Optimally Pacing Budget Spending and Learning

Major Internet advertising platforms offer budget pacing tools as a standard service for advertisers to manage their ad campaigns. Given the inherent non-stationarity in an advertiser's value and also competing advertisers' values over time, a commonly used approach is to learn a target expenditure plan that specifies a target spend as a function of time, and then run a controller that tracks this plan. This raises the question: how many historical samples are required to learn a good expenditure plan? We study this question by considering an advertiser repeatedly participating in $T$ second-price auctions, where the tuple of her value and the highest competing bid is drawn from an unknown time-varying distribution. The advertiser seeks to maximize her total utility subject to her budget constraint. Prior work has shown the sufficiency of $T\log T$ samples per distribution to achieve the optimal $O(\sqrt{T})$-regret. We dramatically improve this state-of-the-art and show that just one sample per distribution is enough to achieve the near-optimal $\tilde O(\sqrt{T})$-regret, while still being robust to noise in the sampling distributions.

This paper addresses the problem of sequential decision-making under learning budget constraints. Such settings naturally arise in applications like managing a portfolio of bandit or reinforcement learning (RL) algorithms. We propose a novel UCB-type algorithm, M-LCB, designed to manage a pool of $K$ self-learning experts in a stochastic environment while accounting for a limited per-round learning budget $M$. At each round, M-LCB selects one expert to make a decision and at most $M \le K$ experts to learn. For selection, M-LCB uses confidence bounds constructed from limited prior knowledge about the experts (i.e., mild assumptions) and their observed training losses. We derive anytime regret bounds for M-LCB that scale with the individual regrets of the experts. In particular, if each expert has regret $\tilde O(T^α)$ by round $T$, then M-LCB guarantees an overall regret of $\tilde O\left(\sqrt{KT/M} + (K/M)^{1-α}T^α\right)$ relative to the best expert in hindsight. Finally, we demonstrate the applicability of M-LCB using self-learning experts instantiated as (i) parametric models and (ii) bandit algorithms.

We study constrained online convex optimization with adversarial constraints and conditionally unbiased, finite-variance observations of constraint values and gradients. Under common feasibility, our \LEDGER\ algorithm attains $O(\sqrt T)$ expected regret and $O(\sqrt{T\log(eT)})$ expected budget violation, the largest cumulative overspend over any window. It uses a reflected exponential potential, clipped signed observations, and predictable adaptive regularization, with one feedback triple and one projection per round. Neither a Slater condition, independence between feedback channels, nor an absolute constraint-value bound is needed. A Gaussian testing lower bound proves that the budget rate has optimal horizon dependence under square-root regret at fixed positive noise, including the logarithm. The same obstruction holds for terminal violation, so the logarithm is not a cost of maximizing over windows; an $O(\sqrt T)$ budget bound instead forces linear regret. In contrast, fixed positive Gaussian value noise yields a joint regret--hard-violation lower bound of $Ω(\min\{σ,1\}T/\log^2 T)$, even with exact gradients in one dimension. The hard-violation construction matches arbitrarily many moments while preserving a feasible-endpoint gap and constant endpoint probabilities. Together, the bounds separate uncertainty about hard feasibility from learnable signed budgets. Deterministic restarts remove the horizon input without changing either upper rate.