arXiv subjects

Hadrien Hendrikx

Publications and source records attributed to Hadrien Hendrikx.

At least 19 recordsLinked to original sources

arXiv2026

Convergence of projected stochastic natural gradient variational inference for various step size and sample or batch size schedules

Stochastic natural gradient variational inference (NGVI) is a popular and efficient algorithm for Bayesian inference. Despite empirical success, the convergence of this method is still not fully understood. In this work, we define and study a projected stochastic NGVI when variational distributions form an exponential family. Stochasticity arises when either gradients are intractable expectations or large sums. We prove new non-asymptotic convergence results for combinations of constant or decreasing step sizes and constant or increasing sample/batch sizes. When all hyperparameters are fixed, NGVI is shown to converge geometrically to a neighborhood of the optimum, while we establish convergence to the optimum with rates of the form $\mathcal{O}\left(\frac{1}{T^{\rho}} \right)$, possibly with $\rho \geq 1$, for all other combinations of step size and sample/batch size schedules. These rates apply when the target posterior distribution is close in some sense to the considered exponential family. Our theoretical results extend existing NGVI and stochastic optimization results and provide more flexibility to adjust, in a principled way, step sizes and sample/batch sizes in order to meet speed, resources, or accuracy constraints.

stat.ME↗

arXiv2026

From Inexact Gradients to Byzantine Robustness: Acceleration and Optimization under Similarity

Standard federated learning algorithms are vulnerable to adversarial nodes, a.k.a. Byzantine failures. To solve this issue, robust distributed learning algorithms have been developed, which typically replace parameter averaging by robust aggregations. While generic conditions on these aggregations exist to guarantee the convergence of (Stochastic) Gradient Descent (SGD), the analyses remain rather ad-hoc. This hinders the development of more complex robust algorithms, such as accelerated ones. In this work, we show that Byzantine-robust distributed optimization can, under standard generic assumptions, be cast as a general optimization with inexact gradient oracles (with both additive and multiplicative error terms), an active field of research. This allows for instance to directly show that GD on top of standard robust aggregation procedures obtains optimal asymptotic error in the Byzantine setting. Going further, we propose two optimization schemes to speed up the convergence. The first one is a Nesterov-type accelerated scheme whose proof directly derives from accelerated inexact gradient results applied to our formulation. The second one hinges on Optimization under Similarity, in which the server leverages an auxiliary loss function that approximates the global loss. Both approaches allow to drastically reduce the communication complexity compared to previous methods, as we show theoretically and empirically.

cs.LG↗

arXiv2025

BotaCLIP: Contrastive Learning for Botany-Aware Representation of Earth Observation Data

Foundation models have demonstrated a remarkable ability to learn rich, transferable representations across diverse modalities such as images, text, and audio. In modern machine learning pipelines, these representations often replace raw data as the primary input for downstream tasks. In this paper, we address the challenge of adapting a pre-trained foundation model to inject domain-specific knowledge, without retraining from scratch or incurring significant computational costs. To this end, we introduce BotaCLIP, a lightweight multimodal contrastive framework that adapts a pre-trained Earth Observation foundation model (DOFA) by aligning high-resolution aerial imagery with botanical relev\'es. Unlike generic embeddings, BotaCLIP internalizes ecological structure through contrastive learning with a regularization strategy that mitigates catastrophic forgetting. Once trained, the resulting embeddings serve as transferable representations for downstream predictors. Motivated by real-world applications in biodiversity modeling, we evaluated BotaCLIP representations in three ecological tasks: plant presence prediction, butterfly occurrence modeling, and soil trophic group abundance estimation. The results showed consistent improvements over those derived from DOFA and supervised baselines. More broadly, this work illustrates how domain-aware adaptation of foundation models can inject expert knowledge into data-scarce settings, enabling frugal representation learning.

cs.CV↗

arXiv2024

Exponential Moving Average of Weights in Deep Learning: Dynamics and Benefits

Weight averaging of Stochastic Gradient Descent (SGD) iterates is a popular method for training deep learning models. While it is often used as part of complex training pipelines to improve generalization or serve as a `teacher' model, weight averaging lacks proper evaluation on its own. In this work, we present a systematic study of the Exponential Moving Average (EMA) of weights. We first explore the training dynamics of EMA, give guidelines for hyperparameter tuning, and highlight its good early performance, partly explaining its success as a teacher. We also observe that EMA requires less learning rate decay compared to SGD since averaging naturally reduces noise, introducing a form of implicit regularization. Through extensive experiments, we show that EMA solutions differ from last-iterate solutions. EMA models not only generalize better but also exhibit improved i) robustness to noisy labels, ii) prediction consistency, iii) calibration and iv) transfer learning. Therefore, we suggest that an EMA of weights is a simple yet effective plug-in to improve the performance of deep learning models.

cs.LG↗

arXiv2024

Unified Breakdown Analysis for Byzantine Robust Gossip

In decentralized machine learning, different devices communicate in a peer-to-peer manner to collaboratively learn from each other's data. Such approaches are vulnerable to misbehaving (or Byzantine) devices. We introduce F-RG, a general framework for building robust decentralized algorithms with guarantees arising from robust-sum-like aggregation rules F. We then investigate the notion of *breakdown point*, and show an upper bound on the number of adversaries that decentralized algorithms can tolerate. We introduce a practical robust aggregation rule, coined CS+, such that CS+-RG has a near-optimal breakdown. Other choices of aggregation rules lead to existing algorithms such as ClippedGossip or NNA. We give experimental evidence to validate the effectiveness of CS+-RG and highlight the gap with NNA, in particular against a novel attack tailored to decentralized communications.

math.OC↗

arXiv2024

Byzantine-Robust Gossip: Insights from a Dual Approach

Distributed learning has many computational benefits but is vulnerable to attacks from a subset of devices transmitting incorrect information. This paper investigates Byzantine-resilient algorithms in a decentralized setting, where devices communicate directly in a peer-to-peer manner within a communication network. We leverage the so-called dual approach for decentralized optimization and propose a Byzantine-robust algorithm. We provide convergence guarantees in the average consensus subcase, discuss the potential of the dual approach beyond this subcase, and re-interpret existing algorithms using the dual framework. Lastly, we experimentally show the soundness of our method.

cs.LG↗

arXiv2024

Investigating Variance Definitions for Mirror Descent with Relative Smoothness

Mirror Descent is a popular algorithm, that extends Gradients Descent (GD) beyond the Euclidean geometry. One of its benefits is to enable strong convergence guarantees through smooth-like analyses, even for objectives with exploding or vanishing curvature. This is achieved through the introduction of the notion of relative smoothness, which holds in many of the common use-cases of Mirror descent. While basic deterministic results extend well to the relative setting, most existing stochastic analyses require additional assumptions on the mirror, such as strong convexity (in the usual sense), to ensure bounded variance. In this work, we revisit Stochastic Mirror Descent (SMD) proofs in the (relatively-strongly-) convex and relatively-smooth setting, and introduce a new (less restrictive) definition of variance which can generally be bounded (globally) under mild regularity assumptions. We then investigate this notion in more details, and show that it naturally leads to strong convergence guarantees for stochastic mirror descent. Finally, we leverage this new analysis to obtain convergence guarantees for the Maximum Likelihood Estimator of a Gaussian with unknown mean and variance.

math.OC↗

arXiv2024

The Relative Gaussian Mechanism and its Application to Private Gradient Descent

The Gaussian Mechanism (GM), which consists in adding Gaussian noise to a vector-valued query before releasing it, is a standard privacy protection mechanism. In particular, given that the query respects some L2 sensitivity property (the L2 distance between outputs on any two neighboring inputs is bounded), GM guarantees Rényi Differential Privacy (RDP). Unfortunately, precisely bounding the L2 sensitivity can be hard, thus leading to loose privacy bounds. In this work, we consider a Relative L2 sensitivity assumption, in which the bound on the distance between two query outputs may also depend on their norm. Leveraging this assumption, we introduce the Relative Gaussian Mechanism (RGM), in which the variance of the noise depends on the norm of the output. We prove tight bounds on the RDP parameters under relative L2 sensitivity, and characterize the privacy loss incurred by using output-dependent noise. In particular, we show that RGM naturally adapts to a latent variable that would control the norm of the output. Finally, we instantiate our framework to show tight guarantees for Private Gradient Descent, a problem that naturally fits our relative L2 sensitivity assumption.

cs.LG↗

arXiv2023

Revisiting Gradient Clipping: Stochastic bias and tight convergence guarantees

Gradient clipping is a popular modification to standard (stochastic) gradient descent, at every iteration limiting the gradient norm to a certain value $c >0$. It is widely used for example for stabilizing the training of deep learning models (Goodfellow et al., 2016), or for enforcing differential privacy (Abadi et al., 2016). Despite popularity and simplicity of the clipping mechanism, its convergence guarantees often require specific values of $c$ and strong noise assumptions. In this paper, we give convergence guarantees that show precise dependence on arbitrary clipping thresholds $c$ and show that our guarantees are tight with both deterministic and stochastic gradients. In particular, we show that (i) for deterministic gradient descent, the clipping threshold only affects the higher-order terms of convergence, (ii) in the stochastic setting convergence to the true optimum cannot be guaranteed under the standard noise assumption, even under arbitrary small step-sizes. We give matching upper and lower bounds for convergence of the gradient norm when running clipped SGD, and illustrate these results with experiments.

cs.LG↗

arXiv2023

Beyond spectral gap (extended): The role of the topology in decentralized learning

In data-parallel optimization of machine learning models, workers collaborate to improve their estimates of the model: more accurate gradients allow them to use larger learning rates and optimize faster. In the decentralized setting, in which workers communicate over a sparse graph, current theory fails to capture important aspects of real-world behavior. First, the `spectral gap' of the communication graph is not predictive of its empirical performance in (deep) learning. Second, current theory does not explain that collaboration enables larger learning rates than training alone. In fact, it prescribes smaller learning rates, which further decrease as graphs become larger, failing to explain convergence dynamics in infinite graphs. This paper aims to paint an accurate picture of sparsely-connected distributed optimization. We quantify how the graph topology influences convergence in a quadratic toy problem and provide theoretical results for general smooth and (strongly) convex objectives. Our theory matches empirical observations in deep learning, and accurately describes the relative merits of different graph topologies. This paper is an extension of the conference paper by Vogels et. al. (2022). Code: https://github.com/epfml/topology-in-decentralized-learning.

cs.LG↗

arXiv2022

Beyond spectral gap: The role of the topology in decentralized learning

In data-parallel optimization of machine learning models, workers collaborate to improve their estimates of the model: more accurate gradients allow them to use larger learning rates and optimize faster. We consider the setting in which all workers sample from the same dataset, and communicate over a sparse graph (decentralized). In this setting, current theory fails to capture important aspects of real-world behavior. First, the 'spectral gap' of the communication graph is not predictive of its empirical performance in (deep) learning. Second, current theory does not explain that collaboration enables larger learning rates than training alone. In fact, it prescribes smaller learning rates, which further decrease as graphs become larger, failing to explain convergence in infinite graphs. This paper aims to paint an accurate picture of sparsely-connected distributed optimization when workers share the same data distribution. We quantify how the graph topology influences convergence in a quadratic toy problem and provide theoretical results for general smooth and (strongly) convex objectives. Our theory matches empirical observations in deep learning, and accurately describes the relative merits of different graph topologies.

cs.LG↗

arXiv2022

Asynchronous speedup in decentralized optimization

In decentralized optimization, nodes of a communication network each possess a local objective function, and communicate using gossip-based methods in order to minimize the average of these per-node functions. While synchronous algorithms are heavily impacted by a few slow nodes or edges in the graph (the \emph{straggler problem}), their asynchronous counterparts are notoriously harder to parametrize. Indeed, their convergence properties for networks with heterogeneous communication and computation delays have defied analysis so far. In this paper, we use a \emph{ continuized} framework to analyze asynchronous algorithms in networks with delays. Our approach yields a precise characterization of convergence time and of its dependency on heterogeneous delays in the network. Our continuized framework benefits from the best of both continuous and discrete worlds: the algorithms it applies to are based on event-driven updates. They are thus essentially discrete and hence readily implementable. Yet their analysis is essentially in continuous time, relying in part on the theory of delayed ODEs. Our algorithms moreover achieve an \emph{asynchronous speedup}: their rate of convergence is controlled by the eigengap of the network graph weighted by local delays, instead of the network-wide worst-case delay as in previous analyses. Our methods thus enjoy improved robustness to stragglers.

math.OC↗

arXiv2022

A principled framework for the design and analysis of token algorithms

We consider a decentralized optimization problem, in which $n$ nodes collaborate to optimize a global objective function using local communications only. While many decentralized algorithms focus on \emph{gossip} communications (pairwise averaging), we consider a different scheme, in which a ``token'' that contains the current estimate of the model performs a random walk over the network, and updates its model using the local model of the node it is at. Indeed, token algorithms generally benefit from improved communication efficiency and privacy guarantees. We frame the token algorithm as a randomized gossip algorithm on a conceptual graph, which allows us to prove a series of convergence results for variance-reduced and accelerated token algorithms for the complete graph. We also extend these results to the case of multiple tokens by extending the conceptual graph, and to general graphs by tweaking the communication procedure. The reduction from token to well-studied gossip algorithms leads to tight rates for many token algorithms, and we illustrate their performance empirically.

math.OC↗

arXiv2021

A Continuized View on Nesterov Acceleration for Stochastic Gradient Descent and Randomized Gossip

We introduce the continuized Nesterov acceleration, a close variant of Nesterov acceleration whose variables are indexed by a continuous time parameter. The two variables continuously mix following a linear ordinary differential equation and take gradient steps at random times. This continuized variant benefits from the best of the continuous and the discrete frameworks: as a continuous process, one can use differential calculus to analyze convergence and obtain analytical expressions for the parameters; and a discretization of the continuized process can be computed exactly with convergence rates similar to those of Nesterov original acceleration. We show that the discretization has the same structure as Nesterov acceleration, but with random parameters. We provide continuized Nesterov acceleration under deterministic as well as stochastic gradients, with either additive or multiplicative noise. Finally, using our continuized framework and expressing the gossip averaging problem as the stochastic minimization of a certain energy function, we provide the first rigorous acceleration of asynchronous gossip algorithms.

math.OC↗

arXiv2021

Fast Stochastic Bregman Gradient Methods: Sharp Analysis and Variance Reduction

We study the problem of minimizing a relatively-smooth convex function using stochastic Bregman gradient methods. We first prove the convergence of Bregman Stochastic Gradient Descent (BSGD) to a region that depends on the noise (magnitude of the gradients) at the optimum. In particular, BSGD with a constant step-size converges to the exact minimizer when this noise is zero (\emph{interpolation} setting, in which the data is fit perfectly). Otherwise, when the objective has a finite sum structure, we show that variance reduction can be used to counter the effect of noise. In particular, fast convergence to the exact minimizer can be obtained under additional regularity assumptions on the Bregman reference function. We illustrate the effectiveness of our approach on two key applications of relative smoothness: tomographic reconstruction with Poisson noise and statistical preconditioning for distributed optimization.

math.OC↗

arXiv2021

Asynchrony and Acceleration in Gossip Algorithms

This paper considers the minimization of a sum of smooth and strongly convex functions dispatched over the nodes of a communication network. Previous works on the subject either focus on synchronous algorithms, which can be heavily slowed down by a few slow nodes (the straggler problem), or consider a model of asynchronous operation (Boyd et al., 2006) in which adjacent nodes communicate at the instants of Poisson point processes. We have two main contributions. 1) We propose CACDM (a Continuously Accelerated Coordinate Dual Method), and for the Poisson model of asynchronous operation, we prove CACDM to converge to optimality at an accelerated convergence rate in the sense of Nesterov et Stich, 2017. In contrast, previously proposed asynchronous algorithms have not been proven to achieve such accelerated rate. While CACDM is based on discrete updates, the proof of its convergence crucially depends on a continuous time analysis. 2) We introduce a new communication scheme based on Loss-Networks, that is programmable in a fully asynchronous and decentralized way, unlike the Poisson model of asynchronous operation that does not capture essential aspects of asynchrony such as non-instantaneous communications and computations. Under this Loss-Network model of asynchrony, we establish for CDM (a Coordinate Dual Method) a rate of convergence in terms of the eigengap of the Laplacian of the graph weighted by local effective delays. We believe this eigengap to be a fundamental bottleneck for convergence rates of asynchronous optimization. Finally, we verify empirically that CACDM enjoys an accelerated convergence rate in the Loss-Network model of asynchrony.

cs.DC↗

arXiv2020

Who started this rumor? Quantifying the natural differential privacy guarantees of gossip protocols

Gossip protocols are widely used to disseminate information in massive peer-to-peer networks. These protocols are often claimed to guarantee privacy because of the uncertainty they introduce on the node that started the dissemination. But is that claim really true? Can the source of a gossip safely hide in the crowd? This paper examines, for the first time, gossip protocols through a rigorous mathematical framework based on differential privacy to determine the extent to which the source of a gossip can be traceable. Considering the case of a complete graph in which a subset of the nodes are curious, we study a family of gossip protocols parameterized by a ``muting'' parameter $s$: nodes stop emitting after each communication with a fixed probability $1-s$. We first prove that the standard push protocol, corresponding to the case $s=1$, does not satisfy differential privacy for large graphs. In contrast, the protocol with $s=0$ achieves optimal privacy guarantees but at the cost of a drastic increase in the spreading time compared to standard push, revealing an interesting tension between privacy and spreading time. Yet, surprisingly, we show that some choices of the muting parameter $s$ lead to protocols that achieve an optimal order of magnitude in both privacy and speed. We also confirm empirically that, with appropriate choices of $s$, we indeed obtain protocols that are very robust against concrete source location attacks while spreading the information almost as fast as the standard (and non-private) push protocol.

cs.DC↗

arXiv2020

Dual-Free Stochastic Decentralized Optimization with Variance Reduction

We consider the problem of training machine learning models on distributed data in a decentralized way. For finite-sum problems, fast single-machine algorithms for large datasets rely on stochastic updates combined with variance reduction. Yet, existing decentralized stochastic algorithms either do not obtain the full speedup allowed by stochastic updates, or require oracles that are more expensive than regular gradients. In this work, we introduce a Decentralized stochastic algorithm with Variance Reduction called DVR. DVR only requires computing stochastic gradients of the local functions, and is computationally as fast as a standard stochastic variance-reduced algorithms run on a $1/n$ fraction of the dataset, where $n$ is the number of nodes. To derive DVR, we use Bregman coordinate descent on a well-chosen dual problem, and obtain a dual-free algorithm using a specific Bregman divergence. We give an accelerated version of DVR based on the Catalyst framework, and illustrate its effectiveness with simulations on real data.

math.OC↗