Recent works have derived non - asymptotic upper bounds for convergence of underdamped Langevin MCMC. We revisit these bound and consider introducing scaling terms in the underlying underdamped Langevin equation. In particular, we provide conditions under which an appropria te scaling allows to improve the error bounds in terms of the condition number of the underlying density of interest.

In classical machine learning, the data streamed to the algorithms is assumed to be independent and identically distributed. Otherwise, if the data distribution changes through time, the algorithm risks to remember only the data from the current state of the distribution and forget everything else. Continual learning is a sub-field of machine learning that aims to find automatic learning processes to solve non-iid problems. The main challenges of continual learning are two-fold. Firstly, to detect concept-drift in the distribution and secondly to remember what happened before a concept-drift. In this article, we study a specific case of continual learning approaches: \textit{the regularization method}. It consists of finding a smart regularization term that will protect important parameters from being modified to not forget. We show in this article, that in the context of multi-task learning for classification, this process does not learn to discriminate classes from different tasks. We propose theoretical reasoning to prove this shortcoming and illustrate it with examples and experiments with the "MNIST Fellowship" dataset.

Graph neural network (GNN) has shown superior performance in dealing with graphs, which has attracted considerable research attention recently. However, most of the existing GNN models are primarily designed for graphs in Euclidean spaces. Recent research has proven that the graph data exhibits non-Euclidean latent anatomy. Unfortunately, there was rarely study of GNN in non-Euclidean settings so far. To bridge this gap, in this paper, we study the GNN with attention mechanism in hyperbolic spaces at the first attempt. The research of hyperbolic GNN has some unique challenges: since the hyperbolic spaces are not vector spaces, the vector operations (e.g., vector addition, subtraction, and scalar multiplication) cannot be carried. To tackle this problem, we employ the gyrovector spaces, which provide an elegant algebraic formalism for hyperbolic geometry, to transform the features in a graph; and then we propose the hyperbolic proximity based attention mechanism to aggregate the features. Moreover, as mathematical operations in hyperbolic spaces could be more complicated than those in Euclidean spaces, we further devise a novel acceleration strategy using logarithmic and exponential mappings to improve the efficiency of our proposed model. The comprehensive experimental results on four real-world datasets demonstrate the performance of our proposed hyperbolic graph attention network model, by comparisons with other state-of-the-art baseline methods.

In this paper, we improve the PAC-Bayesian error bound for linear regression derived in Germain et al. [10]. The improvements are twofold. First, the proposed error bound is tighter, and converges to the generalization loss with a well-chosen temperature parameter. Second, the error bound also holds for training data that are not independently sampled. In particular, the error bound applies to certain time series generated by well-known classes of dynamical models, such as ARX models.

Spectral algorithms are graph partitioning algorithms that partition a node set of a graph into groups by using a spectral embedding map. Clustering techniques based on the algorithms are referred to as spectral clustering and are widely used in data analysis. To gain a better understanding of why spectral clustering is successful, Peng et al. (2015) and Kolev and Mehlhorn (2016) studied the behavior of a certain type of spectral algorithm for a class of graphs, called well-clustered graphs. Specifically, they put an assumption on graphs and showed the performance guarantee of the spectral algorithm under it. The algorithm they studied used the spectral embedding map developed by Shi and Malic (2000). In this paper, we improve on their results, giving a better performance guarantee under a weaker assumption. We also evaluate the performance of the spectral algorithm with the spectral embedding map developed by Ng et al. (2002).

In this article, we focus on the analysis of the potential factors driving the spread of influenza, and possible policies to mitigate the adverse effects of the disease. To be precise, we first invoke discrete Fourier transform (DFT) to conclude a yearly periodic regional structure in the influenza activity, thus safely restricting ourselves to the analysis of the yearly influenza behavior. Then we collect a massive number of possible region-wise indicators contributing to the influenza mortality, such as consumption, immunization, sanitation, water quality, and other indicators from external data, with $1170$ dimensions in total. We extract significant features from the high dimensional indicators using a combination of data analysis techniques, including matrix completion, support vector machines (SVM), autoencoders, and principal component analysis (PCA). Furthermore, we model the international flow of migration and trade as a convolution on regional influenza activity, and solve the deconvolution problem as higher-order perturbations to the linear regression, thus separating regional and international factors related to the influenza mortality. Finally, both the original model and the perturbed model are tested on regional examples, as validations of our models. Pertaining to the policy, we make a proposal based on the connectivity data along with the previously extracted significant features to alleviate the impact of influenza, as well as efficiently propagate and carry out the policies. We conclude that environmental features and economic features are of significance to the influenza mortality. The model can be easily adapted to model other types of infectious diseases.

Rapid advancements in the Internet of Things (IoT) have facilitated more efficient deployment of smart environment solutions for specific user requirement. With the increase in the number of IoT devices, it has become difficult for the user to control or operate every individual smart device into achieving some desired goal like optimized power consumption, scheduled appliance running time, etc. Furthermore, existing solutions to automatically adapt the IoT devices are not capable enough to incorporate the user behavior. This paper presents a novel approach to accurately configure IoT devices while achieving the twin objectives of energy optimization along with conforming to user preferences. Our work comprises of unsupervised clustering of devices' data to find the states of operation for each device, followed by probabilistically analyzing user behavior to determine their preferred states. Eventually, we deploy an online reinforcement learning (RL) agent to find the best device settings automatically. Results for three different smart homes' data-sets show the effectiveness of our methodology. To the best of our knowledge, this is the first time that a practical approach has been adopted to achieve the above mentioned objectives without any human interaction within the system.

Adversarial attacks represent a serious menace for learning algorithms and may compromise the security of future autonomous systems. A theorem by Khoury and Hadfield-Menell (KH), provides sufficient conditions to guarantee the robustness of active learning algorithms, but comes with a caveat: it is crucial to know the smallest distance among the classes of the corresponding classification problem. We propose a theoretical framework that allows us to think of active learning as sampling the most promising new points to be classified, so that the minimum distance between classes can be found and the theorem KH used. The complexity of the quantum active learning algorithm is polynomial in the variables used, like the dimension of the space $m$ and the size of the initial training data $n$. On the other hand, if one replicates this approach with a classical computer, we expect that it would take exponential time in $m$, an example of the so-called `curse of dimensionality'.

Differential privacy provides a rigorous framework to quantify data privacy, and has received considerable interest recently. A randomized mechanism satisfying $(\epsilon, \delta)$-differential privacy (DP) roughly means that, except with a small probability $\delta$, altering a record in a dataset cannot change the probability that an output is seen by more than a multiplicative factor $e^{\epsilon} $. A well-known solution to $(\epsilon, \delta)$-DP is the Gaussian mechanism initiated by Dwork et al. [1] in 2006 with an improvement by Dwork and Roth [2] in 2014, where a Gaussian noise amount $\sqrt{2\ln \frac{2}{\delta}} \times \frac{\Delta}{\epsilon}$ of [1] or $\sqrt{2\ln \frac{1.25}{\delta}} \times \frac{\Delta}{\epsilon}$ of [2] is added independently to each dimension of the query result, for a query with $\ell_2$-sensitivity $\Delta$. Although both classical Gaussian mechanisms [1,2] assume $0 < \epsilon \leq 1$, our review finds that many studies in the literature have used the classical Gaussian mechanisms under values of $\epsilon$ and $\delta$ where the added noise amounts of [1,2] do not achieve $(\epsilon,\delta)$-DP. We obtain such result by analyzing the optimal noise amount $\sigma_{DP-OPT}$ for $(\epsilon,\delta)$-DP and identifying $\epsilon$ and $\delta$ where the noise amounts of classical mechanisms are even less than $\sigma_{DP-OPT}$. Since $\sigma_{DP-OPT}$ has no closed-form expression and needs to be approximated in an iterative manner, we propose Gaussian mechanisms by deriving closed-form upper bounds for $\sigma_{DP-OPT}$. Our mechanisms achieve $(\epsilon,\delta)$-DP for any $\epsilon$, while the classical mechanisms [1,2] do not achieve $(\epsilon,\delta)$-DP for large $\epsilon$ given $\delta$. Moreover, the utilities of our mechanisms improve those of [1,2] and are close to that of the optimal yet more computationally expensive Gaussian mechanism.

In my experience, learning to anything useful in computer science has fallen at the strange intersection of theory and practice. It's pretty easy to ignore the amount of depth that the lies under some of the things we code. Machine learning takes that to an extreme, and everyone wants to be a Machine Learning Engineer these days. Elements of Statistical Learning is a fantastic book. If you can get through it you'll know quite a bit, but it doesn't mean much if you're unable to put any of into practice.