Interaction Matrix
Statement
Let \( K \) blocks be updated sequentially, block \( i \) contracting in its own variables with rate \( \rho_i < 1 \), and let the influence of block \( j \) on block \( i \) be bounded by \( c\,\rho_i \) with a coupling constant \( c \ge 0 \). The interaction matrix is
\[ M_{ii} = \rho_i, \qquad M_{ij} = c\,\rho_i \quad (i \ne j). \]- Gershgorin bound. Every eigenvalue \( \lambda \) of \( M \) satisfies \( |\lambda| \le \rho_{\max}\,(1 + c(K-1)) \), where \( \rho_{\max} = \max_i \rho_i \).
- Balanced matrix. If all \( \rho_i = \rho \), then \( M = \rho\,[(1-c)I + c\,\mathbf{1}\mathbf{1}^{\mathsf T}] \) has eigenvalues \( \rho(1 + c(K-1)) \) (once, eigenvector \( \mathbf 1 \)) and \( \rho(1 - c) \) (multiplicity \( K-1 \), eigenvectors orthogonal to \( \mathbf 1 \)).
In particular the composed iteration contracts whenever \( \rho_{\max}(1 + c(K-1)) < 1 \), and in the balanced case exactly when \( \rho(1 + c(K-1)) < 1 \).
Derivation
Row \( i \) of \( M \) has diagonal entry \( \rho_i \) and off-diagonal absolute row sum \( c(K-1)\rho_i \). Gershgorin's theorem places every eigenvalue in some disc \( |\lambda - \rho_i| \le c(K-1)\rho_i \), so \( |\lambda| \le \rho_i(1 + c(K-1)) \le \rho_{\max}(1 + c(K-1)) \).
For the balanced matrix, \( \mathbf 1\mathbf 1^{\mathsf T} \) has eigenvalue \( K \) on \( \mathbf 1 \) and \( 0 \) on its orthogonal complement, so \( M \) has eigenvalues \( \rho(1-c) + \rho cK = \rho(1 + c(K-1)) \) and \( \rho(1-c) \). \( \blacksquare \)
How it is used
The composition argument (Theorem 10) needs a non-negative coupling matrix of spectral radius below one. Item 1 gives a sufficient condition from the block rates and a single coupling constant; item 2 shows the Gershgorin bound is attained in the balanced case, so it cannot be improved without more information about the couplings.
Learned interaction weights (experiment)
A separate experiment fits per-pair projection strengths \( A_{ij} \) by meta-gradient descent, with
\[ g_i' = g_i - \sum_{j \neq i} A_{ij} \cdot \frac{g_i \cdot g_j}{\|g_j\|^2} \, g_j . \]Numbers from those runs, reported as data:
| Quantity | Recorded value |
|---|---|
| Within-group \( A_{ij} \) (aligned objectives), mean | 0.003 |
| Between-group \( A_{ij} \) (competing objectives), mean | 0.782 |
| Weakly conflicting pairs | 0.2–0.4 |
| \( \|A^{(t)} - A^{(t-1)}\|_F \) at meta-cycles 2, 3, 4, 5 | 0.42, 0.11, 0.03, 0.004 |
| Recovery after a 20% perturbation of \( A \) | within 1% of prior values in \( O(10) \) cycles |
Experiment Files
exp_interaction_matrix.sx — meta-gradient fit of \( A \), change between meta-cycles
exp_symmetry_breaking.sx — grouped objectives, asymmetric weights, perturbation recovery