  Empirical Risk Minimization (ERM) is a standard technique in machine
learning, where a model is selected by minimizing a loss function over
constraint set. When the training dataset consists of private information, it
is natural to use a differentially private ERM algorithm, and this problem has
been the subject of a long line of work started with Chaudhuri and Monteleoni
2008. A private ERM algorithm outputs an approximate minimizer of the loss
function and its error can be measured as the difference from the optimal value
of the loss function. When the constraint set is arbitrary, the required error
bounds are fairly well understood \cite{BassilyST14}. In this work, we show
that the geometric properties of the constraint set can be used to derive
significantly better results. Specifically, we show that a differentially
private version of Mirror Descent leads to error bounds of the form
$\tilde{O}(G_{\mathcal{C}}/n)$ for a lipschitz loss function, improving on the
$\tilde{O}(\sqrt{p}/n)$ bounds in Bassily, Smith and Thakurta 2014. Here $p$ is
the dimensionality of the problem, $n$ is the number of data points in the
training set, and $G_{\mathcal{C}}$ denotes the Gaussian width of the
constraint set that we optimize over. We show similar improvements for strongly
convex functions, and for smooth functions. In addition, we show that when the
loss function is Lipschitz with respect to the $\ell_1$ norm and $\mathcal{C}$
is $\ell_1$-bounded, a differentially private version of the Frank-Wolfe
algorithm gives error bounds of the form $\tilde{O}(n^{-2/3})$. This captures
the important and common case of sparse linear regression (LASSO), when the
data $x_i$ satisfies $|x_i|_{\infty} \leq 1$ and we optimize over the $\ell_1$
ball. We show new lower bounds for this setting, that together with known
bounds, imply that all our upper bounds are tight.
