## Abstract

In this paper, we propose and experimentally demonstrate a novel scheme that helps to solve an any-number-armed bandit problem by utilizing two parallel simultaneously-generated chaotic signals and the epsilon (*ɛ*)-greedy strategy. In the proposed scheme, two chaotic signals are experimentally generated, and then processed by an 8-bit analog-to-digital conversion (ADC) with 4 least significant bits (LSBs), to generate two amplitude-distribution-uniform sequences for decision-making. The correspondence between these two random sequences and different arms is established by a mapping rule designed in virtue of the ɛ-greedy-strategy. Based on this, decision-making for an exemplary 5-armed bandit problem is successfully performed, and moreover, the influences of the mapping rule and unknown reward probabilities on the correction decision rate (CDR) performance for the 4-armed to 7-armed bandit problems are investigated. This work provides a novel way for solving the arbitrary-number-armed bandit problem.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Reinforcement learning is a field in machine learning that emphasizes how to take actions in response to changes in the environment to maximize the expected benefits [1–3]. The multi-armed bandit (MAB) problem is a classical problem in reinforcement learning [4,5]. It is extensively occurred in many areas, such as computer gaming [1,2], advertising recommendation [6], and communication channel selection [7], etc. In the MAB problem [8–12]: a player first selects one arm from a slot machine with *K* arms, then the slot machine yields a reward according to the reward probability of the selected arm (different arms have different reward probabilities). At the same time, the player learns from this cycle to improve his next-cycle selection. This procedure is repeated for a certain cycles and the goal of this player is to maximize the total benefits. However, up to present, most of the previously reported literatures were focused on solving the decision-making problem based on algorithms [13,14]. While solutions based on physical implementation, especially based on photonic technologies are lack of investigations, even though it has been confirmed that photonic technology has the potential to increase computing speed and reduce devices cost [15–21].

In recent years, laser chaos generated by external cavity semiconductor laser (ECSL) has been considered as a promising candidate for photonic implementation to solve the MAB problem, due to its wide bandwidth and excellent randomness features [22–26]. For instance, T. Mihana and coworkers proposed a scheme to solve the 2-armed bandit problem, in virtue of the lag chaos synchronization in mutually coupled semiconductor lasers [10]. S. Y. Xiang and colleagues demonstrated a solution to the 2^{2}-armed bandit problem utilizing the coupled semiconductor lasers with a phase-modulated Sagnac loop, and a creative scheme adopting a three-laser-network structure to solve the 2^{3}-armed bandit problem [11,12]. It has been confirmed that the 2* ^{n}*-armed bandit problem can be effectively solved by jointly utilizing the tug-of-war algorithm, the time division multiplexing technology and the laser chaos [8–12]. Nevertheless, in practical, there are many non-2

*-armed MAB problems, and these 2*

^{n}*-armed MAB decision-making schemes are not efficient. It is a holdback for photonic decision-making applications. Therefore, it is valuable to explore universal approaches that support to solve any-number-armed MAB problem. Recently, T. Mihanna*

^{n}*et al.*proposed a laser network decision making scheme for MAB problems, in virtue of the lag synchronization in a ring configuration, and demonstrated that arbitrary-number-armed MAB problems can be solved by increasing the number of lasers [27].

From a different perspective, in this work, we propose and experimentally demonstrate a novel scheme that supports to solve arbitrary-number-armed bandit problem, on the basis of simultaneous low-correlation chaos generation and *ɛ*-greedy strategy. In Section 2, we experimentally demonstrate the generation of two amplitude-distribution-uniform random sequences for decision-making, on the basis of the simultaneous generation of two low-correlation wideband chaotic signals. Section 3 describes the principles of the proposed decision-making scheme. In Section 4, decision-making for an exemplary 5-armed bandit problem is demonstrated, and the CDR performance is discussed. Finally, a brief conclusion is given in Section 5.

## 2. Extraction of random sequences for decision-making

In this section, the simultaneous generation of two low-correlated chaotic signals and the corresponding extraction of two random sequences (*A*(*t*) and *B*(*t*)) for decision making are experimentally demonstrated.

#### 2.1 Simultaneous generation of low-correlation chaotic signals

Figure 1(a) shows the experimental setup for the simultaneous generation of two low-correlation chaotic signals. The optical chaos generation system consists of a conventional ECSL, one self-feedback phase modulation loop (SFPML), and a filtering output module. An initial chaotic signal is generated by the ECSL, and then it is inputted into the SFPML where the chaotic signal is modulated by an electro-optic phase modulator (PM), and subsequently passed through a dispersion component (DC). After that, the signal is split into two parts, one is fed back, then photo-detected and used as the driving signal of the PM, while the other part is entered into the filtering module. In the experiment, the bias current of ECSL is 16.5 mA that is 1.5 times the threshold current and the central wavelength is 1551.33 nm, and the feedback delay time of the ECSL is 100.2 ns. The DC is constructed with a dispersion compensation fiber with a dispersion value of 638 ps/nm. With the spectrum expansion effect of the phase modulation and the phase-to-intensity conversion of DC in the SFPML, the spectrum of the initial chaotic signal can be significantly expanded and the TDS can be efficiently suppressed [22,24]. The PM modulation depth is 2.2. A VOA is used to control the injection power of the photodetector (30 GHz bandwidth). The maximum power gain of the RF amplifier is 38 dB with a bandwidth of 18 GHz. The feedback delay time of the SFPML is 24.3 ns. In the filtering output module, two optical tunable filters (OTF) with central wavelengths of 1551.1 nm and 1551.7 nm are adopted to generate two chaotic signals referred to as Output A and Output B. The 3-dB bandwidths of the OTFs are 0.2 nm. A 100 GS/s digital oscilloscope with four 25-GHz bandwidth channel is used to record and observe relevant electrical signal information. The optical spectra of the initial ECSL-generated chaos and the SFPML-output chaos, as well as those of Output *A* and Output *B* are shown in Fig. 1(b). It is indicated that the optical spectrum of the ECSL chaotic signal (dark line) is narrow. While after passing through the SFPML, the optical spectrum is broadened significantly (red line). After the parallel non-overlapping filtering (blue lines), two chaotic signals is simultaneously outputted. It is worth mentioning that the cross-correlation coefficient between Output *A* and Output *B* is as low as 0.14 (see Fig. 1(c)). Moreover, as shown in Fig. 1(d), with a properly large central wavelength gap between OTF1 and OTF2 (larger than 0.2 nm), the cross-correlation coefficients between output A and output B can always be maintained at a relatively low level (smaller than 0.15). Here the cross-correlation is calculated by the method defined in [28,29].

Figure 2 presents the temporal waveforms, the auto correlation function (ACF) and power spectra, for the initial ECSL-generated chaotic signal, and (Output A and Output B). Here the bandwidth is defined as the span between direct current component and the frequency that contains 80% of the energy in the power spectrum [30]. For the initial ECSL-generated chaotic signal (first column), the power is concentrated nearby the oscillation relaxation frequency, as such the bandwidth of the initial chaotic signal is only 5.82 GHz. On the other hand, there is an obvious time-delay signature (TDS) appearing at the ECSL feedback time (100.2 ns), which indicates that there is a periodicity in the chaotic signal [31]. While for the Output A and Output B, comparing with the initial chaotic signal, the spectra are much flatter and the bandwidths are enhanced to 13.52 GHz and 13.55 GHz, respectively. Moreover, the noise-like temporal waveforms and ACF curves with no TDS are observed. That is, both of the bandwidth and randomness of the output chaotic signals are enhanced simultaneously.

#### 2.2 Extraction of random sequences for decision-making

A 8-bit analog-to-digital conversion (ADC) with 4 reserved least significant bits (LSBs) is adopted to extract two random sequences *A*(*t*), *B*(*t*), (*t* = 1, 2, 3… 1000000) from Output A and Output B, in virtue of the traditional random number extraction method [32–34]. The sampling rate of ADC is 10GS/s. In Section 3, decision-making scheme are introduced, based on the values and the probability distributions characteristic of {*A*(*t*), *B*(*t*)}. As shown in Fig. 3(a) and 3(b), the probability distributions of *A*(*t*) and *B*(*t*) are uniform. That is, the randomness of *A*(*t*) and *B*(*t*) are excellent. Moreover, Fig. 3(c) shows the joint probability distribution of {*A*(*t*), *B*(*t*)}. It is indicated that each probability for {*A*(*t*), *B*(*t*)} = {*X*, *Y*} (*X*, *Y *= 0, 1, 2…15) is approximately 1/256. By calculating the cross-correlation function of *A*(*t*) and *B*(*t*), the results indicate that the cross-correlation coefficients are always lower than 0.03, this means the cross correlation between *A*(*t*) and *B*(*t*) is low.

## 3. Principles of the decision-making scheme

Firstly, a few related concepts involved in the problem is briefly introduced, before introducing the decision-making scheme to the multi-armed bandit problem [8,10,12]. It is assumed that the slot machine has *K* arms (namely Arm1, Arm2, …, Arm*K*) with different unknown reward probabilities (*P*_{1}, *P*_{2},* …*, *P _{K}*), where

*K*is a positive integer. The player makes

*M*cycles of selection, and in each cycle he selects one of the

*K*arms. Whether the player can get a reward in each cycle depends on the reward probability of the selected arm. For instance, assuming that the Arm

*i*(

*i*=1, 2, …,

*K*) is selected, the probability that the slot machine yields a reward is

*P*, and the probability of yielding no reward is 1-

_{i}*P*. The MAB problem can be interpreted as finding a decision-making strategy to maximize the total benefits, or referred to as how to find the best arm (the arm with the highest reward probability) within a certain cycles.

_{i}Figure 4 shows the mapping rule we proposed to solve the *K*-armed bandit problem. Here the mapping rule is designed on the basis of epsilon-greedy strategy [5]. The epsilon-greedy strategy is an easy-realized and extensively-adopted decision-making strategy, in which the player selects the current optimal arm with a probability of 1-*ɛ* or randomly select other *K* arms with a probability of *ɛ*/*K*. In the proposed scheme, a set {*A*(*t*), *B*(*t*)} consisting of 256 elements is firstly constructed, where the values of the elements *A*(*t*), *B*(*t*) are 0 to 15. Then, this set is divided into *K*+1 subsets, where each of the first *K* subsets contains *a* elements, while the last subset (*K*+1 subset) contains 256-*K*a* elements. The value of the elements in each subset satisfy the following relationship:

If the values of {*A*(*t*), *B*(*t*)} extracted from chaotic signals are in the range of the subset *i* (one of the first *K* subsets), the Arm*i* is selected. Otherwise, they would be in the range of the (*K*+1)-th subset, and the current optimal arm which refers to as the arm with the largest average yield-reward after the previous selections is selected.

Figure 5 presents the overall flow of the decision-making process, the entire decision-making process contains *M* cycles. The player starts from the 1-st cycle and selects an arm according to the mapping rule in each cycle. (The best arm is selected after *M* cycles). The workflow of each cycle is introduced by taking the *t*-th cycle (*t* = 1, 2, …, *M*) as an example: Firstly, the random values of *A*(*t*) and *B*(*t*) for the *t*-th cycle are abstracted from the two low-correlation chaos signals. Then, the corresponding arm in the multi-armed slot machine is selected, according to the mapping rule. Finally, the average yield-reward of each arm is updated and the value of *a* is adjusted, according to some specific information, such as whether the slot machine feedback rewards, which arm has been selected and the number of cycles *t* that has been executed. This procedure is equivalent to updating a new mapping rule. The average yield-reward of each arm is updated as follows:

Here, *V _{i}* (

*t*) is the average yield-reward of Arm

*i*(

*i*= 1, 2, 3…

*K*) in the

*t*-th cycle,

*N*(

_{i}*t*) is the total number of times that the Arm

*i*is selected in the previous

*t*cycles, and

*R*(

_{i}*t*) represents the reward yielded by the slot machine, and its value is 0 or 1. Specifically, if the slot machine yields a reward, then

*R*(

_{i}*t*) = 1, otherwise

*R*(

_{i}*t*) = 0. In the experiment, the initial value of the number of selections for each arm

*N*(0) = 1. To guarantee that only one arm is selected as the current optimal arm in each decision-making cycle, the initial values of the average reward

_{i}*V*(0) are randomly set as floating-point numbers that are not equal to each other and in a narrow range approximate to 1, here it is set in the range 0.95<

_{i}*V*(0) < 1.

_{i}## 4. Experimental results and discussion

In this section, several exemplary decision-making experiments aimed on different MAB problems are carried out to confirm the feasibility and investigate the performance of the proposed scheme, based on the extracted random sequences *A*(*t*), *B*(*t*) and the designed mapping rule. The statistically correct decision ratio (CDR) [6–10] is adopted to evaluate the performance of the proposed decision-making scheme. A CDR value above 0.9 means that the decision is correct, and the shorter the time for CDR converging to greater than 0.9, the higher the decision-making efficiency. The calculation for the CDR of the *t*-th cycle is performed by:

In this equation, *L* represents the number of repeated trials in each cycle. For the *j*-th trials in the *t*-th cycle, if the best arm is selected, then Δ(*j*, *t*) = 1; if one of the other arms is selected, then Δ(*j*, *t*) = 0. It is worthy noticing that Eq. (4) also reflects the average probability of that the player selects the best arm in the *t*-cycle. In this work, *L* is set to 1000, and the number of cycles for each decision-making experiment is set to 200 (*M* = 200).

#### 4.1 Decision-making for an exemplary 5-armed bandit problem

To confirm the feasibility and evaluate the performances of the proposed decision-making scheme, an exemplary non-2* ^{n}*-armed band problem, namely 5-arm bandit problem (

*K*= 5), is taken as an example. The reward probabilities of the arms are set to 0.5, 0.4, 0.8, 0.3 and 0.6, respectively.

First, the influence of the experimental parameter *a* on the CDR value is analyzed. As shown in Fig. 6(a), the smaller the value of *a*, the higher the CDR value can be converged to. This is because that, as the value of *a* decreases, the probability of selecting the current optimal arm in each cycle will increase. After a certain number of selections, the current optimal arm becomes the best arm for the final decision. It is also found that the theoretical maximum value of CDR (CDR_{high}) can be evaluated by:

Equation (5) is derived as follows: If in the (*t*−1)-th cycle of all repeated experiments, the current optimal arm has been converged to the best arm (represented by Arm*i*), then in the mapping rule of the *t*-th cycle, there are two subsets that corresponds to the best arm, namely, subset *i* and subset *K*+1. The probability that the values of *A*(*t*) and *B*(*t*) are in the range of these two subsets are about *a*/256 and (256-*Ka*)/256, respectively. Consequently, the probability of that the best arm is selected in the *t*-cycle is about *a*/256+(256-*Ka*)/256. Equation (5) indicates that *a* should be less than 6.4, in order to guarantee that the CDR can reach 0.9 in the demonstrated 5-armed bandit problem decision-making. Nevertheless, it is worth noting that when the smaller the value of *a*, the larger the number of selection cycles that CDR convergences to 0.9. In Fig. 6(b), when *a* is set to 1 (dark curve), the CDR can only reach 0.87 at *t*=200, but in fact it can reach 0.9 at *t*=381. In addition, by dynamically decreasing the value of *a*, the value of CDR can be quickly converged to 0.9. As shown by the brown curve in Fig. 6(b), where in the first 50 selection cycles, the value of *a* is monotonically decreased to 2 and then *a* is fixed at 1 since the 51-st selection cycle, the value of CDR is converged to 0.9 at *t* = 90 cycle. It is much faster than the case with a fixed *a*, although when *t*<40 the CDR values are smaller than those of the case with fixed *a*. This is because when the initial value of *a* is relatively large, the arm selection action is relatively random, as such the CDR values for the dynamic varying *a* scenario are smaller than those of the scenario with a fixed *a*. However, as the value of *a* gradually decreases, the selection probability of the current optimal arm increases correspondingly, and consequently, the correct decision can be made quickly. In the following discussions, the values of *a* are set in this way.

To more intuitively show the evolution of the decision-making scheme, Fig. 7 presents the average yield-reward curve of each arm in one trial of Fig. 6(b). It is observed that when *t* < 20, the average yield-reward value of each arm changes frequently. This is because in these cases, the value of *a* is large, and the probability of each arm being selected is similar. As the number of selection cycle increases and the value of *a* decreases, almost only Arm3 (red curve) and Arm5 (blue curve) keep changing frequently, which means that the player selects these two arms with a high reward probability to maximize the total benefits. When *t* >120, the average yield-reward value of Arm3 remains higher than those of the other arms, as thus Arm3 is always decided as the current optimal arm.

Figure 8 shows the CDR performance of the proposed decision-making scheme in the exemplary 5-armed bandit problem with different reward probabilities. It is demonstrated that, when only changing the position of the best arm (from Arm1 to Arm5), similar CDR trends can be obtained. This illustrates that some operations in the decision-making scheme, such as naming and numbering the arms, mapping the numbers and the fixed interval one by one, would not cause significant CDR evolution trend differences. Figure 8(b) indicates that the decision-making efficiency is closely related with the reward probability differences of the arms. When the reward probability of the best arm is significantly higher than those of the other arms (Fig. 8(b) blue curve), the CDR value can reach 0.9 more quickly than that of other cases.

In general, the proposed scheme can achieve fast and correct decision-making in solving the multi-armed bandit problem.

#### 4.2 Decision-making for other number-armed bandit problems

Comparing with the existing photonic decision-making scheme, the proposed photonic decision-making scheme is feasible to solve the non-2* ^{n}*-armed bandit problems. When switching from the 5-armed bandit problem to other arbitrary-number-armed bandit problems, by simply adjusting the parameters

*K*and

*a*in the mapping rule, the proposed decision-making scheme can show excellent decision-making performance. As shown in Fig. 9, the evolution curves of CDR for the 4-armed, 5-armed, 6-armed and 7-armed MAB problems are demonstrated. It is observed that these MAB problems can be solved successfully, and the CDR reaches 0.9 at the 58, 89, 104, and 133 cycles, respectively.

Furthermore, Fig. 10 presents the influence of the number of arms (*K*) on the number of cycle where the CDR reaches 0.9 for the first time (*N*_{CDR=0.9}) [27]. Here the number of arms ranges from 4 to 9, the reward probabilities of all ordinary arm (non-optimum arms) are set as 0.4, and the max number of cycles for each decision-making experiment is set as 1000 (*M *= 1000). It is indicated that, the larger the number of arms, the larger the *N*_{CDR=0.9}. Moreover, the overall variation trends of *N*_{CDR=0.9} are approximately linear versus *K*, and the slope of the linear variation is closely related with the reward probability. When the reward probability of the best arm is significantly higher than those of the other arms (red curve), it is easier to make a correct decision, as such the slope of *N*_{CDR=0.9} is smaller than those of other cases. This linear relationship between *N*_{CDR=0.9} and *K* is beneficial to estimating the minimum number of cycles for finding the best arm.

In addition, other *K*-armed bandit problems (especially those with large *K* values, such as *K* > 256) can be solved by expanding the space of random sequences in the mapping rule. Here the space of random sequences refers to the number of different values for the random sequence {*A*(*t*), *B*(*t*)}, and it is set as 256 in the abovementioned discussions. The potential approaches include reserving more significant bits when extracting *A*(*t*) and *B*(*t*) (e.g. reserving 5 or more least significant bits), or expanding the random sequence space to multiple (larger than 2) dimensions by extracting the multiple random number sequences on the basis of multiple-channel low-correlation chaos generation.

## 5. Conclusion

In conclusion, a novel decision-making scheme to solve arbitrary-number-armed bandit problem is proposed and experimentally demonstrated, on the basis of the parallel generation of two low-correlation chaotic signals and the epsilon-greedy strategy. Two low-correlation wideband TDS-suppressed chaotic signals that are originated from an ECSL are simultaneously generated by self-feedback phase modulation and parallel filtering. Based on this, two random sequences with uniform distributions are extracted from the simultaneously-generated chaotic signals utilizing an 8-bit ADC with 4-LSBs. With the random sequences, the mapping rules for arm selection are designed. We successfully perform the decision making in the exemplary cases of 4, 5, 6, 7-armed bandit problems with CDR>0.9. The experimental results show that fast decision making can be realized in the proposed scheme. This work presents an efficient scheme to solve arbitrary-armed bandit problems, which may pave the development of photonic decision-making.

## Funding

Sichuan Province Science and Technology Support Program (2021JDJQ0023); National Natural Science Foundation of China (61671119); Science and Technology Commission of Shanghai Municipality (SKLSFO2020-05); Fundamental Research Funds for the Central Universities (ZYGX2019J003).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

The data used to support the findings of this study are available from the corresponding author upon request.

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