wanyu/IteraTeR_full_sent huggingface dataset:
Original:  Letting R \doteq b-a, standard variance-agnostic MCMC mean-estimators run the chain for%DIFDELCMD < \cal O%%% (\frac{TR^{2}{\varepsilon^{2}}) steps, when given as inputan (often loose) upper-bound T on the relaxation time \tau}  _{ %DIFDELCMD < \rm %%% rel } . When an upper-bound V on the stationary variance v_{\pi \doteq \mathbb{V}_{\pi}[f] is known, }%DIFDELCMD < \tilde{\cal O}\bigl(%%% \frac{TR{\varepsilon}+\frac{TV}{\varepsilon^{2}}}%DIFDELCMD < \bigr) %%% steps suffice. We introduce the DYNAmic Mcmc Inter-Trace variance Estimation (DynaMITE) algorithm for mean-estimation. We define the inter-trace variance v_{T ,  DynaMITE , and requires no bound on v_{\pi}.
Revised:  Letting R \doteq b-a, standard variance-agnostic MCMC mean-estimators run the chain for%DIFDELCMD < \cal O%%% (\frac{TR^{2}{\varepsilon^{2}}) steps, when given as inputan (often loose) upper-bound T on the relaxation time \tau} _{ %DIFDELCMD < \rm %%% rel } . When an upper-bound V on the stationary variance v_{\pi \doteq \mathbb{V}_{\pi}[f] is known, }%DIFDELCMD < \tilde{\cal O}\bigl(%%% \frac{TR{\varepsilon}+\frac{TV}{\varepsilon^{2}}}%DIFDELCMD < \bigr) %%% , DynaMITE , and requires no bound on v_{\pi}.
Edit intent: meaning-changed
Original:  Receiving only an upper-bound }{\cal   on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without a priori bounds on v_{\pi, the variance of f, or the trace variance v_{T}.
Revised:  Receiving only an upper-bound }{\cal T on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without a priori bounds on v_{\pi, the variance of f, or the trace variance v_{T}.
Edit intent: fluency
Original:  Receiving only an upper-bound }{\cal   on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without a priori bounds on v_{\pi, the variance of f, or the trace variance v_{T}.
Revised:  Receiving only an upper-bound }{\cal _{rel on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without a priori bounds on v_{\pi, the variance of f, or the trace variance v_{T}.
Edit intent: meaning-changed
Original:  Receiving only an upper-bound }{\cal   on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without a priori bounds on v_{\pi, the variance of f, or the trace variance v_{T}.
Revised:  Receiving only an upper-bound }{\cal on \tau_{rel}, DynaMITE } estimates the mean %DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without a priori bounds on v_{\pi, the variance of f, or the trace variance v_{T}.
Edit intent: clarity
Original:  Receiving only an upper-bound }{\cal   on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without a priori bounds on v_{\pi, the variance of f, or the trace variance v_{T}.
Revised:  Receiving only an upper-bound }{\cal on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% of f in\tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without a priori bounds on v_{\pi, the variance of f, or the trace variance v_{T}.
Edit intent: clarity
Original:  Receiving only an upper-bound }{\cal   on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without a priori bounds on v_{\pi, the variance of f, or the trace variance v_{T}.
Revised:  Receiving only an upper-bound }{\cal on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( {\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without a priori bounds on v_{\pi, the variance of f, or the trace variance v_{T}.
Edit intent: meaning-changed
Original:  Receiving only an upper-bound }{\cal   on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without a priori bounds on v_{\pi, the variance of f, or the trace variance v_{T}.
Revised:  Receiving only an upper-bound }{\cal on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without a priori bounds on v_{\pi, the variance of f, or the trace variance v_{T}.
Edit intent: fluency
Original:  Receiving only an upper-bound }{\cal   on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without a priori bounds on v_{\pi, the variance of f, or the trace variance v_{T}.
Revised:  Receiving only an upper-bound }{\cal on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \frac{\tau_{rel\cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without a priori bounds on v_{\pi, the variance of f, or the trace variance v_{T}.
Edit intent: fluency
Original:  Receiving only an upper-bound }{\cal   on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without a priori bounds on v_{\pi, the variance of f, or the trace variance v_{T}.
Revised:  Receiving only an upper-bound }{\cal on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without bounds on v_{\pi, the variance of f, or the trace variance v_{T}.
Edit intent: clarity
Original:  Receiving only an upper-bound }{\cal   on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without a priori bounds on v_{\pi, the variance of f, or the trace variance v_{T}.
Revised:  Receiving only an upper-bound }{\cal on \tau_{rel}, DynaMITE } estimates the mean within \varepsilon additive error within%DIFDELCMD < \tilde{\cal O}%%% \tilde{\cal{O}} \bigl( \frac{TR{\varepsilon} } \smash{\frac{{\cal T}_{rel} R}{\varepsilon}} + \frac{\tau_{\rm rel v_{\tau\rm rel}}{\varepsilon^{2}}} \cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}} \bigr) steps, without a priori , the variance of f, or the trace variance v_{T}.
Edit intent: clarity
Original:  When \epsilon is small, }  the dominating term is \tau_{\rm rel v _{\tau\rm rel, thus the complexity of DynaMITE principally depends on the a priori unknown \tau\SUBSCRIPT}%DIFDELCMD < {%%%  \rm  .
Revised:  When \epsilon is small, } a priori bounds on the dominating term is \tau_{\rm rel v _{\tau\rm rel, thus the complexity of DynaMITE principally depends on the a priori unknown \tau\SUBSCRIPT}%DIFDELCMD < {%%% \rm .
Edit intent: meaning-changed
Original:  When \epsilon is small, }  the dominating term is \tau_{\rm rel v _{\tau\rm rel, thus the complexity of DynaMITE principally depends on the a priori unknown \tau\SUBSCRIPT}%DIFDELCMD < {%%%  \rm  .
Revised:  When \epsilon is small, } the stationary variance v _{\tau\rm rel, thus the complexity of DynaMITE principally depends on the a priori unknown \tau\SUBSCRIPT}%DIFDELCMD < {%%% \rm .
Edit intent: meaning-changed
Original:  When \epsilon is small, }  the dominating term is \tau_{\rm rel v _{\tau\rm rel, thus the complexity of DynaMITE principally depends on the a priori unknown \tau\SUBSCRIPT}%DIFDELCMD < {%%%  \rm  .
Revised:  When \epsilon is small, } the dominating term is \tau_{\rm rel v , thus the complexity of DynaMITE principally depends on the a priori unknown \tau\SUBSCRIPT}%DIFDELCMD < {%%% \rm .
Edit intent: coherence
Original:  When \epsilon is small, }  the dominating term is \tau_{\rm rel v _{\tau\rm rel, thus the complexity of DynaMITE principally depends on the a priori unknown \tau\SUBSCRIPT}%DIFDELCMD < {%%%  \rm  .
Revised:  When \epsilon is small, } the dominating term is \tau_{\rm rel v _{\tau\rm rel, thus the complexity of DynaMITE principally depends on the a priori unknown \tau\SUBSCRIPT}%DIFDELCMD < {%%% _{\pi \rm .
Edit intent: meaning-changed
Original:  When \epsilon is small, }  the dominating term is \tau_{\rm rel v _{\tau\rm rel, thus the complexity of DynaMITE principally depends on the a priori unknown \tau\SUBSCRIPT}%DIFDELCMD < {%%%  \rm  .
Revised:  When \epsilon is small, } the dominating term is \tau_{\rm rel v _{\tau\rm rel, thus the complexity of DynaMITE principally depends on the a priori unknown \tau\SUBSCRIPT}%DIFDELCMD < {%%% \rm trv .
Edit intent: clarity
Original:  When \epsilon is small, }  the dominating term is \tau_{\rm rel v _{\tau\rm rel, thus the complexity of DynaMITE principally depends on the a priori unknown \tau\SUBSCRIPT}%DIFDELCMD < {%%%  \rm  .
Revised:  When \epsilon is small, } the dominating term is \tau_{\rm rel v _{\tau\rm rel, thus the complexity of DynaMITE principally depends on the a priori unknown \tau\SUBSCRIPT}%DIFDELCMD < {%%% \rm ^{(\tau rel).
Edit intent: meaning-changed
Original:  Thus we depend minimally on the tightness of }{\cal  , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to  is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal   to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm  )}(}{\cal  .
Revised:  Thus we depend minimally on the tightness of }{\cal T , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm )}(}{\cal .
Edit intent: clarity
Original:  Thus we depend minimally on the tightness of }{\cal  , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to  is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal   to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm  )}(}{\cal  .
Revised:  Thus we depend minimally on the tightness of }{\cal _{mix, as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm )}(}{\cal .
Edit intent: meaning-changed
Original:  Thus we depend minimally on the tightness of }{\cal  , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to  is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal   to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm  )}(}{\cal  .
Revised:  Thus we depend minimally on the tightness of }{\cal , as the complexity is dominated by \tau_{rel}}\rm{trv}^{(\tau{rel)} as \varepsilon }\to is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm )}(}{\cal .
Edit intent: meaning-changed
Original:  Thus we depend minimally on the tightness of }{\cal  , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to  is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal   to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm  )}(}{\cal  .
Revised:  Thus we depend minimally on the tightness of }{\cal , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to 0. Note that bounding \tau_{\rm rel is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm )}(}{\cal .
Edit intent: meaning-changed
Original:  Thus we depend minimally on the tightness of }{\cal  , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to  is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal   to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm  )}(}{\cal  .
Revised:  Thus we depend minimally on the tightness of }{\cal , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal T to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm )}(}{\cal .
Edit intent: fluency
Original:  Thus we depend minimally on the tightness of }{\cal  , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to  is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal   to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm  )}(}{\cal  .
Revised:  Thus we depend minimally on the tightness of }{\cal , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal _{rel to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm )}(}{\cal .
Edit intent: meaning-changed
Original:  Thus we depend minimally on the tightness of }{\cal  , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to  is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal   to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm  )}(}{\cal  .
Revised:  Thus we depend minimally on the tightness of }{\cal , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal to \tau\SUBSCRIPT}{ rel} , simply by exploiting properties of the inter-trace variance. To compare our method to known variance-aware bounds, we show{\rm )}(}{\cal .
Edit intent: meaning-changed
Original:  Thus we depend minimally on the tightness of }{\cal  , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to  is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal   to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm  )}(}{\cal  .
Revised:  Thus we depend minimally on the tightness of }{\cal , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm trv )}(}{\cal .
Edit intent: fluency
Original:  Thus we depend minimally on the tightness of }{\cal  , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to  is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal   to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm  )}(}{\cal  .
Revised:  Thus we depend minimally on the tightness of }{\cal , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm ^{(\tau{rel)}(}{\cal .
Edit intent: meaning-changed
Original:  Thus we depend minimally on the tightness of }{\cal  , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to  is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal   to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm  )}(}{\cal  .
Revised:  Thus we depend minimally on the tightness of }{\cal , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm )}(}{\cal M .
Edit intent: clarity
Original:  Thus we depend minimally on the tightness of }{\cal  , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to  is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal   to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm  )}(}{\cal  .
Revised:  Thus we depend minimally on the tightness of }{\cal , as the complexity is dominated by \tau_{rel}}\rm{trv})} as \varepsilon }\to is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on }{\cal to \tau\SUBSCRIPT}{ rel} and v_{\tau\rm rel{\rm )}(}{\cal ,f) \leq v_{\pi.
Edit intent: meaning-changed
Original:  Furthermore, we show when f's image is distributed (semi)symmetrically on }{\cal  {\rm  })}(}{\cal   = o(v _{\pi} ), and we identify two cases to demonstrate it.
Revised:  Furthermore, we show when f's image is distributed (semi)symmetrically on }{\cal M {\rm })}(}{\cal = o(v _{\pi} ), and we identify two cases to demonstrate it.
Edit intent: meaning-changed
Original:  Furthermore, we show when f's image is distributed (semi)symmetrically on }{\cal  {\rm  })}(}{\cal   = o(v _{\pi} ), and we identify two cases to demonstrate it.
Revised:  Furthermore, we show when f's image is distributed (semi)symmetrically on }{\cal 's traces, we have{\rm })}(}{\cal = o(v _{\pi} ), and we identify two cases to demonstrate it.
Edit intent: meaning-changed
Original:  Furthermore, we show when f's image is distributed (semi)symmetrically on }{\cal  {\rm  })}(}{\cal   = o(v _{\pi} ), and we identify two cases to demonstrate it.
Revised:  Furthermore, we show when f's image is distributed (semi)symmetrically on }{\cal {\rm trv })}(}{\cal = o(v _{\pi} ), and we identify two cases to demonstrate it.
Edit intent: meaning-changed
Original:  Furthermore, we show when f's image is distributed (semi)symmetrically on }{\cal  {\rm  })}(}{\cal   = o(v _{\pi} ), and we identify two cases to demonstrate it.
Revised:  Furthermore, we show when f's image is distributed (semi)symmetrically on }{\cal {\rm ^{({\tau{rel})}(}{\cal = o(v _{\pi} ), and we identify two cases to demonstrate it.
Edit intent: meaning-changed
Original:  Furthermore, we show when f's image is distributed (semi)symmetrically on }{\cal  {\rm  })}(}{\cal   = o(v _{\pi} ), and we identify two cases to demonstrate it.
Revised:  Furthermore, we show when f's image is distributed (semi)symmetrically on }{\cal {\rm })}(}{\cal M = o(v _{\pi} ), and we identify two cases to demonstrate it.
Edit intent: meaning-changed
Original:  Furthermore, we show when f's image is distributed (semi)symmetrically on }{\cal  {\rm  })}(}{\cal   = o(v _{\pi} ), and we identify two cases to demonstrate it.
Revised:  Furthermore, we show when f's image is distributed (semi)symmetrically on }{\cal {\rm })}(}{\cal ,f) = o(v _{\pi} ), and we identify two cases to demonstrate it.
Edit intent: meaning-changed
Original:  Furthermore, we show when f's image is distributed (semi)symmetrically on }{\cal  {\rm  })}(}{\cal   = o(v _{\pi} ), and we identify two cases to demonstrate it.
Revised:  Furthermore, we show when f's image is distributed (semi)symmetrically on }{\cal {\rm })}(}{\cal = o(v _{\pi} ), and we identify two cases to demonstrate it.
Edit intent: fluency
Original:  Furthermore, we show when f's image is distributed (semi)symmetrically on }{\cal  {\rm  })}(}{\cal   = o(v _{\pi} ), and we identify two cases to demonstrate it. Furthermore,it always holds that v_{\tau\rm rel \leq 2v_{\pi}, thus the worst-case complexity of DynaMITE is }%DIFDELCMD < \tilde{\cal O}%%% (\frac{TR{\varepsilon} +\frac{\tau_{\rm rel} v_{\pi}}{\varepsilon^{2}}), improving the dependence of classical methods on the loose bounds T and V}  .
Revised:  Furthermore, we show when f's image is distributed (semi)symmetrically on }{\cal {\rm })}(}{\cal = o(v _{\pi} \leq 2v_{\pi}, thus the worst-case complexity of DynaMITE is }%DIFDELCMD < \tilde{\cal O}%%% (\frac{TR{\varepsilon} +\frac{\tau_{\rm rel} v_{\pi}}{\varepsilon^{2}}), improving the dependence of classical methods on the loose bounds T and V} .
Edit intent: coherence
Original:  Furthermore,it always holds that v_{\tau\rm rel \leq 2v_{\pi}, thus the worst-case complexity of DynaMITE is }%DIFDELCMD < \tilde{\cal O}%%% (\frac{TR{\varepsilon} +\frac{\tau_{\rm rel} v_{\pi}}{\varepsilon^{2}}), improving the dependence of classical methods on the loose bounds T and V}  .
Revised:  Furthermore,it always holds that v_{\tau\rm rel \leq 2v_{\pi}, thus the worst-case complexity of DynaMITE is }%DIFDELCMD < \tilde{\cal O}%%% {\varepsilon} +\frac{\tau_{\rm rel} v_{\pi}}{\varepsilon^{2}}), improving the dependence of classical methods on the loose bounds T and V} .
Edit intent: coherence