# Gradient Descent

Gradient Descent is an algorithm for online linear regression. In the simplest case, the algorithm predicts a sequence of numbers . The prediction space is . The master's prediction is the weighted average of the input vectors . At each step the weights are updated by the rule, which depends on the gradient of a loss function: .

For the bounded signal and convex twice differentiable by the second argument loss function the Gradient Descent algorithm can achieve

,

where is the loss of any linear function of , and is a bound for the derivative of the loss function: . The best . Here us the norm of the experts:

The Gradient Descent algorithm was intestigated for this problem by Cesa-Bianchi et al. (1994), and is also known as Least Mean Squares algorithm for the square-loss function.

### Bibliography

- N. Cesa-Bianchi, P.M. Long, and M.K. Warmuth: Worst-case quadratic loss bounds for prediction using linear functions and gradient descent.
*IEEE Transactions on Neural Networks*, 7(3):604-619, 1996.