55 research outputs found
Bounds on Depth of Decision Trees Derived from Decision Rule Systems
Systems of decision rules and decision trees are widely used as a means for
knowledge representation, as classifiers, and as algorithms. They are among the
most interpretable models for classifying and representing knowledge. The study
of relationships between these two models is an important task of computer
science. It is easy to transform a decision tree into a decision rule system.
The inverse transformation is a more difficult task. In this paper, we study
unimprovable upper and lower bounds on the minimum depth of decision trees
derived from decision rule systems depending on the various parameters of these
systems
Comparative Analysis of Deterministic and Nondeterministic Decision Trees for Decision Tables from Closed Classes
In this paper, we consider classes of decision tables with many-valued
decisions closed under operations of removal of columns, changing of decisions,
permutation of columns, and duplication of columns. We study relationships
among three parameters of these tables: the complexity of a decision table (if
we consider the depth of decision trees, then the complexity of a decision
table is the number of columns in it), the minimum complexity of a
deterministic decision tree, and the minimum complexity of a nondeterministic
decision tree. We consider rough classification of functions characterizing
relationships and enumerate all possible seven types of the relationships
Construction and optimization of partial decision rules
Tematyka pracy związana jest z badaniem algorytmów zachłannych dla konstruowania i optymalizacji częściowych (przybliżonych) reguł decyzyjnych. Przedstawione w pracy badania dotyczące częściowych reguł decyzyjnych opierają się na wynikach badan uzyskanych dla problemu częściowego pokrycia zbioru.
Zostało udowodnione, ze biorąc pod uwagę pewne założenia dotyczące klasy NP, algorytm zachłanny pozwala uzyskać wyniki, bliskie wynikom uzyskiwanym przez najlepsze przybliżone wielomianowe algorytmy, dla minimalizacji długości częściowych reguł decyzyjnych oraz minimalizacji całkowitej wagi atrybutów tworzących częściową regułę decyzyjną. Na podstawie danych uzyskanych podczas pracy algorytmu zachłannego, dokonano oszacowania najlepszych górnych i dolnych granic minimalnej złożoności częściowych
reguł decyzyjnych. Teoretyczne i eksperymentalne wyniki badan pokazały możliwości wykorzystania tych granic w praktycznych zastosowaniach. Dokonano także oszacowania granicy dokładności algorytmu zachłannego dla
generowania częściowych reguł decyzyjnych, która nie zależy od liczby wierszy w rozważanej tablicy decyzyjnej.
Biorąc pod uwagę pewne założenia dotyczące liczby wierszy i kolumn w tablicach decyzyjnych udowodniono, ze dla większości binarnych tablic decyzyjnych istnieją tylko krótkie, nieredukowalne częściowe reguły decyzyjne.
Wyniki przeprowadzonych eksperymentów pozwoliły potwierdzić 0.5-hipoteze: dla większości tablic decyzyjnych algorytm zachłanny w każdej iteracji, podczas generowania częściowej reguły wybiera atrybut, który pozwala oddzielić przynajmniej 50% wierszy jeszcze nie oddzielonych. W przypadku klasyfikacji okazało się, że dokładność klasyfikatorów opartych na częściowych regułach decyzyjnych jest często lepsza, niż dokładność klasyfikatorów opartych na dokładnych regułach decyzyjnych
Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems
Multi-armed bandit problems are the most basic examples of sequential
decision problems with an exploration-exploitation trade-off. This is the
balance between staying with the option that gave highest payoffs in the past
and exploring new options that might give higher payoffs in the future.
Although the study of bandit problems dates back to the Thirties,
exploration-exploitation trade-offs arise in several modern applications, such
as ad placement, website optimization, and packet routing. Mathematically, a
multi-armed bandit is defined by the payoff process associated with each
option. In this survey, we focus on two extreme cases in which the analysis of
regret is particularly simple and elegant: i.i.d. payoffs and adversarial
payoffs. Besides the basic setting of finitely many actions, we also analyze
some of the most important variants and extensions, such as the contextual
bandit model.Comment: To appear in Foundations and Trends in Machine Learnin
PAC-Bayesian inductive and transductive learning
We present here a PAC-Bayesian point of view on adaptive supervised
classification. Using convex analysis, we show how to get local measures of the
complexity of the classification model involving the relative entropy of
posterior distributions with respect to Gibbs posterior measures. We discuss
relative bounds, comparing two classification rules, to show how the margin
assumption of Mammen and Tsybakov can be replaced with some empirical measure
of the covariance structure of the classification model. We also show how to
associate to any posterior distribution an {\em effective temperature} relating
it to the Gibbs prior distribution with the same level of expected error rate,
and how to estimate this effective temperature from data, resulting in an
estimator whose expected error rate adaptively converges according to the best
possible power of the sample size. Then we introduce a PAC-Bayesian point of
view on transductive learning and use it to improve on known Vapnik's
generalization bounds, extending them to the case when the sample is
independent but not identically distributed. Eventually we review briefly the
construction of Support Vector Machines and show how to derive generalization
bounds for them, measuring the complexity either through the number of support
vectors or through transductive or inductive margin estimates
Finite-time bounds for fitted value iteration
In this paper we develop a theoretical analysis of the performance of sampling-based fitted value iteration (FVI) to solve infinite state-space, discounted-reward Markovian decision processes (MDPs) under the assumption that a generative model of the environment is available. Our main results come in the form of finite-time bounds on the performance of two versions of sampling-based FVI.The convergence rate results obtained allow us to show that both versions of FVI are well behaving in the sense that by using a sufficiently large number of samples for a large class of MDPs, arbitrary good performance can be achieved with high probability.An important feature of our proof technique is that it permits the study of weighted -norm performance bounds. As a result, our technique applies to a large class of function-approximation methods (e.g., neural networks, adaptive regression trees, kernel machines, locally weighted learning), and our bounds scale well with the effective horizon of the MDP. The bounds show a dependence on the stochastic stability properties of the MDP: they scale with the discounted-average concentrability of the future-state distributions. They also depend on a new measure of the approximation power of the function space, the inherent Bellman residual, which reflects how well the function space is ``aligned'' with the dynamics and rewards of the MDP.The conditions of the main result, as well as the concepts introduced in the analysis, are extensively discussed and compared to previous theoretical results.Numerical experiments are used to substantiate the theoretical findings
Pac-Bayesian Supervised Classification: The Thermodynamics of Statistical Learning
This monograph deals with adaptive supervised classification, using tools
borrowed from statistical mechanics and information theory, stemming from the
PACBayesian approach pioneered by David McAllester and applied to a conception
of statistical learning theory forged by Vladimir Vapnik. Using convex analysis
on the set of posterior probability measures, we show how to get local measures
of the complexity of the classification model involving the relative entropy of
posterior distributions with respect to Gibbs posterior measures. We then
discuss relative bounds, comparing the generalization error of two
classification rules, showing how the margin assumption of Mammen and Tsybakov
can be replaced with some empirical measure of the covariance structure of the
classification model.We show how to associate to any posterior distribution an
effective temperature relating it to the Gibbs prior distribution with the same
level of expected error rate, and how to estimate this effective temperature
from data, resulting in an estimator whose expected error rate converges
according to the best possible power of the sample size adaptively under any
margin and parametric complexity assumptions. We describe and study an
alternative selection scheme based on relative bounds between estimators, and
present a two step localization technique which can handle the selection of a
parametric model from a family of those. We show how to extend systematically
all the results obtained in the inductive setting to transductive learning, and
use this to improve Vapnik's generalization bounds, extending them to the case
when the sample is made of independent non-identically distributed pairs of
patterns and labels. Finally we review briefly the construction of Support
Vector Machines and show how to derive generalization bounds for them,
measuring the complexity either through the number of support vectors or
through the value of the transductive or inductive margin.Comment: Published in at http://dx.doi.org/10.1214/074921707000000391 the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
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