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Article overview
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Fast Rates with Unbounded Losses | Peter D. Grünwald
; Nishant A. Mehta
; | Date: |
1 May 2016 | Abstract: | We present new excess risk bounds for randomized and deterministic estimators
for general unbounded loss functions including log loss and squared loss. Our
bounds are expressed in terms of the information complexity and hold under the
recently introduced $v$-central condition, allowing for high-probability
bounds, and its weakening, the $v$-pseudoprobability convexity condition,
allowing for bounds in expectation even under heavy-tailed distributions. The
parameter $v$ determines the achievable rate and is akin to the exponent in the
Tsybakov margin condition and the Bernstein condition for bounded losses, which
the $v$-conditions generalize; favorable $v$ in combination with small
information complexity leads to $ ilde{O}(1/n)$ rates. While these fast rate
conditions control the lower tail of the excess loss, the upper tail is
controlled by a new type of witness-of-badness condition which allows us to
connect the excess risk to a generalized R’enyi divergence, generalizing
previous results connecting Hellinger distance to KL divergence. | Source: | arXiv, 1605.0252 | Services: | Forum | Review | PDF | Favorites |
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