Flatness and Gradient Alignment Are Both Necessary: Spectral-Aware Gradient-Aligned Exploration for Multi-Distribution Learning
Sharpness-aware and gradient-alignment methods have been shown to improve generalization, however each family of methods targets a single geometric property of the loss landscape, while ignoring the other. In this paper, we show that this omission is structurally unavoidable and that both flatness and gradient alignment should be considered in multi-distribution learning settings. Specifically, we derive an excess-risk decomposition that yields two additive leading-order terms: (i) an alignment term, controlled by the trace of $\bar{H}^{-1}Σ_g$ and (ii) a curvature term, controlled by $\bar{H}$, where $\bar{H}$ is the average Hessian and $Σ_g$ is the covariance of the gradient across dist...
| Authors | Aristotelis Ballas, Christos Diou |
|---|---|
| Primary category | cs.LG |
| Published | 8 May 2026 |
| Last updated at source | 15 July 2026 |
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