ATM Algorithm on the BBOB 2009 Noiseless Function Testbed Benjamin - - PowerPoint PPT Presentation
Benchmarking The ATM Algorithm on the BBOB 2009 Noiseless Function Testbed Benjamin Bodner Brown University Providence, RI, USA BBOB Workshop GECCO 2019 Prague 4/1/2020 Benchmarking the ATM Algorithm on the BBOB 2009 Noiseless Function
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Brown University Providence, RI, USA
BBOB Workshop GECCO 2019 Prague
Motivation Intuition
BBOB Noiseless BBOB Large-scale Internal runtime
Parameters & main equations Parameter adaptation Resource allocation
Recent progress Goals moving forward Conclusions
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very high-dimensional settings
Image from: https://towardsdatascience.com/why- deep-learning-is-needed-over-traditional- machine-learning-1b6a99177063
Optimization Algorithms
have 10^5- 10^8
[Devlin et al. 2019]
Deep Learning Physical Sciences
Image from GOMC: https://gomc- wsu.github.io/Manual/index.html 12/7/2019 Benchmarking the ATM Algorithm on the BBOB 2009 Noiseless Function Testbed 3 of 34
Deep Learning Gradient-based
create many difficulties
[Shalev-Shwartz et al. 2017]
Current ways of mitigating these issues Vanishing gradients Getting stuck in local minima Hyperparameter tuning
Image from: https://towardsdatascience.com/gradient-descent- algorithm-and-its-variants-10f652806a3
Noise
Image from [He et al. 2015]),
Architecture Design Regularization
Image from: Srivastava, Nitish, et al. 2014
[Sutskever 2013]
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Do not always work
Image by Thomas Splettstoesser: https://www.behance.net/gallery/10952399/Protein- Folding-Funnel Image from GOMC: https://gomc-wsu.github.io/Manual/index.html
numbers of local minima
[Nichita 2002]
Image from: https://en.wikibooks.org/wiki/Structural_Bioch emistry/Proteins/Protein_Folding_Problem
Physical Sciences
quasi-Newton methods can be slow
global minima [Hao et al. 2015]
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Interacting Particles Protein Folding
Existing algorithms have been highly successful in these settings Characteristics intentionally designed into the BBOB function testbeds
[BIPOP CMA-ES, Hansen 2009]
Covariance matrices and Hessians limit their scalability capabilities Key components and operations are usually of order D^2
Images from: Finck, Hansen, Ros, Auger 2015 12/7/2019 Benchmarking the ATM Algorithm on the BBOB 2009 Noiseless Function Testbed 6 of 34
Eliminate the use of D^2 objects and operations Adaptive Two Mode (ATM) Algorithm
A black box optimization algorithm which
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Directional distribution Isotropic distribution
and interactions between the modes Uses a combination of two kinds of search distributions / “modes”
Exploitation Exploration
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Start from isotropic distribution If sample leads to improvement: Suggest samples in that direction If new samples also lead to improvement: Sample in same direction at exponentially increasing amplitude Once no more “good” samples are found: Start over with the isotropic search (using an evolutionary strategy)
Best Sample Best Sample from last step Regular Sample
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There are (currently) 8 parameters which play several roles in the ATM algorithm:
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𝑗𝑔 𝑌𝑐𝑓𝑡𝑢𝑢 − 𝑌𝑐𝑓𝑡𝑢𝑢−1
2 > 𝛦𝑌𝑛𝑗𝑜 2
: 𝑒𝑝 𝑒 += 1, 𝑠 = 0 𝑓𝑚𝑡𝑓: 𝑒𝑝 𝑠 += 1, 𝑒 = 0 𝑆 = 𝑆𝑛𝑏𝑦 exp 𝐻𝑠 sin 𝑛𝑝𝑒 𝜌𝑠 2 𝑈
𝑠
, 𝜌 2 − 1 𝐸 = 𝑆𝑛𝑏𝑦exp 𝐻𝑒𝑒 − 𝐸𝑒𝑠
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𝑻 = 𝛾𝑻 + 1 − 𝛾 𝑛𝑓𝑏𝑜 𝑷 − 𝑃𝐻𝑐𝑓𝑡𝑢 𝒀 − 𝒀𝐻𝑐𝑓𝑡𝑢𝑢
2
𝑩 = 𝛽 𝑻 + 𝛽2
𝑃𝑄 = (𝑛𝑓𝑏𝑜 Δ𝑃𝑄𝑐𝑓𝑡𝑢 + 𝑛𝑗𝑜 Δ𝑃𝑄𝑐𝑓𝑡𝑢 ) 2
𝛦𝑃𝑄𝑐𝑓𝑡𝑢 = 𝐶𝑓𝑡𝑢 𝑑ℎ𝑏𝑜𝑓 𝑗𝑜 𝑢ℎ𝑓 𝑢𝑠𝑣𝑓 𝑝𝑐𝑘𝑓𝑑𝑢𝑗𝑤𝑓 𝑔𝑣𝑜𝑑𝑢𝑗𝑝𝑜, 𝑔𝑝𝑣𝑜𝑒 𝑐𝑧 𝑢ℎ𝑓 𝑞𝑏𝑠𝑏𝑛𝑢𝑓𝑠 𝑡𝑓𝑢
Changing characteristics at different stages Different functions Need for online parameter tuning
another Two-Mode algorithm
reflect the “success” at the task of minimizing the true objective function
How to do this?
+
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New parameter sets Changing local search space Good chance for unsuitable sets
+
Proposal Fewer resources to “bad” parameter sets more resources to better ones
Resources allocated to parameter set Performance of parameter set
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𝑶𝑢+1 = 𝑶𝑢 − 𝐿 𝑁−1 𝜠𝑷𝑸𝒄𝒇𝒕𝒖𝒖 − K0M−1 𝐎t − 𝐎0
𝑶𝒖 = 𝑆𝑓𝑡𝑝𝑣𝑠𝑑𝑓 𝑏𝑚𝑚𝑝𝑑𝑏𝑢𝑗𝑝𝑜 𝑤𝑓𝑑𝑢𝑝𝑠 𝑏𝑢 𝑗𝑢𝑓𝑠𝑏𝑢𝑗𝑝𝑜 𝑢
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K = 𝑛 − 1 ∗ 𝐿 −𝐿 ⋯ −𝐿 −𝐿 𝑛 − 1 ∗ 𝐿 ⋯ −𝐿 ⋮ ⋮ ⋱ ⋮ −𝐿 −𝐿 ⋯ 𝑛 − 1 ∗ 𝐿 𝐿0 = 𝑙0 𝐽 𝑁 = 𝜈 𝐽
𝑶𝑢+1 = 𝑶𝑢 − 𝐿 𝑁−1 𝜠𝑷𝑸𝒄𝒇𝒕𝒖𝒖 − K0M−1 𝐎t − 𝐎0
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Resource allocation Parameter Set3 Parameter Set4 Evaluate Samples Parameter Set1 Parameter Set2 Values of
function
Suggestions for samples Repeat
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Sum Of Different Powers - f14 Sharp Ridge - f13 Rotated Ellipse - f10
Succeeds at solving:
non-separable functions
and/or noisy One of the best
separable functions subset (f1-5) +Large budget
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around the global minima which are convex
(“large” = comparable to 𝑆𝑛𝑏𝑦)
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rotated and ill conditioned functions
rotated and noisy/ multimodal functions
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Budget = 3000D
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Internal runtime of the ATM algorithm scales linearly as a function of the number of variables in the search space
Results from timing experiment:
Total Runtime – Evaluation time
averaged over 3 runs
1.0 sec internal runtime
CMA (pip install CMA) BFGS (scipy.optimize) Internal Runtime as a Function of Number of Variables in Search Space Internal Runtime (seconds) Number of variables in search space (NU) ATM Nelder Mead (scipy.optimize) L-BFGS-B (scipy.optimize)
updated by a moving average rule
is improved on rotated functions
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Iterations log10 Δ𝑔 Convergence Plot Rotated Ellipse f10 Convergence Plot Sharp Ridge f13 Original ATM New Version
Improve performance on rotated and ill-conditioned functions (without using DxD objects)
Increase performance in noisy environments – use averaging and moving mean
Add second population with weak restart conditions – for multimodal functions Make the ATM more user friendly and customizable
For more information see: https://github.com/BjBodner/ATM-optimization-algorithm
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high-dimensional problems
needed
with size of the search space:
rotated functions e.g., ill-conditioned and/or noisy functions
separable functions
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Dr Brenda Rubenstein, Brown University, Providence, RI, USA for her guidance in developing this algorithm. Her contributions and encouragement were essential in advancing this project forward and getting it to its current form. Dr Eran Triester Ben-Gurion University, Beersheva, Israel for his
Working with him is significantly helping improve the performance
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Testbed . GECCO '09 Proceedings of the 11th Annual Conference Companion on Genetic and Evolutionary Computation Conference: Late Breaking Papers Pages 2389-2396
by free-energy optimization along principal components of natural structural
by direct minimization of Gibbs free energy with a global optimization method. Computers and Chemical Engineering 26 (2002) 1703/1724
Bidirectional Transformers for Language Understanding. arXiv:1810.04805
Volume70,Pages3067-3075 2017.
in deep learning. In Proceedings of the 30 the International Conference on Machine Learning-Volume28,I CML13, III1139-III-1147 (JMLR.org,2013)
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Email: benjamin_bodner@brown_edu For more information see: https://github.com/BjBodner/ATM-optimization-algorithm