Run 887ed549

kind=qa

Roughly, a test suite should isolate the four pieces the paper claims credit for: (1) the rate, (2) avoiding Hessian inversions, (3) the barrier-metric geometry mattering, and (4) bias control as the barrier parameter shrinks.

**Rate verification.** Run on synthetic linearly-constrained bilevel instances with closed-form or near-closed-form solutions so a true stationarity measure is computable. Log-log plot stationarity vs. outer iterations $K$, fit the slope, and check it matches $-2/3$ deterministic and $-2/5$ with injected upper-level noise. Sweep noise variance; the deterministic rate should kick in as noise → 0.

**Hessian-free benefit.** Head-to-head against standard hypergradient baselines that do require lower-level Hessian inversions or linear solves — AID, ITD/unrolling, conjugate-gradient-based implicit differentiation. Compare both per-iteration cost and wall-clock to a fixed stationarity tolerance, sweeping lower-level dimension $n$ and number of polyhedral inequalities $m$ so the linear-solve cost is visibly punitive on the baselines.

**Barrier-metric ablation.** Same algorithm, but replace the Dikin-geometry step-size/trust-region schedule with vanilla Euclidean step sizes tuned by grid search. On instances whose lower-level optimum sits near a face of the polytope, the Euclidean variant should either stall, oscillate, or blow up while the barrier-metric variant stays well-behaved. Plot distance-to-boundary of iterates vs. iteration to confirm iterates stay in the "locally well-behaved region" claimed.

**Active-set-change stress test.** Construct instances where the lower-level optimum crosses face boundaries as the upper variable moves — a regime where the unsmoothed upper objective is genuinely nonsmooth. Show the barrier-smoothed proxy gradient still produces descent while a naïve differentiation-through-KKT approach fails or produces inconsistent hypergradients.

**Bias control vs. $\mu$.** Vary the barrier parameter on a schedule and on fixed values; measure the gap between the barrier-smoothed stationary point and a stationary point of the true constrained bilevel objective. The reported quantitative bias bound predicts a specific decay in $\mu$ — verify the empirical bias scales the same way.

**Realistic applications.** At least one bilevel problem that is not a toy: e.g., hyperparameter optimization for a regularized lower-level problem with simplex/box constraints, a Stackelberg pricing problem with linear feasibility, or meta-learning with constrained inner updates. The point is to show the rate and cost advantages survive outside the synthetic regime.

**Sanity checks worth adding.** (i) Robustness to barrier-parameter mis-specification, (ii) sensitivity to polytope conditioning (long thin polytopes vs. well-rounded ones, since Dikin geometry is conditioning-aware), and (iii) verification that the only second-order object actually used is the explicit barrier Hessian — i.e., no hidden linear solves on the lower-level Hessian.

(Caveat: I'm reasoning from the abstract and summary you stored, not the full paper, so the bias-decay rate and the exact noise model would need to be read off Theorems 1–2 before designing the exact $\mu$- and noise-schedules.)

<<<DONE>>>

success

Roughly, a test suite should isolate the four pieces the paper claims credit for: (1) the rate, (2) avoiding Hessian inversions, (3) the barrier-metric geometry mattering, and (4) bias control as the barrier parameter shrinks.

**Rate verification.** Run on synthetic linearly-constrained bilevel instances with closed-form or near-closed-form solutions so a true stationarity measure is computable. Log-log plot stationarity vs. outer iterations $K$, fit the slope, and check it matches $-2/3$ deterministic and $-2/5$ with injected upper-level noise. Sweep noise variance; the deterministic rate should kick in as noise → 0.

**Hessian-free benefit.** Head-to-head against standard hypergradient baselines that do require lower-level Hessian inversions or linear solves — AID, ITD/unrolling, conjugate-gradient-based implicit differentiation. Compare both per-iteration cost and wall-clock to a fixed stationarity tolerance, sweeping lower-level dimension $n$ and number of polyhedral inequalities …