Sven: Singular Value Descent as a Computationally Efficient Natural Gradient Method
arXiv:2604.01279v1 Announce Type: new 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 meth
arXiv:2604.01279v1 Announce Type: new 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.
Executive Summary
This article introduces Sven, a novel optimization algorithm for neural networks that leverages the natural decomposition of loss functions into individual data points, utilizing the Moore-Penrose pseudoinverse and truncated singular value decomposition to achieve computationally efficient parameter updates. Sven significantly outperforms standard first-order methods, including Adam, and remains competitive with LBFGS at a fraction of the wall-time cost. The primary challenge to scaling Sven lies in its memory overhead, which the authors propose to mitigate via several strategies. Sven's potential applications extend beyond machine learning benchmarks, with natural applications in scientific computing settings where custom loss functions can be decomposed into multiple conditions. The algorithm's performance is demonstrated on regression tasks, showcasing its ability to converge faster and to a lower final loss.
Key Points
- ▸ Sven exploits the natural decomposition of loss functions into individual data points
- ▸ Uses the Moore-Penrose pseudoinverse and truncated singular value decomposition for efficient parameter updates
- ▸ Achieves significant performance gains over standard first-order methods, including Adam
- ▸ Competitive with LBFGS at a fraction of the wall-time cost
- ▸ Primary challenge to scaling lies in memory overhead
Merits
Computationally Efficient
Sven's use of truncated singular value decomposition incurs a computational overhead of only a factor of k relative to stochastic gradient descent, offering significant performance gains.
Improved Convergence
Sven demonstrates faster convergence and lower final loss compared to standard first-order methods, including Adam.
Demerits
Memory Overhead
The primary challenge to scaling Sven lies in its memory overhead, which can be mitigated via proposed strategies but remains a significant limitation.
Expert Commentary
This article presents a significant contribution to the field of optimization algorithms for neural networks, offering a novel and computationally efficient approach to parameter updates. Sven's performance gains and improved convergence rates make it a promising candidate for real-world applications. However, the primary challenge to scaling Sven lies in its memory overhead, which must be mitigated via proposed strategies. As the field of artificial intelligence continues to evolve, the development of efficient and effective optimization algorithms like Sven will play a crucial role in the integration of AI into various industries.
Recommendations
- ✓ Further research is needed to fully explore Sven's potential applications and limitations.
- ✓ The proposed strategies for mitigating memory overhead should be tested and validated in practical settings.
Sources
Original: arXiv - cs.LG
