Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes

Nonparametric causal effects based on incremental propensity score interventions

Most work in causal inference considers deterministic interventions that set each unit's treatment to some fixed value. However, under positivity violations these interventions can lead to non-identification, inefficiency, and effects with little practical relevance. Further, corresponding effects in longitudinal studies are highly sensitive to the curse of dimensionality, resulting in widespread use of unrealistic parametric models. We propose a novel solution to these problems: incremental interventions that shift propensity score values rather than set treatments to fixed values. Incremental interventions have several crucial advantages. First, they avoid positivity assumptions entirely. Second, they require no parametric assumptions and yet still admit a simple characterization of longitudinal effects, independent of the number of timepoints. For example, they allow longitudinal effects to be visualized with a single curve instead of lists of coefficients. After characterizing these incremental interventions and giving identifying conditions for corresponding effects, we also develop general efficiency theory, propose efficient nonparametric estimators that can attain fast convergence rates even when incorporating flexible machine learning, and propose a bootstrap-based confidence band and simultaneous test of no treatment effect. Finally we explore finite-sample performance via simulation, and apply the methods to study time-varying sociological effects of incarceration on entry into marriage

On the Gold Standard for Security of Universal Steganography

While symmetric-key steganography is quite well understood both in the information-theoretic and in the computational setting, many fundamental questions about its public-key counterpart resist persistent attempts to solve them. The computational model for public-key steganography was proposed by von Ahn and Hopper in EUROCRYPT 2004. At TCC 2005, Backes and Cachin gave the first universal public-key stegosystem - i.e. one that works on all channels - achieving security against replayable chosen-covertext attacks (SS-RCCA) and asked whether security against non-replayable chosen-covertext attacks (SS-CCA) is achievable. Later, Hopper (ICALP 2005) provided such a stegosystem for every efficiently sampleable channel, but did not achieve universality. He posed the question whether universality and SS-CCA-security can be achieved simultaneously. No progress on this question has been achieved since more than a decade. In our work we solve Hopper's problem in a somehow complete manner: As our main positive result we design an SS-CCA-secure stegosystem that works for every memoryless channel. On the other hand, we prove that this result is the best possible in the context of universal steganography. We provide a family of 0-memoryless channels - where the already sent documents have only marginal influence on the current distribution - and prove that no SS-CCA-secure steganography for this family exists in the standard non-look-ahead model.Comment: EUROCRYPT 2018, llncs styl

Selective machine learning of doubly robust functionals

While model selection is a well-studied topic in parametric and nonparametric regression or density estimation, selection of possibly high-dimensional nuisance parameters in semiparametric problems is far less developed. In this paper, we propose a selective machine learning framework for making inferences about a finite-dimensional functional defined on a semiparametric model, when the latter admits a doubly robust estimating function and several candidate machine learning algorithms are available for estimating the nuisance parameters. We introduce two new selection criteria for bias reduction in estimating the functional of interest, each based on a novel definition of pseudo-risk for the functional that embodies the double robustness property and thus is used to select the pair of learners that is nearest to fulfilling this property. We establish an oracle property for a multi-fold cross-validation version of the new selection criteria which states that our empirical criteria perform nearly as well as an oracle with a priori knowledge of the pseudo-risk for each pair of candidate learners. We also describe a smooth approximation to the selection criteria which allows for valid post-selection inference. Finally, we apply the approach to model selection of a semiparametric estimator of average treatment effect given an ensemble of candidate machine learners to account for confounding in an observational study

Dynamically optimal treatment allocation using Reinforcement Learning

Devising guidance on how to assign individuals to treatment is an important goal in empirical research. In practice, individuals often arrive sequentially, and the planner faces various constraints such as limited budget/capacity, or borrowing constraints, or the need to place people in a queue. For instance, a governmental body may receive a budget outlay at the beginning of a year, and it may need to decide how best to allocate resources within the year to individuals who arrive sequentially. In this and other examples involving inter-temporal trade-offs, previous work on devising optimal policy rules in a static context is either not applicable, or sub-optimal. Here we show how one can use offline observational data to estimate an optimal policy rule that maximizes expected welfare in this dynamic context. We allow the class of policy rules to be restricted for legal, ethical or incentive compatibility reasons. The problem is equivalent to one of optimal control under a constrained policy class, and we exploit recent developments in Reinforcement Learning (RL) to propose an algorithm to solve this. The algorithm is easily implementable with speedups achieved through multiple RL agents learning in parallel processes. We also characterize the statistical regret from using our estimated policy rule by casting the evolution of the value function under each policy in a Partial Differential Equation (PDE) form and using the theory of viscosity solutions to PDEs. We find that the policy regret decays at a $n^{-1/2}$ rate in most examples; this is the same rate as in the static case.Comment: 67 page

Foundations of Declarative Data Analysis Using Limit Datalog Programs

Motivated by applications in declarative data analysis, we study $\mathit{Datalog}_{\mathbb{Z}}$---an extension of positive Datalog with arithmetic functions over integers. This language is known to be undecidable, so we propose two fragments. In $\mathit{limit}~\mathit{Datalog}_{\mathbb{Z}}$ predicates are axiomatised to keep minimal/maximal numeric values, allowing us to show that fact entailment is coNExpTime-complete in combined, and coNP-complete in data complexity. Moreover, an additional $\mathit{stability}$ requirement causes the complexity to drop to ExpTime and PTime, respectively. Finally, we show that stable $\mathit{Datalog}_{\mathbb{Z}}$ can express many useful data analysis tasks, and so our results provide a sound foundation for the development of advanced information systems.Comment: 23 pages; full version of a paper accepted at IJCAI-17; v2 fixes some typos and improves the acknowledgment

A Bayesian view of doubly robust causal inference

In causal inference confounding may be controlled either through regression adjustment in an outcome model, or through propensity score adjustment or inverse probability of treatment weighting, or both. The latter approaches, which are based on modelling of the treatment assignment mechanism and their doubly robust extensions have been difficult to motivate using formal Bayesian arguments, in principle, for likelihood-based inferences, the treatment assignment model can play no part in inferences concerning the expected outcomes if the models are assumed to be correctly specified. On the other hand, forcing dependency between the outcome and treatment assignment models by allowing the former to be misspecified results in loss of the balancing property of the propensity scores and the loss of any double robustness. In this paper, we explain in the framework of misspecified models why doubly robust inferences cannot arise from purely likelihood-based arguments, and demonstrate this through simulations. As an alternative to Bayesian propensity score analysis, we propose a Bayesian posterior predictive approach for constructing doubly robust estimation procedures. Our approach appropriately decouples the outcome and treatment assignment models by incorporating the inverse treatment assignment probabilities in Bayesian causal inferences as importance sampling weights in Monte Carlo integration.Comment: Author's original version. 21 pages, including supplementary materia