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Lion (Evolved Sign Momentum), a new optimizer discovered through program search, has shown promising results in training large AI models. It performs comparably or favorably to AdamW but with greater memory efficiency. As we can expect from the results of a random search program, Lion incorporates elements from several existing algorithms, including signed momentum, decoupled weight decay, Polak, and Nesterov momentum, but does not fit into any existing category of theoretically grounded optimizers. Thus, even though Lion appears to perform well as a general-purpose optimizer for a wide range of tasks, its theoretical basis remains uncertain. This lack of theoretical clarity limits opportunities to further enhance and expand Lion's efficacy. This work aims to demystify Lion. Based on both continuous-time and discrete-time analysis, we demonstrate that Lion is a theoretically novel and principled approach for minimizing a general loss function $f(x)$ while enforcing a bound constraint $\norm{x}_\infty \leq 1/\lambda$. Lion achieves this through the incorporation of decoupled weight decay, where $\lambda$ represents the weight decay coefficient. Our analysis is made possible by the development of a new Lyapunov function for the Lion updates. It applies to a broader family of Lion-$\phi$ algorithms, where the $\text{sign}(\cdot)$ operator in Lion is replaced by the subgradient of a convex function $\phi$, leading to the solution of a general composite optimization problem of $\min_x f(x) + \phi^*(x)$. Our findings provide valuable insights into the dynamics of Lion and pave the way for further improvements and extensions of Lion-related algorithms. 

We present rectified flow, a surprisingly simple approach to learning (neural) ordinary differential equation (ODE) models to transport between two empirically observed distributions π0 and π1, hence providing a unified solution to generative modeling and domain transfer, among various other tasks involving distribution transport. The idea of rectified flow is to learn the ODE to follow the straight paths connecting the points drawn from π0 and π1 as much as possible. This is achieved by solving a straightforward nonlinear least squares optimization problem, which can be easily scaled to large models without introducing extra parameters beyond standard supervised learning. The straight paths are special and preferred because they are the shortest paths between two points, and can be simulated exactly without time discretization and hence yield computationally efficient models. We show that the procedure of learning a rectified flow from data, called rectification, turns an arbitrary coupling of π0 and π1 to a new deterministic coupling with provably non-increasing convex transport costs. In addition, recursively applying rectification allows us to obtain a sequence of flows with increasingly straight paths, which can be simulated accurately with coarse time discretization in the inference phase. In empirical studies, we show that rectified flow performs superbly on image generation, image-to-image translation, and domain adaptation. In particular, on image generation and translation, our method yields nearly straight flows that give high quality results even with a single Euler discretization step. 

Diffusion models have achieved promising results on generative learning recently. However, because diffusion processes are most naturally applied on the uncon- strained Euclidean space Rd, key challenges arise for developing diffusion based models for learning data on constrained and structured domains. We present a simple and unified framework to achieve this that can be easily adopted to various types of domains, including product spaces of any type (be it bounded/unbounded, continuous/discrete, categorical/ordinal, or their mix). In our model, the diffu- sion process is driven by a drift force that is a sum of two terms: one singular force designed by Doob’s h-transform that ensures all outcomes of the process to belong to the desirable domain, and one non-singular neural force field that is trained to make sure the outcome follows the data distribution statistically. Ex- periments show that our methods perform superbly on generating tabular data, images, semantic segments and 3D point clouds. Code is available at https: //github.com/gnobitab/ConstrainedDiffusionBridge. 

We propose a family of First Hitting Diffusion Models (FHDM), deep generative models that generate data with a diffusion process that terminates at a random first hitting time. This yields an extension of the standard fixed-time diffusion models that terminate at a pre-specified deterministic time. Although standard diffusion models are designed for continuous unconstrained data, FHDM is natu- rally designed to learn distributions on continuous as well as a range of discrete and structure domains. Moreover, FHDM enables instance-dependent terminate time and accelerates the diffusion process to sample higher quality data with fewer diffusion steps. Technically, we train FHDM by maximum likelihood estimation on diffusion trajectories augmented from observed data with conditional first hitting processes (i.e., bridge) derived based on Doob’s h-transform, deviating from the commonly used time-reversal mechanism. We apply FHDM to generate data in various domains such as point cloud (general continuous distribution), climate and geographical events on earth (continuous distribution on the sphere), unweighted graphs (distribution of binary matrices), and segmentation maps of 2D images (high-dimensional categorical distribution). We observe considerable improvement compared with the state-of-the-art approaches in both quality and speed. 

Bilevel optimization (BO) is useful for solving a variety of important machine learning problems including but not limited to hyperparameter optimization, meta- learning, continual learning, and reinforcement learning. Conventional BO methods need to differentiate through the low-level optimization process with implicit dif- ferentiation, which requires expensive calculations related to the Hessian matrix. There has been a recent quest for first-order methods for BO, but the methods pro- posed to date tend to be complicated and impractical for large-scale deep learning applications. In this work, we propose a simple first-order BO algorithm that de- pends only on first-order gradient information, requires no implicit differentiation, and is practical and efficient for large-scale non-convex functions in deep learning. We provide a non-asymptotic convergence analysis of the proposed method to stationary points for non-convex objectives and present empirical results that show its superior practical performance. 

As intelligent agents become autonomous over longer periods of time, they may eventually be- come lifelong counterparts to specific people. If so, it may be common for a user to want the agent to master a task temporarily but later on to forget the task due to privacy concerns. How- ever enabling an agent to forget privately what the user specified without degrading the rest of the learned knowledge is a challenging problem. With the aim of addressing this challenge, this paper formalizes this continual learning and private unlearning (CLPU) problem. The pa- per further introduces a straightforward but exactly private solution, CLPU-DER++, as the first step towards solving the CLPU problem, along with a set of carefully designed benchmark prob- lems to evaluate the effectiveness of the proposed solution. 

Stein’s method compares probability distributions through the study of a class of linear operators called Stein operators. While mainly studied in probability and used to underpin theoretical statistics, Stein’s method has led to significant advances in computational statistics in recent years. The goal of this survey is to bring together some of these recent developments, and in doing so, to stimulate further research into the successful field of Stein’s method and statistics. The topics we discuss include tools to benchmark and compare sampling methods such as approximate Markov chain Monte Carlo, deterministic alternatives to sampling methods, control variate techniques, parameter estimation and goodness-of-fit testing. 

Finding diverse and representative Pareto solutions from the Pareto front is a key challenge in multi-objective optimization (MOO). In this work, we propose a novel gradient-based algorithm for profiling Pareto front by using Stein variational gradient descent (SVGD). We also provide a counterpart of our method based on Langevin dynamics. Our methods iteratively update a set of points in a parallel fashion to push them towards the Pareto front using multiple gradient descent, while encouraging the diversity between the particles by using the repulsive force mechanism in SVGD, or diffusion noise in Langevin dynamics. Compared with existing gradient-based methods that require predefined preference functions, our method can work efficiently in high dimensional problems, and can obtain more diverse solutions evenly distributed in the Pareto front. Moreover, our methods are theoretically guaranteed to converge to the Pareto front. We demonstrate the effectiveness of our method, especially the SVGD algorithm, through extensive experiments, showing its superiority over existing gradient-based algorithms. 

Sampling-based inference and learning techniques, especially Bayesian inference, provide an essential approach to handling uncertainty in machine learning (ML). As these techniques are increasingly used in daily life, it becomes essential to safeguard the ML systems with various trustworthy-related constraints, such as fairness, safety, interpretability. Mathematically, enforcing these constraints in probabilistic inference can be cast into sampling from intractable distributions subject to general nonlinear constraints, for which practical efficient algorithms are still largely missing. In this work, we propose a family of constrained sampling algorithms which generalize Langevin Dynamics (LD) and Stein Variational Gradient Descent (SVGD) to incorporate a moment constraint specified by a general nonlinear function. By exploiting the gradient flow structure of LD and SVGD, we derive two types of algorithms for handling constraints, including a primal-dual gradient approach and the constraint controlled gradient descent approach. We investigate the continuous-time mean-field limit of these algorithms and show that they have O(1/t) convergence under mild conditions. Moreover, the LD variant converges linearly assuming that a log Sobolev like inequality holds. Various numerical experiments are conducted to demonstrate the efficiency of our algorithms in trustworthy settings. 

We propose MaxUp, an embarrassingly simple, highly effective technique for improving the generalization performance of machine learning models, especially deep neural networks. The idea is to generate a set of augmented data with some random perturbations or transforms and minimize the maximum, or worst case loss over the augmented data. By doing so, we implicitly introduce a smoothness or robustness regularization against the random perturbations, and hence improve the generation performance. For example, in the case of Gaussian perturbation, MaxUp is asymptotically equivalent to using the gradient norm of the loss as a penalty to encourage smoothness. We test MaxUp on a range of tasks, including image classification, language modeling, and adversarial certification, on which MaxUp consistently outperforms the existing best baseline methods, without introducing substantial computational overhead. In particular, we improve ImageNet classification from the state-of-the-art top-1 accuracy 85.5% without extra data to 85.8%. Code will be released soon.