Theoretical Analysis of Adam Optimizer in the Presence of Gradient Skewness

The Adam optimizer has become a cornerstone in deep learning, widely adopted for its adaptive learning rates and momentumbased updates. However, its behavior under non-standard conditions, particularly skewed gradient distributions, remains underexplored. This paper presents a novel theoretical analysis of the Adam optimizer in the presence of skewed gradients, a scenario frequently encountered in real-world applications due to imbalanced datasets or inherent problem characteristics. We extend the standard convergence analysis of Adam to explicitly account for gradient skewness, deriving new bounds that characterize the optimizer’s performance under these conditions. Our main contributions include: (1) a formal proof of Adam’s convergence under skewed gradient distributions, (2) quantitative error bounds that capture the impact of skewness on optimization outcomes, and (3) insights into how skewness affects Adam’s adaptive learning rate mechanism. We demonstrate that gradient skewness can lead to biased parameter updates and potentially slower convergence compared to scenarios with symmetric distributions. Additionally, we provide practical recommendations for mitigating these effects, including adaptive gradient clipping and distribution-aware hyperparameter tuning. Our findings bridge a critical gap between Adam’s empirical success and its theoretical underpinnings, offering valuable insights for practitioners dealing with non-standard optimization landscapes in deep learning.