## 3 Learning with Corrupted Data
### Observations  
Figure 3 illustrates the relationship between the data-missing ratio and model performance. Both experiments (NLP-SL and Signal-RL) exhibit an initial gradual performance decline, followed by a steeper descent that culminates in a sharp performance drop as the data-missing ratio approaches 1.0. For reference, the figure includes the performance of an optimized fixed-timing signal as a benchmark. The RL-trained signal outperforms the fixed-timing signal when the data-missing ratio is below 0.8. Tables 2 and 3 provide the detailed data for these experiments. In Figure 4, the \(x\)-axis is changed to represent (1--corruption ratio) and fit the curve to the function in Equation (2). The fitted parameters for the NLP-SL experiment are \(a=0.475\), \(\lambda=3.517\), where \(\lambda\) represents the decay rate that controls the curve's steepness. The goodness-of-fit analysis results in \(R^{2}=0.995\), indicating that the exponential cumulative distribution function (CDF) is an excellent model for the observed performance. Similarly, for the Signal-RL experiment, the fitted-curve parameters are \(a=395.8\), \(\lambda=7.493\), with \(R^{2}=0.956\). This striking coincidence reveals an important and universal rule in machine learning: the diminishing return of data. The decay rate \(b\) reflects the task's nature, with the RL task demonstrating a much larger decay rate. This implies that RL tasks are more sensitive to data corruption compared to NLP tasks. An explanation for Equation (2) is provided at the end of this section. \[S=a(1-e^{-\lambda(1-p)}) \tag{2}\]  
where parameter \(a=\frac{S_{0}}{1-e^{-\lambda}}\); and \(S_{0}\) is the model score when corruption ratio \(p=0\) (i.e., no corruption). The Signal-RL model scores under inserting-noise and masking-region types of data corruption are shown in Figure 5. From Figure 5a, results show that inserting noise is significantly more detrimental than data missing. The model's performance deteriorates much more rapidly as the noise level increases, falling below the fixed-timing signal performance as soon as the noise level exceeds 10%. Additionally, the model scores become unstable, as indicated by the oscillations in Figure 5a. The training process also becomes unstable, as shown in Figure 6.  
Masking region is a unique type of data-missing corruption. Information about vehicles farther away from the intersection is less critical. By gradually discarding less important data, it is observed that the model score only experiences a sharp decline when the masking ratio \(p\) exceeds 0.7. This observation leads to an empirical rule: 30% of the data is critical and determines the model's performance, while the remaining 70% can be missing without significantly affecting the model's performance. The exact numbers may not hold for other tasks. And for many tasks (e.g., previous NLP tasks), this kind of screening is not feasible. Figure 3: Model performance across varying data-missing ratios. The \(x\)-axis indicates the data-missing ratio, while the \(y\)-axis shows the model score—normalized accuracy for NLP-SL and normalized negative queue length for Signal-RL. (**a**) NLP-SL experiment; (**b**) Signal-RL experiment. In both cases, model performance initially declines gradually, followed by a steeper drop and a sharp collapse as the missing ratio approaches 1.0. The RL-trained signal outperforms the fixed-timing signal when the data-missing ratio is less than 0.8. Figure 4: Relationship between model performance and data corruption ratio, fitted by Equation (2). The \(x\)-axis represents (\(1\)—corruption ratio), and the \(y\)-axis shows the model score—normalized accuracy for NLP-SL and normalized negative queue length for Signal-RL. (**a**) NLP-SL experiment: Curve-fitting parameters are \(a=0.475\), \(\lambda=3.517\), with \(R^{2}=0.995\). (**b**) Signal-RL experiment: Curve-fitting parameters are \(a=395.8\), \(\lambda=7.493\), with \(R^{2}=0.956\). Both experiments exhibit diminishing returns as data quality improves.