The development of efficient sampling algorithms catering to non-Euclidean geometries has been a challenging endeavor, as discretization techniques which succeed in the Euclidean setting do not readily carry over to more general settings. We develop a non-Euclidean analog of the recent proximal sampler of [LST21], which naturally induces regularization by an object known as the log-Laplace transform (LLT) of a density. We prove new mathematical properties (with an algorithmic flavor) of the LLT, such as strong convexity-smoothness duality and an isoperimetric inequality, which are used to prove a mixing time on our proximal sampler matching [LST21] under a warm start. As our main application, we show our warm-started sampler improves the value oracle complexity of differentially private convex optimization in $\ell_p$ and Schatten-$p$ norms for $p \in [1, 2]$ to match the Euclidean setting [GLL22], while retaining state-of-the-art excess risk bounds [GLLST23]. We find our investigation of the LLT to be a promising proof-of-concept of its utility as a tool for designing samplers, and outline directions for future exploration.

It is very difficult to forecast the production rate of oil wells as the output of a single well is sensitive to various uncertain factors, which implicitly or explicitly show the influence of the static, temporal and spatial properties on the oil well production. In this study, a novel machine learning model is constructed to fuse the static geological information, dynamic well production history, and spatial information of the adjacent water injection wells. There are 3 basic modules in this stacking model, which are regarded as the encoders to extract the features from different types of data. One is Multi-Layer Perceptron, which is to analyze the static geological properties of the reservoir that might influence the well production rate. The other two are both LSTMs, which have the input in the form of two matrices rather than vectors, standing for the temporal and the spatial information of the target well. The difference of the two modules is that in the spatial information processing module we take into consideration the time delay of water flooding response, from the injection well to the target well. In addition, we use Symbolic Transfer Entropy to prove the superiorities of the stacking model from the perspective of Causality Discovery. It is proved theoretically and practically that the presented model can make full use of the model structure to integrate the characteristics of the data and the experts' knowledge into the process of machine learning, greatly improving the accuracy and generalization ability of prediction.

Since its foundations, more than one hundred years ago, the field of structural biology has strived to understand and analyze the properties of molecules and their interactions by studying the structure that they take in 3D space. However, a fundamental challenge with this approach has been the dynamic nature of these particles, which forces us to model not a single but a whole distribution of structures for every molecular system. This thesis proposes Intrinsic Diffusion Modeling, a novel approach to this problem based on combining diffusion generative models with scientific knowledge about the flexibility of biological complexes. The knowledge of these degrees of freedom is translated into the definition of a manifold over which the diffusion process is defined. This manifold significantly reduces the dimensionality and increases the smoothness of the generation space allowing for significantly faster and more accurate generative processes. We demonstrate the effectiveness of this approach on two fundamental tasks at the basis of computational chemistry and biology: molecular conformer generation and molecular docking. In both tasks, we construct the first deep learning method to outperform traditional computational approaches achieving an unprecedented level of accuracy for scalable programs.

Solving high-dimensional Bayesian inverse problems (BIPs) with the variational inference (VI) method is promising but still challenging. The main difficulties arise from two aspects. First, VI methods approximate the posterior distribution using a simple and analytic variational distribution, which makes it difficult to estimate complex spatially-varying parameters in practice. Second, VI methods typically rely on gradient-based optimization, which can be computationally expensive or intractable when applied to BIPs involving partial differential equations (PDEs). To address these challenges, we propose a novel approximation method for estimating the high-dimensional posterior distribution. This approach leverages a deep generative model to learn a prior model capable of generating spatially-varying parameters. This enables posterior approximation over the latent variable instead of the complex parameters, thus improving estimation accuracy. Moreover, to accelerate gradient computation, we employ a differentiable physics-constrained surrogate model to replace the adjoint method. The proposed method can be fully implemented in an automatic differentiation manner. Numerical examples demonstrate two types of log-permeability estimation for flow in heterogeneous media. The results show the validity, accuracy, and high efficiency of the proposed method.

The recently emerged spectral clustering surpasses conventional clustering methods by detecting clusters of any shape without the convexity assumption. Unfortunately, with a computational complexity of $O(n^3)$, it was infeasible for multiple real applications, where $n$ could be large. This stimulates researchers to propose the approximate spectral clustering (ASC). However, most of ASC methods assumed that the number of clusters $k$ was known. In practice, manual setting of $k$ could be subjective or time consuming. The proposed algorithm has two relevance metrics for estimating $k$ in two vital steps of ASC. One for selecting the eigenvectors spanning the embedding space, and the other to discover the number of clusters in that space. The algorithm used a growing neural gas (GNG) approximation, GNG is superior in preserving input data topology. The experimental setup demonstrates the efficiency of the proposed algorithm and its ability to compete with similar methods where $k$ was set manually.

An important problem in network analysis is predicting a node attribute using both network covariates, such as graph embedding coordinates or local subgraph counts, and conventional node covariates, such as demographic characteristics. While standard regression methods that make use of both types of covariates may be used for prediction, statistical inference is complicated by the fact that the nodal summary statistics are often dependent in complex ways. We show that under a mild joint exchangeability assumption, a network analog of conformal prediction achieves finite sample validity for a wide range of network covariates. We also show that a form of asymptotic conditional validity is achievable. The methods are illustrated on both simulated networks and a citation network dataset.

In this paper, we propose energy-based sample adaptation at test time for domain generalization. Where previous works adapt their models to target domains, we adapt the unseen target samples to source-trained models. To this end, we design a discriminative energy-based model, which is trained on source domains to jointly model the conditional distribution for classification and data distribution for sample adaptation. The model is optimized to simultaneously learn a classifier and an energy function. To adapt target samples to source distributions, we iteratively update the samples by energy minimization with stochastic gradient Langevin dynamics. Moreover, to preserve the categorical information in the sample during adaptation, we introduce a categorical latent variable into the energy-based model. The latent variable is learned from the original sample before adaptation by variational inference and fixed as a condition to guide the sample update. Experiments on six benchmarks for classification of images and microblog threads demonstrate the effectiveness of our proposal. Deep neural networks are vulnerable to domain shifts and suffer from lack of generalization on test samples that do not resemble the ones in the training distribution (Recht et al., 2019; Zhou et al., 2021; Krueger et al., 2021; Shen et al., 2022). To deal with the domain shifts, domain generalization has been proposed (Muandet et al., 2013; Gulrajani & Lopez-Paz, 2020; Cha et al., 2021). Domain generalization strives to learn a model exclusively on source domains in order to generalize well on unseen target domains. The major challenge stems from the large domain shifts and the unavailability of any target domain data during training. To address the problem, domain invariant learning has been widely studied, e.g., (Motiian et al., 2017; Zhao et al., 2020; Nguyen et al., 2021), based on the assumption that invariant representations obtained on source domains are also valid for unseen target domains. However, since the target data is inaccessible during training, it is likely an "adaptivity gap" (Dubey et al., 2021) exists between representations from the source and target domains. Therefore, recent works try to adapt the classification model with target samples at test time by further fine-tuning model parameters (Sun et al., 2020; Wang et al., 2021) or by introducing an extra network module for adaptation (Dubey et al., 2021). Rather than adapting the model to target domains, Xiao et al. (2022) adapt the classifier for each sample at test time. Nevertheless, a single sample would not be able to adjust the whole model due to the large number of model parameters and the limited information contained in the sample.

A fundamental challenge for multi-task learning is that different tasks may conflict with each other when they are solved jointly, and a cause of this phenomenon is conflicting gradients during optimization. Recent works attempt to mitigate the influence of conflicting gradients by directly altering the gradients based on some criteria. However, our empirical study shows that "gradient surgery" cannot effectively reduce the occurrence of conflicting gradients. In this paper, we take a different approach to reduce conflicting gradients from the root. In essence, we investigate the task gradients w.r.t. each shared network layer, select the layers with high conflict scores, and turn them to task-specific layers. Our experiments show that such a simple approach can greatly reduce the occurrence of conflicting gradients in the remaining shared layers and achieve better performance, with only a slight increase in model parameters in many cases. Our approach can be easily applied to improve various state-of-the-art methods including gradient manipulation methods and branched architecture search methods. Given a network architecture (e.g., ResNet18), it only needs to search for the conflict layers once, and the network can be modified to be used with different methods on the same or even different datasets to gain performance improvement. Multi-task learning (MTL) is a learning paradigm in which multiple different but correlated tasks are jointly trained with a shared model (Caruana, 1997), in the hope of achieving better performance with an overall smaller model size than learning each task independently. By discovering shared structures across tasks and leveraging domain-specific training signals of related tasks, MTL can achieve efficiency and effectiveness.

Gradient boosting machines (GBMs) based on decision trees consistently demonstrate state-of-the-art results on regression and classification tasks with tabular data, often outperforming deep neural networks. However, these models do not provide well-calibrated predictive uncertainties, which prevents their use for decision making in high-risk applications. The Bayesian treatment is known to improve predictive uncertainty calibration, but previously proposed Bayesian GBM methods are either computationally expensive, or resort to crude approximations. Variational inference is often used to implement Bayesian neural networks, but is difficult to apply to GBMs, because the decision trees used as weak learners are non-differentiable. In this paper, we propose to implement Bayesian GBMs using variational inference with soft decision trees, a fully differentiable alternative to standard decision trees introduced by Irsoy et al. Our experiments demonstrate that variational soft trees and variational soft GBMs provide useful uncertainty estimates, while retaining good predictive performance. The proposed models show higher test likelihoods when compared to the state-of-the-art Bayesian GBMs in 7/10 tabular regression datasets and improved out-of-distribution detection in 5/10 datasets.

Segmenting unseen objects is a crucial ability for the robot since it may encounter new environments during the operation. Recently, a popular solution is leveraging RGB-D features of large-scale synthetic data and directly applying the model to unseen real-world scenarios. However, the domain shift caused by the sim2real gap is inevitable, posing a crucial challenge to the segmentation model. In this paper, we emphasize the adaptation process across sim2real domains and model it as a learning problem on the BatchNorm parameters of a simulation-trained model. Specifically, we propose a novel non-parametric entropy objective, which formulates the learning objective for the test-time adaptation in an open-world manner. Then, a cross-modality knowledge distillation objective is further designed to encourage the test-time knowledge transfer for feature enhancement. Our approach can be efficiently implemented with only test images, without requiring annotations or revisiting the large-scale synthetic training data. Besides significant time savings, the proposed method consistently improves segmentation results on the overlap and boundary metrics, achieving state-of-the-art performance on unseen object instance segmentation.