Wasserstein Task Embedding for Measuring Task Similarities

Measuring similarities between different tasks is critical in a broad spectrum of machine learning problems, including transfer, multi-task, continual, and meta-learning. Most current approaches to measuring task similarities are architecture-dependent: 1) relying on pre-trained models, or 2) training networks on tasks and using forward transfer as a proxy for task similarity. In this paper, we leverage the optimal transport theory and define a novel task embedding for supervised classification that is model-agnostic, training-free, and capable of handling (partially) disjoint label sets. In short, given a dataset with ground-truth labels, we perform a label embedding through multi-dimensional scaling and concatenate dataset samples with their corresponding label embeddings. Then, we define the distance between two datasets as the 2-Wasserstein distance between their updated samples. Lastly, we leverage the 2-Wasserstein embedding framework to embed tasks into a vector space in which the Euclidean distance between the embedded points approximates the proposed 2-Wasserstein distance between tasks. We show that the proposed embedding leads to a significantly faster comparison of tasks compared to related approaches like the Optimal Transport Dataset Distance (OTDD). Furthermore, we demonstrate the effectiveness of our proposed embedding through various numerical experiments and show statistically significant correlations between our proposed distance and the forward and backward transfer between tasks.

PRANC: Pseudo RAndom Networks for Compacting deep models

Communication becomes a bottleneck in various distributed Machine Learning settings. Here, we propose a novel training framework that leads to highly efficient communication of models between agents. In short, we train our network to be a linear combination of many pseudo-randomly generated frozen models. For communication, the source agent transmits only the `seed' scalar used to generate the pseudo-random `basis' networks along with the learned linear mixture coefficients. Our method, denoted as PRANC, learns almost $100\times$ fewer parameters than a deep model and still performs well on several datasets and architectures. PRANC enables 1) efficient communication of models between agents, 2) efficient model storage, and 3) accelerated inference by generating layer-wise weights on the fly. We test PRANC on CIFAR-10, CIFAR-100, tinyImageNet, and ImageNet-100 with various architectures like AlexNet, LeNet, ResNet18, ResNet20, and ResNet56 and demonstrate a massive reduction in the number of parameters while providing satisfactory performance on these benchmark datasets. The code is available \href{https://github.com/UCDvision/PRANC}{https://github.com/UCDvision/PRANC}

Sparsity and Heterogeneous Dropout for Continual Learning in the Null Space of Neural Activations

Continual/lifelong learning from a non-stationary input data stream is a cornerstone of intelligence. Despite their phenomenal performance in a wide variety of applications, deep neural networks are prone to forgetting their previously learned information upon learning new ones. This phenomenon is called "catastrophic forgetting" and is deeply rooted in the stability-plasticity dilemma. Overcoming catastrophic forgetting in deep neural networks has become an active field of research in recent years. In particular, gradient projection-based methods have recently shown exceptional performance at overcoming catastrophic forgetting. This paper proposes two biologically-inspired mechanisms based on sparsity and heterogeneous dropout that significantly increase a continual learner's performance over a long sequence of tasks. Our proposed approach builds on the Gradient Projection Memory (GPM) framework. We leverage K-winner activations in each layer of a neural network to enforce layer-wise sparse activations for each task, together with a between-task heterogeneous dropout that encourages the network to use non-overlapping activation patterns between different tasks. In addition, we introduce Continual Swiss Roll as a lightweight and interpretable -- yet challenging -- synthetic benchmark for continual learning. Lastly, we provide an in-depth analysis of our proposed method and demonstrate a significant performance boost on various benchmark continual learning problems.

Teaching Networks to Solve Optimization Problems

Leveraging machine learning to optimize the optimization process is an emerging field which holds the promise to bypass the fundamental computational bottleneck caused by traditional iterative solvers in critical applications requiring near-real-time optimization. The majority of existing approaches focus on learning data-driven optimizers that lead to fewer iterations in solving an optimization. In this paper, we take a different approach and propose to replace the iterative solvers altogether with a trainable parametric set function that outputs the optimal arguments/parameters of an optimization problem in a single feed-forward. We denote our method as, Learning to Optimize the Optimization Process (LOOP). We show the feasibility of learning such parametric (set) functions to solve various classic optimization problems, including linear/nonlinear regression, principal component analysis, transport-based core-set, and quadratic programming in supply management applications. In addition, we propose two alternative approaches for learning such parametric functions, with and without a solver in the-LOOP. Finally, we demonstrate the effectiveness of our proposed approach through various numerical experiments.

SLOSH: Set LOcality Sensitive Hashing via Sliced-Wasserstein Embeddings

Learning from set-structured data is an essential problem with many applications in machine learning and computer vision. This paper focuses on non-parametric and data-independent learning from set-structured data using approximate nearest neighbor (ANN) solutions, particularly locality-sensitive hashing. We consider the problem of set retrieval from an input set query. Such retrieval problem requires: 1) an efficient mechanism to calculate the distances/dissimilarities between sets, and 2) an appropriate data structure for fast nearest neighbor search. To that end, we propose Sliced-Wasserstein set embedding as a computationally efficient "set-2-vector" mechanism that enables downstream ANN, with theoretical guarantees. The set elements are treated as samples from an unknown underlying distribution, and the Sliced-Wasserstein distance is used to compare sets. We demonstrate the effectiveness of our algorithm, denoted as Set-LOcality Sensitive Hashing (SLOSH), on various set retrieval datasets and compare our proposed embedding with standard set embedding approaches, including Generalized Mean (GeM) embedding/pooling, Featurewise Sort Pooling (FSPool), and Covariance Pooling and show consistent improvement in retrieval results. The code for replicating our results is available here: \href{https://github.com/mint-vu/SLOSH}{https://github.com/mint-vu/SLOSH}.

Lifelong Learning with Sketched Structural Regularization

Preventing catastrophic forgetting while continually learning new tasks is an essential problem in lifelong learning. Structural regularization (SR) refers to a family of algorithms that mitigate catastrophic forgetting by penalizing the network for changing its "critical parameters" from previous tasks while learning a new one. The penalty is often induced via a quadratic regularizer defined by an \emph{importance matrix}, e.g., the (empirical) Fisher information matrix in the Elastic Weight Consolidation framework. In practice and due to computational constraints, most SR methods crudely approximate the importance matrix by its diagonal. In this paper, we propose \emph{Sketched Structural Regularization} (Sketched SR) as an alternative approach to compress the importance matrices used for regularizing in SR methods. Specifically, we apply \emph{linear sketching methods} to better approximate the importance matrices in SR algorithms. We show that sketched SR: (i) is computationally efficient and straightforward to implement, (ii) provides an approximation error that is justified in theory, and (iii) is method oblivious by construction and can be adapted to any method that belongs to the structural regularization class. We show that our proposed approach consistently improves various SR algorithms' performance on both synthetic experiments and benchmark continual learning tasks, including permuted-MNIST and CIFAR-100.

Set Representation Learning with Generalized Sliced-Wasserstein Embeddings

An increasing number of machine learning tasks deal with learning representations from set-structured data. Solutions to these problems involve the composition of permutation-equivariant modules (e.g., self-attention, or individual processing via feed-forward neural networks) and permutation-invariant modules (e.g., global average pooling, or pooling by multi-head attention). In this paper, we propose a geometrically-interpretable framework for learning representations from set-structured data, which is rooted in the optimal mass transportation problem. In particular, we treat elements of a set as samples from a probability measure and propose an exact Euclidean embedding for Generalized Sliced Wasserstein (GSW) distances to learn from set-structured data effectively. We evaluate our proposed framework on multiple supervised and unsupervised set learning tasks and demonstrate its superiority over state-of-the-art set representation learning approaches.

Wasserstein Embedding for Graph Learning

We present Wasserstein Embedding for Graph Learning (WEGL), a novel and fast framework for embedding entire graphs in a vector space, in which various machine learning models are applicable for graph-level prediction tasks. We leverage new insights on defining similarity between graphs as a function of the similarity between their node embedding distributions. Specifically, we use the Wasserstein distance to measure the dissimilarity between node embeddings of different graphs. Different from prior work, we avoid pairwise calculation of distances between graphs and reduce the computational complexity from quadratic to linear in the number of graphs. WEGL calculates Monge maps from a reference distribution to each node embedding and, based on these maps, creates a fixed-sized vector representation of the graph. We evaluate our new graph embedding approach on various benchmark graph-property prediction tasks, showing state-of-the-art classification performance, while having superior computational efficiency.

Radon cumulative distribution transform subspace modeling for image classification

We present a new supervised image classification method for problems where the data at hand conform to certain deformation models applied to unknown prototypes or templates. The method makes use of the previously described Radon Cumulative Distribution Transform (R-CDT) for image data, whose mathematical properties are exploited to express the image data in a form that is more suitable for machine learning. While certain operations such as translation, scaling, and higher-order transformations are challenging to model in native image space, we show the R-CDT can capture some of these variations and thus render the associated image classification problems easier to solve. The method is simple to implement, non-iterative, has no hyper-parameters to tune, it is computationally efficient, and provides competitive accuracies to state-of-the-art neural networks for many types of classification problems, especially in a learning with few labels setting. Furthermore, we show improvements with respect to neural network-based methods in terms of computational efficiency (it can be implemented without the use of GPUs), number of training samples needed for training, as well as out-of-distribution generalization. The Python code for reproducing our results is available at https://github.com/rohdelab/rcdt_ns_classifier.

Statistical and Topological Properties of Sliced Probability Divergences

The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a `base divergence' between one-dimensional random projections of the two measures. However, the computational and statistical consequences of such a technique have not yet been well-established. In this paper, we aim at bridging this gap and derive some properties of sliced divergence functions. First, we show that slicing preserves the metric axioms and the weak continuity of the divergence, implying that the sliced divergence will share similar topological properties. We then precise the results in the case where the base divergence belongs to the class of integral probability metrics. On the other hand, we establish that, under mild conditions, the sample complexity of the sliced divergence does not depend on the dimension, even when the base divergence suffers from the curse of dimensionality. We finally apply our general results to the Wasserstein distance and Sinkhorn divergences, and illustrate our theory on both synthetic and real data experiments.