第 27888 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":[1,2],"linear":[1,3],"point":[0,2],"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":[1,2],"linear":[1,3],"point":[0,2],"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, 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(2, 2)

GD ORIGINAL ORIGINAL gradient d*x coordinate1 = 4

GD ORIGINAL ORIGINAL gradient plus b coordinate1 := sum(4, 3)

GD ORIGINAL ORIGINAL gradient plus b coordinate1 = 7

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(1, 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(2, 4)

GD ORIGINAL ORIGINAL diagonal times square coordinate1 = 8

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 := div(8, 2)

GD ORIGINAL ORIGINAL quadratic divide2 coordinate1 = 4

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

GD ORIGINAL ORIGINAL linear product coordinate1 = 6

GD ORIGINAL ORIGINAL coordinate objective sum1 := sum(4, 6)

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, [1, 2], [1, 3], [0, 2]]

derived = [[1, 2], [1, 3], [0, 2], [1, 7], 10]

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

GD safe ORIGINAL2 over largest diagonal = 1

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(0, 1/4)

GD safe ORIGINAL step1 ORIGINAL next coordinate0 = -1/4

GD safe ORIGINAL step1 ORIGINAL alpha times gradient1 := mul(1/4, 7)

GD safe ORIGINAL step1 ORIGINAL alpha times gradient1 = 7/4

GD safe ORIGINAL step1 ORIGINAL next coordinate1 := sub(2, 7/4)

GD safe ORIGINAL step1 ORIGINAL next coordinate1 = 1/4

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

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate0 = -1/4

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate0 := sum(-1/4, 1)

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(2, 1/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient d*x coordinate1 = 1/2

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate1 := sum(1/2, 3)

GD safe ORIGINAL step1 next recomputation ORIGINAL gradient plus b coordinate1 = 7/2

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 := mul(-1/4, -1/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective SQUARE FIRST coordinate0 = 1/16

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

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate0 = 1/16

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate0 := div(1/16, 2)

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate0 = 1/32

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate0 := mul(1, -1/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate0 = -1/4

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum0 := sum(1/32, -1/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum0 = -7/32

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum0 := sum(0, -7/32)

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum0 = -7/32

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(2, 1/16)

GD safe ORIGINAL step1 next recomputation ORIGINAL diagonal times square coordinate1 = 1/8

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate1 := div(1/8, 2)

GD safe ORIGINAL step1 next recomputation ORIGINAL quadratic divide2 coordinate1 = 1/16

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate1 := mul(3, 1/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL linear product coordinate1 = 3/4

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum1 := sum(1/16, 3/4)

GD safe ORIGINAL step1 next recomputation ORIGINAL coordinate objective sum1 = 13/16

GD safe ORIGINAL step1 next recomputation ORIGINAL objective running sum1 := sum(-7/32, 13/16)

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

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

derived = [[1, 2], [1, 3], [-1/4, 1/4], [3/4, 7/2], 19/32]

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

derived = [f(x-alpha*g)-f(x)=-alpha*sum_i(1-alpha*d_i/2)*g_i^2, [[0, [0, 2], 10, [1, 7]], [1, [-1/4, 1/4], 19/32, [3/4, 7/2]]], 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

(2, 1/4, 1, 1, [[0, [0, 2], 10, [1, 7]], [1, [-1/4, 1/4], 19/32, [3/4, 7/2]]], 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。