Resource-Constrained Adaptive Inference for Sequential Pricing

Technical Summary: Resource-Constrained Adaptive Inference for Sequential Pricing

Problem Formulation
The paper addresses the challenge of performing valid statistical inference on demand sensitivity parameters in dynamic pricing systems operating under hard resource constraints (e.g., inventory limits for airline seats or hotel rooms). In standard adaptive pricing, a controller learns demand while setting prices to maximize revenue. However, resource constraints introduce a unique failure mode for inference: the remaining resource state can render a specific target price neighborhood $[p^\sharp - h, p^\sharp + h]$ infeasible entirely.

Unlike traditional adaptive inference problems where the obstacle is low propensity (rare visits) leading to unstable inverse-density weights, resource constraints can cause support exclusion. In this regime, the target price is not merely visited rarely; it is removed from the feasible set for a significant portion of the horizon. Consequently, the target effect $\beta_j(p^\sharp)$ becomes locally unobserved. The paper formalizes that under support exclusion, no reweighting of logged data can recover the target parameter, as the realized experiment lacks the necessary support. The inferential target is a low-dimensional coordinate of a high-dimensional sparse demand model, estimated from data generated by a controller that must simultaneously manage revenue (regret) and resource feasibility.

Methodology
The authors propose a framework that integrates the control policy with the statistical design, treating the "realized information clock" as the governing quantity for inference rather than the nominal time horizon.

1. Target-Aware Pricing Controller:
   The controller employs a boundary-attracted re-solving mechanism. It repeatedly solves a fluid approximation of the pricing problem but incorporates a "target-reserve guard." This guard ensures that if a target price band is feasible, the controller maintains a continuous logging density over that band.

   • Certified Logging: The controller checks a "full-support certificate" at each round. If the entire target band lies within the resolved feasible set, the controller assigns a known, continuous density to that band.
   • Randomization Budget: The controller allocates a randomization budget $v_t \asymp t^{-\gamma}$ (or $1/t$ at the logarithmic endpoint) to ensure exploration. This budget is used to maintain target-band mass $\alpha^\sharp_t$ when the resource state threatens to exclude the target.

2. Support-Aware Debiased Estimation:
   The estimator is a localized inverse-probability weighted (IPW) debiased construction, augmented with optional centered control variates.

   • Localized IPW: The estimator uses a kernel-weighted IPW score, but only on rounds where the full-support certificate is active ($\chi^\sharp_t = 1$). Rounds with clipped support are logged for diagnostics but excluded from the root-effective-horizon estimation block.
   • Centered Augmentation (CVD): An optional step subtracts a predictable component from the score using history-only regression (control variates). This reduces variance if predictable score components exist but cannot create support where none exists.
   • Information Clock: The width of the confidence interval is governed by a realized information clock $I_{j,T} = N_T^2 / Q_{j,T}$, where $N_T$ is the effective horizon (stopping time due to resource exhaustion) and $Q_{j,T}$ is the realized quadratic variation of the corrected score.

3. Diagnostic-Gated Reporting:
   The paper introduces a "Support-CI" protocol. A fixed-target confidence interval is reported only if realized diagnostics pass: sufficient full-support rounds, adequate effective sample size (ESS), and bounded max-score ratios. If the resource state collapses the target support, the method abstains from reporting a fixed-target interval, potentially offering a supported feasible-price estimand instead.

Key Contributions

1. Formalization of Support-Exclusion Failure: The paper isolates support exclusion as a distinct failure mode where the target parameter is locally unidentifiable due to resource constraints, even if every realized action has a logged density. It proves that without recurrent support, no uniformly shrinking confidence interval exists.
2. Realized Information Clock: It introduces a new metric for inference precision that replaces the nominal horizon with target-local information. This clock depends on the target-band mass $\alpha^\sharp_t$ and the resource stopping time, showing that polynomial target mass yields polynomial convergence rates, while a pure $1/t$ exploration branch may fail to shrink fixed-target intervals without additional local movement.
3. Inference-Aware Re-solving: The authors design a controller that treats target support as an online design constraint. By certifying local support and logging continuous densities, the controller separates control information (for low regret) from local excitation information (for inference).
4. Regret-Information Accounting: The paper demonstrates that regret control and valid inference obey different clocks. Logarithmic exposure may suffice for low regret but remain insufficient for fixed-target inference if the target mass decays too quickly or is excluded by resource constraints.

Results

• Theoretical Guarantees: Under conditions of recurrent full support and sparse nuisance estimation, the proposed estimator achieves asymptotic normality with a variance governed by the realized information clock. The regret is bounded, scaling with the exploration budget and boundary-buffer costs.
• Experimental Validation:
  • Calibration: In regimes where the target support is preserved (non-collapse), the method produces calibrated confidence intervals (coverage $\approx 95\%$) with widths shrinking at the predicted rates.
  • Information Clock Slopes: Experiments confirm that polynomial target mass ($t^{-\gamma}$) leads to information growth of $T^{1-\gamma}$, while the $1/t$ endpoint results in a bounded information clock, preventing interval shrinkage.
  • Diagnostic Abstention: In tight-budget scenarios where the resource state collapses the target support, the diagnostics correctly identify the failure, and the method abstains from reporting a fixed-target interval, avoiding invalid inference.
  • Variance Reduction: The centered augmentation (CVD) reduces RMSE and interval width in regimes with predictable score components, provided the underlying support remains stable.

Significance and Claims
The paper claims that valid inference in resource-constrained environments cannot be separated from the controller that generated the data. The controller determines not only which prices are profitable but also which price effects remain statistically visible.

• Joint Design: The work argues for a joint design where the controller actively preserves the "local experiment" required for sparse inference, rather than treating inference as a post-hoc analysis of a revenue-maximizing policy.
• Regime Dependence: The guarantees are regime-dependent. In scarcity regimes where the target neighborhood leaves the realized path, the method correctly identifies the information collapse and reports the target as unsupported, rather than returning a spurious nominal interval.
• Distinction from Prior Work: Unlike standard adaptive inference or bandit literature where the action set remains available, this work highlights that feasibility itself is part of the statistical design under resource constraints. The overlap condition is a property of the controlled state, not just the logging policy.

The authors maintain modesty regarding efficiency, noting that their results provide a valid inference framework under specific support conditions rather than a general minimax efficiency bound for all constrained settings. The primary contribution is the mechanism-level understanding of how resource constraints induce support exclusion and the design of a controller that mitigates this to enable valid inference.