Optimal Hidden-Target Learning for Online Inventory Optimization on General Convex Sets

2026-06-12

AI Summary

This work addresses the challenge in online inventory optimization where cumulative inventory leads to a history-dependent feasible decision set, particularly under general convex capacity constraints that hinder the attainment of optimal decisions. To tackle this, the authors propose a “hidden target” mechanism: the online learner selects a target and projects it onto the current feasible order set, thereby reducing the high-dimensional state-dependent problem to one-dimensional queue control. Leveraging a norm-alignment principle, the method achieves—for the first time—the first multi-logarithmic static regret bound for strongly convex losses over general convex sets, along with a dynamic regret bound that matches the theoretical lower bound. Notably, the dependence on demand probabilities improves from inverse linear to inverse square root. Empirical evaluations demonstrate the approach’s superiority on both synthetic and real-world inventory datasets.

Abstract

Online inventory optimization (OIO) is online convex optimization with physical memory: inventory carryover makes the feasible action set depend on the past. A natural principle, used in stochastic inventory learning and recently in OIO under a single linear capacity constraint, is to maintain a hidden target chosen by an online learner and implement its projection onto the currently feasible order-up-to set. We prove that this simple principle is optimal for OIO on arbitrary bounded convex capacity sets. With online gradient descent as the base learner, the method improves the best known regret guarantee for OIO on general convex sets from inverse to inverse-square-root dependence on the common-demand probability, and we prove a matching lower bound. The same principle gives the first polylogarithmic regret guarantee for strongly convex losses and the first dynamic regret guarantee adapting to Euclidean path variation on general convex capacity sets. The analysis introduces a norm alignment principle: the right state variable is the distance from the hidden target to the feasible set, measured in the same norm as the projection. Under norm alignment, this distance evolves pathwise as a scalar queue, with target movement as arrival and common demand as service. This reduction to one-dimensional queue control resolves the state dependence and extends the guarantees to general convex capacity sets, beyond the reach of prior productwise approaches. Experiments on synthetic and real-world inventory data corroborate the theory.

Problem

• online inventory optimization
• convex capacity sets
• hidden-target learning
• regret minimization
• state-dependent constraints

Innovation

• hidden-target learning
• norm alignment
• online inventory optimization
• dynamic regret
• convex capacity sets