Scaling Direct Feedback Learning with Jacobian Alignment Guarantees

Deep neural networks rely on backpropagation (BP) for optimization, but its strictly sequential backward pass hinders parallelism and scalability. Direct Feedback Alignment (DFA) has been proposed as a promising approach for parallel learning of deep neural networks, relying on fixed random projections to enable layer-wise parallel updates, but fails on deep convolutional networks, and performs poorly on modern transformer architectures. We introduce GrAPE (Gradient-Aligned Projected Error), a hybrid feedback-alignment method that (i) estimates rank-1 Jacobians via forward-mode JVPs and (ii) aligns each layer’s feedback matrix by minimizing a local cosine-alignment loss. To curb drift in very deep models, GrAPE performs infrequent BP anchor steps on a single mini-batch, preserving mostly parallel updates. We show that the forward-gradient estimator has strictly positive expected cosine with the true Jacobian. We relate this estimator-level guarantee to a standard stochastic-approximation result under a positive expected-cosine condition on the update direction, providing theoretical support for GrAPE’s alignment objective. Empirically, GrAPE consistently outperforms prior alternatives to BP, enabling the training of modern architectures, closing a large fraction of the gap to BP while retaining layer-parallel updates for the vast majority of steps.