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Single-Sample Prophet Inequalities via Greedy-Ordered Selection

2022-01-01
Constantine Caramanis , Paul Dütting , Matthew Faw +6 more
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Single-Sample Prophet Inequalities via Greedy-Ordered SelectionConstantine Caramanis, Paul Dütting, Matthew Faw, Federico Fusco, Philip … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Single-Sample Prophet Inequalities via Greedy-Ordered SelectionConstantine Caramanis, Paul Dütting, Matthew Faw, Federico Fusco, Philip Lazos, Stefano Leonardi, Orestis Papadigenopoulos, Emmanouil Pountourakis, and Rebecca ReiffenhäuserConstantine Caramanis, Paul Dütting, Matthew Faw, Federico Fusco, Philip Lazos, Stefano Leonardi, Orestis Papadigenopoulos, Emmanouil Pountourakis, and Rebecca Reiffenhäuserpp.1298 - 1325Chapter DOI:https://doi.org/10.1137/1.9781611977073.54PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single choice problem [Rubinstein et al., 2020], most existing SSPI results were obtained via an elegant, but inherently lossy reduction to order-oblivious secretary (OOS) policies [Azar et al., 2014]. Motivated by this discrepancy, we develop an intuitive and versatile greedy-based technique that yields SSPIs directly rather than through the reduction to OOSs. Our results can be seen as generalizing and unifying a number of existing results in the area of prophet and secretary problems. Our algorithms significantly improve on the competitive guarantees for a number of interesting scenarios (including general matching with edge arrivals, bipartite matching with vertex arrivals, and certain matroids), and capture new settings (such as budget additive combinatorial auctions). Complementing our algorithmic results, we also consider mechanism design variants. Finally, we analyze the power and limitations of different SSPI approaches by providing a partial converse to the reduction from SSPI to OOS given by Azar et al. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771
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Single-Sample Prophet Inequalities Revisited.

2021-03-24
Constantine Caramanis , Matthew Faw , Orestis Papadigenopoulos +1 more
The study of the prophet inequality problem in the limited information regime was initiated by Azar et al. [SODA'14] in the pursuit of prior-independent posted-price mechanisms. As they show, $O(1)$-competitive … The study of the prophet inequality problem in the limited information regime was initiated by Azar et al. [SODA'14] in the pursuit of prior-independent posted-price mechanisms. As they show, $O(1)$-competitive policies are achievable using only a single sample from the distribution of each agent. A notable portion of their results relies on reducing the design of single-sample prophet inequalities (SSPIs) to that of order-oblivious secretary (OOS) policies. The above reduction comes at the cost of not fully utilizing the available samples. However, to date, this is essentially the only method for proving SSPIs for many combinatorial sets. Very recently, Rubinstein et al. [ITCS'20] give a surprisingly simple algorithm which achieves the optimal competitive ratio for the single-choice SSPI problem $-$ a result which is unobtainable going through the reduction to secretary problems. Motivated by this discrepancy, we study the competitiveness of simple SSPI policies directly, without appealing to results from OOS literature. In this direction, we first develop a framework for analyzing policies against a greedy-like prophet solution. Using this framework, we obtain the first SSPI for general (non-bipartite) matching environments, as well as improved competitive ratios for transversal and truncated partition matroids. Second, motivated by the observation that many OOS policies for matroids decompose the problem into independent rank-$1$ instances, we provide a meta-theorem which applies to any matroid satisfying this partition property. Leveraging the recent results by Rubinstein et al., we obtain improved competitive guarantees (most by a factor of $2$) for a number of matroids captured by the reduction of Azar et al. Finally, we discuss applications of our SSPIs to the design of mechanisms for multi-dimensional limited information settings with improved revenue and welfare guarantees.

Single-Sample Prophet Inequalities via Greedy-Ordered Selection

2021-01-01
Constantine Caramanis , Paul Dütting , Matthew Faw +6 more
We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single … We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single choice problem [Rubinstein et al., 2020], most existing SSPI results were obtained via an elegant, but inherently lossy, reduction to order-oblivious secretary (OOS) policies [Azar et al., 2014]. Motivated by this discrepancy, we develop an intuitive and versatile greedy-based technique that yields SSPIs directly rather than through the reduction to OOSs. Our results can be seen as generalizing and unifying a number of existing results in the area of prophet and secretary problems. Our algorithms significantly improve on the competitive guarantees for a number of interesting scenarios (including general matching with edge arrivals, bipartite matching with vertex arrivals, and certain matroids), and capture new settings (such as budget additive combinatorial auctions). Complementing our algorithmic results, we also consider mechanism design variants. Finally, we analyze the power and limitations of different SSPI approaches by providing a partial converse to the reduction from SSPI to OOS given by Azar et al.

Mix and Match: An Optimistic Tree-Search Approach for Learning Models from Mixture Distributions

2019-01-01
Matthew Faw , Rajat Sen , Karthikeyan Shanmugam +2 more
We consider a covariate shift problem where one has access to several different training datasets for the same learning problem and a small validation set which possibly differs from all … We consider a covariate shift problem where one has access to several different training datasets for the same learning problem and a small validation set which possibly differs from all the individual training distributions. This covariate shift is caused, in part, due to unobserved features in the datasets. The objective, then, is to find the best mixture distribution over the training datasets (with only observed features) such that training a learning algorithm using this mixture has the best validation performance. Our proposed algorithm, ${\sf Mix\&Match}$, combines stochastic gradient descent (SGD) with optimistic tree search and model re-use (evolving partially trained models with samples from different mixture distributions) over the space of mixtures, for this task. We prove simple regret guarantees for our algorithm with respect to recovering the optimal mixture, given a total budget of SGD evaluations. Finally, we validate our algorithm on two real-world datasets.

Single-Sample Prophet Inequalities Revisited

2021-01-01
Constantine Caramanis , Matthew Faw , Orestis Papadigenopoulos +1 more
The study of the prophet inequality problem in the limited information regime was initiated by Azar et al. [SODA'14] in the pursuit of prior-independent posted-price mechanisms. As they show, $O(1)$-competitive … The study of the prophet inequality problem in the limited information regime was initiated by Azar et al. [SODA'14] in the pursuit of prior-independent posted-price mechanisms. As they show, $O(1)$-competitive policies are achievable using only a single sample from the distribution of each agent. A notable portion of their results relies on reducing the design of single-sample prophet inequalities (SSPIs) to that of order-oblivious secretary (OOS) policies. The above reduction comes at the cost of not fully utilizing the available samples. However, to date, this is essentially the only method for proving SSPIs for many combinatorial sets. Very recently, Rubinstein et al. [ITCS'20] give a surprisingly simple algorithm which achieves the optimal competitive ratio for the single-choice SSPI problem $-$ a result which is unobtainable going through the reduction to secretary problems. Motivated by this discrepancy, we study the competitiveness of simple SSPI policies directly, without appealing to results from OOS literature. In this direction, we first develop a framework for analyzing policies against a greedy-like prophet solution. Using this framework, we obtain the first SSPI for general (non-bipartite) matching environments, as well as improved competitive ratios for transversal and truncated partition matroids. Second, motivated by the observation that many OOS policies for matroids decompose the problem into independent rank-$1$ instances, we provide a meta-theorem which applies to any matroid satisfying this partition property. Leveraging the recent results by Rubinstein et al., we obtain improved competitive guarantees (most by a factor of $2$) for a number of matroids captured by the reduction of Azar et al. Finally, we discuss applications of our SSPIs to the design of mechanisms for multi-dimensional limited information settings with improved revenue and welfare guarantees.
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Beyond Uniform Smoothness: A Stopped Analysis of Adaptive SGD

2023-01-01
Matthew Faw , Litu Rout , Constantine Caramanis +1 more
This work considers the problem of finding a first-order stationary point of a non-convex function with potentially unbounded smoothness constant using a stochastic gradient oracle. We focus on the class … This work considers the problem of finding a first-order stationary point of a non-convex function with potentially unbounded smoothness constant using a stochastic gradient oracle. We focus on the class of $(L_0,L_1)$-smooth functions proposed by Zhang et al. (ICLR'20). Empirical evidence suggests that these functions more closely captures practical machine learning problems as compared to the pervasive $L_0$-smoothness. This class is rich enough to include highly non-smooth functions, such as $\exp(L_1 x)$ which is $(0,\mathcal{O}(L_1))$-smooth. Despite the richness, an emerging line of works achieves the $\widetilde{\mathcal{O}}(\frac{1}{\sqrt{T}})$ rate of convergence when the noise of the stochastic gradients is deterministically and uniformly bounded. This noise restriction is not required in the $L_0$-smooth setting, and in many practical settings is either not satisfied, or results in weaker convergence rates with respect to the noise scaling of the convergence rate. We develop a technique that allows us to prove $\mathcal{O}(\frac{\mathrm{poly}\log(T)}{\sqrt{T}})$ convergence rates for $(L_0,L_1)$-smooth functions without assuming uniform bounds on the noise support. The key innovation behind our results is a carefully constructed stopping time $\tau$ which is simultaneously "large" on average, yet also allows us to treat the adaptive step sizes before $\tau$ as (roughly) independent of the gradients. For general $(L_0,L_1)$-smooth functions, our analysis requires the mild restriction that the multiplicative noise parameter $\sigma_1 < 1$. For a broad subclass of $(L_0,L_1)$-smooth functions, our convergence rate continues to hold when $\sigma_1 \geq 1$. By contrast, we prove that many algorithms analyzed by prior works on $(L_0,L_1)$-smooth optimization diverge with constant probability even for smooth and strongly-convex functions when $\sigma_1 > 1$.

The Power of Adaptivity in SGD: Self-Tuning Step Sizes with Unbounded Gradients and Affine Variance

2022-01-01
Matthew Faw , Isidoros Tziotis , Constantine Caramanis +3 more
We study convergence rates of AdaGrad-Norm as an exemplar of adaptive stochastic gradient methods (SGD), where the step sizes change based on observed stochastic gradients, for minimizing non-convex, smooth objectives. … We study convergence rates of AdaGrad-Norm as an exemplar of adaptive stochastic gradient methods (SGD), where the step sizes change based on observed stochastic gradients, for minimizing non-convex, smooth objectives. Despite their popularity, the analysis of adaptive SGD lags behind that of non adaptive methods in this setting. Specifically, all prior works rely on some subset of the following assumptions: (i) uniformly-bounded gradient norms, (ii) uniformly-bounded stochastic gradient variance (or even noise support), (iii) conditional independence between the step size and stochastic gradient. In this work, we show that AdaGrad-Norm exhibits an order optimal convergence rate of $\mathcal{O}\left(\frac{\mathrm{poly}\log(T)}{\sqrt{T}}\right)$ after $T$ iterations under the same assumptions as optimally-tuned non adaptive SGD (unbounded gradient norms and affine noise variance scaling), and crucially, without needing any tuning parameters. We thus establish that adaptive gradient methods exhibit order-optimal convergence in much broader regimes than previously understood.
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Beyond Uniform Smoothness: A Stopped Analysis of Adaptive SGD

2023-01-01
Matthew Faw , Litu Rout , Constantine Caramanis +1 more
This work considers the problem of finding a first-order stationary point of a non-convex function with potentially unbounded smoothness constant using a stochastic gradient oracle. We focus on the class … This work considers the problem of finding a first-order stationary point of a non-convex function with potentially unbounded smoothness constant using a stochastic gradient oracle. We focus on the class of $(L_0,L_1)$-smooth functions proposed by Zhang et al. (ICLR'20). Empirical evidence suggests that these functions more closely captures practical machine learning problems as compared to the pervasive $L_0$-smoothness. This class is rich enough to include highly non-smooth functions, such as $\exp(L_1 x)$ which is $(0,\mathcal{O}(L_1))$-smooth. Despite the richness, an emerging line of works achieves the $\widetilde{\mathcal{O}}(\frac{1}{\sqrt{T}})$ rate of convergence when the noise of the stochastic gradients is deterministically and uniformly bounded. This noise restriction is not required in the $L_0$-smooth setting, and in many practical settings is either not satisfied, or results in weaker convergence rates with respect to the noise scaling of the convergence rate. We develop a technique that allows us to prove $\mathcal{O}(\frac{\mathrm{poly}\log(T)}{\sqrt{T}})$ convergence rates for $(L_0,L_1)$-smooth functions without assuming uniform bounds on the noise support. The key innovation behind our results is a carefully constructed stopping time $\tau$ which is simultaneously "large" on average, yet also allows us to treat the adaptive step sizes before $\tau$ as (roughly) independent of the gradients. For general $(L_0,L_1)$-smooth functions, our analysis requires the mild restriction that the multiplicative noise parameter $\sigma_1 < 1$. For a broad subclass of $(L_0,L_1)$-smooth functions, our convergence rate continues to hold when $\sigma_1 \geq 1$. By contrast, we prove that many algorithms analyzed by prior works on $(L_0,L_1)$-smooth optimization diverge with constant probability even for smooth and strongly-convex functions when $\sigma_1 > 1$.

Single-Sample Prophet Inequalities via Greedy-Ordered Selection

2022-01-01
Constantine Caramanis , Paul Dütting , Matthew Faw +6 more
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Single-Sample Prophet Inequalities via Greedy-Ordered SelectionConstantine Caramanis, Paul Dütting, Matthew Faw, Federico Fusco, Philip … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Single-Sample Prophet Inequalities via Greedy-Ordered SelectionConstantine Caramanis, Paul Dütting, Matthew Faw, Federico Fusco, Philip Lazos, Stefano Leonardi, Orestis Papadigenopoulos, Emmanouil Pountourakis, and Rebecca ReiffenhäuserConstantine Caramanis, Paul Dütting, Matthew Faw, Federico Fusco, Philip Lazos, Stefano Leonardi, Orestis Papadigenopoulos, Emmanouil Pountourakis, and Rebecca Reiffenhäuserpp.1298 - 1325Chapter DOI:https://doi.org/10.1137/1.9781611977073.54PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single choice problem [Rubinstein et al., 2020], most existing SSPI results were obtained via an elegant, but inherently lossy reduction to order-oblivious secretary (OOS) policies [Azar et al., 2014]. Motivated by this discrepancy, we develop an intuitive and versatile greedy-based technique that yields SSPIs directly rather than through the reduction to OOSs. Our results can be seen as generalizing and unifying a number of existing results in the area of prophet and secretary problems. Our algorithms significantly improve on the competitive guarantees for a number of interesting scenarios (including general matching with edge arrivals, bipartite matching with vertex arrivals, and certain matroids), and capture new settings (such as budget additive combinatorial auctions). Complementing our algorithmic results, we also consider mechanism design variants. Finally, we analyze the power and limitations of different SSPI approaches by providing a partial converse to the reduction from SSPI to OOS given by Azar et al. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771
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The Power of Adaptivity in SGD: Self-Tuning Step Sizes with Unbounded Gradients and Affine Variance

2022-01-01
Matthew Faw , Isidoros Tziotis , Constantine Caramanis +3 more
We study convergence rates of AdaGrad-Norm as an exemplar of adaptive stochastic gradient methods (SGD), where the step sizes change based on observed stochastic gradients, for minimizing non-convex, smooth objectives. … We study convergence rates of AdaGrad-Norm as an exemplar of adaptive stochastic gradient methods (SGD), where the step sizes change based on observed stochastic gradients, for minimizing non-convex, smooth objectives. Despite their popularity, the analysis of adaptive SGD lags behind that of non adaptive methods in this setting. Specifically, all prior works rely on some subset of the following assumptions: (i) uniformly-bounded gradient norms, (ii) uniformly-bounded stochastic gradient variance (or even noise support), (iii) conditional independence between the step size and stochastic gradient. In this work, we show that AdaGrad-Norm exhibits an order optimal convergence rate of $\mathcal{O}\left(\frac{\mathrm{poly}\log(T)}{\sqrt{T}}\right)$ after $T$ iterations under the same assumptions as optimally-tuned non adaptive SGD (unbounded gradient norms and affine noise variance scaling), and crucially, without needing any tuning parameters. We thus establish that adaptive gradient methods exhibit order-optimal convergence in much broader regimes than previously understood.
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Single-Sample Prophet Inequalities Revisited.

2021-03-24
Constantine Caramanis , Matthew Faw , Orestis Papadigenopoulos +1 more
The study of the prophet inequality problem in the limited information regime was initiated by Azar et al. [SODA'14] in the pursuit of prior-independent posted-price mechanisms. As they show, $O(1)$-competitive … The study of the prophet inequality problem in the limited information regime was initiated by Azar et al. [SODA'14] in the pursuit of prior-independent posted-price mechanisms. As they show, $O(1)$-competitive policies are achievable using only a single sample from the distribution of each agent. A notable portion of their results relies on reducing the design of single-sample prophet inequalities (SSPIs) to that of order-oblivious secretary (OOS) policies. The above reduction comes at the cost of not fully utilizing the available samples. However, to date, this is essentially the only method for proving SSPIs for many combinatorial sets. Very recently, Rubinstein et al. [ITCS'20] give a surprisingly simple algorithm which achieves the optimal competitive ratio for the single-choice SSPI problem $-$ a result which is unobtainable going through the reduction to secretary problems. Motivated by this discrepancy, we study the competitiveness of simple SSPI policies directly, without appealing to results from OOS literature. In this direction, we first develop a framework for analyzing policies against a greedy-like prophet solution. Using this framework, we obtain the first SSPI for general (non-bipartite) matching environments, as well as improved competitive ratios for transversal and truncated partition matroids. Second, motivated by the observation that many OOS policies for matroids decompose the problem into independent rank-$1$ instances, we provide a meta-theorem which applies to any matroid satisfying this partition property. Leveraging the recent results by Rubinstein et al., we obtain improved competitive guarantees (most by a factor of $2$) for a number of matroids captured by the reduction of Azar et al. Finally, we discuss applications of our SSPIs to the design of mechanisms for multi-dimensional limited information settings with improved revenue and welfare guarantees.

Single-Sample Prophet Inequalities via Greedy-Ordered Selection

2021-01-01
Constantine Caramanis , Paul Dütting , Matthew Faw +6 more
We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single … We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single choice problem [Rubinstein et al., 2020], most existing SSPI results were obtained via an elegant, but inherently lossy, reduction to order-oblivious secretary (OOS) policies [Azar et al., 2014]. Motivated by this discrepancy, we develop an intuitive and versatile greedy-based technique that yields SSPIs directly rather than through the reduction to OOSs. Our results can be seen as generalizing and unifying a number of existing results in the area of prophet and secretary problems. Our algorithms significantly improve on the competitive guarantees for a number of interesting scenarios (including general matching with edge arrivals, bipartite matching with vertex arrivals, and certain matroids), and capture new settings (such as budget additive combinatorial auctions). Complementing our algorithmic results, we also consider mechanism design variants. Finally, we analyze the power and limitations of different SSPI approaches by providing a partial converse to the reduction from SSPI to OOS given by Azar et al.

Single-Sample Prophet Inequalities Revisited

2021-01-01
Constantine Caramanis , Matthew Faw , Orestis Papadigenopoulos +1 more
The study of the prophet inequality problem in the limited information regime was initiated by Azar et al. [SODA'14] in the pursuit of prior-independent posted-price mechanisms. As they show, $O(1)$-competitive … The study of the prophet inequality problem in the limited information regime was initiated by Azar et al. [SODA'14] in the pursuit of prior-independent posted-price mechanisms. As they show, $O(1)$-competitive policies are achievable using only a single sample from the distribution of each agent. A notable portion of their results relies on reducing the design of single-sample prophet inequalities (SSPIs) to that of order-oblivious secretary (OOS) policies. The above reduction comes at the cost of not fully utilizing the available samples. However, to date, this is essentially the only method for proving SSPIs for many combinatorial sets. Very recently, Rubinstein et al. [ITCS'20] give a surprisingly simple algorithm which achieves the optimal competitive ratio for the single-choice SSPI problem $-$ a result which is unobtainable going through the reduction to secretary problems. Motivated by this discrepancy, we study the competitiveness of simple SSPI policies directly, without appealing to results from OOS literature. In this direction, we first develop a framework for analyzing policies against a greedy-like prophet solution. Using this framework, we obtain the first SSPI for general (non-bipartite) matching environments, as well as improved competitive ratios for transversal and truncated partition matroids. Second, motivated by the observation that many OOS policies for matroids decompose the problem into independent rank-$1$ instances, we provide a meta-theorem which applies to any matroid satisfying this partition property. Leveraging the recent results by Rubinstein et al., we obtain improved competitive guarantees (most by a factor of $2$) for a number of matroids captured by the reduction of Azar et al. Finally, we discuss applications of our SSPIs to the design of mechanisms for multi-dimensional limited information settings with improved revenue and welfare guarantees.
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Mix and Match: An Optimistic Tree-Search Approach for Learning Models from Mixture Distributions

2019-01-01
Matthew Faw , Rajat Sen , Karthikeyan Shanmugam +2 more
We consider a covariate shift problem where one has access to several different training datasets for the same learning problem and a small validation set which possibly differs from all … We consider a covariate shift problem where one has access to several different training datasets for the same learning problem and a small validation set which possibly differs from all the individual training distributions. This covariate shift is caused, in part, due to unobserved features in the datasets. The objective, then, is to find the best mixture distribution over the training datasets (with only observed features) such that training a learning algorithm using this mixture has the best validation performance. Our proposed algorithm, ${\sf Mix\&amp;Match}$, combines stochastic gradient descent (SGD) with optimistic tree search and model re-use (evolving partially trained models with samples from different mixture distributions) over the space of mixtures, for this task. We prove simple regret guarantees for our algorithm with respect to recovering the optimal mixture, given a total budget of SGD evaluations. Finally, we validate our algorithm on two real-world datasets.
Coauthor Papers Together
Constantine Caramanis 7
Orestis Papadigenopoulos 4
Emmanouil Pountourakis 4
Sanjay Shakkottai 3
Paul Dütting 2
Stefano Leonardi 2
Rebecca Reiffenhäuser 2
Philip Lazos 2
Federico Fusco 2
Litu Rout 1
Aryan Mokhtari 1
Karthikeyan Shanmugam 1
Rachel Ward 1
Rajat Sen 1
Isidoros Tziotis 1

Bayesian Combinatorial Auctions: Expanding Single Buyer Mechanisms to Many Buyers

2014-01-01
Saeed Alaei
We present a general framework for approximately reducing the mechanism design problem for multiple agents to single agent subproblems in the context of Bayesian combinatorial auctions. Our framework can be … We present a general framework for approximately reducing the mechanism design problem for multiple agents to single agent subproblems in the context of Bayesian combinatorial auctions. Our framework can be applied to any setting which roughly satisfies the following assumptions: (i) agents' types are distributed independently (not necessarily identically), (ii) objective function is additively separable over the agents, and (iii) there are no interagent constraints except for the supply constraints (i.e., that the total allocation of each item should not exceed the supply). Our framework is general in the sense that it makes no direct assumption about agents' valuations, type distributions, or single agent constraints (e.g., budget, incentive compatibility, etc.). We present two generic multiagent mechanisms which use single agent mechanisms as black boxes. If an $\alpha$-approximate single agent mechanism is available for each agent, and assuming no agent ever demands more than $\frac{1}{k}$ of all units of each item, our generic multiagent mechanisms are $\gamma_{k}\alpha$-approximations of the optimal multiagent mechanism, where $\gamma_{k}$ is a constant which is at least $1-\frac{1}{\sqrt{k+3}}$. As a byproduct of our construction, we present a generalization of prophet inequalities where both gambler and prophet are allowed to pick $k$ numbers each to receive a reward equal to their sum. Finally, we use our framework to obtain multiagent mechanisms with improved approximation factor for several settings from the literature.
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Prophet Secretary for Combinatorial Auctions and Matroids

2018-01-01
Soheil Ehsani , MohammadTaghi Hajiaghayi , Thomas Keßelheim +1 more
The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items … The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg [33] and Feldman et al. [17] show that for adversarial arrival order of random variables the optimal prophet inequalities give a 1/2-approximation. For many settings, however, it's conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the 1/2-approximation and obtain (1 – 1/e)-approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan [45] and Esfandiari et al. [15] who worked in the special cases where we can fully control the arrival order or when there is only a single item.Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.
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Comparison of Threshold Stop Rules and Maximum for Independent Nonnegative Random Variables

1984-11-01
Ester Samuel‐Cahn
Let $X_i \geq 0$ be independent, $i = 1, \cdots, n$, and $X^\ast_n = \max(X_1, \cdots, X_n)$. Let $t(c) (s(c))$ be the threshold stopping rule for $X_1, \cdots, X_n$, defined … Let $X_i \geq 0$ be independent, $i = 1, \cdots, n$, and $X^\ast_n = \max(X_1, \cdots, X_n)$. Let $t(c) (s(c))$ be the threshold stopping rule for $X_1, \cdots, X_n$, defined by $t(c) = \text{smallest} i$ for which $X_i \geq c(s(c) = \text{smallest} i$ for which $X_i > c), = n$ otherwise. Let $m$ be a median of the distribution of $X^\ast_n$. It is shown that for every $n$ and $\underline{X}$ either $EX^\ast_n \leq 2EX_{t(m)}$ or $EX^\ast_n \leq 2EX_{s(m)}$. This improves previously known results, [1], [4]. Some results for i.i.d. $X_i$ are also included.
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Matroid prophet inequalities

2012-05-19
Robert Kleinberg , S. Matthew Weinberg
Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent … Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a "prophet" who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the prophet are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case of rank-one matroids. Generalizing the result still further, we show that under an intersection of $p$ matroid constraints, the prophet's reward exceeds the gambler's by a factor of at most $O(p)$, and this factor is also tight.
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The two-sided game of googol and sample-based prophet inequalities

2020-01-05
José R. Correa , Andrés Cristi , Boris Epstein +1 more
The secretary problem or the game of Googol are classic models for online selection problems that have received significant attention in the last five decades. In this paper we consider … The secretary problem or the game of Googol are classic models for online selection problems that have received significant attention in the last five decades. In this paper we consider a variant of the problem and explore its connections to data-driven online selection. Specifically, we are given n cards with arbitrary nonnegative numbers written on both sides. The cards are randomly placed on n consecutive positions on a table, and for each card, the visible side is also selected at random. The player sees the visible side of all cards and wants to select the card with the maximum hidden value. To this end, the player flips the first card, sees its hidden value and decides whether to pick it or drop it and continue with the next card. We study algorithms for two natural objectives. In the first one, similar to the secretary problem, the player wants to maximize the probability of selecting the maximum hidden value. We show that this can be done with probability at least 0.45292. In the second objective, similar to the prophet inequality, the player wants to maximize the expectation of the selected hidden value. Here we show a guarantee of at least 0.63518 with respect to the expected maximum hidden value. Our algorithms result from combining three basic strategies. One is to stop whenever we see a value larger than the initial n visible numbers. The second one is to stop the first time the last flipped card's value is the largest of the currently n visible numbers in the table. And the third one is similar to the latter but to stop it additionally requires that the last flipped value is larger than the value on the other side of its card. We apply our results to the prophet secretary problem with unknown distributions, but with access to a single sample from each distribution. In particular, our guarantee improves upon 1 − 1/e for this problem, which is the currently best known guarantee and only works for the i.i.d. prophet inequality with samples.

Beyond matroids: secretary problem and prophet inequality with general constraints

2016-06-10
Aviad Rubinstein
We study generalizations of the ``Prophet Inequality'' and ``Secretary Problem'', where the algorithm is restricted to an arbitrary downward-closed set system. For 0,1 values, we give O(n)-competitive algorithms for both … We study generalizations of the ``Prophet Inequality'' and ``Secretary Problem'', where the algorithm is restricted to an arbitrary downward-closed set system. For 0,1 values, we give O(n)-competitive algorithms for both problems. This is close to the Omega(n/log n) lower bound due to Babaioff, Immorlica, and Kleinberg. For general values, our results translate to O(log(n) log(r))-competitive algorithms, where r is the cardinality of the largest feasible set. This resolves (up to the O(loglog(n) log(r)) factor) an open question posed to us by Bobby Kleinberg.
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Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Nonstochastic Inputs

2020-01-01
Paul Dütting , Michal Feldman , Thomas Keßelheim +1 more
We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold … We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms and is used to derive new and improved results for combinatorial markets (with and without complements), multidimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees.
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Prophet Inequalities for I.I.D. Random Variables from an Unknown Distribution

2019-06-17
José R. Correa , Paul Dütting , Felix Fischer +1 more
A central object in optimal stopping theory is the single-choice prophet inequality for independent, identically distributed random variables: given a sequence of random variables X1, ..., Xn drawn independently from … A central object in optimal stopping theory is the single-choice prophet inequality for independent, identically distributed random variables: given a sequence of random variables X1, ..., Xn drawn independently from a distribution F, the goal is to choose a stopping time τ so as to maximize α such that for all distributions F we have E[Xτ]≥α•E[maxt Xt]. What makes this problem challenging is that the decision whether τ=t may only depend on the values of the random variables X1, ..., Xt and on the distribution F. For a long time the best known bound for the problem had been α≥1-1/e≅0.632, but quite recently a tight bound of α≅0.745 was obtained. The case where F is unknown, such that the decision whether τ=t may depend only on the values of the random variables X1, ..., Xt, is equally well motivated but has received much less attention. A straightforward guarantee for this case of α≥1-1/e≅0.368 can be derived from the solution to the secretary problem, where an arbitrary set of values arrive in random order and the goal is to maximize the probability of selecting the largest value. We show that this bound is in fact tight. We then investigate the case where the stopping time may additionally depend on a limited number of samples from~F, and show that even with o(n) samples α≥1/e. On the other hand, n samples allow for a significant improvement, while O(n2) samples are equivalent to knowledge of the distribution: specifically, with n samples α≥1-1/e≅0.632 and α≥ln(2)≅0.693, and with O(n2) samples α≥0.745-ε for any ε>0.
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The Simulated Greedy Algorithm for Several Submodular Matroid Secretary Problems

2015-07-05
Tengyu Ma , Bo Tang , Yajun Wang
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Sample-Driven Optimal Stopping: From the Secretary Problem to the i.i.d. Prophet Inequality

2023-04-17
José R. Correa , Andrés Cristi , Boris Epstein +1 more
We take a unifying approach to single selection optimal stopping problems with random arrival order and independent sampling of items. In the problem we consider, a decision maker (DM) initially … We take a unifying approach to single selection optimal stopping problems with random arrival order and independent sampling of items. In the problem we consider, a decision maker (DM) initially gets to sample each of N items independently with probability p, and can observe the relative rankings of these sampled items. Then, the DM faces the remaining items in an online fashion, observing the relative rankings of all revealed items. While scanning the sequence the DM makes irrevocable stop/continue decisions and her reward for stopping the sequence facing the item with rank i is Y i . The goal of the DM is to maximize her reward. We start by studying the case in which the values Y i are known to the DM, and then move to the case in which these values are adversarial. For the former case we are able to recover several classic results in the area, thus giving a unifying framework for single selection optimal stopping. For the latter, we pin down the optimal algorithm, obtaining the optimal competitive ratios for all values of p. Funding: This work was partially supported by The Center for Mathematical Modeling at the University of Chile (ANID FB210005), Grant Anillo Information and Computation in Market Design (ANID ACT210005), FONDECYT 1220054 and 1181180, and a Meta Research PhD Fellowship.
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Algorithms for Secretary Problems on Graphs and Hypergraphs

2009-01-01
Nitish Korula , Martin Pál
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Matroid Secretary Problem in the Random-Assignment Model

2013-01-01
José A. Soto
The matroid secretary problem admits several variants according to the order in which the matroid's elements are presented and how the elements are assigned weights. As the main result of … The matroid secretary problem admits several variants according to the order in which the matroid's elements are presented and how the elements are assigned weights. As the main result of this article, we devise the first constant competitive algorithm for the model in which both the order and the weight assignment are selected uniformly at random, achieving a competitive ratio of approximately $5.7187$. This result is based on the nontrivial fact that every matroid can be approximately decomposed into uniformly dense minors. Based on a preliminary version of this work, Oveis Gharan and Vondrák [Proceedings of the 19th Annual European Symposium on Algorithms, ESA, 2011, pp. 335--346] devised a $40e/(e-1)$-competitive algorithm for the stronger random-assignment adversarial-order model. In this article we present an alternative algorithm achieving a competitive ratio of $16e/(e-1)$. As additional results, we obtain new algorithms for the standard model of the matroid secretary problem: the adversarial-assignment random-order model. We present an $O(\log r)$-competitive algorithm for general matroids which, unlike previous ones, uses only comparisons among seen elements. We also present constant competitive algorithms for various matroid classes, such as column-sparse representable matroids and low-density matroids. The latter class includes, as a special case, the duals of graphic matroids.
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Optimistic Optimization of a Deterministic Function without the Knowledge of its Smoothness

2011-12-12
Rémi Munos
We consider a global optimization problem of a deterministic function f in a semi-metric space, given a finite budget of n evaluations. The function f is assumed to be locally … We consider a global optimization problem of a deterministic function f in a semi-metric space, given a finite budget of n evaluations. The function f is assumed to be locally smooth (around one of its global maxima) with respect to a semi-metric l We describe two algorithms based on optimistic exploration that use a hierarchical partitioning of the space at all scales. A first contribution is an algorithm, DOO, that requires the knowledge of l. We report a finite-sample performance bound in terms of a measure of the quantity of near-optimal states. We then define a second algorithm, SOO, which does not require the knowledge of the semi-metric l under which f is smooth, and whose performance is almost as good as DOO optimally-fitted.

A Stochastic Gradient Method with an Exponential Convergence Rate for Finite Training Sets

2012-01-01
Nicolas Le Roux , Mark Schmidt , Francis Bach
We propose a new stochastic gradient method for optimizing the sum of a finite set of smooth functions, where the sum is strongly convex. While standard stochastic gradient methods converge … We propose a new stochastic gradient method for optimizing the sum of a finite set of smooth functions, where the sum is strongly convex. While standard stochastic gradient methods converge at sublinear rates for this problem, the proposed method incorporates a memory of previous gradient values in order to achieve a linear convergence rate. In a machine learning context, numerical experiments indicate that the new algorithm can dramatically outperform standard algorithms, both in terms of optimizing the training error and reducing the test error quickly.
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Black-box optimization of noisy functions with unknown smoothness

2015-01-01
Jean-Bastien Grill , Michal Vaľko , Rémi Munos
We study the problem of black-box optimization of a function f of any dimension, given function evaluations perturbed by noise. The function is assumed to be locally smooth around one … We study the problem of black-box optimization of a function f of any dimension, given function evaluations perturbed by noise. The function is assumed to be locally smooth around one of its global optima, but this smoothness is unknown. Our contribution is an adaptive optimization algorithm, POO or parallel optimistic optimization, that is able to deal with this setting. POO performs almost as well as the best known algorithms requiring the knowledge of the smoothness. Furthermore, POO works for a larger class of functions than what was previously considered, especially for functions that are difficult to optimize, in a very precise sense. We provide a finite-time analysis of POO's performance, which shows that its error after n evaluations is at most a factor of √ln n away from the error of the best known optimization algorithms using the knowledge of the smoothness.
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Algorithmic stability for adaptive data analysis

2016-06-10
Raef Bassily , Kobbi Nissim , Adam Smith +3 more
Adaptivity is an important feature of data analysis - the choice of questions to ask about a dataset often depends on previous interactions with the same dataset. However, statistical validity … Adaptivity is an important feature of data analysis - the choice of questions to ask about a dataset often depends on previous interactions with the same dataset. However, statistical validity is typically studied in a nonadaptive model, where all questions are specified before the dataset is drawn. Recent work by Dwork et al. (STOC, 2015) and Hardt and Ullman (FOCS, 2014) initiated a general formal study of this problem, and gave the first upper and lower bounds on the achievable generalization error for adaptive data analysis.
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Beating 1-1/e for ordered prophets

2017-06-15
Melika Abolhassani , Soheil Ehsani , Hossein Esfandiari +3 more
Hill and Kertz studied the prophet inequality on iid distributions [The Annals of Probability 1982]. They proved a theoretical bound of 1 - 1/e on the approximation factor of their … Hill and Kertz studied the prophet inequality on iid distributions [The Annals of Probability 1982]. They proved a theoretical bound of 1 - 1/e on the approximation factor of their algorithm. They conjectured that the best approximation factor for arbitrarily large n is 1/1+1/e ≃ 0.731. This conjecture remained open prior to this paper for over 30 years. In this paper we present a threshold-based algorithm for the prophet inequality with n iid distributions. Using a nontrivial and novel approach we show that our algorithm is a 0.738-approximation algorithm. By beating the bound of 1/1+1/e, this refutes the conjecture of Hill and Kertz. Moreover, we generalize our results to non-uniform distributions and discuss its applications in mechanism design.
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Climbing a shaky ladder: Better adaptive risk estimation.

2017-06-08
Moritz Hardt
We revisit the \emph{leaderboard problem} introduced by Blum and Hardt (2015) in an effort to reduce overfitting in machine learning benchmarks. We show that a randomized version of their Ladder … We revisit the \emph{leaderboard problem} introduced by Blum and Hardt (2015) in an effort to reduce overfitting in machine learning benchmarks. We show that a randomized version of their Ladder algorithm achieves leaderboard error O(1/n^{0.4}) compared with the previous best rate of O(1/n^{1/3}). Short of proving that our algorithm is optimal, we point out a major obstacle toward further progress. Specifically, any improvement to our upper bound would lead to asymptotic improvements in the general adaptive estimation setting as have remained elusive in recent years. This connection also directly leads to lower bounds for specific classes of algorithms. In particular, we exhibit a new attack on the leaderboard algorithm that both theoretically and empirically distinguishes between our algorithm and previous leaderboard algorithms.

Multi-Fidelity Black-Box Optimization with Hierarchical Partitions

2018-07-03
Rajat Sen , Kirthevasan Kandasamy , Sanjay Shakkottai

Adaptive black-box optimization got easier: HCT only needs local smoothness

2018-10-01
Xuedong Shang , Emilie Kaufmann , Michal Vaľko
Hierarchical bandits is an approach for global optimization of extremely irregular functions. This paper provides new elements regarding POO, an adaptive meta-algorithm that does not require the knowledge of local … Hierarchical bandits is an approach for global optimization of extremely irregular functions. This paper provides new elements regarding POO, an adaptive meta-algorithm that does not require the knowledge of local smoothness of the target function. We first highlight the fact that the subroutine algorithm used in POO should have a small regret under the assumption of local smoothness with respect to the chosen partitioning, which is unknown if it is satisfied by the standard subroutine HOO. In this work, we establish such regret guarantee for HCT, which is another hierarchical optimistic optimization algorithm that needs to know the smoothness. This confirms the validity of POO. We show that POO can be used with HCT as a subroutine with a regret upper bound that matches the one of best-known algorithms using the knowledge of smoothness up to a √ log n factor.
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Tight Analyses for Non-Smooth Stochastic Gradient Descent

2018-01-01
Nicholas J. A. Harvey , Christopher Liaw , Yaniv Plan +1 more
Consider the problem of minimizing functions that are Lipschitz and strongly convex, but not necessarily differentiable. We prove that after $T$ steps of stochastic gradient descent, the error of the … Consider the problem of minimizing functions that are Lipschitz and strongly convex, but not necessarily differentiable. We prove that after $T$ steps of stochastic gradient descent, the error of the final iterate is $O(\log(T)/T)$ with high probability. We also construct a function from this class for which the error of the final iterate of deterministic gradient descent is $Ω(\log(T)/T)$. This shows that the upper bound is tight and that, in this setting, the last iterate of stochastic gradient descent has the same general error rate (with high probability) as deterministic gradient descent. This resolves both open questions posed by Shamir (2012). An intermediate step of our analysis proves that the suffix averaging method achieves error $O(1/T)$ with high probability, which is optimal (for any first-order optimization method). This improves results of Rakhlin (2012) and Hazan and Kale (2014), both of which achieved error $O(1/T)$, but only in expectation, and achieved a high probability error bound of $O(\log \log(T)/T)$, which is suboptimal. We prove analogous results for functions that are Lipschitz and convex, but not necessarily strongly convex or differentiable. After $T$ steps of stochastic gradient descent, the error of the final iterate is $O(\log(T)/\sqrt{T})$ with high probability, and there exists a function for which the error of the final iterate of deterministic gradient descent is $Ω(\log(T)/\sqrt{T})$.

Agnostic Federated Learning

2019-01-01
Mehryar Mohri , Gary Sivek , Ananda Theertha Suresh
A key learning scenario in large-scale applications is that of federated learning, where a centralized model is trained based on data originating from a large number of clients. We argue … A key learning scenario in large-scale applications is that of federated learning, where a centralized model is trained based on data originating from a large number of clients. We argue that, with the existing training and inference, federated models can be biased towards different clients. Instead, we propose a new framework of agnostic federated learning, where the centralized model is optimized for any target distribution formed by a mixture of the client distributions. We further show that this framework naturally yields a notion of fairness. We present data-dependent Rademacher complexity guarantees for learning with this objective, which guide the definition of an algorithm for agnostic federated learning. We also give a fast stochastic optimization algorithm for solving the corresponding optimization problem, for which we prove convergence bounds, assuming a convex loss function and hypothesis set. We further empirically demonstrate the benefits of our approach in several datasets. Beyond federated learning, our framework and algorithm can be of interest to other learning scenarios such as cloud computing, domain adaptation, drifting, and other contexts where the training and test distributions do not coincide.

Combinatorial Prophet Inequalities

2017-01-01
Aviad Rubinstein , Sahil Singla
We introduce a novel framework of Prophet Inequalities for combinatorial valuation functions. For a (non-monotone) submodular objective function over an arbitrary matroid feasibility constraint, we give an O(1)-competitive algorithm. For … We introduce a novel framework of Prophet Inequalities for combinatorial valuation functions. For a (non-monotone) submodular objective function over an arbitrary matroid feasibility constraint, we give an O(1)-competitive algorithm. For a monotone subadditive objective function over an arbitrary downward- closed feasibility constraint, we give an O(log n log2 r)- competitive algorithm (where r is the cardinality of the largest feasible subset).Inspired by the proof of our subadditive prophet inequality, we also obtain an O(log n · log2 r)-competitive algorithm for the Secretary Problem with a monotone subadditive objective function subject to an arbitrary downward-closed feasibility constraint. Even for the special case of a cardinality feasibility constraint, our algorithm circumvents an lower bound by Bateni, Hajiaghayi, and Zadimoghaddam [10] in a restricted query model.En route to our submodular prophet inequality, we prove a technical result of independent interest: we show a variant of the Correlation Gap Lemma [14, 1] for nonmonotone submodular functions.
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Combinatorial Auctions via Posted Prices

2014-12-22
Michal Feldman , Nick Gravin , Brendan Lucier
We study anonymous posted price mechanisms for combinatorial auctions in a Bayesian framework. In a posted price mechanism, item prices are posted, then the consumers approach the seller sequentially in … We study anonymous posted price mechanisms for combinatorial auctions in a Bayesian framework. In a posted price mechanism, item prices are posted, then the consumers approach the seller sequentially in an arbitrary order, each purchasing her favorite bundle from among the unsold items at the posted prices. These mechanisms are simple, transparent and trivially dominant strategy incentive compatible (DSIC).We show that when agent preferences are fractionally subadditive (which includes all submodular functions), there always exist prices that, in expectation, obtain at least half of the optimal welfare. Our result is constructive: given black-box access to a combinatorial auction algorithm A, sample access to the prior distribution, and appropriate query access to the sampled valuations, one can compute, in polytime, prices that guarantee at least half of the expected welfare of A. As a corollary, we obtain the first polytime (in n and m) constant-factor DSIC mechanism for Bayesian submodular combinatorial auctions, given access to demand query oracles. Our results also extend to valuations with complements, where the approximation factor degrades linearly with the level of complementarity.
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Prophet Inequalities with Limited Information

2013-12-18
Pablo Azar , Robert Kleinberg , S. Matthew Weinberg
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2014 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Prophet Inequalities with Limited InformationPablo D. Azar, Robert Kleinberg, and S. Matthew WeinbergPablo D. … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2014 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Prophet Inequalities with Limited InformationPablo D. Azar, Robert Kleinberg, and S. Matthew WeinbergPablo D. Azar, Robert Kleinberg, and S. Matthew Weinbergpp.1358 - 1377Chapter DOI:https://doi.org/10.1137/1.9781611973402.100PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract In the classical prophet inequality, a gambler observes a sequence of stochastic rewards V1, …, Vn and must decide, for each reward Vi, whether to keep it and stop the game or to forfeit the reward forever and reveal the next value Vi. The gambler's goal is to obtain a constant fraction of the expected reward that the optimal offline algorithm would get. Recently, prophet inequalities have been generalized to settings where the gambler can choose k items, and, more generally, where he can choose any independent set in a matroid. However, all the existing algorithms require the gambler to know the distribution from which the rewards V1, …, Vn are drawn. The assumption that the gambler knows the distribution from which V1, …, Vn are drawn is very strong. Instead, we work with the much simpler assumption that the gambler only knows a few samples from this distribution. We construct the first single-sample prophet inequalities for many settings of interest, whose guarantees all match the best possible asymptotically, even with full knowledge of the distribution. Specifically, we provide a novel single-sample algorithm when the gambler can choose any k elements whose analysis is based on random walks with limited correlation. In addition, we provide a black-box method for converting specific types of solutions to the related secretary problem to single-sample prophet inequalities, and apply it to several existing algorithms. Finally, we provide a constant-sample prophet inequality for constant-degree bipartite matchings. In addition, we apply these results to design the first posted-price and multi-dimensional auction mechanisms with limited information in settings with asymmetric bidders. Connections between prophet inequalities and posted-price mechanisms are already known, but applying the existing framework requires knowledge of the underlying distributions, as well as the so-called "virtual values" even when the underlying prophet inequalities do not. We therefore provide an extension of this framework that bypasses virtual values altogether, allowing our mechanisms to take full advantage of the limited information required by our new prophet inequalities. Previous chapter Next chapter RelatedDetails Published:2014ISBN:978-1-61197-338-9eISBN:978-1-61197-340-2 https://doi.org/10.1137/1.9781611973402Book Series Name:ProceedingsBook Code:PRDA14Book Pages:viii + 1885
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The Ladder: A Reliable Leaderboard for Machine Learning Competitions

2015-02-16
Avrim Blum , Moritz Hardt
The organizer of a machine learning competition faces problem of maintaining an accurate that faithfully represents quality of best submission of each competing team. What makes this estimation problem particularly … The organizer of a machine learning competition faces problem of maintaining an accurate that faithfully represents quality of best submission of each competing team. What makes this estimation problem particularly challenging is its sequential and adaptive nature. As participants are allowed to repeatedly evaluate their submissions on leaderboard, they may begin to overfit to holdout data that supports leaderboard. Few theoretical results give actionable advice on how to design a reliable leaderboard. Existing approaches therefore often resort to poorly understood heuristics such as limiting bit precision of answers and rate of re-submission. In this work, we introduce a notion of leaderboard accuracy tailored to format of a competition. We introduce a natural algorithm called the Ladder and demonstrate that it simultaneously supports strong theoretical guarantees in a fully adaptive model of estimation, withstands practical adversarial attacks, and achieves high utility on real submission files from an actual competition hosted by Kaggle. Notably, we are able to sidestep a powerful recent hardness result for adaptive risk estimation that rules out algorithms such as ours under a seemingly very similar notion of accuracy. On a practical note, we provide a completely parameter-free variant of our algorithm that can be deployed in a real competition with no tuning required whatsoever.

Metric-Optimized Example Weights

2018-01-01
Sen Zhao , Mahdi Milani Fard , Harikrishna Narasimhan +1 more
Real-world machine learning applications often have complex test metrics, and may have training and test data that are not identically distributed. Motivated by known connections between complex test metrics and … Real-world machine learning applications often have complex test metrics, and may have training and test data that are not identically distributed. Motivated by known connections between complex test metrics and cost-weighted learning, we propose addressing these issues by using a weighted loss function with a standard loss, where the weights on the training examples are learned to optimize the test metric on a validation set. These metric-optimized example weights can be learned for any test metric, including black box and customized ones for specific applications. We illustrate the performance of the proposed method on diverse public benchmark datasets and real-world applications. We also provide a generalization bound for the method.

Online Contention Resolution Schemes

2015-12-21
Moran Feldman , Ola Svensson , Rico Zenklusen
We introduce a new rounding technique designed for online optimization problems, which is related to contention resolution schemes, a technique initially introduced in the context of submodular function maximization. Our … We introduce a new rounding technique designed for online optimization problems, which is related to contention resolution schemes, a technique initially introduced in the context of submodular function maximization. Our rounding technique, which we call online contention resolution schemes (OCRSs), is applicable to many online selection problems, including Bayesian online selection, oblivious posted pricing mechanisms, and stochastic probing models. It allows for handling a wide set of constraints, and shares many strong properties of offline contention resolution schemes. In particular, OCRSs for different constraint families can be combined to obtain an OCRS for their intersection. Moreover, we can approximately maximize submodular functions in the online settings we consider.We, thus, get a broadly applicable framework for several online selection problems, which improves on previous approaches in terms of the types of constraints that can be handled, the objective functions that can be dealt with, and the assumptions on the strength of the adversary. Furthermore, we resolve two open problems from the literature; namely, we present the first constant-factor constrained oblivious posted price mechanism for matroid constraints, and the first constant-factor algorithm for weighted stochastic probing with deadlines.
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A Framework for the Secretary Problem on the Intersection of Matroids

2018-01-01
Moran Feldman , Ola Svensson , Rico Zenklusen
The secretary problem became one of the most prominent online selection problems due to its numerous applications in online mechanism design. The task is to select a maximum weight subset … The secretary problem became one of the most prominent online selection problems due to its numerous applications in online mechanism design. The task is to select a maximum weight subset of elements subject to given constraints, where elements arrive one-by-one in random order, revealing a weight upon arrival. The decision whether to select an element has to be taken immediately after its arrival. The different applications that map to the secretary problem ask for different constraint families to be handled. The most prominent ones are matroid constraints, which both capture many relevant settings and admit strongly competitive secretary algorithms. However, dealing with more involved constraints proved to be much more difficult, and strong algorithms are known only for a few specific settings. In this paper, we present a general framework for dealing with the secretary problem over the intersection of several matroids. This framework allows us to combine and exploit the large set of matroid secretary algorithms known in the literature. As one consequence, we get constant-competitive secretary algorithms over the intersection of any constant number of matroids whose corresponding (single-)matroid secretary problems are currently known to have a constant-competitive algorithm. Moreover, we show that our results extend to submodular objectives.MSC codesmatroid secretary problemmatroid intersectiononline algorithms
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Optimization Methods for Large-Scale Machine Learning

2018-01-01
Léon Bottou , Frank E. Curtis , Jorge Nocedal
This paper provides a review and commentary on the past, present, and future of numerical optimization algorithms in the context of machine learning applications. Through case studies on text classification … This paper provides a review and commentary on the past, present, and future of numerical optimization algorithms in the context of machine learning applications. Through case studies on text classification and the training of deep neural networks, we discuss how optimization problems arise in machine learning and what makes them challenging. A major theme of our study is that large-scale machine learning represents a distinctive setting in which the stochastic gradient (SG) method has traditionally played a central role while conventional gradient-based nonlinear optimization techniques typically falter. Based on this viewpoint, we present a comprehensive theory of a straightforward, yet versatile SG algorithm, discuss its practical behavior, and highlight opportunities for designing algorithms with improved performance. This leads to a discussion about the next generation of optimization methods for large-scale machine learning, including an investigation of two main streams of research on techniques that diminish noise in the stochastic directions and methods that make use of second-order derivative approximations.
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Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Non-Stochastic Inputs

2017-10-01
Paul Dütting , Michal Feldman , Thomas Keßelheim +1 more
We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold … We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms, and is used to derive new and improved results for combinatorial markets (with and without complements), multi-dimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees.
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Strong Algorithms for the Ordinal Matroid Secretary Problem

2018-01-01
José A. Soto , Abner Turkieltaub , Víctor Verdugo
In the ordinal Matroid Secretary Problem (MSP), elements from a weighted matroid are presented in random order to an algorithm that must incrementally select a large weight independent set. However, … In the ordinal Matroid Secretary Problem (MSP), elements from a weighted matroid are presented in random order to an algorithm that must incrementally select a large weight independent set. However, the algorithm can only compare pairs of revealed elements without using its numerical value. An algorithm is α probability-competitive if every element from the optimum appears with probability 1/α in the output. We present a technique to design algorithms with strong probability-competitive ratios, improving the guarantees for almost every matroid class considered in the literature: e.g., we get ratios of 4 for graphic matroids (improving on 2e by Korula and Pál [ICALP 2009]) and of 5.19 for laminar matroids (improving on 9.6 by Ma et al. [THEOR COMPUT SYST 2016]). We also obtain new results for superclasses of k column sparse matroids, for hypergraphic matroids, certain gammoids and graph packing matroids, and a probability-competitive algorithm for uniform matroids of rank ρ based on Kleinberg's utility-competitive algorithm [SODA 2005] for that class. Our second contribution are algorithms for the ordinal MSP on arbitrary matroids of rank ρ. We devise an O(log ρ) probability-competitive algorithm and an O(log log ρ) ordinal-competitive algorithm, a weaker notion of competitiveness but stronger than the utility variant. These are based on the O(log log ρ) utility-competitive algorithm by Feldman et al. [SODA 2015].
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Model Similarity Mitigates Test Set Overuse

2019-01-01
Horia Mania , J. J. Miller , Ludwig Schmidt +2 more
Excessive reuse of test data has become commonplace in today's machine learning workflows. Popular benchmarks, competitions, industrial scale tuning, among other applications, all involve test data reuse beyond guidance by … Excessive reuse of test data has become commonplace in today's machine learning workflows. Popular benchmarks, competitions, industrial scale tuning, among other applications, all involve test data reuse beyond guidance by statistical confidence bounds. Nonetheless, recent replication studies give evidence that popular benchmarks continue to support progress despite years of extensive reuse. We proffer a new explanation for the apparent longevity of test data: Many proposed models are similar in their predictions and we prove that this similarity mitigates overfitting. Specifically, we show empirically that models proposed for the ImageNet ILSVRC benchmark agree in their predictions well beyond what we can conclude from their accuracy levels alone. Likewise, models created by large scale hyperparameter search enjoy high levels of similarity. Motivated by these empirical observations, we give a non-asymptotic generalization bound that takes similarity into account, leading to meaningful confidence bounds in practical settings.

Competitive Analysis with a Sample and the Secretary Problem

2019-12-23
Haim Kaplan , David Naori , Danny Raz
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)Competitive Analysis with a Sample and the Secretary ProblemHaim Kaplan, David Naori, and Danny RazHaim … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)Competitive Analysis with a Sample and the Secretary ProblemHaim Kaplan, David Naori, and Danny RazHaim Kaplan, David Naori, and Danny Razpp.2082 - 2095Chapter DOI:https://doi.org/10.1137/1.9781611975994.128PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We extend the standard online worst-case model to accommodate past experience which is available to the online player in many practical scenarios. We do this by revealing a random sample of the adversarial input to the online player ahead of time. The online player competes with the expected optimal value on the part of the input that arrives online. Our model bridges between existing online stochastic models (e.g., items are drawn i.i.d. from a distribution) and the online worst-case model. We also extend in a similar manner (by revealing a sample) the online random-order model. We study the classical secretary problem in our new models. In the worst-case model we present a simple online algorithm with optimal competitive-ratio for any sample size. In the random-order model, we also give a simple online algorithm with an almost tight competitive-ratio for small sample sizes. Interestingly, we prove that for a large enough sample, no algorithm can be simultaneously optimal both in the worst-cast and random-order models. Previous chapter Next chapter RelatedDetails Published:2020eISBN:978-1-61197-599-4 https://doi.org/10.1137/1.9781611975994Book Series Name:ProceedingsBook Code:PRDA20Book Pages:xxii + 3011
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An $O(\log \log m)$ Prophet Inequality for Subadditive Combinatorial Auctions

2022-05-19
Paul Dütting , Thomas Keßelheim , Brendan Lucier
Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case … Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here $n$ agents with subadditive valuation functions compete for the assignment of $m$ items. The goal is to find an allocation of the items that maximizes the total value of the assignment. The question is whether there exists a prophet inequality for this problem that significantly beats the best known approximation factor of $O(\log m)$. We make major progress on this question by providing an $O(\log \log m)$ prophet inequality. Our proof goes through a novel primal-dual approach. It is also constructive, resulting in an online policy that takes the form of static and anonymous item prices that can be computed in polynomial time given appropriate query access to the valuations. As an application of our approach, we construct a simple and incentive compatible mechanism based on posted prices that achieves an $O(\log \log m)$ approximation to the optimal revenue for subadditive valuations under an item-independence assumption.
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Prophet Inequalities with Linear Correlations and Augmentations

2020-07-09
Nicole Immorlica , Sahil Singla , Bo Waggoner
In a classical online decision problem, a decision-maker who is trying to maximize her value inspects a sequence of arriving items to learn their values (drawn from known distributions), and … In a classical online decision problem, a decision-maker who is trying to maximize her value inspects a sequence of arriving items to learn their values (drawn from known distributions), and decides when to stop the process by taking the current item. The goal is to prove a "prophet inequality": that she can do approximately as well as a prophet with foreknowledge of all the values. In this work, we investigate this problem when the values are allowed to be correlated. Since non-trivial guarantees are impossible for arbitrary correlations, we consider a natural "linear" correlation structure introduced by Bateni et al. [ESA'15] as a generalization of the common-base value model of Chawla et al. [GEB'15].
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Online Stochastic Max-Weight Matching: Prophet Inequality for Vertex and Edge Arrival Models

2020-07-09
Tomer Ezra , Michal Feldman , Nick Gravin +1 more
We provide prophet inequality algorithms for online weighted matching in general (non-bipartite) graphs, under two well-studied arrival models, namely edge arrival and vertex arrival. The weight of each edge is … We provide prophet inequality algorithms for online weighted matching in general (non-bipartite) graphs, under two well-studied arrival models, namely edge arrival and vertex arrival. The weight of each edge is drawn independently from an a-priori known probability distribution. Under edge arrival, the weight of each edge is revealed upon arrival, and the algorithm decides whether to include it in the matching or not. Under vertex arrival, the weights of all edges from the newly arriving vertex to all previously arrived vertices are revealed, and the algorithm decides which of these edges, if any, to include in the matching. To study these settings, we introduce a novel unified framework of batched prophet inequalities that captures online settings where elements arrive in batches; in particular it captures matching under the two aforementioned arrival models. Our algorithms rely on the construction of suitable online contention resolution schemes (OCRS). We first extend the framework of OCRS to batched-OCRS, we then establish a reduction from batched prophet inequality to batched OCRS, and finally we construct batched OCRSs with selectable ratios of 0.337 and 0.5 for edge and vertex arrival models, respectively. Both results improve the state of the art for the corresponding settings. For vertex arrival, our result is tight. Interestingly, pricing-based prophet inequalities with comparable competitive ratios are unknown.
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Unknown I.I.D. prophets: better bounds, streaming algorithms, and a new impossibility

2020-07-12
José R. Correa , Paul Dütting , Felix Fischer +2 more
A prophet inequality states, for some α ∈ [0, 1], that the expected value achievable by a gambler who sequentially observes random variables X1, . . . , Xn and … A prophet inequality states, for some α ∈ [0, 1], that the expected value achievable by a gambler who sequentially observes random variables X1, . . . , Xn and selects one of them is at least an α fraction of the maximum value in the sequence. We obtain three distinct improvements for a setting that was first studied by Correa et al. (EC, 2019) and is particularly relevant to modern applications in algorithmic pricing. In this setting, the random variables are i.i.d. from an unknown distribution and the gambler has access to an additional βn samples for some β ≥ 0. We first give improved lower bounds on α for a wide range of values of β; specifically, α ≥ (1 + β)/e when β ≤ 1/(e − 1), which is tight, and α ≥ 0.648 when β = 1, which improves on a bound of around 0.635 due to Correa et al. (SODA, 2020). Adding to their practical appeal, specifically in the context of algorithmic pricing, we then show that the new bounds can be obtained even in a streaming model of computation and thus in situations where the use of relevant data is complicated by the sheer amount of data available. We finally establish that the upper bound of 1/e for the case without samples is robust to additional information about the distribution, and applies also to sequences of i.i.d. random variables whose distribution is itself drawn, according to a known distribution, from a finite set of known candidate distributions. This implies a tight prophet inequality for exchangeable sequences of random variables, answering a question of Hill and Kertz (Contemporary Mathematics, 1992), but leaves open the possibility of better guarantees when the number of candidate distributions is small, a setting we believe is of strong interest to applications.

An O(log log m) prophet inequality for subadditive combinatorial auctions

2020-12-02
Paul Dütting , Thomas Keßelheim , Brendan Lucier
We survey the main results from [Dütting, Kesselheim, and Lucier 2020]: 1 a simple posted-price mechanism for subadditive combinatorial auctions with m items that achieves an O (log log m … We survey the main results from [Dütting, Kesselheim, and Lucier 2020]: 1 a simple posted-price mechanism for subadditive combinatorial auctions with m items that achieves an O (log log m ) approximation to the optimal welfare, plus a variant with entry fees that approximates revenue. These are based on a novel subadditive prophet inequality.
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Constrained-Order Prophet Inequalities

2021-01-01
Makis Arsenis , Odysseas Drosis , Robert Kleinberg
Free order prophet inequalities bound the ratio between the expected value obtained by two parties each selecting one value from a set of independent random variables: a "prophet" who knows … Free order prophet inequalities bound the ratio between the expected value obtained by two parties each selecting one value from a set of independent random variables: a "prophet" who knows the value of each variable and may select the maximum one, and a "gambler" who is free to choose the order in which to observe the values but must select one of them immediately after observing it, without knowing what values will be sampled for the unobserved variables. It is known that the gambler can always ensure an expected payoff at least 0.669 … times as great as that of the prophet. In fact, even if the gambler uses a threshold stopping rule, meaning there is a fixed threshold value such that the gambler rejects every sample below the threshold and accepts every sample above it, the threshold can always be chosen so that the gambler-to-prophet ratio is at least . … In contrast, if the gambler must observe the values in a predetermined order, the tight bound for the gambler-to-prophet ratio is 1/2.In this work we investigate a model that interpolates between these two extremes. We assume there is a predefined set of permutations of the set indexing the random variables, and the gambler is free to choose the order of observation to be any one of these predefined permutations. Surprisingly, we show that even when only two orderings are allowed — namely, the forward and reverse orderings — the gambler-to-prophet ratio improves to …, the inverse of the golden ratio. As the number of allowed permutations grows beyond 2, a striking "double plateau" phenomenon emerges: after increasing from 0.5 to φ–1 when two permutations are allowed, the gambler-to-prophet ratio achievable by threshold stopping rules does not exceed φ–1 + o(1) until the number of allowed permutations grows to O(log n). The ratio reaches for a suitably chosen set of O(poly(∊–1) · log n) permutations and does not exceed even when the full set of n! permutations is allowed.
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Strong Algorithms for the Ordinal Matroid Secretary Problem

2021-02-23
José A. Soto , Abner Turkieltaub , Víctor Verdugo
In the ordinal matroid secretary problem (MSP), candidates do not reveal numerical weights, but the decision maker can still discern if a candidate is better than another. An algorithm [Formula: … In the ordinal matroid secretary problem (MSP), candidates do not reveal numerical weights, but the decision maker can still discern if a candidate is better than another. An algorithm [Formula: see text] is probability-competitive if every element from the optimum appears with probability [Formula: see text] in the output. This measure is stronger than the standard utility competitiveness. Our main result is the introduction of a technique based on forbidden sets to design algorithms with strong probability-competitive ratios on many matroid classes. We improve upon the guarantees for almost every matroid class considered in the MSP literature. In particular, we achieve probability-competitive ratios of 4 for graphic matroids and of [Formula: see text] for laminar matroids. Additionally, we modify Kleinberg’s [Formula: see text] utility-competitive algorithm for uniform matroids of rank [Formula: see text] in order to obtain a [Formula: see text] probability-competitive algorithm. We also contribute algorithms for the ordinal MSP on arbitrary matroids.
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Prophet secretary through blind strategies

2020-08-03
José R. Correa , Raimundo Saona , Bruno Ziliotto
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Online Weighted Bipartite Matching with a Sample.

2021-04-12
Haim Kaplan , David Naori , Danny Raz
We study the classical online bipartite matching problem: One side of the graph is known and vertices of the other side arrive online. It is well known that when the … We study the classical online bipartite matching problem: One side of the graph is known and vertices of the other side arrive online. It is well known that when the graph is edge-weighted, and vertices arrive in an adversarial order, no online algorithm has a nontrivial competitive-ratio. To bypass this hurdle we modify the rules such that the adversary still picks the graph but has to reveal a random part (say half) of it to the player. The remaining part is given to the player in an adversarial order. This models practical scenarios in which the online algorithm has some history to learn from. This way of modeling a history was formalized recently by the authors (SODA 20) and was called the AOS model (for Adversarial Online with a Sample). It allows developing online algorithms for the secretary problem that compete even when the secretaries arrive in an adversarial order. Here we use the same model to attack the much more challenging matching problem. We analyze a natural algorithmic framework that decides how to match an arriving vertex $v$ by applying an offline matching algorithm to $v$ and the sample. We get roughly $1/4$ of the maximum weight by applying the offline greedy matching algorithm to the sample and $v$. Our analysis ties the performance of this algorithm to the performance of the offline greedy matching on the online part and we also prove that it is tight. Surprisingly, when replacing greedy with an optimal algorithm for maximum matching, no constant competitive-ratio can be guaranteed when the size of the sample is comparable to the size of the online part. However, when the sample is quadratic in the size of the online part, we do get a competitive-ratio of $1/e$.

Efficient two-sided markets with limited information

2021-06-15
Paul Dütting , Federico Fusco , Philip Lazos +2 more
A celebrated impossibility result by Myerson and Satterthwaite (1983) shows that any truthful mechanism for two-sided markets that maximizes social welfare must run a deficit, resulting in a necessity to … A celebrated impossibility result by Myerson and Satterthwaite (1983) shows that any truthful mechanism for two-sided markets that maximizes social welfare must run a deficit, resulting in a necessity to relax welfare efficiency and the use of approximation mechanisms. Such mechanisms in general make extensive use of the Bayesian priors. In this work, we investigate a question of increasing theoretical and practical importance: how much prior information is required to design mechanisms with near-optimal approximations?
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The Two-Sided Game of Googol and Sample-Based Prophet Inequalities

2019-12-23
José R. Correa , Andrés Cristi , Boris Epstein +1 more
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)The Two-Sided Game of Googol and Sample-Based Prophet InequalitiesJosé R. Correa, Andrés Cristi, Boris Epstein, … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)The Two-Sided Game of Googol and Sample-Based Prophet InequalitiesJosé R. Correa, Andrés Cristi, Boris Epstein, and José A. SotoJosé R. Correa, Andrés Cristi, Boris Epstein, and José A. Sotopp.2066 - 2081Chapter DOI:https://doi.org/10.1137/1.9781611975994.127PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract The secretary problem or the game of Googol are classic models for online selection problems that have received significant attention in the last five decades. In this paper we consider a variant of the problem and explore its connections to data-driven online selection. Specifically, we are given n cards with arbitrary nonnegative numbers written on both sides. The cards are randomly placed on n consecutive positions on a table, and for each card, the visible side is also selected at random. The player sees the visible side of all cards and wants to select the card with the maximum hidden value. To this end, the player flips the first card, sees its hidden value and decides whether to pick it or drop it and continue with the next card. We study algorithms for two natural objectives. In the first one, similar to the secretary problem, the player wants to maximize the probability of selecting the maximum hidden value. We show that this can be done with probability at least 0.45292. In the second objective, similar to the prophet inequality, the player wants to maximize the expectation of the selected hidden value. Here we show a guarantee of at least 0.63518 with respect to the expected maximum hidden value. Our algorithms result from combining three basic strategies. One is to stop whenever we see a value larger than the initial n visible numbers. The second one is to stop the first time the last flipped card's value is the largest of the currently n visible numbers in the table. And the third one is similar to the latter but to stop it additionally requires that the last flipped value is larger than the value on the other side of its card. We apply our results to the prophet secretary problem with unknown distributions, but with access to a single sample from each distribution. In particular, our guarantee improves upon 1 – 1/e for this problem, which is the currently best known guarantee and only works for the i.i.d. prophet inequality with samples. Previous chapter Next chapter RelatedDetails Published:2020eISBN:978-1-61197-599-4 https://doi.org/10.1137/1.9781611975994Book Series Name:ProceedingsBook Code:PRDA20Book Pages:xxii + 3011
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Efficient Two-Sided Markets with Limited Information

2020-01-01
Paul Dütting , Federico Fusco , Philip Lazos +2 more
A celebrated impossibility result by Myerson and Satterthwaite (1983) shows that any truthful mechanism for two-sided markets that maximizes social welfare must run a deficit, resulting in a necessity to … A celebrated impossibility result by Myerson and Satterthwaite (1983) shows that any truthful mechanism for two-sided markets that maximizes social welfare must run a deficit, resulting in a necessity to relax welfare efficiency and the use of approximation mechanisms. Such mechanisms in general make extensive use of the Bayesian priors. In this work, we investigate a question of increasing theoretical and practical importance: how much prior information is required to design mechanisms with near-optimal approximations? Our first contribution is a more general impossibility result stating that no meaningful approximation is possible without any prior information, expanding the famous impossibility result of Myerson and Satterthwaite. Our second contribution is that one {\em single sample} (one number per item), arguably a minimum-possible amount of prior information, from each seller distribution is sufficient for a large class of two-sided markets. We prove matching upper and lower bounds on the best approximation that can be obtained with one single sample for subadditive buyers and additive sellers, regardless of computational considerations. Our third contribution is the design of computationally efficient blackbox reductions that turn any one-sided mechanism into a two-sided mechanism with a small loss in the approximation, while using only one single sample from each seller. On the way, our blackbox-type mechanisms deliver several interesting positive results in their own right, often beating even the state of the art that uses full prior information.
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Optimal Single-Choice Prophet Inequalities from Samples

2019-01-01
Aviad Rubinstein , Jack Z. Wang , S. Matthew Weinberg
We study the single-choice Prophet Inequality problem when the gambler is given access to samples. We show that the optimal competitive ratio of $1/2$ can be achieved with a single … We study the single-choice Prophet Inequality problem when the gambler is given access to samples. We show that the optimal competitive ratio of $1/2$ can be achieved with a single sample from each distribution. When the distributions are identical, we show that for any constant $\varepsilon > 0$, $O(n)$ samples from the distribution suffice to achieve the optimal competitive ratio ($\approx 0.745$) within $(1+\varepsilon)$, resolving an open problem of Correa, D\"utting, Fischer, and Schewior.
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Prophet inequality for bipartite matching: merits of being simple and non adaptive

2019-01-01
Nick Gravin , Hongao Wang
We consider Bayesian online selection problem of a matching in bipartite graphs, i.e., online weighted matching problem with edge arrivals where online algorithm knows distributions of weights, that corresponds to … We consider Bayesian online selection problem of a matching in bipartite graphs, i.e., online weighted matching problem with edge arrivals where online algorithm knows distributions of weights, that corresponds to the intersection of two matroids in [Kleinberg and Wienberg STOC 12] model. We consider a simple class of non adaptive vertex-additive policies that assign static prices to all vertices in the graph and accept each edge only if its weight exceeds the sum of the prices of the edge's endpoints. We show existence of a vertex-additive policy with the expected payoff of at least one third of the prophet's payoff and present gradient decent type algorithm that quickly converges to the desired vector of vertex prices. This improves the adaptive online policy of [Kleinberg and Wienberg STOC 12] for the intersection of two matroids in two ways: our policy is non adaptive and has better approximation guarantee of $3$ instead of previous guarantee of $5.82$ against the prophet. We give a complementary lower bound of $2.25$ for any online algorithm in the bipartite matching setting.
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Composite objective mirror descent

2010-12-01
John C. Duchi , Shai Shalev‐Shwartz , Yoram Singer +1 more
We present a new method for regularized convex optimization and analyze it under both online and stochastic optimization settings. In addition to unifying previously known firstorder algorithms, such as the … We present a new method for regularized convex optimization and analyze it under both online and stochastic optimization settings. In addition to unifying previously known firstorder algorithms, such as the projected gradient method, mirror descent, and forwardbackward splitting, our method yields new analysis and algorithms. We also derive specific instantiations of our method for commonly used regularization functions, such as l1, mixed norm, and trace-norm.

Sample-driven optimal stopping: From the secretary problem to the i.i.d. prophet inequality

2020-01-01
José R. Correa , Andrés Cristi , Boris Epstein +1 more
We take a unifying approach to single selection optimal stopping problems with random arrival order and independent sampling of items. In the problem we consider, a decision maker (DM) initially … We take a unifying approach to single selection optimal stopping problems with random arrival order and independent sampling of items. In the problem we consider, a decision maker (DM) initially gets to sample each of $N$ items independently with probability $p$, and can observe the relative rankings of these sampled items. Then, the DM faces the remaining items in an online fashion, observing the relative rankings of all revealed items. While scanning the sequence the DM makes irrevocable stop/continue decisions and her reward for stopping the sequence facing the item with rank $i$ is $Y_i$. The goal of the DM is to maximize her reward. We start by studying the case in which the values $Y_i$ are known to the DM, and then move to the case in which these values are adversarial. For the former case, we write the natural linear program that captures the performance of an algorithm, and take its continuous limit. We prove a structural result about this continuous limit, which allows us to reduce the problem to a relatively simple real optimization problem. We establish that the optimal algorithm is given by a sequence of thresholds $t_1\le t_2\le\cdots$ such that the DM should stop if seeing an item with current ranking $i$ after time $t_i$. Additionally we are able to recover several classic results in the area such as those for secretary problem and the minimum ranking problem. For the adversarial case, we obtain a similar linear program with an additional stochastic dominance constraint. Using the same machinery we are able to pin down the optimal competitive ratios for all values of $p$. Notably, we prove that as $p$ approaches 1, our guarantee converges linearly to 0.745, matching that of the i.i.d.~prophet inequality. Also interesting is the case $p=1/2$, where our bound evaluates to $0.671$, which improves upon the state of the art.
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