I-Ratio Theorem
Theorem Statement
Let \( g_1, g_2, \ldots, g_K \) be gradient vectors of \( K \geq 2 \) objectives. Define the interaction ratio:
\[ I(\theta) \;=\; \frac{\displaystyle\sum_{i < j} g_i \cdot g_j}{\displaystyle\sum_{i} \|g_i\|^2} \]Then \( I(\theta) = -\dfrac{1}{2} \) if and only if the system is at equilibrium, i.e., \( \displaystyle\left\|\sum_{i} g_i\right\|^2 = 0 \).
Proof
Expand the squared norm of the total gradient:
\[ \left\|\sum_{i} g_i\right\|^2 = \sum_{i} \|g_i\|^2 + 2\sum_{i < j} g_i \cdot g_j \]At equilibrium, \( \left\|\sum_{i} g_i\right\|^2 = 0 \), so:
\[ 0 = \sum_{i} \|g_i\|^2 + 2\sum_{i < j} g_i \cdot g_j \] \[ \frac{\sum_{i < j} g_i \cdot g_j}{\sum_{i} \|g_i\|^2} = -\frac{1}{2} \]The converse follows by reversing the algebra. \( \blacksquare \)
Significance
The I-ratio provides a single scalar diagnostic for multi-objective equilibrium. Unlike loss values (which are scale-dependent), \( I = -\frac{1}{2} \) is a universal constant that holds regardless of the number of objectives, their magnitudes, or the dimensionality of parameter space.
- Works for any \( K \geq 2 \) objectives
- Scale-invariant (ratio of same-unit quantities)
- Computable in \( O(K^2 d) \) time from gradient inner products
- Deviation \( |I + 0.5| \) measures distance from equilibrium
Empirical Evidence
| Test Suite | Cases | Pass | Max Error |
|---|---|---|---|
| Structured problems, K = 2..20 | 138 | 138/138 | \( 2.22 \times 10^{-16} \) |
| Random problems (statistical) | 70 | 70/70 | \( 2.22 \times 10^{-16} \) |
| Cross-domain (5 domains) | 5 | 5/5 | Machine precision |
The maximum observed error of \( 2.22 \times 10^{-16} \) is exactly IEEE 754 double-precision machine epsilon. The identity is algebraically exact.
Experiment Files
exp_iratio_proof.sx — 138 structured test cases across K = 2 to 20 objectives
exp_iratio_proof_statistical.sx — 70 randomly generated multi-objective problems
exp_iratio_applications.sx — Cross-domain validation in 5 application areas