Adaptive Symmetric Reward Noising

• Adaptive Symmetric Reward Noising is a reinforcement learning technique that mitigates reward corruption by injecting or correcting symmetric (unbiased) noise based on local empirical statistics.
• It employs methodologies like surrogate reward estimation via confusion matrix inversion and adaptive Gaussian noise injection to debias reward signals in both tabular and deep RL settings.
• The approach enhances exploration and reduces training brittleness, though it incurs increased variance and sample complexity, demanding a careful balance in noisy environments.

Adaptive Symmetric Reward Noising is a principled methodology in reinforcement learning (RL) for mitigating the adverse effects of reward noise by either injecting or correcting symmetric (unbiased) noise in the reward signal, with adaptations based on local empirical statistics. This approach encompasses both explicit reward perturbation schemes for improved robustness and adaptive noise correction procedures for debiasing corrupted rewards. Adaptive symmetric reward noising is significant in both tabular and deep RL domains, especially where reward sensors, feedback models, or symbolic abstractions are subject to systematic or random error.

1. Formalization and Motivation

Adaptive symmetric reward noising arises in environments where the observed reward $\tilde{r}$ is a stochastic, noisy transformation of the true reward $r$. In simple settings, noise is symmetric: for discrete rewards, each signal is flipped to another value with a fixed probability, and this probability does not depend on the direction (i.e., $$P(\tilde{r} = r_j | r = r_i) = P(\tilde{r} = r_i | r = r_j)$$ for $i \neq j$). More generally, the noise process can be represented by a confusion matrix $C$, with entries $c_{j,k} = P(\tilde{r} = R_k \mid r = R_j)$.

The motivation for adaptive symmetric reward noising is to achieve unbiased learning and improve exploration without sacrificing the expected return. Without correction, RL algorithms suffer from reward bias and brittle training effects such as the "Boring Areas Trap" and the "Manipulative Consultant" problem (Vivanti et al., 2019). By injecting or correcting symmetric noise, these artifacts are mitigated and policy learning becomes more robust.

2. Methodologies for Adaptive Symmetric Reward Noising

2.1. Surrogate Rewards via Confusion Matrix Inversion

A key solution is the surrogate reward, computed by inverting the confusion matrix $C$. For binary rewards with flip rates $e_+,e_-$: $$\hat{r}(s,a,s') = \begin{cases} \frac{(1-e_-) r_+ - e_+ r_-}{1 - e_+ - e_-} & \text{if } \tilde{r} = r_+ \ \frac{(1-e_+) r_- - e_- r_+}{1 - e_+ - e_-} & \text{if } \tilde{r} = r_- \end{cases}$$ In the multi-valued case, use: $\hat{\mathbf{r}} = C^{-1} \mathbf{r}$ Such estimators are unbiased ($E_{\tilde{r}|r}[\hat{r}] = r$); variance is increased, but the bias is removed. Adaptive estimation of $C$ is performed online—recent reward samples are aggregated, and the confusion matrix is re-estimated dynamically via majority voting or other aggregation rules (Wang et al., 2018).

2.2. Adaptive Symmetric Reward Injection

ASRN is an explicit adaptive scheme where zero-mean Gaussian noise with state-dependent variance is added to observed rewards: $r'_t \sim \mathcal{N}(r_t, N_b^2)$ with $N_b = \sqrt{S_{\text{max}}^2 - S_b^2}$, where $S_b$ is the empirical reward standard deviation in region (or bin) $b$ and $S_{\text{max}}$ is the maximal observed standard deviation. This mechanism ensures that subregions of the state space with low intrinsic reward variance receive proportionally more injected noise, equalizing exploration pressure across the space (Vivanti et al., 2019).

2.3. Variance Estimation Subtasks

Some architectures extend deep RL models by adding a variance estimation branch. The reward is modeled by a Gaussian distribution whose mean is the predicted value and whose variance is learned by the network. The actor branch consumes both the usual features and those from the variance predictor, allowing decisions to take local uncertainty into account, which stabilizes training under noisy rewards (Suzuki et al., 2021).

2.4. Label Noise Correction (Preference Modeling)

In RLHF and preference optimization, symmetric loss functions ($\ell(z) + \ell(-z)$ is constant) in reward modeling yield rank-preserving reward functions under symmetric label noise. Symmetric Preference Optimization (SymPO) achieves robustness to corrupted feedback without requiring explicit noise rate estimation, performing implicit adaptive symmetric reward noising in policy optimization (Nishimori et al., 30 May 2025).

3. Theoretical Guarantees and Convergence

Corrected surrogate rewards and adaptive noise estimation preserve the unbiasedness property in expected value. Under standard conditions, algorithms utilizing surrogate rewards converge almost surely to the optimal policy in the noisy environment (Wang et al., 2018). Sample complexity increases with noise, scaling as

$$O\left(\frac{|\mathcal{S}||\mathcal{A}| T}{\epsilon^2 (1-\gamma)^2 \det(C)^2} \log\frac{|\mathcal{S}||\mathcal{A}| T}{\delta} \right)$$

where $\det(C)$ reflects the cost of adapting to the noise (a smaller determinant indicates higher noise, requiring more samples). The variance of the surrogate reward—although larger—does not impair the consistency of policy learning.

4. Empirical Validation and Application Domains

Adaptive symmetric reward noising is validated on both toy and complex environments:

Approach	Key Environment	Performance under Noise
Surrogate rewards	Atari/Ctrl tasks	Up to 84.6% avg. PPO score increase (10–30% error) (Wang et al., 2018)
ASRN	Two-Armed Bandit, AirSim	Prevents Boring Areas Trap; improves DQN driving duration (Vivanti et al., 2019)
SymPO (Symmetric Loss)	MNIST/RLHF datasets	Rank-preserving reward; robust under noisy labels (Nishimori et al., 30 May 2025)

Notably, methods that adaptively inject noise (ASRN) outperform both uniform noise injection and baseline RL algorithms. SymPO exhibits strong robustness to high label noise, maintaining performance even under severe preference corruption.

5. Extensions and Variants

Adaptive symmetric reward noising connects to additional areas:

• Noisy Symbolic Abstractions: Reward Machine state uncertainty can be framed as a noisy abstraction problem; recurrent belief modeling over symbolic states provides similar adaptive correction mechanisms (Li et al., 2022).
• Robust Policy Optimization: Symmetric RL losses, incorporating reverse cross-entropy terms, reduce instability and gradient variance, especially under RLHF with noisy reward models (Byun et al., 27 May 2024).
• Neuromorphic Hardware Algorithms: Noise‐based reward-modulated learning rules, utilizing stochastic neurons and eligibility traces, enable gradient-free, locally computable updates—suitable for hardware implementation—in settings with noisy or delayed rewards (Fernández et al., 31 Mar 2025).
• Noise-Corrected Group Policy Optimization: In RLHF, group-based normalization schemes (GRPO, Dr.GRPO) combined with label-noise correction admit unbiased gradient estimates under Bernoulli noise, amplifying robustness to human or model error in reward signals (mansouri et al., 21 Oct 2025).

6. Limitations, Open Problems, and Future Directions

The primary trade-off in adaptive symmetric reward noising is increased variance. In high-dimensional environments, variance may slow convergence, and careful balance between bias correction and variance control is necessary. Assumptions of symmetric, stationary noise may not always hold; environments with state-dependent or adversarial noise require more sophisticated estimation (potentially via temporally or instance-adaptive models).

Promising research avenues include:

• Integration of adaptive symmetric noise correction with advanced variance reduction methods and actor–critic algorithms.
• Extension to real-world RLHF pipelines, where non-symmetric, instance-dependent corruption is prevalent.
• Formal connection between reward-punishment symmetry and robustness in universal intelligence measures (Alexander et al., 2021).

7. Conceptual Implications

The algebraic symmetry established for intelligence measures and reward functions under dualization provides principled motivation for symmetric noising and noise correction strategies. By dynamically adapting to changing reward noise and ensuring symmetric treatment, RL algorithms can approach unbiased evaluation and optimal exploration without sacrificing alignment to the underlying task. Adaptive symmetric reward noising thus constitutes a theoretically grounded, empirically effective toolkit for learning robust policies in noisy, uncertain, and complex environments.