Lady Gaga's dogs were targeted by thieves Wednesday night ... thieves who shot her dog walker and stole 2 of her French Bulldogs ... law enforcement sources tell TMZ, and we're told Gaga is offering $500,000 for the return of her dogs, "no questions asked."

The dog walker had three of Gaga's Bulldogs out in Hollywood just before 10 PM when one gunman -- and possibly more -- came upon him. We don't know what was said, but the dog walker was shot and the gunman made off with 2 of the dogs, named Koji and Gustav.

There are reports there were only 2 Bulldogs being walked with one taken and one escaped that was later found, but our law enforcement sources tell us 3 dogs were being walked and 2 were taken by the gunman. We're told the 3rd dog had run away but was later recovered.

Gaga's been in Rome working on a new movie, her bodyguard was seen picking up the recovered Bulldog. Sources close to Gaga tell us she's extremely upset and is offering a half-a-million dollars to anyone who has her 2 dogs ... again, no questions asked.  We're told anyone who has the dogs can use this email -- KojiandGustav@gmail.com -- to retrieve the reward.

The dog walker was taken to the hospital and we're told he's expected to fully recover.

Law enforcement does not know if the target was Lady Gaga's dogs specifically. French Bulldogs are in demand and expensive, so our sources say it's possible the gunman did not know the dogs were owned by Lady Gaga.

The gunman is still on the loose.