Anchor-MoE: A Mean-Anchored Mixture of Experts For Probabilistic Regression

Abstract

Regression under uncertainty is fundamental across science and engineering. We present an Anchored Mixture of Experts (Anchor-MoE), a model that handles both probabilistic and point regression. For simplicity, we use a tuned gradient-boosting model to furnish the anchor mean; however, any off-the-shelf point regressor can serve as the anchor. The anchor prediction is projected into a latent space, where a learnable metric-window kernel scores locality and a soft router dispatches each sample to a small set of mixture-density-network experts; the experts produce a heteroscedastic correction and predictive variance. We train by minimizing negative log-likelihood, and on a disjoint calibration split fit a post-hoc linear map on predicted means to improve point accuracy. On the theory side, assuming a H\"older smooth regression function of order~$\alpha$ and fixed Lipschitz partition-of-unity weights with bounded overlap, we show that Anchor-MoE attains the minimax-optimal $L^2$ risk rate $O\!\big(N^{-2\alpha/(2\alpha+d)}\big)$. In addition, the CRPS test generalization gap scales as $\widetilde{O}\!\Big(\sqrt{(\log(Mh)+P+K)/N}\Big)$; it is logarithmic in $Mh$ and scales as the square root in $P$ and $K$. Under bounded-overlap routing, $K$ can be replaced by $k$, and any dependence on a latent dimension is absorbed into $P$. Under uniformly bounded means and variances, an analogous $\widetilde{O}\!\big(\sqrt{(\log(Mh)+P+K)/N}\big)$ scaling holds for the test NLL up to constants. Empirically, across standard UCI regressions, Anchor-MoE consistently matches or surpasses the strong NGBoost baseline in RMSE and NLL; on several datasets it achieves new state-of-the-art probabilistic regression results on our benchmark suite. Code is available at https://github.com/BaozhuoSU/Probabilistic_Regression.