Adversarial robustness is a critical property in a variety of modern machine learning applications. While it has been the subject of several recent theoretical studies, many important questions related to adversarial robustness are still open. In this work, we study a fundamental question regarding Bayes optimality for adversarial robustness. We provide general sufficient conditions under which the existence of a Bayes optimal classifier can be guaranteed for adversarial robustness. Our results can provide a useful tool for a subsequent study of surrogate losses in adversarial robustness and their consistency properties.

The standard approach to fitting an autoregressive spike train model is to maximize the likelihood for one-step prediction. This maximum likelihood estimation (MLE) often leads to models that perform poorly when generating samples recursively for more than one time step. Moreover, the generated spike trains can fail to capture important features of the data and even show diverging firing rates. To alleviate this, we propose to directly minimize the divergence between neural recorded and model generated spike trains using spike train kernels. We develop a method that stochastically optimizes the maximum mean discrepancy induced by the kernel. Experiments performed on both real and synthetic neural data validate the proposed approach, showing that it leads to well-behaving models.

We propose to jointly analyze experts' eye movements and verbal narrations to discover important and interpretable knowledge patterns to better understand their decision-making processes. The discovered patterns can further enhance data-driven statistical models by fusing experts' domain knowledge to support complex human-machine collaborative decision-making. Our key contribution is a novel dynamic Bayesian nonparametric model that assigns latent knowledge patterns into key phases involved in complex decision-making. Each phase is characterized by a unique distribution of word topics discovered from verbal narrations and their dynamic interactions with eye movement patterns, indicating experts' special perceptual behavior within a given decision-making stage. A new split-merge-switch sampler is developed to efficiently explore the posterior state space with an improved mixing rate.

We propose a new sparse Granger-causal learning framework for temporal event data. We focus on a specific class of point processes called the Hawkes process. We begin by pointing out that most of the existing sparse causal learning algorithms for the Hawkes process suffer from a singularity in maximum likelihood estimation. As a result, their sparse solutions can appear only as numerical artifacts. In this paper, we propose a mathematically well-defined sparse causal learning framework based on a cardinality-regularized Hawkes process, which remedies the pathological issues of existing approaches.

In this paper, we provide a method to learn the directed structure of a Bayesian network using data. The data is accessed by making conditional probability queries to a black-box model. We introduce a notion of simplicity of representation of conditional probability tables for the nodes in the Bayesian network, that we call low rankness''. We connect this notion to the Fourier transformation of real valued set functions and propose a method which learns the exact directed structure of alow rank Bayesian network using very few queries. We formally prove that our method correctly recovers the true directed structure, runs in polynomial time and only needs polynomial samples with respect to the number of nodes.

Score-based diffusion models synthesize samples by reversing a stochastic process that diffuses data to noise, and are trained by minimizing a weighted combination of score matching losses. The log-likelihood of score-based diffusion models can be tractably computed through a connection to continuous normalizing flows, but log-likelihood is not directly optimized by the weighted combination of score matching losses. We show that for a specific weighting scheme, the objective upper bounds the negative log-likelihood, thus enabling approximate maximum likelihood training of score-based diffusion models. We empirically observe that maximum likelihood training consistently improves the likelihood of score-based diffusion models across multiple datasets, stochastic processes, and model architectures. Our best models achieve negative log-likelihoods of 2.83 and 3.76 bits/dim on CIFAR-10 and ImageNet 32\times 32 without any data augmentation, on a par with state-of-the-art autoregressive models on these tasks.

While distributional reinforcement learning (DistRL) has been empirically effective, the question of when and why it is better than vanilla, non-distributional RL has remained unanswered.This paper explains the benefits of DistRL through the lens of small-loss bounds, which are instance-dependent bounds that scale with optimal achievable cost.Particularly, our bounds converge much faster than those from non-distributional approaches if the optimal cost is small.As warmup, we propose a distributional contextual bandit (DistCB) algorithm, which we show enjoys small-loss regret bounds and empirically outperforms the state-of-the-art on three real-world tasks.In online RL, we propose a DistRL algorithm that constructs confidence sets using maximum likelihood estimation. We prove that our algorithm enjoys novel small-loss PAC bounds in low-rank MDPs.As part of our analysis, we introduce the \ell_1 distributional eluder dimension which may be of independent interest. Then, in offline RL, we show that pessimistic DistRL enjoys small-loss PAC bounds that are novel to the offline setting and are more robust to bad single-policy coverage.

Detecting out-of-distribution (OOD) samples is vital for developing machine learning based models for critical safety systems. Common approaches for OOD detection assume access to some OOD samples during training which may not be available in a real-life scenario. Instead, we utilize the {\em predictive normalized maximum likelihood} (pNML) learner, in which no assumptions are made on the tested input. We derive an explicit expression of the pNML and its generalization error, denoted as the regret, for a single layer neural network (NN). We show that this learner generalizes well when (i) the test vector resides in a subspace spanned by the eigenvectors associated with the large eigenvalues of the empirical correlation matrix of the training data, or (ii) the test sample is far from the decision boundary.

We study the problem of learning generalized linear models under adversarial corruptions.We analyze a classical heuristic called the \textit{iterative trimmed maximum likelihood estimator} which is known to be effective against \textit{label corruptions} in practice. Under label corruptions, we prove that this simple estimator achieves minimax near-optimal risk on a wide range of generalized linear models, including Gaussian regression, Poisson regression and Binomial regression. Finally, we extend the estimator to the much more challenging setting of \textit{label and covariate corruptions} and demonstrate its robustness and optimality in that setting as well.

To analyze a collection of interconnected documents, relational topic models (RTMs) have been developed to describe both the link structure and document content, exploring their underlying relationships via a single-layer latent representation with limited expressive capability. To better utilize the document network, we first propose graph Poisson factor analysis (GPFA) that constructs a probabilistic model for interconnected documents and also provides closed-form Gibbs sampling update equations, moving beyond sophisticated approximate assumptions of existing RTMs. Extending GPFA, we develop a novel hierarchical RTM named graph Poisson gamma belief network (GPGBN), and further introduce two different Weibull distribution based variational graph auto-encoders for efficient model inference and effective network information aggregation. Experimental results demonstrate that our models extract high-quality hierarchical latent document representations, leading to improved performance over baselines on various graph analytic tasks.