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The derivation of gradient of the marginal likelihood is given in http://www.gaussianprocess.org/gpml/chapters/RW5.pdf

But the gradient for the most commonly used covariance function, squared exponential covariance, is not explicitly given.

I am implementing the Rprop algorithm in http://ml.informatik.uni-freiburg.de/_media/publications/blumesann2013.pdf for learning hyper-parameters sigma (signal variance) and h (length). Alas, my implementation is not working well. I have derived the gradients but I am not sure if they are correct.

Can someone point me to a good tutorial / article that explicitly give the expressions for the hyper parameter gradients?

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We are looking to maximise the log probability of $lnP(y|x, \theta)$:

$$\ln P(y|x, \theta) = -\frac{1}{2}\ln|K| - \frac{1}{2}y^tK^{-1}y - \frac{N}{2}\ln{2\pi}$$

The three components can be seen as balancing the complexity of the GP (to avoid overfit) and the data fit, with a constant on the end. So the gradient is

$$\frac{\partial}{\partial\theta_i} \log P(y|x, \theta) = \frac{1}{2}y^TK^{-1}\frac{\partial K}{\partial\theta_i}K^{-1}y^T -\frac{1}{2}\mathrm{tr}\left(K^{-1}\frac{\partial K}{\partial\theta_i}\right)$$

So all we need to know is $\frac{\partial K}{\partial\theta_i}$ to be able to solve it. I think you got this far but I wasn't sure so I thought I would recap.

For the case of the RBF/expodentiated quadratic (never call it squared exponential as this is actually incorrect) kernel, under the following formulation:

$$K(x,x') = \sigma^2\exp\left(\frac{-(x-x')^T(x-x')}{2l^2}\right)$$

The derivatives with respect to the hyperparameters are as follows:

$$\frac{\partial K}{\partial\sigma} = 2\sigma\exp\left(\frac{-(x-x')^T(x-x')}{2l^2}\right)$$

$$\frac{\partial K}{\partial l} = \sigma^2\exp\left(\frac{-(x-x')^T(x-x')}{2l^2}\right) \frac{(x-x')^T(x-x')}{l^3}$$

However, often GP libraries use the notation:

$$K(x,x') = \sigma\exp\left(\frac{-(x-x')^T(x-x')}{l}\right)$$

where $\sigma$ and $l$ is confined to only to positive real numbers. Let $l=exp(\theta_1)$ and $\sigma=exp(2\theta_2)$, then by passing in $a,b$ we know are values will conform to this rule. In this case the derivatives are:

$$\frac{\partial K}{\partial\theta_1} = \sigma\exp\left(\frac{-(x-x')^T(x-x')}{2l^2}\right)\left(\frac{(x-x')^T(x-x')}{l^2}\right)$$

$$\frac{\partial K}{\partial \theta_2} = 2 \sigma\exp\left(\frac{-(x-x')^T(x-x')}{2l^2}\right)$$

There is is interesting work carried out by the likes of Mike Osborne looking at marginalising out hyper parameters. However as far as I am aware I think it is only appropriate for low numbers of parameters and isn't incorporated in standard libraries yet. Worth a look all the same.

Another not is that the optimisation space is multimodal the majority of the time so if you are using convex optimisation be sure to use a fare few initialisations.

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  • $\begingroup$ Many thanks! This is very helpful. $\endgroup$ – aaronqli Dec 20 '14 at 19:52
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    $\begingroup$ jpro is right, your answer for $\frac{dK}{dl}$ is incorrect $\endgroup$ – George Aug 25 '16 at 19:27
  • $\begingroup$ Ah sorry about that - I'll fix it when I get home not to confuse people in the future $\endgroup$ – j__ Aug 26 '16 at 13:21
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I believe that the derivative of $\frac{\partial K}{\partial l}$, as it was given by j_f is not correct. I think that the correct one is the following (i present the derivation step by step):

$K(x,x') = \sigma^2\exp\big(\frac{-(x-x')^T(x-x')}{2l^2}\big)$
I now call $g(l)=\big(\frac{-(x-x')^T(x-x')}{2l^2}\big)$. So $K=\sigma^2exp\big( g(l) \big)$.

$\frac{\partial K}{\partial l} = \frac{\partial K}{\partial g} \frac{\partial g}{\partial l} = \sigma^2\exp\big(\frac{-(x-x')^T(x-x')}{2l^2}\big) \frac{\partial g}{\partial l}$.

With simple calculations, I finally get:

$\frac{\partial K}{\partial l} = \sigma^2\exp\big(\frac{-(x-x')^T(x-x')}{2l^2}\big) \frac{(x-x')^T(x-x')}{l^3}$.

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Maybe there is also an error in the gradient formulation, because in Rasmussen&Williams - Gaussian Process for Machine Learning, p.114, eq 5.9, it is expressed as:

Partial deivatives log marginal likelihood w.r.t. hyperparameters

where the 2 terms have different signs and the y targets vector is transposed just the first time.

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