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+--[[ 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