Complexity of simple nonlogarithmic loss functions
The loss complexity for nonlogarithmic loss functions is defined analogously to the stochastic complexity for logarithmic loss functions such that its mean provides an achievable lower bound for estimation, the mean taken with respect to the worst case data generating distribution. The loss complexity also provides a lower bound for the worst case mean prediction error for all predictors. For the important alpha-loss functions \y - (y) over cap \(alpha), where y - (y) over cap denotes the prediction or fitting error and alpha is in the interval [1, 2], an accurate asymptotic formula for the loss complexity is given.
complexity alpha-loss functions maximum entropy min-max bounds prediction bound