  A lexicographic maximum of a set $X \subseteq \mathbb{R}^n$ is a vector in
$X$ whose smallest component is as large as possible, and subject to that
requirement, whose second smallest component is as large as possible, and so on
for the third smallest component, etc. Lexicographic maximization has numerous
practical and theoretical applications, including fair resource allocation,
analyzing the implicit regularization of learning algorithms, and
characterizing refinements of game-theoretic equilibria. We prove that a
minimizer in $X$ of the exponential loss function $L_c(\mathbf{x}) = \sum_i
\exp(-c x_i)$ converges to a lexicographic maximum of $X$ as $c \rightarrow
\infty$, provided that $X$ is stable in the sense that a well-known iterative
method for finding a lexicographic maximum of $X$ cannot be made to fail simply
by reducing the required quality of each iterate by an arbitrarily tiny degree.
Our result holds for both near and exact minimizers of the exponential loss,
while earlier convergence results made much stronger assumptions about the set
$X$ and only held for the exact minimizer. We are aware of no previous results
showing a connection between the iterative method for computing a lexicographic
maximum and exponential loss minimization. We show that every convex polytope
is stable, but that there exist compact, convex sets that are not stable. We
also provide the first analysis of the convergence rate of an exponential loss
minimizer (near or exact) and discover a curious dichotomy: While the two
smallest components of the vector converge to the lexicographically maximum
values very quickly (at roughly the rate $\frac{\log n}{c}$), all other
components can converge arbitrarily slowly.
