arXiv subjects

Ioannis Panageas

Publications and source records attributed to Ioannis Panageas.

At least 55 records · Page 3Linked to original sources

arXiv2023

Zero-sum Polymatrix Markov Games: Equilibrium Collapse and Efficient Computation of Nash Equilibria

The works of (Daskalakis et al., 2009, 2022; Jin et al., 2022; Deng et al., 2023) indicate that computing Nash equilibria in multi-player Markov games is a computationally hard task. This fact raises the question of whether or not computational intractability can be circumvented if one focuses on specific classes of Markov games. One such example is two-player zero-sum Markov games, in which efficient ways to compute a Nash equilibrium are known. Inspired by zero-sum polymatrix normal-form games (Cai et al., 2016), we define a class of zero-sum multi-agent Markov games in which there are only pairwise interactions described by a graph that changes per state. For this class of Markov games, we show that an $ε$-approximate Nash equilibrium can be found efficiently. To do so, we generalize the techniques of (Cai et al., 2016), by showing that the set of coarse-correlated equilibria collapses to the set of Nash equilibria. Afterwards, it is possible to use any algorithm in the literature that computes approximate coarse-correlated equilibria Markovian policies to get an approximate Nash equilibrium.

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arXiv2023

Towards convergence to Nash equilibria in two-team zero-sum games

Contemporary applications of machine learning in two-team e-sports and the superior expressivity of multi-agent generative adversarial networks raise important and overlooked theoretical questions regarding optimization in two-team games. Formally, two-team zero-sum games are defined as multi-player games where players are split into two competing sets of agents, each experiencing a utility identical to that of their teammates and opposite to that of the opposing team. We focus on the solution concept of Nash equilibria (NE). We first show that computing NE for this class of games is $\textit{hard}$ for the complexity class ${\mathrm{CLS}}$. To further examine the capabilities of online learning algorithms in games with full-information feedback, we propose a benchmark of a simple -- yet nontrivial -- family of such games. These games do not enjoy the properties used to prove convergence for relevant algorithms. In particular, we use a dynamical systems perspective to demonstrate that gradient descent-ascent, its optimistic variant, optimistic multiplicative weights update, and extra gradient fail to converge (even locally) to a Nash equilibrium. On a brighter note, we propose a first-order method that leverages control theory techniques and under some conditions enjoys last-iterate local convergence to a Nash equilibrium. We also believe our proposed method is of independent interest for general min-max optimization.

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arXiv2022

On Scrambling Phenomena for Randomly Initialized Recurrent Networks

Recurrent Neural Networks (RNNs) frequently exhibit complicated dynamics, and their sensitivity to the initialization process often renders them notoriously hard to train. Recent works have shed light on such phenomena analyzing when exploding or vanishing gradients may occur, either of which is detrimental for training dynamics. In this paper, we point to a formal connection between RNNs and chaotic dynamical systems and prove a qualitatively stronger phenomenon about RNNs than what exploding gradients seem to suggest. Our main result proves that under standard initialization (e.g., He, Xavier etc.), RNNs will exhibit \textit{Li-Yorke chaos} with \textit{constant} probability \textit{independent} of the network's width. This explains the experimentally observed phenomenon of \textit{scrambling}, under which trajectories of nearby points may appear to be arbitrarily close during some timesteps, yet will be far away in future timesteps. In stark contrast to their feedforward counterparts, we show that chaotic behavior in RNNs is preserved under small perturbations and that their expressive power remains exponential in the number of feedback iterations. Our technical arguments rely on viewing RNNs as random walks under non-linear activations, and studying the existence of certain types of higher-order fixed points called \textit{periodic points} that lead to phase transitions from order to chaos.

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arXiv2022

Optimistic Mirror Descent Either Converges to Nash or to Strong Coarse Correlated Equilibria in Bimatrix Games

We show that, for any sufficiently small fixed $ε> 0$, when both players in a general-sum two-player (bimatrix) game employ optimistic mirror descent (OMD) with smooth regularization, learning rate $η= O(ε^2)$ and $T = Ω(\text{poly}(1/ε))$ repetitions, either the dynamics reach an $ε$-approximate Nash equilibrium (NE), or the average correlated distribution of play is an $Ω(\text{poly}(ε))$-strong coarse correlated equilibrium (CCE): any possible unilateral deviation does not only leave the player worse, but will decrease its utility by $Ω(\text{poly}(ε))$. As an immediate consequence, when the iterates of OMD are bounded away from being Nash equilibria in a bimatrix game, we guarantee convergence to an exact CCE after only $O(1)$ iterations. Our results reveal that uncoupled no-regret learning algorithms can converge to CCE in general-sum games remarkably faster than to NE in, for example, zero-sum games. To establish this, we show that when OMD does not reach arbitrarily close to a NE, the (cumulative) regret of both players is not only negative, but decays linearly with time. Given that regret is the canonical measure of performance in online learning, our results suggest that cycling behavior of no-regret learning algorithms in games can be justified in terms of efficiency.

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arXiv2022

Efficiently Computing Nash Equilibria in Adversarial Team Markov Games

Computing Nash equilibrium policies is a central problem in multi-agent reinforcement learning that has received extensive attention both in theory and in practice. However, provable guarantees have been thus far either limited to fully competitive or cooperative scenarios or impose strong assumptions that are difficult to meet in most practical applications. In this work, we depart from those prior results by investigating infinite-horizon \emph{adversarial team Markov games}, a natural and well-motivated class of games in which a team of identically-interested players -- in the absence of any explicit coordination or communication -- is competing against an adversarial player. This setting allows for a unifying treatment of zero-sum Markov games and Markov potential games, and serves as a step to model more realistic strategic interactions that feature both competing and cooperative interests. Our main contribution is the first algorithm for computing stationary $ε$-approximate Nash equilibria in adversarial team Markov games with computational complexity that is polynomial in all the natural parameters of the game, as well as $1/ε$. The proposed algorithm is particularly natural and practical, and it is based on performing independent policy gradient steps for each player in the team, in tandem with best responses from the side of the adversary; in turn, the policy for the adversary is then obtained by solving a carefully constructed linear program. Our analysis leverages non-standard techniques to establish the KKT optimality conditions for a nonlinear program with nonconvex constraints, thereby leading to a natural interpretation of the induced Lagrange multipliers. Along the way, we significantly extend an important characterization of optimal policies in adversarial (normal-form) team games due to Von Stengel and Koller (GEB `97).

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arXiv2022

Accelerated Multiplicative Weights Update Avoids Saddle Points almost always

We consider non-convex optimization problems with constraint that is a product of simplices. A commonly used algorithm in solving this type of problem is the Multiplicative Weights Update (MWU), an algorithm that is widely used in game theory, machine learning and multi-agent systems. Despite it has been known that MWU avoids saddle points, there is a question that remains unaddressed:"Is there an accelerated version of MWU that avoids saddle points provably?" In this paper we provide a positive answer to above question. We provide an accelerated MWU based on Riemannian Accelerated Gradient Descent, and prove that the Riemannian Accelerated Gradient Descent, thus the accelerated MWU, almost always avoid saddle points.

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arXiv2022

On Last-Iterate Convergence Beyond Zero-Sum Games

Most existing results about \emph{last-iterate convergence} of learning dynamics are limited to two-player zero-sum games, and only apply under rigid assumptions about what dynamics the players follow. In this paper we provide new results and techniques that apply to broader families of games and learning dynamics. First, we use a regret-based analysis to show that in a class of games that includes constant-sum polymatrix and strategically zero-sum games, dynamics such as \emph{optimistic mirror descent (OMD)} have \emph{bounded second-order path lengths}, a property which holds even when players employ different algorithms and prediction mechanisms. This enables us to obtain $O(1/\sqrt{T})$ rates and optimal $O(1)$ regret bounds. Our analysis also reveals a surprising property: OMD either reaches arbitrarily close to a Nash equilibrium, or it outperforms the \emph{robust price of anarchy} in efficiency. Moreover, for potential games we establish convergence to an $ε$-equilibrium after $O(1/ε^2)$ iterations for mirror descent under a broad class of regularizers, as well as optimal $O(1)$ regret bounds for OMD variants. Our framework also extends to near-potential games, and unifies known analyses for distributed learning in Fisher's market model. Finally, we analyze the convergence, efficiency, and robustness of \emph{optimistic gradient descent (OGD)} in general-sum continuous games.

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arXiv2021

Frequency-Domain Representation of First-Order Methods: A Simple and Robust Framework of Analysis

Motivated by recent applications in min-max optimization, we employ tools from nonlinear control theory in order to analyze a class of "historical" gradient-based methods, for which the next step lies in the span of the previously observed gradients within a time horizon. Specifically, we leverage techniques developed by Hu and Lessard (2017) to build a frequency-domain framework which reduces the analysis of such methods to numerically-solvable algebraic tasks, establishing linear convergence under a class of strongly monotone and co-coercive operators. On the applications' side, we focus on the Optimistic Gradient Descent (OGD) method, which augments the standard Gradient Descent with an additional past-gradient in the optimization step. The proposed framework leads to a simple and sharp analysis of OGD -- and generalizations thereof -- under a much broader regime of parameters. Notably, this characterization directly extends under adversarial noise in the observed value of the gradient. Moreover, our frequency-domain framework provides an exact quantitative comparison between simultaneous and alternating updates of OGD. An interesting byproduct is that OGD -- and variants thereof -- is an instance of PID control, arguably one of the most influential algorithms of the last century; this observation sheds more light to the stabilizing properties of "optimism".

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arXiv2021

Independent Natural Policy Gradient Always Converges in Markov Potential Games

Multi-agent reinforcement learning has been successfully applied to fully-cooperative and fully-competitive environments, but little is currently known about mixed cooperative/competitive environments. In this paper, we focus on a particular class of multi-agent mixed cooperative/competitive stochastic games called Markov Potential Games (MPGs), which include cooperative games as a special case. Recent results have shown that independent policy gradient converges in MPGs but it was not known whether Independent Natural Policy Gradient converges in MPGs as well. We prove that Independent Natural Policy Gradient always converges in the last iterate using constant learning rates. The proof deviates from the existing approaches and the main challenge lies in the fact that Markov Potential Games do not have unique optimal values (as single-agent settings exhibit) so different initializations can lead to different limit point values. We complement our theoretical results with experiments that indicate that Natural Policy Gradient outperforms Policy Gradient in routing games and congestion games.

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arXiv2020

Fast Convergence of Langevin Dynamics on Manifold: Geodesics meet Log-Sobolev

Sampling is a fundamental and arguably very important task with numerous applications in Machine Learning. One approach to sample from a high dimensional distribution $e^{-f}$ for some function $f$ is the Langevin Algorithm (LA). Recently, there has been a lot of progress in showing fast convergence of LA even in cases where $f$ is non-convex, notably [53], [39] in which the former paper focuses on functions $f$ defined in $\mathbb{R}^n$ and the latter paper focuses on functions with symmetries (like matrix completion type objectives) with manifold structure. Our work generalizes the results of [53] where $f$ is defined on a manifold $M$ rather than $\mathbb{R}^n$. From technical point of view, we show that KL decreases in a geometric rate whenever the distribution $e^{-f}$ satisfies a log-Sobolev inequality on $M$.

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arXiv2020

Better Depth-Width Trade-offs for Neural Networks through the lens of Dynamical Systems

The expressivity of neural networks as a function of their depth, width and type of activation units has been an important question in deep learning theory. Recently, depth separation results for ReLU networks were obtained via a new connection with dynamical systems, using a generalized notion of fixed points of a continuous map $f$, called periodic points. In this work, we strengthen the connection with dynamical systems and we improve the existing width lower bounds along several aspects. Our first main result is period-specific width lower bounds that hold under the stronger notion of $L^1$-approximation error, instead of the weaker classification error. Our second contribution is that we provide sharper width lower bounds, still yielding meaningful exponential depth-width separations, in regimes where previous results wouldn't apply. A byproduct of our results is that there exists a universal constant characterizing the depth-width trade-offs, as long as $f$ has odd periods. Technically, our results follow by unveiling a tighter connection between the following three quantities of a given function: its period, its Lipschitz constant and the growth rate of the number of oscillations arising under compositions of the function $f$ with itself.

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arXiv2020

Efficient Statistics for Sparse Graphical Models from Truncated Samples

In this paper, we study high-dimensional estimation from truncated samples. We focus on two fundamental and classical problems: (i) inference of sparse Gaussian graphical models and (ii) support recovery of sparse linear models. (i) For Gaussian graphical models, suppose $d$-dimensional samples ${\bf x}$ are generated from a Gaussian $N(μ,Σ)$ and observed only if they belong to a subset $S \subseteq \mathbb{R}^d$. We show that $μ$ and $Σ$ can be estimated with error $ε$ in the Frobenius norm, using $\tilde{O}\left(\frac{\textrm{nz}(Σ^{-1})}{ε^2}\right)$ samples from a truncated $\mathcal{N}(μ,Σ)$ and having access to a membership oracle for $S$. The set $S$ is assumed to have non-trivial measure under the unknown distribution but is otherwise arbitrary. (ii) For sparse linear regression, suppose samples $({\bf x},y)$ are generated where $y = {\bf x}^\top{Ω^*} + \mathcal{N}(0,1)$ and $({\bf x}, y)$ is seen only if $y$ belongs to a truncation set $S \subseteq \mathbb{R}$. We consider the case that $Ω^*$ is sparse with a support set of size $k$. Our main result is to establish precise conditions on the problem dimension $d$, the support size $k$, the number of observations $n$, and properties of the samples and the truncation that are sufficient to recover the support of $Ω^*$. Specifically, we show that under some mild assumptions, only $O(k^2 \log d)$ samples are needed to estimate $Ω^*$ in the $\ell_\infty$-norm up to a bounded error. For both problems, our estimator minimizes the sum of the finite population negative log-likelihood function and an $\ell_1$-regularization term.

stat.ML↗

arXiv2020

On the Analysis of EM for truncated mixtures of two Gaussians

Motivated by a recent result of Daskalakis et al. 2018, we analyze the population version of Expectation-Maximization (EM) algorithm for the case of \textit{truncated} mixtures of two Gaussians. Truncated samples from a $d$-dimensional mixture of two Gaussians $\frac{1}{2} \mathcal{N}(\vecμ, \vecΣ)+ \frac{1}{2} \mathcal{N}(-\vecμ, \vecΣ)$ means that a sample is only revealed if it falls in some subset $S \subset \mathbb{R}^d$ of positive (Lebesgue) measure. We show that for $d=1$, EM converges almost surely (under random initialization) to the true mean (variance $σ^2$ is known) for any measurable set $S$. Moreover, for $d>1$ we show EM almost surely converges to the true mean for any measurable set $S$ when the map of EM has only three fixed points, namely $-\vecμ, \vec{0}, \vecμ$ (covariance matrix $\vecΣ$ is known), and prove local convergence if there are more than three fixed points. We also provide convergence rates of our findings. Our techniques deviate from those of Daskalakis et al. 2017, which heavily depend on symmetry that the untruncated problem exhibits. For example, for an arbitrary measurable set $S$, it is impossible to compute a closed form of the update rule of EM. Moreover, arbitrarily truncating the mixture, induces further correlations among the variables. We circumvent these challenges by using techniques from dynamical systems, probability and statistics; implicit function theorem, stability analysis around the fixed points of the update rule of EM and correlation inequalities (FKG).

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arXiv2020

Convergence to Second-Order Stationarity for Non-negative Matrix Factorization: Provably and Concurrently

Non-negative matrix factorization (NMF) is a fundamental non-convex optimization problem with numerous applications in Machine Learning (music analysis, document clustering, speech-source separation etc). Despite having received extensive study, it is poorly understood whether or not there exist natural algorithms that can provably converge to a local minimum. Part of the reason is because the objective is heavily symmetric and its gradient is not Lipschitz. In this paper we define a multiplicative weight update type dynamics (modification of the seminal Lee-Seung algorithm) that runs concurrently and provably avoids saddle points (first order stationary points that are not second order). Our techniques combine tools from dynamical systems such as stability and exploit the geometry of the NMF objective by reducing the standard NMF formulation over the non-negative orthant to a new formulation over (a scaled) simplex. An important advantage of our method is the use of concurrent updates, which permits implementations in parallel computing environments.

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arXiv2020

Logistic-Regression with peer-group effects via inference in higher order Ising models

Spin glass models, such as the Sherrington-Kirkpatrick, Hopfield and Ising models, are all well-studied members of the exponential family of discrete distributions, and have been influential in a number of application domains where they are used to model correlation phenomena on networks. Conventionally these models have quadratic sufficient statistics and consequently capture correlations arising from pairwise interactions. In this work we study extensions of these to models with higher-order sufficient statistics, modeling behavior on a social network with peer-group effects. In particular, we model binary outcomes on a network as a higher-order spin glass, where the behavior of an individual depends on a linear function of their own vector of covariates and some polynomial function of the behavior of others, capturing peer-group effects. Using a {\em single}, high-dimensional sample from such model our goal is to recover the coefficients of the linear function as well as the strength of the peer-group effects. The heart of our result is a novel approach for showing strong concavity of the log pseudo-likelihood of the model, implying statistical error rate of $\sqrt{d/n}$ for the Maximum Pseudo-Likelihood Estimator (MPLE), where $d$ is the dimensionality of the covariate vectors and $n$ is the size of the network (number of nodes). Our model generalizes vanilla logistic regression as well as the peer-effect models studied in recent works, and our results extend these results to accommodate higher-order interactions.

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arXiv2020

Multiplicative Weights Update as a Distributed Constrained Optimization Algorithm: Convergence to Second-order Stationary Points Almost Always

Non-concave maximization has been the subject of much recent study in the optimization and machine learning communities, specifically in deep learning. Recent papers Ge et al, Lee et al (and references therein) indicate that first order methods work well and avoid saddle points. Results as in Lee et al, however, are limited to the \textit{unconstrained} case or for cases where the critical points are in the interior of the feasibility set, which fail to capture some of the most interesting applications. In this paper we focus on \textit{constrained} non-concave maximization. We analyze a variant of a well-established algorithm in machine learning called Multiplicative Weights Update (MWU) for the maximization problem $\max_{\mathbf{x} \in D} P(\mathbf{x})$, where $P$ is non-concave, twice continuously differentiable and $D$ is a product of simplices. We show that MWU converges almost always for small enough stepsizes to critical points that satisfy the second order KKT conditions. We combine techniques from dynamical systems as well as taking advantage of a recent connection between Baum Eagon inequality and MWU (Palaiopanos et al).

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arXiv2019

Depth-Width Trade-offs for ReLU Networks via Sharkovsky's Theorem

Understanding the representational power of Deep Neural Networks (DNNs) and how their structural properties (e.g., depth, width, type of activation unit) affect the functions they can compute, has been an important yet challenging question in deep learning and approximation theory. In a seminal paper, Telgarsky highlighted the benefits of depth by presenting a family of functions (based on simple triangular waves) for which DNNs achieve zero classification error, whereas shallow networks with fewer than exponentially many nodes incur constant error. Even though Telgarsky's work reveals the limitations of shallow neural networks, it does not inform us on why these functions are difficult to represent and in fact he states it as a tantalizing open question to characterize those functions that cannot be well-approximated by smaller depths. In this work, we point to a new connection between DNNs expressivity and Sharkovsky's Theorem from dynamical systems, that enables us to characterize the depth-width trade-offs of ReLU networks for representing functions based on the presence of generalized notion of fixed points, called periodic points (a fixed point is a point of period 1). Motivated by our observation that the triangle waves used in Telgarsky's work contain points of period 3 - a period that is special in that it implies chaotic behavior based on the celebrated result by Li-Yorke - we proceed to give general lower bounds for the width needed to represent periodic functions as a function of the depth. Technically, the crux of our approach is based on an eigenvalue analysis of the dynamical system associated with such functions.

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arXiv2019

Regression from Dependent Observations

The standard linear and logistic regression models assume that the response variables are independent, but share the same linear relationship to their corresponding vectors of covariates. The assumption that the response variables are independent is, however, too strong. In many applications, these responses are collected on nodes of a network, or some spatial or temporal domain, and are dependent. Examples abound in financial and meteorological applications, and dependencies naturally arise in social networks through peer effects. Regression with dependent responses has thus received a lot of attention in the Statistics and Economics literature, but there are no strong consistency results unless multiple independent samples of the vectors of dependent responses can be collected from these models. We present computationally and statistically efficient methods for linear and logistic regression models when the response variables are dependent on a network. Given one sample from a networked linear or logistic regression model and under mild assumptions, we prove strong consistency results for recovering the vector of coefficients and the strength of the dependencies, recovering the rates of standard regression under independent observations. We use projected gradient descent on the negative log-likelihood, or negative log-pseudolikelihood, and establish their strong convexity and consistency using concentration of measure for dependent random variables.

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