Piecewise-Linear Activations or Analytic Activation Functions: Which Produce More Expressive Neural Networks?
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Many currently available universal approximation theorems affirm that deep feedforward networks defined using any suitable activation function can approximate any integrable function locally in L^1-norm. Though different approximation rates are available for deep neural networks defined using other classes of activation functions, there is little explanation for the empirically confirmed advantage that ReLU networks exhibit over their classical (e.g. sigmoidal) counterparts. Our main result demonstrates that deep networks with piecewise linear activation (e.g. ReLU or PReLU) are fundamentally more expressive than deep feedforward networks with analytic (e.g. sigmoid, Swish, GeLU, or Softplus). More specifically, we construct a strict refinement of the topology on the space L^1_loc(ℝ^d,ℝ^D) of locally Lebesgue-integrable functions, in which the set of deep ReLU networks with (bilinear) pooling NN^ReLU + Pool is dense (i.e. universal) but the set of deep feedforward networks defined using any combination of analytic activation functions with (or without) pooling layers NN^ω+Pool is not dense (i.e. not universal). Our main result is further explained by quantitatively demonstrating that this "separation phenomenon" between the networks in NN^ReLU+Pool and those in NN^ω+Pool by showing that the networks in NN^ReLU are capable of approximate any compactly supported Lipschitz function while simultaneously approximating its essential support; whereas, the networks in NN^ω+pool cannot.
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