Preconditioned Stochastic Optimization Methods for Ill-Conditioned Large-Scale Convex Optimization Problems in Machine Learning

Core Concepts

Introducing PROMISE, a suite of sketching-based preconditioned stochastic gradient algorithms for fast convergence on ill-conditioned large-scale convex optimization problems in machine learning.

Abstract

The paper introduces PROMISE, a suite of sketching-based preconditioned stochastic gradient algorithms that deliver fast convergence on ill-conditioned large-scale convex optimization problems in machine learning. The algorithms include SketchySVRG, SketchySAGA, and SketchyKatyusha, each with theoretical analysis and default hyperparameter values. Empirical results show the superiority of these algorithms over popular tuned stochastic gradient optimizers. The paper also introduces the notion of quadratic regularity to establish linear convergence even with infrequent updates to the preconditioner.

Stats

Ill-conditioned problems are common in large-scale machine learning as datasets grow. Default hyperparameter values outperform popular tuned stochastic gradient optimizers. The condition number for many ML problems is typically on the order of 10^4 to 10^8. Stochastic second-order methods have difficulties delivering fast local-linear convergence without vanishing noise in gradient estimates. The proposed methods achieve linear convergence with lazy updates to the preconditioner.

Quotes

"Using default hyperparameter values, they outperform or match popular tuned stochastic gradient optimizers." "The speed of linear convergence is determined by the quadratic regularity ratio." "Our methods avoid the usual theory-practice gap: our theoretical advances yield practical algorithms."

Key Insights Distilled From

PROMISE

by Zachary Fran... at arxiv.org 03-15-2024

https://arxiv.org/pdf/2309.02014.pdf

Deeper Inquiries

How can the concept of quadratic regularity be applied to other optimization problems

The concept of quadratic regularity can be applied to other optimization problems by providing a framework for analyzing the convergence properties of optimization algorithms. By introducing the notion of quadratic regularity, researchers can establish linear convergence guarantees for various optimization methods even when the preconditioner is updated infrequently. This allows for a deeper understanding of how different factors such as curvature estimates and preconditioning impact the convergence rates of optimization algorithms in different problem settings. Furthermore, by considering the quadratic regularity ratio as a measure of how well-behaved an objective function is with respect to its Hessian, researchers can develop more efficient and effective optimization techniques tailored to specific problem structures. The application of quadratic regularity extends beyond stochastic gradient methods and can provide insights into improving convergence rates in a wide range of optimization problems across different domains.

What are the potential drawbacks or limitations of using sketching-based preconditioned stochastic gradient algorithms

While sketching-based preconditioned stochastic gradient algorithms offer several advantages such as fast convergence on ill-conditioned large-scale convex optimization problems and minimal hyperparameter tuning requirements, there are potential drawbacks and limitations associated with their use: Computational Overhead: Implementing sketching techniques may introduce additional computational overhead due to the need for random projections or subsampling operations during each iteration. This could lead to increased runtime complexity compared to traditional optimization methods. Accuracy Trade-offs: Sketching-based approaches rely on approximations obtained through random projections or subsampling, which may result in loss of accuracy compared to exact computations using full gradients or Hessians. This trade-off between accuracy and efficiency needs careful consideration based on specific problem requirements. Algorithm Sensitivity: The performance of sketching-based algorithms can be sensitive to factors such as batch sizes, sampling strategies, and choice of sketch dimensions. Suboptimal parameter selections could affect algorithm stability and convergence behavior. Limited Generalizability: While effective in certain scenarios like machine learning tasks with large datasets and high-dimensional features, sketching-based preconditioned stochastic gradient algorithms may not generalize well across all types of optimization problems or domains.

How do these findings impact current practices in machine learning optimization techniques

These findings have significant implications for current practices in machine learning optimization techniques: Improved Efficiency: The development of PROMISE (Preconditioned Stochastic Optimization Methods) suite offers practitioners faster convergence on ill-conditioned large-scale convex optimization problems without extensive hyperparameter tuning efforts. Enhanced Performance: By incorporating scalable curvature estimates through sketching-based preconditioners like SSN, NySSN, SASSN-C/R into popular stochastic gradient optimizers (SVRG, SAGA), PROMISE demonstrates superior performance over existing methods. 3Automated Hyperparameter Selection: Default hyperparameters provided by PROMISE along with automated learning rate computation based on estimated smoothness constants enable out-of-the-box usage without manual parameter tuning. 4Theoretical Advances: Introducing concepts like quadratic regularity provides theoretical foundations for understanding linear convergence properties even under lazy updates conditions - bridging theory-practice gaps in modern machine learning optimizations methodologies.