Gabe Guo

Elon Litman, Gabe Guo

We present a non-asymptotic theory of generalization in deep learning where the empirical neural tangent kernel partitions the output space. In directions corresponding to signal, error dissipates rapidly; in the vast orthogonal dimensions corresponding to noise, the kernel's near-zero eigenvalues trap residual error in a test-invisible reservoir. Within the signal channel, minibatch SGD ensures that coherent population signal accumulates via fast linear drift, while idiosyncratic memorization is suppressed into a slow, diffusive random walk. We prove generalization survives even when the kernel evolves $\mathcal{O}(1)$ in operator norm, the full feature-learning regime. This theory naturally explains disparate phenomena in deep learning theory, such as benign overfitting, double descent, implicit bias, and grokking. Lastly, we derive an exact population-risk objective from a single training run with no validation data, for any architecture, loss, or optimizer, and prove that it measures precisely the noise in the signal channel. This objective reduces in practice to an SNR preconditioner on top of Adam, adding one state vector at no extra cost; it accelerates grokking by $5 \times$, suppresses memorization in PINNs and implicit neural representations, and improves DPO fine-tuning under noisy preferences while staying $3 \times$ closer to the reference policy.

Judah Goldfeder, Quinten Roets, Gabe Guo et al.

Inferring the exact parameters of a neural network with only query access is an NP-Hard problem, with few practical existing algorithms. Solutions would have major implications for security, verification, interpretability, and understanding biological networks. The key challenges are the massive parameter space, and complex non-linear relationships between neurons. We resolve these challenges using two insights. First, we observe that almost all networks used in practice are produced by random initialization and first order optimization, an inductive bias that drastically reduces the practical parameter space. Second, we present a novel query generation algorithm that produces maximally informative samples, letting us untangle the non-linear relationships efficiently. We demonstrate reconstruction of a hidden network containing over 1.5 million parameters, and of one 7 layers deep, the largest and deepest reconstructions to date, with max parameter difference less than 0.0001, and illustrate robustness and scalability across a variety of architectures, datasets, and training procedures.

Gabe Guo, Thanawat Sornwanee, Lutong Hao et al.

Generating continuous-time, continuous-space stochastic processes (e.g., videos, weather forecasts) conditioned on partial observations (e.g., first and last frames) is a fundamental challenge. Existing approaches, (e.g., diffusion models), suffer from key limitations: (1) noise-to-data evolution fails to capture structural similarity between states close in physical time and has unstable integration in low-step regimes; (2) random noise injected is insensitive to the physical process's time elapsed, resulting in incorrect dynamics; (3) they overlook conditioning on arbitrary subsets of states (e.g., irregularly sampled timesteps, future observations). We propose ABC: Any-Subset Autoregressive Models via Non-Markovian Diffusion Bridges in Continuous Time and Space. Crucially, we model the process with one continual SDE whose time variable and intermediate states track the real time and process states. This has provable advantages: (1) the starting point for generating future states is the already-close previous state, rather than uninformative noise; (2) random noise injection scales with physical time elapsed, encouraging physically plausible dynamics with similar time-adjacent states. We derive SDE dynamics via changes-of-measure on path space, yielding another advantage: (3) path-dependent conditioning on arbitrary subsets of the state history and/or future. To learn these dynamics, we derive a path- and time-dependent extension of denoising score matching. Our experiments show ABC's superiority to competing methods on multiple domains, including video generation and weather forecasting.

Gabe Guo, Stefano Ermon

In arbitrary-order language models, it is an open question how to sample tokens in parallel from the correct joint distribution. With discrete diffusion models, the more tokens they generate in parallel, the less their predicted distributions adhere to the originally learned data distribution, as they rely on a conditional independence assumption that only works with infinitesimally small timesteps. We find that a different class of models, any-subset autoregressive models (AS-ARMs), holds the solution. As implied by the name, AS-ARMs can generate tokens in any order, and in parallel. Moreover, AS-ARMs support parallelized joint probability density estimation, allowing them to correct their own parallel-generated token distributions, via our Any-Subset Speculative Decoding (ASSD) algorithm. ASSD provably enables generation of tokens from the correct joint distribution, with the number of neural network calls upper bounded by the number of tokens predicted. We empirically verify that ASSD speeds up language generation, without sacrificing quality. Furthermore, we provide a mathematically justified scheme for training AS-ARMs for generation, and show that AS-ARMs achieve state-of-the-art performance among sub-200M parameter models on infilling benchmark tasks, and nearly match the performance of models 50X larger on code generation. Our theoretical and empirical results indicate that the once-forgotten AS-ARMs are a promising direction of language modeling.

Elon Litman, Gabe Guo

We generalize the attention mechanism by viewing it through the lens of Entropic Optimal Transport, revealing that standard attention corresponds to a transport problem regularized by an implicit uniform prior. We introduce Generalized Optimal transport Attention with Trainable priors (GOAT), a new attention mechanism that replaces this naive assumption with a learnable, continuous prior. This prior maintains full compatibility with optimized kernels such as FlashAttention. GOAT also provides an EOT-based explanation of attention sinks and materializes a solution for them, avoiding the representational trade-offs of standard attention. Finally, by absorbing spatial information into the core attention computation, GOAT learns an extrapolatable prior that combines the flexibility of learned positional embeddings with the length generalization of fixed encodings.

Gabe Guo, Tristan Saidi, Maxwell Terban et al.

A major challenge in materials science is the determination of the structure of nanometer sized objects. Here we present a novel approach that uses a generative machine learning model based on diffusion processes that is trained on 45,229 known structures. The model factors both the measured diffraction pattern as well as relevant statistical priors on the unit cell of atomic cluster structures. Conditioned only on the chemical formula and the information-scarce finite-size broadened powder diffraction pattern, we find that our model, PXRDnet, can successfully solve simulated nanocrystals as small as 10 angstroms across 200 materials of varying symmetry and complexity, including structures from all seven crystal systems. We show that our model can successfully and verifiably determine structural candidates four out of five times, with average error among these candidates being only 7% (as measured by post-Rietveld refinement R-factor). Furthermore, PXRDnet is capable of solving structures from noisy diffraction patterns gathered in real-world experiments. We suggest that data driven approaches, bootstrapped from theoretical simulation, will ultimately provide a path towards determining the structure of previously unsolved nano-materials.

Gabe Guo, Judah Goldfeder, Ling Lan et al.

The revolution in materials in the past century was built on a knowledge of the atomic arrangements and the structure-property relationship. The sine qua non for obtaining quantitative structural information is single crystal crystallography. However, increasingly we need to solve structures in cases where the information content in our input signal is significantly degraded, for example, due to orientational averaging of grains, finite size effects due to nanostructure, and mixed signals due to sample heterogeneity. Understanding the structure property relationships in such situations is, if anything, more important and insightful, yet we do not have robust approaches for accomplishing it. In principle, machine learning (ML) and deep learning (DL) are promising approaches since they augment information in the degraded input signal with prior knowledge learned from large databases of already known structures. Here we present a novel ML approach, a variational query-based multi-branch deep neural network that has the promise to be a robust but general tool to address this problem end-to-end. We demonstrate the approach on computed powder x-ray diffraction (PXRD), along with partial chemical composition information, as input. We choose as a structural representation a modified electron density we call the Cartesian mapped electron density (CMED), that straightforwardly allows our ML model to learn material structures across different chemistries, symmetries and crystal systems. When evaluated on theoretically simulated data for the cubic and trigonal crystal systems, the system achieves up to $93.4\%$ average similarity with the ground truth on unseen materials, both with known and partially-known chemical composition information, showing great promise for successful structure solution even from degraded and incomplete input data.