In this paper, we propose the first computationally efficient projection-free algorithm for the bandit convex optimization (BCO). We show that our algorithm achieves a sublinear regret of $ O(nT^{4/5}) $ (where $ T $ is the horizon and $ n $ is the dimension) for any bounded convex functions with uniformly bounded gradients. We also evaluate the performance of our algorithm against prior art on both synthetic and real data sets for portfolio selection and multiclass classification problems.

We address the problem of regret minimization in logistic contextual bandits, where a learner decides among sequential actions or arms given their respective contexts to maximize binary rewards. Using a fast inference procedure with Polya-Gamma distributed augmentation variables, we propose an improved version of Thompson Sampling, a Bayesian formulation of contextual bandits with near-optimal performance. Our approach, Polya-Gamma augmented Thompson Sampling (PG-TS), achieves state-of-the-art performance on simulated and real data. PG-TS explores the action space efficiently and exploits high-reward arms, quickly converging to solutions of low regret. Its explicit estimation of the posterior distribution of the context feature covariance leads to substantial empirical gains over approximate approaches. PG-TS is the first approach to demonstrate the benefits of Polya-Gamma augmentation in bandits and to propose an efficient Gibbs sampler for approximating the analytically unsolvable integral of logistic contextual bandits.

The Kullback-Leilber divergence from model to data is a classic goodness of fit measure but can be intractable in many cases. In this paper, we estimate the ratio function between a data density and a model density with the help of Stein operator. The estimated density ratio allows us to compute the likelihood ratio function which is a surrogate to the actual Kullback-Leibler divergence from model to data. By minimizing this surrogate, we can perform model fitting and inference from either frequentist or Bayesian point of view. This paper discusses methods, theories and algorithms for performing such tasks. Our theoretical claims are verified by experiments and examples are given demonstrating the usefulness of our methods.

Sequential learning of multiple tasks in artificial neural networks using gradient descent leads to catastrophic forgetting, whereby previously learned knowledge is erased during learning of new, disjoint knowledge. Here, we propose a new approach to sequential learning which leverages the recent discovery of adversarial examples. We use adversarial subspaces from previous tasks to enable learning of new tasks with less interference. We apply our method to sequentially learning to classify digits 0, 1, 2 (task 1), 4, 5, 6, (task 2), and 7, 8, 9 (task 3) in MNIST (disjoint MNIST task). We compare and combine our Adversarial Direction (AD) method with the recently proposed Elastic Weight Consolidation (EWC) method for sequential learning. We train each task for 20 epochs, which yields good initial performance (99.24% correct task 1 performance). After training task 2, and then task 3, both plain gradient descent (PGD) and EWC largely forget task 1 (task 1 accuracy 32.95% for PGD and 41.02% for EWC), while our combined approach (AD EWC) still achieves 94.53% correct on task 1. We obtain similar results with a much more difficult disjoint CIFAR10 task, which to our knowledge had not been attempted before (70.10% initial task 1 performance, 67.73% after learning tasks 2 and 3 for AD EWC, while PGD and EWC both fall to chance level). Our results suggest that AD EWC can provide better sequential learning performance than either PGD or EWC.

Numerous methods have been proposed in the statistics and machine learning literature to sum up information and represent data by condensed and simpler to understand quantities. Among those methods, Principal Component Analysis (PCA) aims at identifying the maximal variance axes of data. This serves as a way to represent data in a more compact fashion and hopefully reveal as well as possible their variability. PCA has been introduced by Pearson (1901) and Spearman (1904) and further developed by Hotelling (1933). This is one of the most widely used procedures in multivariate exploratory analysis targeting dimension reduction or features extraction. Nonetheless, PCA is a linear procedure and the need for more sophisticated nonlinear techniques has led to the notion of principal curve. Principal curves may be seen as a nonlinear generalization of the first principal component. The goal is to obtain a curve which passes "in the middle" of data, as illustrated by Figure 1. This notion has been at the heart of numerous applications in many different domains, such as physics (Brunsdon, 2007; Friedsam and Oren, 1989), character and speech recognition (Kégl and Krzyżak, 2002; Reinhard and Niranjan, 1999), mapping and geology (Banfield and Raftery, 1992; Brunsdon, 2007; Stanford and Raftery, 2000), to name but a few.

Sparsification is becoming more and more relevant with the proliferation of huge data sets. Coresets are a principled way to construct representative weighted subsets of a data set that have matching performance with the full data set for specific problems. However, coreset language neglects the nature of the underlying data distribution, which is often continuous. In this paper, we address this oversight by introducing a notion of measure coresets that generalizes coreset language to arbitrary probability measures. Our definition reveals a surprising connection to optimal transport theory which we leverage to design a coreset for problems with Lipschitz costs. We validate our construction on support vector machine (SVM) training, k-means clustering, k-median clustering, and linear regression and show that we are competitive with previous coreset constructions.

Electroencephalographic (EEG) monitoring of neural activity is widely used for sleep disorder diagnostics and research. The standard of care is to manually classify 30-second epochs of EEG time-domain traces into 5 discrete sleep stages. Unfortunately, this scoring process is subjective and time-consuming, and the defined stages do not capture the heterogeneous landscape of healthy and clinical neural dynamics. This motivates the search for a data-driven and principled way to identify the number and composition of salient, reoccurring brain states present during sleep. To this end, we propose a Hierarchical Dirichlet Process Hidden Markov Model (HDP-HMM), combined with wide-sense stationary (WSS) time series spectral estimation to construct a generative model for personalized subject sleep states. In addition, we employ multitaper spectral estimation to further reduce the large variance of the spectral estimates inherent to finite-length EEG measurements. By applying our method to both simulated and human sleep data, we arrive at three main results: 1) a Bayesian nonparametric automated algorithm that recovers general temporal dynamics of sleep, 2) identification of subject-specific "microstates" within canonical sleep stages, and 3) discovery of stage-dependent sub-oscillations with shared spectral signatures across subjects.

Capsule Networks have shown encouraging results on \textit{defacto} benchmark computer vision datasets such as MNIST, CIFAR and smallNORB. Although, they are yet to be tested on tasks where (1) the entities detected inherently have more complex internal representations and (2) there are very few instances per class to learn from and (3) where point-wise classification is not suitable. Hence, this paper carries out experiments on face verification in both controlled and uncontrolled settings that together address these points. In doing so we introduce \textit{Siamese Capsule Networks}, a new variant that can be used for pairwise learning tasks. The model is trained using contrastive loss with $\ell_2$-normalized capsule encoded pose features. We find that \textit{Siamese Capsule Networks} perform well against strong baselines on both pairwise learning datasets, yielding best results in the few-shot learning setting where image pairs in the test set contain unseen subjects.

We present Sequential Neural Likelihood (SNL), a new method for Bayesian inference in simulator models, where the likelihood is intractable but simulating data from the model is possible. SNL trains an autoregressive flow on simulated data in order to learn a model of the likelihood in the region of high posterior density. A sequential training procedure guides simulations and reduces simulation cost by orders of magnitude. We show that SNL is more robust, more accurate and requires less tuning than related state-of-the-art methods which target the posterior, and discuss diagnostics for assessing calibration, convergence and goodness-of-fit.

Cross-validation is the workhorse of modern applied statistics and machine learning, as it provides a principled framework for selecting the model that maximizes generalization performance. In this paper, we show that the cross-validation risk is differentiable with respect to the hyperparameters and training data for many common machine learning algorithms, including logistic regression, elastic-net regression, and support vector machines. Leveraging this property of differentiability, we propose a cross-validation gradient method (CVGM) for hyperparameter optimization. Our method enables efficient optimization in high-dimensional hyperparameter spaces of the cross-validation risk, the best surrogate of the true generalization ability of our learning algorithm.