Inverse-Probability Algebraic Learning Framework

• The inverse-probability algebraic learning framework is an optimization paradigm that infers probability parameters from observed outputs in probabilistic databases and quantum neural networks.
• It employs algebraic corrections via the pseudo-inverse Jacobian and Tikhonov regularization to achieve rapid, covariant, and stable parameter updates.
• Empirical benchmarks demonstrate superior convergence speed and scalability compared to gradient-based methods, making it effective for complex inverse-probability problems.

The inverse-probability algebraic learning framework is an optimization-based paradigm for parameter inference in probabilistic models, where the objective is to recover the underlying probability parameters from observed, potentially labeled outputs. This framework is applied in both tuple-independent probabilistic databases (PDBs) (Dylla et al., 2016) and quantum neural networks (QNNs) (Seo, 23 Jan 2026), formulating parameter learning as an inverse problem: given marginal probability constraints arising from lineage formulas (in PDBs) or Born-rule statistics (in QNNs), the framework seeks the parameter vector that best explains the observed outputs. Unlike procedural gradient-based methods, recent iterations for QNNs implement an algebraic correction via the pseudo-inverse of the local Jacobian, facilitating rapid and covariant parameter updates.

1. Mathematical Formulation

In tuple-independent PDBs, consider a finite set of base tuples $T = \{t_1, ..., t_n\}$ and a probability vector $p = (p_1, ..., p_n) \in [0,1]^n$ where $p_i$ parameterizes the probability of each tuple. Derived tuples arising from queries are annotated by Boolean lineage formulas $\varphi_j$, and their marginal probabilities are expressed as multilinear polynomials: $$P[\varphi_j(p)] = \sum_{V \subseteq Vars(\varphi_j), V \models \varphi_j} \prod_{i \in V} p_i \prod_{i \in Vars(\varphi_j) \setminus V} (1-p_i),$$ with degree at most $|Vars(\varphi_j)|$.

In QNNs, for each input $x_i$, the quantum state is $$\ket{\psi(x_i;\theta)} = U(\theta) U_{\text{enc}}(x_i) \ket{0}$$ and the measured Born-rule probability is $$p_i(\theta) = |\langle 11 | \psi(x_i;\theta)\rangle|^2$$ for model parameters $\theta \in \mathbb{R}^P$. Collecting the outputs gives a prediction vector $p(\theta)$ targeted to match $p_{\text{target}}$.

2. Optimization Problem and Learning Objectives

The general inverse-probability algebraic learning approach is to minimize a loss function over the parameter space: $$L(p) = \sum_{j=1}^m \ell(P[\varphi_j(p)], y_j), \quad p \in [0,1]^n,$$ where $\ell$ is typically mean-squared error $(a-b)^2$ or cross-entropy.

For QNNs, the local linearization yields a least-squares objective for parameter increments $\Delta\theta$ with Tikhonov regularization: $$\Delta\theta = \arg\min_{\Delta\theta} \|\delta p - J\Delta\theta\|_2^2 + \lambda\|\Delta\theta\|_2^2,$$ where $\delta p = p(\theta) - p_{\text{target}}$, $J$ is the Jacobian matrix of partial derivatives, and $\lambda$ regularizes ill-conditioning.

3. Theoretical Properties and Solution Structure

Hardness and Solution Multiplicity

Deciding the existence of $p$ such that $P[\varphi_j(p)] = y_j$ is NP-hard (reduction from 3SAT) (Dylla et al., 2016). Algebraic geometry bounds (Bezout’s theorem) inform the solution count based on the number and complexity of polynomial constraints.

Convexity and Local Minima

The loss functions in both PDBs and QNNs are non-convex except for trivial cases (Dylla et al., 2016). Multiple local minima may exist, necessitating robust optimization techniques.

Covariance and Uniqueness

For QNNs, the pseudo-inverse algebraic update is covariant under smooth reparameterizations of $\theta$. Tikhonov regularization ensures that inversion is unique and stable (Seo, 23 Jan 2026).

4. Algorithmic Solutions

Stochastic Gradient Descent for PDBs

Parameters $p_i$ are maintained via logit transforms to remain in $(0,1)$. SGD is conducted by iterative updates:

• Random label selection or cyclical label processing;
• Gradient calculation based on $$\frac{\partial P[\varphi_j(p)]}{\partial p_i} = P[\varphi_j | t_i = \text{true}] - P[\varphi_j | t_i = \text{false}]$$;
• Adaptive per-parameter learning rates $\eta_i$, increased or decreased based on objective decrease, and stopping criteria based on loss change thresholds.

Extensions include parallelization for non-overlapping tuple sets, regularization/priors via penalty terms, and lineage formula compilation for computational efficiency (Dylla et al., 2016).

Algebraic Jacobian-Based Step for QNNs

The pseudo-inverse step uses: $$\Delta\theta = - (J^\top J + \lambda I)^{-1} J^\top [p(\theta) - p_{\text{target}}],$$ achieving a direct local solution without explicit learning rate tuning. Algorithmically, J is estimated via parameter-shift rules, full-batch correction is applied, and the update is computed in $O(P^3)$ time per iteration. Practical extensions include logit space computations, mini-batch variants, and alternate regularization (Seo, 23 Jan 2026).

5. Empirical Benchmarking and Performance

PDB Applications

Real-world and synthetic datasets yield scalable and accurate performance:

• UW-CSE: ≥40× faster than TheBeast, 600× faster than ProbLog, identical F₁ with sufficient negatives.
• PRAVDA: matches or exceeds ILP+label-propagation in precision/recall; runtimes of seconds.
• YAGO2: scales to millions of tuples, converges in minutes.
• Synthetic: ≥70×/600× faster than TheBeast/ProbLog.
• Per-tuple adaptive SGD outperforms plain GD and L-BFGS.
• MSE objective is 10–100× faster than logical objectives due to marginal recomputation costs (Dylla et al., 2016).

QNN Applications

In teacher-student benchmarks:

• Algebraic method requires 2–3 steps to reach binary-cross-entropy ≈0.1; GD/Adam require ≈200 steps.
• For regression, reaches MSE ≈$10^{-6}$ in 5 steps (GD/Adam: $10^{-2}$ after 150–200 steps).
• Under finite-shot sampling, error scales optimally as $1/S$; Adam deviates at low shot counts.
• Robustness to dephasing noise: algebraic MSE degrades from $10^{-6}$ to $10^{-3}$ as $p_{\text{deph}}$ increases to 0.05; Adam plateaus at $10^{-2}$ and becomes unstable (Seo, 23 Jan 2026).

6. Extensions, Limitations, and Outlook

Potential extensions encompass multi-class outputs (softmax), hybrid mini-batch algebraic steps for scalability, sophisticated regularization (e.g., truncated SVD), and integration with quantum information metrics.

Limitations include:

• For QNNs, the computation of the full Jacobian is costly ($O(NP)$ evaluations per iteration); scaling to large $P,N$ remains hardware-constrained.
• Method relies on the local linearity of the prediction function; highly nonlinear or deep architectures may require backtracking or incremental steps.
• Extreme ill-conditioning of $J$ necessitates large $\lambda$, hampering convergence speed.

A plausible implication is that, as NISQ quantum hardware matures and supports moderate shot and parameter regimes, algebraic inverse-probability learning may become a preferred alternative to tuned gradient descent methods. For probabilistic databases, the framework provides scalable, end-to-end parameter learning, outperforming SRL and specialized constraint solvers while naturally accommodating priors and database cleaning constraints.

7. Contextual Significance and Future Directions

The inverse-probability algebraic learning framework formalizes parameter learning in probabilistic systems as an algebraic inverse problem, positioning it at the intersection of probabilistic reasoning and optimization. In databases, it bridges confidence computations and probabilistic inference; for QNNs, it offers a hyperparameter-free, robust training strategy resilient to device noise and optimization instability. Continued work may focus on integrating the framework with distributed architectures, exploiting sparsity in lineage structures or quantum circuits, and developing hardware-optimized implementations.