14 research outputs found
Suboptimal Solution Path Algorithm for Support Vector Machine
We consider a suboptimal solution path algorithm for the Support Vector
Machine. The solution path algorithm is an effective tool for solving a
sequence of a parametrized optimization problems in machine learning. The path
of the solutions provided by this algorithm are very accurate and they satisfy
the optimality conditions more strictly than other SVM optimization algorithms.
In many machine learning application, however, this strict optimality is often
unnecessary, and it adversely affects the computational efficiency. Our
algorithm can generate the path of suboptimal solutions within an arbitrary
user-specified tolerance level. It allows us to control the trade-off between
the accuracy of the solution and the computational cost. Moreover, We also show
that our suboptimal solutions can be interpreted as the solution of a
\emph{perturbed optimization problem} from the original one. We provide some
theoretical analyses of our algorithm based on this novel interpretation. The
experimental results also demonstrate the effectiveness of our algorithm.Comment: A shorter version of this paper is submitted to ICML 201
An Exponential Lower Bound on the Complexity of Regularization Paths
For a variety of regularized optimization problems in machine learning,
algorithms computing the entire solution path have been developed recently.
Most of these methods are quadratic programs that are parameterized by a single
parameter, as for example the Support Vector Machine (SVM). Solution path
algorithms do not only compute the solution for one particular value of the
regularization parameter but the entire path of solutions, making the selection
of an optimal parameter much easier.
It has been assumed that these piecewise linear solution paths have only
linear complexity, i.e. linearly many bends. We prove that for the support
vector machine this complexity can be exponential in the number of training
points in the worst case. More strongly, we construct a single instance of n
input points in d dimensions for an SVM such that at least \Theta(2^{n/2}) =
\Theta(2^d) many distinct subsets of support vectors occur as the
regularization parameter changes.Comment: Journal version, 28 Pages, 5 Figure
An Algorithmic Framework for Computing Validation Performance Bounds by Using Suboptimal Models
Practical model building processes are often time-consuming because many
different models must be trained and validated. In this paper, we introduce a
novel algorithm that can be used for computing the lower and the upper bounds
of model validation errors without actually training the model itself. A key
idea behind our algorithm is using a side information available from a
suboptimal model. If a reasonably good suboptimal model is available, our
algorithm can compute lower and upper bounds of many useful quantities for
making inferences on the unknown target model. We demonstrate the advantage of
our algorithm in the context of model selection for regularized learning
problems
Root-finding Approaches for Computing Conformal Prediction Set
Conformal prediction constructs a confidence set for an unobserved response
of a feature vector based on previous identically distributed and exchangeable
observations of responses and features. It has a coverage guarantee at any
nominal level without additional assumptions on their distribution. Its
computation deplorably requires a refitting procedure for all replacement
candidates of the target response. In regression settings, this corresponds to
an infinite number of model fit. Apart from relatively simple estimators that
can be written as pieces of linear function of the response, efficiently
computing such sets is difficult and is still considered as an open problem. We
exploit the fact that, \emph{often}, conformal prediction sets are intervals
whose boundaries can be efficiently approximated by classical root-finding
algorithm. We investigate how this approach can overcome many limitations of
formerly used strategies and we discuss its complexity and drawbacks
Sparse convex optimization methods for machine learning
Diss., Eidgenössische Technische Hochschule ETH Zürich, Nr. 20013, 201