There is a new condition, Sudden Rage Syndrome that is being noted in the Bichon. The article has been reprinted in its entirety with permission from Ms. Richardson. You must be consistent and patient while house training a Bichon. They are very intelligent but Bichons are not always cooperative with house training. House training can be difficult but not impossible.
Crate training makes it much easier to house train the dog.
The main rule to follow is keep the dog confined to the room you are in and give them access, on a regular basis, to the place where you want them to go. Bichons can be paper trained or trained to go outside. Establish a routine and follow it. Bichons can even be trained to go on cue with patience and consistency. Each time they do it where you want it done, praise them, which they love.