Effect of Spherical Parameter Space on Number of Non-Forgetting Tasks and Capacity of Continual Learning

Abstract

We propose a continuous learning formulation for $T$ non-forgetting tasks with the spherical parameter space $\mathbb{S}^{d-1}(\sqrt{d})$ in asymptotic regime $d\rightarrow \infty$. Each task $i$ has an optimum $\bm{\Theta}_i$, minimizing its corresponding loss function $R_i(\bm{w})$ on sphere. We reduce each task $i$ to an acceptable loss region modeled by a geodesic ball $B_{\bm{\Theta}_i}(r_i(d))$ on $\mathbb{S}^{d-1}(\sqrt{d})$ centered at $\bm{\Theta}_i$ with radius $r_i(d)$. To model task dependencies (similarities), task optima $\bm{\Theta}_i$ are distributed by a Neyman-Scott clustered point process $\Phi$. Cluster centers $\{\bm{Y}_j\}_{j=1}^{N_p}$ form a homogeneous Poisson point process on sphere, while intra-cluster task optima follow a von Mises-Fisher distribution with mean $\bm{Y}_j$ and concentration $\kappa$. Each task $t\in \{1, ..., T\}$ learning algorithm is considered to be a channel $Q_{t}(\mathrm{d}\bm{w}_t | \bm{w}_{t-1}, \mathcal{K}_{t-1}(\bm{\theta}_{1:t-1}), \bm{\theta}_t)$, where $\bm{W}_t$ is the learned parameter of task $t$ and $\mathcal{K}_{t-1}(\bm{\Theta}_{1:t-1})=\bigcap_{i=1}^{t-1}B_{\bm{\Theta}_i}(r_i(d))$. Since in continual learning, the learner does not need to reach $\bm{\Theta}_t$ exactly to successfully learn task $t$, to avoid forgetting as well as reliable learning, it only needs for the learned parameter vector $\bm{W}_{t}$ to enter $B_{\bm{\Theta}_t}(r_t(d)) \cap \mathcal{K}_{t-1}(\bm{\Theta}_{1:t-1})$. Using this model, we derive an upper bound on $T$ ensuring $\mathbb{P}[\mathcal{K}_{T}(\bm{\Theta}_{1:T}) \neq \emptyset] \rightarrow 1$ as $d\rightarrow \infty$ and $\kappa/d \rightarrow \alpha$, where $\alpha>0$ is a constant. Finally, for $T$ tasks sampled uniformly from $\Phi$, we derive an upper bound for total information theoretic capacity under the non-forgetting constraint for each task.