So far as I know, there is no measurement that can compare the efficiency of devices that turn electricity into EMR. That's because the EMR spectrum is extremely wide (theoretically infinite), going from wavelengths far shorter than the width of a proton to as long as the universe. The ones that scientists usually deal with run from gamma rays (wavelengths in picometers) to ELF (wavelengths in megameters). That includes visible light, x-rays, all the "radio" waves (microwaves, VHF, FM, etc), audible sound, and lots of other stuff that most people never deal with. Even gravity behaves like a form of EMR, with measurable speed and wave-like action. So you have to specify some spectrum of wavelengths in which the EMR behaves in a way that we can interact with, describe, and measure.
Efficiency is really a measure of how well a thing does what we want it to, and how much energy it spends doing things we don't. We would call incandescent light bulbs inefficient because they use a lot of energy producing heat and not visible light, but they are sometimes used as a heater, in which case they're 100% efficient - every bit of energy going in is turned into heat, with none lost. Even all the visible light, assuming you confine it to a sealed box or something, is turned into heat when it's absorbed by the things it hits.
Speaking of heat, that's probably how you should phrase your question, since most of what we consider inefficiencies are just measuring how much energy gets turned into heat as a byproduct of doing whatever else we wanted. So you could ask, what LED, or what light source, has the least percentage of its electricity turned directly into heat? Even that is a tricky question, since many light sources emit some IR, which is totally absorbed as heat by most surfaces. Still, it would be a better way to measure the kind of "pure efficiency" you're talking about. In that case, I suspect that low pressure sodium, since it produces the most lumens/watt, is right up there with anything else in terms of least heat per watt.
Alex