The indirect effect of an exposure on an outcome through an intermediate variable can be identified by a product of regression coefficients under certain causal and regression modeling assumptions. In this context, the null hypothesis of no indirect effect is a composite null hypothesis, as the null holds if either regression coefficient is zero. A consequence is that traditional hypothesis tests are severely underpowered near the origin (i.e., when both coefficients are small with respect to standard errors). We propose hypothesis tests that (i) preserve level alpha type 1 error, (ii) meaningfully improve power when both true underlying effects are small relative to sample size, and (iii) preserve power when at least one is not. One approach gives a closed-form test that is minimax optimal with respect to local power over the alternative parameter space. Another uses sparse linear programming to produce an approximately optimal test for a Bayes risk criterion. We discuss adaptations for performing large-scale hypothesis testing as well as modifications that yield improved interpretability. We provide an R package that implements the minimax optimal test.
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Dealing with uncertainty in optimization parameters is an important and longstanding challenge. Typically, uncertain parameters are predicted accurately, and then a deterministic optimization problem is solved. However, the decisions produced by this so-called \emph{predict-then-optimize} procedure can be highly sensitive to uncertain parameters. In this work, we contribute to recent efforts in producing \emph{decision-focused} predictions, i.e., to build predictive models that are constructed with the goal of minimizing a \emph{regret} measure on the decisions taken with them. We begin by formulating the exact expected regret minimization as a pessimistic bilevel optimization model. Then, we establish NP-completeness of this problem, even in a heavily restricted case. Using duality arguments, we reformulate it as a non-convex quadratic optimization problem. Finally, we show various computational techniques to achieve tractability. We report extensive computational results on shortest-path instances with uncertain cost vectors. Our results indicate that our approach can improve training performance over the approach of Elmachtoub and Grigas (2022), a state-of-the-art method for decision-focused learning.
The majority of fault-tolerant distributed algorithms are designed assuming a nominal corruption model, in which at most a fraction $f_n$ of parties can be corrupted by the adversary. However, due to the infamous Sybil attack, nominal models are not sufficient to express the trust assumptions in open (i.e., permissionless) settings. Instead, permissionless systems typically operate in a weighted model, where each participant is associated with a weight and the adversary can corrupt a set of parties holding at most a fraction $f_w$ of the total weight. In this paper, we suggest a simple way to transform a large class of protocols designed for the nominal model into the weighted model. To this end, we formalize and solve three novel optimization problems, which we collectively call the weight reduction problems, that allow us to map large real weights into small integer weights while preserving the properties necessary for the correctness of the protocols. In all cases, we manage to keep the sum of the integer weights to be at most linear in the number of parties, resulting in extremely efficient protocols for the weighted model. Moreover, we demonstrate that, on weight distributions that emerge in practice, the sum of the integer weights tends to be far from the theoretical worst case and, sometimes, even smaller than the number of participants. While, for some protocols, our transformation requires an arbitrarily small reduction in resilience (i.e., $f_w = f_n - \epsilon$), surprisingly, for many important problems, we manage to obtain weighted solutions with the same resilience ($f_w = f_n$) as nominal ones. Notable examples include erasure-coded distributed storage and broadcast protocols, verifiable secret sharing, and asynchronous consensus.
Logistic regression is a classical model for describing the probabilistic dependence of binary responses to multivariate covariates. We consider the predictive performance of the maximum likelihood estimator (MLE) for logistic regression, assessed in terms of logistic risk. We consider two questions: first, that of the existence of the MLE (which occurs when the dataset is not linearly separated), and second that of its accuracy when it exists. These properties depend on both the dimension of covariates and on the signal strength. In the case of Gaussian covariates and a well-specified logistic model, we obtain sharp non-asymptotic guarantees for the existence and excess logistic risk of the MLE. We then generalize these results in two ways: first, to non-Gaussian covariates satisfying a certain two-dimensional margin condition, and second to the general case of statistical learning with a possibly misspecified logistic model. Finally, we consider the case of a Bernoulli design, where the behavior of the MLE is highly sensitive to the parameter direction.
Over the past decade, studies of naturalistic language processing where participants are scanned while listening to continuous text have flourished. Using word embeddings at first, then large language models, researchers have created encoding models to analyze the brain signals. Presenting these models with the same text as the participants allows to identify brain areas where there is a significant correlation between the functional magnetic resonance imaging (fMRI) time series and the ones predicted by the models' artificial neurons. One intriguing finding from these studies is that they have revealed highly symmetric bilateral activation patterns, somewhat at odds with the well-known left lateralization of language processing. Here, we report analyses of an fMRI dataset where we manipulate the complexity of large language models, testing 28 pretrained models from 8 different families, ranging from 124M to 14.2B parameters. First, we observe that the performance of models in predicting brain responses follows a scaling law, where the fit with brain activity increases linearly with the logarithm of the number of parameters of the model (and its performance on natural language processing tasks). Second, although this effect is present in both hemispheres, it is stronger in the left than in the right hemisphere. Specifically, the left-right difference in brain correlation follows a scaling law with the number of parameters. This finding reconciles computational analyses of brain activity using large language models with the classic observation from aphasic patients showing left hemisphere dominance for language.
We propose a tamed-adaptive Milstein scheme for stochastic differential equations in which the first-order derivatives of the coefficients are locally H\"older continuous of order $\alpha$. We show that the scheme converges in the $L_2$-norm with a rate of $(1+\alpha)/2$ over both finite intervals $[0, T]$ and the infinite interval $(0, +\infty)$, under certain growth conditions on the coefficients.
Aperiodic autocorrelation is an important indicator of performance of sequences used in communications, remote sensing, and scientific instrumentation. Knowing a sequence's autocorrelation function, which reports the autocorrelation at every possible translation, is equivalent to knowing the magnitude of the sequence's Fourier transform. The phase problem is the difficulty in resolving this lack of phase information. We say that two sequences are equicorrelational to mean that they have the same aperiodic autocorrelation function. Sequences used in technological applications often have restrictions on their terms: they are not arbitrary complex numbers, but come from a more restricted alphabet. For example, binary sequences involve terms equal to only $+1$ and $-1$. We investigate the necessary and sufficient conditions for two sequences to be equicorrelational, where we take their alphabet into consideration. There are trivial forms of equicorrelationality arising from modifications that predictably preserve the autocorrelation, for example, negating a binary sequence or reversing the order of its terms. By a search of binary sequences up to length $44$, we find that nontrivial equicorrelationality among binary sequences does occur, but is rare. An integer $n$ is said to be equivocal when there are binary sequences of length $n$ that are nontrivially equicorrelational; otherwise $n$ is unequivocal. For $n \leq 44$, we found that the unequivocal lengths are $1$--$8$, $10$, $11$, $13$, $14$, $19$, $22$, $23$, $26$, $29$, $37$, and $38$. We pose open questions about the finitude of unequivocal numbers and the probability of nontrivial equicorrelationality occurring among binary sequences.
We prove explicit uniform two-sided bounds for the phase functions of Bessel functions and of their derivatives. As a consequence, we obtain new enclosures for the zeros of Bessel functions and their derivatives in terms of inverse values of some elementary functions. These bounds are valid, with a few exceptions, for all zeros and all Bessel functions with non-negative indices. We provide numerical evidence showing that our bounds either improve or closely match the best previously known ones.
We introduce an algebraic concept of the frame for abstract conditional independence (CI) models, together with basic operations with respect to which such a frame should be closed: copying and marginalization. Three standard examples of such frames are (discrete) probabilistic CI structures, semi-graphoids and structural semi-graphoids. We concentrate on those frames which are closed under the operation of set-theoretical intersection because, for these, the respective families of CI models are lattices. This allows one to apply the results from lattice theory and formal concept analysis to describe such families in terms of implications among CI statements. The central concept of this paper is that of self-adhesivity defined in algebraic terms, which is a combinatorial reflection of the self-adhesivity concept studied earlier in context of polymatroids and information theory. The generalization also leads to a self-adhesivity operator defined on the hyper-level of CI frames. We answer some of the questions related to this approach and raise other open questions. The core of the paper is in computations. The combinatorial approach to computation might overcome some memory and space limitation of software packages based on polyhedral geometry, in particular, if SAT solvers are utilized. We characterize some basic CI families over 4 variables in terms of canonical implications among CI statements. We apply our method in information-theoretical context to the task of entropic region demarcation over 5 variables.
Methods for analyzing representations in neural systems are increasingly popular tools in neuroscience and mechanistic interpretability. Measures comparing neural activations across conditions, architectures, and species give scalable ways to understand information transformation within different neural networks. However, recent findings show that some metrics respond to spurious signals, leading to misleading results. Establishing benchmark test cases is thus essential for identifying the most reliable metric and potential improvements. We propose that compositional learning in recurrent neural networks (RNNs) can provide a test case for dynamical representation alignment metrics. Implementing this case allows us to evaluate if metrics can identify representations that develop throughout learning and determine if representations identified by metrics reflect the network's actual computations. Building both attractor and RNN based test cases, we show that the recently proposed Dynamical Similarity Analysis (DSA) is more noise robust and reliably identifies behaviorally relevant representations compared to prior metrics (Procrustes, CKA). We also demonstrate how such test cases can extend beyond metric evaluation to study new architectures. Specifically, testing DSA in modern (Mamba) state space models suggests that these models, unlike RNNs, may not require changes in recurrent dynamics due to their expressive hidden states. Overall, we develop test cases that showcase how DSA's enhanced ability to detect dynamical motifs makes it highly effective for identifying ongoing computations in RNNs and revealing how networks learn tasks.
We present accurate and mathematically consistent formulations of a diffuse-interface model for two-phase flow problems involving rapid evaporation. The model addresses challenges including discontinuities in the density field by several orders of magnitude, leading to high velocity and pressure jumps across the liquid-vapor interface, along with dynamically changing interface topologies. To this end, we integrate an incompressible Navier-Stokes solver combined with a conservative level-set formulation and a regularized, i.e., diffuse, representation of discontinuities into a matrix-free adaptive finite element framework. The achievements are three-fold: First, we propose mathematically consistent definitions for the level-set transport velocity in the diffuse interface region by extrapolating the velocity from the liquid or gas phase. They exhibit superior prediction accuracy for the evaporated mass and the resulting interface dynamics compared to a local velocity evaluation, especially for strongly curved interfaces. Second, we show that accurate prediction of the evaporation-induced pressure jump requires a consistent, namely a reciprocal, density interpolation across the interface, which satisfies local mass conservation. Third, the combination of diffuse interface models for evaporation with standard Stokes-type constitutive relations for viscous flows leads to significant pressure artifacts in the diffuse interface region. To mitigate these, we propose to introduce a correction term for such constitutive model types. Through selected analytical and numerical examples, the aforementioned properties are validated. The presented model promises new insights in simulation-based prediction of melt-vapor interactions in thermal multiphase flows such as in laser-based powder bed fusion of metals.
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