Estimating weights in the synthetic control method involves an optimization procedure that simultaneously selects and aligns control units in order to closely match the treated unit. However, this simultaneous selection and alignment of control units may lead to a loss of efficiency in the synthetic control method. Another concern arising from the aforementioned procedure is its susceptibility to under-fitting due to imperfect pretreatment fit. It is not uncommon for the linear combination, using nonnegative weights, of pre-treatment period outcomes for the control units to inadequately approximate the pre-treatment outcomes for the treated unit. To address both of these issues, this paper proposes a simple and effective method called Synthetic Matching Control (SMC). The SMC method begins by performing the univariate linear regression to establish a proper match between the pre-treatment periods of the control units and the treated unit. Subsequently, a SMC estimator is obtained by synthesizing (taking a weighted average) the matched controls. To determine the weights in the synthesis procedure, we propose an approach that utilizes a criterion of unbiased risk estimator. Theoretically, we show that the synthesis way is asymptotically optimal in the sense of achieving the lowest possible squared error. Extensive numerical experiments highlight the advantages of the SMC method.
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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.
We study the fundamental limits of matching pursuit, or the pure greedy algorithm, for approximating a target function by a sparse linear combination of elements from a dictionary. When the target function is contained in the variation space corresponding to the dictionary, many impressive works over the past few decades have obtained upper and lower bounds on the error of matching pursuit, but they do not match. The main contribution of this paper is to close this gap and obtain a sharp characterization of the decay rate of matching pursuit. Specifically, we construct a worst case dictionary which shows that the existing best upper bound cannot be significantly improved. It turns out that, unlike other greedy algorithm variants, the converge rate is suboptimal and is determined by the solution to a certain non-linear equation. This enables us to conclude that any amount of shrinkage improves matching pursuit in the worst case.
Synthetic control methods (SCMs) have become a crucial tool for causal inference in comparative case studies. The fundamental idea of SCMs is to estimate counterfactual outcomes for a treated unit by using a weighted sum of observed outcomes from untreated units. The accuracy of the synthetic control (SC) is critical for estimating the causal effect, and hence, the estimation of SC weights has been the focus of much research. In this paper, we first point out that existing SCMs suffer from an implicit endogeneity problem, which is the correlation between the outcomes of untreated units and the error term in the model of a counterfactual outcome. We show that this problem yields a bias in the causal effect estimator. We then propose a novel SCM based on density matching, assuming that the density of outcomes of the treated unit can be approximated by a weighted average of the densities of untreated units (i.e., a mixture model). Based on this assumption, we estimate SC weights by matching moments of treated outcomes and the weighted sum of moments of untreated outcomes. Our proposed method has three advantages over existing methods. First, our estimator is asymptotically unbiased under the assumption of the mixture model. Second, due to the asymptotic unbiasedness, we can reduce the mean squared error for counterfactual prediction. Third, our method generates full densities of the treatment effect, not only expected values, which broadens the applicability of SCMs. We provide experimental results to demonstrate the effectiveness of our proposed method.
In the context of the high-dimensional Gaussian linear regression for ordered variables, we study the variable selection procedure via the minimization of the penalized least-squares criterion. We focus on model selection where the penalty function depends on an unknown multiplicative constant commonly calibrated for prediction. We propose a new proper calibration of this hyperparameter to simultaneously control predictive risk and false discovery rate. We obtain non-asymptotic theoretical bounds on the False Discovery Rate with respect to the hyperparameter and we provide an algorithm to calibrate it. It is based on completely observable quantities in view of applications. Our algorithm is validated by an extensive simulation study and is compared with some existing variable selection procedures. Finally, we propose a study to generalize our approach in complete variable selection.
The Fourier transform, serving as an explicit decomposition method for visual signals, has been employed to explain the out-of-distribution generalization behaviors of Convolutional Neural Networks (CNNs). Previous research and empirical studies have indicated that the amplitude spectrum plays a decisive role in CNN recognition, but it is susceptible to disturbance caused by distribution shifts. On the other hand, the phase spectrum preserves highly-structured spatial information, which is crucial for visual representation learning. In this paper, we aim to clarify the relationships between Domain Generalization (DG) and the frequency components by introducing a Fourier-based structural causal model. Specifically, we interpret the phase spectrum as semi-causal factors and the amplitude spectrum as non-causal factors. Building upon these observations, we propose Phase Match (PhaMa) to address DG problems. Our method introduces perturbations on the amplitude spectrum and establishes spatial relationships to match the phase components. Through experiments on multiple benchmarks, we demonstrate that our proposed method achieves state-of-the-art performance in domain generalization and out-of-distribution robustness tasks.
Least-squares programming is a popular tool in robotics due to its simplicity and availability of open-source solvers. However, certain problems like sparse programming in the $\ell_0$- or $\ell_1$-norm for time-optimal control are not equivalently solvable. In this work, we propose a non-linear hierarchical least-squares programming (NL-HLSP) for time-optimal control of non-linear discrete dynamic systems. We use a continuous approximation of the heaviside step function with an additional term that avoids vanishing gradients. We use a simple discretization method by keeping states and controls piece-wise constant between discretization steps. This way, we obtain a comparatively easily implementable NL-HLSP in contrast to direct transcription approaches of optimal control. We show that the NL-HLSP indeed recovers the discrete time-optimal control in the limit for resting goal points. We confirm the results in simulation for linear and non-linear control scenarios.
Convergence rate analyses of random walk Metropolis-Hastings Markov chains on general state spaces have largely focused on establishing sufficient conditions for geometric ergodicity or on analysis of mixing times. Geometric ergodicity is a key sufficient condition for the Markov chain Central Limit Theorem and allows rigorous approaches to assessing Monte Carlo error. The sufficient conditions for geometric ergodicity of the random walk Metropolis-Hastings Markov chain are refined and extended, which allows the analysis of previously inaccessible settings such as Bayesian Poisson regression. The key technical innovation is the development of explicit drift and minorization conditions for random walk Metropolis-Hastings, which allows explicit upper and lower bounds on the geometric rate of convergence. Further, lower bounds on the geometric rate of convergence are also developed using spectral theory. The existing sufficient conditions for geometric ergodicity, to date, have not provided explicit constraints on the rate of geometric rate of convergence because the method used only implies the existence of drift and minorization conditions. The theoretical results are applied to random walk Metropolis-Hastings algorithms for a class of exponential families and generalized linear models that address Bayesian Regression problems.
Synthetic control methods (SCMs) have become a crucial tool for causal inference in comparative case studies. The fundamental idea of SCMs is to estimate counterfactual outcomes for a treated unit by using a weighted sum of observed outcomes from untreated units. The accuracy of the synthetic control (SC) is critical for estimating the causal effect, and hence, the estimation of SC weights has been the focus of much research. In this paper, we first point out that existing SCMs suffer from an implicit endogeneity problem, which is the correlation between the outcomes of untreated units and the error term in the model of a counterfactual outcome. We show that this problem yields a bias in the causal effect estimator. We then propose a novel SCM based on density matching, assuming that the density of outcomes of the treated unit can be approximated by a weighted average of the densities of untreated units (i.e., a mixture model). Based on this assumption, we estimate SC weights by matching moments of treated outcomes and the weighted sum of moments of untreated outcomes. Our proposed method has three advantages over existing methods. First, our estimator is asymptotically unbiased under the assumption of the mixture model. Second, due to the asymptotic unbiasedness, we can reduce the mean squared error for counterfactual prediction. Third, our method generates full densities of the treatment effect, not only expected values, which broadens the applicability of SCMs. We provide experimental results to demonstrate the effectiveness of our proposed method.
In the domain generalization literature, a common objective is to learn representations independent of the domain after conditioning on the class label. We show that this objective is not sufficient: there exist counter-examples where a model fails to generalize to unseen domains even after satisfying class-conditional domain invariance. We formalize this observation through a structural causal model and show the importance of modeling within-class variations for generalization. Specifically, classes contain objects that characterize specific causal features, and domains can be interpreted as interventions on these objects that change non-causal features. We highlight an alternative condition: inputs across domains should have the same representation if they are derived from the same object. Based on this objective, we propose matching-based algorithms when base objects are observed (e.g., through data augmentation) and approximate the objective when objects are not observed (MatchDG). Our simple matching-based algorithms are competitive to prior work on out-of-domain accuracy for rotated MNIST, Fashion-MNIST, PACS, and Chest-Xray datasets. Our method MatchDG also recovers ground-truth object matches: on MNIST and Fashion-MNIST, top-10 matches from MatchDG have over 50% overlap with ground-truth matches.
Learning from a few examples remains a key challenge in machine learning. Despite recent advances in important domains such as vision and language, the standard supervised deep learning paradigm does not offer a satisfactory solution for learning new concepts rapidly from little data. In this work, we employ ideas from metric learning based on deep neural features and from recent advances that augment neural networks with external memories. Our framework learns a network that maps a small labelled support set and an unlabelled example to its label, obviating the need for fine-tuning to adapt to new class types. We then define one-shot learning problems on vision (using Omniglot, ImageNet) and language tasks. Our algorithm improves one-shot accuracy on ImageNet from 87.6% to 93.2% and from 88.0% to 93.8% on Omniglot compared to competing approaches. We also demonstrate the usefulness of the same model on language modeling by introducing a one-shot task on the Penn Treebank.
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