## 3 Learning with Corrupted Data
### Explanation  
An explanation for the observed pattern in Figure 4 and Equation (2) is provided here. For both NLP and DRL tasks, the model relies on recognizing patterns (e.g., semantic patterns, queuing patterns) in the data. When the data are completely corrupted, the critical patterns necessary for performance are entirely lost. As corruption decreases, the model rapidly recovers key patterns, resulting in a steep improvement in performance. However, the marginal utility of additional clean data diminishes, leading to a saturation of performance.  
First, some basic probability theory is introduced. The probabilities of observing rare events in a large number of trials converge to the Poisson distribution [29]. When \(n\) (the number of trials) is large, \(p^{\prime}\) (the probability of success per trial) is small and the expected number of successes \(\lambda=n\cdot p^{\prime}\) is finite, the binomial distribution approximates the Poisson distribution:  
\[P(K=k)\approx\frac{\lambda^{k}e^{-\lambda}}{k!}, \tag{3}\]  
Figure 6: Training instability under noise-insertion corruption. The \(x\)-axis represents training episodes, and the \(y\)-axis shows episode return. From (**a–d**), the observation noise ratio increases from 0.0 to 0.4, leading to progressively more unstable training dynamics.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Data missing ratio** & 0.0 & 0.1 & 0.2 & 0.3 & 0.4 \\ \hline
**Model score mean** & 409.86 & 403.30 & 396.40 & 398.40 & 381.74 \\ \hline
**Model score std** & 3.83 & 5.04 & 7.92 & 8.97 & 8.44 \\ \hline
**Data missing ratio** & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\ \hline
**Model score mean** & 376.33 & 371.30 & 348.58 & 292.54 & 231.56 \\ \hline
**Model score std** & 9.62 & 10.05 & 13.25 & 26.44 & 47.59 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Mean and standard deviation of Signal-RL model scores at different data-missing ratios; plotted in Figure 3b.  
where \(\lambda=n\cdot p^{\prime}\) is the rate parameter, representing the expected number of successes. The exponential term \(e^{-\lambda}\) in the Poisson distribution describes the probability of observing 0 successes.  
Next, the above theory is applied to our pattern-recognition problem. In both experiments, the dataset size \(n\) is large, making the discovery of a pattern from an individual sample a rare event. In our experiments, each pattern has an equal probability \(p\) of being corrupted. Let \(x=1-p\). A pattern can be recovered multiple times from samples in the dataset with probability \(P(K=k)\). The probability of failing to recover such a pattern with corruption level \(p\) is \(e^{-\lambda x}\), where \(\lambda\) is the pattern appearance rate given no corruption. As \(x\) increases, the probability of failing to recover a pattern have derivative \(-\lambda e^{-\lambda x}\). In other words, the probability of identifying such a pattern increase by \(\lambda e^{-\lambda x}\) marginally. Suppose that model score \(S\) is proportional to the number of patterns identified. Then, as \(x\) increases, the rate of \(S\) is proportional to \(\lambda e^{-\lambda x}\). This leads to Equation (4), where \(a\) is coefficient for this linear relation. Its solution corresponds to Equation (2). It can be shown that \(\lambda e^{-\lambda x}=\lambda(a-S)\), so Equation (4) actually describes a dynamic system where the rate of change in performance depends on the difference between the system's current performance and its limit.  
\[\frac{dS}{dx}=a\lambda e^{-\lambda x} \tag{4}\]  
where \(S\) is the model score; \(x=1-p\) and \(p\) is corruption rate; \(b\) is the pattern appearance rate; and \(a\) is the model score when there is no corruption.