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Many problems in statistics and machine learning can be formulated as model selection problems, where the goal is to choose an optimal parsimonious model among a set of candidate models. It is typical to conduct model selection by penalizing the objective function via information criteria (IC), as with the pioneering work by Akaike and Schwarz. Via recent work, we propose a generalized IC framework to consistently estimate general loss-based learning problems. In this work, we propose a consistent estimation method for Generalized Linear Model (GLM) regressions by utilizing the recent IC developments. We advance the generalized IC framework by proposing model selection problems, where the model set consists of a potentially uncountable set of models. In addition to theoretical expositions, our proposal introduces a computational procedure for the implementation of our methods in the finite sample setting, which we demonstrate via an extensive simulation study.

We consider the problem of estimating probability density functions based on sample data, using a finite mixture of densities from some component class. To this end, we introduce the $h$-lifted Kullback--Leibler (KL) divergence as a generalization of the standard KL divergence and a criterion for conducting risk minimization. Under a compact support assumption, we prove an $\mathcal{O}(1/{\sqrt{n}})$ bound on the expected estimation error when using the $h$-lifted KL divergence, which extends the results of Rakhlin et al. (2005, ESAIM: Probability and Statistics, Vol. 9) and Li and Barron (1999, Advances in Neural Information ProcessingSystems, Vol. 12) to permit the risk bounding of density functions that are not strictly positive. We develop a procedure for the computation of the corresponding maximum $h$-lifted likelihood estimators ($h$-MLLEs) using the Majorization-Maximization framework and provide experimental results in support of our theoretical bounds.

Model selection is a ubiquitous problem that arises in the application of many statistical and machine learning methods. In the likelihood and related settings, it is typical to use the method of information criteria (IC) to choose the most parsimonious among competing models by penalizing the likelihood-based objective function. Theorems guaranteeing the consistency of IC can often be difficult to verify and are often specific and bespoke. We present a set of results that guarantee consistency for a class of IC, which we call PanIC (from the Greek root 'pan', meaning 'of everything'), with easily verifiable regularity conditions. The PanIC are applicable in any loss-based learning problem and are not exclusive to likelihood problems. We illustrate the verification of regularity conditions for model selection problems regarding finite mixture models, least absolute deviation and support vector regression, and principal component analysis, and we demonstrate the effectiveness of the PanIC for such problems via numerical simulations. Furthermore, we present new sufficient conditions for the consistency of BIC-like estimators and provide comparisons of the BIC to PanIC.

An on-chip microwave circulator that is compatible with superconducting devices is a key element for scale-up of superconducting circuits. Previous approaches to integrating circulators on chip involve either external driving that requires extra microwave lines or a strong magnetic field that would compromise superconductivity. Here we report the first proof-of-principle realisation of a passive on-chip circulator which is made from a superconducting loop interrupted by three notionally-identical Josephson junctions and is tuned with only DC control fields. Our experimental results shows evidence for nonreciprocal scattering, and excellent agreement with theoretical simulations. We also present a detailed analysis of quasiparticle tunneling in our device using a hidden Markov model. By reducing the junction asymmetry and utilising the known methods of protection from quasiparticles, we anticipate that Josephson-loop circulator will become ubiquitous in superconducting circuits.

Model selection, via penalized likelihood type criteria, is a standard task in many statistical inference and machine learning problems. Progress has led to deriving criteria with asymptotic consistency results and an increasing emphasis on introducing non-asymptotic criteria. We focus on the problem of modeling non-linear relationships in regression data with potential hidden graph-structured interactions between the high-dimensional predictors, within the mixture of experts modeling framework. In order to deal with such a complex situation, we investigate a block-diagonal localized mixture of polynomial experts (BLoMPE) regression model, which is constructed upon an inverse regression and block-diagonal structures of the Gaussian expert covariance matrices. We introduce a penalized maximum likelihood selection criterion to estimate the unknown conditional density of the regression model. This model selection criterion allows us to handle the challenging problem of inferring the number of mixture components, the degree of polynomial mean functions, and the hidden block-diagonal structures of the covariance matrices, which reduces the number of parameters to be estimated and leads to a trade-off between complexity and sparsity in the model. In particular, we provide a strong theoretical guarantee: a finite-sample oracle inequality satisfied by the penalized maximum likelihood estimator with a Jensen-Kullback-Leibler type loss, to support the introduced non-asymptotic model selection criterion. The penalty shape of this criterion depends on the complexity of the considered random subcollection of BLoMPE models, including the relevant graph structures, the degree of polynomial mean functions, and the number of mixture components., Comment: Corrected typos. Extended results from arXiv:2104.02640

Mixture of experts (MoE) are a popular class of statistical and machine learning models that have gained attention over the years due to their flexibility and efficiency. In this work, we consider Gaussian-gated localized MoE (GLoME) and block-diagonal covariance localized MoE (BLoME) regression models to present nonlinear relationships in heterogeneous data with potential hidden graph-structured interactions between high-dimensional predictors. These models pose difficult statistical estimation and model selection questions, both from a computational and theoretical perspective. This paper is devoted to the study of the problem of model selection among a collection of GLoME or BLoME models characterized by the number of mixture components, the complexity of Gaussian mean experts, and the hidden block-diagonal structures of the covariance matrices, in a penalized maximum likelihood estimation framework. In particular, we establish non-asymptotic risk bounds that take the form of weak oracle inequalities, provided that lower bounds for the penalties hold. The good empirical behavior of our models is then demonstrated on synthetic and real datasets., Comment: To appear, Electronic Journal of Statistics, 2022

Mixture of experts (MoE) models are widely applied for conditional probability density estimation problems. We demonstrate the richness of the class of MoE models by proving denseness results in Lebesgue spaces, when inputs and outputs variables are both compactly supported. We further prove an almost uniform convergence result when the input is univariate. Auxiliary lemmas are proved regarding the richness of the soft-max gating function class, and their relationships to the class of Gaussian gating functions., Comment: Corrected typos. Added new Section 6. Summary and conclusions

Autoregressive models are a class of time series models that are important in both applied and theoretical statistics. Typically, inferential devices such as confidence sets and hypothesis tests for time series models require nuanced asymptotic arguments and constructions. We present a simple alternative to such arguments that allow for the construction of finite sample valid inferential devices, using a data splitting approach. We prove the validity of our constructions, as well as the validity of related sequential inference tools. A set of simulation studies are presented to demonstrate the applicability of our methodology.

Wasserman et al. (2020, PNAS, vol. 117, pp. 16880-16890) constructed estimator agnostic and finite-sample valid confidence sets and hypothesis tests, using split-data likelihood ratio-based statistics. We demonstrate that the same approach extends to the use of split-data composite likelihood ratios as well, and thus establish universal methods for conducting multivariate inference when the data generating process is only known up to marginal and conditional relationships between the coordinates. Always-valid sequential inference is also considered.

Big Data has become an ever more commonplace setting that is encountered by data analysts. In the Big Data setting, analysts are faced with very large numbers of observations as well as data that arrive as a stream, both of which are phenomena that many traditional statistical techniques are unable to contend with. Unfortunately, many of these traditional techniques are useful and cannot be discarded. One such technique is the Kolmogorov-Smirnov (KS) test for goodness-of-fit (GoF). A Big Data and stream-appropriate KS-type test is derived via the chunked-and-averaged (CA) estimator paradigm. The new test is termed the CAKS GoF test. The CAKS test statistic is proved to be asymptotically normal, allowing for the large sample testing of GoF. Furthermore, theoretical results demonstrate that the CAKS test is consistent against both fixed alternatives, where the null and the true data generating distribution are a fixed distance apart, and alternatives that approach the null at a slow enough rate. Numerical results demonstrate that the CAKS test is effective in identifying deviation in the distribution with respect to changes in mean, variance, and shape. Furthermore, it is found that the CAKS test is faster than the KS test, for large numbers of observation, and can be applied to sample sizes of 10^{9} and beyond.