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Numerical evaluations of the memory capacity (MC) of recurrent neural networks reported in the literature often contradict well-established theoretical bounds. In this paper, we study the case of linear echo state networks, for which the total memory capacity has been proven to be equal to the rank of the corresponding Kalman controllability matrix. We shed light on various reasons for the inaccurate numerical estimations of the memory, and we show that these issues, often overlooked in the recent literature, are of an exclusively numerical nature. More explicitly, we prove that when the Krylov structure of the linear MC is ignored, a gap between the theoretical MC and its empirical counterpart is introduced. As a solution, we develop robust numerical approaches by exploiting a result of MC neutrality with respect to the input mask matrix. Simulations show that the memory curves that are recovered using the proposed methods fully agree with the theory.

Companies offering web services routinely run randomized online experiments to estimate the causal impact associated with the adoption of new features and policies on key performance metrics of interest. These experiments are used to estimate a variety of effects: the increase in click rate due to the repositioning of a banner, the impact on subscription rate as a consequence of a discount or special offer, etc. In these settings, even effects whose sizes are very small can have large downstream impacts. The simple difference in means estimator (Splawa-Neyman et al., 1990) is still the standard estimator of choice for many online A/B testing platforms due to its simplicity. This method, however, can fail to detect small effects, even when the experiment contains thousands or millions of observational units. As a by-product of these experiments, however, large amounts of additional data (covariates) are collected. In this paper, we discuss benefits, costs and risks of allowing experimenters to leverage more complicated estimators that make use of covariates when estimating causal effects of interest. We adapt a recently proposed general-purpose algorithm for the estimation of causal effects with covariates to the setting of online A/B tests. Through this paradigm, we implement several covariate-adjusted causal estimators. We thoroughly evaluate their performance at scale, highlighting benefits and shortcomings of different methods. We show on real experiments how "covariate-adjusted" estimators can (i) lead to more precise quantification of the causal effects of interest and (ii) fix issues related to imbalance across treatment arms - a practical concern often overlooked in the literature. In turn, (iii) these more precise estimates can reduce experimentation time, cutting cost and helping to streamline decision-making processes, allowing for faster adoption of beneficial interventions.

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.

We present new results on average causal effects in settings with unmeasured exposure-outcome confounding. Our results are motivated by a class of estimands, e.g., frequently of interest in medicine and public health, that are currently not targeted by standard approaches for average causal effects. We recognize these estimands as queries about the average causal effect of an intervening variable. We anchor our introduction of these estimands in an investigation of the role of chronic pain and opioid prescription patterns in the opioid epidemic, and illustrate how conventional approaches will lead unreplicable estimates with ambiguous policy implications. We argue that our altenative effects are replicable and have clear policy implications, and furthermore are non-parametrically identified by the classical frontdoor formula. As an independent contribution, we derive a new semiparametric efficient estimator of the frontdoor formula with a uniform sample boundedness guarantee. This property is unique among previously-described estimators in its class, and we demonstrate superior performance in finite-sample settings. Theoretical results are applied with data from the National Health and Nutrition Examination Survey.

In this paper, we find a sample complexity bound for learning a simplex from noisy samples. Assume a dataset of size $n$ is given which includes i.i.d. samples drawn from a uniform distribution over an unknown simplex in $\mathbb{R}^K$, where samples are assumed to be corrupted by a multi-variate additive Gaussian noise of an arbitrary magnitude. We prove the existence of an algorithm that with high probability outputs a simplex having a $\ell_2$ distance of at most $\varepsilon$ from the true simplex (for any $\varepsilon>0$). Also, we theoretically show that in order to achieve this bound, it is sufficient to have $n\ge\left(K^2/\varepsilon^2\right)e^{\Omega\left(K/\mathrm{SNR}^2\right)}$ samples, where $\mathrm{SNR}$ stands for the signal-to-noise ratio. This result solves an important open problem and shows as long as $\mathrm{SNR}\ge\Omega\left(K^{1/2}\right)$, the sample complexity of the noisy regime has the same order to that of the noiseless case. Our proofs are a combination of the so-called sample compression technique in \citep{ashtiani2018nearly}, mathematical tools from high-dimensional geometry, and Fourier analysis. In particular, we have proposed a general Fourier-based technique for recovery of a more general class of distribution families from additive Gaussian noise, which can be further used in a variety of other related problems.

Assessing causal effects in the presence of unmeasured confounding is a challenging problem. Although auxiliary variables, such as instrumental variables, are commonly used to identify causal effects, they are often unavailable in practice due to stringent and untestable conditions. To address this issue, previous researches have utilized linear structural equation models to show that the causal effect can be identifiable when noise variables of the treatment and outcome are both non-Gaussian. In this paper, we investigate the problem of identifying the causal effect using auxiliary covariates and non-Gaussianity from the treatment. Our key idea is to characterize the impact of unmeasured confounders using an observed covariate, assuming they are all Gaussian. The auxiliary covariate can be an invalid instrument or an invalid proxy variable. We demonstrate that the causal effect can be identified using this measured covariate, even when the only source of non-Gaussianity comes from the treatment. We then extend the identification results to the multi-treatment setting and provide sufficient conditions for identification. Based on our identification results, we propose a simple and efficient procedure for calculating causal effects and show the $\sqrt{n}$-consistency of the proposed estimator. Finally, we evaluate the performance of our estimator through simulation studies and an application.

According to ICH Q8 guidelines, the biopharmaceutical manufacturer submits a design space (DS) definition as part of the regulatory approval application, in which case process parameter (PP) deviations within this space are not considered a change and do not trigger a regulatory post approval procedure. A DS can be described by non-linear PP ranges, i.e., the range of one PP conditioned on specific values of another. However, independent PP ranges (linear combinations) are often preferred in biopharmaceutical manufacturing due to their operation simplicity. While some statistical software supports the calculation of a DS comprised of linear combinations, such methods are generally based on discretizing the parameter space - an approach that scales poorly as the number of PPs increases. Here, we introduce a novel method for finding linear PP combinations using a numeric optimizer to calculate the largest design space within the parameter space that results in critical quality attribute (CQA) boundaries within acceptance criteria, predicted by a regression model. A precomputed approximation of tolerance intervals is used in inequality constraints to facilitate fast evaluations of this boundary using a single matrix multiplication. Correctness of the method was validated against different ground truths with known design spaces. Compared to stateof-the-art, grid-based approaches, the optimizer-based procedure is more accurate, generally yields a larger DS and enables the calculation in higher dimensions. Furthermore, a proposed weighting scheme can be used to favor certain PPs over others and therefore enabling a more dynamic approach to DS definition and exploration. The increased PP ranges of the larger DS provide greater operational flexibility for biopharmaceutical manufacturers.

Stratification in both the design and analysis of randomized clinical trials is common. Despite features in automated randomization systems to re-confirm the stratifying variables, incorrect values of these variables may be entered. These errors are often detected during subsequent data collection and verification. Questions remain about whether to use the mis-reported initial stratification or the corrected values in subsequent analyses. It is shown that the likelihood function resulting from the design of randomized clinical trials supports the use of the corrected values. New definitions are proposed that characterize misclassification errors as `ignorable' and `non-ignorable'. Ignorable errors may depend on the correct strata and any other modeled baseline covariates, but they are otherwise unrelated to potential treatment outcomes. Data management review suggests most misclassification errors are arbitrarily produced by distracted investigators, so they are ignorable or at most weakly dependent on measured and unmeasured baseline covariates. Ignorable misclassification errors may produce a small increase in standard errors, but other properties of the planned analyses are unchanged (e.g., unbiasedness, confidence interval coverage). It is shown that unbiased linear estimation in the absence of misclassification errors remains unbiased when there are non-ignorable misclassification errors, and the corresponding confidence intervals based on the corrected strata values are conservative.

Understanding epistasis (genetic interaction) may shed some light on the genomic basis of common diseases, including disorders of maximum interest due to their high socioeconomic burden, like schizophrenia. Distance correlation is an association measure that characterises general statistical independence between random variables, not only the linear one. Here, we propose distance correlation as a novel tool for the detection of epistasis from case-control data of single-nucleotide polymorphisms (SNPs). On the methodological side, we highlight the derivation of the explicit asymptotic distribution of the test statistic. We show that this is the only way to obtain enough computational speed for the method to be used in practice, in a scenario where the resampling techniques found in the literature are impractical. Our simulations show satisfactory calibration of significance, as well as comparable or better power than existing methodology. We conclude with the application of our technique to a schizophrenia genetics dataset, obtaining biologically sound insights.

While existing work in robust deep learning has focused on small pixel-level $\ell_p$ norm-based perturbations, this may not account for perturbations encountered in several real world settings. In many such cases although test data might not be available, broad specifications about the types of perturbations (such as an unknown degree of rotation) may be known. We consider a setup where robustness is expected over an unseen test domain that is not i.i.d. but deviates from the training domain. While this deviation may not be exactly known, its broad characterization is specified a priori, in terms of attributes. We propose an adversarial training approach which learns to generate new samples so as to maximize exposure of the classifier to the attributes-space, without having access to the data from the test domain. Our adversarial training solves a min-max optimization problem, with the inner maximization generating adversarial perturbations, and the outer minimization finding model parameters by optimizing the loss on adversarial perturbations generated from the inner maximization. We demonstrate the applicability of our approach on three types of naturally occurring perturbations -- object-related shifts, geometric transformations, and common image corruptions. Our approach enables deep neural networks to be robust against a wide range of naturally occurring perturbations. We demonstrate the usefulness of the proposed approach by showing the robustness gains of deep neural networks trained using our adversarial training on MNIST, CIFAR-10, and a new variant of the CLEVR dataset.