Information processing device, information processing method, and information processing program

Abstract

In order to provide a solution to an optimization problem suitably for both stochastic and adversarial models without reference to a priori information as to which of these models should be assumed, an information processing device (1) is provided with a selection unit (11) that selects, in a given round t∈[T], a subset Xt∈[n] of a set [n] with reference to the observed value of an objective function in round t-1, and an output means that outputs information indicating the subset Xt∈[n] selected by the selection unit, wherein the selection unit (11) selects the subset Xt∈[n] such that the asymptotic behavior of the expected value of the regret ∑t∈[T]ft(Xt)-∑t∈[T]ft(X*), which is expressed using the observed value ft(Xt) of the objective function in each round t∈[T] and a solution X* to be compared, is bound from above by an upper limit A(Δ, n, C) that depends at least on the gap index Δ for the stochastic model and a disturbance index C indicating the adversarial disturbance for the stochastic model.

Description

[Technical field]

[0001] The present invention relates to an information processing device, an information processing method, and an information processing program that provide a solution to an online optimization problem. [Background technology]

[0002] There is known an algorithm for optimizing an index (also called online optimization) under a condition where a function expressing the index to be optimized can be changed sequentially (for example, Non-Patent Document 1). [Prior art documents] [Non-patent literature]

[0003] [Non-Patent Document 1] E. Hazan and S. Kale, 'Online Submodular Minimization', Journal of Machine Learning Research 13 (2012) 2903-2922 Summary of the Invention [Problem to be solved by the invention]

[0004] In the method described in Non-Patent Document 1, regret Σ t∈[T] f t (X t )-min X∈S {Σ t∈[T] f t (X)} is O(nT 1 / 2 ) subsets X1,X2,…,X T is derived.

[0005] Meanwhile, it is known that online optimization problems can be roughly classified into the following two models, and a suitable optimization algorithm may differ depending on which model is assumed. A stochastic model that assumes that the objective function follows a stationary probability distribution Adversarial models in which the objective function varies non-stationarily (adversarially) However, in an online optimization problem, it is generally difficult to obtain a priori information regarding whether a probabilistic model or an adversarial model should be assumed. For this reason, a technology that can provide a suitable solution to an optimization problem for both a probabilistic model and an adversarial model without referring to a priori information regarding whether a probabilistic model or an adversarial model should be assumed has been desired.

[0006] One aspect of the present invention has been made in consideration of the above problems, and one example of its objective is to realize a technology that can provide a solution to an optimization problem appropriately for both a probabilistic model and an adversarial model, without referring to a priori information regarding whether a probabilistic model or an adversarial model should be assumed. [Means for solving the problem]

[0007] In one embodiment of the present invention, an information processing device calculates a set [n]= { 1,2,...,n } (n is any natural number) t ⊆ A selection means for selecting [n] by referring to the observed value of the objective function in round t-1; and a subset X t ⊆ and an output means for outputting information indicating the observed value f of the objective function in each round t ∈ [T]. t (X t ) and the comparison solution X * The regret Σ expressed using t∈[T] f t (X t )-Σ t∈[T] f t (X * The subset X is bounded from above by an upper limit A(Δ, n, C) that depends at least on the gap index Δ in the probabilistic model and the disturbance index C indicating the hostile disturbance of the probabilistic model. t⊆ Select [n] 。

[0008] In one embodiment of the present invention, in a certain round t ∈ [T] (T is an arbitrary natural number), a set [n]= { 1,2,...,n } (n is any natural number) t ⊆ A selection means for selecting [n] by referring to the observed value of the objective function in round t-1; and a subset X t ⊆ and an output unit for outputting information indicating [n], wherein the selection unit is For each i ∈ [n], the learning rate λ ti and the cumulative subgradient G ti Using and, n-dimensional vector x t ∈[0,1] n of

number

[0009] In one embodiment of the present invention, an information processing device calculates a set [n]= { 1,2,...,n } (n is any natural number) t ⊆ A selection means for selecting [n] by referring to the observed value of the objective function in round t-1; and a subset X t ⊆ An output means for outputting information indicating [n]; The selection means comprises: For each i ∈ [n], the learning rate λ t and the cumulative subgradient ^G ti and function ζ

number

number

number

[0010] In one aspect of the information processing method of the present invention, in a certain round t ∈ [T] (T is an arbitrary natural number), { 1,2,...,n } (n is any natural number) t ⊆ [n] by referring to the observed value of the objective function in round t-1; and t ⊆ and outputting information indicating the observed value f of the objective function in each round t ∈ [T] in the selecting step. t (X t ) and the optimal solution X * The regret Σ expressed using t∈[T] f t (X t )-Σ t∈[T] f t (X * The subset X is bounded from above by an upper limit A(Δ, n, C) that depends at least on the gap index Δ in the probabilistic model and the disturbance index C indicating the hostile disturbance of the probabilistic model. t ⊆ Select [n].

[0011] An information processing program according to an aspect of the present invention is a program for causing a computer to execute a program for processing a set [n]={ 1,2,...,n } (n is any natural number) t ⊆ [n] by referring to the observed value of the objective function in round t-1; t ⊆ and outputting information indicating the observed value f of the objective function in each round t ∈ [T] in the selection process. t (X t ) and the optimal solution X * The regret Σ expressed using t∈[T] f t (X t )-Σ t∈[T] f t (X * The subset X is bounded from above by an upper limit A(Δ, n, C) that depends at least on the gap index Δ in the probabilistic model and the disturbance index C indicating the hostile disturbance of the probabilistic model. t ⊆ Select [n]. Effect of the Invention

[0012] According to one aspect of the present invention, a solution to an optimization problem can be provided suitably for both a probabilistic model and an adversarial model without referring to a priori information regarding whether a probabilistic model or an adversarial model should be assumed. [Brief description of the drawings]

[0013] [Figure 1] 1 is a block diagram showing a configuration of an information processing device according to a first exemplary embodiment. [Diagram 2] 1 is a flow chart showing a flow of an information processing method according to a first exemplary embodiment. [Diagram 3] 1A to 1C are diagrams for explaining effects achieved by the information processing device according to the first exemplary embodiment. [Figure 4] FIG. 11 is a block diagram showing a configuration of an information processing device according to a second exemplary embodiment. [Diagram 5]FIG. 11 is a flow diagram showing a process flow in an information processing system according to a second exemplary embodiment. [Figure 6] FIG. 11 is a flow diagram illustrating a first processing example executed by an information processing device according to a second exemplary embodiment. [Figure 7] FIG. 11 is a flow diagram illustrating a second processing example executed by an information processing device according to a second exemplary embodiment. [Figure 8] FIG. 11 is a diagram showing an example of a display screen displayed by an information processing device according to a second exemplary embodiment. [Figure 9] FIG. 1 is a block diagram showing a configuration of a computer that functions as an information processing device according to each exemplary embodiment. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0014] Example 1 A first exemplary embodiment of the present invention will be described in detail with reference to the drawings. This exemplary embodiment is a basic form of the exemplary embodiments described later.

[0015] <Overview of information processing device 1> The information processing device 1 according to this exemplary embodiment is, in brief, an information processing device that performs optimization of an index (also called online optimization) under a condition where a function expressing the index to be optimized can change sequentially. In other words, the information processing device 1 is an information processing device that provides a solution to an online optimization problem.

[0016] As an example, the information processing device 1 determines an action in a certain round, and obtains an observation value related to the result obtained by the action. Then, the information processing device 1 repeats the process of determining an action to be executed in the next round by referring to the value of the observation value. Here, it is assumed that the relationship between the action and the result is expressed by an unknown objective function in which the action is an argument and the result is a function value. Therefore, by referring to the action determined in a certain round and the result obtained by the action, the information processing device 1 obtains (local) information related to the objective function.

[0017] An example of the above-mentioned behavior is the setting of prices for one or more products, and the observed value relating to the result is the actual sales amount obtained by the pricing, or the actual loss amount caused by the pricing (target sales amount minus actual sales amount), but this is not intended to limit the present exemplary embodiment. Also, an example of the index to be optimized is: Regret, which is the difference (or expected value) between the actual total sales amount and the total sales amount that would be made if ideal (optimal) actions were taken, or Regret is the difference (or expected value) between the total amount of actual losses and the total amount of losses that would be incurred if you had taken the ideal (optimal) action. However, this is not intended to limit the present exemplary embodiment.

[0018] Moreover, in this exemplary embodiment, the online optimization problem solved by the information processing device 1 is positioned as follows, for example.

[0019] (Feasible region) An example of the online optimization problem solved by the information processing device 1 is an online optimization problem classified into a combination set (subset family). Here, in the online optimization problem classified into the combination set, weighted sum or submodularity is assumed as the property of the objective function. In addition, in the online optimization problem classified into the combination set, a combination set having multiple elements is targeted, so that, as an example, combinations of prices of multiple products can be handled.

[0020] (Feedback information amount) The online optimization problem solved by the information processing device 1 is roughly classified into online optimization problems based on the following two settings: Here, the Full-Information setting provides a larger amount of feedback information regarding the objective function than the Bandit-Feedback setting. Full-Information Setting: In a certain round t, a subset X tAfter selecting (a certain action), the objective function f for any subset X (other actions) t The value of f t It is possible to refer to (X). Bandit-Feedback Setting: In round t, subset X t After selecting (a certain action), the selected subset X t Objective function f for (a certain action) t The value of f t (X t ) and the objective function f for the subset X (other actions) other than the selected subset t The value of f t It is not possible to refer to (X).

[0021] (Applicable models) Online optimization problems are broadly classified into the following two models: A stochastic model that assumes that the objective function follows a stationary probability distribution Adversarial models in which the objective function varies non-stationarily (adversarially) As described later, the information processing device 1 · Stochastic model with adversarial corruption The system is configured to be able to quantitatively evaluate the disturbance index C, and executes an algorithm that can be applied to any of a probabilistic model, an adversarial model, and an adversarial disturbance to a probabilistic model.

[0022] Therefore, the information processing device 1 can suitably solve the optimization problem for both the probabilistic model and the adversarial model using a hybrid algorithm (Best of Both Worlds Algorithm) that can be suitably applied to both the probabilistic model and the adversarial model without referring to a priori information regarding whether the probabilistic model or the adversarial model should be assumed.

[0023] <Configuration of information processing device 1> Next, the configuration of the information processing device 1 according to this exemplary embodiment will be described. Fig. 1 is a block diagram showing the configuration of the information processing device 1 according to this exemplary embodiment. As shown in Fig. 1, the information processing device 1 includes a selection unit 11 and an output unit 12.

[0024] The selection unit 11 selects a set [n]= { 1,2,...,n } (n is any natural number) t ⊆ [n] is selected with reference to the observed value of the objective function in round t-1.

[0025] The output unit 12 outputs the subset X selected by the selection unit 11. t ⊆ Outputs information indicating [n].

[0026] Here, the subset X t is a subset that defines the behavior in round t, and as an example, has the meaning of a subset that includes product identification information as an element, but this does not limit the present exemplary embodiment. Also, as an example of the objective function, there is an objective function that defines the relationship between the behavior and the sales amount, or an objective function that defines the relationship between the behavior and the loss amount, but this does not limit the present exemplary embodiment.

[0027] Moreover, each element of the set [n]={1, 2, . . . , n} (n is any natural number) according to this exemplary embodiment has a one-to-one correspondence with each element of any set S consisting of n elements. Therefore, the process described in this exemplary embodiment can be applied to any set S consisting of n elements.

[0028] In addition, the selection unit 11 according to the present exemplary embodiment selects the observed value f of the objective function in each round t ∈ [T]. t (X t ) and the comparison solution X * The regret Σ expressed using t∈[T] f t (X t)-Σ t∈[T] f t (X * The subset X is bounded from above by an upper limit A(Δ, n, C) that depends at least on the gap index Δ in the probabilistic model and the disturbance index C indicating the hostile disturbance of the probabilistic model. t ⊆ [n] is selected. Note that the solution to be compared is, for example, an optimal solution that minimizes an objective function that represents loss.

[0029] In addition, when using an objective function that represents sales, the sign of the regret may be reversed. In other words, the regret Σ t∈[T] f t (X t )-Σ t∈[T] f t (X * The subset X is bounded from above by an upper limit A(Δ, n, C) that depends at least on the gap index Δ in the probabilistic model and the disturbance index C indicating the hostile disturbance of the probabilistic model. t ⊆ It may also be expressed as selecting [n].

[0030] In this exemplary embodiment, the objective function f t For example, f is a function that satisfies submodularity. That is, the objective function f t For any subsets X,Y such that X,Y ⊆ [n], the inequality f t (X∩Y)+f t (X∪Y)≦f t (X)+f t (Y) However, this is not a limitation of this exemplary embodiment.

[0031] Also, the output unit 12 outputs the subset X t ⊆ Although the output destination of the information indicating [n] is not limited to this exemplary embodiment, as an example, the output unit 12 may include a display panel, and the information indicating the subset Xt The output unit 12 may display information indicating the subset X t The information indicating the price of a product or the like may be provided and the information may be displayed on the other device, or the information may be referenced to automatically update the price of the product or the like.

[0032] The gap index is calculated by the objective function f t The objective function f f follows the probability distribution D. t Expected value of

number

number

[0033] The disturbance index C is calculated by the objective function f t and the time-dependent objective function f t ' and

number

[0034] In the information processing device 1 configured as above, the expected value of regret is restricted from above by an upper limit value A(Δ, n, C). Here, the upper limit value is defined so as to include both the probabilistic model and the adversarial model, and is expressed using the disturbance index C indicating the adversarial disturbance of the probabilistic model, as described above. Therefore, according to the information processing device 1 configured as above, it is possible to suitably solve the optimization problem for both the probabilistic model and the adversarial model.

[0035] <Information processing method> Next, the flow of the information processing method S1 executed by the information processing device 1 will be described with reference to Fig. 2. Fig. 2 is a flow diagram showing the flow of the information processing method S1 executed by the information processing device 1.

[0036] (Step S11) In step S11, the selection unit 11 selects a set [n]= { 1,2,...,n } (n is any natural number) t ⊆ [n] is selected with reference to the observed value of the objective function in round t-1.

[0037] (Step S12) In step S12, the output unit 12 outputs the subset X selected by the selection unit 11. t ⊆ Outputs information indicating [n].

[0038] Here, in step S11, the selection unit 11 selects the observed value f of the objective function in each round t ∈ [T]. t (X t ) and the comparison solution X * The regret Σ expressed using t∈[T] f t (X t )-Σ t∈[T] f t (X * The subset X is bounded from above by an upper limit A(Δ, n, C) that depends at least on the gap index Δ in the probabilistic model and the disturbance index C indicating the hostile disturbance of the probabilistic model. t ⊆ [n] is selected. Note that the solution to be compared is, for example, an optimal solution that minimizes the objective function representing the loss, as described above.

[0039] In the information processing method S1 configured as above, the expected value of regret is restricted from above by an upper limit value A(Δ, n, C). Here, the upper limit value is defined so as to include both the probabilistic model and the adversarial model, and is expressed using the disturbance index C indicating the adversarial disturbance of the probabilistic model, as described above. Therefore, according to the information processing method S1 as described above, the optimization problem can be suitably solved for both the probabilistic model and the adversarial model.

[0040] (Comparison of the algorithm based on the information processing method S1 with other algorithms) 3 is a diagram showing the effect of the algorithm by the information processing method S1 according to the present exemplary embodiment. The first row from the top of FIG. 3 is a diagram showing the relationship between the loss (objective function) and time (round) in a probabilistic model (more specifically, a situation in which a probabilistic model is preferably applied, the same applies below), and the second row from the top of FIG. 3 shows the effect of each algorithm applied to the probabilistic model. -Algorithms dedicated to adversarial models - Algorithms dedicated to probabilistic models · Algorithm S1 (algorithm by information processing method S1) FIG. 13 is a schematic diagram showing the relationship between regret and time according to the present invention.

[0041] As shown in the first row from the top of Fig. 3, in the probabilistic model, the objective function is assumed to follow an unknown distribution. Here, the unknown distribution is, as an example, a uniform probability distribution that is independent of time (stationary). In such a situation, as shown in the first row from the top of Fig. 3, the regret by the algorithm dedicated to the adversarial model is larger than the regret by the algorithm dedicated to the probabilistic model. On the other hand, in the algorithm S1, the regret is restricted from above by the upper limit value A(Δ,n,C), so the regret by the algorithm S1 is smaller than the regret by the algorithm dedicated to the adversarial model.

[0042] On the other hand, the third row from the top of Fig. 3 is a schematic diagram showing the relationship between the loss (objective function) and time (round) in the adversarial model (more specifically, the situation in which the adversarial model is preferably applied, the same below), and the fourth row from the top of Fig. 3 shows the relationship between each algorithm applied to the adversarial model. -Algorithms dedicated to adversarial models - Algorithms dedicated to probabilistic models · Algorithm S1 (algorithm by information processing method S1) FIG. 13 is a schematic diagram showing the relationship between regret and time according to the present invention.

[0043] As shown in the third row of Figure 3, the adversarial model assumes that the objective function behaves non-stationary. More specifically, the objective function f t Let us consider the sequence of selected subsets (X1, X2, . . . , X t-1 ) In such a situation, as shown in the fourth row from the top of Figure 3, the regret by the algorithm dedicated to probabilistic models is larger than the regret by the algorithm dedicated to adversarial models. On the other hand, in algorithm S1, the regret is restricted from above by the upper limit value A(Δ,n,C), so the regret by algorithm S1 is smaller than the regret by the algorithm dedicated to probabilistic models.

[0044] In this way, according to the information processing device 1 and the information processing method S1 described above, it is possible to suitably solve the optimization problem for both the probabilistic model and the adversarial model.

[0045] Exemplary embodiment 2 A second exemplary embodiment of the present invention will be described in detail with reference to the drawings. Note that components having the same functions as those described in the first exemplary embodiment are given the same reference numerals, and the description thereof will be omitted as appropriate.

[0046] <Overview of Information Processing System 100A> The configuration of an information processing system 100A according to this exemplary embodiment will be described with reference to FIG. 4. FIG. 4 is a block diagram showing the configuration of the information processing system 100A. As shown in FIG. 4, the information processing system 100A includes an information processing device 1A and a terminal device 2A. Also, as shown in FIG. 4, the information processing device 1A and the terminal device 2A are configured to be able to communicate with each other via a network N. Here, the specific configuration of the network N does not limit this exemplary embodiment, but as an example, a wireless LAN (Local Area Network), a wired LAN, a WAN (Wide Area Network), a public line network, a mobile data communication network, or a combination of these networks can be used.

[0047] <Configuration of information processing device 1A> As shown in FIG. 4, the information processing device 1A includes a control unit 10A, a storage unit 17A, and a communication unit 19A.

[0048] The storage unit 17A stores observed values ​​OB of the objective function received from the terminal device 2A, which will be described later. The storage unit 17A also stores a selected subset SB selected by the selection unit 11.

[0049] The communication unit 19A communicates with devices external to the information processing device 1A. As an example, the communication unit 19A communicates with the terminal device 2A. The communication unit 19A transmits data provided from the control unit 10A to the terminal device 2A, and provides data received from the terminal device 2A to the control unit 10A.

[0050] (Control unit 10A) The control unit 10A includes a selection unit 11, an output unit 12, and an acquisition unit 13, as shown in FIG.

[0051] The acquisition unit 13 acquires an objective function f tThe observation value OB of the objective function is acquired from the terminal device 2A via the communication unit 19A. The acquisition unit 13 stores the acquired observation value OB of the objective function in the storage unit 17A. Here, the information that the acquisition unit 13 can acquire may differ depending on whether the Full-Information setting or the Bandit-Feedback setting is used.

[0052] In the Full-Information setting, the acquisition unit 13 acquires the subset X t The observed values ​​of the objective function obtained after outputting ⊆ Observed value f of the objective function for [n] t (X) can be obtained.

[0053] On the other hand, in the Bandit-Feedback setting, the acquisition unit 13 receives the subset X t The observed values ​​of the objective function obtained after outputting the selected subset X t The observed value of the objective function for f t (X t ), but a subset X other than the selected subset can be obtained. ⊆ Observed value f of the objective function for [n] t (X) is not available.

[0054] Here, the subset X t is a subset that defines the behavior in round t, as in the first exemplary embodiment, and has the meaning of a subset that includes product identification information as an element, but this does not limit the present exemplary embodiment. Also, examples of the objective function include an objective function that defines the relationship between the behavior and the sales amount, or an objective function that defines the relationship between the behavior and the loss amount, but this does not limit the present exemplary embodiment.

[0055] In a certain round, the selection unit 11 selects a set [n]= { 1,2,...,n } (n is any natural number) t ⊆[n] is selected with reference to the observed value of the objective function in round t-1.

[0056] Here, the selection unit 11 according to the present exemplary embodiment selects the observed value f of the objective function in each round t ∈ [T]. t (X t ) and the comparison solution X * The regret Σ expressed using t∈[T] f t (X t )-Σ t∈[T] f t (X * The subset X is bounded from above by an upper limit A(Δ, n, C) that depends at least on the gap index Δ in the probabilistic model and the disturbance index C indicating the hostile disturbance of the probabilistic model. t ⊆ [n] is selected. Note that the above-mentioned comparison solution is, for example, an optimal solution that minimizes an objective function that represents a loss, as in the exemplary embodiment 1. More specific processing by the selection unit 11 will be described later.

[0057] The output unit 12 outputs the subset X selected by the selection unit 11. t ⊆ Here, the output unit 12 outputs information indicating the subset X t ⊆ Although the output destination of the information indicating [n] is not limited to this exemplary embodiment, as an example, the output unit 12 can be configured to transmit the information to the terminal device 2A via the communication unit 19A. Also, the output unit 12 can be configured to have a display panel, and to display the information indicating the subset X t The information indicating the above may be displayed.

[0058] <Configuration of Terminal Device 2A> 4, the terminal device 2A includes a control unit 20A, a display unit 27A, an input receiving unit 28A, and a communication unit 29A. As an example, the terminal device can be specifically realized as an information processing terminal or the like installed in a store, but this is not intended to limit the present exemplary embodiment.

[0059] The communication unit 29A communicates with devices external to the terminal device 2A. As an example, the communication unit 29A communicates with the information processing device 1A. The communication unit 29A transmits data provided from the control unit 20A to the information processing device 1A, and provides data received from the information processing device 1A to the control unit 20A.

[0060] The display unit 27A displays the display data supplied from the control unit 20A. As an example, the display unit 27A displays the subset X selected by the selection unit 11 of the information processing device 1A and supplied to the terminal device 2A. t Displays information indicating the.

[0061] The input receiving unit 28A receives various inputs to the terminal device 2A. As an example, the input receiving unit 28A receives an observation value of the objective function in each round t. Then, the input receiving unit 28A supplies the received observation value to the control unit 20A. The supplied observation value is transmitted to the information processing device 1A via the communication unit 29A and acquired by the acquisition unit 13 described above. The input receiving unit 28A may be configured to receive the observation value via a user operation, or may be configured to automatically acquire the observation value.

[0062] Although the specific configuration of the input reception unit 28A does not limit this exemplary embodiment, as an example, the input reception unit 28A may be configured to include input devices such as a keyboard and a touch pad. Also, the input reception unit 28A may be configured to include a data scanner that reads data via electromagnetic waves such as infrared rays and radio waves.

[0063] (Control unit 20A) The control unit 20A includes a behavior executing unit 21 and an observed value acquiring unit 22, as shown in FIG.

[0064] The action execution unit 21 selects the subset X selected by the selection unit 11 of the information processing device 1A in each round t. t Obtain information indicating the obtained subset X t As an example, the subset X tRefer to the information indicating the subset X t The behavior execution unit 21 updates the prices of one or more products indicated by the subset X t In this configuration, the display unit 27A may generate display data showing the number of characters, supply the generated display data to the display unit 27A, and display the display data. In this configuration, the user may select one or more characters by referring to the display data displayed by the display unit 27A. Multiple Update the price of a product.

[0065] The observed value acquiring unit 22 acquires the observed value of the objective function after the action executing unit 21 executes the action via the input receiving unit 28. The observed value of the objective function acquired by the observed value acquiring unit 22 is supplied to the information processing device 1A via the communication unit 29A and acquired by the acquiring unit 13 of the information processing device 1A.

[0066] <Processing flow by information processing system 100A> Next, the flow of the information processing method S100A (also referred to as algorithm 1) by the information processing system 100A according to this exemplary embodiment will be described with reference to Fig. 5. In the following description, each step in round t-1 is marked with (t-1), and each step in round t is marked with (t) to distinguish between the different rounds.

[0067] (Step S23(t-1)) As shown in FIG. 5, in step S23(t−1), the terminal device 2A calculates the observed value f t to the information processing device 1A. Here, the observation values ​​that the terminal device 2A can provide may differ depending on whether the above-mentioned Full-Information setting or Bandit-Feedback setting is used.

[0068] In the Full-Information setting, in step S23(t-1), the terminal device 2A outputs the subset X t-1 The observed values ​​of the objective function obtained after outputting ⊆ Observed value f of the objective function for [n] t-1(X) can be provided.

[0069] On the other hand, in the Bandit-Feedback setting, in step S23(t-1), the terminal device 2A outputs the subset X t-1 The observed values ​​of the objective function obtained after outputting the selected subset X t-1 The observed value of the objective function for f t-1 (X t-1 ), but a subset X other than the selected subset can be obtained. ⊆ Observed value f of the objective function for [n] t-1 (X) is not available.

[0070] (Step S13(t-1)) Next, in step S13(t-1), the acquisition unit 13 of the information processing device 1A acquires the observed value f of the objective function provided by the terminal device 2A in step S23(t-1). t-1 Get the.

[0071] (Step S11(t)) Next, in step S11(t), the selection unit 11 of the information processing device 1A selects the observed value f of the objective function acquired by the acquisition unit in step S13(t-1). t-1 Refer to the subset X t ⊆ The process performed by the selection unit 11 will be described in more detail later.

[0072] (Step S12(t)) Next, in step S12(t), the output unit 12 outputs the subset X selected by the selection unit 11 in step S11(t). t ⊆ Output information indicating [n]. The output subset X t ⊆ The information indicating [n] is transmitted to the terminal device 2A via the communication unit 19A.

[0073] (Step S21(t)) Next, in step S21(t), the action execution unit 21 of the terminal device 2A outputs the subset X t ⊆ The action execution unit 21 executes an action according to the information indicating [n]. The specific processing by the action execution unit 21 has been described above, and therefore a description thereof will be omitted here.

[0074] (Step S22(t)) Subsequently, in step S22(t), the observed value acquisition unit 22 of the terminal device 2A acquires observed values ​​of the objective function obtained after the action taken by the action execution unit 21 in step S21(t).

[0075] (Step S23(t)) Subsequently, in step S23(t), the terminal device 2A provides the observed value of the objective function acquired in step S22(t) to the information processing device 1A.

[0076] Thereafter, as shown in FIG. 5, rounds in which the above-mentioned steps are executed are repeated.

[0077] <Processing example 1 by information processing device 1A: Full-Information Settings > Next, a flow of processing example 1 (algorithm 1) by the information processing device 1A will be described with reference to Fig. 6. Processing example 1 shows processing by the information processing device 1A in the Full-Information setting.

[0078] (Step S101) First, in step S101, the selection unit 11 initializes various parameters used in the process. As an example, the selection unit 11 sets the cumulative sub-gradient G1 in the first round as follows: G 1i =0∈R n Here, i is an index such that i∈[n], and [n] is the set of natural numbers [n] = { 1,2,...,n } (n is any natural number).

[0079] (Step S102) Step S102 is the start point of a loop process represented by a loop variable t (t=1, 2, . . . , T) (T is an arbitrary natural number). Here, the loop variable t is an index indicating the round number.

[0080] (Step S111A) In step S111A, the selection unit 11 selects the vector x t ∈[0,1] n of,

number

number

number

[0081] (Step S112A) Subsequently, in step S112A, the selection unit 11 performs the following for all i ∈ [n-1]: x tσ(i) ≦x tσ(i+1) A substitution σ t :Calculate [n]→[n].

[0082] (Step S113A) Next, in step S113A, the selection unit 11 selects a random variable u t In other words, the selection unit 11 determines the value of the variable u according to a uniform probability distribution on [0,1]. t Determine the value of

[0083] (Step S114A) Next, in step S114A, the selection unit 11 selects the subset X t of, X t ={i∈[n]|x ti ≧u t} is determined to satisfy the following.

[0084] (Step S12) Next, in step S12, the output unit 12 outputs the subset X selected by the selection unit 11 in step S114A. t The output subset X t is supplied to the terminal device 2A, and in the environment of the terminal device 2A, the subset X t The corresponding action is taken.

[0085] (Step S13) Next, in step S13, the acquisition unit 13 calculates the objective function f t Observed value f t In this step, the acquisition unit 13 acquires the subset X t The observed values ​​of the objective function obtained after outputting ⊆ Observed value f of the objective function for [n] t (X) can be obtained.

[0086] (Step S115A) Next, in step S115A, the selection unit 11 selects the subgradient g t ∈R d of,

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[0087] (Step S116A) Next, in step S116A, the selection unit 11 selects the cumulative subgradient G t of, G t+1 =G t +g t Update by.

[0088] (Step S103) Step S103 is the end of the loop process represented by the loop variable t.

[0089] (Theorem 1) The inventors of the present application have obtained regrets by the above processing example 1 (algorithm 1).

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[0090] Here, the parameter Δ≧0 represents the suboptimality gap in the probabilistic model. More specifically, the parameter Δ is the functiont The objective function f when f follows an unknown distribution D t Expected value of

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[0091] Moreover, the parameter C represents a disturbance index, and the disturbance index C is calculated by the objective function f t and the time-dependent objective function f t ' and

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[0092] In the information processing device 1A configured as above, by executing algorithm 1 in the Full-Information setting, the expected value of regret is restricted from above by an upper limit value A(Δ, n, C) that depends on Δ, n, and C. Therefore, according to the information processing device 1 configured as above, in the Full-Information setting, it is possible to suitably solve the optimization problem for both the probabilistic model and the adversarial model.

[0093] <Processing Example 2 by Information Processing Device 1A: Bandit-Feedback Setting> Next, a flow of a processing example 2 (algorithm 2) by the information processing device 1A will be described with reference to Fig. 7. The processing example 2 shows a process by the information processing device 1A in the Bandit-Feedback setting.

[0094] (Step S101) First, in step S101, the selection unit 11 initializes various parameters used in the process. As an example, the selection unit 11 sets the cumulative subgradient ^G1 in the first round as follows: ^G 1i =0∈R n Here, i is an index such that i∈[n], and [n] is the set of natural numbers [n] = { 1,2,...,n } (n is any natural number). Also, ^G ti is "G" with a hat "^". ti " It represents.

[0095] (Step S102) Step S102 is the start point of a loop process represented by a loop variable t (t=1, 2, . . . , T) (T is an arbitrary natural number). Here, the loop variable t is an index indicating the round number.

[0096] (Step S111B) In step S111B, the selection unit 11 selects a learning rate λ for each i ∈ [n]. t and the cumulative subgradient ^G ti and function ζ

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[0097] (Step S112B) Subsequently, in step S112B, the selection unit 11 performs the following for all i ∈ [n-1]: x tσ(i) ≦x tσ(i+1) A substitution σ t :Calculate [n]→[n].

[0098] (Step S113B) In step S113B, the selection unit 11 selects the probability

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[0099] (Step S114B) Step S11 4 In B, the selection unit 11 selects a subset X t of, X t = σ t ([i t ])={σ t (j)|j∈[i t ]} is determined to satisfy the following.

[0100] (Step S12) Next, in step S12, the output unit 12 outputs the subset X selected by the selection unit 11 in step S114B. t The output subset X t is supplied to the terminal device 2A, and in the environment of the terminal device 2A, the subset X t The corresponding action is taken.

[0101] (Step S13) Next, in step S13, the acquisition unit 13 calculates the objective function f t Observed value f tIn this step, the acquisition unit 13 acquires the subset X t The observed values ​​of the objective function obtained after outputting the selected subset X t The observed value of the objective function for f t (X t ), but a subset X other than the selected subset can be obtained. ⊆ Observed value f of the objective function for [n] t (X) is not available.

[0102] (Step S115B) Next, in step S115B, the selection unit 11 selects the subgradient ^g t ∈R d of,

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[0103] (Step S116B) Next, in step S116B, the selection unit 11 selects the cumulative subgradient ^G t of, ^G t+1 =^G t +^g t Update by.

[0104] (Step S103) Step S103 is the end of the loop process represented by the loop variable t.

[0105] (Theorem 2) The inventors of the present application have obtained regrets by the above processing example 2 (algorithm 2).

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[0106] In the information processing device 1A configured as above, by executing algorithm 2 in the Bandit-Feedback setting, the expected value of regret is restricted from above by an upper limit value A(Δ, n, C) that depends on Δ, n, and C. Therefore, according to the information processing device 1 configured as above, in the Bandit-Feedback setting, it is possible to suitably solve the optimization problem for both the probabilistic model and the adversarial model.

[0107] (Relationship with Robust Extension) In the following, the relationship between the above-mentioned Algorithm 1 and Algorithm 2 and Lovasz extension will be described.

[0108] Function f:2 [n] →R Given, the Robesian extension of a function f ~f:[0,1]n→R is given as follows, where "~f" stands for "f" with a tilde:

[0109] first, x = (x1,x2,...,x n ) T ∈[0,1] n and u∈[0,1] For x i Let H be the set of indexes i such that ≧u. u (x). That is, H u (x), H u (x) = {i∈[n]|x i ≧u} This H is defined as u Using (x), the Robesian expansion ~f(x) is

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[0110] From the above definition, for any x∈[0,1] n , and for any i ∈ [n-1], x σ(i) ≦x σ(i+1) For any permutation σ:[n]→[n] that satisfies, the Roveress extension ~f(x) is

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[0111] This gives us the subgradient g(σ)∈R of the Robust extension ∼f(x). n teeth,

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[0112] The subgradient of the Roveress extension .about.f(x) is used in both Algorithm 1 and Algorithm 2, as described above.

[0113] (Relationship with the FTRL algorithm) In the following, the relationship between the above-mentioned algorithm 1 and algorithm 2 and the FTRL (Follow The Regularized Leader) algorithm will be described.

[0114] The FTRL algorithm is n It is a general and elegant approach to online convex optimization on a subset Ω of . The update rule in the FTRL algorithm is

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[0115] The above-mentioned Algorithm 1 uses the regularization term ψt This corresponds to the following definition:

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[0116] On the other hand, Algorithm 2 uses the regularization term ψ in the FTRL algorithm. t This corresponds to the following definition:

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[0117] (Example of display by information processing system 1A) Next, a display example by the information processing system 1A will be described with reference to Fig. 8. Fig. 8 is a diagram showing a display example by the information processing system 1A. The example shown in Fig. 8 is a display example in which a function indicating (the total of) the sales amount of a plurality of products is used as the objective function, and one round is set to one day. That is, in the example shown in Fig. 8, the selection unit 11 of the information processing device 1A selects a subset X t is selected by referring to the observed values ​​(sales amount) of the objective function up to the day before the given day (round t-1).

[0118] Then, as shown in Fig. 8, the display unit 27A of the terminal device 2A displays each observed value of the objective function (sales amount in Fig. 8) for each round (day in Fig. 8). Furthermore, in the example shown in Fig. 8, the display unit 27A of the terminal device 2A displays information (prices of products A to C) about the subset selected in round t.

[0119] By performing such a display, the information processing system 1A can present the sales amount and the price of the product to the user.

[0120] [Application example] The above-described information processing devices 1 and 1A can be applied to various problems, examples of which are given below.

[0121] (Minimum time routing problem) Let us assume that the action is to select a route from one point to another. For example, suppose that there are n-1 relay points from one point to another, and that there are m selectable routes in each section. In this situation, the action plan (selected subset) X t If =[0, 2, 1, ···], it indicates that route 0 is selected in the first section, route 2 is selected in the second section, and route 1 is selected in the third section.

[0122] And the objective function f t is action measure X tis input, and the time required to travel the route indicated by the action plan is output. In this case, by applying the optimization method described above, it is possible to derive an optimal route setting for arriving at the other point from the one point in the shortest possible time.

[0123] (Retail) Let the discount on the price of beer from each company at a certain store be an action. For example, action measure (selected subset) X t If = [0, 2, 1, ···], the first element indicates that the price of beer from company A is the list price, the second element indicates that the price of beer from company B is 10% higher than the list price, and the third element indicates that the price of beer from company C is 10% discounted from the list price.

[0124] And the objective function f t is action measure X t The input is the price of each company's beer, and the output is the result of applying action measure X to the price of each company's beer. In this case, by applying the optimization method described above, it is possible to derive the optimal price setting for each company's beer price at the above store.

[0125] (Investment Portfolio) We will explain the case where it is applied to the investment behavior of investors. In this case, the investment (purchase, capital increase), sale, and holding of multiple financial products (stock brands, etc.) that an investor holds or intends to hold are called behavioral measure X. t For example, let action policy (selected subset) X t = [1, 0, 2, ···], the first element indicates an additional investment in Company A's stock, the second element indicates holding (neither buying nor selling) Company B's debt, and the third element indicates the sale of Company C's stock.

[0126] And the objective function f t is action measure X t The input is behavioral policy X for the investment behavior of each company in financial products. t In this case, by applying the optimization method described above, it is possible to derive the optimal investment behavior of the investor for each stock.

[0127] (Clinical Trials) This section explains the case where this is applied to the medication behavior for a clinical trial of a certain drug at a pharmaceutical company. In this case, the amount of medication and avoidance of medication are behavioral measures X t For example, let action policy (selected subset) X t = [1, 0, 2, ...], the first element indicates that subject A will be given dose 1, the second element indicates that subject B will not be given dose, and the third element indicates that subject C will be given dose 2.

[0128] And the objective function f t is action measure X t The input is the behavioral policy X t In this case, by applying the optimization method described above, it is possible to derive optimal medication behavior for each subject in the clinical trial of the pharmaceutical company.

[0129] (Web Marketing) We will explain the case where this is applied to advertising behavior (marketing measures) of an operating company of an electronic commerce site. In this case, advertising (online (banner) advertising, advertising by e-mail, direct mail, sending discount coupons by e-mail, etc.) to multiple customers for the products or services that the operating company is trying to sell is action measure X. t For example, let action policy (selected subset) X t = [1, 0, 2, ...], the first element indicates a banner advertisement for customer A, the second element indicates no advertisement for customer B, and the third element indicates sending a discount coupon by email to customer C.

[0130] And the objective function f t is action measure X t Input the following and apply the behavioral measure X to the advertising behavior of each customer: tThe result of applying the above is output. Here, the execution result may be whether or not a banner advertisement was clicked, the purchase amount, the purchase probability, or the expected value of the purchase amount. In this case, by applying the optimization method of this embodiment, it is possible to derive the optimal advertising behavior for each customer in the operating company.

[0131] [Software implementation example] Some or all of the functions of the information processing devices 1, 1A and the terminal device 2A may be realized by hardware such as an integrated circuit (IC chip), or may be realized by software.

[0132] In the latter case, the information processing devices 1, 1A and the terminal device 2A are realized by, for example, a computer that executes instructions of a program, which is software that realizes each function. An example of such a computer (hereinafter, referred to as computer C) is shown in FIG. 9 The computer C includes at least one processor C1 and at least one memory C2. The memory C2 stores a program P for causing the computer C to operate as the information processing device 1, 1A and the terminal device 2A. In the computer C, the processor C1 reads the program P from the memory C2 and executes it to realize the functions of the information processing device 1, 1A and the terminal device 2A.

[0133] The processor C1 may be, for example, a central processing unit (CPU), a graphic processing unit (GPU), a digital signal processor (DSP), a micro processing unit (MPU), a floating point number processing unit (FPU), a physics processing unit (PPU), a microcontroller, or a combination of these. The memory C2 may be, for example, a flash memory, a hard disk drive (HDD), a solid state drive (SSD), or a combination of these.

[0134] The computer C may further include a RAM (Random Access Memory) for expanding the program P during execution and for temporarily storing various data. The computer C may further include a communication interface for transmitting and receiving data to and from other devices. The computer C may further include an input / output interface for connecting input / output devices such as a keyboard, a mouse, a display, and a printer.

[0135] Furthermore, the program P can be recorded on a non-transitory tangible recording medium M that can be read by the computer C. Such a recording medium M can be, for example, a tape, a disk, a card, a semiconductor memory, or a programmable logic circuit. The computer C can obtain the program P via such a recording medium M. Furthermore, the program P can be transmitted via a transmission medium. Such a transmission medium can be, for example, a communication network or broadcast waves. The computer C can also obtain the program P via such a transmission medium.

[0136] [Additional Note 1] The present invention is not limited to the above-described embodiment, and various modifications are possible within the scope of the claims. For example, embodiments obtained by appropriately combining the technical means disclosed in the above-described embodiment are also included in the technical scope of the present invention.

[0137] [Additional Note 2] Some or all of the above-described embodiments can be described as follows. However, the present invention is not limited to the following described aspects.

[0138] (Appendix 1) In a certain round t∈[T] (T is any natural number), set [n] = { 1,2,...,n } (n is any natural number) t⊆ A selection means for selecting [n] by referring to the observed value of the objective function in round t-1; The subset X selected by the selection means t ⊆ An output means for outputting information indicating [n]; Equipped with The selection means is The observed value f of the objective function in each round t ∈ [T] t (X t ) and the comparison solution X * The regret Σ expressed using t∈[T] f t (X t )-Σ t∈[T] f t (X * The subset X is bounded from above by an upper limit A(Δ, n, C) that depends at least on the gap index Δ in the probabilistic model and the disturbance index C indicating the hostile disturbance of the probabilistic model. t ⊆ Select [n] 23. An information processing apparatus comprising:

[0139] (Appendix 2) The gap index is calculated by the objective function f t The objective function f f follows the probability distribution D. t Expected value of

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[0140] (Appendix 3) The output means outputs the subset X t After outputting, any subset X ⊆ The observed value f of the objective function for [n] t Further comprising an acquisition means capable of acquiring (X); The selection means selects the observed value f of the objective function acquired by the acquisition means. t (X) can be referenced, The upper limit A(Δ, n, C) is

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[0141] (Appendix 4) The selection means is For each i ∈ [n], the learning rate λ ti and the cumulative subgradient G ti Using and, n-dimensional vector x t ∈[0,1] n of

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[0142] (Appendix 5) The pre-output means outputs the subset X t After outputting, the selected subset X t The observed value f of the objective function for t (X t ) further comprising an acquisition means capable of acquiring the The selection means is The selected subset X t The observed value f of the objective function for t (X t ), and A subset X other than the selected subset ⊆ The observed value f of the objective function for [n] t It is not possible to refer to (X), The upper limit A(Δ, n, C) is

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[0143] (Appendix 6) The selection means is For each i ∈ [n], the learning rate λ t and the cumulative subgradient ^G ti and function ζ

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[0144] (Appendix 7) In a certain round t∈[T] (T is any natural number), set [n] = { 1,2,...,n } (n is any natural number) t ⊆A selection means for selecting [n] by referring to the observed value of the objective function in round t-1; The subset X selected by the selection means t ⊆ An output means for outputting information indicating [n]; Equipped with The selection means is For each i ∈ [n], the learning rate λ ti and the cumulative subgradient G ti Using and, n-dimensional vector x t ∈[0,1] n of

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[0145] (Appendix 8) In a certain round t∈[T] (T is any natural number), set [n] = { 1,2,...,n} (n is any natural number) t ⊆ A selection means for selecting [n] by referring to the observed value of the objective function in round t-1; The subset X selected by the selection means t ⊆ An output means for outputting information indicating [n]; Equipped with The selection means is For each i ∈ [n], the learning rate λ t and the cumulative subgradient ^G ti and function ζ

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[0146] (Appendix 9) In a certain round t∈[T] (T is any natural number), set [n] = { 1,2,...,n } (n is any natural number) t ⊆ [n] is selected with reference to the observed value of the objective function in round t-1; The selected subset X t ⊆ and outputting information indicating [n], In the selecting step, The observed value f of the objective function in each round t ∈ [T] t (X t ) and the optimal solution X * The regret Σ expressed using t∈[T] f t (X t )-Σ t∈[T] f t (X * The subset X is bounded from above by an upper limit A(Δ, n, C) that depends at least on the gap index Δ in the probabilistic model and the disturbance index C indicating the hostile disturbance of the probabilistic model. t ⊆ Select [n] 23. An information processing method comprising:

[0147] (Appendix 10) To your computer In a certain round t∈[T] (T is any natural number), set [n] = { 1,2,...,n } (n is any natural number) t⊆ A process of selecting [n] by referring to the observed value of the objective function in round t-1; The selected subset X t ⊆ A process to output information indicating [n] Run the command, In the selection process, The observed value f of the objective function in each round t ∈ [T] t (X t ) and the optimal solution X * The regret Σ expressed using t∈[T] f t (X t )-Σ t∈[T] f t (X * The subset X is bounded from above by an upper limit A(Δ, n, C) that depends at least on the gap index Δ in the probabilistic model and the disturbance index C indicating the hostile disturbance of the probabilistic model. t ⊆ Select [n] 2. An information processing program comprising:

[0148] (Appendix 11) A computer-readable recording medium having the program according to claim 10 recorded thereon.

[0149] (Appendix 12) At least one processor is provided, and the processor selects a set [n]= { 1,2,...,n } (n is any natural number) t ⊆ A process of selecting [n] by referring to the observed value of the objective function in round t-1; The selected subset X t ⊆ A process to output information indicating [n] Run In the selection process, The observed value f of the objective function in each round t ∈ [T] t (Xt ) and the optimal solution X * The regret Σ expressed using t∈[T] f t (X t )-Σ t∈[T] f t (X * The subset X is bounded from above by an upper limit A(Δ, n, C) that depends at least on the gap index Δ in the probabilistic model and the disturbance index C indicating the hostile disturbance of the probabilistic model. t ⊆ Select [n] Information processing device .

[0150] The information processing device may further include a memory, and the memory may store a program for causing the processor to execute the selecting process and the outputting process. The program may be recorded in a computer-readable, non-transitory, tangible recording medium. [Explanation of symbols]

[0151] 1,1A Information processing equipment 11 Selection unit (selection means) 12 Output section (output means) 13 Acquisition Department

Claims

1. In a certain round t ∈ [T] (T is any natural number), a subset X of the set [n] = {1, 2, ..., n} (n is any natural number) ｔ a selection means for selecting ⊆[n] by referring to the observed value of the objective function in round t−1; The subset X selected by the selection means ｔ an output means for outputting information indicating ⊆[n]; Equipped with The selection means is The observed value f of the objective function in each round t ∈ [T] ｔ (X ｔ ) and comparison solution X ＊ The regret Σ expressed using ｔ∈［Ｔ］ f ｔ (X ｔ ) -Σ ｔ∈［Ｔ］ f ｔ (X ＊ The subset X is bounded from above by an upper limit A(Δ, n, C) that depends at least on a gap index Δ in the probabilistic model and a disturbance index C indicating an adversarial disturbance of the probabilistic model. ｔ Select ⊆ [n], The method further includes an acquisition unit capable of acquiring an observed value f t (X) of the objective function for any subset X ⊆ [n] after the output unit outputs the subset X t in round t, The selection means is capable of referring to the observed value f t (X) of the objective function acquired by the acquisition means, The upper limit A(Δ, n, C) is [0010] It is expressed as 23. An information processing apparatus comprising:

2. The gap index is calculated by the objective function f ｔ The objective function f ｔ Expected value of [0025] Using [0030] It is expressed as, The disturbance index C is calculated by the objective function f ｔ and the time-dependent objective function f ｔ ' and [0045] It is expressed as The information processing device according to claim 1 .

3. The selection means is For each i ∈ [n], the learning rate λ ｔｉ and the cumulative subgradient G ｔｉ Using the above, n-dimensional vector x ｔ ∈[0, 1] ｎ of [0050] A vector calculation step of calculating For all i ∈ [n−1], x ｔσ（ｉ） ≦x ｔσ（ｉ＋１） A substitution σ ｔ : A substitution calculation step for calculating [n] → [n]; A random variable u uniformly distributed on [0, 1] ｔ a random variable determination step for determining the value of subset X ｔ , X ｔ = {i∈[n]|x ｔｉ ≧u ｔ }; and The observed value of the objective function f ｔ An acquisition step of acquiring (X); Objective function f ｔ The subgradient g ｔ ∈R ｄ A subgradient calculation step of calculating Cumulative subgradient G ｔ , G ｔ＋１ = G ｔ +g ｔ and an update step of updating by In each round, The information processing device according to claim 1 .

4. A selection means for selecting a subset X t ⊆ [n] of a set [n] = {1, 2, ..., n} (n is any natural number) in a certain round t ∈ [T] (T is any natural number) by referring to an observed value of an objective function in round t-1; an output means for outputting information indicating the subset X t ⊆ [n] selected by the selection means; Equipped with The selection means is The subset X t ⊆ [n] is selected so that the asymptotic behavior of the expected value of regret Σ t ∈ [T] ft (X t ) - Σ t ∈ [T] ft (X * ), expressed using the observed value ft (X t ) of the objective function in each round t ∈ [T] and the comparison solution X * , is bounded from above by an upper limit value A (Δ, n, C) that depends at least on a gap index Δ in the probabilistic model and a disturbance index C indicating an adversarial disturbance of the probabilistic model; The output means outputs the subset X ｔ After outputting, the selected subset X ｔ The observed value f of the objective function for ｔ (X ｔ ) further comprising an acquisition means capable of acquiring the The selection means is The selected subset X ｔ The observed value f of the objective function for ｔ (X ｔ ) and The observed value f of the objective function for a subset X ⊆ [n] other than the selected subset ｔ It is not possible to refer to (X), The upper limit A(Δ, n, C) is [006] An information processing device expressed as such.

5. The selection means is For each i ∈ [n], the learning rate λ ｔ and the cumulative subgradient ^G ｔｉ and function ζ [0070] Using the above, n-dimensional vector x ｔ ∈[0, 1] ｎ , x ｔｉ = ζ(^G ｔｉ / λ ｔ ) a vector calculation step of calculating the vector For all i ∈ [n−1], x ｔσ（ｉ） ≦x ｔσ（ｉ＋１） A substitution σ ｔ : a substitution calculation step of calculating [n] → [n]; probability [0080] [0097] According to the index i ｔ an index selection step of selecting ∈{0, 1, ..., n}; subset X ｔ , X ｔ = {σ ｔ (j) | j ∈ [i ｔ ]}; and The observed value of the objective function f ｔ (X ｔ ) and Objective function f ｔ The subgradient of ^g ｔ ∈R ｄ A subgradient calculation step of calculating Cumulative subgradient ^G ｔｉ ^G ｔ＋１ =^G ｔ +^g ｔ and an update step of updating by In each round, The information processing device according to claim 4.

6. In a certain round t ∈ [T] (T is any natural number), a subset X of the set [n] = {1, 2, ..., n} (n is any natural number) ｔ a selection means for selecting ⊆[n] by referring to the observed value of the objective function in round t−1; The subset X selected by the selection means ｔ an output means for outputting information indicating ⊆[n]; Equipped with The selection means is For each i ∈ [n], the learning rate λ ｔｉ and the cumulative subgradient G ｔｉ Using the above, n-dimensional vector x ｔ ∈[0, 1] ｎ of [0089] A vector calculation step of calculating For all i ∈ [n−1], x ｔσ（ｉ） ≦x ｔσ（ｉ＋１） A substitution σ ｔ : A substitution calculation step for calculating [n] → [n]; A random variable u uniformly distributed on [0, 1] ｔ a random variable determination step for determining the value of subset X ｔ , X ｔ = {i∈[n]|x ｔｉ ≧u ｔ }; and Objective function f ｔ An acquisition step of acquiring a value of (X); Objective function f ｔ The subgradient g ｔ ∈R ｄ A subgradient calculation step of calculating Cumulative subgradient G ｔ , G ｔ＋１ = G ｔ +g ｔ and an update step of updating by An information processing device that executes the above in each round.

7. In a certain round t ∈ [T] (T is any natural number), a subset X of the set [n] = {1, 2, ..., n} (n is any natural number) ｔ a selection means for selecting ⊆[n] by referring to the observed value of the objective function in round t−1; The subset X selected by the selection means ｔ an output means for outputting information indicating ⊆[n]; Equipped with The selection means is For each i ∈ [n], the learning rate λ ｔ and the cumulative subgradient ^G ｔｉ and function ζ ##EQU00011## Using the above, n-dimensional vector x ｔ ∈[0, 1] ｎ , x ｔｉ = ζ(^G ｔｉ / λ ｔ ) a vector calculation step of calculating the vector For all i ∈ [n−1], x ｔσ（ｉ） ≦x ｔσ（ｉ＋１） A substitution σ ｔ : a substitution calculation step of calculating [n] → [n]; probability ##EQU00012## ##EQU00013## According to the index i ｔ an index selection step of selecting ∈{0, 1, ..., n}; subset X ｔ , X ｔ = {σ ｔ (j) | j ∈ [i ｔ ]}; and Objective function f ｔ (X ｔ ) value; Objective function f ｔ The subgradient of ^g ｔ ∈R ｄ A subgradient calculation step of calculating Cumulative subgradient ^G ｔｉ ^G ｔ＋１ =^G ｔ +^g ｔ Update by update step and, An information processing device that executes the above in each round.

8. One or more processors, In a certain round t ∈ [T] (T is any natural number), a subset X of the set [n] = {1, 2, ..., n} (n is any natural number) ｔ ⊆[n] with reference to the observed value of the objective function in round t−1; The selected subset X ｔ and outputting information indicating ⊆[n]; In the selecting step, The observed value f of the objective function in each round t ∈ [T] ｔ (X ｔ ) and optimal solution X ＊ The regret Σ expressed using ｔ∈［Ｔ］ f ｔ (X ｔ ) -Σ ｔ∈［Ｔ］ f ｔ (X ＊ The subset X is bounded from above by an upper limit A(Δ, n, C) that depends at least on a gap index Δ in the probabilistic model and a disturbance index C indicating an adversarial disturbance of the probabilistic model. ｔ Select ⊆ [n], The method further includes an acquisition step capable of acquiring an observed value f t (X) of the objective function for any subset X ⊆ [n] after outputting the subset X t in round t in the output step, In the selecting step, the observed value f t (X) of the objective function acquired in the acquiring step can be referenced; The upper limit A(Δ, n, C) is ##EQU00014## It is expressed as 23. An information processing method comprising:

9. One or more processors, In a certain round t ∈ [T] (T is any natural number), a subset X t ⊆ [n] of a set [n] = {1, 2, ..., n} (n is any natural number) is selected by referring to the observed value of the objective function in round t-1; and outputting information indicative of the selected subset X t ⊆ [n]; In the selecting step, The subset X t ⊆ [n] is selected so that the asymptotic behavior of the expected value of regret Σ t ∈ [T] ft (X t )-Σ t ∈ [T] ft (X * ), expressed using the observed value ft (X t ) of the objective function in each round t ∈ [T] and the optimal solution X * , is bounded from above by an upper limit value A (Δ, n, C) that depends at least on a gap index Δ in the probabilistic model and a disturbance index C indicating an adversarial disturbance of the probabilistic model; The method further includes an acquisition step capable of acquiring an observed value f t (X t ) of the objective function for a selected subset X t after outputting the subset X t in round t in the output step, In the selecting step, It is possible to refer to the observed value f t (X t ) of the objective function for the selected subset X t , and It is not possible to refer to the observed value f t (X) of the objective function for a subset X ⊆ [n] other than the selected subset; The upper limit A(Δ, n, C) is ##EQU00015## It is expressed as 23. An information processing method comprising:

10. To your computer In a certain round t ∈ [T] (T is any natural number), a subset X of the set [n] = {1, 2, ..., n} (n is any natural number) ｔ ⊆[n] by referring to the observed value of the objective function in round t−1; The selected subset X ｔ ⊆[n]. Run the command, In the selection process, The observed value f of the objective function in each round t ∈ [T] ｔ (X ｔ ) and optimal solution X ＊ The regret Σ expressed using ｔ∈［Ｔ］ f ｔ (X ｔ ) -Σ ｔ∈［Ｔ］ f ｔ (X ＊ The subset X is bounded from above by an upper limit A(Δ, n, C) that depends at least on a gap index Δ in the probabilistic model and a disturbance index C indicating an adversarial disturbance of the probabilistic model. ｔ Select ⊆ [n], After outputting the subset X t in round t in the output process, further executing an acquisition process capable of acquiring an observed value f t (X) of the objective function for any subset X ⊆ [n]; In the selection process, the observed value f t (X) of the objective function acquired in the acquisition process can be referenced; The upper limit A(Δ, n, C) is ##EQU00016## It is expressed as 2. An information processing program comprising:

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