Local Rademacher Complexity for Multi-Label Learning

© 1992-2012 IEEE. We analyze the local Rademacher complexity of empirical risk minimization-based multi-label learning algorithms, and in doing so propose a new algorithm for multi-label learning. Rather than using the trace norm to regularize the multi-label predictor, we instead minimize the tail sum of the singular values of the predictor in multi-label learning. Benefiting from the use of the local Rademacher complexity, our algorithm, therefore, has a sharper generalization error bound. Compared with methods that minimize over all singular values, concentrating on the tail singular values results in better recovery of the low-rank structure of the multi-label predictor, which plays an important role in exploiting label correlations. We propose a new conditional singular value thresholding algorithm to solve the resulting objective function. Moreover, a variance control strategy is employed to reduce the variance of variables in optimization. Empirical studies on real-world data sets validate our theoretical results and demonstrate the effectiveness of the proposed algorithm for multi-label learning

Semi-supervised Vector-valued Learning: From Theory to Algorithm

Vector-valued learning, where the output space admits a vector-valued structure, is an important problem that covers a broad family of important domains, e.g. multi-label learning and multi-class classification. Using local Rademacher complexity and unlabeled data, we derive novel data-dependent excess risk bounds for learning vector-valued functions in both the kernel space and linear space. The derived bounds are much sharper than existing ones, where convergence rates are improved from $\mathcal{O}(1/\sqrt{n})$ to $\mathcal{O}(1/\sqrt{n+u}),$ and $\mathcal{O}(1/n)$ in special cases. Motivated by our theoretical analysis, we propose a unified framework for learning vector-valued functions, incorporating both local Rademacher complexity and Laplacian regularization. Empirical results on a wide number of benchmark datasets show that the proposed algorithm significantly outperforms baseline methods, which coincides with our theoretical findings

Optimistic Bounds for Multi-output Prediction

We investigate the challenge of multi-output learning, where the goal is to learn a vector-valued function based on a supervised data set. This includes a range of important problems in Machine Learning including multi-target regression, multi-class classification and multi-label classification. We begin our analysis by introducing the self-bounding Lipschitz condition for multi-output loss functions, which interpolates continuously between a classical Lipschitz condition and a multi-dimensional analogue of a smoothness condition. We then show that the self-bounding Lipschitz condition gives rise to optimistic bounds for multi-output learning, which are minimax optimal up to logarithmic factors. The proof exploits local Rademacher complexity combined with a powerful minoration inequality due to Srebro, Sridharan and Tewari. As an application we derive a state-of-the-art generalization bound for multi-class gradient boosting