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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2505.07046 |
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| _version_ | 1866916732092809216 |
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| author | Thomas, Stephen |
| author_facet | Thomas, Stephen |
| contents | We present SKA-SGD (Streaming Krylov-Accelerated Stochastic Gradient Descent), a novel optimization approach that accelerates convergence for ill-conditioned problems by projecting stochastic gradients onto a low-dimensional Krylov subspace. Directly inspired by recent advances in s-step Conjugate Gradient methods with streaming Gauss-Seidel Gram solvers \cite{dambra2025sstep}, our method extends these techniques to the stochastic optimization domain. Our approach combines three key innovations: (1) projection coefficients computed via a single streaming Gauss-Seidel iteration, which is mathematically equivalent to Modified Gram-Schmidt orthogonalization; (2) a Chebyshev polynomial basis for constructing the Krylov subspace, providing superior numerical stability; and (3) efficient implementation for AMD GPUs using HIP. We prove that our streaming approach achieves a backward error near machine precision with $O(s^2)$ complexity rather than $O(s^3)$, where $s$ is the Krylov subspace dimension. Experimental results demonstrate that SKA-SGD significantly outperforms standard SGD and Adam in convergence rate and final error, particularly for problems with condition numbers exceeding $10^3$. GPU performance analysis reveals a crossover point where communication-avoiding benefits outweigh computational overhead, typically occurring at moderate scale ($p \approx 64$ processors) for problem sizes $n \geq 10^6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07046 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Streaming Krylov-Accelerated Stochastic Gradient Descent Thomas, Stephen Numerical Analysis Machine Learning We present SKA-SGD (Streaming Krylov-Accelerated Stochastic Gradient Descent), a novel optimization approach that accelerates convergence for ill-conditioned problems by projecting stochastic gradients onto a low-dimensional Krylov subspace. Directly inspired by recent advances in s-step Conjugate Gradient methods with streaming Gauss-Seidel Gram solvers \cite{dambra2025sstep}, our method extends these techniques to the stochastic optimization domain. Our approach combines three key innovations: (1) projection coefficients computed via a single streaming Gauss-Seidel iteration, which is mathematically equivalent to Modified Gram-Schmidt orthogonalization; (2) a Chebyshev polynomial basis for constructing the Krylov subspace, providing superior numerical stability; and (3) efficient implementation for AMD GPUs using HIP. We prove that our streaming approach achieves a backward error near machine precision with $O(s^2)$ complexity rather than $O(s^3)$, where $s$ is the Krylov subspace dimension. Experimental results demonstrate that SKA-SGD significantly outperforms standard SGD and Adam in convergence rate and final error, particularly for problems with condition numbers exceeding $10^3$. GPU performance analysis reveals a crossover point where communication-avoiding benefits outweigh computational overhead, typically occurring at moderate scale ($p \approx 64$ processors) for problem sizes $n \geq 10^6$. |
| title | Streaming Krylov-Accelerated Stochastic Gradient Descent |
| topic | Numerical Analysis Machine Learning |
| url | https://arxiv.org/abs/2505.07046 |