We study a general version of the problem of online learning under binary feedback: there is a hidden function f : \mathcal{X \rightarrow Y in a certain hypothesis class H. A learner is given adversarially chosen inputs (contexts) x_t \in X and is asked to submit a guess y_t \in Y for the value f(x_t). Upon guessing the learner incurs a certain loss L(y_t, f(x_t)) and learns whether y_t \leq f(x_t) or y_t > f(x_t). The special case where H is the class of linear functions over the unit ball has been studied in a series of papers. We both generalize and improve these results. We provide a }  O(d ^2) regret bound where d is the covering dimension of the hypothesis class. The algorithms are based on a novel technique which we call Steiner potential since in the linear case it reduces to controlling the value of the Steiner polynomial of a convex region at various scales. We also show that this new technique provides optimal regret (up to log factors) in the linear case (i. e. the original contextual search problem), improving the previously known bound of \rightarrow Y in a certain hypothesis class H. We provide a generic algorithm with } O(d ^4) to O(d \log d) . Finally, we extend these results to a \tilde{O} noisy feedback model, where each round our feedback is flipped with  fixed probability p < 1/2.
 \rightarrow Y in a certain hypothesis class H. A learner is given adversarially chosen inputs (contexts) x_t \in X and is asked to submit a guess y_t \in Y for the value f(x_t). Upon guessing the learner incurs a certain loss L(y_t, f(x_t)) and learns whether y_t \leq f(x_t) or y_t > f(x_t). The special case where H is the class of linear functions over the unit ball has been studied in a series of papers. We both generalize and improve these results. We provide a } In the contextual pricing problem a seller repeatedly obtains products described by an adversarially chosen feature vector in \mathbb{R O(d \log \log T + d \log d) regret which matches the \Omega(d \log \log T) lower bound up to the d \log d additive factor. If we replace pricing loss by the symmetric loss, we obtain an algorithm with nearly optimal regret of O(d \log d) matching the \Omega(d) lower bound up to \log d. These algorithms are based on a novel technique of bounding the value of the Steiner polynomial of a convex region at various scales. The Steiner polynomial is a degree d polynomial with intrinsic volumes as the coefficients. We also study a generalized version of contextual search where the hidden linear function over the Euclidean space is replaced by a hidden function f : \mathcal{X \rightarrow Y in a certain hypothesis class H. We provide a generic algorithm with } O(d ^2) regret where d is the covering dimension of this class. This leads in particular to a \tilde{O}(s^2) regret algorithm for linear contextual search if the linear function is guaranteed to be s-sparse. Finally we also extend our results to the noisy feedback model, where each round our feedback is flipped with a fixed probability p < 1/2.
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We study a general version of the problem of online learning under binary feedback: there is a hidden function f : \mathcal{X
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In the contextual pricing problem a seller repeatedly obtains products described by an adversarially chosen feature vector in \mathbb{R
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^2) regret bound where d is the covering dimension of the hypothesis class. The
\log \log T + d \log d) regret which matches the \Omega(d \log \log T) lower bound up to the d \log d additive factor. If we replace pricing loss by the symmetric loss, we obtain an algorithm with nearly optimal regret of O(d \log d) matching the \Omega(d) lower bound up to \log d. These
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which we call Steiner potential since in the linear case it reduces to controlling the
of bounding the
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We also show that this new technique provides optimal regret (up to log factors) in the linear case (i. e. the original contextual search problem), improving the previously known bound of
The Steiner polynomial is a degree d polynomial with intrinsic volumes as the coefficients. We also study a generalized version of contextual search where the hidden linear function over the Euclidean space is replaced by a hidden function f : \mathcal{X
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^4) to O(d \log d) . Finally, we extend these results to
^2) regret where d is the covering dimension of this class. This leads in particular to
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(s^2) regret algorithm for linear contextual search if the linear function is guaranteed to be s-sparse. Finally we also extend our results to the
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