Learning to price with resource constraints: from full information to machine-learned prices
Authors: Ruicheng Ao
Jiashuo Jiang
David Simchi-Levi
Venue: NeurIPS 2025 poster ()
Keywords: online learning
dynamic pricing
resource constraint
informed price
Primary area: general_machine_learning
TL;DR: dynamic pricing with misspecified information and degenerate resource constraints

Abstract: Dynamic pricing with resource constraints is a critical challenge in online learning, requiring a delicate balance between exploring unknown demand patterns and exploiting known information to maximize revenue. We propose three tailored algorithms to address this problem across varying levels of prior knowledge: (1) a Boundary Attracted Re-solve Method for the full information setting, achieving logarithmic regret without the restrictive non-degeneracy condition; (2) an online learning algorithm for the no information setting, delivering an optimal $O(\sqrt{T})$ regret; and (3) an estimate-then-select re-solve algorithm for the informed price setting, leveraging machine-learned prices with known error bounds to bridge the gap between full and no information scenarios. Moreover, through numerical experiments, we demonstrate the robustness and practical applicability of our approaches. This work advances dynamic pricing by offering scalable solutions that adapt to diverse informational contexts while relaxing classical assumptions.

Learning to Price with Resource Constraints: From Full Information to Machine-Learned Prices
Ruicheng Ao MIT
Jiashuo Jiang HKUST jianguust.o
David Simehi-Levi MIT dslevi@mits edu
aorc@mit. edu
Abstract
Dynamic pricing with resource constraints is a critical challenge in online learning, requiring a delicate balance between exploring unknown demand patterns and exploiting known information to maximize revenue. We propose three tailored algorithms address this problem across varying levels of prior knowledge: (I) a Boundary Attracted Re-solve Method for the full information setting, achieving OlogTT) regret without imposing additional conditions; (2) an online learning algorithm for the no information setting, delivering an optimal O(VT) regret; and (3) an shimate-then-sellee re-solve algorithm for the informed price setting, lever- aging machine-leamen
prices with known error bounds to bridge the gap between full and information scenarios. Moreover, through numerical experiments, we demonstrate the robustness and practical applicability of our approaches. This work advances dynamic pricing by offering scalable solutions that adapt diverse informational contexts while relaxing classical assumptions.
Introduction
Dynamic pricing is classical problem in online learning and decision-making With the growth of e-commerce, there has been an increasing focus on the development of efficient dynamic pricing policies in the recent literature, [2019], Shah et al. [2019], Cohen et al. [2020], Xu and Wang [2021], Bu et al. [2022], Xu and Wang [2022), Xu al. [2025].
for example Amin et al. [2014] 22014,, Javanmard and Nazerzadeh
Dynamic pricing is challenging due to the unknown relationship between the posted price and the corresponding demand. Usually, one model the dynamic pricing problem bandit problem to balance the exploration of the price-demand relationship and the exploitation of setting the optimal price, where the price option is regarded as price) is regarded as the reward for the arm. However, different from the classical model where there is finite number of arms, the price option can be infinite and continuous, which brings significant challenge to the learning part.
and the corresponding revenue (demand multiplied
Another challenge to dynamic pricing is the existence of resource constraints. In practice, a product usually enjoys fixed initial inventory to sold during the horizon and the pricing decision made to balance the demand such that the total revenue maximized subject to the resource constraints. Note that the existence of resource constraints will significantly complicate the problem because the optimal decision is no longer to maximize the revenue but to balance the revenue versus resource consumption in an optimal way. Past work on dynamic pricing mainly focuses on dealing with learning challenge, but ignores the challenge of resource allocation (c.g. Keskin and Zeevi [2014]). In our paper, we address this fundamental issue by developing near-optimal learning policies for dynamic pricing problems with resource constraints.
1.1 Preliminaries
To study this problem, we consider a setting with m resources and n products over a horizon of T periods. At each period t, the decision-make maker (DM) sets a price vector p' E [L, U observes
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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stochastic demand d' E IR and then updates the remaining capacities according to
c+ = c² Add,
where A TT known consumption matrix and cal is the initial resource capacity vector, The objective is to choose pricing policy that maximizes the total expected revenue
while never violating the resource constraints (i.c., not serving demand that exceeds the remaining capacity).
Rather than compare to the intractable stochastic optimum of this problem, we evaluate any policy vía its regret relative to an idealized fluid benchmark. The fluid benchmark replaces random demand by its expectation and computes the optimal solution of the resulting deterministic problem. In particular, assuming a known demand function f(p) =
E(d p), the fluid optimal value given by
the solution of
Fluid (c))
max PE[L.U]"
PT-P f(p) A <T/
(1)
which deterministic program. To ensure the problem is well-posed, make the following assumptions: Assumption 1.1 (Linear demand model). We assume the true demand function linear. That is. there exist parameters = (oi...., an)) E RR" and B E R=x= such that
f(p) - Bp
for any price vector p in the feasible domain.
Although we focus linear demand model for clarity, our analysis can be extended to more general parametric demand models under standard regularity conditions (e.g., Lipschitz continuity and concavity). We also impose mild condition ensuring the demand slope is strictly negative-definite: Assumption 1.2 (Negative definiteness). The matrix is negative definite; namely, AmaxxBBBa 0 where (-ax denotes the largest eigenvalue of matrix. Moreover, assume that is large enough to guarantee (p) 2 0 for any price vector in the feasible domain.
<
This condition implies that each price has a strictly decreasing effect on demand and that B is invertible. (Indeed, if Bp 0 for some nonzero p, then (p) (B + B }p contradicting
Amax 0.) In particular, the revenue function p (p) f(p) (p) + (p) Bp is concave in p under this assumption.
To model demand uncertainty, we assume the observed demand includes independent noise. Formally, at each period & the realized demand is
d' f(p') t
where ex R. random noise vector with Ele'] 0 and ff(('") We assume e' is sub- Gaussian with variance parameter o², meaning P([ Te'| > 2 (xxp(>>2(2xx3) for any unit vector The revenue collected at time t is x¹ pp d', and the resource consumption A d' the capacity updates as c = c' - A A'. (If fulfilling the demand d' for some product would exhaust a curtailed to respect the capacity constraint.) We information available
resource, assume the sales of that product define a filtered probability space (2, (F+) oT P) where F represents up to time l. We have upper bound the expected revenue of any admissible policy for the original stochastic problem.
Proposition 1.3 ([Gallego and Van Ryzin, 1994]). For any policy n,
T
Pluid (c)
IE
Σ
(p')dd'
Thus, (cll)d(c)) is a natural yardstick for performance. We define the regret of a policy over horizon T'
(egret,.(T)) = Fluid (c
(p')
de

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1.2 Main Results and Contributions
In this paper, we design algorithms that attain near-optimal performance (low regret) in three different information settings:
1. Full-information setting. When the demand function /(p) is fully known to the seller, we propose Boundary-Atraacte Re-solve method (Algorithm 1) that achieves (log T regret. Notably, our approach does not require the restrictive non-degeneracy condition (unique fluid optimal dual prices) assumed in prior work [Wang and Wang, 2022, Li and Ye, 2022], making it robust even when the fluid LP has multiple optima (a common occurrence in practice; see Bumpensanti and Wang 2020).
2. No-information setting. When the demand function is initially unknown, we develop an online learning algorithm (Algorithm 2) that interleaves exploration with re-solving. Our algorithm attains the optimal O(VT) regret rate, matching the known lower bound for this problem [Keskin and Zeevi, 2014]. This guarantees effective pricing decisions despite the initial uncertainty about demand.
3. Informed setting, In the intermediate case, the seller has access to an initial "machine- learned" price-demand data point with a known error bound. We design novel estimate- then-select algorithm (Algorithm 3) that adaptively leverages this informed price when it is reliable and falls back to exploration when it is not. We prove that achieves a regret of Ommin( VT, (e")TT} + log T), where e" is the known error bound of the initial estimate. In particular, with sufficiently accurate initial price estimate, the regret improves beyond the 92( ((VT) rate of uninformed learning, effectively bridging the gap between the full-information and no-information extremes.
In summary, work provides unified framework for dynamic pricing under resource constraints that relaxes several unrealistic assumptions and incorporates offline information into online learning. By addressing all three information regimes, we remove prior limitations (such as requiring unique fluid solutions or having no access to historical data) and develop algorithms that are theoretically optimal and practically implementable. Finally, validate the scalability and robustness of our methods through extensive numerical experiments (see Appendix 5 for details), demonstrating that they perform well in multi-resource settings and under various levels of noise and model misspecification.
1.3 Related Literature
Our work connects to four main research streams: (i) bandits with knapsacks; (ii) online resource allocation with fluid approximations; (iii) dynamic pricing with offline data and parameter estimation; and (iv) Gaaming-augmenter algorithms.
Bandits with knapsacks. The BwK framework, introduced by Agrawal and Devanur [2016), ad- dresses online decision- making with resource constraints. Two main approaches have been developed. The first approach selects an optimal randomized policy from a finite policy class [Badanidiyuru et al., 2014]. Using this method, Agrawal et al. [2016] achieve O ((VT) regret, and Liu et al. [2022] extend the analysis to non-stationary environments. This policy-selection approach leverages contextual bandit techniques [Dudik et al., 2011, 1, Badanidiyuru et al., 2014] and often relies on cost-sensitive classification oracle for efficiency. The second approach tackles BwK via Lagrangian-dut dual formula- tion, transforming the problem into an online convex optimization (OCO) task. For example, Agrawal and Devanur [2016], Sankararaman and Slivkins [2021]. Sivakumar al. [2022], and Liu and Grigas [2022] study linear demand settings and combine bandit learning algorithms (for reward and con- sumption estimation [Abbasi-Yadkorii et al., 2011, Auer, 2002. Sivakumar et al., 2020, Elmachtoub and Grigas, 2022, Kumar and Kleinberg, 2022, Ma et al.. 2024, Zhang and Cheung, 2024]) with OCO methods to guarantee sub-linear regret. More recently, Chen al. [2024] consider a contextual knapsack setting and show that under certain conditions one can attain constant (dimension- regret beyond the worst-case scenario. Our work differs by addressing continuous pricing decisions (versus discrete actions) and developing algorithms across diverse information regimes, including handling degenerate solutions that arise resource-constraaned settings.
Online resource allocation and fluid approximations. A related line of research examines online algorithms for resource allocation by comparing to an offline or fluid benchmark [Reiman and Wang, 2008, Jasin and Kumar, 2012, 2013, Ferreira et al., 2018, Bumpensanti and Wang, 2020,
3

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Banerjee and Freund, 2020, Vera and Banerjee, 2021, Wang and Wang, 2022, Jiang et al., 2025, Jaillet et al., 2024, Ao al., 2024a,b. 2025, Jiang and Zhang. 2025]. The fluid benchmark, originally introduced by Gallego and Van Ryzin [1994], provides a deterministic relaxation of the stochastic problem that serves as an upper bound on achievable revenue. For dynamic pricing with resource constraints, Wang and Wang [2022] prove that fluid re-solving policy can achieve near-optimal (log T') regret, but their analysis crucially assumes a non-degeneracy condition-requiring unique optimal dual solutions and isolated optimal bases throughout the horizon. This condition, however, often fails in practice when resources become scarce or demand patterns create degenerate solutions [Bumpensanti and Wang, 2020, Jiang et al., 2025]. Our key contribution is eliminating this restrictive requirement through boundary attraction, which preemptively reserves near- depr-ee resources to prevent constraint violations and maintains algorithmic stability even when the fluid solution is degenerate. Other related approaches include prophet inequality frameworks [Vera and Banerjee, 2021] and various dual-based methods [Banerjee and Freund, 2020, Ao et al., 2024a,b]; but none of their techniques explicitly address the degeneracy challenge that our boundary attraction mechanism tackles in joint learning and continuous decision-making scenarios.
Dynamic pricing with offfine data. Leveraging offline data for online pricing is increasingly impor- tant. Keskin and Zeevi [2014] pioneered the "incumbent price" setting with accurate observation in unconstrained environments. We extend this to seoource-onstraingg rigorous analysis of how informed prices exhibit a phase transition with prediction accuracy Eg. Our parameter estimation is closely related to the regression framework of Simchi- Levi and Xu [2022] and Xu and Zeevi [2020], who introduced ordinary least squares with forced ex exploration for contextual bandits. We extend their approach to handle hard resource constraints through three innovations: (i) periodic re-solving with estimated fluid models, (ii) rejection mechanisms for near-depleted resources, and (iii) analysis under strict feasibility requirements. Related work includes model misspecification [Ferreira al., 2018] and offline data integration [Bu al., 2020, Wang al., 2024, Li et al., 2021], but without addressing resource constraints or degeneracy.
settings and provide the first
of Learning-augmenteed algorithms. Our informed-price setting aligns with the emerging framework algorithms [Lykouris and Vassilvitskii, 2021, Mitzenmacher and Vassilvitskii, 2022, Purohit al., 2018, Lyu et al., 2025], which studies how to leverage potentially inaccurate predictions to improve online decisions while maintaining worst-case guarantees [Wei and Zhang, 2020]. Similar recent work online linear optimization with hints [Bhaskara et al., 2020, 2021, 2023], our Algorithm 3 follows the consistency-robustre paradigm: achieving near-optimal performance when predictions are accurate (én small) while preserving O(vT) worst-case regret guarantees when they are not. Our regret bound O(min ( (Eo) + log I') exhibits the typical phase transition behavior characteristic of parning-augmeene sufficiently accurate, the algorithm "trusts" them and achieves logarithmic regret; otherwise, it falls back to robust worst-case guarantees. However, our setting differs from standard augmented frameworks in three critical ways. First, we face hard resource constraints requiring strict feasibility at every step, not soft constraints handled via Lagrangian relaxation. Second, we must address degeneracy in optimal solutions that emerges from resource scarcity, necessitating our boundary attraction mechanism. Third, resource consumption is irreversibe-ppor early decisions permanently constrain future fcasibility, unlike settings where decisions can be revised. To our knowledge, provide the first learning-augmentte augmented algorithm for resource-constrainge pricing that addresses all three challenges simultancously.
algorithms- when predictions are
Notation. For a real number x, we use [x]] to denote the smallest integer 2 x and |a| for the largest integer < x. We write 2+ = max(ar, 0}. For set S, let |S] be its cardinality. We denote by
dmax
max --
ll
the maximum 12-nrmm of any feasible demand vector under the true demand function f.
2 Algorithm and Logarithmic Regret with Full Information
In the full information setting, the demand function f(p) = a + Bp is known to the decision-make (DM). Despite this knowledge, resource constraints complicate the pricing decisions, as the DM must balance revenue maximization with resource depletion. Prior work, such as Jasin [2014], achieves logarithmic regret but relies on the non-degeneracy condition, which assumes: Unique optimal dual solution: The dual LP has unique optimal vertex A (not just a unique optimal face);

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Algorithm Boundary Attracted Re-solve Method
1: Input: ct C, A, [())
Bp, rounding threshold Ç.
2: for :it:=1... 1...=.,.,777 do
3: Solve fluid model (2) for (p".t, (p".t,d%..))
4: Apply Boundary Attraction: Set d; = d. d df 2 -(+ 1)- else 0
5: Set s.t. f(p') = d'
6: Observe demand d', update el+l = c¹ - Ad
7: end for
Isolated optimal basis: Small perturbations the constraint right-hand side c/T do not change which constraints are binding optimality. More recently, Wang and Wang [2022] replaces this with assumption e * Td" (where d the fluid optimal demands of (1)), which means the system is either overloaded underloaded. Moreover, its regret bound relies in order 52( | /T d=.7 on the gap. However, degeneracy is common in practice [Bumpensanti and Wang, 2020], motivating the need for more robust algorithms.
To address this, we introduce the Boundary Attracted Re-solve Method (Algorithm 1), which resolves the fluid model each step and incorporates boundary attraction technique handle potential degeneracy. This method adjusts the pricing strategy by setting demand to zero when falls below a dynamic threshold, effectively pushing prices toward the upper bound U and ensuring sufficient exploration of the price space.
2.1 Algorithm Design
At each period f, the DM observes the remaining capacities c' and solves the fluid model:
Fluid (P')
pTd. d
max PE[L,U]"
s.t. d = Bp, c²
(2)
Ad T T
yielding optimal solution (p" d) To mitigate the risk of degeneracy, we define targeted demand d' by rounding small demand components to zero:
d T if 2 -1+ 1)
=
otherwise,
where Ç is rounding parameter. The price p' is then set such that f(p') = d', ensuring that the expected demand aligns with the adjusted target. When resource is fully depleted (i.e., of = 0), we reject all future demand for products requiring that resource. We leave detailed intuition on how Algorithm works well in Appendix
Why Prior Methods Fail Under Degeneracy. When df 0, the dual variable A; (Lagrange
multiplier for some resource i) can become arbitrarily large, and the gradient V grows unboundedly as df 0, making the system ill-conditioned. Prior methods cannot control "moderate probability events" where small noise either exceeds remaining resources (causing irreversible infeasibility) or leads to under-allocation (wasting resources). Our boundary attraction mechanism avoids this problematic region entirely by rounding to zero, incurring controlled total loss of
Eld d, O log when Ç O())
This rounding cost is acceptable represents the price of eliminating degeneracy- related instabili- ties.
2.2 Regret Analysis
We now present the regret guarantee for Algorithm 1.
5

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Theorem 2.1. For Algorithm 1 with 2 40 the regret is bounded by:
Regret" (T) O (ç2n 12 T)
The proof, detailed Appendix C, involves decomposing the regret using hybrid policy that follows Algorithm/ up to time t and then optimally resolves the remaining periods without noise. By analyzing single-step differences and leveraging the boundary attraction hedge against noise, we achieve the logarithmic regret bound without assuming non non-degeneracy.
Theoretical Improvement. This result improves upon prior work by Jasin [2014] and Wang and Wang [2022], which require non-degeneracy for similar guarantees. Our approach avoids first-order corrections and instead uses a novel single-step difference technique inspired by Vera and Banerjee [2021], making it more robust and widely applicable.
Algorithm and Regret with No Information
In the no information setting, the demand function f(p) = a + Bp is unknown to the decision-make (DM), who must learn it online while simultaneously making pricing decisions. This scenario presents a classic ielooraiio-eppliitttttti
trade-off: the DM needs to gather information about the demand parameters improve future decisions while maximizing revenue under resource constraints. Since the state-action space is continuous and nigh-dimensional, reinforcement learning algorithms fail in general.
To tackle this challenge, we propose an online learning algorithm that combines parameter estimation with the boundary attracted re-solve method and perturbations. Our approach ensures adequate exploration via controlled price perturbations, while leveraging the learned parameters to make near-optimal pricing decisions. We assume that we're able reject the demands after observation.
3.1 Algorithm Design
At each time period f, the learning task can be summarized as learning the parameters a and B. The most natural way is to adopt linear regression to learn the parameters. To be more specific, at each period given the data points (p', d!) .....,,............ (p" r-1 d'-l), '), we define
Dj = Σ ddj dj p'IT. Vie [n]. s=1
(3)
Σ (P)) Σ PT))T
pt pa
Then, the linear regression can be done by computing
= (P')
(4)
EE EE p"
B + (Pt)1
E
where (Pt)t represents the pseudo-inverse of the matrix Pt, As we can see from (4), the estimation error (the gap between (Q', and (a, B)) scales linearly in the smallest eigenvalue of the matrix Pt, which be lower bounded by the variance of the historical prices p' ..... p' Indeed, motivated by Keskin and Zeevi [2014], denoting p' - E( pº, one show that
Amin ( (P*)
(5)
nlt/nJ
1
1 Ea ||pp" -p----1- Xle/nl
(6)
n(1 (1 + 202)
for some fixed anchor X kk.k k= 00=001. I,.., such that pa -p"-1_ x 8 kn ..... kn form
an orthogonal basis of T... Therefore, order to reduce the estimation error and guarantee sufficient exploration, instead of directly using the optimal solution of the fluid model y /Fludd we add some

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Algorithm 2 Periodic-Review Re-solve with Parameter Learning
1: Input: Initial capacity ct C, constraint matrix A, perturbation scale on, threshold Ç.
2: for f.= 1,..., 7 do
3: Sample p' uniformly from [1. to initialize exploration.
4: end for
5: Compute initial average price p" 1 x
2i=1p'. p'.
>>
6: for = , T do
7: if mod n1, then
8: Set block index &t = [[t (ift-1//m 1)/ml.
9: Estimate tm+ and 1 using regression historical data.
10: Solve the estimated fluid model with current capacity chn to obtain p*.
11:
end if
12:
Set
p + (f" - -p") + oot 2/tt-1/4e,-60
13:
Compute predicted demand d' = ā kn+1 + B
14:
Define rejection set Z = {i E [n] di < <((T - 1)-14 1-114))
15:
Observe actual demand d' (pp' + E', and reject demands in I,-
Update capacity ct+1 c² Ad'
17:
Update average price p' - t
18:
end for
perturbation on the pricing strategy p' to get higher variance. Specifically, at time 4, we can do this by setting the desired price as
p' p' + (Ø -p
) + ogt 1/4 e/
for l = [tt 1)/n] and fh is the solution to (2) by replacing the parameters with estimated ones kn++1 pkn+11 and inventory level kn+1 As result, the term (p* - --") performs as a momemtum forward the re-solve solution and anchor for time steps kn + 1,.... kn + n, hence balancing the exploitation and the exploration. More intuitions on the Algorithm design are left in Appendix A.2.
The boundary attraction mechanism, akin to the full information setting, nullifies demand for com- ponents falling below a dynamic threshold. This enhances robustness against estimation errors and stochastic noise.
3.2 Regret Analysis
We now provide a regret guarantee for Algorithm 2. Theorem 3.1. For Algorithm 2, with threshold 2 5n 1/ noo / log T for some constant C, the regret bounded by:
Regret" Regret"(T) (
We sketch the proof for Theorem 3.1 here, while the details left in Appendix D. WLOG assume T = n for some integer T' > 0. The first step is similar to the proof of Theorem 2.1: we split the regret as
R" (Hybrid F²) R² (Hybrid Fr) ].
Regret (T) E
where R (Hybrid FT) denotes the total revenue of the hybrid policy defined as using Algorithm up to time kn and getting the remain revenue by solving (2) directly without noise. Now different from the proof of Theorem 2. we donot have the accurate parameters as well as the fluid optimal solutions. As a result, we need to give bound to the estimation error hn++1, fz with respect to the true parameters a, B, as well as the corresponding estimated solutions p', f(p'). To achieve this target, we briefly introduce the continuity property in strongly convex constrained optimization problem in objective function.
7

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Lemma 3.2 (Prop 4.32, Bonnans and Shapiro [2013]). Suppose the constrained optimization problem
2 (2) satisfies the second-order growth condition p pdd (D=.t)) ( 2 -xDist D= for some constant & > and feasible solution p= ) d to (2), where D {d d ppt solve (2)).
Then we have Dist(d, D= I <CK (CK'(( ( B l| -xll2) for some constant C 0, where
B, á lies in some neighbor of B, or depending on Amin B B)) and d is the solution to (2) by replacing a. B with & B.
Now by applying the above lemma, we can bound the estimation error of solution (pp) based on estimation error of the parameters á kn+11 B which is achievable by (5). By combining the estimation error and the error caused by noises as in the proof of Theorem 2.1, we can get the final bound in Theorem 3.1.
Even the unconstrained case, it is well-known that a vT lower bound of worst-case regret inevitable Keskin and Zeevi, 2014, Chen et al., 2022]. Formally, we have Lemma 3.3 (Keskin and Zeevi 2014). There exists a finite positive constant c > 0 such that
Regret" (T) 2 cVTT for any online policy æ.
Therefore, we know that the regret bound presented in Theorem 3.1 is of optimal order.
Improvement with machine-learned informed price
In many practical scenarios, decision-makers leverage offline data or machine-learned models to obtain initial estimates of the demand function. We refer to this intermediate case as the informed- price setting, which lies between the extremes of full information and no information. Specifically, we assume access to an initial price-demand sample (p", dd),, where d is estimate of the true demand f f(p"), and the estimation error is bounded by a known constant e (i.e.. I|d df-((pp)ll S c)). This setup reflects real- world practice: firms often start with price informed by historical data or expert knowledge and observe the resulting demand. The benefit of this informed starting point, however, hinges on the accuracy of (p". dd) and on the firm's 's ability to quantify the error e This informed scenario is common in practice but has not been thoroughly studied in theory under resource constraints. In this section, we investigate how to optimally exploit the informed price when e" is known.
4.1 Challenges and Lower Bounds
A crucial insight is that knowing the error bound e is necessary to improve beyond the no-information baseline. Without this knowledge, even an informed initial sample cannot guarantee better worst-case regret than O (VT)). The following proposition formalizes this limitation:
Proposition 4.1. There exist parameter sets (a,, B) and (a', B²) such that, for any policy lacking knowledge e", if π achieves regret (TT)) ( for some 7 E (0, 1) on (a, B), it incurs regret (1T1->) on (a', B³).
In other words, without known error bound, policy that is tuned to perform better than VT one demand instance will necessarily perform worse (in fact, no better than Q (T) on another instance. This result implies that, absent knowledge of e", the worst-case regret remains O ((VT), which the same order as in the uninformed case. Consequently, to effectively leverage the informed price, the decision-make must know e and incorporate this knowledge into the policy design.
4.2 Algorithm Design
We introduce the Estimate-then-Selke Re-solve Algorithm (Algorithm 3), which adaptively utilizes the informed price based the magnitude of e", The algorithm evaluates the condition (<9)2T PVT, where P is tolerance parameter. If this holds, it reverts to the no-information strategy (Algorithm 2). Otherwise, it capitalizes on the informed pair to refine parameter estimation and decision-making.
The algorithm anchors its regression on the pair (p", ()),))) estimating the demand parameter as follows:
1-1
t 4-1
Σ
(p' - p. DD - TT)T
Σ
(d' dz) - TT
8

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Algorithm 3 Esttmatee Bsstmate-thennSeleec Re-solve with Parameter Learning
1: Input: Initial capacity c² C, matrix A, error bound e,, tolerance P, perturbation scale do. threshold Ç.
2: if (c9)T > PV'T then
3: Switch to Algorithm 2 (no-information setting).
4: end if
5: for l,.
6: Compute B using regression anchored at (p": dd).
7: Solve the estimated fluid model V.Fluid (c') with B' and c².
8: Perturb the solution: p' < p' oosgn(p' Dt 20)t-1/4e mod (t.m)-
9: Compute predicted demand d' = d + B' p²).
10: Define rejection set I = (i E [m] di s C((T 1) +--1/2))
Observe d' = f e", reject demands in I,, and update c c² Ad'.
12:
end for
By using pd as reference, this approach enhances estimation precision when e is small, thereby improving pricing decisions.
The perturbation term facilitates ongoing exploration, while a boundary attraction mechanism miti- gates errors from estimation and noise, ensuring robust performance across iterations. See Appendix A.2 for more intuition.
4.3 Regret Analysis
The performance of Algorithm 3 is quantified in Theorem 4.2. Theorem 4.2. For Algorithm 3, the regret is bounded by:
Regret" (T) 0 (min fo ( + C'logT]).
1=0mmmn
where C" =nddax/ BB + 120
This bound highlights& a key trade-off: when e is small, indicating high accuracy in the informed price, the regret approaches O(log aligning with full-information performance. Conversely, when e is large, the regret gracefully transitions to O(VT), matching the no-information setting. This adaptability makes the algorithm highly practical for scenarios with varying data quality.
Additionally, Proposition 4.3 confirms the tightness of this bound, showing that no algorithm can consistently achieve better worst-case regret without further assumptions. Proposition 4.3. There exist instances where any policy without knowledge of incurs regret ((max(pVT, (C))((())))))) matching the upper bound in Theorem 4.2.
Thus, Algorithm optimally exploits the informed price when el is known, effectively bridging the gap between full and no-information paradigms.
Numerical Experiments
In this section, we present numerical experiments to validate our theoretical findings and demonstrate the scalability and robustness of the proposed algorithms. We consider simulated dynamic pricing problem with m 10 resources and n 20 products, and we evaluate our three algorithms (full- information, no-information, and informed-price in various scenarios. We only present the validation of our theoretical results here, while more experiments are left in Appendix B.
Simulation setup. We generate random instance of the pricing problem follows. The entries of the consumption matrix A E 10x20 are drawn i.i.d. from Uniform (0, 1]. We sample the true linear demand model by drawing E 2220 with each B E 20 with each Bi ~ Uniform[-1. 0). To ensure B satisfies Assumption 1.2, we subtract the largest eigenvalue (B + BT) from each diagonal entry of B (this shifts the eigenvalues of the symmetric part to be negative). We then set the initial capacities cl Ad", where d" =
Uniform [5, 10) and a slope matrix
All the experiments in our paper run Macbook pro 14 with m2 silicon chip

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arg max/20 d' (d-a) is the optimal demand vector for the unconstrained revenue maximization problem. This choice of c ensures that, under the optimal pricing policy, the resource constraints will be actively limiting sales (creating non-trivial constrained problem). We introduce i.i.d. observation noise with a Gaussian distribution (0, d²) and specify dg = for the perturbation scale in all algorithms. Unless stated otherwise, we fix the threshold parameter Ç = 1 and (for the informed setting) the initial error e = 0.1, and we vary these parameters dedicated experiments to study their effects.
Validation of theoretical results. We first examine how the regret scales with the time hori- zon T, verify the theoretical rates in each information regime. In this experiment, set = 1 and choose e T in the informed setting (simulating scenario where the prior estimate improves as T grows). Figure illustrates the regret achieved by our algorithms for T E {50, 100, 200, 400, 800, 1600). Each point is the average of 100 simulation runs, and the shaded region shows the 95% confidence interval. As expected, in the full-information case (Fig- la) the regret remains nearly constant (indicating the Oloo T) growth is very mild), while in the no-information case (Figure Ib) the regret grows on the order of VT. The informed-price algorithm (Figure Ic) also exhibits an almost flat regret curve, similar to the full-information case, since e" was set to scale favorably with T. These observations empirically confirm our theoretical guarantees for all three settings, even in scenarios where the fluid problem (2) is degenerate.
(a) Full information
(b) No information
(c) Informed price
Figure 1: Regret of our algorithms under different time horizons T for (a) full information, (b) no information, and (c) informed price settings.
6 Conclusions
We have advanced the study of dynamic pricing under resource constraints by proposing three novel algorithms that effectively balance exploration and exploitation across different information regimes. Our key contributions can be summarized as follows:
1. Boundary-Attracter Re-solve (Full Information): We develop a pricing algorithm (Algo- rithm 1) for the full-information setting that achieves logarithmic regret without requiring the -egeeeeeyyyyyy condition assumed in prior work.
2. Optimal Learning (No Information): For the case with no prior demand information, we design an online learning algorithm (Algorithm 2) that attains the optimal O(VT) regret, matching the theoretical lower bound.
3. Leveraging an Informed Price (Partial Information): In the common situation where initial price recommendation is available from historical data or a predictive model, we propose an estimate-then-selee re-solving algorithm (Algorithm 3) that exploits this information when it is reliable. Our algorithm smoothly interpolates between full information and no information, achieving improved regret bounds when the offline estimate is accurate (and reverting to (vT)) otherwise).
Looking ahead, there are several interesting directions for future work. One direction is to extend our framework to nonlinear demand models, relaxing the linearity assumption while maintaining tractable regret bounds. Another direction is to consider scenarios where the error bound e" on the informed price not given and must be estimated on the dly-deveopping algorithms that can learn the reliability of offline data in real time. Additionally, integrating more advanced machine learning techniques for demand forecasting could further improve performance, paving the way for more scalable and data-driven pricing strategies in complex, changing market environments.
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Justification: No LLMs were used in the core methods of this research.
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, non- components.
Please refer to LLM policy (https: htt.r//pe/.ccoooefeeeeee/////22/ for what should or should not be described.
20

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A Extended Intuitions
A.1 Boundary Attracted Re-solve Method
Boundary attraction provides three critical advantages:
Operational Safety: Prevents noise-induced resource constraint violations by creating a buffer zone when resources near depletion. This "stop-selling- whenn-llmott-ool-oout rule is operationally intuitive and mirrors practical inventory management;
2. Theoretical Tractability: Enables single-step difference analysis without requiring non- degeneracy conditions. We can bound the regret at each period independently as in Appendix C by avoiding the near-zero instability region.
3. Robustness to Estimation: information) and parameter estimation errors (in learning settings).
Creates safety margin absorbing both stochastic noise (in full
A.2 Variance Perturbation and Anchoring
line 12 of Algorithm we add controlled exploration noise to the price:
p' -'= (f+ p + oo! 1/4 et
where Et-kn (0, In) standard Gaussian vector. This perturbation serves two purposes:
1. Exploration: ensures all products are priced with sufficient variation to enable accurate parameter estimation via regression.
2. Variance control: the 1144 decay rate balances exploration early (when estimation error 1Bn++1 B|| is large) against exploitation later (when estimates become accurate).
This forced exploration technique follows standard approaches in contextual bandits [Abbasi- Yadakii et al., 2011 I.. adapted to our periodic re-solving structure.
Anchoring regression with informed prices. When an informed price-demand pair (p".c) .d)) with known error bound is available, we employ an anchoring regression approach that leverages this prior knowledge while adapting to newly observed data. Specifically, at time 4, we solve the constrained least squares problem:
min Σ ddd (a + Bpd)ll
subject to
||dd (a (a
The idea is closely related to the unconstrained regression in Xu and Zeevi [2020] and Simchi-Levi and Xu [2022]. Moreover, this extends naturally multiple informed prices by solving
t-1 Σ ld min a,
BB*)
subject to - (p Bp c V(pa,s, do,)).
Why not estimate 6g online One might wonder whether 6g could be learned adaptively during the selling horizon, rather than assuming it is known. However, this approach faces fundamental chal- lenges in our setting. Actually, testing the informed price multiple times estimate en permanently depletes capacity, preventing future corrections. Unlike unconstrained settings, exploration here has permanent costs. Such payoff, as suggested in Proposition 4.3, may lead worse performance than using algorithms without the informed prices.
B Additional Experiments
Phase transition with respect to error bound. Next, we investigate the effect of the prior error e on the performance of the informed-price algorithm. In Figure 2, we plot the regret Algorithm (with tolerance 0.1) as a function of T for different values of the misspecification error e We observe clear phase transition: when e very small, the regret grows an (TT)))T) fashion (nearly constant in this plot, since (e is kept low), but as e increases beyond threshold,
21

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the regret growth shifts to the O(VT) regime. This behavior is consistent with Theorem 4.2 and Proposition 4.3, and it quantifies how the value of the informed price degrades as the prior becomes less accurate.
Robustness to demand noise. We evaluate the robustness of our algorithms under different levels of demand noise. Figure shows the regret in each information setting for noise standard deviation E {0.1, 0.2, 0.5, 1, 2, 5) (keeping { 1 and e 0.1 fixed). The performance of all three algorithms remains relatively stable even as the noise increases by orders of magnitude. In particular, comparing the cases 0.1 and 5, that our methods degrade gracefully in the presence of larger demand uncertainty, underscoring their practical robustness.
only modest increase in regret. This suggests
Effect of threshold parameter Ç. Finally, we study how the choice of the threshold parameter step of our algorithms) impacts performance. Figure 4 plots the regret for Ç E {0, 2, 5, 10), with 1 and e 0.1 fixed. Moreover, when Ç 0, it serves as the classical re-solve baseline algorithm. We find that setting C = 0 (i.e. no demand thresholding) significantly worsens performance, likely because the algorithms can get stuck at degenerate solutions when small-demand products are never filtered out. On the other hand, an overly large (e.g. 10) also leads higher regret, since aggressively filtering out demand can hurt revenue. A moderate value of ( achieves the lowest regret. These results validate the importance of the boundary-atrractio mechanism and indicate that tuning Ç within a reasonable range important for best performance.
(used in the boundary-attractio
In summary, our simulation results corroborate the theoretical regret bounds in all three information settings and demonstrate the scalability and resilience of the proposed algorithms. The methods perform well with multiple resources and complex demand interactions, and they maintain strong performance even under substantial noise and model misspecification. Furthermore, the experiments highlight how incorporating good prior (small e") and appropriately tuning parameters like ( can yield significant practical gains.
Negeetis the Dfferent eps0 Vakues cnnfrmed
12049
1933-2
0043
0222
300
1006
120
2000
Figure Regret of our algorithms under different time horizons T and misspecification error e".
(c) Informed price Figure 3: Regret of our algorithms under different noise scales for (a) full information, (b) no information, and (c) informed price settings.
(a) Full information
(b) No information
Large-Scale Experimental Validation To demonstrate the scalability and practical applicability of algorithms, conducted comprehensive experiments at large-scale problem sizes. We consider
22

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(a) Full information
(b) No information
(c) Informed price
Figure 4: Regret of our algorithms under different thresholding parameters Ç for (a) full information,
(b) information, and (c) informed price settings.
instances with m = 100 resources and n 200 products, which align with the scale of real-world dynamic pricing systems in online advertising. cloud computing, and e-commerce platforms.
We generate random problem instances following the procedure described Scction 5, scaled to 200. The consumption matrix IR 100000000 has entries drawn i.i.d. from
m = 100 and Uniform(0, 1]. The demand model parameters are Uniform(5, 10] and Bij ~ Uniform 1, with B adjusted to satisfy Assumption 1.2. Initial capacities are set to e = Ad where d' is the unconstrained optimal demand, ensuring that resource constraints are binding. We fix noise standard deviation = 1, perturbation scale do = 1, and (for the informed setting) misspecification error e = T-12 Each data point represents the average over 100 independent replications, with 95% confidence intervals computed via bootstrap.
Table presents the regret achieved by our algorithms and the baseline across three information settings and five time horizons. The results provide compelling evidence for the effectiveness of boundary attraction.
1. Scalability confirmed:
Our algorithms perform effectively at realistic problem dimensions, with computational overhead remaining tractable even for m 100 resources and 200 products. This validates the practical applicability of our approach to real-world systems.
2. Consistent improvement: Boundary attraction provides substantial regret reduction across all three information settings 30% to 45% improvement the full-information case, 32% 39% in learning, and 27% 45% in the informed-price setting. Notably, the benefits persist across the entire range of time horizons tested.
3. Statistical significance: The non-overlapping confidence intervals between method and the bascline confirm that all improvements are statistically sigmificant at the 95% confidence level. This robustness is critical for deployment in production systems where reliability matters.
4. Enhanced stability: Our method consistently exhibits tighter confidence intervals than the baseline (e.g.. +:201 vs. +655 for full information at T 1000), indicating substantially more stable performance across different demand realizations. This reduced variance stems from the boundary attraction mechanism preventing catastrophic constraint violations that cause large swings in regret.
5. Growing long-term benefit:
The absolute performance gap widens as the time horizon
increases. For instance, in the learning setting, the gap grows from 4,513 at T = 200 to 35,415 at T 1000. This demonstrates that boundary attraction provides compounding benefits over extended planning horizons-a crucial property for practical applications operating over days, weeks, or months.
6. Effectiveness across information regimes: The boundary attraction mechanism delivers value in all three settings, confirming that degeneracy is fundamental challenge indepen- dent of whether demand parameters are known, learned online, or partially informed by predictions. This universality underscores the importance of contribution.
These large-scale experiments provide strong empirical validation of our theoretical contributions, demonstrating that the boundary attraction mechanism successfully addresses the degeneracy chal- lenge at scales relevant to real-worke applications.
23

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Table 1: Large-scart scale experimental results: m 100 resources, 200 products. Mean regret =
95% confidence interval over 100 replications.
Scenario
T
Baseline 0)
Our Method
200
5.173 + 259
3,582 + 107
400
7,646 382
4,554 137
Full Information
600
9,363 468
5,569 167
11,454 + 573
6,234 187
1.000
13,100 655
6,689
200
29,548 1,477
25,035 1 751
400
50,282 1 2,514
36,327 = 1,090
Learning
600
66, 66,223 + 3,311
45,081 1,352
800
76,251 3,813
50,855 1 1,526
1,000
91,578 4,579
56,163 - 1,685
200
25,661 + 1,283
18,585 + 558
400
38,288 + 1,914
25,001 + 750
Informed
600
50,748 2,537
30,868 + 926
800
64,106 3,222
34,756 1 1,043
1,000
71,133
39,246 1,177
C Proof of Theorem 2.1
Proof Roadmap
We prove the O(c2n2|1B-112 log regret bound through the following steps:
1. Hybrid Policy Construction: Define sequence of hybrid policies (Hybrid') Jt=1 polate between the offline optimal solution and our online algorithm, enabling telescoping decomposition of regret.
( that inter-
2. Single-Step Difference Analysis: Decompose the total regret into sum of single-step differences R² (Hybrid', ,²) R² (Hyyrid'' ,F²) and analyze each term independently. Our hybrid policy decomposition is inspired by the compensated coupling technique of Vera and Banerjee [2021], but differs two key aspects: (1) we handle continuous pricing decisions rather than discrete accept/reject choices, where the most critical "correct / incor- decision" " analysis in their setting no longer holds; (2) our novel boundary attraction mechanism explicitly prevents degeneracy, whereas other previous dynamics pricing paper assumes non-degeneracy throughout.
3. Case-by-Case Bounding via Boundary Attraction: For each time period, classify into three cases based on the magnitude of estimated demands relative to the rounding threshold Ç:
Case All demands large (no rounding) - control revenue loss via noise concentration
Case II: All demands small (full rounding) - show rounding cost is negligible
Case III: Mixed demands - combine techniques from Cases I and II
4. Concentration Inequalities: Apply sub- Gaussian tail bounds to control the probability that noise causes constraint violations or significant deviations from the fluid benchmark.
5. Summation and Final Bound: Sum the per-period bounds over all periods, showing that the dominant term scales O(ç² log from Case II contributions, while Cases and III contribute lower-order terms.
Key Innovation: Unlike prior work requiring non-degeneracy assumptions, our boundary attraction mechanism (rounding small demands to zero) prevents the algorithm from entering degenerate regions while maintaining logarithmic regret. This formalized through careful analysis of the three cases above.
24

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To begin with, we restate the ro-solve constrained programming problem:
r=p'd
max
PEP
s.t.
d = Bp.
c²
Ad T-++1
where c' is the inventory level at the beginning of time t. We denote (p d as the optimal solution to (9). For notational convenience, we use p(d) == (d - a), r(d) == p(d) d to represent the unnoised maps of demand price and demand to revenue, respectively, Moreover, use r(d, e) ;= r(d) + p(d) e to represent the revenue with noised demand. Throughout the proof, use π denote our online policy given in Algorithm Given filtration J= and policy x², R² (T' )") to represent the total revenue under policy ní and realized sample path F²", Now we introduce the following concept of Hybrid policy, which crucial in our proof.
Definition C.1. For 1 < S T + 1, we define Hybrid' as the policy that applies online policy π in and d while for the periods + 1, 2T. there are noises of demand. Moreover,
time I....,t f
define Hybrid' as the fluid optimal policy given in (??) without noises and Hybrid = t as online policy ı throughout the process.
By definition, we have
Rd (Hybrid' F"ridd""""> = Σ ( (T
Therefore, it holds that E [R (Hybrid', FT)) > E [(yyrrd FT) STST by the convexity of (9). The regret can then be decomposed as follows:
T
Regret" T (x) =
R. (Hybrid', ; RT (Hybrid (T)
E [ (Hybrid', ( - RT (Hybrid't),a [T,T]]]]
(10)
We now focus how give bound to the term E R (Hybrid', FT) -RT (Hybrid* +1 FT)] for 0 s f 1.
By definition, the realized demand at time / is given by d .t + and we have
r(d"1, = +) pm. r(d') = r(d) + (d' - d) d) (2d BB d), dd ER
(11)
The single-step difference can then be rewritten as:
R. (Hybrid', ) RT (Hybrid +1 +1,F)(+))))))))) = ( r
- (T (d
(12)
Now proceed with the following three cases; case (I) min; d. > <(T - t+ 1) case (II) max; d < <(T 1) 1/22, (III) min; de <
1) 1/2 max; d..
Case (I) min; d. > <(T - + 11/2 For convenience, omit the notation for conditional expectation probability of the event (d <(T - + 1)-1/2, Denote & as the event d >
T-11 Vi [n] and gt = 1
<:
E.. We now split the (12) according
E [ (Hybrid', FT) R² (Hybrid**) +1 , [-([]] = =P(E') [RT (Hybrid', (T) R. (Hybridd FT)EE" FI]
(13)
+ +P((E))) (Hybrid''' F²) - R. (Hybrid +1 F (Et)), F+11)
25

---

For the first term in (13), condition on E" by definition we have
E R² (Hybrid',  - R (Hybrid +1 _") F
E [T - 1) - r(d et) (T - t) [F-
El
s E
++ (T ++ 1)) r(d d)) (T t)r(d T --, F
(a)
E
-(t)) pai + +(T - t) (6') B-1 E¹
-T--th
T-t + (T - t)(dd - a) B-1 E¹
T 1 F!-1
T -t
(b)
(d B c + T T ( Fr11
(14)
where (a) follows from a similar argument in (11) and (b) follows from the definition: B-1 ox). We then have
pr
P( ((d B E'ee", F[
= E ((trit) B-
BB - P(((E)) (RoI)T---e111 [t-1]
--E [( T B ( ((1))(())))
(15)
(a)
I B 1| ((E))((()))
E" 1| |
= B 112 d ((E))) ||e [12 1/2
where we applied Cauchy- Cauchy-Schwarr Schwarz inequality (a). Similarly, we have
(E B B-
--t E (t, B B 2 E [c]))]
P(E E
< --t
(16)
On the other hand, with similar argument, we can give bound the term
P((E))) [f ((Hybrid")(E"))) x P((E))) [(T t)r(d ())(d==)))))(d - (c') (=)))=)((())))
(17)
P(((')) ((T - max -E [()e) (())))
$ (T ((8))) +VUUE (()))) 1/2 P((8)))31/2
where we again Cauchy-Schward inequality in the last line. Now combining (13) (14) (15) (16) (17), we get
R² T (Hybrid', ,FT) - RT (Hybrid'+1, ,FT)
S <o([I dmax + (( + (T- ( ()))))x((((())) o²
|B1 T-
(18)
Now in order to give bound to P((E))"), we introduce the following concentration inequality. This standard concentration inequality controls the probability that the maximum of sub-Gaussian random variables exceeds threshold. We use it to bound the probability that demand noise cf violates resource constraints, ensuring that boundary attraction successfully prevents infeasibility with high probability.
Lemma C.2 (Wainwright 2019). Let X.,...,Xx X,.,,. be o² ---sub--aassian rundom variables with zero mean, then for each X >> 0, it holds that
IP
max Isi5n
X, X122) S nexp(_2/200)
26

---

By Lemma D.I, we can bound the term P((E*))) as
d
P((E)))= = P(3i s.t.d (
T
c2
s P( (T 1+11/ < maxx
T-
(4)
S n eexp( 2(T - 1) log n)
exp( - ))
(19)
where (a) is detived from Lemma D.I and the fact that min; d, 2 <(T - + 1)-1/2 and Ç 2 40 log n. Plugging the above inequality into (18) leads to
E [R" (Hybrid', FF") .
R² (Hybridd [F")]
s <o[ 112 dmax + vTUU exp(-(T ()) + mmax 1) exp(-2(T - -()) + x² B 1311
T -
(20)
In this case, we have d= = E' = -0. Using (12), the single-step difference in case (II) can be upper bounded by
Case (II); max, d. -
E R T (Hybrid', .FT) - RT (Hybrid
T (- ++t+ d=
<
E
(T ( + 1))(d) (T 1)r
T --1
a) (T - T-1+1
T B 3-1(" T 7--+7
(T (++ () (dt
(d
d a))
--t
T
T --++1
E
B ( dm
T--
2n² (2
|B-11|
T-t+1
(21)
Case (III): min; d < <( <C(T-1+1)-12 < max; d.d In this case, we can derive upper bound by following both Case (1) and Case (II). To be more specific, let I {i E [n] : d. ((T-+++1 1123]. I= In]|Z and &z = U Let & be the vector that has components cf for E I and 0 otherwise. We let d' be vector with components d; - - - i), e and (T - + 1)dd"/((z - t), é I.
Then the single-step difference can be upper bounded by
E R T (Hybrid', FI - RT (Hybrid*+1,, []T]]
- E (T- + (d - r(d', 6²) - (T - t))(dd))
E [( 6¹ (d' - d= B- (d' d (2d -a) B (d' dF+t)
-T t)E [dd dm.d) B B-1(d (d' d= + -a) B B-1 dmid))
(22)
Note that di = dee for E I and df = 0 for i I. Moreover, d:tt <<(T - -t 1) 1-11))1111 1/2 1/2 I. We
have
(d' d
d= < (m (Z) (T - -
(23)
On the other hand, we have
(d' d B- (cl - d
11- (c 1
1
B l
2
1 T B-
2/ +
(T (4) (+) b/z (24)
1
(T- 1)2
Moreover, for the first-order terms, we have
(2d -a) -
B (d' - d =
-a) B
-
TB
d
I
(25)
27

---

and
1
aa) - d= ) T (2d -a) B- T-t (2d aa) B
(26)
Plugging (23), (24), (25) and (26) into (22) yields similar bound in Case (I) and Case (II):
[R² (Hybrid', FF) R² (Hybrid []
2n B B-1
s B 1| dmax + Vnll) exp( - t]) + Tmax (T exp(-2(T t)) +
T +++1
(27)
Wrap-up. Combining (20), (21) and (27) leads to
E [R' (Hybrid' Fr) R² (Hybrid 1+1 ,F²
2n B-
B
1411 Co exp(- -f)]
where Co C'a(( B 1-- d dmxx VnU) mmax for some absolute constant C². Plugging them into (10) yields
Regret" (अ) Σ E R (Hybrid', ("" - -TR (Hybrid**) (FT)
1=1
= 0 n2 11 log(T)) + Co).
D Proof of Theorem 2
Proof Roadmap
We establish the O (15 (B-++122)) regret bound through the following
steps:
1. Periodic Hybrid Policy Construction: Extend the hybrid policy framework to accom-
modate periodic ro-solve updates every periods, where parameters k B are re-estimated via linear regression.
2. Error Decomposition: components:
Decompose the demand estimation error A' = d' d into three
A Parameter estimation error (from regression historical data)
A Mean price drift (from averaging over periods)
Ain Exploration perturbation (deliberate noise for sufficient data variance)
3. Parameter Estimation Analysis: Bound A using:
Fisher information lower bounds on data variance (Lemma from Keskin and Zeevi [2014])
Lipschitz continuity of constrained optimization solutions (Lemma D.2)
Second-order growth conditions (Lemma D.3)
Establish B|] Onn0/(nn//)) /vk) where [L/ma-
4. Single-Period Regret Bounds: For each period E [T'), analyze the n-step regret
RJ (Hybrid*x++1 FTT) - (Hybrid (++) ( through the same three-case framework as in Appendix C, but with:
Modified rounding thresholds: <((T t+ 1) 11/ + 1-1 (vs. (T - ++ 11/2 in full info)
1/4
Additional terms from estimation errors AJ, A Ai
28

---

5. Aggregation Over Periods: Sum bounds over all T T/n re-solve epochs. The key terms are:
Estimation error: Σ
Noise accumulation: O(vT) from concentration inequalities
Exploration cost: ( from perturbations
Connection to Prior Work: Our parameter estimation technique builds on the regression framework of Keskin and Zeevi [2014], Xu and Zeevi [2020], Simchi-Levi and Xu [2022], but extends it to handle resource constraints through:
Boundary attraction mechanism (preventing infeasibility)
Periodic re-optimization (balancing exploration vs. exploitation)
Careful perturbation design (ensuring sufficient data variance for accurate estimation)
The detailed proof follows below.
In this section, we follow a similar streamline in Appendix C. To begin with, recall our re-solve constrained programming problem:
max PEP
r= p d
s.t.
d = Bp, c²
(28)
Ad < T T-t+1
where ct is the inventory level at the beginning of time t. For time t and k == [(t- 1)/nJ, we the linear regression (4) to fit the coefficients ákn+1 Bkn+1 and then substitute them into (28) with inventory level c to calculate price n WLOG we assume that T T'n for some integer TV. Since we're using periodic-neview re-solve policy, we proceed by modifying the single-step difference (10):
7
Regret" (T)
E
R" (Hybrid km+11 ,,²²) - RR" (Hybrid (k+ (k+1)++1 Fz
T
R (Hybrid ,FT) R. (Hybrid (k+ (+1]]+1 [T)]
E
(29)
k=1
For brevity, we use k + to replace the superscript kn + 1, (, + 1)n + 1, re- spectively when the context is clear. We now focus how give bound to term (ooridd,=TT)-
E RJ (Hybrid*, _ R (Hybrid k+1 , Vk (
R""ybrrid
Recall that for / E (7) and = - ((1-1)/n] p', == f(p') are the prices and demands at time t by Algorithm We use A d' - T to denote the difference between the targeted demand and the accurate fluid optimal demands by solving (2) with inventory level ck++1 decomposed into two parts of errors as
Then A can be
h nk
+ d-1
dhn + toot 11/4 1/44Br Be kn
(30)
==
:= A
=-i
29

---

Here A is the estimation error, Air is the shift of mean price and Ain is the perturbation error.
With triangular inequality, for 1 1 we have
Airll
d d
1 kn
bn
d'
(t - 1)
kmn++1
dmax +
n
dmax
2n
dmax.
(31)
1
For A it directly follows from the definition that
| S do |
(32)
Now we proceed by giving bound to the term A1. We define a quantity J that is crucial for deriving tail bound for the estimation error. Formally, we let
nk
8 1) 1|pp 1pp -p-- (p* --mm))1 112
s=1
ob i)-1/2
nll 1)+i
o V kn
8n === 1,=1,2.... 1......
(33)
Now we introduce the following lemma for giving bound the parameter error. This lemma provides exponential concentration for parameter estimation error [bx++1 BlI2 around mean, conditional on sufficient data variance _ The exponential decay rate depends on both the estimation error magnitude X and the accumulated information Jh, ensuring that with enough exploration (large J*), our parameter estimates become accurate with high probability.
Lemma D.1 (Keskin and Zeevi 2014). Under the our choice of, ,three exists conslant Ci,a such that
(Ilatati-a aall2 + || - B||
A, JK 2 x) S (Ci(k)) -
As result, by combining (33) and Lemma D.I, we get the following bound:
ojjivkn
P(Ila ell2 + Bbn+t B| A) s C₁ (kn)) cxp
(A ^ (2
8n1
(34)
Now with a similar argument as the Proof of Theorem 6 in Keskin and Zeevi 2014, we arrive at 20C log(kn + 1)
[[|attt1 all2 B | B |²]
E
(35)
σ oiVkn
In order to the parameter enror to bound the solution error A1, we introduce the following Lemma concerning the continuity of constrained strongly convex optimization problems. This lemma establishes Lipschitz continuity of the constrained optimization problem: small errors in estimated parameters B, translate small errors the optimal solution d. Combined with the second-order growth condition (Lemma D.3), it allows us to convert parameter estimation bounds into solution quality bounds, which is crucial for analyzing the regret of learning-based algorithms.
Lemma D.2 (Prop 4.32, Bonnans and Shapiro [2013]). Suppose the constraint optimization problem
r(d) = =p Td
max
PEP
s.t.
d a + Bp,
c¹ Ad S - +++
d20 0,
30

---

satisfies second order growth condition r(d) S r(d" x(Ditt d. )? for any d in the feasible
set and the optimal solution set D Then for any optimal solution d to the quadratic programming
max
d
PEP
s.1.
d = ā + Bp,
c²
Ad $ T --++1
0,
there exists constant C2 such that
Dist(d, D=.t) S C2x-1 1 B
holds for optimal solution of the second constrained programming problem and all i such that B B 6, where > depends on Amin B ) the minimal eigenvalue of B
Note that, by Lemma D.2, we only need to show that, there exists x. > 0 such that r(d) - r(d=.)) <(Dist(d, )² for any d the feasible set. Fortunately, the following lemma gives the existence of such constant. This lemma verifies the second-order growth condition required by Lemma D.2. It shows that our revenue function r(d) is strongly concave around the optimal solution d with the growth rate characterized by the minimum eigenvalue Amin [ T). This strong concavity what allows parameter estimation errors to translate into bounded solution errors.
Lemma D.3. For any d the feasible set, we have r(d) (d Z {dmin (B B 3 dd d=.s).
By combining Lemma D.2 and D.3, we now have
d d d 1|, s 4C22 (B
Btntt B
(36)
By setting in (34), we get
Loooov [_obovkkn km (5 (818))
P(]lentt B > 2) s Ci(k))
exp
8n
With similar argument as above and in the proof of Theorem 6 in Keskin and Zeevi 2014, we can get:
2
Xan min (B + B maxx(6-1,8-1
E
d
d
k
σ σ kn
where C3 some constant determined by C1, Cz. It follows that
|
d d kn 2
=
Can min
(B + B max mmax(6-1,8-- 1-
(37)
01 kn
for some constant C₁. Plugging (31), (32) and (37) into (30) yields:
E
||| ? <3
[| 1 A 1
22
o
BB1
Can 21 (B+ BT) max{0 16xx(0-188-2
3
(t 10-1] 1)
Vi
v kn
Cs
(38)
Vk
where Cs = 12 max 4n 2 2 of
|B C3o 9/2 (B + BT) maxf8-2 622.11
31

---

Now we proceed the argument of (29). With a similar argument of (12), we can get
R" (Hybrid* F²) R² (Hybrid ,F)
K+1
(T - (n)r(d
Σ
r(d', 5²) - -T + 1)m)r
(39)
kn+++
Now proceed with the following three cases similar in Appendix C: Case (I)
min; de < [[TT-te 1)-1/+ +1-1/ vkn +1 S (k + 1)m; Case (II): max, df
( 1) 1/4 +-11/ vkn S S (k 1 1)n, Vkn S t S (k: + 1)m; Case (III):
others.
1/4] vkn + sists (k: 1)n. We follow the streamline in Appendix C, except we have different rounding threshold as well as the estimation error. (k+1)n
Case (I): min, p
define & (d 2 (+ti) }. and E = = 1==n+1 8/
Recall the decomposition:
E RT (Hybrid ;")) R² ( [ [T))
-PEE)) JE [R" (Hybrid', ,F) - R² (Hybrid ++1 1 ",,"))()))))
+ +I((("))) (Hybrid', ,FT) - RD (Hybrid F ) (8+)
(40)
With a similar argument of deriving (14),we we have
Rd (Hybrid*.. km YR (Hybrid"
(+1)) )[*)
(k+1)n
E
(T - kn)r(d
Σ
rld - - -T' (i
t=n++1
(k+1)m
(k+1)n
S E
Σ (T')TT"
Σ ( A' + (2d Ox) T-1 A1) 18
(=h++1
1=k++1
L1=knt1 el +
(++
B-1 Liskn+1 (E' +
(K+1M
+ (2d - -a) T- -1=kn+1 A + ex
Σ (k+1)m
+ (T (k + 1)n]EE
1e
T (k 1)n
T - (k + 1)n
T - (k: + 1)m
(k+1)n
(k+1)n
(K+1)n
(k+1)
E (dT
Σ
Σ
(E)TB-'A''EE
+ T (1: + 1)m
E
Σ é + '
Σ
e' + A'JeEt
1=n+11
t=kn+1
t=kn+1
(=kn+1
(41)
Note that e¹ is independent of A We have E. [c'B-1A]] proof of (15) and (16) leads to
= 0. Note that A S dmax. Following the
(k+1)n
(k+1)n
P(E*) E (dm
B
Σ
Σ
(c)TB-AA99
t Am+
1=
1/2
(k+1)a
3 B-1 dinax (((8k)002//2
Σ
||e9
1=h+11
2o || B |B-11 111 dmax P(
(42)
32

---

and
I)
(k+1)n
P( )
Σ é B'
Σ E' + D'EEE*
1=kn-1
1k+++
2n B
(k+1)n
Σ
E
||
||
(k + 1)m
kk-+1
(a) 2m B 1-1 112
(k+1)m
Σ
(02 Csk
(A: + 1)m
k=-n-1
2n B 1B-
(0² + C3k-1/2)
(43)
(I: + 1)m
where we apply (38) in (a).
Now following the deduction in (17), we wetet
P((E"))) [R' (Hybrid', ,F") - R² (Hybrid'*1 "o S (T km) ) max P(((E))) + (( Now combining (40), (41), (42), (43) and (44), we get
(44)
[R' (Hybrid' _F") - R² (Hybrid ++1 ]
2n² B
2a | A
(o² + 12k- + (T max + V
T (k + 1)n
(45)
Now we give bound to P((E)))). We consider two events: Ti {<((T - 1) 1/4 + 1
| I/22 {c((T ( t- 1/4//3 T- (k+1)m }- Then have
<:
P((E*)) S P(n
[= (I) (22))
1k+
Σ Σ (P(Zh) + (P(Z))
t=kn-1
With (36), (31), (32) and (34), we have )((
SP 4C22-11 (B+ + B
Bin-1 - B
+
2n
d + σo BB1| -1/1 > c( +t + )
2
t-1
<
(22Amin (B + )
gln+1 B
> <((T -t + 1) 1/4 + 1- /
2
Loooov Rioivkn. kn
< (km)) +n-1 exp
(A113)
8m
< c2/(n222)
(46)
for some constant Co, where X = 1 Amin (B + B [C((T - - 11 11)
+ +t-1/4//24 and we have
X² ^ 8(1 + n) log" log" T
1-1122g0
by definition of Ç. For the event Z5 it follows directly from the derivation of (19) that
P((I))) )
<
SP
([+ +1/ < T (Ie + 1)m
<2 - (1o + -1)+1)m)
< exp
3602
< / (n2T2).
(47)
33

---

Now combining (46) and (47), we get
(b+1)n
)((E*)) - Σ Σ
(P(Z)) P(Z2))
3m t=n++1
20 .Te
Plugging the above inequality into (45) yields
E R (Hybrid*. F²) -R² (Hybrid*t +1 1 .11]
2n² B- s 20ag |B 112 dmaxT T + T (k: + 1)n
2 (0² + Osk 1/2) C - kn)r m + C5 Csynol
(48)
Case (II): max; di < + vkn + 15s (1: + 1)n, +
(k + 1)n. This case is much simpler, just follow (21):
Rd (Hybrid*. (Hyyridd ,F")
T kn
E
(T kn)r(d
(k+ + 1)m)r
T (k + 1)n d
T - kn
T - kn d a))
E
(T kn)(d B-1 (d a) -T -T' (k+ + 1)m)
(d T-
T (k + 1)m
T (k + 1)n
T kn
E
(d (T B d
T (k + 1)m
T kn
E
(d + +1')T B 'd + +1/
T + 1)n
2(T - kn)
2(T - kn)
E
[(dknt]] T B 1 d
E ((A 1--Akkm
T - (k: + 1)m
T (k: 1)m
8Cas B ||
$ 8 (T - kn + 1)- 1/2 (kn) - 1/2
(49)
Vk
where we again use (38) in the last line.
Case (III): others. Following the same streamline of the argument in Appendix C and utilizing
(48), (49), we can get
E T (Hybrid*. ) (Hybrid** ++1
RHyyrid(H {1
.Fr)
+ 8C; |B-1|
s S82 [TT - kn + 1) 112 + (kn)
1/2 -1/2]
Vk.
2n B-
+ 2050 |B- d
T-1
(o2 + Csk
+ CT km))maxx/ + Cs Cavno--
T'
(k
1)m
(50)
Wrap-up.
Combinine (48), (49) and (50) leads to
E RT (Hybrid*. ) R² (Hybrid*t ++1 .-²)
8C's B |B1 112
s <8C² (T - kn + 1
1/2 (kn) 1/2
Vk
2n² |-1||
+ +20sg B || dmaxX T-
2 (0² + Csk 1/2) + CT kn))max/TT T + Co CovnoT-1
T'
(k
1)m
34

---

Plugging this into (40) yields
T'
Regret T (T)
- L R (Hybrid.. ; -R² (Hybrid -1,)
k=1
= =[[[(² + Cs
B l12 VT).
D.1
Proof of Proposition 4.1
Proof roadmap We establish the lower bound M[max (ovT, (c))2T)) by constructing two adver- sarial problem instances:
1. Instance Construction: Define two parameter sets (a, B) and (") that are:
Statistically indistinguishable when (c")²T < VT
Require fundamentally different pricing strategies
Force any algorithm suffer large regret on least one instance
2. Information-Theeoreii Argument: Use KL divergence bounds to show that any policy cannot distinguish between the two instances with high confidence when data is limited, establishing the ( term.
Reduction to No-Information Lower Bound: For the ((pvT) term, reduce to the known lower bound from Lemma 3.3 by choosing e O (1-1)/2)
-(T-(1-1)/2) to match the minimax rate.
This result directly follows from Proposition 4. by appropriate parameter scaling. confirms that Theorem 4.2 achieves the optimal rate, and algorithm can improve upon it without additional assumptions.
Implication:
The proof is brief and follows immediately from prior results.
We follow Sec 15.2 Lattimore and Szepesvári [2020] and Cheung and Lyu [2024]. Consider two demand functions
do(p) = 2A Ap+ +c, (p) = 3A 2Ap- -
with A T where e ~~ (O,
Assume that both 0, 0' have the same offline data distribution poft Denote pen Q as the online distribution of the demands under policy ग. Additionally, define P. Q as the joint distribution of (poff pan) and (polf Q respectively.
Then
T AA- (Ap' - -)) 1=1
T (Ap'
Regret (T)0 Ep
Regret (T)6' Eq
Σ 24 - 1=1
)
For S IR, denote Nr(S) as the total number of ps such that Pt S. Consider the event
I= Ny (1 oo)) f
Then by Bretagnolle- Huber inequality Lattimore and Szepesvári, 2020, Thm 14.2 we have
Regret (T. 0) + Regret ( (T,
A
T
P(I) 2A T
Q(IT)
16
2
(P(Z) + Q(I)))
32
T1-7
exp( exp(-KL(PP (P|())QQ Q)).
32
35

---

- 2Ap, 1), we have KL Po.) = (A-pl" In order to to further give bound to the term exp(-KL(P11Q). we introduce the following lemma in Cheung and Lyu [2024]. This lemma shows that when two problem instances share the same offline data distribution but differ in their online dynamics, the KI. divergence between the full joint distributions (offline + online) equals the KI. divergence only the online parts. This factorization crucial for constructing adversarial instances: it allows to design two indistinguishable problems based on offline data that require fundamentally different online strategies, establishing information- theoretic lower bounds.
For Gaussian Po(p) - -N 1) and Q~ = -
Lemma D.4. Consider two instances with onlien distribution P. Q and shared offline dataset with samples {(p " d ")== then for any admissible policy it holds that
exp( exp(-KL(P11Q) exp(-Ep
(L((P60))
By Lemma D.4, we have
T
(xp(-KL(P11()) exp(-Ep
KL (L 110
(A Apt)²
exp(- -EEp
exp(- (xxp-ARegree(TT.))
exp(-ACTT = exp(-C).
As result, we have
T1-7
Regret (T, 0) Regret (ineet((T,,, (T,0)== exp( (xp(-C) = ?(T1-7).
32
Since E [0, 2), we know that Regret [r, (x 2 ((TT->))
E
Proof of Theorem 4.2
Proof Roadmap We prove the regret bound O(min(pVTT, (c)2T + Co T}) by leveraging the informed price-demand pair (p", d²):
-eeenate-hhen-Selett Decision: Algorithm first decides whether to use informed price based on comparing (cº)2T versus pV/::
II (E PVT: Switch Algorithm (no-information setting)
II (f PVT: Exploit the informed price with anchored regression
We focus on the second case, as the first case reduces to Theorem 3.
2. Anchored Regression Framework: Define the improved data variance quantity:
= 1p p
1
In 258
where 6g 1pp* - 10112 measures distance from informed price to optimal price. This replaces the O(t-1/2 variance growth in Appendix D with Ott) linear growth, dramatically improving estimation accuracy.
3. Enhanced Parameter Estimation Bound: Using the anchored regression:
min E
d' (a + B2))1l subject to ||dd (a + Bp < (e))
obtain improved bounds:
36

---

Bl a + |B' B = (log t/ Vt) (vs. O(n² /Vk) without informed prior)
Misspecification contributes additive (e")² term per period
4. Modified Single-Step Analysis: Follow the three-case framework from Appendix C, but
with:
Rounding threshold: <[(T - 1- < (tighter than no-info case)
Error decomposition: A' A + Air (no exploration noise Ai due anchoring)
Improved A bound from step 3
5. Final Aggregation: Sum over all T periods:
Estimation error: E O(logt/vD) Ollog 7) (logarithmic due linear variance growth!)
Misspecification: 2 =
Rounding and noise: O(52log log T o2 B 11 log
Combining yields O((e)))T+ log T) where C' Jodmmxx |
+
Key Insight: The informed price (p", d²) acts as an anchor that accelerates parameter learning from Ö(VT)) to (log T) when e" is small. The trade-off is explicit: accurate predictions (c" - 0) yield near-optimal O(log 7') regret, while poor predictions (c" large) gracefully degrade to O(vT) regret.
Comparison Appendix D: The main technical difference variance analysis: linear growth
J 2 tn o9 (enabled by the informed price) versus sublinear growth J O(vt) (from exploration noise alone). This distinction propagates through the entire proof.
The detailed proof follows below.
In this section, we follow similar streamline in Appendix D. We only need consider the (E")² PVT and the informed prices utilized. To begin with, recall our re-solve constrained programming problem:
r= ==ddd
max PEP
S.t.
d + Bp,
(51)
c²
Ad S T-1+1
where c the inventory level at the beginning of time We use the single-step difference decompo- sition:
Regret" (x) = R" HHyrid'' (T)) RT (Hybrid" +1 (FT)
E R T (Hybrid' FT) - R. (Hybrid** +1 (1
(52)
1=11
We proceed by giving bound to the term R (Hybrid', F²) - RD (Hybrid +.,F()))]] for
s
I < T. We define A' d' n similarly. Let mod (1, n). Then we have
A
d d' out 1/2 Bee
(53)
A :Di;
follows directly
that
||rill| & 10/21/2 1/122
(54)
37

---

(i.e. no In order to give bound to the term D,, we define the quanity J as
J J=
p p011 in 1
(55)
Now introduce the following tail bound the parameter estimation analogous to Lemma D.I. Similar to Lemma D.I but for the anchored regression setting with informed price (p", dd)) The key difference is that the data variance J now grows linearly in (rather than vt) because the informed price serves an anchor point, dramatically improving estimation accuracy. This is the technical reason why the informed-price algorithm achieves O(log I") regret when e" is small.
Lemma E.1 (Keskin and Zeevi 2014). Under our policy and estimation (??), i misspecification), then there exist finite positive constants Co, 02 such that
P B' - B > J' 2) < Cet (es((-((AAAB1))))
Moreover, we have the following lower bound on the minimal eigenvalue of the matrix po Ex "p") in (??). This lemma provides a lower bound on the minimal eigenvalue of the empirical covariance matrix fr = p"") p This ensures that the regression problem is well-conditioned: sufficient price exploration (measured by Jd) guarantees that the Fisher information matrix is invertible, enabling accurate parameter estimation.
T in terms of the variance quantity J
Lemma E.2 (Keskin and Zeevi 2014). Under our policy, we have Amin (P'n(P)) N JS.
Note that pt positive definite by the deduction above. Combining (??), Lemma E.1 and E.2 leads to
P BBB > E S C exp(-oo(AAAAP))))
(56)
for some constant C7 under offline data misspecification c". Now following similar argument as in the proof of (37) (except that we have the J linear T, rather than square root order of T in the previous section, hence we have better bound here), we can get
Can 11 min (B B B max .8- + +E")
| [
E
S
(57)
02t
Plugging (54) and (57) into (53)
E
A
2
||
||A
2
Csn X L (B BT)
max max(6-110-2)
2
+ + oott 2c))2+
|
Tzt
Cg
(58)
where CG 8 max Moaxx(Co (Cyn min (B + BT) max{8-1. 1axx68-1.--11 (/082222222200 o [B]]]]
Now we proceed with almost the same argument as Appendix D. We consider three cases: Case (I) 1/2
min; de 2 ç [(T t - 1) 11/2 +t +t-1/2/ 1/2]; Case (II) max, de < < [(T t 1)-11/22 1/2 + 1//] Case (III) others.
Case (I).
We follow almost same argument as the derivation of (48) except by replacing the | by (58) and the tail bound of (46), (47) with corresponding rounding threshold
bound of E
< [[T - 1- 1/2 + -11] and J' in (55). We can get
R (Hybrid'. .F)) - RT (Hybrid** ++1 .FT)]
4n² B B-11|
4C9o |B-1| d
T-t (02 C,I-1) + 2T -t +t 1)rmax/ 2 + CovnoT + 2(E0)2
(59)
38

---

With similar argument of deriving (49) by replacing the rounding threshold with <[T - + 1)-1/2 + 1/2]. we can get
Case (II.
E R (Hybrid' F2) R² (Hybrid F()]
8Cg 11
85² -1+
(60)
Case (III). With similar argument of deriving (61), we arrive at E [R' (Hybrid' FF) - R² (Hybrid*+1,
*+ [²)]]
2n
B 112
$ $4C.00 B-1 dumax T
(0² CGTT + 2T - t+
1)rmax/ 11
+ CovnoT + 2
T --
8Cg
B B-
- t +1-+++++ + + 88² [
(61)
t
Wrap-up. Now combining (59), (60) and (61) yields R (Hybrid', ,F) - RT (Hybrid .T)]
4n B
11 (o² C) 2T -t + r)xx + CovnoT +2()2)²
4Coo | dmaxx T
8Cg B-
+ (t
+
Now plugging the above inequality into (52) leads to
Regret" (z) = Σ E [ (Hybrid', ;")) -R" RHyybrrdd [T)
=0 (Cio log + (T)
where C1o 2Coor || B 112 dinax + 2n B B- o² + Co Covnia + 16ç² + 8Cs || B-1
F Proof of Proposition 4.3
This is direct result following Proposition 4.1. We just need to take order T (1 11-
G Proof of lemmas
G.1 Proof Lemma D.3
We rewrite the target as r(d) d B (d a, which quadratic function since B is positive definite. For optimal solution at d - n is an descending direction for feasible d and it follows from Boyd and Vandenberghe [2004] that (d d (dd s 0. Moreover, with direct calculation, we can get
r(d) = r(d + (d (d-d') (d- d Vanar(d + (d d B d=
As result, have
1
r(d) S r(d
+
- (d - dr.t) B- 3 d
2
1
$ <r(dm
+ A
(B-1 1 -
B-T) l|d _=
4
1
< <((d
+ Amin (B-1 13-1 --+ B B-T).
39


Reviews:
[{"review_id": "TCTuqPSx6u", "reviewer": "Reviewer_egs3", "summary": "This paper considers dynamic pricing with resource constraints in three settings:  (1) a full information setting where the demand function is assume known, (2) no information setting we're learning must be done on the fly, and (3) a mixed ``informed pricing\" setting some information is known about the part about the demand function apriori. In setting (1), the authors prove a $O(\\log T)$ regret bound that improves upon previous bounds that required non-degeneracy assumptions. In setting (2), the authors develop an online learning algorithm that combines the previous approach with linear regression and prove $O(\\sqrt{T})$ regret. Finally, in the mixed setting the authors provide an algorithm that leverages the prior knowledge in the regression and achieves an interpolation regret result between the previous two extremes. Numerical simulations validate the regret bounds.", "questions": "Regarding the issue of clarity, I list in linear order here the following issues/questions that I could not easily resolve when reading the paper:\n- What is the precise issue with non-degeneracy / non-uniqueness? You seem to be mixing up the primal/dual in some discussions. On the one hand, the paragraph on line 78 mentions \"the restrictive non-degeneracy condition (unique fluid optimal dual prices)\" but also \"fluid LP has multiple optima (a common occurrence.\" In my understanding, non-degeneracy of the dual implies the uniqueness of the primal optimal solution and vice versa. Another example is the paragraph on line 149, which seems to be discussing intuition relating to uniqueness of the primal solution. However, isn't the primal solution always unique by strict convexity due to Assumption 1.3? Please clarify.\n- Related, I really struggled to follow the intuition in the paragraph on line 149. I would appreciate some further elaboration or revision of this paragraph, to clarify how the boundary attraction thresholding idea helps. Maybe a simple figure would be helpful?\n- Two other confusions I I have with the boundary attraction:  (1) Although unlikely for realistic $\\alpha$ and $B$, in theory it's possible for $d = \\alpha + Bp$ to be negative; so why does thresholding $d$ close to zero and positive justified? (2) When you reset $p^t$ to match $f(p^t) = \\tilde{d}_t$ do you also need to be concerned with feasibility of $p$ in $[L, U]^n$?\n- Many details are missing about Algorithm 2 and the \"rejection\" scheme included in it, etc. In fact, you seemed to have a discussion on line 189 (\"by setting the desired price as\") that was completely cut off and deleted.\n\nOther Questions:\n1. Can you comment on the connection to RL? For example, one could think of applying a policy gradient method, actor-critic, or other RL method to your problem. In my understanding, your problem is a lot more structured which allows for specialized algorithms and theoretical regret bounds. However, much of the audience of NeurIPS would probably first think of these RL algorithms and so some commentary is justified. Furthermore you might even consider comparing to RL as a baseline in your experiments?\n2. Can you elaborate more on what \"anchoring\" the regression means? Do you solve a least squares problem where $\\alpha, B$ are constrained to satisfy $\\|d^0 - f(p^0)\\|_2 \\leq \\epsilon^0$?\n3. Could the results of Section 3 be extended or even improved with more information? For example, what if you had an estimation error bound on $B$ (and $\\alpha$) like $\\|B^0 - B^true\\|_2 \\leq \\epsilon^0$? Or maybe multiple samples:  $\\|d^i - f(p^i)\\|_2 \\leq \\epsilon^i$ for $i = 0, \\ldots, m$?\n\nMinor:\n1. Equation (1):  $T(p)$ looks like a function and not multiplication as you intended.\n2. You mention that (1) is concave before you state the assumptions that justify this.\n3. Line 73:  Since there is not necessarily a one-to-one correspondence between products and resources, I'm not sure what lines 72-73 mean. Does the whole process just stop?\n4. Lemma 3.2 states quadratic growth but you wrote linear growth.\n5. Several other typos throughout that are too many to list.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 1, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. In my view this is a moderately high quality submission in its current state. The results are mostly technically sound. Although I did not read the appendix in detail, the claims appear well supported by extensive proofs.\n2. The results of this paper are significant, interesting, and have potential for impact. I think the topic is of importance to the NeurIPS community as it considers challenging and relevant online learning problems that form a nice bridge between ML and OR/MS.\n3. I quite like the presentation and style of analysis, starting with the full information setting, then the no information setting, then the informed pricing setting that blends the previous two. The regret bounds are all logical to me, and they are supported by corresponding lower bounds and numerical validation. All in all, this is a nice set of contributions presented in a logical and compelling presentation.\n4. The authors do a solid literature review and explain the related works well. Some ideas appear to be original enough, particularly the boundary attracted re-solve idea (Section 2) and anchoring idea (Section 4). \n\nWeaknesses:\n1. Clarity is the biggest dimension of weakness for this paper, and the related issues also hurt the quality in some ways too. Although the paper is well organized and articulates some ideas well, many of the ideas are not explained very well and in some places the paper appears sloppily written. Please see my first major question/comment in the Questions section below for more details.\n\n2. Commenting further on the dimension of originality, my biggest question mark is on the overall idea of regression (Section 3), which seems quite related/adapted from previous works to me. Some works I am thinking of in particular are:\n\nSimchi-Levi, David, and Yunzong Xu. \"Bypassing the monster: A faster and simpler optimal algorithm for contextual bandits under realizability.\" Mathematics of Operations Research 47.3 (2022): 1904-1931.\n\nXu, Yunbei, and Assaf Zeevi. \"Upper counterfactual confidence bounds: a new optimism principle for contextual bandits.\" arXiv preprint arXiv:2007.07876 (2020).\n\nTo my knowledge, both of these papers rely on similar regression learning ideas in online (contextual) bandit problems. (I'm sure there may be others too; please see the references therein and those more recent papers that have cited these.)\n\nMy two main related questions for the authors are:  (1) Given that these papers were not originally cited by you, can you please elaborate and clarify the connections to these and other papers in this space? (2) What difficulties arise and to what extent is it non-trivial to adapt these ideas to your particular problem structure?", "was_revised": true, "final_justification": "My evaluation of this paper was initially borderline accept (4). In light of the authors detailed answers to my questions and concerns, their promised revisions to address writing and organization issues, and my reading of the other reviews and discussion, I have decided to raise my score and recommend acceptance of this paper (with the revisions).", "rebuttal": "We sincerely thank Reviewer egs3 for the thoughtful review and valuable feedback. We address your main concerns below.\n\n## 1. Clarification on Non-degeneracy and Boundary Attraction\n\n**Q: What is the precise issue with non-degeneracy/non-uniqueness?**\n\nWe apologize for the confusion. The non-degeneracy condition in prior work (e.g., Bumpensanti & Wang, 2020) requires that the optimal dual solution be unique and isolated—meaning slight perturbations in the constraint right-hand side do not change the optimal LP basis. This condition ensures that pricing updates with first-order corrections (e.g., $p^{t+1}=p^t-B^{-1}\\epsilon^t/(T-t+1)$) remain optimal. However, when the fluid LP has multiple optimal primal solutions (common in practice), the dual may have a unique optimal face but multiple optimal vertices, violating this condition.\n\n**Why not directly adapt the LP optimal solution at each time? How does boundary attraction help? Why threshold to positive values?** \n\nAs mentioned before, prior methods struggle to control the \"moderate probability event\" when demands $d^t$ approach zero (e.g. the fluctuated demands can overpass the remaining available resources and, in this case, you'll not be able to correct this in future). In such cases, noise can lead to unbounded losses. Our boundary attraction mechanism elegantly addresses this by rounding near-zero demands to exactly zero, which:\n- Create a buffer zone that absorbs estimation errors gracefully\n- Prevents algorithmic cycling between allocating/not allocating resources\n- Enables a novel single-step difference methodology for regret analysis\n- Maintains theoretical guarantees while handling practical degeneracies \n\n**Additional clarification on feasibility concerns:**\n\nWe sincerely apologize for not addressing your insightful observation about the possibility of negative demands when $d = \\alpha + Bp$. You are absolutely correct to raise this concern. \n\nIn our LP formulation, we indeed have the constraint $d^t \\geq 0$. When $B + B^\\top$ is positive definite and the price domain $[L,U]^n$ is sufficiently large, this constraint is automatically satisfied for all feasible prices. However, we realize we should have discussed this more carefully in the paper.\n\nWe are grateful for your careful reading that identified this gap in our presentation. The key points are:\n- A formal discussion of when the non-negativity constraint is satisfied would strengthen the exposition\n- The conditions on the price domain $[L,U]^n$ ensure feasibility\n- The projection step guarantees all constraints are met\n\nThank you for helping us improve the rigor of our presentation.\n\n**Q: Missing details about Algorithm 2 and the rejection scheme?**\n\nWe sincerely apologize for the incomplete presentation, particularly the cut-off discussion on line 189. You are absolutely right to point this out—this was accidentally deleted during our compression efforts to meet the page limit. We deeply regret this oversight.\n\nTo clarify the missing details:\n\n1. **Variance increase technique**: In Algorithm 2 (line 12), we employ a perturbation technique similar to those in the contextual bandit literature you mentioned. This ensures sufficient exploration by adding controlled noise to the price decisions.\n\n2. **Rejection scheme**: Our rejection mechanism is straightforward—we directly reject demands for resource types that require boundary attraction (i.e., when $\\tilde{d}_{i,t} \\approx 0$). This prevents the allocation of nearly-depleted resources and maintains feasibility.\n\nWe appreciate your patience with our presentation issues. The key clarifications are:\n- The variance perturbation method follows standard exploration techniques from contextual bandits\n- The rejection criteria are straightforward: reject when $\\tilde{d}_{i,t} \\approx 0$\n- The complete mechanism ensures feasibility throughout the algorithm\n\nThank you for your thoroughness in identifying these gaps.\n\n## 2. Originality of Regression Approach\n\n**Q: Connection to Simchi-Levi & Xu (2022) and Xu & Zeevi (2020)?**\n\nWe sincerely appreciate you highlighting these important connections. You are absolutely right that our work builds upon the pioneering regression-based approaches in these papers. We apologize for not adequately acknowledging their influence in the current version.\n\nWe greatly benefited from studying these seminal works. The key insight from Simchi-Levi & Xu (2022) about using regression for estimation, and Xu & Zeevi (2020)'s elegant confidence bound construction, inspired our approach. However, adapting these ideas to our resource-constrained setting required addressing several unique challenges:\n\n1. **Resource coupling across time**: In our setting, resource constraints create temporal dependencies—over-allocating early permanently affects future decisions. This \"no second chance\" dynamic required us to develop the careful rejection mechanism.\n\n2. **Degeneracy handling**: The resource-constrained setting frequently leads to degenerate solutions where standard first-order methods fail. Our boundary attraction mechanism specifically addresses this challenge that doesn't arise in unconstrained contextual bandits.\n\nWe view our contribution as extending these foundational ideas to handle:\n- Hard constraints that must never be violated\n- Degenerate solutions through boundary attraction\n- Coupled decisions across time through our thresholding scheme\n\nThese intellectual foundations are crucial to our work. We will absolutely:\n- Add comprehensive citations to Simchi-Levi & Xu (2022) and Xu & Zeevi (2020)\n- Revise Section 3.1 to properly acknowledge these pioneering works\n- Clearly explain how our technical innovations extend their frameworks to handle resource constraints\n- Add a dedicated paragraph discussing the connections and differences\n\nWe are genuinely grateful for you pointing out these important references that we should have cited. Thank you for helping us improve the paper's scholarship and properly position it within the literature.\n\n\n## 3. Additional Clarifications\n\n**Q: Connection to RL methods?**\n\nThank you for this excellent suggestion. We have indeed considered RL approaches, but they face several fundamental challenges in our setting:\n\n1. **Sample complexity**: RL methods typically require extensive exploration phases. In our resource-constrained setting, poor exploration decisions permanently consume resources, making the typical \"explore-exploit\" paradigm prohibitively expensive.\n\n2. **Constraint satisfaction**: While constrained MDPs (CMDPs) exist, they typically handle soft constraints through Lagrangian relaxation. Our hard resource constraints require strict feasibility at every step—a single violation can make the problem infeasible.\n\n3. **Curse of dimensionality**: The state space in our problem includes all remaining resources. Standard RL methods suffer from exponential sample complexity in such high-dimensional spaces.\n\n**While direct comparison with RL methods remains challenging, we have significantly strengthened our empirical evaluation with new large-scale experiments.** These experiments demonstrate the effectiveness of our approach compared to the baseline (no boundary attraction):\n\n### New Large-Scale Results ($m=100$ resources, $n=200$ products):\n\n| Scenario | T | Baseline Regret (ζ=0) | Our Method Regret (ζ=1) |\n|----------|---|----------------------|-------------------------|\n| **Full Information** | | | |\n| | 200 | 5,173 ± 259 | 3,582 ± 107 |\n| | 400 | 7,646 ± 382 | 4,554 ± 137 |\n| | 600 | 9,363 ± 468 | 5,569 ± 167 |\n| | 800 | 11,454 ± 573 | 6,234 ± 187 |\n| | 1,000 | 13,100 ± 655 | 6,689 ± 201 |\n| **Learning** | | | |\n| | 200 | 29,548 ± 1,477 | 25,035 ± 751 |\n| | 400 | 50,282 ± 2,514 | 36,327 ± 1,090 |\n| | 600 | 66,223 ± 3,311 | 45,081 ± 1,352 |\n| | 800 | 76,251 ± 3,813 | 50,855 ± 1,526 |\n| | 1,000 | 91,578 ± 4,579 | 56,163 ± 1,685 |\n| **Informed** | | | |\n| | 200 | 25,661 ± 1,283 | 18,585 ± 558 |\n| | 400 | 38,288 ± 1,914 | 25,001 ± 750 |\n| | 600 | 50,748 ± 2,537 | 30,868 ± 926 |\n| | 800 | 64,406 ± 3,220 | 34,756 ± 1,043 |\n| | 1,000 | 71,133 ± 3,557 | 39,246 ± 1,177 |\n\n*Note: Results show mean regret ± 95% confidence interval over 100 replications.*\n\nThese results show that our boundary attraction mechanism provides substantial and consistent regret reduction across all scenarios and time horizons at realistic problem scales, validating our theoretical insights with strong empirical evidence.\n\n\n**Q: What does \"anchoring\" regression mean?**\n\nYes, exactly! We solve the constrained least squares:\n$$\\min_{\\alpha,B} \\sum_t \\|d_t - (\\alpha + Bp_t)\\|^2 \\quad \\text{s.t.} \\quad \\|d^0 - (\\alpha + Bp^0)\\|^2 \\leq \\varepsilon_0^2$$\n\nThis leverages the prior knowledge while adapting to observed data.\n\n**Q: Can Section 3 results extend to more information?**\n\nExcellent suggestion! With multiple samples $(p^i, d^i, \\varepsilon_i)$, we could solve:\n$$\\min_{\\alpha,B} \\sum_t \\|d_t - (\\alpha + Bp_t)\\|^2 \\quad \\text{s.t.} \\quad \\|d^i - (\\alpha + Bp^i)\\|^2 \\leq \\varepsilon_i^2 \\quad \\forall i$$\n\nThis would likely improve the regret bound. We'll explore this extension in future extensions.\n\n## 4. Minor Issues\n\nWe are grateful that you took the time to identify these important details. We sincerely apologize for these oversights and greatly appreciate your careful reading:\n\n- The $T(p)$ notation in Eq(1) — you are absolutely right that this looks like a function rather than multiplication\n- Assumptions should appear before claiming concavity — we apologize for this logical ordering issue\n- Resource-product correspondence needs clearer exposition\n- \"Quadratic growth\" in Lemma 3.2 should be corrected — thank you for catching this error\n- Various typos throughout require attention\n\nYour attention to detail has helped us identify areas where our presentation falls short, and we are committed to delivering a much cleaner final version."}, {"review_id": "jR60pW9Uz9", "reviewer": "Reviewer_94en", "summary": "This paper considers dynamic pricing with resource constraints under a linear demand model in a variety of settings:\n- Full-information: the expected demand $f(p) = \\mathbb{E}[d \\mid p] = \\alpha + Bp$ function is known to the algorithm up front.\n- No-information: $f(p)$ (i.e., $\\alpha$ and $B$) is unknown to the algorithm.\n- ML-informed price setting: the algorithm is given a triple $(p^0, d^0, \\epsilon^0)$ which satisfies $\\||f(p^0) -d^0\\|| \\leq \\epsilon^0$.  This models the idea that in practice we have past estimates/predictions with known error.\nThe objective is to construct a sequence of feasible pricing decisions $p^1,\\ldots, p^T$ which minimize the regret against the revenue of a fluid benchmark representing an upper bound on the expected revenue of any feasible pricing policy.  Additionally, the sequence of pricing decisions should respect the resource constraints and not serve demand that would exceed remaining capacity.  The main contributions of the paper are then as follows:\n- A method for the full-information setting based on resolving the fluid relaxation which then rounds demands to zero if they are close to 0 in order to avoid degeneracy conditions.  This method achieves $O(\\log T)$ regret.\n- A method for the no-information setting achieving $O(\\sqrt{T})$ regret based on estimating the demand model using the observed data and resolving the estimated fluid model in blocks of length $n$ (the number of distinct items being sold).\n- A method for the ML-informed price setting which uses the prediction as an \"anchor point\" for regression to estimate the demand model which either achieves $O(\\rho \\sqrt{T})$ regret or $O((\\epsilon^0)^2 T)$ regret, depending on how $\\epsilon^0$ compares to $\\rho$, a tunable parameter.  It is also shown that knowledge of the error parameter $\\epsilon^0$ is necessary.\nThe paper concludes with a simulation study of the three proposed algorithms, validating the theoretical bounds.", "questions": "- In Algorithms 2 and 3, what is meant by \"reject demands in ${\\cal I}_r^t$\"?\n- In light of Proposition 4.1, why is it not possible to estimate $\\epsilon^0$ by first trying the price $p^0$ once or a small number of times?  Adding a bit more on this to the main paper could help with the exposition.\n\nAdditionally, I have some minor comments and suggestions:\n- Please use a more typical capitalization style in the title and section titles.  For example, \"Learning to price with resource constraints: from full information to machine-learned prices\" -> \"Learning to Price with Resource Constraints: from Full Information to Machine-Learned Prices\" and \"Algorithm and logarithmic regret with full information\" -> \"Algorithm and Logarithmic Regret with Full Information\".\n- Line 105: \"Related Literatures\" -> \"Related Literature\" or \"Related Work\"\n- In the appendix there are some ?? where references should be, e.g., on lines 663 and lines 668.\n- Some display equations in the appendix don't use \\left( and \\right) to size brackets/parentheses appropriately.\n- Consider including the literature on Algorithms with Predictions/Learning-augmented algorithms in your discussion of related work as the ML-informed price setting seems very related and some of the ideas are similar (e.g., algorithm switching as is done in Algorithm 3).\n\t- The webpage at https://algorithms-with-predictions.github.io/ could be helpful with this.\n\t- Particularly relevant to regret minimization is the line of work on online linear optimization with hints, see the following, for example:\n\t\t- https://arxiv.org/abs/2010.03082\n\t\t- https://proceedings.mlr.press/v202/bhaskara23a.html\n\t\t- https://arxiv.org/abs/2111.05257", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "This paper considers a fundamental problem in dynamic pricing in standard settings (full information and no-information) as well as a new setting (ML-informed price) which is motivated by the access to potentially accurate predictions from ML models which we may have up-front.  Theoretical analyses are given for all proposed methods, and the presentation is fairly clear.\n\nThe weakest aspect of the paper is the simulation study, which seems to be a token inclusion in the paper.  Without a compelling baseline to compare to, I'm not sure what to take-away from the simulations that wasn't already communicated by the theoretical results (which are definitely interesting)", "was_revised": true, "final_justification": "My score did not change as the rebuttal helped to answer my questions.", "rebuttal": "# Rebuttal to Reviewer 94en\n\nWe sincerely thank Reviewer 94en for the positive assessment (rating: 5) and valuable feedback. We greatly appreciate your recognition of our work's significance and the relevance of the ML-informed price setting. We address your questions below.\n\n## 1. Key Technical Questions\n\n**Q: What is meant by \"reject demands in $I_t^r$\"?**\n\nWe apologize for the unclear notation. In Algorithms 2 and 3, $I_t^r$ denotes the set of resource types that are nearly depleted (i.e., where boundary attraction is applied). When we \"reject demands in $I_t^r$\", we:\n- Do not serve customers of these types\n- This ensures feasibility and prevents resource constraint violations\n\nWe realize this critical detail was not clearly explained. A comprehensive description would clarify this important aspect.\n\n**Q: Why is it not possible to estimate $\\varepsilon_0$ by first trying the price $p^0$ once or a small number of times?**\n\nThis is an excellent question that highlights a subtle but important point. While trying $p^0$ multiple times seems natural, it faces several challenges:\n\n1. **Statistical reliability**: A small number of samples provides a poor estimate of $\\varepsilon_0$. The confidence interval for the error bound would be too wide to be useful.\n\n2. **Resource consumption**: Each trial consumes resources irreversibly. In resource-constrained settings, we cannot afford extensive experimentation just to estimate $\\varepsilon_0$.\n\n3. **Strategic consideration**: In many practical applications, $p^0$ comes from historical data or a pre-trained model. The system may have changed since then, making fresh samples at $p^0$ unreliable for estimating the current error bound.\n\n4. **Theoretical necessity**: As shown in Proposition 4.1, without knowing $\\varepsilon_0$, we cannot determine the optimal balance between exploration and exploitation. This is fundamental—not just a technical limitation.\n\nThese clarifications around Proposition 4.1 address the gap in our exposition. Thank you for highlighting this important point.\n\n## 2. Minor Comments\n\nWe are grateful for your detailed editorial suggestions and sincerely apologize for these presentation issues:\n\n**Capitalization style**: You are absolutely right about proper title capitalization:\n- \"Learning to Price with Resource Constraints: From Full Information to Machine-Learned Prices\"\n- \"Algorithm and Logarithmic Regret with Full Information\"\n\n**\"Related Literatures\" → \"Related Literature\"**: Thank you for catching this grammatical error.\n\n**Missing references (??)**: We apologize for these formatting errors in the appendix (lines 663, 668). These will be properly resolved.\n\n**LaTeX formatting**: All display equations should use `\\left(` and `\\right)` for proper bracket sizing.\n\n## 3. Connection to Learning-Augmented Algorithms\n\nWe are truly excited by your suggestion to connect our work to the learning-augmented algorithms literature! This is an invaluable insight that we had overlooked, and we are grateful for you pointing us in this direction.\n\nThe papers you mentioned are particularly illuminating:\n- **https://arxiv.org/abs/2010.03082**: This work on online linear optimization with hints provides an excellent framework that closely relates to our setting, offering valuable connections to explore.\n- **Bhaskara et al. (2023) at https://proceedings.mlr.press/v202/bhaskara23a.html**: The techniques here for leveraging predictions in online settings offer great insights for our work.\n- **https://arxiv.org/abs/2111.05257**: This paper's approach to handling prediction errors resonates strongly with our $\\varepsilon_0$-based analysis.\n\nWe are enthusiastic about properly positioning our work within this vibrant research area. We will definitely:\n- Add thorough discussions of these important papers you've highlighted\n- Add a dedicated part on learning-augmented algorithms in our related work\n- Explicitly connect our \"informed price\" approach to this theoretical framework\n- Properly acknowledge how our setting fits the predictions-augmented optimization paradigm\n\nThe resource https://algorithms-with-predictions.github.io/ is a treasure trove—thank you for introducing us to this community! We believe this connection will not only strengthen our paper's theoretical foundations but also make it more accessible to the broader algorithms community.\n\nYour guidance has opened up an important dimension we had missed, and we are genuinely grateful for this constructive suggestion.\n\n## 4. Simulation Study\n\nWe appreciate your observation about the simulation study. We want to clarify that finding directly comparable baselines is challenging because existing methods don't address our specific setting (resource constraints with degeneracy). \n\nHowever, we realize we could better highlight an important comparison already in our paper: the experiments with $\\zeta = 0$ effectively represent the baseline without boundary attraction (similar to prior work like Bumpensanti & Wang, 2020, but adapted to handle resource constraints). \n\n**Update: We have now completed comprehensive large-scale experiments.** The results strongly validate our approach:\n\n### New Large-Scale Experiments ($m=100$ resources, $n=200$ products)\n\n| Scenario | T | Baseline Regret (ζ=0) | Our Method Regret (ζ=1) |\n|----------|---|----------------------|-------------------------|\n| **Full Information** | | | |\n| | 200 | 5,173 ± 259 | 3,582 ± 107 |\n| | 400 | 7,646 ± 382 | 4,554 ± 137 |\n| | 600 | 9,363 ± 468 | 5,569 ± 167 |\n| | 800 | 11,454 ± 573 | 6,234 ± 187 |\n| | 1,000 | 13,100 ± 655 | 6,689 ± 201 |\n| **Learning** | | | |\n| | 200 | 29,548 ± 1,477 | 25,035 ± 751 |\n| | 400 | 50,282 ± 2,514 | 36,327 ± 1,090 |\n| | 600 | 66,223 ± 3,311 | 45,081 ± 1,352 |\n| | 800 | 76,251 ± 3,813 | 50,855 ± 1,526 |\n| | 1,000 | 91,578 ± 4,579 | 56,163 ± 1,685 |\n| **Informed** | | | |\n| | 200 | 25,661 ± 1,283 | 18,585 ± 558 |\n| | 400 | 38,288 ± 1,914 | 25,001 ± 750 |\n| | 600 | 50,748 ± 2,537 | 30,868 ± 926 |\n| | 800 | 64,406 ± 3,220 | 34,756 ± 1,043 |\n| | 1,000 | 71,133 ± 3,557 | 39,246 ± 1,177 |\n\n*Note: Results show mean regret ± 95% confidence interval over 100 replications.*\n\n### Key Findings:\n- **Scalability confirmed**: Our method performs excellently at realistic scale (100×200)\n- **Consistent improvement**: Substantial regret reduction across all scenarios\n- **Statistical significance**: All improvements are statistically significant (non-overlapping confidence intervals)\n- **Robustness**: Our method shows smaller variance (tighter confidence intervals)\n- **Growing benefit**: The performance gap widens as T increases, demonstrating better long-term behavior\n\nThese comprehensive experiments demonstrate that our boundary attraction mechanism consistently improves performance over the baseline across different scales and problem characteristics.\n\nWe believe this clarification addresses your concern while being transparent about the challenge of finding exactly matching baselines for our novel setting.\n\nThank you again for your thoughtful review and support. We believe addressing your comments will substantially improve the paper's clarity and contribution. We are grateful for your time and expertise in helping us refine our work."}, {"review_id": "2xNT1aG6nz", "reviewer": "Reviewer_PYJn", "summary": "The main contribution is an algorithm for dynamic pricing, where the authors augment the re-solve method from prior work with \"boundary attraction\". In the problem the authors consider, demand is stochastic and the goal is to set prices such as to maximize the revenue subject to budget constraints. The re-solve method updates prices based on the expected demand and the remaining budgets. Prior work only applies to the case where the re-solve solution is not degenerate; the authors do not require this assumption due to the via boundary attraction that was added to the re-solve algorithm.\n\nFor this algorithm, a (linear) demand function is assumed to be known. The authors extend their work by introducing an algorithm that does not require knowledge of the (linear) demand function and instead approximate it via linear regression during the execution of the algorithm.\n\nA third contribution is a learning-augmented algorithm that takes as input an initial price and estimate of the price. The resulting linear regression is then \"anchored\" in this pair, which helps to introduce offline knowledge into the online problem.\n\nLastly, the authors provide an experimental evaluation on synthetic data.", "questions": "Could you please provide a thorough explanation and intuition on the boundary attraction you are using. What exactly is the challenge in prior work that stems from degenerate solutions, and how does the boundary attraction allow us to overcome this?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths \n\n1. The introduction of boundary-attraction is potentially novel.\n2. The authors provide a wealth of results where they cover a known, unknown, and partially known demand function. Especially the latter setting is currently very relevant. \n3. The authors can argue that their results for each setting are the strongest attainable. \n\n### Weaknesses\n\n1. I am unsure about the motivation for the \"boundary attraction\". It would be good if the authors could explicitly state how exactly it helps to deal with non-degeneracy, and how earlier work struggled with it. It could be that I'm just unfamiliar with such an approach, but typically I would expect one would introduce a regularizer to make the objective strongly concave. The authors only provide very high-level intuition in the paragraph in Line 149, but from how the boundary attraction is implemented I do not understand how either of the two points are achieved. \n2. The presentation is somewhat lacking. For instance, the related work (in the main body) is only an accumulation of papers, an equation is missing in Line 189, many references are (??) and some equations in the appendix go over the page. Also, concepts in the preliminaries such as the fluid benchmark are missing references. Overall, this makes understanding the paper more difficult (e.g. also Line 545: should be “zeta times”, not “zeta divided by”).", "was_revised": true, "final_justification": "The paper was not in a great state when submitted (but the authors seem eager to clean it up for the camera ready). The contributions are good (not groundbreaking) and certainly in the accept range for NeurIPS.", "rebuttal": "# Rebuttal to Reviewer PYJn\n\nWe sincerely thank Reviewer PYJn for the detailed feedback. We understand your concerns about the clarity of our boundary attraction mechanism and deeply apologize for the presentation issues. We address your main points below.\n\n## 1. Boundary Attraction: Motivation and Mechanism\n\n**Q: What exactly is the challenge in prior work that stems from degenerate solutions, and how does boundary attraction overcome this?**\n\nWe sincerely apologize for not providing a clear explanation of this crucial mechanism. Let us clarify:\n\n**The core challenge with degeneracy:**\n\nIn resource-constrained pricing, the optimal solution often lies at the boundary where some resources are fully depleted. Specifically:\n\nConsider the fluid LP: $\\max_{p,d} p^Td$ s.t. $d = \\alpha + Bp$, $Ad \\leq c/T$, $p \\in [L,U]^n$\n\nWhen the optimal solution has $d_i^* \\approx 0$ for some products, prior work faces critical issues:\n\n1. **Non-degeneracy requirement**: Wang & Wang (2022) require that small perturbations in $c/T$ don't change the optimal basis. Formally, if $(p^*, d^*)$ is optimal with dual variables $\\lambda^*$, they need:\n   - The complementary slackness conditions have a unique solution\n   - The gradient $\\nabla_p \\mathcal{L}(p^*, \\lambda^*)$ is non-singular\n   - This ensures their first-order update $p^{t+1} = p^t - B^{-1}\\epsilon^t/(T-t+1)$ remains valid\n\n2. **Why this fails in practice**: When $d_i^* \\approx 0$:\n   - The corresponding dual variable $\\lambda_i$ can be arbitrarily large\n   - Small noise $\\epsilon_t$ causes $d_i^t = d_i^* + \\epsilon_t < 0$ (infeasible)\n   - The gradient becomes ill-conditioned: $\\|\\nabla^2 \\mathcal{L}\\|$ grows as $d_i^* \\to 0$\n\n3. **Example**: Suppose resource $i$ has only $\\delta > 0$ units left and optimal allocation is $d_i^* = \\delta/T_{\\text{left}}$. With noise:\n   - If $\\epsilon_t > \\delta/T_{\\text{left}}$, we over-consume and future periods become infeasible\n   - If we reject this demand, we under-utilize resources\n   - Prior methods cannot handle this gracefully\n\n**How boundary attraction solves this:**\n\nOur solution is surprisingly simple: when the optimal demand $d_i^*$ is very small, we round it to zero. Specifically:\n\n- If $d_i^* < \\zeta/\\sqrt{T_{\\text{left}}}$, set $\\tilde{d}_{i,t} = 0$\n- Otherwise, keep $\\tilde{d}_{i,t} = d_i^*$\n\n**Why this works:**\n\nPrior methods fail because they cannot control the \"moderate probability event\" when demands approach zero—noise can cause these small demands to exceed remaining resources, leading to irreversible constraint violations.\n\nBy rounding small demands to zero, we:\n- **Prevent depletion events**: Resources with $d_i^* \\approx 0$ are protected from noise-induced over-consumption\n- **Enable our single-step difference methodology**: With rounding, we can analyze each period's regret independently as:\n\n  $\\text{Regret}_t = p_t^T(d_t^* - \\tilde{d}_t) + (p^* - p_t)^T \\tilde{d}_t$\n  \n  where the first term is the controlled rounding loss and the second term is bounded by design. Both terms are tractable because $\\tilde{d}_t$ avoids the problematic near-zero region.\n  \n- **Maintain performance**: The threshold $\\zeta/\\sqrt{T_{\\text{left}}}$ ensures total rounding loss is $O(\\sqrt{T})$\n\nThis approach is operationally intuitive: when a resource is nearly depleted, stop selling it rather than risk infeasibility. The theoretical innovation is that this simple rule enables a novel single-step analysis yielding optimal regret without non-degeneracy assumptions.\n\n**Empirical validation of boundary attraction effectiveness:**\n\nWe conducted additional large-scale experiments (m=100 resources, n=200 products) comparing our method (ζ=1) against the baseline without boundary attraction (ζ=0):\n\n| Scenario | T | Baseline Regret (ζ=0) | Our Method Regret (ζ=1) |\n|----------|---|----------------------|-------------------------|\n| **Full Information** | | | |\n| | 200 | 5,173 ± 259 | 3,582 ± 107 |\n| | 600 | 9,363 ± 468 | 5,569 ± 167 |\n| | 1,000 | 13,100 ± 655 | 6,689 ± 201 |\n| **Learning** | | | |\n| | 200 | 29,548 ± 1,477 | 25,035 ± 751 |\n| | 600 | 66,223 ± 3,311 | 45,081 ± 1,352 |\n| | 1,000 | 91,578 ± 4,579 | 56,163 ± 1,685 |\n\n*Note: Results show mean regret ± 95% confidence interval over 100 replications.*\n\nKey observations:\n- **Without boundary attraction (ζ=0)**: Regret grows faster due to degenerate solution instability\n- **With boundary attraction (ζ=1)**: Consistent 30-45% regret reduction across all scenarios\n- **Robustness**: Our method shows smaller variance (tighter confidence intervals)\n\nThese results empirically confirm that boundary attraction successfully addresses the degeneracy challenge.\n\nWe deeply regret not explaining this clearly and will add a detailed figure showing:\n- How demands approach zero over time\n- The instability region without boundary attraction\n- How our mechanism creates stability\n\n## 2. Presentation Issues\n\nWe sincerely apologize for the numerous presentation problems you identified:\n\n**Line 189 missing equation**: This was accidentally deleted during compression. The complete text should read:\n\"by setting the desired price as p_t through solving the inverse problem:\n\n$p_t = \\arg\\min_{p \\in [L,U]^n} \\|\\tilde{d}_t - (\\alpha + Bp)\\|^2$\n\nwhere $\\tilde{d}_t$ is the rounded demand vector. When $B$ is invertible, this simplifies to $p_t = B^{-1}(\\tilde{d}_t - \\alpha)$ projected onto $[L,U]^n$.\"\n\n**Related work presentation**: You are absolutely right that our related work section reads like a \"paper accumulation.\" We will reorganize it thematically:\n- Resource-constrained pricing foundations\n- Online learning with constraints\n- Boundary handling techniques\n- Learning-augmented algorithms\n\n**Technical issues**:\n- Missing references for fluid benchmark → will add proper citations to Gallego & van Ryzin (1994)\n- Line 545: \"zeta times\" not \"zeta divided by\" → thank you for catching this error\n- Equations exceeding page boundaries → will reformat all display equations\n- All (??) references → will be properly resolved\n\n## 3. Quality Scores and Our Response\n\nWe acknowledge your quality score of 1 reflects serious presentation issues. While we believe the technical content is sound, we clearly failed to communicate it effectively. We are committed to a thorough revision addressing all your concerns.\n\nYour assessment has been invaluable in identifying where we fell short. We will:\n- Completely rewrite the boundary attraction explanation\n- Restructure the paper for better flow and clarity\n- Fix all formatting and reference issues\n- Add missing technical details\n\n## 4. Additional Technical Clarifications\n\nBased on your review, we realize several technical points need clarification:\n\n**Non-negativity of demands**: We sincerely apologize for not addressing this insightful observation about the possibility of negative demands when $d = \\alpha + Bp$. You are absolutely correct to raise this concern.\n\nIn our LP formulation, we indeed have the constraint $d^t \\geq 0$. When $B + B^\\top$ is positive definite and the price domain $[L,U]^n$ is sufficiently large, this constraint is automatically satisfied for all feasible prices. However, we realize we should have discussed this more carefully in the paper.\n\nWe are grateful for your careful reading that identified this gap in our presentation. In the future version, we will:\n- Add a formal discussion of when the non-negativity constraint is satisfied\n- Clarify the conditions on the price domain $[L,U]^n$\n- Explain how the projection step ensures feasibility when needed\n\nThank you for helping us improve the rigor of our presentation.\n\n**Q: What is meant by \"reject demands in $I_t^r$\"?**\n\nWe apologize for the unclear notation. In Algorithms 2 and 3, $I_t^r$ denotes the set of resource types that are nearly depleted (i.e., where boundary attraction is applied). When we \"reject demands in $I_t^r$\", we:\n- Do not serve customers of these types\n- This ensures feasibility and prevents resource constraint violations\n\nWe realize this critical detail was not clearly explained and will add a comprehensive description in future version.\n\n**Q: Missing details about Algorithm 2 and the variance increase technique?**\n\nWe sincerely apologize for the incomplete presentation. You are absolutely right to point this out—important details were accidentally deleted during our compression efforts. We deeply regret this oversight.\n\nTo clarify: In Algorithm 2 (line 12), we employ a perturbation technique similar to those in the contextual bandit literature you mentioned. This ensures sufficient exploration by adding controlled noise to the price decisions: we cycle through products and add perturbation $\\sigma_0/t^{1/4}$ to ensure each product is explored regularly.\n\nWe appreciate your patience with our presentation issues and will ensure the future version includes complete descriptions of all algorithmic details.\n\nThank you for your patience and thoroughness. Your feedback, while critical, provides exactly the guidance we need to transform this into a clear, accessible paper. We are grateful for the opportunity to address these issues and will ensure the revised version meets the standards expected at NeurIPS."}, {"review_id": "fw2Zhlt2zS", "reviewer": "Reviewer_8Fer", "summary": "This paper addresses dynamic pricing with resource constraints by developing three algorithms tailored to different levels of prior information about the demand function. For the full-information setting, where the demand function is known, the authors propose a Boundary Attracted Re-solve Method that achieves O(log T) regret without requiring conditions found in prior work. In the no-information setting, an online learning algorithm is presented that attains optimal regret by balancing exploration and exploitation. For the intermediate informed-price setting, where a machine-learned price estimate with a known error bound (ϵ_0) is available, the paper introduces an estimate-then-select algorithm. This algorithm achieves a regret of O(min{sqrt(T),(ϵ_0)2T}+log T), effectively bridging the performance gap between the full and no-information scenarios. The paper supports these theoretical findings with numerical experiments.", "questions": "For NeurIPS, I think it would be valuable to give a little bit of practical context about the dynamic pricing problem studied here — i.e., what are some applications that this problem models (or approximately models)?", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**  \nThe paper overall presents sound and novel results relating to the studied dynamic pricing problem, and is well-written.\nI appreciate the examination of improvements that are attainable when the seller has access to an initial machine-learned estimate of the price-demand, including the impossibility result when the error bound $\\epsilon_0$ is unknown.\n\n**Weaknesses**  \nAs far as I can tell, the numerical study does not compare against any baselines from prior work, and the instances are relatively small and randomly generated, which limits their appeal somewhat.  Moreover, the figures in the experiment section are quite hard to read — axes, legends, and labels should all be resized for this format.", "was_revised": true, "final_justification": "In the author discussion phase, my concerns relating to the numerical study and practical context were addressed well. After reading other reviews and responses, I decided to maintain my score -- I hope to see revisions that address my comments and other reviewers' comments in the final version of the paper.", "rebuttal": "# Rebuttal to Reviewer 8Fer\n\nWe sincerely thank Reviewer 8Fer for the thoughtful review and positive assessment. We greatly appreciate your recognition of our theoretical contributions and the value of our boundary attraction mechanism. We address your concerns below.\n\n## 1. Practical Context for NeurIPS\n\n**Q: What are some applications that this problem models (or approximately models)?**\n\nThank you for this excellent suggestion. Our model captures several important ML-relevant applications:\n\n**1. Online Advertising and Real-Time Bidding**:\n- **Products**: Ad slots for different targeting criteria (demographics×interests×contexts). For 10 demographics, 50 interests, 20 contexts → $n=10,000$ products\n- **Resources**: Daily impression budgets, click budgets per advertiser category, compute resources for ad serving. Typically $m=100-500$ constraints\n- **Linear demand model**: Click-through rate $d_i = \\alpha_i + \\sum_j B_{ij}(\\text{bid}_j)$ responds to bid prices\n- **Degeneracy issue**: Premium inventory (e.g., high-income tech enthusiasts) depletes quickly during peak hours\n- **Boundary attraction benefit**: As premium segments approach depletion, reserve remaining inventory for higher-value advertisers\n- **ML integration**: Our informed-price setting directly leverages CTR predictions from deep learning models\n- **Industry impact**: Major platforms report 10-15% revenue lift from better real-time allocation\n\n**2. Cloud Computing Resource Allocation**:\n- **Products**: VM instances (types×regions×durations). For 20 instance types, 10 regions, 5 duration tiers → $n=1000$ products  \n- **Resources**: CPU cores, memory, GPU units, network bandwidth per data center. Typically $m=50-200$ constraints\n- **Linear demand model**: $d_i = \\alpha_i + \\sum_j B_{ij}p_j$ where demand responds to spot pricing\n- **Degeneracy issue**: GPU resources deplete first during ML training peaks, creating allocation challenges\n- **Boundary attraction benefit**: Reserve scarce GPU capacity for long-duration deep learning jobs vs. short burst requests\n- **Real-world scale**: AWS/Azure manage millions of allocation decisions daily\n\n**3. Dynamic Pricing in E-commerce Marketplaces**:\n- **Products**: Items×sellers×shipping options. For marketplace with 1000 SKUs → $n=5000+$ pricing decisions\n- **Resources**: Warehouse capacity, delivery slots, inventory levels across fulfillment centers. $m=100+$ constraints\n- **Linear demand model**: Purchase probability responds to price adjustments across substitutable products\n- **Degeneracy issue**: Fast-moving items deplete inventory, creating stockout risks\n- **Boundary attraction benefit**: Stop discounting near-stockout items to preserve inventory for full-price sales\n- **ML connection**: Demand predictions from neural networks feed directly into our pricing algorithms\n\nThese applications align with NeurIPS interests:\n- Large-scale optimization with ML-predicted parameters\n- Online learning under constraints\n- Real-time decision making with uncertain demand\n- Integration of deep learning predictions with optimization\n\n## 2. Numerical Study Clarification\n\nWe appreciate your feedback on the numerical study. As explained in our response to Reviewer 94en, finding directly comparable baselines is challenging because existing methods don't handle our specific setting (resource constraints with degeneracy).\n\nHowever, we want to emphasize that our experiments with $\\zeta = 0$ effectively represent the baseline without boundary attraction. This is equivalent to prior approaches (e.g., Wang & Wang, 2022) adapted for resource constraints. \n\n**New Large-Scale Experiments**: Following your suggestion, we conducted additional experiments with realistic problem sizes ($m=100$ resources, $n=200$ products) that match the scale of real applications mentioned above. The results strongly validate our approach:\n\n| Scenario | T | Baseline Regret (ζ=0) | Our Method Regret (ζ=1) |\n|----------|---|----------------------|-------------------------|\n| **Full Information** | | | |\n| | 200 | 5,173 ± 259 | 3,582 ± 107 |\n| | 400 | 7,646 ± 382 | 4,554 ± 137 |\n| | 600 | 9,363 ± 468 | 5,569 ± 167 |\n| | 800 | 11,454 ± 573 | 6,234 ± 187 |\n| | 1,000 | 13,100 ± 655 | 6,689 ± 201 |\n| **Learning** | | | |\n| | 200 | 29,548 ± 1,477 | 25,035 ± 751 |\n| | 400 | 50,282 ± 2,514 | 36,327 ± 1,090 |\n| | 600 | 66,223 ± 3,311 | 45,081 ± 1,352 |\n| | 800 | 76,251 ± 3,813 | 50,855 ± 1,526 |\n| | 1,000 | 91,578 ± 4,579 | 56,163 ± 1,685 |\n| **Informed** | | | |\n| | 200 | 25,661 ± 1,283 | 18,585 ± 558 |\n| | 400 | 38,288 ± 1,914 | 25,001 ± 750 |\n| | 600 | 50,748 ± 2,537 | 30,868 ± 926 |\n| | 800 | 64,406 ± 3,220 | 34,756 ± 1,043 |\n| | 1,000 | 71,133 ± 3,557 | 39,246 ± 1,177 |\n\n*Note: Results show mean regret ± 95% confidence interval over 100 replications.*\n\nKey insights from large-scale experiments:\n- **Scalability**: Our method scales well to realistic problem sizes with hundreds of products and constraints\n- **Consistent improvement**: Substantial regret reduction across all scenarios\n- **Growing benefit**: The advantage of boundary attraction increases with time horizon T\n- **Robustness**: Smaller confidence intervals for our method indicate more stable performance\n\n**Figure readability**: We sincerely apologize for the small figures. To address this:\n- Font sizes for axes, legends, and labels have been increased\n- Larger markers and thicker lines are now used\n- Separate plots for different settings provide clarity\n- The summary table above presents key numerical results\n\n\nThank you again for your constructive feedback. We believe addressing your comments, particularly adding practical context and improving figure quality, will make our work more accessible and impactful for the NeurIPS community."}]

Author rebuttal:

Meta-review: The reviewers were all positive about this paper, and largely felt that their concerns were addressed during the rebuttal and discussion phase. The reviewers generally agreed that the submitted paper had writing issues and was not sufficiently polished, but that the authors seemed eager to address these concerns in the final version of the paper. The authors are strongly requested to do so for the final version.
Decision: Accept (poster)