Benjamin L. Badger

Benjamin L. Badger

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.

Benjamin L. Badger

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.

Benjamin L. Badger

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.

Benjamin L. Badger

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 input information is lost. In support of this idea we observe poor input representation accuracy in transformers and more accurate representation in what we term masked mixers, which replace self-attention with masked convolutions. The masked mixer learns causal language modeling more efficiently than early transformer implementations and even outperforms optimized, current transformers when training on small ($n_{ctx}<512$) but not larger context windows. Evidence is presented for the hypothesis that differences in transformer and masked mixer training efficiencies for various tasks are best predicted by input representation accuracy, or equivalently global invertibility. We hypothesize that the information loss exhibited by transformers would be more detrimental to retrieval than generation, as the former is more closely approximated by a bijective and thus invertible function. We find that masked mixers are more effective retrieval models both when the pretrained embedding model is unchanged as well as when the embedding model is modified via cosine similarity-based InfoNCE loss minimization. A small masked mixer is shown to outperform a large and near state-of-the-art transformer-based retrieval model, despite the latter being trained with many orders of magnitude more data and compute.

Benjamin L. Badger

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.

Benjamin L. Badger

Over the last two decades, language modeling has experienced a shift from predominantly recurrent architectures that process tokens sequentially during training and inference to non-recurrent models that process sequence elements in parallel during training, which results in greater training efficiency and stability at the expense of lower inference throughput. Here we introduce the Structured Recurrent Mixer, an architecture that allows for algebraic conversion between a sequence parallel representation at train time and a recurrent representation at inference, notably without the need for specialized kernels or device-specific memory management. We show experimentally that this dual representation allows for greater training efficiency, higher input information capacity, and larger inference throughput and concurrency when compared to other linear complexity models. We postulate that recurrent models are poorly suited to extended sequence length scaling for information-rich inputs typical of language, but are well suited to scaling in the sample (batch) dimension due to their constant memory per sample. We provide Mojo/MAX inference implementations of SRMs exhibiting 12x the throughput and 170x the concurrency of similarly powerful Transformers inferenced on vLLM, increases characteristic of Pytorch implementations resulting in a 30\% increase in compute-constant GSM8k Pass@k. We conclude by demonstrating that SRMs are effective reinforcement learning training candidates.

Benjamin L. Badger

The ability of machine learning models to store input information in hidden layer vector embeddings, analogous to the concept of `memory', is widely employed but not well characterized. We find that language model embeddings typically contain relatively little input information regardless of data and compute scale during training. In contrast, embeddings from autoencoders trained for input regeneration are capable of nearly perfect memory formation. The substitution of memory embeddings for token sequences leads to substantial computational efficiencies, motivating the introduction of a parallelizable encoder-decoder memory model architecture. Upon causal training these models contain information-poor embeddings incapable of arbitrary information access, but by combining causal and information retention objective functions they learn to form and decode information-rich memories. Training can be further streamlined by freezing a high fidelity encoder followed by a curriculum training approach where decoders first learn to process memories and then learn to additionally predict next tokens. We introduce the perspective that next token prediction training alone is poorly suited for accurate memory formation as the objective itself is non-invertible, motivating the use of combined objective functions for models where the entire input is not exposed.

Benjamin L. Badger, Matthew Neligeorge

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.

Benjamin L. Badger, Ethan Roland

Transformer-based large language models are in some respects limited by the quadratic time and space computational complexity of attention. We introduce the Toeplitz MLP Mixer (TMM), a transformer-like architecture that swaps attention for triangular-masked Toeplitz matrix multiplication over the sequence dimension resulting in $\mathcal{O} (dn \log n)$ time and $\mathcal O(dn)$ space complexity during training and $\mathcal O(dn)$ time and space at inference prefill. Despite the lack of sophisticated input modulation or state maintenance present in other sub-quadratic architectures, TMMs yield greater training efficiency in terms of loss achieved per compute and device memory. We demonstrate that TMMs are capable of retaining more input information resulting in improved copying ability, which we argue results from a lack of architectural biases. Consistent with higher input information retention, TMMs exhibit superior information retrieval and in-context learning benchmark accuracy compared to comparable architectures. We conclude with an analysis from the perspective of operator index theory and show that, counterintuitively, trained Toeplitz layers of causal non-invertible models are more likely to be invertible or nearly so than models that are actually invertible over their inputs.