Online Dimensionality Reduction
Sammanfattning: In this thesis, we investigate online dimensionality reduction methods, wherethe algorithms learn by sequentially acquiring data. We focus on two specificalgorithm design problems in (i) recommender systems and (ii) heterogeneousclustering from binary user feedback. (i) For recommender systems, we consider a system consisting of m users and n items. In each round, a user,selected uniformly at random, arrives to the system and requests a recommendation. The algorithm observes the user id and recommends an itemfrom the item set. A notable restriction here is that the same item cannotbe recommended to the same user more than once, a constraint referred toas a no-repetition constraint. We study this problem as a variant of themulti-armed bandit problem and analyze regret with the various structurespertaining to items and users. We devise fundamental limits of regret andalgorithms that can achieve the limits order-wise. The analysis explicitlyhighlights the importance of each component of regret. For example, we candistinguish the regret due to the no-repetition constraint, that generated tolearn the statistics of user’s preference for an item, and that generated tolearn the low-dimensional space of the users and items were shown. (ii) Inthe clustering with binary feedback problem, the objective is to classify itemssolely based on limited user feedback. More precisely, users are just askedsimple questions with binary answers. A notable difficulty stems from theheterogeneity in the difficulty in classifying the various items (some itemsrequire more feedback to be classified than others). For this problem, wederive fundamental limits of the cluster recovery rates for both offline andonline algorithms. For the offline setting, we devise a simple algorithm thatachieves the limit order-wise. For the online setting, we propose an algorithm inspired by the lower bound. For both of the problems, we evaluatethe proposed algorithms by inspecting their theoretical guarantees and usingnumerical experiments performed on the synthetic and non-synthetic dataset.
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