Recent advances in diffusion models have shown remarkable potential in the conditional generation of novel molecules. These models can be guided in two ways: (i) explicitly, through additional features representing the condition, or (ii) implicitly, using a property predictor. However, training property predictors or conditional diffusion models requires an abundance of labeled data and is inherently challenging in real-world applications. We propose a novel approach that attenuates the limitations of acquiring large labeled datasets by leveraging domain knowledge from quantum chemistry as a non-differentiable oracle to guide an unconditional diffusion model. Instead of relying on neural networks, the oracle provides accurate guidance in the form of estimated gradients, allowing the diffusion process to sample from a conditional distribution specified by quantum chemistry. We show that this results in more precise conditional generation of novel and stable molecular structures. Our experiments demonstrate that our method: (1) significantly reduces atomic forces, enhancing the validity of generated molecules when used for stability optimization; (2) is compatible with both explicit and implicit guidance in diffusion models, enabling joint optimization of molecular properties and stability; and (3) generalizes effectively to molecular optimization tasks beyond stability optimization.

This paper introduces the Bidirectional Clustered MPPI (BiC-MPPI) algorithm, a novel trajectory optimization method aimed at enhancing goal-directed guidance within the Model Predictive Path Integral (MPPI) framework. BiC-MPPI incorporates bidirectional dynamics approximations and a new guide cost mechanism, improving both trajectory planning and goal-reaching performance. By leveraging forward and backward rollouts, the bidirectional approach ensures effective trajectory connections between initial and terminal states, while the guide cost helps discover dynamically feasible paths. Experimental results demonstrate that BiC-MPPI outperforms existing MPPI variants in both 2D and 3D environments, achieving higher success rates and competitive computation times across 900 simulations on a modified BARN dataset for autonomous navigation. GitHub: https://github.com/i-ASL/BiC-MPPI

Decentralized Federated Learning (DFL) surpasses Centralized Federated Learning (CFL) in terms of faster training, privacy preservation, and light communication, making it a promising alternative in the field of federated learning. However, DFL still exhibits significant disparities with CFL in terms of generalization ability such as rarely theoretical understanding and degraded empirical performance due to severe inconsistency. In this paper, we enhance the consistency of DFL by developing an opposite lookahead enhancement technique (Ole), yielding OledFL to optimize the initialization of each client in each communication round, thus significantly improving both the generalization and convergence speed. Moreover, we rigorously establish its convergence rate in non-convex setting and characterize its generalization bound through uniform stability, which provides concrete reasons why OledFL can achieve both the fast convergence speed and high generalization ability. Extensive experiments conducted on the CIFAR10 and CIFAR100 datasets with Dirichlet and Pathological distributions illustrate that our OledFL can achieve up to 5\% performance improvement and 8$\times$ speedup, compared to the most popular DFedAvg optimizer in DFL.

Fine-tuning language models (LMs) with the Adam optimizer often demands excessive memory, limiting accessibility. The "in-place" version of Stochastic Gradient Descent (IP-SGD) and Memory-Efficient Zeroth-order Optimizer (MeZO) have been proposed to address this. However, IP-SGD still requires substantial memory, and MeZO suffers from slow convergence and degraded final performance due to its zeroth-order nature. This paper introduces Addax, a novel method that improves both memory efficiency and performance of IP-SGD by integrating it with MeZO. Specifically, Addax computes zeroth- or first-order gradients of data points in the minibatch based on their memory consumption, combining these gradient estimates to update directions. By computing zeroth-order gradients for data points that require more memory and first-order gradients for others, Addax overcomes the slow convergence of MeZO and the excessive memory requirement of IP-SGD. Additionally, the zeroth-order gradient acts as a regularizer for the first-order gradient, further enhancing the model's final performance. Theoretically, we establish the convergence of Addax under mild assumptions, demonstrating faster convergence and less restrictive hyper-parameter choices than MeZO. Our experiments with diverse LMs and tasks show that Addax consistently outperforms MeZO regarding accuracy and convergence speed while having a comparable memory footprint. When fine-tuning OPT-13B with one A100 GPU, on average, Addax outperforms MeZO in accuracy/F1 score by 14% and runs 15x faster while using memory similar to MeZO. In our experiments on the larger OPT-30B model, on average, Addax outperforms MeZO in terms of accuracy/F1 score by >16 and runs 30x faster on a single H100 GPU. Moreover, Addax surpasses the performance of standard fine-tuning approaches, such as IP-SGD and Adam, in most tasks with significantly less memory requirement.

The problem of determining the configuration of points from partial distance information, known as the Euclidean Distance Geometry (EDG) problem, is fundamental to many tasks in the applied sciences. In this paper, we propose two algorithms grounded in the Riemannian optimization framework to address the EDG problem. Our approach formulates the problem as a low-rank matrix completion task over the Gram matrix, using partial measurements represented as expansion coefficients of the Gram matrix in a non-orthogonal basis. For the first algorithm, under a uniform sampling with replacement model for the observed distance entries, we demonstrate that, with high probability, a Riemannian gradient-like algorithm on the manifold of rank-$r$ matrices converges linearly to the true solution, given initialization via a one-step hard thresholding. This holds provided the number of samples, $m$, satisfies $m \geq \mathcal{O}(n^{7/4}r^2 \log(n))$. With a more refined initialization, achieved through resampled Riemannian gradient-like descent, we further improve this bound to $m \geq \mathcal{O}(nr^2 \log(n))$. Our analysis for the first algorithm leverages a non-self-adjoint operator and depends on deriving eigenvalue bounds for an inner product matrix of restricted basis matrices, leveraging sparsity properties for tighter guarantees than previously established. The second algorithm introduces a self-adjoint surrogate for the sampling operator. This algorithm demonstrates strong numerical performance on both synthetic and real data. Furthermore, we show that optimizing over manifolds of higher-than-rank-$r$ matrices yields superior numerical results, consistent with recent literature on overparameterization in the EDG problem.

A typical manipulation task consists of a manipulator equipped with a gripper to grasp and move an object with constraints on the motion of the hand-held object, which may be due to the nature of the task itself or from object-environment contacts. In this paper, we study the problem of computing joint torques and grasping forces for time-optimal motion of an object, while ensuring that the grasp is not lost and any constraints on the motion of the object, either due to dynamics, environment contact, or no-slip requirements, are also satisfied. We present a second-order cone program (SOCP) formulation of the time-optimal trajectory planning problem that considers nonlinear friction cone constraints at the hand-object and object-environment contacts. Since SOCPs are convex optimization problems that can be solved optimally in polynomial time using interior point methods, we can solve the trajectory optimization problem efficiently. We present simulation results on three examples, including a non-prehensile manipulation task, which shows the generality and effectiveness of our approach.

Reinforcement Learning from Human Feedback (RLHF) has emerged as a pivotal tool for aligning large language models (LLMs) with human preferences. Direct Preference Optimization (DPO), one of the most popular approaches, formulates RLHF as a policy optimization problem without explicitly estimating the reward function. It overcomes the stability and efficiency issues of two-step approaches, which typically involve first estimating the reward function and then optimizing the policy via proximal policy optimization (PPO). Since RLHF is essentially an optimization problem, and it is well-known that momentum techniques can accelerate optimization both theoretically and empirically, a natural question arises: Can RLHF be accelerated by momentum? This paper answers this question in the affirmative. In detail, we first show that the iterative preference optimization method can be viewed as a proximal point method. Based on this observation, we propose a general Accelerated Preference Optimization (APO) framework, which unifies many existing preference optimization algorithms and employs Nesterov's momentum technique to speed up the alignment of LLMs. Theoretically, we demonstrate that APO can achieve a faster convergence rate than the standard iterative preference optimization methods, including DPO and Self-Play Preference Optimization (SPPO). Empirically, we show the superiority of APO over DPO, iterative DPO, and other strong baselines for RLHF on the AlpacaEval 2.0 benchmark.

The minimum sum-of-squares clustering problem (MSSC), also known as $k$-means clustering, refers to the problem of partitioning $n$ data points into $k$ clusters, with the objective of minimizing the total sum of squared Euclidean distances between each point and the center of its assigned cluster. We propose an efficient algorithm for solving large-scale MSSC instances, which combines column generation (CG) with dynamic constraint aggregation (DCA) to effectively reduce the number of constraints considered in the CG master problem. DCA was originally conceived to reduce degeneracy in set partitioning problems by utilizing an aggregated restricted master problem obtained from a partition of the set partitioning constraints into disjoint clusters. In this work, we explore the use of DCA within a CG algorithm for MSSC exact solution. Our method is fine-tuned by a series of ablation studies on DCA design choices, and is demonstrated to significantly outperform existing state-of-the-art exact approaches available in the literature.

Many recent developments for robots to represent environments have focused on photorealistic reconstructions. This paper particularly focuses on generating sequences of images from the photorealistic Gaussian Splatting models, that match instructions that are given by user-inputted language. We contribute a novel framework, SplaTraj, which formulates the generation of images within photorealistic environment representations as a continuous-time trajectory optimization problem. Costs are designed so that a camera following the trajectory poses will smoothly traverse through the environment and render the specified spatial information in a photogenic manner. This is achieved by querying a photorealistic representation with language embedding to isolate regions that correspond to the user-specified inputs. These regions are then projected to the camera's view as it moves over time and a cost is constructed. We can then apply gradient-based optimization and differentiate through the rendering to optimize the trajectory for the defined cost. The resulting trajectory moves to photogenically view each of the specified objects. We empirically evaluate our approach on a suite of environments and instructions, and demonstrate the quality of generated image sequences.

Zero-order (ZO) optimization is a powerful tool for dealing with realistic constraints. On the other hand, the gradient-tracking (GT) technique proved to be an efficient method for distributed optimization aiming to achieve consensus. However, it is a first-order (FO) method that requires knowledge of the gradient, which is not always possible in practice. In this work, we introduce a zero-order distributed optimization method based on a one-point estimate of the gradient tracking technique. We prove that this new technique converges with a single noisy function query at a time in the non-convex setting. We then establish a convergence rate of $O(\frac{1}{\sqrt[3]{K}})$ after a number of iterations K, which competes with that of $O(\frac{1}{\sqrt[4]{K}})$ of its centralized counterparts. Finally, a numerical example validates our theoretical results.