Skip to content

ADR-0011: Stratum-Aware Locality and Hybrid Optimization Strategy

Status

Accepted

Context

Axis 3 optimization trajectory testing revealed a fundamental property of the stratum-aware approach (Method C): within a stratum where the objective is invariant to a design variable, the gradient is exactly zero. This is mathematically correct but prevents gradient-based optimization from crossing stratum boundaries.

Concrete example: In the disjoint stratum, union_area = pi*r1^2 + pi*r2^2 is independent of center positions c1, c2. Method (C) correctly returns d(area)/d(c2_x) = 0. Gradient descent cannot cross from disjoint to intersecting by moving centers.

Method (A) smoothing provides a weak but non-zero gradient signal in this case because its smooth-min kernel extends beyond the exact boundary, creating a "shadow" of the intersecting stratum's area function into the disjoint regime.

This is structurally isomorphic to the visibility problem in differentiable rendering: fully occluded objects have zero gradient w.r.t. camera parameters. SoftRas (2019) solved this with soft rasterization; nvdiffrast (2020) with edge sampling. Both are hybrid approaches combining global signal with boundary precision.

Decision

Method (A) and Method (C) are complementary, not competing strategies. BRepAX will provide a hybrid framework that combines:

  • Method (A) for cross-stratum exploration (global gradient signal via smoothing)
  • Method (C) for within-stratum precision (exact analytical gradients)

Gate 2 Reinterpretation

The original Gate 2 criterion ("Method (C) converges to cross-stratum optimum") was based on an incomplete understanding of stratum-aware gradient structure. Revised:

  • Gate 2a (within-stratum precision): Method (C) reaches analytical optimum with position error 7e-6 vs Method (A)'s 0.021. PASS.
  • Gate 2b (cross-stratum behavior): Gradient zero in stratum-invariant directions is a mathematical property, not a defect. REINTERPRETED.

Narrative Adjustment

  • Previous: "Method (C) is superior to Method (A)"
  • Revised: "Method (A) and (C) are complementary; BRepAX provides a coherent framework for combining both"

This parallels the differentiable rendering trajectory: SoftRas (global, approximate) was not replaced by nvdiffrast (precise, local) but complemented by it. BRepAX occupies the same structural position for CAD geometry.

Method (B) Reconsideration

The TOI correction (Method B, deferred in ADR-0009) was designed precisely for boundary-crossing gradient computation. The Axis 3 finding strengthens the case for Method (B) implementation in future work: it provides the mathematical machinery to transfer gradient information across stratum boundaries via the implicit function theorem.

Re-evaluation trigger: when a use case requires single-method cross-stratum optimization without a coarse-to-fine schedule.

Empirical Confirmation

3D benchmark testing confirmed the hybrid strategy with three findings:

  1. Disjoint→intersecting stall validated. intersect_volume_stratum with two disjoint spheres (d=2.5) returns gradient exactly zero. Optimization stalls at step 0. Method A is the only path to cross-stratum exploration in this direction.

  2. Within-stratum optimization works. Adam optimizer on sphere-sphere and cylinder-sphere objectives converges to < 1% of analytical optimum despite near-tangent STE error of 25-542%. Gradient direction (sign) is always correct.

  3. Contained↔intersecting crossing is smooth. Method C handles the contained-intersecting boundary without stalling, unlike the disjoint-intersecting boundary. This asymmetry is expected: contained→intersecting involves a continuous volume change, while disjoint→intersecting involves a volume jump from zero.

The three-problem decomposition (Problem A: disjoint stall, Problem B: STE near-tangent bias, Problem C: Path C frozen topology) confirms that the hybrid framework addresses Problem A while Method C alone handles Problems B and C adequately for practical optimization.

See tests/benchmarks/test_gradient_accuracy_3d.py for data.

Consequences

  • Hybrid optimizer design becomes a first-class concern (not an afterthought)
  • boolean/__init__.py union_area API remains method-agnostic; hybrid scheduling is handled at the optimizer level
  • Method (B) regains relevance as a potential bridge between (A) and (C)
  • The differentiable rendering analogy provides a well-studied template for the hybrid design