The advanced money paid to get a rented house is a debt in the hands of landlord and getting benefit out of debt is unlawful and interest. It is mentioned in Hadith:
کل قرض جر نفعاً فھو رباً
“Every debt fetching benefit is usury.”
In the case mentioned above, paying less rent against debt is benefit free from compensation which is unlawful; rather you should decide a rent as per the location and value of the house.
Allah knows Best!
Darul Uloom Deoband