Why Adam Can Beat SGD: Second-Moment Normalization Yields Sharper Tails

Research paper establishes the first theoretical separation between Adam and SGD optimization algorithms, proving Adam achieves better high-probability convergence guarantees. The study provides mathematical backing for Adam's superior empirical performance through second-moment normalization analysis.

• →Adam optimizer theoretically proven to outperform SGD with better convergence behavior under bounded variance conditions.
• →Study establishes first rigorous theoretical explanation for Adam's superior empirical performance in machine learning applications.
• →Adam achieves δ^(-1/2) dependence on confidence parameter versus SGD's δ^(-1) dependence in high-probability guarantees.
• →Research uses stopping-time and martingale analysis to distinguish the two optimization methods mathematically.
• →Findings bridge the gap between theoretical guarantees and observed empirical performance differences.