- Research Article
- Open Access
Multiagent -Learning for Aloha-Like Spectrum Access in Cognitive Radio Systems
EURASIP Journal on Wireless Communications and Networking volume 2010, Article number: 876216 (2010)
An Aloha-like spectrum access scheme without negotiation is considered for multiuser and multichannel cognitive radio systems. To avoid collisions incurred by the lack of coordination, each secondary user learns how to select channels according to its experience. Multiagent reinforcement leaning (MARL) is applied for the secondary users to learn good strategies of channel selection. Specifically, the framework of -learning is extended from single user case to multiagent case by considering other secondary users as a part of the environment. The dynamics of the -learning are illustrated using a Metrick-Polak plot, which shows the traces of -values in the two-user case. For both complete and partial observation cases, rigorous proofs of the convergence of multiagent -learning without communications, under certain conditions, are provided using the Robins-Monro algorithm and contraction mapping, respectively. The learning performance (speed and gain in utility) is evaluated by numerical simulations.
In recent years, cognitive radio has attracted intensive studies in the community of wireless communications. It allows users without license (called secondary users) to access licensed frequency bands when licensed users (called primary users) are not present. Therefore, the cognitive radio technique can substantially alleviate the problem of underutilization of frequency spectrum [1, 2].
The following two problems are key to cognitive radio systems.
Resource mining, that is, how to detect the available resource (the frequency bands that are not being used by primary users); usually it is done by spectrum sensing.
Resource allocation, that is, how to allocate the available resource to different secondary users.
Substantial work has been done for the resource mining problem. Many signal processing techniques have been applied to sense the frequency spectrum , for example, cyclostationary feature , quickest change detection , and collaborative spectrum sensing . Meanwhile, a significant amount of research has been conducted for the resource allocation in cognitive radio systems [7, 8]. Typically, it is assumed that secondary users exchange information about available spectrum resources and then negotiate on the resource allocation according to their own requirements of traffic (since the same resource cannot be shared by different secondary users if orthogonal transmission is assumed). These studies typically apply theories in economics, for example, game theory, bargaining theory, or microeconomics.
However, in many applications of cognitive radio, such a negotiation-based resource allocation may incur significant overhead. In traditional wireless communication systems, the available resource is almost fixed (even if we consider the fluctuation of channel quality incurred by fading, the change of available resource is usually very slow and thus can be considered stationary). Therefore, the negotiation need not be carried out frequently, and the negotiation result can be applied for a long period of data communication, thus incurring only tolerable overhead. However, in many cognitive radio systems, the resource may change very rapidly since the activity of primary users may be highly dynamic. Therefore, the available resource needs to be updated very frequently, and the data communication period between two spectrum sensing periods should be fairly short since minimum violation to primary users should be guaranteed. In such a situation, the negotiation of resource allocation may be highly inefficient since a substantial portion of time needs to be used for the negotiation. To alleviate such an inefficiency, high-speed transceivers need to be used to minimize the time consumed on negotiation. Particularly, the turn-around time that is, the time needed to switch from receiving (transmitting) to transmitting (receiving) should be very small, which is a substantial challenge to hardware design.
Motivated by the above discussion and observation, in this paper, we study the problem of spectrum access without negotiation in multiuser and multichannel cognitive radio systems. The spectrum access without negotiation is achieved by applying the framework of reinforcement learning. In such a scheme, each secondary user senses channels and then chooses an idle frequency channel to transmit data, as if no other secondary user exists. If two secondary users choose the same channel for data transmission, they will collide with each other and the corresponding data packets cannot be decoded. Such a procedure is illustrated in Figure 1, where three secondary users access an access point via four channels. Note that such a scheme is similar to Aloha  where no explicit collision avoidance is applied. We can also apply techniques similar to -persistent Carrier Sensing Multiple Access (CSMA) that is, each secondary user transmits with probability when it finds an available channel. However, it is beyond the scope of this paper. In the Aloha-like approach, since there is no mutual communication among these secondary users, collision is unavoidable. However, the secondary users can try to learn collision avoidance, as well as channel qualities (we assume that the secondary users have no a priori information about the channel qualities), according to their experience. In such a context, the learning procedure includes not only the available frequency spectrum but also the behavior of other secondary users.
To accomplish the learning of Aloha-like spectrum access, multiagent reinforcement learning (MARL)  is applied in this paper. One challenge of MARL in our context is that the secondary users do not know the payoffs (thus do not know the strategies) of other secondary users in each stage; thus the environment of each secondary user, including other secondary users, is nonstationary and may not assure the convergence of learning. Due to the assumption that there is no mutual communication between different secondary users, many traditional MARL techniques like fictitious play [11, 12] and Nash-Q learning  cannot be used since they need information exchange among players (e.g., exchanging their action information). To alleviate the lack of mutual communication, we extend the principle of single-agent -learning, that is, evaluating the values of different state-action pairs in an incremental way, to the multiagent situation without information exchange. By applying the theory of stochastic approximation , which has been used in many studies on wireless networks [15, 16], we will prove the main result of this paper, that is, the learning converges to a stationary point regardless of the initial strategies (Propositions 1 and 2).
Some studies on reinforcement learning in cognitive radio networks have been done [17–19]. In  and , the studies are focused on the resource competition in a spectrum auction system, where the channel allocation is determined by the spectrum regulator, which is different from this paper in which no regulator exists. Reference  discusses correlated equilibrium and achieves it by no-regret learning; that is, minimizing the gap between the current reward and optimal reward. In this approach, mutual communication is needed among the secondary users. However, in our study, no intersecondary-user communication is assumed.
Note that the study in this paper has subtle similarities to the evolutionary game theory , which has been successfully applied in the cooperative spectrum sensing in cognitive radio systems . Both our study and the evolutionary game focus on the dynamics of strategy changes of users. However, there is a key difference between the two studies. The evolutionary game theory assumes pure strategies for the players (e.g., cooperate or free-ride in cooperative spectrum sensing ) and studies the proportions of players using different pure strategies. The key equation in the evolutionary game theory, called replicator equation, describes the dynamics of the corresponding proportions. In contrast to the evolutionary game, the players in our study use mixed strategies and the basic (16) describes the dynamics of the -values for different channels. Although the convergence is proved by studying ordinary different equations in both studies, the proof is significantly different since the equations have totally different expressions.
The remainder of this paper is organized as follows. In Section 2, the system model is introduced. Basic elements of the game and the proposed multiagent -learning for fully observable case (i.e., each secondary user can sense all channels) are introduced in Section 3. The corresponding convergence of -learning is proved in Section 4. The -learning for partially observable case (i.e., each secondary user can sense a subset of the channels) is discussed in Section 5. Numerical results are provided in Section 6, while conclusions are drawn in Section 7.
2. System Model
We consider active secondary users accessing licensed frequency channels. (When there are more than channels, there is less competition; thus making the problem easier. We do not consider the case when the number of channels is less than the number of secondary users since a typical cognitive radio system can provide sufficient channels. Meanwhile, the proposed algorithm can also be applied to all possible cases of ) We index the secondary users, as well as the channels, by integers 1, 2,, . For simplicity, we denote by the set of users (channels) different from user (channel) .
The following assumptions are made throughout this paper.
The secondary users are sufficiently close to each other such that they share the same activity of primary users. There is no communication among these secondary users, thus excluding the possibility of negotiation.
We assume that the activity of primary users over each channel is a Markov chain (A more reasonable model for the activity of primary users is the semi-Markov chain. The corresponding analysis is more tedious but similar to that in this paper. Therefore, for simplicity of analysis, we consider only Markov chain in this paper)with states (busy: the channel is occupied by primary users and cannot be used by secondary users) and (idle: there is no primary user over this channel). We denote by the state of channel in the sensing period of the th spectrum access period. For channel , the transition probability from state to state (resp., from state to state ) is denoted by (resp., ). We assume that the Markov chains for the channels are mutually independent. We also assume perfect spectrum sensing and do not consider possible errors of spectrum sensing.
We assume that the channel state transition probabilities, as well as the channel rewards, are unknown with the secondary users at the beginning. They are fixed throughout the game, unless otherwise noted. Therefore, the secondary users need to learn the channel properties.
The timing structure of spectrum sensing and data transmission is illustrated in Figure 2, where data is transmitted after the spectrum sensing period. We assume that each secondary user is able to sense only one channel during the spectrum sensing period and transmit over only one channel during the data transmission period.
In Sections 3 and 4, we consider the case in which all secondary users have full knowledge of channel states in the previous spectrum access period (complete observation). Note that this does not contradict the assumption that a secondary user can sense only one channel during the spectrum sensing period since the secondary user can continue to sense other channels during the data transmission period (suppose that the signal from primary users can be well distinguished from that from secondary users, e.g., using different cyclostationary features ). If we consider the set of channel states in the previous spectrum access period as the system state, denoted by at spectrum access period , then the previous assumption implies a completely observable system state, which substantially simplifies the analysis. In Section 5, we will also study the case in which secondary users cannot continue to sense during the data transmission period (partial observation); thus each secondary user has only partial observations about the system state.
3. Game and -Learning
In this section, we introduce the game associated to the learning procedure and the application of -learning to the Aloha-like spectrum access problem. Note that in this section and Section 3, we assume that each secondary user knows all channel states in the previous time slot, that is, the completely observable case.
3.1. Game of Aloha-Like Spectrum Access
The Aloha-like spectrum access problem is essentially an game. When secondary user transmits over an idle channel , it receives reward (e.g., channel capacity or successful transmission probability), if no other secondary user transmits over this channel, and reward 0, if one or more other secondary users are transmitting over this channel, since collision will happen. We assume that the reward does not change with time. When channels change slowly, the learning algorithm proposed in this paper can also be applied to track the change of channels. When channels change very fast, it is impossible for secondary users to learn. Since there is no explicit information exchange among secondary users, the collision avoidance is completely based on the received reward. The payoff matrices for the case of are given in Figure 3. Note that the actions, denoted by for user at time , in the game are the selections of channels. Obviously, the diagonal elements in the payoff matrices are all zero since collision yields zero reward.
It is well known that Nash equilibrium means the strategies such that unilaterally changing strategy incurs the degradation of its own performance. Mathematically, a Nash equilibrium means a set of strategies , where is the strategy of player , which satisfy
where means the reward of player and means the strategies of all players except player .
It is easy to verify that there are multiple Nash equilibrium points in the game. Obviously, orthogonal transmission strategies, that is, , , are pure equilibria. The reason is the following. If a secondary user changes its strategy and transmits over other channels with nonzero probability, those transmission will collide with other secondary users (recall that, for the Nash equilibrium, all other secondary users do not change their strategies) and incurs performance degradation. The orthogonal channel assignment can be achieved in the following approach: let all secondary users sense the channel randomly at the very beginning; once a secondary user finds an idle channel, it will access this channel forever; after a random number of rounds, all secondary users will find different channels, thus achieving the orthogonal transmission. We call this scheme the simple orthogonal channel assignment since it is simple and fast. However, in this scheme, the different rewards of different channels are ignored. As will be seen in the numerical simulation results, the proposed learning procedure can significant outperform the simple orthogonal channel assignment.
We define the -function as the expected reward in one time slot (since the channel states are completely known to the secondary users and are not controlled by the secondary users, each secondary user needs to consider only the expected reward in one time slot, that is, a myopic strategy) of each action under different states; that is, for secondary user and system state , the -value of choosing channel is given by
where is the reward obtained by secondary user , which is dependent on the action, as well as the system state, and the expectation is over the randomness of other users' actions, as well as the primary users' occupancies.
In contrast to fictitious play , which is deterministic, the action in -learning is random. We assign nonzero probabilities for all channels such that all channels will be explored. Such an exploration guarantees that good channels will not be missed during the learning procedure. We consider Boltzmann distribution  for random exploration, that is,
where is called temperature, which controls the randomness of exploration. Obviously, the smaller is (the colder), the more focused the actions are. When , each user chooses only the channel having the largest -value.
When secondary user selects channel and the system state is , the expected reward is given by
since secondary user () chooses channel with probability (collision happens and secondary user receives no reward) and channel is idle with probability ; then the product in (4) is the probability that no other secondary user accesses channel .
3.4. Updating -Values
In the procedure of -learning, the -functions are updated after each spectrum access using the following rule:
where is a step factor (when channel is not selected by user , we set ), is the reward of secondary user and is the characteristic function of the event that channel is selected by secondary user at the th spectrum access. Note that this is the standard -learning without considering the future states. An intuitive explanation for (5) is that, once channel is accessed, the corresponding -value is updated by combining the old value and the new reward; if channel is not chosen, we keep the old value by setting . Our study is focused on the dynamics of (5). To assure convergence, we assume that
as well as
Note that, in a typical stochastic game setting and -learning, the updating rule in (5) should consider the reward of the future and add a discounted term of the future reward to the right hand side of (5). However, in this paper, the optimal strategy is myopic since we assume that the system state is known, and thus the secondary users' actions do not affect the system state. For the case of partial observation (i.e., each secondary user knows only the state of a single channel), the action does change each secondary user's state (typically the belief of system state), and the future reward should be included in the right hand side of (5), which will be discussed in Section 5.
3.5. Stationary Point
The -values for different users are mutually coupled and all -values change if one -value is changed since the strategy of the corresponding user is changed, thus changing the expected rewards of other users. We define -values satisfying the following equations as a stationary point
Note that the stationarity is only in the statistical sense since the -values can fluctuate around the stationary point due to the randomness of exploration. Obviously, as , the stationary point converges to a Nash equilibrium point. However, we are still not sure about the existence of such a stationary point. The following lemma assures the existence of stationary point. The proof is given in Appendix .
For sufficiently small , there exists at least one stationary point satisfying (8).
4. Convergence of -learning without Information Exchange
In this section, we study the convergence of the proposed -learning algorithm. First, we provide an intuitive explanation for the convergence in case. Then, we apply the tools of stochastic approximation and ordinary differential equation (ODE) to prove the convergence rigorously.
4.1. Intuition on Convergence
As will be shown in Proposition 1, the updating rule of -values in (5) will converge to a stationary equilibrium point close to Nash equilibrium. Before the rigorous proof, we provide an intuitive explanation for the convergence using the geometric argument proposed in .
The intuitive explanation is provided in Figure 4 for the case of (we call it Metrick-Polak plot since it was originally proposed by A. Metrick and B. Polak in ). For simplicity, we ignore the indices of state and assume that both channels are idle. The axes are and , respectively. As labeled in the figure, the plane is divided into four regions separated by two lines and , in which the dynamics of -learning are different. We discuss these four regions separately.
Region I: in this region, ; therefore, secondary user 1 prefers visiting channel 1; meanwhile, secondary user 2 prefers accessing channel 2 since ; then, with large probability, the strategies will converge to a stationary point in which secondary users 1 and 2 access channels 1 and 2, respectively.
Region II: in this region, both secondary users prefer accessing channel 1, thus causing many collisions. Therefore, both and will be reduced until entering either region I or region III.
Region III: similar to region I.
Region IV: similar to region II.
Then, we observe that the points in Regions II and IV are unstable and will move into Region I or III with large probability. In Regions I and III, the strategy will move close to the stationary point with large probability. Therefore, regardless where the initial point is, the updating rule in (5) will converge to a stationary point with large probability.
4.2. Stochastic Approximation-Based Convergence
In this section, we prove the convergence of the -learning of the proposed Aloha-like spectrum access with Boltzman distributed exploration. First, we find the equivalence between the updating rule (5) and Robbins-Monro iteration  for solving an equation with unknown expression (a brief introduction is provided in Appendix ). Then, we apply a conclusion in stochastic approximation  to relate the dynamics of the updating rule to an ODE and prove the convergence of the ODE.
4.2.1. Robbins-Monro Iteration
At a stationary point, the expected values of -functions satisfy the equations in (8). For system state , define
Then, (8) can be rewritten as
and (function mod means the remainder of dividing integer with integer )
with convention . Obviously, is the probability that channel can be used by secondary user without collision with other secondary users, when the current system state is .
Then, the updating rule in (5) is equivalent to solving (8) (the expression of the equation is unknown since the rewards, channel transition probabilities, as well as the strategies of other users, are all unknown) using Robbins-Monro algorithm , that is,
where is the vector of all step factors, is the vector of rewards obtained at spectrum access period and is a random observation on function contaminated by noise, that is,
where , is noise and (recall that means the reward of secondary user at time )
Obviously, since the expectation of the difference between the reward and the expected reward is equal to 0. Therefore, the observation is a Martingale difference.
4.2.2. ODE and Convergence
The procedure of Robbins-Monro algorithm (i.e., the updating of -value) is the stochastic approximation of the solution of the equation. It is well known that the convergence of such a procedure can be characterized by an ODE. Since the noise in (14) is a Martingale difference, it is easy to verify the conditions in Theorem 1 in Appendix and obtain the following lemma (the proof is given in Appendix ).
With probability 1, the sequence , , converges to some limit set of the ODE
What remains to do is to analyze the convergence property of the ODE (16). We obtain the following lemma by applying Lyapunov function. The proof is given in Appendix .
If a stationary point determined by (10) exists, the solution of ODE (16) converges to the stationary point for sufficiently large .
Combining Lemmas 1, 2, and 3, we obtain the main result in this paper.
Suppose that a stationary point determined by (10) exists. For any system state and sufficiently large , the -learning converges to a stationary point with probability 1.
Note that a sufficiently small guarantees the existence of stationary point and a sufficiently large assures the convergence of the learning procedure. However, they do not conflict since they are not necessary conditions. As we found in our simulations, we can always choose a suitable to guarantee the existence of the stationary point and the convergence.
5. -Learning with Partial Observations
In this section, we remove the assumption that all secondary users know all channel states in the previous spectrum access period and assume that each secondary user knows the state of only the channel sensed in the previous spectrum access period; thus making the system state partially observable. The difficulties of analyzing such a scenario are given below:
The system state is partially observable.
The game is imperfectly monitored, that is, each player does not know other players' actions.
The game has incomplete information, that is, each player does not know the strategies of other players, as well as their beliefs on the system state.
Note that the latter two difficulties are common for both the complete and partial observation cases. However, the imperfect monitoring and incomplete information add much more difficulty in the partial observation case. In this section, we formulate the -learning algorithm and then prove the convergence under certain conditions.
5.1. State Definition
It is well known that, in partially observable Markov decision process (POMDP) problems, the belief on the system state, that is, ( is the observation history before period ), can play the role of system state. Due to the special structure of the game, we can define the state of secondary user at period as
where , , means the number of consecutive periods during which channel has not been sensed before period (e.g., if the last time that channel was sensed by secondary user is time slot , .) and is the state of channel in the last time when it is sensed before period .
5.2. Learning in the POMDP Game
For the purpose of learning, we define the objective function for user as the discounted sum of rewards in each spectrum access period with discount factor , that is,
Then, to maximize the objectively function, the corresponding -learning strategy is given by 
where is uniquely determined by and , and is the step factor dependent on the time, channel, user, and belief state. Note that is the system state in the next time slot, which is random. Intuitively, the new -value is updated by combining the old value and the new estimation, which is the sum of the new reward and discounted old -value.
Similarly to the complete information situation, we have the following proposition which states the convergence of the learning procedure with partial information and large . The proof is given in Appendix . Note that numerical simulation shows that small also results in convergence. However, we are still unable to prove it rigorously.
When is sufficiently large, the learning procedure in (19) converges.
6. Numerical Results
In this section, we use numerical simulations to demonstrate the theoretical results obtained in previous sections. For the fully observable case, we use the following step factor:
where is the initial learning factor. A similar step factor is used for the partially observable case. In Sections 6.1, 6.2, and 6.3, we consider the fully observable case and, in Section 6.4, we consider the partially observable case. Note that, in all simulations, we initialize the -values by choosing uniformly random variables in the interval .
Figures 5 and 6 show the dynamics of versus (recall that and ) of several typical trajectories for the state of both channels being idle when . We assume that for all and . Note that in Figure 5 and in Figure 6. We observe that the trajectories move from unstable regions (II and IV in Figure 4) to stable regions (I and III in Figure 4). We also observe that the trajectories for smaller temperature is smoother since less explorations are carried out.
Figure 7 shows the evolution of the probability of choosing channel 1 when , and both channels are idle. We observe that both secondary users prefer channel 1 at the beginning and soon secondary user 1 intends to choose channel 2, thus avoiding collisions.
6.2. CDF of Rewards
In this subsection, we consider the performance of reward averaged over all system states. When , we set and for the three channels, respectively. When , we use the first two channels in the case of . The rewards of different channels for different secondary users are randomly generated with a uniform distribution between . The CDF curves of performance gain, defined as the difference of average rewards after and before the learning procedure, are plotted in Figure 8 for both and . Note that the CDF curves are obtained from 100 realizations of learning procedure. From a CDF curve, we can read the distribution of the performance gains. For example, for the curve , performance gain 0.4 in the horizontal axis corresponds to 0.6 in the vertical axis; this means that around 60% of the secondary users obtain performance gain less than 0.4. We observe that when , most performance gains are positive. However, when , a small portion of the performance gains are negative, that is, the performance is decreased after the learning procedure. Such a performance reduction is reasonable since Nash equilibrium may not be Pareto optimal. We also plotted the average performance gains versus different and in Figure 9. We observe that larger results in worse performance gain. When is small, smaller yields better performance, but decreases faster than larger when increases. The performance gain over the simple orthogonal channel assignment scheme is given in Figure 10. We observe that the learning procedure generates a much better performance than the simple orthogonal channel assignment.
6.3. Learning Speed
We define the stopping time of learning as the time that the relative fluctuation of average reward, which is obtained from 2000 spectrum access periods using the current -values, has been below 5 percent for successive 5 time slots. That is, compute the relative fluctuation at time slot using
where is the vector containing all -values and the norm is 2-norm. Then, when is smaller than 0.05 for 5 consecutive times, we claim that the learning is completed. Then, the learning delay is the time spent before the stopping time. Obviously, the smaller the learning delay is, the faster the learning is. Figures 11 and 12 show the delays of learning, which characterizes the learning speed, for different learning factor and different temperature , respectively, when . The original values are randomly selected. When the probabilities of choosing channel 1 are larger than 0.95 for one secondary user and smaller than 0.05 for the other secondary user, we claim that the learning procedure is completed. We observe that larger learning factor results in smaller delay while smaller yields faster learning procedure.
The speed of learning is compared for , and in Figure 13 (both and are fixed). We observe that, for more than 90% of the realizations, the learning can be completed within 20 spectrum access periods. However, the learning procedure may last for a long period of time for some situations. We can notice that the learning speeds are similar for cases and . We also observe that, when is much larger (), the increase of delay is not significant.
6.4. Time-Varying Channel Rewards
In previous simulations, the rewards of successfully accessing channels, are assumed to be constant. In practical systems, they may change with time since wireless channels are usually dynamic. In Figure 14, we show the CDF of performance gains (the configuration is the same as that in Figure 8) when channel changes slowly. We used a simple model for channel reward, which is given by
where is a random variable uniformly distributed between 0 and 1. From Figure 14, we observe that the learning algorithm still improves the performance significantly.
6.5. Partial Observation Case
Figure 15 shows the performance gain of learning in the case of partial observations. We adopt the -learning mechanism introduced in Section 5. Note that there are infinitely many belief states since a channel could be unsensed for an infinite period of time. For computational simplicity, we set all to (recall that is the period of time that channel has not been sensed by user before time ). From Figure 15, we observe that the performance is actually degraded for around 40% () or 50% () cases. However, the amplitude of performance degradation is averagely less than the amplitude of performance gain. We also observe that the performance gain is decreased when is increased from 2 to 3.
The learning delay for the partial observation case is shown in Figure 16, where the simulation setup is similar to that of Figure 13. Again, we observe that the learning speeds of and are similar to each other.
We have discussed a learning procedure for Aloha-like spectrum access without negotiation in cognitive radio systems. During the learning, each secondary user considers the channel and other secondary users as its environment, updates its -values, and takes the best action. An intuitive explanation for the convergence of learning is provided using Metrick-Polak plot. By applying the theory of stochastic approximation and ODE, we have shown the convergence of learning under certain conditions. We also extended the case of full observations to the case of partial observations. Numerical results show that secondary users can learn to avoid collision quickly. The performance after the learning is significantly better than that before the learning and that using a simple scheme to achieve a Nash equilibrium. Note that our study is one extreme of the resource allocation problem since no negotiation is considered, while the other extreme is full negotiation to achieve optimal performance. Our future work will be the intermediate case; that is, limited negotiation for resource allocation.
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This work was supported by the National Science Foundation under grant CCF-0830451.
A. Stochastic Approximation
For being self-contained, we briefly introduce the theory of stochastic approximation and cite the conclusion used for proving Lemma 2.
Essentially, stochastic approximation is used to solve an equation with unknown expression and noisy observations. Consider equation
where is the unknown variable and the expression of function is unknown. Denote by the solution to this equation (we assume that there is only one solution to the equation). Suppose that for and for . We have a series of noisy observations of , denoted by . Then, we can approximate the solution iteratively in the following way (called Robbins-Monro algorithm).
where is the step for the th iteration.
The convergence of (A.2) is deeply related to a "mean" ODE, which is given by
The following Theorem (part of Theorem in ) discloses the relationship between the convergence in (A.2) and the mean ODE in (A.3).
If the following assumptions are satisfied, converges to a limit set in which all points satisfy with probability 1:
Noise is a martingale difference, that is,(A.4)
There exists a continuously differentiable function such that and is a constant on the limit set .
B. Proof of Lemma 1
For simplicity, we fix one system state since the -learning procedures for different state are mutually independent when the system state is fully observable and the action of each secondary user does not affect the system state. Consider a Nash equilibrium point, at which there is no collision. Without loss of generality, we assume that secondary users use channels , respectively.
Now, we choose a set of such that
Then, we can always choose a sufficiently small such that
since the right hand sides of (B.2) and (B.3) converge to and 0, respectively, as .
Then, we carry out the following iterations, that is, the -values of the th iteration is given by, ,
Next, we show that, , increases while () decreases during the iterations by carrying out inductions on . For the first iteration, is increased while () is decreased due to the conditions (B.2) and (B.3). Suppose that, in the th iteration, the conclusion holds. Then, in the th iteration, is increased due to the expression of the right hand side of (B.4) and the assumptions and (). For the same reason, is decreased in the th iteration. This concludes the induction.
Now, we have shown that is a monotonically increasing sequence while () is a monotonically decreasing sequence. Since all sequences are bounded ( and ()), all sequences converge to their limits, which is the stationary point. This concludes the proof.
C. Proof of Lemma 2
We verify the conditions in Theorem 1 one by one.
Condition (A): This is obvious since is upper bounded by (recall that is the difference between the instantaneous reward and the -value).
Condition (B): The martingale difference noise has been proved right after (14).
Condition (C): The function is given by(C.1)
where is defined in (12). We only need to verify the continuity of . Obviously, each element in is differentiable with respect to . Therefore, is not only continuous but also differentiable.
Condition (D): It is guaranteed by (7).
Condition (E): The function can be defined as the integral of . It is continuously differentiable since is continuous. It is a constant on the limit set since there is only one point at the limit set.
D. Proof of Lemma 3
We apply Lyapunov's method to analyze the convergence of the ODE in (16). We define the Lyapunov function as
where is the expected reward of secondary user at period .
Then, we examine the derivative of the Lyapunov function with respect to time , that is,
where we applied the ODE (16).
Then, we focus on the computation of .
For secondary user and channel , we have
where the first equation is due to the definition of and the second equation is due to the rule of the derivative of products.
We consider the derivative in (D.4), that is,
where the last equation is obtained from ODE (16).
Substituting (D.5) into (D.4), we obtain
Combining (D.2) and (D.6), we have
where the coefficient is given by
if and , and
if and . When , .
It is easy to verify
Therefore, when is sufficiently large, we have
Then, we have
Therefore, when is sufficiently large, the derivative of the Lyapunov function is strictly negative, which implies that the ODE (16) converges to a stationary point. This concludes the proof.
We can actually obtain a stronger conclusion from the last part of the proof, that is, the convergence can be assured if
E. Proof of Proposition 2
We define a mapping from all -values to another set of -values, which is given by
where is determined by and and is the average reward when secondary user chooses channel . Note that is a function of all -values.
What we need to prove is that is a contraction mapping. Once this is proved, the remainder part is exactly the same as the proof of the convergence of -learning in . Therefore, we focus on the analysis on the mapping .
We consider two sets of -values, denoted by and , respectively. Considering the difference after the mapping between the two sets of -values, we have
where means the average reward when the -values are . Then, we have
We discuss the two terms in (E.3) separately. For the first term, we have
where is the set of states of secondary users except user and is the probability of the set of states conditioned on the state of secondary user , .
When is sufficiently large, we have
where is a polynomial of order .
Then, it is easy to verify that (E.4) can be rewritten as
where and are both polynomials of smaller order than . Note that the coefficients of both polynomials are independent of the -values.
Then, we have
Then, we can always take a sufficiently large such that
Now, we turn to the second term in (E.3). Without loss of generality, we assume . We have
where, in the first inequality, we define as
Due to symmetry, we have
Combining (E.8) and (E.11), we have
Therefore, is a contraction mapping under the norm . This concludes the proof.
Note that, in contrast to the stochastic approximation approach for the proof of the convergence in the complete observation case, we used a different approach to prove the convergence of the learning with partial observations since it is difficult to apply the stochastic approximation in the partial observation case. Although the stochastic approximation approach is slightly more complicated, we can find a finite value for in (D.14) to assure the convergence. For the contraction mapping approach, we are still unable to find such a finite value for .
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Cite this article
Li, H. Multiagent -Learning for Aloha-Like Spectrum Access in Cognitive Radio Systems. J Wireless Com Network 2010, 876216 (2010). https://doi.org/10.1155/2010/876216
- Nash Equilibrium
- Cognitive Radio
- Primary User
- Performance Gain
- Secondary User