Sourya Dey

Sourya Dey, Eric Davis

We present DLKoopman -- a software package for Koopman theory that uses deep learning to learn an encoding of a nonlinear dynamical system into a linear space, while simultaneously learning the linear dynamics. While several previous efforts have either restricted the ability to learn encodings, or been bespoke efforts designed for specific systems, DLKoopman is a generalized tool that can be applied to data-driven learning and optimization of any dynamical system. It can either be trained on data from individual states (snapshots) of a system and used to predict its unknown states, or trained on data from trajectories of a system and used to predict unknown trajectories for new initial states. DLKoopman is available on the Python Package Index (PyPI) as 'dlkoopman', and includes extensive documentation and tutorials. Additional contributions of the package include a novel metric called Average Normalized Absolute Error for evaluating performance, and a ready-to-use hyperparameter search module for improving performance.

Michael Robinson, Sourya Dey, Shauna Sweet

Large language models encode the correlational structure present in natural language by fitting segments of utterances (tokens) into a high dimensional ambient latent space upon which the models then operate. We assert that in order to develop a foundational, first-principles understanding of the behavior and limitations of large language models, it is crucial to understand the topological and geometric structure of this token subspace. In this article, we present estimators for the dimension and Ricci scalar curvature of the token subspace, and apply it to three open source large language models of moderate size: GPT2, LLEMMA7B, and MISTRAL7B. In all three models, using these measurements, we find that the token subspace is not a manifold, but is instead a stratified manifold, where on each of the individual strata, the Ricci curvature is significantly negative. We additionally find that the dimension and curvature correlate with generative fluency of the models, which suggest that these findings have implications for model behavior.

Michael Robinson, Sourya Dey, Tony Chiang

A full understanding of the behavior of a large language model (LLM) requires our grasp of its input token space. If this space differs from our assumptions, our comprehension of and conclusions about the LLM will likely be flawed. We elucidate the structure of the token embeddings both empirically and theoretically. We present a novel statistical test assuming that the neighborhood around each token has a relatively flat and smooth structure as the null hypothesis. Failing to reject the null is uninformative, but rejecting it at a specific token $ψ$ implies an irregularity in the token subspace in a $ψ$-neighborhood, $B(ψ)$. The structure assumed in the null is a generalization of a manifold with boundary called a \emph{smooth fiber bundle} (which can be split into two spatial regimes -- small and large radius), so we denote our new hypothesis test as the ``fiber bundle hypothesis.'' By running our test over several open-source LLMs, each with unique token embeddings, we find that the null is frequently rejected, and so the evidence suggests that the token subspace is not a fiber bundle and hence also not a manifold. As a consequence of our findings, when an LLM is presented with two semantically equivalent prompts, if one prompt contains a token implicated by our test, the response to that prompt will likely exhibit less stability than the other.

Sourya Dey, Walt Woods

This paper presents LAGOON -- an open source platform for understanding the complex ecosystems of Open Source Software (OSS) communities. The platform currently utilizes spatiotemporal graphs to store and investigate the artifacts produced by these communities, and help analysts identify bad actors who might compromise an OSS project's security. LAGOON provides ingest of artifacts from several common sources, including source code repositories, issue trackers, mailing lists and scraping content from project websites. Ingestion utilizes a modular architecture, which supports incremental updates from data sources and provides a generic identity fusion process that can recognize the same community members across disparate accounts. A user interface is provided for visualization and exploration of an OSS project's complete sociotechnical graph. Scripts are provided for applying machine learning to identify patterns within the data. While current focus is on the identification of bad actors in the Python community, the platform's reusability makes it easily extensible with new data and analyses, paving the way for LAGOON to become a comprehensive means of assessing various OSS-based projects and their communities.

Sourya Dey, Saikrishna C. Kanala, Keith M. Chugg et al.

We present Deep-n-Cheap -- an open-source AutoML framework to search for deep learning models. This search includes both architecture and training hyperparameters, and supports convolutional neural networks and multi-layer perceptrons. Our framework is targeted for deployment on both benchmark and custom datasets, and as a result, offers a greater degree of search space customizability as compared to a more limited search over only pre-existing models from literature. We also introduce the technique of 'search transfer', which demonstrates the generalization capabilities of the models found by our framework to multiple datasets. Deep-n-Cheap includes a user-customizable complexity penalty which trades off performance with training time or number of parameters. Specifically, our framework results in models offering performance comparable to state-of-the-art while taking 1-2 orders of magnitude less time to train than models from other AutoML and model search frameworks. Additionally, this work investigates and develops various insights regarding the search process. In particular, we show the superiority of a greedy strategy and justify our choice of Bayesian optimization as the primary search methodology over random / grid search.

Michael Robinson, Sourya Dey, Taisa Kushner

This article presents a general and flexible method for prompting a large language model (LLM) to reveal its (hidden) token input embedding up to homeomorphism. Moreover, this article provides strong theoretical justification -- a mathematical proof for generic LLMs -- for why this method should be expected to work. With this method in hand, we demonstrate its effectiveness by recovering the token subspace of Llemma-7B. The results of this paper apply not only to LLMs but also to general nonlinear autoregressive processes.

Sourya Dey, Diandian Chen, Zongyang Li et al.

We demonstrate an FPGA implementation of a parallel and reconfigurable architecture for sparse neural networks, capable of on-chip training and inference. The network connectivity uses pre-determined, structured sparsity to significantly reduce complexity by lowering memory and computational requirements. The architecture uses a notion of edge-processing, leading to efficient pipelining and parallelization. Moreover, the device can be reconfigured to trade off resource utilization with training time to fit networks and datasets of varying sizes. The combined effects of complexity reduction and easy reconfigurability enable significantly greater exploration of network hyperparameters and structures on-chip. As proof of concept, we show implementation results on an Artix-7 FPGA.

Sourya Dey, Peter A. Beerel, Keith M. Chugg

We propose a class of interleavers for a novel deep neural network (DNN) architecture that uses algorithmically pre-determined, structured sparsity to significantly lower memory and computational requirements, and speed up training. The interleavers guarantee clash-free memory accesses to eliminate idle operational cycles, optimize spread and dispersion to improve network performance, and are designed to ease the complexity of memory address computations in hardware. We present a design algorithm with mathematical proofs for these properties. We also explore interleaver variations and analyze the behavior of neural networks as a function of interleaver metrics.

Sourya Dey

We designed a multilayer perceptron neural network to predict the price of a football (soccer) player using data on more than 15,000 players from the football simulation video game FIFA 2017. The network was optimized by experimenting with different activation functions, number of neurons and layers, learning rate and its decay, Nesterov momentum based stochastic gradient descent, L2 regularization, and early stopping. Simultaneous exploration of various aspects of neural network training is performed and their trade-offs are investigated. Our final model achieves a top-5 accuracy of 87.2% among 119 pricing categories, and places any footballer within 6.32% of his actual price on average.