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Since their introduction in Abadie and Gardeazabal (2003), Synthetic Control (SC) methods have quickly become one of the leading methods for estimating causal effects in observational studies in settings with panel data. Formal discussions often motivate SC methods by the assumption that the potential outcomes were generated by a factor model. Here we study SC methods from a design-based perspective, assuming a model for the selection of the treated unit(s) and period(s). We show that the standard SC estimator is generally biased under random assignment. We propose a Modified Unbiased Synthetic Control (MUSC) estimator that guarantees unbiasedness under random assignment and derive its exact, randomization-based, finite-sample variance. We also propose an unbiased estimator for this variance. We document in settings with real data that under random assignment, SC-type estimators can have root mean-squared errors that are substantially lower than that of other common estimators. We show that such an improvement is weakly guaranteed if the treated period is similar to the other periods, for example, if the treated period was randomly selected. While our results only directly apply in settings where treatment is assigned randomly, we believe that they can complement model-based approaches even for observational studies.

The exponential increase in the amount of available data makes taking advantage of them without violating users' privacy one of the fundamental problems of computer science. This question has been investigated thoroughly under the framework of differential privacy. However, most of the literature has not focused on settings where the amount of data is so large that we are not even able to compute the exact answer in the non-private setting (such as in the streaming setting, sublinear-time setting, etc.). This can often make the use of differential privacy unfeasible in practice. In this paper, we show a general approach for making Monte-Carlo randomized approximation algorithms differentially private. We only need to assume the error $R$ of the approximation algorithm is sufficiently concentrated around $0$ (e.g.\ $\mathbb{E}[|R|]$ is bounded) and that the function being approximated has a small global sensitivity $\Delta$. Specifically, if we have a randomized approximation algorithm with sufficiently concentrated error which has time/space/query complexity $T(n,\rho)$ with $\rho$ being an accuracy parameter, we can generally speaking get an algorithm with the same accuracy and complexity $T(n,\Theta(\epsilon \rho))$ that is $\epsilon$-differentially private.

Let $G$ be a graph, which represents a social network, and suppose each node $v$ has a threshold value $\tau(v)$. Consider an initial configuration, where each node is either positive or negative. In each discrete time step, a node $v$ becomes/remains positive if at least $\tau(v)$ of its neighbors are positive and negative otherwise. A node set $\mathcal{S}$ is a Target Set (TS) whenever the following holds: if $\mathcal{S}$ is fully positive initially, all nodes in the graph become positive eventually. We focus on a generalization of TS, called Timed TS (TTS), where it is permitted to assign a positive state to a node at any step of the process, rather than just at the beginning. We provide graph structures for which the minimum TTS is significantly smaller than the minimum TS, indicating that timing is an essential aspect of successful target selection strategies. Furthermore, we prove tight bounds on the minimum size of a TTS in terms of the number of nodes and maximum degree when the thresholds are assigned based on the majority rule. We show that the problem of determining the minimum size of a TTS is NP-hard and provide an Integer Linear Programming formulation and a greedy algorithm. We evaluate the performance of our algorithm by conducting experiments on various synthetic and real-world networks. We also present a linear-time exact algorithm for trees.

Grammar compression is a general compression framework in which a string $T$ of length $N$ is represented as a context-free grammar of size $n$ whose language contains only $T$. In this paper, we focus on studying the limitations of algorithms and data structures operating on strings in grammar-compressed form. Previous work focused on proving lower bounds for grammars constructed using algorithms that achieve the approximation ratio $\rho=\mathcal{O}(\text{polylog }N)$. Unfortunately, for the majority of grammar compressors, $\rho$ is either unknown or satisfies $\rho=\omega(\text{polylog }N)$. In their seminal paper, Charikar et al. [IEEE Trans. Inf. Theory 2005] studied seven popular grammar compression algorithms: RePair, Greedy, LongestMatch, Sequential, Bisection, LZ78, and $\alpha$-Balanced. Only one of them ($\alpha$-Balanced) is known to achieve $\rho=\mathcal{O}(\text{polylog }N)$. We develop the first technique for proving lower bounds for data structures and algorithms on grammars that is fully general and does not depend on the approximation ratio $\rho$ of the used grammar compressor. Using this technique, we first prove that $\Omega(\log N/\log \log N)$ time is required for random access on RePair, Greedy, LongestMatch, Sequential, and Bisection, while $\Omega(\log\log N)$ time is required for random access to LZ78. All these lower bounds hold within space $\mathcal{O}(n\text{ polylog }N)$ and match the existing upper bounds. We also generalize this technique to prove several conditional lower bounds for compressed computation. For example, we prove that unless the Combinatorial $k$-Clique Conjecture fails, there is no combinatorial algorithm for CFG parsing on Bisection (for which it holds $\rho=\tilde{\Theta}(N^{1/2})$) that runs in $\mathcal{O}(n^c\cdot N^{3-\epsilon})$ time for all constants $c>0$ and $\epsilon>0$. Previously, this was known only for $c<2\epsilon$.

We consider the best-k-arm identification problem for multi-armed bandits, where the objective is to select the exact set of k arms with the highest mean rewards by sequentially allocating measurement effort. We characterize the necessary and sufficient conditions for the optimal allocation using dual variables. Remarkably these optimality conditions lead to the extension of top-two algorithm design principle (Russo, 2020), initially proposed for best-arm identification. Furthermore, our optimality conditions induce a simple and effective selection rule dubbed information-directed selection (IDS) that selects one of the top-two candidates based on a measure of information gain. As a theoretical guarantee, we prove that integrated with IDS, top-two Thompson sampling is (asymptotically) optimal for Gaussian best-arm identification, solving a glaring open problem in the pure exploration literature (Russo, 2020). As a by-product, we show that for k > 1, top-two algorithms cannot achieve optimality even when the algorithm has access to the unknown "optimal" tuning parameter. Numerical experiments show the superior performance of the proposed top-two algorithms with IDS and considerable improvement compared with algorithms without adaptive selection.

We assume to be given structural equations over discrete variables inducing a directed acyclic graph, namely, a structural causal model, together with data about its internal nodes. The question we want to answer is how we can compute bounds for partially identifiable counterfactual queries from such an input. We start by giving a map from structural casual models to credal networks. This allows us to compute exact counterfactual bounds via algorithms for credal nets on a subclass of structural causal models. Exact computation is going to be inefficient in general given that, as we show, causal inference is NP-hard even on polytrees. We target then approximate bounds via a causal EM scheme. We evaluate their accuracy by providing credible intervals on the quality of the approximation; we show through a synthetic benchmark that the EM scheme delivers accurate results in a fair number of runs. In the course of the discussion, we also point out what seems to be a neglected limitation to the trending idea that counterfactual bounds can be computed without knowledge of the structural equations. We also present a real case study on palliative care to show how our algorithms can readily be used for practical purposes.

Two observational methods are currently being used to monitor post-deployment vaccine effectiveness: the obvious crude method comparing rate testing positive per head of vaccinated population with that rate per head of unvaccinated population; and the test-negative case control (TNCC) method. The two methods give very different results. We want to know whether either method is reliable. We assume either a homogeneous population or one partitioned into two homogeneous subsets which differ only in their not-directly-observable healthcare-seeking behaviour including probability of getting vaccinated. We first consider uniform independent priors on the probabilities of being hospitalised conditional on subset, vaccination status, and infection status. We simulate from the resulting model and observe the TNCC estimate, the crude estimate, and the Bayesian central 95% confidence interval on vaccine effectiveness represented as log ratio of odds ratios for infection with and without vaccination. With these wide open priors, even when the population is homogeneous, the Bayesian 95% confidence interval typically has a width of nearly 4 nats (55-fold), implying too much uncertainty for the data collected to be of any use in monitoring effectiveness. There do exist some tight priors under which the data is useful: some lead to TNCC being more accurate while with others the crude estimate is more accurate. Thus using only data from those spontaneously choosing to be tested, we find that neither method is reliably better than the other, and indeed that the desired information is not present in this data. We conclude that effective monitoring of vaccine effectiveness and side-effects requires either strong information on the population's behaviour, or ongoing randomised controlled trials (RCTs), rather than just choosing whichever of TNCC and crude estimate gives the result we prefer to find.

Fingerprinting arguments, first introduced by Bun, Ullman, and Vadhan (STOC 2014), are the most widely used method for establishing lower bounds on the sample complexity or error of approximately differentially private (DP) algorithms. Still, there are many problems in differential privacy for which we don't know suitable lower bounds, and even for problems that we do, the lower bounds are not smooth, and usually become vacuous when the error is larger than some threshold. In this work, we present a simple method to generate hard instances by applying a padding-and-permuting transformation to a fingerprinting code. We illustrate the applicability of this method by providing new lower bounds in various settings: 1. A tight lower bound for DP averaging in the low-accuracy regime, which in particular implies a new lower bound for the private 1-cluster problem introduced by Nissim, Stemmer, and Vadhan (PODS 2016). 2. A lower bound on the additive error of DP algorithms for approximate k-means clustering, as a function of the multiplicative error, which is tight for a constant multiplication error. 3. A lower bound for estimating the top singular vector of a matrix under DP in low-accuracy regimes, which is a special case of DP subspace estimation studied by Singhal and Steinke (NeurIPS 2021). Our main technique is to apply a padding-and-permuting transformation to a fingerprinting code. However, rather than proving our results using a black-box access to an existing fingerprinting code (e.g., Tardos' code), we develop a new fingerprinting lemma that is stronger than those of Dwork et al. (FOCS 2015) and Bun et al. (SODA 2017), and prove our lower bounds directly from the lemma. Our lemma, in particular, gives a simpler fingerprinting code construction with optimal rate (up to polylogarithmic factors) that is of independent interest.

In uncertainty quantification, variance-based global sensitivity analysis quantitatively determines the effect of each input random variable on the output by partitioning the total output variance into contributions from each input. However, computing conditional expectations can be prohibitively costly when working with expensive-to-evaluate models. Surrogate models can accelerate this, yet their accuracy depends on the quality and quantity of training data, which is expensive to generate (experimentally or computationally) for complex engineering systems. Thus, methods that work with limited data are desirable. We propose a diffeomorphic modulation under observable response preserving homotopy (D-MORPH) regression to train a polynomial dimensional decomposition surrogate of the output that minimizes the number of training data. The new method first computes a sparse Lasso solution and uses it to define the cost function. A subsequent D-MORPH regression minimizes the difference between the D-MORPH and Lasso solution. The resulting D-MORPH surrogate is more robust to input variations and more accurate with limited training data. We illustrate the accuracy and computational efficiency of the new surrogate for global sensitivity analysis using mathematical functions and an expensive-to-simulate model of char combustion. The new method is highly efficient, requiring only 15% of the training data compared to conventional regression.

We introduce a new quantum algorithm for computing the Betti numbers of a simplicial complex. In contrast to previous quantum algorithms that work by estimating the eigenvalues of the combinatorial Laplacian, our algorithm is an instance of the generic Incremental Algorithm for computing Betti numbers that incrementally adds simplices to the simplicial complex and tests whether or not they create a cycle. In contrast to existing quantum algorithms for computing Betti numbers that work best when the complex has close to the maximal number of simplices, our algorithm works best for sparse complexes. To test whether a simplex creates a cycle, we introduce a quantum span-program algorithm. We show that the query complexity of our span program is parameterized by quantities called the effective resistance and effective capacitance of the boundary of the simplex. Unfortunately, we also prove upper and lower bounds on the effective resistance and capacitance, showing both quantities can be exponentially large with respect to the size of the complex, implying that our algorithm would have to run for exponential time to exactly compute Betti numbers. However, as a corollary to these bounds, we show that the spectral gap of the combinatorial Laplacian can be exponentially small. As the runtime of all previous quantum algorithms for computing Betti numbers are parameterized by the inverse of the spectral gap, our bounds show that all quantum algorithms for computing Betti numbers must run for exponentially long to exactly compute Betti numbers. Finally, we prove some novel formulas for effective resistance and effective capacitance to give intuition for these quantities.