  We consider the problem of performing linear regression over a stream of
$d$-dimensional examples, and show that any algorithm that uses a subquadratic
amount of memory exhibits a slower rate of convergence than can be achieved
without memory constraints. Specifically, consider a sequence of labeled
examples $(a_1,b_1), (a_2,b_2)\ldots,$ with $a_i$ drawn independently from a
$d$-dimensional isotropic Gaussian, and where $b_i = \langle a_i, x\rangle +
\eta_i,$ for a fixed $x \in \mathbb{R}^d$ with $\|x\|_2 = 1$ and with
independent noise $\eta_i$ drawn uniformly from the interval
$[-2^{-d/5},2^{-d/5}].$ We show that any algorithm with at most $d^2/4$ bits of
memory requires at least $\Omega(d \log \log \frac{1}{\epsilon})$ samples to
approximate $x$ to $\ell_2$ error $\epsilon$ with probability of success at
least $2/3$, for $\epsilon$ sufficiently small as a function of $d$. In
contrast, for such $\epsilon$, $x$ can be recovered to error $\epsilon$ with
probability $1-o(1)$ with memory $O\left(d^2 \log(1/\epsilon)\right)$ using $d$
examples. This represents the first nontrivial lower bounds for regression with
super-linear memory, and may open the door for strong memory/sample tradeoffs
for continuous optimization.
