Weight perturbation (WP) estimates learning directions from random parameter perturbations and a scalar change in loss, without backpropagating derivatives. We examine how batch size, training duration, and estimator stabilization affect its attainable accuracy in fully connected MNIST networks. In the shallow configuration, large-batch training reaches approximately 97\% test accuracy, compared with an early backpropagation (BP) reference near 97.5--98\%. The same comparison exposes a substantial efficiency gap: under a ten-epoch budget, shallow WP reaches 90.2\%, whereas the reported BP reference reaches 97.4\%. Epoch-indexed curves show diminishing returns from increasing batch size beyond 4,000. Learning-rate reduction and averaging independent perturbation-based updates also stabilize deeper networks, although their recorded results do not establish BP-level accuracy. We distinguish these empirical findings from a finite-horizon approximation argument: under stated regularity conditions, population WP can approximate a fixed gradient-descent trajectory arbitrarily closely at any fixed finite depth. Positive classification margins then suffice to preserve the reference predictions on a finite test set with arbitrarily high probability. Together, the results identify near-BP accuracy as a configuration- and budget-dependent possibility, rather than evidence of equal efficiency or an unconditional accuracy guarantee across depth.

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