Yifan Chen

Yifan Chen, Tianning Xu, Dilek Hakkani-Tur et al.

Multiple sampling-based methods have been developed for approximating and accelerating node embedding aggregation in graph convolutional networks (GCNs) training. Among them, a layer-wise approach recursively performs importance sampling to select neighbors jointly for existing nodes in each layer. This paper revisits the approach from a matrix approximation perspective, and identifies two issues in the existing layer-wise sampling methods: suboptimal sampling probabilities and estimation biases induced by sampling without replacement. To address these issues, we accordingly propose two remedies: a new principle for constructing sampling probabilities and an efficient debiasing algorithm. The improvements are demonstrated by extensive analyses of estimation variance and experiments on common benchmarks. Code and algorithm implementations are publicly available at https://github.com/ychen-stat-ml/GCN-layer-wise-sampling .

Gaofeng Cheng, Yifan Chen, Runyan Yang et al.

The conversation scenario is one of the most important and most challenging scenarios for speech processing technologies because people in conversation respond to each other in a casual style. Detecting the speech activities of each person in a conversation is vital to downstream tasks, like natural language processing, machine translation, etc. People refer to the detection technology of "who speak when" as speaker diarization (SD). Traditionally, diarization error rate (DER) has been used as the standard evaluation metric of SD systems for a long time. However, DER fails to give enough importance to short conversational phrases, which are short but important on the semantic level. Also, a carefully and accurately manually-annotated testing dataset suitable for evaluating the conversational SD technologies is still unavailable in the speech community. In this paper, we design and describe the Conversational Short-phrases Speaker Diarization (CSSD) task, which consists of training and testing datasets, evaluation metric and baselines. In the dataset aspect, despite the previously open-sourced 180-hour conversational MagicData-RAMC dataset, we prepare an individual 20-hour conversational speech test dataset with carefully and artificially verified speakers timestamps annotations for the CSSD task. In the metric aspect, we design the new conversational DER (CDER) evaluation metric, which calculates the SD accuracy at the utterance level. In the baseline aspect, we adopt a commonly used method: Variational Bayes HMM x-vector system, as the baseline of the CSSD task. Our evaluation metric is publicly available at https://github.com/SpeechClub/CDER_Metric.

Mengting Ai, Tianxin Wei, Yifan Chen et al.

Fine-tuning a pre-trained language model (PLM) emerges as the predominant strategy in many natural language processing applications. However, this process is known to be expensive, especially on edge devices with low computing power. While general approaches (e.g. quantization and distillation) have been widely studied to reduce the compute/memory of PLM fine-tuning, one-shot compression techniques specifically designed for fine-tuning remain largely unexplored. In this paper, we investigate the neural tangent kernel (NTK)--which reveals the gradient descent dynamics of neural networks--of the multilayer perceptrons (MLP) modules in a PLM and propose to coin a lightweight PLM through NTK-approximating MLP fusion. By incorporating NTK into the compression process, MLP Fusion not only preserves the original model's output but also maintains its training dynamics. To achieve this, we reconsider the MLP as a bundle of sub-MLPs and cluster them into a given number of centroids, which can then be restored as a compressed MLP and surprisingly well approximate the NTK of the original PLM. Our approach is applicable to both standard MLP modules and Mixture-of-Experts (MoE) modules in PLMs, demonstrating its scalability and versatility. Additionally, we provide theoretical derivations to demonstrate how the proposed compression preserves the NTK. Extensive experiments of PLM fine-tuning on both natural language understanding and generation tasks are provided to verify the effectiveness of MLP fusion. Our code is available at https://github.com/weitianxin/MLP_Fusion.

Yu Sun, Zihui Wu, Yifan Chen et al.

Estimating high-quality images while also quantifying their uncertainty are two desired features in an image reconstruction algorithm for solving ill-posed inverse problems. In this paper, we propose plug-and-play Monte Carlo (PMC) as a principled framework for characterizing the space of possible solutions to a general inverse problem. PMC is able to incorporate expressive score-based generative priors for high-quality image reconstruction while also performing uncertainty quantification via posterior sampling. In particular, we develop two PMC algorithms that can be viewed as the sampling analogues of the traditional plug-and-play priors (PnP) and regularization by denoising (RED) algorithms. To improve the sampling efficiency, we introduce weighted annealing into these PMC algorithms, further developing two additional annealed PMC algorithms (APMC). We establish a theoretical analysis for characterizing the convergence behavior of PMC algorithms. Our analysis provides non-asymptotic stationarity guarantees in terms of the Fisher information, fully compatible with the joint presence of weighted annealing, potentially non-log-concave likelihoods, and imperfect score networks. We demonstrate the performance of the PMC algorithms on multiple representative inverse problems with both linear and nonlinear forward models. Experimental results show that PMC significantly improves reconstruction quality and enables high-fidelity uncertainty quantification.

Yifan Chen, Devamanyu Hazarika, Mahdi Namazifar et al.

The massive amount of trainable parameters in the pre-trained language models (PLMs) makes them hard to be deployed to multiple downstream tasks. To address this issue, parameter-efficient transfer learning methods have been proposed to tune only a few parameters during fine-tuning while freezing the rest. This paper looks at existing methods along this line through the \textit{kernel lens}. Motivated by the connection between self-attention in transformer-based PLMs and kernel learning, we propose \textit{kernel-wise adapters}, namely \textit{Kernel-mix}, that utilize the kernel structure in self-attention to guide the assignment of the tunable parameters. These adapters use guidelines found in classical kernel learning and enable separate parameter tuning for each attention head. Our empirical results, over a diverse set of natural language generation and understanding tasks, show that our proposed adapters can attain or improve the strong performance of existing baselines.

Yifan Chen, Devamanyu Hazarika, Mahdi Namazifar et al.

Prefix-tuning, or more generally continuous prompt tuning, has become an essential paradigm of parameter-efficient transfer learning. Using a large pre-trained language model (PLM), prefix-tuning can obtain strong performance by training only a small portion of parameters. In this paper, we propose to understand and further develop prefix-tuning through the kernel lens. Specifically, we make an analogy between \textit{prefixes} and \textit{inducing variables} in kernel methods and hypothesize that \textit{prefixes} serving as \textit{inducing variables} would improve their overall mechanism. From the kernel estimator perspective, we suggest a new variant of prefix-tuning -- \textit{inducer-tuning}, which shares the exact mechanism as prefix-tuning while leveraging the residual form found in adapter-tuning. This mitigates the initialization issue in prefix-tuning. Through comprehensive empirical experiments on natural language understanding and generation tasks, we demonstrate that inducer-tuning can close the performance gap between prefix-tuning and fine-tuning.

Yifan Chen, Daniel Zhengyu Huang, Jiaoyang Huang et al.

Sampling a target probability distribution with an unknown normalization constant is a fundamental challenge in computational science and engineering. Recent work shows that algorithms derived by considering gradient flows in the space of probability measures open up new avenues for algorithm development. This paper makes three contributions to this sampling approach by scrutinizing the design components of such gradient flows. Any instantiation of a gradient flow for sampling needs an energy functional and a metric to determine the flow, as well as numerical approximations of the flow to derive algorithms. Our first contribution is to show that the Kullback-Leibler divergence, as an energy functional, has the unique property (among all f-divergences) that gradient flows resulting from it do not depend on the normalization constant of the target distribution. Our second contribution is to study the choice of metric from the perspective of invariance. The Fisher-Rao metric is known as the unique choice (up to scaling) that is diffeomorphism invariant. As a computationally tractable alternative, we introduce a relaxed, affine invariance property for the metrics and gradient flows. In particular, we construct various affine invariant Wasserstein and Stein gradient flows. Affine invariant gradient flows are shown to behave more favorably than their non-affine-invariant counterparts when sampling highly anisotropic distributions, in theory and by using particle methods. Our third contribution is to study, and develop efficient algorithms based on Gaussian approximations of the gradient flows; this leads to an alternative to particle methods. We establish connections between various Gaussian approximate gradient flows, discuss their relation to gradient methods arising from parametric variational inference, and study their convergence properties both theoretically and numerically.

José A. Carrillo, Yifan Chen, Daniel Zhengyu Huang et al.

The dynamics of probability density functions has been extensively studied in science and engineering to understand physical phenomena and facilitate algorithmic design. Of particular interest are dynamics that can be formulated as gradient flows of energy functionals under the Wasserstein metric. The development of functional inequalities, such as the log-Sobolev inequality, plays a pivotal role in analyzing the convergence of these dynamics. The goal of this paper is to parallel the success of techniques using functional inequalities, for dynamics that are gradient flows under the Fisher-Rao metric, with various $f$-divergences as energy functionals. Such dynamics take the form of a nonlocal differential equation, for which existing analysis critically relies on using the explicit solution formula in special cases. We provide a comprehensive study on functional inequalities and the relevant geodesic convexity for Fisher-Rao gradient flows under minimal assumptions. A notable feature of the obtained functional inequalities is that they do not depend on the log-concavity or log-Sobolev constants of the target distribution. Consequently, the convergence rate of the dynamics (assuming well-posed) is uniform across general target distributions, making them potentially desirable dynamics for posterior sampling applications in Bayesian inference.

Yu Li, Yifan Chen, Gongye Liu et al.

Layout generation is a foundation task of graphic design, which requires the integration of visual aesthetics and harmonious expression of content delivery. However, existing methods still face challenges in generating precise and visually appealing layouts, including blocking, overlapping, small-sized, or spatial misalignment. We found that these methods overlook the crucial balance between learning content-aware and graphic-aware features. This oversight results in their limited ability to model the graphic structure of layouts and generate reasonable layout arrangements. To address these challenges, we introduce LayoutDiT, an effective framework that balances content and graphic features to generate high-quality, visually appealing layouts. Specifically, we first design an adaptive factor that optimizes the model's awareness of the layout generation space, balancing the model's performance in both content and graphic aspects. Secondly, we introduce a graphic condition, the saliency bounding box, to bridge the modality difference between images in the visual domain and layouts in the geometric parameter domain. In addition, we adapt a diffusion transformer model as the backbone, whose powerful generative capability ensures the quality of layout generation. Benefiting from the properties of diffusion models, our method excels in constrained settings without introducing additional constraint modules. Extensive experimental results demonstrate that our method achieves superior performance in both constrained and unconstrained settings, significantly outperforming existing methods.

Haitao Li, Jiaying Ye, Yiran Hu et al.

Legal case documents play a critical role in judicial proceedings. As the number of cases continues to rise, the reliance on manual drafting of legal case documents is facing increasing pressure and challenges. The development of large language models (LLMs) offers a promising solution for automating document generation. However, existing benchmarks fail to fully capture the complexities involved in drafting legal case documents in real-world scenarios. To address this gap, we introduce CaseGen, the benchmark for multi-stage legal case documents generation in the Chinese legal domain. CaseGen is based on 500 real case samples annotated by legal experts and covers seven essential case sections. It supports four key tasks: drafting defense statements, writing trial facts, composing legal reasoning, and generating judgment results. To the best of our knowledge, CaseGen is the first benchmark designed to evaluate LLMs in the context of legal case document generation. To ensure an accurate and comprehensive evaluation, we design the LLM-as-a-judge evaluation framework and validate its effectiveness through human annotations. We evaluate several widely used general-domain LLMs and legal-specific LLMs, highlighting their limitations in case document generation and pinpointing areas for potential improvement. This work marks a step toward a more effective framework for automating legal case documents drafting, paving the way for the reliable application of AI in the legal field. The dataset and code are publicly available at https://github.com/CSHaitao/CaseGen.

Mingchen Jiang, Peng Xu, Xichen Ye et al.

Distributional data have become increasingly prominent in modern signal processing, highlighting the necessity of computing optimal transport (OT) maps across multiple probability distributions. Nevertheless, recent studies on neural OT methods predominantly focused on the efficient computation of a single map between two distributions. To address this challenge, we introduce a novel approach to learning transport maps for new empirical distributions. Specifically, we employ the transformer architecture to produce embeddings from distributional data of varying length; these embeddings are then fed into a hypernetwork to generate neural OT maps. Various numerical experiments were conducted to validate the embeddings and the generated OT maps. The model implementation and the code are provided on https://github.com/jiangmingchen/HOTET.

Xiaoyuan Zhang, Xi Lin, Yichi Zhang et al.

Multiobjective optimization (MOO) is prevalent in numerous applications, in which a Pareto front (PF) is constructed to display optima under various preferences. Previous methods commonly utilize the set of Pareto objectives (particles on the PF) to represent the entire PF. However, the empirical distribution of the Pareto objectives on the PF is rarely studied, which implicitly impedes the generation of diverse and representative Pareto objectives in previous methods. To bridge the gap, we suggest in this paper constructing \emph{uniformly distributed} Pareto objectives on the PF, so as to alleviate the limited diversity found in previous MOO approaches. We are the first to formally define the concept of ``uniformity" for an MOO problem. We optimize the maximal minimal distances on the Pareto front using a neural network, resulting in both asymptotically and non-asymptotically uniform Pareto objectives. Our proposed method is validated through experiments on real-world and synthetic problems, which demonstrates the efficacy in generating high-quality uniform Pareto objectives and the encouraging performance exceeding existing state-of-the-art methods. The detailed model implementation and the code are scheduled to be open-sourced upon publication.

Zhong Yi Wan, Ricardo Baptista, Yi-fan Chen et al.

We introduce a two-stage probabilistic framework for statistical downscaling using unpaired data. Statistical downscaling seeks a probabilistic map to transform low-resolution data from a biased coarse-grained numerical scheme to high-resolution data that is consistent with a high-fidelity scheme. Our framework tackles the problem by composing two transformations: (i) a debiasing step via an optimal transport map, and (ii) an upsampling step achieved by a probabilistic diffusion model with a posteriori conditional sampling. This approach characterizes a conditional distribution without needing paired data, and faithfully recovers relevant physical statistics from biased samples. We demonstrate the utility of the proposed approach on one- and two-dimensional fluid flow problems, which are representative of the core difficulties present in numerical simulations of weather and climate. Our method produces realistic high-resolution outputs from low-resolution inputs, by upsampling resolutions of 8x and 16x. Moreover, our procedure correctly matches the statistics of physical quantities, even when the low-frequency content of the inputs and outputs do not match, a crucial but difficult-to-satisfy assumption needed by current state-of-the-art alternatives. Code for this work is available at: https://github.com/google-research/swirl-dynamics/tree/main/swirl_dynamics/projects/probabilistic_diffusion.

Baojun Che, Yifan Chen, Zhenghao Huan et al.

This paper is concerned with the approximation of probability distributions known up to normalization constants, with a focus on Bayesian inference for large-scale inverse problems in scientific computing. In this context, key challenges include costly repeated evaluations of forward models, multimodality, and inaccessible gradients for the forward model. To address them, we develop a variational inference framework that combines Fisher-Rao natural gradient with specialized quadrature rules to enable derivative free updates of Gaussian mixture variational families. The resulting method, termed Derivative Free Gaussian Mixture Variational Inference (DF-GMVI), guarantees covariance positivity and affine invariance, offering a stable and efficient framework for approximating complex posterior distributions. The effectiveness of DF-GMVI is demonstrated through numerical experiments on challenging scenarios, including distributions with multiple modes, infinitely many modes, and curved modes in spaces with up to 100 dimensions. The method's practicality is further demonstrated in a large-scale application, where it successfully recovers the initial conditions of the Navier-Stokes equations from solution data at positive times.

Yifan Chen, Bamdad Hosseini, Houman Owhadi et al.

The article presents a systematic study of the problem of conditioning a Gaussian random variable $ξ$ on nonlinear observations of the form $F \circ φ(ξ)$ where $φ: \mathcal{X} \to \mathbb{R}^N$ is a bounded linear operator and $F$ is nonlinear. Such problems arise in the context of Bayesian inference and recent machine learning-inspired PDE solvers. We give a representer theorem for the conditioned random variable $ξ\mid F\circ φ(ξ)$, stating that it decomposes as the sum of an infinite-dimensional Gaussian (which is identified analytically) as well as a finite-dimensional non-Gaussian measure. We also introduce a novel notion of the mode of a conditional measure by taking the limit of the natural relaxation of the problem, to which we can apply the existing notion of maximum a posteriori estimators of posterior measures. Finally, we introduce a variant of the Laplace approximation for the efficient simulation of the aforementioned conditioned Gaussian random variables towards uncertainty quantification.

Yifan Chen, Eric Vanden-Eijnden, Jiawei Xu

We study the design of interpolation schedules in the stochastic interpolants framework for flow and diffusion-based generative models. We show that while all scalar interpolation schedules achieve identical statistical efficiency under Kullback-Leibler divergence in path space after optimal diffusion coefficient tuning, their numerical efficiency can differ substantially. This observation motivates focusing on numerical properties of the resulting drift fields rather than statistical criteria for schedule design. We propose averaged squared Lipschitzness minimization as a principled criterion for numerical optimization, providing an alternative to kinetic energy minimization used in optimal transport approaches. A transfer formula is derived that enables conversion between different schedules at inference time without retraining neural networks. For Gaussian distributions, our optimized schedules achieve exponential improvements in Lipschitz constants over standard linear schedules, while for Gaussian mixtures, they reduce mode collapse in few-step sampling. We also validate our approach on high-dimensional invariant distributions from stochastic Allen-Cahn equations and Navier-Stokes equations, demonstrating robust performance improvements across resolutions.

Yifan Chen, Eric Vanden-Eijnden

Flow-based generative models can face significant challenges when modeling scientific data with multiscale Fourier spectra, often producing large errors in fine-scale features. We address this problem within the framework of stochastic interpolants, via principled design of noise distributions and interpolation schedules. The key insight is that the noise should not be smoother than the target data distribution -- measured by Fourier spectrum decay rates -- to ensure bounded drift fields near the initial time. For Gaussian and near-Gaussian distributions whose fine-scale structure is known, we show that spectrum-matched noise improves numerical efficiency compared to standard white-noise approaches. For complex non-Gaussian distributions, we develop scale-adaptive interpolation schedules that address the numerical ill-conditioning arising from rougher-than-data noise. Numerical experiments on synthetic Gaussian random fields and solutions to the stochastic Allen-Cahn and Navier-Stokes equations validate our approach and demonstrate its ability to generate high-fidelity samples at lower computational cost than traditional approaches.

Yifan Chen, Daniel Zhengyu Huang, Jiaoyang Huang et al.

In this paper, we study efficient approximate sampling for probability distributions known up to normalization constants. We specifically focus on a problem class arising in Bayesian inference for large-scale inverse problems in science and engineering applications. The computational challenges we address with the proposed methodology are: (i) the need for repeated evaluations of expensive forward models; (ii) the potential existence of multiple modes; and (iii) the fact that gradient of, or adjoint solver for, the forward model might not be feasible. While existing Bayesian inference methods meet some of these challenges individually, we propose a framework that tackles all three systematically. Our approach builds upon the Fisher-Rao gradient flow in probability space, yielding a dynamical system for probability densities that converges towards the target distribution at a uniform exponential rate. This rapid convergence is advantageous for the computational burden outlined in (i). We apply Gaussian mixture approximations with operator splitting techniques to simulate the flow numerically; the resulting approximation can capture multiple modes thus addressing (ii). Furthermore, we employ the Kalman methodology to facilitate a derivative-free update of these Gaussian components and their respective weights, addressing the issue in (iii). The proposed methodology results in an efficient derivative-free sampler flexible enough to handle multi-modal distributions: Gaussian Mixture Kalman Inversion (GMKI). The effectiveness of GMKI is demonstrated both theoretically and numerically in several experiments with multimodal target distributions, including proof-of-concept and two-dimensional examples, as well as a large-scale application: recovering the Navier-Stokes initial condition from solution data at positive times.

Zehui Yang, Yifan Chen, Lei Luo et al.

This paper introduces a high-quality rich annotated Mandarin conversational (RAMC) speech dataset called MagicData-RAMC. The MagicData-RAMC corpus contains 180 hours of conversational speech data recorded from native speakers of Mandarin Chinese over mobile phones with a sampling rate of 16 kHz. The dialogs in MagicData-RAMC are classified into 15 diversified domains and tagged with topic labels, ranging from science and technology to ordinary life. Accurate transcription and precise speaker voice activity timestamps are manually labeled for each sample. Speakers' detailed information is also provided. As a Mandarin speech dataset designed for dialog scenarios with high quality and rich annotations, MagicData-RAMC enriches the data diversity in the Mandarin speech community and allows extensive research on a series of speech-related tasks, including automatic speech recognition, speaker diarization, topic detection, keyword search, text-to-speech, etc. We also conduct several relevant tasks and provide experimental results to help evaluate the dataset.

Yifan Chen, Qi Zeng, Dilek Hakkani-Tur et al.

Transformer-based models are not efficient in processing long sequences due to the quadratic space and time complexity of the self-attention modules. To address this limitation, Linformer and Informer are proposed to reduce the quadratic complexity to linear (modulo logarithmic factors) via low-dimensional projection and row selection respectively. These two models are intrinsically connected, and to understand their connection, we introduce a theoretical framework of matrix sketching. Based on the theoretical analysis, we propose Skeinformer to accelerate self-attention and further improve the accuracy of matrix approximation to self-attention with three carefully designed components: column sampling, adaptive row normalization and pilot sampling reutilization. Experiments on the Long Range Arena (LRA) benchmark demonstrate that our methods outperform alternatives with a consistently smaller time/space footprint.

Yifan Chen, Qi Zeng, Heng Ji et al.

Transformers are expensive to train due to the quadratic time and space complexity in the self-attention mechanism. On the other hand, although kernel machines suffer from the same computation bottleneck in pairwise dot products, several approximation schemes have been successfully incorporated to considerably reduce their computational cost without sacrificing too much accuracy. In this work, we leverage the computation methods for kernel machines to alleviate the high computational cost and introduce Skyformer, which replaces the softmax structure with a Gaussian kernel to stabilize the model training and adapts the Nyström method to a non-positive semidefinite matrix to accelerate the computation. We further conduct theoretical analysis by showing that the matrix approximation error of our proposed method is small in the spectral norm. Experiments on Long Range Arena benchmark show that the proposed method is sufficient in getting comparable or even better performance than the full self-attention while requiring fewer computation resources.

Yifan Chen, Bamdad Hosseini, Houman Owhadi et al.

We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian processes. The proposed approach: (1) provides a natural generalization of collocation kernel methods to nonlinear PDEs and IPs; (2) has guaranteed convergence for a very general class of PDEs, and comes equipped with a path to compute error bounds for specific PDE approximations; (3) inherits the state-of-the-art computational complexity of linear solvers for dense kernel matrices. The main idea of our method is to approximate the solution of a given PDE as the maximum a posteriori (MAP) estimator of a Gaussian process conditioned on solving the PDE at a finite number of collocation points. Although this optimization problem is infinite-dimensional, it can be reduced to a finite-dimensional one by introducing additional variables corresponding to the values of the derivatives of the solution at collocation points; this generalizes the representer theorem arising in Gaussian process regression. The reduced optimization problem has the form of a quadratic objective function subject to nonlinear constraints; it is solved with a variant of the Gauss--Newton method. The resulting algorithm (a) can be interpreted as solving successive linearizations of the nonlinear PDE, and (b) in practice is found to converge in a small number of iterations (2 to 10), for a wide range of PDEs. Most traditional approaches to IPs interleave parameter updates with numerical solution of the PDE; our algorithm solves for both parameter and PDE solution simultaneously. Experiments on nonlinear elliptic PDEs, Burgers' equation, a regularized Eikonal equation, and an IP for permeability identification in Darcy flow illustrate the efficacy and scope of our framework.

Yifan Chen, Yun Yang

With appropriately chosen sampling probabilities, sampling-based random projection can be used to implement large-scale statistical methods, substantially reducing computational cost while maintaining low statistical error. However, computing optimal sampling probabilities is often itself expensive, and in practice one typically resorts to suboptimal schemes. This generally leads to increased time and space costs, as more subsamples are required and the resulting projection matrices become larger, thereby making the inference procedure more computationally demanding. In this paper, we extend the framework of sampling-based random projection and propose a new projection method, \emph{accumulative sub-sampling}. By carefully accumulating multiple such projections, accumulative sub-sampling improves statistical efficiency while controlling the effective matrix size throughout the statistical computation. On the theoretical side, we quantify how the quality of the subsampling scheme affects the error in approximating matrix products and positive semidefinite matrices, and show how the proposed accumulation strategy mitigates this effect. Moreover, we apply our method to statistical models involving intensive matrix operations, such as eigendecomposition in spectral clustering and matrix inversion in kernel ridge regression, and demonstrate that reducing the effective matrix size leads to substantial computational savings. Numerical experiments across a range of problems further show that our approach consistently improves computational efficiency compared to existing random projection baselines under suboptimal sampling schemes.

Yifan Chen, Houman Owhadi, Andrew M. Stuart

Gaussian process regression has proven very powerful in statistics, machine learning and inverse problems. A crucial aspect of the success of this methodology, in a wide range of applications to complex and real-world problems, is hierarchical modeling and learning of hyperparameters. The purpose of this paper is to study two paradigms of learning hierarchical parameters: one is from the probabilistic Bayesian perspective, in particular, the empirical Bayes approach that has been largely used in Bayesian statistics; the other is from the deterministic and approximation theoretic view, and in particular the kernel flow algorithm that was proposed recently in the machine learning literature. Analysis of their consistency in the large data limit, as well as explicit identification of their implicit bias in parameter learning, are established in this paper for a Matérn-like model on the torus. A particular technical challenge we overcome is the learning of the regularity parameter in the Matérn-like field, for which consistency results have been very scarce in the spatial statistics literature. Moreover, we conduct extensive numerical experiments beyond the Matérn-like model, comparing the two algorithms further. These experiments demonstrate learning of other hierarchical parameters, such as amplitude and lengthscale; they also illustrate the setting of model misspecification in which the kernel flow approach could show superior performance to the more traditional empirical Bayes approach.

Yifan Chen, Yevgeniy Vorobeychik

Despite the considerable success of convolutional neural networks in a broad array of domains, recent research has shown these to be vulnerable to small adversarial perturbations, commonly known as adversarial examples. Moreover, such examples have shown to be remarkably portable, or transferable, from one model to another, enabling highly successful black-box attacks. We explore this issue of transferability and robustness from two dimensions: first, considering the impact of conventional $l_p$ regularization as well as replacing the top layer with a linear support vector machine (SVM), and second, the value of combining regularized models into an ensemble. We show that models trained with different regularizers present barriers to transferability, as does partial information about the models comprising the ensemble.

Yifan Chen, Wuchen Li

We study the Wasserstein natural gradient in parametric statistical models with continuous sample spaces. Our approach is to pull back the $L^2$-Wasserstein metric tensor in the probability density space to a parameter space, equipping the latter with a positive definite metric tensor, under which it becomes a Riemannian manifold, named the Wasserstein statistical manifold. In general, it is not a totally geodesic sub-manifold of the density space, and therefore its geodesics will differ from the Wasserstein geodesics, except for the well-known Gaussian distribution case, a fact which can also be validated under our framework. We use the sub-manifold geometry to derive a gradient flow and natural gradient descent method in the parameter space. When parametrized densities lie in $\bR$, the induced metric tensor establishes an explicit formula. In optimization problems, we observe that the natural gradient descent outperforms the standard gradient descent when the Wasserstein distance is the objective function. In such a case, we prove that the resulting algorithm behaves similarly to the Newton method in the asymptotic regime. The proof calculates the exact Hessian formula for the Wasserstein distance, which further motivates another preconditioner for the optimization process. To the end, we present examples to illustrate the effectiveness of the natural gradient in several parametric statistical models, including the Gaussian measure, Gaussian mixture, Gamma distribution, and Laplace distribution.

Yifan Chen, Yuejiao Sun, Wotao Yin

Many optimization algorithms converge to stationary points. When the underlying problem is nonconvex, they may get trapped at local minimizers and occasionally stagnate near saddle points. We propose the Run-and-Inspect Method, which adds an "inspect" phase to existing algorithms that helps escape from non-global stationary points. The inspection samples a set of points in a radius $R$ around the current point. When a sample point yields a sufficient decrease in the objective, we move there and resume an existing algorithm. If no sufficient decrease is found, the current point is called an approximate $R$-local minimizer. We show that an $R$-local minimizer is globally optimal, up to a specific error depending on $R$, if the objective function can be implicitly decomposed into a smooth convex function plus a restricted function that is possibly nonconvex, nonsmooth. For high-dimensional problems, we introduce blockwise inspections to overcome the curse of dimensionality while still maintaining optimality bounds up to a factor equal to the number of blocks. Our method performs well on a set of artificial and realistic nonconvex problems by coupling with gradient descent, coordinate descent, EM, and prox-linear algorithms.