Return to Question

I am reading a paper on Bayesian optimization which aims at selecting the batch of points $x_0,x_1,..,x_k$ to sample to obtain maxima or minima of the GP. So the first batch element is normal acquisition function and optimizing the argmax will lead to first sampling element as follows:

$${x}_{1}= \textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0})$$ $ \mathcal{\alpha} (\mathbf{x}; I_{0})$ is just symbolic term for any acquisition function such as Expected Improvement, Probability of Improvement etc.

For second element of the batch we need to marginalize (integrate out previous element) as well as optimize (argmax for new element) as follows $${x}_{2}= \textrm{argmax} \int \mathcal{\alpha} (\mathbf{x}; I_{1})p(y_{1}|x_{1},I_0)p(x_{1}|I_0)dx_1dy_1$$ where, $ p(y_{1}|x_{1},I_0)$ is predictive distribution of the GP

$p(x_{1}|I_0)$ = $\delta (x_{1}-\textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0}))$

Similarly for the third element of the batch $x_3$ we need to marginalize the first two and optimize and for kth element marginalize all the previous k-1 and then optimize as shown by equation 2 in the attached paper.

Now as per the post here marginalization makes sense because we are using predictive distribution of the GP and when we predict the future point we need to account for uncertainty based on previous predictions. but what I dont understand is what is the role of the delta function here? This post provides intuition of using delta function but I still cant make a connection.

$$\delta (x_{1}-\textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0}))$$

Shouldn't the above expression evaluate to $\delta (0)$?

Further Clarification:

In simple sequential Bayesian optimization we have the following scenerio:

Here we will sample ,real experiment,at the suggested point by the argmax of acquisition function i.e 0.7 and then include in result in our data and use this updated data to fit the GP and use optimization of the acquisition function (argmax) again to suggest next point to sample.

However we can see that "maybe" the "next sample" the acquisition function will "suggest" could be the second best "mode" of multi-modal acquisition function above which is roughly around 0.38. What shall be the mathematics to get second, third and fourth ... modes of the acquisition function in one go and without running experiment each time, to form a batch of sampling points. This can be done by suppressing the acquisition function at 0.7 we can do so by taking **GP Prediction at 0.7 and insert it to our dataset $D$ as an "uncertain data", hence the marginalization of uncertain data, after doing so the acquisition function will might look like this:

Now we can optimize and obtain the second element which is centered around 0.38 (without actually having to run a real experiment) and repeat the process to obtain third element i.e. third mode of the multi-modal acquisition function by marginalizing the first two and optimizing the result and so on.

I am actually confused on how the mathematics of delta function is enabling us to achieve the above behaviour. Is delta function shifting the prior to the next best maxima/mode of the acquisition function? I understand the process I am just trying to make sense of the mathematics.

I am reading a paper on Bayesian optimization which aims at selecting the batch of points $x_0,x_1,..,x_k$ to sample to obtain maxima or minima of the GP. So the first batch element is normal acquisition function and optimizing the argmax will lead to first sampling element as follows:

$${x}_{1}= \textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0})$$ $ \mathcal{\alpha} (\mathbf{x}; I_{0})$ is just symbolic term for any acquisition function such as Expected Improvement, Probability of Improvement etc.

For second element of the batch we need to marginalize (integrate out previous element) as well as optimize (argmax for new element) as follows $${x}_{2}= \textrm{argmax} \int \mathcal{\alpha} (\mathbf{x}; I_{1})p(y_{1}|x_{1},I_0)p(x_{1}|I_0)dx_1dy_1$$ where, $ p(y_{1}|x_{1},I_0)$ is predictive distribution of the GP

$p(x_{1}|I_0)$ = $\delta (x_{1}-\textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0}))$

Similarly for the third element of the batch $x_3$ we need to marginalize the first two and optimize and for kth element marginalize all the previous k-1 and then optimize as shown by equation 2 in the attached paper.

Now as per the post here marginalization makes sense because we are using predictive distribution of the GP and when we predict the future point we need to account for uncertainty based on previous predictions. but what I dont understand is what is the role of the delta function here? This post provides intuition of using delta function but I still cant make a connection.

$$\delta (x_{1}-\textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0}))$$

Shouldn't the above expression evaluate to $\delta (0)$?

I am reading a paper on Bayesian optimization which aims at selecting the batch of points $x_0,x_1,..,x_k$ to sample to obtain maxima or minima of the GP. So the first batch element is normal acquisition function and optimizing the argmax will lead to first sampling element as follows:

$${x}_{1}= \textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0})$$ $ \mathcal{\alpha} (\mathbf{x}; I_{0})$ is just symbolic term for any acquisition function such as Expected Improvement, Probability of Improvement etc.

For second element of the batch we need to marginalize (integrate out previous element) as well as optimize (argmax for new element) as follows $${x}_{2}= \textrm{argmax} \int \mathcal{\alpha} (\mathbf{x}; I_{1})p(y_{1}|x_{1},I_0)p(x_{1}|I_0)dx_1dy_1$$ where, $ p(y_{1}|x_{1},I_0)$ is predictive distribution of the GP

$p(x_{1}|I_0)$ = $\delta (x_{1}-\textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0}))$

Similarly for the third element of the batch $x_3$ we need to marginalize the first two and optimize and for kth element marginalize all the previous k-1 and then optimize as shown by equation 2 in the attached paper.

Now as per the post here marginalization makes sense because we are using predictive distribution of the GP and when we predict the future point we need to account for uncertainty based on previous predictions. but what I dont understand is what is the role of the delta function here? This post provides intuition of using delta function but I still cant make a connection.

$$\delta (x_{1}-\textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0}))$$

Shouldn't the above expression evaluate to $\delta (0)$?

Further Clarification:

In simple sequential Bayesian optimization we have the following scenerio:

Here we will sample ,real experiment,at the suggested point by the argmax of acquisition function i.e 0.7 and then include in result in our data and use this updated data to fit the GP and use optimization of the acquisition function (argmax) again to suggest next point to sample.

However we can see that "maybe" the "next sample" the acquisition function will "suggest" could be the second best "mode" of multi-modal acquisition function above which is roughly around 0.38. What shall be the mathematics to get second, third and fourth ... modes of the acquisition function in one go and without running experiment each time, to form a batch of sampling points. This can be done by suppressing the acquisition function at 0.7 we can do so by taking **GP Prediction at 0.7 and insert it to our dataset $D$ as an "uncertain data", hence the marginalization of uncertain data, after doing so the acquisition function will might look like this:

Now we can optimize and obtain the second element which is centered around 0.38 (without actually having to run a real experiment) and repeat the process to obtain third element i.e. third mode of the multi-modal acquisition function by marginalizing the first two and optimizing the result and so on.

I am actually confused on the role of prior as suggested by @dizq22 that uses delta function. Is delta function shifting the prior to the next best maxima/mode of the acquisition function? I understand the process I am just trying to make sense of the mathematics.

I am reading a paper on Bayesian optimization which aims at selecting the batch of points $x_0,x_1,..,x_k$ to sample to obtain maxima or minima of the GP. So the first batch element is normal acquisition function and optimizing the argmax will lead to first sampling element as follows:

$${x}_{1}= \textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0})$$ $ \mathcal{\alpha} (\mathbf{x}; I_{0})$ is just symbolic term for any acquisition function such as Expected Improvement, Probability of Improvement etc.

For second element of the batch we need to marginalize (integrate out previous element) as well as optimize (argmax for new element) as follows $${x}_{2}= \textrm{argmax} \int \mathcal{\alpha} (\mathbf{x}; I_{1})p(y_{1}|x_{1},I_0)p(x_{1}|I_0)dx_1dy_1$$ where, $ p(y_{1}|x_{1},I_0)$ is predictive distribution of the GP

$p(x_{1}|I_0)$ = $\delta (x_{1}-\textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0}))$

Similarly for the third element of the batch $x_3$ we need to marginalize the first two and optimize and for kth element marginalize all the previous k-1 and then optimize as shown by equation 2 in the attached paper.

Now as per the post here marginalization makes sense because we are using predictive distribution of the GP and when we predict the future point we need to account for uncertainty based on previous predictions. but what I dont understand is what is the role of the delta function here? This post provides intuition of using delta function but I still cant make a connection.

$$\delta (x_{1}-\textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0}))$$

Shouldn't the above expression evaluate to $\delta (0)$?

Further Clarification:

In simple sequential Bayesian optimization we have the following scenerio:

Here we will sample ,real experiment,at the suggested point by the argmax of acquisition function i.e 0.7 and then include in result in our data and use this updated data to fit the GP and use optimization of the acquisition function (argmax) again to suggest next point to sample.

However we can see that "maybe" the "next sample" the acquisition function will "suggest" could be the second best "mode" of multi-modal acquisition function above which is roughly around 0.38. What shall be the mathematics to get second, third and fourth ... modes of the acquisition function in one go and without running experiment each time, to form a batch of sampling points. This can be done by suppressing the acquisition function at 0.7 we can do so by taking **GP Prediction at 0.7 and insert it to our dataset $D$ as an "uncertain data", hence the marginalization of uncertain data, after doing so the acquisition function will might look like this:

Now we can optimize and obtain the second element which is centered around 0.38 (without actually having to run a real experiment) and repeat the process to obtain third element i.e. third mode of the multi-modal acquisition function by marginalizing the first two and optimizing the result and so on.

I am actually confused on how the mathematics of delta function is enabling us to achieve the above behaviour. Is delta function shifting the prior to the next best maxima/mode of the acquisition function? I understand the process I am just trying to make sense of the mathematics.

I am reading a paper on Bayesian optimization which aims at selecting the batch of points $x_0,x_1,..,x_k$ to sample to obtain maxima or minima of the GP. So the first batch element is normal acquisition function and optimizing the argmax will lead to first sampling element as follows:

$${x}_{1}= \textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0})$$ $ \mathcal{\alpha} (\mathbf{x}; I_{0})$ is just symbolic term for any acquisition function such as Expected Improvement, Probability of Improvement etc.

For second element of the batch we need to marginalize (integrate out previous element) as well as optimize (argmax for new element) as follows $${x}_{2}= \textrm{argmax} \int \mathcal{\alpha} (\mathbf{x}; I_{1})p(y_{1}|x_{1},I_0)p(x_{1}|I_0)dx_1dy_1$$ where, $ p(y_{1}|x_{1},I_0)$ is predictive distribution of the GP

$p(x_{1}|I_0)$ = $\delta (x_{1}-\textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0}))$

Similarly for the third element of the batch $x_3$ we need to marginalize the first two and optimize and for kth element marginalize all the previous k-1 and then optimize as shown by equation 2 in the attached paper.

Now as per the post here marginalization makes sense because we are using predictive distribution of the GP and when we predict the future point we need to account for uncertainty based on previous predictions. but what I dont understand is what is the role of the delta function here? This post provides intuition of using delta function but I still cant make a connection.

$$\delta (x_{1}-\textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0}))$$

Shouldn't the above expression evaluate to $\delta (0)$?

Further Clarification:

In simple sequential Bayesian optimization we have the following scenerio:

Here we will sample ,real experiment,at the suggested point by the acquisition function i.e 0.7 and then include in result in our data and use this updated data to fit the GP and use acquisition function again to suggest next point to sample.

However we can see that "maybe" the "next sample" the acquisition function will "suggest" could be the second best "mode" of multi-modal acquistion function above. What shall be the mathematics to say get second, third and fourth ... modes of the acquisition function to form a batch and then sample the batch at once. This can be done by suppressing the acquisition function at 0.7 we can do so by taking GP Prediction at 0.7 and insert it to our dataset $D$ as an "uncertain data" hence the marginalization of uncertain data after doing so the acquisition function will might look like this:

Now we can optimize and obtain the second element of the batch (without actually having to run a real experiment) and repeat the process to obtain third element i.e. third mode of the multi-modal acquisition function.

I am actually confused on the role of prior as suggested by @dizq22 that uses delta function. Is delta function shifting the prior to the next best maxima/mode of the acquisition function? I understand the process I am just trying to make sense of the mathematics.

I am reading a paper on Bayesian optimization which aims at selecting the batch of points $x_0,x_1,..,x_k$ to sample to obtain maxima or minima of the GP. So the first batch element is normal acquisition function and optimizing the argmax will lead to first sampling element as follows:

$${x}_{1}= \textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0})$$ $ \mathcal{\alpha} (\mathbf{x}; I_{0})$ is just symbolic term for any acquisition function such as Expected Improvement, Probability of Improvement etc.

For second element of the batch we need to marginalize (integrate out previous element) as well as optimize (argmax for new element) as follows $${x}_{2}= \textrm{argmax} \int \mathcal{\alpha} (\mathbf{x}; I_{1})p(y_{1}|x_{1},I_0)p(x_{1}|I_0)dx_1dy_1$$ where, $ p(y_{1}|x_{1},I_0)$ is predictive distribution of the GP

$p(x_{1}|I_0)$ = $\delta (x_{1}-\textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0}))$

Similarly for the third element of the batch $x_3$ we need to marginalize the first two and optimize and for kth element marginalize all the previous k-1 and then optimize as shown by equation 2 in the attached paper.

Now as per the post here marginalization makes sense because we are using predictive distribution of the GP and when we predict the future point we need to account for uncertainty based on previous predictions. but what I dont understand is what is the role of the delta function here? This post provides intuition of using delta function but I still cant make a connection.

$$\delta (x_{1}-\textrm{argmax} \ \mathcal{\alpha} (\mathbf{x}; I_{0}))$$

Shouldn't the above expression evaluate to $\delta (0)$?

Further Clarification:

In simple sequential Bayesian optimization we have the following scenerio:

Here we will sample ,real experiment,at the suggested point by the argmax of acquisition function i.e 0.7 and then include in result in our data and use this updated data to fit the GP and use optimization of the acquisition function (argmax) again to suggest next point to sample.

However we can see that "maybe" the "next sample" the acquisition function will "suggest" could be the second best "mode" of multi-modal acquisition function above which is roughly around 0.38. What shall be the mathematics to get second, third and fourth ... modes of the acquisition function in one go and without running experiment each time, to form a batch of sampling points. This can be done by suppressing the acquisition function at 0.7 we can do so by taking **GP Prediction at 0.7 and insert it to our dataset $D$ as an "uncertain data", hence the marginalization of uncertain data, after doing so the acquisition function will might look like this:

Now we can optimize and obtain the second element which is centered around 0.38 (without actually having to run a real experiment) and repeat the process to obtain third element i.e. third mode of the multi-modal acquisition function by marginalizing the first two and optimizing the result and so on.

I am actually confused on the role of prior as suggested by @dizq22 that uses delta function. Is delta function shifting the prior to the next best maxima/mode of the acquisition function? I understand the process I am just trying to make sense of the mathematics.