Normalizing flows are popular generative learning methods that train an invertible function to transform a simple prior distribution into a complicated target distribution. Here we generalize the framework by introducing Stochastic Normalizing Flows (SNF) - an arbitrary sequence of deterministic invertible functions and stochastic processes such as Markov Chain Monte Carlo (MCMC) or Langevin Dynamics. This combination can be powerful as adding stochasticity to a flow helps overcoming expressiveness limitations of a chosen deterministic invertible function, while the trainable flow transformations can improve the sampling efficiency over pure MCMC. Key to our approach is that we can match a marginal target density without having to marginalize out the stochasticity of traversed paths. Invoking ideas from nonequilibrium statistical mechanics, we introduce a training method that only uses conditional path probabilities. We can turn an SNF into a Boltzmann Generator that samples asymptotically unbiased from a given target density by importance sampling of these paths. We illustrate the representational power, sampling efficiency and asymptotic correctness of SNFs on several benchmarks.

ABSTRACT We consider statistical learning problems, when the distribution P ′ of the training observations Z ′ 1,..., Z′ n differs from the distribution P involved in the risk one seeks to minimize (referred to as the test distribution) but is still defined on the same measurable space as P and dominates it. In the unrealistic case where the likelihood ratio Φ(z) dP/dP ′ (z) is known, one may straightforwardly extends the Empirical Risk Minimization (ERM) approach to this specific transfer learning setup using the same idea as that behind Importance Sampling, by minimizing a weighted version of the empirical risk functional computed from the'biased' training data Zi ′ with weights Φ(Zi ′). Although the importance function Φ(z) is generally unknown in practice, we show that, in various situations frequently encountered in practice, it takes a simple form and can be directly estimated from the Zi ′ 's and some auxiliary information on the statistical population P. By means of linearization techniques, we then prove that the generalization capacity of the approach aforementioned is preserved when plugging the resulting estimates of the Φ(Zi ′)'s into the weighted empirical risk. Beyond these theoretical guarantees, numerical results provide strong empirical evidence of the relevance of the approach promoted in this article. Keywords: Statistical Learning Theory, Importance Sampling, Transfer Learning. 1 Introduction Prediction problems are of major importance in statistical learning. The main paradigm of predictive learning is Empirical Risk Minimization (ERM in abbreviated form), see e.g. In the standard setup, Z is a random variable (r.v. in short) that takes its values in a feature space Z with distribution P, Θ is a parameter space and l: Θ Z R is a (measurable) loss function. The risk is then defined by: θ Θ, R P (θ) E P [l(θ, Z)], (1) and more generally for any measure Q on Z: R Q (θ) l(θ, z)dQ(z).

We address the problem of the achievable regret rates with online logistic regression. We derive lower bounds with logarithmic regret under $L_1$, $L_2$, and $L_\infty$ constraints on the parameter values. The bounds are dominated by $d/2 \log T$, where $T$ is the horizon and $d$ is the dimensionality of the parameter space. We show their achievability for $d=o(T^{1/3})$ in all these cases with Bayesian methods, that achieve them up to a $d/2 \log d$ term. Interesting different behaviors are shown for larger dimensionality. Specifically, on the negative side, if $d = \Omega(\sqrt{T})$, any algorithm is guaranteed regret of $\Omega(d \log T)$ (greater than $\Omega(\sqrt{T})$) under $L_\infty$ constraints on the parameters (and the example features). On the positive side, under $L_1$ constraints on the parameters, there exist algorithms that can achieve regret that is sub-linear in $d$ for the asymptotically larger values of $d$. For $L_2$ constraints, it is shown that for large enough $d$, the regret remains linear in $d$ but no longer logarithmic in $T$. Adapting the redundancy-capacity theorem from information theory, we demonstrate a principled methodology based on grids of parameters to derive lower bounds. Grids are also utilized to derive some upper bounds. Our results strengthen results by Kakade and Ng (2005) and Foster et al. (2018) for upper bounds for this problem, introduce novel lower bounds, and adapt a methodology that can be used to obtain such bounds for other related problems. They also give a novel characterization of the asymptotic behavior when the dimension of the parameter space is allowed to grow with $T$. They additionally establish connections to the information theory literature, demonstrating that the actual regret for logistic regression depends on the richness of the parameter class, where even within this problem, richer classes lead to greater regret.

Networks are a fundamental and flexible way of representing various complex systems. Many domains such as communication, citation, procurement, biology, social media, and transportation can be modeled as a set of entities and their relationships. Temporal networks are a specialization of general networks where the temporal evolution of the system is as important to understand as the structure of the entities and relationships. We present the Independent Temporal Motif (ITeM) to characterize temporal graphs from different domains. The ITeMs are edge-disjoint temporal motifs that can be used to model the structure and the evolution of the graph. For a given temporal graph, we produce a feature vector of ITeM frequencies and apply this distribution to the task of measuring the similarity of temporal graphs. We show that ITeM has higher accuracy than other motif frequency-based approaches. We define various metrics based on ITeM that reveal salient properties of a temporal network. We also present importance sampling as a method for efficiently estimating the ITeM counts. We evaluate our approach on both synthetic and real temporal networks.

Variational Autoencoders (VAEs) represent the given data in a low-dimensional latent space, which is generally assumed to be Euclidean. This assumption naturally leads to the common choice of a standard Gaussian prior over continuous latent variables. Recent work has, however, shown that this prior has a detrimental effect on model capacity, leading to subpar performance. We propose that the Euclidean assumption lies at the heart of this failure mode. To counter this, we assume a Riemannian structure over the latent space, which constitutes a more principled geometric view of the latent codes, and replace the standard Gaussian prior with a Riemannian Brownian motion prior. We propose an efficient inference scheme that does not rely on the unknown normalizing factor of this prior. Finally, we demonstrate that this prior significantly increases model capacity using only one additional scalar parameter.

We propose a hierarchical correlation clustering method that extends the well-known correlation clustering to produce hierarchical clusters. We then investigate embedding the respective hierarchy to be used for (tree preserving) embedding and feature extraction. We study the connection of such an embedding to single linkage embedding and minimax distances, and in particular study minimax distances for correlation clustering. Finally, we demonstrate the performance of our methods on several UCI and 20 newsgroup datasets.

We study the parameter estimation problem in mixture models with observational nonidentifiability: the full model (also containing hidden variables) is identifiable, but the marginal (observed) model is not. Hence global maxima of the marginal likelihood are (infinitely) degenerate and predictions of the marginal likelihood are not unique. We show how to generalize the marginal likelihood by introducing an effective temperature, and making it similar to the free energy. This generalization resolves the observational nonidentifiability, since its maximization leads to unique results that are better than a random selection of one degenerate maximum of the marginal likelihood or the averaging over many such maxima. The generalized likelihood inherits many features from the usual likelihood, e.g. it holds the conditionality principle, and its local maximum can be searched for via suitably modified expectation-maximization method. The maximization of the generalized likelihood relates to entropy optimization.

In recent years we see a rapidly growing line of research which shows learnability of various models via common neural network algorithms. Yet, besides a very few outliers, these results show learnability of models that can be learned using linear methods. Namely, such results show that learning neural-networks with gradient-descent is competitive with learning a linear classifier on top of a data-independent representation of the examples. This leaves much to be desired, as neural networks are far more successful than linear methods. Furthermore, on the more conceptual level, linear models don't seem to capture the ``deepness" of deep networks. In this paper we make a step towards showing leanability of models that are inherently non-linear. We show that under certain distributions, sparse parities are learnable via gradient decent on depth-two network. On the other hand, under the same distributions, these parities cannot be learned efficiently by linear methods.

Neural networks utilize the softmax as a building block in classification tasks, which contains an overconfidence problem and lacks an uncertainty representation ability. As a Bayesian alternative to the softmax, we consider a random variable of a categorical probability over class labels. In this framework, the prior distribution explicitly models the presumed noise inherent in the observed label, which provides consistent gains in generalization performance in multiple challenging tasks. The proposed method inherits advantages of Bayesian approaches that achieve better uncertainty estimation and model calibration. Our method can be implemented as a plug-and-play loss function with negligible computational overhead compared to the softmax with the cross-entropy loss function.

There is a growing trend of applying machine learning methods to medical datasets in order to predict patients' future status. Although some of these methods achieve high performance, challenges still exist in comparing and evaluating different models through their interpretable information. Such analytics can help clinicians improve evidence-based medical decision making. In this work, we develop a visual analytics system that compares multiple models' prediction criteria and evaluates their consistency. Using our system, knowledge can be generated on how differently each model made the predictions and how confidently we can rely on each model's prediction for a certain patient. Through a case study of a publicly available clinical dataset, we demonstrate the effectiveness of our visual analytics system to assist clinicians and researchers in comparing and quantitatively evaluating different machine learning methods.