Competitive regret

In decision theory, competitive regret is the relative regret compared to an oracle with limited or unlimited power in the process of distribution estimation.

Competitive regret to the oracle with full power

Consider estimating a discrete probability distribution on a discrete set based on data , the regret of an estimator[1] is defined as

where is the set of all possible probability distribution, and

where is the Kullback–Leibler divergence between and .

Competitive regret to the oracle with limited power

Oracle with partial information

The oracle is restricted to have access to partial information of the true distribution by knowing the location of in the parameter space up to a partition.[1] Given a partition of the parameter space, and suppose the oracle knows the subset where the true . The oracle will have regret as

The competitive regret to the oracle will be

Oracle with partial information

The oracle knows exactly , but can only choose the estimator among natural estimators. A natural estimator assigns equal probability to the symbols which appear the same number of time in the sample.[1] The regret of the oracle is

and the competitive regret is

Example

For the estimator proposed in Acharya et al.(2013),[2]

Here denotes the k-dimensional unit simplex surface. The partition denotes the permutation class on , where and are partitioned into the same subset if and only if is a permutation of .

References

  1. Orlitsky, Alon; Suresh, Ananda Theertha. (2015), Competitive Distribution Estimation, arXiv:1503.07940, Bibcode:2015arXiv150307940O
  2. Acharya, Jayadev; Jafarpour, Ashkan; Orlitsky, Alon; Suresh, Ananda Theertha (2013), "Optimal probability estimation with applications to prediction and classification", Proceedings of the 26th Annual Conference on Learning Theory (COLT)
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