This work addresses a key limitation in current federated learning approaches, which predominantly focus on homogeneous tasks, neglecting the task diversity on local devices. We propose a principled integration of multi-task learning using multi-output Gaussian processes (MOGP) at the local level and federated learning at the global level. MOGP handles correlated classification and regression tasks, offering a Bayesian non-parametric approach that naturally quantifies uncertainty. The central server aggregates the posteriors from local devices, updating a global MOGP prior redistributed for training local models until convergence. Challenges in performing posterior inference on local devices are addressed through the P\'{o}lya-Gamma augmentation technique and mean-field variational inference, enhancing computational efficiency and convergence rate. Experimental results on both synthetic and real data demonstrate superior predictive performance, OOD detection, uncertainty calibration and convergence rate, highlighting the method's potential in diverse applications. Our code is publicly available at https://github.com/JunliangLv/task_diversity_BFL.
With the advancement of pre-trained vision-language (VL) models, enhancing the alignment between visual and linguistic modalities in downstream tasks has emerged as a critical challenge. Different from existing fine-tuning methods that add extra modules to these two modalities, we investigate whether the frozen model can be fine-tuned by customized noise. Our approach is motivated by the scientific study of beneficial noise, namely Positive-incentive Noise (Pi-noise or $\pi$-noise) , which quantitatively analyzes the impact of noise. It therefore implies a new scheme to learn beneficial noise distribution that can be employed to fine-tune VL models. Focusing on few-shot classification tasks based on CLIP, we reformulate the inference process of CLIP and apply variational inference, demonstrating how to generate $\pi$-noise towards visual and linguistic modalities. Then, we propose Positive-incentive Noise Injector (PiNI), which can fine-tune CLIP via injecting noise into both visual and text encoders. Since the proposed method can learn the distribution of beneficial noise, we can obtain more diverse embeddings of vision and language to better align these two modalities for specific downstream tasks within limited computational resources. We evaluate different noise incorporation approaches and network architectures of PiNI. The evaluation across 11 datasets demonstrates its effectiveness.
We explore the use of Residual Vector Quantization (RVQ) for high-fidelity generation in vector-quantized generative models. This quantization technique maintains higher data fidelity by employing more in-depth tokens. However, increasing the token number in generative models leads to slower inference speeds. To this end, we introduce ResGen, an efficient RVQ-based discrete diffusion model that generates high-fidelity samples without compromising sampling speed. Our key idea is a direct prediction of vector embedding of collective tokens rather than individual ones. Moreover, we demonstrate that our proposed token masking and multi-token prediction method can be formulated within a principled probabilistic framework using a discrete diffusion process and variational inference. We validate the efficacy and generalizability of the proposed method on two challenging tasks across different modalities: conditional image generation} on ImageNet 256x256 and zero-shot text-to-speech synthesis. Experimental results demonstrate that ResGen outperforms autoregressive counterparts in both tasks, delivering superior performance without compromising sampling speed. Furthermore, as we scale the depth of RVQ, our generative models exhibit enhanced generation fidelity or faster sampling speeds compared to similarly sized baseline models. The project page can be found at https://resgen-genai.github.io
Large scale Bayesian nonparametrics (BNP) learner such as stochastic variational inference (SVI) can handle datasets with large class number and large training size at fractional cost. Like its predecessor, SVI rely on the assumption of conjugate variational posterior to approximate the true posterior. A more challenging problem is to consider large scale learning on non-conjugate posterior. Recent works in this direction are mostly associated with using Monte Carlo methods for approximating the learner. However, these works are usually demonstrated on non-BNP related task and less complex models such as logistic regression, due to higher computational complexity. In order to overcome the issue faced by SVI, we develop a novel approach based on the recently proposed variational maximization-maximization (VMM) learner to allow large scale learning on non-conjugate posterior. Unlike SVI, our VMM learner does not require closed-form expression for the variational posterior expectatations. Our only requirement is that the variational posterior is differentiable. In order to ensure convergence in stochastic settings, SVI rely on decaying step-sizes to slow its learning. Inspired by SVI and Adam, we propose the novel use of decaying step-sizes on both gradient and ascent direction in our VMM to significantly improve its learning. We show that our proposed methods is compatible with ResNet features when applied to large class number datasets such as MIT67 and SUN397. Finally, we compare our proposed learner with several recent works such as deep clustering algorithms and showed we were able to produce on par or outperform the state-of-the-art methods in terms of clustering measures.
Normalizing flow-based variational inference (flow VI) is a promising approximate inference approach, but its performance remains inconsistent across studies. Numerous algorithmic choices influence flow VI's performance. We conduct a step-by-step analysis to disentangle the impact of some of the key factors: capacity, objectives, gradient estimators, number of gradient estimates (batchsize), and step-sizes. Each step examines one factor while neutralizing others using insights from the previous steps and/or using extensive parallel computation. To facilitate high-fidelity evaluation, we curate a benchmark of synthetic targets that represent common posterior pathologies and allow for exact sampling. We provide specific recommendations for different factors and propose a flow VI recipe that matches or surpasses leading turnkey Hamiltonian Monte Carlo (HMC) methods.
Estimating the uncertainty of responses from Large Language Models (LLMs) remains a critical challenge. While recent Bayesian methods have demonstrated effectiveness in quantifying uncertainty through low-rank weight updates, they typically require complex fine-tuning or post-training procedures. In this paper, we propose Training-Free Bayesianization (TFB), a simple yet theoretically grounded framework that efficiently transforms trained low-rank adapters into Bayesian ones without additional training. TFB systematically searches for the maximally acceptable level of variance in the weight posterior, constrained within a family of low-rank isotropic Gaussian distributions. Our theoretical analysis shows that under mild conditions, this search process is equivalent to KL-regularized variational optimization, a generalized form of variational inference. Through comprehensive experiments, we show that TFB achieves superior uncertainty estimation and generalization compared to existing methods while eliminating the need for complex Bayesianization training procedures. Code will be available at https://github.com/Wang-ML-Lab/bayesian-peft.
Clustering based on vibration responses, such as transmissibility functions (TFs), is promising in structural anomaly detection, but most existing approaches struggle with determining the optimal cluster number and handling high-dimensional streaming data, while their shallow structures also make them sensitive to manually-engineered feature quality. To bridge this gap, this work proposes the Dirichlet process-deep generative model-integrated incremental learning (DPGIIL) for clustering by combining the advantages of deep generative models (DGMs) in representation learning and the Dirichlet process mixture model (DPMM) in identifying distinct patterns in observed data. By introducing a DPMM prior into the latent space of DGMs, DPGIIL automatically captures dissimilarities in extracted latent representations, enabling both generative modeling and clustering. Within the context of variational Bayesian inference, a lower bound on the log marginal likelihood of DPGIIL, tighter than the evidence lower bound given sufficient training data, is derived analytically, which enables the joint optimization of DGM and DPMM parameters, thereby allowing the DPMM to regularize the DGM's feature extraction process. Additionally, a greedy split-merge scheme-based coordinate ascent variational inference method is devised to accelerate the optimization. The summary statistics of the DPMM, along with the network parameters, are used to retain information about previous data for incremental learning. Notably, this study uses variational autoencoder (VAE) within DPGIIL as an illustrative example, while this framework is adaptable to other DGMs. Two case studies show that the proposed method outperforms some state-of-the-art approaches in structural anomaly detection and clustering, while also dynamically generating new clusters to indicate the emergence of new structural conditions for online monitoring.
Recently, there has been an increasing interest in performing post-hoc uncertainty estimation about the predictions of pre-trained deep neural networks (DNNs). Given a pre-trained DNN via back-propagation, these methods enhance the original network by adding output confidence measures, such as error bars, without compromising its initial accuracy. In this context, we introduce a novel family of sparse variational Gaussian processes (GPs), where the posterior mean is fixed to any continuous function when using a universal kernel. Specifically, we fix the mean of this GP to the output of the pre-trained DNN, allowing our approach to effectively fit the GP's predictive variances to estimate the DNN prediction uncertainty. Our approach leverages variational inference (VI) for efficient stochastic optimization, with training costs that remain independent of the number of training points, scaling efficiently to large datasets such as ImageNet. The proposed method, called fixed mean GP (FMGP), is architecture-agnostic, relying solely on the pre-trained model's outputs to adjust the predictive variances. Experimental results demonstrate that FMGP improves both uncertainty estimation and computational efficiency when compared to state-of-the-art methods.
Tensor decompositions play a crucial role in numerous applications related to multi-way data analysis. By employing a Bayesian framework with sparsity-inducing priors, Bayesian Tensor Ring (BTR) factorization offers probabilistic estimates and an effective approach for automatically adapting the tensor ring rank during the learning process. However, previous BTR method employs an Automatic Relevance Determination (ARD) prior, which can lead to sub-optimal solutions. Besides, it solely focuses on continuous data, whereas many applications involve discrete data. More importantly, it relies on the Coordinate-Ascent Variational Inference (CAVI) algorithm, which is inadequate for handling large tensors with extensive observations. These limitations greatly limit its application scales and scopes, making it suitable only for small-scale problems, such as image/video completion. To address these issues, we propose a novel BTR model that incorporates a nonparametric Multiplicative Gamma Process (MGP) prior, known for its superior accuracy in identifying latent structures. To handle discrete data, we introduce the P\'olya-Gamma augmentation for closed-form updates. Furthermore, we develop an efficient Gibbs sampler for consistent posterior simulation, which reduces the computational complexity of previous VI algorithm by two orders, and an online EM algorithm that is scalable to extremely large tensors. To showcase the advantages of our model, we conduct extensive experiments on both simulation data and real-world applications.
Diffusion probabilistic models learn to remove noise that is artificially added to the data during training. Novel data, like images, may then be generated from Gaussian noise through a sequence of denoising operations. While this Markov process implicitly defines a joint distribution over noise-free data, it is not simple to condition the generative process on masked or partial images. A number of heuristic sampling procedures have been proposed for solving inverse problems with diffusion priors, but these approaches do not directly approximate the true conditional distribution imposed by inference queries, and are often ineffective for large masked regions. Moreover, many of these baselines cannot be applied to latent diffusion models which use image encodings for efficiency. We instead develop a hierarchical variational inference algorithm that analytically marginalizes missing features, and uses a rigorous variational bound to optimize a non-Gaussian Markov approximation of the true diffusion posterior. Through extensive experiments with both pixel-based and latent diffusion models of images, we show that our VIPaint method significantly outperforms previous approaches in both the plausibility and diversity of imputations, and is easily generalized to other inverse problems like deblurring and superresolution.
This tutorial investigates the convergence of statistical mechanics and learning theory, elucidating the potential enhancements in machine learning methodologies through the integration of foundational principles from physics. The tutorial delves into advanced techniques like entropy, free energy, and variational inference which are utilized in machine learning, illustrating their significant contributions to model efficiency and robustness. By bridging these scientific disciplines, we aspire to inspire newer methodologies in researches, demonstrating how an in-depth comprehension of physical systems' behavior can yield more effective and dependable machine learning models, particularly in contexts characterized by uncertainty.
Variational empirical Bayes (VEB) methods provide a practically attractive approach to fitting large, sparse, multiple regression models. These methods usually use coordinate ascent to optimize the variational objective function, an approach known as coordinate ascent variational inference (CAVI). Here we propose alternative optimization approaches based on gradient-based (quasi-Newton) methods, which we call gradient-based variational inference (GradVI). GradVI exploits a recent result from Kim et. al. [arXiv:2208.10910] which writes the VEB regression objective function as a penalized regression. Unfortunately the penalty function is not available in closed form, and we present and compare two approaches to dealing with this problem. In simple situations where CAVI performs well, we show that GradVI produces similar predictive performance, and GradVI converges in fewer iterations when the predictors are highly correlated. Furthermore, unlike CAVI, the key computations in GradVI are simple matrix-vector products, and so GradVI is much faster than CAVI in settings where the design matrix admits fast matrix-vector products (e.g., as we show here, trendfiltering applications) and lends itself to parallelized implementations in ways that CAVI does not. GradVI is also very flexible, and could exploit automatic differentiation to easily implement different prior families. Our methods are implemented in an open-source Python software, GradVI (available from https://github.com/stephenslab/gradvi ).
We present a generative modeling approach based on the variational inference framework for likelihood-free simulation-based inference. The method leverages latent variables within variational autoencoders to efficiently estimate complex posterior distributions arising from stochastic simulations. We explore two variations of this approach distinguished by their treatment of the prior distribution. The first model adapts the prior based on observed data using a multivariate prior network, enhancing generalization across various posterior queries. In contrast, the second model utilizes a standard Gaussian prior, offering simplicity while still effectively capturing complex posterior distributions. We demonstrate the efficacy of these models on well-established benchmark problems, achieving results comparable to flow-based approaches while maintaining computational efficiency and scalability.
We propose a Bayesian neural network-based continual learning algorithm using Variational Inference, aiming to overcome several drawbacks of existing methods. Specifically, in continual learning scenarios, storing network parameters at each step to retain knowledge poses challenges. This is compounded by the crucial need to mitigate catastrophic forgetting, particularly given the limited access to past datasets, which complicates maintaining correspondence between network parameters and datasets across all sessions. Current methods using Variational Inference with KL divergence risk catastrophic forgetting during uncertain node updates and coupled disruptions in certain nodes. To address these challenges, we propose the following strategies. To reduce the storage of the dense layer parameters, we propose a parameter distribution learning method that significantly reduces the storage requirements. In the continual learning framework employing variational inference, our study introduces a regularization term that specifically targets the dynamics and population of the mean and variance of the parameters. This term aims to retain the benefits of KL divergence while addressing related challenges. To ensure proper correspondence between network parameters and the data, our method introduces an importance-weighted Evidence Lower Bound term to capture data and parameter correlations. This enables storage of common and distinctive parameter hyperspace bases. The proposed method partitions the parameter space into common and distinctive subspaces, with conditions for effective backward and forward knowledge transfer, elucidating the network-parameter dataset correspondence. The experimental results demonstrate the effectiveness of our method across diverse datasets and various combinations of sequential datasets, yielding superior performance compared to existing approaches.
We present LazyDINO, a transport map variational inference method for fast, scalable, and efficiently amortized solutions of high-dimensional nonlinear Bayesian inverse problems with expensive parameter-to-observable (PtO) maps. Our method consists of an offline phase in which we construct a derivative-informed neural surrogate of the PtO map using joint samples of the PtO map and its Jacobian. During the online phase, when given observational data, we seek rapid posterior approximation using surrogate-driven training of a lazy map [Brennan et al., NeurIPS, (2020)], i.e., a structure-exploiting transport map with low-dimensional nonlinearity. The trained lazy map then produces approximate posterior samples or density evaluations. Our surrogate construction is optimized for amortized Bayesian inversion using lazy map variational inference. We show that (i) the derivative-based reduced basis architecture [O'Leary-Roseberry et al., Comput. Methods Appl. Mech. Eng., 388 (2022)] minimizes the upper bound on the expected error in surrogate posterior approximation, and (ii) the derivative-informed training formulation [O'Leary-Roseberry et al., J. Comput. Phys., 496 (2024)] minimizes the expected error due to surrogate-driven transport map optimization. Our numerical results demonstrate that LazyDINO is highly efficient in cost amortization for Bayesian inversion. We observe one to two orders of magnitude reduction of offline cost for accurate posterior approximation, compared to simulation-based amortized inference via conditional transport and conventional surrogate-driven transport. In particular, LazyDINO outperforms Laplace approximation consistently using fewer than 1000 offline samples, while other amortized inference methods struggle and sometimes fail at 16,000 offline samples.
Trajectory prediction for multi-agents in complex scenarios is crucial for applications like autonomous driving. However, existing methods often overlook environmental biases, which leads to poor generalization. Additionally, hardware constraints limit the use of large-scale data across environments, and continual learning settings exacerbate the challenge of catastrophic forgetting. To address these issues, we propose the Continual Causal Intervention (C$^{2}$INet) method for generalizable multi-agent trajectory prediction within a continual learning framework. Using variational inference, we align environment-related prior with posterior estimator of confounding factors in the latent space, thereby intervening in causal correlations that affect trajectory representation. Furthermore, we store optimal variational priors across various scenarios using a memory queue, ensuring continuous debiasing during incremental task training. The proposed C$^{2}$INet enhances adaptability to diverse tasks while preserving previous task information to prevent catastrophic forgetting. It also incorporates pruning strategies to mitigate overfitting. Comparative evaluations on three real and synthetic complex datasets against state-of-the-art methods demonstrate that our proposed method consistently achieves reliable prediction performance, effectively mitigating confounding factors unique to different scenarios. This highlights the practical value of our method for real-world applications.
Despite the dominant role of deep models in machine learning, limitations persist, including overconfident predictions, susceptibility to adversarial attacks, and underestimation of variability in predictions. The Bayesian paradigm provides a natural framework to overcome such issues and has become the gold standard for uncertainty estimation with deep models, also providing improved accuracy and a framework for tuning critical hyperparameters. However, exact Bayesian inference is challenging, typically involving variational algorithms that impose strong independence and distributional assumptions. Moreover, existing methods are sensitive to the architectural choice of the network. We address these issues by focusing on a stochastic relaxation of the standard feed-forward rectified neural network and using sparsity-promoting priors on the weights of the neural network for increased robustness to architectural design. Thanks to Polya-Gamma data augmentation tricks, which render a conditionally linear and Gaussian model, we derive a fast, approximate variational inference algorithm that avoids distributional assumptions and independence across layers. Suitable strategies to further improve scalability and account for multimodality are considered.
Model-based reinforcement learning refers to a set of approaches capable of sample-efficient decision making, which create an explicit model of the environment. This model can subsequently be used for learning optimal policies. In this paper, we propose a temporal Gaussian Mixture Model composed of a perception model and a transition model. The perception model extracts discrete (latent) states from continuous observations using a variational Gaussian mixture likelihood. Importantly, our model constantly monitors the collected data searching for new Gaussian components, i.e., the perception model performs a form of structure learning (Smith et al., 2020; Friston et al., 2018; Neacsu et al., 2022) as it learns the number of Gaussian components in the mixture. Additionally, the transition model learns the temporal transition between consecutive time steps by taking advantage of the Dirichlet-categorical conjugacy. Both the perception and transition models are able to forget part of the data points, while integrating the information they provide within the prior, which ensure fast variational inference. Finally, decision making is performed with a variant of Q-learning which is able to learn Q-values from beliefs over states. Empirically, we have demonstrated the model's ability to learn the structure of several mazes: the model discovered the number of states and the transition probabilities between these states. Moreover, using its learned Q-values, the agent was able to successfully navigate from the starting position to the maze's exit.
This work addresses the fundamental linear inverse problem in compressive sensing (CS) by introducing a new type of regularizing generative prior. Our proposed method utilizes ideas from classical dictionary-based CS and, in particular, sparse Bayesian learning (SBL), to integrate a strong regularization towards sparse solutions. At the same time, by leveraging the notion of conditional Gaussianity, it also incorporates the adaptability from generative models to training data. However, unlike most state-of-the-art generative models, it is able to learn from a few compressed and noisy data samples and requires no optimization algorithm for solving the inverse problem. Additionally, similar to Dirichlet prior networks, our model parameterizes a conjugate prior enabling its application for uncertainty quantification. We support our approach theoretically through the concept of variational inference and validate it empirically using different types of compressible signals.
Bayes' rule naturally allows for inference refinement in a streaming fashion, without the need to recompute posteriors from scratch whenever new data arrives. In principle, Bayesian streaming is straightforward: we update our prior with the available data and use the resulting posterior as a prior when processing the next data chunk. In practice, however, this recipe entails i) approximating an intractable posterior at each time step; and ii) encapsulating results appropriately to allow for posterior propagation. For continuous state spaces, variational inference (VI) is particularly convenient due to its scalability and the tractability of variational posteriors. For discrete state spaces, however, state-of-the-art VI results in analytically intractable approximations that are ill-suited for streaming settings. To enable streaming Bayesian inference over discrete parameter spaces, we propose streaming Bayes GFlowNets (abbreviated as SB-GFlowNets) by leveraging the recently proposed GFlowNets -- a powerful class of amortized samplers for discrete compositional objects. Notably, SB-GFlowNet approximates the initial posterior using a standard GFlowNet and subsequently updates it using a tailored procedure that requires only the newly observed data. Our case studies in linear preference learning and phylogenetic inference showcase the effectiveness of SB-GFlowNets in sampling from an unnormalized posterior in a streaming setting. As expected, we also observe that SB-GFlowNets is significantly faster than repeatedly training a GFlowNet from scratch to sample from the full posterior.