| Terms | Error | Output Layer Delta | |
| Regression | $$T = \mathbb{R}^{N\times K}$$ | $$\frac{1}{NK} \sum_{n=1}^N \sum_{k=1}^K (T_{n,k} - Y_{n,k})^2$$ | $$\frac{-2}{NK} (T_{n,k} - Y_{n,k}) \\ \text{ for each } n \text{ and } k \text{ or, }\\ \frac{-2}{NK} (T - Y)$$ |
| Classification | $$\begin{align*} T &= \mathbb{C}^{N\times 1}\\ \mathit{Tiv}^{N\times K} &= \text{indvars}(T)\\ \mathit{Ysm}^{N\times K} &= \frac{e^Y}{\sum_{k=1}^K e^{Y_{*,k}}} \end{align*}$$ | $$\begin{align*} -\log \left( (\prod_{n=1}^N \prod_{k=1}^K \mathit{Ysm}_{n,k}^{\mathit{Tiv}_{n,k}})^{\frac{1}{NK}} \right )\\ = -\frac{1}{NK} \sum_{n=1}^N \sum_{k=1}^K \mathit{Tiv}_{n,k} \log(\mathit{Ysm}_{n,k}) \end{align*}$$ | $$-\frac{1}{NK} (\mathit{Tiv}_{n,k} - \mathit{Ysm}_{n,k}) \\ \text{ for each } n \text{ and } k \text{ or, }\\ -\frac{1}{NK} (\mathit{Tiv} - \mathit{Ysm})$$ |
| Reinforcement Learning | $$\mathit{T}_n = r_{n+1} + \gamma Y_{n+1} \\ \text{ for sequential samples } n \\ \text{ where } Y_{n+1} \text{ is } Q_{n+1} \text{ value.}$$ | $$\frac{1}{N} \sum_{n=1}^N (\mathit{T}_n - Y_n)^2$$ | $$\frac{-2}{N} (\mathit{T}_n - Y_n) \\ \text{ for each } n \text{ or, }\\ \frac{-2}{N} (\mathit{T} - Y)$$ |