Grad-GradaGrad? A Non-Monotone Adaptive Stochastic Gradient Method (2206.06900v1)

Abstract: The classical AdaGrad method adapts the learning rate by dividing by the square root of a sum of squared gradients. Because this sum on the denominator is increasing, the method can only decrease step sizes over time, and requires a learning rate scaling hyper-parameter to be carefully tuned. To overcome this restriction, we introduce GradaGrad, a method in the same family that naturally grows or shrinks the learning rate based on a different accumulation in the denominator, one that can both increase and decrease. We show that it obeys a similar convergence rate as AdaGrad and demonstrate its non-monotone adaptation capability with experiments.

• The paper introduces GradaGrad, a non-monotone adaptive method that improves upon AdaGrad by allowing learning rates to both increase and decrease, overcoming its monotonic limitations.
• The method integrates a novel inner product of consecutive gradients into its update rule, enhancing convergence in high-dimensional stochastic optimization.
• Experimental results show GradaGrad achieves robust performance with minimal hyperparameter tuning, making it effective in both convex and deep learning applications.

Grad-GradaGrad? A Non-Monotone Adaptive Stochastic Gradient Method

In the field of stochastic optimization, the challenge of tuning learning rates emerges as a central issue, particularly in the context of high-dimensional machine learning problems. The paper "Grad-GradaGrad? A Non-Monotone Adaptive Stochastic Gradient Method" addresses this challenge by proposing GradaGrad, an innovative adaptation of the AdaGrad algorithm that allows for non-monotone adjustment of the learning rate.

AdaGrad and Its Limitations

AdaGrad is an established method in adaptive learning rates that effectively scales the step size using the cumulative sum of squared gradients. However, a fundamental limitation of AdaGrad is its monotonic nature: the learning rate can only decrease over time as the denominator in the step size formula continually increases. This characteristic necessitates careful and often labor-intensive tuning of the initial learning rate hyperparameter. Moreover, AdaGrad is not equipped to increase the learning rate if the initial setting was suboptimal and results in slow convergence.

Introduction to GradaGrad

GradaGrad is designed to overcome the monotone adaptation limitation of AdaGrad by introducing a novel adaptive strategy that enables the learning rate to both increase and decrease based on the problem's characteristics. This method incorporates an inner product term between consecutive gradients into the step size adjustment formula, offering a mechanism for swift adaptation to changing optimization landscapes. The proposed update rule introduces a non-monotonic adjustment by employing the expression:

$$A_{i,k+1} \propto \sqrt{\sum_{t=0}^k \left(g_{i,t}^2 - \rho g_{i,t}g_{i,t-1}\right)}$$

where $\rho$ is a constant. This approach allows the learning rate to grow when consecutive gradients are positively correlated and contract otherwise, thus offering a more responsive adaptation to the dynamics of the problem.

Implementation and Convergence

The paper details the mathematical foundation required to ensure GradaGrad maintains a competitive convergence rate when compared to traditional AdaGrad implementations. By reparametrizing the learning rate and integrating a stabilization term to control exponential growth in uncontrolled settings, GradaGrad achieves stability across various problem types.

The paper also presents a simplified scalar variant alongside a more sophisticated diagonal version, the latter incorporating momentum to further enhance convergence rates. Consistency with theoretical bounds where the learning rate adapts according to the gradient correlation ensures that GradaGrad matches, and in some cases outpaces, the practical performance of benchmark methods on standard datasets.

Experimental Results

The efficacy of GradaGrad is illustrated through a series of experiments. The method demonstrates competitive performance on established benchmarks without requiring extensive hyperparameter tuning, showing robustness in both convex and complex deep learning scenarios. Crucially, GradaGrad displayed the unique capability to automatically adapt to poor initial learning rate settings, highlighting its potential as a drop-in replacement for AdaGrad in varied contexts.

Implications and Future Directions

The introduction of GradaGrad opens multiple avenues for future exploration. On the theoretical side, further analysis into the long-term behavior of non-monotonic adaptive methods could yield improvements in convergence rates or stability refinements. Practically, extending GradaGrad to other stochastic gradient methods, such as those with variance reduction or adaptive preconditioning, may further enhance its applicability in high-stakes machine learning applications.

Overall, the development of GradaGrad signals a notable advancement in stochastic optimization techniques by addressing inherent limitations of monotonic adaptive methods and introducing a flexible approach that is resilient to hyperparameter selection challenges. This progression plays a critical role in the ongoing endeavor to streamline and optimize the practical implementation of large-scale machine learning systems.