{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-gradient-descent",
  "semantic_family_id": "sf-gradient-descent",
  "type": "algorithm",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "梯度下降法"
  },
  "summary": {
    "zh-Hant": "在固定Euclidean coordinates中以正步長沿負梯度更新；下降與收斂仍需目標及步長假設。"
  },
  "statement": {
    "zh-Hant": "對可微f:R^d到R及明示alpha_k>0，unconstrained gradient descent更新x_{k+1}=x_k-alpha_k grad f(x_k)。本輪exact slice固定coordinate order，使用positive diagonal quadratic f(x)=1/2 sum d_i x_i^2+sum b_i x_i，逐步重算gradient、iterate與objective。若0<alpha<2/L且L=max d_i，有限trace可逐項核objective nonincrease；這仍不是一般函數或未驗證無限序列的收斂證書。positive step本身不保證下降，zero gradient可出現在maximum或saddle，且unconstrained step不得冒充projected或constrained descent。"
  },
  "formula": {
    "tex": "G(f,x,a)=y",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>G</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow><mo>=</mo><mi>y</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "G",
          "glyph": "G"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "f",
            "glyph": "f"
          },
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          },
          {
            "type": "symbol",
            "name": "a",
            "glyph": "a"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "y",
        "glyph": "y"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "8ee26333423b907e5af5c01e1ffff276b754e74b8e258d42699dbc443d6d4ef5"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "positive diagonal與linear vectors arity相同，coordinate order固定；所有資料為bounded exact rationals。"
    },
    {
      "id": "a2",
      "zh-Hant": "one-step只要求alpha>0；safe trace另核0<alpha<2/L。"
    },
    {
      "id": "a3",
      "zh-Hant": "只處理unconstrained deterministic gradient descent，不含projection、stochastic gradient或line search。"
    }
  ],
  "symbols": [
    {
      "token": "G",
      "role": "gradient_descent_update",
      "role_zh": "梯度下降更新",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "differentiable_objective",
      "role_zh": "可微目標",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "current_vector",
      "role_zh": "目前向量",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "positive_step_size",
      "role_zh": "正步長",
      "scope": "representative_formula"
    },
    {
      "token": "y",
      "role": "next_vector",
      "role_zh": "下一向量",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-gradient-vector",
      "title_zh": "梯度向量",
      "reason_zh": "負梯度更新。"
    },
    {
      "id": "mko-optimization-problem",
      "title_zh": "最佳化問題",
      "reason_zh": "目標與minimization語義。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "iterate trace。"
    },
    {
      "id": "mko-real-vector-space",
      "title_zh": "實向量空間",
      "reason_zh": "Euclidean vector update與內積。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "coordinate order、objective與step size全部明示。",
      "safe diagonal quadratic trace逐步驗證，不以alpha>0代替下降條件。",
      "stationary、local/global optimum及convergence是分離責任。"
    ],
    "common_misconception_zh": "只因alpha>0就宣稱下降，或把zero gradient及有限trace當global optimum與收斂證明。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪不宣稱一般nonconvex／convex convergence rate，也不包含projection或stochastic updates。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入L-smooth convex descent lemma、strong convexity rate與projected gradient。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-gradient-descent.md 整理；2026-10-15 脈衝 R03（使用者手動接續 2026-11-19 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}