wanyu/IteraTeR_full_sent huggingface dataset:
Original:  As a consequence, this new {class of networks} overcome  the curse of dimensionality in approximation power when the variation of \omega_f(r) as r\rightarrow 0 is moderate (e.g., \omega_f(r) %DIFDELCMD < \le %%% {\lesssim} r^\alpha for H{\"o}lder continuous functions), since the major term to be concerned in our approximation rate is {essentially d times a function of N and L independent of d within the modulus of continuity. }
Revised:  As a consequence, this new {class of networks} overcome s the curse of dimensionality in approximation power when the variation of \omega_f(r) as r\rightarrow 0 is moderate (e.g., \omega_f(r) %DIFDELCMD < \le %%% {\lesssim} r^\alpha for H{\"o}lder continuous functions), since the major term to be concerned in our approximation rate is {essentially d times a function of N and L independent of d within the modulus of continuity. }
Edit intent: fluency
Original:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with a width \max\{d,\, 5N+13\} and a depth 64dL+3 can uniformly approximate a H \"o%DIFDELCMD < }%%% lder function f on [0,1]^d with an approximation rate 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the H \"o%DIFDELCMD < }%%% lder order and constant, respectively.
Revised:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with width \max\{d,\, 5N+13\} and a depth 64dL+3 can uniformly approximate a H \"o%DIFDELCMD < }%%% lder function f on [0,1]^d with an approximation rate 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the H \"o%DIFDELCMD < }%%% lder order and constant, respectively.
Edit intent: fluency
Original:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with a width \max\{d,\, 5N+13\} and a depth 64dL+3 can uniformly approximate a H \"o%DIFDELCMD < }%%% lder function f on [0,1]^d with an approximation rate 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the H \"o%DIFDELCMD < }%%% lder order and constant, respectively.
Revised:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with a width \max\{d,\, 5N+13\} and depth 64dL+3 can uniformly approximate a H \"o%DIFDELCMD < }%%% lder function f on [0,1]^d with an approximation rate 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the H \"o%DIFDELCMD < }%%% lder order and constant, respectively.
Edit intent: fluency
Original:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with a width \max\{d,\, 5N+13\} and a depth 64dL+3 can uniformly approximate a H \"o%DIFDELCMD < }%%% lder function f on [0,1]^d with an approximation rate 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the H \"o%DIFDELCMD < }%%% lder order and constant, respectively.
Revised:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with a width \max\{d,\, 5N+13\} and a depth 64dL+3 can uniformly approximate a \"o%DIFDELCMD < }%%% lder function f on [0,1]^d with an approximation rate 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the H \"o%DIFDELCMD < }%%% lder order and constant, respectively.
Edit intent: fluency
Original:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with a width \max\{d,\, 5N+13\} and a depth 64dL+3 can uniformly approximate a H \"o%DIFDELCMD < }%%% lder function f on [0,1]^d with an approximation rate 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the H \"o%DIFDELCMD < }%%% lder order and constant, respectively.
Revised:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with a width \max\{d,\, 5N+13\} and a depth 64dL+3 can uniformly approximate a H %DIFDELCMD < }%%% lder function f on [0,1]^d with an approximation rate 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the H \"o%DIFDELCMD < }%%% lder order and constant, respectively.
Edit intent: fluency
Original:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with a width \max\{d,\, 5N+13\} and a depth 64dL+3 can uniformly approximate a H \"o%DIFDELCMD < }%%% lder function f on [0,1]^d with an approximation rate 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the H \"o%DIFDELCMD < }%%% lder order and constant, respectively.
Revised:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with a width \max\{d,\, 5N+13\} and a depth 64dL+3 can uniformly approximate a H \"o%DIFDELCMD < }%%% H\"older function f on [0,1]^d with an approximation rate 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the H \"o%DIFDELCMD < }%%% lder order and constant, respectively.
Edit intent: meaning-changed
Original:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with a width \max\{d,\, 5N+13\} and a depth 64dL+3 can uniformly approximate a H \"o%DIFDELCMD < }%%% lder function f on [0,1]^d with an approximation rate 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the H \"o%DIFDELCMD < }%%% lder order and constant, respectively.
Revised:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with a width \max\{d,\, 5N+13\} and a depth 64dL+3 can uniformly approximate a H \"o%DIFDELCMD < }%%% lder function f on [0,1]^d with an approximation error 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the H \"o%DIFDELCMD < }%%% lder order and constant, respectively.
Edit intent: clarity
Original:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with a width \max\{d,\, 5N+13\} and a depth 64dL+3 can uniformly approximate a H \"o%DIFDELCMD < }%%% lder function f on [0,1]^d with an approximation rate 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the H \"o%DIFDELCMD < }%%% lder order and constant, respectively.
Revised:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with a width \max\{d,\, 5N+13\} and a depth 64dL+3 can uniformly approximate a H \"o%DIFDELCMD < }%%% lder function f on [0,1]^d with an approximation rate 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the \"o%DIFDELCMD < }%%% lder order and constant, respectively.
Edit intent: fluency
Original:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with a width \max\{d,\, 5N+13\} and a depth 64dL+3 can uniformly approximate a H \"o%DIFDELCMD < }%%% lder function f on [0,1]^d with an approximation rate 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the H \"o%DIFDELCMD < }%%% lder order and constant, respectively.
Revised:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with a width \max\{d,\, 5N+13\} and a depth 64dL+3 can uniformly approximate a H \"o%DIFDELCMD < }%%% lder function f on [0,1]^d with an approximation rate 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the H %DIFDELCMD < }%%% lder order and constant, respectively.
Edit intent: fluency
Original:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with a width \max\{d,\, 5N+13\} and a depth 64dL+3 can uniformly approximate a H \"o%DIFDELCMD < }%%% lder function f on [0,1]^d with an approximation rate 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the H \"o%DIFDELCMD < }%%% lder order and constant, respectively.
Revised:  For any hyper-parameters N\inN^+ and L\inN^+, it is shown that Floor-ReLU networks with a width \max\{d,\, 5N+13\} and a depth 64dL+3 can uniformly approximate a H \"o%DIFDELCMD < }%%% lder function f on [0,1]^d with an approximation rate 3\lambda d^{\alpha/2}N^{-\alphaL}, where \alpha \in(0,1] and \lambda are the H \"o%DIFDELCMD < }%%% H\"older order and constant, respectively.
Edit intent: meaning-changed
Original:  As a consequence, this new class of networks overcome s the curse of dimensionality in approximation power when the variation of \omega_f(r) as r \rightarrow \to 0 is moderate (e.g., \omega_f(r) \lesssim r^\alpha for H \"o%DIFDELCMD < }%%% lder continuous functions), since the major term to be concerned in our approximation rate is essentially \sqrt{d} times a function of N and L independent of d within the modulus of continuity.
Revised:  As a consequence, this new class of networks s the curse of dimensionality in approximation power when the variation of \omega_f(r) as r \rightarrow \to 0 is moderate (e.g., \omega_f(r) \lesssim r^\alpha for H \"o%DIFDELCMD < }%%% lder continuous functions), since the major term to be concerned in our approximation rate is essentially \sqrt{d} times a function of N and L independent of d within the modulus of continuity.
Edit intent: clarity
Original:  As a consequence, this new class of networks overcome s the curse of dimensionality in approximation power when the variation of \omega_f(r) as r \rightarrow \to 0 is moderate (e.g., \omega_f(r) \lesssim r^\alpha for H \"o%DIFDELCMD < }%%% lder continuous functions), since the major term to be concerned in our approximation rate is essentially \sqrt{d} times a function of N and L independent of d within the modulus of continuity.
Revised:  As a consequence, this new class of networks overcome overcomes the curse of dimensionality in approximation power when the variation of \omega_f(r) as r \rightarrow \to 0 is moderate (e.g., \omega_f(r) \lesssim r^\alpha for H \"o%DIFDELCMD < }%%% lder continuous functions), since the major term to be concerned in our approximation rate is essentially \sqrt{d} times a function of N and L independent of d within the modulus of continuity.
Edit intent: fluency
Original:  As a consequence, this new class of networks overcome s the curse of dimensionality in approximation power when the variation of \omega_f(r) as r \rightarrow \to 0 is moderate (e.g., \omega_f(r) \lesssim r^\alpha for H \"o%DIFDELCMD < }%%% lder continuous functions), since the major term to be concerned in our approximation rate is essentially \sqrt{d} times a function of N and L independent of d within the modulus of continuity.
Revised:  As a consequence, this new class of networks overcome s the curse of dimensionality in approximation power when the variation of \omega_f(r) as r \to 0 is moderate (e.g., \omega_f(r) \lesssim r^\alpha for H \"o%DIFDELCMD < }%%% lder continuous functions), since the major term to be concerned in our approximation rate is essentially \sqrt{d} times a function of N and L independent of d within the modulus of continuity.
Edit intent: meaning-changed
Original:  As a consequence, this new class of networks overcome s the curse of dimensionality in approximation power when the variation of \omega_f(r) as r \rightarrow \to 0 is moderate (e.g., \omega_f(r) \lesssim r^\alpha for H \"o%DIFDELCMD < }%%% lder continuous functions), since the major term to be concerned in our approximation rate is essentially \sqrt{d} times a function of N and L independent of d within the modulus of continuity.
Revised:  As a consequence, this new class of networks overcome s the curse of dimensionality in approximation power when the variation of \omega_f(r) as r \rightarrow \to 0 is moderate (e.g., \omega_f(r) \lesssim r^\alpha for \"o%DIFDELCMD < }%%% lder continuous functions), since the major term to be concerned in our approximation rate is essentially \sqrt{d} times a function of N and L independent of d within the modulus of continuity.
Edit intent: clarity
Original:  As a consequence, this new class of networks overcome s the curse of dimensionality in approximation power when the variation of \omega_f(r) as r \rightarrow \to 0 is moderate (e.g., \omega_f(r) \lesssim r^\alpha for H \"o%DIFDELCMD < }%%% lder continuous functions), since the major term to be concerned in our approximation rate is essentially \sqrt{d} times a function of N and L independent of d within the modulus of continuity.
Revised:  As a consequence, this new class of networks overcome s the curse of dimensionality in approximation power when the variation of \omega_f(r) as r \rightarrow \to 0 is moderate (e.g., \omega_f(r) \lesssim r^\alpha for H %DIFDELCMD < }%%% lder continuous functions), since the major term to be concerned in our approximation rate is essentially \sqrt{d} times a function of N and L independent of d within the modulus of continuity.
Edit intent: clarity
Original:  As a consequence, this new class of networks overcome s the curse of dimensionality in approximation power when the variation of \omega_f(r) as r \rightarrow \to 0 is moderate (e.g., \omega_f(r) \lesssim r^\alpha for H \"o%DIFDELCMD < }%%% lder continuous functions), since the major term to be concerned in our approximation rate is essentially \sqrt{d} times a function of N and L independent of d within the modulus of continuity.
Revised:  As a consequence, this new class of networks overcome s the curse of dimensionality in approximation power when the variation of \omega_f(r) as r \rightarrow \to 0 is moderate (e.g., \omega_f(r) \lesssim r^\alpha for H \"o%DIFDELCMD < }%%% H\"older continuous functions), since the major term to be concerned in our approximation rate is essentially \sqrt{d} times a function of N and L independent of d within the modulus of continuity.
Edit intent: clarity
Original:  As a consequence, this new class of networks overcome s the curse of dimensionality in approximation power when the variation of \omega_f(r) as r \rightarrow \to 0 is moderate (e.g., \omega_f(r) \lesssim r^\alpha for H \"o%DIFDELCMD < }%%% lder continuous functions), since the major term to be concerned in our approximation rate is essentially \sqrt{d} times a function of N and L independent of d within the modulus of continuity.
Revised:  As a consequence, this new class of networks overcome s the curse of dimensionality in approximation power when the variation of \omega_f(r) as r \rightarrow \to 0 is moderate (e.g., \omega_f(r) \lesssim r^\alpha for H \"o%DIFDELCMD < }%%% lder continuous functions), since the major term to be considered in our approximation rate is essentially \sqrt{d} times a function of N and L independent of d within the modulus of continuity.
Edit intent: clarity
Original:  Informally, we say that a point (x^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: R^d \times R^d \rightarrow R if y^\star is a second-order local maximum for f(x^\star,\cdot) in a sense of Nesterov and, roughly, x^\star is a local minimum for a greedy optimization version of the function \max_y f(x,y) which can be efficiently estimated using greedy algorithms.
Revised:  Informally, we say that a point (x^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: R^d \times R^d \rightarrow R if y^\star is a second-order local maximum for f(x^\star,\cdot) and, roughly, x^\star is a local minimum for a greedy optimization version of the function \max_y f(x,y) which can be efficiently estimated using greedy algorithms.
Edit intent: meaning-changed
Original:  The existence follows from an algorithm that converges from any starting point to such a point in a number of gradient and function evaluations that is polynomial in \frac{1{\epsilon}} {\varepsilon}} , the dimension d, and the bounds on f and its first three derivatives.
Revised:  The existence follows from an algorithm that converges from any starting point to such a point in a number of gradient and function evaluations that is polynomial in {\epsilon}} {\varepsilon}} , the dimension d, and the bounds on f and its first three derivatives.
Edit intent: meaning-changed
Original:  The existence follows from an algorithm that converges from any starting point to such a point in a number of gradient and function evaluations that is polynomial in \frac{1{\epsilon}} {\varepsilon}} , the dimension d, and the bounds on f and its first three derivatives.
Revised:  The existence follows from an algorithm that converges from any starting point to such a point in a number of gradient and function evaluations that is polynomial in \frac{1{\epsilon}} \frac{1{\varepsilon}} , the dimension d, and the bounds on f and its first three derivatives.
Edit intent: fluency
Original: Min-max optimization , with a nonconvex-nonconcave objective function f: R^d \times R^d \rightarrow R , arises in many areas, including optimization, economics, and deep learning.
Revised: Min-max optimization , with a nonconvex-nonconcave objective function f: R^d \times R^d \rightarrow R is an important model for robustness in an adversarial setting, with applications to many areas including optimization, economics, and deep learning.
Edit intent: meaning-changed
Original:  The nonconvexity-nonconcavity of f means that the problem of finding a global \varepsilon-min-max point cannot be solved in \mathrm{poly(d, 1{\varepsilon}) evaluations of f.
Revised:  In many of these applications f may be nonconvex-nonconcave, and finding a global \varepsilon-min-max point cannot be solved in \mathrm{poly(d, 1{\varepsilon}) evaluations of f.
Edit intent: meaning-changed
Original:  The nonconvexity-nonconcavity of f means that the problem of finding a global \varepsilon-min-max point cannot be solved in \mathrm{poly(d, 1{\varepsilon}) evaluations of f.
Revised:  The nonconvexity-nonconcavity of f means that the problem of finding a global (d, 1{\varepsilon}) evaluations of f.
Edit intent: clarity
Original:  Thus, most algorithms seek to obtain a certain notion of local }  min-max point where, roughly speaking, each player optimizes her payoff in a local sense .
Revised:  Thus, most algorithms seek to obtain a certain notion of local } min-max point may be computationally intractable. There is a long line of work that seeks computationally tractable algorithms for alternatives to the min-max point where, roughly speaking, each player optimizes her payoff in a local sense .
Edit intent: meaning-changed
Original:  However, the classes of local min-max solutions which prior algorithms seek are only guaranteed to exist under very strong assumptions on f, such as convexity or monotonicity. We propose a notion of a greedy equilibrium pointfor min-max optimization and prove the existence of such a point for any function such that it and its first three derivatives are bounded. Informally , we say that a point (x^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: \mathbb{R^d \times \mathbb{R}^d \rightarrow \mathbb{R} }  if y^\star is a second-order local maximum for f(x^\star,\cdot) and, roughly, x^\star is a local minimum for a greedy optimization version of the function \max _y f(x, y ) which can be efficiently estimated using greedy algorithms.
Revised:  However, the classes of local min-max solutions which prior algorithms seek are only guaranteed to exist under very strong assumptions on f, such as convexity , monotonicity, or special properties of the starting point. In this paper, we propose an optimization model, the \varepsilon-greedy adversarial equilibrium, which can serve as a computationally tractable alternative to the min-max optimization model. Roughly , we say that a point (x^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: \mathbb{R^d \times \mathbb{R}^d \rightarrow \mathbb{R} } if y^\star is a second-order local maximum for f(x^\star,\cdot) and, roughly, x^\star is a local minimum for a greedy optimization version of the function \max _y f(x, y ) which can be efficiently estimated using greedy algorithms.
Edit intent: meaning-changed
Original:  Informally , we say that a point (x^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: \mathbb{R^d \times \mathbb{R}^d \rightarrow \mathbb{R} }  if y^\star is a second-order local maximum for f(x^\star,\cdot) and, roughly, x^\star is a local minimum for a greedy optimization version of the function \max _y f(x, y ) which can be efficiently estimated using greedy algorithms.
Revised:  Informally , we say a point (x^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: \mathbb{R^d \times \mathbb{R}^d \rightarrow \mathbb{R} } if y^\star is a second-order local maximum for f(x^\star,\cdot) and, roughly, x^\star is a local minimum for a greedy optimization version of the function \max _y f(x, y ) which can be efficiently estimated using greedy algorithms.
Edit intent: fluency
Original:  Informally , we say that a point (x^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: \mathbb{R^d \times \mathbb{R}^d \rightarrow \mathbb{R} }  if y^\star is a second-order local maximum for f(x^\star,\cdot) and, roughly, x^\star is a local minimum for a greedy optimization version of the function \max _y f(x, y ) which can be efficiently estimated using greedy algorithms.
Revised:  Informally , we say that a point (x^\star, y^\star) is an \varepsilon-greedy ^d \times \mathbb{R}^d \rightarrow \mathbb{R} } if y^\star is a second-order local maximum for f(x^\star,\cdot) and, roughly, x^\star is a local minimum for a greedy optimization version of the function \max _y f(x, y ) which can be efficiently estimated using greedy algorithms.
Edit intent: clarity
Original:  Informally , we say that a point (x^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: \mathbb{R^d \times \mathbb{R}^d \rightarrow \mathbb{R} }  if y^\star is a second-order local maximum for f(x^\star,\cdot) and, roughly, x^\star is a local minimum for a greedy optimization version of the function \max _y f(x, y ) which can be efficiently estimated using greedy algorithms.
Revised:  Informally , we say that a point (x^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: \mathbb{R^d \times \mathbb{R}^d \rightarrow \mathbb{R} } adversarial equilibrium if y^\star is a second-order local maximum for f(x^\star,\cdot) and, roughly, x^\star is a local minimum for a greedy optimization version of the function \max _y f(x, y ) which can be efficiently estimated using greedy algorithms.
Edit intent: meaning-changed
Original:  Informally , we say that a point (x^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: \mathbb{R^d \times \mathbb{R}^d \rightarrow \mathbb{R} }  if y^\star is a second-order local maximum for f(x^\star,\cdot) and, roughly, x^\star is a local minimum for a greedy optimization version of the function \max _y f(x, y ) which can be efficiently estimated using greedy algorithms.
Revised:  Informally , we say that a point (x^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: \mathbb{R^d \times \mathbb{R}^d \rightarrow \mathbb{R} } if y^\star is an \varepsilon-approximate local maximum for f(x^\star,\cdot) and, roughly, x^\star is a local minimum for a greedy optimization version of the function \max _y f(x, y ) which can be efficiently estimated using greedy algorithms.
Edit intent: meaning-changed
Original:  Informally , we say that a point (x^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: \mathbb{R^d \times \mathbb{R}^d \rightarrow \mathbb{R} }  if y^\star is a second-order local maximum for f(x^\star,\cdot) and, roughly, x^\star is a local minimum for a greedy optimization version of the function \max _y f(x, y ) which can be efficiently estimated using greedy algorithms.
Revised:  Informally , we say that a point (x^\star, y^\star) is an \varepsilon-greedy min-max equilibrium point of a function f: \mathbb{R^d \times \mathbb{R}^d \rightarrow \mathbb{R} } if y^\star is a second-order local maximum for f(x^\star,\cdot) , and x^\star is a local minimum for a greedy optimization version of the function \max _y f(x, y ) which can be efficiently estimated using greedy algorithms.
Edit intent: coherence