Optimal Convergence Rates for the Proximal Bundle Method

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 Hölder 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 complete the paper with a new parallelizable variant of the bundle method that attains near-optimal rates without prior knowledge of function parameters. These results improve on the limited existing convergence rates and provide a unified analysis approach across problem settings and algorithmic details. Numerical experiments support our findings and illustrate the effectiveness of the parallel bundle method.



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