We provide novel bounds on average treatment effects (on the treated) that are valid under an unconfoundedness assumption. Our bounds are designed to be robust in challenging situations, for example, when the conditioning variables take on a large number of different values in the observed sample, or when the overlap condition is violated. This robustness is achieved by only using limited "pooling" of information across observations. Namely, the bounds are constructed as sample averages over functions of the observed outcomes such that the contribution of each outcome only depends on the treatment status of a limited number of observations. No information pooling across observations leads to so-called "Manski bounds", while unlimited information pooling leads to standard inverse propensity score weighting. We explore the intermediate range between these two extremes and provide corresponding inference methods. We show in Monte Carlo experiments and through an empirical application that our bounds are indeed robust and informative in practice.
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Unobserved confounding is one of the main challenges when estimating causal effects. We propose a causal reduction method that, given a causal model, replaces an arbitrary number of possibly high-dimensional latent confounders with a single latent confounder that takes values in the same space as the treatment variable, without changing the observational and interventional distributions the causal model entails. This allows us to estimate the causal effect in a principled way from combined data without relying on the common but often unrealistic assumption that all confounders have been observed. We apply our causal reduction in three different settings. In the first setting, we assume the treatment and outcome to be discrete. The causal reduction then implies bounds between the observational and interventional distributions that can be exploited for estimation purposes. In certain cases with highly unbalanced observational samples, the accuracy of the causal effect estimate can be improved by incorporating observational data. Second, for continuous variables and assuming a linear-Gaussian model, we derive equality constraints for the parameters of the observational and interventional distributions. Third, for the general continuous setting (possibly nonlinear or non-Gaussian), we parameterize the reduced causal model using normalizing flows, a flexible class of easily invertible nonlinear transformations. We perform a series of experiments on synthetic data and find that in several cases the number of interventional samples can be reduced when adding observational training samples without sacrificing accuracy.
The heterogeneity of treatment effect lies at the heart of precision medicine. Randomized controlled trials are gold-standard for treatment effect estimation but are typically underpowered for heterogeneous effects. While large observational studies have high predictive power but are often confounded due to lack of randomization of treatment. We show that the observational study, even subject to hidden confounding, may empower trials in estimating the heterogeneity of treatment effect using the notion of confounding function. The confounding function summarizes the impact of unmeasured confounders on the difference in the potential outcomes between the treated and untreated groups accounting for the observed covariates, which is unidentifiable based only on the observational study. Coupling the randomized trial and observational study, we show that the heterogeneity of treatment effect and confounding function are nonparametrically identifiable. We then derive the semiparametric efficient scores and the rate-doubly robust integrative estimators of the heterogeneity of treatment effect and confounding function under parametric structural models. We clarify the conditions under which the integrative estimator of the heterogeneity of treatment effect is strictly more efficient than the trial estimator. We illustrate the integrative estimators via simulation and an application.
The methodological development of this paper is motivated by the need to address the following scientific question: does the issuance of heat alerts prevent adverse health effects? Our goal is to address this question within a causal inference framework in the context of time series data. A key challenge is that causal inference methods require the overlap assumption to hold: each unit (i.e., a day) must have a positive probability of receiving the treatment (i.e., issuing a heat alert on that day). In our motivating example, the overlap assumption is often violated: the probability of issuing a heat alert on a cool day is zero. To overcome this challenge, we propose a stochastic intervention for time series data which is implemented via an incremental time-varying propensity score (ItvPS). The ItvPS intervention is executed by multiplying the probability of issuing a heat alert on day $t$ -- conditional on past information up to day $t$ -- by an odds ratio $\delta_t$. First, we introduce a new class of causal estimands that relies on the ItvPS intervention. We provide theoretical results to show that these causal estimands can be identified and estimated under a weaker version of the overlap assumption. Second, we propose nonparametric estimators based on the ItvPS and derive an upper bound for the variances of these estimators. Third, we extend this framework to multi-site time series using a meta-analysis approach. Fourth, we show that the proposed estimators perform well in terms of bias and root mean squared error via simulations. Finally, we apply our proposed approach to estimate the causal effects of increasing the probability of issuing heat alerts on each warm-season day in reducing deaths and hospitalizations among Medicare enrollees in $2,837$ U.S. counties.
We propose a definition for the average indirect effect of a binary treatment in the potential outcomes model for causal inference under cross-unit interference. Our definition is analogous to the standard definition of the average direct effect, and can be expressed without needing to compare outcomes across multiple randomized experiments. We show that the proposed indirect effect satisfies a decomposition theorem whereby, in a Bernoulli trial, the sum of the average direct and indirect effects always corresponds to the effect of a policy intervention that infinitesimally increases treatment probabilities. We also consider a number of parametric models for interference, and find that our non-parametric indirect effect remains a natural estimand when re-expressed in the context of these models.
We consider the problem of in-order packet transmission over a cascade of packet-erasure links with acknowledgment (ACK) signals, interconnected by relays. We treat first the case of transmitting a single packet, in which ACKs are unnecessary, over links with independent identically distributed erasures. For this case, we derive tight upper and lower bounds on the probability of arrive failure within an allowed end-to-end communication delay over a given number of links. When the number of links is commensurate with the allowed delay, we determine the maximal ratio between the two -- coined information velocity -- for which the arrive-failure probability decays to zero; we further derive bounds on the arrive-failure probability when the ratio is below the information velocity, determine the exponential arrive-failure decay rate, and extend the treatment to links with different erasure probabilities. We then elevate all these results for a stream of packets with independent geometrically distributed interarrival times, and prove that the information velocity and the exponential decay rate remain the same for any stationary ergodic arrival process and for deterministic interarrival times. We demonstrate the significance of the derived fundamental limits -- the information velocity and the arrive-failure exponential decay rate -- by comparing them to simulation results.
Statistical tests for dataset shift are susceptible to false alarms: they are sensitive to minor differences when there is in fact adequate sample coverage and predictive performance. We propose instead a framework to detect adverse dataset shifts based on outlier scores, $\texttt{D-SOS}$ for short. $\texttt{D-SOS}$ holds that the new (test) sample is not substantively worse than the reference (training) sample, and not that the two are equal. The key idea is to reduce observations to outlier scores and compare contamination rates at varying weighted thresholds. Users can define what $\it{worse}$ means in terms of relevant notions of outlyingness, including proxies for predictive performance. Compared to tests of equal distribution, our approach is uniquely tailored to serve as a robust metric for model monitoring and data validation. We show how versatile and practical $\texttt{D-SOS}$ is on a wide range of real and simulated data.
Imitation learning enables agents to reuse and adapt the hard-won expertise of others, offering a solution to several key challenges in learning behavior. Although it is easy to observe behavior in the real-world, the underlying actions may not be accessible. We present a new method for imitation solely from observations that achieves comparable performance to experts on challenging continuous control tasks while also exhibiting robustness in the presence of observations unrelated to the task. Our method, which we call FORM (for "Future Observation Reward Model") is derived from an inverse RL objective and imitates using a model of expert behavior learned by generative modelling of the expert's observations, without needing ground truth actions. We show that FORM performs comparably to a strong baseline IRL method (GAIL) on the DeepMind Control Suite benchmark, while outperforming GAIL in the presence of task-irrelevant features.
Despite their overwhelming capacity to overfit, deep neural networks trained by specific optimization algorithms tend to generalize well to unseen data. Recently, researchers explained it by investigating the implicit regularization effect of optimization algorithms. A remarkable progress is the work (Lyu&Li, 2019), which proves gradient descent (GD) maximizes the margin of homogeneous deep neural networks. Except GD, adaptive algorithms such as AdaGrad, RMSProp and Adam are popular owing to their rapid training process. However, theoretical guarantee for the generalization of adaptive optimization algorithms is still lacking. In this paper, we study the implicit regularization of adaptive optimization algorithms when they are optimizing the logistic loss on homogeneous deep neural networks. We prove that adaptive algorithms that adopt exponential moving average strategy in conditioner (such as Adam and RMSProp) can maximize the margin of the neural network, while AdaGrad that directly sums historical squared gradients in conditioner can not. It indicates superiority on generalization of exponential moving average strategy in the design of the conditioner. Technically, we provide a unified framework to analyze convergent direction of adaptive optimization algorithms by constructing novel adaptive gradient flow and surrogate margin. Our experiments can well support the theoretical findings on convergent direction of adaptive optimization algorithms.
Counterfactual explanations are usually generated through heuristics that are sensitive to the search's initial conditions. The absence of guarantees of performance and robustness hinders trustworthiness. In this paper, we take a disciplined approach towards counterfactual explanations for tree ensembles. We advocate for a model-based search aiming at "optimal" explanations and propose efficient mixed-integer programming approaches. We show that isolation forests can be modeled within our framework to focus the search on plausible explanations with a low outlier score. We provide comprehensive coverage of additional constraints that model important objectives, heterogeneous data types, structural constraints on the feature space, along with resource and actionability restrictions. Our experimental analyses demonstrate that the proposed search approach requires a computational effort that is orders of magnitude smaller than previous mathematical programming algorithms. It scales up to large data sets and tree ensembles, where it provides, within seconds, systematic explanations grounded on well-defined models solved to optimality.
Imitation learning seeks to circumvent the difficulty in designing proper reward functions for training agents by utilizing expert behavior. With environments modeled as Markov Decision Processes (MDP), most of the existing imitation algorithms are contingent on the availability of expert demonstrations in the same MDP as the one in which a new imitation policy is to be learned. In this paper, we study the problem of how to imitate tasks when there exist discrepancies between the expert and agent MDP. These discrepancies across domains could include differing dynamics, viewpoint, or morphology; we present a novel framework to learn correspondences across such domains. Importantly, in contrast to prior works, we use unpaired and unaligned trajectories containing only states in the expert domain, to learn this correspondence. We utilize a cycle-consistency constraint on both the state space and a domain agnostic latent space to do this. In addition, we enforce consistency on the temporal position of states via a normalized position estimator function, to align the trajectories across the two domains. Once this correspondence is found, we can directly transfer the demonstrations on one domain to the other and use it for imitation. Experiments across a wide variety of challenging domains demonstrate the efficacy of our approach.
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