More Like this

1. Analyzing Convergence in Quantum Neural Networks: Deviations from Neural Tangent Kernels

   Xuchen You ; Shouvanik Chakrabarti ; Boyang Chen ; Xiaodi Wu ( January 2023 , Proceedings of the 40th International Conference on Machine Learning)

   A quantum neural network (QNN) is a parameterized mapping efficiently implementable on near- term Noisy Intermediate-Scale Quantum (NISQ) computers. It can be used for supervised learn- ing when combined with classical gradient-based optimizers. Despite the existing empirical and theoretical investigations, the convergence of QNN training is not fully understood. Inspired by the success of the neural tangent kernels (NTKs) in probing into the dynamics of classical neural net- works, a recent line of works proposes to study over-parameterized QNNs by examining a quantum version of tangent kernels. In this work, we study the dynamics of QNNs and show that contrary to popular belief it is qualitatively different from that of any kernel regression: due to the unitarity of quantum operations, there is a non- negligible deviation from the tangent kernel regression derived at the random initialization. As a result of the deviation, we prove the at-most sub- linear convergence for QNNs with Pauli measurements, which is beyond the explanatory power of any kernel regression dynamics. We then present the actual dynamics of QNNs in the limit of over- parameterization. The new dynamics capture the change of convergence rate during training, and implies that the range of measurements is crucial to the fast QNN convergence. 

   more » « less
2. Exponentially Many Local Minima in Quantum Neural Networks

   You, Xuchen ; Wu, Xiaodi ( January 2021 , Proceedings of the 38th International Conference on Machine Learning)

   null (Ed.)

   Quantum Neural Networks (QNNs), or the so-called variational quantum circuits, are important quantum applications both because of their similar promises as classical neural networks and because of the feasibility of their implementation on near-term intermediate-size noisy quantum machines (NISQ). However, the training task of QNNs is challenging and much less understood. We conduct a quantitative investigation on the landscape of loss functions of QNNs and identify a class of simple yet extremely hard QNN instances for training. Specifically, we show for typical under-parameterized QNNs, there exists a dataset that induces a loss function with the number of spurious local minima depending exponentially on the number of parameters. Moreover, we show the optimality of our construction by providing an almost matching upper bound on such dependence. While local minima in classical neural networks are due to non-linear activations, in quantum neural networks local minima appear as a result of the quantum interference phenomenon. Finally, we empirically confirm that our constructions can indeed be hard instances in practice with typical gradient-based optimizers, which demonstrates the practical value of our findings. 

   more » « less
3. QTROJAN: A Circuit Backdoor Against Quantum Neural Networks

   https://doi.org/10.1109/ICASSP49357.2023.10096293  

   Chu, Cheng ; Jiang, Lei ; Swany, Martin ; Chen, Fan ( June 2023 , IEEE International Conference on Acoustics, Speech and Signal Processing)

   We propose a circuit-level backdoor attack, QTrojan, against Quantum Neural Networks (QNNs) in this paper. QTrojan is implemented by a few quantum gates inserted into the variational quantum circuit of the victim QNN. QTrojan is much stealthier than a prior Data-Poisoning-based Backdoor Attack (DPBA) since it does not embed any trigger in the inputs of the victim QNN or require access to original training datasets. Compared to a DPBA, QTrojan improves the clean data accuracy by 21% and the attack success rate by 19.9%. 

   more » « less
4. A Local Convergence Theory for Mildly Over-Parameterized Two-Layer Neural Network

   Zhou, Mo ; Ge, Rong ; Jin, Chi ( July 2021 , COLT)

   While over-parameterization is widely believed to be crucial for the success of optimization for the neural networks, most existing theories on over-parameterization do not fully explain the reason -- they either work in the Neural Tangent Kernel regime where neurons don't move much, or require an enormous number of neurons. In practice, when the data is generated using a teacher neural network, even mildly over-parameterized neural networks can achieve 0 loss and recover the directions of teacher neurons. In this paper we develop a local convergence theory for mildly over-parameterized two-layer neural net. We show that as long as the loss is already lower than a threshold (polynomial in relevant parameters), all student neurons in an over-parameterized two-layer neural network will converge to one of teacher neurons, and the loss will go to 0. Our result holds for any number of student neurons as long as it is at least as large as the number of teacher neurons, and our convergence rate is independent of the number of student neurons. A key component of our analysis is the new characterization of local optimization landscape -- we show the gradient satisfies a special case of Lojasiewicz property which is different from local strong convexity or PL conditions used in previous work. 

   more » « less
5. The Recurrent Neural Tangent Kernel

   Alemohammad, Sina ; Wang, Zichao ; Balestriero, Randall ; Baraniuk, Richard ( May 2021 , The International Conference on Learning Representations)

   The study of deep neural networks (DNNs) in the infinite-width limit, via the so-called neural tangent kernel (NTK) approach, has provided new insights into the dynamics of learning, generalization, and the impact of initialization. One key DNN architecture remains to be kernelized, namely, the recurrent neural network (RNN). In this paper we introduce and study the Recurrent Neural Tangent Kernel (RNTK), which provides new insights into the behavior of overparametrized RNNs. A key property of the RNTK should greatly benefit practitioners is its ability to compare inputs of different length. To this end, we characterize how the RNTK weights different time steps to form its output under different initialization parameters and nonlinearity choices. A synthetic and 56 real-world data experiments demonstrate that the RNTK offers significant performance gains over other kernels, including standard NTKs, across a wide array of data sets. 

   more » « less