Abstract: Conservative Contextual Bandits (CCBs) address safety in sequential decision making by requiring that an agent's policy, along with minimizing regret, also satisfies a safety constraint: the performance is not worse than a baseline policy (e.g., the policy that the company has in production) by more than $(1+\alpha)$ factor. Prior work developed UCB-stylealgorithms for this problem in the multi-armed (Wu et al., 2016) and contextuallinear (Kazerouni et al., 2017) settings.However, in practice the cost of the armsis often a non-linear function, and therefore existing UCB algorithms are ineffective in such settings. In this paper, we consider CCBs beyond the linear case and develop two algorithms $\mathtt{C\text{-}SquareCB}$ and $\mathtt{C\text{-}FastCB}$, using Inverse Gap Weighting (IGW) based exploration and an online regression oracle. We show that the safety constraint is satisfied in high probability and that the regret for $\mathtt{C\text{-}SquareCB}$ is sub-linear in horizon $T$, while the the regret for $\mathtt{C\text{-}FastCB}$ is first-order and is sub-linear in $L^*$, the cumulative loss of the optimal policy. Subsequently, we use a neural network for function approximation and online gradient descent as the regression oracle to provide $\tilde{\mathcal{O}}\big(\sqrt{KT} + K/\alpha\big)$ and $\tilde{\mathcal{O}}\big(\sqrt{KL^*} + K (1 + 1/\alpha)\big)$ regret bounds respectively. Finally, we demonstrate the efficacy of our algorithms on real world data, and show that they significantly outperform the existing baseline while maintaining the performance guarantee.