Nonasymptotic estimates for Stochastic Gradient Langevin Dynamics under local conditions in nonconvex optimization

Zhang, Ying; Akyıldız, Ömer Deniz; Damoulas, Theo; Sabanis, Sotirios

doi:10.48550/arxiv.1910.02008

Cited by 14 publications

(23 citation statements)

References 20 publications

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“…In this section, we start with placing assumptions on stochastic gradients H(θ, ε) as defined in (8). We note that these assumptions are the most relaxed conditions to prove the convergence of Langevin dynamics to this date, see, e.g., Zhang et al (2019), Chau et al (2021). We first need to assume that sufficient moments of the distribution r ε exists.…”

Section: Convergence Rates Of Solaismentioning

confidence: 98%

“…This will lead to a global optimizer θ ⋆ which will give the best possible proposal in terms of minimizing the MSE of the importance sampler. We use stochastic gradient Langevin dynamics (SGLD) (Zhang et al, 2019) and its underdamped counterpart stochastic gradient Hamiltonian Monte Carlo (SGHMC) (Akyildiz and Sabanis, 2020) for global optimization. We summarize the algorithms in the next section.…”

Section: Adaptation As Global Nonconvex Optimizationmentioning

confidence: 99%

“…We can finally state the convergence result of the SGLD for non-convex optimization from Zhang et al (2019). Theorem 6.…”

Section: Convergence Rates Of Solaismentioning

confidence: 99%

“…Specifically, we use stochastic gradient Langevin dynamics (SGLD) (Welling and Teh, 2011) and its underdamped counterpart, stochastic gradient Hamiltonian Monte Carlo (SGHMC) (Chen et al, 2014), for non-convex optimization. Recently, global convergence of these algorithms for nonconvex optimization were shown in several works, see, e.g., Raginsky et al (2017), Xu et al (2018), Erdogdu et al (2018, Zhang et al (2019), Akyildiz and Sabanis (2020), Gao et al (2021), , . We leverage these results for proving that optimizing a general non-convex χ 2 -divergence leads to a global convergence result for the resulting AIS schemes.…”

Section: Introductionmentioning

confidence: 97%

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Convergence rates for optimised adaptive importance samplers

Akyıldız

Mı́guez

2021

Stat Comput

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Adaptive importance samplers are adaptive Monte Carlo algorithms to estimate expectations with respect to some target distribution which adapt themselves to obtain better estimators over a sequence of iterations. Although it is straightforward to show that they have the same $$\mathcal {O}(1/\sqrt{N})$$ O ( 1 / N ) convergence rate as standard importance samplers, where N is the number of Monte Carlo samples, the behaviour of adaptive importance samplers over the number of iterations has been left relatively unexplored. In this work, we investigate an adaptation strategy based on convex optimisation which leads to a class of adaptive importance samplers termed optimised adaptive importance samplers (OAIS). These samplers rely on the iterative minimisation of the $$\chi ^2$$ χ 2 -divergence between an exponential family proposal and the target. The analysed algorithms are closely related to the class of adaptive importance samplers which minimise the variance of the weight function. We first prove non-asymptotic error bounds for the mean squared errors (MSEs) of these algorithms, which explicitly depend on the number of iterations and the number of samples together. The non-asymptotic bounds derived in this paper imply that when the target belongs to the exponential family, the $$L_2$$ L 2 errors of the optimised samplers converge to the optimal rate of $$\mathcal {O}(1/\sqrt{N})$$ O ( 1 / N ) and the rate of convergence in the number of iterations are explicitly provided. When the target does not belong to the exponential family, the rate of convergence is the same but the asymptotic $$L_2$$ L 2 error increases by a factor $$\sqrt{\rho ^\star } > 1$$ ρ ⋆ > 1 , where $$\rho ^\star - 1$$ ρ ⋆ - 1 is the minimum $$\chi ^2$$ χ 2 -divergence between the target and an exponential family proposal.

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Section: Convergence Rates Of Solaismentioning

confidence: 98%

Section: Adaptation As Global Nonconvex Optimizationmentioning

confidence: 99%

“…We can finally state the convergence result of the SGLD for non-convex optimization from Zhang et al (2019). Theorem 6.…”

Section: Convergence Rates Of Solaismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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Convergence rates for optimised adaptive importance samplers

Akyıldız

Mı́guez

2021

Stat Comput

Self Cite

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“…Remark 3.4. Langevin sampling method has been studied under the (strongly) convex potential assumption [15,16,17,11,12,8,13]; the dissipativity condition for the drift term [34,33,40]; the local convexity condition for the potential function outside a ball [17,9,30,3]. However, these conditions may not hold for models with multiple modes, for example, Gaussian mixtures, where their potentials are not convex and the log Sobolev inequality may not be satisfied.…”

mentioning

confidence: 99%

Convergence Analysis of Schr{ö}dinger-F{ö}llmer Sampler without Convexity

Jiao

Kang

Liu

et al. 2021

Preprint

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Schrödinger-Föllmer sampler (SFS) [26], is a novel and efficient approach for sampling from possibly unnormalized distributions without ergodicity. SFS is based on the Euler-Maruyama discretization of Schrödinger-Föllmer diffusion process dXt = −∇U (Xt, t) dt + dBt, t ∈ [0, 1], X 0 = 0 on the unit interval, which transports the degenerate distribution at time zero to the target distribution at time one. In [26], the consistency of SFS is established under a restricted assumption that the potential U (x, t) is uniformly (on t) strongly convex (on x). In this paper we provide a nonasymptotic error bound of SFS in Wasserstein distance under some smooth and bounded conditions on the density ratio of the target distribution over the standard normal distribution, but without requiring the strongly convexity of the potential.

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Multi-index antithetic stochastic gradient algorithm

2023

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Stochastic Gradient Algorithms (SGAs) are ubiquitous in computational statistics, machine learning and optimisation. Recent years have brought an influx of interest in SGAs, and the non-asymptotic analysis of their bias is by now well-developed. However, relatively little is known about the optimal choice of the random approximation (e.g mini-batching) of the gradient in SGAs as this relies on the analysis of the variance and is problem specific. While there have been numerous attempts to reduce the variance of SGAs, these typically exploit a particular structure of the sampled distribution by requiring a priori knowledge of its density’s mode. In this paper, we construct a Multi-index Antithetic Stochastic Gradient Algorithm (MASGA) whose implementation is independent of the structure of the target measure. Our rigorous theoretical analysis demonstrates that for log-concave targets, MASGA achieves performance on par with Monte Carlo estimators that have access to unbiased samples from the distribution of interest. In other words, MASGA is an optimal estimator from the mean square error-computational cost perspective within the class of Monte Carlo estimators. To illustrate the robustness of our approach, we implement MASGA also in some simple non-log-concave numerical examples, however, without providing theoretical guarantees on our algorithm’s performance in such settings.

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Nonasymptotic estimates for Stochastic Gradient Langevin Dynamics under local conditions in nonconvex optimization

Cited by 14 publications

References 20 publications

Convergence rates for optimised adaptive importance samplers

Convergence rates for optimised adaptive importance samplers

Convergence Analysis of Schr{ö}dinger-F{ö}llmer Sampler without Convexity

Multi-index antithetic stochastic gradient algorithm

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