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.

Test like you Train in Implicit Deep Learning

Implicit deep learning has recently gained popularity with applications ranging from meta-learning to Deep Equilibrium Networks (DEQs). In its general formulation, it relies on expressing some components of deep learning pipelines implicitly, typically via a root equation called the inner problem. In practice, the solution of the inner problem is approximated during training with an iterative procedure, usually with a fixed number of inner iterations. During inference, the inner problem needs to be solved with new data. A popular belief is that increasing the number of inner iterations compared to the one used during training yields better performance. In this paper, we question such an assumption and provide a detailed theoretical analysis in a simple setting. We demonstrate that overparametrization plays a key role: increasing the number of iterations at test time cannot improve performance for overparametrized networks. We validate our theory on an array of implicit deep-learning problems. DEQs, which are typically overparametrized, do not benefit from increasing the number of iterations at inference while meta-learning, which is typically not overparametrized, benefits from it.

Infeasible Deterministic, Stochastic, and Variance-Reduction Algorithms for Optimization under Orthogonality Constraints

Orthogonality constraints naturally appear in many machine learning problems, from Principal Components Analysis to robust neural network training. They are usually solved using Riemannian optimization algorithms, which minimize the objective function while enforcing the constraint. However, enforcing the orthogonality constraint can be the most time-consuming operation in such algorithms. Recently, Ablin & Peyr\'e (2022) proposed the Landing algorithm, a method with cheap iterations that does not enforce the orthogonality constraint but is attracted towards the manifold in a smooth manner. In this article, we provide new practical and theoretical developments for the landing algorithm. First, the method is extended to the Stiefel manifold, the set of rectangular orthogonal matrices. We also consider stochastic and variance reduction algorithms when the cost function is an average of many functions. We demonstrate that all these methods have the same rate of convergence as their Riemannian counterparts that exactly enforce the constraint. Finally, our experiments demonstrate the promise of our approach to an array of machine-learning problems that involve orthogonality constraints.

A Near-Optimal Algorithm for Bilevel Empirical Risk Minimization

Bilevel optimization problems, which are problems where two optimization problems are nested, have more and more applications in machine learning. In many practical cases, the upper and the lower objectives correspond to empirical risk minimization problems and therefore have a sum structure. In this context, we propose a bilevel extension of the celebrated SARAH algorithm. We demonstrate that the algorithm requires $\mathcal{O}((n+m)^{\frac12}\varepsilon^{-1})$ gradient computations to achieve $\varepsilon$-stationarity with $n+m$ the total number of samples, which improves over all previous bilevel algorithms. Moreover, we provide a lower bound on the number of oracle calls required to get an approximate stationary point of the objective function of the bilevel problem. This lower bound is attained by our algorithm, which is therefore optimal in terms of sample complexity.

Monge, Bregman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps

Optimal transport (OT) theory focuses, among all maps $T:\mathbb{R}^d\rightarrow \mathbb{R}^d$ that can morph a probability measure onto another, on those that are the ``thriftiest'', i.e. such that the averaged cost $c(x, T(x))$ between $x$ and its image $T(x)$ be as small as possible. Many computational approaches have been proposed to estimate such Monge maps when $c$ is the $\ell_2^2$ distance, e.g., using entropic maps [Pooladian'22], or neural networks [Makkuva'20, Korotin'20]. We propose a new model for transport maps, built on a family of translation invariant costs $c(x, y):=h(x-y)$, where $h:=\tfrac{1}{2}\|\cdot\|_2^2+\tau$ and $\tau$ is a regularizer. We propose a generalization of the entropic map suitable for $h$, and highlight a surprising link tying it with the Bregman centroids of the divergence $D_h$ generated by $h$, and the proximal operator of $\tau$. We show that choosing a sparsity-inducing norm for $\tau$ results in maps that apply Occam's razor to transport, in the sense that the displacement vectors $\Delta(x):= T(x)-x$ they induce are sparse, with a sparsity pattern that varies depending on $x$. We showcase the ability of our method to estimate meaningful OT maps for high-dimensional single-cell transcription data, in the $34000$-$d$ space of gene counts for cells, without using dimensionality reduction, thus retaining the ability to interpret all displacements at the gene level.

Benchopt: Reproducible, efficient and collaborative optimization benchmarks

Numerical validation is at the core of machine learning research as it allows to assess the actual impact of new methods, and to confirm the agreement between theory and practice. Yet, the rapid development of the field poses several challenges: researchers are confronted with a profusion of methods to compare, limited transparency and consensus on best practices, as well as tedious re-implementation work. As a result, validation is often very partial, which can lead to wrong conclusions that slow down the progress of research. We propose Benchopt, a collaborative framework to automate, reproduce and publish optimization benchmarks in machine learning across programming languages and hardware architectures. Benchopt simplifies benchmarking for the community by providing an off-the-shelf tool for running, sharing and extending experiments. To demonstrate its broad usability, we showcase benchmarks on three standard learning tasks: $\ell_2$-regularized logistic regression, Lasso, and ResNet18 training for image classification. These benchmarks highlight key practical findings that give a more nuanced view of the state-of-the-art for these problems, showing that for practical evaluation, the devil is in the details. We hope that Benchopt will foster collaborative work in the community hence improving the reproducibility of research findings.