Optimization without retraction on the random generalized Stiefel manifold

Optimization over the set of matrices that satisfy $X^\top B X = I_p$, referred to as the generalized Stiefel manifold, appears in many applications involving sampled covariance matrices such as canonical correlation analysis (CCA), independent component analysis (ICA), and the generalized eigenvalue problem (GEVP). Solving these problems is typically done by iterative methods, such as Riemannian approaches, which require a computationally expensive eigenvalue decomposition involving fully formed $B$. We propose a cheap stochastic iterative method that solves the optimization problem while having access only to a random estimate of the feasible set. Our method does not enforce the constraint in every iteration exactly, but instead it produces iterations that converge to a critical point on the generalized Stiefel manifold defined in expectation. The method has lower per-iteration cost, requires only matrix multiplications, and has the same convergence rates as its Riemannian counterparts involving the full matrix $B$. Experiments demonstrate its effectiveness in various machine learning applications involving generalized orthogonality constraints, including CCA, ICA, and GEVP.

Enhancing Hypergradients Estimation: A Study of Preconditioning and Reparameterization

Bilevel optimization aims to optimize an outer objective function that depends on the solution to an inner optimization problem. It is routinely used in Machine Learning, notably for hyperparameter tuning. The conventional method to compute the so-called hypergradient of the outer problem is to use the Implicit Function Theorem (IFT). As a function of the error of the inner problem resolution, we study the error of the IFT method. We analyze two strategies to reduce this error: preconditioning the IFT formula and reparameterizing the inner problem. We give a detailed account of the impact of these two modifications on the error, highlighting the role played by higher-order derivatives of the functionals at stake. Our theoretical findings explain when super efficiency, namely reaching an error on the hypergradient that depends quadratically on the error on the inner problem, is achievable and compare the two approaches when this is impossible. Numerical evaluations on hyperparameter tuning for regression problems substantiate our theoretical findings.

Careful with that Scalpel: Improving Gradient Surgery with an EMA

Beyond minimizing a single training loss, many deep learning estimation pipelines rely on an auxiliary objective to quantify and encourage desirable properties of the model (e.g. performance on another dataset, robustness, agreement with a prior). Although the simplest approach to incorporating an auxiliary loss is to sum it with the training loss as a regularizer, recent works have shown that one can improve performance by blending the gradients beyond a simple sum; this is known as gradient surgery. We cast the problem as a constrained minimization problem where the auxiliary objective is minimized among the set of minimizers of the training loss. To solve this bilevel problem, we follow a parameter update direction that combines the training loss gradient and the orthogonal projection of the auxiliary gradient to the training gradient. In a setting where gradients come from mini-batches, we explain how, using a moving average of the training loss gradients, we can carefully maintain this critical orthogonality property. We demonstrate that our method, Bloop, can lead to much better performances on NLP and vision experiments than other gradient surgery methods without EMA.

Specialized Language Models with Cheap Inference from Limited Domain Data

Large language models have emerged as a versatile tool but are challenging to apply to tasks lacking large inference budgets and large in-domain training sets. This work formalizes these constraints and distinguishes four important variables: the pretraining budget (for training before the target domain is known), the specialization budget (for training after the target domain is known), the inference budget, and the in-domain training set size. Across these settings, we compare different approaches from the machine learning literature. Limited by inference cost, we find better alternatives to the standard practice of training very large vanilla transformer models. In particular, we show that hyper-networks and mixture of experts have better perplexity for large pretraining budgets, while small models trained on importance sampled datasets are attractive for large specialization budgets.

Understanding the Regularity of Self-Attention with Optimal Transport

Transformers and their multi-head attention mechanism have completely changed the machine learning landscape in just a few years, by outperforming state-of-art models in a wide range of domains. Still, little is known about their robustness from a theoretical perspective. We tackle this problem by studying the local Lipschitz constant of self-attention, that provides an attack-agnostic way of measuring the robustness of a neural network. We adopt a measure-theoretic framework, by viewing inputs as probability measures equipped with the Wasserstein distance. This allows us to generalize attention to inputs of infinite length, and to derive an upper bound and a lower bound on the Lipschitz constant of self-attention on compact sets. The lower bound significantly improves prior results, and grows more than exponentially with the radius of the compact set, which rules out the possibility of obtaining robustness guarantees without any additional constraint on the input space. Our results also point out that measures with a high local Lipschitz constant are typically made of a few diracs, with a very unbalanced distribution of mass. Finally, we analyze the stability of self-attention under perturbations that change the number of tokens, which appears to be a natural question in the measure-theoretic framework. In particular, we show that for some inputs, attacks that duplicate tokens before perturbing them are more efficient than attacks that simply move tokens. We call this phenomenon mass splitting.

MultiView Independent Component Analysis with Delays

Linear Independent Component Analysis (ICA) is a blind source separation technique that has been used in various domains to identify independent latent sources from observed signals. In order to obtain a higher signal-to-noise ratio, the presence of multiple views of the same sources can be used. In this work, we present MultiView Independent Component Analysis with Delays (MVICAD). This algorithm builds on the MultiView ICA model by allowing sources to be delayed versions of some shared sources: sources are shared across views up to some unknown latencies that are view- and source-specific. Using simulations, we demonstrate that MVICAD leads to better unmixing of the sources. Moreover, as ICA is often used in neuroscience, we show that latencies are age-related when applied to Cam-CAN, a large-scale magnetoencephalography (MEG) dataset. These results demonstrate that the MVICAD model can reveal rich effects on neural signals without human supervision.

Adaptive Training Distributions with Scalable Online Bilevel Optimization

Large neural networks pretrained on web-scale corpora are central to modern machine learning. In this paradigm, the distribution of the large, heterogeneous pretraining data rarely matches that of the application domain. This work considers modifying the pretraining distribution in the case where one has a small sample of data reflecting the targeted test conditions. We propose an algorithm motivated by a recent formulation of this setting as an online, bilevel optimization problem. With scalability in mind, our algorithm prioritizes computing gradients at training points which are likely to most improve the loss on the targeted distribution. Empirically, we show that in some cases this approach is beneficial over existing strategies from the domain adaptation literature but may not succeed in other cases. We propose a simple test to evaluate when our approach can be expected to work well and point towards further research to address current limitations.

A Challenge in Reweighting Data with Bilevel Optimization

In many scenarios, one uses a large training set to train a model with the goal of performing well on a smaller testing set with a different distribution. Learning a weight for each data point of the training set is an appealing solution, as it ideally allows one to automatically learn the importance of each training point for generalization on the testing set. This task is usually formalized as a bilevel optimization problem. Classical bilevel solvers are based on a warm-start strategy where both the parameters of the models and the data weights are learned at the same time. We show that this joint dynamic may lead to sub-optimal solutions, for which the final data weights are very sparse. This finding illustrates the difficulty of data reweighting and offers a clue as to why this method is rarely used in practice.

How to Scale Your EMA

Preserving training dynamics across batch sizes is an important tool for practical machine learning as it enables the trade-off between batch size and wall-clock time. This trade-off is typically enabled by a scaling rule, for example, in stochastic gradient descent, one should scale the learning rate linearly with the batch size. Another important tool for practical machine learning is the model Exponential Moving Average (EMA), which is a model copy that does not receive gradient information, but instead follows its target model with some momentum. This model EMA can improve the robustness and generalization properties of supervised learning, stabilize pseudo-labeling, and provide a learning signal for Self-Supervised Learning (SSL). Prior works have treated the model EMA separately from optimization, leading to different training dynamics across batch sizes and lower model performance. In this work, we provide a scaling rule for optimization in the presence of model EMAs and demonstrate its validity across a range of architectures, optimizers, and data modalities. We also show the rule's validity where the model EMA contributes to the optimization of the target model, enabling us to train EMA-based pseudo-labeling and SSL methods at small and large batch sizes. For SSL, we enable training of BYOL up to batch size 24,576 without sacrificing performance, optimally a 6$\times$ wall-clock time reduction.

Learning Costs for Structured Monge Displacements

Optimal transport theory has provided machine learning with several tools to infer a push-forward map between densities from samples. While this theory has recently seen tremendous methodological developments in machine learning, its practical implementation remains notoriously difficult, because it is plagued by both computational and statistical challenges. Because of such difficulties, existing approaches rarely depart from the default choice of estimating such maps with the simple squared-Euclidean distance as the ground cost, $c(x,y)=\|x-y\|^2_2$. We follow a different path in this work, with the motivation of \emph{learning} a suitable cost structure to encourage maps to transport points along engineered features. We extend the recently proposed Monge-Bregman-Occam pipeline~\citep{cuturi2023monge}, that rests on an alternative cost formulation that is also cost-invariant $c(x,y)=h(x-y)$, but which adopts a more general form as $h=\tfrac12 \ell_2^2+\tau$, where $\tau$ is an appropriately chosen regularizer. We first propose a method that builds upon proximal gradient descent to generate ground truth transports for such structured costs, using the notion of $h$-transforms and $h$-concave potentials. We show more generally that such a method can be extended to compute $h$-transforms for entropic potentials. We study a regularizer that promotes transport displacements in low-dimensional spaces, and propose to learn such a basis change using Riemannian gradient descent on the Stiefel manifold. We show that these changes lead to estimators that are more robust and easier to interpret.