Information-theoretic lower bounds for convex optimization with erroneous oracles
We consider the problem of optimizing convex and concave functions with access to an erroneous zeroth-order oracle. In particular, for a given function x → f (x) we consider optimization when one is given access to absolute error oracles that return values in [f (x) − , f (x) + ] or relative error oracles that return value in
, for some > 0. We show stark information theoretic impossibility results for minimizing convex functions and maximizing concave functions over polytopes in this model.
Added post-rebuttal: Thanks for addressing my technical concerns, and I've raised my score to 6. Note that the \delta,\epsilon issue still seems a bit weird (for example, theorem 4.2, why not just take delta=0.5, getting the strongest possible statement?).============================================The paper studies lower bounds for zero-order optimization, where the returned function values have some arbitrary bounded error. Previous results provided algorithms when the error scales down linearly in the dimension n. In contrast, here it is shown that if the error scales more slowly than n^{-1/2}, then no non-trivial optimization is possible. This is shown in several settings, considering both absolute and relative errors, and for several different optimization domains. The proof technique is based on constructing two functions which agree almost everywhere on the domain, yet have quite different optima. I am a bit undecided regarding this paper. On one hand, the basic question is scientifically interesting, and the answer is somewhat surprising at first, and potentially useful. On the other hand, there are several weaknesses:- Beyond the first interesting result (theorem 3.1), the other results are variations of the same idea for various specific domains (whose relevance for zero-order optimization is unclear), and I didn't feel they provide anything conceptually new. I think the paper would have been stronger if it contained more genuinely different results. For example, is it possible to generalize the impossibility results here to general families of domains? What is the error in the oracle values is between n^{-1/2} and n^{-1} (at which point existing algorithms do work)? What if the oracle error isn't arbitrary but has some structure?- The connection of the paper to machine learning is somewhat tenuous (in previous papers, which considered a noisy oracle, one can at least argue that such an oracle reflects uncertainty due to data sampling).- Although I believe the results, the proofs are sloppy at places and some parts need to be fixed (see technical comments below). - The text is wider than the required format (maybe because of an erroneous use of \fullpage in the latex?) This means that the paper will require some non-trivial editing to get to the page limit.  Specific Comments ----------------- - All the proofs rely on extending a Chernoff bound for a fixed query to a sequence of several queries, using a union bound. However, after the first query, the other queries are no longer fixed, but rather conditioned on previous queries. So, one cannot apply a Chernoff bound as-is, and a more delicate argument is required.- In theorem 3.1, one should also require delta < 1, no? Otherwise the proof doesn't seem to work. Similarly, in theorem 4.2, I think delta must be < 0.5 (otherwise, in the proof, f and g would be the same function).- In all the proofs, there is an implicit assumption that n^{delta} and such numbers are integers (e.g. splitting [n] to sets of this size), which is not necessarily the case.- In the proof of theorem 4.2, the definition of the polytope domain is missing (and is apparently not the same as in theorem 4.1, since the optimal values of f and g would be different than those stated in the proof). - In the proof of theorem 3.1, the fact that the function value is the sum of negatively correlated random variables needs to be more carefully justified, either by a reference or by explicitly showing why they satisfy the definition of negative correlation from section 2.- The results of section 4 appear to be stronger the larger we pick epsilon,delta. So, why not show the result for some large constant epsilon,delta.?- Section 5 is extremely short and appears to be just an advertisement to something which appears in supplementary material. This makes the paper not self-contained. I would recommend either providing some formal result (possibly without the proof), or dropping this section. - line 319: "We again we select"