Optimization of the best element of the set of expressions - what kind of the optimization problem this is?
Let the following expression describes the conditional probability of the translation of the source sequence to the target sequence (e.g. for the sentence-to-sentence translation in the neural machine translation, where xi and yj are the words in the source and target language respectively):
![P(y_{1}^{N_{T}}, theta) = prod_{i=1}^{N_{T}} P(y_{i}|y_{
This probability depends on the parameters theta of the translator function (e.g. the weights and biases of the neural network) and this probability is used as the objective function to be maximized, to find optimal parameters theta* and optimal choices yj that gives the maximum of this probability. This can be solved using gradient methods, backpropagation. That is fine.
But I am trying to consider the case when the neural machine translator generates multiple target sentences for the one source sentence. Some sentences can be complete junk, some sentences can be the parapharses of the target sentence and one sentence can be the optimal choice. So - I would like to optimize the parameters theta in such manner that the following function achieves maximum:
![max_{theta} big{P(u_{1}^{N_{T}}, theta), P(s_{1}^{N_{T}}, theta), P(t_{1}^{N_{T}}, theta), ... big}]
So - optimizer has multiple sub-objectives and the optimizer shold find one sub-objective who can achieve the maximum over all the other sub-objective and over all the theta parameters and choices of target translations u, s, t.
What kind of optimization problem this is? How to apply gradient methods to such max-function? Where to start?
optimization nonlinear-optimization neural-networks programming
asked Dec 26 at 13:59
TomR
259312
add a comment |
Let the following expression describes the conditional probability of the translation of the source sequence to the target sequence (e.g. for the sentence-to-sentence translation in the neural machine translation, where xi and yj are the words in the source and target language respectively):
![P(y_{1}^{N_{T}}, theta) = prod_{i=1}^{N_{T}} P(y_{i}|y_{
This probability depends on the parameters theta of the translator function (e.g. the weights and biases of the neural network) and this probability is used as the objective function to be maximized, to find optimal parameters theta* and optimal choices yj that gives the maximum of this probability. This can be solved using gradient methods, backpropagation. That is fine.
But I am trying to consider the case when the neural machine translator generates multiple target sentences for the one source sentence. Some sentences can be complete junk, some sentences can be the parapharses of the target sentence and one sentence can be the optimal choice. So - I would like to optimize the parameters theta in such manner that the following function achieves maximum:
![max_{theta} big{P(u_{1}^{N_{T}}, theta), P(s_{1}^{N_{T}}, theta), P(t_{1}^{N_{T}}, theta), ... big}]
So - optimizer has multiple sub-objectives and the optimizer shold find one sub-objective who can achieve the maximum over all the other sub-objective and over all the theta parameters and choices of target translations u, s, t.
What kind of optimization problem this is? How to apply gradient methods to such max-function? Where to start?
optimization nonlinear-optimization neural-networks programming
asked Dec 26 at 13:59
TomR
259312
add a comment |
Let the following expression describes the conditional probability of the translation of the source sequence to the target sequence (e.g. for the sentence-to-sentence translation in the neural machine translation, where xi and yj are the words in the source and target language respectively):
![P(y_{1}^{N_{T}}, theta) = prod_{i=1}^{N_{T}} P(y_{i}|y_{
This probability depends on the parameters theta of the translator function (e.g. the weights and biases of the neural network) and this probability is used as the objective function to be maximized, to find optimal parameters theta* and optimal choices yj that gives the maximum of this probability. This can be solved using gradient methods, backpropagation. That is fine.
But I am trying to consider the case when the neural machine translator generates multiple target sentences for the one source sentence. Some sentences can be complete junk, some sentences can be the parapharses of the target sentence and one sentence can be the optimal choice. So - I would like to optimize the parameters theta in such manner that the following function achieves maximum:
![max_{theta} big{P(u_{1}^{N_{T}}, theta), P(s_{1}^{N_{T}}, theta), P(t_{1}^{N_{T}}, theta), ... big}]
So - optimizer has multiple sub-objectives and the optimizer shold find one sub-objective who can achieve the maximum over all the other sub-objective and over all the theta parameters and choices of target translations u, s, t.
What kind of optimization problem this is? How to apply gradient methods to such max-function? Where to start?
optimization nonlinear-optimization neural-networks programming
asked Dec 26 at 13:59
TomR
259312
Let the following expression describes the conditional probability of the translation of the source sequence to the target sequence (e.g. for the sentence-to-sentence translation in the neural machine translation, where xi and yj are the words in the source and target language respectively):
![P(y_{1}^{N_{T}}, theta) = prod_{i=1}^{N_{T}} P(y_{i}|y_{
This probability depends on the parameters theta of the translator function (e.g. the weights and biases of the neural network) and this probability is used as the objective function to be maximized, to find optimal parameters theta* and optimal choices yj that gives the maximum of this probability. This can be solved using gradient methods, backpropagation. That is fine.
But I am trying to consider the case when the neural machine translator generates multiple target sentences for the one source sentence. Some sentences can be complete junk, some sentences can be the parapharses of the target sentence and one sentence can be the optimal choice. So - I would like to optimize the parameters theta in such manner that the following function achieves maximum:
![max_{theta} big{P(u_{1}^{N_{T}}, theta), P(s_{1}^{N_{T}}, theta), P(t_{1}^{N_{T}}, theta), ... big}]
So - optimizer has multiple sub-objectives and the optimizer shold find one sub-objective who can achieve the maximum over all the other sub-objective and over all the theta parameters and choices of target translations u, s, t.
What kind of optimization problem this is? How to apply gradient methods to such max-function? Where to start?
optimization nonlinear-optimization neural-networks programming
optimization nonlinear-optimization neural-networks programming
asked Dec 26 at 13:59
TomR
259312
asked Dec 26 at 13:59
TomR
259312
edited Dec 26 at 14:14
asked Dec 26 at 13:59
TomR
259312
asked Dec 26 at 13:59
TomR
259312
asked Dec 26 at 13:59
TomR
259312
259312
add a comment |
add a comment |
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3052961%2foptimization-of-the-best-element-of-the-set-of-expressions-what-kind-of-the-op%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3052961%2foptimization-of-the-best-element-of-the-set-of-expressions-what-kind-of-the-op%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
