Efficiency of a voltage regulator
Let's say I have an LDO that create 5V at 2A with an input of 10V at 1A...

You have 10 W input, and 10 W output, therefore you have 100% efficiency.

...with an efficiency of 50%

This is a contradiction. In the first part of your sentence you gave numbers that show the efficiency is 100%.
But, a linear regulator (the LDO being a sub-type of linear regulators whose capabilities aren't important in the scenario you presented) could not produce the numbers you stated. A linear regulator requires the same input current as output current (plus a few microamps or milliamps to run the regulator circuit itself).
So realistic numbers for a linear regulator to get 5 V, 2 A output would be 10 V, 2.001 A input. And those numbers do give an efficiency close to 50% (20 W input, 10 W output).
If you can reduce the input voltage, it won't affect the input current significantly and the efficiency will improve.
If you want efficiency with a large voltage difference between input and output, consider using a buck switching converter instead of a linear regulator.