{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,3,12]],"date-time":"2023-03-12T13:49:36Z","timestamp":1678628976185},"reference-count":36,"publisher":"Association for Computing Machinery (ACM)","issue":"4","funder":[{"name":"European Research Council Starting Grant","award":["803084\/5"]},{"DOI":"10.13039\/501100003977","name":"Israel Science Foundation","doi-asserted-by":"crossref","award":["715\/16"]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["J. ACM"],"published-print":{"date-parts":[[2021,8]]},"abstract":"\n We consider the adversarial convex bandit problem and we build the first poly(\n T<\/jats:italic>\n )-time algorithm with poly(\n n<\/jats:italic>\n ) \u221a\n T<\/jats:italic>\n -regret for this problem. To do so, we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves \u00d5(\n n<\/jats:italic>\n 9.5<\/jats:sup>\n \u221a\n T<\/jats:italic>\n )-regret, and we show that a simple variant of this algorithm can be run in poly(\n n<\/jats:italic>\n log (\n T<\/jats:italic>\n ))-time per step (for polytopes with polynomially many constraints) at the cost of an additional poly(\n n<\/jats:italic>\n )\n T<\/jats:italic>\n o(1)<\/jats:sup>\n factor in the regret. These results improve upon the \u00d5(\n n<\/jats:italic>\n 11<\/jats:sup>\n \u221a\n T<\/jats:italic>\n -regret and exp (poly(\n T<\/jats:italic>\n ))-time result of the first two authors and the log (\n T<\/jats:italic>\n )\n \n poly(\n n<\/jats:italic>\n )\n <\/jats:sup>\n \u221a\n T<\/jats:italic>\n -regret and log(\n T<\/jats:italic>\n )\n \n poly(\n n<\/jats:italic>\n )\n <\/jats:sup>\n -time result of Hazan and Li. Furthermore, we conjecture that another variant of the algorithm could achieve \u00d5(\n n<\/jats:italic>\n 1.5<\/jats:sup>\n \u221a\n T<\/jats:italic>\n )-regret, and moreover that this regret is unimprovable (the current best lower bound being \u03a9 (\n n<\/jats:italic>\n \u221a\n T<\/jats:italic>\n ) and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order\n n<\/jats:italic>\n 3<\/jats:sup>\n \/ \u025b\n 2<\/jats:sup>\n .\n <\/jats:p>","DOI":"10.1145\/3453721","type":"journal-article","created":{"date-parts":[[2021,6,30]],"date-time":"2021-06-30T19:11:05Z","timestamp":1625080265000},"page":"1-35","update-policy":"http:\/\/dx.doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["Kernel-based Methods for Bandit Convex Optimization"],"prefix":"10.1145","volume":"68","author":[{"given":"S\u00e9bastien","family":"Bubeck","sequence":"first","affiliation":[{"name":"Microsoft Research, USA"}]},{"given":"Ronen","family":"Eldan","sequence":"additional","affiliation":[{"name":"Weizmann Institute of Science, Israel"}]},{"given":"Yin Tat","family":"Lee","sequence":"additional","affiliation":[{"name":"University of Washington, USA"}]}],"member":"320","published-online":{"date-parts":[[2021,6,30]]},"reference":[{"key":"e_1_2_1_1_1","volume-title":"Proceedings of the 21st Conference on Learning Theory (COLT\u201908)","author":"Abernethy J.","unstructured":"J. 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