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PENG Bo 4 years ago committed by GitHub
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@ -49,7 +49,7 @@ Compare with the **simplified formula** for RWKV-2 (the parallel mode, looks sim
![F[\mathrm{t}+1]=\sigma(\mathbf{R}x[\mathrm{t}]) \cdot \frac{\sum_{\mathrm{i}=0}^{\mathrm{t}} \exp (\mathbf{W} \cdot(\mathrm{t}-\mathrm{i})) \cdot \exp (\mathbf{K}F[\mathrm{i}]) \cdot(\mathbf{V}F[\mathrm{i}])}{\sum_{\mathrm{i}=0}^{\mathrm{t}} \exp (\mathbf{W} \cdot(\mathrm{t}-\mathrm{i})) \cdot \exp (\mathbf{K }F[\mathrm{i}])}](https://render.githubusercontent.com/render/math?math=%5Ccolor%7Bblack%7D%5Cdisplaystyle+F%5B%5Cmathrm%7Bt%7D%2B1%5D%3D%5Csigma%28%5Cmathbf%7BR%7Dx%5B%5Cmathrm%7Bt%7D%5D%29+%5Ccdot+%5Cfrac%7B%5Csum_%7B%5Cmathrm%7Bi%7D%3D0%7D%5E%7B%5Cmathrm%7Bt%7D%7D+%5Cexp+%28%5Cmathbf%7BW%7D+%5Ccdot%28%5Cmathrm%7Bt%7D-%5Cmathrm%7Bi%7D%29%29+%5Ccdot+%5Cexp+%28%5Cmathbf%7BK%7DF%5B%5Cmathrm%7Bi%7D%5D%29+%5Ccdot%28%5Cmathbf%7BV%7DF%5B%5Cmathrm%7Bi%7D%5D%29%7D%7B%5Csum_%7B%5Cmathrm%7Bi%7D%3D0%7D%5E%7B%5Cmathrm%7Bt%7D%7D+%5Cexp+%28%5Cmathbf%7BW%7D+%5Ccdot%28%5Cmathrm%7Bt%7D-%5Cmathrm%7Bi%7D%29%29+%5Ccdot+%5Cexp+%28%5Cmathbf%7BK+%7DF%5B%5Cmathrm%7Bi%7D%5D%29%7D) ![F[\mathrm{t}+1]=\sigma(\mathbf{R}x[\mathrm{t}]) \cdot \frac{\sum_{\mathrm{i}=0}^{\mathrm{t}} \exp (\mathbf{W} \cdot(\mathrm{t}-\mathrm{i})) \cdot \exp (\mathbf{K}F[\mathrm{i}]) \cdot(\mathbf{V}F[\mathrm{i}])}{\sum_{\mathrm{i}=0}^{\mathrm{t}} \exp (\mathbf{W} \cdot(\mathrm{t}-\mathrm{i})) \cdot \exp (\mathbf{K }F[\mathrm{i}])}](https://render.githubusercontent.com/render/math?math=%5Ccolor%7Bblack%7D%5Cdisplaystyle+F%5B%5Cmathrm%7Bt%7D%2B1%5D%3D%5Csigma%28%5Cmathbf%7BR%7Dx%5B%5Cmathrm%7Bt%7D%5D%29+%5Ccdot+%5Cfrac%7B%5Csum_%7B%5Cmathrm%7Bi%7D%3D0%7D%5E%7B%5Cmathrm%7Bt%7D%7D+%5Cexp+%28%5Cmathbf%7BW%7D+%5Ccdot%28%5Cmathrm%7Bt%7D-%5Cmathrm%7Bi%7D%29%29+%5Ccdot+%5Cexp+%28%5Cmathbf%7BK%7DF%5B%5Cmathrm%7Bi%7D%5D%29+%5Ccdot%28%5Cmathbf%7BV%7DF%5B%5Cmathrm%7Bi%7D%5D%29%7D%7B%5Csum_%7B%5Cmathrm%7Bi%7D%3D0%7D%5E%7B%5Cmathrm%7Bt%7D%7D+%5Cexp+%28%5Cmathbf%7BW%7D+%5Ccdot%28%5Cmathrm%7Bt%7D-%5Cmathrm%7Bi%7D%29%29+%5Ccdot+%5Cexp+%28%5Cmathbf%7BK+%7DF%5B%5Cmathrm%7Bi%7D%5D%29%7D)
Here R, K, V are trainable matrices, and W is a trainable vector (time-decay factor for each channel). The R, K, V are trainable matrices, and W is a trainable vector (time-decay factor for each channel).
In GPT, the contribution of F[i] to F[t+1] is weighted by ![ \exp (\mathbf{Q}x[\mathrm{t}] * \mathbf{K}F[\mathrm{i}]) ](https://render.githubusercontent.com/render/math?math=%5Ccolor%7Bblack%7D%5Cdisplaystyle++%5Cexp+%28%5Cmathbf%7BQ%7Dx%5B%5Cmathrm%7Bt%7D%5D+%2A+%5Cmathbf%7BK%7DF%5B%5Cmathrm%7Bi%7D%5D%29+). In GPT, the contribution of F[i] to F[t+1] is weighted by ![ \exp (\mathbf{Q}x[\mathrm{t}] * \mathbf{K}F[\mathrm{i}]) ](https://render.githubusercontent.com/render/math?math=%5Ccolor%7Bblack%7D%5Cdisplaystyle++%5Cexp+%28%5Cmathbf%7BQ%7Dx%5B%5Cmathrm%7Bt%7D%5D+%2A+%5Cmathbf%7BK%7DF%5B%5Cmathrm%7Bi%7D%5D%29+).
@ -64,7 +64,7 @@ Here comes the punchline: we can rewrite it into a RNN (recursive formula). Note
![F[2]=\sigma(\mathbf{R }x[1]) \cdot \frac{ \exp (\mathbf{K }F[1]) \cdot(\mathbf{V }F[1])+\exp (\mathbf{W} ) \cdot \exp (\mathbf{K }F[0]) \cdot(\mathbf{V }F[0])}{ \exp (\mathbf{K }F[1])+\exp (\mathbf{W} ) \cdot \exp (\mathbf{K }F[0])}](https://render.githubusercontent.com/render/math?math=%5Ccolor%7Bblack%7D%5Cdisplaystyle+F%5B2%5D%3D%5Csigma%28%5Cmathbf%7BR+%7Dx%5B1%5D%29+%5Ccdot+%5Cfrac%7B+%5Cexp+%28%5Cmathbf%7BK+%7DF%5B1%5D%29+%5Ccdot%28%5Cmathbf%7BV+%7DF%5B1%5D%29%2B%5Cexp+%28%5Cmathbf%7BW%7D+%29+%5Ccdot+%5Cexp+%28%5Cmathbf%7BK+%7DF%5B0%5D%29+%5Ccdot%28%5Cmathbf%7BV+%7DF%5B0%5D%29%7D%7B+%5Cexp+%28%5Cmathbf%7BK+%7DF%5B1%5D%29%2B%5Cexp+%28%5Cmathbf%7BW%7D+%29+%5Ccdot+%5Cexp+%28%5Cmathbf%7BK+%7DF%5B0%5D%29%7D) ![F[2]=\sigma(\mathbf{R }x[1]) \cdot \frac{ \exp (\mathbf{K }F[1]) \cdot(\mathbf{V }F[1])+\exp (\mathbf{W} ) \cdot \exp (\mathbf{K }F[0]) \cdot(\mathbf{V }F[0])}{ \exp (\mathbf{K }F[1])+\exp (\mathbf{W} ) \cdot \exp (\mathbf{K }F[0])}](https://render.githubusercontent.com/render/math?math=%5Ccolor%7Bblack%7D%5Cdisplaystyle+F%5B2%5D%3D%5Csigma%28%5Cmathbf%7BR+%7Dx%5B1%5D%29+%5Ccdot+%5Cfrac%7B+%5Cexp+%28%5Cmathbf%7BK+%7DF%5B1%5D%29+%5Ccdot%28%5Cmathbf%7BV+%7DF%5B1%5D%29%2B%5Cexp+%28%5Cmathbf%7BW%7D+%29+%5Ccdot+%5Cexp+%28%5Cmathbf%7BK+%7DF%5B0%5D%29+%5Ccdot%28%5Cmathbf%7BV+%7DF%5B0%5D%29%7D%7B+%5Cexp+%28%5Cmathbf%7BK+%7DF%5B1%5D%29%2B%5Cexp+%28%5Cmathbf%7BW%7D+%29+%5Ccdot+%5Cexp+%28%5Cmathbf%7BK+%7DF%5B0%5D%29%7D)
Therefore it's easy to verify: Therefore it's straightforward to verify:
![F[t+1]=\sigma(\mathbf{R }x[t]) \cdot \frac{\exp (\mathbf{K}F[\mathrm{t}]) \cdot(\mathbf{V}F[\mathrm{t}])+\exp (\mathbf{W}) \cdot A[\mathrm{t}]}{ \exp (\mathbf{K}F[\mathrm{t}])+\exp (\mathbf{W}) \cdot B[\mathrm{t}]}](https://render.githubusercontent.com/render/math?math=%5Ccolor%7Bblack%7D%5Cdisplaystyle+F%5Bt%2B1%5D%3D%5Csigma%28%5Cmathbf%7BR+%7Dx%5Bt%5D%29+%5Ccdot+%5Cfrac%7B%5Cexp+%28%5Cmathbf%7BK%7DF%5B%5Cmathrm%7Bt%7D%5D%29+%5Ccdot%28%5Cmathbf%7BV%7DF%5B%5Cmathrm%7Bt%7D%5D%29%2B%5Cexp+%28%5Cmathbf%7BW%7D%29+%5Ccdot+A%5B%5Cmathrm%7Bt%7D%5D%7D%7B+%5Cexp+%28%5Cmathbf%7BK%7DF%5B%5Cmathrm%7Bt%7D%5D%29%2B%5Cexp+%28%5Cmathbf%7BW%7D%29+%5Ccdot+B%5B%5Cmathrm%7Bt%7D%5D%7D) ![F[t+1]=\sigma(\mathbf{R }x[t]) \cdot \frac{\exp (\mathbf{K}F[\mathrm{t}]) \cdot(\mathbf{V}F[\mathrm{t}])+\exp (\mathbf{W}) \cdot A[\mathrm{t}]}{ \exp (\mathbf{K}F[\mathrm{t}])+\exp (\mathbf{W}) \cdot B[\mathrm{t}]}](https://render.githubusercontent.com/render/math?math=%5Ccolor%7Bblack%7D%5Cdisplaystyle+F%5Bt%2B1%5D%3D%5Csigma%28%5Cmathbf%7BR+%7Dx%5Bt%5D%29+%5Ccdot+%5Cfrac%7B%5Cexp+%28%5Cmathbf%7BK%7DF%5B%5Cmathrm%7Bt%7D%5D%29+%5Ccdot%28%5Cmathbf%7BV%7DF%5B%5Cmathrm%7Bt%7D%5D%29%2B%5Cexp+%28%5Cmathbf%7BW%7D%29+%5Ccdot+A%5B%5Cmathrm%7Bt%7D%5D%7D%7B+%5Cexp+%28%5Cmathbf%7BK%7DF%5B%5Cmathrm%7Bt%7D%5D%29%2B%5Cexp+%28%5Cmathbf%7BW%7D%29+%5Ccdot+B%5B%5Cmathrm%7Bt%7D%5D%7D)

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