{
  "schema_version": 1,
  "builder_plan": {
    "scenario": "You already need task-specific training and want to reduce its memory or compute cost.",
    "change": "Change only the training technique on a small controlled run after checking prerequisites.",
    "baseline": "Your current training recipe with the same model, data split, and evaluation.",
    "steps": [
      "Locate the paper's implementation and check its license, model support, data needs, and hardware requirements. Stop here if you cannot obtain the required artifacts.",
      "Save a small set of representative product tasks and expected outcomes. Keep evaluation tasks out of any training or tuning, and choose quality and cost limits before running.",
      "Run the baseline and the proposed change on identical inputs. Keep other settings fixed and repeat paired runs to expose variation.",
      "Save per-task outputs, failures, timings, and all usage costs. Review regressions before expanding the test."
    ],
    "measure": "Held-out quality, peak memory, training time, and full compute cost to the target quality.",
    "decision": "Keep the change only if it meets your preselected quality and latency limits and improves the metric you care about after setup and operating costs. Otherwise retain the baseline. This is an exploration plan, not a validated recommendation."
  },
  "builder_review": null,
  "arxiv_id": "2609.16760",
  "title": "Turn-level Multiscale Density Ratio Estimation for LLM Agents",
  "summary": "Replaces single-turn alignment objectives (PPO, DPO, GRPO) with a turn-level weighted, asymmetric token-level objective for multi-turn agent post-training. Costs: requires preference or positive/negative trajectory data across multi-turn tasks, a modified training loop, and post-training compute that on one 24 GB GPU realistically limits model size (roughly 7B-class with parameter-efficient methods or heavy quantization). Can fail because abstract-level claims of competitive performance provide no verified magnitude, turn-weighting hyperparameters may not transfer to a different tool-use distribution, and small teams bear the engineering cost of integrating a nonstandard objective over mature GRPO/DPO tooling.",
  "rationale": "The method is plausibly applicable at small scale for teams already doing multi-turn agent fine-tuning, but with no quantified gains and a custom objective to implement, it warrants workload-specific validation rather than immediate adoption.",
  "technique": "Turn-level multiscale density ratio estimation for alignment",
  "family": "treino_eficiente",
  "recommendation": "testar",
  "published": "2026-09-15",
  "as_of": "2026-10-18",
  "editorial_status": "indexed",
  "source_url": "https://arxiv.org/abs/2609.16760",
  "independent_implementations": 0,
  "total_implementations": 0,
  "stars": 0,
  "citations": 0,
  "claims": [
    {
      "claim_id": "claim-01",
      "statement": "Replaces single-turn alignment objectives (PPO, DPO, GRPO) with a turn-level weighted, asymmetric token-level objective for multi-turn agent post-training. Costs: requires preference or positive/negative trajectory data across multi-turn tasks, a modified training loop, and post-training compute that on one 24 GB GPU realistically limits model size (roughly 7B-class with parameter-efficient methods or heavy quantization). Can fail because abstract-level claims of competitive performance provide no verified magnitude, turn-weighting hyperparameters may not transfer to a different tool-use distribution, and small teams bear the engineering cost of integrating a nonstandard objective over mature GRPO/DPO tooling.",
      "basis": "inferred",
      "result": "",
      "baseline": "",
      "conditions": "",
      "source_url": "https://arxiv.org/abs/2609.16760",
      "source_page": null,
      "source_excerpt": ""
    }
  ],
  "exposure_map": [
    {
      "dimension": "quality",
      "basis": "not_evaluated",
      "finding": "",
      "interpretation": "",
      "limitation": "",
      "next_check": "",
      "source_url": "",
      "source_page": null,
      "source_excerpt": ""
    },
    {
      "dimension": "compute",
      "basis": "not_evaluated",
      "finding": "",
      "interpretation": "",
      "limitation": "",
      "next_check": "",
      "source_url": "",
      "source_page": null,
      "source_excerpt": ""
    },
    {
      "dimension": "latency",
      "basis": "not_evaluated",
      "finding": "",
      "interpretation": "",
      "limitation": "",
      "next_check": "",
      "source_url": "",
      "source_page": null,
      "source_excerpt": ""
    },
    {
      "dimension": "operations",
      "basis": "not_evaluated",
      "finding": "",
      "interpretation": "",
      "limitation": "",
      "next_check": "",
      "source_url": "",
      "source_page": null,
      "source_excerpt": ""
    },
    {
      "dimension": "compatibility",
      "basis": "not_evaluated",
      "finding": "",
      "interpretation": "",
      "limitation": "",
      "next_check": "",
      "source_url": "",
      "source_page": null,
      "source_excerpt": ""
    },
    {
      "dimension": "security",
      "basis": "not_evaluated",
      "finding": "",
      "interpretation": "",
      "limitation": "",
      "next_check": "",
      "source_url": "",
      "source_page": null,
      "source_excerpt": ""
    },
    {
      "dimension": "data_and_training",
      "basis": "not_evaluated",
      "finding": "",
      "interpretation": "",
      "limitation": "",
      "next_check": "",
      "source_url": "",
      "source_page": null,
      "source_excerpt": ""
    },
    {
      "dimension": "reproducibility",
      "basis": "not_evaluated",
      "finding": "",
      "interpretation": "",
      "limitation": "",
      "next_check": "",
      "source_url": "",
      "source_page": null,
      "source_excerpt": ""
    }
  ],
  "risks": [],
  "minimum_test": [],
  "open_questions": [],
  "independent_tests": [],
  "lab_reproductions": [],
  "report_available": false,
  "validation_tier": "unknown",
  "evidence_tier": "unknown",
  "infrastructure_basis": "unknown",
  "equations": [
    {
      "anchor": "S4.E16",
      "label": "(16)",
      "section": "4 Methodology",
      "role": "proposed_method",
      "latex": "\\mathcal{L}_{tlm-DRE} \\\\ =\\mathbb{E}_{(\\boldsymbol{x},\\boldsymbol{y}_{w},\\boldsymbol{y}_{l}\\sim\\mathcal{D})}\\{[\\partial f(\\hat{r}_{\\theta}(\\boldsymbol{y}_{l}))\\hat{r}_{\\theta}(\\boldsymbol{y}_{l})-f(\\hat{r}_{\\theta}(\\boldsymbol{y}_{l}))] \\\\ -\\partial f(\\hat{r}_{\\theta}(\\boldsymbol{y}_{w}))\\prod\\limits_{i=1}^{t}\\frac{\\pi^{\\omega_{i}-1}_{c}(\\boldsymbol{y_{w,a_{i}}}\\|\\boldsymbol{x},\\boldsymbol{c}_{w,i-1})}{\\pi^{\\omega_{i}-1}_{ref}(\\boldsymbol{y_{w,a_{i}}}\\|\\boldsymbol{x},\\boldsymbol{c}_{w,i-1})}\\} \\\\ \\approx\\mathbb{E}_{(\\boldsymbol{x},\\boldsymbol{y}_{w},\\boldsymbol{y}_{l}\\sim\\mathcal{D})}\\{[\\partial f(\\hat{r}_{\\theta}(\\boldsymbol{y}_{l}))\\hat{r}_{\\theta}(\\boldsymbol{y}_{l})-f(\\hat{r}_{\\theta}(\\boldsymbol{y}_{l}))] \\\\ -\\partial f(\\hat{r}_{\\theta}(\\boldsymbol{y}_{w}))\\prod\\limits_{i=1}^{t}\\frac{1}{\\pi^{\\omega_{i}-1}_{ref}(\\boldsymbol{y_{w,a_{i}}}\\|\\boldsymbol{x},\\boldsymbol{c}_{w,i-1})}\\}.",
      "mathml": "<math display=\"block\"><msub><mi>ℒ</mi><mrow><mrow><mi>t</mi><mo lspace=\"0em\" rspace=\"0em\">​</mo><mi>l</mi><mo lspace=\"0em\" rspace=\"0em\">​</mo><mi>m</mi></mrow><mo>−</mo><mrow><mi>D</mi><mo lspace=\"0em\" rspace=\"0em\">​</mo><mi>R</mi><mo lspace=\"0em\" rspace=\"0em\">​</mo><mi>E</mi></mrow></mrow></msub></math><math display=\"block\"><mrow><mo>=</mo><mi>𝔼</mi><msub><mrow></mrow><mrow><mo stretchy=\"false\">(</mo><mi>𝒙</mi><mo>,</mo><msub><mi>𝒚</mi><mi>w</mi></msub><mo>,</mo><mrow><msub><mi>𝒚</mi><mi>l</mi></msub><mo>∼</mo><mi>𝒟</mi></mrow><mo stretchy=\"false\">)</mo></mrow></msub><mo stretchy=\"false\">{</mo><mo stretchy=\"false\">[</mo><mo lspace=\"0em\">∂</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><msub><mrow></mrow><mi>θ</mi></msub><mo stretchy=\"false\">(</mo><mi>𝒚</mi><msub><mrow></mrow><mi>l</mi></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><msub><mrow></mrow><mi>θ</mi></msub><mo stretchy=\"false\">(</mo><mi>𝒚</mi><msub><mrow></mrow><mi>l</mi></msub><mo stretchy=\"false\">)</mo><mo>−</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><msub><mrow></mrow><mi>θ</mi></msub><mo stretchy=\"false\">(</mo><mi>𝒚</mi><msub><mrow></mrow><mi>l</mi></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">]</mo></mrow></math><math display=\"block\"><mrow><mo rspace=\"0.055em\">−</mo><mo>∂</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><msub><mrow></mrow><mi>θ</mi></msub><mo stretchy=\"false\">(</mo><mi>𝒚</mi><msub><mrow></mrow><mi>w</mi></msub><mo stretchy=\"false\">)</mo><mo rspace=\"0.167em\" stretchy=\"false\">)</mo><mstyle displaystyle=\"true\"><mo movablelimits=\"false\">∏</mo></mstyle><msub><mrow></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow></msub><msup><mrow></mrow><mi>t</mi></msup><mstyle displaystyle=\"true\"><mfrac><mrow><mi>π</mi><msup><mrow></mrow><mrow><msub><mi>ω</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow></mrow><mi>c</mi></msub><mo stretchy=\"false\">(</mo><mi>𝒚</mi><msub><mrow></mrow><mrow><mi>𝒘</mi><mo mathvariant=\"bold\">,</mo><msub><mi>𝒂</mi><mi>𝒊</mi></msub></mrow></msub><mo lspace=\"0em\" rspace=\"0.167em\">∥</mo><mi>𝒙</mi><mo>,</mo><mi>𝒄</mi><msub><mrow></mrow><mrow><mi>w</mi><mo>,</mo><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></mrow></msub><mo stretchy=\"false\">)</mo></mrow><mrow><mi>π</mi><msup><mrow></mrow><mrow><msub><mi>ω</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow></mrow><mrow><mi>r</mi><mo lspace=\"0em\" rspace=\"0em\">​</mo><mi>e</mi><mo lspace=\"0em\" rspace=\"0em\">​</mo><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>𝒚</mi><msub><mrow></mrow><mrow><mi>𝒘</mi><mo mathvariant=\"bold\">,</mo><msub><mi>𝒂</mi><mi>𝒊</mi></msub></mrow></msub><mo lspace=\"0em\" rspace=\"0.167em\">∥</mo><mi>𝒙</mi><mo>,</mo><mi>𝒄</mi><msub><mrow></mrow><mrow><mi>w</mi><mo>,</mo><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></mrow></msub><mo stretchy=\"false\">)</mo></mrow></mfrac></mstyle><mo stretchy=\"false\">}</mo></mrow></math><math display=\"block\"><mrow><mo>≈</mo><mi>𝔼</mi><msub><mrow></mrow><mrow><mo stretchy=\"false\">(</mo><mi>𝒙</mi><mo>,</mo><msub><mi>𝒚</mi><mi>w</mi></msub><mo>,</mo><mrow><msub><mi>𝒚</mi><mi>l</mi></msub><mo>∼</mo><mi>𝒟</mi></mrow><mo stretchy=\"false\">)</mo></mrow></msub><mo stretchy=\"false\">{</mo><mo stretchy=\"false\">[</mo><mo lspace=\"0em\">∂</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><msub><mrow></mrow><mi>θ</mi></msub><mo stretchy=\"false\">(</mo><mi>𝒚</mi><msub><mrow></mrow><mi>l</mi></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><msub><mrow></mrow><mi>θ</mi></msub><mo stretchy=\"false\">(</mo><mi>𝒚</mi><msub><mrow></mrow><mi>l</mi></msub><mo stretchy=\"false\">)</mo><mo>−</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><msub><mrow></mrow><mi>θ</mi></msub><mo stretchy=\"false\">(</mo><mi>𝒚</mi><msub><mrow></mrow><mi>l</mi></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">]</mo></mrow></math><math display=\"block\"><mrow><mo rspace=\"0.055em\">−</mo><mo>∂</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><msub><mrow></mrow><mi>θ</mi></msub><mo stretchy=\"false\">(</mo><mi>𝒚</mi><msub><mrow></mrow><mi>w</mi></msub><mo stretchy=\"false\">)</mo><mo rspace=\"0.167em\" stretchy=\"false\">)</mo><mstyle displaystyle=\"true\"><mo movablelimits=\"false\">∏</mo></mstyle><msub><mrow></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow></msub><msup><mrow></mrow><mi>t</mi></msup><mstyle displaystyle=\"true\"><mfrac><mn>1</mn><mrow><mi>π</mi><msup><mrow></mrow><mrow><msub><mi>ω</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow></mrow><mrow><mi>r</mi><mo lspace=\"0em\" rspace=\"0em\">​</mo><mi>e</mi><mo lspace=\"0em\" rspace=\"0em\">​</mo><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>𝒚</mi><msub><mrow></mrow><mrow><mi>𝒘</mi><mo mathvariant=\"bold\">,</mo><msub><mi>𝒂</mi><mi>𝒊</mi></msub></mrow></msub><mo lspace=\"0em\" rspace=\"0.167em\">∥</mo><mi>𝒙</mi><mo>,</mo><mi>𝒄</mi><msub><mrow></mrow><mrow><mi>w</mi><mo>,</mo><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></mrow></msub><mo stretchy=\"false\">)</mo></mrow></mfrac></mstyle><mo stretchy=\"false\">}</mo><mo lspace=\"0em\">.</mo></mrow></math>",
      "context": "The distribution shows a significant concentration on the candidate tokens of the chosen trajectory in the language space. Since <math display=\"inline\"><msub><mi>π</mi><mi>c</mi></msub></math> is the ideal policy to be approximated from the parameterized policy’s manifold, we might as well assume that <math display=\"inline\"><mrow><msub><mi>π</mi><mi>c</mi></msub><mrow><mo stretchy=\"false\">(</mo><msub><mi>𝒚</mi><msub><mi>𝒂</mi><mi>𝒊</mi></msub></msub><mo lspace=\"0em\" rspace=\"0.167em\">∥</mo><mi>𝒙</mi><mo>,</mo><msub><mi>𝒄</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy=\"false\">)</mo></mrow><mo>∼</mo><mn>1</mn></mrow></math> uniformly, leading to the final tlm-DRE loss for training:",
      "headline": "",
      "explanation": "",
      "objective": "",
      "symbols": {},
      "evidence": "",
      "evidence_url": "",
      "source_url": ""
    },
    {
      "anchor": "S4.E14",
      "label": "(14)",
      "section": "4 Methodology",
      "role": "loss",
      "latex": "\\mathop{\\mathrm{min}}\\limits_{\\theta}D_{\\omega,f}(\\hat{r}^{*}\\|\\hat{r}_{\\theta}) \\\\ =\\sum_{\\boldsymbol{y}}\\pi_{ref}(\\partial f(\\hat{r}_{\\theta})\\hat{r}_{\\theta}-f(\\hat{r}_{\\theta})) \\\\ -\\sum_{\\boldsymbol{y}_{w}}\\pi_{c}\\partial f(\\hat{r}_{\\theta})\\prod\\limits_{i=1}^{t}\\frac{\\pi^{\\omega_{i}-1}_{c}(\\boldsymbol{y_{a_{i}}}\\|\\boldsymbol{x},\\boldsymbol{c}_{i-1})}{\\pi^{\\omega_{i}-1}_{ref}(\\boldsymbol{y_{a_{i}}}\\|\\boldsymbol{x},\\boldsymbol{c}_{i-1})},",
      "mathml": "<math display=\"block\"><mrow><mo movablelimits=\"false\" rspace=\"0.167em\">min</mo><msub><mrow></mrow><mi>θ</mi></msub><mi>D</mi><msub><mrow></mrow><mrow><mi>ω</mi><mo>,</mo><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><msup><mrow></mrow><mo>∗</mo></msup><mo lspace=\"0em\" rspace=\"0.167em\">∥</mo><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><msub><mrow></mrow><mi>θ</mi></msub><mo stretchy=\"false\">)</mo></mrow></math><math display=\"block\"><mrow><mphantom></mphantom><mo>=</mo><mrow><mstyle displaystyle=\"true\"><munder><mo movablelimits=\"false\">∑</mo><mi>𝒚</mi></munder></mstyle><mrow><msub><mi>π</mi><mrow><mi>r</mi><mo lspace=\"0em\" rspace=\"0em\">​</mo><mi>e</mi><mo lspace=\"0em\" rspace=\"0em\">​</mo><mi>f</mi></mrow></msub><mo lspace=\"0em\" rspace=\"0em\">​</mo><mrow><mo stretchy=\"false\">(</mo><mrow><mrow><mo lspace=\"0em\">∂</mo><mrow><mrow><mi>f</mi><mo>⁡</mo><mrow><mo stretchy=\"false\">(</mo><msub><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><mi>θ</mi></msub><mo stretchy=\"false\">)</mo></mrow></mrow><mo lspace=\"0em\" rspace=\"0em\">​</mo><msub><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><mi>θ</mi></msub></mrow></mrow><mo>−</mo><mrow><mi>f</mi><mo>⁡</mo><mrow><mo stretchy=\"false\">(</mo><msub><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><mi>θ</mi></msub><mo stretchy=\"false\">)</mo></mrow></mrow></mrow><mo stretchy=\"false\">)</mo></mrow></mrow></mrow></mrow></math><math display=\"block\"><mrow><mo>−</mo><mstyle displaystyle=\"true\"><mo movablelimits=\"false\">∑</mo></mstyle><msub><mrow></mrow><msub><mi>𝒚</mi><mi>w</mi></msub></msub><mi>π</mi><msub><mrow></mrow><mi>c</mi></msub><mo lspace=\"0em\">∂</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>r</mi><mo>^</mo></mover><msub><mrow></mrow><mi>θ</mi></msub><mo rspace=\"0.167em\" stretchy=\"false\">)</mo><mstyle displaystyle=\"true\"><mo movablelimits=\"false\">∏</mo></mstyle><msub><mrow></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow></msub><msup><mrow></mrow><mi>t</mi></msup><mstyle displaystyle=\"true\"><mfrac><mrow><mi>π</mi><msup><mrow></mrow><mrow><msub><mi>ω</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow></mrow><mi>c</mi></msub><mo stretchy=\"false\">(</mo><mi>𝒚</mi><msub><mrow></mrow><msub><mi>𝒂</mi><mi>𝒊</mi></msub></msub><mo lspace=\"0em\" rspace=\"0.167em\">∥</mo><mi>𝒙</mi><mo>,</mo><mi>𝒄</mi><msub><mrow></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy=\"false\">)</mo></mrow><mrow><mi>π</mi><msup><mrow></mrow><mrow><msub><mi>ω</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow></mrow><mrow><mi>r</mi><mo lspace=\"0em\" rspace=\"0em\">​</mo><mi>e</mi><mo lspace=\"0em\" rspace=\"0em\">​</mo><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>𝒚</mi><msub><mrow></mrow><msub><mi>𝒂</mi><mi>𝒊</mi></msub></msub><mo lspace=\"0em\" rspace=\"0.167em\">∥</mo><mi>𝒙</mi><mo>,</mo><mi>𝒄</mi><msub><mrow></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy=\"false\">)</mo></mrow></mfrac></mstyle><mo>,</mo></mrow></math>",
      "context": "Due to the fact that the possibility of the un-chosen trajectory by policy <math display=\"inline\"><msub><mi>π</mi><mi>c</mi></msub></math> is vanishingly small and <math display=\"inline\"><mrow><mi>ω</mi><mo>∈</mo><mrow><mo stretchy=\"false\">[</mo><mrow><msub><mi>ω</mi><mi>L</mi></msub><mo>,</mo><msub><mi>ω</mi><mi>U</mi></msub></mrow><mo stretchy=\"false\">]</mo></mrow></mrow></math>, Substituting (13)into (12) the equation yields:",
      "headline": "",
      "explanation": "",
      "objective": "",
      "symbols": {},
      "evidence": "",
      "evidence_url": "",
      "source_url": ""
    },
    {
      "anchor": "S3.E5",
      "label": "(5)",
      "section": "3 Notations and Preliminaries",
      "role": "baseline",
      "latex": "\\mathcal{L}_{DIL} =-\\mathbb{E}_{(\\boldsymbol{x},\\boldsymbol{y}_{w},\\boldsymbol{y}_{l}\\sim\\mathcal{D})}\\{\\partial f(r_{\\theta}(\\boldsymbol{y}_{w}))- \\\\ [\\partial f(r_{\\theta}(\\boldsymbol{y}_{l}))r_{\\theta}(\\boldsymbol{y}_{l})-f(r_{\\theta}(\\boldsymbol{y}_{l}))]\\}.",
      "mathml": "<math display=\"block\"><mrow><msub><mi>ℒ</mi><mrow><mi>D</mi><mo lspace=\"0em\" rspace=\"0em\">​</mo><mi>I</mi><mo lspace=\"0em\" rspace=\"0em\">​</mo><mi>L</mi></mrow></msub><mrow><mo rspace=\"0em\">=</mo><mo lspace=\"0em\">−</mo><mi>𝔼</mi><msub><mrow></mrow><mrow><mo stretchy=\"false\">(</mo><mi>𝒙</mi><mo>,</mo><msub><mi>𝒚</mi><mi>w</mi></msub><mo>,</mo><mrow><msub><mi>𝒚</mi><mi>l</mi></msub><mo>∼</mo><mi>𝒟</mi></mrow><mo stretchy=\"false\">)</mo></mrow></msub><mo stretchy=\"false\">{</mo><mo lspace=\"0em\">∂</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>r</mi><msub><mrow></mrow><mi>θ</mi></msub><mo stretchy=\"false\">(</mo><mi>𝒚</mi><msub><mrow></mrow><mi>w</mi></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo>−</mo></mrow></mrow></math><math display=\"block\"><mrow><mo stretchy=\"false\">[</mo><mo lspace=\"0em\">∂</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>r</mi><msub><mrow></mrow><mi>θ</mi></msub><mo stretchy=\"false\">(</mo><mi>𝒚</mi><msub><mrow></mrow><mi>l</mi></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mi>r</mi><msub><mrow></mrow><mi>θ</mi></msub><mo stretchy=\"false\">(</mo><mi>𝒚</mi><msub><mrow></mrow><mi>l</mi></msub><mo stretchy=\"false\">)</mo><mo>−</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>r</mi><msub><mrow></mrow><mi>θ</mi></msub><mo stretchy=\"false\">(</mo><mi>𝒚</mi><msub><mrow></mrow><mi>l</mi></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">]</mo><mo stretchy=\"false\">}</mo><mo lspace=\"0em\">.</mo></mrow></math>",
      "context": "Removing the irrelevant constant <math display=\"inline\"><mi>θ</mi></math> and under the assumption that <math display=\"inline\"><mrow><msub><mi>𝒚</mi><mi>l</mi></msub><mo>∼</mo><msub><mi>π</mi><mrow><mi>r</mi><mo lspace=\"0em\" rspace=\"0em\">​</mo><mi>e</mi><mo lspace=\"0em\" rspace=\"0em\">​</mo><mi>f</mi></mrow></msub><mrow><mo stretchy=\"false\">(</mo><msub><mi>𝒚</mi><mi>l</mi></msub><mo lspace=\"0em\" rspace=\"0.167em\">∥</mo><mi>𝒙</mi><mo stretchy=\"false\">)</mo></mrow></mrow></math>, we then get the loss function of the DIL method:",
      "headline": "",
      "explanation": "",
      "objective": "",
      "symbols": {},
      "evidence": "",
      "evidence_url": "",
      "source_url": ""
    }
  ],
  "equations_status": "selected",
  "core_kind": "formula",
  "equations_fetched_at": "2026-10-15",
  "equations_source_url": "",
  "equations_source_sha256": "",
  "exposures_reviewed_at": "",
  "exposures_source_sha256": ""
}