ICML 2022 | meta proposes a robust multi-objective Bayesian optimization method to effectively deal with input noise

author |  Yangzequn

Company |  Renmin University of China

Research direction |  Multimodal learning

Paper title ：

Robust Multi-Objective Bayesian Optimization Under Input Noise

Thesis link ：

https://arxiv.org/abs/2202.07549

Project links ：

https://github.com/facebookresearch/robust_mobo

This article is about facebook Published in ICML 2022 A piece of work for , It theoretically analyzes the multi-objective Bayesian Optimization with input noise .

introduction

This paper deals with the input noise problem of multi-objective optimization , Combined with Bayesian Optimization and Pareto optimization, the global multi-objective VaR is designed and optimized , To solve the problem of black box constraint sensitive to input noise . Bayesian optimization by adjusting design parameters , Black box performance indicators with high evaluation cost can be optimized . Although many methods have been proposed to optimize a single target under input noise , However, there is still a lack of methods to solve the practical problem that multiple targets are sensitive to input disturbances .

In this work , The author proposes the first robust multi-objective Bayesian optimization method to deal with input noise . The author formalizes the goal as a risk measure to optimize an uncertain goal , That is, multivariable value at risk （MVaR）. Due to direct optimization MVaR In many cases, it is computationally infeasible , The author proposes an extensible 、 A theory based approach to use random scales to optimize MVaR. Experimentally speaking , This method is significantly better than other methods in data set , And effectively realize the optimal Luban design .

▲ chart 1： The simple data set shows that the optimal value of multi-objective optimization of non robust design is sensitive to input noise , And it gives a demonstration of the selection of the best set .

Here, through the graph 1 Let's sort out the questions raised by the author ： On the left , Non robust design （ violet ） And robust design （ green ） The nominal value of is represented by a square . The plus sign indicates the target value of each design under zero mean Gaussian input noise , The standard deviation is 0.1. You can see , Although non robust design may obtain locally better results , But its instability under input disturbance , Easy to lead to worse performance ; And robust design The result of is small for the input disturbance , Insensitive to input noise . 

The graph in the middle is for non robust and robust design MVaR Description of the set , The triangle represents the distribution of input noise , Every design MVaR Discrete approximation of sets . Without considering noise , violet A square can correspond to a better value ; But after being disturbed, its risk （MVaR） more , It is difficult to be robust to input noise . Therefore, multivariable risk value can be used to characterize the stability of the solution . The right figure is a description of the selection strategies of different risk sets , The hypothetical method gives MVaR aggregate , The optimal set of risks is different objectives MVaR The best set on the union of sets .

background

Multi objective optimization makes trade-offs between multiple black box functions , The goal is to identify the Pareto boundary of the optimal trade-off and the Pareto set of the corresponding optimal design . Consider maximizing black box functions ： among , Is the number of targets , Is the tight search space . The above definition can lead to the definition of Pareto domination and Pareto boundary . If the vector Pareto dominates , Write it down as , If and only if also Satisfy .

Pareto is the best （Pareto optimality） It is a state that cannot be improved , It is impossible to improve some individuals or preference criteria without damaging any other individuals or criteria . If there is Pareto optimal improvement in a state , Then it is called Pareto dominated . If a state is not dominated by Pareto , Then it is called Pareto optimal or Pareto efficient , In optimization problems, it can be regarded as the best . The set of such best points is called Pareto boundary . As you can see from the picture below ,A and B Is the point on the Pareto boundary , They are all for C The domination of .

▲ chart 2： Here is an example of Pareto boundary . Points in the set represent feasible choices , Here, it is considered that a smaller value is better , The red line represents the Pareto boundary , The sample points above are Pareto efficient . spot C At the same time A Sum point B control , So it is not on the Pareto boundary . spot A Sum point B Not strictly controlled by any other point , So it's on the boundary .

Defined here Pareto boundary of ：

if PARETO The elements in satisfy the additional constraints of the black box , Then the corresponding optimal design set is considered as Pareto boundary . Then , By defining measures （ Hypervolume and hypervolume increment ） To measure the quality of different Pareto boundaries , That is, the measure value of the region contained within the Pareto boundary .

Method

First of all, we need to define the risk . Because the expected risk measurement may not always be consistent with the real robustness goal , Therefore, probabilistic risk analysis is used here , And get the following definition ：

The definition of value at risk is given above , It gets in the noise A lower bound of , bring At least The probability of falling is greater than In the domain of , And call it probabilistic risk , To measure the noise of a single target .

And for Multivariate Value-at-Risk（MVaR） for , be-all All targets are evaluated in the same noisy samples . The author designs on the boundary of multiple targets, which is the Pareto boundary of different targets , It can be written in the following form ：

thus , It defines the global risk across the design space , Is to adopt a series of points （ See the picture 1 The triangle of ）, Robust approximation of Pareto boundary in multi-objective situation , This is also one of the important contributions of this article .

▲ chart 3： For Graphs 1 in toy Data MVaR The building process of

This paper proposes MARS Method , By introducing Chebyshev scaling VaR and MVaR The relationship between , And can be used to MVaR Set to estimate . Here's the picture 3 It's the graph 1 Simple data MVaR The process of building a collection , The black dot in the left figure indicates that the standard deviation is 0.1, The mean for 0 The function value of Gaussian input disturbance , The background is an outline showing Chebyshev scaling values across the target space . The graph in the middle shows the probability density and of Chebyshev scaling Chebyshev scaled var , The probability mass to the right of the black line is equal to . The right figure shows the relationship constructed by the theorem proved in the text , take VaR Mapping to MVaR in , The green triangle represents MVaR Discrete approximation of sets .

The main result

▲ chart 4： stay 4 Evaluation on different noise data sets

▲  chart 5： Although non robust design in noiseless targets （Nominal Values） It is feasible , But it is located near the boundary of the feasible area in the design space , In the case of input disturbance, the constraint in the black box will be violated , Make the obtained solution infeasible

▲  surface 1： The running time of each iteration of Bayesian Optimization of different algorithms

chart 4 It shows the changes with the training process , The performance of each algorithm , The overall situation is adopted here MVaR And design MVaR The gap between HV The logarithm of is used as the evaluation index , It can explain the design MVaR Can we approach the overall situation . Under the intervention of input noise , The non robust method is significantly weaker than the robust method , The author's method is due to other comparative methods . chart 5 It shows that in the real data set , Choose the benefits of robust design and non robust design respectively . It can be seen here that through MVaR The learned design is closer to the target value , The solution obtained by non robust design is more likely to fall into the infeasible region . surface 1 It shows MARS-based The advantages of the method in running time .

Summary and reflection

In this work , The author combines the properties of Bayesian Optimization and multi-objective fusion , The input noise is analyzed from the distribution level , It is designed MVaR Risk and find Pareto optimality of multi-objective risk , A good combination of the characteristics of the two methods , The idea is relatively simple and reasonable . For the optimization of other multi-source targets , For example, multimodal 、 Multiple perspectives 、 For multi task learning , This method guides us to start from the perspective of data disturbance risk , To analyze the potential input noise problem in the method .

meanwhile , Because the method is simple but difficult to express intuitively , The author used less space to achieve a clear exposition of the method , Then, through a lot of foreshadowing, it clearly expounds the background and main contributions of the article , A large number of proofs are given in the appendix to illustrate the correctness of the lemma . meanwhile , The author explains the main problems and methods through only two images , It clearly shows the robustness problem in the multi-objective situation .

At the method level , This method uses a series of points to estimate the distribution boundary , It is based on anchor Methods , The idea of estimating data distribution through anchor selection is similar , The two describe the distribution boundary and the distribution situation respectively . Later, for noise （ Input noise 、 Label noise ） In this case, we should think deeply about the relationship between the two .

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