arXiv subjects

William Lu

Publications and source records attributed to William Lu.

5 recordsLinked to original sources

arXiv2026

On the Oracle Complexity of Interpolation-Based Gradient Descent

Recent work on first-order optimizers for empirical risk minimization (ERM) has suggested that smoothness of ERM loss functions in the training data, rather than in the optimization parameters, can be leveraged to improve the oracle complexity of gradient descent (GD) methods. In this paper, we propose an inexact gradient method, piecewise polynomial interpolation-based gradient descent (PPI-GD), which approximates the full gradient in each iteration by querying the first-order oracle at equidistant points in the data domain to construct polynomial interpolants of the resulting gradient samples over appropriately sized patches of the data domain. We analyze the oracle complexity of PPI-GD for strongly convex and non-convex loss functions when the data space dimension is bounded by a polylogarithmic function of the number of training samples, and find it to outperform several GD variants in key regimes when the loss function is sufficiently smooth. Furthermore, our analysis extends several techniques from the error analysis of bicubic spline interpolants to the setting of $d$-variate tensor product polynomial interpolants which may be of independent interest in interpolation analysis.

cs.LG↗

arXiv2026

Doeblin Curves

Recent research on Doeblin coefficients has shed light on their usefulness as a multi-way generalization of the Dobrushin contraction coefficient for TV distance, in a separate vein from their classic role in the theory of Markov chain ergodicity. However, strong conditions, such as being bounded away from 0, are typically necessary for Doeblin coefficients to establish the existence of information contraction. Building on recently formulated concepts of nonlinear information contraction, we aim to propose a finer-grained Doeblin-based characterization of multi-way contraction behavior which yields non-vacuous contraction guarantees even for channels whose Doeblin coefficient is 0. To this end, we introduce the notion of a Doeblin curve -- a nonlinear function which quantifies the contraction behavior of a Markov kernel on collections of input distributions at specific levels of divergence and power. Through the course of our analysis, we develop a new variational characterization of Doeblin coefficients, present several properties of Doeblin curves, define several versions of power-constrained Doeblin curves, and derive upper and lower bounds using our aforementioned variational characterization. We then utilize these results in diverse areas, including generalization bounds for noisy iterative optimization, error bounds for reliable computation with noisy circuits, and differential privacy guarantees for online iterative algorithms. In particular, we extend results in these areas to broader domains or group settings, leveraging Doeblin curves to reveal finer-grained contraction phenomena than Doeblin coefficients.

cs.IT↗

arXiv2023

On the Stability of Expressive Positional Encodings for Graphs

Designing effective positional encodings for graphs is key to building powerful graph transformers and enhancing message-passing graph neural networks. Although widespread, using Laplacian eigenvectors as positional encodings faces two fundamental challenges: (1) \emph{Non-uniqueness}: there are many different eigendecompositions of the same Laplacian, and (2) \emph{Instability}: small perturbations to the Laplacian could result in completely different eigenspaces, leading to unpredictable changes in positional encoding. Despite many attempts to address non-uniqueness, most methods overlook stability, leading to poor generalization on unseen graph structures. We identify the cause of instability to be a ``hard partition'' of eigenspaces. Hence, we introduce Stable and Expressive Positional Encodings (SPE), an architecture for processing eigenvectors that uses eigenvalues to ``softly partition'' eigenspaces. SPE is the first architecture that is (1) provably stable, and (2) universally expressive for basis invariant functions whilst respecting all symmetries of eigenvectors. Besides guaranteed stability, we prove that SPE is at least as expressive as existing methods, and highly capable of counting graph structures. Finally, we evaluate the effectiveness of our method on molecular property prediction, and out-of-distribution generalization tasks, finding improved generalization compared to existing positional encoding methods. Our code is available at \url{https://github.com/Graph-COM/SPE}.

cs.LG↗

arXiv2023

Permutation Capacity Region of Adder Multiple-Access Channels

Point-to-point permutation channels are useful models of communication networks and biological storage mechanisms and have received theoretical attention in recent years. Propelled by relevant advances in this area, we analyze the permutation adder multiple-access channel (PAMAC) in this work. In the PAMAC network model, $d$ senders communicate with a single receiver by transmitting $p$-ary codewords through an adder multiple-access channel whose output is subsequently shuffled by a random permutation block. We define a suitable notion of permutation capacity region $\mathcal{C}_\mathsf{perm}$ for this model, and establish that $\mathcal{C}_\mathsf{perm}$ is the simplex consisting of all rate $d$-tuples that sum to $d(p - 1) / 2$ or less. We achieve this sum-rate by encoding messages as i.i.d. samples from categorical distributions with carefully chosen parameters, and we derive an inner bound on $\mathcal{C}_\mathsf{perm}$ by extending the concept of time sharing to the permutation channel setting. Our proof notably illuminates various connections between mixed-radix numerical systems and coding schemes for multiple-access channels. Furthermore, we derive an alternative inner bound on $\mathcal{C}_\mathsf{perm}$ for the binary PAMAC by analyzing the root stability of the probability generating function of the adder's output distribution. Using eigenvalue perturbation results, we obtain error bounds on the spectrum of the probability generating function's companion matrix, providing quantitative estimates of decoding performance. Finally, we obtain a converse bound on $\mathcal{C}_\mathsf{perm}$ matching our achievability result.

cs.IT↗

arXiv2022

Deep object detection for waterbird monitoring using aerial imagery

Monitoring of colonial waterbird nesting islands is essential to tracking waterbird population trends, which are used for evaluating ecosystem health and informing conservation management decisions. Recently, unmanned aerial vehicles, or drones, have emerged as a viable technology to precisely monitor waterbird colonies. However, manually counting waterbirds from hundreds, or potentially thousands, of aerial images is both difficult and time-consuming. In this work, we present a deep learning pipeline that can be used to precisely detect, count, and monitor waterbirds using aerial imagery collected by a commercial drone. By utilizing convolutional neural network-based object detectors, we show that we can detect 16 classes of waterbird species that are commonly found in colonial nesting islands along the Texas coast. Our experiments using Faster R-CNN and RetinaNet object detectors give mean interpolated average precision scores of 67.9% and 63.1% respectively.

cs.CV↗