Abstract: We study lifelong learning in linear bandits, where a learner interacts with a sequence of linear bandit tasks whose parameters lie in an $m$-dimensional subspace of $\mathbb{R}^d$, thereby sharing a low-rank representation. Current literature typically assumes that the tasks are diverse; that is, their parameters uniformly span the $m$-dimensional subspace. This assumption allows the low-rank representation to be learned before all tasks are revealed, which can be unrealistic in real-world applications. In this work, we consider a setting extending beyond the task diversity assumption. We present an algorithm that can efficiently learn and transfer low-rank representations. When facing $N$ tasks each played over $\tau$ rounds, our algorithm achieves a regret guarantee of $\tilde{O}\left (Nm \sqrt{\tau} + N^{\frac{2}{3}} \tau^{\frac{2}{3}} d m^{\frac13} \right)$ under mild assumptions. This improves upon the baseline $\tilde{O} \left (Nd \sqrt{\tau}\right)$ that does not leverage the low-rank structure. We demonstrate empirically on synthetic data that our algorithm outperforms baseline algorithms that rely on the task diversity assumption.