Approximate Computation via Le Cam Simulability
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908741483364352 |
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| author | Akdemir, Deniz |
| author_facet | Akdemir, Deniz |
| contents | We propose a decision-theoretic framework for computational complexity, complementary to classical theory: moving from syntactic exactness (Turing / Shannon) to semantic simulability (Le Cam). While classical theory classifies problems by the cost of exact solution, modern computation often seeks only decision-valid approximations. We introduce a framework where "computation" is viewed as the efficient simulation of a target statistical experiment within a bounded risk distortion (Le Cam deficiency).
We formally define computational deficiency ($δ_{\text{poly}}$) and use it to construct the complexity class LeCam-P (Decision-Robust Polynomial Time), characterizing problems that may be syntactically hard but semantically easy to approximate. We show that classical Karp reductions can be viewed as zero-deficiency simulations, and that approximate reductions correspond to bounded deficiency. Furthermore, we establish the No-Free-Transfer Inequality, showing that strictly invariant representations inevitably destroy decision-relevant information. This framework offers a statistical perspective on approximation theory, bridging the gap between algorithmic complexity and decision theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_24860 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximate Computation via Le Cam Simulability Akdemir, Deniz Statistics Theory Computational Complexity Information Theory We propose a decision-theoretic framework for computational complexity, complementary to classical theory: moving from syntactic exactness (Turing / Shannon) to semantic simulability (Le Cam). While classical theory classifies problems by the cost of exact solution, modern computation often seeks only decision-valid approximations. We introduce a framework where "computation" is viewed as the efficient simulation of a target statistical experiment within a bounded risk distortion (Le Cam deficiency). We formally define computational deficiency ($δ_{\text{poly}}$) and use it to construct the complexity class LeCam-P (Decision-Robust Polynomial Time), characterizing problems that may be syntactically hard but semantically easy to approximate. We show that classical Karp reductions can be viewed as zero-deficiency simulations, and that approximate reductions correspond to bounded deficiency. Furthermore, we establish the No-Free-Transfer Inequality, showing that strictly invariant representations inevitably destroy decision-relevant information. This framework offers a statistical perspective on approximation theory, bridging the gap between algorithmic complexity and decision theory. |
| title | Approximate Computation via Le Cam Simulability |
| topic | Statistics Theory Computational Complexity Information Theory |
| url | https://arxiv.org/abs/2512.24860 |