Document

**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

We study the complexity of computing stationary Nash equilibrium (NE) in n-player infinite-horizon general-sum stochastic games. We focus on the problem of computing NE in such stochastic games when each player is restricted to choosing a stationary policy and rewards are discounted. First, we prove that computing such NE is in PPAD (in addition to clearly being PPAD-hard). Second, we consider turn-based specializations of such games where at each state there is at most a single player that can take actions and show that these (seemingly-simpler) games remain PPAD-hard. Third, we show that under further structural assumptions on the rewards computing NE in such turn-based games is possible in polynomial time. Towards achieving these results we establish structural facts about stochastic games of broader utility, including monotonicity of utilities under single-state single-action changes and reductions to settings where each player controls a single state.

Yujia Jin, Vidya Muthukumar, and Aaron Sidford. The Complexity of Infinite-Horizon General-Sum Stochastic Games. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 76:1-76:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jin_et_al:LIPIcs.ITCS.2023.76, author = {Jin, Yujia and Muthukumar, Vidya and Sidford, Aaron}, title = {{The Complexity of Infinite-Horizon General-Sum Stochastic Games}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {76:1--76:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.76}, URN = {urn:nbn:de:0030-drops-175791}, doi = {10.4230/LIPIcs.ITCS.2023.76}, annote = {Keywords: complexity, stochastic games, general-sum games, Nash equilibrium} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Designing efficient dynamic graph algorithms against an adaptive adversary is a major goal in the field of dynamic graph algorithms and has witnessed many exciting recent developments in, e.g., dynamic matching (Wajc STOC'20) and decremental shortest paths (Chuzhoy and Khanna STOC'19). Compared to other graph primitives (e.g. spanning trees and matchings), designing such algorithms for graph spanners and (more broadly) graph sparsifiers poses a unique challenge since there is no fast deterministic algorithm known for static computation and the lack of a way to adjust the output slowly (known as "small recourse/replacements").
This paper presents the first non-trivial efficient adaptive algorithms for maintaining many sparsifiers against an adaptive adversary. Specifically, we present algorithms that maintain
1) a polylog(n)-spanner of size Õ(n) in polylog(n) amortized update time,
2) an O(k)-approximate cut sparsifier of size Õ(n) in Õ(n^{1/k}) amortized update time, and
3) a polylog(n)-approximate spectral sparsifier in polylog(n) amortized update time. Our bounds are the first non-trivial ones even when only the recourse is concerned. Our results hold even against a stronger adversary, who can access the random bits previously used by the algorithms and the amortized update time of all algorithms can be made worst-case by paying sub-polynomial factors. Our spanner result resolves an open question by Ahmed et al. (2019) and our results and techniques imply additional improvements over existing results, including (i) answering open questions about decremental single-source shortest paths by Chuzhoy and Khanna (STOC'19) and Gutenberg and Wulff-Nilsen (SODA'20), implying a nearly-quadratic time algorithm for approximating minimum-cost unit-capacity flow and (ii) de-amortizing a result of Abraham et al. (FOCS'16) for dynamic spectral sparsifiers.
Our results are based on two novel techniques. The first technique is a generic black-box reduction that allows us to assume that the graph is initially an expander with almost uniform-degree and, more importantly, stays as an almost uniform-degree expander while undergoing only edge deletions. The second technique is called proactive resampling: here we constantly re-sample parts of the input graph so that, independent of an adversary’s computational power, a desired structure of the underlying graph can be always maintained. Despite its simplicity, the analysis of this sampling scheme is far from trivial, because the adversary can potentially create dependencies between the random choices used by the algorithm. We believe these two techniques could be useful for developing other adaptive algorithms.

Aaron Bernstein, Jan van den Brand, Maximilian Probst Gutenberg, Danupon Nanongkai, Thatchaphol Saranurak, Aaron Sidford, and He Sun. Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bernstein_et_al:LIPIcs.ICALP.2022.20, author = {Bernstein, Aaron and van den Brand, Jan and Probst Gutenberg, Maximilian and Nanongkai, Danupon and Saranurak, Thatchaphol and Sidford, Aaron and Sun, He}, title = {{Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {20:1--20:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.20}, URN = {urn:nbn:de:0030-drops-163611}, doi = {10.4230/LIPIcs.ICALP.2022.20}, annote = {Keywords: dynamic graph algorithm, adaptive adversary, spanner, sparsifier} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Box-simplex games are a family of bilinear minimax objectives which encapsulate graph-structured problems such as maximum flow [Sherman, 2017], optimal transport [Arun Jambulapati et al., 2019], and bipartite matching [Sepehr Assadi et al., 2022]. We develop efficient near-linear time, high-accuracy solvers for regularized variants of these games. Beyond the immediate applications of such solvers for computing Sinkhorn distances, a prominent tool in machine learning, we show that these solvers can be used to obtain improved running times for maintaining a (fractional) ε-approximate maximum matching in a dynamic decremental bipartite graph against an adaptive adversary. We give a generic framework which reduces this dynamic matching problem to solving regularized graph-structured optimization problems to high accuracy. Through our reduction framework, our regularized box-simplex game solver implies a new algorithm for dynamic decremental bipartite matching in total time Õ(m ⋅ ε^{-3}), from an initial graph with m edges and n nodes. We further show how to use recent advances in flow optimization [Chen et al., 2022] to improve our runtime to m^{1 + o(1)} ⋅ ε^{-2}, thereby demonstrating the versatility of our reduction-based approach. These results improve upon the previous best runtime of Õ(m ⋅ ε^{-4}) [Aaron Bernstein et al., 2020] and illustrate the utility of using regularized optimization problem solvers for designing dynamic algorithms.

Arun Jambulapati, Yujia Jin, Aaron Sidford, and Kevin Tian. Regularized Box-Simplex Games and Dynamic Decremental Bipartite Matching. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 77:1-77:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{jambulapati_et_al:LIPIcs.ICALP.2022.77, author = {Jambulapati, Arun and Jin, Yujia and Sidford, Aaron and Tian, Kevin}, title = {{Regularized Box-Simplex Games and Dynamic Decremental Bipartite Matching}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {77:1--77:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.77}, URN = {urn:nbn:de:0030-drops-164181}, doi = {10.4230/LIPIcs.ICALP.2022.77}, annote = {Keywords: bipartite matching, decremental matching, dynamic algorithms, continuous optimization, box-simplex games, primal-dual method} }

Document

**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

We show that standard extragradient methods (i.e. mirror prox [Arkadi Nemirovski, 2004] and dual extrapolation [Yurii Nesterov, 2007]) recover optimal accelerated rates for first-order minimization of smooth convex functions. To obtain this result we provide fine-grained characterization of the convergence rates of extragradient methods for solving monotone variational inequalities in terms of a natural condition we call relative Lipschitzness. We further generalize this framework to handle local and randomized notions of relative Lipschitzness and thereby recover rates for box-constrained 𝓁_∞ regression based on area convexity [Jonah Sherman, 2017] and complexity bounds achieved by accelerated (randomized) coordinate descent [Zeyuan {Allen Zhu} et al., 2016; Yurii Nesterov and Sebastian U. Stich, 2017] for smooth convex function minimization.

Michael B. Cohen, Aaron Sidford, and Kevin Tian. Relative Lipschitzness in Extragradient Methods and a Direct Recipe for Acceleration. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 62:1-62:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cohen_et_al:LIPIcs.ITCS.2021.62, author = {Cohen, Michael B. and Sidford, Aaron and Tian, Kevin}, title = {{Relative Lipschitzness in Extragradient Methods and a Direct Recipe for Acceleration}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {62:1--62:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.62}, URN = {urn:nbn:de:0030-drops-136011}, doi = {10.4230/LIPIcs.ITCS.2021.62}, annote = {Keywords: Variational inequalities, minimax optimization, acceleration, 𝓁\underline∞ regression} }

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RANDOM

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

We give a deterministic, nearly logarithmic-space algorithm that given an undirected graph G, a positive integer r, and a set S of vertices, approximates the conductance of S in the r-step random walk on G to within a factor of 1+epsilon, where epsilon>0 is an arbitrarily small constant. More generally, our algorithm computes an epsilon-spectral approximation to the normalized Laplacian of the r-step walk.
Our algorithm combines the derandomized square graph operation [Eyal Rozenman and Salil Vadhan, 2005], which we recently used for solving Laplacian systems in nearly logarithmic space [Murtagh et al., 2017], with ideas from [Cheng et al., 2015], which gave an algorithm that is time-efficient (while ours is space-efficient) and randomized (while ours is deterministic) for the case of even r (while ours works for all r). Along the way, we provide some new results that generalize technical machinery and yield improvements over previous work. First, we obtain a nearly linear-time randomized algorithm for computing a spectral approximation to the normalized Laplacian for odd r. Second, we define and analyze a generalization of the derandomized square for irregular graphs and for sparsifying the product of two distinct graphs. As part of this generalization, we also give a strongly explicit construction of expander graphs of every size.

Jack Murtagh, Omer Reingold, Aaron Sidford, and Salil Vadhan. Deterministic Approximation of Random Walks in Small Space. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 42:1-42:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{murtagh_et_al:LIPIcs.APPROX-RANDOM.2019.42, author = {Murtagh, Jack and Reingold, Omer and Sidford, Aaron and Vadhan, Salil}, title = {{Deterministic Approximation of Random Walks in Small Space}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {42:1--42:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.42}, URN = {urn:nbn:de:0030-drops-112577}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.42}, annote = {Keywords: random walks, space complexity, derandomization, spectral approximation, expander graphs} }

Document

**Published in:** LIPIcs, Volume 93, 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)

This work provides a simplified proof of the statistical minimax
optimality of (iterate averaged) stochastic gradient descent (SGD), for
the special case of least squares. This result is obtained by
analyzing SGD as a stochastic process and by sharply characterizing
the stationary covariance matrix of this process. The finite rate optimality characterization captures the
constant factors and addresses model mis-specification.

Prateek Jain, Sham M. Kakade, Rahul Kidambi, Praneeth Netrapalli, Venkata Krishna Pillutla, and Aaron Sidford. A Markov Chain Theory Approach to Characterizing the Minimax Optimality of Stochastic Gradient Descent (for Least Squares). In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 2:1-2:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{jain_et_al:LIPIcs.FSTTCS.2017.2, author = {Jain, Prateek and Kakade, Sham M. and Kidambi, Rahul and Netrapalli, Praneeth and Pillutla, Venkata Krishna and Sidford, Aaron}, title = {{A Markov Chain Theory Approach to Characterizing the Minimax Optimality of Stochastic Gradient Descent (for Least Squares)}}, booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)}, pages = {2:1--2:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-055-2}, ISSN = {1868-8969}, year = {2018}, volume = {93}, editor = {Lokam, Satya and Ramanujam, R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.2}, URN = {urn:nbn:de:0030-drops-83941}, doi = {10.4230/LIPIcs.FSTTCS.2017.2}, annote = {Keywords: Stochastic Gradient Descent, Minimax Optimality, Least Squares Regression} }

Document

**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

Understanding the singular value spectrum of an n x n matrix A is a fundamental task in countless numerical computation and data analysis applications. In matrix multiplication time, it is possible to perform a full SVD of A and directly compute the singular values \sigma_1,...,\sigma_n. However, little is known about algorithms that break this runtime barrier.
Using tools from stochastic trace estimation, polynomial approximation, and fast linear system solvers, we show how to efficiently isolate different ranges of A's spectrum and approximate the number of singular values in these ranges. We thus effectively compute an approximate histogram of the spectrum, which can stand in for the true singular values in many applications.
We use our histogram primitive to give the first algorithms for approximating a wide class of symmetric matrix norms and spectral sums faster than the best known runtime for matrix multiplication. For example, we show how to obtain a (1 + \epsilon) approximation to the Schatten 1-norm (i.e. the nuclear or trace norm) in just ~ O((nnz(A)n^{1/3} + n^2)\epsilon^{-3}) time for A with uniform row sparsity or \tilde O(n^{2.18} \epsilon^{-3}) time for dense matrices. The runtime scales smoothly for general Schatten-p norms, notably becoming \tilde O (p nnz(A) \epsilon^{-3}) for any real p >= 2.
At the same time, we show that the complexity of spectrum approximation is inherently tied to fast matrix multiplication in the small \epsilon regime. We use fine-grained complexity to give conditional lower bounds for spectrum approximation, showing that achieving milder \epsilon dependencies in our algorithms would imply triangle detection algorithms for general graphs running in faster than state of the art matrix multiplication time. This further implies, through a reduction of (Williams & William, 2010), that highly accurate spectrum approximation algorithms running in subcubic time can be used to give subcubic time matrix multiplication. As an application of our bounds, we show that precisely computing all effective resistances in a graph in less than matrix multiplication time is likely difficult, barring a major algorithmic breakthrough.

Cameron Musco, Praneeth Netrapalli, Aaron Sidford, Shashanka Ubaru, and David P. Woodruff. Spectrum Approximation Beyond Fast Matrix Multiplication: Algorithms and Hardness. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 8:1-8:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{musco_et_al:LIPIcs.ITCS.2018.8, author = {Musco, Cameron and Netrapalli, Praneeth and Sidford, Aaron and Ubaru, Shashanka and Woodruff, David P.}, title = {{Spectrum Approximation Beyond Fast Matrix Multiplication: Algorithms and Hardness}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {8:1--8:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.8}, URN = {urn:nbn:de:0030-drops-83397}, doi = {10.4230/LIPIcs.ITCS.2018.8}, annote = {Keywords: spectrum approximation, matrix norm computation, fine-grained complexity, linear algebra} }

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