Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Adwords with Unknown Budgets and Beyond

Rajan Udwani

TL;DR

This work considers a variation of the classic Adwords problem where the online algorithm does not know the advertisers' budgets a priori and the budget of an advertiser is revealed to the algorithm only when it is exceeded and achieves the best possible performance guarantee for deterministic online matching in the presence of multi-channel traffic.

Abstract

In the classic Adwords problem introduced by Mehta et al.\ (2007), we have a bipartite graph between advertisers and queries. Each advertiser has a maximum budget that is known a priori. Queries are unknown a priori and arrive sequentially. When a query arrives, advertisers make bids and we (immediately and irrevocably) decide which (if any) Ad to display based on the bids and advertiser budgets. The winning advertiser for each query pays their bid up to their remaining budget. Our goal is to maximize total budget utilized without any foreknowledge of the arrival sequence (which could be adversarial). We consider the setting where the online algorithm does not know the advertisers' budgets a priori and the budget of an advertiser is revealed to the algorithm only when it is exceeded. A naïve greedy algorithm is 0.5 competitive for this setting and finding an algorithm with better performance remained an open problem. We show that no deterministic algorithm has competitive ratio better than 0.5 and give the first (randomized) algorithm with strictly better performance guarantee. We show that the competitive ratio of our algorithm is at least 0.522 but also strictly less than $(1-1/e)$. We present novel applications of budget oblivious algorithms in search ads and beyond. In particular, we show that our algorithm achieves the best possible performance guarantee for deterministic online matching in the presence of multi-channel traffic (Manshadi et al. (2022)).

Adwords with Unknown Budgets and Beyond

TL;DR

This work considers a variation of the classic Adwords problem where the online algorithm does not know the advertisers' budgets a priori and the budget of an advertiser is revealed to the algorithm only when it is exceeded and achieves the best possible performance guarantee for deterministic online matching in the presence of multi-channel traffic.

Abstract

In the classic Adwords problem introduced by Mehta et al.\ (2007), we have a bipartite graph between advertisers and queries. Each advertiser has a maximum budget that is known a priori. Queries are unknown a priori and arrive sequentially. When a query arrives, advertisers make bids and we (immediately and irrevocably) decide which (if any) Ad to display based on the bids and advertiser budgets. The winning advertiser for each query pays their bid up to their remaining budget. Our goal is to maximize total budget utilized without any foreknowledge of the arrival sequence (which could be adversarial). We consider the setting where the online algorithm does not know the advertisers' budgets a priori and the budget of an advertiser is revealed to the algorithm only when it is exceeded. A naïve greedy algorithm is 0.5 competitive for this setting and finding an algorithm with better performance remained an open problem. We show that no deterministic algorithm has competitive ratio better than 0.5 and give the first (randomized) algorithm with strictly better performance guarantee. We show that the competitive ratio of our algorithm is at least 0.522 but also strictly less than . We present novel applications of budget oblivious algorithms in search ads and beyond. In particular, we show that our algorithm achieves the best possible performance guarantee for deterministic online matching in the presence of multi-channel traffic (Manshadi et al. (2022)).

Paper Structure

This paper contains 32 sections, 30 theorems, 148 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Preliminaries
  5. Upper Bound for Deterministic Algorithms
  6. Randomized Budget Oblivious Algorithm and Analysis
  7. Overview of the Analysis of $f\textsc{-GPG}$
  8. Main Challenge.
  9. Lower Bound on Competitive Ratio of $f\textsc{-GPG}$ (Theorem \ref{['guafalg']})
  10. Notation and Definitions.
  11. Establishing \ref{['dual2']} for the Candidate Solution.
  12. From Fractional to Integral Algorithm (Theorem \ref{['main']})
  13. Upper Bound on Competitive Ratio of GPG
  14. Extensions and Applications
  15. Online Matching with Multi-channel Traffic
  16. ...and 17 more sections

Key Result

Theorem 1

Every deterministic budget oblivious online algorithm for Adwords has competitive ratio at most 0.5, even on instances with binary bids and large budgets.

Figures (1)

  • Figure 1: Plot of $\alpha(x)= \left[1-g(x)+ \int_0^x \left( \min\left\{g(y),\frac{g(x)-g(y)}{1-g(y)} \right\}\right)dy\right]$ for $g(x)=e^{1.15(x-1)}$.

Theorems & Definitions (55)

  • Example 3.1
  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • proof
  • ...and 45 more