Study of the nonlinear evolution deep neural network (DNN) parameters undergo during training has uncovered regimes of distinct dynamical behavior. While a detailed understanding of these phenomena has the potential to advance improvements in training efficiency and robustness, the lack of methods for identifying when DNN models have equivalent dynamics limits the insight that can be gained from prior work. Topological conjugacy, a notion from dynamical systems theory, provides a precise definition of dynamical equivalence, offering a possible route to address this need. However, topological conjugacies have historically been challenging to compute. By leveraging advances in Koopman operator theory, we develop a framework for identifying conjugate and non-conjugate training dynamics. To validate our approach, we demonstrate that comparing Koopman eigenvalues can correctly identify a known equivalence between online mirror descent and online gradient descent. We then utilize our approach to: (a) identify non-conjugate training dynamics between shallow and wide fully connected neural networks; (b) characterize the early phase of training dynamics in convolutional neural networks; (c) uncover non-conjugate training dynamics in Transformers that do and do not undergo grokking. Our results, across a range of DNN architectures, illustrate the flexibility of our framework and highlight its potential for shedding new light on training dynamics.

A critical question about Large Language Models (LLMs) is whether their apparent deficiency in mathematical reasoning is inherent, or merely a result of insufficient exposure to high-quality mathematical data. To explore this, we developed an automated method for generating high-quality, supervised mathematical datasets. The method carefully mutates existing math problems, ensuring both diversity and validity of the newly generated problems. This is achieved by a neuro-symbolic data generation framework combining the intuitive informalization strengths of LLMs, and the precise symbolic reasoning of math solvers along with projected Markov chain Monte Carlo sampling in the highly-irregular symbolic space. Empirical experiments demonstrate the high quality of data generated by the proposed method, and that the LLMs, specifically LLaMA-2 and Mistral, when realigned with the generated data, surpass their state-of-the-art counterparts.

Self-play via online learning is one of the premier ways to solve large-scale twoplayer zero-sum games, both in theory and practice. Particularly popular algorithms include optimistic multiplicative weights update (OMWU) and optimistic gradient-descent-ascent (OGDA). While both algorithms enjoy O(1/T) ergodic convergence to Nash equilibrium in two-player zero-sum games, OMWU offers several advantages including logarithmic dependence on the size of the payoff matrix and Õ(1/T) convergence to coarse correlated equilibria even in generalsum games. However, in terms of last-iterate convergence in two-player zero-sum games, an increasingly popular topic in this area, OGDA guarantees that the duality gap shrinks at a rate of (1/ T), while the best existing last-iterate convergence for OMWU depends on some game-dependent constant that could be arbitrarily large. This begs the question: is this potentially slow last-iterate convergence an inherent disadvantage of OMWU, or is the current analysis too loose? Somewhat surprisingly, we show that the former is true. More generally, we prove that a broad class of algorithms that do not forget the past quickly all suffer the same issue: for any arbitrarily small δ > 0, there exists a 2 2 matrix game such that the algorithm admits a constant duality gap even after 1/δ rounds. This class of algorithms includes OMWU and other standard optimistic follow-the-regularized-leader algorithms.

Functional magnetic resonance imaging (fMRI) is an indispensable tool in modern neuroscience, providing a non-invasive window into whole-brain dynamics at millimeter-scale spatial resolution. However, fMRI is constrained by issues such as high operation costs and immobility. With the rapid advancements in crossmodality synthesis and brain decoding, the use of deep neural networks has emerged as a promising solution for inferring whole-brain, high-resolution fMRI features directly from electroencephalography (EEG), a more widely accessible and portable neuroimaging modality. Nonetheless, the complex projection from neural activity to fMRI hemodynamic responses and the spatial ambiguity of EEG pose substantial challenges both in modeling and interpretability. Relatively few studies to date have developed approaches for EEG-fMRI translation, and although they have made significant strides, the inference of fMRI signals in a given study has been limited to a small set of brain areas and to a single condition (i.e., either resting-state or a specific task). The capability to predict fMRI signals in other brain areas, as well as to generalize across conditions, remain critical gaps in the field.

Despite extensive research on adversarial training strategies to improve robustness, the decisions of even the most robust deep learning models can still be quite sensitive to imperceptible perturbations, creating serious risks when deploying them for high-stakes real-world applications. While detecting such cases may be critical, evaluating a model's vulnerability at a per-instance level using adversarial attacks is computationally too intensive and unsuitable for real-time deployment scenarios. The input space margin is the exact score to detect non-robust samples and is intractable for deep neural networks. This paper introduces the concept of margin consistency - a property that links the input space margins and the logit margins in robust models - for efficient detection of vulnerable samples. First, we establish that margin consistency is a necessary and sufficient condition to use a model's logit margin as a score for identifying non-robust samples. Next, through comprehensive empirical analysis of various robustly trained models on CIFAR10 and CIFAR100 datasets, we show that they indicate high margin consistency with a strong correlation between their input space margins and the logit margins. Then, we show that we can effectively and confidently use the logit margin to detect brittle decisions with such models. Finally, we address cases where the model is not sufficiently margin-consistent by learning a pseudo-margin from the feature representation. Our findings highlight the potential of leveraging deep representations to assess adversarial vulnerability in deployment scenarios efficiently.

Modelling partial differential equations (PDEs) is of crucial importance in science and engineering, and it includes tasks ranging from forecasting to inverse problems, such as data assimilation. However, most previous numerical and machine learning approaches that target forecasting cannot be applied out-of-the-box for data assimilation. Recently, diffusion models have emerged as a powerful tool for conditional generation, being able to flexibly incorporate observations without retraining. In this work, we perform a comparative study of score-based diffusion models for forecasting and assimilation of sparse observations. In particular, we focus on diffusion models that are either trained in a conditional manner, or conditioned after unconditional training. We address the shortcomings of existing models by proposing 1) an autoregressive sampling approach, that significantly improves performance in forecasting, 2) a new training strategy for conditional score-based models that achieves stable performance over a range of history lengths, and 3) a hybrid model which employs flexible pre-training conditioning on initial conditions and flexible posttraining conditioning to handle data assimilation. We empirically show that these modifications are crucial for successfully tackling the combination of forecasting and data assimilation, a task commonly encountered in real-world scenarios.

Without privacy constraints, the standard estimators for this task are U-statistics, which commonly arise in a wide range of problems, including nonparametric signed rank tests, symmetry testing, uniformity testing, and subgraph counts in random networks, and are the unique minimum variance unbiased estimators under mild conditions. Despite the recent outpouring of interest in private mean estimation, privatizing U-statistics has received little attention. While existing private mean estimation algorithms can be applied in a black-box manner to obtain confidence intervals, we show that they can lead to suboptimal private error, e.g., constant-factor inflation in the leading term, or even Θ(1/n) rather than O(1/n

Federated learning faces challenges due to the heterogeneity in data volumes and distributions at different clients, which can compromise model generalization ability to various distributions. Existing approaches to address this issue based on group distributionally robust optimization (GDRO) often lead to high communication and sample complexity. To this end, this work introduces algorithms tailored for communication-efficient Federated Group Distributionally Robust Optimization (FGDRO).

High-level visual brain regions contain subareas in which neurons appear to respond more strongly to examples of a particular semantic category, like faces or bodies, rather than objects. However, recent work has shown that while this finding holds on average, some out-of-category stimuli also activate neurons in these regions. This may be due to visual features common among the preferred class also being present in other images. Here, we propose a deep-learning-based approach for visualizing these features. For each neuron, we identify relevant visual features driving its selectivity by modelling responses to images based on latent activations of a deep neural network.

The emergence of Large Language Models (LLMs) has significantly advanced natural language processing, but these models often generate factually incorrect information, known as "hallucination". Initial retrieval-augmented generation (RAG) methods like the "Retrieve-Read" framework was inadequate for complex reasoning tasks. Subsequent prompt-based RAG strategies and Supervised Fine-Tuning (SFT) methods improved performance but required frequent retraining and risked altering foundational LLM capabilities.