Moonwalk: Inverse-Forward Differentiation
arXiv:2402.14212v4 Announce Type: replace-cross Abstract: Backpropagation's main limitation is its need to store intermediate activations (residuals) during the forward pass, which restricts the depth of trainable networks. This raises a fundamental question: can we avoid storing these activations? We address this by revisiting the structure of gradient computation. Backpropagation computes gradients through a sequence of vector-Jacobian products, an operation that is generally irreversible. The lost information lies in the cokernel of each layer's Jacobian. We define submersive networks -- networks whose layer Jacobians have trivial cokernels -- in which gradients can be reconstructed exactly in a forward sweep without storing activations. For non-submersive layers, we introduce fragmental gradient checkpointing, which records only the minimal subset of residuals necessary to restore the cotangents erased by the Jacobian. Central to our approach is a novel operator, the vector-inverse