An Adaptive Stochastic Gradient Method with Non-negative Gauss-Newton Stepsizes (2407.04358v1)

Abstract: We consider the problem of minimizing the average of a large number of smooth but possibly non-convex functions. In the context of most machine learning applications, each loss function is non-negative and thus can be expressed as the composition of a square and its real-valued square root. This reformulation allows us to apply the Gauss-Newton method, or the Levenberg-Marquardt method when adding a quadratic regularization. The resulting algorithm, while being computationally as efficient as the vanilla stochastic gradient method, is highly adaptive and can automatically warmup and decay the effective stepsize while tracking the non-negative loss landscape. We provide a tight convergence analysis, leveraging new techniques, in the stochastic convex and non-convex settings. In particular, in the convex case, the method does not require access to the gradient Lipshitz constant for convergence, and is guaranteed to never diverge. The convergence rates and empirical evaluations compare favorably to the classical (stochastic) gradient method as well as to several other adaptive methods.

• The paper introduces NGN, a novel approach that adapts stepsizes based on local curvature without requiring the gradient's Lipschitz constant.
• It reformulates the loss function to ensure non-negative estimates, guaranteeing stability and convergence for convex, strongly convex, and non-convex problems.
• Empirical evaluations reveal that NGN outperforms standard SGD and other adaptive methods by improving training speed, stability, and robustness across various deep learning benchmarks.

An Adaptive Stochastic Gradient Method with Non-negative Gauss-Newton Stepsizes

The paper "An Adaptive Stochastic Gradient Method with Non-negative Gauss-Newton Stepsizes" by Antonio Orvieto and Lin Xiao introduces a novel optimization method that integrates the principles of stochastic gradient descent (SGD) and Gauss-Newton methods. This hybrid approach aims to leverage the computational efficiency of SGD while incorporating adaptive stepsizes to enhance performance in convex, strongly convex, and non-convex optimization tasks.

Methodology and Theoretical Contributions

The proposed method, abbreviated as NGN (Non-negative Gauss-Newton), tackles the optimization of the average of a large number of smooth, non-negative, and possibly non-convex functions. The central innovation in this method is the use of adaptive stepsizes derived from a reformulation of the loss function as the composition of its square and its real-valued square root, which aligns with the Gauss-Newton method.

Derivation and Properties

1. NGN Stepsize:
   • The stepsize for NGN is computed automatically without requiring the Lipschitz constant of the gradient, which is often unknown and hard to estimate.
   • The stepsize adapts based on the local curvature of the loss landscape, which allows it to automatically warm up and decay, enhancing stability and convergence.
2. Non-negative Loss Landscape:
   • By expressing the loss function as the square of its root, the authors ensure a non-negative estimation, a crucial aspect for maintaining stability in optimization.
3. Range of Stepsize:
   • The authors provide formal bounds for the stepsizes, ensuring they lie within a range that guarantees convergence without divergence, even for large values of the hyperparameter $\sigma$.
4. Connections to Polyak Stepsizes:
   • The NGN stepsize is shown to interpolate between the Polyak stepsize and a small constant stepsize, providing a theoretical foundation for its adaptive behavior.

Convergence Analysis

1. Convex and Strongly Convex Settings:
   • In the convex case, NGN is proved to converge to a neighborhood of the solution without requiring knowledge of the Lipschitz constant. The convergence rate is shown to be $O(1/\sqrt{K})$ with decreasing $\sigma$.
   • For strongly convex functions, NGN ensures linear convergence to the solution's neighborhood, with the rate of convergence being adaptive to the specific problem constants.
2. Non-convex Setting:
   • For non-convex problems, the NGN stepsize achieves similar performance to SGD with constant stepsizes. The convergence rate is $O(1/\sqrt{K})$ with decreasing $\sigma$, under the assumption that the gradient's Lipschitz constant is known.

Empirical Evaluation

The authors conducted extensive experiments to validate the performance of NGN against standard SGD, Adam, and other adaptive methods like Adagrad and stochastic Polyak stepsizes (SPS). The empirical results across various neural network architectures and tasks, including SVHN, CIFAR10, and Imagenet, demonstrate that NGN outperforms these methods in terms of training loss and exhibits substantial robustness to hyperparameter tuning.

• Performance with Adaptive Stepsizes:
  • NGN showed strong numerical results in maintaining high performance across different hyperparameter settings, significantly improving training stability and speed.
  • The effective stepsize in NGN displayed a warm-up and decay behavior that aligns with practical practices in deep learning, such as learning rate schedules.

Implications and Future Directions

The introduction of NGN provides a potent tool for large-scale machine learning problems where adaptive stepsizes are crucial for optimization efficiency. Theoretical guarantees of convergence without necessitating the Lipschitz constant make NGN an appealing choice over classical SGD and its variants.

Practical Applications

• Deep Learning:
  • In training deep neural networks, NGN's adaptive stepsizes can lead to faster convergence and better utilization of computational resources compared to methods like Adam, which require additional memory overhead due to second-order statistics.

Theoretical Extensions

• Momentum Integration:
  • Future work could explore incorporating momentum into NGN, potentially improving its performance further. Recent works on momentum with stochastic Polyak stepsizes could provide a foundation for this exploration.
• Block-diagonal Variants:
  • Investigating block-coordinate versions of NGN, where each coordinate or block of coordinates has its own adaptive stepsize, could enhance performance for models with varying parameter curvatures, such as transformers.

Conclusion

The NGN method proposed by Orvieto and Xiao stands out for its innovative use of non-negative Gauss-Newton stepsizes, providing robust, adaptive optimization without the need for precise knowledge of gradient Lipschitz constants. Its favorable empirical performance and theoretical guarantees position it as a significant contribution to the field of stochastic optimization and gradient-based learning methods. Future work on momentum integration and block-diagonal variants could further unlock its potential, particularly in specialized deep learning contexts.