Inference on the conditional mean function (CMF) is central to tasks from adaptive experimentation to optimal treatment assignment and algorithmic fairness auditing. In this work, we provide a novel asymptotic anytime-valid test for a CMF global null (e.g., that all conditional means are zero) and contrasts between CMFs, enabling experimenters to make high confidence decisions at any time during the experiment beyond a minimum sample size. We provide mild conditions under which our tests achieve (i) asymptotic type-I error guarantees, (i) power one, and, unlike past tests, (iii) optimal sample complexity relative to a Gaussian location testing. By inverting our tests, we show how to construct function-valued asymptotic confidence sequences for the CMF and contrasts thereof. Experiments on both synthetic and real-world data show our method is well-powered across various distributions while preserving the nominal error rate under continuous monitoring.

Many machine learning tasks that involve predicting an output response can be solved by training a weighted regression model.

Stochastic processes are random variables with values in some space of paths. However, reducing a stochastic process to a path-valued random variable ignores its filtration, i.e. the flow of information carried by the process through time. By conditioning the process on its filtration, we introduce a family of higher order kernel mean embeddings (KMEs) that generalizes the notion of KME and captures additional information related to the filtration. We derive empirical estimators for the associated higher order maximum mean discrepancies (MMDs) and prove consistency. We then construct a filtration-sensitive kernel two-sample test able to pick up information that gets missed by the standard MMD test. In addition, leveraging our higher order MMDs we construct a family of universal kernels on stochastic processes that allows to solve real-world calibration and optimal stopping problems in quantitative finance (such as the pricing of American options) via classical kernel-based regression methods. Finally, adapting existing tests for conditional independence to the case of stochastic processes, we design a causaldiscovery algorithm to recover the causal graph of structural dependencies among interacting bodies solely from observations of their multidimensional trajectories.

In this work, we introduce the Assisted Learning framework for organizations to assist each other in supervised learning tasks without revealing anyorganization'salgorithm,data,oreventask.

We thank the reviewers for their constructive suggestions. A summary of the added discussion is provided point-by-point below. Reviewer 1: How much more work is needed versus linear regression. The technical details are thus different (and more involved) here. We have included these derivations for completeness.

The implementations of the experiments on ABC and FTDC datasets are similar. For the stability analysis, we are interested in the norm of term 1. In Section E.1, we briefly discuss the motivation behind studying age prediction and PCA-based statistical analysis in this context. In Section E.2, we provide additional details on cortical thickness data acquisition. In Section E.3, we report the results for stability analysis of VNNs and PCA-regression models for FTDC100 ( In Section E.4, we study the stability of VNNs on two simulated In Section E.5, we include additional figures A promising application of brain age prediction is early detection of neurodegenerative diseases (such as Alzheimer's, Huntingson's disease) which may manifest themselves as error in age prediction in pathological contexts by machine learning models trained E.4 Stability of VNNs on Synthetic Data We consider two settings for synthetic data.

By combining robust regression and prior information, we develop an effective robust regression method that can resist adaptive adversarial attacks. Due to the widespread existence of noise and data corruption, it is necessary to recover the true regression parameters when a certain proportion of the response variables have been corrupted. Methods to overcome this problem often involve robust least-squaresregression.

By combining robust regression and prior information, we develop an effective robust regression method that can resist adaptive adversarial attacks. Due to the widespread existence of noise and data corruption, it is necessary to recover the true regression parameters when a certain proportion of the response variables have been corrupted. Methods to overcome this problem often involve robust least-squaresregression.