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1. Batchman and Robin: Batched and Non-batched Branching for Interactive ZK

   Yibin Yang, David Heath (November 2023, ACM CCS 2023)

   Cas Cremers and Engin Kirda (Ed.)

   Vector Oblivious Linear Evaluation (VOLE) supports fast and scalable interactive Zero-Knowledge (ZK) proofs. Despite recent improvements to VOLE-based ZK, compiling proof statements to a control-flow oblivious form (e.g., a circuit) continues to lead to expensive proofs. One useful setting where this inefficiency stands out is when the statement is a disjunction of clauses $$\mathcal{L}_1 \lor \cdots \lor \mathcal{L}_B$$. Typically, ZK requires paying the price to handle all $$B$$ branches. Prior works have shown how to avoid this price in communication, but not in computation. Our main result, $$\mathsf{Batchman}$$, is asymptotically and concretely efficient VOLE-based ZK for batched disjunctions, i.e. statements containing $$R$$ repetitions of the same disjunction. This is crucial for, e.g., emulating CPU steps in ZK. Our prover and verifier complexity is only $$\bigO(RB+R|\C|+B|\C|)$$, where $$|\C|$$ is the maximum circuit size of the $$B$$ branches. Prior works' computation scales in $$RB|\C|$$. For non-batched disjunctions, we also construct a VOLE-based ZK protocol, $$\mathsf{Robin}$$, which is (only) communication efficient. For small fields and for statistical security parameter $$\lambda$$, this protocol's communication improves over the previous state of the art ($$\mathsf{Mac'n'Cheese}$$, Baum et al., CRYPTO'21) by up to factor $$\lambda$$. Our implementation outperforms prior state of the art. E.g., we achieve up to $$6\times$$ improvement over $$\mathsf{Mac'n'Cheese}$$ (Boolean, single disjunction), and for arithmetic batched disjunctions our experiments show we improve over $$\mathsf{QuickSilver}$$ (Yang et al., CCS'21) by up to $$70\times$$ and over $$\mathsf{AntMan}$$ (Weng et al., CCS'22) by up to $$36\times$$. 

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2. Batched Dueling Bandits

   Agarwal, Arpit; Ghuge, Rohan; Nagarajan, Viswanath (January 2022, International Conference on Machine Learning)
3. Optimal Batched Linear Bandits

   Ren, Xuanfei; Jin, Tianyuan; Xu, Pan (May 2024, Proceedings of The 41st International Conference on Machine Learning (ICML))

   We introduce the E$^4$ algorithm for the batched linear bandit problem, incorporating an Explore-Estimate-Eliminate-Exploit framework. With a proper choice of exploration rate, we prove E$^4$ achieves the finite-time minimax optimal regret with only $$O(\log\log T)$$ batches, and the asymptotically optimal regret with only $$3$$ batches as $$T\rightarrow\infty$$, where $$T$$ is the time horizon. We further prove a lower bound on the batch complexity of linear contextual bandits showing that any asymptotically optimal algorithm must require at least $$3$$ batches in expectation as $$T\rightarrow\infty$$, which indicates E$^4$ achieves the asymptotic optimality in regret and batch complexity simultaneously. To the best of our knowledge, E$^4$ is the first algorithm for linear bandits that simultaneously achieves the minimax and asymptotic optimality in regret with the corresponding optimal batch complexities. In addition, we show that with another choice of exploration rate E$^4$ achieves an instance-dependent regret bound requiring at most $$O(\log T)$$ batches, and maintains the minimax optimality and asymptotic optimality. We conduct thorough experiments to evaluate our algorithm on randomly generated instances and the challenging \textit{End of Optimism} instances \citep{lattimore2017end} which were shown to be hard to learn for optimism based algorithms. Empirical results show that E$^4$ consistently outperforms baseline algorithms with respect to regret minimization, batch complexity, and computational efficiency. 

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4. Batched Nonparametric Contextual Bandits

   Jiang, Rong; Ma, Cong (June 2024, arXivorg)
5. Batched Nonparametric Contextual Bandits

   https://doi.org/10.1109/TIT.2025.3555071  

   Jiang, Rong; Ma, Cong (June 2025, IEEE Transactions on Information Theory)