Lorenz Conditioned Networks

Lorenz Conditioned Networks (LCN) extends Pareto Conditioned Networks (PCN) by replacing the dominance criterion used to select and condition on past episodes. Where PCN uses Pareto dominance, LCN instead uses Lorenz dominance: each return’s objective values are sorted in increasing order and cumulatively summed (its Lorenz vector), and one return Lorenz-dominates another if its Lorenz vector is at least as high in every position. Lorenz dominance favors returns that distribute reward more equitably across objectives.

The lcn_lambda parameter controls λ-Lorenz dominance, an interpolation between strict Lorenz dominance and ordinary dominance computed on the sorted returns:

fv = lcn_lambda * sorted(returns) + (1 - lcn_lambda) * lorenz_vector(returns)

lcn_lambda=0 recovers strict Lorenz dominance (maximizes fairness), while values closer to 1 relax the fairness bias towards rewarding raw performance on the best-performing sorted objectives. Pick one of the two distance_ref options to choose between the two:

• "nondominated" (default): strict Lorenz dominance.

• "lambda_lorenz": λ-Lorenz dominance, requires lcn_lambda to be set.

For more details, see:

Michailidis, D., Röpke, W., Roijers, D. M., Ghebreab, S., & Santos, F. P. (2026). Scalable Multi-Objective Reinforcement Learning with Fairness Guarantees using Lorenz Dominance. Journal of Artificial Intelligence Research, 85. https://jair.org/index.php/jair/article/view/19862