Title: Contextual Dynamic Pricing with Heterogeneous Buyers
Venue: NeurIPS 2025
Accepted: yes
Keywords: dynamic pricing

bandits

contextual
Abstract: We initiate the study of contextual dynamic pricing with a heterogeneous population of buyers, where a seller repeatedly posts prices (over $T$ rounds) that depend on the observable $d$-dimensional context and receives binary purchase feedback. Unlike prior work assuming homogeneous buyer types, in our setting the buyer's valuation type is drawn from an unknown distribution with finite support size $K_{\star}$. We develop a contextual pricing algorithm based on optimistic posterior sampling with regret $\widetilde{O}(K_{\star}\sqrt{dT})$, which we prove to be tight in $d$ and $T$ up to logarithmic terms. Finally, we refine our analysis for the non-contextual pricing case, proposing a variance-aware zooming algorithm that achieves the optimal dependence on $K_{\star}$.

Contextual Dynamic Pricing with Heterogeneous Buyers

Abstract

Weinitiate the study of contextual dynamic pricing with a heterogeneous population of buyers, where a seller repeatedly posts prices (over T rounds) that depend on the observable d -dimensional context and receives binary purchase feedback. Unlike prior work assuming homogeneous buyer types, in our setting the buyer's valuation type is drawn from an unknown distribution with finite support size K ⋆ . We develop a contextual pricing algorithm based on optimistic posterior sampling with regret ˜ O ( K ⋆ √ dT ) , which we prove to be tight in d and T up to logarithmic terms. Finally, we refine our analysis for the non-contextual pricing case, proposing a variance-aware zooming algorithm that achieves the optimal dependence on K ⋆ .

1 Introduction

In online learning for contextual pricing, a learner (aka seller) repeatedly sets prices for different products with the goal of maximizing revenue through interactions with agents (aka buyers or customers). Concretely, in each round t = 1 , . . . , T , nature selects a product with a d -dimensional feature representation u t (context) and the seller selects a price p t ≥ 0 . In the simplest variant, the linear valuation model , customers have a fixed intrinsic valuation model (type) that is unknown to the learner; this has a d -dimensional representation θ ⋆ whose coordinates reflect the valuation that each product feature adds, i.e., the customer's valuation is v t = ⟨ θ ⋆ , u t ⟩ + ε t where ε t is a noise term. The customer makes a purchase only when their valuation is higher than the price, i.e., v t ≥ p t . The learner's goal is to maximize revenue, i.e., the sum of the prices in rounds when purchases occur. An equivalent objective is to minimize regret , which is measured against a benchmark that always selects the customer's valuation as the price for the given round.

Before outlining our contributions, we highlight the unique challenge of online learning in contextual pricing. One key difficulty is that the learner faces both an infinite action space (i.e., all possible prices) and a discontinuous revenue function; indeed, even small price increases can deter a buyer from purchasing, hence causing sharp revenue loss for the learner. However, the problem offers a richer feedback structure than classical multi-armed bandits: a non-purchase indicates that all higher prices would also be rejected by the buyer, while a purchase confirms that all lower prices would be accepted too. The two primary approaches from the literature to tackle this problem involve estimating the unknown parameter θ ⋆ through online regression or multi-dimensional binary search (see Section 1.1 for further discussion).

A crucial limitation for both approaches is that they require all customers to behave homogeneously according to a single type θ ⋆ ; see the related work section for results robust to small deviations from this assumption. Moving beyond this homogeneity assumption, we pose the following question:

How can one design contextual pricing algorithms with a heterogeneous population of customers?

1.1 Our Contribution

Our setting (Section 2). To study contextual pricing with a heterogeneous buyer population, we assume that the type θ t in round t is drawn from a fixed, unknown distribution D ⋆ . When D ⋆ is supported on a single type θ ⋆ , we recover the homogeneous setting. In our setup, the number of distinct buyer types K ⋆ = | supp D ⋆ | reflects the degree of heterogeneity . We assume that K ⋆ > 1 throughout the paper.

There are several obstacles to applying existing algorithms from the literature. First, canonical contextual pricing algorithms based on regression either compete against (simple) linear policies or assume context-independent and identically distributed (i.i.d.) valuation noise. In contrast, the optimal policy in our setting may best respond based on a context-dependent type rather than a fixed type, and the stochasticity due to heterogeneity is inherently context-dependent and thus non-i.i.d. Second, given that the buyer types are not observable , one cannot connect the observed feedback to shrinkage of type-dependent uncertainty sets; this rules out running canonical multi-dimensional binary search / contextual pricing algorithms for each buyer type in parallel. Third, since in our setting there is a continuum of actions , any canonical contextual bandits algorithm whose regret scales with the discretized action count (e.g., EXP4) will suffer suboptimal performance.

Our contextual pricing algorithm (Section 3). To tackle the above challenges, we employ recent advances in the contextual bandit literature that attain a better scaling with the number of actions, thus evading the shortcomings of EXP4 with naïve discretization. In particular, we build on the optimistic posterior sampling (OPS) approach [29] which, in our setting, maintains a posterior µ t over all candidate type distributions. We call these candidate type distributions models and refer to their (possibly infinite) family as D . At a high level, in every round, OPS best responds to a model sampled from µ t . As typical in online learning, the posterior update penalizes models that disagree with the observed feedback ( model mismatch ) aiming to converge to the model D ⋆ . To encourage exploration in the absence of full information, this penalty is reduced by an optimism bias term that rewards models with the highest potential to positively contribute to the revenue. The OPS approach enables regret bounds of √ T · c · log |D| , scaling with a disagreement coefficient c that measures the per-context structural complexity of the reward functions and captures the tension between exploration and exploitation. This coefficient is always bounded by the number of actions but can be much smaller in general.

Our main technical contributions in adapting OPS to heterogeneous contextual pricing are twofold. First, to bound the disagreement coefficient c , we observe that, for any fixed context, the aggregate demand function induced by D ⋆ has at most K ⋆ 'jumps', 1 thus creating K ⋆ +1 intervals. Over each interval, we bound the disagreement coefficient by a factor of 2 . Combining these arguments with a novel decomposition lemma for the disagreement coefficient of functions with K ⋆ breakpoints, we show that c ≤ 2( K ⋆ +1) . When K ⋆ is known, we apply a variant of OPS over a finite covering of the class D containing all possible distributions over K ⋆ types, of log cardinality dK ⋆ log T . Second, to extend our sublinear regret guarantee to the infinite model class D , we modify OPS to conservatively perturb its recommended prices (which cannot overly impact regret due to one-sided Lipschitzness of the revenue function). We then construct a coupling between the actual trajectory of OPS and one where D ⋆ belongs to the finite cover, allowing us to transfer regret bounds. Finally, we adapt to unknown K ⋆ by initializing OPS with a non-uniform prior over models. These technical contributions enable us to show a regret guarantee of ˜ O ( K ⋆ √ dT ) . Finally, we show that this guarantee is optimal (up to logarithmic terms) with respect to the dependence on both the contextual dimension d and the time horizon T , establishing a lower bound of Ω( √ K ⋆ dT ) for sufficiently large T = Ω( dK 3 ⋆ ) .

Non-contextual improvements (Section 4). The above upper and lower bounds raise a natural question on the optimal dependence of the regret on the number of buyer types K ⋆ ; we resolve this

question in the non-contextual version of the problem ( d = 1 ) by providing an algorithm with an upper bound of ˜ O ( √ K ⋆ T ) . Our algorithm, ZoomV , combines zooming (i.e., adaptive discretization) methods from Lipschitz bandits [13] with variance-aware confidence intervals [2]. Our analysis shows that the regret of ZoomV scales with a novel variance-aware zooming dimension that can be significantly smaller than the standard measure of complexity for Lipschitz bandits. For pricing, this variance adaptation unlocks our ˜ O (min { √ K ⋆ T, T 2 / 3 } ) bound (versus O ( T 2 / 3 ) , obtained via the standard zooming analysis).

We note that the non-contextual version of pricing for heterogeneous buyers was previously studied by [4], who establish a matching upper and lower bound if all types are 'well separated' from each other. Although our algorithmic approach towards this guarantee is very different, our result can be viewed as a strengthening of theirs by removing this separation assumption.

Stronger type observability (Section 5). Finally, we consider contextual pricing under the assumption that the learner can identify each arriving type, i.e., where the learner observes ex-post information about the sampled type θ t . We analyze two observability models: one where the learner receives a discrete identifier z t ∈ [ K ⋆ ] -under which a computationally efficient pricing algorithm matches the ˜ O ( K ⋆ √ dT ) regret bound of OPS - and another where the full type vector θ t ∈ B d is observed - for which we reduce the dependence on K ⋆ and d to achieve regret ˜ O ( √ min { K ⋆ , d } T ) . These results demonstrate how richer feedback reduces complexity in dynamic pricing.

Related work. We briefly review three existing lines of work, deferring full discussion to Appendix A. The closest to ours is contextual pricing/search , where a learner interacts with nature to learn a hidden vector θ ⋆ ∈ R d while receiving single-bit feedback [5, 18, 21, 17, 12, 11, 6, 19]. Under an appropriate pricing loss, this setting reduces to ours with a homogeneous buyer population, i.e., K ⋆ = 1 . However, the deterministic nature of the feedback leads to aggressive search policies via cutting planes that do not lend themselves to the heterogeneous case. Although some works tolerate i.i.d., context-independent valuation noise [12, 11, 6, 19], their methods do not treat our non-i.i.d., context-dependent noise due to heterogeneity. Next is the case of non-contextual dynamic pricing where d = 1 , originally treated with multi-armed bandits methods by [14]. The closest non-contextual work is [4], whose finite-types model reduces to ours with d = 1 . We improve upon their regret bound, but neither our improvement nor the existing methods generalize readily to the contextual setting. Finally, our setting relates to Lipschitz bandits . Although revenue is not fully Lipschitz, it satisfies a one-sided Lipschitzness , enabling the use of zooming [13] when d = 1 (see, e.g., [24]). We successfully refine these for the non-contextual case, but contextual variants of zooming [26, 15] scale with complexity parameters which admit no direct bounds for heterogeneous pricing.

2 Setup and Preliminaries

Notation. Let ∥ · ∥ and ⟨· , ·⟩ denote the Euclidean norm and inner product on R d . Let S d -1 , B d ⊆ R d denote the unit sphere and ball, respectively. Let ∆( S ) denote the set of all probability measures on a measurable set S ⊆ R d , and let supp( D ) denote the support of D ∈ ∆( R d ) . We use ∆ k ( S ) for those D ∈ ∆( S ) with | supp( D ) | ≤ k . For a positive integer m , let [ m ] := { 1 , 2 , . . . , m } .

Problem setup. We consider T rounds of repeated interaction between a seller, a population of buyers, and an adversary. At each round t ∈ [ T ] , the seller posts a price p t ∈ [0 , 1] for an item to be sold and a buyer, sampled from the population, decides whether or not to buy the item based on their valuation v t ∈ [0 , 1] . We denote the indicator of their purchase by y t = 1 { v t ≥ p t } . The valuation of the buyer is determined by two factors: their type θ t , which encodes their intrinsic preferences, and an external context u t , which describes the current item to be sold and any relevant environmental factors. The learner does not know θ t , but they do know u t . We employ a linear valuation model, supposing that θ t and u t lie in d -dimensional spaces Θ ⊆ [0 , 1] d and U ⊆ S d -1 , respectively, and take v t = ⟨ θ t , u t ⟩ . We assume that ⟨ θ, u ⟩ ∈ [0 , 1] for all θ ∈ Θ and u ∈ U . We impose no further assumptions on the contexts, allowing them to be generated (potentially adaptively) by the adversary. On the other hand, we assume that each θ t is sampled independently from a fixed distribution D ⋆ ∈ ∆(Θ) that describes the buyer population, unknown to the seller. All together, the following occur at each round t ∈ [ T ] :

the adversary selects a context u t ∈ U ;

a buyer arrives with type θ t ∈ Θ sampled independently from D ⋆ , with valuation v t = ⟨ u t , θ t ⟩ ;

the seller observes u t and posts price p t ∈ [0 , 1] for the item;

the seller observes the purchase decision y t = 1 { v t ≥ p t } and receives revenue p t y t .

Benchmark. The seller's goal is to maximize their cumulative revenue compared to that which they could have achieved with knowledge of D ⋆ . To express this concisely, we introduce some additional notation. Each distribution Q over valuations in [0 , 1] induces the following:

a demand function dem Q ( p ) := P v ∼ Q [ v ≥ p ] ,

an expected revenue function rev Q ( p ) := p · dem Q ( p ) ,

a revenue-maximizing best response br Q := arg max p ∈ [0 , 1] rev Q ( p ) (breaking ties arbitrarily),

and a gap function gap Q ( p ) := rev Q ( br Q ) -rev Q ( p ) .

Once we restrict to a fixed context u ∈ U , each type θ ∈ Θ induces valuation v = ⟨ u, θ ⟩ . Thus, each type distribution D ∈ ∆(Θ) induces a projected valuation distribution Q = proj ( D,u ) ∈ ∆([0 , 1]) , defined as the law of ⟨ u, θ ⟩ when θ ∼ D . We then set dem D ( p, u ) := dem Q ( p ) , rev D ( p, u ) := rev Q ( p ) , br D ( u ) := br Q , and gap D ( p, u ) := gap Q ( p ) , accordingly. We abbreviate a subscript of D ⋆ by ' ⋆ ' alone, writing, e.g., dem ⋆ ( p, u ) and br ⋆ ( u ) . These details are summarized in Table 1.

A seller policy A is a (potentially randomized) map from a history { u τ , p τ , y τ } t -1 τ =1 and the current context u t to a posted price p t . An adversary policy B is a (potentially randomized) map from a history { u τ , θ τ , p τ , y τ } τ ∈ [ t -1] to the next context u t . We then define the seller's pricing regret by

where { u t , p t } t ∈ [ T ] are selected according to A and B . We will omit the policies from the subscript when clear from context. We focus on controlling the pricing regret in expectation, and will say that A satisfies a regret bound f ( T ) if E [ R A , B ( T )] ≤ f ( T ) for all B .

Our guarantees will scale with context dimension d and the degree of heterogeneity , which we quantify via the support size K ⋆ := | supp( D ⋆ ) | (that may be infinite). We do not assume that K ⋆ is known to the seller. Designing an effective seller policy is challenging because D ⋆ , K ⋆ , and the realized buyer types are unknown to the seller, who must carefully balance exploration and exploitation given only the current context and the history of purchase outcomes.

Basic pricing facts. Finally, we provide some basic properties of the pricing problem, with proofs in Appendix B. Essential for this work is the one-sided Lipschitzness of the expected revenue function. This is a consequence of the monotonicity of demand functions, and it has previously been used to apply techniques from Lipschitz bandits to non-contextual pricing [24].

Lemma 2.1 (One-sided Lipschitzness) . Fix any distribution Q ∈ ∆([0 , 1]) and let 0 ≤ p < p ′ ≤ 1 . We then have rev Q ( p ′ ) -rev Q ( p ) ≤ dem Q ( p )( p ′ -p ) ≤ p ′ -p .

x

Throughout this work, we must handle distributional uncertainty over value distributions. To compare two distributions P, Q ∈ P ([0 , 1]) , we employ the Levy metric defined by

This quantity is at most 1 and equals the side length of the largest square which can be inscribed between the graphs of dem P and dem Q (equivalently, the CDFs of P and Q ). For type distributions D,D ′ ∈ P (Θ) , we use the Levy distance between their projected value distributions, taking d L ( D,D ′ ) := sup u ∈ S d -1 d L ( proj ( D,u ) , proj ( D ′ , u )) . We use this metric because, if D and D ′ are close under d L , then there exists a policy which performs well on both of them; this motivates the use of the Lévy metric throughout the dynamic pricing literature (see, e.g., 23).

Lemma 2.2 (Pricing implication of Lévy metric bound) . Suppose that D,D ′ ∈ ∆(Θ) satisfy d L ( D,D ′ ) < ε . Then the conservative best-response policy π ( u ) = max { br D ( u ) -ε, 0 } satisfies rev D ( π ( u ) , u ) ≥ rev D ( br D ( u ) , u ) -ε and rev D ′ ( π ( u ) , u ) ≥ rev D ′ ( br D ′ ( u ) , u ) -3 ε for all u ∈ U .

3 Contextual Algorithm with Optimal Dependence on d and T

We now develop statistically efficient (albeit computationally inefficient) algorithms for contextual pricing. In Section 3.1, we treat the simpler setting where D ⋆ belongs to a finite model class D and K ⋆ is known. In Section 3.2, we remove these two assumptions, achieving regret ˜ O ( K ⋆ √ dT ) . We also provide a regret lower bound of Ω( √ K ⋆ dT ) , even if K ⋆ is known, thus proving the optimality of our regret bound's dependence on d and T . Omitted proofs appear in Appendix C.

3.1 Warm-Up: Heterogeneous Contextual Pricing with a Finite Model Class

As a warm-up, we consider pricing when D ⋆ belongs to a known, finite model class.

Assumption 3.1. Assume that D ⋆ ∈ D , where D ⊆ ∆(Θ) is finite and known to the seller.

This realizability assumption simplifies our analysis, and the resulting algorithm extends naturally to infinite classes. We employ optimistic posterior sampling ( OPS ), originally studied for contextual bandits by [29] under the name 'Feel-Good Thompson Sampling.' In our instantiation for contextual pricing, OPS (Algorithm 1) maintains a posterior distribution over models, initialized at prior µ 1 ∈ ∆( D ) . At round t with context u t ∈ U , we sample a model D t ∼ µ t , play the best-response price p t = br D t ( u t ) , and observe purchase feedback y t . Then, for each candidate model D ∈ D , we update its posterior weight µ t +1 ( D ) according to the loss ℓ λ ( proj ( D,u t ) , p t , y t ) , defined by

The 'model mismatch' penalty captures the extent to which the observed demand y t differs from that predicted by D when p t is played. In particular, as a function of D ∈ D , the expected model mismatch E y t [( y t -dem D ( p t , u t )) 2 | p t ] is minimized only by those models D which make the same prediction as D ⋆ , i.e., those for which dem D ( p t , u t ) = dem ⋆ ( p t , u t ) . On the other hand, the 'optimism bias' reduces the loss for models which have the potential to provide large revenue, ensuring that we perform sufficient exploration.

ALGORITHM 1: OPS : Contextual Pricing with a Finite Model Class

The requirement of known K ⋆ is imposed for simplicity and will be removed in Section 3.2. To prove Theorem 3.2, we employ a disagreement coefficient that controls the per-context complexity of balancing exploration and exploitation. In general, for an arbitrary measurable space X and function class F : X → R , we define the disagreement coefficient of F by

In our setting, each f will measure the discrepancy between the demand function predicted by some model D and that of the true model D ⋆ , for a fixed context (full details will appear shortly). While not the most primitive complexity measure, variants of this quantity have been successfully used to analyze a wide variety of structured bandits and active learning problems (see Remark 3.7). The δ 2 /ε 2 scaling was historically chosen so that dis ( F ) can be directly bounded by the domain size |X| . For our application, X = [0 , 1] is the (infinite) price set and each function f ∈ F , induced by a model D ∈ D and context u ∈ U , measures the discrepancy between the demand functions of D and D ⋆ after projection onto u , i.e., f ( p ) = dem D ( p, u ) -dem ⋆ ( p, u ) . In particular, we set

By this definition, if dis ( D , D ⋆ ) is small and the seller plays price p ∼ ν when faced with context u , it is unlikely for a model D to disagree with D ⋆ at p if it is close to D ⋆ under the L 2 ( ν ) norm, i.e., if E q ∼ ν [( dem D ( q, u ) -dem ⋆ ( q, u )) 2 ] is small. In particular, playing p ∼ ν guarantees that

with probability at least 1 -ε 2 δ 2 dis ( D , D ⋆ ) . Since OPS penalizes models with substantial least squares loss, while incentivizing exploration via its optimism bonus, we are able to show the following.

Lemma 3.3. Under Assumption 3.1 with optimism strength λ > 0 , OPS (Algorithm 1) achieves regret ˜ O ( λ dis ( D , D ⋆ ) T +log( |D| ) /λ ) .

The proof in Appendix C.3 combines the OPS analysis of [29] with a decoupling lemma due to [8]. To control dis ( D , D ⋆ ) , we show that each function of the form dem D ( · , u ) -dem ⋆ ( · , u ) can be decomposed into K ⋆ + 1 non-increasing pieces. In Appendix C.4, we prove the following disagreement coefficient bound for non-increasing functions.

Lemma 3.4. Let F : [0 , 1] → R be the set of nonincreasing functions. Then dis ( F ) ≤ 2 .

Next, we examine the useful notion of composite function classes. A function class G : Z → R is called an N -composite of F : X → R if there exists a disjoint partition Z = Z 1 ∪ · · · ∪ Z N and mappings { h i : Z i →X} i ∈ [ N ] such that each g ∈ G can be decomposed as g ( x ) = f i ( h i ( x )) for all x ∈ Z i and i ∈ [ N ] , for some choice of { f i : X → R } i ∈ [ N ] . We show the following in Appendix C.5.

Lemma 3.5. If G is an N -composite of F , then dis ( G ) ≤ N dis ( F ) .

With these results in hand, we bound dis ( D , D ⋆ ) = O ( K ⋆ ) and prove the theorem. Even though the action space is infinite, the disagreement coefficient matches that which would arise with K ⋆ actions.

Proof of Theorem 3.2. For each u ∈ U , the function dem ⋆ ( · , u ) is piecewise constant with K ⋆ +1 sections, since jumps can only occur at the projections of the K ⋆ types. For any D ∈ D , the demand dem D ( · , u ) is monotonic, since increasing price always reduces demand. Thus, dem D ( · , u ) -dem ⋆ ( · , u ) is non-increasing on each of the K ⋆ +1 sections, and so the function classes defining dis ( D , D ⋆ ) are ( K ⋆ +1) -composites of the non-increasing function class. Applying Lemmas 3.4 and 3.5 then implies that dis ( D , D ⋆ ) ≤ 2( K ⋆ +1) . The theorem then follows by the regret bound of Lemma 3.3 and our choice of λ .

Remark 3.6 (Comparison to Thompson sampling) . Standard Thompson sampling corresponds to the alternative choice of log losses: ℓ ( Q,p,y ) = log P z ∼ Ber( dem Q ( p )) [ z = y ] = y log dem Q ( p ) + (1 -y ) log(1 -dem Q ( p )) . In comparison, OPS uses the squared loss (this is not essential but simplifies analysis) and an optimism bias towards models under which the seller can attain large revenue. This

is crucial for obtaining frequentist (rather than Bayesian) regret bounds, as outlined in [29]. One appealing aspect of the log loss is that it eliminates models which predict that the observed feedback would never occur, mirroring the elimination-based methods for contextual pricing when K ⋆ = 1 . Thus, one natural question (beyond the current scope) is whether OPS with log loss achieves regret scaling logarithmically in T when K ⋆ = 1 .

Remark 3.7 (Relation with existing results) . Variants of the disagreement coefficient and the related Alexander capacity are well-studied in the active learning and empirical process theory literature [9]. The version above was first considered by [7]. [8] proved a regret bound which translates to ˜ O ( √ dis ( D ) T log |D| ) in our setting, matching Theorem 3.2. However, the estimation-to-decisions (E2D) meta-algorithm which they employ is non-constructive, hence we apply OPS instead. In [29], the regret of OPS is controlled by a distinct 'decoupling coefficient. ' Our proof of Lemma 3.3 shows that a (slightly modified) decoupling coefficient is bounded by the disagreement coefficient.

3.2 The General Case

We now seek to eliminate the assumptions that D ⋆ belongs to a finite class D and that the support size K ⋆ is known to the seller. For the first point, we loosen the requirement that D ⋆ belongs to D and take D to be a large, finite cover of the full distribution space ∆(Θ) . Then, we replace the uniform prior µ 1 with a non-uniform prior that places less weight on models with large support sizes. Ultimately, this will enable a choice of optimism strength λ that is independent of K ⋆ , achieving our second goal. Unfortunately, if D ⋆ is close but not equal to a model in D , our analysis of OPS fails.

To remedy this, we employ perturbed OPS ( POPS , Algorithm 2), an OPS variant with conservatively perturbed and discretized prices. This modified algorithm and its analysis require some new notation. Given a value distribution Q ∈ ∆([0 , 1]) , define the ε -smoothed demand function dem ε Q by

Similarly, we let rev ε Q ( p ) := p dem ε Q ( p ) . Define contextual extensions dem ε D ( p, u ) and dem ε D ( p, u ) as in the non-smoothed case. For discretization, write P ε := ε N ∩ [0 , 1] for prices which are multiples of ε and let br ε Q := arg max p ∈P ε rev ε ( p ) (lifting to br ε D ( u ) as in the non-smoothed case).

Now, at each round t ∈ [ T ] , POPS samples a model D t ∼ µ t from the current posterior µ t ∈ ∆( D ) and computes its (discretized) best response ˆ p t = br ε D t ( u t ) . Instead of posting price ˆ p t directly, POPS posts p t = max { ˆ p t -δ t , 0 } , where δ t ∼ Unif([0 , ε ]) is a small random perturbation. Due to this perturbation and discretization, we employ the modified loss ℓ ε λ ( proj ( D t , u t ) , ˆ p t , y t ) , where

The perturbations allow us, in the analysis of POPS , to couple its trajectory when run with D ⋆ ̸∈ D to a trajectory where D ⋆ ∈ D . The discretization is needed to bound a modified disagreement coefficient which appears in the analysis. All together, we obtain the following.

ALGORITHM 2: Perturbed OPS ( POPS ) for Contextual Pricing with Infinite Model Class

Theorem 3.8. With appropriately tuned parameters, POPS (Algorithm 2) achieves regret ˜ O ( K ⋆ √ dT ) without prior knowledge of K ⋆ . Moreover, even for known K ⋆ > 1 and stochastic contexts, no contextual pricing policy can achieve expected regret o ( √ K ⋆ dT ) for all instances if T ≥ 8 dK 3 ⋆ log(2 d ) .

Our analysis views the perturbation at Step 5 as being performed by nature, rather than the seller. Treating the seller's action as ˆ p t , they then observe a purchase ( y t = 1 ) with probability

justifying the definitions above. Through this lens, POPS can viewed as OPS for an alternative, smoothed demand model. To bound regret, we apply a three-step argument.

First, we show that POPS maintains our OPS regret bound when D ⋆ ∈ D . This requires bounding a modified decoupling coefficient and is the only step where discretization is used. A direct application of the previous OPS analysis provides a regret bound with respect to a smoothed and discretized benchmark. Fortunately, one-sided Lipschitzness of revenue (Lemma 2.1) ensures that this modified regret benchmark is within O ( εT ) of the original benchmark, as we prove in Appendix C.6.

Lemma 3.9. Under Assumption 3.1, using prior µ 1 ∈ ∆( D ) , discretization error ε ∈ [0 , 1) , and optimism strength λ > 0 , POPS (Algorithm 2) achieves regret ˜ O ( λK ⋆ T +log(1 /µ 1 ( D ⋆ )) /λ + εT ) .

Next, we show that, if there exists D ∈ D whose smoothed demand function uniformly approximates that of D ⋆ , then the trajectory of POPS under D ⋆ can be coupled with that under D , such that the trajectories coincide with high probability. See Appendix C.7 for the proof.

Lemma 3.10. If there exists D ∈ D for which ∥ dem ε D -dem ε D ⋆ ∥ ∞ ≤ ε , then the trajectory { u t , ˆ p t , y t } T t =1 of POPS with type distribution D ⋆ can be coupled with that { u ′ t , ˆ p ′ t , y ′ t } T t =1 of POPS with type distribution D , such that the two trajectories are identical with probability 1 -εT .

Finally, we show that, to obtain a uniform ε -cover of the smoothed demand functions, it suffices to find a O ( ε 2 ) -cover of the type distributions under the Lévy metric d L , as defined in (1). Moreover, we show that the family of all type distributions with support size at most K , ∆ K (Θ) , admits an appropriately small Lévy cover. For notation, we write N ( X, d , τ ) for the size of the smallest subset X ′ ⊆ X which covers set X under metric d up to accuracy τ (i.e., for each x ∈ X , there exists x ′ ∈ X ′ such that d( x, x ′ ) ≤ τ ). A proof of the following appears in Appendix C.8.

Lemma 3.11. If D,D ′ ∈ ∆(Θ) satisfy d L ( D,D ′ ) ≤ ε 2 / 2 , then ∥ dem ε D -dem ε D ′ ∥ ∞ ≤ ε . Moreover, we have log N ( ∆ K (Θ) , d L , ε ) = ˜ O ( dK log 1 /ε ) .

In Appendix C.9, we combine these lemmas to prove the upper bound of Theorem 3.8. For the lower bound in Appendix C.10, we modify a construction from [4] for the non-contextual case, so that it can be tensored into d -dimensions without leaking information between orthogonal contexts.

Remark 3.12 (Bayesian analysis) . Consider the Bayesian setting where D ⋆ is sampled from a known prior µ ∈ ∆( D ) (keeping D finite for simplicity). Then, Lemma 3.9 with µ 1 = µ and ε = 0 implies that OPS , with µ 1 set to µ at Step 2, achieves Bayesian regret ˜ O ( λK ⋆ T + E D ⋆ ∼ µ [ -log µ ( D ⋆ )] /λ ) = ˜ O ( λK ⋆ T + H ( µ ) /λ ) , where H is Shannon entropy. For known K ⋆ , λ can be tuned to achieve regret ˜ O (√ K ⋆ TH ( µ ) ) , matching the (non-Bayesian) bound of Theorem 3.2 with H ( µ ) instead of log |D| .

Remark 3.13 (Misspecified/noisy setting) . We note that POPS is inherently robust to small misspecifications and noise. Indeed, if D ⋆ does not have support size K ⋆ but is within Lévy distance δ of the family ∆ K ⋆ (Θ) , then the proof of Theorem 3.8 still goes through if we choose ε ← T -2 + √ δ , incurring an additive regret overhead of √ δT 2 . The same overhead applies (up to logarithmic factors) in the noisy model where valuations are subject to mean-zero δ 2 -sub-Gaussian noise, since the associated convolution can only perturb demands by ˜ O ( δ ) under the Lévy metric. We do not attempt to optimize this overhead but note that one is better off using EXP4 (as below) when δ ≫ 1 / poly( T ) .

Remark 3.14 (Large K ⋆ ) . Our lower bound above can be restated as ˜ Ω(min { √ K ⋆ dT,d 1 / 3 T 2 / 3 } ) . When K ⋆ = ˜ Ω(( T/d ) 1 / 3 ) and the second term is active, POPS no longer has an advantage over EXP4 with naïvely discretized actions. In particular, one can run EXP4 with a policy cover of log cardinality ˜ O ( dε -2 ) and price set { ε, 2 ε, . . . , 1 } , incurring regret overhead εT due to discretization. This gives a regret bound of ˜ O ( √ Td 2 ε -3 + εT ) , which balances out to d 2 / 5 T 4 / 5 after tuning. Characterizing the optimal regret in this large K ⋆ regime is an interesting question beyond the current scope.

4 Non-Contextual Refinements via Zooming

Our results from Section 3 leave a key open question: what is the optimal regret dependence on K ⋆ ? We resolve this question for the non-contextual setting, where d = 1 and, without loss of generality, u t ≡ 1 for all t . To do so, we employ adaptive discretization (aka zooming ) methods from Lipschitz

bandits [13] with novel variance-aware confidence intervals and achieve a regret bound of ˜ O ( √ K ⋆ T ) . Throughout, we label the K ⋆ types supp( D ⋆ ) = { θ (1) < θ (2) < · · · < θ ( K ⋆ ) } and set θ (0) = 0 .

Our algorithm, ZoomV , mirrors standard zooming [13] with two key adjustments. First, since revenue is only one-sided Lipschitz, we use a dyadic price selection rule inspired by [24]. Second, our confidence intervals incorporate empirical variance, a method previously used for variance-aware K -armed bandits [2]. In more detail, ZoomV maintains a set S of active prices in [0 , 1] and a varianceaware confidence interval for the expected revenue at each p ∈ S . Each active price 'covers' an interval of neighboring, larger prices, with the width of this covering interval scaling proportionally to that of the confidence interval. The intuition is that a small increase in price can only marginally increase expected revenue, so it is not worth exploring such covered prices. Initially, every price in [0 , 1] is covered by some point in S . At each round, ZoomV optimistically chooses a price in S and updates its confidence and covering intervals. If after the update there exists an uncovered price, then we add a new point to S which covers it, maintaining the invariant that every price is covered.

Theorem 4.1. ZoomV (Algorithm 4) achieves regret ˜ O ( min { √ K ⋆ T, T 2 / 3 }) for non-contextual pricing, without knowledge of K ⋆ . This is minimax optimal up to logarithmic factors when K ⋆ > 1 .

To bound regret, we employ a variance-aware zooming dimension which controls its performance. For comparison, we first recall the definition of the standard zooming dimension, which characterizes a certain complexity of the expected reward function rev ⋆ ( p ) . For each δ > 0 , write X δ := { p ∈ [0 , 1] : gap ⋆ ( p ) ≤ δ } for the set of δ -approximate revenue maximizers. Write N ( X,δ ) := N ( X, | · | , δ ) for the smallest δ -covering of a set X ⊆ R . Then, for each c > 0 , define the zooming dimension

Standard zooming techniques imply that ZoomV achieves regret c 1 / (2+ z ) T 1 -1 / (2+ z ) when ZoomDim ( c ) ≤ z , even with confidence intervals that do not incorporate empirical variance. Since the price interval [0 , 1] is one-dimensional, one can bound ZoomDim ( c ) ≤ 1 for c = O (1) , giving regret ˜ O ( T 2 / 3 ) . Moreover, the set X δ of approximate revenue maximizers is contained in the union of K ⋆ intervals preceding the unknown types, where the interval corresponding to type θ ( i ) has width δ/ dem ⋆ ( θ ( i ) ) ≤ δ/ dem ⋆ ( θ ( K ⋆ ) ) . This implies ZoomDim ( c ) = 0 and gives regret ˜ O ( √ cT ) , but only for c = O ( K ⋆ / dem ⋆ ( θ ( K ⋆ ) )) , which may be arbitrarily large for worst-case instances.

To remedy this, we incorporate variance, writing σ 2 ( p ) = p 2 dem ⋆ ( p )(1 -dem ⋆ ( p )) for the revenue variance when p is played. For the problematic types above with low demand, variance is also low, and the confidence intervals of ZoomV are designed to adapt to this. Specifically, our proof in Appendix D.2 shows that the regret of ZoomV scales according to a variance-aware zooming dimension , defined as follows. First define the variance-weighted covering number N v ( X,δ ) := inf {∑ x ∈ X ′ σ 2 ( x ) : X ′ is a δ -cover of X } . Then, for each c > 0 , we set

Note that ZoomDimV ( c ) ≤ ZoomDim ( c ) , since σ 2 ( p ) ≤ 1 . Moreover, we show ZoomDim (10 K ⋆ ) = 0 , implying the desired ˜ O ( √ K ⋆ T ) regret. The lower bound follows from Theorem 3.8 with d = 1 .

Remark 4.2 (Comparison to [4]) . The non-contextual setting was previously studied by [4], whose Algorithm 1 achieves regret ˜ O ( √ K ⋆ T )+ V ( V +1) , where V = max i ∈ [ K ⋆ ] ( θ ( K ⋆ ) ) 4 ( θ ( i ) -θ ( i -1) ) -5 . They maintain a set of intervals which contain all types with substantial probability mass, gradually refining these intervals until they are all of width O ( T -1 / 2 ) , at which point they employ UCB over the intervals' left endpoints. Unfortunately, the instance-dependent term V ( V +1) can blow up to infinity for worst-case realizations of D ⋆ ∈ ∆ K ⋆ ([0 , 1]) , in contrast to our guarantee.

5 Improved Performance with Ex-Post Type Observability

We now study dynamic pricing with heterogeneous buyer types under the additional assumption that the learner can identify each arriving type. That is, after setting price p t , the learner observes the purchase feedback 1 {⟨ u t , θ t ⟩ ≥ p t } and some information about the sampled type θ t . We consider the two models of observability. In the first, the learner only observes an identifier z t ∈ [ K ⋆ ] that specifies which of the K ⋆ candidate types was drawn. In practice, the learner need not know K ⋆ a

priori . Here, we design an algorithm that matches the ˜ O ( K ⋆ √ dT ) regret bound of POPS and can be implemented efficiently, using a contextual search algorithm for K ⋆ = 1 as a subroutine. In the second model, the learner observes the full type embedding θ t ∈ Θ . Here, we show that best-responding to a simple plug-in estimate for D ⋆ achieves an improved regret bound of ˜ O ( √ min { K ⋆ , d } T ) .

Observed type identifiers. Our algorithm for the model where the learner only observes the identifier uses a K ⋆ = 1 contextual search algorithm, ProjectedVolume [18], as a subroutine. We maintain a separate instance of this ProjectedVolume algorithm for each observed type and keep track of the empirical type frequencies along with the number of times we've explored each type. It then adaptively chooses which types to explore (or exploit) based on confidence estimates for the type distribution. We present the full algorithm and prove the following regret bound in Appendix E.1.

Theorem 5.1. Consider contextual dynamic pricing with ex-post type observability where the learner observes which type z t ∈ [ K ⋆ ] arrived. Then, Algorithm 5 achieves regret ˜ O ( K ⋆ √ dT ) and takes no more than time poly( K ⋆ , d, T ) per round.

Observed type vectors. If the full type vector θ t is revealed at the end of each round, we can achieve improved regret with a simpler algorithm. Indeed, writing ˆ D t = 1 t ∑ t τ =1 δ θ t for the empirical type distribution after round t , we take each p t as the best response to ˆ D t -1 along the current context.

Theorem 5.2. Consider contextual dynamic pricing with ex-post type observability where the learner observes θ t ∈ Θ at the end of each round. Then the algorithm which plays p 1 = 1 / 2 and best response p t = br ˆ D t -1 ( u t ) for remaining rounds achieves regret ˜ O ( √ min { K ⋆ , d } T ) . Each price can be computed in time poly( K ⋆ , d ) .

The proof in Appendix E.2 uses VC dimension bounds to show that the empirical revenue function rev ˆ D t converges uniformly in both arguments (price and context) to the true revenue function rev ⋆ .

6 Discussion

In this work, we have introduced contextual dynamic pricing with heterogeneous buyers. Our main algorithm achieves a regret bound of ˜ O ( K ⋆ √ dT ) , optimal up to a O ( √ K ⋆ ) factor and logarithmic terms. Our analysis bounds the disagreement coefficient by leveraging a novel decomposition lemma for aggregate demand functions with K ⋆ breakpoints, thereby ensuring an efficient explorationexploitation tradeoff. Additionally, we propose a variance-aware zooming algorithm for the noncontextual pricing case, improving regret dependence on K ⋆ by incorporating adaptive discretization methods from the Lipschitz bandits literature. Finally, under stronger observability assumptions on the buyers' types, we develop efficient algorithms that significantly reduce regret to ˜ O ( √ min { K ⋆ , d } T ) , demonstrating the potential benefits of richer feedback in dynamic pricing settings.

There are several natural open questions, the first revolving around computation . The run time of POPS scales with the size of the discretized model class, which is exponential in K ⋆ and d . It would be interesting to see if there is a way to alleviate this exponential dependence, while achieving similar regret bounds. The second question is around the optimal dependence on K ⋆ for the general, contextual case. While in Section 4 we showed how to optimize the dependence of our bounds on K ⋆ it is unclear how to scale this approach for the contextual version of the problem. One starting point could be the results on Zooming techniques for contextual bandits (see e.g., 26). Finally, it would be interesting to see if our results can be applied to more broad families of settings where a learner tries to learn from heterogeneous agents while obtaining only single-bit feedback: for example, it is unknown if the approach presented in this work generalizes to general contextual search settings (i.e., with ε -ball or symmetric loss) or if it generalizes for settings that share some core properties with pricing, but differ in the fundamental techniques used to address them (see e.g., 10).

Acknowledgments and Disclosure of Funding

The authors thank Jason Gaitonde and Oliver Richardson for helpful discussions on high-dimensional probability and algorithm design. We are also grateful to the Simons Institute for the Theory of Computing, as this work started during the Fall'22 semester-long program on Data-Driven Decision Processes.

References

K. Amin, A. Rostamizadeh, and U. Syed. Repeated contextual auctions with strategic buyers. Advances in Neural Information Processing Systems , 27, 2014.

J.-Y. Audibert, R. Munos, and C. Szepesvári. Exploration-exploitation tradeoff using variance estimates in multi-armed bandits. Theoretical Computer Science , 410(19):1876-1902, 2009.

S. Boucheron, G. Lugosi, and P. Massart. Concentration Inequalities: A Nonasymptotic Theory of Independence . Oxford University Press, 02 2013.

N. Cesa-Bianchi, T. Cesari, and V. Perchet. Dynamic pricing with finitely many unknown valuations. In A. Garivier and S. Kale, editors, Algorithmic Learning Theory (ALT) , 2019.

M. C. Cohen, I. Lobel, and R. Paes Leme. Feature-based dynamic pricing. Management Science , 66(11):4921-4943, 2020.

J. Fan, Y. Guo, and M. Yu. Policy optimization using semiparametric models for dynamic pricing. Journal of the American Statistical Association , 119(545):552-564, 2024.

D. Foster, A. Rakhlin, D. Simchi-Levi, and Y. Xu. Instance-dependent complexity of contextual bandits and reinforcement learning: A disagreement-based perspective. In Proceedings of Thirty Fourth Conference on Learning Theory (COLT) , 2021.

D. J. Foster, S. M. Kakade, J. Qian, and A. Rakhlin. The statistical complexity of interactive decision making. arXiv preprint arXiv:2112.13487 , 2021.

S. Hanneke et al. Theory of disagreement-based active learning. Foundations and Trends® in Machine Learning , 7(2-3):131-309, 2014.

C.-J. Ho, A. Slivkins, and J. W. Vaughan. Adaptive contract design for crowdsourcing markets: Bandit algorithms for repeated principal-agent problems. In Proceedings of the fifteenth ACM conference on Economics and computation , pages 359-376, 2014.

A. Javanmard. Perishability of data: dynamic pricing under varying-coefficient models. Journal of Machine Learning Research , 18(53):1-31, 2017.

A. Javanmard and H. Nazerzadeh. Dynamic pricing in high-dimensions. Journal of Machine Learning Research , 20(9):1-49, 2019.

R. Kleinberg, A. Slivkins, and E. Upfal. Multi-armed bandits in metric spaces. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing , STOC '08, page 681-690, New York, NY, USA, 2008. Association for Computing Machinery. ISBN 9781605580470. doi: 10.1145/1374376.1374475. URL https://doi.org/10.1145/1374376.1374475 .

R. D. Kleinberg and F. T. Leighton. The value of knowing a demand curve: Bounds on regret for online posted-price auctions. In Symposium on Foundations of Computer Science (FOCS) , pages 594-605, 2003.

A. Krishnamurthy, J. Langford, A. Slivkins, and C. Zhang. Contextual bandits with continuous actions: Smoothing, zooming, and adapting. Journal of Machine Learning Research , 21(137): 1-45, 2020.

A. Krishnamurthy, T. Lykouris, C. Podimata, and R. Schapire. Contextual search in the presence of irrational agents. In ACM SIGACT Symposium on Theory of Computing (STOC) , 2021.

A. Liu, R. P. Leme, and J. Schneider. Optimal contextual pricing and extensions. In Symposium on Discrete Algorithms (SODA) , 2021.

I. Lobel, R. Paes Leme, and A. Vladu. Multidimensional binary search for contextual decisionmaking. Operations Research , 66(5):1346-1361, 2018.

Y. Luo, W. W. Sun, and Y. Liu. Distribution-free contextual dynamic pricing. Mathematics of Operations Research , 49(1):599-618, 2024.

A. Maurer and M. Pontil. Empirical bernstein bounds and sample variance penalization. Conference on Learning Theory (COLT) , 2009.

R. Paes Leme and J. Schneider. Contextual search via intrinsic volumes. SIAM Journal on Computing , 51(4):1096-1125, 2022.

R. Paes Leme, C. Podimata, and J. Schneider. Corruption-robust contextual search through density updates. In Conference on Learning Theory (COLT) , volume 178, 2022.

R. Paes Leme, B. Sivan, Y. Teng, and P. Worah. Pricing query complexity of revenue maximization. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) , pages 399-415. SIAM, 2023.

C. Podimata and A. Slivkins. Adaptive discretization for adversarial lipschitz bandits. In Conference on Learning Theory , pages 3788-3805. PMLR, 2021.

S. Shalev-Shwartz and S. Ben-David. Understanding machine learning: From theory to algorithms . Cambridge university press, 2014.

A. Slivkins. Contextual bandits with similarity information. In Proceedings of the 24th annual Conference On Learning Theory , pages 679-702. JMLR Workshop and Conference Proceedings, 2011.

A. Slivkins et al. Introduction to multi-armed bandits. Foundations and Trends® in Machine Learning , 12(1-2):1-286, 2019.

J. Xu and Y.-X. Wang. Towards agnostic feature-based dynamic pricing: Linear policies vs linear valuation with unknown noise. In International Conference on Artificial Intelligence and Statistics , pages 9643-9662. PMLR, 2022.

T. Zhang. Feel-good thompson sampling for contextual bandits and reinforcement learning. SIAM Journal on Mathematics of Data Science , 4(2):834-857, 2022.

S. Zuo. Corruption-robust lipschitz contextual search. In C. Vernade and D. Hsu, editors, Proceedings of The 35th International Conference on Algorithmic Learning Theory , volume 237 of Proceedings of Machine Learning Research , pages 1234-1254. PMLR, 25-28 Feb 2024.

Appendix

A Extended Discussion on Related Work

Our work relates closely to three lines of work: (I) contextual pricing/search; (II) non-contextual pricing; and (III) Lipschitz bandits.

(I) Contextual pricing/search. The closest line to our work is contextual pricing/search . In contextual search, there is a repeated interaction between a learner and nature, where the learner is trying to learn a hidden vector θ ⋆ ∈ R d over time while receiving only single-bit feedback. Mathematically, at each round t ∈ [ T ] , the learner receives a (potentially adversarially chosen) context u t ∈ R d and decides to query y t ∈ R . The learner receives feedback σ t = sgn( ⟨ u t , θ ⋆ ⟩ -y t ) ∈ {-1 , +1 } and incurs loss ℓ t ( y t , ⟨ u t , θ ⋆ ⟩ ) [5]. Notably, the learner does not observe ℓ t ( y t , ⟨ u t , θ ⋆ ⟩ ) ; only the binary feedback σ t . When the loss function ℓ t corresponds to the lost revenue as a result of posting price y t (i.e., the 'pricing loss'), this setting reduces to ours with a homogeneous buyer population, i.e., K ⋆ = 1 . The contextual search literature has also considered two other loss functions: the symmetric/absolute loss ℓ t ( y t , ⟨ u t , θ ⋆ ⟩ ) := | y t -⟨ u t , θ ⋆ ⟩| and the ε -ball loss ℓ t ( y t , ⟨ u t , θ ⋆ ⟩ ) = 1 {| y t -⟨ u t , θ ⋆ ⟩| > ε } , which are motivated by settings other than pricing.

There have been two approaches in the literature for learning in contextual pricing and contextual search (for homogeneous agents/buyers). The first approach (e.g., 5, 18, 21, 17) employs a version of multidimensional binary search: specifically, the algorithms maintain a 'knowledge set' with all the possible values of θ ⋆ which are 'consistent' with the feedback that nature has given thus far. Similar to traditional binary search, the query point is chosen to be the point that (given the nature's feedback) will eliminate roughly half of the current knowledge set. As the knowledge set shrinks, the learner ends up with a small knowledge set for the possible values of θ ⋆ ; this is enough to guarantee sublinear regret. The series of works in [5, 18, 21, 17] optimized regret bounds for the three different loss functions (i.e., symmetric, ε -ball, and pricing). The specific algorithms were different at each paper, but they all maintained a 'binary search' flavor. Most of the algorithms employing a multidimensional binary search approach can be 'robustified' to very little noise in the agents' responses; since the learner will irrevocably shrink the knowledge set according to the feedback received from nature, they can only afford very few mistakes.

The second approach (e.g., 12, 11, 6, 19) focuses exclusively on pricing settings. This approach uses regression-based algorithms for learning the correct price and needs to assume stochastic noise in the buyers' responses. There have also been works studying other aspects of contextual pricing (e.g., strategic agents [1] and unknown noise distribution [28]). Apart from the methodological differences with our work, both streams of literature focus on a homogeneous agent population and cannot be readily adapted for a heterogeneous population setting.

Moving closer to the heterogeneous agents problem, [16] studied 'corruption-robust' contextual search, where the agent population is mostly homogeneous, except for C = o ( T ) corrupted agent responses. Their regret bounds were subsequently strengthened by [22], but the latter approach only works for contextual search with absolute and ε -ball loss and does not cover the pricing loss. This model has been also studied with a Lipschitz target function [30]. Learning with corruptions can be seen as a first step towards learning from heterogeneous agents, but the approaches above do not scale appropriately for truly heterogeneous agent populations. In contrast, we focus on fully heterogeneous settings, where we do not constrain the number or the size of the different buyer types.

(II) Non-contextual pricing. The special case of d = 1 , where there is no context to inform decisionmaking, was introduced by [14], who studied non-contextual dynamic pricing for a homogeneous, stochastic, and adversarial buyer population. For the adversarial buyer population, the authors assumed that there can be T different valuations and showed tight regret bounds of ˜ O ( T 2 / 3 ) . In contrast, in our setting, we assume that users are 'clustered' in K ⋆ types), and so the lower bound of Ω( T 2 / 3 ) of Kleinberg and Leighton [14] does not apply.

The closest to our work is the work of [4], who consider pricing a heterogeneous agent population with an unknown number of types, but the types are still limited to be less than o ( T ) . Throughout the paper, we discuss how their bounds relate to ours for the special case of d = 1 . None of the aforementioned techniques readily generalize to contextual pricing settings.

(III) Lipschitz bandits. Finally, our work is related to the literature on Lipschitz bandits. Although the pricing loss is not fully Lipschitz, it has recently been observed that it satisfies a one-sided Lipschitzness . This allows us to leverage techniques from adaptive discretization [13] to obtain improved bounds for d = 1 . Zooming had previously been applied to pricing (see, e.g., 24), but these algorithms are insufficient for the K ⋆ -types setting. Indeed, their performance scales with a zooming dimension ZoomDim that is too large here. On the other hand, ZoomV uses variance-aware confidence intervals so that its performance scales with a smaller, variance-aware zooming dimension ZoomDimV . In particular, while ZoomDim can equal 1 for worst-case instances, we show that ZoomDimV is 0 (with a lower-order scaling constant at most K ⋆ ).

Finally, [15] consider contextual bandits with continuous action spaces, which encompasses the setting of this work. Their regret bounds cover the case where, for a fixed context, the expected reward is Lipschitz in the learner's action. Although their analysis can be adapted to the one-sided Lipschitz setting of pricing, their results either require stochastic contexts or incur large regret due to naïve discretization. Even in the stochastic case, their regret bound scales with a policy zooming coefficient that does not appear to admit a useful bound in terms of K ⋆ .

B Proofs for Section 2

B.1 One-Sided Lipschitzness of Revenue (Proof of Lemma 2.1)

We simply bound rev Q ( p ′ ) -rev Q ( p ) = p ′ dem Q ( p ′ ) -p dem Q ( p ) ≤ dem Q ( p )( p ′ -p ) ≤ p ′ -p , using monotonicity of demand functions.

B.2 Pricing Implication of Lévy Metric Bound (Proof of Lemma 2.2)

By the definition of d L , it suffices to prove the lemma when d = 1 . The first revenue lower bound holds by Lemma 2.1. For the second, omitting the context u and writing [ x ] + = max { x, 0 } , we use the Lévy metric guarantee to bound

as desired.

C Proofs for Section 3

To unify analysis of Sections 3.1 and 3.2, we introduce a more general problem setup and algorithm.

C.1 Generalized Problem Setup

To start, we replace the price set [0 , 1] with a subset P ⊆ [0 , 1] , which will remain [0 , 1] in Section 3.1 but will be restricted to a finite set for Section 3.2. Then, instead of selecting a type distribution D ⋆ , we have the adversary choose a demand function f ⋆ which maps price p ∈ P and context u ∈ U to a purchase probability f ⋆ ( p, u ) ∈ [0 , 1] . Then, at round t , if the adversary selects context u t ∈ U and the learner posts price p t ∈ P , purchase decision y t ∈ { 0 , 1 } is sampled independently from Ber( f ⋆ ( p t , u t )) . This abstracts away our previous notions of buyer types and values and will also model the smoothed environment of Section 3.2. We impose the corresponding notion of realizability.

Setting C.1 (realizability, general) . Under the setup described above, the demand function f ⋆ belongs to a known, finite class F of measurable functions from P × U to [0 , 1] .

Often, we shall fix a context and consider univariate (non-contextual) demand functions. Given a univariate demand function g : [0 , 1] → [0 , 1] , we define the corresponding revenue function rev g ( p ) = p · g ( p ) , best-response br g = arg max p ∈P rev g (breaking ties arbitrarily), and gap gap g ( p ) = rev g ( br g ) -rev g ( p ) . For a contextual demand function f : [0 , 1] × U → [0 , 1] and a context u ∈ U , write proj ( f, u ) for the induced univariate demand function p ↦→ f ( p, u ) . We define rev f ( p, u ) = rev proj ( f,u ) ( p ) , br f ( u ) = br proj ( f,u ) , gap f ( p, u ) = gap proj ( f,u ) ( p ) , and proj ( F , u ) := { proj ( f, u ) : f ∈ F} , along with rev ⋆ := rev f ⋆ , br ⋆ := br f ⋆ , and gap ⋆ := gap f ⋆ . Finally, we define

generalizing the definition in Section 3.1. Regret is defined as the sum of gaps ∑ T t =1 gap ⋆ ( p t , u t ) .

C.2 Generalized OPS and its Regret Guarantee

We now present a extension of OPS and POPS to the generalized setup of Appendix C.1. Both can be recovered for appropriate choices of F and P , which we will discuss later. First, for a univariate demand function g : P → [0 , 1] , price p , and purchase decision y , we define loss

We now adapt OPS to this setting, introducing GOPS (Algorithm 3).

ALGORITHM 3: GOPS : Generalized OPS for Contextual Pricing with Finite Model Class

We prove the following regret bound.

Lemma C.2. Under Setting C.1, with model prior µ 1 ∈ ∆( F ) and optimism strength λ ≥ 4 /T , GOPS (Algorithm 3) achieves regret 25 λ ( dis ( F , f ⋆ ) ∨ 1) T log 2 ( T ) + log(1 /µ 1 ( f ⋆ )) /λ .

Our proof employs the following decoupling lemma.

Lemma C.3 (8) . Let G be a finite family of univariate demand functions and fix g ⋆ ∈ G . Then, for any ν ∈ ∆( G ) and γ > 0 , we have

This is simply Lemma E.2 of [8] with function class { g -g ⋆ : g ∈ G} and ∆ → 0 . In our proof, G and g ⋆ will be the projections of F and f ⋆ onto a fixed context u ∈ U .

The remainder of our analysis is a slight modification to that of [29], which we provide for completeness. For each round t ∈ [ T ] of OPS, we adopt the following notation:

history up to round t : S t := { u τ , f τ , p τ , y τ } t τ =1 ,

true univariate demand function: g ⋆ t := proj ( f ⋆ , u t ) ,

univariate demand function posterior: ν t := proj ( µ t , u t ) := Law f ∼ µ t ( proj ( f, u t )) ,

sampled univariate demand function: g t := proj ( f t , u t ) , so that p t = br g t ,

independently sampled univariate demand function (for analysis): ˜ g t ∼ ν t ,

regret: REG t := rev ⋆ ( br ⋆ ( u t ) , u t ) -rev ⋆ ( p t , u t ) = rev g ⋆ t ( p t ) -rev g ⋆ t ( p t ) ,

least-squares errors: LS t ( g ) := ( g ( p t ) -g ⋆ ( p t )) 2 ,

'feel-good' (optimism) bonuses: FG t ( g ) := rev g ( br g ) -rev g ⋆ t ( br g ⋆ t ) ,

loss discrepancies: ∆ L t ( g ) := ℓ λ ( g, p t , y t ) -ℓ λ ( g ⋆ t , p t , y t ) ,

potential function: Z t := E S t log E f ∼ Unif( F ) exp ( -∑ t τ =1 ∆ L t ( proj ( f, u t )) ) .

Our proof requires several supporting lemmas. The first is a basic concentration result.

Lemma C.4. For c ≥ 0 and a random variable X supported on [0 , 1] , we have log E exp( -cX ) ≤ ( 1 2 c 2 -c ) E X . For X supported on [ a, b ] , we have log E exp( cX ) ≤ c E X + 1 8 ( b -a ) 2 c 2 .

Proof. For the first inequality, we bound

The second inequality is exactly Hoeffding's lemma.

The next lemma mirrors Lemma 4 of [29]. This is a consequence of the definition of ∆ L t and the sub-Gaussianity of its components.

Lemma C.5. For round t of GOPS (Algorithm 3), we have

We note that this lemma does not rely on how p t is selected.

Proof. Let g ∼ ν t and y ∼ Law( y t | u t , p t ) = Ber( g t ( p t )) be independent. Let ε = y -g ⋆ t ( p t ) denote the discrepancy between the observed and expected demand. Since demands lie in [0 , 1] , Lemma C.4 with X = ε gives

Moreover, we have

Combining with (4) gives

Therefore, we have

where the last inequality follows by Hölder's inequality. For the first term, we use Lemma C.4 with X = LS t ( g ) and c = 3 4 to bound

For the second term, we apply the lemma with X = FG t ( g ) and c = 3 λ to obtain

Combining, we have

implying the lemma.

Our last helper lemma mirrors Lemma 5 of [29].

Lemma C.6. For round t of GOPS (Algorithm 3), we have

This lemma also does not rely on how prices are chosen.

Proof. Recall that µ 1 = Unif( F ) . Defining W t ( f | S t ) := exp ( -∑ t τ =1 ∆ L τ ( proj ( f, u τ )) ) , we have Z t = E S t log E f ∼ µ 1 W t ( f | S t ) . Note that

We then bound

where (a) uses Jensen's inequality and (b) uses Lemma C.4. Rearranging gives the lemma.

Now, we return to Lemma C.2, where we will finally incorporate our price selection rule and the decoupling lemma.

Proof of Lemma C.2. For each round t ∈ [ T ] , we recall that p t = br g t and decompose

Conditioning on S t -1 and u t , we apply Lemma C.3 with γ = 1 4 λ to obtain

Taking expectations over S t -1 and u t , we bound

using Lemma C.6. Summing over t ∈ [ T ] and noting that Z 0 = 0 , we bound

Moreover, by realizability, we have

Combining, we obtain

as desired.

C.3 Base Regret Bound for OPS (Proof of Lemma 3.3)

Under the general setup of Appendix C.1, we take F to be the class of demand functions induced by D , set P = [0 , 1] , and fix µ 1 = Unif( F ) . By these choices, GOPS coincides exactly with OPS , as does our notion of regret. Thus, Lemma C.2 gives the desired regret bound of

C.4 Disagreement Coefficient Bound for Non-increasing Functions (Proof of Lemma 3.4)

Fixing f ∈ F , ν ∈ ∆([0 , 1]) , and p ∈ [0 , 1] , suppose that E q ∼ ν [ f ( q ) 2 ] ≤ ε 2 and | f ( p ) | > δ . If f ( p ) > δ , then f ( q ) > δ for all q ≤ p by monotonicity. Thus, P q ∼ ν ( q ≤ p ) δ 2 ≤ E q ∼ ν [ f ( q ) 2 ] ≤ ε 2 . Otherwise, if f ( p ) < -δ , we analogously have P q ∼ ν ( q ≥ p ) δ 2 ≤ E q ∼ ν [ f ( q ) 2 ] ≤ ε 2 . Thus, for ν ∈ ∆( X ) , we have

Plugging this into the definition of dis finishes the proof.

C.5 Disagreement Coefficient Bound for Composite Classes (Proof of Lemma 3.5)

For any distribution ν ∈ ∆( Z ) , write ν i = h i ◦ ν | Z i for law of h i ( p ) when p ∼ ν , conditioned on p ∈ Z i , and let µ ν ( i ) = P p ∼ ν ( p ∈ Z i ) . We then bound

as desired. Here, the first inequality uses that E q ∼ ν [ g ( q ) 2 ] ≥ µ ν ( i ) E q ∼ ν i [ f ( h i ( x )] for some f ∈ F , and the second uses that ν i ∈ ∆( X ) and the definition of dis .

C.6 Base Regret Bound for POPS (Proof of Lemma 3.9)

POPS as generalized OPS. We observe that POPS is an instance of GOPS (Algorithm 3), with discretized price set P = P ε and smoothed demand function class F = { dem ε D : D ∈ D} (where

each f ∈ F is viewed as a function on P ε ×U rather than [0 , 1] ×U ). Here, we view each ˆ p t as the posted price instead of p t . Indeed, taking f ⋆ = dem ε ⋆ , we have

and, for value distribution Q ∈ ∆([0 , 1]) with smoothed demand function g = dem ε Q , we have

where ℓ λ on the right hand size is defined in (3). We do note, however, that the regret benchmark with smoothed demands and discretized prices differs slightly from the original benchmark.

Fixing the regret benchmark. Applying Lemma C.2 for this choice of P and F , we have that

Note that

so the regret bound above measures cumulative revenue in line with our original regret definition. Although the benchmark does not match that in the original definition, we have | rev ε ⋆ ( br ε ⋆ ( u t ) , u t ) -rev ⋆ ( br ⋆ ( u t ) , u t ) | = O ( ε ) for all t ∈ [ T ] due to one-sided Lipschitzness (Lemma 2.1). Consequently, the regret of POPS is bounded by

It remains to bound the disagreement coefficient by O ( K ⋆ ) , giving the lemma. Our argument below mirrors that in the proof of Theorem 3.2 but takes into account the smoothing and discretization.

Bounded disagreement coefficient. For each u ∈ U , the function dem ε ⋆ ( · , u ) with domain P ε is piecewise constant with O ( K ⋆ ) sections. Indeed, the unsmoothed demand function dem ⋆ ( · , u ) is piecewise constant with O ( K ⋆ ) sections, and smoothing can only introduce new sections at the O ( K ⋆ ) prices in P ε that are within distance ε of a previous section boundary. Moreover, smoothing preserves monotonicity of demand functions. Hence, the function classes defining dis ( F , f ⋆ ) are O ( K ⋆ ) -composites of the nonincreasing function class. Applying Lemmas 3.4 and 3.5 then implies that dis ( F , f ⋆ ) = O ( K ⋆ ) , giving the lemma.

C.7 Trajectory Coupling (Proof of Lemma 3.10)

Fix any round t with context u t and best-response price ˆ p t . Since ∥ f ε D -f ε D ⋆ ∥ ∞ ≤ ε , feedback y t coincides with that which would have been obtained if D ⋆ = D with probability at least 1 -ε , conditioned on u t and ˆ p t . Since the update to µ t is only a function of u t , ˆ p t , and y t (notably, not the realized price p t ), we can iterate through all rounds and apply a union bound to obtain the lemma.

C.8 Metric Entropy Bound (Proof of Lemma 3.11)

For part one, fix D,D ′ ∈ ∆(Θ) with d L ( D,D ′ ) ≤ ε 2 / 2 . Then, for all u ∈ U and ˆ p ∈ [0 , 1] , we have

Note that the maximum is unneeded since proj ( D,u ) is supported on [0 , 1] and places no mass on negative values. Writing Q = proj ( D,u ) and Q ′ = proj ( D ′ , u ) , we then have

Writing τ = ε 2 / 2 , we further have

where the last step uses the fact that d L ( Q,Q ′ ) ≤ τ . A symmetric argument gives the reverse bound. Consequently, we have | f ε D (ˆ p, u ) -f ε D ′ (ˆ p, u ) | ≤ 1 ε · 2 τ = ε , as desired.

For part two, let C = { C 1 , C 2 , . . . , C n } denote the intersection of the standard partition of R d into cubes of side length ε/ √ d with Θ ⊆ [0 , 1] d . Denote the lexicographically smallest vertex of each C i by c i , and note that log( n ) = O ( d log( d/ε )) . Given any D ∈ ∆ K (Θ) , we define the initial discretization ˆ D 0 = ∑ n i =1 D ( C i ) δ c i . We obtain the final discretized measure ˆ D by rounding each weight to a neighboring multiple of ε/K (choice doesn't matter so long as we maintain unit mass, this is always possible). Then, for any context u ∈ S d -1 and price p ∈ [0 , 1] , we have

using that each cube in C has diameter ε and that the mass in each cube was perturbed by at most ε/K (so with K cubes the probability of any event is shifted by at most ε ). Thus, d L ( D, ˆ D ) ≤ ε . Using balls and bins, we can thus bound

C.9 Proof of Theorem 3.8, Upper Bound

Upper bound. Fix ε = T -2 and λ = √ d/T . We now construct D and µ 1 . Write M = ⌈ log T ⌉ , and, for i = 1 , . . . , M , take D i to be a minimal ( ε 2 / 2) -cover of ∆ 2 i (Θ) under the Lévy metric. By Lemma 3.11, we have log |D i | = ˜ O (2 i d log 1 /ε ) . Now set D = D 1 ∪ · · · ∪ D M and take µ 1 ( D ) ∝ (2 i |D i | ) -1 for D ∈ D i . This ensures that log(1 /µ 1 ( D )) = ˜ O (2 i d log T ) for D ∈ D i .

Assume without loss of generality that M ≥ log K ⋆ ; otherwise, the regret bound is vacuous. Then, there exists ˆ D ∈ D ⌈ log K ⋆ ⌉ ⊆ D such that d L ( ˆ D,D ⋆ ) ≤ ε 2 / 2 and ∥ dem ε D -dem ε D ⋆ ∥ ∞ ≤ ε , again using Lemma 3.11. Thus, by Lemma 3.10, the realized trajectory of POPS { u t , ˆ p t , p t , y t } T t =1 can be coupled with an alternative trajectory { u ′ t , ˆ p ′ t , p ′ t , y ′ t } T t =1 of POPS with type distribution ˆ D , such that { u t , ˆ p t , y t } T t =1 = { u ′ t , ˆ p ′ t , y ′ t } T t =1 with probability at least 1 -εT . By Lemma 3.9, we have

At this point, we can use the coupling guarantee and the bound d L ( ˆ D,D ⋆ ) ≤ ε 2 to show that the left hand side above and the true expected regret differ by O ( ε 2 T ) = O (1) , giving the theorem.

Specifically, by the coupling guarantee, we have

Since d L ( ˆ D,D ⋆ ) ≤ ε 2 , Lemma 2.2 implies that rev ⋆ ( br ⋆ ( u ′ t ) , u ′ t ) ≤ rev ˆ D ( br ˆ D ( u ′ t ) , u ′ t ) + O ( ε 2 ) for all rounds t . We further have

Combining the above, we obtain

as desired.

C.10 Proof of Theorem 3.8, Lower Bound

Previously, [4] gave a lower bound of Ω( √ K ⋆ T ) for the non-contextual case. We now modify their one-dimensional construction so that it can be cleanly tensored into d dimensions, when K ⋆ ≥ 4 .

One-dimensional construction ( K ⋆ ≥ 4 ). Starting in 1D, we define valuations 1 2 = v 1 ≤ · · · ≤ v K ⋆ = 1 by v i := 1 2 + i -1 4 K ⋆ -2 i -2 . Define the base distribution Q 0 on { v 1 , . . . , v K ⋆ } by

Observe that each valuation v i has the same expected revenue of 1 / 2 . Indeed, we compute

and

Now, for j ∈ { 2 , . . . , K ⋆ -1 } , we define distribution Q j by slightly lowering the probability of v j -1 and increasing the probability of v j by the some small ε > 0 , to be determined. That is, we define

which is well-defined so long as ε ≤ 1 2 K ⋆ -2 . In contrast to the construction in [4], our Q j distributions share the same multiset of probability weights, differing only in the locations of the perturbed valuations. This allows us to tensor these problem instances into a d -dimensional instance without leaking information between instances.

Moreover, the hardness result of [4] is maintained, which applies even if K ⋆ is known to the learner. The proof is quite similar, so we defer it to Appendix C.11.

Lemma C.7 (Regret of one-dimensional family, K ⋆ ≥ 4 ) . Fix any number of types K ⋆ ≥ 4 and T ≥ K 3 ⋆ . Let A be any algorithm for non-contextual pricing. Then, for tuned ε and an instance Q ⋆ drawn uniformly at random from { Q j } j ∈{ 2 ,...,K ⋆ -1 } , A suffers expected regret at least Ω( √ K ⋆ T ) .

Tensoring one-dimensional instances. To extend these one-dimensional distributions into d dimensions, define the base distribution D 0 ∈ ∆([0 , 1] d ) as follows. For i ∈ [ K ⋆ ] , let θ i := [ v i , . . . , v i ] ∈ R d , w i := Q 0 ( v i ) , and take D 0 := ∑ K ⋆ i =1 w i δ θ i . That is, θ i ∈ [0 , 1] d has all entries equal to v i and w i is the probability D 0 places on θ i , taken to match Q 0 at v i . By design, the marginal distribution of D 0 along each coordinate is Q 0 . Now, for a selection j = ( j 1 , . . . , j d ) ∈ { 2 , . . . , K ⋆ -1 } d , define the perturbed instance D j by starting from D 0 and modifying it as follows:

Adjust the probabilities w 1 and w 2 by w 1 ← w 1 -ε and w 2 ← w 2 + ε .

For each dimension ℓ ∈ [ d ] , permute the ℓ th coordinates of the θ i vectors so that the marginal distribution of their ℓ th coordinates coincides with Q j ℓ .

Specifically, we define D j := ∑ K ⋆ i =1 ˜ w i δ ˜ θ j,i where

and ˜ θ j,i [ ℓ ] := v σ ( i ) , for any permutation σ of [ K ⋆ ] such that σ ( j ℓ -1) = 1 , σ ( j ℓ ) = 2 , and σ ( K ⋆ ) = K ⋆ . Simply swapping j ℓ -1 and 1 and j ℓ with 2 works unless j ℓ = 3 , in which case one can send 2 → 1 , 3 → 2 , and 1 → 3 . As claimed above, this construction ensures that for each dimension ℓ ∈ [ d ] , the marginal distribution of D j is Q j ℓ . Indeed, for each i ∈ [ K ⋆ ] , we have

Lower bounding the regret. For the contextual setting, we sample a selection j = ( j 1 , . . . , j d ) ∈ { 2 , . . . , K ⋆ -1 } d uniformly at random and set D ⋆ to the perturbed instance D j . We consider stochastic contexts, where each u t is the standard basis vector along coordinate ℓ t sampled uniformly at random from [ d ] . Now, fix any contextual pricing policy A for this this randomized environment. Our high-level intuition is that, for each coordinate ℓ ∈ [ d ] , the sub-environment during the roughly T/d rounds when ℓ t = ℓ mirrors that of Lemma C.7, and so we incur regret Ω( √ K ⋆ · T/d ) during such rounds. Summing over ℓ ∈ [ d ] then gives the lower bound.

To formalize this, note that each coordinate is sampled at least T ′ = ⌊ T/ 2 d ⌋ times under an event E with probability at least 1 -1 /T . This follows by a Chernoff bound a union bound over coordinates

as desired.

Small K ⋆ . If K ⋆ = 2 , a simpler one-dimensional construction suffices. We set v 1 = 1 / 4 , v 2 = 1 / 2 , Q 0 ( v 1 ) = Q 0 ( v 2 ) = 1 / 2 , so that dem Q 0 ( v 1 ) = 1 , dem Q 0 ( v 1 ) = 1 / 2 , and rev Q 0 ( v 1 ) = rev Q 0 ( v 2 ) = 1 / 4 . We then define Q ± ( v 1 ) = 1 / 2 ∓ ε and Q ± ( v 2 ) = 1 / 2 ± ε , so that rev Q ± ( v 1 ) = 1 / 4 and rev Q ± ( v 2 ) = 1 / 4 ± ε/ 2 . Moreover, taking ε = 1 / √ T (valid so long as T ≥ 4 , which we assumed), we can simply employ the standard 2 -armed bandits lower bound (e.g., using the same techniques as in Appendix C.11) to show that no algorithm can achieve regret o ( √ T ) for both instances. Our argument above, tensoring one-dimensional instances and obtaining a contextual lower bound, still goes through, since Q + and Q -share the same set of probability weights, giving a lower bound of Ω( √ Td ) for a worst-case instance. For K ⋆ = 3 , we can easily tweak the K ⋆ = 2 construction to place negligible mass at v 3 = 0 , and the lower bound still holds.

C.11 Non-contextual Lower Bound (Proof of Lemma C.7)

We first recall some basic information theory. Write KL( p ∥ q ) := E p [log(d p/ d q )] for the KullbackLeibler divergence between distributions p and q on the same domain X . When p and q are Bernoulli distributions with success probabilities a, b ∈ [0 , 1] , we write KL( a ∥ b ) = KL( p ∥ q ) .

Fact C.8 (Pinsker's inequality) . For p, q ∈ ∆( X ) and M ≥ 0 , we have

̸

since T ≥ 8 d log(2 d ) . Then, for each ℓ ∈ [ d ] , there is a natural policy A ℓ induced for the noncontextual setting of Lemma C.7 with time horizon T ′ . To start, A ℓ samples a valuation distribution ˜ Q ℓ ′ uniformly at random from { Q j } j ∈{ 2 ,...,K ⋆ -1 } for each ℓ ′ = ℓ . Further, it instantiates a simulated copy of A and a counter τ , initialized at 1 , tracking the round of this simulation. Then, for round t ′ = 1 , . . . , T ′ , A ℓ performs the following:

If τ > T , play p t ′ = 1 for remaining rounds and terminate.

Otherwise, sample ℓ τ uniformly from [ d ] .

Submit the associated context as u τ to A and receive suggested price ˜ p τ .

If ℓ τ = ℓ , play price p t ′ = ˜ p τ , submit the purchase feedback ˜ y τ = y t ′ to A , increment τ ← τ +1 , and continue to the next round.

Otherwise, submit ˜ y τ ∼ Ber( x ) where x is the demand of ˜ Q ℓ τ at ˜ p τ , increment τ ← τ +1 , and return to Step 1.

By design, if A ℓ is run under the setting of Lemma C.7, its simulated copy of A experiences feedback indistinguishable from that described in our setting above. Thus, writing T ℓ for the rounds of our initial setting where ℓ t = ℓ and conditioning on E (under which |T ℓ | ≥ T ′ ), we have

since T ′ ≥ K 3 ⋆ and T ≥ 2 d . We then compute

Fact C.9 (Bernoulli KL bound) . For a, δ ∈ [0 , 1] such that a + δ ∈ (0 , 1) , we have

Fact C.10 (KL chain rule) . For distributions p, q over sequences X n =( X 1 , . . . , X n ) ∈X n ,

where p ( X i | X 1 , . . . , X i -1 ) denotes the conditional distribution of X i under p given X 1 , . . . , X i -1 .

̸

Next, we observe that, if the buyer valuations are drawn from Q j , the price v j achieves expected revenue at least 1 / 2 + Ω( ε ) whereas all other v i = v j achieve revenue 1 / 2 . Indeed, the demand function of Q j is identical to Q 0 at all v i = v j , and at v j we have

̸

Thus, gap Q j ( v i ) ≥ ε/ 2 · 1 { i = j } . Also, for i > 1 , Eq. (5) implies that 1 / 2 ≤ dem Q 0 ( v i ) ≤ 5 / 6 . So, imposing that our perturbation size ε is less than 1 / 12 , we have that

for all i, j ∈ { 2 , . . . , K ⋆ -1 } .

̸

Now, let J be drawn uniformly at random from { 2 , . . . , K ⋆ -1 } , so that Q J coincides with the random instance from the lemma statement. Conditioned on J , let the valuations V T = ( V 1 , . . . , V T ) be drawn i.i.d. from Q J , i.e., V T ∼ Q T J . Under this set up, v J is the unique optimal price, and playing v i for i = J incurs regret Ω( ε ) . We will be comparing to the alternative world where V T ∼ Q T 0 and all arms have equal expected revenue of 1 / 2 . All expectations under this alternative will be clearly denoted as such.

Without loss of generality, we assume that the fixed pricing algorithm A is deterministic (since we may condition on any internal randomness of A ). Applying A to instance Q J induces a random sequence of prices P 1 , . . . , P T , where P t each is a function of the previous purchase decisions Y τ = 1 { V τ ≥ P τ } for τ ∈ [ t -1] . Without loss of generality, we may assume that each P t belongs to { v 2 , . . . , v K ⋆ -1 } , since rounding up to the nearest element of this set can only increase expected revenue. Thus, defining the empirical frequencies

we have ∑ K ⋆ -1 j =2 N j = T . Conditioned on J , the regret of A over the T rounds is

̸

Thus, we have in expectation over J and V T that

Wenext control E J,V T [ N J ] using the KL chain rule and Pinsker's inequality. Fixing j ∈ { 2 , . . . , K ⋆ -1 } , write q j ( · ) and q 0 ( · ) for the induced distributions on the entire purchase sequence Y T = ( Y 1 , . . . , Y T ) under Q j and Q 0 , respectively. By the chain rule, we compute

In short, q 0 and q j are only distinguishable for rounds in which A selects price v j , and even in these rounds their divergence is bounded. Now, since N j is a deterministic function of Y T for fixed A ,

Taking an expectation over J then gives

Since ∑ K ⋆ -1 j =2 N j = T and J is uniform over { 2 , . . . , K ⋆ -1 } , K ⋆ ≥ 4 , it follows that

Plugging this into (7) gives

We now set ε = 1 3 √ 24 √ ( K ⋆ -2) /T . Our construction required that ε ≤ 1 / (2 K ⋆ -2) and ε ≤ 1 / 12 , which are satisfied under our assumption that T ≥ K 3 ⋆ . This yields a final lower bound of

as desired.

D Proofs for Section 4

To fully specify the algorithm, we introduce the following notation:

Since our analysis exclusively reasons about the true type distribution D ⋆ , we abbreviate dem = dem ⋆ , rev = rev ⋆ , gap = gap ⋆ , and br = br ⋆ .

For each price p ∈ [0 , 1] and round t ∈ [ T ] , we write

T t ( p ) := { τ ∈ [ t -1] : p τ = p } for the set of previous rounds where p was played,

n t ( p ) := |T t ( p ) | for the count of these rounds,

µ t ( p ) := 1 n t ( p ) ∑ τ ∈T t ( p ) py t for the average revenue during these rounds,

V t ( p ) := 1 n t ( p ) -1 ∑ τ ∈T t ( p ) ( py t -µ t ( p )) 2 for the sample variance, and

σ 2 ( p ) := p 2 dem ( p )(1 -dem ( p )) for the population variance (unknown to the seller).

When n t ( p ) = 0 , we set µ t ( p ) = 0 = V t ( p ) = 0 . When n t ( p ) = 1 , take V t ( p ) = ∞ .

Defining confidence radius r t ( p ) := √ 10 V t ( p ) log T n t ( p ) + 12 log( T ) n t ( p ) -1 (taken as + ∞ if n t ( p ) ≤ 1 ), a variant of Bernstein's inequality shows that | µ t ( p ) -rev ( p ) | ≤ r t ( p ) w.h.p. (see Lemma D.3).

ALGORITHM 4: ZoomV : Variance-Aware Zooming for Non-Contextual Pricing

Write UCB t ( p ) := µ t ( p ) + r t ( p ) , so that rev ( p ) ≤ UCB t ( p ) w.h.p.

We say a price p is covered by q ∈ S if p ∈ [ q, q + r t ( q )] and q is the largest active price no greater than p , i.e., q = max { q ′ ∈ S : q ′ ≤ p } . One-sided Lipschitzness of the revenue function (Lemma 2.1) implies that rev ( p ) -rev ( q ) ≤ r t ( q ) w.h.p.

Define the index of a price q ∈ S as index t ( q ) := UCB t ( q ) + r t ( q ) . Each price p covered by some q ∈ S satisfies rev ( p ) ≤ index t ( q ) .

D.1 Main Regret Bound for ZoomV (Proof of Theorem 4.1)

As mentioned, the lower bound follows by that in Theorem 3.8 when d = 1 . For the upper bound, we prove a generic regret bound in Appendix D.2 depending on the variance-aware zooming dimension.

Next, we prove the zooming dimension bounds claimed in Section 4.

Lemma D.2. We have ZoomDimV (10 K ⋆ ) = 0 and ZoomDimV (10) ≤ ZoomDim (10) ≤ 1 .

Proof. For each δ > 0 and type i ∈ [ K ⋆ ] , let X ( i ) δ denote the set of activated arms p with gap ⋆ ( p ) ≤ δ that lie in the interval ( θ ( i -1) , θ ( i ) ] , to the left of type i . Since revenue is linearly increasing within each such interval, with slope d i = dem ( θ ( i ) ) , the gap condition requires that each p ∈ X ( i ) δ also satisfies p ≥ θ ( i ) -δd -1 i . Moreover, for p ∈ X ( i ) δ , we have σ 2 ( p ) ≤ d i . Thus, we obtain

implying the first bound. For the second, we note that N ( X δ , δ/ 20) ≤ N ([0 , 1] , δ/ 20) ≤ 20 δ -1 .

Combining the two lemmas gives the theorem. Indeed, the √ K ⋆ T bound follows by Lemma D.1 with c = 10 K ⋆ and z = 0 , using Lemma D.2. Similarly, the T 2 / 3 bound follows by taking c = 10 and z = 1 .

D.2 Base Regret Bound for ZoomV (Proof of Lemma D.1)

We begin with a few helper lemmas. Throughout, we assume that T ≥ 3 (otherwise the regret bound holds trivially). Our proofs mirror those of similar lemmas in [27], with small adjustments to handle the variance-adjusted confidence radii and the dyadic price selection rule.

Lemma D.3 (Concentration) . Write E clean for the event that

for all t ∈ [ T ] and for all p ∈ [0 , 1] . Then P ( E clean ) ≥ 1 -8 T -2 .

Proof. For fixed p ∈ [0 , 1] and t ∈ [ T ] , Theorems 10 and 11 of Maurer and Pontil [20] imply that

with probability 1 -8 T -5 . We note that similar bounds appear in [2], which inspired our adjustments to the confidence intervals. Under this event, we further bound

Taking a union bound over t , the above must hold for all t ∈ [ T ] with probability at least 1 -8 T -4 . Now, the same Chernoff bound argument used in Claim 4.13 of Slivkins et al. [27] implies that | µ t ( p ) -rev ( p ) | ≤ r t ( p ) for all p ∈ [0 , 1] and t ∈ [ T ] with probability at least 1 -8 T -2 . One technical observation is that Claim 4.13 requires that the set of arms every played by the algorithm is finite. This holds for ZoomV due to our dyadic arm activation rule.

Lemma D.4 (Covering invariant) . At the beginning of each round, every price p ≥ 1 /T is covered by some active arm.

Proof. At round 1, r t ( q ) = ∞ for all active arms q ∈ S , and so all arms larger than 1 /T are covered by our choice of S . Now, suppose that the lemma holds up to round t , and that playing p t causes a price p ∈ R to become uncovered. Then, by the definition of covering, we must have

where q is the nearest active price to the right of p t , selected at Step 6. First, verify that the added price, p ′ := ( p t + q ) / 2 , is less than p . Since r t +1 ( p t ) must be less than one, we must have n t +1 ( p ) -1 = n t ( p ) ≥ 12 . Thus, we can bound

Consequently, one can show that

Combining, we find that p ′ = ( p t + q ) / 2 ≤ p t + 1 2 r t ( p t ) < p t + r t +1 ( p t ) < p , as desired. Moreover, p ′ could not have already been active at round t ; otherwise, p t would not have been covering p . Finally, once p ′ is added, it covers p since r t +1 ( p ′ ) = ∞ .

Lemma D.5 (Gap bound) . Condition on E clean . Then gap ( p ) ≤ 5 r t ( p ) and n t ( p ) ≤ 252 σ 2 ( p ) log T gap ( p ) 2 + 504 log T gap ( p ) for all p ∈ [0 , 1] and t ∈ [ T ] .

Proof. Write ˆ p = max { br , 1 T } , and fix any price p ∈ [0 , 1] . Consider some round t at which p is played. By Lemma D.4, we know that ˆ p was covered at the beginning of round t by some price q ∈ S , and that p had a higher index than q . Hence,

Moreover, index t ( p ) ≤ µ t ( p ) + 2 r t ( p ) ≤ rev ( p ) + 3 r t ( p ) . Thus,

Now, if n t ( p ) ≤ 12 , gap ( p ) ≤ 1 < r t +1 ( p ) . Otherwise, the bound above implies that gap ( p ) ≤ 5 r t +1 ( p ) . Since r t ( p ) only changes when p is played and gap ( p ) ≤ 5 r t ( p ) when t = 1 , this guarantee holds for all t . For the other bound, we apply concentration to obtain

Rearranging and solving the quadratic inequality in n t ( p ) gives the stated result.

Compared to the standard gap bound lemma for zooming (see, e.g., Lemma 4.14 of 27), Lemma D.5 is adapted to the variance of each price p . We can now prove the regret bound.

Lemma D.6 (Active arm separation) . Conditioned on E clean , consider any three consecutive active arms x < y < z which did not belong to S at initialization. Then z -x > 1 10 min { gap ( x ) , gap ( y ) } .

Proof. If y was activated before z , then z must have been added as the midpoint of active arms y and y +2( z -y ) = 2 z -y at round τ z , when y must have not covered 2 z -y . Thus, by Lemma D.5, we would have 2( z -x ) > 2( z -y ) = (2 z -y ) -y > 1 5 gap ( y ) . On the other hand, if y was activated after z , then it must have been added as the midpoint of x and z at round τ y , when x must have not covered z . Again, by Lemma D.5, this would imply z -x > 1 5 ∆( y ) .

Proof of Lemma D.1 We freely condition on E clean , since the complement has negligible probability O ( T -2 ) . For each δ > 0 , let Y δ ⊆ X 2 δ denote the set of activated prices p with gap ( p ) ∈ [ δ, 2 δ ) . In what follows, we say that two prices are adjacent if they are neighboring within Y δ . Note that at most O (log T ) of the prices in Y δ were activated at initialization. Consider the set Y 0 δ which, for each such price, contains this price and up to two neighboring prices, such that the remaining prices Y δ \ Y 0 δ can be split into triples of neighboring prices. We then decompose

where p i, 1 , p i, 2 , p i, 3 are neighboring for each i ∈ [ n ] . By Lemma D.6, we have p i, 3 -p i, 1 > δ/ 10 for all i ∈ [ n ] , and so p i,k -p j,k > δ/ 10 for all k ∈ [3] whenever i < j -1 . Thus, we can partition Y δ \ Y 0 δ into at most 6 packings, each of which has separation at least δ/ 10 . Of course any ( δ/ 10) -packing of Y δ ⊆ X 2 δ is contained within a ( δ/ 5) -cover of X 2 δ . Consequently, we have

Noting that a ( δ/ 10) -packing within [0 , 1] can have cardinality at most 10 δ -1 , we further bound | Y δ | = O (log T + δ -1 ) . Thus, by Lemma D.5, the regret incurred due to posting prices in Y δ is at most

Now we sum over δ = 1 / 2 , 1 / 4 , . . . , α , where α will be tuned later, giving a total regret bound of

Taking α = ( c log( T ) /T ) 1 / (2+ z ) , we obtain the desired bound of ˜ O ( c 1 / (2+ z ) T 1 -1 / (2+ z ) ) .

E Proofs for Section 5

We first recall some results from Vapnik-Chervonenkis (VC) theory. Given distributions D,D ′ over a finite domain X , define the total variation distance ∥ D -D ′ ∥ TV := sup A ⊆X | D ( A ) -D ′ ( A ) | .

Lemma E.1 (Section 28.1 of 25) . Fix a finite set X , a function family F ⊆ { 0 , 1 } X , a distribution D ∈ ∆( X ) , and δ ∈ (0 , 1) . Then, for X 1 , . . . , X n sampled i.i.d. from D , we have

with probability at least 1 -δ , where V is the VC dimension of F .

Lemma E.2 (Theorem 9.3 of 25) . The family F = { 0 , 1 } X has VC dimension |X| , and the family of linear thresholds over R d has VC dimension d +1 . The former result implies that, under the setting above, we have

with probability at least 1 -δ .

We also recall Bernstein's inequality for the case of i.i.d. Bernoulli random variables.

Lemma E.3 (Theorem 2.10 of 3) . For i.i.d. X 1 , . . . , X n ∼ Ber( p ) and δ > 0 ,

with probability at least 1 -δ .

We now turn to the main proofs.

E.1 Observed Type Identifiers (Proof of Theorem 5.1)

To state our result for the first setting, we introduce an ε -ball performance metric for contextual search. This is a slight strengthening of the standard ε -ball metric ∑ T t =1 1 {| p t -v t | > ε } .

Definition E.4 (Strong ε -ball regret) . Let A be a contextual pricing policy which, at round t ∈ [ T ] with context u t , outputs a price p t ∈ [0 , 1] along with a confidence width w t ∈ [0 , 1] . For ε ∈ (0 , 1] , we say that A achieves strong ε -ball regret R ( T ) for contextual search if, when K ⋆ = 1 and D ⋆ = δ θ ⋆ , we have | p t -v t | ≤ w t for each round t and ∑ T t =1 1 { w t > ε } ≤ R ( T ) , where v t = ⟨ u t , θ ⋆ ⟩ .

That is, A produces ε -accurate estimates for the true values, outside of up to R ( T ) rounds for exploration, and it can identify when these estimates are accurate. In practice, this tends to require that A maintain a confidence set around θ ⋆ whose width, when projected onto the current context, is greater than ε for at most R ( T ) rounds. Fortunately, there are existing efficient algorithms which achieve low ε -ball regret.

Lemma E.5 (18) . For ε ∈ (0 , 1] , there exists a contextual search algorithm ProjectedVolume ( ε ) , based on the ellipsoid method, with strong ε -ball regret O ( d log( d/ε )) and running time poly( d, 1 /ε ) per round. 2

We now present our algorithm (Algorithm 5), which uses ProjectedVolume as a subroutine.

Overview of Algorithm 5 We maintain a set I of observed types, initially empty, and an accuracy ε (tuned to minimize regret). We will initialize an independent copy A i of ProjectedVolume for each i added to I . Since these copies are simulated, we are free to query the price price ( A i , u t ) and confidence width width ( A i , u t ) for a context u t without updating A i . Moreover, for each i ∈ I , the algorithm maintains a frequency count n t ( i ) , recording the number of rounds which we have followed the recommended price of A i , along with an exploration count m t ( i ) , recording the number of rounds which we have played the price of A i due to its lack of confidence along the current context. At each round t , we perform the following:

If there is an observed type i ∈ I such that width ( A i , u t ) > ε and that its number of exploration plays m t ( i ) is below a threshold of εT/K ⋆ , the algorithm plays price ( A i , u t ) , observes the outcome and the realized type z t , and updates A i if z t = i . In addition, we increment n t ( i ) and m t ( i ) .

Otherwise, it defines active set S = { i ∈ I : width ( A i , u t ) ≤ ε } , computes for each i ∈ S the score

and plays p t = max { price ( A i ∗ , u t ) -ε, 0 } where i ∗ ∈ arg max i ∈S { F ( i ) · price ( A i , u t ) } . It then observes y t and z t and updates the frequency count n t +1 ( z t ) . Here, F is an estimate for the demand at the price suggested by A i , and so i ⋆ is an estimate for the revenue maximizing type. We pull back the price recommended by A i ⋆ by ε to avoid issues due to estimation error.

Bounding exploration regret. Write T 1 for the set of exploration rounds. By design, an exploration round is one in which some type i is used with width ( A i , u t ) > ε and exploration counter satisfying m t ( i ) < εT . Trivially, |T 1 | ≤ εT , so we can incur regret at most εT = ˜ O ( √ dT ) during exploration.

Bounding mass of types which saturate exploration threshold. Next, consider any type i that has been explored sufficiently so that m T ( i ) = εT after time T ; denote by S ′ the set of all such types. We will show that D ⋆ plays small mass on S ′ . Fix i ∈ S ′ and write T 1 ,i for the exploration rounds where we follow A i . Conditioned on T 1 ,i , we note that X t = 1 { z t = i } , t ∈ T 1 ,i , are i.i.d. Bernoulli random variables with Pr( X t = 1) = D ⋆ ( i ) . Defining

our guarantee for ProjectedVolume (Lemma E.5) and the width condition for exploration imply that S i = O ( d log( d/ε )) . On the other hand, by Bernstein's inequality (Lemma E.3), we have

with probability at least 1 -1 /T . Since m T ( i ) = εT = √ dT log( T ) , the dominant term is m T ( i ) D ⋆ ( i ) for T greater than a sufficiently large constant. Thus, we deduce that

Summing over all types in S ′ (of which there are at most K ⋆ ) and taking a union bound, the total mass in S ′ is at most

with probability at least 1 -K ⋆ /T . We condition on this bound holding for the remainder of the proof, since doing so contributes a negligible K ⋆ to the regret. We also condition on the event that, for each round t ∈ [ T ] the empirical frequencies of (all) types deviate from their true masses by at most O (√ K ⋆ log( T ) /t ) in total variation. This is permissible by Lemma E.2 and a union bound over rounds.

Bounding exploitation regret. Fix an exploitation round t , and recall the set of accurately estimated types

Write v 1 , . . . , v K ⋆ ∈ [0 , 1] for the true values at round t . By our construction and the ε -ball guarantee for ProjectedVolume , A i returns a price that is an ε -accurate estimate of v i , for each i ∈ S . Moreover, by our analysis above, the mass on types outside of S is quite small. We thus bound

Consequently, we bound rev ⋆ ( p t , u t ) + ˜ O ( K ⋆ √ d log( T ) /t ) from below by

or equivalently,

All together, we see that playing p t incurs regret at most ˜ O ( K ⋆ √ d log( T ) /t ) . Summing over exploitation rounds and adding the exploration regret gives a total bound of

as desired.

E.2 Observed Type Vectors (Proof of Theorem 5.2)

We first show that rev ˆ D τ concentrates tightly around rev ⋆ , using a simple VC bound.

Lemma E.6. Fix D ∈ ∆ K (Θ) and let ˆ D t be the empirical measure of t i.i.d. samples from D . We then have

with probability at least 1 -δ .

Proof. We compute

where F is the space of linear threshold functions f p,u : supp( D ) → { 0 , 1 } given by f p,u ( θ ) = 1 {⟨ u, θ ⟩ ≥ p } . The result then follows by Lemma E.2.

Now, our best response policy ensures that, at each round t > 1 , we have

Consequently, regret is at most

as desired.

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: We summarize our key contributions and results, which all have proofs.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [NA]

Justification: This is a theoretical work and we point out all necessary assumptions for the applicability of the results

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate "Limitations" section in their paper.

The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: We provide the full set of assumptions and a complete proof.

Guidelines:

The answer NA means that the paper does not include theoretical results.

All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.

All assumptions should be clearly stated or referenced in the statement of any theorems.

The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.

Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.

Theorems and Lemmas that the proof relies upon should be properly referenced.

4. Experimental result reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [NA]

Justification: The paper does not contain any experimental work.

Guidelines:

The answer NA means that the paper does not include experiments.

If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.

Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.

While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example

(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.

(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).

(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [NA]

Justification: The paper does not contain any experimental work.

Guidelines:

The answer NA means that paper does not include experiments requiring code.

Please see the NeurIPS code and data submission guidelines ( https://nips.cc/ public/guides/CodeSubmissionPolicy ) for more details.

While we encourage the release of code and data, we understand that this might not be possible, so 'No' is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).

The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines ( https: //nips.cc/public/guides/CodeSubmissionPolicy ) for more details.

The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.

The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).

Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

6. Experimental setting/details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [NA]

Justification: The paper does not contain any experimental work.

Guidelines:

The answer NA means that the paper does not include experiments.

The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment statistical significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [NA]

Justification: The paper does not contain any experimental work.

Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)

The assumptions made should be given (e.g., Normally distributed errors).

It should be clear whether the error bar is the standard deviation or the standard error of the mean.

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments compute resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [NA]

Justification: The paper does not contain any experimental work.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

9. Code of ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: We abide by the Code of Ethics.

Guidelines:

The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [NA]

Justification: This is a theoretical work without any clear and immediate social impact.

Guidelines:

The answer NA means that there is no societal impact of the work performed.

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to

generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: The paper poses no such risks.

Guidelines:

The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.

We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [NA]

Justification: The paper does not use existing assets.

Guidelines:

The answer NA means that the paper does not use existing assets.

The authors should cite the original paper that produced the code package or dataset.

The authors should state which version of the asset is used and, if possible, include a URL.

The name of the license (e.g., CC-BY 4.0) should be included for each asset.

For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.

If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

If this information is not available online, the authors are encouraged to reach out to the asset's creators.

13. New assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [NA]

Justification: The paper does not release new assets.

Guidelines:

The answer NA means that the paper does not release new assets.

Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.

The paper should discuss whether and how consent was obtained from people whose asset is used.

At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

14. Crowdsourcing and research with human subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: The paper does not release new assets

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.

According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.

15. Institutional review board (IRB) approvals or equivalent for research with human subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: The paper does not involve crowdsourcing nor research with human subjects.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.

We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.

For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

16. Declaration of LLM usage

Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.

Answer: [NA]

Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components.

Guidelines:

The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.

Please refer to our LLM policy ( https://neurips.cc/Conferences/2025/LLM ) for what should or should not be described.

Peer reviews:
comments: SUMMARY:
In this paper the authors propose algorithms for posting prices when buyers have unknown, diverse preferences and the seller sees some buyer-specific context. It learns these preferences over time to make better pricing decisions. The methods work both when the number of buyer types is known or unknown and adapt to different feedback levels, achieving near-optimal performance in all cases.

STRENGTHS AND WEAKNESSES:
To me, this paper makes an interesting contribution to the community working in this field, and I cannot identify any real weaknesses. The algorithms are simple, and the analysis is non-trivial. Furthermore, I believe that modifying the zooming algorithm to achieve tighter results in the presence of one-sided Lipschitzness is an interesting idea. As a minor suggestion, the authors could include a table summarizing all the notation, as it is quite annoying to search for symbols across different pages.

QUESTIONS:
Can these techniques be extended to different (and more general) settings? Can the author provide a short discussion on this point?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-23

comments: SUMMARY:
This paper studies the contextual dynamic pricing problem. 
In this problem, in each round, an unknown type $\theta_t$ is drawn from an unknown distribution, and a context $u_t$ is adversarially chosen. 
The value is then $v_t = \theta_t \cdot u_t$, and the agent should pick a price $p_t$. 
The problem is to maximize the total revenue. 
The paper proposes an algorithm with a regret of $\tilde{O}(\sqrt{K_* T})$ when $K_*$ is known, and another algorithm with regret $\tilde{O}(K_*\sqrt{T})$ when $K_*$ is unknown. 
Here $K_*$ is the support size of the type distribution. 
The paper also considers the non-contextual case and the model with ex-post type observations and presents corresponding algorithms.

STRENGTHS AND WEAKNESSES:
Overall, the results given in this paper are interesting. 
Nevertheless, I think the paper has a lot to improve in its presentation. 
In general, the paper is written in a too technical and notation-heavy manner with many technical definitions, while the intuitions of the given algorithms/analysis are lacking. 
This makes the paper hard to follow, and also makes it hard to evaluate the contributions of this work. 
For example, for Section 3, I highly recommend that the authors put technical details into the appendix and present more intuition on the presented analysis (e.g., the disagreement coefficient), and address how the underlying techniques are related to previous works. 
In Sections 4 and 5, since the two algorithms discussed are not even given in the main body, the pure analysis seems too dry and hard to follow.

QUESTIONS:
In section 3.2, what do you mean by "take $\mathcal D$ to be a large, finite cover of the full distribution space"? 
What is $\mathcal D$ formally?

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 2
IMPACT: 3
ORIGINALITY: 2
DATE: 2025-06-23

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the well-established problem of contextual dynamic pricing, assuming the existence of several buyers. Formally, the valuation $v_t$ at every round is a linear function of a context, known by the agent, and a vector of coefficients $\\theta_t$, drawn in an i.i.d. fashion with respect to the distribution $D$ with finite support, in which each element represents a different buyer. By employing a Thompson Sampling-inspired algorithm, the paper manages to bound the expected regret by $K_{\\star}\\sqrt{dT}$ where $K_{\\star}$ is the number of buyers, $d$ the dimension of the context space and $T$ the number of rounds. Concurrently, they prove a lower bound for the same model of the order of $\\sqrt{K_{\\star}dT}$. The question of which is the right dependency with respect to $K_{\\star}$ is answered in the paper for the non-contextual case, for which a bound of the order of $\\mathcal{O}(\\min\\{\\sqrt{K_{\\star}T}, T^{2/3}\\})$ is obtained. Lastly, they provide some additional results for the case in which the learner can rely on some additional information, specifically, after every round, it has access to either the type of the buyer or the whole parameter vector $\\theta_t$.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper investigates the well-known problem of dynamic pricing, focusing on the less-studied setting of multiple buyers, therefore the results are novel and of interest to the community. Furthermore, the paper is written in a clear and precise way where all the assumptions needed are clearly stated and the results, together with the steps to achieve them, are well-motivated and easy to follow.   \n\nThe weaknesses of this work are the theoretical assumptions posed, which represent a limitation to the applicability of the results stated. Specifically, the setting analysed by this paper only considers the case in which the number of buyers is finite, by assuming that the distribution with which the vectors $\\theta_t$ are drawn has bounded support.\n\nI also want to point out that in [1], they also study the problem of contextual dynamic pricing with multiple buyers. While the model they consider is different and therefore the results they obtain are not comparable, I think it should be mentioned nonetheless.\n\n---\n[1]  Contextual Pricing for Lipschitz Buyers, J. Mao et al., NeurIPS 2018\n\nQUESTIONS:\n1.  On the line of what I stated in the Strengths and Weaknesses section, I wonder if the assumption on the boundedness of support of the buyers could be relaxed, for instance, assuming that only a finite amount of them are chosen with probability bigger than a threshold or assuming concentrated zones of buyers for which the corresponding parameters $\\theta$ are similar. \n\n\n2. The paper leaves the question of which is the correct dependency on $K_{\\star}$ in the general case open. Do you think it is possible to generalize the analysis of Section 4 to the contextual case, or do you have any insights about what the right rate could be?\n\nLIMITATIONS:\nThe limitations have been addressed.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers the contextual pricing setting with a twist: the incoming agents/buyers have heterogeneous types that are sampled from an unknown distribution with support $K^* $. Prior works considered the buyers to have fixed but unknown type. Clearly, the optimal context-dependent pricing strategy depends on knowing the unknown type distribution (this is the offline optimal against which regret is defined). The work proposes algorithms for this setting based optimal posterior sampling. When the distribution is known to be from a given set, and $K^* $ is known, a $\\tilde{O}(K^*\\sqrt{dT})$ bound is given (Theorem 3.2) that is tight upto the $K$ factor (lower bound is for $\\sqrt{K}$). This guarantees are maintained when assumptions about knowing $K$ and distribution set are relaxed to an extent. The work also focuses on tightening the bound for $K$ for non-contextual settings (via zooming) and the setting where agent types are observed after each round (called ex-post observability)\n\nSTRENGTHS AND WEAKNESSES:\nOverall I am in the middle with this paper\n\n### Strengths\nThe overall model of the paper is fairly attractive and natural. Agents having heterogenous types from an unknown distribution is very natural and standard in many dynamic pricing settings. The paper also answers the natural questions of learning under both type feedback (ex-post observability) and just realized actions. The technical work in the paper is fairly competent. \n\n### Weaknesses\nMy main criticism of the work is it seems to not mention or hand-wave some important claims. There's also parts of the work that seem a bit unfocused. I expand below.\n\nIn section 3.2, you assume that $\\mathcal{D}$ is not known but the algorithm uses a a finite large cover of the whole distribution space $\\Delta(|\\Theta|)$. This clearly is an epsilon cover, the size of which scales exponentially in d. So it is not clear to me whether the algorithm 2 is computationally efficient/feasible since we need to sample and update from this very large cover. Further, this also relies on the support size being discrete - this isn't a big issue, but it inconsistent with line 150 where you claim that $K^* $ may be infinite. I don't see any results in the paper that support this. \n\nI also find the results on the non-contextual setting that leverages the zooming algorithm also a bit unfocused. The work is ostensibly about the contextual pricing setting, so it's uncertain what considering the non-contextual setting as a major section in the work achieves. This is akin to pricing a single item to a heterogenous set of buyers that arrive online - there have been relevant works here.\n\nQUESTIONS:\nSee above.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies dynamic pricing policy optimization in the presence of $K^\\star$ heterogeneous buyers, each associated with an unknown parameter $\\theta_t$. The authors build upon the recently proposed Optimistic Posterior Sampling (OPS) approach to develop their algorithm. They establish a near-optimal regret bound of $O(K^\\star \\sqrt{dT})$, which is optimal in both the dimension $d$ and the time horizon $T$. Furthermore, they derive a sharper upper bound of $O(\\sqrt{K^\\star T})$ in the special case where covariates are not involved.\n\nSTRENGTHS AND WEAKNESSES:\nStrength: This paper explores a novel and important setting in dynamic pricing, where buyers are heterogeneous and not drawn from a single population. The authors propose a new algorithm along with theoretical guarantees that are near-optimal, making a valuable contribution to the literature.\n\nWeakness: The writing could be improved for clarity. In particular, the paper introduces many non-standard notations, which would benefit from clearer explanations. It would be helpful if each assumption or definition were followed by an intuitive explanation or example to aid reader understanding.\n\nQUESTIONS:\n1. The authors model the buyer's valuation as $v_t = \\mathbf{u}_t^\\top \\theta_t $, implying that the customer's value is fully determined by the observed covariates. It would be more realistic and flexible to include a noise term in this formulation—for example, $v_t = \\mathbf{u}_t^\\top \\theta_t + Z_t $, where $Z_t $ is an i.i.d. random noise term. This would better capture potential randomness in customer behavior not explained by covariates.\n\n2. As mentioned in the weakness section, the clarity of the paper could be significantly improved by refining the notation. For example, the function $\\mathrm{proj}(\\cdot)$ is defined but not clearly explained, and some notations like $\\mathrm{dem}_Q$ may be confusing to readers. Additionally, when citing results or introducing new concepts, it would be helpful to provide more detailed explanations or context. Without this, readers may find it difficult to follow the technical developments.\n\n3. It would be beneficial to include more real data experiments to demonstrate the effectiveness of this proposed method in reality and compare it with existing methods.\n\nLIMITATIONS:\nSee the questions.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-04"}