# Local Batch Normalization-Aided CNN Model for RSSI-Based Fingerprint Indoor Positioning
## 3 Proposed LBN-Aided Positioning Scheme  
### Local Batch Normalization  
BN has been widely recognized for its effectiveness in enhancing the performance of positioning systems across various network architectures, such as the improved residual network (ResNet) [28] and the multi-path res-inception (MPRI) network [29]. However, despite its strengths, BN has limitations. Its performance can degrade when faced with uneven data distributions [30] or small batch sizes [22]. Additionally, BN's reliance on batch-level statistics becomes problematic in non-independent and identically distributed (non-IID) data settings [31]. These limitations are especially evident in scenarios like RSSI-based positioning, where the data exhibit non-IID characteristics [32] due to spatial dependencies, temporal variations, and environmental noise. This highlights the need for more robust normalization techniques. Figure 1 illustrates RSSI heatmaps for two WiFi access points, revealing spatial dependencies and uneven signal distributions that reflect the non-IID nature of the data. To address these challenges, [PERSON] et al. introduced local batch normalization (FedBN) for federated learning, a specific adaptation of batch normalization (BN) designed to mitigate feature shift [31]. Unlike traditional BN, which aggregates normalization parameters across all clients, FedBN retains parameters locally on each client, ensuring the normalization process aligns with the specific data distribution of each client. This approach has proven effective in mitigating feature heterogeneity, improving both convergence rates and model performance in federated learning scenarios [31]. The key aspects are as follows:  
1. Compute Local Batch Normalization Statistics: Adhering to the BN formulation detailed in Section 2.1, FedBN modifies the normalization procedure by computing the mean and variance locally for each individual client. Specifically, the mean and variance are calculated as follows: \[\mu_{c}=\frac{1}{m}\sum_{i=1}^{m}x_{i},\sigma_{c}^{2}=\frac{1}{m}\sum_{i=1}^{ m}(x_{i}-\mu_{c})^{2}\] (7)
2. Normalize Using Local Statistics: Each client normalizes activations using its local statistics to ensure consistency within its own feature distribution:  
Figure 1: Heatmap visualization of RSSI distributions from two WiFi access points. (**a**) RSSI heatmap for WiFi AP 1 across 74 measurement points. (**b**) RSSI heatmap for WiFi AP 2 across 74 measurement points. \[\hat{x}_{i}=\frac{x_{i}-\mu_{c}}{\sqrt{\sigma_{c}^{2}+\epsilon}} \tag{8}\]
3. Scale and Shift Using Learnable Parameters: Following the standard BN formulation, each client incorporates learnable scale and shift parameters to restore the network's representational capacity: \[y_{i}=\gamma_{c}\hat{x}_{i}+\beta_{c},\] (9)
4. Global Model Aggregation (Excluding BN Parameters)  
FedBN modifies the federated learning model aggregation process by ensuring that only non-BN parameters are synchronized across clients. The BN parameters \(\mu_{c}\), \(\sigma_{c}^{2}\) remain client-specific and are excluded from model averaging. This adaptation ensures that each client retains its own batch statistics. By doing so, it allows FedBN to effectively handle feature shift while still enabling clients to benefit from federated model training. Variable Definitions:  
\(x_{i}\) is the input of the \(i\)-th sample in a mini-batch. \(\hat{x}_{i}\) is the normalized activation of the input. \(m\) is the number of samples in the mini-batch,  
\(\mu_{c}\) is the mean of the batch normalization layer computed locally on client \(c\). \(\sigma_{c}^{2}\) is the variance of the batch normalization layer computed locally on client \(c\). \(\epsilon\) is a small constant added to prevent division by zero. \(y_{i}\) is the final output of the FedBN process for the \(i\)-th sample. \(\gamma_{c}\) and \(\beta_{c}\) are learnable parameters that scale and shift the normalized data, respectively. FedBN's principles provide a valuable foundation for addressing distributional disparities in RSSI-based indoor positioning.