Precision-Recall analysis abounds in applications of binary classification where true negatives do not add value and hence should not affect assessment of the classifier's performance. Perhaps inspired by the many advantages of receiver operating characteristic (ROC) curves and the area under such curves for accuracy-based performance assessment, many researchers have taken to report Precision-Recall (PR) curves and associated areas as performance metric. We demonstrate in this paper that this practice is fraught with difficulties, mainly because of incoherent scale assumptions -- e.g., the area under a PR curve takes the arithmetic mean of precision values whereas the $F_{\beta}$ score applies the harmonic mean. We show how to fix this by plotting PR curves in a different coordinate system, and demonstrate that the new Precision-Recall-Gain curves inherit all key advantages of ROC curves. In particular, the area under Precision-Recall-Gain curves conveys an expected $F_1$ score on a harmonic scale, and the convex hull of a Precision-Recall-Gain curve allows us to calibrate the classifier's scores so as to determine, for each operating point on the convex hull, the interval of $\beta$ values for which the point optimises $F_{\beta}$. We demonstrate experimentally that the area under traditional PR curves can easily favour models with lower expected $F_1$ score than others, and so the use of Precision-Recall-Gain curves will result in better model selection.

We propose an exploratory approach to statistical model criticism using maximum mean discrepancy (MMD) two sample tests. Typical approaches to model criticism require a practitioner to select a statistic by which to measure discrepancies between data and a statistical model. MMD two sample tests are instead constructed as an analytic maximisation over a large space of possible statistics and therefore automatically select the statistic which most shows any discrepancy. We demonstrate on synthetic data that the selected statistic, called the witness function, can be used to identify where a statistical model most misrepresents the data it was trained on. We then apply the procedure to real data where the models being assessed are restricted Boltzmann machines, deep belief networks and Gaussian process regression and demonstrate the ways in which these models fail to capture the properties of the data they are trained on.

In this paper, we propose a novel parameter estimator for probabilistic models on discrete space. The proposed estimator is derived from minimization of homogeneous divergenceand can be constructed without calculation of the normalization constant, which is frequently infeasible for models in the discrete space. We investigate statisticalproperties of the proposed estimator such as consistency and asymptotic normality, and reveal a relationship with the information geometry. Some experiments show that the proposed estimator attains comparable performance tothe maximum likelihood estimator with drastically lower computational cost.

The goal of precipitation nowcasting is to predict the future rainfall intensity in a local region over a relatively short period of time. Very few previous studies have examined this crucial and challenging weather forecasting problem from the machine learning perspective. In this paper, we formulate precipitation nowcasting as a spatiotemporal sequence forecasting problem in which both the input and the prediction target are spatiotemporal sequences. By extending the fully connected LSTM (FC-LSTM) to have convolutional structures in both the input-to-state and state-to-state transitions, we propose the convolutional LSTM (ConvLSTM) and use it to build an end-to-end trainable model for the precipitation nowcasting problem. Experiments show that our ConvLSTM network captures spatiotemporal correlations better and consistently outperforms FC-LSTM and the state-of-the-art operational ROVER algorithm for precipitation nowcasting.

The framework of online learning with memory naturally captures learning problems with temporal effects, and was previously studied for the experts setting. In this work we extend the notion of learning with memory to the general Online Convex Optimization (OCO) framework, and present two algorithms that attain low regret. The first algorithm applies to Lipschitz continuous loss functions, obtaining optimal regret bounds for both convex and strongly convex losses. The second algorithm attains the optimal regret bounds and applies more broadly to convex losses without requiring Lipschitz continuity, yet is more complicated to implement. We complement the theoretic results with two applications: statistical arbitrage in finance, and multi-step ahead prediction in statistics.

One approach to improving the running time of kernel-based methods is to build a small sketch of the kernel matrix and use it in lieu of the full matrix in the machine learning task of interest. Here, we describe a version of this approach that comes with running time guarantees as well as improved guarantees on its statistical performance.By extending the notion of \emph{statistical leverage scores} to the setting of kernel ridge regression, we are able to identify a sampling distribution that reduces the size of the sketch (i.e., the required number of columns to be sampled) to the \emph{effective dimensionality} of the problem. This latter quantity is often much smaller than previous bounds that depend on the \emph{maximal degrees of freedom}. We give an empirical evidence supporting this fact. Our second contribution is to present a fast algorithm to quickly compute coarse approximations to thesescores in time linear in the number of samples. More precisely, the running time of the algorithm is $O(np^2)$ with $p$ only depending on the trace of the kernel matrix and the regularization parameter. This is obtained via a variant of squared length sampling that we adapt to the kernel setting. Lastly, we discuss how this new notion of the leverage of a data point captures a fine notion of the difficulty of the learning problem.

The objective in extreme multi-label learning is to train a classifier that can automatically taga novel data point with the most relevant subset of labels from an extremely large label set. Embedding based approaches attempt to make training and prediction tractable by assuming that the training label matrix is low-rank and reducing the effective number of labels by projecting the high dimensional label vectors onto a low dimensional linear subspace. Still, leading embedding approaches havebeen unable to deliver high prediction accuracies, or scale to large problems as the low rank assumption is violated in most real world applications. In this paper we develop the SLEEC classifier to address both limitations. The main technical contribution in SLEEC is a formulation for learning a small ensemble oflocal distance preserving embeddings which can accurately predict infrequently occurring(tail) labels. This allows SLEEC to break free of the traditional low-rank assumption and boost classification accuracy by learning embeddings which preserve pairwise distances between only the nearest label vectors. We conducted extensive experiments on several real-world, as well as benchmark datasets and compared our method against state-of-the-art methods for extreme multi-labelclassification. Experiments reveal that SLEEC can make significantly moreaccurate predictions then the state-of-the-art methods including both embedding-based (by as much as 35%) as well as tree-based (by as much as 6%) methods. SLEEC can also scale efficiently to data sets with a million labels which are beyond the pale of leading embedding methods.

We study the problem of Robust Least Squares Regression (RLSR) where several response variables can be adversarially corrupted. More specifically, for a data matrix X \in \R^{p x n} and an underlying model w*, the response vector is generated as y = X'w* + b where b \in n is the corruption vector supported over at most C.n coordinates. Existing exact recovery results for RLSR focus solely on L1-penalty based convex formulations and impose relatively strict model assumptions such as requiring the corruptions b to be selected independently of X.In this work, we study a simple hard-thresholding algorithm called TORRENT which, under mild conditions on X, can recover w* exactly even if b corrupts the response variables in an adversarial manner, i.e. both the support and entries of b are selected adversarially after observing X and w*. Our results hold under deterministic assumptions which are satisfied if X is sampled from any sub-Gaussian distribution. Finally unlike existing results that apply only to a fixed w*, generated independently of X, our results are universal and hold for any w* \in \R^p.Next, we propose gradient descent-based extensions of TORRENT that can scale efficiently to large scale problems, such as high dimensional sparse recovery. and prove similar recovery guarantees for these extensions. Empirically we find TORRENT, and more so its extensions, offering significantly faster recovery than the state-of-the-art L1 solvers. For instance, even on moderate-sized datasets (with p = 50K) with around 40% corrupted responses, a variant of our proposed method called TORRENT-HYB is more than 20x faster than the best L1 solver.

To capture the interdependencies between labels in multi-label classification problems, classifier chain (CC) tries to take the multiple labels of each instance into account under a deterministic high-order Markov Chain model. Since its performance is sensitive to the choice of label order, the key issue is how to determine the optimal label order for CC. In this work, we first generalize the CC model over a random label order. Then, we present a theoretical analysis of the generalization error for the proposed generalized model. Based on our results, we propose a dynamic programming based classifier chain (CC-DP) algorithm to search the globally optimal label order for CC and a greedy classifier chain (CC-Greedy) algorithm to find a locally optimal CC. Comprehensive experiments on a number of real-world multi-label data sets from various domains demonstrate that our proposed CC-DP algorithm outperforms state-of-the-art approaches and the CC-Greedy algorithm achieves comparable prediction performance with CC-DP.

We study the problem of black-box optimization of a function $f$ of any dimension, given function evaluations perturbed by noise. The function is assumed to be locally smooth around one of its global optima, but this smoothness is unknown. Our contribution is an adaptive optimization algorithm, POO or parallel optimistic optimization, that is able to deal with this setting. POO performs almost as well as the best known algorithms requiring the knowledge of the smoothness. Furthermore, POO works for a larger class of functions than what was previously considered, especially for functions that are difficult to optimize, in a very precise sense. We provide a finite-time analysis of POO's performance, which shows that its error after $n$ evaluations is at most a factor of $\sqrt{\ln n}$ away from the error of the best known optimization algorithms using the knowledge of the smoothness.