De-randomized Decision Rule

Updated 26 July 2025

De-randomized decision rules are deterministic strategies that replicate the statistical behavior of randomized methods, ensuring almost-sure outcomes in settings like game theory and machine learning.
They utilize methodologies such as game-theoretic probability, algorithmic compression, and modular arithmetic games to enforce event compliance and mirror the performance of stochastic processes.
These rules enhance reproducibility and auditability in decision-making processes, though challenges remain in high-dimensional, non-convex environments where approximation errors can occur.

A de-randomized decision rule is a deterministic procedure constructed to replicate the almost-sure outcomes or statistical properties of a randomized rule, eliminating extrinsic randomization in scenarios such as game-theoretic probability, statistical decision-making, complexity theory, distributionally robust optimization, mechanism design, and certification of robustness in machine learning models. De-randomization transforms an originally random or sampling-based process into an explicit, reproducible function or algorithmic rule, often underpinned by theoretical guarantees that match those of the stochastic counterpart.

1. Conceptual Foundations

The central aim of de-randomized decision rules is to obtain deterministic procedures that yield the same performance—whether in covering almost-sure events, achieving statistical optimality, controlling error rates, or preserving fairness—as their randomized analogues. In game-theoretic probability, de-randomization serves to demonstrate that for any event occurring with probability one under some random strategy, there exists an explicit deterministic strategy that achieves the same outcome (Miyabe et al., 2014). In mechanism design and algorithmic optimization, de-randomization produces procedures that can be externally audited and replicated, replacing traditional external or cryptographic sources of randomness with deterministic selection—sometimes via structural games between agents (Walsh, 2023).

Theoretical underpinnings of de-randomization include:

Construction of deterministic strategies that “simulate” the distributional or probabilistic effects of randomization.
Topological or geometric representation theorems that ensure the existence (and sometimes construction) of such deterministic strategies in highly abstract settings.
Approximation theorems guaranteeing that deterministic or de-randomized compressions of complex rules closely match the output distribution or event compliance of the original random process.

2. Methodologies for De-randomization

Game-Theoretic Probability

The most explicit framework for de-randomization is provided in game-theoretic probability, where the classic three-step procedure is as follows (Miyabe et al., 2014):

Selection of a randomized strategy: Start with a measure-theoretic or stochastic strategy that almost surely achieves the event of interest. For instance, Reality in a coin-tossing game might declare $I_n = 1$ with probability $p_n$ and $I_n = 0$ otherwise.
Forcing strategy construction: Construct a deterministic Skeptic’s strategy that “forces” the same event (e.g., ensures capital explodes if the event fails), typically using explicit capital process manipulation.

Example: In the coin-tossing game, set

$M_n = -2^{-b_n-1}, \quad b_n = \#\{k < n : x_k = 1\}$

and analyze capital increments to guarantee

$\sum_{n=1}^\infty p_n < \infty \implies \sum_{n=1}^\infty I_n < \infty.$

Reality’s deterministic emulation: Reverse-engineer a deterministic strategy for Reality by “simulating” the capital dynamics such that the event holds and Skeptic’s capital remains bounded. These selection rules often involve waiting times until a trigger (e.g., nonzero Skeptic bet), counters, and conditional deterministic selections that mimic previously random events.

Algorithmic and Statistical De-Randomization

Scoring Rules, Decision Lists, and Compression: In learning and algorithmic contexts, de-randomization often means sparsification and integer rounding of scoring functions. For example, the select-regress-and-round framework (Jung et al., 2017) yields integer-weighted checklists (simple rules) whose empirical performance mirrors that of stochastic or complex models, underpinned by bounds relating the impact of rounding-induced noise in decision thresholds to classification error rates.
Approximation of Boolean Functions: De-randomization of decision lists or DNF circuits involves compressing wide or dense rule lists into thin, shallow, and sparse yet functionally similar forms (Lovett et al., 2019). Here, random restriction lemmas provide sharp bounds for the degree of simplification attainable with mild restriction, enabling explicit deterministic rules to approximate randomized processes to any desired $\epsilon$ -level.

Mechanism Design and Strategic Games

In social choice and algorithmic mechanism domains, de-randomization is achieved by replacing randomized tie-breaking or allocations with deterministic but strategically structured games, such as modular arithmetic games or parity games among agents (Walsh, 2023). The equilibria of these games yield distributions over outcomes that are statistically identical to those induced by randomization, but the procedure itself is deterministic from an external perspective.

3. Representative Applications

Game-Theoretic Probability and Strong Laws

Coin-Tossing Game: Deterministic strategies for Reality can be constructed (using explicit capital process formulas and event-dependent triggers) to guarantee compliance with almost sure events, such as the Borel–Cantelli lemma analog (Miyabe et al., 2014).
Unbounded Forecasting: The method derandomizes Kolmogorov's pathwise argument for the strong law of large numbers (SLLN), producing explicit deterministic rules that either strongly enforce or prevent SLLN, depending on target event formulation.

Statistical Decision Making Under Partial Identification

In scenarios where policy choices depend on partially identified structural parameters (e.g. in regression discontinuity designs for social welfare maximization), minimax regret criteria lead to deterministic threshold rules whenever the identified set is sufficiently narrow relative to stochastic noise, yielding de-randomized optimal rules in finite samples (Yata, 2021).

Robust Machine Learning and Certification

Decision Stump Ensembles: De-randomized smoothing for ensemble classifiers leverages the structure of decision stumps to compute, via dynamic programming, the exact distribution of the aggregated output under input randomization. This yields deterministic, certified robustness guarantees even in the presence of adversarial perturbations (Horváth et al., 2022). No Monte Carlo sampling is required.
Malware Detection: For sequential data such as binaries, window ablation and majority voting produce de-randomized smoothed classifiers (e.g., DRSM) with certified robustness to contiguous adversarial byte manipulations, as only a bounded number of windows can be influenced per attack (Saha et al., 2023).

Structural Complexity and Counting vs. Decision

The status of de-randomized decision rules in complexity theory is intimately connected with class separations (e.g., RP versus P). For subclasses of $\#\text{P}$ whose decision versions lie in P or RP, deterministic (de-randomized) rules enable fully polynomial randomized approximation schemes, and the potential for fully de-randomizing decision rules characterizes separation or collapse phenomena among counting complexity classes (Bakali, 2018).

Mechanism Design

De-randomized mechanisms, constructed by replacing exogenous coin tossing with modular arithmetic or parity games among agents, preserve normatively attractive properties (fairness, efficiency), lead to deterministic rules, and at equilibrium induce agents to act sincerely on the primary economic problem. Responsiveness is enhanced in peer selection by ensuring that every agent retains the ability to affect the outcome through their game action (Walsh, 2023).

4. Limitations, Structural Assumptions, and Theoretical Guarantees

The existence, structure, and performance of de-randomized decision rules are mediated by the properties of the underlying problem:

Knowledge of a suitable randomized baseline: The method typically presumes that a randomized strategy is well understood and delivers the target property (e.g. SLLN, optimality, or coverage).
Game structure and collateral duties: In game-theoretic probability, explicit de-randomization relies on perfect information, pathwise capital constraints, and event compliance properties inherent to the capital processes (Miyabe et al., 2014).
Convexity and extremal structure: In decentralized control and stochastic teams, the ability to replace randomized strategies with pure ones depends critically on convexity assumptions. In many infinite decision-maker settings, only randomized symmetric rules attain optimality; deterministic rules may be strictly suboptimal (Sanjari et al., 2020).
Technical conditions for approximation: In circuit and list compression, the degree to which de-randomization is possible (i.e., the size of the sparse approximant) depends on random restriction parameters, input dimension, and the total influence structure (Lovett et al., 2019).
Finite sample and identification strength: For statistical decision problems, de-randomized (nonrandomized, threshold-type) rules are optimal when the uncertainty about the key parameter is small relative to the sample noise; weak identification or near-certain data necessitates randomized rule retention (Yata, 2021).

5. Algorithms and Explicit Constructions

Context	De-randomization Methodology	Key Formula or Step
Game-theoretic probability	Capital process simulation, event forcing	For Reality: see explicit $I_n$ formulas (Miyabe et al., 2014)
Decision stump ensembles	DP-based PDF/CDF computation	$\bar{F}_M(z) = \sum_{t\leq z M\Delta} \text{pdf}[d][t]$
Statistical testing (panel)	Averaging randomized test function	$Q_{n,T,B}(\alpha) = \frac{1}{B} \sum_{b=1}^B \mathbb{I}\{ Z_{n,T}^{(b)} \le c(\alpha) \}$ (Massacci et al., 23 Jul 2025)
Mechanism design	Modular arithmetic, parity games	$j = (\sum_i a_i) \bmod n$ (random seed in voting) (Walsh, 2023)

Critical features:

Explicit dependence on auxiliary statistics (e.g. capital, counters, or weighted means).
Iterated DP tables (in stump ensembles) for eliminating sampling variance.
Averaging over replications (statistical testing) to “wash out” auxiliary randomness in the reporting of test outcomes and ensure reproducibility (Massacci et al., 23 Jul 2025).
Game-theoretic equilibrium selection (mechanisms) ensuring the outcome replicates the randomized mechanism's property under rational play.

6. Empirical Performance, Scalability, and Adoption

Numerical and simulation studies, as reported in the primary literature, demonstrate the following:

Game-theoretic constructions guarantee deterministic event compliance (such as realized path satisfying SLLN or Borel–Cantelli events) without recourse to external randomization, with complexity tied to the explicit rules and auxiliary counters.
Select-regress-and-round rules (weighted checklists) yield classification accuracy within one percentage point of full logistic regression or random forests in a variety of UCI tasks, while remaining interpretable and reproducible (Jung et al., 2017).
Statistical de-randomization for asset pricing tests ensures exact nominal size and high power, under minimal assumptions and with no need to estimate or invert high-dimensional covariance matrices, as confirmed by extensive Monte Carlo evaluation (Massacci et al., 23 Jul 2025).
Robustness certification for tree ensembles: Deterministic smoothing achieves a fourfold increase in certified accuracy (e.g., certified accuracy at $\ell_2$ radius of 0.8 for MNIST increases from 23.0% to 89.6%) over prior randomized smoothing approaches (Horváth et al., 2022).

Scalability is facilitated by:

Parallel computation of decision rules across partitions (in DRO and robust learning).
Dynamic programming for efficient aggregation (in ensemble smoothing).
Marginal estimation for high-dimensional statistical testing (unit-by-unit OLS in factor models).

7. Impact and Ongoing Challenges

De-randomized decision rules have reshaped the theoretical understanding of randomization necessity across mathematical probability, algorithmic learning, economic mechanism design, and robust statistical inference:

Theoretical impact: The possible collapse of randomized to deterministic classes in complexity theory (e.g., RP = P if certain counting class inclusions hold) exposes the deep connection between the feasibility of de-randomized rules and core open questions about computation (Bakali, 2018).
Methodological innovation: The three-step game-theoretic construction, modular arithmetic games, and dynamic programming for exact certification have provided blueprints for constructing deterministic analogues of randomized methods in broadly different disciplines (Miyabe et al., 2014, Walsh, 2023, Horváth et al., 2022).
Practical consequences and uptake: Reproducibility and auditability, fairness and efficiency, and the ability to work under minimal assumptions have led to adoption of de-randomized rules in settings ranging from asset pricing tests to robust machine learning classifiers and policy evaluations under partial identification.

Challenges remain, particularly in high-dimensional, non-convex, or information-constrained environments, where de-randomization may be infeasible or entail a non-negligible approximation error. In decentralized stochastic teams, relaxation to randomized symmetric policies appears structurally unavoidable without strong additional assumptions (Sanjari et al., 2020). Moreover, the explicit construction of implementable de-randomized rules can be technically delicate, especially as the underlying stochastic or event logic grows more intricate.

Summary

De-randomized decision rules provide a general strategy for replacing randomization in decision-making, optimization, statistical inference, learning, and mechanism design with explicit, often interpretable, reproducible procedures. The technical literature establishes methodologies, analyzes theoretical guarantees, and demonstrates empirical validity for deterministic rules matched to their randomized counterparts, while delineating the contexts where de-randomization is structurally feasible or fundamentally limited.