Bibliographic Information: Starnes, A., Zhang, G., Reshniak, V., & Webster, C. (2024). Anisotropic Gaussian Smoothing for Gradient-based Optimization. arXiv preprint arXiv:2411.11747v1.
Research Objective: This paper introduces a novel family of optimization algorithms—AGS-GD, AGS-SGD, and AGS-Adam—that leverage anisotropic Gaussian smoothing to enhance traditional gradient-based optimization methods and address the challenge of escaping suboptimal local minima.
Methodology: The authors propose replacing the standard gradient in GD, SGD, and Adam with a non-local gradient derived from averaging function values using anisotropic Gaussian smoothing. This technique adapts the smoothing directionality based on the underlying function's properties, aligning better with complex loss landscapes. The anisotropy is computed by adjusting the Gaussian distribution's covariance matrix, allowing for directional smoothing tailored to the gradient's behavior. The paper provides detailed convergence analyses for these algorithms, extending results from both unsmoothed and isotropic Gaussian smoothing cases to the more general anisotropic smoothing, applicable to both convex and non-convex, L-smooth functions.
Key Findings: The research demonstrates that AGS algorithms effectively mitigate the impact of minor fluctuations in the loss landscape, enabling them to approach global minima more effectively. The convergence analyses prove that AGS algorithms converge to a noisy ball in the stochastic setting, with its size determined by the smoothing parameters.
Main Conclusions: The authors conclude that anisotropic Gaussian smoothing offers a promising approach to enhancing traditional gradient-based optimization methods. The proposed AGS algorithms demonstrate improved convergence properties and a greater ability to escape suboptimal local minima.
Significance: This research contributes to the field of optimization by introducing a novel technique for improving the performance of gradient-based algorithms. The proposed AGS algorithms have the potential to impact various domains, including machine learning, deep learning, and other areas where optimization plays a crucial role.
Limitations and Future Research: The paper acknowledges the computational complexity of calculating smoothed functions or their gradients as a practical challenge. The authors suggest exploring efficient numerical methods, such as Monte Carlo estimation, for approximating the smoothed gradient. Future research directions include investigating the relationship between smoothing parameter selection and algorithm performance across different problem domains and further exploring the application of AGS algorithms in various practical optimization tasks.
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by Andrew Starn... ב- arxiv.org 11-19-2024
https://arxiv.org/pdf/2411.11747.pdfשאלות מעמיקות