Learning Pareto Stationary Fronts via Single-Pass Backpropagation

Elina Rojin Celik; Marcos Medeiros Raimundo; Isabel Valera

cs.LGOct 5, 2026

Learning Pareto Stationary Fronts via Single-Pass Backpropagation

Authors: Elina Rojin Celik, Marcos Medeiros Raimundo, Isabel Valera

Organizations: Department of Computer Science Saarland University, Germany · Instituto de Computação Universidade Estadual de Campinas, Brazil

Abstract

We propose MOSEL (Multi-Objective Stackelberg Efficient Learning), a framework for a posteriori multi-objective optimization (MOO) in deep neural networks that recovers a full front of Pareto stationary solutions at the computational cost of standard single-objective training. MOSEL reformulates the problem as a bilevel optimization problem that leverages network modularity to decouple representation learning from objective-preference alignment. Casting the bilevel optimization problem as a Stackelberg game enables solving the original a posteriori MOO problem in a single forward-backward pass. As a result, MOSEL matches the time and memory efficiency of standard single-objective training while enabling scalable Pareto stationary front learning. Empirically, MOSEL uncovers diverse and optimal Pareto frontiers in strongly conflicting settings (e.g., fairness-accuracy). Remarkably, even in weakly conflicting regimes such as multi-task learning, it consistently converges to solutions closer to the utopia point, outperforming both standard single-objective training and specialized multi-task learning methods. These results highlight the broader potential of a posteriori MOO learning as a pathway to efficiently learn more diverse and robust representations, ultimately improving generalization.

Figures & tables

Figure 1: Overview of MOSEL. (a) MOSEL partitions the NN into a representation learning network, which steers

\boldsymbol{z}

toward Pareto optimal solutions, and a preference alignment network conditioning on

\boldsymbol{r}\sim\text{Dir}(\boldsymbol{\alpha})

to cover diverse trade-offs. (b) Illustration on ResNet-34 [ 20 ] :

\boldsymbol{z}

is extracted at the network split point and modulated via a FiLM layer [ 50 ] , which maps

\boldsymbol{r}

to a scaling and shift factor per feature channel, steering the final predictions

\hat{y}(\boldsymbol{x};\boldsymbol{r})

toward the desired objective trade-off.

Dataset	Method	HV Loss $\uparrow$	HV Error $\uparrow$	# PO sol. (Loss) $\uparrow$	# PO sol. (Error) $\uparrow$
Adult	MOSEL-LS	0.939 $\pm$ 0.007	0.849 $\pm$ 0.012	22.8 $\pm$ 1.2	21.2 $\pm$ 1.5
	MOSEL-CS	0.819 $\pm$ 0.017	0.776 $\pm$ 0.015	23.8 $\pm$ 0.7	22.0 $\pm$ 1.4
	COSMOS	0.926 $\pm$ 0.014	0.830 $\pm$ 0.010	20.2 $\pm$ 0.7	17.4 $\pm$ 1.9
	Classifier	0.906 $\pm$ 0.025	0.618 $\pm$ 0.016	8.0 $\pm$ 0.6	6.2 $\pm$ 0.7
COMPAS	MOSEL-LS	0.920 $\pm$ 0.019	0.568 $\pm$ 0.008	15.6 $\pm$ 1.4	14.6 $\pm$ 1.4
	MOSEL-CS	0.930 $\pm$ 0.007	0.560 $\pm$ 0.005	15.2 $\pm$ 1.6	14.2 $\pm$ 1.6

Table 1: Fairness-Performance Trade-offs. Relative hypervolume and number of Pareto optimal solutions on the fairness benchmarks, computed on training loss and test error. Mean

\pm

std over five runs; best entries per column shaded.

Figure 2: Fairness-Performance Trade-offs. Test-time Pareto fronts, plotting classification error against fairness violation. The grey dotted lines indicate zero unfairness and the best accuracy achievable by an unconstrained classifier; their intersection defines the utopia point—an ideal, yet unattainable solution toward which all methods aim to converge.

Figure 3: Two-task setting. Test-time Pareto fronts on the error, showing the trade-offs between task-specific errors. Grey dotted lines indicate the best achievable single-task performance on each task; their intersection marks the single-task baseline. In this weakly conflicting setting, positive transfer allows methods to surpass this baseline, with MOSEL achieving the largest improvements.

Method	$\Delta$ Param.	Time ( $\times$ )	CPU ( $\times$ )
Equal weight	17,495,136	$248.4\pm 45.7$ min	$11.15\pm 0.25$ MB
MOSEL-LS	+672	$\times 1.0\pm 0.1$	$\times 0.97\pm 0.02$
MOSEL-CS	+672	$\times 1.0\pm 0.2$	$\times 0.98\pm 0.02$
COSMOS	+1,072	$\times 1.1\pm 0.1$	$\times 0.99\pm 0.02$
PCGrad	+0	$\times 9.8\pm 2.2$	$\times 1.06\pm 0.01$
MGDA	+0	$\times 3.0\pm 0.5$	$\times 0.99\pm 0.03$

Table 5

Figure 4: Scalability to number of objectives on CelebA . Per-task accuracy improvement over the single-task baseline (gray column) for the most correlated (left) and least correlated (right) task sets, as the number of jointly optimized tasks increases. Results are averaged over 5 seeds.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Hypervolume computation

Figure 6: Test-time Pareto fronts on the fairness use case, showing training objectives (BCE loss vs DEO). The grey dotted lines mark zero unfairness and the best BCE loss achievable by an unconstrained classifier; their intersection defines the utopia point, the ideal but infeasible solution, toward which all methods aim to converge.

Dataset	Method	$\Delta$ #Param.	Time (rel.)	CPU Mem (rel.)	GPU Mem (rel.)
Adult	MOSEL-LS	+50	$1.2\pm 0.2$	$1.10\pm 0.07$	–
	MOSEL-CS	+50	$1.2\pm 0.1$	$1.06\pm 0.06$	–
	COSMOS	+120	$1.2\pm 0.2$	$1.09\pm 0.03$	–
	Classifier	6,891	$44.4\pm 4.4$ s	$10.30\pm 0.34$ MB	–
COMPAS	MOSEL-LS	+50	$1.5\pm 0.7$	$1.00\pm 0.03$	–
	MOSEL-CS	+50	$1.5\pm 0.7$	$0.96\pm 0.03$	–

Appendix

Table 2: Computational overhead on fairness datasets relative to classifier (bottom); # Param. are absolute counts, time and CPU are multiplicative factors. Mean

\pm

std over five runs.

	Multi-MNIST		Multi-Fashion		Multi-Fashion+MNIST		CelebA
Method	Acc TL	Acc BR	Acc TL	Acc BR	Acc TL	Acc BR	Acc Goatee	Acc Mustache
MOSEL-LS	0.919 $\pm$ 0.001	0.901 $\pm$ 0.002	0.829 $\pm$ 0.003	0.821 $\pm$ 0.001	0.930 $\pm$ 0.001	0.851 $\pm$ 0.003	0.968 $\pm$ 0.002	0.964 $\pm$ 0.001
MOSEL-CS	0.910 $\pm$ 0.002	0.895 $\pm$ 0.002	0.817 $\pm$ 0.004	0.812 $\pm$ 0.003	0.916 $\pm$ 0.003	0.845 $\pm$ 0.003	0.970 $\pm$ 0.001	0.966 $\pm$ 0.001
COSMOS	0.907 $\pm$ 0.003	0.886 $\pm$ 0.002	0.824 $\pm$ 0.004	0.821 $\pm$ 0.004	0.915 $\pm$ 0.004	0.837 $\pm$ 0.005	0.960 $\pm$ 0.005	0.959 $\pm$ 0.004
PCGrad	0.914 $\pm$ 0.006	0.894 $\pm$ 0.004	0.814 $\pm$ 0.004	0.828 $\pm$ 0.001	0.928 $\pm$ 0.003	0.845 $\pm$ 0.007	0.954 $\pm$ 0.000	0.961 $\pm$ 0.000
MGDA	0.912 $\pm$ 0.009	0.897 $\pm$ 0.006	0.822 $\pm$ 0.004	0.820 $\pm$ 0.002	0.919 $\pm$ 0.002	0.852 $\pm$ 0.002	0.967 $\pm$ 0.001	0.961 $\pm$ 0.000

Appendix

Table 3: Per-task accuracy at the solution closest to the utopia point on the test Pareto front, across all MTL benchmarks. As joint training enables positive transfer, we take

(0,0)

as the utopia point and select the solution minimizing Euclidean distance to it, which can lie beyond the single-task baseline. TL and BR denote top-left and bottom-right tasks on the synthetic benchmarks. Mean

\pm

std over five independent runs; best results per column shaded.

Figure 7: Test-time Pareto fronts on the MTL use case, showing the trade-offs between task-specific (Binary) Cross-Entropy losses. Grey dotted lines indicate the best achievable single-task performance on each task; their intersection marks the single-task baseline. In this weakly conflicting setting, positive transfer allows methods to surpass this baseline, with MOSEL achieving the largest improvements.

Dataset	Method	$\Delta$ #Param.	Time (rel.)	CPU Mem (rel.)	GPU Mem (rel.)
Multi-MNIST	MOSEL-LS	+4,320	$1.1\pm 0.2$	$1.15\pm 0.05$	$1.01\pm 0.00$
	MOSEL-CS	+4,320	$1.2\pm 0.2$	$1.09\pm 0.09$	$1.00\pm 0.00$
	COSMOS	+708	$1.3\pm 0.2$	$1.07\pm 0.02$	$1.09\pm 0.03$
	PCGrad ∗	+0	$6.2\pm 0.6$	$1.22\pm 0.04$	$1.00\pm 0.00$
	MGDA	+0	$2.0\pm 0.4$	$1.04\pm 0.06$	$1.12\pm 0.02$
	GradNorm	+0	1.9 $\pm$ 0.2	1.07 $\pm$ 0.05	1.00 $\pm$ 0.00

Appendix

Table 4: Computational overhead on MTL datasets relative to equal weight training (bottom); # Param. are absolute counts, time and CPU are multiplicative factors. Mean

\pm

std over five runs.

Dataset	Method	HV Loss $\uparrow$	HV Error $\uparrow$	# PS sol. (Loss) $\uparrow$	# PS sol. (Error) $\uparrow$
Multi-MNIST	MOSEL-LS	0.330 $\pm$ 0.010	0.441 $\pm$ 0.008	10.6 $\pm$ 0.8	9.0 $\pm$ 0.6
	MOSEL-CS	0.318 $\pm$ 0.008	0.426 $\pm$ 0.009	17.8 $\pm$ 0.4	16.8 $\pm$ 1.2
	COSMOS	0.303 $\pm$ 0.008	0.418 $\pm$ 0.007	20.0 $\pm$ 1.1	19.0 $\pm$ 3.0
	PCGrad	0.274 $\pm$ 0.017	0.425 $\pm$ 0.010	2.2 $\pm$ 0.7	2.8 $\pm$ 0.4
	MGDA	0.267 $\pm$ 0.026	0.409 $\pm$ 0.034	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0
	GradNorm	0.281 $\pm$ 0.010	0.410 $\pm$ 0.016	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0

Appendix

Table 5: Relative hypervolume and number of Pareto stationary solutions on the MTL benchmarks, computed on training loss and test error. Mean

\pm

std over five runs; best entries per column shaded.

Dataset	Scalarization	$\alpha$	HV $\uparrow$	HV MCR $\uparrow$	Obj 1 -best (mean)	Obj 2 -best (mean)
Adult		0.5	0.939 $\pm$ 0.007	0.849 $\pm$ 0.012	(0.143,0.106)	(0.211,0.009)
	LS	0.8	0.912 $\pm$ 0.011	0.838 $\pm$ 0.019	(0.144,0.105)	(0.202,0.014)
		1.2	0.816 $\pm$ 0.015	0.774 $\pm$ 0.009	(0.143,0.105)	(0.186,0.030)
		0.5	0.819 $\pm$ 0.017	0.776 $\pm$ 0.015	(0.142,0.106)	(0.188,0.030)
	CS	0.8	0.702 $\pm$ 0.019	0.683 $\pm$ 0.022	(0.142,0.105)	(0.172,0.049)
		1.2	0.524 $\pm$ 0.023	0.525 $\pm$ 0.025	(0.142,0.106)	(0.153,0.079)

Appendix

Table 6: Ablation on Dirichlet parameter

\alpha

. Relative hypervolume on two strongly and two weakly conflicting datasets computed on training loss (HV) and test error (HV MCR). Mean

\pm

std over five runs. Mean extreme solutions (Obj 1 -best, Obj 2 -best) are reported for varying

\alpha

and scalarization.

Dataset	RL / PA	HV $\uparrow$	HV MCR $\uparrow$	Obj 1 -best	Obj 2 -best
CelebA	3 / 35	0.213	0.396	(0.075,0.080)	(0.076,0.079)
	7 / 31	0.238	0.406	(0.070,0.077)	(0.071,0.076)
	11 / 27	0.232	0.401	(0.070,0.079)	(0.071,0.078)
	17 / 21	0.242	0.415	(0.068,0.079)	(0.071,0.076)
	23 / 15	0.242	0.415	(0.068,0.077)	(0.069,0.077)
	31 / 7	0.238	0.422	(0.069,0.078)	(0.071,0.076)

Appendix

Table 7: Ablation on split point. The split is represented by the number of allocated blocks to representation learning (RL) and preference alignment (PA) network respectively. We report obtained HV in training and evaluation space as well as extreme points.

Figure 8: Ablation study on CelebA: runtime and peak memory for increasing numbers of objectives, for correlated and uncorrelated features.

			Fairness extreme		Accuracy extreme
RL / PA	HV $\uparrow$	HV MCR $\uparrow$	Acc@MinDEO(%) $\uparrow$	MinDEO $\downarrow$	DEO@MaxAcc $\downarrow$	MaxAcc(%) $\uparrow$
5 / 1	0.423	0.479	88.68	0.0000	0.037	92.22
4 / 2	0.419	0.477	88.69	0.0001	0.035	92.17
3 / 3	0.419	0.475	88.70	0.0002	0.036	92.12

Appendix

Table 8: Results on DistilBERT text-classification. Results show HV and extreme points for the front for different split points (splits between transformer blocks)

Epoch	MOSEL-LS Max	Equal Weight Max	$\Delta$ Max
0	5.54 $\pm$ 0.86	15.89 $\pm$ 5.91	-10.35
10	0.52 $\pm$ 0.03	0.42 $\pm$ 0.04	+0.09
30	0.38 $\pm$ 0.02	0.30 $\pm$ 0.05	+0.08
50	0.36 $\pm$ 0.02	0.28 $\pm$ 0.05	+0.08
70	0.37 $\pm$ 0.01	0.28 $\pm$ 0.05	+0.09
99	0.36 $\pm$ 0.02	0.27 $\pm$ 0.04	+0.09

Appendix

Table 9: Per-epoch maximum objective loss (mean

\pm

std over 5 seeds).

\Delta

is defined as difference between MOSEL and Equal Weight baseline; negative values indicate lower (better) worst-case loss for MOSEL. After an initial transient, MOSEL maintains a small, constant gap with tighter cross-seed variance, confirming stable optimization throughout training.

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Interactive Pareto navigation for deep multi-task learning

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Open paper AskPDF arXiv

In multi-task learning, handling an increasing number of objectives can quickly become challenging, both in terms of the computational resources and the decision maker's capacity to choose appropriate trade-offs. A widely used approach is thus to aggregate the individual losses in a single loss function by a weighted sum. This often fails to capture either the decision maker's preferences as a result of the shape of the Pareto front, or requires multiple adjustments and computations which becomes prohibitively expensive in deep learning applications. To address these issues, we introduce a novel framework, Preference Pareto Exploration (PPE), which enforces the decision maker's preferences while accounting for the geometry of the Pareto set in an interactive exploration process. PPE is based on a predictor-corrector method that performs predictor steps tangential to the manifold of Pareto-optimal solutions, following the decision maker's preference. The subsequent corrector step results in a new trade-off reflecting this preference. To avoid explicit Hessian computations when characterizing the tangent space of the manifold, we employ a Krylov subspace method that relies solely on matrix-vector products. These products can be efficiently obtained via automatic differentiation, ensuring both efficiency and robustness throughout the optimization process. The method's functionality and performance are demonstrated using both toy problems and examples from deep learning.

Abstract

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