smbot/xnes.lua

-- Exponential Natural Evolution Strategies
-- http://people.idsia.ch/~juergen/xNES2010gecco.pdf
-- not to be confused with the Nintendo Entertainment System.

local assert = assert
local exp = math.exp
local floor = math.floor
local ipairs = ipairs
local log = math.log
local max = math.max
local pairs = pairs
local pow = math.pow
local sqrt = math.sqrt
local unpack = table.unpack or unpack

local Base = require "Base"

local nn = require "nn"
local normal = nn.normal
local zeros = nn.zeros

local util = require "util"
local argsort = util.argsort

local Xnes = Base:extend()

local function dot_mv(mat, vec, out)
    -- treats matrix as a matrix.
    -- treats vec as a column vector, flattened.
    assert(#mat.shape == 2)
    local d0, d1 = unpack(mat.shape)
    assert(d1 == #vec)

    local out_shape = {d0}
    if out == nil then
        out = zeros(out_shape)
    else
        assert(d0 == #out, "given output is the wrong size")
    end

    for i=1, d0 do
        local sum = 0
        for j=1, d1 do
            sum = sum + mat[(i - 1) * d1 + j] * vec[j]
        end
        out[i] = sum
    end

    return out
end

local function make_utility(popsize, out)
    local utility = out or {}
    local temp = log(popsize / 2 + 1)
    for i=1, popsize do utility[i] = max(0, temp - log(i)) end
    local sum = 0
    for _, v in ipairs(utility) do sum = sum + v end
    for i, v in ipairs(utility) do utility[i] = v / sum - 1 / popsize end
    return utility
end

local function make_covars(dims, sigma, out)
    local covars = out or zeros{dims, dims}
    local c = sigma / dims
    -- simplified form of the determinant of the matrix we're going to create:
    local det = pow(1 - c, dims - 1) * (c * (dims - 1) + 1)
    -- multiplying by this constant makes the determinant 1:
    local m = pow(1 / det, 1 / dims)

    local filler = c * m
    for i=1, #covars do covars[i] = filler end
    -- diagonals:
    for i=1, dims do covars[i + dims * (i - 1)] = m end

    return covars
end

function Xnes:init(dims, popsize, learning_rate, sigma)
    -- heuristic borrowed from CMA-ES:
    self.dims = dims
    self.popsize = popsize or 4 + (3 * floor(log(dims)))
    self.learning_rate = learning_rate or 3/5 * (3 + log(dims)) / (dims * sqrt(dims))
    self.sigma = sigma or 1

    self.utility = make_utility(self.popsize)

    self.mean = zeros{dims}
    -- note: this is technically the co-standard-deviation.
    -- you can imagine the "s" standing for "sqrt" if you like.
    self.covars = make_covars(self.dims, self.sigma, self.covars)

    --self.log_sigma = log(self.sigma)
    --self.log_covars = zeros{dims, dims}
    --for i, v in ipairs(self.covars) do self.log_covars[i] = log(v) end
end

function Xnes:params(new_mean, new_covars)
    if new_mean ~= nil then
        assert(#self.mean == #new_mean, "new parameters have the wrong size")
        for i, v in ipairs(new_mean) do self.mean[i] = v end
    end
    if new_covars ~= nil then
        -- TODO: assert determinant of new_covars is 1.
        error("TODO")
    end
    return self.mean
end

function Xnes:ask_once(asked, noise)
    asked = asked or zeros(self.dims)
    noise = noise or {}

    for i=1, self.dims do noise[i] = normal() end
    noise.shape = {#noise}

    dot_mv(self.covars, noise, asked)
    for i, v in ipairs(asked) do asked[i] = self.mean[i] + self.sigma * v end

    return asked, noise
end

function Xnes:ask(asked, noise)
    -- return a list of parameters for the user to score,
    -- and later pass to :tell().
    if asked == nil then
        asked = {}
        for i=1, self.popsize do asked[i] = zeros(self.dims) end
    end
    if noise == nil then
        noise = {}
        for i=1, self.popsize do noise[i] = zeros(self.dims) end
    end
    for i=1, self.popsize do self:ask_once(asked[i], noise[i]) end
    self.noise = noise
    return asked, noise
end

function Xnes:tell(scored, noise)
    local noise = noise or self.noise
    assert(noise, "missing noise argument")

    local arg = argsort(scored, function(a, b) return a > b end)

    local g_delta = zeros{self.dims}
    for p=1, self.popsize do
        local noise_p = noise[arg[p]]
        for i=1, self.dims do
            g_delta[i] = g_delta[i] + self.utility[p] * noise_p[i]
        end
    end

    local g_covars = zeros{self.dims, self.dims}
    local traced = 0
    for p=1, self.popsize do
        local noise_p = noise[arg[p]]
        for i=1, self.dims do
            for j=1, self.dims do
                local ind = (i - 1) * self.dims + j
                local zzt = noise_p[i] * noise_p[j] - (i == j and 1 or 0)
                local temp = self.utility[p] * zzt
                g_covars[ind] = g_covars[ind] + temp
                traced = traced + temp
            end
        end
    end

    local g_sigma = traced / self.dims

    for i=1, self.dims do
        local ind = (i - 1) * self.dims + i
        g_covars[ind] = g_covars[ind] - g_sigma
    end

    -- finally, update according to the gradients.

    local dotted = dot_mv(self.covars, g_delta)
    for i, v in ipairs(self.mean) do
        self.mean[i] = v + self.sigma * dotted[i]
    end

    --[[
    --self.log_sigma = self.log_sigma + self.learning_rate / 2 * g_sigma
    for i, v in ipairs(self.log_covars) do
        self.log_covars[i] = v + lr * g_covars[i]
    end
    --]]

    local lr = self.learning_rate * 0.5
    self.sigma = self.sigma * exp(lr * g_sigma)
    for i, v in ipairs(self.covars) do
        self.covars[i] = v * exp(lr * g_covars[i])
    end

    -- bookkeeping:
    --self.sigma = exp(self.log_sigma)
    --for i, v in ipairs(self.log_covars) do self.covars[i] = exp(v) end
    self.noise = nil
end

return {
    dot_mv = dot_mv,
    make_utility = make_utility,
    make_covars = make_covars,

    Xnes = Xnes,
}