There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently—those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers--the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C9O processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.

Fast parallel algorithms for short-range molecular dynamics

Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

Planning Algorithms: Introductory Material

Modern mobile devices have access to a wealth of data suitable for learning models, which in turn can greatly improve the user experience on the device. For example, language models can improve speech recognition and text entry, and image models can automatically select good photos. However, this rich data is often privacy sensitive, large in quantity, or both, which may preclude logging to the data center and training there using conventional approaches. We advocate an alternative that leaves the training data distributed on the mobile devices, and learns a shared model by aggregating locally-computed updates. We term this decentralized approach Federated Learning. 
We present a practical method for the federated learning of deep networks based on iterative model averaging, and conduct an extensive empirical evaluation, considering five different model architectures and four datasets. These experiments demonstrate the approach is robust to the unbalanced and non-IID data distributions that are a defining characteristic of this setting. Communication costs are the principal constraint, and we show a reduction in required communication rounds by 10-100x as compared to synchronized stochastic gradient descent.

Communication-Efficient Learning of Deep Networks from Decentralized Data

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Convex Analysisの二,三の進展について

Proportional response dynamics in the Fisher market

Empirical Risk Minimization (ERM) is a standard technique in machine learning, where a model is selected by minimizing a loss function over constraint set. When the training dataset consists of private information, it is natural to use a differentially private ERM algorithm, and this problem has been the subject of a long line of work started with Chaudhuri and Monteleoni 2008. A private ERM algorithm outputs an approximate minimizer of the loss function and its error can be measured as the difference from the optimal value of the loss function. When the constraint set is arbitrary, the required error bounds are fairly well understood \cite{BassilyST14}. In this work, we show that the geometric properties of the constraint set can be used to derive significantly better results. Specifically, we show that a differentially private version of Mirror Descent leads to error bounds of the form $\tilde{O}(G_{\mathcal{C}}/n)$ for a lipschitz loss function, improving on the $\tilde{O}(\sqrt{p}/n)$ bounds in Bassily, Smith and Thakurta 2014. Here $p$ is the dimensionality of the problem, $n$ is the number of data points in the training set, and $G_{\mathcal{C}}$ denotes the Gaussian width of the constraint set that we optimize over. We show similar improvements for strongly convex functions, and for smooth functions. In addition, we show that when the loss function is Lipschitz with respect to the $\ell_1$ norm and $\mathcal{C}$ is $\ell_1$-bounded, a differentially private version of the Frank-Wolfe algorithm gives error bounds of the form $\tilde{O}(n^{-2/3})$. This captures the important and common case of sparse linear regression (LASSO), when the data $x_i$ satisfies $|x_i|_{\infty} \leq 1$ and we optimize over the $\ell_1$ ball. We show new lower bounds for this setting, that together with known bounds, imply that all our upper bounds are tight.

Private Empirical Risk Minimization Beyond the Worst Case: The Effect of the Constraint Set Geometry.

In this paper, we show that the proportional response dynamics, a utility based distributed dynamics, converges to the market equilibrium in the Fisher market with constant elasticity of substitution (CES) utility functions. By the proportional response dynamics, each buyer allocates his budget proportional to the utility he receives from each good in the previous time period. Unlike the tâtonnement process and its variants, the proportional response dynamics is a large step discrete dynamics, and the buyers do not solve any optimization problem at each step. In addition, the goods are always cleared and assigned to the buyers proportional to their bids at each step. Despite its simplicity, the dynamics converges fast for strictly concave CES utility functions, matching the best upperbound of computing the market equilibrium via solving a global convex optimization problem.

/pdf/proportional-response-dynamics-in-the-fisher-market-4b9znvhkuk.pdf

Proportional Response Dynamics in the Fisher Market

We develop a class of new kinetic data structures for collision detection between moving convex polytopes; the performance of these structures is sensitive to the separation of the polytopes during their motion. For two convex polygons in the plane, let D be the maximum diameter of the polygons, and let s be the minimum distance between them during their motion. Our separation certificate changes O(log(D/s)) times when the relative motion of the two polygons is a translation along a straight line or convex curve, O( √ D/s) for translation along an algebraic trajectory, and O(D/s) for algebraic rigid motion (translation and rotation). Each certificate update is performed in O(log(D/s)) time. Variants of these data structures are also shown that exhibit hysteresis—after a separation certificate fails, the new certificate cannot fail again until the objects have moved by some constant fraction of their current separation. We can then bound the number of events by the combinatorial size of a certain cover of the motion path by balls.

Separation-sensitive collision detection for convex objects

3D object detection in autonomous driving aims to reason “what” and “where” the objects of interest present in a 3D world. Following the conventional wisdom of previous 2D object detection, existing methods often adopt the canonical Cartesian coordinate system with perpendicular axis. However, we conjugate that this does not fit the nature of the ego car’s perspective, as each onboard camera perceives the world in shape of wedge intrinsic to the imaging geometry with radical (non perpendicular) axis. Hence, in this paper we advocate the exploitation of the Polar coordinate system and propose a new Polar Transformer (PolarFormer) for more accurate 3D object detection in the bird’s-eye-view (BEV) taking as input only multi-camera 2D images. Specifically, we design a cross-attention based Polar detection head without restriction to the shape of input structure to deal with irregular Polar grids. For tackling the unconstrained object scale variations along Polar’s distance dimension, we further introduce a multi-scale Polar representation learning strategy. As a result, our model can make best use of the Polar representation rasterized via attending to the corresponding image observation in a sequence-to-sequence fashion subject to the geometric constraints. Thorough experiments on the nuScenes dataset demonstrate that our PolarFormer outperforms significantly state-of-the-art 3D object detection alternatives.

Li Zhang

Papers

Proportional response dynamics in the Fisher market

Private Empirical Risk Minimization Beyond the Worst Case: The Effect of the Constraint Set Geometry.

Proportional Response Dynamics in the Fisher Market

Separation-sensitive collision detection for convex objects

PolarFormer: Multi-camera 3D Object Detection with Polar Transformers