The LM317 circuit your playing with is what they call a 'buck' regulator. Meaning it cuts the source voltage down to what it needed by the load to be to be at the amount of current it's set for.
For the circuit to work properly, your supply voltage always has to be at or higher than the voltage the load wants.
So going with the above posted numbers for that diode, if you wanted to run your diode at 1000mA, the forward voltage would be 4.2v. That means your supply voltage to the regulator would have to stay higher than 4.2v to keep the diode running at 1000mA. If the regulator input voltage sagged below 4.2v, a buck only regulator would source less voltage and current as the battery voltage dropped. Meaning it would get dimmer and dimmer as the batteries died.