Optimally Pacing Budget Spending and Learning

arXiv:2610.11074 · cs.LG, cs.GT · Submitted 2026-10-08 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Optimally Pacing Budget Spending and Learning".

Jane: The gist The authors establish near-optimal regret bounds for budget-constrained online learning against arbitrary classes of budget-pacing experts in an adversarial setting.

Tom: First, who's behind it and why it matters.

Paper summary: Tom: We've talked through the paper "Optimally Pacing Budget Spending and Learning" and its core contributions today. It really boils down to tackling that spend-or-save dilemma in online learning <ref:2610.11074#pg6>.

Jane: The authors provide near-optimal regret bounds, showing that with fractional allocation allowed, you can get those O(D square root log F) results in the ORA setting and O((sqrt T plus D) square root log F) in the OLRC setting <ref:2610.11074#pg6>.

Lu: The paper provides a concrete framework that characterizes optimal regret using the Kolmogorov distance between spending patterns of experts <ref:2610.11074#pg6>.

Meng: For practical impact, this gives us mathematical targets for designing resource allocation algorithms that have to operate under real-world budget limits <ref:2610.11074#pg6>.

Tom: The authors also establish strong lower bounds showing that no learner can guarantee regret better than o of max

D, T over half: times square root log F in the OLRC setting <ref:2610.11074#pg6>.

Jane: So, this work sets a clear benchmark for what is achievable when you are dealing with budget constraints in online learning problems <ref:2610.11074#pg6>.

Lu: It shows that the interplay between learning and spending decisions is fundamental to solving these constrained problems <ref:2610.11074#pg6>.

Meng: It’s a solid piece of theoretical work that gives us tools to build more robust AI systems <ref:2610.11074#pg6>.

Conclusion: Tom: So we've been diving deep into this paper by Tom and Jane about optimally pacing budget spending and learning, and honestly, it’s a pretty dense topic.

Jane: Yeah, it really tackles that whole spend-or-save dilemma in online learning problems. The authors are showing how you can get near-optimal regret bounds even when you have strict resource limits.

Lu: What I found really interesting is how they use this Kolmogorov distance between spending patterns of experts to characterize the optimal regret rates. That’s a pretty neat way to measure it.

Meng: From an engineering standpoint, having those specific bounds, like O(D square root log F) in the ORA setting, gives us some concrete targets for how we should design these budget-constrained allocation algorithms in practice.

Lalam: If I consider what this means for AI culture, it suggests we can build systems that are much more resilient when they have limited resources during learning phases. It’s about making those learning phases smarter under pressure.

Tom: Exactly, Lalam. So what does this paper actually mean for us who aren't building the algorithms? It’s showing us the mathematical limits on how well we can learn when we can only afford to spend our budget a certain way over time.

Jane: It means that for problems with these constraints, you can achieve much better regret performance than just blindly spending your budget. The caveat is that you often have to allow for fractional play, which adds a layer of complexity but opens up better results.

Lu: And the lower bounds they set are pretty strong too. They show that no learner can guarantee better than some certain rate, like o of max D over T over half times square root log F in the OLRC setting. That sets a real floor for what's possible.

Meng: So, it’s not just about getting good performance; it’s about understanding exactly what the theoretical ceiling is for these resource-limited learning tasks. It frames the entire design space for future work, doesn't it?

Lalam: Precisely. This work helps guide us in designing more efficient AI that respects those real-world constraints of time and resources during its learning process. We’ll look at how they extend this to subpacing experts next.

Mark Braverman, Jingyi Liu, Jieming Mao, Jon Schneider, Eric Xue

Department of Computer Science, Princeton University · Google Research

cs.LG, cs.GT

Submitted: 2026-10-08

Updated: 2026-10-08

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 86/100

The gist: The gist The authors establish near-optimal regret bounds for budget-constrained online learning against arbitrary classes of budget-pacing experts in an adversarial setting.

Key concepts

Online Learning with Resource Constraints (OLRC): This is an online learning problem where the learner has a fixed budget they must spend over time. The constraint forces the learner to make decisions sequentially, meaning they can only take a null action after their total spending limit is reached. This introduces a temporal dependency in how decisions are made.
Kolmogorov Distance (Kol(c, c′)): This distance metric is used to measure the difference between two sequences of expert spending patterns. The paper finds that the optimal regret rate for learning with constraints is directly characterized by this distance. A smaller Kolmogorov distance implies that the experts' spending behaviors are more similar, leading to better performance.
Fractional Play: This strategy involves allowing the learner to allocate its budget fractionally across different experts rather than committing fully to one expert at a time. When combined with fractional allocation, algorithms can achieve specific near-optimal regret bounds in both the standard and constrained learning settings.

Summary

The gist The authors establish near-optimal regret bounds for budget-constrained online learning against arbitrary classes of budget-pacing experts in an adversarial setting.

Learning with Constraints

The paper addresses the problem of online learning problems with resource constraints (OLRC), which forces a learner to only take the null action once its fixed budget is depleted, introducing a temporal coupling of decisions across the time horizon<ref:2610.11074#pg6>. The core difficulty lies in simultaneously learning which action is better and deciding what mixture of experts to follow that guarantees both no-regret and budget feasibility<ref:2610.11074#pg4>. This complexity is summarized by the “spend-or-save" dilemma, where any algorithm satisfying the budget constraint must incur a worst-case regret of omega(T)<ref:2610.11074#pg6> and learnability depends on controlling the variability of experts’ spending patterns<ref:2610.11074#pg6>.

Key Findings and Characterization

The authors introduce the Kolmogorov distance, Kol(c, c′), as a more relevant quantity than Earth Mover’s Distance (EMD) for characterizing optimal regret rates in learning settings with resource constraints<ref:2610.11074#pg6>. They show that the optimal regret is characterized by this distance between spending patterns of experts<ref:2610.11074#pg6>. Specifically, they establish that a regret bound of either O(D√log F) in the ORA setting or O((√T + D)√log F) in the OLRC setting can be achieved when allocation is allowed to be fractional<ref:2610.11074#pg6> and fractional play is used<ref:2610.11074#pg6>.

Optimal Algorithms and Guarantees

The authors present an algorithm with a regret guarantee of O(D) and is O(D)-pacing against the baseline b in a toy example involving alternating red and blue balls<ref:2610.11074#pg6>. More generally, they prove that for fractional play, the algorithms achieve RegT = O(D√log F) in the ORA setting and RegT = O((√T + D)√log F) in the OLRC setting<ref:2610.11074#pg6> when allocation is allowed to be fractional<ref:2610.11074#pg6>.

Extensions for Subpacing Experts

The paper provides three modular extensions to generalize the results, including extending the comparator class from pacing to subpacing experts<ref:2610.11074#pg6> and requiring the learner itself to be subpacing<ref:2610.11074#pg6>. Extension 3 converts any resulting fractional controller to sampled play with pathwise budget feasibility and high-probability guarantees<ref:2610.11074#pg6>.

Lower Bounds

The paper establishes strong lower bounds, showing that for any two spending patterns s, s′ in [0, 1]T such that Pst = t s′t = B and for any algorithm that spends at most B in expectation, there exists a set of experts F and a sequence of rewards and costs such that the algorithm incurs expected regret at least Kol(s, s′)/4<ref:2610.11074#pg6>. This lower bound holds even if the algorithm can observe the rewards and costs of the current round before making its decision<ref:2610.11074#pg6>.

Composition of Extensions

A corollary shows that applying Extension 3 to an algorithm with a deterministic fractional regret bound does not change this deterministic bound, but it adds a term related to the concentration radius<ref:2610.11074#pg6>. Furthermore, if Extension 2 is used, the realized sampled resource-i cost path is (Wi + Γi(δ))-subpacing for every i in [m]<ref:2610.11074#pg6>.

Conclusion

The final results show that the sampled learner satisfies every hard budget on every realization and, with probability at least 1 − δ, simultaneously for every expert f, RegfT ≤ Regfrac +Γr(δ) + max i∈[m] ηiΓi(δ)<ref:2610.11074#pg6>. The realized sampled resource-i path is (Wi + Γi(δ))-subpacing for every i in [m]<ref:2610.11074#pg6>. The optimal rates are derived by optimizing the regularization parameters, yielding bounds such as RegfT = Oem(Dα,∞ p log F + αmax + Vmix) with pre-decision feedback<ref:2610.11074#pg6> and RegfT = Oem((√T + D)p log F + Dη,∞ p log F log(m + 1)) without pre-decision feedback<ref:2610.11074#pg6>. The lower bounds show that no learner can guarantee regret o(max[D, T1/2] · √log F) in the OLRC setting<ref:2610.11074#pg6>.

Improvements for AI systems

Bold header: Enhanced Budget Pacing for Resource Allocation Systems

The improved system can achieve near-optimal regret bounds of O(D√log F + √T log F) in the ORA setting by employing fractional play with a controller that chooses an action distribution over experts, specifically by solving the minimax problem defined in Algorithm 2.

Bold header: Robust Learning Under Budget Constraints

The system can guarantee a worst-case regret of O((√T + D)√log F) in the OLRC setting when fractional allocation is allowed, providing tighter bounds than previously known for tasks where experts exhibit variability in their spending patterns.

Bold header: Adaptation to Subpacing Experts

By implementing Extension 1, the system can handle experts whose spending patterns are D-subpacing with respect to b, ensuring that the regret bound remains optimal up to constant factors even when expert pacing is relaxed.

Bold header: High-Probability Budget Feasibility in Sampled Play

The system can operate in a sampled play setting, guaranteeing every hard budget on every realization and achieving a regret bound of Regfrac + Γr(δ) + max i∈[m] ηiΓi(δ) with probability at least 1 − δ.

Bold header: Subpacing Guarantee for Realized Paths

When using Extension 2, the system can ensure that the realized sampled path has the same W plus a deviation of this order, meaning the realized resource-i cost path is (Wi + Γi(δ))-subpacing for every i ∈ [m].

Abstract

We establish near-optimal regret bounds for budget-constrained online learning against arbitrary classes of budget-pacing experts in the adversarial setting. In particular, given any class of F experts and a candidate budget pacing schedule, we provide a full-information algorithm which obtains regret O(D sqrt F + sqrt T F) against all experts whose cumulative spending stays within distance D of this schedule, matching lower bounds established by Braverman et al. (2025). We additionally show that our technique extends to various problems in online resource allocation, where the learner gets to see the rewards and costs of the current options available to them, and establish O(D sqrt F) regret bounds when fractional allocation is allowed. This is the first algorithm we are aware of which can achieve o(sqrt T) guarantees for such tasks.