When Can Nonconvex Optimization Problems be Solved with Gradient Descent? A Few Case Studies

Gradient descent and related algorithms are ubiquitously used to solve optimization problems arising in machine learning and signal processing. In many cases, these problems are nonconvex yet such simple algorithms are still effective. In an attempt to better understand this phenomenon, we study a number of nonconvex problems, proving that they can be solved efficiently with gradient descent. We will consider complete, orthogonal dictionary learning, and present a geometric analysis allowing us to obtain efficient convergence rate for gradient descent that hold with high probability. We also show that similar geometric structure is present in other nonconvex problems such as generalized phase retrieval. Turning next to neural networks, we will also calculate conditions on certain classes of networks under which signals and gradients propagate through the network in a stable manner during the initial stages of training. Initialization schemes derived using these calculations allow training recurrent networks on long sequence tasks, and in the case of networks with low precision activation functions they make explicit a tradeoff between the reduction in precision and the maximal depth of a model that can be trained with gradient descent. We finally consider manifold classification with a deep feed-forward neural network, for a particularly simple configuration of the manifolds. We provide an end-to-end analysis of the training process, proving that under certain conditions on the architectural hyperparameters of the network, it can successfully classify any point on the manifolds with high probability given a sufficient number of independent samples from the manifold, in a timely manner. Our analysis relates the depth and width of the network to its fitting capacity and statistical regularity respectively in early stages of training

On quantum backpropagation, information reuse, and cheating measurement collapse

The success of modern deep learning hinges on the ability to train neural networks at scale. Through clever reuse of intermediate information, backpropagation facilitates training through gradient computation at a total cost roughly proportional to running the function, rather than incurring an additional factor proportional to the number of parameters - which can now be in the trillions. Naively, one expects that quantum measurement collapse entirely rules out the reuse of quantum information as in backpropagation. But recent developments in shadow tomography, which assumes access to multiple copies of a quantum state, have challenged that notion. Here, we investigate whether parameterized quantum models can train as efficiently as classical neural networks. We show that achieving backpropagation scaling is impossible without access to multiple copies of a state. With this added ability, we introduce an algorithm with foundations in shadow tomography that matches backpropagation scaling in quantum resources while reducing classical auxiliary computational costs to open problems in shadow tomography. These results highlight the nuance of reusing quantum information for practical purposes and clarify the unique difficulties in training large quantum models, which could alter the course of quantum machine learning.Comment: 29 pages, 2 figure

Suppressing quantum errors by scaling a surface code logical qubit

Practical quantum computing will require error rates that are well below what is achievable with physical qubits. Quantum error correction offers a path to algorithmically-relevant error rates by encoding logical qubits within many physical qubits, where increasing the number of physical qubits enhances protection against physical errors. However, introducing more qubits also increases the number of error sources, so the density of errors must be sufficiently low in order for logical performance to improve with increasing code size. Here, we report the measurement of logical qubit performance scaling across multiple code sizes, and demonstrate that our system of superconducting qubits has sufficient performance to overcome the additional errors from increasing qubit number. We find our distance-5 surface code logical qubit modestly outperforms an ensemble of distance-3 logical qubits on average, both in terms of logical error probability over 25 cycles and logical error per cycle ($2.914\%\pm 0.016\%$ compared to $3.028\%\pm 0.023\%$). To investigate damaging, low-probability error sources, we run a distance-25 repetition code and observe a $1.7\times10^{-6}$ logical error per round floor set by a single high-energy event ($1.6\times10^{-7}$ when excluding this event). We are able to accurately model our experiment, and from this model we can extract error budgets that highlight the biggest challenges for future systems. These results mark the first experimental demonstration where quantum error correction begins to improve performance with increasing qubit number, illuminating the path to reaching the logical error rates required for computation.Comment: Main text: 6 pages, 4 figures. v2: Update author list, references, Fig. S12, Table I