Towards Optimal Inventory Control under Censored Demand: A Biased Sample-Average Approximation Approach

Yuxuan Han; Xiaoyu Fan; Jiawei Zhang; Zhengyuan Zhou

stat.MLSep 30, 2026

Towards Optimal Inventory Control under Censored Demand: A Biased Sample-Average Approximation Approach

Authors: Yuxuan Han, Xiaoyu Fan, Jiawei Zhang, Zhengyuan Zhou

Organizations: Stern School of Business, New York University

Abstract

We study data-driven multi-period lost-sales inventory control under censored demand, where a stockout reveals only that demand exceeded the stocking level. We develop a unified, model-based framework for policy learning from censored data, built on a new cost decomposition for base-stock policies and a biased sample-average approximation (SAA) approach. The cost decomposition allows us to propose a new coverage condition under which censored observations are informative enough for sample-efficient policy learning. Guided by this coverage condition, we design two biased SAA algorithms: an upper-biased one that achieves near-optimal sample complexity under the offline coverage condition, and a lower-biased one that actively generates the required coverage and achieves near-optimal regret online. More broadly, this biased SAA approach provides a general principle for implementing pessimism and optimism under censored feedback, which may be of independent interest.

Figures & tables

Setup	Work	Stationary demand	Censored observations	Leading-order Result
Offline sub-optimality gap with $N$ trajectories	Qin et al. (2023)	✗	✗	$\mathcal{O}(MT^{3/2}/\sqrt{N})$
	Xie et al. (2024)	✗	✗	$\mathcal{O}({MT}/{\sqrt{N}})$
	This work	✗	✓	$\widetilde{\mathcal{O}}(MT/\sqrt{\mathcal{N}^{\star}})$
	This work	✓	✓	$\widetilde{\mathcal{O}}(MT/\sqrt{\mathcal{N}_{\mathsf{agg}}^{\star}})$
Online episodic regret over $K$ episodes	This work	✗	✓	$\widetilde{\Theta}(MT\sqrt{K})$
Online episodic regret over $K$ episodes	This work	✓	✓	$\widetilde{\Theta}(M\sqrt{KT})$

Table 1: Comparison of sub-optimality and regret guarantees for learning multi-period inventory policies.

\mathcal{N}^{\star}

and

\mathcal{N}_{\mathsf{agg}}^{\star}

denote the effective sample sizes and its time-aggregated version under censored observations, where

\mathcal{N}^{\star}\geq N

and

\mathcal{N}_{\mathsf{agg}}^{\star}\geq NT

in the uncensored setting.

Notation	Interpretation	Notation	Interpretation
$\bm{F}_{t}\in\mathbb{R}^{M}$	$\big[\bm{F}_{t}\big]_{j}=F_{t}(j)$ for $j\in[M-1]_{+}$	$\bm{W}_{t}^{\star}\in\mathbb{R}^{M+1}$	Order-up-to value under the optimal policy
$\bm{D}_{t}^{\star}\in\mathbb{R}^{M}$	Discrete derivative of $\bm{W}_{t}^{\star}$	$\bm{A}_{t}(s)\in\mathbb{R}^{M\times M}$	$[\bm{A}_{t}(s)]_{ij}=\mu_{t,i-j}\bm{1}\{i\geq j\geq s\}$
$\bm{c}_{t}\in\mathbb{R}^{M}$	$\big[\bm{c}_{t}\big]_{j}=(h_{t}+b_{t})F_{t}(j)-b_{t}$	$\bm{q}_{t}\in\mathbb{R}^{M}$	$[\bm{q}_{t}]_{j}=\sigma_{t}\bm{1}\{j\in[\alpha_{t},\beta_{t})\}\mathbb{P}(x_{t}\leq j\mid x_{1}=0)$
$\bm{u}_{t}\in\mathbb{R}^{M}$	$\big[\bm{u}_{t}\big]_{j}=\sigma_{t}\bm{1}\{j\in[\alpha_{t},\beta_{t})\}\mathbb{P}(x_{t}^{\star}\leq j\mid x_{1}=0)$	$\bm{v}_{t}\in\mathbb{R}^{M}$	$[\bm{v}_{t}]_{j}=\mathbb{P}(y_{t}^{\star}\leq j\mid x_{1}=0)-\mathbb{P}(y_{t}\leq j\mid x_{1}=0)$

Table 2: Summary of notations in Section 3 , where the estimated versions

\hat{\bm{W}}_{t}

\hat{\bm{D}}_{t}

\hat{\bm{A}}_{t}(s)

, and

\hat{\bm{c}}_{t}

are defined analogously by replacing

\bm{F}_{t}

with

\hat{\bm{F}}_{t}

and

\bm{\mu}_{t}

with

\hat{\bm{\mu}}_{t}

\hat{F}^{(k+1)}_{t}(j):=\max\big\{\hat{F}^{(k)}_{t}(j),\ F^{\mathrm{LCB},(k+1)}_{t}(j)\big\},

(6.4)

Algorithm 2 DP-LCB

Figure 1: Panel (a) plots the DP-UCB gap versus the number of trajectories

N

; panel (b) plots the adequate-coverage curves versus the effective sample size

\mathcal{N}^{\star}

. Experiments are conducted under a seasonal truncated-Poisson instance with

M=30,T=6

and

s_{t}^{\star}\in\{7,\dots,17\}

, with

150

repetitions. In each trajectory, the logging policy draws

B_{t}\sim\mathrm{Uniform}\{0,\dots,U\}

each period and orders up to

y_{t}^{b}=\max\{x_{t},B_{t}\}

Figure 2: Performance gap under partial coverage. DP-UCB is compared with vanilla DP and SAIL on an instance with

M=30

and

T=12

over

200

repetitions; the demand distributions are detailed in Appendix C . Panel (a) plots the sub-optimality gap versus

N

with

N\in\{128,192,256,512,1024\}

. Panel (b) plots the distribution of learned first-period base-stock levels at

N=192

across

200

repetitions.

Figure 3: Online regret under censored feedback. DP-LCB is compared with vanilla DP and SAIL-CE on an instance with

M=30

and

T=12

over

60

repetitions, with demand distributions detailed in Appendix C . Panel (a) plots cumulative regret versus the number of episodes

K

, up to

K=2048

. Panel (b) shows how the mean value of the learned first-period base-stock levels changes across episodes, over

60

repetitions.

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May 14, 2026, cs.LGJ/K move · Enter open · S save

In-Context Learning for Data-Driven Censored Inventory Control

Sohom Mukherjee, Anh-Duy Pham, Richard Pibernik, Yunbei Xu

Julius-Maximilians-Universität Würzburg · National University of Singapore Zaragoza Logistics Center

Inventory Control Adaptive Sampling Generative Control Policies In-Context Learning Chronos

Open paper AskPDF arXiv

We study inventory control with decision-dependent censoring, focusing on the censored or repeated newsvendor (R-NV), where each order quantity determines whether demand is fully observed or censored by sales. Existing approaches based on parametric Thompson sampling (TS) can be brittle under prior mismatch, while offline imputation methods need not transfer to online learning. Motivated by the predictive view of decision making, we combine these ideas by taking oracle actions on learned completions of latent demand. We propose in-context generative posterior sampling (ICGPS), which uses modern generative models that are meta-trained offline and deployed online by in-context autoregressive generation. Theoretically, we show that the Bayesian regret of ICGPS with a learned completion kernel is bounded by the Bayesian regret of a TS benchmark with the ideal completion kernel plus a deployment penalty scaling as

\sqrt{T}

times the square root of the completion mismatch. This yields a plug-in template for operational problems with known TS regret bounds. For R-NV, we derive sublinear Bayesian regret by reducing censored feedback to bandit convex optimization feedback. We also show that, under reasonable coverage and stability assumptions, the online completion mismatch is controlled by the offline censored predictive mismatch, so offline predictive quality transfers to online performance. Practically, we instantiate ICGPS with ChronosFlow, which combines a frozen time-series transformer backbone with a trainable conditional normalizing-flow head for fast censoring-consistent sampling. In benchmark experiments, ChronosFlow-ICGPS matches correctly specified TS, outperforms myopic and UCB-style baselines, and is robust to prior mismatch and distribution shift. ChronosFlow-ICGPS also performs well for the real-world SuperStore dataset, especially under heavy censoring.

Abstract

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