In this chapter we take a look at the universal approximation question for stochastic feedforward neural networks. In contrast to deterministic networks, which represent mappings from a set of inputs to a set of outputs, stochastic networks represent mappings from a set of inputs to a set of probability distributions over the set of outputs. In particular, even if the sets of inputs and outputs are finite, the class of stochastic mappings in question is not finite. Moreover, while for a deterministic function the values of all output variables can be computed independently of each other given the values of the inputs, in the stochastic setting the values of the output variables may need to be correlated, which requires that their values are computed jointly. A prominent class of stochastic feedforward networks which has played a key role in the resurgence of deep learning are deep belief networks. The representational power of these networks has been studied mainly in the generative setting, as models of probability distributions without an input, or in the discriminative setting for the special case of deterministic mappings. We study the representational power of deep sigmoid belief networks in terms of compositions of linear transformations of probability distributions, Markov kernels, that can be expressed by the layers of the network. We investigate different types of shallow and deep architectures, and the minimal number of layers and units per layer that are sufficient and necessary in order for the network to be able to approximate any given stochastic mapping from the set of inputs to the set of outputs arbitrarily well.

With the increasing availability of AI-based decision support, there is an increasing need for their certification by both AI manufacturers and notified bodies, as well as the pragmatic (real-world) validation of these systems. Therefore, there is the need for meaningful and informative ways to assess the performance of AI systems in clinical practice. Common metrics (like accuracy scores and areas under the ROC curve) have known problems and they do not take into account important information about the preferences of clinicians and the needs of their specialist practice, like the likelihood and impact of errors and the complexity of cases. In this paper, we present a new accuracy measure, the H-accuracy (Ha), which we claim is more informative in the medical domain (and others of similar needs) for the elements it encompasses. We also provide proof that the H-accuracy is a generalization of the balanced accuracy and establish a relation between the H-accuracy and the Net Benefit. Finally, we illustrate an experimentation in two user studies to show the descriptive power of the Ha score and how complementary and differently informative measures can be derived from its formulation (a Python script to compute Ha is also made available).

Despite the significant advances in recent years, Generative Adversarial Networks (GANs) are still notoriously hard to train. In this paper, we propose three novel curriculum learning strategies for training GANs. All strategies are first based on ranking the training images by their difficulty scores, which are estimated by a state-of-the-art image difficulty predictor. Our first strategy is to divide images into gradually more difficult batches. Our second strategy introduces a novel curriculum loss function for the discriminator that takes into account the difficulty scores of the real images. Our third strategy is based on sampling from an evolving distribution, which favors the easier images during the initial training stages and gradually converges to a uniform distribution, in which samples are equally likely, regardless of difficulty. We compare our curriculum learning strategies with the classic training procedure on two tasks: image generation and image translation. Our experiments indicate that all strategies provide faster convergence and superior results. For example, our best curriculum learning strategy applied on spectrally normalized GANs (SNGANs) fooled human annotators in thinking that generated CIFAR-like images are real in 25.0% of the presented cases, while the SNGANs trained using the classic procedure fooled the annotators in only 18.4% cases. Similarly, in image translation, the human annotators preferred the images produced by the Cycle-consistent GAN (CycleGAN) trained using curriculum learning in 40.5% cases and those produced by CycleGAN based on classic training in only 19.8% cases, 39.7% cases being labeled as ties.

The aim of clinical effectiveness research using repositories of electronic health records is to identify what health interventions 'work best' in real-world settings. Since there are several reasons why the net benefit of intervention may differ across patients, current comparative effectiveness literature focuses on investigating heterogeneous treatment effect and predicting whether an individual might benefit from an intervention. The majority of this literature has concentrated on the estimation of the effect of treatment on binary outcomes. However, many medical interventions are evaluated in terms of their effect on future events, which are subject to loss to follow-up. In this study, we describe a framework for the estimation of heterogeneous treatment effect in terms of differences in time-to-event (survival) probabilities. We divide the problem into three phases: (1) estimation of treatment effect conditioned on unique sets of the covariate vector; (2) identification of features important for heterogeneity using an ensemble of non-parametric variable importance methods; and (3) estimation of treatment effect on the reference classes defined by the previously selected features, using one-step Targeted Maximum Likelihood Estimation. We conducted a series of simulation studies and found that this method performs well when either sample size or event rate is high enough and the number of covariates contributing to the effect heterogeneity is moderate. An application of this method to a clinical case study was conducted by estimating the effect of oral anticoagulants on newly diagnosed non-valvular atrial fibrillation patients using data from the UK Clinical Practice Research Datalink.

We introduce the notion of pointwise coverage to measure the explainability properties of machine learning classifiers. An explanation for a prediction is a definably simple region of the feature space sharing the same label as the prediction, and the coverage of an explanation measures its size or generalizability. With this notion of explanation, we investigate whether or not there is a natural characterization of the most explainable classifier. According with our intuitions, we prove that the binary linear classifier is uniquely the most explainable classifier up to negligible sets.

--We present a way to use T opological Data Analysis (TDA) for machine learning tasks on grayscale images. We apply persistent homology to generate a wide range of topological features using a point cloud obtained from an image, its natural grayscale filtration, and different filtrations defined on the binarized image. We show that this topological machine learning pipeline can be used as a highly relevant dimensionality reduction by applying it to the MNIST digits dataset. We conduct a feature selection and study their correlations while providing an intuitive interpretation of their importance, which is relevant in both machine learning and TDA. Finally, we show that we can classify digit images while reducing the size of the feature set by a factor 5 compared to the grayscale pixel value features and maintain similar accuracy. I NTRODUCTION Topological Data Analysis (TDA) [1] applies techniques from algebraic topology to study and extract topological and geometric information on the shape of data. In this paper, we use persistent homology [2], a tool from TDA that extracts features representing the numbers of connected components, cycles, and voids and their birth and death during an iterative process called a filtration. Each of those features is summarized as a point in a persistence diagram .

Federated Averaging (FedAvg) serves as the fundamental framework in Federated Learning (FL) settings. However, we argue that 1) the multiple steps of local updating will result in gradient biases and 2) there is an inconsistency between the target distribution and the optimization objectives following the training paradigm in FedAvg. To tackle these problems, we first propose an unbiased gradient aggregation algorithm with the keep-trace gradient descent and gradient evaluation strategy. Then we introduce a meta updating procedure with a controllable meta training set to provide a clear and consistent optimization objective. Experimental results demonstrate that the proposed methods outperform compared ones with various network architectures in both the IID and non-IID FL settings.

Markov chain Monte Carlo (MCMC) methods and neural networks are instrumental in tackling inferential and prediction problems. However, Bayesian inference based on joint use of MCMC methods and of neural networks is limited. This paper reviews the main challenges posed by neural networks to MCMC developments, including lack of parameter identifiability due to weight symmetries, prior specification effects, and consequently high computational cost and convergence failure. Population and manifold MCMC algorithms are combined to demonstrate these challenges via multilayer perceptron (MLP) examples and to develop case studies for assessing the capacity of approximate inference methods to uncover the posterior covariance of neural network parameters. Some of these challenges, such as high computational cost arising from the application of neural networks to big data and parameter identifiability arising from weight symmetries, stimulate research towards more scalable approximate MCMC methods or towards MCMC methods in reduced parameter spaces.

Graph Convolutional Network (GCN) has attracted intensive interests recently. One major limitation of GCN is that it often cannot benefit from using a deep architecture, while traditional CNN and an alternative Graph Neural Network architecture, namely GraphCNN, often achieve better quality with a deeper neural architecture. How can we explain this phenomenon? In this paper, we take the first step towards answering this question. We first conduct a systematic empirical study on the accuracy of GCN, GraphCNN, and ResNet-18 on 2D images and identified relative importance of different factors in architectural design. This inspired a novel theoretical analysis on the mutual information between the input and the output after l GCN/ GraphCNN layers. We identified regimes in which GCN suffers exponentially fast "information lose" and show that GraphCNN requires a much weaker condition for similar behavior to happen. Extending convolutional neural networks (CNN) over images to a graph has attracted intense interest recently. One early attempt is the GCN model proposed by Kipf & Welling (2016a). However, when applying GCN to many practical applications, one discrepancy lingers -- although traditional CNN usually gets higher accuracy when it goes deeper, GCN, as a natural extension of CNN, does not seem to benefit much from going deeper by stacking multiple layers together. This phenomenon has been the focus of multiple recent papers (Li et al., 2018; 2019; Oono & Suzuki, 2019). On the theoretical side, Li et al. (2018) and Oono & Suzuki (2019) identified the problem as oversmoothing -- under certain conditions, when multiple GCN layers are stacked together, the output will converge to a region that is independent of weights and inputs. On the empirical side, Li et al. (2019) showed that many techniques that were designed to train a deep CNN, e.g., the skip connections in ResNet (He et al., 2016a), can also make it easier for GCN to go deeper.

In this paper, we formalise order-robust optimisation as an instance of online learning minimising simple regret, and propose VROOM, a zero'th order optimisation algorithm capable of achieving vanishing regret in non-stationary environments, while recovering favorable rates under stochastic reward-generating processes. Our results are the first to target simple regret definitions in adversarial scenarios unveiling a challenge that has been rarely considered in prior work.