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We construct a general framework for deriving error bounds for conicfeasibility problems. In particular, our approach allows one to work with conesthat fail to be amenable or even to have computable projections, two previouslychallenging barriers. For the purpose, we first show how error bounds may beconstructed using objects called one-step facial residual functions. Then, wedevelop several tools to compute these facial residual functions even in theabsence of closed form expressions for the projections onto the cones. Wedemonstrate the use and power of our results by computing tight error boundsfor the exponential cone feasibility problem. Interestingly, we discover anatural example for which the tightest error bound is related to theBoltzmann-Shannon entropy. We were also able to produce an example of sets forwhich a H\"{o}lderian error bound holds but the supremum of the set ofadmissible exponents is not itself an admissible exponent.

Error bounds, facial residual functions and applications to the exponential cone