多任务深度学习的算子理论泛化界
Operator-Theoretic Generalization Bounds for Multitask Deep Learning
- Free University of Bozen–Bolzano（博尔扎诺自由大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对多任务深度学习，通过算子理论推导泛化界，分析不同假设空间的复杂度，构建跨任务共享算子学习并验证相关代理。
AI中文摘要：
我们通过将网络层表示为向量值再生核希尔伯特空间（RKHS）上的Koopman复合算子，推导了深度多输出函数类的算子理论泛化界。在向量值Sobolev RKHS中，我们针对可逆且宽度扩张的单射架构，推导了Rademacher复杂度界。这些估计将由任务矩阵的迹表示的输出耦合贡献，与线性映射生成的分层算子范数、Sobolev符号比、行列式因子及限制常数分离开来。随后，我们分析了一种不同的一维布朗/Cameron–Martin regime。利用向量值布朗RKHS的精确锚定导数范数表征，我们得到了保域标量线性映射和锚定微分同胚激活的分层界；对应的因子分别按|W_l|^{1/2}和||σ_l'||_∞^{1/2}缩放，且不涉及Sobolev光滑指数。由于Sobolev和布朗结果涉及不同的假设空间，二者均未被断言在整体上优于对方。我们还构建了跨任务的共享算子学习，证明了有限秩表示定理，推导了平方损失的精确有限维问题，并在学习到的算子独立于目标样本获取时，建立了目标迁移界。合成数据与MNIST研究检验了受Sobolev启发和布朗启发的稳定复杂度代理；这些经验代理并非对秩亏架构所证明界的评估。
英文摘要：
We develop operator-theoretic generalization bounds for deep multi-output function classes by representing network layers as Koopman composition operators on vector-valued reproducing kernel Hilbert spaces. In vector-valued Sobolev RKHSs, we derive Rademacher complexity bounds for invertible and width-expanding injective architectures. The estimates separate the output-coupling contribution, represented by the trace of the task matrix, from the layerwise operator norms, Sobolev symbol ratios, determinant factors, and restriction constants generated by the linear maps. We then analyze a distinct one-dimensional Brownian/Cameron--Martin regime. Using the exact anchored derivative-norm characterization of the vector-valued Brownian RKHS, we obtain layerwise bounds for domain-preserving scalar linear maps and anchored diffeomorphic activations; the corresponding factors scale as $|W_l|^{1/2}$ and $\|σ_l'\|_\infty^{1/2}$, respectively, and do not involve Sobolev smoothness exponents. Because the Sobolev and Brownian results concern different hypothesis spaces, neither is asserted to dominate the other uniformly. We additionally formulate shared operator learning across tasks, prove a finite-rank representer theorem, derive the exact finite-dimensional problem for squared loss, and establish a target-transfer bound when the learned operator is obtained independently of the target sample. Synthetic and MNIST studies examine stabilized Sobolev-inspired and Brownian-inspired complexity proxies; these empirical proxies are not evaluations of the proved bounds for rank-deficient architectures.
