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Interference occurs when the potential outcomes of a unit depend on the treatments assigned to other units. That is frequently the case in many domains, such as in the social sciences and infectious disease epidemiology. Often, the interference structure is represented by a network, which is typically assumed to be given and accurate. However, correctly specifying the network can be challenging, as edges can be censored, the structure can change over time, and contamination between clusters may exist. Building on the exposure mapping framework, we derive the bias arising from estimating causal effects under a misspecified interference structure. To address this problem, we propose a novel estimator that uses multiple networks simultaneously and is unbiased if one of the networks correctly represents the interference structure, thus providing robustness to the network specification. Additionally, we propose a sensitivity analysis that quantifies the impact of a postulated misspecification mechanism on the causal estimates. Through simulation studies, we illustrate the bias from assuming an incorrect network and show the bias-variance tradeoff of our proposed network-misspecification-robust estimator. We demonstrate the utility of our methods in two real examples.

The parameters of a machine learning model are typically learned by minimizing a loss function on a set of training data. However, this can come with the risk of overtraining; in order for the model to generalize well, it is of great importance that we are able to find the optimal parameter for the model on the entire population -- not only on the given training sample. In this paper, we construct valid confidence sets for this optimal parameter of a machine learning model, which can be generated using only the training data without any knowledge of the population. We then show that studying the distribution of this confidence set allows us to assign a notion of confidence to arbitrary regions of the parameter space, and we demonstrate that this distribution can be well-approximated using bootstrapping techniques.

Multi-task learning has attracted much attention due to growing multi-purpose research with multiple related data sources. Moreover, transduction with matrix completion is a useful method in multi-label learning. In this paper, we propose a transductive matrix completion algorithm that incorporates a calibration constraint for the features under the multi-task learning framework. The proposed algorithm recovers the incomplete feature matrix and target matrix simultaneously. Fortunately, the calibration information improves the completion results. In particular, we provide a statistical guarantee for the proposed algorithm, and the theoretical improvement induced by calibration information is also studied. Moreover, the proposed algorithm enjoys a sub-linear convergence rate. Several synthetic data experiments are conducted, which show the proposed algorithm out-performs other existing methods, especially when the target matrix is associated with the feature matrix in a nonlinear way.

We study the statistical inference of nonlinear stochastic approximation algorithms utilizing a single trajectory of Markovian data. Our methodology has practical applications in various scenarios, such as Stochastic Gradient Descent (SGD) on autoregressive data and asynchronous Q-Learning. By utilizing the standard stochastic approximation (SA) framework to estimate the target parameter, we establish a functional central limit theorem for its partial-sum process, $\boldsymbol{\phi}_T$. To further support this theory, we provide a matching semiparametric efficient lower bound and a non-asymptotic upper bound on its weak convergence, measured in the L\'evy-Prokhorov metric. This functional central limit theorem forms the basis for our inference method. By selecting any continuous scale-invariant functional $f$, the asymptotic pivotal statistic $f(\boldsymbol{\phi}_T)$ becomes accessible, allowing us to construct an asymptotically valid confidence interval. We analyze the rejection probability of a family of functionals $f_m$, indexed by $m \in \mathbb{N}$, through theoretical and numerical means. The simulation results demonstrate the validity and efficiency of our method.

Differential machine learning (DML) is a recently proposed technique that uses samplewise state derivatives to regularize least square fits to learn conditional expectations of functionals of stochastic processes as functions of state variables. Exploiting the derivative information leads to fewer samples than a vanilla ML approach for the same level of precision. This paper extends the methodology to parametric problems where the processes and functionals also depend on model and contract parameters, respectively. In addition, we propose adaptive parameter sampling to improve relative accuracy when the functionals have different magnitudes for different parameter sets. For calibration, we construct pricing surrogates for calibration instruments and optimize over them globally. We discuss strategies for robust calibration. We demonstrate the usefulness of our methodology on one-factor Cheyette models with benchmark rate volatility specification with an extra stochastic volatility factor on (two-curve) caplet prices at different strikes and maturities, first for parametric pricing, and then by calibrating to a given caplet volatility surface. To allow convenient and efficient simulation of processes and functionals and in particular the corresponding computation of samplewise derivatives, we propose to specify the processes and functionals in a low-code way close to mathematical notation which is then used to generate efficient computation of the functionals and derivatives in TensorFlow.

Statistical power is a measure of the replicability of a categorical hypothesis test. Formally, it is the probability of detecting an effect, if there is a true effect present in the population. Hence, optimizing statistical power as a function of some parameters of a hypothesis test is desirable. However, for most hypothesis tests, the explicit functional form of statistical power for individual model parameters is unknown; but calculating power for a given set of values of those parameters is possible using simulated experiments. These simulated experiments are usually computationally expensive. Hence, developing the entire statistical power manifold using simulations can be very time-consuming. We propose a novel genetic algorithm-based framework for learning statistical power manifolds. For a multiple linear regression $F$-test, we show that the proposed algorithm/framework learns the statistical power manifold much faster as compared to a brute-force approach as the number of queries to the power oracle is significantly reduced. We also show that the quality of learning the manifold improves as the number of iterations increases for the genetic algorithm. Such tools are useful for evaluating statistical power trade-offs when researchers have little information regarding a priori best guesses of primary effect sizes of interest or how sampling variability in non-primary effects impacts power for primary ones.

Synthetic control (SC) methods are commonly used to estimate the treatment effect on a single treated unit in panel data settings. An SC is a weighted average of control units built to match the treated unit, with weights typically estimated by regressing (summaries of) pre-treatment outcomes and measured covariates of the treated unit to those of the control units. However, it has been established that in the absence of a good fit, such regression estimator will generally perform poorly. In this paper, we introduce a proximal causal inference framework to formalize identification and inference for both the SC and ultimately the treatment effect on the treated, based on the observation that control units not contributing to the construction of an SC can be repurposed as proxies of latent confounders. We view the difference in the post-treatment outcomes between the treated unit and the SC as a time series, which opens the door to various time series methods for treatment effect estimation. The proposed framework can accommodate nonlinear models, which allows for binary and count outcomes that are understudied in the SC literature. We illustrate with simulation studies and an application to evaluation of the 1990 German Reunification.

This paper is concerned with the statistical analysis of matrix-valued time series. These are data collected over a network of sensors (typically a set of spatial locations) along time, where a vector of features is observed per time instant per sensor. Thus each sensor is characterized by a vectorial time series. We would like to identify the dependency structure among these sensors and represent it by a graph. When there is only one feature per sensor, the vector auto-regressive models have been widely adapted to infer the structure of Granger causality. The resulting graph is referred to as causal graph. Our first contribution is then extending VAR models to matrix-variate models to serve the purpose of graph learning. Secondly, we propose two online procedures respectively in low and high dimensions, which can update quickly the estimates of coefficients when new samples arrive. In particular in high dimensional regime, a novel Lasso-type is introduced and we develop its homotopy algorithms for the online learning. We also provide an adaptive tuning procedure for the regularization parameter. Lastly, we consider that, the application of AR models onto data usually requires detrending the raw data, however, this step is forbidden in online context. Therefore, we augment the proposed AR models by incorporating trend as extra parameter, and then adapt the online algorithms to the augmented data models, which allow us to simultaneously learn the graph and trend from streaming samples. In this work, we consider primarily the periodic trend. Numerical experiments using both synthetic and real data are performed, whose results support the effectiveness of the proposed methods.

Recent work on algorithmic fairness has largely focused on the fairness of discrete decisions, or classifications. While such decisions are often based on risk score models, the fairness of the risk models themselves has received considerably less attention. Risk models are of interest for a number of reasons, including the fact that they communicate uncertainty about the potential outcomes to users, thus representing a way to enable meaningful human oversight. Here, we address fairness desiderata for risk score models. We identify the provision of similar epistemic value to different groups as a key desideratum for risk score fairness. Further, we address how to assess the fairness of risk score models quantitatively, including a discussion of metric choices and meaningful statistical comparisons between groups. In this context, we also introduce a novel calibration error metric that is less sample size-biased than previously proposed metrics, enabling meaningful comparisons between groups of different sizes. We illustrate our methodology - which is widely applicable in many other settings - in two case studies, one in recidivism risk prediction, and one in risk of major depressive disorder (MDD) prediction.

Classic algorithms and machine learning systems like neural networks are both abundant in everyday life. While classic computer science algorithms are suitable for precise execution of exactly defined tasks such as finding the shortest path in a large graph, neural networks allow learning from data to predict the most likely answer in more complex tasks such as image classification, which cannot be reduced to an exact algorithm. To get the best of both worlds, this thesis explores combining both concepts leading to more robust, better performing, more interpretable, more computationally efficient, and more data efficient architectures. The thesis formalizes the idea of algorithmic supervision, which allows a neural network to learn from or in conjunction with an algorithm. When integrating an algorithm into a neural architecture, it is important that the algorithm is differentiable such that the architecture can be trained end-to-end and gradients can be propagated back through the algorithm in a meaningful way. To make algorithms differentiable, this thesis proposes a general method for continuously relaxing algorithms by perturbing variables and approximating the expectation value in closed form, i.e., without sampling. In addition, this thesis proposes differentiable algorithms, such as differentiable sorting networks, differentiable renderers, and differentiable logic gate networks. Finally, this thesis presents alternative training strategies for learning with algorithms.