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\contentsline {section}{\numberline {1}Introduction}{1}{section.1}
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\contentsline {section}{\numberline {2}Related work}{3}{section.2}
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\contentsline {paragraph}{Extra-gradient method.}{3}{section*.3}
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\contentsline {paragraph}{Finding Nash equilibria in non-convex settings.}{3}{section*.4}
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\contentsline {paragraph}{Opponent shaping and gradient adjustment.}{3}{section*.5}
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\contentsline {section}{\numberline {3}Solving convex games with partial first-order information}{3}{section.3}
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\contentsline {subsection}{\numberline {3.1}Solving convex $n$-player games with gradients}{3}{subsection.3.1}
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\contentsline {paragraph}{Nash equilibria.}{4}{section*.6}
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\contentsline {paragraph}{First-order methods and extrapolation.}{4}{section*.7}
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\contentsline {paragraph}{Computational caveats.}{5}{section*.8}
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\contentsline {subsection}{\numberline {3.2}Partial extrapolation and update for extra-gradient}{5}{subsection.3.2}
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\contentsline {paragraph}{Motivation.}{5}{section*.9}
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\contentsline {paragraph}{Variance reduction for player noise.}{6}{section*.10}
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\contentsline {paragraph}{Sampling strategies.}{6}{section*.11}
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\contentsline {section}{\numberline {4}Sharp analysis of convergence rates}{6}{section.4}
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\contentsline {paragraph}{Importance sampling.}{7}{section*.12}
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\contentsline {section}{\numberline {5}Applications}{7}{section.5}
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\contentsline {subsection}{\numberline {5.1}Random quadratic games}{8}{subsection.5.1}
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\contentsline {paragraph}{Experiments.}{8}{section*.14}
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\contentsline {paragraph}{Results.}{8}{section*.15}
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\contentsline {paragraph}{Spectral effect of sampling.}{9}{section*.16}
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\contentsline {subsection}{\numberline {5.2}Generative adversarial networks (WGAN-GP + ResNet)}{9}{subsection.5.2}
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\contentsline {paragraph}{Results.}{9}{section*.19}
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\contentsline {paragraph}{Interpretation.}{9}{section*.20}
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\contentsline {section}{\numberline {6}Discussion and conclusion}{10}{section.6}
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\contentsline {section}{\numberline {7}Acknowledgements}{10}{section.7}
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\contentsline {section}{\numberline {A}Existing results}{13}{appendix.A}
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\contentsline {subsection}{\numberline {A.1}Mirror-prox}{13}{subsection.A.1}
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\contentsline {paragraph}{Setting and notations.}{13}{section*.22}
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\contentsline {subsection}{\numberline {A.2}Link between convex games and variational inequalities}{14}{subsection.A.2}
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\contentsline {subsection}{\numberline {A.3}Convergence rates for the stochastic mirror-prox}{15}{subsection.A.3}
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\contentsline {subsection}{\numberline {A.4}Player randomness as noise}{16}{subsection.A.4}
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\contentsline {section}{\numberline {B}Proofs and mirror-setting algorithms}{17}{appendix.B}
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\contentsline {subsection}{\numberline {B.1}Useful lemmas}{17}{subsection.B.1}
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\contentsline {subsection}{\numberline {B.2}Doubly-stochastic mirror-prox}{19}{subsection.B.2}
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\contentsline {subsubsection}{\numberline {B.2.1}Algorithm}{19}{subsubsection.B.2.1}
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\contentsline {paragraph}{Notation.}{19}{section*.23}
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\contentsline {paragraph}{Differences with~\autoref {alg:doubly_stoch}}{19}{section*.24}
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\contentsline {subsubsection}{\numberline {B.2.2}Convergence rate under Assumption~\ref {ass:non-smooth} (non-smoothness)}{19}{subsubsection.B.2.2}
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\contentsline {subsubsection}{\numberline {B.2.3}Convergence rate under Assumption~\ref {ass:smooth} (smoothness)}{22}{subsubsection.B.2.3}
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\contentsline {subsection}{\numberline {B.3}Doubly-stochastic mirror-prox with importance sampling}{24}{subsection.B.3}
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\contentsline {subsubsection}{\numberline {B.3.1}Setting}{24}{subsubsection.B.3.1}
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\contentsline {subsubsection}{\numberline {B.3.2}Analysis}{24}{subsubsection.B.3.2}
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\contentsline {subsection}{\numberline {B.4}Doubly-stochastic mirror-prox with variance reduction}{25}{subsection.B.4}
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\contentsline {subsubsection}{\numberline {B.4.1}Algorithm}{25}{subsubsection.B.4.1}
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\contentsline {subsubsection}{\numberline {B.4.2}Convergence rate under Assumption~\ref {ass:smooth} (smoothness)}{26}{subsubsection.B.4.2}
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\contentsline {paragraph}{Preliminaries.}{26}{section*.25}
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\contentsline {paragraph}{Proof.}{27}{section*.26}
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\contentsline {section}{\numberline {C}Spectral convergence analysis for non-constrained 2-player games}{36}{appendix.C}
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\contentsline {subsection}{\numberline {C.1}Recursion operator for the different sampling schemes}{36}{subsection.C.1}
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\contentsline {paragraph}{Full extrapolation and update.}{36}{section*.27}
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\contentsline {paragraph}{Cyclic sampling.}{36}{section*.28}
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\contentsline {paragraph}{Random sampling.}{36}{section*.29}
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\contentsline {subsection}{\numberline {C.2}Convergence behavior through spectral analysis}{37}{subsection.C.2}
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\contentsline {subsection}{\numberline {C.3}Empirical distributions of the spectral radii}{38}{subsection.C.3}
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\contentsline {paragraph}{Experiment.}{38}{section*.31}
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\contentsline {paragraph}{Results and interpretation.}{38}{section*.32}
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\contentsline {section}{\numberline {D}Experimental results and details}{40}{appendix.D}
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\contentsline {subsection}{\numberline {D.1}Quadratic games}{40}{subsection.D.1}
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\contentsline {paragraph}{Generation of random matrices.}{40}{section*.33}
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\contentsline {paragraph}{Parameters for quadratic games.}{40}{section*.35}
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\contentsline {paragraph}{Grids.}{40}{section*.37}
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\contentsline {paragraph}{Limitations in skew non-noisy games.}{41}{section*.38}
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\contentsline {subsection}{\numberline {D.2}Generative adversarial networks}{41}{subsection.D.2}
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\contentsline {paragraph}{Models and loss.}{41}{section*.39}
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\contentsline {paragraph}{Grids.}{41}{section*.40}
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