Two-Loop Direction

Use this skill when an optimizer has L-BFGS curvature memory and needs a quasi-Newton descent direction without materializing a Hessian matrix. Do not use it when curvature pairs fail the positive-curvature condition.

Inputs

• Current gradient vector.
• Ordered correction memory (s_i, y_i) from oldest to newest.
• Optional scalar initial inverse-Hessian scale.

Outputs

• Search direction p = -H g.
• Diagnostics with pair count, effective scale, and descent dot product.

Workflow

1. Run the first loop from newest pair to oldest and store alpha values.
2. Apply scalar initial scaling; use the newest pair scale (s^T y)/(y^T y) when none is supplied.
3. Run the second loop from oldest pair to newest.
4. Negate the resulting inverse-Hessian product.
5. Verify the result is a descent direction on convex positive-curvature tests.

Validation

Run python tests/test_two_loop_direction.py or use the generated skill tree validator with tests enabled.

Limitations

The skill computes directions only; it does not update memory or perform line search.