Master-thesis-NAP/ModernBERT-DAPT-Embed-DAPT-Math-v2
172
1---2language:3- en4license: apache-2.05tags:6- sentence-transformers7- sentence-similarity8- feature-extraction9- generated_from_trainer10- dataset_size:7987611- loss:MultipleNegativesRankingLoss12base_model: Master-thesis-NAP/ModernBert-DAPT-math13widget:14- source_sentence: What is the error estimate for the difference between the exact15 solution and the local oscillation decomposition (LOD) solution in terms of the16 $L_0$ norm?17 sentences:18 - '\label{thm1}19 20 Suppose $\kappa$ and $\bar a$ are as above. Then $|\Pcut(\bar a)| \leq 2^\kappa$.21 Indeed if22 23 $2^\kappa=\aleph_\alpha,$ then $|\Pcut(\bar a)| \leq |\alpha+1|^2$.'24 - "\\cite{kyushu}\n For every discrete group $\\G$ and every 2-dimensional representation\25 \ $\\varrho$ of $\\G$, $\\varrho-$equivariant functions for $\\G$ always exist."26 - "\\label{Corollary}\n Let Assumptions~\\ref{assum_1} and~\\ref{assump2} be\27 \ satisfied. Let $u$ be the solution of~\\eqref{WeakForm} and let $u_{H,k}$ be\28 \ the LOD solution of~\\eqref{local_probelm }. Then we have \n \\begin{equation}\\\29 label{L2Estimate}\n \\|u-I_Hu_{H,k}\\|_0\\lesssim \\|u-I_Hu\\|_0+\\|u-u_{H,k}\\\30 |_0 +H|u-u_{H,k}|_1.\n \\end{equation}\n %\\[\\|u-I_Hu_{H,k}\\|_0\\lesssim\31 \ H |u|_1 +|u-u_{H,k}|_1.\\]"32- source_sentence: Does the theorem imply that the rate of convergence of the sequence33 $T_{m,j}(E)$ to $T_{m+k_n,j+k_n}(E)$ is exponential in the distance between $m$34 and $j$, and that this rate is bounded by a constant $C$ times an exponential35 decay factor involving the parameter $\gamma$?36 sentences:37 - "\\label{thm:weibull}\nSuppose random variable $X$ follows Weibull distribution,\38 \ and $E(X^i)$ denotes the $i$-th moment of $X$. Then the random variable $X$\39 \ satisfy the following inequality: \n\\begin{equation}\\label{eq:moments}\n \40 \ E(X^n)^{\\frac{1}{n}} \\geq E(X^m)^{\\frac{1}{m}},\n\\end{equation}\nwhere\41 \ $n > m$."42 - "\\label{lem1}\n\t\tFor all $m,j\\in\\Z$, we have\n\t\t\\begin{equation*}\n\t\43 \t|| T_{m,j} (E)-T_{m+k_n,j+k_n}(E)||\\leq C e^{-\\gamma k_n} e^{(\\mathcal\44 \ L(E)+\\varepsilon) |m-j|}. \n\t\t\\end{equation*}"45 - If the problem \eqref{eq:Model-based_Program} is convex, then under the primal-dual46 dynamics \eqref{eq:PDD}-\eqref{eq:AlgebraicConstruction}, the system \eqref{eq:Input-OutputMap}47 asymptotically converges to a steady state that is the optimal solution of \eqref{eq:Model-based_Program}.48- source_sentence: What is the rate of convergence for the total error in the given49 problem, assuming the conditions in Theorem~\ref{convergence-rates} are met?50 sentences:51 - "\\label{convergence-rates}\nUnder the assumptions of Theorem~\\ref{well-posedness}.\52 \ Given $(\\bu,{p},\\bzeta,\\varphi)\\in (\\bH^{s_1+1}(\\Omega)\\cap \\bV_1)\\\53 times (\\text{H}^{s_1}(\\Omega)\\cap Q_{b_1}) \\times (\\bH^{s_2}\\cap \\bV_2)\54 \ \\times (\\text{H}^{s_2}\\cap Q_{b_2})$, $(\\bu_h,{p}_h,\\bzeta_h,\\varphi_h)\\\55 in \\bV_1^{h,k_1}\\times Q_1^{h,k_1}\\times \\bV_2^{h,k_2}\\times Q_2^{h,k_2}$\56 \ be the respective solutions of the continuous and discrete problems, with the\57 \ data satisfying $\\fb\\in \\bH^{s_1-1}\\cap \\bQ_{b_1}$ and $g\\in H^{s_2}(\\\58 Omega)\\cap Q_{b_2}$. If $\\overline{C}_1 \\sqrt{M} L_\\ell + \\overline{C}_2^2\59 \ \\sqrt{M^3} L_\\bbM\\sqrt{2\\mu} (\\norm{\\varphi_D}_{1/2,\\Gamma_D} + \\\60 norm{g}_{0,\\Omega}) < 1/2.$ Then, the total error $\\overline{\\textnormal{e}}_h:=\\\61 norm{(\\bu-\\bu_h,{p}-{p}_h, \\bzeta-\\bzeta_h,\\varphi-\\varphi_h)}_{\\bV_1\\\62 times Q_{1} \\times \\bV_2\\times Q_2}$ decays with the following rate for $s:=\63 \ \\min \\left\\{s_1,s_2\\right\\}$\n \\begin{align*}\\label{convergence-rate}\n\64 \ \\overline{\\textnormal{e}}_h &\\lesssim h^{ s} (|\\fb|_{s_1-1,\\bQ_{b_1}}\65 \ + |\\bu|_{s_1+1,\\bV_1} + |{p}|_{s_1,Q_{b_1}} + |g|_{s_2,Q_{b_2}} + |\\bzeta|_{s_2,\\\66 bV_2}+|\\varphi|_{s_2,Q_{b_2}}).\n \\end{align*}"67 - "\\label{thm}\nFor vector linear secure aggregation defined above, the optimal\68 \ total key rate is \n\\begin{eqnarray}\n R_{Z_{\\Sigma}}^* %= \\left\\{R_{Z_{\\\69 Sigma}}: R_{Z_{\\Sigma}} \\geq \n = \\mbox{rank} \\left( \\left[ \\mathbf{F}\70 \ ; \\mathbf{G} \\right] \\right)\n - \\mbox{rank} \\left( \\mathbf{F} \\\71 right) = \\mbox{rank}({\\bf G} | {\\bf F}).\n %\\right\\}.\n% \\\\ \\\72 mbox{rank}\n\\end{eqnarray}"73 - "The process $Y(t)$, $t\\geq 0,$ is called Markov branching process with\r\nnon-homogeneous\74 \ Poisson immigration (MBPNPI)."75- source_sentence: Is the local time of the horizontal component of the Peano curve76 ever greater than 1?77 sentences:78 - "[Divergence Theorem or Gauss-Green Theorem for Surfaces in $\\R^3$]\n\t\\label{thm:surface_int}\n\79 \t Let $\\Sigma \\subset \\Omega\\subseteq\\R^3$ be a bounded smooth surface.\n\80 \t Further, $\\bb a:\\Sigma\\to\\R^3$ is a continuously differentiable\81 \ vector field that is either defined on the\n\t\t\t\t\tboundary $\\partial\\\82 Sigma$ or has a bounded continuous extension to this boundary.\n\t Like\83 \ in \\eqref{eq:decomp} it may be decomposed into tangential and normal components\n\84 \t\t\t\t\tas follows $\\bb a = \\bb a^\\shortparallel + a_\\nu\\bs\\nu_\\Sigma$.\85 \ By $\\dd l$ we denote the line element on \n\t\t\t\t\tthe curve $\\partial \\\86 Sigma$. We assume that the curve is continuous and consists of finitely many\n\87 \t\t\t\t\tsmooth pieces.\n\t Then the following divergence formula for\88 \ surface integrals holds\n\t %\n\t \\begin{align}\n\t \89 \ %\n\t \\int\\limits_\\Sigma \\left[\\nabla_\\Sigma\\cdot\\bb a^\\\90 shortparallel\\right](\\x)\\;\\dd S\n\t\t\t\t\t\t\t= \\int\\limits_{\\partial\\\91 Sigma} \\left[\\bb a\\cdot\\bs\\nu_{\\partial\\Sigma}\\right](\\x)\\,\\dd l .\n\92 \t \\label{eq:surface_div}\n\t %\n\t \\end{align}\n\93 \t\t\t\t\t%\n\t\t\t\t\tFrom this we obtain the formula\n\t\t\t\t\t%\n\t \94 \ \\begin{align}\n\t %\n\t \\int\\limits_\\Sigma \\left[\\\95 nabla_\\Sigma\\cdot\\bb a\\right](\\x)\\;\\dd S\n\t\t\t\t\t\t\t= \\int\\limits_{\\\96 partial\\Sigma} \\left[\\bb a\\cdot\\bs\\nu_{\\partial\\Sigma}\\right](\\x)\\\97 ,\\dd l \n\t\t\t\t\t\t\t-\\int\\limits_\\Sigma\\left[ 2\\kappa_Ma_\\nu\\right](\\\98 x)\\;\\dd S.\n\t \\label{eq:surface_div_2}\n\t %\n\t \99 \ \\end{align}\n\t %"100 - There exists local time of the horizontal component $x$ of the Peano curve. Moreover,101 this local time attains values no greater than $1$.102 - "[Werner-Young's inequality]\\label{Young op-op}\nSuppose $S\\in \\cS^p$ and $T\\\103 in \\cS^q$ with $1+r^{-1}=p^{-1}+q^{-1}$.\nThen $S\\star T\\in L^r(\\R^{2d})$\104 \ and\n\\begin{align*}\n \\|S\\star T\\|_{L^{r}}\\leq \\|S\\|_{\\cS^p}\\|T\\\105 |_{\\cS^q}.\n\\end{align*}"106- source_sentence: What is the meaning of the identity containment $1_x:x\to x$ in107 the context of the bond system?108 sentences:109 - "\\label{lem:opt_lin}\nConsider the optimization problem\n\\begin{equation}\\\110 label{eq:max_tr_lem}\n\\begin{aligned}\n \\max_{\\bs{U}}&\\;\\; \\Re\\{\\mrm{tr}(\\\111 bs{U}^\\mrm{H}\\bs{B}) \\}\\\\\n \\mrm{s.t. \\;\\;}& \\bs{U}\\in \\mathcal{U}(N),\n\112 \\end{aligned}\n\\end{equation}\nwhere $\\bs{B}$ may be an arbitrary $N\\times\113 \ N$ matrix with singular value decomposition (SVD) $\\bs{B}=\\bs{U}_{\\bs{B}}\\\114 bs{S}_{\\bs{B}}\\bs{V}_{\\bs{B}}^\\mrm{H}$. The solution to \\eqref{eq:max_tr_lem}\115 \ is given by\n\\begin{equation}\\label{eq:sol_max}\n \\bs{U}_\\mrm{opt} =\116 \ \\bs{U}_{\\bs{B}}^\\mrm{H}\\bs{V}_{\\bs{B}}.\n\\end{equation}\n\\begin{skproof}\n\117 \ A formal proof, which may be included in the extended version, can be obtained\118 \ by defining the Riemannian gradient over the unitary group and finding the stationary\119 \ point where it vanishes. However, an intuitive argument is that the solution\120 \ to \\eqref{eq:max_tr_lem} is obtained by positively combining the singular values\121 \ of $\\bs{B}$, leading to \\eqref{eq:sol_max}.\n\\end{skproof}"122 - '\label{AM_BA_lem1}123 124 Let $$\Omega =\left\{a={{\left(k_1x_1+k_2,\dots,k_1x_n+k_2\right)}}\mid k_1, k_2\in125 \mathbb{R}\right\} .$$ Then ${\displaystyle\underset{a\in \Omega}{\operatorname{argmin}}126 {J_{\alpha }}(a)=\overline{a}\ },$ where $\overline{a}=\left(\overline{a}_1,\dots,\overline{a}_n\right)$,127 $$\overline{a}_i=\frac{1}{n}\sum^n_{j =1}{y_j},\quad\forall i=1,\dots,n.$$ In128 other words, on the class of lines $J_{\alpha }\left(a\right)$ reaches a minimum129 on a straight line parallel to the $Ox$ axis. So, this is the average line for130 the ordinates of all points of set $X$.'131 - "A \\emph{bond system} is a tuple $(B,C,s,t,1,\\cdot)$, where $B$ is a set of\132 \ \\emph{bonds}, $C$ is a set of \\emph{content} relations, and $s,t:C\\to B$\133 \ are \\emph{source} and \\emph{target} functions. For $c\\in C$ with $s(c)=x$\134 \ and $t(c)=y$, we write $x\\xrightarrow{c}y$ or $c:x\\to y$, indicating that\135 \ $x$ \\emph{contains} $y$. Each bond $x\\in B$ has an \\emph{identity} containment\136 \ $1_x:x\\to x$, meaning every bond trivially contains itself. For $c:x\\to y$\137 \ and $c':y\\to z$, their composition is $cc':x\\to z$. These data must satisfy:\n\138 \ \\begin{enumerate}\n \\item Identity laws: For each $c:x\\to y$, $1_x\139 \ c= c=c1_y$\n \\item Associativity: For $c:x\\to y$, $c':y\\to z$, $c'':z\\\140 to w$, $c(c'c'')=(cc')c''$\n \\item Anti-symmetry: For $c:x\\to y$ and\141 \ $c':y\\to x$, $x=y$\n \\item Left cancellation: For $c,c':x\\to y$ and\142 \ $c'':y\\to z$, if $cc''=c'c''$, then $c=c'$\n \\end{enumerate}"143pipeline_tag: sentence-similarity144library_name: sentence-transformers145metrics:146- cosine_accuracy@1147- cosine_accuracy@3148- cosine_accuracy@5149- cosine_accuracy@10150- cosine_precision@1151- cosine_precision@3152- cosine_precision@5153- cosine_precision@10154- cosine_recall@1155- cosine_recall@3156- cosine_recall@5157- cosine_recall@10158- cosine_ndcg@10159- cosine_mrr@10160- cosine_map@100161model-index:162- name: ModernBERT DAPT Embed DAPT Math163 results:164 - task:165 type: information-retrieval166 name: Information Retrieval167 dataset:168 name: TESTING169 type: TESTING170 metrics:171 - type: cosine_accuracy@1172 value: 0.868020304568528173 name: Cosine Accuracy@1174 - type: cosine_accuracy@3175 value: 0.9183202584217812176 name: Cosine Accuracy@3177 - type: cosine_accuracy@5178 value: 0.9325103830179973179 name: Cosine Accuracy@5180 - type: cosine_accuracy@10181 value: 0.9495846792801107182 name: Cosine Accuracy@10183 - type: cosine_precision@1184 value: 0.868020304568528185 name: Cosine Precision@1186 - type: cosine_precision@3187 value: 0.6118674050146131188 name: Cosine Precision@3189 - type: cosine_precision@5190 value: 0.49353945546838945191 name: Cosine Precision@5192 - type: cosine_precision@10193 value: 0.34758883248730965194 name: Cosine Precision@10195 - type: cosine_recall@1196 value: 0.04186710795480722197 name: Cosine Recall@1198 - type: cosine_recall@3199 value: 0.08315252408701693200 name: Cosine Recall@3201 - type: cosine_recall@5202 value: 0.1073909448198794203 name: Cosine Recall@5204 - type: cosine_recall@10205 value: 0.14207392775097807206 name: Cosine Recall@10207 - type: cosine_ndcg@10208 value: 0.4493273991613623209 name: Cosine Ndcg@10210 - type: cosine_mrr@10211 value: 0.8963655316764447212 name: Cosine Mrr@10213 - type: cosine_map@100214 value: 0.16376932233660765215 name: Cosine Map@100216---217 218# ModernBERT DAPT Embed DAPT Math219 220This 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.221 222## Model Details223 224### Model Description225- **Model Type:** Sentence Transformer226- **Base model:** [Master-thesis-NAP/ModernBert-DAPT-math](https://huggingface.co/Master-thesis-NAP/ModernBert-DAPT-math) <!-- at revision a30384f91d764c272e6b740c256d5581325ea4bb -->227- **Maximum Sequence Length:** 8192 tokens228- **Output Dimensionality:** 768 dimensions229- **Similarity Function:** Cosine Similarity230<!-- - **Training Dataset:** Unknown -->231- **Language:** en232- **License:** apache-2.0233 234### Model Sources235 236- **Documentation:** [Sentence Transformers Documentation](https://sbert.net)237- **Repository:** [Sentence Transformers on GitHub](https://github.com/UKPLab/sentence-transformers)238- **Hugging Face:** [Sentence Transformers on Hugging Face](https://huggingface.co/models?library=sentence-transformers)239 240### Full Model Architecture241 242```243SentenceTransformer(244 (0): Transformer({'max_seq_length': 8192, 'do_lower_case': False}) with Transformer model: ModernBertModel 245 (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})246 (2): Normalize()247)248```249 250## Usage251 252### Direct Usage (Sentence Transformers)253 254First install the Sentence Transformers library:255 256```bash257pip install -U sentence-transformers258```259 260Then you can load this model and run inference.261```python262from sentence_transformers import SentenceTransformer263 264# Download from the 🤗 Hub265model = SentenceTransformer("Master-thesis-NAP/ModernBERT-DAPT-Embed-DAPT-Math-v2")266# Run inference267sentences = [268 'What is the meaning of the identity containment $1_x:x\\to x$ in the context of the bond system?',269 "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}",270 '\\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}',271]272embeddings = model.encode(sentences)273print(embeddings.shape)274# [3, 768]275 276# Get the similarity scores for the embeddings277similarities = model.similarity(embeddings, embeddings)278print(similarities.shape)279# [3, 3]280```281 282<!--283### Direct Usage (Transformers)284 285<details><summary>Click to see the direct usage in Transformers</summary>286 287</details>288-->289 290<!--291### Downstream Usage (Sentence Transformers)292 293You can finetune this model on your own dataset.294 295<details><summary>Click to expand</summary>296 297</details>298-->299 300<!--301### Out-of-Scope Use302 303*List how the model may foreseeably be misused and address what users ought not to do with the model.*304-->305 306## Evaluation307 308### Metrics309 310#### Information Retrieval311 312* Dataset: `TESTING`313* Evaluated with [<code>InformationRetrievalEvaluator</code>](https://sbert.net/docs/package_reference/sentence_transformer/evaluation.html#sentence_transformers.evaluation.InformationRetrievalEvaluator)314 315| Metric | Value |316|:--------------------|:-----------|317| cosine_accuracy@1 | 0.868 |318| cosine_accuracy@3 | 0.9183 |319| cosine_accuracy@5 | 0.9325 |320| cosine_accuracy@10 | 0.9496 |321| cosine_precision@1 | 0.868 |322| cosine_precision@3 | 0.6119 |323| cosine_precision@5 | 0.4935 |324| cosine_precision@10 | 0.3476 |325| cosine_recall@1 | 0.0419 |326| cosine_recall@3 | 0.0832 |327| cosine_recall@5 | 0.1074 |328| cosine_recall@10 | 0.1421 |329| **cosine_ndcg@10** | **0.4493** |330| cosine_mrr@10 | 0.8964 |331| cosine_map@100 | 0.1638 |332 333<!--334## Bias, Risks and Limitations335 336*What are the known or foreseeable issues stemming from this model? You could also flag here known failure cases or weaknesses of the model.*337-->338 339<!--340### Recommendations341 342*What are recommendations with respect to the foreseeable issues? For example, filtering explicit content.*343-->344 345## Training Details346 347### Training Dataset348 349#### Unnamed Dataset350 351* Size: 79,876 training samples352* Columns: <code>anchor</code> and <code>positive</code>353* Approximate statistics based on the first 1000 samples:354 | | anchor | positive |355 |:--------|:-----------------------------------------------------------------------------------|:------------------------------------------------------------------------------------|356 | type | string | string |357 | 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> |358* Samples:359 | anchor | positive |360 |:---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|:--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|361 | <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> |362 | <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> |363 | <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> |364* Loss: [<code>MultipleNegativesRankingLoss</code>](https://sbert.net/docs/package_reference/sentence_transformer/losses.html#multiplenegativesrankingloss) with these parameters:365 ```json366 {367 "scale": 20.0,368 "similarity_fct": "cos_sim"369 }370 ```371 372### Training Hyperparameters373#### Non-Default Hyperparameters374 375- `eval_strategy`: epoch376- `per_device_train_batch_size`: 16377- `per_device_eval_batch_size`: 16378- `gradient_accumulation_steps`: 8379- `learning_rate`: 2e-05380- `num_train_epochs`: 4381- `lr_scheduler_type`: cosine382- `warmup_ratio`: 0.1383- `bf16`: True384- `tf32`: True385- `load_best_model_at_end`: True386- `optim`: adamw_torch_fused387- `batch_sampler`: no_duplicates388 389#### All Hyperparameters390<details><summary>Click to expand</summary>391 392- `overwrite_output_dir`: False393- `do_predict`: False394- `eval_strategy`: epoch395- `prediction_loss_only`: True396- `per_device_train_batch_size`: 16397- `per_device_eval_batch_size`: 16398- `per_gpu_train_batch_size`: None399- `per_gpu_eval_batch_size`: None400- `gradient_accumulation_steps`: 8401- `eval_accumulation_steps`: None402- `torch_empty_cache_steps`: None403- `learning_rate`: 2e-05404- `weight_decay`: 0.0405- `adam_beta1`: 0.9406- `adam_beta2`: 0.999407- `adam_epsilon`: 1e-08408- `max_grad_norm`: 1.0409- `num_train_epochs`: 4410- `max_steps`: -1411- `lr_scheduler_type`: cosine412- `lr_scheduler_kwargs`: {}413- `warmup_ratio`: 0.1414- `warmup_steps`: 0415- `log_level`: passive416- `log_level_replica`: warning417- `log_on_each_node`: True418- `logging_nan_inf_filter`: True419- `save_safetensors`: True420- `save_on_each_node`: False421- `save_only_model`: False422- `restore_callback_states_from_checkpoint`: False423- `no_cuda`: False424- `use_cpu`: False425- `use_mps_device`: False426- `seed`: 42427- `data_seed`: None428- `jit_mode_eval`: False429- `use_ipex`: False430- `bf16`: True431- `fp16`: False432- `fp16_opt_level`: O1433- `half_precision_backend`: auto434- `bf16_full_eval`: False435- `fp16_full_eval`: False436- `tf32`: True437- `local_rank`: 0438- `ddp_backend`: None439- `tpu_num_cores`: None440- `tpu_metrics_debug`: False441- `debug`: []442- `dataloader_drop_last`: False443- `dataloader_num_workers`: 0444- `dataloader_prefetch_factor`: None445- `past_index`: -1446- `disable_tqdm`: False447- `remove_unused_columns`: True448- `label_names`: None449- `load_best_model_at_end`: True450- `ignore_data_skip`: False451- `fsdp`: []452- `fsdp_min_num_params`: 0453- `fsdp_config`: {'min_num_params': 0, 'xla': False, 'xla_fsdp_v2': False, 'xla_fsdp_grad_ckpt': False}454- `tp_size`: 0455- `fsdp_transformer_layer_cls_to_wrap`: None456- `accelerator_config`: {'split_batches': False, 'dispatch_batches': None, 'even_batches': True, 'use_seedable_sampler': True, 'non_blocking': False, 'gradient_accumulation_kwargs': None}457- `deepspeed`: None458- `label_smoothing_factor`: 0.0459- `optim`: adamw_torch_fused460- `optim_args`: None461- `adafactor`: False462- `group_by_length`: False463- `length_column_name`: length464- `ddp_find_unused_parameters`: None465- `ddp_bucket_cap_mb`: None466- `ddp_broadcast_buffers`: False467- `dataloader_pin_memory`: True468- `dataloader_persistent_workers`: False469- `skip_memory_metrics`: True470- `use_legacy_prediction_loop`: False471- `push_to_hub`: False472- `resume_from_checkpoint`: None473- `hub_model_id`: None474- `hub_strategy`: every_save475- `hub_private_repo`: None476- `hub_always_push`: False477- `gradient_checkpointing`: False478- `gradient_checkpointing_kwargs`: None479- `include_inputs_for_metrics`: False480- `include_for_metrics`: []481- `eval_do_concat_batches`: True482- `fp16_backend`: auto483- `push_to_hub_model_id`: None484- `push_to_hub_organization`: None485- `mp_parameters`: 486- `auto_find_batch_size`: False487- `full_determinism`: False488- `torchdynamo`: None489- `ray_scope`: last490- `ddp_timeout`: 1800491- `torch_compile`: False492- `torch_compile_backend`: None493- `torch_compile_mode`: None494- `include_tokens_per_second`: False495- `include_num_input_tokens_seen`: False496- `neftune_noise_alpha`: None497- `optim_target_modules`: None498- `batch_eval_metrics`: False499- `eval_on_start`: False500- `use_liger_kernel`: False501- `eval_use_gather_object`: False502- `average_tokens_across_devices`: False503- `prompts`: None504- `batch_sampler`: no_duplicates505- `multi_dataset_batch_sampler`: proportional506 507</details>508 509### Training Logs510<details><summary>Click to expand</summary>511 512| Epoch | Step | Training Loss | TESTING_cosine_ndcg@10 |513|:---------:|:--------:|:-------------:|:----------------------:|514| 0.0160 | 10 | 20.2777 | - |515| 0.0320 | 20 | 19.6613 | - |516| 0.0481 | 30 | 18.8588 | - |517| 0.0641 | 40 | 17.5525 | - |518| 0.0801 | 50 | 15.1065 | - |519| 0.0961 | 60 | 10.8128 | - |520| 0.1122 | 70 | 7.0698 | - |521| 0.1282 | 80 | 4.532 | - |522| 0.1442 | 90 | 3.5143 | - |523| 0.1602 | 100 | 2.3256 | - |524| 0.1762 | 110 | 1.4688 | - |525| 0.1923 | 120 | 1.0081 | - |526| 0.2083 | 130 | 0.949 | - |527| 0.2243 | 140 | 0.9709 | - |528| 0.2403 | 150 | 0.8403 | - |529| 0.2564 | 160 | 0.8749 | - |530| 0.2724 | 170 | 0.7955 | - |531| 0.2884 | 180 | 0.6587 | - |532| 0.3044 | 190 | 0.5832 | - |533| 0.3204 | 200 | 0.5376 | - |534| 0.3365 | 210 | 0.608 | - |535| 0.3525 | 220 | 0.4639 | - |536| 0.3685 | 230 | 0.6611 | - |537| 0.3845 | 240 | 0.5589 | - |538| 0.4006 | 250 | 0.5845 | - |539| 0.4166 | 260 | 0.4392 | - |540| 0.4326 | 270 | 0.4746 | - |541| 0.4486 | 280 | 0.4517 | - |542| 0.4647 | 290 | 0.4034 | - |543| 0.4807 | 300 | 0.4437 | - |544| 0.4967 | 310 | 0.4339 | - |545| 0.5127 | 320 | 0.4445 | - |546| 0.5287 | 330 | 0.3793 | - |547| 0.5448 | 340 | 0.3591 | - |548| 0.5608 | 350 | 0.4694 | - |549| 0.5768 | 360 | 0.4668 | - |550| 0.5928 | 370 | 0.4121 | - |551| 0.6089 | 380 | 0.4688 | - |552| 0.6249 | 390 | 0.387 | - |553| 0.6409 | 400 | 0.3748 | - |554| 0.6569 | 410 | 0.2997 | - |555| 0.6729 | 420 | 0.3756 | - |556| 0.6890 | 430 | 0.2993 | - |557| 0.7050 | 440 | 0.3514 | - |558| 0.7210 | 450 | 0.3646 | - |559| 0.7370 | 460 | 0.308 | - |560| 0.7531 | 470 | 0.3612 | - |561| 0.7691 | 480 | 0.2845 | - |562| 0.7851 | 490 | 0.2792 | - |563| 0.8011 | 500 | 0.2204 | - |564| 0.8171 | 510 | 0.2757 | - |565| 0.8332 | 520 | 0.2674 | - |566| 0.8492 | 530 | 0.3753 | - |567| 0.8652 | 540 | 0.3546 | - |568| 0.8812 | 550 | 0.3166 | - |569| 0.8973 | 560 | 0.2656 | - |570| 0.9133 | 570 | 0.3215 | - |571| 0.9293 | 580 | 0.2559 | - |572| 0.9453 | 590 | 0.4629 | - |573| 0.9613 | 600 | 0.31 | - |574| 0.9774 | 610 | 0.3601 | - |575| 0.9934 | 620 | 0.2391 | - |576| 1.0 | 625 | - | 0.4229 |577| 1.0080 | 630 | 0.2507 | - |578| 1.0240 | 640 | 0.1852 | - |579| 1.0401 | 650 | 0.1836 | - |580| 1.0561 | 660 | 0.1487 | - |581| 1.0721 | 670 | 0.1495 | - |582| 1.0881 | 680 | 0.1567 | - |583| 1.1041 | 690 | 0.1497 | - |584| 1.1202 | 700 | 0.1632 | - |585| 1.1362 | 710 | 0.1997 | - |586| 1.1522 | 720 | 0.182 | - |587| 1.1682 | 730 | 0.1884 | - |588| 1.1843 | 740 | 0.1766 | - |589| 1.2003 | 750 | 0.1477 | - |590| 1.2163 | 760 | 0.181 | - |591| 1.2323 | 770 | 0.092 | - |592| 1.2483 | 780 | 0.1506 | - |593| 1.2644 | 790 | 0.1305 | - |594| 1.2804 | 800 | 0.1533 | - |595| 1.2964 | 810 | 0.2306 | - |596| 1.3124 | 820 | 0.1861 | - |597| 1.3285 | 830 | 0.1157 | - |598| 1.3445 | 840 | 0.1054 | - |599| 1.3605 | 850 | 0.1696 | - |600| 1.3765 | 860 | 0.1327 | - |601| 1.3925 | 870 | 0.1485 | - |602| 1.4086 | 880 | 0.1395 | - |603| 1.4246 | 890 | 0.1021 | - |604| 1.4406 | 900 | 0.1283 | - |605| 1.4566 | 910 | 0.102 | - |606| 1.4727 | 920 | 0.1825 | - |607| 1.4887 | 930 | 0.1395 | - |608| 1.5047 | 940 | 0.157 | - |609| 1.5207 | 950 | 0.1444 | - |610| 1.5368 | 960 | 0.1317 | - |611| 1.5528 | 970 | 0.146 | - |612| 1.5688 | 980 | 0.1809 | - |613| 1.5848 | 990 | 0.1368 | - |614| 1.6008 | 1000 | 0.2036 | - |615| 1.6169 | 1010 | 0.1292 | - |616| 1.6329 | 1020 | 0.1306 | - |617| 1.6489 | 1030 | 0.1473 | - |618| 1.6649 | 1040 | 0.1595 | - |619| 1.6810 | 1050 | 0.1471 | - |620| 1.6970 | 1060 | 0.1869 | - |621| 1.7130 | 1070 | 0.1445 | - |622| 1.7290 | 1080 | 0.157 | - |623| 1.7450 | 1090 | 0.1382 | - |624| 1.7611 | 1100 | 0.157 | - |625| 1.7771 | 1110 | 0.1073 | - |626| 1.7931 | 1120 | 0.0864 | - |627| 1.8091 | 1130 | 0.1312 | - |628| 1.8252 | 1140 | 0.1644 | - |629| 1.8412 | 1150 | 0.1366 | - |630| 1.8572 | 1160 | 0.1257 | - |631| 1.8732 | 1170 | 0.127 | - |632| 1.8892 | 1180 | 0.1494 | - |633| 1.9053 | 1190 | 0.1516 | - |634| 1.9213 | 1200 | 0.1709 | - |635| 1.9373 | 1210 | 0.1717 | - |636| 1.9533 | 1220 | 0.1044 | - |637| 1.9694 | 1230 | 0.1551 | - |638| 1.9854 | 1240 | 0.1303 | - |639| 2.0 | 1250 | 0.1081 | 0.4392 |640| 2.0160 | 1260 | 0.0572 | - |641| 2.0320 | 1270 | 0.0504 | - |642| 2.0481 | 1280 | 0.0535 | - |643| 2.0641 | 1290 | 0.0512 | - |644| 2.0801 | 1300 | 0.0539 | - |645| 2.0961 | 1310 | 0.0462 | - |646| 2.1122 | 1320 | 0.0611 | - |647| 2.1282 | 1330 | 0.0989 | - |648| 2.1442 | 1340 | 0.0462 | - |649| 2.1602 | 1350 | 0.061 | - |650| 2.1762 | 1360 | 0.0557 | - |651| 2.1923 | 1370 | 0.0622 | - |652| 2.2083 | 1380 | 0.0744 | - |653| 2.2243 | 1390 | 0.0531 | - |654| 2.2403 | 1400 | 0.0507 | - |655| 2.2564 | 1410 | 0.0533 | - |656| 2.2724 | 1420 | 0.0676 | - |657| 2.2884 | 1430 | 0.0706 | - |658| 2.3044 | 1440 | 0.0452 | - |659| 2.3204 | 1450 | 0.0415 | - |660| 2.3365 | 1460 | 0.0562 | - |661| 2.3525 | 1470 | 0.0487 | - |662| 2.3685 | 1480 | 0.0614 | - |663| 2.3845 | 1490 | 0.045 | - |664| 2.4006 | 1500 | 0.0529 | - |665| 2.4166 | 1510 | 0.048 | - |666| 2.4326 | 1520 | 0.059 | - |667| 2.4486 | 1530 | 0.0593 | - |668| 2.4647 | 1540 | 0.0631 | - |669| 2.4807 | 1550 | 0.0506 | - |670| 2.4967 | 1560 | 0.058 | - |671| 2.5127 | 1570 | 0.0896 | - |672| 2.5287 | 1580 | 0.0522 | - |673| 2.5448 | 1590 | 0.035 | - |674| 2.5608 | 1600 | 0.0677 | - |675| 2.5768 | 1610 | 0.0538 | - |676| 2.5928 | 1620 | 0.0485 | - |677| 2.6089 | 1630 | 0.0575 | - |678| 2.6249 | 1640 | 0.0571 | - |679| 2.6409 | 1650 | 0.0761 | - |680| 2.6569 | 1660 | 0.0582 | - |681| 2.6729 | 1670 | 0.0366 | - |682| 2.6890 | 1680 | 0.0445 | - |683| 2.7050 | 1690 | 0.0519 | - |684| 2.7210 | 1700 | 0.0506 | - |685| 2.7370 | 1710 | 0.0637 | - |686| 2.7531 | 1720 | 0.0618 | - |687| 2.7691 | 1730 | 0.0433 | - |688| 2.7851 | 1740 | 0.0503 | - |689| 2.8011 | 1750 | 0.0541 | - |690| 2.8171 | 1760 | 0.0443 | - |691| 2.8332 | 1770 | 0.0634 | - |692| 2.8492 | 1780 | 0.0586 | - |693| 2.8652 | 1790 | 0.0497 | - |694| 2.8812 | 1800 | 0.0444 | - |695| 2.8973 | 1810 | 0.0397 | - |696| 2.9133 | 1820 | 0.0483 | - |697| 2.9293 | 1830 | 0.0441 | - |698| 2.9453 | 1840 | 0.0758 | - |699| 2.9613 | 1850 | 0.0988 | - |700| 2.9774 | 1860 | 0.0566 | - |701| 2.9934 | 1870 | 0.0497 | - |702| 3.0 | 1875 | - | 0.4466 |703| 3.0080 | 1880 | 0.0388 | - |704| 3.0240 | 1890 | 0.0278 | - |705| 3.0401 | 1900 | 0.0231 | - |706| 3.0561 | 1910 | 0.0482 | - |707| 3.0721 | 1920 | 0.0416 | - |708| 3.0881 | 1930 | 0.052 | - |709| 3.1041 | 1940 | 0.0403 | - |710| 3.1202 | 1950 | 0.0384 | - |711| 3.1362 | 1960 | 0.0288 | - |712| 3.1522 | 1970 | 0.0368 | - |713| 3.1682 | 1980 | 0.0301 | - |714| 3.1843 | 1990 | 0.029 | - |715| 3.2003 | 2000 | 0.0332 | - |716| 3.2163 | 2010 | 0.0307 | - |717| 3.2323 | 2020 | 0.0502 | - |718| 3.2483 | 2030 | 0.0474 | - |719| 3.2644 | 2040 | 0.0383 | - |720| 3.2804 | 2050 | 0.0392 | - |721| 3.2964 | 2060 | 0.0308 | - |722| 3.3124 | 2070 | 0.0479 | - |723| 3.3285 | 2080 | 0.0448 | - |724| 3.3445 | 2090 | 0.0478 | - |725| 3.3605 | 2100 | 0.0249 | - |726| 3.3765 | 2110 | 0.03 | - |727| 3.3925 | 2120 | 0.0284 | - |728| 3.4086 | 2130 | 0.0323 | - |729| 3.4246 | 2140 | 0.0379 | - |730| 3.4406 | 2150 | 0.0221 | - |731| 3.4566 | 2160 | 0.0354 | - |732| 3.4727 | 2170 | 0.0332 | - |733| 3.4887 | 2180 | 0.0287 | - |734| 3.5047 | 2190 | 0.0382 | - |735| 3.5207 | 2200 | 0.0342 | - |736| 3.5368 | 2210 | 0.0381 | - |737| 3.5528 | 2220 | 0.056 | - |738| 3.5688 | 2230 | 0.0426 | - |739| 3.5848 | 2240 | 0.0465 | - |740| 3.6008 | 2250 | 0.0372 | - |741| 3.6169 | 2260 | 0.0345 | - |742| 3.6329 | 2270 | 0.0459 | - |743| 3.6489 | 2280 | 0.0368 | - |744| 3.6649 | 2290 | 0.0349 | - |745| 3.6810 | 2300 | 0.059 | - |746| 3.6970 | 2310 | 0.0275 | - |747| 3.7130 | 2320 | 0.0305 | - |748| 3.7290 | 2330 | 0.0406 | - |749| 3.7450 | 2340 | 0.0456 | - |750| 3.7611 | 2350 | 0.0311 | - |751| 3.7771 | 2360 | 0.0428 | - |752| 3.7931 | 2370 | 0.0308 | - |753| 3.8091 | 2380 | 0.0345 | - |754| 3.8252 | 2390 | 0.0378 | - |755| 3.8412 | 2400 | 0.0322 | - |756| 3.8572 | 2410 | 0.0236 | - |757| 3.8732 | 2420 | 0.0383 | - |758| 3.8892 | 2430 | 0.0295 | - |759| 3.9053 | 2440 | 0.0273 | - |760| 3.9213 | 2450 | 0.0286 | - |761| 3.9373 | 2460 | 0.0366 | - |762| 3.9533 | 2470 | 0.0285 | - |763| 3.9694 | 2480 | 0.0335 | - |764| 3.9854 | 2490 | 0.0278 | - |765| **3.995** | **2496** | **-** | **0.4493** |766 767* The bold row denotes the saved checkpoint.768</details>769 770### Framework Versions771- Python: 3.11.12772- Sentence Transformers: 4.1.0773- Transformers: 4.51.3774- PyTorch: 2.6.0+cu124775- Accelerate: 1.6.0776- Datasets: 2.14.4777- Tokenizers: 0.21.1778 779## Citation780 781### BibTeX782 783#### Sentence Transformers784```bibtex785@inproceedings{reimers-2019-sentence-bert,786 title = "Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks",787 author = "Reimers, Nils and Gurevych, Iryna",788 booktitle = "Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing",789 month = "11",790 year = "2019",791 publisher = "Association for Computational Linguistics",792 url = "https://arxiv.org/abs/1908.10084",793}794```795 796#### MultipleNegativesRankingLoss797```bibtex798@misc{henderson2017efficient,799 title={Efficient Natural Language Response Suggestion for Smart Reply},800 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},801 year={2017},802 eprint={1705.00652},803 archivePrefix={arXiv},804 primaryClass={cs.CL}805}806```807 808<!--809## Glossary810 811*Clearly define terms in order to be accessible across 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