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In this paper, we address learning problems for high dimensional data.Previously, oblivious random projection based approaches that project highdimensional features onto a random subspace have been used in practice fortackling high-dimensionality challenge in machine learning. Recently, variousnon-oblivious randomized reduction methods have been developed and deployed forsolving many numerical problems such as matrix product approximation, low-rankmatrix approximation, etc. However, they are less explored for the machinelearning tasks, e.g., classification. More seriously, the theoretical analysisof excess risk bounds for risk minimization, an important measure ofgeneralization performance, has not been established for non-obliviousrandomized reduction methods. It therefore remains an open problem what is thebenefit of using them over previous oblivious random projection basedapproaches. To tackle these challenges, we propose an algorithmic framework foremploying non-oblivious randomized reduction method for general empirical riskminimizing in machine learning tasks, where the original high-dimensionalfeatures are projected onto a random subspace that is derived from the datawith a small matrix approximation error. We then derive the first excess riskbound for the proposed non-oblivious randomized reduction approach withoutrequiring strong assumptions on the training data. The established excess riskbound exhibits that the proposed approach provides much better generalizationperformance and it also sheds more insights about different randomizedreduction approaches. Finally, we conduct extensive experiments on bothsynthetic and real-world benchmark datasets, whose dimension scales to$O(10^7)$, to demonstrate the efficacy of our proposed approach.

Efficient Non-oblivious Randomized Reduction for Risk Minimization with Improved Excess Risk Guarantee