Browsing by Author "Lu, Chi-Jen"

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Bounds on the price of anarchy for a more general class of directed graphs in opinion formation games
(2016-11)
Chen, Po-An
;
Chen, Yi-Le
;
Lu, Chi-Jen
;
資訊工程學系
;
資訊管理與財務金融系註：原資管所+財金所
In opinion formation games with directed graphs, a bounded price of anarchy is only known for weighted Eulerian graphs. Thus, we bound the price of anarchy for a more general class of directed graphs with conditions intuitively meaning that each node does not influence the others more than she is influenced, where the bounds depend on such difference (in a ratio). We also show that there exists an example just slightly violating the conditions with an unbounded price of anarchy. (C) 2016 Elsevier B.V. All rights reserved.
Complexity of Hard-Core Set Proofs
(2011-03-01)
Lu, Chi-Jen
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Tsai, Shi-Chun
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Wu, Hsin-Lung
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資訊工程學系
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Department of Computer Science
We study a fundamental result of Impagliazzo (FOCS'95) known as the hard-core set lemma. Consider any function f : {0,1}(n) -> {0,1} which is "mildly hard", in the sense that any circuit of size s must disagree with f on at least a delta fraction of inputs. Then, the hard-core set lemma says that f must have a hard-core set H of density delta on which it is "extremely hard", in the sense that any circuit of size s' = O(s/1/epsilon(2) log 1/epsilon(delta)))) must disagree wiht f on at least (1 - epsilon)/2 fraction of inputs from H. There are three issues of the lemma which we would like to address: the loss of circuit size, the need of non-uniformity, and its inapplicability to a low-level complexity class. We introduce two models of hard-core set proofs, a strongly black-box one and a weakly black-box one, and show that those issues are unavoidable in such models. First, we show that using any strongly black-box proof, one can only prove the hardness of a hard-core set for smaller circuits of size at most s' = O(s/1/epsilon(2) log 1/delta)). Next, we show that any weakly black-box proof must be inherently non-uniform-to have a hard-core set for a class G of functions, we need to start from the assumption that f is hard against a non-uniform complexity class with Omega(1/epsilon log vertical bar G vertical bar) bits of advice. Finally, we show that weakly black-box proofs in general cannot be realized in a low-level complexity class such as AC (0)[p]-the assumption that f is hard for AC (0)[p] is not sufficient to guarantee the existence of a hard-core set.
Deterministic extractors for independent-symbol sources
(2006)
Lee, Chia-Jung
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Lu, Chi-Jen
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Tsai, Shi-Chun
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資訊工程學系
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Department of Computer Science
In this paper, we consider the task of deterministically extracting randomness from sources consisting of a sequence of n independent symbols from {0, 1}(d). The only randomness guarantee on such a source is that the whole source has min-entropy k. We give an explicit deterministic extractor which can extract Omega(log k-log d-log log(1/epsilon)) bits with error epsilon, for any n, d, k is an element of N and epsilon G (0, 1). For sources with a larger min-entropy, we can extract even more randomness. When k >= n(1/2+gamma) for any constant gamma is an element of (0, 1/2), we can extract m = k - O(d log(1/epsilon)) bits with any error epsilon >= 2(-Omega)(n(gamma)). When k >= log(c)n, for some constant c > 0, we can extract m = k - d(1/epsilon)(O(1)) bits with any error epsilon >= k(-Omega(1)). Our results generalize those of Kamp & Zuckerman and Gabizon et al. which only work for bit-fixing sources (with d = I and each bit of the source being either fixed or perfectly random). Moreover, we show the existence of a non-explicit deterministic extractor which can extract m = k-O(log(1/epsilon)) bits whenever k = w(d+log(n/epsilon)). Finally, we show that even to extract from bit-fixing sources, any extractor, seeded or not, must suffer an entropy loss k-m = Q (log (1/epsilon)). This generalizes a lower bound of Radhakrishnan & Ta-Shma with respect to general sources.
Deterministic Extractors for Independent-Symbol Sources
(2010-12-01)
Lee, Chia-Jung
;
Lu, Chi-Jen
;
Tsai, Shi-Chun
;
資訊工程學系
;
Department of Computer Science
In this paper, we consider the task of deterministically extracting randomness from sources consisting of a sequence of n independent symbols from {0, 1}(d). The only randomness guarantee on such a source is that the whole source has min-entropy k. We give an explicit deterministic extractor which extract Omega(log k - log log(1/epsilon)) bits with error epsilon, for any, n, d, k is an element of N and epsilon is an element of (0, 1). For sources with a larger min-entropy, we can extract even more randomness. When k >= n(1/2+gamma), for any constant gamma is an element of (1, 1/2), we can extract m = k - O(d log(1/epsilon)) bits with any error epsilon >= 2(-Omega(n gamma)). When k >= log(c) n, for some constant c > 0 , we can extract m = k - (1/epsilon)(O(1)) bits with any error epsilon >= k(-Omega(1)). Our results generalize those of Kamp and Zuckerman and Gabizon et al. which only work for bit-fixing sources (with d = 1 and each bit of the source being either fixed or perfectly random). Moreover, we show the existence of a nonexplicit deterministic extractor which can extract m = k - O(log(1/epsilon)) bits whenever k = omega(d + log(n/epsilon)). Finally, we show that even to extract from bit-fixing sources, any extractor, seeded or not, must suffer an entropy loss k - m = Omega(log(1/epsilon)). This generalizes a lower bound of Radhakrishnan and Ta-Shma on extracting from general sources.
Extracting Computational Entropy and Learning Noisy Linear Functions
(2009)
Lee, Chia-Jung
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Lu, Chi-Jen
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Tsai, Shi-Chun
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資訊工程學系
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Department of Computer Science
We study the task of deterministically extracting randomness from sources containing computational entropy. The sources we consider have the form of a conditional distribution (f(X)vertical bar X), for some function f and some distribution X, and we say that such a source has computational min-entropy k if any circuit of size 2(k) can only predict f(x) correctly with probability at most 2(-k) given input x sampled from X. We first show that it is impossible to have a seedless extractor to extract from one single source of this kind. Then we show that it becomes possible if we are allowed a seed which is weakly random (instead of perfectly random) but contains some statistical min-entropy, or even a seed which is not random at all but contains some computational min-entropy. This can be seen as a step toward extending the study of multi-source extractors from the traditional, statistical setting to a computational setting. We reduce the task of constructing such extractors to a problem in learning theory: learning linear functions under arbitrary distribution with adversarial noise. For this problem, we provide a learning algorithm, which may have interest of its own.
Extracting Computational Entropy and Learning Noisy Linear Functions
(2011-08-01)
Lee, Chia-Jung
;
Lu, Chi-Jen
;
Tsai, Shi-Chun
;
資訊工程學系
;
Department of Computer Science
We study the task of deterministically extracting randomness from sources containing computational entropy. The sources we consider have the form of a conditional distribution (f(X)vertical bar X), for some function f and some distribution X, and we say that such a source has computational min-entropy if any circuit of size 2(k) can only predict f(x) correctly with probability at most 2(-k) given input sampled from X. We first show that it is impossible to have a seedless extractor to extract from one single source of this kind. Then we show that it becomes possible if we are allowed a seed which is weakly random (instead of perfectly random) but contains some statistical min-entropy, or even a seed which is not random at all but contains some computational min-entropy. This can be seen as a step toward extending the study of multisource extractors from the traditional, statistical setting to a computational setting. We reduce the task of constructing such extractors to a problem in computational learning theory: learning linear functions under arbitrary distribution with adversarial noise, and we provide a learning algorithm for this problem. In fact, this problem is a well-recognized one in computational learning theory and variants of this problem have been studied intensively before. Thus, in addition to its application to extractors, our learning algorithm also has independent interest of its own, and it can be considered as the main technical contribution of this paper.
Generalized mirror descents in congestion games
(2016-12)
Chen, Po-An
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Lu, Chi-Jen
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資訊管理與財務金融系註：原資管所+財金所
;
Department of Information Management and Finance
Different types of dynamics have been studied in repeated game play, and one of them which has received much attention recently consists of those based on "no-regret" algorithms from the area of machine learning. It is known that dynamics based on generic no-regret algorithms may not converge to Nash equilibria in general, but to a larger set of outcomes, namely coarse correlated equilibria. Moreover, convergence results based on generic no-regret algorithms typically use a weaker notion of convergence: the convergence of the average plays instead of the actual plays. Some work has been done showing that when using a specific no-regret algorithm, the well-known multiplicative updates algorithm, convergence of actual plays to equilibria can be shown and better quality of outcomes in terms of the price of anarchy can be reached for atomic congestion games and load balancing games. Are there more cases of natural no-regret dynamics that perform well in suitable classes of games in terms of convergence and quality of outcomes that the dynamics converge to? We answer this question positively in the bulletin-board model by showing that when employing the mirror-descent algorithm, a well-known generic no-regret algorithm, the actual plays converge quickly to equilibria in nonatomic congestion games. This gives rise to a family of algorithms, including the multiplicative updates algorithm and the gradient descent algorithm as well as many others. Furthermore, we show that our dynamics achieves good bounds on the outcome quality in terms of the price-of-anarchy type of measures with two different social costs: the average individual cost and the maximum individual cost. Finally, the bandit model considers a probably more realistic and prevalent setting with only partial information, in which at each time step each player only knows the cost of her own currently played strategy, but not any costs of unplayed strategies. For the class of atomic congestion games, we propose a family of bandit algorithms based on the mirror descent algorithms previously presented, and show that when each player individually adopts such a bandit algorithm, their joint (mixed) strategy profile quickly converges with implications. (C) 2016 Elsevier B.V. All rights reserved.
Generalized Mirror Descents in Congestion Games with Splittable Flows
(2014-01-01)
Chen, Po-An
;
Lu, Chi-Jen
;
交大名義發表
;
National Chiao Tung University
Different types of dynamics have been studied in repeated game play, and one of them which has received much attention recently consists of those based on "no-regret" algorithms from the area of machine learning. It is known that dynamics based on generic no-regret algorithms may not converge to Nash equilibria in general, but to a larger set of outcomes, namely coarse correlated equilibria. Moreover, convergence results based on generic no-regret algorithms typically use a weaker notion of convergence: the convergence of the average plays instead of the actual plays. Some work has been done showing that when using a specific no-regret algorithm, the well-known multiplicative updates algorithm, convergence of actual plays to equilibria can be shown and better quality of outcomes can be reached for atomic congestion games and load balancing games. Are there more cases of natural no-regret dynamics that perform well in suitable classes of games in terms of convergence and quality of outcomes? We answer this question positively by showing that when each player individually employs the mirror-descent algorithm, a well-known generic no-regret algorithm, the actual plays converge quickly to equilibria in nonatomic congestion games. This gives rise to a family of algorithms, including the multiplicative updates algorithm and the gradient descent algorithm as well as many others. Furthermore, we show that our dynamics achieves good bounds on the quality of outcomes measured by two different social costs: the average individual cost and the maximum individual cost.
Impossibility results on weakly black-box hardness amplification
(2007)
Lu, Chi-Jen
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Tsai, Shi-Chun
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Wu, Hsin-Lung
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資訊工程學系
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Department of Computer Science
We study the task of hardness amplification which transforms a hard function into a harder one. It is known that in a high complexity class such as exponential time, one can convert worst-case hardness into average-case hardness. However, in a lower complexity class such as NP or sub-exponential time, the existence of such an amplification procedure remains unclear. We consider a class of hardness amplifications called weakly black-box hardness amplification, in which the initial hard function is only used as a black box to construct the harder function. We show that if an amplification procedure in TIME(t) can amplify hardness beyond an O(t) factor, then it must basically embed in itself a hard function computable in TIME(t). As a result, it is impossible to have such a hardness amplification with hardness measured against TIME(t). Furthermore, we show that, for any k is an element of N, if an amplification procedure in Sigma P-k can amplify hardness beyond a polynomial factor, then it must basically embed a hard function in Sigma P-k. This in turn implies the impossibility of having such hardness amplification with hardness measured against Sigma P-k/poly.
Improved hardness amplification in NP
(2007-02-12)
Lu, Chi-Jen
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Tsai, Shi-Chun
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Wu, Hsin-Lung
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資訊工程學系
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Department of Computer Science
We study the problem of hardness amplification in NP. We prove that if there is a balanced function in NP such that any circuit of size s(n) = 2(Omega(n)) fails to compute it on a 1/poly(n) fraction of inputs, then there is a function in NP such that any circuit of size s(n) fails to compute it on a 1/2 - 1/s'(n) fraction of inputs, with s'(n) = 2(Omega(n2/13)). This improves the result of Healy et al. (STOC'04), which only achievess s' (n) = 2(Omega(n1/2)) for the case with s(n) = 2(Omega(n)). (c) 2006 Elsevier B.V. All rights reserved.
On the complexity of hardness amplification
(2008-10-01)
Lu, Chi-Jen
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Tsai, Shi-Chun
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Wu, Hsin-Lung
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資訊工程學系
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Department of Computer Science
For delta is an element of (0, 1) and k, n is an element of N, we study the task of transforming a hard function f : {0, 1}(n) -> {0, 1}, with which any small circuit disagrees on (1 - delta)/2 fraction of the input, into a harder function f', with which any small circuit disagrees on (1 - delta(k))/2 fraction of the input. First, we show that such hardness amplification, when carried out in some black-box way, must require a high complexity. In particular, it cannot be realized by a circuit of depth d and size 2(o(k1/d)) or by a nondeterministic circuit of size o(k/log k) (and arbitrary depth) for any delta is an element of (0, 1). This extends the result of Viola, which only works when (1 - delta)/2 is small enough. Furthermore, we show that even without any restriction on the complexity of the amplification procedure, such a black-box hardness amplification must be inherently nonuniform in the following sense. To guarantee the hardness of the resulting function f', even against uniform machines, one has to start with a function f, which is hard against nonuniform algorithms with Omega(k log(1/delta)) bits of advice. This extends the result of Trevisan and Vadhan, which only addresses the case with (1 - delta)/2 = 2(-n). Finally, we derive similar lower bounds for any black-box construction of a pseudorandom generator (PRG) from a hard function. To prove our results, we link the task of hardness amplifications and PRG constructions, respectively, to some type of error-reduction codes, and then we establish lower bounds for such codes, which we hope could find interest in both coding theory and complexity theory.
Playing Congestion Games with Bandit Feedbacks
(2015-01-01)
Chen, Po-An
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Lu, Chi-Jen
;
交大名義發表
;
National Chiao Tung University
Almost all convergence results from each player adopting specific "no-regret" learning algorithms such as multiplicative updates or the more general mirror-descent algorithms in repeated games are only known in the more generous information model, in which each player is assumed to have access to the costs of all possible choices, even the unchosen ones, at each time step. This assumption in general may seem too strong, while a more realistic one is captured by the bandit model, in which each player at each time step is restricted to know only the cost of her currently chosen path, but not any of the unchosen ones. Can convergence still be achieved in such a more challenging bandit model? We answer this question positively. While existing bandit algorithms do not seem to work here, we develop a new family of bandit algorithms based on the mirror-descent algorithm with such a guarantee in atomic congestion games.