704 research outputs found
Non-strongly-convex smooth stochastic approximation with convergence rate O(1/n)
We consider the stochastic approximation problem where a convex function has
to be minimized, given only the knowledge of unbiased estimates of its
gradients at certain points, a framework which includes machine learning
methods based on the minimization of the empirical risk. We focus on problems
without strong convexity, for which all previously known algorithms achieve a
convergence rate for function values of O(1/n^{1/2}). We consider and analyze
two algorithms that achieve a rate of O(1/n) for classical supervised learning
problems. For least-squares regression, we show that averaged stochastic
gradient descent with constant step-size achieves the desired rate. For
logistic regression, this is achieved by a simple novel stochastic gradient
algorithm that (a) constructs successive local quadratic approximations of the
loss functions, while (b) preserving the same running time complexity as
stochastic gradient descent. For these algorithms, we provide a non-asymptotic
analysis of the generalization error (in expectation, and also in high
probability for least-squares), and run extensive experiments on standard
machine learning benchmarks showing that they often outperform existing
approaches
Approximately counting semismooth integers
An integer is -semismooth if where is an integer with
all prime divisors and is 1 or a prime . arge quantities of
semismooth integers are utilized in modern integer factoring algorithms, such
as the number field sieve, that incorporate the so-called large prime variant.
Thus, it is useful for factoring practitioners to be able to estimate the value
of , the number of -semismooth integers up to , so that
they can better set algorithm parameters and minimize running times, which
could be weeks or months on a cluster supercomputer. In this paper, we explore
several algorithms to approximate using a generalization of
Buchstab's identity with numeric integration.Comment: To appear in ISSAC 2013, Boston M
Testing for Homogeneity with Kernel Fisher Discriminant Analysis
We propose to investigate test statistics for testing homogeneity in
reproducing kernel Hilbert spaces. Asymptotic null distributions under null
hypothesis are derived, and consistency against fixed and local alternatives is
assessed. Finally, experimental evidence of the performance of the proposed
approach on both artificial data and a speaker verification task is provided
- …