Collective Stability in Structured Prediction: Appendix

Note that the above does not require independence. To prove Theorem 1, it therefore suffices to bound ∑n i=1 α 2 i . Kontorovich & Ramanan (2008, Remark 2.1) showed that, if f is c-Lipschitz with respect to the Hamming metric, then ∑n i=1 α 2 i ≤ nc ‖Θ π n‖ 2 ∞. (Though the published results only prove this for countable spaces, Kontorovich later extended this analysis to continuous spaces in his thesis (2007).) If f is c-Lipschitz with respect to the normalized Hamming metric, then ∑n i=1 α 2 i ≤ c ‖Θ π n‖ 2 ∞ /n, which completes the proof.