Remove the structural asymmetry of one-sided ALM penalties

Construct a defeasible-prior mechanism that removes the structural asymmetry of an Augmented Lagrangian constraint by preventing weight displaced from a wrongly forbidden edge from being absorbed by its unconstrained reverse direction.

Background

AdaptiveRelax combined with covariance matching improves recovery of wrongly forbidden edges but remains unsuccessful in most trials: the correct edge is recovered in 24.7% of trials, while the reverse edge is active in 41.1%. The paper attributes this residual failure to the ALM term penalizing only the forbidden direction and leaving the reverse direction unconstrained.

Neither of the four fitting objectives nor either relaxation operator tested addresses this directional asymmetry. The authors identify designing a mechanism that directly removes it as the principal unresolved direction of the work.

References

Closing this gap requires breaking the ALM constraint's structural asymmetry---penalizing only one direction of a covered pair---which none of the four fitting objectives or either relaxation operator we tested addresses; we do not have a design for this and name it as the paper's principal open direction.

— Guide, Not Bind: Why Defeasible Priors Fail in Augmented Lagrangian Causal Discovery  (2609.03442 - Sundararaman et al., 3 Sep 2026) in Limitations, item 1; reiterated in Section 7, Conclusion

Building a mechanism that removes this asymmetry directly, rather than compensating for its consequences one objective at a time, is the next necessary condition---and, unlike the four this paper derives and tests, we do not yet know how to state it in closed form.

— Guide, Not Bind: Why Defeasible Priors Fail in Augmented Lagrangian Causal Discovery  (2609.03442 - Sundararaman et al., 3 Sep 2026) in Section 7, Conclusion