Computing Jacobians with automatic differentiation is ubiquitous in many scientific domains such as machine learning, computational fluid dynamics, robotics and finance. Even small savings in the number of computations or memory usage in Jacobian computations can already incur massive savings in energy consumption and runtime. While there exist many methods that allow for such savings, they generally trade computational efficiency for approximations of the exact Jacobian.In this paper, we present a novel method to optimize the number of necessary multiplications for Jacobian computation by leveraging deep reinforcement learning (RL) and a concept called cross-country elimination while still computing the exact Jacobian. Cross-country elimination is a framework for automatic differentiation that phrases Jacobian accumulation as ordered elimination of all vertices on the computational graph where every elimination incurs a certain computational cost.Finding the optimal elimination order that minimizes the number of necessary multiplications can be seen as a single player game which in our case is played by an RL agent.We demonstrate that this method achieves up to 33% improvements over state-of-the-art methods on several relevant tasks taken from relevant domains.Furthermore, we show that these theoretical gains translate into actual runtime improvements by providing a cross-country elimination interpreter in JAX that can execute the obtained elimination orders.

Square-root Kalman filters propagate state covariances in Cholesky-factor form for numerical stability, and are a natural target for gradient-based parameter learning in state-space models. Their core operation, triangularization of a matrix $M \in \mathbb{R}^{n \times m}$, is computed via a QR decomposition in practice, but naively differentiating through it causes two problems: the semi-orthogonal factor is non-unique when $m > n$, yielding undefined gradients; and the standard Jacobian formula involves inverses, which diverges when $M$ is rank-deficient. Both are resolved by the observation that all filter outputs relevant to learning depend on the input matrix only through the Gramian $MM^\top$, so the composite loss is smooth in $M$ even where the triangularization is not. We derive a closed-form chain-rule directly from the differential of this Gramian identity, prove it exact for the Kalman log-marginal likelihood and filtered moments, and extend it to rank-deficient inputs via a two-component decomposition: a column-space term based on the Moore--Penrose pseudoinverse, and a null-space correction for perturbations outside the column space of $M$.

For iterative algorithms, implicit differentiation alleviates this issue but requires custom implementation of Jacobian evaluation. In this paper, we study one-step differentiation, also known as Jacobian-free backpropagation, a method as easy as automatic differentiation and as efficient as implicit differentiation for fast algorithms (e.g., superlinear

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Firstly, many machine learning models use optimization algorithms which require access to derivatives of the model.

Optimization is an integral part of most machine learning systems and most numerical optimization schemes relyonthecomputation ofderivatives.

While existing automatic differentiation (AD) frameworks allow flexibly composing model architectures, theydonotprovide thesame flexibility forcomposing learning algorithms--everything has to be implemented in terms of backpropagation.