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Researcher profile

Daniel M. Roy

Daniel M. Roy contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
Machine Learningmath.STStatistics TheoryArtificial IntelligenceComputationInformation TheoryLogic in Computer Sciencemath.ITmath.PRNeural and Evolutionary ComputingProgramming Languages

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Published work

10 published item(s)

preprint2026arXiv

AI co-mathematician: Accelerating mathematicians with agentic AI

We introduce the AI co-mathematician, a workbench for mathematicians to interactively leverage AI agents to pursue open-ended research. The AI co-mathematician is optimized to provide holistic support for the exploratory and iterative reality of mathematical workflows, including ideation, literature search, computational exploration, theorem proving and theory building. By providing an asynchronous, stateful workspace that manages uncertainty, refines user intent, tracks failed hypotheses, and outputs native mathematical artifacts, the system mirrors human collaborative workflows. In early tests, the AI co-mathematician helped researchers solve open problems, identify new research directions, and uncover overlooked literature references. Besides demonstrating a highly interactive paradigm for AI-assisted mathematical discovery, the AI co-mathematician also achieves state of the art results on hard problem-solving benchmarks, including scoring 48% on FrontierMath Tier 4, a new high score among all AI systems evaluated.

0 citations0 reviewsNo code linkedOpen access
preprint2023arXiv

Probabilistic programming interfaces for random graphs: Markov categories, graphons, and nominal sets

We study semantic models of probabilistic programming languages over graphs, and establish a connection to graphons from graph theory and combinatorics. We show that every well-behaved equational theory for our graph probabilistic programming language corresponds to a graphon, and conversely, every graphon arises in this way. We provide three constructions for showing that every graphon arises from an equational theory. The first is an abstract construction, using Markov categories and monoidal indeterminates. The second and third are more concrete. The second is in terms of traditional measure theoretic probability, which covers 'black-and-white' graphons. The third is in terms of probability monads on the nominal sets of Gabbay and Pitts. Specifically, we use a variation of nominal sets induced by the theory of graphs, which covers Erdős-Rényi graphons. In this way, we build new models of graph probabilistic programming from graphons.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Relaxing the I.I.D. Assumption: Adaptively Minimax Optimal Regret via Root-Entropic Regularization

We consider prediction with expert advice when data are generated from distributions varying arbitrarily within an unknown constraint set. This semi-adversarial setting includes (at the extremes) the classical i.i.d. setting, when the unknown constraint set is restricted to be a singleton, and the unconstrained adversarial setting, when the constraint set is the set of all distributions. The Hedge algorithm -- long known to be minimax (rate) optimal in the adversarial regime -- was recently shown to be simultaneously minimax optimal for i.i.d. data. In this work, we propose to relax the i.i.d. assumption by seeking adaptivity at all levels of a natural ordering on constraint sets. We provide matching upper and lower bounds on the minimax regret at all levels, show that Hedge with deterministic learning rates is suboptimal outside of the extremes, and prove that one can adaptively obtain minimax regret at all levels. We achieve this optimal adaptivity using the follow-the-regularized-leader (FTRL) framework, with a novel adaptive regularization scheme that implicitly scales as the square root of the entropy of the current predictive distribution, rather than the entropy of the initial predictive distribution. Finally, we provide novel technical tools to study the statistical performance of FTRL along the semi-adversarial spectrum.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Understanding Generalization via Leave-One-Out Conditional Mutual Information

We study the mutual information between (certain summaries of) the output of a learning algorithm and its $n$ training data, conditional on a supersample of $n+1$ i.i.d. data from which the training data is chosen at random without replacement. These leave-one-out variants of the conditional mutual information (CMI) of an algorithm (Steinke and Zakynthinou, 2020) are also seen to control the mean generalization error of learning algorithms with bounded loss functions. For learning algorithms achieving zero empirical risk under 0-1 loss (i.e., interpolating algorithms), we provide an explicit connection between leave-one-out CMI and the classical leave-one-out error estimate of the risk. Using this connection, we obtain upper and lower bounds on risk in terms of the (evaluated) leave-one-out CMI. When the limiting risk is constant or decays polynomially, the bounds converge to within a constant factor of two. As an application, we analyze the population risk of the one-inclusion graph algorithm, a general-purpose transductive learning algorithm for VC classes in the realizable setting. Using leave-one-out CMI, we match the optimal bound for learning VC classes in the realizable setting, answering an open challenge raised by Steinke and Zakynthinou (2020). Finally, in order to understand the role of leave-one-out CMI in studying generalization, we place leave-one-out CMI in a hierarchy of measures, with a novel unconditional mutual information at the root. For 0-1 loss and interpolating learning algorithms, this mutual information is observed to be precisely the risk.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Bootstrap estimators for the tail-index and for the count statistics of graphex processes

Graphex processes resolve some pathologies in traditional random graph models, notably, providing models that are both projective and allow sparsity. Most of the literature on graphex processes study them from a probabilistic point of view. Techniques for inferring the parameter of these processes -- the so-called \textit{graphon} -- are still marginal; exceptions are a few papers considering parametric families of graphons. Nonparametric estimation remains unconsidered. In this paper, we propose estimators for a selected choice of functionals of the graphon. Our estimators originate from the subsampling theory for graphex processes, hence can be seen as a form of bootstrap procedure.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

In Search of Robust Measures of Generalization

One of the principal scientific challenges in deep learning is explaining generalization, i.e., why the particular way the community now trains networks to achieve small training error also leads to small error on held-out data from the same population. It is widely appreciated that some worst-case theories -- such as those based on the VC dimension of the class of predictors induced by modern neural network architectures -- are unable to explain empirical performance. A large volume of work aims to close this gap, primarily by developing bounds on generalization error, optimization error, and excess risk. When evaluated empirically, however, most of these bounds are numerically vacuous. Focusing on generalization bounds, this work addresses the question of how to evaluate such bounds empirically. Jiang et al. (2020) recently described a large-scale empirical study aimed at uncovering potential causal relationships between bounds/measures and generalization. Building on their study, we highlight where their proposed methods can obscure failures and successes of generalization measures in explaining generalization. We argue that generalization measures should instead be evaluated within the framework of distributional robustness.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Information-Theoretic Generalization Bounds for SGLD via Data-Dependent Estimates

In this work, we improve upon the stepwise analysis of noisy iterative learning algorithms initiated by Pensia, Jog, and Loh (2018) and recently extended by Bu, Zou, and Veeravalli (2019). Our main contributions are significantly improved mutual information bounds for Stochastic Gradient Langevin Dynamics via data-dependent estimates. Our approach is based on the variational characterization of mutual information and the use of data-dependent priors that forecast the mini-batch gradient based on a subset of the training samples. Our approach is broadly applicable within the information-theoretic framework of Russo and Zou (2015) and Xu and Raginsky (2017). Our bound can be tied to a measure of flatness of the empirical risk surface. As compared with other bounds that depend on the squared norms of gradients, empirical investigations show that the terms in our bounds are orders of magnitude smaller.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Linear Mode Connectivity and the Lottery Ticket Hypothesis

We study whether a neural network optimizes to the same, linearly connected minimum under different samples of SGD noise (e.g., random data order and augmentation). We find that standard vision models become stable to SGD noise in this way early in training. From then on, the outcome of optimization is determined to a linearly connected region. We use this technique to study iterative magnitude pruning (IMP), the procedure used by work on the lottery ticket hypothesis to identify subnetworks that could have trained in isolation to full accuracy. We find that these subnetworks only reach full accuracy when they are stable to SGD noise, which either occurs at initialization for small-scale settings (MNIST) or early in training for large-scale settings (ResNet-50 and Inception-v3 on ImageNet).

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Tight Bounds on Minimax Regret under Logarithmic Loss via Self-Concordance

We consider the classical problem of sequential probability assignment under logarithmic loss while competing against an arbitrary, potentially nonparametric class of experts. We obtain tight bounds on the minimax regret via a new approach that exploits the self-concordance property of the logarithmic loss. We show that for any expert class with (sequential) metric entropy $\mathcal{O}(γ^{-p})$ at scale $γ$, the minimax regret is $\mathcal{O}(n^{p/(p+1)})$, and that this rate cannot be improved without additional assumptions on the expert class under consideration. As an application of our techniques, we resolve the minimax regret for nonparametric Lipschitz classes of experts.

0 citations0 reviewsNo code linkedOpen access
preprint2017arXiv

Sequential Monte Carlo as Approximate Sampling: bounds, adaptive resampling via $\infty$-ESS, and an application to Particle Gibbs

Sequential Monte Carlo (SMC) algorithms were originally designed for estimating intractable conditional expectations within state-space models, but are now routinely used to generate approximate samples in the context of general-purpose Bayesian inference. In particular, SMC algorithms are often used as subroutines within larger Monte Carlo schemes, and in this context, the demands placed on SMC are different: control of mean-squared error is insufficient---one needs to control the divergence from the target distribution directly. Towards this goal, we introduce the conditional adaptive resampling particle filter, building on the work of Gordon, Salmond, and Smith (1993), Andrieu, Doucet, and Holenstein (2010), and Whiteley, Lee, and Heine (2016). By controlling a novel notion of effective sample size, the $\infty$-ESS, we establish the efficiency of the resulting SMC sampling algorithm, providing an adaptive resampling extension of the work of Andrieu, Lee, and Vihola (2013). We apply our results to arrive at new divergence bounds for SMC samplers with adaptive resampling as well as an adaptive resampling version of the Particle Gibbs algorithm with the same geometric-ergodicity guarantees as its nonadaptive counterpart.

0 citations0 reviewsNo code linkedOpen access