Auditing Marketing Budget Allocation with Hindsight Regret

We study budget pacing in repeated first-price auctions when an advertiser's private-value distributions change over time and the stationary competing-bid distribution is unknown. We ask how a feasible expenditure plan should enter online bid shading, learning, and hard budget control. We establish a plan-to-performance decomposition for a plan-driven projected-dual policy. The policy uses any feasible expenditure plan as a soft target, learns an unknown stationary competing-bid CDF from thresholds revealed after each auction, and enforces the campaign budget on every sample path. Against a distribution-informed expected-budget fluid benchmark, the uniform-plan reward gap is $O(\sqrt T)+O(\mathcal W_T)$, where $\mathcal W_T$ measures heterogeneity in private-value distributions. With a supplied feasible plan, the global gap decomposes into a one-sided $O(\sqrt{T})$ fixed-plan execution term and a plan-mismatch term bounded by $(b/2a)PlanError$. The same analysis provides guarantees for strict and relaxed period-cap comparators, exact recovery of the global benchmark under a specific allowance vector, and separate lower bounds establishing the necessity of the temporal-heterogeneity and Plan Error terms. An upstream planner can translate forecasts or managerial priorities into a feasible spending trajectory, while the online controller adapts bids using realized thresholds and expenditures. The guarantee is modular: it evaluates the final normalized or projected plan through $PlanError$. A specific forecasting model can be linked to the guarantee by establishing how its primitive estimation errors propagate to this plan-quality metric.

We study finite-horizon budget allocation as a closed-loop economic control problem and evaluate receding-horizon Model Predictive Control (MPC) relative to reactive budgeting policies. Budgets are allocated periodically under execution noise and operational constraints, while return efficiency may evolve over time. Using a controlled simulation framework motivated by digital marketing, we compare reactive pacing to MPC across environments with increasing degrees of non-stationarity. Our results show that non-stationarity alone does not justify predictive control. When return dynamics are stationary or evolve through unpredictable stochastic drift, MPC offers no systematic advantage over reactive baselines. By contrast, when return efficiency exhibits predictable structure over the planning horizon, that is captured through an underlying model, MPC consistently outperforms reactive budgeting by exploiting intertemporal trade-offs.

Treatment allocation under budget constraints is a central challenge in digital advertising: advertisers must decide which users to show ads to while spending a limited budget wisely. The standard approach follows a two-stage offline pipeline - first collect historical data to estimate heterogeneous treatment effects (HTE), then solve a constrained optimization to allocate the budget. This works well with abundant data, but fails in cold-start settings such as new campaigns, new markets, or new customer segments 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, making treatment decisions one user at a time. BCCB unifies three components into a single sequential process: learning individual-level ad effectiveness, exploring users whose response is uncertain, and pacing the budget over time. We evaluated on the Criteo Uplift dataset, a large-scale advertising dataset from a real randomized controlled trial. Our key finding is a data-efficiency crossover: offline methods require approximately 10,000 historical observations to produce reliable results, while BCCB operates effectively from the very first user. Furthermore, BCCB exhibits 3-5x lower performance variance between runs, making it more practical for real campaign planning. Among purely online methods, BCCB consistently outperforms standard Thompson Sampling, budgeted Thompson Sampling, and greedy HTE estimation across all budget levels tested.