Pedram Bakhtiarifard

Pedram Bakhtiarifard, Sophia N. Wilson, Mahmoud Afifi et al.

Training large-scale deep neural networks (DNNs) is resource-intensive, making model compression a practical necessity. The widely accepted ''learning as compression'' hypothesis posits that training induces structure in network weights, which enables compression. Measuring this structure through Kolmogorov-Chaitin-Solomonoff (KCS) complexity is appealing, but existing estimators based on the Coding Theorem Method (CTM) and the Block Decomposition Method (BDM) are limited to small binary objects and do not scale to modern DNNs. We introduce the Quantized Block Decomposition method (QuBD), which extends algorithmic complexity estimation to any $k$-ary object. QuBD first quantizes the network weights to a finite alphabet, then estimates the KCS complexity by aggregating per bit-plane CTM estimates. We show theoretically that QuBD yields a strictly tighter estimation gap with respect to true KCS complexity than binarization-based methods. Using QuBD, we study how the algorithmic complexity of neural network weights evolves during training, showing that it decreases as models learn, scales with data budget, increases during overfitting, follows the delayed generalization observed during grokking, and correlates with generalization performance. We further show that algorithmic information resides predominantly in the most significant bit-planes, which can serve as a practical diagnostic for determining appropriate post-training quantization levels. This work offers novel insights into learning mechanisms in DNNs by providing the first scalable, tractable estimates of KCS complexity for large, non-binary objects such as DNN weights.

Pedram Bakhtiarifard, Tong Chen, Jonathan Wenshøj et al.

Large-scale deep learning models are well-suited for compression. Methods like pruning, quantization, and knowledge distillation have been used to achieve massive reductions in the number of model parameters, with marginal performance drops across a variety of architectures and tasks. This raises the central question: \emph{Why are deep neural networks suited for compression?} In this work, we take up the perspective of algorithmic complexity to explain this behavior. We hypothesize that the parameters of trained models have more structure and, hence, exhibit lower algorithmic complexity compared to the weights at (random) initialization. Furthermore, that model compression methods harness this reduced algorithmic complexity to compress models. Although an unconstrained parameterization of model weights, $\mathbf{w} \in \mathbb{R}^n$, can represent arbitrary weight assignments, the solutions found during training exhibit repeatability and structure, making them algorithmically simpler than a generic program. To this end, we formalize the Kolmogorov complexity of $\mathbf{w}$ by $\mathcal{K}(\mathbf{w})$. We introduce a constrained parameterization $\widehat{\mathbf{w}}$, that partitions parameters into blocks of size $s$, and restricts each block to be selected from a set of $k$ reusable motifs, specified by a reuse pattern (or mosaic). The resulting method, $\textit{Mosaic-of-Motifs}$ (MoMos), yields algorithmically simpler model parameterization compared to unconstrained models. Empirical evidence from multiple experiments shows that the algorithmic complexity of neural networks, measured using approximations to Kolmogorov complexity, can be reduced during training. This results in models that perform comparably with unconstrained models while being algorithmically simpler.