第 27826 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":[1,2],"linear":[-1,-2],"point":[-2,-3],"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":[1,2],"linear":[-1,-2],"point":[-2,-3],"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(1, -2)

GD ORIGINAL ORIGINAL gradient d*x coordinate0 = -2

GD ORIGINAL ORIGINAL gradient plus b coordinate0 := sum(-2, -1)

GD ORIGINAL ORIGINAL gradient plus b coordinate0 = -3

GD ORIGINAL ORIGINAL gradient d*x coordinate1 := mul(2, -3)

GD ORIGINAL ORIGINAL gradient d*x coordinate1 = -6

GD ORIGINAL ORIGINAL gradient plus b coordinate1 := sum(-6, -2)

GD ORIGINAL ORIGINAL gradient plus b coordinate1 = -8

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 := mul(-2, -2)

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 = 4

GD ORIGINAL ORIGINAL diagonal times square coordinate0 := mul(1, 4)

GD ORIGINAL ORIGINAL diagonal times square coordinate0 = 4

GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 := div(4, 2)

GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 = 2

GD ORIGINAL ORIGINAL linear product coordinate0 := mul(-1, -2)

GD ORIGINAL ORIGINAL linear product coordinate0 = 2

GD ORIGINAL ORIGINAL coordinate objective sum0 := sum(2, 2)

GD ORIGINAL ORIGINAL coordinate objective sum0 = 4

GD ORIGINAL ORIGINAL objective running sum0 := sum(0, 4)

GD ORIGINAL ORIGINAL objective running sum0 = 4

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 := mul(-3, -3)

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate1 = 9

GD ORIGINAL ORIGINAL diagonal times square coordinate1 := mul(2, 9)

GD ORIGINAL ORIGINAL diagonal times square coordinate1 = 18

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 := div(18, 2)

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 = 9

GD ORIGINAL ORIGINAL linear product coordinate1 := mul(-2, -3)

GD ORIGINAL ORIGINAL linear product coordinate1 = 6

GD ORIGINAL ORIGINAL coordinate objective sum1 := sum(9, 6)

GD ORIGINAL ORIGINAL coordinate objective sum1 = 15

GD ORIGINAL ORIGINAL objective running sum1 := sum(4, 15)

GD ORIGINAL ORIGINAL objective running sum1 = 19

source roles = [GD ORIGINAL, [1, 2], [-1, -2], [-2, -3]]

derived = [[1, 2], [-1, -2], [-2, -3], [-3, -8], 19]

GD ORIGINAL one step ORIGINAL alpha times gradient0 := mul(1/4, -3)

GD ORIGINAL one step ORIGINAL alpha times gradient0 = -3/4

GD ORIGINAL one step ORIGINAL next coordinate0 := sub(-2, -3/4)

GD ORIGINAL one step ORIGINAL next coordinate0 = -5/4

GD ORIGINAL one step ORIGINAL alpha times gradient1 := mul(1/4, -8)

GD ORIGINAL one step ORIGINAL alpha times gradient1 = -2

GD ORIGINAL one step ORIGINAL next coordinate1 := sub(-3, -2)

GD ORIGINAL one step ORIGINAL next coordinate1 = -1

GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate0 := mul(1, -5/4)

GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate0 = -5/4

GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate0 := sum(-5/4, -1)

GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate0 = -9/4

GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate1 := mul(2, -1)

GD ORIGINAL one step next recomputation ORIGINAL gradient d*x coordinate1 = -2

GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate1 := sum(-2, -2)

GD ORIGINAL one step next recomputation ORIGINAL gradient plus b coordinate1 = -4

GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate0 := mul(-5/4, -5/4)

GD ORIGINAL one step next recomputation ORIGINAL objective SQUARE FIRST coordinate0 = 25/16

GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate0 := mul(1, 25/16)

GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate0 = 25/16

GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate0 := div(25/16, 2)

GD ORIGINAL one step next recomputation ORIGINAL quadratic divide2 coordinate0 = 25/32

GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate0 := mul(-1, -5/4)

GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate0 = 5/4

GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum0 := sum(25/32, 5/4)

GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum0 = 65/32

GD ORIGINAL one step next recomputation ORIGINAL objective running sum0 := sum(0, 65/32)

GD ORIGINAL one step next recomputation ORIGINAL objective running sum0 = 65/32

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

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

GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate1 := mul(2, 1)

GD ORIGINAL one step next recomputation ORIGINAL diagonal times square coordinate1 = 2

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

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

GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate1 := mul(-2, -1)

GD ORIGINAL one step next recomputation ORIGINAL linear product coordinate1 = 2

GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum1 := sum(1, 2)

GD ORIGINAL one step next recomputation ORIGINAL coordinate objective sum1 = 3

GD ORIGINAL one step next recomputation ORIGINAL objective running sum1 := sum(65/32, 3)

GD ORIGINAL one step next recomputation ORIGINAL objective running sum1 = 161/32

source roles = [GD ORIGINAL one step next recomputation, [[1, 1], [2, 1]], [[-1, 1], [-2, 1]], [[-5, 4], [-1, 1]]]

derived = [[1, 2], [-1, -2], [-5/4, -1], [-9/4, -4], 161/32]

source roles = [[-2, -3], [-3, -8], 1/4]

derived = [[-5/4, -1], 19, 161/32, 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

([-2, -3], [-3, -8], 1/4, [-5/4, -1], 19, 161/32, 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。