Q-Newton: Hybrid Quantum-Classical Scheduling for Accelerating Neural Network Training with Newton's Gradient Descent
This addresses the problem of slow convergence in deep learning optimization for researchers and practitioners, but it is incremental as it builds on existing quantum linear solver algorithms with scheduling improvements.
The paper tackles the slow matrix inversion bottleneck in Newton's gradient descent for neural network training by proposing Q-Newton, a hybrid quantum-classical scheduler that uses quantum linear solvers for exponential acceleration, potentially reducing total training time compared to optimizers like SGD.
Optimization techniques in deep learning are predominantly led by first-order gradient methodologies, such as SGD. However, neural network training can greatly benefit from the rapid convergence characteristics of second-order optimization. Newton's GD stands out in this category, by rescaling the gradient using the inverse Hessian. Nevertheless, one of its major bottlenecks is matrix inversion, which is notably time-consuming in $O(N^3)$ time with weak scalability. Matrix inversion can be translated into solving a series of linear equations. Given that quantum linear solver algorithms (QLSAs), leveraging the principles of quantum superposition and entanglement, can operate within a $\text{polylog}(N)$ time frame, they present a promising approach with exponential acceleration. Specifically, one of the most recent QLSAs demonstrates a complexity scaling of $O(d\cdotκ\log(N\cdotκ/ε))$, depending on: {size~$N$, condition number~$κ$, error tolerance~$ε$, quantum oracle sparsity~$d$} of the matrix. However, this also implies that their potential exponential advantage may be hindered by certain properties (i.e. $κ$ and $d$). We propose Q-Newton, a hybrid quantum-classical scheduler for accelerating neural network training with Newton's GD. Q-Newton utilizes a streamlined scheduling module that coordinates between quantum and classical linear solvers, by estimating & reducing $κ$ and constructing $d$ for the quantum solver. Our evaluation showcases the potential for Q-Newton to significantly reduce the total training time compared to commonly used optimizers like SGD. We hypothesize a future scenario where the gate time of quantum machines is reduced, possibly realized by attoseconds physics. Our evaluation establishes an ambitious and promising target for the evolution of quantum computing.