*hyperparameters*. These are numbers like weight decay magnitude, Gaussian kernel width, and so forth. The algorithm doesn't set them, instead, it's up to you to determine their values. If you don't set these parameters to "good" values the algorithm doesn't work. So what do you do? Well, here is a list of everything I've seen people do, listed in order of most to least common:

**Guess and Check**: Listen to your gut, pick numbers that feel good and see if they work. Keep doing this until you are tired of doing it.**Grid Search**: Ask your computer to try a bunch of values spread evenly over some range.**Random Search**: Ask your computer to try a bunch of values by picking them randomly.**Bayesian Optimization**: Use a tool like MATLAB's bayesopt to automatically pick the best parameters, then find out Bayesian Optimization has more hyperparameters than your machine learning algorithm, get frustrated, and go back to using guess and check or grid search.**Local Optimization With a Good Initial Guess**: This is what MITIE does, it uses the BOBYQA algorithm with a well chosen starting point. Since BOBYQA only finds the nearest*local*optima the success of this method is heavily dependent on a good starting point. In MITIE's case we know a good starting point, but this isn't a general solution since usually you won't know a good starting point. On the plus side, this kind of method is extremely good at finding a local optima. I'll have more to say on this later.

The vast majority of people just do guess and check. That sucks and there should be something better. We all want some black-box optimization strategy like Bayesian optimization to be useful, but in my experience, if you don't set its hyperparameters to the right values it doesn't work as well as an expert doing guess and check. Everyone I know who has used Bayesian optimization has had the same experience. Ultimately, if I think I can do better hyperparameter selection manually then that's what I'm going to do, and most of my colleagues feel the same way. The end result is that I don't use automated hyperparameter selection tools most of the time, and that bums me out. I badly want a parameter-free global optimizer that I can trust to do hyperparameter selection.

So I was very excited when I encountered the paper

\[ \newcommand{\norm}[1]{\left\lVert#1\right\rVert} U(x) = \min_{i=1\dots t} (f(x_i) + k \cdot \norm{x-x_i}_2 ) \] Where $k$ is the Lipschitz constant for $f(x)$. Therefore, it is trivially true that $U(x) \geq f(x), \forall x$, by the definition of the Lipschitz constant. The authors go on to suggest a simple algorithm, called LIPO, that picks points at random, checks if the upper bound for the new point is better than the best point seen so far, and if so selects it as the next point to evaluate. For example, the figure below shows a plot of a simple $f(x)$ in red with a plot of its associated upper bound $U(x)$ in green. In this case $U(x)$ is defined by 4 points, indicated here with little black squares.

It shouldn't take a lot of imagination to see how the upper bound helps you pick good points to evaluate. For instance, if you selected the max upper bound as the next iterate you would already get pretty close to the global maximizer. The authors go on to prove a bunch of nice properties of this method. In particular, they both prove mathematically and show empirically that the method is better than random search in a number of non-trivial situations. This is a fairly strong statement considering how competitive random hyperparameter search turns out to be relative to competing hyperparameter optimization methods. They also compare the method to other algorithms like Bayesian optimization and show that it's competitive.

But you are probably thinking: "Hold on a second, we don't know the value of the Lipschitz constant $k$!". This isn't a big deal since it's easily estimated, for instance, by setting $k$ to the largest observed slope of $f(x)$ before each iteration. That's equivalent to solving the following easy problem:

\begin{align}

\min_{k} & \quad k^2 \\

\text{s.t.} & \quad U(x_i) \geq f(x_i), \quad \forall i \in [1\dots t] \\

& \quad k \geq 0

\end{align} Malherbe et al. test a variant of this $k$ estimation approach and show it works well.

This is great. I love this paper. It's proposing a global optimization method called LIPO that is both parameter free and provably better than random search. It's also really simple. Reading this paper gives you one of those "duah" moments where you wonder why you didn't think of this a long time ago. That's the mark of a great paper. So obviously I was going to add some kind of LIPO algorithm to dlib, which I did in the recent dlib v19.8 release.

However, if you want to use LIPO in practice there are some issues that need to be addressed. The rest of this blog post discusses these issues and how the dlib implementation addresses them. First, if $f(x)$ is noisy or discontinuous even a little it's not going to work reliably since $k$ will be infinity. This happens in real world situations all the time. For instance, evaluating a binary classifier against the 0-1 loss gives you an objective function with little discontinuities anywhere samples switch their predicted class. You could cross your fingers and run LIPO anyway, but you run the very real risk of two $x$ samples closely straddling a discontinuity and causing the estimated $k$ to explode. Second, not all hyperparameters are equally important, some hardly matter while small changes in others drastically affect the output of $f(x)$. So it would be nice if each hyperparameter got its own $k$. You can address these problems by defining the upper bound $U(x)$ as follows:

\[ U(x) = \min_{i=1\dots t} \left[ f(x_i) + \sqrt{\sigma_i +(x-x_i)^\intercal K (x-x_i)} \ \right] \] Now each sample from $f(x)$ has its own noise term, $\sigma_i$, which should be 0 most of the time unless $x_i$ is really close to a discontinuity or there is some stochasticity. Here, $K$ is a diagonal matrix that contains our "per hyperparameter Lipschitz $k$ terms". With this formulation, setting each $\sigma$ to 0 and $K=k^2I$ gives the same $U(x)$ as suggested by Malherbe et al., but if we let them take more general values we can deal with the above mentioned problems.

Just like before, we can find the parameters of $U(x)$ by solving an optimization problem:

\begin{align}

\min_{K,\sigma} & \quad \norm{K}^2_F + 10^6 \sum_{i=1}^t {\sigma_i^2} &\\

\text{s.t.} & \quad U(x_i) \geq f(x_i), & \quad \forall i \in [1\dots t] \\

& \quad \sigma_i \geq 0 & \quad \forall i \in [1\dots t] \\

& \quad K_{i,j} \geq 0 & \quad \forall i,j \in [1\dots d] \\

& \quad \text{K is a diagonal matrix}

\end{align} The $10^6$ penalty on $\sigma^2$ causes most $\sigma$ terms to be exactly 0. The behavior of the whole algorithm is insensitive to the particular penalty value used here, so long as it's reasonably large the $\sigma$ values will be 0 most of the time while still preventing $k$ from becoming infinite, which is the behavior we want. It's also possible to rewrite this as a big quadratic programming problem and solve it with a dual coordinate descent method. I'm not going into the details here. It's all in the dlib code for those really interested. The TL;DR is that it turns out to be easy to solve using well known methods and it fixes the infinite $k$ problem.

The final issue that needs to be addressed is LIPO's terrible convergence in the area of a local maximizer. So while it's true that LIPO is great at getting onto the tallest peak of $f(x)$, once you are there it does not make very rapid progress towards the optimal location (i.e. the very top of the peak). This is a problem shared by many derivative free optimization algorithms, including MATLAB's Bayesian optimization tool. Fortunately, not all methods have this limitation. In particular, the late and great Michael J. D. Powell wrote a series of papers on how to apply classic trust region methods to derivative free optimization. These methods fit a quadratic surface around the best point seen so far and then take the next iterate to be the maximizer of that quadratic surface within some distance of the current best point. So we "trust" this local quadratic model to be accurate within some small region around the best point, hence the name "trust region". The BOBYQA method I mentioned above is one of these methods and it has excellent convergence to the nearest local optima, easily finding local optima to full floating point precision in a very small number of steps.

We can fix LIPO's convergence problem by combining these two methods, LIPO will explore $f(x)$ and quickly find a point on the biggest peak. Then a Powell style trust region method can efficiently find the exact maximizer of that peak. The simplest way to combine these two things is to alternate between them, which is what dlib does. On even iterations we pick the next $x$ according to our upper bound while on odd iterations we pick the next $x$ according to the trust region model. I've also used a slightly different version of LIPO that I'm calling MaxLIPO. Recall that Malherbe et al. suggest selecting any point with an upper bound larger than the current best objective value. However, I've found that selecting the maximum upper bounding point on each iteration is slightly better. This alternative version, MaxLIPO, is therefore what dlib uses. You can see this hybrid of MaxLIPO and a trust region method in action in the following video:

In the video, the red line is the function to be optimized and we are looking for the maximum point. Every time the algorithm samples a point from the function we note it with a little box. The state of the solver is determined by the global upper bound $U(x)$ and the local quadratic model used by the trust region method. Therefore, we draw the upper bounding model as well as the current local quadratic model so you can see how they evolve as the optimization proceeds. We also note the location of the best point seen so far by a little vertical line.

You can see that the optimizer is alternating between picking the maximum upper bounding point and the maximum point according to the quadratic model. As the optimization proceeds, the upper bound becomes progressively more accurate, helping to find the best peak to investigate, while the quadratic model quickly finds a high precision maximizer on whatever peak it currently rests. These two things together allow the optimizer to find the true global maximizer to high precision (within $\pm{10^{-9}}$ in this case) by the time the video concludes.

So I was very excited when I encountered the paper

*Global optimization of Lipschitz functions*by Cédric Malherbe and Nicolas Vayatis in this year's international conference on machine learning. In this paper, they propose a very simple*parameter-free and provably correct method*for finding the $x \in \mathbb{R}^d$ that maximizes a function, $f(x)$, even if $f(x)$ has many local maxima. The key idea in their paper is to maintain a piecewise linear upper bound of $f(x)$ and use that to decide which $x$ to evaluate at each step of the optimization. So if you already evaluated the points $x_1, x_2, \cdots, x_t$ then you can define a simple upper bound on $f(x)$ like this:\[ \newcommand{\norm}[1]{\left\lVert#1\right\rVert} U(x) = \min_{i=1\dots t} (f(x_i) + k \cdot \norm{x-x_i}_2 ) \] Where $k$ is the Lipschitz constant for $f(x)$. Therefore, it is trivially true that $U(x) \geq f(x), \forall x$, by the definition of the Lipschitz constant. The authors go on to suggest a simple algorithm, called LIPO, that picks points at random, checks if the upper bound for the new point is better than the best point seen so far, and if so selects it as the next point to evaluate. For example, the figure below shows a plot of a simple $f(x)$ in red with a plot of its associated upper bound $U(x)$ in green. In this case $U(x)$ is defined by 4 points, indicated here with little black squares.

It shouldn't take a lot of imagination to see how the upper bound helps you pick good points to evaluate. For instance, if you selected the max upper bound as the next iterate you would already get pretty close to the global maximizer. The authors go on to prove a bunch of nice properties of this method. In particular, they both prove mathematically and show empirically that the method is better than random search in a number of non-trivial situations. This is a fairly strong statement considering how competitive random hyperparameter search turns out to be relative to competing hyperparameter optimization methods. They also compare the method to other algorithms like Bayesian optimization and show that it's competitive.

But you are probably thinking: "Hold on a second, we don't know the value of the Lipschitz constant $k$!". This isn't a big deal since it's easily estimated, for instance, by setting $k$ to the largest observed slope of $f(x)$ before each iteration. That's equivalent to solving the following easy problem:

\begin{align}

\min_{k} & \quad k^2 \\

\text{s.t.} & \quad U(x_i) \geq f(x_i), \quad \forall i \in [1\dots t] \\

& \quad k \geq 0

\end{align} Malherbe et al. test a variant of this $k$ estimation approach and show it works well.

This is great. I love this paper. It's proposing a global optimization method called LIPO that is both parameter free and provably better than random search. It's also really simple. Reading this paper gives you one of those "duah" moments where you wonder why you didn't think of this a long time ago. That's the mark of a great paper. So obviously I was going to add some kind of LIPO algorithm to dlib, which I did in the recent dlib v19.8 release.

However, if you want to use LIPO in practice there are some issues that need to be addressed. The rest of this blog post discusses these issues and how the dlib implementation addresses them. First, if $f(x)$ is noisy or discontinuous even a little it's not going to work reliably since $k$ will be infinity. This happens in real world situations all the time. For instance, evaluating a binary classifier against the 0-1 loss gives you an objective function with little discontinuities anywhere samples switch their predicted class. You could cross your fingers and run LIPO anyway, but you run the very real risk of two $x$ samples closely straddling a discontinuity and causing the estimated $k$ to explode. Second, not all hyperparameters are equally important, some hardly matter while small changes in others drastically affect the output of $f(x)$. So it would be nice if each hyperparameter got its own $k$. You can address these problems by defining the upper bound $U(x)$ as follows:

\[ U(x) = \min_{i=1\dots t} \left[ f(x_i) + \sqrt{\sigma_i +(x-x_i)^\intercal K (x-x_i)} \ \right] \] Now each sample from $f(x)$ has its own noise term, $\sigma_i$, which should be 0 most of the time unless $x_i$ is really close to a discontinuity or there is some stochasticity. Here, $K$ is a diagonal matrix that contains our "per hyperparameter Lipschitz $k$ terms". With this formulation, setting each $\sigma$ to 0 and $K=k^2I$ gives the same $U(x)$ as suggested by Malherbe et al., but if we let them take more general values we can deal with the above mentioned problems.

Just like before, we can find the parameters of $U(x)$ by solving an optimization problem:

\begin{align}

\min_{K,\sigma} & \quad \norm{K}^2_F + 10^6 \sum_{i=1}^t {\sigma_i^2} &\\

\text{s.t.} & \quad U(x_i) \geq f(x_i), & \quad \forall i \in [1\dots t] \\

& \quad \sigma_i \geq 0 & \quad \forall i \in [1\dots t] \\

& \quad K_{i,j} \geq 0 & \quad \forall i,j \in [1\dots d] \\

& \quad \text{K is a diagonal matrix}

\end{align} The $10^6$ penalty on $\sigma^2$ causes most $\sigma$ terms to be exactly 0. The behavior of the whole algorithm is insensitive to the particular penalty value used here, so long as it's reasonably large the $\sigma$ values will be 0 most of the time while still preventing $k$ from becoming infinite, which is the behavior we want. It's also possible to rewrite this as a big quadratic programming problem and solve it with a dual coordinate descent method. I'm not going into the details here. It's all in the dlib code for those really interested. The TL;DR is that it turns out to be easy to solve using well known methods and it fixes the infinite $k$ problem.

The final issue that needs to be addressed is LIPO's terrible convergence in the area of a local maximizer. So while it's true that LIPO is great at getting onto the tallest peak of $f(x)$, once you are there it does not make very rapid progress towards the optimal location (i.e. the very top of the peak). This is a problem shared by many derivative free optimization algorithms, including MATLAB's Bayesian optimization tool. Fortunately, not all methods have this limitation. In particular, the late and great Michael J. D. Powell wrote a series of papers on how to apply classic trust region methods to derivative free optimization. These methods fit a quadratic surface around the best point seen so far and then take the next iterate to be the maximizer of that quadratic surface within some distance of the current best point. So we "trust" this local quadratic model to be accurate within some small region around the best point, hence the name "trust region". The BOBYQA method I mentioned above is one of these methods and it has excellent convergence to the nearest local optima, easily finding local optima to full floating point precision in a very small number of steps.

We can fix LIPO's convergence problem by combining these two methods, LIPO will explore $f(x)$ and quickly find a point on the biggest peak. Then a Powell style trust region method can efficiently find the exact maximizer of that peak. The simplest way to combine these two things is to alternate between them, which is what dlib does. On even iterations we pick the next $x$ according to our upper bound while on odd iterations we pick the next $x$ according to the trust region model. I've also used a slightly different version of LIPO that I'm calling MaxLIPO. Recall that Malherbe et al. suggest selecting any point with an upper bound larger than the current best objective value. However, I've found that selecting the maximum upper bounding point on each iteration is slightly better. This alternative version, MaxLIPO, is therefore what dlib uses. You can see this hybrid of MaxLIPO and a trust region method in action in the following video:

In the video, the red line is the function to be optimized and we are looking for the maximum point. Every time the algorithm samples a point from the function we note it with a little box. The state of the solver is determined by the global upper bound $U(x)$ and the local quadratic model used by the trust region method. Therefore, we draw the upper bounding model as well as the current local quadratic model so you can see how they evolve as the optimization proceeds. We also note the location of the best point seen so far by a little vertical line.

You can see that the optimizer is alternating between picking the maximum upper bounding point and the maximum point according to the quadratic model. As the optimization proceeds, the upper bound becomes progressively more accurate, helping to find the best peak to investigate, while the quadratic model quickly finds a high precision maximizer on whatever peak it currently rests. These two things together allow the optimizer to find the true global maximizer to high precision (within $\pm{10^{-9}}$ in this case) by the time the video concludes.

The Holder Table Test Functionfrom https://en.wikipedia.org/wiki/File:Holder_table_function.pdf |

Now let's do an experiment to see how this hybrid of MaxLIPO and Powell's trust region method (TR) compares to MATLAB's Bayesian optimization tool with its default settings. I ran both algorithms on the Holder table test function 100 times and plotted the average error with one standard deviation error bars. So the plot below shows $f(x^\star)-f(x_i)$, the difference between the true global optimum and the best solution found so far, as a function of the number of calls to $f(x)$. You can see that MATLAB's BayesOpt stalls out at an accuracy of about $\pm{10^{-3}}$ while our hybrid method (MaxLIPO+TR, the new method in dlib) quickly approaches full floating point precision of around $\pm{10^{-17}}$.

I also reran some of the tests from Figure 5 of the LIPO paper. The results are shown in the table below. In these experiments I compared the performance of LIPO with and without the trust region solver (LIPO+TR and LIPO). Additionally, to verify that LIPO is better than pure random search I tested a version of the algorithm that alternates between pure random search and the trust region solver (PRS+TR) rather than alternating between a LIPO method and a trust region solver (LIPO+TR and MaxLIPO+TR). Pure random search (PRS) is also included for reference. Finally, the new algorithm implemented in dlib, MaxLIPO+TR, is included as well. In each test I ran the algorithm 1000 times and recorded the mean and standard deviation of the number of calls to $f(x)$ required to reach a particular solution accuracy. For instance, $\epsilon=0.01$ means that $f(x^\star)-f(x_i) \leq 0.01$, while "target 99%" uses the "target" metric from Malherbe's paper, which for most tests corresponds to an $\epsilon > 0.1$. Tests that took too long to execute are noted with a - symbol.

The key points to notice about these results are that the addition of a trust region method allows LIPO to reach much higher solution accuracy. It also makes the algorithm run faster. Recall that LIPO works internally by using random search of $U(x)$. Therefore, the number of calls LIPO makes to $U(x)$ is at least as many as PRS would require when searching $f(x)$. So for smaller $\epsilon$ it becomes very expensive to execute LIPO. For instance, I wasn't able to get results for LIPO, by itself, at accuracies better than $0.1$ on any of the test problems since it took too long to execute. However, with a trust region method the combined algorithm can easily achieve high precision solutions. The other significant detail is that, for tests with many local optima, all methods combining LIPO with TR are much better than PRS+TR. This is most striking on ComplexHolder, which is a version of the HolderTable test function with additional high frequency sinusoidal noise that significantly increases the number of local optima. On ComplexHolder, LIPO based methods require about an order of magnitude fewer calls to $f(x)$ than PRS+TR, further justifying the claims by Malherbe et al. of the superiority of LIPO relative to pure random search.

The new method in dlib, MaxLIPO+TR, fares the best in all my tests. What is remarkable about this method is its simplicity. In particular,

**MaxLIPO+TR doesn't have any hyperparameters, making it very easy to use.**I've been using it for a while now for hyperparameter optimization and have been very pleased. It's the first black-box hyperparameter optimization algorithm I've had enough confidence in to use on real problems.

Finally, here is an example of how you can use this new optimizer from Python:

def holder_table(x0,x1): return -abs(sin(x0)*cos(x1)*exp(abs(1-sqrt(x0*x0+x1*x1)/pi))) x,y = dlib.find_min_global(holder_table, [-10,-10], # Lower bound constraints on x0 and x1 respectively [10,10], # Upper bound constraints on x0 and x1 respectively 80) # The number of times find_min_global() will call holder_table()

```
```

Or in C++11:

auto holder_table = [](doublex0,doublex1){return -abs(sin(x0)*cos(x1)*exp(abs(1-sqrt(x0*x0+x1*x1)/pi)));}; // obtain result.x and result.y auto result = find_min_global(holder_table,{-10,-10}, // lower bounds{10,10}, // upper bounds max_function_calls(80));

Both of these methods find holder_table's global optima to about 12 digits of precision in about 0.1 seconds. The C++ API exposes a wide range of ways to call the solver, including optimizing multiple functions at a time and adding integer constraints. See the documentation for full details.

## 99 comments :

The first best read (including the references) in my 2018. Thanks!

Thanks :)

Holy Cow, this is amazing. Thanks for sharing it in such an infromative and detailed manner. Thanks for making 2018 cool again.

No problem :)

Fantastic work! I enormously appreciate that you take the time to make things work for real, combining multiple methods if necessary.

I'm quite interested in knowing the practical limitations of this approach, you mentioned discontinuities and noise, when applied to the objective functions of neural networks. Would it work for a some NN models - what maximum network size? Is the dimensionality of the function a problem?

Thank you!

Thanks :)

You definitely can't use this to find the parameters of a NN. I wouldn't attempt to optimize functions with more than 10s of parameters with a derivative free optimizer. Even that is pushing it. You really need to know something about the structure of your objective function if you want to optimize something with large numbers of variables. In particular, you need to know the gradient at least.

Consider for example that a 10000 dimensional objective function requires 10001 evaluations to even estimate the gradient. That's a huge problem for a derivative free optimizer.

Great read!! Thank you!

One question: Would it be feasible to enforce some restrictions on the parameters e.g. take only integer values or support categorical features? I guess I could apply them inside the objective function (e.g. through rounding and by transforming numeric values to categories), but would this make sense?

excellent work.I compare the algrithm with AdamGradient and rms method ,find it is much better optimizer when the effictive value range is narrow.

i test the function 1/(1+exp(100*x))+1/(1+exp(-100*(x-0.05))), and get the resault following:

global optimal inputs: 0.0

global optimal output: 0.5066928509242848

adam optimal inputs: nan

adam optimal output: nan

here is my code:

import tensorflow as tf

import dlib

import math

def test_fun(x0):

return 1/(1+math.exp(100*x0))+1/(1+math.exp(-100*(x0-0.05)))

def test_fun_tf(x0):

return 1/(1+tf.exp(100*x0))+1/(1+tf.exp(-100*(x0-0.05)))

x,y = dlib.find_min_global(test_fun,[-6],[6],10)

print("global optimal inputs: {}".format(x[0]))

print("global optimal output: {}".format(y))

t_x0=tf.Variable(tf.random_uniform((),-2,2))

cost=test_fun_tf(t_x0)

opt=tf.train.AdamOptimizer(1e-6).minimize(cost)

with tf.Session() as sess:

sess.run(tf.global_variables_initializer())

for i in range(10):

loss,_,x=sess.run([cost,opt,t_x0])

print("adam optimal inputs: {}".format(x))

print("adam optimal output: {}".format(loss))

The code has an option to say some variables are integers. This just removes those variables from consideration by the trust region solver, meaning their optimization is then up to just LIPO, which is most likely what you want to do.

The variabeles are all float. Actually, when I expand the trust region, adam gonna work. But still not good as global method.

----function-----

def test_fun(x0):

return 1/(1+math.exp(10*x0))+1/(1+math.exp(-10*(x0-0.05)))

def test_fun_tf(x0):

return 1/(1+tf.exp(10*x0))+1/(1+tf.exp(-10*(x0-0.05)))

----result-----

global optimal inputs: 0.02497796370782311

global optimal output: 0.875646999714699

adam optimal inputs: 0.052661534398794174

adam optimal output: 0.8779600858688354

Hi happy new year

i wanted to build dlib with cuda and cudnn

using cmake it says

"Building a CUDA test project to see if your compiler is compatible with CUDA...

CUDA was found but your compiler failed to compile a simple CUDA program so dlib isn't going to use CUDA "

i am using mingw530_32

what is wrong with this compiler?

You have to use a compiler supported by CUDA. Like visual studio 2015. You should refer to the NVIDIA documentation for complete details.

This is a great write-up and excellent analysis! Thank you.

Thanks :)

Btw, is MaxLIPO without finding k equivalent to a GP with acquisition function? Looks very similar...

Yeah, I'm sure there is a way to write it as a GP, like by selecting the right sparse kernel or something. I haven't thought about how you would do it though.

I'm not sure I follow your question. The optimization problem you need to solve to find k is a quadratic program so you can use any quadratic program solver. Maybe you are asking about the sqrt() in the constraints? If you just square both sides of the constraints with sqrt() in them you get linear constraints.

Btw, have you seen this: https://github.com/chrisstroemel/Simple looks quite similar and kinda cool.

No, haven't seen that. Seems like a reasonable thing to do. There are a lot of these partitioning algorithms that don't get as much love as they probably deserve.

Interesting and elegant work. I'll definitely try it out on a real problem.

For the Python API call

find_min_global, in addition to the optimum parameters and value, is there a way to get all the parameters the algorithm has tried and their corresponding objective values?Thanks.

You can add whatever kind of logging code you want to the function find_min_global() calls. You define that function so you can put any code in it that you like.

Makes sense. Thanks Davis.

Nice explanation!. Are you using uniform distribution when randomly drawing the next candidate for evaluation?

Thanks. Yes, that's what it does.

I am wondering how this method scales as the search space gets very big. I am worried that the number of constraints in the QP will get too large. What approaches do you recommend to solve this problem?

It's fine. Solving the QP is not the bottleneck since constraints become inactive and stay inactive and can be pruned. The bigger issue is the overall quadratic runtime due to evaluations of U(x) having linear runtime.

Anyway, try calling it. It's plenty fast and there is a lot of room in the code to make it run even faster. But for hyper-parameter search it's way faster than it needs to be already.

Hi, I'm reading the paper "Global optimization of Lipschitz functions" recently and have a question about it. Hope you can shed some light on it.

In the AdaLIPO algorithm, the last line of Step 2 in Figure 4, they set kt_hat as the infimum of the set {ki}. According to this, the resulting kt_hat will only satisfy the Lipschitz inequality for a particular Xi paired with all the other Xj (where j != i). It does NOT satisfy the Lipschitz inequality for all pairs of (Xi, Xj) since one only pick the infimum (minimum). Am I reading it wrong or do they have a good reason for this that I'm missing?

Thanks for your attention.

No, that's not what it says. It says they pick the smallest k that satisfies the Lipschitz condition everywhere.

I just realized that my confusion originated from their notation. In that formula, they have both k_i and X_i, but these are actually not the same "i"s.

Hello Davis,

Great work with the library. This could be a major relief for anyone who has to deal with parameter tuning.

I have a question for you. From reading your post, the method seems to be a

sequential, direction based method. Is that correct? We are looking to automate some of the model building process and hyper-parameter tuning is one of them and so I wanted to understand if it is possible to distribute the computations of the algorithm across nodes.So, given the nature of the algorithm, is it possible to have a distributed version of this?

Yes, you can use it with parallel function evaluations. I specifically made the API in a way that supports that. Read the documentation here http://dlib.net/optimization.html#global_function_search and also click on the more details button and read the documentation. There is a lot of discussion of this topic there. The entire API is designed around the goal of supporting parallel function evaluation.

Hi Davis,

You mentioned that in your MaxLIPO version of LIPO you select the maximum upper bounding point, i.e., you are maximizing U(x). How exactly do you do this? Thank you.

If you square both sides of the constraints in the QP shown in the post you get a QP in canonical form and can use any QP solver you want to solve it. That gives you U(x).

Super interesting. I was going to try to use this to optimize some neural network hyperparameters (LR, momentum, etc) -- but I was wondering how well you'd expect this to work when there's variation in the function evaluation. Any thoughts?

Thanks

That's exactly what this optimizer is for. So it will work fine.

Hi Davis,

Thanks for the great work

I am doing a port to scala so that it can be used in spark env, would you mind helping me understand some of the issues:

https://github.com/davisking/dlib/blob/master/dlib/global_optimization/upper_bound_function.h#L217 trainer.force_last_weight_to_1(true);

any reason to set this? I am using liblinear-java, which does not have force_last_weight_to_1

https://github.com/bwaldvogel/liblinear-java/

Thanks!

Yes, you need that.

Why don't you just call dlib? It's not hard to call C++ from spark.

I would love to, however, in the spark mode, each model may run on one or multiple executors, not sure how to wrap it with function evaluation

Great blog Davis and very intriguing! The solver part of delegating it back to solve a similar SVM problem is so elegant!

One question though, assuming after we estimate out a good U(x) with minimized K in each step; how should we decide the next evaluation point? Looks to me that in one-dimensional case, all candidate points are within a fixed set: {x|x=(xi + xj) / 2}, so it's only O(n^2) complexity to query U(x). How about higher dimensional cases? Can we still only evaluate those boundary points? Or maybe we need to do random sampling?

I just evaluate 5000 random points and take the best. I thought about how you might try to optimize U(x) exactly, but I'm not sure there is an efficient way. And random search of U(x) is plenty good enough.

Hi Davis! Thank you very much for this impressive algorithm. I was recently struggling with Bayesian Optimization and its parameters, so I definitively got what you were talking about in the first paragraphs of this article.

However, I'm wondering if it possible to save the "current state" of the optimization process. BayesOpt (the C++ lib) has a very nice functionality that allows to save the current sampled points as well as the parameters of the optimization, being able to resume it later; I think that something similar could be possible in this case saving the current bounds.

I haven't explored DLib fully (I just discovered it some weeks ago), so I don't know if this is currently implemented, but it would be very useful if it is not.

Thanks!

Thanks, glad you like it.

Yes, you can do this. See the extended discussion here: http://dlib.net/dlib/global_optimization/global_function_search_abstract.h.html#global_function_search. That's the C++ interface. There is a python interface with essentially an identical API as well.

Hi

Excellent work, thank you for sharing.

Two questions:

1) Can this be combined with hyperband https://archive.is/CDVR8 ?

2) How does this compare to HORD / pysot (https://archive.is/YgRvr)? Do you have any comment on that approach?

Greetings from London

You could combine the trust region strategy with pretty much any other derivative free solver. I used LIPO here because I find LIPO the most compelling. I have not compared it to HORD.

Is it possible to use this brilliant optimizer in the training of neural network of machine learning? Is this optimzer is something that could replace traditional optimizer like Adam,AdaGrad...?

No, it's not going to be reasonable to use this in place of adam or sgd.

Hi Davis

Apologies in advance if some of these is very basic.

RE:

"You could combine the trust region strategy with pretty much any other derivative free solver."

Just to be 100% clear: "the trust region strategy" is *itself* derivative-free (looking at BOBYQA definition) - is my understanding right?

RE:

Q: "Is it possible to use this brilliant optimizer in the training of neural network of machine learning? Is this optimzer is something that could replace traditional optimizer like Adam,AdaGrad...?"

A: "No, it's not going to be reasonable to use this in place of adam or sgd."

Is this true only of neural network hyperparam optimisations, or more in genearl of machine learning hyperparam optimisation problems?

Would you mind expanding on this?

Reiterating my gratitude for your excellent work.

Yes, it's all derivative free.

This kind of algorithm only works for problems with a relatively small number of parameters. There is no way it could possibly optimize a problem with 10s of thousands of parameters (or more, some DNNs have millions to billions of parameters). This is true for any derivative free algorithm. It's just not going to work.

But if you have something like 5 parameters you want to optimize then it's great.

How can I pass dynamic array to bound1 and bound2? All my approaches are failing...

Hi Davis!

Thank you for your amazing algorithm!

I have question for your suggestion using it for hyperparameter tuning:

"Shouldn't we know first that the neural network we are using is

a Lipschitz function with input its hyperparameters?"

There is already work done proving that neural nets are Lipschitz functions,

but not with input their hyperparameters. At least I cannot find any work

that proves so...

Thanks!

Is there an easy way to visualise the optimisation/surrogate surface? Something similar to what one gets out of skopt plot_objective, plot_evaluations functions?

Hi Davis! Thank you very much for the sharing.

I am trying to use your algorithm to tune a Machine Learning program that takes a day to get one data point (each with a specific set of hyper-parameters) . I've already a few data point and would like to know if there is an option in your function to incorporate these data points to predefine the upper bound U(x) (rather than starting from a random point)?

Also, is there an option to print out/save the value of the parameter x_i and result y obtained in each iteration so that I can decide whether I should iterate more.

My calculation just takes so long to run and those features will help a lot!

Thank you.

Sorry for the late reply, for some reason I haven't been getting notifications of new commends on the blog.

Yes, you can do all those more complex use cases by using the global_function_search class directly rather than using find_min_global(). See the documentation for global_function_search.

No, there is no built in tool to visualize the surrogate surfaces, if you want to do that it's on you. I had hacked together something to make the video from this blog post, but the code is not readily reusable, so I'm not sharing it. It would be easier for someone to write it from scratch than figure out how to use that bit of hacky code.

The extensions I made to the algorithm make it work with things that are not lipschitz. There is a whole discussion in the blog post about functions that are discontinuous. Discontinuous functions are not Lipschitz.

Hi Davis,

For a global optimization algorithm, this seems extremely easy to use, and seems to converge fast (from your tests upto 5 dimensions). I have a CFD optimization problem with 10 geometric design variables, and am considering applying this method for finding the global maximum in efficiency. However, since the CFD's are quite computationally expensive, I am trying to be sure it would work well in higher dimensions. Do you have some test results in the range of 6-10 dimensions that you can share with us?

Thanks :)

I don't have any numbers to share, but I've used it plenty on problems of 10 dimension and it works fine. There isn't anything special about 5 vs 10 dimensions.

Hello, I have some trouble using the dlib.find_min_global in my NN model...

I have setup a class that creates a model object and attaches it to 'self'. Then, a method is called where this model object is trained using self.model_obj.fit_generator from Keras.

I have made it in such a way that this method needs additional parameters like learning rate, and others, which I want to approximate using this dlib function.

However, there is a 'self' argument in the method by the nature of my class... and I cant seem to find a way to have dlib.find_min_global ignore this first argument in its optimizaiton. Do you have an idea maybe?

For now I have set all the additional parameters (put in a dict) as a global variable. It definitely solve my problem.. but it doesnt 'feel' that great.

Could you think of a better solution? (where "better" here is quite arbitrary)

I was going to steal the wonderful video for an interview presentation, but it now appears to be corrupted... downloading didn't work either :(

The video is still there when I look at it. It's hosted on dlib.net. There are even two versions, both of which seem fine: http://dlib.net/find_max_global_example.mp4 and http://dlib.net/find_max_global_example.webm

As for zwep's question, use a closure or lambda function.

Thanks Davis, the MP4 version is working well 👍

Sweet :)

I got the job, by the way \o/

For reference and until I do more investigation and a write-up, this optimizer seems to work very well with XGBoost optimizing over five hyperparameters. It can usually find an excellent candidate for a global optimum in considerably less than 50 function calls.

Awesome :)

In the animated (video) version of the algorithm, how do you obtain the location of the 4th point?

The first few points are just picked randomly.

What is the stopping criteria of the algorithm?

There isn't any stopping criteria. You tell it how many times it is allowed to call the objective function and it runs that many calls and returns the best thing its found.

Isn't that a hyperparameter the user has to set? I though the method was supposed to be parameter free as far as the user is concerned Or is that somehow hard-coded in the algorithm or maybe it has some strategy to determine the number of necessary iterations?

It means you tell it how long you are willing to wait.

It’s impossible, in general, to know if you have found the optimal solution. So you run as long as your problem allows and get the best solution available.

Hi again Davis :)

> The first few points are just picked randomly.

Is there any way, using the Python binding, to specify a seed for this random process so that the subsequent search is (presumably) repeatable?

Gah, I can see now that the seed appears to be fixed. Can it be specified anywhere that might make it easy to add into the Python binding?

Yes. This is all described in the docshttp://dlib.net/dlib/global_optimization/global_function_search_abstract.h.html#global_function_search

Hi Davis,

Great work & article.

I was trying to use the python API on a problem with ~18 variables, but I get an error msg saying "Functions being optimized must take between 1 and 15 scalar arguments."

Is there a way around this ? very much appreciate your help (the optimizer does very well on 14 variable problem .. should probably work fine with a few more).

Thank you

The C++ API lets you use as many variables as you want by passing a vector of variables. The Python API takes user defined functions with explicit scalar arguments and is only overloaded for up to 15 scalars though. The python API should probably also allow giving a list of parameters in addition to explicit positional arguments too. Maybe I'll update that later. For the moment I just increased the overloading to allow up to 35 parameters and it's updated on github now.

Thank you ! appreciate the quick response

Hi Davis,

Would the latest version with 35 parameters be available on pypi ? pip install makes life much easier when integrating with other tools.

Totally understand if you prefer not to push it to pypi for now.

Thank you

It will be on pypi eventually when the next dlib version is released. For now you can just run python setup.py install and it will do the same thing.

Hello Davis,

Thank you so much for this post. Could you please help me understand whether I can use this method with nested hyperparameters? i.e., constraining a child hyperparameter to be active and evaluated only when the parent hyperparameter is active. Thank you very much.

Yeah, it should work alright for that.

I wanted to let you know that find_min_global is currently outperforming a dozen popular global optimizers.

https://github.com/microprediction/optimizer-elo-ratings/tree/main/results/leaderboards/overall

I'm really impressed and grateful for this library.

Sweet, glad to hear it :)

Thank you for lipo. I am using `from lipo import GlobalOptimizer` in Python and was wondering if there is any documentation for `categories` and `evaluations`. My guess is that evaluations sets some fixed points to evaluate at first but what is categories for?

That lipo package is news to me. I think you should use dlib directly if you want to use the optimizer in dlib (which is what this post is about).

Hello Davis,

thank you for your great software! I am seeing from the documentation that find_min_global allows concurrent function calls on multiple threads: can this be done also with its Python version? Thank you.

No, you can't do it with python because python doesn't support threading.

Hi Davis,

I've been using BO to minimize a response surface which is from a surrogate model approximated by Gaussian Process Regression. The Acquisition function I adopted is Expected Improvement. The score I compare is the RMSE. The RMSE keeps decreasing with each step until a point then the RMSE is actually getting bigger. It might be that the hyperparams are not "good" enough.

I find this article after some Q&A on stackexchange. I wonder if dlib can be used with the surrogate model (from GPR) which does not have an exact expression? like the one in the python example: "-abs(sin(x0)*cos(x1)*exp(abs(1-sqrt(x0*x0+x1*x1)/pi)))"?

Or is BO the only option if I'm using a surrogate model? Thank you.

Yeah no deriviatives or or anything like that are required. You just need to provide the result of the function. E.g. dlib will say "what's the objective function output for inputs (1,2,3,4)?" and your code has to say "it's 5.678".

For future readers who have difficulties in install dlib:

-- Using CMake version: 3.24.1

-- Compiling dlib version: 19.24.0

...

Successfully installed dlib-19.24.0

YEAH!!

I fixed the installation on Windows 10. Only install Visual Studio is not sufficient, you have to install C++ explicitly by "Modify" then in the "Individual Components" tab of VS installer, under the name of "MSVC v140 - VS2015 C++ Build Tools (v14.00)" and needs to download 800 MB.

I also read from a thread that

"Use the link to Visual C++ 2015 Build Tools. That will install Visual C++ 14.0 without installing Visual Studio." >> see https://stackoverflow.com/questions/44951456/pip-error-microsoft-visual-c-14-0-is-required

========

Above is for Windows, still not sure the installation on Mac system.

Davis, thank you for making such high-quality work freely available for us all.

As a response to your post a while back,

"I just evaluate 5000 random points and take the best. I thought about how you might try to optimize U(x) exactly, but I'm not sure there is an efficient way. And random search of U(x) is plenty good enough."Here is a neat post about low discrepancy quasi-random sequences which may be worth investigating.

http://extremelearning.com.au/unreasonable-effectiveness-of-quasirandom-sequences/

I particularly like the elegance of their generation and they have really nice scaling properties in higher dimensions.

Quote from his site,

"The new R_d sequence is the only d-dimensional low discrepancy quasirandom sequence that does not require any selection of basis parameters."Thanks again for the great work

Jacques

Yeah, I should make this use a low discrepancy sequence. That would totally be a bit better :D

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