Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\|\hat x - \mathbf x_\star\| \le
\epsilon_0\|\mathbf{x}_{\star}\|$, where $\mathbf x_{\star} =
\mathbf{A}^{-1}\mathbf{b}$ is the optimal solution, and $\epsilon_0$ a small
constant. Setting $\kappa =1/\epsilon$, this lower bound implies the minimax
rate of $T = \Omega(\lambda_1(\mathbf{A})\|\mathbf
x_\star\|^2/\sqrt{\epsilon})$ 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.

11
The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
22
The categories of this paper are: chao-dyn
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
6
The categories of this paper are: astro-ph
23
The categories of this paper are: solv-int
21
The categories of this paper are: adap-org
24
The categories of this paper are: dg-ga
18
The categories of this paper are: econ