3 research outputs found
Extended Formulations via Decision Diagrams
We propose a general algorithm of constructing an extended formulation for
any given set of linear constraints with integer coefficients. Our algorithm
consists of two phases: first construct a decision diagram that somehow
represents a given constraint matrix, and then build an equivalent
set of linear constraints over variables. That is, the size of
the resultant extended formulation depends not explicitly on the number of
the original constraints, but on its decision diagram representation.
Therefore, we may significantly reduce the computation time for optimization
problems with integer constraint matrices by solving them under the extended
formulations, especially when we obtain concise decision diagram
representations for the matrices. We can apply our method to -norm
regularized hard margin optimization over the binary instance space
, which can be formulated as a linear programming problem with
constraints with -valued coefficients over variables, where
is the size of the given sample. Furthermore, introducing slack variables over
the edges of the decision diagram, we establish a variant formulation of soft
margin optimization. We demonstrate the effectiveness of our extended
formulations for integer programming and the -norm regularized soft margin
optimization tasks over synthetic and real datasets
Boosting over non-deterministic ZDDs
We propose a new approach to large-scale machine learning, learning over compressed data: First compress the training data somehow and then employ various machine learning algorithms on the compressed data, with the hope that the computation time is significantly reduced when the training data is well compressed. As the first step, we consider a variant of the Zero-Suppressed Binary Decision Diagram (ZDD) as the data structure for representing the training data, which is a generalization of the ZDD by incorporating non-determinism. For the learning algorithm to be employed, we consider boosting algorithm called AdaBoost∗ and its precursor AdaBoost. In this work, we give efficient implementations of the boosting algorithms whose running times (per iteration) are linear in the size of the given ZDD
Boosting over non-deterministic ZDDs
We propose a new approach to large-scale machine learning, learning over compressed data: First compress the training data somehow and then em-ploy various machine learning algorithms on the compressed data, with the hope that the computation time is signi_cantly reduced when the training data is well compressed. As a _rst step toward this approach, we consider a variant of the Zero-Suppressed Binary Decision Diagram (ZDD) as the data structure for representing the training data, which is a generalization of the ZDD by incorporating non-determinism. For the learning algorithm to be employed, we consider a boosting algorithm called AdaBoost_ and its precursor AdaBoost. In this paper, we give efficient implementations of the boosting algorithms whose running times (per iteration) are linear in the size of the given ZDD