11institutetext: Department of Decision Sciences, University of South Africa,
Pretoria, South Africa
11email<EMAIL_ADDRESS>22institutetext: School of Computer Science and
Applied Mathematics, University of the Witwatersrand, Johannesburg, South
Africa
22email<EMAIL_ADDRESS>
# A Continuous Optimisation Benchmark Suite from Neural Network Regression

Katherine M. Malan 11 0000-0002-6070-2632 Christopher W. Cleghorn 22
0000-0002-7860-0650

###### Abstract

Designing optimisation algorithms that perform well in general requires
experimentation on a range of diverse problems. Training neural networks is an
optimisation task that has gained prominence with the recent successes of deep
learning. Although evolutionary algorithms have been used for training neural
networks, gradient descent variants are by far the most common choice with
their trusted good performance on large-scale machine learning tasks. With
this paper we contribute CORNN (Continuous Optimisation of Regression tasks
using Neural Networks), a large suite for benchmarking the performance of any
continuous black-box algorithm on neural network training problems. Using a
range of regression problems and neural network architectures, problem
instances with different dimensions and levels of difficulty can be created.
We demonstrate the use of the CORNN Suite by comparing the performance of
three evolutionary and swarm-based algorithms on over 300 problem instances,
showing evidence of performance complementarity between the algorithms. As a
baseline, the performance of the best population-based algorithm is
benchmarked against a gradient-based approach. The CORNN suite is shared as a
public web repository to facilitate easy integration with existing
benchmarking platforms.

###### Keywords:

Benchmark suite Unconstrained continuous optimisation Neural network
regression.

##  1 Introduction

The importance of a good set of benchmark problem instances is a critical
component of a meaningful benchmarking study in optimisation [4]. As a
consequence of the No Free Lunch Theorems for optimisation [49], if an
algorithm is tuned to improve its performance on one class of problems it will
most likely perform worse on other problems [22]. Therefore, to develop
algorithms that perform well in general or are able to adapt to new scenarios,
a wide range of different problem instances are needed for experimental
algorithm design.

Most of the benchmark problems available for continuous optimisation are
artificial, so performance achieved through tuning algorithms on these
problems cannot be assumed to transfer to real-world problems. On the other
hand, testing algorithms on real-world problems is not always feasible and has
the disadvantage of not covering the wide range of problem characteristics
needed for different problem scenarios. To bridge this gap, we introduce a
benchmark suite from the real-world domain of neural network (NN) training
that includes some of the advantages of artificial benchmark problems.

Metaheuristics frequently suffer from the curse of dimensionality with
performance degrading as the number of decision variables increases [30, 31,
37, 46]. This is in part due to the increased complexity of the problem, but
also due to the exponential growth in the size of the search space [30]. The
training of NNs presents an ideal context for high-dimensional optimisation as
even a medium-sized network will have hundreds of weights to optimise. Most NN
training studies use a limited number of problem instances (classification
datasets or regression problems), which brings into question the
generalisability of the results. For example, in six studies using population-
based algorithms for NN training [15, 34, 33, 35, 43, 6], the number of
problem instances used for testing ranged from a single real-world instance
[15] to eight classification or regression problems [33, 35]. To facilitate
the generalisability of NN training studies, we provide a suite of hundreds of
problem instances that can easily be re-used for benchmarking algorithm
performance.

Stochastic gradient descent [29] is the default approach to training NNs with
its trusted good performance on large-scale learning problems [10].
Population-based algorithms have been proposed for training NNs [15, 34, 33,
35, 43, 6], but they are seldom used in practice. One of the challenges is
that the search space of NN weights is unbounded and algorithms such as
particle swarm optimisation may fail due to high weight magnitudes leading to
hidden unit saturation [39, 40]. The one domain where population-based
metaheuristics have shown competitive results compared to gradient-based
methods is in deep reinforcement learning tasks [45]. More benchmarking of
gradient-free methods against gradient-based methods is needed to highlight
the possible benefits of different approaches.

This paper proposes CORNN, Continuous Optimisation of Regression tasks using
Neural Networks, a software repository of problem instances, including
benchmarking of population-based algorithms against a gradient-based algorithm
(Adam). Over 300 regression tasks are formed from 54 regression functions with
different network architectures generating problem instances with dimensions
ranging from 41 to 481. The source code and datasets associated with the
benchmark suite are publicly available at github.com/CWCleghornAI/CORNN.

##  2 Continuous Optimisation Benchmark Suites

When benchmarking algorithms, it is usually not practical to use real-world
problems, due to the limited range of problem instances available and the
domain knowledge required in constructing the problems. Individual real-world
problem instances also do not effectively test the limits of an algorithm
because they will not usually cover all the problem characteristics of
interest [41]. Artificial benchmark suites and problem instance generators
have therefore become popular alternatives for testing optimisation
algorithms. In the continuous optimisation domain, the most commonly used
artificial benchmark suites include the ACM Genetic and Evolutionary
Computation Conference (GECCO) BBOB suites [13] and IEEE Congress on
Evolutionary Computation (CEC) suites [14]. These artificial suites have been
criticised for having no direct link to real-world settings [17], resulting in
a disconnect between the performance of algorithms on benchmarks and real-
world problems [47].

To address the limitations of artificial benchmarks, suites that are based on
real-world problems or involve tasks that are closer to real-world problems
have been proposed, such as from the domains of electroencephalography (EEG)
[21], clustering [20], crane boom design [18], and games [48]. The CORNN Suite
proposed in this paper extends these sets to include the class of problems for
solving NN regression tasks. These tasks are unique in that the decision
variables are unbounded and the scenario makes it possible to benchmark black-
box algorithms against gradient-based approaches.

##  3 Neural Network Training Landscapes

Training NNs involves adjusting weights on the connections between neurons to
minimise the error of the network on some machine learning task. Since weight
values are real numbers, the search space is continuous with dimension equal
to the number of adjustable weights in the network. Training NNs is known to
be NP-complete even for very small networks [9] and the properties of error
landscapes are still poorly understood [12] with conflicting theoretical
claims on the presence and number of local minima [2, 3, 23, 32]. Some studies
have suggested that these landscapes have large flat areas with valleys that
radiate outwards [11, 19, 36, 28] and a prevalence of saddle points rather
than local minima [12, 16]. Saddle points present a challenge for search,
because they are generally surrounded by high error plateaus [16] and, being
stationary points, can create the illusion of being local optima. It has,
however, also been found that failure of gradient-based deep learning is not
necessarily related to an abundance of saddle points, but rather to aspects
such as the level of informativeness of gradients, signal-to-noise ratios and
flatness in activation functions [42].

To better understand the nature of NN error landscapes, investigations are
needed into the behaviour of different algorithms on a wide range of problems.
The CORNN Suite can be used to complement existing suites or as a starting
point for this kind of analysis.

##  4 The CORNN Benchmark Suite

With sufficient neurons, NNs are able to model an arbitrary mathematical
function [7, 25], so are a suitable model for solving complex regression
problems. The optimisation task of the CORNN Suite involves fitting a fully-
connected feed-forward NN to a real-valued function, $f(\mathbf{x})$. Each
network has an $n$-dimensional real-valued input,$\mathbf{x}$, and a single
real-valued output, which is the prediction of the target value,
$f(\mathbf{x})$. The CORNN Suite uses 54 two-dimensional functions as the
basis for regression fitting tasks. These functions are specified on the CORNN
Suite repository on github
111https://github.com/CWCleghornAI/CORNN/blob/main/CORNN_RegressionFunctions.pdf.
The functions cover a range of characteristics with respect to modality,
separability, differentiability, ruggedness, and so on. Note, however, that
although these characteristics will no doubt have some effect on the
difficulty of the regression task, we cannot assume that the features of the
functions relate to the characteristics of the higher dimensional search space
of NN weights for fitting the functions.

###  4.1 Training and Test Sets

Datasets were generated for each of the 54 functions as follows: 5000
$(x_{1},x_{2})$ pairs were sampled from a uniform random distribution of the
function’s domain. A value of 5000 was used to be large enough to represent
the actual function, while still allowing for reasonable computational time
for simulation runs. The true output for each $(x_{1},x_{2})$ pair was
calculated using the mathematical function and stored as the target variable.
Each dataset was split randomly into training (75% of samples) and testing
(25% of samples) sets. Two forms of preprocessing were performed on the CORNN
Suite datasets:(1) Input values were normalised to the range $[-1,1]$ using
the domain of each function; (2) To compare results of problem instances with
different output ranges, output values were normalised using simple min-max
scaling based on the training data to the range $(0,1)$.

In addition, the CORNN Suite’s implementation allows for the use of custom
datasets; either generated from analytic functions or existing datasets.

###  4.2 Neural Network Models

The architecture used in the CORNN Suite is a fully connected feed-forward
network with 2 inputs; 1-, 3-, or 5-hidden layers, each with 10 neurons plus a
bias unit; and 1 output neuron. This results in 41, 261, and 481 weights to be
optimised for the 1-, 3-, and 5-layer networks respectively. Each architecture
uses one of two hidden layer activation functions: the conventional hyperbolic
tangent (Tanh) and the rectified linear unit (ReLU). ReLU is currently the
most commonly used activation function in deep learning [29], but Tanh has
been recommended above ReLU for reinforcement learning tasks [1]. The output
layer uses a linear activation function in all cases. These six topologies are
referred to as Tanh1, Tanh3, Tanh5, ReLU1, ReLU3, and ReLU5, specifying the
activation function and number of hidden layers. The CORNN Suite therefore
consists of $54\times 6=324$ problem instances, since each function has six NN
models for fitting the function. Note that the CORNN Suite’s implementation
allows for complete customisation of architectures to create any desired
topology for further analysis.

###  4.3 Performance Evaluation

Performance of an algorithm is measured using mean squared error (MSE) of the
trained model on the test set given a set budget of function evaluations. Note
that in the analysis presented in this paper, no evidence of overfitting was
observed. If overfitting becomes a consideration as more specialised
optimisers are developed/considered, it may become necessary to hold out a
portion of the training set to employ techniques such as early stopping etc.
When using hold-out training instances, the total number of function
evaluations should be seen as the maximum number of times any one training
instance has been used. If no hold-out instances are used this measurement is
equivalent to the number of full passes of the training data. A similar
consideration should be made if an optimiser requires hyper-parameter tuning;
in such cases hold-out instances from the training set should be used for
tuning and not the test set instances.

###  4.4 Implementation Details

The CORNN Suite was developed in Python 3 using PyTorch. The user selects a
regression task and a model architecture, after which the library constructs a
problem instance object with a callable function to which the user can pass a
candidate solution for evaluation on either the training set during
optimisation, or on the test set after optimisation. A user of CORNN therefore
does not have to concern themselves with any data processing or NN
computation. The complexity of the problem instances are abstracted away to
the point where a user of CORNN can just work with an objective function after
setup.

The GitHub repository provides installation instructions with a detailed
example of how to construct and use a CORNN problem instance. The suite is
easily extended beyond the $324$ problem instances presented in this paper to
include other regression tasks and/or NN architectures through reflection. The
datasets for all $54$ regression tasks are also provided in CSV format, but
when using the CORNN Suite it is not necessary to directly interact with these
files.

##  5 Experimentation and Results

To demonstrate CORNN, we provide results of metaheuristics and contrast these
with a gradient-based method. The aim is not to compare algorithms, but to
provide a use-case of the suite. No tuning of algorithm parameters was done,
so the results are not representative of the best performance of the
algorithms.

###  5.1 Experimental Setup

The algorithms used in this study were: particle swarm optimisation (PSO)
[26], differential evolution (DE) [44], covariance matrix adaptation evolution
strategy (CMA-ES) [24], Adam [27], and random search. For the population-based
algorithms, standard versions and hyperparameters defined in the Nevergrad [5]
library were used to facilitate reproducibility222Nevergrad version 0.4.2, PSO
with optimizers.PSO, DE with optimizers.TwoPointsDE, CMA-ES with
optimizers.CMA, and random search with optimizers.RandomSearch. Adam was used
for the gradient-based approach (PyTorch [38] implementation with default
parameters). Each algorithm had a function evaluation (FE) budget of $5000$
per problem instance, where an FE is defined as one complete pass through the
training dataset. We used full batch learning, but the suite is not limited to
this approach. Optimisation runs were repeated $30$ times for each
algorithm/problem instance pair.

###  5.2 Analysis of Population-based Algorithms

The first set of results contrasts the performance of the three population-
based algorithms: CMA-ES, DE and PSO against random search. We only present
the performance on the testing datasets, because we found no evidence of
overfitting by any of the algorithms on the problem set. Each algorithm is
given a performance score at each evaluation using the following scoring
mechanism per problem instance against each competing algorithm:

  * •

1 point is awarded for a draw (when there is no statistically significant
difference based on a two-tailed Mann-Whitney U test with 95% confidence).

  * •

3 points are awarded for a win and 0 for a loss. In the absence of a draw, we
determine whether a win or loss occurred using one-tailed Mann-Whitney U tests
(with 95% confidence).

This results in a maximum score for an algorithm on a single instance of 9, or
$3(n-1)$ in general, where $n$ is the number of algorithms. The scores per
instance were normalised to the range $[0,1]$.

Figure 1 plots the normalised mean performance score over all $54$ problem
instances for the full budget of $5000$ evaluations for the six NN models.
Solid lines denote the mean performance score with shaded bands depicting the
standard deviation around the mean. Two general observations from Figure 1 are
that the three metaheuristics all performed significantly better than random
search and that no single algorithm performed the best on all NN models. On
the Tanh models (plots in the left column of Figure 1), CMA-ES performed the
best on the 1-layer network, while DE performed the best on the 3- and 5-layer
networks after the full budget of evaluations. On the ReLU models (plots in
the right column of Figure 1), PSO ultimately performed the best on all three
models. CMA-ES was the quickest to find relatively good solutions, but
converged to solutions that were inferior to those ultimately found by PSO.

(a) 1-layer Tanh model (b) 1-layer ReLU model (c) 3-layer Tanh model (d)
3-layer ReLU model

(e) 5-layer Tanh model (f) 5-layer ReLU model

Figure 1: Summarised relative performance on the test set of a 1-, 3- and
5-layer Tanh and ReLU models over $54$ function datasets. Lines denote mean
performance with shaded bands denoting the standard deviation around the mean.

Figure 2 visualises the performance of the algorithms on individual problem
instances (the 3-layer models are omitted to save space). Each dot represents
a single problem instance (54 of them in each column) and the vertical
position corresponds to the average normalised MSE on the testing set from 30
runs of the algorithm after 5000 evaluations. Algorithms are plotted in a
different colour for clarity. The final column (purple dots) shows the results
of the best performing algorithm for each problem instance. In these plots,
better performing algorithms have a concentration of dots closer to the
horizontal axis (lower MSE values).

For all models, random search (red dots) performs worse than other algorithms.
Contrasting Figure 2a and 2c, shows that for Tanh, the performance across all
algorithms deteriorates (fewer dots lower down) as the number of layers
increase from 1 to 5, indicating an increase in problem difficulty. The same
can be observed for ReLU architectures in Figures 2b to 2d. Note that in
Figure 2d, only a single dot is shown for random search, because the other MSE
values are above the range plotted on the graph. The very poor results of
random search on ReLU5 indicates that these instances are more challenging
overall than the other architectures.

(a) 1-layer Tanh model

(b) 1-layer ReLU

(c) 5-layer Tanh

(d) 5-layer ReLU

Figure 2: Performance per function for each optimiser on two Tanh and ReLU
models

Figure 2 shows interesting outliers in the individual high dots at
approximately 0.25 MSE. This corresponds to function 34, Periodic, on which
most algorithms perform markedly worse than on the other problem instances. In
contrast, the lowest dots in Figure 2c correspond to function 20, Easom, on
which all algorithms (except random search) achieved close to 0.00 MSE. Figure
3 plots these two functions, clearly illustrating why it was easier for the
search algorithms to fit models to Easom than to Periodic.

(a) Easom: easiest regression function (b) Periodic: hardest regression
function

Figure 3: Two regression tasks on which all algorithms performed the best and
the worst.

###  5.3 Analysis of Individual Problem Instances

The next set of results highlights the range of difficulty of problems in the
CORNN Suite for population-based algorithms in relation to a gradient-based
approach. As a baseline, we provide the results of Adam, a form of gradient-
descent with adaptive learning-rate that is popular for training deep NNs.

Figures 4 to 7 show violin plots of the distribution of MSE values from 30
runs on each of the 54 problem instances for Tanh1, Tanh5, Relu1 and Relu5,
respectively. The blue violins represent the performance of the best
population-based algorithm (of the three discussed in Section 5.2) on the
problem instance, while the red violins represent the performance of Adam.
Note that most of the violins for Adam are very small due to the small
variance in the performance over the 30 runs – except for the random initial
weights, Adam is a deterministic algorithm. The median MSE values appear as
tiny white dots in the centre of each violin and the maximal extent of the
violins are cropped to reflect the actual range of the data.

The functions are sorted from left to right by the difference between the
median MSE of the two approaches. For example, in Figure 4, the first function
on the left is function 26 (Himmelblau), where the median MSE of the best
population-based algorithm was slightly lower than the median MSE of Adam.
From about the eighth function onwards, it can however be seen that Adam out-
performed the best population-based algorithm. The superior performance of
Adam is even more marked for the 5-layer Tanh model (Figure 5), with function
43 (Schwefel 2.22) resulting in the worst relative performance of the
population-based algorithms. The most difficult function to fit for both Adam
and population-based algorithms is evident by the high MSE values on function
34 (Periodic).

Figure 4: Best performing population based optimiser compared to Adam on the
1-layer Tanh model

Figure 5: Best performing population based optimiser compared to Adam on the
5-layer Tanh model

Figures 6 to Figure 7 show slightly better relative performance of the
population based algorithms on the ReLU architectures compared to the Tanh
architectures. On the left of Figure 7 we can see that the median MSE of the
best population based algorithm is lower than Adam on about the first nine
functions (37, 8, 47, 54, 25, 43, 46, and 52).

Figure 6: Best performing population based optimiser compared to Adam on the
1-layer ReLU model

Figure 7: Best performing population based optimiser compared to Adam on the
5-layer ReLU model

In this way, these results provide a ranking of CORNN problem instances from
the easiest to the hardest in terms of relative performance against gradient-
based techniques. Future studies can focus on black-box algorithm development
to reduce the gap in performance compared to gradient-based approaches. The
CORNN Suite can be used in different types of analysis and not just in the way
illustrated in this paper. For example, to simulate cases where the analytical
gradient is not available for using gradient-based techniques, black-box
optimisers can be benchmarked against one another to investigate the
effectiveness on NN training tasks.

##  6 Discussion

The CORNN Suite complements existing benchmark sets with NN training tasks
that can be used to benchmark the performance of any continuous black-box
algorithm. Ideally, a benchmarking suite should be [4]: (1) diverse, (2)
representative, (3) scalable and tunable, and (4) should have known solutions
/ best performance. The problem instances of the CORNN Suite are diverse as
demonstrated by the wide range of performances by different algorithms. In
addition, the problems are representative of real-world problems in being
continuous (which is more common in real-world settings than combinatorial
problems [8]), with computationally expensive evaluation and involving the
real-world task of NN training. Problems are scalable and tunable through the
selection of different NN models coupled with different regression tasks. For
each problem instance the theoretical optimal solution is known (where MSE = 0
on the test set). However, given a fixed number of neurons in a model, we
cannot rely on the universal approximator theorem [25] to guarantee the
existence of an optimal solution of weights that will result in an error of
zero. In addition to the theoretical minimum, we provide the performance of
Adam as a baseline against which alternative algorithms can be benchmarked.

##  7 Conclusion

The CORNN Suite is an easy-to-use set of unbounded continuous optimisation
problems from NN training for benchmarking optimisation algorithms that can be
used on its own or as an extension to existing benchmark problem sets. An
advantage of the suite is that black-box optimisation algorithms can be
benchmarked against gradient-based algorithms to better understand the
strengths and weaknesses of different approaches.

The results in this paper provide an initial baseline for further studies. We
have found that although Adam in general performed better, population-based
algorithms did out-perform Adam on a limited set of problem instances. Further
studies could analyse the characteristics of these instances using landscape
analysis to better understand which NN training tasks are better suited to
population-based approaches than gradient-based approaches. The CORNN Suite
can also be used to try to improve population-based algorithms on NN training
tasks. It would be interesting to analyse whether parameter configurations
from tuning on the CORNN Suite can be transferred to other contexts to improve
black-box metaheuristic algorithm performance on NN training tasks in general.

## Acknowledgments

This work was supported by the National Research Foundation, South Africa,
under Grant 120837. The authors acknowledge the use of the High Performance
Computing Cluster of the University of South Africa. The authors also
acknowledge the contribution of Tobias Bester for his initial implementation
of the underlying regression functions.

## References

  * [1] Andrychowicz, M., Raichuk, A., Stanczyk, P., Orsini, M., Girgin, S., Marinier, R., Hussenot, L., Geist, M., Pietquin, O., Michalski, M., Gelly, S., Bachem, O.: What matters in on-policy reinforcement learning? A large-scale empirical study. CoRR abs/2006.05990 (2020), https://arxiv.org/abs/2006.05990
  * [2] Auer, P., Herbster, M., Warmuth, M.K.: Exponentially many local minima for single neurons. In: Advances in Neural Information Processing Systems 9 (NIPS 1996) (1996), http://papers.nips.cc/paper/1028-exponentially-many-local-minima-for-single-neurons.pdf
  * [3] Baldi, P., Hornik, K.: Neural networks and principal component analysis: Learning from examples without local minima. Neural Networks 2(1), 53–58 (January 1989). https://doi.org/10.1016/0893-6080(89)90014-2, https://doi.org/10.1016/0893-6080(89)90014-2
  * [4] Bartz-Beielstein, T., Doerr, C., van den Berg, D., Bossek, J., Chandrasekaran, S., Eftimov, T., Fischbach, A., Kerschke, P., Cava, W.L., Lopez-Ibanez, M., Malan, K.M., Moore, J.H., Naujoks, B., Orzechowski, P., Volz, V., Wagner, M., Weise, T.: Benchmarking in optimization: Best practice and open issues. arXiv 2007.03488v2 (2020)
  * [5] Bennet, P., Doerr, C., Moreau, A., Rapin, J., Teytaud, F., Teytaudt, O.: Nevergrad: Black-box optimization platform. ACM SIGEVOlution 14(1), 8–15 (Apr 2021). https://doi.org/10.1145/3460310.3460312
  * [6] den Bergh, F.V., Engelbrecht, A.: Cooperative learning in neural networks using particle swarm optimizers. South African Computer Journal 2000(26), 84–90 (Nov 2000)
  * [7] Bishop, C.M.: Neural Networks for Pattern Recognition. Oxford University Press (1995)
  * [8] van der Blom, K., Deist, T.M., Tušar, T., Marchi, M., Nojima, Y., Oyama, A., Volz, V., Naujoks, B.: Towards realistic optimization benchmarks. In: Proceedings of the 2020 Genetic and Evolutionary Computation Conference Companion. pp. 293–294. ACM (Jul 2020). https://doi.org/10.1145/3377929.3389974
  * [9] Blum, A.L., Rivest, R.L.: Training a 3-node neural network is NP-complete. Neural Networks 5(1), 117–127 (Jan 1992). https://doi.org/10.1016/s0893-6080(05)80010-3
  * [10] Bottou, L., Bousquet, O.: The tradeoffs of large scale learning. In: Optimization for Machine Learning, chap. 13, pp. 351–368. The MIT Press (2012)
  * [11] Chaudhari, P., Choromanska, A., Soatto, S., LeCun, Y., Baldassi, C., Borgs, C., Chayes, J., Sagun, L., Zecchina, R.: Entropy-SGD: biasing gradient descent into wide valleys. Journal of Statistical Mechanics: Theory and Experiment 2019(12), 124018 (Dec 2019). https://doi.org/10.1088/1742-5468/ab39d9
  * [12] Choromanska, A., Henaff, M., Mathieu, M., Arous, G.B., LeCun, Y.: The loss surfaces of multilayer networks. In: Proceedings of the 18th International Conference on Artificial Intelligence and Statistics. pp. 192–204 (2015)
  * [13] COCO: Black-box optimisation benchmarking (BBOB) (2021), https://coco.gforge.inria.fr
  * [14] C.T.Yue, K.V.Price, P.N.Suganthan, J.J.Liang, M.Z.Ali, B.Y.Qu, N.H.Awad, P.Biswas, P.: IEEE CEC Bound Constrained benchmark suite (2020), https://github.com/P-N-Suganthan/2020-Bound-Constrained-Opt-Benchmark
  * [15] Das, G., Pattnaik, P.K., Padhy, S.K.: Artificial neural network trained by particle swarm optimization for non-linear channel equalization. Expert Systems with Applications 41(7), 3491–3496 (Jun 2014). https://doi.org/10.1016/j.eswa.2013.10.053
  * [16] Dauphin, Y.N., Pascanu, R., Gulcehre, C., Cho, K., Ganguli, S., Bengio, Y.: Identifying and attacking the saddle point problem in high-dimensional non-convex optimization. In: Ghahramani, Z., Welling, M., Cortes, C., Lawrence, N.D., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems 27, pp. 2933–2941. Curran Associates, Inc. (2014)
  * [17] Fischbach, A., Bartz-Beielstein, T.: Improving the reliability of test functions generators. Applied Soft Computing 92, 106315 (Jul 2020). https://doi.org/10.1016/j.asoc.2020.106315
  * [18] Fleck, P., Entner, D., Münzer, C., Kommenda, M., Prante, T., Schwarz, M., Hächl, M., Affenzeller, M.: Box-type boom design using surrogate modeling: Introducing an industrial optimization benchmark. In: Computational Methods in Applied Sciences, pp. 355–370. Springer International Publishing (Sep 2019)
  * [19] Gallagher, M.R.: Multi-layer perceptron error surfaces: Visualization, structure and modelling. Ph.D. thesis, University of Queensland, Australia (2000)
  * [20] Gallagher, M.: Towards improved benchmarking of black-box optimization algorithms using clustering problems. Soft Computing 20(10), 3835–3849 (Mar 2016). https://doi.org/10.1007/s00500-016-2094-1, https://doi.org/10.1007/s00500-016-2094-1
  * [21] Goh, S.K., Tan, K.C., Al-Mamun, A., Abbass, H.A.: Evolutionary big optimization (BigOpt) of signals. In: 2015 IEEE Congress on Evolutionary Computation (CEC). IEEE (May 2015). https://doi.org/10.1109/cec.2015.7257307, https://doi.org/10.1109/cec.2015.7257307
  * [22] Haftka, R.T.: Requirements for papers focusing on new or improved global optimization algorithms. Structural and Multidisciplinary Optimization 54(1), 1–1 (2016)
  * [23] Hamey, L.G.: XOR has no local minima: A case study in neural network error surface analysis. Neural Networks 11(4), 669–681 (June 1998). https://doi.org/10.1016/s0893-6080(97)00134-2
  * [24] Hansen, N., Ostermeier, A.: Adapting arbitrary normal mutation distributions in evolution strategies: The covariance matrix adaptation. In: Proceedings of the IEEE Congress on Evolutionary Computation. pp. 312–317. IEEE Press, Piscataway, NJ (1996)
  * [25] Huang, G.B., Chen, L., Siew, C.K.: Universal approximation using incremental constructive feedforward networks with random hidden nodes. IEEE Transactions on Neural Networks 17(4), 879–892 (Jul 2006). https://doi.org/10.1109/tnn.2006.875977
  * [26] Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of the IEEE International Joint Conference on Neural Networks. pp. 1942–1948. IEEE Press, Piscataway, NJ (1995)
  * [27] Kingma, D., Ba, J.: Adam: A method for stochastic optimization. ArXiv abs/1412.6980 (2014)
  * [28] Kordos, M., Duch, W.: A survey of factors influencing MLP error surface. Control and Cybernetics 33, 611–631 (2004)
  * [29] LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436–444 (May 2015). https://doi.org/10.1038/nature14539
  * [30] Lozano, M., Molina, D., Herrera, F.: Editorial scalability of evolutionary algorithms and other metaheuristics for large-scale continuous optimization problems. Soft Computing 15(11), 2085–2087 (Sep 2010). https://doi.org/10.1007/s00500-010-0639-2
  * [31] Mahdavi, S., Shiri, M.E., Rahnamayan, S.: Metaheuristics in large-scale global continues optimization: A survey. Information Sciences 295, 407–428 (Feb 2015). https://doi.org/10.1016/j.ins.2014.10.042
  * [32] Mehta, D., Zhao, X., Bernal, E.A., Wales, D.J.: Loss surface of XOR artificial neural networks. Physical Review E 97(5) (May 2018). https://doi.org/10.1103/physreve.97.052307, https://doi.org/10.1103/physreve.97.052307
  * [33] Mirjalili, S.: How effective is the grey wolf optimizer in training multi-layer perceptrons. Applied Intelligence 43(1), 150–161 (Jan 2015). https://doi.org/10.1007/s10489-014-0645-7
  * [34] Mirjalili, S., Hashim, S.Z.M., Sardroudi, H.M.: Training feedforward neural networks using hybrid particle swarm optimization and gravitational search algorithm. Applied Mathematics and Computation 218(22), 11125–11137 (Jul 2012). https://doi.org/10.1016/j.amc.2012.04.069
  * [35] Mousavirad, S.J., Schaefer, G., Jalali, S.M.J., Korovin, I.: A benchmark of recent population-based metaheuristic algorithms for multi-layer neural network training. In: Proceedings of the 2020 Genetic and Evolutionary Computation Conference Companion. ACM (Jul 2020). https://doi.org/10.1145/3377929.3398144
  * [36] N.S.Keskar, D.Mudigere, J.Nocedal, M.Smelyanskiy, P.T.P.Tang: On large-batch training for deep learning: Generalization gap and sharp minima. In: Proceedings of the International Conference for Learning Representations (2017)
  * [37] Oldewage, E.T.: The perils of particle swarm optimization in high dimensional problem spaces. Master’s thesis, University of Pretoria, South Africa (2017), http://hdl.handle.net/2263/66233
  * [38] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., Desmaison, A., Kopf, A., Yang, E., DeVito, Z., Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B., Fang, L., Bai, J., Chintala, S.: Pytorch: An imperative style, high-performance deep learning library. In: Advances in Neural Information Processing Systems 32, pp. 8024–8035. Curran Associates, Inc. (2019)
  * [39] Rakitianskaia, A., Engelbrecht, A.: Training high-dimensional neural networks with cooperative particle swarm optimiser. In: 2014 International Joint Conference on Neural Networks (IJCNN). IEEE (Jul 2014). https://doi.org/10.1109/ijcnn.2014.6889933
  * [40] Rakitianskaia, A., Engelbrecht, A.: Saturation in PSO neural network training: Good or evil? In: 2015 IEEE Congress on Evolutionary Computation (CEC). IEEE (May 2015). https://doi.org/10.1109/cec.2015.7256883
  * [41] Rardin, R.L., Uzsoy, R.: Experimental evaluation of heuristic optimization algorithms: A tutorial. Journal of Heuristics 7(3), 261–304 (2001). https://doi.org/10.1023/a:1011319115230
  * [42] Shalev-Shwartz, S., Shamir, O., Shammah, S.: Failures of gradient-based deep learning. In: Proceedings of the 34th International Conference on Machine Learning. pp. 3067–3075. PMLR (06–11 Aug 2017)
  * [43] Socha, K., Blum, C.: An ant colony optimization algorithm for continuous optimization: application to feed-forward neural network training. Neural Computing and Applications 16(3), 235–247 (Mar 2007). https://doi.org/10.1007/s00521-007-0084-z
  * [44] Storn, R., Price, K.: Differential evolution: A simple evolution strategy for fast optimization. Journal of Global Optimization 11, 341–359 (1997)
  * [45] Such, F., Madhavan, V., Conti, E., Lehman, J., Stanley, K.O., Clune, J.: Deep neuroevolution: Genetic algorithms are a competitive alternative for training deep neural networks for reinforcement learning. ArXiv abs/1712.06567 (2018)
  * [46] Tang, K., Li, X., Suganthan, P.N., Yang, Z., Weise, T.: Benchmark functions for the CEC 2010 Special Session and Competition on Large-Scale Global Optimization. Tech. rep., Nature Inspired Computation and Applications Laboratory (2009), https://titan.csit.rmit.edu.au/~e46507/publications/lsgo-cec10.pdf
  * [47] Tangherloni, A., Spolaor, S., Cazzaniga, P., Besozzi, D., Rundo, L., Mauri, G., Nobile, M.S.: Biochemical parameter estimation vs. benchmark functions: A comparative study of optimization performance and representation design. Applied Soft Computing 81, 105494 (Aug 2019). https://doi.org/10.1016/j.asoc.2019.105494
  * [48] Volz, V., Naujoks, B., Kerschke, P., Tušar, T.: Single- and Multi-Objective Game-Benchmark for Evolutionary Algorithms. In: Proceedings of the Genetic and Evolutionary Computation Conference. pp. 647 – 655. ACM (2019)
  * [49] Wolpert, D.H., Macready, W.G.: No Free Lunch Theorems for Optimization. IEEE Transactions on Evolutionary Computation 1(1), 67–82 (Apr 1997)


# Sensitive group actions on regular curves of almost $\leq n$ order

Suhua Wang & Enhui Shi & Hui Xu & Zhiwen Xie  Zhangjiagang Campus, Jiangsu
University of Science and Technology, Zhangjiagang 215600, P. R. China
<EMAIL_ADDRESS>School of Mathematical Sciences, Soochow University,
Suzhou 215006, P. R. China<EMAIL_ADDRESS>School of Mathematical Sciences,
Soochow University, Suzhou 215006, P. R. China<EMAIL_ADDRESS>School of Mathematical Sciences, Soochow University, Suzhou 215006, P. R.
China<EMAIL_ADDRESS>
###### Abstract.

Let $X$ be a regular curve and $n$ be a positive integer such that for every
nonempty open set $U\subset X$, there is a nonempty connected open set
$V\subset U$ with the cardinality $|\partial_{X}(V)|\leq n$. We show that if
$X$ admits a sensitive action of a group $G$, then $G$ contains a free
subsemigroup and the action has positive geometric entropy. As a corollary,
$X$ admits no sensitive nilpotent group action.

###### Key words and phrases:

sensitivity, regular curve, ping pong game, geometric entropy, nilpotent group

###### 2010 Mathematics Subject Classification:

37B05

##  1\. Introduction

Sensitivity is usually regarded as an important character of chaotic systems.
Many systems imply sensitivity, such as nontrivial weak mixing systems, non-
equicontinuous minimal systems, and topologically transitive systems with
dense minimal points [8]. In the definitions of Devaney’s chaos and Ruelle-
Taken’s chaos (also called Auslander-Yorke’s chaos), the notion of sensitivity
is a key ingredient [3, 6]. One may consult [2, 17, 31, 34, 36] for some very
interesting studies around various forms of sensitivity.

The notion of expansivity is closely related to the theory of structure
stability in differential dynamical systems, which is stronger than
sensitivity. It is well known that hyperbolic automorphisms on torus and full
shifts on symbolic spaces are expansive. Expansive algebraic actions by
$\mathbb{Z}^{d}$ or amenable groups are also intensively studied [4, 18].

Which space can admit an expansive $\mathbb{Z}$ action is an extensively
studied question. It is well known that the Cantor set, solenoids, and each
compact orientable surface of positive genus admit expansive $\mathbb{Z}$
actions [30, 35]. However, the circle $\mathbb{S}^{1}$ and the $2$-sphere
$\mathbb{S}^{2}$ admit no expansive $\mathbb{Z}$ actions [9]. A famous result
due to Mañé [22] says that if a compact metric space $X$ admit an expansive
$\mathbb{Z}$ action, then $X$ is of finite dimension. One may refer to [12,
15, 24, 25] for recent progresses on the study of existence of expansive
$\mathbb{Z}$-actions on $1$-dimensional continua.

T. Ward once asked whether the circle $\mathbb{S}^{1}$ can admit an expansive
nilpotent group action [16]. This question was solved negatively by C.
Connell, A. Furman and S. Hurder in an unpublished paper [5] using the ping
pong game technique. More strong results by G. Margulis can be found in [23].
These stimulate the following

######  Question 1.1.

Given a group $G$ and a continuum $X$, can $G$ act on $X$
expansively/sensitively?

The answer towards this question certainly depends on the topology of the
phase space and involves the algebraic structure of the acting group. For
abelian group $G$, some people showed the nonexistence of expansive or
sensitive $G$ actions on some locally connected continua [19, 21]. Relying on
the semi ping pong technique as in [5], Shi and Wang showed the nonexistence
of expansive nilpotent group actions on Peano continua with a free dendrite
[32].

The purpose of this paper is to study sensitive group actions on regular
curves. Regular curves are a class of $1$-dimensional continua, which are
natural generalization of graphs and dendrites. Some well-known curves such as
the triangular Sierpinski curve and Apollonius curves are regular. We will
introduce the notion of regular curve of almost $\leq n$ order (see Section 2
for the definition), which covers many important continua including the
mentioned as above. The dynamics of continuous maps or homeomorphisms on
regular curves were also studied very recently [13, 14, 28, 29].

The following is the main theorem we obtained in this paper. Noting that a
regular curve of almost $\leq n$ order may have no free dendrites, such as the
triangular Sierpinski curve, the present theorem is not implied by the results
in [32] even in the case of expansive actions.

######  Theorem 1.2.

Let $X$ be a regular curve of almost $\leq n$ order for some positive integer
$n$. Let $G$ be a group acting on $X$ sensitively. Then $G$ contains a free
noncommutative subsemigroup and the action has positive geometric entropy. In
particular, $G$ cannot be nilpotent.

The idea of the proof is to use ping pong game technique as in [5]. However,
since the topology of regular curves is more complicated than that of the
circle and sensitivity is weaker than expansivity, we have to develop some new
ideas to overcome these difficulties. As a supplement to the main theorem, we
remark that the solvable group $\mathbb{Z}\ltimes\mathbb{Z}_{2}$ can act on
the closed interval $[0,1]$ expansively [33].

One may expect that if a continuum $X$ admits no expansive $\mathbb{Z}$
actions, then it admits no expansive nilpotent group actions. However, this is
not true in general. According to Mañé’s results in [22], if $X$ is of
infinite dimension, then it admits no expansive $\mathbb{Z}$ actions; but
there do exist an expansive $\mathbb{Z}^{2}$ action on the infinite product of
circles $\mathbb{T}^{\infty}$ [33]. Moreover, Mouron constructed for each
positive integer $n$, a continuum $X$ which admits an expansive
$\mathbb{Z}^{n+1}$ action but admits no expansive $\mathbb{Z}^{n}$ actions
[26]. These examples show the essential differences between expansive
$\mathbb{Z}$ actions and expansive $\mathbb{Z}^{n}$ actions with $n\geq 2$. Up
to now, for nilpotent group $G$, the answer to Question 1.1 only concentrates
on the case that $X$ is locally connected one dimensional continua. Few is
known when $X$ is not locally connected or is of dimension $>1$.

The main theorem of this paper is only a partial answer to the following
question.

######  Question 1.3.

Do there exist sensitive nilpotent group actions on regular curves?

Here, we remark that H. Kato proved that Suslinian continua admit no expansive
homeomorphisms in [10]. This implies that there are no expansive $\mathbb{Z}$
actions on regular curves as every regular curve is a Suslinian continuum. In
addition, Mai and Shi constructed a sensitive homeomorphism on a Suslinian
continuum in [20], which answered a question proposed by Kato in [11]. So, the
above question is not true if we replace “regular curves” by “Suslinian
continua”. These motivate us to ask the following question.

######  Question 1.4.

Do there exist expansive nilpotent group actions on Suslinian continua?

In Section 2, we will introduce some general notions and facts around group
actions and the topology of continua. We show a convergence property of
subcontinua contained in regular curves in Section 3 and obtained in Section 4
a relation between sensitivity and the existence of transitive open subsystems
for group actions on regular curves. Based on these preparations, we prove in
the last section the existence of semi ping pong for any sensitive group
actions on regular curves of almost $\leq n$ order.

##  2\. Preliminaries

###  2.1. General notions around group actions

Let $X$ be a compact metric space and let ${\rm Homeo}(X)$ be the
homeomorphism group of $X$. Suppose $G$ is a group. A group homomorphism
$\phi:G\rightarrow{\rm Homeo}(X)$ is called a continuous action of $G$ on $X$;
we use the symbol $(X,G,\phi)$ to denote this action. For brevity, we usually
use $gx$ or $g(x)$ instead of $\phi(g)(x)$ and use $(X,G)$ instead of
$(X,G,\phi)$ if no confusion occurs.

For a subset $A$ of $X$, we denote by $\overline{A}$ the closure of $A$ in
$X$. The diameter of $A$ is defined by ${\rm diam}(A)={\rm
sup}\\{d(x,y):x,y\in A\\}$, where $d$ is the metric on $X$. For $x\in X$, the
orbit of $x$ is the set $Gx\equiv\\{gx:g\in G\\}$; $K\subset X$ is called
$G$-invariant if $Gx\subset K$ for every $x\in K$. If $K$ is $G$-invariant, we
naturally get the restriction action of $G$ on $K$, which is denoted by
$(K,G|_{K})$. An action $(X,G)$ is called topologically transitive (or
transitive for brevity) if for any nonempty open subsets $U$ and $V$ of $X$,
there is some $g\in G$ such that $g(U)\cap V\neq\emptyset$; is called point
transitive if the closure ${\overline{Gx}}=X$ for some $x\in X$, and $x$ is
called a transitive point. It is well known that when $X$ is a compact metric
space and $G$ is countable, the definitions of topological transitivity and
point transitivity are equivalent. $(X,G)$ is called sensitive if for some
$c>0$ and for every nonempty open set $U$ in $X$ there is $g\in G$ such that
the diameter ${\rm diam}(g(U))>c$, where $c$ is called a sensitivity constant
of $(X,G)$.

###  2.2. Geometric entropy and semi ping pong

A group $G$ is called nilpotent if there is a normal series of subgroups
$G=G_{0}\rhd G_{1}\rhd\cdots\rhd G_{n}=\\{e\\}$ with $G_{i+1}=[G_{i},G]$ for
each $i$. Nilpotent groups contain no free subsemigroups. Let $(X,G)$ be an
action of group $G$ on space $X$. A tuple $(A,B_{1},B_{2},g_{1},g_{2})$ is
called a semi ping pong if $A,B_{1},B_{2}\subset X$ and $g_{1},g_{2}\in G$
such that $B_{1}\subset A$, $B_{2}\subset A$, $B_{1}\cap B_{2}=\emptyset$,
$g_{1}(A)\subset B_{1}$, and $g_{2}(A)\subset B_{2}$ (see Fig. 1). It is well
known and easy to check that if $(A,B_{1},B_{2},g_{1},g_{2})$ is a semi ping
pong, then the semigroup $S(g_{1},g_{2})$ generated by $g_{1}$ and $g_{2}$ is
free. This implies immediately that $G$ cannot be nilpotent. In some sense,
the notion of semi ping pong can be viewed as a generalization of horseshoe
for group actions, the existence of which means a kind of complexity of the
system.

The notion of geometric entropy was introduced by Ghys, Langevin, and Walczak
in [7] for group actions, which is a generalization of topological entropy for
systems generated by a single map. Let $G$ be a finitely generated group, and
let $S$ be a symmetric generating set of $G$ such that the identity is
contained in $S$ and $s\in S$ if and only if $s^{-1}\in S$. For an element
$g\in G$, if there exist $s_{1},s_{2},\cdots,s_{n}\in S$ such that
$g=s_{1}\cdots s_{n}$, then we say that $g$ has word length $||g||\leq n$. Let
$(X,G,\phi)$ be a group action. Given $\epsilon>0$ and $n>0$, two distinct
points $x,y\in X$ are called $(n,\epsilon)$-separated if there exists some
$g\in G$ such that $||g||\leq n$ and $d(gx,gy)>\epsilon$. A finite subset of
$X$ is called an $(n,\epsilon)$-separated subset of $X$ if any two distinct
points in the set are $(n,\epsilon)$-separated. Denote by $s(\phi,\epsilon,n)$
the maximum cardinality of all $(n,\epsilon)$-separated subsets of $X$. The
geometric entropy $h(\phi)$ of a group action $(X,G,\phi)$ is defined as
follows:

$h(\phi)=\lim_{\epsilon\to 0}\left[\limsup_{n\to\infty}\ \frac{\log
s(\phi,\epsilon,n)}{n}\right].$

The positivity of geometric entropy is also a description of the complexity of
the action. The existence of semi ping pong implies the positivity of
geometric entropy for group actions on compact metric spaces.

Fig.1. The semi ping pong

###  2.3. Notions and results in continuum theory

Recall that a continuum means a compact connected metric space. A continuum
$X$ is called a regular curve if for every $x\in X$ and every open
neighborhood $U$ of $x$, there is an open neighborhood $V$ of $x$ contained in
$U$ such that $|\partial_{X}(V)|$ (the cardinality of the boundary
$\partial_{X}(V)$ of $V$ in $X$) is finite; if the
cardinality$|\partial_{X}(V)|\leq n$ for some previously fixed positive
integer $n$, then $X$ is called a regular curve of order $\leq n$. A regular
curve $X$ is said to be of almost $\leq n$ order for some fixed positive
integer $n$, if for every nonempty open set $U$ of $X$, there is an open set
$V\subset U$ such that $|\partial_{X}(V)|\leq n$. Clearly, any regular curve
of $\leq n$ order is almost $\leq n$ order by the definition. It is easy to
check that dendrites are regular curves of almost $\leq 2$ order; and the
triangular Sierpinski curve (see Fig. 2) is a regular curve of almost $\leq 3$
order. However, there are many dendrites not of finite order, such as the
infinite star (see Fig. 3) which is the union of countable many arcs with
diameters tending to $0$ and with one endpoint being as a common intersecting
point. Regular curves are known to be locally connected.

Fig. 2. The triangular Sierpinski curve Fig. 3. The infinite star

Let $(X,d)$ be a continuum. For every $x\in X$, $B(x,\epsilon)$ denotes the
open ball with center $x$ and radius $\epsilon$. Let $2^{X}$ be the family of
all nonempty closed subsets of $X$, which is called the hyperspace of $X$. Let
$C(X)$ be the set of all subcontinua of $X$. For each $\epsilon>0$ and $A\in
2^{X}$, let $N_{d}(\epsilon,A)=\\{x\in X:d(x,a)<\epsilon\ {\rm for\ some\
}a\in A\\}$. For $A,B\in 2^{X}$, define $d_{H}(A,B)={\rm
inf}\\{\epsilon>0:A\subset N_{d}(\epsilon,B){\rm\ and\ }B\subset
N_{d}(\epsilon,A)\\}$. Then $d_{H}$ is a metric on $2^{X}$ and is called the
Hausdorff metric on it, with respect to which $2^{X}$ and $C(X)$ are compact
metric spaces (see e.g. [27]).

The following theorem is known as the Boundary Bumping Theorem (see e.g. [27,
p. 73]) which will be used in the sequel.

######  Theorem 2.1.

Let $X$ be a continuum and let $U$ be a nonempty proper open subset of $X$. If
$K$ is a component of $\overline{U}$, then
$K\cap\partial_{X}(U)\not=\emptyset.$

##  3\. Topology of regular curves

######  Lemma 3.1.

Let $X$ be a regular curve and $U$ be a connected open subset of $X$ with
$|\partial_{X}(U)|=n$ for some positive integer $n$. If ${\rm
diam}(U)>\epsilon$ for some $\epsilon>0$, then there is some connected open
set $V\subset U$, such that $d(V,\partial_{X}(U))\geq\epsilon/4n$ and ${\rm
diam}(V)\geq\epsilon/4n$.

###### Proof.

Let $\partial_{X}(U)=\\{e_{1},e_{2},\cdots,e_{n}\\}$. We claim that
$U\setminus\bigcup_{i=1}^{n}B(e_{i},\epsilon/2n)\neq\emptyset$. In fact, if
$U\subset\bigcup_{i=1}^{n}B(e_{i},\epsilon/2n)$, then for any two distinct
points $a,b\in U$, by the connectivity of $U$ there are finite
$B(e_{i_{1}},\epsilon/2n),\cdots,B(e_{i_{m}},\epsilon/2n)$ ($m\leq n$) such
that $a\in B(e_{i_{1}},\epsilon/2n)$, $b\in B(e_{i_{m}},\epsilon/2n)$ and
$B(e_{i_{k}},\epsilon/2n)\cap B(e_{i_{k+1}},\epsilon/2n)\neq\emptyset$ for
$1\leq k\leq m-1$. Thus we have

$\displaystyle d(a,b)\leq$ $\displaystyle
d(a,e_{i_{1}})+d(e_{i_{1}},e_{i_{2}})+\cdots+d(e_{i_{m-1}},e_{i_{m}})+d(e_{i_{m}},b)$
$\displaystyle<$
$\displaystyle\frac{\epsilon}{2n}+\frac{\epsilon}{n}\cdot(m-1)+\frac{\epsilon}{2n}$
$\displaystyle\leq$ $\displaystyle\epsilon.$

It follows that ${\rm diam}(U)\leq\epsilon$, which is a contradiction. Hence
$U\setminus\bigcup_{i=1}^{n}B(e_{i},\epsilon/2n)\neq\emptyset$.

Take a point $x\in U\setminus\bigcup_{i=1}^{n}B(e_{i},\epsilon/2n)$. Then
$d(x,\partial_{X}(U))\geq\epsilon/2n$, and hence

$d(B(x,\epsilon/4n),\partial_{X}(U))\geq\epsilon/4n.$

Let $V$ be the component of $U\cap B(x,\epsilon/4n)$ which contains $x$. Then
$d(V,\partial_{X}(U))\geq\epsilon/4n$. Let $W=U\cap B(x,\epsilon/4n).$ Since
$U$ is connected,
$\emptyset\not=\partial_{X}(W)\subset\partial_{X}(B(x,\epsilon/4n))$. This
together with Theorem 2.1 implies
$\emptyset\not=\partial_{X}(V)\subset\partial_{X}(B(x,\epsilon/4n))$. So,
${\rm diam}(V)={\rm diam}(\overline{V})\geq\epsilon/4n$. ∎

######  Proposition 3.2.

Let $X$ be a regular curve and let $(U_{i})_{i=1}^{\infty}$ be a sequence of
connected open subsets of $X$ with $|\partial_{X}(U_{i})|=n$ for some positive
integer $n$ and for each $i$. Suppose that there is some $\epsilon>0$ with
${\rm diam}(U_{i})>\epsilon$ for each $i$. Then there are a nonempty open
subset $W$ of $X$ and infinitely many $i$’s such that $W$ is contained in
$U_{i}$.

###### Proof.

For each $i$, it follows from Lemma 3.1 that there is a connected open subset
$W_{i}\subset U_{i}$ with $d(W_{i},\partial_{X}(U_{i}))\geq\epsilon/4n$ and
${\rm diam}(W_{i})\geq\epsilon/4n$. By the compactness of $2^{X}$ and $C(X)$,
there are subsequences $(\overline{W_{i_{k}}})$ and
$(\partial_{X}(U_{i_{k}}))$ such that $(\overline{W_{i_{k}}})$ converges to a
subcontinuum $A$, and

$None$
$d_{H}(\partial_{X}(U_{i_{k_{1}}}),\partial_{X}(U_{i_{k_{2}}}))<\epsilon/4n,\forall
k_{1}\not=k_{2}.$

Take a point $z\in A$. Then there exists a connected open neighborhood $Q$ of
$z$ such that $\partial_{X}(Q)$ is finite and ${\rm diam}(Q)<\epsilon/4n$.
Since $(\overline{W_{i_{k}}})$ converges to $A$, there exists a positive
integer $N$ such that $W_{i_{k}}\cap Q\neq\emptyset$ for each $k\geq N$.
Noting that ${\rm diam}(W_{i_{k}})\geq\epsilon/4n$ and ${\rm
diam}(Q)<\epsilon/4n$, we have $W_{i_{k}}\nsubseteq Q$. Thus
$W_{i_{k}}\cap\partial_{X}(Q)\neq\emptyset$ by the connectivity of
$W_{i_{k}}$. Hence, there exist a point $p\in\partial_{X}(Q)$ and infinitely
many $k$’s such that $p\in W_{i_{k}}$. Passing to a subsequence if necessary,
we may suppose that $p\in W_{i_{k}}$ for each $k\geq N$.

Let $W=W_{i_{N}}$. To complete the proof, we only need to show that $W\subset
U_{i_{k}}$ for all $k\geq N$. Otherwise, there is some $k^{\prime}\geq N$ with
$W\nsubseteq U_{i_{k^{\prime}}}$. Since $W$ is connected and $p\in W\cap
U_{i_{k^{\prime}}}$, there is a point $e\in
W\cap\partial_{X}(U_{i_{k^{\prime}}})$. By $(3.1)$, there is
$e^{\prime}\in\partial_{X}(U_{i_{N}})$ such that
$d(e,e^{\prime})<\epsilon/4n$. Then we have $d(W,e^{\prime})\leq
d(e,e^{\prime})<\epsilon/4n$. This contradicts the assumption that
$d(W,\partial_{X}(U_{i_{N}}))\geq\epsilon/4n$ at the beginning. ∎

##  4\. Sensitivity and transitive open subsystem

######  Lemma 4.1.

Let $X$ be a regular curve. Then for every constant $c>0$ there is a finite
set $A\subset X$ such that the diameter of each component of $X\setminus A$ is
not greater than $c$.

###### Proof.

Since $X$ is a regular curve, for every $x\in X$, there is an open
neighborhood $V_{x}$ of $x$ such that $\partial_{X}(V_{x})$ is finite and
${\rm diam}(V_{x})\leq c$. Then the family $\mathcal{V}=\\{V_{x}:x\in X\\}$ is
an open cover of $X$. By the compactness of $X$, there is a finite subcover
$\\{V_{x_{i}}\\}_{i=1}^{n}\subset\mathcal{V}$ of $X$. Thus
$\bigcup_{i=1}^{n}\partial_{X}(V_{x_{i}})$ is finite, which is denoted by $A$.
For each component $C$ of $X\setminus A$, there exists some $V_{x_{i}}$ such
that $C\cap V_{x_{i}}\neq\emptyset$. Since $C$ is connected and
$C\cap\partial_{X}(V_{x_{i}})=\emptyset$, $C\subset V_{x_{i}}$. This implies
that ${\rm diam}(C)\leq c$. ∎

The following proposition discusses the relations between sensitivity and the
existence of transitive open subsystems for group actions on regular curves.

######  Proposition 4.2.

Let $X$ be a regular curve and let $G$ be a group acting on $X$ sensitively.
Then there is a $G$-invariant open set $V$ in $X$ such that the restriction
$(V,G|_{V})$ is transitive.

###### Proof.

Let $c$ be a sensitivity constant of the action $(X,G)$. By Lemma 4.1, there
is a finite set $A\subset X$ such that each component of $X\setminus A$ has
diameter $\leq c$. We claim that $\overline{G(A)}=\bigcup_{a\in
A}\overline{Ga}$ has non-empty interior. Otherwise, $\overline{G(A)}$ is
nowhere dense in $X$. Thus $X\setminus\overline{G(A)}$ is a $G$-invariant
dense open subset of $X$. Noting that $X$ is locally connected, each component
$U$ of $X\setminus\overline{G(A)}$ is open in $X$, and for each $g\in G$,
$g(U)\subset X\setminus\overline{G(A)}$. This implies that ${\rm
diam}(g(U))\leq c$, which contradicts the assumption that $c$ is a sensitivity
constant of $(X,G)$. Thus the claim holds.

By the claim, there is an $a\in A$ such that ${\rm
Int}(\overline{Ga})\neq\emptyset$. Let $V={\rm Int}(\overline{Ga})$. Then $V$
is a $G$-invariant open set and $(V,G|_{V})$ is transitive. ∎

##  5\. Existence of semi ping pong

Now let us prove the main theorem 1.2. According to the discussions in Section
2.2, we need only to show the existence of semi ping pong.

###### Proof.

By Proposition 4.2, there is a $G$-invariant open subset $V\subset X$ such
that $(V,G|_{V})$ is transitive. Let $x_{0}$ be a transitive point of
$(V,G|_{V})$. Noting that $X$ is a regular curve of almost $\leq n$ order, for
each $m\in\mathbb{N}$, there is a nonempty open set $V_{m}\subset
B(x_{0},1/m)\cap V$ with $|\partial_{X}(V_{m})|\leq n$. By sensitivity, there
exists some $g_{m}\in G$ such that ${\rm diam}(g_{m}(V_{m}))>c$, where $c$ is
a sensitivity constant of $(X,G)$. Clearly, $|\partial_{X}(g_{m}(V_{m}))|\leq
n$ for every $m\in\mathbb{N}$.

Applying Proposition 3.2 to $(g_{m}(V_{m}))_{m=1}^{\infty}$, we get a
subsequence $(g_{m_{k}}(V_{m_{k}}))_{k=1}^{\infty}$ and a nonempty open subset
$D\subset X$ such that $D\subset\bigcap g_{m_{k}}(V_{m_{k}})$. Since
$g_{m_{k}}^{-1}(D)\subset V_{m_{k}}\subset B(x_{0},1/m_{k})$ for each $k$, we
have

$None$ $\lim_{k\to\infty}g_{m_{k}}^{-1}(D)=\\{x_{0}\\}.$

Choose two nonempty open subsets $B_{1},B_{2}\subset D$ with $B_{1}\cap
B_{2}=\emptyset$. Since $x_{0}$ is a transitive point of $(V,G|_{V})$, there
are $f,h\in G$ such that $f(x_{0})\in B_{1}$ and $h(x_{0})\in B_{2}$.
According to $(5.1)$ and the continuity of $f$ and $h$, there are $k_{1}$ and
$k_{2}$ such that $fg_{m_{k_{1}}}^{-1}(D)\subset B_{1}$ and
$hg_{m_{k_{2}}}^{-1}(D)\subset B_{2}$. Thus
$\\{D,B_{1},B_{2},fg_{m_{k_{1}}}^{-1},hg_{m_{k_{2}}}^{-1}\\}$ becomes a semi
ping pong. Thus we complete the proof. ∎

### Acknowledgements

The work is supported by NSFC (No. 11771318, 11790274).

## References

  * [1]
  * [2] Akin, E., Kolyada, S.: Li-Yorke sensitivity. Nonlinearity, 16, 1421-1433 (2003)
  * [3] Auslander, J., Yorke, J. A.: Interval maps, factors of maps and chaos. Tohoku Math. J., 32, 177-188 (1980)
  * [4] Chung, N., Li, H. F.: Homoclinic groups, IE groups, and expansive algebraic actions. Invent. Math., 199, 805-858 (2015)
  * [5] Connell, C., Furman, A., Hurder, S.: Expansive maps of the circle. http://homepages.math.uic.edu/~hurder/publications
  * [6] Devaney, R. L.: An Introduction to Chaotic Dynamical Systems, 2nd edn, Addison-Wesley Publishing Company, Redwood City, 1989
  * [7] Ghys, É., Langevin, R., Walczak, P.: Entropie géométrique des feuilletages. Acta Math., 160, 105-142 (1988)
  * [8] Glasner, E., Weiss, B.: Sensitive dependence on initial conditions. Nonlinearity, 6, 1067–1075 (1993)
  * [9] Hiraide, K.: There are no expansive homeomorphisms on $\mathbb{S}^{2}$. Dynamical systems and singular phenomena, 214-220 (Kyoto, 1986), World Sci. Adv. Ser. Dynam. Systems, 2, World Sci. Publishing, Singapore, 1987.
  * [10] Kato, H.: The nonexistence of expansive homeomorphisms of Suslinian continua. J. Math. Soc. Japan, 42, 631-637 (1990)
  * [11] Kato, H.: Continuum-wise expansive homeomorphisms. Canad. J. Math., 45, 576-598 (1993)
  * [12] Kato, H.: The nonexistence of expansive homeomorphisms of chainable continua. Fund. Math., 149, 119-126 (1996)
  * [13] Kato, H.: Topological entropy of piecewise embedding maps on regular curves. Ergodic Theory Dynam. Systems, 26, 1115-1125 (2006)
  * [14] Kato, H.: Topological entropy of maps on regular curves. Topology Appl., 154, 1027-1031 (2007)
  * [15] Kato, H., Mouron, C.: Hereditarily indecomposable compacta do not admit expansive homeomorphisms. Proc. Amer. Math. Soc., 136, 3689-3696 (2008)
  * [16] Katok, A., Llavé, R., Pesin, Y., Weiss, H: Smooth ergodic theory and its applications, Proceedings of the AMS Summer Research Institute held at University of Washington, Seattle, WA, July 26-August 13, 1999, Proceedings of Symposia in Pure Mathematics, Vol. 69, American Mathmatical Society, Providence, RI, 2001.
  * [17] Li, J., Tu, S., Ye, X. D.: Mean equicontinuity and mean sensitivity. Ergodic Theory Dynam. Systems, 35, 2587-2612 (2015)
  * [18] Lind, D., Schmidt, K.: Homoclinic points of algebraic ${\mathbb{Z}}^{d}$-actions. J. Amer. Math. Soc., 12, 953-980 (1999)
  * [19] Mai, J. H., Shi, E. H.: The nonexistence of expansive commutative group actions on Peano continua having free dendrites. Topology Appl., 155, 33-38 (2007)
  * [20] Mai, J. H., Shi, E. H.: The nonexistence of sensitive commutative group actions on graphs. Science in China, Ser. A: Mathematics, 50, 1197-1204 (2007)
  * [21] Mai, J. H., Shi, E. H.: A class of spaces that admit no sensitive commutative group actions. Fund. Math., 217, 1-12 (2012)
  * [22] Mañé, R.: Expansive homeomorphisms and topological dimension. Trans. Amer. Math. Soc., 252, 313-319 (1979)
  * [23] Margulis, G.: Free subgroups of the homeomorphism group of the circle. C. R. Acad. Sci. Paris Sér. I Math., 331, 669-674 (2000)
  * [24] Mouron, C.: Tree-like continua do not admit expansive homeomorphisms. Proc. Amer. Math. Soc., 130, 3409-3413 (2002)
  * [25] Mouron, C.: The Solenoids are the only circle-like continua that admit expansive homeomorphisms. Fund. Math., 205, 237-264 (2009)
  * [26] Mouron, C.: Continua that admit expansive $\mathbb{Z}^{n+1}$ actions but not expansive $\mathbb{Z}^{n}$ actions. Houston J. Math., 36, 167-180 (2010)
  * [27] Nadler, Jr., S. B.: Continuum Theory: An Introduction, Marcel Dekker Inc., New York, 1992
  * [28] Naghmouchi, I.: Homeomorphisms of regular curves. J. Difference Equ. Appl., 23, 1485-1490 (2017)
  * [29] Naghmouchi, I.: Dynamics of homeomorphisms of regular curves. Colloq. Math., 162, 263-277 (2020)
  * [30] O’ Brien, T., Reddy, W.: Each compact orientable surface of positive genus admits an expansive homeomorphism. Pacific J. Math., 35, 737-741 (1970)
  * [31] Shao, S., Ye, X. D.: A non-PI minimal system is Li-Yorke sensitive. Proc. Amer. Math. Soc., 146, 1105-1112 (2018)
  * [32] Shi, E. H., Wang, S. H.: The ping-pong game, geometric entropy and expansiveness for group actions on Peano continua having free dendrites. Fund. Math., 203, 21-37 (2009)
  * [33] Shi, E. H., Zhou, L. Z.: The nonexistence of expansive $\mathbb{Z}^{d}$ actions on graphs. Acta Math. Sin. (Engl. Ser.), 21, 1509-1514 (2005).
  * [34] Subrahmonian Moothathu, T. K.: Stronger forms of sensitivity for dynamical systems. Nonlinearity, 20, 2115-2126 (2007)
  * [35] Williams, R. F.: A note on unstable homeomorphisms. Proc. Amer. Math. Soc., 6, 308-309 (1955)
  * [36] Ye, X. D., Yu, T.: Sensitivity proximal extension and higher order almost automorphy. Trans. Amer. Math. Soc., 370, 3639-3662 (2018)


# Optimal work fluctuations for finite-time and weak processes

Pierre Nazé<EMAIL_ADDRESS>Universidade Estadual Paulista, 14800-090,
Araraquara, São Paulo, Brazil

###### Abstract

The optimal protocols for the irreversible work achieve their maximum
usefulness if their work fluctuations are the smallest ones. In this work, for
classical and isothermal processes subjected to finite-time and weak drivings,
I show that the optimal protocol for the irreversible work is the same for the
variance of work. This conclusion is based on the fluctuation-dissipation
relation $\overline{W}=\Delta F+\beta\sigma_{W}^{2}/2$, extended now to
finite-time and weak drivings. To illustrate it, I analyze a white noise
overdamped Brownian motion subjected to an anharmonic stiffening trap for fast
processes. By contrast with the already known results in the literature for
classical systems, the linear-response theory approach of the work
probabilistic distribution is not a Gaussian reduction.

##  I Introduction

The optimization of the thermodynamic work of driving processes is a practical
example where averages and fluctuations work side-by-side. Finding a protocol
to which the external parameter of the system leads to the minimal value of
the average of the thermodynamic work has its utmost value when its
fluctuations are equally minimal.

Few remarkable works have been done in the last decades unveiling relations
between those concepts involving averages and fluctuations of the
thermodynamic work. In the classical realm, Jarzynski [1] found out from its
famous equality that systems with time-dependent quadratic potentials,
starting its driving process in contact with a heat bath of temperature
$\beta^{-1}$, obey the following fluctuation-dissipation relation

$\overline{W}=\Delta F+\frac{\beta}{2}\sigma_{W}^{2},$ (1)

where $\overline{W}$ is the average work, $\sigma_{W}^{2}$ is the variance of
the work, and $\Delta F$ is the difference of Helmholtz free energy between
the final and initial equilibrium states of the process. Some years later,
Speck and Seifert [2] deduced the same result, but now for slowly-varying
processes, that is, processes whose rate is not fast enough when compared to
the relaxation rate of the system. Very recently, in the quantum realm, Miller
and coauthors found out that such fluctuation-dissipation relation fails when
the coherence of quantum systems is added [3]. From the point-of-view of
optimization, in these regimes where the fluctuation-dissipation relation
holds, the minimal work is the precisest one.

The objective of this work is to derive the same result for isothermal,
finite-time, and weak driving processes, where the rate of the process is
arbitrary and the perturbation of the external parameter is small compared to
its initial value. To accomplish that, I generalize the fluctuation-
dissipation relation (1) to this regime using linear-response theory. To
illustrate such a relation, I analyze the case of the overdamped Brownian
motion subjected to an anharmonic stiffening trap and white noise for fast
processes. Finally, as a main consequence of our findings, the Euler-Lagrange
equation that determines the optimal protocol for the minimization of the
variance of the work is the same as the one that minimizes the irreversible
work.

##  II Preliminaries

I start defining notations and developing the main concepts to be used in this
work. This section is based on the technical introductory section of Ref. [4].

Consider a classical system with a Hamiltonian
$\mathcal{H}(\mbox{\boldmath$z$}(\mbox{\boldmath$z_{0}$},t)),\lambda(t))$,
where $\mbox{\boldmath$z$}(\mbox{\boldmath$z_{0}$},t)$ is a point in the phase
space $\Gamma$ evolved from the initial point $z_{0}$ until time $t$, with
$\lambda(t)$ being a time-dependent external parameter. During a switching
time $\tau$, the external parameter is changed from $\lambda_{0}$ to
$\lambda_{0}+\delta\lambda$, with the system being in contact with a heat bath
of temperature $\beta\equiv{(k_{B}T)}^{-1}$, where $k_{B}$ is Boltzmann’s
constant. The average work performed on the system during this interval of
time is

$\overline{W}\equiv\int_{0}^{\tau}\left\langle\overline{\partial_{\lambda}\mathcal{H}}(t)\right\rangle_{0}\dot{\lambda}(t)dt,$
(2)

where $\partial_{\lambda}$ is the partial derivative in respect to $\lambda$
and the superscripted dot the total time derivative. The generalized force
$\left\langle\overline{\partial_{\lambda}\mathcal{H}}\right\rangle_{0}$ is
calculated using the averaging $\overline{\cdot}$ over the stochastic path and
the averaging $\langle\cdot\rangle_{0}$ over the initial canonical ensemble.
The external parameter can be expressed as

$\lambda(t)=\lambda_{0}+g(t)\delta\lambda,$ (3)

where to satisfy the initial conditions of the external parameter the protocol
$g(t)$ must satisfy the following boundary conditions

$g(0)=0,\quad g(\tau)=1.$ (4)

Linear-response theory aims to express average quantities until the first-
order of some perturbation parameter considering how this perturbation affects
the observable to be averaged and the probabilistic distribution [5]. In our
case, we consider that the parameter does not considerably change during the
process, $|g(t)\delta\lambda/\lambda_{0}|\ll 1$, for all $t\in[0,\tau]$ and
$\lambda_{0}\neq 0$. The generalized force can be approximated until the first
order as

$\begin{split}\left\langle\overline{\partial_{\lambda}\mathcal{H}}(t)\right\rangle_{0}=&\,\left\langle\partial_{\lambda}\mathcal{H}\right\rangle_{0}-\delta\lambda\widetilde{\Psi}_{0}g(t)\\\
&+\delta\lambda\int_{0}^{t}\Psi_{0}(t-t^{\prime})\dot{g}(t^{\prime})dt^{\prime},\end{split}$
(5)

where

$\Psi_{0}(t)=\beta\left\langle\partial_{\lambda}\mathcal{H}(0)\overline{\partial_{\lambda}\mathcal{H}}(t)\right\rangle_{0}-\mathcal{C}$
(6)

is the relaxation function and
$\widetilde{\Psi}_{0}\equiv\Psi_{0}(0)-\left\langle\partial_{\lambda\lambda}^{2}\mathcal{H}\right\rangle_{0}$
[5]. The constant $\mathcal{C}$ is calculated to vanish the relaxation
function for long times [5]. Combining Eqs. (2) and (5), the average work
performed at the linear response of the generalized force is

$\begin{split}\overline{W}=&\,\delta\lambda\left\langle\partial_{\lambda}\mathcal{H}\right\rangle_{0}-\frac{\delta\lambda^{2}}{2}\widetilde{\Psi}_{0}\\\
&+\delta\lambda^{2}\int_{0}^{\tau}\int_{0}^{t}\Psi_{0}(t-t^{\prime})\dot{g}(t^{\prime})\dot{g}(t)dt^{\prime}dt.\end{split}$
(7)

We observe that the double integral on Eq. (7) vanishes for long switching
times [6], which indicates that the other terms are the contribution of the
difference of Helmholtz’s free energy. The irreversible work $W_{\rm irr}$ is
therefore

$\begin{split}W_{\rm
irr}=\frac{\delta\lambda^{2}}{2}\int_{0}^{\tau}\int_{0}^{\tau}\Psi_{0}(t-t^{\prime})\dot{g}(t^{\prime})\dot{g}(t)dt^{\prime}dt,\end{split}$
(8)

where the symmetric property of the relaxation function was used [5]. The
regime where such expression holds is the finite-time and weak processes,
where the ratio $\delta\lambda/\lambda_{0}\ll 1$, while $\tau_{R}/\tau$ is
arbitrary.

##  III Optimization problem

Consider the irreversible work rewritten in terms of the protocols $g(t)$
instead of its derivative

$\displaystyle W_{\rm irr}=$
$\displaystyle\frac{\delta\lambda^{2}}{2}\Psi_{0}(0)+\delta\lambda^{2}\int_{0}^{\tau}\dot{\Psi}_{0}(\tau-t)g(t)dt$
(9)
$\displaystyle-\frac{\delta\lambda^{2}}{2}\int_{0}^{\tau}\int_{0}^{\tau}\ddot{\Psi}_{0}(t-t^{\prime})g(t)g(t^{\prime})dtdt^{\prime}.$
(10)

Using the calculus of variations, one can derive the Euler-Lagrange equation
that furnishes the optimal protocol $g^{*}(t)$ that minimizes the irreversible
work [4]

$\int_{0}^{\tau}\ddot{\Psi}_{0}(t-t^{\prime})g^{*}(t^{\prime})dt^{\prime}=\dot{\Psi}_{0}(\tau-t).$
(11)

In particular, the optimal irreversible work will be [4]

$W_{\rm
irr}^{*}=\frac{\delta\lambda^{2}}{2}\Psi_{0}(0)+\frac{\delta\lambda^{2}}{2}\int_{0}^{\tau}\dot{\Psi}_{0}(\tau-t)g^{*}(t)dt.$
(12)

The objective of this work is to derive the linear response version of the
variance of the work

$\sigma_{W}^{2}=\langle\overline{W^{2}}\rangle_{0}-\langle\overline{W}\rangle_{0}^{2}$
(13)

and to find an Euler-Lagrange equation of such a functional. In this way, the
Euler-Lagrange equation will furnish the optimal protocol that will minimize
the variance of the work. I shall accomplish it by using an extension to
finite-time and weak processes of the fluctuation-dissipation relation (1).

##  IV Fluctuation-dissipation relation

To calculate the linear response version of the variance of the work, I remark
that such quantity must be of second-order in the driving strength
$\delta\lambda$. This occurs since the work starts in the first order and must
be squared to calculate its variation. Therefore, it can be approximated as
(see Appendix for details)

$\displaystyle\left\langle\overline{W^{2}}\right\rangle_{0}$
$\displaystyle\approx\int_{0}^{\tau}\int_{0}^{\tau}\left\langle\partial_{\lambda}\mathcal{H}(0)\overline{\partial_{\lambda}\mathcal{H}}(t-t^{\prime})\right\rangle_{0}\dot{\lambda}(t)\dot{\lambda}(t^{\prime})dtdt^{\prime},$
(14)

To calculate $\left\langle\overline{W}\right\rangle_{0}^{2}$, the same
reasoning is followed

$\displaystyle\left\langle\overline{W}\right\rangle_{0}^{2}$
$\displaystyle\approx\left\langle\partial_{\lambda}\mathcal{H}(0)\right\rangle_{0}^{2}.$
(15)

Therefore, by rewriting the variance of the work in terms of the relaxation
function, we have

$\sigma_{W}^{2}\approx\frac{1}{\beta}\int_{0}^{\tau}\int_{0}^{\tau}\Psi_{0}(t-t^{\prime})\dot{\lambda}(t)\dot{\lambda}(t^{\prime})dtdt^{\prime}+\mathcal{C}-\left\langle\partial_{\lambda}\mathcal{H}(0)\right\rangle_{0}^{2}$
(16)

The first term on the right-hand side is $2W_{\rm irr}/\beta$, while the
second one, because the relaxation function decorrelates for long times, is
(see Appendix for details)

$\displaystyle\mathcal{C}-\left\langle\partial_{\lambda}\mathcal{H}(0)\right\rangle_{0}^{2}$
$\displaystyle=0,$ (17)

This immediately leads to the extended version of the work fluctuation-
dissipation theorem

$\overline{W}=\Delta F+\frac{\beta}{2}\sigma_{W}^{2},$ (18)

which now holds for finite-time and weak processes. Observe that such a
relation indicates that the work ceases to fluctuate for large switching
times, as demonstrated in Ref. [7]. Similar expressions for the first and
second moments were also discussed in this article.

In the following I present an example that treats a case outside the
hypotheses of Jarzynski’s and Seifert’s works [1, 2]: a stochastic model with
a time-dependent non-quadratic potential subjected to fast processes.

##  V Example

I am going to illustrate the relation (1) for a white noise overdamped
Brownian particle, subjected to an anharmonic stiffening trap. First, I
consider an overdamped Brownian particle in contact with a heat bath, whose
dynamics are governed by the following Langevin equation

$\dot{x}(t)+\frac{1}{\gamma}\partial_{x}\mathcal{V}(x(t),\lambda(t))=\eta(t),$
(19)

where $x(t)$ is its position at the instant $t$, $\gamma$ is the damping
coefficient, $\lambda(t)$ is the control parameter and $\eta(t)$ is a Gaussian
white noise characterized by

$\overline{\eta(t)}=0,\quad\overline{\eta(t)\eta(t^{\prime})}=\frac{2}{\gamma\beta}\delta(t-t^{\prime}).$
(20)

The time-dependent potential will be a quartic anharmonic stiffening trap

$\mathcal{V}(x(t),\lambda(t))=\frac{\lambda(t)}{4}x^{4}(t).$ (21)

Figure 1: Verification of the work fluctuation-dissipation theorem of Eq. (1)
for processes with $\tau=10^{-1},10^{-2},10^{-3}$. The black dots correspond
to the values of $\overline{W}$ and the red squares to the values of $\Delta
F+\beta\sigma_{W}^{2}/2$. We considered $\gamma=1$, $\beta=1$, $\lambda_{0}=1$
and $\delta\lambda=0.01$. The agreement is satisfactory.

To verify the work fluctuation-dissipation theorem, we numerically compare the
average work $\overline{W}$ with the sum of $\Delta F$ and
$\beta\sigma_{W}^{2}/2$. To do so, we consider a linear protocol, given by

$g(t)=\frac{t}{\tau},$ (22)

and use the value of $\Delta F$ furnished by its analytical expression

$\Delta
F=\frac{1}{4\beta}\ln{\left(1+\frac{\delta\lambda}{\lambda_{0}}\right)}.$ (23)

Repeating $N_{\rm exp}$ experiments of sampling $N$ initial conditions in the
canonical ensemble of the system for a stochastic process with a number of
divisions $N_{d}$ of the switching time $\tau$, I collect the data for the
analysis [8]. To achieve convergence in at least three significant digits in
the relation (1), we use $N_{\rm exp}=10$, $N=2\times 10^{5}$ and $N_{d}=200$,
for processes where $\tau=10^{-1},10^{-2},10^{-3}$. It was also used
$\lambda_{0}=1$, $\delta\lambda=0.01$, $\gamma=1$ and $\beta=1$. It was
assumed that in the approximation of 1% in the perturbation linear-response
theory would be in its range of validity, according to previous works [9, 10,
11]. The results of plotting the averages of $\overline{W}$ and $\Delta
F+\beta\sigma_{W}^{2}/2$ with respective standard deviations are depicted in
Fig. 1. The agreement is satisfactory. The independence of the irreversible
work on $\tau$ is indicative that the system is performing a fast process
since no calculation of relaxation function was indeed made and a comparison
with the relaxation time was not done [4].

##  VI Non-Gaussian probabilistic distribution

By contrast with Ref. [1, 2], the linear-response approach used in this
approximation of the work probabilistic distribution is not a reduction to a
Gaussian one. Indeed, Fig. 2 depicts the work probabilistic distribution for a
process with $\tau=0.01,\lambda_{0}=1,\delta\lambda=0.01,\gamma=1,\beta=1$.
For this particular case, we have a non-Gaussian distribution, where its
skewness is $\mu_{3}=3.941$, which indicates a long right tail and non-
Gaussianity.

Figure 2: Non-Gaussian work probabilistic distribution for a process with
$\tau=0.01,\lambda_{0}=1,\delta\lambda=0.01,\gamma=1,\beta=1$. The positive
value of the skewness $\mu_{3}$ indicates the long right tail of the graphic.

##  VII Optimal work fluctuations

Ever since it holds the fluctuation-dissipation relation for finite-time and
weak processes, the functional of the variance of the work is the same (up to
a factor $\beta/2$) as the one of the irreversible work. Therefore, the Euler-
Lagrange equation that describes its optimization is the same as Eq. (11).
This means that, for finite-time and weak processes, if one achieves the
minimal work spent, one does it in the precisest way possible.

Consider as an example the white noise overdamped Brownian motion subjected to
a harmonic stiffening trap. By looking at the optimal protocol that minimizes
the irreversible work in Ref. [4], the optimal protocol that minimizes the
variance of work will be

$g^{*}(t)=\frac{t+\tau_{R}}{\tau+2\tau_{R}},$ (24)

with jumps at the beginning and end of the process, where $\tau_{R}$ is its
relaxation time. The optimal work variance will be

$\sigma_{W}^{2*}=\frac{1}{\beta^{2}}\frac{{(\delta\lambda/\lambda_{0})}^{2}}{\tau+2\tau_{R}}.$
(25)

For this case, the variance is increased for fast processes, $\tau/\tau_{R}\ll
1$, or high temperatures, $\beta\ll 1$. As predicted by [7], the variance
vanishes for slowly-varying processes, $\tau/\tau_{R}\gg 1$.

##  VIII Final remarks

In this work, using linear-response theory, I have extended the fluctuation-
dissipation relation (1) for finite-time and weak processes. To illustrate
such a result, I analyzed the example of the white noise overdamped Brownian
motion subjected to an anharmonic stiffening trap. Such an extension shows
that the optimal protocol that minimizes the irreversible work is the same
that minimizes the variance of the work. By contrast with previous results in
the literature for classical systems, the linear-response approach is not a
Gaussian reduction in the work probabilistic distribution. This work opens the
door for fluctuation-dissipation relations in higher orders of nonlinear
response.

## References

  * Jarzynski [1997] C. Jarzynski, Physical Review Letters 78, 2690 (1997).
  * Speck and Seifert [2004] T. Speck and U. Seifert, Physical Review E 70, 066112 (2004).
  * Miller  _et al._ [2019] H. J. D. Miller, M. Scandi, J. Anders, and M. Perarnau-Llobet, Physical Review Letters 123, 230603 (2019).
  * Nazé  _et al._ [2022] P. Nazé, S. Deffner, and M. V. S. Bonança, Journal of Physics Communications 6, 083001 (2022).
  * Kubo  _et al._ [2012] R. Kubo, M. Toda, and N. Hashitsume,  _Statistical physics II: nonequilibrium statistical mechanics_, Vol. 31 (Springer Science & Business Media, 2012).
  * Nazé and Bonança [2020] P. Nazé and M. V. S. Bonança, Journal of Statistical Mechanics: Theory and Experiment 2020, 013206 (2020).
  * Yi and Talkner [2021] J. Yi and P. Talkner, Phys. Rev. E 104, L062102 (2021).
  * Tomé and De Oliveira [2015] T. Tomé and M. J. De Oliveira,  _Stochastic dynamics and irreversibility_ (Springer, 2015).
  * Acconcia and Bonança [2015] T. V. Acconcia and M. V. S. Bonança, Phys. Rev. E 91, 042141 (2015).
  * Nazé  _et al._ [2022] P. Nazé, M. V. S. Bonança, and S. Deffner, Entropy 24, 10.3390/e24050666 (2022).
  * Kamizaki  _et al._ [2022] L. P. Kamizaki, M. V. S. Bonança, and S. R. Muniz, Phys. Rev. E 106, 064123 (2022).

##  Appendix A Fluctuation-dissipation relation

I derive first the second-order of $\left\langle\overline{W^{2}}\right\rangle$
in terms of $\delta\lambda$. One has

$\displaystyle\left\langle\overline{W^{2}}\right\rangle_{0}$
$\displaystyle=\left\langle\overline{\left(\int_{0}^{\tau}\partial_{\lambda}\mathcal{H}(t)\dot{\lambda}(t)dt\right)^{2}}\right\rangle_{0}$
(26)
$\displaystyle=\left\langle\overline{\int_{0}^{\tau}\int_{0}^{\tau}\partial_{\lambda}\mathcal{H}(t_{1})\partial_{\lambda}\mathcal{H}(t_{2})\dot{\lambda}(t_{1})\dot{\lambda}(t_{2})dt_{1}dt_{2}}\right\rangle_{0}$
(27)
$\displaystyle=\int_{0}^{\tau}\int_{0}^{\tau}\left\langle\overline{\partial_{\lambda}\mathcal{H}(t_{1})\partial_{\lambda}\mathcal{H}(t_{2})}\right\rangle_{0}\dot{\lambda}(t_{1})\dot{\lambda}(t_{2})dt_{1}dt_{2}$
(28)
$\displaystyle=\int_{0}^{\tau}\int_{0}^{\tau}\left\langle\overline{e^{\mathcal{L}t_{1}}\partial_{\lambda}\mathcal{H}(0)e^{\mathcal{L}t_{2}}\partial_{\lambda}\mathcal{H}(0)}\right\rangle_{0}\dot{\lambda}(t_{1})\dot{\lambda}(t_{2})dt_{1}dt_{2}$
(29)
$\displaystyle\approx\int_{0}^{\tau}\int_{0}^{\tau}\left\langle\overline{e^{\mathcal{L}_{0}t_{1}}\partial_{\lambda}\mathcal{H}(0)e^{\mathcal{L}_{0}t_{2}}\partial_{\lambda}\mathcal{H}(0)}\right\rangle_{0}\dot{\lambda}(t_{1})\dot{\lambda}(t_{2}),$
(30)

where, in the time-evolution operator $e^{\mathcal{L}t}$ of the perturbed
solution, I used its non-perturbed solution version

$e^{\mathcal{L}t}\approx e^{\mathcal{L}_{0}t}$ (31)

to furnish the second-order expansion of $\langle\overline{W^{2}}\rangle_{0}$.
Therefore

$\left\langle\overline{W^{2}}\right\rangle_{0}\approx\int_{0}^{\tau}\int_{0}^{\tau}\left\langle\overline{\partial_{\lambda}\mathcal{H}(0)e^{\mathcal{L}_{0}(t_{1}-t_{2})}\partial_{\lambda}\mathcal{H}(0)}\right\rangle_{0}\dot{\lambda}(t_{1})\dot{\lambda}(t_{2})dt_{1}dt_{2}.$
(32)

Using the same argument, one has for
$\left\langle\overline{W}\right\rangle_{0}^{2}$

$\displaystyle\left\langle\overline{W}\right\rangle_{0}^{2}$
$\displaystyle=\left(\int_{0}^{\tau}\left\langle\overline{\partial_{\lambda}\mathcal{H}(t)}\right\rangle_{0}\dot{\lambda}(t)dt\right)^{2}$
(33)
$\displaystyle=\int_{0}^{\tau}\int_{0}^{\tau}\left\langle\overline{\partial_{\lambda}\mathcal{H}(t_{1})}\right\rangle_{0}\left\langle\overline{\partial_{\lambda}\mathcal{H}(t_{2})}\right\rangle_{0}\dot{\lambda}(t_{1})\dot{\lambda}(t_{2})dt_{1}dt_{2}$
(34)
$\displaystyle\approx\int_{0}^{\tau}\int_{0}^{\tau}\left\langle\partial_{\lambda}\mathcal{H}(0)\right\rangle_{0}\left\langle\partial_{\lambda}\mathcal{H}(0)\right\rangle_{0}\dot{\lambda}(t_{1})\dot{\lambda}(t_{2})dt_{1}dt_{2}$
(35)
$\displaystyle=\int_{0}^{\tau}\int_{0}^{\tau}\left\langle\partial_{\lambda}\mathcal{H}(0)\right\rangle_{0}^{2}\dot{\lambda}(t_{1})\dot{\lambda}(t_{2})dt_{1}dt_{2}$
(36)
$\displaystyle=\left\langle\partial_{\lambda}\mathcal{H}(0)\right\rangle_{0}^{2}.$
(37)

Finally, the difference
$\mathcal{C}-\left\langle\partial_{\lambda}\mathcal{H}(0)\right\rangle_{0}^{2}$
is given by

$\displaystyle\mathcal{C}-\left\langle\partial_{\lambda}\mathcal{H}(0)\right\rangle_{0}^{2}$
$\displaystyle=\left\langle\partial_{\lambda}\mathcal{H}(0)\partial_{\lambda}\mathcal{H}(\infty)\right\rangle_{0}-\left\langle\partial_{\lambda}\mathcal{H}(0)\right\rangle_{0}^{2}$
(38)
$\displaystyle=\left\langle\partial_{\lambda}\mathcal{H}(0)\right\rangle_{0}\left\langle\partial_{\lambda}\mathcal{H}(\infty)\right\rangle_{0}-\left\langle\partial_{\lambda}\mathcal{H}(0)\right\rangle_{0}^{2}$
(39)
$\displaystyle=\left\langle\partial_{\lambda}\mathcal{H}(0)\right\rangle_{0}^{2}-\left\langle\partial_{\lambda}\mathcal{H}(0)\right\rangle_{0}^{2}$
(40) $\displaystyle=0.$ (41)