What kind of dog was it?

The Seattle Times headline:

Bear attacks man on Lewis-McChord base

and the story begins

Washington wildlife agents are trying to trap a bear that attacked a man while walking his dog on Joint Base Lewis-McChord.

What kind of dog do black bears favor, I wonder?

More seriously, "while" doesn't work like that. If you don't specify the subject of the participle (the -ing form here), it's presumed to take the same one as its matrix clause. That's the clause the while-clause is subordinate to. Compare

A man saw a bear while walking his dog.
Agents are looking for a man that saw a bear while walking his dog.

So, in the story, who's walking the dog?

A bear attacked a man while walking his dog.
Agents are looking for a bear that attacked a man while walking his dog.

Yeah. It's the bear. To make it the man, you have to repeat him (with a pronoun probably) and turn the participle into a full verb phrase by adding the auxiliary verb:

Agents are trying to trap a bear that attacked a man while he was walking his dog.

And now, of course, anybody could be walking the dog:

Agents are trying to trap a bear that attacked a man while it was walking its dog.
Agents are trying to trap a bear that attacked a man while they were walking their (or his) dog.
Agents are trying to trap a bear that attacked a man while his girlfriend was walking his (or her) dog.
Agents are trying to trap a bear that attacked a man while some random dog-owning tourist was walking his dog.

ps - The dog went into woods then came running out, chased by the bear. As the bear ran by the man, it took a swipe at him and continued chasing the dog. The dog was later found OK off the military base.