\lambda
    \norm{{\mbox{${\mbox{$\mathbf{w}$}}$}}-
    {\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau-1}}^2.$$

Generalisation for Continual Learning
-------------------------------------

We define within-task generalisation as the ability of the agent to acquire new knowlege and outside-task generalisation as its ability to preserve the acquired knowledge.

Consider a loss function $l: {\mbox{$\mathbb{R}$}}\times {\mbox{$\mathbb{R}$}}\rightarrow {\mbox{$\mathbb{R}$}}$. The population loss over the distribution ${\mbox{$\mathcal{D}$}}$, and the empirical loss over $n$ samples $D = \{({\mbox{${\mbox{$\mathbf{x}$}}$}}_i, y_i), i \in [n] \}$ from the same distribution ${\mbox{$\mathcal{D}$}}$ are defined as: $$\begin{aligned}
        L_D(f) &= {\mathbb{E}}_{({\mbox{${\mbox{$\mathbf{x}$}}$}}, y) \sim {\mbox{$\mathcal{D}$}}} [l(f({\mbox{${\mbox{$\mathbf{x}$}}$}}), y)], &\\
        L_S(f) &= \frac{1}{n} \sum_{i=1}^n l(f({\mbox{${\mbox{$\mathbf{x}$}}$}}_i), y_i).
    \end{aligned}$$

Let ${\mbox{$\mathcal{T}$}}_1, \ldots {\mbox{$\mathcal{T}$}}_T$ a sequence of tasks, and ${\mbox{$\mathcal{D}$}}_1, \ldots {\mbox{$\mathcal{D}$}}_T$ their corresponding distributions.\
Let $f_1^\star, \ldots f_T^\star$ the trained models at each task. Let $\tau \in [T]$ fixed.\
We define:

-   within-task generalisation of the task ${\mbox{$\mathcal{T}$}}_\tau$ as $L_{D_\tau}(f_\tau^\star)$.

-   outside-task generalisation of the task ${\mbox{$\mathcal{T}$}}_\tau$ with respect to a task ${\mbox{$\mathcal{T}$}}_{\tau^\prime}$, where $\tau^\prime < \tau$ as $L_{D_\tau}(f_{\tau^\prime}^\star)$.

In practice, several works also tracked these metrics in their experiments (@Kirkpatrick2016EWC, @farajtabar2019orthogonal).

Neural Tangent Kernel
---------------------

In their seminal paper, @Jacot2018NTK established the connection between deep networks and kernel methods by introducing the Neural Tangent Kernel (NTK). They showed that at the infinite width limit, the kernel remains constant throughout training. 