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We consider a general problem of finding a strategy that minimizes theexponential moment of a given cost function, with an emphasis on its relationto the more common criterion of minimization the expectation of the firstmoment of the same cost function. In particular, our main result is a theoremthat gives simple sufficient conditions for a strategy to be optimum in theexponential moment sense. This theorem may be useful in various situations, andapplication examples are given. We also examine the asymptotic regime andinvestigate universal asymptotically optimum strategies in light of theaforementioned sufficient conditions, as well as phenomena of irregularities,or phase transitions, in the behavior of the asymptotic performance, which canbe viewed and understood from a statistical-mechanical perspective. Finally, wepropose a new route for deriving lower bounds on exponential moments of certaincost functions (like the square error in estimation problems) on the basis ofwell known lower bounds on their expectations.

On optimum strategies for minimizing the exponential moments of a given cost function