Generalization to the Natural Gradient Descent

Abstract

Optimization problem, which is aimed at finding the global minimal value of a given cost function, is one of the central problem in science and engineering. Various numerical methods have been proposed to solve this problem, among which the Gradient Descent (GD) method is the most popular one due to its simplicity and efficiency. However, the GD method suffers from two main issues: the local minima and the slow convergence especially near the minima point. The Natural Gradient Descent(NGD), which has been proved as one of the most powerful method for various optimization problems in machine learning, tensor network, variational quantum algorithms and so on, supplies an efficient way to accelerate the convergence. Here, we give a unified method to extend the NGD method to a more general situation which keeps the fast convergence by looking for a more suitable metric through introducing a 'proper' reference Riemannian manifold. Our method generalizes the NDG, and may give more insight of the optimization methods.

Efficient learning of mixed-state tomography for photonic quantum walk

“…By contrast, we both theoretically and experimentally realized the NDO tomography of a 2 × n open QW system and the reconstruction of the mixed state with the largest number of matrix elements, up to 62 2 , surpassing all previously reported results. Furthermore, this experimental study is also the first time to verify the effectiveness of our self-developed GNGD optimization algorithm (38) for accelerating training on the long-standing challenging task of experimental learning of open QW. The GNGD optimizer can be applied to various neuralnetwork architectures for effectively speeding up the gradientbased tomography of the quantum state (65,66) and the quantum process (67)(68)(69).…”

“…, between the target and the reconstructed distributions. The process starts with the network parameters being initialized to random values, and in each optimization iteration i, the parameters are updated according to the GNGD procedure (38):…”

“…As the total QW step N increases, the learning of mixed states with the dimension of 4(N + 1) 2 demands the complicated training of a larger neural network to keep its high expressive power. Thus, we introduce the GNGD procedure (38) to enhance the training efficiency of NDO. To better demonstrate the performance of GNGD for addressing complex networks of NDO, we extend the number of discrete time steps N = 30, and a 30-step Hadamard walk is experimentally realized.…”

“…b θ , c θ , and d θ are the vectors of dimension d representing the biases coupled to the visible, hidden, and ancillary neurons, respectively. To accelerate the convergence efficiency of training, the GNGD method emphasizes introducing a proper reference space (38). Here, the space span by the density matrix ρ θ (α, β) is selected as the reference manifold.…”

“…We achieve this by parameterizing the complex-valued matrix elements using a neural density operator (NDO) (37) because such an effective ansatz has been shown to have better performance than standard maximum likelihood (MaxLik) estimation in terms of the reconstruction fidelity and the number of measurements required. For training NDO to maximize the measurement data likelihood more efficiently, the generalized natural gradient descent (GNGD) procedure, recently developed in our group (38), is adopted to achieve about an order of magnitude of convergence rate than traditional gradient descent (GD) procedure. The effective mixed-state tomography of open QW, with a lower requirement of the number of measurements and training iterations for high-fidelity reconstructions, is benchmarked on synthetic datasets.…”