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.
@cite_5 , the authors proposed new methods for constructing sparse random sign matrix for dimensionality reduction. In the ensuing section, we name our algorithm of CBRAP with Random Sign (RS) matrix (abbreviated as CBRAP.RS ).
@cite_5
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There are two main algorithmic approaches to sparse signal recovery: geometric and combinatorial. The geometric approach utilizes geometric properties of the measurement matrix Phi. A notable example is the Restricted Isometry Property, which states that the mapping Phi preserves the Euclidean norm of sparse signals; it is known that random dense matrices satisfy this constraint with high probability. On the other hand, the combinatorial approach utilizes sparse matrices, interpreted as adjacency matrices of sparse (possibly random) graphs, and uses combinatorial techniques to recover an approximation to the signal. In this paper we present a unification of these two approaches. To this end, we extend the notion of Restricted Isometry Property from the Euclidean lscr2 norm to the Manhattan lscr1 norm. Then we show that this new lscr1 -based property is essentially equivalent to the combinatorial notion of expansion of the sparse graph underlying the measurement matrix. At the same time we show that the new property suffices to guarantee correctness of both geometric and combinatorial recovery algorithms. As a result, we obtain new measurement matrix constructions and algorithms for signal recovery which, compared to previous algorithms, are superior in either the number of measurements or computational efficiency of decoders. Recovering low-rank and sparse matrices from incomplete or corrupted observations is an important problem in statistics, machine learning, computer vision, as well as signal and image processing. In theory, this problem can be solved by the natural convex joint mixed relaxations (i.e., l 1 -norm and trace norm) under certain conditions. However, all current provable algorithms suffer from superlinear per-iteration cost, which severely limits their applicability to large-scale problems. In this paper, we propose a scalable, provable and structured robust bilinear factorization (RBF) method to recover low-rank and sparse matrices from missing and grossly corrupted data, i.e., robust matrix completion (RMC), or incomplete and grossly corrupted measurements, i.e., compressive principal component pursuit (CPCP). Specifically, we first present two small-scale matrix trace norm regularized bilinear factorization models for RMC and CPCP problems, in which repetitively calculating SVD of a large-scale matrix is replaced by updating two much smaller factor matrices. Then, we apply the alternating direction method of multipliers (ADMM) to efficiently solve the RMC problems. Finally, we provide the convergence analysis of our algorithm, and extend it to address general CPCP problems. Experimental results verified both the efficiency and effectiveness of our method compared with the state-of-the-art methods. We shall show that if the restricted isometry constant (RIC) @math of the measurement matrix @math satisfies @math then the greedy algorithm Orthogonal Matching Pursuit(OMP) will succeed. That is, OMP can recover every @math -sparse signal @math in @math iterations from @math . Moreover, we shall show the upper bound of RIC is sharp in the following sense. For any given @math , we shall construct a matrix @math with the RIC @math such that OMP may not recover some @math -sparse signal @math in @math iterations. [1] conjectured the existence of a sharp threshold on model parameters for community detection in sparse random graphs drawn from the stochastic block model. Mossel, Neeman and Sly [2] established the negative part of the conjecture, proving impossibility of non-trivial reconstruction below the threshold. In this work we solve the positive part of the conjecture. To that end we introduce a modified adjacency matrix B which counts self-avoiding paths of a given length e between pairs of nodes. We then prove that for logarithmic length e, the leading eigenvectors of this modified matrix provide a non-trivial reconstruction of the underlying structure, thereby settling the conjecture. A key step in the proof consists in establishing a weak Ramanujan property of the constructed matrix B. Namely, the spectrum of B consists in two leading eigenvalues ρ(B), λ2 and n -- 2 eigenvalues of a lower order O(ne √ρ(B) for all e 0, ρ(B) denoting B's spectral radius.