Machine Learning (stat.ML)

Abstract: We propose theoretical analyses of a modified natural gradient descent method in the neural network function space based on the eigendecompositions of neural tangent kernel and Fisher information matrix. We firstly present analytical expression for the function learned by this modified natural gradient under the assumptions of Gaussian distribution and infinite width limit. Thus, we explicitly derive the generalization error of the learned neural network function using theoretical methods from eigendecomposition and statistics theory. By decomposing of the total generalization error attributed to different eigenspace of the kernel in function space, we propose a criterion for balancing the errors stemming from training set and the distribution discrepancy between the training set and the true data. Through this approach, we establish that modifying the training direction of the neural network in function space leads to a reduction in the total generalization error. Furthermore, We demonstrate that this theoretical framework is capable to explain many existing results of generalization enhancing methods. These theoretical results are also illustrated by numerical examples on synthetic data.

Abstract: We study a principal component analysis problem under the spiked Wishart model in which the structure in the signal is captured by a class of union-of-subspace models. This general class includes vanilla sparse PCA as well as its variants with graph sparsity. With the goal of studying these problems under a unified statistical and computational lens, we establish fundamental limits that depend on the geometry of the problem instance, and show that a natural projected power method exhibits local convergence to the statistically near-optimal neighborhood of the solution. We complement these results with end-to-end analyses of two important special cases given by path and tree sparsity in a general basis, showing initialization methods and matching evidence of computational hardness. Overall, our results indicate that several of the phenomena observed for vanilla sparse PCA extend in a natural fashion to its structured counterparts.

Abstract: The development of Machine Learning is experiencing growing interest from the general public, and in recent years there have been numerous press articles questioning its objectivity: racism, sexism, \dots Driven by the growing attention of regulators on the ethical use of data in insurance, the actuarial community must rethink pricing and risk selection practices for fairer insurance. Equity is a philosophy concept that has many different definitions in every jurisdiction that influence each other without currently reaching consensus. In Europe, the Charter of Fundamental Rights defines guidelines on discrimination, and the use of sensitive personal data in algorithms is regulated. If the simple removal of the protected variables prevents any so-called `direct' discrimination, models are still able to `indirectly' discriminate between individuals thanks to latent interactions between variables, which bring better performance (and therefore a better quantification of risk, segmentation of prices, and so on). After introducing the key concepts related to discrimination, we illustrate the complexity of quantifying them. We then propose an innovative method, not yet met in the literature, to reduce the risks of indirect discrimination thanks to mathematical concepts of linear algebra. This technique is illustrated in a concrete case of risk selection in life insurance, demonstrating its simplicity of use and its promising performance.