arXiv subjects

Sam Davanloo Tajbakhsh

Publications and source records attributed to Sam Davanloo Tajbakhsh.

9 recordsLinked to original sources

arXiv2024

Nonasymptotic Analysis of Accelerated Methods With Inexact Oracle Under Absolute Error Bound

Performance analysis of first-order algorithms with inexact oracles has gained recent attention due to various emerging applications in which obtaining exact gradients is impossible or computationally expensive. Previous research has demonstrated that the performance of accelerated first-order methods is more sensitive to gradient errors compared with non-accelerated ones. This paper investigates the nonasymptotic convergence bound of two accelerated methods with inexact gradients to solve deterministic smooth convex problems. Performance Estimation Problem (PEP) is used as the primary tool to analyze the convergence bounds of the underlying algorithms. By finding an analytical solution to PEP, we derive novel convergence bounds of the Generalized Optimized Gradient Method (GOGM) and Generalized Fast Gradient Method (GFGM) with inexact gradient oracles following the absolute error bound. Next, we analyze the tradeoff between the vanishing term and the accumulated error in the convergence bound that guides finding the optimal stepsize. Furthermore, we determine the optimal strategy to set the gradient inexactness along iterations, ensuring that the accumulated error remains subordinate to the vanishing term. Finally, we establish a lower bound for accumulated error for a subclass of first-order methods satisfying two properties, which then motivated us to develop PEP solutions satisfying this lower bound for GOGM and GFGM methods.

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arXiv2024

Fitting ARMA Time Series Models without Identification: A Proximal Approach

Fitting autoregressive moving average (ARMA) time series models requires model identification before parameter estimation. Model identification involves determining the order of the autoregressive and moving average components which is generally performed by inspection of the autocorrelation and partial autocorrelation functions or other offline methods. In this work, we regularize the parameter estimation optimization problem with a non-smooth hierarchical sparsity-inducing penalty based on two path graphs that allow performing model identification and parameter estimation simultaneously. A proximal block coordinate descent algorithm is then proposed to solve the underlying optimization problem efficiently. The resulting model satisfies the required stationarity and invertibility conditions for ARMA models. Numerical results supporting the proposed method are also presented.

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arXiv2024

Riemannian Stochastic Gradient Method for Nested Composition Optimization

This work considers optimization of composition of functions in a nested form over Riemannian manifolds where each function contains an expectation. This type of problems is gaining popularity in applications such as policy evaluation in reinforcement learning or model customization in meta-learning. The standard Riemannian stochastic gradient methods for non-compositional optimization cannot be directly applied as stochastic approximation of inner functions create bias in the gradients of the outer functions. For two-level composition optimization, we present a Riemannian Stochastic Composition Gradient Descent (R-SCGD) method that finds an approximate stationary point, with expected squared Riemannian gradient smaller than $ε$, in $O(ε^{-2})$ calls to the stochastic gradient oracle of the outer function and stochastic function and gradient oracles of the inner function. Furthermore, we generalize the R-SCGD algorithms for problems with multi-level nested compositional structures, with the same complexity of $O(ε^{-2})$ for the first-order stochastic oracle. Finally, the performance of the R-SCGD method is numerically evaluated over a policy evaluation problem in reinforcement learning.

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arXiv2023

Stochastic Optimization Algorithms for Problems with Controllable Biased Oracles

Motivated by emerging applications in machine learning, we consider an optimization problem in a general form where the gradient of the objective function is available through a biased stochastic oracle. We assume a bias-control parameter can reduce the bias magnitude; however, a lower bias requires more computation/samples. For instance, in two applications on stochastic composition optimization and policy optimization for infinite-horizon Markov decision processes, we show that the bias follows a power law and exponential decay, respectively, as functions of their corresponding bias control parameters. For problems with such gradient oracles, the paper proposes stochastic algorithms that adjust the bias-control parameter throughout the iterations. We analyze the nonasymptotic performance of the proposed algorithms in the nonconvex regime and establish their sample or bias-control computational complexities to obtain a stationary point in expectation or with high probability. Finally, we numerically evaluate the performance of the proposed algorithms over three applications.

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arXiv2023

Stochastic Composition Optimization of Functions without Lipschitz Continuous Gradient

In this paper, we study stochastic optimization of two-level composition of functions without Lipschitz continuous gradient. The smoothness property is generalized by the notion of relative smoothness which provokes the Bregman gradient method. We propose three Stochastic Composition Bregman Gradient algorithms for the three possible relatively smooth compositional scenarios and provide their sample complexities to achieve an $ε$-approximate stationary point. For the smooth of relatively smooth composition, the first algorithm requires $O(ε^{-2})$ calls to the stochastic oracles of the inner function value and gradient as well as the outer function gradient. When both functions are relatively smooth, the second algorithm requires $O(ε^{-3})$ calls to the inner function value stochastic oracle and $O(ε^{-2})$ calls to the inner and outer functions gradients stochastic oracles. We further improve the second algorithm by variance reduction for the setting where just the inner function is smooth. The resulting algorithm requires $O(ε^{-5/2})$ calls to the inner function value stochastic oracle, $O(ε^{-3/2})$ calls to the inner function gradient and $O(ε^{-2})$ calls to the outer function gradient stochastic oracles. Finally, we numerically evaluate the performance of these three algorithms over two different examples.

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arXiv2022

Riemannian Stochastic Variance-Reduced Cubic Regularized Newton Method for Submanifold Optimization

We propose a stochastic variance-reduced cubic regularized Newton algorithm to optimize the finite-sum problem over a Riemannian submanifold of the Euclidean space. The proposed algorithm requires a full gradient and Hessian update at the beginning of each epoch while it performs stochastic variance-reduced updates in the iterations within each epoch. The iteration complexity of $O(ε^{-3/2})$ to obtain an $(ε,\sqrtε)$-second-order stationary point, i.e., a point with the Riemannian gradient norm upper bounded by $ε$ and minimum eigenvalue of Riemannian Hessian lower bounded by $-\sqrtε$, is established when the manifold is embedded in the Euclidean space. Furthermore, the paper proposes a computationally more appealing modification of the algorithm which only requires an inexact solution of the cubic regularized Newton subproblem with the same iteration complexity. The proposed algorithm is evaluated and compared with three other Riemannian second-order methods over two numerical studies on estimating the inverse scale matrix of the multivariate t-distribution on the manifold of symmetric positive definite matrices and estimating the parameter of a linear classifier on the Sphere manifold.

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arXiv2020

A first-order optimization algorithm for statistical learning with hierarchical sparsity structure

In many statistical learning problems, it is desired that the optimal solution conforms to an a priori known sparsity structure represented by a directed acyclic graph. Inducing such structures by means of convex regularizers requires nonsmooth penalty functions that exploit group overlapping. Our study focuses on evaluating the proximal operator of the Latent Overlapping Group lasso developed by Jacob et al. (2009). We implemented an Alternating Direction Method of Multiplier with a sharing scheme to solve large-scale instances of the underlying optimization problem efficiently. In the absence of strong convexity, global linear convergence of the algorithm is established using the error bound theory. More specifically, the paper contributes to establishing primal and dual error bounds when the nonsmooth component in the objective function does not have a polyhedral epigraph. We also investigate the effect of the graph structure on the speed of convergence of the algorithm. Detailed numerical simulation studies over different graph structures supporting the proposed algorithm and two applications in learning are provided.

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arXiv2020

On the Theoretical Guarantees for Parameter Estimation of Gaussian Random Field Models: A Sparse Precision Matrix Approach

Iterative methods for fitting a Gaussian Random Field (GRF) model via maximum likelihood (ML) estimation requires solving a nonconvex optimization problem. The problem is aggravated for anisotropic GRFs where the number of covariance function parameters increases with the dimension. Even evaluation of the likelihood function requires $O(n^3)$ floating point operations, where $n$ denotes the number of data locations. In this paper, we propose a new two-stage procedure to estimate the parameters of second-order stationary GRFs. First, a convex likelihood problem regularized with a weighted $\ell_1$-norm, utilizing the available distance information between observation locations, is solved to fit a sparse precision (inverse covariance) matrix to the observed data. Second, the parameters of the covariance function are estimated by solving a least squares problem. Theoretical error bounds for the solutions of stage I and II problems are provided, and their tightness are investigated.

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arXiv2017

Generalized Sparse Precision Matrix Selection for Fitting Multivariate Gaussian Random Fields to Large Data Sets

We present a new method for estimating multivariate, second-order stationary Gaussian Random Field (GRF) models based on the Sparse Precision matrix Selection (SPS) algorithm, proposed by Davanloo et al. (2015) for estimating scalar GRF models. Theoretical convergence rates for the estimated between-response covariance matrix and for the estimated parameters of the underlying spatial correlation function are established. Numerical tests using simulated and real datasets validate our theoretical findings. Data segmentation is used to handle large data sets.

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