  We give lower bounds on the amount of memory required by one-pass streaming
algorithms for solving several natural learning problems. In a setting where
examples lie in $\{0,1\}^d$ and the optimal classifier can be encoded using
$\kappa$ bits, we show that algorithms which learn using a near-minimal number
of examples, $\tilde O(\kappa)$, must use $\tilde \Omega( d\kappa)$ bits of
space. Our space bounds match the dimension of the ambient space of the
problem's natural parametrization, even when it is quadratic in the size of
examples and the final classifier. For instance, in the setting of $d$-sparse
linear classifiers over degree-2 polynomial features, for which
$\kappa=\Theta(d\log d)$, our space lower bound is $\tilde\Omega(d^2)$. Our
bounds degrade gracefully with the stream length $N$, generally having the form
$\tilde\Omega\left(d\kappa \cdot \frac{\kappa}{N}\right)$.
  Bounds of the form $\Omega(d\kappa)$ were known for learning parity and other
problems defined over finite fields. Bounds that apply in a narrow range of
sample sizes are also known for linear regression. Ours are the first such
bounds for problems of the type commonly seen in recent learning applications
that apply for a large range of input sizes.
