Know Your Limits: Entropy Estimation Modeling for Compression and Generalization

Language prediction is constrained by informational entropy intrinsic to language, such that there exists a limit to how accurate any language model can become and equivalently a lower bound to language compression. The most efficient language compression algorithms today are causal (next token prediction) large language models, but the use of these models to form accurate estimates of language entropy is currently computationally infeasible. We introduce encoder-augmented causal decoder model architectures that exhibit superior training efficiency characteristics and achieve higher compression than causal transformers even when trained on modest hardware. We demonstrate how entropy estimates can be obtained on a per-token basis, and show that the generalization of models trained to approach the entropy of their training data necessarily exceeds the generalization of models trained to minimize loss beyond this value. We show empirically that causal models trained to approach but not exceed estimated per-token entropies exhibit greater generalization than models trained without taking entropy into account.

Masked Mixers for Language Generation and Retrieval

Attention mechanisms that confer selective focus on a strict subset of input elements are nearly ubiquitous in language models today. We posit there to be downside to the use of attention: most information present in the input is necessarily lost. In support of this idea we observe poor input representation accuracy in transformers, but find more accurate representation in what we term masked mixers which replace self-attention with masked convolutions. Applied to TinyStories the masked mixer learns causal language tasks more efficiently than early transformer implementations and somewhat less efficiently than optimized, current implementations. The most efficient learning algorithm observed for this dataset is a transformer-masked mixer hybrid, suggesting that these models learn in an orthogonal manner. We hypothesized that the information loss exhibited by transformers would be much more detrimental to retrieval than generation, and to test this we introduce an efficient training approach for retrieval models based on existing generative model embeddings. With this method, embeddings from masked mixers are found to result in far better summary-to-story retrieval compared to embeddings from transformers.

Adversarial Examples from Dimensional Invariance

Adversarial examples have been found for various deep as well as shallow learning models, and have at various times been suggested to be either fixable model-specific bugs, or else inherent dataset feature, or both. We present theoretical and empirical results to show that adversarial examples are approximate discontinuities resulting from models that specify approximately bijective maps $f: \Bbb R^n \to \Bbb R^m; n \neq m$ over their inputs, and this discontinuity follows from the topological invariance of dimension.

Why Deep Learning Generalizes

Very large deep learning models trained using gradient descent are remarkably resistant to memorization given their huge capacity, but are at the same time capable of fitting large datasets of pure noise. Here methods are introduced by which models may be trained to memorize datasets that normally are generalized. We find that memorization is difficult relative to generalization, but that adding noise makes memorization easier. Increasing the dataset size exaggerates the characteristics of that dataset: model access to more training samples makes overfitting easier for random data, but somewhat harder for natural images. The bias of deep learning towards generalization is explored theoretically, and we show that generalization results from a model's parameters being attracted to points of maximal stability with respect to that model's inputs during gradient descent.

Depth and Representation in Vision Models

Deep learning models develop successive representations of their input in sequential layers, the last of which maps the final representation to the output. Here we investigate the informational content of these representations by observing the ability of convolutional image classification models to autoencode the model's input using embeddings existing in various layers. We find that the deeper the layer, the less accurate that layer's representation of the input is before training. Inaccurate representation results from non-uniqueness in which various distinct inputs give approximately the same embedding. Non-unique representation is a consequence of both exact and approximate non-invertibility of transformations present in the forward pass. Learning to classify natural images leads to an increase in representation clarity for early but not late layers, which instead form abstract images. Rather than simply selecting for features present in the input necessary for classification, deep layer representations are found to transform the input so that it matches representations of the training data such that arbitrary inputs are mapped to manifolds learned during training. This work provides support for the theory that the tasks of image recognition and input generation are inseparable even for models trained exclusively to classify.

Small Language Models for Tabular Data

Supervised deep learning is most commonly applied to difficult problems defined on large and often extensively curated datasets. Here we demonstrate the ability of deep representation learning to address problems of classification and regression from small and poorly formed tabular datasets by encoding input information as abstracted sequences composed of a fixed number of characters per input field. We find that small models have sufficient capacity for approximation of various functions and achieve record classification benchmark accuracy. Such models are shown to form useful embeddings of various input features in their hidden layers, even if the learned task does not explicitly require knowledge of those features. These models are also amenable to input attribution, allowing for an estimation of the importance of each input element to the model output as well as of which inputs features are effectively embedded in the model. We present a proof-of-concept for the application of small language models to mixed tabular data without explicit feature engineering, cleaning, or preprocessing, relying on the model to perform these tasks as part of the representation learning process.