Journal articles: 'Gradient bound'

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ARONSSON, GUNNAR. "INTERPOLATION UNDER A GRADIENT BOUND." Journal of the Australian Mathematical Society 87, no. 01 (August 2009): 19. http://dx.doi.org/10.1017/s1446788709000044.

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Chang, Ting-Jui, and Shahin Shahrampour. "On Online Optimization: Dynamic Regret Analysis of Strongly Convex and Smooth Problems." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 8 (May 18, 2021): 6966–73. http://dx.doi.org/10.1609/aaai.v35i8.16858.

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The regret bound of dynamic online learning algorithms is often expressed in terms of the variation in the function sequence (V_T) and/or the path-length of the minimizer sequence after T rounds. For strongly convex and smooth functions, Zhang et al. (2017) establish the squared path-length of the minimizer sequence (C*_{2,T}) as a lower bound on regret. They also show that online gradient descent (OGD) achieves this lower bound using multiple gradient queries per round. In this paper, we focus on unconstrained online optimization. We first show that a preconditioned variant of OGD achieves O(min{C*_T,C*_{2,T}}) with one gradient query per round (C*_T refers to the normal path-length). We then propose online optimistic Newton (OON) method for the case when the first and second order information of the function sequence is predictable. The regret bound of OON is captured via the quartic path-length of the minimizer sequence (C*_{4,T}), which can be much smaller than C*_{2,T}. We finally show that by using multiple gradients for OGD, we can achieve an upper bound of O(min{C*_{2,T},V_T}) on regret.

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Li, Dong, Fan Wang, and Kai Yang. "An improved gradient bound for 2D MBE." Journal of Differential Equations 269, no. 12 (December 2020): 11165–71. http://dx.doi.org/10.1016/j.jde.2020.08.045.

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De Silva, Daniela, and David Jerison. "A gradient bound for free boundary graphs." Communications on Pure and Applied Mathematics 64, no. 4 (December 13, 2010): 538–55. http://dx.doi.org/10.1002/cpa.20354.

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Hao, Jia, Winfield Zhao, Jeong Min Oh, and Keyue Shen. "A Pillar-Free Diffusion Device for Studying Chemotaxis on Supported Lipid Bilayers." Micromachines 12, no. 10 (October 16, 2021): 1254. http://dx.doi.org/10.3390/mi12101254.

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Chemotactic cell migration plays a crucial role in physiological and pathophysiological processes. In tissues, cells can migrate not only through extracellular matrix (ECM), but also along stromal cell surfaces via membrane-bound receptor–ligand interactions to fulfill critical functions. However, there remains a lack of models recapitulating chemotactic migration mediated through membrane-bound interactions. Here, using micro-milling, we engineered a multichannel diffusion device that incorporates a chemoattractant gradient and a supported lipid bilayer (SLB) tethered with membrane-bound factors that mimics stromal cell membranes. The chemoattractant channels are separated by hydrogel barriers from SLB in the cell loading channel, which enable precise control of timing and profile of the chemokine gradients applied on cells interacting with SLB. The hydrogel barriers are formed in pillar-free channels through a liquid pinning process, which eliminates complex cleanroom-based fabrications and distortion of chemoattractant gradient by pillars in typical microfluidic hydrogel barrier designs. As a proof-of-concept, we formed an SLB tethered with ICAM-1, and demonstrated its lateral mobility and different migratory behavior of Jurkat T cells on it from those on immobilized ICAM-1, under a gradient of chemokine CXCL12. Our platform can thus be widely used to investigate membrane-bound chemotaxis such as in cancer, immune, and stem cells.

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Wang, Zhengxing, Yuke Wang, Shumao Wang, Bin Li, and Hu Wang. "Effect of Longitudinal Gradient on 3D Face Stability of Circular Tunnel in Undrained Clay." Advances in Civil Engineering 2020 (August 19, 2020): 1–12. http://dx.doi.org/10.1155/2020/5846151.

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The longitudinal gradient existed in shield-driven tunnel crossing river or channel has a longitudinal gradient, which is often ignored in most stability analyses of the tunnel face. Considering the influence of the longitudinal gradient into A(a) continuous velocity field, the present paper, conducting a limit analysis of the tunnel face in undrained clay, adopted to yield the upper-bound solutions of the limit pressure supporting on a three-dimensional tunnel face. The least upper bounds of the collapse and blow-out pressures can be obtained by conducting an optimization procedure. These upper-bound solutions are given in the design charts, which provide a simple way to assess the range of the limit pressure in practice. The influence of the longitudinal gradient becomes more significant with the increase of γD/su and C/D. The blow-out pressure for tunneling in a downward movement could be overestimated and the collapse pressure for tunneling in an upward movement could be conversely underestimated, with ignoring the influence of the longitudinal gradient.

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Bovier, Anton. "Sharp upper bounds on perfect retrieval in the Hopfield model." Journal of Applied Probability 36, no. 3 (September 1999): 941–50. http://dx.doi.org/10.1239/jap/1032374647.

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We prove a sharp upper bound on the number of patterns that can be stored in the Hopfield model if the stored patterns are required to be fixed points of the gradient dynamics. We also show corresponding bounds on the one-step convergence of the sequential gradient dynamics. The bounds coincide with the known lower bounds and confirm the heuristic expectations. The proof is based on a crucial idea of Loukianova (1997) using the negative association properties of some random variables arising in the analysis.

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Bovier, Anton. "Sharp upper bounds on perfect retrieval in the Hopfield model." Journal of Applied Probability 36, no. 03 (September 1999): 941–50. http://dx.doi.org/10.1017/s0021900200017708.

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We prove a sharp upper bound on the number of patterns that can be stored in the Hopfield model if the stored patterns are required to be fixed points of the gradient dynamics. We also show corresponding bounds on the one-step convergence of the sequential gradient dynamics. The bounds coincide with the known lower bounds and confirm the heuristic expectations. The proof is based on a crucial idea of Loukianova (1997) using the negative association properties of some random variables arising in the analysis.

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Albin, Nathan, Sergio Conti, and Vincenzo Nesi. "Improved bounds for composites and rigidity of gradient fields." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2084 (June 13, 2007): 2031–48. http://dx.doi.org/10.1098/rspa.2007.1863.

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We determine an improved lower bound for the conductivity of three-component composite materials. Our bound is strictly larger than the well-known Hashin–Shtrikman bound outside the regime where the latter is known to be optimal. The main ingredient of our result is a new quantitative rigidity estimate for gradient fields in two dimensions.

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Kuusi, Tuomo, and Giuseppe Mingione. "The Wolff gradient bound for degenerate parabolic equations." Journal of the European Mathematical Society 16, no. 4 (2014): 835–92. http://dx.doi.org/10.4171/jems/449.

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Lin, Chih-Jen. "Projected Gradient Methods for Nonnegative Matrix Factorization." Neural Computation 19, no. 10 (October 2007): 2756–79. http://dx.doi.org/10.1162/neco.2007.19.10.2756.

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Nonnegative matrix factorization (NMF) can be formulated as a minimization problem with bound constraints. Although bound-constrained optimization has been studied extensively in both theory and practice, so far no study has formally applied its techniques to NMF. In this letter, we propose two projected gradient methods for NMF, both of which exhibit strong optimization properties. We discuss efficient implementations and demonstrate that one of the proposed methods converges faster than the popular multiplicative update approach. A simple Matlab code is also provided.

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Rousseau-Rizzi, Raphaël, and Kerry Emanuel. "An Evaluation of Hurricane Superintensity in Axisymmetric Numerical Models." Journal of the Atmospheric Sciences 76, no. 6 (June 1, 2019): 1697–708. http://dx.doi.org/10.1175/jas-d-18-0238.1.

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Abstract Potential intensity (PI) is an analytical bound on steady, inviscid, axisymmetric hurricane wind speed. Studies have shown that simulated hurricane azimuthal wind speed can greatly exceed a PI bound on the maximum gradient wind. This disparity is called superintensity (SI) and has been attributed to the contribution of the unbalanced flow to the azimuthal wind. The goals of this study are 1) to introduce a new surface wind PI (PIs), based on a differential Carnot cycle and bounding the magnitude of the surface winds; 2) to evaluate SI in numerical simulations with respect to diagnostic PI bounds on gradient wind (PIg), azimuthal wind (PIa), and surface wind (PIs); and 3) to evaluate the validity of each PI bound based on the SI computations. Here, we define superintensity as the normalized amount by which each version of PI is exceeded by the quantity it bounds. Axisymmetric tropical cyclone simulations are performed while varying the parameterized turbulent mixing as a way of estimating SI in the inviscid limit. As the mixing length decreases, all three bounded wind speeds increase similarly from a sub-PI state to a marginally superintense state. This shows that all three forms of PI evaluated here are good approximations to their respective metrics in numerical simulations.

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Ghose, Debraj, Katherine Jacobs, Samuel Ramirez, Timothy Elston, and Daniel Lew. "Chemotactic movement of a polarity site enables yeast cells to find their mates." Proceedings of the National Academy of Sciences 118, no. 22 (May 28, 2021): e2025445118. http://dx.doi.org/10.1073/pnas.2025445118.

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How small eukaryotic cells can interpret dynamic, noisy, and spatially complex chemical gradients to orient growth or movement is poorly understood. We address this question using Saccharomyces cerevisiae, where cells orient polarity up pheromone gradients during mating. Initial orientation is often incorrect, but polarity sites then move around the cortex in a search for partners. We find that this movement is biased by local pheromone gradients across the polarity site: that is, movement of the polarity site is chemotactic. A bottom-up computational model recapitulates this biased movement. The model reveals how even though pheromone-bound receptors do not mimic the shape of external pheromone gradients, nonlinear and stochastic effects combine to generate effective gradient tracking. This mechanism for gradient tracking may be applicable to any cell that searches for a target in a complex chemical landscape.

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Zhang, Junzi, Jongho Kim, Brendan O'Donoghue, and Stephen Boyd. "Sample Efficient Reinforcement Learning with REINFORCE." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 12 (May 18, 2021): 10887–95. http://dx.doi.org/10.1609/aaai.v35i12.17300.

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Policy gradient methods are among the most effective methods for large-scale reinforcement learning, and their empirical success has prompted several works that develop the foundation of their global convergence theory. However, prior works have either required exact gradients or state-action visitation measure based mini-batch stochastic gradients with a diverging batch size, which limit their applicability in practical scenarios. In this paper, we consider classical policy gradient methods that compute an approximate gradient with a single trajectory or a fixed size mini-batch of trajectories under soft-max parametrization and log-barrier regularization, along with the widely-used REINFORCE gradient estimation procedure. By controlling the number of "bad" episodes and resorting to the classical doubling trick, we establish an anytime sub-linear high probability regret bound as well as almost sure global convergence of the average regret with an asymptotically sub-linear rate. These provide the first set of global convergence and sample efficiency results for the well-known REINFORCE algorithm and contribute to a better understanding of its performance in practice.

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CHENG, WANYOU, and ERBAO CAO. "AN ADAPTIVE GRADIENT ALGORITHM FOR LARGE-SCALE NONLINEAR BOUND CONSTRAINED OPTIMIZATION." Asia-Pacific Journal of Operational Research 30, no. 03 (June 2013): 1340005. http://dx.doi.org/10.1142/s0217595913400058.

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In this paper, an adaptive gradient algorithm (AGM) for box constrained optimization is developed. The algorithm is based on an active set identification technique and consists of a nonmonotone gradient projection step, a conjugate gradient step and a rule for branching between the two steps. We show that the method is globally convergent under appropriate conditions. Numerical experiments are presented using bound constrained problems in the CUTEr test problem library.

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Yang, Liu, and Deng Cai. "AdaDB: An adaptive gradient method with data-dependent bound." Neurocomputing 419 (January 2021): 183–89. http://dx.doi.org/10.1016/j.neucom.2020.07.070.

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Laurence, Peter, and Edward Stredulinsky. "A Gradient Bound for the Grad-Kruskal-Kulsrud Functional." Mathematical Research Letters 1, no. 3 (1994): 377–87. http://dx.doi.org/10.4310/mrl.1994.v1.n3.a9.

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Jacopin, Gwénolé, Mehran Shahmohammadi, Jean-Daniel Ganière, and Benoît Deveaud. "Hopping process of bound excitons under an energy gradient." Applied Physics Letters 104, no. 4 (January 27, 2014): 042109. http://dx.doi.org/10.1063/1.4863319.

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Laurence, Peter, and Edward W. Stredulinsky. "A gradient bound for the Grad-Kruskal-Kulsrud functional." Communications on Pure and Applied Mathematics 49, no. 3 (March 1996): 237–84. http://dx.doi.org/10.1002/(sici)1097-0312(199603)49:3<237::aid-cpa2>3.0.co;2-e.

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Wu, Jindan, Zhengwei Mao, Huaping Tan, Lulu Han, Tanchen Ren, and Changyou Gao. "Gradient biomaterials and their influences on cell migration." Interface Focus 2, no. 3 (March 21, 2012): 337–55. http://dx.doi.org/10.1098/rsfs.2011.0124.

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Cell migration participates in a variety of physiological and pathological processes such as embryonic development, cancer metastasis, blood vessel formation and remoulding, tissue regeneration, immune surveillance and inflammation. The cells specifically migrate to destiny sites induced by the gradually varying concentration (gradient) of soluble signal factors and the ligands bound with the extracellular matrix in the body during a wound healing process. Therefore, regulation of the cell migration behaviours is of paramount importance in regenerative medicine. One important way is to create a microenvironment that mimics the in vivo cellular and tissue complexity by incorporating physical, chemical and biological signal gradients into engineered biomaterials. In this review, the gradients existing in vivo and their influences on cell migration are briefly described. Recent developments in the fabrication of gradient biomaterials for controlling cellular behaviours, especially the cell migration, are summarized, highlighting the importance of the intrinsic driving mechanism for tissue regeneration and the design principle of complicated and advanced tissue regenerative materials. The potential uses of the gradient biomaterials in regenerative medicine are introduced. The current and future trends in gradient biomaterials and programmed cell migration in terms of the long-term goals of tissue regeneration are prospected.

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Sturm, Karl-Theodor. "Distribution-Valued Ricci Bounds for Metric Measure Spaces, Singular Time Changes, and Gradient Estimates for Neumann Heat Flows." Geometric and Functional Analysis 30, no. 6 (November 20, 2020): 1648–711. http://dx.doi.org/10.1007/s00039-020-00554-0.

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AbstractWe will study metric measure spaces $$(X,\mathsf{d},{\mathfrak {m}})$$ ( X , d , m ) beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds $$\mathsf{BE}_1(\kappa ,\infty )$$ BE 1 ( κ , ∞ ) for which we prove the equivalence with sharp gradient estimates, the class of which will be preserved under time changes with arbitrary $$\psi \in \mathrm {Lip}_b(X)$$ ψ ∈ Lip b ( X ) , and which are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets $$Y\subset X$$ Y ⊂ X . In the latter case, the distribution-valued Ricci bound will be given by the signed measure $$\kappa = k\,{\mathfrak {m}}_Y + \ell \,\sigma _{\partial Y}$$ κ = k m Y + ℓ σ ∂ Y where k denotes a variable synthetic lower bound for the Ricci curvature of X and $$\ell $$ ℓ denotes a lower bound for the “curvature of the boundary” of Y, defined in purely metric terms. We also present a new localization argument which allows us to pass on the RCD property to arbitrary open subsets of RCD spaces. And we introduce new synthetic notions for boundary curvature, second fundamental form, and boundary measure for subsets of RCD spaces.

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Chinnayan Kannan, Pandiyarajan, and Jan Genzer. "UV- and thermally-active small bi-functional gelator for creating gradient polymer network coatings." Biointerphases 18, no. 1 (January 2023): 011001. http://dx.doi.org/10.1116/6.0002268.

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We present a versatile one-pot synthesis method for creating surface-anchored orthogonal gradient networks using a small bi-functional gelator, 4-azidosulfonylphenethyltrimethoxysilane (4-ASPTMS). The sulfonyl azide (SAz) group of 4-ASPTMS is UV (≤254 nm) and thermally active (≥100 °C) and, thus, enables us to vary the cross-link density in orthogonal directions by controlling the activation of SAz groups via UV and temperature means. We deposit a thin layer (∼200 nm) of a mixture comprising ∼90% precursor polymer and ∼10% 4-ASPTMS in a silicon wafer. Upon UV irradiation or annealing the layers, SAz releases nitrogen by forming singlet and triplet nitrenes that concurrently react with any C–H bond in the vicinity leading to sulfonamide cross-links. Condensation among trimethoxy groups in the bulk connects 4-ASPTMS units and completes the cross-linking. Simultaneously, 4-ASPTMS near the substrate reacts with surface-bound –OH motifs that anchor the cross-linked polymer chains to the substrate. We demonstrate the generation of orthogonal gradient network coatings exhibiting cross-link density (or stiffness) gradients in orthogonal directions using such a simple process.

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Lv, Shao-Gao. "Refined Generalization Bounds of Gradient Learning over Reproducing Kernel Hilbert Spaces." Neural Computation 27, no. 6 (June 2015): 1294–320. http://dx.doi.org/10.1162/neco_a_00739.

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Gradient learning (GL), initially proposed by Mukherjee and Zhou ( 2006 ) has been proved to be a powerful tool for conducting variable selection and dimensional reduction simultaneously. This approach presents a nonparametric version of a gradient estimator with positive definite kernels without estimating the true function itself, so that the proposed version has wide applicability and allows for complex effects between predictors. In terms of theory, however, existing generalization bounds for GL depend on capacity-independent techniques, and the capacity of kernel classes cannot be characterized completely. Thus, this letter considers GL estimators that minimize the empirical convex risk. We prove generalization bounds for such estimators with rates that are faster than previous results. Moreover, we provide a novel upper bound for Rademacher chaos complexity of order two, which also plays an important role in general pairwise-type estimations, including ranking and score problems.

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Fercoq, Olivier, and Zheng Qu. "Adaptive restart of accelerated gradient methods under local quadratic growth condition." IMA Journal of Numerical Analysis 39, no. 4 (March 5, 2019): 2069–95. http://dx.doi.org/10.1093/imanum/drz007.

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Abstract By analyzing accelerated proximal gradient methods under a local quadratic growth condition, we show that restarting these algorithms at any frequency gives a globally linearly convergent algorithm. This result was previously known only for long enough frequencies. Then as the rate of convergence depends on the match between the frequency and the quadratic error bound, we design a scheme to automatically adapt the frequency of restart from the observed decrease of the norm of the gradient mapping. Our algorithm has a better theoretical bound than previously proposed methods for the adaptation to the quadratic error bound of the objective. We illustrate the efficiency of the algorithm on Lasso, regularized logistic regression and total variation denoising problems.

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Wang, Di, and Jinhui Xu. "Differentially Private Empirical Risk Minimization with Smooth Non-Convex Loss Functions: A Non-Stationary View." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 1182–89. http://dx.doi.org/10.1609/aaai.v33i01.33011182.

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In this paper, we study the Differentially Private Empirical Risk Minimization (DP-ERM) problem with non-convex loss functions and give several upper bounds for the utility in different settings. We first consider the problem in low-dimensional space. For DP-ERM with non-smooth regularizer, we generalize an existing work by measuring the utility using ℓ2 norm of the projected gradient. Also, we extend the error bound measurement, for the first time, from empirical risk to population risk by using the expected ℓ2 norm of the gradient. We then investigate the problem in high dimensional space, and show that by measuring the utility with Frank-Wolfe gap, it is possible to bound the utility by the Gaussian Width of the constraint set, instead of the dimensionality p of the underlying space. We further demonstrate that the advantages of this result can be achieved by the measure of ℓ2 norm of the projected gradient. A somewhat surprising discovery is that although the two kinds of measurements are quite different, their induced utility upper bounds are asymptotically the same under some assumptions. We also show that the utility of some special non-convex loss functions can be reduced to a level (i.e., depending only on log p) similar to that of convex loss functions. Finally, we test our proposed algorithms on both synthetic and real world datasets and the experimental results confirm our theoretical analysis.

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Qian, Zhongmin. "Gradient estimates and heat kernel estimates." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 125, no. 5 (1995): 975–90. http://dx.doi.org/10.1017/s0308210500022599.

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In the first part of this paper, Yau's estimates for positive L-harmonic functions and Li and Yau's gradient estimates for the positive solutions of a general parabolic heat equation on a complete Riemannian manifold are obtained by the use of Bakry and Emery's theory. In the second part we establish a heat kernel bound for a second-order differential operator which has a bounded and measurable drift, using Girsanov's formula.

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Aviles, Patricio, and Yoshikazu Giga. "On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg–Landau type energy for gradient fields." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 129, no. 1 (1999): 1–17. http://dx.doi.org/10.1017/s0308210500027438.

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A defect energy Jβ, which measures jump discontinuities of a unit length gradient field, is studied. The number β indicates the power of the jumps of the gradient fields that appear in the density of Jβ. It is shown that Jβ for β = 3 is lower semicontinuous (on the space of unit gradient fields belonging to BV) in L1-convergence of gradient fields. A similar result holds for the modified energy , which measures only a particular type of defect. The result turns out to be very subtle, since with β > 3 is not lower semicontinuous, as is shown in this paper. The key idea behind semicontinuity is a duality representation for J3 and . The duality representation is also important for obtaining a lower bound by using J3+ for the relaxation limit of the Ginzburg–Landau type energy for gradient fields. The lower bound obtained here agrees with the conjectured value of the relaxation limit.

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Camley, Brian A., and Wouter-Jan Rappel. "Cell-to-cell variation sets a tissue-rheology–dependent bound on collective gradient sensing." Proceedings of the National Academy of Sciences 114, no. 47 (November 7, 2017): E10074—E10082. http://dx.doi.org/10.1073/pnas.1712309114.

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When a single cell senses a chemical gradient and chemotaxes, stochastic receptor–ligand binding can be a fundamental limit to the cell’s accuracy. For clusters of cells responding to gradients, however, there is a critical difference: Even genetically identical cells have differing responses to chemical signals. With theory and simulation, we show collective chemotaxis is limited by cell-to-cell variation in signaling. We find that when different cells cooperate, the resulting bias can be much larger than the effects of ligand–receptor binding. Specifically, when a strongly responding cell is at one end of a cell cluster, cluster motion is biased toward that cell. These errors are mitigated if clusters average measurements over times long enough for cells to rearrange. In consequence, fluid clusters are better able to sense gradients: We derive a link between cluster accuracy, cell-to-cell variation, and the cluster rheology. Because of this connection, increasing the noisiness of individual cell motion can actually increase the collective accuracy of a cluster by improving fluidity.

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Carlon, André Gustavo, Henrique Machado Kroetz, André Jacomel Torii, Rafael Holdorf Lopez, and Leandro Fleck Fadel Miguel. "Risk optimization using the Chernoff bound and stochastic gradient descent." Reliability Engineering & System Safety 223 (July 2022): 108512. http://dx.doi.org/10.1016/j.ress.2022.108512.

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Yu, Zhensheng, Jing Sun, and Yi Qin. "A multivariate spectral projected gradient method for bound constrained optimization." Journal of Computational and Applied Mathematics 235, no. 8 (February 2011): 2263–69. http://dx.doi.org/10.1016/j.cam.2010.10.023.

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Arnaudon, Marc, Anton Thalmaier, and Feng-Yu Wang. "Equivalent Harnack and gradient inequalities for pointwise curvature lower bound." Bulletin des Sciences Mathématiques 138, no. 5 (July 2014): 643–55. http://dx.doi.org/10.1016/j.bulsci.2013.11.001.

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Vollebregt, Edwin A. H. "The Bound-Constrained Conjugate Gradient Method for Non-negative Matrices." Journal of Optimization Theory and Applications 162, no. 3 (December 13, 2013): 931–53. http://dx.doi.org/10.1007/s10957-013-0499-x.

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Syed Shahul Hameed, A., and Narendran Rajagopalan. "SPGD: Search Party Gradient Descent Algorithm, a Simple Gradient-Based Parallel Algorithm for Bound-Constrained Optimization." Mathematics 10, no. 5 (March 2, 2022): 800. http://dx.doi.org/10.3390/math10050800.

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Nature-inspired metaheuristic algorithms remain a strong trend in optimization. Human-inspired optimization algorithms should be more intuitive and relatable. This paper proposes a novel optimization algorithm inspired by a human search party. We hypothesize the behavioral model of a search party searching for a treasure. Motivated by the search party’s behavior, we abstract the “Divide, Conquer, Assemble” (DCA) approach. The DCA approach allows us to parallelize the traditional gradient descent algorithm in a strikingly simple manner. Essentially, multiple gradient descent instances with different learning rates are run parallelly, periodically sharing information. We call it the search party gradient descent (SPGD) algorithm. Experiments performed on a diverse set of classical benchmark functions show that our algorithm is good at optimizing. We believe our algorithm’s apparent lack of complexity will equip researchers to solve problems efficiently. We compare the proposed algorithm with SciPy’s optimize library and it is found to be competent with it.

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Ching, Joshua, and Florica C. Cîrstea. "Gradient estimates for nonlinear elliptic equations with a gradient-dependent nonlinearity." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 3 (January 30, 2019): 1361–76. http://dx.doi.org/10.1017/prm.2018.133.

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AbstractIn this paper, we obtain gradient estimates of the positive solutions to weightedp-Laplacian type equations with a gradient-dependent nonlinearity of the form0.1$${\rm div }( \vert x \vert ^\sigma \vert \nabla u \vert ^{p-2}\nabla u) = \vert x \vert ^{-\tau }u^q \vert \nabla u \vert ^m\quad {\rm in}\;\Omega^*: = \Omega {\rm \setminus }\{ 0\} .$$Here,$\Omega \subseteq {\open R}^N$denotes a domain containing the origin with$N\ges 2$, whereas$m,q\in [0,\infty )$,$1<p\les N+\sigma $and$q>\max \{p-m-1,\sigma +\tau -1\}$. The main difficulty arises from the dependence of the right-hand side of (0.1) onx,uand$ \vert \nabla u \vert $, without any upper bound restriction on the powermof$ \vert \nabla u \vert $. Our proof of the gradient estimates is based on a two-step process relying on a modified version of the Bernstein's method. As a by-product, we extend the range of applicability of the Liouville-type results known for (0.1).

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Aussem, Alex. "Sufficient Conditions for Error Backflow Convergence in Dynamical Recurrent Neural Networks." Neural Computation 14, no. 8 (August 1, 2002): 1907–27. http://dx.doi.org/10.1162/089976602760128063.

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This article extends previous analysis of the gradient decay to a class of discrete-time fully recurrent networks, called dynamical recurrent neural networks, obtained by modeling synapses as finite impulse response (FIR) filters instead of multiplicative scalars. Using elementary matrix manipulations, we provide an upper bound on the norm of the weight matrix, ensuring that the gradient vector, when propagated in a reverse manner in time through the error-propagation network, decays exponentially to zero. This bound applies to all recurrent FIR architecture proposals, as well as fixed-point recurrent networks, regardless of delay and connectivity. In addition, we show that the computational overhead of the learning algorithm can be reduced drastically by taking advantage of the exponential decay of the gradient.

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Rajadas, J. N., A. Chattopadhyay, N. Pagaldipti, and S. Zhang. "Shape optimization of turbine blades with the integration of aerodynamics and heat transfer." Mathematical Problems in Engineering 4, no. 1 (1998): 21–42. http://dx.doi.org/10.1155/s1024123x98000702.

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A multidisciplinary optimization procedure, with the integration of aerodynamic and heat transfer criteria, has been developed for the design of gas turbine blades. Two different optimization formulations have been used. In the first formulation, the maximum temperature in the blade section is chosen as the objective function to be minimized. An upper bound constraint is imposed on the blade average temperature and a lower bound constraint is imposed on the blade tangential force coefficient. In the second formulation, the blade average and maximum temperatures are chosen as objective functions. In both formulations, bounds are imposed on the velocity gradients at several points along the surface of the airfoil to eliminate leading edge velocity spikes which deteriorate aerodynamic performance. Shape optimization is performed using the blade external and coolant path geometric parameters as design variables. Aerodynamic analysis is performed using a panel code. Heat transfer analysis is performed using the finite element method. A gradient based procedure in conjunction with an approximate analysis technique is used for optimization. The results obtained using both optimization techniques are compared with a reference geometry. Both techniques yield significant improvements with the multiobjective formulation resulting in slightly superior design.

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Marin, B. P. "The control by ΔµH+ of the tonoplast-bound H+-translocating adenosine triphosphatase from rubber-tree (Hevea brasiliensis) latex." Biochemical Journal 229, no. 2 (July 15, 1985): 459–67. http://dx.doi.org/10.1042/bj2290459.

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The relationship between tonoplast-bound ATPase activity and the magnitude of the electrochemical proton gradient has been investigated on tightly sealed vesicles prepared from rubber-tree (Hevea brasiliensis) latex. A variety of methods have been used to modify, either alone or together, the two components of the electrochemical proton gradient (delta mu H+). When the delta pH component was decreased either by titration with (NH4)2SO4 or by addition of protonophores or nigericin in the presence of K+, ATPase activity was stimulated. On the other hand, when the delta psi component was decreased either by addition of lipophilic cations or by addition of valinomycin in the presence of K+, ATPase activity decreased. It is concluded that activity of the tonoplast-bound ATPase is regulated by changes in the electrochemical proton gradient across the tonoplast, so that, once the maximum proton gradient is established across the tonoplast, any perturbation of the equilibrium state should result in the increased rate of ATP hydrolysis as the enzyme attempts to re-establish the initial gradient.

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Cao, Yuan, and Quanquan Gu. "Generalization Error Bounds of Gradient Descent for Learning Over-Parameterized Deep ReLU Networks." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 04 (April 3, 2020): 3349–56. http://dx.doi.org/10.1609/aaai.v34i04.5736.

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Empirical studies show that gradient-based methods can learn deep neural networks (DNNs) with very good generalization performance in the over-parameterization regime, where DNNs can easily fit a random labeling of the training data. Very recently, a line of work explains in theory that with over-parameterization and proper random initialization, gradient-based methods can find the global minima of the training loss for DNNs. However, existing generalization error bounds are unable to explain the good generalization performance of over-parameterized DNNs. The major limitation of most existing generalization bounds is that they are based on uniform convergence and are independent of the training algorithm. In this work, we derive an algorithm-dependent generalization error bound for deep ReLU networks, and show that under certain assumptions on the data distribution, gradient descent (GD) with proper random initialization is able to train a sufficiently over-parameterized DNN to achieve arbitrarily small generalization error. Our work sheds light on explaining the good generalization performance of over-parameterized deep neural networks.

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Liao, Xuanzhi, Shahnorbanun Sahran, Azizi Abdullah, and Syaimak Abdul Shukor. "AdaCB: An Adaptive Gradient Method with Convergence Range Bound of Learning Rate." Applied Sciences 12, no. 18 (September 19, 2022): 9389. http://dx.doi.org/10.3390/app12189389.

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Adaptive gradient descent methods such as Adam, RMSprop, and AdaGrad achieve great success in training deep learning models. These methods adaptively change the learning rates, resulting in a faster convergence speed. Recent studies have shown their problems include extreme learning rates, non-convergence issues, as well as poor generalization. Some enhanced variants have been proposed, such as AMSGrad, and AdaBound. However, the performances of these alternatives are controversial and some drawbacks still occur. In this work, we proposed an optimizer called AdaCB, which limits the learning rates of Adam in a convergence range bound. The bound range is determined by the LR test, and then two bound functions are designed to constrain Adam, and two bound functions tend to a constant value. To evaluate our method, we carry out experiments on the image classification task, three models including Smallnet, Network IN Network, and Resnet are trained on CIFAR10 and CIFAR100 datasets. Experimental results show that our method outperforms other optimizers on CIFAR10 and CIFAR100 datasets with accuracies of (82.76%, 53.29%), (86.24%, 60.19%), and (83.24%, 55.04%) on Smallnet, Network IN Network and Resnet, respectively. The results also indicate that our method maintains a faster learning speed, like adaptive gradient methods, in the early stage and achieves considerable accuracy, like SGD (M), at the end.

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Fung, Y. T. "Unstable Waves of Jet Flows With Density Inhomogeneity." Journal of Fluids Engineering 111, no. 3 (September 1, 1989): 238–42. http://dx.doi.org/10.1115/1.3243636.

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Instability of axisymmetric jet flows of a fluid having a radius-dependent density is investigated. The necessary condition for the existence of unstable waves depends not only on the velocity profile but also on the density gradient as well. Large density gradients, positive or negative, have stabilizing effects. The semicircle theorem for amplified waves is valid in this case. It is shown by considering the top-hat type velocity profile that the velocity-dependent semicircle bound is the best possible.

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Moore, Kenneth, and Eric Woolgar. "Bakry–Émery Ricci curvature, X-minimal hypersurfaces, and near horizon geometries." Journal of Mathematical Physics 64, no. 2 (February 1, 2023): 022504. http://dx.doi.org/10.1063/5.0113859.

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Motivated by the extreme black hole near horizon geometry equation and the Ellis–Ehlers equation of mathematical cosmology, we prove a Bakry–Émery generalization of a theorem of Frankel that closed minimal hypersurfaces in a complete manifold with a suitable curvature bound must intersect. We do not assume that the Bakry–Émery vector field is of gradient type. We also present splitting theorems of warped product type for manifolds bounded by hypersurfaces obeying Bakry–Émery curvature bounds.

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Deshmukh, Sharief. "Almost Ricci solitons isometric to spheres." International Journal of Geometric Methods in Modern Physics 16, no. 05 (May 2019): 1950073. http://dx.doi.org/10.1142/s0219887819500737.

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We find a characterization of a sphere using a compact gradient almost Ricci soliton and the lower bound on the integral of Ricci curvature in the direction of potential field. Also, we use Poisson equation on a compact gradient almost Ricci soliton to find a characterization of the unit sphere.

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Yuan, Jianjun, and Andrew Lamperski. "Trading-Off Static and Dynamic Regret in Online Least-Squares and Beyond." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 04 (April 3, 2020): 6712–19. http://dx.doi.org/10.1609/aaai.v34i04.6149.

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Recursive least-squares algorithms often use forgetting factors as a heuristic to adapt to non-stationary data streams. The first contribution of this paper rigorously characterizes the effect of forgetting factors for a class of online Newton algorithms. For exp-concave and strongly convex objectives, the algorithms achieve the dynamic regret of max{O(log T),O(√TV)}, where V is a bound on the path length of the comparison sequence. In particular, we show how classic recursive least-squares with a forgetting factor achieves this dynamic regret bound. By varying V, we obtain a trade-off between static and dynamic regret. In order to obtain more computationally efficient algorithms, our second contribution is a novel gradient descent step size rule for strongly convex functions. Our gradient descent rule recovers the order optimal dynamic regret bounds described above. For smooth problems, we can also obtain static regret of O(T1-β) and dynamic regret of O(Tβ V*), where β ∈ (0,1) and V* is the path length of the sequence of minimizers. By varying β, we obtain a trade-off between static and dynamic regret.

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Atkinson, J. F., E. Eric Adams, and D. R. F. Harleman. "Double-Diffusive Fluxes in a Salt Gradient Solar Pond." Journal of Solar Energy Engineering 110, no. 1 (February 1, 1988): 17–22. http://dx.doi.org/10.1115/1.3268231.

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The possible influence of double-diffusive stratification on the vertical transport of salt and heat in a mixed-layer simulation model for a salt gradient solar pond is examined. The study is concerned primarily with the interfacial fluxes across the boundary between the gradient zone and upper convecting zone of solar ponds, though the arguments presented should be applicable to other “diffusive” interfaces as well. In the absence of mechanical stirring in the upper convecting zone (e.g., by wind), double diffusive instabilities could govern the vertical flux of heat and salt by adjusting interfacial gradients of temperature and salinity which control transport by molecular diffusion. Because these gradients are generally too sharp to be resolved by numerical models, the fluxes can either be modeled directly or be parameterized by grid-dependent “effective diffusivities.” It is shown that when mechanical stirring is present in the mixed layer, double-diffusive instabilities will not be allowed to grow in the interfacial boundary layer region. Thus, double-diffusive fluxes become important only in the absence of stirring and, in effect, provide a lower bound to the fluxes that would be expected across the interface.

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Tatkiewicz, Witold I., Joaquin Seras-Franzoso, Elena Garcia-Fruitós, Esther Vazquez, A. R. Kyvik, Judith Guasch, Antonio Villaverde, Jaume Veciana, and Imma Ratera. "Surface-Bound Gradient Deposition of Protein Nanoparticles for Cell Motility Studies." ACS Applied Materials & Interfaces 10, no. 30 (July 10, 2018): 25779–86. http://dx.doi.org/10.1021/acsami.8b06821.

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Xiao, Yunhai, and Qingjie Hu. "Subspace Barzilai-Borwein Gradient Method for Large-Scale Bound Constrained Optimization." Applied Mathematics and Optimization 58, no. 2 (May 23, 2008): 275–90. http://dx.doi.org/10.1007/s00245-008-9038-9.

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Ecker, Klaus. "An interior gradient bound for solutions of equations of capillary type." Archive for Rational Mechanics and Analysis 92, no. 2 (June 1986): 137–51. http://dx.doi.org/10.1007/bf00251254.

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Tseng, Paul. "Approximation accuracy, gradient methods, and error bound for structured convex optimization." Mathematical Programming 125, no. 2 (August 17, 2010): 263–95. http://dx.doi.org/10.1007/s10107-010-0394-2.

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Zhang, Qi S., and Meng Zhu. "Li-Yau gradient bound for collapsing manifolds under integral curvature condition." Proceedings of the American Mathematical Society 145, no. 7 (January 6, 2017): 3117–26. http://dx.doi.org/10.1090/proc/13418.

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Xiao, Yun-Hai, Qing-Jie Hu, and Zengxin Wei. "Modified active set projected spectral gradient method for bound constrained optimization." Applied Mathematical Modelling 35, no. 7 (July 2011): 3117–27. http://dx.doi.org/10.1016/j.apm.2010.09.011.

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Journal articles on the topic 'Gradient bound'

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