arXiv subjects

Ali Jadbabaie

Publications and source records attributed to Ali Jadbabaie.

At least 73 records · Page 4Linked to original sources

arXiv2020

Resilient Monotone Sequential Maximization

Applications in machine learning, optimization, and control require the sequential selection of a few system elements, such as sensors, data, or actuators, to optimize the system performance across multiple time steps. However, in failure-prone and adversarial environments, sensors get attacked, data get deleted, and actuators fail. Thence, traditional sequential design paradigms become insufficient and, in contrast, resilient sequential designs that adapt against system-wide attacks, deletions, or failures become important. In general, resilient sequential design problems are computationally hard. Also, even though they often involve objective functions that are monotone and (possibly) submodular, no scalable approximation algorithms are known for their solution. In this paper, we provide the first scalable algorithm, that achieves the following characteristics: system-wide resiliency, i.e., the algorithm is valid for any number of denial-of-service attacks, deletions, or failures; adaptiveness, i.e., at each time step, the algorithm selects system elements based on the history of inflicted attacks, deletions, or failures; and provable approximation performance, i.e., the algorithm guarantees for monotone objective functions a solution close to the optimal. We quantify the algorithm's approximation performance using a notion of curvature for monotone (not necessarily submodular) set functions. Finally, we support our theoretical analyses with simulated experiments, by considering a control-aware sensor scheduling scenario, namely, sensing-constrained robot navigation.

stat.ML↗

arXiv2020

Robust and Adaptive Sequential Submodular Optimization

Emerging applications of control, estimation, and machine learning, ranging from target tracking to decentralized model fitting, pose resource constraints that limit which of the available sensors, actuators, or data can be simultaneously used across time. Therefore, many researchers have proposed solutions within discrete optimization frameworks where the optimization is performed over finite sets. By exploiting notions of discrete convexity, such as submodularity, the researchers have been able to provide scalable algorithms with provable suboptimality bounds. In this paper, we consider such problems but in adversarial environments, where in every step a number of the chosen elements in the optimization is removed due to failures/attacks. Specifically, we consider for the first time a sequential version of the problem that allows us to observe the failures and adapt, while the attacker also adapts to our response. We call the novel problem Robust Sequential submodular Maximization (RSM). Generally, the problem is computationally hard and no scalable algorithm is known for its solution. However, in this paper we propose Robust and Adaptive Maximization (RAM), the first scalable algorithm. RAM runs in an online fashion, adapting in every step to the history of failures. Also, it guarantees a near-optimal performance, even against any number of failures among the used elements. Particularly, RAM has both provable per-instance a priori bounds and tight and/or optimal a posteriori bounds. Finally, we demonstrate RAM's near-optimality in simulations across various application scenarios, along with its robustness against several failure types, from worst-case to random.

math.OC↗

arXiv2020

A General Framework for Distributed Inference with Uncertain Models

This paper studies the problem of distributed classification with a network of heterogeneous agents. The agents seek to jointly identify the underlying target class that best describes a sequence of observations. The problem is first abstracted to a hypothesis-testing framework, where we assume that the agents seek to agree on the hypothesis (target class) that best matches the distribution of observations. Non-Bayesian social learning theory provides a framework that solves this problem in an efficient manner by allowing the agents to sequentially communicate and update their beliefs for each hypothesis over the network. Most existing approaches assume that agents have access to exact statistical models for each hypothesis. However, in many practical applications, agents learn the likelihood models based on limited data, which induces uncertainty in the likelihood function parameters. In this work, we build upon the concept of uncertain models to incorporate the agents' uncertainty in the likelihoods by identifying a broad set of parametric distribution that allows the agents' beliefs to converge to the same result as a centralized approach. Furthermore, we empirically explore extensions to non-parametric models to provide a generalized framework of uncertain models in non-Bayesian social learning.

cs.AI↗

arXiv2020

Gradient-Based Empirical Risk Minimization using Local Polynomial Regression

In this paper, we consider the problem of empirical risk minimization (ERM) of smooth, strongly convex loss functions using iterative gradient-based methods. A major goal of this literature has been to compare different algorithms, such as gradient descent (GD) or stochastic gradient descent (SGD), by analyzing their rates of convergence to $ε$-approximate solutions. For example, the oracle complexity of GD is $O(n\log(ε^{-1}))$, where $n$ is the number of training samples. When $n$ is large, this can be expensive in practice, and SGD is preferred due to its oracle complexity of $O(ε^{-1})$. Such standard analyses only utilize the smoothness of the loss function in the parameter being optimized. In contrast, we demonstrate that when the loss function is smooth in the data, we can learn the oracle at every iteration and beat the oracle complexities of both GD and SGD in important regimes. Specifically, at every iteration, our proposed algorithm performs local polynomial regression to learn the gradient of the loss function, and then estimates the true gradient of the ERM objective function. We establish that the oracle complexity of our algorithm scales like $\tilde{O}((p ε^{-1})^{d/(2η)})$ (neglecting sub-dominant factors), where $d$ and $p$ are the data and parameter space dimensions, respectively, and the gradient of the loss function belongs to a $η$-Hölder class with respect to the data. Our proof extends the analysis of local polynomial regression in non-parametric statistics to provide interpolation guarantees in multivariate settings, and also exploits tools from the inexact GD literature. Unlike GD and SGD, the complexity of our method depends on $d$ and $p$. However, when $d$ is small and the loss function exhibits modest smoothness in the data, our algorithm beats GD and SGD in oracle complexity for a very broad range of $p$ and $ε$.

cs.LG↗

arXiv2020

A Distributed Cubic-Regularized Newton Method for Smooth Convex Optimization over Networks

We propose a distributed, cubic-regularized Newton method for large-scale convex optimization over networks. The proposed method requires only local computations and communications and is suitable for federated learning applications over arbitrary network topologies. We show a $O(k^{{-}3})$ convergence rate when the cost function is convex with Lipschitz gradient and Hessian, with $k$ being the number of iterations. We further provide network-dependent bounds for the communication required in each step of the algorithm. We provide numerical experiments that validate our theoretical results.

math.OC↗

arXiv2020

Complexity of Finding Stationary Points of Nonsmooth Nonconvex Functions

We provide the first non-asymptotic analysis for finding stationary points of nonsmooth, nonconvex functions. In particular, we study the class of Hadamard semi-differentiable functions, perhaps the largest class of nonsmooth functions for which the chain rule of calculus holds. This class contains examples such as ReLU neural networks and others with non-differentiable activation functions. We first show that finding an $ε$-stationary point with first-order methods is impossible in finite time. We then introduce the notion of $(δ, ε)$-stationarity, which allows for an $ε$-approximate gradient to be the convex combination of generalized gradients evaluated at points within distance $δ$ to the solution. We propose a series of randomized first-order methods and analyze their complexity of finding a $(δ, ε)$-stationary point. Furthermore, we provide a lower bound and show that our stochastic algorithm has min-max optimal dependence on $δ$. Empirically, our methods perform well for training ReLU neural networks.

math.OC↗

arXiv2020

GAT-GMM: Generative Adversarial Training for Gaussian Mixture Models

Generative adversarial networks (GANs) learn the distribution of observed samples through a zero-sum game between two machine players, a generator and a discriminator. While GANs achieve great success in learning the complex distribution of image, sound, and text data, they perform suboptimally in learning multi-modal distribution-learning benchmarks including Gaussian mixture models (GMMs). In this paper, we propose Generative Adversarial Training for Gaussian Mixture Models (GAT-GMM), a minimax GAN framework for learning GMMs. Motivated by optimal transport theory, we design the zero-sum game in GAT-GMM using a random linear generator and a softmax-based quadratic discriminator architecture, which leads to a non-convex concave minimax optimization problem. We show that a Gradient Descent Ascent (GDA) method converges to an approximate stationary minimax point of the GAT-GMM optimization problem. In the benchmark case of a mixture of two symmetric, well-separated Gaussians, we further show this stationary point recovers the true parameters of the underlying GMM. We numerically support our theoretical findings by performing several experiments, which demonstrate that GAT-GMM can perform as well as the expectation-maximization algorithm in learning mixtures of two Gaussians.

cs.LG↗

arXiv2020

Robust Federated Learning: The Case of Affine Distribution Shifts

Federated learning is a distributed paradigm that aims at training models using samples distributed across multiple users in a network while keeping the samples on users' devices with the aim of efficiency and protecting users privacy. In such settings, the training data is often statistically heterogeneous and manifests various distribution shifts across users, which degrades the performance of the learnt model. The primary goal of this paper is to develop a robust federated learning algorithm that achieves satisfactory performance against distribution shifts in users' samples. To achieve this goal, we first consider a structured affine distribution shift in users' data that captures the device-dependent data heterogeneity in federated settings. This perturbation model is applicable to various federated learning problems such as image classification where the images undergo device-dependent imperfections, e.g. different intensity, contrast, and brightness. To address affine distribution shifts across users, we propose a Federated Learning framework Robust to Affine distribution shifts (FLRA) that is provably robust against affine Wasserstein shifts to the distribution of observed samples. To solve the FLRA's distributed minimax problem, we propose a fast and efficient optimization method and provide convergence guarantees via a gradient Descent Ascent (GDA) method. We further prove generalization error bounds for the learnt classifier to show proper generalization from empirical distribution of samples to the true underlying distribution. We perform several numerical experiments to empirically support FLRA. We show that an affine distribution shift indeed suffices to significantly decrease the performance of the learnt classifier in a new test user, and our proposed algorithm achieves a significant gain in comparison to standard federated learning and adversarial training methods.

cs.LG↗

arXiv2020

Estimation of Skill Distributions

In this paper, we study the problem of learning the skill distribution of a population of agents from observations of pairwise games in a tournament. These games are played among randomly drawn agents from the population. The agents in our model can be individuals, sports teams, or Wall Street fund managers. Formally, we postulate that the likelihoods of game outcomes are governed by the Bradley-Terry-Luce (or multinomial logit) model, where the probability of an agent beating another is the ratio between its skill level and the pairwise sum of skill levels, and the skill parameters are drawn from an unknown skill density of interest. The problem is, in essence, to learn a distribution from noisy, quantized observations. We propose a simple and tractable algorithm that learns the skill density with near-optimal minimax mean squared error scaling as $n^{-1+\varepsilon}$, for any $\varepsilon>0$, when the density is smooth. Our approach brings together prior work on learning skill parameters from pairwise comparisons with kernel density estimation from non-parametric statistics. Furthermore, we prove minimax lower bounds which establish minimax optimality of the skill parameter estimation technique used in our algorithm. These bounds utilize a continuum version of Fano's method along with a covering argument. We apply our algorithm to various soccer leagues and world cups, cricket world cups, and mutual funds. We find that the entropy of a learnt distribution provides a quantitative measure of skill, which provides rigorous explanations for popular beliefs about perceived qualities of sporting events, e.g., soccer league rankings. Finally, we apply our method to assess the skill distributions of mutual funds. Our results shed light on the abundance of low quality funds prior to the Great Recession of 2008, and the domination of the industry by more skilled funds after the financial crisis.

stat.ML↗

arXiv2020

FedPAQ: A Communication-Efficient Federated Learning Method with Periodic Averaging and Quantization

Federated learning is a distributed framework according to which a model is trained over a set of devices, while keeping data localized. This framework faces several systems-oriented challenges which include (i) communication bottleneck since a large number of devices upload their local updates to a parameter server, and (ii) scalability as the federated network consists of millions of devices. Due to these systems challenges as well as issues related to statistical heterogeneity of data and privacy concerns, designing a provably efficient federated learning method is of significant importance yet it remains challenging. In this paper, we present FedPAQ, a communication-efficient Federated Learning method with Periodic Averaging and Quantization. FedPAQ relies on three key features: (1) periodic averaging where models are updated locally at devices and only periodically averaged at the server; (2) partial device participation where only a fraction of devices participate in each round of the training; and (3) quantized message-passing where the edge nodes quantize their updates before uploading to the parameter server. These features address the communications and scalability challenges in federated learning. We also show that FedPAQ achieves near-optimal theoretical guarantees for strongly convex and non-convex loss functions and empirically demonstrate the communication-computation tradeoff provided by our method.

cs.LG↗

arXiv2020

Deterministic and Randomized Actuator Scheduling With Guaranteed Performance Bounds

In this paper, we investigate the problem of actuator selection for linear dynamical systems. We develop a framework to design a sparse actuator schedule for a given large-scale linear system with guaranteed performance bounds using deterministic polynomial-time and randomized approximately linear-time algorithms. First, we introduce systemic controllability metrics for linear dynamical systems that are monotone and homogeneous with respect to the controllability Gramian. We show that several popular and widely used optimization criteria in the literature belong to this class of controllability metrics. Our main result is to provide a polynomial-time actuator schedule that on average selects only a constant number of actuators at each time step, independent of the dimension, to furnish a guaranteed approximation of the controllability metrics in comparison to when all actuators are in use. Our results naturally apply to the dual problem of sensor selection, in which we provide a guaranteed approximation to the observability Gramian. We illustrate the effectiveness of our theoretical findings via several numerical simulations using benchmark examples.

eess.SY↗

arXiv2020

LQG Control and Sensing Co-Design

We investigate a Linear-Quadratic-Gaussian (LQG) control and sensing co-design problem, where one jointly designs sensing and control policies. We focus on the realistic case where the sensing design is selected among a finite set of available sensors, where each sensor is associated with a different cost (e.g., power consumption). We consider two dual problem instances: sensing-constrained LQG control, where one maximizes control performance subject to a sensor cost budget, and minimum-sensing LQG control, where one minimizes sensor cost subject to performance constraints. We prove no polynomial time algorithm guarantees across all problem instances a constant approximation factor from the optimal. Nonetheless, we present the first polynomial time algorithms with per-instance suboptimality guarantees. To this end, we leverage a separation principle, that partially decouples the design of sensing and control. Then, we frame LQG co-design as the optimization of approximately supermodular set functions; we develop novel algorithms to solve the problems; and we prove original results on the performance of the algorithms, and establish connections between their suboptimality and control-theoretic quantities. We conclude the paper by discussing two applications, namely, sensing-constrained formation control and resource-constrained robot navigation.

math.OC↗

arXiv2020

Sensor Placement for Optimal Kalman Filtering: Fundamental Limits, Submodularity, and Algorithms

In this paper, we focus on sensor placement in linear dynamic estimation, where the objective is to place a small number of sensors in a system of interdependent states so to design an estimator with a desired estimation performance. In particular, we consider a linear time-variant system that is corrupted with process and measurement noise, and study how the selection of its sensors affects the estimation error of the corresponding Kalman filter over a finite observation interval. Our contributions are threefold: First, we prove that the minimum mean square error of the Kalman filter decreases only linearly as the number of sensors increases. That is, adding extra sensors so to reduce this estimation error is ineffective, a fundamental design limit. Similarly, we prove that the number of sensors grows linearly with the system's size for fixed minimum mean square error and number of output measurements over an observation interval; this is another fundamental limit, especially for systems where the system's size is large. Second, we prove that the logdet of the error covariance of the Kalman filter, which captures the volume of the corresponding confidence ellipsoid, with respect to the system's initial condition and process noise is a supermodular and non-increasing set function in the choice of the sensor set. Therefore, it exhibits the diminishing returns property. Third, we provide efficient approximation algorithms that select a small number sensors so to optimize the Kalman filter with respect to this estimation error ---the worst-case performance guarantees of these algorithms are provided as well. Finally, we illustrate the efficiency of our algorithms using the problem of surface-based monitoring of CO2 sequestration sites studied in Weimer et al. (2008).

math.OC↗

arXiv2020

Sensing-Constrained LQG Control

Linear-Quadratic-Gaussian (LQG) control is concerned with the design of an optimal controller and estimator for linear Gaussian systems with imperfect state information. Standard LQG assumes the set of sensor measurements, to be fed to the estimator, to be given. However, in many problems, arising in networked systems and robotics, one may not be able to use all the available sensors, due to power or payload constraints, or may be interested in using the smallest subset of sensors that guarantees the attainment of a desired control goal. In this paper, we introduce the sensing-constrained LQG control problem, in which one has to jointly design sensing, estimation, and control, under given constraints on the resources spent for sensing. We focus on the realistic case in which the sensing strategy has to be selected among a finite set of possible sensing modalities. While the computation of the optimal sensing strategy is intractable, we present the first scalable algorithm that computes a near-optimal sensing strategy with provable sub-optimality guarantees. To this end, we show that a separation principle holds, which allows the design of sensing, estimation, and control policies in isolation. We conclude the paper by discussing two applications of sensing-constrained LQG control, namely, sensing-constrained formation control and resource-constrained robot navigation.

math.OC↗

arXiv2020

A Separation Theorem for Joint Sensor and Actuator Scheduling with Guaranteed Performance Bounds

We study the problem of jointly designing a sparse sensor and actuator schedule for linear dynamical systems while guaranteeing a control/estimation performance that approximates the fully sensed/actuated setting. We further prove a separation principle, showing that the problem can be decomposed into finding sensor and actuator schedules separately. However, it is shown that this problem cannot be efficiently solved or approximated in polynomial, or even quasi-polynomial time for time-invariant sensor/actuator schedules; instead, we develop deterministic polynomial-time algorithms for a time-varying sensor/actuator schedule with guaranteed approximation bounds. Our main result is to provide a polynomial-time joint actuator and sensor schedule that on average selects only a constant number of sensors and actuators at each time step, irrespective of the dimension of the system. The key idea is to sparsify the controllability and observability Gramians while providing approximation guarantees for Hankel singular values. This idea is inspired by recent results in theoretical computer science literature on sparsification.

eess.SY↗

arXiv2020

Network Inference from Consensus Dynamics with Unknown Parameters

We explore the problem of inferring the graph Laplacian of a weighted, undirected network from snapshots of a single or multiple discrete-time consensus dynamics, subject to parameter uncertainty, taking place on the network. Specifically, we consider three problems in which we assume different levels of knowledge about the diffusion rates, observation times, and the input signal power of the dynamics. To solve these underdetermined problems, we propose a set of algorithms that leverage the spectral properties of the observed data and tools from convex optimization. Furthermore, we provide theoretical performance guarantees associated with these algorithms. We complement our theoretical work with numerical experiments, that demonstrate how our proposed methods outperform current state-of-the-art algorithms and showcase their effectiveness in recovering both synthetic and real-world networks.

cs.SI↗

arXiv2020

Why gradient clipping accelerates training: A theoretical justification for adaptivity

We provide a theoretical explanation for the effectiveness of gradient clipping in training deep neural networks. The key ingredient is a new smoothness condition derived from practical neural network training examples. We observe that gradient smoothness, a concept central to the analysis of first-order optimization algorithms that is often assumed to be a constant, demonstrates significant variability along the training trajectory of deep neural networks. Further, this smoothness positively correlates with the gradient norm, and contrary to standard assumptions in the literature, it can grow with the norm of the gradient. These empirical observations limit the applicability of existing theoretical analyses of algorithms that rely on a fixed bound on smoothness. These observations motivate us to introduce a novel relaxation of gradient smoothness that is weaker than the commonly used Lipschitz smoothness assumption. Under the new condition, we prove that two popular methods, namely, \emph{gradient clipping} and \emph{normalized gradient}, converge arbitrarily faster than gradient descent with fixed stepsize. We further explain why such adaptively scaled gradient methods can accelerate empirical convergence and verify our results empirically in popular neural network training settings.

math.OC↗

arXiv2019

Competitive Contagion with Sparse Seeding

This paper studies a strategic model of marketing and product diffusion in social networks. We consider two firms offering substitutable products which can improve their market share by seeding the key individuals in the market. Consumers update their consumption level for each of the two products as the best response to the consumption of their neighbors in the previous period. This results in linear update dynamics for the product consumption. Each consumer receives externality from the consumption of each neighbor where the strength of the externality is higher for consumption of the products of the same firm. We represent the above setting as a duopoly game between the firms and introduce a novel framework that allows for sparse seeding to emerge as an equilibrium strategy. We then study the effect of the network structure on the optimal seeding strategies and the extent to which the strategies can be sparsified. In particular, we derive conditions under which near Nash equilibrium strategies can asymptotically lead to sparse seeding in large populations. The results are illustrated using a core-periphery network.

cs.SI↗