Manuscript Psy03-C@1 · Psy03-C-v1.tex · section · d3380b90

A complete recurrent comparison

Noise, parameter, and control certificates › A complete recurrent comparison · lines 905–940

Psy03-C@1#eq:fullresidual

For a comparison interface, allow both input and hidden perturbations explicitly:

u_{t} = g_{e} (x_{t} + ξ E_{t}) + τ G_{t}^{in},

v_{t} = a F_{θ} (u_{t}, h_{t - 1}) + aσ G_{t}^{hid} + a β b_{p} 1 - o_{p} 1, h_{t} = φ (v_{t}) .

The source's ordinary and biased paths are particular cases. Using both perturbation types simultaneously is a separately declared experiment.

For primed controls under the same primitive draws, put

U_{t} = η_{t} = \abs g_{e} - g_{e}^{'} \norm x_{t} + ξ^{'} E_{t}_{2} + g_{e} \abs ξ - ξ^{'} \norm E_{t}_{2} + \abs τ - τ^{'} \norm G_{t}^{in}_{2}, a L_{x} U_{t} + a ϵ_{F} (u_{t}^{'}, h_{t - 1}^{'}) + \abs a - a^{'} \norm F_{θ^{'}} (u_{t}^{'}, h_{t - 1}^{'})_{2} + \abs aσ - a^{'} σ^{'} \norm G_{t}^{hid}_{2} + (\abs a β b_{p} - a^{'} β^{'} b_{p^{'}} + 0.15 \abs p - p^{'}) H + \norm φ (v_{t}^{'}) - φ^{'} (v_{t}^{'})_{2} .

Displayed formulas in this passage

Psy03-C@1#eq:fullresidual · line 923
$U_{t} = η_{t} = \abs g_{e} - g_{e}^{'} \norm x_{t} + ξ^{'} E_{t}_{2} + g_{e} \abs ξ - ξ^{'} \norm E_{t}_{2} + \abs τ - τ^{'} \norm G_{t}^{in}_{2}, a L_{x} U_{t} + a ϵ_{F} (u_{t}^{'}, h_{t - 1}^{'}) + \abs a - a^{'} \norm F_{θ^{'}} (u_{t}^{'}, h_{t - 1}^{'})_{2} + \abs aσ - a^{'} σ^{'} \norm G_{t}^{hid}_{2} + (\abs a β b_{p} - a^{'} β^{'} b_{p^{'}} + 0.15 \abs p - p^{'}) H + \norm φ (v_{t}^{'}) - φ^{'} (v_{t}^{'})_{2} .$
unlabeled · line 909
$u_{t} = g_{e} (x_{t} + ξ E_{t}) + τ G_{t}^{in},$
unlabeled · line 912
$v_{t} = a F_{θ} (u_{t}, h_{t - 1}) + aσ G_{t}^{hid} + a β b_{p} 1 - o_{p} 1, h_{t} = φ (v_{t}) .$