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Probabilistic Methods

Aleksandr Karakulev, Dave Zachariah, Prashant Singh

We present an adaptive approach for robust learning from corrupted training sets. We identify corrupted and non-corrupted samples with latent Bernoulli variables and thus formulate the learning problem as maximization of the likelihood where latent variables are marginalized. The resulting problem is solved via variational inference, using an efficient Expectation-Maximization based method. The proposed approach improves over the state-of-the-art by automatically inferring the corruption level, while adding minimal computational overhead. We demonstrate our robust learning method and its parameter-free nature on a wide variety of machine learning tasks including online learning and deep learning where it adapts to different levels of noise and maintains high prediction accuracy.

Optimization

Junyi FAN, Yuxuan Han, Zijian Liu, Jian-Feng Cai, Yang Wang, Zhengyuan Zhou

The Bures-Wasserstein (BW) gradient descent method has gained considerable attention in various domains, including Gaussian barycenter, matrix recovery and variational inference problems, due to its alignment with the Wasserstein geometry of normal distributions. Despite its popularity, existing convergence analysis are often contingent upon specific loss functions, and the exploration of constrained settings within this framework remains limited. In this work, we make an attempt to bridge this gap by providing a general convergence rate guarantee for BW gradient descent when the Euclidean strong convexity of the loss and the constraints is assumed. In an effort to advance practical implementations, we also derive a closed-form solution for the projection onto BW distance-constrained sets, which enables the fast implementation of projected BW gradient descent for problems that arise in the constrained barycenter and distributionally robust optimization literature. Experimental results demonstrate significant improvements in computational efficiency and convergence speed, underscoring the efficacy of our method in practical scenarios.

Probabilistic Methods

Kirill Neklyudov, Rob Brekelmans, Alexander Tong, Lazar Atanackovic, qiang liu, Alireza Makhzani

The dynamical formulation of the optimal transport can be extended through various choices of the underlying geometry (*kinetic energy*), and the regularization of density paths (*potential energy*). These combinations yield different variational problems (*Lagrangians*), encompassing many variations of the optimal transport problem such as the Schrödinger bridge, unbalanced optimal transport, and optimal transport with physical constraints, among others. In general, the optimal density path is unknown, and solving these variational problems can be computationally challenging. We propose a novel deep learning based framework approaching all of these problems from a unified perspective. Leveraging the dual formulation of the Lagrangians, our method does not require simulating or backpropagating through the trajectories of the learned dynamics, and does not need access to optimal couplings. We showcase the versatility of the proposed framework by outperforming previous approaches for the single-cell trajectory inference, where incorporating prior knowledge into the dynamics is crucial for correct predictions.

Probabilistic Methods · Variational Inference

Alessandro Mastrototaro, Jimmy Olsson

Being the most classical generative model for serial data, state-space models (SSM) are fundamental in AI and statistical machine learning. In SSM, any form of parameter learning or latent state inference typically involves the computation of complex latent-state posteriors. In this work, we build upon the variational sequential Monte Carlo (VSMC) method, which provides computationally efficient and accurate model parameter estimation and Bayesian latent-state inference by combining particle methods and variational inference. While standard VSMC operates in the offline mode, by re-processing repeatedly a given batch of data, we distribute the approximation of the gradient of the VSMC surrogate ELBO in time using stochastic approximation, allowing for online learning in the presence of streams of data. This results in an algorithm, online VSMC, that is capable of performing efficiently, entirely on-the-fly, both parameter estimation and particle proposal adaptation. In addition, we provide rigorous theoretical results describing the algorithm's convergence properties as the number of data tends to infinity as well as numerical illustrations of its excellent convergence properties and usefulness also in batch-processing settings.

Deep Learning · Large Language Models

Dyah Adila, Shuai Zhang, Boran Han, Yuyang Wang

The question-answering (QA) capabilities of foundation models are highly sensitive to prompt variations, rendering their performance susceptible to superficial, non-meaning-altering changes. This vulnerability often stems from the model's preference or bias towards specific input characteristics, such as option position or superficial image features in multi-modal settings. We propose to rectify this bias directly in the model's internal representation. Our approach, SteerFair, finds the bias direction in the model's representation space and steers activation values away from it during inference. Specifically, we exploit the observation that bias often adheres to simple association rules, such as the spurious association between the first option and correctness likelihood. Next, we construct demonstrations of these rules from unlabeled samples and use them to identify the bias directions. We empirically show that SteerFair significantly reduces instruction-tuned model performance variance across prompt modifications on three benchmark tasks. Remarkably, our approach surpasses a supervised baseline with 100 labels by an average of 10.86% accuracy points and 12.95 score points and matches the performance with 500 labels.

Probabilistic Methods · Bayesian Models and Methods

Yirui Liu, Xinghao Qiao, Yulong Pei, Liying Wang

This paper introduces the Deep Functional Factor Model (DF2M), a Bayesian nonparametric model designed for analysis of high-dimensional functional time series. DF2M is built upon the Indian Buffet Process and the multi-task Gaussian Process, incorporating a deep kernel function that captures non-Markovian and nonlinear temporal dynamics. Unlike many black-box deep learning models, DF2M offers an explainable approach to utilizing neural networks by constructing a factor model and integrating deep neural networks within the kernel function. Additionally, we develop a computationally efficient variational inference algorithm to infer DF2M. Empirical results from four real-world datasets demonstrate that DF2M provides better explainability and superior predictive accuracy compared to conventional deep learning models for high-dimensional functional time series.

Applications · Neuroscience, Cognitive Science

Yididiya Nadew, Xuhui Fan, Christopher J Quinn

Gaussian process factor analysis (GPFA) is a latent variable modeling technique commonly used to identify smooth, low-dimensional latent trajectories underlying high-dimensional neural recordings. Specifically, researchers model spiking rates as Gaussian observations, resulting in tractable inference. Recently, GPFA has been extended to model spike count data. However, due to the non-conjugacy of the likelihood, the inference becomes intractable. Prior works rely on either black-box inference techniques, numerical integration or polynomial approximations of the likelihood to handle intractability. To overcome this challenge, we propose a conditionally-conjugate Gaussian process factor analysis (ccGPFA) resulting in both analytically and computationally tractable inference for modeling neural activity from spike count data. In particular, we develop a novel data augmentation based method that renders the model conditionally conjugate. Consequently, our model enjoys the advantage of simple closed-form updates using a variational EM algorithm. Furthermore, due to its conditional conjugacy, we show our model can be readily scaled using sparse Gaussian Processes and accelerated inference via natural gradients. To validate our method, we empirically demonstrate its efficacy through experiments.

Deep Learning · Generative Models and Autoencoders

Zalan Fabian, Berk Tinaz, Mahdi Soltanolkotabi

Inverse problems arise in a multitude of applications, where the goal is to recover a clean signal from noisy and possibly (non)linear observations. The difficulty of a reconstruction problem depends on multiple factors, such as the ground truth signal structure, the severity of the degradation and the complex interactions between the above. This results in natural sample-by-sample variation in the difficulty of a reconstruction problem. Our key observation is that most existing inverse problem solvers lack the ability to adapt their compute power to the difficulty of the reconstruction task, resulting in subpar performance and wasteful resource allocation. We propose a novel method, *severity encoding*, to estimate the degradation severity of corrupted signals in the latent space of an autoencoder. We show that the estimated severity has strong correlation with the true corruption level and can provide useful hints on the difficulty of reconstruction problems on a sample-by-sample basis. Furthermore, we propose a reconstruction method based on latent diffusion models that leverages the predicted degradation severities to fine-tune the reverse diffusion sampling trajectory and thus achieve sample-adaptive inference times. Our framework, Flash-Diffusion, acts as a wrapper that can be combined with any latent diffusion-based baseline solver, imbuing it with sample-adaptivity and acceleration. We perform experiments on both linear and nonlinear inverse problems and demonstrate that our technique greatly improves the performance of the baseline solver and achieves up to $10\times$ acceleration in mean sampling speed.

Probabilistic Methods · Variational Inference

Johannes Zenn, Robert Bamler

Differentiable annealed importance sampling (DAIS), proposed by Geffner & Domke (2021) and Zhang et al. (2021), allows optimizing, among others, over the initial distribution of AIS. In this paper, we show that, in the limit of many transitions, DAIS minimizes the symmetrized KL divergence (Jensen-Shannon divergence) between the initial and target distribution. Thus, DAIS can be seen as a form of variational inference (VI) in that its initial distribution is a parametric fit to an intractable target distribution. We empirically evaluate the usefulness of the initial distribution as a variational distribution on synthetic and real-world data, observing that it often provides more accurate uncertainty estimates than standard VI (optimizing the reverse KL divergence), importance weighted VI, and Markovian score climbing (optimizing the forward KL divergence).

Deep Learning

XUE YU, Muchen Li, Yan Leng, Renjie Liao

In network games, individuals interact strategically within network environments to maximize their utilities. However, obtaining network structures is challenging. In this work, we propose an unsupervised learning model, called data-dependent gated-prior graph variational autoencoder (GPGVAE), that infers the underlying latent interaction type (strategic complement vs. substitute) among individuals and the latent network structure based on their observed actions. Specially, we propose a spectral graph neural network (GNN) based encoder to predict the interaction type and a data-dependent gated prior that models network structures conditioned on the interaction type. We further propose a Transformer based mixture of Bernoulli encoder of network structures and a GNN based decoder of game actions. We systematically study the Monte Carlo gradient estimation methods and effectively train our model in a stage-wise fashion. Extensive experiments across various synthetic and real-world network games demonstrate that our model achieves state-of-the-art performances in inferring network structures and well captures interaction types.

Probabilistic Methods · Variational Inference

Kaiwen Wu, Jacob Gardner

Stochastic natural gradient variational inference (NGVI) is a popular posterior inference method with applications in various probabilistic models. Despite its wide usage, little is known about the non-asymptotic convergence rate in the *stochastic* setting. We aim to lessen this gap and provide a better understanding. For conjugate likelihoods, we prove the first $\mathcal{O}(\frac{1}{T})$ non-asymptotic convergence rate of stochastic NGVI. The complexity is no worse than stochastic gradient descent (a.k.a. black-box variational inference) and the rate likely has better constant dependency that leads to faster convergence in practice. For non-conjugate likelihoods, we show that stochastic NGVI with the canonical parameterization implicitly optimizes a non-convex objective. Thus, a global convergence rate of $\mathcal{O}(\frac{1}{T})$ is unlikely without some significant new understanding of optimizing the ELBO using natural gradients.

Social Aspects · Accountability, Transparency and Interpretability

Wenshuo Wang, Lucas Janson, Lihua Lei, Aaditya Ramdas

Inferring variable importance is the key goal of many scientific studies, where researchers seek to learn the effect of a feature $X$ on the outcome $Y$ in the presence of confounding variables $Z$. Focusing on classification problems, we define the expected total variation (ETV), which is an intuitive and deterministic measure of variable importance that does not rely on any model assumption. We then introduce algorithms for statistical inference on the ETV under design-based/model-X assumptions. We name our method Total Variation Floodgate in reference to its shared high-level structure with the Floodgate method of Zhang & Janson (2020). The algorithms we introduce can leverage any user-specified regression function and produce asymptotic lower confidence bounds for the ETV. We show the effectiveness of our algorithms with simulations and a case study in conjoint analysis on the US general election.

Sumanth Varambally, Yian Ma, Rose Yu

Discovering causal relationships from time series data is significant in fields such as finance, climate science, and neuroscience. However, contemporary techniques rely on the simplifying assumption that data originates from the same causal model, while in practice, data is heterogeneous and can stem from different causal models. In this work, we relax this assumption and perform causal discovery from time series data originating from *a mixture of causal models*. We propose a general variational inference-based framework called MCD to infer the underlying causal models as well as the mixing probability of each sample. Our approach employs an end-to-end training process that maximizes an evidence-lower bound for the data likelihood. We present two variants: MCD-Linear for linear relationships and independent noise, and MCD-Nonlinear for nonlinear causal relationships and history-dependent noise. We demonstrate that our method surpasses state-of-the-art benchmarks in causal discovery tasks through extensive experimentation on synthetic and real-world datasets, particularly when the data emanates from diverse underlying causal graphs. Theoretically, we prove the identifiability of such a model under some mild assumptions. Implementation is available at [https://github.com/Rose-STL-Lab/MCD](https://github.com/Rose-STL-Lab/MCD).

Deep Learning · Sequential Models, Time series

Maksim Sinelnikov, Manuel Haussmann, Harri Lähdesmäki

Longitudinal data are important in numerous fields, such as healthcare, sociology and seismology, but real-world datasets present notable challenges for practitioners because they can be high-dimensional, contain structured missingness patterns, and measurement time points can be governed by an unknown stochastic process. While various solutions have been suggested, the majority of them have been designed to account for only one of these challenges. In this work, we propose a flexible and efficient latent-variable model that is capable of addressing all these limitations. Our approach utilizes Gaussian processes to capture correlations between samples and their associated missingness masks as well as to model the underlying point process. We construct our model as a variational autoencoder together with deep neural network parameterised decoder and encoder models, and develop a scalable amortised variational inference approach for efficient model training. We demonstrate competitive performance using both simulated and real datasets.

Probabilistic Methods · Variational Inference

Yash Patel, Declan McNamara, Jackson Loper, Jeffrey Regier, Ambuj Tewari

Amortized variational inference is an often employed framework in simulation-based inference that produces a posterior approximation that can be rapidly computed given any new observation. Unfortunately, there are few guarantees about the quality of these approximate posteriors. We propose Conformalized Amortized Neural Variational Inference (CANVI), a procedure that is scalable, easily implemented, and provides guaranteed marginal coverage. Given a collection of candidate amortized posterior approximators, CANVI constructs conformalized predictors based on each candidate, compares the predictors using a metric known as predictive efficiency, and returns the most efficient predictor. CANVI ensures that the resulting predictor constructs regions that contain the truth with a user-specified level of probability. CANVI is agnostic to design decisions in formulating the candidate approximators and only requires access to samples from the forward model, permitting its use in likelihood-free settings. We prove lower bounds on the predictive efficiency of the regions produced by CANVI and explore how the quality of a posterior approximation relates to the predictive efficiency of prediction regions based on that approximation. Finally, we demonstrate the accurate calibration and high predictive efficiency of CANVI on a suite of simulation-based inference benchmark tasks and an important scientific task: analyzing galaxy emission spectra.

Probabilistic Methods · Variational Inference

Xinlei Niu, Christian Walder, Jing Zhang, Charles Martin

We propose the stochastic optimal path which solves the classical optimal path problem by a probability-softening solution. This unified approach transforms a wide range of DP problems into directed acyclic graphs in which all paths follow a Gibbs distribution. We show the equivalence of the Gibbs distribution to a message-passing algorithm by the properties of the Gumbel distribution and give all the ingredients required for variational Bayesian inference of a latent path, namely Bayesian dynamic programming (BDP). We demonstrate the usage of BDP in the latent space of variational autoencoders (VAEs) and propose the BDP-VAE which captures structured sparse optimal paths as latent variables. This enables end-to-end training for generative tasks in which models rely on unobserved structural information. At last, we validate the behavior of our approach and showcase its applicability in two real-world applications: text-to-speech and singing voice synthesis. Our implementation code is available at https://github.com/XinleiNIU/LatentOptimalPathsBayesianDP.

Probabilistic Methods · Variational Inference

Yunfei Long, Zilin Tian, Liguo Zhang, Huosheng Xu

Mean-field variational inference (MFVI) methods provide computationally cheap approximations to the posterior of Bayesian Neural Networks (BNNs) when compared to alternatives like MCMC. However, applying MFVI to BNNs encounters limitations due to the Monte Carlo sampling problem. This problem stems from two main issues. *First*, most samples do not accurately represent the most probable weights. *Second*, random sampling from variational distributions introduces high variance in gradient estimates, which can hinder the optimization process, leading to slow convergence or even failure. In this paper, we introduce a novel sampling method called *Reparameterized Importance Sampling* (RIS) to estimate the first moment in neural networks, reducing variance during feed-forward propagation. We begin by analyzing the generalized form of the optimal proposal distribution and presenting an inexpensive approximation. Next, we describe the sampling process from the proposal distribution as a transformation that combines exogenous randomness with the variational parameters. Our experimental results demonstrate the effectiveness of the proposed RIS method in three critical aspects: improved convergence, enhanced predictive performance, and successful uncertainty estimation for out-of-distribution data.

Deep Learning · Self-Supervised Learning

Biao Liu, Ning Xu, Xiangyu Fang, Xin Geng

Semi-supervised multi-label learning (SSMLL) aims to address the challenge of limited labeled data availability in multi-label learning (MLL) by leveraging unlabeled data to improve the model's performance. Due to the difficulty of estimating the reliable label correlation on minimal multi-labeled data, previous SSMLL methods fail to unlash the power of the correlation among multiple labels to improve the performance of the predictive model in SSMLL. To deal with this problem, we propose a novel SSMLL method named PCLP where the correlation-induced label prior is inferred to enhance the pseudo-labeling instead of dirtily estimating the correlation among labels. Specifically, we construct the correlated label prior probability distribution using structural causal model (SCM), constraining the correlations of generated pseudo-labels to conform to the prior, which can be integrated into a variational label enhancement framework optimized by both labeled and unlabeled instances in a unified manner. Theoretically, we demonstrate the accuracy of the generated pseudo-labels and guarantee the learning consistency of the proposed method. Comprehensive experiments on several benchmark datasets have validated the superiority of the proposed method.

Probabilistic Methods · Gaussian Processes

Ying Li, Zhidi Lin, Feng Yin, Michael Minyi Zhang

Gaussian process latent variable models (GPLVMs) are a versatile family of unsupervised learning models commonly used for dimensionality reduction. However, common challenges in modeling data with GPLVMs include inadequate kernel flexibility and improper selection of the projection noise, leading to a type of model collapse characterized by vague latent representations that do not reflect the underlying data structure. This paper addresses these issues by, first, theoretically examining the impact of projection variance on model collapse through the lens of a linear GPLVM. Second, we tackle model collapse due to inadequate kernel flexibility by integrating the spectral mixture (SM) kernel and a differentiable random Fourier feature (RFF) kernel approximation, which ensures computational scalability and efficiency through off-the-shelf automatic differentiation tools for learning the kernel hyperparameters, projection variance, and latent representations within the variational inference framework. The proposed GPLVM, named *advised*RFLVM, is evaluated across diverse datasets and consistently outperforms various salient competing models, including state-of-the-art variational autoencoders (VAEs) and other GPLVM variants, in terms of informative latent representations and missing data imputation.

Probabilistic Methods · Variational Inference

Chengrui Li, Weihan Li, Yule Wang, Anqi Wu

The partially observable generalized linear model (POGLM) is a powerful tool for understanding neural connectivities under the assumption of existing hidden neurons. With spike trains only recorded from visible neurons, existing works use variational inference to learn POGLM meanwhile presenting the difficulty of learning this latent variable model. There are two main issues: (1) the sampled Poisson hidden spike count hinders the use of the pathwise gradient estimator in VI; and (2) the existing design of the variational model is neither expressive nor time-efficient, which further affects the performance. For (1), we propose a new differentiable POGLM, which enables the pathwise gradient estimator, better than the score function gradient estimator used in existing works. For (2), we propose the forward-backward message-passing sampling scheme for the variational model. Comprehensive experiments show that our differentiable POGLMs with our forward-backward message passing produce a better performance on one synthetic and two real-world datasets. Furthermore, our new method yields more interpretable parameters, underscoring its significance in neuroscience.