wanyu/IteraTeR_v2 huggingface dataset:
Original:  Finally, under the additional (restrictive) assumption that the MDP is weakly-communicating, we show that Simplex-PI and Howard's PI terminate after at most n(m-1)%DIFDELCMD < \left( %DIFDELCMD < \lceil %%% \tau_t \log n \tau_t%DIFDELCMD < \rceil %%% +%DIFDELCMD < \lceil %%% \tau_r \log n \tau_r%DIFDELCMD < \rceil \right) %DIFDELCMD < %%% = \tilde O(nm (\tau_t+\tau_r)) iterations.
Revised:  Finally, under the additional (restrictive) assumption that the MDP is weakly-communicating, we show that Simplex-PI and Howard's PI terminate after at most n(m-1)%DIFDELCMD < \left( %DIFDELCMD < \lceil %%% %DIFDELCMD < \rceil %%% +%DIFDELCMD < \lceil %%% \tau_r \log n \tau_r%DIFDELCMD < \rceil \right) %DIFDELCMD < %%% = \tilde O(nm (\tau_t+\tau_r)) iterations.
Edit intent: fluency
Original:  Finally, under the additional (restrictive) assumption that the MDP is weakly-communicating, we show that Simplex-PI and Howard's PI terminate after at most n(m-1)%DIFDELCMD < \left( %DIFDELCMD < \lceil %%% \tau_t \log n \tau_t%DIFDELCMD < \rceil %%% +%DIFDELCMD < \lceil %%% \tau_r \log n \tau_r%DIFDELCMD < \rceil \right) %DIFDELCMD < %%% = \tilde O(nm (\tau_t+\tau_r)) iterations.
Revised:  Finally, under the additional (restrictive) assumption that the MDP is weakly-communicating, we show that Simplex-PI and Howard's PI terminate after at most n(m-1)%DIFDELCMD < \left( %DIFDELCMD < \lceil %%% \tau_t \log n \tau_t%DIFDELCMD < \rceil %%% %DIFDELCMD < \lceil %%% \tau_r \log n \tau_r%DIFDELCMD < \rceil \right) %DIFDELCMD < %%% = \tilde O(nm (\tau_t+\tau_r)) iterations.
Edit intent: coherence
Original:  Finally, under the additional (restrictive) assumption that the MDP is weakly-communicating, we show that Simplex-PI and Howard's PI terminate after at most n(m-1)%DIFDELCMD < \left( %DIFDELCMD < \lceil %%% \tau_t \log n \tau_t%DIFDELCMD < \rceil %%% +%DIFDELCMD < \lceil %%% \tau_r \log n \tau_r%DIFDELCMD < \rceil \right) %DIFDELCMD < %%% = \tilde O(nm (\tau_t+\tau_r)) iterations.
Revised:  Finally, under the additional (restrictive) assumption that the MDP is weakly-communicating, we show that Simplex-PI and Howard's PI terminate after at most n(m-1)%DIFDELCMD < \left( %DIFDELCMD < \lceil %%% \tau_t \log n \tau_t%DIFDELCMD < \rceil %%% +%DIFDELCMD < \lceil %%% %DIFDELCMD < \rceil \right) %DIFDELCMD < %%% = \tilde O(nm (\tau_t+\tau_r)) iterations.
Edit intent: fluency
Original:  Finally, under the additional (restrictive) assumption that the MDP is weakly-communicating, we show that Simplex-PI and Howard's PI terminate after at most n(m-1)%DIFDELCMD < \left( %DIFDELCMD < \lceil %%% \tau_t \log n \tau_t%DIFDELCMD < \rceil %%% +%DIFDELCMD < \lceil %%% \tau_r \log n \tau_r%DIFDELCMD < \rceil \right) %DIFDELCMD < %%% = \tilde O(nm (\tau_t+\tau_r)) iterations.
Revised:  Finally, under the additional (restrictive) assumption that the MDP is weakly-communicating, we show that Simplex-PI and Howard's PI terminate after at most n(m-1)%DIFDELCMD < \left( %DIFDELCMD < \lceil %%% \tau_t \log n \tau_t%DIFDELCMD < \rceil %%% +%DIFDELCMD < \lceil %%% \tau_r \log n \tau_r%DIFDELCMD < \rceil \right) %DIFDELCMD < %%% \tilde O(nm (\tau_t+\tau_r)) iterations.
Edit intent: clarity
Original: Given a Markov Decision Process (MDP) with n states and m actions per state , we study the number of iterations needed by Policy Iteration (PI) algorithms to converge to the optimal\gamma-discounted optimal policy.
Revised: Given a Markov Decision Process (MDP) with n states and a totalnumber m of actions , we study the number of iterations needed by Policy Iteration (PI) algorithms to converge to the optimal\gamma-discounted optimal policy.
Edit intent: clarity
Original: Given a Markov Decision Process (MDP) with n states and m actions per state , we study the number of iterations needed by Policy Iteration (PI) algorithms to converge to the optimal\gamma-discounted optimal policy.
Revised: Given a Markov Decision Process (MDP) with n states and m actions per state , we study the number of iterations needed byPolicy Iteration (PI) algorithms to converge to the optimal\gamma-discounted optimal policy.
Edit intent: clarity
Original: Given a Markov Decision Process (MDP) with n states and m actions per state , we study the number of iterations needed by Policy Iteration (PI) algorithms to converge to the optimal\gamma-discounted optimal policy.
Revised: Given a Markov Decision Process (MDP) with n states and m actions per state , we study the number of iterations needed by Policy Iteration (PI) algorithms to converge to the optimal\gamma-discounted policy.
Edit intent: clarity
Original:  We consider two variations of PI: Howard' s PI that changes the actions in all states with a positive advantage,and Simplex-PI that only changes the action in the state with maximal advantage .
Revised:  We consider two variations of PI: Howard' sPI that changes the actions in all states with a positive advantage,and Simplex-PI that only changes the action in the state with maximal advantage .
Edit intent: fluency
Original:  We consider two variations of PI: Howard' s PI that changes the actions in all states with a positive advantage,and Simplex-PI that only changes the action in the state with maximal advantage .
Revised:  We consider two variations of PI: Howard' s PI that changes the actions in all states with a positive advantage,and Simplex-PI that only changes the action in the state with maximaladvantage .
Edit intent: fluency
Original:  We show that Howard's PI terminates after at most O (\frac{n m{1-\gamma} \log(1{1-\gamma})) } \left({1-\gamma}\log}\left({1-\gamma}}\right)\right) iterations, improving by a factor O(\log n) a result by Hansen et al.
Revised:  We show that Howard's PI terminates after at most O {1-\gamma} \log(1{1-\gamma})) } \left({1-\gamma}\log}\left({1-\gamma}}\right)\right) iterations, improving by a factor O(\log n) a result by Hansen et al.
Edit intent: clarity
Original:  We show that Howard's PI terminates after at most O (\frac{n m{1-\gamma} \log(1{1-\gamma})) } \left({1-\gamma}\log}\left({1-\gamma}}\right)\right) iterations, improving by a factor O(\log n) a result by Hansen et al.
Revised:  We show that Howard's PI terminates after at most O (\frac{n m{1-\gamma} \log(1{1-\gamma})) } \left(\frac{m{1-\gamma}\log}\left({1-\gamma}}\right)\right) iterations, improving by a factor O(\log n) a result by Hansen et al.
Edit intent: fluency
Original:  We show that Howard's PI terminates after at most O (\frac{n m{1-\gamma} \log(1{1-\gamma})) } \left({1-\gamma}\log}\left({1-\gamma}}\right)\right) iterations, improving by a factor O(\log n) a result by Hansen et al.
Revised:  We show that Howard's PI terminates after at most O (\frac{n m{1-\gamma} \log(1{1-\gamma})) } \left({1-\gamma}\log}\left(\frac{1{1-\gamma}}\right)\right) iterations, improving by a factor O(\log n) a result by Hansen et al.
Edit intent: fluency
Original: (2013) , while Simplex-PI terminates after at most O (\frac{n^2 m{1-\gamma} \log(1{1-\gamma})) } \left({1-\gamma}\log}\left({1-\gamma}}\right)\right) iterations, improving by a factor O(\log n) a result by Ye (2011).
Revised: (2013) , while Simplex-PI terminates after at most O (\frac{n^2 m{1-\gamma} \log(1{1-\gamma})) } \left({1-\gamma}\log}\left(\frac{1{1-\gamma}}\right)\right) iterations, improving by a factor O(\log n) a result by Ye (2011).
Edit intent: fluency
Original:  Under some structural assumptions of the MDP, we then consider bounds that are independent of the discount factor \gamma: given a measure of the maximal transient time \tau_t and the maximal time \tau_r to revisit states in recurrent classes under all policies\emph{ \emph{  , we show that Simplex-PI terminates after at most \tilde O ( \left( n^3 m^2 \tau _t \tau_r)\right) iterations.
Revised:  Under some structural assumptions of the MDP, we then consider bounds thatare independent of the discount factor \gamma: given a measure of the maximal transient time \tau_t and the maximal time \tau_r to revisit states in recurrent classes under all policies\emph{ \emph{ , we show that Simplex-PI terminates after at most \tilde O ( \left( n^3 m^2 \tau _t \tau_r)\right) iterations.
Edit intent: fluency
Original:  Under some structural assumptions of the MDP, we then consider bounds that are independent of the discount factor \gamma: given a measure of the maximal transient time \tau_t and the maximal time \tau_r to revisit states in recurrent classes under all policies\emph{ \emph{  , we show that Simplex-PI terminates after at most \tilde O ( \left( n^3 m^2 \tau _t \tau_r)\right) iterations.
Revised:  Under some structural assumptions of the MDP, we then consider bounds that are independent of the discount factor \gamma: given a measure of the maximal transient time \tau_t and the maximal time \tau_r to revisit states in recurrent classes under all policies\emph{ and\emph{ , we show that Simplex-PI terminates after at most \tilde O ( \left( n^3 m^2 \tau _t \tau_r)\right) iterations.
Edit intent: coherence
Original:  Under some structural assumptions of the MDP, we then consider bounds that are independent of the discount factor \gamma: given a measure of the maximal transient time \tau_t and the maximal time \tau_r to revisit states in recurrent classes under all policies\emph{ \emph{  , we show that Simplex-PI terminates after at most \tilde O ( \left( n^3 m^2 \tau _t \tau_r)\right) iterations.
Revised:  Under some structural assumptions of the MDP, we then consider bounds that are independent of the discount factor \gamma: given a measure of the maximal transient time \tau_t and the maximal time \tau_r to revisit states in recurrent classes under all policies\emph{ \emph{ , we show that Simplex-PI terminates after at most \tilde O \left( n^3 m^2 \tau _t \tau_r)\right) iterations.
Edit intent: fluency
Original:  Under some structural assumptions of the MDP, we then consider bounds that are independent of the discount factor \gamma: given a measure of the maximal transient time \tau_t and the maximal time \tau_r to revisit states in recurrent classes under all policies\emph{ \emph{  , we show that Simplex-PI terminates after at most \tilde O ( \left( n^3 m^2 \tau _t \tau_r)\right) iterations.
Revised:  Under some structural assumptions of the MDP, we then consider bounds that are independent of the discount factor \gamma: given a measure of the maximal transient time \tau_t and the maximal time \tau_r to revisit states in recurrent classes under all policies\emph{ \emph{ , we show that Simplex-PI terminates after at most \tilde O ( \left( n^3 m^2 \tau \_t \tau\_r\right) iterations.
Edit intent: fluency
Original:  This generalizes a recent result for deterministic MDPs by Post & Ye (2012) , in which \tau _t \le n and \tau_r \le n  . We explain why similar results seem hard to derive for Howard's PI.
Revised:  This extends arecent result for deterministic MDPs by Post & Ye (2012) , in which \tau _t \le n and \tau_r \le n . We explain why similar results seem hard to derive for Howard's PI.
Edit intent: clarity
Original:  This generalizes a recent result for deterministic MDPs by Post & Ye (2012) , in which \tau _t \le n and \tau_r \le n  . We explain why similar results seem hard to derive for Howard's PI.
Revised:  This generalizes a recent result for deterministic MDPs by Post & Ye (2012) , in which \tau \_t \le n and \tau_r \le n . We explain why similar results seem hard to derive for Howard's PI.
Edit intent: fluency
Original:  This generalizes a recent result for deterministic MDPs by Post & Ye (2012) , in which \tau _t \le n and \tau_r \le n  . We explain why similar results seem hard to derive for Howard's PI.
Revised:  This generalizes a recent result for deterministic MDPs by Post & Ye (2012) , in which \tau _t \le n and \tau_r \le n . We explain whysimilar results seem hard to derive for Howard's PI.
Edit intent: fluency
Original:  Finally, under the additional (restrictive) assumption that the state space is partitioned in two sets, respectively states that are transient and recurrent for all policies, we show that Simplex-PI and Howard's PI  terminate after at most \tilde O( nm (\tau_t + \tau_r ))iterations.
Revised:  Finally, underthe additional (restrictive) assumption that the state space is partitioned in two sets, respectively states that are transient and recurrent for all policies, we show that Simplex-PI and Howard's PI terminate after at most \tilde O( nm (\tau_t + \tau_r ))iterations.
Edit intent: fluency
Original:  Finally, under the additional (restrictive) assumption that the state space is partitioned in two sets, respectively states that are transient and recurrent for all policies, we show that Simplex-PI and Howard's PI  terminate after at most \tilde O( nm (\tau_t + \tau_r ))iterations.
Revised:  Finally, under the additional (restrictive) assumption that the state space ispartitioned in two sets, respectively states that are transient and recurrent for all policies, we show that Simplex-PI and Howard's PI terminate after at most \tilde O( nm (\tau_t + \tau_r ))iterations.
Edit intent: fluency
Original:  Finally, under the additional (restrictive) assumption that the state space is partitioned in two sets, respectively states that are transient and recurrent for all policies, we show that Simplex-PI and Howard's PI  terminate after at most \tilde O( nm (\tau_t + \tau_r ))iterations.
Revised:  Finally, under the additional (restrictive) assumption that the state space is partitioned in two sets, respectively states that are transient andrecurrent for all policies, we show that Simplex-PI and Howard's PI terminate after at most \tilde O( nm (\tau_t + \tau_r ))iterations.
Edit intent: fluency
Original:  In this paper, we establish an upper bound for this approach, by showing that  SSM does not hold on T_{ saw%DIFDELCMD < \MBLOCKRIGHTBRACE%%% (%DIFDELCMD < \Z%%%   ^2) when \lambda>3.4.
Revised:  In this paper, we establish an upper bound for this approach, by showing that SSM does not hold on T_{ %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (%DIFDELCMD < \Z%%% ^2) when \lambda>3.4.
Edit intent: fluency
Original:  In this paper, we establish an upper bound for this approach, by showing that  SSM does not hold on T_{ saw%DIFDELCMD < \MBLOCKRIGHTBRACE%%% (%DIFDELCMD < \Z%%%   ^2) when \lambda>3.4.
Revised:  In this paper, we establish an upper bound for this approach, by showing that SSM does not hold on T_{ saw%DIFDELCMD < \MBLOCKRIGHTBRACE%%% %DIFDELCMD < \Z%%% ^2) when \lambda>3.4.
Edit intent: fluency
Original: We introduce persistent contextual neural networks (PCNNs) as a probabilistic model for learning symbolic datasequences, aimed at discovering complex algorithmic dependencies in the sequence . PCNNs are similar to recurrent neural networks but feature an architecture inspired by finite automata and a modified time evolution to better model memory effects.
Revised: Recurrent neural networks are powerful models for sequential data, able to represent complex dependencies in the sequence . PCNNs are similar to recurrent neural networks but feature an architecture inspired by finite automata and a modified time evolution to better model memory effects.
Edit intent: clarity
Original:  An effective training procedure using a gradient ascent in a metric inspired by Riemannian geometry is developed : this produces an algorithm independent from design choices such as the encoding of parameters and unit activities.
Revised:  An effective training procedure using a gradient ascent in a metric inspired by Riemannian geometry : this produces an algorithm independent from design choices such as the encoding of parameters and unit activities.
Edit intent: clarity
Original:  Here we introduce an effective training procedure using a gradient ascent in a metric inspired by Riemannian geometry : this produces an algorithm independent from design choices such as the encoding of parameters and unit activities.
Revised:  Here we introduce a training procedure using a gradient ascent in a metric inspired by Riemannian geometry : this produces an algorithm independent from design choices such as the encoding of parameters and unit activities.
Edit intent: clarity
Original:  Here we introduce an effective training procedure using a gradient ascent in a metric inspired by Riemannian geometry : this produces an algorithm independent from design choices such as the encoding of parameters and unit activities.
Revised:  Here we introduce an effective training procedure using a gradient ascent in a Riemannian metric : this produces an algorithm independent from design choices such as the encoding of parameters and unit activities.
Edit intent: clarity
Original:  We also introduce persistent contextual neural networks (PCNNs) as a variant of recurrent neural networks . PCNNs feature an architecture inspired by finite automata and a modified time evolution to better model long-distance effects.
Revised:  We also introduce persistent contextual neural networks (PCNNs) as a variant of recurrent neural networks . PCNNs feature an architecture inspired by finite automata and a modified time evolution to better model long-distance effects.
Edit intent: fluency
Original:  We also introduce persistent contextual neural networks (PCNNs) as a variant of recurrent neural networks . PCNNs feature an architecture inspired by finite automata and a modified time evolution to better model long-distance effects.
Revised:  We also introduce persistent contextual neural networks (PCNNs) as a variant of recurrent neural networks with an architecture inspired by finite automata and a modified time evolution to better model long-distance effects.
Edit intent: clarity
Original:  PCNNs are demonstrated to effectively capture a variety of complex algorithmic constraints on hard synthetic problems: basic block nesting as in context-free grammars (an important feature of natural languages, but difficult to learn), intersections of multiple independent Markov-type relations, or long-distance relationships such as the distant-XOR problem.
Revised:  PCNNs are demonstrated to effectively capture a variety of structures in synthetic problems: basic block nesting as in context-free grammars (an important feature of natural languages, but difficult to learn), intersections of multiple independent Markov-type relations, or long-distance relationships such as the distant-XOR problem.
Edit intent: clarity
Original:  PCNNs are demonstrated to effectively capture a variety of complex algorithmic constraints on hard synthetic problems: basic block nesting as in context-free grammars (an important feature of natural languages, but difficult to learn), intersections of multiple independent Markov-type relations, or long-distance relationships such as the distant-XOR problem.
Revised:  PCNNs are demonstrated to effectively capture a variety of complex algorithmic constraints on hard synthetic problems: basic block nesting as in context-free grammars (an important feature of natural languages, but difficult to learn), intersections of multiple independent Markov- type relations, or long-distance relationships such as the distant-XOR problem.
Edit intent: fluency
Original:  PCNNs trained this way are demonstrated to effectively capture a variety of structures in synthetic problems: basic block nesting as in context-free grammars (an important feature of natural languages, but difficult to learn), intersections of multiple independent Markov- type relations, or long-distance relationships such as the distant-XOR problem.
Revised:  PCNNs trained this way are demonstrated to effectively capture a variety of structures in synthetic problems: basic block nesting as in context-free grammars (an important feature of natural languages, but difficult to learn), intersections of multiple independent Markov-type relations, or long-distance relationships such as the distant-XOR problem.
Edit intent: fluency
Original: In this paper, an SPH method based on the SPH--ALE formulation is used for modelling two-phase flows with large density ratios and realistic sound speeds.
Revised: In this paper, an SPH method based on the SPH-ALE formulation is used for modelling two-phase flows with large density ratios and realistic sound speeds.
Edit intent: fluency
Original: We consider a general heterogeneous network in which, besides a general propagation effects (shadowing and-or fading), individual base stations may use different emitted powers and be subject to different parameters of the Hata-like propagation loss model (path-loss exponent and constant) , e.g. due todifferent antenna heights.
Revised: We consider a general heterogeneous network in which, besides general propagation effects (shadowing and-or fading), individual base stations may use different emitted powers and be subject to different parameters of the Hata-like propagation loss model (path-loss exponent and constant) , e.g. due todifferent antenna heights.
Edit intent: fluency
Original: We consider a general heterogeneous network in which, besides a general propagation effects (shadowing and-or fading), individual base stations may use different emitted powers and be subject to different parameters of the Hata-like propagation loss model (path-loss exponent and constant) , e.g. due todifferent antenna heights.
Revised: We consider a general heterogeneous network in which, besides a general propagation effects (shadowing and/or fading), individual base stations may use different emitted powers and be subject to different parameters of the Hata-like propagation loss model (path-loss exponent and constant) , e.g. due todifferent antenna heights.
Edit intent: fluency
Original: We consider a general heterogeneous network in which, besides a general propagation effects (shadowing and-or fading), individual base stations may use different emitted powers and be subject to different parameters of the Hata-like propagation loss model (path-loss exponent and constant) , e.g. due todifferent antenna heights.
Revised: We consider a general heterogeneous network in which, besides a general propagation effects (shadowing and-or fading), individual base stations can have different emitting powers and be subject to different parameters of the Hata-like propagation loss model (path-loss exponent and constant) , e.g. due todifferent antenna heights.
Edit intent: clarity
Original: We consider a general heterogeneous network in which, besides a general propagation effects (shadowing and-or fading), individual base stations may use different emitted powers and be subject to different parameters of the Hata-like propagation loss model (path-loss exponent and constant) , e.g. due todifferent antenna heights.
Revised: We consider a general heterogeneous network in which, besides a general propagation effects (shadowing and-or fading), individual base stations may use different emitted powers and be subject to different parameters of Hata-like propagation loss model (path-loss exponent and constant) , e.g. due todifferent antenna heights.
Edit intent: fluency
Original: We consider a general heterogeneous network in which, besides a general propagation effects (shadowing and-or fading), individual base stations may use different emitted powers and be subject to different parameters of the Hata-like propagation loss model (path-loss exponent and constant) , e.g. due todifferent antenna heights.
Revised: We consider a general heterogeneous network in which, besides a general propagation effects (shadowing and-or fading), individual base stations may use different emitted powers and be subject to different parameters of the Hata-like path-loss models (path-loss exponent and constant) , e.g. due todifferent antenna heights.
Edit intent: fluency