PT$\mathrm{L}^{p}$: Partial Transport $\mathrm{L}^{p}$ Distances

Optimal transport and its related problems, including optimal partial transport, have proven to be valuable tools in machine learning for computing meaningful distances between probability or positive measures. This success has led to a growing interest in defining transport-based distances that allow for comparing signed measures and, more generally, multi-channeled signals. Transport $\mathrm{L}^{p}$ distances are notable extensions of the optimal transport framework to signed and possibly multi-channeled signals. In this paper, we introduce partial transport $\mathrm{L}^{p}$ distances as a new family of metrics for comparing generic signals, benefiting from the robustness of partial transport distances. We provide theoretical background such as the existence of optimal plans and the behavior of the distance in various limits. Furthermore, we introduce the sliced variation of these distances, which allows for rapid comparison of generic signals. Finally, we demonstrate the application of the proposed distances in signal class separability and nearest neighbor classification.

Equivariant vs. Invariant Layers: A Comparison of Backbone and Pooling for Point Cloud Classification

Learning from set-structured data, such as point clouds, has gained significant attention from the community. Geometric deep learning provides a blueprint for designing effective set neural networks by incorporating permutation symmetry. Of our interest are permutation invariant networks, which are composed of a permutation equivariant backbone, permutation invariant global pooling, and regression/classification head. While existing literature has focused on improving permutation equivariant backbones, the impact of global pooling is often overlooked. In this paper, we examine the interplay between permutation equivariant backbones and permutation invariant global pooling on three benchmark point cloud classification datasets. Our findings reveal that: 1) complex pooling methods, such as transport-based or attention-based poolings, can significantly boost the performance of simple backbones, but the benefits diminish for more complex backbones, 2) even complex backbones can benefit from pooling layers in low data scenarios, 3) surprisingly, the choice of pooling layers can have a more significant impact on the model's performance than adjusting the width and depth of the backbone, and 4) pairwise combination of pooling layers can significantly improve the performance of a fixed backbone. Our comprehensive study provides insights for practitioners to design better permutation invariant set neural networks.

Characterizing Out-of-Distribution Error via Optimal Transport

Out-of-distribution (OOD) data poses serious challenges in deployed machine learning models, so methods of predicting a model's performance on OOD data without labels are important for machine learning safety. While a number of methods have been proposed by prior work, they often underestimate the actual error, sometimes by a large margin, which greatly impacts their applicability to real tasks. In this work, we identify pseudo-label shift, or the difference between the predicted and true OOD label distributions, as a key indicator to this underestimation. Based on this observation, we introduce a novel method for estimating model performance by leveraging optimal transport theory, Confidence Optimal Transport (COT), and show that it provably provides more robust error estimates in the presence of pseudo-label shift. Additionally, we introduce an empirically-motivated variant of COT, Confidence Optimal Transport with Thresholding (COTT), which applies thresholding to the individual transport costs and further improves the accuracy of COT's error estimates. We evaluate COT and COTT on a variety of standard benchmarks that induce various types of distribution shift -- synthetic, novel subpopulation, and natural -- and show that our approaches significantly outperform existing state-of-the-art methods with an up to 3x lower prediction error.

Sharing Lifelong Reinforcement Learning Knowledge via Modulating Masks

Lifelong learning agents aim to learn multiple tasks sequentially over a lifetime. This involves the ability to exploit previous knowledge when learning new tasks and to avoid forgetting. Modulating masks, a specific type of parameter isolation approach, have recently shown promise in both supervised and reinforcement learning. While lifelong learning algorithms have been investigated mainly within a single-agent approach, a question remains on how multiple agents can share lifelong learning knowledge with each other. We show that the parameter isolation mechanism used by modulating masks is particularly suitable for exchanging knowledge among agents in a distributed and decentralized system of lifelong learners. The key idea is that the isolation of specific task knowledge to specific masks allows agents to transfer only specific knowledge on-demand, resulting in robust and effective distributed lifelong learning. We assume fully distributed and asynchronous scenarios with dynamic agent numbers and connectivity. An on-demand communication protocol ensures agents query their peers for specific masks to be transferred and integrated into their policies when facing each task. Experiments indicate that on-demand mask communication is an effective way to implement distributed lifelong reinforcement learning and provides a lifelong learning benefit with respect to distributed RL baselines such as DD-PPO, IMPALA, and PPO+EWC. The system is particularly robust to connection drops and demonstrates rapid learning due to knowledge exchange.

Predicting Out-of-Distribution Error with Confidence Optimal Transport

Out-of-distribution (OOD) data poses serious challenges in deployed machine learning models as even subtle changes could incur significant performance drops. Being able to estimate a model's performance on test data is important in practice as it indicates when to trust to model's decisions. We present a simple yet effective method to predict a model's performance on an unknown distribution without any addition annotation. Our approach is rooted in the Optimal Transport theory, viewing test samples' output softmax scores from deep neural networks as empirical samples from an unknown distribution. We show that our method, Confidence Optimal Transport (COT), provides robust estimates of a model's performance on a target domain. Despite its simplicity, our method achieves state-of-the-art results on three benchmark datasets and outperforms existing methods by a large margin.

Linear Optimal Partial Transport Embedding

Optimal transport (OT) has gained popularity due to its various applications in fields such as machine learning, statistics, and signal processing. However, the balanced mass requirement limits its performance in practical problems. To address these limitations, variants of the OT problem, including unbalanced OT, Optimal partial transport (OPT), and Hellinger Kantorovich (HK), have been proposed. In this paper, we propose the Linear optimal partial transport (LOPT) embedding, which extends the (local) linearization technique on OT and HK to the OPT problem. The proposed embedding allows for faster computation of OPT distance between pairs of positive measures. Besides our theoretical contributions, we demonstrate the LOPT embedding technique in point-cloud interpolation and PCA analysis.

Linear optimal partial transport embedding

Optimal transport (OT) has gained popularity due to its various applications in fields such as machine learning, statistics, and signal processing. However, the balanced mass requirement limits its performance in practical problems. To address these limitations, variants of the OT problem, including unbalanced OT, Optimal partial transport (OPT), and Hellinger Kantorovich (HK), have been proposed. In this paper, we propose the Linear optimal partial transport (LOPT) embedding, which extends the (local) linearization technique on OT and HK to the OPT problem. The proposed embedding allows for faster computation of OPT distance between pairs of positive measures. Besides our theoretical contributions, we demonstrate the LOPT embedding technique in point-cloud interpolation and PCA analysis.

A Domain-Agnostic Approach for Characterization of Lifelong Learning Systems

Despite the advancement of machine learning techniques in recent years, state-of-the-art systems lack robustness to "real world" events, where the input distributions and tasks encountered by the deployed systems will not be limited to the original training context, and systems will instead need to adapt to novel distributions and tasks while deployed. This critical gap may be addressed through the development of "Lifelong Learning" systems that are capable of 1) Continuous Learning, 2) Transfer and Adaptation, and 3) Scalability. Unfortunately, efforts to improve these capabilities are typically treated as distinct areas of research that are assessed independently, without regard to the impact of each separate capability on other aspects of the system. We instead propose a holistic approach, using a suite of metrics and an evaluation framework to assess Lifelong Learning in a principled way that is agnostic to specific domains or system techniques. Through five case studies, we show that this suite of metrics can inform the development of varied and complex Lifelong Learning systems. We highlight how the proposed suite of metrics quantifies performance trade-offs present during Lifelong Learning system development - both the widely discussed Stability-Plasticity dilemma and the newly proposed relationship between Sample Efficient and Robust Learning. Further, we make recommendations for the formulation and use of metrics to guide the continuing development of Lifelong Learning systems and assess their progress in the future.

Sliced Optimal Partial Transport

Optimal transport (OT) has become exceedingly popular in machine learning, data science, and computer vision. The core assumption in the OT problem is the equal total amount of mass in source and target measures, which limits its application. Optimal Partial Transport (OPT) is a recently proposed solution to this limitation. Similar to the OT problem, the computation of OPT relies on solving a linear programming problem (often in high dimensions), which can become computationally prohibitive. In this paper, we propose an efficient algorithm for calculating the OPT problem between two non-negative measures in one dimension. Next, following the idea of sliced OT distances, we utilize slicing to define the sliced OPT distance. Finally, we demonstrate the computational and accuracy benefits of the sliced OPT-based method in various numerical experiments. In particular, we show an application of our proposed Sliced-OPT in noisy point cloud registration.

Lifelong Reinforcement Learning with Modulating Masks

Lifelong learning aims to create AI systems that continuously and incrementally learn during a lifetime, similar to biological learning. Attempts so far have met problems, including catastrophic forgetting, interference among tasks, and the inability to exploit previous knowledge. While considerable research has focused on learning multiple input distributions, typically in classification, lifelong reinforcement learning (LRL) must also deal with variations in the state and transition distributions, and in the reward functions. Modulating masks, recently developed for classification, are particularly suitable to deal with such a large spectrum of task variations. In this paper, we adapted modulating masks to work with deep LRL, specifically PPO and IMPALA agents. The comparison with LRL baselines in both discrete and continuous RL tasks shows competitive performance. We further investigated the use of a linear combination of previously learned masks to exploit previous knowledge when learning new tasks: not only is learning faster, the algorithm solves tasks that we could not otherwise solve from scratch due to extremely sparse rewards. The results suggest that RL with modulating masks is a promising approach to lifelong learning, to the composition of knowledge to learn increasingly complex tasks, and to knowledge reuse for efficient and faster learning.