Method

replicator_nashmtl is a two-level weighting method.

Inner level: Nash bargaining

At every optimization step, Nash-MTL computes a bargaining-based compromise over task gradients. This produces the raw nash_weights, which preserve the step-level fairness and conflict-handling behavior of the original method.

Outer level: replicator scheduling

A slower replicator-style scheduler maintains task shares over time. Its role is not to override Nash, but to respond to slower training dynamics such as:

• which tasks are underperforming relative to others
• which tasks are improving more slowly
• which tasks may need more long-run support

The preferred toy signal is qm9_proxy, which blends:

• current relative task loss
• EMA-smoothed under-improvement
• relative task imbalance

Final weighting rule

The final optimization weights come from combining the two levels:

• nash_weights supply the stepwise bargaining solution
• replicator_shares supply the slower adaptive allocation
• final_weights are formed by modulating Nash with the scheduler state

This hierarchy matters. The method is meant to stay recognizably Nash-MTL while gaining a controlled outer adaptation mechanism.

Why the repaired version is more stable

The old failure mode came from an overly aggressive scheduler on the hard init=1 case. The repaired setup stabilizes that behavior through:

• smoother improvement normalization
• clipping and payoff shaping
• gentler replicator learning dynamics
• nonzero uniform mixing
• moderated modulation strength

Final toy settings

--scheduler-signal qm9_proxy
--debug-init-preset representative
--replicator-update-every 1
--replicator-lr 0.05
--replicator-uniform-mix 0.05
--replicator-modulation-strength 0.3
--toy-payoff-temperature 0.1
--toy-payoff-gain 5.0
--n-epochs 220

What to inspect during runs

The most useful outputs are:

• replicator_shares
• nash_weights
• final_weights

Together they show whether changes are coming from the scheduler, the bargaining layer, or the interaction between them.