This work studies heterogeneous Multi-Objective Reinforcement Learning (MORL), where objectives can differ sharply in temporal frequency. Such heterogeneity allows dense objectives to dominate learning, while sparse long-horizon rewards receive weak credit assignment, leading to poor sample efficiency. We propose a Parallel Reward Integration with Symmetry (PRISM) algorithm that enforces reflectional symmetry as an inductive bias in aligning reward channels. PRISM introduces ReSymNet, a theory-motivated model that reconciles temporal-frequency mismatches across objectives, using residual blocks to learn a scaled opportunity value that accelerates exploration while preserving the optimal policy. We also propose SymReg, a reflectional equivariance regulariser that enforces agent mirroring and constrains policy search to a reflection-equivariant subspace. This restriction provably reduces hypothesis complexity and improves generalisation. Across MuJoCo benchmarks, PRISM consistently outperforms both a sparse-reward baseline and an oracle trained with full dense rewards, improving Pareto coverage and distributional balance: it achieves hypervolume gains exceeding 100% over the baseline and up to 32% over the oracle. The code is at https://github.com/EVIEHub/PRISM.
Figures & tables
Figure 1: Reflectional symmetry in a two-legged agent. The left panel shows a transition from state s to s′ under action a , whereas the right panel shows the reflected transition, where states and actions are transformed by Lg and Kg , respectively.
Figure 2: Overview of ReSymNet.
Figure 3: The obtained hypervolume for various levels of sparsity amongst various dimensions.
Figure 4: The approximated Pareto front for dense rewards (blue dots) and sparse rewards (orange dots) for the first reward objective.
Environment
Metric
Oracle
Baseline
PRISM
Mo-hopper-v5
HV ( ×107 )
1.30 ± 0.13
0.84 ± 0.05
1.58 ± 0.05
EUM
129.04 ± 7.96
97.64 ± 4.18
147.43 ± 2.61
VO
59.07 ± 3.45
43.36 ± 1.61
66.66 ± 1.40
Mo-walker2d-v5
HV ( ×104 )
4.21 ± 0.11
3.34 ± 0.16
4.77 ± 0.07
EUM
107.58 ± 2.86
82.13 ± 4.34
120.43 ± 1.64
VO
53.22 ± 1.39
39.18 ± 2.49
59.35 ± 0.80
Table 1: Experimental results. We report the average hypervolume (HV), Expected Utility Metric (EUM), and Variance Objective (VO) over 10 trials, with the standard error shown in grey. The largest (best) values are in bold font.
Figure 5: The dense (blue line) and shaped rewards (orange line) over time for mo-walker2d-v5 and the first reward objective.
Appendix figures & tables13 assets
Supplementary material from the paper’s appendix.
Appendix
Symbol
Description
S
State space
A
Action space
P(s′∣s,a)
Transition probability
r(s,a)∈RL
Vector-valued reward with L objectives
γ∈[0,1)
Discount factor
π:S→A
Policy mapping
Appendix
Table 2: Notation.
State Space
Action Space
Reward Space
Mo-hopper-v5
S∈R11
A∈R3
R∈R3
Mo-walker2d-v5
S∈R17
A∈R6
R∈R2
Mo-halfcheetah-v5
S∈R17
A∈R6
R∈R2
Mo-swimmer-v5
S∈R8
A∈R2
R∈R2
Appendix
Table 3: Considered MuJoCo environments.
Index
Observation Component
Type
Symmetry
0
z-coordinate of the torso
position
Asymmetric
1
angle of the torso
angle
Asymmetric
2
angle of the thigh joint
angle
Symmetric
3
angle of the leg joint
angle
Symmetric
4
angle of the foot joint
angle
Symmetric
5
velocity of the x-coordinate of the torso
velocity
Asymmetric
Appendix
Table 4: Reflectional symmetry partition for mo-hopper-v5 observation space.
Index
Observation Component
Type
Symmetry
0
z-coordinate of the torso
position
Asymmetric
1
angle of the torso
angle
Asymmetric
2
angle of the thigh joint
angle
Symmetric
3
angle of the leg joint
angle
Symmetric
4
angle of the foot joint
angle
Symmetric
5
angle of the left thigh joint
angle
Symmetric
Appendix
Table 5: Reflectional symmetry partition for mo-walker2d-v5 observation space.
Index
Observation Component
Type
Symmetry
0
z-coordinate of the front tip
position
Asymmetric
1
angle of the front tip
angle
Asymmetric
2
angle of the back thigh
angle
Symmetric
3
angle of the back shin
angle
Symmetric
4
angle of the back foot
angle
Symmetric
5
angle of the front thigh
angle
Symmetric
Appendix
Table 6: Reflectional symmetry partition for mo-halfcheetah-v5 observation space.
Index
Observation Component
Type
Symmetry
0
angle of the front tip
angle
Asymmetric
1
angle of the first rotor
angle
Symmetric
2
angle of the second rotor
angle
Symmetric
3
velocity of the tip along the x-axis
velocity
Asymmetric
4
velocity of the tip along the y-axis
velocity
Symmetric
5
angular velocity of the front tip
angular velocity
Asymmetric
Appendix
Table 7: Reflectional symmetry partition for mo-swimmer-v5 observation space.
PRISM
Initial collection N
1000
Expert collection E
1000
Number of refinements IR
2
Timesteps per cycle M
100,000
Epochs
1000
Learning rate
0.005
Appendix
Table 8: Hyperparameters for ReSymNet.
Mo-hopper-v5
Mo-walker2d-v5
Mo-halfcheetah-v5
Mo-swimmer-v5
λ
0.01
1
0.01
0.005
Appendix
Table 9: SymReg hyperparameter.
Figure 6: The approximated Pareto front for dense rewards (blue dots), sparse rewards (orange dots), and shaped rewards (green dots). Sparsity is imposed on the first reward objective.
Environment
Metric
uniform
random
Mo-hopper-v5
HV ( ×107 )
1.38 ± 0.08
0.49 ± 0.06
EUM
135.19 ± 5.30
65.22 ± 6.63
VO
63.90 ± 2.34
29.62 ± 3.68
Mo-walker2d-v5
HV ( ×104 )
4.67 ± 0.07
1.18 ± 0.10
EUM
116.72 ± 2.11
16.52 ± 4.98
VO
56.22 ± 1.01
3.77 ± 2.46
Appendix
Table 11: ReSymNet ablation study results. We report the average hypervolume (HV), Expected Utility Metric (EUM), and Variance Objective (VO) over 10 trials, with the standard error shown in grey. w/o is the abbreviation of without.
Environment
Metric
Oracle
Baseline
PRISM
Mo-hopper-v5
HV ( ×107 )
1.30 ± 0.13
1.19 ± 0.10
1.51 ± 0.11
EUM
129.04 ± 7.96
124.82 ± 7.21
142.89 ± 7.38
VO
59.07 ± 3.45
56.21 ± 3.20
67.58 ± 3.31
Mo-walker2d-v5
HV ( ×104 )
4.21 ± 0.11
3.16 ± 0.13
4.59 ± 0.14
EUM
107.58 ± 2.86
85.95 ± 3.27
114.62 ± 2.80
VO
53.22 ± 1.39
41.29 ± 1.49
54.84 ± 1.25
Appendix
Table 12: Experimental results on the control cost objective. We report the average hypervolume (HV), Expected Utility Metric (EUM), and Variance Objective (VO) over 10 trials, with the standard error shown in grey. The largest (best) values are in bold font.
Figure 7: The obtained hypervolume for various levels of sparsity for PRISM.
Environment
Metric
Oracle
Baseline
PRISM
Mo-hopper-v5
HV ( ×107 )
1.65 ± 0.10
0.67 ± 0.04
1.65 ± 0.07
EUM
151.45 ± 5.87
85.87 ± 3.17
148.19 ± 4.26
VO
72.26 ± 2.90
41.21 ± 1.44
70.24 ± 2.51
Mo-walker2d-v5
HV ( ×104 )
5.93 ± 0.10
3.20 ± 0.23
5.61 ± 0.10
EUM
141.88 ± 2.38
76.41 ± 6.47
132.67 ± 2.26
VO
67.63 ± 1.17
35.64 ± 3.91
63.19 ± 1.75
Appendix
Table 13: Experimental results of GPI-PD. We report the average hypervolume (HV), Expected Utility Metric (EUM), and Variance Objective (VO) over 10 trials, with the standard error shown in grey. The largest (best) values are in bold font.
Multi-objective reinforcement learning (MORL) is a powerful tool to learn Pareto-optimal policy families across conflicting objectives. However, unlike traditional RL algorithms, existing MORL algorithms do not effectively leverage large-scale parallelization to concurrently simulate thousands of environments, resulting in vastly increased computation time. Ultimately, this has limited MORL's application towards complex multi-objective robotics problems. To address these challenges, we present 1) MORLAX, a new GPU-native, fast MORL algorithm, and 2) MO-Playground, a pip-installable playground of GPU-accelerated multi-objective environments. Together, MORLAX and MO-Playground approximate Pareto sets within minutes, offering 25-270x speed-ups compared to legacy CPU-based approaches whilst achieving superior Pareto front hypervolumes. We demonstrate the versatility of our approach by implementing a custom BRUCE humanoid robot environment using MO-Playground and learning Pareto-optimal locomotion policies across 6 realistic objectives for BRUCE, such as smoothness, efficiency and arm swinging.
Neil Janwani, Ellen Novoseller, Vernon J. Lawhern +1
Georgia Institute of Technology, Atlanta, GA USA · DEVCOM Army Research Laboratory, MD, USA
Many sequential decision-making tasks involve optimizing multiple conflicting objectives, requiring policies that adapt to different user preferences. In multi-objective reinforcement learning (MORL), one widely studied approach} addresses this by training a single policy network conditioned on preference-weighted rewards. In this paper, we explore a novel algorithmic perspective: leveraging reward-free reinforcement learning (RFRL) for MORL. While RFRL has historically been studied independently of MORL, it learns optimal policies for any possible reward function, making it a natural fit for MORL's challenge of handling unknown user preferences. We propose using the RFRL's training objective as an auxiliary task to enhance MORL, enabling more effective knowledge sharing beyond the multi-objective reward function given at training time. To this end, we adapt a state-of-the-art RFRL algorithm to the MORL setting and introduce a preference-guided exploration strategy that focuses learning on relevant parts of the environment. Through extensive experiments and ablation studies, we demonstrate that our approach significantly outperforms the state-of-the-art MORL methods across diverse MO-Gymnasium tasks, achieving superior performance and data efficiency. This work provides the first systematic adaptation of RFRL to MORL, demonstrating its potential as a scalable and empirically effective solution to multi-objective policy learning.
Ying-Tu Chen, Wei Hung, Bing-Shu Wu +2
National Yang Ming Chiao Tung University, Hsinchu, Taiwan · Massachusetts Institute of Technology
Recent advances in deep reinforcement learning (RL) have shown that improving neural network architectures can yield substantial gains in sample efficiency and asymptotic performance without altering the underlying algorithms. In contrast, work on multi-objective reinforcement learning (MORL), which aims to discover a set of policies that balance trade-offs among conflicting objectives, has predominantly focused on algorithmic innovations, leaving the area of architectures underexplored. While the optimal policies and value functions can differ significantly depending on the trade-offs, MORL algorithms commonly represent them with simple feedforward networks conditioned on the trade-off. This raises the question of whether the performance of the algorithms could be improved with more expressive function approximators. In this paper, we integrate recent advances in neural network design: (i) observation and feature normalization, (ii) weight normalization, and (iii) modeling of distributional returns with an entropy-regularized MORL algorithm. The empirical results across standard continuous control benchmarks demonstrate that these changes substantially improve the quality of the produced solution sets without requiring major changes to the underlying algorithm.
Adam Štafa, Santeri Heiskanen, Petr Novotný +1
Faculty of Informatics, Masaryk University · Department of Electrical Engineering and Automation, Aalto University