Unlocking the Potential of Transformers in Time Series Forecasting with Sharpness-Aware Minimization and Channel-Wise Attention

Transformer-based architectures achieved breakthrough performance in natural language processing and computer vision, yet they remain inferior to simpler linear baselines in multivariate long-term forecasting. To better understand this phenomenon, we start by studying a toy linear forecasting problem for which we show that transformers are incapable of converging to their true solution despite their high expressive power. We further identify the attention of transformers as being responsible for this low generalization capacity. Building upon this insight, we propose a shallow lightweight transformer model that successfully escapes bad local minima when optimized with sharpness-aware optimization. We empirically demonstrate that this result extends to all commonly used real-world multivariate time series datasets. In particular, SAMformer surpasses the current state-of-the-art model TSMixer by 14.33% on average, while having ~4 times fewer parameters. The code is available at https://github.com/romilbert/samformer.

Characterising Gradients for Unsupervised Accuracy Estimation under Distribution Shift

Estimating test accuracy without access to the ground-truth test labels under varying test environments is a challenging, yet extremely important problem in the safe deployment of machine learning algorithms. Existing works rely on the information from either the outputs or the extracted features of neural networks to formulate an estimation score correlating with the ground-truth test accuracy. In this paper, we investigate--both empirically and theoretically--how the information provided by the gradients can be predictive of the ground-truth test accuracy even under a distribution shift. Specifically, we use the norm of classification-layer gradients, backpropagated from the cross-entropy loss after only one gradient step over test data. Our key idea is that the model should be adjusted with a higher magnitude of gradients when it does not generalize to the test dataset with a distribution shift. We provide theoretical insights highlighting the main ingredients of such an approach ensuring its empirical success. Extensive experiments conducted on diverse distribution shifts and model structures demonstrate that our method significantly outperforms state-of-the-art algorithms.

Leveraging Ensemble Diversity for Robust Self-Training in the Presence of Sample Selection Bias

Self-training is a well-known approach for semi-supervised learning. It consists of iteratively assigning pseudo-labels to unlabeled data for which the model is confident and treating them as labeled examples. For neural networks, softmax prediction probabilities are often used as a confidence measure, despite the fact that they are known to be overconfident, even for wrong predictions. This phenomenon is particularly intensified in the presence of sample selection bias, i.e., when data labeling is subject to some constraint. To address this issue, we propose a novel confidence measure, called $\mathcal{T}$-similarity, built upon the prediction diversity of an ensemble of linear classifiers. We provide the theoretical analysis of our approach by studying stationary points and describing the relationship between the diversity of the individual members and their performance. We empirically demonstrate the benefit of our confidence measure for three different pseudo-labeling policies on classification datasets of various data modalities.

Understanding deep neural networks through the lens of their non-linearity

The remarkable success of deep neural networks (DNN) is often attributed to their high expressive power and their ability to approximate functions of arbitrary complexity. Indeed, DNNs are highly non-linear models, and activation functions introduced into them are largely responsible for this. While many works studied the expressive power of DNNs through the lens of their approximation capabilities, quantifying the non-linearity of DNNs or of individual activation functions remains an open problem. In this paper, we propose the first theoretically sound solution to track non-linearity propagation in deep neural networks with a specific focus on computer vision applications. Our proposed affinity score allows us to gain insights into the inner workings of a wide range of different architectures and learning paradigms. We provide extensive experimental results that highlight the practical utility of the proposed affinity score and its potential for long-reaching applications.

Revisiting invariances and introducing priors in Gromov-Wasserstein distances

Gromov-Wasserstein distance has found many applications in machine learning due to its ability to compare measures across metric spaces and its invariance to isometric transformations. However, in certain applications, this invariance property can be too flexible, thus undesirable. Moreover, the Gromov-Wasserstein distance solely considers pairwise sample similarities in input datasets, disregarding the raw feature representations. We propose a new optimal transport-based distance, called Augmented Gromov-Wasserstein, that allows for some control over the level of rigidity to transformations. It also incorporates feature alignments, enabling us to better leverage prior knowledge on the input data for improved performance. We present theoretical insights into the proposed metric. We then demonstrate its usefulness for single-cell multi-omic alignment tasks and a transfer learning scenario in machine learning.

Beyond invariant representation learning: linearly alignable latent spaces for efficient closed-form domain adaptation

Optimal transport (OT) is a powerful geometric tool used to compare and align probability measures following the least effort principle. Among many successful applications of OT in machine learning (ML), domain adaptation (DA) -- a field of study where the goal is to transfer a classifier from one labelled domain to another similar, yet different unlabelled or scarcely labelled domain -- has been historically among the most investigated ones. This success is due to the ability of OT to provide both a meaningful discrepancy measure to assess the similarity of two domains' distributions and a mapping that can project source domain data onto the target one. In this paper, we propose a principally new OT-based approach applied to DA that uses the closed-form solution of the OT problem given by an affine mapping and learns an embedding space for which this solution is optimal and computationally less complex. We show that our approach works in both homogeneous and heterogeneous DA settings and outperforms or is on par with other famous baselines based on both traditional OT and OT in incomparable spaces. Furthermore, we show that our proposed method vastly reduces computational complexity.

Meta Optimal Transport

We study the use of amortized optimization to predict optimal transport (OT) maps from the input measures, which we call Meta OT. This helps repeatedly solve similar OT problems between different measures by leveraging the knowledge and information present from past problems to rapidly predict and solve new problems. Otherwise, standard methods ignore the knowledge of the past solutions and suboptimally re-solve each problem from scratch. Meta OT models surpass the standard convergence rates of log-Sinkhorn solvers in the discrete setting and convex potentials in the continuous setting. We improve the computational time of standard OT solvers by multiple orders of magnitude in discrete and continuous transport settings between images, spherical data, and color palettes. Our source code is available at http://github.com/facebookresearch/meta-ot.

Unbalanced CO-Optimal Transport

Optimal transport (OT) compares probability distributions by computing a meaningful alignment between their samples. CO-optimal transport (COOT) takes this comparison further by inferring an alignment between features as well. While this approach leads to better alignments and generalizes both OT and Gromov-Wasserstein distances, we provide a theoretical result showing that it is sensitive to outliers that are omnipresent in real-world data. This prompts us to propose unbalanced COOT for which we provably show its robustness to noise in the compared datasets. To the best of our knowledge, this is the first such result for OT methods in incomparable spaces. With this result in hand, we provide empirical evidence of this robustness for the challenging tasks of heterogeneous domain adaptation with and without varying proportions of classes and simultaneous alignment of samples and features across single-cell measurements.