Tuning hyperparameters of learning algorithms is hard because gradients are usually unavailable. We compute exact gradients of cross-validation performance with respect to all hyperparameters by chaining derivatives backwards through the entire training procedure. These gradients allow us to optimize thousands of hyperparameters, including step-size and momentum schedules, weight initialization distributions, richly parameterized regularization schemes, and neural network architectures. We compute hyperparameter gradients by exactly reversing the dynamics of stochastic gradient descent with momentum.

This paper presents the first theoretical results showing that stable identification of overcomplete $\mu$-coherent dictionaries $\Phi \in \mathbb{R}^{d\times K}$ is locally possible from training signals with sparsity levels $S$ up to the order $O(\mu^{-2})$ and signal to noise ratios up to $O(\sqrt{d})$. In particular the dictionary is recoverable as the local maximum of a new maximisation criterion that generalises the K-means criterion. For this maximisation criterion results for asymptotic exact recovery for sparsity levels up to $O(\mu^{-1})$ and stable recovery for sparsity levels up to $O(\mu^{-2})$ as well as signal to noise ratios up to $O(\sqrt{d})$ are provided. These asymptotic results translate to finite sample size recovery results with high probability as long as the sample size $N$ scales as $O(K^3dS \tilde \varepsilon^{-2})$, where the recovery precision $\tilde \varepsilon$ can go down to the asymptotically achievable precision. Further, to actually find the local maxima of the new criterion, a very simple Iterative Thresholding and K (signed) Means algorithm (ITKM), which has complexity $O(dKN)$ in each iteration, is presented and its local efficiency is demonstrated in several experiments.

Folded concave penalization methods have been shown to enjoy the strong oracle property for high-dimensional sparse estimation. However, a folded concave penalization problem usually has multiple local solutions and the oracle property is established only for one of the unknown local solutions. A challenging fundamental issue still remains that it is not clear whether the local optimum computed by a given optimization algorithm possesses those nice theoretical properties. To close this important theoretical gap in over a decade, we provide a unified theory to show explicitly how to obtain the oracle solution via the local linear approximation algorithm. For a folded concave penalized estimation problem, we show that as long as the problem is localizable and the oracle estimator is well behaved, we can obtain the oracle estimator by using the one-step local linear approximation. In addition, once the oracle estimator is obtained, the local linear approximation algorithm converges, namely it produces the same estimator in the next iteration. The general theory is demonstrated by using four classical sparse estimation problems, that is, sparse linear regression, sparse logistic regression, sparse precision matrix estimation and sparse quantile regression.

Deep neural networks (DNNs) have recently been achieving state-of-the-art performance on a variety of pattern-recognition tasks, most notably visual classification problems. Given that DNNs are now able to classify objects in images with near-human-level performance, questions naturally arise as to what differences remain between computer and human vision. A recent study revealed that changing an image (e.g. of a lion) in a way imperceptible to humans can cause a DNN to label the image as something else entirely (e.g. mislabeling a lion a library). Here we show a related result: it is easy to produce images that are completely unrecognizable to humans, but that state-of-the-art DNNs believe to be recognizable objects with 99.99% confidence (e.g. labeling with certainty that white noise static is a lion). Specifically, we take convolutional neural networks trained to perform well on either the ImageNet or MNIST datasets and then find images with evolutionary algorithms or gradient ascent that DNNs label with high confidence as belonging to each dataset class. It is possible to produce images totally unrecognizable to human eyes that DNNs believe with near certainty are familiar objects, which we call "fooling images" (more generally, fooling examples). Our results shed light on interesting differences between human vision and current DNNs, and raise questions about the generality of DNN computer vision.

Truly complex stochastic processes--the infinitary processes [1] whose mutual information between past and future diverges--arise in many physical and biological systems [2-5], such as those in critical states. They are implicated in many natural phenomena, from the geophysics of earthquakes [6] and physiological measurements of neural avalanches [7] to semantics in natural language [8] and cascading failure in power transmission grids [9]. Their apparent infinite memory makes empirical estimation and modeling particularly challenging. The difficulty is reflected in the computational complexity of inference [10]: the resources required to predict and model them diverge in sample size, in memory for storing model parameters, and in memory required for prediction. Resource scaling, an analog of the venerable technique of finite-size scaling in statistical mechanics, suggests that for infinitary processes we look for statistical signatures that track divergences. Since resource divergences are sensitive to a process's inherent randomness and organization, one hopes that their scaling forms are uniquely revealing indicators of process complexity and can guide the selection of appropriate models. To date, though, there are few tractable constructions with which to explore possible general relationships between prediction, complexity, and learning for infinitary processes.

The general goal here is to choose groups (clusters) of objects so as to maximize homogeneity within clusters and minimize homogeneity across clusters. The clustering problem has been addressed by researchers in many disciplines. A few well-known methods are metric based e.g. K-means (MacQueen et al., 1967), hierarchical clustering (Ward, 1963), clustering based on principal components, spectral clustering (Ng et al., 2002) and so on (Jain and Dubes, 1988; Ozawa, 1985). Traditional clustering methods are complemented by methods based on a probability model where one assumes a data generating distribution (e.g., Gaussian) and infers clustering configurations that maximize certain objective function (Banfield and Raftery, 1993; Fraley and Raftery, 1998, 2002, 2006; MacCullagh and Yang, 2008). A modelbased clustering can be useful in addressing challenges posed by traditional clustering methods. This is because a probability model allows the number of clusters to be treated as a parameter in the model, and can be embedded in a Bayesian framework providing quantification of uncertainty in the number of clusters and clustering configurations.

We consider the problem of supervised learning with convex loss functions and propose a new form of iterative regularization based on the subgradient method. Unlike other regularization approaches, in iterative regularization no constraint or penalization is considered, and generalization is achieved by (early) stopping an empirical iteration. We consider a nonparametric setting, in the framework of reproducing kernel Hilbert spaces, and prove finite sample bounds on the excess risk under general regularity conditions. Our study provides a new class of efficient regularized learning algorithms and gives insights on the interplay between statistics and optimization in machine learning.

Recently theoretical guarantees have been obtained for matrix completion in the non-uniform sampling regime. In particular, if the sampling distribution aligns with the underlying matrix's leverage scores, then with high probability nuclear norm minimization will exactly recover the low rank matrix. In this article, we analyze the scenario in which the non-uniform sampling distribution may or may not not align with the underlying matrix's leverage scores. Here we explore learning the parameters for weighted nuclear norm minimization in terms of the empirical sampling distribution. We provide a sufficiency condition for these learned weights which provide an exact recovery guarantee for weighted nuclear norm minimization. It has been established that a specific choice of weights in terms of the true sampling distribution not only allows for weighted nuclear norm minimization to exactly recover the low rank matrix, but also allows for a quantifiable relaxation in the exact recovery conditions. In this article we extend this quantifiable relaxation in exact recovery conditions for a specific choice of weights defined analogously in terms of the empirical distribution as opposed to the true sampling distribution. To accomplish this we employ a concentration of measure bound and a large deviation bound. We also present numerical evidence for the healthy robustness of the weighted nuclear norm minimization algorithm to the choice of empirically learned weights. These numerical experiments show that for a variety of easily computable empirical weights, weighted nuclear norm minimization outperforms unweighted nuclear norm minimization in the non-uniform sampling regime.

The Libra Toolkit is a collection of algorithms for learning and inference with discrete probabilistic models, including Bayesian networks, Markov networks, dependency networks, and sum-product networks. Compared to other toolkits, Libra places a greater emphasis on learning the structure of tractable models in which exact inference is efficient. It also includes a variety of algorithms for learning graphical models in which inference is potentially intractable, and for performing exact and approximate inference. Libra is released under a 2-clause BSD license to encourage broad use in academia and industry.

A discourse typically involves numerous entities, but few are mentioned more than once. Distinguishing those that die out after just one mention (singleton) from those that lead longer lives (coreferent) would dramatically simplify the hypothesis space for coreference resolution models, leading to increased performance. To realize these gains, we build a classifier for predicting the singleton/coreferent distinction. The models feature representations synthesize linguistic insights about the factors affecting discourse entity lifespans (especially negation, modality, and attitude predication) with existing results about the benefits of surface (part-of-speech and n-gram-based) features for coreference resolution. The model is effective in its own right, and the feature representations help to identify the anchor phrases in bridging anaphora as well. Furthermore, incorporating the model into two very different state-of-the-art coreference resolution systems, one rule-based and the other learning-based, yields significant performance improvements.