Any-Time Inference

• Any-time inference is a family of algorithms that yield progressively refined, accurate predictions as computation continues.
• The methods guarantee interruptibility and monotonic improvement, allowing valid inferences even when computation is halted early.
• They are applied in statistical tests, graphical models, and deep learning to support robust, resource-adaptive decision-making.

Any-time inference refers to a family of algorithms and statistical procedures that provide valid, progressively improving inferences or predictions which can be interrupted at arbitrary points in computation—delivering outputs whose quality increases monotonically with additional computation or data. In distinction to classical fixed-sample methods, which only guarantee inferential validity at predetermined sample sizes or after full algorithmic completion, any-time inference ensures statistical or computational guarantees hold uniformly over time, supporting resource-adaptive or real-time decision-making under dynamically varying constraints.

1. Core Definitions and Principles

A canonical anytime inference algorithm guarantees that, for each interruption or budget level $t$, the output $A(K, Q, t)$ (e.g., an interval estimate or partial solution to a query $Q$ given knowledge base $K$) refines as $t$ increases, with monotonic improvement in accuracy, tightness or confidence. Formally, such an algorithm must deliver:

• Interruptibility: For any $t\geq 0$, a sound (though possibly loose) approximation is available.
• Monotonicity: The solution quality never degrades as computation continues; the error or width of inferential intervals decreases in $t$.
• Convergence: As $t\to\infty$, the output converges to the exact inference, i.e., $\lim_{t\to\infty} A(K, Q, t) = A^*(K,Q)$.
• Uniform validity: For statistical inference, coverage or error control persists for arbitrary stopping times, even under data-dependent or opportunistic monitoring.

Typical metrics include width of confidence intervals, lower/upper error bounds, or quality/accuracy indices over computational budgets or streaming data sequences (Pittarelli, 2013, Lindon et al., 2020, Martin, 2024, Ferreira et al., 2017, Jazbec et al., 2023, Kilian et al., 23 May 2025).

2. Generic Algorithmic Frameworks

Anytime inference algorithms arise in a range of computational and inferential settings, including probabilistic graphical models, valuation algebras, sequential hypothesis testing, and deep learning. Common methodologies include:

• Incremental inference in logic and graphical models: Algorithms such as the Frisch–Haddawy anytime deduction system and its generalizations for valuation algebras (Pittarelli, 2013, Dasgupta et al., 2016) iteratively apply local inference rules, incrementally tightening bounds on query probabilities, and returning interval estimates whose widths decrease with progressing rule applications.
• Sequential updating and confidence sequences: In statistical inference, nonnegative supermartingales (often constructed via likelihood ratios or e-values) and their mixtures provide confidence sequences—intervals that contain the target parameter uniformly over all sample sizes—with Ville's inequality ensuring control of Type I error under optional stopping (Lindon et al., 2020, Martin, 2024, Kilian et al., 23 May 2025, Dixit et al., 2023, Cho et al., 8 Feb 2026).
• Factor-set and partial-solution propagation: In graphical modeling, set-based message-passing algorithms propagate collections of factors or summary statistics to provide lower and upper bounds on marginal MAP or belief calculations, refining these bounds iteratively as more compute is devoted or as factor sets expand (Maua et al., 2012, Ferreira et al., 2017).
• Interruptible and monotonic deep models: Early-exit neural architectures, hierarchical ensembles, or supernets provide per-layer or per-ensemble outputs that can be halted at any depth or ensemble size, outputting predictions whose accuracy is controlled by computation spent, with monotonicity in quality enforced via post-hoc modifications (e.g., product-of-experts) or hierarchical distillation (Jazbec et al., 2023, Ruiz et al., 2020, Chen et al., 2022).

3. Statistical Anytime Validity and Confidence Sequences

The statistical aspect of anytime inference revolves around techniques that allow hypothesis testing, interval estimation, and broader uncertainty quantification to remain valid regardless of when analysis is halted (Lindon et al., 2020, Martin, 2024, Cho et al., 8 Feb 2026, Dixit et al., 2023, Kilian et al., 23 May 2025). Key tools include:

• E-processes and Ville’s inequality: A sequence $\{E_\theta^n\}_{n\ge0}$ is an e-process for parameter $A(K, Q, t)$0 if under $A(K, Q, t)$1, every stopped process $A(K, Q, t)$2 (with $A(K, Q, t)$3 any stopping time) satisfies $A(K, Q, t)$4. Then, $A(K, Q, t)$5 for all $A(K, Q, t)$6. This enables sequential tests and confidence sets (or confidence sequences) that are uniformly valid over all $A(K, Q, t)$7 and arbitrary stopping rules (Martin, 2024, Lindon et al., 2020, Dixit et al., 2023, Kilian et al., 23 May 2025).
• Mixture martingales and regularization: By constructing convex or mixture supermartingales over a parameter space, uniformly valid tests for composite hypotheses or function-valued parameters can be derived, often yielding function-valued confidence sequences for the conditional mean or CATE functions (Cho et al., 8 Feb 2026).
• Regularized e-processes: Incorporation of partial prior information (as credal sets, possibility contours, or knowledge-based regularizers) can reduce expected stopping times and interval widths while maintaining strong anytime guarantees, via generalized Ville inequalities (Martin, 2024, Kilian et al., 23 May 2025).
• Confidence sequence construction: For any-time estimation of parameters, interval sets $A(K, Q, t)$8 are defined such that $A(K, Q, t)$9, and are often derived via inverting mixture martingale bounds (Lindon et al., 2020, Kilian et al., 23 May 2025, Martin, 2024).

4. Anytime Inference in Probabilistic and AI Systems

Resource-bounded inference and decision-making in knowledge bases, probabilistic databases, and AI models have adopted anytime principles:

• Interval-propagation in probabilistic logic: Systems proceed by applying inference rules that shrink the possible interval for a query as more logical connections are explored or as more premises are incorporated (Pittarelli, 2013).
• Belief propagation and set-based searches: Algorithms such as Anytime Exact Belief Propagation and marginal MAP via factor-set elimination iteratively propagate bounds or factor sets across graphical structures. At every refinement, current bounds can be reported, and, when time or computation is halted, the best available solution (with associated provable bounds) is available (Ferreira et al., 2017, Maua et al., 2012).
• Valuation algebras and generic frameworks: Dasgupta & Abramsky’s algebraic extension allows the construction of fully generic anytime algorithms applicable to a variety of domains, including probabilistic potentials, DNF, and distributive lattices, with formal monotonicity, convergence, and complexity guarantees (Dasgupta et al., 2016).

5. Machine Learning and Deep Model Realizations

Anytime inference in modern machine learning and vision is realized via network architectures and training regularizations designed for interruptible operation:

• Early-exit networks and conditional monotonicity: Auxiliary classifiers are attached at intermediate layers, with post-hoc product-of-experts (PoE) assembly ensuring per-input prediction quality is non-decreasing across exits. This method achieves strict conditional monotonicity, crucial for reliability in safety-sensitive or latency-constrained deployment (Jazbec et al., 2023).
• Hierarchical neural ensembles and distillation: Construction of tree-structured neural ensembles, with shared blocks and hierarchical distillation targets, enables dynamic adjustment of inference cost/accuracy by varying the number of evaluated ensemble leaves; ensemble diversity and performance are maintained across all compute budgets (Ruiz et al., 2020).
• Supernet approaches for structured prediction tasks: For image super-resolution, width-slimmable supernets trained over a spectrum of sub-network capacities utilize data-adaptive patch routing based on complexity heuristics (e.g., edge-to-PSNR lookup tables), delivering budget-aware, anytime trade-offs on a per-patch basis, without retraining (Chen et al., 2022).

6. Empirical Performance and Operational Impact

Anytime inference unlocks principled early-stopping, continuous monitoring, and deployment under resource or time limits, with quantifiable accuracy-resource curves:

• Sequential testing and streaming data: In A/B/n experiments, multi-arm conversion analysis, or outlier detection, anytime-valid tests and confidence sequences empower practitioners to monitor experiments continuously and halt confidently upon reaching decision or coverage thresholds, without inflating error rates (Lindon et al., 2020, Banerjee et al., 2023).
• Function inference under nonparametric settings: GAAVI constructs function-valued confidence sequences and tests for the conditional mean function with type I error controlled across all monitoring times, achieving power and sample complexity close to the information-theoretic optimum for corresponding parametric problems (Cho et al., 8 Feb 2026).
• Robustness to optional stopping and prior information: Regularized e-processes and Bayes-assisted prediction-powered approaches capitalize on prior knowledge or auxiliary predictions for narrower intervals and faster inference, yet always retain strong frequentist (time-uniform) coverage (Martin, 2024, Kilian et al., 23 May 2025).

7. Limitations and Ongoing Research

While anytime inference methods are increasingly mature, open challenges remain:

• Calibration and empirical accuracy: Some post-hoc or product-based approaches may require further fine-tuning for probability calibration or suffer underdegeneracy for small label sets (Jazbec et al., 2023).
• Computational trade-offs: Fully generic anytime algorithms may face exponential worst-case complexity unless problem structure (e.g., bounded treewidth, local factorization) is exploited (Maua et al., 2012, Dasgupta et al., 2016).
• Scope of side information: Classical anytime-valid inference typically excludes high-dimensional side information or covariate-adjustment; recent advances with regularized e-processes and streaming lifelong learning are closing this gap (Banerjee et al., 2023, Martin, 2024, Kilian et al., 23 May 2025).
• Generalization and scalability: Continuous or infinite-parameter function-valued confidence sequences require advanced martingale and mixture approaches for computational feasibility at scale (Cho et al., 8 Feb 2026).

Anytime inference thus provides a foundational paradigm for resource-adaptive, robust, and interruptible computation and statistical inference, serving as a unifying principle across multiple disciplines in statistical science, machine learning, and AI.