2,128 research outputs found
Explicit Learning Curves for Transduction and Application to Clustering and Compression Algorithms
Inductive learning is based on inferring a general rule from a finite data
set and using it to label new data. In transduction one attempts to solve the
problem of using a labeled training set to label a set of unlabeled points,
which are given to the learner prior to learning. Although transduction seems
at the outset to be an easier task than induction, there have not been many
provably useful algorithms for transduction. Moreover, the precise relation
between induction and transduction has not yet been determined. The main
theoretical developments related to transduction were presented by Vapnik more
than twenty years ago. One of Vapnik's basic results is a rather tight error
bound for transductive classification based on an exact computation of the
hypergeometric tail. While tight, this bound is given implicitly via a
computational routine. Our first contribution is a somewhat looser but explicit
characterization of a slightly extended PAC-Bayesian version of Vapnik's
transductive bound. This characterization is obtained using concentration
inequalities for the tail of sums of random variables obtained by sampling
without replacement. We then derive error bounds for compression schemes such
as (transductive) support vector machines and for transduction algorithms based
on clustering. The main observation used for deriving these new error bounds
and algorithms is that the unlabeled test points, which in the transductive
setting are known in advance, can be used in order to construct useful data
dependent prior distributions over the hypothesis space
On PAC-Bayesian Bounds for Random Forests
Existing guarantees in terms of rigorous upper bounds on the generalization
error for the original random forest algorithm, one of the most frequently used
machine learning methods, are unsatisfying. We discuss and evaluate various
PAC-Bayesian approaches to derive such bounds. The bounds do not require
additional hold-out data, because the out-of-bag samples from the bagging in
the training process can be exploited. A random forest predicts by taking a
majority vote of an ensemble of decision trees. The first approach is to bound
the error of the vote by twice the error of the corresponding Gibbs classifier
(classifying with a single member of the ensemble selected at random). However,
this approach does not take into account the effect of averaging out of errors
of individual classifiers when taking the majority vote. This effect provides a
significant boost in performance when the errors are independent or negatively
correlated, but when the correlations are strong the advantage from taking the
majority vote is small. The second approach based on PAC-Bayesian C-bounds
takes dependencies between ensemble members into account, but it requires
estimating correlations between the errors of the individual classifiers. When
the correlations are high or the estimation is poor, the bounds degrade. In our
experiments, we compute generalization bounds for random forests on various
benchmark data sets. Because the individual decision trees already perform
well, their predictions are highly correlated and the C-bounds do not lead to
satisfactory results. For the same reason, the bounds based on the analysis of
Gibbs classifiers are typically superior and often reasonably tight. Bounds
based on a validation set coming at the cost of a smaller training set gave
better performance guarantees, but worse performance in most experiments
Generalization Error in Deep Learning
Deep learning models have lately shown great performance in various fields
such as computer vision, speech recognition, speech translation, and natural
language processing. However, alongside their state-of-the-art performance, it
is still generally unclear what is the source of their generalization ability.
Thus, an important question is what makes deep neural networks able to
generalize well from the training set to new data. In this article, we provide
an overview of the existing theory and bounds for the characterization of the
generalization error of deep neural networks, combining both classical and more
recent theoretical and empirical results
Meta learning of bounds on the Bayes classifier error
Meta learning uses information from base learners (e.g. classifiers or
estimators) as well as information about the learning problem to improve upon
the performance of a single base learner. For example, the Bayes error rate of
a given feature space, if known, can be used to aid in choosing a classifier,
as well as in feature selection and model selection for the base classifiers
and the meta classifier. Recent work in the field of f-divergence functional
estimation has led to the development of simple and rapidly converging
estimators that can be used to estimate various bounds on the Bayes error. We
estimate multiple bounds on the Bayes error using an estimator that applies
meta learning to slowly converging plug-in estimators to obtain the parametric
convergence rate. We compare the estimated bounds empirically on simulated data
and then estimate the tighter bounds on features extracted from an image patch
analysis of sunspot continuum and magnetogram images.Comment: 6 pages, 3 figures, to appear in proceedings of 2015 IEEE Signal
Processing and SP Education Worksho
Information Losses in Neural Classifiers from Sampling
This paper considers the subject of information losses arising from the
finite datasets used in the training of neural classifiers. It proves a
relationship between such losses as the product of the expected total variation
of the estimated neural model with the information about the feature space
contained in the hidden representation of that model. It then bounds this
expected total variation as a function of the size of randomly sampled datasets
in a fairly general setting, and without bringing in any additional dependence
on model complexity. It ultimately obtains bounds on information losses that
are less sensitive to input compression and in general much smaller than
existing bounds. The paper then uses these bounds to explain some recent
experimental findings of information compression in neural networks which
cannot be explained by previous work. Finally, the paper shows that not only
are these bounds much smaller than existing ones, but that they also correspond
well with experiments.Comment: To be published in IEEE TNNL
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