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1 for k =1,2,... do
2 fori∈1..n do
3t=2k+Tw
4=JrA（t）
5=T-1（t-x）
6rk=B(zk)+∈k
7
8yn+1=B（+1）+ek // ek is unknown noise term n+1
9w+1=w- 1 )n+1
101x） i∈1..(n+1) n+1 ≥i=1
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\\quad \\quad \\quad \\quad \\quad } \\\\ & { \\leq 6 \\bar { \\rho } ^ { 2 } \\big ( 2 ( N + 1 ) L ^ { 2 } \\| p ^ { k } - p ^ { * } \\| ^ { 2 } + \\| w _ { n + 1 } ^ { k } - w _ { n + 1 } ^ { * } \\| ^ { 2 } \\big ) } \\\\ & { \\qquad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad + 1 8 \\bar { \\rho } ^ { 2 } ( N + 1 ) \\| B ( z ^ { * } ) \\| ^ { 2 } + 3 \\bar { \\rho } ^ { 2 } N } \\\\ & { \\leq 1 8 \\bar { \\rho } ^ { 2 } ( N + 1 ) \\big ( ( L ^ { 2 } + 1 ) \\| p ^ { k } - p ^ { * } \\| ^ { 2 } + \\| B ( z ^ { * } ) \\| ^ { 2 } \\big ) + 3 \\bar { \\rho } ^ { 2 } N } \\end{array}" }, { "category_id": 14, "poly": [ 468, 272, 1230, 272, 1230, 458, 468, 458 ], "score": 0.94, "latex": "\\begin{array} { r l } & { \\| B ( x _ { n + 1 } ^ { k } ) \\| ^ { 2 } = \\| B ( z ^ { k } ) + B ( x _ { n + 1 } ^ { k } ) - B ( z ^ { k } ) \\| ^ { 2 } } \\\\ & { \\qquad \\leq 2 \\| B ( z ^ { k } ) \\| ^ { 2 } + 2 \\| B ( x _ { n + 1 } ^ { k } ) - B ( z ^ { k } ) \\| ^ { 2 } } \\\\ & { \\qquad \\leq 2 \\| B ( z ^ { k } ) \\| ^ { 2 } + 2 L ^ { 2 } \\| x _ { n + 1 } ^ { k } - z ^ { k } \\| ^ { 2 } } \\\\ & { \\qquad \\leq 4 L ^ { 2 } \\| p ^ { k } - p ^ { * } \\| ^ { 2 } + 4 \\| B ( z ^ { * } ) \\| ^ { 2 } + 2 L ^ { 2 } \\| x _ { n + 1 } ^ { k } - z ^ { k } \\| ^ { 2 } } \\end{array}" }, { "category_id": 14, "poly": [ 447, 836, 1250, 836, 1250, 933, 447, 933 ], "score": 0.93, "latex": "\\begin{array} { r l } & { \\mathbb { E } \\big [ \\| x _ { n + 1 } ^ { k } - z ^ { k } \\| ^ { 2 } | \\mathcal { F } _ { k } \\big ] \\leq \\mathbb { E } \\big [ 3 \\overline { { \\rho } } ^ { 2 } \\big ( \\| B ( z ^ { k } ) \\| ^ { 2 } + \\| \\epsilon ^ { k } \\| ^ { 2 } + \\| w _ { n + 1 } ^ { k } \\| ^ { 2 } \\big ) | \\mathcal { F } _ { k } \\big ] } \\\\ & { \\qquad \\leq 3 \\overline { { \\rho } } ^ { 2 } \\big ( ( N + 1 ) \\| B ( z ^ { k } ) \\| ^ { 2 } + \\| w _ { n + 1 } ^ { k } \\| ^ { 2 } + N \\big ) . } \\end{array}" }, { "category_id": 13, "poly": [ 691, 1295, 864, 1295, 864, 1330, 691, 1330 ], "score": 0.93, "latex": "w _ { n + 1 } ^ { * } = B ( z ^ { * } )" }, { "category_id": 14, "poly": [ 537, 648, 1161, 648, 1161, 789, 537, 789 ], "score": 0.93, "latex": "\\begin{array} { r l } & { \\| x _ { n + 1 } ^ { k } - z ^ { k } \\| ^ { 2 } = \\rho _ { k } ^ { 2 } \\| B ( z ^ { k } ) + \\epsilon ^ { k } - w _ { n + 1 } ^ { k } \\| ^ { 2 } } \\\\ & { \\qquad \\leq \\overline { { \\rho } } ^ { 2 } \\| B ( z ^ { k } ) + \\epsilon ^ { k } - w _ { n + 1 } ^ { k } \\| ^ { 2 } } \\\\ & { \\qquad \\leq 3 \\overline { { \\rho } } ^ { 2 } ( \\| B ( z ^ { k } ) \\| ^ { 2 } + \\| \\epsilon ^ { k } \\| ^ { 2 } + \\| w _ { n + 1 } ^ { k } \\| ^ { 2 } ) . } \\end{array}" }, { "category_id": 14, "poly": [ 310, 1571, 1400, 1571, 1400, 2039, 310, 2039 ], "score": 0.92, "latex": "\\begin{array} { r l r } { { \\sum _ { i = 1 } ^ { n } y _ { i } ^ { k } \\bigg \\| ^ { 2 } = \\bigg \\| \\sum _ { i = 1 } ^ { n } ( \\tau ^ { - 1 } ( z ^ { k } - x _ { i } ^ { k } ) + w _ { i } ^ { k } ) \\bigg \\| ^ { 2 } } } \\\\ & { } & { \\leq 2 \\bigg \\| \\tau ^ { - 1 } \\sum _ { i = 1 } ^ { n } ( z ^ { k } - x _ { i } ^ { k } ) \\bigg \\| ^ { 2 } + 2 \\bigg \\| \\sum _ { i = 1 } ^ { n } w _ { i } ^ { k } \\bigg \\| ^ { 2 } } \\\\ & { } & { \\leq 2 \\pi \\tau ^ { - 2 } \\sum _ { i = 1 } ^ { n } \\| z ^ { k } - x _ { i } ^ { k } \\| ^ { 2 } + 2 \\bigg \\| \\sum _ { i = 1 } ^ { n } w _ { i } ^ { k } \\bigg \\| ^ { 2 } } \\\\ & { } & { \\leq 4 \\pi ^ { 2 } \\tau ^ { - 2 } \\| z ^ { k } - z ^ { * } \\| ^ { 2 } + 4 \\pi \\tau ^ { - 2 } \\sum _ { i = 1 } ^ { n } \\| z ^ { * } - x _ { i } ^ { k } \\| ^ { 2 } + 4 m \\sum _ { i = 1 } ^ { n } \\| w _ { i } ^ { k } - w _ { i } ^ { * } \\| ^ { 2 } + 4 \\bigg \\| \\sum _ { i = 1 } ^ { n } w _ { i } ^ { * } \\bigg \\| ^ { 2 } } \\\\ & { } & { \\leq 4 \\pi ^ { 2 } ( \\tau ^ { - 2 } + 1 ) \\| y ^ { k } - p ^ { * } \\| ^ { 2 } + 4 \\pi \\tau ^ { - 2 } \\sum _ { i = 1 } ^ { n } \\| z ^ { * } - x _ { i } ^ { k 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k } \\| ^ { 2 } + \\tau ^ { - 1 } \\sum _ { i = 1 } ^ { n } \\mathbb { E } \\| z ^ { k } - x _ { i } ^ { k } \\| ^ { 2 } + \\rho \\mathbb { E } \\| B ( z ^ { k } ) - w _ { n + 1 } ^ { k } \\| ^ { 2 } ." }, { "category_id": 14, "poly": [ 319, 557, 1340, 557, 1340, 756, 319, 756 ], "score": 0.92, "latex": "\\begin{array} { l } { \\displaystyle \\frac { 1 } { K } \\sum _ { j = 1 } ^ { K } \\mathbb { E } [ G _ { j } ] \\leq \\frac { 8 L ^ { 3 } \\exp \\left( C _ { f } ( C _ { 1 } + C _ { 3 } ) \\right) } { C _ { f } \\operatorname* { m i n } \\{ \\tau , \\tau ^ { - 1 } \\} K } \\left( \\| p ^ { 1 } - p ^ { * } \\| ^ { 2 } + \\frac { C _ { f } C _ { 2 } + C _ { 4 } } { C _ { f } C _ { 1 } + C _ { 3 } } \\right) \\mathrm { ~ } f o r ~ K < ( 2 L ) ^ { 4 } } \\\\ { \\displaystyle \\frac { 1 } { K } \\sum _ { j = 1 } ^ { K } \\mathbb { E } [ G _ { j } ] \\leq \\frac { \\exp \\left( C _ { f } ( C _ { 1 } + C _ { 3 } ) \\right) } { C _ { f } \\operatorname* { m i n } \\{ \\tau , \\tau ^ { - 1 } 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