We present a maximum entropy inverse reinforcement learning (IRL) approach for improving the sample quality of diffusion generative models, especially when the number of generation time steps is small. Similar to how IRL trains a policy based on the reward function learned from expert demonstrations, we train (or fine-tune) a diffusion model using the log probability density estimated from training data. Since we employ an energy-based model (EBM) to represent the log density, our approach boils down to the joint training of a diffusion model and an EBM. Our IRL formulation, named Diffusion by Maximum Entropy IRL (DxMI), is a minimax problem that reaches equilibrium when both models converge to the data distribution. The entropy maximization plays a key role in DxMI, facilitating the exploration of the diffusion model and ensuring the convergence of the EBM. We also propose Diffusion by Dynamic Programming (DxDP), a novel reinforcement learning algorithm for diffusion models, as a subroutine in DxMI. DxDP makes the diffusion model update in DxMI efficient by transforming the original problem into an optimal control formulation where value functions replace back-propagation in time. Our empirical studies show that diffusion models fine-tuned using DxMI can generate high-quality samples in as few as 4 and 10 steps. Additionally, DxMI enables the training of an EBM without MCMC, stabilizing EBM training dynamics and enhancing anomaly detection performance.

We develop a novel method for carrying out model selection for Bayesian autoencoders (BAEs) by means of prior hyper-parameter optimization. Inspired by the common practice of type-II maximum likelihood optimization and its equivalence to Kullback-Leibler divergence minimization, we propose to optimize the distributional sliced-Wasserstein distance (DSWD) between the output of the autoencoder and the empirical data distribution. The advantages of this formulation are that we can estimate the DSWD based on samples and handle high-dimensional problems. We carry out posterior estimation of the BAE parameters via stochastic gradient Hamiltonian Monte Carlo and turn our BAE into a generative model by fitting a flexible Dirichlet mixture model in the latent space. Thanks to this approach, we obtain a powerful alternative to variational autoencoders, which are the preferred choice in modern application of autoencoders for representation learning with uncertainty. We evaluate our approach qualitatively and quantitatively using a vast experimental campaign on a number of unsupervised learning tasks and show that, in small-data regimes where priors matter, our approach provides state-of-the-art results, outperforming multiple competitive baselines.

Wasserstein barycenters provide a geometrically meaningful way to aggregate probability distributions, built on the theory of optimal transport. They are difficult to compute in practice, however, leading previous work to restrict their supports to finite sets of points. Leveraging a new dual formulation for the regularized Wasserstein barycenter problem, we introduce a stochastic algorithm that constructs a continuous approximation of the barycenter. We establish strong duality and use the corresponding primal-dual relationship to parametrize the barycenter implicitly using the dual potentials of regularized transport problems. The resulting problem can be solved with stochastic gradient descent, which yields an efficient online algorithm to approximate the barycenter of continuous distributions given sample access. We demonstrate the effectiveness of our approach and compare against previous work on synthetic examples and real-world applications.

Machine learning systems often acquire biases by leveraging undesired features in the data, impacting accuracy variably across different sub-populations of the data. However, our current understanding of bias formation mostly focuses on the initial and final stages of learning, leaving a gap in knowledge regarding the transient dynamics. To address this gap, this paper explores the evolution of bias in a teacher-student setup that models different data sub-populations with a Gaussian-mixture model. We provide an analytical description of the stochastic gradient descent dynamics of a linear classifier in this setup, which we prove to be exact in high dimension.Notably, our analysis identifies different properties of the sub-populations that drive bias at different timescales and hence shows a shifting preference of our classifier during training. By applying our general solution to fairness and robustness, we delineate how and when heterogeneous data and spurious features can generate and amplify bias.

Multivariate regression (or multi-task learning) concerns the task of predicting the value of multiple responses from a set of covariates. In this article, we propose a convex optimization formulation for high-dimensional multivariate linear regression under a general error covariance structure. The main difficulty with simultaneous estimation of the regression coefficients and the error covariance matrix lies in the fact that the negative log-likelihood function is not convex. To overcome this difficulty, a new parameterization is proposed, under which the negative log-likelihood function is proved to be convex. For faster computation, two other alternative loss functions are also considered, and proved to be convex under the proposed parameterization. This new parameterization is also useful for covariate-adjusted Gaussian graphical modeling in which the inverse of the error covariance matrix is of interest. A joint non-asymptotic analysis of the regression coefficients and the error covariance matrix is carried out under the new parameterization. In particular, we show that the proposed method recovers the oracle estimator under sharp scaling conditions, and rates of convergence in terms of vector $\ell_\infty$ norm are also established. Empirically, the proposed methods outperform existing high-dimensional multivariate linear regression methods that are based on either minimizing certain non-convex criteria or certain two-step procedures.

Annealed importance sampling (AIS) and related algorithms are highly effective tools for marginal likelihood estimation, but are not fully differentiable due to the use of Metropolis-Hastings correction steps. Differentiability is a desirable property as it would admit the possibility of optimizing marginal likelihood as an objective using gradient-based methods. To this end, we propose Differentiable AIS (DAIS), a variant of AIS which ensures differentiability by abandoning the Metropolis-Hastings corrections. As a further advantage, DAIS allows for mini-batch gradients. We provide a detailed convergence analysis for Bayesian linear regression which goes beyond previous analyses by explicitly accounting for the sampler not having reached equilibrium.

Langevin algorithms are gradient descent methods augmented with additive noise, and are widely used in Markov Chain Monte Carlo (MCMC) sampling, optimization, and machine learning. In recent years, the non-asymptotic analysis of Langevin algorithms for non-convex learning has been extensively explored. For constrained problems with non-convex losses over a compact convex domain with IID data variables, the projected Langevin algorithm achieves a deviation of $O(T^{-1/4} (\log T)^{1/2})$ from its target distribution \cite{lamperski2021projected} in $1$-Wasserstein distance. In this paper, we obtain a deviation of $O(T^{-1/2} \log T)$ in $1$-Wasserstein distance for non-convex losses with $L$-mixing data variables and polyhedral constraints (which are not necessarily bounded). This improves on the previous bound for constrained problems and matches the best-known bound for unconstrained problems.

Intrusion detection is a form of anomalous activity detection in communication network traffic. Continual learning (CL) approaches to the intrusion detection task accumulate old knowledge while adapting to the latest threat knowledge. Previous works have shown the effectiveness of memory replay-based CL approaches for this task. In this work, we present two novel contributions to improve the performance of CL-based network intrusion detection in the context of class imbalance and scalability. First, we extend class balancing reservoir sampling (CBRS), a memory-based CL method, to address the problems of severe class imbalance for large datasets. Second, we propose a novel approach titled perturbation assistance for parameter approximation (PAPA) based on the Gaussian mixture model to reduce the number of \textit{virtual stochastic gradient descent (SGD) parameter} computations needed to discover maximally interfering samples for CL. We demonstrate that the proposed approaches perform remarkably better than the baselines on standard intrusion detection benchmarks created over shorter periods (KDDCUP'99, NSL-KDD, CICIDS-2017/2018, UNSW-NB15, and CTU-13) and a longer period with distribution shift (AnoShift). We also validated proposed approaches on standard continual learning benchmarks (SVHN, CIFAR-10/100, and CLEAR-10/100) and anomaly detection benchmarks (SMAP, SMD, and MSL). Further, the proposed PAPA approach significantly lowers the number of virtual SGD update operations, thus resulting in training time savings in the range of 12 to 40\% compared to the maximally interfered samples retrieval algorithm.

Adam has become one of the most favored optimizers in deep learning problems. Despite its success in practice, numerous mysteries persist regarding its theoretical understanding. In this paper, we study the implicit bias of Adam in linear logistic regression. Specifically, we show that when the training data are linearly separable, the iterates of Adam converge towards a linear classifier that achieves the maximum $\ell_\infty$-margin in direction. Notably, for a general class of diminishing learning rates, this convergence occurs within polynomial time. Our result shed light on the difference between Adam and (stochastic) gradient descent from a theoretical perspective.

Random Reshuffling (RR) is an algorithm for minimizing finite-sum functions that utilizes iterative gradient descent steps in conjunction with data reshuffling. Often contrasted with its sibling Stochastic Gradient Descent (SGD), RR is usually faster in practice and enjoys significant popularity in convex and non-convex optimization. The convergence rate of RR has attracted substantial attention recently and, for strongly convex and smooth functions, it was shown to converge faster than SGD if 1) the stepsize is small, 2) the gradients are bounded, and 3) the number of epochs is large. We remove these 3 assumptions, improve the dependence on the condition number from $\kappa^2$ to $\kappa$ (resp.\