Edoardo M. Airoldi

Amir Ghasemian, Homa Hosseinmardi, Aram Galstyan et al.

Most real-world networks are incompletely observed. Algorithms that can accurately predict which links are missing can dramatically speedup the collection of network data and improve the validity of network models. Many algorithms now exist for predicting missing links, given a partially observed network, but it has remained unknown whether a single best predictor exists, how link predictability varies across methods and networks from different domains, and how close to optimality current methods are. We answer these questions by systematically evaluating 203 individual link predictor algorithms, representing three popular families of methods, applied to a large corpus of 548 structurally diverse networks from six scientific domains. We first show that individual algorithms exhibit a broad diversity of prediction errors, such that no one predictor or family is best, or worst, across all realistic inputs. We then exploit this diversity via meta-learning to construct a series of "stacked" models that combine predictors into a single algorithm. Applied to a broad range of synthetic networks, for which we may analytically calculate optimal performance, these stacked models achieve optimal or nearly optimal levels of accuracy. Applied to real-world networks, stacked models are also superior, but their accuracy varies strongly by domain, suggesting that link prediction may be fundamentally easier in social networks than in biological or technological networks. These results indicate that the state-of-the-art for link prediction comes from combining individual algorithms, which achieves nearly optimal predictions. We close with a brief discussion of limitations and opportunities for further improvement of these results.

Panos Toulis, Thibaut Horel, Edoardo M. Airoldi

The need for parameter estimation with massive datasets has reinvigorated interest in stochastic optimization and iterative estimation procedures. Stochastic approximations are at the forefront of this recent development as they yield procedures that are simple, general, and fast. However, standard stochastic approximations are often numerically unstable. Deterministic optimization, on the other hand, increasingly uses proximal updates to achieve numerical stability in a principled manner. A theoretical gap has thus emerged. While standard stochastic approximations are subsumed by the framework of Robbins and Monro (1951), there is no such framework for stochastic approximations with proximal updates. In this paper, we conceptualize a proximal version of the classical Robbins-Monro procedure. Our theoretical analysis demonstrates that the proposed procedure has important stability benefits over the classical Robbins-Monro procedure, while it retains the best known convergence rates. Exact implementations of the proximal Robbins-Monro procedure are challenging, but we show that approximate implementations lead to procedures that are easy to implement, and still dominate classical procedures by achieving numerical stability, practically without tradeoffs. Moreover, approximate proximal Robbins-Monro procedures can be applied even when the objective cannot be calculated analytically, and so they generalize stochastic proximal procedures currently in use.

Dustin Tran, Panos Toulis, Edoardo M. Airoldi

We develop methods for parameter estimation in settings with large-scale data sets, where traditional methods are no longer tenable. Our methods rely on stochastic approximations, which are computationally efficient as they maintain one iterate as a parameter estimate, and successively update that iterate based on a single data point. When the update is based on a noisy gradient, the stochastic approximation is known as standard stochastic gradient descent, which has been fundamental in modern applications with large data sets. Additionally, our methods are numerically stable because they employ implicit updates of the iterates. Intuitively, an implicit update is a shrinked version of a standard one, where the shrinkage factor depends on the observed Fisher information at the corresponding data point. This shrinkage prevents numerical divergence of the iterates, which can be caused either by excess noise or outliers. Our sgd package in R offers the most extensive and robust implementation of stochastic gradient descent methods. We demonstrate that sgd dominates alternative software in runtime for several estimation problems with massive data sets. Our applications include the wide class of generalized linear models as well as M-estimation for robust regression.

Guillaume W. Basse, Edoardo M. Airoldi

We consider the problem of how to assign treatment in a randomized experiment, in which the correlation among the outcomes is informed by a network available pre-intervention. Working within the potential outcome causal framework, we develop a class of models that posit such a correlation structure among the outcomes. Then we leverage these models to develop restricted randomization strategies for allocating treatment optimally, by minimizing the mean square error of the estimated average treatment effect. Analytical decompositions of the mean square error, due both to the model and to the randomization distribution, provide insights into aspects of the optimal designs. In particular, the analysis suggests new notions of balance based on specific network quantities, in addition to classical covariate balance. The resulting balanced, optimal restricted randomization strategies are still design unbiased, in situations where the model used to derive them does not hold. We illustrate how the proposed treatment allocation strategies improve on allocations that ignore the network structure, with extensive simulations.

Daniel L. Sussman, Alexander Volfovsky, Edoardo M. Airoldi

The recent explosion in the amount and dimensionality of data has exacerbated the need of trading off computational and statistical efficiency carefully, so that inference is both tractable and meaningful. We propose a framework that provides an explicit opportunity for practitioners to specify how much statistical risk they are willing to accept for a given computational cost, and leads to a theoretical risk-computation frontier for any given inference problem. We illustrate the tradeoff between risk and computation and illustrate the frontier in three distinct settings. First, we derive analytic forms for the risk of estimating parameters in the classical setting of estimating the mean and variance for normally distributed data and for the more general setting of parameters of an exponential family. The second example concentrates on computationally constrained Hodges-Lehmann estimators. We conclude with an evaluation of risk associated with early termination of iterative matrix inversion algorithms in the context of linear regression.

Dustin Tran, David M. Blei, Edoardo M. Airoldi

We develop a general variational inference method that preserves dependency among the latent variables. Our method uses copulas to augment the families of distributions used in mean-field and structured approximations. Copulas model the dependency that is not captured by the original variational distribution, and thus the augmented variational family guarantees better approximations to the posterior. With stochastic optimization, inference on the augmented distribution is scalable. Furthermore, our strategy is generic: it can be applied to any inference procedure that currently uses the mean-field or structured approach. Copula variational inference has many advantages: it reduces bias; it is less sensitive to local optima; it is less sensitive to hyperparameters; and it helps characterize and interpret the dependency among the latent variables.

Panos Toulis, Dustin Tran, Edoardo M. Airoldi

Iterative procedures for parameter estimation based on stochastic gradient descent allow the estimation to scale to massive data sets. However, in both theory and practice, they suffer from numerical instability. Moreover, they are statistically inefficient as estimators of the true parameter value. To address these two issues, we propose a new iterative procedure termed averaged implicit SGD (AI-SGD). For statistical efficiency, AI-SGD employs averaging of the iterates, which achieves the optimal Cramér-Rao bound under strong convexity, i.e., it is an optimal unbiased estimator of the true parameter value. For numerical stability, AI-SGD employs an implicit update at each iteration, which is related to proximal operators in optimization. In practice, AI-SGD achieves competitive performance with other 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.

Aviv Tamar, Panos Toulis, Shie Mannor et al.

In reinforcement learning, the TD($λ$) algorithm is a fundamental policy evaluation method with an efficient online implementation that is suitable for large-scale problems. One practical drawback of TD($λ$) is its sensitivity to the choice of the step-size. It is an empirically well-known fact that a large step-size leads to fast convergence, at the cost of higher variance and risk of instability. In this work, we introduce the implicit TD($λ$) algorithm which has the same function and computational cost as TD($λ$), but is significantly more stable. We provide a theoretical explanation of this stability and an empirical evaluation of implicit TD($λ$) on typical benchmark tasks. Our results show that implicit TD($λ$) outperforms standard TD($λ$) and a state-of-the-art method that automatically tunes the step-size, and thus shows promise for wide applicability.

Panos Toulis, Edoardo M. Airoldi

Stochastic gradient descent procedures have gained popularity for parameter estimation from large data sets. However, their statistical properties are not well understood, in theory. And in practice, avoiding numerical instability requires careful tuning of key parameters. Here, we introduce implicit stochastic gradient descent procedures, which involve parameter updates that are implicitly defined. Intuitively, implicit updates shrink standard stochastic gradient descent updates. The amount of shrinkage depends on the observed Fisher information matrix, which does not need to be explicitly computed; thus, implicit procedures increase stability without increasing the computational burden. Our theoretical analysis provides the first full characterization of the asymptotic behavior of both standard and implicit stochastic gradient descent-based estimators, including finite-sample error bounds. Importantly, analytical expressions for the variances of these stochastic gradient-based estimators reveal their exact loss of efficiency. We also develop new algorithms to compute implicit stochastic gradient descent-based estimators for generalized linear models, Cox proportional hazards, M-estimators, in practice, and perform extensive experiments. Our results suggest that implicit stochastic gradient descent procedures are poised to become a workhorse for approximate inference from large data sets

Elham Azizi, James E. Galagan, Edoardo M. Airoldi

A standard technique for understanding underlying dependency structures among a set of variables posits a shared conditional probability distribution for the variables measured on individuals within a group. This approach is often referred to as module networks, where individuals are represented by nodes in a network, groups are termed modules, and the focus is on estimating the network structure among modules. However, estimation solely from node-specific variables can lead to spurious dependencies, and unverifiable structural assumptions are often used for regularization. Here, we propose an extended model that leverages direct observations about the network in addition to node-specific variables. By integrating complementary data types, we avoid the need for structural assumptions. We illustrate theoretical and practical significance of the model and develop a reversible-jump MCMC learning procedure for learning modules and model parameters. We demonstrate the method accuracy in predicting modular structures from synthetic data and capability to learn influence structures in twitter data and regulatory modules in the Mycobacterium tuberculosis gene regulatory network.