arXiv subjects

Davoud Ataee Tarzanagh

Publications and source records attributed to Davoud Ataee Tarzanagh.

28 records · Page 2Linked to original sources

arXiv2022

FedNest: Federated Bilevel, Minimax, and Compositional Optimization

Standard federated optimization methods successfully apply to stochastic problems with single-level structure. However, many contemporary ML problems -- including adversarial robustness, hyperparameter tuning, and actor-critic -- fall under nested bilevel programming that subsumes minimax and compositional optimization. In this work, we propose \fedblo: A federated alternating stochastic gradient method to address general nested problems. We establish provable convergence rates for \fedblo in the presence of heterogeneous data and introduce variations for bilevel, minimax, and compositional optimization. \fedblo introduces multiple innovations including federated hypergradient computation and variance reduction to address inner-level heterogeneity. We complement our theory with experiments on hyperparameter \& hyper-representation learning and minimax optimization that demonstrate the benefits of our method in practice. Code is available at https://github.com/ucr-optml/FedNest.

cs.LG↗

arXiv2022

Grassmannian Optimization for Online Tensor Completion and Tracking with the t-SVD

We propose a new fast streaming algorithm for the tensor completion problem of imputing missing entries of a low-tubal-rank tensor using the tensor singular value decomposition (t-SVD) algebraic framework. We show the t-SVD is a specialization of the well-studied block-term decomposition for third-order tensors, and we present an algorithm under this model that can track changing free submodules from incomplete streaming 2-D data. The proposed algorithm uses principles from incremental gradient descent on the Grassmann manifold of subspaces to solve the tensor completion problem with linear complexity and constant memory in the number of time samples. We provide a local expected linear convergence result for our algorithm. Our empirical results are competitive in accuracy but much faster in compute time than state-of-the-art tensor completion algorithms on real applications to recover temporal chemo-sensing and MRI data under limited sampling.

eess.SP↗

arXiv2021

Certainty Equivalent Quadratic Control for Markov Jump Systems

Real-world control applications often involve complex dynamics subject to abrupt changes or variations. Markov jump linear systems (MJS) provide a rich framework for modeling such dynamics. Despite an extensive history, theoretical understanding of parameter sensitivities of MJS control is somewhat lacking. Motivated by this, we investigate robustness aspects of certainty equivalent model-based optimal control for MJS with quadratic cost function. Given the uncertainty in the system matrices and in the Markov transition matrix is bounded by $ε$ and $η$ respectively, robustness results are established for (i) the solution to coupled Riccati equations and (ii) the optimal cost, by providing explicit perturbation bounds which decay as $\mathcal{O}(ε+ η)$ and $\mathcal{O}((ε+ η)^2)$ respectively.

math.OC↗

arXiv2021

Solving a class of non-convex min-max games using adaptive momentum methods

Adaptive momentum methods have recently attracted a lot of attention for training of deep neural networks. They use an exponential moving average of past gradients of the objective function to update both search directions and learning rates. However, these methods are not suited for solving min-max optimization problems that arise in training generative adversarial networks. In this paper, we propose an adaptive momentum min-max algorithm that generalizes adaptive momentum methods to the non-convex min-max regime. Further, we establish non-asymptotic rates of convergence for the proposed algorithm when used in a reasonably broad class of non-convex min-max optimization problems. Experimental results illustrate its superior performance vis-a-vis benchmark methods for solving such problems.

math.OC↗

arXiv2021

A Newton-Type Active Set Method for Nonlinear Optimization with Polyhedral Constraints

A Newton-type active set algorithm for large-scale minimization subject to polyhedral constraints is proposed. The algorithm consists of a gradient projection step, a second-order Newton-type step in the null space of the constraint matrix, and a set of rules for branching between the two steps. We show that the proposed method asymptotically takes the Newton step when the active constraints are linearly independent and a strong second-order sufficient optimality condition holds. We also show that the method has a quadratic rate of convergence under standard conditions. Numerical experiments are presented illustrating the performance of the algorithm on the CUTEst and on a specific class of problems for which finding second-order stationary points is critical.

math.OC↗

arXiv2020

Adaptive First-and Zeroth-order Methods for Weakly Convex Stochastic Optimization Problems

In this paper, we design and analyze a new family of adaptive subgradient methods for solving an important class of weakly convex (possibly nonsmooth) stochastic optimization problems. Adaptive methods that use exponential moving averages of past gradients to update search directions and learning rates have recently attracted a lot of attention for solving optimization problems that arise in machine learning. Nevertheless, their convergence analysis almost exclusively requires smoothness and/or convexity of the objective function. In contrast, we establish non-asymptotic rates of convergence of first and zeroth-order adaptive methods and their proximal variants for a reasonably broad class of nonsmooth \& nonconvex optimization problems. Experimental results indicate how the proposed algorithms empirically outperform stochastic gradient descent and its zeroth-order variant for solving such optimization problems.

math.OC↗

arXiv2019

DADAM: A Consensus-based Distributed Adaptive Gradient Method for Online Optimization

Adaptive gradient-based optimization methods such as \textsc{Adagrad}, \textsc{Rmsprop}, and \textsc{Adam} are widely used in solving large-scale machine learning problems including deep learning. A number of schemes have been proposed in the literature aiming at parallelizing them, based on communications of peripheral nodes with a central node, but incur high communications cost. To address this issue, we develop a novel consensus-based distributed adaptive moment estimation method (\textsc{Dadam}) for online optimization over a decentralized network that enables data parallelization, as well as decentralized computation. The method is particularly useful, since it can accommodate settings where access to local data is allowed. Further, as established theoretically in this work, it can outperform centralized adaptive algorithms, for certain classes of loss functions used in applications. We analyze the convergence properties of the proposed algorithm and provide a dynamic regret bound on the convergence rate of adaptive moment estimation methods in both stochastic and deterministic settings. Empirical results demonstrate that \textsc{Dadam} works also well in practice and compares favorably to competing online optimization methods.

cs.LG↗

arXiv2019

Online Distributed Estimation of Principal Eigenspaces

Principal components analysis (PCA) is a widely used dimension reduction technique with an extensive range of applications. In this paper, an online distributed algorithm is proposed for recovering the principal eigenspaces. We further establish its rate of convergence and show how it relates to the number of nodes employed in the distributed computation, the effective rank of the data matrix under consideration, and the gap in the spectrum of the underlying population covariance matrix. The proposed algorithm is illustrated on low-rank approximation and $\boldsymbol{k}$-means clustering tasks. The numerical results show a substantial computational speed-up vis-a-vis standard distributed PCA algorithms, without compromising learning accuracy.

stat.ML↗

arXiv2018

Fast Randomized Algorithms for t-Product Based Tensor Operations and Decompositions with Applications to Imaging Data

Tensors of order three or higher have found applications in diverse fields, including image and signal processing, data mining, biomedical engineering and link analysis, to name a few. In many applications that involve for example time series or other ordered data, the corresponding tensor has a distinguishing orientation that exhibits a low tubal structure. This has motivated the introduction of the tubal rank and the corresponding tubal singular value decomposition in the literature. In this work, we develop randomized algorithms for many common tensor operations, including tensor low-rank approximation and decomposition, together with tensor multiplication. The proposed tubal focused algorithms employ a small number of lateral and/or horizontal slices of the underlying 3-rd order tensor, that come with {\em relative error guarantees} for the quality of the obtained solutions. The performance of the proposed algorithms is illustrated on diverse imaging applications, including mass spectrometry data and image and video recovery from incomplete and noisy data. The results show both good computational speed-up vis-a-vis conventional completion algorithms and good accuracy.

cs.DS↗

arXiv2018

Estimation of Graphical Models through Structured Norm Minimization

Estimation of Markov Random Field and covariance models from high-dimensional data represents a canonical problem that has received a lot of attention in the literature. A key assumption, widely employed, is that of {\em sparsity} of the underlying model. In this paper, we study the problem of estimating such models exhibiting a more intricate structure comprising simultaneously of {\em sparse, structured sparse} and {\em dense} components. Such structures naturally arise in several scientific fields, including molecular biology, finance, and political science. We introduce a general framework based on a novel structured norm that enables us to estimate such complex structures from high-dimensional data. The resulting optimization problem is convex and we introduce a linearized multi-block alternating direction method of multipliers (ADMM) algorithm to solve it efficiently. We illustrate the superior performance of the proposed framework on a number of synthetic data sets generated from both random and structured networks. Further, we apply the method to a number of real data sets and discuss the results.

math.OC↗