115:. Unlike traditional SGD, SGLD can be used for Bayesian learning as a sampling method. SGLD may be viewed as Langevin dynamics applied to posterior distributions, but the key difference is that the likelihood gradient terms are minibatched, like in SGD. SGLD, like Langevin dynamics, produces samples from a posterior distribution of parameters based on available data. First described by Welling and Teh in 2011, the method has applications in many contexts which require optimization, and is most notably applied in machine learning problems.
74:
996:, a task in which the method provides a distribution over model parameters. By introducing information about the variance of these parameters, SGLD characterizes the generalizability of these models at certain points in training. Additionally, obtaining samples from a posterior distribution permits uncertainty quantification by means of confidence intervals, a feature which is not possible using traditional stochastic gradient descent.
1016:, consisting of a single leapfrog step proposal rather than a series of steps. Since SGLD can be formulated as a modification of both stochastic gradient descent and MCMC methods, the method lies at the intersection between optimization and sampling algorithms; the method maintains SGD's ability to quickly converge to regions of low cost while providing samples to facilitate posterior inference.
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Recent contributions have proven upper bounds on mixing times for both the traditional
Langevin algorithm and the Metropolis adjusted Langevin algorithm. Released in Ma et al., 2018, these bounds define the rate at which the algorithms converge to the true posterior distribution, defined formally as:
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SGLD is applicable in any optimization context for which it is desirable to quickly obtain posterior samples instead of a maximum a posteriori mode. In doing so, the method maintains the computational efficiency of stochastic gradient descent when compared to traditional
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1263:{\displaystyle {\frac {p(\mathbf {\theta } ^{t}\mid \mathbf {\theta } ^{t+1})p^{*}\left(\mathbf {\theta } ^{t}\right)}{p\left(\mathbf {\theta } ^{t+1}\mid \mathbf {\theta } ^{t}\right)p^{*}(\mathbf {\theta } ^{t+1})}}<u,\ u\sim {\mathcal {U}}}
331:
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For early iterations of the algorithm, each parameter update mimics
Stochastic Gradient Descent; however, as the algorithm approaches a local minimum or maximum, the gradient shrinks to zero and the chain produces samples surrounding the
323:
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refer to the mixing rates of the
Unadjusted Langevin Algorithm and the Metropolis Adjusted Langevin Algorithm respectively. These bounds are important because they show computational complexity is polynomial in dimension
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allowing for posterior inference. This process generates approximate samples from the posterior as by balancing variance from the injected
Gaussian noise and stochastic gradient computation.
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2102:{\displaystyle \tau _{MALA}(\varepsilon ,p^{0})\leq {\mathcal {O}}\left(e^{16LR^{2}}\kappa ^{3/2}d^{1/2}\left(d\ln \kappa +\ln \left({\frac {1}{\varepsilon }}\right)\right)^{3/2}\right)}
683:{\displaystyle \Delta \theta _{t}={\frac {\varepsilon _{t}}{2}}\left(\nabla \log p(\theta _{t})+{\frac {N}{n}}\sum _{i=1}^{n}\nabla \log p(x_{t_{i}}\mid \theta _{t})\right)+\eta _{t}}
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models. Like stochastic gradient descent, SGLD is an iterative optimization algorithm which uses minibatching to create a stochastic gradient estimator, as used in SGD to optimize a
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1898:{\displaystyle \tau _{ULA}(\varepsilon ,p^{0})\leq {\mathcal {O}}\left(e^{32LR^{2}}\kappa ^{2}{\frac {d}{\varepsilon ^{2}}}\ln \left({\frac {d}{\varepsilon ^{2}}}\right)\right)}
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494:{\displaystyle \Delta \theta _{t}={\frac {\varepsilon _{t}}{2}}\left(\nabla \log p(\theta _{t})+\sum _{i=1}^{N}\nabla \log p(x_{t_{i}}\mid \theta _{t})\right)+\eta _{t}}
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rejection rate is zero, and thus a MH rejection step becomes necessary. The resulting algorithm, dubbed the
Metropolis Adjusted Langevin algorithm, requires the step:
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1501:{\displaystyle \tau (\varepsilon ;p^{0})=\min \left\{k\mid \left\|p^{k}-p^{*}\right\|_{\mathrm {V} }\leq \varepsilon \right\}}
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953:{\displaystyle \sum _{t=1}^{\infty }\varepsilon _{t}=\infty \quad \sum _{t=1}^{\infty }\varepsilon _{t}^{2}<\infty }
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SGLD can be applied to the optimization of non-convex objective functions, shown here to be a sum of
Gaussians.
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Stochastic gradient
Langevin dynamics uses a modified update procedure with minibatched likelihood terms:
2328:; Sagun, Levent; Zecchina, Riccardo (2017). "Entropy-sgd: Biasing gradient descent into wide valleys".
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such that they do not approach zero asymptotically, SGLD fails to produce samples for which the
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318:{\displaystyle p(\theta \mid X)\propto p(\theta )\prod _{i=1}^{N}p(x_{i}\mid \theta )}
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norm. Under some regularity conditions of an L-Lipschitz smooth objective function
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of the objective function. In practice, SGLD can be applied to the training of
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while providing additional information regarding the landscape around the
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Kennedy, A. D. (1990). "The theory of hybrid stochastic algorithms".
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Proceedings of the 28th
International Conference on Machine Learning
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Probabilistic
Methods in Quantum Field Theory and Quantum Gravity
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is a normal distribution centered one gradient descent step from
771:{\displaystyle \eta _{t}\sim {\mathcal {N}}(0,\varepsilon _{t})}
18:
2292:"Bayesian Learning via Stochastic Gradient Langevin Dynamics"
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If gradient computations are exact, SGLD reduces down to the
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229:, Langevin dynamics samples from the posterior distribution
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is the likelihood of the data given the parameter vector
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and sampling technique composed of characteristics from
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1321:{\displaystyle p(\theta ^{t}\mid \theta ^{t+1})}
2412:Proceedings of the National Academy of Sciences
2402:Ma, Y. A.; Chen, Y.; Jin, C.; Flammarion, N.;
1008:algorithm, first coined in the literature of
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2408:"Sampling can be faster than optimization"
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1388:Mixing rates and algorithmic convergence
1012:. This algorithm is also a reduction of
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2312:Chaudhari, Pratik; Choromanska, Anna;
2264:{\displaystyle {\mathcal {O}}(\log d)}
1739:{\displaystyle \kappa ={\frac {L}{m}}}
1542:{\displaystyle \varepsilon \in (0,1)}
222:{\displaystyle X=\{x_{i}\}_{i=1}^{N}}
82:Stochastic gradient Langevin dynamics
7:
2397:
2395:
2376:Handbook of Markov Chain Monte Carlo
2290:Welling, Max; Teh, Yee Whye (2011).
2285:
2283:
2483:Optimization algorithms and methods
1603:is the posterior distribution, and
16:Optimization and sampling technique
2353:. Plenum Press. pp. 209–223.
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1000:Variants and associated algorithms
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860:satisfy the following conditions:
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14:
1549:is an arbitrary error tolerance,
1039:{\displaystyle \varepsilon _{t}}
853:{\displaystyle \varepsilon _{t}}
806:{\displaystyle p(x\mid \theta )}
23:
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1746:, we have mixing rate bounds:
1690:outside of a region of radius
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104:, a mathematical extension of
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2174:{\displaystyle \tau _{MALA}}
1384:is our target distribution.
123:Given some parameter vector
100:optimization algorithm, and
2138:{\displaystyle \tau _{ULA}}
1348:{\displaystyle \theta ^{t}}
172:, and a set of data points
94:Stochastic gradient descent
2504:
1646:{\displaystyle ||*||_{TV}}
1377:{\displaystyle p(\theta )}
165:{\displaystyle p(\theta )}
966:maximum a posteriori mode
143:, its prior distribution
2473:Computational statistics
2488:Stochastic optimization
2435:10.1073/pnas.1820003116
1014:Hamiltonian Monte Carlo
826:{\displaystyle \theta }
719:is a positive integer,
325:by updating the chain:
136:{\displaystyle \theta }
38:, as no other articles
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2226:
2225:{\displaystyle LR^{2}}
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712:{\displaystyle n<N}
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1596:{\displaystyle p^{*}}
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1010:lattice field theory
1006:Langevin Monte Carlo
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2418:(42): 20881–20885.
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1048:Metropolis Hastings
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778:is Gaussian noise,
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113:objective function
106:molecular dynamics
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57:for suggestions.
47:to this page from
2385:978-1-4200-7941-8
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102:Langevin dynamics
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110:differentiable
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55:Find link tool
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2404:Jordan, M. I.
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994:Deep Learning
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32:This article
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90:optimization
85:
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62:January 2019
59:
33:
2318:LeCun, Yann
972:Application
2467:Categories
2425:1811.08413
2335:1611.01838
2301:: 681–688.
2275:References
53:; try the
40:link to it
2253:
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43:. Please
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