Authors: Beatrix M. G. Nielsen, Emanuele Marconato, Luigi Gresele, Andrea Dittadi, Simon Buchholz
Abstract
For a broad family of discriminative models that includes autoregressive language models, identifiability results imply that if two models induce the same conditional distributions, then their internal representations are equal up to an invertible linear transformation. We ask whether an analogous conclusion holds approximately when the distributions are close instead of equal. Building on the observation of Nielsen et al. (2025) that closeness in KL divergence need not imply high linear representational similarity, we study a distributional distance based on logit differences and show that closeness in this distance does yield linear similarity guarantees. Specifically, we define a representational dissimilarity measure based on the models' identifiability class and prove that it is bounded by the logit distance. We further show that, when model probabilities are bounded away from zero, KL divergence upper-bounds logit distance; yet the resulting bound fails to provide nontrivial control in practice. As a consequence, KL-based distillation can match a teacher's predictions while failing to preserve linear representational properties, such as linear-probe recoverability of human-interpretable concepts. In distillation experiments on synthetic and image datasets, logit-distance distillation yields students with higher linear representational similarity and better preservation of the teacher's linearly recoverable concepts.
A simple way to improve the performance of almost any machine learning model is not to train a single but several models with diverse algorithms which will make slightly distinct kinds of predictions and errors on the same data, and thus improve the average predictions and robustness. However, making predictions using a whole ensemble of models is cumbersome and computationally too expensive to allow deployment to a large number of users, especially if the models are large neural nets. In response to this, we introduce a layer and point wise projection mapping, which maps student and teacher representations into an aligned high-dimensional embedding space during training process. The proposed approach combined with LoRA injection reduces the student model trainable parameters to less than 1% of the teacher model, while significantly improving word error rate (WER) compared to other distillation methods, as demonstrated in ablation studies. Unlike a mixture of experts, our method can be trained rapidly and in parallel.
Knowledge distillation is usually framed as a choice of what to match in the teacher - its logits, hidden features, or sample relations - which presupposes that the teacher's representation has absolute coordinates to match. It does not: a pretrained representation is identifiable only up to an orthogonal-and-isotropic-scaling equivalence class, so a student should learn the teacher's equivalence class, not its features. The organizing fact is that capability is the teacher's output function, a class invariant that factors through the quotient by the class action, so an objective recovers capability exactly when it is defined there. This makes absolute feature matching ill-posed, and admissible supervision a matter of targeting class invariants (Gram structure, CKA, principal subspaces) or aligning coordinates first, unifying feature matching, relational distillation, alignment, and grafting in one geometric account. We validate our framework on Qwen2.5 and Llama-3.1. A restoration study recovers a corrupted model's representation (CKA ~ 0.99) but not its capability, and an ablation isolates the cause: output-function (logit) matching drives capability, while matching hidden representations aligns geometry without restoring function. Recovery is confined to the corpus-covered region, and a graft study confirms that boundary overlap predicts transplant success but is necessary, not sufficient.
Logit-based knowledge distillation for autoregressive language models usually aligns teacher and student next-token distributions over the entire vocabulary. However, this global objective overlooks relative preferences among likely token alternatives. Existing local approaches often select candidate tokens from either the teacher or the student alone. Teacher-only selection can miss tokens that the student considers likely, while student-only selection can rely on an inaccurate ranking early in training. We propose Adaptive Local Relational Alignment (ALRA), a position-specific framework combining student proposals with teacher guidance. At each valid prediction position, the student proposes likely tokens, while the teacher's most probable token is included as an anchor. ALRA adjusts the number of selected tokens according to how broadly the teacher distributes probability within this candidate set relative to the current batch. Adaptive Local Divergence retains the mass-matching term and separately matches the relative token distributions within the selected and remaining vocabulary regions. Unlike the exact full-vocabulary decomposition, it replaces the teacher-mass coefficients of the two conditional terms with unit coefficients, preventing either term from being downweighted solely because its region has low teacher probability. Student-Weighted Pairwise Relational Alignment emphasizes high-probability token pairs with small student probability gaps and gives less weight to unlikely or clearly separated pairs. Experiments on The Pile with randomly initialized 200M- and 500M-parameter students across nine zero-shot benchmarks yield average accuracies of 36.62% and 37.40%. ALRA exceeds the strongest competing distillation baseline by 0.94 and 0.83 percentage points and improves over pre-training without distillation by 2.31 and 2.91 points, respectively.
Quang Hoang Trung, Quang Huu Hieu, Nguyen Van Hoang Phuc +1