On the Randomized Complexity of Minimizing a Convex Quadratic Function

Minimizing a convex, quadratic objective of the form $f{\mathbf{A},\mathbf{b}}(x) := \frac{1}{2}x^\top \mathbf{A} x - \langle \mathbf{b}, x \rangle$ for $\mathbf{A} \succ 0 $ is a fundamental problem in machine learning and optimization. In this work, we prove gradient-query complexity lower bounds for minimizing convex quadratic functions which apply to both deterministic and \emph{randomized} algorithms. Specifically, for $\kappa > 1$, we exhibit a distribution over $(\mathbf{A},\mathbf{b})$ with condition number $\mathrm{cond}(\mathbf{A}) \le \kappa$, such that any \emph{randomized} algorithm requires $\Omega(\sqrt{\kappa})$ gradient queries to find a solution $\hat x$ for which $|\hat x - \mathbf x\star| \le \epsilon0|\mathbf{x}{\star}|$, where $\mathbf x{\star} = \mathbf{A}^{{-1}\mathbf{b}$} is the optimal solution, and $\epsilon0$ a small constant. Setting $\kappa =1/\epsilon$, this lower bound implies the minimax rate of $T = \Omega(\lambda1(\mathbf{A})|\mathbf x\star|^{2/\sqrt{\epsilon})$} queries required to minimize an arbitrary convex quadratic function up to error $f(\hat{x}) - f(\mathbf x\star) \le \epsilon$. Our lower bound holds for a distribution derived from classical ensembles in random matrix theory, and relies on a careful reduction from adaptively estimating a planted vector $\mathbf u$ in a deformed Wigner model. A key step in deriving sharp lower bounds is demonstrating that the optimization error $\mathbf x\star - \hat x$ cannot align too closely with $\mathbf{u}$. To this end, we prove an upper bound on the cosine between $\mathbf x_\star - \hat x$ and $\mathbf u$ in terms of the MMSE of estimating the plant $\mathbf u$ in a deformed Wigner model. We then bound the MMSE by carefully modifying a result due to Lelarge and Miolane 2016, which rigorously establishes a general replica-symmetric formula for planted matrix models.