Symbolic regression, a task discovering the formula best fitting the given data, is typically based on the heuristical search. These methods usually update candidate formulas to obtain new ones with lower prediction errors iteratively. However, since formulas with similar function shapes may have completely different symbolic forms, the prediction error does not decrease monotonously as the search approaches the target formula, causing the low recovery rate of existing methods. To solve this problem, we propose a novel search objective based on the minimum description length, which reflects the distance from the target and decreases monotonically as the search approaches the correct form of the target formula. To estimate the minimum description length of any input data, we design a neural network, MDLformer, which enables robust and scalable estimation through large-scale training. With the MDLformer's output as the search objective, we implement a symbolic regression method, SR4MDL, that can effectively recover the correct mathematical form of the formula. Extensive experiments illustrate its excellent performance in recovering formulas from data. Our method successfully recovers around 50 formulas across two benchmark datasets comprising 133 problems, outperforming state-of-the-art methods by 43.92%.

Sparse Autoencoders (SAEs) have emerged as a useful tool for interpreting the internal representations of neural networks. However, naively optimising SAEs for reconstruction loss and sparsity results in a preference for SAEs that are extremely wide and sparse. We present an information-theoretic framework for interpreting SAEs as lossy compression algorithms for communicating explanations of neural activations. We appeal to the Minimal Description Length (MDL) principle to motivate explanations of activations which are both accurate and concise. We further argue that interpretable SAEs require an additional property, "independent additivity": features should be able to be understood separately. We demonstrate an example of applying our MDL-inspired framework by training SAEs on MNIST handwritten digits and find that SAE features representing significant line segments are optimal, as opposed to SAEs with features for memorised digits from the dataset or small digit fragments. We argue that using MDL rather than sparsity may avoid potential pitfalls with naively maximising sparsity such as undesirable feature splitting and that this framework naturally suggests new hierarchical SAE architectures which provide more concise explanations.

Rather than performing inefficient local iterative routing between adjacent capsule layers, we propose an alternative global view based on representing the inherent uncertainty in part-object assignment. In our formulation, the local routing iterations are replaced with variational inference of part-object connections in a probabilistic capsule network, leading to a significant speedup without sacrificing performance. In this way, global context is also considered when routing capsules by introducing global latent variables that have direct influence on the objective function, and are updated discriminatively in accordance with the minimum description length (MDL) principle. We focus on enhancing capsule network properties, and perform a thorough evaluation on pose-aware tasks, observing improvements in performance over previous approaches whilst being more computationally efficient.

A major challenge in designing efficient statistical supervised learning algorithms is finding representations that perform well not only on available training samples but also on unseen data. While the study of representation learning has spurred much interest, most existing such approaches are heuristic; and very little is known about theoretical generalization guarantees. For example, the information bottleneck method seeks a good generalization by finding a minimal description of the input that is maximally informative about the label variable, where minimality and informativeness are both measured by Shannon's mutual information. In this paper, we establish a compressibility framework that allows us to derive upper bounds on the generalization error of a representation learning algorithm in terms of the Minimum Description Length'' (MDL) of the labels or the latent variables (representations). Rather than the mutual information between the encoder's input and the representation, which is often believed to reflect the algorithm's generalization capability in the related literature but in fact, falls short of doing so, our new bounds involve the "multi-letter" relative entropy between the distribution of the representations (or labels) of the training and test sets and a fixed prior.

Graph pooling is an essential part of deep graph representation learning. We introduce MapEqPool, a principled pooling operator that takes the inherent hierarchical structure of real-world graphs into account. MapEqPool builds on the map equation, an information-theoretic objective function for community detection based on the minimum description length principle which naturally implements Occam's razor and balances between model complexity and fit. We demonstrate MapEqPool's competitive performance with an empirical comparison against various baselines across standard graph classification datasets.

Obtaining compositional mappings is important for the model to generalize well compositionally. To better understand when and how to encourage the model to learn such mappings, we study their uniqueness through different perspectives. Specifically, we first show that the compositional mappings are the simplest bijections through the lens of coding length (i.e., an upper bound of their Kolmogorov complexity). This property explains why models having such mappings can generalize well. We further show that the simplicity bias is usually an intrinsic property of neural network training via gradient descent. That partially explains why some models spontaneously generalize well when they are trained appropriately.

We estimate the Kolmogorov complexity of melodies in Irish traditional dance music using Lempel-Ziv compression. The "tunes" of the music are presented in so-called "ABC notation" as simply a sequence of letters from an alphabet: We have no rhythmic variation, with all notes being of equal length. Our estimation of algorithmic complexity can be used to distinguish "simple" or "easy" tunes (with more repetition) from "difficult" ones (with less repetition) which should prove useful for students learning tunes. We further present a comparison of two tune categories (reels and jigs) in terms of their complexity.

We study the task of conducting structured reasoning as generating a reasoning graph from natural language input using large language models (LLMs). Previous approaches have explored various prompting schemes, yet they suffer from error propagation due to the autoregressive nature and single-pass-based decoding, which lack error correction capability. Additionally, relying solely on a single sample may result in the omission of true nodes and edges. To counter this, we draw inspiration from self-consistency (SC), which involves sampling a diverse set of reasoning chains and taking the majority vote as the final answer. To tackle the substantial challenge of applying SC on generated graphs, we propose MIDGARD (MInimum Description length Guided Aggregation of Reasoning in Directed acyclic graph) that leverages Minimum Description Length (MDL)-based formulation to identify consistent properties among the different graph samples generated by an LLM. This formulation helps reject properties that appear in only a few samples, which are likely to be erroneous, while enabling the inclusion of missing elements without compromising precision. Our method demonstrates superior performance than comparisons across various structured reasoning tasks, including argument structure extraction, explanation graph generation, inferring dependency relations among actions for everyday tasks, and semantic graph generation from natural texts.

We present an algorithm for skill discovery from expert demonstrations. The algorithm first utilizes Large Language Models (LLMs) to propose an initial segmentation of the trajectories. Following that, a hierarchical variational inference framework incorporates the LLM-generated segmentation information to discover reusable skills by merging trajectory segments. To further control the trade-off between compression and reusability, we introduce a novel auxiliary objective based on the Minimum Description Length principle that helps guide this skill discovery process. Our results demonstrate that agents equipped with our method are able to discover skills that help accelerate learning and outperform baseline skill learning approaches on new long-horizon tasks in BabyAI, a grid world navigation environment, as well as ALFRED, a household simulation environment.

A fundamental problem associated with the task of network reconstruction from dynamical or behavioral data consists in determining the most appropriate model complexity in a manner that prevents overfitting, and produces an inferred network with a statistically justifiable number of edges. The status quo in this context is based on $L_{1}$ regularization combined with cross-validation. However, besides its high computational cost, this commonplace approach unnecessarily ties the promotion of sparsity with weight "shrinkage". This combination forces a trade-off between the bias introduced by shrinkage and the network sparsity, which often results in substantial overfitting even after cross-validation. In this work, we propose an alternative nonparametric regularization scheme based on hierarchical Bayesian inference and weight quantization, which does not rely on weight shrinkage to promote sparsity. Our approach follows the minimum description length (MDL) principle, and uncovers the weight distribution that allows for the most compression of the data, thus avoiding overfitting without requiring cross-validation. The latter property renders our approach substantially faster to employ, as it requires a single fit to the complete data. As a result, we have a principled and efficient inference scheme that can be used with a large variety of generative models, without requiring the number of edges to be known in advance. We also demonstrate that our scheme yields systematically increased accuracy in the reconstruction of both artificial and empirical networks. We highlight the use of our method with the reconstruction of interaction networks between microbial communities from large-scale abundance samples involving in the order of $10^{4}$ to $10^{5}$ species, and demonstrate how the inferred model can be used to predict the outcome of interventions in the system.