For many inference problems in statistics and econometrics, the unknown parameter is identified by a set of moment conditions. A generic method of solving moment conditions is the Generalized Method of Moments (GMM). However, classical GMM estimation is potentially very sensitive to outliers. Robustified GMM estimators have been developed in the past, but suffer from several drawbacks: computational intractability, poor dimension-dependence, and no quantitative recovery guarantees in the presence of a constant fraction of outliers. In this work, we develop the first computationally efficient GMM estimator (under intuitive assumptions) that can tolerate a constant $\epsilon$ fraction of adversarially corrupted samples, and that has an $\ell_2$ recovery guarantee of $O(\sqrt{\epsilon})$. To achieve this, we draw upon and extend a recent line of work on algorithmic robust statistics for related but simpler problems such as mean estimation, linear regression and stochastic optimization. As two examples of the generality of our algorithm, we show how our estimation algorithm and assumptions apply to instrumental variables linear and logistic regression. Moreover, we experimentally validate that our estimator outperforms classical IV regression and two-stage Huber regression on synthetic and semi-synthetic datasets with corruption.

Deep neural networks tend to underestimate uncertainty and produce overly confident predictions. Recently proposed solutions, such as MC Dropout and SDENet, require complex training and/or auxiliary out-of-distribution data. We propose a simple solution by extending the time-tested iterative reweighted least square (IRLS) in generalised linear regression. We use two sub-networks to parametrise the prediction and uncertainty estimation, enabling easy handling of complex inputs and nonlinear response. The two sub-networks have shared representations and are trained via two complementary loss functions for the prediction and the uncertainty estimates, with interleaving steps as in a cooperative game. Compared with more complex models such as MC-Dropout or SDE-Net, our proposed network is simpler to implement and more robust (insensitive to varying aleatoric and epistemic uncertainty).

Researchers have been facing a difficult problem that data generation mechanisms could be influenced by internal or external factors leading to the training and test data with quite different distributions, consequently traditional classification or regression from the training set is unable to achieve satisfying results on test data. In this paper, we address this nontrivial domain generalization problem by finding a central subspace in which domain-based covariance is minimized while the functional relationship is simultaneously maximally preserved. We propose a novel variance measurement for multiple domains so as to minimize the difference between conditional distributions across domains with solid theoretical demonstration and supports, meanwhile, the algorithm preserves the functional relationship via maximizing the variance of conditional expectations given output. Furthermore, we also provide a fast implementation that requires much less computation and smaller memory for large-scale matrix operations, suitable for not only domain generalization but also other kernel-based eigenvalue decompositions. To show the practicality of the proposed method, we compare our methods against some well-known dimension reduction and domain generalization techniques on both synthetic data and real-world applications. We show that for small-scale datasets, we are able to achieve better quantitative results indicating better generalization performance over unseen test datasets. For large-scale problems, the proposed fast implementation maintains the quantitative performance but at a substantially lower computational cost.

It is a well-known fact that nonconvex optimization is computationally intractable in the worst case. As a result, theoretical analysis of optimization algorithms such as gradient descent often focuses on local convergence to stationary points where the gradient norm is zero or negligible. In this work, we examine the disconnect between the existing theoretical analysis of gradient-based algorithms and actual practice. Specifically, we provide numerical evidence that in large-scale neural network training, such as in ImageNet, ResNet, and WT103 TransformerXL models, the Neural Network weight variables do not converge to stationary points where the gradient of the loss function vanishes. Remarkably, however, we observe that while weights do not converge to stationary points, the value of the loss function converges. Inspired by this observation, we propose a new perspective based on ergodic theory of dynamical systems. We prove convergence of the distribution of weight values to an approximate invariant measure (without smoothness and assumptions) that explains how the training loss can stabilize without weights necessarily converging to stationary points. We further discuss how this perspective can better align the theory with empirical observations.

Stochastic gradient descent (SGD) has been demonstrated to generalize well in many deep learning applications. In practice, one often runs SGD with a geometrically decaying stepsize, i.e., a constant initial stepsize followed by multiple geometric stepsize decay, and uses the last iterate as the output. This kind of SGD is known to be nearly minimax optimal for classical finite-dimensional linear regression problems (Ge et al., 2019), and provably outperforms SGD with polynomially decaying stepsize in terms of the statistical minimax rates. However, a sharp analysis for the last iterate of SGD with decaying step size in the overparameterized setting is still open. In this paper, we provide problem-dependent analysis on the last iterate risk bounds of SGD with decaying stepsize, for (overparameterized) linear regression problems. In particular, for SGD with geometrically decaying stepsize (or tail geometrically decaying stepsize), we prove nearly matching upper and lower bounds on the excess risk. Our results demonstrate the generalization ability of SGD for a wide class of overparameterized problems, and can recover the minimax optimal results up to logarithmic factors in the classical regime. Moreover, we provide an excess risk lower bound for SGD with polynomially decaying stepsize and illustrate the advantage of geometrically decaying stepsize in an instance-wise manner, which complements the minimax rate comparison made in previous work.

We describe an implicit sparsity-inducing mechanism based on minimization over a family of kernels: \begin{equation*} \min_{\beta, f}~\widehat{\mathbb{E}}[L(Y, f(\beta^{1/q} \odot X)] + \lambda_n \|f\|_{\mathcal{H}_q}^2~~\text{subject to}~~\beta \ge 0, \end{equation*} where $L$ is the loss, $\odot$ is coordinate-wise multiplication and $\mathcal{H}_q$ is the reproducing kernel Hilbert space based on the kernel $k_q(x, x') = h(\|x-x'\|_q^q)$, where $\|\cdot\|_q$ is the $\ell_q$ norm. Using gradient descent to optimize this objective with respect to $\beta$ leads to exactly sparse stationary points with high probability. The sparsity is achieved without using any of the well-known explicit sparsification techniques such as penalization (e.g., $\ell_1$), early stopping or post-processing (e.g., clipping). As an application, we use this sparsity-inducing mechanism to build algorithms consistent for feature selection.

Understanding logistic function is an important prerequisite to understanding logistic regression. So let's start by understanding what logistic function is. Logistic function is a type of sigmoid function that squishes values between 0 and 1. Although sigmoid function is an umbrella term for logistic and other functions, the term is often used to refer to logistic function. For instance, the logistic function is commonly known as sigmoid activation function in the context of neural networks.

Finding the top python packages and libraries that aren't only popular, but get the job done isn't easy. Here's a list to help you out. Out of all the Python scientific libraries and packages available, which ones are not only popular but the most useful in getting the job done? To help you filter down a list of libraries and packages worth adding to your data science toolbox, we have compiled our top picks for aspiring and practicing data scientists. But you'll also want to know how to best use these tools for tricky, real-world data problems.

Concrete is the most commonly used material in civil engineering. That is why lots of research and experiments are done on concrete. In this experiment, it is tried to understand how the compressive strength will be according to the materials in the concrete. Concrete has many properties like shear strength, tensile strength. Compressive strength is one of the most important properties.

We've only seen Supervised learning algorithms in my prior articles. In this post, we will learn about K-means and begin with an example 2D dataset. Following that, we will apply the K-means algorithm to compress images by reducing the number of colors in a picture to only those that are common in that image. In Unsupervised learning, our algorithm will learn from unlabelled data instead from a labelled data. So, what is Unsupervised learning?