Cubature-Embedded Digital-Net Minibatches

Integrate this at the minibatch sampler for an objective L(theta)=E_{u~Unif([0,1]^d)}[ell(theta;u)], where u contains continuous augmentation parameters, bounded latent variables, random mask probabilities, or other auxiliary randomness. Generate an N=2^m base-2 Sobol or digital-net tensor U with shape [N,d], preferably using a digital scrambling that preserves projected stratification. Choose p so that 2^p is compatible with the batch size and prepare a one-dimensional equal-weight cubature table C of length 2^p; initially use midpoint nodes c_r=(r+0.5)/2^p, then test a higher-polynomial-exactness table. Compute A=floor(2^p U), gather X=C[A], and pass X into the existing augmentation or loss code. If the important coordinates are S, inspect the tuples A[:,S] and verify that they cover the full product grid; do not assume this property for an arbitrary Sobol implementation. For Gaussian variables, apply a fixed inverse-normal CDF after X, while interpreting the cubature guarantee on the underlying uniform variables. Pseudocode is: generate U; A=floor(2^p*U); X=gather(C,A); loss=mean(ell(theta,X)); backpropagate; update parameters. The paper supplies the substitution and exact full-grid criterion. Estimate projected coverage, gradient variance, and sampling overhead empirically. The first experiment should train a small MLP on MNIST or CIFAR-10 with two to four random continuous augmentation parameters, comparing iid-uniform, ordinary Sobol, and transformed-Sobol sampling at identical batch size, optimizer, and number of examples. Use at least ten seeds and measure gradient variance, loss versus examples, validation accuracy, and wall-clock cost. Success means lower gradient variance and faster validation-loss reduction at equal examples without significant sampler overhead.