Thompson sampling is a widely used strategy for sequential decision-making under uncertainty, offering a principled way to balance exploration and exploitation. This strategy involves maximizing a target objective under a single sample drawn from the current belief over unknown quantities. Conventionally, this optimization is performed once the belief has been inferred from data, and it becomes computationally burdensome when many Thompson samples are required, especially in continuous action spaces. In this work, we introduce an approach that amortizes this process ahead of time, before any data is collected, so that an approximation of the full Thompson sampling distribution is available immediately once observations arrive. We employ this approach to solve the multi-round setting, involving sequential data acquisition and Thompson-sample recommendation across rounds. Beyond accelerating Thompson sampling, our approach enables efficient estimation of expected information gain about the optimal action. This can be used to select more informative Thompson samples or to obtain maximally informative designs with optimization. Combining this estimate of expected information gain with immediate regret supports information-directed sampling (IDS), which can outperform Thompson sampling by explicitly balancing these two quantities. We demonstrate our approach on two simulator-based design problems in synthetic biology, where the simulators have intractable likelihoods.

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