Noise can speed convergence in Markov chains

The Exact Stochastic Process of the Haploid Multi-Allelic Wright-Fisher Mutation Model

Diffusion models are widely applied in population genetics, but their approximate solutions may not accurately capture the exact stochastic process. Nevertheless, this practice was necessary due to computing limitations, particularly for large populations. In this article, we develop the exact Markov chain algebra (MCA) for a discrete haploid multi-allelic Wright-Fisher model (MA-WFM) with a full mutation matrix to address this challenge. A special case of nonzero mutations between multiple alleles have not been captured by previous bi-allelic models. We formulated the mean allele frequencies for asymptotic equilibrium analytically for the tri- and quad-allelic case. We also evaluated the exact time-dependent Markov model numerically, presenting it concisely in terms of diffusion variables. The convergence with increasing population size to a diffusion limit is demonstrated for the population composition distribution. Our model shows that there will never be exact irreversible extinction when there are nonzero mutation rates into each allele and never be an exact irreversible fixation when there are nonzero mutation rates out of each allele. We only present results where there is no complete extinction and no complete fixation. Finally, we present detailed computations for the full Markov process, exposing the behavior near the boundaries for the compositional domains, which are non-singular boundaries according to diffusion theory.

Generalized fluctuation-dissipation theorem for non-Markovian reaction networks

Intracellular biochemical networks often display large fluctuations in the molecule numbers or the concentrations of reactive species, making molecular approaches necessary for system descriptions. For Markovian reaction networks, the fluctuation-dissipation theorem (FDT) has been well established and extensively used in fast evaluation of fluctuations in reactive species. For non-Markovian reaction networks, however, the similar FDT has not been established so far. Here, we present a generalized FDT (gFDT) for a large class of non-Markovian reaction networks where general intrinsic-event waiting-time distributions account for the effect of intrinsic noise and general stochastic reaction delays represent the impact of extrinsic noise from environmental perturbations. The starting point is a generalized chemical master equation (gCME), which describes the probabilistic behavior of an equivalent Markovian reaction network and identifies the structure of the original non-Markovian reaction network in terms of stoichiometries and effective transition rates (extensions of common reaction propensity functions). From this formulation follows directly the solution of the linear noise approximation of the stationary gCME for all the components in the non-Markovian reaction network. While the gFDT can quickly trace noisy sources in non-Markovian reaction networks, example analysis verifies its effectiveness.

Noise can speed backpropagation learning and deep bidirectional pretraining

We show that the backpropagation algorithm is a special case of the generalized Expectation-Maximization (EM) algorithm for iterative maximum likelihood estimation. We then apply the recent result that carefully chosen noise can speed the average convergence of the EM algorithm as it climbs a hill of probability or log-likelihood. Then injecting such noise can speed the average convergence of the backpropagation algorithm for both the training and pretraining of multilayer neural networks. The beneficial noise adds to the hidden and visible neurons and related parameters. The noise also applies to regularized regression networks. This beneficial noise is just that noise that makes the current signal more probable. We show that such noise also tends to improve classification accuracy. The geometry of the noise-benefit region depends on the probability structure of the neurons in a given layer. The noise-benefit region in noise space lies above the noisy-EM (NEM) hyperplane for classification and involves a hypersphere for regression. Simulations demonstrate these noise benefits using MNIST digit classification. The NEM noise benefits substantially exceed those of simply adding blind noise to the neural network. We further prove that the noise speed-up applies to the deep bidirectional pretraining of neural-network bidirectional associative memories (BAMs) or their functionally equivalent restricted Boltzmann machines. We then show that learning with basic contrastive divergence also reduces to generalized EM for an energy-based network probability. The optimal noise adds to the input visible neurons of a BAM in stacked layers of trained BAMs. Global stability of generalized BAMs guarantees rapid convergence in pretraining where neural signals feed back between contiguous layers. Bipolar coding of inputs further improves pretraining performance.

Noise can speed Markov chain Monte Carlo estimation and quantum annealing

Carefully injected noise can speed the average convergence of Markov chain Monte Carlo (MCMC) estimates and simulated annealing optimization. This includes quantum annealing and the MCMC special case of the Metropolis-Hastings algorithm. MCMC seeks the solution to a computational problem as the equilibrium probability density of a reversible Markov chain. The algorithm must cycle through a long burn-in phase until it reaches equilibrium because the Markov samples are statistically correlated. The special injected noise reduces this burn-in period in MCMC. A related theorem shows that it reduces the cooling time in simulated annealing. Simulations showed that optimal noise gave a 76% speed-up in finding the global minimum in the Schwefel optimization benchmark. The noise-boosted simulations found the global minimum in 99.8% of trials compared with only 95.4% of trials in noiseless simulated annealing. Simulations also showed that the noise boost is robust to accelerated cooling schedules and that noise decreased convergence times by more than 32% under aggressive geometric cooling. Molecular dynamics simulations showed that optimal noise gave a 42% speed-up in finding the minimum potential energy configuration of an eight-argon-atom gas system with a Lennard-Jones 12-6 potential. The annealing speed-up also extends to quantum Monte Carlo implementations of quantum annealing. Noise improved ground-state energy estimates in a 1024-spin simulated quantum annealing simulation by 25.6%. The quantum noise flips spins along a Trotter ring. The noisy MCMC algorithm brings each Markov step closer on average to equilibrium if an inequality holds between two expectations. Gaussian or Cauchy jump probabilities reduce the noise-benefit inequality to a simple quadratic inequality. Simulations show that noise-boosted simulated annealing is more likely than noiseless annealing to sample high probability regions of the search space and to accept solutions that increase the search breadth.

Cooperation dynamics in networked geometric Brownian motion

Recent works suggest that pooling and sharing may constitute a fundamental mechanism for the evolution of cooperation in well-mixed fluctuating environments. The rationale is that, by reducing the amplitude of fluctuations, pooling and sharing increases the steady-state growth rate at which individuals self-reproduce. However, in reality interactions are seldom realized in a well-mixed structure, and the underlying topology is in general described by a complex network. Motivated by this observation, we investigate the role of the network structure on the cooperative dynamics in fluctuating environments, by developing a model for networked pooling and sharing of resources undergoing a geometric Brownian motion. The study reveals that, while in general cooperation increases the individual steady state growth rates (i.e., is evolutionary advantageous), the interplay with the network structure may yield large discrepancies in the observed individual resource endowments. We comment possible biological and social implications and discuss relations to econophysics.

Noise-enhanced convolutional neural networks

Injecting carefully chosen noise can speed convergence in the backpropagation training of a convolutional neural network (CNN). The Noisy CNN algorithm speeds training on average because the backpropagation algorithm is a special case of the generalized expectation-maximization (EM) algorithm and because such carefully chosen noise always speeds up the EM algorithm on average. The CNN framework gives a practical way to learn and recognize images because backpropagation scales with training data. It has only linear time complexity in the number of training samples. The Noisy CNN algorithm finds a special separating hyperplane in the network's noise space. The hyperplane arises from the likelihood-based positivity condition that noise-boosts the EM algorithm. The hyperplane cuts through a uniform-noise hypercube or Gaussian ball in the noise space depending on the type of noise used. Noise chosen from above the hyperplane speeds training on average. Noise chosen from below slows it on average. The algorithm can inject noise anywhere in the multilayered network. Adding noise to the output neurons reduced the average per-iteration training-set cross entropy by 39% on a standard MNIST image test set of handwritten digits. It also reduced the average per-iteration training-set classification error by 47%. Adding noise to the hidden layers can also reduce these performance measures. The noise benefit is most pronounced for smaller data sets because the largest EM hill-climbing gains tend to occur in the first few iterations. This noise effect can assist random sampling from large data sets because it allows a smaller random sample to give the same or better performance than a noiseless sample gives.

Noise-benefit forbidden-interval theorems for threshold signal detectors based on cross correlations

We show that the main forbidden interval theorems of stochastic resonance hold for a correlation performance measure. Earlier theorems held only for performance measures based on mutual information or the probability of error detection. Forbidden interval theorems ensure that a threshold signal detector benefits from deliberately added noise if the average noise does not lie in an interval that depends on the threshold value. We first show that this result holds for correlation for all finite-variance noise and for all forms of infinite-variance stable noise. A second forbidden-interval theorem gives necessary and sufficient conditions for a local noise benefit in a bipolar signal system when the noise comes from a location-scale family. A third theorem gives a general condition for a local noise benefit for arbitrary signals with finite second moments and for location-scale noise. This result also extends forbidden intervals to forbidden bands of parameters. A fourth theorem gives necessary and sufficient conditions for a local noise benefit when both the independent signal and noise are normal. A final theorem derives necessary and sufficient conditions for forbidden bands when using arrays of threshold detectors for arbitrary signals and location-scale noise.

3D genome reconstruction from chromosomal contacts

A computational challenge raised by chromosome conformation capture (3C) experiments is to reconstruct spatial distances and three-dimensional genome structures from observed contacts between genomic loci. We propose a two-step algorithm, ShRec3D, and assess its accuracy using both in silico data and human genome-wide 3C (Hi-C) data. This algorithm avoids convergence issues, accommodates sparse and noisy contact maps, and is orders of magnitude faster than existing methods.