Local Decision Boundary

The local decision boundary exists in the model's full input space, which may contain hundreds or thousands of dimensions. While humans can only visualize 2D or 3D separations, the true boundary is a complex hyperplane or non-linear manifold. The surrogate model projects this high-dimensional structure onto an interpretable representation—such as superpixels or bag-of-words—making it comprehensible at the cost of losing some geometric fidelity.

• The boundary's dimensionality equals the number of input features
• Local linear approximations assume the manifold is smooth within the neighborhood
• Curvature and sharp turns in the boundary indicate regions where linear surrogates will fail

The local decision boundary is only meaningful within a tightly constrained neighborhood around the instance of interest. Beyond this region, the boundary may curve, twist, or exhibit entirely different behavior. The exponential kernel enforces this constraint by assigning higher weights to perturbed samples closer to the original instance, ensuring the surrogate model learns only the local geometry rather than the global decision surface.

• Kernel width controls the effective radius of the neighborhood
• Too large a neighborhood captures non-local behavior, reducing local fidelity
• Too small a neighborhood suffers from high variance due to insufficient samples

For a local surrogate to be valid, the decision boundary must be approximately linear within the sampled neighborhood. Complex models like deep neural networks often produce highly non-linear boundaries globally, but locally they can be well-approximated by a sparse linear model or decision tree. This property is the foundational assumption of LIME and similar additive feature attribution methods.

• The boundary is treated as a hyperplane separating classes in the local region
• Lasso regression enforces sparsity by zeroing out irrelevant feature coefficients
• Non-linear boundaries require more expressive surrogates, sacrificing interpretability

The local decision boundary's shape determines how the model's predictions change when features are perturbed. By generating a synthetic neighborhood through random masking or alteration of input features, the explanation system probes the boundary's orientation and steepness. Features that cause large prediction changes when perturbed lie orthogonal to the boundary and receive high feature importance scores.

• Perturbation sampling strategy varies by data modality
• Text: token masking creates documents on both sides of the boundary
• Images: superpixel segmentation defines the atomic units for perturbation
• Tabular: value perturbation or discretization explores the local feature space

The local decision boundary's inherent complexity creates a fundamental tension: a highly faithful surrogate may be too complex for humans to understand, while a simple linear boundary may miss important non-linearities. This fidelity-interpretability trade-off is managed by choosing the surrogate model class and regularization strength to balance accuracy against comprehensibility.

• Sparse linear models prioritize interpretability with few features
• Decision trees capture axis-aligned splits but may require deeper structures
• Explanation regularization explicitly penalizes complexity in the surrogate objective
• The optimal balance depends on the use case: debugging requires fidelity, compliance requires simplicity

The local decision boundary should yield consistent explanations across multiple perturbation sampling runs. If the identified important features change dramatically with different random seeds, the explanation stability is low, indicating that the boundary is either too complex for a linear approximation or the neighborhood is poorly defined. OptiLIME and Bayesian LIME address this by optimizing kernel width and providing uncertainty estimates.

• Instability signals a highly curved or fragmented local boundary
• Bayesian ridge regression provides confidence intervals for feature importance
• Repeated sampling with different seeds tests explanation robustness
• Stable explanations are essential for high-stakes decisions requiring auditability

A measure of how accurately an interpretable surrogate model mimics the black-box model's predictions within the immediate neighborhood of the instance being explained. High local fidelity means the surrogate's simple linear boundary closely matches the complex true boundary near the point of interest. This metric is critical because a surrogate that perfectly replicates global behavior may fail to capture the specific local curvature that determined an individual prediction.

A hyperparameter controlling the effective size of the local neighborhood by determining how quickly sample weights decay with distance from the original instance. A narrow kernel width enforces strict locality, capturing fine-grained boundary details but risking instability. A wide kernel width smooths over boundary irregularities, improving stability but potentially missing the precise local curvature that drove the prediction.

A distance-based weighting function used in LIME to assign higher importance to perturbed samples closer to the original instance. The weight of a sample decays exponentially with its distance, enforcing the locality constraint. This ensures the surrogate model focuses on approximating the decision boundary immediately surrounding the instance rather than fitting the global prediction function.

A non-parametric regression method that fits a simple model to a localized subset of data, weighting points by their proximity to the target instance. This forms the statistical backbone of the LIME algorithm. By solving a weighted least-squares problem, the method produces a linear approximation of the decision boundary that is valid only in the immediate vicinity of the query point.

The fundamental balancing act between using a highly accurate but complex surrogate and a simple, human-understandable model. A complex surrogate can closely track the twists of the local decision boundary but defeats the purpose of explanation. A linear surrogate is interpretable but may miss non-linear boundary segments. This trade-off is managed through regularization and kernel width tuning.

A proximity measure used in LIME for text data that calculates similarity based on the angle between TF-IDF vector representations. Unlike Euclidean distance, cosine distance ignores differences in document length, focusing purely on the directional alignment of word usage patterns. This defines the local neighborhood in the high-dimensional text space where the decision boundary is approximated.