arXiv subjects

Simon Lacoste-Julien

Publications and source records attributed to Simon Lacoste-Julien.

At least 19 recordsLinked to original sources

arXiv2026

Lloyd's $K$-Means Clustering Algorithm Is Frank-Wolfe in Disguise

Lloyd's $K$-means algorithm, also known as na\"{i}ve $K$-means, is a widely used ad hoc optimization heuristic, designed to minimize the sum of squared errors (SSE) across all $K$-partitions of a dataset via iterative cluster refinement. In this work, we establish a novel connection between Lloyd's algorithm and the Frank-Wolfe (FW) algorithm, a prominent first-order method for projection-free optimization. We demonstrate that Lloyd's algorithm is a special case of FW. Leveraging recent advances in FW methods for concave objectives, we derive a non-asymptotic $\mathcal{O}(1/t)$ convergence rate to a local minimum of the SSE objective. To account for empty clusters, an outcome possible under Lloyd's greedy assignment, we develop an FW variant for semismooth objectives while retaining the same convergence rate that is solely controlled by the initial SSE value. We illustrate our findings with a simulation study for spherical Gaussian mixtures and a real-world image segmentation dataset.

stat.ML↗

arXiv2026

Accelerated and Stable Convergence with Anchored Generalized Optimistic Method

We study first-order methods for solving monotone variational inequalities arising in min-max optimization. Classical approaches such as the extragradient method rely on two gradient queries per iteration, which limits their analysis and applicability in the online and stochastic settings. We propose a family of Generalized Optimistic Methods with Anchoring (GOMA), which combine two-time-scale optimistic updates with an anchoring term inspired by Halpern iteration. In the deterministic setting, GOMA achieves the optimal accelerated last-iterate rate $O(1/k^2)$ on the squared gradient norm for monotone Lipschitz operators. In the stochastic setting with unbounded variance, a simplified single-call variant of GOMA achieves a last-iterate convergence rate of $O(1/\sqrt{k})$ on the squared gradient norm. To the best of our knowledge, this is the first such guarantee for stochastic monotone Lipschitz variational inequalities in the unconstrained setting without variance reduction or growing batches.

math.OC↗

arXiv2026

Unsupervised Causal Abstractions Discovery

Causal abstractions formalize when a high-level structural causal model (SCM) captures the interventional behavior of a lower-level SCM. Existing applications of this notion largely follow a hypothesis-testing paradigm: an expert proposes a candidate high-level model and then evaluates if the low-level system implements it. We study the complementary problem of learning a high-level model directly from low-level measurements. Our contributions leverage hypotheses from low-rank causal discovery, and can be summarized as follows: (1) we show that observations generated by a low-rank graph induce latents that form a causal abstraction, (2) we provide identifiability results about these latents, and (3) we propose a practical objective to learn this high-level SCM.

cs.LG↗

arXiv2026

The Role of Causal Features in Strategic Classification for Robustness and Alignment

In strategic classification, an institution (e.g., a bank) anticipates adaptation from users who change their features to increase utility in a classification task (e.g., loan repayment). Since a key challenge is the distribution shift induced by users, we turn to causal models, which have been shown to bound the worst-case out-of-distribution (OOD) risk, and establish several new results that link causality and strategic classification. First, we show that causal classification leads to optimal classification error after any sufficiently large adaptation, when the noise is bounded in a certain way. Second, when these assumptions do not hold, we show OOD cross-entropy risk of optimal classifiers decomposes into an OOD bias term and a term arising from not using all observable features, allowing us to understand when causal classifiers have an advantage. Finally, we show that the use of causal features can allow alignment of long-term incentives between institutions and users, contrasting with previous work that highlights social costs of such approaches. We validate our theory empirically on synthetic data, finding that our results predict behavior in practice.

cs.LG↗

arXiv2026

Reparametrizing Shampoo and SOAP for Subspace Basis Updates and BFloat16 Storage

Shampoo-based methods, such as KL-Shampoo and SOAP, have demonstrated strong performance in training neural networks and rely on QR decomposition. Because existing QR implementations require single-precision (FP32) arithmetic and remain computationally expensive, these methods become time- and memory-intensive when their preconditioning matrices are large. Moreover, using BFloat16 (BFP16) storage to reduce memory usage can degrade the performance of Shampoo-based methods. We propose a reparametrization of the preconditioner that supports BFP16 storage and forms a complete basis by combining updated basis vectors with unchanged ones. By updating only part of the basis through QR decomposition in a subspace, our approach reduces computational overhead while mitigating the performance degradation caused by BFP16 storage. Our approach applies broadly to Shampoo-based methods that employ QR decomposition, including KL-Shampoo, SOAP, and KL-SOAP. In particular, it improves the performance of SOAP and KL-SOAP under BFP16 storage, enabling KL-SOAP to match or exceed KL-Shampoo. Overall, our approach makes Shampoo-based methods more memory- and time-efficient.

cs.LG↗

arXiv2026

Position: Adopt Constraints Over Fixed Penalties in Deep Learning

Recent efforts to develop trustworthy AI systems have increased interest in learning problems with explicit requirements, or constraints. In deep learning, however, such problems are often handled through fixed weighted-sum penalization: the constraints are added to the task loss with fixed coefficients, and the resulting scalarized objective is minimized. This position paper argues that fixed penalization is often ill-suited for deep learning problems with non-negotiable requirements for several reasons. First, in non-convex settings, the penalized and constrained problems are generally not equivalent, so solving the former need not solve the latter. Second, fixed penalization weakens hard requirements into soft penalties to be traded off against task performance. Third, choosing penalty coefficients to indirectly solve the constrained problem often involves costly trial and error, because changing them alters the penalized objective itself, and hence can mean solving the wrong problem altogether. We therefore argue that, when a deep learning problem specifies non-negotiable requirements, the constrained formulation itself should be the starting point, not the surrogate problem defined by fixed penalization. The appropriate solution strategy should then be chosen based on the problem's structure and scale.

cs.LG↗

arXiv2026

Layerwise LQR for Geometry-Aware Optimization of Deep Networks

Geometry-aware optimizers such as Newton and natural gradient can improve conditioning in deep learning, but scalable variants such as K-FAC, Shampoo, and related preconditioners usually impose structural approximations early, often discarding cross-layer interactions induced by the network computation. We introduce Layerwise LQR (LLQR), a framework for learning structured inverse preconditioners under a global layerwise optimal-control objective. The starting point is an exact equivalence: the steepest-descent step under a broad class of divergence-induced quadratic models--including Newton, Gauss-Newton, Fisher/natural-gradient, and intermediate-layer metrics--can be written as a finite-horizon Linear Quadratic Regulator (LQR) problem. This formulation serves as a reference that exposes the layerwise dynamics and cost matrices encoding the original dense geometry. We then derive a scalable relaxation that learns diagonal, (E-)Kronecker-factored, or other structured inverse preconditioners by minimizing the LQR objective and reusing them across iterations. The resulting optimizer wraps standard methods while retaining a principled connection to second-order geometry, without forming or inverting the global curvature matrix. Experiments on ResNets and Transformers show that LLQR improves optimization dynamics and often translates these gains into improved final test performance, while adding only modest wall-clock overhead. It establishes LLQR as a practical framework for geometry-aware second-order methods and a reference for evaluating scalable approximations.

cs.LG↗

arXiv2026

Stop Probing, Start Coding: Why Linear Probes and Sparse Autoencoders Fail at Compositional Generalisation

The linear representation hypothesis states that neural network activations encode high-level concepts as linear mixtures. However, under superposition, this encoding is a projection from a higher-dimensional concept space into a lower-dimensional activation space, and a linear decision boundary in the concept space need not remain linear after projection. In this setting, classical sparse coding methods with per-sample iterative inference leverage compressed sensing guarantees to recover latent factors. Sparse autoencoders (SAEs), on the other hand, amortise sparse inference into a fixed encoder, introducing a systematic gap. We show this amortisation gap persists across training set sizes, latent dimensions, and sparsity levels, causing SAEs to fail under out-of-distribution (OOD) compositional shifts. Through controlled experiments that decompose the failure, we identify dictionary learning -- not the inference procedure -- as the binding constraint: SAE-learned dictionaries point in substantially wrong directions, and replacing the encoder with per-sample FISTA on the same dictionary does not close the gap. An oracle baseline proves the problem is solvable with a good dictionary at all scales tested. Our results reframe the SAE failure as a dictionary learning challenge, not an amortisation problem, and point to scalable dictionary learning as the key open problem for sparse inference under superposition.

cs.LG↗

arXiv2026

Tight Lower Bounds and Improved Convergence in Performative Prediction

Performative prediction is a framework accounting for the shift in the data distribution induced by the prediction of a model deployed in the real world. Ensuring rapid convergence to a stable solution where the data distribution remains the same after the model deployment is crucial, especially in evolving environments. This paper extends the Repeated Risk Minimization (RRM) framework by utilizing historical datasets from previous retraining snapshots, yielding a class of algorithms that we call Affine Risk Minimizers and enabling convergence to a performatively stable point for a broader class of problems. We introduce a new upper bound for methods that use only the final iteration of the dataset and prove for the first time the tightness of both this new bound and the previous existing bounds within the same regime. We also prove that utilizing historical datasets can surpass the lower bound for last iterate RRM, and empirically observe faster convergence to the stable point on various performative prediction benchmarks. We offer at the same time the first lower bound analysis for RRM within the class of Affine Risk Minimizers, quantifying the potential improvements in convergence speed that could be achieved with other variants in our framework.

cs.LG↗

arXiv2026

Dual Optimistic Ascent (PI Control) is the Augmented Lagrangian Method in Disguise

Constrained optimization is a powerful framework for enforcing requirements on neural networks. These constrained deep learning problems are typically solved using first-order methods on their min-max Lagrangian formulation, but such approaches often suffer from oscillations and can fail to find all local solutions. While the Augmented Lagrangian method (ALM) addresses these issues, practitioners often favor dual optimistic ascent schemes (PI control) on the standard Lagrangian, which perform well empirically but lack formal guarantees. In this paper, we establish a previously unknown equivalence between these approaches: dual optimistic ascent on the Lagrangian is equivalent to gradient descent-ascent on the Augmented Lagrangian. This finding allows us to transfer the robust theoretical guarantees of the ALM to the dual optimistic setting, proving it converges linearly to all local solutions. Furthermore, the equivalence provides principled guidance for tuning the optimism hyper-parameter. Our work closes a critical gap between the empirical success of dual optimistic methods and their theoretical foundation in the single-step, first-order regime commonly used in constrained deep learning.

cs.LG↗

arXiv2025

Operationalizing Quantized Disentanglement

Recent theoretical work established the unsupervised identifiability of quantized factors under any diffeomorphism. The theory assumes that quantization thresholds correspond to axis-aligned discontinuities in the probability density of the latent factors. By constraining a learned map to have a density with axis-aligned discontinuities, we can recover the quantization of the factors. However, translating this high-level principle into an effective practical criterion remains challenging, especially under nonlinear maps. Here, we develop a criterion for unsupervised disentanglement by encouraging axis-aligned discontinuities. Discontinuities manifest as sharp changes in the estimated density of factors and form what we call cliffs. Following the definition of independent discontinuities from the theory, we encourage the location of the cliffs along a factor to be independent of the values of the other factors. We show that our method, Cliff, outperforms the baselines on all disentanglement benchmarks, demonstrating its effectiveness in unsupervised disentanglement.

cs.LG↗

arXiv2025

Understanding Adam Requires Better Rotation Dependent Assumptions

Despite its widespread adoption, Adam's advantage over Stochastic Gradient Descent (SGD) lacks a comprehensive theoretical explanation. This paper investigates Adam's sensitivity to rotations of the parameter space. We observe that Adam's performance in training transformers degrades under random rotations of the parameter space, indicating a crucial sensitivity to the choice of basis in practice. This reveals that conventional rotation-invariant assumptions are insufficient to capture Adam's advantages theoretically. To better understand the rotation-dependent properties that benefit Adam, we also identify structured rotations that preserve or even enhance its empirical performance. We then examine the rotation-dependent assumptions in the literature and find that they fall short in explaining Adam's behaviour across various rotation types. In contrast, we verify the orthogonality of the update as a promising indicator of Adam's basis sensitivity, suggesting it may be the key quantity for developing rotation-dependent theoretical frameworks that better explain its empirical success.

cs.LG↗

arXiv2025

Bias Analysis in Unconditional Image Generative Models

The widespread adoption of generative AI models has raised growing concerns about representational harm and potential discriminatory outcomes. Yet, despite growing literature on this topic, the mechanisms by which bias emerges - especially in unconditional generation - remain disentangled. We define the bias of an attribute as the difference between the probability of its presence in the observed distribution and its expected proportion in an ideal reference distribution. In our analysis, we train a set of unconditional image generative models and adopt a commonly used bias evaluation framework to study bias shift between training and generated distributions. Our experiments reveal that the detected attribute shifts are small. We find that the attribute shifts are sensitive to the attribute classifier used to label generated images in the evaluation framework, particularly when its decision boundaries fall in high-density regions. Our empirical analysis indicates that this classifier sensitivity is often observed in attributes values that lie on a spectrum, as opposed to exhibiting a binary nature. This highlights the need for more representative labeling practices, understanding the shortcomings through greater scrutiny of evaluation frameworks, and recognizing the socially complex nature of attributes when evaluating bias.

cs.CV↗

arXiv2025

Cooper: A Library for Constrained Optimization in Deep Learning

Cooper is an open-source package for solving constrained optimization problems involving deep learning models. Cooper implements several Lagrangian-based first-order update schemes, making it easy to combine constrained optimization algorithms with high-level features of PyTorch such as automatic differentiation, and specialized deep learning architectures and optimizers. Although Cooper is specifically designed for deep learning applications where gradients are estimated based on mini-batches, it is suitable for general non-convex continuous constrained optimization. Cooper's source code is available at https://github.com/cooper-org/cooper.

cs.LG↗

arXiv2025

Accelerating Training with Neuron Interaction and Nowcasting Networks

Neural network training can be accelerated when a learnable update rule is used in lieu of classic adaptive optimizers (e.g. Adam). However, learnable update rules can be costly and unstable to train and use. Recently, Jang et al. (2023) proposed a simpler approach to accelerate training based on weight nowcaster networks (WNNs). In their approach, Adam is used for most of the optimization steps and periodically, only every few steps, a WNN nowcasts (predicts near future) parameters. We improve WNNs by proposing neuron interaction and nowcasting (NiNo) networks. In contrast to WNNs, NiNo leverages neuron connectivity and graph neural networks to more accurately nowcast parameters. We further show that in some networks, such as Transformers, modeling neuron connectivity accurately is challenging. We address this and other limitations, which allows NiNo to accelerate Adam training by up to 50% in vision and language tasks.

cs.LG↗

arXiv2025

Performative Prediction on Games and Mechanism Design

Agents often have individual goals which depend on a group's actions. If agents trust a forecast of collective action and adapt strategically, such prediction can influence outcomes non-trivially, resulting in a form of performative prediction. This effect is ubiquitous in scenarios ranging from pandemic predictions to election polls, but existing work has ignored interdependencies among predicted agents. As a first step in this direction, we study a collective risk dilemma where agents dynamically decide whether to trust predictions based on past accuracy. As predictions shape collective outcomes, social welfare arises naturally as a metric of concern. We explore the resulting interplay between accuracy and welfare, and demonstrate that searching for stable accurate predictions can minimize social welfare with high probability in our setting. By assuming knowledge of a Bayesian agent behavior model, we then show how to achieve better trade-offs and use them for mechanism design.

cs.LG↗

arXiv2025

Feasible Learning

We introduce Feasible Learning (FL), a sample-centric learning paradigm where models are trained by solving a feasibility problem that bounds the loss for each training sample. In contrast to the ubiquitous Empirical Risk Minimization (ERM) framework, which optimizes for average performance, FL demands satisfactory performance on every individual data point. Since any model that meets the prescribed performance threshold is a valid FL solution, the choice of optimization algorithm and its dynamics play a crucial role in shaping the properties of the resulting solutions. In particular, we study a primal-dual approach which dynamically re-weights the importance of each sample during training. To address the challenge of setting a meaningful threshold in practice, we introduce a relaxation of FL that incorporates slack variables of minimal norm. Our empirical analysis, spanning image classification, age regression, and preference optimization in large language models, demonstrates that models trained via FL can learn from data while displaying improved tail behavior compared to ERM, with only a marginal impact on average performance.

cs.LG↗

arXiv2024

Promoting Exploration in Memory-Augmented Adam using Critical Momenta

Adaptive gradient-based optimizers, notably Adam, have left their mark in training large-scale deep learning models, offering fast convergence and robustness to hyperparameter settings. However, they often struggle with generalization, attributed to their tendency to converge to sharp minima in the loss landscape. To address this, we propose a new memory-augmented version of Adam that encourages exploration towards flatter minima by incorporating a buffer of critical momentum terms during training. This buffer prompts the optimizer to overshoot beyond narrow minima, promoting exploration. Through comprehensive analysis in simple settings, we illustrate the efficacy of our approach in increasing exploration and bias towards flatter minima. We empirically demonstrate that it can improve model performance for image classification on ImageNet and CIFAR10/100, language modelling on Penn Treebank, and online learning tasks on TinyImageNet and 5-dataset. Our code is available at \url{https://github.com/chandar-lab/CMOptimizer}.

cs.LG↗