Bias- and Variance-Aware Probabilistic Rounding Error Analysis for Floating-Point Arithmetic
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866918376600764416 |
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| author | Bhola, Sahil Duraisamy, Karthik |
| author_facet | Bhola, Sahil Duraisamy, Karthik |
| contents | Probabilistic rounding error analysis can yield much sharper bounds than classical worst-case theory, but existing results typically rely on zero-mean rounding errors and often leave the confidence parameter implicit. This work revisits probabilistic rounding error analysis in a moment-aware setting. We first derive a confidence-calibrated reformulation of the Higham and Mary [16] bound that makes its confidence parameter explicit. We then introduce a variance-informed probabilistic backward error bound based on the first two moments of $\log(1+δ)$, where $δ$ is the relative rounding error. This allows the analysis to accommodate biased rounding error models rather than relying on a zero-mean assumption. To illustrate this framework, we study both a uniform model and a log-space $\operatorname{Beta}$ model for rounding errors, the latter of which provides a simple way to represent bias. This perspective shows that the growth of probabilistic rounding error bounds is not universal: near-zero-mean regimes recover $\sqrt{n}$-like behavior, while biased models can exhibit faster accumulation. $\texttt{CUDA}$ experiments in single and half precision on dot products, sparse matrix-vector products, and a stochastic boundary-value problem show that the proposed framework is especially useful in low-precision regimes where deterministic bounds are overly conservative and where bias-aware modeling better matches observed error growth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_12556 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bias- and Variance-Aware Probabilistic Rounding Error Analysis for Floating-Point Arithmetic Bhola, Sahil Duraisamy, Karthik Computation 65G50, 97N20, 65F99, 65C99 Probabilistic rounding error analysis can yield much sharper bounds than classical worst-case theory, but existing results typically rely on zero-mean rounding errors and often leave the confidence parameter implicit. This work revisits probabilistic rounding error analysis in a moment-aware setting. We first derive a confidence-calibrated reformulation of the Higham and Mary [16] bound that makes its confidence parameter explicit. We then introduce a variance-informed probabilistic backward error bound based on the first two moments of $\log(1+δ)$, where $δ$ is the relative rounding error. This allows the analysis to accommodate biased rounding error models rather than relying on a zero-mean assumption. To illustrate this framework, we study both a uniform model and a log-space $\operatorname{Beta}$ model for rounding errors, the latter of which provides a simple way to represent bias. This perspective shows that the growth of probabilistic rounding error bounds is not universal: near-zero-mean regimes recover $\sqrt{n}$-like behavior, while biased models can exhibit faster accumulation. $\texttt{CUDA}$ experiments in single and half precision on dot products, sparse matrix-vector products, and a stochastic boundary-value problem show that the proposed framework is especially useful in low-precision regimes where deterministic bounds are overly conservative and where bias-aware modeling better matches observed error growth. |
| title | Bias- and Variance-Aware Probabilistic Rounding Error Analysis for Floating-Point Arithmetic |
| topic | Computation 65G50, 97N20, 65F99, 65C99 |
| url | https://arxiv.org/abs/2404.12556 |