Pub Date : 2022-12-19DOI: https://dl.acm.org/doi/10.1145/3565019
James B. Orlin, László Végh
We present an O(nm) algorithm for all-pairs shortest paths computations in a directed graph with n nodes, m arcs, and nonnegative integer arc costs. This matches the complexity bound attained by Thorup [31] for the all-pairs problems in undirected graphs. The main insight is that shortest paths problems with approximately balanced directed cost functions can be solved similarly to the undirected case. The algorithm finds an approximately balanced reduced cost function in an O(m√ n log n) preprocessing step. Using these reduced costs, every shortest path query can be solved in O(m) time using an adaptation of Thorup’s component hierarchy method. The balancing result can also be applied to the ℓ∞-matrix balancing problem.
{"title":"Directed Shortest Paths via Approximate Cost Balancing","authors":"James B. Orlin, László Végh","doi":"https://dl.acm.org/doi/10.1145/3565019","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3565019","url":null,"abstract":"<p>We present an <i>O(nm)</i> algorithm for all-pairs shortest paths computations in a directed graph with <i>n</i> nodes, <i>m</i> arcs, and nonnegative integer arc costs. This matches the complexity bound attained by Thorup [31] for the all-pairs problems in undirected graphs. The main insight is that shortest paths problems with approximately balanced directed cost functions can be solved similarly to the undirected case. The algorithm finds an approximately balanced reduced cost function in an <i>O(m</i>√ <i>n</i> log <i>n</i>) preprocessing step. Using these reduced costs, every shortest path query can be solved in <i>O(m)</i> time using an adaptation of Thorup’s component hierarchy method. The balancing result can also be applied to the ℓ<sub>∞</sub>-matrix balancing problem.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-19DOI: https://dl.acm.org/doi/10.1145/3566050
Haotian Jiang
Given a separation oracle SO for a convex function f defined on ℝn that has an integral minimizer inside a box with radius R, we show how to find an exact minimizer of f using at most
O(n (n log log (n)/log (n) + log (R))) calls to SO and poly (n, log (R)) arithmetic operations, or
O(n log (nR) calls to SO and exp (O(n)) ⋅ poly (log (R)) arithmetic operations.
When the set of minimizers of f has integral extreme points, our algorithm outputs an integral minimizer of f. This improves upon the previously best oracle complexity of O(n2 (n + log (R))) for polynomial time algorithms and O(n2 log (nR) for exponential time algorithms obtained by [Grötschel, Lovász and Schrijver, Prog. Comb. Opt. 1984, Springer 1988] over thirty years ago. Our improvement on Grötschel, Lovász and Schrijver’s result generalizes to the setting where the set of minimizers of f is a rational polyhedron with bounded vertex complexity.
For the Submodular Function Minimization problem, our result immediately implies a strongly polynomial algorithm that makes at most O(n3 log log (n)/log (n)) calls to an evaluation oracle, and an exponential time algorithm that makes at most O(n2 log (n)) calls to an evaluation oracle. These improve upon the previously best O(n3 log2(n)) oracle complexity for strongly polynomial algorithms given in [Lee, Sidford and Wong, FOCS 2015] and [Dadush, Végh and Zambelli, SODA 2018], and an exponential time algorithm with oracle complexity O(n3 log (n)) given in the former work.
Our result is achieved via a reduction to the Shortest Vector Problem in lattices. We show how an approximately shortest vector of an auxiliary lattice can be used to effectively reduce the dimension of the problem. Our analysis of the oracle complexity is based on a potential function that simultaneously captures the size of the search set and the density of the lattice, which we analyze via tools from convex geometry and lattice theory.
{"title":"Minimizing Convex Functions with Rational Minimizers","authors":"Haotian Jiang","doi":"https://dl.acm.org/doi/10.1145/3566050","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3566050","url":null,"abstract":"<p>Given a separation oracle SO for a convex function <i>f</i> defined on ℝ<sup>n</sup> that has an integral minimizer inside a box with radius <i>R</i>, we show how to find an exact minimizer of <i>f</i> using at most\u0000<p><ul><li><p><i>O(n (n</i> log log <i>(n)/</i>log <i>(n)</i> + log (<i>R</i>))) calls to SO and poly (<i>n</i>, log (<i>R</i>)) arithmetic operations, or</p></li><li><p><i>O(n</i> log <i>(nR)</i> calls to SO and exp (<i>O(n)</i>) ⋅ poly (log <i>(R)</i>) arithmetic operations.</p></li></ul></p></p><p>When the set of minimizers of <i>f</i> has integral extreme points, our algorithm outputs an integral minimizer of <i>f</i>. This improves upon the previously best oracle complexity of <i>O(n</i><sup>2</sup> (<i>n</i> + log (<i>R</i>))) for polynomial time algorithms and <i>O(n</i><sup>2</sup> log (<i>nR</i>) for exponential time algorithms obtained by [Grötschel, Lovász and Schrijver, Prog. Comb. Opt. 1984, Springer 1988] over thirty years ago. Our improvement on Grötschel, Lovász and Schrijver’s result generalizes to the setting where the set of minimizers of <i>f</i> is a rational polyhedron with bounded vertex complexity.</p><p>For the Submodular Function Minimization problem, our result immediately implies a strongly polynomial algorithm that makes at most <i>O(n</i><sup>3</sup> log log (<i>n</i>)/log (<i>n</i>)) calls to an evaluation oracle, and an exponential time algorithm that makes at most <i>O(n</i><sup>2</sup> log (<i>n</i>)) calls to an evaluation oracle. These improve upon the previously best <i>O(n</i><sup>3</sup> log<sup>2</sup>(<i>n</i>)) oracle complexity for strongly polynomial algorithms given in [Lee, Sidford and Wong, FOCS 2015] and [Dadush, Végh and Zambelli, SODA 2018], and an exponential time algorithm with oracle complexity <i>O(n</i><sup>3</sup> log (<i>n</i>)) given in the former work.</p><p>Our result is achieved via a reduction to the Shortest Vector Problem in lattices. We show how an approximately shortest vector of an auxiliary lattice can be used to effectively reduce the dimension of the problem. Our analysis of the oracle complexity is based on a potential function that simultaneously captures the size of the search set and the density of the lattice, which we analyze via tools from convex geometry and lattice theory.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-19DOI: https://dl.acm.org/doi/10.1145/3559103
Joachim Kock
We present a formalism for Petri nets based on polynomial-style finite-set configurations and etale maps. The formalism supports both a geometric semantics in the style of Goltz and Reisig (processes are etale maps from graphs) and an algebraic semantics in the style of Meseguer and Montanari, in terms of free coloured props, and allows the following unification: for P a Petri net, the Segal space of P-processes is shown to be the free coloured prop-in-groupoids on P. There is also an unfolding semantics à la Winskel, which bypasses the classical symmetry problems: with the new formalism, every Petri net admits a universal unfolding, which in turn has associated an event structure and a Scott domain. Since everything is encoded with explicit sets, Petri nets and their processes have elements. In particular, individual-token semantics is native. (Collective-token semantics emerges from rather drastic quotient constructions à la Best–Devillers, involving taking π0 of the groupoids of states.)
我们提出了一种基于多项式型有限集构形和线性映射的Petri网的形式。该形式主义既支持Goltz和Reisig风格的几何语义(过程是从图中生成的映射),也支持Meseguer和Montanari风格的代数语义,就自由彩色支柱而言,并允许以下统一:对于P a Petri网,P-过程的Segal空间被证明是P上的群中的自由彩色支柱。在新的形式主义中,每个Petri网都承认一个普遍展开,这反过来又将事件结构和斯科特域联系起来。因为所有东西都是用显式集合编码的,所以Petri网和它们的过程都有元素。特别是,单个令牌语义是本地的。(集体令牌语义来自于相当激烈的商构造(例如Best-Devillers),包括取状态群类群的π0。)
{"title":"Whole-grain Petri Nets and Processes","authors":"Joachim Kock","doi":"https://dl.acm.org/doi/10.1145/3559103","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3559103","url":null,"abstract":"<p>We present a formalism for Petri nets based on polynomial-style finite-set configurations and etale maps. The formalism supports both a geometric semantics in the style of Goltz and Reisig (processes are etale maps from graphs) and an algebraic semantics in the style of Meseguer and Montanari, in terms of free coloured props, and allows the following unification: for <monospace>P</monospace> a Petri net, the Segal space of <monospace>P</monospace>-processes is shown to be the free coloured prop-in-groupoids on <monospace>P</monospace>. There is also an unfolding semantics à la Winskel, which bypasses the classical symmetry problems: with the new formalism, every Petri net admits a universal unfolding, which in turn has associated an event structure and a Scott domain. Since everything is encoded with explicit sets, Petri nets and their processes have elements. In particular, individual-token semantics is native. (Collective-token semantics emerges from rather drastic quotient constructions à la Best–Devillers, involving taking π<sub>0</sub> of the groupoids of states.)</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-19DOI: https://dl.acm.org/doi/10.1145/3566049
Gilad Asharov, Ilan Komargodski, Wei-Kai Lin, Kartik Nayak, Enoch Peserico, Elaine Shi
Oblivious RAM (ORAM), first introduced in the ground-breaking work of Goldreich and Ostrovsky (STOC ’87 and J. ACM ’96) is a technique for provably obfuscating programs’ access patterns, such that the access patterns leak no information about the programs’ secret inputs. To compile a general program to an oblivious counterpart, it is well-known that Ω (log N) amortized blowup in memory accesses is necessary, where N is the size of the logical memory. This was shown in Goldreich and Ostrovksy’s original ORAM work for statistical security and in a somewhat restricted model (the so-called balls-and-bins model), and recently by Larsen and Nielsen (CRYPTO ’18) for computational security.
A long-standing open question is whether there exists an optimal ORAM construction that matches the aforementioned logarithmic lower bounds (without making large memory word assumptions, and assuming a constant number of CPU registers). In this article, we resolve this problem and present the first secure ORAM with O(log N) amortized blowup, assuming one-way functions. Our result is inspired by and non-trivially improves on the recent beautiful work of Patel et al. (FOCS ’18) who gave a construction with O(log N⋅ log log N) amortized blowup, assuming one-way functions.
One of our building blocks of independent interest is a linear-time deterministic oblivious algorithm for tight compaction: Given an array of n elements where some elements are marked, we permute the elements in the array so that all marked elements end up in the front of the array. Our O(n) algorithm improves the previously best-known deterministic or randomized algorithms whose running time is O(n ⋅ log n) or O(n ⋅ log log n), respectively.
{"title":"OptORAMa: Optimal Oblivious RAM","authors":"Gilad Asharov, Ilan Komargodski, Wei-Kai Lin, Kartik Nayak, Enoch Peserico, Elaine Shi","doi":"https://dl.acm.org/doi/10.1145/3566049","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3566049","url":null,"abstract":"<p>Oblivious RAM (ORAM), first introduced in the ground-breaking work of Goldreich and Ostrovsky (STOC ’87 and J. ACM ’96) is a technique for provably obfuscating programs’ access patterns, such that the access patterns leak no information about the programs’ secret inputs. To compile a general program to an oblivious counterpart, it is well-known that Ω (log <i>N</i>) amortized blowup in memory accesses is necessary, where <i>N</i> is the size of the logical memory. This was shown in Goldreich and Ostrovksy’s original ORAM work for statistical security and in a somewhat restricted model (the so-called <i>balls-and-bins</i> model), and recently by Larsen and Nielsen (CRYPTO ’18) for computational security.</p><p>A long-standing open question is whether there exists an <i>optimal</i> ORAM construction that matches the aforementioned logarithmic lower bounds (without making large memory word assumptions, and assuming a constant number of CPU registers). In this article, we resolve this problem and present the first secure ORAM with <i>O</i>(log <i>N</i>) amortized blowup, assuming one-way functions. Our result is inspired by and non-trivially improves on the recent beautiful work of Patel et al. (FOCS ’18) who gave a construction with <i>O</i>(log <i>N</i>⋅ log log <i>N</i>) amortized blowup, assuming one-way functions. </p><p>One of our building blocks of independent interest is a linear-time deterministic oblivious algorithm for tight compaction: Given an array of <i>n</i> elements where some elements are marked, we permute the elements in the array so that all marked elements end up in the front of the array. Our <i>O</i>(<i>n</i>) algorithm improves the previously best-known deterministic or randomized algorithms whose running time is <i>O</i>(<i>n</i> ⋅ log <i>n</i>) or <i>O</i>(<i>n</i> ⋅ log log <i>n</i>), respectively.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-24DOI: https://dl.acm.org/doi/10.1145/3561047
Guy Blanc, Jane Lange, Mingda Qiao, Li-Yang Tan
We give an nO(log log n)-time membership query algorithm for properly and agnostically learning decision trees under the uniform distribution over { ± 1}n. Even in the realizable setting, the previous fastest runtime was nO(log n), a consequence of a classic algorithm of Ehrenfeucht and Haussler.
Our algorithm shares similarities with practical heuristics for learning decision trees, which we augment with additional ideas to circumvent known lower bounds against these heuristics. To analyze our algorithm, we prove a new structural result for decision trees that strengthens a theorem of O’Donnell, Saks, Schramm, and Servedio. While the OSSS theorem says that every decision tree has an influential variable, we show how every decision tree can be “pruned” so that every variable in the resulting tree is influential.
{"title":"Properly Learning Decision Trees in almost Polynomial Time","authors":"Guy Blanc, Jane Lange, Mingda Qiao, Li-Yang Tan","doi":"https://dl.acm.org/doi/10.1145/3561047","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3561047","url":null,"abstract":"<p>We give an <i>n</i><sup><i>O</i>(log log <i>n</i>)</sup>-time membership query algorithm for properly and agnostically learning decision trees under the uniform distribution over { ± 1}<sup><i>n</i></sup>. Even in the realizable setting, the previous fastest runtime was <i>n</i><sup><i>O</i>(log <i>n</i>)</sup>, a consequence of a classic algorithm of Ehrenfeucht and Haussler.</p><p>Our algorithm shares similarities with practical heuristics for learning decision trees, which we augment with additional ideas to circumvent known lower bounds against these heuristics. To analyze our algorithm, we prove a new structural result for decision trees that strengthens a theorem of O’Donnell, Saks, Schramm, and Servedio. While the OSSS theorem says that every decision tree has an influential variable, we show how every decision tree can be “pruned” so that <i>every</i> variable in the resulting tree is influential.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-24DOI: https://dl.acm.org/doi/10.1145/3556972
Avinatan Hassidim, Haim Kaplan, Yishay Mansour, Yossi Matias, Uri Stemmer
A streaming algorithm is said to be adversarially robust if its accuracy guarantees are maintained even when the data stream is chosen maliciously, by an adaptive adversary. We establish a connection between adversarial robustness of streaming algorithms and the notion of differential privacy. This connection allows us to design new adversarially robust streaming algorithms that outperform the current state-of-the-art constructions for many interesting regimes of parameters.
{"title":"Adversarially Robust Streaming Algorithms via Differential Privacy","authors":"Avinatan Hassidim, Haim Kaplan, Yishay Mansour, Yossi Matias, Uri Stemmer","doi":"https://dl.acm.org/doi/10.1145/3556972","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3556972","url":null,"abstract":"<p>A streaming algorithm is said to be <i>adversarially robust</i> if its accuracy guarantees are maintained even when the data stream is chosen maliciously, by an <i>adaptive adversary</i>. We establish a connection between adversarial robustness of streaming algorithms and the notion of <i>differential privacy</i>. This connection allows us to design new adversarially robust streaming algorithms that outperform the current state-of-the-art constructions for many interesting regimes of parameters.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates. A natural approach to leverage these quantum computers is interleaving them with classical computers. Understanding the capabilities and limits of this hybrid approach is an essential topic in quantum computation. Most notably, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Therefore, it seems possible that quantum polylogarithmic depth is as powerful as quantum polynomial depth in the presence of classical computation. Indeed, Jozsa conjectured that “Any quantum polynomial-time algorithm can be implemented with only O(log n) quantum depth interspersed with polynomial-time classical computations.” This can be formalized as asserting the equivalence of BQP and “BQNCBPP.” However, Aaronson conjectured that “there exists an oracle separation between BQP and BPPBQNC.” BQNCBPP and BPPBQNC are two natural and seemingly incomparable ways of hybrid classical-quantum computation. In this work, we manage to prove Aaronson’s conjecture and in the meantime prove that Jozsa’s conjecture, relative to an oracle, is false. In fact, we prove a stronger statement that for any depth parameter d, there exists an oracle that separates quantum depth d and 2d+1 in the presence of classical computation. Thus, our results show that relative to oracles, doubling the quantum circuit depth does make the hybrid model more powerful, and this cannot be traded by classical computation.
{"title":"On the Need for Large Quantum Depth","authors":"Nai-Hui Chia, Kai-Min Chung, C. Lai","doi":"10.1145/3570637","DOIUrl":"https://doi.org/10.1145/3570637","url":null,"abstract":"Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates. A natural approach to leverage these quantum computers is interleaving them with classical computers. Understanding the capabilities and limits of this hybrid approach is an essential topic in quantum computation. Most notably, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Therefore, it seems possible that quantum polylogarithmic depth is as powerful as quantum polynomial depth in the presence of classical computation. Indeed, Jozsa conjectured that “Any quantum polynomial-time algorithm can be implemented with only O(log n) quantum depth interspersed with polynomial-time classical computations.” This can be formalized as asserting the equivalence of BQP and “BQNCBPP.” However, Aaronson conjectured that “there exists an oracle separation between BQP and BPPBQNC.” BQNCBPP and BPPBQNC are two natural and seemingly incomparable ways of hybrid classical-quantum computation. In this work, we manage to prove Aaronson’s conjecture and in the meantime prove that Jozsa’s conjecture, relative to an oracle, is false. In fact, we prove a stronger statement that for any depth parameter d, there exists an oracle that separates quantum depth d and 2d+1 in the presence of classical computation. Thus, our results show that relative to oracles, doubling the quantum circuit depth does make the hybrid model more powerful, and this cannot be traded by classical computation.","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75401752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-18DOI: https://dl.acm.org/doi/10.1145/3563772
Gabriele Farina, Andrea Celli, Alberto Marchesi, Nicola Gatti
The existence of simple uncoupled no-regret learning dynamics that converge to correlated equilibria in normal-form games is a celebrated result in the theory of multi-agent systems. Specifically, it has been known for more than 20 years that when all players seek to minimize their internal regret in a repeated normal-form game, the empirical frequency of play converges to a normal-form correlated equilibrium. Extensive-form (that is, tree-form) games generalize normal-form games by modeling both sequential and simultaneous moves, as well as imperfect information. Because of the sequential nature and presence of private information in the game, correlation in extensive-form games possesses significantly different properties than in normal-form games, many of which are still open research directions. Extensive-form correlated equilibrium (EFCE) has been proposed as the natural extensive-form counterpart to the classical notion of correlated equilibrium in normal-form games. Compared to the latter, the constraints that define the set of EFCEs are significantly more complex, as the correlation device (a.k.a. mediator) must take into account the evolution of beliefs of each player as they make observations throughout the game. Due to that significant added complexity, the existence of uncoupled learning dynamics leading to an EFCE has remained a challenging open research question for a long time. In this article, we settle that question by giving the first uncoupled no-regret dynamics that converge to the set of EFCEs in n-player general-sum extensive-form games with perfect recall. We show that each iterate can be computed in time polynomial in the size of the game tree, and that, when all players play repeatedly according to our learning dynamics, the empirical frequency of play after T game repetitions is proven to be a ( O(1/sqrt {T}) )-approximate EFCE with high probability, and an EFCE almost surely in the limit.
{"title":"Simple Uncoupled No-regret Learning Dynamics for Extensive-form Correlated Equilibrium","authors":"Gabriele Farina, Andrea Celli, Alberto Marchesi, Nicola Gatti","doi":"https://dl.acm.org/doi/10.1145/3563772","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3563772","url":null,"abstract":"<p>The existence of simple uncoupled no-regret learning dynamics that converge to correlated equilibria in normal-form games is a celebrated result in the theory of multi-agent systems. Specifically, it has been known for more than 20 years that when all players seek to minimize their <i>internal</i> regret in a repeated normal-form game, the empirical frequency of play converges to a normal-form correlated equilibrium. Extensive-form (that is, tree-form) games generalize normal-form games by modeling both sequential and simultaneous moves, as well as imperfect information. Because of the sequential nature and presence of private information in the game, correlation in extensive-form games possesses significantly different properties than in normal-form games, many of which are still open research directions. Extensive-form correlated equilibrium (EFCE) has been proposed as the natural extensive-form counterpart to the classical notion of correlated equilibrium in normal-form games. Compared to the latter, the constraints that define the set of EFCEs are significantly more complex, as the correlation device (a.k.a. mediator) must take into account the evolution of beliefs of each player as they make observations throughout the game. Due to that significant added complexity, the existence of uncoupled learning dynamics leading to an EFCE has remained a challenging open research question for a long time. In this article, we settle that question by giving the first uncoupled no-regret dynamics that converge to the set of EFCEs in <i>n</i>-player general-sum extensive-form games with perfect recall. We show that each iterate can be computed in time polynomial in the size of the game tree, and that, when all players play repeatedly according to our learning dynamics, the empirical frequency of play after <i>T</i> game repetitions is proven to be a ( O(1/sqrt {T}) )-approximate EFCE with high probability, and an EFCE almost surely in the limit.</p>","PeriodicalId":50022,"journal":{"name":"Journal of the ACM","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2022-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-17DOI: https://dl.acm.org/doi/10.1145/3556971
Matthew Fahrbach, Zhiyi Huang, Runzhou Tao, Morteza Zadimoghaddam
Online bipartite matching is one of the most fundamental problems in the online algorithms literature. Karp, Vazirani, and Vazirani (STOC 1990) gave an elegant algorithm for unweighted bipartite matching that achieves an optimal competitive ratio of 1-1/e . Aggarwal et al. (SODA 2011) later generalized their algorithm and analysis to the vertex-weighted case. Little is known, however, about the most general edge-weighted problem aside from the trivial 1/2-competitive greedy algorithm. In this article, we present the first online algorithm that breaks the long-standing 1/2 barrier and achieves a competitive ratio of at least 0.5086. In light of the hardness result of Kapralov, Post, and Vondrák (SODA 2013), which restricts beating a 1/2 competitive ratio for the more general monotone submodular welfare maximization problem, our result can be seen as strong evidence that edge-weighted bipartite matching is strictly easier than submodular welfare maximization in an online setting.
The main ingredient in our online matching algorithm is a novel subroutine called online correlated selection (OCS), which takes a sequence of pairs of vertices as input and selects one vertex from each pair. Instead of using a fresh random bit to choose a vertex from each pair, the OCS negatively correlates decisions across different pairs and provides a quantitative measure on the level of correlation. We believe our OCS technique is of independent interest and will find further applications in other online optimization problems.
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Pub Date : 2022-11-17DOI: https://dl.acm.org/doi/10.1145/3557045
Nicole Immorlica, Karthik Sankararaman, Robert Schapire, Aleksandrs Slivkins
We consider Bandits with Knapsacks (henceforth, BwK), a general model for multi-armed bandits under supply/budget constraints. In particular, a bandit algorithm needs to solve a well-known knapsack problem: find an optimal packing of items into a limited-size knapsack. The BwK problem is a common generalization of numerous motivating examples, which range from dynamic pricing to repeated auctions to dynamic ad allocation to network routing and scheduling. While the prior work on BwK focused on the stochastic version, we pioneer the other extreme in which the outcomes can be chosen adversarially. This is a considerably harder problem, compared to both the stochastic version and the “classic” adversarial bandits, in that regret minimization is no longer feasible. Instead, the objective is to minimize the competitive ratio: the ratio of the benchmark reward to algorithm’s reward.
We design an algorithm with competitive ratio O(log T) relative to the best fixed distribution over actions, where T is the time horizon; we also prove a matching lower bound. The key conceptual contribution is a new perspective on the stochastic version of the problem. We suggest a new algorithm for the stochastic version, which builds on the framework of regret minimization in repeated games and admits a substantially simpler analysis compared to prior work. We then analyze this algorithm for the adversarial version, and use it as a subroutine to solve the latter.
Our algorithm is the first “black-box reduction” from bandits to BwK: it takes an arbitrary bandit algorithm and uses it as a subroutine. We use this reduction to derive several extensions.
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