  We study convergence rates of the classic proximal bundle method for a
variety of nonsmooth convex optimization problems. We show that, without any
modification, this algorithm adapts to converge faster in the presence of
smoothness or a H\"older growth condition. Our analysis reveals that with a
constant stepsize, the bundle method is adaptive, yet it exhibits suboptimal
convergence rates. We overcome this shortcoming by proposing nonconstant
stepsize schemes with optimal rates. These schemes use function information
such as growth constants, which might be prohibitive in practice. We provide a
parallelizable variant of the bundle method that can be applied without prior
knowledge of function parameters while maintaining near-optimal rates. The
practical impact of this scheme is limited since we incur a (parallelizable)
log factor in the complexity. These results improve on the scarce existing
convergence rates and provide a unified analysis approach across problem
settings and algorithmic details. Numerical experiments support our findings.
