alt Hacker NewsI know a bit about this field. This conjecture reads as somewhat more niche than the cyclic double cover conjecture recently proved by OpenAI, but nevertheless represents a real contribution.
You want to know how long it takes to solve an optimization problem, in this case over convex, lipschitz functions. (The restriction to a spherical domain is not really a restriction, you can just change variables for any bounded domain.) Anyway, showing upper bounds on time complexity is "easy" because it's just the runtime of your algorithm. Showing (nontrivial) lower bounds is usually much harder because it requires constraining all algorithms.
This proof apparently shows that the lower bound time complexity is equal to the time complexity of an existing 30-year old algorithm: it requires Omega(d^2) function evaluations to solve over this class of functions.
My gut says likely implies that d is the minimal number of evaluations if you have a gradient oracle because you can approximate a gradient with d function evaluations, but I'm not sure how hard it is to make that rigorous.
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It should be noted that optimization of a convex bounded lipschitz function is exactly what most modern statistical learning (AI) models are based on.
Yes, order d is the minimal number of evaluations of gradients needed for the same problem! That has actually been known since 1979 (Nemirovsky and Yudin showed that), and there are methods with the same complexity so this question in the gradient model has been solved for a long time. "because you can approximate a gradient with d function evaluations" was exactly why d^2 made sense as a lower bound for this case! Basically, the lower bound question can also be thought about as "can you do better than approxing a gradient?", so this result says no.
Any implications for discrete/integer optimization?
Sometimes I read a comment on HN that is so advanced that it's just as readable to me as Greek. Love reading it just to see someone work though!