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Abstract We describe a stochastic, dynamical system capable of inference and learning in a probabilistic latent variable model. The most challenging problem in such models—sampling the posterior distribution over latent variables—is proposed to be solved by harnessing natural sources of stochasticity inherent in electronic and neural systems. We demonstrate this idea for a sparse coding model by deriving a continuous-time equation for inferring its latent variables via Langevin dynamics. The model parameters are learned by simultaneously evolving according to another continuous-time equation, thus bypassing the need for digital accumulators or a global clock. Moreover, we show that Langevin dynamics lead to an efficient procedure for sampling from the posterior distribution in the L0 sparse regime, where latent variables are encouraged to be set to zero as opposed to having a small L1 norm. This allows the model to properly incorporate the notion of sparsity rather than having to resort to a relaxed version of sparsity to make optimization tractable. Simulations of the proposed dynamical system on both synthetic and natural image data sets demonstrate that the model is capable of probabilistically correct inference, enabling learning of the dictionary as well as parameters of the prior.
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This content will become publicly available on November 1, 2026
The Price Equation Reveals a Universal Force–Metric–Bias Law of Algorithmic Learning and Natural Selection
Diverse learning algorithms, optimization methods, and natural selection share a common mathematical structure despite their apparent differences. Here, I show that a simple notational partitioning of change by the Price equation reveals a universal force–metric–bias (FMB) law: Δθ=Mf+b+ξ. The force f drives improvement in parameters, Δθ, in proportion to the slope of performance with respect to the parameters. The metric M rescales movement by inverse curvature. The bias b adds momentum or changes in the frame of reference. The noise ξ enables exploration. This framework unifies natural selection, Bayesian updating, Newton’s method, stochastic gradient descent, stochastic Langevin dynamics, Adam optimization, and most other algorithms as special cases of the same underlying process. The Price equation also reveals why Fisher information, Kullback–Leibler divergence, and d’Alembert’s principle arise naturally in learning dynamics. By exposing this common structure, the FMB law provides a principled foundation for understanding, comparing, and designing learning algorithms across disciplines.
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- Award ID(s):
- 2325755
- PAR ID:
- 10664444
- Publisher / Repository:
- MDPI
- Date Published:
- Journal Name:
- Entropy
- Volume:
- 27
- Issue:
- 11
- ISSN:
- 1099-4300
- Page Range / eLocation ID:
- 1129
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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