18,178 research outputs found
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
Network Lasso: Clustering and Optimization in Large Graphs
Convex optimization is an essential tool for modern data analysis, as it
provides a framework to formulate and solve many problems in machine learning
and data mining. However, general convex optimization solvers do not scale
well, and scalable solvers are often specialized to only work on a narrow class
of problems. Therefore, there is a need for simple, scalable algorithms that
can solve many common optimization problems. In this paper, we introduce the
\emph{network lasso}, a generalization of the group lasso to a network setting
that allows for simultaneous clustering and optimization on graphs. We develop
an algorithm based on the Alternating Direction Method of Multipliers (ADMM) to
solve this problem in a distributed and scalable manner, which allows for
guaranteed global convergence even on large graphs. We also examine a
non-convex extension of this approach. We then demonstrate that many types of
problems can be expressed in our framework. We focus on three in particular -
binary classification, predicting housing prices, and event detection in time
series data - comparing the network lasso to baseline approaches and showing
that it is both a fast and accurate method of solving large optimization
problems
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
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