#### Authors

Pedro Domingos, Geoff Hulten

#### Abstract

We propose the following general method for scaling learning algorithms to arbitrarily large data sets. Consider the model Mii learned by the algorithm using ni examples in step i (ii = (nl , ... ,nm)) , and the model Moo that would be learned using in(cid:173) finite examples. Upper-bound the loss L(Mii' M oo ) between them as a function of ii, and then minimize the algorithm's time com(cid:173) plexity f(ii) subject to the constraint that L(Moo , Mii ) be at most f with probability at most 8. We apply this method to the EM algorithm for mixtures of Gaussians. Preliminary experiments on a series of large data sets provide evidence of the potential of this approach.

1 An Approach to Large-Scale Learning

Large data sets make it possible to reliably learn complex models. On the other hand, they require large computational resources to learn from. While in the past the factor limiting the quality of learnable models was typically the quantity of data available, in many domains today data is super-abundant, and the bottleneck is t he time required to process it. Many algorithms for learning on large data sets have been proposed, but in order to achieve scalability they generally compromise the quality of the results to an unspecified degree. We believe this unsatisfactory state of affairs is avoidable, and in this paper we propose a general method for scaling learning algorithms to arbitrarily large databases without compromising the quality of the results. Our method makes it possible to learn in finite time a model that is essentially indistinguishable from the one that would be obtained using infinite data.

Consider the simplest possible learning problem: estimating the mean of a random variable x. If we have a very large number of samples, most of them are probably superfluous. If we are willing to accept an error of at most f with probability at most 8, Hoeffding bounds [4] (for example) tell us that, irrespective of the distribution of x, only n = ~(R/f)2 1n (2/8) samples are needed, where R is x's range. We propose to extend this type of reasoning beyond learning single parameters, to learning complex models. The approach we propose consists of three steps:

1. Derive an upper bound on the relative loss between the finite-data and infinite-data models, as a function of the number of samples used in each step of the finite-data algorithm.

2. Derive an upper bound on the time complexity of the learning algorithm,

as a function of the number of samples used in each step.

1. Minimize the time bound (via the number of samples used in each step)

subject to target limits on the loss.

In this paper we exemplify this approach using the EM algorithm for mixtures of Gaussians. In earlier papers we applied it (or an earlier version of it) to decision tree induction [2J and k-means clustering [3J. Despite its wide use, EM has long been criticized for its inefficiency (see discussion following Dempster et al. [1]), and has been considered unsuitable for large data sets [8J. Many approaches to speeding it up have been proposed (see Thiesson et al. [6J for a survey) . Our method can be seen as an extension of progressive sampling approaches like Meek et al. [5J: rather than minimize the total number of samples needed by the algorithm, we minimize the number needed by each step, leading to potentially much greater savings; and we obtain guarantees that do not depend on unverifiable extrapolations of learning curves.

2 A Loss Bound for EM

In a mixture of Gaussians model, each D-dimensional data point Xj is assumed to have been independently generated by the following process: 1) randomly choose a mixture component k; 2) randomly generate a point from it according to a Gaussian distribution with mean f-Lk and covariance matrix ~k. In this paper we will restrict ourselves to the case where the number K of mixture components and the probabil(cid:173) ity of selection P(f-Lk) and covariance matrix for each component are known. Given a training set S = {Xl, ... , X N }, the learning goal is then to find the maximum(cid:173) likelihood estimates of the means f-Lk. The EM algorithm [IJ accomplishes this by, starting from some set of initial means, alternating until convergence between esti(cid:173) mating the probability p(f-Lk IXj) that each point was generated by each Gaussian (the Estep), and computing the ML estimates of the means ilk = 2::;':1 WjkXj / 2::f=l Wjk (the M step), where Wjk = p(f-Lklxj) from the previous E step. In the basic EM algorithm, all N examples in the training set are used in each iteration. The goal in this paper is to speed up EM by using only ni < N examples in the ith itera(cid:173) tion, while guaranteeing that the means produced by the algorithm do not differ significantly from those that would be obtained with arbitrarily large N.

Let Mii = (ill , . . . , ilK) be the vector of mean estimates obtained by the finite-data EM algorithm (i.e., using ni examples in iteration i), and let Moo = (f-L1, ... ,f-LK) be the vector obtained using infinite examples at each iteration. In order to proceed, we need to quantify the difference between Mii and Moo . A natural choice is the sum of the squared errors between corresponding means, which is proportional to the negative log-likelihood of the finite-data means given the infinite-data ones:

L(Mii' Moo ) = L Ililk - f-Lkl12 = L L lilkd -