3,894 research outputs found
Training Support Vector Machines Using Frank-Wolfe Optimization Methods
Training a Support Vector Machine (SVM) requires the solution of a quadratic
programming problem (QP) whose computational complexity becomes prohibitively
expensive for large scale datasets. Traditional optimization methods cannot be
directly applied in these cases, mainly due to memory restrictions.
By adopting a slightly different objective function and under mild conditions
on the kernel used within the model, efficient algorithms to train SVMs have
been devised under the name of Core Vector Machines (CVMs). This framework
exploits the equivalence of the resulting learning problem with the task of
building a Minimal Enclosing Ball (MEB) problem in a feature space, where data
is implicitly embedded by a kernel function.
In this paper, we improve on the CVM approach by proposing two novel methods
to build SVMs based on the Frank-Wolfe algorithm, recently revisited as a fast
method to approximate the solution of a MEB problem. In contrast to CVMs, our
algorithms do not require to compute the solutions of a sequence of
increasingly complex QPs and are defined by using only analytic optimization
steps. Experiments on a large collection of datasets show that our methods
scale better than CVMs in most cases, sometimes at the price of a slightly
lower accuracy. As CVMs, the proposed methods can be easily extended to machine
learning problems other than binary classification. However, effective
classifiers are also obtained using kernels which do not satisfy the condition
required by CVMs and can thus be used for a wider set of problems
A Divide-and-Conquer Solver for Kernel Support Vector Machines
The kernel support vector machine (SVM) is one of the most widely used
classification methods; however, the amount of computation required becomes the
bottleneck when facing millions of samples. In this paper, we propose and
analyze a novel divide-and-conquer solver for kernel SVMs (DC-SVM). In the
division step, we partition the kernel SVM problem into smaller subproblems by
clustering the data, so that each subproblem can be solved independently and
efficiently. We show theoretically that the support vectors identified by the
subproblem solution are likely to be support vectors of the entire kernel SVM
problem, provided that the problem is partitioned appropriately by kernel
clustering. In the conquer step, the local solutions from the subproblems are
used to initialize a global coordinate descent solver, which converges quickly
as suggested by our analysis. By extending this idea, we develop a multilevel
Divide-and-Conquer SVM algorithm with adaptive clustering and early prediction
strategy, which outperforms state-of-the-art methods in terms of training
speed, testing accuracy, and memory usage. As an example, on the covtype
dataset with half-a-million samples, DC-SVM is 7 times faster than LIBSVM in
obtaining the exact SVM solution (to within relative error) which
achieves 96.15% prediction accuracy. Moreover, with our proposed early
prediction strategy, DC-SVM achieves about 96% accuracy in only 12 minutes,
which is more than 100 times faster than LIBSVM
Robustness Verification of Support Vector Machines
We study the problem of formally verifying the robustness to adversarial
examples of support vector machines (SVMs), a major machine learning model for
classification and regression tasks. Following a recent stream of works on
formal robustness verification of (deep) neural networks, our approach relies
on a sound abstract version of a given SVM classifier to be used for checking
its robustness. This methodology is parametric on a given numerical abstraction
of real values and, analogously to the case of neural networks, needs neither
abstract least upper bounds nor widening operators on this abstraction. The
standard interval domain provides a simple instantiation of our abstraction
technique, which is enhanced with the domain of reduced affine forms, which is
an efficient abstraction of the zonotope abstract domain. This robustness
verification technique has been fully implemented and experimentally evaluated
on SVMs based on linear and nonlinear (polynomial and radial basis function)
kernels, which have been trained on the popular MNIST dataset of images and on
the recent and more challenging Fashion-MNIST dataset. The experimental results
of our prototype SVM robustness verifier appear to be encouraging: this
automated verification is fast, scalable and shows significantly high
percentages of provable robustness on the test set of MNIST, in particular
compared to the analogous provable robustness of neural networks
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