arXiv subjects
Publications and source records attributed to Andreas Maurer.
The expected supremum of a Gaussian process indexed by the image of an index set under a function class is bounded in terms of separate properties of the index set and the function class. The bound is relevant to the estimation of nonlinear transformations or the analysis of learning algorithms whenever hypotheses are chosen from composite classes, as is the case for multi-layer models.
We investigate the use of sparse coding and dictionary learning in the context of multitask and transfer learning. The central assumption of our learning method is that the tasks parameters are well approximated by sparse linear combinations of the atoms of a dictionary on a high or infinite dimensional space. This assumption, together with the large quantity of available data in the multitask and transfer learning settings, allows a principled choice of the dictionary. We provide bounds on the generalization error of this approach, for both settings. Numerical experiments on one synthetic and two real datasets show the advantage of our method over single task learning, a previous method based on orthogonal and dense representation of the tasks and a related method learning task grouping.
From concentration inequalities for the suprema of Gaussian or Rademacher processes an inequality is derived. It is applied to sharpen existing and to derive novel bounds on the empirical Rademacher complexities of unit balls in various norms appearing in the context of structured sparsity and multitask dictionary learning or matrix factorization. A key role is played by the largest eigenvalue of the data covariance matrix.
Trace norm regularization is a popular method of multitask learning. We give excess risk bounds with explicit dependence on the number of tasks, the number of examples per task and properties of the data distribution. The bounds are independent of the dimension of the input space, which may be infinite as in the case of reproducing kernel Hilbert spaces. A byproduct of the proof are bounds on the expected norm of sums of random positive semidefinite matrices with subexponential moments.
We show that the thermal subadditivity of entropy provides a common basis to derive a strong form of the bounded difference inequality and related results as well as more recent inequalities applicable to convex Lipschitz functions, random symmetric matrices, shortest travelling salesmen paths and weakly self-bounding functions. We also give two new concentration inequalities.
We present a data dependent generalization bound for a large class of regularized algorithms which implement structured sparsity constraints. The bound can be applied to standard squared-norm regularization, the Lasso, the group Lasso, some versions of the group Lasso with overlapping groups, multiple kernel learning and other regularization schemes. In all these cases competitive results are obtained. A novel feature of our bound is that it can be applied in an infinite dimensional setting such as the Lasso in a separable Hilbert space or multiple kernel learning with a countable number of kernels.
We give improved constants for data dependent and variance sensitive confidence bounds, called empirical Bernstein bounds, and extend these inequalities to hold uniformly over classes of functionswhose growth function is polynomial in the sample size n. The bounds lead us to consider sample variance penalization, a novel learning method which takes into account the empirical variance of the loss function. We give conditions under which sample variance penalization is effective. In particular, we present a bound on the excess risk incurred by the method. Using this, we argue that there are situations in which the excess risk of our method is of order 1/n, while the excess risk of empirical risk minimization is of order 1/sqrt/{n}. We show some experimental results, which confirm the theory. Finally, we discuss the potential application of our results to sample compression schemes.
We prove existence and uniqueness of the minimizer for the average geodesic distance to the points of a geodesically convex set on the sphere. This implies a corresponding existence and uniqueness result for an optimal algorithm for halfspace learning, when data and target functions are drawn from the uniform distribution.
We prove general exponential moment inequalities for averages of [0,1]-valued iid random variables and use them to tighten the PAC Bayesian Theorem. The logarithmic dependence on the sample count in the enumerator of the PAC Bayesian bound is halved.