439 research outputs found
A Primer on Bayesian Neural Networks: Review and Debates
Neural networks have achieved remarkable performance across various problem
domains, but their widespread applicability is hindered by inherent limitations
such as overconfidence in predictions, lack of interpretability, and
vulnerability to adversarial attacks. To address these challenges, Bayesian
neural networks (BNNs) have emerged as a compelling extension of conventional
neural networks, integrating uncertainty estimation into their predictive
capabilities.
This comprehensive primer presents a systematic introduction to the
fundamental concepts of neural networks and Bayesian inference, elucidating
their synergistic integration for the development of BNNs. The target audience
comprises statisticians with a potential background in Bayesian methods but
lacking deep learning expertise, as well as machine learners proficient in deep
neural networks but with limited exposure to Bayesian statistics. We provide an
overview of commonly employed priors, examining their impact on model behavior
and performance. Additionally, we delve into the practical considerations
associated with training and inference in BNNs.
Furthermore, we explore advanced topics within the realm of BNN research,
acknowledging the existence of ongoing debates and controversies. By offering
insights into cutting-edge developments, this primer not only equips
researchers and practitioners with a solid foundation in BNNs, but also
illuminates the potential applications of this dynamic field. As a valuable
resource, it fosters an understanding of BNNs and their promising prospects,
facilitating further advancements in the pursuit of knowledge and innovation.Comment: 65 page
PAC-Bayesian Theory Meets Bayesian Inference
We exhibit a strong link between frequentist PAC-Bayesian risk bounds and the
Bayesian marginal likelihood. That is, for the negative log-likelihood loss
function, we show that the minimization of PAC-Bayesian generalization risk
bounds maximizes the Bayesian marginal likelihood. This provides an alternative
explanation to the Bayesian Occam's razor criteria, under the assumption that
the data is generated by an i.i.d distribution. Moreover, as the negative
log-likelihood is an unbounded loss function, we motivate and propose a
PAC-Bayesian theorem tailored for the sub-gamma loss family, and we show that
our approach is sound on classical Bayesian linear regression tasks.Comment: Published at NIPS 2015
(http://papers.nips.cc/paper/6569-pac-bayesian-theory-meets-bayesian-inference
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
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 for the learnt predictors, based on part of the data used to learn the 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 using the whole data set for learning a predictor and certifying its risk on any unseen data (from the same distribution as the training data) potentially without the need for holding out test data
A review of domain adaptation without target labels
Domain adaptation has become a prominent problem setting in machine learning
and related fields. This review asks the question: how can a classifier learn
from a source domain and generalize to a target domain? We present a
categorization of approaches, divided into, what we refer to as, sample-based,
feature-based and inference-based methods. Sample-based methods focus on
weighting individual observations during training based on their importance to
the target domain. Feature-based methods revolve around on mapping, projecting
and representing features such that a source classifier performs well on the
target domain and inference-based methods incorporate adaptation into the
parameter estimation procedure, for instance through constraints on the
optimization procedure. Additionally, we review a number of conditions that
allow for formulating bounds on the cross-domain generalization error. Our
categorization highlights recurring ideas and raises questions important to
further research.Comment: 20 pages, 5 figure
Bayesian Gaussian Process Models: PAC-Bayesian Generalisation Error Bounds and Sparse Approximations
Non-parametric models and techniques enjoy a growing popularity in the field of machine learning, and among these Bayesian inference for Gaussian process (GP) models has recently received significant attention. We feel that GP priors should be part of the standard toolbox for constructing models relevant to machine learning in the same way as parametric linear models are, and the results in this thesis help to remove some obstacles on the way towards this goal. In the first main chapter, we provide a distribution-free finite sample bound on the difference between generalisation and empirical (training) error for GP classification methods. While the general theorem (the PAC-Bayesian bound) is not new, we give a much simplified and somewhat generalised derivation and point out the underlying core technique (convex duality) explicitly. Furthermore, the application to GP models is novel (to our knowledge). A central feature of this bound is that its quality depends crucially on task knowledge being encoded faithfully in the model and prior distributions, so there is a mutual benefit between a sharp theoretical guarantee and empirically well-established statistical practices. Extensive simulations on real-world classification tasks indicate an impressive tightness of the bound, in spite of the fact that many previous bounds for related kernel machines fail to give non-trivial guarantees in this practically relevant regime. In the second main chapter, sparse approximations are developed to address the problem of the unfavourable scaling of most GP techniques with large training sets. Due to its high importance in practice, this problem has received a lot of attention recently. We demonstrate the tractability and usefulness of simple greedy forward selection with information-theoretic criteria previously used in active learning (or sequential design) and develop generic schemes for automatic model selection with many (hyper)parameters. We suggest two new generic schemes and evaluate some of their variants on large real-world classification and regression tasks. These schemes and their underlying principles (which are clearly stated and analysed) can be applied to obtain sparse approximations for a wide regime of GP models far beyond the special cases we studied here
PAC-Bayesian Computation
Risk bounds, which are also called generalisation bounds in the statistical learning literature, are important objects of study because they give some information on the expected error that a predictor may incur on randomly chosen data points. In classical statistical learning, the analyses focus on individual hypotheses, and the aim is deriving risk bounds that are valid for the data-dependent hypothesis output by some learning method. Often, however, such risk bounds are valid uniformly over a hypothesis class, which is a consequence of the methods used to derive them, namely the theory of uniform convergence of empirical processes. This is a source of looseness of these classical kinds of bounds which has lead to debates and criticisms, and motivated the search of alternative methods to derive tighter bounds.
The PAC-Bayes analysis focuses on distributions over hypotheses and randomised predictors defined by such distributions. Other prediction schemes can be devised based on a distribution over hypotheses, however, the randomised predictor is a typical starting point. Lifting the analysis to distributions over hypotheses, rather than individual hypotheses, makes available sharp analysis tools, which arguably account for the tightness of PAC-Bayes bounds. Two main uses of PAC-Bayes bounds are (1) risk certification, and (2) cost function derivation. The first consists of evaluating numerical risk certificates for the distributions over hypotheses learned by some method, while the second consists of turning a PAC-Bayes bound into a training objective, to learn a distribution by minimising the bound. This thesis revisits both kinds of uses of PAC-Bayes bounds. We contribute results on certifying the risk of randomised kernel and neural network classifiers, adding evidence to the success of PAC-Bayes bounds at delivering tight certificates. This thesis proposes the name “PAC-Bayesian Computation” as a generic name to encompass the class of methods that learn a distribution over hypotheses by minimising a PAC-Bayes bound (i.e. the second use case described above: cost function derivation), and reports an interesting case of PAC-Bayesian Computation leading to self-certified learning: we develop a learning and certification strategy that uses all the available data to produce a predictor together with a tight risk certificate, as demonstrated with randomised neural network classifiers on two benchmark data sets (MNIST, CIFAR-10)
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