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In this work we connect two notions: That of the nonparametric mode of a probability measure, defined by asymptotic small ball probabilities, and that of the Onsager--Machlup functional, a generalized density also defined via asymptotic small ball probabilities. We show that in a separable Hilbert space setting and under mild conditions on the likelihood, the modes of a Bayesian posterior distribution based upon a Gaussian prior agree with the minimizers of its Onsager--Machlup functional. We apply this result to inverse problems and derive conditions on the forward mapping under which this variational characterization of posterior modes holds. Our results show rigorously that in the limit case of infinite-dimensional data corrupted by additive Gaussian or Laplacian noise, nonparametric MAP estimation is equivalent to Tikhonov--Phillips regularization. In comparison with the work of Dashti, Law, Stuart, and Voss (2013), the assumptions on the likelihood are relaxed so that they cover in particular the important case of Gaussian process noise. We illustrate our results by applying them to a severely ill-posed linear problem with Laplacian noise, where we express the MAP estimator analytically and study its rate of convergence.

Searching for a path between two nodes in a graph is one of the most well-studied and fundamental problems in computer science. In numerous domains such as robotics, AI, or biology, practitioners develop search heuristics to accelerate their pathfinding algorithms. However, it is a laborious and complex process to hand-design heuristics based on the problem and the structure of a given use case. Here we present PHIL (Path Heuristic with Imitation Learning), a novel neural architecture and a training algorithm for discovering graph search and navigation heuristics from data by leveraging recent advances in imitation learning and graph representation learning. At training time, we aggregate datasets of search trajectories and ground-truth shortest path distances, which we use to train a specialized graph neural network-based heuristic function using backpropagation through steps of the pathfinding process. Our heuristic function learns graph embeddings useful for inferring node distances, runs in constant time independent of graph sizes, and can be easily incorporated in an algorithm such as A* at test time. Experiments show that PHIL reduces the number of explored nodes compared to state-of-the-art methods on benchmark datasets by 58.5\% on average, can be directly applied in diverse graphs ranging from biological networks to road networks, and allows for fast planning in time-critical robotics domains.

In social, medical, and behavioral research we often encounter datasets with a multilevel structure and multiple correlated dependent variables. These data are frequently collected from a study population that distinguishes several subpopulations with different (i.e. heterogeneous) effects of an intervention. Despite the frequent occurrence of such data, methods to analyze them are less common and researchers often resort to either ignoring the multilevel and/or heterogeneous structure, analyzing only a single dependent variable, or a combination of these. These analysis strategies are suboptimal: Ignoring multilevel structures inflates Type I error rates, while neglecting the multivariate or heterogeneous structure masks detailed insights. To analyze such data comprehensively, the current paper presents a novel Bayesian multilevel multivariate logistic regression model. The clustered structure of multilevel data is taken into account, such that posterior inferences can be made with accurate error rates. Further, the model shares information between different subpopulations in the estimation of average and conditional average multivariate treatment effects. To facilitate interpretation, multivariate logistic regression parameters are transformed to posterior success probabilities and differences between them. A numerical evaluation compared our framework to less comprehensive alternatives and highlighted the need to model the multilevel structure: Treatment comparisons based on the multilevel model had targeted Type I error rates, while single-level alternatives resulted in inflated Type I errors. A re-analysis of the Third International Stroke Trial data illustrated how incorporating a multilevel structure, assessing treatment heterogeneity, and combining dependent variables contributed to an in-depth understanding of treatment effects.

This work studies the problem of transfer learning under the functional linear regression model framework, which aims to improve the estimation and prediction of the target model by leveraging the information from related source models. We measure the relatedness between target and source models using Reproducing Kernel Hilbert Spaces (RKHS) norm, allowing the type of information being transferred to be interpreted by the structural properties of the spaces. Two transfer learning algorithms are proposed: one transfers information from source tasks when we know which sources to use, while the other one aggregates multiple transfer learning results from the first algorithm to achieve robust transfer learning without prior information about the sources. Furthermore, we establish the optimal convergence rates for the prediction risk in the target model, making the statistical gain via transfer learning mathematically provable. The theoretical analysis of the prediction risk also provides insights regarding what factors are affecting the transfer learning effect, i.e. what makes source tasks useful to the target task. We demonstrate the effectiveness of the proposed transfer learning algorithms on extensive synthetic data as well as real financial data application.

Gaussian mixtures are commonly used for modeling heavy-tailed error distributions in robust linear regression. Combining the likelihood of a multivariate robust linear regression model with a standard improper prior distribution yields an analytically intractable posterior distribution that can be sampled using a data augmentation algorithm. When the response matrix has missing entries, there are unique challenges to the application and analysis of the convergence properties of the algorithm. Conditions for geometric ergodicity are provided when the incomplete data have a "monotone" structure. In the absence of a monotone structure, an intermediate imputation step is necessary for implementing the algorithm. In this case, we provide sufficient conditions for the algorithm to be Harris ergodic. Finally, we show that, when there is a monotone structure and intermediate imputation is unnecessary, intermediate imputation slows the convergence of the underlying Monte Carlo Markov chain, while post hoc imputation does not. An R package for the data augmentation algorithm is provided.

We address the problem of integrating data from multiple observational and interventional studies to eventually compute counterfactuals in structural causal models. We derive a likelihood characterisation for the overall data that leads us to extend a previous EM-based algorithm from the case of a single study to that of multiple ones. The new algorithm learns to approximate the (unidentifiability) region of model parameters from such mixed data sources. On this basis, it delivers interval approximations to counterfactual results, which collapse to points in the identifiable case. The algorithm is very general, it works on semi-Markovian models with discrete variables and can compute any counterfactual. Moreover, it automatically determines if a problem is feasible (the parameter region being nonempty), which is a necessary step not to yield incorrect results. Systematic numerical experiments show the effectiveness and accuracy of the algorithm, while hinting at the benefits of integrating heterogeneous data to get informative bounds in case of unidentifiability.

This paper considers the problem of kernel regression and classification with possibly unobservable response variables in the data, where the mechanism that causes the absence of information is unknown and can depend on both predictors and the response variables. Our proposed approach involves two steps: In the first step, we construct a family of models (possibly infinite dimensional) indexed by the unknown parameter of the missing probability mechanism. In the second step, a search is carried out to find the empirically optimal member of an appropriate cover (or subclass) of the underlying family in the sense of minimizing the mean squared prediction error. The main focus of the paper is to look into the theoretical properties of these estimators. The issue of identifiability is also addressed. Our methods use a data-splitting approach which is quite easy to implement. We also derive exponential bounds on the performance of the resulting estimators in terms of their deviations from the true regression curve in general Lp norms, where we also allow the size of the cover or subclass to diverge as the sample size n increases. These bounds immediately yield various strong convergence results for the proposed estimators. As an application of our findings, we consider the problem of statistical classification based on the proposed regression estimators and also look into their rates of convergence under different settings. Although this work is mainly stated for kernel-type estimators, they can also be extended to other popular local-averaging methods such as nearest-neighbor estimators, and histogram estimators.

In many modern applications of deep learning the neural network has many more parameters than the data points used for its training. Motivated by those practices, a large body of recent theoretical research has been devoted to studying overparameterized models. One of the central phenomena in this regime is the ability of the model to interpolate noisy data, but still have test error lower than the amount of noise in that data. arXiv:1906.11300 characterized for which covariance structure of the data such a phenomenon can happen in linear regression if one considers the interpolating solution with minimum $\ell_2$-norm and the data has independent components: they gave a sharp bound on the variance term and showed that it can be small if and only if the data covariance has high effective rank in a subspace of small co-dimension. We strengthen and complete their results by eliminating the independence assumption and providing sharp bounds for the bias term. Thus, our results apply in a much more general setting than those of arXiv:1906.11300, e.g., kernel regression, and not only characterize how the noise is damped but also which part of the true signal is learned. Moreover, we extend the result to the setting of ridge regression, which allows us to explain another interesting phenomenon: we give general sufficient conditions under which the optimal regularization is negative.

We hypothesize that due to the greedy nature of learning in multi-modal deep neural networks, these models tend to rely on just one modality while under-fitting the other modalities. Such behavior is counter-intuitive and hurts the models' generalization, as we observe empirically. To estimate the model's dependence on each modality, we compute the gain on the accuracy when the model has access to it in addition to another modality. We refer to this gain as the conditional utilization rate. In the experiments, we consistently observe an imbalance in conditional utilization rates between modalities, across multiple tasks and architectures. Since conditional utilization rate cannot be computed efficiently during training, we introduce a proxy for it based on the pace at which the model learns from each modality, which we refer to as the conditional learning speed. We propose an algorithm to balance the conditional learning speeds between modalities during training and demonstrate that it indeed addresses the issue of greedy learning. The proposed algorithm improves the model's generalization on three datasets: Colored MNIST, Princeton ModelNet40, and NVIDIA Dynamic Hand Gesture.

In structure learning, the output is generally a structure that is used as supervision information to achieve good performance. Considering the interpretation of deep learning models has raised extended attention these years, it will be beneficial if we can learn an interpretable structure from deep learning models. In this paper, we focus on Recurrent Neural Networks (RNNs) whose inner mechanism is still not clearly understood. We find that Finite State Automaton (FSA) that processes sequential data has more interpretable inner mechanism and can be learned from RNNs as the interpretable structure. We propose two methods to learn FSA from RNN based on two different clustering methods. We first give the graphical illustration of FSA for human beings to follow, which shows the interpretability. From the FSA's point of view, we then analyze how the performance of RNNs are affected by the number of gates, as well as the semantic meaning behind the transition of numerical hidden states. Our results suggest that RNNs with simple gated structure such as Minimal Gated Unit (MGU) is more desirable and the transitions in FSA leading to specific classification result are associated with corresponding words which are understandable by human beings.