A matrix $A$ is said to have the $\ell_p$-Restricted Isometry Property ($\ell_p$-RIP) if for all vectors $x$ of up to some sparsity $k$, $\|{Ax}\|_p$ is roughly proportional to $\|{x}\|_p$. We study this property for $m \times n$ matrices of rank proportional to $n$ and $k = \Theta(n)$. In this parameter regime, $\ell_p$-RIP matrices are closely connected to Euclidean sections, and are "real analogs" of testing matrices for locally testable codes. It is known that with high probability, random dense $m\times n$ matrices (e.g., with i.i.d. $\pm 1$ entries) are $\ell_2$-RIP with $k \approx m/\log n$, and sparse random matrices are $\ell_p$-RIP for $p \in [1,2)$ when $k, m = \Theta(n)$. However, when $m = \Theta(n)$, sparse random matrices are known to not be $\ell_2$-RIP with high probability. Against this backdrop, we show that sparse matrices cannot be $\ell_2$-RIP in our parameter regime. On the other hand, for $p \neq 2$, we show that every $\ell_p$-RIP matrix must be sparse. Thus, sparsity is incompatible with $\ell_2$-RIP, but necessary for $\ell_p$-RIP for $p \neq 2$. Under a suitable interpretation, our negative result about $\ell_2$-RIP gives an impossibility result for a certain continuous analog of "$c^3$-LTCs": locally testable codes of constant rate, constant distance and constant locality that were constructed in recent breakthroughs.
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Gaussianization is a simple generative model that can be trained without backpropagation. It has shown compelling performance on low dimensional data. As the dimension increases, however, it has been observed that the convergence speed slows down. We show analytically that the number of required layers scales linearly with the dimension for Gaussian input. We argue that this is because the model is unable to capture dependencies between dimensions. Empirically, we find the same linear increase in cost for arbitrary input $p(x)$, but observe favorable scaling for some distributions. We explore potential speed-ups and formulate challenges for further research.
In many modern statistical problems, the limited available data must be used both to develop the hypotheses to test, and to test these hypotheses-that is, both for exploratory and confirmatory data analysis. Reusing the same dataset for both exploration and testing can lead to massive selection bias, leading to many false discoveries. Selective inference is a framework that allows for performing valid inference even when the same data is reused for exploration and testing. In this work, we are interested in the problem of selective inference for data clustering, where a clustering procedure is used to hypothesize a separation of the data points into a collection of subgroups, and we then wish to test whether these data-dependent clusters in fact represent meaningful differences within the data. Recent work by Gao et al. [2022] provides a framework for doing selective inference for this setting, where a hierarchical clustering algorithm is used for producing the cluster assignments, which was then extended to k-means clustering by Chen and Witten [2022]. Both these works rely on assuming a known covariance structure for the data, but in practice, the noise level needs to be estimated-and this is particularly challenging when the true cluster structure is unknown. In our work, we extend this work to the setting of noise with unknown variance, and provide a selective inference method for this more general setting. Empirical results show that our new method is better able to maintain high power while controlling Type I error when the true noise level is unknown.
In stochastic zeroth-order optimization, a problem of practical relevance is understanding how to fully exploit the local geometry of the underlying objective function. We consider a fundamental setting in which the objective function is quadratic, and provide the first tight characterization of the optimal Hessian-dependent sample complexity. Our contribution is twofold. First, from an information-theoretic point of view, we prove tight lower bounds on Hessian-dependent complexities by introducing a concept called energy allocation, which captures the interaction between the searching algorithm and the geometry of objective functions. A matching upper bound is obtained by solving the optimal energy spectrum. Then, algorithmically, we show the existence of a Hessian-independent algorithm that universally achieves the asymptotic optimal sample complexities for all Hessian instances. The optimal sample complexities achieved by our algorithm remain valid for heavy-tailed noise distributions, which are enabled by a truncation method.
Dynamic Sparse Training (DST) is a rapidly evolving area of research that seeks to optimize the sparse initialization of a neural network by adapting its topology during training. It has been shown that under specific conditions, DST is able to outperform dense models. The key components of this framework are the pruning and growing criteria, which are repeatedly applied during the training process to adjust the network's sparse connectivity. While the growing criterion's impact on DST performance is relatively well studied, the influence of the pruning criterion remains overlooked. To address this issue, we design and perform an extensive empirical analysis of various pruning criteria to better understand their effect on the dynamics of DST solutions. Surprisingly, we find that most of the studied methods yield similar results. The differences become more significant in the low-density regime, where the best performance is predominantly given by the simplest technique: magnitude-based pruning. The code is provided at //github.com/alooow/fantastic_weights_paper
Pruning deep neural networks is a widely used strategy to alleviate the computational burden in machine learning. Overwhelming empirical evidence suggests that pruned models retain very high accuracy even with a tiny fraction of parameters. However, relatively little work has gone into characterising the small pruned networks obtained, beyond a measure of their accuracy. In this paper, we use the sparse double descent approach to identify univocally and characterise pruned models associated with classification tasks. We observe empirically that, for a given task, iterative magnitude pruning (IMP) tends to converge to networks of comparable sizes even when starting from full networks with sizes ranging over orders of magnitude. We analyse the best pruned models in a controlled experimental setup and show that their number of parameters reflects task difficulty and that they are much better than full networks at capturing the true conditional probability distribution of the labels. On real data, we similarly observe that pruned models are less prone to overconfident predictions. Our results suggest that pruned models obtained via IMP not only have advantageous computational properties but also provide a better representation of uncertainty in learning.
For safety and robustness of AI systems, we introduce topological parallax as a theoretical and computational tool that compares a trained model to a reference dataset to determine whether they have similar multiscale geometric structure. Our proofs and examples show that this geometric similarity between dataset and model is essential to trustworthy interpolation and perturbation, and we conjecture that this new concept will add value to the current debate regarding the unclear relationship between overfitting and generalization in applications of deep-learning. In typical DNN applications, an explicit geometric description of the model is impossible, but parallax can estimate topological features (components, cycles, voids, etc.) in the model by examining the effect on the Rips complex of geodesic distortions using the reference dataset. Thus, parallax indicates whether the model shares similar multiscale geometric features with the dataset. Parallax presents theoretically via topological data analysis [TDA] as a bi-filtered persistence module, and the key properties of this module are stable under perturbation of the reference dataset.
Pre-trained Language Models (PLMs) have achieved great success in various Natural Language Processing (NLP) tasks under the pre-training and fine-tuning paradigm. With large quantities of parameters, PLMs are computation-intensive and resource-hungry. Hence, model pruning has been introduced to compress large-scale PLMs. However, most prior approaches only consider task-specific knowledge towards downstream tasks, but ignore the essential task-agnostic knowledge during pruning, which may cause catastrophic forgetting problem and lead to poor generalization ability. To maintain both task-agnostic and task-specific knowledge in our pruned model, we propose ContrAstive Pruning (CAP) under the paradigm of pre-training and fine-tuning. It is designed as a general framework, compatible with both structured and unstructured pruning. Unified in contrastive learning, CAP enables the pruned model to learn from the pre-trained model for task-agnostic knowledge, and fine-tuned model for task-specific knowledge. Besides, to better retain the performance of the pruned model, the snapshots (i.e., the intermediate models at each pruning iteration) also serve as effective supervisions for pruning. Our extensive experiments show that adopting CAP consistently yields significant improvements, especially in extremely high sparsity scenarios. With only 3% model parameters reserved (i.e., 97% sparsity), CAP successfully achieves 99.2% and 96.3% of the original BERT performance in QQP and MNLI tasks. In addition, our probing experiments demonstrate that the model pruned by CAP tends to achieve better generalization ability.
The growing energy and performance costs of deep learning have driven the community to reduce the size of neural networks by selectively pruning components. Similarly to their biological counterparts, sparse networks generalize just as well, if not better than, the original dense networks. Sparsity can reduce the memory footprint of regular networks to fit mobile devices, as well as shorten training time for ever growing networks. In this paper, we survey prior work on sparsity in deep learning and provide an extensive tutorial of sparsification for both inference and training. We describe approaches to remove and add elements of neural networks, different training strategies to achieve model sparsity, and mechanisms to exploit sparsity in practice. Our work distills ideas from more than 300 research papers and provides guidance to practitioners who wish to utilize sparsity today, as well as to researchers whose goal is to push the frontier forward. We include the necessary background on mathematical methods in sparsification, describe phenomena such as early structure adaptation, the intricate relations between sparsity and the training process, and show techniques for achieving acceleration on real hardware. We also define a metric of pruned parameter efficiency that could serve as a baseline for comparison of different sparse networks. We close by speculating on how sparsity can improve future workloads and outline major open problems in the field.
Catastrophic forgetting refers to the tendency that a neural network "forgets" the previous learned knowledge upon learning new tasks. Prior methods have been focused on overcoming this problem on convolutional neural networks (CNNs), where the input samples like images lie in a grid domain, but have largely overlooked graph neural networks (GNNs) that handle non-grid data. In this paper, we propose a novel scheme dedicated to overcoming catastrophic forgetting problem and hence strengthen continual learning in GNNs. At the heart of our approach is a generic module, termed as topology-aware weight preserving~(TWP), applicable to arbitrary form of GNNs in a plug-and-play fashion. Unlike the main stream of CNN-based continual learning methods that rely on solely slowing down the updates of parameters important to the downstream task, TWP explicitly explores the local structures of the input graph, and attempts to stabilize the parameters playing pivotal roles in the topological aggregation. We evaluate TWP on different GNN backbones over several datasets, and demonstrate that it yields performances superior to the state of the art. Code is publicly available at \url{//github.com/hhliu79/TWP}.
The area of Data Analytics on graphs promises a paradigm shift as we approach information processing of classes of data, which are typically acquired on irregular but structured domains (social networks, various ad-hoc sensor networks). Yet, despite its long history, current approaches mostly focus on the optimization of graphs themselves, rather than on directly inferring learning strategies, such as detection, estimation, statistical and probabilistic inference, clustering and separation from signals and data acquired on graphs. To fill this void, we first revisit graph topologies from a Data Analytics point of view, and establish a taxonomy of graph networks through a linear algebraic formalism of graph topology (vertices, connections, directivity). This serves as a basis for spectral analysis of graphs, whereby the eigenvalues and eigenvectors of graph Laplacian and adjacency matrices are shown to convey physical meaning related to both graph topology and higher-order graph properties, such as cuts, walks, paths, and neighborhoods. Next, to illustrate estimation strategies performed on graph signals, spectral analysis of graphs is introduced through eigenanalysis of mathematical descriptors of graphs and in a generic way. Finally, a framework for vertex clustering and graph segmentation is established based on graph spectral representation (eigenanalysis) which illustrates the power of graphs in various data association tasks. The supporting examples demonstrate the promise of Graph Data Analytics in modeling structural and functional/semantic inferences. At the same time, Part I serves as a basis for Part II and Part III which deal with theory, methods and applications of processing Data on Graphs and Graph Topology Learning from data.
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