I-Ratio Identity
Statement
Let \( g_1, g_2, \ldots, g_K \) be gradient vectors of \( K \geq 2 \) objectives, and write \( D = \sum_i \|g_i\|^2 \). Define the interaction ratio
\[ I(\theta) \;=\; \frac{\displaystyle\sum_{i < j} g_i \cdot g_j}{D}. \]Then
\[ \left\|\sum_{i} g_i\right\|^2 \;=\; D\,(1 + 2I). \]Consequently \( I \in \left[-\tfrac{1}{2},\, \tfrac{K-1}{2}\right] \): the lower end is attained exactly when the gradients sum to zero, the upper end when all gradients coincide.
Derivation
Expand the squared norm of the total gradient:
\[ \left\|\sum_{i} g_i\right\|^2 = \sum_{i} \|g_i\|^2 + 2\sum_{i < j} g_i \cdot g_j = D + 2DI. \]Non-negativity of the left side gives \( 1 + 2I \ge 0 \). Cauchy–Schwarz on each pair gives \( \sum_{i
Use as a diagnostic
\( I \) is a scale-free scalar computable from the \( O(K^2) \) gradient inner products. The deviation \( |I + \tfrac12| \) measures, relative to \( D \), how far the summed gradient is from zero; it says nothing about whether that point is a minimum, a saddle, or a stable point of any composed update. The same quantity appears as the balance residual \( B = 1 + 2I \) on the B-flow page.
- Defined for any \( K \geq 2 \) objectives
- Scale-invariant (ratio of same-unit quantities)
- Computable in \( O(K^2 d) \) time from gradient inner products
Implementation checks
The checks below verify that the implementation computes the identity to double precision. They test the arithmetic, not any property of equilibria.
| Test suite | Cases | Pass | Max error |
|---|---|---|---|
| Structured problems, K = 2..20 | 138 | 138 | \( 2.22 \times 10^{-16} \) |
| Random problems | 70 | 70 | \( 2.22 \times 10^{-16} \) |
| Five illustrative domains | 5 | 5 | \( 2.22 \times 10^{-16} \) |
The maximum error equals IEEE 754 double-precision epsilon, as expected for an exact algebraic identity.
Experiment Files
exp_iratio_proof.sx — 138 structured cases across K = 2 to 20 objectives
exp_iratio_proof_statistical.sx — 70 randomly generated multi-objective problems
exp_iratio_applications.sx — the diagnostic computed in five illustrative settings