Learning Bounds for Greedy Approximation with Explicit Feature Maps from Multiple Kernels
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.
### Post-Rebuttal ###  I have read the rebuttal. Thank you very much for the clarifications.  ##################  # Summary The paper builds on [1] and proposes an algorithm for feature selection from an a priori specified finite dictionary/set of features. In particular, [1, Algorithm 3] propose an approach for minimization of the expected loss of a linear predictor that aims at finding a `good' sparse solution. The main idea of the algorithm from [1] is to iteratively add features by picking a previously unselected feature that amounts to the largest reduction in the expected risk. Then, a linear model is trained using the extended feature representation and afterwards the whole process is repeated. The authors use pretty much the same idea and take a large dictionary of features to represent the data. Following this, they run Algorithm 3 from [1] to pick `informative' features and generate a sparse feature representation. In particular, the step 2 from Algorithm 1 proposed by the authors can be shown to be `almost equivalent' to the first step of Algorithm 3 in [1].  The approach is motivated by the problems with scalability of kernel methods and related to learning with multiple kernels. In particular, a positive definite Mercer kernel can be represented as an inner product of possibly countably infinitely many features/eigenfunctions. For multiple kernels, it is possible to concatenate the feature representations corresponding to different kernels. For a Mercer kernel, the authors aim to find subsets of the features that produce a good approximation of the inner product defined by it.  There are two theoretical results: i) A bound on the Rademacher complexity of the hypothesis space defined by a dictionary of pre-selected features. This result builds on [3] and [4] and it is a rather straightforward derivation, following the previous results. ii) A bound on the expected risk of the estimator generated by the proposed algorithm. There are two terms that are bounded when deriving this result. The first term bounds the difference between the risk of an estimator generated by Algorithm 1 and the best estimator over the hypothesis space defined by a dictionary of features. The most important result for this term follows from [1]. The second term bounds the difference between the expected risk of the best estimator over the hypothesis space defined by the dictionary and that over the combination of multiple kernels with possibly infinitely many features. To the best of my knowledge, this piece is novel and interesting. Here it is important to note that similar bounds (finite vs. infinite dictionary) have been considered in [5] for random Fourier features.  The bound on the expected risk consists of three terms: i) a sample complexity term accounting for the concentration with respect to the number of instances (the standard \sqrt{n} rate), ii) an approximation term that accounts for the effect of adding more features and extending the feature representation (approximation properties improve exponentially, quite interesting), and iii) a spectral property term that accounts for the coefficient mass present over the (possibly infinitely many) remaining features defined on the whole feature space.   The empirical evaluation is performed using a dictionary of random Fourier features and the approach is compared to 5 baselines. The first baseline is the random Fourier feature approximation of the Gaussian kernel. The second and the third baselines sample a large pool of features and then perform feature selection using a scoring function and kernel alignment, respectively. The fourth approach is learning with the standard Gaussian kernel and the fifth one is multiple kernel learning with a combination of Gaussian and linear kernels. Overall the results show an improvement over the competing baselines and good test prediction times.  # Comments - The paper is well-written and easy to follow. The derivations are complete and understandable. The provided proofs are also legible and cite all the auxiliary results from previous work. - I recognize that there is a subtle difference between Algorithm 1 proposed by the authors and Algorithm 3 from [1]. In particular, [1] consider feature selection with respect to the expected risk and state that the result holds for the empirical risk, as well. The authors, on the other hand, consider empirical risk when picking a new feature and this makes their result from Theorem 2 novel and interesting. - Related to the previous point, Algorithm 1 should include a regularization term with a hyperparameter controlling the model complexity? - A limitation of the paper is the fact that the total number of features in the dictionary needs to be finite. A number of approaches have been developed for boosting features that work with infinite dictionaries [6-10]. I also think that approaches considered in [6, 8, 10] are quite related to the goal of this paper. Moreover, it would be very interesting to compare the effectiveness of feature selection to an approach like [9]. For a dictionary of features selected by sampling from a Gaussian distribution with thick tails the feature selection could be competitive with [9], where a mixture of Gaussian distributions models the spectral density of a stationary kernel and outputs a feature representation. - The spectral term from the bound in Theorem 2 deserves some more space. I do see that the spectral decay of the kernel can affect the eigenvalue decay but in this particular case we have coefficients of the model. It would be instructive to assume some decay (e.g., polynomial or exponential) and produce an estimate or rate for this term (this should be related to the remark on the Parseval's theorem). - Related to the previous point, the paper makes a strong assumption that a large dictionary of features specified a priori covers the `important' coefficients/features. In general, this is quite difficult to satisfy and the bound indicates that the concentration might not happen at all if the dictionary is not well-equipped to model the conditional dependence of interest. - A minor issue with the empirical evaluation is the fact that the bandwidth parameter was fixed and not cross-validated. The gap between the proposed approach and other baselines might be `over emphasized' as a result of this. - In Table 2, why is there such a large difference in test time predictions between LKRF, EERF, and the proposed approach? Are they not predicting with the same number of features? - Line 97, citation is needed for Parseval's theorem.   # References +[1] S. Shalev-Shwartz, N. Srebro, T. Zhang -- Trading accuracy for sparsity in optimization problems with sparsity constraints (SIAM Journal of Optimization, 2010) +[2] M. Mohri, A. Rostamizadeh, A. Talwalkar -- Foundations of machine learning (MIT Press, 2012) +[3] P. Bartlett, S. Mendelson -- Rademacher and Gaussian complexities: risk bounds and structural results (JMLR, 2002) +[4] C. Cortes, M. Mohri, A. Rostamizadeh -- Generalization bounds for learning kernels (ICML, 2010) --- -[5] A. Rahimi, B. Recht -- Uniform approximation of functions with random bases (IEEE, 2008) -[6] F. Huang, J. T. Ash, J. Langford, R. E. Shapire -- Learning deep ResNet blocks sequentially using boosting theory (ICML, 2018) -[7] C. Cortes, M. Mohri, U. Syed -- Deep boosting (ICML, 2014) -[8] C. Cortes, X. Gonzalvo, V. Kuznetsov, M. Mohri, S. Yang -- AdaNet: adaptive structural learning of artificial neural networks (ICML, 2017) -[9] Z. Yang, A. Smola, L. Song, A. G. Wilson -- A la carte-learning fast kernels (AISTATS, 2015) -[10] D. Oglic, T. Gaertner -- Greedy feature construction (NIPS, 2016)