We consider the adversarial convex bandit problem and we build the first poly( T )-time algorithm with poly( n ) √ T -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 Õ( n 9.5 √ T )-regret, and we show that a simple variant of this algorithm can be run in poly( n log ( T ))-time per step (for polytopes with polynomially many constraints) at the cost of an additional poly( n ) T o(1) factor in the regret. These results improve upon the Õ( n 11 √ T -regret and exp (poly( T ))-time result of the first two authors and the log ( T ) poly( n ) √ T -regret and log( T ) poly( n ) -time result of Hazan and Li. Furthermore, we conjecture that another variant of the algorithm could achieve Õ( n 1.5 √ T )-regret, and moreover that this regret is unimprovable (the current best lower bound being Ω ( n √ T ) 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 3 / ɛ 2 .

中文翻译：

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基于核的强盗凸优化方法

我们考虑对抗性凸老虎机问题，并构建第一个 poly(吨)-时间算法与 poly(n) √吨- 后悔这个问题。为此，我们在无导数优化文献中引入了三个新思想：（i）核方法，（ii）伯努利卷积的泛化，以及（iii）指数权重的新退火计划（随着学习率的增加）。我们算法的基本版本实现了Õ(n 9.5√吨)-遗憾，我们证明了这个算法的一个简单变体可以在 poly(n日志 （吨))-每一步的时间（对于具有多项式约束的多面体）以额外的 poly(n)吨 o(1)后悔的因素。这些结果在 Õ(n 11√吨-遗憾和 exp（聚（吨))-前两位作者的时间结果和日志(吨)聚(n)√吨-遗憾并记录（吨)聚(n)- Hazan 和 Li 的时间结果。此外，我们推测该算法的另一种变体可以实现 Õ(n 1.5√吨)-遗憾，而且这种遗憾是无法改善的（当前最佳下限为 Ω (n√吨)，它是用线性函数实现的)。对于零阶随机凸优化的更简单情况，这对应于最优查询复杂度是有序的猜想n 3/ ɛ2.