Agent-based methods allow for defining simple rules that generate complex group behaviors. The governing rules of such models are typically set a priori and parameters are tuned from observed behavior trajectories. Instead of making simplifying assumptions across all anticipated scenarios, inverse reinforcement learning provides inference on the short-term (local) rules governing long term behavior policies by using properties of a Markov decision process. We use the computationally efficient linearly-solvable Markov decision process to learn the local rules governing collective movement for a simulation of the self propelled-particle (SPP) model and a data application for a captive guppy population. The estimation of the behavioral decision costs is done in a Bayesian framework with basis function smoothing. We recover the true costs in the SPP simulation and find the guppies value collective movement more than targeted movement toward shelter.

We study the convergence rates of empirical Bayes posterior distributions for nonparametric and high-dimensional inference. We show that as long as the hyperparameter set is discrete, the empirical Bayes posterior distribution induced by the maximum marginal likelihood estimator can be regarded as a variational approximation to a hierarchical Bayes posterior distribution. This connection between empirical Bayes and variational Bayes allows us to leverage the recent results in the variational Bayes literature, and directly obtains the convergence rates of empirical Bayes posterior distributions from a variational perspective. For a more general hyperparameter set that is not necessarily discrete, we introduce a new technique called "prior decomposition" to deal with prior distributions that can be written as convex combinations of probability measures whose supports are low-dimensional subspaces. This leads to generalized versions of the classical "prior mass and testing" conditions for the convergence rates of empirical Bayes. Our theory is applied to a number of statistical estimation problems including nonparametric density estimation and sparse linear regression.

Learning the optimal ordering of content is an important challenge in website design. The learning to rank (LTR) framework models this problem as a sequential problem of selecting lists of content and observing where users decide to click. Most previous work on LTR assumes that the user considers each item in the list in isolation, and makes binary choices to click or not on each. We introduce a multinomial logit (MNL) choice model to the LTR framework, which captures the behaviour of users who consider the ordered list of items as a whole and make a single choice among all the items and a no-click option. Under the MNL model, the user favours items which are either inherently more attractive, or placed in a preferable position within the list. We propose upper confidence bound algorithms to minimise regret in two settings - where the position dependent parameters are known, and unknown. We present theoretical analysis leading to an $\Omega(\sqrt{T})$ lower bound for the problem, an $\tilde{O}(\sqrt{T})$ upper bound on regret for the known parameter version. Our analyses are based on tight new concentration results for Geometric random variables, and novel functional inequalities for maximum likelihood estimators computed on discrete data.

Reconstructing parameters of physical models is an important task in science. Usually, such problems are severely under determined and sophisticated reconstruction techniques are necessary. Whereas classical regularisation methods focus on finding just the most desirable or most likely solution of an inverse problem, more recent methods focus on analyzing the complete distribution of possible parameters. In particular, this provides us with a way to quantify how trustworthy the obtained solution is. Among the most popular methods for uncertainty quantification are Bayesian methods [16], which build up on evaluating the posterior using Bayes theorem, and Markov Chain Monte Carlo (MCMC) [38].

We solve the problem of 6-DoF localisation and 3D dense reconstruction in spatial environments as approximate Bayesian inference in a deep generative approach which combines learned with engineered models. This principled treatment of uncertainty and probabilistic inference overcomes the shortcoming of current state-of-the-art solutions to rely on heavily engineered, heterogeneous pipelines. Variational inference enables us to use neural networks for system identification, while a differentiable raycaster is used for the emission model. This ensures that our model is amenable to end-to-end gradient-based optimisation. We evaluate our approach on realistic unmanned aerial vehicle flight data, nearing the performance of a state-of-the-art visual inertial odometry system. The applicability of the learned model to downstream tasks such as generative prediction and planning is investigated.

We formulate meta learning using information theoretic concepts such as mutual information and the information bottleneck. The idea is to learn a stochastic representation or encoding of the task description, given by a training or support set, that is highly informative about predicting the validation set. By making use of variational approximations to the mutual information, we derive a general and tractable framework for meta learning. We particularly develop new memorybased meta learning algorithms based on Gaussian processes and derive extensions that combine memory and gradient-based meta learning. We demonstrate our method on few-shot regression and classification by using standard benchmarks such as Omniglot, mini-Imagenet and Augmented Omniglot. Such systems require training deep neural networks from a set of tasks drawn from a common distribution, where each task is described by a small amount of experience, typically divided into a training or support set and a validation set. By sharing information across tasks the neural network can learn to rapidly adapt to new tasks and generalize from few examples at test time. Several few-shot learning algorithms use memory-based (Vinyals et al., 2016; Ravi & Larochelle, 2017) or gradient-based procedures (Finn et al., 2017; Nichol et al., 2018), with the gradient-based model agnostic meta learning algorithm (MAML) by Finn et al. (2017) being very influential in the literature. Despite the success of specific schemes, one fundamental issue in meta learning is concerned with deriving unified principles that can allow to relate different approaches and invent new schemes.

Introduced by Kiefer and Wolfowitz \cite{KW56}, the nonparametric maximum likelihood estimator (NPMLE) is a widely used methodology for learning mixture odels and empirical Bayes estimation. Sidestepping the non-convexity in mixture likelihood, the NPMLE estimates the mixing distribution by maximizing the total likelihood over the space of probability measures, which can be viewed as an extreme form of overparameterization. In this paper we discover a surprising property of the NPMLE solution. Consider, for example, a Gaussian mixture model on the real line with a subgaussian mixing distribution. Leveraging complex-analytic techniques, we show that with high probability the NPMLE based on a sample of size $n$ has $O(\log n)$ atoms (mass points), significantly improving the deterministic upper bound of $n$ due to Lindsay \cite{lindsay1983geometry1}. Notably, any such Gaussian mixture is statistically indistinguishable from a finite one with $O(\log n)$ components (and this is tight for certain mixtures). Thus, absent any explicit form of model selection, NPMLE automatically chooses the right model complexity, a property we term \emph{self-regularization}. Extensions to other exponential families are given. As a statistical application, we show that this structural property can be harnessed to bootstrap existing Hellinger risk bound of the (parametric) MLE for finite Gaussian mixtures to the NPMLE for general Gaussian mixtures, recovering a result of Zhang \cite{zhang2009generalized}.

Machine learning models suffer from overfitting, which is caused by a lack of labeled data. To tackle this problem, we proposed a framework of regularization methods, called density-fixing, that can be used commonly for supervised and semi-supervised learning. Our proposed regularization method improves the generalization performance by forcing the model to approximate the class's prior distribution or the frequency of occurrence. This regularization term is naturally derived from the formula of maximum likelihood estimation and is theoretically justified. We further provide the several theoretical analyses of the proposed method including asymptotic behavior. Our experimental results on multiple benchmark datasets are sufficient to support our argument, and we suggest that this simple and effective regularization method is useful in real-world machine learning problems.

Tensor rank learning for canonical polyadic decomposition (CPD) has long been deemed as an essential but challenging problem. In particular, since the tensor rank controls the complexity of the CPD model, its inaccurate learning would cause overfitting to noise or underfitting to the signal sources, and even destroy the interpretability of model parameters. However, the optimal determination of a tensor rank is known to be a non-deterministic polynomial-time hard (NP-hard) task. Rather than exhaustively searching for the best tensor rank via trial-and-error experiments, Bayesian inference under the Gaussian-gamma prior was introduced in the context of probabilistic CPD modeling and it was shown to be an effective strategy for automatic tensor rank determination. This triggered flourishing research on other structured tensor CPDs with automatic tensor rank learning. As the other side of the coin, these research works also reveal that the Gaussian-gamma model does not perform well for high-rank tensors or/and low signal-to-noise ratios (SNRs). To overcome these drawbacks, in this paper, we introduce a more advanced generalized hyperbolic (GH) prior to the probabilistic CPD model, which not only includes the Gaussian-gamma model as a special case, but also provides more flexibilities to adapt to different levels of sparsity. Based on this novel probabilistic model, an algorithm is developed under the framework of variational inference, where each update is obtained in a closed-form. Extensive numerical results, using synthetic data and real-world datasets, demonstrate the excellent performance of the proposed method in learning both low as well as high tensor ranks even for low SNR cases.

Machine Learning Tutorial - Expectation-Maximization (EM) Algorithm In Machine Learning, covers the EM algorithm along with the problem of latent variables in maximum likelihood and Gaussian mixture model. You'll learn: What is EM Algorithm In Machine Learning? This Edureka video on'EM Algorithm In Machine Learning' covers the EM algorithm along with the problem of latent variables in maximum likelihood and Gaussian mixture model.