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ReLU and Softplus Activations

July 28, 2020

ReLU is one of the commonly used activations for artificial neural networks, and softplus can viewed as its smooth version.

ReLU(x)=max(0,x)           softplusβ(x)=1βlog(1+eβx),\text{ReLU}(x) = \max(0, x) \ \ \ \ \ \ \ \ \ \ \ \text{softplus}_\beta(x) = \frac{1}{\beta}\log(1 + e^{\beta x}),

where β\beta is a parameter that one can specify; it is typically set to one. The figure illustrates how soffplus becomes closer to ReLU when using different β\beta values.

Dombrowski et al. (2019)'s Theorem 2 says that for one-layer neuron network with ReLU activations g(x)=max(0,wTx)g(x) = \max(0, w^Tx) and and its softplus version gβ(x)=softplusβ(wTx)g_\beta(x) = \text{softplus}_\beta(w^Tx), the following equality holds:

Eϵpβ[g(xϵ)]=gβ/w(x).\mathbb{E}_{\epsilon \sim p_\beta}[\nabla g(x - \epsilon)] = \nabla g_{\beta/\|w\|} (x).

The gradient wrt. to the input of the softplus network is the expectation of the gradient of the ReLU network when the input is perturbed by the noise ϵ\epsilon.

In the following, I state the proof that is provided in the supplement of the paper.

Let assume for a moment that xx is scalar. We first start by showing that Eϵpβ[g(xϵ)]=softplusβ(x)\mathbb{E}_{\epsilon \sim p_\beta} [ g(x-\epsilon) ] = \text{softplus}_\beta(x). To do this, we write

softplusβ(x)=p(ϵ)g(xϵ)dϵ,(1)\text{softplus}_\beta(x) = \int_{-\infty}^{\infty} p(\epsilon) g(x-\epsilon) \text{d}\epsilon \tag{1},

where p(ϵ)p(\epsilon) is implicitly defined. Differentiating the equation both, we get

σβ(x)=p(ϵ)1[xϵ>0]dϵ=xp(ϵ)dϵ.\begin{aligned} \sigma_\beta(x) &= \int_{-\infty}^{\infty} p(\epsilon) \mathbb 1_{[x-\epsilon > 0]} \text{d}\epsilon \\ &= \int_{-\infty}^{x} p(\epsilon) \text{d}\epsilon. \end{aligned}

Applying another differentiaion yields

β(eβϵ/2+eβϵ/2)2pβ(x)=p(x). \underbrace{\frac{\beta}{(e^{\beta \epsilon /2} + e^{-\beta \epsilon / 2})^2}}_{\triangleq p_\beta(x)} = p(x).

Substituting pβp_\beta in (1), we get

softplusβ(x)=pβ(ϵ)g(xϵ)dϵ=Eϵpβ[g(xϵ)].(2)\begin{aligned} \text{softplus}_\beta(x) &= \int_{-\infty}^{\infty} p_\beta(\epsilon) g(x-\epsilon) \text{d}\epsilon \\ &= \mathbb{E}_{\epsilon \sim p_\beta} [ g(x-\epsilon)]. \tag{2} \end{aligned}

Consider x,ϵRd\mathbf x, \mathbf \epsilon \in \Reals^d. We assume that pβ(ϵ)=i=1dpβ(ϵi)p_\beta(\mathbf \epsilon) = \prod_{i=1}^d p_{\beta}(\epsilon_i); that is the noise vector ϵ\mathbf \epsilon consists of dd independent noise components. Furthermore, we choose

ϵ=ϵpw^+i=1dϵ0(i)w0^(i),\mathbf \epsilon = \epsilon_p \hat{\mathbf w} + \sum_{i=1}^d \epsilon_0^{(i)} \hat{\mathbf{w}_0}^{(i)},

s.t. i[d],w^Tw0^(i)=0\forall i \in [d], \hat{\mathbf w}^T \hat{\mathbf{w}_0}^{(i)} = 0 and w=ww^\mathbf w = \| \mathbf w \| \hat{\mathbf{w}}. Then, from (2), it follows that

Eϵpβ[g(xϵ)]=Eϵpβ[ReLU(wT(xϵ))]=Eϵpβ[ReLU(wTxwTϵ)]=Eϵpβ[ReLU(wTxϵpw)].\begin{aligned} \mathbb{E}_{\epsilon \sim p_\beta} [ g(\mathbf x- \mathbf \epsilon)] &= \mathbb{E}_{\epsilon \sim p_\beta} [ \text{ReLU}(\mathbf w^T(\mathbf x- \mathbf \epsilon))] \\ &= \mathbb{E}_{\epsilon \sim p_\beta} [ \text{ReLU}(\mathbf w^T\mathbf x- \mathbf w^T \mathbf \epsilon)] \\ &= \mathbb{E}_{\epsilon \sim p_\beta} [ \text{ReLU}(\mathbf w^T\mathbf x- \epsilon_p \| \mathbf w \|)]. \end{aligned}

Consider ϵ~ϵpw\tilde{\mathbf \epsilon} \triangleq \mathbf \epsilon_p \| \mathbf w \|. The expectation can be rewritten as

Eϵpβ[ReLU(wTxϵpw)]=pβ(ϵ)ReLU(wTxϵpw)dϵ=pβ(ϵ~w)(1w)pβw(ϵ)~ReLU(wTxϵ~)dϵ~=Eϵ~pβw[ReLU(wTxϵ~)]=softplusβw(x).\begin{aligned} \mathbb{E}_{\epsilon \sim p_\beta} &[ \text{ReLU}(\mathbf w^T\mathbf x- \epsilon_p \| \mathbf w \|)] = \int_{-\infty}^{\infty} p_\beta(\mathbf \epsilon) \text{ReLU}(\mathbf w^T\mathbf x- \epsilon_p \| \mathbf w \|) \text{d} \mathbf \epsilon \\ &= \int_{-\infty}^{\infty} \underbrace{p_\beta\bigg(\frac{\tilde{\mathbf \epsilon}}{\| \mathbf w \|}\bigg) \bigg( \frac{1}{\| \mathbf w \|} \bigg)}_{p_\frac{\beta}{\|\mathbf w\|}(\tilde{\mathbf \epsilon)}} \text{ReLU}(\mathbf w^T\mathbf x- \tilde\mathbf{\epsilon}) \text{d} \tilde {\mathbf \epsilon} \\ &= \mathbb E_{\tilde{\mathbf \epsilon} \sim p_{\frac{\beta}{\| \mathbf w \|}}} [ \text{ReLU}(\mathbf w^T\mathbf x- \tilde\mathbf{\epsilon})] \\ &= \text{softplus}_{\frac{\beta}{\| \mathbf w \|}}(x). \end{aligned}

Differentiating both sides yields the desired result. ◼️

Why does this result matter?

In the paper, Dombrowski et al. (2019) show that attribution (explanation) maps can be arbitrarily manipulated. They argue that this is because the output manifold of the ReLU neural network has a large curvature, and it causes gradients wrt. to the input to be highly unstable when the input is slightly perturbed.

They show that one can prevent such manipulations by replacing ReLU with the softplus activation. Based on the theorem 2, they argue and empirically show that doing so has the same effect (attribution maps) as SmoothGrad, in which the attribution map is averaged from several maps of the input perturbed by some noise.

Reference

Dombrowski et al. (2019) "Explanations can be manipulated and geometry is to blame"

Appendix

From Softplus to Sigmoid

Consider Softplus1(x)=log(1+exp(x))\text{Softplus}_1(x) = \log(1+\exp(x)). The derivative is

ddxSoftplus1(x)=11+exp(x)exp(x)Sigmoid(x).\begin{aligned} \frac{\text{d}}{\text{d}x} \text{Softplus}_1(x) = \underbrace{\frac{1}{1+\exp(x)}\exp(x)}_{\text{Sigmoid}(x)}. \end{aligned}