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One Step Closer to Unbiased Aleatoric Uncertainty Estimation
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Shows that variance attenuation overestimates aleatoric uncertainty and proposes Denoising Variance Attenuation (DVA), which jointly infers normalized per-sample noise under zero-mean, unit-variance constraints to get a closer estimate of true data noise.
Problem
Variance attenuation (VA) is the popular way to estimate aleatoric (data) uncertainty alongside epistemic uncertainty, but its aleatoric estimates are biased upward, so true data noise is not recovered accurately and disentanglement is evaluated only by regression error.
Approach
Instead of fitting a predicted variance directly to residuals, DVA augments each training label (or input) with a learnable normalized noise term, constrained to zero mean and unit variance and optimized with projected/normalized gradient descent, so that denoised targets are fit and the noise scale is estimated separately; the prediction model is not retrained.
Key results
- Under homoscedastic label noise, DVA + BNN gives 1.12 vs VA + BNN 2.41 at a^2=0.5 and 9.03 vs 10.35 at a^2=8.0 (target a^2).
- On the dynamical system with noise a^2=1.0, DVA + BNN estimates 1.66 vs MSE estimation 2.69 (target 1.0).
- On depth estimation with a^2=0.25 label noise, DVA + VPD estimates 0.281 vs VA 0.344 (lower is closer to target).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace VA with DVA | Toy regression with BNN · Homoscedastic label noise, a²=0.5, 1.0, 2.0, 8.0 | distance of estimated variance from true variance: 2.41±0.20 -> 1.12±0.20; 3.20±0.36 -> 1.87±0.25; 4.02±0.26 -> 2.54±0.31; 10.35±0.32 -> 9.03±0.65 | Helped |
| Replace VA with DVA | Toy regression with ensemble · Homoscedastic label noise, a²=0.5, 1.0, 2.0, 8.0 | distance of estimated variance from true variance: 2.53± 1.21 -> 1.77±0.81; 2.60±0.68 -> 2.28±0.61; 4.72±0.41 -> 3.48±0.49; 10.54±1.09 -> 9.34±0.30 | Helped |
| Replace VA with DVA | Toy regression with BNN · Heteroscedastic label noise | squared mean difference to true variance: a²=0.5: 0.65±0.17 -> 0.45±0.26; a²=1.0: 0.32±0.06 -> 0.40±0.61; a²=2.0: 0.36±0.17 -> 0.32±0.17; a²=8.0: 0.51±0.18 -> 0.3 | Mixed |
| Replace VA with DVA | Toy regression with ensemble · Heteroscedastic label noise | squared mean difference to true variance: 1.97± 0.86 -> 1.42±1.17; 1.30± 0.44 -> 0.82±0.42; 1.97±0.24 -> 1.00±0.28; 1.32±0.32 -> 1.23±0.39 | Helped |
| Use DVA instead of MSE estimation | Neural ODE with BNN · Dynamical system, homoscedastic label noise | distance of estimated variance from true variance: 1.87±0.56 -> 1.06±0.04; 2.69±0.87 -> 1.66±0.10; 4.00±1.73 -> 2.86±0.19; 11.63±4.82 -> 10.55±0.45 | Helped |
| Replace VA with DVA | VPD depth estimation model · NYU Depth v2, homoscedastic synthetic label noise | distance of estimated uncertainty from true noise variance: 0.105 ± 0.001 -> 0.021 ± 0.000; 0.344 ± 0.001 -> 0.281 ± 0.001; 1.093 ± 0.001 -> 1.056 ± 0.006 | Helped |
| Apply DVA to homoscedastic input noise | Toy regression · Homoscedastic input noise | noise estimation accuracy | Hurt |
Surprising findings
- DVA still overestimates the true noise variance, with the residual bias being Var[y - mu(x)] in theory.
- Input-noise denoising greatly overestimates noise levels (e.g. 75.21 vs target 8.0 for DVA + BNN).
Limitations
- Assumes zero-mean noise and needs large sample sizes for the law-of-large-numbers normalization.
- Cannot separate input and label noise in general; only works for label noise or dynamical systems where they share a source.
- Adds extra variables and memory cost; experiments limited to regression, not classification.
Experimental setup
Datasets: synthetic 1D regression data f(x)=x(1+sin(x)) synthetic dynamical system trajectories NYU Depth v2 APPA-REAL
Benchmarks: synthetic toy regression synthetic dynamical system NYU Depth v2 (noise-injected) APPA-REAL
Metrics: squared difference between estimated and true noise variance estimated aleatoric variance vs true variance
Baselines: VA + BNN VA + ensemble MSE estimation + BNN
Builds on: variance attenuation deep ensembles Bayesian neural networks projected gradient descent attack neural ODE
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