In this talk we provide improved estimates on the exponential
convergence rate of stochastic gradient descent (SG) for smooth strongly
convex objectives, in the regime where all stochastic estimates share an
optimum and so such an exponential rate is possible.   We then show that
by incorporating importance sampling -- perturbing the uniform row
selection rule in the direction of sampling estimates proportionally to
the Lipschitz constant of their gradients -- the convergence rate of SG
can be improved dramatically from depending on the average squared
condition number to depending on the average (unsquared) condition number
of the system.  We finish with several open questions.