equations-source.json

{
  "renderer": "matplotlib mathtext, STIX fonts, vector paths",
  "formulas": {
    "decision-distribution": [
      "p_d(a_i\\mid x,A)=\\frac{\\exp(z_{d,i})}{\\sum_{j=1}^{K}\\exp(z_{d,j})}"
    ],
    "joint-depth-objective": [
      "\\mathcal{L}_{\\mathrm{decision}}=0.5\\,\\mathrm{CE}(q,p_{\\mathrm{low}})+0.5\\,\\mathrm{CE}(q,p_{\\mathrm{high}})",
      "\\qquad +0.1\\,\\mathrm{KL}\\!\\left(\\mathrm{stopgrad}(p_{\\mathrm{high}})\\,\\Vert\\,p_{\\mathrm{low}}\\right)"
    ],
    "dcrl-objective": [
      "\\mathcal{L}_{\\mathrm{DCRL}}=\\frac{1}{|\\mathcal{D}|}\\sum_{d\\in\\mathcal{D}}\\left[\\mathrm{NLL}(q,p_d)+\\lambda_B\\,\\mathrm{Brier}(q,p_d)\\right]",
      "\\qquad +\\lambda_{\\mathrm{ref}}\\,\\mathrm{KL}(p_{\\mathrm{high}}\\,\\Vert\\,p_{\\mathrm{reference}})",
      "\\qquad +\\lambda_U\\,\\mathcal{L}_{\\mathrm{utility}}(p_{\\mathrm{low}})"
    ],
    "expected-utility": [
      "\\mathcal{L}_{\\mathrm{utility}}=-\\sum_{a\\in A(x)}p_{\\mathrm{low}}(a)\\,u(a)"
    ],
    "candidate-projection": [
      "z_C=W[C]h_d+b[C]=(Wh_d+b)[C]"
    ],
    "projection-cost": [
      "\\mathrm{Full\\ vocabulary}:\\ O(HV)\\qquad\\mathrm{Candidate\\ rows}:\\ O(HK)"
    ]
  }
}