ReLU Network Approximation in Terms of Intrinsic Parameters
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This paper studies the approximation error of ReLU networks in terms of the number of intrinsic parameters (i.e., those depending on the target function f). First, we prove by construction that, for any Lipschitz continuous function f on [0,1]^d with a Lipschitz constant λ>0, a ReLU network with n+2 intrinsic parameters can approximate f with an exponentially small error 5λ√(d) 2^-n measured in the L^p-norm for p∈ [1,∞). More generally for an arbitrary continuous function f on [0,1]^d with a modulus of continuity ω_f(·), the approximation error is ω_f(√(d) 2^-n)+2^-n+2ω_f(√(d)). Next, we extend these two results from the L^p-norm to the L^∞-norm at a price of 3^d n+2 intrinsic parameters. Finally, by using a high-precision binary representation and the bit extraction technique via a fixed ReLU network independent of the target function, we design, theoretically, a ReLU network with only three intrinsic parameters to approximate Hölder continuous functions with an arbitrarily small error.
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