Being a persistent or persevering learner?

I'd say a diligent, or hardworking student.

EDIT: I'm a US American, and learner is a funny word --- I can't tell you offhand what the rule is, but it appears in sentences like "He's a slow learner". When I Google for "learner", Google responds with "a fast learner", which is ok too. So in my opinion, learner/student is one of those word pairs where they're each used in slightly different situations.

Similar to how in the US, student driver is far more common, and in the UK, it's learner driver.

Between the two options presented, "persistent learner" sounds more natural to me but has some slight connotations that make it less than ideal. In particular, persistence and perseverance when applied to people typically mean "continuing despite adversity". In other words, rather than generally being "somebody who works hard at studying" a persistent learner is specifically "somebody who has a hard time studying but does so anyway", or "somebody who spends a lot of time studying because they need more time to learn the same amount".

If you want to describe someone who studies a lot or works hard at it, the best adjective would be "studious". "Studious learner" is a bit redundant though, if you're combining with a noun like "learner" that already implies that studying is being done then the adjective "diligent" is better, with the best phrase I can think of being "diligent student" assuming that the person is in fact a student.

"Persistent" vs "Perservering" refers to two different kinds of adversity.

"Persistent" is for ordinary adversity, like partying or romantic distractions.

"Perservering" means extraordinary or overwhelming adversity, like oppression or discrimination.

• Santa Cruz has a reputation as a party school, but Kate was a persistent student - never missing a class and carrying a 4.0 average.
• Leads on the case were few and far between, but Hercule Poirot was nothing if not persistent."
• None of the women on the science team were taken seriously - yet they proved to be the better scientists; they had each perservered through worse simply to get here.
• The German doomed their nuclear program, having long decried the science as "Jewish physics", and directing SS inquisitions against those who perservered in its study.

Somebody seems to have coined the phrase 'persistent learner', but persistent is normally used when someone keeps on doing something undesirable, like a child pestering a parent for sweets, or in a phrase like 'persistent cough'.

OK, the dictionary definition doesn't specify a negative implication, though an old (1934) edition of the Concise Oxford in my possession defines persist as Continue firmly or obstinately (in opinion, course, doing) esp. against remonstrance &c.

To my way of thinking, you persist in a bad course, persevere in a good one.