In linear regression we wish to estimate the optimum linear least squares predictor for a distribution over d-dimensional input points and real-valued responses, based on a small sample. Under standard random design analysis, where the sample is drawn i.i.d. from the input distribution, the least squares solution for that sample can be viewed as the natural estimator of the optimum. Unfortunately, this estimator almost always incurs an undesirable bias coming from the randomness of the input points. In this paper we show that it is possible to draw a non-i.i.d. sample of input points such that, regardless of the response model, the least squares solution is an unbiased estimator of the optimum. Moreover, this sample can be produced efficiently by augmenting a previously drawn i.i.d. sample with an additional set of d points drawn jointly from the input distribution rescaled by the squared volume spanned by the points. Motivated by this, we develop a theoretical framework for studying volume-rescaled sampling, and in the process prove a number of new matrix expectation identities. We use them to show that for any input distribution and $\epsilon>0$ there is a random design consisting of $O(d\log d+ d/\epsilon)$ points from which an unbiased estimator can be constructed whose square loss over the entire distribution is with high probability bounded by $1+\epsilon$ times the loss of the optimum. We provide efficient algorithms for generating such unbiased estimators in a number of practical settings and support our claims experimentally.

The exclusive or (xor) function is one of the simplest examples that illustrate why nonlinear feedforward networks are superior to linear regression for machine learning applications. We review the xor representation and approximation problems and discuss their solutions in terms of probabilistic logic and associative copula functions. After briefly reviewing the specification of feedforward networks, we compare the dynamics of learned error surfaces with different activation functions such as RELU and tanh through a set of colorful three-dimensional charts. The copula representations extend xor from Boolean to real values, thereby providing a convenient way to demonstrate the concept of cross-validation on in-sample and out-sample data sets. Our approach is pedagogical and is meant to be a machine learning prolegomenon. Keywords: machine learning; neural networks; probabilistic logic; copulas; error surfaces; xor.

We consider the problem of estimating the common mean of independently sampled data, where samples are drawn in a possibly non-identical manner from symmetric, unimodal distributions with a common mean. This generalizes the setting of Gaussian mixture modeling, since the number of distinct mixture components may diverge with the number of observations. We propose an estimator that adapts to the level of heterogeneity in the data, achieving near-optimality in both the i.i.d. setting and some heterogeneous settings, where the fraction of ``low-noise'' points is as small as $\frac{\log n}{n}$. Our estimator is a hybrid of the modal interval, shorth, and median estimators from classical statistics; however, the key technical contributions rely on novel empirical process theory results that we derive for independent but non-i.i.d. data. In the multivariate setting, we generalize our theory to mean estimation for mixtures of radially symmetric distributions, and derive minimax lower bounds on the expected error of any estimator that is agnostic to the scales of individual data points. Finally, we describe an extension of our estimators applicable to linear regression. In the multivariate mean estimation and regression settings, we present computationally feasible versions of our estimators that run in time polynomial in the number of data points.

Kernel-based machine learning approaches are gaining increasing interest for exploring and modeling large dataset in recent years. Gaussian process (GP) is one example of such kernel-based approaches, which can provide very good performance for nonlinear modeling problems. In this work, we first propose a grey-box modeling approach to analyze the forces in cross country skiing races. To be more precise, a disciplined set of kinetic motion model formulae is combined with data-driven Gaussian process regression model, which accounts for everything unknown in the system. Then, a modeling approach is proposed to analyze the kinetic flow of both individual and clusters of skiers. The proposed approaches can be generally applied to use cases where positioned trajectories and kinetic measurements are available. The proposed approaches are evaluated using data collected from the Falun Nordic World Ski Championships 2015, in particular the Men's cross country $4\times10$ km relay. Forces during the cross country skiing races are analyzed and compared. Velocity models for skiers at different competition stages are also evaluated. Finally, the comparisons between the grey-box and black-box approach are carried out, where the grey-box approach can reduce the predictive uncertainty by $30\%$ to $40\%$.

In recent years, there have been significant efforts on mitigating unethical demographic biases in machine learning methods. However, very little is done for kernel methods. In this paper, we propose a new fair kernel regression method via fair feature embedding (FKR-F$^2$E) in kernel space. Motivated by prior works on feature selection in kernel space and feature processing for fair machine learning, we propose to learn fair feature embedding functions that minimize demographic discrepancy of feature distributions in kernel space. Compared to the state-of-the-art fair kernel regression method and several baseline methods, we show FKR-F$^2$E achieves significantly lower prediction disparity across three real-world data sets.

The Alternating Direction Method of Multipliers (ADMM) has now days gained tremendous attentions for solving large-scale machine learning and signal processing problems due to the relative simplicity. However, the two-block structure of the classical ADMM still limits the size of the real problems being solved. When one forces a more-than-two-block structure by variable-splitting, the convergence speed slows down greatly as observed in practice. Recently, a randomly assembled cyclic multi-block ADMM (RAC-MBADMM) was developed by the authors for solving general convex and nonconvex quadratic optimization problems where the number of blocks can go greater than two so that each sub-problem has a smaller size and can be solved much more efficiently. In this paper, we apply this method to solving few selected machine learning problems related to convex quadratic optimization, such as Linear Regression, LASSO, Elastic-Net, and SVM. We prove that the algorithm would converge in expectation linearly under the standard statistical data assumptions. We use our general-purpose solver to conduct multiple numerical tests, solving both synthetic and large-scale bench-mark problems. Our results show that RAC-MBADMM could significantly outperform, in both solution time and quality, other optimization algorithms/codes for solving these machine learning problems, and match up the performance of the best tailored methods such as Glmnet or LIBSVM. In certain problem regions RAC-MBADMM even achieves a superior performance than that of the tailored methods.

Federated learning enables a large amount of edge computing devices to learn a centralized model while keeping all local data on edge devices. As a leading algorithm in this setting, Federated Averaging (\texttt{FedAvg}) runs Stochastic Gradient Descent (SGD) in parallel on a small subset of the total devices and averages the sequences only once in a while. Despite its simplicity, it lacks theoretical guarantees in the federated setting. In this paper, we analyze the convergence of \texttt{FedAvg} on non-iid data. We investigate the effect of different sampling and averaging schemes, which are crucial especially when data are unbalanced. We prove a concise convergence rate of $\mathcal{O}(\frac{1}{T})$ for \texttt{FedAvg} with proper sampling and averaging schemes in convex problems, where $T$ is the total number of steps. Our results show that heterogeneity of data slows down the convergence, which is intrinsic in the federated setting. Low device participation rate can be achieved without severely harming the optimization process in federated learning. We show that there is a trade-off between communication efficiency and convergence rate. We analyze the necessity of learning rate decay by taking a linear regression as an example. Our work serves as a guideline for algorithm design in applications of federated learning, where heterogeneity and unbalance of data are the common case.

We study the application of dynamic pricing to insurance. We view this as an online revenue management problem where the insurance company looks to set prices to optimize the long-run revenue from selling a new insurance product. We develop two pricing models: an adaptive Generalized Linear Model (GLM) and an adaptive Gaussian Process (GP) regression model. Both balance between exploration, where we choose prices in order to learn the distribution of demands & claims for the insurance product, and exploitation, where we myopically choose the best price from the information gathered so far. The performance of the pricing policies is measured in terms of regret: the expected revenue loss caused by not using the optimal price. As is commonplace in insurance, we model demand and claims by GLMs. In our adaptive GLM design, we use the maximum quasi-likelihood estimation (MQLE) to estimate the unknown parameters. We show that, if prices are chosen with suitably decreasing variability, the MQLE parameters eventually exist and converge to the correct values, which in turn implies that the sequence of chosen prices will also converge to the optimal price. In the adaptive GP regression model, we sample demand and claims from Gaussian Processes and then choose selling prices by the upper confidence bound rule. We also analyze these GLM and GP pricing algorithms with delayed claims. Although similar results exist in other domains, this is among the first works to consider dynamic pricing problems in the field of insurance. We also believe this is the first work to consider Gaussian Process regression in the context of insurance pricing. These initial findings suggest that online machine learning algorithms could be a fruitful area of future investigation and application in insurance.

An $\varepsilon$-coreset for Least-Mean-Squares (LMS) of a matrix $A\in{\mathbb{R}}^{n\times d}$ is a small weighted subset of its rows that approximates the sum of squared distances from its rows to every affine $k$-dimensional subspace of ${\mathbb{R}}^d$, up to a factor of $1\pm\varepsilon$. Such coresets are useful for hyper-parameter tuning and solving many least-mean-squares problems such as low-rank approximation ($k$-SVD), $k$-PCA, Lassso/Ridge/Linear regression and many more. Coresets are also useful for handling streaming, dynamic and distributed big data in parallel. With high probability, non-uniform sampling based on upper bounds on what is known as importance or sensitivity of each row in $A$ yields a coreset. The size of the (sampled) coreset is then near-linear in the total sum of these sensitivity bounds. We provide algorithms that compute provably \emph{tight} bounds for the sensitivity of each input row. It is based on two ingredients: (i) iterative algorithm that computes the exact sensitivity of each point up to arbitrary small precision for (non-affine) $k$-subspaces, and (ii) a general reduction of independent interest from computing sensitivity for the family of affine $k$-subspaces in ${\mathbb{R}}^d$ to (non-affine) $(k+1)$- subspaces in ${\mathbb{R}}^{d+1}$. Experimental results on real-world datasets, including the English Wikipedia documents-term matrix, show that our bounds provide significantly smaller and data-dependent coresets also in practice. Full open source is also provided.

Many people think that artificial intelligence and machine learning are recent phenomena. However, these techniques and ideas actually go back deep into human history. Machine learning has always been an important tool for data mining for humanity, it was given different names in different eras. The key to machine learning is not machines but mathematics. There is nothing special about silicon and electricity.