423 research outputs found
Divide and Conquer Kernel Ridge Regression: A Distributed Algorithm with Minimax Optimal Rates
We establish optimal convergence rates for a decomposition-based scalable
approach to kernel ridge regression. The method is simple to describe: it
randomly partitions a dataset of size N into m subsets of equal size, computes
an independent kernel ridge regression estimator for each subset, then averages
the local solutions into a global predictor. This partitioning leads to a
substantial reduction in computation time versus the standard approach of
performing kernel ridge regression on all N samples. Our two main theorems
establish that despite the computational speed-up, statistical optimality is
retained: as long as m is not too large, the partition-based estimator achieves
the statistical minimax rate over all estimators using the set of N samples. As
concrete examples, our theory guarantees that the number of processors m may
grow nearly linearly for finite-rank kernels and Gaussian kernels and
polynomially in N for Sobolev spaces, which in turn allows for substantial
reductions in computational cost. We conclude with experiments on both
simulated data and a music-prediction task that complement our theoretical
results, exhibiting the computational and statistical benefits of our approach
Sharp analysis of low-rank kernel matrix approximations
We consider supervised learning problems within the positive-definite kernel
framework, such as kernel ridge regression, kernel logistic regression or the
support vector machine. With kernels leading to infinite-dimensional feature
spaces, a common practical limiting difficulty is the necessity of computing
the kernel matrix, which most frequently leads to algorithms with running time
at least quadratic in the number of observations n, i.e., O(n^2). Low-rank
approximations of the kernel matrix are often considered as they allow the
reduction of running time complexities to O(p^2 n), where p is the rank of the
approximation. The practicality of such methods thus depends on the required
rank p. In this paper, we show that in the context of kernel ridge regression,
for approximations based on a random subset of columns of the original kernel
matrix, the rank p may be chosen to be linear in the degrees of freedom
associated with the problem, a quantity which is classically used in the
statistical analysis of such methods, and is often seen as the implicit number
of parameters of non-parametric estimators. This result enables simple
algorithms that have sub-quadratic running time complexity, but provably
exhibit the same predictive performance than existing algorithms, for any given
problem instance, and not only for worst-case situations
TIGER: A Tuning-Insensitive Approach for Optimally Estimating Gaussian Graphical Models
We propose a new procedure for estimating high dimensional Gaussian graphical
models. Our approach is asymptotically tuning-free and non-asymptotically
tuning-insensitive: it requires very few efforts to choose the tuning parameter
in finite sample settings. Computationally, our procedure is significantly
faster than existing methods due to its tuning-insensitive property.
Theoretically, the obtained estimator is simultaneously minimax optimal for
precision matrix estimation under different norms. Empirically, we illustrate
the advantages of our method using thorough simulated and real examples. The R
package bigmatrix implementing the proposed methods is available on the
Comprehensive R Archive Network: http://cran.r-project.org/
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