Plan-Driven Adaptive Bidding for First-Price Auctions with Budget Constraints under Nonstationarity

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(√ T)+O(mathcal WT), where mathcal WT measures heterogeneity in private-value distributions. With a supplied feasible plan, the global gap decomposes into a one-sided O(√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.