693 research outputs found
Tighter risk certificates for neural networks
This paper presents an empirical study regarding training probabilistic
neural networks using training objectives derived from PAC-Bayes bounds. In the
context of probabilistic neural networks, the output of training is a
probability distribution over network weights. We present two training
objectives, used here for the first time in connection with training neural
networks. These two training objectives are derived from tight PAC-Bayes
bounds. We also re-implement a previously used training objective based on a
classical PAC-Bayes bound, to compare the properties of the predictors learned
using the different training objectives. We compute risk certificates that are
valid on any unseen examples for the learnt predictors. We further experiment
with different types of priors on the weights (both data-free and
data-dependent priors) and neural network architectures. Our experiments on
MNIST and CIFAR-10 show that our training methods produce competitive test set
errors and non-vacuous risk bounds with much tighter values than previous
results in the literature, showing promise not only to guide the learning
algorithm through bounding the risk but also for model selection. These
observations suggest that the methods studied here might be good candidates for
self-certified learning, in the sense of certifying the risk on any unseen data
without the need for data-splitting protocols.Comment: Preprint under revie
Data-Dependent Stability of Stochastic Gradient Descent
We establish a data-dependent notion of algorithmic stability for Stochastic
Gradient Descent (SGD), and employ it to develop novel generalization bounds.
This is in contrast to previous distribution-free algorithmic stability results
for SGD which depend on the worst-case constants. By virtue of the
data-dependent argument, our bounds provide new insights into learning with SGD
on convex and non-convex problems. In the convex case, we show that the bound
on the generalization error depends on the risk at the initialization point. In
the non-convex case, we prove that the expected curvature of the objective
function around the initialization point has crucial influence on the
generalization error. In both cases, our results suggest a simple data-driven
strategy to stabilize SGD by pre-screening its initialization. As a corollary,
our results allow us to show optimistic generalization bounds that exhibit fast
convergence rates for SGD subject to a vanishing empirical risk and low noise
of stochastic gradient
Learning Stochastic Majority Votes by Minimizing a PAC-Bayes Generalization Bound
We investigate a stochastic counterpart of majority votes over finite ensembles of classifiers, and study its generalization properties. While our approach holds for arbitrary distributions, we instantiate it with Dirichlet distributions: this allows for a closed-form and differentiable expression for the expected risk, which then turns the generalization bound into a tractable training objective.The resulting stochastic majority vote learning algorithm achieves state-of-the-art accuracy and benefits from (non-vacuous) tight generalization bounds, in a series of numerical experiments when compared to competing algorithms which also minimize PAC-Bayes objectives -- both with uninformed (data-independent) and informed (data-dependent) priors
Bayesian fairness
We consider the problem of how decision making can be fair when the
underlying probabilistic model of the world is not known with certainty. We
argue that recent notions of fairness in machine learning need to explicitly
incorporate parameter uncertainty, hence we introduce the notion of {\em
Bayesian fairness} as a suitable candidate for fair decision rules. Using
balance, a definition of fairness introduced by Kleinberg et al (2016), we show
how a Bayesian perspective can lead to well-performing, fair decision rules
even under high uncertainty.Comment: 13 pages, 8 figures, to appear at AAAI 201
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