Why Line Search when you can Plane Search? SO-Friendly Neural Networks allow Per-Iteration Optimization of Learning and Momentum Rates for Every Layer (2406.17954v1)

Abstract: We introduce the class of SO-friendly neural networks, which include several models used in practice including networks with 2 layers of hidden weights where the number of inputs is larger than the number of outputs. SO-friendly networks have the property that performing a precise line search to set the step size on each iteration has the same asymptotic cost during full-batch training as using a fixed learning. Further, for the same cost a planesearch can be used to set both the learning and momentum rate on each step. Even further, SO-friendly networks also allow us to use subspace optimization to set a learning rate and momentum rate for each layer on each iteration. We explore augmenting gradient descent as well as quasi-Newton methods and Adam with line optimization and subspace optimization, and our experiments indicate that this gives fast and reliable ways to train these networks that are insensitive to hyper-parameters.

• The paper introduces SO-friendly networks that enable per-iteration plane search to efficiently adjust learning and momentum rates.
• It demonstrates that plane search matches the computational cost of fixed learning rates while reducing hyper-parameter tuning.
• Empirical results show that subspace optimization methods lead to faster convergence in logistic regression and two-layer neural networks.

An Expert Overview of "Why Line Search When You Can Plane Search?"

The paper authored by Betty Shea and Mark Schmidt explores an innovative approach to optimizing neural networks by leveraging subspace optimization (SO) in place of traditional line search (LS) methods. Their work focuses on a specific class of neural networks termed SO-friendly networks. These networks demonstrate that using plane search to optimize learning and momentum rates per iteration can match the asymptotic computational costs of using a fixed learning rate, thus potentially providing significant performance improvements.

Key Insights and Contributions

Subspace Optimization-Friendly Neural Networks

The primary contribution of the paper is the identification and formalization of SO-friendly neural networks. This class includes networks with two hidden layers where the number of inputs significantly exceeds the number of outputs. Such networks enable efficient use of SO to dynamically adjust learning and momentum rates per layer without incurring additional asymptotic computational costs compared to fixed-rate methods. This attribute is particularly vital in full-batch training scenarios.

Line and Plane Search: Conceptual Shift

Under traditional LS methods, algorithms like L-BFGS or standard GD with momentum (GD+M) set a fixed learning rate, which often necessitates a costly hyper-parameter search for optimal performance. However, SO-friendly networks permit a shift towards plane search, where both learning and momentum rates are tuned per iteration "for free".

The authors demonstrate this through augmentations of gradient and quasi-Newton methods with line optimization (LO) and subspace optimization (SO). Numerical comparisons indicate that incorporating LO and SO significantly improves the convergence rates and robustness of GD, quasi-Newton methods, and Adam optimizer concerning hyper-parameter sensitivity.

Experimental Validation

To support their claims, the authors conduct extensive empirical analysis across multiple datasets and network configurations:

• Logistic Regression: The experimental results show that using LO and SO can drastically outperform traditional fixed-step-size approaches, demonstrating faster and more reliable convergence for logistic regression.
• Neural Networks: Different configurations for two-layer neural networks underscore that SO methods consistently outshine LS methods, both in terms of speed and final convergence. Notably, methods optimizing per-layer step sizes suggest significant yet sometimes variable performance gains, especially under fused scenarios where L2 regularization is applied.

Practical and Theoretical Implications

Practical Implications:

1. Training Efficiency: By harnessing LO and SO, practitioners can achieve faster convergence without the burdensome hyper-parameter searches typically associated with traditional methods. This efficiency gain can translate to shorter training times and potentially lower computational costs, critical in large-scale ML applications.
2. Per-Layer Optimization: The capacity to optimize learning and momentum rates per layer can lead to more finely-tuned models, especially in settings where differing learning dynamics across layers can be exploited.

Theoretical Implications:

1. Convergence Properties: The paper implicitly challenges the sufficiency and efficiency of LS in modern ML optimization, particularly in deep learning contexts. It provides a robust foundation for rethinking step size adjustments in gradient-based optimization.
2. SO-Friendly Networks: While the work currently restricts SO efficiency to a specific class of networks, it paves the way for further research into broader classes of models and problem structures where SO can be similarly applied efficaciously.

Future Directions in AI

The promising results and conceptual shifts presented in this work suggest several avenues for future research:

1. Stochastic Training: Extending SO methods to stochastic gradient descent (SGD) could be explored. Potential performance benefits could be substantial for large-scale data scenarios, where full-batch methods are impractical.
2. Deep Network Architectures: Investigating deeper and more complex architectures, beyond two-layer networks, to identify under what conditions and configurations SO can be efficient.
3. Alternative Optimization Methods: Beyond GD and Adam, future investigations could explore the integration of SO with other modern optimization algorithms, potentially unveiling new classes of optimization methods tailored for broader applicability.

In summary, this paper's thorough exploration and validation of using plane search instead of line search propose a noteworthy advancement in the optimization of certain neural networks. Their findings are poised to influence future research and practice in ML, potentially offering a more efficient and robust pathway for training complex models.