Residual correlation predicts benefits of joint uncertainty in multi-target regression
Residual Correlation as a Diagnostic for Joint-Uncertainty Gains from GP Coregionalisation
Machine Learning
Summary
When predicting multiple related things at once, linking their predictions can sometimes help. The authors found that simply looking at how these things are usually correlated doesn't tell you when linking helps. Instead, looking at the leftover correlation after making separate predictions reveals when sharing information improves the certainty of the predictions together. They also created a quick way to measure this leftover correlation, which predicts improvements better than other methods. Their new approach improves how well the uncertainty of multiple predictions is captured without changing the individual accuracy much.
What this means in practice
- •For machine learning engineers: Improve multi-output regression models by using residual correlation diagnostics to decide when to couple predictions for better joint uncertainty quantification.
- •For computer vision developers: Enhance keypoint regression models by incorporating residual-correlation structures to boost joint uncertainty estimation without impacting individual predictions.
Authors
Fangqin Zhou, Joaquin Vanschoren
Abstract
In multi-target regression, correlated targets are often coupled through multi-output Gaussian processes with an intrinsic model of coregionalisation (GP-ICM), assuming that sharing statistical strength improves overall performance. In practice, the benefits are inconsistent. Across the settings studied, we find that the main benefit of coregionalisation is joint uncertainty quantification rather than point prediction. Raw target correlation does not predict when coupling helps; in the separable GP-ICM settings studied here, residual correlation, the cross-target dependence left unexplained by independent per-target predictors, is the strongest predictor of joint-uncertainty gains. We introduce a lightweight diagnostic, $D_{\rm logdet}=-\frac{1}{2}\log\det R_{\rm res}$, which represents the idealised joint negative log-likelihood (NLL) gain from modelling a full rather than diagonal residual covariance and is computable from independent GPs alone. Across a controlled synthetic study, 16 multi-target benchmarks, and frozen transformer and convolutional neural network representations for keypoint regression, point prediction remains largely unchanged ($ΔR^2\approx 0$). In contrast, $D_{\rm logdet}$ strongly predicts observed ICM NLL improvements ($ρ_s=-0.83$, $p<0.001$), outperforming heuristics such as the feature-to-sample ratio. We also propose Residual-ICM, which preserves independent marginal variances while adding residual-correlation structure to the joint covariance. Residual-ICM achieves the best average joint NLL among the compared methods, while the diagnostic indicates when covariance coupling is likely to be useful. The diagnostic is specific to global Gaussian residual dependence, the structure captured by separable coregionalisation.