Hemant Tyagi

Continuum Armed Bandit Problem of Few Variables in High Dimensions

We consider the stochastic and adversarial settings of continuum armed bandits where the arms are indexed by [0,1] d . The reward functions r:[0,1] d → ℝ are assumed to intrinsically depend on at most k coordinate variables implying \(r(x_1,\dots,x_d) = g(x_{i_1},\dots,x_{i_k})\) for distinct and unknown i 1,…,i k ∈ {1,…,d} and some locally Holder continuous g:[0,1] k → ℝ with exponent α ∈ (0,1]. Firstly, assuming (i 1,…,i k ) to be fixed across time, we propose a simple modification of the CAB1 algorithm where we construct the discrete set of sampling points to obtain a bound of \(O(n^{\frac{\alpha+k}{2\alpha+k}} (\log n)^{\frac{\alpha}{2\alpha+k}} C(k,d))\) on the regret, with C(k,d) depending at most polynomially in k and sub-logarithmically in d. The construction is based on creating partitions of {1,…,d} into k disjoint subsets and is probabilistic, hence our result holds with high probability. Secondly we extend our results to also handle the more general case where (i 1,…,i k ) can change over time and derive regret bounds for the same.

On Two Continuum Armed Bandit Problems in High Dimensions

We consider the problem of continuum armed bandits where the arms are indexed by a compact subset of ℝd

Tangent space estimation for smooth embeddings of Riemannian manifolds

\mathbb {R}^{d}

Learning non-parametric basis independent models from point queries via low-rank methods

. For large d, it is well known that mere smoothness assumptions on the reward functions lead to regret bounds that suffer from the curse of dimensionality. A typical way to tackle this in the literature has been to make further assumptions on the structure of reward functions. In this work we assume the reward functions to be intrinsically of low dimension k ≪ d and consider two models: (i) The reward functions depend on only an unknown subset of k coordinate variables and, (ii) a generalization of (i) where the reward functions depend on an unknown k dimensional subspace of ℝd

Efficient Sampling for Learning Sparse Additive Models in High Dimensions

\mathbb {R}^{d}

Learning Sparse Additive Models with Interactions in High Dimensions

. By placing suitable assumptions on the smoothness of the rewards we derive randomized algorithms for both problems that achieve nearly optimal regret bounds in terms of the number of rounds n.