arXiv subjects

Evgeny Andriyash

Publications and source records attributed to Evgeny Andriyash.

16 recordsLinked to original sources

arXiv2020

Undirected Graphical Models as Approximate Posteriors

The representation of the approximate posterior is a critical aspect of effective variational autoencoders (VAEs). Poor choices for the approximate posterior have a detrimental impact on the generative performance of VAEs due to the mismatch with the true posterior. We extend the class of posterior models that may be learned by using undirected graphical models. We develop an efficient method to train undirected approximate posteriors by showing that the gradient of the training objective with respect to the parameters of the undirected posterior can be computed by backpropagation through Markov chain Monte Carlo updates. We apply these gradient estimators for training discrete VAEs with Boltzmann machines as approximate posteriors and demonstrate that undirected models outperform previous results obtained using directed graphical models. Our implementation is available at https://github.com/QuadrantAI/dvaess .

stat.ML↗

arXiv2019

A Path Towards Quantum Advantage in Training Deep Generative Models with Quantum Annealers

The development of quantum-classical hybrid (QCH) algorithms is critical to achieve state-of-the-art computational models. A QCH variational autoencoder (QVAE) was introduced in Ref. [1] by some of the authors of this paper. QVAE consists of a classical auto-encoding structure realized by traditional deep neural networks to perform inference to, and generation from, a discrete latent space. The latent generative process is formalized as thermal sampling from either a quantum or classical Boltzmann machine (QBM or BM). This setup allows quantum-assisted training of deep generative models by physically simulating the generative process with quantum annealers. In this paper, we have successfully employed D-Wave quantum annealers as Boltzmann samplers to perform quantum-assisted, end-to-end training of QVAE. The hybrid structure of QVAE allows us to deploy current-generation quantum annealers in QCH generative models to achieve competitive performance on datasets such as MNIST. The results presented in this paper suggest that commercially available quantum annealers can be deployed, in conjunction with well-crafted classical deep neutral networks, to achieve competitive results in unsupervised and semisupervised tasks on large-scale datasets. We also provide evidence that our setup is able to exploit large latent-space (Q)BMs, which develop slowly mixing modes. This expressive latent space results in slow and inefficient classical sampling, and paves the way to achieve quantum advantage with quantum annealing in realistic sampling applications.

quant-ph↗

arXiv2019

PixelVAE++: Improved PixelVAE with Discrete Prior

Constructing powerful generative models for natural images is a challenging task. PixelCNN models capture details and local information in images very well but have limited receptive field. Variational autoencoders with a factorial decoder can capture global information easily, but they often fail to reconstruct details faithfully. PixelVAE combines the best features of the two models and constructs a generative model that is able to learn local and global structures. Here we introduce PixelVAE++, a VAE with three types of latent variables and a PixelCNN++ for the decoder. We introduce a novel architecture that reuses a part of the decoder as an encoder. We achieve the state of the art performance on binary data sets such as MNIST and Omniglot and achieve the state of the art performance on CIFAR-10 among latent variable models while keeping the latent variables informative.

cs.CV↗

arXiv2019

Improved Gradient-Based Optimization Over Discrete Distributions

In many applications we seek to maximize an expectation with respect to a distribution over discrete variables. Estimating gradients of such objectives with respect to the distribution parameters is a challenging problem. We analyze existing solutions including finite-difference (FD) estimators and continuous relaxation (CR) estimators in terms of bias and variance. We show that the commonly used Gumbel-Softmax estimator is biased and propose a simple method to reduce it. We also derive a simpler piece-wise linear continuous relaxation that also possesses reduced bias. We demonstrate empirically that reduced bias leads to a better performance in variational inference and on binary optimization tasks.

stat.ML↗

arXiv2019

Quantum Variational Autoencoder

Variational autoencoders (VAEs) are powerful generative models with the salient ability to perform inference. Here, we introduce a quantum variational autoencoder (QVAE): a VAE whose latent generative process is implemented as a quantum Boltzmann machine (QBM). We show that our model can be trained end-to-end by maximizing a well-defined loss-function: a 'quantum' lower-bound to a variational approximation of the log-likelihood. We use quantum Monte Carlo (QMC) simulations to train and evaluate the performance of QVAEs. To achieve the best performance, we first create a VAE platform with discrete latent space generated by a restricted Boltzmann machine (RBM). Our model achieves state-of-the-art performance on the MNIST dataset when compared against similar approaches that only involve discrete variables in the generative process. We consider QVAEs with a smaller number of latent units to be able to perform QMC simulations, which are computationally expensive. We show that QVAEs can be trained effectively in regimes where quantum effects are relevant despite training via the quantum bound. Our findings open the way to the use of quantum computers to train QVAEs to achieve competitive performance for generative models. Placing a QBM in the latent space of a VAE leverages the full potential of current and next-generation quantum computers as sampling devices.

quant-ph↗

arXiv2018

DVAE#: Discrete Variational Autoencoders with Relaxed Boltzmann Priors

Boltzmann machines are powerful distributions that have been shown to be an effective prior over binary latent variables in variational autoencoders (VAEs). However, previous methods for training discrete VAEs have used the evidence lower bound and not the tighter importance-weighted bound. We propose two approaches for relaxing Boltzmann machines to continuous distributions that permit training with importance-weighted bounds. These relaxations are based on generalized overlapping transformations and the Gaussian integral trick. Experiments on the MNIST and OMNIGLOT datasets show that these relaxations outperform previous discrete VAEs with Boltzmann priors. An implementation which reproduces these results is available at https://github.com/QuadrantAI/dvae .

stat.ML↗

arXiv2018

DVAE++: Discrete Variational Autoencoders with Overlapping Transformations

Training of discrete latent variable models remains challenging because passing gradient information through discrete units is difficult. We propose a new class of smoothing transformations based on a mixture of two overlapping distributions, and show that the proposed transformation can be used for training binary latent models with either directed or undirected priors. We derive a new variational bound to efficiently train with Boltzmann machine priors. Using this bound, we develop DVAE++, a generative model with a global discrete prior and a hierarchy of convolutional continuous variables. Experiments on several benchmarks show that overlapping transformations outperform other recent continuous relaxations of discrete latent variables including Gumbel-Softmax (Maddison et al., 2016; Jang et al., 2016), and discrete variational autoencoders (Rolfe 2016).

cs.LG↗

arXiv2018

Observation of topological phenomena in a programmable lattice of 1,800 qubits

The celebrated work of Berezinskii, Kosterlitz and Thouless in the 1970s revealed exotic phases of matter governed by topological properties of low-dimensional materials such as thin films of superfluids and superconductors. Key to this phenomenon is the appearance and interaction of vortices and antivortices in an angular degree of freedom---typified by the classical XY model---due to thermal fluctuations. In the 2D Ising model this angular degree of freedom is absent in the classical case, but with the addition of a transverse field it can emerge from the interplay between frustration and quantum fluctuations. Consequently a Kosterlitz-Thouless (KT) phase transition has been predicted in the quantum system by theory and simulation. Here we demonstrate a large-scale quantum simulation of this phenomenon in a network of 1,800 in situ programmable superconducting flux qubits arranged in a fully-frustrated square-octagonal lattice. Essential to the critical behavior, we observe the emergence of a complex order parameter with continuous rotational symmetry, and the onset of quasi-long-range order as the system approaches a critical temperature. We use a simple but previously undemonstrated approach to statistical estimation with an annealing-based quantum processor, performing Monte Carlo sampling in a chain of reverse quantum annealing protocols. Observations are consistent with classical simulations across a range of Hamiltonian parameters. We anticipate that our approach of using a quantum processor as a programmable magnetic lattice will find widespread use in the simulation and development of exotic materials.

quant-ph↗

arXiv2017

Global warming: Temperature estimation in annealers

Sampling from a Boltzmann distribution is NP-hard and so requires heuristic approaches. Quantum annealing is one promising candidate. The failure of annealing dynamics to equilibrate on practical time scales is a well understood limitation, but does not always prevent a heuristically useful distribution from being generated. In this paper we evaluate several methods for determining a useful operational temperature range for annealers. We show that, even where distributions deviate from the Boltzmann distribution due to ergodicity breaking, these estimates can be useful. We introduce the concepts of local and global temperatures that are captured by different estimation methods. We argue that for practical application it often makes sense to analyze annealers that are subject to post-processing in order to isolate the macroscopic distribution deviations that are a practical barrier to their application.

quant-ph↗

arXiv2017

Can quantum Monte Carlo simulate quantum annealing?

Recent theoretical and experimental studies have suggested that quantum Monte Carlo (QMC) simulation can behave similarly to quantum annealing (QA). The theoretical analysis was based on calculating transition rates between local minima, in the large spin limit using WentzelKramers-Brillouin (WKB) approximation, for highly symmetric systems of ferromagnetically coupled qubits. The rate of transition was observed to scale the same in QMC and incoherent quantum tunneling, implying that there might be no quantum advantage of QA compared to QMC other than a prefactor. Quantum annealing is believed to provide quantum advantage through large scale superposition and entanglement and not just incoherent tunneling. Even for incoherent tunneling, the scaling similarity with QMC observed above does not hold in general. Here, we compare incoherent tunneling and QMC escape using perturbation theory, which has much wider validity than WKB approximation. We show that the two do not scale the same way when there are multiple homotopy-inequivalent paths for tunneling. We demonstrate through examples that frustration can generate an exponential number of tunneling paths, which under certain conditions can lead to an exponential advantage for incoherent tunneling over classical QMC escape. We provide analytical and numerical evidence for such an advantage and show that it holds beyond perturbation theory.

quant-ph↗

arXiv2016

Benchmarking Quantum Hardware for Training of Fully Visible Boltzmann Machines

Quantum annealing (QA) is a hardware-based heuristic optimization and sampling method applicable to discrete undirected graphical models. While similar to simulated annealing, QA relies on quantum, rather than thermal, effects to explore complex search spaces. For many classes of problems, QA is known to offer computational advantages over simulated annealing. Here we report on the ability of recent QA hardware to accelerate training of fully visible Boltzmann machines. We characterize the sampling distribution of QA hardware, and show that in many cases, the quantum distributions differ significantly from classical Boltzmann distributions. In spite of this difference, training (which seeks to match data and model statistics) using standard classical gradient updates is still effective. We investigate the use of QA for seeding Markov chains as an alternative to contrastive divergence (CD) and persistent contrastive divergence (PCD). Using $k=50$ Gibbs steps, we show that for problems with high-energy barriers between modes, QA-based seeds can improve upon chains with CD and PCD initializations. For these hard problems, QA gradient estimates are more accurate, and allow for faster learning. Furthermore, and interestingly, even the case of raw QA samples (that is, $k=0$) achieved similar improvements. We argue that this relates to the fact that we are training a quantum rather than classical Boltzmann distribution in this case. The learned parameters give rise to hardware QA distributions closely approximating classical Boltzmann distributions that are hard to train with CD/PCD.

quant-ph↗

arXiv2016

Degeneracy, degree, and heavy tails in quantum annealing

Both simulated quantum annealing and physical quantum annealing have shown the emergence of "heavy tails" in their performance as optimizers: The total time needed to solve a set of random input instances is dominated by a small number of very hard instances. Classical simulated annealing, in contrast, does not show such heavy tails. Here we explore the origin of these heavy tails, which appear for inputs with high local degeneracy---large isoenergetic clusters of states in Hamming space. This category includes the low-precision Chimera-structured problems studied in recent benchmarking work comparing the D-Wave Two quantum annealing processor with simulated annealing. On similar inputs designed to suppress local degeneracy, performance of a quantum annealing processor on hard instances improves by orders of magnitude at the 512-qubit scale, while classical performance remains relatively unchanged. Simulations indicate that perturbative crossings are the primary factor contributing to these heavy tails, while sensitivity to Hamiltonian misspecification error plays a less significant role in this particular setting.

quant-ph↗

arXiv2016

Quantum Boltzmann Machine

Inspired by the success of Boltzmann Machines based on classical Boltzmann distribution, we propose a new machine learning approach based on quantum Boltzmann distribution of a transverse-field Ising Hamiltonian. Due to the non-commutative nature of quantum mechanics, the training process of the Quantum Boltzmann Machine (QBM) can become nontrivial. We circumvent the problem by introducing bounds on the quantum probabilities. This allows us to train the QBM efficiently by sampling. We show examples of QBM training with and without the bound, using exact diagonalization, and compare the results with classical Boltzmann training. We also discuss the possibility of using quantum annealing processors like D-Wave for QBM training and application.

quant-ph↗

arXiv2010

Bound state transformation walls

In four dimensional N=2 supergravity theories, BPS bound states near marginal stability are described by configurations of widely separated constituents with nearly parallel central charges. When the vacuum moduli can be dialed adiabatically until the central charges become anti -parallel, a paradox arises. We show that this paradox is always resolved by the existence of "bound state transformation walls" across which the nature of the bound state changes, although the index does not jump. We find that there are two distinct phenomena that can take place on these walls, which we call recombination and conjugation. The latter is associated to the presence of singularities at finite distance in moduli space. Consistency of conjugation and wall-crossing rules near these singularities leads to new constraints on the BPS spectrum. Singular loci supporting massless vector bosons are particularly subtle in this respect. We argue that the spectrum at such loci necessarily contains massless magnetic monopoles, and that bound states around them transform by intricate hybrids of conjugation and recombination.

hep-th↗

arXiv2010

Wall-crossing from supersymmetric galaxies

We give an elementary physical derivation of the Kontsevich-Soibelman wall crossing formula, valid for any theory with a 4d N=2 supergravity description. Our argument leads to a slight generalization of the formula, which relates monodromy to the BPS spectrum.

hep-th↗

arXiv2008

Ample D4-D2-D0 Decay

We study the wall-crossing behavior of the index of BPS states for D4-D2-D0 brane systems on a Calabi-Yau 3-fold at large radius and point out that not only is the ``BPS index at large radius'' chamber-dependent, but that the changes in the index can be large in the sense that they dominate single-centered black hole entropy. We discuss implications for the weak coupling OSV conjecture. We also analyze the near horizon limit of multicentered solutions, introduced in arXiv:0802.2257, for these particular configurations and comment on a general criterion, conjectured in arXiv:0802.2257, which identifies those multicentered solutions whose near horizon limit corresponds to a geometry with a single asymptotic AdS_3 x S^2 boundary.

hep-th↗