Bayesian inference is widely used in practice due to its ability to assess epistemic uncertainty (EU) in predictions. However, its computational complexity necessitates the use of approximation methods, such as variational inference (VI). When estimating EU within the VI framework, metrics such as the variance of the posterior predictive distribution and conditional mutual information are commonly employed. Despite their practical importance, these metrics lack comprehensive theoretical analysis. In this paper, we investigate these EU metrics by providing their novel relationship to excess risk, which allows for a convergence analysis based on PAC-Bayesian theory. Based on these analyses, we then demonstrate that some existing objective functions of VI regularize EU metrics in different ways leading to different performance in EU evaluation. Finally, we propose a novel objective function for VI that directly optimizes both prediction and EU under the PAC-Bayesian framework. Experimental results indicate that our algorithm significantly improves EU estimation compared to existing VI methods.

Nonparametric estimation using uniform-width binning is a standard approach for evaluating the calibration performance of machine learning models. However, existing theoretical analyses of the bias induced by binning are limited to binary classification, creating a significant gap with practical applications such as multiclass classification. Additionally, many parametric recalibration algorithms lack theoretical guarantees for their generalization performance. To address these issues, we conduct a generalization analysis of calibration error using the probably approximately correct Bayes framework. This approach enables us to derive the first optimizable upper bound for generalization error in the calibration context. On the basis of our theory, we propose a generalization-aware recalibration algorithm. Numerical experiments show that our algorithm enhances the performance of Gaussian process-based recalibration across various benchmark datasets and models.

Latent variables (LVs) play a crucial role in encoder-decoder models by enabling effective data compression, prediction, and generation. Although their theoretical properties, such as generalization, have been extensively studied in supervised learning, similar analyses for unsupervised models such as variational autoencoders (VAEs) remain insufficiently underexplored. In this work, we extend information-theoretic generalization analysis to vector-quantized (VQ) VAEs with discrete latent spaces, introducing a novel data-dependent prior to rigorously analyze the relationship among LVs, generalization, and data generation. We derive a novel generalization error bound of the reconstruction loss of VQ-VAEs, which depends solely on the complexity of LVs and the encoder, independent of the decoder. Additionally, we provide the upper bound of the 2-Wasserstein distance between the distributions of the true data and the generated data, explaining how the regularization of the LVs contributes to the data generation performance.

Bayesian inference is often used to quantify uncertainty. Several recent analyses have rigorously decomposed uncertainty in prediction by Bayesian inference into two types: the inherent randomness in the data generation process and the variability due to lack of data respectively. Existing studies have analyzed these uncertainties from an information-theoretic perspective, assuming the model is well-specified and treating the model parameters as latent variables. However, such information-theoretic uncertainty analysis fails to account for a widely believed property of uncertainty known as sensitivity between test and training data. This means that if the test data is similar to the training data in some sense, the uncertainty will be smaller. In this study, we study such sensitivity using a new decomposition of uncertainty. Our analysis successfully defines such sensitivity using information-theoretic quantities. Furthermore, we extend the existing analysis of Bayesian meta-learning and show the novel sensitivities among tasks for the first time.

While the expected calibration error (ECE), which employs binning, is widely adopted to evaluate the calibration performance of machine learning models, theoretical understanding of its estimation bias is limited. In this paper, we present the first comprehensive analysis of the estimation bias in the two common binning strategies, uniform mass and uniform width binning. Our analysis establishes upper bounds on the bias, achieving an improved convergence rate. Moreover, our bounds reveal, for the first time, the optimal number of bins to minimize the estimation bias. We further extend our bias analysis to generalization error analysis based on the information-theoretic approach, deriving upper bounds that enable the numerical evaluation of how small the ECE is for unknown data. Experiments using deep learning models show that our bounds are nonvacuous thanks to this information-theoretic generalization analysis approach.

We provide novel information-theoretic generalization bounds for stochastic gradient Langevin dynamics (SGLD) under the assumptions of smoothness and dissipativity, which are widely used in sampling and non-convex optimization studies. Our bounds are time-independent and decay to zero as the sample size increases, regardless of the number of iterations and whether the step size is fixed. Unlike previous studies, we derive the generalization error bounds by focusing on the time evolution of the Kullback--Leibler divergence, which is related to the stability of datasets and is the upper bound of the mutual information between output parameters and an input dataset. Additionally, we establish the first information-theoretic generalization bound when the training and test loss are the same by showing that a loss function of SGLD is sub-exponential. This bound is also time-independent and removes the problematic step size dependence in existing work, leading to an improved excess risk bound by combining our analysis with the existing non-convex optimization error bounds.

Since the Bayesian inference works poorly under model misspecification, various solutions have been explored to counteract the shortcomings. Recently proposed predictive Bayes (PB) that directly optimizes the Kullback Leibler divergence between the empirical distribution and the approximate predictive distribution shows excellent performances not only under model misspecification but also for over-parametrized models. However, its behavior and superiority are still unclear, which limits the applications of PB. Specifically, the superiority of PB has been shown only in terms of the predictive test log-likelihood and the performance in the sense of parameter estimation has not been investigated yet. Also, it is not clear why PB is superior with misspecified and over-parameterized models. In this paper, we clarify these ambiguities by studying PB in the framework of risk-seeking optimization. To achieve this, first, we provide a consistency theory for PB and then present intuition of robustness of PB to model misspecification using a response function theory. Thereafter, we theoretically and numerically show that PB has an implicit regularization effect that leads to flat local minima in over-parametrized models.

In many scientific fields, various phenomena are modeled by ordinary differential equations (ODEs). Parameters in ODEs are generally unknown and hard to measure directly. Since analytical solutions for ODEs can rarely be obtained, statistical methods are often used to infer parameters from experimental observations. Among many existing methods, Gaussian process-based gradient matching has been explored extensively. However, the existing method cannot be scaled to a massive dataset. Given $N$ data points, existing algorithms show $\mathcal{O}(N^3)$ computational cost. In this paper, we propose a novel algorithm using the state space reformulation of Gaussian processes. More specifically, we reformulate Gaussian process gradient matching as a special state-space model problem, then approximate its posterior distribution by a novel Rao-Blackwellization filtering, which enjoys $\mathcal{O}(N)$ computational cost. Moreover, our algorithm is expressed as closed forms, it is 1000 times more faster than existing methods measured in wall clock time.

Recently, many methods for optimization and sampling have been developed by designing continuous dynamics followed by discretization. The dynamics that have been used for optimization have their corresponding underlying functionals to be minimized. On the other hand, a wider class of dynamics have been studied for sampling, which is not necessarily limited to functional minimization. For example, dynamics perturbed with skew-symmetric matrices, which cannot be seen as minimization of functionals, have been widely used to reduce asymptotic variance. Following this success in sampling, exploring such perturbed dynamics in the context of optimization can open a new avenue to optimization algorithm design. In this work, we introduce a perturbation technique for sampling into optimization for strongly convex functions. We show that perturbation applied to the gradient flow yields rapid convergence in optimization for strongly convex functions. Based on this continuous dynamics, we propose an optimization algorithm for strongly convex functions with a novel discretization framework that combines the Euler method with the leapfrog method which is used in the Hamilton Monte Carlo method. Our numerical experiments show that the perturbation technique is useful for optimization.

Bayesian model averaging, obtained as the expectation of a likelihood function by a posterior distribution, has been widely used for prediction, evaluation of uncertainty, and model selection. Various approaches have been developed to efficiently capture the information in the posterior distribution; one such approach is the optimization of a set of models simultaneously with interaction to ensure the diversity of the individual models in the same way as ensemble learning. A representative approach is particle variational inference (PVI), which uses an ensemble of models as an empirical approximation for the posterior distribution. PVI iteratively updates each model with a repulsion force to ensure the diversity of the optimized models. However, despite its promising performance, a theoretical understanding of this repulsion and its association with the generalization ability remains unclear. In this paper, we tackle this problem in light of PAC-Bayesian analysis. First, we provide a new second-order Jensen inequality, which has the repulsion term based on the loss function. Thanks to the repulsion term, it is tighter than the standard Jensen inequality. Then, we derive a novel generalization error bound and show that it can be reduced by enhancing the diversity of models. Finally, we derive a new PVI that optimizes the generalization error bound directly. Numerical experiments demonstrate that the performance of the proposed PVI compares favorably with existing methods in the experiment.

Posterior distribution approximation is a central task in Bayesian inference. Stochastic gradient Langevin dynamics (SGLD) and its extensions have been practically used and theoretically studied. While SGLD updates a single particle at a time, ensemble methods that update multiple particles simultaneously have been recently gathering attention. Compared with the naive parallel-chain SGLD that updates multiple particles independently, ensemble methods update particles with their interactions. Thus, these methods are expected to be more particle-efficient than the naive parallel-chain SGLD because particles can be aware of other particles’ behavior through their interactions. Although ensemble methods numerically demonstrated their superior performance, no theoretical guarantee exists to assure such particle-efficiency and it is unclear whether those ensemble methods are really superior to the naive parallel-chain SGLD in the non-asymptotic settings. To cope with this problem, we propose a novel ensemble method that uses a non-reversible Markov chain for the interaction, and we present a non-asymptotic theoretical analysis for our method. Our analysis shows that, for the first time, the interaction causes a faster convergence rate than the naive parallel-chain SGLD in the non-asymptotic setting if the discretization error is appropriately controlled. Numerical experiments show that we can control the discretization error by tuning the interaction appropriately.

In Bayesian inference, the posterior distributions are difficult to obtain analytically for complex models such as neural networks. Variational inference usually uses a parametric distribution for approximation, from which we can easily draw samples. Recently discrete approximation by particles has attracted attention because of its high expression ability. An example is Stein variational gradient descent (SVGD), which iteratively optimizes particles. Although SVGD has been shown to be computationally efficient empirically, its theoretical properties have not been clarified yet and no finite sample bound of the convergence rate is known. Another example is the Stein points (SP) method, which minimizes kernelized Stein discrepancy directly. Althoughafinitesampleboundisassuredtheoretically, SP is computationally inefficient empirically, especially in high-dimensional problems. In this paper, we propose a novel method named maximum mean discrepancy minimization by the Frank-Wolfe algorithm (MMD-FW), which minimizes MMD in a greedy way by the FW algorithm. Our method is computationally efficient empirically and we show that its finite sample convergence bound is in a linear order in finite dimensions.

Robustness to outliers is a central issue in real-world machine learning applications. While replacing a model to a heavy-tailed one (e.g., from Gaussian to Student-t) is a standard approach for robustification, it can only be applied to simple models. In this paper, based on Zellner’s optimization and variational formulation of Bayesian inference, we propose an outlier-robust pseudo-Bayesian variational method by replacing the Kullback-Leibler divergence used for data fitting to a robust divergence such as the beta- and gamma-divergences. An advantage of our approach is that superior but complex models such as deep networks can also be handled. We theoretically prove that, for deep networks with ReLU activation functions, the influence function in our proposed method is bounded, while it is unbounded in the ordinary variational inference. This implies that our proposed method is robust to both of input and output outliers, while the ordinary variational method is not. We experimentally demonstrate that our robust variational method outperforms ordinary variational inference in regression and classification with deep networks.

Exponential family distributions are highly useful in machine learning since their calculation can be performed efficiently through natural parameters. The exponential family has recently been extended to the t-exponential family, which contains Student-t distributions as family members and thus allows us to handle noisy data well. However, since the t-exponential family is defined by the deformed exponential, an efficient learning algorithm for the t-exponential family such as expectation propagation (EP) cannot be derived in the same way as the ordinary exponential family. In this paper, we borrow the mathematical tools of q-algebra from statistical physics and show that the pseudo additivity of distributions allows us to perform calculation of t-exponential family distributions through natural parameters. We then develop an expectation propagation (EP) algorithm for the t-exponential family, which provides a deterministic approximation to the posterior or predictive distribution with simple moment matching. We finally apply the proposed EP algorithm to the Bayes point machine and Student-t process classification, and demonstrate their performance numerically.