Learning-Augmented Online Bidding in Stochastic Settings
Authors: Spyros Angelopoulos
Bertrand Simon
Venue: NeurIPS 2025 poster ()
Keywords: Learning-augmented algorithms
online algorithms
competitive analysis
randomized algorithms
distributional advice
Primary area: theory

Abstract: Online bidding is a classic optimization problem, with several applications in online decision-making, the design of interruptible systems, and the analysis of approximation algorithms. In this work, we study online bidding under learning-augmented settings that incorporate stochasticity, in either the prediction oracle or the algorithm itself. In the first part, we study bidding under distributional predictions, and find Pareto-optimal algorithms that offer the best-possible tradeoff between the consistency and the robustness of the algorithm. In the second part, we study the power and limitations of randomized bidding algorithms, by presenting upper and lower bounds on the consistency/robustness tradeoffs. Previous works focused predominantly on oracles that do not leverage stochastic information on the quality of the prediction, and deterministic algorithms.

Learning-Augmented Online Bidding in Stochastic Settings
Spyros Angelopoulos
Bertrand Simon
CNRS and International Laboratory
Université Grenoble Alpes
on Learning Systems
CNRS, INRIA, Grenoble INP, LIG
Montreal, Canada
Grenoble, France
spyros. pypross.ngellppuloos@pp.........
bertrand. s simon@cnrs..r
Abstract
Online bidding is a classic optimization problem, with several applications online madisiinmmaiin,,,. the design of interruptible systems, and the analysis of approximation algorithms. In this work, we study online bidding under learning- augmented settings that incorporate stochasticity, in either the prediction oracle or the algorithm itself. the first part, we study bidding under distributional predictions, and find Pareto-optimal algorithms that offer the best-possible tradeoff between the consistency and the robustness of the algorithm. In the second part, we study the power and limitations of randomized bidding algorithms, by presenting upper and lower bounds on the consistency/robustness tradeoffs. Previous works focused predominantly on oracles that do not leverage stochastic information on the quality of the prediction, and deterministic algorithms.
Introduction
Recent advancements machine learning have led to the development of powerful tools for efficiently learning patterns in various types of data. These advances spearheaded novel computational framework, which the algorithm designer has the ability to integrate a prediction oracle into the algorithm's design, theoretical analysis, and empirical evaluation. This paradigm shift has led the emergence of the field of learning augmented algorithms, which aims to leverage ML approaches towards the development of more efficient algorithms with data-driven capabilities.
Learning-augmenteed algorithms have witnessed significant growth in recent years, starting with the seminal works [45 and [46). They have been particularly impactful in the context of sequential and online decision making, which the algorithm must act on incoming pieces the input, adapt its strategy judiciously at appropriately chosen time steps. The defining characteristic of such settings that the algorithm operates in a state of incomplete information about the input, thus making the application of ML techniques particularly appealing. Specifically, they allow for more nuanced, and beyond the worst-case performance evaluation than the standard, and often overly pessimistic approaches such competitive analysis 23
In this work, we focus on simple, yet important problem of incomplete information. known as online bidding. In this problem, faced with some unknown target (or "threshold") sequence bids until one greater than or equal to u, paying the sum of its bids up to that point. Formally, the objective is to find an increasing sequence X = (zz)s20 of positive numbers of
player submits
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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minimum competirive ratio, defined as
cost(X, u)
r( r(X) sup =21
where cost( X,,) u) =
Σ 1=0
J. : 11-1 <u u S xz.
4
Despite its seeming simplicity, online bidding has many applications in optimization problems such as incremental clustering and online k-median (25], load balancing [33, searching for hidden target in the infinite line [18], and the design of algorithms with interruptible capabilities [47]; refer to
the survey [28] for a detailed discussion of several applications. As a concrete application, consider the setting in which user must submit a job for processing in cloud service: here, the bidding strategy defines the increasing walltimes with which the user submits the job [12]. More broadly, the problem abstracts the process of finding increasingly better estimates of optimal solutions both online and NP-hard optimization problems, a process that is informally known as "guess and double". It is thus not surprising that online bidding and related competitive sequencing problems were among the earliest studied with a learning-augmenn
lens. We discuss several related results in Section L.2
Previous studies of online bidding focused on deterministic settings, i.e., the bidding sequence is output by a deterministic process and, likewise, on deterministic prediction oracles, i.e., oracles that do not provide any probabilistic information on the quality of the prediction. In contrast, very little is known about bidding under stochastic settings. There are two main motivating reasons:
In many realistic applications, the prediction has an inherently distributional nature 330 For instance, in the related problem of contract scheduling [7. the system designer may rely available historical data about the estimated interruption time (i.e., the target, in the bidding terminology). Hence the question: Can we find strategies that efficiently leverage a stochastic prediction oracle, while remaining robust against adversarial predictions?
It is known that randomization can help improve the competitive ratio of standard online bid- ding [27 Randomized algorithms also have additional benefits: they remain much more robust to small prediction errors, and tend to be much less brittle than deterministic algorithms. as demon- strated in 333 21). Hence the question: What the power and the limitations of randomized, learning-auge augmented bidding algorithms?
1.1 Contribution
Motivated by the discussion above, in this work we study online bidding under two distinct stochastic settings. First, in Section 2 we consider stochastic prediction oracles that provide distributional information on the target. We show how to compute a strategy that attains the best-possible tradeoff between the consistency (informally, the expected performance assuming that the target is generated according to the predicted distribution) and the robustness (i.e., the performance assuming an adversarial prediction). Using the terminology of serming-augmenter computation, we obtain provably Pareto-optimal strategy. Previous work only addressed the case in which the robustness is equal to the best-possible competitive ratio, namely r = 4 77]. a case for which optimal strategies have a very simple structure. In our work, instead, we compute the entire Pareto frontier between the consistency and the robustness, which is much more challenging task since optimal strategies have a complex structure and no clear characterization. Indeed, unlike the deterministic setting where simple geometric strategies suffice, in the stochastic setting such strategies can be extremely inefficient even on trivial distributions. Hence one needs to search a much larger space of candidate strategies which have no obvious structure. To get around this difficulty. we find a suitable "partial" strategy, i.e., one whose bids are within the support of the prediction, minimizes the expected cost, and can be extended to fully robust, infinite strategy. This can be formulated solution yields the optimal strategy. We also provide a computationally efficient approximation of the Pareto front, with provable performance and run-time guarantees.
family of linear programs (LPs), whose
Our second main contribution, discussed in Section 3 the study of randomized bidding strategies. Here, in order to allow a comparison with the known results, all of which have focused on deterministic -ugnentd strategies, we assume a single-valued oracle that predicts the target's 's value. We first present and analyze a randomized strategy whose consistency/robustnes tradeoff provably improves upon the best deterministic one. However, this improvement rapidly dissipates as function of the robustness requirement We complement this result with a lower bound that applies to all

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randomized algorithms, and which establishes that r' increases, the gap between the deterministic and the randomized consistency rapidly diminishes. Randomization poses new challenges, particularly in the context of negative results: it is not obvious how to show lower bounds on randomized tradeoffs, unlike traditional competitive analysis where Yao's principle (50] is readily applicable, and unlike the deterministic setting in which r-robust algorithms have a very clear characterization. Our approach relies on the use of scalarization, in that we prove lower bound on a weighted combination of the two objectives. This is turn allows us to obtain a negative result on their tradeoff, via the minimax theorem (or Yao's s principle) (50).
In Section 4 we extend our approaches to other settings and problems. First, we show how our LP- based approach can be applied a dynamic setting in which the oracle provides updated predictions in an on-line fashion, and the (robust) caaming-augmeent relative to each incoming prediction. To our knowledge, this is the first study of competitive sequencing problems under dynamic predictions. Moreover, we show that our results can be extended to the problem of searching for hidden target in the infinite line, which is a well-known problem from search theory and operations research (see, e.g., Chapter 8 in [)), and extend previous works that did not address robustness issues, We conclude, Section with an experimental evaluation of our Pareto-optimal algorithm which demonstrates the performance improvements in practice.
algorithm must maintain optimal consistency
1.2 Related work
Learning-augmentin algorithms. Algorithms with predictions have been studied in large variety of online problems, such as rent-and-buy problems [36), scheduling [41], caching [45], matching [Л]. packing (37], covering [16] and optimal stopping [32] This paradigm has also applications outside online computation, e.g., in improving the time complexity of sorting [15), graph problems [14), and data structures 43). The above are only some representative works and we refer to the online repository [ for a comprehensive listing. We note that the vast majority of these works do not assume stochastic oracles. The distributional prediction framework was introduced in [30]. and has been further applied in the context of search trees [I], online matching [24. and 1-max search [20].
Pareto-optimal algorithms. Several studies have focused the consistency-rouusinee
trade-offs
of deterministic Pareto-optimality has been limited problems such as ski rental 44 and search games [10].
algorithms e.g [48, 5 16 26 2). The study of randomized
For some problems, including online bidding, deterministic Pareto-optimal solutions suffer from brittleness, in that their performance shows marked degradation for very small prediction error [33 A randomized smoothening can help mitigate brittleness, as shown in (21]. but their results still compare to deterministic algorithms. In contrast, our randomized setting, we compare against randomized algorithms instead of deterministic ones.
Online bidding and competitive sequencing problems. For deterministic strategies, folklore result that goes back to studies of linear search [18] shows that the optimal competitive ratio is equal 4, and is achieved by simple strategy of the form X = Randomization can help improve the competitive ratio to e, whose tightness was proven in [2 using a complex application of dual-fitting the LP formulation. In regards to learning-augmentee
algorithms, and single- valued deterministic oracles, [8] gave Pareto-optimal algorithms,, whereas 4 388 gave performance guarantees function of the prediction error. Concerning distributional oracles, the only related previous work is [7]. which obtained the best consistency of 4-robust strategies. Note that in these studies, Pareto-optimality
attained by simple, geometric strategies of the form X (Ab');20, where b is chosen appropriately as function of the required robustness r. Related competitive sequencing problems such as searching on an infinite line were studied (5). No results are known about randomization in auggnnged bidding, even for single-valued oracles.
2 Pareto-optimal bidding with distributional predictions
In this section, we present and analyze deterministic Pareto-optimal algorithms in the distributional prediction model. We begin with the setting in which the prediction μ is a discrete distribution supported on k points, where point i is defined as (pe, Ms), with μ4 E R22,, such that Σ P: 1. We assume, without loss of generality, that 14 S Uúj for all < j. At the end of the section, we will show how to extend the result to general distributions; we will also discuss time-complcxity issues.
3

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The main challenge we face is that, unlike the deterministic setting where simple geometric strategies are optimal, in the stochastic setting such strategies can be extremely inefficient even on trivial distributions, we show in Proposition [8 Appendix D Hence one needs to search much larger space of candidate strategies which have no obvious structure. Our approach consists in restricting the search space to fight r-extendable partial strategies (Lemma which are specific finite strategies restricted to the domain of μ that can be extended to full strategies. We will rely on a linear programming (LP) to find the best partial strategy, by defining the concept of a configurations (see Definition D. By showing that the configuration space is sufficiently restricted, we can find the best strategy by solving the series of LPs that correspond to each possible configuration.
Given a distributional prediction μ and a strategy X, we define the consistency of X as
(1) [ (cost(X,=)) Ezzu(=]]
cons(X, (x,
following 30 and 77. The robustness of is the same as its worst-case competitive ratio, and is thus given by ① By observing that the competitive ratio is maximized for targets which are infinitesimally larger than any of the bids, a strategy X is r-robust if and only if it satisfies
I5-1 S T. 24
2z. for all i 2 0.
Hence our problem is formulated as follows: Given a robustness requirement E N+, and prediction μ,, we seek an r-robust strategy X that minimizes (ons(X, p)). We will assume that r is constant, independent of other parameters.
The following definition introduces the concept of a configuration, which is central in the design and analysis of our strategy, that it allows us to express the requirements of the problem using linear constraints.
Definition 1. Given a strategy X and a prediction μs, we define the configuration of X according to the vector (Ji,..., ja) E N such that
IJz < p4 S Fijtt, for all i E [1,
(4)
Furthermore, we say that the vector E Nk is a valid configuration, relative to robustness requirement and prediction μs, if there exists r-robust strategy X with configuration, j.
The number of valid configurations is bounded from above according to the following proposition.
Proposition 2. The number of valid configurations of an r-robust strategy is Olog" pa)-
Our algorithm will compute a partial, i.e., finite strategy, which must then be extended appropriately to fully r-robust, infinite strategy. The following definition formalizes this requirement.
Definition 3. Given an increasing sequence Y = (y:), that defines a partial strategy of + bids, and increasing sequence Z = (2x) o- call the strategy with sequence (yu.... V. 20, 21, 22 ...) the extension of Y based on Z. We say that Y r-extendable if there exists a sequence Z such that the extension of Y based Z is valid, r-robust bidding strategy. Last, say that Y is tightly r-extendable if it is r-extendable by a sequence 7. for which
4 =
Σ ==
U/
(5)
where 2-1 is defined to be equal to yt-
We also formalize what it means for partial strategy to be r-robust:
Definition 4. partial strategy is (partially) r-robust if r-robust assuming the target is smaller than its last bid.
The following lemma allows to tightness criterion for extendability.
Lemma 5. A partial strategy Y = (y.) is r-extendable if and only if it is tightly r-extendable.
The next lemma provides necessary and sufficient condition under which partial strategy is r-extendable. Its proof relies on finding the conditions for which linear recurrence solution that formulates the tight r-extension yields an increasing, unbounded infinite sequence.

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Algorithm Algorithm for computing the Pareto-optimal strategy.
Input:
Robustness requirement r, distributional prediction = (().. Output: An r-robust strategy of optimal consistency.
For each configuration find the solution zj to the LP 2j. if exists.
2: Let x* = (zg.... + +1) be the solution of optimal objective value, for configuration.j j*, among
all solutions found in Step 1. 3: Return the strategy
(z) .... 2zz 80, 21....)
where Z = =(i))20 is the tight r-extension of (zg...., ,......,..) +5)z)
Lemma 6. A partial strategy Y (y:)i is r-extendable if and only Y is partially r-robust and it satisfies the condition
Σ Ei V/
T
vr(r 4)
2
Algorithm T computes the Pareto-optimal strategy. It consists of three steps. In steps and 2, computes a partial strategy of minimum expected cost, which is partially r-robust and is guaranteed to be r-extendable. In step 3, it applies the tight r-extension, which yields an r-robust bidding strategy that is Pareto-optimal. Given a configuration j (j1, fkk, we define by Pj the following family of LPs.
In this family of LPs, the objective func- tion (6) is by definition the expected cost of the partial strategy (zo...., f1++1) assum- ing a target drawn according to μ., which optimizes the consistency. Constraint guar- antees that this partial strategy is r-robust, whereas (8) guarantees that the strategy is r-extendable, as stipulated in Lemma 6 Constraint (9) describes the configuration j. whereas IT is an initialization condition that is necessary for monotonicity. The fol- lowing theorem formalizes the intuition be- hind the LP formulation, and establishes the correctness of our algorithm.
Pj min
Pi
2q
(6)
1=1
i-1
subj. to
5; S rT;
Fj E [1, + +1]
1=0
(7)
in -1
+
vr(r 4)
Fj
2
88)
Tj S μ4 i [L,
(9)
x, < 2141, E [0,
(10)
r.
(11)
Theorem 7. Algorithm computes Pareto-optimal r-robust strategy.
In regards the time complexity, from Proposition it follows that Step of Algorithm requires solving Olog" p&) LPs, each of which takes time polynomial in the size of the input. For constant the algorithm is polynomial-time as l becomes large. To mitigate the effect on the complexity, but also order to handle any general distribution μ. we can apply quantization rule to with exponentially spaced levels. This allows to reduce the runtime complexity, at the expense of a small degradation on the approximation of the consistency. The following theorem shows that we can approximate the Pareto frontier to an arbitrary degree of precision in time polynomial in the range the distributional.
though a realisneuripstic implementation can be time-consuming
Theorem 8. Let y be distributional prediction with support [m, M). There exists quanti- zation of μ. for which Algorithm | runs in time polynomial in max{M//m logm), m) and yields independent of
et -approxximation of the optimal consistency, for any constant
This result is useful especially if the prediction has locality, i.e., M is not vastly larger than m. This occurs, for instance, in the related application of contract scheduling, as discussed in (9] In particular, if M /m is bounded by constant, the algorithm of Theorem 8 is polynomial-time hence PTAS.
5

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3 Randomized learning-uugmentdd bidding
In this section, we consider different stochastic setting. Namely, we do not assume any known stochastic properties of the prediction oracle, but we allow the bidding algorithm access to random bits. That it, the elements of the bidding sequence X are random variables, and so is the cost at which X finds the target. Given a prediction û on the target, the consistency and the robustness of the randomized strategy X are defined
cons(X) E|cost ü)] and rob(X) sup
Elcost(X, u
(12)
ú
In the standard setting of no predictions, there is a simple randomized strategy of the form R (et+*)s, with U10 1), of competitive ratio equal e. This is optimal as shown in [27] using complex proof based on linear programming and dual fitting techniques. The following proposition
shows that the algorithm is essentially tight across all targets, result which will be useful in interpreting the performance of our learning augmented algorithm.
Proposition 9. The randomized algorithm R. (eft*), with s E U(0,1) satisfies Elcost([X, /)]/u] = O(1/u), for all u.
In the remainder of the section, we provide both upper and lower bounds the performance of randomized aumning-auggentee strategies.
3.1 Upper bound
We propose a parameterized algorithm with parameters 5 E [0, 1) and a > which, informally, regulate the amount of the desired randomness and the geometric step of the strategy, respectively. 6
Let À E [1, a), and j E N be such that â Aa namely j = [log, â 81, and X ariz. We detine by R5 the randomized strategy with bids r; Aa where E U|ō, 1). The following theorem quantifies the consistency and the robustness of this strategy.
ala-aá)
Theorem 10. Strategy R5 has consistency at most daa-l)(1-6)hee and robustness at most
a(a-a" 11(1(18))mmd
R5,a interpolates between two extreme strategies. Specifically, for 5 0, it has consistency and robustness that are both equal to a/ In a, which is minimized for a = e.. In this case, the strategy is equivalent to the competitively optimal one, and from Proposition 9 it does not exhibit any consistency/robustnes tradeoff (it optimizes, however, the robustness). In the other extreme, for 5 - 1, simple limit evaluations show that rob( (R1.)) ) a² a²/(a- (a-1)) 1), whereas cons (R1,a) a/ (a). From the study of deterministic bidding strategies [8], this is the best possible consistency/robuumeess tradcoff that can be attained by deterministic strategy (and note that a² /(a(aa (a-((aa) 1) is again expected, since as 8 + 1, the strategy does not use any randomness.
4, for all a). This
Define now the strategy R = Ræ-.a- in which the parameters 8", a" are chosen in optimal manner. Namely, they are the solutions the optimization problem
a(a - a²)
max a subject to 1a - () lna Sr,
(13)
then from Theorem 10 it follows that R achieves the best-possible consistency among all r-robust strategies in the class Rs.. Hence, R* has at least good consistency as the deterministic Pareto-optimal strategy. For instance, for r = 4, R² has consistency approximately equal to 1.724, which is attained at 8 0.9. However, quickly approaches 1. For example, for r = 4.5, 5* = 0.95, and for r = 5, 5* ~ 0.99. This implies that any benefit that H attains quickly dissipates as r increases, and randomization is not helpful. In the next section we show that this is not just deficiency of Rt" but that any randomized strategy has consistency that approaches the best deterministic one, as a function of the robustness r.
increases, numerical solutions of (13) show that 8*
3.2 Lower bound
In this section, we prove a lower bound on the consistency of any randomized r-robust strategy, with 2 e. We first begin with alternative proof that is the best randomized competitive ratio in

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the no-prediction setting. A generalization of this proof will be likewise useful in the prediction setting, namely in the proof of Lemma 12] We adapt an approach due to Gal [35, who studied the randomized linear search problem. For any fixed, bur arbitrarily small ε > 0, define R = exp (-)
Let Dr be the distribution on the target with density e/t, for all 1 t < R. and has probability atom of mass ε distribution D.. Since the domain of De is bounded, X is of the form (z:) for some finite n. We show that for targets drawn from D., the expected performance ratio of X is at least (1 E)e. From
R. Let X (xx) denote a deterministic bidding strategy, against the target
the minimax theorem, we then obtain that the optimal competitive ratio is arbitrarily close to
Theorem 11. For any & > and Dg defined as above, it holds that D.
cosn X.ai)
2 (1 e)e.
We now move to the -aggiingeegmeeteed setting. Consider again the distribution of targets De, and suppose that the prediction of the oracle is û R. Let X (æ:)² denote deterministic strategy, for some n that depends on E, then the consistency of X
cons(X) cost(X, X, à
1 R E=0
Z4
(14)
which does not depend on distributional assumptions, whereas for the robustness, we have the following useful series of inequalities following the proof of Theorem (see Appendix B
72
T2
(1 E) R(((+1
rob(X)
2 E( 1)
II
(15)
In R!/(+41)'
81-1
T
The following is the main technical result that places a lower bound on the scalarized objective.
R#
Lemma 12. Given e O, T denote the quantity
For any X O, it holds that
In R
1
rob(X) + A ccons(X ( (1-
A(1 T. TF(TT) - 8(e) -8(e)
where F(T) = (I( + In and lim d(E) =0
3.0
randcmized lower bound
randcmized upper bound deterministic tight bound
2.5
consistency
2.0
1.-
10
3
5
robustness
Figure 1: Comparison between the deterministic tight upper bound the consistency, and the randomized upper and lower bound of Theorems 10 and
We can now prove our lower bound with the help of Lemma 12 specifically, by considering both sufficiently small and sufficiently large values of the parameter
Theorem 13. Any randomized learning-augmente augmented algorithm of robustness at most r has consistency at least + (145)F(()) for any arbitrarily small, but constant & > O.
Figure compares the consistency of the best deterministic strategy and the lower bound of Theo- rem 13 function of the robustness r.
7

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Further applications
4.1 Dynamic predictions
The methodology of (eacioon]allovy us to obtain Pareto-optimal algorithms for online bidding and related problems, in a dynamic setting in which the oracle can provide multiple predictions in on-line manner. We will use, as illustration, the abstraction of contract scheduling [47], a problem that is closely related to online bidding, and which offers a more intuitive interpretation of the dynamic setting. In the standard version of the problem (without predictions), we seek a schedule X that is defined as an increasing sequence
(xx))=o- Here, I; is the length of the i-th contract that is executed in this schedule, and the objective is to minimize the expression
r(X) = sup T ((X,T)' where T) is the length of largest contract completed in X by time T..
Contract scheduling has several applications in real-time systems and interruptible computation [5]] 24] The learning-ause augmented version in which (single) prediction on the interruption I' is given ahead of time is identical to bidding, that both problems have the same Pareto front 99.
Consider now dynamic version of contract scheduling. where the oracle provides progressively updated, i.e., improved predictions. Let denote the desired robustness requirement, then our setting can be formulated inductively as follows. Initially, the oracle provides its first prediction on the interruption time, say úi, and seek an r-robust deterministic schedule that minimizes the consistency relative to this prediction. Let X2 be this schedule. Inductively, let X,, denote the updated schedule that the algorithm decides to follow after the oracle outputs the m-th prediction ūmm- Suppose that a new prediction um with um+11 üm becomes available at the complction of contract in Xm. say time Tm. The objective is then define an updated schedule Xm+1 which starts its execution at time T,, remains r-robust, and minimizes the consistency relative to um+11
We can compute each updated schedule X,, by applying an LP-based approach along the lines of Algorithm I Moreover, each schedule Xm can be computed efficiently in time Oloo üm)) We refer Appendix C for details and the proof of the following theorem:
Theorem 14. In the dynamic setting, there exists algorithm for computing incremental Pareto- optimal schedule of contracts, that runs in time polynomial the size of each prediction.
4.2 Linear search
The results of Sections 2 can be extended to another well-known problem, namely searching for hidden target the infinite line. This is fundamental problem from the theory of search games that goes back to works of Beck [17] and Bellman [9). and has been studied extensively in TCS and OR; see [40] for a recent survey. this problem, a mobile searcher initially placed at some point O of the infinite line, must locate an immobile hider that hides at some unknown position on the line. The objective is to devise a search strategy that minimizes the competitive ratio, defined by sup,, COst(S, / h). where cost(S, h) is the traveled distance of searcher that follows S the first time it finds h. and [h] is the distance of h from O. The best deterministic competitive ratio is equal to 9 [18), whereas the best randomized competitive ratio is 1 ++ min In deterministic strategies for a single positional prediction on the hider were given in 5
1+a xs 4.6 [35). Parcto-optimal,
We obtain Pareto-optimal strategies for distributional prediction on the position of the hider, thus generalizing the result of Alpern and Gal (Section 8.7 expected search time, but without any robustness guarantees. Furthermore, we can extend the approach of Section to randomized search strategies with a positional prediction. In comparison to online bidding, which "I "1-dimensional" the inherent difficulty in linear search is that the searcher operates in a "2-dimensional" space defined by the two halflines. We refer to Appendix C for details.
(3) who gave strategy of minimum
Experimental evaluation
We evaluate experimentally our main, Pareto-optimal algorithm, namely Algorithm Of Section 2
Datasets The distributional predictions y are composed pairs (1). ... (Ju,, p&). We fix k 4, and the support of μ is in the interval [1, 10 The p; values are either uniform, i.c., all equal to
8

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30
Algorithm zeta zela
Algorithm
zeta
2.5
2.5
Compistency
Consistency
2.0
1.5
1.0
Robustness
7
8
9
10
11
12
Robustness
(a) Gaussian 14, σ 2000
(b) Gaussian J4, 4000
Figure 2: Empirical consistency function of the robustness requirement
3.0
3.0
Algorithm zeta
Algoritmm
zeta
2.5
2.5
Consisency
Consistecyy
2.0
2.0
1.5
1.5
1.0
1.0
10
11
12
10
11
12
Robustness
Robustness
//2
p + pm
(b) μ2
Co p
Figure 3: Empirical consistency when μ is comprised of two equiprobable points, with μ) 5000
and #2 generated as specified the caption. (a) is adversarial for p = r/2 and (b) for p = C2-
1/k, drawn uniformly in [0, 1] then scaled so that they sum to The values of μr are either drawn uniformly in [1; 10 or from Gaussian distribution centered at the middle of this range, and truncated to the range. This Gaussian distribution is either "narrow", with a standard deviation 4000. In total, we consider two settings for the values of and
= 2000, or "spread", with three setting for the values of pts leading to six different datasets. Furthermore, each dataset is composed of 10 iid samples. We report results on datasets in which the pt are drawn uniformly and ii; follows Gaussian distribution, and refer the Appendix for further experiments and discussion.
Algorithms We compare Algorithm to three heuristics of the form (A (x-p")22o. Such heuristics are r- competitive if and only if [2) where Çi -r Vr(r - 4)) and (2 = [r++ Vr(r- 4)) B. The three chosen heuristies correspond to p {S1, rr, (2), that is, to the extrema and the middle of the feasibility range for p. and À chosen, for each heuristic, so as to minimize the expected cost over μ. The choice of the base p controls how fast the heuristic increases its geometric bids. In particular, the heuristic with {2 optimizes the consistency assuming a singled-valued prediction [8]
Results For each of the 6 datasets, we consider integer values of within [4, 12] and report, for each algorithm, the average consistency as well as the standard deviation over the 10 samples (crror bars). The computation was
AMD EPYC 7302 processor using 8 threads. The average runtime for each of the 540 instances was 1 1.8s. Figure Souureddes depicts the obtained results.
The experiments show that the heuristic with p Ça is very inefficient, and its consistency increases near-linearly with (Appendix 55. This is because its bids increase very slowly as function of r. We thus focus on the remaining algorithms. For all r. Algorithm [ has better consistency than all heuristics, thus confirming Theorem 7 also has better variance, as observed the error bars, since it outputs optimal solutions to each instance. In contrast, increasing the heuristic base increases their variance, since the bids grow more rapidly, thus rendering the heuristic less flexible. This also explains why the performance gap increases with the standard deviation of μ in the dataset.
Analysis

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Furthermore, as shown in Fi for each heuristic there exist very simple predictions comprised of only two equiprobable points for which the heuristic has performance that much inferior to that of our algorithm. This can be explained by the fact that, in the known heuristies, the bids follow fixed geometric rule and cannot adapt to the specificity of the prediction, even when the latter is extremely simple. In particular, for large the computations showed that the consistency of A Algorithm monotonically approaches I, whereas the consistency of the heuristics tends to 2. This gap is large, because it implies that the geometric heuristics pay cost equal to that of for finding the largest point, thus getting no benefit whatsoever from the distributional prediction. In Appendix [ we demonstrate that this gap is unbounded for large r, even for simple predictions over two points.
6 Conclusion
We gave the first systematic study of online bidding in stochastic settings where either the prediction is distributional, or the algorithm is randomized. Our results quantify the power and limitations of randomness, an issue that remains underexplored in the literature, and is bound to become more prevalent in the broader study of eearning-augmentt
algorithms. Our techniques may be applicable to several further extensions, e.g, in searching of a hidden target in a star environment [39] or in layered graphs B4 They can also apply to settings outside learning-auge augmented computation, as we demonstrated in Section 4.1 Here, an interesting direction for future work is to apply LP-based approaches towards the study of barely random algorithms for online bidding or linear search. For instance, what is the best competitive ratio that can be achieved with only one random bit? This is
motivated by recent studies of limited randomness other well-known problems, e.g., k-server 22
7 Acknowledgments
We are thankful to Thomas Lidbetter for helpful discussions on the randomized setting.
This work was supported by the grant Research Agency (ANR).
PREDICTIONS from the French National
References
[1] Repository of works on algorithms with predictions.
https://
algori teoorittmswwiii www.--tteeeepiiinn....ee www.iiiciiinniittu...bom github.io 2025. Accessed: 2025-01-01.
[2] Matteo Almanza, Flavio Chierichetti, Silvio Lattanzi, Alessandro Panconesi, and Giuseppe Re. Online facility location with multiple advice. Advances in Neural Information Processing Systems, 14:4661-46673 2021.
[3] Steve Alper and Shmuel Gal. The theory of search games and rendezvous. Kluwer Academic Publishers, 2003.
[4] Keerti Anand, Rong Ge, Amit Kumar, and Debmalya Panigrahi. A regression approach to learning-augmented online algorithms. Advances in Neural Information Processing Systems, 34:30504-30517, 2021.
[5] Spyros Angelopoulos. Online search with a hint. Inf. Comput., 295(Part B):105091, 2023.
[6] Spyros Angelopoulos, Diogo Arsénio, and Shahin Kamali. Competitive sequencing with noisy advice. CoRR. abs/2111.05288. 2021.
[7] Spyros Angelopoulos, Marein Bienkowski, Christoph Dürr, and Bertrand Simon. Contract scheduling with distributional and multiple advice. In Proceedings of the Thirty-Thrr Interna- tional Joint Conference on Artificial Intelligence, IJCAI, pages 3652-3660 2024.
[8] Spyros Angelopoulos, Christoph Dürr, Shendan Jin, Shahin Kamali, and Mare Renault. Online computation with untrusted advice. Journal of Computer and System Sciences, 144:103545, 2024.
[9] Spyros Angelopoulos and Shahin Kamali. Contract scheduling with predictions. J. Artif. Intell. Res., 77:395-426 2023.
10

---

[10] Spyros Angelopoulos, Thomas Lidbetter, and Konstantinos Panagiotou. Search games with predictions, arXiv preprint arXiv:2401.01149 2024.
[11] Antonios Antoniadis, Themis Gouleakis, Pieter Kleer, and Pavel Kolev, Secretary and online matching problems with machine learned advice. Advances in Neural Information Processing Systems, 1944933-7944, 2020.
[12] Guillaume Aupy, Ana Gainaru, Valentin Honoré, Padma Raghavan, Yves Robert, and Hongyang Sun. Reservation strategies for stochastic jobs. In 2019 IEEE International Parallel and Distributed Processing Symposium (IPDPS), pages 166-175. IEEE, 2019.
[13] Yossi Azar. On-line load balancing. Online algorithms: the state of the art, pages 178-195, 2005.
[14] Yossi Azar, Debmalya Panigrahi, and Noam Touitou. Online graph algorithms with predictions. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 35-66. SIAM, 2022.
[15] Xingjian Bai and Christian Coester. Sorting with predictions. Advances in Neural Information Processing Systems, 36:26563- 2023.
[16] Etienne Bamas, Andreas Maggiori, and Ola Svensson. The primal-dual method for learning augmented algorithms. Advances in Neural Information Processing Systems, 33:20083-20094, 2020.
[17] Anatole Beck. On the linear search problem. Naval Research Logistics, 2:221-223 1964.
[18] Anatole Beck and Donald Newman. Yet more on the linear search problem. Israel journal of mathematics, 8(4):419-429, 1970.
[19] Richard Bellman. An optimal search problem. SIAM Review, 5:274, 1963.
[20] Ziyad Benomar, Lorenzo Croissant, Vianney Perchet, and Spyros Angelopoulos. Pareto- optimality, smoothness, and stochasticity in rearning-augmentee one-max-search arXiv preprint arXiv:2502.05770, 2025.
[21] Ziyad Benomar and Vianney Perchet. On tradeoffs in learning- algorithms, January 2025. arXiv:2501.12770 [cs].
[22] Daniel S. Bernstein, Lev Finkelstein, and Shlomo Zilberstein. Contract algorithms and robots on rays: Unifying two scheduling problems. In Proceedings of the 18th International Joint Conference Artificial Intelligence (IJCAI), pages 1211-1217, 2003.
[23] Allan Borodin and Ran El-Yan El-Yaniv. Online computation and competitive analysis. Cambridge University Press, 1998.
[24] Clément L Canonne, Kenny Chen, and Julián Mestre. With little help from my friends: Exploiting probability distribution advice in algorithm design. arXiv preprint 044499, 2025.
[25] Moses Charikar, Chandra Chekuri, Tomás Feder, and Rajeev Motwani. Incremental clustering and dynamic information retrieval. In Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pages 626-635, 1997.
[26] Nicolas Christianson, Junxuan Shen, and Adam Wierman. Optimal robustness-consistenes tradeoff's for learning- metrical task systems. In AISTATS, volume 206 of Proceedings of Machine Learning Research, pages 9377-9399. PMLR, 2023.
[27] Marck Chrobak, Claire Kenyon, John Noga, and Neal E Young. Incremental medians via online bidding. Algorúthmica, 50:455-478, 2008.
[28] Marek Chrobak and Claire Kenyon-Mathieu. Competitiveness via doubling. ACM SIGACT News, 37(4):115-126, 2006.

---

[29] Romain Cosson and Laurent Massoulié. Barely random algorithms and collective metrical task systems. In Conference on Neural Information Processing Systems, 2024.
[30] Ilias Diakonikolas, Vasilis Kontonis, Christos Tzamos, Ali Vakilian, and Nikos Zarifis. Learning online algorithms with distributional advice. In International Conference on Machine Learning. pages 2687-2696. PMLR, 2021.
[31] Michael Dinitz, Sungjin Im. Thomas Lavastida, Benjamin Moseley, Aidin Niaparast, and Sergei Vassilvitskii. Binary search with distributional predictions. In Advances in Neural Information Processing Systems 38: Annual Conference on Neural Information Processing Systems 2024, NeurlPS, 2024.
[32] Paul Dütting, Silvio Lattanzi, Renato Paes Leme, and Sergei Vassilvitskii. Secretaries with advice. Mathematics of Operations Research, 49(2):856-879) 2024.
[33] Alex Elenter, Spyros Angelopoulos, Christoph Dürr, and Yanni Lefki. Overcoming brittleness in pareto-optimal learning augmented algorithms. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024.
[34] Amos Fiat, Dean P Foster, Howard Karloff, Yuval Rabani, Yiftach Ravid, and Sundar Vish- wanathan. Competitive algorithms for layered graph traversal. SIAM Journal on Computing, 28(2):447-455 462, 1998.
[35] Shmuel Gal. Search Games. Academic Press, 1980.
[36] Sreenivas Gollapudi and Debmalya Panigrahi. Online algorithms for rent-or-buy with expert advice. In Proceedings of the 36th International Conference on Muchine Learning, ICML, volume 97 of Proceedings of Machine Learning Research, pages 2319-2327, PMLR, 2019.
[37] Sungjin Im, Ravi Kumar, Mahshid Montazer Qaem, and Manish Purohit. Online knapsack with frequency predictions. Advances in Neural Information Processing Systems, 34:2733-2743, 2021.
[38] Sungjin Im, Benjamin Moseley, Chenyang Xu, and Ruilong Zhang. Online state exploration: Competitive worst case and learning- augmented algorithms. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pages 333-348, Springer, 2023.
[39] Patrick Jaillet and Matthew Stafford. Online searching. Operations Research, 49:234-244, 1993.
[40] Evangelos Kranakis. A survey of the impact of knowledge on the competitive ratio in linear search. In Stabilization, Safety, and Security of Distributed Systems: 26th International Symposium, volume 14931, page 23. Springer Nature, 2025.
[41] Silvio Lattanzi, Thomas Lavastida, Benjamin Moseley, and Sergei Vassilvitskii. Online schedul- ing via learned weights. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithmss, pages 1859-1877. SIAM, 2020.
[42] Russell Lee, Bo Sun, Mohammad Hajiesmaili, and John C. S. Lui. Online search with pre- dictions: Pareto-optimal algorithm and its applications in energy markets. In The 15th ACM International Conference on Future and Sustainable Energy Systems, e-Energy 2024, Singapore, June 4-7, 2024, pages 50-71. ACM, 2024.
[43] Honghao Lin, Tian Luo, and David Woodruff. Learning augmented binary search trees. In International Conference on Machine Learning, pages 13431-13440. PMLR, 2022.
[44] Alejandro López-Ortiz, Spyros Angelopoulos, and Angele Hamel. Optimal scheduling of contract algorithms for anytime problem-solving. J. Artif. Intell. Res., 51:533-55:: 2014.
[45] Thodoris Lykouris and Sergei Vassilvitskii. Competitive caching with machine learned advice.
J. ACM, 68(4):24:122:255 2021.
[46] Manish Purohit, Zoya Svitkina, and Ravi Kumar. Improving online algorithms via ML predic- tions. In Proceedings of the 31 Annual Conference on Neural Information Processing Systems (NIPS), pages 9661-9670, 2018
12

---

[47] Stuart J. Russell and Shlomo Zilberstein. Composing real-time systems. In Proceedings of the 12th International Joint Conference on Artificial Intelligence (IJCAI), pages 212 2222 1991.
[48] Bo Sun, Russell Lee, Mohammad Hajiesmaili, Adam Wierman, and Danny Tsang. Pareto- Optimal
Algorithms for Online Conversion Problems. In Advances in Neural Information Processing Systems, volume 34, pages 10339-10350. Curran Associates, Inc., 2021.
[49] Alexander Wei and Fred Zhang. Optimal robustness consessteec trade-offs for learning- augmented online algorithms. In Proceedings of the 33rd Conference on Neural Information Processing Systems (NeurIPS), 2020.
[50] Andrew Chi-Chin Yao. Probabilistic computations: Toward a unified measure of complexity, In 18th Annual Symposium on Foundations of Computer Science (sfes 1977), pages 222-227. IEEE Computer Society, 1977.
[51] Shlomo Zilberstein and Stuart Russell. Optimal composition of real-time systems. Artif. Intell., 82(1-2):181-2111 1996.
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Appendix
A Proofs from Section
Proof of Proposition 2 Let X = (m denote an r-robust schedule, which thus satisfies 3 From previous studies of this linear recurrence inequality [6]. know that for any 1, there exists an index l such that 27 2 u, where / = (log, u), and
V
G
Note that (1 is constant, since r is assumed be constant. From Definition | it follows that the i-th component of a valid configuration has value O(log (s).. which concludes the proof.
We note that this bound is tight, in that from 66 it holds that E 22(logc, (u)), where C2 = 1 1
Proof of Lemma 5 Suppose that is r-extendable, then there exists a strategy Z that r-extends Y. Let j E N be the smallest integer such that 2; < F2;-1 Ejec Zj, and let 5 = (r2:-1 Ejer 2// > 1. Define the strategy
Z' = (yo.... Vi, 20 2, 8zi, 8zi+1, 82442....)
then we remark that Z' is also r-robust. Indeed, for > i, we have, as Z r-extends Y
8zg S 8(rzg-1
Σ
1/j
2j)
151
j<q
11(zzg1))
Σ jSl
2j
823.
1=
Hence, Z' is r-robust and satisfies (5) for i 1. By repeating this process iteratively, we obtain sequence that converges, at the limit, to the the tight r-extension of Y.
For the opposite direction, it suffices to note that tightly r-extendable strategy is r-extendable by definition.
Proof of Lemma Consider partially r-robust partial strategy Y (y:))=o Note that if Y is not partially r-robust, it cannot be r-extendable. By scaling all values Vi. we can assume, without loss of generality, that Σl=o // r, this simplifies the formulas. From Lemma 5 Y is r-extendable if and only it has a tight r-extension, say Z. We will describe explicitly the constraints that must be imposed on Z, in order for the sequence Y U Z to be valid tight r-extension of Y.
The first bid of Z, namely 2p. must satisfy the condition 31 E' Vt + 2g
2g. hence
have 2p = r (() - 1). The second bid, 21, must satisfy r 2g = Ei= Vi + 2g + 21 = r + 21 hence 21 = r(=0 w) For each subsequent bid 2;. with é >> 1, it must hold that 2, r(2,-1 2;-2).
which can be extended to all E N with the initial conditions = y/1 and 2-2 = 1. We have thus defined linear recurrence relation, whose solution is given by the expression
2a-1 Zn-2 p' (r' ((r r')" + (r r'r) (r 2yn ((r + r'y"- (r-r)))),
(16)
where r' 4 <r.
The series z describes a valid strategy if and only if is increasing and unbounded. We consider two cases.
21

---

Suppose first that r' <r- 2z,. For large enough, the terms of the expression of Zn-2 in (r + r') dominate the ones in (r ---rr) r')", so 2n-2 becomes negative and is not a valid strategy.
Next, suppose that r' 2 r- 2tjt. and we consider two further cases. Suppose first that r < 2y-- In this case, both terms of (T6) are positive,
is increasing and valid strategy. For the next subcase, suppose that r > 2yr, then it follows that the second term of (16) is negative. Since
r' 2 r 2y,, we also have (r+r')" (r- (r-r²)" (r + +r)" (r - r') 2n-2 2- -r 2yx) ((r + r')" + (r - rr)"),, which is increasing in and unbounded, so is valid strategy. Consequently, is valid strategy if and only if
2y,, which is equivalent to:
2y1 2
T'
vr(r 4)
i== Vi
2r
2r(r + vr(r 4))
r+ Vr(r - 4)
y/ - vr(r 4)
x2 - p2 4r
2
Therefore, we have shown that partially r-robust strategy Y = (wi)== admits an r-extension Z if
and only Limn -tvr(r-4) which completes the proof.
In this proof we assumed the general case r > 4. For the recurrence relation slightly different, since the characteristic polynomial has a double root; we do not show details, since this case is studied in [7].
Proof of Theorrm 7 Let Y = (V:)i20 denote a Pareto-optimal strategy, i.e., a strategy that is r-robust
and has minimum consistency. Let = (4.. be the configuration of Y according μ.. Since Y r-robust, the sequence Y' (y/s )mm partially r-robust, as well as r-extendable. From Lemma 6
this implies that Y' satisfies all the LP constraints 7 (8) and Hence, in Step 2, Algorithm
computes partial strategy x for some configuration j for which C' : D i-1 pi 2g T.g is at
most the consistency of Y. Furthermore, since x* satisfies (8), from Lemmas and 6 is aggressively r-extendable. Therefore, we conclude that the strategy returned by the algorithm is r-robust, and has consistency most that of Y, hence it is Pareto-optimal.
Proof of Theorem 8 Let Fj, denote the cumulative distribution function of the prediction p. Since μ has support in the interval [m, M] C R>1, we have (m)) 0 and F, = 1. We can assume, for simplicity, that M > e m hence M /m (l1), as we focus the asymptotic complexity of the algorithm.
We define the distribution ji quantization of into k [clog(M/m)) points (or "levels") ..... Au of value J4: et/c. Note that jıg m clog(M/m) 2 M. The probability pt associated with P4 is defined to be equal to F(Z,) 1 (assuming (oo = m). Intuitively, f is obtained from j by shifting probability mass to the right, by a maximum multiplicative factor of e//c, and results in O(log(M/m)) points.
Consider running Algorithm [ on j.. The space of all configurations is now more restricted, in comparison to the setting of Section [ Specifically, as discussed in the proof of Proposition] 2 we
know that for any r-robust strategy X (z:) the first index i such that X; 2 fo = m e e/c must satisfy O(log m). Hence, we can restrict the execution of Algorithm I on configurations
j j= (Ji...., jic) for which ji O(lo m)) Similarly, we can also restrict for which jı: - f1
Olog(M//m). This results total of configurations that is at most
o 1og m
(10g
& I ------))
2=2
We now bound [ (Jr - f--1) using the fact that there exists a constant C 2 e for which Σt=
)t-1) ja- j1 C [oog (M/ m,], and the AM-GM inequality:
22

---

k-1
1 ( - j-1) z
jk - 11 k -
(AM-GM inequality)
[olog M1 1-1
Cllog M
m
(definition of l and upper bound on jk: - j1)
Iclog M.
m
2o 2o M
20 log m
(follows from log Af > and C > <>> 1)
2 log m
M.
40
2c log M
M
#2
M
O(1)
The complexity to solve each I.P in Algorithm T is polynomial in the number of variables and
constraints, polynomial in l- Ollog(M/m) and jk O(log m log(M/m)),, hence the total complexity is polynomial in O M m log m).
Regarding the consistency the strategy X returned by this procedure, we have:
Eg cost X, 2)]
cons(X,
(definition of consistency)
E Ex~ua(>
E u cost X, 2]]
(follows from the definition of and monotonicity of cost)
E Ex [=]
cons(X (X, pi) Espgle]
(definition of consistency)
Exzule [z]
S cons(X, (1) 1/c
(follows from the definition jt)
The first inequality comes from the fact that probability masses have only be moved to the right from ja Ji and the function = -> cost( X, =) is non-decreasing. The second is the definition of cons(X, n.). And the last one comes from fact that no probability mass has been shifted by a multiplicative factor larger than t/e from μ to p.
Therefore, as Algorithm Dofers the best consistency / robustness tradeoff, this procedure results in consistency that within a factor at most e/c from the best consistency achievable in the worst-case, assuming robustness r.
B Proofs from Section
Proof of Proposition where j E N. Then will either locate u iteration j with probability 5, or in iteration j + with probability d. Hence
9 Let E [0, be such that for the unknown target u, it holds that es+5
+++++
j+2+
E/cost( X, u]]
Pr(s 2) + E
1s S Pr(s {8)
634
e
(Bie*ls 2 6](1 + elAee" < 800) o(-)
e (e 1)
(c- の
x (1 8) + c(e² -1) 0 o
ed 1))
8
= =e- 0
which completes the proof.
23

---

Proof of Theorem 10 To evaluate the consistency, note that Rs locates u at iteration j. regardless of the random outcome of s. Therefore.
E (s, = XE +++ - S Aa a - a
(1- 1l-
which establishes the consistency of Ro since ú = Aa
To bound the robustness, let u denote an arbitrary target, and let j E [0, 1) such that a = Aa where & E N. We consider two cases:
Case 1: μ ك d: In this case, R locates at iteration k. Hence, similarly the calculations above, we obtain that
E Eloost( Rs.a. u)]
a x+1 (a - a²) #gd au (a 1)(1 8) Ina (a 1)(1 - 8) In
a(a - ad)
rob ( = sup (0.8.8
I sup
Aab
Case 2: μ. > 8. In this case, Rš,a locates the target either at iteration k. if 2 μ. or at iteration & + 1, otherwise. Hence
1
EETA a k+++1
k+s+2
rob(Rs,a)
a pb Pr(s 2 1) + E[A a
ur p))
Aak=u
1
(a at + aaaa" - -d²))
(a 1)(1 In
a(a a²)
1)(1 8) Ina"
Let G denote the cdf of De. We also define x-1 = 1, for initialization purposes. Since any target drawn from D, is bounded by R, can assume that X is of the form X ( for some n such that In = R. We have that
Proof of Theorem 77
cost(X, u)
E.
T dG(y))
(by definition)
11
i=
==
32
dG(y)
(rearranging terms)
1=0
11-1
V
SR dGG(y) = E (J 3 dy + 1 )
26-1
12
W
2 2
(AM-GM inequality)
T/-1
1=0
R441
(1 5)
(from the definition of and telscoping product)
In Ru
2 (1 E)e.
(since x/ In is minimized at
Proof of Lemma 12
Let us denote by Pr the ratio 0i/wi-1 and by m the ratio 1/n.
From
15
it follows that
n-1
A
E(rob(X) + Acons(X)] =
=
Σ
pi +
:
Σ
pi + ppp
+
A +
(17)
Pn
Pn.
1=0
24

---

For further convenience of notation, let us denote a == Pn. then from the AM-GM inequality and reasoning along the lines of (15). Eq.
gives
T n-1 R
E [(oob(X) + Acons(X)) 2 61
+ ca
A+ Ea
+
X+ A
T
(18)
a
a
by recalling that I denotes the quantity
In 12
Define f(a) = (1 - =ETT + sa A. arbitrarily close to f(a"), where
We will prove that as - 0, f has a minimum that is
m(1 m)
(19)
To see why this suffices to complete the proof, note first that
RTh S e Wo(T)
from the definition of the Lambert function, and more precisely its right branch. From the definition of R, the above inequality implies that
m S -
Wo(T),
(20)
and note that
lim m 0. 6
(21)
From 19
and 20 it will then follow that
a' T (T)
Moreover, as & - 0, ((a) is arbitrarily close to the function
g(a) = (1 8)T A
hence s(a)) is arbitrarily close to g(a"), ), and from (18) this completes the proof. Note that d(e) can be defined as function that comprised by infinitesimally small terms, due the approximations that we will introduce in what follows.
Next, we find the extreme points of g(a). The derivative of g with respect to a is equal to
(a(a) (1 ()- + a
To find the local extreme points, we solve ga(a) = 0. Since g' is continuous, 0, local extrema are arbitrarily close to the solution of the equation
(1 =)mT +
0.
a
The solution a" of this quadratic equation, considering that is
(1 - E + ((11 - e)mm))² + 4sA
25
and note that O, a" approaches (1 e)mT/s, hence establishing 119 Last, this value
corresponds to the unique maximum of g. This is because g" (a) m(1 e) 2 > 0.
O
Proof of Theorem 13 By way of contradiction, suppose that there exists a randomized r-robust strategy X with consistency most C = 1/(rWo(r)), then for every À > 0
rob(X) + Acons(X) $ AC.
(22)
25

---

Define
Rott
S
In ROFT
then recalling the definition of T, we have
T R m S.
(23)
Let ç > O be any fixed constant. We consider two cases:
Case 1: S 2 (1 + f)r. In this case, from 15 we have that E(rob( > (1 + E)r. This, however, contradicts Yao's principle from 22 and any sufficiently small value of A.
Case 2: < (1+ gr.. From 23 and 21
we obtain that T is arbitrarily close, but smaller than S,
- 0. For sufficiently large e.g.
r Lemma 12 yields
E[rob(X)) Acons(XX) (1 E)T r² ( Two((T))
(1 e1 + E)r +
(1+
(1+ ErrWo((1 +tr)
where the second inequality follows from the monotonicity of the RHS, assuming that T < r2 (otherwise, the lower bound follows trivially).
C Proofs and omitted details from Section 4
C.1 Contract scheduling with dynamic predictions
We show how to compute, inductively, each updated schedule in the dynamic setting. Recall that fist schedule, X1, can be computed as the Pareto-optinal schedule for prediction i 9 Let Xm denote the schedule computed for prediction üm from the discussion of the setting in Section 4.1 Xm begins the execution of its first contract at time Tm-1-
We describe how to compute schedule Xm, for prediction úm From Proposition 2 any r-robust schedule can complete at most (log úm)) contracts by time üm Therefore, it suffices solve the following family of LPs Pj, for E Ollg ùm):
Pj = max Tj
(24)
subj. to
ST:-11
Tm-1 + Σ xb
i [1.5]
(25)
k=
i
T Tm-1 +
Dj I im
(26)
i=U
+ r(r 4)
Tm-1 +
5
Iz
(27)
1=0
< E E [0, - 1]
(28)
To S r last 1ast((m-1))
Tm-1-
(29)
The objective of the LP is to maximize zy: here, j is the index of the largest contract that completes by time ün. Maximizing v, minimizes the consistency of the schedule. Constraint (25) guarantees the r-robustness of the schedule: this is because the worst-case interruptions occur right before the
completion of contract [47]. Note that the time starts Tn 1. which is the time that ú, available
and the first contract of X, will be scheduled. Constraint 26 states that 2ú is indeed completed by
time â..- Constraint 27 guarantees that the final schedule can be r-extendable, and constraint 28 captures the monotonicity of the contracts. Last, (29 is initialization constraint. Here, the notation
26

---

denotes the last contract that was executed in Xm-1 before dm was issued, i.e., the last contract executed before Tm-
To complete the computation of Xm, after solving the above family of LPs and computing a partial schedule of the form (1)))....).)) (for the optimal index j"), it suffices to augment this partial
schedule with its tight r-extension, as in step of Algorithm [
C.2 I Linear search
We discuss how the approach of Sections
2 and
can be applied to the linear search problem.
C.2.1 Pareto-optimal search
Let us assume the convention that the left half-line (i.e., the half line to the left of O) is mapped to the negative reals, whereas the right halfline is mapped to the positive reals. search strategy X can be described as sequence of the form
X = ((-1)(+6 2.)120, where x, > and E 10.
Consider target hiding the point (-1)8'u a where 8' E [0, Then X discovers the target cost
i-1
cost(X, u)
_
2j : < cusa x, and (-1)6tí (-1)8²
j=0
where the factor 2 accounts for the searcher moving away from O, then returning to O, in each iteration.
With the above definitions in place, we can define configurations in a manner similar to Definition For given prediction = Pi).......... (Hi, Pk)) with μ4; I, we first list the orderings in which search strategy may find each which are at most 2 Indeed, initially and after each μı is found by the searcher. there are at most two possibilities for the next μy: the one directly to the right of μæ or to its left. Let a be such a valid ordering, so X first finds Mo(1), then (a(() until (a(k)- A configuration of X according to and is defined as the vector (J1....;Jk) E N* such that, for all i E [1, k):
(-1pi-1t5 x1 Mo(1) & (-1)):+1+ Fjtt1 if (o())
(- (-1) > Ho(l) 2 (-1) T1+1 if Ha(() 0.
Furthermore, we note that strategy is r-robust if and only if for all i it holds that
Fi41 S
xj
Note that the definition is matching the one for bidding, by replacing r with p - Hence, the tight y-extension of a strategy has the same properties as in Lemma 5 Therefore, Lemma holds as well, by replacing with p. It follows that partial strategy
11 w)) r-robust if and
only if is partially r-robust and E i < 1 plp 4).
We can therefore design an algorithm analogous to Algorithm which first lists all possibilities for and the configurations. Then, it finds the best corresponding r-robust partial solution
5 E {0, which can r-extended, if it exists. This computation done through family of LPs similar that for the bidding problem:
27

---

Pass == min
Σ pi
(Mo(o) +
Σ Zqg
==1
9=0
i-1
subj. to
I; S
:i-1
Σ
Ty E [1.ja +
1=0
d++1
p+ p(p 4)
Σ 33 5
Ij++1
(- 1) 843 -1 f/ S Mo(1) S 1i+t1, i E [1, and (o(i) > 0
(- 1-1)8+h-11 01-1 2 Ma(i) - i [1, k] and Ma(i) 0
S 1422 E [0.
1; 2 0, [0, j++]
zo r.
C.2.2 Randomized search
We now show how extend the approach of Section B to the problem of linear search.
First, for the standard competitive analysis without predictions, we note that there are two different ways of defining and analyzing randomized strategies of optimal competitive ratio. The first concerns the setting with the assumption that the target cannot hide at a distance less than unit from O (otherwise no constant-competitiv strategies exist). Here, efficient strategies are of the form IT X = ((-1)>af+*) = where is a random bit {0, 1}, and is chosen u.a.r in [0, 1]. The second setting assumes that the search can start with infinitesimally small oscillations around O: this "bi- infinite" setting does not require a random starting choice between the left and right halflines, and an efficient strategy if of the form X ((-1)'a' "1)"))'*))))))))"")))))"")) +>*=a)+++ + where is chosen u.a.r. in [0, 2]. Both settings lead to the same optimal competitive ratio of
1+a In
= min
~ 4.6.
(30)
For convenience, in what follows we will assume the bi-infinite model, under which the cost expressions are easier to derive.
Upper bound We first obtain randomized strategy, parameterized over E [0, 2] and 1. Let denote the predicted target, and let j E N, À (1,a) be such that [â] 1 Suppose also, without loss generality that ů is in the right halfline. We define Rs,a as the randomized strategy (-1)d++) Aa*+*) )+ where s is chosen u.a.r. in [6. 2), and d has the same parity as j.
2²-a" Theorem 15. R5 has consistency at most + 2 11)-- and robustness at most
6² 2 2aa- ()(-))(28)))))) Ina"
Proof. By definition, the strategy discovers à at iteration j. hence its consistency is bounded as follows:
á 1
(ons)R 11+2
a ai+s Ele*]
1
a² 5
=1 2
1)a8 (2 8) Ina
a²
5
-1+2 2 a (2 (a 1) Ina
'This setting is analyzed in [Ka,, Ming- Yang. John H. Reif, and Stephen R. Tate. "Searching in an unknown environment: An optimal randomized algorithm for the cow-path problem." Information and computation 131.1 (1996): 63-79]. The work is rediscovery of original result due Gal [35).
28

---

To bound the robustness, we can generalize the competitive analysis of the randomized q- -competitive strategy. More precisely, following the lines of the proof of Lemma 8.5 in 3 we obtain that
a²
2
rob( (a.() =1+
dx
a 5 2
a² á
=1
(a 1)(2 8) Im
As in the case of bidding, R5,a interpolates between two extreme strategies. Specifically, for has consistency and robustness that are both equal to 1 - (1 + aa/ In which has a unique minimizer q², as in (30). In this case, the strategy is equivalent to the competitively optimal one.
In the other extreme, for ē + 2, simple limit evaluations show that rob( R2,a) = 1 + 2a²/ - 1), whereas cons co.s( the best possible consisteney/roouutness tradeoff that can be attained by a deterministic strategy.
+ 2/ 1). From the study of deterministic search strategies [5]. this is
Lower bound We show how to extend the approach of Section 3.2 to the lincar search problem. Recall that, without predictions, the optimal competitive ratio is expressed by (30). For the proof that 9" is optimal, we refer to Theorem 8.6 in B1 which is based hyperbolic pdf very similar to the online bidding case: the main difference is that targets may hide in either half-line, so the pdf is divided by a factor of 2, to reflect this. Using the same distribution on targets, and a prediction for target at distance ú R, we have the following expressions on the robustness and the consistency of a deterministic strategy X ((-1) )
distribution
targets with
rob(X) 1 en
Ti-1
i=0
1 +1 Sn +
T/-1
= =1+5n 2/n2
R1/n
=1+en+ +
In Rl/m
1 + Rl/n
= =1+ - 8)
(31)
In R2/nn
and
2n-
cons(X) X)
-R 2x
(32)
R
Fn-1
The above expressions are the counterpart (15) and (1),, respectively, for the linear search. Eq 33 follows directly from the proof of (3) whereas (32) follows by assuming, without loss of generality, that the searcher locates the target at distance at the n - 1-th iteration. This is because the target will be located either at the n-th or the (n 1)-th iteration, and the latter minimizes the cost, hence the consistency as well.
With this setup in place, we obtain the following lemma, whose proof follows the same lines as Lemma
R For any X >> 0, it holds that Lemma 16. Given O, let T denote the quantity
In R
rob(X) + X ccons() 2 21tee+ X(1
TWo(T) - 8(E)
where Wo is the principal branch of the Lambert function, and lim 5(E) =0.
Last, Lemma 16 allows show the following lower bound.
Theorem 17. Any randomized ctarning-augment algorithm of robustness at most r has consistency
at least + 2 (144) where Wol {=))) for any arbitrarily small, but constant & > 0.
29

---

Algorithm zeta_1
Algorithm
10
2eta
zeta
2.5
Consistency
r/2
Consistency
2.0
1.5
1.0
5
10
11
12
10
11 12
Robustness
Robustness
(a)
(b)
Figure 4; Empirical consistency as a function of the robustness requirement r, where μ4 and pj are drawn i.idd following uniform distributions. Fig.(a) depicts the same values as (b), but has wider y-axis so as to illustrate the relative performance of the heuristic with base (1-
Proof sketch. The proof is analogously to that of Theorem 12 with the difference that S and T defined slightly differently, namely
R
R
and
In R=
In R=
Furthermore, from
31
the cases are defined relative to whether
1 + R
+
Sr.
In R
iec. based on whether S < (1 Wol
D Further experimental results and discussion
In this section, we report additional experimental results.
Figure[deepice the empirical consistency of the algorithms for the dataset where (4, and p; are drawn from uniform distributions defined in Section Figure highlights the empirical consistency of the heuristic Ç1. The plot shows that the consistency increases near-linearly with i.e., this heuristic very inefficient. Hence we focus the comparison to other algorithms. The results depicted in Figure 4bbb are comparable to the ones in the main paper for a spread Gaussian distribution. The standard deviation of the heuristic results larger, as seen in the error bars, which attributed the fact that the uniform distribution has larger variance than the gaussian.
Figure F depicts experimental results for the remaining three datasets, in which all points have the same probability mass. Compared to datasets where p; are chosen i.i.d, the performance gap between the heuristics and Algorithm iis slightly smaller. The standard deviation of the heuristics is likewise smaller, since the i.i.d. predictions involve more randomness than the equiprobable predictions.
We conclude with result that supports the experimental findings reported at the end of Section 4 Namely, we will show that for large class of geometric heuristics, which includes those with bases &2 and r/2, the gap between their consistency and that of our algorithm becomes arbitrarily large, as the robustness r increases.
Formally, we say that a geometric algorithm is r-increasing if is of the form {} p²) 1:20 where depends only on r, tends to oo as > oo, and may depend on r, and μ. By definition, this class contains the heuristics with bases such as r/2, and (2.
Proposition shows that r-increasing geometric algorithms may have an arbitrarily large consistency, even on very simple prediction composed of two points. This is in contrast to Algorithm I whose consistency tends
0..
30

---

3.0
3,0
Algorithm 1
Algorithm zeta_2
zeta_1 1
2.5
2eta_2
Consistency
1/2
1
2.0
2.0
1.5
1.5
1.0
Robustness
Robustness
(a) Uniform
(b) Gaussian μis 7 4000
Algorithm
zeta
2.5
Consistency
1.0
5
9
10
11
12
Robustness
(c) Gaussian μt: a 2000
Figure 5: Empirical consistency as function of the robustness requirement when Pt are equal, and the μ4 are drawn from the stated distributions.
Proposition 18. For any r-increasing geometric algorithm, and any R 1, there exists robustness requirement t and a prediction distribution over two points such that the consistency of the algorithu over μ least R, whereas the consistency of Algorithm is at most 1/R.
Proof. Consider an r-increasing geometric algorithm, with base (which is a function of r). Let 2R and r be such that p 2 and 2 2 these choices are possible, by the definition of r-increasing strategies. Let μ be defined by (J1,, (2)
(1, A) and (P1,72) = (1 1/-,, 1 A). Since 2 A there are only two possibilities for the sequence of bids (I:):20 output by the algorithm: either there is a bid in jun (4g not. In the first case, Ti+1 is at least p#d A?, and the expected cost of the algorithm
at least P2 2 A, by considering only contribution of μ22 in the expected cost. In the second case, the first bid Iy that is larger than μi is at least 142, hence the
expected cost of the algorithm over i is at least #2 = In both cases, the expected cost of the algorithm is at least A 2.R.
Consider now partial strategy whose first bids are equal in and #2. respectively. This partial strategy is r-extendable, as 2 2 and the conditions of Lemma are satisfied. Thus, there exists an r-robust strategy of expected cost equal (1 3 1 A (#1 + M2) = A + + 2. Since 1 the above strategy has consistency at most + 1/R, and so does Algorithm
Egogl [2] since it is Parcto-optimal. In contrast, the consistency of X is at least 2R 2-1/A > R.
The above results shows that any geometric strategy will have inferior performance: Either it is r-increasing, thus inefficient according to Proposition|] [8 it is not, in which case is inefficient even on single-point predictions. The latter follows from [8) which showed that for single-point predictions the consistency is equal to p/ (p 1).
31


Reviews:
[{"review_id": "4SPEFtoWBm", "reviewer": "Reviewer_M7pr", "summary": "The authors study the online bidding problem where we submit a sequence of bids until one bid exceeds a fixed but hidden value. The goal is to minimize the total sum of bids (including the final bid that exceeds the hidden value). The problem is typically studied under competitive analysis, and the authors transfer this to the learning-augmented setting where we want to bound the consistency (which describes how well we are doing compared to the prediction) and robustness (which is still the competitive ratio).\n\nThe paper contains two algorithms for different settings. In the first setting, a stochastic prediction with finite support is provided to the algorithm. Through a structural analysis of the problem, the authors manage to provide a Pareto-optimal algorithm which for any given robustness outputs a sequence of bids that achieve the best possible consistency. \n\nThe second algorithm is in the setting where the prediction is a fixed value, but the algorithm is allowed to run a randomized sequence and attain robustness and consistency in expectation. As in the prior work in the non-learning-augmented setting, this allows for slightly improved results. The authors extend these prior methods to the learning-augmented setting.\n\nFinally, they complement their algorithms with an experimental evaluation.", "questions": "As it stands, the algorithm considers only two cases, namely when the prediction is totally accurate or totally adversarial. But it would be interesting to obtain guarantees that smoothly trade off the two cases, e.g., as a function of the EMD.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths\n\n1. The paper provides a complete contribution as they provide upper and lower bounds for the settings they consider. Especially in learning-augmented algorithms, such theoretical results are valuable as they are typically hard to obtain.\n2. The authors provide fundamental theoretical contributions to the online bidding problem (such as Definition 1, Proposition 2, or Lemma 6) which may be valuable beyond learning-augmentation.\n3. The paper is very well-written and easy to follow.\n\n### Weaknesses\n\n1. The algorithms do not address the setting where the prediction is stochastic and the algorithm is randomized, but this seems like a natural extension.\n2. Experiments on real-world data would be great.", "was_revised": true, "final_justification": "As outlined in my review, I believe the problem and solution is interesting and worth studying and the paper well written. Since the techniques are not groundbreaking, I am giving a score of 5.", "rebuttal": "We are thankful to the reviewer for their constructive comments and helpful feedback. \n\n**Response to questions**\n\n1. It may indeed be possible to obtain such guarantees via an EMD-based analysis, by generalizing the approach of [7], which studied the contract scheduling problem with distributional predictions for robustness $r=4$. But to obtain a general, theoretical result for Pareto-optimal algorithms will require some realistic assumptions on the distribution (e.g., that is has continuous, smooth and bounded density). Otherwise, we know from [7] that any Pareto-optimal algorithm is brittle against single-point predictions, and also against single-point distributional predictions with very small error (measured by the EMD). While this is an interesting theoretical problem, we should note that the consistency/robustness tradeoffs that one obtains from distributional prediction analysis are much more robust, in practice, than single-point oracles and \"deterministic\" analysis. \n\nConcerning the future study of an extension with double-sided stochasticity, there are several challenges. From the upper bound point of view, the difficulty is finding the structure of optimal solutions, a task that is already difficult even with one-side stochasticity, as shown in our LP-based approach. From the lower bound point of view, the difficulty is that the proof must be based on an entire family of adversarial distributions that should be tailored to the stochastic oracle; again this task is highly complex even for one adversarial distribution, as shown in the proof of our single-prediction bound. But in the context of our lower bound of Theorem 13,  any improvement due to randomization quickly dissipates as $r$ increases. This implies that studying the full model will be a hard theoretical problem, but will not lead to significant improvements to our Algorithm 1, under a worst-case analysis. To make such a future study more accessible, one could start with a limited randomness model, in which the algorithm has access to only a few random bits: this limited randomness can be incorporated into the LP, at the expense of a much more complex proof approach."}, {"review_id": "gMnAiIADk4", "reviewer": "Reviewer_88Mf", "summary": "The paper considers the online bidding problem, in which an agent submits a sequence of bids until the bid exceeds an unknown target threshold. The goal is to minimize the sum of bids until the target threshold is reached. For this problem, the best–possible competitive ratios are $4$ for deterministic algorithms and $e$ for randomized algorithms.\n\nIn this paper, the authors consider two learning-augmented settings for online bidding. First, they consider deterministic algorithms with access to distributional predictions, i.e., predictions on the distribution from which the unknown target threshold is drawn. Then, they consider randomized algorithms with access to non-distributional predictions, i.e., a single value prediction on the unknown target threshold. \n\nFor the first setting, the authors give algorithms that achieve the best-possible consistency for each robustness requirement. Here, the consistency of an algorithm is the competitive ratio (of expectations) that the algorithm achieves if the threshold is indeed drawn from the predicted distribution and the robustness is the competitive ratio for arbitrary thresholds. The algorithm is based on first computing a partial strategy (finite bidding sequence) that is consistent, and robust if the finite bidding sequence reaches the unknown threshold. Then, they show how to extend the partial strategy to a full one that achieves the target robustness even if the partial sequence does not reach the unknown thresholds. The main technical contribution here is a characterization of robust and extendentable partial strategies. With the help of this characterization, the pareto-optimal strategy can be computed by solving a family of linear programs.\n\nIn the second setting, the authors give lower and upper bounds on the best-possible trade-off between consistency and robustness. Interestingly, the results show that the benefit of randomization is largest for small robustness requirements and decreases with increasing requirements.", "questions": "* Do you have any insights on the brittleness of your algorithm? I.e., what happens if the predicted distribution is only slightly wrong? \n* Can you elaborate on the choice of $k=4$ in the empirical section? What is the reasoning behind fixing $k$ at $4$?\n* Can you elaborate on the main new technical ideas in your paper compared to previous works on (learning-augmented) online bidding?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\n* The considered prediction model of distributional predictions seems highly relevant and has, thus far, received significantly less attention than non-distributional predictions. More positive results on these types of predictions could be impactful for the future direction of the field.\n* The presented results on distributional predictions resolve the question for deterministic pareto-optimal algorithms along the complete pareto frontier.\n* Even though it might not be the main result of the paper, I found the insights on randomized algorithms interesting. In particular, the observation that the benefit of randomization decreases with an increasing robustness requirement is very nice. To my knowledge, results of this type are not very common in the learning-augmented algorithm field.\n\nWeaknesses:\n* The paper is missing a discussion of error measures: What happens if the unknown threshold is not drawn from $\\mu$ but from a distribution “similar” to $\\mu$? Does the algorithm immediately revert to the robustness case or are improved guarantees still possible? \n* The choice of the parameter $k=4$ in the empirical part of the paper seems a bit arbitrary. It is not clear to me if this choice is a good representative of distributional predictions that might show up in practice. Similar to the previous point, experiments where the threshold is drawn from a slightly different distribution would be quite interesting.\n* The algorithms of Theorem 7 and 8 have an exponential resp. pseudo polynomial running time. A formal justification of these running times, for example in the form of hardness results, is missing.\n* Figure 1 in Section 3 compares the deterministic upper bound with the randomised lower bound. It would be nice to also integrate the upper bound of Theorem 10 to put the algorithmic results into perspective.", "was_revised": true, "final_justification": "In my opinion, the paper is a strong contribution. The study of distributional predictions is a trending topic in learning-augmented algorithms, making this paper highly relevant. As outlined in my original review, the paper characterises the complete Pareto-frontier for the trade-off between consistency and robustness for deterministic algorithms with distributional predictions, and gives a very nice lower bound for randomised algorithms with deterministic predictions. My main concern regarding the brittleness has been adequately addressed in the author's rebuttal. Hence, I would like to increase my estimation from 4 to 5.\n\nI do not share the concerns about the running time of the presented algorithms. While I don't have any insights on whether the running time is practical or not, I think that PTAS-type guarantees as given in Theorem 8 are well established from a theoretical point of view. In my opinion, resolving the Pareto-frontier is a strong contribution either way.", "rebuttal": "We are thankful to the reviewer for their constructive comments and helpful feedback. \n\n**Response to questions**\n\n1. Brittleness is mainly a byproduct of \"single-point\" prediction oracles, combined with deterministic algorithms and deterministic analysis. In stochastic settings, brittleness intuitively is much less of an issue, as was shown formally in [20]. Notwithstanding this observation, there are various ways we can enhance Algorithm 1 to make it more robust to prediction errors. First, note that we can detect a potentially brittle solution by observing whether any of the bids is very close to one of the $k$ probability masses. Then we can rerun the algorithm using one of the following two approaches: either based on a \"tolerance\" parameter $\\delta$, in which each of the $k$ \"critical\" bids must be at least a function of $\\delta$ larger than the corresponding $\\mu_i$ (this action can be encoded in the LP); or by introducing some randomized \"perturbation\" of the solution, and specifically of the size of the $k$ critical bids, along the lines of [20]. In what concerns the randomized setting, the strategy of Theorem 10 is inherently more robust to brittleness thanks to randomization.\n\n    Furthermore, in regards to error measures, one may be able to obtain an EMD-based analysis, by generalizing the approach of [7], which studied the contract scheduling problem with distributional predictions for robustness $r=4$. But to obtain a general result for Pareto-optimal algorithms will require some realistic assumptions on the distribution (e.g., that is has continuous, smooth and bounded density). Otherwise, we know from [7] that any Pareto-optimal algorithm is brittle against single-point predictions, and also against  single-point distributional predictions with very small error (measured by the EMD). \n\n&nbsp;\n\n2. Considering that our algorithm is guaranteed to be optimal on *all* distributions, our goal was to evaluate it, in comparison to known baselines, on distributions that do not need to be overly complex. As shown in the experiments, even a relatively small value such as $k = 4$ allows us to draw very useful conclusions about the comparative performance of all algorithms.\n\n&nbsp;\n\n3. Compared to previous work that focused on deterministic models and algorithms, our main novel technical ideas are as follows:\n\n- For distributional predictions (Section 2): The main challenge in Section 2 is that, unlike the deterministic setting where simple geometric strategies of the form $(\\lambda b^i)_i$ suffice, in the stochastic setting such strategies can be extremely inefficient even on trivial distributions. Hence one needs to search a much larger space of candidate strategies which have no obvious structure. We introduce a novel approach based on Linear Programming formulations to achieve this goal. But to do so, we need to address two other technical points: 1) Prune the solution space: this is achieved by using \"tight\" strategies as defined in (5), and by relying on Proposition 2; 2) Show necessary and sufficient conditions so as to extend a partial, finite strategy to a full, infinite strategy with robustness guarantees : this is accomplished in Lemmas 5 and 6. All these are novel approaches that have applications in other settings such as contract scheduling with dynamic predictions and search games for a hidden target, as we show in Section 4.\n\n\n- For randomized algorithms (Section 3), the main challenge is the lower bound for two reasons: First, Yao’s principle (the main tool for randomized lower bounds) does not immediately apply, since we have two objectives in a tradeoff relation; Second, while deterministic $r$-robust strategies have a very clear characterization and known properties from previous work, there is no known characterization of the structure of  *randomized* $r$-robust strategies. We introduce two novel ideas:  A lower bound on a linear combination of the objectives ($rob+\\lambda \\cdot cons$), in Theorem 12, which allows us to give lower bounds on the trade-off between the two objectives; and an explicit, adversarial target distribution that allows us to properly quantify both consistency and robustness. In contrast, past work on randomized bidding without prediction used a very complex approach based on LP duality [26] which cannot incorporate predictions in any obvious way. Last, concerning the upper bound of Theorem 10, the novelty is to trade randomness for variability in performance: this allows us to obtain actual consistency/robustness tradeoffs, whereas with full randomness an algorithm has the same performance across all targets, and cannot use the prediction to the benefit of the consistency."}, {"review_id": "eIbOMF2Pzt", "reviewer": "Reviewer_C7xM", "summary": "In this paper, the authors studied the online bidding problem in an learning-augmented setting. The paper comprises of two parts. In the first part, the authors first analyzed online bidding with distributional predictions, identifying Pareto-optimal algorithms that optimally balance consistency and robustness. In the second part, they then investigated randomized bidding algorithms, providing upper and lower bounds on their performance tradeoffs.", "questions": "See weaknesses.", "limitations": "See weaknesses.", "rating": 3, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strength: \n- The paper builds on prior work in online bidding, which primarily focused on deterministic settings. It offers a comprehensive discussion by incorporating stochasticity in both the prediction oracle and the algorithm. The theoretical results appear sound, although I have not checked the proofs in detail.\n\nWeakness:\n- For Section 2, it would be helpful if the authors could elaborate on the main challenges or novel aspects of their proof techniques. As presented, the theoretical results are somewhat hard to digest.\n- The paper notes that a practical implementation of Algorithm 1 may become computationally expensive as $k$ grows, and proposes a quantization rule to address this. Have the authors implemented this approach in practice? If so, a runtime comparison would be informative.\n- From the numerical experiments, it seems that both the $\\zeta_2$ and $r/2$ heuristics perform reasonably well in terms of consistency, albeit slightly worse than the proposed algorithm. Given that these heuristics are likely much easier to implement and computationally cheaper, it would be helpful if the authors could provide some discussion comparing their practical tradeoffs.\n- Overall, while the theoretical results are interesting, the paper currently lacks a clear discussion of how the insights from these results translate to practical relevance. It would be helpful if the authors could comment on how the theoretical intuitions might inform or impact real-world applications.", "was_revised": true, "final_justification": "I appreciate the authors' response. I will keep my score due to concerns related to the practicality of the algorithm and I feel that the paper lacks runtime results from empirical evaluations. Nonetheless, I'm also open to discussions with the AC and other reviewers.", "rebuttal": "We are thankful to the reviewer for their constructive comments and helpful feedback. \n\n**Response to questions**\n\n1. The main challenge in Section 2 is that, unlike the deterministic setting where simple geometric strategies of the form $(\\lambda b^i)_{i \\geq 0}$ suffice, in the stochastic setting such strategies can be extremely inefficient even on trivial distributions. Hence one needs to search a much larger space of candidate strategies which have no obvious structure. We introduce a novel approach based on Linear Programming  formulations to achieve this goal. But to do so, we need to address two technical difficulties: 1) Prune the solution space: this is achieved by using \"tight\" strategies as defined in (5), and by relying on Proposition 2; 2) Show necessary and sufficient conditions so as to extend a partial, finite strategy to a full, infinite strategy with robustness guarantees : this is accomplished in Lemmas 5 and 6. All these are novel approaches that have applications to other settings, such as contract scheduling with dynamic predictions and search games for a hidden target, as we show in Section 4.\n\n2. Theorem 8 provides strict, theoretical guarantees both on the performance of the quantized algorithm and its runtime, where the latter does not depend anymore on $k$. As we note in lines 181-183, in practical settings when we expect the distributional advice to have reasonably small dynamic range (otherwise the oracle gives very inefficient information), the algorithm has provably fast runtime guarantees.\n\n3.  Figure 3 shows that even for very simple distributions over only two points, the  $\\zeta_2$ and $r/2$ heuristics can be as bad as almost 2-consistent, whereas our solution is 1-consistent, i.e., they are almost 100% worse than our solution. But the improvement we obtain can be much more dramatic, as we mention in lines 337-338, and as we show in Proposition 18 in the Appendix: namely as $r$ increases, the gap between these heuristics and our optimal solution is provably *unbounded*, even on distributions defined over two points. Hence, the two heuristics are provably *very inefficient* even for simple, yet adversarial data sets. \n\n4.  For a concrete application of online bidding, consider the example of job submissions to busy data centers, such as the Amazon AWS computing platform. A user submits a job to be processed by the platform, but does not know in advance how much time it will require for its completion. The user thus proposes a *walltime* $t$, and pays cost equal to $t$. If the job is not completed within time $t$, the user must resubmit the job with a new walltime $t'>t$, and the total cost payed by the user is the sum of all walltimes until the job is finally completed. How should the user decide the sequence of submitted walltimes so as to minimize the ratio between the cost it pays, and the optimal cost that would be paid if the required time was known to the user in advance? This is precisely the formulation of online bidding, and it is at the core of scheduling problems in cloud computing, see for example [Aupy et al, 2019] . \n\nWe will add a discussion of this application, as an example of the many applications we already mentioned in lines 32-37 and  Section 4 (that is, interruptible systems, search problems,  incremental clustering etc).\n \n\n[Aupy et al, 2019] Aupy, Guillaume, Ana Gainaru, Valentin Honoré, Padma Raghavan, Yves Robert, and Hongyang Sun. \"Reservation strategies for stochastic jobs.\" In 2019 IEEE International Parallel and Distributed Processing Symposium (IPDPS), pp. 166-175. IEEE, 2019."}, {"review_id": "BtZofloMQa", "reviewer": "Reviewer_dCyV", "summary": "This paper investigates the online bidding problem under a stochastic prediction oracle or stochastic algorithms. Given a stochastic prediction oracle, this paper proposes an algorithm to find the Pareto-optimal strategy. Moreover, this paper investigates the lower/upper bounds for stochastic algorithms.", "questions": "What is the potential difficulty of studying the stochastic algorithm with a stochastic prediction oracle?", "limitations": "Yes", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths\n- The writing is clear.\n- This paper gives a systematic study of online bidding in stochastic settings, including both upper bounds and lower bounds.\n- Empirical results justify the efficiency of Algorithm 1.\n- The paper gives an algorithm to find the entire Pareto frontier.\n\n## Weaknesses\n- There is no empirical comparison between the stochastic algorithm proposed in this paper and the deterministic algorithm.\n- The computational complexity of Algorithm 1 is exponential in $k$.\n- The experiments are restricted to some synthetic data, which cannot justify why the stochastic setting is reasonable.", "was_revised": true, "final_justification": "From the latest discussion with the authors, the time complexity of Theorem 8 seems to be exponential with respect to the consistency approximation. However, I am not sure whether this is a big issue in this area. Therefore, I keep my rating but change the confidence to 1.", "rebuttal": "We are thankful to the reviewer for their constructive comments and helpful feedback. \n\n**Response to questions**\n\n1. From the upper bound point of view, the difficulty is finding the structure of optimal solutions, a task that is already difficult even with one-side stochasticity, as shown in our LP-based approach. From the lower bound point of view, the difficulty is that the proof must be based on an entire family of adversarial distributions that should be tailored to the stochastic oracle; again this task is highly complex even for one adversarial distribution, as shown in the proof of our single-prediction bound.  One should also take into consideration that the competitive ratio of this problem is a constant, hence the margins for theoretical improvements are quite small. It should be emphasized that our lower bound of Section 3 shows that any improvement due to randomization quickly dissipates as $r$ increases. This implies that studying the full model will be a hard theoretical problem, but will not lead to significant improvements to our Algorithm 1, under a worst-case analysis. To make such a future study more accessible, one could start with a limited randomness model, in which the algorithm has access to only a few random bits: this limited randomness can be incorporated into the LP, at the expense of a much more complex proof approach.\n\n**Additional comments**\n\n1. Our main contribution in Section 3 is the lower bound of Theorem 13, which shows that the improvement that can be obtained from randomization becomes marginal, as $r$ increases. \n\n2. In regards to the complexity of Algorithm 1: If $k$ is too large, then Theorem 8 shows how to reduce complexity via distributional quantization, thus obtaining improved, non-trivial runtime guarantees. We also refer to the discussion in lines 181-183. The runtime analysis in this theorem depends on the dynamic range of the distribution, but is independent of $k$.\n\n3. There are no established real benchmarks for online bidding, hence we used synthetic data which suffice to show a large performance gap between our algorithm and state-of-the-art heuristics, considering that our Algorithm 1 is theoretically optimal on *any* distribution. We note that even for concrete applications that can be modeled as the online bidding problem, the experimental analysis is often based on robust synthetic data. Consider, e.g., the problem of job submissions to busy data centers, such as the Amazon AWS computing platform, which can be modeled as an application of online bidding (See point 4 in the response to Reviewer C7xM for a formal definition of the setting). In [Aupy et al, 2019], the experimental analysis is based, likewise, on synthetic data, using distributions that can adequately simulate real settings. Our aim was to obtain an algorithm of optimal performance under any current, or future stochastic oracle that may become available. This is the broader objective in the emerging subfield of learning-augmented algorithms with distributional predictions.\n\n[Aupy et al, 2019] Aupy, Guillaume, Ana Gainaru, Valentin Honoré, Padma Raghavan, Yves Robert, and Hongyang Sun. \"Reservation strategies for stochastic jobs.\" In 2019 IEEE International Parallel and Distributed Processing Symposium (IPDPS), pp. 166-175. IEEE, 2019."}]

Author rebuttal:

Meta-review: This paper studies an online bidding problem where the bidders submit a sequence of bids until one bid exceeds a fixed but hidden value. The goal is to propose a learning-augmented algorithm that not only achieves a high competitive ratio (robustness) but also a a good performance when the prediction is accurate (consistency). 

Two settings are considered in this paper. In the first setting, a stochastic prediction with finite support is provided to the algorithm. A Pareto-optimal algorithm is proposed to output a sequence of bids that achieve the best possible consistency for any given robustness. In the second setting, the prediction is a fixed value, but the algorithm is allowed to be randomized. The authors extend prior methods in the non-learning-augmented setting to the learning-augmented setting and show improved results. Finally, the algorithms are complemented with an experimental evaluation.

All reviewers are unanimously supportive of this paper. It studies an interesting problem and provides sound solutions. Some concerns include the running time of the algorithm (e.g. exponential in 1/\varepsilon, possibly with a large degree in the polynomial-time), and that the contributions in the second setting is a bit incremental. Therefore, I would like to recommend acceptance as a poster.
Decision: Accept (poster)