Fast Stochastic Ordinal Embedding with Variance Reduction and Adaptive Step Size

Learning representation from relative similarity comparisons, often called ordinal embedding, gains rising attention in recent years. Most of the existing methods are based on semi-definite programming (\textit{SDP}), which is generally time-consuming and degrades the scalability, especially confronting large-scale data. To overcome this challenge, we propose a stochastic algorithm called \textit{SVRG-SBB}, which has the following features: i) achieving good scalability via dropping positive semi-definite (\textit{PSD}) constraints as serving a fast algorithm, i.e., stochastic variance reduced gradient (\textit{SVRG}) method, and ii) adaptive learning via introducing a new, adaptive step size called the stabilized Barzilai-Borwein (\textit{SBB}) step size. Theoretically, under some natural assumptions, we show the $\boldsymbol{O}(\frac{1}{T})$ rate of convergence to a stationary point of the proposed algorithm, where $T$ is the number of total iterations. Under the further Polyak-\L{}ojasiewicz assumption, we can show the global linear convergence (i.e., exponentially fast converging to a global optimum) of the proposed algorithm. Numerous simulations and real-world data experiments are conducted to show the effectiveness of the proposed algorithm by comparing with the state-of-the-art methods, notably, much lower computational cost with good prediction performance.