谱表示学习的锐利速率与一行修正
Sharp Regret Bounds and a Task-Covariance Correction for Spectral Representation Learning
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中文总结 AI 辅助
本文证明各向同性假设错误，提出对齐系数刻画迁移风险，给出匹配双边速率，并设计一行重加权修正方法，在受控数据上将遗憾从0.86降至0.003。
中文摘要 AI 辅助
一个自监督编码器被训练一次、冻结，并通过轻量级探针在训练时无人命名的任务上重复使用；实践者的问题是，现成的特征何时足够好，何时需要修正。典型相关分析、HGR最大相关以及谱对比损失的总体最优都返回交叉视图依赖算子的前$k$奇异子空间，其合理性由各向同性论证：如果任务先验没有方向偏好，该子空间是普遍最优的。我们证明各向同性是错误的假设。先验仅通过任务协方差$\Lambda=\mathbb{E}[\Delta\Delta^\top]$及其在算子主导奇异方向上的压缩进入迁移风险；重要的不是$\Lambda$是否各向同性，而是其偏好方向是否与算子的谱顺序一致。我们证明了匹配的双边速率——最坏情况遗憾恰好为$1-1/\kappa(\Lambda)$，对于对齐系数$A_k$细化为$1-A_k$，局部化到前$2k$子空间，在谱间隙下变为二阶，且无法通过任何任务无关表示改进——并说明为何对齐是普遍的：高维下不相干的偏好相互抵消，$T$个多样任务迫使$\alpha=\widetilde O(\sqrt{d_x/T})$，这定量解释了为何任务多样性而非对称性使自监督特征可迁移。主导统计成本为$O(kd_x^2)$，当它们发出未对齐信号时，对正样本项的一行重新加权可证明恢复精确最优性。结果是一个诊断，从少量标注预算回答实践者的问题，并在任务库无法支持所请求宽度时拒绝；在受控数据上，它将遗憾从$0.86$降至$0.003$，在CIFAR-100编码器上，它正确预测无需修正。
英文摘要
Spectral features can remain optimal under strongly uneven task preferences when they retain the directions most useful to the tasks. In a local-task model, expected probing gain depends on the task prior only through its covariance $Λ$: with $B$ recording dependence between views, $k$ spectral features span the leading eigenspace of $BB^\top$, while task-optimal features span that of $BΛB^\top$. We derive alignment-dependent regret bounds, matching worst-case lower bounds for a flat leading spectrum, and bounds using the leading $2k$ directions with a spectral-tail term; a spectral gap makes the bound quadratic in small anisotropy. We also bound the imbalance from random preference patterns and from averaging independent tasks with an isotropic population covariance. With known $Λ$, changing one term of the spectral contrastive loss selects task-optimal features; with a labelled task bank, we propose a diagnostic and test correction without retraining. Correction reduces synthetic held-out regret from $0.858$ to $0.003$ with $2000$ tasks when less task-relevant directions dominate, and on CIFAR-100 lowers empirical regret by $0.031$--$0.053$ on held-out fine-label tasks using the same images. With a small task bank, however, correction can hurt.
发表机构
- The University of Hong Kong（香港大学）
机构由 AI 辅助整理，请以论文原文为准。
