  We design accelerated algorithms with improved rates for several fundamental
classes of optimization problems. Our algorithms all build upon techniques
related to the analysis of primal-dual extragradient methods via relative
Lipschitzness proposed recently by [CST21].
  (1) Separable minimax optimization. We study separable minimax optimization
problems $\min_x \max_y f(x) - g(y) + h(x, y)$, where $f$ and $g$ have
smoothness and strong convexity parameters $(L^x, \mu^x)$, $(L^y, \mu^y)$, and
$h$ is convex-concave with a $(\Lambda^{xx}, \Lambda^{xy},
\Lambda^{yy})$-blockwise operator norm bounded Hessian. We provide an algorithm
with gradient query complexity $\tilde{O}\left(\sqrt{\frac{L^{x}}{\mu^{x}}} +
\sqrt{\frac{L^{y}}{\mu^{y}}} + \frac{\Lambda^{xx}}{\mu^{x}} +
\frac{\Lambda^{xy}}{\sqrt{\mu^{x}\mu^{y}}} +
\frac{\Lambda^{yy}}{\mu^{y}}\right)$. Notably, for convex-concave minimax
problems with bilinear coupling (e.g.\ quadratics), where $\Lambda^{xx} =
\Lambda^{yy} = 0$, our rate matches a lower bound of [ZHZ19].
  (2) Finite sum optimization. We study finite sum optimization problems
$\min_x \frac{1}{n}\sum_{i\in[n]} f_i(x)$, where each $f_i$ is $L_i$-smooth and
the overall problem is $\mu$-strongly convex. We provide an algorithm with
gradient query complexity $\tilde{O}\left(n + \sum_{i\in[n]}
\sqrt{\frac{L_i}{n\mu}} \right)$. Notably, when the smoothness bounds
$\{L_i\}_{i\in[n]}$ are non-uniform, our rate improves upon accelerated SVRG
[LMH15, FGKS15] and Katyusha [All17] by up to a $\sqrt{n}$ factor.
  (3) Minimax finite sums. We generalize our algorithms for minimax and finite
sum optimization to solve a natural family of minimax finite sum optimization
problems at an accelerated rate, encapsulating both above results up to a
logarithmic factor.
