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Non-Stationary Reinforcement Learning Ruihao Zhu MIT IDSS Joint work with Wang Chi Cheung (NUS) and David Simchi-Levi (MIT) 1 / 18 Epidemic Control A DM iteratively: 1. Pick a measure to contain the virus. 2. See the corresponding outcome.

  1. Non-Stationary Reinforcement Learning Ruihao Zhu MIT IDSS Joint work with Wang Chi Cheung (NUS) and David Simchi-Levi (MIT) 1 / 18

  2. Epidemic Control A DM iteratively: 1. Pick a measure to contain the virus. 2. See the corresponding outcome. Goal : Minimize the total infected cases. 2 / 18

  3. Epidemic Control A DM iteratively: 1. Pick a measure to contain the virus. 2. See the corresponding outcome. Goal : Minimize the total infected cases. Challenges: ◮ Uncertainty: effectiveness of each measure is unknown. ◮ Bandit feedback: no feedback for un-chosen measures. ◮ Non-stationarity: virus might mutate throughout. 2 / 18

  4. Epidemic Control The DM’s action could have long-term impact . ◮ Quarantine lockdown stem the spread of virus to elsewhere, but also delayed key supplies from getting in. 3 / 18

  5. Model Model epidemic control by a Markov decision process (MDP) (Nowzari et al. 15, Kiss et al. 17) . For each time step t = 1 , . . . , T , ◮ Observe the current state s t = { 1 , 2 } , and receive a reward. For example r (1) = 1 and r (2) = 0 . ◮ Pick an action a t ∈ { B , G } , and transition to the next state s t +1 ∼ p t ( ·| s t , a t ) (unknown). 4 / 18

  6. Model Model epidemic control by a Markov decision process (MDP) (Nowzari et al. 15, Kiss et al. 17) . For each time step t = 1 , . . . , T , ◮ Observe the current state s t = { 1 , 2 } , and receive a reward. For example r (1) = 1 and r (2) = 0 . ◮ Pick an action a t ∈ { B , G } , and transition to the next state s t +1 ∼ p t ( ·| s t , a t ) (unknown). 4 / 18

  7. Model Model epidemic control by a Markov decision process (MDP) (Nowzari et al. 15, Kiss et al. 17) . For each time step t = 1 , . . . , T , ◮ Observe the current state s t = { 1 , 2 } , and receive a reward. For example r (1) = 1 and r (2) = 0 . ◮ Pick an action a t ∈ { B , G } , and transition to the next state s t +1 ∼ p t ( ·| s t , a t ) (unknown). 4 / 18

  8. Model cont’d ◮ Task: Design a reward-maximizing policy π. For every time step t : π t : { 1 , 2 } → { B , G } ◮ Dynamic regret (Besbes et al. 15) :   �� T � � T  − E dym-reg T = E t =1 r ( s t ( π ∗ )) t =1 r ( s t ( π )) . ���� knows p t ’s ◮ Variation budget: � p 1 − p 2 � + � p 2 − p 3 � + . . . + � p T − 1 − p T � ≤ B p . 5 / 18

  9. Model cont’d ◮ Task: Design a reward-maximizing policy π. For every time step t : π t : { 1 , 2 } → { B , G } ◮ Dynamic regret (Besbes et al. 15) :   �� T � � T  − E dym-reg T = E t =1 r ( s t ( π ∗ )) t =1 r ( s t ( π )) . ���� knows p t ’s ◮ Variation budget: � p 1 − p 2 � + � p 2 − p 3 � + . . . + � p T − 1 − p T � ≤ B p . 5 / 18

  10. Model cont’d ◮ Task: Design a reward-maximizing policy π. For every time step t : π t : { 1 , 2 } → { B , G } ◮ Dynamic regret (Besbes et al. 15) :   �� T � � T  − E dym-reg T = E t =1 r ( s t ( π ∗ )) t =1 r ( s t ( π )) . ���� knows p t ’s ◮ Variation budget: � p 1 − p 2 � + � p 2 − p 3 � + . . . + � p T − 1 − p T � ≤ B p . 5 / 18

  11. Diameter of a MDP cont’d ◮ If the DM leaves state 1, she has to come back to state 1 to collect samples. 6 / 18

  12. Diameter of a MDP cont’d ◮ If the DM leaves state 1, she has to come back to state 1 to collect samples. ◮ The longer it takes to commute between states, the harder the learning process. 6 / 18

  13. Diameter of a MDP cont’d ◮ If the DM leaves state 1, she has to come back to state 1 to collect samples. ◮ The longer it takes to commute between states, the harder the learning process. Definition ( (Jaksch et al. 10) Informal) Diameter = max { E [min. time(1 → 2)] , E [min. time(2 → 1)] } 6 / 18

  14. Diameter of a MDP cont’d ◮ If the DM leaves state 1, she has to come back to state 1 to collect samples. ◮ The longer it takes to commute between states, the harder the learning process. Definition ( (Jaksch et al. 10) Informal) Diameter = max { E [min. time(1 → 2)] , E [min. time(2 → 1)] } Example. Diameter = max { 1 / 0 . 8 , 1 / 0 . 1 } = 10 . 6 / 18

  15. Existing Works Stationary Non-stationary Multi-armed bandit OFU* Forgetting + OFU † Reinforcement learning OFU ‡ ? (Forgetting + OFU) * Auer et al. 03 † Besbes et al. 14, Cheung et al. 19 ‡ Jaksch et al. 10, Agrawal and Jia 20 7 / 18

  16. UCB for Stationary RL 1. Suppose at time t , N t (1 , B ) = 10 : 5 × (1 , B ) → 1 , 5 × (1 , B ) → 2 8 / 18

  17. UCB for Stationary RL 1. Suppose at time t , N t (1 , B ) = 10 : 5 × (1 , B ) → 1 , 5 × (1 , B ) → 2 N t (2 , B ) = 10 : 5 × (2 , B ) → 1 , 5 × (2 , B ) → 2 8 / 18

  18. UCB for Stationary RL 1. Suppose at time t , N t (1 , B ) = 10 : 5 × (1 , B ) → 1 , 5 × (1 , B ) → 2 N t (2 , B ) = 10 : 5 × (2 , B ) → 1 , 5 × (2 , B ) → 2 Empirical state transition distribution: 8 / 18

  19. UCB for Stationary RL 1. Suppose at time t , N t (1 , B ) = 10 : 5 × (1 , B ) → 1 , 5 × (1 , B ) → 2 N t (2 , B ) = 10 : 5 × (2 , B ) → 1 , 5 × (2 , B ) → 2 Empirical state transition distribution: 2. Confidence intervals: √ � ˆ p t ( ·| 1 , B ) − p ( ·| 1 , B ) � ≤ c t (1 , B ) := C / 10 √ � ˆ p t ( ·| 2 , B ) − p ( ·| 2 , B ) � ≤ c t (2 , B ) := C / 10 8 / 18

  20. UCB for Stationary RL 3. UCB of reward: find the ˚ p that maximizes Pr(visiting state 1) within the confidence interval. 4. Execute the optimal policy w.r.t. the UCB until some termination criteria are met. 9 / 18

  21. UCB for Stationary RL 3. UCB of reward: find the ˚ p that maximizes Pr(visiting state 1) within the confidence interval. 4. Execute the optimal policy w.r.t. the UCB until some termination criteria are met. 9 / 18

  22. UCB for RL cont’d Regret analysis: ◮ LCB of diameter: find the ˚ p that maximizes Pr(commuting) within the confidence interval. �� � ◮ Regret ∝ LCB × ( s , a ) c t ( s , a ) . ◮ Under stationarity, LCB of diameter ≤ Diameter( p ) . Theorem Denote D := Diameter ( p ) , the regret of the UCB algorithm is √ O ( D T ) . ◮ Summary: UCB of reward + LCB of diameter ⇒ low regret. 10 / 18

  23. UCB for RL cont’d Regret analysis: ◮ LCB of diameter: find the ˚ p that maximizes Pr(commuting) within the confidence interval. �� � ◮ Regret ∝ LCB × ( s , a ) c t ( s , a ) . ◮ Under stationarity, LCB of diameter ≤ Diameter( p ) . Theorem Denote D := Diameter ( p ) , the regret of the UCB algorithm is √ O ( D T ) . ◮ Summary: UCB of reward + LCB of diameter ⇒ low regret. 10 / 18

  24. UCB for RL cont’d Regret analysis: ◮ LCB of diameter: find the ˚ p that maximizes Pr(commuting) within the confidence interval. �� � ◮ Regret ∝ LCB × ( s , a ) c t ( s , a ) . ◮ Under stationarity, LCB of diameter ≤ Diameter( p ) . Theorem Denote D := Diameter ( p ) , the regret of the UCB algorithm is √ O ( D T ) . ◮ Summary: UCB of reward + LCB of diameter ⇒ low regret. 10 / 18

  25. UCB for RL cont’d Regret analysis: ◮ LCB of diameter: find the ˚ p that maximizes Pr(commuting) within the confidence interval. �� � ◮ Regret ∝ LCB × ( s , a ) c t ( s , a ) . ◮ Under stationarity, LCB of diameter ≤ Diameter( p ) . Theorem Denote D := Diameter ( p ) , the regret of the UCB algorithm is √ O ( D T ) . ◮ Summary: UCB of reward + LCB of diameter ⇒ low regret. 10 / 18

  26. UCB for RL cont’d Regret analysis: ◮ LCB of diameter: find the ˚ p that maximizes Pr(commuting) within the confidence interval. �� � ◮ Regret ∝ LCB × ( s , a ) c t ( s , a ) . ◮ Under stationarity, LCB of diameter ≤ Diameter( p ) . Theorem Denote D := Diameter ( p ) , the regret of the UCB algorithm is √ O ( D T ) . ◮ Summary: UCB of reward + LCB of diameter ⇒ low regret. 10 / 18

  27. SWUCB for RL According to (Cheung et al. 19) : ◮ SWUCB for RL: UCB for RL with W most recent samples. 11 / 18

  28. SWUCB for RL According to (Cheung et al. 19) : ◮ SWUCB for RL: UCB for RL with W most recent samples. ◮ The perils of drift: Under non-stationarity, LCB of diameter ≫ Diameter( p s ) for all s ∈ [ T ] . 11 / 18

  29. Perils of Non-Stationarity in RL Non-stationarity: The DM faces time-varying environment. 12 / 18

  30. Perils of Non-Stationarity in RL Non-stationarity: The DM faces time-varying environment. Bandit feedback: The DM is not seeing everything. 12 / 18

  31. Perils of Non-Stationarity in RL Non-stationarity: The DM faces time-varying environment. Bandit feedback: The DM is not seeing everything. Collected data: { (1 , B ) → 1 , (2 , B ) → 2 } 12 / 18

  32. Perils of Non-Stationarity in RL Non-stationarity: The DM faces time-varying environment. Bandit feedback: The DM is not seeing everything. Collected data: { (1 , B ) → 1 , (2 , B ) → 2 } Empirical state transition ˆ p t : 12 / 18

  33. Perils of Non-Stationarity in RL Non-stationarity: The DM faces time-varying environment. Bandit feedback: The DM is not seeing everything. Collected data: { (1 , B ) → 1 , (2 , B ) → 2 } Empirical state transition ˆ p t : Diameter explodes! 12 / 18

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