Known-Subspace Rate Ladder for Streaming Low-Rank Regression with Fast Coefficient Drift

We study streaming low-rank regression when the relevant subspace is fixed but its coefficients can change abruptly. This setting exposes a plasticity–stability tradeoff: decaying steps learn stationary segments well but recover slowly after jumps, while constant steps react quickly but retain a noise floor. We prove a known-subspace rate ladder for expected dynamic regret under the same linear-Gaussian sample feedback. Monotone decaying-step stochastic gradient suffers linear regret under coefficient jumps, whereas a clipped fixed-window estimator achieves $O(T^{3/4})$ regret when the number of jumps grows as $\sqrt{T}$. An adaptive estimator that combines generalized likelihood ratio (GLR) change-point detection with clipped least squares achieves $O(\sqrt{T}\log T)$ on a well-spaced class with $\Theta (\sqrt{T})$ jumps, and a van Trees argument gives a matching $\Omega (\sqrt{T}\log T)$ lower bound on the same class. Charging each change by its own size, the ratio of the two bounds depends only on the condition number and the signal-to-noise ratio, uniformly in the minimum jump size, whereas a size-independent charge grows as the squared amplitude-to-jump ratio and is linear in $T$ for the smallest jumps the spacing admits. When the subspace is fixed but unknown, running the same detector on half of the samples projected onto a spectral-moment estimate built from the other half keeps the $O(\sqrt{T}\log T)$ rate at an additive cost of order $rd{\log }^2(dT)$ for learning the subspace when $d\ge \log (dT)$, and a matching lower bound shows that this cost is necessary up to the factor $\log (dT)$ when $d=O(T^{1/4})$; a blockwise Fantope extension covers moving subspaces at the $T^{3/4}$ scale. In synthetic experiments, a fixed window cuts the regret of a theorem-class decaying-step learner $11.3$-fold at $T=50,000$; the analyzed GLR rule, run exactly on equally spaced jumps of size at least $0.5$ with isotropic Gaussian covariates at high signal-to-noise ratio, grows at close to the oracle's rate and has lower regret at $T\ge 50,000$ than every tuned baseline; and a suffix-retaining GLR variant has $17$–$37\%$ lower regret than the preregistered tuned competitor with the lowest tuning regret at $T\in \{50,000,100,000\}$ on equally and unequally spaced change paths with mixed jump sizes, three covariate designs and noise levels $0.1$–$1$.