Causal identification is at the core of the causal inference literature, where complete algorithms have been proposed to identify causal queries of interest. The validity of these algorithms hinges on the restrictive assumption of having access to a correctly specified causal structure. In this work, we study the setting where a probabilistic model of the causal structure is available. Specifically, the edges in a causal graph exist with uncertainties which may, for example, represent degree of belief from domain experts. Alternatively, the uncertainty about an edge may reflect the confidence of a particular statistical test. The question that naturally arises in this setting is: Given such a probabilistic graph and a specific causal effect of interest, what is the subgraph which has the highest plausibility and for which the causal effect is identifiable? We show that answering this question reduces to solving an NP-complete combinatorial optimization problem which we call the edge ID problem. We propose efficient algorithms to approximate this problem and evaluate them against both real-world networks and randomly generated graphs.

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Sparse recovery is among the most well-studied problems in learning theory and high-dimensional statistics. In this work, we investigate the statistical and computational landscapes of sparse recovery with $\ell_\infty$ error guarantees. This variant of the problem is motivated by \emph{variable selection} tasks, where the goal is to estimate the support of a $k$-sparse signal in $\mathbb{R}^d$. Our main contribution is a provable separation between the \emph{oblivious} (``for each'') and \emph{adaptive} (``for all'') models of $\ell_\infty$ sparse recovery. We show that under an oblivious model, the optimal $\ell_\infty$ error is attainable in near-linear time with $\approx k\log d$ samples, whereas in an adaptive model, $\gtrsim k^2$ samples are necessary for any algorithm to achieve this bound. This establishes a surprising contrast with the standard $\ell_2$ setting, where $\approx k \log d$ samples suffice even for adaptive sparse recovery. We conclude with a preliminary examination of a \emph{partially-adaptive} model, where we show nontrivial variable selection guarantees are possible with $\approx k\log d$ measurements.

Theorem 4.2.Thefollowingholdsfor Algorithm 2: withprobabilityatleast1 , forallt T hP Pt,Ci 32 log ( 3e / ) ( C)2 t+ 1 1 , where = (C) Theempirical implementation condition allowsusCt, with specified components, 7 1: Experimentsonsyntheticdata.

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We consider the problem of minimizing a continuous function given given access to a natural quantum generalization of a stochastic gradient oracle. We provide two new methods for the special case of minimizing a Lipschitz convex function. Each method obtains a dimension versus accuracy trade-off which is provably unachievable classically and we prove that one method is asymptotically optimal in low-dimensional settings. Additionally, we provide quantum algorithms for computing a critical point of a smooth non-convex function at rates not known to be achievable classically. To obtain these results we build upon the quantum multivariate mean estimation result of Cornelissen et al. [25] and provide a general quantum variance reduction technique of independent interest.

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