Lotka–Volterra model with mutations and generative adversarial networks

Abstract

A model of population genetics of the Lotka–Volterra type with mutations on a statistical manifold is introduced. Mutations in the model are described by diffusion on a statistical manifold with a generator in the form of a Laplace–Beltrami operator with a Fisher–Rao metric, that is, the model combines population genetics and information geometry. This model describes a generalization of the model of machine learning theory, the model of generative adversarial network (GAN), to the case of populations of generative adversarial networks. The introduced model describes the control of overfitting for generating adversarial networks.

Appendix

Maximum likelihood method

We consider a parametric family of probability distributions $p(x,\theta)$, $x\in X$, with a parameter $\theta$. Let $\{x_i\}$ be a sample of independent trials in $X$ (we assume the existence of a probability distribution $q$ on $X$ that generates such a sample). The likelihood function has the form of the product of probability densities for events from the sample

$$L(\{x_i\},\theta)=\prod_{i=1}^n p(x_i,\theta).$$

A maximum likelihood estimate for $\theta$,

$$\theta_0=\arg \max L(\{x_i\},\theta)=\arg \max \frac{1}{n} \log L(\{x_i\},\theta)=\arg \max \frac{1}{n} \sum_{i=1}^n \log p(x_i,\theta),$$

takes the form of an empirical sum. We assume that the maximum likelihood distribution $p(x,\theta_0)$ approximates the unknown distribution $q(x)$.

Overfitting is a well-known problem in learning theory: fitting $p(x,\theta)$ to the training sample $\{x_i\}$ can have a high likelihood but a low likelihood on the control sample $\{x'_i\}$ (some other sample generated by the distribution $q$). The control of overfitting in learning theory usually reduces to regularization (additions to the empirical sum that change the form of the likelihood function, or another kind of the empirical risk functional).

Game theory

A game is a function of several variables called players. A game takes values equal to a set of real numbers, one for each player:

$$(v_1,\dots,v_n)(s_1,\dots,s_n).$$

The $i$th number $v_i$ in the set is called the payoff for the $i$th player. The variable $s_i$ for each player takes values called (pure) strategies, and each player has its own set of strategies.

Example: two players, a zero-sum game (the sum of the players payoffs is zero).

A mixed strategy for the $i$th player: the probability distribution $p_i(s_j^i)$ for this player’s strategies, the index $j$ enumerates different strategies for the $i$th player. The payoff of the $i$th player for a set of mixed strategies

$$\langle v_i\rangle=\sum_{j_1,\dots,j_n} p_1(s_{j_1}^1)\dots p_n(s_{j_n}^n) v_i(s_{j_1}^1,\dots,s_{j_n}^n)$$

is the average of the payoff of the $i$th player over mixed strategies for all players. Here, the index $j_k$ ranges the values of possible strategies $s_k$ for the $k$th player: $s_k\in\{s_{1}^k,\dots,s_{m_k}^k\}$.

Nash equilibrium: A set of strategies in which no participant can increase the payoff by changing their strategy if the other participants do not change their strategies. It exists in the class of mixed strategies (may not exist in the class of pure strategies).

Maximin: the largest payoff that a given player can obtain without knowing the actions of other players,

$$\underline{v_i}=\max_{s_i}\min_{s_{-i}}v_i(s_i,s_{-i}),$$

where $s_i$ is the strategy of the $i$th player, the $s_{-i}$ are other players’ strategies, and $v_i$ is the payoff of the $i$th player.

Minimax: The smallest payoff that other players can force without knowing the player’s action,

$$\overline{v_i}=\min_{s_{-i}}\max_{s_i}v_i(s_i,s_{-i}).$$

The minimax is not less than the maximin, $\overline{v_i}\ge \underline{v_i}$.

The minimax for two-player zero-sum games is the same as the Nash equilibrium.

Lotka–Volterra model

This model describes the population dynamics of two species (predator and prey) by a system of two ODEs

$$\frac{dx}{dt}=\alpha x-\beta xy, \qquad \frac{dy}{dt}=-\gamma y+\delta xy,$$

where $x$ is the prey population and $y$ is the predator population. All constants are positive and the nonlinear term describes the interaction between the predator and prey. For an ecological niche of a finite volume, the first equation becomes ($N>0$)

$$\frac{dx}{dt}=\alpha \biggl(x-\frac{x^2}{N}\biggr)-\beta xy.$$

The Lotka–Volterra model with mutations generalizes the above model as follows. There are prey $x_i$ and predators $y_m$, and the equations of population dynamics have the form

$$\begin{aligned} \, &\frac{dx_i}{dt}=\sum_j A_{ij} x_j - x_i \sum_{n}B_{in} y_n, \\ &\frac{dy_m}{dt}=\sum_{n} C_{mn} y_n + y_m \sum_{j} B_{jm} x_j. \end{aligned}$$

The matrix $A$ has the following meaning: the diagonal part describes the reproduction of prey, the off-diagonal part describes mutations (the matrix elements are positive). The matrix $C$ has the following meaning: its diagonal part (with negative matrix elements) describes the extinction of predators, the off-diagonal part describes mutations (these matrix elements are positive). The $B$ matrix describes the predator–prey interaction (the matrix elements are positive).

For a bounded ecological niche, the first equation takes the form

$$\frac{dx_i}{dt}=\sum_j A_{ij} x_j - \frac{1}{N} x_i \sum_{ij} A_{ij} x_j - x_i \sum_{n}B_{in} y_n.$$

In the absence of predators ($y_n=0$), this equation takes the form of the equation of Eigen’s model [8], which describes the competition of genotypes in a limited ecological niche in the presence of mutations.

Information geometry [9]–[14]

A statistical manifold is a manifold of parameters of a parametric probability distribution, or a manifold whose points are probability distributions on $X$ (a smooth dependence of the distribution on parameters is assumed).

The (nonsymmetric) Kullback–Leibler distance between two probability distributions on $X$ is defined as

$$D(p|q)=\int_{X}p(x)\log\frac{p(x)}{q(x)}\,dx.$$

The Fisher–Rao metric on the statistical manifold has the form

$$g_{ij}(\theta)=\int_{X}\frac{\partial \log p(x,\theta)}{\partial \theta_i}\frac{\partial \log p(x,\theta)}{\partial \theta_j} p(x,\theta)\,dx.$$

The Kullback–Leibler distance expansion for small parameter differences $\Delta\theta=\theta-\theta_0$ is related to the Fisher metric:

$$D(p(\theta_0)|p(\theta))=\frac{1}{2}\sum_{ij}g_{ij}(\theta_0)\Delta\theta^i\Delta\theta^j.$$

The Laplace–Beltrami operator on a statistical manifold with a parameter $\theta$ has the following form, where $g_{ij}$ is the Fisher metric:

$$\Delta_{\theta}= \frac{1}{\sqrt{g}}\sum_{i}\frac{\partial}{\partial \theta_i}\biggl(\sqrt{g}\sum_{j}g^{ij}\frac{\partial}{\partial \theta_j}\biggr).$$