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language:
- en
license: apache-2.0
tags:
- sentence-transformers
- sentence-similarity
- feature-extraction
- generated_from_trainer
- dataset_size:1140
- loss:MatryoshkaLoss
- loss:MultipleNegativesRankingLoss
base_model: NovaSearch/stella_en_400M_v5
widget:
- source_sentence: Prove that x + |x - 7| ≥ 7
sentences:
- "The subtyping relationship between `Iterable[Pair[A, Y]]` and `Map[A, Y]` can\
\ be understood by examining the covariance and structure of these types.\n\n\
1. **Covariance**: Both `Iterable` and `Pair` are covariant in their type parameters.\
\ This means that if `A` is a supertype of `B`, then `Iterable[Pair[A, Y]]` can\
\ be treated as a subtype of `Iterable[Pair[B, Y]]`.\n\n2. **Map's Structure**:\
\ The `Map` class extends `Iterable[Pair[U, V]]`, where `U` is invariant and `V`\
\ is covariant. Therefore, `Map[A, Y]` is considered an `Iterable[Pair[A, Y]]`.\
\ \n\n3. **Type Compatibility**: Since `Map[A, Y]` is an `Iterable[Pair[A, Y]]`,\
\ it follows that any instance of `Map[A, Y]` can be treated as an instance of\
\ `Iterable[Pair[A, Y]]`. \n\n4. **Subtyping**: Given that `Iterable[Pair[A, Y]]`\
\ can be seen as an interface that describes a broader category of collections,\
\ while `Map[A, Y]` is a specific implementation of that interface, we conclude\
\ that `Iterable[Pair[A, Y]]` is a supertype of `Map[A, Y]`. Thus, `Iterable[Pair[A,\
\ Y]]` can be assigned to a variable of type `Map[A, Y]`, making the relationship\
\ `Iterable[Pair[A, Y]] <: Map[A, Y]`.\n\nTherefore, the correct subtyping relationship\
\ is `Iterable[Pair[A, Y]] <: Map[A, Y]`."
- "To implement a function that inserts an element into a sorted list while maintaining\
\ the sorted order, we can use recursion. The idea is to check the first element\
\ of the list and decide whether to insert the new element before it or to continue\
\ the recursion with the rest of the list.\n\nHere’s how you can implement the\
\ `insert` function in Scala:\n\n```scala\ndef insert(elem: Int, list: List[Int]):\
\ List[Int] = list match {\n case Nil => List(elem) // If the list is empty,\
\ return a new list with the element\n case head :: tail =>\n if (elem <=\
\ head) {\n elem :: list // If the element is less than or equal to the head,\
\ insert it at the front\n } else {\n head :: insert(elem, tail) // Otherwise,\
\ keep the head and recurse on the tail\n }\n}\n```\n\n### Explanation:\n1.\
\ **Base Case**: \n - If the list is empty (`Nil`), we return a new list that\
\ contains only the element `elem`.\n \n2. **Recursive Case**:\n - The list\
\ is matched against its head and tail using pattern matching. `head` is the first\
\ element, and `tail` is the remainder of the list.\n - If `elem` is less than\
\ or equal to `head`, we prepend `elem` to the list (using `::`), resulting in\
\ a new list that maintains the sorted order.\n - If `elem` is greater than\
\ `head`, we keep `head` in the resulting list and recursively call `insert` with\
\ `elem` and `tail`. This continues until we find the correct position for `elem`.\n\
\n### Example Usage:\n```scala\nval sortedList = List(1, 3, 5, 7)\nval newElement\
\ = 4\nval newList = insert(newElement, sortedList)\n// newList will be List(1,\
\ 3, 4, 5, 7)\n```\n\nThis recursive approach effectively ensures that the newly\
\ inserted element maintains the sorted property of the list."
- "To prove the inequality \\( x + |x - 7| \\geq 7 \\), we will analyze it by considering\
\ two cases based on the definition of the absolute value function.\n\n### Step\
\ 1: Understanding the Absolute Value\n\nThe absolute value function \\( |x -\
\ 7| \\) can be expressed in terms of piecewise functions:\n\n\\[\n|x - 7| = \n\
\\begin{cases} \n7 - x & \\text{if } x < 7 \\\\\nx - 7 & \\text{if } x \\geq 7\
\ \n\\end{cases}\n\\]\n\n### Step 2: Case 1: \\( x < 7 \\)\n\nIn this case, we\
\ have:\n\n\\[\n|x - 7| = 7 - x\n\\]\n\nSubstituting this into the inequality\
\ \\( x + |x - 7| \\):\n\n\\[\nx + |x - 7| = x + (7 - x) = 7\n\\]\n\nNow, we need\
\ to check if:\n\n\\[\n7 \\geq 7\n\\]\n\nThis statement is true. Therefore, for\
\ all \\( x < 7 \\), the inequality \\( x + |x - 7| \\geq 7 \\) holds.\n\n###\
\ Step 3: Case 2: \\( x \\geq 7 \\)\n\nIn this case, we have:\n\n\\[\n|x - 7|\
\ = x - 7\n\\]\n\nSubstituting this into the inequality \\( x + |x - 7| \\):\n\
\n\\[\nx + |x - 7| = x + (x - 7) = 2x - 7\n\\]\n\nNow, we need to check if:\n\n\
\\[\n2x - 7 \\geq 7\n\\]\n\nTo solve this inequality, we can rearrange it:\n\n\
\\[\n2x \\geq 14 \\\\\nx \\geq 7\n\\]\n\nThis statement is also true for all \\\
( x \\geq 7 \\).\n\n### Conclusion\n\nCombining both cases, we find that the inequality\
\ \\( x + |x - 7| \\geq 7 \\) holds true for all \\( x \\in \\mathbb{R} \\).\n\
\nThus, we have proven that:\n\n\\[\nx + |x - 7| \\geq 7 \\quad \\forall x \\\
in \\mathbb{R}.\n\\]"
- source_sentence: 'Assume you are working on SuperQuiz, a trendy app that lets everyone
design quizzes and share them with friends! SuperQuiz recently hired a new CEO,
who wants to improve the development practices using modern methods. However,
this CEO has no engineering background, so the suggested improvements are well
intentioned but not always feasible. The latest CEO suggestion is this:
"Continuous Integration is a modern best practice. We must adopt it, so that the
code in our repository never has bugs. From now on, all branches in the SuperQuiz
repository must have continuous integration enabled, and at the end of each day
all branches must pass all tests."
Propose (in 1-2 sentences) a compromise that achieves the CEO''s true objective:'
sentences:
- To effectively implement continuous integration while maintaining practicality,
we should set it up for the main branch of the SuperQuiz repository. This approach
will help ensure that the code merged into the main branch is thoroughly tested,
thereby reducing the likelihood of bugs in the final product while allowing developers
to work on feature branches without the burden of daily testing requirements.
- "When removing a method from a project due to its potential for misuse and replacing\
\ it with a more user-friendly alternative, consider the following structured\
\ steps for upcoming releases:\n\n### 1. **Evaluate the Impact of Removal**\n\
\ - **Dependency Analysis**: Identify all components, modules, or external libraries\
\ that depend on the method.\n - **Usage Tracking**: Analyze how frequently\
\ the method is used across the codebase and document its current usage patterns.\n\
\n### 2. **Design the Replacement**\n - **Feature Comparison**: Ensure the new\
\ method covers all functionalities of the old method while being easier to use.\n\
\ - **User Experience**: Focus on enhancing usability and reducing the likelihood\
\ of errors with the new method.\n\n### 3. **Update Documentation**\n - **API\
\ Documentation**: Revise the API documentation to remove references to the old\
\ method and provide comprehensive details about the new method.\n - **Migration\
\ Guide**: Create a clear migration guide outlining how to transition from the\
\ old method to the new one.\n\n### 4. **Deprecate the Old Method**\n - **Deprecation\
\ Notice**: Mark the old method as deprecated, ensuring it remains available for\
\ a defined period while encouraging users to transition.\n - **Warnings**:\
\ Implement warnings in the code to alert developers using the old method about\
\ its deprecation.\n\n### 5. **Refactor Codebase**\n - **Replace Instances**:\
\ Update all instances of the old method in the codebase with the new method,\
\ ensuring that functionality remains intact.\n - **Testing**: Write and run\
\ unit tests to verify that the new method behaves as expected and does not introduce\
\ new bugs.\n\n### 6. **Communicate Changes**\n - **Release Notes**: Include\
\ detailed information about the removal and replacement in the release notes\
\ for users.\n - **Community Announcement**: If applicable, communicate the\
\ changes through relevant channels (e.g., newsletters, forums, or social media).\n\
\n### 7. **Monitor Adoption**\n - **Feedback Mechanism**: Establish a way for\
\ users to provide feedback on the new method, allowing for quick identification\
\ of any issues.\n - **Usage Metrics**: Track the adoption rate of the new method\
\ versus the old one to gauge the success of the transition.\n\n### 8. **Plan\
\ for Future Releases**\n - **Versioning Strategy**: Decide on a versioning\
\ strategy to handle the removal (e.g., major version bump) and inform users of\
\ the changes.\n - **Continuous Improvement**: Gather feedback post-release\
\ and refine the new method as needed based on user experience.\n\nBy following\
\ these steps, you can ensure a smooth transition from the old method to the new\
\ one while minimizing disruption for users and maintaining the integrity of the\
\ project."
- "Draft: \n\nTo solve the problem of finding a non-negative vertex potential \\\
( p(v) \\) for each vertex \\( v \\in V \\) in a graph \\( G = (V, E) \\), we\
\ can formulate it as a linear program and utilize a polynomial-time separation\
\ oracle to verify feasibility. Below are the detailed steps to achieve this.\n\
\n### Step 1: Formulate the Linear Program\nWe will set up a linear program with\
\ variables corresponding to the potentials of each vertex:\n\n- **Objective Function**:\n\
\ \\[\n \\text{Maximize } Z = \\sum_{v \\in V} p(v)\n \\]\n\n- **Constraints**:\n\
\ For every non-empty subset \\( S \\subset V \\):\n \\[\n \\sum_{v \\in S}\
\ p(v) \\leq |E(S, \\bar{S})|\n \\]\n\n- **Non-negativity Constraints**:\n \\\
[\n p(v) \\geq 0 \\quad \\text{for all } v \\in V\n \\]\n\n### Step 2: Design\
\ a Separation Oracle\nTo check whether a given potential vector \\( p^* \\) is\
\ feasible, we need to verify the constraints efficiently. We will construct a\
\ function \\( f(S) \\) for each non-empty subset \\( S \\subset V \\):\n\\[\n\
f(S) = |E(S, \\bar{S})| - \\sum_{v \\in S} p^*(v)\n\\]\n\nNow, we need to check\
\ if \\( f(S) \\geq 0 \\) holds for all \\( \\emptyset \\subset S \\subset V \\\
). If \\( \\min_{\\emptyset \\subseteq S \\subset V} f(S) < 0 \\), then \\( p^*\
\ \\) is not feasible, and we can output a violated constraint.\n\n### Step 3:\
\ Submodularity of \\( f(S) \\)\nThe function \\( f(S) \\) is a submodular function\
\ as it is the sum of a cut function (which is submodular) and a linear term (which\
\ is trivially submodular). The key insight is that identifying violated constraints\
\ is equivalent to minimizing the submodular function \\( f(S) \\).\n\n### Step\
\ 4: Solve the Submodular Function Minimization\nSince we do not allow \\( S =\
\ V \\) in our solutions, we can solve the submodular minimization for the smaller\
\ ground sets \\( V \\setminus \\{v_1\\}, V \\setminus \\{v_2\\}, \\ldots, V \\\
setminus \\{v_n\\} \\). Each of these instances can be solved using polynomial-time\
\ algorithms for submodular function minimization.\n\n### Conclusion\nBy constructing\
\ a feasible region using the LP and developing a separation oracle that confirms\
\ whether a given potential is feasible or outputs a violated constraint, we can\
\ solve the original problem in polynomial time using the Ellipsoid method.\n\n\
Answer:\n\nTo design a polynomial-time algorithm for the problem of finding a\
\ vertex potential \\( p(v) \\) in a graph \\( G = (V, E) \\), we can use linear\
\ programming combined with a separation oracle:\n\n1. **Linear Program Formulation**:\n\
\ We define the following linear program:\n - **Objective**:\n \\[\n \
\ \\text{Maximize } Z = \\sum_{v \\in V} p(v)\n \\]\n - **Constraints**:\n\
\ For every non-empty subset \\( S \\subset V \\):\n \\[\n \\sum_{v\
\ \\in S} p(v) \\leq |E(S, \\bar{S})|\n \\]\n - **Non-negativity**:\n \
\ \\[\n p(v) \\geq 0 \\quad \\text{for all } v \\in V\n \\]\n\n2. **Separation\
\ Oracle**:\n To check if a potential vector \\( p^* \\) is feasible, we define:\n\
\ \\[\n f(S) = |E(S, \\bar{S})| - \\sum_{v \\in S} p^*(v)\n \\]\n We need\
\ to determine if \\( \\min_{\\emptyset \\subseteq S \\subset V} f(S) < 0 \\).\
\ If it is, \\( p^* \\) is not feasible.\n\n3. **Submodular Function**:\n The\
\ function \\( f(S) \\) is submodular, and we can solve the minimization problem\
\ for the smaller sets \\( V \\setminus \\{v_1\\}, V \\setminus \\{v_2\\}, \\\
ldots, V \\setminus \\{v_n\\} \\) to find any violations efficiently.\n\n4. **Polynomial\
\ Time**:\n The submodular function minimization can be solved in polynomial\
\ time, enabling us to check feasibility of \\( p^* \\) efficiently. As a result,\
\ this method allows us to apply the Ellipsoid method to solve the linear program\
\ in polynomial time.\n\nJustification:\n\nThe approach leverages linear programming\
\ to maximize the vertex potentials while adhering to the specified constraints.\
\ By formulating the problem in terms of a separation oracle, we can efficiently\
\ verify the feasibility of potential solutions. The use of submodular functions\
\ is crucial because they have well-established properties that allow us to find\
\ violations quickly. This guarantees that the algorithm runs in polynomial time\
\ due to the efficiency of the submodular minimization process. Overall, this\
\ method combines theoretical insights from combinatorial optimization with practical\
\ computational techniques, leading to a robust solution to the problem."
- source_sentence: 'Let $A \in \mathbb{R}^{m\times n}$, $b\in \mathbb{R}^m$ and $c\in
\mathbb{R}^n$. Consider the following linear program with $n$ variables: \begin{align*}
\textbf{maximize} \hspace{0.8cm} & c^Tx \\ \textbf{subject to}\hspace{0.8cm} &
Ax =b \\ \hspace{0.8cm} & x \geq 0 \end{align*} Show that any extreme point $x^*$
has at most $m$ non-zero entries, i.e., $|\{i: x^*_i > 0 \}| \leq m$. \\[-0.2cm]
\noindent \emph{Hint: what happens if the columns corresponding to non-zero entries
in $x^*$ are linearly dependent?}\\[-0.2cm] {\small (If you are in a good mood
you can prove the following stronger statement: $x^*$ is an extreme point if and
only if the columns of $A$ corresponding to non-zero entries of $x^*$ are linearly
independent.)}'
sentences:
- 'To show that any extreme point \( x^* \) of the given linear program has at most
\( m \) non-zero entries, we proceed with the following steps:
1. **Definition of Extreme Points**: An extreme point \( x^* \) is one that cannot
be represented as a convex combination of other feasible points. In the context
of the linear program, this means that if \( x^* \) has more than \( m \) non-zero
entries, it can potentially be expressed as such a combination.
2. **Identifying Non-Zero Columns**: Let \( S = \{ i : x_i^* > 0 \} \). If \(
|S| > m \), then we have more than \( m \) non-zero variables in \( x^* \). The
corresponding columns of matrix \( A \) associated with these non-zero entries
are denoted as \( A_S \).
3. **Linear Dependence**: Since there are more than \( m \) columns in \( A_S
\), and given that the maximum rank of any set of vectors (columns) is limited
by the number of rows (which is \( m \)), the columns of \( A_S \) must be linearly
dependent when \( |S| > m \).
4. **Feasibility Condition**: The linear dependence implies that we can find a
non-trivial combination of these columns that sums to zero, allowing us to perturb
some components of \( x^* \) while still satisfying the equality constraint \(
Ax = b \). This contradicts the property of extreme points.
5. **Conclusion**: Therefore, if an extreme point has more than \( m \) non-zero
entries, it cannot be an extreme point due to linear dependence among its corresponding
columns in \( A \). Thus, we conclude that any extreme point \( x^* \) must have
at most \( m \) non-zero entries, i.e., \( |\{i: x_i^* > 0\}| \leq m \).'
- 'The sentences produced by the recognizer, such as "A was salmon outer the does"
and "I Thomas at mice not the spoon," exhibit a lack of grammatical coherence
and semantic meaning. The issues with these sentences can be categorized as follows:
1. **Grammatical Errors**: The sentences contain phrases that do not follow the
rules of English grammar. For example, "A was salmon outer the does" lacks a clear
subject-verb-object structure, and "I Thomas at mice not the spoon" is not a coherent
expression.
2. **Semantic Incoherence**: Even if the sentences were grammatically correct,
they still do not convey meaningful ideas. Words are used in ways that do not
align with their typical meanings or relations, leading to nonsensical phrases.
To improve the quality of the recognizer and select the correct sequences of words,
the following Natural Language Processing (NLP) techniques can be employed:
1. **Language Modeling**: Utilizing statistical or neural language models (e.g.,
n-grams, recurrent neural networks, transformers) can help evaluate the probability
of word sequences. These models can predict the likelihood of a sequence of words
based on the context and structure of the language.
2. **Syntactic Parsing**: Implementing syntactic parsers can help identify the
grammatical structure of sentences. This will allow the system to filter out those
that do not conform to acceptable grammatical rules.
3. **Semantic Analysis**: Techniques such as word embeddings (e.g., Word2Vec,
GloVe) or transformer-based models (e.g., BERT) can be used to assess the semantic
similarity of phrases or to identify word meanings in context. This can help in
selecting sentences that are not only grammatically correct but also semantically
meaningful.
4. **Contextual Constraints**: Incorporating contextual information from the surrounding
text or the specific domain of the input can help guide the selection process,
ensuring that the output is relevant and coherent.
### Required Resources:
- **Training Data**: A large corpus of correctly structured and meaningful English
sentences for training language models and parsers.
- **Computational Power**: Access to sufficient computational resources (e.g.,
GPUs) for training deep learning models, especially for complex transformer architectures.
- **NLP Libraries and Frameworks**: Utilizing libraries such as NLTK, spaCy, or
Hugging Face Transformers for implementing various NLP techniques.
- **Expertise in NLP**: Knowledge of linguistic principles, machine learning,
and natural language processing methodologies will be essential to design and
implement an effective solution.
By applying these techniques and leveraging the necessary resources, the company
can significantly improve the quality and coherence of the outputs from their
hand-written document recognition system.'
- 'A Part-of-Speech (PoS) tagger is designed to categorize words in a given text
into their respective grammatical roles, such as nouns, verbs, adjectives, etc.
This task is complex due to two primary challenges: first, lexical ambiguity arises
when a single word can serve multiple grammatical functions depending on its context
(e.g., "lead" can be a noun or a verb), which complicates the tagging process
and can result in numerous potential interpretations as sentence length increases.
Second, the presence of out-of-vocabulary words—those not found in the tagger''s
training data—presents a further difficulty, as it necessitates the development
of strategies for effectively assigning tags to these unfamiliar terms. This often
requires the implementation of linguistic heuristics and context-based inference,
which adds an additional layer of complexity to the tagging process.'
- source_sentence: "One of your colleagues has recently taken over responsibility\
\ for a legacy codebase, a library currently used by some of your customers. Before\
\ making functional changes, your colleague found a bug caused by incorrect use\
\ of the following method in the codebase:\n\npublic class User {\n /** Indicates\
\ whether the user’s browser, if any, has JavaScript enabled. */\n public boolean\
\ hasJavascriptEnabled() { … }\n\n // … other methods, such as getName(), getAge(),\
\ ...\n}\n\nYour colleague believes that this is a bad API. You are reviewing\
\ the pull request your colleague made to fix this bug. After some discussion\
\ and additional commits to address feedback, the pull request is ready. You can\
\ either \"squash\" the pull request into a single commit, or leave the multiple\
\ commits as they are. Explain in 1 sentence whether you should \"squash\" and\
\ why."
sentences:
- "**Answer:** A\n\n**Plan:**\n1. Establish that the given linear programming problem\
\ is a bipartite network flow problem and identify the relevant properties of\
\ bipartite graphs.\n2. Use the property of total unimodularity of the constraint\
\ matrix in the linear program to show that any basic feasible solution (extreme\
\ point solution) is integral.\n3. Explain how the degree bounds \\(b(v)\\) affect\
\ the structure of the feasible region in the linear program.\n4. Conclude that\
\ since all extreme points of the feasible region are integral, the solution must\
\ also be integral.\n\n**Steps:**\n\n1. **Understanding the Problem Context:**\n\
\ The linear programming formulation given corresponds to the min-cost perfect\
\ \\(b\\)-matching problem in a bipartite graph. The objective is to minimize\
\ the cost of the edges while satisfying the degree constraints for each vertex\
\ in the graph, where \\(b(v)\\) indicates how many edges must be incident to\
\ vertex \\(v\\).\n\n2. **Total Unimodularity of the Constraint Matrix:**\n \
\ The matrix representing the constraints in this linear program is derived from\
\ the incidence matrix of the bipartite graph. Each row of the matrix corresponds\
\ to a vertex \\(v\\) and contains coefficients that indicate the edges incident\
\ to that vertex. In a bipartite graph, this incidence matrix is totally unimodular.\n\
\ - A matrix is said to be totally unimodular if every square submatrix has\
\ a determinant of \\(0\\), \\(1\\), or \\(-1\\). This property ensures that every\
\ basic feasible solution (which corresponds to an extreme point in the feasible\
\ region) of the linear program can be expressed with integer values.\n\n3. **Impact\
\ of Degree Bounds \\(b(v)\\):**\n Since \\(b(v)\\) are all non-negative integers,\
\ the constraints \\( \\sum_{e\\in E: v \\in e} x_e = b(v) \\) for all vertices\
\ \\(v\\) ensure that the solution must balance the contributions of edge variables\
\ \\(x_e\\) to meet the exact degree requirements. The total unimodularity of\
\ the matrix guarantees that these constraints will yield integer solutions for\
\ the variables \\(x_e\\).\n\n4. **Conclusion on Integral Solutions:**\n Given\
\ that the linear program's constraint matrix is totally unimodular and the constraints\
\ are linear combinations of the \\(x_e\\) variables that must equal the integer\
\ degree bounds \\(b(v)\\), we conclude that all extreme point solutions of this\
\ linear program are integral. Thus, if the input graph \\(G = (V, E)\\) is bipartite,\
\ every extreme point solution to the linear programming relaxation is indeed\
\ integral.\n\nAfter reasoning through the properties of the linear program and\
\ the implications of total unimodularity in a bipartite context, I confirm that\
\ the argument is valid and the conclusion is correct."
- "To differentiate the objective function with respect to \\( b_u \\), we start\
\ with the given objective:\n\n\\[\nJ = \\frac{1}{2} \\sum_{(u, m)} \\left(f_{um}\
\ - r_{um}\\right)^2 + \\frac{\\lambda}{2} \\left[ \\sum_{u \\in \\mathbf{U}}\
\ (b_u^2 + \\|\\mathbf{v}_u\\|^2) + \\sum_{m \\in \\mathbf{M}} (b_m^2 + \\|\\\
mathbf{w}_m\\|^2) \\right]\n\\]\n\nwhere \\( f_{um} = \\langle \\mathbf{v}_u,\
\ \\mathbf{w}_m \\rangle + b_u + b_m \\).\n\nTo find the optimal \\( b_u \\),\
\ we focus only on the terms that involve \\( b_u \\). The relevant part of the\
\ objective that includes \\( b_u \\) is:\n\n\\[\nJ_u = \\frac{1}{2} \\sum_{m\
\ \\in N(u)} \\left( \\left\\langle \\mathbf{v}_u, \\mathbf{w}_m \\right\\rangle\
\ + b_u + b_m - r_{um} \\right)^2 + \\frac{\\lambda}{2} b_u^2\n\\]\n\nwhere \\\
( N(u) \\) is the set of movies rated by user \\( u \\).\n\nNow, to differentiate\
\ \\( J_u \\) with respect to \\( b_u \\):\n\n1. **Differentiate the first term**:\n\
\ \\[\n \\frac{\\partial J_u}{\\partial b_u} = \\sum_{m \\in N(u)} \\left(\
\ \\left\\langle \\mathbf{v}_u, \\mathbf{w}_m \\right\\rangle + b_u + b_m - r_{um}\
\ \\right)\n \\]\n\n2. **Differentiate the regularization term**:\n \\[\n\
\ \\frac{\\partial}{\\partial b_u} \\left( \\frac{\\lambda}{2} b_u^2 \\right)\
\ = \\lambda b_u\n \\]\n\nCombining these results, we get:\n\n\\[\n\\frac{\\\
partial J_u}{\\partial b_u} = \\sum_{m \\in N(u)} \\left( \\left\\langle \\mathbf{v}_u,\
\ \\mathbf{w}_m \\right\\rangle + b_u + b_m - r_{um} \\right) + \\lambda b_u\n\
\\]\n\nSetting the derivative to zero to find the optimal \\( b_u \\):\n\n\\[\n\
\\sum_{m \\in N(u)} \\left( \\left\\langle \\mathbf{v}_u, \\mathbf{w}_m \\right\\\
rangle + b_u + b_m - r_{um} \\right) + \\lambda b_u = 0\n\\]\n\nRearranging gives:\n\
\n\\[\n\\sum_{m \\in N(u)} \\left\\langle \\mathbf{v}_u, \\mathbf{w}_m \\right\\\
rangle + |N(u)| b_u + \\sum_{m \\in N(u)} b_m - \\sum_{m \\in N(u)} r_{um} + \\\
lambda b_u = 0\n\\]\n\nCombine the terms involving \\( b_u \\):\n\n\\[\n\\left(\
\ |N(u)| + \\lambda \\right) b_u = \\sum_{m \\in N(u)} r_{um} - \\sum_{m \\in\
\ N(u)} \\left\\langle \\mathbf{v}_u, \\mathbf{w}_m \\right\\rangle - \\sum_{m\
\ \\in N(u)} b_m\n\\]\n\nNow, solving for \\( b_u \\):\n\n\\[\nb_u = \\frac{1}{|N(u)|\
\ + \\lambda} \\left( \\sum_{m \\in N(u)} r_{um} - \\sum_{m \\in N(u)} \\left\\\
langle \\mathbf{v}_u, \\mathbf{w}_m \\right\\rangle - \\sum_{m \\in N(u)} b_m\
\ \\right)\n\\]\n\nThis is the optimal value of \\( b_u \\) when all other parameters\
\ are fixed. The important point here is that the regularization term contributes\
\ to the denominator, ensuring that the bias \\( b_u \\) is adjusted appropriately\
\ based on the number of ratings \\( |N(u)| \\) and the regularization strength\
\ \\( \\lambda \\)."
- Yes, you should squash the pull request into a single commit to ensure that the
commit history is clean and focused on the final, complete change, avoiding confusion
from intermediate commits that may not represent valid states of the code.
- source_sentence: 'Suppose we use the Simplex method to solve the following linear
program: \begin{align*} \textbf{maximize} \hspace{0.8cm} & 2x_1 - x_2 \\ \textbf{subject
to}\hspace{0.8cm} & x_1 - x_2 + s_1 = 1 \\ \hspace{0.8cm} & \hspace{0.85cm}x_1 +
s_2 = 4 \\ \hspace{0.8cm} & \hspace{0.85cm} x_2 + s_3 = 2 \\ \hspace{0.8cm}
&\hspace{-0.8cm} x_1,\: x_2, \:s_1, \:s_2, \:s_3 \geq 0 \end{align*} At the current
step, we have the following Simplex tableau: \begin{align*} \hspace{1cm} x_1 &=
1 + x_2 - s_1 \\ s_2 &= 3 -x_2 + s_1 \\ s_3 &= 2 -x_2 \\ \cline{1-2} z &= 2
+ x_2 - 2s_1 \end{align*} Write the tableau obtained by executing one iteration
(pivot) of the Simplex method starting from the above tableau.'
sentences:
- "To analyze whether we need to introduce constraints on the number of Byzantine\
\ processes in a non-synchronous environment, we first need to understand the\
\ properties of Byzantine consistent broadcast (BCB) and the implications of Byzantine\
\ processes on these properties.\n\n### Definitions:\n- Let \\( N \\) be the total\
\ number of processes in the system.\n- Let \\( F \\) be the maximum number of\
\ Byzantine processes.\n\n### Properties of Byzantine Consistent Broadcast:\n\
1. **Validity**: If the designated sender \\( S \\) is correct, then every correct\
\ process eventually delivers the message.\n2. **No duplication**: Every correct\
\ process delivers at most one message.\n3. **Integrity**: If a correct process\
\ delivers a message, and \\( S \\) is correct, then \\( S \\) has previously\
\ broadcast the message.\n4. **Consistency**: No two correct processes deliver\
\ different messages.\n\n### Analyzing Byzantine Processes:\nIn a non-synchronous\
\ environment, messages may take an unpredictable amount of time to be delivered,\
\ and processes may operate independently without a global clock. This introduces\
\ challenges when Byzantine processes (which can behave arbitrarily) are present.\n\
\n#### Hypothetical Scenarios:\n1. **Scenario with \\( N = 3F \\)**:\n - Suppose\
\ \\( N = 3F \\), meaning there are exactly three times as many processes as there\
\ are Byzantine processes. For example, if \\( F = 1 \\), then \\( N = 3 \\).\
\ In this case, the processes can be divided into:\n - Correct process A\n\
\ - Correct process B\n - Byzantine process C\n\n In this scenario,\
\ if the Byzantine process (C) decides to send a different message than what the\
\ correct sender (S) sends, it can mislead the correct processes (A and B). Both\
\ A and B may receive different messages if they are not able to distinguish between\
\ the correct and Byzantine processes.\n\n2. **Potential Violation of Properties**:\n\
\ - **Validity**: If the correct process A receives a message from the Byzantine\
\ process C, it may not be able to determine if the message is valid. This could\
\ lead to a situation where A does not deliver the correct message broadcasted\
\ by S.\n - **Consistency**: If A and B receive different messages due to the\
\ Byzantine behavior of C, this violates the consistency property, as A delivers\
\ message \\( m_1 \\) while B delivers \\( m_2 \\).\n\n### Mathematical Relationships:\n\
To ensure the properties of BCB hold, particularly the consistency property, we\
\ must have more correct processes than Byzantine processes. The established consensus\
\ in distributed systems is that the maximum number of Byzantine processes \\\
( F \\) can be tolerated if and only if:\n\n$$ N > 3F $$\n\nor, more precisely,\n\
\n$$ N = 3F + 1 $$\n\n### Conclusion:\nYes, we must introduce constraints on the\
\ number of Byzantine processes in non-synchronous environments to maintain the\
\ integrity and consistency properties of Byzantine consistent broadcast. Specifically,\
\ the condition \\( N = 3F + 1 \\) must be satisfied to ensure that correct processes\
\ can always outvote or ignore the Byzantine processes, thereby ensuring that\
\ the system remains reliable and consistent even in the presence of failures\
\ or malicious behavior. This constraint is crucial for the robustness of the\
\ BCB protocol in distributed systems."
- "To execute one iteration (pivot) of the Simplex method from the given tableau,\
\ we need to follow these key steps:\n\n1. **Identify the entering variable:**\
\ This is the variable that will increase to improve the objective function. In\
\ our case, we look at the coefficients of the objective function row (z-row).\
\ We want to maximize z, so we will select the variable with the highest positive\
\ coefficient. In this tableau, the coefficients for \\(x_1\\), \\(x_2\\), and\
\ the slack variables \\(s_1\\), \\(s_2\\), and \\(s_3\\) in the objective function\
\ are \\(2\\), \\(1\\), \\(-2\\), \\(0\\), and \\(0\\), respectively. The variable\
\ \\(x_1\\) has the highest positive coefficient of \\(2\\), so it will be our\
\ entering variable.\n\n2. **Identify the leaving variable:** Next, we need to\
\ determine which variable will leave the basis. This is done using the minimum\
\ ratio test. We look at the constraints and find the ratios of the current solution\
\ values to the coefficients of the entering variable in each constraint that\
\ has a positive coefficient for that variable.\n\n From the current tableau:\n\
\ - For the first equation, \\( x_1 = 1 + x_2 - s_1 \\): The coefficient of\
\ \\(x_1\\) is \\(1\\). If we set \\(x_2 = 0\\) and \\(s_1 = 0\\), we can solve\
\ for the right-hand side (RHS), which is \\(1\\). The ratio is \\( \\frac{1}{1}\
\ = 1\\).\n - For the second equation, \\(s_2 = 3 - x_2 + s_1\\): The coefficient\
\ of \\(x_1\\) is \\(1\\). The RHS is \\(3\\), so the ratio is again \\( \\frac{3}{1}\
\ = 3\\).\n - For the third equation, \\(s_3 = 2 - x_2\\): The coefficient of\
\ \\(x_1\\) is \\(0\\), so we can't use this equation.\n\n The minimum ratio\
\ is \\(1\\) from the first equation, so \\(s_1\\) will leave the basis.\n\n3.\
\ **Perform the pivot operation:** We pivot on the intersection of the entering\
\ variable \\(x_1\\) and the leaving variable \\(s_1\\). We want to express all\
\ variables in terms of the new basic variable \\(x_1\\).\n\n The tableau before\
\ pivoting is:\n\n \\[\n \\begin{array}{c|c|c|c|c|c}\n & x_1 & x_2 & s_1\
\ & s_2 & s_3 \\\\\n \\hline\n x_1 & 1 & 1 & -1 & 0 & 0 \\\\\n s_2 & 0 &\
\ 1 & 1 & 1 & 0 \\\\\n s_3 & 0 & 0 & 0 & 0 & 2 \\\\\n \\hline\n z & 0 &\
\ 1 & 2 & 0 & 0 \\\\\n \\end{array}\n \\]\n\n Now we perform row operations\
\ to update the tableau:\n\n - **Row for \\(x_1\\)**: This will remain unchanged\
\ as it becomes our new basic variable.\n - **Row for \\(s_2\\)**: We will subtract\
\ the \\(0\\) multiplied by row \\(x_1\\).\n - **Row for \\(s_3\\)**: This row\
\ will also not change as \\(x_1\\) does not appear.\n - **Row for \\(z\\)**:\
\ We need to replace the \\(z\\) row to reflect the new basis. The new coefficient\
\ of \\(z\\) will be \\(2\\) because we have removed \\(s_1\\) from the basis.\n\
\nAfter performing these operations, we can represent the new tableau:\n\n\\[\n\
\\begin{array}{c|c|c|c|c|c}\n & x_1 & x_2 & s_2 & s_3 \\\\\n \\hline\n x_1\
\ & 1 & 1 & -1 & 0 \\\\\n s_2 & 0 & 1 & 1 & 1 \\\\\n s_3 & 0 & 0 & 0 & 2 \\\
\\\n \\hline\n z & 0 & 1 & 2 & 0 \\\\\n\\end{array}\n\\]\n\nHere, we have\
\ successfully updated the tableau after one pivot step of the Simplex method.\
\ The new solution reflects a state where \\(x_1\\) is now a basic variable. We\
\ will repeat this process until we find the optimal solution."
- "Certainly! Let's reevaluate the proof that \\( w(S_\\ell) = \\max_{T \\in \\\
mathcal{I}: |T| = \\ell} w(T) \\) for all \\( \\ell = 1, \\ldots, k \\) in a more\
\ rigorous manner.\n\n### Overview of the Proof Structure\n\nWe need to show that\
\ for each \\( \\ell \\), the greedy algorithm produces a set \\( S_\\ell \\)\
\ of elements with the maximum weight among all independent sets of size \\( \\\
ell \\) in a matroid \\( \\mathcal{M} = (E, \\mathcal{I}) \\) with a weight function\
\ \\( w: E \\rightarrow \\mathbb{R} \\).\n\n### Step 1: Base Case\n\nFor \\( \\\
ell = 1 \\):\n- \\( S_1 = \\{s_1\\} \\) is the element with the highest weight.\n\
- The maximum weight of any independent set of size 1 is indeed \\( \\max_{e \\\
in E} w(e) \\), and since \\( s_1 \\) is chosen to be the element with the maximum\
\ weight, we have:\n\n\\[\nw(S_1) = w(s_1) = \\max_{T \\in \\mathcal{I}: |T| =\
\ 1} w(T)\n\\]\n\nThis base case holds true.\n\n### Step 2: Induction Hypothesis\n\
\nNow we assume that for some \\( \\ell \\) where \\( 1 \\leq \\ell < k \\):\n\
\n\\[\nw(S_\\ell) = \\max_{T \\in \\mathcal{I}: |T| = \\ell} w(T)\n\\]\n\nThis\
\ means that the greedy choice up to \\( \\ell \\) produces a maximum weight independent\
\ set of size \\( \\ell \\).\n\n### Step 3: Induction Step\n\nWe need to prove\
\ that:\n\n\\[\nw(S_{\\ell + 1}) = \\max_{T \\in \\mathcal{I}: |T| = \\ell + 1}\
\ w(T)\n\\]\n\nLet \\( S_{\\ell + 1} = \\{s_1, s_2, \\ldots, s_{\\ell + 1}\\}\
\ \\).\n\n#### Constructing the Set \\( S_{\\ell + 1} \\)\n\nTo show this equality,\
\ we will analyze any independent set \\( T \\) of size \\( \\ell + 1 \\).\n\n\
1. **Case 1: \\( s_{\\ell + 1} \\notin T \\)**\n\n If \\( s_{\\ell + 1} \\)\
\ is not in \\( T \\), then \\( T \\) must consist of elements among \\( \\{s_1,\
\ s_2, \\ldots, s_\\ell\\} \\) and possibly other elements from \\( E \\). Since\
\ \\( S_\\ell \\) consists of the elements with the highest weights, we can state:\n\
\n \\[\n w(T) \\leq w(S_\\ell)\n \\]\n\n By the induction hypothesis,\
\ this means:\n\n \\[\n w(T) \\leq \\max_{T' \\in \\mathcal{I}: |T'| = \\\
ell} w(T') = w(S_\\ell)\n \\]\n\n2. **Case 2: \\( s_{\\ell + 1} \\in T \\)**\n\
\n In this case, we can remove \\( s_{\\ell + 1} \\) from \\( T \\) to obtain\
\ an independent set \\( T' \\) of size \\( \\ell \\). Thus, we have:\n\n \\\
[\n w(T) = w(T') + w(s_{\\ell + 1})\n \\]\n\n Since \\( T' \\) is an independent\
\ set, we can apply the induction hypothesis, which gives us:\n\n \\[\n w(T')\
\ \\leq w(S_\\ell)\n \\]\n\n Therefore:\n\n \\[\n w(T) = w(T') + w(s_{\\\
ell + 1}) \\leq w(S_\\ell) + w(s_{\\ell + 1})\n \\]\n\n### Conclusion from the\
\ Induction Step\n\nTo compare \\( w(S_{\\ell + 1}) \\) with \\( w(T) \\), we\
\ note:\n\n- \\( S_{\\ell + 1} \\) contains the highest weight elements, and thus:\n\
\n\\[\nw(S_{\\ell + 1}) = w(S_\\ell) + w(s_{\\ell + 1})\n\\]\n\nSince \\( s_{\\\
ell + 1} \\) is the highest weight among the remaining elements, we establish\
\ the following:\n\n- If \\( s_{\\ell + 1} \\) is included in \\( T \\), we have\
\ \\( w(T) \\leq w(S_{\\ell + 1}) \\).\n- If \\( s_{\\ell + 1} \\) is not included,\
\ \\( w(T) \\leq w(S_\\ell) \\) which is already less than or equal to \\( w(S_{\\\
ell + 1}) \\).\n\nThus, we conclude that:\n\n\\[\nw(S_{\\ell + 1}) = \\max_{T\
\ \\in \\mathcal{I}: |T| = \\ell + 1} w(T)\n\\]\n\n### Final Remarks\n\nThe proof\
\ holds without any assumptions being violated. Each step has been justified based\
\ on the properties of the matroid and the greedy choice made at each step, ensuring\
\ that the maximum weight independent set is correctly identified for all sizes\
\ from 1 to \\( k \\).\n\nThis completes the proof with careful consideration\
\ of all cases and steps involved. Thank you for your patience!"
pipeline_tag: sentence-similarity
library_name: sentence-transformers
metrics:
- cosine_accuracy@1
- cosine_accuracy@3
- cosine_accuracy@5
- cosine_accuracy@10
- cosine_precision@1
- cosine_precision@3
- cosine_precision@5
- cosine_precision@10
- cosine_recall@1
- cosine_recall@3
- cosine_recall@5
- cosine_recall@10
- cosine_ndcg@10
- cosine_mrr@10
- cosine_map@100
model-index:
- name: BGE base Financial Matryoshka
results:
- task:
type: information-retrieval
name: Information Retrieval
dataset:
name: dim 768
type: dim_768
metrics:
- type: cosine_accuracy@1
value: 0.2771929824561403
name: Cosine Accuracy@1
- type: cosine_accuracy@3
value: 0.8807017543859649
name: Cosine Accuracy@3
- type: cosine_accuracy@5
value: 0.9298245614035088
name: Cosine Accuracy@5
- type: cosine_accuracy@10
value: 0.9824561403508771
name: Cosine Accuracy@10
- type: cosine_precision@1
value: 0.2771929824561403
name: Cosine Precision@1
- type: cosine_precision@3
value: 0.29356725146198825
name: Cosine Precision@3
- type: cosine_precision@5
value: 0.1859649122807017
name: Cosine Precision@5
- type: cosine_precision@10
value: 0.09824561403508769
name: Cosine Precision@10
- type: cosine_recall@1
value: 0.2771929824561403
name: Cosine Recall@1
- type: cosine_recall@3
value: 0.8807017543859649
name: Cosine Recall@3
- type: cosine_recall@5
value: 0.9298245614035088
name: Cosine Recall@5
- type: cosine_recall@10
value: 0.9824561403508771
name: Cosine Recall@10
- type: cosine_ndcg@10
value: 0.6591255563949661
name: Cosine Ndcg@10
- type: cosine_mrr@10
value: 0.551246170983013
name: Cosine Mrr@10
- type: cosine_map@100
value: 0.552368028142022
name: Cosine Map@100
- task:
type: information-retrieval
name: Information Retrieval
dataset:
name: dim 512
type: dim_512
metrics:
- type: cosine_accuracy@1
value: 0.2807017543859649
name: Cosine Accuracy@1
- type: cosine_accuracy@3
value: 0.8771929824561403
name: Cosine Accuracy@3
- type: cosine_accuracy@5
value: 0.9298245614035088
name: Cosine Accuracy@5
- type: cosine_accuracy@10
value: 0.9824561403508771
name: Cosine Accuracy@10
- type: cosine_precision@1
value: 0.2807017543859649
name: Cosine Precision@1
- type: cosine_precision@3
value: 0.2923976608187134
name: Cosine Precision@3
- type: cosine_precision@5
value: 0.1859649122807017
name: Cosine Precision@5
- type: cosine_precision@10
value: 0.09824561403508769
name: Cosine Precision@10
- type: cosine_recall@1
value: 0.2807017543859649
name: Cosine Recall@1
- type: cosine_recall@3
value: 0.8771929824561403
name: Cosine Recall@3
- type: cosine_recall@5
value: 0.9298245614035088
name: Cosine Recall@5
- type: cosine_recall@10
value: 0.9824561403508771
name: Cosine Recall@10
- type: cosine_ndcg@10
value: 0.6589926958168313
name: Cosine Ndcg@10
- type: cosine_mrr@10
value: 0.5511487050960734
name: Cosine Mrr@10
- type: cosine_map@100
value: 0.5522544998860789
name: Cosine Map@100
- task:
type: information-retrieval
name: Information Retrieval
dataset:
name: dim 256
type: dim_256
metrics:
- type: cosine_accuracy@1
value: 0.2771929824561403
name: Cosine Accuracy@1
- type: cosine_accuracy@3
value: 0.8736842105263158
name: Cosine Accuracy@3
- type: cosine_accuracy@5
value: 0.9368421052631579
name: Cosine Accuracy@5
- type: cosine_accuracy@10
value: 0.9894736842105263
name: Cosine Accuracy@10
- type: cosine_precision@1
value: 0.2771929824561403
name: Cosine Precision@1
- type: cosine_precision@3
value: 0.2912280701754386
name: Cosine Precision@3
- type: cosine_precision@5
value: 0.18736842105263155
name: Cosine Precision@5
- type: cosine_precision@10
value: 0.09894736842105262
name: Cosine Precision@10
- type: cosine_recall@1
value: 0.2771929824561403
name: Cosine Recall@1
- type: cosine_recall@3
value: 0.8736842105263158
name: Cosine Recall@3
- type: cosine_recall@5
value: 0.9368421052631579
name: Cosine Recall@5
- type: cosine_recall@10
value: 0.9894736842105263
name: Cosine Recall@10
- type: cosine_ndcg@10
value: 0.6595162220662993
name: Cosine Ndcg@10
- type: cosine_mrr@10
value: 0.5497257031467556
name: Cosine Mrr@10
- type: cosine_map@100
value: 0.5503770544576845
name: Cosine Map@100
- task:
type: information-retrieval
name: Information Retrieval
dataset:
name: dim 128
type: dim_128
metrics:
- type: cosine_accuracy@1
value: 0.29473684210526313
name: Cosine Accuracy@1
- type: cosine_accuracy@3
value: 0.8701754385964913
name: Cosine Accuracy@3
- type: cosine_accuracy@5
value: 0.9333333333333333
name: Cosine Accuracy@5
- type: cosine_accuracy@10
value: 0.9859649122807017
name: Cosine Accuracy@10
- type: cosine_precision@1
value: 0.29473684210526313
name: Cosine Precision@1
- type: cosine_precision@3
value: 0.2900584795321637
name: Cosine Precision@3
- type: cosine_precision@5
value: 0.18666666666666662
name: Cosine Precision@5
- type: cosine_precision@10
value: 0.09859649122807015
name: Cosine Precision@10
- type: cosine_recall@1
value: 0.29473684210526313
name: Cosine Recall@1
- type: cosine_recall@3
value: 0.8701754385964913
name: Cosine Recall@3
- type: cosine_recall@5
value: 0.9333333333333333
name: Cosine Recall@5
- type: cosine_recall@10
value: 0.9859649122807017
name: Cosine Recall@10
- type: cosine_ndcg@10
value: 0.6666560081123373
name: Cosine Ndcg@10
- type: cosine_mrr@10
value: 0.5603745474798105
name: Cosine Mrr@10
- type: cosine_map@100
value: 0.5612992076149971
name: Cosine Map@100
- task:
type: information-retrieval
name: Information Retrieval
dataset:
name: dim 64
type: dim_64
metrics:
- type: cosine_accuracy@1
value: 0.27017543859649124
name: Cosine Accuracy@1
- type: cosine_accuracy@3
value: 0.8456140350877193
name: Cosine Accuracy@3
- type: cosine_accuracy@5
value: 0.9333333333333333
name: Cosine Accuracy@5
- type: cosine_accuracy@10
value: 0.9789473684210527
name: Cosine Accuracy@10
- type: cosine_precision@1
value: 0.27017543859649124
name: Cosine Precision@1
- type: cosine_precision@3
value: 0.2818713450292398
name: Cosine Precision@3
- type: cosine_precision@5
value: 0.18666666666666662
name: Cosine Precision@5
- type: cosine_precision@10
value: 0.09789473684210524
name: Cosine Precision@10
- type: cosine_recall@1
value: 0.27017543859649124
name: Cosine Recall@1
- type: cosine_recall@3
value: 0.8456140350877193
name: Cosine Recall@3
- type: cosine_recall@5
value: 0.9333333333333333
name: Cosine Recall@5
- type: cosine_recall@10
value: 0.9789473684210527
name: Cosine Recall@10
- type: cosine_ndcg@10
value: 0.6492874099033707
name: Cosine Ndcg@10
- type: cosine_mrr@10
value: 0.5396101364522413
name: Cosine Mrr@10
- type: cosine_map@100
value: 0.5411050628327516
name: Cosine Map@100
---
# BGE base Financial Matryoshka
This is a [sentence-transformers](https://www.SBERT.net) model finetuned from [NovaSearch/stella_en_400M_v5](https://huggingface.co/NovaSearch/stella_en_400M_v5) on the json dataset. It maps sentences & paragraphs to a 1024-dimensional dense vector space and can be used for semantic textual similarity, semantic search, paraphrase mining, text classification, clustering, and more.
## Model Details
### Model Description
- **Model Type:** Sentence Transformer
- **Base model:** [NovaSearch/stella_en_400M_v5](https://huggingface.co/NovaSearch/stella_en_400M_v5) <!-- at revision dcae70d3f2b4aaee36afc3cde638ca4614497aec -->
- **Maximum Sequence Length:** 512 tokens
- **Output Dimensionality:** 1024 dimensions
- **Similarity Function:** Cosine Similarity
- **Training Dataset:**
- json
- **Language:** en
- **License:** apache-2.0
### Model Sources
- **Documentation:** [Sentence Transformers Documentation](https://sbert.net)
- **Repository:** [Sentence Transformers on GitHub](https://github.com/UKPLab/sentence-transformers)
- **Hugging Face:** [Sentence Transformers on Hugging Face](https://huggingface.co/models?library=sentence-transformers)
### Full Model Architecture
```
SentenceTransformer(
(0): Transformer({'max_seq_length': 512, 'do_lower_case': False}) with Transformer model: NewModel
(1): Pooling({'word_embedding_dimension': 1024, 'pooling_mode_cls_token': False, 'pooling_mode_mean_tokens': True, 'pooling_mode_max_tokens': False, 'pooling_mode_mean_sqrt_len_tokens': False, 'pooling_mode_weightedmean_tokens': False, 'pooling_mode_lasttoken': False, 'include_prompt': True})
(2): Dense({'in_features': 1024, 'out_features': 1024, 'bias': True, 'activation_function': 'torch.nn.modules.linear.Identity'})
)
```
## Usage
### Direct Usage (Sentence Transformers)
First install the Sentence Transformers library:
```bash
pip install -U sentence-transformers
```
Then you can load this model and run inference.
```python
from sentence_transformers import SentenceTransformer
# Download from the 🤗 Hub
model = SentenceTransformer("cristiano-sartori/stella_finetuned")
# Run inference
sentences = [
'Suppose we use the Simplex method to solve the following linear program: \\begin{align*} \\textbf{maximize} \\hspace{0.8cm} & 2x_1 - x_2 \\\\ \\textbf{subject to}\\hspace{0.8cm} & x_1 - x_2 + s_1 = 1 \\\\ \\hspace{0.8cm} & \\hspace{0.85cm}x_1 + s_2 = 4 \\\\ \\hspace{0.8cm} & \\hspace{0.85cm} x_2 + s_3 = 2 \\\\ \\hspace{0.8cm} &\\hspace{-0.8cm} x_1,\\: x_2, \\:s_1, \\:s_2, \\:s_3 \\geq 0 \\end{align*} At the current step, we have the following Simplex tableau: \\begin{align*} \\hspace{1cm} x_1 &= 1 + x_2 - s_1 \\\\ s_2 &= 3 -x_2 + s_1 \\\\ s_3 &= 2 -x_2 \\\\ \\cline{1-2} z &= 2 + x_2 - 2s_1 \\end{align*} Write the tableau obtained by executing one iteration (pivot) of the Simplex method starting from the above tableau.',
"To execute one iteration (pivot) of the Simplex method from the given tableau, we need to follow these key steps:\n\n1. **Identify the entering variable:** This is the variable that will increase to improve the objective function. In our case, we look at the coefficients of the objective function row (z-row). We want to maximize z, so we will select the variable with the highest positive coefficient. In this tableau, the coefficients for \\(x_1\\), \\(x_2\\), and the slack variables \\(s_1\\), \\(s_2\\), and \\(s_3\\) in the objective function are \\(2\\), \\(1\\), \\(-2\\), \\(0\\), and \\(0\\), respectively. The variable \\(x_1\\) has the highest positive coefficient of \\(2\\), so it will be our entering variable.\n\n2. **Identify the leaving variable:** Next, we need to determine which variable will leave the basis. This is done using the minimum ratio test. We look at the constraints and find the ratios of the current solution values to the coefficients of the entering variable in each constraint that has a positive coefficient for that variable.\n\n From the current tableau:\n - For the first equation, \\( x_1 = 1 + x_2 - s_1 \\): The coefficient of \\(x_1\\) is \\(1\\). If we set \\(x_2 = 0\\) and \\(s_1 = 0\\), we can solve for the right-hand side (RHS), which is \\(1\\). The ratio is \\( \\frac{1}{1} = 1\\).\n - For the second equation, \\(s_2 = 3 - x_2 + s_1\\): The coefficient of \\(x_1\\) is \\(1\\). The RHS is \\(3\\), so the ratio is again \\( \\frac{3}{1} = 3\\).\n - For the third equation, \\(s_3 = 2 - x_2\\): The coefficient of \\(x_1\\) is \\(0\\), so we can't use this equation.\n\n The minimum ratio is \\(1\\) from the first equation, so \\(s_1\\) will leave the basis.\n\n3. **Perform the pivot operation:** We pivot on the intersection of the entering variable \\(x_1\\) and the leaving variable \\(s_1\\). We want to express all variables in terms of the new basic variable \\(x_1\\).\n\n The tableau before pivoting is:\n\n \\[\n \\begin{array}{c|c|c|c|c|c}\n & x_1 & x_2 & s_1 & s_2 & s_3 \\\\\n \\hline\n x_1 & 1 & 1 & -1 & 0 & 0 \\\\\n s_2 & 0 & 1 & 1 & 1 & 0 \\\\\n s_3 & 0 & 0 & 0 & 0 & 2 \\\\\n \\hline\n z & 0 & 1 & 2 & 0 & 0 \\\\\n \\end{array}\n \\]\n\n Now we perform row operations to update the tableau:\n\n - **Row for \\(x_1\\)**: This will remain unchanged as it becomes our new basic variable.\n - **Row for \\(s_2\\)**: We will subtract the \\(0\\) multiplied by row \\(x_1\\).\n - **Row for \\(s_3\\)**: This row will also not change as \\(x_1\\) does not appear.\n - **Row for \\(z\\)**: We need to replace the \\(z\\) row to reflect the new basis. The new coefficient of \\(z\\) will be \\(2\\) because we have removed \\(s_1\\) from the basis.\n\nAfter performing these operations, we can represent the new tableau:\n\n\\[\n\\begin{array}{c|c|c|c|c|c}\n & x_1 & x_2 & s_2 & s_3 \\\\\n \\hline\n x_1 & 1 & 1 & -1 & 0 \\\\\n s_2 & 0 & 1 & 1 & 1 \\\\\n s_3 & 0 & 0 & 0 & 2 \\\\\n \\hline\n z & 0 & 1 & 2 & 0 \\\\\n\\end{array}\n\\]\n\nHere, we have successfully updated the tableau after one pivot step of the Simplex method. The new solution reflects a state where \\(x_1\\) is now a basic variable. We will repeat this process until we find the optimal solution.",
"Certainly! Let's reevaluate the proof that \\( w(S_\\ell) = \\max_{T \\in \\mathcal{I}: |T| = \\ell} w(T) \\) for all \\( \\ell = 1, \\ldots, k \\) in a more rigorous manner.\n\n### Overview of the Proof Structure\n\nWe need to show that for each \\( \\ell \\), the greedy algorithm produces a set \\( S_\\ell \\) of elements with the maximum weight among all independent sets of size \\( \\ell \\) in a matroid \\( \\mathcal{M} = (E, \\mathcal{I}) \\) with a weight function \\( w: E \\rightarrow \\mathbb{R} \\).\n\n### Step 1: Base Case\n\nFor \\( \\ell = 1 \\):\n- \\( S_1 = \\{s_1\\} \\) is the element with the highest weight.\n- The maximum weight of any independent set of size 1 is indeed \\( \\max_{e \\in E} w(e) \\), and since \\( s_1 \\) is chosen to be the element with the maximum weight, we have:\n\n\\[\nw(S_1) = w(s_1) = \\max_{T \\in \\mathcal{I}: |T| = 1} w(T)\n\\]\n\nThis base case holds true.\n\n### Step 2: Induction Hypothesis\n\nNow we assume that for some \\( \\ell \\) where \\( 1 \\leq \\ell < k \\):\n\n\\[\nw(S_\\ell) = \\max_{T \\in \\mathcal{I}: |T| = \\ell} w(T)\n\\]\n\nThis means that the greedy choice up to \\( \\ell \\) produces a maximum weight independent set of size \\( \\ell \\).\n\n### Step 3: Induction Step\n\nWe need to prove that:\n\n\\[\nw(S_{\\ell + 1}) = \\max_{T \\in \\mathcal{I}: |T| = \\ell + 1} w(T)\n\\]\n\nLet \\( S_{\\ell + 1} = \\{s_1, s_2, \\ldots, s_{\\ell + 1}\\} \\).\n\n#### Constructing the Set \\( S_{\\ell + 1} \\)\n\nTo show this equality, we will analyze any independent set \\( T \\) of size \\( \\ell + 1 \\).\n\n1. **Case 1: \\( s_{\\ell + 1} \\notin T \\)**\n\n If \\( s_{\\ell + 1} \\) is not in \\( T \\), then \\( T \\) must consist of elements among \\( \\{s_1, s_2, \\ldots, s_\\ell\\} \\) and possibly other elements from \\( E \\). Since \\( S_\\ell \\) consists of the elements with the highest weights, we can state:\n\n \\[\n w(T) \\leq w(S_\\ell)\n \\]\n\n By the induction hypothesis, this means:\n\n \\[\n w(T) \\leq \\max_{T' \\in \\mathcal{I}: |T'| = \\ell} w(T') = w(S_\\ell)\n \\]\n\n2. **Case 2: \\( s_{\\ell + 1} \\in T \\)**\n\n In this case, we can remove \\( s_{\\ell + 1} \\) from \\( T \\) to obtain an independent set \\( T' \\) of size \\( \\ell \\). Thus, we have:\n\n \\[\n w(T) = w(T') + w(s_{\\ell + 1})\n \\]\n\n Since \\( T' \\) is an independent set, we can apply the induction hypothesis, which gives us:\n\n \\[\n w(T') \\leq w(S_\\ell)\n \\]\n\n Therefore:\n\n \\[\n w(T) = w(T') + w(s_{\\ell + 1}) \\leq w(S_\\ell) + w(s_{\\ell + 1})\n \\]\n\n### Conclusion from the Induction Step\n\nTo compare \\( w(S_{\\ell + 1}) \\) with \\( w(T) \\), we note:\n\n- \\( S_{\\ell + 1} \\) contains the highest weight elements, and thus:\n\n\\[\nw(S_{\\ell + 1}) = w(S_\\ell) + w(s_{\\ell + 1})\n\\]\n\nSince \\( s_{\\ell + 1} \\) is the highest weight among the remaining elements, we establish the following:\n\n- If \\( s_{\\ell + 1} \\) is included in \\( T \\), we have \\( w(T) \\leq w(S_{\\ell + 1}) \\).\n- If \\( s_{\\ell + 1} \\) is not included, \\( w(T) \\leq w(S_\\ell) \\) which is already less than or equal to \\( w(S_{\\ell + 1}) \\).\n\nThus, we conclude that:\n\n\\[\nw(S_{\\ell + 1}) = \\max_{T \\in \\mathcal{I}: |T| = \\ell + 1} w(T)\n\\]\n\n### Final Remarks\n\nThe proof holds without any assumptions being violated. Each step has been justified based on the properties of the matroid and the greedy choice made at each step, ensuring that the maximum weight independent set is correctly identified for all sizes from 1 to \\( k \\).\n\nThis completes the proof with careful consideration of all cases and steps involved. Thank you for your patience!",
]
embeddings = model.encode(sentences)
print(embeddings.shape)
# [3, 1024]
# Get the similarity scores for the embeddings
similarities = model.similarity(embeddings, embeddings)
print(similarities.shape)
# [3, 3]
```
<!--
### Direct Usage (Transformers)
<details><summary>Click to see the direct usage in Transformers</summary>
</details>
-->
<!--
### Downstream Usage (Sentence Transformers)
You can finetune this model on your own dataset.
<details><summary>Click to expand</summary>
</details>
-->
<!--
### Out-of-Scope Use
*List how the model may foreseeably be misused and address what users ought not to do with the model.*
-->
## Evaluation
### Metrics
#### Information Retrieval
* Dataset: `dim_768`
* Evaluated with [<code>InformationRetrievalEvaluator</code>](https://sbert.net/docs/package_reference/sentence_transformer/evaluation.html#sentence_transformers.evaluation.InformationRetrievalEvaluator) with these parameters:
```json
{
"truncate_dim": 768
}
```
| Metric | Value |
|:--------------------|:-----------|
| cosine_accuracy@1 | 0.2772 |
| cosine_accuracy@3 | 0.8807 |
| cosine_accuracy@5 | 0.9298 |
| cosine_accuracy@10 | 0.9825 |
| cosine_precision@1 | 0.2772 |
| cosine_precision@3 | 0.2936 |
| cosine_precision@5 | 0.186 |
| cosine_precision@10 | 0.0982 |
| cosine_recall@1 | 0.2772 |
| cosine_recall@3 | 0.8807 |
| cosine_recall@5 | 0.9298 |
| cosine_recall@10 | 0.9825 |
| **cosine_ndcg@10** | **0.6591** |
| cosine_mrr@10 | 0.5512 |
| cosine_map@100 | 0.5524 |
#### Information Retrieval
* Dataset: `dim_512`
* Evaluated with [<code>InformationRetrievalEvaluator</code>](https://sbert.net/docs/package_reference/sentence_transformer/evaluation.html#sentence_transformers.evaluation.InformationRetrievalEvaluator) with these parameters:
```json
{
"truncate_dim": 512
}
```
| Metric | Value |
|:--------------------|:----------|
| cosine_accuracy@1 | 0.2807 |
| cosine_accuracy@3 | 0.8772 |
| cosine_accuracy@5 | 0.9298 |
| cosine_accuracy@10 | 0.9825 |
| cosine_precision@1 | 0.2807 |
| cosine_precision@3 | 0.2924 |
| cosine_precision@5 | 0.186 |
| cosine_precision@10 | 0.0982 |
| cosine_recall@1 | 0.2807 |
| cosine_recall@3 | 0.8772 |
| cosine_recall@5 | 0.9298 |
| cosine_recall@10 | 0.9825 |
| **cosine_ndcg@10** | **0.659** |
| cosine_mrr@10 | 0.5511 |
| cosine_map@100 | 0.5523 |
#### Information Retrieval
* Dataset: `dim_256`
* Evaluated with [<code>InformationRetrievalEvaluator</code>](https://sbert.net/docs/package_reference/sentence_transformer/evaluation.html#sentence_transformers.evaluation.InformationRetrievalEvaluator) with these parameters:
```json
{
"truncate_dim": 256
}
```
| Metric | Value |
|:--------------------|:-----------|
| cosine_accuracy@1 | 0.2772 |
| cosine_accuracy@3 | 0.8737 |
| cosine_accuracy@5 | 0.9368 |
| cosine_accuracy@10 | 0.9895 |
| cosine_precision@1 | 0.2772 |
| cosine_precision@3 | 0.2912 |
| cosine_precision@5 | 0.1874 |
| cosine_precision@10 | 0.0989 |
| cosine_recall@1 | 0.2772 |
| cosine_recall@3 | 0.8737 |
| cosine_recall@5 | 0.9368 |
| cosine_recall@10 | 0.9895 |
| **cosine_ndcg@10** | **0.6595** |
| cosine_mrr@10 | 0.5497 |
| cosine_map@100 | 0.5504 |
#### Information Retrieval
* Dataset: `dim_128`
* Evaluated with [<code>InformationRetrievalEvaluator</code>](https://sbert.net/docs/package_reference/sentence_transformer/evaluation.html#sentence_transformers.evaluation.InformationRetrievalEvaluator) with these parameters:
```json
{
"truncate_dim": 128
}
```
| Metric | Value |
|:--------------------|:-----------|
| cosine_accuracy@1 | 0.2947 |
| cosine_accuracy@3 | 0.8702 |
| cosine_accuracy@5 | 0.9333 |
| cosine_accuracy@10 | 0.986 |
| cosine_precision@1 | 0.2947 |
| cosine_precision@3 | 0.2901 |
| cosine_precision@5 | 0.1867 |
| cosine_precision@10 | 0.0986 |
| cosine_recall@1 | 0.2947 |
| cosine_recall@3 | 0.8702 |
| cosine_recall@5 | 0.9333 |
| cosine_recall@10 | 0.986 |
| **cosine_ndcg@10** | **0.6667** |
| cosine_mrr@10 | 0.5604 |
| cosine_map@100 | 0.5613 |
#### Information Retrieval
* Dataset: `dim_64`
* Evaluated with [<code>InformationRetrievalEvaluator</code>](https://sbert.net/docs/package_reference/sentence_transformer/evaluation.html#sentence_transformers.evaluation.InformationRetrievalEvaluator) with these parameters:
```json
{
"truncate_dim": 64
}
```
| Metric | Value |
|:--------------------|:-----------|
| cosine_accuracy@1 | 0.2702 |
| cosine_accuracy@3 | 0.8456 |
| cosine_accuracy@5 | 0.9333 |
| cosine_accuracy@10 | 0.9789 |
| cosine_precision@1 | 0.2702 |
| cosine_precision@3 | 0.2819 |
| cosine_precision@5 | 0.1867 |
| cosine_precision@10 | 0.0979 |
| cosine_recall@1 | 0.2702 |
| cosine_recall@3 | 0.8456 |
| cosine_recall@5 | 0.9333 |
| cosine_recall@10 | 0.9789 |
| **cosine_ndcg@10** | **0.6493** |
| cosine_mrr@10 | 0.5396 |
| cosine_map@100 | 0.5411 |
<!--
## Bias, Risks and Limitations
*What are the known or foreseeable issues stemming from this model? You could also flag here known failure cases or weaknesses of the model.*
-->
<!--
### Recommendations
*What are recommendations with respect to the foreseeable issues? For example, filtering explicit content.*
-->
## Training Details
### Training Dataset
#### json
* Dataset: json
* Size: 1,140 training samples
* Columns: <code>anchor</code> and <code>positive</code>
* Approximate statistics based on the first 1000 samples:
| | anchor | positive |
|:--------|:------------------------------------------------------------------------------------|:-----------------------------------------------------------------------------------|
| type | string | string |
| details | <ul><li>min: 5 tokens</li><li>mean: 169.58 tokens</li><li>max: 512 tokens</li></ul> | <ul><li>min: 3 tokens</li><li>mean: 375.1 tokens</li><li>max: 512 tokens</li></ul> |
* Samples:
| anchor | positive |
|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| <code>In the following let $\kappa_{1}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)$ and $\kappa_{2}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)$ be two valid kernels. Show that the following is also valid kernel: $\kappa\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\kappa_{1}\left(\mathbf{x}, \mathbf{x}^{\prime}\right) \kappa_{2}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)$.</code> | <code>To show that the product of two kernels, \(\kappa(\mathbf{x}, \mathbf{x}^{\prime}) = \kappa_1(\mathbf{x}, \mathbf{x}^{\prime}) \kappa_2(\mathbf{x}, \mathbf{x}^{\prime})\), is also a valid kernel, we can use the property of positive semi-definiteness.<br><br>### Step 1: Understanding Kernels<br>A function \(\kappa(\mathbf{x}, \mathbf{x}^{\prime})\) is a kernel if it is symmetric (i.e., \(\kappa(\mathbf{x}, \mathbf{x}^{\prime}) = \kappa(\mathbf{x}^{\prime}, \mathbf{x})\)) and positive semi-definite (PSD). A function is PSD if for any finite set of points \(\{\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n\}\) and any real coefficients \(c_1, c_2, \ldots, c_n\), the following holds:<br>\[<br>\sum_{i=1}^n \sum_{j=1}^n c_i c_j \kappa(\mathbf{x}_i, \mathbf{x}_j} \geq 0.<br>\]<br><br>### Step 2: Properties of \(\kappa_1\) and \(\kappa_2\)<br>Given that \(\kappa_1\) and \(\kappa_2\) are kernels, we know:<br>1. They are symmetric: \(\kappa_1(\mathbf{x}, \mathbf{x}^{\prime}) = \kappa_1(\mathbf{x}^{\prime}, \mathbf{x})\) and...</code> |
| <code>The first annotator rated {a} reviews as positive and the rest as negative. The second annotator rated {b} reviews as positive and the rest as negative. 80 reviews were rated as positive by both annotators. Compute the quality of the above reference using Cohen's Kappa.Give your answer as a numerical value to three decimal places.</code> | <code>To compute Cohen's Kappa, we need to understand the components required for the calculation. Cohen's Kappa (\( \kappa \)) is a statistic that measures inter-rater agreement for categorical items. The formula for Cohen's Kappa is:<br><br>\[<br>\kappa = \frac{P_o - P_e}{1 - P_e}<br>\]<br><br>Where:<br>- \( P_o \) is the observed agreement (the proportion of times both annotators agreed).<br>- \( P_e \) is the expected agreement by chance.<br><br>Let's break down the problem step by step:<br><br>1. **Understanding the Data**:<br> - Let \( N \) be the total number of reviews. <br> - The first annotator rated \( a \) reviews as positive and \( N - a \) as negative.<br> - The second annotator rated \( b \) reviews as positive and \( N - b \) as negative.<br> - We know that both annotators agreed on 80 reviews being positive.<br><br>2. **Observed Agreement \( P_o \)**:<br> - The total number of reviews is \( N \).<br> - The observed agreement \( P_o \) is the proportion of reviews where both annotators agreed on the rating. <br> - Therefore...</code> |
| <code>Consider the (toy) grammar $G$ consisting of the following rules:<br>R1: S --> NP VP<br>R2: NP --> NN<br>R3: NP --> Det NN<br>R4: NN --> N<br>R5: NN --> NN NN<br>R6: NN --> NN PNP<br>R7: PNP --> Prep NP<br>R8: VP --> V<br>R9: VP --> Adv V<br>Precisely define the type of grammar G is corresponding to (for that, consider at least the following aspects: dependency-based vs. constituency-based, position in the Chomsky hierarchy, and CNF).<br>Justify your answer for each of the aspects you will be mentioning.</code> | <code>To analyze the grammar \( G \) you've provided, we need to consider several aspects: whether it is dependency-based or constituency-based, its position in the Chomsky hierarchy, and whether it is in Chomsky Normal Form (CNF).<br><br>### 1. Dependency-based vs. Constituency-based<br><br>**Definition**: <br>- **Dependency-based grammar** focuses on the relationships between words in a sentence, where a word is dependent on another word (its head).<br>- **Constituency-based grammar** organizes sentences into nested structures (constituents) that can be replaced with single units.<br><br>**Analysis**:<br>The grammar \( G \) is **constituency-based** because it uses hierarchical structures (Non-terminal symbols like \( S, NP, VP, NN \)) to define the grammatical relationships. The rules show that phrases (like Noun Phrases and Verb Phrases) are made up of smaller constituents. For instance, \( S \) is defined as consisting of a noun phrase \( NP \) and a verb phrase \( VP \). This clearly indicates a constituency str...</code> |
* Loss: [<code>MatryoshkaLoss</code>](https://sbert.net/docs/package_reference/sentence_transformer/losses.html#matryoshkaloss) with these parameters:
```json
{
"loss": "MultipleNegativesRankingLoss",
"matryoshka_dims": [
768,
512,
256,
128,
64
],
"matryoshka_weights": [
1,
1,
1,
1,
1
],
"n_dims_per_step": -1
}
```
### Training Hyperparameters
#### Non-Default Hyperparameters
- `eval_strategy`: epoch
- `per_device_train_batch_size`: 2
- `per_device_eval_batch_size`: 16
- `gradient_accumulation_steps`: 16
- `learning_rate`: 2e-05
- `num_train_epochs`: 4
- `lr_scheduler_type`: cosine
- `warmup_ratio`: 0.1
- `bf16`: True
- `tf32`: False
- `load_best_model_at_end`: True
- `optim`: adamw_torch_fused
- `batch_sampler`: no_duplicates
#### All Hyperparameters
<details><summary>Click to expand</summary>
- `overwrite_output_dir`: False
- `do_predict`: False
- `eval_strategy`: epoch
- `prediction_loss_only`: True
- `per_device_train_batch_size`: 2
- `per_device_eval_batch_size`: 16
- `per_gpu_train_batch_size`: None
- `per_gpu_eval_batch_size`: None
- `gradient_accumulation_steps`: 16
- `eval_accumulation_steps`: None
- `torch_empty_cache_steps`: None
- `learning_rate`: 2e-05
- `weight_decay`: 0.0
- `adam_beta1`: 0.9
- `adam_beta2`: 0.999
- `adam_epsilon`: 1e-08
- `max_grad_norm`: 1.0
- `num_train_epochs`: 4
- `max_steps`: -1
- `lr_scheduler_type`: cosine
- `lr_scheduler_kwargs`: {}
- `warmup_ratio`: 0.1
- `warmup_steps`: 0
- `log_level`: passive
- `log_level_replica`: warning
- `log_on_each_node`: True
- `logging_nan_inf_filter`: True
- `save_safetensors`: True
- `save_on_each_node`: False
- `save_only_model`: False
- `restore_callback_states_from_checkpoint`: False
- `no_cuda`: False
- `use_cpu`: False
- `use_mps_device`: False
- `seed`: 42
- `data_seed`: None
- `jit_mode_eval`: False
- `use_ipex`: False
- `bf16`: True
- `fp16`: False
- `fp16_opt_level`: O1
- `half_precision_backend`: auto
- `bf16_full_eval`: False
- `fp16_full_eval`: False
- `tf32`: False
- `local_rank`: 0
- `ddp_backend`: None
- `tpu_num_cores`: None
- `tpu_metrics_debug`: False
- `debug`: []
- `dataloader_drop_last`: False
- `dataloader_num_workers`: 0
- `dataloader_prefetch_factor`: None
- `past_index`: -1
- `disable_tqdm`: False
- `remove_unused_columns`: True
- `label_names`: None
- `load_best_model_at_end`: True
- `ignore_data_skip`: False
- `fsdp`: []
- `fsdp_min_num_params`: 0
- `fsdp_config`: {'min_num_params': 0, 'xla': False, 'xla_fsdp_v2': False, 'xla_fsdp_grad_ckpt': False}
- `fsdp_transformer_layer_cls_to_wrap`: None
- `accelerator_config`: {'split_batches': False, 'dispatch_batches': None, 'even_batches': True, 'use_seedable_sampler': True, 'non_blocking': False, 'gradient_accumulation_kwargs': None}
- `deepspeed`: None
- `label_smoothing_factor`: 0.0
- `optim`: adamw_torch_fused
- `optim_args`: None
- `adafactor`: False
- `group_by_length`: False
- `length_column_name`: length
- `ddp_find_unused_parameters`: None
- `ddp_bucket_cap_mb`: None
- `ddp_broadcast_buffers`: False
- `dataloader_pin_memory`: True
- `dataloader_persistent_workers`: False
- `skip_memory_metrics`: True
- `use_legacy_prediction_loop`: False
- `push_to_hub`: False
- `resume_from_checkpoint`: None
- `hub_model_id`: None
- `hub_strategy`: every_save
- `hub_private_repo`: None
- `hub_always_push`: False
- `gradient_checkpointing`: False
- `gradient_checkpointing_kwargs`: None
- `include_inputs_for_metrics`: False
- `include_for_metrics`: []
- `eval_do_concat_batches`: True
- `fp16_backend`: auto
- `push_to_hub_model_id`: None
- `push_to_hub_organization`: None
- `mp_parameters`:
- `auto_find_batch_size`: False
- `full_determinism`: False
- `torchdynamo`: None
- `ray_scope`: last
- `ddp_timeout`: 1800
- `torch_compile`: False
- `torch_compile_backend`: None
- `torch_compile_mode`: None
- `include_tokens_per_second`: False
- `include_num_input_tokens_seen`: False
- `neftune_noise_alpha`: None
- `optim_target_modules`: None
- `batch_eval_metrics`: False
- `eval_on_start`: False
- `use_liger_kernel`: False
- `eval_use_gather_object`: False
- `average_tokens_across_devices`: False
- `prompts`: None
- `batch_sampler`: no_duplicates
- `multi_dataset_batch_sampler`: proportional
</details>
### Training Logs
| Epoch | Step | Training Loss | dim_768_cosine_ndcg@10 | dim_512_cosine_ndcg@10 | dim_256_cosine_ndcg@10 | dim_128_cosine_ndcg@10 | dim_64_cosine_ndcg@10 |
|:-------:|:-------:|:-------------:|:----------------------:|:----------------------:|:----------------------:|:----------------------:|:---------------------:|
| 0.2807 | 10 | 0.1249 | - | - | - | - | - |
| 0.5614 | 20 | 0.8091 | - | - | - | - | - |
| 0.8421 | 30 | 0.0235 | - | - | - | - | - |
| 1.0 | 36 | - | 0.6463 | 0.6516 | 0.6468 | 0.6425 | 0.6363 |
| 1.1123 | 40 | 0.0293 | - | - | - | - | - |
| 1.3930 | 50 | 0.0474 | - | - | - | - | - |
| 1.6737 | 60 | 0.0062 | - | - | - | - | - |
| 1.9544 | 70 | 0.0022 | - | - | - | - | - |
| 2.0 | 72 | - | 0.6535 | 0.6501 | 0.6488 | 0.6488 | 0.6454 |
| 2.2246 | 80 | 0.9144 | - | - | - | - | - |
| 2.5053 | 90 | 0.0139 | - | - | - | - | - |
| 2.7860 | 100 | 0.0019 | - | - | - | - | - |
| 3.0 | 108 | - | 0.6533 | 0.6582 | 0.6523 | 0.6651 | 0.6499 |
| 3.0561 | 110 | 0.3805 | - | - | - | - | - |
| 3.3368 | 120 | 0.0075 | - | - | - | - | - |
| 3.6175 | 130 | 0.0035 | - | - | - | - | - |
| 3.8982 | 140 | 0.0012 | - | - | - | - | - |
| **4.0** | **144** | **-** | **0.6591** | **0.659** | **0.6595** | **0.6667** | **0.6493** |
* The bold row denotes the saved checkpoint.
### Framework Versions
- Python: 3.12.8
- Sentence Transformers: 4.1.0
- Transformers: 4.52.4
- PyTorch: 2.7.0+cu126
- Accelerate: 1.3.0
- Datasets: 3.6.0
- Tokenizers: 0.21.0
## Citation
### BibTeX
#### Sentence Transformers
```bibtex
@inproceedings{reimers-2019-sentence-bert,
title = "Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks",
author = "Reimers, Nils and Gurevych, Iryna",
booktitle = "Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing",
month = "11",
year = "2019",
publisher = "Association for Computational Linguistics",
url = "https://arxiv.org/abs/1908.10084",
}
```
#### MatryoshkaLoss
```bibtex
@misc{kusupati2024matryoshka,
title={Matryoshka Representation Learning},
author={Aditya Kusupati and Gantavya Bhatt and Aniket Rege and Matthew Wallingford and Aditya Sinha and Vivek Ramanujan and William Howard-Snyder and Kaifeng Chen and Sham Kakade and Prateek Jain and Ali Farhadi},
year={2024},
eprint={2205.13147},
archivePrefix={arXiv},
primaryClass={cs.LG}
}
```
#### MultipleNegativesRankingLoss
```bibtex
@misc{henderson2017efficient,
title={Efficient Natural Language Response Suggestion for Smart Reply},
author={Matthew Henderson and Rami Al-Rfou and Brian Strope and Yun-hsuan Sung and Laszlo Lukacs and Ruiqi Guo and Sanjiv Kumar and Balint Miklos and Ray Kurzweil},
year={2017},
eprint={1705.00652},
archivePrefix={arXiv},
primaryClass={cs.CL}
}
```
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