arXiv subjects

Raghu Bollapragada

Publications and source records attributed to Raghu Bollapragada.

At least 19 recordsLinked to original sources

arXiv2026

Negative Curvature Methods with High-Probability Complexity Guarantees for Stochastic Nonconvex Optimization

This paper develops negative curvature methods for continuous nonlinear unconstrained optimization in stochastic settings, in which function, gradient, and Hessian information is available only through probabilistic oracles, i.e., oracles that return approximations of a certain accuracy and reliability. We introduce conditions on these oracles and design a two-step framework that systematically combines gradient and negative curvature steps. The framework employs an early-stopping mechanism to guarantee sufficient progress and uses an adaptive mechanism based on an Armijo-type criterion to select the step sizes for both steps. We establish high-probability iteration-complexity guarantees for attaining second-order stationary points, deriving explicit tail bounds that quantify the convergence neighborhood and its dependence on oracle noise. Importantly, these bounds match deterministic rates up to noise-dependent terms, and the framework recovers the deterministic results as a special case. Finally, numerical experiments demonstrate the practical benefits of exploiting negative curvature directions even in the presence of noise.

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arXiv2025

On the Convergence and Complexity of Proximal Gradient and Accelerated Proximal Gradient Methods under Adaptive Gradient Estimation

In this paper, we propose a proximal gradient method and an accelerated proximal gradient method for solving composite optimization problems, where the objective function is the sum of a smooth and a convex, possibly nonsmooth, function. We consider settings where the smooth component is either a finite-sum function or an expectation of a stochastic function, making it computationally expensive or impractical to evaluate its gradient. To address this, we utilize gradient estimates within the proximal gradient framework. Our methods dynamically adjust the accuracy of these estimates, increasing it as the iterates approach a solution, thereby enabling high-precision solutions with minimal computational cost. We analyze the methods when the smooth component is nonconvex, convex, or strongly convex, using a biased gradient estimate. In all cases, the methods achieve the optimal iteration complexity for first-order methods. When the gradient estimate is unbiased, we further refine the analysis to show that the methods simultaneously achieve optimal iteration complexity and optimal complexity in terms of the number of stochastic gradient evaluations. Finally, we validate our theoretical results through numerical experiments.

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arXiv2025

On the fast convergence of minibatch heavy ball momentum

Simple stochastic momentum methods are widely used in machine learning optimization, but their good practical performance is at odds with an absence of theoretical guarantees of acceleration in the literature. In this work, we aim to close the gap between theory and practice by showing that stochastic heavy ball momentum retains the fast linear rate of (deterministic) heavy ball momentum on quadratic optimization problems, at least when minibatching with a sufficiently large batch size. The algorithm we study can be interpreted as an accelerated randomized Kaczmarz algorithm with minibatching and heavy ball momentum. The analysis relies on carefully decomposing the momentum transition matrix, and using new spectral norm concentration bounds for products of independent random matrices. We provide numerical illustrations demonstrating that our bounds are reasonably sharp.

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arXiv2025

Retrospective Approximation Sequential Quadratic Programming for Stochastic Optimization with General Deterministic Nonlinear Constraints

In this paper, we propose a framework based on the Retrospective Approximation (RA) paradigm to solve optimization problems with a stochastic objective function and general nonlinear deterministic constraints. This framework sequentially constructs increasingly accurate approximations of the true problems which are solved to a specified accuracy via a deterministic solver, thereby decoupling the uncertainty from the optimization. Such frameworks retain the advantages of deterministic optimization methods, such as fast convergence, while achieving the optimal performance of stochastic methods without the need to redesign algorithmic components. For problems with general nonlinear equality constraints, we present a framework that can employ any deterministic solver and analyze its theoretical work complexity. We then present an instance of the framework that employs a deterministic Sequential Quadratic Programming (SQP) method and that achieves optimal complexity in terms of gradient evaluations and linear system solves for this class of problems. For problems with general nonlinear constraints, we present an RA-based algorithm that employs an SQP method with robust subproblems. Finally, we demonstrate the empirical performance of the proposed framework on multi-class logistic regression problems and benchmark instances from the CUTEst test set, comparing its results to established methods from the literature.

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arXiv2025

On the Convergence and Complexity of the Stochastic Central Finite-Difference Based Gradient Estimation Methods

This paper presents an algorithmic framework for solving unconstrained stochastic optimization problems using only stochastic function evaluations. We employ central finite-difference based gradient estimation methods to approximate the gradients and dynamically control the accuracy of these approximations by adjusting the sample sizes used in stochastic realizations. We analyze the theoretical properties of the proposed framework on nonconvex functions. Our analysis yields sublinear convergence results to the neighborhood of the solution, and establishes the optimal worst-case iteration complexity ($\mathcal{O}(ε^{-1})$) and sample complexity ($\mathcal{O}(ε^{-2})$) for each gradient estimation method to achieve an $ε$-accurate solution. Finally, we demonstrate the performance of the proposed framework and the quality of the gradient estimation methods through numerical experiments on nonlinear least squares problems.

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arXiv2024

Efficient Mathematical Programming Formulation and Algorithmic Framework for Optimal Camera Placement

Optimal camera placement plays a crucial role in applications such as surveillance, environmental monitoring, and infrastructure inspection. Even highly abstracted versions of this problem are NP-hard due to the high-dimensional continuous domain of camera configurations (i.e., positions and orientations) and difficulties in efficiently and accurately calculating camera coverage. In this paper, we present a novel framework for optimal camera placement that uses integer programming and adaptive sampling strategies to maximize coverage, given a limited camera budget. We develop a modified maximum k-coverage formulation and two adaptive sampling strategies, Explore and Exploit (E&E) and Target Uncovered Spaces (TUS), that iteratively add new camera configurations to the candidate set in order to improve the solution. E&E focuses on local search around camera configurations chosen in previous iterations, whereas TUS focuses specifically on covering regions that were previously uncovered. We first conduct theoretical analysis to provide bounds on the probability of finding an optimal solution and expected sampling needs, while ensuring monotonic improvements in coverage. Then, we conduct a detailed numerical analysis over different environments. Results show that E&E achieves coverage improvements of 3.3-16.0% over all baseline random sampling approaches, while maintaining manageable computational times. Meanwhile, TUS performs well in open environments and with tight camera budgets, achieving gains of 6.9-9.1% in such conditions. Compared to the baseline, our approach achieves similar coverage using only 30-70% of the sampling budget, demonstrating its computational efficiency. Through a case study, we obtain insights into optimal camera placement decisions for a typical indoor surveillance application.

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arXiv2024

Exploiting Negative Curvature in Conjunction with Adaptive Sampling: Theoretical Results and a Practical Algorithm

In this paper, we propose algorithms that exploit negative curvature for solving noisy nonlinear nonconvex unconstrained optimization problems. We consider both deterministic and stochastic inexact settings, and develop two-step algorithms that combine directions of negative curvature and descent directions to update the iterates. Under reasonable assumptions, we prove second-order convergence results and derive complexity guarantees for both settings. To tackle large-scale problems, we develop a practical variant that utilizes the conjugate gradient method with negative curvature detection and early stopping to compute a step, a simple adaptive step size scheme, and a strategy for selecting the sample sizes of the gradient and Hessian approximations as the optimization progresses. Numerical results on two machine learning problems showcase the efficacy and efficiency of the practical method.

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arXiv2024

Fast Unconstrained Optimization via Hessian Averaging and Adaptive Gradient Sampling Methods

We consider minimizing finite-sum and expectation objective functions via Hessian-averaging based subsampled Newton methods. These methods allow for gradient inexactness and have fixed per-iteration Hessian approximation costs. The recent work (Na et al. 2023) demonstrated that Hessian averaging can be utilized to achieve fast $\mathcal{O}\left(\sqrt{\tfrac{\log k}{k}}\right)$ local superlinear convergence for strongly convex functions in high probability, while maintaining fixed per-iteration Hessian costs. These methods, however, require gradient exactness and strong convexity, which poses challenges for their practical implementation. To address this concern we consider Hessian-averaged methods that allow gradient inexactness via norm condition based adaptive-sampling strategies. For the finite-sum problem we utilize deterministic sampling techniques which lead to global linear and sublinear convergence rates for strongly convex and nonconvex functions respectively. In this setting we are able to derive an improved deterministic local superlinear convergence rate of $\mathcal{O}\left(\tfrac{1}{k}\right)$. For the %expected risk expectation problem we utilize stochastic sampling techniques, and derive global linear and sublinear rates for strongly convex and nonconvex functions, as well as a $\mathcal{O}\left(\tfrac{1}{\sqrt{k}}\right)$ local superlinear convergence rate, all in expectation. We present novel analysis techniques that differ from the previous probabilistic results. Additionally, we propose scalable and efficient variations of these methods via diagonal approximations and derive the novel diagonally-averaged Newton (Dan) method for large-scale problems. Our numerical results demonstrate that the Hessian averaging not only helps with convergence, but can lead to state-of-the-art performance on difficult problems such as CIFAR100 classification with ResNets.

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arXiv2024

Modified Line Search Sequential Quadratic Methods for Equality-Constrained Optimization with Unified Global and Local Convergence Guarantees

In this paper, we propose a method that has foundations in the line search sequential quadratic programming paradigm for solving general nonlinear equality constrained optimization problems. The method employs a carefully designed modified line search strategy that utilizes second-order information of both the objective and constraint functions, as required, to mitigate the Maratos effect. Contrary to classical line search sequential quadratic programming methods, our proposed method is endowed with global convergence and local superlinear convergence guarantees. Moreover, we extend the method and analysis to the setting in which the constraint functions are deterministic but the objective function is stochastic or can be represented as a finite-sum. We also design and implement a practical inexact matrix-free variant of the method. Finally, numerical results illustrate the efficiency and efficacy of the method.

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arXiv2024

Derivative-Free Optimization via Adaptive Sampling Strategies

In this paper, we present a novel derivative-free optimization framework for solving unconstrained stochastic optimization problems. Many problems in fields ranging from simulation optimization to reinforcement learning involve settings where only stochastic function values are obtained via an oracle with no available gradient information, necessitating the usage of derivative-free optimization methodologies. Our approach includes estimating gradients using stochastic function evaluations and integrating adaptive sampling techniques to control the accuracy in these stochastic approximations. We consider various gradient estimation techniques including standard finite difference, Gaussian smoothing, sphere smoothing, randomized coordinate finite difference, and randomized subspace finite difference methods. We provide theoretical convergence guarantees for our framework and analyze the worst-case iteration and sample complexities associated with each gradient estimation method. Finally, we demonstrate the empirical performance of the methods on logistic regression and nonlinear least squares problems.

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arXiv2024

A Retrospective Approximation Approach for Smooth Stochastic Optimization

Stochastic Gradient (SG) is the defacto iterative technique to solve stochastic optimization (SO) problems with a smooth (non-convex) objective $f$ and a stochastic first-order oracle. SG's attractiveness is due in part to its simplicity of executing a single step along the negative subsampled gradient direction to update the incumbent iterate. In this paper, we question SG's choice of executing a single step as opposed to multiple steps between subsample updates. Our investigation leads naturally to generalizing SG into Retrospective Approximation (RA) where, during each iteration, a "deterministic solver" executes possibly multiple steps on a subsampled deterministic problem and stops when further solving is deemed unnecessary from the standpoint of statistical efficiency. RA thus rigorizes what is appealing for implementation -- during each iteration, "plug in" a solver, e.g., L-BFGS line search or Newton-CG, as is, and solve only to the extent necessary. We develop a complete theory using relative error of the observed gradients as the principal object, demonstrating that almost sure and $L_1$ consistency of RA are preserved under especially weak conditions when sample sizes are increased at appropriate rates. We also characterize the iteration and oracle complexity (for linear and sub-linear solvers) of RA, and identify a practical termination criterion leading to optimal complexity rates. To subsume non-convex $f$, we present a certain "random central limit theorem" that incorporates the effect of curvature across all first-order critical points, demonstrating that the asymptotic behavior is described by a certain mixture of normals. The message from our numerical experiments is that the ability of RA to incorporate existing second-order deterministic solvers in a strategic manner might be important from the standpoint of dispensing with hyper-parameter tuning.

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arXiv2023

An Adaptive Sampling Augmented Lagrangian Method for Stochastic Optimization with Deterministic Constraints

The primary goal of this paper is to provide an efficient solution algorithm based on the augmented Lagrangian framework for optimization problems with a stochastic objective function and deterministic constraints. Our main contribution is combining the augmented Lagrangian framework with adaptive sampling, resulting in an efficient optimization methodology validated with practical examples. To achieve the presented efficiency, we consider inexact solutions for the augmented Lagrangian subproblems, and through an adaptive sampling mechanism, we control the variance in the gradient estimates. Furthermore, we analyze the theoretical performance of the proposed scheme by showing equivalence to a gradient descent algorithm on a Moreau envelope function, and we prove sublinear convergence for convex objectives and linear convergence for strongly convex objectives with affine equality constraints. The worst-case sample complexity of the resulting algorithm, for an arbitrary choice of penalty parameter in the augmented Lagrangian function, is $O(ε^{-3-δ})$, where $ε> 0$ is the expected error of the solution and $δ> 0$ is a user-defined parameter. If the penalty parameter is chosen to be $O(ε^{-1})$, we demonstrate that the result can be improved to $O(ε^{-2})$, which is competitive with the other methods employed in the literature. Moreover, if the objective function is strongly convex with affine equality constraints, we obtain $O(ε^{-1}\log(1/ε))$ complexity. Finally, we empirically verify the performance of our adaptive sampling augmented Lagrangian framework in machine learning optimization and engineering design problems, including topology optimization of a heat sink with environmental uncertainty.

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arXiv2023

A Flexible Gradient Tracking Algorithmic Framework for Decentralized Optimization

In decentralized optimization over networks, each node in the network has a portion of the global objective function and the aim is to collectively optimize this function. Gradient tracking methods have emerged as a popular alternative for solving such problems due to their strong theoretical guarantees and robust empirical performance. These methods perform two operations (steps) at each iteration: (1) compute local gradients at each node, and (2) communicate local information with neighboring nodes. The complexity of these two steps can vary significantly across applications. In this work, we present a flexible gradient tracking algorithmic framework designed to balance the composition of communication and computation steps over the optimization process using a randomized scheme. The proposed framework is general, unifies gradient tracking methods, and recovers classical gradient tracking methods as special cases. We establish convergence guarantees in expectation and illustrate how the complexity of communication and computation steps can be balanced using the provided flexibility. Finally, we illustrate the performance of the proposed methods on quadratic and logistic regression problems, and compare against popular algorithms from the literature.

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arXiv2023

Balancing Communication and Computation in Gradient Tracking Algorithms for Decentralized Optimization

Gradient tracking methods have emerged as one of the most popular approaches for solving decentralized optimization problems over networks. In this setting, each node in the network has a portion of the global objective function, and the goal is to collectively optimize this function. At every iteration, gradient tracking methods perform two operations (steps): $(1)$ compute local gradients, and $(2)$ communicate information with local neighbors in the network. The complexity of these two steps varies across different applications. In this paper, we present a framework that unifies gradient tracking methods and is endowed with flexibility with respect to the number of communication and computation steps. We establish unified theoretical convergence results for the algorithmic framework with any composition of communication and computation steps, and quantify the improvements achieved as a result of this flexibility. The framework recovers the results of popular gradient tracking methods as special cases, and allows for a direct comparison of these methods. Finally, we illustrate the performance of the proposed methods on quadratic functions and binary classification problems.

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arXiv2023

Adaptive Consensus: A network pruning approach for decentralized optimization

We consider network-based decentralized optimization problems, where each node in the network possesses a local function and the objective is to collectively attain a consensus solution that minimizes the sum of all the local functions. A major challenge in decentralized optimization is the reliance on communication which remains a considerable bottleneck in many applications. To address this challenge, we propose an adaptive randomized communication-efficient algorithmic framework that reduces the volume of communication by periodically tracking the disagreement error and judiciously selecting the most influential and effective edges at each node for communication. Within this framework, we present two algorithms: Adaptive Consensus (AC) to solve the consensus problem and Adaptive Consensus based Gradient Tracking (AC-GT) to solve smooth strongly convex decentralized optimization problems. We establish strong theoretical convergence guarantees for the proposed algorithms and quantify their performance in terms of various algorithmic parameters under standard assumptions. Finally, numerical experiments showcase the effectiveness of the framework in significantly reducing the information exchange required to achieve a consensus solution.

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arXiv2023

A Stochastic Gradient Tracking Algorithm for Decentralized Optimization With Inexact Communication

Decentralized optimization is typically studied under the assumption of noise-free transmission. However, real-world scenarios often involve the presence of noise due to factors such as additive white Gaussian noise channels or probabilistic quantization of transmitted data. These sources of noise have the potential to degrade the performance of decentralized optimization algorithms if not effectively addressed. In this paper, we focus on the noisy communication setting and propose an algorithm that bridges the performance gap caused by communication noise while also mitigating other challenges like data heterogeneity. We establish theoretical results of the proposed algorithm that quantify the effect of communication noise and gradient noise on the performance of the algorithm. Notably, our algorithm achieves the optimal convergence rate for minimizing strongly convex, smooth functions in the context of inexact communication and stochastic gradients. Finally, we illustrate the superior performance of the proposed algorithm compared to its state-of-the-art counterparts on machine learning problems using MNIST and CIFAR-10 datasets.

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arXiv2023

An Adaptive Sampling Sequential Quadratic Programming Method for Equality Constrained Stochastic Optimization

This paper presents a methodology for using varying sample sizes in sequential quadratic programming (SQP) methods for solving equality constrained stochastic optimization problems. The first part of the paper deals with the delicate issue of dynamic sample selection in the evaluation of the gradient in conjunction with inexact solutions to the SQP subproblems. Under reasonable assumptions on the quality of the employed gradient approximations and the accuracy of the solutions to the SQP subproblems, we establish global convergence results for the proposed method. Motivated by these results, the second part of the paper describes a practical adaptive inexact stochastic sequential quadratic programming (PAIS-SQP) method. We propose criteria for controlling the sample size and the accuracy in the solutions of the SQP subproblems based on estimates of the variance in the stochastic gradient approximations obtained as the optimization progresses. Finally, we demonstrate the performance of the practical method on a subset of the CUTE problems and constrained classification tasks.

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arXiv2022

Scalable Unidirectional Pareto Optimality for Multi-Task Learning with Constraints

Multi-objective optimization (MOO) problems require balancing competing objectives, often under constraints. The Pareto optimal solution set defines all possible optimal trade-offs over such objectives. In this work, we present a novel method for Pareto-front learning: inducing the full Pareto manifold at train-time so users can pick any desired optimal trade-off point at run-time. Our key insight is to exploit Fritz-John Conditions for a novel guided double gradient descent strategy. Evaluation on synthetic benchmark problems allows us to vary MOO problem difficulty in controlled fashion and measure accuracy vs. known analytic solutions. We further test scalability and generalization in learning optimal neural model parameterizations for Multi-Task Learning (MTL) on image classification. Results show consistent improvement in accuracy and efficiency over prior MTL methods as well as techniques from operations research.

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