I have a circuit where I want to have a constant current of 2mA through a variable resistor. I've been told that I could probably use a LM317 as a current regulator, with one resistor on the ouput. But I've read some places that the LM317 takes minimum 5-10mA load to function correctly.
How can I achieve a constant current output of 2mA when I don't know the resistance of the variable resistor?
The input voltage is about 2.755V. Output voltage doesn't matter, just the current.
Here's an image to my feeble attempt at a schematic: 
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