Budget-Constrained Causal Bandits: Bridging Uplift Modeling and Sequential Decision-Making

arXiv:2604.26169v2 Announce Type: replace Abstract: Treatment allocation under budget constraints is a central challenge in digital advertising. The standard approach trains an offline uplift model on historical data, then solves a constrained optimization to allocate budget. This fails in cold-start settings where little historical data exists. We propose Budget-Constrained Causal Bandits (BCCB), an online framework that learns which users respond to ads while simultaneously spending the budget. BCCB unifies three components: learning individual-level treatment effects, exploring users whose response is uncertain, and pacing the budget over time. We derive the per-arrival decision rule as the KKT condition of a Lagrangian relaxation of the budgeted causal-allocation objective, providing a principled foundation for the algorithm. We evaluate on the Criteo Uplift dataset using 20 random seeds with paired statistical tests. Our central finding is a data-efficiency crossover at n = 7,500 historical observations (paired one-sided t-test, p = 0.043): below this threshold, offline pipelines either fail or produce unreliable allocations, while BCCB operates from the first user. BCCB exhibits 2-4x lower run-to-run variance than offline methods and outperforms all four online baselines (Thompson Sampling, budgeted Thompson Sampling, HTE Greedy, and Uplifting Bandits) at every budget level tested (p < 0.001). These results give practitioners a concrete decision rule for choosing between offline and online paradigms.

Abstract:Treatment allocation under budget constraints is a central challenge in digital advertising. The standard approach trains an offline uplift model on historical data, then solves a constrained optimization to allocate budget. This fails in cold-start settings where little historical data exists. We propose Budget-Constrained Causal Bandits (BCCB), an online framework that learns which users respond to ads while simultaneously spending the budget. BCCB unifies three components: learning individual-level treatment effects, exploring users whose response is uncertain, and pacing the budget over time. We derive the per-arrival decision rule as the KKT condition of a Lagrangian relaxation of the budgeted causal-allocation objective, providing a principled foundation for the algorithm. We evaluate on the Criteo Uplift dataset using 20 random seeds with paired statistical tests. Our central finding is a data-efficiency crossover at n = 7,500 historical observations (paired one-sided t-test, p = 0.043): below this threshold, offline pipelines either fail or produce unreliable allocations, while BCCB operates from the first user. BCCB exhibits 2-4x lower run-to-run variance than offline methods and outperforms all four online baselines (Thompson Sampling, budgeted Thompson Sampling, HTE Greedy, and Uplifting Bandits) at every budget level tested (p < 0.001). These results give practitioners a concrete decision rule for choosing between offline and online paradigms.

17 pages, 5 tables. v2: Expanded to 20 random seeds with paired statistical tests; formal crossover test at n=7,500 (p=0.043); added Uplifting Bandits (Hsieh et al., NeurIPS 2022) as fifth baseline; Lagrangian derivation of decision rule with Proposition 1; hyperparameter sensitivity analysis; corrected mathematical claims

Machine Learning (cs.LG); Econometrics (econ.EM); Machine Learning (stat.ML)

arXiv:2604.26169 [cs.LG]

(or arXiv:2604.26169v2 [cs.LG] for this version)

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Budget-Constrained Causal Bandits: Bridging Uplift Modeling and Sequential Decision-Making

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