2,163 research outputs found
Surrogate regret bounds for generalized classification performance metrics
We consider optimization of generalized performance metrics for binary
classification by means of surrogate losses. We focus on a class of metrics,
which are linear-fractional functions of the false positive and false negative
rates (examples of which include -measure, Jaccard similarity
coefficient, AM measure, and many others). Our analysis concerns the following
two-step procedure. First, a real-valued function is learned by minimizing
a surrogate loss for binary classification on the training sample. It is
assumed that the surrogate loss is a strongly proper composite loss function
(examples of which include logistic loss, squared-error loss, exponential loss,
etc.). Then, given , a threshold is tuned on a separate
validation sample, by direct optimization of the target performance metric. We
show that the regret of the resulting classifier (obtained from thresholding
on ) measured with respect to the target metric is
upperbounded by the regret of measured with respect to the surrogate loss.
We also extend our results to cover multilabel classification and provide
regret bounds for micro- and macro-averaging measures. Our findings are further
analyzed in a computational study on both synthetic and real data sets.Comment: 22 page
Calibrated Surrogate Losses for Classification with Label-Dependent Costs
We present surrogate regret bounds for arbitrary surrogate losses in the
context of binary classification with label-dependent costs. Such bounds relate
a classifier's risk, assessed with respect to a surrogate loss, to its
cost-sensitive classification risk. Two approaches to surrogate regret bounds
are developed. The first is a direct generalization of Bartlett et al. [2006],
who focus on margin-based losses and cost-insensitive classification, while the
second adopts the framework of Steinwart [2007] based on calibration functions.
Nontrivial surrogate regret bounds are shown to exist precisely when the
surrogate loss satisfies a "calibration" condition that is easily verified for
many common losses. We apply this theory to the class of uneven margin losses,
and characterize when these losses are properly calibrated. The uneven hinge,
squared error, exponential, and sigmoid losses are then treated in detail.Comment: 33 pages, 7 figure
Online and Stochastic Gradient Methods for Non-decomposable Loss Functions
Modern applications in sensitive domains such as biometrics and medicine
frequently require the use of non-decomposable loss functions such as
precision@k, F-measure etc. Compared to point loss functions such as
hinge-loss, these offer much more fine grained control over prediction, but at
the same time present novel challenges in terms of algorithm design and
analysis. In this work we initiate a study of online learning techniques for
such non-decomposable loss functions with an aim to enable incremental learning
as well as design scalable solvers for batch problems. To this end, we propose
an online learning framework for such loss functions. Our model enjoys several
nice properties, chief amongst them being the existence of efficient online
learning algorithms with sublinear regret and online to batch conversion
bounds. Our model is a provable extension of existing online learning models
for point loss functions. We instantiate two popular losses, prec@k and pAUC,
in our model and prove sublinear regret bounds for both of them. Our proofs
require a novel structural lemma over ranked lists which may be of independent
interest. We then develop scalable stochastic gradient descent solvers for
non-decomposable loss functions. We show that for a large family of loss
functions satisfying a certain uniform convergence property (that includes
prec@k, pAUC, and F-measure), our methods provably converge to the empirical
risk minimizer. Such uniform convergence results were not known for these
losses and we establish these using novel proof techniques. We then use
extensive experimentation on real life and benchmark datasets to establish that
our method can be orders of magnitude faster than a recently proposed cutting
plane method.Comment: 25 pages, 3 figures, To appear in the proceedings of the 28th Annual
Conference on Neural Information Processing Systems, NIPS 201
- …