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Application of Quantum Annealing to Training of Deep Neural Networks
Steven H. Adachi, Maxwell P. Henderson
TL;DR
Conventional generative training for Deep Belief Networks can be slow because Gibbs sampling mixes slowly. This paper uses quantum samples from a D-Wave machine to train Restricted Boltzmann Machines and reports comparable or better accuracy with fewer training iterations on coarse-grained MNIST.
Problem
Generative Deep Belief Network training commonly uses Contrastive Divergence, but slow Gibbs-sampling mixing can make it time-consuming.
Method
The approach estimates Restricted Boltzmann Machine model expectations using quantum sampling from a D-Wave machine during generative Deep Belief Network training.
Results
Comparable or better accuracy was achieved with significantly fewer generative-training iterations than conventional Contrastive Divergence training on coarse-grained MNIST.
Takeaways & Limitations
Quantum sampling-based training can produce high-accuracy networks despite faulty qubits and missing hardware connections in the tested setting.
Takeaways & Limitations
Intrinsic control errors and coupling leakage can skew statistics when multiple qubits are chained to overcome limited connectivity.
Abstract
from arXiv · showhide
In Deep Learning, a well-known approach for training a Deep Neural Network starts by training a generative Deep Belief Network model, typically using Contrastive Divergence (CD), then fine-tuning the weights using backpropagation or other discriminative techniques. However, the generative training can be time-consuming due to the slow mixing of Gibbs sampling. We investigated an alternative approach that estimates model expectations of Restricted Boltzmann Machines using samples from a D-Wave quantum annealing machine. We tested this method on a coarse-grained version of the MNIST data set. In our tests we found that the quantum sampling-based training approach achieves comparable or better accuracy with significantly fewer iterations of generative training than conventional CD-based training. Further investigation is needed to determine whether similar improvements can be achieved for other data sets, and to what extent these improvements can be attributed to quantum effects.
1 Introduction
Conventional DBN training can be slowed by Gibbs-sampling mixing, motivating quantum sampling for RBM expectations. The proposed approach addresses hardware constraints and reports comparable or better accuracy with fewer generative-training iterations, while leaving scalability and quantum-specific causes unresolved.
- Generative DBN training typically precedes discriminative fine-tuning, but slow Gibbs-sampling mixing can make it time-consuming.
- Quantum annealing is proposed as an alternative sampler because superposition and tunneling may explore some energy landscapes more efficiently.
- D-Wave RBM performance can be impaired by limited qubit connectivity and, to a lesser extent, parameter-setting noise.
- Prior D-Wave work found fewer generative-training iterations were needed to reach the same Kullback-Leibler divergence as classical training.
- The proposed method trains RBMs with up to 32 visible and 32 hidden nodes on a 512-qubit machine with 8 faulty qubits, achieving comparable or better accuracy with significantly fewer generative iterations.
- Further work is needed to test other data sets, separate quantum effects from other factors, and characterize scaling as RBM size increases.
- Quantum sampling may also provide an alternative for inference and sampling, where exact inference is often intractable and Markov-chain methods can mix slowly.
2 Conventional Approach for Training Deep Belief Networks
A DBN is trained by stacking RBMs and maximizing data log-likelihood, but exact model expectations become intractable as networks grow. Contrastive Divergence approximates the gradient through one-step reconstruction, yet convergence can be slow because of Gibbs-sampling noise and mixing.
- A DBN is constructed by stacking RBMs, each with stochastic binary visible and hidden layers connected bipartitely without intra-layer connections.
- Generative training seeks weights and biases that maximize the log-likelihood of observed training data.
- The data-clamped expectation is efficiently computed, whereas the model expectation becomes intractable as visible and hidden node counts increase.
- Contrastive Divergence starts from a training vector, samples hidden units, reconstructs visible units, and samples hidden units again.
- CD updates use a one-step reconstruction approximation rather than directly computing the intractable model expectation.
- CD updates do not follow the exact gradient, converge to it only as repeated sampling steps increase, and may require many iterations because Gibbs sampling is noisy and mixes slowly.
3 Quantum Sampling-Based Approach to Training Deep Neural Networks
The proposed training approach uses quantum annealing to generate RBM samples and estimate model expectations. It describes how this quantum sampler is incorporated into an overall DNN training procedure.
- The approach uses a quantum annealing machine to implement an RBM and generate samples for estimating model expectations.
- The section presents the overall training approach and explains how quantum annealing is used within it.
3.1 Overall training approach
Generative training proceeds one RBM at a time in a hybrid classical/quantum architecture, with quantum samples estimating model expectations for gradient updates before optional discriminative training.
- Generative Training (aka Pre-Training): The outer training loop runs classically, while a quantum annealing machine generates samples to estimate model expectations in the gradient updates.This replaces conventional sampling for the intractable model-expectation terms.
- Generative Training (aka Pre-Training): Each RBM is initialized with random weights and biases and trained sequentially within the DBN.Updates use the stated gradient formulas for weights and biases.
- Discriminative Training: When truth labels are available, the generative phase is called pre-training before discriminative training.Discriminative training is performed classically and can initialize the final layer using least-squares regression.
- Overall Training Approach: The overall approach combines generative and discriminative training stages.Figure 2 depicts the complete training approach.
3.2 Quantum sampling approach
The approach uses quantum annealing to sample RBM model expectations for generative DBN training, while addressing hardware connectivity, faulty qubits, parameter noise, and calibration issues. It embeds RBMs onto D-Wave hardware, averages samples across gauges, and selects the effective inverse-temperature scale according to RBM size.
- Quantum annealing and Boltzmann sampling: Quantum annealing samples excited states that are conjectured to approximately follow a Boltzmann distribution, enabling estimates of RBM model expectations.The sample average from repeated anneals estimates expectations such as ⟨v_i h_j⟩_model, ⟨v_i⟩_model, and ⟨h_j⟩_model.
- Mapping RBMs onto the quantum annealing hardware graph: The method addresses sparse hardware connectivity by mapping each RBM visible and hidden node to a chain of physical qubits.Visible nodes map to vertical-qubit chains and hidden nodes to horizontal-qubit chains, with ferromagnetic couplings enforcing agreement within chains.
- Mapping RBMs onto the quantum annealing hardware graph: A 512-qubit Vesuvius chip can represent a full 32x32 bipartite graph ideally, while eight faulty qubits prevent 32 of 1024 visible-to-hidden connections from being implemented on the real chip.The approach assumes the unavailable connection weights are zero.
- Mapping RBMs onto the quantum annealing hardware graph: Majority voting with a threshold below 1 can reduce sampling noise by retaining more annealing samples, and was slightly more accurate than strict chain enforcement in practice.A threshold r accepts configurations when more than fraction r of chain qubits agree, using the majority value for the chain.
- Correcting for parameter noise: Gauge transformations partially mitigate intrinsic control errors by reversing coupling signs, and the method averages samples over four gauges.This targets leakage effects associated with chained qubits and parameter-setting noise.
- Selecting a value for β_eff: The optimal β_eff decreases as RBM size increases: 4.5 for 5x5 and 8x8, 3 for 12x12 through 16x16, and 2 for 32x32.Even at optimal settings, quantum model-expectation estimates retain error, but the paper compares this with noise from Gibbs sampling in Contrastive Divergence.
4 Experimental Results
The experiments compare classical and quantum sampling-based training on coarse-grained MNIST using matched DBN architectures and varying generative and discriminative training iterations. Quantum training reaches higher accuracy with substantially fewer iterations in the reported comparisons.
- Dataset: The coarse-grained MNIST task is more difficult than original MNIST because image ambiguity increases after downsampling.The authors caution that state-of-the-art accuracy above 99.7% on real MNIST should not be expected for CG-MNIST.
- Experimental setup: 32/32/32/10 DBNs were trained on ten 6,000-image MNIST subsets using coarse-grained 32-super-pixel inputs.The coarse-grained data were produced by averaging 4x4 pixel blocks after cropping the original images and discarding four corners.
- Experimental setup: Training-set and test-set accuracy were averaged over 10 trials while pre-training iterations ranged from 1 to 50.Figures 7–10 compare accuracy after 100, 200, 400, and 800 backpropagation iterations; error bars represent ±1 standard deviation.
- Results: Quantum sampling-based training appears to require significantly fewer pre-training and backpropagation iterations to reach a given accuracy than classical training.Both approaches improve as the numbers of pre-training and backpropagation iterations increase.
- Results: 20 pre-training and 100 backpropagation iterations produced more accurate quantum networks than classical networks trained with 50 pre-training and 800 backpropagation iterations.The comparison is reported between the 100- and 800-backpropagation-iteration results.
5 Discussion
The discussion reports that quantum sampling-based training can reduce both generative and discriminative training needed for comparable post-training accuracy. The authors emphasize that the empirical advantage does not yet establish quantum speedup and that scaling remains unresolved.
- Performance: Quantum sampling-based training could significantly reduce generative and discriminative iterations needed to achieve the same post-training accuracy as classical training.This conclusion is based on the reported empirical results.
- Interpretation: The reduced generative-training iterations support, but do not establish, the hypothesis that quantum sampling mixes faster than classical Gibbs sampling.The authors state that further theoretical and experimental work is needed to attribute the behavior to quantum effects.
- Robustness: The method remained accurate despite processor imperfections, including 32 missing connections out of 1,024 possible connections in the mapped 32x32 RBM.The authors report that quantum training worked as well as, and in fact better than, classical training in this setting.
- Scaling: A 64x64 RBM could be mapped onto a square 2048-qubit chip under the current 8-qubit unit-cell architecture.The authors expect larger chips and greater connectivity to increase trainable RBM-layer sizes.
6 Conclusions
The paper presents quantum-annealing sampling as an alternative component for training deep neural networks and reports comparable post-training accuracy with fewer training iterations on coarse-grained MNIST. The authors identify broader-data, quantum-attribution, and scaling questions for future work.
- Conclusion: The proposed approach maps RBMs onto quantum hardware while addressing limited qubit connectivity and faulty qubits.It uses quantum-annealing samples during DNN training and was tested on coarse-grained MNIST.
- Conclusion: Comparable post-training accuracy to classical Contrastive Divergence training was achieved with fewer generative and discriminative training iterations.The reported conclusion is limited to the tested coarse-grained MNIST setting.
- Limitations: Further investigation is needed across other data sets, for quantum-effect attribution, and for performance scaling as RBM size increases.These boundaries are explicitly identified by the authors as unresolved.
- Broader implications: Quantum sampling could potentially provide an alternative for sampling and inference where exact inference is intractable or conventional samplers mix slowly.The authors present this as a possible source of applications beyond deep learning.