The proposition makes use of the following observation: For the discriminator defined in (1), the norm of gradient for wt is upper bounded by k wtDθ(x)k F kxk LY

The upper bound of gradient's Frobenius norm for spectrally-normalized discriminators follows directly. As lw(x) is a linear transformation, we have lcw(x) = c lw(x), and lw(cx) = c lw(x). Moreover, since ReLU and leaky ReLU is linear in R+ and R region, we have ai(cx) = c ai(x). In this section we discuss the gradients with respect the actual parameter wi. From Eq. (12) in [30] we know wtDθ(x) = A, we know that w0tDθ(x) F, otl(x)Dθ(x), and kotl (x)k have upper bounds. From Theorem 1.1 in [44] we know that if wt is initialized with i.i.d random variables from uniform or Gaussian distribution, E kwtkspis lower bounded away from zero at initialization. So k wtDθ(x)kF is upper bounded at initialization. Moreover, we observe empirically that kwtksp is usually increasing during training. Therefore, k wtDθ(x)kF is typically upper bounded during training as well. The following proposition states that spectral normalization also gives an upper bound on kHwi(Dθ)(x)ksp for networks with ReLU or leaky ReLU internal activations.