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We consider minimizing a smooth and strongly convex objective function using a stochastic Newton method. At each iteration, the algorithm is given an oracle access to a stochastic estimate of the Hessian matrix. The oracle model includes popular algorithms such as the Subsampled Newton and Newton Sketch, which can efficiently construct stochastic Hessian estimates for many tasks. Despite using second-order information, these existing methods do not exhibit superlinear convergence, unless the stochastic noise is gradually reduced to zero during the iteration, which would lead to a computational blow-up in the per-iteration cost. We address this limitation with Hessian averaging: instead of using the most recent Hessian estimate, our algorithm maintains an average of all past estimates. This reduces the stochastic noise while avoiding the computational blow-up. We show that this scheme enjoys local $Q$-superlinear convergence with a non-asymptotic rate of $(\Upsilon\sqrt{\log (t)/t}\,)^{t}$, where $\Upsilon$ is proportional to the level of stochastic noise in the Hessian oracle. A potential drawback of this (uniform averaging) approach is that the averaged estimates contain Hessian information from the global phase of the iteration, i.e., before the iterates converge to a local neighborhood. This leads to a distortion that may substantially delay the superlinear convergence until long after the local neighborhood is reached. To address this drawback, we study a number of weighted averaging schemes that assign larger weights to recent Hessians, so that the superlinear convergence arises sooner, albeit with a slightly slower rate. Remarkably, we show that there exists a universal weighted averaging scheme that transitions to local convergence at an optimal stage, and still enjoys a superlinear convergence~rate nearly (up to a logarithmic factor) matching that of uniform Hessian averaging.

Escaping from saddle points and finding local minimum is a central problem in nonconvex optimization. Perturbed gradient methods are perhaps the simplest approach for this problem. However, to find $(\epsilon, \sqrt{\epsilon})$-approximate local minima, the existing best stochastic gradient complexity for this type of algorithms is $\tilde O(\epsilon^{-3.5})$, which is not optimal. In this paper, we propose LENA (Last stEp shriNkAge), a faster perturbed stochastic gradient framework for finding local minima. We show that LENA with stochastic gradient estimators such as SARAH/SPIDER and STORM can find $(\epsilon, \epsilon_{H})$-approximate local minima within $\tilde O(\epsilon^{-3} + \epsilon_{H}^{-6})$ stochastic gradient evaluations (or $\tilde O(\epsilon^{-3})$ when $\epsilon_H = \sqrt{\epsilon}$). The core idea of our framework is a step-size shrinkage scheme to control the average movement of the iterates, which leads to faster convergence to the local minima.

Transformer architectures show spectacular performance on NLP tasks and have recently also been used for tasks such as image completion or image classification. Here we propose to use a sequential image representation, where each prefix of the complete sequence describes the whole image at reduced resolution. Using such Fourier Domain Encodings (FDEs), an auto-regressive image completion task is equivalent to predicting a higher resolution output given a low-resolution input. Additionally, we show that an encoder-decoder setup can be used to query arbitrary Fourier coefficients given a set of Fourier domain observations. We demonstrate the practicality of this approach in the context of computed tomography (CT) image reconstruction. In summary, we show that Fourier Image Transformer (FIT) can be used to solve relevant image analysis tasks in Fourier space, a domain inherently inaccessible to convolutional architectures.

Recently, Graph Neural Networks (GNNs) have been applied for scheduling jobs over clusters, achieving better performance than hand-crafted heuristics. Despite their impressive performance, concerns remain over whether these GNN-based job schedulers meet users' expectations about other important properties, such as strategy-proofness, sharing incentive, and stability. In this work, we consider formal verification of GNN-based job schedulers. We address several domain-specific challenges such as networks that are deeper and specifications that are richer than those encountered when verifying image and NLP classifiers. We develop vegas, the first general framework for verifying both single-step and multi-step properties of these schedulers based on carefully designed algorithms that combine abstractions, refinements, solvers, and proof transfer. Our experimental results show that vegas achieves significant speed-up when verifying important properties of a state-of-the-art GNN-based scheduler compared to previous methods.

Ethereum Improvement Proposal (EIP) 1559 was recently implemented to transform Ethereum's transaction fee market. EIP-1559 utilizes an algorithmic update rule with a constant learning rate to estimate a base fee. The base fee reflects prevailing network conditions and hence provides a more reliable oracle for current gas prices. Using on-chain data from the period after its launch, we evaluate the impact of EIP-1559 on the user experience and market performance. Our empirical findings suggest that although EIP-1559 achieves its goals on average, short-term behavior is marked by intense, chaotic oscillations in block sizes (as predicted by our recent theoretical dynamical system analysis [1]) and slow adjustments during periods of demand bursts (e.g., NFT drops). Both phenomena lead to unwanted inter-block variability in mining rewards. To address this issue, we propose an alternative base fee adjustment rule in which the learning rate varies according to an additive increase, multiplicative decrease (AIMD) update scheme. Our simulations show that the latter robustly outperforms the EIP-1559 protocol under various demand scenarios. These results provide evidence that variable learning rate mechanisms may constitute a promising alternative to the default EIP-1559-based format and contribute to the ongoing discussion on the design of more efficient transaction fee markets.

Model predictive control (MPC) has been used widely in power electronics due to its simple concept, fast dynamic response, and good reference tracking. However, it suffers from parametric uncertainties, since it directly relies on the mathematical model of the system to predict the optimal switching states to be used at the next sampling time. As a result, uncertain parameters lead to an ill-designed MPC. Thus, this paper offers a model-free control strategy on the basis of artificial neural networks (ANNs), for mitigating the effects of parameter mismatching while having a little negative impact on the inverter's performance. This method includes two related stages. First, MPC is used as an expert to control the studied converter in order to provide a dataset, while, in the second stage, the obtained dataset is utilized to train the proposed ANN. The case study herein is based on a four-level three-cell flying capacitor inverter. In this study, MATLAB/Simulink is used to simulate the performance of the proposed method, taking into account various operating conditions. Afterward, the simulation results are reported in comparison with the conventional MPC scheme, demonstrating the superior performance of the proposed control strategy in terms of robustness against parameters mismatch and low total harmonic distortion (THD), especially when changes occur in the system parameters, compared to the conventional MPC. Furthermore, the experimental validation of the proposed method is provided based on the Hardware-in-the-Loop (HIL) simulation using the C2000TM-microcontrollerLaunchPadXL TMS320F28379D kit, demonstrating the applicability of the ANN-based control strategy to be implemented on a DSP controller.

Cyclic motions are fundamental patterns in robotic applications including industrial manipulation and legged robot locomotion. This paper proposes an approach for the online modulation of cyclic motions in robotic applications. For this purpose, we present an integrated programmable Central Pattern Generator (CPG) for the online generation of the reference joint trajectory of a robotic system out of a library of desired periodic motions. The reference trajectory is then followed by the lower-level controller of the robot. The proposed CPG generates a smooth reference joint trajectory convergence to the desired one while preserving the position and velocity joint limits of the robot. The integrated programmable CPG consists of one novel bounded output programmable oscillator. We design the programmable oscillator for encoding the desired multidimensional periodic trajectory as a stable limit cycle. We also use the state transformation method to ensure that the oscillator's output and its first-time derivative preserve the joint position and velocity limits of the robot. With the help of Lyapunov-based arguments, We prove that the proposed CPG provides the global stability and convergence of the desired trajectory. The effectiveness of the proposed integrated CPG for trajectory generation is shown in a passive rehabilitation scenario on the Kuka iiwa robot arm, and also in a walking simulation on a seven-link bipedal robot.

Low-rank matrix estimation under heavy-tailed noise is challenging, both computationally and statistically. Convex approaches have been proven statistically optimal but suffer from high computational costs, especially since robust loss functions are usually non-smooth. More recently, computationally fast non-convex approaches via sub-gradient descent are proposed, which, unfortunately, fail to deliver a statistically consistent estimator even under sub-Gaussian noise. In this paper, we introduce a novel Riemannian sub-gradient (RsGrad) algorithm which is not only computationally efficient with linear convergence but also is statistically optimal, be the noise Gaussian or heavy-tailed. Convergence theory is established for a general framework and specific applications to absolute loss, Huber loss, and quantile loss are investigated. Compared with existing non-convex methods, ours reveals a surprising phenomenon of dual-phase convergence. In phase one, RsGrad behaves as in a typical non-smooth optimization that requires gradually decaying stepsizes. However, phase one only delivers a statistically sub-optimal estimator which is already observed in the existing literature. Interestingly, during phase two, RsGrad converges linearly as if minimizing a smooth and strongly convex objective function and thus a constant stepsize suffices. Underlying the phase-two convergence is the smoothing effect of random noise to the non-smooth robust losses in an area close but not too close to the truth. Lastly, RsGrad is applicable for low-rank tensor estimation under heavy-tailed noise where a statistically optimal rate is attainable with the same phenomenon of dual-phase convergence, and a novel shrinkage-based second-order moment method is guaranteed to deliver a warm initialization. Numerical simulations confirm our theoretical discovery and showcase the superiority of RsGrad over prior methods.

One of the most important problems in system identification and statistics is how to estimate the unknown parameters of a given model. Optimization methods and specialized procedures, such as Empirical Minimization (EM) can be used in case the likelihood function can be computed. For situations where one can only simulate from a parametric model, but the likelihood is difficult or impossible to evaluate, a technique known as the Two-Stage (TS) Approach can be applied to obtain reliable parametric estimates. Unfortunately, there is currently a lack of theoretical justification for TS. In this paper, we propose a statistical decision-theoretical derivation of TS, which leads to Bayesian and Minimax estimators. We also show how to apply the TS approach on models for independent and identically distributed samples, by computing quantiles of the data as a first step, and using a linear function as the second stage. The proposed method is illustrated via numerical simulations.

Since deep neural networks were developed, they have made huge contributions to everyday lives. Machine learning provides more rational advice than humans are capable of in almost every aspect of daily life. However, despite this achievement, the design and training of neural networks are still challenging and unpredictable procedures. To lower the technical thresholds for common users, automated hyper-parameter optimization (HPO) has become a popular topic in both academic and industrial areas. This paper provides a review of the most essential topics on HPO. The first section introduces the key hyper-parameters related to model training and structure, and discusses their importance and methods to define the value range. Then, the research focuses on major optimization algorithms and their applicability, covering their efficiency and accuracy especially for deep learning networks. This study next reviews major services and toolkits for HPO, comparing their support for state-of-the-art searching algorithms, feasibility with major deep learning frameworks, and extensibility for new modules designed by users. The paper concludes with problems that exist when HPO is applied to deep learning, a comparison between optimization algorithms, and prominent approaches for model evaluation with limited computational resources.