Tree-Path KL Divergence (TP-KL)

• Tree-Path KL Divergence (TP-KL) is a hierarchy-aware regularization method that enforces consistency across coarse-to-fine levels in classification tasks.
• It quantifies the KL divergence between the predicted joint probability distribution and the ground-truth taxonomic path, ensuring structural validity.
• When combined with techniques like LoRA adapters and HiSCE, TP-KL significantly improves full-path accuracy while reducing taxonomic inconsistency errors.

Tree-Path KL Divergence (TP-KL) is a hierarchy-aware regularization objective designed to enforce vertical coherence in hierarchical classification tasks, particularly in the fine-tuning of large multimodal models such as Vision-LLMs (VLMs). TP-KL quantifies and penalizes deviations of a model’s predicted probabilistic path through a taxonomy tree from the ground-truth path, ensuring consistency across coarse-to-fine levels with minimal parameter overhead. This approach is foundational to recent parameter-efficient fine-tuning frameworks for structured label spaces, enabling robust adaptation of models to real-world tasks with hierarchical taxonomies (Li et al., 25 Dec 2025).

1. Motivation and Problem Context

Hierarchical classification tasks appear naturally in domains where category labels form tree-structured taxonomies—e.g., biological species (order, family, genus, species), medical diagnosis (supercategory, subcategory, disease), or fine-grained object recognition (manufacturer, family, variant). Standard fine-tuning approaches for VLMs treat labels as flat categories, either training classifiers per leaf node or applying cross-entropy at the deepest level. This strategy ignores structural relations, often resulting in inconsistent predictions across levels (e.g., predicting a species incompatible with its assigned family). Full-model fine-tuning is computationally demanding and still fails to guarantee structural validity along the taxonomic path. TP-KL was introduced to explicitly enforce coherence along the ground-truth path, aligning model predictions vertically across all levels (Li et al., 25 Dec 2025).

2. Mathematical Formulation of TP-KL Divergence

The central object of TP-KL is the Kullback–Leibler (KL) divergence between the predicted joint probability distribution traversing the hierarchy (P) and the ground-truth path distribution (Y). This is computed as follows:

For a taxonomy of depth $L$ where each level $l$ contains $C_l$ classes, the procedure is:

• Compute normalized similarity logits between image embedding $\mathbf{v}$ and the set of text embeddings $\{\mathbf{t}_c^{(l)}\}_{c=1}^{C_l}$ for every level:

$$z^{(l)}_c = \frac{\mathbf{v}^\top \mathbf{t}^{(l)}_c}{\|\mathbf{v}\|\,\|\mathbf{t}^{(l)}_c\|},\qquad c=1,\ldots,C_l$$

• Apply temperature scaling and log-softmax at each level:

$$\hat{\mathbf{z}}^{(l)} = \mathrm{LogSoftmax}(z^{(l)} / \tau)$$

• Concatenate all levels’ log-probabilities into one vector:

$$\hat{\mathbf{z}} = [\hat{\mathbf{z}}^{(1)}; \dots; \hat{\mathbf{z}}^{(L)}]$$

• Define the predicted distribution via softmax:

$$P = \mathrm{Softmax}(\hat{\mathbf{z}}) \in \Delta^{\sum_{l=1}^L C_l - 1}$$

• Construct the ground-truth path vector $Y$ as a concatenated one-hot indicator over the true path $(y^{(1)},\dots,y^{(L)})$:

$$Y = \tfrac{1}{L}[\,\mathbf{1}_{y^{(1)}}; \dots; \mathbf{1}_{y^{(L)}}\,]$$

• Calculate the Tree-Path KL Divergence (TP-KL) loss:

$$\mathcal{L}_{\mathrm{TP\text{-}KL}} = \mathrm{KL}(P\;\|\;Y) = \sum_{i=1}^{\sum C_l} P_i \log\, \frac{P_i}{Y_i}$$

This objective encourages the model’s predicted distribution to allocate probability mass only along the ground-truth label path traversing the taxonomy. The vertical alignment is strictly enforced across all levels, penalizing any allocation of probability to off-path nodes (Li et al., 25 Dec 2025).

3. Integration in Hierarchy-Aware Fine-Tuning Frameworks

TP-KL is implemented as an auxiliary regularization term within a composite loss function combining standard cross-entropy (CE) and often Hierarchy-Sibling Smoothed Cross-Entropy (HiSCE), for horizontal (sibling) consistency:

$$\mathcal{L}_{\mathrm{total}} = \mathcal{L}_{\mathrm{CE}} + \lambda_1\,\mathcal{L}_{\mathrm{TP\text{-}KL}} + \lambda_2\,\mathcal{L}_{\mathrm{HiSCE}}$$

Where $\lambda_1$ and $\lambda_2$ are hyperparameters (typically $\in[0.5, 2]$), with TP-KL generally set equal or slightly above HiSCE for maximal vertical consistency. TP-KL leverages the shared multimodal embedding space, making it lightweight when used alongside parameter-efficient modules such as LoRA adapters (Li et al., 25 Dec 2025).

4. Empirical Impact and Evaluation Metrics

Empirical studies demonstrate that integrating TP-KL markedly improves Full-Path Accuracy (FPA)—the fraction of samples for which all hierarchical levels are correct—and reduces Tree-based Inconsistency Error (TICE)—the rate of invalid taxonomic paths in predictions. In controlled ablations on benchmarks such as CUB-200-2011 and FGVC-Aircraft, increasing the TP-KL loss weight produces substantial gains in hierarchical metrics, e.g., FPA rising from 50.2% (CE only) to 72.0% (joint TP-KL+HiSCE), while TICE drops from 21.9% to 7.5%. In all tested domains, joint TP-KL+HiSCE optimization yields superior performance compared to flat or sibling-only regularization, with only 0.5% trainable parameter overhead when using LoRA (Li et al., 25 Dec 2025).

Dataset	CE only FPA	TP-KL+HiSCE FPA	CE only TICE	TP-KL+HiSCE TICE
CUB-200-2011	50.2%	72.9%	21.9%	5.9%
FGVC-Aircraft	38.3%	61.5%	17.9%	8.5%

TP-KL also produces dendrogram-structured label embeddings that visually follow the target taxonomy after fine-tuning, suggesting effective semantic restructuring of the representation space.

5. Theoretical Properties and Design Implications

TP-KL is unique in its strict vertical coherence enforcement. By concatenating multi-level logits into one path-structured distribution, KL divergence penalizes both coarse- and fine-level misallocations proportionally. Unlike label smoothing or consistency regularizers, TP-KL guarantees that probability is distributed only along taxonomically valid paths, preventing any form of "drift" across incompatible ancestors or off-branch siblings. Its primary trade-off is the risk of over-penalizing near-miss fine-grained classes if used in the absence of sibling smoothing; empirical results support using TP-KL jointly with HiSCE for best performance (Li et al., 25 Dec 2025).

6. Practical Implementation and Best Practices

• Use TP-KL as a regularizer with weight $\lambda_1 \approx 1$–$2$; tune via validation set grid search.
• Concatenate per-level logits; normalization via temperature scaling ($\tau$) is crucial—set $\tau$ based on validation curves.
• When using LoRA adapters for efficient fine-tuning of large VLMs, TP-KL does not inflate parameter counts nor require modification of encoder architectures.
• Always combine with horizontal consistency regularizers (e.g., HiSCE) to prevent over-constraining and improve fine-grained discriminability.
• TP-KL performance is robust across a range of hierarchical benchmark datasets and demonstrates rapid convergence and stability.

7. Related Frameworks and Extensions

TP-KL belongs to a family of hierarchy-aware regularization losses designed for structured output spaces. Related methods in purely vision or NLP domains include:

• Weighted Tree-Path KL applied to orthogonal subspaces for feature mapping (Hier-COS) (Sani et al., 10 Mar 2025).
• Jensen–Shannon divergence losses for inter-level consistency (Hierarchy-Aware Features) (Garg et al., 2022).
• Layer-wise guided training protocols mapping hierarchy levels to model layers for incremental representation refinement (Manginas et al., 2020).

A plausible implication is that TP-KL and its variants can be generalized to arbitrary hierarchical graphs, DAGs, or even probabilistic taxonomies by extending the ground-truth path representation and loss construction.

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TP-KL is a formally principled, lightweight, and effective hierarchy-regularization loss for structured output fine-tuning, providing robust vertical coherence and state-of-the-art hierarchical consistency when combined with parameter-efficient adaptation techniques in VLMs and related architectures (Li et al., 25 Dec 2025).