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Abstract The detection of orbital eccentricity for a binary black hole system via gravitational waves is a key signature to distinguish between the possible binary origins. The identification of eccentricity has been difficult so far due to the limited availability of eccentric gravitational waveforms over the full range of black hole masses and eccentricities. Here we evaluate the eccentricity of five black hole mergers detected by the LIGO and Virgo observatories using theTEOBResumS-DALI,TEOBResumS-GIOTTO, andTEOBResumSPmodels. This analysis studies eccentricities up to 0.6 at the reference frequency of 5 Hz and incorporates higher-order gravitational-wave modes critical to model emission from highly eccentric orbits. The binaries have been selected due to previous hints of eccentricity or due to their unusual mass and spin. While other studies found marginal evidence for eccentricity for some of these events, our analyses do not favor the incorporation of eccentricity compared to the quasi-circular case. While lacking the eccentric evidence of other analyses, we find our analyses marginally shifts the posterior in multiple parameters for several events when allowing eccentricity to be nonzero. 

We give the first reconstruction algorithm for decision trees: given queries to a function f that is opt-close to a size-s decision tree, our algorithm provides query access to a decision tree T where: - T has size S := s^O((log s)²/ε³); - dist(f,T) ≤ O(opt)+ε; - Every query to T is answered with poly((log s)/ε)⋅ log n queries to f and in poly((log s)/ε)⋅ n log n time. This yields a tolerant tester that distinguishes functions that are close to size-s decision trees from those that are far from size-S decision trees. The polylogarithmic dependence on s in the efficiency of our tester is exponentially smaller than that of existing testers. Since decision tree complexity is well known to be related to numerous other boolean function properties, our results also provide a new algorithm for reconstructing and testing these properties. 

We initiate the study of a fundamental question concerning adversarial noise models in statistical problems where the algorithm receives i.i.d. draws from a distribution D. The definitions of these adversaries specify the {\sl type} of allowable corruptions (noise model) as well as {\sl when} these corruptions can be made (adaptivity); the latter differentiates between oblivious adversaries that can only corrupt the distribution D and adaptive adversaries that can have their corruptions depend on the specific sample S that is drawn from D. We investigate whether oblivious adversaries are effectively equivalent to adaptive adversaries, across all noise models studied in the literature, under a unifying framework that we introduce. Specifically, can the behavior of an algorithm A in the presence of oblivious adversaries always be well-approximated by that of an algorithm A′ in the presence of adaptive adversaries? Our first result shows that this is indeed the case for the broad class of {\sl statistical query} algorithms, under all reasonable noise models. We then show that in the specific case of {\sl additive noise}, this equivalence holds for {\sl all} algorithms. Finally, we map out an approach towards proving this statement in its fullest generality, for all algorithms and under all reasonable noise models. 

We study the problem of certification: given queries to a function f : {0,1}n → {0,1} with certificate complexity ≤ k and an input x⋆, output a size-k certificate for f’s value on x⋆. For monotone functions, a classic local search algorithm of Angluin accomplishes this task with n queries, which we show is optimal for local search algorithms. Our main result is a new algorithm for certifying monotone functions with O(k8 logn) queries, which comes close to matching the information-theoretic lower bound of Ω(k logn). The design and analysis of our algorithm are based on a new connection to threshold phenomena in monotone functions. We further prove exponential-in-k lower bounds when f is non-monotone, and when f is monotone but the algorithm is only given random examples of f. These lower bounds show that assumptions on the structure of f and query access to it are both necessary for the polynomial dependence on k that we achieve. 

We design an algorithm for finding counterfactuals with strong theoretical guarantees on its performance. For any monotone model f:Xd→{0,1} and instance x⋆, our algorithm makes S(f)O(Δf(x⋆))⋅logd {queries} to f and returns an {\sl optimal} counterfactual for x⋆: a nearest instance x′ to x⋆ for which f(x′)≠f(x⋆). Here S(f) is the sensitivity of f, a discrete analogue of the Lipschitz constant, and Δf(x⋆) is the distance from x⋆ to its nearest counterfactuals. The previous best known query complexity was dO(Δf(x⋆)), achievable by brute-force local search. We further prove a lower bound of S(f)Ω(Δf(x⋆))+Ω(logd) on the query complexity of any algorithm, thereby showing that the guarantees of our algorithm are essentially optimal. 

Using the framework of boosting, we prove that all impurity-based decision tree learning algorithms, including the classic ID3, C4.5, and CART, are highly noise tolerant. Our guarantees hold under the strongest noise model of nasty noise, and we provide near-matching upper and lower bounds on the allowable noise rate. We further show that these algorithms, which are simple and have long been central to everyday machine learning, enjoy provable guarantees in the noisy setting that are unmatched by existing algorithms in the theoretical literature on decision tree learning. Taken together, our results add to an ongoing line of research that seeks to place the empirical success of these practical decision tree algorithms on firm theoretical footing. 

We consider the problem of explaining the predictions of an arbitrary blackbox model f: given query access to f and an instance x, output a small set of x's features that in conjunction essentially determines f(x). We design an efficient algorithm with provable guarantees on the succinctness and precision of the explanations that it returns. Prior algorithms were either efficient but lacked such guarantees, or achieved such guarantees but were inefficient. We obtain our algorithm via a connection to the problem of {\sl implicitly} learning decision trees. The implicit nature of this learning task allows for efficient algorithms even when the complexity of~f necessitates an intractably large surrogate decision tree. We solve the implicit learning problem by bringing together techniques from learning theory, local computation algorithms, and complexity theory. Our approach of “explaining by implicit learning” shares elements of two previously disparate methods for post-hoc explanations, global and local explanations, and we make the case that it enjoys advantages of both.