This work tackles the problem of characterizing and understanding the decision boundaries of neural networks with piecewise linear non-linearity activations. We use tropical geometry, a new development in the area of algebraic geometry, to characterize the decision boundaries of a simple neural network of the form (Affine, ReLU, Affine). Our main finding is that the decision boundaries are a subset of a tropical hypersurface, which is intimately related to a polytope formed by the convex hull of two zonotopes. The generators of these zonotopes are functions of the neural network parameters. This geometric characterization provides new perspective to three tasks. Specifically, we propose a new tropical perspective to the lottery ticket hypothesis, where we see the effect of different initializations on the tropical geometric representation of a network's decision boundaries. Moreover, we use this characterization to propose a new set of tropical regularizers, which directly deal with the decision boundaries of a network. We investigate the use of these regularizers in neural network pruning (by removing network parameters that do not contribute to the tropical geometric representation of the decision boundaries) and in generating adversarial input attacks (by producing input perturbations that explicitly perturb the decision boundaries' geometry and ultimately change the network's prediction).

Optimization-based meta-learning algorithms are a powerful class of methods for learning-to-learn applications such as few-shot learning. They tackle the limited availability of training data by leveraging the experience gained from previously observed tasks. However, when the complexity of the tasks distribution cannot be captured by a single set of shared meta-parameters, existing methods may fail to fully adapt to a target task. We address this issue with a novel perspective on conditional meta-learning based on structured prediction. We propose task-adaptive structured meta-learning (TASML), a principled estimator that weighs meta-training data conditioned on the target task to design tailored meta-learning objectives. In addition, we introduce algorithmic improvements to tackle key computational limitations of existing methods. Experimentally, we show that TASML outperforms state-of-the-art methods on benchmark datasets both in terms of accuracy and efficiency. An ablation study quantifies the individual contribution of model components and suggests useful practices for meta-learning.

In this paper, we take a statistical approach to the problem of recovering a function from low-resolution measurements taken with respect to an arbitrary basis, by regarding the function of interest as a realization of a random field. We introduce an infinite-dimensional framework for high-resolution estimation of a random field from its low-resolution indirect measurements as well as the high-resolution measurements of training observations by merging the existing frameworks of generalized sampling and functional principal component analysis. We study the statistical performance of the resulting estimation procedure and show that high-resolution recovery is indeed possible provided appropriate low-rank and angle conditions hold and provided the training set is sufficiently large relative to the desired resolution. We also consider sparse representations of the principle components, which can reduce the required size of the training set. Furthermore, the effectiveness of the proposed procedure is investigated in various numerical examples.

Overparameterized deep networks have the capacity to memorize training data with zero training error. Even after memorization, the training loss continues to approach zero, making the model overconfident and the test performance degraded. Since existing regularizers do not directly aim to avoid zero training loss, they often fail to maintain a moderate level of training loss, ending up with a too small or too large loss. We propose a direct solution called flooding that intentionally prevents further reduction of the training loss when it reaches a reasonably small value, which we call the flooding level. Our approach makes the loss float around the flooding level by doing mini-batched gradient descent as usual but gradient ascent if the training loss is below the flooding level. This can be implemented with one line of code, and is compatible with any stochastic optimizer and other regularizers. With flooding, the model will continue to "random walk" with the same non-zero training loss, and we expect it to drift into an area with a flat loss landscape that leads to better generalization. We experimentally show that flooding improves performance and as a byproduct, induces a double descent curve of the test loss.

Optimal transport is a foundational problem in optimization, that allows to compare probability distributions while taking into account geometric aspects. Its optimal objective value, the Wasserstein distance, provides an important loss between distributions that has been used in many applications throughout machine learning and statistics. Recent algorithmic progress on this problem and its regularized versions have made these tools increasingly popular. However, existing techniques require solving an optimization problem to obtain a single gradient of the loss, thus slowing down first-order methods to minimize the sum of losses, that require many such gradient computations. In this work, we introduce an algorithm to solve a regularized version of this problem of Wasserstein estimators, with a time per step which is sublinear in the natural dimensions of the problem. We introduce a dual formulation, and optimize it with stochastic gradient steps that can be computed directly from samples, without solving additional optimization problems at each step. Doing so, the estimation and computation tasks are performed jointly. We show that this algorithm can be extended to other tasks, including estimation of Wasserstein barycenters. We provide theoretical guarantees and illustrate the performance of our algorithm with experiments on synthetic data.

In this paper, we study the formalism of unsupervised multi-class domain adaptation (multi-class UDA), which underlies some recent algorithms whose learning objectives are only motivated empirically. A Multi-Class Scoring Disagreement (MCSD) divergence is presented by aggregating the absolute margin violations in multi-class classification; the proposed MCSD is able to fully characterize the relations between any pair of multi-class scoring hypotheses. By using MCSD as a measure of domain distance, we develop a new domain adaptation bound for multi-class UDA as well as its data-dependent, probably approximately correct bound, which naturally suggest adversarial learning objectives to align conditional feature distributions across the source and target domains. Consequently, an algorithmic framework of Multi-class Domain-adversarial learning Networks (McDalNets) is developed, whose different instantiations via surrogate learning objectives either coincide with or resemble a few of recently popular methods, thus (partially) underscoring their practical effectiveness. Based on our same theory of multi-class UDA, we also introduce a new algorithm of Domain-Symmetric Networks (SymmNets), which is featured by a novel adversarial strategy of domain confusion and discrimination. SymmNets afford simple extensions that work equally well under the problem settings of either closed set, partial, or open set UDA. We conduct careful empirical studies to compare different algorithms of McDalNets and our newly introduced SymmNets. Experiments verify our theoretical analysis and show the efficacy of our proposed SymmNets. We make our implementation codes publicly available.

Machine learning pipelines often rely on optimization procedures to make discrete decisions (e.g. sorting, picking closest neighbors, finding shortest paths or optimal matchings). Although these discrete decisions are easily computed in a forward manner, they cannot be used to modify model parameters using first-order optimization techniques because they break the back-propagation of computational graphs. In order to expand the scope of learning problems that can be solved in an end-to-end fashion, we propose a systematic method to transform a block that outputs an optimal discrete decision into a differentiable operation. Our approach relies on stochastic perturbations of these parameters, and can be used readily within existing solvers without the need for ad hoc regularization or smoothing. These perturbed optimizers yield solutions that are differentiable and never locally constant. The amount of smoothness can be tuned via the chosen noise amplitude, whose impact we analyze. The derivatives of these perturbed solvers can be evaluated efficiently. We also show how this framework can be connected to a family of losses developed in structured prediction, and describe how these can be used in unsupervised and supervised learning, with theoretical guarantees. We demonstrate the performance of our approach on several machine learning tasks in experiments on synthetic and real data.

Graph convolution networks have attracted many attentions and several graph auto-encoder based clustering models are developed for attributed graph clustering. However, most existing approaches separate clustering and optimization of graph auto-encoder into two individual steps. In this paper, we propose a graph convolution network based clustering model, namely, Embedding Graph Auto-Encoder with JOint Clustering via Adjacency Sharing (\textit{EGAE-JOCAS}). As for the embedded model, we develop a novel joint clustering method, which combines relaxed k-means and spectral clustering and is applicable for the learned embedding. The proposed joint clustering shares the same adjacency within graph convolution layers. Two parts are optimized simultaneously through performing SGD and taking close-form solutions alternatively to ensure a rapid convergence. Moreover, our model is free to incorporate any mechanisms (e.g., attention) into graph auto-encoder. Extensive experiments are conducted to prove the superiority of EGAE-JOCAS. Sufficient theoretical analyses are provided to support the results.

Adversarial training methods typically align distributions by solving two-player games. However, in most current formulations, even if the generator aligns perfectly with data, a sub-optimal discriminator can still drive the two apart. Absent additional regularization, the instability can manifest itself as a never-ending game. In this paper, we introduce a family of objectives by leveraging pairwise discriminators, and show that only the generator needs to converge. The alignment, if achieved, would be preserved with any discriminator. We provide sufficient conditions for local convergence; characterize the capacity balance that should guide the discriminator and generator choices; and construct examples of minimally sufficient discriminators. Empirically, we illustrate the theory and the effectiveness of our approach on synthetic examples. Moreover, we show that practical methods derived from our approach can better generate higher-resolution images.

Adversarial training (AT) is one of the most effective defenses to improve the adversarial robustness of deep learning models. In order to promote the reliability of the adversarially trained models, we propose to boost AT via incorporating hypersphere embedding (HE), which can regularize the adversarial features onto compact hypersphere manifolds. We formally demonstrate that AT and HE are well coupled, which tunes up the learning dynamics of AT from several aspects. We comprehensively validate the effectiveness and universality of HE by embedding it into the popular AT frameworks including PGD-AT, ALP, and TRADES, as well as the FreeAT and FastAT strategies. In experiments, we evaluate our methods on the CIFAR-10 and ImageNet datasets, and verify that integrating HE can consistently enhance the performance of the models trained by each AT framework with little extra computation.