  We study the generalization performance of $\text{full-batch}$ optimization
algorithms for stochastic convex optimization: these are first-order methods
that only access the exact gradient of the empirical risk (rather than
gradients with respect to individual data points), that include a wide range of
algorithms such as gradient descent, mirror descent, and their regularized
and/or accelerated variants. We provide a new separation result showing that,
while algorithms such as stochastic gradient descent can generalize and
optimize the population risk to within $\epsilon$ after $O(1/\epsilon^2)$
iterations, full-batch methods either need at least $\Omega(1/\epsilon^4)$
iterations or exhibit a dimension-dependent sample complexity.
