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Probably Approximately Correct Maximum A Posteriori Inference

Matthew Shorvon, Frederik Mallmann-Trenn, David S. Watson

TL;DR

This paper tackles the intractability of MAP inference by introducing PAC-MAP, a framework that delivers probably approximately correct MAP solutions with provable guarantees under user-specified budgets. By leveraging probabilistic circuits to provide efficient samplers and likelihood calculations, the authors derive information-theoretic tractability conditions based on min-entropy $H_∞(\bm Q|\bm e)$ and design uniformly optimal algorithms, along with Pareto-optimal, budget-based certificates when exact guarantees are infeasible. They develop both binary and continuous-data strategies, including adaptive variants like smooth-PAC-MAP and warm-start integrations, and demonstrate competitive performance against established approximate MAP solvers on large benchmark datasets. The work integrates PAC guarantees with PC-based inference to produce scalable, certifiable MAP estimates, offering a principled path for robust probabilistic reasoning under computational limits and opening avenues for extensions to MCMC and reinforcement learning settings.

Abstract

Computing the conditional mode of a distribution, better known as the $\mathit{maximum\ a\ posteriori}$ (MAP) assignment, is a fundamental task in probabilistic inference. However, MAP estimation is generally intractable, and remains hard even under many common structural constraints and approximation schemes. We introduce $\mathit{probably\ approximately\ correct}$ (PAC) algorithms for MAP inference that provide provably optimal solutions under variable and fixed computational budgets. We characterize tractability conditions for PAC-MAP using information theoretic measures that can be estimated from finite samples. Our PAC-MAP solvers are efficiently implemented using probabilistic circuits with appropriate architectures. The randomization strategies we develop can be used either as standalone MAP inference techniques or to improve on popular heuristics, fortifying their solutions with rigorous guarantees. Experiments confirm the benefits of our method in a range of benchmarks.

Probably Approximately Correct Maximum A Posteriori Inference

TL;DR

This paper tackles the intractability of MAP inference by introducing PAC-MAP, a framework that delivers probably approximately correct MAP solutions with provable guarantees under user-specified budgets. By leveraging probabilistic circuits to provide efficient samplers and likelihood calculations, the authors derive information-theoretic tractability conditions based on min-entropy and design uniformly optimal algorithms, along with Pareto-optimal, budget-based certificates when exact guarantees are infeasible. They develop both binary and continuous-data strategies, including adaptive variants like smooth-PAC-MAP and warm-start integrations, and demonstrate competitive performance against established approximate MAP solvers on large benchmark datasets. The work integrates PAC guarantees with PC-based inference to produce scalable, certifiable MAP estimates, offering a principled path for robust probabilistic reasoning under computational limits and opening avenues for extensions to MCMC and reinforcement learning settings.

Abstract

Computing the conditional mode of a distribution, better known as the (MAP) assignment, is a fundamental task in probabilistic inference. However, MAP estimation is generally intractable, and remains hard even under many common structural constraints and approximation schemes. We introduce (PAC) algorithms for MAP inference that provide provably optimal solutions under variable and fixed computational budgets. We characterize tractability conditions for PAC-MAP using information theoretic measures that can be estimated from finite samples. Our PAC-MAP solvers are efficiently implemented using probabilistic circuits with appropriate architectures. The randomization strategies we develop can be used either as standalone MAP inference techniques or to improve on popular heuristics, fortifying their solutions with rigorous guarantees. Experiments confirm the benefits of our method in a range of benchmarks.
Paper Structure (34 sections, 6 theorems, 31 equations, 3 figures, 4 tables, 3 algorithms)

This paper contains 34 sections, 6 theorems, 31 equations, 3 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Oracle PAC-MAP
  4. Binary data
  5. Continuous data
  6. Practical Considerations
  7. Circuit Architectures
  8. PAC-MAP Variants
  9. Experiments
  10. Illustration
  11. Benchmark
  12. Warm Starts
  13. Relaxing PAC Guarantees
  14. Discussion
  15. Conclusion
  16. ...and 19 more sections

Key Result

Lemma 1

Fix any $\varepsilon, \delta \in (0,1)$. Define the $\varepsilon$-superlevel set: with corresponding mass $\mu_\varepsilon = \sum_{\bm q \in G_\varepsilon} p(\bm q \mid \bm e)$. Then Alg. 0 PAC-identifies $\bm q^*$ in $\Theta(\mu_\varepsilon^{-1} \log\delta^{-1})$ samples.

Figures (3)

  • Figure 1: Schematic illustration of PAC-MAP. (A) Lower and upper bounds approach the MAP probability $p^*$ as $m$ increases. (B) Maximum failure probability approaches $\delta$ as $m$ increases.
  • Figure 2: Average percentage increase in MAP probability across ten trials per dataset when using AMP outputs as a warm start for smooth-PAC-MAP. Whiskers represent standard errors.
  • Figure 3: Pareto frontiers computed by budget-PAC-MAP on three datasets where $10^6$ samples were insufficient for target guarantees. Estimated MAP probabilities are given at the top of the graph.

Theorems & Definitions (6)

  • Lemma 1: Discovery complexity
  • Theorem 1: Certification complexity
  • Corollary 1.1
  • Theorem 2: Uniform optimality
  • Theorem 3: Pareto optimality
  • Lemma 2: Identifiability