Bounded Precision-Geometry Scaling for Robust Multi-Task Learning under Loss Scale Mismatch

Krishna Subedi

cs.LGAug 21, 2026

Bounded Precision-Geometry Scaling for Robust Multi-Task Learning under Loss Scale Mismatch

Authors: Krishna Subedi

Organizations: Neryva

Abstract

Multi-task learning often combines losses that span several orders of magnitude, causing homoscedastic uncertainty weighting to degrade severely. We propose Bounded Precision-Geometry Scaling (BPGS), a method that maps each task's log-variance through a bounded sigmoid parameterisation anchored to detached batch loss statistics, and decouples network optimisation from uncertainty optimisation. Its normalised task weights are provably invariant to uniform rescaling under non-degenerate loss scales. We evaluate BPGS on synthetic stress tests and three real-world benchmarks: NYUv2 dense prediction, Yeast multi-label classification, and RF1 multi-target regression. Under pure loss rescaling from $\times 1$ to $\times 1000$ , its macro score changes from 0.777 to 0.778, whereas Kendall weighting drops from 0.780 to 0.637; $\ell_1$ -normalising Kendall's weights does not close the gap. On NYUv2, BPGS records the lowest depth absolute relative error (0.223), depth RMSE (0.790), and total loss (1.891) among all compared methods, including Nash-MTL. Sensitivity studies on batch size and calibration show small variation across the tested ranges, and runtime overhead relative to Kendall is under 1%. BPGS posts the highest Yeast micro-F1 (0.616) and is competitive on RF1, though PCGrad leads RMSE and MAE there. These findings establish BPGS as a scale-robust alternative to homoscedastic uncertainty weighting, notably effective when loss-scale disparities dominate multi-task optimisation.

Figures & tables

Panel of Figure 1 — Figure 1: Schematic overview of BPGS. The shared encoder $f_{w}$ yields $T$ task losses per batch. A stop-gradient branch extracts log-loss statistics $\mu(L)$ and $\bar{\varsigma}(L)$ , which anchor a bounded chart that restricts each task’s log-variance to a finite interval (§ 3.4 ). Precisions derived from the chart are $\ell_{1}$ -normalised into task weights $\alpha_{i}$ . Training is split: the network pass minimises a weighted loss with detached weights, while a separate uncertainty pass adjusts the $\theta_{i}$ coordinates with detached losses. First-batch auto-calibration sets the initial $\theta_{i}$ values; equations and design details appear in § 3.2 – 3.4 .

Method	$\times{1}$	$\times{10}$	$\times{100}$	$\times{1000}$
Kendall	$\mathbf{0.780\pm 0.007}$	$0.769\pm 0.013$	$0.714\pm 0.017$	$0.637\pm 0.015$
UWSO	$0.622\pm 0.073$	$0.674\pm 0.036$	$0.688\pm 0.035$	$0.681\pm 0.014$
BPGS	$0.777\pm 0.012$	$\mathbf{0.784\pm 0.008}$	$\mathbf{0.781\pm 0.001}$	$\mathbf{0.778\pm 0.006}$

Table 1: Pure loss rescaling: Macro Score across scale factors. Mean

\pm

std over 3 seeds.

Figure 2: Synthetic robustness diagnostics under scale mismatch. Left column: scale stress, where performance is evaluated as task scales become increasingly imbalanced. Right column: pure loss rescaling, where only the reported magnitude changes. BPGS remains nearly invariant under pure rescaling and retains the strongest macro score at the largest scale-stress factors.

Method	mIoU $\uparrow$	Abs Rel $\downarrow$	RMSE $\downarrow$	Angle $\downarrow$	Within 11.25 ∘ $\uparrow$	Total Loss $\downarrow$	$\Delta_{M}$ $\uparrow$
Static	$\mathbf{0.341\pm 0.006}$	$0.239\pm 0.000$	$0.829\pm 0.011$	$35.934\pm 2.055$	$0.169\pm 0.009$	$1.992\pm 0.033$	$0.000\pm 0.000$
Kendall	$0.281\pm 0.013$	$0.229\pm 0.001$	$\underline{0.808\pm 0.003}$	$28.629\pm 0.431$	$0.231\pm 0.007$	$1.977\pm 0.029$	$0.022\pm 0.000$
UWSO	$0.294\pm 0.007$	$\underline{0.228\pm 0.003}$	$0.815\pm 0.012$	$\mathbf{26.110\pm 0.383}$	$\mathbf{0.271\pm 0.007}$	$\underline{1.936\pm 0.023}$	$\underline{0.059\pm 0.000}$
BPGS	$\underline{0.312\pm 0.001}$	$\mathbf{0.223\pm 0.002}$	$\mathbf{0.790\pm 0.010}$	$\underline{26.851\pm 0.334}$	$\underline{0.257\pm 0.005}$	$\mathbf{1.891\pm 0.014}$	$\mathbf{0.078\pm 0.000}$
Nash-MTL	$0.252\pm 0.027$	$0.242\pm 0.009$	$0.856\pm 0.029$	$30.180\pm 1.010$	$0.211\pm 0.011$	$2.071\pm 0.070$	$\approx-0.038$

Table 2: NYUv2 dense prediction benchmark. All methods trained for 120 epochs. Final-epoch metrics reported as mean

\pm

std over 3 seeds.

\Delta_{M}

is computed against the Static baseline; the Nash-MTL value is derived from the reported mean metrics and is shown approximately.

Method	mIoU $\uparrow$	Abs Rel $\downarrow$	RMSE $\downarrow$	Angle $\downarrow$	Within 11.25 ∘ $\uparrow$	Total Loss $\downarrow$
Kendall	$\underline{0.137\pm 0.006}$	$0.294\pm 0.010$	$\underline{0.990\pm 0.016}$	$38.681\pm 0.936$	$0.130\pm 0.005$	$2.546\pm 0.035$
BPGS (canonical)	$0.128\pm 0.008$	$\mathbf{0.283\pm 0.005}$	$\mathbf{0.988\pm 0.043}$	$\mathbf{33.834\pm 1.068}$	$\mathbf{0.161\pm 0.015}$	$\mathbf{2.488\pm 0.075}$
BPGS (BA-fixed)	$\mathbf{0.143\pm 0.006}$	$0.291\pm 0.005$	$0.994\pm 0.022$	$39.097\pm 0.478$	$0.130\pm 0.004$	$\underline{2.524\pm 0.047}$
BPGS (SL-auto)	$0.058\pm 0.006$	$0.331\pm 0.003$	$1.305\pm 0.015$	$\underline{34.661\pm 1.310}$	$\underline{0.147\pm 0.024}$	$6.142\pm 1.332$
BPGS (SL-fixed)	$0.120\pm 0.007$	$\underline{0.287\pm 0.006}$	$1.013\pm 0.025$	$35.183\pm 0.963$	$0.142\pm 0.010$	$2.566\pm 0.051$

Table 3: Ablation study: BPGS variants vs. Kendall uncertainty weighting on NYUv2. All methods trained for 60 epochs. Mean

\pm

std over 3 seeds.

Method	Macro-F1 $\uparrow$	Micro-F1 $\uparrow$	Hamming Acc $\uparrow$	$R^{2}$ $\uparrow$	RMSE $\downarrow$	MAE $\downarrow$
Static	$0.430\pm 0.008$	$\underline{0.610\pm 0.009}$	$0.772\pm 0.003$	$-0.453\pm 0.127$	$29.875\pm 0.485$	$23.376\pm 0.451$
Kendall	$\underline{0.431\pm 0.003}$	$0.609\pm 0.001$	$0.768\pm 0.002$	$-0.439\pm 0.164$	$\underline{29.801\pm 0.821}$	$\underline{23.219\pm 0.790}$
UWSO	$0.142\pm 0.014$	$0.486\pm 0.008$	$0.767\pm 0.002$	$\mathbf{-0.408\pm 0.193}$	$30.144\pm 1.137$	$23.449\pm 1.230$
PCGrad	$0.430\pm 0.007$	$0.604\pm 0.010$	$0.769\pm 0.002$	$-0.432\pm 0.107$	$\mathbf{29.596\pm 0.385}$	$\mathbf{23.098\pm 0.368}$
GradNormProxy	$0.408\pm 0.028$	$0.597\pm 0.014$	$\underline{0.772\pm 0.001}$	$-0.508\pm 0.248$	$30.033\pm 0.482$	$23.380\pm 0.582$
BPGS	$\mathbf{0.436\pm 0.006}$	$\mathbf{0.616\pm 0.008}$	$\mathbf{0.773\pm 0.003}$	$\underline{-0.429\pm 0.213}$	$30.158\pm 0.952$	$23.548\pm 0.908$

Table 4: Real-data transfer benchmarks. Final-epoch task metrics reported as mean

\pm

std over 3 seeds. Higher is better for Yeast metrics and RF1

R^{2}

; lower is better for RF1 RMSE and MAE.

Appendix figures & tables21 assets

Supplementary material from the paper’s appendix.

Appendix

Method	$\times{1}$	$\times{10}$	$\times{100}$	$\times{1000}$
Kendall	$0.738\pm 0.017$	$0.607\pm 0.038$	$0.522\pm 0.030$	$0.499\pm 0.025$
UWSO	$0.690\pm 0.034$	$0.604\pm 0.040$	$0.525\pm 0.035$	$0.494\pm 0.024$
BPGS	$\mathbf{0.743\pm 0.018}$	$\mathbf{0.616\pm 0.039}$	$\mathbf{0.547\pm 0.032}$	$\mathbf{0.543\pm 0.032}$

Appendix

Table 5: Synthetic scale stress: Macro Score across scale factors. Mean

\pm

std over 10 seeds.

Method	Scale Stress	Loss Rescaling
Kendall	$32.3\%$	$18.4\%$
UWSO	$28.5\%$	$\mathbf{-9.5\%}$
BPGS	$\mathbf{27.0\%}$	$0.0\%$

Appendix

Table 6: Relative Macro Score change (%) from

\times 1\to\times 1000

. Positive values indicate degradation; negative values indicate improvement.

Method	Clean		Noisy		Conflict
	Macro	Worst	Macro	Worst	Macro	Worst
Kendall	$0.687\pm 0.010$	$0.267\pm 0.056$	$\mathbf{0.659\pm 0.007}$	$\mathbf{0.242\pm 0.045}$	$\mathbf{0.664\pm 0.012}$	$\mathbf{0.211\pm 0.041}$
UWSO	$0.437\pm 0.028$	$0.000\pm 0.000$	$0.438\pm 0.020$	$0.000\pm 0.000$	$0.433\pm 0.013$	$0.000\pm 0.000$
PCGrad	$\mathbf{0.688\pm 0.016}$	$0.274\pm 0.055$	$0.656\pm 0.007$	$0.241\pm 0.046$	$0.660\pm 0.011$	$0.205\pm 0.048$
BPGS	$0.679\pm 0.017$	$\mathbf{0.301\pm 0.052}$	$0.641\pm 0.013$	$0.176\pm 0.076$	$0.652\pm 0.012$	$0.198\pm 0.063$

Appendix

Table 7: Heterogeneous mixed stress: Macro Score and Worst-Task Score across regimes. Mean

\pm

std over 3 seeds.

Scale	BPGS	Kendall	Kendall+L1
$\times 1$	$0.789\pm 0.018$	$0.788\pm 0.014$	$0.794\pm 0.017$
$\times 10$	$0.791\pm 0.015$	$0.781\pm 0.023$	$0.780\pm 0.022$
$\times 100$	$0.788\pm 0.016$	$0.734\pm 0.037$	$0.742\pm 0.029$
$\times 1000$	$0.785\pm 0.017$	$0.658\pm 0.042$	$0.689\pm 0.034$
$\Delta(\times 1\!\to\!\times 1000)$	$-0.004$	$-0.130$	$-0.105$

Appendix

Table 8: Normalization ablation on pure loss rescaling. Macro score (mean

\pm

std over seeds 42, 123, and 999). L1-normalizing Kendall improves the largest-scale result but does not reproduce the BPGS scale sensitivity.

Figure 3: Synthetic scale-stress curves. As the scale factor increases, all methods degrade, but BPGS retains the strongest macro score at every tested factor and tracks the strongest worst-task score at the larger stress levels.

Figure 4: Pure loss-rescaling curves. BPGS remains nearly invariant across the entire rescaling range, while Kendall degrades sharply at larger factors and UWSO improves only from a substantially weaker starting point.

Figure 5: Relative degradation comparison between synthetic scale stress and pure loss rescaling. The figure shows that BPGS has the smallest macro-score degradation under scale stress and essentially no macro-score degradation under pure rescaling.

Figure 6: Heterogeneous mixed-stress summary. BPGS is not the best method in every regime, but it achieves the strongest worst-task score in the clean regime and remains well above the degenerate UWSO worst-task score across all regimes.

Figure 7: NYUv2 ablation validation curves by task. The stateless auto-calibrated variant fails on segmentation and remains weak on depth, while the batch-aware variants maintain stronger trajectories throughout training.

Figure 8: Task-weight dynamics for representative stable BPGS variants on the NYUv2 ablation. The batch-aware variants keep task weights in distinct but controlled regimes, whereas the stateless fixed variant shows a different long-run allocation pattern, especially for the surface-normal task.

Figure 9: NYUv2 per-task performance comparison. The figure makes the trade-off pattern visually explicit: Static is strongest on segmentation, UWSO is strongest on surface normals, and BPGS is strongest on both depth metrics.

Figure 10: NYUv2 performance heatmap across reported metrics. Darker cells indicate stronger relative performance after accounting for metric direction. BPGS concentrates its gains on depth and total loss, while Static and UWSO remain stronger on some other tasks.

Figure 11: NYUv2 validation training curves by task. BPGS tracks the strongest depth trajectory and remains competitive on segmentation and surface normals over the full 120-epoch budget.

Figure 12: NYUv2

\Delta_{M}

comparison against the Static baseline. BPGS attains the strongest aggregate improvement among the adaptive methods evaluated in the main benchmark.

Figure 13: Yeast training curves. BPGS tracks the strongest final micro-F1 and macro-F1 among the compared methods, while UWSO remains substantially weaker throughout training.

Figure 14: RF1 training curves. The methods remain closer than in the synthetic stress studies, which is consistent with the paper’s claim that RF1 supports competitiveness rather than clear superiority.

Metric	Split on	Split off
$\theta_{\max}$ , epoch 0	$3.082\pm 0.017$	$3.109\pm 0.011$
$\theta_{\max}$ , final	$2.438\pm 0.014$	$3.993\pm 0.192$
Segmentation mIoU	$0.1983$	$0.1897$
Depth RMSE	$0.8433$	$0.8544$
Total loss	$2.1278$	$2.1419$

Appendix

Table 9: Stop-gradient ablation on the NYUv2 50% subset (three seeds, 60 epochs). Removing the stop-gradient changes the uncertainty dynamics and lowers segmentation mIoU.

Metric	bs=4	bs=8	bs=16	CV
Total loss	2.1468	2.1226	2.1507	0.58%
Depth Abs Rel	0.2411	0.2421	0.2451	0.70%
Depth RMSE	0.8643	0.8403	0.8490	1.17%
Normal angle	29.400	29.223	29.754	0.75%
Segmentation mIoU	0.1955	0.1932	0.1810	3.35%
$\theta_{\max}$ , final	2.396	2.441	2.589	–

Appendix

Table 10: Batch-size sensitivity of canonical BPGS on the NYUv2 50% subset (three seeds per batch size, 60 epochs). Values are seed-averaged final metrics.

Metric	1001	2001	3001	CV
Total loss	2.1145	2.1040	2.1090	0.20%
Depth Abs Rel	0.2385	0.2392	0.2411	0.46%
Depth RMSE	0.8348	0.8302	0.8325	0.23%
Normal angle	28.9205	28.8975	29.1069	0.32%
Segmentation mIoU	0.1912	0.1866	0.1952	1.84%
$\theta_{\max}$ , final	2.4451	2.4382	2.4380	0.3% spread

Appendix

Table 11: First-batch calibration sensitivity on the NYUv2 50% subset. Only the loader seed changes; the model seed is fixed.

Quantity	BPGS	Kendall	Difference
Per-epoch time (s)	40.979	40.916	+0.15%
Peak GPU memory (MB)	3368.36	3339.34	+0.87%
Total time, 60 epochs (s)	2461.6	2457.9	+3.7
Trainable parameters	18,869,143	18,869,140	+3

Appendix

Table 12: Computational overhead on the NYUv2 50% subset (three seeds, 60 epochs).

Dataset	$T$	max $\|z_{i}\|/\tau_{T}$	final cap
NYUv2	3	0.9327	$\leq 0.84$
Yeast	14	0.7017	$\leq 0.69$
RF1	8	0.8937	$\leq 0.89$

Appendix

Table 13: Closest observed approach to the bounded-chart radius across logged main-paper runs. The maximum occurs at initialization; no run reaches the boundary.

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