Distinguishing two classes of candidate models is a fundamental and practically important problem in statistical inference. Error rate control is crucial to the logic but, in complex nonparametric settings, such guarantees can be difficult to achieve, especially when the stopping rule that determines the data collection process is not available. In this paper we develop a novel e-value construction that leverages the so-called predictive recursion (PR) algorithm designed to recursively fit nonparametric mixture models. The resulting PRe-value affords anytime valid inference uniformly over stopping rules and is shown to be efficient in the sense that it achieves the maximal growth rate under the alternative relative to the mixture model being fit by PR. In the special case of testing the density for log-concavity, the PRe-value test is shown empirically to be significantly more efficient than a recently proposed anytime valid test based on universal inference.
相關內容
Model-based optimization (MBO) is increasingly applied to design problems in science and engineering. A common scenario involves using a fixed training set to train models, with the goal of designing new samples that outperform those present in the training data. A major challenge in this setting is distribution shift, where the distributions of training and design samples are different. While some shift is expected, as the goal is to create better designs, this change can negatively affect model accuracy and subsequently, design quality. Despite the widespread nature of this problem, addressing it demands deep domain knowledge and artful application. To tackle this issue, we propose a straightforward method for design practitioners that detects distribution shifts. This method trains a binary classifier using knowledge of the unlabeled design distribution to separate the training data from the design data. The classifier's logit scores are then used as a proxy measure of distribution shift. We validate our method in a real-world application by running offline MBO and evaluate the effect of distribution shift on design quality. We find that the intensity of the shift in the design distribution varies based on the number of steps taken by the optimization algorithm, and our simple approach can identify these shifts. This enables users to constrain their search to regions where the model's predictions are reliable, thereby increasing the quality of designs.
This paper considers the extension of data-enabled predictive control (DeePC) to nonlinear systems via general basis functions. Firstly, we formulate a basis functions DeePC behavioral predictor and we identify necessary and sufficient conditions for equivalence with a corresponding basis functions multi-step identified predictor. The derived conditions yield a dynamic regularization cost function that enables a well-posed (i.e., consistent) basis functions formulation of nonlinear DeePC. To optimize computational efficiency of basis functions DeePC we further develop two alternative formulations that use a simpler, sparse regularization cost function and ridge regression, respectively. Consistency implications for Koopman DeePC as well as several methods for constructing the basis functions representation are also indicated. The effectiveness of the developed consistent basis functions DeePC formulations is illustrated on a benchmark nonlinear pendulum state-space model, for both noise free and noisy data.
An iterative algorithm for high-dimensional linear models with both sparse and non-sparse structures
Numerous practical medical problems often involve data that possess a combination of both sparse and non-sparse structures. Traditional penalized regularizations techniques, primarily designed for promoting sparsity, are inadequate to capture the optimal solutions in such scenarios. To address these challenges, this paper introduces a novel algorithm named Non-sparse Iteration (NSI). The NSI algorithm allows for the existence of both sparse and non-sparse structures and estimates them simultaneously and accurately. We provide theoretical guarantees that the proposed algorithm converges to the oracle solution and achieves the optimal rate for the upper bound of the $l_2$-norm error. Through simulations and practical applications, NSI consistently exhibits superior statistical performance in terms of estimation accuracy, prediction efficacy, and variable selection compared to several existing methods. The proposed method is also applied to breast cancer data, revealing repeated selection of specific genes for in-depth analysis.
Causal representation learning algorithms discover lower-dimensional representations of data that admit a decipherable interpretation of cause and effect; as achieving such interpretable representations is challenging, many causal learning algorithms utilize elements indicating prior information, such as (linear) structural causal models, interventional data, or weak supervision. Unfortunately, in exploratory causal representation learning, such elements and prior information may not be available or warranted. Alternatively, scientific datasets often have multiple modalities or physics-based constraints, and the use of such scientific, multimodal data has been shown to improve disentanglement in fully unsupervised settings. Consequently, we introduce a causal representation learning algorithm (causalPIMA) that can use multimodal data and known physics to discover important features with causal relationships. Our innovative algorithm utilizes a new differentiable parametrization to learn a directed acyclic graph (DAG) together with a latent space of a variational autoencoder in an end-to-end differentiable framework via a single, tractable evidence lower bound loss function. We place a Gaussian mixture prior on the latent space and identify each of the mixtures with an outcome of the DAG nodes; this novel identification enables feature discovery with causal relationships. Tested against a synthetic and a scientific dataset, our results demonstrate the capability of learning an interpretable causal structure while simultaneously discovering key features in a fully unsupervised setting.
We propose three test criteria each of which is appropriate for testing, respectively, the equivalence hypotheses of symmetry, of homogeneity, and of independence, with multivariate data. All quantities have the common feature of involving weighted--type distances between characteristic functions and are convenient from the computational point of view if the weight function is properly chosen. The asymptotic behavior of the tests under the null hypothesis is investigated, and numerical studies are conducted in order to examine the performance of the criteria in finite samples.
Difficult, in particular NP-complete, optimization problems are traditionally solved approximately using search heuristics. These are usually slowed down by the rugged landscapes encountered, because local minima arrest the search process. Cover-encoding maps were devised to circumvent this problem by transforming the original landscape to one that is free of local minima and enriched in near-optimal solutions. By definition, these involve the mapping of the original (larger) search space into smaller subspaces, by processes that typically amount to a form of coarse-graining. In this paper, we explore the details of this coarse-graining using formal arguments, as well as concrete examples of cover-encoding maps, that are investigated analytically as well as computationally. Our results strongly suggest that the coarse-graining involved in cover-encoding maps bears a strong resemblance to that encountered in renormalisation group schemes. Given the apparently disparate nature of these two formalisms, these strong similarities are rather startling, and suggest deep mathematical underpinnings that await further exploration.
Permutation tests enable testing statistical hypotheses in situations when the distribution of the test statistic is complicated or not available. In some situations, the test statistic under investigation is multivariate, with the multiple testing problem being an important example. The corresponding multivariate permutation tests are then typically based on a suitableone-dimensional transformation of the vector of partial permutation p-values via so called combining functions. This paper proposes a new approach that utilizes the optimal measure transportation concept. The final single p-value is computed from the empirical center-outward distribution function of the permuted multivariate test statistics. This method avoids computation of the partial p-values and it is easy to be implemented. In addition, it allows to compute and interpret contributions of the components of the multivariate test statistic to the non-conformity score and to the rejection of the null hypothesis. Apart from this method, the measure transportation is applied also to the vector of partial p-values as an alternative to the classical combining functions. Both techniques are compared with the standard approaches using various practical examples in a Monte Carlo study. An application on a functional data set is provided as well.
Block majorization-minimization (BMM) is a simple iterative algorithm for nonconvex constrained optimization that sequentially minimizes majorizing surrogates of the objective function in each block coordinate while the other coordinates are held fixed. BMM entails a large class of optimization algorithms such as block coordinate descent and its proximal-point variant, expectation-minimization, and block projected gradient descent. We establish that for general constrained nonconvex optimization, BMM with strongly convex surrogates can produce an $\epsilon$-stationary point within $O(\epsilon^{-2}(\log \epsilon^{-1})^{2})$ iterations and asymptotically converges to the set of stationary points. Furthermore, we propose a trust-region variant of BMM that can handle surrogates that are only convex and still obtain the same iteration complexity and asymptotic stationarity. These results hold robustly even when the convex sub-problems are inexactly solved as long as the optimality gaps are summable. As an application, we show that a regularized version of the celebrated multiplicative update algorithm for nonnegative matrix factorization by Lee and Seung has iteration complexity of $O(\epsilon^{-2}(\log \epsilon^{-1})^{2})$. The same result holds for a wide class of regularized nonnegative tensor decomposition algorithms as well as the classical block projected gradient descent algorithm. These theoretical results are validated through various numerical experiments.
Two-level stochastic optimization formulations have become instrumental in a number of machine learning contexts such as continual learning, neural architecture search, adversarial learning, and hyperparameter tuning. Practical stochastic bilevel optimization problems become challenging in optimization or learning scenarios where the number of variables is high or there are constraints. In this paper, we introduce a bilevel stochastic gradient method for bilevel problems with nonlinear and possibly nonconvex lower-level constraints. We also present a comprehensive convergence theory that addresses both the lower-level unconstrained and constrained cases and covers all inexact calculations of the adjoint gradient (also called hypergradient), such as the inexact solution of the lower-level problem, inexact computation of the adjoint formula (due to the inexact solution of the adjoint equation or use of a truncated Neumann series), and noisy estimates of the gradients, Hessians, and Jacobians involved. To promote the use of bilevel optimization in large-scale learning, we have developed new low-rank practical bilevel stochastic gradient methods (BSG-N-FD and~BSG-1) that do not require second-order derivatives and, in the lower-level unconstrained case, dismiss any matrix-vector products.
The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191