Convergence Analysis of RMSProp and Adam Optimizers for Generalized-smooth Non-convex Optimization with Affine Noise Variance

核心概念

This paper provides the first tight convergence analyses for RMSProp and Adam optimizers in non-convex optimization under the most relaxed assumptions of coordinate-wise generalized smoothness and affine noise variance.

要約

The paper focuses on the convergence analysis of two popular adaptive optimizers, RMSProp and Adam, under the most relaxed assumptions of coordinate-wise generalized (L0, L1)-smoothness and affine noise variance.

Key highlights:

1. For RMSProp, the authors address several major challenges such as the dependence between stepsize and gradient, potential unbounded gradients, and additional error terms due to (L0, L1)-smoothness. They develop novel techniques to bound these terms and show that RMSProp with proper hyperparameters converges to an ε-stationary point with an iteration complexity of O(ε^-4), matching the lower bound.

2. For Adam, the authors face additional challenges due to the mismatch between gradient and first-order momentum. They develop a new upper bound on the first-order term in the descent lemma and show that Adam with proper hyperparameters also converges to an ε-stationary point with an iteration complexity of O(ε^-4).

3. The results improve upon prior work by considering the more practical and challenging coordinate-wise (L0, L1)-smooth objectives and the refined affine noise variance assumption, which better capture the training of neural networks.

4. The authors' analyses are comprehensive, providing detailed technical insights to address the key challenges, and the final convergence results match the optimal lower bound, demonstrating the tightness of their analyses.

統計

None.

引用

None.

抽出されたキーインサイト

Convergence Guarantees for RMSProp and Adam in Generalized-smooth Non-convex Optimization with Affine Noise Variance

by Qi Zhang,Yi ... 場所 arxiv.org 04-03-2024

https://arxiv.org/pdf/2404.01436.pdf

深掘り質問

How can the convergence analysis be extended to other adaptive optimizers beyond RMSProp and Adam

The convergence analysis presented in the paper for RMSProp and Adam can be extended to other adaptive optimizers by following a similar approach. The key lies in adapting the analysis to the specific characteristics and update rules of the optimizer in question. For instance, if we consider an optimizer that incorporates both adaptive learning rates and momentum, similar techniques can be applied to analyze its convergence properties under the (L0, L1)-smoothness and affine noise variance assumptions. By carefully examining the interplay between the optimizer's components, such as the adaptive learning rates and momentum terms, and considering the impact of the refined assumptions on the optimization process, one can extend the convergence analysis to a broader range of adaptive optimizers.

What are the potential implications of the refined (L0, L1)-smoothness and affine noise variance assumptions on the design and tuning of deep learning models

The refined (L0, L1)-smoothness and affine noise variance assumptions have significant implications for the design and tuning of deep learning models.

Model Performance: These assumptions provide a more accurate characterization of the optimization landscape for neural networks, especially in scenarios where traditional smoothness assumptions like L-smoothness do not hold. By considering the coordinate-wise (L0, L1)-smoothness, the optimization process can better adapt to the varying Lipschitz constants across different dimensions, leading to improved convergence properties and potentially better model performance.

Hyperparameter Tuning: The assumptions of affine noise variance and (L0, L1)-smoothness can guide hyperparameter tuning strategies. For example, when designing the learning rate schedules or momentum terms for deep learning models, the understanding of how these assumptions affect the convergence guarantees can help in selecting appropriate hyperparameters that enhance optimization efficiency.

Generalization and Robustness: By incorporating these refined assumptions into the optimization process, deep learning models may exhibit improved generalization and robustness properties. The consideration of coordinate-wise smoothness and refined noise assumptions can lead to more stable optimization trajectories, potentially reducing the risk of overfitting and improving the model's ability to generalize to unseen data.

Are there any connections between the technical insights developed in this paper and the underlying optimization landscape of neural networks

The technical insights developed in the paper have several connections to the underlying optimization landscape of neural networks:

Dependence between Stepsize and Gradient: The analysis of the dependence between stepsize and gradient, as discussed in the paper, sheds light on the challenges faced in optimizing non-convex loss functions. Understanding how the adaptive update mechanisms interact with the gradient estimates can provide valuable insights into the dynamics of optimization in neural networks.

Unbounded Gradients and Noise Variance: The exploration of potential unbounded gradients due to refined gradient noise assumptions highlights the importance of robust optimization strategies in the presence of noise. By addressing these challenges, the paper contributes to a deeper understanding of how noise variance affects the convergence properties of optimization algorithms in neural networks.

Mismatch between Gradient and Momentum: The analysis of the mismatch between gradient and first-order momentum in Adam provides insights into the optimization dynamics of adaptive optimizers. By bounding the first-order terms and considering the impact of bias terms, the paper offers a nuanced perspective on how momentum affects the convergence behavior of optimization algorithms in the context of neural network training.