Dynamic Linear Models (DLMs) are commonly employed for time series analysis due to their versatile structure, simple recursive updating, ability to handle missing data, and probabilistic forecasting. However, the options for count time series are limited: Gaussian DLMs require continuous data, while Poisson-based alternatives often lack sufficient modeling flexibility. We introduce a novel semiparametric methodology for count time series by warping a Gaussian DLM. The warping function has two components: a (nonparametric) transformation operator that provides distributional flexibility and a rounding operator that ensures the correct support for the discrete data-generating process. We develop conjugate inference for the warped DLM, which enables analytic and recursive updates for the state space filtering and smoothing distributions. We leverage these results to produce customized and efficient algorithms for inference and forecasting, including Monte Carlo simulation for offline analysis and an optimal particle filter for online inference. This framework unifies and extends a variety of discrete time series models and is valid for natural counts, rounded values, and multivariate observations. Simulation studies illustrate the excellent forecasting capabilities of the warped DLM. The proposed approach is applied to a multivariate time series of daily overdose counts and demonstrates both modeling and computational successes.
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Bilevel optimization has arisen as a powerful tool in modern machine learning. However, due to the nested structure of bilevel optimization, even gradient-based methods require second-order derivative approximations via Jacobian- or/and Hessian-vector computations, which can be costly and unscalable in practice. Recently, Hessian-free bilevel schemes have been proposed to resolve this issue, where the general idea is to use zeroth- or first-order methods to approximate the full hypergradient of the bilevel problem. However, we empirically observe that such approximation can lead to large variance and unstable training, but estimating only the response Jacobian matrix as a partial component of the hypergradient turns out to be extremely effective. To this end, we propose a new Hessian-free method, which adopts the zeroth-order-like method to approximate the response Jacobian matrix via taking difference between two optimization paths. Theoretically, we provide the convergence rate analysis for the proposed algorithms, where our key challenge is to characterize the approximation and smoothness properties of the trajectory-dependent estimator, which can be of independent interest. This is the first known convergence rate result for this type of Hessian-free bilevel algorithms. Experimentally, we demonstrate that the proposed algorithms outperform baseline bilevel optimizers on various bilevel problems. Particularly, in our experiment on few-shot meta-learning with ResNet-12 network over the miniImageNet dataset, we show that our algorithm outperforms baseline meta-learning algorithms, while other baseline bilevel optimizers do not solve such meta-learning problems within a comparable time frame.
Multiple hypothesis testing has been widely applied to problems dealing with high-dimensional data, e.g., selecting significant variables and controlling the selection error rate. The most prevailing measure of error rate used in the multiple hypothesis testing is the false discovery rate (FDR). In recent years, local false discovery rate (fdr) has drawn much attention, due to its advantage of accessing the confidence of individual hypothesis. However, most methods estimate fdr through p-values or statistics with known null distributions, which are sometimes not available or reliable. Adopting the innovative methodology of competition-based procedures, e.g., knockoff filter, this paper proposes a new approach, named TDfdr, to local false discovery rate estimation, which is free of the p-values or known null distributions. Simulation results demonstrate that TDfdr can accurately estimate the fdr with two competition-based procedures. In real data analysis, the power of TDfdr on variable selection is verified on two biological datasets.
While deep learning has outperformed other methods for various tasks, theoretical frameworks that explain its reason have not been fully established. To address this issue, we investigate the excess risk of two-layer ReLU neural networks in a teacher-student regression model, in which a student network learns an unknown teacher network through its outputs. Especially, we consider the student network that has the same width as the teacher network and is trained in two phases: first by noisy gradient descent and then by the vanilla gradient descent. Our result shows that the student network provably reaches a near-global optimal solution and outperforms any kernel methods estimator (more generally, linear estimators), including neural tangent kernel approach, random feature model, and other kernel methods, in a sense of the minimax optimal rate. The key concept inducing this superiority is the non-convexity of the neural network models. Even though the loss landscape is highly non-convex, the student network adaptively learns the teacher neurons.
This paper investigates the problem of regret minimization in linear time-varying (LTV) dynamical systems. Due to the simultaneous presence of uncertainty and non-stationarity, designing online control algorithms for unknown LTV systems remains a challenging task. At a cost of NP-hard offline planning, prior works have introduced online convex optimization algorithms, although they suffer from nonparametric rate of regret. In this paper, we propose the first computationally tractable online algorithm with regret guarantees that avoids offline planning over the state linear feedback policies. Our algorithm is based on the optimism in the face of uncertainty (OFU) principle in which we optimistically select the best model in a high confidence region. Our algorithm is then more explorative when compared to previous approaches. To overcome non-stationarity, we propose either a restarting strategy (R-OFU) or a sliding window (SW-OFU) strategy. With proper configuration, our algorithm is attains sublinear regret $O(T^{2/3})$. These algorithms utilize data from the current phase for tracking variations on the system dynamics. We corroborate our theoretical findings with numerical experiments, which highlight the effectiveness of our methods. To the best of our knowledge, our study establishes the first model-based online algorithm with regret guarantees under LTV dynamical systems.
Analyzing time series in the frequency domain enables the development of powerful tools for investigating the second-order characteristics of multivariate stochastic processes. Parameters like the spectral density matrix and its inverse, the coherence or the partial coherence, encode comprehensively the complex linear relations between the component processes of the multivariate system. In this paper, we develop inference procedures for such parameters in a high-dimensional, time series setup. In particular, we first focus on the derivation of consistent estimators of the coherence and, more importantly, of the partial coherence which possess manageable limiting distributions that are suitable for testing purposes. Statistical tests of the hypothesis that the maximum over frequencies of the coherence, respectively, of the partial coherence, do not exceed a prespecified threshold value are developed. Our approach allows for testing hypotheses for individual coherences and/or partial coherences as well as for multiple testing of large sets of such parameters. In the latter case, a consistent procedure to control the false discovery rate is developed. The finite sample performance of the inference procedures proposed is investigated by means of simulations and applications to the construction of graphical interaction models for brain connectivity based on EEG data are presented.
Estimation and variable selection in joint mean and dispersion models applied to mixture experiments
In industrial experiments, controlling variability is of paramount importance to ensure product quality. Classical regression models for mixture experiments are widely used in industry, however, when the assumption of constant variance is not satisfied, the building of procedures that allow minimizing the variability becomes necessary and other methods of statistical modeling should be considered. In this article, we use the class of generalized linear models (GLMs) to build statistical models in mixture experiments. The GLMs class is general and very flexible, generalizing some of the most important probability distributions, and allows modeling the variability through the methodology of the joint modeling of mean and dispersion (JMMD). This paper shows how the JMMD can be used to obtain models for mean and variance in mixture experiments. We give a comprehensive understanding of the procedures for estimating parameters and selecting variables in the JMMD. The variable selection procedure was adapted for the case of mixture experiments, where the verification of constant dispersion is ensured by the existence of only the constant term in the dispersion model; the absence of the constant term or the existence of any other term in the dispersion model implies non-constant dispersion. A simulation study, considering the most common case of Normal distribution, was used to verify the effectiveness of the proposed variable selection procedure. A practical example from the Food Industry was used to illustrate the proposed methodology.
The synthetic control method has become a widely popular tool to estimate causal effects with observational data. Despite this, inference for synthetic control methods remains challenging. Often, inferential results rely on linear factor model data generating processes. In this paper, we characterize the conditions on the factor model primitives (the factor loadings) for which the statistical risk minimizers are synthetic controls (in the simplex). Then, we propose a Bayesian alternative to the synthetic control method that preserves the main features of the standard method and provides a new way of doing valid inference. We explore a Bernstein-von Mises style result to link our Bayesian inference to the frequentist inference. For linear factor model frameworks we show that a maximum likelihood estimator (MLE) of the synthetic control weights can consistently estimate the predictive function of the potential outcomes for the treated unit and that our Bayes estimator is asymptotically close to the MLE in the total variation sense. Through simulations, we show that there is convergence between the Bayes and frequentist approach even in sparse settings. Finally, we apply the method to re-visit the study of the economic costs of the German re-unification. The Bayesian synthetic control method is available in the bsynth R-package.
Gradient based meta-learning methods are prone to overfit on the meta-training set, and this behaviour is more prominent with large and complex networks. Moreover, large networks restrict the application of meta-learning models on low-power edge devices. While choosing smaller networks avoid these issues to a certain extent, it affects the overall generalization leading to reduced performance. Clearly, there is an approximately optimal choice of network architecture that is best suited for every meta-learning problem, however, identifying it beforehand is not straightforward. In this paper, we present MetaDOCK, a task-specific dynamic kernel selection strategy for designing compressed CNN models that generalize well on unseen tasks in meta-learning. Our method is based on the hypothesis that for a given set of similar tasks, not all kernels of the network are needed by each individual task. Rather, each task uses only a fraction of the kernels, and the selection of the kernels per task can be learnt dynamically as a part of the inner update steps. MetaDOCK compresses the meta-model as well as the task-specific inner models, thus providing significant reduction in model size for each task, and through constraining the number of active kernels for every task, it implicitly mitigates the issue of meta-overfitting. We show that for the same inference budget, pruned versions of large CNN models obtained using our approach consistently outperform the conventional choices of CNN models. MetaDOCK couples well with popular meta-learning approaches such as iMAML. The efficacy of our method is validated on CIFAR-fs and mini-ImageNet datasets, and we have observed that our approach can provide improvements in model accuracy of up to 2% on standard meta-learning benchmark, while reducing the model size by more than 75%.
We present a novel counterfactual framework for both Zero-Shot Learning (ZSL) and Open-Set Recognition (OSR), whose common challenge is generalizing to the unseen-classes by only training on the seen-classes. Our idea stems from the observation that the generated samples for unseen-classes are often out of the true distribution, which causes severe recognition rate imbalance between the seen-class (high) and unseen-class (low). We show that the key reason is that the generation is not Counterfactual Faithful, and thus we propose a faithful one, whose generation is from the sample-specific counterfactual question: What would the sample look like, if we set its class attribute to a certain class, while keeping its sample attribute unchanged? Thanks to the faithfulness, we can apply the Consistency Rule to perform unseen/seen binary classification, by asking: Would its counterfactual still look like itself? If ``yes'', the sample is from a certain class, and ``no'' otherwise. Through extensive experiments on ZSL and OSR, we demonstrate that our framework effectively mitigates the seen/unseen imbalance and hence significantly improves the overall performance. Note that this framework is orthogonal to existing methods, thus, it can serve as a new baseline to evaluate how ZSL/OSR models generalize. Codes are available at //github.com/yue-zhongqi/gcm-cf.
High spectral dimensionality and the shortage of annotations make hyperspectral image (HSI) classification a challenging problem. Recent studies suggest that convolutional neural networks can learn discriminative spatial features, which play a paramount role in HSI interpretation. However, most of these methods ignore the distinctive spectral-spatial characteristic of hyperspectral data. In addition, a large amount of unlabeled data remains an unexploited gold mine for efficient data use. Therefore, we proposed an integration of generative adversarial networks (GANs) and probabilistic graphical models for HSI classification. Specifically, we used a spectral-spatial generator and a discriminator to identify land cover categories of hyperspectral cubes. Moreover, to take advantage of a large amount of unlabeled data, we adopted a conditional random field to refine the preliminary classification results generated by GANs. Experimental results obtained using two commonly studied datasets demonstrate that the proposed framework achieved encouraging classification accuracy using a small number of data for training.
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