In this paper, we introduce the weighted-average quantile regression framework, $\int_0^1 q_{Y|X}(u)\psi(u)du = X'\beta$, where $Y$ is a dependent variable, $X$ is a vector of covariates, $q_{Y|X}$ is the quantile function of the conditional distribution of $Y$ given $X$, $\psi$ is a weighting function, and $\beta$ is a vector of parameters. We argue that this framework is of interest in many applied settings and develop an estimator of the vector of parameters $\beta$. We show that our estimator is $\sqrt T$-consistent and asymptotically normal with mean zero and easily estimable covariance matrix, where $T$ is the size of available sample. We demonstrate the usefulness of our estimator by applying it in two empirical settings. In the first setting, we focus on financial data and study the factor structures of the expected shortfalls of the industry portfolios. In the second setting, we focus on wage data and study inequality and social welfare dependence on commonly used individual characteristics.

Before we look at any code, we should understand a little about the math behind a regression model. As mentioned, regression models can have multiple input variables, or features, but for this article, we will use a single feature for simplicity. Regression analysis involves making a guess at what type of function would fit your dataset the best, whether that be a line, an nth degree polynomial, a logarithmic function, etc. Regression models assume the dataset follows this form: Here, x and y are our feature and response at observation i, and e is an error term. The goal of the regression model is to estimate the function, f, so that it most closely fits the dataset (neglecting the error term). The function, f, is the guess we make about what type of function would best fit our dataset.

Sparse linear regression with ill-conditioned Gaussian random designs is widely believed to exhibit a statistical/computational gap, but there is surprisingly little formal evidence for this belief, even in the form of examples that are hard for restricted classes of algorithms. Recent work has shown that, for certain covariance matrices, the broad class of Preconditioned Lasso programs provably cannot succeed on polylogarithmically sparse signals with a sublinear number of samples. However, this lower bound only shows that for every preconditioner, there exists at least one signal that it fails to recover successfully. This leaves open the possibility that, for example, trying multiple different preconditioners solves every sparse linear regression problem. In this work, we prove a stronger lower bound that overcomes this issue. For an appropriate covariance matrix, we construct a single signal distribution on which any invertibly-preconditioned Lasso program fails with high probability, unless it receives a linear number of samples. Surprisingly, at the heart of our lower bound is a new positive result in compressed sensing. We show that standard sparse random designs are with high probability robust to adversarial measurement erasures, in the sense that if $b$ measurements are erased, then all but $O(b)$ of the coordinates of the signal are still information-theoretically identifiable. To our knowledge, this is the first time that partial recoverability of arbitrary sparse signals under erasures has been studied in compressed sensing.

Multiplying the weights with the adjacency matrix means that all the feature vectors of all (1-hop) neighboring nodes are summed and aggregated for every node. However, the feature vector of the node itself is not included. To address this, Kipf and Welling [4] add the identity matrix to the adjacency matrix and denote this new matrix  A I. Multiplication of the adjacency matrix will also change the scale of the feature vectors.

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We analyse the convergence of the proximal gradient algorithm for convex composite problems in the presence of gradient and proximal computational inaccuracies. We derive new tighter deterministic and probabilistic bounds that we use to verify a simulated (MPC) and a synthetic (LASSO) optimization problems solved on a reduced-precision machine in combination with an inaccurate proximal operator. We also show how the probabilistic bounds are more robust for algorithm verification and more accurate for application performance guarantees. Under some statistical assumptions, we also prove that some cumulative error terms follow a martingale property. And conforming to observations, e.g., in \cite{schmidt2011convergence}, we also show how the acceleration of the algorithm amplifies the gradient and proximal computational errors.

Simulated annealing (SA) is a stochastic global optimisation technique applicable to a wide range of discrete and continuous variable problems. Despite its simplicity, the development of an effective SA optimiser for a given problem hinges on a handful of carefully handpicked components; namely, neighbour proposal distribution and temperature annealing schedule. In this work, we view SA from a reinforcement learning perspective and frame the proposal distribution as a policy, which can be optimised for higher solution quality given a fixed computational budget. We demonstrate that this Neural SA with such a learnt proposal distribution, parametrised by small equivariant neural networks, outperforms SA baselines on a number of problems: Rosenbrock's function, the Knapsack problem, the Bin Packing problem, and the Travelling Salesperson problem. We also show that Neural SA scales well to large problems - generalising to significantly larger problems than the ones seen during training - while achieving comparable performance to popular off-the-shelf solvers and other machine learning methods in terms of solution quality and wall-clock time.

A treatment regime is a rule that assigns a treatment to patients based on their covariate information. Recently, estimation of the optimal treatment regime that yields the greatest overall expected clinical outcome of interest has attracted a lot of attention. In this work, we consider estimation of the optimal treatment regime with electronic medical record data under a semi-supervised setting. Here, data consist of two parts: a set of `labeled' patients for whom we have the covariate, treatment and outcome information, and a much larger set of `unlabeled' patients for whom we only have the covariate information. We proposes an imputation-based semi-supervised method, utilizing `unlabeled' individuals to obtain a more efficient estimator of the optimal treatment regime. The asymptotic properties of the proposed estimators and their associated inference procedure are provided. Simulation studies are conducted to assess the empirical performance of the proposed method and to compare with a fully supervised method using only the labeled data. An application to an electronic medical record data set on the treatment of hypotensive episodes during intensive care unit (ICU) stays is also given for further illustration.

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