A recent literature considers causal inference using noisy proxies for unobserved confounding factors. The proxies are divided into two sets that are independent conditional on the confounders. One set of proxies are `negative control treatments' and the other are `negative control outcomes'. Existing work applies to low-dimensional settings with a fixed number of proxies and confounders. In this work we consider linear models with many proxy controls and possibly many confounders. A key insight is that if each group of proxies is strictly larger than the number of confounding factors, then a matrix of nuisance parameters has a low-rank structure and a vector of nuisance parameters has a sparse structure. We can exploit the rank-restriction and sparsity to reduce the number of free parameters to be estimated. The number of unobserved confounders is not known a priori but we show that it is identified, and we apply penalization methods to adapt to this quantity. We provide an estimator with a closed-form as well as a doubly-robust estimator that must be evaluated using numerical methods. We provide conditions under which our doubly-robust estimator is uniformly root-$n$ consistent, asymptotically centered normal, and our suggested confidence intervals have asymptotically correct coverage. We provide simulation evidence that our methods achieve better performance than existing approaches in high dimensions, particularly when the number of proxies is substantially larger than the number of confounders.
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Models with intractable normalising functions have numerous applications ranging from network models to image analysis to spatial point processes. Because the normalising constants are functions of the parameters of interest, standard Markov chain Monte Carlo cannot be used for Bayesian inference for these models. A number of algorithms have been developed for such models. Some have the posterior distribution as the asymptotic distribution. Other "asymptotically inexact" algorithms do not possess this property. There is limited guidance for evaluating approximations based on these algorithms, and hence it is very hard to tune them. We propose two new diagnostics that address these problems for intractable normalising function models. Our first diagnostic, inspired by the second Bartlett identity, is, in principle, applicable in most any likelihood-based context where misspecification is of concern. We develop an approximate version that is applicable to intractable normalising function problems. Our second diagnostic is a Monte Carlo approximation to a kernel Stein discrepancy-based diagnostic introduced by Gorham and Mackey (2017). We provide theoretical justification for our methods and apply them to several algorithms in the context of challenging simulated and real data examples including an Ising model, an exponential random graph model, and a Markov point process.
We consider an optimal control problem for the steady-state Kirchhoff equation, a prototype for nonlocal partial differential equations, different from fractional powers of closed operators. Existence and uniqueness of solutions of the state equation, existence of global optimal solutions, differentiability of the control-to-state map and first-order necessary optimality conditions are established. The aforementioned results require the controls to be functions in $H^1$ and subject to pointwise upper and lower bounds. In order to obtain the Newton differentiability of the optimality conditions, we employ a Moreau-Yosida-type penalty approach to treat the control constraints and study its convergence. The first-order optimality conditions of the regularized problems are shown to be Newton diffentiable, and a generalized Newton method is detailed. A discretization of the optimal control problem by piecewise linear finite elements is proposed and numerical results are presented.
In the analyses of cluster-randomized trials, a standard approach for covariate adjustment and handling within-cluster correlations is the mixed-model analysis of covariance (ANCOVA). The mixed-model ANCOVA makes stringent assumptions, including normality, linearity, and a compound symmetric correlation structure, which may be challenging to verify and may not hold in practice. When mixed-model ANCOVA assumptions are violated, the validity and efficiency of the model-based inference for the average treatment effect are currently unclear. In this article, we prove that the mixed-model ANCOVA estimator for the average treatment effect is consistent and asymptotically normal under arbitrary misspecification of its working model. Under equal randomization, we further show that the model-based variance estimator for the mixed-model ANCOVA estimator remains consistent, clarifying that the confidence interval given by standard software is asymptotically valid even under model misspecification. Beyond robustness, we also provide a caveat that covariate adjustment via mixed-model ANCOVA may lead to precision loss compared to no adjustment when the covariance structure is misspecified, and describe when a cluster-level ANCOVA becomes more efficient. These results hold under both simple and stratified randomization, and are further illustrated via simulations as well as analyses of three cluster-randomized trials.
Optimal Markov Decision Process policies for problems with finite state and action space are identified through a partial ordering by comparing the value function across states. This is referred to as state-based optimality. This paper identifies when such optimality guarantees some form of system-based optimality as measured by a scalar. Four such system-based metrics are introduced. Uni-variate empirical distributions of these metrics are obtained through simulation as to assess whether theoretically optimal policies provide a statistically significant advantage. This has been conducted using a Student's t-test, Welch's $t$-test and a Mann-Whitney $U$-test. The proposed method is applied to a common problem in queuing theory: admission control.
We investigate the optimal design of experimental studies that have pre-treatment outcome data available. The average treatment effect is estimated as the difference between the weighted average outcomes of the treated and control units. A number of commonly used approaches fit this formulation, including the difference-in-means estimator and a variety of synthetic-control techniques. We propose several methods for choosing the set of treated units in conjunction with the weights. Observing the NP-hardness of the problem, we introduce a mixed-integer programming formulation which selects both the treatment and control sets and unit weightings. We prove that these proposed approaches lead to qualitatively different experimental units being selected for treatment. We use simulations based on publicly available data from the US Bureau of Labor Statistics that show improvements in terms of mean squared error and statistical power when compared to simple and commonly used alternatives such as randomized trials.
Training of generative adversarial networks (GANs) is known for its difficulty to converge. This paper first confirms analytically one of the culprits behind this convergence issue: the lack of convexity in GANs objective functions, hence the well-posedness problem of GANs models. Then, it proposes a stochastic control approach for hyper-parameters tuning in GANs training. In particular, it presents an optimal solution for adaptive learning rate which depends on the convexity of the objective function, and builds a precise relation between improper choices of learning rate and explosion in GANs training. Finally, empirical studies demonstrate that training algorithms incorporating this selection methodology outperform standard ones.
We present Path Integral Sampler~(PIS), a novel algorithm to draw samples from unnormalized probability density functions. The PIS is built on the Schr\"odinger bridge problem which aims to recover the most likely evolution of a diffusion process given its initial distribution and terminal distribution. The PIS draws samples from the initial distribution and then propagates the samples through the Schr\"odinger bridge to reach the terminal distribution. Applying the Girsanov theorem, with a simple prior diffusion, we formulate the PIS as a stochastic optimal control problem whose running cost is the control energy and terminal cost is chosen according to the target distribution. By modeling the control as a neural network, we establish a sampling algorithm that can be trained end-to-end. We provide theoretical justification of the sampling quality of PIS in terms of Wasserstein distance when sub-optimal control is used. Moreover, the path integrals theory is used to compute importance weights of the samples to compensate for the bias induced by the sub-optimality of the controller and time-discretization. We experimentally demonstrate the advantages of PIS compared with other start-of-the-art sampling methods on a variety of tasks.
Large linear systems of saddle-point type have arisen in a wide variety of applications throughout computational science and engineering. The discretizations of distributed control problems have a saddle-point structure. The numerical solution of saddle-point problems has attracted considerable interest in recent years. In this work, we propose a novel Braess-Sarazin multigrid relaxation scheme for finite element discretizations of the distributed control problems, where we use the stiffness matrix obtained from the five-point finite difference method for the Laplacian to approximate the inverse of the mass matrix arising in the saddle-point system. We apply local Fourier analysis to examine the smoothing properties of the Braess-Sarazin multigrid relaxation. From our analysis, the optimal smoothing factor for Braess-Sarazin relaxation is derived. Numerical experiments validate our theoretical results. The relaxation scheme considered here shows its high efficiency and robustness with respect to the regularization parameter and grid size.
Although reinforcement learning methods can achieve impressive results in simulation, the real world presents two major challenges: generating samples is exceedingly expensive, and unexpected perturbations can cause proficient but narrowly-learned policies to fail at test time. In this work, we propose to learn how to quickly and effectively adapt online to new situations as well as to perturbations. To enable sample-efficient meta-learning, we consider learning online adaptation in the context of model-based reinforcement learning. Our approach trains a global model such that, when combined with recent data, the model can be be rapidly adapted to the local context. Our experiments demonstrate that our approach can enable simulated agents to adapt their behavior online to novel terrains, to a crippled leg, and in highly-dynamic environments.
In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.
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