# Nullspace Inverse-Loss Identification

- ID: 2478
- Canonical URL: https://synthcore.org/idea/2478/nullspace-inverse-loss-identification
- API JSON: https://synthcore.org/api/idea/2478.json
- API Markdown: https://synthcore.org/api/idea/2478.md
- Verification status: unverified
- Source: [arXiv:2608.17939](https://arxiv.org/abs/2608.17939)
- Category: theory
- Solves: stability, generalization
- ML areas: optimizer, training-dynamics, regularization
- Math tags: control-theory, optimization, linear-algebra, stochastic-processes, dynamical-systems
- Ratings: usefulness 6/10; difficulty 6/10; novelty 8/10

## Idea description

Use a window of observed neural-network update trajectories to identify the set of local quadratic objectives and preconditioners that are consistent with the observed optimizer behavior. Rather than selecting one arbitrary curvature model, retain the nullspace of compatible parameters and use its dimension or smallest singular value as an identifiability and stability diagnostic.

## Mathematical statement

Consider a local stochastic quadratic model for module i, with state z in R^d, action u_i in R^{p_i}, drift dz=(Az+sum_j B_j u_j)dt, and diffusion G(z,u)dW where G=Cz+sum_j D_j u_j. Under linear feedback u_j=-K_jz, define A_K=A-sum_j B_jK_j and C_K=C-sum_j D_jK_j. For quadratic value V_i(z)=z^T P_i z, the stochastic HJB condition is the coupled Riccati equation 0=Q_i+K_i^T R_{ii}K_i+A_K^T P_i+P_i A_K+C_K^T P_i C_K, together with stationarity involving S_i=R_{ii}+D_i^T P_i D_i. Observed K and known or fitted dynamics produce linear homogeneous constraints M_i theta_i=0 after vectorizing unknown cost parameters theta_i, while admissibility imposes Q_i positive semidefinite and R_{ii} positive definite. The compatible set is Theta_i=ker(M_i) intersected with the admissible set, and its dimension is the number of unidentifiable objective directions.

## Key formulas

- $$d z_t=\left(Az_t+\sum_{j=1}^{N}B_j u_{j,t}\right)dt+\left(Cz_t+\sum_{j=1}^{N}D_j u_{j,t}\right)dW_t,\qquad u_{j,t}=-K_jz_t,$$
- $$A_K=A-\sum_jB_jK_j,\qquad C_K=C-\sum_jD_jK_j,$$
- $$0=Q_i+K_i^{\mathsf T}R_{ii}K_i+A_K^{\mathsf T}P_i+P_iA_K+C_K^{\mathsf T}P_iC_K,$$
- $$\widehat{\Theta}_i=\{\theta_i:\|M_i\theta_i\|_2\leq\varepsilon,\ Q_i\succeq0,\ R_{ii}\succ0\},\qquad \dim(\ker M_i)=n_{\theta_i}-\operatorname{rank}(M_i).$$

## Implementation notes

Integrate this as an offline diagnostic or optimizer-design module, not as a replacement for backpropagation. First choose a parameter block of a small MLP, flatten its parameters into z_t, and interpret the observed update as an action u_t=z_{t+1}-z_t. For a short training window, fit a local linear stochastic model z_{t+1}-z_t approximately equal to A z_t+B u_t plus residual noise; for Adam, either use the raw update as u_t or use the preconditioned gradient before the learning-rate multiplication. Assume a candidate local quadratic objective z^TQz+u^TRu and construct M by vectorizing the stationarity and Riccati/HJB residuals evaluated on the fitted A,B,C,D and observed feedback K. Pseudocode: collect (z_t,u_t); estimate A,B and conditional noise covariance; estimate K by least squares from u_t approximately equal to -Kz_t; build M from the vectorized residual equations; compute the SVD M=U Sigma V^T; retain singular vectors with singular value below epsilon; sample theta from this nullspace and project Q to the positive-semidefinite cone and R to the positive-definite cone. The Riccati structure comes from the paper, whereas A,B,C,D,K and noise statistics are estimated empirically, so estimation error must be measured separately. The first cheap experiment should use a two-layer MLP on MNIST and a synthetic quadratic-regression task, comparing SGD, momentum, and Adam over 10 trajectory windows. The quantitative prediction is that the smallest singular values of M decrease when trajectories visit fewer independent parameter directions, and every sampled admissible theta should have residual norm at the noise floor. Deliberately reduce the window rank; the measured ambiguity dimension should equal d_theta-rank(M) within one dimension, while adding diverse perturbations should increase rank and reduce ambiguity.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.