  The spectral bundle method proposed by Helmberg and Rendl is well established
for solving large-scale semidefinite programs (SDP) thanks to its low per
iteration computational complexity and strong practical performance. In this
paper, we revisit this classic method showing it achieves sublinear convergence
rates in terms of both primal and dual SDPs under merely strong duality,
complementing previous guarantees on primal-dual convergence. Moreover, we show
the method speeds up to linear convergence if (1) structurally, the SDP admits
strict complementarity, and (2) algorithmically, the bundle method captures the
rank of the optimal solutions. Such complementary and low rank structure is
prevalent in many modern and classical applications. The linear convergent
result is established via an eigenvalue approximation lemma which might be of
independent interest. Numerically, we confirm our theoretical findings that the
spectral bundle method, for modern and classical applications, speeds up under
these conditions. Finally, we show that the spectral bundle method combined
with a recent matrix sketching technique is able to solve an SDP with billions
of decision variables in a matter of minutes.
