We consider the problem of designing minimax estimators for estimating the parameters of a probability distribution. Unlike classical approaches such as the MLE and minimum distance estimators, we consider an algorithmic approach for constructing such estimators. We view the problem of designing minimax estimators as finding a mixed strategy Nash equilibrium of a zero-sum game. By leveraging recent results in online learning with non-convex losses, we provide a general algorithm for finding a mixed-strategy Nash equilibrium of general non-convex non-concave zero-sum games. Our algorithm requires access to two subroutines: (a) one which outputs a Bayes estimator corresponding to a given prior probability distribution, and (b) one which computes the worst-case risk of any given estimator. Given access to these two subroutines, we show that our algorithm outputs both a minimax estimator and a least favorable prior. To demonstrate the power of this approach, we use it to construct provably minimax estimators for classical problems such as estimation in the finite Gaussian sequence model, and linear regression.

Estimating parameters of mixture model has wide applications Determining the number of components in a finite mixture model ranging from classification problems to estimating of complex distributions. is crucial in many application areas such as financial data [16, 31, 35], Most of the current literature on estimating the parameters biomedical studies [17, 36] and low-frequency accident occurrence of the mixture densities are based on iterative Expectation Maximization prediction [27, 32]. Existing literature have witnessed numerous (EM) type algorithms which require the use of either computational methods, and in particular Markov Chain Monte taking expectations over the latent label variables or generating Carlo methods [7, 14, 33] and EM algorithms [20-22] have been samples from the conditional distribution of such latent labels using used with a lot of success. However, either these methods are computationally the Bayes rule. Moreover, when the number of components is demanding and/or these methods are developed under unknown, the problem becomes computationally more demanding the assumption of data being generated from mixtures of densities due to well-known label switching issues [28]. In this paper, we from the exponential family, in part because the family of exponential propose a robust and quick approach based on change-point methods distribution has a sufficient statistic of constant dimension (i.e., to determine the number of mixture components that works the dimension of the sufficient statistic remains fixed for any sample for almost any location-scale families even when the components size) and so the updates of the data augmentation type algorithm are heavy tailed (e.g., Cauchy). We present several numerical illustrations involve their smaller dimensional sufficient statistics [11, 12, 24].

Outlier detection is an important data mining task with numerous practical applications such as intrusion detection, credit card fraud detection, and video surveillance. However, given a specific complicated task with big data, the process of building a powerful deep learning based system for outlier detection still highly relies on human expertise and laboring trials. Although Neural Architecture Search (NAS) has shown its promise in discovering effective deep architectures in various domains, such as image classification, object detection, and semantic segmentation, contemporary NAS methods are not suitable for outlier detection due to the lack of intrinsic search space, unstable search process, and low sample efficiency. To bridge the gap, in this paper, we propose AutoOD, an automated outlier detection framework, which aims to search for an optimal neural network model within a predefined search space. Specifically, we firstly design a curiosity-guided search strategy to overcome the curse of local optimality. A controller, which acts as a search agent, is encouraged to take actions to maximize the information gain about the controller's internal belief. We further introduce an experience replay mechanism based on self-imitation learning to improve the sample efficiency. Experimental results on various real-world benchmark datasets demonstrate that the deep model identified by AutoOD achieves the best performance, comparing with existing handcrafted models and traditional search methods.

Matrix square roots and their inverses arise frequently in machine learning, e.g., when sampling from high-dimensional Gaussians $\mathcal{N}(\mathbf 0, \mathbf K)$ or whitening a vector $\mathbf b$ against covariance matrix $\mathbf K$. While existing methods typically require $O(N^3)$ computation, we introduce a highly-efficient quadratic-time algorithm for computing $\mathbf K^{1/2} \mathbf b$, $\mathbf K^{-1/2} \mathbf b$, and their derivatives through matrix-vector multiplication (MVMs). Our method combines Krylov subspace methods with a rational approximation and typically achieves $4$ decimal places of accuracy with fewer than $100$ MVMs. Moreover, the backward pass requires little additional computation. We demonstrate our method's applicability on matrices as large as $50,\!000 \times 50,\!000$ - well beyond traditional methods - with little approximation error. Applying this increased scalability to variational Gaussian processes, Bayesian optimization, and Gibbs sampling results in more powerful models with higher accuracy.

Gaussian process-based latent variable models are flexible and theoretically grounded tools for nonlinear dimension reduction, but generalizing to non-Gaussian data likelihoods within this nonlinear framework is statistically challenging. Here, we use random features to develop a family of nonlinear dimension reduction models that are easily extensible to non-Gaussian data likelihoods; we call these random feature latent variable models (RFLVMs). By approximating a nonlinear relationship between the latent space and the observations with a function that is linear with respect to random features, we induce closed-form gradients of the posterior distribution with respect to the latent variable. This allows the RFLVM framework to support computationally tractable nonlinear latent variable models for a variety of data likelihoods in the exponential family without specialized derivations. Our generalized RFLVMs produce results comparable with other state-of-the-art dimension reduction methods on diverse types of data, including neural spike train recordings, images, and text data.

Classifiers are often tested on relatively small data sets, which should lead to uncertain performance metrics. Nevertheless, these metrics are usually taken at face value. We present an approach to quantify the uncertainty of classification performance metrics, based on a probability model of the confusion matrix. Application of our approach to classifiers from the scientific literature and a classification competition shows that uncertainties can be surprisingly large and limit performance evaluation. In fact, some published classifiers are likely to be misleading. The application of our approach is simple and requires only the confusion matrix. It is agnostic of the underlying classifier. Our method can also be used for the estimation of sample sizes that achieve a desired precision of a performance metric.

In the Logistic Regression for Machine Learning using Python blog, I have introduced the basic idea of the logistic function. We have discussed the cost function. And in the iterative method, we focus on the Gradient descent optimization method. Now so in this section, we are going to introduce the Maximum Likelihood cost function. And we would like to maximize this cost function.

Dependability assurance of systems embedding machine learning(ML) components---so called learning-enabled systems (LESs)---is a key step for their use in safety-critical applications. In emerging standardization and guidance efforts, there is a growing consensus in the value of using assurance cases for that purpose. This paper develops a quantitative notion of assurance that an LES is dependable, as a core component of its assurance case, also extending our prior work that applied to ML components. Specifically, we characterize LES assurance in the form of assurance measures: a probabilistic quantification of confidence that an LES possesses system-level properties associated with functional capabilities and dependability attributes. We illustrate the utility of assurance measures by application to a real world autonomous aviation system, also describing their role both in i) guiding high-level, runtime risk mitigation decisions and ii) as a core component of the associated dynamic assurance case.

As humans, our goals and our environment are persistently changing throughout our lifetime based on our experiences, actions, and internal and external drives. In contrast, typical reinforcement learning problem set-ups consider decision processes that are stationary across episodes. Can we develop reinforcement learning algorithms that can cope with the persistent change in the former, more realistic problem settings? While on-policy algorithms such as policy gradients in principle can be extended to non-stationary settings, the same cannot be said for more efficient off-policy algorithms that replay past experiences when learning. In this work, we formalize this problem setting, and draw upon ideas from the online learning and probabilistic inference literature to derive an off-policy RL algorithm that can reason about and tackle such lifelong non-stationarity. Our method leverages latent variable models to learn a representation of the environment from current and past experiences, and performs off-policy RL with this representation. We further introduce several simulation environments that exhibit lifelong non-stationarity, and empirically find that our approach substantially outperforms approaches that do not reason about environment shift.

Various structured output prediction problems (e.g., sequential tagging) involve constraints over the output space. By identifying these constraints, we can filter out infeasible solutions and build an accountable model. To this end, we present a general integer linear programming (ILP) framework for mining constraints from data. We model the inference of structured output prediction as an ILP problem. Then, given the coefficients of the objective function and the corresponding solution, we mine the underlying constraints by estimating the outer and inner polytopes of the feasible set. We verify the proposed constraint mining algorithm in various synthetic and real-world applications and demonstrate that the proposed approach successfully identifies the feasible set at scale. In particular, we show that our approach can learn to solve 9x9 Sudoku puzzles and minimal spanning tree problems from examples without providing the underlying rules. We also demonstrate results on hierarchical multi-label classification and conduct a theoretical analysis on how close the mined constraints are from the ground truth.