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Online Bidding Algorithms with Strict Return on Spend (ROS) Constraint

Rahul Vaze, Abhishek Sinha

TL;DR

The paper analyzes auto-bidding under a strict ROSC constraint in an online setting with i.i.d. inputs and unknown allocation/payments. It proves an inherent barrier: any online algorithm that strictly satisfies ROSC cannot achieve sub-linear regret, even for simple input structures, and characterizes trade-offs between regret and ROSC violations. Focusing on structured regimes, it introduces an algorithm with regret $O( oot 2{T} olinebreak[0]{ ext{log} T})$ under repeated identical auctions with threshold allocations, and a $1/2$-competitive Learn-Alg for a discretized setting, both while meeting ROSC with high probability. The results clarify the fundamental limits of ROSC-constrained online bidding and provide practical strategies when ROSC is allowed to be violated or when problem structure is leveraged. Together, the work highlights the tension between optimal offline benchmarks and strict constraint satisfaction in online advertising settings, offering both impossibility results and near-optimal algorithms under specific conditions.

Abstract

Auto-bidding problem under a strict return-on-spend constraint (ROSC) is considered, where an algorithm has to make decisions about how much to bid for an ad slot depending on the revealed value, and the hidden allocation and payment function that describes the probability of winning the ad-slot depending on its bid. The objective of an algorithm is to maximize the expected utility (product of ad value and probability of winning the ad slot) summed across all time slots subject to the total expected payment being less than the total expected utility, called the ROSC. A (surprising) impossibility result is derived that shows that no online algorithm can achieve a sub-linear regret even when the value, allocation and payment function are drawn i.i.d. from an unknown distribution. The problem is non-trivial even when the revealed value remains constant across time slots, and an algorithm with regret guarantee that is optimal up to logarithmic factor is derived.

Online Bidding Algorithms with Strict Return on Spend (ROS) Constraint

TL;DR

The paper analyzes auto-bidding under a strict ROSC constraint in an online setting with i.i.d. inputs and unknown allocation/payments. It proves an inherent barrier: any online algorithm that strictly satisfies ROSC cannot achieve sub-linear regret, even for simple input structures, and characterizes trade-offs between regret and ROSC violations. Focusing on structured regimes, it introduces an algorithm with regret under repeated identical auctions with threshold allocations, and a -competitive Learn-Alg for a discretized setting, both while meeting ROSC with high probability. The results clarify the fundamental limits of ROSC-constrained online bidding and provide practical strategies when ROSC is allowed to be violated or when problem structure is leveraged. Together, the work highlights the tension between optimal offline benchmarks and strict constraint satisfaction in online advertising settings, offering both impossibility results and near-optimal algorithms under specific conditions.

Abstract

Auto-bidding problem under a strict return-on-spend constraint (ROSC) is considered, where an algorithm has to make decisions about how much to bid for an ad slot depending on the revealed value, and the hidden allocation and payment function that describes the probability of winning the ad-slot depending on its bid. The objective of an algorithm is to maximize the expected utility (product of ad value and probability of winning the ad slot) summed across all time slots subject to the total expected payment being less than the total expected utility, called the ROSC. A (surprising) impossibility result is derived that shows that no online algorithm can achieve a sub-linear regret even when the value, allocation and payment function are drawn i.i.d. from an unknown distribution. The problem is non-trivial even when the revealed value remains constant across time slots, and an algorithm with regret guarantee that is optimal up to logarithmic factor is derived.

Paper Structure

This paper contains 18 sections, 20 theorems, 51 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Prior Work
  3. Related Problems
  4. Our Contributions
  5. Problem Formulation
  6. WarmUp
  7. Lower Bound on Regret of Any Online Algorithm that strictly satisfies the ROSC
  8. Repeated Identical Auction Setting $v_t=v >0\ \forall \ t.$
  9. Lower Bound on Regret for Any Online Algorithm $\mathcal{A}$
  10. Lower Bound on Regret for algorithm Feng
  11. Algorithm with regret $O(\sqrt{T \log T})$ in the Repeated Identical Auction strictly satisfying the ROSC
  12. $1/2$-Competitive Algorithm for Problem \ref{['eq:prob']}
  13. Conclusions
  14. Proof of Theorem \ref{['thm:lbunivGenV']}
  15. Proof of Theorem \ref{['thm:lbuniv']}
  16. ...and 3 more sections

Key Result

Lemma 4

$\mathcal{R}_{\mathcal{A}_s}(T) =\Omega(T)$ regret even when $v_t=v\ \forall \ t$.

Figures (3)

  • Figure 1: Bid Profiles for algorithm $\mathcal{A}_b$ and $\mathcal{A}_c$ for Example \ref{['exm:1']}.
  • Figure 2: Input used to prove Theorem \ref{['thm:lbunivGenV']}.
  • Figure 3: Input used to prove Theorem \ref{['thm:lbuniv']}.

Theorems & Definitions (36)

  • Remark 1
  • Definition 1
  • Remark 2
  • Definition 2
  • Example 3
  • Lemma 4
  • Theorem 5
  • Corollary 6
  • Remark 3
  • Definition 7
  • ...and 26 more