IA Scholar Query: Polymatroid Prophet Inequalities.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgFri, 18 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Prophet-Inequalities over Time
https://scholar.archive.org/work/uyaa43jfbbhejn5lajfha6luju
In this paper, we introduce an over-time variant of the well-known prophet-inequality with i.i.d. random variables. Instead of stopping with one realized value at some point in the process, we decide for each step how long we select the value. Then we cannot select another value until this period is over. The goal is to maximize the expectation of the sum of selected values. We describe the structure of the optimal stopping rule and give upper and lower bounds on the prophet-inequality. - Which, in online algorithms terminology, corresponds to bounds on the competitive ratio of an online algorithm. We give a surprisingly simple algorithm with a single threshold that results in a prophet-inequality of ≈ 0.396 for all input lengths n. Additionally, as our main result, we present a more advanced algorithm resulting in a prophet-inequality of ≈ 0.598 when the number of steps tends to infinity. We complement our results by an upper bound that shows that the best possible prophet-inequality is at most 1/φ≈ 0.618, where φ denotes the golden ratio. As part of the proof, we give an advanced bound on the weighted mediant.Andreas Abels, Elias Pitschmann, Daniel Schmandwork_uyaa43jfbbhejn5lajfha6lujuFri, 18 Nov 2022 00:00:00 GMTProphet Inequalities via the Expected Competitive Ratio
https://scholar.archive.org/work/rmv7k3mbdngh5hfdppj5hqcqwm
We consider prophet inequalities under general downward-closed constraints. In a prophet inequality problem, a decision-maker sees a series of online elements and needs to decide immediately and irrevocably whether or not to select each element upon its arrival, subject to an underlying feasibility constraint. Traditionally, the decision-maker's expected performance has been compared to the expected performance of the prophet, i.e., the expected offline optimum. We refer to this measure as the Ratio of Expectations (or, in short, RoE). However, a major limitation of the RoE measure is that it only gives a guarantee against what the optimum would be on average, while, in theory, algorithms still might perform poorly compared to the realized ex-post optimal value. Hence, we study alternative performance measures. In particular, we suggest the Expected Ratio (or, in short, EoR), which is the expectation of the ratio between the value of the algorithm and the value of the prophet. This measure yields desirable guarantees, e.g., a constant EoR implies achieving a constant fraction of the ex-post offline optimum with constant probability. Moreover, in the single-choice setting, we show that the EoR is equivalent (in the worst case) to the probability of selecting the maximum, a well-studied measure in the literature. This is no longer the case for combinatorial constraints (beyond single-choice), which is the main focus of this paper. Our main goal is to understand the relation between RoE and EoR in combinatorial settings. Specifically, we establish a two-way black-box reduction: for every feasibility constraint, the RoE and the EoR are at most a constant factor apart. This implies a wealth of EoR results in multiple settings where RoE results are known.Tomer Ezra and Stefano Leonardi and Rebecca Reiffenhäuser and Matteo Russo and Alexandros Tsigonias-Dimitriadiswork_rmv7k3mbdngh5hfdppj5hqcqwmTue, 08 Nov 2022 00:00:00 GMT"Who Is Next in Line?" On the Significance of Knowing the Arrival Order in Bayesian Online Settings
https://scholar.archive.org/work/yvhw5csz7bh57b4lv3veb7lmpi
We introduce a new measure for the performance of online algorithms in Bayesian settings, where the input is drawn from a known prior, but the realizations are revealed one-by-one in an online fashion. Our new measure is called order-competitive ratio. It is defined as the worst case (over all distribution sequences) ratio between the performance of the best order-unaware and order-aware algorithms, and quantifies the loss that is incurred due to lack of knowledge of the arrival order. Despite the growing interest in the role of the arrival order on the performance of online algorithms, this loss has been overlooked thus far. We study the order-competitive ratio in the paradigmatic prophet inequality problem, for the two common objective functions of (i) maximizing the expected value, and (ii) maximizing the probability of obtaining the largest value; and with respect to two families of algorithms, namely (i) adaptive algorithms, and (ii) single-threshold algorithms. We provide tight bounds for all four combinations, with respect to deterministic algorithms. Our analysis requires new ideas and departs from standard techniques. In particular, our adaptive algorithms inevitably go beyond single-threshold algorithms. The results with respect to the order-competitive ratio measure capture the intuition that adaptive algorithms are stronger than single-threshold ones, and may lead to a better algorithmic advice than the classical competitive ratio measure.Tomer Ezra, Michal Feldman, Nick Gravin, Zhihao Gavin Tangwork_yvhw5csz7bh57b4lv3veb7lmpiFri, 04 Nov 2022 00:00:00 GMTAn Improved Lower Bound for Matroid Intersection Prophet Inequalities
https://scholar.archive.org/work/d6k35drd4vf5hou3362ozq4aiq
We consider prophet inequalities subject to feasibility constraints that are the intersection of q matroids. The best-known algorithms achieve a Θ(q)-approximation, even when restricted to instances that are the intersection of q partition matroids, and with i.i.d. Bernoulli random variables. The previous best-known lower bound is Θ(√(q)) due to a simple construction of [Kleinberg-Weinberg STOC 2012] (which uses i.i.d. Bernoulli random variables, and writes the construction as the intersection of partition matroids). We establish an improved lower bound of q^1/2+Ω(1/loglog q) by writing the construction of [Kleinberg-Weinberg STOC 2012] as the intersection of asymptotically fewer partition matroids. We accomplish this via an improved upper bound on the product dimension of a graph with p^p disjoint cliques of size p, using recent techniques developed in [Alon-Alweiss European Journal of Combinatorics 2020].Raghuvansh R. Saxena, Santhoshini Velusamy, S. Matthew Weinbergwork_d6k35drd4vf5hou3362ozq4aiqMon, 12 Sep 2022 00:00:00 GMTA Nonparametric Framework for Online Stochastic Matching with Correlated Arrivals
https://scholar.archive.org/work/2kvfyovgyndbzcd3jfu2h62m7q
The design of online policies for stochastic matching and revenue management settings is usually bound by the Bayesian prior that the demand process is formed by a fixed-length sequence of queries with unknown types, each drawn independently. This assumption of serial independence implies that the demand of each type, i.e., the number of queries of a given type, has low variance and is approximately Poisson-distributed. Thus, matching policies are often based on "fluid" LPs that only use the expectations of these distributions. This paper explores alternative stochastic models for online matching that allow for nonparametric, higher variance demand distributions. We propose two new models, INDEP and CORREL, that relax the serial independence assumption in different ways by combining a nonparametric distribution for the demand with standard assumptions on the arrival patterns -- adversarial or random-order. In our INDEP model, the demand for each type follows an arbitrary distribution, while being mutually independent across different types. In our CORREL model, the total demand follows an arbitrary distribution, and conditional on the sequence length, the type of each query is drawn independently. In both settings, we show that the fluid LP relaxation based on only expected demands can be an arbitrarily bad benchmark for algorithm design. We develop tighter LP relaxations for the INDEP and CORREL models that leverage the exact distribution of the demand, leading to matching algorithms that achieve constant-factor performance guarantees under adversarial and random-order arrivals. More broadly, our paper provides a data-driven framework for expressing demand uncertainty (i.e., variance and correlations) in online stochastic matching models.Ali Aouad, Will Mawork_2kvfyovgyndbzcd3jfu2h62m7qSun, 07 Aug 2022 00:00:00 GMTOptimal Algorithms for Free Order Multiple-Choice Secretary
https://scholar.archive.org/work/imm5gwhcizbqrokk5lahmcf644
Suppose we are given integer k ≤ n and n boxes labeled 1,..., n by an adversary, each containing a number chosen from an unknown distribution. We have to choose an order to sequentially open these boxes, and each time we open the next box in this order, we learn its number. If we reject a number in a box, the box cannot be recalled. Our goal is to accept the k largest of these numbers, without necessarily opening all boxes. This is the free order multiple-choice secretary problem. Free order variants were studied extensively for the secretary and prophet problems. Kesselheim, Kleinberg, and Niazadeh KKN (STOC'15) initiated a study of randomness-efficient algorithms (with the cheapest order in terms of used random bits) for the free order secretary problems. We present an algorithm for free order multiple-choice secretary, which is simultaneously optimal for the competitive ratio and used amount of randomness. I.e., we construct a distribution on orders with optimal entropy Θ(loglog n) such that a deterministic multiple-threshold algorithm is 1-O(√(log k/k))-competitive. This improves in three ways the previous best construction by KKN, whose competitive ratio is 1 - O(1/k^1/3) - o(1). Our competitive ratio is (near)optimal for the multiple-choice secretary problem; it works for exponentially larger parameter k; and our algorithm is a simple deterministic multiple-threshold algorithm, while that in KKN is randomized. We also prove a corresponding lower bound on the entropy of optimal solutions for the multiple-choice secretary problem, matching entropy of our algorithm, where no such previous lower bound was known. We obtain our algorithmic results with a host of new techniques, and with these techniques we also improve significantly the previous results of KKN about constructing entropy-optimal distributions for the classic free order secretary.Mohammad Taghi Hajiaghayi, Dariusz R. Kowalski, Piotr Krysta, Jan Olkowskiwork_imm5gwhcizbqrokk5lahmcf644Thu, 21 Jul 2022 00:00:00 GMTHindsight Learning for MDPs with Exogenous Inputs
https://scholar.archive.org/work/ic7txolewnbongfrgeyh2pjwlu
We develop a reinforcement learning (RL) framework for applications that deal with sequential decisions and exogenous uncertainty, such as resource allocation and inventory management. In these applications, the uncertainty is only due to exogenous variables like future demands. A popular approach is to predict the exogenous variables using historical data and then plan with the predictions. However, this indirect approach requires high-fidelity modeling of the exogenous process to guarantee good downstream decision-making, which can be impractical when the exogenous process is complex. In this work we propose an alternative approach based on hindsight learning which sidesteps modeling the exogenous process. Our key insight is that, unlike Sim2Real RL, we can revisit past decisions in the historical data and derive counterfactual consequences for other actions in these applications. Our framework uses hindsight-optimal actions as the policy training signal and has strong theoretical guarantees on decision-making performance. We develop an algorithm using our framework to allocate compute resources for real-world Microsoft Azure workloads. The results show our approach learns better policies than domain-specific heuristics and Sim2Real RL baselines.Sean R. Sinclair, Felipe Frujeri, Ching-An Cheng, Adith Swaminathanwork_ic7txolewnbongfrgeyh2pjwluWed, 13 Jul 2022 00:00:00 GMTEntropy-Optimal Algorithms for Multiple-Choice Secretary
https://scholar.archive.org/work/3mmmh2pyinhzrcno2tajpcmk44
In this paper, we study the problem for the entropy of both admissible and optimal distributions of permutations to the multiple-choice secretary problem. In particular, we construct a distribution with entropy Θ(loglog n) such that a deterministic threshold-based algorithm gives a nearly-optimal competitive ratio 1-O(logk/k^1/3) for k=O((log n)^3/14). Our error is simultaneously nearly-optimal and with optimal entropy Θ(loglog n). Our result improves in two ways the previous best construction by Kesselheim, Kleinberg and Niazadeh (KKN) [STOC'15] whose competitive ratio is 1 - O(1/k^1/3) - o(1). First, our solution works for an exponentially larger range of parameters k, as in the work of KKN k=O((logloglog n)^ϵ). Second, our algorithm is a simple deterministic single-threshold algorithm (only drawing a permutation from a stochastic uniform distribution), while the algorithm in KKN uses additional randomness. We also prove a new corresponding lower bound for entropy of optimal solutions to the k-secretary problem, matching the entropy of our algorithm. We further show the strength of our techniques by obtaining fine-grained results for optimal distributions of permutations for the classic secretary problem. For optimal entropy Θ(loglog n), we precisely characterize the success probability of uniform distributions that is below, and close to, 1/e, and construct such distributions in polynomial time. We prove even higher entropy Θ(log(n)) suffices for a success probability above 1/e, but, no uniform probability distribution with small support and entropy strictly less than log (n) can have success probability above 1/e. For maximum entropy Θ(n log(n)), improving upon a result of Samuels from 1981, we find the precise formula for the optimal success probability of any secretary algorithm.Mohammad Taghi Hajiaghayi and Dariusz R. Kowalski and Piotr Krysta and Jan Olkowskiwork_3mmmh2pyinhzrcno2tajpcmk44Tue, 12 Apr 2022 00:00:00 GMTStatic pricing for multi-unit prophet inequalities
https://scholar.archive.org/work/gkm2akttjjbx7emec34bk6gozi
We study a pricing problem where a seller has k identical copies of a product, buyers arrive sequentially, and the seller prices the items aiming to maximize social welfare. When k=1, this is the so called "prophet inequality" problem for which there is a simple pricing scheme achieving a competitive ratio of 1/2. On the other end of the spectrum, as k goes to infinity, the asymptotic performance of both static and adaptive pricing is well understood. We provide a static pricing scheme for the small-supply regime: where k is small but larger than 1. Prior to our work, the best competitive ratio known for this setting was the 1/2 that follows from the single-unit prophet inequality. Our pricing scheme is easy to describe as well as practical – it is anonymous, non-adaptive, and order-oblivious. We pick a single price that equalizes the expected fraction of items sold and the probability that the supply does not sell out before all customers are served; this price is then offered to each customer while supply lasts. This extends an approach introduced by Samuel-Cahn for the case of k=1. This pricing scheme achieves a competitive ratio that increases gradually with the supply and approaches to 1 at the optimal rate. Astonishingly, for k<20, it even outperforms the state-of-the-art adaptive pricing for the small-k regime.Shuchi Chawla, Nikhil Devanur, Thodoris Lykouriswork_gkm2akttjjbx7emec34bk6goziWed, 22 Dec 2021 00:00:00 GMT