Chaos and unpredictability are traditionally synonymous, yet large-scale machine learning methods recently have demonstrated a surprising ability to forecast chaotic systems well beyond typical predictability horizons. However, recent works disagree on whether specialized methods grounded in dynamical systems theory, such as reservoir computers or neural ordinary differential equations, outperform general-purpose large-scale learning methods such as transformers or recurrent neural networks. These prior studies perform comparisons on few individually-chosen chaotic systems, thereby precluding robust quantification of how statistical modeling choices and dynamical invariants of different chaotic systems jointly determine empirical predictability. Here, we perform the largest to-date comparative study of forecasting methods on the classical problem of forecasting chaos: we benchmark 24 state-of-the-art forecasting methods on a crowdsourced database of 135 low-dimensional systems with 17 forecast metrics. We find that large-scale, domain-agnostic forecasting methods consistently produce predictions that remain accurate up to two dozen Lyapunov times, thereby accessing a new long-horizon forecasting regime well beyond classical methods. We find that, in this regime, accuracy decorrelates with classical invariant measures of predictability like the Lyapunov exponent. However, in data-limited settings outside the long-horizon regime, we find that physics-based hybrid methods retain a comparative advantage due to their strong inductive biases.

The convergence of many numerical optimization techniques is highly dependent on the initial guess given to the solver. To address this issue, we propose a novel approach that utilizes tensor methods to initialize existing optimization solvers near global optima. Our method does not require access to a database of good solutions. We first transform the cost function, which depends on both task parameters and optimization variables, into a probability density function. Unlike existing approaches, the joint probability distribution of the task parameters and optimization variables is approximated using the Tensor Train model, which enables efficient conditioning and sampling. We treat the task parameters as random variables, and for a given task, we generate samples for decision variables from the conditional distribution to initialize the optimization solver. Our method can produce multiple solutions (when they exist) faster than existing methods. We first evaluate the approach on benchmark functions for numerical optimization that are hard to solve using gradient-based optimization solvers with a naive initialization. The results show that the proposed method can generate samples close to global optima and from multiple modes. We then demonstrate the generality and relevance of our framework to robotics by applying it to inverse kinematics with obstacles and motion planning problems with a 7-DoF manipulator.

We study the problem of learning hierarchical polynomials over the standard Gaussian distribution with three-layer neural networks. We specifically consider target functions of the form $h = g \circ p$ where $p : \mathbb{R}^d \rightarrow \mathbb{R}$ is a degree $k$ polynomial and $g: \mathbb{R} \rightarrow \mathbb{R}$ is a degree $q$ polynomial. This function class generalizes the single-index model, which corresponds to $k=1$, and is a natural class of functions possessing an underlying hierarchical structure. Our main result shows that for a large subclass of degree $k$ polynomials $p$, a three-layer neural network trained via layerwise gradient descent on the square loss learns the target $h$ up to vanishing test error in $\widetilde{\mathcal{O}}(d^k)$ samples and polynomial time. This is a strict improvement over kernel methods, which require $\widetilde \Theta(d^{kq})$ samples, as well as existing guarantees for two-layer networks, which require the target function to be low-rank. Our result also generalizes prior works on three-layer neural networks, which were restricted to the case of $p$ being a quadratic. When $p$ is indeed a quadratic, we achieve the information-theoretically optimal sample complexity $\widetilde{\mathcal{O}}(d^2)$, which is an improvement over prior work~\citep{nichani2023provable} requiring a sample size of $\widetilde\Theta(d^4)$. Our proof proceeds by showing that during the initial stage of training the network performs feature learning to recover the feature $p$ with $\widetilde{\mathcal{O}}(d^k)$ samples. This work demonstrates the ability of three-layer neural networks to learn complex features and as a result, learn a broad class of hierarchical functions.

This optimization paradigm can naturally be used to formulate fundamental problems in interpretable supervised learning (e.g., sparse regression and decision trees), in unsupervised learning (e.g., clustering), and beyond; BackboneLearn solves the aforementioned problems faster than exact methods and with higher accuracy than commonly used heuristics. The package is built in Python and is user-friendly and easily extensible: users can directly implement a backbone algorithm for their MIO problem at hand.

Quantum kernel methods are promising candidates for achieving a practical quantum advantage for certain machine learning tasks. Similar to classical machine learning, an exact form of a quantum kernel is expected to have a great impact on the model performance. In this work we combine all trace-induced quantum kernels, including the commonly-used global fidelity and local projected quantum kernels, into a common framework. We show how generalized trace-induced quantum kernels can be constructed as combinations of the fundamental building blocks we coin "Lego" kernels, which impose an inductive bias on the resulting quantum models. We relate the expressive power and generalization ability to the number of non-zero weight Lego kernels and propose a systematic approach to increase the complexity of a quantum kernel model, leading to a new form of the local projected kernels that require fewer quantum resources in terms of the number of quantum gates and measurement shots. We show numerically that models based on local projected kernels can achieve comparable performance to the global fidelity quantum kernel. Our work unifies existing quantum kernels and provides a systematic framework to compare their properties.

In this work we consider the problem of numerical integration, i.e., approximating integrals with respect to a target probability measure using only pointwise evaluations of the integrand. We focus on the setting in which the target distribution is only accessible through a set of $n$ i.i.d. observations, and the integrand belongs to a reproducing kernel Hilbert space. We propose an efficient procedure which exploits a small i.i.d. random subset of $m

We study private empirical risk minimization (ERM) problem for losses satisfying the $(\gamma,\kappa)$-Kurdyka-{\L}ojasiewicz (KL) condition. The Polyak-{\L}ojasiewicz (PL) condition is a special case of this condition when $\kappa=2$. Specifically, we study this problem under the constraint of $\rho$ zero-concentrated differential privacy (zCDP). When $\kappa\in[1,2]$ and the loss function is Lipschitz and smooth over a sufficiently large region, we provide a new algorithm based on variance reduced gradient descent that achieves the rate $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrt{\rho}}\big)^\kappa\big)$ on the excess empirical risk, where $n$ is the dataset size and $d$ is the dimension. We further show that this rate is nearly optimal. When $\kappa \geq 2$ and the loss is instead Lipschitz and weakly convex, we show it is possible to achieve the rate $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrt{\rho}}\big)^\kappa\big)$ with a private implementation of the proximal point method. When the KL parameters are unknown, we provide a novel modification and analysis of the noisy gradient descent algorithm and show that this algorithm achieves a rate of $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrt{\rho}}\big)^{\frac{2\kappa}{4-\kappa}}\big)$ adaptively, which is nearly optimal when $\kappa = 2$. We further show that, without assuming the KL condition, the same gradient descent algorithm can achieve fast convergence to a stationary point when the gradient stays sufficiently large during the run of the algorithm. Specifically, we show that this algorithm can approximate stationary points of Lipschitz, smooth (and possibly nonconvex) objectives with rate as fast as $\tilde{O}\big(\frac{\sqrt{d}}{n\sqrt{\rho}}\big)$ and never worse than $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrt{\rho}}\big)^{1/2}\big)$. The latter rate matches the best known rate for methods that do not rely on variance reduction.

Learning and forecasting stochastic time series is essential in various scientific fields. However, despite the proposals of nonlinear filters and deep-learning methods, it remains challenging to capture nonlinear dynamics from a few noisy samples and predict future trajectories with uncertainty estimates while maintaining computational efficiency. Here, we propose a fast algorithm to learn and forecast nonlinear dynamics from noisy time series data. A key feature of the proposed model is kernel functions applied to projected lines, enabling fast and efficient capture of nonlinearities in the latent dynamics. Through empirical case studies and benchmarking, the model demonstrates its effectiveness in learning and forecasting complex nonlinear dynamics, offering a valuable tool for researchers and practitioners in time series analysis.

Robotic capacities in object manipulation are incomparable to those of humans. Besides years of learning, humans rely heavily on the richness of information from physical interaction with the environment. In particular, tactile sensing is crucial in providing such rich feedback. Despite its potential contributions to robotic manipulation, tactile sensing is less exploited; mainly due to the complexity of the time series provided by tactile sensors. In this work, we propose a method for assessing grasp stability using tactile sensing. More specifically, we propose a methodology to extract task-relevant features and design efficient classifiers to detect object slippage with respect to individual fingertips. We compare two classification models: support vector machine and logistic regression. We use highly sensitive Uskin tactile sensors mounted on an Allegro hand to test and validate our method. Our results demonstrate that the proposed method is effective in slippage detection in an online fashion.

Diffusion models have achieved state-of-the-art performance in generative modeling tasks across various domains. Prior works on time series diffusion models have primarily focused on developing conditional models tailored to specific forecasting or imputation tasks. In this work, we explore the potential of task-agnostic, unconditional diffusion models for several time series applications. We propose TSDiff, an unconditionally-trained diffusion model for time series. Our proposed self-guidance mechanism enables conditioning TSDiff for downstream tasks during inference, without requiring auxiliary networks or altering the training procedure. We demonstrate the effectiveness of our method on three different time series tasks: forecasting, refinement, and synthetic data generation. First, we show that TSDiff is competitive with several task-specific conditional forecasting methods (predict). Second, we leverage the learned implicit probability density of TSDiff to iteratively refine the predictions of base forecasters with reduced computational overhead over reverse diffusion (refine). Notably, the generative performance of the model remains intact -- downstream forecasters trained on synthetic samples from TSDiff outperform forecasters that are trained on samples from other state-of-the-art generative time series models, occasionally even outperforming models trained on real data (synthesize).