Chebyshev-Exact Acceleration under Hessian Variation, I: Sine-Jacobi Method

Abstract

We study finite-horizon one-gradient realizations with the Chebyshev minimax terminal residual on $[\mu,L]$. Under time-dependent Hessian perturbations, the terminal first variation is governed by a time-ordered spectral kernel $K_s(\lambda,\nu)$; its sharp $\ell_2$ gain is $A_N$. For the prefix-exact Chebyshev recurrence, $$ A_N^{\rm pref} =\frac{\epsilon_N^\star}{L-\mu} \left(4N^2+16\sum_{m=1}^{N-1}m^2\right)^{1/2} =\frac4{\sqrt3}\frac{N^{3/2}}{L-\mu}\epsilon_N^\star(1+o(1)), $$ and this is sharp in the causal two-term class with Chebyshev exactness at every prefix. For terminal-only exactness, Jacobi coordinates give $P_N=2^{1-N}T_N$: the spectrum is fixed at the midpoint Chebyshev nodes, while the spectral weights parametrize the realizations. The sine weights give a final-exact Jacobi method with the same terminal residual and $$ A_N(J_N^{\sin}) =2\sqrt{c_{\sin}}\frac{N^{3/2}}{L-\mu}\epsilon_N^\star(1+o(1)), \; 2\sqrt{c_{\sin}}\approx2.137936<4/\sqrt3. $$ Thus the Chebyshev terminal polynomial does not determine the first-order Hessian-drift gain. The experiments show the finite-horizon effect: lower stochastic curvature overhead, larger admissible-block frontiers, accurate time-varying quadratic predictions, and lower restart cost on an endpoint-coupled smooth strongly convex GLM.