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Salakhutdinov, Ruslan; Kolter, Zico; Heller, Katherine; Weller, Adrian; Oliver, Nuria; Scarlett, Jonathan; Berkenkamp, Felix (Ed.)
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Salakhutdinov, Ruslan; Kolter, Zico; Heller, Katherine; Weller, Adrian; Oliver, Nuria; Scarlett, Jonathan; Berkenkamp, Felix (Ed.)
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Salakhutdinov, Ruslan; Kolter, Zico; Heller, Katherine; Weller, Adrian; Oliver, Nuria; Scarlett, Jonathan; Berkenkamp, Felix (Ed.)Replica exchange stochastic gradient Langevin dynamics (reSGLD) is an effective sampler for non-convex learning in large-scale datasets. However, the simulation may encounter stagnation issues when the high-temperature chain delves too deeply into the distribution tails. To tackle this issue, we propose reflected reSGLD (r2SGLD): an algorithm tailored for constrained non-convex exploration by utilizing reflection steps within a bounded domain. Theoretically, we observe that reducing the diameter of the domain enhances mixing rates, exhibiting a quadratic behavior. Empirically, we test its performance through extensive experiments, including identifying dynamical systems with physical constraints, simulations of constrained multi-modal distributions, and image classification tasks. The theoretical and empirical findings highlight the crucial role of constrained exploration in improving the simulation efficiency.more » « less
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Kiyavash, Negar; Mooij, Joris M (Ed.)
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We establish the global existence of weak martingale solutions to the simplified stochastic Ericksen–Leslie system modeling the nematic liquid crystal flow driven by Wiener-type noises on the two-dimensional bounded domains. The construction of solutions is based on the convergence of Ginzburg–Landau approximations. To achieve such a convergence, we first utilize the concentration-cancellation method for the Ericksen stress tensor fields based on a Pohozaev type argument, and then the Skorokhod compactness theorem, which is built upon uniform energy estimates.more » « less
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In this paper, we will establish the global existence of a suitable weak solution to the Erickson–Leslie system modelling hydrodynamics of nematic liquid crystal flows with kinematic transports for molecules of various shapes in $${\mathbb{R}}^{3}$$, which is smooth away from a closed set of (parabolic) Hausdorff dimension at most $$\frac{15}{7}$$more » « less
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