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We study the problem of learning unknown parameters in stochastic interacting particle systems with polynomial drift, interaction and diffusion functions from the path of one single particle in the system. Our estimator is obtained by solving a linear system which is constructed by imposing appropriate conditions on the moments of the invariant distribution of the mean field limit and on the quadratic variation of the process. Our approach is easy to implement as it only requires the approximation of the moments via the ergodic theorem and the solution of a low-dimensional linear system. Moreover, we prove that our estimator is asymptotically unbiased in the limits of infinite data and infinite number of particles (mean field limit). In addition, we present several numerical experiments that validate the theoretical analysis and show the effectiveness of our methodology to accurately infer parameters in systems of interacting particles.

Multivariate Hawkes processes are temporal point processes extensively applied to model event data with dependence on past occurrences and interaction phenomena. In the generalised nonlinear model, positive and negative interactions between the components of the process are allowed, therefore accounting for so-called excitation and inhibition effects. In the nonparametric setting, learning the temporal dependence structure of Hawkes processes is often a computationally expensive task, all the more with Bayesian estimation methods. In general, the posterior distribution in the nonlinear Hawkes model is non-conjugate and doubly intractable. Moreover, existing Monte-Carlo Markov Chain methods are often slow and not scalable to high-dimensional processes in practice. Recently, efficient algorithms targeting a mean-field variational approximation of the posterior distribution have been proposed. In this work, we unify existing variational Bayes inference approaches under a general framework, that we theoretically analyse under easily verifiable conditions on the prior, the variational class, and the model. We notably apply our theory to a novel spike-and-slab variational class, that can induce sparsity through the connectivity graph parameter of the multivariate Hawkes model. Then, in the context of the popular sigmoid Hawkes model, we leverage existing data augmentation technique and design adaptive and sparsity-inducing mean-field variational methods. In particular, we propose a two-step algorithm based on a thresholding heuristic to select the graph parameter. Through an extensive set of numerical simulations, we demonstrate that our approach enjoys several benefits: it is computationally efficient, can reduce the dimensionality of the problem by selecting the graph parameter, and is able to adapt to the smoothness of the underlying parameter.

Variational Bayes methods are a scalable estimation approach for many complex state space models. However, existing methods exhibit a trade-off between accurate estimation and computational efficiency. This paper proposes a variational approximation that mitigates this trade-off. This approximation is based on importance densities that have been proposed in the context of efficient importance sampling. By directly conditioning on the observed data, the proposed method produces an accurate approximation to the exact posterior distribution. Because the steps required for its calibration are computationally efficient, the approach is faster than existing variational Bayes methods. The proposed method can be applied to any state space model that has a closed-form measurement density function and a state transition distribution that belongs to the exponential family of distributions. We illustrate the method in numerical experiments with stochastic volatility models and a macroeconomic empirical application using a high-dimensional state space model.

Variational autoencoders employ an amortized inference model to approximate the posterior of latent variables. However, such amortized variational inference faces two challenges: (1) the limited posterior expressiveness of fully-factorized Gaussian assumption and (2) the amortization error of the inference model. We present a novel approach that addresses both challenges. First, we focus on ReLU networks with Gaussian output and illustrate their connection to probabilistic PCA. Building on this observation, we derive an iterative algorithm that finds the mode of the posterior and apply full-covariance Gaussian posterior approximation centered on the mode. Subsequently, we present a general framework named Variational Laplace Autoencoders (VLAEs) for training deep generative models. Based on the Laplace approximation of the latent variable posterior, VLAEs enhance the expressiveness of the posterior while reducing the amortization error. Empirical results on MNIST, Omniglot, Fashion-MNIST, SVHN and CIFAR10 show that the proposed approach significantly outperforms other recent amortized or iterative methods on the ReLU networks.

Marketing is an important mechanism to increase user engagement and improve platform revenue, and heterogeneous causal learning can help develop more effective strategies. Most decision-making problems in marketing can be formulated as resource allocation problems and have been studied for decades. Existing works usually divide the solution procedure into two fully decoupled stages, i.e., machine learning (ML) and operation research (OR) -- the first stage predicts the model parameters and they are fed to the optimization in the second stage. However, the error of the predicted parameters in ML cannot be respected and a series of complex mathematical operations in OR lead to the increased accumulative errors. Essentially, the improved precision on the prediction parameters may not have a positive correlation on the final solution due to the side-effect from the decoupled design. In this paper, we propose a novel approach for solving resource allocation problems to mitigate the side-effects. Our key intuition is that we introduce the decision factor to establish a bridge between ML and OR such that the solution can be directly obtained in OR by only performing the sorting or comparison operations on the decision factor. Furthermore, we design a customized loss function that can conduct direct heterogeneous causal learning on the decision factor, an unbiased estimation of which can be guaranteed when the loss converges. As a case study, we apply our approach to two crucial problems in marketing: the binary treatment assignment problem and the budget allocation problem with multiple treatments. Both large-scale simulations and online A/B Tests demonstrate that our approach achieves significant improvement compared with state-of-the-art.

The matrix-based R\'enyi's entropy allows us to directly quantify information measures from given data, without explicit estimation of the underlying probability distribution. This intriguing property makes it widely applied in statistical inference and machine learning tasks. However, this information theoretical quantity is not robust against noise in the data, and is computationally prohibitive in large-scale applications. To address these issues, we propose a novel measure of information, termed low-rank matrix-based R\'enyi's entropy, based on low-rank representations of infinitely divisible kernel matrices. The proposed entropy functional inherits the specialty of of the original definition to directly quantify information from data, but enjoys additional advantages including robustness and effective calculation. Specifically, our low-rank variant is more sensitive to informative perturbations induced by changes in underlying distributions, while being insensitive to uninformative ones caused by noises. Moreover, low-rank R\'enyi's entropy can be efficiently approximated by random projection and Lanczos iteration techniques, reducing the overall complexity from $\mathcal{O}(n^3)$ to $\mathcal{O}(n^2 s)$ or even $\mathcal{O}(ns^2)$, where $n$ is the number of data samples and $s \ll n$. We conduct large-scale experiments to evaluate the effectiveness of this new information measure, demonstrating superior results compared to matrix-based R\'enyi's entropy in terms of both performance and computational efficiency.

Reinforcement learning (RL) problems over general state and action spaces are notoriously challenging. In contrast to the tableau setting, one can not enumerate all the states and then iteratively update the policies for each state. This prevents the application of many well-studied RL methods especially those with provable convergence guarantees. In this paper, we first present a substantial generalization of the recently developed policy mirror descent method to deal with general state and action spaces. We introduce new approaches to incorporate function approximation into this method, so that we do not need to use explicit policy parameterization at all. Moreover, we present a novel policy dual averaging method for which possibly simpler function approximation techniques can be applied. We establish linear convergence rate to global optimality or sublinear convergence to stationarity for these methods applied to solve different classes of RL problems under exact policy evaluation. We then define proper notions of the approximation errors for policy evaluation and investigate their impact on the convergence of these methods applied to general-state RL problems with either finite-action or continuous-action spaces. To the best of our knowledge, the development of these algorithmic frameworks as well as their convergence analysis appear to be new in the literature.

In this paper, we propose an inverse-kinematics controller for a class of multi-robot systems in the scenario of sampled communication. The goal is to make a group of robots perform trajectory tracking in a coordinated way when the sampling time of communications is much larger than the sampling time of low-level controllers, disrupting theoretical convergence guarantees of standard control design in continuous time. Given a desired trajectory in configuration space which is precomputed offline, the proposed controller receives configuration measurements, possibly via wireless, to re-compute velocity references for the robots, which are tracked by a low-level controller. We propose joint design of a sampled proportional feedback plus a novel continuous-time feedforward that linearizes the dynamics around the reference trajectory: this method is amenable to distributed communication implementation where only one broadcast transmission is needed per sample. Also, we provide closed-form expressions for instability and stability regions and convergence rate in terms of proportional gain $k$ and sampling period $T$. We test the proposed control strategy via numerical simulations in the scenario of cooperative aerial manipulation of a cable-suspended load using a realistic simulator (Fly-Crane). Finally, we compare our proposed controller with centralized approaches that adapt the feedback gain online through smart heuristics, and show that it achieves comparable performance.

This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.

Recently, graph neural networks (GNNs) have revolutionized the field of graph representation learning through effectively learned node embeddings, and achieved state-of-the-art results in tasks such as node classification and link prediction. However, current GNN methods are inherently flat and do not learn hierarchical representations of graphs---a limitation that is especially problematic for the task of graph classification, where the goal is to predict the label associated with an entire graph. Here we propose DiffPool, a differentiable graph pooling module that can generate hierarchical representations of graphs and can be combined with various graph neural network architectures in an end-to-end fashion. DiffPool learns a differentiable soft cluster assignment for nodes at each layer of a deep GNN, mapping nodes to a set of clusters, which then form the coarsened input for the next GNN layer. Our experimental results show that combining existing GNN methods with DiffPool yields an average improvement of 5-10% accuracy on graph classification benchmarks, compared to all existing pooling approaches, achieving a new state-of-the-art on four out of five benchmark data sets.