Bayesian Learning Rule (BLR)

• Bayesian Learning Rule is a unified framework that casts learning as approximate Bayesian inference, using candidate distributions and natural gradient updates.
• It leverages mirror descent and information geometry to recover classical algorithms and extend to high-dimensional, constraint-satisfying optimization.
• The framework supports diverse applications including deep learning, control, and rule-based inference, promoting robust and interpretable decision-making.

The Bayesian Learning Rule (BLR) is a unifying algorithmic principle that formulates learning as approximate Bayesian inference with candidate distributions optimized via information-geometric descent. It provides a precise variational structure behind classical and modern optimization algorithms, extends Bayesian reasoning to high-dimensional and deep models, enables principled handling of constraints (including on positive-definite matrices), and supports various forms of rule-based, distributed, or continual learning. This article synthesizes the mathematical foundations, algorithmic consequences, and key applications of the BLR, referencing core research developments.

1. Foundational Formulation and Unification

The BLR operationalizes learning as variational minimization of a composite objective: $$L(q) = \mathbb{E}_{q}[f(\theta)] + D_{\mathrm{KL}}(q(\theta) \,\|\, p(\theta))$$ where:

• $q(\theta)$ is an approximate posterior from a candidate family (e.g., exponential family, mixtures)
• $f(\theta)$ is a task-specific loss, often the negative log-likelihood
• $D_{\mathrm{KL}}$ is the Kullback-Leibler divergence to the prior $p(\theta)$.

The BLR updates the candidate distribution parameters via a natural gradient: $$\lambda_{t+1} = \lambda_t - \rho_t \cdot F(\lambda_t)^{-1} \nabla_\lambda L(q)$$ where $\lambda$ is the natural parameter, $F$ the Fisher information matrix, and $\rho_t$ the learning rate (Khan et al., 2021). This mirror-descent update accounts for the statistical manifold geometry, contrasting with standard Euclidean updates.

The BLR framework is flexible: different choices of $q(\theta)$ and approximations to $\mathbb{E}_q[f(\theta)]$ yield a range of algorithms:

• Delta function: stochastic gradient descent (SGD)
• Gaussian with adaptive precision: Newton's method, RMSprop, Adam
• Mixtures/Multimodal: robust optimization or uncertainty-aware inference

Notably, the BLR recovers classical algorithms (ridge regression, Kalman filter) as well as diverse deep learning optimizers, and even forms the basis for Dropout and posterior averaging techniques.

2. Natural Gradient, Mirror Descent, and Information Geometry

By structuring the update as a natural gradient, the BLR ensures that steps are compatible with the underlying probabilistic model geometry—particularly vital for distributions parameterized on non-Euclidean manifolds (e.g., covariance matrices, probability simplices).

The duality between natural and expectation parameters, and their relation via the log-partition function in exponential families, enables mirror descent updates with respect to Bregman divergences induced by the log-normalizer. For exponential families,

$$q(\theta) = h(\theta) \exp\big(\lambda^T T(\theta) - A(\lambda)\big)$$

so that Bregman divergence $D_A(\eta_2 \| \eta_1)$ directly characterizes the KL divergence between distributions parameterized by $\eta_1, \eta_2$.

This principle extends to the dynamic mirror descent setting in control (Cho et al., 2022), where BLR and mirror descent (with appropriate Bregman divergence) become algorithmically equivalent, clarifying the linkage between Bayesian and online learning formulations.

3. Handling Constraints and Riemannian Extensions

When candidate distribution parameters are subject to constraints (such as positive-definiteness in covariance matrices), the naïve BLR update may impair feasibility. This is addressed by recasting the step as an inexact Riemannian gradient descent along geodesics on the parameter manifold, incorporating a second-order correction with Christoffel symbols: $$\eta^{c_i} \leftarrow \eta^{c_i} - t\, \tilde{\nabla}_{c_i} - \frac{t^2}{2} \sum_{a_i,b_i} \Gamma_{\,a_i b_i}^{c_i} \tilde{\nabla}_{a_i} \tilde{\nabla}_{b_i}$$ for block-coordinate natural parameterizations (BCN) (Lin et al., 2020). This update intrinsically preserves the feasible set, eliminates the need for line-search or unconstrained reparameterizations, and enables efficient, stable optimization in high dimensions for Gaussian, gamma, and mixture models.

The Lie-Group BLR further generalizes the approach, parameterizing candidate distributions via group actions and performing updates through exponential maps on the group, ensuring the candidate always remains within the desired manifold (Kıral et al., 2023). In neural networks, this enables the design of biologically plausible algorithms with built-in invariances and constraints, such as maintaining the sign of weights according to Dale’s law.

4. Stochasticity, Uncertainty, and Bayesian Discrete Optimization

The BLR provides a rigorous framework for learning not just point estimates but full probability distributions (mean-field or richer). In binary neural networks, for instance, the BLR trains a Bernoulli distribution over weights by:

• Defining the variational posterior as $q(w) = \prod_j p_j^{(1+w_j)/2}(1-p_j)^{(1-w_j)/2}$
• Executing natural gradient-based updates on the (unconstrained) natural parameters $\lambda_j$
• Employing reparameterization tricks (e.g., Gumbel–softmax relaxations) for differentiability and gradient estimation (Meng et al., 2020)
• Providing uncertainty estimates that are critical for continual learning (e.g., via Bayesian regularization across task sequences)

For models with discrete or binary synapses (e.g., in RBMs), BLR prevents the need for heuristic clipping—since natural parameters span $\mathbb{R}$—achieving both robustness and principled optimization (Meng, 2020).

5. Rule-Based Learning, Explainability, and Domain Priors

The BLR extends to rule-based and interpretable learning settings by integrating logical rules or expert knowledge as structured priors or penalties:

• In “falling rule lists”, monotonic risk constraints are enforced by parameterizing risk scores as products and logarithms of constrained factors (Wang et al., 2014).
• In fuzzy or rule-based Bayesian learning, binary rule indicators or grammar-derived rule bases are incorporated as latent variables or penalty terms in the Bayesian posterior; MCMC or genetic programming is used for inference and rule extraction, enabling both uncertainty quantification and interpretable model structures (Pan et al., 2016, Botsas et al., 2022).
• Healthcare stratification, risk grouping, and knowledge-guided regression benefit from these approaches by translating domain-specific rules into effective probabilistic models.

6. Distributed, Social, and Dynamic Bayesian Learning

BLR-inspired principles underpin models of networked and sequential learning:

• The “Bayesian without Recall” paradigm models decentralized learning with agents sharing only partial past information, leading to log-linear (memoryless) Bayesian updates and tractable convergence analysis in networks (Rahimian et al., 2016).
• In dynamic inference, the optimal Bayesian learning rule is formalized for sequential estimation where current estimates influence future observations; dynamic programming on the augmented (belief, observation) state provides the optimal policy for updating beliefs and actions (Xu et al., 2022).
• Information-theoretic generalizations use BLR to ground performance bounds (e.g., Bayesian regret) via KL and Wasserstein distances, linking the cost of learning vs. knowing with the divergence between belief-induced and true outcome distributions (Gouverneur et al., 2022).

7. Extensions to Model-Based and Likelihood-Free Learning

Recent developments advance the use of BLR in high-dimensional, model-based, or likelihood-free settings:

• In deep model-based RL, parameter uncertainty over neural network simulators is captured by a generalized Bayesian (scoring-rule or prequential) posterior, implemented via sequential Monte Carlo samplers with gradient-based Markov kernels for scalability (Roy et al., 16 Dec 2024).
• The use of alternative scoring rules (e.g., energy scores) in the absence of tractable likelihoods enables BLR-inspired posterior inference under loose regularity. The framework supports improved exploration and policy learning, such as expected Thompson sampling (ETS), which maximizes performance over the averaged posterior rather than a single sample.
• For architecture search (BaLeNAS), the BLR is employed to learn a distribution over architectures, with natural-gradient variational inference ensuring both exploration and implicit regularization of Hessian-based stability (Zhang et al., 2021).

8. Practical Implications and Algorithmic Design

The BLR prescribes a principled methodology for algorithm construction:

• By tuning the candidate posterior family and the degree of approximation in expectation computations, one can generate or generalize a wide spectrum of algorithms (as shown in (Khan et al., 2021) and related works).
• Incorporating geometric information yields stable, constraint-satisfying steps (vital for Bayesian neural network training, adaptive optimizers, and constrained variational inference).
• In reinforcement learning and control, BLR provides a lens for embedding sequential exploration-exploitation tradeoffs, integrating prior knowledge, and devising tractable approximations to Bayesian RL (e.g., via MC sampling or scoring-rule posteriors) (Wang et al., 2012, Titsias et al., 2018, Roy et al., 16 Dec 2024).

9. Theoretical Significance and Future Directions

The BLR underpins rigorous asymptotic results (e.g., Bernstein–von Mises theorems for generalized posteriors), supports performance guarantees, and enables global search beyond greedy or point-estimate-based procedures. Future research is anticipated in extending posterior classes (beyond exponential families), integrating richer temporal and structural regularization, and exploring dynamic uncertainty quantification in high-dimensional, nonstationary, and adversarial environments.

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In summary, the Bayesian Learning Rule establishes a comprehensive and mathematically coherent framework for learning—spanning optimization, inference, control, and rule-based AI—by casting the learning process itself as sequential, geometrically informed, and distributional Bayesian updating. It connects the design of practical algorithms to information geometry and variational Bayesian principles, supporting robust, interpretable, and scalable decision-making across a diverse array of settings.