Domain adaptation (DA) is a statistical learning problem that arises when the distribution of the source data used to train a model differs from that of the target data used to evaluate the model. While many DA algorithms have demonstrated considerable empirical success, blindly applying these algorithms can often lead to worse performance on new datasets. To address this, it is crucial to clarify the assumptions under which a DA algorithm has good target performance. In this work, we focus on the assumption of the presence of conditionally invariant components (CICs), which are relevant for prediction and remain conditionally invariant across the source and target data. We demonstrate that CICs, which can be estimated through conditional invariant penalty (CIP), play three prominent roles in providing target risk guarantees in DA. First, we propose a new algorithm based on CICs, importance-weighted conditional invariant penalty (IW-CIP), which has target risk guarantees beyond simple settings such as covariate shift and label shift. Second, we show that CICs help identify large discrepancies between source and target risks of other DA algorithms. Finally, we demonstrate that incorporating CICs into the domain invariant projection (DIP) algorithm can address its failure scenario caused by label-flipping features. We support our new algorithms and theoretical findings via numerical experiments on synthetic data, MNIST, CelebA, Camelyon17, and DomainNet datasets.

Deep neural networks (DNN) have been widely used in scalar-on-image regression to predict an outcome variable from imaging predictors. However, training DNN typically requires large sample sizes for accurate prediction, and the resulting models often lack interpretability. In this work, we propose a novel Bayesian nonlinear scalar-on-image regression framework with a spatially varying single-layer neural network (SV-NN) prior. The SV-NN is constructed using a single hidden layer neural network with its weights generated by the soft-thresholded Gaussian process. Our framework enables the selection of interpretable image regions while achieving high prediction accuracy with limited training samples. The SV-NN offers large prior support for the imaging effect function, facilitating efficient posterior inference on image region selection and automatic network structures determination. We establish the posterior consistency for model parameters and selection consistency for image regions when the number of voxels/pixels grows much faster than the sample size. To ensure computational efficiency, we develop a stochastic gradient Langevin dynamics (SGLD) algorithm for posterior inference. We evaluate our method through extensive comparisons with state-of-the-art deep learning approaches, analyzing multiple real datasets, including task fMRI data from the Adolescent Brain Cognitive Development (ABCD) study.

We study the mixing time of the Metropolis-adjusted Langevin algorithm (MALA) for sampling from a log-smooth and strongly log-concave distribution. We establish its optimal minimax mixing time under a warm start. Our main contribution is two-fold. First, for a $d$-dimensional log-concave density with condition number $\kappa$, we show that MALA with a warm start mixes in $\tilde O(\kappa \sqrt{d})$ iterations up to logarithmic factors. This improves upon the previous work on the dependency of either the condition number $\kappa$ or the dimension $d$. Our proof relies on comparing the leapfrog integrator with the continuous Hamiltonian dynamics, where we establish a new concentration bound for the acceptance rate. Second, we prove a spectral gap based mixing time lower bound for reversible MCMC algorithms on general state spaces. We apply this lower bound result to construct a hard distribution for which MALA requires at least $\tilde\Omega(\kappa \sqrt{d})$ steps to mix. The lower bound for MALA matches our upper bound in terms of condition number and dimension. Finally, numerical experiments are included to validate our theoretical results.