Answers.Body: <p>From a dynamical systems perspective this makes sense because based on where you start, your solution/how the physical system acts should be different.</p>
  <p>I would argue that this may be a good thing to have for AI too. I am imagining an RL system (whose continuous versions of control can be formalized with continuous DP principles via pde's) like a robot navigating a room. If the third condition (3) didn't hold, then you would be enacting the same policy regardless of where you start. If it is a stationary setting where a repeat of states could occur (a ground hogs day), then one may argue that they do not want (3). Technically though it could satisfy the third condition if one changes continuously by staying constant with respect to initial conditions.</p>
  
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Answers.CreationDate: 2022-02-06T02:02:03.277Z
Answers.LastActivityDate: 2022-02-06T02:02:03.277Z
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Body: <p>I was reading the following link (<a href="https://en.wikipedia.org/wiki/Well-posed_problem" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Well-posed_problem</a>) on &quot;Well Posed Problems&quot;. Supposedly, if a problem is &quot;Well Posed&quot;, it must meet the following conditions:</p>
  <ul>
  <li><p>a solution exists</p>
  </li>
  <li><p>the solution is unique</p>
  </li>
  <li><p>the solution's behavior changes continuously with the initial conditions</p>
  </li>
  </ul>
  <p>I can understand why the first two conditions are desirable for a problem to be &quot;Well Posed&quot;, but I am having trouble understanding the last condition.</p>
  <p>For instance, if &quot;the solution's behavior changes continuously with the initial conditions&quot; - isn't this a bad thing? Would we not want the &quot;solution's behavior <strong>NOT TO CHANGE</strong> continuously with the initial conditions&quot;?</p>
  <p>If the solution's behavior can change with the initial conditions, would this not result in the problem having the ability to be &quot;chaotic and unpredictable&quot;, thus displaying &quot;bad behavior&quot; and as a result - &quot;Ill Posed&quot;?</p>
  <p>Thanks!</p>
  
Tags: <optimization>
Title: Are Problems in AI Usually "Ill Posed"?
UserDisplayName: stats_noob
ThreadText: # Are Problems in AI Usually "Ill Posed"?
  
  (Asked by: username_0 on 2022-02-06)
  
  I was reading the following link (<https://en.wikipedia.org/wiki/Well-posed_problem>) on "Well Posed Problems". Supposedly, if a problem is "Well Posed", it must meet the following conditions:
  
  * a solution exists
  * the solution is unique
  * the solution's behavior changes continuously with the initial conditions
  
  I can understand why the first two conditions are desirable for a problem to be "Well Posed", but I am having trouble understanding the last condition.
  
  For instance, if "the solution's behavior changes continuously with the initial conditions" - isn't this a bad thing? Would we not want the "solution's behavior **NOT TO CHANGE** continuously with the initial conditions"?
  
  If the solution's behavior can change with the initial conditions, would this not result in the problem having the ability to be "chaotic and unpredictable", thus displaying "bad behavior" and as a result - "Ill Posed"?
  
  Thanks!
  
  **Tags:** optimization
  
  
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  ## 1 answers
  
  ### Answer by username_1 on 2022-02-06. Score: 1
  
  From a dynamical systems perspective this makes sense because based on where you start, your solution/how the physical system acts should be different.
  
  I would argue that this may be a good thing to have for AI too. I am imagining an RL system (whose continuous versions of control can be formalized with continuous DP principles via pde's) like a robot navigating a room. If the third condition (3) didn't hold, then you would be enacting the same policy regardless of where you start. If it is a stationary setting where a repeat of states could occur (a ground hogs day), then one may argue that they do not want (3). Technically though it could satisfy the third condition if one changes continuously by staying constant with respect to initial conditions.
  
  
  ---
  
  