We consider the distributed stochastic optimization problem where $n$ agents want to minimize a global function given by the sum of agents' local functions, and focus on the heterogeneous setting when agents' local functions are defined over non-i.i.d. data sets. We study the Local SGD method, where agents perform a number of local stochastic gradient steps and occasionally communicate with a central node to improve their local optimization tasks. We analyze the effect of local steps on the convergence rate and the communication complexity of Local SGD. In particular, instead of assuming a fixed number of local steps across all communication rounds, we allow the number of local steps during the $i$-th communication round, $H_i$, to be different and arbitrary numbers. Our main contribution is to characterize the convergence rate of Local SGD as a function of $\{H_i\}_{i=1}^R$ under various settings of strongly convex, convex, and nonconvex local functions, where $R$ is the total number of communication rounds. Based on this characterization, we provide sufficient conditions on the sequence $\{H_i\}_{i=1}^R$ such that Local SGD can achieve linear speed-up with respect to the number of workers. Furthermore, we propose a new communication strategy with increasing local steps superior to existing communication strategies for strongly convex local functions. On the other hand, for convex and nonconvex local functions, we argue that fixed local steps are the best communication strategy for Local SGD and recover state-of-the-art convergence rate results. Finally, we justify our theoretical results through extensive numerical experiments.

We propose the new task of K principal concept identification for dataset summarizarion. The objective is to find a set of K concepts that best explain the variation within the dataset. Concepts are high-level human interpretable terms such as "tiger", "kayaking" or "happy". The K concepts are selected from a (potentially long) input list of candidates, which we denote the concept-bank. The concept-bank may be taken from a generic dictionary or constructed by task-specific prior knowledge. An image-language embedding method (e.g. CLIP) is used to map the images and the concept-bank into a shared feature space. To select the K concepts that best explain the data, we formulate our problem as a K-uncapacitated facility location problem. An efficient optimization technique is used to scale the local search algorithm to very large concept-banks. The output of our method is a set of K principal concepts that summarize the dataset. Our approach provides a more explicit summary in comparison to selecting K representative images, which are often ambiguous. As a further application of our method, the K principal concepts can be used to classify the dataset into K groups. Extensive experiments demonstrate the efficacy of our approach.

We present a flow-based approach to the optimal transport (OT) problem between two continuous distributions $\pi_0,\pi_1$ on $\mathbb{R}^d$, of minimizing a transport cost $\mathbb{E}[c(X_1-X_0)]$ in the set of couplings $(X_0,X_1)$ whose marginal distributions on $X_0,X_1$ equals $\pi_0,\pi_1$, respectively, where $c$ is a cost function. Our method iteratively constructs a sequence of neural ordinary differentiable equations (ODE), each learned by solving a simple unconstrained regression problem, which monotonically reduce the transport cost while automatically preserving the marginal constraints. This yields a monotonic interior approach that traverses inside the set of valid couplings to decrease the transport cost, which distinguishes itself from most existing approaches that enforce the coupling constraints from the outside. The main idea of the method draws from rectified flow, a recent approach that simultaneously decreases the whole family of transport costs induced by convex functions $c$ (and is hence multi-objective in nature), but is not tailored to minimize a specific transport cost. Our method is a single-object variant of rectified flow that guarantees to solve the OT problem for a fixed, user-specified convex cost function $c$.

Reinforcement Learning (RL) has witnessed great strides for quadruped locomotion, with continued progress in the reliable sim-to-real transfer of policies. However, it remains a challenge to reuse a policy on another robot, which could save time for retraining. In this work, we present a framework for zero-shot policy retargeting wherein diverse motor skills can be transferred between robots of different shapes and sizes. The new framework centers on a planning-and-control pipeline that systematically integrates RL and Model Predictive Control (MPC). The planning stage employs RL to generate a dynamically plausible trajectory as well as the contact schedule, avoiding the combinatorial complexity of contact sequence optimization. This information is then used to seed the MPC to stabilize and robustify the policy roll-out via a new Hybrid Kinodynamic (HKD) model that implicitly optimizes the foothold locations. Hardware results show an ability to transfer policies from both the A1 and Laikago robots to the MIT Mini Cheetah robot without requiring any policy re-tuning.

Specialized motions such as jumping are often achieved on quadruped robots by solving a trajectory optimization problem once and executing the trajectory using a tracking controller. This approach is in parallel with Model Predictive Control (MPC) strategies that commonly control regular gaits via online re-planning. In this work, we present a nonlinear MPC (NMPC) technique that unlocks on-the-fly re-planning of specialized motion skills and regular locomotion within a unified framework. The NMPC reasons about a hybrid kinodynamic model, and is solved using a variant of a constrained Differential Dynamic Programming (DDP) solver. The proposed NMPC enables the robot to perform a variety of agile skills like jumping, bounding, and trotting, and the rapid transition between these skills. We evaluated the proposed algorithm with three challenging motion sequences that combine multiple agile skills, on two quadruped platforms, Unitree A1, and MIT Mini Cheetah, showing its effectiveness and generality.

From optimal transport to robust dimensionality reduction, a plethora of machine learning applications can be cast into the min-max optimization problems over Riemannian manifolds. Though many min-max algorithms have been analyzed in the Euclidean setting, it has proved elusive to translate these results to the Riemannian case. Zhang et al. [2022] have recently shown that geodesic convex concave Riemannian problems always admit saddle-point solutions. Inspired by this result, we study whether a performance gap between Riemannian and optimal Euclidean space convex-concave algorithms is necessary. We answer this question in the negative-we prove that the Riemannian corrected extragradient (RCEG) method achieves last-iterate convergence at a linear rate in the geodesically strongly-convex-concave case, matching the Euclidean result. Our results also extend to the stochastic or non-smooth case where RCEG and Riemanian gradient ascent descent (RGDA) achieve near-optimal convergence rates up to factors depending on curvature of the manifold.

Recently, Supervised Contrastive Learning (SCL) has been shown to achieve excellent performance in most classification tasks. In SCL, a neural network is trained to optimize two objectives: pull an anchor and positive samples together in the embedding space, and push the anchor apart from the negatives. However, these two different objectives may conflict, requiring trade-offs between them during optimization. In this work, we formulate the SCL problem as a Multi-Objective Optimization problem for the fine-tuning phase of RoBERTa language model. Two methods are utilized to solve the optimization problem: (i) the linear scalarization (LS) method, which minimizes a weighted linear combination of pertask losses; and (ii) the Exact Pareto Optimal (EPO) method which finds the intersection of the Pareto front with a given preference vector. We evaluate our approach on several GLUE benchmark tasks, without using data augmentations, memory banks, or generating adversarial examples. The empirical results show that the proposed learning strategy significantly outperforms a strong competitive contrastive learning baseline

Dynamic movement primitives are widely used for learning skills which can be demonstrated to a robot by a skilled human or controller. While their generalization capabilities and simple formulation make them very appealing to use, they possess no strong guarantees to satisfy operational safety constraints for a task. In this paper, we present constrained dynamic movement primitives (CDMP) which can allow for constraint satisfaction in the robot workspace. We present a formulation of a non-linear optimization to perturb the DMP forcing weights regressed by locally-weighted regression to admit a Zeroing Barrier Function (ZBF), which certifies workspace constraint satisfaction. We demonstrate the proposed CDMP under different constraints on the end-effector movement such as obstacle avoidance and workspace constraints on a physical robot. A video showing the implementation of the proposed algorithm using different manipulators in different environments could be found here https://youtu.be/hJegJJkJfys.

This paper describes a revision of the classic Lazy Probabilistic Roadmaps algorithm (Lazy PRM), that results from pairing PRM and a novel Branch-and-Cut (BC) algorithm. Cuts are dynamically generated constraints that are imposed on minimum cost paths over the geometric graphs selected by PRM. Cuts eliminate paths that cannot be mapped into smooth plans that satisfy suitably defined kinematic constraints. We generate candidate smooth plans by fitting splines to vertices in minimum-cost path. Plans are validated with a recently proposed algorithm that maps them into finite traces, without need to choose a fixed discretization step. Trace elements exactly describe when plans cross constraint boundaries modulo arithmetic precision. We evaluate several planners using our methods over the recently proposed BARN benchmark, and we report evidence of the scalability of our approach.

We present a novel procedure for optimization based on the combination of efficient quantized tensor train representation and a generalized maximum matrix volume principle. We demonstrate the applicability of the new Tensor Train Optimizer (TTOpt) method for various tasks, ranging from minimization of multidimensional functions to reinforcement learning. Our algorithm compares favorably to popular evolutionary-based methods and outperforms them by the number of function evaluations or execution time, often by a significant margin.