More Like this

1. Condition number bounds for causal inference

   Gordon, S.L. ; Kumar, V. M. ; Schulman, L. J. ; Srivastava, P. ( January 2021 , Proceedings of Machine Learning Research)

   de Campos, C. ; Maathuis, M. H. (Ed.)

   An important achievement in the field of causal inference was a complete characterization of when a causal effect, in a system modeled by a causal graph, can be determined uniquely from purely observational data. The identification algorithms resulting from this work produce exact symbolic expressions for causal effects, in terms of the observational probabilities. More recent work has looked at the numerical properties of these expressions, in particular using the classical notion of the condition number. In its classical interpretation, the condition number quantifies the sensitivity of the output values of the expressions to small numerical perturbations in the input observational probabilities. In the context of causal identification, the condition number has also been shown to be related to the effect of certain kinds of uncertainties in the structure of the causal graphical model. In this paper, we first give an upper bound on the condition number for the interesting case of causal graphical models with small “confounded components”. We then develop a tight characterization of the condition number of any given causal identification problem. Finally, we use our tight characterization to give a specific example where the condition number can be much lower than that obtained via generic boundsmore »on the condition number, and to show that even “equivalent” expressions for causal identification can behave very differently with respect to their numerical stability properties.« less
2. HOGWILD!-Gibbs can be PanAccurate

   Daskalakis, Constantinos ; Dikkala, Nishanth ; Jayanti, Siddhartha ( January 2018 , 32nd Annual Conference on Neural Information Processing Systems)

   Asynchronous Gibbs sampling has been recently shown to be fast-mixing and an accurate method for estimating probabilities of events on a small number of variables of a graphical model satisfying Dobrushin's condition~\cite{DeSaOR16}. We investigate whether it can be used to accurately estimate expectations of functions of {\em all the variables} of the model. Under the same condition, we show that the synchronous (sequential) and asynchronous Gibbs samplers can be coupled so that the expected Hamming distance between their (multivariate) samples remains bounded by O(τlogn), where n is the number of variables in the graphical model, and τ is a measure of the asynchronicity. A similar bound holds for any constant power of the Hamming distance. Hence, the expectation of any function that is Lipschitz with respect to a power of the Hamming distance, can be estimated with a bias that grows logarithmically in n. Going beyond Lipschitz functions, we consider the bias arising from asynchronicity in estimating the expectation of polynomial functions of all variables in the model. Using recent concentration of measure results, we show that the bias introduced by the asynchronicity is of smaller order than the standard deviation of the function value already present in the truemore »model. We perform experiments on a multi-processor machine to empirically illustrate our theoretical findings.« less
3. HOGWILD!-Gibbs can be PanAccurate

   Daskalakis, Constantinos ; Dikkala, Nishanth ; Jayanti, Siddhartha ( January 2018 , Journal of neural information processing)

   Asynchronous Gibbs sampling has been recently shown to be fast-mixing and an accurate method for estimating probabilities of events on a small number of variables of a graphical model satisfying Dobrushin's condition~\cite{DeSaOR16}. We investigate whether it can be used to accurately estimate expectations of functions of {\em all the variables} of the model. Under the same condition, we show that the synchronous (sequential) and asynchronous Gibbs samplers can be coupled so that the expected Hamming distance between their (multivariate) samples remains bounded by O(τlogn), where n is the number of variables in the graphical model, and τ is a measure of the asynchronicity. A similar bound holds for any constant power of the Hamming distance. Hence, the expectation of any function that is Lipschitz with respect to a power of the Hamming distance, can be estimated with a bias that grows logarithmically in n. Going beyond Lipschitz functions, we consider the bias arising from asynchronicity in estimating the expectation of polynomial functions of all variables in the model. Using recent concentration of measure results, we show that the bias introduced by the asynchronicity is of smaller order than the standard deviation of the function value already present in the truemore »model. We perform experiments on a multi-processor machine to empirically illustrate our theoretical findings.« less
4. Interactive rank testing by betting

   Duan, Boyan ; Ramdas, Aaditya ; Wasserman, Larry ( April 2022 , First Conference on Causal Learning and Reasoning, PMLR)

   Scholkopf, Bernhard ; Uhler, Caroline ; Zhang, Kun (Ed.)

   In order to test if a treatment is perceptibly different from a placebo in a randomized experiment with covariates, classical nonparametric tests based on ranks of observations/residuals have been employed (eg: by Rosenbaum), with finite-sample valid inference enabled via permutations. This paper proposes a different principle on which to base inference: if — with access to all covariates and outcomes, but without access to any treatment assignments — one can form a ranking of the subjects that is sufficiently nonrandom (eg: mostly treated followed by mostly control), then we can confidently conclude that there must be a treatment effect. Based on a more nuanced, quantifiable, version of this principle, we design an interactive test called i-bet: the analyst forms a single permutation of the subjects one element at a time, and at each step the analyst bets toy money on whether that subject was actually treated or not, and learns the truth immediately after. The wealth process forms a real-valued measure of evidence against the global causal null, and we may reject the null at level if the wealth ever crosses 1= . Apart from providing a fresh “game-theoretic” principle on which to base the causal conclusion, the i-bet has othermore »statistical and computational benefits, for example (A) allowing a human to adaptively design the test statistic based on increasing amounts of data being revealed (along with any working causal models and prior knowledge), and (B) not requiring permutation resampling, instead noting that under the null, the wealth forms a nonnegative martingale, and the type-1 error control of the aforementioned decision rule follows from a tight inequality by Ville. Further, if the null is not rejected, new subjects can later be added and the test can be simply continued, without any corrections (unlike with permutation p-values). Numerical experiments demonstrate good power under various heterogeneous treatment effects. We first describe i-bet test for two-sample comparisons with unpaired data, and then adapt it to paired data, multi-sample comparison, and sequential settings; these may be viewed as interactive martingale variants of the Wilcoxon, Kruskal-Wallis, and Friedman tests.« less
5. High-performance symbolic-numerics via multiple dispatch

   https://doi.org/10.1145/3511528.3511535  

   Gowda, Shashi ; Ma, Yingbo ; Cheli, Alessandro ; Gwóźzdź, Maja ; Shah, Viral B. ; Edelman, Alan ; Rackauckas, Christopher ( September 2021 , ACM Communications in Computer Algebra)

   As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To this end, we developed Symbolics.jl, an extendable symbolic system which uses dynamic multiple dispatch to change behavior depending on the domain needs. In this work we detail an underlying abstract term interface which allows for speed without sacrificing generality. We show that by formalizing a generic API on actions independent of implementation, we can retroactively add optimized data structures to our system without changing the pre-existing term rewriters. We showcase how this can be used to optimize term construction and give a 113x acceleration on general symbolic transformations. Further, we show that such a generic API allows for complementary term-rewriting implementations. Exploiting this feature, we demonstrate the ability to swap between classical term-rewriting simplifiers and e-graph-based term-rewriting simplifiers. We illustrate how this symbolic system improves numerical computing tasks by showcasing an e-graph ruleset which minimizes the number of CPU cycles during expression evaluation, and demonstratemore »how it simplifies a real-world reaction-network simulation to halve the runtime. Additionally, we show a reaction-diffusion partial differential equation solver which is able to be automatically converted into symbolic expressions via multiple dispatch tracing, which is subsequently accelerated and parallelized to give a 157x simulation speedup. Together, this presents Symbolics.jl as a next-generation symbolic-numeric computing environment geared towards modeling and simulation.« less