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In online learning from non-stationary data streams, it is necessary to learnrobustly to outliers and to adapt quickly to changes in the underlying datagenerating mechanism. In this paper, we refer to the former attribute of onlinelearning algorithms as robustness and to the latter as adaptivity. There is anobvious tradeoff between the two attributes. It is a fundamental issue toquantify and evaluate the tradeoff because it provides important information onthe data generating mechanism. However, no previous work has considered thetradeoff quantitatively. We propose a novel algorithm called the stochasticapproximation-based robustness-adaptivity algorithm (SRA) to evaluate thetradeoff. The key idea of SRA is to update parameters of distribution orsufficient statistics with the biased stochastic approximation scheme, whiledropping data points with large values of the stochastic update. We address therelation between the two parameters: one is the step size of the stochasticapproximation, and the other is the threshold parameter of the norm of thestochastic update. The former controls the adaptivity and the latter does therobustness. We give a theoretical analysis for the non-asymptotic convergenceof SRA in the presence of outliers, which depends on both the step size andthreshold parameter. Because SRA is formulated on the majorization-minimizationprinciple, it is a general algorithm that includes many algorithms, such as theonline EM algorithm and stochastic gradient descent. Empirical experiments forboth synthetic and real datasets demonstrated that SRA was superior to previousmethods.

Online Robust and Adaptive Learning from Data Streams