Mirror Descent Meets Fixed Share (and feels no regret)

Mirror descent with an entropic regularizer is known to achieve shifting regretbounds that are logarithmic in the dimension. This is done using either a carefullydesigned projection or by a weight sharing technique. Via a novel unified analysis,we show that these two approaches deliver essentially equivalent bounds on a no-tion of regret generalizing shifting, adaptive, discounted, and other related regrets.Our analysis also captures and extends the generalized weight sharing techniqueof Bousquet and Warmuth, and can be refined in several ways, including improve-ments for small losses and adaptive tuning of parameters.