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Causal mediation analysis concerns the pathways through which a treatment affects an outcome. While most of the mediation literature focuses on settings with a single mediator, a flourishing line of research has examined settings involving multiple mediators, under which path-specific effects (PSEs) are often of interest. We consider estimation of PSEs when the treatment effect operates through K(\geq1) causally ordered, possibly multivariate mediators. In this setting, the PSEs for many causal paths are not nonparametrically identified, and we focus on a set of PSEs that are identified under Pearl's nonparametric structural equation model. These PSEs are defined as contrasts between the expectations of 2^{K+1} potential outcomes and identified via what we call the generalized mediation functional (GMF). We introduce an array of regression-imputation, weighting, and "hybrid" estimators, and, in particular, two K+2-robust and locally semiparametric efficient estimators for the GMF. The latter estimators are well suited to the use of data-adaptive methods for estimating their nuisance functions. We establish the rate conditions required of the nuisance functions for semiparametric efficiency. We also discuss how our framework applies to several estimands that may be of particular interest in empirical applications. The proposed estimators are illustrated with a simulation study and an empirical example.

We observe $n$ pairs of independent random variables $X_{1}=(W_{1},Y_{1}),\ldots,X_{n}=(W_{n},Y_{n})$ and assume, although this might not be true, that for each $i\in\{1,\ldots,n\}$, the conditional distribution of $Y_{i}$ given $W_{i}$ belongs to a given exponential family with real parameter $\theta_{i}^{\star}=\boldsymbol{\theta}^{\star}(W_{i})$ the value of which is an unknown function $\boldsymbol{\theta}^{\star}$ of the covariate $W_{i}$. Given a model $\boldsymbol{\overline\Theta}$ for $\boldsymbol{\theta}^{\star}$, we propose an estimator $\boldsymbol{\widehat \theta}$ with values in $\boldsymbol{\overline\Theta}$ the construction of which is independent of the distribution of the $W_{i}$. We show that $\boldsymbol{\widehat \theta}$ possesses the properties of being robust to contamination, outliers and model misspecification. We establish non-asymptotic exponential inequalities for the upper deviations of a Hellinger-type distance between the true distribution of the data and the estimated one based on $\boldsymbol{\widehat \theta}$. We deduce a uniform risk bound for $\boldsymbol{\widehat \theta}$ over the class of H\"olderian functions and we prove the optimality of this bound up to a logarithmic factor. Finally, we provide an algorithm for calculating $\boldsymbol{\widehat \theta}$ when $\boldsymbol{\theta}^{\star}$ is assumed to belong to functional classes of low or medium dimensions (in a suitable sense) and, on a simulation study, we compare the performance of $\boldsymbol{\widehat \theta}$ to that of the MLE and median-based estimators. The proof of our main result relies on an upper bound, with explicit numerical constants, on the expectation of the supremum of an empirical process over a VC-subgraph class. This bound can be of independent interest.

Model-free deep reinforcement learning (RL) has been successfully applied to challenging continuous control domains. However, poor sample efficiency prevents these methods from being widely used in real-world domains. We address this problem by proposing a novel model-free algorithm, Realistic Actor-Critic(RAC), which aims to solve trade-offs between value underestimation and overestimation by learning a policy family concerning various confidence-bounds of Q-function. We construct uncertainty punished Q-learning(UPQ), which uses uncertainty from the ensembling of multiple critics to control estimation bias of Q-function, making Q-functions smoothly shift from lower- to higher-confidence bounds. With the guide of these critics, RAC employs Universal Value Function Approximators (UVFA) to simultaneously learn many optimistic and pessimistic policies with the same neural network. Optimistic policies generate effective exploratory behaviors, while pessimistic policies reduce the risk of value overestimation to ensure stable updates of policies and Q-functions. The proposed method can be incorporated with any off-policy actor-critic RL algorithms. Our method achieve 10x sample efficiency and 25\% performance improvement compared to SAC on the most challenging Humanoid environment, obtaining the episode reward $11107\pm 475$ at $10^6$ time steps. All the source codes are available at //github.com/ihuhuhu/RAC.

Numerous studies have demonstrated that deep neural networks are easily misled by adversarial examples. Effectively evaluating the adversarial robustness of a model is important for its deployment in practical applications. Currently, a common type of evaluation is to approximate the adversarial risk of a model as a robustness indicator by constructing malicious instances and executing attacks. Unfortunately, there is an error (gap) between the approximate value and the true value. Previous studies manually design attack methods to achieve a smaller error, which is inefficient and may miss a better solution. In this paper, we establish the tightening of the approximation error as an optimization problem and try to solve it with an algorithm. More specifically, we first analyze that replacing the non-convex and discontinuous 0-1 loss with a surrogate loss, a necessary compromise in calculating the approximation, is one of the main reasons for the error. Then we propose AutoLoss-AR, the first method for searching loss functions for tightening the approximation error of adversarial risk. Extensive experiments are conducted in multiple settings. The results demonstrate the effectiveness of the proposed method: the best-discovered loss functions outperform the handcrafted baseline by 0.9%-2.9% and 0.7%-2.0% on MNIST and CIFAR-10, respectively. Besides, we also verify that the searched losses can be transferred to other settings and explore why they are better than the baseline by visualizing the local loss landscape.

First, we analyze the variance of the Cross Validation (CV)-based estimators used for estimating the performance of classification rules. Second, we propose a novel estimator to estimate this variance using the Influence Function (IF) approach that had been used previously very successfully to estimate the variance of the bootstrap-based estimators. The motivation for this research is that, as the best of our knowledge, the literature lacks a rigorous method for estimating the variance of the CV-based estimators. What is available is a set of ad-hoc procedures that have no mathematical foundation since they ignore the covariance structure among dependent random variables. The conducted experiments show that the IF proposed method has small RMS error with some bias. However, surprisingly, the ad-hoc methods still work better than the IF-based method. Unfortunately, this is due to the lack of enough smoothness if compared to the bootstrap estimator. This opens the research for three points: (1) more comprehensive simulation study to clarify when the IF method win or loose; (2) more mathematical analysis to figure out why the ad-hoc methods work well; and (3) more mathematical treatment to figure out the connection between the appropriate amount of "smoothness" and decreasing the bias of the IF method.

In this paper, new results in random matrix theory are derived which allow us to construct a shrinkage estimator of the global minimum variance (GMV) portfolio when the shrinkage target is a random object. More specifically, the shrinkage target is determined as the holding portfolio estimated from previous data. The theoretical findings are applied to develop theory for dynamic estimation of the GMV portfolio, where the new estimator of its weights is shrunk to the holding portfolio at each time of reconstruction. Both cases with and without overlapping samples are considered in the paper. The non-overlapping samples corresponds to the case when different data of the asset returns are used to construct the traditional estimator of the GMV portfolio weights and to determine the target portfolio, while the overlapping case allows intersections between the samples. The theoretical results are derived under weak assumptions imposed on the data-generating process. No specific distribution is assumed for the asset returns except from the assumption of finite $4+\varepsilon$, $\varepsilon>0$, moments. Also, the population covariance matrix with unbounded spectrum can be considered. The performance of new trading strategies is investigated via an extensive simulation. Finally, the theoretical findings are implemented in an empirical illustration based on the returns on stocks included in the S\&P 500 index.

In this paper we estimate the mean-variance portfolio in the high-dimensional case using the recent results from the theory of random matrices. We construct a linear shrinkage estimator which is distribution-free and is optimal in the sense of maximizing with probability $1$ the asymptotic out-of-sample expected utility, i.e., mean-variance objective function for different values of risk aversion coefficient which in particular leads to the maximization of the out-of-sample expected utility and to the minimization of the out-of-sample variance. One of the main features of our estimator is the inclusion of the estimation risk related to the sample mean vector into the high-dimensional portfolio optimization. The asymptotic properties of the new estimator are investigated when the number of assets $p$ and the sample size $n$ tend simultaneously to infinity such that $p/n \rightarrow c\in (0,+\infty)$. The results are obtained under weak assumptions imposed on the distribution of the asset returns, namely the existence of the $4+\varepsilon$ moments is only required. Thereafter we perform numerical and empirical studies where the small- and large-sample behavior of the derived estimator is investigated. The suggested estimator shows significant improvements over the existent approaches including the nonlinear shrinkage estimator and the three-fund portfolio rule, especially when the portfolio dimension is larger than the sample size. Moreover, it is robust to deviations from normality.

Precision medicine aims to tailor treatment decisions according to patients' characteristics. G-estimation and dynamic weighted ordinary least squares (dWOLS) are double robust statistical methods that can be used to identify optimal adaptive treatment strategies. They require both a model for the outcome and a model for the treatment and are consistent if at least one of these models is correctly specified. It is underappreciated that these methods additionally require modeling all existing treatment-confounder interactions to yield consistent estimators. Identifying partially adaptive treatment strategies that tailor treatments according to only a few covariates, ignoring some interactions, may be preferable in practice. It has been proposed to combine inverse probability weighting and G-estimation to address this issue, but we argue that the resulting estimator is not expected to be double robust. Building on G-estimation and dWOLS, we propose alternative estimators of partially adaptive strategies and demonstrate their double robustness. We investigate and compare the empirical performance of six estimators in a simulation study. As expected, estimators combining inverse probability weighting with either G-estimation or dWOLS are biased when the treatment model is incorrectly specified. The other estimators are unbiased if either the treatment or the outcome model are correctly specified and have similar standard errors. Using data maintained by the Centre des Maladies du Sein, the methods are illustrated to estimate a partially adaptive treatment strategy for tailoring hormonal therapy use in breast cancer patients according to their estrogen receptor status and body mass index. R software implementing our estimators is provided.

Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.

Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): finding mappings of request graphs (describing the workloads) onto a substrate graph (describing the physical infrastructure). In the offline setting, the two natural objectives are profit maximization, i.e., embedding a maximal number of request graphs subject to the resource constraints, and cost minimization, i.e., embedding all requests at minimal overall cost. The VNEP can be seen as a generalization of classic routing and call admission problems, in which requests are arbitrary graphs whose communication endpoints are not fixed. Due to its applications, the problem has been studied intensively in the networking community. However, the underlying algorithmic problem is hardly understood. This paper presents the first fixed-parameter tractable approximation algorithms for the VNEP. Our algorithms are based on randomized rounding. Due to the flexible mapping options and the arbitrary request graph topologies, we show that a novel linear program formulation is required. Only using this novel formulation the computation of convex combinations of valid mappings is enabled, as the formulation needs to account for the structure of the request graphs. Accordingly, to capture the structure of request graphs, we introduce the graph-theoretic notion of extraction orders and extraction width and show that our algorithms have exponential runtime in the request graphs' maximal width. Hence, for request graphs of fixed extraction width, we obtain the first polynomial-time approximations. Studying the new notion of extraction orders we show that (i) computing extraction orders of minimal width is NP-hard and (ii) that computing decomposable LP solutions is in general NP-hard, even when restricting request graphs to planar ones.