Asymptotic learning curves of kernel methods: empirical data v.s. Teacher-Student paradigm

How many training data are needed to learn a supervised task? It is often observed that the generalization error decreases as $n^{-\beta}$ where $n$ is the number of training examples and $\beta$ an exponent that depends on both data and algorithm. In this work we measure $\beta$ when applying kernel methods to real datasets. For MNIST we find $\beta\approx 0.4$ and for CIFAR10 $\beta\approx 0.1$, for both regression and classification tasks, and for Gaussian or Laplace kernels. To rationalize the existence of non-trivial exponents that can be independent of the specific kernel used, we study the Teacher-Student framework for kernels. In this scheme, a Teacher generates data according to a Gaussian random field, and a Student learns them via kernel regression. With a simplifying assumption -- namely that the data are sampled from a regular lattice -- we derive analytically $\beta$ for translation invariant kernels, using previous results from the kriging literature. Provided that the Student is not too sensitive to high frequencies, $\beta$ depends only on the smoothness and dimension of the training data. We confirm numerically that these predictions hold when the training points are sampled at random on a hypersphere. Overall, the test error is found to be controlled by the magnitude of the projection of the true function on the kernel eigenvectors whose rank is larger than $n$. Using this idea we predict relate the exponent $\beta$ to an exponent $a$ describing how the coefficients of the true function in the eigenbasis of the kernel decay with rank. We extract $a$ from real data by performing kernel PCA, leading to $\beta\approx0.36$ for MNIST and $\beta\approx0.07$ for CIFAR10, in good agreement with observations. We argue that these rather large exponents are possible due to the small effective dimension of the data.Comment: We added (i) the prediction of the exponent $\beta$ for real data using kernel PCA; (ii) the generalization of our results to non-Gaussian data from reference [11] (Bordelon et al., "Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks"

A Noise-Robust Fast Sparse Bayesian Learning Model

This paper utilizes the hierarchical model structure from the Bayesian Lasso in the Sparse Bayesian Learning process to develop a new type of probabilistic supervised learning approach. The hierarchical model structure in this Bayesian framework is designed such that the priors do not only penalize the unnecessary complexity of the model but will also be conditioned on the variance of the random noise in the data. The hyperparameters in the model are estimated by the Fast Marginal Likelihood Maximization algorithm which can achieve sparsity, low computational cost and faster learning process. We compare our methodology with two other popular learning models; the Relevance Vector Machine and the Bayesian Lasso. We test our model on examples involving both simulated and empirical data, and the results show that this approach has several performance advantages, such as being fast, sparse and also robust to the variance in random noise. In addition, our method can give out a more stable estimation of variance of random error, compared with the other methods in the study.Comment: 15 page

The Neural Representation Benchmark and its Evaluation on Brain and Machine

A key requirement for the development of effective learning representations is their evaluation and comparison to representations we know to be effective. In natural sensory domains, the community has viewed the brain as a source of inspiration and as an implicit benchmark for success. However, it has not been possible to directly test representational learning algorithms directly against the representations contained in neural systems. Here, we propose a new benchmark for visual representations on which we have directly tested the neural representation in multiple visual cortical areas in macaque (utilizing data from [Majaj et al., 2012]), and on which any computer vision algorithm that produces a feature space can be tested. The benchmark measures the effectiveness of the neural or machine representation by computing the classification loss on the ordered eigendecomposition of a kernel matrix [Montavon et al., 2011]. In our analysis we find that the neural representation in visual area IT is superior to visual area V4. In our analysis of representational learning algorithms, we find that three-layer models approach the representational performance of V4 and the algorithm in [Le et al., 2012] surpasses the performance of V4. Impressively, we find that a recent supervised algorithm [Krizhevsky et al., 2012] achieves performance comparable to that of IT for an intermediate level of image variation difficulty, and surpasses IT at a higher difficulty level. We believe this result represents a major milestone: it is the first learning algorithm we have found that exceeds our current estimate of IT representation performance. We hope that this benchmark will assist the community in matching the representational performance of visual cortex and will serve as an initial rallying point for further correspondence between representations derived in brains and machines.Comment: The v1 version contained incorrectly computed kernel analysis curves and KA-AUC values for V4, IT, and the HT-L3 models. They have been corrected in this versio