This paper introduces a variational approximation framework using direct optimization of what is known as the {\it scale invariant Alpha-Beta divergence} (sAB divergence). This new objective encompasses most variational objectives that use the Kullback-Leibler, the R{\'e}nyi or the gamma divergences. It also gives access to objective functions never exploited before in the context of variational inference. This is achieved via two easy to interpret control parameters, which allow for a smooth interpolation over the divergence space while trading-off properties such as mass-covering of a target distribution and robustness to outliers in the data. Furthermore, the sAB variational objective can be optimized directly by repurposing existing methods for Monte Carlo computation of complex variational objectives, leading to estimates of the divergence instead of variational lower bounds. We show the advantages of this objective on Bayesian models for regression problems.

The non-stationary nature of electroencephalography (EEG) signals makes an EEG-based brain-computer interface (BCI) a dynamic system, thus improving its performance is a challenging task. In addition, it is well-known that due to non-stationarity based covariate shifts, the input data distributions of EEG-based BCI systems change during inter- and intra-session transitions, which poses great difficulty for developments of online adaptive data-driven systems. Ensemble learning approaches have been used previously to tackle this challenge. However, passive scheme based implementation leads to poor efficiency while increasing high computational cost. This paper presents a novel integration of covariate shift estimation and unsupervised adaptive ensemble learning (CSE-UAEL) to tackle non-stationarity in motor-imagery (MI) related EEG classification. The proposed method first employs an exponentially weighted moving average model to detect the covariate shifts in the common spatial pattern features extracted from MI related brain responses. Then, a classifier ensemble was created and updated over time to account for changes in streaming input data distribution wherein new classifiers are added to the ensemble in accordance with estimated shifts. Furthermore, using two publicly available BCI-related EEG datasets, the proposed method was extensively compared with the state-of-the-art single-classifier based passive scheme, single-classifier based active scheme and ensemble based passive schemes. The experimental results show that the proposed active scheme based ensemble learning algorithm significantly enhances the BCI performance in MI classifications.

In recent years, many variance reduced algorithms for empirical risk minimization have been introduced. In contrast to vanilla SGD, these methods converge linearly on strong convex problems. To obtain the variance reduction, current methods either require frequent passes over the full data to recompute gradients---without making any progress during this time (like in SVRG), or they require memory of the same size as the input problem (like SAGA). In this work, we propose k-SVRG, an algorithm that interpolates between those two extremes: it makes best use of the available memory and in turn does avoid full passes over the data without making progress. We prove linear convergence of k-SVRG on strongly convex problems and convergence to stationary points on non-convex problems. Numerical experiments show the effectiveness of our method.

In reinforcement learning, temporal difference (TD) is the most direct algorithm to learn the value function of a policy. For large or infinite state spaces, exact representations of the value function are usually not available, and it must be approximated by a function in some parametric family. However, with \emph{nonlinear} parametric approximations (such as neural networks), TD is not guaranteed to converge to a good approximation of the true value function within the family, and is known to diverge even in relatively simple cases. TD lacks an interpretation as a stochastic gradient descent of an error between the true and approximate value functions, which would provide such guarantees. We prove that approximate TD is a gradient descent provided the current policy is \emph{reversible}. This holds even with nonlinear approximations. A policy with transition probabilities $P(s,s')$ between states is reversible if there exists a function $\mu$ over states such that $\frac{P(s,s')}{P(s',s)}=\frac{\mu(s')}{\mu(s)}$. In particular, every move can be undone with some probability. This condition is restrictive; it is satisfied, for instance, for a navigation problem in any unoriented graph. In this case, approximate TD is exactly a gradient descent of the \emph{Dirichlet norm}, the norm of the difference of \emph{gradients} between the true and approximate value functions. The Dirichlet norm also controls the bias of approximate policy gradient. These results hold even with no decay factor ($\gamma=1$) and do not rely on contractivity of the Bellman operator, thus proving stability of TD even with $\gamma=1$ for reversible policies.

This study presents an extension of the Gaussian process regression model for multiple-input multiple-output forecasting. This approach allows modelling the cross-dependencies between a given set of input variables and generating a vectorial prediction. Making use of the existing correlations in international tourism demand to all seventeen regions of Spain, the performance of the proposed model is assessed in a multiple-step-ahead forecasting comparison. The results of the experiment in a multivariate setting show that the Gaussian process regression model significantly improves the forecasting accuracy of a multi-layer perceptron neural network used as a benchmark. The results reveal that incorporating the connections between different markets in the modelling process may prove very useful to refine predictions at a regional level.

Clustering is ubiquitous in data analysis, including analysis of time series. It is inherently subjective: different users may prefer different clusterings for a particular dataset. Semi-supervised clustering addresses this by allowing the user to provide examples of instances that should (not) be in the same cluster. This paper studies semi-supervised clustering in the context of time series. We show that COBRAS, a state-of-the-art semi-supervised clustering method, can be adapted to this setting. We refer to this approach as COBRAS-TS. An extensive experimental evaluation supports the following claims: (1) COBRAS-TS far outperforms the current state of the art in semi-supervised clustering for time series, and thus presents a new baseline for the field; (2) COBRAS-TS can identify clusters with separated components; (3) COBRAS-TS can identify clusters that are characterized by small local patterns; (4) a small amount of semi-supervision can greatly improve clustering quality for time series; (5) the choice of the clustering algorithm matters (contrary to earlier claims in the literature).

In this paper we show that the computational complexity of the Iterative Thresholding and K-Residual-Means (ITKrM) algorithm for dictionary learning can be significantly reduced by using dimensionality reduction techniques based on the Johnson-Lindenstrauss Lemma. We introduce the Iterative Compressed-Thresholding and K-Means (IcTKM) algorithm for fast dictionary learning and study its convergence properties. We show that IcTKM can locally recover a generating dictionary with low computational complexity up to a target error $\tilde{\varepsilon}$ by compressing $d$-dimensional training data into $m < d$ dimensions, where $m$ is proportional to $\log d$ and inversely proportional to the distortion level $\delta$ incurred by compressing the data. Increasing the distortion level $\delta$ reduces the computational complexity of IcTKM at the cost of an increased recovery error and reduced admissible sparsity level for the training data. For generating dictionaries comprised of $K$ atoms, we show that IcTKM can stably recover the dictionary with distortion levels up to the order $\delta \leq O(1/\sqrt{\log K})$. The compression effectively shatters the data dimension bottleneck in the computational cost of the ITKrM algorithm. For training data with sparsity levels $S \leq O(K^{2/3})$, ITKrM can locally recover the dictionary with a computational cost that scales as $O(d K \log(\tilde{\varepsilon}^{-1}))$ per training signal. We show that for these same sparsity levels the computational cost can be brought down to $O(\log^5 (d) K \log(\tilde{\varepsilon}^{-1}))$ with IcTKM, a significant reduction when high-dimensional data is considered. Our theoretical results are complemented with numerical simulations which demonstrate that IcTKM is a powerful, low-cost algorithm for learning dictionaries from high-dimensional data sets.

k-Nearest Neighbors is one of the most fundamental but effective classification models. In this paper, we propose two families of models built on a sequence to sequence model and a memory network model to mimic the k-Nearest Neighbors model, which generate a sequence of labels, a sequence of out-of-sample feature vectors and a final label for classification, and thus they could also function as oversamplers. We also propose 'out-of-core' versions of our models which assume that only a small portion of data can be loaded into memory. Computational experiments show that our models outperform k-Nearest Neighbors, a feed-forward neural network and a memory network, due to the fact that our models must produce additional output and not just the label. As an oversample on imbalanced datasets, the sequence to sequence kNN model often outperforms Synthetic Minority Over-sampling Technique and Adaptive Synthetic Sampling.

Apache Spark has become one of the most commonly used and supported open-source tools for machine learning and data science. In this post, I'll help you get started using Apache Spark's spark.ml Our data is from the Kaggle competition: Housing Values in Suburbs of Boston. AGE -- proportion of owner-occupied units built prior to 1940. BLACK -- 1000(Bk -- 0.63)² where Bk is the proportion of blacks by town. This is the target variable.

Sometimes we may have prior knowledge that we want to group the data into a given number of clusters. Other times we may wish to investigate what may be a good number of clusters. In the example below we look at changing the number of clusters between 1 and 100 and measure the average distance points are from their closest cluster centre (kmeans.transform Looking at the results we may decide that up to about 10 clusters may be useful, but after that there are diminishing returns of adding further clusters.