Expected Shortfall (ES), also known as superquantile or Conditional Value-at-Risk, has been recognized as an important measure in risk analysis and stochastic optimization, and is also finding applications beyond these areas. In finance, it refers to the conditional expected return of an asset given that the return is below some quantile of its distribution. In this paper, we consider a recently proposed joint regression framework that simultaneously models the quantile and the ES of a response variable given a set of covariates, for which the state-of-the-art approach is based on minimizing a joint loss function that is non-differentiable and non-convex. This inevitably raises numerical challenges and limits its applicability for analyzing large-scale data. Motivated by the idea of using Neyman-orthogonal scores to reduce sensitivity with respect to nuisance parameters, we propose a statistically robust (to highly skewed and heavy-tailed data) and computationally efficient two-step procedure for fitting joint quantile and ES regression models. With increasing covariate dimensions, we establish explicit non-asymptotic bounds on estimation and Gaussian approximation errors, which lay the foundation for statistical inference. Finally, we demonstrate through numerical experiments and two data applications that our approach well balances robustness, statistical, and numerical efficiencies for expected shortfall regression.
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This paper characterizes the impact of covariate serial dependence on the non-asymptotic estimation error bound of penalized regressions (PRs). Focusing on the direct relationship between the degree of cross-correlation between covariates and the estimation error bound of PRs, we show that orthogonal or weakly cross-correlated stationary AR processes can exhibit high spurious correlations caused by serial dependence. We provide analytical results on the distribution of the sample cross-correlation in the case of two orthogonal Gaussian AR(1) processes, and extend and validate them through an extensive simulation study. Furthermore, we introduce a new procedure to mitigate spurious correlations in a time series setting, applying PRs to pre-whitened (ARMA filtered) time series. We show that under mild assumptions our procedure allows both to reduce the estimation error and to develop an effective forecasting strategy. The estimation accuracy of our proposal is validated through additional simulations, as well as an empirical application to a large set of monthly macroeconomic time series relative to the Euro Area.
With the modern software and online platforms to collect massive amount of data, there is an increasing demand of applying causal inference methods at large scale when randomized experimentation is not viable. Weighting methods that directly incorporate covariate balancing have recently gained popularity for estimating causal effects in observational studies. These methods reduce the manual efforts required by researchers to iterate between propensity score modeling and balance checking until a satisfied covariate balance result. However, conventional solvers for determining weights lack the scalability to apply such methods on large scale datasets in companies like Snap Inc. To address the limitations and improve computational efficiency, in this paper we present scalable algorithms, DistEB and DistMS, for two balancing approaches: entropy balancing and MicroSynth. The solvers have linear time complexity and can be conveniently implemented in distributed computing frameworks such as Spark, Hive, etc. We study the properties of balancing approaches at different scales up to 1 million treated units and 487 covariates. We find that with larger sample size, both bias and variance in the causal effect estimation are significantly reduced. The results emphasize the importance of applying balancing approaches on large scale datasets. We combine the balancing approach with a synthetic control framework and deploy an end-to-end system for causal impact estimation at Snap Inc.
We study the interplay between the data distribution and Q-learning-based algorithms with function approximation. We provide a unified theoretical and empirical analysis as to how different properties of the data distribution influence the performance of Q-learning-based algorithms. We connect different lines of research, as well as validate and extend previous results. We start by reviewing theoretical bounds on the performance of approximate dynamic programming algorithms. We then introduce a novel four-state MDP specifically tailored to highlight the impact of the data distribution in the performance of Q-learning-based algorithms with function approximation, both online and offline. Finally, we experimentally assess the impact of the data distribution properties on the performance of two offline Q-learning-based algorithms under different environments. According to our results: (i) high entropy data distributions are well-suited for learning in an offline manner; and (ii) a certain degree of data diversity (data coverage) and data quality (closeness to optimal policy) are jointly desirable for offline learning.
We study a multi-objective multi-armed bandit problem in a dynamic environment. The problem portrays a decision-maker that sequentially selects an arm from a given set. If selected, each action produces a reward vector, where every element follows a piecewise-stationary Bernoulli distribution. The agent aims at choosing an arm among the Pareto optimal set of arms to minimize its regret. We propose a Pareto generic upper confidence bound (UCB)-based algorithm with change detection to solve this problem. By developing the essential inequalities for multi-dimensional spaces, we establish that our proposal guarantees a regret bound in the order of $\gamma_T\log(T/{\gamma_T})$ when the number of breakpoints $\gamma_T$ is known. Without this assumption, the regret bound of our algorithm is $\gamma_T\log(T)$. Finally, we formulate an energy-efficient waveform design problem in an integrated communication and sensing system as a toy example. Numerical experiments on the toy example and synthetic and real-world datasets demonstrate the efficiency of our policy compared to the current methods.
When estimating a Global Average Treatment Effect (GATE) under network interference, units can have widely different relationships to the treatment depending on a combination of the structure of their network neighborhood, the structure of the interference mechanism, and how the treatment was distributed in their neighborhood. In this work, we introduce a sequential procedure to generate and select graph- and treatment-based covariates for GATE estimation under regression adjustment. We show that it is possible to simultaneously achieve low bias and considerably reduce variance with such a procedure. To tackle inferential complications caused by our feature generation and selection process, we introduce a way to construct confidence intervals based on a block bootstrap. We illustrate that our selection procedure and subsequent estimator can achieve good performance in terms of root mean squared error in several semi-synthetic experiments with Bernoulli designs, comparing favorably to an oracle estimator that takes advantage of regression adjustments for the known underlying interference structure. We apply our method to a real world experimental dataset with strong evidence of interference and demonstrate that it can estimate the GATE reasonably well without knowing the interference process a priori.
We present an efficient semiparametric variational method to approximate the Gibbs posterior distribution of Bayesian regression models, which predict the data through a linear combination of the available covariates. Remarkable cases are generalized linear mixed models, support vector machines, quantile and expectile regression. The variational optimization algorithm we propose only involves the calculation of univariate numerical integrals, when no analytic solutions are available. Neither differentiability, nor conjugacy, nor elaborate data-augmentation strategies are required. Several generalizations of the proposed approach are discussed in order to account for additive models, shrinkage priors, dynamic and spatial models, providing a unifying framework for statistical learning that cover a wide range of applications. The properties of our semiparametric variational approximation are then assessed through a theoretical analysis and an extensive simulation study, in which we compare our proposal with Markov chain Monte Carlo, conjugate mean field variational Bayes and Laplace approximation in terms of signal reconstruction, posterior approximation accuracy and execution time. A real data example is then presented through a probabilistic load forecasting application on the US power load consumption data.
Recent advances in Transformer architectures have empowered their empirical success in a variety of tasks across different domains. However, existing works mainly focus on predictive accuracy and computational cost, without considering other practical issues, such as robustness to contaminated samples. Recent work by Nguyen et al., (2022) has shown that the self-attention mechanism, which is the center of the Transformer architecture, can be viewed as a non-parametric estimator based on kernel density estimation (KDE). This motivates us to leverage a set of robust kernel density estimation methods for alleviating the issue of data contamination. Specifically, we introduce a series of self-attention mechanisms that can be incorporated into different Transformer architectures and discuss the special properties of each method. We then perform extensive empirical studies on language modeling and image classification tasks. Our methods demonstrate robust performance in multiple scenarios while maintaining competitive results on clean datasets.
We investigate a simple objective for nonlinear instrumental variable (IV) regression based on a kernelized conditional moment restriction (CMR) known as a maximum moment restriction (MMR). The MMR objective is formulated by maximizing the interaction between the residual and the instruments belonging to a unit ball in a reproducing kernel Hilbert space (RKHS). First, it allows us to simplify the IV regression as an empirical risk minimization problem, where the risk functional depends on the reproducing kernel on the instrument and can be estimated by a U-statistic or V-statistic. Second, based on this simplification, we are able to provide the consistency and asymptotic normality results in both parametric and nonparametric settings. Lastly, we provide easy-to-use IV regression algorithms with an efficient hyper-parameter selection procedure. We demonstrate the effectiveness of our algorithms using experiments on both synthetic and real-world data.
We consider the differentially private estimation of multiple quantiles (MQ) of a distribution from a dataset, a key building block in modern data analysis. We apply the recent non-smoothed Inverse Sensitivity (IS) mechanism to this specific problem. We establish that the resulting method is closely related to the recently published ad hoc algorithm JointExp. In particular, they share the same computational complexity and a similar efficiency. We prove the statistical consistency of these two algorithms for continuous distributions. Furthermore, we demonstrate both theoretically and empirically that this method suffers from an important lack of performance in the case of peaked distributions, which can degrade up to a potentially catastrophic impact in the presence of atoms. Its smoothed version (i.e. by applying a max kernel to its output density) would solve this problem, but remains an open challenge to implement. As a proxy, we propose a simple and numerically efficient method called Heuristically Smoothed JointExp (HSJointExp), which is endowed with performance guarantees for a broad class of distributions and achieves results that are orders of magnitude better on problematic datasets.
Although understanding and characterizing causal effects have become essential in observational studies, it is challenging when the confounders are high-dimensional. In this article, we develop a general framework $\textit{CausalEGM}$ for estimating causal effects by encoding generative modeling, which can be applied in both binary and continuous treatment settings. Under the potential outcome framework with unconfoundedness, we establish a bidirectional transformation between the high-dimensional confounders space and a low-dimensional latent space where the density is known (e.g., multivariate normal distribution). Through this, CausalEGM simultaneously decouples the dependencies of confounders on both treatment and outcome and maps the confounders to the low-dimensional latent space. By conditioning on the low-dimensional latent features, CausalEGM can estimate the causal effect for each individual or the average causal effect within a population. Our theoretical analysis shows that the excess risk for CausalEGM can be bounded through empirical process theory. Under an assumption on encoder-decoder networks, the consistency of the estimate can be guaranteed. In a series of experiments, CausalEGM demonstrates superior performance over existing methods for both binary and continuous treatments. Specifically, we find CausalEGM to be substantially more powerful than competing methods in the presence of large sample sizes and high dimensional confounders. The software of CausalEGM is freely available at //github.com/SUwonglab/CausalEGM.
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