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Evan Chen

Publications and source records attributed to Evan Chen.

At least 37 records · Page 2Linked to original sources
arXiv2025

Semi-Lie arithmetic fundamental lemma for the full spherical Hecke algebra

As an analog to the Jacquet-Rallis fundamental lemma that appears in the relative trace formula approach to the Gan-Gross-Prasad conjectures, the arithmetic fundamental lemma was proposed by Wei Zhang and used in an approach to the arithmetic Gan-Gross-Prasad conjectures. The Jacquet-Rallis fundamental lemma was recently generalized by Spencer Leslie to a statement holding for the full spherical Hecke algebra. In the same spirit, Li, Rapoport, and Zhang have recently formulated a conjectural generalization of the arithmetic fundamental lemma to the full spherical Hecke algebra. This paper formulates another analogous conjecture for the semi-Lie version of the arithmetic fundamental lemma proposed by Yifeng Liu. Then this paper produces explicit formulas for particular cases of the weighted orbital integrals in the two conjectures mentioned above, and proves the first non-trivial case of the conjecture.

math.NT↗
arXiv2025

Differentially-Private Multi-Tier Federated Learning: A Formal Analysis and Evaluation

While federated learning (FL) eliminates the transmission of raw data over a network, it is still vulnerable to privacy breaches from the communicated model parameters. Differential privacy (DP) is often employed to address such issues. However, the impact of DP on FL in multi-tier networks -- where hierarchical aggregations couple noise injection decisions at different tiers, and trust models are heterogeneous across subnetworks -- is not well understood. To fill this gap, we develop \underline{M}ulti-Tier \underline{F}ederated Learning with \underline{M}ulti-Tier \underline{D}ifferential \underline{P}rivacy ({\tt M$^2$FDP}), a DP-enhanced FL methodology for jointly optimizing privacy and performance over such networks. One of the key principles of {\tt M$^2$FDP} is to adapt DP noise injection across the established edge/fog computing hierarchy (e.g., edge devices, intermediate nodes, and other tiers up to cloud servers) according to the trust models in different subnetworks. We conduct a comprehensive analysis of the convergence behavior of {\tt M$^2$FDP} under non-convex problem settings, revealing conditions on parameter tuning under which the training process converges sublinearly to a finite stationarity gap that depends on the network hierarchy, trust model, and target privacy level. We show how these relationships can be employed to develop an adaptive control algorithm for {\tt M$^2$FDP} that tunes properties of local model training to minimize energy, latency, and the stationarity gap while meeting desired convergence and privacy criterion. Subsequent numerical evaluations demonstrate that {\tt M$^2$FDP} obtains substantial improvements in these metrics over baselines for different privacy budgets and system configurations.

cs.NI↗
arXiv2025

Gradient Correction in Federated Learning with Adaptive Optimization

In federated learning (FL), model training performance is strongly impacted by data heterogeneity across clients. Client-drift compensation methods have recently emerged as a solution to this issue, introducing correction terms into local model updates. To date, these methods have only been considered under stochastic gradient descent (SGD)-based model training, while modern FL frameworks also employ adaptive optimizers (e.g., Adam) for improved convergence. However, due to the complex interplay between first and second moments found in most adaptive optimization methods, naively injecting correction terms can lead to performance degradation in heterogeneous settings. In this work, we propose {\tt FAdamGC}, the first algorithm to integrate drift compensation into adaptive federated optimization. The key idea of {\tt FAdamGC} is injecting a pre-estimation correction term that aligns with the moment structure of adaptive methods. We provide a rigorous convergence analysis of our algorithm under non-convex settings, showing that {\tt FAdamGC} results in better rate and milder assumptions than naively porting SGD-based correction algorithms into adaptive optimizers. Our experimental results demonstrate that {\tt FAdamGC} consistently outperform existing methods in total communication and computation cost across varying levels of data heterogeneity, showing the efficacy of correcting gradient information in federated adaptive optimization.

cs.LG↗
arXiv2024

FrontierMath: A Benchmark for Evaluating Advanced Mathematical Reasoning in AI

We introduce FrontierMath, a benchmark of hundreds of original, exceptionally challenging mathematics problems crafted and vetted by expert mathematicians. The questions cover most major branches of modern mathematics -- from computationally intensive problems in number theory and real analysis to abstract questions in algebraic geometry and category theory. Solving a typical problem requires multiple hours of effort from a researcher in the relevant branch of mathematics, and for the upper end questions, multiple days. FrontierMath uses new, unpublished problems and automated verification to reliably evaluate models while minimizing risk of data contamination. Current state-of-the-art AI models solve under 2% of problems, revealing a vast gap between AI capabilities and the prowess of the mathematical community. As AI systems advance toward expert-level mathematical abilities, FrontierMath offers a rigorous testbed that quantifies their progress.

cs.AI↗
arXiv2024

Hierarchical Federated Learning with Multi-Timescale Gradient Correction

While traditional federated learning (FL) typically focuses on a star topology where clients are directly connected to a central server, real-world distributed systems often exhibit hierarchical architectures. Hierarchical FL (HFL) has emerged as a promising solution to bridge this gap, leveraging aggregation points at multiple levels of the system. However, existing algorithms for HFL encounter challenges in dealing with multi-timescale model drift, i.e., model drift occurring across hierarchical levels of data heterogeneity. In this paper, we propose a multi-timescale gradient correction (MTGC) methodology to resolve this issue. Our key idea is to introduce distinct control variables to (i) correct the client gradient towards the group gradient, i.e., to reduce client model drift caused by local updates based on individual datasets, and (ii) correct the group gradient towards the global gradient, i.e., to reduce group model drift caused by FL over clients within the group. We analytically characterize the convergence behavior of MTGC under general non-convex settings, overcoming challenges associated with couplings between correction terms. We show that our convergence bound is immune to the extent of data heterogeneity, confirming the stability of the proposed algorithm against multi-level non-i.i.d. data. Through extensive experiments on various datasets and models, we validate the effectiveness of MTGC in diverse HFL settings. The code for this project is available at \href{https://github.com/wenzhifang/MTGC}{https://github.com/wenzhifang/MTGC}.

cs.LG↗
arXiv2024

A Hierarchical Gradient Tracking Algorithm for Mitigating Subnet-Drift in Fog Learning Networks

Federated learning (FL) encounters scalability challenges when implemented over fog networks that do not follow FL's conventional star topology architecture. Semi-decentralized FL (SD-FL) has proposed a solution for device-to-device (D2D) enabled networks that divides model cooperation into two stages: at the lower stage, D2D communications is employed for local model aggregations within subnetworks (subnets), while the upper stage handles device-server (DS) communications for global model aggregations. However, existing SD-FL schemes are based on gradient diversity assumptions that become performance bottlenecks as data distributions become more heterogeneous. In this work, we develop semi-decentralized gradient tracking (SD-GT), the first SD-FL methodology that removes the need for such assumptions by incorporating tracking terms into device updates for each communication layer. Our analytical characterization of SD-GT reveals upper bounds on convergence for non-convex, convex, and strongly-convex problems. We show how the bounds enable the development of an optimization algorithm that navigates the performance-efficiency trade-off by tuning subnet sampling rate and D2D rounds for each global training interval. Our subsequent numerical evaluations demonstrate that SD-GT obtains substantial improvements in trained model quality and communication cost relative to baselines in SD-FL and gradient tracking on several datasets.

cs.NI↗
arXiv2024

Differentially-Private Multi-Tier Federated Learning

While federated learning (FL) eliminates the transmission of raw data over a network, it is still vulnerable to privacy breaches from the communicated model parameters. In this work, we propose Multi-Tier Federated Learning with Multi-Tier Differential Privacy (M^2FDP), a DP-enhanced FL methodology for jointly optimizing privacy and performance in hierarchical networks. One of the key concepts of M^2FDP is to extend the concept of HDP towards Multi-Tier Differential Privacy (MDP), while also adapting DP noise injection at different layers of an established FL hierarchy -- edge devices, edge servers, and cloud servers -- according to the trust models within particular subnetworks. We conduct a comprehensive analysis of the convergence behavior of M^2FDP, revealing conditions on parameter tuning under which the training process converges sublinearly to a finite stationarity gap that depends on the network hierarchy, trust model, and target privacy level. Subsequent numerical evaluations demonstrate that M^2FDP obtains substantial improvements in these metrics over baselines for different privacy budgets, and validate the impact of different system configurations.

cs.LG↗
arXiv2023

Taming Subnet-Drift in D2D-Enabled Fog Learning: A Hierarchical Gradient Tracking Approach

Federated learning (FL) encounters scalability challenges when implemented over fog networks. Semi-decentralized FL (SD-FL) proposes a solution that divides model cooperation into two stages: at the lower stage, device-to-device (D2D) communications is employed for local model aggregations within subnetworks (subnets), while the upper stage handles device-server (DS) communications for global model aggregations. However, existing SD-FL schemes are based on gradient diversity assumptions that become performance bottlenecks as data distributions become more heterogeneous. In this work, we develop semi-decentralized gradient tracking (SD-GT), the first SD-FL methodology that removes the need for such assumptions by incorporating tracking terms into device updates for each communication layer. Analytical characterization of SD-GT reveals convergence upper bounds for both non-convex and strongly-convex problems, for a suitable choice of step size. We employ the resulting bounds in the development of a co-optimization algorithm for optimizing subnet sampling rates and D2D rounds according to a performance-efficiency trade-off. Our subsequent numerical evaluations demonstrate that SD-GT obtains substantial improvements in trained model quality and communication cost relative to baselines in SD-FL and gradient tracking on several datasets.

cs.NI↗
arXiv2022

Cross-Resolution Flow Propagation for Foveated Video Super-Resolution

The demand of high-resolution video contents has grown over the years. However, the delivery of high-resolution video is constrained by either computational resources required for rendering or network bandwidth for remote transmission. To remedy this limitation, we leverage the eye trackers found alongside existing augmented and virtual reality headsets. We propose the application of video super-resolution (VSR) technique to fuse low-resolution context with regional high-resolution context for resource-constrained consumption of high-resolution content without perceivable drop in quality. Eye trackers provide us the gaze direction of a user, aiding us in the extraction of the regional high-resolution context. As only pixels that falls within the gaze region can be resolved by the human eye, a large amount of the delivered content is redundant as we can't perceive the difference in quality of the region beyond the observed region. To generate a visually pleasing frame from the fusion of high-resolution region and low-resolution region, we study the capability of a deep neural network of transferring the context of the observed region to other regions (low-resolution) of the current and future frames. We label this task a Foveated Video Super-Resolution (FVSR), as we need to super-resolve the low-resolution regions of current and future frames through the fusion of pixels from the gaze region. We propose Cross-Resolution Flow Propagation (CRFP) for FVSR. We train and evaluate CRFP on REDS dataset on the task of 8x FVSR, i.e. a combination of 8x VSR and the fusion of foveated region. Departing from the conventional evaluation of per frame quality using SSIM or PSNR, we propose the evaluation of past foveated region, measuring the capability of a model to leverage the noise present in eye trackers during FVSR. Code is made available at https://github.com/eugenelet/CRFP.

cs.CV↗
arXiv2020

Meta-rPPG: Remote Heart Rate Estimation Using a Transductive Meta-Learner

Remote heart rate estimation is the measurement of heart rate without any physical contact with the subject and is accomplished using remote photoplethysmography (rPPG) in this work. rPPG signals are usually collected using a video camera with a limitation of being sensitive to multiple contributing factors, e.g. variation in skin tone, lighting condition and facial structure. End-to-end supervised learning approach performs well when training data is abundant, covering a distribution that doesn't deviate too much from the distribution of testing data or during deployment. To cope with the unforeseeable distributional changes during deployment, we propose a transductive meta-learner that takes unlabeled samples during testing (deployment) for a self-supervised weight adjustment (also known as transductive inference), providing fast adaptation to the distributional changes. Using this approach, we achieve state-of-the-art performance on MAHNOB-HCI and UBFC-rPPG.

cs.CV↗
arXiv2017

Multiplicative and Exponential Variations of Orthomorphisms of Cyclic Groups

An orthomorphism is a permutation $\sigma$ of $\{1, \dots, n-1\}$ for which $x + \sigma(x) \mod n$ is also a permutation on $\{1, \dots, n-1\}$. Eberhard, Manners, Mrazovi\'c, showed that the number of such orthomorphisms is $(\sqrt{e} + o(1)) \cdot \frac{n!^2}{n^n}$ for odd $n$ and zero otherwise. In this paper we prove two analogs of these results where $x+\sigma(x)$ is replaced by $x \sigma(x)$ (a "multiplicative orthomorphism") or with $x^{\sigma(x)}$ (an "exponential orthomorphism"). Namely, we show that no multiplicative orthomorphisms exist for $n > 2$ but that exponential orthomorphisms exist whenever $n$ is twice a prime $p$ such that $p-1$ is squarefree. In the latter case we then estimate the number of exponential orthomorphisms.

math.CO↗
arXiv2017

A Family of Partially Ordered Sets with Small Balance Constant

Given a finite poset $\mathcal P$ and two distinct elements $x$ and $y$, we let $\operatorname{pr}_{\mathcal P}(x \prec y)$ denote the fraction of linear extensions of $\mathcal P$ in which $x$ precedes $y$. The balance constant $\delta(\mathcal P)$ of $\mathcal P$ is then defined by \[ \delta(\mathcal P) = \max_{x \neq y \in \mathcal P} \min \left\{ \operatorname{pr}_{\mathcal P}(x \prec y), \operatorname{pr}_{\mathcal P}(y \prec x) \right\}. \] The $1/3$-$2/3$ conjecture asserts that $\delta(\mathcal P) \ge \frac13$ whenever $\mathcal P$ is not a chain, but except from certain trivial examples it is not known when equality occurs, or even if balance constants can approach $1/3$. In this paper we make some progress on the conjecture by exhibiting a sequence of posets with balance constants approaching $\frac{1}{32}(93-\sqrt{6697}) \approx 0.3488999$, answering a question of Brightwell. These provide smaller balance constants than any other known nontrivial family.

math.CO↗
arXiv2017

Schur-Concavity for Avoidance of Increasing Subsequences in Block-Ascending Permutations

For integers $a_1, \dots, a_n \ge 0$ and $k \ge 1$, let $\mathcal L_{k+2}(a_1, \dots, a_n)$ denote the set of permutations of $\{1, \dots, a_1+\dots+a_n\}$ whose descent set is contained in $\{a_1, a_1+a_2, \dots, a_1+\dots+a_{n-1}\}$, and which avoids the pattern $12\dots(k+2)$. We exhibit some bijections between such sets, most notably showing that $\# \mathcal L_{k+2} (a_1, \dots, a_n)$ is symmetric in the $a_i$ and is in fact Schur-concave. This generalizes a set of equivalences observed by Mei and Wang.

math.CO↗
arXiv2016

The 26 Wilf-equivalence classes of length five quasi-consecutive patterns

We present two families of Wilf-equivalences for consecutive and quasi-consecutive vincular patterns. These give new proofs of the classification of consecutive patterns of length $4$ and $5$. We then prove additional equivalences to explicitly classify all quasi-consecutive patterns of length $5$ into 26 Wilf-equivalence classes.

math.CO↗
arXiv2016

Linear polychromatic colorings of hypercube faces

A coloring of the $\ell$-dimensional faces of $Q_n$ is called $d$-polychromatic if every embedded $Q_d$ has every color on at least one face. Denote by $p^\ell(d)$ the maximum number of colors such that any $Q_n$ can be colored in this way. We provide a new lower bound on $p^\ell(d)$ for $\ell > 1$.

math.CO↗
arXiv2016

Avoiding algebraic integers of bounded house in orbits of rational functions over cyclotomic closures

Let $k$ be a number field with cyclotomic closure $k^{\mathrm{cyc}}$, and let $h \in k^{\mathrm{cyc}}(x)$. For $A \ge 1$ a real number, we show that \[ \{ \alpha \in k^{\mathrm{cyc}} : h(\alpha) \in \overline{\mathbb Z} \text{ has house at most } A \} \] is finite for many $h$. We also show that for many such $h$ the same result holds if $h(\alpha)$ is replaced by orbits $h(h(\cdots h(\alpha)))$. This generalizes a result proved by Ostafe that concerns avoiding roots of unity, which is the case $A=1$.

math.NT↗