Sphere-Jacobian Performative Optimizer

Integrate this at the optimizer step, not inside the network forward pass. The model has parameters \(\theta\); the environment must expose an observable low-dimensional distribution statistic \(\beta\), such as class frequencies, click-rate bins, demand moments, user-state proportions, or a learned environment embedding. First compute the ordinary gradient \(g_{\theta,t}\) on data generated by the current deployed model. Then estimate the response Jacobian using paired environment rollouts. Pseudocode: (1) sample \(u_i\sim\mathrm{Uniform}(S^{n-1})\), or sample Gaussian vectors and normalize them; (2) for every direction, temporarily deploy \(\theta_t+\delta u_i\), collect \(b_{\min}\) environment samples, and compute \(\hat\beta_i^+\); (3) deploy \(\theta_t-\delta u_i\), collect an independently seeded matched batch, and compute \(\hat\beta_i^-\); (4) for each coordinate \(j\), form \(v_j=\frac{n}{b_{\mathrm{sp}}}\sum_i((\hat\beta_i^+-\hat\beta_i^-)_j/(2\delta))u_i\), and stack \(v_j\) as \(\widehat J^\top\); (5) estimate \(g_{\beta,t}=\nabla_\beta\mathbb{E}_{z\sim\mathcal D_\beta}[\ell(z;\theta_t)]\), either analytically through a known simulator or by fitting a small differentiable surrogate \(r_\phi(\beta,\theta)\) to held-out rollout losses; (6) update \(\theta\leftarrow\Pi_\Theta(\theta-\alpha(g_\theta+\widehat J^\top g_\beta))\). Estimate \(\sigma_{\max}^2\) from repeated rollout batches and estimate \(\|\widehat J\|_F\) directly; tune \(\delta\) by minimizing the empirical proxy consisting of the decreasing noise term \(1/\delta^2\) and increasing finite-difference term \(\delta^4\). For a cheap first experiment, use a two-layer MNIST or CIFAR-10 classifier connected to a synthetic feedback environment in which predicted class probabilities alter the next-round class mixture, for example \(\beta(\theta)=\mathrm{softmax}(A\,\bar p_\theta)\) with noisy sampled labels. Compare SGD/Adam, an oracle full-Jacobian optimizer, and the proposed estimator at matched environment rollouts. Start with \(n\) reduced by optimizing a low-rank adapter rather than all weights, \(b_{\mathrm{sp}}\in\{4,8,16\}\), and several \(\delta\) values. Success means lower true post-deployment performative loss, reduced oscillation of the induced class distribution, and a smaller gap to the oracle at equal rollout budget; a useful target is at least 20% lower final performative loss or materially faster loss descent than standard SGD.