Basis Learn (BL1, BL2, and BL3)

A family of communication-efficient second-order distributed and federated optimization algorithms (BL1, BL2, and BL3) for minimizing f(x) = (1/n) sum_{i=1}^n f_i(x) that transforms each client i's local Hessian nabla^2 f_i(x) into a custom matrix basis {B_i^{jl}} of R^{d x d} or the symmetric matrix space S^d via nabla^2 f_i(x) = sum_{j,l} h_{jl}^i(nabla^2 f_i(x)) B_i^{jl} before applying lossy matrix compression. When local generalized linear model (GLM) data vectors {a_{ij}}_{j=1}^m lie in an r-dimensional subspace G_i subseteq R^d spanned by {v_{it}}_{t=1}^r ( transmitted once to the server at cost r d floats without revealing raw data a_{ij}), the outer-product basis V_{itl} = v_{it} v_{il}^T concentrates all non-zero Hessian coefficients into an r x r block gamma_{itl}(x) = (1/m) sum_{j=1}^m phi_{ij}''(a_{ij}^T x) alpha_{ijt} alpha_{ijl} (r^2 << d^2 floats). BL1 integrates custom-basis Hessian learning L_i^{k+1} = L_i^k + alpha C_i^k(h^i(nabla^2 f_i(z^k)) - L_i^k) with bidirectional model/gradient compression, while BL2 and BL3 extend basis learning to partial client participation using Frobenius-error shifting (BL2) or positive semidefinite basis decomposition (BL3).