arXiv subjects

Cameron J. Hogan

Publications and source records attributed to Cameron J. Hogan.

5 recordsLinked to original sources

arXiv2026

Adapt or Forget: Provable Tradeoffs Between Adam and SGD in Nonstationary Optimization

We provide a theoretical analysis of Adam under non-stationary stochastic objectives, separating two regimes: Euclidean tracking under adaptive strong monotonicity of the Adam-preconditioned mean-gradient operator, and high-probability projected stationarity guarantees under general $L$-smooth objectives. In the tracking regime, we derive finite-time expected and high-probability bounds that decompose sharply into four components: initialization, objective drift, a first-moment tracking error governed by $β_1$, and a preconditioner perturbation governed by $β_2$. We characterize the burn-in time required for the transient terms to decay to the asymptotic tracking bound under constant and step-decay schedules. We also prove a high-probability bound on the average projected stationarity gap for Adam under distribution shift. Across both analyses, our bounds reveal a noise--drift tradeoff: in noise-dominated regimes, first-moment averaging and adaptive preconditioning can yield favorable upper guarantees, whereas in drift-dominated regimes, stale first-moment information and preconditioner perturbations can enlarge Adam's tracking guarantee, potentially allowing vanilla SGD to attain a smaller tracking error. Our explicit $(β_1,β_2,ε)$-dependent bounds identify mechanisms through which adaptive step-sizing can help or hurt under nonstationarity and provide theoretical explanations consistent with Adam's empirical instability and stabilization under distribution shift.

stat.ML↗

arXiv2026

On the Provable Suboptimality of Momentum SGD in Nonstationary Stochastic Optimization

In this paper, we provide a comprehensive theoretical analysis of Stochastic Gradient Descent (SGD) and its momentum variants (Polyak Heavy-Ball and Nesterov) for tracking time-varying optima under strong convexity and smoothness. Our finite-time bounds reveal a sharp decomposition of tracking error into transient, noise-induced, and drift-induced components. This decomposition exposes a fundamental trade-off: while momentum is often used as a gradient-smoothing heuristic, under distribution shift it incurs an explicit drift-amplification penalty that diverges as the momentum parameter $β$ approaches 1, yielding systematic tracking lag. We complement these upper bounds with minimax lower bounds under gradient-variation constraints, proving this momentum-induced tracking penalty is not an analytical artifact but an information-theoretic barrier: in drift-dominated regimes, momentum is unavoidably worse because stale-gradient averaging forces systematic lag. Our results provide theoretical grounding for the empirical instability of momentum in dynamic settings and precisely delineate regime boundaries where vanilla SGD provably outperforms its accelerated counterparts.

stat.ML↗

arXiv2025

Quantum Geometry of Data

We demonstrate how Quantum Cognition Machine Learning (QCML) encodes data as quantum geometry. In QCML, features of the data are represented by learned Hermitian matrices, and data points are mapped to states in Hilbert space. The quantum geometry description endows the dataset with rich geometric and topological structure - including intrinsic dimension, quantum metric, and Berry curvature - derived directly from the data. QCML captures global properties of data, while avoiding the curse of dimensionality inherent in local methods. We illustrate this on a number of synthetic and real-world examples. Quantum geometric representation of QCML could advance our understanding of cognitive phenomena within the framework of quantum cognition.

cs.LG↗

arXiv2024

Robust estimation of the intrinsic dimension of data sets with quantum cognition machine learning

We propose a new data representation method based on Quantum Cognition Machine Learning and apply it to manifold learning, specifically to the estimation of intrinsic dimension of data sets. The idea is to learn a representation of each data point as a quantum state, encoding both local properties of the point as well as its relation with the entire data. Inspired by ideas from quantum geometry, we then construct from the quantum states a point cloud equipped with a quantum metric. The metric exhibits a spectral gap whose location corresponds to the intrinsic dimension of the data. The proposed estimator is based on the detection of this spectral gap. When tested on synthetic manifold benchmarks, our estimates are shown to be robust with respect to the introduction of point-wise Gaussian noise. This is in contrast to current state-of-the-art estimators, which tend to attribute artificial ``shadow dimensions'' to noise artifacts, leading to overestimates. This is a significant advantage when dealing with real data sets, which are inevitably affected by unknown levels of noise. We show the applicability and robustness of our method on real data, by testing it on the ISOMAP face database, MNIST, and the Wisconsin Breast Cancer Dataset.

stat.ML↗

arXiv2022

Data-adaptive Transfer Learning for Translation: A Case Study in Haitian and Jamaican

Multilingual transfer techniques often improve low-resource machine translation (MT). Many of these techniques are applied without considering data characteristics. We show in the context of Haitian-to-English translation that transfer effectiveness is correlated with amount of training data and relationships between knowledge-sharing languages. Our experiments suggest that for some languages beyond a threshold of authentic data, back-translation augmentation methods are counterproductive, while cross-lingual transfer from a sufficiently related language is preferred. We complement this finding by contributing a rule-based French-Haitian orthographic and syntactic engine and a novel method for phonological embedding. When used with multilingual techniques, orthographic transformation makes statistically significant improvements over conventional methods. And in very low-resource Jamaican MT, code-switching with a transfer language for orthographic resemblance yields a 6.63 BLEU point advantage.

cs.CL↗