Incremental learning

Suppose we have a sequence of learning rates $\alpha: \mathbb{N} \rightarrow \mathbb{R}$. Then a value $V$ incrementally learns some random variable $X$ if for $t \in \mathbb{N}$ with samples $x_t$ from $X$ we set

$$V_t = V_{t-1}(1 - \alpha_t) + \alpha x_t$$

then let $V = \lim_{t \rightarrow \infty} V_t$. This is denoted $V \leftarrow^{\alpha} X$.