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Nonparametric Contextual Pricing and Inventory Learning under Censored Demand

Zean Han, Jing Liang, Ruihan Lin, Zezhen Ding, Jiheng Zhang

arXiv:2608.30944v1cs.LG

TL;DR

The paper studies how to learn contextual pricing and stocking policies when inventory censors demand and no parametric demand model is assumed. It proposes MCK-UCB, combining inventory profiling, local contextual pooling, and calibrated price learning from sales observed while serving customers. The algorithm achieves matching minimax rates for the stated Lipschitz and twice-smooth price classes, with faster rates when expected profit varies more smoothly with price.

  • Problem

    The paper asks how a seller can learn context-dependent pricing and stocking when demand is censored by inventory decisions and no parametric demand model is available.

  • Method

    MCK-UCB combines inventory profiling, local contextual pooling, and calibrated price learning using censored sales collected during service.

  • Results

    The algorithm achieves matching minimax rates for the stated Lipschitz and twice-smooth price classes, with strictly faster rates under smoother expected-profit variation.

  • Takeaways & Limitations

    Visible censored sales can support joint contextual pricing and stocking without a separate exploration phase or recovering all demand hidden by stockouts.

  • Takeaways & Limitations

    The model has no inventory carryover or lead times and excludes arbitrary price-dependent noise distributions.

Abstract

from arXiv · show

In online retailing, when a product sells out, a retailer often sees only the units sold, not how many customers would have bought it had inventory been available. However, the inventory level determines how much demand is revealed, and this information can influence subsequent decisions and future profits. We study an online selling problem in which, in each round, the seller observes a market context and then makes pricing and stocking decisions based on censored sales data from previous rounds. The challenge is to learn a context-dependent pricing and stocking policy without assuming a particular formula for demand or observing realized profit. To overcome this difficulty, we propose a Mean-Calibrated Kernel UCB (MCK-UCB) algorithm that turns each incomplete sales record into a reliable guide for both inventory and price decisions, using data from past rounds with similar market conditions. This design allows us to learn while serving customers, without a separate exploration phase or the need to recover all demand hidden by stockouts. We prove the minimax optimality of the proposed algorithm, with strictly faster rates when expected profit varies more smoothly with price. Comprehensive numerical experiments have been conducted to confirm the effectiveness of the proposed algorithm.

1 INTRODUCTION

The paper studies contextual pricing and stocking when sales are censored by inventory decisions and demand is nonparametric. It introduces an observable, context-aware learning approach and establishes minimax-optimal regret rates.

  • Problem: The seller must jointly choose prices and inventory across changing market contexts while stockouts hide unmet demand.Inventory decisions also affect the information available for future pricing and stocking decisions.
  • Problem: The central question is how to learn a context-dependent pricing and stocking policy from censored sales without a parametric demand model.
  • Contributions: The observable learning signal uses stockouts and leftover inventory for stocking, while visible sales provide a reference for comparing prices.The method avoids recovering demand hidden by stockouts.
  • Contributions: MCK-UCB pools censored observations across nearby contexts, profiles inventory when needed, and learns prices from sales collected during service.It does not require a separate demand-revealing experiment for every context.
  • Contributions: The paper proves minimax-optimal regret rates for the proposed contextual censored-sales learning problem.

2 PROBLEM SETUP

The model captures repeated contextual price-and-inventory decisions with censored sales and no inventory carryover. Its assumptions enable local context sharing, within-context price calibration, and inventory profiling from stockouts.

  • Operational timing: The operational benchmark is per-period contextual decision-making with perishable stock or replenishment and no inventory carryover or lead times.
  • Model: At each period, the seller observes a context, chooses price and inventory, and observes only sales capped by available inventory.Potential demand depends on context and price, while the demand surface and context-specific noise laws are unknown.
  • Censored feedback: Stockouts hide excess demand and the lost-sales component of profit, while nonstockout sales reveal realized demand.Inventory therefore controls both current operations and future information.
  • Contextual learning: Contexts cannot be pooled indiscriminately, but nearby contexts can share information because demand varies smoothly across context.The policy must also account for context-dependent prices and inventory levels.
  • Assumptions: Conditional on a fixed context, changing price shifts demand location without changing the residual noise shape.This shared-noise structure supplies a common calibration reference across prices, but excludes arbitrary price-dependent noise laws.

3 MEAN-CALIBRATED KERNEL UCB

MCK-UCB converts censored sales into calibrated signals for inventory and price learning. It profiles inventory locally, pools observations through a contextual kernel, and compares prices using a corrected optimistic value.

  • Algorithm design: The method combines an observable sales-based score, an inventory profiler, and a contextual UCB rule for calibrated price comparison.
  • Mean calibration: The sales proxy differs from profit by bD, preserving inventory maximizers while requiring calibration to compare prices.The price-dependent proxy shift is removed using a visible demand quantile, leaving a common context-specific offset.
  • Inventory profiling: Inventory profiling uses stockout indicators to find levels above a lower demand quantile, making that quantile visible in censored sales.
  • Contextual UCB: After profiling, the UCB layer combines kernel estimates of the observable proxy and calibration quantile from the same sales history.Observations are pooled locally across contexts rather than globally.
  • Asynchronous learning: Each context cell progresses independently from profiling to price learning, so frequently visited cells need not wait for rare cells.
  • Resolution choices: Inventory uses coarser profiling resolutions than price learning, reducing profiling effort while preserving the calibrated target for UCB.

4 MAIN RESULTS

MCK-UCB achieves minimax regret rates up to logarithmic factors for contextual censored-sales pricing and inventory learning. The rates worsen with context dimension, improve with contextual smoothness, and become faster when optimized value is twice smooth in price.

  • MCK-UCB achieves the minimax regret rate, up to logarithmic factors, for both Lipschitz and twice-smooth price-value classes.The result holds for fixed b, hc > 0.
  • Larger context dimension increases learning cost, while larger H¨older smoothness allows stronger information sharing across nearby contexts.
  • Twice-smooth optimized value improves the regret exponent because a coarser price grid achieves the same approximation accuracy.
  • The regret decomposition combines inventory profiling, local contextual approximation and inventory error, contextual price-learning uncertainty, and price-grid approximation error.
  • Matching lower bounds show that the costs of censored demand and contextual variation cannot all be avoided.

5 EXPERIMENTS

Experiments evaluate MCK-UCB across horizons, context dimensions, and smoothness levels using mean cumulative regret and confidence bands. All six finite-sample fits are sublinear, while the study cautions that the experiment is narrower than the theory.

  • Five repetitions across horizons from 1000 to 100000 evaluate six combinations of context dimension d and H¨older exponent α.The tested pairs are (1, 0.5), (1, 0.75), (1, 1), (2, 0.5), (2, 0.75), and (2, 1).
  • Figure 1 reports mean cumulative regret with 95% confidence bands and dashed T^β(d,α) log T scaling guides.The guides use β(d, α) = (d + 2α)/(d + 3α).
  • 0.886, 0.885, 0.893, 0.842, 0.832, and 0.824 are the fitted log–log slopes across the six panels.All six finite-sample fits are sublinear.
  • The experiment shows that the policy can learn from sales-only feedback in a context-dependent environment, but remains narrower than the theory.

6 RELATED WORK

The paper combines contextual information with online joint pricing–inventory learning under censored demand, distinguishing its setting from prior work that typically includes only some of these elements. Related literatures address censored inventory learning, contextual bandits, partial monitoring, and contextual pricing separately or under different feedback assumptions.

  • The paper’s setting requires context, joint pricing–inventory learning, and censored online feedback simultaneously.
  • Censored pricing and inventory learning: Prior censored pricing–inventory studies consider nonparametric lost-sales learning in context-free settings, using high-inventory exploration, spline estimation, reward differences, or structured search.
  • Censored pricing and inventory learning: Feature-based inventory control uses censored sales for stocking decisions, but this setting additionally requires the same sale to support price comparison through calibration.
  • Contextual and offline pricing–inventory learning: Contextual pricing–inventory studies use rich historical data available before policy deployment, unlike the online learning setting considered here.
  • Partial monitoring and contextual bandits: Contextual bandit and partial-monitoring work generally assumes unbiased reward or loss feedback, whereas this problem has an endogenous inventory threshold and observes neither reward nor gradient.
  • Contextual dynamic pricing: Contextual dynamic-pricing studies address covariates, nonparametric actions, or multimodal rewards, but do not jointly model endogenous inventory censoring.

7 CONCLUSION

Under a nonparametric contextual location model, MCK-UCB learns joint pricing and stocking from censored sales by combining inventory profiling, local contextual pooling, and calibrated price learning. It achieves matching minimax rates for Lipschitz and twice-smooth price classes, while the result is limited to a per-period setting and a shared-noise demand structure.

  • 7 CONCLUSION: MCK-UCB combines inventory profiling, local contextual pooling, and calibrated price learning to support joint pricing and stocking from censored sales.The profiler adaptively refines inventory estimates, while calibrated proxies support price comparison without recovering demand hidden by stockouts.
  • 7 CONCLUSION: Matching minimax rates hold for the stated Lipschitz and twice-smooth price classes.The upper and lower bounds match for the corresponding context-conditioned shared-noise classes.
  • 7 CONCLUSION: The model excludes inventory carryover and lead times and does not cover arbitrary price-dependent noise distributions.Extensions to dynamic inventory states, weaker feedback, and more general conditional demand laws remain future directions.
  • B.3 UPPER AND LOWER BOUNDS: Twice-smooth price values permit a distinct upper-bound specialization using a price mesh of Δp = √ε.The smooth-price specialization has grid cardinality O(ε^-1/2) and approximation error O(ε).

C EXPERIMENTAL DETAILS

The experiments evaluate MCK-UCB in controlled contextual environments, compare it with noncontextual online baselines, and test sensitivity to smoothness misspecification. The implementation uses fixed tuning conventions, common random numbers, and benchmark-specific regret measurements.

  • Smoothness mismatch: When the true cusp exponent is fixed, bα = 1 has the lowest mean regret for every tested dimension and environmental smoothness.This finite-sample diagnostic does not establish a general smoothness-adaptation guarantee.
  • Baseline comparison: Figure 2 varies optimized reward and context-optimal price across contexts, making context information decision relevant in the baseline instance.The instance uses explicit surfaces w(x), m(x), and λ(x, p).
  • Baseline comparison: MCK-UCB uses contextual kernel estimates, whereas CWZ and SALA pool away the context as noncontextual baselines.The comparison supplies all methods with common exogenous context and residual-shock trajectories.
  • Baseline comparison: 2570.1 mean cumulative regret was obtained by MCK-UCB at T = 8000, compared with 4957.7 for CWZ and 5482.1 for SALA.The comparison averages eight repetitions on a two-dimensional instance.
  • Implementation and theory: The proof and implementation combine inventory profiling, calibrated proxies, contextual kernel UCB, and fixed meshes without requiring realized-profit observations.Profiling uses coarse context cells and √ε-spaced price anchors before completed cells switch to calibrated kernel UCB.

E PROOFS FOR THE PROXY AND PROFILING RESULTS

These proofs establish that censored sales can be transformed into a calibrated objective whose maximizers agree with those of the true reward, then show that profiling produces sufficiently safe inventories. Context regularity transfers across prices and supports the resulting guarantees.

  • Context regularity: Shared-noise and density conditions imply that demand quantiles, critical inventories, and optimized values vary Hölder-continuously with context.The proof obtains uniform regularity for zρ(x), zp(x), qρ(x, p), and y∗x(p).
  • Price discretization: A √ε price mesh contains O(ε^-1/2) points while retaining O(ε) discretization loss under the bounded-second-derivative condition.The same order applies at boundary maximizers because endpoints belong to the grid.
  • Proxy identification: The calibrated objective differs from the true objective by an additive term independent of inventory, so both have the same maximizers and inventory gaps.This follows from the sample identity and conditional expectation argument in Proposition 3.
  • Inventory profiling: The inventory profiler estimates critical inventory levels at coarse context cells and price anchors, then transfers each anchor estimate to nearby decision prices.The nearest-anchor price distance is at most √ε.
  • Inventory profiling: The profiled inventory is within O(√ε) of the critical inventory, incurs O(ε) value loss, and creates a positive margin above the censoring quantile.That margin makes the relevant lower-tail sales observations uncensored.

F PROOFS FOR KERNEL CONCENTRATION AND COUNTING

The concentration proofs control sequential kernel proxy estimates and weighted empirical quantiles under adaptive decisions. A deterministic counting argument then bounds the cumulative contribution of context-price observations without requiring deterministic effective sample-size lower bounds.

  • Kernel concentration: The sequential kernel proxy bound handles data-dependent effective sample sizes using conditional martingale concentration.The stochastic error scales with the logarithmic factor divided by the effective sample size, with no deterministic lower bound on that size.
  • Quantile concentration: The calibrated-value confidence proof controls weighted empirical quantiles on an uncensored neighborhood around the target quantile.A finite grid, monotonicity, and bounded CDF variation extend the result from grid points to the full neighborhood.
  • Kernel counting: Kernel counting partitions the context space into cells and lower-bounds same-cell kernel weights, converting cumulative confidence radii into a deterministic counting expression.The argument applies to every predictable adaptive interleaving policy and price sequence.
  • Quantile concentration: The concentration argument uses exponential supermartingales and union bounds over time-price queries, dyadic weight levels, and CDF grid points.The resulting failure probability is bounded by δ.
  • Kernel concentration: Profiling completion ensures that UCB observations use certified inventories, allowing Proposition 3 and the profiling lemma to support proxy concentration.Only completed-cell observations enter the UCB histories.

G COMPLETE PROOFS OF THE CONTEXTUAL UPPER BOUNDS

The upper-bound proof combines profiling, calibrated optimism, kernel counting, and discretization into a regret decomposition. Choosing the bandwidth according to context dimension and smoothness yields the stated sublinear rates, with a faster rate under smoother price dependence.

  • Lipschitz-price upper bound: The Lipschitz-price rate uses exponent r = (d + 2α)/(d + 3α), with bandwidth h = T^-1/(d+3α) up to logarithmic factors.The profiling term is lower order under this choice.
  • Regret decomposition: The per-round regret decomposition separates optimism loss, proxy-value loss, and inventory-profiling loss.The first two terms are controlled by the contextual confidence argument, while profiling contributes at most Cε.
  • Regret decomposition: The expected regret bound sums UCB confidence radii, profiling rounds, discretization terms, and failure-event contributions.The failure event contributes at most 2Rmax/T to expected average regret under the stated choice of δ.
  • Smooth-price upper bound: The smooth-price rate uses exponent rsm = (2d + 3α)/(2d + 5α), with bandwidth h = T^-2/(2d+5α) up to logarithmic factors.The profiling term remains lower order for this bandwidth.
  • Smooth-price upper bound: Under bounded second derivatives in price, a √ε mesh gives Kdec = O(ε^-1/2) and O(ε) price approximation loss.This changes the bandwidth balance and improves the smooth-price exponent.

H PROOFS FOR THE LOWER BOUND CONSTRUCTION

The lower-bound construction builds smooth context–price perturbations around a flat-value baseline and verifies that every resulting instance remains in the model class. The proof establishes regularity, feasibility, interior optimal prices, and the required inventory-loss and visibility properties.

  • Baseline construction: The construction starts from a smooth compactly supported mean-zero density and a baseline whose value is flat on P0 and lower outside P1.The baseline is extended with smoothness and derivative bounds while preserving strict slack margins.
  • Packing construction: The normalized packing uses pairwise-disjoint context cells and price supports, with Mh proportional to h^-d and K∆ proportional to w^-1.Under the uniform context law, the cell and core volumes are h^d and (h/2)^d, respectively.
  • Packing construction: Each perturbation is localized to one context cell and one price set, reaches height ∆ on its price plateau, and vanishes outside that price support.The bump functions provide the required smoothness and disjoint-support properties.
  • Model-class verification: Choosing ∆ = c∆h^α and sufficiently small constants preserves nonnegative demand, context Hölder regularity, and price regularity uniformly over the packing.The Lipschitz construction sets w = ∆, while the twice-smooth construction uses its corresponding smooth price width.
  • Model-class verification: The constant noise family satisfies the shared-noise and calibration conditions, while density bounds ensure a unique critical inventory root and quadratic inventory loss.The proof also establishes the required derivative bounds and keeps optimal prices in the interior of P0.
  • Stability verification: Censoring preserves the observation-distance bound because it is a measurable image of demand, completing the stability argument for neighboring packed instances.Neighboring instances differ only within one context cell and price support, with location differences bounded by C∆.

I COMPLETE DIRECT-SUM PROOF OF THE LOWER BOUND

The direct-sum lower-bound proof converts localized indistinguishability into regret across context cells. A randomized-policy lower bound follows by combining cellwise information bounds, packing structure, and Yao’s principle.

  • Hard-instance setup: The proof considers Lipschitz and twice-smooth perturbation families and reduces randomized policies to deterministic ones by conditioning on internal randomness and applying Yao’s principle.Every instance in the chosen family belongs to the claimed model class by the preceding stability lemma.
  • Visit-count decomposition: The first visit count includes the transition shell because shell observations also carry information, whereas core counts are reserved for regret lower bounds.This separates information acquisition from the visits used to certify price suboptimality.
  • Information bound: For a fixed context cell, the reference experiment differs from alternative labels only when the context is in that cell and the price is in the corresponding support.Adaptive Hellinger tensorization controls the accumulated information in these localized rounds.
  • Information aggregation: Disjoint price supports imply that at most two label-events can occur on a sample path with a positive core-visit count.This combinatorial restriction is used to aggregate the cellwise testing bounds.
  • Regret conversion: On the event that the policy selects prices outside the active support, at least half of the relevant core visits incur a price gap of ∆, while inventory suboptimality is nonnegative.The optimized value is C0 + ∆, and prices outside the active support have gap at least ∆.
  • Regret lower bound: Under the uniform prior over cell labels, summing the cellwise Bayes bounds yields Eω∼Unif(Ω)[RegT] ≥ cMhThd∆ ≥ cT∆, and Yao’s principle gives a worst-instance bound.The argument uses independent labels and localization of the conditional observation kernel to each context cell.
  • Rate calculation: The information balance is evaluated separately in the Lipschitz regime, where w = ∆ and K∆ ≍ ∆^-1, and in the twice-smooth regime.The resulting bounds cover both price-regularity cases, with constants enlarged to handle finitely many smaller horizons.
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