Obviously, Emperor Palpatine isn't likely to send Darth Vader on sensitivity training for his casual killing of several Imperial military leaders, and he is deeply feared himself (see Moff Jerjerrod's reaction to his planned appearance on the Second Death Star). But in the novelisations, or anywhere else, is there any evidence of his opinion?
Does he approve? Does he scowl at the murder of his most experienced tactical officers? Does he even know?
The best evidence I'm aware of regarding Palpatine's opinion on killing Imperial officers comes from the canon novel Lords of the Sith.
In the novel, Palpatine's shuttle was shot down by rebels of the Free Ryloth movement and crash-landed on Ryloth. The shuttle carried Palpatine, Vader, and several Royal Guardsmen. Despite crashing on the home planet of the rebel movement and of several dangerous predators such as lyleks, the Emperor ordered the execution of a Royal Guardsman for forgetting to put on his seat belt before the crash landing:
The leader of the guards knelt and checked his fellow. “He’s unconscious, my Emperor. He didn’t strap himself in and was thrown about during the landing.”
The wounded guard groaned. His gloved hand opened and closed.
“Kill him,” his Master said.
The leader of the Royal Guard, conditioned to obey any order of the Emperor instantly without question, did not hesitate. He stood, drew his heavy blaster, and shot his comrade once in the head, leaving a dark, smoking hole in his helmet.
“And now there are four [dead Imperials],” his Master said.
Vader did not miss the point. He turned to face his Master, his respirator loud and steady.
The Emperor shook his head with false regret. “He was stupid. And stupidity, like nostalgia, is weakness. I cannot abide weakness in those close to me. It’s a shame, really. But sometimes we must make hard choices.
Lords of the Sith, p. 148
Granted, this was not an instance in which a high-ranking Imperial officer (the "brass") was killed. Nonetheless, the Emperor's Royal Guards were selected from among the best and most experienced Imperial soldiers and therefore represented a significant training investment on the part of the Empire, just like an admiral such as Ozzel. Furthermore, the Emperor's explanation of his order to Vader explains his philosophy -- the Emperor does not abide stupidity, weakness, or failure in those close to him, whether they be his bodyguards or his admirals.
Based on this incident, Vader's practice of executing high-ranking Imperial officers who failed him was likely a method he learned from the Emperor himself. Vader wasn't lying when he said the Emperor is not as forgiving as he is.
As for whether or not the Emperor knew of Vader's executions, elsewhere we have encountered the Emperor interacting with the Imperial brass on even a "seemingly trivial matter":
The Emperor spent a long moment studying [Imperial Security Bureau Deputy Director] Ison and [Vice Admiral] Rancit, stretching out with his powers to discern alignments, configurations, some syzygy of events. Then his thoughts turned to Vader and Tarkin. He appreciated how well they were working together, but he began to wonder if they were perhaps too close to the details of the dissidents’ scheme to recognize their ultimate objective. One needed to have a safe remove, as he felt he had, gazing into the 3-D representation of the galaxy he had made his own. How Plagueis would have mocked him for allowing himself to become personally involved in such a seemingly trivial matter; but then his Master had never foreseen that his onetime apprentice would become Emperor.
Tarkin, p. 185
If the Emperor interacted with the Imperial brass on such matters, it stands to reason that he would be aware of the execution of the commander of Vader's flagship Executor (Admiral Ozzel). And if he knew of one Vader's executions, he probably knew of the others (especially since he seems to have encouraged them).
Null's answer is good, but I have offered a differing view of Vader and Palpatine's relationship here: Could Vader really have crushed the Rebels on Hoth?
Simply put, I don't think Palpatine approved of every killing, but must publicly support Vader. There is significant evidence that Vader is somewhat dim, and possibly makes poor choices during his 'corrections'.
In one of the comic books, Palpatine tells Vader to give Grand Admiral Tarkin "free reign" to run his particular military operations as he sees fit. Aside from Tarkin, there doesn't seem to be any other Imperial officer that Vader has been told to "leave alone" or to "spare". If you notice, Tarkin actually orders Vader to release Admiral Motti in A New Hope, but Vader kills Admiral Ozzel in The Empire Strikes Back. So, essentially, every Imperial officer, but Tarkin, was fair game to Vader to what he wanted with.
