--[[ An implementation of Adam https://arxiv.org/abs/1412.6980

ARGS:

- 'opfunc' : a function that takes a single input (X), the point
             of a evaluation, and returns f(X) and df/dX
- 'x'      : the initial point
- 'config` : a table with configuration parameters for the optimizer
- 'config.learningRate'      : learning rate
- `config.learningRateDecay` : learning rate decay
- 'config.beta1'             : first moment coefficient
- 'config.beta2'             : second moment coefficient
- 'config.epsilon'           : for numerical stability
- 'config.weightDecay'       : weight decay
- 'state'                    : a table describing the state of the optimizer; after each
                              call the state is modified

RETURN:
- `x`     : the new x vector
- `f(x)`  : the function, evaluated before the update

]]

function optim.adam(opfunc, x, config, state)
   -- (0) get/update state
   local config = config or {}
   local state = state or config
   local lr = config.learningRate or 0.001
   local lrd = config.learningRateDecay or 0

   local beta1 = config.beta1 or 0.9
   local beta2 = config.beta2 or 0.999
   local epsilon = config.epsilon or 1e-8
   local wd = config.weightDecay or 0

   -- (1) evaluate f(x) and df/dx
   local fx, dfdx = opfunc(x)

   -- (2) weight decay
   if wd ~= 0 then
      dfdx:add(wd, x)
   end

   -- Initialization
   state.t = state.t or 0
   -- Exponential moving average of gradient values
   state.m = state.m or x.new(dfdx:size()):zero()
   -- Exponential moving average of squared gradient values
   state.v = state.v or x.new(dfdx:size()):zero()
   -- A tmp tensor to hold the sqrt(v) + epsilon
   state.denom = state.denom or x.new(dfdx:size()):zero()

   -- (3) learning rate decay (annealing)
   local clr = lr / (1 + state.t*lrd)

   state.t = state.t + 1

   -- Decay the first and second moment running average coefficient
   state.m:mul(beta1):add(1-beta1, dfdx)
   state.v:mul(beta2):addcmul(1-beta2, dfdx, dfdx)

   state.denom:copy(state.v):sqrt():add(epsilon)

   local biasCorrection1 = 1 - beta1^state.t
   local biasCorrection2 = 1 - beta2^state.t
   local stepSize = clr * math.sqrt(biasCorrection2)/biasCorrection1
   -- (4) update x
   x:addcdiv(-stepSize, state.m, state.denom)

   -- return x*, f(x) before optimization
   return x, {fx}
end