Boost converter confusion
In a switching supply including a boost converter, the output power will equal the input power minus losses.  

So suppose you have a 2V 1A input.  A good boost converter might be 90% efficient, so 2W of input would result in 1.8W of output power.

If your LED forward voltage needed is say 3V, you could get 600mA out of the boost converter.  That's plenty of current to power a typical LED, and in actual practice you would want to boost to a higher voltage and limit the current with a resistor.  (Or build a current regulating converter rather than a voltage regulator.)  Your diagram doesn't show any control scheme, so it's hard to tell what you have in mind.

So you do get less current out of the boost than you put in, but if there's enough output power to power your LED you should be fine.