第 27897 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-safe-quadratic-trace；明示資料={"diagonal":[1,3],"linear":[0,-2],"point":[-1,-3],"step_size":[1,4],"steps":1}；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-safe-quadratic-trace；明示資料={"diagonal":[1,3],"linear":[0,-2],"point":[-1,-3],"step_size":[1,4],"steps":1}；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-safe-quadratic-trace]

derived = [[largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope]]

GD ORIGINAL ORIGINAL gradient d*x coordinate0 := mul(1, -1)

GD ORIGINAL ORIGINAL gradient d*x coordinate0 = -1

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

GD ORIGINAL ORIGINAL gradient plus b coordinate0 = -1

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

GD ORIGINAL ORIGINAL gradient d*x coordinate1 = -9

GD ORIGINAL ORIGINAL gradient plus b coordinate1 := sum(-9, -2)

GD ORIGINAL ORIGINAL gradient plus b coordinate1 = -11

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 := mul(-1, -1)

GD ORIGINAL ORIGINAL objective SQUARE FIRST coordinate0 = 1

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

GD ORIGINAL ORIGINAL diagonal times square coordinate0 = 1

GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 := div(1, 2)

GD ORIGINAL ORIGINAL quadratic divide2 coordinate0 = 1/2

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

GD ORIGINAL ORIGINAL linear product coordinate0 = 0

GD ORIGINAL ORIGINAL coordinate objective sum0 := sum(1/2, 0)

GD ORIGINAL ORIGINAL coordinate objective sum0 = 1/2

GD ORIGINAL ORIGINAL objective running sum0 := sum(0, 1/2)

GD ORIGINAL ORIGINAL objective running sum0 = 1/2

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(3, 9)

GD ORIGINAL ORIGINAL diagonal times square coordinate1 = 27

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 := div(27, 2)

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 = 27/2

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

GD ORIGINAL ORIGINAL linear product coordinate1 = 6

GD ORIGINAL ORIGINAL coordinate objective sum1 := sum(27/2, 6)

GD ORIGINAL ORIGINAL coordinate objective sum1 = 39/2

GD ORIGINAL ORIGINAL objective running sum1 := sum(1/2, 39/2)

GD ORIGINAL ORIGINAL objective running sum1 = 20

source roles = [GD ORIGINAL, [1, 3], [0, -2], [-1, -3]]

derived = [[1, 3], [0, -2], [-1, -3], [-1, -11], 20]

GD safe ORIGINAL2 over largest diagonal := div(2, 3)

GD safe ORIGINAL2 over largest diagonal = 2/3

GD safe ORIGINAL step1 ORIGINAL alpha times gradient0 := mul(1/4, -1)

GD safe ORIGINAL step1 ORIGINAL alpha times gradient0 = -1/4

GD safe ORIGINAL step1 ORIGINAL next coordinate0 := sub(-1, -1/4)

GD safe ORIGINAL step1 ORIGINAL next coordinate0 = -3/4

GD safe ORIGINAL step1 ORIGINAL alpha times gradient1 := mul(1/4, -11)

GD safe ORIGINAL step1 ORIGINAL alpha times gradient1 = -11/4

GD safe ORIGINAL step1 ORIGINAL next coordinate1 := sub(-3, -11/4)

GD safe ORIGINAL step1 ORIGINAL next coordinate1 = -1/4

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate0 := mul(1, -3/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate0 = -3/4

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate0 := sum(-3/4, 0)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate0 = -3/4

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate1 := mul(3, -1/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate1 = -3/4

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate1 := sum(-3/4, -2)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate1 = -11/4

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 := mul(-3/4, -3/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 = 9/16

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate0 := mul(1, 9/16)

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate0 = 9/16

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate0 := div(9/16, 2)

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate0 = 9/32

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate0 := mul(0, -3/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate0 = 0

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum0 := sum(9/32, 0)

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum0 = 9/32

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum0 := sum(0, 9/32)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum0 = 9/32

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 := mul(-1/4, -1/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1 = 1/16

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate1 := mul(3, 1/16)

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate1 = 3/16

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate1 := div(3/16, 2)

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate1 = 3/32

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate1 := mul(-2, -1/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate1 = 1/2

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum1 := sum(3/32, 1/2)

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum1 = 19/32

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum1 := sum(9/32, 19/32)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum1 = 7/8

source roles = [GD safe ORIGINAL step1 next recomputation, [[1, 1], [3, 1]], [[0, 1], [-2, 1]], [[-3, 4], [-1, 4]]]

derived = [[1, 3], [0, -2], [-3/4, -1/4], [-3/4, -11/4], 7/8]

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

derived = [f(x-alpha*g)-f(x)=-alpha*sum_i(1-alpha*d_i/2)*g_i^2, [[0, [-1, -3], 20, [-1, -11]], [1, [-3/4, -1/4], 7/8, [-3/4, -11/4]]], 是, 否]

逐步反向核對

• S041 · 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。
• 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 — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum1，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum1，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate1，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate1，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate1，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate1，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum0，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum0，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate0，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate0，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate0，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate1，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate1，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate0，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate0，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 ORIGINAL next coordinate1，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 ORIGINAL alpha times gradient1，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 ORIGINAL next coordinate0，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL step1 ORIGINAL alpha times gradient0，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 GD safe ORIGINAL2 over largest diagonal，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

(3, 1/4, 2/3, 1, [[0, [-1, -3], 20, [-1, -11]], [1, [-3/4, -1/4], 7/8, [-3/4, -11/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)

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。