What does regret mean?

My dog is an idiot.  And I'm not just talking about your run-of-the-mill dog type behavior, I mean his brain turns off the instant he smells something or sees something he wants to chase.  He's some sort of Labrador mix, and I have been working for most of his 6+ years under the delusion that, like other Lab-type dogs I've had, that he is going to calm down and become the really wonderful companion sort of animal those dogs were after their exuberant puppy-hood.
But he's not, he's apparently got some sort of hound DNA in there and he's a chaser, a nose to the ground, I'm not stopping until it climbs a tree, chaser.
Zeke's life with us started with the promise of a "free" puppy.  I know darn well there's no such thing as a "free" puppy, there are only puppies with no up front cost, usually because the owners of said "free" puppies are desperate to get rid of them.  His mother was a nice enough black labbish dog, who the owners claimed was a purebred, despite what the white markings on her chest and feet would seem to indicate.  Now I'm not a dog snob, my best friend growing up was a mutt of such indeterminate parentage even guessing at pedigree was laughable.  And I'm also not a moron when it comes to dogs, I have managed to more or less train the aforementioned mutt (with my Dad's help), I have trained our one "high class" pedigreed animal, a Chocolate Lab, to the point where she was absolutely trustworthy around small children and the only negative to having her around was in being a tripping hazard and the occasional episode of flatulence.
In fact, we got Zeke with the reasonable expectation that he would be socialized into the pack by Maggie and mirror her truly wonderful family style.  Also, we had kids who held puppies and pretty much had to have one.  I knew when I said we could go "look at the puppies" that we were going to get a puppy. I went in, at least partially, with my eyes open.
But knowing better only makes regret even worse.
Zeke is a nice enough animal, and that's really his saving grace, otherwise he has very few redeeming qualities.  He smells, and it's not a smell that can be cured with regular bathing.  He is irredeemably wild, even if it is in a friendly, exuberant way.  He is both defiant and painfully submissive, which is weird, because it's hard to know if you're going too far or not far enough with correction.  I am a believer in the form of discipline that dogs use on their own pups, a firm grip on the scruff of the neck and bit of a shake.  It worked wonders with Maggie, she clearly deferred to my dominance, and (after the puppy years) I very rarely had to yell at her or discipline her in any way, and I could pretty much trust her to do what she was supposed to do, at least until we got Zeke, he was actually a bad influence on her, pretty much the opposite of what I had in mind.
Zeke's not getting the hint.  He knows when he's gone wrong, but he doesn't change his behavior.  It's been getting to me temper wise, because it's been over six years.  In that time, I have come to regret our decision to get him in the first place.  I have never regretted owning an animal before, even the cat that we had to hide during our apartment years.  He's not dangerous, in fact, if anything he's too friendly with other people.  He's a good companion on walks and he is always pretty happy, but here's the thing: I feel like I have utterly failed as his pack leader.
My failure started the minute I decided to bring him home.  It continued when I didn't really sink the time into training him that he needed, because he was difficult and seemed oblivious, and because I had human children that needed more attention and because I thought my dear Maggie would get him in line (I think she regretted my decision more than me).
I believe that owning an animal is a lifetime commitment.  For better or worse, I am stuck with Zeke and Zeke is stuck with me.  I'm not giving up, because for his own well being he needs to get it under control, or else I'm going to be scraping him off of someone's bumper.  So I'm going to try a different method, and if that doesn't work, we'll go for something else.
The reality is that it's not going to be all that long before old age calms Zeke down, but in the mean time I am treating this as a lesson in regret, and living with regret.
You can regret all sorts of things, some of them are big life decisions others are just dogs or a tattoo, but the spiritual process is kind of the same, you have to accept, and love and grow somehow or other.  Or else you're just stuck.  Zeke and I have been stuck for too long.