Sven: Singular Value Descent as a Computationally Efficient Natural Gradient Method

TL;DR

Sven is a novel optimization algorithm for neural networks that efficiently approximates natural gradient descent by exploiting data point decompositions, outperforming standard methods in regression tasks.

Contribution

The paper introduces Sven, a scalable natural gradient method using truncated SVD, which improves training speed and convergence over traditional first-order optimizers.

Findings

Sven outperforms Adam in regression tasks, converging faster and to lower loss.

Sven's computational overhead is only a factor of k compared to SGD.

It approximates natural gradient descent in over-parametrized regimes.

Abstract

We introduce Sven (Singular Value dEsceNt), a new optimization algorithm for neural networks that exploits the natural decomposition of loss functions into a sum over individual data points, rather than reducing the full loss to a single scalar before computing a parameter update. Sven treats each data point's residual as a separate condition to be satisfied simultaneously, using the Moore-Penrose pseudoinverse of the loss Jacobian to find the minimum-norm parameter update that best satisfies all conditions at once. In practice, this pseudoinverse is approximated via a truncated singular value decomposition, retaining only the $k$ most significant directions and incurring a computational overhead of only a factor of $k$ relative to stochastic gradient descent. This is in comparison to traditional natural gradient methods, which scale as the square of the number of parameters. We show that Sven can be understood as a natural gradient method generalized to the over-parametrized regime, recovering natural gradient descent in the under-parametrized limit. On regression tasks, Sven significantly outperforms standard first-order methods including Adam, converging faster and to a lower final loss, while remaining competitive with LBFGS at a fraction of the wall-time cost. We discuss the primary challenge to scaling, namely memory overhead, and propose mitigation strategies. Beyond standard machine learning benchmarks, we anticipate that Sven will find natural application in scientific computing settings where custom loss functions decompose into several conditions.