We propose a novel method that solves global optimization problems in two steps: (1) perform a (exponential) power-$N$ transformation to the not-necessarily differentiable objective function $f$ and get $f_N$, and (2) optimize the Gaussian-smoothed $f_N$ with stochastic approximations. Under mild conditions on $f$, for any $\delta>0$, we prove that with a sufficiently large power $N_\delta$, this method converges to a solution in the $\delta$-neighborhood of $f$'s global optimum point. The convergence rate is $O(d^2\sigma^4\varepsilon^{-2})$, which is faster than both the standard and single-loop homotopy methods if $\sigma$ is pre-selected to be in $(0,1)$. In most of the experiments performed, our method produces better solutions than other algorithms that also apply smoothing techniques.
In complex physical systems, conventional differential equations often fall short in capturing non-local and memory effects, as they are limited to local dynamics and integer-order interactions. This study introduces a stepwise data-driven framework for discovering fractional differential equations (FDEs) directly from data. FDEs, known for their capacity to model non-local dynamics with fewer parameters than integer-order derivatives, can represent complex systems with long-range interactions. Our framework applies deep neural networks as surrogate models for denoising and reconstructing sparse and noisy observations while using Gaussian-Jacobi quadrature to handle the challenges posed by singularities in fractional derivatives. To optimize both the sparse coefficients and fractional order, we employ an alternating optimization approach that combines sparse regression with global optimization techniques. We validate the framework across various datasets, including synthetic anomalous diffusion data, experimental data on the creep behavior of frozen soils, and single-particle trajectories modeled by L\'{e}vy motion. Results demonstrate the framework's robustness in identifying the structure of FDEs across diverse noise levels and its capacity to capture integer-order dynamics, offering a flexible approach for modeling memory effects in complex systems.
Complex dynamical systems, such as particle accelerators, often require complicated and time-consuming tuning procedures for optimal performance. It may also be required that these procedures estimate the optimal system parameters, which govern the dynamics of a spatiotemporal beam -- this can be a high-dimensional optimization problem. To address this, we propose a Classifier-pruned Bayesian Optimization-based Latent space Tuner (CBOL-Tuner), a framework for efficient exploration within a temporally-structured latent space. The CBOL-Tuner integrates a convolutional variational autoencoder (CVAE) for latent space representation, a long short-term memory (LSTM) network for temporal dynamics, a dense neural network (DNN) for parameter estimation, and a classifier-pruned Bayesian optimizer (C-BO) to adaptively search and filter the latent space for optimal solutions. CBOL-Tuner demonstrates superior performance in identifying multiple optimal settings and outperforms alternative global optimization methods.
In modern deep neural networks, the learning dynamics of the individual neurons is often obscure, as the networks are trained via global optimization. Conversely, biological systems build on self-organized, local learning, achieving robustness and efficiency with limited global information. We here show how self-organization between individual artificial neurons can be achieved by designing abstract bio-inspired local learning goals. These goals are parameterized using a recent extension of information theory, Partial Information Decomposition (PID), which decomposes the information that a set of information sources holds about an outcome into unique, redundant and synergistic contributions. Our framework enables neurons to locally shape the integration of information from various input classes, i.e. feedforward, feedback, and lateral, by selecting which of the three inputs should contribute uniquely, redundantly or synergistically to the output. This selection is expressed as a weighted sum of PID terms, which, for a given problem, can be directly derived from intuitive reasoning or via numerical optimization, offering a window into understanding task-relevant local information processing. Achieving neuron-level interpretability while enabling strong performance using local learning, our work advances a principled information-theoretic foundation for local learning strategies.
The design of both FIR and IIR digital filters is a multi-variable optimization problem, where traditional algorithms fail to obtain optimal solutions. A modified Shuffled Frog Leaping Algorithm (SFLA) is here proposed for the design of FIR and IIR discrete-time filters as close as possible to the desired filter frequency response. This algorithm can be considered a type of memetic algorithm. In this paper, simulations prove the obtained filters outperform those designed using the traditional bilinear Z transform (BZT) method with elliptic approximation. Besides, results are close to, and even slightly better, than those reported in recent bio-inspired approaches using algorithms such as particle swarm optimization (PSO), differential evolution (DE) and regularized global optimization (RGA).
Implications of uncertain objective functions and permutative symmetry of traditional deep learning architectures are discussed. It is shown that traditional architectures are polluted by an astronomical number of equivalent global and local optima. Uncertainty of the objective makes local optima unattainable, and, as the size of the network grows, the global optimization landscape likely becomes a tangled web of valleys and ridges. Some remedies which reduce or eliminate ghost optima are discussed including forced pre-pruning, re-ordering, ortho-polynomial activations, and modular bio-inspired architectures.
The extension of elastic optical networks (EON) to multi-band transmission (MB-EON) shows promise in enhancing spectral efficiency, throughput, and long-term cost-effectiveness for telecom operators. However, designing MB-EON networks introduces complex challenges, notably the optimization of physical parameters like optical power and quality of transmission (QoT). Frequency-dependent characteristics of fiber, such as loss, dispersion, and nonlinear effects, alongside inter-channel stimulated Raman scattering, pose significant hurdles when extending beyond the L+C (LC) band to a continuous spectrum over 100 nm. In this study, we propose a span-by-span methodology for optimal power allocation, introducing two hyper-accelerated power optimization (HPO) strategies: flat launch power (FLP) and flat received power (FRP). These approaches significantly expedite network power optimization while preserving the stability of running services. Our comparative analysis of FLP and FRP models reveals that while FRP has a minimal effect on capacity (increasing less than 10 Tbps for an L+C+S (LCS) system over 100 km), it improves flatness and GSNR/OSNR metrics in the S-band by approximately 2/0 dB and 2.5/6 dB, respectively. A network-wide analysis across various topologies shows that the FRP technique enhances minimum GSNR, contributing to a throughput increase of 12% to 75%, depending on network scale, at a 1% bandwidth blocking rate. Lastly, our application of HPO in MB-EON for both local and global power optimization demonstrates that while both approaches offer comparable performance, global optimization is simpler and more cost-effective for large-scale networks.
We study episodic linear mixture MDPs with the unknown transition and adversarial rewards under full-information feedback, employing dynamic regret as the performance measure. We start with in-depth analyses of the strengths and limitations of the two most popular methods: occupancy-measure-based and policy-based methods. We observe that while the occupancy-measure-based method is effective in addressing non-stationary environments, it encounters difficulties with the unknown transition. In contrast, the policy-based method can deal with the unknown transition effectively but faces challenges in handling non-stationary environments. Building on this, we propose a novel algorithm that combines the benefits of both methods. Specifically, it employs (i) an occupancy-measure-based global optimization with a two-layer structure to handle non-stationary environments; and (ii) a policy-based variance-aware value-targeted regression to tackle the unknown transition. We bridge these two parts by a novel conversion. Our algorithm enjoys an $\widetilde{\mathcal{O}}(d \sqrt{H^3 K} + \sqrt{HK(H + \bar{P}_K)})$ dynamic regret, where $d$ is the feature dimension, $H$ is the episode length, $K$ is the number of episodes, $\bar{P}_K$ is the non-stationarity measure. We show it is minimax optimal up to logarithmic factors by establishing a matching lower bound. To the best of our knowledge, this is the first work that achieves near-optimal dynamic regret for adversarial linear mixture MDPs with the unknown transition without prior knowledge of the non-stationarity measure.
Token generation speed is critical to power the next wave of AI inference applications. GPUs significantly underperform during token generation due to synchronization overheads at kernel boundaries, utilizing only 21% of their peak memory bandwidth. While recent dataflow architectures mitigate these overheads by enabling aggressive fusion of decoder layers into a single kernel, they too leave performance on the table due to synchronization penalties at layer boundaries. This paper presents kernel looping, a specialized global optimization technique which exploits an optimization opportunity brought by combining the unique layer-level fusion possible in modern dataflow architectures with the repeated layer structure found in language models. Kernel looping eliminates synchronization costs between consecutive calls to the same kernel by transforming these calls into a single call to a modified kernel containing a pipelined outer loop. We evaluate kernel looping on the SambaNova SN40L Reconfigurable Dataflow Unit (RDU), a commercial dataflow accelerator for AI. Experiments demonstrate that kernel looping speeds up the decode phase of a wide array of powerful open-source models by up to 2.2$\times$ on SN40L. Kernel looping allows scaling of decode performance over multiple SN40L sockets, achieving speedups of up to 2.5$\times$. Finally, kernel looping enables SN40L to achieve over 90% of peak performance on 8 and 16 sockets and achieve a speedup of up to 3.7$\times$ over DGX H100. Kernel looping, as well as the models evaluated in this paper, are deployed in production in a commercial AI inference cloud.
Bayesian optimization devolves the global optimization of a costly objective function to the global optimization of a sequence of acquisition functions. This inner-loop optimization can be catastrophically difficult if it involves posterior sample paths, especially in higher dimensions. We introduce an efficient global optimization strategy for posterior samples based on global rootfinding. It provides gradient-based optimizers with two sets of judiciously selected starting points, designed to combine exploration and exploitation. The number of starting points can be kept small without sacrificing optimization quality. Remarkably, even with just one point from each set, the global optimum is discovered most of the time. The algorithm scales practically linearly to high dimensions, breaking the curse of dimensionality. For Gaussian process Thompson sampling (GP-TS), we demonstrate remarkable improvement in both inner- and outer-loop optimization, surprisingly outperforming alternatives like EI and GP-UCB in most cases. Our approach also improves the performance of other posterior sample-based acquisition functions, such as variants of entropy search. Furthermore, we propose a sample-average formulation of GP-TS, which has a parameter to explicitly control exploitation and can be computed at the cost of one posterior sample. Our implementation is available at https://github.com/UQUH/TSRoots .
This study investigates the application of Bayesian Optimization (BO) for the hyperparameter tuning of neural networks, specifically targeting the enhancement of Convolutional Neural Networks (CNN) for image classification tasks. Bayesian Optimization is a derivative-free global optimization method suitable for expensive black-box functions with continuous inputs and limited evaluation budgets. The BO algorithm leverages Gaussian Process regression and acquisition functions like Upper Confidence Bound (UCB) and Expected Improvement (EI) to identify optimal configurations effectively. Using the Ax and BOTorch frameworks, this work demonstrates the efficiency of BO in reducing the number of hyperparameter tuning trials while achieving competitive model performance. Experimental outcomes reveal that BO effectively balances exploration and exploitation, converging rapidly towards optimal settings for CNN architectures. This approach underlines the potential of BO in automating neural network tuning, contributing to improved accuracy and computational efficiency in machine learning pipelines.
Global minimization is a fundamental challenge in optimization, especially in machine learning, where finding the global minimum of a function directly impacts model performance and convergence. This article introduces a novel optimization method that we called Super Gradient Descent, designed specifically for one-dimensional functions, guaranteeing convergence to the global minimum for any k-Lipschitz function defined on a closed interval [a, b]. Our approach addresses the limitations of traditional optimization algorithms, which often get trapped in local minima. In particular, we introduce the concept of global gradient which offers a robust solution for precise and well-guided global optimization. By focusing on the global minimization problem, this work bridges a critical gap in optimization theory, offering new insights and practical advancements in different optimization problems in particular Machine Learning problems like line search.
Identifying dynamical systems characterized by nonlinear parameters presents significant challenges in deriving mathematical models that enhance understanding of physics. Traditional methods, such as Sparse Identification of Nonlinear Dynamics (SINDy) and symbolic regression, can extract governing equations from observational data; however, they also come with distinct advantages and disadvantages. This paper introduces a novel method within the SINDy framework, termed ADAM-SINDy, which synthesizes the strengths of established approaches by employing the ADAM optimization algorithm. This facilitates the simultaneous optimization of nonlinear parameters and coefficients associated with nonlinear candidate functions, enabling precise parameter estimation without requiring prior knowledge of nonlinear characteristics such as trigonometric frequencies, exponential bandwidths, or polynomial exponents, thereby addressing a key limitation of SINDy. Through an integrated global optimization, ADAM-SINDy dynamically adjusts all unknown variables in response to data, resulting in an adaptive identification procedure that reduces the sensitivity to the library of candidate functions. The performance of the ADAM-SINDy methodology is demonstrated across a spectrum of dynamical systems, including benchmark coupled nonlinear ordinary differential equations such as oscillators, chaotic fluid flows, reaction kinetics, pharmacokinetics, as well as nonlinear partial differential equations (wildfire transport). The results demonstrate significant improvements in identifying parameterized dynamical systems and underscore the importance of concurrently optimizing all parameters, particularly those characterized by nonlinear parameters. These findings highlight the potential of ADAM-SINDy to extend the applicability of the SINDy framework in addressing more complex challenges in dynamical system identification.
Bayesian optimization relies on iteratively constructing and optimizing an acquisition function. The latter turns out to be a challenging, non-convex optimization problem itself. Despite the relative importance of this step, most algorithms employ sampling- or gradient-based methods, which do not provably converge to global optima. This work investigates mixed-integer programming (MIP) as a paradigm for \textit{global} acquisition function optimization. Specifically, our Piecewise-linear Kernel Mixed Integer Quadratic Programming (PK-MIQP) formulation introduces a piecewise-linear approximation for Gaussian process kernels and admits a corresponding MIQP representation for acquisition functions. We analyze the theoretical regret bounds of the proposed approximation, and empirically demonstrate the framework on synthetic functions, constrained benchmarks, and a hyperparameter tuning task.
We propose a zero-shot text-driven 3D shape deformation system that deforms an input 3D mesh of a manufactured object to fit an input text description. To do this, our system optimizes the parameters of a deformation model to maximize an objective function based on the widely used pre-trained vision language model CLIP. We find that CLIP-based objective functions exhibit many spurious local optima; to circumvent them, we parameterize deformations using a novel deformation model called BoxDefGraph which our system automatically computes from an input mesh, the BoxDefGraph is designed to capture the object aligned rectangular/circular geometry features of most manufactured objects. We then use the CMA-ES global optimization algorithm to maximize our objective, which we find to work better than popular gradient-based optimizers. We demonstrate that our approach produces appealing results and outperforms several baselines.
K-means clustering is a cornerstone of data mining, but its efficiency deteriorates when confronted with massive datasets. To address this limitation, we propose a novel heuristic algorithm that leverages the Variable Neighborhood Search (VNS) metaheuristic to optimize K-means clustering for big data. Our approach is based on the sequential optimization of the partial objective function landscapes obtained by restricting the Minimum Sum-of-Squares Clustering (MSSC) formulation to random samples from the original big dataset. Within each landscape, systematically expanding neighborhoods of the currently best (incumbent) solution are explored by reinitializing all degenerate and a varying number of additional centroids. Extensive and rigorous experimentation on a large number of real-world datasets reveals that by transforming the traditional local search into a global one, our algorithm significantly enhances the accuracy and efficiency of K-means clustering in big data environments, becoming the new state of the art in the field.
Symbolic regression (SR) methods have been extensively investigated to explore explicit algebraic Reynolds stress models (EARSM) for turbulence closure of Reynolds-averaged Navier-Stokes (RANS) equations. The deduced EARSM can be readily implemented in existing computational fluid dynamic (CFD) codes and promotes the identification of physically interpretable turbulence models. The existing SR methods, such as genetic programming, sparse regression, or artificial neural networks, require user-defined functional operators, a library of candidates, or complex optimization algorithms. In this work, a novel framework using LLMs to automatically discover algebraic expressions for correcting the RSM is proposed. The direct observation of Reynolds stress and the indirect output of the CFD simulation are both involved in the training process to guarantee data consistency and avoid numerical stiffness. Constraints of functional complexity and convergence are supplementally imposed in the objective function on account of the tremendous flexibility of LLMs. The evolutionary search is employed for global optimization. The proposed method is performed for separated flow over periodic hills at Re = 10,595. The generalizability of the discovered model is verified on a set of 2D turbulent separated flow configurations with different Reynolds numbers and geometries. It is demonstrated that the corrective RANS can improve the prediction for both the Reynolds stress and mean velocity fields. Compared with algebraic models discovered by other works, the discovered model performs better in accuracy and generalization capability. The proposed approach provides a promising paradigm for using LLMs to improve turbulence modeling for a given class of flows.
With the rapid advances in diffusion models, generating decent images from text prompts is no longer challenging. The key to text-to-image generation is how to optimize the results of a text-to-image generation model so that they can be better aligned with human intentions or prompts. Existing optimization methods commonly treat the entire image uniformly and conduct global optimization. These methods overlook the fact that when viewing an image, the human visual system naturally prioritizes attention toward salient areas, often neglecting less or non-salient regions. That is, humans are likely to neglect optimizations in non-salient areas. Consequently, although model retaining is conducted under the guidance of additional large and multimodality models, existing methods, which perform uniform optimizations, yield sub-optimal results. To address this alignment challenge effectively and efficiently, we propose Saliency Guided Optimization Of Diffusion Latents (SGOOL). We first employ a saliency detector to mimic the human visual attention system and mark out the salient regions. To avoid retraining an additional model, our method directly optimizes the diffusion latents. Besides, SGOOL utilizes an invertible diffusion process and endows it with the merits of constant memory implementation. Hence, our method becomes a parameter-efficient and plug-and-play fine-tuning method. Extensive experiments have been done with several metrics and human evaluation. Experimental results demonstrate the superiority of SGOOL in image quality and prompt alignment.
Bayesian optimization (BO) is an attractive machine learning framework for performing sample-efficient global optimization of black-box functions. The optimization process is guided by an acquisition function that selects points to acquire in each round of BO. In batched BO, when multiple points are acquired in parallel, commonly used acquisition functions are often high-dimensional and intractable, leading to the use of sampling-based alternatives. We propose a statistical physics inspired acquisition function for BO with Gaussian processes that can natively handle batches. Batched Energy-Entropy acquisition for BO (BEEBO) enables tight control of the explore-exploit trade-off of the optimization process and generalizes to heteroskedastic black-box problems. We demonstrate the applicability of BEEBO on a range of problems, showing competitive performance to existing methods.
Thompson sampling (TS) is a simple, effective stochastic policy in Bayesian decision making. It samples the posterior belief about the reward profile and optimizes the sample to obtain a candidate decision. In continuous optimization, the posterior of the objective function is often a Gaussian process (GP), whose sample paths have numerous local optima, making their global optimization challenging. In this work, we introduce an efficient global optimization strategy for GP-TS that carefully selects starting points for gradient-based multi-start optimizers. It identifies all local optima of the prior sample via univariate global rootfinding, and optimizes the posterior sample using a differentiable, decoupled representation. We demonstrate remarkable improvement in the global optimization of GP posterior samples, especially in high dimensions. This leads to dramatic improvements in the overall performance of Bayesian optimization using GP-TS acquisition functions, surprisingly outperforming alternatives like GP-UCB and EI.