Training Deep Learning Models with Norm-Constrained LMOs (2502.07529v2)

Abstract: In this work, we study optimization methods that leverage the linear minimization oracle (LMO) over a norm-ball. We propose a new stochastic family of algorithms that uses the LMO to adapt to the geometry of the problem and, perhaps surprisingly, show that they can be applied to unconstrained problems. The resulting update rule unifies several existing optimization methods under a single framework. Furthermore, we propose an explicit choice of norm for deep architectures, which, as a side benefit, leads to the transferability of hyperparameters across model sizes. Experimentally, we demonstrate significant speedups on nanoGPT training using our algorithm, Scion, without any reliance on Adam. The proposed method is memory-efficient, requiring only one set of model weights and one set of gradients, which can be stored in half-precision. The code is available at https://github.com/LIONS-EPFL/scion .

• The paper proposes a novel stochastic family of optimization algorithms utilizing norm-constrained Linear Minimization Oracles (LMOs) for training deep neural networks, applicable to both constrained and unconstrained problems.
• Theoretical analysis shows optimal convergence rates for the proposed methods like uSCG (O(n Gʻ -1/4)), and experimental results on nanoGPT demonstrate significant training speedups and memory efficiency.
• This framework simplifies hyperparameter tuning across models of varying sizes and establishes a path towards more efficient and scalable deep learning methodologies.

Training Deep Learning Models with Norm-Constrained LMOs: Methodologies and Insights

The paper "Training Deep Learning Models with Norm-Constrained Linear Minimization Oracles (LMOs)" proposes a novel approach to training deep learning models by leveraging norm-constrained LMOs. The authors thoroughly investigate the use of these oracles in optimizing the training of neural networks, offering both theoretical and experimental evaluations.

Summary of Core Contributions

The essence of this paper is the development of a stochastic family of optimization algorithms that utilize LMOs to adapt to the geometry of the problem at hand. These algorithms are not only applicable to constrained optimization problems — as traditionally employed in conditional gradient methods — but also to unconstrained ones. This adaptability is shown to unify several existing optimization methods within a single cohesive framework.

The authors emphasize a specific choice of norm constraints, particularly advantageous for deep learning architectures. This choice facilitates hyperparameter transferability across models of varying sizes, which is often a significant challenge in scalable deep learning applications.

Theoretical Insights and Algorithmic Framework

The paper introduces two methods: the unconstrained Stochastic Conditional Gradient (uSCG) method and a revisitation of the Stochastic Conditional Gradient (SCG) method for non-convex objectives. These methods incorporate momentum, enabling better handling of stochastic gradients.

Convergence Analysis

The theoretical analysis shows that these methods yield optimal convergence rates. For non-convex and stochastic problems, uSCG achieves an $O(n^{-1/4})$ convergence rate, which is acknowledged as the order-optimal rate under these conditions. SCG, adapted for the constrained setting, provides explicit norm control, establishing a theoretical guarantee that extends beyond traditional usage of such methods. Notably, these methods ensure that the parameter norms remain bounded by effectively leveraging norm constraints.

Numerical Validation and Experimental Results

One of the standout experimental results is the application of these methods to the nanoGPT architecture. The authors report substantial speedups in training without resorting to the commonly used Adam optimizer, which is notable considering nanoGPT's complexity. Furthermore, the proposed methods demonstrate significant memory efficiency by requiring only one set of model weights and gradients, which can be stored in half-precision, thereby reducing the memory footprint during large-scale model training.

Practical Implications and Future Directions

The implications of this research are significant for deep learning practitioners. The proposed norm-constrained LMO-based optimization framework simplifies the tuning of hyperparameters across models with different widths, potentially reducing the need for extensive hyperparameter searches in practice. This transferability enhances model efficiency when scaling up neural network architectures.

For future work, exploring the extension of these methods to more complex neural architectures and application domains would be insightful. Additionally, further investigation into how different norm choices affect model behaviors, such as generalization and robustness, may provide deeper insights.

Conclusion

This paper offers a robust framework for exploiting problem geometry through norm-constrained LMOs, achieving both theoretical and practical advancements in optimizing neural network training. It marks a progressive shift towards more efficient and scalable deep learning methodologies, with broad applications across various neural network architectures and problem domains. It's an important step in understanding how altering optimization strategies can significantly benefit the training efficiency and generalization capabilities of modern deep learning models.