AdaGrad stepsizes: Sharp convergence over nonconvex landscapes (1806.01811v8)

Abstract: Adaptive gradient methods such as AdaGrad and its variants update the stepsize in stochastic gradient descent on the fly according to the gradients received along the way; such methods have gained widespread use in large-scale optimization for their ability to converge robustly, without the need to fine-tune the stepsize schedule. Yet, the theoretical guarantees to date for AdaGrad are for online and convex optimization. We bridge this gap by providing theoretical guarantees for the convergence of AdaGrad for smooth, nonconvex functions. We show that the norm version of AdaGrad (AdaGrad-Norm) converges to a stationary point at the $\mathcal{O}(\log(N)/\sqrt{N})$ rate in the stochastic setting, and at the optimal $\mathcal{O}(1/N)$ rate in the batch (non-stochastic) setting -- in this sense, our convergence guarantees are 'sharp'. In particular, the convergence of AdaGrad-Norm is robust to the choice of all hyper-parameters of the algorithm, in contrast to stochastic gradient descent whose convergence depends crucially on tuning the step-size to the (generally unknown) Lipschitz smoothness constant and level of stochastic noise on the gradient. Extensive numerical experiments are provided to corroborate our theory; moreover, the experiments suggest that the robustness of AdaGrad-Norm extends to state-of-the-art models in deep learning, without sacrificing generalization.

AdaGrad Stepsizes: Sharp Convergence over Nonconvex Landscapes

This paper addresses the theoretical underpinnings of the AdaGrad algorithm within the context of nonconvex optimization landscapes, which are prevalent in machine learning applications, particularly in deep learning. The exposition and analysis by Ward, Wu, and Bottou rigorously extend the convergence guarantees of AdaGrad beyond the classical online and convex optimization paradigms to smooth, nonconvex functions.

AdaGrad and its variants adjust the stepsize dynamically using cumulative gradient information. This behavior has facilitated robust convergence across diverse optimization contexts without the necessity of finely adjusting the stepsize schedule manually. Despite its empirical success, AdaGrad's theoretical convergence assurances had remained restricted to convex settings, a gap this paper aims to bridge.

Key Findings and Contributions

1. Convergence Guarantees for Nonconvex Optimization:
   • The authors establish that the norm variant of AdaGrad, termed AdaGrad-Norm, converges to a stationary point at an $\mathcal{O}(\log(N)/\sqrt{N})$ rate in stochastic settings and $\mathcal{O}(1/N)$ in deterministic (batch) settings. These rates are consistent with the optimal bounds for gradient descent methods tailored with precise knowledge of problem parameters.
2. Robustness to Hyperparameters:
   • A significant finding is the robustness of AdaGrad-Norm to its hyperparameters, contrasting sharply with conventional SGD methods which require careful tuning of stepsizes based on the unknown Lipschitz smoothness constant and gradient noise level. AdaGrad-Norm adjusts autonomously to these conditions, maintaining consistent convergence behavior.
3. Theoretical Framework:
   • The paper provides novel proofs handling the dependency between the dynamically adjusted stepsizes and the gradient norms, a correlation omitted in standard analysis for fixed step-sizes. The methods involve bounding the expectation terms and adapting classical descent lemmas for stochastic environments with inherent noise.
4. Extensive Numerical Experiments:
   • The paper enriches its theoretical contributions with empirical evidence across synthetic and real-world datasets, including deep learning architectures on ImageNet. AdaGrad-Norm's reliability and efficiency in practical scenarios are demonstrated, validating its theoretical resilience.

Implications and Future Directions

The establishment of sharp convergence rates for AdaGrad-Norm opens up new avenues for its deployment in large-scale nonconvex optimization tasks, a common scenario in training deep learning models. By reducing the dependence on hyperparameter tuning, AdaGrad-Norm offers a more scalable and user-friendly optimization paradigm.

The results also pose intriguing questions about further refining the constants in convergence bounds and possibly extending these findings to a broader family of adaptive gradient methods, including those which feature momentum terms or coordinate-specific adaptations like Adam. Such extensions could potentially cater to even more complex optimization landscapes prevalent in state-of-the-art neural networks applications.

Overall, this paper makes a substantial addition to the theoretical landscape of adaptive optimization by thoroughly exploring AdaGrad-Norm's potential in nonconvex domains, reinforcing its empirical utility with robust theoretical backing. Further exploration could also involve addressing the adaptive methods in asynchronous or distributed settings, which are increasingly relevant in large-scale machine learning deployments.