Masahiro Fujisawa

Futoshi Futami, Masahiro Fujisawa

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.

Takeshi Koshizuka, Masahiro Fujisawa, Yusuke Tanaka et al.

In this paper, we explores the expressivity and trainability of the Fourier Neural Operator (FNO). We establish a mean-field theory for the FNO, analyzing the behavior of the random FNO from an edge of chaos perspective. Our investigation into the expressivity of a random FNO involves examining the ordered-chaos phase transition of the network based on the weight distribution. This phase transition demonstrates characteristics unique to the FNO, induced by mode truncation, while also showcasing similarities to those of densely connected networks. Furthermore, we identify a connection between expressivity and trainability: the ordered and chaotic phases correspond to regions of vanishing and exploding gradients, respectively. This finding provides a practical prerequisite for the stable training of the FNO. Our experimental results corroborate our theoretical findings.

Masahiro Fujisawa, Masaki Adachi, Michael A. Osborne

Despite the importance of aligning language models with human preferences, crowd-sourced human feedback is often noisy -- for example, preferring less desirable responses -- posing a fundamental challenge to alignment. A truly robust alignment objective should yield identical model parameters even under severe label noise, a property known as redescending. We prove that no existing alignment methods satisfy this property. To address this, we propose Hölder-DPO, the first principled alignment loss with a provable redescending property, enabling estimation of the clean data distribution from noisy feedback. The aligned model estimates the likelihood of clean data, providing a theoretically grounded metric for dataset valuation that identifies the location and fraction of mislabels. This metric is gradient-free, enabling scalable and automated human feedback valuation without costly manual verification or clean validation dataset. Hölder-DPO achieves state-of-the-art robust alignment performance while accurately detecting mislabels in controlled datasets. Finally, applied to Anthropic HH-RLHF dataset, it reveals substantial noise levels and removing these mislabels significantly improves alignment performance across methods. The code is available at https://github.com/ma921/HolderDPO.

Futoshi Futami, Masahiro Fujisawa

Information-theoretic generalization bounds based on the supersample construction are a central tool for algorithm-dependent generalization analysis in the batch i.i.d.~setting. However, existing supersample conditional mutual information (CMI) bounds do not directly apply to sequential decision-making problems such as online learning, streaming active learning, and bandits, where data are revealed adaptively and the learner evolves along a causal trajectory. To address this limitation, we develop a sequential supersample framework that separates the learner filtration from a proof-side enlargement used for ghost-coordinate comparisons. Under a row-wise exchangeability assumption, the sequential generalization gap is controlled by sequential CMI, a sum of roundwise selector--loss information terms. We also establish a Bernstein-type refinement that yields faster rates under suitable variance conditions. The selector-SCMI proof strategy applies to online learning, streaming active learning with importance weighting, and stochastic multi-armed bandits.

Yuanchao Xu, Fengyi Li, Masahiro Fujisawa et al.

We propose Koopman Spectral Wasserstein Gradient Descent (KSWGD), a generative modeling framework that combines operator-theoretic spectral analysis with optimal transport. The novel insight is that the spectral structure required for accelerated Wasserstein gradient descent can be directly estimated from trajectory data via Koopman operator approximation which can eliminate the need for explicit knowledge of the target potential or neural network training. We provide rigorous convergence analysis and establish connection to Feynman-Kac theory that clarifies the method's probabilistic foundation. Experiments across diverse settings, including compact manifold sampling, metastable multi-well systems, image generation, and high dimensional stochastic partial differential equation, demonstrate that KSWGD consistently achieves faster convergence than other existing methods while maintaining high sample quality.

Masahiro Fujisawa, Futoshi Futami

Calibration of predicted probabilities is critical for reliable machine learning, yet it is poorly understood how standard training procedures yield well-calibrated models. This work provides the first theoretical proof that canonical $L_{2}$-regularized empirical risk minimization directly controls the smooth calibration error (smCE) without post-hoc correction or specialized calibration-promoting regularizer. We establish finite-sample generalization bounds for smCE based on optimization error, regularization strength, and the Rademacher complexity. We then instantiate this theory for models in reproducing kernel Hilbert spaces, deriving concrete guarantees for kernel ridge and logistic regression. Our experiments confirm these specific guarantees, demonstrating that $L_{2}$-regularized ERM can provide a well-calibrated model without boosting or post-hoc recalibration. The source code to reproduce all experiments is available at https://github.com/msfuji0211/erm_calibration.

Futoshi Futami, Masahiro Fujisawa

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.

Masaki Adachi, Masahiro Fujisawa, Michael A Osborne

Despite the significance of probabilistic time-series forecasting models, their evaluation metrics often involve intractable integrations. The most widely used metric, the continuous ranked probability score (CRPS), is a strictly proper scoring function; however, its computation requires approximation. We found that popular CRPS estimators--specifically, the quantile-based estimator implemented in the widely used GluonTS library and the probability-weighted moment approximation--both exhibit inherent estimation biases. These biases lead to crude approximations, resulting in improper rankings of forecasting model performance when CRPS values are close. To address this issue, we introduced a kernel quadrature approach that leverages an unbiased CRPS estimator and employs cubature construction for scalable computation. Empirically, our approach consistently outperforms the two widely used CRPS estimators.

Futoshi Futami, Masahiro Fujisawa

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.

Masahiro Fujisawa, Futoshi Futami

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.

Masahiro Fujisawa, Takeshi Teshima, Issei Sato et al.

Approximate Bayesian computation (ABC) is a likelihood-free inference method that has been employed in various applications. However, ABC can be sensitive to outliers if a data discrepancy measure is chosen inappropriately. In this paper, we propose to use a nearest-neighbor-based $γ$-divergence estimator as a data discrepancy measure. We show that our estimator possesses a suitable theoretical robustness property called the redescending property. In addition, our estimator enjoys various desirable properties such as high flexibility, asymptotic unbiasedness, almost sure convergence, and linear-time computational complexity. Through experiments, we demonstrate that our method achieves significantly higher robustness than existing discrepancy measures.

Masahiro Fujisawa, Issei Sato

We propose a variance reduction framework for variational inference using the Multilevel Monte Carlo (MLMC) method. Our framework is built on reparameterized gradient estimators and "recycles" parameters obtained from past update history in optimization. In addition, our framework provides a new optimization algorithm based on stochastic gradient descent (SGD) that adaptively estimates the sample size used for gradient estimation according to the ratio of the gradient variance. We theoretically show that, with our method, the variance of the gradient estimator decreases as optimization proceeds and that a learning rate scheduler function helps improve the convergence. We also show that, in terms of the \textit{signal-to-noise} ratio, our method can improve the quality of gradient estimation by the learning rate scheduler function without increasing the initial sample size. Finally, we confirm that our method achieves faster convergence and reduces the variance of the gradient estimator compared with other methods through experimental comparisons with baseline methods using several benchmark datasets.