The problem of synchronization over a group $\mathcal{G}$ aims to estimate a collection of group elements $G^*_1, \dots, G^*_n \in \mathcal{G}$ based on noisy observations of a subset of all pairwise ratios of the form $G^*_i {G^*_j}^{-1}$. Such a problem has gained much attention recently and finds many applications across a wide range of scientific and engineering areas. In this paper, we consider the class of synchronization problems in which the group is a closed subgroup of the orthogonal group. This class covers many group synchronization problems that arise in practice. Our contribution is fivefold. First, we propose a unified approach for solving this class of group synchronization problems, which consists of a suitable initialization step and an iterative refinement step based on the generalized power method, and show that it enjoys a strong theoretical guarantee on the estimation error under certain assumptions on the group, measurement graph, noise, and initialization. Second, we formulate two geometric conditions that are required by our approach and show that they hold for various practically relevant subgroups of the orthogonal group. The conditions are closely related to the error-bound geometry of the subgroup -- an important notion in optimization. Third, we verify the assumptions on the measurement graph and noise for standard random graph and random matrix models. Fourth, based on the classic notion of metric entropy, we develop and analyze a novel spectral-type estimator. Finally, we show via extensive numerical experiments that our proposed non-convex approach outperforms existing approaches in terms of computational speed, scalability, and/or estimation error.

We consider the problem of detecting multiple changes in multiple independent time series. The search for the best segmentation can be expressed as a minimization problem over a given cost function. We focus on dynamic programming algorithms that solve this problem exactly. When the number of changes is proportional to data length, an inequality-based pruning rule encoded in the PELT algorithm leads to a linear time complexity. Another type of pruning, called functional pruning, gives a close-to-linear time complexity whatever the number of changes, but only for the analysis of univariate time series. We propose a few extensions of functional pruning for multiple independent time series based on the use of simple geometric shapes (balls and hyperrectangles). We focus on the Gaussian case, but some of our rules can be easily extended to the exponential family. In a simulation study we compare the computational efficiency of different geometric-based pruning rules. We show that for small dimensions (2, 3, 4) some of them ran significantly faster than inequality-based approaches in particular when the underlying number of changes is small compared to the data length.

Policy optimization methods are powerful algorithms in Reinforcement Learning (RL) for their flexibility to deal with policy parameterization and ability to handle model misspecification. However, these methods usually suffer from slow convergence rates and poor sample complexity. Hence it is important to design provably sample efficient algorithms for policy optimization. Yet, recent advances for this problems have only been successful in tabular and linear setting, whose benign structures cannot be generalized to non-linearly parameterized policies. In this paper, we address this problem by leveraging recent advances in value-based algorithms, including bounded eluder-dimension and online sensitivity sampling, to design a low-switching sample-efficient policy optimization algorithm, LPO, with general non-linear function approximation. We show that, our algorithm obtains an $\varepsilon$-optimal policy with only $\widetilde{O}(\frac{\text{poly}(d)}{\varepsilon^3})$ samples, where $\varepsilon$ is the suboptimality gap and $d$ is a complexity measure of the function class approximating the policy. This drastically improves previously best-known sample bound for policy optimization algorithms, $\widetilde{O}(\frac{\text{poly}(d)}{\varepsilon^8})$. Moreover, we empirically test our theory with deep neural nets to show the benefits of the theoretical inspiration.

Magneto-static finite element (FE) simulations make numerical optimization of electrical machines very time-consuming and computationally intensive during the design stage. In this paper, we present the application of a hybrid data-and physics-driven model for numerical optimization of permanent magnet synchronous machines (PMSM). Following the data-driven supervised training, deep neural network (DNN) will act as a meta-model to characterize the electromagnetic behavior of PMSM by predicting intermediate FE measures. These intermediate measures are then post-processed with various physical models to compute the required key performance indicators (KPIs), e.g., torque, shaft power, and material costs. We perform multi-objective optimization with both classical FE and a hybrid approach using a nature-inspired evolutionary algorithm. We show quantitatively that the hybrid approach maintains the quality of Pareto results better or close to conventional FE simulation-based optimization while being computationally very cheap.

Machine learning is a prevalent approach to tame the complexity of design space exploration for domain-specific architectures. Using ML for design space exploration poses challenges. First, it's not straightforward to identify the suitable algorithm from an increasing pool of ML methods. Second, assessing the trade-offs between performance and sample efficiency across these methods is inconclusive. Finally, lack of a holistic framework for fair, reproducible, and objective comparison across these methods hinders progress of adopting ML-aided architecture design space exploration and impedes creating repeatable artifacts. To mitigate these challenges, we introduce ArchGym, an open-source gym and easy-to-extend framework that connects diverse search algorithms to architecture simulators. To demonstrate utility, we evaluate ArchGym across multiple vanilla and domain-specific search algorithms in designing custom memory controller, deep neural network accelerators, and custom SoC for AR/VR workloads, encompassing over 21K experiments. Results suggest that with unlimited samples, ML algorithms are equally favorable to meet user-defined target specification if hyperparameters are tuned; no solution is necessarily better than another (e.g., reinforcement learning vs. Bayesian methods). We coin the term hyperparameter lottery to describe the chance for a search algorithm to find an optimal design provided meticulously selected hyperparameters. The ease of data collection and aggregation in ArchGym facilitates research in ML-aided architecture design space exploration. As a case study, we show this advantage by developing a proxy cost model with an RMSE of 0.61% that offers a 2,000-fold reduction in simulation time. Code and data for ArchGym is available at https://bit.ly/ArchGym.

To accelerate distributed training, many gradient compression methods have been proposed to alleviate the communication bottleneck in synchronous stochastic gradient descent (S-SGD), but their efficacy in real-world applications still remains unclear. In this work, we first evaluate the efficiency of three representative compression methods (quantization with Sign-SGD, sparsification with Top-k SGD, and low-rank with Power-SGD) on a 32-GPU cluster. The results show that they cannot always outperform well-optimized S-SGD or even worse due to their incompatibility with three key system optimization techniques (all-reduce, pipelining, and tensor fusion) in S-SGD. To this end, we propose a novel gradient compression method, called alternate compressed Power-SGD (ACP-SGD), which alternately compresses and communicates low-rank matrices. ACP-SGD not only significantly reduces the communication volume, but also enjoys the three system optimizations like S-SGD. Compared with Power-SGD, the optimized ACP-SGD can largely reduce the compression and communication overheads, while achieving similar model accuracy. In our experiments, ACP-SGD achieves an average of 4.06x and 1.43x speedups over S-SGD and Power-SGD, respectively, and it consistently outperforms other baselines across different setups (from 8 GPUs to 64 GPUs and from 1Gb/s Ethernet to 100Gb/s InfiniBand).

In many applications, the labeled data at the learner's disposal is subject to privacy constraints and is relatively limited. To derive a more accurate predictor for the target domain, it is often beneficial to leverage publicly available labeled data from an alternative domain, somewhat close to the target domain. This is the modern problem of supervised domain adaptation from a public source to a private target domain. We present two $(\epsilon, \delta)$-differentially private adaptation algorithms for supervised adaptation, for which we make use of a general optimization problem, recently shown to benefit from favorable theoretical learning guarantees. Our first algorithm is designed for regression with linear predictors and shown to solve a convex optimization problem. Our second algorithm is a more general solution for loss functions that may be non-convex but Lipschitz and smooth. While our main objective is a theoretical analysis, we also report the results of several experiments first demonstrating that the non-private versions of our algorithms outperform adaptation baselines and next showing that, for larger values of the target sample size or $\epsilon$, the performance of our private algorithms remains close to that of the non-private formulation.

Federated bilevel optimization has attracted increasing attention due to emerging machine learning and communication applications. The biggest challenge lies in computing the gradient of the upper-level objective function (i.e., hypergradient) in the federated setting due to the nonlinear and distributed construction of a series of global Hessian matrices. In this paper, we propose a novel communication-efficient federated hypergradient estimator via aggregated iterative differentiation (AggITD). AggITD is simple to implement and significantly reduces the communication cost by conducting the federated hypergradient estimation and the lower-level optimization simultaneously. We show that the proposed AggITD-based algorithm achieves the same sample complexity as existing approximate implicit differentiation (AID)-based approaches with much fewer communication rounds in the presence of data heterogeneity. Our results also shed light on the great advantage of ITD over AID in the federated/distributed hypergradient estimation. This differs from the comparison in the non-distributed bilevel optimization, where ITD is less efficient than AID. Our extensive experiments demonstrate the great effectiveness and communication efficiency of the proposed method.

We analyze Newton's method with lazy Hessian updates for solving general possibly non-convex optimization problems. We propose to reuse a previously seen Hessian for several iterations while computing new gradients at each step of the method. This significantly reduces the overall arithmetical complexity of second-order optimization schemes. By using the cubic regularization technique, we establish fast global convergence of our method to a second-order stationary point, while the Hessian does not need to be updated each iteration. For convex problems, we justify global and local superlinear rates for lazy Newton steps with quadratic regularization, which is easier to compute. The optimal frequency for updating the Hessian is once every $d$ iterations, where $d$ is the dimension of the problem. This provably improves the total arithmetical complexity of second-order algorithms by a factor $\sqrt{d}$.

The traditional Neural Network-development process requires substantial expert knowledge and relies heavily on intuition and trial-and-error. Neural Architecture Search (NAS) frameworks were introduced to robustly search for network topologies, as well as facilitate the automated development of Neural Networks. While some optimization approaches -- such as Genetic Algorithms -- have been extensively explored in the NAS context, other Metaheuristic Optimization algorithms have not yet been investigated. In this study, we evaluate the viability of Artificial Bee Colony optimization for Neural Architecture Search. Our proposed framework, HiveNAS, outperforms existing state-of-the-art Swarm Intelligence-based NAS frameworks in a fraction of the time.