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What it means to have a higher cost for a local minima than the global minima?
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In the following lines, I do not understand what it means to have a high cost for local minima than a global minima?
Local minima can be problematic if they have high cost in comparison to the global minimum. One can construct small neural networks, even without hidden units, that have local minima with higher cost than the global minimum. If local minima with high cost are common, this could pose a serious problem for gradient-based optimization algorithms.
neural-networks deep-learning optimization
asked Sep 7 at 0:46
samra irshad
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In the following lines, I do not understand what it means to have a high cost for local minima than a global minima?
Local minima can be problematic if they have high cost in comparison to the global minimum. One can construct small neural networks, even without hidden units, that have local minima with higher cost than the global minimum. If local minima with high cost are common, this could pose a serious problem for gradient-based optimization algorithms.
neural-networks deep-learning optimization
asked Sep 7 at 0:46
samra irshad
947
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
In the following lines, I do not understand what it means to have a high cost for local minima than a global minima?
Local minima can be problematic if they have high cost in comparison to the global minimum. One can construct small neural networks, even without hidden units, that have local minima with higher cost than the global minimum. If local minima with high cost are common, this could pose a serious problem for gradient-based optimization algorithms.
neural-networks deep-learning optimization
asked Sep 7 at 0:46
samra irshad
947
In the following lines, I do not understand what it means to have a high cost for local minima than a global minima?
Local minima can be problematic if they have high cost in comparison to the global minimum. One can construct small neural networks, even without hidden units, that have local minima with higher cost than the global minimum. If local minima with high cost are common, this could pose a serious problem for gradient-based optimization algorithms.
neural-networks deep-learning optimization
asked Sep 7 at 0:46
samra irshad
947
asked Sep 7 at 0:46
samra irshad
947
asked Sep 7 at 0:46
samra irshad
947
asked Sep 7 at 0:46
samra irshad
947
947
add a comment |Â
add a comment |Â
1 Answer
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An optimization algorithm's goal is to minimize a cost function $C$. Some optimization techniques (such as gradient-based methods) are only guaranteed to find a local minimum rather than the global minimum of $C$. So if $C$ looks like the picture below, a gradient-based method might find the local minimum $x_0$, which has much higher cost than the global minimum $x^*$.
In this example, you could avoid the problem by running the optimization several times with random starting points. But as the quotation mentions, this is not effective when there are many high-cost local minima.
answered Sep 7 at 0:57
tddevlin
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Many thanks for this cute illustration!
– samra irshad
Sep 7 at 2:14
Tiny (perhaps straightforward) addition from my side, in case the last line of your quote is still unclear: Many algorithms (especially gradient-based ones, as you mention) try to find the global minimum by finding the direction in model space in which $C(x)$ decreases most rapidly. This explains why it can get stuck in $C(x_0)$ in the illustration above, that direction does not exist. This is what the authors meant in "If local minima with high cost are common, this could pose a serious problem for gradient-based optimization algorithms.". There are many such places to get stuck.
– Floris SA
Sep 7 at 9:33
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
12
down vote
accepted
An optimization algorithm's goal is to minimize a cost function $C$. Some optimization techniques (such as gradient-based methods) are only guaranteed to find a local minimum rather than the global minimum of $C$. So if $C$ looks like the picture below, a gradient-based method might find the local minimum $x_0$, which has much higher cost than the global minimum $x^*$.
In this example, you could avoid the problem by running the optimization several times with random starting points. But as the quotation mentions, this is not effective when there are many high-cost local minima.
answered Sep 7 at 0:57
tddevlin
1,067415
2
Many thanks for this cute illustration!
– samra irshad
Sep 7 at 2:14
Tiny (perhaps straightforward) addition from my side, in case the last line of your quote is still unclear: Many algorithms (especially gradient-based ones, as you mention) try to find the global minimum by finding the direction in model space in which $C(x)$ decreases most rapidly. This explains why it can get stuck in $C(x_0)$ in the illustration above, that direction does not exist. This is what the authors meant in "If local minima with high cost are common, this could pose a serious problem for gradient-based optimization algorithms.". There are many such places to get stuck.
– Floris SA
Sep 7 at 9:33
add a comment |Â
up vote
12
down vote
accepted
An optimization algorithm's goal is to minimize a cost function $C$. Some optimization techniques (such as gradient-based methods) are only guaranteed to find a local minimum rather than the global minimum of $C$. So if $C$ looks like the picture below, a gradient-based method might find the local minimum $x_0$, which has much higher cost than the global minimum $x^*$.
In this example, you could avoid the problem by running the optimization several times with random starting points. But as the quotation mentions, this is not effective when there are many high-cost local minima.
answered Sep 7 at 0:57
tddevlin
1,067415
2
Many thanks for this cute illustration!
– samra irshad
Sep 7 at 2:14
Tiny (perhaps straightforward) addition from my side, in case the last line of your quote is still unclear: Many algorithms (especially gradient-based ones, as you mention) try to find the global minimum by finding the direction in model space in which $C(x)$ decreases most rapidly. This explains why it can get stuck in $C(x_0)$ in the illustration above, that direction does not exist. This is what the authors meant in "If local minima with high cost are common, this could pose a serious problem for gradient-based optimization algorithms.". There are many such places to get stuck.
– Floris SA
Sep 7 at 9:33
add a comment |Â
up vote
12
down vote
accepted
up vote
12
down vote
accepted
An optimization algorithm's goal is to minimize a cost function $C$. Some optimization techniques (such as gradient-based methods) are only guaranteed to find a local minimum rather than the global minimum of $C$. So if $C$ looks like the picture below, a gradient-based method might find the local minimum $x_0$, which has much higher cost than the global minimum $x^*$.
In this example, you could avoid the problem by running the optimization several times with random starting points. But as the quotation mentions, this is not effective when there are many high-cost local minima.
answered Sep 7 at 0:57
tddevlin
1,067415
An optimization algorithm's goal is to minimize a cost function $C$. Some optimization techniques (such as gradient-based methods) are only guaranteed to find a local minimum rather than the global minimum of $C$. So if $C$ looks like the picture below, a gradient-based method might find the local minimum $x_0$, which has much higher cost than the global minimum $x^*$.
In this example, you could avoid the problem by running the optimization several times with random starting points. But as the quotation mentions, this is not effective when there are many high-cost local minima.
answered Sep 7 at 0:57
tddevlin
1,067415
answered Sep 7 at 0:57
tddevlin
1,067415
answered Sep 7 at 0:57
tddevlin
1,067415
answered Sep 7 at 0:57
tddevlin
1,067415
1,067415
2
Many thanks for this cute illustration!
– samra irshad
Sep 7 at 2:14
Tiny (perhaps straightforward) addition from my side, in case the last line of your quote is still unclear: Many algorithms (especially gradient-based ones, as you mention) try to find the global minimum by finding the direction in model space in which $C(x)$ decreases most rapidly. This explains why it can get stuck in $C(x_0)$ in the illustration above, that direction does not exist. This is what the authors meant in "If local minima with high cost are common, this could pose a serious problem for gradient-based optimization algorithms.". There are many such places to get stuck.
– Floris SA
Sep 7 at 9:33
add a comment |Â
2
Many thanks for this cute illustration!
– samra irshad
Sep 7 at 2:14
Tiny (perhaps straightforward) addition from my side, in case the last line of your quote is still unclear: Many algorithms (especially gradient-based ones, as you mention) try to find the global minimum by finding the direction in model space in which $C(x)$ decreases most rapidly. This explains why it can get stuck in $C(x_0)$ in the illustration above, that direction does not exist. This is what the authors meant in "If local minima with high cost are common, this could pose a serious problem for gradient-based optimization algorithms.". There are many such places to get stuck.
– Floris SA
Sep 7 at 9:33
2
2
Many thanks for this cute illustration!
– samra irshad
Sep 7 at 2:14
Many thanks for this cute illustration!
– samra irshad
Sep 7 at 2:14
Tiny (perhaps straightforward) addition from my side, in case the last line of your quote is still unclear: Many algorithms (especially gradient-based ones, as you mention) try to find the global minimum by finding the direction in model space in which $C(x)$ decreases most rapidly. This explains why it can get stuck in $C(x_0)$ in the illustration above, that direction does not exist. This is what the authors meant in "If local minima with high cost are common, this could pose a serious problem for gradient-based optimization algorithms.". There are many such places to get stuck.
– Floris SA
Sep 7 at 9:33
Tiny (perhaps straightforward) addition from my side, in case the last line of your quote is still unclear: Many algorithms (especially gradient-based ones, as you mention) try to find the global minimum by finding the direction in model space in which $C(x)$ decreases most rapidly. This explains why it can get stuck in $C(x_0)$ in the illustration above, that direction does not exist. This is what the authors meant in "If local minima with high cost are common, this could pose a serious problem for gradient-based optimization algorithms.". There are many such places to get stuck.
– Floris SA
Sep 7 at 9:33
add a comment |Â
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