I propose to formalize the main minimax lower-bound tools from Chapter 15 of Wainwright’s High-Dimensional Statistics. This would complement StatsMLlib’s estimation upper bounds with tools for proving statistical optimality. In particular, the existing linear_minimax_rate establishes an upper bound; a matching lower bound would complete the optimality argument.
The proposed scope is:
- Foundations: statistical models, measurable estimators, loss functions, minimax risk, and the reduction from estimation to finite hypothesis testing.
- Divergence tools: reuse Mathlib’s existing infrastructure and add the missing results relating total variation, KL divergence, Hellinger distance, and testing error.
- Le Cam’s method: the two-point lower bound, followed by its mixture/convex-hull extension.
- Fano’s method: the information-theoretic inequality and its minimax consequences, progressing to local-packing, Gaussian-entropy, and Yang–Barron bounds.
- Applications: begin with Gaussian mean estimation, then develop lower bounds for linear and sparse regression.
I suggest placing the statistical framework under StatsMLlib/Statistics/Minimax/, keeping generic probability and divergence lemmas in the appropriate foundational modules, and reusing the existing covering/packing API.
The contribution would proceed through small PRs, starting with the risk framework, estimation-to-testing reduction, and Le Cam’s two-point method with one worked example. Each stage would include explicit assumptions and constants, complete proofs, and references to the corresponding results in the book.
I propose to formalize the main minimax lower-bound tools from Chapter 15 of Wainwright’s High-Dimensional Statistics. This would complement StatsMLlib’s estimation upper bounds with tools for proving statistical optimality. In particular, the existing linear_minimax_rate establishes an upper bound; a matching lower bound would complete the optimality argument.
The proposed scope is:
I suggest placing the statistical framework under
StatsMLlib/Statistics/Minimax/, keeping generic probability and divergence lemmas in the appropriate foundational modules, and reusing the existing covering/packing API.The contribution would proceed through small PRs, starting with the risk framework, estimation-to-testing reduction, and Le Cam’s two-point method with one worked example. Each stage would include explicit assumptions and constants, complete proofs, and references to the corresponding results in the book.