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Register a new userLearning Bounds for Greedy Approximation with Explicit Feature Maps from Multiple Kernels
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Nonlinear kernels can be approximated using finite-dimensional feature maps for efficient risk minimization. Due to the inherent trade-off between the dimension of the (mapped) feature space and the approximation accuracy, the key problem is to identify promising (explicit) features leading to a satisfactory out-of-sample performance. In this work, we tackle this problem by efficiently choosing such features from multiple kernels in a greedy fashion. Our method sequentially selects these explicit features from a set of candidate features using a correlation metric. We establish an out-of-sample error bound capturing the trade-off between the error in terms of explicit features (approximation error) and the error due to spectral properties of the best model in the Hilbert space associated to the combined kernel (spectral error). The result verifies that when the (best) underlying data model is sparse enough, i.e., the spectral error is negligible, one can control the test error with a small number of explicit features, that can scale poly-logarithmically with data. Our empirical results show that given a fixed number of explicit features, the method can achieve a lower test error with a smaller time cost, compared to the state-of-the-art in data-dependent random features.
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By transferring knowledge learned from seen/previous tasks, meta learning aims to generalize well to unseen/future tasks. Existing meta-learning approaches have shown promising empirical performance on various multiclass classification problems, but few provide theoretical analysis on the classifiers generalization ability on future tasks. In this paper, under the assumption that all classification tasks are sampled from the same meta-distribution, we leverage margin theory and statistical learning theory to establish three margin-based transfer bounds for meta-learning based multiclass classification (MLMC). These bounds reveal that the expected error of a given classification algorithm for a future task can be estimated with the average empirical error on a finite number of previous tasks, uniformly over a class of preprocessing feature maps/deep neural networks (i.e. deep feature embeddings). To validate these bounds, instead of the commonly-used cross-entropy loss, a multi-margin loss is employed to train a number of representative MLMC models. Experiments on three benchmarks show that these margin-based models still achieve competitive performance, validating the practical value of our margin-based theoretical analysis.
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