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Documents authored by Panageas, Ioannis

Document
DOI: 10.4230/LIPIcs.ITCS.2019.27
Last-Iterate Convergence: Zero-Sum Games and Constrained Min-Max Optimization

Authors: Constantinos Daskalakis and Ioannis Panageas

Published in: LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Abstract
Motivated by applications in Game Theory, Optimization, and Generative Adversarial Networks, recent work of Daskalakis et al [Daskalakis et al., ICLR, 2018] and follow-up work of Liang and Stokes [Liang and Stokes, 2018] have established that a variant of the widely used Gradient Descent/Ascent procedure, called "Optimistic Gradient Descent/Ascent (OGDA)", exhibits last-iterate convergence to saddle points in unconstrained convex-concave min-max optimization problems. We show that the same holds true in the more general problem of constrained min-max optimization under a variant of the no-regret Multiplicative-Weights-Update method called "Optimistic Multiplicative-Weights Update (OMWU)". This answers an open question of Syrgkanis et al [Syrgkanis et al., NIPS, 2015]. The proof of our result requires fundamentally different techniques from those that exist in no-regret learning literature and the aforementioned papers. We show that OMWU monotonically improves the Kullback-Leibler divergence of the current iterate to the (appropriately normalized) min-max solution until it enters a neighborhood of the solution. Inside that neighborhood we show that OMWU becomes a contracting map converging to the exact solution. We believe that our techniques will be useful in the analysis of the last iterate of other learning algorithms.
Cite as

Constantinos Daskalakis and Ioannis Panageas. Last-Iterate Convergence: Zero-Sum Games and Constrained Min-Max Optimization. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{daskalakis_et_al:LIPIcs.ITCS.2019.27,
  author =	{Daskalakis, Constantinos and Panageas, Ioannis},
  title =	{{Last-Iterate Convergence: Zero-Sum Games and Constrained Min-Max Optimization}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{27:1--27:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{124},
  editor =	{Blum, Avrim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.27},
  URN =		{urn:nbn:de:0030-drops-101204},
  doi =		{10.4230/LIPIcs.ITCS.2019.27},
  annote =	{Keywords: No regret learning, Zero-sum games, Convergence, Dynamical Systems, KL divergence}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2017.2
Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions

Authors: Ioannis Panageas and Georgios Piliouras

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract
Given a twice continuously differentiable cost function f, we prove that the set of initial conditions so that gradient descent converges to saddle points where \nabla^2 f has at least one strictly negative eigenvalue, has (Lebesgue) measure zero, even for cost functions f with non-isolated critical points, answering an open question in [Lee, Simchowitz, Jordan, Recht, COLT 2016]. Moreover, this result extends to forward-invariant convex subspaces, allowing for weak (non-globally Lipschitz) smoothness assumptions. Finally, we produce an upper bound on the allowable step-size.
Cite as

Ioannis Panageas and Georgios Piliouras. Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 2:1-2:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{panageas_et_al:LIPIcs.ITCS.2017.2,
  author =	{Panageas, Ioannis and Piliouras, Georgios},
  title =	{{Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{2:1--2:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.2},
  URN =		{urn:nbn:de:0030-drops-81640},
  doi =		{10.4230/LIPIcs.ITCS.2017.2},
  annote =	{Keywords: Gradient Descent, Center-stable manifold, Saddle points, Hessian}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2017.16
Mutation, Sexual Reproduction and Survival in Dynamic Environments

Authors: Ruta Mehta, Ioannis Panageas, Georgios Piliouras, Prasad Tetali, and Vijay V. Vazirani

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract
A new approach to understanding evolution [Valiant, JACM 2009], namely viewing it through the lens of computation, has already started yielding new insights, e.g., natural selection under sexual reproduction can be interpreted as the Multiplicative Weight Update (MWU) Algorithm in coordination games played among genes [Chastain, Livnat, Papadimitriou, Vazirani, PNAS 2014]. Using this machinery, we study the role of mutation in changing environments in the presence of sexual reproduction. Following [Wolf, Vazirani, Arkin, J. Theor. Biology], we model changing environments via a Markov chain, with the states representing environments, each with its own fitness matrix. In this setting, we show that in the absence of mutation, the population goes extinct, but in the presence of mutation, the population survives with positive probability. On the way to proving the above theorem, we need to establish some facts about dynamics in games. We provide the first, to our knowledge, polynomial convergence bound for noisy MWU in a coordination game. Finally, we also show that in static environments, sexual evolution with mutation converges, for any level of mutation.
Cite as

Ruta Mehta, Ioannis Panageas, Georgios Piliouras, Prasad Tetali, and Vijay V. Vazirani. Mutation, Sexual Reproduction and Survival in Dynamic Environments. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 16:1-16:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{mehta_et_al:LIPIcs.ITCS.2017.16,
  author =	{Mehta, Ruta and Panageas, Ioannis and Piliouras, Georgios and Tetali, Prasad and Vazirani, Vijay V.},
  title =	{{Mutation, Sexual Reproduction and Survival in Dynamic Environments}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{16:1--16:29},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.16},
  URN =		{urn:nbn:de:0030-drops-81655},
  doi =		{10.4230/LIPIcs.ITCS.2017.16},
  annote =	{Keywords: Evolution, Non-linear dynamics, Mutation}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2017.140
Opinion Dynamics in Networks: Convergence, Stability and Lack of Explosion

Authors: Tung Mai, Ioannis Panageas, and Vijay V. Vazirani

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Abstract
Inspired by the work of Kempe et al. [Kempe, Kleinberg, Oren, Slivkins, EC 2013], we introduce and analyze a model on opinion formation; the update rule of our dynamics is a simplified version of that of [Kempe, Kleinberg, Oren, Slivkins, EC 2013]. We assume that the population is partitioned into types whose interaction pattern is specified by a graph. Interaction leads to population mass moving from types of smaller mass to those of bigger mass. We show that starting uniformly at random over all population vectors on the simplex, our dynamics converges point-wise with probability one to an independent set. This settles an open problem of [Kempe, Kleinberg, Oren, Slivkins, EC 2013], as applicable to our dynamics. We believe that our techniques can be used to settle the open problem for the Kempe et al. dynamics as well. Next, we extend the model of Kempe et al. by introducing the notion of birth and death of types, with the interaction graph evolving appropriately. Birth of types is determined by a Bernoulli process and types die when their population mass is less than epsilon (a parameter). We show that if the births are infrequent, then there are long periods of "stability" in which there is no population mass that moves. Finally we show that even if births are frequent and "stability" is not attained, the total number of types does not explode: it remains logarithmic in 1/epsilon.
Cite as

Tung Mai, Ioannis Panageas, and Vijay V. Vazirani. Opinion Dynamics in Networks: Convergence, Stability and Lack of Explosion. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 140:1-140:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{mai_et_al:LIPIcs.ICALP.2017.140,
  author =	{Mai, Tung and Panageas, Ioannis and Vazirani, Vijay V.},
  title =	{{Opinion Dynamics in Networks: Convergence, Stability and Lack of Explosion}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{140:1--140:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.140},
  URN =		{urn:nbn:de:0030-drops-74440},
  doi =		{10.4230/LIPIcs.ICALP.2017.140},
  annote =	{Keywords: Opinion Dynamics, Convergence, Jacobian, Center-stable Manifold}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2016.63
Mixing Time of Markov Chains, Dynamical Systems and Evolution

Authors: Ioannis Panageas and Nisheeth K. Vishnoi

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract
In this paper we study the mixing time of evolutionary Markov chains over populations of a fixed size (N) in which each individual can be one of m types. These Markov chains have the property that they are guided by a dynamical system from the m-dimensional probability simplex to itself. Roughly, given the current state of the Markov chain, which can be viewed as a probability distribution over the m types, the next state is generated by applying this dynamical system to this distribution, and then sampling from it N times. Many processes in nature, from biology to sociology, are evolutionary and such chains can be used to model them. In this study, the mixing time is of particular interest as it determines the speed of evolution and whether the statistics of the steady state can be efficiently computed. In a recent result [Panageas, Srivastava, Vishnoi, Soda, 2016], it was suggested that the mixing time of such Markov chains is connected to the geometry of this guiding dynamical system. In particular, when the dynamical system has a fixed point which is a global attractor, then the mixing is fast. The limit sets of dynamical systems, however, can exhibit more complex behavior: they could have multiple fixed points that are not necessarily stable, periodic orbits, or even chaos. Such behavior arises in important evolutionary settings such as the dynamics of sexual evolution and that of grammar acquisition. In this paper we prove that the geometry of the dynamical system can also give tight mixing time bounds when the dynamical system has multiple fixed points and periodic orbits. We show that the mixing time continues to remain small in the presence of several unstable fixed points and is exponential in N when there are two or more stable fixed points. As a consequence of our results, we obtain a phase transition result for the mixing time of the sexual/grammar model mentioned above. We arrive at the conclusion that in the interesting parameter regime for these models, i.e., when there are multiple stable fixed points, the mixing is slow. Our techniques strengthen the connections between Markov chains and dynamical systems and we expect that the tools developed in this paper should have a wider applicability.
Cite as

Ioannis Panageas and Nisheeth K. Vishnoi. Mixing Time of Markov Chains, Dynamical Systems and Evolution. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 63:1-63:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{panageas_et_al:LIPIcs.ICALP.2016.63,
  author =	{Panageas, Ioannis and Vishnoi, Nisheeth K.},
  title =	{{Mixing Time of Markov Chains, Dynamical Systems and Evolution}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{63:1--63:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.63},
  URN =		{urn:nbn:de:0030-drops-62213},
  doi =		{10.4230/LIPIcs.ICALP.2016.63},
  annote =	{Keywords: Markov chains, Mixing time, Dynamical Systems, Evolutionary dynamics, Language evolution}
}
Document
DOI: 10.4230/LIPIcs.ESA.2016.65
The Computational Complexity of Genetic Diversity

Authors: Ruta Mehta, Ioannis Panageas, Georgios Piliouras, and Sadra Yazdanbod

Published in: LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Abstract
A key question in biological systems is whether genetic diversity persists in the long run under evolutionary competition, or whether a single dominant genotype emerges. Classic work by [Kalmus, J. og Genetics, 1945] has established that even in simple diploid species (species with chromosome pairs) diversity can be guaranteed as long as the heterozygous (having different alleles for a gene on two chromosomes) individuals enjoy a selective advantage. Despite the classic nature of the problem, as we move towards increasingly polymorphic traits (e.g., human blood types) predicting diversity (and its implications) is still not fully understood. Our key contribution is to establish complexity theoretic hardness results implying that even in the textbook case of single locus (gene) diploid models, predicting whether diversity survives or not given its fitness landscape is algorithmically intractable. Our hardness results are structurally robust along several dimensions, e.g., choice of parameter distribution, different definitions of stability/persistence, restriction to typical subclasses of fitness landscapes. Technically, our results exploit connections between game theory, nonlinear dynamical systems, and complexity theory and establish hardness results for predicting the evolution of a deterministic variant of the well known multiplicative weights update algorithm in symmetric coordination games; finding one Nash equilibrium is easy in these games. In the process we characterize stable fixed points of these dynamics using the notions of Nash equilibrium and negative semidefiniteness. This as well as hardness results for decision problems in coordination games may be of independent interest. Finally, we complement our results by establishing that under randomly chosen fitness landscapes diversity survives with significant probability. The full version of this paper is available at http://arxiv.org/abs/1411.6322.
Cite as

Ruta Mehta, Ioannis Panageas, Georgios Piliouras, and Sadra Yazdanbod. The Computational Complexity of Genetic Diversity. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 65:1-65:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{mehta_et_al:LIPIcs.ESA.2016.65,
  author =	{Mehta, Ruta and Panageas, Ioannis and Piliouras, Georgios and Yazdanbod, Sadra},
  title =	{{The Computational Complexity of Genetic Diversity}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{65:1--65:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Sankowski, Piotr and Zaroliagis, Christos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.65},
  URN =		{urn:nbn:de:0030-drops-64073},
  doi =		{10.4230/LIPIcs.ESA.2016.65},
  annote =	{Keywords: Dynamical Systems, Stability, Complexity, Optimization, Equilibrium}
}
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