Low-Power Optimal Strategy for Witsenhausen Counterexample
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908515630579712 |
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| author | Zhao, Mengyuan Treust, Maël Le Oechtering, Tobias J. |
| author_facet | Zhao, Mengyuan Treust, Maël Le Oechtering, Tobias J. |
| contents | We discuss the Witsenhausen counterexample from the perspective of varying power budgets and propose a low-power estimation (LoPE) strategy. Specifically, our LoPE approach designs the first decision-maker (DM) a quantization step function of the Gaussian source state, making the target system state a piecewise linear function of the source with slope one. This approach contrasts with Witsenhausen's original two-point strategy, which instead designs the system state itself to be a binary step. While the two-point strategy can outperform the linear strategy in estimation cost, it, along with its multi-step extensions, typically requires a substantial power budget. Analogous to Binary Phase Shift Keying (BPSK) communication for Gaussian channels, we show that the binary LoPE strategy attains first-order optimality in the low-power regime, matching the performance of the linear strategy as the power budget increases from zero. Our analysis also provides an interpretation of the previously observed near-optimal sloped step function ("sawtooth") structure to the Witsenhausen counterexample: In the low-power regime, power saving is prioritized, in which case the LoPE strategy dominates, making the system state a piecewise linear function with slope close to one. Conversely, in the high-power regime, setting the system state as a step function with the slope approaching zero facilitates accurate estimation. Hence, the sawtooth solution can be seen as a combination of both strategies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02381 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Low-Power Optimal Strategy for Witsenhausen Counterexample Zhao, Mengyuan Treust, Maël Le Oechtering, Tobias J. Optimization and Control We discuss the Witsenhausen counterexample from the perspective of varying power budgets and propose a low-power estimation (LoPE) strategy. Specifically, our LoPE approach designs the first decision-maker (DM) a quantization step function of the Gaussian source state, making the target system state a piecewise linear function of the source with slope one. This approach contrasts with Witsenhausen's original two-point strategy, which instead designs the system state itself to be a binary step. While the two-point strategy can outperform the linear strategy in estimation cost, it, along with its multi-step extensions, typically requires a substantial power budget. Analogous to Binary Phase Shift Keying (BPSK) communication for Gaussian channels, we show that the binary LoPE strategy attains first-order optimality in the low-power regime, matching the performance of the linear strategy as the power budget increases from zero. Our analysis also provides an interpretation of the previously observed near-optimal sloped step function ("sawtooth") structure to the Witsenhausen counterexample: In the low-power regime, power saving is prioritized, in which case the LoPE strategy dominates, making the system state a piecewise linear function with slope close to one. Conversely, in the high-power regime, setting the system state as a step function with the slope approaching zero facilitates accurate estimation. Hence, the sawtooth solution can be seen as a combination of both strategies. |
| title | Low-Power Optimal Strategy for Witsenhausen Counterexample |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2509.02381 |