Defect-Regularized Particle Optimizer

Use this as a particle-based optimizer or compact ensemble with P=4 to 16 independently initialized parameter copies. At every training step, draw one minibatch per particle, compute stochastic gradients g_i = grad_theta L_Bi(theta_i), and update theta_i <- theta_i - h g_i. Select K random particle pairs from the P(P-1)/2 possible pairs. For each pair compute delta_ij = theta_i-theta_j and q_ij = alpha||delta_ij||^2 - dot(g_i-g_j,delta_ij). Accumulate max(0,q_ij)/(alpha||delta_ij||^2+epsilon) as the empirical defect loss. The lowest-cost version treats g_i and g_j as detached tensors and uses the defect only to adapt h or trigger particle repulsion; a differentiable regularizer requires higher-order autodiff and should initially be applied to only one or two pairs per step. A practical alternative is a shared-direction estimator: evaluate gradients at theta_i and theta_i + rho u, then use those two points in the same formula. Initialize alpha from the median positive directional secant, alpha = median(dot(g(theta+rho u)-g(theta),rho u)/rho^2), clipped to [1e-6,1]. Set lambda so the defect term is initially 1-10% of the supervised loss, and optionally reduce h when the batch-average defect exceeds a threshold. The paper supplies the defect definition and its additive effect on pairwise Euler stability; pair sampling, detached gradients, and lambda selection are engineering approximations. First test on CIFAR-10 with a four-particle CNN, comparing independent SGD, SWA, and ordinary deep ensembles at equal total examples and parameter-update FLOPs. Measure mean accuracy, ensemble negative log-likelihood, calibration error, pairwise parameter distance, loss-spike frequency, and wall-clock cost. Success means lower validation NLL or better calibration at matched accuracy with no more than 10% extra overhead. Failure is particle collapse, excessive higher-order autodiff cost, or no improvement over independent training.