Preventing Gradient Attenuation in Lipschitz Constrained Convolutional Networks

Lipschitz constraints under L2 norm on deep neural networks are useful for provable adversarial robustness bounds, stable training, and Wasserstein distance estimation. While heuristic approaches such as the gradient penalty have seen much practical success, it is challenging to achieve similar practical performance while provably enforcing a Lipschitz constraint. In principle, one can design Lipschitz constrained architectures using the composition property of Lipschitz functions, but Anil et al. recently identified a key obstacle to this approach: gradient norm attenuation. They showed how to circumvent this problem in the case of fully connected networks by designing each layer to be gradient norm preserving. We extend their approach to train scalable, expressive, provably Lipschitz convolutional networks. In particular, we present the Block Convolution Orthogonal Parameterization (BCOP), an expressive parameterization of orthogonal convolution operations. We show that even though the space of orthogonal convolutions is disconnected, the largest connected component of BCOP with 2n channels can represent arbitrary BCOP convolutions over n channels. Our BCOP parameterization allows us to train large convolutional networks with provable Lipschitz bounds. Empirically, we find that it is competitive with existing approaches to provable adversarial robustness and Wasserstein distance estimation.

TimbreTron: A WaveNet(CycleGAN(CQT(Audio))) Pipeline for Musical Timbre Transfer

In this work, we address the problem of musical timbre transfer, where the goal is to manipulate the timbre of a sound sample from one instrument to match another instrument while preserving other musical content, such as pitch, rhythm, and loudness. In principle, one could apply image-based style transfer techniques to a time-frequency representation of an audio signal, but this depends on having a representation that allows independent manipulation of timbre as well as high-quality waveform generation. We introduce TimbreTron, a method for musical timbre transfer which applies "image" domain style transfer to a time-frequency representation of the audio signal, and then produces a high-quality waveform using a conditional WaveNet synthesizer. We show that the Constant Q Transform (CQT) representation is particularly well-suited to convolutional architectures due to its approximate pitch equivariance. Based on human perceptual evaluations, we confirmed that TimbreTron recognizably transferred the timbre while otherwise preserving the musical content, for both monophonic and polyphonic samples.

Sorting out Lipschitz function approximation

Training neural networks subject to a Lipschitz constraint is useful for generalization bounds, provable adversarial robustness, interpretable gradients, and Wasserstein distance estimation. By the composition property of Lipschitz functions, it suffices to ensure that each individual affine transformation or nonlinear activation function is 1-Lipschitz. The challenge is to do this while maintaining the expressive power. We identify a necessary property for such an architecture: each of the layers must preserve the gradient norm during backpropagation. Based on this, we propose to combine a gradient norm preserving activation function, GroupSort, with norm-constrained weight matrices. We show that norm-constrained GroupSort architectures are universal Lipschitz function approximators. Empirically, we show that norm-constrained GroupSort networks achieve tighter estimates of Wasserstein distance than their ReLU counterparts and can achieve provable adversarial robustness guarantees with little cost to accuracy.

Training Deep Networks with Synthetic Data: Bridging the Reality Gap by Domain Randomization

We present a system for training deep neural networks for object detection using synthetic images. To handle the variability in real-world data, the system relies upon the technique of domain randomization, in which the parameters of the simulator$-$such as lighting, pose, object textures, etc.$-$are randomized in non-realistic ways to force the neural network to learn the essential features of the object of interest. We explore the importance of these parameters, showing that it is possible to produce a network with compelling performance using only non-artistically-generated synthetic data. With additional fine-tuning on real data, the network yields better performance than using real data alone. This result opens up the possibility of using inexpensive synthetic data for training neural networks while avoiding the need to collect large amounts of hand-annotated real-world data or to generate high-fidelity synthetic worlds$-$both of which remain bottlenecks for many applications. The approach is evaluated on bounding box detection of cars on the KITTI dataset.