modeling contexts, the XGBoost algorithm reigns supreme. It offers efficiency and effectivity good points over different tree-based strategies and different boosting implementations. The XGBoost algorithm features a laundry record of hyperparameters, though often solely a subset is chosen in the course of the hyperparameter tuning course of. In my expertise, I’ve at all times used a grid search methodology utilizing k-fold cross-validation to determine the optimum mixture of hyperparameters, though there are different strategies for hyperparameter tuning with the hyperopt library that may search the hyperparameter house extra systematically.
By my work constructing XGBoost fashions throughout totally different tasks, I got here throughout the nice useful resource Effective XGBoost by Matt Harrison, a textbook masking XGBoost, together with find out how to tune hyperparameters. Chapter 12 of the e-book is devoted to tuning hyperparameters utilizing the hyperopt library; nonetheless, there have been some pure questions that arose upon studying the part. The introduction to the chapter offers a high-level overview of how utilizing hyperopt and Bayesian optimization offers a extra guided method for tuning hyperparameters in comparison with grid search. Nevertheless, I used to be curious, what’s going on right here below the hood?
As well as, as is the case with many tutorials about tuning XGBoost hyperparameters, the ranges for the hyperparameters appeared considerably arbitrary. Harrison explains that he pulled the record of hyperparameters to be tuned from a chat that information scientist Bradley Boehmke gave (here). Each Harrison and Boehmke present tutorials for utilizing hyperopt with the identical set of hyperparameters, though they use barely totally different search areas for locating an optimum mixture. In Boehmke’s case, his search house is way bigger; for instance, he recommends that the utmost depth for every tree (max_depth) be allowed to range between 1 and 100. Harrison had narrowed the ranges he presents in his e-book considerably, however these two circumstances led to the query: What’s the marginal acquire in comparison with the marginal enhance in time from growing the hyperparameter search house when tuning XGBoost fashions?
The aim of this text is centered on these two questions. First, we are going to discover how hyperopt works when tuning hyperparameters at a barely deeper stage to assist acquire some instinct for what’s going on below the hood. Second, we are going to discover the tradeoff between giant search areas and narrower search areas in a rigorous approach. I hope to reply these questions in order that this can be utilized as a useful resource for understanding hyperparameter tuning sooner or later.
All code for the challenge may be discovered on my GitHub web page right here: https://github.com/noahswan19/XGBoost-Hyperparameter-Analysis
hyperopt with Tree-Structured Parzen Estimators for Hyperparameter Tuning
Within the chapter of his textbook masking hyperopt, Harrison describes the method of utilizing hyperopt for hyperparameter tuning as utilizing “Bayesian optimization” to determine sequential hyperparameter mixtures to strive in the course of the tuning course of.
The high-level description makes it clear why utilizing hyperopt is a superior methodology to the grid search methodology, however I used to be curious how that is applied. What is definitely happening once we run the fmin perform utilizing the Tree-Structured Parzen (TPE) estimator algorithm?
Sequential Mannequin-Primarily based Optimization
To begin with, the TPE algorithm originates from a paper written in 2011 by James Bergstra, Rémi Bardenet, Yoshua Bengio, and Balázs Kégl, the authors of the hyperopt bundle referred to as “Algorithms for Hyper-Parameter Optimization”. The paper begins by introducing Sequential Mannequin-Primarily based Optimization (SMBO) algorithms, the place the TPE algorithm is one model of a broader SMBO methodology. SMBOs present a scientific approach to decide on the following hyperparameters to guage, avoiding the brute-force nature of grid search and the inefficiency of random search. It includes creating a “surrogate” mannequin for the underlying mannequin we’re optimizing for (i.e. XGBoost in our case), which we are able to use direct the seek for optimum hyperparameters in a approach that’s computationally cheaper than evaluating the underlying mannequin. The algorithm for an SMBO is described within the following picture:
There’s loads of symbols right here, so let’s break down each:
- x* and x: x* represents the hyperparameter mixture that’s being examined in a given trial, and x represents a common hyperparameter mixture.
- f: That is the “health perform” which is the underlying mannequin that we’re optimizing. Inside this algorithm, f(x*) is mapping a hyperparameter mixture x* to efficiency of this mix on a validation information set.
- M_0: The M phrases within the algorithm correspond to the “surrogate” mannequin we’re utilizing to approximate f. Since f is often costly to run, we are able to use a less expensive estimation, M, to assist determine which hyperparameter mixtures will seemingly enhance efficiency.
- H: The curly H corresponds to the historical past of hyperparameters searched thus far. It’s up to date upon each iteration. It is usually used to develop an up to date surrogate mannequin after every iteration.
- T: This corresponds to the variety of trials we use for hyperparameter tuning. That is fairly self-explanatory, however this corresponds to the max_evals argument within the fmin perform from hyperopt.
- S: The S corresponds to the criterion used to select a set of hyperparameter mixtures to test given a surrogate mannequin. Within the hyperopt implementation of the TPE algorithm, S corresponds to the Anticipated Enchancment (EI) criterion, described within the following picture.
Every iteration, some variety of doable hyperparameter mixtures are drawn (within the python hyperopt bundle, that is set to 24 by default). We are going to focus on in a bit how TPE signifies how these 24 are drawn. These 24 hyperparameter mixtures are evaluated utilizing the EI criterion and surrogate mannequin (which is cheap) to determine the one mixture that’s almost certainly to have the best efficiency. That is the place we see the advantages of the surrogate mannequin: as a substitute of coaching and evaluating 24 XGBoost fashions to guage the one greatest hyperparameter mixture, we are able to approximate this with a computationally cheap surrogate mannequin. Because the identify would counsel, the components above corresponds to the anticipated efficiency enchancment of a hyperparameter mixture x:
- max(y*-y,0): This represents the precise enchancment in efficiency for a hyperparameter mixture x. y* corresponds to the most effective validation loss we’ve attained up to now; we’re aiming to attenuate the validation loss, so we’re searching for values of y which might be lower than y*. This implies we wish to maximize EI in our algorithm.
- p_M(x|y): That is the piece of the criterion that will likely be approximated utilizing the surrogate mannequin and the piece the place TPE will slot in. That is the chance density for doable values for y given a hyperparameter mixture x.
So every spherical, we take a set of 24 hyperparameter mixtures, then proceed with the one which maximizes the EI criterion, which makes use of our surrogate mannequin M.
The place does the TPE algorithm slot in
The important thing piece of the SMBO algorithm that may range throughout implementations is the surrogate mannequin or how we approximate the success of hyperparameter features. Utilizing the EI criterion, the surrogate mannequin is required to estimate the density perform p(y|x). The paper talked about above introduces one methodology referred to as the Gaussian Course of Strategy which fashions p(y|x) instantly, however the TPE method (which is extra typically used for XGBoost hyperparameter optimization) as a substitute approximates p(x|y) and p(y). This method follows from Bayes theorem:
The TPE algorithm splits p(x|y) right into a piecewise mixture of two distributions:
- l(x) if y < y*
- g(x) if y ≥ y*
These two distributions have an intuitive understanding: l(x) refers back to the distribution of hyperparameters related to fashions which have a decrease loss (higher) than the most effective mannequin thus far whereas g(x) refers back to the distribution of hyperparameters related to fashions which have a better loss (worse) than the most effective mannequin thus far. This expression for p(y|x) is substituted into the equation for EI within the paper, and a mathematical derivation (that will be too verbose to interrupt down fully right here) arrives at the truth that maximizing EI is equal to selecting factors which might be extra seemingly below l(x) and fewer seemingly below g(x).
So how does this work in apply? When utilizing hyperopt, we use the fmin perform and provide the tpe.counsel algorithm to specify we need to use the TPE algorithm. We provide an area of hyperparameters the place every parameter is related to a uniform or log-uniform distribution. These preliminary distributions are used to initialize l(x) and g(x) and supply a previous distribution for l(x) and g(x) whereas working with a small variety of preliminary trials. By default (parameter n_startup_jobs in tpe.counsel), hyperopt runs 20 trials by randomly sampling hyperparameter mixtures from the distributions offered for the house parameter of fmin. For every of the 20 trials, an XGBoost mannequin is run and a validation loss obtained.
The 20 observations are then break up in order that two subsets are used to construct non-parametric densities for l(x) and g(x). Subsequent observations are used to replace these distributions. The densities are estimated utilizing a non-parametric methodology (which I’m not certified to explain absolutely) involving the prior distributions for every hyperparameter (that we specified) and particular person distributions for every statement from the trial historical past. Observations are break up into subsets utilizing a technique that modifications with the variety of complete trials run; the “n” observations with the bottom loss are used for l(x) with the remaining observations used for g(x). The “n” is set by multiplying a parameter gamma (default for tpe.counsel is 0.25) by the sq. root of the variety of trials and rounding up; nonetheless, a most for “n” is ready at 25 so l(x) will likely be parameterized with at most 25 values. If we use the default setting for tpe.counsel, then the most effective two observations (0.25 * sqrt(20) = 1.12 rounds to 2) from the preliminary trials are used to parameterize l(x) with the remaining 18 used for g(x). The 0.25 worth refers back to the gamma parameter in tpe.counsel which may be modified if desired. Wanting again to the pseudocode for the SMBO algorithm and the components for EI, if n observations are used to parameterize l(x), then the (n+1)th statement is the brink worth y*.
As soon as l(x) and g(x) are instantiated utilizing the start trials, we are able to transfer ahead with every analysis of our goal perform for the variety of max_evals that we specify for fmin. For every iteration, a set of candidate hyperparameter mixtures (24 by default in tpe.counsel however may be specified with n_EI_candidates) is generated by taking random attracts from l(x). Every of those mixtures is evaluated utilizing the ratio l(x)/g(x); the mixture that maximizes this ratio is chosen as the mixture for use for the iteration. The ratio will increase for hyperparameters mixtures which might be both (1) prone to be related to low losses or (2) unlikely to be related to excessive losses (which drives exploration). This course of of selecting the most effective candidate corresponds to utilizing the surrogate mannequin with the EI as mentioned when trying on the pseudocode for an SMBO.
An XGBoost mannequin is then skilled with the highest candidate for the iteration; a loss worth is obtained, and the information level (x*, f(x*)) is used to replace the surrogate mannequin (l(x) and g(x)) to proceed optimization.
Marginal Impact of Hyperparameter Tuning
So now, with a background on how the hyperopt library can be utilized within the hyperparameter tuning course of, we transfer to the query of how utilizing wider distributions impacts mannequin efficiency. When trying to check the efficiency of fashions skilled on giant search areas towards these skilled on narrower search areas, the speedy query is find out how to create the narrower search house. For instance, the presentation from Boehmke advises utilizing a uniform distribution from 1 to 100 for the max_depth hyperparameter. XGBoost fashions are likely to generalize higher when combining a lot of weak learners, however does that imply we slender the distribution to a minimal of 1 and a most of fifty? We might have some kind of common understanding from work others have achieved to intuitively slender the house, however can we discover a option to analytically slender the search house?
The answer proposed on this article includes working a set of shorter hyperparameter tuning trials to slender the search house based mostly on shallow searches of a wider hyperparameter house. The broader search house we use comes from slide 20 of Boehmke’s aforementioned presentation (here). As an alternative of working hyperparameter tuning on a large search house for 1,000 rounds of hyperparameter testing, we’ll run 20 unbiased trials with 25 rounds of hyperparameter testing every. We are going to slender the search house utilizing percentile values for every hyperparameter utilizing the trial outcomes. With the percentiles, we are going to run a ultimate seek for 200 rounds utilizing the narrower hyperparameter search house, the place the distribution we offer for every hyperparameter is given a most and minimal from the percentile values we see within the trials.
For instance, say we run our 20 trials and get 20 optimum values for max_depth utilizing the shallow search. We select to slender the search house for max_depth from the uniform distribution from 1 to 100 to the uniform distribution from the tenth percentile worth for max_depth from our trials to the ninetieth percentile worth for max_depth. We are going to run a few totally different fashions altering up the percentiles we use to check aggressive narrowing methods.
Fashions produced utilizing the trial-based methodology require 700 evaluations of hyperparameter mixtures (500 from the trials and 200 from the ultimate analysis). We are going to examine the efficiency of those fashions towards one tuned for 1,000 hyperparameter evaluations on the broader house and one tuned for 700 hyperparameter evaluations on the broader house. We’re curious as as to if this methodology of narrowing the hyperparameter search house will result in quicker convergence towards the optimum hyperparameter mixture or if this narrowing negatively impacts outcomes.
We check this methodology on a job from a previous challenge involving simulated tennis match outcomes (extra data within the article I wrote here). A part of the challenge concerned constructing post-match win chance fashions utilizing high-level details about every match and statistics for a given participant within the match that adopted a truncated regular distribution; that is the duty used to check the hyperparameter tuning methodology right here. Extra details about the precise job may be discovered within the article and within the code linked firstly of the article. At a excessive stage, we are attempting to take details about what occurred within the match to foretell a binary win/loss for the match; one may use a post-match win chance mannequin to determine any gamers that could be overperforming their statistical efficiency, who could be candidates for regression. To coach every XGBoost mannequin, we use log loss/cross-entropy loss because the loss perform. The information for the duty comes from Jeff Sackmann’s GitHub web page right here: https://github.com/JeffSackmann/tennis_atp. Anybody inquisitive about tennis or tennis information ought to take a look at his GitHub and wonderful web site, tennisabstract.com.
For this job and our methodology, we have now six fashions, two skilled on the total search house and 4 skilled on a narrower house. These are titled as follows within the charts:
- “Full Search”: That is the mannequin skilled for 1000 hyperparameter evaluations throughout the total hyperparameter search house.
- “XX-XX Percentile”: These fashions are these skilled on a narrower search house for 200 evaluations after the five hundred rounds of trial evaluations on the total hyperparameter search house. The “10–90 Percentile” mannequin for instance trains on a hyperparameter search house the place the distribution for every hyperparameter is set by the tenth percentile and ninetieth percentile values from the 20 trials.
- “Shorter Search”: That is the mannequin skilled for 700 hyperparameter evaluations throughout the total hyperparameter search house. We use this to check the efficiency of the trial methodology towards the broader search house when allotting the identical variety of hyperparameter evaluations to each strategies.
A log of coaching the fashions is included on the GitHub page linked on the high of the article which incorporates the hyperparameters discovered at every step of the method given the random seeds used together with the time it took to run every mannequin on my laptop computer. It additionally offers the outcomes of the 20 trials run so to know how every narrowed search house can be parameterized. These occasions are listed beneath:
- Full Search: ~6000 seconds
- 10–90 Percentile: ~4300 seconds (~3000 seconds for trials, ~1300 for narrower search)
- 20–80 Percentile: ~3800 seconds (~3000 seconds for trials, ~800 for narrower search)
- 30–70 Percentile: ~3650 seconds (~3000 seconds for trials, ~650 for narrower search)
- 40–60 Percentile: ~3600 seconds (~3000 seconds for trials, ~600 for narrower search)
- Shorter Search: ~4600 seconds
The timing doesn’t scale 1:1 with the variety of complete evaluations use; the trial methodology fashions are likely to take much less time to coach given the identical variety of evaluations, with narrower searches taking even much less time. The subsequent query is whether or not this time-saving impacts mannequin efficiency in any respect. We’ll start by validation log loss throughout the fashions.
Little or no distinguishes the log losses throughout the fashions, however we’ll zoom in a little bit bit to get a visible have a look at the variations. We current the total vary y-axis first to contextualize the minor variations within the log losses.
Okay so doing higher, however we’ll zoom in yet another time to see the pattern most clearly.
We discover that the 20–80 Percentile mannequin attains the most effective validation log loss, barely higher than the Full Search and Shorter Search strategies. The opposite percentile fashions all carry out barely worse than the broader search fashions, however the variations are minor throughout the board. We are going to look now on the variations in accuracy between the fashions.
As with the log losses, we see very minor variations and select to zoom in to see a extra definitive pattern.
The Full Search mannequin attains the most effective accuracy of any mannequin, however the 10–90 Percentile and 20–80 Percentile each beat out the Shorter Search mannequin over the identical variety of evaluations. That is the form of tradeoff I hoped to determine with the caveat that that is task-specific and on a really small scale.
The outcomes utilizing log loss and accuracy counsel the opportunity of a doable efficiency-performance commerce off when selecting how huge to make the XGBoost hyperparameter search house. We discovered that fashions skilled on a narrower search house can outperform or examine to fashions skilled on wider search areas whereas taking much less time to coach general.
Additional Work
The code offered within the prior part ought to present modularity to run this check towards totally different duties with out issue; the outcomes for this classification job may range from these of others. Altering the variety of evaluations run when exploring the hyperparameter search house or the variety of trials run to get percentile ranges may present different conclusions from these discovered right here. This work additionally assumed the set of hyperparameters to tune; one other query I’d be inquisitive about exploring can be the marginal impact of together with further hyperparameters to tune (i.e., colsample_bylevel) on the efficiency of an XGBoost mannequin.
References
(used with permission)
[2] M. Harrison, Effective XGBoost (2023), MetaSnake
[3] B. Boehmke, “Advanced XGBoost Hyperparameter Tuning on Databricks” (2021), GitHub
[4] J. Bergstra, R. Bardenet, Y. Bengio, B. Kégl, “Algorithms for Hyper-Parameter Optimization” (2011), NeurIPS 2011
