##### Department of Mathematics,

University of California San Diego

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### Stochastic Systems Seminar

## Yoav Freund

#### UCSD, Computer Science and Electrical Engineering

## Boosting and Brownian motion

##### Abstract:

The computational task that lies in the core of many machine learning problems is the minimization of a cost function called the training error. This problem is frequently solved by local search algorithms such as gradient descent. The training error can usually be expressed as a sum over many terms, each corresponding to the loss of the model on a single training example. We show that the iteratively minimizing a cost function of this form by local search is closely related to the following game: Imagine you are a shepherd in charge of a large herd of sheep and your goal is to concentrate the sheep in a particular small area by nightfall. Your influence on the sheep movements is represented by vectors which define the direction in which you want each sheep to move. The lengths of the vectors correspond to the fraction of your ''energy'' that you spend on moving the particular sheep, and these lengths sum to one. The sheep then have to respond by moving in a way that has a slight correlation with the influence direction on average. We characterize the min/max solution to this game and show that by taking the appropriate small-step/continuous-time limit, this solution can be characterized by a stochastic differential equation. By solving this differential equation we re-derive some known boosting as well as design some new ones with desirable properties.

Host:

### April 26, 2006

### 4:00 PM

### AP&M 7218

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