| --- |
| language: |
| - en |
| license: apache-2.0 |
| tags: |
| - sentence-transformers |
| - sentence-similarity |
| - feature-extraction |
| - generated_from_trainer |
| - dataset_size:79876 |
| - loss:MultipleNegativesRankingLoss |
| base_model: Master-thesis-NAP/ModernBert-DAPT-math |
| widget: |
| - source_sentence: What is the error estimate for the difference between the exact |
| solution and the local oscillation decomposition (LOD) solution in terms of the |
| $L_0$ norm? |
| sentences: |
| - '\label{thm1} |
| |
| Suppose $\kappa$ and $\bar a$ are as above. Then $|\Pcut(\bar a)| \leq 2^\kappa$. |
| Indeed if |
| |
| $2^\kappa=\aleph_\alpha,$ then $|\Pcut(\bar a)| \leq |\alpha+1|^2$.' |
| - "\\cite{kyushu}\n For every discrete group $\\G$ and every 2-dimensional representation\ |
| \ $\\varrho$ of $\\G$, $\\varrho-$equivariant functions for $\\G$ always exist." |
| - "\\label{Corollary}\n Let Assumptions~\\ref{assum_1} and~\\ref{assump2} be\ |
| \ satisfied. Let $u$ be the solution of~\\eqref{WeakForm} and let $u_{H,k}$ be\ |
| \ the LOD solution of~\\eqref{local_probelm }. Then we have \n \\begin{equation}\\\ |
| label{L2Estimate}\n \\|u-I_Hu_{H,k}\\|_0\\lesssim \\|u-I_Hu\\|_0+\\|u-u_{H,k}\\\ |
| |_0 +H|u-u_{H,k}|_1.\n \\end{equation}\n %\\[\\|u-I_Hu_{H,k}\\|_0\\lesssim\ |
| \ H |u|_1 +|u-u_{H,k}|_1.\\]" |
| - source_sentence: Does the theorem imply that the rate of convergence of the sequence |
| $T_{m,j}(E)$ to $T_{m+k_n,j+k_n}(E)$ is exponential in the distance between $m$ |
| and $j$, and that this rate is bounded by a constant $C$ times an exponential |
| decay factor involving the parameter $\gamma$? |
| sentences: |
| - "\\label{thm:weibull}\nSuppose random variable $X$ follows Weibull distribution,\ |
| \ and $E(X^i)$ denotes the $i$-th moment of $X$. Then the random variable $X$\ |
| \ satisfy the following inequality: \n\\begin{equation}\\label{eq:moments}\n \ |
| \ E(X^n)^{\\frac{1}{n}} \\geq E(X^m)^{\\frac{1}{m}},\n\\end{equation}\nwhere\ |
| \ $n > m$." |
| - "\\label{lem1}\n\t\tFor all $m,j\\in\\Z$, we have\n\t\t\\begin{equation*}\n\t\ |
| \t|| T_{m,j} (E)-T_{m+k_n,j+k_n}(E)||\\leq C e^{-\\gamma k_n} e^{(\\mathcal\ |
| \ L(E)+\\varepsilon) |m-j|}. \n\t\t\\end{equation*}" |
| - If the problem \eqref{eq:Model-based_Program} is convex, then under the primal-dual |
| dynamics \eqref{eq:PDD}-\eqref{eq:AlgebraicConstruction}, the system \eqref{eq:Input-OutputMap} |
| asymptotically converges to a steady state that is the optimal solution of \eqref{eq:Model-based_Program}. |
| - source_sentence: What is the rate of convergence for the total error in the given |
| problem, assuming the conditions in Theorem~\ref{convergence-rates} are met? |
| sentences: |
| - "\\label{convergence-rates}\nUnder the assumptions of Theorem~\\ref{well-posedness}.\ |
| \ Given $(\\bu,{p},\\bzeta,\\varphi)\\in (\\bH^{s_1+1}(\\Omega)\\cap \\bV_1)\\\ |
| times (\\text{H}^{s_1}(\\Omega)\\cap Q_{b_1}) \\times (\\bH^{s_2}\\cap \\bV_2)\ |
| \ \\times (\\text{H}^{s_2}\\cap Q_{b_2})$, $(\\bu_h,{p}_h,\\bzeta_h,\\varphi_h)\\\ |
| in \\bV_1^{h,k_1}\\times Q_1^{h,k_1}\\times \\bV_2^{h,k_2}\\times Q_2^{h,k_2}$\ |
| \ be the respective solutions of the continuous and discrete problems, with the\ |
| \ data satisfying $\\fb\\in \\bH^{s_1-1}\\cap \\bQ_{b_1}$ and $g\\in H^{s_2}(\\\ |
| Omega)\\cap Q_{b_2}$. If $\\overline{C}_1 \\sqrt{M} L_\\ell + \\overline{C}_2^2\ |
| \ \\sqrt{M^3} L_\\bbM\\sqrt{2\\mu} (\\norm{\\varphi_D}_{1/2,\\Gamma_D} + \\\ |
| norm{g}_{0,\\Omega}) < 1/2.$ Then, the total error $\\overline{\\textnormal{e}}_h:=\\\ |
| norm{(\\bu-\\bu_h,{p}-{p}_h, \\bzeta-\\bzeta_h,\\varphi-\\varphi_h)}_{\\bV_1\\\ |
| times Q_{1} \\times \\bV_2\\times Q_2}$ decays with the following rate for $s:=\ |
| \ \\min \\left\\{s_1,s_2\\right\\}$\n \\begin{align*}\\label{convergence-rate}\n\ |
| \ \\overline{\\textnormal{e}}_h &\\lesssim h^{ s} (|\\fb|_{s_1-1,\\bQ_{b_1}}\ |
| \ + |\\bu|_{s_1+1,\\bV_1} + |{p}|_{s_1,Q_{b_1}} + |g|_{s_2,Q_{b_2}} + |\\bzeta|_{s_2,\\\ |
| bV_2}+|\\varphi|_{s_2,Q_{b_2}}).\n \\end{align*}" |
| - "\\label{thm}\nFor vector linear secure aggregation defined above, the optimal\ |
| \ total key rate is \n\\begin{eqnarray}\n R_{Z_{\\Sigma}}^* %= \\left\\{R_{Z_{\\\ |
| Sigma}}: R_{Z_{\\Sigma}} \\geq \n = \\mbox{rank} \\left( \\left[ \\mathbf{F}\ |
| \ ; \\mathbf{G} \\right] \\right)\n - \\mbox{rank} \\left( \\mathbf{F} \\\ |
| right) = \\mbox{rank}({\\bf G} | {\\bf F}).\n %\\right\\}.\n% \\\\ \\\ |
| mbox{rank}\n\\end{eqnarray}" |
| - "The process $Y(t)$, $t\\geq 0,$ is called Markov branching process with\r\nnon-homogeneous\ |
| \ Poisson immigration (MBPNPI)." |
| - source_sentence: Is the local time of the horizontal component of the Peano curve |
| ever greater than 1? |
| sentences: |
| - "[Divergence Theorem or Gauss-Green Theorem for Surfaces in $\\R^3$]\n\t\\label{thm:surface_int}\n\ |
| \t Let $\\Sigma \\subset \\Omega\\subseteq\\R^3$ be a bounded smooth surface.\n\ |
| \t Further, $\\bb a:\\Sigma\\to\\R^3$ is a continuously differentiable\ |
| \ vector field that is either defined on the\n\t\t\t\t\tboundary $\\partial\\\ |
| Sigma$ or has a bounded continuous extension to this boundary.\n\t Like\ |
| \ in \\eqref{eq:decomp} it may be decomposed into tangential and normal components\n\ |
| \t\t\t\t\tas follows $\\bb a = \\bb a^\\shortparallel + a_\\nu\\bs\\nu_\\Sigma$.\ |
| \ By $\\dd l$ we denote the line element on \n\t\t\t\t\tthe curve $\\partial \\\ |
| Sigma$. We assume that the curve is continuous and consists of finitely many\n\ |
| \t\t\t\t\tsmooth pieces.\n\t Then the following divergence formula for\ |
| \ surface integrals holds\n\t %\n\t \\begin{align}\n\t \ |
| \ %\n\t \\int\\limits_\\Sigma \\left[\\nabla_\\Sigma\\cdot\\bb a^\\\ |
| shortparallel\\right](\\x)\\;\\dd S\n\t\t\t\t\t\t\t= \\int\\limits_{\\partial\\\ |
| Sigma} \\left[\\bb a\\cdot\\bs\\nu_{\\partial\\Sigma}\\right](\\x)\\,\\dd l .\n\ |
| \t \\label{eq:surface_div}\n\t %\n\t \\end{align}\n\ |
| \t\t\t\t\t%\n\t\t\t\t\tFrom this we obtain the formula\n\t\t\t\t\t%\n\t \ |
| \ \\begin{align}\n\t %\n\t \\int\\limits_\\Sigma \\left[\\\ |
| nabla_\\Sigma\\cdot\\bb a\\right](\\x)\\;\\dd S\n\t\t\t\t\t\t\t= \\int\\limits_{\\\ |
| partial\\Sigma} \\left[\\bb a\\cdot\\bs\\nu_{\\partial\\Sigma}\\right](\\x)\\\ |
| ,\\dd l \n\t\t\t\t\t\t\t-\\int\\limits_\\Sigma\\left[ 2\\kappa_Ma_\\nu\\right](\\\ |
| x)\\;\\dd S.\n\t \\label{eq:surface_div_2}\n\t %\n\t \ |
| \ \\end{align}\n\t %" |
| - There exists local time of the horizontal component $x$ of the Peano curve. Moreover, |
| this local time attains values no greater than $1$. |
| - "[Werner-Young's inequality]\\label{Young op-op}\nSuppose $S\\in \\cS^p$ and $T\\\ |
| in \\cS^q$ with $1+r^{-1}=p^{-1}+q^{-1}$.\nThen $S\\star T\\in L^r(\\R^{2d})$\ |
| \ and\n\\begin{align*}\n \\|S\\star T\\|_{L^{r}}\\leq \\|S\\|_{\\cS^p}\\|T\\\ |
| |_{\\cS^q}.\n\\end{align*}" |
| - source_sentence: What is the meaning of the identity containment $1_x:x\to x$ in |
| the context of the bond system? |
| sentences: |
| - "\\label{lem:opt_lin}\nConsider the optimization problem\n\\begin{equation}\\\ |
| label{eq:max_tr_lem}\n\\begin{aligned}\n \\max_{\\bs{U}}&\\;\\; \\Re\\{\\mrm{tr}(\\\ |
| bs{U}^\\mrm{H}\\bs{B}) \\}\\\\\n \\mrm{s.t. \\;\\;}& \\bs{U}\\in \\mathcal{U}(N),\n\ |
| \\end{aligned}\n\\end{equation}\nwhere $\\bs{B}$ may be an arbitrary $N\\times\ |
| \ N$ matrix with singular value decomposition (SVD) $\\bs{B}=\\bs{U}_{\\bs{B}}\\\ |
| bs{S}_{\\bs{B}}\\bs{V}_{\\bs{B}}^\\mrm{H}$. The solution to \\eqref{eq:max_tr_lem}\ |
| \ is given by\n\\begin{equation}\\label{eq:sol_max}\n \\bs{U}_\\mrm{opt} =\ |
| \ \\bs{U}_{\\bs{B}}^\\mrm{H}\\bs{V}_{\\bs{B}}.\n\\end{equation}\n\\begin{skproof}\n\ |
| \ A formal proof, which may be included in the extended version, can be obtained\ |
| \ by defining the Riemannian gradient over the unitary group and finding the stationary\ |
| \ point where it vanishes. However, an intuitive argument is that the solution\ |
| \ to \\eqref{eq:max_tr_lem} is obtained by positively combining the singular values\ |
| \ of $\\bs{B}$, leading to \\eqref{eq:sol_max}.\n\\end{skproof}" |
| - '\label{AM_BA_lem1} |
| |
| Let $$\Omega =\left\{a={{\left(k_1x_1+k_2,\dots,k_1x_n+k_2\right)}}\mid k_1, k_2\in |
| \mathbb{R}\right\} .$$ Then ${\displaystyle\underset{a\in \Omega}{\operatorname{argmin}} |
| {J_{\alpha }}(a)=\overline{a}\ },$ where $\overline{a}=\left(\overline{a}_1,\dots,\overline{a}_n\right)$, |
| $$\overline{a}_i=\frac{1}{n}\sum^n_{j =1}{y_j},\quad\forall i=1,\dots,n.$$ In |
| other words, on the class of lines $J_{\alpha }\left(a\right)$ reaches a minimum |
| on a straight line parallel to the $Ox$ axis. So, this is the average line for |
| the ordinates of all points of set $X$.' |
| - "A \\emph{bond system} is a tuple $(B,C,s,t,1,\\cdot)$, where $B$ is a set of\ |
| \ \\emph{bonds}, $C$ is a set of \\emph{content} relations, and $s,t:C\\to B$\ |
| \ are \\emph{source} and \\emph{target} functions. For $c\\in C$ with $s(c)=x$\ |
| \ and $t(c)=y$, we write $x\\xrightarrow{c}y$ or $c:x\\to y$, indicating that\ |
| \ $x$ \\emph{contains} $y$. Each bond $x\\in B$ has an \\emph{identity} containment\ |
| \ $1_x:x\\to x$, meaning every bond trivially contains itself. For $c:x\\to y$\ |
| \ and $c':y\\to z$, their composition is $cc':x\\to z$. These data must satisfy:\n\ |
| \ \\begin{enumerate}\n \\item Identity laws: For each $c:x\\to y$, $1_x\ |
| \ c= c=c1_y$\n \\item Associativity: For $c:x\\to y$, $c':y\\to z$, $c'':z\\\ |
| to w$, $c(c'c'')=(cc')c''$\n \\item Anti-symmetry: For $c:x\\to y$ and\ |
| \ $c':y\\to x$, $x=y$\n \\item Left cancellation: For $c,c':x\\to y$ and\ |
| \ $c'':y\\to z$, if $cc''=c'c''$, then $c=c'$\n \\end{enumerate}" |
| 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: ModernBERT DAPT Embed DAPT Math |
| results: |
| - task: |
| type: information-retrieval |
| name: Information Retrieval |
| dataset: |
| name: TESTING |
| type: TESTING |
| metrics: |
| - type: cosine_accuracy@1 |
| value: 0.868020304568528 |
| name: Cosine Accuracy@1 |
| - type: cosine_accuracy@3 |
| value: 0.9183202584217812 |
| name: Cosine Accuracy@3 |
| - type: cosine_accuracy@5 |
| value: 0.9325103830179973 |
| name: Cosine Accuracy@5 |
| - type: cosine_accuracy@10 |
| value: 0.9495846792801107 |
| name: Cosine Accuracy@10 |
| - type: cosine_precision@1 |
| value: 0.868020304568528 |
| name: Cosine Precision@1 |
| - type: cosine_precision@3 |
| value: 0.6118674050146131 |
| name: Cosine Precision@3 |
| - type: cosine_precision@5 |
| value: 0.49353945546838945 |
| name: Cosine Precision@5 |
| - type: cosine_precision@10 |
| value: 0.34758883248730965 |
| name: Cosine Precision@10 |
| - type: cosine_recall@1 |
| value: 0.04186710795480722 |
| name: Cosine Recall@1 |
| - type: cosine_recall@3 |
| value: 0.08315252408701693 |
| name: Cosine Recall@3 |
| - type: cosine_recall@5 |
| value: 0.1073909448198794 |
| name: Cosine Recall@5 |
| - type: cosine_recall@10 |
| value: 0.14207392775097807 |
| name: Cosine Recall@10 |
| - type: cosine_ndcg@10 |
| value: 0.4493273991613623 |
| name: Cosine Ndcg@10 |
| - type: cosine_mrr@10 |
| value: 0.8963655316764447 |
| name: Cosine Mrr@10 |
| - type: cosine_map@100 |
| value: 0.16376932233660765 |
| name: Cosine Map@100 |
| --- |
| |
| # ModernBERT DAPT Embed DAPT Math |
|
|
| This is a [sentence-transformers](https://www.SBERT.net) model finetuned from [Master-thesis-NAP/ModernBert-DAPT-math](https://huggingface.co/Master-thesis-NAP/ModernBert-DAPT-math). It maps sentences & paragraphs to a 768-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:** [Master-thesis-NAP/ModernBert-DAPT-math](https://huggingface.co/Master-thesis-NAP/ModernBert-DAPT-math) <!-- at revision a30384f91d764c272e6b740c256d5581325ea4bb --> |
| - **Maximum Sequence Length:** 8192 tokens |
| - **Output Dimensionality:** 768 dimensions |
| - **Similarity Function:** Cosine Similarity |
| <!-- - **Training Dataset:** Unknown --> |
| - **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': 8192, 'do_lower_case': False}) with Transformer model: ModernBertModel |
| (1): Pooling({'word_embedding_dimension': 768, '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): Normalize() |
| ) |
| ``` |
|
|
| ## 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("Master-thesis-NAP/ModernBERT-DAPT-Embed-DAPT-Math-v2") |
| # Run inference |
| sentences = [ |
| 'What is the meaning of the identity containment $1_x:x\\to x$ in the context of the bond system?', |
| "A \\emph{bond system} is a tuple $(B,C,s,t,1,\\cdot)$, where $B$ is a set of \\emph{bonds}, $C$ is a set of \\emph{content} relations, and $s,t:C\\to B$ are \\emph{source} and \\emph{target} functions. For $c\\in C$ with $s(c)=x$ and $t(c)=y$, we write $x\\xrightarrow{c}y$ or $c:x\\to y$, indicating that $x$ \\emph{contains} $y$. Each bond $x\\in B$ has an \\emph{identity} containment $1_x:x\\to x$, meaning every bond trivially contains itself. For $c:x\\to y$ and $c':y\\to z$, their composition is $cc':x\\to z$. These data must satisfy:\n \\begin{enumerate}\n \\item Identity laws: For each $c:x\\to y$, $1_x c= c=c1_y$\n \\item Associativity: For $c:x\\to y$, $c':y\\to z$, $c'':z\\to w$, $c(c'c'')=(cc')c''$\n \\item Anti-symmetry: For $c:x\\to y$ and $c':y\\to x$, $x=y$\n \\item Left cancellation: For $c,c':x\\to y$ and $c'':y\\to z$, if $cc''=c'c''$, then $c=c'$\n \\end{enumerate}", |
| '\\label{lem:opt_lin}\nConsider the optimization problem\n\\begin{equation}\\label{eq:max_tr_lem}\n\\begin{aligned}\n \\max_{\\bs{U}}&\\;\\; \\Re\\{\\mrm{tr}(\\bs{U}^\\mrm{H}\\bs{B}) \\}\\\\\n \\mrm{s.t. \\;\\;}& \\bs{U}\\in \\mathcal{U}(N),\n\\end{aligned}\n\\end{equation}\nwhere $\\bs{B}$ may be an arbitrary $N\\times N$ matrix with singular value decomposition (SVD) $\\bs{B}=\\bs{U}_{\\bs{B}}\\bs{S}_{\\bs{B}}\\bs{V}_{\\bs{B}}^\\mrm{H}$. The solution to \\eqref{eq:max_tr_lem} is given by\n\\begin{equation}\\label{eq:sol_max}\n \\bs{U}_\\mrm{opt} = \\bs{U}_{\\bs{B}}^\\mrm{H}\\bs{V}_{\\bs{B}}.\n\\end{equation}\n\\begin{skproof}\n A formal proof, which may be included in the extended version, can be obtained by defining the Riemannian gradient over the unitary group and finding the stationary point where it vanishes. However, an intuitive argument is that the solution to \\eqref{eq:max_tr_lem} is obtained by positively combining the singular values of $\\bs{B}$, leading to \\eqref{eq:sol_max}.\n\\end{skproof}', |
| ] |
| embeddings = model.encode(sentences) |
| print(embeddings.shape) |
| # [3, 768] |
| |
| # 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: `TESTING` |
| * Evaluated with [<code>InformationRetrievalEvaluator</code>](https://sbert.net/docs/package_reference/sentence_transformer/evaluation.html#sentence_transformers.evaluation.InformationRetrievalEvaluator) |
|
|
| | Metric | Value | |
| |:--------------------|:-----------| |
| | cosine_accuracy@1 | 0.868 | |
| | cosine_accuracy@3 | 0.9183 | |
| | cosine_accuracy@5 | 0.9325 | |
| | cosine_accuracy@10 | 0.9496 | |
| | cosine_precision@1 | 0.868 | |
| | cosine_precision@3 | 0.6119 | |
| | cosine_precision@5 | 0.4935 | |
| | cosine_precision@10 | 0.3476 | |
| | cosine_recall@1 | 0.0419 | |
| | cosine_recall@3 | 0.0832 | |
| | cosine_recall@5 | 0.1074 | |
| | cosine_recall@10 | 0.1421 | |
| | **cosine_ndcg@10** | **0.4493** | |
| | cosine_mrr@10 | 0.8964 | |
| | cosine_map@100 | 0.1638 | |
| |
| <!-- |
| ## 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 |
| |
| #### Unnamed Dataset |
| |
| * Size: 79,876 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: 9 tokens</li><li>mean: 38.48 tokens</li><li>max: 142 tokens</li></ul> | <ul><li>min: 5 tokens</li><li>mean: 210.43 tokens</li><li>max: 924 tokens</li></ul> | |
| * Samples: |
| | anchor | positive | |
| |:---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|:--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |
| | <code>What is the limit of the proportion of 1's in the sequence $a_n$ as $n$ approaches infinity, given that $0 \leq 3g_n -2n \leq 4$?</code> | <code>Let $g_n$ be the number of $1$'s in the sequence $a_1 a_2 \cdots a_n$.<br>Then <br>\begin{equation}<br>0 \leq 3g_n -2n \leq 4<br>\label{star}<br>\end{equation}<br>for all $n$, and hence<br>$\lim_{n \rightarrow \infty} g_n/n = 2/3$.<br>\label{thm1}</code> | |
| | <code>Does the statement of \textbf{ThmConjAreTrue} imply that the maximum genus of a locally Cohen-Macaulay curve in $\mathbb{P}^3_{\mathbb{C}}$ of degree $d$ that does not lie on a surface of degree $s-1$ is always equal to $g(d,s)$?</code> | <code>\label{ThmConjAreTrue}<br>Conjectures \ref{Conj1} and \ref{Conj2} are true.<br>As a consequence, <br>if either $d=s \geq 1$ or $d \geq 2s+1 \geq 3$, <br>the maximum genus of a locally Cohen-Macaulay curve in $\mathbb{P}^3_{\mathbb{C}}$ of degree $d$ that does not lie on a surface of degree $s-1$ is equal to $g(d,s)$.</code> | |
| | <code>\\emph{Is the statement \emph{If $X$ is a compact Hausdorff space, then $X$ is normal}, proven in the first isomorphism theorem for topological groups, or is it a well-known result in topology?}</code> | <code>}<br>\newcommand{\ep}{</code> | |
| * Loss: [<code>MultipleNegativesRankingLoss</code>](https://sbert.net/docs/package_reference/sentence_transformer/losses.html#multiplenegativesrankingloss) with these parameters: |
| ```json |
| { |
| "scale": 20.0, |
| "similarity_fct": "cos_sim" |
| } |
| ``` |
| |
| ### Training Hyperparameters |
| #### Non-Default Hyperparameters |
| |
| - `eval_strategy`: epoch |
| - `per_device_train_batch_size`: 16 |
| - `per_device_eval_batch_size`: 16 |
| - `gradient_accumulation_steps`: 8 |
| - `learning_rate`: 2e-05 |
| - `num_train_epochs`: 4 |
| - `lr_scheduler_type`: cosine |
| - `warmup_ratio`: 0.1 |
| - `bf16`: True |
| - `tf32`: True |
| - `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`: 16 |
| - `per_device_eval_batch_size`: 16 |
| - `per_gpu_train_batch_size`: None |
| - `per_gpu_eval_batch_size`: None |
| - `gradient_accumulation_steps`: 8 |
| - `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`: True |
| - `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} |
| - `tp_size`: 0 |
| - `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 |
| <details><summary>Click to expand</summary> |
| |
| | Epoch | Step | Training Loss | TESTING_cosine_ndcg@10 | |
| |:---------:|:--------:|:-------------:|:----------------------:| |
| | 0.0160 | 10 | 20.2777 | - | |
| | 0.0320 | 20 | 19.6613 | - | |
| | 0.0481 | 30 | 18.8588 | - | |
| | 0.0641 | 40 | 17.5525 | - | |
| | 0.0801 | 50 | 15.1065 | - | |
| | 0.0961 | 60 | 10.8128 | - | |
| | 0.1122 | 70 | 7.0698 | - | |
| | 0.1282 | 80 | 4.532 | - | |
| | 0.1442 | 90 | 3.5143 | - | |
| | 0.1602 | 100 | 2.3256 | - | |
| | 0.1762 | 110 | 1.4688 | - | |
| | 0.1923 | 120 | 1.0081 | - | |
| | 0.2083 | 130 | 0.949 | - | |
| | 0.2243 | 140 | 0.9709 | - | |
| | 0.2403 | 150 | 0.8403 | - | |
| | 0.2564 | 160 | 0.8749 | - | |
| | 0.2724 | 170 | 0.7955 | - | |
| | 0.2884 | 180 | 0.6587 | - | |
| | 0.3044 | 190 | 0.5832 | - | |
| | 0.3204 | 200 | 0.5376 | - | |
| | 0.3365 | 210 | 0.608 | - | |
| | 0.3525 | 220 | 0.4639 | - | |
| | 0.3685 | 230 | 0.6611 | - | |
| | 0.3845 | 240 | 0.5589 | - | |
| | 0.4006 | 250 | 0.5845 | - | |
| | 0.4166 | 260 | 0.4392 | - | |
| | 0.4326 | 270 | 0.4746 | - | |
| | 0.4486 | 280 | 0.4517 | - | |
| | 0.4647 | 290 | 0.4034 | - | |
| | 0.4807 | 300 | 0.4437 | - | |
| | 0.4967 | 310 | 0.4339 | - | |
| | 0.5127 | 320 | 0.4445 | - | |
| | 0.5287 | 330 | 0.3793 | - | |
| | 0.5448 | 340 | 0.3591 | - | |
| | 0.5608 | 350 | 0.4694 | - | |
| | 0.5768 | 360 | 0.4668 | - | |
| | 0.5928 | 370 | 0.4121 | - | |
| | 0.6089 | 380 | 0.4688 | - | |
| | 0.6249 | 390 | 0.387 | - | |
| | 0.6409 | 400 | 0.3748 | - | |
| | 0.6569 | 410 | 0.2997 | - | |
| | 0.6729 | 420 | 0.3756 | - | |
| | 0.6890 | 430 | 0.2993 | - | |
| | 0.7050 | 440 | 0.3514 | - | |
| | 0.7210 | 450 | 0.3646 | - | |
| | 0.7370 | 460 | 0.308 | - | |
| | 0.7531 | 470 | 0.3612 | - | |
| | 0.7691 | 480 | 0.2845 | - | |
| | 0.7851 | 490 | 0.2792 | - | |
| | 0.8011 | 500 | 0.2204 | - | |
| | 0.8171 | 510 | 0.2757 | - | |
| | 0.8332 | 520 | 0.2674 | - | |
| | 0.8492 | 530 | 0.3753 | - | |
| | 0.8652 | 540 | 0.3546 | - | |
| | 0.8812 | 550 | 0.3166 | - | |
| | 0.8973 | 560 | 0.2656 | - | |
| | 0.9133 | 570 | 0.3215 | - | |
| | 0.9293 | 580 | 0.2559 | - | |
| | 0.9453 | 590 | 0.4629 | - | |
| | 0.9613 | 600 | 0.31 | - | |
| | 0.9774 | 610 | 0.3601 | - | |
| | 0.9934 | 620 | 0.2391 | - | |
| | 1.0 | 625 | - | 0.4229 | |
| | 1.0080 | 630 | 0.2507 | - | |
| | 1.0240 | 640 | 0.1852 | - | |
| | 1.0401 | 650 | 0.1836 | - | |
| | 1.0561 | 660 | 0.1487 | - | |
| | 1.0721 | 670 | 0.1495 | - | |
| | 1.0881 | 680 | 0.1567 | - | |
| | 1.1041 | 690 | 0.1497 | - | |
| | 1.1202 | 700 | 0.1632 | - | |
| | 1.1362 | 710 | 0.1997 | - | |
| | 1.1522 | 720 | 0.182 | - | |
| | 1.1682 | 730 | 0.1884 | - | |
| | 1.1843 | 740 | 0.1766 | - | |
| | 1.2003 | 750 | 0.1477 | - | |
| | 1.2163 | 760 | 0.181 | - | |
| | 1.2323 | 770 | 0.092 | - | |
| | 1.2483 | 780 | 0.1506 | - | |
| | 1.2644 | 790 | 0.1305 | - | |
| | 1.2804 | 800 | 0.1533 | - | |
| | 1.2964 | 810 | 0.2306 | - | |
| | 1.3124 | 820 | 0.1861 | - | |
| | 1.3285 | 830 | 0.1157 | - | |
| | 1.3445 | 840 | 0.1054 | - | |
| | 1.3605 | 850 | 0.1696 | - | |
| | 1.3765 | 860 | 0.1327 | - | |
| | 1.3925 | 870 | 0.1485 | - | |
| | 1.4086 | 880 | 0.1395 | - | |
| | 1.4246 | 890 | 0.1021 | - | |
| | 1.4406 | 900 | 0.1283 | - | |
| | 1.4566 | 910 | 0.102 | - | |
| | 1.4727 | 920 | 0.1825 | - | |
| | 1.4887 | 930 | 0.1395 | - | |
| | 1.5047 | 940 | 0.157 | - | |
| | 1.5207 | 950 | 0.1444 | - | |
| | 1.5368 | 960 | 0.1317 | - | |
| | 1.5528 | 970 | 0.146 | - | |
| | 1.5688 | 980 | 0.1809 | - | |
| | 1.5848 | 990 | 0.1368 | - | |
| | 1.6008 | 1000 | 0.2036 | - | |
| | 1.6169 | 1010 | 0.1292 | - | |
| | 1.6329 | 1020 | 0.1306 | - | |
| | 1.6489 | 1030 | 0.1473 | - | |
| | 1.6649 | 1040 | 0.1595 | - | |
| | 1.6810 | 1050 | 0.1471 | - | |
| | 1.6970 | 1060 | 0.1869 | - | |
| | 1.7130 | 1070 | 0.1445 | - | |
| | 1.7290 | 1080 | 0.157 | - | |
| | 1.7450 | 1090 | 0.1382 | - | |
| | 1.7611 | 1100 | 0.157 | - | |
| | 1.7771 | 1110 | 0.1073 | - | |
| | 1.7931 | 1120 | 0.0864 | - | |
| | 1.8091 | 1130 | 0.1312 | - | |
| | 1.8252 | 1140 | 0.1644 | - | |
| | 1.8412 | 1150 | 0.1366 | - | |
| | 1.8572 | 1160 | 0.1257 | - | |
| | 1.8732 | 1170 | 0.127 | - | |
| | 1.8892 | 1180 | 0.1494 | - | |
| | 1.9053 | 1190 | 0.1516 | - | |
| | 1.9213 | 1200 | 0.1709 | - | |
| | 1.9373 | 1210 | 0.1717 | - | |
| | 1.9533 | 1220 | 0.1044 | - | |
| | 1.9694 | 1230 | 0.1551 | - | |
| | 1.9854 | 1240 | 0.1303 | - | |
| | 2.0 | 1250 | 0.1081 | 0.4392 | |
| | 2.0160 | 1260 | 0.0572 | - | |
| | 2.0320 | 1270 | 0.0504 | - | |
| | 2.0481 | 1280 | 0.0535 | - | |
| | 2.0641 | 1290 | 0.0512 | - | |
| | 2.0801 | 1300 | 0.0539 | - | |
| | 2.0961 | 1310 | 0.0462 | - | |
| | 2.1122 | 1320 | 0.0611 | - | |
| | 2.1282 | 1330 | 0.0989 | - | |
| | 2.1442 | 1340 | 0.0462 | - | |
| | 2.1602 | 1350 | 0.061 | - | |
| | 2.1762 | 1360 | 0.0557 | - | |
| | 2.1923 | 1370 | 0.0622 | - | |
| | 2.2083 | 1380 | 0.0744 | - | |
| | 2.2243 | 1390 | 0.0531 | - | |
| | 2.2403 | 1400 | 0.0507 | - | |
| | 2.2564 | 1410 | 0.0533 | - | |
| | 2.2724 | 1420 | 0.0676 | - | |
| | 2.2884 | 1430 | 0.0706 | - | |
| | 2.3044 | 1440 | 0.0452 | - | |
| | 2.3204 | 1450 | 0.0415 | - | |
| | 2.3365 | 1460 | 0.0562 | - | |
| | 2.3525 | 1470 | 0.0487 | - | |
| | 2.3685 | 1480 | 0.0614 | - | |
| | 2.3845 | 1490 | 0.045 | - | |
| | 2.4006 | 1500 | 0.0529 | - | |
| | 2.4166 | 1510 | 0.048 | - | |
| | 2.4326 | 1520 | 0.059 | - | |
| | 2.4486 | 1530 | 0.0593 | - | |
| | 2.4647 | 1540 | 0.0631 | - | |
| | 2.4807 | 1550 | 0.0506 | - | |
| | 2.4967 | 1560 | 0.058 | - | |
| | 2.5127 | 1570 | 0.0896 | - | |
| | 2.5287 | 1580 | 0.0522 | - | |
| | 2.5448 | 1590 | 0.035 | - | |
| | 2.5608 | 1600 | 0.0677 | - | |
| | 2.5768 | 1610 | 0.0538 | - | |
| | 2.5928 | 1620 | 0.0485 | - | |
| | 2.6089 | 1630 | 0.0575 | - | |
| | 2.6249 | 1640 | 0.0571 | - | |
| | 2.6409 | 1650 | 0.0761 | - | |
| | 2.6569 | 1660 | 0.0582 | - | |
| | 2.6729 | 1670 | 0.0366 | - | |
| | 2.6890 | 1680 | 0.0445 | - | |
| | 2.7050 | 1690 | 0.0519 | - | |
| | 2.7210 | 1700 | 0.0506 | - | |
| | 2.7370 | 1710 | 0.0637 | - | |
| | 2.7531 | 1720 | 0.0618 | - | |
| | 2.7691 | 1730 | 0.0433 | - | |
| | 2.7851 | 1740 | 0.0503 | - | |
| | 2.8011 | 1750 | 0.0541 | - | |
| | 2.8171 | 1760 | 0.0443 | - | |
| | 2.8332 | 1770 | 0.0634 | - | |
| | 2.8492 | 1780 | 0.0586 | - | |
| | 2.8652 | 1790 | 0.0497 | - | |
| | 2.8812 | 1800 | 0.0444 | - | |
| | 2.8973 | 1810 | 0.0397 | - | |
| | 2.9133 | 1820 | 0.0483 | - | |
| | 2.9293 | 1830 | 0.0441 | - | |
| | 2.9453 | 1840 | 0.0758 | - | |
| | 2.9613 | 1850 | 0.0988 | - | |
| | 2.9774 | 1860 | 0.0566 | - | |
| | 2.9934 | 1870 | 0.0497 | - | |
| | 3.0 | 1875 | - | 0.4466 | |
| | 3.0080 | 1880 | 0.0388 | - | |
| | 3.0240 | 1890 | 0.0278 | - | |
| | 3.0401 | 1900 | 0.0231 | - | |
| | 3.0561 | 1910 | 0.0482 | - | |
| | 3.0721 | 1920 | 0.0416 | - | |
| | 3.0881 | 1930 | 0.052 | - | |
| | 3.1041 | 1940 | 0.0403 | - | |
| | 3.1202 | 1950 | 0.0384 | - | |
| | 3.1362 | 1960 | 0.0288 | - | |
| | 3.1522 | 1970 | 0.0368 | - | |
| | 3.1682 | 1980 | 0.0301 | - | |
| | 3.1843 | 1990 | 0.029 | - | |
| | 3.2003 | 2000 | 0.0332 | - | |
| | 3.2163 | 2010 | 0.0307 | - | |
| | 3.2323 | 2020 | 0.0502 | - | |
| | 3.2483 | 2030 | 0.0474 | - | |
| | 3.2644 | 2040 | 0.0383 | - | |
| | 3.2804 | 2050 | 0.0392 | - | |
| | 3.2964 | 2060 | 0.0308 | - | |
| | 3.3124 | 2070 | 0.0479 | - | |
| | 3.3285 | 2080 | 0.0448 | - | |
| | 3.3445 | 2090 | 0.0478 | - | |
| | 3.3605 | 2100 | 0.0249 | - | |
| | 3.3765 | 2110 | 0.03 | - | |
| | 3.3925 | 2120 | 0.0284 | - | |
| | 3.4086 | 2130 | 0.0323 | - | |
| | 3.4246 | 2140 | 0.0379 | - | |
| | 3.4406 | 2150 | 0.0221 | - | |
| | 3.4566 | 2160 | 0.0354 | - | |
| | 3.4727 | 2170 | 0.0332 | - | |
| | 3.4887 | 2180 | 0.0287 | - | |
| | 3.5047 | 2190 | 0.0382 | - | |
| | 3.5207 | 2200 | 0.0342 | - | |
| | 3.5368 | 2210 | 0.0381 | - | |
| | 3.5528 | 2220 | 0.056 | - | |
| | 3.5688 | 2230 | 0.0426 | - | |
| | 3.5848 | 2240 | 0.0465 | - | |
| | 3.6008 | 2250 | 0.0372 | - | |
| | 3.6169 | 2260 | 0.0345 | - | |
| | 3.6329 | 2270 | 0.0459 | - | |
| | 3.6489 | 2280 | 0.0368 | - | |
| | 3.6649 | 2290 | 0.0349 | - | |
| | 3.6810 | 2300 | 0.059 | - | |
| | 3.6970 | 2310 | 0.0275 | - | |
| | 3.7130 | 2320 | 0.0305 | - | |
| | 3.7290 | 2330 | 0.0406 | - | |
| | 3.7450 | 2340 | 0.0456 | - | |
| | 3.7611 | 2350 | 0.0311 | - | |
| | 3.7771 | 2360 | 0.0428 | - | |
| | 3.7931 | 2370 | 0.0308 | - | |
| | 3.8091 | 2380 | 0.0345 | - | |
| | 3.8252 | 2390 | 0.0378 | - | |
| | 3.8412 | 2400 | 0.0322 | - | |
| | 3.8572 | 2410 | 0.0236 | - | |
| | 3.8732 | 2420 | 0.0383 | - | |
| | 3.8892 | 2430 | 0.0295 | - | |
| | 3.9053 | 2440 | 0.0273 | - | |
| | 3.9213 | 2450 | 0.0286 | - | |
| | 3.9373 | 2460 | 0.0366 | - | |
| | 3.9533 | 2470 | 0.0285 | - | |
| | 3.9694 | 2480 | 0.0335 | - | |
| | 3.9854 | 2490 | 0.0278 | - | |
| | **3.995** | **2496** | **-** | **0.4493** | |
| |
| * The bold row denotes the saved checkpoint. |
| </details> |
| |
| ### Framework Versions |
| - Python: 3.11.12 |
| - Sentence Transformers: 4.1.0 |
| - Transformers: 4.51.3 |
| - PyTorch: 2.6.0+cu124 |
| - Accelerate: 1.6.0 |
| - Datasets: 2.14.4 |
| - Tokenizers: 0.21.1 |
| |
| ## 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", |
| } |
| ``` |
| |
| #### 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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