project:hillclimb_dp:draft
Differences
This shows you the differences between two versions of the page.
|
project:hillclimb_dp:draft [2014/04/02 16:49] taolei [Justification of the Algorithm] |
project:hillclimb_dp:draft [2014/05/02 14:57] (current) taolei [Justification of the Algorithm] |
||
|---|---|---|---|
| Line 109: | Line 109: | ||
| Unfortunately, it is not clear if there is an efficient way to compute this loss function by finding $y$ and $y'$. Instead, we could use the standard structural learning loss function, which corresponds to condition (1): | Unfortunately, it is not clear if there is an efficient way to compute this loss function by finding $y$ and $y'$. Instead, we could use the standard structural learning loss function, which corresponds to condition (1): | ||
| $$ | $$ | ||
| - | \mathcal{L}(\hat{x_i},\hat{y_i}) = S(\hat{x_i}, \tilde{y}) + \text{err}(\tilde{y}) - S(\hat{x_i}, \hat{y_i}) \quad\quad\quad\quad (1o) | + | \mathcal{L}(\hat{x_i},\hat{y_i}) = \max_{\tilde{y}}\left\lbrace S(\hat{x_i}, \tilde{y}) + \text{err}(\tilde{y})\right\rbrace - S(\hat{x_i}, \hat{y_i}) \quad\quad (1o) |
| $$ | $$ | ||
| where $\tilde{y}$ is the output of the decoding algorithm. | where $\tilde{y}$ is the output of the decoding algorithm. | ||
| - | + | The connections between the two loss functions are the following: condition (1) is a necessary condition of condition (2), and (1o) is the lower bound of sum of loss (2o) for each two consecutive $(y, y')$ on any "climbing" path from $\tilde{y}$ to $\hat{y}$. Thus, by minimizing (1o) we are making the lower bound of (2o) close to zero. If this lower bound is **//tight//** (because $err(\tilde{y})$ is small and the path from $\tilde{y}$ to $\hat{y}$ is short), | |
| - | + | ||
| - | **What we currently do**: We therefore minimize (1b) instead of (2b). The connections between the two loss functions are the following: condition (1) is a necessary condition of condition (2), and (3) is the lower bound of sum of loss (4) for each two consecutive $(y, y')$ on any "climbing" path from $\tilde{y}$ to $\hat{y}$. | + | |
project/hillclimb_dp/draft.1396471747.txt.gz · Last modified: 2014/04/02 16:49 by taolei
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 3.0 Unported
