On-line Linear Optimization

An on-line linear optimization problem is defined as the following repeated game between the learner (player) and the environment (adversary, or Reality). Let $\mathcal{K} \in \mathbb{R}^n$ be a compact closed convex set. The protocol of the game is

FOR {$t=1,2,\dots,T$}
Player chooses $x_t \in \mathcal{K}$
Adversary independently chooses $f_t \in \mathbb{R}^n$
Player suffers loss $f'_t x_t$, and observes some feedback.

The goal of the Player is to minimize his regret $R_T = \sum_{t=1}^T f'_t x_t - \min_{x^* \in \mathcal{K}} \sum_{t=1}^T f'_t x^*$. In the full information setting of the game, the player may observe the entire function $f_t$ as his feedback, and can exploit this in making decisions. The other setting is a Bandit setting, when the player observes only a scalar value $f'_t x_t$. In practice this means, that we do not know "what would be if we changed our action", but only current feedback. Such a setting is useful for a various range of applications.

To prove the theoretical bound for the regret, Follow the Leader algorithm is sometimes used, and other ones like Follow the regularized leader.