We analyse adversarial bandit convex optimisation with an adversary that is restricted to playing functions of the form f  (x) = g  (\langle x, \theta\rangle) for convex g  : \mathbb R \to \mathbb R and  \theta \in \mathbb R^d  . We provide a short information-theoretic proof that the minimax regret is at most O(d n \log(  diam  \mathcal K))  where n is the number of interactions, d the dimension and diam(\mathcal K) is the diameter of the constraint set . Hence, this class of functions is at most logarithmically harder than the linear case .
We analyse adversarial bandit convex optimisation with an adversary that is restricted to playing functions of the form f _t (x) = g _t (\langle x, \theta\rangle) for convex g _t : \mathbb R \to \mathbb R and unknown \theta \in \mathbb R^d that is homogeneous over time . We provide a short information-theoretic proof that the minimax regret is at most O(d n \log( n diam ( \mathcal K)) ) where n is the number of interactions, d the dimension and diam(\mathcal K) is the diameter of the constraint set .
A
_t
A
_t
A
_t
A
unknown
A
that is homogeneous over time
A
n
A
(
A
)
D
. Hence, this class of functions is at most logarithmically harder than the linear case