101 research outputs found
Online Combinatorial Linear Optimization via a Frank-Wolfe-based Metarounding Algorithm
Metarounding is an approach to convert an approximation algorithm for linear
optimization over some combinatorial classes to an online linear optimization
algorithm for the same class. We propose a new metarounding algorithm under a
natural assumption that a relax-based approximation algorithm exists for the
combinatorial class. Our algorithm is much more efficient in both theoretical
and practical aspects
Fractures in Duchenne Muscular Dystrophy: chiefly about their causes
Among 148 Duchenne muscular dystrophy children, nineteen cases (twenty-six fractures) were associated with bone fractures involving twelve femurs, nine humeri, four tibiae and one metatarsal bone. Seven out of twenty-six cases had experienced fractures twice. The causes of fractures were falls in fifteen cases, collision against surrounding objects in five, body position change in four and unknown in two. Femoral fractures were dominant during the wheelchair-bound phase, while, humeral fractures were dominant during the ambulatory phase. As these children lack sitting and standing balance as well as normal muscular power, we have to be careful of falls to prevent bone fractures when they are in a sitting or standing posture. Most of these fractures seem to be prevented if careful attention was paid during rehabilitation exercise, transfer and body position change etc
Boosting as Frank-Wolfe
Some boosting algorithms, such as LPBoost, ERLPBoost, and C-ERLPBoost, aim to
solve the soft margin optimization problem with the -norm
regularization. LPBoost rapidly converges to an -approximate solution
in practice, but it is known to take iterations in the worst case,
where is the sample size. On the other hand, ERLPBoost and C-ERLPBoost are
guaranteed to converge to an -approximate solution in
iterations. However, the
computation per iteration is very high compared to LPBoost.
To address this issue, we propose a generic boosting scheme that combines the
Frank-Wolfe algorithm and any secondary algorithm and switches one to the other
iteratively. We show that the scheme retains the same convergence guarantee as
ERLPBoost and C-ERLPBoost. One can incorporate any secondary algorithm to
improve in practice. This scheme comes from a unified view of boosting
algorithms for soft margin optimization. More specifically, we show that
LPBoost, ERLPBoost, and C-ERLPBoost are instances of the Frank-Wolfe algorithm.
In experiments on real datasets, one of the instances of our scheme exploits
the better updates of the secondary algorithm and performs comparably with
LPBoost
Boosting-based Construction of BDDs for Linear Threshold Functions and Its Application to Verification of Neural Networks
Understanding the characteristics of neural networks is important but
difficult due to their complex structures and behaviors. Some previous work
proposes to transform neural networks into equivalent Boolean expressions and
apply verification techniques for characteristics of interest. This approach is
promising since rich results of verification techniques for circuits and other
Boolean expressions can be readily applied. The bottleneck is the time
complexity of the transformation. More precisely, (i) each neuron of the
network, i.e., a linear threshold function, is converted to a Binary Decision
Diagram (BDD), and (ii) they are further combined into some final form, such as
Boolean circuits. For a linear threshold function with variables, an
existing method takes time to construct an ordered BDD of
size consistent with some variable ordering. However, it
is non-trivial to choose a variable ordering producing a small BDD among
candidates.
We propose a method to convert a linear threshold function to a specific form
of a BDD based on the boosting approach in the machine learning literature. Our
method takes time and outputs BDD of size
, where is the margin of some
consistent linear threshold function. Our method does not need to search for
good variable orderings and produces a smaller expression when the margin of
the linear threshold function is large. More precisely, our method is based on
our new boosting algorithm, which is of independent interest. We also propose a
method to combine them into the final Boolean expression representing the
neural network
Decision Diagrams for Solving a Job Scheduling Problem Under Precedence Constraints
We consider a job scheduling problem under precedence constraints, a classical problem for a single processor and multiple jobs to be done. The goal is, given processing time of n fixed jobs and precedence constraints over jobs, to find a permutation of n jobs that minimizes the total flow time, i.e., the sum of total wait time and processing times of all jobs, while satisfying the precedence constraints. The problem is an integer program and is NP-hard in general. We propose a decision diagram pi-MDD, for solving the scheduling problem exactly. Our diagram is suitable for solving linear optimization over permutations with precedence constraints. We show the effectiveness of our approach on the experiments on large scale artificial scheduling problems
- …