In the training of over-parameterized model functions via gradient descent, sometimes the parameters do not change significantly and remain close to their initial values. This phenomenon is called lazy training, and motivates consideration of the linear approximation of the model function around the initial parameters. In the lazy regime, this linear approximation imitates the behavior of the parameterized function whose associated kernel, called the tangent kernel, specifies the training performance of the model. Lazy training is known to occur in the case of (classical) neural networks with large widths. In this paper, we show that the training of geometrically local parameterized quantum circuits enters the lazy regime for large numbers of qubits. More precisely, we prove bounds on the rate of changes of the parameters of such a geometrically local parameterized quantum circuit in the training process, and on the precision of the linear approximation of the associated quantum model function; both of these bounds tend to zero as the number of qubits grows. We support our analytic results with numerical simulations.
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In the coming years, quantum networks will allow quantum applications to thrive thanks to the new opportunities offered by end-to-end entanglement of qubits on remote hosts via quantum repeaters. On a geographical scale, this will lead to the dawn of the Quantum Internet. While a full-blown deployment is yet to come, the research community is already working on a variety of individual enabling technologies and solutions. In this paper, with the guidance of extensive simulations, we take a broader view and investigate the problems of Quality of Service (QoS) and provisioning in the context of quantum networks, which are very different from their counterparts in classical data networks due to some of their fundamental properties. Our work leads the way towards a new class of studies that will allow the research community to better understand the challenges of quantum networks and their potential commercial exploitation.
Quantum communications is a promising technology that will play a fundamental role in the design of future networks. In fact, significant efforts are being dedicated by both the quantum physics and the classical communications communities on developing new architectures, solutions, and practical implementations of quantum communication networks (QCNs). Although these efforts led to various advances in today's technologies, there still exists a non-trivial gap between the research efforts of the two communities on designing and optimizing the QCN performance. For instance, most prior works by the classical communications community ignore important quantum physics-based constraints when designing QCNs. For example, many works on entanglement distribution do not account for the decoherence of qubits inside quantum memories and, thus, their designs become impractical since they assume an infinite quantum states' lifetime. In this paper, we introduce a novel framework, dubbed physics-informed QCNs, for designing and analyzing the performance of QCNs, by relying on the quantum physics principles that underly the different QCN components. The need of the proposed approach is then assessed and its fundamental role in designing practical QCNs is analyzed across various open research areas. Moreover, we identify novel physics-informed performance metrics and controls that enable QCNs to leverage the state-of-the-art advancements in quantum technologies to enhance their performance. Finally, we analyze multiple pressing challenges and open research directions in QCNs that must be treated using a physics-informed approach to lead practically viable results. Ultimately, this work attempts to bridge the gap between the classical communications and the quantum physics communities in the area of QCNs to foster the development of future communication networks (6G and beyond, and the quantum Internet).
In this work a quantum analogue of Bayesian statistical inference is considered. Based on the notion of instrument, we propose a sequential measurement scheme from which observations needed for statistical inference are obtained. We further put forward a quantum analogue of Bayes rule, which states how the prior normal state of a quantum system updates under those observations. We next generalize the fundamental notions and results of Bayesian statistics according to the quantum Bayes rule. It is also note that our theory retains the classical one as its special case. Finally, we investigate the limit of posterior normal state as the number of observations tends to infinity.
Distributed machine learning (ML) can bring more computational resources to bear than single-machine learning, thus enabling reductions in training time. Distributed learning partitions models and data over many machines, allowing model and dataset sizes beyond the available compute power and memory of a single machine. In practice though, distributed ML is challenging when distribution is mandatory, rather than chosen by the practitioner. In such scenarios, data could unavoidably be separated among workers due to limited memory capacity per worker or even because of data privacy issues. There, existing distributed methods will utterly fail due to dominant transfer costs across workers, or do not even apply. We propose a new approach to distributed fully connected neural network learning, called independent subnet training (IST), to handle these cases. In IST, the original network is decomposed into a set of narrow subnetworks with the same depth. These subnetworks are then trained locally before parameters are exchanged to produce new subnets and the training cycle repeats. Such a naturally "model parallel" approach limits memory usage by storing only a portion of network parameters on each device. Additionally, no requirements exist for sharing data between workers (i.e., subnet training is local and independent) and communication volume and frequency are reduced by decomposing the original network into independent subnets. These properties of IST can cope with issues due to distributed data, slow interconnects, or limited device memory, making IST a suitable approach for cases of mandatory distribution. We show experimentally that IST results in training times that are much lower than common distributed learning approaches.
Momentum methods, including heavy-ball~(HB) and Nesterov's accelerated gradient~(NAG), are widely used in training neural networks for their fast convergence. However, there is a lack of theoretical guarantees for their convergence and acceleration since the optimization landscape of the neural network is non-convex. Nowadays, some works make progress towards understanding the convergence of momentum methods in an over-parameterized regime, where the number of the parameters exceeds that of the training instances. Nonetheless, current results mainly focus on the two-layer neural network, which are far from explaining the remarkable success of the momentum methods in training deep neural networks. Motivated by this, we investigate the convergence of NAG with constant learning rate and momentum parameter in training two architectures of deep linear networks: deep fully-connected linear neural networks and deep linear ResNets. Based on the over-parameterization regime, we first analyze the residual dynamics induced by the training trajectory of NAG for a deep fully-connected linear neural network under the random Gaussian initialization. Our results show that NAG can converge to the global minimum at a $(1 - \mathcal{O}(1/\sqrt{\kappa}))^t$ rate, where $t$ is the iteration number and $\kappa > 1$ is a constant depending on the condition number of the feature matrix. Compared to the $(1 - \mathcal{O}(1/{\kappa}))^t$ rate of GD, NAG achieves an acceleration over GD. To the best of our knowledge, this is the first theoretical guarantee for the convergence of NAG to the global minimum in training deep neural networks. Furthermore, we extend our analysis to deep linear ResNets and derive a similar convergence result.
In this paper, we introduce $\mathsf{CO}_3$, an algorithm for communication-efficiency federated Deep Neural Network (DNN) training.$\mathsf{CO}_3$ takes its name from three processing applied steps which reduce the communication load when transmitting the local gradients from the remote users to the Parameter Server.Namely:(i) gradient quantization through floating-point conversion, (ii) lossless compression of the quantized gradient, and (iii) quantization error correction.We carefully design each of the steps above so as to minimize the loss in the distributed DNN training when the communication overhead is fixed.In particular, in the design of steps (i) and (ii), we adopt the assumption that DNN gradients are distributed according to a generalized normal distribution.This assumption is validated numerically in the paper. For step (iii), we utilize an error feedback with memory decay mechanism to correct the quantization error introduced in step (i). We argue that this coefficient, similarly to the learning rate, can be optimally tuned to improve convergence. The performance of $\mathsf{CO}_3$ is validated through numerical simulations and is shown having better accuracy and improved stability at a reduced communication payload.
Momentum methods, such as heavy ball method~(HB) and Nesterov's accelerated gradient method~(NAG), have been widely used in training neural networks by incorporating the history of gradients into the current updating process. In practice, they often provide improved performance over (stochastic) gradient descent~(GD) with faster convergence. Despite these empirical successes, theoretical understandings of their accelerated convergence rates are still lacking. Recently, some attempts have been made by analyzing the trajectories of gradient-based methods in an over-parameterized regime, where the number of the parameters is significantly larger than the number of the training instances. However, the majority of existing theoretical work is mainly concerned with GD and the established convergence result of NAG is inferior to HB and GD, which fails to explain the practical success of NAG. In this paper, we take a step towards closing this gap by analyzing NAG in training a randomly initialized over-parameterized two-layer fully connected neural network with ReLU activation. Despite the fact that the objective function is non-convex and non-smooth, we show that NAG converges to a global minimum at a non-asymptotic linear rate $(1-\Theta(1/\sqrt{\kappa}))^t$, where $\kappa > 1$ is the condition number of a gram matrix and $t$ is the number of the iterations. Compared to the convergence rate $(1-\Theta(1/{\kappa}))^t$ of GD, our result provides theoretical guarantees for the acceleration of NAG in neural network training. Furthermore, our findings suggest that NAG and HB have similar convergence rate. Finally, we conduct extensive experiments on six benchmark datasets to validate the correctness of our theoretical results.
Remarkable progress has been achieved in synthesizing photo-realistic images with generative adversarial neural networks (GANs). Recently, GANs are utilized as the training sample generator when obtaining or storing real training data is expensive even infeasible. However, traditional GANs generated images are not as informative as the real training samples when being used to train deep neural networks. In this paper, we propose a novel method to synthesize Informative Training samples with GAN (IT-GAN). Specifically, we freeze a pre-trained GAN model and learn the informative latent vectors that corresponds to informative training samples. The synthesized images are required to preserve information for training deep neural networks rather than visual reality or fidelity. Experiments verify that the deep neural networks can learn faster and achieve better performance when being trained with our IT-GAN generated images. We also show that our method is a promising solution to dataset condensation problem.
The performance of a quantum information processing protocol is ultimately judged by distinguishability measures that quantify how distinguishable the actual result of the protocol is from the ideal case. The most prominent distinguishability measures are those based on the fidelity and trace distance, due to their physical interpretations. In this paper, we propose and review several algorithms for estimating distinguishability measures based on trace distance and fidelity. The algorithms can be used for distinguishing quantum states, channels, and strategies (the last also known in the literature as "quantum combs"). The fidelity-based algorithms offer novel physical interpretations of these distinguishability measures in terms of the maximum probability with which a single prover (or competing provers) can convince a verifier to accept the outcome of an associated computation. We simulate many of these algorithms by using a variational approach with parameterized quantum circuits. We find that the simulations converge well in both the noiseless and noisy scenarios, for all examples considered. Furthermore, the noisy simulations exhibit a parameter noise resilience.
Deep graph neural networks (GNNs) have achieved excellent results on various tasks on increasingly large graph datasets with millions of nodes and edges. However, memory complexity has become a major obstacle when training deep GNNs for practical applications due to the immense number of nodes, edges, and intermediate activations. To improve the scalability of GNNs, prior works propose smart graph sampling or partitioning strategies to train GNNs with a smaller set of nodes or sub-graphs. In this work, we study reversible connections, group convolutions, weight tying, and equilibrium models to advance the memory and parameter efficiency of GNNs. We find that reversible connections in combination with deep network architectures enable the training of overparameterized GNNs that significantly outperform existing methods on multiple datasets. Our models RevGNN-Deep (1001 layers with 80 channels each) and RevGNN-Wide (448 layers with 224 channels each) were both trained on a single commodity GPU and achieve an ROC-AUC of $87.74 \pm 0.13$ and $88.14 \pm 0.15$ on the ogbn-proteins dataset. To the best of our knowledge, RevGNN-Deep is the deepest GNN in the literature by one order of magnitude. Please visit our project website //www.deepgcns.org/arch/gnn1000 for more information.
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