SPRKD: Effective Knowledge Distillation for Deep Neural Networks via Saddle Region Approximation

Modern deep neural networks are potent catalysts for scientific and industrial impact, yet excessive parameter counts impede deployment in low-compute settings such as hospital equipment and energy infrastructure. Predominant knowledge distillation (KD) methods favor replication: smaller students mimic teacher output logits, yet empirically yield low task performance, hamper convergence, and act merely as regularization rather than substantive knowledge transfer. We propose Saddle Point Recruitment for Knowledge Distillation (SPRKD), reframing distillation from replication to employing teachers as optimization-curvature and domain proxies, characterizing saddle points as regions of strong further-descent potential via embedding and basin-fractal properties. Using Hessian eigenvalue spectral density (ESD), SPRKD identifies low-loss saddle regions for student re-exploration; weak-teacher ensembles are aggregated into an Approximated Saddle Region (ASR), re-parameterized into the student via Transfer Learning by Injection, and approached with exponentially decaying Euclidean transformations, Negative Hessian Eigensteps, and Gaussian perturbations. On malaria blood smear classification with a 6,430-parameter CNN distilled from a weak 25,546-parameter teacher, SPRKD reaches 94.8% validation accuracy, outperforming Response KD by 24.70 percentage points (McNemar p = 6.3e-87) and matching scratch-trained baselines of the same architecture to statistical equivalence (p = 1.0). Across MNIST, CIFAR-100, and TinyImageNet, SPRKD exceeds scratch-trained baselines by up to 8 percentage points on preliminary benchmarks. Hessian ESD and 2-D loss landscape analysis show convergence to wider minima with substantially smaller Hessian trace and spectral radius than Response KD and control students, indicating smoother descent and greater noise robustness.

Neural Network Quantization by Learning Low-Loss Subspaces

Neural network quantization aims to find a discrete representation of parameters that preserves the performance of a full-precision (FP) model as faithfully as possible. Enforcing discrete constraints perturbs parameters away from a well-optimized minimum, generally resulting in performance degradation. Recent studies indicate that low-loss FP solutions are not isolated, but instead belong to connected low-loss subspaces of the loss landscape, where the loss maintains nearly the same minimum value. Models sampled from these subspaces are diverse and retain high accuracy. This raises the question: can a quantized model be constructed to lie within a low-loss subspace of the FP model, thereby automatically preserving performance? We address this question by learning quantization-aware linear paths in weight space optimized to minimize loss. We demonstrate that the midpoint of the resulting subspace is, by design, quantization-friendly and that its direct quantization yields performance comparable to that of quantization-aware training. The proposed procedure offers a novel perspective on weight quantization and, in contrast to conventional methods, neither relies on the straight-through estimator nor involves explicit discretization during training.

Understanding Quantization-Aware Training: Gradients at Quantized Weights Bias to the Low-Loss Basin

Post-training quantization (PTQ) converts a trained full-precision model into low-bit weights without task-level retraining, while quantization-aware training (QAT) incorporates quantization into the training loop. Although PTQ is efficient and often accurate at moderate bitwidths, it can fail sharply at aggressive bitwidths; QAT is more expensive but can often recover the lost accuracy. We propose a unified geometric framework that explains both PTQ failure and QAT recovery. We model full-precision training as following a low-loss \emph{river} inside a wider \emph{valley}: a normal neighborhood of the river forms a nearly flat \emph{basin}, while leaving this basin incurs a sharp loss increase. When the quantization grid is comparable to the basin width, local PTQ objectives, including rounding and Hessian-based second-order reconstruction, can select a high-loss deployed quantized point outside the basin even when nearby low-loss quantized points exist. In this regime, straight-through-estimator-based QAT has a useful bias: it evaluates gradients at the deployed quantized weights while updating latent full-precision weights, causing the gradient to sense the valley wall and acquire an inward component that steers subsequent quantized iterates back into the basin. We formalize this mechanism through a local landscape model, construct a geometric PTQ failure mode, and prove finite-time QAT recovery under local quantizer-compatibility assumptions. Experiments across vision and language models under multiple neural-network quantization schemes corroborate the predicted basin-crossing failure of PTQ and the corresponding recovery mechanism of QAT.