We study the problem of sparse tensor principal component analysis: given a tensor $\pmb Y = \pmb W + \lambda x^{\otimes p}$ with $\pmb W \in \otimes^p\mathbb{R}^n$ having i.i.d. Gaussian entries, the goal is to recover the $k$-sparse unit vector $x \in \mathbb{R}^n$. The model captures both sparse PCA (in its Wigner form) and tensor PCA. For the highly sparse regime of $k \leq \sqrt{n}$, we present a family of algorithms that smoothly interpolates between a simple polynomial-time algorithm and the exponential-time exhaustive search algorithm. For any $1 \leq t \leq k$, our algorithms recovers the sparse vector for signal-to-noise ratio $\lambda \geq \tilde{\mathcal{O}} (\sqrt{t} \cdot (k/t)^{p/2})$ in time $\tilde{\mathcal{O}}(n^{p+t})$, capturing the state-of-the-art guarantees for the matrix settings (in both the polynomial-time and sub-exponential time regimes). Our results naturally extend to the case of $r$ distinct $k$-sparse signals with disjoint supports, with guarantees that are independent of the number of spikes. Even in the restricted case of sparse PCA, known algorithms only recover the sparse vectors for $\lambda \geq \tilde{\mathcal{O}}(k \cdot r)$ while our algorithms require $\lambda \geq \tilde{\mathcal{O}}(k)$. Finally, by analyzing the low-degree likelihood ratio, we complement these algorithmic results with rigorous evidence illustrating the trade-offs between signal-to-noise ratio and running time. This lower bound captures the known lower bounds for both sparse PCA and tensor PCA. In this general model, we observe a more intricate three-way trade-off between the number of samples $n$, the sparsity $k$, and the tensor power $p$.
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Although pretrained Transformers such as BERT achieve high accuracy on in-distribution examples, do they generalize to new distributions? We systematically measure out-of-distribution (OOD) generalization for various NLP tasks by constructing a new robustness benchmark with realistic distribution shifts. We measure the generalization of previous models including bag-of-words models, ConvNets, and LSTMs, and we show that pretrained Transformers' performance declines are substantially smaller. Pretrained transformers are also more effective at detecting anomalous or OOD examples, while many previous models are frequently worse than chance. We examine which factors affect robustness, finding that larger models are not necessarily more robust, distillation can be harmful, and more diverse pretraining data can enhance robustness. Finally, we show where future work can improve OOD robustness.
The model parameters of convolutional neural networks (CNNs) are determined by backpropagation (BP). In this work, we propose an interpretable feedforward (FF) design without any BP as a reference. The FF design adopts a data-centric approach. It derives network parameters of the current layer based on data statistics from the output of the previous layer in a one-pass manner. To construct convolutional layers, we develop a new signal transform, called the Saab (Subspace Approximation with Adjusted Bias) transform. It is a variant of the principal component analysis (PCA) with an added bias vector to annihilate activation's nonlinearity. Multiple Saab transforms in cascade yield multiple convolutional layers. As to fully-connected (FC) layers, we construct them using a cascade of multi-stage linear least squared regressors (LSRs). The classification and robustness (against adversarial attacks) performances of BP- and FF-designed CNNs applied to the MNIST and the CIFAR-10 datasets are compared. Finally, we comment on the relationship between BP and FF designs.
Graph Neural Networks (GNNs) for representation learning of graphs broadly follow a neighborhood aggregation framework, where the representation vector of a node is computed by recursively aggregating and transforming feature vectors of its neighboring nodes. Many GNN variants have been proposed and have achieved state-of-the-art results on both node and graph classification tasks. However, despite GNNs revolutionizing graph representation learning, there is limited understanding of their representational properties and limitations. Here, we present a theoretical framework for analyzing the expressive power of GNNs in capturing different graph structures. Our results characterize the discriminative power of popular GNN variants, such as Graph Convolutional Networks and GraphSAGE, and show that they cannot learn to distinguish certain simple graph structures. We then develop a simple architecture that is provably the most expressive among the class of GNNs and is as powerful as the Weisfeiler-Lehman graph isomorphism test. We empirically validate our theoretical findings on a number of graph classification benchmarks, and demonstrate that our model achieves state-of-the-art performance.
Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.
This paper addresses the problem of formally verifying desirable properties of neural networks, i.e., obtaining provable guarantees that neural networks satisfy specifications relating their inputs and outputs (robustness to bounded norm adversarial perturbations, for example). Most previous work on this topic was limited in its applicability by the size of the network, network architecture and the complexity of properties to be verified. In contrast, our framework applies to a general class of activation functions and specifications on neural network inputs and outputs. We formulate verification as an optimization problem (seeking to find the largest violation of the specification) and solve a Lagrangian relaxation of the optimization problem to obtain an upper bound on the worst case violation of the specification being verified. Our approach is anytime i.e. it can be stopped at any time and a valid bound on the maximum violation can be obtained. We develop specialized verification algorithms with provable tightness guarantees under special assumptions and demonstrate the practical significance of our general verification approach on a variety of verification tasks.
This work considers the problem of provably optimal reinforcement learning for episodic finite horizon MDPs, i.e. how an agent learns to maximize his/her long term reward in an uncertain environment. The main contribution is in providing a novel algorithm --- Variance-reduced Upper Confidence Q-learning (vUCQ) --- which enjoys a regret bound of $\widetilde{O}(\sqrt{HSAT} + H^5SA)$, where the $T$ is the number of time steps the agent acts in the MDP, $S$ is the number of states, $A$ is the number of actions, and $H$ is the (episodic) horizon time. This is the first regret bound that is both sub-linear in the model size and asymptotically optimal. The algorithm is sub-linear in that the time to achieve $\epsilon$-average regret for any constant $\epsilon$ is $O(SA)$, which is a number of samples that is far less than that required to learn any non-trivial estimate of the transition model (the transition model is specified by $O(S^2A)$ parameters). The importance of sub-linear algorithms is largely the motivation for algorithms such as $Q$-learning and other "model free" approaches. vUCQ algorithm also enjoys minimax optimal regret in the long run, matching the $\Omega(\sqrt{HSAT})$ lower bound. Variance-reduced Upper Confidence Q-learning (vUCQ) is a successive refinement method in which the algorithm reduces the variance in $Q$-value estimates and couples this estimation scheme with an upper confidence based algorithm. Technically, the coupling of both of these techniques is what leads to the algorithm enjoying both the sub-linear regret property and the asymptotically optimal regret.
Image foreground extraction is a classical problem in image processing and vision, with a large range of applications. In this dissertation, we focus on the extraction of text and graphics in mixed-content images, and design novel approaches for various aspects of this problem. We first propose a sparse decomposition framework, which models the background by a subspace containing smooth basis vectors, and foreground as a sparse and connected component. We then formulate an optimization framework to solve this problem, by adding suitable regularizations to the cost function to promote the desired characteristics of each component. We present two techniques to solve the proposed optimization problem, one based on alternating direction method of multipliers (ADMM), and the other one based on robust regression. Promising results are obtained for screen content image segmentation using the proposed algorithm. We then propose a robust subspace learning algorithm for the representation of the background component using training images that could contain both background and foreground components, as well as noise. With the learnt subspace for the background, we can further improve the segmentation results, compared to using a fixed subspace. Lastly, we investigate a different class of signal/image decomposition problem, where only one signal component is active at each signal element. In this case, besides estimating each component, we need to find their supports, which can be specified by a binary mask. We propose a mixed-integer programming problem, that jointly estimates the two components and their supports through an alternating optimization scheme. We show the application of this algorithm on various problems, including image segmentation, video motion segmentation, and also separation of text from textured images.
We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.
The Deep Q-Network proposed by Mnih et al. [2015] has become a benchmark and building point for much deep reinforcement learning research. However, replicating results for complex systems is often challenging since original scientific publications are not always able to describe in detail every important parameter setting and software engineering solution. In this paper, we present results from our work reproducing the results of the DQN paper. We highlight key areas in the implementation that were not covered in great detail in the original paper to make it easier for researchers to replicate these results, including termination conditions and gradient descent algorithms. Finally, we discuss methods for improving the computational performance and provide our own implementation that is designed to work with a range of domains, and not just the original Arcade Learning Environment [Bellemare et al., 2013].
Labeled Latent Dirichlet Allocation (LLDA) is an extension of the standard unsupervised Latent Dirichlet Allocation (LDA) algorithm, to address multi-label learning tasks. Previous work has shown it to perform in par with other state-of-the-art multi-label methods. Nonetheless, with increasing label sets sizes LLDA encounters scalability issues. In this work, we introduce Subset LLDA, a simple variant of the standard LLDA algorithm, that not only can effectively scale up to problems with hundreds of thousands of labels but also improves over the LLDA state-of-the-art. We conduct extensive experiments on eight data sets, with label sets sizes ranging from hundreds to hundreds of thousands, comparing our proposed algorithm with the previously proposed LLDA algorithms (Prior--LDA, Dep--LDA), as well as the state of the art in extreme multi-label classification. The results show a steady advantage of our method over the other LLDA algorithms and competitive results compared to the extreme multi-label classification algorithms.
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