Layerwise Conditioned Backpropagation: A Curvature-Aware Reparameterization of the Backward Pass with Convergence Guarantees

Backpropagation is less a single algorithm than a pipeline of choices: how the error signal is propagated, how the weight gradient is assembled, and how the update is applied. This paper revisits three consecutive steps and proposes small, mathematically transparent modifications that improve gradient scaling and conditioning without changing the represented function class. The resulting method, Conditioned Backpropagation(CBP), combines (i) a layerwise gradient-norm equalization that counters the geometric depth dependence of the backpropagated error; (ii) an activation-centering reparameterization that removes the dominant rank-one mean term from the per-layer curvature; and (iii) a damped diagonal preconditioner that is positive-definite by construction. The composite operator is a bounded positive-definite preconditioner, so the method inherits standard nonconvex, Polyak–ojasiewicz, and stochastic convergence guarantees at the per-step cost of ordinary backpropagation. No prior method composes these three repairs into one operator with a joint boundedness and positive-definiteness guarantee. Two further results, both new, concern equalization. On a block-structured strongly convex model, and for the curvature-equalizing target that the implemented gradient-energy equalizer approximates up to a quantified heterogeneity factor, equalization makes the convergence rate depth-uniform; the bounded-clip version that is actually run stays depth-uniform up to a clip-determined depth and retains a constant-factor improvement beyond it. Controlled experiments, run over ten or more seeds with paired significance tests, confirm the mechanisms: Equalization compresses an order-of-magnitude per-layer gradient disparity, centering cuts the top curvature eigenvalue about threefold and yields the lowest training loss, and the configurations combining centering with the damped preconditioner, including the full method, converge fastest. The effects persist on MNIST and on CIFAR-10 with a small residual convolutional network, at a measured per-iteration overhead below about twice that of Adam. Generalization is comparable across methods, and no end-to-end depth-scaling advantage is claimed, keeping the contribution focused on optimization geometry.