Confidence-Weighted Regression Method

• Confidence-Weighted Regression Method is a framework that integrates uncertainty quantification into regression outputs using methods like dual-head architectures and weighted intervals.
• It combines diverse techniques including kernel-weighted sample construction, online recalibration, ensemble methods, and shrinkage approaches to enhance predictive accuracy and reliability.
• Empirical evaluations in simulation and high-dimensional settings demonstrate marked improvements in stability, error reduction, and model adaptability.

The confidence-weighted regression method encompasses a diverse set of statistical and machine learning techniques designed to estimate model parameters, predictions, or actions, while quantifying and leveraging confidence or uncertainty related to those estimates. Confidence weighting integrates uncertainty measures—often derived from classification scores, model variance, kernel-weighted samples, or prediction intervals—with regression outputs to produce valid predictions, tight confidence intervals, or robust decisions. This paradigm finds application in settings ranging from online learning and high-dimensional regression to autonomous decision-making systems and domain adaptation.

1. Dual-Head Confidence-Weighted Regression Architectures

Recent developments in autonomous driving and imitation learning utilize dual-head neural architectures in which a regression head produces continuous control outputs (e.g., steering angle), and a parallel classification head estimates discrete confidence scores over binned action classes (Delavari et al., 2 Mar 2025). This design provides actionable confidence signals for each prediction. The methodology proceeds as follows:

• Raw sensor input (image $I$) is encoded via a backbone (e.g., ResNet-50).
• Regression head outputs continuous action $y_{cont}$.
• Classification head predicts probability vector $y_{disc}$ over $N$ bins; confidence is given by $\max p$ and uncertainty by entropy $H = -\sum_i p_i \log p_i$.
• Correction logic adapts the regression output according to confidence and regression-classification alignment:
  • High confidence and agreement: use $y_{cont}$.
  • High confidence but disagreement: sample uniformly from the most confident bin.
  • Low confidence, low entropy, and misalignment: sample from $\mathcal{N}(y_{cont}, \sigma^2)$ with $\sigma$ determined by class probabilities.
  • Low confidence, high entropy: retain base regression output.

Training employs a multi-task loss: $$\mathcal{L} = \lambda_1 \, \text{MSE}(y_{cont}, y_{true}) + \lambda_2\, \text{SparseCatCrossEntropy}(y_{disc}, y_{bin(true)}),$$ with balanced weights ($\lambda_1=\lambda_2=0.5$).

Empirical evaluation in closed-loop CARLA simulations demonstrates substantial improvements in trajectory accuracy, stability, and reduced error variance compared to regression-only baselines—reducing Fréchet distance from 25.99 to 8.93 (two-turn routes), and curve length deviation from 1.48 to 0.60. Confidence-driven corrections generalize across maneuvers and are effective for rare or ambiguous cases.

2. Confidence-Weighted Sample and Interval Construction

Construction of confidence intervals in regression often exploits confidence-weighted statistics. For local quantile inference, the weighted quantile (WQ) method (Jang et al., 2023) utilizes kernel weighting to upweight samples near a covariate of interest $x_0$: $L_i = K\left(\frac{x_0-X_i}{h}\right),$ yielding a weighted empirical distribution: $$\tilde{Q}_n(y) = \frac{1}{n}\sum_{i=1}^n \frac{L_i}{\sum_j L_j} I(Y_i \leq y)$$ and associated quantile estimate $\tilde{\theta}_p$. Confidence intervals are formed via normal approximation to the weighted CDF, achieving semiparametric efficiency and asymptotically optimal coverage as soon as effective sample size $n_{\text{eff}} \geq 10-20$.

Alternative rejection-based schemes offer finite-sample distribution-free coverage but at the cost of conservativeness (wider intervals) due to reduced effective sample utilization. The WQ method is applicable under minimal distributional assumptions, challenging classical conditional inference paradigms.

3. Confidence-Weighted Online and Ensemble Regression

Online learning frameworks employ confidence-weighted mechanisms for adaptive prediction in adversarial or non-stationary environments (Deshpande et al., 2023, Guille-Escuret et al., 27 Jan 2024). Key approaches include:

• Residual Interval Inversion (RII): Constructs finite-sample valid confidence regions for regression coefficients by aggregating the containment of test point predictions within residual intervals defined via arbitrary predictors. The confidence region $\Theta_\alpha$ contains all $\theta$ verifying $C(\theta)\geq k_{n_{te}}(\alpha,b)$, where $b$ quantifies the minimal probability of interval containment. The region’s MILP formulation enables robust optimization and finite-sample hypothesis testing, with the distinctive property that regions may be empty (indicating model misspecification).
• Online recalibration algorithms: Employ discretized CDF bins, recalibrating probabilistic forecasts post hoc to enforce marginal calibration, ensuring that, e.g., 80% confidence intervals contain the true response 80% of the time, even in adversarial data streams. Regret with respect to any baseline model is provably bounded under proper scoring rules.

In ensemble settings, confidence-weighted logistic regression aggregates human and machine judgments, weighting predictors by their associated confidence levels (magnitude), with the sign encoding choice direction (Yáñez et al., 15 Aug 2024): $$p_x = \frac{1}{1 + e^{-(\beta_I + \sum_k \beta_k x_k)}}$$ where $x_k$ is the signed confidence per teammate $k$, fitted via maximum likelihood. Integration outperforms individuals if confidences are well-calibrated and error profiles are diverse.

4. Confidence-Weighted Expectation and Reparametrization Invariance

Confidence-weighted estimation offers a prior-free, reparametrization-invariant mechanism for probabilistic inference (Pijlman, 2017). Letting $\alpha(\vec{x},\tau)$ denote the fraction of likelihood above the observed data for parameter $\tau$, expectation values of an observable $\mathcal{O}(\tau)$ are computed as: $$\left< \mathcal{O} \right>_c = \frac{1}{K} \int_{N(\vec{x},\alpha)\neq 0} d\alpha \; \frac{1}{N(\vec{x},\alpha)} \sum_{i=1}^{N(\alpha,\vec{x})} \mathcal{O}(\tau_i(\vec{x},\alpha)),$$ with equal weighting for parameter sets contributing identical confidence.

Contrasting with Bayesian methods, which require priors possibly violating reparametrization invariance, confidence-weighted approaches base uncertainty and expectation solely on data and likelihood structure. Numerical studies demonstrate convergence to Bayesian estimates with a flat prior in low-dimensional cases, but divergence otherwise, especially in multi-parameter models.

5. Confidence Ellipsoids and Bands in Regression

Weighted ellipsoidal confidence sets in regression arise in mixture models with unknown label origin, with nonparametric and parametric estimation methods available (Miroshnichenko et al., 2018). Weighted least squares estimators exploit known mixture probabilities via minimax weighting to estimate component coefficients, constructing ellipsoidal regions as

$$B^{\mathrm{LS}_\alpha} = \left\{ \beta : n(\beta - \hat{b})^\top \hat{V}_n^{-1} (\beta - \hat{b}) \leq Q_{\chi^2_d}(1-\alpha) \right\},$$

where $\hat{V}_n$ is the estimated covariance matrix.

For functional regression, confidence bands are constructed around PCA-based estimators by simulating the distribution of $n\|\hat{b}-b\|^2$ under resampling, thereby covering the slope function at most $1-\tau_2$ fraction of points with probability $\geq 1-\tau_1$ (Imaizumi et al., 2016). Bandwidth selection is based on $L^2$ risk, with undersmoothing recommended for proper inference.

Simultaneous bands in nonparametric regression with missing covariates utilize inverse selection probability weighting, achieving oracally efficient coverage (Cai et al., 2020).

$$\hat{m}(x, \hat{\pi}) \pm (nh)^{-1/2} r_n^{1/2} \hat{d}_n^{1/2}(x) \left( b_h + a_h^{-1} q_\alpha \right)$$

Here, $r_n$ corrects for observed cases, and plug-in variance estimates ensure robustness to moderate model misspecification.

6. High-Dimensional Confidence Sets and Shrinkage Methods

Honest and adaptive confidence sets for high-dimensional linear regression are constructed through projection onto strong signal coordinates, combined with Stein shrinkage for weak signals (Zhou et al., 2019). The resulting ellipsoid

$$C = \left\{ \mu \in \mathbb{R}^n : \frac{\|P_A \mu - \hat{\mu}_A\|^2}{n r_A^2} + \frac{\|P_A^\perp \mu - \hat{\mu}_\perp\|^2}{n r_\perp^2} \leq 1 \right\}$$

is honest (coverage $\geq 1-\alpha$) over all $\beta$ and adapts its diameter to signal sparsity and strength, achieving rate $n^{-1/4}$ for sparse or weakly signaled models.

7. Confidence Weighting in Model Transfer and Domain Adaptation

Confidence weighting is also employed in transferring knowledge from complex models to simple, interpretable ones. The ProfWeight method (Dhurandhar et al., 2018) attaches linear probes to intermediate layers, computes per-sample confidence profiles, and increases the training weight of samples classified with high confidence at lower layers by the teacher network: $$w_i = \frac{1}{|I|} \sum_{u \in I} P_u(R_u(x_i))[y_i]$$ Retraining the simple model with these weights yields substantial improvements in test accuracy under memory-limited or interpretable deployment constraints.

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In summary, confidence-weighted regression methods unify a broad range of inference, learning, and decision-making strategies in regression settings by systematically quantifying, exploiting, and calibrating uncertainty and confidence. They contribute to statistical validity, robustness to model misspecification, domain adaptability, interpretable uncertainty quantification, and safety improvements across contemporary applications.