A Novel Stochastic Optimizer: Filtering Informed Newton-like and Derivative-free Evolutionary Recursion (FINDER) for Large-Scale Optimization

Concepts de base

This paper introduces FINDER, a novel stochastic optimizer that combines the advantages of quasi-Newton methods and noise-assisted global search, demonstrating its effectiveness in high-dimensional optimization problems, including deep network training.

Résumé

• Bibliographic Information: Sumana, U., Mamajiwala, M., Saxena, M., Tyagi, A., & Roy, D. (2024). Stochastic Quasi-Newton Optimization in Large Dimensions Including Deep Network Training. arXiv preprint arXiv:2410.14270.

• Research Objective: This paper proposes a new stochastic optimizer, FINDER, designed to address the challenges of high-dimensional, non-convex optimization problems, particularly in the context of deep network training.

• Methodology: FINDER leverages stochastic filtering principles to approximate the inverse Hessian of the objective function, mimicking the behavior of quasi-Newton methods without explicitly calculating the Hessian. It employs a derivative-free approach, utilizing an ensemble Kalman filter framework to update the search particles. The algorithm incorporates several simplifications and enhancements, including a diagonal approximation of the inverse Hessian and a history-dependent update rule, to ensure scalability and efficiency in large-scale problems.

• Key Findings: FINDER demonstrates competitive performance compared to the Adam optimizer across a range of benchmark functions and deep learning tasks. It exhibits faster local convergence due to its quasi-Newton-like update scheme while maintaining the ability to escape local optima through its stochastic nature. The authors demonstrate FINDER's effectiveness in training various deep neural networks, including Physics-Informed Neural Networks (PINNs), for problems like the Burgers' equation, 2D elasticity, and strain gradient plasticity.

• Main Conclusions: FINDER presents a promising approach for high-dimensional optimization, particularly in machine learning applications. Its ability to combine rapid local convergence with global search capabilities makes it suitable for complex, non-convex optimization landscapes.

• Significance: This research contributes to the field of optimization by introducing a novel algorithm that addresses the limitations of existing methods in high-dimensional spaces. FINDER's application to deep network training, especially for PINNs, highlights its potential impact on solving complex scientific and engineering problems.

• Limitations and Future Research: While FINDER shows promising results, further investigation into its hyperparameter tuning and exploration of more sophisticated variance reduction techniques could enhance its performance. Additionally, comparing FINDER with a wider range of state-of-the-art optimizers on diverse large-scale problems would provide a more comprehensive evaluation of its capabilities.

Stats

The Adam optimizer uses a constant learning rate of 10^-3 and default hyperparameters: β1 = 0.9, β2 = 0.999, ϵ = 10^-8.
FINDER uses the following hyperparameters: p = 5, θ = 0.9, γ = 1, cs = 0.1, cα = 0.01.
For the Burgers’ equation and 2D elasticity problems, FINDER uses ζ1 = ζ2 = 10^-4.
For the strain gradient plasticity problem, FINDER starts with ζ1 = ζ2 = 0.1 and reduces them to 10^-4 after reaching a low loss value.

Citations

"Our present aim is to harness the rapid local convergence of a Newton-like strategy whilst simultaneously allowing a noise-aided exploration of the search space."
"The inherent trade-off between faster convergence and quality of solution underscores the complexity of optimization challenges."
"An optimization method, with twin abilities of linear scaling across dimensions and good performance at global search, remains largely elusive."

Idées clés tirées de

Stochastic Quasi-Newton Optimization in Large Dimensions Including Deep Network Training

by Uttam Suman,... à arxiv.org 10-21-2024

https://arxiv.org/pdf/2410.14270.pdf

Questions plus approfondies

How does the performance of FINDER compare to other established stochastic optimizers like RMSprop or Adadelta in deep learning tasks beyond those presented in the paper?

While the paper focuses on comparing FINDER with Adam, a direct comparison with RMSprop or Adadelta on deep learning tasks is not provided. However, we can infer potential advantages and disadvantages based on FINDER's design and the known characteristics of these established optimizers:
Potential Advantages of FINDER:

Quasi-Newton Updates: FINDER's strength lies in its approximation of the inverse Hessian, potentially leading to faster convergence in regions with a well-defined curvature. This could be advantageous compared to RMSprop and Adadelta, which primarily rely on exponentially weighted moving averages of past squared gradients.
Noise-Assisted Exploration: The inherent stochasticity in FINDER, stemming from its filtering framework, might offer better exploration capabilities compared to RMSprop and Adadelta, especially in landscapes with plateaus or saddle points.
Potential Disadvantages of FINDER:

Computational Overhead: FINDER's reliance on ensemble simulations and matrix operations introduces computational overhead, potentially making it slower than RMSprop or Adadelta, especially in very high-dimensional settings.
Hyperparameter Sensitivity: The performance of FINDER might be sensitive to the choice of hyperparameters like ζ1, ζ2, and γ, requiring careful tuning.
In conclusion, FINDER's performance relative to RMSprop or Adadelta would depend on the specific deep learning task and the dataset's characteristics. Empirical evaluations on diverse tasks are needed to draw definitive conclusions.

While FINDER incorporates noise for exploration, could its performance be hindered in high-dimensional spaces where the curse of dimensionality might limit its exploration capabilities?

Yes, FINDER's performance could be hindered in high-dimensional spaces due to the curse of dimensionality, despite its noise-assisted exploration. Here's why:

Sparse Sampling: In high dimensions, the number of particles used in FINDER's ensemble might become insufficient to effectively sample the vast search space. This can lead to a limited exploration and a higher chance of getting stuck in local optima.
Ineffective Diffusion: The diffusion process governed by the matrix Rt might become inefficient in exploring the high-dimensional space. The fixed spread, even with adaptive scaling, might not be sufficient to escape local optima effectively.
Potential Mitigation Strategies:

Increased Ensemble Size: Increasing the number of particles in the ensemble could improve exploration but at the cost of higher computational burden.
Adaptive Noise Scaling: Implementing adaptive noise scaling mechanisms, potentially inspired by techniques like simulated annealing or evolutionary strategies, could help FINDER better navigate the high-dimensional landscape.
Dimensionality Reduction: Exploring dimensionality reduction techniques or feature selection methods before applying FINDER could alleviate the curse of dimensionality to some extent.
In summary, while FINDER incorporates noise for exploration, careful consideration of the curse of dimensionality is crucial in high-dimensional optimization problems. Further research and modifications to the algorithm might be necessary to enhance its effectiveness in such scenarios.

Given the increasing use of deep learning in scientific discovery, could FINDER's success in training PINNs pave the way for developing more efficient and robust solvers for complex physical simulations?

FINDER's success in training PINNs indeed holds promising implications for developing more efficient and robust solvers for complex physical simulations. Here's how:

Handling Stiffness and Non-linearities: The paper demonstrates FINDER's capability to handle the stiff nature and non-linearities inherent in problems like strain gradient plasticity. This suggests its potential for solving complex physical simulations often characterized by such challenges.
Derivative-Free Nature: FINDER's derivative-free approach could be particularly beneficial in scenarios where obtaining analytical gradients of the loss function is difficult or computationally expensive. This is often the case in complex physical simulations involving intricate constitutive models or multi-physics interactions.
Improved Convergence: The quasi-Newton updates in FINDER, leveraging an approximation of the inverse Hessian, could lead to faster convergence compared to traditional gradient-based optimizers, potentially reducing the computational cost of these simulations.
Potential Impact on Scientific Discovery:

Accelerated Material Design: FINDER's ability to efficiently solve problems like strain gradient plasticity could accelerate the design of new materials with enhanced properties.
Enhanced Understanding of Physical Phenomena: By providing more efficient solvers, FINDER could enable researchers to explore a wider range of physical parameters and scenarios, leading to a deeper understanding of complex phenomena.
Improved Predictive Capabilities: More robust and efficient solvers could lead to more accurate and reliable predictions in fields like climate modeling, fluid dynamics, and structural analysis.
However, challenges remain:

Scalability to Higher Dimensions: Extending FINDER's efficiency to even larger-scale simulations involving millions or billions of degrees of freedom requires further research and development.
Integration with Existing Simulation Frameworks: Seamless integration of FINDER with existing simulation software and workflows is crucial for its widespread adoption in scientific communities.
In conclusion, FINDER's success in training PINNs represents a significant step towards developing more efficient and robust solvers for complex physical simulations. Further research and development in this direction could have a transformative impact on scientific discovery and engineering applications.