When solving inverse problems, it is increasingly popular to use pre-trained diffusion models as plug-and-play priors. This framework can accommodate different forward models without re-training while preserving the generative capability of diffusion models. Despite their success in many imaging inverse problems, most existing methods rely on privileged information such as derivative, pseudo-inverse, or full knowledge about the forward model. This reliance poses a substantial limitation that restricts their use in a wide range of problems where such information is unavailable, such as in many scientific applications. To address this issue, we propose Ensemble Kalman Diffusion Guidance (EnKG) for diffusion models, a derivative-free approach that can solve inverse problems by only accessing forward model evaluations and a pre-trained diffusion model prior. We study the empirical effectiveness of our method across various inverse problems, including scientific settings such as inferring fluid flows and astronomical objects, which are highly non-linear inverse problems that often only permit black-box access to the forward model.

Ads supply personalization aims to balance the revenue and user engagement, two long-term objectives in social media ads, by tailoring the ad quantity and density. In the industry-scale system, the challenge for ads supply lies in modeling the counterfactual effects of a conservative supply treatment (e.g., a small density change) over an extended duration. In this paper, we present a streamlined framework for personalized ad supply. This framework optimally utilizes information from data collection policies through the doubly robust learning. Consequently, it significantly improves the accuracy of long-term treatment effect estimates. Additionally, its low-complexity design not only results in computational cost savings compared to existing methods, but also makes it scalable for billion-scale applications. Through both offline experiments and online production tests, the framework consistently demonstrated significant improvements in top-line business metrics over months. The framework has been fully deployed to live traffic in one of the world's largest social media platforms.

We present differentially private (DP) algorithms for bilevel optimization, a problem class that received significant attention lately in various machine learning applications. These are the first DP algorithms for this task that are able to provide any desired privacy, while also avoiding Hessian computations which are prohibitive in large-scale settings. Under the well-studied setting in which the upper-level is not necessarily convex and the lower-level problem is strongly-convex, our proposed gradient-based $(\epsilon,\delta)$-DP algorithm returns a point with hypergradient norm at most $\widetilde{\mathcal{O}}\left((\sqrt{d_\mathrm{up}}/\epsilon n)^{1/2}+(\sqrt{d_\mathrm{low}}/\epsilon n)^{1/3}\right)$ where $n$ is the dataset size, and $d_\mathrm{up}/d_\mathrm{low}$ are the upper/lower level dimensions. Our analysis covers constrained and unconstrained problems alike, accounts for mini-batch gradients, and applies to both empirical and population losses.

A prevalent belief among optimization specialists is that linear convergence of gradient descent is contingent on the function growing quadratically away from its minimizers. In this work, we argue that this belief is inaccurate. We show that gradient descent with an adaptive stepsize converges at a local (nearly) linear rate on any smooth function that merely exhibits fourth-order growth away from its minimizer. The adaptive stepsize we propose arises from an intriguing decomposition theorem: any such function admits a smooth manifold around the optimal solution -- which we call the ravine -- so that the function grows at least quadratically away from the ravine and has constant order growth along it. The ravine allows one to interlace many short gradient steps with a single long Polyak gradient step, which together ensure rapid convergence to the minimizer. We illustrate the theory and algorithm on the problems of matrix sensing and factorization and learning a single neuron in the overparameterized regime.

In this paper, we present a novel framework for the analysis of Riemann Hypothesis [27], which is composed of three key components: a) probabilistic modeling with cross entropy optimization and reasoning; b) the application of the law of large numbers; c) the application of mathematical inductions. The analysis is mainly conducted by virtue of probabilistic modeling of cross entropy optimization and reasoning with rare event simulation techniques. The application of the law of large numbers [2, 3, 6] and the application of mathematical inductions make the analysis of Riemann Hypothesis self-contained and complete to make sure that the whole complex plane is covered as conjectured in Riemann Hypothesis. We also discuss the method of enhanced top-p sampling with large language models (LLMs) for reasoning, where next token prediction is not just based on the estimated probabilities of each possible token in the current round but also based on accumulated path probabilities among multiple top-k chain of thoughts (CoTs) paths. The probabilistic modeling of cross entropy optimization and reasoning may suit well with the analysis of Riemann Hypothesis as Riemann Zeta functions are inherently dealing with the sums of infinite components of a complex number series. We hope that our analysis in this paper could shed some light on some of the insights of Riemann Hypothesis. The framework and techniques presented in this paper, coupled with recent developments with chain of thought (CoT) or diagram of thought (DoT) reasoning in large language models (LLMs) with reinforcement learning (RL) [1, 7, 18, 21, 24, 34, 39-41], could pave the way for eventual proof of Riemann Hypothesis [27].

Constrained Reinforcement Learning (RL) has emerged as a significant research area within RL, where integrating constraints with rewards is crucial for enhancing safety and performance across diverse control tasks. In the context of heating systems in the buildings, optimizing the energy efficiency while maintaining the residents' thermal comfort can be intuitively formulated as a constrained optimization problem. However, to solve it with RL may require large amount of data. Therefore, an accurate and versatile simulator is favored. In this paper, we propose a novel building simulator I4B which provides interfaces for different usages and apply a model-free constrained RL algorithm named constrained Soft Actor-Critic with Linear Smoothed Log Barrier function (CSAC-LB) to the heating optimization problem. Benchmarking against baseline algorithms demonstrates CSAC-LB's efficiency in data exploration, constraint satisfaction and performance.

This paper investigates distributed zeroth-order optimization for smooth nonconvex problems. We propose a novel variance-reduced gradient estimator, which randomly renovates one orthogonal direction of the true gradient in each iteration while leveraging historical snapshots for variance correction. By integrating this estimator with gradient tracking mechanism, we address the trade-off between convergence rate and sampling cost per zeroth-order gradient estimation that exists in current zeroth-order distributed optimization algorithms, which rely on either the 2-point or $2d$-point gradient estimators. We derive a convergence rate of $\mathcal{O}(d^{\frac{5}{2}}/m)$ for smooth nonconvex functions in terms of sampling number $m$ and problem dimension $d$. Numerical simulations comparing our algorithm with existing methods confirm the effectiveness and efficiency of the proposed gradient estimator.

Similarity matrix serves as a fundamental tool at the core of numerous downstream machine-learning tasks. However, missing data is inevitable and often results in an inaccurate similarity matrix. To address this issue, Similarity Matrix Completion (SMC) methods have been proposed, but they suffer from high computation complexity due to the Singular Value Decomposition (SVD) operation. To reduce the computation complexity, Matrix Factorization (MF) techniques are more explicit and frequently applied to provide a low-rank solution, but the exact low-rank optimal solution can not be guaranteed since it suffers from a non-convex structure. In this paper, we introduce a novel SMC framework that offers a more reliable and efficient solution. Specifically, beyond simply utilizing the unique Positive Semi-definiteness (PSD) property to guide the completion process, our approach further complements a carefully designed rank-minimization regularizer, aiming to achieve an optimal and low-rank solution. Based on the key insights that the underlying PSD property and Low-Rank property improve the SMC performance, we present two novel, scalable, and effective algorithms, SMCNN and SMCNmF, which investigate the PSD property to guide the estimation process and incorporate nonconvex low-rank regularizer to ensure the low-rank solution. Theoretical analysis ensures better estimation performance and convergence speed. Empirical results on real-world datasets demonstrate the superiority and efficiency of our proposed methods compared to various baseline methods.

The Shortest-Path Problem in Graph of Convex Sets (SPP in GCS) is a recently developed optimization framework that blends discrete and continuous decision making. Many relevant problems in robotics, such as collision-free motion planning, can be cast and solved as an SPP in GCS, yielding lower-cost solutions and faster runtimes than state-of-the-art algorithms. In this paper, we are motivated by motion planning of robot arms that must operate swiftly in static environments. We consider a multi-query extension of the SPP in GCS, where the goal is to efficiently precompute optimal paths between given sets of initial and target conditions. Our solution consists of two stages. Offline, we use semidefinite programming to compute a coarse lower bound on the problem's cost-to-go function. Then, online, this lower bound is used to incrementally generate feasible paths by solving short-horizon convex programs.

We address the challenge of real-time planning of minimum-time trajectories over multiple waypoints, onboard multirotor UAVs. Previous works demonstrated that achieving a truly time-optimal trajectory is computationally too demanding to enable frequent replanning during agile flight, especially on less powerful flight computers. Our approach overcomes this stumbling block by utilizing a point-mass model with a novel iterative thrust decomposition algorithm, enabling the UAV to use all of its collective thrust, something previous point-mass approaches could not achieve. The approach enables gravity and drag modeling integration, significantly reducing tracking errors in high-speed trajectories, which is proven through an ablation study. When combined with a new multi-waypoint optimization algorithm, which uses a gradient-based method to converge to optimal velocities in waypoints, the proposed method generates minimum-time multi-waypoint trajectories within milliseconds. The proposed approach, which we provide as open-source package, is validated both in simulation and in real-world, using Nonlinear Model Predictive Control. With accelerations of up to 3.5g and speeds over 100 km/h, trajectories generated by the proposed method yield similar or even smaller tracking errors than the trajectories generated for a full multirotor model.