arXiv subjects

Mehrdad Mahdavi

Publications and source records attributed to Mehrdad Mahdavi.

At least 55 records · Page 3Linked to original sources

arXiv2020

Federated Learning with Compression: Unified Analysis and Sharp Guarantees

In federated learning, communication cost is often a critical bottleneck to scale up distributed optimization algorithms to collaboratively learn a model from millions of devices with potentially unreliable or limited communication and heterogeneous data distributions. Two notable trends to deal with the communication overhead of federated algorithms are gradient compression and local computation with periodic communication. Despite many attempts, characterizing the relationship between these two approaches has proven elusive. We address this by proposing a set of algorithms with periodical compressed (quantized or sparsified) communication and analyze their convergence properties in both homogeneous and heterogeneous local data distribution settings. For the homogeneous setting, our analysis improves existing bounds by providing tighter convergence rates for both strongly convex and non-convex objective functions. To mitigate data heterogeneity, we introduce a local gradient tracking scheme and obtain sharp convergence rates that match the best-known communication complexities without compression for convex, strongly convex, and nonconvex settings. We complement our theoretical results and demonstrate the effectiveness of our proposed methods by several experiments on real-world datasets.

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arXiv2020

Adaptive Personalized Federated Learning

Investigation of the degree of personalization in federated learning algorithms has shown that only maximizing the performance of the global model will confine the capacity of the local models to personalize. In this paper, we advocate an adaptive personalized federated learning (APFL) algorithm, where each client will train their local models while contributing to the global model. We derive the generalization bound of mixture of local and global models, and find the optimal mixing parameter. We also propose a communication-efficient optimization method to collaboratively learn the personalized models and analyze its convergence in both smooth strongly convex and nonconvex settings. The extensive experiments demonstrate the effectiveness of our personalization schema, as well as the correctness of established generalization theories.

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arXiv2020

Online Structured Meta-learning

Learning quickly is of great importance for machine intelligence deployed in online platforms. With the capability of transferring knowledge from learned tasks, meta-learning has shown its effectiveness in online scenarios by continuously updating the model with the learned prior. However, current online meta-learning algorithms are limited to learn a globally-shared meta-learner, which may lead to sub-optimal results when the tasks contain heterogeneous information that are distinct by nature and difficult to share. We overcome this limitation by proposing an online structured meta-learning (OSML) framework. Inspired by the knowledge organization of human and hierarchical feature representation, OSML explicitly disentangles the meta-learner as a meta-hierarchical graph with different knowledge blocks. When a new task is encountered, it constructs a meta-knowledge pathway by either utilizing the most relevant knowledge blocks or exploring new blocks. Through the meta-knowledge pathway, the model is able to quickly adapt to the new task. In addition, new knowledge is further incorporated into the selected blocks. Experiments on three datasets demonstrate the effectiveness and interpretability of our proposed framework in the context of both homogeneous and heterogeneous tasks.

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arXiv2020

Local SGD with Periodic Averaging: Tighter Analysis and Adaptive Synchronization

Communication overhead is one of the key challenges that hinders the scalability of distributed optimization algorithms. In this paper, we study local distributed SGD, where data is partitioned among computation nodes, and the computation nodes perform local updates with periodically exchanging the model among the workers to perform averaging. While local SGD is empirically shown to provide promising results, a theoretical understanding of its performance remains open. We strengthen convergence analysis for local SGD, and show that local SGD can be far less expensive and applied far more generally than current theory suggests. Specifically, we show that for loss functions that satisfy the Polyak-Łojasiewicz condition, $O((pT)^{1/3})$ rounds of communication suffice to achieve a linear speed up, that is, an error of $O(1/pT)$, where $T$ is the total number of model updates at each worker. This is in contrast with previous work which required higher number of communication rounds, as well as was limited to strongly convex loss functions, for a similar asymptotic performance. We also develop an adaptive synchronization scheme that provides a general condition for linear speed up. Finally, we validate the theory with experimental results, running over AWS EC2 clouds and an internal GPU cluster.

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arXiv2020

Efficient Fair Principal Component Analysis

It has been shown that dimension reduction methods such as PCA may be inherently prone to unfairness and treat data from different sensitive groups such as race, color, sex, etc., unfairly. In pursuit of fairness-enhancing dimensionality reduction, using the notion of Pareto optimality, we propose an adaptive first-order algorithm to learn a subspace that preserves fairness, while slightly compromising the reconstruction loss. Theoretically, we provide sufficient conditions that the solution of the proposed algorithm belongs to the Pareto frontier for all sensitive groups; thereby, the optimal trade-off between overall reconstruction loss and fairness constraints is guaranteed. We also provide the convergence analysis of our algorithm and show its efficacy through empirical studies on different datasets, which demonstrates superior performance in comparison with state-of-the-art algorithms. The proposed fairness-aware PCA algorithm can be efficiently generalized to multiple group sensitive features and effectively reduce the unfairness decisions in downstream tasks such as classification.

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arXiv2019

On the Convergence of Local Descent Methods in Federated Learning

In federated distributed learning, the goal is to optimize a global training objective defined over distributed devices, where the data shard at each device is sampled from a possibly different distribution (a.k.a., heterogeneous or non i.i.d. data samples). In this paper, we generalize the local stochastic and full gradient descent with periodic averaging-- originally designed for homogeneous distributed optimization, to solve nonconvex optimization problems in federated learning. Although scant research is available on the effectiveness of local SGD in reducing the number of communication rounds in homogeneous setting, its convergence and communication complexity in heterogeneous setting is mostly demonstrated empirically and lacks through theoretical understating. To bridge this gap, we demonstrate that by properly analyzing the effect of unbiased gradients and sampling schema in federated setting, under mild assumptions, the implicit variance reduction feature of local distributed methods generalize to heterogeneous data shards and exhibits the best known convergence rates of homogeneous setting both in general nonconvex and under {\pl}~ condition (generalization of strong-convexity). Our theoretical results complement the recent empirical studies that demonstrate the applicability of local GD/SGD to federated learning. We also specialize the proposed local method for networked distributed optimization. To the best of our knowledge, the obtained convergence rates are the sharpest known to date on the convergence of local decant methods with periodic averaging for solving nonconvex federated optimization in both centralized and networked distributed optimization.

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arXiv2017

Learning Feature Nonlinearities with Non-Convex Regularized Binned Regression

For various applications, the relations between the dependent and independent variables are highly nonlinear. Consequently, for large scale complex problems, neural networks and regression trees are commonly preferred over linear models such as Lasso. This work proposes learning the feature nonlinearities by binning feature values and finding the best fit in each quantile using non-convex regularized linear regression. The algorithm first captures the dependence between neighboring quantiles by enforcing smoothness via piecewise-constant/linear approximation and then selects a sparse subset of good features. We prove that the proposed algorithm is statistically and computationally efficient. In particular, it achieves linear rate of convergence while requiring near-minimal number of samples. Evaluations on synthetic and real datasets demonstrate that algorithm is competitive with current state-of-the-art and accurately learns feature nonlinearities. Finally, we explore an interesting connection between the binning stage of our algorithm and sparse Johnson-Lindenstrauss matrices.

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arXiv2016

Sketching Meets Random Projection in the Dual: A Provable Recovery Algorithm for Big and High-dimensional Data

Sketching techniques have become popular for scaling up machine learning algorithms by reducing the sample size or dimensionality of massive data sets, while still maintaining the statistical power of big data. In this paper, we study sketching from an optimization point of view: we first show that the iterative Hessian sketch is an optimization process with preconditioning, and develop accelerated iterative Hessian sketch via the searching the conjugate direction; we then establish primal-dual connections between the Hessian sketch and dual random projection, and apply the preconditioned conjugate gradient approach on the dual problem, which leads to the accelerated iterative dual random projection methods. Finally to tackle the challenges from both large sample size and high-dimensionality, we propose the primal-dual sketch, which iteratively sketches the primal and dual formulations. We show that using a logarithmic number of calls to solvers of small scale problem, primal-dual sketch is able to recover the optimum of the original problem up to arbitrary precision. The proposed algorithms are validated via extensive experiments on synthetic and real data sets which complements our theoretical results.

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arXiv2016

Train and Test Tightness of LP Relaxations in Structured Prediction

Structured prediction is used in areas such as computer vision and natural language processing to predict structured outputs such as segmentations or parse trees. In these settings, prediction is performed by MAP inference or, equivalently, by solving an integer linear program. Because of the complex scoring functions required to obtain accurate predictions, both learning and inference typically require the use of approximate solvers. We propose a theoretical explanation to the striking observation that approximations based on linear programming (LP) relaxations are often tight on real-world instances. In particular, we show that learning with LP relaxed inference encourages integrality of training instances, and that tightness generalizes from train to test data.

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arXiv2014

Matrix Factorization with Explicit Trust and Distrust Relationships

With the advent of online social networks, recommender systems have became crucial for the success of many online applications/services due to their significance role in tailoring these applications to user-specific needs or preferences. Despite their increasing popularity, in general recommender systems suffer from the data sparsity and the cold-start problems. To alleviate these issues, in recent years there has been an upsurge of interest in exploiting social information such as trust relations among users along with the rating data to improve the performance of recommender systems. The main motivation for exploiting trust information in recommendation process stems from the observation that the ideas we are exposed to and the choices we make are significantly influenced by our social context. However, in large user communities, in addition to trust relations, the distrust relations also exist between users. For instance, in Epinions the concepts of personal "web of trust" and personal "block list" allow users to categorize their friends based on the quality of reviews into trusted and distrusted friends, respectively. In this paper, we propose a matrix factorization based model for recommendation in social rating networks that properly incorporates both trust and distrust relationships aiming to improve the quality of recommendations and mitigate the data sparsity and the cold-start users issues. Through experiments on the Epinions data set, we show that our new algorithm outperforms its standard trust-enhanced or distrust-enhanced counterparts with respect to accuracy, thereby demonstrating the positive effect that incorporation of explicit distrust information can have on recommender systems.

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arXiv2014

Exploiting Smoothness in Statistical Learning, Sequential Prediction, and Stochastic Optimization

In the last several years, the intimate connection between convex optimization and learning problems, in both statistical and sequential frameworks, has shifted the focus of algorithmic machine learning to examine this interplay. In particular, on one hand, this intertwinement brings forward new challenges in reassessment of the performance of learning algorithms including generalization and regret bounds under the assumptions imposed by convexity such as analytical properties of loss functions (e.g., Lipschitzness, strong convexity, and smoothness). On the other hand, emergence of datasets of an unprecedented size, demands the development of novel and more efficient optimization algorithms to tackle large-scale learning problems. The overarching goal of this thesis is to reassess the smoothness of loss functions in statistical learning, sequential prediction/online learning, and stochastic optimization and explicate its consequences. In particular we examine how smoothness of loss function could be beneficial or detrimental in these settings in terms of sample complexity, statistical consistency, regret analysis, and convergence rate, and investigate how smoothness can be leveraged to devise more efficient learning algorithms.

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arXiv2014

Recovering the Optimal Solution by Dual Random Projection

Random projection has been widely used in data classification. It maps high-dimensional data into a low-dimensional subspace in order to reduce the computational cost in solving the related optimization problem. While previous studies are focused on analyzing the classification performance of using random projection, in this work, we consider the recovery problem, i.e., how to accurately recover the optimal solution to the original optimization problem in the high-dimensional space based on the solution learned from the subspace spanned by random projections. We present a simple algorithm, termed Dual Random Projection, that uses the dual solution of the low-dimensional optimization problem to recover the optimal solution to the original problem. Our theoretical analysis shows that with a high probability, the proposed algorithm is able to accurately recover the optimal solution to the original problem, provided that the data matrix is of low rank or can be well approximated by a low rank matrix.

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arXiv2014

Excess Risk Bounds for Exponentially Concave Losses

The overarching goal of this paper is to derive excess risk bounds for learning from exp-concave loss functions in passive and sequential learning settings. Exp-concave loss functions encompass several fundamental problems in machine learning such as squared loss in linear regression, logistic loss in classification, and negative logarithm loss in portfolio management. In batch setting, we obtain sharp bounds on the performance of empirical risk minimization performed in a linear hypothesis space and with respect to the exp-concave loss functions. We also extend the results to the online setting where the learner receives the training examples in a sequential manner. We propose an online learning algorithm that is a properly modified version of online Newton method to obtain sharp risk bounds. Under an additional mild assumption on the loss function, we show that in both settings we are able to achieve an excess risk bound of $O(d\log n/n)$ that holds with a high probability.

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arXiv2014

Binary Excess Risk for Smooth Convex Surrogates

In statistical learning theory, convex surrogates of the 0-1 loss are highly preferred because of the computational and theoretical virtues that convexity brings in. This is of more importance if we consider smooth surrogates as witnessed by the fact that the smoothness is further beneficial both computationally- by attaining an {\it optimal} convergence rate for optimization, and in a statistical sense- by providing an improved {\it optimistic} rate for generalization bound. In this paper we investigate the smoothness property from the viewpoint of statistical consistency and show how it affects the binary excess risk. We show that in contrast to optimization and generalization errors that favor the choice of smooth surrogate loss, the smoothness of loss function may degrade the binary excess risk. Motivated by this negative result, we provide a unified analysis that integrates optimization error, generalization bound, and the error in translating convex excess risk into a binary excess risk when examining the impact of smoothness on the binary excess risk. We show that under favorable conditions appropriate choice of smooth convex loss will result in a binary excess risk that is better than $O(1/\sqrt{n})$.

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arXiv2014

Beating the Minimax Rate of Active Learning with Prior Knowledge

Active learning refers to the learning protocol where the learner is allowed to choose a subset of instances for labeling. Previous studies have shown that, compared with passive learning, active learning is able to reduce the label complexity exponentially if the data are linearly separable or satisfy the Tsybakov noise condition with parameter $κ=1$. In this paper, we propose a novel active learning algorithm using a convex surrogate loss, with the goal to broaden the cases for which active learning achieves an exponential improvement. We make use of a convex loss not only because it reduces the computational cost, but more importantly because it leads to a tight bound for the empirical process (i.e., the difference between the empirical estimation and the expectation) when the current solution is close to the optimal one. Under the assumption that the norm of the optimal classifier that minimizes the convex risk is available, our analysis shows that the introduction of the convex surrogate loss yields an exponential reduction in the label complexity even when the parameter $κ$ of the Tsybakov noise is larger than $1$. To the best of our knowledge, this is the first work that improves the minimax rate of active learning by utilizing certain priori knowledge.

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arXiv2013

An Efficient Primal-Dual Prox Method for Non-Smooth Optimization

We study the non-smooth optimization problems in machine learning, where both the loss function and the regularizer are non-smooth functions. Previous studies on efficient empirical loss minimization assume either a smooth loss function or a strongly convex regularizer, making them unsuitable for non-smooth optimization. We develop a simple yet efficient method for a family of non-smooth optimization problems where the dual form of the loss function is bilinear in primal and dual variables. We cast a non-smooth optimization problem into a minimax optimization problem, and develop a primal dual prox method that solves the minimax optimization problem at a rate of $O(1/T)$ {assuming that the proximal step can be efficiently solved}, significantly faster than a standard subgradient descent method that has an $O(1/\sqrt{T})$ convergence rate. Our empirical study verifies the efficiency of the proposed method for various non-smooth optimization problems that arise ubiquitously in machine learning by comparing it to the state-of-the-art first order methods.

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arXiv2013

MixedGrad: An O(1/T) Convergence Rate Algorithm for Stochastic Smooth Optimization

It is well known that the optimal convergence rate for stochastic optimization of smooth functions is $O(1/\sqrt{T})$, which is same as stochastic optimization of Lipschitz continuous convex functions. This is in contrast to optimizing smooth functions using full gradients, which yields a convergence rate of $O(1/T^2)$. In this work, we consider a new setup for optimizing smooth functions, termed as {\bf Mixed Optimization}, which allows to access both a stochastic oracle and a full gradient oracle. Our goal is to significantly improve the convergence rate of stochastic optimization of smooth functions by having an additional small number of accesses to the full gradient oracle. We show that, with an $O(\ln T)$ calls to the full gradient oracle and an $O(T)$ calls to the stochastic oracle, the proposed mixed optimization algorithm is able to achieve an optimization error of $O(1/T)$.

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arXiv2013

Online Stochastic Optimization with Multiple Objectives

In this paper we propose a general framework to characterize and solve the stochastic optimization problems with multiple objectives underlying many real world learning applications. We first propose a projection based algorithm which attains an $O(T^{-1/3})$ convergence rate. Then, by leveraging on the theory of Lagrangian in constrained optimization, we devise a novel primal-dual stochastic approximation algorithm which attains the optimal convergence rate of $O(T^{-1/2})$ for general Lipschitz continuous objectives.

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