1 Introduction
Online advertising has emerged as the dominant marketing channel in many parts of the world. According to some estimates, in the year 2019, more than 450 billion USD were spent on online ads, which accounts for over 60% of overall expenditure on ads
^{1}^{1}1Digital advertising spending worldwide from 2019 to 2024. https://www.statista.com/statistics/237974/onlineadvertisingspendingworldwide/. Internet search is a prominent channel for online advertisement. In this medium, also called search ads, advertisements are displayed along side online search results for key words relevant to the advertiser. In 2020 alone, Google made a revenue of over 104 billion USD from “search & other”, which accounts for 70% of their total revenue from advertising and exceeds half the total revenue of parent company Alphabet ^{2}^{2}2Alphabet Year in Review 2020. https://abc.xyz/investor/. Given the significance of search advertisement, there is a wealth of work that studies the problem from different points of view. In this paper, we are interested in the following viewpoints.From the platform’s point of view: A crucial problem is deciding which (if any) ads to show for each search query before future queries are realized. The Adwords model introduced by Mehta et al. (2007), captures the key elements of this problem. For simplicity, the model describes the problem of showing at most one ad per query^{3}^{3}3See Section 6 in Mehta et al. (2007) for the generalization to multiple ads..
Adwords problem: At the beginning of the planning period (typically a day), the platform has a set of advertisers along with their maximum budgets . Queries arrive sequentially on the platform and when a query arrives, advertisers make bids . Given the bids and advertiser budgets, the platform decides immediately which ad to display along side the search results for the query. The chosen advertiser pays their bid but only up to their remaining budget i.e., advertisers do not make a total payment exceeding their budget. The objective of the platform is to maximize the total advertiser budgets utilized without any foreknowledge of the arrival sequence (which could be adversarial).
We measure the performance of an online allocation algorithm for this problem by evaluating the competitive ratio i.e., the worst case relative performance gap between the online algorithm and (optimal) offline algorithm that knows all the queries and advertiser bids.
The main algorithm design goal for this problem and its many variations is to perform better than the naïve greedy algorithm that, for each query, shows the ad with highest bid and (nonzero) available budget. Mehta et al. (2007) proposed the prominent bid pricing algorithm for this problem (formally discussed later on), that achieves the best possible competitive ratio of . In comparison, the greedy algorithm has a competitive ratio of 0.5.
From advertiser’s point of view: The goal is often to target their customers through multiple ad campaigns and marketing channels. Starting with an overall budget, an advertiser must determine a good distribution of their budget to individual ad campaigns. For search ads, advertisers must also determine the bids for relevant key words. While these decisions play a crucial role in the success of their ad campaigns, determining a good distribution of budgets between diverse options and deciding optimal bids for specific campaigns can be an incredibly challenging task for any advertiser.
Many advertisers rely on automated bidding and budget management tools. The motivation behind these tools is to improve performance for advertisers while simplifying the usage of ad platforms (Aggarwal et al. 2019). In using a tool to manage their portfolio, the advertiser decides the overall budget, creates a portfolio of ad campaigns, and specifies a high level performance goal for the portfolio. Using these specifications, the tool aims to automatically determine (over time) a good budget distribution for the portfolio, as well as, bids for key words in each campaign.
1.1 Adwords with Unknown Budgets: Motivation and Problem Description
While automated management of ad portfolio has become increasing prominent, to the best of our knowledge, platforms typically optimize a daily budget distribution for the portfolio and do not perform realtime adjustments during the day to this distribution based on live performance of campaigns in the portfolio ^{4}^{4}4Google Search Ads 360 Help. https://support.google.com/searchads/answer/7657341. Platforms with inhouse tools for automated budget optimization (such as Search Ads 360 by Google) may, in fact, have the crosscampaign data necessary to perform adjustments to budget distribution in realtime. This can have tangible benefits. For the sake of illustration, consider the following stylized example.
Example:
Consider an advertiser with a portfolio composed of two search ad campaigns labeled
. In the absence of a budget constraint, let the total expenditure (per day) in campaign 1 be a Bernoulli random variable
. Similarly, let represent the maximum expenditure (per day) in campaign 2. Now, suppose we have a total daily budget of 1 that needs to be distributed between the two campaigns. When and are identically distributed, splitting the budget evenly between the two campaigns is always optimal. When the mean of (and ) is , this strategy uses % of the total budget in expectation.Now, suppose that instead of fixing the daily budget the platform performs realtime budget adjustments using live crosscampaign data. Suppose that i.e., the expenditures in the campaigns are in perfect negative correlation. Using this knowledge in conjunction with live data, the platform can always adjust budgets on the fly to obtain a budget utilization of 100%. In general, by realtime budget adjustments the platform can achieve a utilization of %%.
We believe that an important rationale behind the current practice of fixing budgets at the start of each day, stems from the importance of fixed budgets in algorithms for ad allocation. For instance, consider the bid pricing algorithm of Mehta et al. (2007) that we mentioned earlier. This algorithm decides ad allocation greedily based on bid prices that account for the fraction of remaining budgets. Formally, given remaining budget on arrival of query , the algorithm computes bid prices
and shows the ad with the largest bid price. Thus, foreknowledge of daily budgets for all advertisers is essential for defining the algorithm. Similarly, most (if not all) algorithms for ad allocation in the literature (see Mehta et al. (2013), Alaei et al. (2012), Devanur et al. (2019), Devanur and Hayes (2009), Mirrokni et al. (2012)), rely critically on the knowledge of (fixed) budgets for each advertiser.
The naïve greedy algorithm is the one exception to this. For each query, greedy shows the ad with highest bid and (nonzero) available budget. Therefore, it is budget oblivious i.e., does not require any advance information about budgets except the knowledge of which advertisers are still participating (have nonzero remaining budget). Recall that the central goal in designing online algorithms for ad allocation is to find algorithms that are provably better than greedy. This motivates us to consider the following problem.
Is there a budget oblivious online algorithm for Adwords that outperforms greedy?
A budget oblivious algorithm easily adapts to realtime changes in budget since it makes (randomized) ad allocation decisions for each query using only the bids and the knowledge of which advertisers have nonzero remaining budget.
Another motivation: Remarkably, this problem has a strong connection to the setting of online matching with stochastic rewards (Mehta and Panigrahi 2012). In fact, this connection previously prompted Mehta et al. (2013) to raise the same question in a different context (see Open Question 20 in Mehta et al. (2013)). We describe this relationship in more detail in Section 2.1.1.
1.2 Overview of Our Contributions
We give the first budget oblivious algorithm for ad allocation that provably outperforms greedy.
Theorem (Informal).
There exists a randomized competitive algorithm for Adwords with unknown budgets.
Our algorithm (defined in Section 3) is a natural generalization of the PerturbedGreedy algorithm of Aggarwal et al. (2011) (further details in Section 2.1). Proving a performance guarantee for this family of algorithms faces many obstacles due to combinatorial interactions between the time varying nature of bids and randomness inherent in the algorithm, as well as, its budget oblivious nature. To address these challenges, our analysis takes an indirect path where we first show the desired guarantee for a relaxed (fractional) version of the algorithm that is not budget oblivious. To establish a performance guarantee for the relaxed algorithm, we employ a recently developed LP free approach from Goyal et al. (2021) and Goyal and Udwani (2020) which, in turn, builds on the classic randomized primaldual approach of Devanur et al. (2013). Even with this approach, proving a guarantee better than 0.5 for the relaxed algorithm requires novel structural insights into the problem (due to the timevarying bids and randomness in the algorithm). Finally, we relate the expected performance of the original algorithm with its budget aware relaxation. This step is made challenging by the fact that our original algorithm is budget oblivious and its decisions may differ substantially from the budget aware relaxation.
Outline for rest of the paper: Section 2 discusses the assumption of small bids and presents a resource allocation version of the Adwords setting that we use in the rest of the paper. We show our results for this generalization. In Section 2.1, we discuss related work in online matching and resource allocation. In Section 3, we first present our algorithm and main result, followed by a discussion of the key bottlenecks in proving the result. This is followed by the formal analysis in Section 3.2. Finally, Section 4 concludes our discussion with some intriguing open problems.
2 Preliminaries
The following resource allocation setting is equivalent to the Adwords problem.
Online Budgeted Allocation (OBA): Consider a complete bipartite graph with vertex set . Vertices , called resources, have budgets and per unit rewards . Resource and their rewards and budgets are known to us. The remaining vertices , called arrivals, are unknown a priori and arrive sequentially. When a vertex arrives, we see their bids
. The bid vector
indicates that arrival is interested in up to units of resource , for every . Every arrival is interested in at most one resource. Given the bids for arrival , we must immediately and irrevocably match the arrival to at most one resource. If arrival is matched to resource , units of are consumed and we receive a reward per unit of consumed with the caveat that the total reward/revenue from can not exceed budget . In other words, the total reward from is capped at . The goal is to decide the allocation/matching for arrivals without any knowledge of future arrivals and such that the total reward is maximized.The Adwords problem is an instance of OBA where resources correspond to ads, arrivals to queries, and arrival bids represent the bid of advertisers for query . The per units rewards are set to 1 in Adwords for every . On the other hand, an instance of OBA with bids , budgets , and per unit revenues , is equivalent to the Adwords setting with scaled bids and budgets .
To understand a crucial but often implicit assumption in the Adwords setting, we define the bidtobudget ratio,
In the Adwords setting, one typically assumes that . This is also called the small bid assumption. This assumption is in line with the practice of search ads, where individual bids are typically much smaller than the overall budget. Note that the guarantee for the bid pricing algorithm of Mehta et al. (2007) holds only in the small bid regime. Even in this regime, no online algorithm can have competitive ratio better than . The focus of this paper is on OBA with unknown budgets in the small bid regime.
2.1 Related Work
Let us start with the classic setting of online bipartite matching. This problem can be viewed as a special case of OBA where every resource has unit budget and bids are binary. A bid of 1 denotes an edge in the bipartite graph and bid of 0 denotes the absence of an edge. All per unit rewards are identically 1 and the goal is to find the largest matching. Karp et al. (1990) introduced this problem and showed (among other results) that randomly ranking resources at the start and then matching every arrival to the best ranked unmatched resource is a competitive algorithm for this setting. In fact, this algorithm, called Ranking, achieves the best possible guarantee for the problem. The analysis of Ranking was clarified and considerably simplified by Birnbaum and Mathieu (2008) and Goel and Mehta (2008).
Aggarwal et al. (2011), proposed the Perturbed Greedy algorithm that is competitive for vertex weighted case where per unit rewards can be arbitrary. Kalyanasundaram and Pruhs (2000) considered the problem of online matching where the budget of every resource can be more than 1 and showed that as , the natural (deterministic) algorithm that balances the budget used across resources is competitive. Generalizing this setting, Mehta et al. (2007) introduced the Adwords problem and proposed the bid pricing based algorithm for Adwords under the small bid assumption. Buchbinder et al. (2007) gave a primaldual analysis for this algorithm. Subsequently, Devanur et al. (2013) proposed the randomized primaldual framework that can be used to show all the aforementioned results in an elegant and unified way. It is worth noting that the Adwords/OBA setting (without the small bid assumption), generalizes each of the settings discussed above. The budgetaware greedy algorithm that matches every arrival according to the following rule,
where is the remaining budget of on arrival of , is unconditionally 0.5 competitive for OBA. Recently, Huang et al. (2020) gave the first algorithm with competitive ratio better than 0.5 for general OBA/Adwords i.e., without the small bids assumption.
While the body of work discussed above considers an adversarial arrival sequence, there is also a wealth of work on online matching and resource allocation in stochastic and hybrid/mixed models of arrival (for example, Goel and Mehta (2008), Feldman et al. (2009), Devanur and Hayes (2009), Karande et al. (2011), Manshadi et al. (2012), Alaei et al. (2012), Devanur et al. (2019), Mirrokni et al. (2012)). For a comprehensive review of these settings, we refer to Mehta et al. (2013).
2.1.1 Online Matching with Stochastic Rewards
Introduced by Mehta and Panigrahi (2012)
, this problem generalizes online bipartite matching by associating a probability of success
with every edge . When a match is made i.e., edge is chosen, it succeeds independently with this probability. If the match fails the arrival departs but the resource is available for future rematch. While the general case of this problem is challenging and unresolved, a well studied case (motivated by applications) is when edge probabilities are vanishingly small i.e., . Mehta and Panigrahi (2012) showed an equivalence between the vanishing probabilities case and the following instance of Adwords with unknown budgets: (i) Bidsi.e., bids are equal to the edge probabilities and missing edges have a bid of 0. (ii) Budgets are unknown but it is known that the budget of each resource is independently sampled and follows the exponential distribution with unit mean.
Subsequent work (Mehta et al. 2015, Goyal and Udwani 2020, Huang and Zhang 2020), further generalized and used this connection to show new results for the stochastic rewards setting with vanishing probabilities. Using this connection, our result for Adwords with unknown budgets yields a 0.522 competitive algorithm for stochastic rewards with vanishing probabilities. We note that this result does not improve the stateoftheart for this problem in terms of the competitive ratio guarantee^{5}^{5}5In particular, Goyal and Udwani (2020) give an algorithm with guarantee of 0.596.. However, it shows that a natural generalization of the Perturbed Greedy algorithm, which yields the optimal guarantee for the setting where all edge probabilities are 1 (more generally, decomposable (Goyal and Udwani 2020)), also has a guarantee better than 0.5 for the well studied case of vanishing probabilities.
3 Budget Oblivious Algorithm and Analysis
Consider the following family of randomized algorithms with parameter .
Observe that Algorithm 1 matches every arrival greedily based on randomized bid prices . The uniform random variables , called seeds, are sampled independently for each . Therefore, ties between bid prices of any two (different) resources occur with a probability of 0. Algorithm 1 replaces the budget dependent factor with a random quantity, resulting in a budget oblivious algorithm.
Note that, Algorithm 1 is a generalization of the Perturbed Greedy (PG) algorithm in Aggarwal et al. (2011). Recall that the PG was designed for vertex weighted online bipartite matching that, in terms of OBA, corresponds to instances where bids are binary and all budgets equal 1. In this special case, when , Algorithm 1 collapses to the PG algorithm.
Due to challenges outlined in Section 3.1, we prove a performance guarantee for Algorithm 1 by first analyzing its fractional relaxation.
Description of Algorithm 2: In the fractional setting time runs continuously from . In other words, arrival is matched fractionally during the time period . Set is the set of resources with available budget at time . Taking an infinitesimal viewpoint, at time we match fraction of arrival to resource . This is the resource with the maximum bid price out of all resources in . Similar to Algorithm 1, ties between bid prices occur with a probability of 0. Now, the allocation of to uses of resource ’s budget and earns a reward of . At time the process of fractionally matching arrival ends and we start matching arrival . The total amount of ’s budget that is allocated to arrival between is given by . Thus, the total reward of Algorithm 2 given seed is .
Observe that Algorithm 2 is not budget oblivious. However, this is not of concern as the algorithm is only used as an intermediate step to analyze Algorithm 1. We show the following guarantee for Algorithm 1 and Algorithm 2.
Theorem 1.
Theorem 2.
We start by highlighting some of the key challenges with analyzing the algorithms above. This is followed by an overview of our analysis and approach for tackling the obstacles presented below. We present the main analysis in Section 3.2.
3.1 Challenges in Analysis
We illustrate the main challenges at a high level via a simple example. For more details and nuanced examples, see Appendices 5 and 6. We shall use the following labels, Algorithm 1 is Alg and Algorithm 2 is . Also, we use to denote the random seeds in Alg and .
Example 3.1.
Consider a snippet of an instance with resources and and per unit rewards . We focus on consecutive arrivals and . Let bids , and . Consider an execution of Alg with and the random seed for resource fixed at . Suppose that for every value of seed , resource is not matched to any arrival prior to and only unit of resource ’s budget is available . Observe that,

.

.
Consider the matching generated by Alg as we decrease from to 0. For , Alg matches to and this uses up all of ’s budget. Consequently, is matched to for . For , is matched to and to . Thus, the number of units of resource matched in Alg decreases from 2 to 1 as decreases. This is somewhat surprising as the bid price of resource 1 increases monotonically as we decrease . At a high level, there are two main reasons behind this occurrence.
The first reason is that, in general, bid prices vary with both time (due to timevarying bids) and seed values . In the example above, even though bid price of resource increases everywhere as decreases, the time variation in bid ensures that at arrival the bid price of resource 2 is always higher than that of resource 1. This is the first high level challenge in analyzing the performance of Alg.
The second reason is unknown budgets, due to which we may overestimate the number of units of a resource available. In the example above, recall that only 1 unit of resource is available at . For , we have i.e., at arrival , resource has a higher bid price than resource 2 if we account for the number of remaining units of each resource. However, Alg, being ignorant of the budget, computes a higher bid price for resource at .
Overview of Our Analysis.
To address these challenges we first isolate them. We accomplish the isolation by switching over to the fractional algorithm. Notice that the second challenge disappears in the fractional version; uses budgets and never overestimates the bid price of a resource. However, the variation of bid prices with both time and seed makes it challenging to analyze even (more details in Appendices 5 and 6). So to prove Theorem 2, we adopt an LP free analysis approach that was developed recently in Goyal et al. (2021) and Goyal and Udwani (2020). It is worth noting that previous work uses the LP free approach to handle stochastic elements in online allocation problems. While our underlying problem is deterministic, the increased flexibility of the LP free framework resolves, in a very natural way, some of the difficulty we face in analyzing (for an example, see Appendix 5).
To prove Theorem 1, one possibility would be to establish that the behavior of Alg is very close to on every sample path. This is where the second challenge comes to the fore. Since Alg is oblivious to budgets but is not, there can be a substantial difference between the output of these algorithms for the same seed (see Example 6.1 in Appendix 6). We overcome this challenge by comparing Alg not with on the same instance but with the performance of on a modified instance where the budgets of resources are (slightly) higher. Next, we introduce some new notation before presenting the main analysis.
Notation.
Let the number of resources
. We extend the definition of bids to every moment
as follows, where for some arrival .We use Alg to refer to Algorithm 1 as well as its expected reward. denotes the matching generated by Alg when the random seed is given by . Let denote the set of arrivals matched to in matching . Overloading notation, we also use to denote the total reward of Alg with seed and to denote the total budget of used in . Let denote the total budget of allocated to arrivals in i.e.,
Similarly, let denote the fractional Algorithm 2 and also its expected total reward. Let denote the fractional matching as well as the total reward generated by given seed . Let denote the set of resources available at time in with seed . Notice that for every and , there is at most one interval such that every moment is matched to and every other moment i.e., , is not matched to . . We call the interval a segment. Since each segment corresponds to a unique resource, there are at most segments in the interval for every . Thus, for every , there are at most segments in ^{6}^{6}6It can be shown that there are at most segments in all but the loose bound of suffices.. Let denote the union of all segments matched to in as well as the total budget of used in . Let denote the total budget of allocated prior to time in . Notice that
Finally, let Opt refer to the optimal offline algorithm as well as its total reward. Since there are no unknowns in the offline problem i.e., budgets and bids are all known, Opt is deterministic. Let denote the set of arrivals matched to as well as the total fraction of ’s budget that is matched in Opt. More generally, we use the notation , for every moment such that for some arrival . Note that .
3.2 Analysis
Consider the following linear system in variables and ,
(1)  
(2)  
(3) 
The following lemma is a special case of Lemma 5 in Goyal et al. (2021). For the sake of completeness, we include a proof.
Proof.
In the following, we prove Theorem 2 by finding a feasible solution to the system (1)(3) with a suitably large value . Recall that denotes the set of resources available at time in . To define the candidate solution, we first define,
(4)  
(5) 
The candidate solution is,
(6) 
For the above candidate solution, constraints (3) are obviously satisfied. Also by definition, constraint (1) holds with equality. It remains to show that inequalities (2) are satisfied. In fact, we show a stronger statement as described in the next lemma.
Lemma 4.
Proof.
The lemma follows by taking expectation over on both sides of (7). ∎
For the analysis that follows, fix an arbitrary resource and seed . For brevity, we suppress dependence on from notation and highlight only the dependence on seed . So is the matching generated by when it is executed with seed . Similarly, denotes the total amount of ’s budget allocated in . Let and . Let denote the set of resources available at in . For every , we define critical threshold such that
Set if no such value exists and if the set is empty. Due to the monotonicity of , we have a unique value of .
Lemma 5.
Given and seed , for every , we have
Proof.
Given and , consider an arbitrary seed and a moment . By definition of , we have . It remains to show that . We claim that this follows from,
To see this, observe that given the claim above we have,
It remains to show that .
For the sake of contradiction, let be such that . Let be the earliest time such that there exists a resource i.e., () is available at in but unavailable at in . This occurs only if is matched at some time in . but not matched at in . Now, the following statements are true.

. This follows from the definition of and the fact that .

. Follows from and the fact that .
Thus, . Now, at every moment, picks an available resource with highest bid price. So if is matched to in , then it must also be matched to in (the bid prices of do not change). This contradicts the definition of . ∎
Recall that a segment in is a contiguous time interval such that every moment in the interval is matched to the same resource. As we discussed, there are a finite number of segments in . Observe that every moment in a given segment has the same critical threshold value. We define the set and notice that the set has finitely many distinct values (at most one value per segment). Let
i.e., is the cumulative bid on from segments in that have critical threshold value and correspond to arrivals in . Observe that,
Let
We also define the set that includes all moments in with critical threshold at least i.e.,
By definition, .
Corollary 6.
Given and seed , we have .
Proof.
Follows from Lemma 5 and the definition of set and values . ∎
Lemma 7.
Given and seed , let indicate the event . Then, for every , we have
Proof.
Consider and and fix an arbitrary seed . Recall that, denotes both, the set of moments matched to in , as well as, the total budget of used in . When bids for every , it can be shown that , and (it can be shown that) this implies a competitive ratio of for . In general, may be strictly smaller than with constant probability (see Example 6.2). In fact, there exist examples where for some . Keeping this in mind, we now consider two cases. Combining inequalities derived in individual cases gives us the desired.
Case II: () i.e., resource is available at every moment in . First, we establish a refined lower bound on , followed by a lower bound on .
Since is available at every moment in , we have ,
Combining this with the lower bound from Lemma 5, we get
(8) 
Next, from the definition of and Lemma 8 (shown subsequently), we have
(9)  
For a given , we could be in the worse of the two cases above. Thus, we have the following combined lower bound,
∎
Lemma 8.
Consider a resource and seed vector such that . Then, we have .
Proof.
Given resource and seed , for every , define
Using Lemma 5 we have, . Now, fix a seed such that . Observe that the main claim follows from the following upper and lower bounds on .
Proof of lower bound: Since , we have that is available at every moment in . Thus, in , every is matched to a resource such that,
From this, we have for every ,
where we used the following facts (i) For , we have and (ii) For and , we have .
Integrating over all segments in , we get
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