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In spite of the large literature on reinforcement learning (RL) algorithms for partially observable Markov decision processes (POMDPs), a complete theoretical understanding is still lacking. In a partially observable setting, the history of data available to the agent increases over time so most practical algorithms either truncate the history to a finite window or compress it using a recurrent neural network leading to an agent state that is non-Markovian. In this paper, it is shown that in spite of the lack of the Markov property, recurrent Q-learning (RQL) converges in the tabular setting. Moreover, it is shown that the quality of the converged limit depends on the quality of the representation which is quantified in terms of what is known as an approximate information state (AIS). Based on this characterization of the approximation error, a variant of RQL with AIS losses is presented. This variant performs better than a strong baseline for RQL that does not use AIS losses. It is demonstrated that there is a strong correlation between the performance of RQL over time and the loss associated with the AIS representation.

Parameter inference for ordinary differential equations (ODEs) is of fundamental importance in many scientific applications. While ODE solutions are typically approximated by deterministic algorithms, new research on probabilistic solvers indicates that they produce more reliable parameter estimates by better accounting for numerical errors. However, many ODE systems are highly sensitive to their parameter values. This produces deep local minima in the likelihood function -- a problem which existing probabilistic solvers have yet to resolve. Here, we show that a Bayesian filtering paradigm for probabilistic ODE solution can dramatically reduce sensitivity to parameters by learning from the noisy ODE observations in a data-adaptive manner. Our method is applicable to ODEs with partially unobserved components and with arbitrary non-Gaussian noise. Several examples demonstrate that it is more accurate than existing probabilistic ODE solvers, and even in some cases than the exact ODE likelihood.

We present an illustrative study in which we use a mixture of regressions model to improve on an ill-fitting simple linear regression model relating log brain mass to log body mass for 100 placental mammalian species. The slope of the model is of particular scientific interest because it corresponds to a constant that governs a hypothesized allometric power law relating brain mass to body mass. We model these data using an anchored Bayesian mixture of regressions model, which modifies the standard Bayesian Gaussian mixture by pre-assigning small subsets of observations to given mixture components with probability one. These observations (called anchor points) break the relabeling invariance (or label-switching) typical of exchangeable models. In the article, we develop a strategy for selecting anchor points using tools from case influence diagnostics. We compare the performance of three anchoring methodson the allometric data and in simulated settings.

Since the rise of fair machine learning as a critical field of inquiry, many different notions on how to quantify and measure discrimination have been proposed in the literature. Some of these notions, however, were shown to be mutually incompatible. Such findings make it appear that numerous different kinds of fairness exist, thereby making a consensus on the appropriate measure of fairness harder to reach, hindering the applications of these tools in practice. In this paper, we investigate one of these key impossibility results that relates the notions of statistical and predictive parity. Specifically, we derive a new causal decomposition formula for the fairness measures associated with predictive parity, and obtain a novel insight into how this criterion is related to statistical parity through the legal doctrines of disparate treatment, disparate impact, and the notion of business necessity. Our results show that through a more careful causal analysis, the notions of statistical and predictive parity are not really mutually exclusive, but complementary and spanning a spectrum of fairness notions through the concept of business necessity. Finally, we demonstrate the importance of our findings on a real-world example.

Demand for reliable statistics at a local area (small area) level has greatly increased in recent years. Traditional area-specific estimators based on probability samples are not adequate because of small sample size or even zero sample size in a local area. As a result, methods based on models linking the areas are widely used. World Bank focused on estimating poverty measures, in particular poverty incidence and poverty gap called FGT measures, using a simulated census method, called ELL, based on a one-fold nested error model for a suitable transformation of the welfare variable. Modified ELL methods leading to significant gain in efficiency over ELL also have been proposed under the one-fold model. An advantage of ELL and modified ELL methods is that distributional assumptions on the random effects in the model are not needed. In this paper, we extend ELL and modified ELL to two-fold nested error models to estimate poverty indicators for areas (say a state) and subareas (say counties within a state). Our simulation results indicate that the modified ELL estimators lead to large efficiency gains over ELL at the area level and subarea level. Further, modified ELL method retaining both area and subarea estimated effects in the model (called MELL2) performs significantly better in terms of mean squared error (MSE) for sampled subareas than the modified ELL retaining only estimated area effect in the model (called MELL1).

Spatial point patterns are a commonly recorded form of data in ecology, medicine, astronomy, criminology, epidemiology and many other application fields. One way to understand their second order dependence structure is via their spectral density function. However, unlike time series analysis, for point processes such approaches are currently underutilized. In part, this is because the interpretation of the spectral representation of point patterns is challenging. In this paper, we demonstrate how to band-pass filter point patterns, thus enabling us to explore the spectral representation of point patterns in space by isolating the signal corresponding to certain sets of wavenumbers.

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincar\'e ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincar\'e ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.

The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.

Deep Learning algorithms have achieved the state-of-the-art performance for Image Classification and have been used even in security-critical applications, such as biometric recognition systems and self-driving cars. However, recent works have shown those algorithms, which can even surpass the human capabilities, are vulnerable to adversarial examples. In Computer Vision, adversarial examples are images containing subtle perturbations generated by malicious optimization algorithms in order to fool classifiers. As an attempt to mitigate these vulnerabilities, numerous countermeasures have been constantly proposed in literature. Nevertheless, devising an efficient defense mechanism has proven to be a difficult task, since many approaches have already shown to be ineffective to adaptive attackers. Thus, this self-containing paper aims to provide all readerships with a review of the latest research progress on Adversarial Machine Learning in Image Classification, however with a defender's perspective. Here, novel taxonomies for categorizing adversarial attacks and defenses are introduced and discussions about the existence of adversarial examples are provided. Further, in contrast to exisiting surveys, it is also given relevant guidance that should be taken into consideration by researchers when devising and evaluating defenses. Finally, based on the reviewed literature, it is discussed some promising paths for future research.

This paper addresses the difficulty of forecasting multiple financial time series (TS) conjointly using deep neural networks (DNN). We investigate whether DNN-based models could forecast these TS more efficiently by learning their representation directly. To this end, we make use of the dynamic factor graph (DFG) from that we enhance by proposing a novel variable-length attention-based mechanism to render it memory-augmented. Using this mechanism, we propose an unsupervised DNN architecture for multivariate TS forecasting that allows to learn and take advantage of the relationships between these TS. We test our model on two datasets covering 19 years of investment funds activities. Our experimental results show that our proposed approach outperforms significantly typical DNN-based and statistical models at forecasting their 21-day price trajectory.