Betting for Sim-to-Real Performance Evaluation

Abstract: This paper studies the problem of robot performance evaluation, focusing on how to obtain accurate and efficient estimates of real-world behavior under severe constraints on physical experimentation. Such estimates are essential for benchmarking algorithms, comparing design alternatives, validating controllers, and supporting certification or regulatory decision-making, yet real-world testing with physical robots is often expensive, time-consuming, and safety-limited. To mitigate the scarcity of real-world trials, sim-to-real methodologies are commonly employed, using low-cost simulators to inform, supplement, or prioritize physical experiments. Departing from (and complementary to) existing approaches in variance reduction (e.g., importance-sampling variants) or bias-correction (e.g., through prediction-powered inference or learned control variates), we examine this performance-evaluation problem through the lens of betting. We establish theoretical conditions under which a betting mechanism can yield accurate and efficient estimates (provably outperforming the Monte Carlo estimator) and we characterize how such bets should be constructed. We further develop theoretically grounded yet practically implementable approximations of the ideal bet, and we provide concrete decision rules that diagnose when these approximate betting strategies are working as intended. We demonstrate the effectiveness of the proposed methods using both synthetic examples and cross-fidelity computational simulators. Notably, we also showcase an illustrative case in which a group of synthetic distributions are used to infer the real-world pick-and-place accuracy of a robotic manipulator, a seemingly unconventional sim-to-real transfer that becomes natural and feasible under the proposed betting perspective. Programs for reproducing empirical results are available at https://github.com/ISUSAIL/Bet4Sim2Real.

• The paper introduces a novel sequential betting framework to efficiently estimate real-world performance using simulation-derived predictive bets.
• It applies the Kelly criterion for variance-optimal weighting, leading to significant improvements in sample efficiency over traditional Monte Carlo techniques.
• Empirical studies across robotic tasks demonstrate rapid convergence and robustness of the bet-weighted estimator, even under mismatched simulator conditions.

Betting-Based Sim-to-Real Performance Evaluation Framework

Problem Definition and Theoretical Foundations

The paper "Betting for Sim-to-Real Performance Evaluation" (2604.24018) rigorously formalizes the problem of evaluating the real-world performance of fixed policies and controllers in robotics under severe constraints on physical experimentation. The central objective is to efficiently estimate the expected value $\mu = \mathbb{E}_{x \sim P}[\psi(x)]$ for a bounded scoring function $\psi$, with $P$ representing the underlying real-world distribution. Traditional Monte Carlo approaches, though theoretically complete, are often inefficient due to the high cost and rarity of informative real-world samples, especially in safety-critical or low-frequency event regimes.

Previous paradigms emphasize simulator fidelity for variance reduction (e.g., importance sampling) and bias correction (e.g., prediction-powered inference, control variates). Both rely on reducing the discrepancy between simulators and the real world, implicitly prioritizing close alignment. However, these methods fail to exploit the scalability and diversity of modern simulators, which afford vast, inexpensive synthetic data not directly translatable into improved real-world inference.

This paper introduces a fundamentally distinct approach: the sim-to-real performance estimation problem is reformulated as a sequential betting problem. Rather than seeking fidelity, the framework leverages predictive advantage, allowing informative but misspecified simulators to contribute to performance estimation via mathematically grounded bets. The abstract sequential betting algorithm (Algorithm 1) outputs a bet-weighted estimator, placing adaptive bets informed by simulation on whether the real outcome will be above or below a running mean.

Kelly Principle and Efficient Estimation

The pivotal theoretical insight links optimal betting to statistical efficiency. The paper proves that adopting the classical Kelly criterion for sequential betting—stakes chosen proportional to predictive advantage divided by variance—leads to variance-optimal weighting for estimation, coinciding with inverse-variance weighted estimators (Theorem 2). This equivalence demonstrates that maximizing wealth growth in betting directly translates to improved sample-efficiency and estimation accuracy in sim-to-real performance evaluation.

However, the ideal Kelly bets require knowledge of the real outcome distribution, which is not accessible in practice. The solution developed is a practical universal-portfolio–based approximation (Algorithm 2): a bank of diverse simulators, each proposing distinct predictions and associated variances, is maintained and adaptively reweighted based on log-score evidence from real-world outcomes. The framework dynamically aggregates simulator predictions, computing mixture moments and Kelly-style betting fractions to control risk (parameterized by the Kelly fraction $\lambda$).

Figure 1: Extended performance comparison results among various methods discussed in Section~\ref{sec:synthetic}, demonstrating superior estimation error performance for Kelly-style betting over Monte Carlo across diverse synthetic tasks ($\eta = 5.0$).

Diagnostic and Guarantee via Wealth Process

A rigorous statistical guarantee is provided: sustained wealth growth in betting implies operation in a regime where predictive advantage is present, and thus the bet-weighted estimator statistically outperforms Monte Carlo (Theorem 3). Under a no-edge null hypothesis (no informative predictive signal), the wealth process forms an $e$-process, bounding the probability of observing large wealth purely by chance.

Empirical Studies and Numerical Results

The framework is empirically validated across synthetic distributions and real robotic tasks, including manipulator pick-and-place accuracy and reinforcement learning-based humanoid locomotion tracking. In synthetic scenarios, ideal Kelly betting achieves win rates of 70–100% against Monte Carlo, while practical approximations with diverse simulator banks achieve 60–80% depending on coverage density and alignment.

Figure 2: A combined illustration of different variants of banks of Sim distributions, showing the distributional coverage across task-relevant domains.

Figure 3: Real_6 distribution variants employed for benchmarking and tracking experimental performance across the pick-and-place manipulation task.

In manipulator evaluation, synthetic expert banks derived from artificially parameterized distributions produced effective bet-weighted estimates, converging rapidly to real outcomes, even in the absence of a physically matched simulator. In domain-randomized locomotion tracking, a bank of MuJoCo simulators with deliberately varying parameters demonstrated fast adaptation, with Kelly approximations outperforming both Monte Carlo and the recently proposed prediction-powered inference methods under equal simulator budgets.

Figure 4: Extended experiments on another policy evaluated as described in Section~\ref{sec:real_pnp}, illustrating robustness of betting-based inference across manipulator policy variations.

Figure 5: A win-rate comparison with SureSim~\cite{badithela2025reliable} on the locomotion task, indicating superior estimation accuracy for Kelly-style betting even under paired sim–real settings.

Implications, Practical and Theoretical

Practically, the betting-based approach enables the exploitation of simulation abundance and diversity, even with simulators that are biased or deliberately mismatched. It elevates predictive edge rather than fidelity as the source of statistical efficiency—simulator diversity and domain randomization, traditionally used for policy robustness, now become theoretically justified for performance evaluation. The method generalizes to arbitrarily large banks of simulators, scalable to high-dimensional domains and tasks where matching simulators are unavailable.

Theoretically, the paper unifies efficient estimation and information-theoretic betting, demonstrating that Kelly optimality is sufficient for variance reduction in mean estimation and that universal mixture strategies are robust approximations in practice. Diagnostic wealth processes provide non-asymptotic evidence for estimator advantage, offering statistical safety in deployment. Limitations include reliance on i.i.d. sample assumptions and a lack of formal regret bounds for the practical approximation, suggesting directions for future work in non-stationary environments and integration with bias-correction protocols.

Figure 6: Extending the case studies to optimal importance sampling reveals asymptotic bias and limits in practical sample efficiency compared to sequential Kelly betting.

Conclusion

The betting lens on sim-to-real performance evaluation offers a fundamental shift in methodology, replacing simulator fidelity with predictive advantage and enabling the theoretical and practical exploitation of simulation diversity. The framework achieves provable accuracy gains over classical Monte Carlo, is robust to simulator mismatch, and is adaptable to modern robotic workflows. Future developments will extend guarantees to non-i.i.d. regimes, integrate bias-correction mechanisms, and scale empirical validation to large-scale perception and high-dimensional tasks.