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1. Feasible Q-Learning for Average Reward Reinforcement Learning

   Ying, Jin; Ramki, Gummadi; Zhengyuan, Zhou; Blanchet, Jose (May 2024, Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR)

   Dasgupta, Sanjoy; Mandt, Stephan; Li, Yingzhen (Ed.)

   Average reward reinforcement learning (RL) provides a suitable framework for capturing the objective (i.e. long-run average reward) for continuing tasks, where there is often no natural way to identify a discount factor. However, existing average reward RL algorithms with sample complexity guarantees are not feasible, as they take as input the (unknown) mixing time of the Markov decision process (MDP). In this paper, we make initial progress towards addressing this open problem. We design a feasible average-reward $$Q$$-learning framework that requires no knowledge of any problem parameter as input. Our framework is based on discounted $$Q$$-learning, while we dynamically adapt the discount factor (and hence the effective horizon) to progressively approximate the average reward. In the synchronous setting, we solve three tasks: (i) learn a policy that is $$\epsilon$$-close to optimal, (ii) estimate optimal average reward with $$\epsilon$$-accuracy, and (iii) estimate the bias function (similar to $$Q$$-function in discounted case) with $$\epsilon$$-accuracy. We show that with carefully designed adaptation schemes, (i) can be achieved with $$\tilde{O}(\frac{SA t_{\mathrm{mix}}^{8}}{\epsilon^{8}})$$ samples, (ii) with $$\tilde{O}(\frac{SA t_{\mathrm{mix}}^5}{\epsilon^5})$$ samples, and (iii) with $$\tilde{O}(\frac{SA B}{\epsilon^9})$$ samples, where $$t_\mathrm{mix}$$ is the mixing time, and $B > 0$ is an MDP-dependent constant. To our knowledge, we provide the first finite-sample guarantees that are polynomial in $$S, A, t_{\mathrm{mix}}, \epsilon$$ for a feasible variant of $$Q$$-learning. That said, the sample complexity bounds have tremendous room for improvement, which we leave for the community’s best minds. Preliminary simulations verify that our framework is effective without prior knowledge of parameters as input. 

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2. Feasible $$Q$$-Learning for Average Reward Reinforcement Learning

   Ying, Jin; Ramki, Gummadi; Zhou, Zhengyuan; Blanchet, Jose (May 2024, Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR)

   Dasgupta, Sanjoy; Mandt, Stephan; Li, Yingzhen (Ed.)

   Average reward reinforcement learning (RL) provides a suitable framework for capturing the objective (i.e. long-run average reward) for continuing tasks, where there is often no natural way to identify a discount factor. However, existing average reward RL algorithms with sample complexity guarantees are not feasible, as they take as input the (unknown) mixing time of the Markov decision process (MDP). In this paper, we make initial progress towards addressing this open problem. We design a feasible average-reward $$Q$$-learning framework that requires no knowledge of any problem parameter as input. Our framework is based on discounted $$Q$$-learning, while we dynamically adapt the discount factor (and hence the effective horizon) to progressively approximate the average reward. In the synchronous setting, we solve three tasks: (i) learn a policy that is $$\epsilon$$-close to optimal, (ii) estimate optimal average reward with $$\epsilon$$-accuracy, and (iii) estimate the bias function (similar to $$Q$$-function in discounted case) with $$\epsilon$$-accuracy. We show that with carefully designed adaptation schemes, (i) can be achieved with $$\tilde{O}(\frac{SA t_{\mathrm{mix}}^{8}}{\epsilon^{8}})$$ samples, (ii) with $$\tilde{O}(\frac{SA t_{\mathrm{mix}}^5}{\epsilon^5})$$ samples, and (iii) with $$\tilde{O}(\frac{SA B}{\epsilon^9})$$ samples, where $$t_\mathrm{mix}$$ is the mixing time, and $B > 0$ is an MDP-dependent constant. To our knowledge, we provide the first finite-sample guarantees that are polynomial in $$S, A, t_{\mathrm{mix}}, \epsilon$$ for a feasible variant of $$Q$$-learning. That said, the sample complexity bounds have tremendous room for improvement, which we leave for the community’s best minds. Preliminary simulations verify that our framework is effective without prior knowledge of parameters as input. 

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3. Feasible Q-Learning for Average Reward Reinforcement Learning

   Ying, Jin; Ramki, Gummadi; Zhou, Zhengyuan; Blanchet, Jose (May 2024, Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR)

   Dasgupta, Sanjoy; Mandt, Stephan; Li, Yingzhen (Ed.)

   Average reward reinforcement learning (RL) provides a suitable framework for capturing the objective (i.e. long-run average reward) for continuing tasks, where there is often no natural way to identify a discount factor. However, existing average reward RL algorithms with sample complexity guarantees are not feasible, as they take as input the (unknown) mixing time of the Markov decision process (MDP). In this paper, we make initial progress towards addressing this open problem. We design a feasible average-reward $$Q$$-learning framework that requires no knowledge of any problem parameter as input. Our framework is based on discounted $$Q$$-learning, while we dynamically adapt the discount factor (and hence the effective horizon) to progressively approximate the average reward. In the synchronous setting, we solve three tasks: (i) learn a policy that is $$\epsilon$$-close to optimal, (ii) estimate optimal average reward with $$\epsilon$$-accuracy, and (iii) estimate the bias function (similar to $$Q$$-function in discounted case) with $$\epsilon$$-accuracy. We show that with carefully designed adaptation schemes, (i) can be achieved with $$\tilde{O}(\frac{SA t_{\mathrm{mix}}^{8}}{\epsilon^{8}})$$ samples, (ii) with $$\tilde{O}(\frac{SA t_{\mathrm{mix}}^5}{\epsilon^5})$$ samples, and (iii) with $$\tilde{O}(\frac{SA B}{\epsilon^9})$$ samples, where $$t_\mathrm{mix}$$ is the mixing time, and $B > 0$ is an MDP-dependent constant. To our knowledge, we provide the first finite-sample guarantees that are polynomial in $$S, A, t_{\mathrm{mix}}, \epsilon$$ for a feasible variant of $$Q$$-learning. That said, the sample complexity bounds have tremendous room for improvement, which we leave for the community’s best minds. Preliminary simulations verify that our framework is effective without prior knowledge of parameters as input. 

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4. 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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5. On the derivation of the wave kinetic equation for NLS

   https://doi.org/10.1017/fmp.2021.6  

   Deng, Yu; Hani, Zaher (January 2021, Forum of Mathematics, Pi)

   Abstract A fundamental question in wave turbulence theory is to understand how the wave kinetic equation describes the long-time dynamics of its associated nonlinear dispersive equation. Formal derivations in the physics literature, dating back to the work of Peierls in 1928, suggest that such a kinetic description should hold (for well-prepared random data) at a large kinetic time scale $$T_{\mathrm {kin}} \gg 1$$ and in a limiting regime where the size L of the domain goes to infinity and the strength $$\alpha $$ of the nonlinearity goes to $$0$$ (weak nonlinearity). For the cubic nonlinear Schrödinger equation, $$T_{\mathrm {kin}}=O\left (\alpha ^{-2}\right )$$ and $$\alpha $$ is related to the conserved mass $$\lambda $$ of the solution via $$\alpha =\lambda ^2 L^{-d}$$ . In this paper, we study the rigorous justification of this monumental statement and show that the answer seems to depend on the particular scaling law in which the $$(\alpha , L)$$ limit is taken, in a spirit similar to how the Boltzmann–Grad scaling law is imposed in the derivation of Boltzmann’s equation. In particular, there appear to be two favourable scaling laws: when $$\alpha $$ approaches $$0$$ like $$L^{-\varepsilon +}$$ or like $$L^{-1-\frac {\varepsilon }{2}+}$$ (for arbitrary small $$\varepsilon $$ ), we exhibit the wave kinetic equation up to time scales $$O(T_{\mathrm {kin}}L^{-\varepsilon })$$ , by showing that the relevant Feynman-diagram expansions converge absolutely (as a sum over paired trees). For the other scaling laws, we justify the onset of the kinetic description at time scales $$T_*\ll T_{\mathrm {kin}}$$ and identify specific interactions that become very large for times beyond $$T_*$$ . In particular, the relevant tree expansion diverges absolutely there. In light of those interactions, extending the kinetic description beyond $$T_*$$ toward $$T_{\mathrm {kin}}$$ for such scaling laws seems to require new methods and ideas. 

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