Contextual bandits with linear payoffs, which are also known as linear bandits, provide a powerful alternative for solving practical problems of sequential decisions, e.g., online advertisements. In the era of big data, contextual data usually tend to be high-dimensional, which leads to new challenges for traditional linear bandits mostly designed for the setting of low-dimensional contextual data. Due to the curse of dimensionality, there are two challenges in most of the current bandit algorithms: the first is high time-complexity; and the second is extreme large upper regret bounds with high-dimensional data. In this paper, in order to attack the above two challenges effectively, we develop an algorithm of Contextual Bandits via RAndom Projection ( CBRAP ) in the setting of linear payoffs, which works especially for high-dimensional contextual data. The proposed CBRAP algorithm is time-efficient and flexible, because it enables players to choose an arm in a low-dimensional space, and relaxes the sparsity assumption of constant number of non-zero components in previous work. Besides, we provide a linear upper regret bound for the proposed algorithm, which is associated with reduced dimensions.
One common technique for dimensionality reduction is to perform linear random projection @cite_22 @cite_28 . In this paper, we consider projecting the contextual data of @math onto a low-dimensional space of @math . Without loss of generality, we denote the random projection matrix by @math . Then, we have where @math and @math .
@cite_28 @cite_22
1571401318 2061375390
This paper discusses the topic of dimensionality reduction for @math -means clustering. We prove that any set of @math points in @math dimensions (rows in a matrix @math ) can be projected into @math dimensions, for any @math , in @math time, such that with constant probability the optimal @math -partition of the point set is preserved within a factor of @math . The projection is done by post-multiplying @math with a @math random matrix @math having entries @math or @math with equal probability. A numerical implementation of our technique and experiments on a large face images dataset verify the speed and the accuracy of our theoretical results. We show how to approximate a data matrix @math with a much smaller sketch @math that can be used to solve a general class of constrained k-rank approximation problems to within @math error. Importantly, this class of problems includes @math -means clustering and unconstrained low rank approximation (i.e. principal component analysis). By reducing data points to just @math dimensions, our methods generically accelerate any exact, approximate, or heuristic algorithm for these ubiquitous problems. For @math -means dimensionality reduction, we provide @math relative error results for many common sketching techniques, including random row projection, column selection, and approximate SVD. For approximate principal component analysis, we give a simple alternative to known algorithms that has applications in the streaming setting. Additionally, we extend recent work on column-based matrix reconstruction, giving column subsets that not only cover' a good subspace for @math , but can be used directly to compute this subspace. Finally, for @math -means clustering, we show how to achieve a @math approximation by Johnson-Lindenstrauss projecting data points to just @math dimensions. This gives the first result that leverages the specific structure of @math -means to achieve dimension independent of input size and sublinear in @math . The @math -means clustering algorithm is a ubiquitous tool in data mining and machine learning that shows promising performance. However, its high computational cost has hindered its applications in broad domains. Researchers have successfully addressed these obstacles with dimensionality reduction methods. Recently, [1] develop a state-of-the-art random projection (RP) method for faster @math -means clustering. Their method delivers many improvements over other dimensionality reduction methods. For example, compared to the advanced singular value decomposition based feature extraction approach, [1] reduce the running time by a factor of @math for data matrix @math with @math data points and @math features, while losing only a factor of one in approximation accuracy. Unfortunately, they still require @math for matrix multiplication and this cost will be prohibitive for large values of @math and @math . To break this bottleneck, we carefully build a sparse embedded @math -means clustering algorithm which requires @math ( @math denotes the number of non-zeros in @math ) for fast matrix multiplication. Moreover, our proposed algorithm improves on [1]'s results for approximation accuracy by a factor of one. Our empirical studies corroborate our theoretical findings, and demonstrate that our approach is able to significantly accelerate @math -means clustering, while achieving satisfactory clustering performance. We introduce a new low-distortion embedding of @math into @math ( @math ) called the fast Johnson-Lindenstrauss transform (FJLT). The FJLT is faster than standard random projections and just as easy to implement. It is based upon the preconditioning of a sparse projection matrix with a randomized Fourier transform. Sparse random projections are unsuitable for low-distortion embeddings. We overcome this handicap by exploiting the “Heisenberg principle” of the Fourier transform, i.e., its local-global duality. The FJLT can be used to speed up search algorithms based on low-distortion embeddings in @math and @math . We consider the case of approximate nearest neighbors in @math . We provide a faster algorithm using classical projections, which we then speed up further by plugging in the FJLT. We also give a faster algorithm for searching over the hypercube.