Establish an optimizer-agnostic rate condition

Establish an optimizer-agnostic necessity result for the penalty-growth rate condition governing data-adaptive relaxation, beyond the first-order local approximation around the unconstrained optimum.

Background

The paper derives the penalty-rate condition from a first-order expansion around the unconstrained edge weight and uses it to design AdaptiveRelax. Consequently, the condition is local and depends on the optimizer and the approximation regime.

The authors have not established a Lyapunov-style or two-timescale stochastic-approximation theorem showing that the condition remains necessary independently of the local expansion. They also have not determined how tight it is away from the single schedule used in the experiments.

References

We have neither a Lyapunov-style nor a two-timescale stochastic-approximation argument establishing necessity independent of the local expansion, nor an empirical sensitivity sweep over $\kappa$, $\rho_0$, and $\eta_r$ beyond the single schedule used throughout Section~\ref{sec:empirical}; how tight Eq.~\ref{eq:rate_condition} is away from that schedule is open.

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

Neither method changes how a forbidden-edge prior would be enforced on top of it, so Proposition~\ref{prop:conditions}'s necessary conditions would apply unchanged to either if a sequential-ramping ALM penalty were added; whether a more numerically stable base optimizer narrows the suppression window of Eq.~\ref{eq:window} is an open, testable question we do not address.

— Guide, Not Bind: Why Defeasible Priors Fail in Augmented Lagrangian Causal Discovery  (2609.03442 - Sundararaman et al., 3 Sep 2026) in Section 5, Related Work, paragraph “Optimization stability of differentiable causal discovery”; Limitations, item 9