In this paper, we develop a statistical inference procedure using stochastic gradient descent (SGD)-based confidence intervals. These intervals are of the simplest possible form: $\theta_{N,j} \pm 2\sqrt{}(\gamma/N)$ , where $\theta_N$ is the SGD estimate of model parameters $\theta$ over N data points, and $\gamma$ is the learning rate. This construction relies only on a proper selection of the learning rate to ensure the standard SGD conditions for O(1/n) convergence. The procedure performs well in our empirical evaluations, achieving near-nominal coverage intervals scaling up to 20$\times$ as many parameters as other SGD-based inference methods. We also demonstrate our method’s practical significance on modeling adverse events in emergency general surgery patients using a novel dataset from the Hospital of the University of Pennsylvania.

We propose a residual randomization procedure designed for robust inference using Lasso estimates in the high-dimensional setting. Compared to earlier work that focuses on sub-Gaussian errors, the proposed procedure is designed to work robustly in settings that also include heavy-tailed covariates and errors. Moreover, our procedure can be valid under clustered errors, which is important in practice, but has been largely overlooked by earlier work. Through extensive simulations, we illustrate our method’s wider range of applicability as suggested by theory. In particular, we show that our method outperforms state-of-art methods in challenging, yet more realistic, settings where the distribution of covariates is heavy-tailed or the sample size is small, while it remains competitive in standard, “well behaved" settings previously studied in the literature.

In this paper, we adopt results in nonlinear time series analysis for causal inference in dynamical settings. Our motivation is policy analysis with panel data, particularly through the use of “synthetic control" methods. These methods regress pre-intervention outcomes of the treated unit to outcomes from a pool of control units, and then use the fitted regression model to estimate causal effects post-intervention. In this setting, we propose to screen out control units that have a weak dynamical relationship to the treated unit. In simulations, we show that this method can mitigate bias from “cherry-picking" of control units, which is usually an important concern. We illustrate on real-world applications, including the tobacco legislation example of \citet{Abadie2010}, and Brexit.

Many iterative procedures in stochastic optimization exhibit a transient phase followed by a stationary phase. During the transient phase the procedure converges towards a region of interest, and during the stationary phase the procedure oscillates in that region, commonly around a single point. In this paper, we develop a statistical diagnostic test to detect such phase transition in the context of stochastic gradient descent with constant learning rate. We present theory and experiments suggesting that the region where the proposed diagnostic is activated coincides with the convergence region. For a class of loss functions, we derive a closed-form solution describing such region. Finally, we suggest an application to speed up convergence of stochastic gradient descent by halving the learning rate each time stationarity is detected. This leads to a new variant of stochastic gradient descent, which in many settings is comparable to state-of-art.

Iterative procedures for parameter estimation based on stochastic gradient descent (SGD) allow the estimation to scale to massive data sets. However, they typically suffer from numerical instability, while estimators based on SGD are statistically inefficient as they do not use all the information in the data set. To address these two issues we propose an iterative estimation procedure termed averaged implicit SGD (AI-SGD). For statistical efficiency AI-SGD employs averaging of the iterates, which achieves the Cramer-Rao bound under strong convexity, i.e., it is asymptotically an optimal unbiased estimator of the true parameter value. For numerical stability AI-SGD employs an implicit update at each iteration, which is similar to updates performed by proximal operators in optimization. In practice, AI-SGD achieves competitive performance with state-of-the-art procedures. Furthermore, it is more stable than averaging procedures that do not employ proximal updates, and is simple to implement as it requires fewer tunable hyperparameters than procedures that do employ proximal updates.

Planned experiments are the gold standard in reliably comparing the causal effect of switching from a baseline policy to a new policy. % One critical shortcoming of classical experimental methods, however, is that they typically do not take into account the dynamic nature of response to policy changes. For instance, in an experiment where we seek to understand the effects of a new ad pricing policy on auction revenue, agents may adapt their bidding in response to the experimental pricing changes. Thus, causal effects of the new pricing policy after such adaptation period, the {\em long-term causal effects}, are not captured by the classical methodology even though they clearly are more indicative of the value of the new policy. % Here, we formalize a framework to define and estimate long-term causal effects of policy changes in multiagent economies. Central to our approach is behavioral game theory, which we leverage to formulate the ignorability assumptions that are necessary for causal inference. Under such assumptions we estimate long-term causal effects through a latent space approach, where a behavioral model of how agents act conditional on their latent behaviors is combined with a temporal model of how behaviors evolve over time.

We consider the design of experiments to evaluate treatments that are administered by self-interested agents, each seeking to achieve the highest evaluation and win the experiment. For example, in an advertising experiment, a company wishes to evaluate two marketing agents in terms of their efficacy in viral marketing, and assign a contract to the winner agent. Contrary to traditional experimental design, this problem has two new implications. First, the experiment induces a game among agents, where each agent can select from multiple versions of the treatment it administers. Second, the action of one agent -- selection of treatment version -- may affect the actions of another agent, with the resulting strategic interference complicating the evaluation of agents. An incentive-compatible experiment design is one with an equilibrium where each agent selects its natural action, which is the action that maximizes the performance of the agent without competition (e.g., expected number of conversions if agent is assigned the advertising contract).

We study the statistical properties of stochastic gradient descent (SGD) using explicit and implicit updates for fitting generalized linear models (GLMs). Initially, we develop a computationally efficient algorithm to implement implicit SGD learning of GLMs. Next, we obtain exact formulas for the bias and variance of both updates which leads to two important observations on their comparative statistical properties. First, in small samples, the estimates from the implicit procedure are more biased than the estimates from the explicit one, but their empirical variance is smaller and they are more robust to learning rate misspecification. Second, the two procedures are statistically identical in the limit: they are both unbiased, converge at the same rate and have the same asymptotic variance. Our set of experiments confirm our theory and more broadly suggest that the implicit procedure can be a competitive choice for fitting large-scale models, especially when robustness is a concern.

The broad adoption of social media has generated interest in leveraging peer influence for inducing desired user behavior. Quantifying the causal effect of peer influence presents technical challenges, however, including how to deal with social interference, complex response functions and network uncertainty. In this paper, we extend potential outcomes to allow for interference, we introduce well-defined causal estimands of peer-influence, and we develop two estimation procedures: a frequentist procedure relying on a sequential randomization design that requires knowledge of the network but operates under complicated response functions, and a Bayesian procedure which accounts for network uncertainty but relies on a linear response assumption to increase estimation precision. Our results show the advantages and disadvantages of the proposed methods in a number of situations.

In kidney exchanges, hospitals share patient lists and receive transplantations. A kidney-paired donation (KPD) mechanism needs to promote full sharing of information about donor-patient pairs, and identify a Pareto efficient outcome that also satisfies participation constraints of hospitals. We introduce a random graph model of the KPD exchange and then fully characterize the structure of the efficient outcome and the expected number of transplantations that can be performed. Random graph theory allows early experimental results to be explained analytically, and enables the study of participation incentives in a methodological way. We derive a square-root law between the welfare gains from sharing patient-donor pairs in a central pool and the individual sizes of hospitals, illustrating the urgent need for the nationwide expansion of such programs. Finally, we establish through theoretical and computational analysis that enforcing simple individual rationality constraints on the outcome can mitigate the negative impact of strategic behavior by hospitals.