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On-line learning for very large data sets. (English) Zbl 1091.68063
The learning algorithms optimise an empirical cost function $$C_{n}(\theta)={1\over n}\sum_{i=1}^{n}L(z_{i},\theta)$$. Each term measures the cost associated with running a model with parameter vector $$\theta$$ on independently drawn examples $$z_{i}$$. The authors consider two kinds of optimisation procedures: 1) batch gradient – parameter updates are performed on the basis of the gradient and Hessian information accumulated over the entire training set $$\theta(k)=\theta(k-1)-\Phi_{k}{\partial C_{n}\over\partial\theta}(\theta(k-1))$$, and (2) on-line or stochastic gradient – parameter updates are performed on the basis of a single sample $$z_{t}$$ picked randomly at each iteration $$\theta(t)=\theta(t-1)-{1\over t}\Phi_{t}{\partial L\over\partial\theta}(z_{t},\theta(t-1))$$, where $$\Phi_{t}$$ is an appropriately chosen positive definite symmetric matrix. Very often the examples $$z_{t}$$ are chosen by cycling over a randomly permuted training set. Each cycle is called an epoch. This paper shows that performing a single epoch of a suitable on-line algorithm converges to the true solution of the learning problem asymptotically as fast as any other algorithm.

##### MSC:
 68Q32 Computational learning theory 93E35 Stochastic learning and adaptive control
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