]
Martin Mladenov, Elliot Creager, Omer Ben-Porat, Kevin Swersky, Richard Zemel, and Craig Boutilier.
Optimizing long-term social welfare in recommender systems: A constrained matching approach.
In Hal Daumé III and Aarti Singh, editors, *Proceedings of the 37th International Conference on Machine Learning*, volume 119 of *Proceedings of Machine Learning Research*, pages 6987–6998. PMLR, 13–18 Jul 2020.
URL [https://proceedings.mlr.press/v119/mladenov20a.html](https://proceedings.mlr.press/v119/mladenov20a.html).
- Neu and Bartók [2016]
Gergely Neu and Gábor Bartók.
Importance weighting without importance weights: An efficient algorithm for combinatorial semi-bandits.
*Journal of Machine Learning Research*, 17(154):1–21, 2016.
- Nie et al. [2022]
Guanyu Nie, Mridul Agarwal, Abhishek Kumar Umrawal, Vaneet Aggarwal, and Christopher John Quinn.
An explore-then-commit algorithm for submodular maximization under full-bandit feedback.
In *Uncertainty in Artificial Intelligence*, pages 1541–1551. PMLR, 2022.
- Nie et al. [2023]
Guanyu Nie, Yididiya Y Nadew, Yanhui Zhu, Vaneet Aggarwal, and Christopher John Quinn.
A framework for adapting offline algorithms to solve combinatorial multi-armed bandit problems with bandit feedback.
In *International Conference on Machine Learning*, pages 26166–26198. PMLR, 2023.
- Peters and Lackner [2020]
Dominik Peters and Martin Lackner.
Preferences single-peaked on a circle.
*Journal of Artificial Intelligence Research*, 68:463–502, 2020.
- Rizk et al. [2022]
Geovani Rizk, Igor Colin, Albert Thomas, Rida Laraki, and Yann Chevaleyre.
An $\alpha$-no-regret algorithm for graphical bilinear bandits.
In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho, editors, *Advances in Neural Information Processing Systems*, 2022.
URL [https://openreview.net/forum?id=2zQx2Pxbd7J](https://openreview.net/forum?id=2zQx2Pxbd7J).
- Slivkins [2019]
Aleksandrs Slivkins.
Introduction to multi-armed bandits.
*Foundations and Trends® in Machine Learning*, 12(1-2):1–286, 2019.
- Sliwinski and Elkind [2019]
Jakub Sliwinski and Edith Elkind.
Preferences single-peaked on a tree: Sampling and tree recognition.
In *Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence*. Association for the Advancement of Artificial Intelligence, 2019.
- The Sage Developers [2025]
The Sage Developers.
*SageMath, the Sage Mathematics Software System (Version 10.6.0)*, 2025.
https://www.sagemath.org.
- Yaroslavtsev et al. [2020]
Grigory Yaroslavtsev, Samson Zhou, and Dmitrii Avdiukhin.
”bring your own greedy”+ max: near-optimal 1/2-approximations for submodular knapsack.
In *International Conference on Artificial Intelligence and Statistics*, pages 3263–3274. PMLR, 2020.

## Appendix A Deferred Proofs for Section 3

See [1](https://arxiv.org/html/2510.09425v1#Thmtheorem1)

###### Proof of Lemma 1 .

Recall that under single-peaked preferences, the preferences of each user $u$ are unimodal with a peak at $p(u)$—the reward function $\theta_{u,k}$ is non-decreasing for $k\preceq p(u)$ and non-increasing for $p(u)\preceq k$.

For the first case, assume that $k_{j}\preceq p(u)\preceq k_{j+1}$ for some $1\leq j<m$. For any $\ell<j$, since $k_{\ell}\preceq k_{j}\preceq p(u)$, the non-decreasing property gives $\theta_{u,k_{\ell}}\leq\theta_{u,k_{j}}$. Similarly, for any $\ell>j+1$, since $p(u)\preceq k_{j+1}\preceq k_{\ell}$, the non-increasing property yields $\theta_{u,k_{j+1}}\geq\theta_{u,k_{\ell}}$. Therefore, any arm outside $\{k_{j},k_{j+1}\}$ is dominated by at least one arm in this pair, establishing that $\pi^{\star}(u)\in\{k_{j},k_{j+1}\}$.

For the second case, start with the case of $p(u)\preceq k_{1}$, which implies $p(u)\preceq k_{1}\preceq\cdots\preceq k_{m}$. By the non-increasing property after the peak, we have $\theta_{u,k_{1}}\geq\theta_{u,k_{2}}\geq\cdots\geq\theta_{u,k_{m}}$, which implies $\pi^{\star}(u)=k_{1}$. The case of $k_{m}\preceq p(u)$ is similar, and yields $\pi^{\star}(u)=k_{m}$.
∎

###### Remark 1 .

[Lemma 1](https://arxiv.org/html/2510.09425v1#Thmtheorem1) is robust to the arbitrary choice of the peak when a user has several maximizers.
Suppose $u$ has two peak indices $x<y$ with $\theta_{u,x}=\theta_{u,y}=\max_{k\in[K]}\theta_{u,k}$, and fix a selected set $S=\{k_{1}<k_{2}<\cdots<k_{m}\}$. Then the value of the best matching of $u$ against $S$ is independent of whether we instantiate $p(u)=x$ or $p(u)=y$. Specifically, exactly one of the following holds:

- •
There exists $j\in\{1,\ldots,m-1\}$ with $k_{j}\leq x\leq y\leq k_{j+1}$ and no $k_{\ell}\in(x,y)$. Then by unimodality, any arm outside $\{k_{j},k_{j+1}\}$ is dominated by one of them, so $\pi^{\star}(u)\in\{k_{j},k_{j+1}\}$, regardless of whether $p(u)=x$ or $p(u)=y$.
- •
$y\leq k_{1}$. Values are nonincreasing to the right of the peaks, hence $\pi^{\star}(u)=k_{1}$, regardless of the tie-break.
- •
$k_{m}\leq x$. Values are nondecreasing to the left of the peaks, hence $\pi^{\star}(u)=k_{m}$, regardless of the tie-break.
- •
There exist indices $i\leq j$ with $x\leq k_{i}\leq\cdots\leq k_{j}\leq y$. Since all $k\in[x,y]$ attain the peak value from the SP property, any $k\in\{k_{i},\ldots,k_{j}\}$ is optimal, regardless of the tie-break. The lemma asserts that $\pi^{\star}(u)\in\{k_{i-1},k_{i}\}$ if the peak is set to $x$, and $\pi^{\star}(u)\in\{k_{j},k_{j+1}\}$ if the peak is set to $y$. In either case, the attained value $\max_{k\in S}\theta_{u,k}$ remains unchanged.

In every case, $u$’s attained value is invariant to the choice of the peak.

Figure: Algorithm 4 SP-Matching

See [2](https://arxiv.org/html/2510.09425v1#Thmtheorem2)

###### Proof of Theorem 2 .

Define $\text{OPT}(k,b)$ as the maximum reward achievable when arm $k$ is the rightmost selected arm, the total budget is at most $b$, and we consider only users whose peaks lie in $\{0,1,\ldots,k\}$. Note that the fictive arm 0 contributes zero reward and zero cost. We prove by induction that $F(k,b)=\text{OPT}(k,b)$ for all $k\geq 0$ and $b\geq 0$.

Base case: For the fictive arm, $\text{OPT}(0,b)=0$ for all $b\leq B$ since no users have peaks at arm 0 and the arm contributes zero reward. The algorithm correctly assigns $F(0,b)=0$ in Line ([5](https://arxiv.org/html/2510.09425v1#alg4.l5)).

Inductive step: Assume $F(i,b^{\prime})=\text{OPT}(i,b^{\prime})$ for all $0\leq i<k$ and all budgets $c_{k}\leq b^{\prime}<b$. For any $k\geq 1$ and budget $b\geq c_{k}$, the optimal solution must select some arm $i$ with $0\leq i<k$ as the second-rightmost selected arm (where $i=0$ corresponds to selecting only arm $k$).

By [Lemma 1](https://arxiv.org/html/2510.09425v1#Thmtheorem1), users with peaks in the interval $(i,k]$ are optimally assigned to either arm $i$ or arm $k$, contributing exactly $G_{i,k}$ to the total reward. Users with peaks in $\{0,1,\ldots,i\}$ contribute optimally according to $\text{OPT}(i,b-c_{k})$ by definition. Since the fictive arm 0 has zero cost, the constraint $b\geq c_{i}+c_{k}$ is always satisfiable with $i=0$. Therefore,

$$ $\text{OPT}(k,b)=\max_{\begin{subarray}{c}i:0\leq i<k,\\ b\geq c_{i}+c_{k}\end{subarray}}[\text{OPT}(i,b-c_{k})+G_{i,k}].$ $$

The inductive hypothesis ensures that $\text{OPT}(i,b-c_{k})=F(i,b-c_{k})$, so:

$$ $\text{OPT}(k,b)=\max_{\begin{subarray}{c}i:0\leq i<k,\\ b\geq c_{i}+c_{k}\end{subarray}}[F(i,b-c_{k})+G_{i,k}]=F(k,b).$ $$

After computing $F(k,b)$ for all relevant values, Line ([9](https://arxiv.org/html/2510.09425v1#alg4.l9)) computes

$$ $\max_{k\in[K]}\{F(k,B)+\sum_{u:p(u)>k}P_{u,k}\},$ $$

which accounts for users whose peaks exceed the rightmost selected arm due to the second case in [Lemma 1](https://arxiv.org/html/2510.09425v1#Thmtheorem1). This ensures that the algorithm correctly computes the maximum reward for the given budget $B$. The backtracking reconstructs the optimal matching, from which we can exclude arm 0 if it was somehow included by picking any other arm for that user.

###### Proof of Lemma 1 .

###### Remark 1 .

###### Proof of Theorem 2 .

#### Runtime Analysis:

Computing the $G$ matrix requires $O(K^{2}U)$ time. The dynamic programming fills $O(KB)$ entries, with each entry requiring $O(K)$ operations, yielding $O(K^{2}B)$ time. The remaining steps take $O(KU)$ time. Therefore, the total complexity is $O(K^{2}U+K^{2}B)$.
∎

###### Remark 2 .

SP-Matching is invariant to the arbitrary choice of the peak when a user has several maximizers as well. More specifically, notice that every choice of the last arm $k$ in the subset (Line ([9](https://arxiv.org/html/2510.09425v1#alg4.l9))) corresponds to a subset of selected arms, and a matching that assigns each user based on [Lemma 1](https://arxiv.org/html/2510.09425v1#Thmtheorem1). Thus, according to [Remark 1](https://arxiv.org/html/2510.09425v1#Thmremark1), the attained value of each user in this subset is independent of the choice of the peak. The choice of the peak only affects *when* we account for this user’s contribution to the total value, not the contribution itself. Therefore, running the algorithm with different valid peak indices yields the same final value $V^{\star}$.

###### Remark 2 .

## Appendix B Deferred Proofs for Section 4

See [3](https://arxiv.org/html/2510.09425v1#Thmtheorem3)

###### Proof of Proposition 3 .

We prove each direction separately.

SP $\Rightarrow$ 0-ASP: Let $P$ be an SP matrix with respect to an order $\prec$, user $u\in[U]$, and a triplet of arms $i\prec j\prec\ell$. We divide the analysis into cases based on the relative position of $j$ w.r.t. the user’s peak $p(u)$, leveraging the unimodality of the preference vector $P_{u,(\cdot)}$. If $i\prec j\preceq p(u)$, then $P_{u,j}\geq P_{u,i}$; additionally, if $p(u)\preceq j\prec\ell$, then $P_{u,j}\geq P_{u,\ell}$. In either case, we have

$$ $P_{u,j}\geq\min\{P_{u,i},P_{u,\ell}\},$ $$

which implies that $P$ is 0-ASP with respect to $\prec$.

0-ASP $\Rightarrow$ SP: Let $P$ be a 0-ASP matrix with respect to an order $\prec$, and user $u\in[U]$. Denote $p(u)=\operatorname*{arg\,max}_{k\in[K]}P_{u,k}$. Then, for every arms $i,j$ such that $i\prec j\prec p(u)$, we have

$$ $P_{u,j}\geq\min\{P_{u,i},P_{u,p(u)}\}=P_{u,i},$ $$

Which implies that $P_{u,i}\leq P_{u,j}\leq P_{u,p(u)}$. Similarly, for every arms $i,j$ such that $p(u)\prec i\prec j$ it holds that $P_{u,p(u)}\geq P_{u,i}\geq P_{u,j}$. Thus, $P_{u,(\cdot)}$ is unimodal with peak at $p(u)$, and therefore $P$ is SP with respect to $\prec$.
∎

See [4](https://arxiv.org/html/2510.09425v1#Thmtheorem4)

###### Proof of Lemma 4 .

The proof is via construction. For every $u\in U$, set $p(u)=\operatorname*{arg\,max}_{k\in[K]}\tilde{P}_{u,k}$, and define $P$ as follows:

$$ $P_{u,k}=\begin{cases}\max_{i:i\preceq k}\tilde{P}_{u,i},&k\preceq p(u)\\ \max_{i:k\preceq i}\tilde{P}_{u,i},&p(u)\preceq k\end{cases}.$ $$

It is straightforward to verify that $P$ is SP w.r.t. $\prec$, as $P_{u,(\cdot)}$ is non-decreasing up to $p(u)$ and non-increasing afterward. Now, focus on an entry $(u,k)$ of $P$ such that $k\prec p(u)$. Then, by the construction of $P$, we have:

$$ $P_{u,k}=\max_{i\in[K]:i\preceq k}\tilde{P}_{u,i}\leq\max\{\tilde{P}_{u,k},\max_{i\in[K]:i\prec k}\tilde{P}_{u,k}+\delta\}=\tilde{P}_{u,k}+\delta,$ $$

where the inequality follows from employing Inequality ([(2)](https://arxiv.org/html/2510.09425v1#S4.Ex6)) using $i\prec k\prec p(u)$ for every $i\prec k$. A similar argument holds for entries $(u,k)$ such that $p(u)\prec k$. Therefore, for every pair $(u,k)$ it holds that $P_{u,k}-\tilde{P}_{u,k}\leq\delta$. Additionally, from construction, we always have $P_{u,p(u)}\geq\tilde{P}_{u,p(u)}$. Thus, we can conclude that $\lvert P_{u,k}-\tilde{P}_{u,k}\rvert\leq\delta$ for every $u\in U$ and $k\in[K]$.
∎

See [5](https://arxiv.org/html/2510.09425v1#Thmtheorem5)

###### Proof of Lemma 5 .

Let $\prec_{P}$ be an SP order of $P$. Recall that [Proposition 3](https://arxiv.org/html/2510.09425v1#Thmtheorem3) implies that $P$ is 0-ASP with respect to $\prec_{P}$. We will show that $\prec_{P}$ satisfies all the constraints that can be imposed in Line ([7](https://arxiv.org/html/2510.09425v1#alg1.l7)) of the Extract-Order algorithm. Fix a user $u$, and recall that we denote the sequence of $u$’s sorted preferences as $\tilde{P}_{u,k_{1}^{u}},\tilde{P}_{u,k_{2}^{u}},\ldots,\tilde{P}_{u,k_{K}^{u}}$. Let $m$ be an index such that $\tilde{P}_{u,k_{m}^{u}}>\tilde{P}_{u,k_{m+1}^{u}}+2\varepsilon$. Because $\lVert\tilde{P}-P\rVert_{\infty}\leq\varepsilon$, for all $i\in[m]$ and $j\in[m+1,K]$ it holds that

$$ $P_{u,k_{i}^{u}}-P_{u,k_{j}^{u}}\geq\tilde{P}_{u,k_{i}^{u}}-\tilde{P}_{u,k_{j}^{u}}-2\varepsilon>0.$ (6) $$

Since [Proposition 3](https://arxiv.org/html/2510.09425v1#Thmtheorem3) ensures that $P$ is 0-ASP with respect to $\prec_{P}$, if any arm $k_{j}^{u}$ with $j\in[m+1,K]$ were to appear between two arms $k_{i}^{u},k_{\ell}^{u}$ with $i,\ell\in[m]$ in the order $\prec_{P}$, then the 0-ASP condition would require

$$ $P_{u,k_{j}^{u}}\geq\min\{P_{u,k_{i}^{u}},P_{u,k_{\ell}^{u}}\}.$ $$

However, this violates [(6)](https://arxiv.org/html/2510.09425v1#A2.Ex16). Therefore, the sequence $k_{1}^{u},k_{2}^{u},\ldots,k_{m}^{u}$ must appear contiguously in $\prec_{P}$. Thus, every constraint that might be imposed in Line ([7](https://arxiv.org/html/2510.09425v1#alg1.l7)) is naturally satisfied by $\prec$.

After showing that a feasible order exists, we show that $\tilde{P}$ is $(2K\varepsilon)$-ASP w.r.t. any output order $\prec$. Fix any user $u$ and arms $i\prec j\prec\ell$. We need to show that

$$ $\tilde{P}_{u,j}\geq\min\{\tilde{P}_{u,i},\tilde{P}_{u,\ell}\}-2K\varepsilon.$ $$

Since $i\prec j\prec\ell$, the algorithm couldn’t have imposed a contiguity constraint that contains both $i$ and $\ell$ but excludes $j$. This means that in user $u$’s sorted preference sequence $\tilde{P}_{u,k_{1}^{u}}\geq\cdots\geq\tilde{P}_{u,k_{K}^{u}}$, there is no index $m$ such that $\{k_{1}^{u},\ldots,k_{m}^{u}\}$ contains $\{i,\ell\}\setminus\{j\}$ and $\tilde{P}_{u,k_{m}^{u}}-\tilde{P}_{u,k_{m+1}^{u}}>2\varepsilon$.

Without loss of generality, assume $\tilde{P}_{u,i}\leq\tilde{P}_{u,\ell}$. The difference $\tilde{P}_{u,j}-\tilde{P}_{u,i}$ is bounded by the sum of consecutive gaps in the sorted sequence between the positions of $j$ and $i$. Since no such gap exceeds $2\varepsilon$, and there are at most $K-1$ gaps, we have:

$$ $\tilde{P}_{u,i}-\tilde{P}_{u,j}\leq 2K\varepsilon,$ $$

which completes the proof.
∎

See [6](https://arxiv.org/html/2510.09425v1#Thmtheorem6)

###### Proof of Theorem 6 .

Using Hoeffding’s inequality and taking a union bound over all pairs $(u,k)$, we can ensure that for all $u\in[U],k\in[K]$,

$$ $\lvert\bar{\theta}_{u,k}-\theta_{u,k}\rvert\leq\sqrt{\frac{2\ln T}{N}}\quad$ (7) $$

holds with probability at least $1-\frac{2KU}{T^{4}}$. As the complementary event contributes at most $TU$ to the regret, its effect on the regret is negligible.

Let $\tilde{\Theta}$ be the SP matrix constructed in Line ([4](https://arxiv.org/html/2510.09425v1#alg2.l4)). [Lemma 4](https://arxiv.org/html/2510.09425v1#Thmtheorem4) and [Lemma 5](https://arxiv.org/html/2510.09425v1#Thmtheorem5) ensure that

$$ $\lVert\tilde{\Theta}-\bar{\Theta}\rVert_{\infty}\leq 2K\sqrt{\frac{2\ln T}{N}}.$ (8) $$

Combining [(7)](https://arxiv.org/html/2510.09425v1#A2.Ex20) and [(8)](https://arxiv.org/html/2510.09425v1#A2.Ex21), we obtain:

$$ $\lVert\tilde{\Theta}-\Theta\rVert_{\infty}\leq(2K+1)\sqrt{\frac{2\ln T}{N}}.$ (9) $$

Now, let $\tilde{\pi}$ be the matching computed in Line ([5](https://arxiv.org/html/2510.09425v1#alg2.l5)), and let $\pi^{\star}$ be the optimal matching w.r.t. $\Theta$. Using [(9)](https://arxiv.org/html/2510.09425v1#A2.Ex22), we can bound the regret of a single commitment round ($t>KN$) as follows:

$$ $V(\pi^{\star};\Theta)-V(\tilde{\pi};\Theta)\leq V(\pi^{\star};\tilde{\Theta})-V(\tilde{\pi};\tilde{\Theta})+2U(2K+1)\sqrt{\frac{2\ln T}{N}}\leq 2U(2K+1)\sqrt{\frac{2\ln T}{N}},$ $$

where the last inequality follows the optimality of $\tilde{\pi}$ with respect to $\tilde{\Theta}$. Thus, by dividing into exploration and commitment rounds, we can bound the total regret by

$$ $R_{T}\leq O\left(KNU+T\cdot 2U(2K+1)\sqrt{\frac{2\ln T}{N}}\right).$ $$

Substituting $N=\left\lceil T^{2/3}(\ln T)^{1/3}\right\rceil$ yields the stated regret bound.
∎

###### Proof of Proposition 3 .

###### Proof of Lemma 4 .

###### Proof of Lemma 5 .

###### Proof of Theorem 6 .

## Appendix C Deferred Proofs for Section 5

See [7](https://arxiv.org/html/2510.09425v1#Thmtheorem7)

###### Proof of Lemma 7 .

Assume towards contradiction that there exists a matrix $P\in\mathcal{C}$ and a pair $(u,k)$ such that $P_{u,k}>P^{t}_{u,k}$. By the suggested explicit form of $P^{t}$, there exists an index $k^{\prime}$ with either $k\preceq k^{\prime}\preceq p(u)$ or $p(u)\preceq k^{\prime}\preceq k$ such that

$$ $P^{t}_{u,k}=UCB_{u,k^{\prime}}(t).$ $$

From our assumption, we have $P_{u,k}>UCB_{u,k^{\prime}}$. Since $P\in\mathcal{C}$, [Definition 4](https://arxiv.org/html/2510.09425v1#Thmdefinition4) ensures that $P_{u,k^{\prime}}\leq UCB_{u,k^{\prime}}(t)$. Combining these inequalities yields

$$ $P_{u,k}>UCB_{u,k^{\prime}}(t)\geq P_{u,k^{\prime}}.$ $$

However, the single-peaked property of $P$ with peak at $p(u)$ implies $P_{u,k}\leq P_{u,k^{\prime}}$ (since both arms lie on the same side of the peak), contradicting $P_{u,k}>P_{u,k^{\prime}}$, and the proof is complete.
∎

See [8](https://arxiv.org/html/2510.09425v1#Thmtheorem8)

###### Proof of Theorem 8 .

We condition the analysis on the event that for all rounds $t\in[T]$ and pairs $(u,k)$ it holds that

$$ $LCB_{u,k}(t)\leq\theta_{u,k}\leq UCB_{u,k}(t).$ $$

Due to Hoeffding’s inequality and a union bound over all pairs $(u,k)$ and rounds $t$, this event holds with probability at least $1-\frac{2KU}{T^{3}}$. As the complementary event contributes at most $TU$ to the regret, its effect on the regret is negligible.

Conditioning on this event and summing Inequality ([5](https://arxiv.org/html/2510.09425v1#S5.E5)) over all rounds $t\in[T]$, we obtain

$$ $\displaystyle R_{T}$ $\displaystyle\leq O\left(\sum_{t=1}^{T}\sum_{u\in[U]}2\sqrt{\frac{2\ln T}{n_{u,\pi_{t}(u)}(t)}}\right)$ $\displaystyle=O\left(\sqrt{\ln T}\sum_{u\in[U]}\sum_{k\in[K]}\sum_{t=1}^{n_{u,k}(T)}\frac{1}{\sqrt{t}}\right)$ $\displaystyle=O\left(\sqrt{\ln T}\sum_{u\in[U]}\sum_{k\in[K]}\sqrt{n_{u,k}(T)}\right).$ $$

Using Jensen’s inequality, we get:

$$ $R_{T}\leq O\left(\sqrt{\ln T}\sum_{u\in[U]}\sqrt{K}\sqrt{\sum_{k\in[K]}n_{u,k}(T)}\right)=O\left(U\sqrt{TK\ln T}\right),$ $$

which completes the regret analysis.

For the runtime analysis, observe that each round of [Algorithm 3](https://arxiv.org/html/2510.09425v1#alg3) requires three main computational steps: constructing the maximal matrix $P^{t}$ according to [Lemma 7](https://arxiv.org/html/2510.09425v1#Thmtheorem7), computing the optimal matching via SP-Matching, and updating the empirical statistics. The maximal matrix construction requires $O(UK)$ time to compute all entries. By [Theorem 2](https://arxiv.org/html/2510.09425v1#Thmtheorem2), the SP-Matching algorithm runs in $O(K^{2}U+K^{2}B)$ time. Updating the empirical means and confidence bounds takes $O(U)$ time. Therefore, the per-round runtime is dominated by the SP-Matching call, yielding $O(K^{2}U+K^{2}B)$ time per round.
∎

###### Remark 3 .

Consider the case where some user $u$ has multiple peak indices, and this fact is known to the algorithm beforehand. In this case, the maximal matrix that is consistent with this new information should allocate the same value to all peak indices of user $u$. Additionally, this value is the minimum of the upper confidence bounds of all peak indices. More concretely, given that user $u$ has peak indices $p,p+1,\ldots,p+m$ for some $m\geq 0$, we should set

$$ $P^{t}_{u,k}=\begin{cases}\min_{i\in[0,m]}UCB_{u,p+i}(t),&k\in\{p,p+1,\ldots,p+m\}\\ \min_{i:k\preceq i\preceq p+m}UCB_{u,i}(t),&k\prec p\\ \min_{i:p\preceq i\preceq k}UCB_{u,i}(t),&p+m\prec k\end{cases}.$ $$

However, for every choice of peak index $p+i$ with $i\in[0,m]$, using the original definition of $P^{t}$ from [Lemma 7](https://arxiv.org/html/2510.09425v1#Thmtheorem7) would yield a matrix that upper bounds this modified maximal matrix. Thus, the regret analysis remains unchanged, even if we slightly overestimate the maximal matrix in this scenario and even cast it out of the confidence set. The important properties we used in the analysis–that $P^{t}$ upper bounds any matrix in the confidence set and that it is SP w.r.t. $\prec$–still hold.

###### Proof of Lemma 7 .

###### Proof of Theorem 8 .

###### Remark 3 .

### C.1 Generalization to Unknown Separated Peaks

The assumption of known peaks, which we relied upon in the construction of $P^{t}$, can be relaxed when each user’s peak is sufficiently separated from their second-best option. We formalize this condition through the peak separation gap:

$$ $\Delta_{\text{peak}}=\min_{u\in[U]}\left(\theta_{u,p(u)}-\max_{k:k\neq p(u)}\theta_{u,k}\right)>0.$ $$

Under this separation condition, we develop a two-phase algorithm. We first explore arms in round-robin fashion, maintaining confidence bounds for each user-arm pair and identifying arm $k$ as user $u$’s peak once $LCB_{u,k}(t)>\max_{j\neq k}UCB_{u,j}(t)$. Once all user peaks are identified, we switch to the maximal matrix approach from [Algorithm 3](https://arxiv.org/html/2510.09425v1#alg3) using the identified peaks. The correctness of peak identification follows from standard concentration arguments:

###### Lemma 10 .

The exploration phase correctly identifies all user peaks with probability at least $1-\frac{2KU}{T^{3}}$ after at most $O\left(\frac{K\ln T}{\Delta_{\text{peak}}^{2}}\right)$ exploration rounds.

###### Proof of Lemma 10 .

We condition the analysis on the clean event: for all rounds $t\in[T]$ and pairs $(u,k)$ it holds that

$$ $LCB_{u,k}(t)\leq\theta_{u,k}\leq UCB_{u,k}(t).$ (10) $$

Due to Hoeffding’s inequality and a union bound over all pairs $(u,k)$ and rounds $t$, this event holds with probability at least $1-\frac{2KU}{T^{3}}$.

Under the clean event, no incorrect peak can be identified because any claim that some index $k\neq p(u)$ is the peak would require $LCB_{u,k}(t)>UCB_{u,p(u)}(t)$. Since the clean event guarantees $\theta_{u,k}\leq UCB_{u,k}(t)$ and $\theta_{u,p(u)}\geq LCB_{u,p(u)}(t)$, the above inequality would imply $\theta_{u,k}>\theta_{u,p(u)}$, which contradicts the definition of $p(u)$.

Having established that under the clean event no incorrect peak will be identified, we move on to bound the time until identification of the correct peak. Let $n=\frac{128\ln T}{\Delta_{\text{peak}}^{2}}+1$. For every $u\in[U]$ and $k\neq p(u)$, we have:

$$ $\displaystyle LCB_{u,p(u)}(n)-UCB_{u,k}(n)$ $\displaystyle\geq\bar{\theta}_{u,p(u)}(n)-\bar{\theta}_{u,k}(n)-2\sqrt{\frac{2\ln T}{n-1}}$ $\displaystyle\geq\theta_{u,p(u)}-\theta_{u,k}-4\sqrt{\frac{2\ln T}{n-1}}$ $\displaystyle=\Delta_{\text{peak}}-5\Delta_{\text{peak}}>0,$ $$

where the second-to-last inequality follows from [(10)](https://arxiv.org/html/2510.09425v1#A3.Ex32) and the last inequality follows from the definition of $n$. Thus, with probability at least $1-\frac{2KU}{T^{3}}$ it holds that

$$ $LCB_{u,p(u)}(n)>UCB_{u,k}(n)\quad\forall k\neq p(u)$ $$

for all $u\in[U]$, and the proof is complete.
∎

Importantly, this procedure does not require prior knowledge of $\Delta_{\text{peak}}$—the algorithm naturally adapts to the separation structure. However, the gap value appears in the regret analysis, as smaller separations necessitate longer exploration to achieve reliable identification.

Combining both phases, the total regret becomes:

$$ $O\left(\frac{UK\ln T}{\Delta_{\text{peak}}^{2}}+U\sqrt{TK\ln T}\right),$ $$

where the first term captures the cost of peak identification and the second term corresponds to applying [Algorithm 3](https://arxiv.org/html/2510.09425v1#alg3) for the remaining rounds.

###### Lemma 10 .

###### Proof of Lemma 10 .

## Appendix D Deferred Proofs from Section 6

This section is dedicated to proving [Theorem 9](https://arxiv.org/html/2510.09425v1#Thmtheorem9). As each of the cases and bounds necessitates a different construction, we present and prove them separately in the following lemmas.

###### Lemma 11 .

Any algorithm for the online budgeted matching problem with single-peaked preferences, known SP order, and known peaks must have worst-case expected regret $R_{T}=\Omega(U\sqrt{T})$.

###### Proof of Lemma 11 .

We construct a family of hard instances parameterized by $\omega=(({\mu}_{i},{\nu}_{i}))_{i=1}^{U}\in[1/4,3/4]^{2U}$. For each $\omega$, define instance $I_{\omega}$ with $3U$ users, $4$ Bernoulli arms with order $1\prec 2\prec 3\prec 4$, costs $c_{1}=1$, $c_{2}=3$, $c_{3}=1$, $c_{4}=1$, and budget $B=3$. The expected reward matrix $\Theta^{\omega}$ is:

$$ $\theta^{\omega}_{u,k}=\begin{cases}\mu_{i}&\text{if }u\in\{1,\ldots,U\},k=1\\ 1&\text{if }u\in\{1,\ldots,U\},k=2\\ \nu_{i}&\text{if }u\in\{1,\ldots,U\},k=3\\ 0&\text{if }u\in\{1,\ldots,U\},k=4\\ 0&\text{if }u\in\{U+1,\ldots,3U\},k\neq 4\\ 1&\text{if }u\in\{U+1,\ldots,3U\},k=4\\ \end{cases}.$ $$

Notice that each such instance is PSP, with users $\{1,\ldots,U\}$ having their peaks at arm $2$, and users $\{U+1,\ldots,3U\}$ having their peaks at arm $4$.

The optimal matching in instance $I_{\omega}$ selects arms $1$, $3$, and $4$, whose total cost is exactly $B=3$. For each user $i\in\{1,\dots,U\}$ the assignment is $\pi(i)=\arg\max\{\mu_{i},\nu_{i}\}$, yielding expected reward $\sum_{i=1}^{U}\max\{\mu_{i},\nu_{i}\}$. For each user $u\in\{U+1,\dots,3U\}$ the assignment is $\pi(u)=4$, yielding total reward $2U$. Hence, the optimal expected total reward is

$$ $V^{\star}(\omega)=\sum_{i=1}^{U}\max\{\mu_{i},\nu_{i}\}+2U.$ (11) $$

Consider any algorithm $\mathcal{A}$ that selects matchings $\pi^{1},\ldots,\pi^{T}$ based on observed history. Let $T_{2}=\{t\in[T]:\text{Im}(\pi^{t})=\{2\}\}$, and $T_{1}=T\setminus T_{2}$. When $t\in T_{2}$, all users must be assigned to arm 2, yielding total reward $U$ (zero from users $U+1,\ldots,3U$, one each from users $1,\ldots,U$). Therefore, the algorithm’s expected cumulative reward under instance $\omega$ is:

$$ $\mathbb{E}_{\omega}^{\mathcal{A}}\left[\sum_{t=1}^{T}V(\pi^{t};\Theta^{\omega})\right]=\mathbb{E}_{\omega}^{\mathcal{A}}\left[U\lvert T_{2}\rvert+\sum_{t\in T_{1}}V(\pi^{t};\Theta^{\omega})\right].$ (12) $$

We can further decompose the second term by users and then upper bound the reward users $U+1,\ldots,3U$ receive. For user $i\in[U]$, define $N^{j}_{i}=|\{t:\pi^{t}(2U+i)=j\}|$ for $j\in\{1,3,4\}$. Then,

$$ $\displaystyle\mathbb{E}_{\omega}^{\mathcal{A}}\left[\sum_{t\in T_{1}}V(\pi^{t};\Theta^{\omega})\right]\leq\mathbb{E}_{\omega}^{\mathcal{A}}\left[2U\lvert T_{1}\rvert+\sum_{i=1}^{U}\left(\mu N_{1}^{i}+\nu N_{3}^{i}\right)\right].$ (13) $$

Using [Equation 11](https://arxiv.org/html/2510.09425v1#A4.E11), [Equation 12](https://arxiv.org/html/2510.09425v1#A4.E12) and [(13)](https://arxiv.org/html/2510.09425v1#A4.Ex37), we can lower bound the regret as follows:

$$ $\displaystyle R_{T}(\mathcal{A},\omega)=T\cdot V^{\star}(\omega)-\mathbb{E}_{\omega}^{\mathcal{A}}\left[\sum_{t=1}^{T}V(\pi^{t};\Theta^{\omega})\right]$ $\displaystyle\geq\mathbb{E}_{\omega}^{\mathcal{A}}\left[U\lvert T_{2}\rvert+\lvert T_{1}\rvert\sum_{i=1}^{U}\max\{{\mu}_{i},{\nu}_{i}\}-\sum_{i=1}^{U}\left(\mu N_{1}^{i}+\nu N_{3}^{i}\right)\right].$ (14) $$

Notice that the second and third expressions of Inequality ([14](https://arxiv.org/html/2510.09425v1#A4.E14)) correspond to the regret of $U$ 3-armed bandit problem with arms having means $({\mu}_{i},{\nu}_{i},0)$, played for $\mathbb{E}_{\omega}^{\mathcal{A}}[T_{1}]$ rounds. We will now divide the analysis into cases based on the magnitude of $\mathbb{E}_{\omega}^{\mathcal{A}}[T_{2}]$.

$\mathbb{E}_{\omega}^{\mathcal{A}}[T_{2}]\geq\sqrt{T}$. In this case, the first term in Inequality ([14](https://arxiv.org/html/2510.09425v1#A4.E14)) contributes at least $U\sqrt{T}$ to the regret, and the lower bound holds.

$\mathbb{E}_{\omega}^{\mathcal{A}}[T_{2}]\leq\sqrt{T}$. In this case, we have $\mathbb{E}_{\omega}^{\mathcal{A}}[T_{1}]=\Omega(T)$. By the minimax lower bound for 2-armed bandits [2], which holds here too since the algorithm still needs to differentiate two non-redundant arms for each user, and since the $U$ bandit problems are independent (different users have independent parameters $({\mu}_{i},{\nu}_{i})$), we obtain:

$$ $\displaystyle\inf_{\mathcal{A}}\sup_{\omega}R_{T}(\mathcal{A},\omega)$ $\displaystyle\geq\Omega(U\sqrt{T}).$ $$

This completes the proof of [Lemma 11](https://arxiv.org/html/2510.09425v1#Thmtheorem11).
∎

###### Lemma 12 .

For $K=o(T^{\nicefrac{{1}}{{3}}})$, any algorithm for the online budgeted matching problem with single-peaked preferences, known SP order, and known peaks must have worst-case expected regret $R_{T}=\Omega(\sqrt{TK})$.

###### Proof of Lemma 12 .

Suppose we have $2$ users and $K$ arms, each costing $1$, with a total budget of $B=1$. That is, we can play at most one arm each round and match both users to it. All the distributions in those instances are Bernoulli, and we specify their expectations in $\Theta$. We define the baseline preference matrix $\Theta^{0}$ as follows:

$$ $\theta^{0}_{1,k}=\frac{3}{4}-\frac{\eta(k-1)}{2(K-1)},\quad\theta^{0}_{2,k}=\frac{3}{4}-\frac{\eta(K-k)}{2(K-1)}.$ $$

Notice that $\Theta^{0}$ is PSP, with user 1’s peak being arm $1$ and user 2’s peak being arm $K$.

Now, think of an instance $\nu$ which almost have $\Theta^{0}$ as its preference matrix, except that it has $\theta_{2,1}=\theta^{0}_{2,1}+\Delta=\frac{1}{4}+\Delta$ for some $\Delta\leq\frac{1}{4(K-1)}$ that we will specify later. Notice that $\nu$ is still PSP and has the same peaks as $\Theta^{0}$.

Let $N_{k}$ denote the number of times arm $k$ is pulled over all $T$ rounds, and let $i=\operatorname*{arg\,min}_{k\in[K]}\mathbb{E}_{\nu}[N_{k}]$. Consequently, we must have $\mathbb{E}_{\nu}[N_{k}]\leq\frac{T}{K-1}$.

We will now define our second instance $\nu’$ to have the same preference matrix as $\nu$, except $\theta’_{1,i}=\theta^{0}_{1,i}+2\Delta$. Notice that $\nu’$ is still single-peaked with the same order and peaks as $\nu$.

It is easy to see that the sum of each column of $\Theta^{0}$ is identical; thus, all of the arms yield the same expected reward. Consequently, the optimal arm in $\nu$ (recall that the budget allows only for one arm) is arm $1$, and every other choice yields a regret of $\Delta$. Similarly, the optimal arm in $\nu’$ is arm $i$; choosing arm $1$ yields a regret of $\Delta$, and choosing any other arm yields a regret of $2\Delta$.

The rest of the analysis relies on a change-of-measure argument via Bretagnolle–Huber. Since it is almost identical to the proof in Lattimore and Szepesvári [23, Chapter 15], we omit the details here. It it important to note that in the proof, $\Delta$ is set to be $\Theta\left(\sqrt{\frac{K}{T}}\right)$. Thus, because we assume that $K=o(T^{\nicefrac{{1}}{{3}}})$, the condition $\Delta\leq\frac{1}{4(K-1)}$ we needed for the SP property to hold is satisfied.
∎

###### Lemma 13 .

Any algorithm for $\mathsf{CBR}$ with SP instance and known SP order and unknown peaks must have worst-case expected regret $R_{T}=\Omega(U\sqrt{TK})$.

###### Proof of Lemma 13 .

We reduce from the classical multi-armed bandit lower bound. Consider instances where each user $u$ has a unique peak arm $p(u)\in[K]$ with expected reward $\theta_{u,p(u)}=\frac{1}{2}+\Delta$, while all other arms have identical rewards $\theta_{u,k}=\frac{1}{2}$ for $k\neq p(u)$. These preference profiles are trivially single-peaked w.r.t. any order, as they are constant except at the peak.

Since users’ reward distributions are independent, learning one user’s peak provides no information about others. This reduces to $U$ independent $K$-armed bandit instances. By the standard minimax lower bound, whose construction uses exactly those hard instances (e.g., Slivkins [31, Chapter 2]), each user generates an expected regret of $\Omega(\sqrt{TK})$ over $T$ rounds.

Therefore, the total expected regret is at least

$$ $R_{T}\geq U\cdot\Omega(\sqrt{TK})=\Omega(U\sqrt{TK}).$ $$

∎

###### Lemma 11 .

###### Proof of Lemma 11 .

###### Lemma 12 .

###### Proof of Lemma 12 .

###### Lemma 13 .

###### Proof of Lemma 13 .

## Appendix E General Preferences Analysis

In this section, we analyze the computational and statistical complexity of the problem under general preferences that do not satisfy the single-peaked assumption. Our analysis reveals a sharp computational-statistical tradeoff: while optimal regret bounds are achievable via inefficient algorithms, any computationally efficient algorithm faces inherent limitations due to the underlying combinatorial complexity.

### E.1 Computational Hardness Results

We begin by establishing the computational intractability of the offline optimization problem, which forms the foundation for understanding the online learning challenges.

###### Theorem 14 .

Under general preferences, the offline problem is NP-hard to approximate within a constant factor better than $\left(1-\frac{1}{e}\right)$.

This hardness result has immediate implications for online learning, as any efficient online algorithm with sublinear regret could be leveraged to solve the offline problem beyond known computational limits.

###### Corollary 15 .

For any $U,K,B$ and $T=\text{poly}(U,K,B)$, any online algorithm achieves a regret of $\Omega(T)$, unless $\mathrm{P}=\mathrm{NP}$.

###### Theorem 14 .

###### Corollary 15 .

### E.2 Achieving Optimal Regret

Despite the computational barriers, we show that optimal regret is achievable when computational efficiency is disregarded. This establishes that the challenge lies in computation rather than statistical estimation.

Figure: Algorithm 5 UCB-Based Algorithm

In each round $t$, $\pi_{t}$ is chosen optimistically by selecting the arms and matching that maximize the sum of the upper confidence bounds of the expected rewards. Notice that calculating $\pi_{t}$ requires us to solve the offline problem in each round, which we demonstrated is NP-hard in [Theorem 14](https://arxiv.org/html/2510.09425v1#Thmtheorem14). Besides that, the analysis of the regret of [Algorithm 5](https://arxiv.org/html/2510.09425v1#alg5) is almost the same as the analysis of MvM. The only minor difference is that we need to obtain an equivalent single-regret bound to [Equation 5](https://arxiv.org/html/2510.09425v1#S5.E5); simply using $P$ instead of $P^{t}$ resolves this issue, as it is an element-wise maximal matrix as well. We summarize the regret of [Algorithm 5](https://arxiv.org/html/2510.09425v1#alg5) in the following corollary.

###### Corollary 16 .

There exists an algorithm (Algorithm [5](https://arxiv.org/html/2510.09425v1#alg5)) that achieves regret $O(U\sqrt{KT\log{T}})$, which is optimal up to logarithmic factors.

The optimality claim is supported by a matching lower bound, which stems directly from [Lemma 13](https://arxiv.org/html/2510.09425v1#Thmtheorem13).

###### Corollary 17 .

The regret of any algorithm for the online problem is $\Omega(U\sqrt{KT})$.

###### Corollary 16 .

###### Corollary 17 .

### E.3 Efficient Approximation via Alpha-Regret

To bridge the gap between computational efficiency and performance guarantees, we consider the alpha-regret framework, which compares against the best efficiently computable solution rather than the true optimum.

###### Definition 5 (Alpha Regret) .

For an approximation factor $\alpha\in(0,1]$, the expected cumulative $\alpha$-regret over $T$ rounds is defined as:

$$ $R_{\alpha}(T)=\alpha\cdot T\cdot\max_{\pi\in\Pi}V(\pi;\Theta)-\mathbb{E}\left[\sum_{t=1}^{T}\sum_{u\in[U]}r^{t}_{u,\pi^{t}(u)}\right].$ $$

By reformulating our problem as submodular maximization subject to knapsack constraints, we can leverage existing approximation algorithms to achieve efficient learning with meaningful guarantees.

###### Lemma 18 .

The reformulated reward function $f(S)=\sum_{u\in[U]}\max_{k\in S}\Theta_{u,k}$ is submodular and monotone.

###### Proposition 19 .

There exists an algorithm running in $O(K^{3}U)$ time per-round achieving expected cumulative $\frac{1}{2}$-regret of:

$$ $\mathbb{E}[R_{1/2}(T)]\leq\tilde{O}\left(U(K+\beta)^{4/3}T^{2/3}\right),$ $$

where $\beta=\nicefrac{{B}}{{\min_{k}c_{k}}}$ is the budget-to-minimum-cost ratio.

###### Definition 5 (Alpha Regret) .

###### Lemma 18 .

###### Proposition 19 .

### E.4 Proofs

We now provide detailed proofs for all results stated in the previous subsection.

###### Proof of Theorem 14 .

We prove the result via a reduction from the well-known *Max $\ell$-Cover* problem, which is NP-hard to approximate within a factor better than $(1-\tfrac{1}{e}+\varepsilon)$ for any $\varepsilon>0$ [17].

The Max $\ell$-Cover problem is defined as follows: given a universe $\mathcal{U}$ of $n$ elements and a collection of subsets $\mathcal{S}=\{S_{1},\dots,S_{m}\}$ with each $S_{i}\subseteq\mathcal{U}$, and an integer $\ell$, the goal is to select $\ell$ subsets whose union covers the maximum number of elements in $\mathcal{U}$.

Given an instance of Max $\ell$-Cover, we can construct an instance $\mathcal{I}=\left(T,K,U,\left(D_{u,k}\right),\left(c_{k}\right),B\right)$ of $\mathsf{CBR}$ as follows: set the number of users to be $U=n$, the number of arms to be $K=m$, and define the expected reward matrix $\Theta$ such that $\Theta_{u,k}=1$ if user $u$ is in subset $S_{k}$, and 0 otherwise. The cost for each item is set to $c_{k}=1$, and the budget is set to $B=\ell$.

Notice that the set of all feasible matching $\Pi$ in $\mathcal{I}$ becomes $\Pi=\{\pi:[U]\to[K]\mid|\text{Im}(\pi)|\leq\ell\}$. Additionally, for any feasible matching $\pi$, user $u$ receives reward 1 if and only if $\pi(u)=k$ for some $k$ such that $u\in S_{k}$. The optimal value is therefore:

$$ $\max_{\pi\in\Pi}V(\pi;\Theta)=\max_{\begin{subarray}{c}S\subseteq[K]\\ |S|\leq\ell\end{subarray}}\left|\bigcup_{k\in S}S_{k}\right|,$ $$

which exactly corresponds to the objective of Max $\ell$-Cover: selecting at most $\ell$ subsets to maximize the number of covered elements.

Now, for the hardness argument, suppose there exists a polynomial-time algorithm for the offline problem that achieves an approximation ratio better than $1-\tfrac{1}{e}+\varepsilon$ for some $\varepsilon>0$. Then, this algorithm can be used to approximate Max $\ell$-Cover beyond the known hardness threshold using the above reduction, contradicting the inapproximability result of Feige [17] unless $\mathrm{P}=\mathrm{NP}$. Therefore, the offline problem in our model is NP-hard to approximate within a factor better than $\left(1-\tfrac{1}{e}\right)$.
∎

###### Proof of Corollary 15 .

Assume towards contradiction that there exists an algorithm $\mathcal{A}$ running in time $\text{poly}(U,K,B)$ per round that achieves regret $R_{T}\leq CT^{\rho}$ for some constants $\rho\in[0,1)$ and $C>0$, for horizons $T$ that are polynomial in $U,K,B$. For ease of presentation, we assume $T=U^{\alpha}K^{\beta}B^{\gamma}$ for some $\alpha,\beta,\gamma\geq 1$.

###### Proof of Theorem 14 .

###### Proof of Corollary 15 .

#### Restriction to hard instances.

By [Theorem 14](https://arxiv.org/html/2510.09425v1#Thmtheorem14), the inapproximability result holds already for instances produced by the reduction from Max-$\ell$-Cover [17]. These instances have $\Theta\in\{0,1\}^{U\times K}$ with unit costs, budget $B=\ell$, and at least one entry equal to $1$. Consequently, for such instances we always have $\max_{\pi\in\Pi(\Theta)}V(\pi;\Theta)\geq 1$. Since the hardness persists on this restricted subclass, it suffices to prove the corollary under the additional assumption $\max_{u,k}\theta_{u,k}=1$.

#### Construction of the online instance.

Given such an offline instance with expected reward matrix $\Theta$, we construct a deterministic online instance where the realized reward of each $(u,k)$ pair is fixed to $\theta_{u,k}$. Additionally, we pick an arbitrary arm and duplicate it so that there are

$$ $D=\max\Bigg\{1,\left\lceil(10C)^{\tfrac{1}{\beta(1-\rho)}}U^{\tfrac{\alpha(\rho-1)}{\beta(1-\rho)}}B^{\tfrac{\gamma(\rho-1)}{\beta(1-\rho)}}\right\rceil\Bigg\}$ $$

copies of it. Let $\Theta^{\prime}$ denote the resulting matrix. Duplicating an arm cannot increase the optimal offline value, since multiple users can already be assigned to the same arm at no additional per-round cost. Therefore,

$$ $\max_{\pi\in\Pi(\Theta^{\prime})}V(\pi;\Theta^{\prime})=\max_{\pi\in\Pi(\Theta)}V(\pi;\Theta)=\mathrm{OPT}\geq 1.$ $$

#### Simulating 𝒜 \mathcal{A} .

Run $\mathcal{A}$ for $T^{\prime}=U^{\alpha}D^{\beta}B^{\gamma}$ rounds on the expanded instance, obtaining matchings $\pi^{1},\ldots,\pi^{T^{\prime}}$, and return $\pi^{\text{best}}=\arg\max_{t\in[T^{\prime}]}V(\pi^{t};\Theta^{\prime})$. The regret guarantee yields

$$ $T^{\prime}\cdot\mathrm{OPT}-\sum_{t=1}^{T^{\prime}}V(\pi^{t};\Theta^{\prime})\leq CT^{\prime\rho}.$ $$

Since $V(\pi^{\text{best}};\Theta^{\prime})\geq\tfrac{1}{T^{\prime}}\sum_{t=1}^{T^{\prime}}V(\pi^{t};\Theta^{\prime})$, it follows that

$$ $\mathrm{OPT}-V(\pi^{\text{best}};\Theta^{\prime})\leq CT^{\prime\rho-1}.$ $$

#### Approximation guarantee.

By construction of $D$, we ensured that $CT^{\prime\rho-1}\leq 0.1$. Hence

$$ $\mathrm{OPT}-V(\pi^{\text{best}};\Theta^{\prime})\leq 0.1\leq 0.1\mathrm{OPT}.$ $$

Since any matching on the duplicated instance is equivalent to a matching on the non-duplicated one, without loss of generality, we assume that $\pi^{\text{best}}\in\Pi(\Theta)$. Thus:

$$ $V(\pi^{\text{best}};\Theta)\geq 0.9\,V(\pi^{\star};\Theta).$ $$

#### Contradiction.

The above algorithm runs in $\text{poly}(U,K,B)$ time and yields a $0.9$-approximation to the offline optimum, contradicting the $(1-1/e)$-hardness result from [Theorem 14](https://arxiv.org/html/2510.09425v1#Thmtheorem14) unless $\mathrm{P}=\mathrm{NP}$.
∎

###### Proof of Lemma 18 .

We prove both properties separately.

Submodularity. For $M\subseteq N\subseteq[K]$ and $k\in[K]\setminus N$, we need

$$ $f(M\cup\{k\})-f(M)\geq f(N\cup\{k\})-f(N).$ $$

Let $\theta^{M}_{u}=\max_{j\in M}\Theta_{u,j}$ and $\theta^{N}_{u}=\max_{j\in N}\Theta_{u,j}$. Since $M\subseteq N$, we have $\theta^{M}_{u}\leq\theta^{N}_{u}$. The marginal gains are:

$$ $f(M\cup\{k\})-f(M)=\sum_{u\in[U]}\max\{0,\Theta_{u,k}-\theta^{M}_{u}\},$ $$

$$ $f(N\cup\{k\})-f(N)=\sum_{u\in[U]}\max\{0,\Theta_{u,k}-\theta^{N}_{u}\}.$ $$

For each user $u$, since $\theta^{M}_{u}\leq\theta^{N}_{u}$:

$$ $\max\{0,\Theta_{u,k}-\theta^{M}_{u}\}\geq\max\{0,\Theta_{u,k}-\theta^{N}_{u}\}.$ $$

Summing over all users gives the desired inequality.

Monotonicity. For $M\subseteq N$, since $\max_{k\in M}\Theta_{u,k}\leq\max_{k\in N}\Theta_{u,k}$ for each user $u$, we have:

$$ $f(M)=\sum_{u\in[U]}\max_{k\in M}\Theta_{u,k}\leq\sum_{u\in[U]}\max_{k\in N}\Theta_{u,k}=f(N).$ $$

∎

###### Proof of Proposition 19 .

We establish both the regret guarantee and the runtime complexity of the algorithm.

Regret analysis. We reformulate our problem as submodular maximization over subsets, where we select $S\subseteq[K]$ subject to the budget constraint $\sum_{k\in S}c_{k}\leq B$ and each user receives their most preferred item from the selected subset. The reformulated reward function $f(S)=\sum_{u\in[U]}\max_{k\in S}\Theta_{u,k}$ is submodular and monotone by Lemma [18](https://arxiv.org/html/2510.09425v1#Thmtheorem18). Under knapsack constraints, the Greedy+Max algorithm of Yaroslavtsev et al. [34] achieves a $\frac{1}{2}$-approximation and is $\left(\frac{1}{2},\frac{1}{2}+\tilde{K}+2\beta\right)$-robust in the sense of Nie et al. [28], where $\tilde{K}=\min\{K,B/c_{\min}\}$ and $\beta=B/c_{\min}$. Applying the C-ETC framework of Nie et al. [28] with normalized rewards $\tilde{f}(S)=\frac{1}{U}f(S)$ yields the bound $O(U\cdot(\frac{1}{2}+\tilde{K}+2\beta)^{2/3}(\tilde{K}K)^{1/3}T^{2/3}\log(T)^{1/3})$. To obtain our clean bound, we use $(\frac{1}{2}+\tilde{K}+2\beta)^{2/3}\leq(K+\beta)^{2/3}$ and $(\tilde{K}K)^{1/3}\leq K^{2/3}$, then apply $(K+\beta)^{2/3}\cdot K^{2/3}\leq(K+\beta)^{4/3}$ to get the stated bound.

The runtime analysis stems directly from the algorithm of Nie et al. [28].
∎

###### Proof of Lemma 18 .

###### Proof of Proposition 19 .

## Appendix F Experimental Validation

In this section, we present computational experiments to demonstrate the practicality of our algorithms and provide empirical validation of the theoretical regret bounds. (^1^11All the code needed for reproducing the results presented in this paper is available at [https://github.com/GurKeinan/code-for-Bandits-with-Single-Peaked-Preferences-and-Limited-Resources-paper](https://github.com/GurKeinan/code-for-Bandits-with-Single-Peaked-Preferences-and-Limited-Resources-paper))

### F.1 Simulation Details

We now describe the synthetic setup used to evaluate our algorithms, including instance generation, implementation details, and experimental protocol.

#### Instance generation.

We constructed single-peaked expected reward matrices $\Theta$ using the following procedure: for each user $u$, we first sampled $K$ preference values independently from a uniform distribution over $[0.2,0.9]$. We then selected a random peak location $p(u)\in[K]$ and arranged the values to create a unimodal preference profile—The largest sampled value was assigned to the peak, while the remaining values were sorted and distributed to create an increasing sequence up to the peak and a decreasing sequence afterward.

The actual rewards observed during algorithm execution were drawn from Bernoulli distributions, where $r_{u,k}\sim\text{Ber}(\theta_{u,k})$ for each user-arm pair. We set unit costs for all arms ($c_{k}=1$ for all $k$) and used a budget constraint of $B=\lfloor K/2\rfloor$, allowing selection of approximately half the available arms in each round.

#### Algorithmic implementation.

The implementation of EMC and MvM uses standard Python scientific computing libraries (NumPy, Pandas) without GPU acceleration, as the algorithms are CPU-based. The SP-Matching algorithm, used by both algorithms, was implemented using the dynamic programming approach described in [Algorithm 4](https://arxiv.org/html/2510.09425v1#alg4). Additionally, for the PQ tree implementation, we used the SageMath library [33], a standard Python package for advanced mathematical computations.

#### Experimental setup.

We tested both algorithms on 10 random instances with $U=100$ users and $K=20$ arms. We estimated the expected regret using 10 independent runs for each instance. For the EMC algorithm, we tested time horizons ranging from $T=10^{5}$ to $T=10^{6}$ rounds. To simulate the unknown order setting, we randomly permuted the columns of each generated single-peaked matrix before running the algorithm. For the MvM algorithm, we executed a single run up to $T=10^{5}$ rounds, recording the cumulative regret at each time step to observe the full regret trajectory. This method naturally serves as an upper bound on the regret that would be incurred if the algorithm were run separately for smaller values of $T$. However, we still observe the same predicted asymptotic behavior. Since this algorithm assumes a known order and peaks, we provided the true single-peaked structure directly without column permutation.

The difference in the number of rounds $T$ for the two algorithms stems from the fact that EMC necessitates larger time horizons to ensure a sufficient number of commitment rounds relative to the exploration rounds.

#### Used hardware.

The complete simulation suite was executed on a standard MacBook Pro with 18 GB of RAM and 8 CPU cores. We utilized parallel processing across five cores to accelerate computation. The complete set of experiments required approximately 6 hours of computation time.

### F.2 Results

Figure: (a) EMC Algorithm
Refer to caption: x1.png

Figure [2](https://arxiv.org/html/2510.09425v1#A6.F2) presents log-log plots of cumulative regret versus time for both algorithms. Each plot displays the mean regret computed over all 10 instances (with 10 runs each), along with shaded regions representing the standard deviation. The narrow confidence bands demonstrate the consistency of our results across different problem instances and random seeds. Note that the two plots use different x-axis representations: the EMC plot ([Figure 2(a)](https://arxiv.org/html/2510.09425v1#A6.F2.sf1)) shows $\log(T)$ where each point represents the total regret after $T$ rounds of interaction, while the MvM plot ([Figure 2(b)](https://arxiv.org/html/2510.09425v1#A6.F2.sf2)) shows $\log(t)$ with the cumulative regret trajectory at each round $t$.

#### The EMC algorithm.

As shown in [Figure 2(a)](https://arxiv.org/html/2510.09425v1#A6.F2.sf1), the algorithm exhibits consistent behavior, as evidenced by the nearly indistinguishable standard deviation bands. The fitted slope is approximately 0.694, approaching but not quite reaching the theoretical prediction of $2/3\approx 0.667$ from our $\tilde{O}(UKT^{2/3})$ regret bound. When tested for larger time horizons, the empirical slope approaches the theoretical value of $2/3$, confirming the asymptotic prediction of our analysis.

#### The MvM algorithm.

As shown in [Figure 2(b)](https://arxiv.org/html/2510.09425v1#A6.F2.sf2), the algorithm demonstrates excellent empirical performance with slopes ranging from 0.388 to 0.434 across different instances. All observed slopes fall strictly below the theoretical upper bound of 0.5, corresponding to our $\tilde{O}(U\sqrt{KT})$ regret bound. The variation in slopes across instances reflects the algorithm’s adaptive nature: instances with more challenging preference structures result in steeper regret growth, which corresponds to slower learning. The small standard deviation bands indicate that the algorithm’s performance is stable across multiple runs of the same instance. This gap between empirical performance and theoretical bounds suggests that our analysis may be conservative for typical problem instances.

#### Comparison.

The empirical slopes–0.694 for EMC versus 0.388-0.434 for MvM–highlight the cost of unknown order. While both algorithms achieve sublinear regret, the MvM algorithm’s knowledge of the single-peaked structure enables significantly better asymptotic performance. The EMC algorithm’s performance is hindered by the finite-horizon effects discussed above, but still achieves sublinear regret that would approach the theoretical rate with larger time horizons.

## Peer Reviews

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. For the UCB-based algorithm, the authors claim that solving the optimization problem requires a known SP order. How about using the estimated order from the estimated means and UCB/LCBs? Specifically, though the total order is initially unknown,  the algorithm can construct a partial ordering based on estimations. Can the algorithms solve some optimization problem corresponding to partial ordering to avoid the strong assumption and get an $O(\sqrt{T})$ regret? The authors can discuss this point. 
2. Writing suggestion (optional): When two algorithms are available, usually the paper first presents an algorithm that requires strong assumptions to convey some intuition and then presents a more general one.
soundness: 3
strengths: 1. The paper introduces the single-peaked (SP) structural assumption for budget-constrained matching, turning an otherwise NP-hard problem into one solvable in polynomial time. This may offer a principled blueprint for simplifying other computationally hard online learning settings.
2. The algorithmic design is well structured: SP-MATCHING yields a polynomial-time optimal solution offline; the online setting presents MvM (known SP) and EMC (unknown SP), both built on SP-MATCHING. The theory is tight and carefully argued, with detailed proofs of regret bounds and supporting lemmas in the appendix. The results are complete with both upper bounds and lower bounds provided.
summary: This paper studies a budget-constrained matching problem. The authors first point out that, under general preference structures, the problem is NP-hard even in the offline setting. To overcome this computational barrier, the paper introduces the assumption of single-peaked (SP) preferences, which theoretically ensures tractability. For the offline case, the authors propose a dynamic programming algorithm, SP-MATCHING, which computes the optimal matching in polynomial time under a known SP order. This algorithm provides the theoretical and computational foundation for the subsequent online methods. In the online setting, the paper considers two cases:

1.	When the SP structure is known, the authors propose an optimistic algorithm MvM (Match-via-Maximal) based on maximal matrices, which uses SP-MATCHING each round to compute the optimal matching and achieves sublinear regret $O(\sqrt{T})$.
2. When the SP structure is unknown, the authors propose an explore-then-commit (EMC) method that first learns an approximate preference order from data collected in a round-robin manner via an EXTRACT-ORDER process based on PQ trees, and then executes this policy based on the approximate preference order, achieving regret $O(T^{2/3})$.
The paper further establishes corresponding regret lower bounds for the known and unknown peak settings. Experiments on synthetic data validate the theoretical results.
weaknesses: 1.	Strong assumptions.
The paper relies on the single-peaked (SP) preference assumption, which may be overly idealized in complex real-world environments. Although the paper states that this preference structure is common in various domains, such as recommendation systems (Line 049-052), no reference is provided for justification. The additional requirement of known SP order and user peaks for the MvM algorithm is unrealistic, which limits its applicability. 

2. Insufficient experiments.

This paper introduces a new matching setting with a budget constraint, which appears to have not been studied before. The experiments are also restricted to synthetic data. No real-world applications/datasets have been tested. Moreover, the paper lacks comparisons with general combinatorial bandit baselines, particularly those without SP assumptions but with α-regret guarantees, which would better contextualize the claimed advantages.

---

confidence: 4
contribution: 2
initial_rating: 4
presentation: 3
questions: 1. Could the SP assumption be relaxed to more general structures while retaining computational efficiency?
2. What is the hardness in designing algorithm attaining $O(\sqrt{T})$ regret for online setting with unknown structure? Is it possible to improve the lower bound by constructing instance with this hardness?
soundness: 3
strengths: 1. ​​The paper bridges combinatorial bandits and social choice theory by incorporating SP preferences, transforming an NP-hard problem into a tractable one while maintaining standard regret bounds instead of approximate regret.
2. ​​Both offline and online algorithms are designed with theoretical guarantees. The maximal matrix construction for MvM and PQ-tree-based order recovery in EMC demonstrate creative problem-solving.
3. ​​Experimental results demonstrate theoretical regret rates.
4. The budget constraint modeling and SP assumption aligns well with real-world applications like content recommendation.
summary: This paper addresses an online stochastic matching problem where a learner sequentially matches users to arms under budget constraints, aiming to maximize cumulative reward over $T$ rounds. The key innovation is leveraging single-peaked (SP) preferences to overcome computational intractability, where user preferences are unimodal with respect to a common arm order. This paper studies 1) offline setting, 2) online setting with known preference structure, and 3) online setting with unknown structure.
weaknesses: My major concern is the gaps in bounds​​ of online setting with unknown structure. The gap between EMC's $O(UKT^{2/3})$ regret and the $\Omega{\sqrt{KT}}$ lower bound suggests potential for improved algorithms or tighter analysis.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: See weakness.
soundness: 3
strengths: 1. The single-peaked preference assumption is grounded in social choice theory and circumvents NP-hardness, enabling efficient algorithms with standard regret bounds instead of weaker α-regret.
2. It provides a complete landscape by offering efficient algorithms for both unknown (EMC) and known (MvM) structure settings, while rigid theoretical analysis are given on both algorithms.
3. The PQ-tree-based order extraction method and the concept of a maximal matrix, which enables optimistic planning while preserving structural constraints, are novel.
summary: This paper studies an online stochastic matching problem where a learner must sequentially match users to arms under a budget constraint, with the goal of maximizing cumulative reward. The general problem is NP-hard, but the authors circumvent this computational barrier by imposing a single-peaked preference structure, a well-known concept from social choice theory where each user's utility is unimodal with respect to a common ordering of the arms. For the offline problem, they develop SP-Matching, an efficient dynamic programming algorithm that finds the optimal budgeted matching. Leveraging this, they propose two online algorithms: EMC, an explore-then-commit algorithm for the challenging case of unknown preference structure, which achieves a regret of $\tilde{O}(UKT^{2/3})$ by extracting an approximate order using PQ trees; and MVM, an efficient UCB-like algorithm for the case of known structure, which achieves a tighter regret of $\tilde{O}(U\sqrt{TK})$ by leveraging a novel maximal matrix within its confidence sets. This paper also gives a regret lower bound analysis for both known and unknown peaks cases.
weaknesses: 1. While lower bounds are provided, gaps remain between upper and lower bounds in some settings (e.g., for EMC). The paper does not conclusively determine if the attained rates are optimal for polynomial-time algorithms in the unknown structure case. To my knowledge, the ETC-based algorithm is not an order-optimal algorithm in the classical stochastic MAB problem, so it is not surprising that there exists a gap between the upper and lower bounds. Even though the proposed EXTRACT-ORDER algorithm used in EMC provides insight into the construction of an ASP order.
2. The setting that different users can be matched to the same arm simplifies this problem, while a more common setting is the bipartite matching between users and arms.  


Minor
1. Title of Section 4.3: written as ETC instead of EMC.

---

confidence: 3
contribution: 2
initial_rating: 4
presentation: 3
questions: Is there a specific reason for using the term ``preference'' here? To me, preference suggests a Bradley--Terry--Luce--type model, where the reward of each arm is defined relatively, depending on the assigned counterpart.
soundness: 3
strengths: This is the first work to study the matching bandit setting under single-peaked preferences, where an efficient algorithm can be designed. The proposed method achieves a tight regret bound when the preference structure is known.
summary: This work studies an online stochastic matching problem involving $U$ users and $K$ arms, aiming to maximize the cumulative reward over $T$ rounds under budget constraints. Without assuming any structural property of user preferences, the problem is NP-hard, rendering online learning computationally infeasible. To address this, the authors focus on the case of single-peaked preferences. They first propose an efficient algorithm that achieves a regret of $\tilde{O}(UKT^{2/3})$. When the single-peaked structure is known in advance, they further develop an algorithm that attains a regret of $\tilde{O}(U\sqrt{TK})$.
weaknesses: The main concern lies in the tightness of the regret bound under the unknown structure. The authors propose an ETC-based algorithm that achieves a regret of order $T^{2/3}$, which appears to be suboptimal. Establishing a tight regret bound for the unknown-structure case would be a meaningful goal, as the current result seems incomplete. If achieving a tighter bound is not possible, the authors should at least provide a corresponding lower bound to justify that the $T^{2/3}$ rate is optimal. Moreover, the ETC and UCB algorithms themselves do not appear to be novel.