In this work we initiate the study of the correspondence between p-adic statistical field theories (SFTs) and neural networks (NNs). In general quantum field theories over a p-adic spacetime can be formulated in a rigorous way. Nowadays these theories are considered just mathematical toy models for understanding the problems of the true theories. In this work we show these theories are deeply connected with the deep belief networks (DBNs). Hinton et al. constructed DBNs by stacking several restricted Boltzmann machines (RBMs). The purpose of this construction is to obtain a network with a hierarchical structure (a deep learning architecture). An RBM corresponds to a certain spin glass, we argue that a DBN should correspond to an ultrametric spin glass. A model of such a system can be easily constructed by using p-adic numbers. In our approach, a p-adic SFT corresponds to a p-adic continuous DBN, and a discretization of this theory corresponds to a p-adic discrete DBN. We show that these last machines are universal approximators. In the p-adic framework, the correspondence between SFTs and NNs is not fully developed. We point out several open problems.

Identifying functional connectivity biomarkers of major depressive disorder (MDD) patients is essential to advance understanding of the disorder mechanisms and early intervention. However, due to the small sample size and the high dimension of available neuroimaging data, the performance of existing methods is often limited. Multi-site data could enhance the statistical power and sample size, while they are often subject to inter-site heterogeneity and data-sharing policies. In this paper, we propose a federated joint estimator, NOTEARS-PFL, for simultaneous learning of multiple Bayesian networks (BNs) with continuous optimization, to identify disease-induced alterations in MDD patients. We incorporate information shared between sites and site-specific information into the proposed federated learning framework to learn personalized BN structures by introducing the group fused lasso penalty. We develop the alternating direction method of multipliers, where in the local update step, the neuroimaging data is processed at each local site. Then the learned network structures are transmitted to the center for the global update. In particular, we derive a closed-form expression for the local update step and use the iterative proximal projection method to deal with the group fused lasso penalty in the global update step. We evaluate the performance of the proposed method on both synthetic and real-world multi-site rs-fMRI datasets. The results suggest that the proposed NOTEARS-PFL yields superior effectiveness and accuracy than the comparable methods.

We study the constrained reinforcement learning problem, in which an agent aims to maximize the expected cumulative reward subject to a constraint on the expected total value of a utility function. In contrast to existing model-based approaches or model-free methods accompanied with a `simulator', we aim to develop the first model-free, simulator-free algorithm that achieves a sublinear regret and a sublinear constraint violation even in large-scale systems. To this end, we consider the episodic constrained Markov decision processes with linear function approximation, where the transition dynamics and the reward function can be represented as a linear function of some known feature mapping. We show that $\tilde{\mathcal{O}}(\sqrt{d^3H^3T})$ regret and $\tilde{\mathcal{O}}(\sqrt{d^3H^3T})$ constraint violation bounds can be achieved, where $d$ is the dimension of the feature mapping, $H$ is the length of the episode, and $T$ is the total number of steps. Our bounds are attained without explicitly estimating the unknown transition model or requiring a simulator, and they depend on the state space only through the dimension of the feature mapping. Hence our bounds hold even when the number of states goes to infinity. Our main results are achieved via novel adaptations of the standard LSVI-UCB algorithms. In particular, we first introduce primal-dual optimization into the LSVI-UCB algorithm to balance between regret and constraint violation. More importantly, we replace the standard greedy selection with respect to the state-action function in LSVI-UCB with a soft-max policy. This turns out to be key in establishing uniform concentration for the constrained case via its approximation-smoothness trade-off. We also show that one can achieve an even zero constraint violation while still maintaining the same order with respect to $T$.

To integrate high amounts of renewable energy resources, electrical power grids must be able to cope with high amplitude, fast timescale variations in power generation. Frequency regulation through demand response has the potential to coordinate temporally flexible loads, such as air conditioners, to counteract these variations. Existing approaches for discrete control with dynamic constraints struggle to provide satisfactory performance for fast timescale action selection with hundreds of agents. We propose a decentralized agent trained with multi-agent proximal policy optimization with localized communication. We explore two communication frameworks: hand-engineered, or learned through targeted multi-agent communication. The resulting policies perform well and robustly for frequency regulation, and scale seamlessly to arbitrary numbers of houses for constant processing times.

This article belongs to the series "Probabilistic Deep Learning". This weekly series covers probabilistic approaches to deep learning. The main goal is to extend deep learning models to quantify uncertainty, i.e. know what they do not know. The frequentist approach to statistics is based on the idea of repeated sampling and long-run relative frequency. It involves constructing hypotheses about a population and testing them using sample data.

This paper considers ranking inference of $n$ items based on the observed data on the top choice among $M$ randomly selected items at each trial. This is a useful modification of the Plackett-Luce model for $M$-way ranking with only the top choice observed and is an extension of the celebrated Bradley-Terry-Luce model that corresponds to $M=2$. Under a uniform sampling scheme in which any $M$ distinguished items are selected for comparisons with probability $p$ and the selected $M$ items are compared $L$ times with multinomial outcomes, we establish the statistical rates of convergence for underlying $n$ preference scores using both $\ell_2$-norm and $\ell_\infty$-norm, with the minimum sampling complexity. In addition, we establish the asymptotic normality of the maximum likelihood estimator that allows us to construct confidence intervals for the underlying scores. Furthermore, we propose a novel inference framework for ranking items through a sophisticated maximum pairwise difference statistic whose distribution is estimated via a valid Gaussian multiplier bootstrap. The estimated distribution is then used to construct simultaneous confidence intervals for the differences in the preference scores and the ranks of individual items. They also enable us to address various inference questions on the ranks of these items. Extensive simulation studies lend further support to our theoretical results. A real data application illustrates the usefulness of the proposed methods convincingly.

We introduce a general framework for Markov decision problems under model uncertainty in a discrete-time infinite horizon setting. By providing a dynamic programming principle we obtain a local-to-global paradigm, namely solving a local, i.e., a one time-step robust optimization problem leads to an optimizer of the global (i.e. infinite time-steps) robust stochastic optimal control problem, as well as to a corresponding worst-case measure. Moreover, we apply this framework to portfolio optimization involving data of the S&P 500. We present two different types of ambiguity sets; one is fully data-driven given by a Wasserstein-ball around the empirical measure, the second one is described by a parametric set of multivariate normal distributions, where the corresponding uncertainty sets of the parameters are estimated from the data. It turns out that in scenarios where the market is volatile or bearish, the optimal portfolio strategies from the corresponding robust optimization problem outperforms the ones without model uncertainty, showcasing the importance of taking model uncertainty into account.

We present a novel $Q$-learning algorithm to solve distributionally robust Markov decision problems, where the corresponding ambiguity set of transition probabilities for the underlying Markov decision process is a Wasserstein ball around a (possibly estimated) reference measure. We prove convergence of the presented algorithm and provide several examples also using real data to illustrate both the tractability of our algorithm as well as the benefits of considering distributional robustness when solving stochastic optimal control problems, in particular when the estimated distributions turn out to be misspecified in practice.

Nowadays, a tremendous amount of human communications occur on Internet-based communication infrastructures, like social networks, email, forums, organizational communication platforms, etc. Indeed, the automatic prediction or assessment of individuals' personalities through their written or exchanged text would be advantageous to ameliorate their relationships. To this end, this paper aims to propose KGrAt-Net, which is a Knowledge Graph Attention Network text classifier. For the first time, it applies the knowledge graph attention network to perform Automatic Personality Prediction (APP), according to the Big Five personality traits. After performing some preprocessing activities, it first tries to acquire a knowing-full representation of the knowledge behind the concepts in the input text by building its equivalent knowledge graph. A knowledge graph collects interlinked descriptions of concepts, entities, and relationships in a machine-readable form. Practically, it provides a machine-readable cognitive understanding of concepts and semantic relationships among them. Then, applying the attention mechanism, it attempts to pay attention to the most relevant parts of the graph to predict the personality traits of the input text. We used 2,467 essays from the Essays Dataset. The results demonstrated that KGrAt-Net considerably improved personality prediction accuracies (up to 70.26% on average). Furthermore, KGrAt-Net also uses knowledge graph embedding to enrich the classification, which makes it even more accurate (on average, 72.41%) in APP.

The empirical risk minimization approach to data-driven decision making assumes that we can learn a decision rule from training data drawn under the same conditions as the ones we want to deploy it in. However, in a number of settings, we may be concerned that our training sample is biased, and that some groups (characterized by either observable or unobservable attributes) may be under- or over-represented relative to the general population; and in this setting empirical risk minimization over the training set may fail to yield rules that perform well at deployment. We propose a model of sampling bias called $\Gamma$-biased sampling, where observed covariates can affect the probability of sample selection arbitrarily much but the amount of unexplained variation in the probability of sample selection is bounded by a constant factor. Applying the distributionally robust optimization framework, we propose a method for learning a decision rule that minimizes the worst-case risk incurred under a family of test distributions that can generate the training distribution under $\Gamma$-biased sampling. We apply a result of Rockafellar and Uryasev to show that this problem is equivalent to an augmented convex risk minimization problem. We give statistical guarantees for learning a model that is robust to sampling bias via the method of sieves, and propose a deep learning algorithm whose loss function captures our robust learning target. We empirically validate our proposed method in simulations and a case study on ICU length of stay prediction.