第 27848 題：梯度下降

原題與已知條件

standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[2,4],"linear":[1,-1],"point":[0,-2],"step_size":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。

• G001 — standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={"diagonal":[2,4],"linear":[1,-1],"point":[0,-2],"step_size":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-gradient-descent

逐步推導

source roles = [gd-one-step]

derived = [[point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope]]

GD ORIGINAL ORIGINAL gradient d*x coordinate0 := mul(2, 0)

GD ORIGINAL ORIGINAL gradient d*x coordinate0 = 0

GD ORIGINAL ORIGINAL gradient plus b coordinate0 := sum(0, 1)

GD ORIGINAL ORIGINAL gradient plus b coordinate0 = 1

GD ORIGINAL ORIGINAL gradient d*x coordinate1 := mul(4, -2)

GD ORIGINAL ORIGINAL gradient d*x coordinate1 = -8

GD ORIGINAL ORIGINAL gradient plus b coordinate1 := sum(-8, -1)

GD ORIGINAL ORIGINAL gradient plus b coordinate1 = -9

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 := mul(0, 0)

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 = 0

GD ORIGINAL ORIGINAL diagonal times square coordinate0 := mul(2, 0)

GD ORIGINAL ORIGINAL diagonal times square coordinate0 = 0

GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 := div(0, 2)

GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 = 0

GD ORIGINAL ORIGINAL linear product coordinate0 := mul(1, 0)

GD ORIGINAL ORIGINAL linear product coordinate0 = 0

GD ORIGINAL ORIGINAL coordinate objective sum0 := sum(0, 0)

GD ORIGINAL ORIGINAL coordinate objective sum0 = 0

GD ORIGINAL ORIGINAL objective running sum0 := sum(0, 0)

GD ORIGINAL ORIGINAL objective running sum0 = 0

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 := mul(-2, -2)

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 = 4

GD ORIGINAL ORIGINAL diagonal times square coordinate1 := mul(4, 4)

GD ORIGINAL ORIGINAL diagonal times square coordinate1 = 16

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 := div(16, 2)

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 = 8

GD ORIGINAL ORIGINAL linear product coordinate1 := mul(-1, -2)

GD ORIGINAL ORIGINAL linear product coordinate1 = 2

GD ORIGINAL ORIGINAL coordinate objective sum1 := sum(8, 2)

GD ORIGINAL ORIGINAL coordinate objective sum1 = 10

GD ORIGINAL ORIGINAL objective running sum1 := sum(0, 10)

GD ORIGINAL ORIGINAL objective running sum1 = 10

source roles = [GD ORIGINAL, [2, 4], [1, -1], [0, -2]]

derived = [[2, 4], [1, -1], [0, -2], [1, -9], 10]

GD ORIGINAL one step ORIGINAL alpha times gradient0 := mul(1/4, 1)

GD ORIGINAL one step ORIGINAL alpha times gradient0 = 1/4

GD ORIGINAL one step ORIGINAL next coordinate0 := sub(0, 1/4)

GD ORIGINAL one step ORIGINAL next coordinate0 = -1/4

GD ORIGINAL one step ORIGINAL alpha times gradient1 := mul(1/4, -9)

GD ORIGINAL one step ORIGINAL alpha times gradient1 = -9/4

GD ORIGINAL one step ORIGINAL next coordinate1 := sub(-2, -9/4)

GD ORIGINAL one step ORIGINAL next coordinate1 = 1/4

GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate0 := mul(2, -1/4)

GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate0 = -1/2

GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate0 := sum(-1/2, 1)

GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate0 = 1/2

GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate1 := mul(4, 1/4)

GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate1 = 1

GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate1 := sum(1, -1)

GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate1 = 0

GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate0 := mul(-1/4, -1/4)

GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate0 = 1/16

GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate0 := mul(2, 1/16)

GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate0 = 1/8

GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate0 := div(1/8, 2)

GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate0 = 1/16

GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate0 := mul(1, -1/4)

GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate0 = -1/4

GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum0 := sum(1/16, -1/4)

GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum0 = -3/16

GD ORIGINAL one step next recomputation ORIGINAL objective running sum0 := sum(0, -3/16)

GD ORIGINAL one step next recomputation ORIGINAL objective running sum0 = -3/16

GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate1 := mul(1/4, 1/4)

GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate1 = 1/16

GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate1 := mul(4, 1/16)

GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate1 = 1/4

GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate1 := div(1/4, 2)

GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate1 = 1/8

GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate1 := mul(-1, 1/4)

GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate1 = -1/4

GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum1 := sum(1/8, -1/4)

GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum1 = -1/8

GD ORIGINAL one step next recomputation ORIGINAL objective running sum1 := sum(-3/16, -1/8)

GD ORIGINAL one step next recomputation ORIGINAL objective running sum1 = -5/16

source roles = [GD ORIGINAL one step next recomputation, [[2, 1], [4, 1]], [[1, 1], [-1, 1]], [[-1, 4], [1, 4]]]

derived = [[2, 4], [1, -1], [-1/4, 1/4], [1/2, 0], -5/16]

source roles = [[0, -2], [1, -9], 1/4]

derived = [[-1/4, 1/4], 10, -5/16, 是]

逐步反向核對

• S040 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S039 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S038 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step next recomputation ORIGINAL objective running sum1，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum1，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate1，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate1，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate1，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate1，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step next recomputation ORIGINAL objective running sum0，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum0，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate0，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate0，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate0，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate0，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate1，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate1，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate0，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate0，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step ORIGINAL next coordinate1，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step ORIGINAL alpha times gradient1，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step ORIGINAL next coordinate0，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL one step ORIGINAL alpha times gradient0，不以小數近似替代。
• S018 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S017 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL objective running sum1，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL coordinate objective sum1，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL linear product coordinate1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL quadratic divide2 coordinate1，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL diagonal times square coordinate1，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL objective running sum0，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL coordinate objective sum0，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL linear product coordinate0，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL quadratic divide2 coordinate0，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL diagonal times square coordinate0，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL gradient plus b coordinate1，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL gradient d*x coordinate1，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL gradient plus b coordinate0，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 GD ORIGINAL ORIGINAL gradient d*x coordinate0，不以小數近似替代。
• S001 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。

結論

([0, -2], [1, -9], 1/4, [-1/4, 1/4], 10, -5/16, 是, positive_step_alone_does_not_guarantee_descent)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。