Information Geometry Loss for Time Series Forecasting

Jiayu Fang, Xuande Liu, Sangsha Fang, Ernie Tian, Hongwei Ma, Zhiqi Shao, Junbin Gao
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:29256-29283, 2026.

Abstract

Time series forecasting fundamentally involves learning probability distributions over future observations. However, existing loss functions rely on point-wise Euclidean metrics, neglecting the intrinsic geometric structure of probability distributions. This leads to suboptimal alignment between predicted and true distributions, particularly for uncertainty quantification. We propose InfoGeo Loss, a principled loss function grounded in information geometry that measures distributional discrepancies on statistical manifolds. Our approach comprises three key components: (1) a distribution parameterization module that models predictions with learnable sufficient statistics, (2) a Fisher information metric that quantifies intrinsic distributional distance, and (3) a Bregman divergence component that captures asymmetric prediction errors. We further introduce a natural gradient weighting strategy for efficient optimization on statistical manifolds. Theoretically, we prove statistical consistency and establish convergence guarantees. Extensive experiments on seven datasets with five architectures show that InfoGeo Loss consistently outperforms existing losses, achieving average improvements of 6.8% in MSE and 5.3% in MAE.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-fang26m, title = {Information Geometry Loss for Time Series Forecasting}, author = {Fang, Jiayu and Liu, Xuande and Fang, Sangsha and Tian, Ernie and Ma, Hongwei and Shao, Zhiqi and Gao, Junbin}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {29256--29283}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/fang26m/fang26m.pdf}, url = {https://proceedings.mlr.press/v306/fang26m.html}, abstract = {Time series forecasting fundamentally involves learning probability distributions over future observations. However, existing loss functions rely on point-wise Euclidean metrics, neglecting the intrinsic geometric structure of probability distributions. This leads to suboptimal alignment between predicted and true distributions, particularly for uncertainty quantification. We propose InfoGeo Loss, a principled loss function grounded in information geometry that measures distributional discrepancies on statistical manifolds. Our approach comprises three key components: (1) a distribution parameterization module that models predictions with learnable sufficient statistics, (2) a Fisher information metric that quantifies intrinsic distributional distance, and (3) a Bregman divergence component that captures asymmetric prediction errors. We further introduce a natural gradient weighting strategy for efficient optimization on statistical manifolds. Theoretically, we prove statistical consistency and establish convergence guarantees. Extensive experiments on seven datasets with five architectures show that InfoGeo Loss consistently outperforms existing losses, achieving average improvements of 6.8% in MSE and 5.3% in MAE.} }
Endnote
%0 Conference Paper %T Information Geometry Loss for Time Series Forecasting %A Jiayu Fang %A Xuande Liu %A Sangsha Fang %A Ernie Tian %A Hongwei Ma %A Zhiqi Shao %A Junbin Gao %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-fang26m %I PMLR %P 29256--29283 %U https://proceedings.mlr.press/v306/fang26m.html %V 306 %X Time series forecasting fundamentally involves learning probability distributions over future observations. However, existing loss functions rely on point-wise Euclidean metrics, neglecting the intrinsic geometric structure of probability distributions. This leads to suboptimal alignment between predicted and true distributions, particularly for uncertainty quantification. We propose InfoGeo Loss, a principled loss function grounded in information geometry that measures distributional discrepancies on statistical manifolds. Our approach comprises three key components: (1) a distribution parameterization module that models predictions with learnable sufficient statistics, (2) a Fisher information metric that quantifies intrinsic distributional distance, and (3) a Bregman divergence component that captures asymmetric prediction errors. We further introduce a natural gradient weighting strategy for efficient optimization on statistical manifolds. Theoretically, we prove statistical consistency and establish convergence guarantees. Extensive experiments on seven datasets with five architectures show that InfoGeo Loss consistently outperforms existing losses, achieving average improvements of 6.8% in MSE and 5.3% in MAE.
APA
Fang, J., Liu, X., Fang, S., Tian, E., Ma, H., Shao, Z. & Gao, J.. (2026). Information Geometry Loss for Time Series Forecasting. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:29256-29283 Available from https://proceedings.mlr.press/v306/fang26m.html.

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