{"id": "ch10_lemma_10_1__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\ndefinition 12.1. If \\(\\mathcal{C}\\) is a VC class of sets, then F is VC-subgraph class of functions and, by definition, \\(V C(\\mathcal{F})=V C(\\mathcal{C})\\)\nFor \\(f:\\mathcal{X}\\to\\mathbb{R}\\) , the subgraph of \\(f\\) is\n\\[C_{f}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}:0\\leq t\\leq f(x)\\ o r\\ f(x)\\leq t\\leq 0\\}.\\]\nDefine class of sets \\(\\mathcal{C}=\\{C_{f}:f\\in\\mathcal{F}\\}\\).\nNote that equivalent definition of \\(C_{f}\\) is\n\\[C_{f}^{\\prime}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}: |f(x)|\\geq|t|\\}.\\]\n\ndefinition 12.3. The ε-packing number, \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) , is the marimal cardinality of an E-separated set.\nNote that \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) is decreasing in ε.\n\ndefinition 12.2. Given \\(\\varepsilon > 0\\) and \\(f_{1},\\ldots,f_{N}\\in\\mathcal{F}\\) ， we say that \\(f_{1},\\ldots,f_{N}\\) are E-separated if \\(d(f_{i},f_{j})>\\varepsilon\\) for any \\(i \\neq j\\)\n\nDenote \\[\\triangle_{n}(\\mathcal{C})=\\sup_{\\{x_{1},\\ldots,x_{n}\\}}\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})\\leq 2^{n}.\\] We will see that for some classes, \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\) for \\(n \\leq V\\) and \\(\\triangle_{n}(\\mathcal{C})<2^{n}\\) for \\(n > V\\) for some constant \\(V\\).\n\ndefinition 8.1. If \\(V < \\infty\\) , then C is called a VC class. V is called VC dimension of C.\n\ndefinition 8.0. For any sample \\(\\{x_{1},\\ldots,x_{n}\\}\\), the ways that \\({\\mathcal{C}}\\) intersects with the sample are \\[\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in{\\mathcal{C}}\\}.\\] Let \\[\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})=\\mathrm{card}~\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in\\mathcal{C}\\},\\] the number of different subsets picked out by \\(C\\in{\\mathcal{C}}\\). If \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\), this means we can find \\(\\{x_{1},\\ldots,x_{n}\\}\\) such that \\({\\mathcal{C}}\\) can pick out any subset of it; in this case \\({\\mathcal{C}}\\) shatters \\(\\{x_{1},\\ldots,x_{n}\\}\\). Equivalently, a set \\(A\\) is shattered by \\({\\mathcal{C}}\\) when \\({\\mathcal{C}}\\) picks out every subset of \\(A\\).", "theorems": [{"local_id": "T1", "statement": "Setup. Let \\(0<\\varepsilon\\le 1\\).\n\nAssume \\(\\mathcal{F}\\) is a VC-subgraph class and \\(V C(\\mathcal{F})=V\\) Suppose \\(-1\\leq f(x)\\leq 1\\) for all \\(f\\in{\\mathcal{F}}\\) and \\(x\\in\\mathcal{X}\\) .Let \\(x_{1},\\ldots,x_{n}\\in{\\mathcal{X}}\\) and define \\(\\begin{array}{ r }{d(f,g)=\\frac{1}{n}\\sum_{i=1}^{n}|f(x_{i})-g(x_{i})|}\\end{array}\\) . Then\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]\n(which \\(\\begin{array}{ r }{i s\\le\\left(\\frac{K}{\\varepsilon}\\right)^{V+\\delta}}\\end{array}\\) for some \\(\\delta . \\jmath\\)", "proof": {"steps": [{"n": 1, "claim": "Let \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) and \\(f_{1},\\ldots,f_{m}\\) be ε-separated, i.e.\n\\[{\\frac{1}{n}}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})| >\\varepsilon .\\]"}, {"n": 2, "claim": "Let \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) be constructed in the following way:"}, {"n": 3, "claim": "\\(z_{i}\\) is chosen uniformly from \\(x_{1},\\ldots,x_{n}\\) and \\(t_{i}\\) is uniform on \\([-1,1]\\)\nConsider \\(f_{r}\\) and \\(f_{\\ell}\\) from the ε-packing."}, {"n": 4, "claim": "Let \\(C_{f_{\\tau}}\\) and \\(C_{f_{\\ell}}\\) be subgraphs of \\(f_{r}\\) and \\(f_{\\ell}\\) Then\n\\(\\mathbb{P}\\left(C_{f_{\\tau}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n= P(At least one point \\((z_{i},t_{i})\\) is picked by \\(C_{f_{r}}\\) or \\(C_{f_{\\ell}}\\) but not picked by the other)\n\\(=1-\\mathbb{P}\\) (All points \\((z_{i},t_{i})\\) are picked either by both or by none)\n\\(\\mathbf{\\Sigma}=1-\\mathbb{P}\\left(\\left(z_{i},t_{i}\\right)\\right.\\) is picked either by both or by none)k\nSince \\(z_{i}\\) is drawn uniformly from \\(x_{1},\\ldots,x_{n}\\)\nP \\(((z_{1},t_{1})\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)\n\\[\\begin{array}{ l }{\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{P}\\left(\\left(x_{i},t_{1}\\right)\\mathrm{~i s~p i c k e d~b y~b o t h~}C_{f r},C_{f_{\\ell}}\\mathrm{~o r~b y~n e i t h e r}\\right)}\\\\ {\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\left(1-\\frac{1}{2}|f_{r}(x_{i})-f_{\\ell}(x_{i})|\\right)}\\\\ {\\displaystyle=1-\\frac{1}{2}\\frac{1}{n}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})|}\\\\ {\\displaystyle=1-\\frac{1}{2}d(f_{r},f_{\\ell})\\leq 1-\\varepsilon/2\\leq e^{-\\varepsilon/2}}\\end{array}\\]\nSubstituting,\n\\(\\mathbb{P}\\left(C_{f_{r}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n\\(=1-\\mathbb{P}\\left((z_{1},t_{1})\\right.\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)𝑘\n\\[\\begin{array}{ l }{\\geq 1-\\left(e^{-\\varepsilon/2}\\right)^{k}}\\\\ {=1-e^{-k\\varepsilon/2}}\\end{array}\\]\nThere are (m） ways to choose \\(f_{r}\\) and \\(f_{\\ell},\\) s0\nP (All pairs \\(C_{f_{r}}\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\geq 1-{\\binom{m}{2}}e^{-k\\varepsilon/2}.\\)\nWhat k should we choose so that \\(1-{\\binom{m}{2}}e^{-k\\varepsilon/2}>0?\\) Choose\n\\[k>{\\frac{2}{\\varepsilon}}\\log{\\binom{m}{2}}.\\]\nThen there exist \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) such that all \\(C_{f_{\\ell}}\\) pick out different subsets.But \\(\\{C_{f}:f\\in\\mathcal{F}\\}\\) is \\(\\mathrm{VC}\\) so by Sauer's lemma, we can pick out at most \\(\\left({\\frac{e k}{V}}\\right)^{V}\\) out of these k points, provided \\(k \\geq V\\); if instead \\(k < V\\) then \\(m\\leq 2^{k}<2^{V}\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}\\) and the claimed bound already holds, so that case may be set aside."}, {"n": 5, "claim": "Hence, \\(\\begin{array}{ r }{m\\leq\\left(\\frac{e k}{V}\\right)^{V}}\\end{array}\\) as long as \\(k>{\\frac{2}{\\varepsilon}}\\log\\left({{\\binom{m}{2}}}\\right)\\) and \\(k \\geq V\\), the case \\(k < V\\) having been disposed of above."}, {"n": 6, "claim": "The latter holds for \\(\\begin{array}{ r }{k=\\frac{2}{\\varepsilon}\\log m^{2}}\\end{array}\\) ."}, {"n": 7, "claim": "Therefore,\n\\[m\\leq\\left(\\frac{e}{V}\\frac{2}{\\varepsilon}\\log m^{2}\\right)^{V}=\\left(\\frac{4 e}{V\\varepsilon}\\log m\\right)^{V},\\]\nwhere \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) ."}, {"n": 8, "claim": "Hence, we get\n\\[m^{1/V}\\leq\\frac{4 e}{\\varepsilon}\\log m^{1/V}\\]\nand defining \\(m^{1/V}=s .\\)\n\\[s\\leq{\\frac{4 e}{\\varepsilon}}\\log s .\\]\nNote that \\(\\frac{s}{\\log s}\\) is increasing for \\(s \\geq e\\) and so for large enough s, the inequality will be violated."}, {"n": 9, "claim": "We now check that the inequality is violated for \\(\\begin{array}{ r }{s^{\\prime}=\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}}\\end{array}\\) ."}, {"n": 10, "claim": "Indeed, one can show that\n\\[\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{7}{\\varepsilon}\\right)^{2}>\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)\\]\nsince\n\\[\\frac{49}{8e\\varepsilon}>\\log{\\frac{7}{\\epsilon}}.\\]\nHence, \\(m^{1/V}=s\\leq s^{\\prime}\\) and, thus,\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]"}]}}, {"local_id": "T2", "statement": "Setup. Let \\(X_1,\\dots,X_n\\) be i.i.d. random variables with common law \\(P\\) on a measurable space \\((\\mathcal{X},\\mathcal{A})\\), let \\(\\mathcal{C}\\subseteq\\mathcal{A}\\) be a class of measurable sets, and write \\(P(C)=\\mathbb{P}(X_1\\in C)\\). Let \\(X'_1,\\dots,X'_n\\) (the ghost sample) be an independent copy of \\(X_1,\\dots,X_n\\), i.e. i.i.d. \\(P\\) and independent of \\(X_1,\\dots,X_n\\). Suprema over \\(\\mathcal{C}\\) are assumed measurable; otherwise read \\(\\mathbb{P}\\) as outer probability.\n\nIf \\(t\\geq{\\sqrt{\\frac{2}{n}}}.\\) then\n\\[\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\right)\\leq 2\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\geq t/2\\right).\\]", "proof": {"steps": [{"n": 1, "claim": "Suppose the event\n\\[\\operatorname*{sup}_{C\\in{\\mathcal{C}}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\]\noccurs.Let \\(\\begin{array}{ r }{X=(X_{1},\\ldots,X_{n})\\in\\{\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\}}\\end{array}\\) ."}, {"n": 2, "claim": "Then\n\\[\\exists C_{X}{\\mathrm{~s u c h~t h a t~}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nFor a fixed C,\n\\[\\begin{array}{ r }{\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right|\\geq t/2\\right)=\\mathbb{P}\\left(\\left(\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}\\geq t^{2}/4\\right)}\\\\ {\\leq\\mathrm{~(b y~C h e b y s h e v~s~I n e q)~}\\frac{4\\mathbb{E}\\left(\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}}{t^{2}}}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i,j}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))(I(X_{j}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i=1}^{n}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))^{2}=\\frac{4 n\\mathbb{P}\\left(\\mathcal{C}\\right)\\left(1-\\mathbb{P}(\\mathcal{C})\\right)}{n^{2}t^{2}}\\leq\\frac{1}{n t^{2}}\\leq\\frac{1}{2}}\\end{array}\\]\nsince we chose \\(t\\geq\\sqrt{\\textstyle{\\frac{2}{n}}}.\\)\nSo,\n\\[\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\Bigg|\\exists C_{X}\\right)\\geq 1/2\\]\nif \\(t\\geq\\sqrt{2/n}\\) .Assume that the event\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\]\noccurs."}, {"n": 3, "claim": "Recall that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nHence,it must be that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\geq t/2 .\\]\nWe conclude\n\\[\\begin{array}{ r c l }{\\displaystyle\\frac 12}&{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}(C_{X})\\right|\\le t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right).}\\end{array}\\]\nSince indicators are 0,1-valued,\n\\[\\begin{array}{ r l r }&{}&{\\displaystyle\\frac{1}{2}T\\left(\\underbrace{\\operatorname*{sup}_{\\zeta\\in{\\mathcal C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}(C)\\right|\\ge t}_{\\Im C x}\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C x\\right)\\cdot I\\left(\\exists C x\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X,X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]\nNow, take expectation with respect to \\(X_{i}^{,}\\) s to obtain\n\\[\\begin{array}{ r l r }&{}&{\\mathbb{P}_{X}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\ge t\\right)}\\\\ &{}&{\\displaystyle\\le 2\\cdot\\mathbb{P}_{X,X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]"}]}}, {"local_id": "T3", "statement": "Let \\(\\xi,\\nu\\) - random variables.Assume that\n\\[\\mathbb{P}\\left(\\nu\\geq t\\right)\\leq\\Gamma e^{-\\gamma t}\\]\nwhere \\(\\Gamma\\geq 1,t\\geq 0\\) ，and \\(\\gamma > 0\\) . Furthermore, for all \\(a > 0\\) assume that\n\\[\\mathbb{E}\\phi(\\xi)\\le\\mathbb{E}\\phi(\\nu)\\]\nwhere \\(\\phi(x)=(x-a)_{+}\\) .Then\n\\[\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\Gamma\\cdot e\\cdot e^{-\\gamma t}.\\]", "proof": {"steps": [{"n": 1, "claim": "Since \\(\\phi(x)=(x-a)_{+}\\) ，we have \\(\\phi(\\xi)\\geq\\phi(t)\\) whenever \\(\\xi \\geq t .\\)\n\\[\\begin{array}{ r l }&{\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\mathbb{P}\\left(\\phi(\\xi)\\geq\\phi(t)\\right)}\\\\ &{\\qquad\\leq\\displaystyle\\frac{\\mathbb{E}\\phi(\\xi)}{\\phi(t)}\\leq\\frac{\\mathbb{E}\\phi(\\nu)}{\\phi(t)}=\\frac{\\mathbb{E}\\left(\\nu-a\\right)_{+}}{(t-a)_{+}}}\\end{array}\\]\nFurthermore,\n\\[\\begin{array}{ r l }{\\displaystyle\\mathbb{E}(\\nu-a)_{+}=}&{\\mathbb{E}\\int_{0}^{(\\nu-a)_{+}}\\mathrm{1}d x}\\\\ &{=\\mathbb{E}\\int_{0}^{\\infty}I(x\\leq(\\nu-a)_{+})d x}\\\\ &{=\\int_{0}^{\\infty}\\mathbb{E}I(x\\leq(\\nu-a)_{+})d x}\\\\ &{=\\int_{0}^{\\infty}\\mathbb{P}\\left((\\nu-a)_{+}\\geq x\\right)d x}\\\\ &{=\\int_{0}^{\\infty}\\mathbb{P}\\left(\\nu\\geq a+x\\right)d x}\\\\ &{\\leq\\int_{0}^{\\infty}\\Gamma e^{-\\gamma(a+x)}d x=\\frac{\\Gamma e^{-\\gamma a}}{\\gamma}.}\\end{array}\\]\nHence,\n\\[\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\frac{\\Gamma e^{-\\gamma a}}{\\gamma(t-a)_{+}}=\\frac{\\Gamma\\cdot e\\cdot e^{-\\gamma t}}{1}=\\Gamma\\cdot e\\cdot e^{-\\gamma t}\\]\nwhere we chose optimal \\(\\begin{array}{ r }{a=t-\\frac{1}{\\gamma}}\\end{array}\\) to minimize \\(\\frac{\\Gamma e^{-\\gamma a}}{\\gamma(t-a)_{+}}\\)"}]}}, {"local_id": "T4", "statement": "Let \\(x=(x_{1},\\ldots,x_{n}),x^{\\prime}=(x_{1}^{\\prime},\\ldots,x_{n}^{\\prime})\\) . If for functions \\(\\varphi_{1}(x,x^{\\prime}),\\varphi_{2}(x,x^{\\prime}),\\varphi_{3}(x,x^{\\prime})\\)\n\\[\\begin{array}{ r }{\\mathbb{P}\\left(\\varphi_{1}(x,x^{\\prime})\\geq\\varphi_{2}(x,x^{\\prime})+\\sqrt{\\varphi_{3}(x,x^{\\prime})\\cdot t}\\right)\\leq\\Gamma e^{-\\gamma t}}\\end{array}\\]\nthen\n\\[\\begin{array}{ r }{\\mathbb{P}\\left(\\mathbb{E}_{x^{\\prime}}\\varphi_{1}(x,x^{\\prime})\\geq\\mathbb{E}_{x^{\\prime}}\\varphi_{2}(x,x^{\\prime})+\\sqrt{\\mathbb{E}_{x^{\\prime}}\\varphi_{3}(x,x^{\\prime})\\cdot t}\\right)\\leq\\Gamma\\cdot e\\cdot e^{-\\gamma t}.}\\end{array}\\]\n(i.e. if the inequality holds, then it holds with averaging over one of the copies)", "proof": {"steps": [{"n": 1, "claim": "First, note that \\(\\begin{array}{ r }{\\sqrt{a b}=\\operatorname*{inf}_{\\delta>0}(\\delta a+\\frac{b}{4\\delta})}\\end{array}\\) with \\(\\begin{array}{ r }{\\delta_{*}=\\sqrt{\\frac{b}{4 a}}}\\end{array}\\) achieving the infima."}, {"n": 2, "claim": "Hence,\n\\[\\begin{array}{ c }{{\\{\\varphi_{1}\\geq\\varphi_{2}+\\sqrt{\\varphi_{3}t}\\}=\\{\\exists\\delta>0,\\varphi_{1}\\geq\\varphi_{2}+\\delta\\varphi_{3}+\\displaystyle\\frac{t}{4\\delta}\\}}}\\\\ {{=\\{\\exists\\delta>0,(\\varphi_{1}-\\varphi_{2}-\\delta\\varphi_{3})4\\delta\\geq t\\}}}\\\\ {{=\\{\\underbrace{\\sup_{\\delta>0}(\\varphi_{1}-\\varphi_{2}-\\delta\\varphi_{3})4\\delta}_{\\nu}\\geq t\\}}}\\end{array}\\]\nand similarly\n\\[\\big\\{\\mathbb{E}_{x^{\\prime}}\\varphi_{1}\\geq\\mathbb{E}_{x^{\\prime}}\\varphi_{2}+\\sqrt{\\mathbb{E}_{x^{\\prime}}\\varphi_{3}t}\\big\\}=\\big\\{\\underbrace{\\sup_{\\delta>0}(\\mathbb{E}_{x^{\\prime}}\\varphi_{1}-\\mathbb{E}_{x^{\\prime}}\\varphi_{2}-\\delta\\mathbb{E}_{x^{\\prime}}\\varphi_{3})4\\delta}_{\\xi}\\geq t\\big\\}.\\]\nBy assumption, \\(\\mathbb{P}\\left(\\nu\\geq t\\right)\\leq\\Gamma e^{-\\gamma t}\\) ."}, {"n": 3, "claim": "We want to prove \\(\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\Gamma\\cdot e\\cdot e^{-\\gamma t}\\) ."}, {"n": 4, "claim": "By the previous lemma, we only need to check whether \\(\\mathbb{E}\\phi(\\xi)\\le\\mathbb{E}\\phi(\\nu)\\)\n\\[\\begin{array}{ l }{\\displaystyle\\xi=\\operatorname*{sup}_{\\delta>0}\\mathbb{E}_{x^{\\prime}}(\\varphi_{1}-\\varphi_{2}-\\delta\\varphi_{3})4\\delta}\\\\ {\\displaystyle\\quad\\leq\\mathbb{E}_{x^{\\prime}}\\operatorname*{sup}_{\\delta>0}(\\varphi_{1}-\\varphi_{2}-\\delta\\varphi_{3})4\\delta}\\\\ {\\displaystyle=\\mathbb{E}_{x^{\\prime}}\\nu}\\end{array}\\]\nThus,\n\\[\\phi(\\xi)\\leq\\phi(\\mathbb{E}_{x^{\\prime}}\\nu)\\leq\\mathbb{E}_{x^{\\prime}}\\phi(\\nu)\\]\nby Jensen's inequality (\\(\\phi\\) is convex)."}, {"n": 5, "claim": "Hence,\n\\[\\mathbb{E}\\phi(\\xi)\\le\\mathbb{E}\\mathbb{E}_{x^{\\prime}}\\phi(\\nu)=\\mathbb{E}\\phi(\\nu).\\]"}]}}], "target_local_id": "T2", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch13_proposition_13_2__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nConvex set. A set \\(K\\subseteq\\mathbb{R}^n\\) is convex if the line segment between any two points in \\(K\\) is also contained in \\(K\\). Formally, for all \\(x,y\\in K\\) and all scalars \\(\\gamma\\in[0,1]\\) we have \\(\\gamma x+(1-\\gamma)y\\in K\\).\n\nStrong convexity. A function \\(f\\colon\\Omega\\to\\mathbb{R}\\) is \\emph{\\(\\alpha\\)-strongly convex} if, for all \\(x,y\\in\\Omega\\), the following inequality holds for some \\(\\alpha>0\\): \\[ f(y) \\geqslant f(x) + g(x)^\\top(y-x) + \\frac{\\alpha}{2}\\left\\lVert x-y\\right\\rVert^2 \\]\n\nUniform stability. The uniform stability of an algorithm \\(A\\) is defined as\n\\[\n\\Delta_{\\sup}(A) = \\sup_{S, S' \\in (X\\times Y)^m } \\sup_{i \\in [m]} |\\ell(A(S), z_i') - \\ell(A(S^{(i)}, z_i')|\n\\]\n\nLocal minimum. A point \\(x^*\\) is an unconstrained \\textit{local minimum} if there exists \\(\\epsilon > 0\\) such that \\(f(x^*) \\leqslant f(x)\\) for all \\(x\\) with \\(\\|x - x^*\\| < \\epsilon\\).", "theorems": [{"local_id": "T1", "statement": "If \\(f\\) is in addition \\(\\alpha\\)-strongly convex w.r.t to \\(\\|\\cdot\\|_{[1-\\gamma]}\\), then we get\n\\[\n\\mathbb{E}[f(x_{t+1}) - \\arg\\min_{x \\in \\mathbb{R}^{n}} f(x)] \\leqslant \\left(1 - \\frac{\\alpha}{\\sum_{i=1}^n \\beta_i^\\gamma}\\right)^t (f(x_1) - f(x^*)) \\ .\n\\]", "proof": {"steps": [{"n": 1, "claim": "We need the following lemma:\n\nLemma 12.4. Let \\(f\\) be an \\(\\alpha\\)-strongly convex function w.r.t. a norm \\(\\|\\cdot\\|\\). Then, \\(f(x) - f(x^{*}) \\leqslant \\frac{1}{2\\alpha}\\|\\nabla f(x)\\|_{*}^{2}\\)."}, {"n": 2, "claim": "Lemma 12.4 shows that\n\\[\n\\|\\nabla f(x_s)\\|^{*2}_{[1-\\gamma]} \\geqslant 2 \\alpha \\delta_s \\ .\n\\]\nOn the other hand, Theorem 12.2 showed that\n\\[\n\\delta_{t+1} - \\delta_t \\leqslant -\\frac{\\|\\nabla f(x_t)\\|^{*2}_{[1-\\gamma]}}{2\\sum_{i=1}^n \\beta_i^{\\gamma}}\n\\]\nCombining these two, we get\n\\begin{align*}\n\\delta_{t+1} - \\delta_t &\\leqslant -\\frac{\\alpha \\delta_t}{\\sum_{i=1}^n \\beta_i^{\\gamma}} \\\\\n\\delta_{t+1} &\\leqslant \\delta_t \\left(1 -\\frac{\\alpha}{\\sum_{i=1}^n \\beta_i^{\\gamma}} \\right)~.\n\\end{align*}\nApplying the above inequality recursively and recalling that \\(\\delta_t = \\mathbb{E}[f(x_t) - f(x^*)]\\) gives the result."}]}}, {"local_id": "T2", "statement": "Assume \\(\\ell(w, z)\\) is \\(\\alpha\\)-strongly convex over the domain \\(\\Omega\\) and \\(L\\)-Lipschitz. Let \\(\\hat w_S = \\arg \\min_{w\\in \\Omega} \\frac{1}{m} \\sum_{i=1}^m \\ell(w, z_i)\\) denote the empirical risk minimizer (ERM). Then, ERM satisfies\n\\[\n\\Delta_{\\sup} (\\text{ERM}) \\leqslant \\frac{4L^2}{\\alpha m}\\,.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Denote by \\(\\hat w_S\\) the empirical risk minimizer on a sample \\(S.\\) Fix arbitrary samples \\(S,S'\\) of size \\(m\\) and an index \\(i\\in[m].\\) We need to show that\n\\[\n|(\\ell(\\hat w_{S^{(i)}},z_i')-\\ell(\\hat w_{S},z_i'))|\\leqslant\\frac{4L^2}{\\alpha m}\\,.\n\\]\nOn one hand, by strong convexity we know that\n\\[\nR_S(\\hat w_{S^{(i)}}) - R_S(\\hat w_{S}) \\geqslant \\frac{\\alpha}{2} \\| \\hat w_{S} - \\hat w_{S^{(i)}} \\|^2 \\,.\n\\]\nOn the other hand,\n\\begin{align*}\n&R_S(\\hat w_{S^{(i)}})-R_S(\\hat w_{S})\\\\\n&=\\frac{1}{m}(\\ell(\\hat w_{S^{(i)}},z_i)-\\ell(\\hat w_{S},z_i))+\\frac{1}{m}\\sum_{i\\neq j}(\\ell(\\hat w_{S^{(i)}},z_j)-\\ell(\\hat w_{S},z_j))\\\\\n&=\\frac{1}{m}(\\ell(\\hat w_{S^{(i)}},z_i)-\\ell(\\hat w_{S},z_i))+\\frac{1}{m}(\\ell(\\hat w_{S},z_i')-\\ell(\\hat w_{S^{(i)}},z_i'))+\\left(R_{S^{(i)}}(\\hat w_{S^{(i)}})-R_{S^{(i)}}(\\hat w_{S})\\right)\\\\\n&\\leqslant\\frac{1}{m}|\\ell(\\hat w_{S^{(i)}},z_i)-\\ell(\\hat w_{S},z_i)| +\\frac{1}{m}|(\\ell(\\hat w_{S},z_i')-\\ell(\\hat w_{S^{(i)}},z_i'))|\\\\\n&\\leqslant\\frac{2L}{m}\\|\\hat w_{S^{(i)}}-\\hat w_{S}\\|\\,.\n\\end{align*}\nHere, we used that\n\\[\nR_{S^{(i)}}(\\hat w_{S^{(i)}}) - R_{S^{(i)}}(\\hat w_{S})) \\leqslant 0\n\\]\nand the fact that \\(\\ell\\) is \\(L\\)-Lipschitz."}, {"n": 2, "claim": "Putting it all together \\(\\| \\hat w_{S^{(i)}} - \\hat w_{S} \\| \\leqslant \\frac{4 L}{\\alpha m}\\). Then by the Lipschitz condition we have that\n\\[\n|(\\ell(\\hat w_{S^{(i)}},z_i')-\\ell(\\hat w_{S},z_i'))|\\leqslant L\\|\\hat w_{S^{(i)}}-\\hat w_{S}\\|\\leqslant\\frac{4L^2}{\\alpha m}\\,.\n\\]\nHence, \\(\\Delta_{\\sup}(\\text{ERM}) \\leqslant \\frac{4 L ^ 2}{\\alpha m}\\)."}]}}, {"local_id": "T3", "statement": "Let \\(x^*\\) be an unconstrained local minimum of \\(f\\colon \\mathbb{R}^n \\to \\mathbb{R}\\) and assume \\(f\\) is continuously differentiable (\\(C^1\\)) in an open set containing \\(x^*\\). Then\n\\begin{enumerate}\n    \\item \\(\\nabla f(x^*) = 0\\) (First-Order Necessary Condition)\n    \\item If in addition \\(f\\) is twice continuously differentiable in an open set around \\(x^*\\), then \\(\\nabla^2 f(x^*) \\succeq 0\\). (Second Order Necessary Condition)\n\\end{enumerate}", "proof": {"steps": [{"n": 1, "claim": "Fix any direction \\(d \\in \\mathbb{R}^n\\).\n\\begin{enumerate}\n    \\item \\(g(\\alpha) := f(x^* + \\alpha d)\\). Then\n    \\[\n        0 \\leqslant \\lim_{\\alpha \\to 0} \\frac{f(x^* + \\alpha d) - f(x^*)}{\\alpha} = \\frac{\\partial g(0)}{\\partial \\alpha} = d^\\top \\nabla f(x^*)\n    \\]\n    The inequality follows because \\(x^*\\) is a local minimum, \\(0 \\leqslant f(x^* + \\alpha d) - f(x^*)\\) for sufficiently small \\(\\alpha\\). So, we can construct a sequence with only positive \\(\\alpha\\) that converges to \\(x^*\\) such that each element \\(0 \\leqslant \\frac{f(x^* + \\alpha_n d) - f(x^*)}{\\alpha_n}\\) which implies that statement given that \\(f\\) is locally differentiable. Since \\(d\\) is arbitrary, this implies that \\(\\nabla f(x^*) = 0\\).\n    \\item First we represent \\(f(x^* + \\alpha d) - f(x^*)\\) using the 2nd order Taylor expansion.\n    \\[\n        f(x^* + \\alpha d) - f(x^*) = \\alpha \\nabla f(x^*)^\\top d + \\frac{\\alpha^2}{2} d^\\top \\nabla^2 f(x^*) d + o(\\alpha^2) = \\frac{\\alpha^2}{2} d^\\top \\nabla^2 f(x^*) d + o(\\alpha^2)\n    \\]\n    Now we do the following\n    \\[\n        0 \\leqslant \\lim_{\\alpha \\to 0} \\frac{f(x^* + \\alpha d) - f(x^*)}{\\alpha^2} = \\lim_{\\alpha \\to 0} \\frac{1}{2} d^\\top \\nabla^2 f(x^*) d + \\frac{o(\\alpha^2)}{\\alpha^2} = \\frac{1}{2} d^\\top \\nabla^2 f(x^*) d\n    \\]\n    Because \\(d\\) is arbitrary, this implies that \\(\\nabla^2 f(x^*) \\succeq 0\\) (Positive semidefinite).\n\\end{enumerate}"}]}}, {"local_id": "T4", "statement": "Let \\(\\Omega \\subseteq \\mathbb{R}^d\\) be a closed convex set. Let \\(\\mathcal{A}\\) denote the affine set \\(\\{x : Ax = b\\}\\) for some \\(A \\in \\mathbb{R}^{n \\times d}\\) and \\(b \\in \\mathbb{R}^n\\). Suppose that the set \\(\\operatorname{ri}(\\Omega) \\cap \\mathcal{A}\\) is non-empty. Then for any \\(x \\in \\Omega \\cap \\mathcal{A}\\), \\[\\mathcal{N}_{\\Omega \\cap \\mathcal{A}}(x) = \\mathcal{N}_{\\Omega}(x) + \\{A^\\top \\lambda : \\lambda \\in \\mathbb{R}^n\\}\\,.\\]", "proof": {"steps": [{"n": 1, "claim": "The ``\\(\\supseteq\\)'' assertion is straightforward."}, {"n": 2, "claim": "To see this, suppose \\(z \\in \\Omega \\cap \\mathcal{A}\\) and note that \\(z-x \\in \\operatorname{null}(A)\\) so that \\((z-x)^\\top A^\\top \\lambda = \\lambda^\\top A(z-x) = 0\\) for all \\(\\lambda \\in \\mathbb{R}^n\\)."}, {"n": 3, "claim": "If \\(u \\in \\mathcal{N}_{\\Omega}(x)\\), then \\((z-x)^\\top u \\leqslant 0\\), so for \\(\\lambda \\in \\mathbb{R}^n\\), we have \\[\\langle z-x, u + A^\\top \\lambda \\rangle = \\langle z-x, u \\rangle \\leqslant 0\\] implying that \\(u + A^\\top \\lambda \\in \\mathcal{N}_{\\Omega \\cap \\mathcal{A}}(x)\\)."}, {"n": 4, "claim": "For the reverse inclusion, let \\(v \\in \\mathcal{N}_{\\Omega \\cap \\mathcal{A}}(x)\\)."}, {"n": 5, "claim": "Then we have \\[v^\\top (z-x) \\leqslant 0 \\text{ for all } z \\in \\Omega \\cap \\mathcal{A}.\\] Now define the sets \\[\\begin{aligned} C_1 &= \\left\\{ (y,\\mu) \\in \\mathbb{R}^{d+1} : y = z-x,\\,\\, z \\in \\Omega,\\,\\, \\mu \\leqslant v^\\top y \\right\\}\\\\ C_2 &= \\left\\{ (y,\\mu) \\in \\mathbb{R}^{d+1} : y \\in \\operatorname{ker}(A),\\,\\, \\mu = 0 \\right\\}\\,. \\end{aligned}\\] Note that \\(\\operatorname{ri}(C_1) \\cap C_2 = \\emptyset\\) because otherwise there would exist a \\((\\hat{y},\\hat{\\mu})\\) such that \\[v^\\top \\hat{y} > \\hat{\\mu} = 0\\] and \\(\\hat{y} \\in \\mathcal{T}_{\\Omega \\cap \\mathcal{A}}(x)\\)."}, {"n": 6, "claim": "This would contradict our assumption that \\(v \\in \\mathcal{N}_{\\Omega \\cap \\mathcal{A}}(x)\\)."}, {"n": 7, "claim": "Since their intersection is empty, we can properly separate \\(\\operatorname{ri}(C_1)\\) from \\(C_2\\)."}, {"n": 8, "claim": "Indeed, since \\(C_2\\) is a subspace and \\(C_1\\) has nonempty relative interior, there must be a \\((w,\\gamma)\\) such that \\[\\inf_{(y,\\mu)\\in C_1} \\{ w^\\top y + \\gamma \\mu\\} < \\sup_{(y,\\mu)\\in C_1} \\{ w^\\top y + \\gamma \\mu\\} \\leqslant 0\\] while \\[w^\\top u = 0 \\text{ for all } u \\in \\ker(A).\\] In particular, this means that \\(w = A^\\top \\lambda\\) for some \\(\\lambda \\in \\mathbb{R}^n\\)."}, {"n": 9, "claim": "Now, \\(\\gamma\\) must be nonnegative, as otherwise, \\[\\sup_{(y,\\mu)\\in C_1} \\{ w^\\top y + \\gamma \\mu\\} = \\infty\\] (which can be seen by letting \\(\\mu\\) tend to negative infinity)."}, {"n": 10, "claim": "If \\(\\gamma = 0\\), then \\[\\sup_{y \\in C_1} w^\\top y \\leqslant 0\\] but the set \\(\\{y : w^\\top y = 0\\}\\) does not contain the set \\(\\{z-x : z \\in \\Omega\\}\\) as the infimum is strictly negative."}, {"n": 11, "claim": "This means that the relative interior of \\(\\Omega - \\{x\\}\\) cannot intersect the kernel of \\(A\\) which contradicts our assumptions."}, {"n": 12, "claim": "Thus, we can conclude that \\(\\gamma\\) is strictly positive."}, {"n": 13, "claim": "By homogeneity, we may as well assume that \\(\\gamma = 1\\)."}, {"n": 14, "claim": "To complete the argument, note that we now have \\[(w+v)^\\top (z-x) \\leqslant 0 \\text{ for all } z \\in \\Omega.\\] This means that \\(v+w \\in \\mathcal{N}_{\\Omega}(x)\\) and we have already shown that \\(w = A^\\top \\lambda\\)."}, {"n": 15, "claim": "Thus, \\[v = (v+w)-w \\in \\mathcal{N}_{\\Omega}(x) + \\mathcal{N}_{\\mathcal{A}}(x)\\,.\\]"}]}}, {"local_id": "T5", "statement": "For any function \\(\\varphi(x,z)\\), \\[\\min_x \\max_z \\varphi(x,z) \\geqslant \\max_z \\min_x \\varphi(x,z)\\,.\\]", "proof": {"steps": [{"n": 1, "claim": "The proof is essentially tautological. Note that we always have \\[\\varphi(x,z) \\geqslant \\min_x \\varphi(x,z).\\] Taking the maximization with respect to the second argument verifies that \\[\\max_z \\varphi(x,z) \\geqslant \\max_z \\min_x \\varphi(x,z) \\quad \\forall x\\,.\\] Now, minimizing the left hand side with respect to \\(x\\) proves \\[\\min_x \\max_z \\varphi(x,z) \\geqslant \\max_z \\min_x \\varphi(x,z)\\] which is precisely our assertion."}]}}], "target_local_id": "T4", "topic_domain": "Optimization", "topic_subtopic": "Convex Analysis & Duality", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch13_theorem_13_15__correct", "context": "theorem 13.11: We have for any \\(\\lambda>0\\), with probability at least \\(1-\\delta\\), for all \\(n\\geq 1\\) and \\(f\\in\\mathcal{F}\\): \\[\\left[-\\sum_{i=1}^{n} f(Z_i)-\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_i^{(y)}}e^{-\\lambda f(Z_i)}\\right]\\leq\\inf_{\\epsilon>0}\\left[2n\\epsilon+\\frac{\\ln(M(\\epsilon,\\mathcal{F},\\|\\cdot\\|_\\infty)/\\delta)}{\\lambda}\\right].\\]\n\nShared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nsection 13.1 sequential observation setup. In Section 13.1, data are observed sequentially. At each time \\(t\\), after the history \\(S_{t-1}\\) has been observed, the next observation \\(Z_t\\) is drawn according to a distribution that may depend on that history. The observation is decomposed as \\(Z_t=(Z_t^{(x)},Z_t^{(y)})\\), where \\(Z_t^{(x)}\\) is the component determined by the past history or the chosen action at time \\(t\\), and \\(Z_t^{(y)}\\) is the random response component. The history after time \\(t\\) is \\(S_t=(Z_1,\\ldots,Z_t)\\), and \\(\\mathcal{S}_T\\) denotes the full sequential experiment/history through time \\(T\\). For a function depending on \\(S_t\\), the notation \\(\\mathbb{E}_{Z_t^{(y)}}[\\cdot]\\) means conditional expectation over the response component \\(Z_t^{(y)}\\), conditional on \\(S_{t-1}\\) and on \\(Z_t^{(x)}\\). Stopping times are adapted to the filtration generated by the histories \\(S_t\\).\n\ndefinition 4.1. Let \\((\\mathcal{V},d)\\) be a pseudometric space with metric \\(d(\\cdot,\\cdot)\\). A finite\nset \\(\\mathcal{G}(\\epsilon) \\subset \\mathcal{V}\\) is an \\(\\epsilon\\)-cover (or \\(\\epsilon\\)-net) of \\(\\mathcal{G} \\subset \\mathcal{V}\\) if, for all \\(\\phi \\in \\mathcal{G}\\), there exists \\(\\phi' \\in \\mathcal{G}(\\epsilon)\\) so that \\(d(\\phi', \\phi) \\leq \\epsilon\\). The \\(\\epsilon\\)-covering number of \\(\\mathcal{G}\\) with metric \\(d\\) is the smallest cardinality \\(N(\\epsilon, \\mathcal{G}, d)\\) of such \\(\\mathcal{G}(\\epsilon)\\). The number \\(\\ln N(\\epsilon, \\mathcal{G}, d)\\) is called the \\(\\epsilon\\)-entropy.\nFor a function class \\(\\mathcal{G}\\) with seminorm \\(L_{p}(\\mathcal{D})\\) in (3.19), we denote the corresponding \\(L_{p}(\\mathcal{D})\\)-covering number as \\(N(\\epsilon, \\mathcal{G}, L_{p}(\\mathcal{D}))\\). When \\(1 \\leq p \\leq q\\), we have\n\\[\nN(\\epsilon,\\mathcal{G},L_{p}(\\mathcal{D}))\\leq N(\\epsilon,\\mathcal{G},L_{q}(\\mathcal{D})).\n\\]\nIt is easy to verify the following relationship, which implies that \\( L_{p}(\\mathcal{D}) \\) bracketing cover is a stronger requirement than \\( L_{p}(\\mathcal{D}) \\) cover.\n\nPacking Number. Let \\((\\mathcal{V},d)\\) be a pseudometric space with metric \\(d(\\cdot,\\cdot)\\). A finite subset \\(\\mathcal{G}(\\epsilon)\\subset \\mathcal{G}\\) is an \\(\\epsilon\\)-packing of \\(\\mathcal{G}\\) if \\(d(\\phi,\\phi^{\\prime})>\\epsilon\\) for all \\(\\phi,\\phi^{\\prime}\\in\\mathcal{G}(\\epsilon)\\). The \\(\\epsilon\\)-packing number of \\(\\mathcal{G}\\), denoted by \\(M(\\epsilon,\\mathcal{G},d)\\), is the largest cardinality of \\(\\epsilon\\)-packing of \\(\\mathcal{G}\\).\nThe following results illustrate the equivalence between covering number and packing number. One advantage of using an \\(\\epsilon\\) packing of \\(\\mathcal{G}\\) instead of an \\(\\epsilon\\) cover of \\(\\mathcal{G}\\) is that all members in the \\(\\epsilon\\) packing also belong to \\(\\mathcal{G}\\). There if members in \\(\\mathcal{G}\\) satisfy certain assumptions such as the variance condition, then members of its \\(\\epsilon\\) packing also satisfy such assumptions. For this reason, we will use packing numbers instead of covering numbers in some of the theoretical analysis in later chapters.", "theorems": [{"local_id": "T1", "statement": "Assume that for all \\( f \\in \\mathcal{F} \\) and \\( Z_{t}: -f(Z_{t}) + \\mathbb{E}_{Z_{t}^{(y)}}f(Z_{t}) \\leq b \\), and the following variance condition holds for all \\( f \\in \\mathcal{F} \\) and \\( S_{t} \\):\n\\[\n\\operatorname{Var}_{Z_{t}^{(y)}}[f(Z_{t})]\\leq c_{0}^{2}+c_{1}\\mathbb{E}_{Z_{t}^{(y)}}[f(Z_{t})],\n\\]\nthen with probability at least \\(1 - \\delta\\), for all \\(f \\in \\mathcal{F}\\):\n\\[\n\\begin{array}{l}\\left(1-\\lambda c_{1}\\phi(\\lambda b)\\right)\\sum_{t=1}^{n}\\mathbb{E}_{Z_{t}^{(y)}}[f(Z_{t})]\\leq\\sum_{t=1}^{n}f(Z_{t})+\\lambda n c_{0}^{2}\\phi(\\lambda b)\\\\ +\\inf_{\\epsilon>0}\\left[2 n\\epsilon+\\frac{\\ln(M(\\epsilon,\\mathcal{F},\\|\\cdot\\|_{\\infty})/\\delta)}{\\lambda}\\right],\\\\ \\end{array}\n\\]\nwhere \\(\\phi (z) = (e^{z} - z - 1) / z^{2}\\)", "proof": {"steps": [{"n": 1, "claim": "Note that by employing the moment estimation of Bennett's inequality in Lemma 2.20, we obtain\n\\[\n\\begin{array}{l}\\frac{1}{\\lambda}\\ln\\mathbb{E}_{Z_{t}^{(y)}}e^{-\\lambda f(Z_{t})}\\\\ \\leq-\\mathbb{E}_{Z_{t}^{(y)}}[f(Z_{t})]+\\lambda\\phi(\\lambda b)\\operatorname{Var}_{Z_{t}^{(y)}}[f(Z_{t})]\\\\ \\leq-\\mathbb{E}_{Z_{t}^{(y)}}[f(Z_{t})]+\\lambda\\phi(\\lambda b)[c_{0}^{2}+c_{1}\\mathbb{E}_{Z_{t}^{(y)}}[f(Z_{t})]].\\\\ \\end{array}\n\\]\nWe can now obtain the desired bound from Theorem 13.11."}]}}, {"local_id": "T2", "statement": "Let \\(\\{(X_t, \\epsilon_t)\\}\\) be a filtered sequence in \\(\\mathcal{X} \\times \\mathbb{R}\\) so that \\(\\epsilon_t\\) is conditional zero-mean sub-Gaussian noise: for all \\(\\lambda \\in \\mathbb{R}\\),\n\\[\n\\ln\\mathbb{E}\\left[e^{\\lambda\\epsilon_{t}}\\mid X_{t},\\mathcal{S}_{t-1}\\right]\\leq\\frac{\\lambda^{2}}{2}\\sigma^{2},\n\\]\nwhere \\( S_{t-1} \\) denotes the history data. Assume that \\( Y_t = f_*(X_t) + \\epsilon_t \\), with \\( f_*(x) \\in \\mathcal{F}: \\mathcal{X} \\to \\mathbb{R} \\). Let \\( \\hat{f}_t \\) be an approximate ERM solution for some \\( \\epsilon' > 0 \\):\n\\[\n\\left(\\sum_{s=1}^{t}\\left(\\hat{f}_{t}\\left(X_{s}\\right)-Y_{s}\\right)^{2}\\right)^{1/2}\\leq\\min_{f\\in\\mathcal{F}}\\left(\\sum_{s=1}^{t}\\left(f\\left(X_{s}\\right)-Y_{s}\\right)^{2}\\right)^{1/2}+\\sqrt{t}\\epsilon^{\\prime}.\n\\]\nThen with probability at least \\(1 - \\delta\\), for all \\(t \\geq 0\\):\n\\[\n\\sum_{s=1}^{t}(\\hat{f}_{t}(X_{t})-f_{*}(X_{t}))^{2}\\leq\\inf_{\\epsilon>0}\\left[8 t(\\epsilon+\\epsilon^{\\prime})(\\sigma+2\\epsilon+2\\epsilon^{\\prime})+12\\sigma^{2}\\ln\\frac{2 N(\\epsilon,\\mathcal{F},\\|\\cdot\\|_{\\infty})}{\\delta}\\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\eta = \\sigma^{-2} / 4\\), and for \\(f \\in \\mathcal{F}\\), let\n\\[\n\\phi(f,Z_{t})=\\left[\\left(f(X_{t})-Y_{t}\\right)^{2}-\\left(f_{*}(X_{t})-Y_{t}\\right)^{2}\\right].\n\\]\nLet \\(\\mathcal{F}_{\\epsilon}\\) be an \\(\\epsilon \\| \\cdot \\|_{\\infty}\\) cover of \\(\\mathcal{F}\\) with \\(N = N(\\epsilon, \\mathcal{F}, \\| \\cdot \\|_{\\infty})\\) members."}, {"n": 2, "claim": "Given any \\(f \\in \\mathcal{F}_{\\epsilon}\\),\n\\[\n\\begin{array}{l}\\ln\\mathbb{E}_{Y_{t}|X_{t},S_{t-1}}\\exp(-\\eta\\phi(f,Z_{t}))\\\\ = -\\eta\\left(f\\left(X_{t}\\right)-f_{*}\\left(X_{t}\\right)\\right)^{2}\\\\ +\\ln\\mathbb{E}_{Y_{t}\\mid X_{t},S_{t-1}}\\exp(-2\\eta(f(X_{t})-f_{*}(X_{t}))(f_{*}(X_{t})-Y_{t}))\\\\ \\leq-\\eta(f(X_{t})-f_{*}(X_{t}))^{2}+\\frac{4\\eta^{2}\\sigma^{2}}{2}(f(X_{t})-f_{*}(X_{t}))^{2}\\\\ = -\\frac{\\left(f(X_{t})-f_{*}(X_{t})\\right)^{2}}{8\\sigma^{2}}.\\\\ \\end{array}\n\\]\nThe inequality used the sub-Gaussian noise assumption."}, {"n": 3, "claim": "It follows from Theorem 13.11 that with probability at least \\( 1 - \\delta / 2 \\), for all \\( f \\in \\mathcal{F}_{\\epsilon} \\) and \\( t \\geq 0 \\):"}, {"n": 4, "claim": "\\[\n\\sum_{i=1}^{t}-\\eta\\phi(f,Z_{i})\\leq-\\sum_{i=1}^{t}\\frac{\\left(f\\left(X_{i}\\right)-f_{*}\\left(X_{i}\\right)\\right)^{2}}{8\\sigma^{2}}+\\ln(2 N/\\delta).\\tag{13.5}\n\\]\nUsing Exercise 13.3, we have with probability at least \\(1 - \\delta /2\\), for all \\(t\\geq 0\\)\n\\[\n\\underbrace{\\sum_{i=1}^{t}\\left(f_{*}\\left(X_{i}\\right)-Y_{i}\\right)^{2}}_{A}\\leq 2 t\\sigma^{2}+3\\sigma^{2}\\ln(2/\\delta).\\tag{13.6}\n\\]\nIt follows that with probability at least \\(1 - \\delta\\), for all \\(t \\geq 0\\) and \\(f \\in \\mathcal{F}_{\\epsilon}\\), both (13.5)\nand (13.6) hold."}, {"n": 5, "claim": "Now, let \\( f \\in \\mathcal{F}_{\\epsilon} \\) so that \\( \\| \\hat{f}_t - f \\|_{\\infty} \\leq \\epsilon \\)."}, {"n": 6, "claim": "This implies that\n\\[\n\\begin{array}{l}\\sum_{i=1}^{t}\\left[(f(X_{i})-Y_{i})^{2}-(f_{*}(X_{i})-Y_{i})^{2}\\right]\\leq\\left(\\sqrt{\\sum_{i=1}^{t}(\\hat{f}_{t}(X_{i})-Y_{i})^{2}}+\\sqrt{t}\\epsilon\\right)^{2}-A\\\\ \\leq\\left(\\sqrt{A}+\\sqrt{t}(\\epsilon+\\epsilon^{\\prime})\\right)^{2}-A\\\\ \\stackrel{(a)}{\\leq}(\\epsilon^{\\prime}+\\epsilon)^{2}t+2\\sqrt{t}(\\epsilon^{\\prime}+\\epsilon)\\sigma\\sqrt{2 t+3\\ln(2/\\delta)}\\\\ \\leq 4\\left(\\epsilon^{\\prime}+\\epsilon\\right)^{2}t+3\\sigma t\\left(\\epsilon^{\\prime}+\\epsilon\\right)+\\sigma^{2}\\ln(2/\\delta).\\tag{13.7}\\\\ \\end{array}\n\\]\nIn the above derivations, the first inequality used \\( |f(X_i) - Y_i| \\leq |\\hat{f}_t(X_i) - Y_i| + \\epsilon \\) for all \\( i \\)."}, {"n": 7, "claim": "The second inequality used the fact that \\( \\hat{f}_t \\) is approximate ERM. (a) follows from (13.6)."}, {"n": 8, "claim": "The last inequality used \\( 2a\\sigma \\sqrt{2b + 3c} \\leq 3a\\sigma \\sqrt{b} + 3a^2 + \\sigma^2 c \\)."}, {"n": 9, "claim": "We thus obtain\n\\[\n\\begin{array}{l}\\left(\\sum_{i=1}^{t}(\\hat{f}(X_{i})-f_{*}(X_{i}))^{2}\\right)^{1/2}\\\\ \\leq\\sqrt{\\epsilon^{2}t}+\\left(\\sum_{i=1}^{t}\\left(f\\left(X_{i}\\right)-f_{*}\\left(X_{i}\\right)\\right)^{2}\\right)^{1/2}\\quad(\\text{t r i a n g l e i n e q u a l i t y})\\\\ \\leq\\sqrt{\\epsilon^{2}t}+\\left(8\\sigma^{2}\\ln(2N/\\delta)+8\\sigma^{2}\\sum_{i=1}^{t}\\eta\\phi(f,Z_{i})\\right)^{1/2}\\quad(\\text{f r o m}(13. 5))\\\\ =\\sqrt{\\epsilon^{2}t}+\\left(8\\sigma^{2}\\ln(2N/\\delta)+2\\sum_{i=1}^{t}\\left[\\left(f\\left(X_{i}\\right)-Y_{i}\\right)^{2}-\\left(f_{*}\\left(X_{i}\\right)-Y_{i}\\right)^{2}\\right]\\right)^{1/2}\\\\ \\leq\\sqrt{\\epsilon^{2}t}+\\left(10\\sigma^{2}\\ln(2N/\\delta)+8t\\left(\\epsilon^{\\prime}+\\epsilon\\right)^{2}+6\\sigma t\\left(\\epsilon^{\\prime}+\\epsilon\\right)\\right)^{1/2}\\quad(\\text{f r o m}(13. 7))\\\\ \\leq\\left(12\\sigma^{2}\\ln(2N/\\delta)+16t(\\epsilon+\\epsilon^{\\prime})^{2}+8\\sigma t(\\epsilon+\\epsilon^{\\prime})\\right)^{1/2},\\\\ \\end{array}\n\\]\nwhere the last inequality used \\(\\sqrt{a} +\\sqrt{b}\\leq \\sqrt{6a + 1.2b}\\) ."}, {"n": 10, "claim": "This implies the bound."}]}}, {"local_id": "T3", "statement": "Consider a sequence of real-valued random (measurable) functions \\(\\xi_1(S_1),\\ldots,\\xi_T(S_T)\\). Let \\(\\tau \\leq T\\) be a stopping time so that \\(\\mathbb{1}(t\\leq \\tau)\\) is measurable in \\(S_{t}\\). We have\n\\[\n\\mathbb{E}_{\\mathcal{S}_{T}}\\exp\\left(\\sum_{i=1}^{\\tau}\\xi_{i}-\\sum_{i=1}^{\\tau}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{\\xi_{i}}\\right)=1.\n\\]", "proof": {"steps": [{"n": 1, "claim": "We prove the lemma by induction on \\( T \\). When \\( T = 0 \\), the equality is trivial."}, {"n": 2, "claim": "Assume that the claim holds at \\( T - 1 \\) for some \\( T \\geq 1 \\)."}, {"n": 3, "claim": "Now we will prove the equation at time \\( T \\) using the induction hypothesis."}, {"n": 4, "claim": "Note that \\(\\tilde{\\xi}_i = \\xi_i\\mathbb{1}(i\\leq \\tau)\\) is measurable in \\(S_{i}\\)."}, {"n": 5, "claim": "We have\n\\[\n\\sum_{i=1}^{\\tau}\\xi_{i}-\\sum_{i=1}^{\\tau}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{\\xi_{i}}=\\sum_{i=1}^{T}\\tilde{\\xi}_{i}-\\sum_{i=1}^{T}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{\\tilde{\\xi}_{i}}.\n\\]\nIt follows that\n\\[\n\\begin{array}{l}\\mathbb{E}_{Z_{1},\\dots,Z_{T}}\\exp\\left(\\sum_{i=1}^{\\tau}\\xi_{i}-\\sum_{i=1}^{\\tau}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{\\xi_{i}}\\right)\\\\ =\\mathbb{E}_{Z_{1},\\dots,Z_{T}}\\exp\\left(\\sum_{i=1}^{T}\\tilde{\\xi}_{i}-\\sum_{i=1}^{T}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{\\tilde{\\xi}_{i}}\\right)\\\\ =\\mathbb{E}_{Z_{1},\\ldots,Z_{n-1},Z_{T}^{(x)}}\\left[\\exp\\left(\\sum_{i=1}^{T-1}\\tilde{\\xi}_{i}-\\sum_{i=1}^{T-1}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{\\tilde{\\xi}_{i}}\\right)\\underbrace{\\mathbb{E}_{Z_{T}^{(y)}}\\exp(\\tilde{\\xi}_{T}-\\ln\\mathbb{E}_{Z_{T}^{(y)}}e^{\\tilde{\\xi}_{T}})}_{=1}\\right]\\\\ =\\mathbb{E}_{Z_{1},\\dots,Z_{T-1}}\\exp\\left(\\sum_{i=1}^{T-1}\\tilde{\\xi}_{i}-\\sum_{i=1}^{T-1}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{\\tilde{\\xi}_{i}}\\right)\\\\ =\\mathbb{E}_{Z_{1},\\dots,Z_{T-1}}\\exp\\left(\\sum_{i=1}^{\\min(\\tau,T-1)}\\xi_{i}-\\sum_{i=1}^{\\min(\\tau,T-1)}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{\\xi_{i}}\\right)=1.\\\\ \\end{array}\n\\]\nNote that the last equation follows from the induction hypothesis, and the fact that \\(\\min(\\tau,T-1)\\) is a stopping time \\(\\leq T - 1\\)."}]}}, {"local_id": "T4", "statement": "Consider a sequence of random functions \\(\\xi_1(S_1),\\ldots,\\xi_t(S_t),\\ldots\\) with filtration \\(\\{\\mathcal{F}_t\\}\\). We have for any \\(\\delta \\in (0,1)\\) and \\(\\lambda >0\\):\n\\[\n\\operatorname*{Pr}\\left[\\exists n>0: -\\sum_{i=1}^{n}\\xi_{i}\\geq\\frac{\\ln(1/\\delta)}{\\lambda}+\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda\\xi_{i}}\\right]\\leq\\delta .\n\\]\nMoreover, consider a sequence of \\(\\{z_t\\in \\mathbb{R}\\}\\) adapted to \\(\\{\\mathcal{F}_t\\}\\), and events \\(A_{t}\\) on \\(\\mathcal{F}_t\\):\n\\[\n\\ln\\Pr\\left[\\exists n>0:\\sum_{i=1}^{n}\\xi_{i}\\leq z_{n}\\&\\mathcal{S}_{n}\\in A_{n}\\right]\\leq\\inf_{\\lambda>0}\\sup_{n>0}\\sup_{\\mathcal{S}_{n}\\in A_{n}}\\left[\\lambda z_{n}+\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda\\xi_{i}}\\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "We will prove the result for a finite time sequence \\(\\xi_1(S_1),\\ldots,\\xi_T(S_T)\\)."}, {"n": 2, "claim": "It implies the desired result by letting \\(T\\to \\infty\\)."}, {"n": 3, "claim": "Let\n\\[\n\\xi_{\\tau}(\\lambda)= -\\sum_{i=1}^{\\tau}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda\\xi_{i}}-\\lambda\\sum_{i=1}^{\\tau}\\xi_{i},\n\\]\nwhere \\(\\tau\\) is a stopping time, then we have from Lemma 13.1:"}, {"n": 4, "claim": "\\(\\mathbb{E}e^{\\xi_{\\tau}(\\lambda)} = 1\\). Now for any given sequence of \\(\\tilde{z}_n(S_n)\\) and \\(A_{n}\\), define the stopping time \\(\\tau\\) as either \\(T\\), or the first time step \\(n\\) so that\n\\[\n\\xi_{n}(\\lambda)\\geq-\\tilde{z}_{n}\\left(\\mathcal{S}_{n}\\right)\\&\\mathcal{S}_{n}\\in A_{n}\n\\]\nfor each sequence \\(S_T\\)."}, {"n": 5, "claim": "It follows that\n\\[\n\\begin{array}{l}\\Pr\\left(\\exists n:\\xi_{n}(\\lambda)\\geq-\\tilde{z}_{n}\\left(\\mathcal{S}_{n}\\right)\\&\\mathcal{S}_{n}\\in A_{n}\\right)\\inf_{n>0,\\mathcal{S}_{n}\\in A_{n}}e^{-\\tilde{z}_{n}\\left(\\mathcal{S}_{n}\\right)}\\\\ \\leq\\mathbb{E}\\left[e^{\\xi_{\\tau}(\\lambda)+\\tilde{z}_{\\tau}\\left(\\mathcal{S}_{\\tau}\\right)}\\mathbb{1}\\left(\\mathcal{S}_{\\tau}\\in A_{\\tau}\\right)\\right]\\inf_{n>0,\\mathcal{S}_{n}\\in A_{n}}e^{-\\tilde{z}_{n}\\left(\\mathcal{S}_{n}\\right)}\\\\ \\leq\\mathbb{E}\\left[e^{\\xi_{\\tau}(\\lambda)+\\tilde{z}_{\\tau}\\left(\\mathcal{S}_{\\tau}\\right)}\\mathbb{1}\\left(\\mathcal{S}_{\\tau}\\in A_{\\tau}\\right)e^{-\\tilde{z}_{\\tau}\\left(\\mathcal{S}_{\\tau}\\right)}\\right]\\\\ \\leq\\mathbb{E}e^{\\xi_{\\tau}(\\lambda)}=1.\\\\ \\end{array}\n\\]\nTherefore we obtain\n\\[\n\\begin{array}{l}\\ln\\operatorname*{Pr}\\left[\\exists n>0: -\\lambda\\sum_{i=1}^{n}\\xi_{i}\\geq-\\tilde{z}_{n}(\\mathcal{S}_{n})+\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda\\xi_{i}}\\&\\mathcal{S}_{n}\\in A_{n}\\right]\\\\ \\leq\\sup_{n>0:\\mathcal{S}_{n}\\in A_{n}}\\tilde{z}_{n}(\\mathcal{S}_{n}).\\\\ \\end{array}\n\\]\nLet \\(\\tilde{z}(\\mathcal{S}_n) = \\ln \\delta\\), we obtain the first inequality."}, {"n": 6, "claim": "Let\n\\[\n\\tilde{z}_{n}(\\mathcal{S}_{n})=\\lambda z_{n}+\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda\\xi_{i}},\n\\]\nwe obtain the second inequality."}]}}, {"local_id": "T5", "statement": "We have for any \\(\\lambda >0\\), with probability at least \\(1 - \\delta\\), for all \\(n\\geq 1\\) and \\(f\\in \\mathcal{F}\\):\n\\[\n\\left[-\\sum_{i=1}^{n}f(Z_{i})-\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda f(Z_{i})}\\right]\\leq\\inf_{\\epsilon>0}\\left[2 n\\epsilon+\\frac{\\ln(M(\\epsilon,\\mathcal{F},\\|\\cdot\\|_{\\infty})/\\delta)}{\\lambda}\\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\mathcal{F}_{\\epsilon} \\subset \\mathcal{F}\\) be an \\(\\epsilon\\) maximal packing of \\(\\mathcal{F}\\), with \\(|\\mathcal{F}_{\\epsilon}| \\leq M(\\epsilon, \\mathcal{F}, \\| \\cdot \\|_{\\infty})\\)."}, {"n": 2, "claim": "We obtain from Theorem 13.2, and the uniform bound over \\(\\mathcal{F}_{\\epsilon}\\) that with probability at least \\(1 - \\delta\\):"}, {"n": 3, "claim": "\\[\n\\sup_{f\\in\\mathcal{F}_{\\epsilon}}\\left[-\\sum_{i=1}^{n}f(Z_{i})-\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda f(Z_{i})}\\right]\\leq\\frac{\\ln(M(\\epsilon,\\mathcal{F},\\|\\cdot\\|_{\\infty})/\\delta)}{\\lambda}.\n\\]\nSince \\(\\mathcal{F}_{\\epsilon}\\) is also an \\(\\epsilon L_{\\infty}\\) cover of \\(\\mathcal{F}\\) (see Theorem 5.2), we obtain\n\\[\n\\begin{array}{l}\\sup_{f\\in\\mathcal{F}}\\left[-\\sum_{i=1}^{n}f(Z_{i})-\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda f(Z_{i})}\\right]\\\\ \\leq 2 n\\epsilon+\\sup_{f\\in\\mathcal{F}_{\\epsilon}}\\left[-\\sum_{i=1}^{n}f(Z_{i})-\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda f(Z_{i})}\\right].\\\\ \\end{array}\n\\]\nThis implies the bound."}]}}], "target_local_id": "T2", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch13_theorem_13_1__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nFor \\(f:\\mathcal{X}\\to\\mathbb{R}\\) , the subgraph of \\(f\\) is\n\\[C_{f}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}:0\\leq t\\leq f(x)\\ o r\\ f(x)\\leq t\\leq 0\\}.\\]\nDefine class of sets \\(\\mathcal{C}=\\{C_{f}:f\\in\\mathcal{F}\\}\\).\n\ndefinition 12.1. If \\(\\mathcal{C}\\) is a VC class of sets, then F is VC-subgraph class of functions and, by definition, \\(V C(\\mathcal{F})=V C(\\mathcal{C})\\)\nNote that equivalent definition of \\(C_{f}\\) is\n\\[C_{f}^{\\prime}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}: |f(x)|\\geq|t|\\}.\\]\n\ndefinition 12.3. The ε-packing number, \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) , is the marimal cardinality of an E-separated set.\nNote that \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) is decreasing in ε.\n\ndefinition 12.2. Given \\(\\varepsilon > 0\\) and \\(f_{1},\\ldots,f_{N}\\in\\mathcal{F}\\) ， we say that \\(f_{1},\\ldots,f_{N}\\) are E-separated if \\(d(f_{i},f_{j})>\\varepsilon\\) for any \\(i \\neq j\\)\n\nDenote \\[\\triangle_{n}(\\mathcal{C})=\\sup_{\\{x_{1},\\ldots,x_{n}\\}}\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})\\leq 2^{n}.\\] We will see that for some classes, \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\) for \\(n \\leq V\\) and \\(\\triangle_{n}(\\mathcal{C})<2^{n}\\) for \\(n > V\\) for some constant \\(V\\).\n\ndefinition 8.1. If \\(V < \\infty\\) , then C is called a VC class. V is called VC dimension of C.\n\ndefinition 8.0. For any sample \\(\\{x_{1},\\ldots,x_{n}\\}\\), the ways that \\({\\mathcal{C}}\\) intersects with the sample are \\[\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in{\\mathcal{C}}\\}.\\] Let \\[\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})=\\mathrm{card}~\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in\\mathcal{C}\\},\\] the number of different subsets picked out by \\(C\\in{\\mathcal{C}}\\). If \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\), this means we can find \\(\\{x_{1},\\ldots,x_{n}\\}\\) such that \\({\\mathcal{C}}\\) can pick out any subset of it; in this case \\({\\mathcal{C}}\\) shatters \\(\\{x_{1},\\ldots,x_{n}\\}\\). Equivalently, a set \\(A\\) is shattered by \\({\\mathcal{C}}\\) when \\({\\mathcal{C}}\\) picks out every subset of \\(A\\).", "theorems": [{"local_id": "T1", "statement": "Setup. Let \\(X_1,\\dots,X_n\\) be i.i.d. random variables with common law \\(P\\) on a measurable space \\((\\mathcal{X},\\mathcal{A})\\), let \\(\\mathcal{C}\\subseteq\\mathcal{A}\\) be a class of measurable sets, and write \\(P(C)=\\mathbb{P}(X_1\\in C)\\). Let \\(X'_1,\\dots,X'_n\\) (the ghost sample) be an independent copy of \\(X_1,\\dots,X_n\\), i.e. i.i.d. \\(P\\) and independent of \\(X_1,\\dots,X_n\\). Suprema over \\(\\mathcal{C}\\) are assumed measurable; otherwise read \\(\\mathbb{P}\\) as outer probability.\n\nIf \\(t\\geq{\\sqrt{\\frac{2}{n}}}.\\) then\n\\[\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\right)\\leq 2\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\geq t/2\\right).\\]", "proof": {"steps": [{"n": 1, "claim": "Suppose the event\n\\[\\operatorname*{sup}_{C\\in{\\mathcal{C}}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\]\noccurs.Let \\(\\begin{array}{ r }{X=(X_{1},\\ldots,X_{n})\\in\\{\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\}}\\end{array}\\) ."}, {"n": 2, "claim": "Then\n\\[\\exists C_{X}{\\mathrm{~s u c h~t h a t~}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nFor a fixed C,\n\\[\\begin{array}{ r }{\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right|\\geq t/2\\right)=\\mathbb{P}\\left(\\left(\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}\\geq t^{2}/4\\right)}\\\\ {\\leq\\mathrm{~(b y~C h e b y s h e v~s~I n e q)~}\\frac{4\\mathbb{E}\\left(\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}}{t^{2}}}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i,j}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))(I(X_{j}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i=1}^{n}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))^{2}=\\frac{4 n\\mathbb{P}\\left(\\mathcal{C}\\right)\\left(1-\\mathbb{P}(\\mathcal{C})\\right)}{n^{2}t^{2}}\\leq\\frac{1}{n t^{2}}\\leq\\frac{1}{2}}\\end{array}\\]\nsince we chose \\(t\\geq\\sqrt{\\textstyle{\\frac{2}{n}}}.\\)\nSo,\n\\[\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\Bigg|\\exists C_{X}\\right)\\geq 1/2\\]\nif \\(t\\geq\\sqrt{2/n}\\) .Assume that the event\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\]\noccurs."}, {"n": 3, "claim": "Recall that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nHence,it must be that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\geq t/2 .\\]\nWe conclude\n\\[\\begin{array}{ r c l }{\\displaystyle\\frac 12}&{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}(C_{X})\\right|\\le t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right).}\\end{array}\\]\nSince indicators are 0,1-valued,\n\\[\\begin{array}{ r l r }&{}&{\\displaystyle\\frac{1}{2}T\\left(\\underbrace{\\operatorname*{sup}_{\\zeta\\in{\\mathcal C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}(C)\\right|\\ge t}_{\\Im C x}\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C x\\right)\\cdot I\\left(\\exists C x\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X,X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]\nNow, take expectation with respect to \\(X_{i}^{,}\\) s to obtain\n\\[\\begin{array}{ r l r }&{}&{\\mathbb{P}_{X}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\ge t\\right)}\\\\ &{}&{\\displaystyle\\le 2\\cdot\\mathbb{P}_{X,X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]"}]}}, {"local_id": "T2", "statement": "Let \\(\\xi,\\nu\\) - random variables.Assume that\n\\[\\mathbb{P}\\left(\\nu\\geq t\\right)\\leq\\Gamma e^{-\\gamma t}\\]\nwhere \\(\\Gamma\\geq 1,t\\geq 0\\) ，and \\(\\gamma > 0\\) . Furthermore, for all \\(a > 0\\) assume that\n\\[\\mathbb{E}\\phi(\\xi)\\le\\mathbb{E}\\phi(\\nu)\\]\nwhere \\(\\phi(x)=(x-a)_{+}\\) .Then\n\\[\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\Gamma\\cdot e\\cdot e^{-\\gamma t}.\\]", "proof": {"steps": [{"n": 1, "claim": "Since \\(\\phi(x)=(x-a)_{+}\\) ，we have \\(\\phi(\\xi)\\geq\\phi(t)\\) whenever \\(\\xi \\geq t .\\)\n\\[\\begin{array}{ r l }&{\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\mathbb{P}\\left(\\phi(\\xi)\\geq\\phi(t)\\right)}\\\\ &{\\qquad\\leq\\displaystyle\\frac{\\mathbb{E}\\phi(\\xi)}{\\phi(t)}\\leq\\frac{\\mathbb{E}\\phi(\\nu)}{\\phi(t)}=\\frac{\\mathbb{E}\\left(\\nu-a\\right)_{+}}{(t-a)_{+}}}\\end{array}\\]\nFurthermore,\n\\[\\begin{array}{ r l }{\\displaystyle\\mathbb{E}(\\nu-a)_{+}=}&{\\mathbb{E}\\int_{0}^{(\\nu-a)_{+}}\\mathrm{1}d x}\\\\ &{=\\mathbb{E}\\int_{0}^{\\infty}I(x\\leq(\\nu-a)_{+})d x}\\\\ &{=\\int_{0}^{\\infty}\\mathbb{E}I(x\\leq(\\nu-a)_{+})d x}\\\\ &{=\\int_{0}^{\\infty}\\mathbb{P}\\left((\\nu-a)_{+}\\geq x\\right)d x}\\\\ &{=\\int_{0}^{\\infty}\\mathbb{P}\\left(\\nu\\geq a+x\\right)d x}\\\\ &{\\leq\\int_{0}^{\\infty}\\Gamma e^{-\\gamma(a+x)}d x=\\frac{\\Gamma e^{-\\gamma a}}{\\gamma}.}\\end{array}\\]\nHence,\n\\[\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\frac{\\Gamma e^{-\\gamma a}}{\\gamma(t-a)_{+}}=\\frac{\\Gamma\\cdot e\\cdot e^{-\\gamma t}}{1}=\\Gamma\\cdot e\\cdot e^{-\\gamma t}\\]\nwhere we chose optimal \\(\\begin{array}{ r }{a=t-\\frac{1}{\\gamma}}\\end{array}\\) to minimize \\(\\frac{\\Gamma e^{-\\gamma a}}{\\gamma(t-a)_{+}}\\)"}]}}, {"local_id": "T3", "statement": "The number \\(\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})\\) of subsets picked out by C is bounded by the number of subsets shattered by C.", "proof": {"steps": [{"n": 1, "claim": "Without loss of generality, we restrict C to \\({\\mathcal{C}}: =\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in{\\mathcal{C}}\\}\\) , and we have \\(\\operatorname{card}({\\mathcal{C}})=\\) \\(\\Delta_{n}({\\mathcal{C}},x_{1},\\cdots,x_{n})\\)\nWe will say that C is hereditary if and only if whenever \\(B\\subseteq C\\in{\\mathcal{C}},B\\in{\\mathcal{C}}\\) .If C is hereditary, then every \\(C\\in{\\mathcal{C}}\\) is shattered by C,and the lemma is obvious."}, {"n": 2, "claim": "Otherwise,we will transform \\({\\mathcal{C}}\\to{\\mathcal{C}}^{\\prime}\\) ,hereditary, without changing the cardinality of C and without increasing the number of shattered subsets."}, {"n": 3, "claim": "Define the operators \\(T_{i}\\) for \\(i=1,\\cdots,n\\) as the following,\n\\[\\begin{array}{ r c l }{T_{i}(C)}&{=}&{\\left\\{\\begin{array}{ l l }{C-\\{x_{i}\\}}&{\\mathrm{~i f~}C-\\{x_{i}\\}\\mathrm{~i s~n o t~i n~}\\mathcal{C}}\\\\ {C}&{\\mathrm{~o t h e r w i s e}}\\end{array}\\right.}\\\\ {T_{i}(\\mathcal{C})}&{=}&{\\{T_{i}(C):C\\in\\mathcal{C}\\}.}\\end{array}\\]\nIt follows that card \\(T_{i}({\\mathcal{C}})=\\operatorname{card}{\\mathcal{C}}\\) Moreover, every \\(A\\subseteq\\{x_{1},\\cdots,x_{n}\\}\\) that is shattered by \\(T_{i}(\\mathcal{C})\\) is also shattered by \\({\\mathcal{C}}.{\\mathrm{~I f~}}x_{i}\\notin A .\\) ,then \\(\\forall C\\in\\mathcal{C},A\\bigcap C=A\\bigcap T_{i}(C)\\) , thus C and \\(T_{i}(\\mathcal{C})\\) both or neither shatter A."}, {"n": 4, "claim": "On the other hand, if \\(x_{i}\\in A\\) and A is shattered by \\(T_{i}(\\mathcal{C})\\) , then \\(\\forall B\\subseteq A,\\exists C\\in{\\mathcal{C}}\\) ,such that \\(B\\bigcup\\{x_{i}\\}=A\\bigcap T_{i}(C)\\) This means that \\(x_{i}\\in T_{i}(C)\\) ，and that \\(C\\backslash\\{x_{i}\\}\\in\\mathcal{C}\\) Thus both \\(B\\bigcup\\{x_{i}\\}\\) and \\(B\\backslash\\{x_{i}\\}\\) are picked out by C."}, {"n": 5, "claim": "Since either \\(B=B\\cup\\{x_{i}\\}\\) or \\(B=B\\backslash\\{x_{i}\\}\\) , B is picked out by C."}, {"n": 6, "claim": "Thus A is shattered by C. Apply the operator \\(T=T_{1}\\circ\\dots\\circ T_{n}\\) until \\(T^{k+1}({\\mathcal{C}})=T^{k}({\\mathcal{C}})\\) ."}, {"n": 7, "claim": "This will happen for at most \\(\\scriptstyle\\sum_{C\\in{\\mathcal{C}}}\\operatorname{card}(C)\\) times,since \\(\\begin{array}{ r }{\\sum_{C\\in\\mathcal{C}}\\operatorname{card}(T_{i}(C))<\\sum_{C\\in\\mathcal{C}}\\operatorname{card}(C)\\mathrm{~i f~}T_{i}(\\mathcal{C})\\neq\\mathcal{C}}\\end{array}\\) ，The resulting collection \\(\\scriptstyle{\\mathcal{C}}^{\\prime}\\) is hereditary."}, {"n": 8, "claim": "This proves the lemma."}]}}, {"local_id": "T4", "statement": "Setup. Let \\(0<\\varepsilon\\le 1\\).\n\nAssume \\(\\mathcal{F}\\) is a VC-subgraph class and \\(V C(\\mathcal{F})=V\\) Suppose \\(-1\\leq f(x)\\leq 1\\) for all \\(f\\in{\\mathcal{F}}\\) and \\(x\\in\\mathcal{X}\\) .Let \\(x_{1},\\ldots,x_{n}\\in{\\mathcal{X}}\\) and define \\(\\begin{array}{ r }{d(f,g)=\\frac{1}{n}\\sum_{i=1}^{n}|f(x_{i})-g(x_{i})|}\\end{array}\\) . Then\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]\n(which \\(\\begin{array}{ r }{i s\\le\\left(\\frac{K}{\\varepsilon}\\right)^{V+\\delta}}\\end{array}\\) for some \\(\\delta . \\jmath\\)", "proof": {"steps": [{"n": 1, "claim": "Let \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) and \\(f_{1},\\ldots,f_{m}\\) be ε-separated, i.e.\n\\[{\\frac{1}{n}}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})| >\\varepsilon .\\]"}, {"n": 2, "claim": "Let \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) be constructed in the following way:"}, {"n": 3, "claim": "\\(z_{i}\\) is chosen uniformly from \\(x_{1},\\ldots,x_{n}\\) and \\(t_{i}\\) is uniform on \\([-1,1]\\)\nConsider \\(f_{r}\\) and \\(f_{\\ell}\\) from the ε-packing."}, {"n": 4, "claim": "Let \\(C_{f_{\\tau}}\\) and \\(C_{f_{\\ell}}\\) be subgraphs of \\(f_{r}\\) and \\(f_{\\ell}\\) Then\n\\(\\mathbb{P}\\left(C_{f_{\\tau}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n= P(At least one point \\((z_{i},t_{i})\\) is picked by \\(C_{f_{r}}\\) or \\(C_{f_{\\ell}}\\) but not picked by the other)\n\\(=1-\\mathbb{P}\\) (All points \\((z_{i},t_{i})\\) are picked either by both or by none)\n\\(\\mathbf{\\Sigma}=1-\\mathbb{P}\\left(\\left(z_{i},t_{i}\\right)\\right.\\) is picked either by both or by none)k\nSince \\(z_{i}\\) is drawn uniformly from \\(x_{1},\\ldots,x_{n}\\)\nP \\(((z_{1},t_{1})\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)\n\\[\\begin{array}{ l }{\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{P}\\left(\\left(x_{i},t_{1}\\right)\\mathrm{~i s~p i c k e d~b y~b o t h~}C_{f r},C_{f_{\\ell}}\\mathrm{~o r~b y~n e i t h e r}\\right)}\\\\ {\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\left(1-\\frac{1}{2}|f_{r}(x_{i})-f_{\\ell}(x_{i})|\\right)}\\\\ {\\displaystyle=1-\\frac{1}{2}\\frac{1}{n}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})|}\\\\ {\\displaystyle=1-\\frac{1}{2}d(f_{r},f_{\\ell})\\leq 1-\\varepsilon/2\\leq e^{-\\varepsilon/2}}\\end{array}\\]\nSubstituting,\n\\(\\mathbb{P}\\left(C_{f_{r}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n\\(=1-\\mathbb{P}\\left((z_{1},t_{1})\\right.\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)𝑘\n\\[\\begin{array}{ l }{\\geq 1-\\left(e^{-\\varepsilon/2}\\right)^{k}}\\\\ {=1-e^{-k\\varepsilon/2}}\\end{array}\\]\nThere are (m） ways to choose \\(f_{r}\\) and \\(f_{\\ell},\\) s0\nP (All pairs \\(C_{f_{r}}\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\geq 1-{\\binom{m}{2}}e^{-k\\varepsilon/2}.\\)\nWhat k should we choose so that \\(1-{\\binom{m}{2}}e^{-k\\varepsilon/2}>0?\\) Choose\n\\[k>{\\frac{2}{\\varepsilon}}\\log{\\binom{m}{2}}.\\]\nThen there exist \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) such that all \\(C_{f_{\\ell}}\\) pick out different subsets.But \\(\\{C_{f}:f\\in\\mathcal{F}\\}\\) is \\(\\mathrm{VC}\\) so by Sauer's lemma, we can pick out at most \\(\\left({\\frac{e k}{V}}\\right)^{V}\\) out of these k points, provided \\(k \\geq V\\); if instead \\(k < V\\) then \\(m\\leq 2^{k}<2^{V}\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}\\) and the claimed bound already holds, so that case may be set aside."}, {"n": 5, "claim": "Hence, \\(\\begin{array}{ r }{m\\leq\\left(\\frac{e k}{V}\\right)^{V}}\\end{array}\\) as long as \\(k>{\\frac{2}{\\varepsilon}}\\log\\left({{\\binom{m}{2}}}\\right)\\) and \\(k \\geq V\\), the case \\(k < V\\) having been disposed of above."}, {"n": 6, "claim": "The latter holds for \\(\\begin{array}{ r }{k=\\frac{2}{\\varepsilon}\\log m^{2}}\\end{array}\\) ."}, {"n": 7, "claim": "Therefore,\n\\[m\\leq\\left(\\frac{e}{V}\\frac{2}{\\varepsilon}\\log m^{2}\\right)^{V}=\\left(\\frac{4 e}{V\\varepsilon}\\log m\\right)^{V},\\]\nwhere \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) ."}, {"n": 8, "claim": "Hence, we get\n\\[m^{1/V}\\leq\\frac{4 e}{\\varepsilon}\\log m^{1/V}\\]\nand defining \\(m^{1/V}=s .\\)\n\\[s\\leq{\\frac{4 e}{\\varepsilon}}\\log s .\\]\nNote that \\(\\frac{s}{\\log s}\\) is increasing for \\(s \\geq e\\) and so for large enough s, the inequality will be violated."}, {"n": 9, "claim": "We now check that the inequality is violated for \\(\\begin{array}{ r }{s^{\\prime}=\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}}\\end{array}\\) ."}, {"n": 10, "claim": "Indeed, one can show that\n\\[\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{7}{\\varepsilon}\\right)^{2}>\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)\\]\nsince\n\\[\\frac{49}{8e\\varepsilon}>\\log{\\frac{7}{\\epsilon}}.\\]\nHence, \\(m^{1/V}=s\\leq s^{\\prime}\\) and, thus,\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]"}]}}], "target_local_id": "T4", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch14_theorem_14_15__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\ndefinition 2.11. A sub-Gaussian random variable \\(X\\) has quadratic logarithmic moment generating function for all \\(\\lambda \\in \\mathbb{R}\\):\n\\[\n\\ln\\mathbb{E}e^{\\lambda X}\\leq\\lambda\\mu+\\frac{\\lambda^{2}}{2}b.\\tag{2.10}\n\\]\nUsing (2.10), we can obtain an upper bound of the rate function for sub-Gaussian random variables, which imply the following tail inequality.\n\nIn the setting of general online (nonconvex) optimization, the Gibbs distribution can be defined after time \\(t\\) as follows\n\\[\np_t(w) \\propto p_0(w) \\exp \\left(-\\eta \\sum_{s=1}^{t} \\ell_s(w)\\right), \\tag{14.9}\n\\]\nwhere \\(p_0(w)\\) is a prior on \\(\\Omega\\).\n\nHedge Algorithm. Hedge Algorithm\nInput: \\(T\\), prior \\(p_{0}(w)\\) on \\(\\Omega\\), learning rate \\(\\eta > 0\\)\n1 Randomly draw \\(w_{0} \\sim p_{0}(w)\\)\n2 for \\(t = 1, 2, \\ldots, T\\) do\n3 \\quad Observe loss \\(\\ell_{t}(w_{t-1})\\)\n4 \\quad Randomly draw \\(w_{t} \\sim p_{t}(w)\\) according to \\((14.9)\\)\n\nWe have the following result for the Hedge algorithm, using conditions similar to those of Theorem 7.17.\n\nOnline Gradient Descent. Algorithm 14.3: Online Gradient Descent\nInput: Sequence of loss functions \\(\\ell_1,\\dots,\\ell_T\\) defined on \\(\\Omega\\) Output: \\(\\hat{w}\\)   \n1 Let \\(w_{0} = 0\\)   \n2 for \\(t=1,2,\\ldots,T\\) do   \n3 Observe loss \\(\\ell_t(w_{t - 1})\\)   \n4 Let \\(\\tilde{w}_t = w_{t - 1} - \\eta_t\\nabla \\ell_t(w_{t - 1})\\)   \n5 Let \\(w_{t} = \\arg \\min_{w\\in \\Omega}\\| w - \\tilde{w}_{t}\\|_{2}^{2}\\)   \n6 Let \\(\\hat{w} = T^{-1}\\sum_{t = 1}^{T}w_{t - 1}\\) or \\(\\hat{w} = w_{s}\\) for a random \\(s\\) from O to \\(T - 1\\) Return: \\(\\hat{w}\\)\n\nexample 14.10. Consider the structured-SVM loss of Example 9.32, where\n\\[\n\\phi(w,z)=\\max_{\\ell}[\\gamma(y,\\ell)-w^{\\top}(\\psi(x,y)-\\psi(x,\\ell))].\n\\]\nIf \\( \\| \\psi(x, y) - \\psi(, \\ell) \\|_2 \\leq B \\), then we can take \\( G = B \\). Using Theorem 14.5, we obtain\n\\[\n\\frac{1}{T}\\sum_{t=1}^{T}\\phi(w_{t-1},Z_{t})\\leq\\frac{1}{T}\\sum_{t=1}^{T}\\phi(w,Z_{t})+\\frac{\\|w_{0}-w\\|_{2}^{2}}{2\\eta T}+\\frac{\\eta}{2}B^{2}.\n\\]\nBy taking expectation, and using Jensen's inequality with \\(\\lambda = 1 / (\\eta T)\\), we obtain\n\\[\n\\mathbb{E}_{\\mathcal{S}_{T}}\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(\\hat{w},Z)\\leq\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(w,Z)+\\frac{\\lambda}{2}\\|w_{0}-w\\|_{2}^{2}+\\frac{1}{2\\lambda T}B^{2}.\n\\]\nFor \\( L_{2} \\) regularization (or kernel methods), one can obtain a better bound using strong convexity. Observe that for regularized loss, we take\n\\[\n\\ell_{t}(w)=\\phi\\left(w,Z_{t}\\right)+\\frac{\\lambda}{2}\\|w-w_{0}\\|_{2}^{2}.\\tag{14.8}\n\\]\nIf \\(\\phi (w,z)\\) is convex in \\(w\\), then \\(\\ell_t(w)\\) is \\(\\lambda\\) strongly convex.\nThe following result holds for strongly convex loss functions, and the specific learning rate schedule was proposed by (Shalev-Shwartz et al., 2011) to solve SVMs.\n\ndefinition 6.35. Given \\(\\mathcal{D}\\) and \\(\\mathcal{F}\\), and consider a localization function \\(h: \\mathcal{F} \\to \\mathbb{R}\\) such that \\(b_0 = \\inf_{f \\in \\mathcal{F}} h(f) > -\\infty\\). Define localized function class \\(\\mathcal{F}^h(b) = \\{f \\in \\mathcal{F}: h(f) \\leq b\\}\\) for all \\(b > b_0\\). For any \\(\\alpha > 0\\), the rate function with respect to localization \\(h\\) is defined as\n\\[\n\\bar{r}_{n}^{h}(\\alpha,\\mathcal{F},\\mathcal{D})=\\sup\\left\\{r:r\\leq\\inf_{r^{\\prime}>\\max(r,\\alpha b_{0})}\\epsilon_{n}\\left(\\mathcal{F}^{h}\\big(r^{\\prime}/\\alpha\\big),\\mathcal{D}\\right)\\right\\}.\n\\]\nWe note that the requirement of \\( r' > \\alpha b_0 \\) in Definition 6.35 is only to make sure that \\( \\mathcal{F}^h(r'/\\alpha) \\) is always non-empty, and thus \\( \\epsilon_n(\\mathcal{F}^h(r'/\\alpha), \\mathcal{D}) \\) is well-defined.\n\ndefinition 6.6. Consider any known data-dependent offset function \\( h(w, S_n) \\). Define the uniform convergence complexity of a function class \\( \\mathcal{G} \\) with offset \\( h \\) as\n\\[\n\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\sup_{w\\in\\Omega}[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})].\\tag{6.2}\n\\]\nThis quantity measures the one-sided expected uniform convergence of function class \\(\\mathcal{G}\\) with offset function \\(h(w, S_n)\\).\nWe note that\n\\[\n\\epsilon_{n}(\\mathcal{G},\\mathcal{D})=\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D}),\\qquad \\text{w i t h} h=0.\n\\]\nWe have the following generalization of Theorem 6.2, with a similar proof.", "theorems": [{"local_id": "T1", "statement": "Given any function \\(U(w)\\), we have\n\\[\n\\min_{p\\in\\Delta(\\Omega)}\\left[\\mathbb{E}_{w\\sim p}U(w)+\\mathrm{KL}(p| |p_{0})\\right]= -\\ln\\mathbb{E}_{w\\sim p_{0}}\\exp(-U(w)),\n\\]\nand the solution is achieved by the Gibbs distribution \\( q(w) \\propto p_0(w) \\exp(-U(w)) \\). Here \\( \\Delta(\\Omega) \\) denotes the set of probability distributions on \\( \\Omega \\).", "proof": {"steps": [{"n": 1, "claim": "Let \\(C = \\mathbb{E}_{w\\sim p_0}\\exp (-U(w))\\) . Then we have\n\\[\n\\begin{array}{l}\\mathbb{E}_{w\\sim p}U(w)+\\operatorname{KL}(p| |p_{0})=\\mathbb{E}_{w\\sim p}\\ln\\frac{p(w)}{p_{0}(w)\\exp(-U(w))}\\\\ =\\mathbb{E}_{w\\sim p}\\ln\\frac{p(w)}{Cq(w)}\\geq-\\ln C.\\\\ \\end{array}\n\\]\nThe inequality used the fact that \\(\\mathrm{KL}(p||q) \\geq 0\\), and the equality holds when \\(p(w) = q(w)\\)."}]}}, {"local_id": "T2", "statement": "Assume that for all \\(t\\):\n\\[\n\\sup_{w\\in\\Omega}\\ell_{t}(w)-\\inf_{w\\in\\Omega}\\ell_{t}(w)\\leq M,\n\\]\nthen Algorithm 14.4 has regret\n\\[\n\\sum_{t=1}^{T}\\mathbb{E}_{w_{t-1}\\sim p_{t-1}(\\cdot)}\\ell_{t}(w_{t-1})\\leq\\inf_{p\\in\\Delta(\\Omega)}\\left[\\mathbb{E}_{w\\sim p}\\sum_{t=1}^{T}\\ell_{t}(w)+\\frac{1}{\\eta}\\mathrm{KL}(p| |p_{0})\\right]+\\frac{\\eta TM^{2}}{8},\n\\]\nwhere \\(\\Delta (\\Omega)\\) denotes the set of probability distributions on \\(\\Omega\\)", "proof": {"steps": [{"n": 1, "claim": "Let\n\\[\nZ_{t}= -\\ln\\mathbb{E}_{w\\sim p_{0}}\\exp\\left(-\\eta\\sum_{s=1}^{t}\\ell_{s}(w)\\right)\n\\]\nbe the log-partition function for observations up to time \\( t \\)."}, {"n": 2, "claim": "We have\n\\[\n\\begin{array}{l}Z_{t-1}-Z_{t}=\\ln\\mathbb{E}_{w\\sim p_{t-1}}\\exp(-\\eta\\ell_{t}(w))\\\\ \\leq-\\eta\\mathbb{E}_{w\\sim p_{t-1}}\\ell_{t}(w)+\\frac{\\eta^{2}M^{2}}{8},\\\\ \\end{array}\n\\]\nwhere the first equation is simple algebra, and the inequality follows from the estimate of logarithmic moment generation function in Lemma 2.15."}, {"n": 3, "claim": "By summing\nover \\(t = 1\\) to \\(T\\), and noticing that \\(Z_{0} = 0\\), we obtain\n\\[\n\\sum_{t=1}^{T}\\mathbb{E}_{w_{t-1}\\sim p_{t-1}(\\cdot)}\\ell_{t}(w_{t-1})\\leq\\frac{1}{\\eta}Z_{T}+\\frac{\\eta T M^{2}}{8}.\n\\]\nThe desired bound follows by applying Proposition 7.16 to reformulate the log-partition function \\( Z_{T} \\)."}]}}, {"local_id": "T3", "statement": "The rate function in Definition 6.35 is always non-negative.", "proof": {"steps": [{"n": 1, "claim": "Note that \\(\\mathcal{F}^h (r' / \\alpha)\\neq \\emptyset\\) when \\(r^\\prime >\\max (0,\\alpha b_0)\\). Since \\(\\epsilon_{n}(\\mathcal{F}^{h}(r^{\\prime} / \\alpha),\\mathcal{D})\\) is always non-negative, with \\(r = 0\\), we have \\(r\\leq \\epsilon_n(\\mathcal{F}^h (r' / \\alpha),\\mathcal{D})\\) when \\(r^\\prime >\\) \\(\\max (r,\\alpha b_0)\\)"}]}}, {"local_id": "T4", "statement": "Consider a random variable \\(X \\in [0,1]\\) and \\(\\mathbb{E}X = \\mu\\). We have the following inequality:\n\\[\n\\ln\\mathbb{E}e^{\\lambda X}\\leq\\ln [(1-\\mu)e^{0}+\\mu e^{\\lambda}]\\leq\\lambda\\mu+\\lambda^{2}/8.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\( h_L(\\lambda) = \\mathbb{E}e^{\\lambda X} \\) and \\( h_R(\\lambda) = (1 - \\mu)e^0 + \\mu e^\\lambda \\)."}, {"n": 2, "claim": "We know that \\( h_L(0) = h_R(0) \\). Moreover, when \\( \\lambda \\geq 0 \\):"}, {"n": 3, "claim": "\\[\nh_{L}^{\\prime}(\\lambda)=\\mathbb{E}X e^{\\lambda X}\\leq\\mathbb{E}X e^{\\lambda}=\\mu e^{\\lambda}=h_{R}^{\\prime}(\\lambda),\n\\]\nand similarly \\(h_L'(\\lambda) \\geq h_R'(\\lambda)\\) when \\(\\lambda \\leq 0\\)."}, {"n": 4, "claim": "This proves the first inequality. Now we let\n\\[\nh(\\lambda)=\\ln [(1-\\mu)e^{0}+\\mu e^{\\lambda}].\n\\]\nIt implies that\n\\[\nh^{\\prime}(\\lambda)=\\frac{\\mu e^{\\lambda}}{(1-\\mu)e^{0}+\\mu e^{\\lambda}},\n\\]\nand\n\\[\n\\begin{array}{l}h^{\\prime\\prime}(\\lambda)=\\frac{\\mu e^{\\lambda}}{(1-\\mu)e^{0}+\\mu e^{\\lambda}}-\\frac{(\\mu e^{\\lambda})^{2}}{[(1-\\mu)e^{0}+\\mu e^{\\lambda}]^{2}}\\\\ =\\left|h^{\\prime}(\\lambda)\\right|\\left(1-\\left|h^{\\prime}(\\lambda)\\right|\\right)\\leq 1/4.\\\\ \\end{array}\n\\]\nUsing Taylor expansion, we obtain the inequality \\( h(\\lambda) \\leq h(0) + \\lambda h'(0) + \\lambda^2 / 8 \\), which proves the second inequality."}]}}, {"local_id": "T5", "statement": "Consider the regularized loss function (14.8) with \\( w_0 = 0 \\), where \\( \\phi(w, z) \\) is convex in \\( w \\), and \\( G \\) Lipschitz in \\( w \\). Moreover assume that \\( d(\\Omega) = \\sup \\{ \\| w \\|_2 : w \\in \\Omega \\} \\). If \\( Z_1, \\ldots, Z_T \\sim \\mathcal{D} \\) are independent samples, then we can obtain the following expected oracle inequality for Algorithm 14.3 if we take learning rate in Theorem 14.11:\n\\[\n\\mathbb{E}_{\\mathcal{S}_{T}}\\mathbb{E}_{\\mathcal{D}}\\phi(\\hat{w},Z)+\\frac{\\lambda}{2}\\|\\hat{w}\\|_{2}^{2}\\leq\\inf_{w\\in\\Omega}\\left[\\mathbb{E}_{\\mathcal{D}}\\phi(w,Z)+\\frac{\\lambda}{2}\\|w\\|_{2}^{2}\\right]+\\frac{\\ln(e T)}{2\\lambda T}[G+\\lambda d(\\Omega)]^{2}.\n\\]\nWe can also obtain the following expected oracle inequality for Algorithm 14.3 if we take learning rate in Theorem 14.12 with\n\\[\n\\hat{w}^{\\prime}=\\sum_{t=1}^{T}\\frac{2(t+1)}{T^{2}+3 T}w_{t-1},\n\\]\nthen\n\\[\n\\mathbb{E}_{\\mathcal{S}_{T}}\\mathbb{E}_{\\mathcal{D}}\\phi(\\hat{w}^{\\prime},Z)+\\frac{\\lambda}{2}\\|\\hat{w}^{\\prime}\\|_{2}^{2}\\leq\\inf_{w\\in\\Omega}\\left[\\mathbb{E}_{\\mathcal{D}}\\phi(w,Z)+\\frac{\\lambda}{2}\\|w\\|_{2}^{2}\\right]+\\frac{2[G+\\lambda d(\\Omega)]^{2}}{\\lambda(T+3)}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Note that \\(\\ell_t(w)\\) is \\(\\lambda\\)-strongly convex in \\(w\\)."}, {"n": 2, "claim": "Moreover, it is \\(G + \\lambda d(\\Omega)\\) Lipschitz."}, {"n": 3, "claim": "We simply take expectation and apply Jensen's inequality to obtain the desired bounds."}]}}], "target_local_id": "T2", "topic_domain": "Online Learning & RL", "topic_subtopic": "Online Learning & Regret", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch15_lemma_15_2__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\ndefinition 12.1. If \\(\\mathcal{C}\\) is a VC class of sets, then F is VC-subgraph class of functions and, by definition, \\(V C(\\mathcal{F})=V C(\\mathcal{C})\\)\nFor \\(f:\\mathcal{X}\\to\\mathbb{R}\\) , the subgraph of \\(f\\) is\n\\[C_{f}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}:0\\leq t\\leq f(x)\\ o r\\ f(x)\\leq t\\leq 0\\}.\\]\nDefine class of sets \\(\\mathcal{C}=\\{C_{f}:f\\in\\mathcal{F}\\}\\).\nNote that equivalent definition of \\(C_{f}\\) is\n\\[C_{f}^{\\prime}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}: |f(x)|\\geq|t|\\}.\\]\n\ndefinition 12.3. The ε-packing number, \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) , is the marimal cardinality of an E-separated set.\nNote that \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) is decreasing in ε.\n\ndefinition 12.2. Given \\(\\varepsilon > 0\\) and \\(f_{1},\\ldots,f_{N}\\in\\mathcal{F}\\) ， we say that \\(f_{1},\\ldots,f_{N}\\) are E-separated if \\(d(f_{i},f_{j})>\\varepsilon\\) for any \\(i \\neq j\\)\n\nDenote \\[\\triangle_{n}(\\mathcal{C})=\\sup_{\\{x_{1},\\ldots,x_{n}\\}}\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})\\leq 2^{n}.\\] We will see that for some classes, \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\) for \\(n \\leq V\\) and \\(\\triangle_{n}(\\mathcal{C})<2^{n}\\) for \\(n > V\\) for some constant \\(V\\).\n\ndefinition 8.1. If \\(V < \\infty\\) , then C is called a VC class. V is called VC dimension of C.\n\ndefinition 8.0. For any sample \\(\\{x_{1},\\ldots,x_{n}\\}\\), the ways that \\({\\mathcal{C}}\\) intersects with the sample are \\[\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in{\\mathcal{C}}\\}.\\] Let \\[\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})=\\mathrm{card}~\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in\\mathcal{C}\\},\\] the number of different subsets picked out by \\(C\\in{\\mathcal{C}}\\). If \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\), this means we can find \\(\\{x_{1},\\ldots,x_{n}\\}\\) such that \\({\\mathcal{C}}\\) can pick out any subset of it; in this case \\({\\mathcal{C}}\\) shatters \\(\\{x_{1},\\ldots,x_{n}\\}\\). Equivalently, a set \\(A\\) is shattered by \\({\\mathcal{C}}\\) when \\({\\mathcal{C}}\\) picks out every subset of \\(A\\).", "theorems": [{"local_id": "T1", "statement": "Let \\(\\xi,\\nu\\) - random variables.Assume that\n\\[\\mathbb{P}\\left(\\nu\\geq t\\right)\\leq\\Gamma e^{-\\gamma t}\\]\nwhere \\(\\Gamma\\geq 1,t\\geq 0\\) ，and \\(\\gamma > 0\\) . Furthermore, for all \\(a > 0\\) assume that\n\\[\\mathbb{E}\\phi(\\xi)\\le\\mathbb{E}\\phi(\\nu)\\]\nwhere \\(\\phi(x)=(x-a)_{+}\\) .Then\n\\[\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\Gamma\\cdot e\\cdot e^{-\\gamma t}.\\]", "proof": {"steps": [{"n": 1, "claim": "Since \\(\\phi(x)=(x-a)_{+}\\) ，we have \\(\\phi(\\xi)\\geq\\phi(t)\\) whenever \\(\\xi \\geq t .\\)\n\\[\\begin{array}{ r l }&{\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\mathbb{P}\\left(\\phi(\\xi)\\geq\\phi(t)\\right)}\\\\ &{\\qquad\\leq\\displaystyle\\frac{\\mathbb{E}\\phi(\\xi)}{\\phi(t)}\\leq\\frac{\\mathbb{E}\\phi(\\nu)}{\\phi(t)}=\\frac{\\mathbb{E}\\left(\\nu-a\\right)_{+}}{(t-a)_{+}}}\\end{array}\\]\nFurthermore,\n\\[\\begin{array}{ r l }{\\displaystyle\\mathbb{E}(\\nu-a)_{+}=}&{\\mathbb{E}\\int_{0}^{(\\nu-a)_{+}}\\mathrm{1}d x}\\\\ &{=\\mathbb{E}\\int_{0}^{\\infty}I(x\\leq(\\nu-a)_{+})d x}\\\\ &{=\\int_{0}^{\\infty}\\mathbb{E}I(x\\leq(\\nu-a)_{+})d x}\\\\ &{=\\int_{0}^{\\infty}\\mathbb{P}\\left((\\nu-a)_{+}\\geq x\\right)d x}\\\\ &{=\\int_{0}^{\\infty}\\mathbb{P}\\left(\\nu\\geq a+x\\right)d x}\\\\ &{\\leq\\int_{0}^{\\infty}\\Gamma e^{-\\gamma(a+x)}d x=\\frac{\\Gamma e^{-\\gamma a}}{\\gamma}.}\\end{array}\\]\nHence,\n\\[\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\frac{\\Gamma e^{-\\gamma a}}{\\gamma(t-a)_{+}}=\\frac{\\Gamma\\cdot e\\cdot e^{-\\gamma t}}{1}=\\Gamma\\cdot e\\cdot e^{-\\gamma t}\\]\nwhere we chose optimal \\(\\begin{array}{ r }{a=t-\\frac{1}{\\gamma}}\\end{array}\\) to minimize \\(\\frac{\\Gamma e^{-\\gamma a}}{\\gamma(t-a)_{+}}\\)"}]}}, {"local_id": "T2", "statement": "Setup. Fix \\(f_1,\\dots,f_d:\\mathcal{X}\\to\\mathbb{R}\\) and let \\(\\mathcal{C}=\\left\\{\\left\\{x:\\sum_{k=1}^{d}\\alpha_k f_k(x)>0\\right\\}:\\alpha_1,\\dots,\\alpha_d\\in\\mathbb{R}\\right\\}\\).\n\nThen \\(VC(\\mathcal{C})\\le d\\).", "proof": {"steps": [{"n": 1, "claim": "Observation: For any \\(\\{x_{1},\\dots,x_{d+1}\\}\\) if we cannot shatter \\(\\{x_{1},\\dots,x_{d+1}\\}\\longleftrightarrow\\exists I\\subseteq\\{1\\dots d+1\\}\\) s.t. we cannot pick out \\(\\{x_{i},i\\in I\\}\\) ."}, {"n": 2, "claim": "If we can pick out \\(\\{x_{i},i\\in I\\}\\) , then for some \\(C\\in{\\mathcal{C}}\\) there are \\(\\alpha_{1},\\ldots,\\alpha_{d}\\) s.t."}, {"n": 3, "claim": "\\(\\textstyle\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x)>0\\) for \\(i \\in I\\) and \\(\\begin{array}{ r }{\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x)\\le 0}\\end{array}\\) for \\(i \\notin I\\)."}, {"n": 4, "claim": "Denote\n\\[\\left(\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x_{1}),\\ldots,\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x_{d+1})\\right)=F(\\alpha)\\in\\mathbb{R}^{d+1}.\\]\nBy linearity,\n\\[F(\\alpha)=\\sum_{k=1}^{d}\\alpha_{k}\\left(f_{k}(x_{1}),\\ldots,f(x_{d+1})\\right)=\\sum_{k=1}^{d}\\alpha_{k}F_{k}\\subseteq H\\subset\\mathbb{R}^{d+1}\\]\nand H is a d-dim subspace."}, {"n": 5, "claim": "Hence, \\(\\exists\\phi\\neq 0,\\phi\\cdot h=0,\\forall h\\in H\\) (\\(\\phi\\) orthogonal to H)."}, {"n": 6, "claim": "Let \\(I=\\{i:\\phi_{i}>0\\}\\) where \\(\\phi=\\left(\\phi_{1},\\ldots,\\phi_{d+1}\\right)\\) ."}, {"n": 7, "claim": "If \\(I=\\emptyset\\) then take \\(-\\phi\\) instead of \\(\\phi\\) so that \\(\\phi\\) has positive coordinates.\nClaim: We cannot pick out \\(\\{x_{i},i\\in I\\}\\) .Suppose we can: then \\(\\exists\\alpha_{1},\\dotsc,\\alpha_{d}\\) s.t. \\(\\begin{array}{ r }{\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x_{i})>0}\\end{array}\\) for \\(i\\in I\\) and \\(\\begin{array}{ r }{\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x_{i})\\le 0}\\end{array}\\) for \\(i \\not \\in I\\). But \\(\\phi\\cdot F(\\alpha)=0\\) and so\n\\[\\phi_{1}\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x_{1})+\\ldots+\\phi_{d+1}\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x_{d+1})=0 .\\]\nHence,\n\\[\\sum_{i\\in I}\\underbrace{\\phi_{i}\\left(\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x_{i})\\right)}_{>0}=\\sum_{i\\notin I}\\underbrace{(-\\phi_{i})}_{\\geq 0}\\underbrace{\\left(\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x_{i})\\right)}_{\\leq 0}.\\]\nContradiction."}]}}, {"local_id": "T3", "statement": "Let \\(x=(x_{1},\\ldots,x_{n}),x^{\\prime}=(x_{1}^{\\prime},\\ldots,x_{n}^{\\prime})\\) . If for functions \\(\\varphi_{1}(x,x^{\\prime}),\\varphi_{2}(x,x^{\\prime}),\\varphi_{3}(x,x^{\\prime})\\)\n\\[\\begin{array}{ r }{\\mathbb{P}\\left(\\varphi_{1}(x,x^{\\prime})\\geq\\varphi_{2}(x,x^{\\prime})+\\sqrt{\\varphi_{3}(x,x^{\\prime})\\cdot t}\\right)\\leq\\Gamma e^{-\\gamma t}}\\end{array}\\]\nthen\n\\[\\begin{array}{ r }{\\mathbb{P}\\left(\\mathbb{E}_{x^{\\prime}}\\varphi_{1}(x,x^{\\prime})\\geq\\mathbb{E}_{x^{\\prime}}\\varphi_{2}(x,x^{\\prime})+\\sqrt{\\mathbb{E}_{x^{\\prime}}\\varphi_{3}(x,x^{\\prime})\\cdot t}\\right)\\leq\\Gamma\\cdot e\\cdot e^{-\\gamma t}.}\\end{array}\\]\n(i.e. if the inequality holds, then it holds with averaging over one of the copies)", "proof": {"steps": [{"n": 1, "claim": "First, note that \\(\\begin{array}{ r }{\\sqrt{a b}=\\operatorname*{inf}_{\\delta>0}(\\delta a+\\frac{b}{4\\delta})}\\end{array}\\) with \\(\\begin{array}{ r }{\\delta_{*}=\\sqrt{\\frac{b}{4 a}}}\\end{array}\\) achieving the infima."}, {"n": 2, "claim": "Hence,\n\\[\\begin{array}{ c }{{\\{\\varphi_{1}\\geq\\varphi_{2}+\\sqrt{\\varphi_{3}t}\\}=\\{\\exists\\delta>0,\\varphi_{1}\\geq\\varphi_{2}+\\delta\\varphi_{3}+\\displaystyle\\frac{t}{4\\delta}\\}}}\\\\ {{=\\{\\exists\\delta>0,(\\varphi_{1}-\\varphi_{2}-\\delta\\varphi_{3})4\\delta\\geq t\\}}}\\\\ {{=\\{\\underbrace{\\sup_{\\delta>0}(\\varphi_{1}-\\varphi_{2}-\\delta\\varphi_{3})4\\delta}_{\\nu}\\geq t\\}}}\\end{array}\\]\nand similarly\n\\[\\big\\{\\mathbb{E}_{x^{\\prime}}\\varphi_{1}\\geq\\mathbb{E}_{x^{\\prime}}\\varphi_{2}+\\sqrt{\\mathbb{E}_{x^{\\prime}}\\varphi_{3}t}\\big\\}=\\big\\{\\underbrace{\\sup_{\\delta>0}(\\mathbb{E}_{x^{\\prime}}\\varphi_{1}-\\mathbb{E}_{x^{\\prime}}\\varphi_{2}-\\delta\\mathbb{E}_{x^{\\prime}}\\varphi_{3})4\\delta}_{\\xi}\\geq t\\big\\}.\\]\nBy assumption, \\(\\mathbb{P}\\left(\\nu\\geq t\\right)\\leq\\Gamma e^{-\\gamma t}\\) ."}, {"n": 3, "claim": "We want to prove \\(\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\Gamma\\cdot e\\cdot e^{-\\gamma t}\\) ."}, {"n": 4, "claim": "By the previous lemma, we only need to check whether \\(\\mathbb{E}\\phi(\\xi)\\le\\mathbb{E}\\phi(\\nu)\\)\n\\[\\begin{array}{ l }{\\displaystyle\\xi=\\operatorname*{sup}_{\\delta>0}\\mathbb{E}_{x^{\\prime}}(\\varphi_{1}-\\varphi_{2}-\\delta\\varphi_{3})4\\delta}\\\\ {\\displaystyle\\quad\\leq\\mathbb{E}_{x^{\\prime}}\\operatorname*{sup}_{\\delta>0}(\\varphi_{1}-\\varphi_{2}-\\delta\\varphi_{3})4\\delta}\\\\ {\\displaystyle=\\mathbb{E}_{x^{\\prime}}\\nu}\\end{array}\\]\nThus,\n\\[\\phi(\\xi)\\leq\\phi(\\mathbb{E}_{x^{\\prime}}\\nu)\\leq\\mathbb{E}_{x^{\\prime}}\\phi(\\nu)\\]\nby Jensen's inequality (\\(\\phi\\) is convex)."}, {"n": 5, "claim": "Hence,\n\\[\\mathbb{E}\\phi(\\xi)\\le\\mathbb{E}\\mathbb{E}_{x^{\\prime}}\\phi(\\nu)=\\mathbb{E}\\phi(\\nu).\\]"}]}}, {"local_id": "T4", "statement": "Setup. Let \\(0<\\varepsilon\\le 1\\).\n\nAssume \\(\\mathcal{F}\\) is a VC-subgraph class and \\(V C(\\mathcal{F})=V\\) Suppose \\(-1\\leq f(x)\\leq 1\\) for all \\(f\\in{\\mathcal{F}}\\) and \\(x\\in\\mathcal{X}\\) .Let \\(x_{1},\\ldots,x_{n}\\in{\\mathcal{X}}\\) and define \\(\\begin{array}{ r }{d(f,g)=\\frac{1}{n}\\sum_{i=1}^{n}|f(x_{i})-g(x_{i})|}\\end{array}\\) . Then\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]\n(which \\(\\begin{array}{ r }{i s\\le\\left(\\frac{K}{\\varepsilon}\\right)^{V+\\delta}}\\end{array}\\) for some \\(\\delta . \\jmath\\)", "proof": {"steps": [{"n": 1, "claim": "Let \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) and \\(f_{1},\\ldots,f_{m}\\) be ε-separated, i.e.\n\\[{\\frac{1}{n}}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})| >\\varepsilon .\\]"}, {"n": 2, "claim": "Let \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) be constructed in the following way:"}, {"n": 3, "claim": "\\(z_{i}\\) is chosen uniformly from \\(x_{1},\\ldots,x_{n}\\) and \\(t_{i}\\) is uniform on \\([-1,1]\\)\nConsider \\(f_{r}\\) and \\(f_{\\ell}\\) from the ε-packing."}, {"n": 4, "claim": "Let \\(C_{f_{\\tau}}\\) and \\(C_{f_{\\ell}}\\) be subgraphs of \\(f_{r}\\) and \\(f_{\\ell}\\) Then\n\\(\\mathbb{P}\\left(C_{f_{\\tau}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n= P(At least one point \\((z_{i},t_{i})\\) is picked by \\(C_{f_{r}}\\) or \\(C_{f_{\\ell}}\\) but not picked by the other)\n\\(=1-\\mathbb{P}\\) (All points \\((z_{i},t_{i})\\) are picked either by both or by none)\n\\(\\mathbf{\\Sigma}=1-\\mathbb{P}\\left(\\left(z_{i},t_{i}\\right)\\right.\\) is picked either by both or by none)k\nSince \\(z_{i}\\) is drawn uniformly from \\(x_{1},\\ldots,x_{n}\\)\nP \\(((z_{1},t_{1})\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)\n\\[\\begin{array}{ l }{\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{P}\\left(\\left(x_{i},t_{1}\\right)\\mathrm{~i s~p i c k e d~b y~b o t h~}C_{f r},C_{f_{\\ell}}\\mathrm{~o r~b y~n e i t h e r}\\right)}\\\\ {\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\left(1-\\frac{1}{2}|f_{r}(x_{i})-f_{\\ell}(x_{i})|\\right)}\\\\ {\\displaystyle=1-\\frac{1}{2}\\frac{1}{n}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})|}\\\\ {\\displaystyle=1-\\frac{1}{2}d(f_{r},f_{\\ell})\\leq 1-\\varepsilon/2\\leq e^{-\\varepsilon/2}}\\end{array}\\]\nSubstituting,\n\\(\\mathbb{P}\\left(C_{f_{r}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n\\(=1-\\mathbb{P}\\left((z_{1},t_{1})\\right.\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)𝑘\n\\[\\begin{array}{ l }{\\geq 1-\\left(e^{-\\varepsilon/2}\\right)^{k}}\\\\ {=1-e^{-k\\varepsilon/2}}\\end{array}\\]\nThere are (m） ways to choose \\(f_{r}\\) and \\(f_{\\ell},\\) s0\nP (All pairs \\(C_{f_{r}}\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\geq 1-{\\binom{m}{2}}e^{-k\\varepsilon/2}.\\)\nWhat k should we choose so that \\(1-{\\binom{m}{2}}e^{-k\\varepsilon/2}>0?\\) Choose\n\\[k>{\\frac{2}{\\varepsilon}}\\log{\\binom{m}{2}}.\\]\nThen there exist \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) such that all \\(C_{f_{\\ell}}\\) pick out different subsets.But \\(\\{C_{f}:f\\in\\mathcal{F}\\}\\) is \\(\\mathrm{VC}\\) so by Sauer's lemma, we can pick out at most \\(\\left({\\frac{e k}{V}}\\right)^{V}\\) out of these k points, provided \\(k \\geq V\\); if instead \\(k < V\\) then \\(m\\leq 2^{k}<2^{V}\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}\\) and the claimed bound already holds, so that case may be set aside."}, {"n": 5, "claim": "Hence, \\(\\begin{array}{ r }{m\\leq\\left(\\frac{e k}{V}\\right)^{V}}\\end{array}\\) as long as \\(k>{\\frac{2}{\\varepsilon}}\\log\\left({{\\binom{m}{2}}}\\right)\\) and \\(k \\geq V\\), the case \\(k < V\\) having been disposed of above."}, {"n": 6, "claim": "The latter holds for \\(\\begin{array}{ r }{k=\\frac{2}{\\varepsilon}\\log m^{2}}\\end{array}\\) ."}, {"n": 7, "claim": "Therefore,\n\\[m\\leq\\left(\\frac{e}{V}\\frac{2}{\\varepsilon}\\log m^{2}\\right)^{V}=\\left(\\frac{4 e}{V\\varepsilon}\\log m\\right)^{V},\\]\nwhere \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) ."}, {"n": 8, "claim": "Hence, we get\n\\[m^{1/V}\\leq\\frac{4 e}{\\varepsilon}\\log m^{1/V}\\]\nand defining \\(m^{1/V}=s .\\)\n\\[s\\leq{\\frac{4 e}{\\varepsilon}}\\log s .\\]\nNote that \\(\\frac{s}{\\log s}\\) is increasing for \\(s \\geq e\\) and so for large enough s, the inequality will be violated."}, {"n": 9, "claim": "We now check that the inequality is violated for \\(\\begin{array}{ r }{s^{\\prime}=\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}}\\end{array}\\) ."}, {"n": 10, "claim": "Indeed, one can show that\n\\[\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{7}{\\varepsilon}\\right)^{2}>\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)\\]\nsince\n\\[\\frac{49}{8e\\varepsilon}>\\log{\\frac{7}{\\epsilon}}.\\]\nHence, \\(m^{1/V}=s\\leq s^{\\prime}\\) and, thus,\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]"}]}}], "target_local_id": "T3", "topic_domain": "Probability & Concentration", "topic_subtopic": "Functional & Supremum Concentration", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch17_theorem_17_4__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nContextual Bandit Problem. In contextual bandit, we consider a context space \\(\\mathcal{X}\\) and an action space \\(\\mathcal{A}\\). Given context \\(x \\in \\mathcal{X}\\), we take an action \\(a \\in \\mathcal{A}\\), and observe a reward \\(r \\in \\mathbb{R}\\) that can depend on \\((x, a)\\). The contextual bandit problem is a repeated game: at each time step \\(t\\):\n- The player observes a sample \\( x_{t} \\in \\mathcal{X} \\)\n- The player chooses precisely one action (or arm) \\(a_{t} \\in \\mathcal{A}\\)\n- The reward \\( r_t \\) is revealed.\nNote that in this chapter, we will follow the convention in the bandit literature of using \\(\\mathcal{A}\\) to denote the action space instead of the learning algorithm. We may also introduce the notation of policy for contextual bandits. Using the notation of Section 13.4, a bandit learning algorithm returns a policy \\(\\hat{q}(\\mathcal{S}_{t-1})\\) based on the history \\(\\mathcal{S}_{t-1}\\).\n\ndefinition 17.2. A policy \\(\\pi\\) for contextual bandit is a map \\(\\mathcal{X} \\to \\Delta(\\mathcal{A})\\), where \\(\\Delta(\\mathcal{A})\\) denotes probability measures over \\(\\mathcal{A}\\) with an appropriately defined \\(\\sigma\\)-algebra. One may also write it as a conditional distribution \\(\\pi(a|x)\\), and the policy draws \\(a \\sim \\pi(\\cdot|x)\\) when it observes context \\(x\\). A contextual bandit algorithm \\(\\hat{q}\\) maps historic observations\n\\[\n\\mathcal{S}_{t-1}=\\left\\{\\left(x_{1},a_{1},r_{1}\\right),\\dots,\\left(x_{t-1},a_{t-1},r_{t-1}\\right)\\right\\}\n\\]\nto a policy \\(\\pi_t = \\hat{q}(\\cdot | \\mathcal{S}_{t-1})\\) at each time step \\(t\\), and pulls an arm \\(a_t \\sim \\pi_t(\\cdot | x_t)\\) based on the observation \\(x_t\\). In this chapter, we will also write the history dependent policy as \\(a_t \\sim \\hat{q}(a_t | x_t, \\mathcal{S}_{t-1})\\).\nThe contextual bandit problem includes the multi-armed bandit problem as a special case if we take \\( x = x_0 \\) to be a fixed context for all time steps.\nSimilar to the case of multi-armed bandit problem, we may also consider the adversarial setting with an oblivious adversary as follows. At each time step \\( t \\), we have the information of all rewards \\( [r_t(a) : a \\in \\mathcal{A}] \\), but only reveals the value\nof \\( r_t(a_t) \\) for the chosen arm \\( a_t \\). The goal is to maximize the expected cumulative reward\n\\[\n\\sum_{t=1}^{T}\\mathbb{E}_{a_{t}\\sim\\pi_{t}}[r_{t}(a_{t})].\n\\]\nIf we are given a policy class \\(\\Pi\\), then regret of a contextual bandit algorithm with respect to \\(\\Pi\\) can be written as\n\\[\n\\operatorname{REG}_{T}=\\sup_{\\pi\\in\\Pi}\\sum_{t=1}^{T}\\mathbb{E}_{a_{t}\\sim\\pi}[r_{t}(a_{t})]-\\sum_{t=1}^{T}\\mathbb{E}_{a_{t}\\sim\\pi_{t}}[r_{t}(a_{t})].\\tag{17.1}\n\\]\nIf we consider the stochastic contextual bandit setting with unknown value functions\n\\[\nf_{*}(x,a)=\\mathbb{E}[r|x,a],\\qquad f_{*}(x)=\\max_{a\\in\\mathcal{A}}f(x,a)\n\\]\nthat do not change over time, then the goal becomes to maximize the expected reward\n\\[\n\\sum_{t=1}^{T}\\mathbb{E}_{a_{t}\\sim\\pi_{t}}[f_{*}(x_{t},a_{t})].\n\\]\nThe regret of the algorithm that produces policy sequence \\(\\{\\pi_t\\}\\) is:\n\\[\n\\operatorname{REG}_{T}=\\sum_{t=1}^{T}\\mathbb{E}_{a_{t}\\sim\\pi_{t}}\\left[f_{*}\\left(x_{t}\\right)-f_{*}\\left(x_{t},a_{t}\\right)\\right].\\tag{17.2}\n\\]\n\ndefinition 17.7. Stochastic linear contextual bandit (or stochastic contextual bandit with linear payoff) is a contextual bandit problem, where the reward at each time step \\( t \\) is given by\n\\[\nr_{t}(a)=r_{t}(x_{t},a)=w_{*}^{\\top}\\psi(x_{t},a)+\\epsilon_{t}(x_{t},a),\\tag{17.8}\n\\]\nwhere \\(\\epsilon_t(x, a)\\) is a zero-mean random variable. We assume that \\(\\mathcal{H}\\) is a known inner product space, \\(w_* \\in \\mathcal{H}\\) is the unknown model parameter, and the feature vector \\(\\psi(x, a) \\in \\mathcal{H}\\) is known.\nIn the stochastic linear bandit model, the number of arms can be either infinite or finite, and an UCB style algorithm is presented in Algorithm 17.2. Note that the algorithm selects an arm which has the largest confidence bound adjusted reward. This methodology is referred to as the optimism in the face of uncertainty, and is a direct generalization of upper confidence bound (UCB) for MAB. Therefore we call Algorithm 17.2 linear UCB, although it is also referred to as OFUL (optimism in the face of uncertainty linear bandit) in the literature (Abbasi-yadjkori et al., 2011).\n\nEXP4. Algorithm 17.1: EXP4\nInput: \\(K,T,\\mathcal{G},p_0(\\cdot),\\gamma \\in (0,1],\\eta >0,b\\geq 0\\)   \n1 Let \\(u_{0}(w) = 1\\)   \n2 for \\(t=1,2,\\ldots,T\\) do   \n3 Observe \\(x_{t}\\)   \n4 for \\(a=1,\\dots,K\\) do   \n5 Let \\(\\hat{\\pi}_t(a) = (1 - \\gamma)\\mathbb{E}_{w\\sim p_{t - 1}(w)}\\hat{q}_t(a|w,x_t) + \\gamma /K\\)   \n6 Sample \\(a_{t}\\) according to \\(\\hat{\\pi}_t(\\cdot)\\)   \n7 Pull arm \\(a_{t}\\) and observe reward \\(r_t(a_t)\\in [0,1]\\)   \n8 Let \\(\\hat{r}_t(w,x_t,a_t) = \\hat{q}_t(a_t|w,x_t)(r_t(a_t) - b) / \\hat{\\pi}_t(a_t)\\)   \n9 Let \\(u_{t}(w) = u_{t - 1}(w)\\exp (\\eta \\hat{r}_{t}(w,x_{t},a_{t}))\\)   \n10 Let \\(p_t(w) = p_0(w)u_t(w) / \\mathbb{E}_{w\\sim p_0(w)}u_t(w)\\)\n\nNote that conditioned on the history, the estimator \\(\\hat{r}_t(w,x_t,a_t)\\) is a random estimator that depends on the partial reward \\(r_t(a_t)\\) received for \\(a_t \\sim \\hat{\\pi}_t(a)\\). Moreover, it is an unbiased estimator of the following shifted reward of \\(w\\), according to policy \\(\\hat{q}_t(\\cdot|w,x_t)\\):\n\\[\n\\mathbb{E}_{a_t \\sim \\hat{\\pi}_t} \\hat{r}_t(w,x_t,a_t) = \\mathbb{E}_{a \\sim \\hat{q}_t(a|w,x_t)} (r_t(a) - b). \\tag{17.3}\n\\]\nwhich relies on the full reward vector \\([r_t(a)]\\) at time step \\(t\\) over all arms \\(a\\).\n\nLinear UCB Algorithm. Algorithm 17.2: Linear UCB Algorithm\nInput: \\(\\lambda, T, \\{\\beta_t\\}\\)  \n1 Let \\(A_0 = \\lambda I\\)  \n2 Let \\(w_0 = 0\\)  \n3 Let \\(b_0 = 0\\)  \n4 for \\(t = 1, 2, \\dots, T\\) do  \n5 Observe \\(x_t\\)  \n6 Let \\(a_t \\in \\arg \\max_a \\left[ w_{t-1}^\\top \\psi(x_t, a) + \\beta_{t-1} \\sqrt{\\psi(x_t, a)^\\top A_{t-1}^{-1} \\psi(x_t, a)} \\right]\\)  \n7 Pull arm \\(a_t\\) and observe reward \\(r_t(x_t, a_t)\\)  \n8 Let \\(b_t = b_{t-1} + r_t(x_t, a_t) \\psi(x_t, a_t)\\)  \n9 Let \\(A_t = A_{t-1} + \\psi(x_t, a_t) \\psi(x_t, a_t)^\\top\\)  \n10 Let \\(w_t = A_t^{-1} b_t\\)", "theorems": [{"local_id": "T1", "statement": "Let \\(\\eta = \\gamma / K\\) and \\(b = 0\\). Assume that the uniform random policy belongs to \\(\\Omega\\) and \\(|\\Omega| = N < \\infty\\). Let \\(p_0(w)\\) be the uniform prior over \\(\\Omega\\), then\n\\[\nG_{*}-\\mathbb{E}\\sum_{t=1}^{T}r_{t}(a_{t})\\leq(e-1)\\gamma G_{*}+\\frac{K\\ln N}{\\gamma},\n\\]\nwhere the expectation is with respect to the randomization of the algorithm, and\n\\[\nG_{*}=\\arg\\max_{w}\\sum_{t=1}^{T}\\mathbb{E}\\left.\\mathbb{E}_{a\\sim\\hat{q}_{t}(\\cdot|w,x_{t})}[r_{t}(a)]\\right..\n\\]", "proof": {"steps": [{"n": 1, "claim": "We have \\(\\eta \\hat{r}_t(w,x_t,a_t)\\leq 1\\), and thus \\(c(\\eta,b)=e-2\\)."}, {"n": 2, "claim": "Note that the uniform random policy belongs to \\(\\Omega\\) implies that\n\\[\n\\frac{1}{K}\\sum_{t=1}^{T}\\sum_{a=1}^{K}r_{t}(a)\\leq G_{*}.\n\\]\nWe consider Theorem 17.4, with \\( q \\) defined as \\( q(w) = \\mathbb{1}(w = w_{*}) \\), where \\( w_{*} \\) achieves the maximum of \\( G_{*} \\)."}, {"n": 3, "claim": "This implies\n\\[\n\\mathbb{E}\\sum_{t=1}^{T}r_{t}(a_{t})\\geq(1-\\gamma)\\left[G_{*}-\\frac{K}{\\gamma}\\ln N\\right]-(e-2)\\gamma G_{*}.\n\\]\nThis implies the bound."}]}}, {"local_id": "T2", "statement": "Assume that in the stochastic linear bandit model, \\(r_t(x_t, a_t) \\in [0,1]\\) and \\(\\|w_*\\|_2 \\leq B\\) for some constant \\(B\\). Let \\(\\mu_t(x, a) = \\mathbb{E}_{\\epsilon_t(x,a)} r_t(x,a) = w_*^\\top \\psi(x,a)\\). Let \\(a_*(x) \\in \\arg \\max_a \\mu_t(x,a)\\) be the optimal arm for each context \\(x\\). Then in Algorithm 17.2, with probability at least \\(1 - \\delta\\),\n\\[\n\\mathbb{E}\\sum_{t=1}^{T}[\\mu_{t}(x_{t},a_{*}(x_{t}))-\\mu_{t}(x_{t},a_{t})]\\leq 3\\sqrt{\\ln|A_{T}/\\lambda|\\sum_{t=1}^{T}\\beta_{t-1}^{2}}+2\\ln|A_{T}/\\lambda|,\n\\]\nwhere \\(\\{\\beta_t\\}\\) is any sequence that satisfies (17.9).", "proof": {"steps": [{"n": 1, "claim": "We have for \\(t \\geq 1\\): \\[\n\\begin{array}{l}w_{*}^{\\top}\\psi(x_{t},a_{*}(x_{t}))\\\\ \\leq w_{t-1}^{\\top}\\psi(x_{t},a_{*}(x_{t}))+\\beta_{t-1}\\sqrt{\\psi(x_{t},a_{*}(x_{t}))^{\\top}A_{t-1}^{-1}\\psi(x_{t},a_{*}(x_{t}))}\\\\ \\leq w_{t-1}^{\\top}\\psi(x_{t},a_{t})+\\beta_{t-1}\\sqrt{\\psi(x_{t},a_{t})^{\\top}A_{t-1}^{-1}\\psi(x_{t},a_{t})}\\\\ \\leq w_{*}^{\\top}\\psi(x_{t},a_{t})+2\\beta_{t-1}\\sqrt{\\psi(x_{t},a_{t})^{\\top}A_{t-1}^{-1}\\psi(x_{t},a_{t})},\\\\ \\end{array}\n\\]\nwhere the first and the third inequalities used Lemma 17.8."}, {"n": 2, "claim": "The second inequality is due to the UCB choice of \\( a_{t} \\) in Algorithm 17.2."}, {"n": 3, "claim": "Let \\( E_{t} \\) be the event of \\( \\| \\psi (x_{t},a_{t})\\|_{A_{t - 1}^{- 1}}\\leq 1 \\)."}, {"n": 4, "claim": "Since \\( w_{*}^{\\top}\\psi (x_{t},a)\\in [0,1] \\), we have\n\\[\nw_{*}^{\\top}\\psi(x_{t},a_{*}(x_{t}))-w_{*}^{\\top}\\psi(x_{t},a_{t})\\leq 2\\beta_{t-1}\\|\\psi(x_{t},a_{t})\\|_{A_{t-1}^{-1}}\\mathbb{1}(E_{t})+\\mathbb{1}(E_{t}^{c}).\n\\]\nBy summing over \\( t = 1 \\) to \\( t = T \\), we obtain\n\\[\n\\begin{array}{l}\\sum_{t=1}^{T}\\left[\\mu_{t}\\left(x_{t},a_{*}\\left(x_{t}\\right)\\right)-\\mu_{t}\\left(x_{t},a_{t}\\right)\\right]\\\\ \\leq 2\\sum_{t=1}^{T}\\beta_{t-1}\\|\\psi\\left(x_{t},a_{t}\\right)\\|_{A_{t-1}^{-1}}\\mathbb{1}\\left(E_{t}\\right)+\\sum_{t=1}^{T}\\mathbb{1}\\left(E_{t}^{c}\\right)\\\\ \\leq 2\\sum_{t=1}^{T}\\beta_{t-1}\\sqrt{\\frac{2\\|\\psi(x_{t},a_{t})\\|_{A_{t-1}^{-1}}^{2}}{1+\\|\\psi(x_{t},a_{t})\\|_{A_{t-1}^{-1}}^{2}}}+2\\sum_{t=1}^{T}\\frac{\\psi(x_{t},a_{t})^{\\top}A_{t-1}^{-1}\\psi(x_{t},a_{t})}{1+\\psi(x_{t},a_{t})^{\\top}A_{t-1}^{-1}\\psi(x_{t},a_{t})}\\\\ \\leq 3\\sqrt{\\sum_{t=1}^{T}\\beta_{t-1}^{2}}\\sqrt{\\sum_{t=1}^{T}\\frac{\\psi(x_{t},a_{t})^{\\top}A_{t-1}^{-1}\\psi(x_{t},a_{t})}{1+\\psi(x_{t},a_{t})^{\\top}A_{t-1}^{-1}\\psi(x_{t},a_{t})}}+2\\ln|A_{T}/\\lambda|.\\\\ \\end{array}\n\\]\nThe second inequality used simple algebraic inequalities under \\( E_{t} \\) and \\( E_{t}^{c} \\)."}, {"n": 5, "claim": "The third inequality used the Cauchy-Schwartz inequality and Lemma 13.9."}, {"n": 6, "claim": "We can now apply Lemma 13.9 again to obtain the desired bound."}]}}, {"local_id": "T3", "statement": "Given any function \\(U(w)\\), we have\n\\[\n\\min_{p\\in\\Delta(\\Omega)}\\left[\\mathbb{E}_{w\\sim p}U(w)+\\mathrm{KL}(p| |p_{0})\\right]= -\\ln\\mathbb{E}_{w\\sim p_{0}}\\exp(-U(w)),\n\\]\nand the solution is achieved by the Gibbs distribution \\( q(w) \\propto p_0(w) \\exp(-U(w)) \\). Here \\( \\Delta(\\Omega) \\) denotes the set of probability distributions on \\( \\Omega \\).", "proof": {"steps": [{"n": 1, "claim": "Let \\(C = \\mathbb{E}_{w\\sim p_0}\\exp (-U(w))\\) . Then we have\n\\[\n\\begin{array}{l}\\mathbb{E}_{w\\sim p}U(w)+\\operatorname{KL}(p| |p_{0})=\\mathbb{E}_{w\\sim p}\\ln\\frac{p(w)}{p_{0}(w)\\exp(-U(w))}\\\\ =\\mathbb{E}_{w\\sim p}\\ln\\frac{p(w)}{Cq(w)}\\geq-\\ln C.\\\\ \\end{array}\n\\]\nThe inequality used the fact that \\(\\mathrm{KL}(p||q) \\geq 0\\), and the equality holds when \\(p(w) = q(w)\\)."}]}}, {"local_id": "T4", "statement": "Assume that in the stochastic linear bandit model, \\(\\| w_{*}\\|_{\\mathcal{H}}\\leq B\\) for some constant \\(B\\), and in Algorithm 17.2, assume that \\(\\{\\beta_t\\}\\) is any sequence so that\n\\[\n\\Pr\\left[\\forall 0\\leq t\\leq T:\\beta_{t}\\geq\\sqrt{\\lambda}B+\\left\\|\\sum_{s=1}^{t}\\epsilon_{s}\\left(x_{s},a_{s}\\right)\\psi\\left(x_{s},a_{s}\\right)\\right\\|_{A_{t}^{-1}}\\right]\\geq 1-\\delta .\\tag{17.9}\n\\]\nThen with probability at least \\(1 - \\delta\\), for all \\(t = 0, \\dots, T\\) and \\(u \\in \\mathcal{H}\\):\n\\[\n\\left|u^{\\top}\\left(w_{t}-w_{*}\\right)\\right|\\leq\\beta_{t}\\sqrt{u^{\\top}A_{t}^{-1}u}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "We have\n\\[\n\\begin{array}{l}u^{\\top}\\left(w_{t}-w_{*}\\right)=u^{\\top}A_{t}^{-1}\\sum_{s=1}^{t}r_{s}\\left(x_{s},a_{s}\\right)\\psi\\left(x_{s},a_{s}\\right)-u^{\\top}w_{*}\\\\ =u^{\\top}A_{t}^{-1}\\sum_{s=1}^{t}\\epsilon_{s}\\left(x_{s},a_{s}\\right)\\psi\\left(x_{s},a_{s}\\right)-\\lambda u^{\\top}A_{t}^{-1}w_{*}\\\\ \\leq\\|u\\|_{A_{t}^{-1}}\\left\\|\\sum_{s=1}^{t}\\epsilon_{s}\\left(x_{s},a_{s}\\right)\\psi\\left(x_{s},a_{s}\\right)\\right\\|_{A_{t}^{-1}}+\\lambda\\|u\\|_{A_{t}^{-1}}\\|w_{*}\\|_{A_{t}^{-1}}\\\\ \\leq\\|u\\|_{A_{t}^{-1}}\\left[\\left\\|\\sum_{s=1}^{t}\\epsilon_{s}\\left(x_{s},a_{s}\\right)\\psi\\left(x_{s},a_{s}\\right)\\right\\|_{A_{t}^{-1}}+\\sqrt{\\lambda}B\\right]\\leq\\beta_{t}\\|u\\|_{A_{t}^{-1}}.\\\\ \\end{array}\n\\]\nThe second equality used (17.8)."}, {"n": 2, "claim": "The first inequality used the Cauchy-Schwartz\ninequality."}, {"n": 3, "claim": "The last inequality used the definition of \\(\\beta_{t}\\)."}, {"n": 4, "claim": "This implies the desired bound."}]}}, {"local_id": "T5", "statement": "For any \\(K\\), \\(T \\geq 0\\), and any \\(\\gamma \\in (0,1]\\), \\(\\eta > 0\\) and \\(b \\geq 0\\). Consider any expert class \\(\\mathcal{G} = \\{[\\hat{q}_t(\\cdot | w, x)]_{t=1,2,\\ldots} : w \\in \\Omega\\}\\) with prior \\(p_0(w)\\). Let\n\\[\nR_{T}(w)=\\mathbb{E}\\sum_{t=1}^{T}\\mathbb{E}_{a\\sim\\hat{q}_{t}(\\cdot|w,x_{t})}r_{t}(a)\n\\]\nbe the reward of expert \\( w \\). Then the expected reward of Algorithm 17.1 satisfies:\n\\[\n\\begin{array}{l}\\mathbb{E}\\sum_{t=1}^{T}r_{t}(a_{t})\\geq(1-\\gamma)\\max_{q}\\left[\\mathbb{E}_{w\\sim q}R_{T}(w)-\\frac{1}{\\eta}\\mathrm{KL}(q| |p_{0})\\right]\\\\ -c(\\eta,b)\\eta\\sum_{t=1}^{T}\\sum_{a=1}^{K}|r_{t}(a)-b|,\\\\ \\end{array}\n\\]\nwhere the expectation is with respect to the randomization of the algorithm,\n\\[\nc(\\eta,b)=\\phi(z_{0})\\max(b,1-b),\\quad z_{0}=\\max(0,\\eta(1-b)K/\\gamma),\n\\]\nand \\(\\phi (z) = (e^{z} - 1 - z) / z^{2}\\)", "proof": {"steps": [{"n": 1, "claim": "The proof is similar to that of Theorem 16.17. By the definition of \\(\\hat{\\pi}_t(a_t)\\), we have\n\\[\n\\tilde{\\rho}_{t}\\left(a_{t}\\right)=\\mathbb{E}_{w\\sim p_{t-1}(w)}\\hat{q}_{t}\\left(a_{t}\\mid w,x_{t}\\right)/\\hat{\\pi}_{t}\\left(a_{t}\\right)\\leq 1/(1-\\gamma).\\tag{17.4}\n\\]\nTherefore\n\\[\n\\begin{array}{l}\\mathbb{E}_{w\\sim p_{t-1}(w)}\\hat{r}_{t}(w,x_{t},a_{t})=\\mathbb{E}_{w\\sim p_{t-1}(w)}\\hat{q}_{t}(a_{t}|w,x_{t})[r_{t}(a_{t})-b]/\\hat{\\pi}_{t}(a_{t})\\\\ \\leq\\frac{1}{1-\\gamma}r_{t}\\left(a_{t}\\right)-\\tilde{\\rho}_{t}\\left(a_{t}\\right)b.\\tag{17.5}\\\\ \\end{array}\n\\]\nThe first equation used the definition of \\(\\hat{r}\\)."}, {"n": 2, "claim": "The inequality used (17.4). Moreover\n\\[\n\\begin{array}{l}\\mathbb{E}_{w\\sim p_{t-1}(w)}\\hat{r}_{t}(w,x_{t},a_{t})^{2}\\\\ =\\mathbb{E}_{w\\sim p_{t-1}(w)}\\hat{q}_{t}\\left(a_{t}\\mid w,x_{t}\\right)^{2}\\left(\\left(r_{t}\\left(a_{t}\\right)-b\\right)/\\hat{\\pi}_{t}\\left(a_{t}\\right)\\right)^{2}\\\\ \\leq\\max(b,1-b)\\mathbb{E}_{w\\sim p_{t-1}(w)}\\hat{q}_{t}\\left(a_{t}|w,x_{t}\\right)\\left(\\left|r_{t}\\left(a_{t}\\right)-b\\right| /\\hat{\\pi}_{t}\\left(a_{t}\\right)^{2}\\right)\\\\ \\leq\\frac{\\max(b,1-b)}{1-\\gamma}\\left(\\left|r_{t}\\left(a_{t}\\right)-b\\right| /\\hat{\\pi}_{t}\\left(a_{t}\\right)\\right).\\tag{17.6}\\\\ \\end{array}\n\\]\nThe first equation used the definition of \\(\\hat{r}\\)."}, {"n": 3, "claim": "The first inequality used \\(|r_t(a_t) - b| \\leq \\max(b, 1 - b)\\)."}, {"n": 4, "claim": "The second inequality used (17.4). If we let\n\\[\nW_{t}=\\mathbb{E}_{w\\sim p_{0}(w)}u_{t}(w),\n\\]\nthen\n\\[\n\\begin{array}{l}\\ln\\frac{W_{t}}{W_{t-1}}=\\ln\\mathbb{E}_{w\\sim p_{0}(w)}\\frac{u_{t}(w)}{W_{t-1}}\\\\ =\\ln\\mathbb{E}_{w\\sim p_{t-1}(w)}\\exp(\\eta\\hat{r}_{t}(w,x_{t},a_{t}))\\\\ \\leq\\ln\\mathbb{E}_{w\\sim p_{t-1}(w)}\\left[1+\\left(\\eta\\hat{r}_{t}\\left(w,x_{t},a_{t}\\right)\\right)+\\phi\\left(z_{0}\\right)\\left(\\eta\\hat{r}_{t}\\left(w,x_{t},a_{t}\\right)\\right)^{2}\\right]\\\\ \\leq\\mathbb{E}_{w\\sim p_{t-1}(w)}\\left(\\eta\\hat{r}_{t}\\left(w,x_{t},a_{t}\\right)\\right)+\\phi\\left(z_{0}\\right)\\mathbb{E}_{w\\sim p_{t-1}(w)}\\left(\\eta\\hat{r}_{t}\\left(w,x_{t},a_{t}\\right)\\right)^{2}\\\\ \\leq\\frac{\\eta}{1-\\gamma}r_{t}(a_{t})-\\eta\\tilde{\\rho}_{t}(a_{t})b+\\frac{c(\\eta,b)\\eta^{2}}{(1-\\gamma)}\\frac{|r_{t}(a_{t})-b|}{\\hat{\\pi}_{t}(a_{t})}.\\\\ \\end{array}\n\\]\nThe second equality used the definition of \\( u_{t} \\)."}, {"n": 5, "claim": "The first inequality used \\( \\exp (z) \\leq \\)\n\\(1 + z + \\phi (z_0)z^2\\) when \\(z = \\eta \\hat{r}_t(w,x_t,a_t)\\leq z_0\\); the second inequality used \\(\\ln (1 + z)\\leq\\) \\(z\\); and the third inequality used (17.5) and (17.6)."}, {"n": 6, "claim": "Now we can sum over \\( t = 1 \\) to \\( t = T \\), and obtain:"}, {"n": 7, "claim": "\\[\n\\ln\\frac{W_{T}}{W_{0}}\\leq\\frac{\\eta}{1-\\gamma}\\sum_{t=1}^{T}r_{t}(a_{t})-\\eta b\\sum_{t=1}^{T}\\tilde{\\rho}_{t}(a_{t})+\\frac{c(\\eta,b)\\eta^{2}}{(1-\\gamma)}\\sum_{t=1}^{T}\\frac{|r_{t}(a_{t})-b|}{\\hat{\\pi}_{t}(a_{t})}.\n\\]\nNote that \\( W_0 = 1 \\)."}, {"n": 8, "claim": "We can take expectation with respect to the randomization of the algorithm, and obtain\n\\[\n\\begin{array}{l}\\mathbb{E}\\ln\\mathbb{E}_{w\\sim p_{0}(w)}\\exp\\left(\\eta\\sum_{t=1}^{T}\\hat{r}_{t}(w,x_{t},a_{t})\\right)\\\\ \\leq\\frac{\\eta}{1-\\gamma}\\mathbb{E}\\sum_{t=1}^{T}r_{t}\\left(a_{t}\\right)-\\eta T b+\\frac{c(\\eta,b)\\eta^{2}}{(1-\\gamma)}\\mathbb{E}\\sum_{t=1}^{T}\\sum_{a=1}^{K}\\left|r_{t}(a)-b\\right|.\\tag{17.7}\\\\ \\end{array}\n\\]\nThe derivation also used \\(\\mathbb{E}\\tilde{\\rho}_t(a_t) = \\mathbb{E}\\mathbb{E}_{a_t\\sim \\hat{\\pi}_t}\\tilde{\\rho}_t(a_t) = 1\\) and\n\\[\n\\mathbb{E}\\frac{|r_{t}(a_{t})-b|}{\\hat{\\pi}_{t}(a_{t})}=\\mathbb{E}\\mathbb{E}_{a_{t}\\sim\\hat{\\pi}_{t}}\\frac{|r_{t}(a_{t})-b|}{\\hat{\\pi}_{t}(a_{t})}=\\mathbb{E}\\sum_{a=1}^{K}|r_{t}(a)-b|.\n\\]\nWe can lower bound the left hand side of (17.7) as:"}, {"n": 9, "claim": "\\[\n\\begin{array}{l}\\mathbb{E}\\ln\\mathbb{E}_{w\\sim p_{0}(w)}\\exp\\left(\\eta\\sum_{t=1}^{T}\\hat{r}_{t}(w,x_{t},a_{t})\\right)\\\\ =\\mathbb{E}\\max_{q}\\left[\\mathbb{E}_{w\\sim q}\\eta\\sum_{t=1}^{T}\\hat{r}_{t}(w,x_{t},a_{t})-\\operatorname{KL}(q| |p_{0})\\right]\\\\ \\geq\\max_{q}\\left[\\mathbb{E}_{w\\sim q}\\eta\\mathbb{E}\\sum_{t=1}^{T}\\hat{r}_{t}(w,x_{t},a_{t})-\\mathrm{KL}(q| |p_{0})\\right]\\\\ =\\max_{q}\\left[\\mathbb{E}_{w\\sim q}\\eta [R_{T}(w)-bT]-\\mathrm{KL}(q| |p_{0})\\right].\\\\ \\end{array}\n\\]\nThe first equality used Proposition 7.16."}, {"n": 10, "claim": "The last equation used (17.3) and the definition of \\( R_{T} \\)."}, {"n": 11, "claim": "By plugging this estimate into the left hand side of (17.7), we obtain the desired result."}]}}], "target_local_id": "T5", "topic_domain": "Online Learning & RL", "topic_subtopic": "Bandits", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch18_theorem_18_14__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nassumption 18.12. Given a candidate value function class \\(\\mathcal{F}\\) of functions \\(f = \\{f^{h}(x^{h},a^{h}):\\mathcal{X}\\times \\mathcal{A}\\to \\mathbb{R}\\}\\), with \\(f^{H + 1}(\\cdot) = 0\\). We assume that (realizable assumption)\n\\[\nQ_{*}=f_{*}\\in\\mathcal{F}.\n\\]\nMoreover, we assume that \\( f^1(x^1) \\in [0, 1] \\) and \\( r^h + f^{h+1}(x^{h+1}) \\in [0, 1] \\) (\\( h \\geq 1 \\)).\nBellman error plays an important role in the analysis of reinforcement learning algorithms. For value function based approach to reinforcement learning, we would like to approximate Bellman error by minimizing least squares loss\n\\[\n\\left(f^{h}\\left(x^{h},a^{h}\\right)-r^{h}-f^{h+1}\\left(x^{h+1}\\right)\\right)^{2}.\\tag{18.2}\n\\]\nBy taking conditional expectation with respect to \\((x^{h},a^{h})\\), we obtain\n\\[\n\\begin{array}{l}\\mathbb{E}_{r^{h},x^{h+1}\\mid x^{h},a^{h}}\\left(f^{h}\\left(x^{h},a^{h}\\right)-r^{h}-f^{h+1}\\left(x^{h+1}\\right)\\right)^{2}\\\\ =\\mathcal{E}^{h}(f,x^{h},a^{h})^{2}+\\mathbb{E}_{r^{h},x^{h+1}|x^{h},a^{h}}\\Bigg(\\underbrace{r^{h}+f^{h+1}(x^{h+1})-(\\mathcal{T}^{h}f)(x^{h},a^{h})}_{f \\text{- d e p e n d e n t z e r o - m e a n n o i s e}}\\Bigg)^{2}.\\\\ \\end{array}\n\\]\nSince noise variance depends on \\( f \\), if we use (18.2) to estimate \\( f \\), we will favor \\( f \\) with smaller noise variance, which may not have zero Bellman error. To overcome this issue, one can estimate the noise variance term, and then subtract it from (18.2) to obtain the term \\( \\mathcal{E}^h (f,x^h,a^h)^2 \\). This requires an additional assumption referred to as Bellman completeness (or simply completeness).\n\nPacking Number. Let \\((\\mathcal{V},d)\\) be a pseudometric space with metric \\(d(\\cdot,\\cdot)\\). A finite subset \\(\\mathcal{G}(\\epsilon)\\subset \\mathcal{G}\\) is an \\(\\epsilon\\)-packing of \\(\\mathcal{G}\\) if \\(d(\\phi,\\phi^{\\prime})>\\epsilon\\) for all \\(\\phi,\\phi^{\\prime}\\in\\mathcal{G}(\\epsilon)\\). The \\(\\epsilon\\)-packing number of \\(\\mathcal{G}\\), denoted by \\(M(\\epsilon,\\mathcal{G},d)\\), is the largest cardinality of \\(\\epsilon\\)-packing of \\(\\mathcal{G}\\).\nThe following results illustrate the equivalence between covering number and packing number. One advantage of using an \\(\\epsilon\\) packing of \\(\\mathcal{G}\\) instead of an \\(\\epsilon\\) cover of \\(\\mathcal{G}\\) is that all members in the \\(\\epsilon\\) packing also belong to \\(\\mathcal{G}\\). There if members in \\(\\mathcal{G}\\) satisfy certain assumptions such as the variance condition, then members of its \\(\\epsilon\\) packing also satisfy such assumptions. For this reason, we will use packing numbers instead of covering numbers in some of the theoretical analysis in later chapters.\n\ndefinition 18.10. We say \\( f \\) is a candidate \\( Q \\)-function if \\( f = \\{f^{h}(x^{h},a^{h}) : \\mathcal{X} \\times \\mathcal{A} \\to \\mathbb{R} : h \\in [H + 1]\\} \\), with \\( f^{H + 1}(\\cdot) = 0 \\). Define\n\\[\nf^{h}(x^{h})=\\arg\\max_{a\\in\\mathcal{A}}f^{h}(x^{h},a),\n\\]\nand define its greedy policy \\(\\pi_f\\) as a deterministic policy that satisfies\n\\[\n\\pi_{f}^{h}(x^{h})\\in\\arg\\max_{a\\in\\mathcal{A}}f^{h}(x^{h},a).\n\\]\nGiven an MDP \\(M\\), we also define the Bellman operator of \\(f\\) as\n\\[\n\\left(\\mathcal{T}^{h}f\\right)\\left(x^{h},a^{h}\\right)=\\mathbb{E}_{r^{h},x^{h+1}\\mid x^{h},a^{h}}\\left[r^{h}+f^{h+1}\\left(x^{h+1}\\right)\\right],\n\\]\nand its Bellman error as\n\\[\n\\mathcal{E}^{h}(f,x^{h},a^{h})=f^{h}\\left(x^{h},a^{h}\\right)-\\left(\\mathcal{T}^{h}f\\right)\\left(x^{h},a^{h}\\right),\n\\]\nwhere the conditional expectation is with respect to the MDP \\(M\\).\n\nBellman Completeness. A candidate value function class \\(\\mathcal{F}\\) is complete with respect to another candidate value function class \\(\\mathcal{G}\\) if for any \\(h\\in [H]\\), \\(f\\in \\mathcal{F}\\), there exists \\(g\\in \\mathcal{G}\\) so that for all \\(h\\in [H]\\):\n\\[\ng^{h}(x^{h},a^{h})=(\\mathcal{T}^{h}f)(x^{h},a^{h})=\\mathbb{E}_{r^{h},x^{h+1}|x^{h},a^{h}}\\left[r^{h}+f^{h+1}(x^{h+1})\\right].\n\\]\nWe say \\(\\mathcal{F}\\) is complete if \\(\\mathcal{F}\\) is complete with respect to itself.\nNote that if \\(\\mathcal{F}\\) is complete with respect to \\(\\mathcal{G}\\), then we may use the solution of\n\\[\n\\min_{g^{h}\\in\\mathcal{G}^{h}}\\sum_{s=1}^{t}\\left(g^{h}\\left(x_{s}^{h},a_{s}^{h}\\right)-r_{s}^{h}-f^{h+1}\\left(x_{s}^{h+1}\\right)\\right)^{2}\n\\]\nto estimate \\((\\mathcal{T}^h f)(x^h,a^h)\\), which can be used to cancel the variance term in (18.2). This motivates the following loss function\n\\[\n\\begin{array}{l}L^{h}(f,g,x^{h},a^{h},r^{h},x^{h+1})=\\left[\\left(f^{h}\\left(x^{h},a^{h}\\right)-r^{h}-f^{h+1}\\left(x^{h+1}\\right)\\right)^{2}\\right.\\\\ \\left.-\\left(g^{h}\\left(x^{h},a^{h}\\right)-r^{h}-f^{h+1}\\left(x^{h+1}\\right)\\right)^{2}\\right].\\tag{18.3}\\\\ \\end{array}\n\\]\nThe following result is a counterpart of Theorem 13.15.\n\nLinear MDP. Let \\(\\mathcal{H} = \\{\\mathcal{H}^h\\}\\) be a sequence of vector spaces with inner products \\(\\langle \\cdot, \\cdot \\rangle\\). An MDP \\(M = \\mathrm{MDP}(\\mathcal{X}, \\mathcal{A}, P)\\) is a linear MDP with feature maps \\(\\phi = \\{\\phi^h(x^h, a^h) : \\mathcal{X} \\times \\mathcal{A} \\to \\mathcal{H}^h\\}_{h=1}^H\\) if for all \\(h \\in [H]\\), there exist a map \\(\\nu^h(x^{h+1}) : \\mathcal{X} \\to \\mathcal{H}^h\\) and \\(\\theta^h \\in \\mathcal{H}^h\\), such that\n\\[\nd P^{h}\\left(x^{h+1}\\mid x^{h},a^{h}\\right)=\\left\\langle\\nu^{h}\\left(x^{h+1}\\right),\\phi^{h}\\left(x^{h},a^{h}\\right)\\right\\rangle d\\mu^{h+1}\\left(x^{h+1}\\right),\n\\]\n\\[\n\\mathbb{E}\\left[r^{h}\\mid x^{h},a^{h}\\right]=\\langle\\theta^{h},\\phi^{h}\\left(x^{h},a^{h}\\right)\\rangle .\n\\]\nHere \\(\\langle \\cdot, \\cdot \\rangle\\) denotes the inner product in \\(\\mathcal{H}^h\\) for different \\(h\\), and the conditional probability measure \\(dP^h(\\cdot | x^h, a^h)\\) is absolute continuous with respect to a measure \\(d\\mu^{h+1}(\\cdot)\\) with density \\(\\langle \\nu^h(x^{h+1}), \\phi^h(x^h, a^h) \\rangle\\). In general, we assume that \\(\\nu^h(\\cdot)\\) and \\(\\theta^h\\) are unknown. We may assume \\(\\phi(\\cdot)\\) to be either known or unknown.\n\ndefinition 18.8. The optimal value functions starting at step \\( h \\) are given by\n\\[\nQ_{*}^{h}(x^{h},a^{h})=\\sup_{\\pi}Q_{\\pi}^{h}(x^{h},a^{h}),\\qquad V_{*}^{h}(x^{h})=\\sup_{\\pi}V_{\\pi}^{h}(x^{h}).\n\\]\nWe also define the optimal value as \\( V_{*} = \\mathbb{E}_{x^{1}}V_{*}^{1}(x^{1}) \\).\nIn the following, for notation simplicity, we assume that the solution to\n\\[\n\\operatorname*{argmax}_{a\\in\\mathcal{A}}Q_{*}^{h}(x^{h},a)\n\\]\ncan be achieved by some \\( a \\in \\mathcal{A} \\) so that greedy policy in the Theorem 18.9 can be properly defined. The following theorem describes key properties of optimal value functions and optimal policy for reinforcement learning. The proof is straightforward, and we leave it as an exercise.\n\nV-type Bellman Eluder Coefficient. Given a candidate \\(Q\\) function class \\(\\mathcal{F}\\), its V-type Bellman eluder coefficient \\(\\mathrm{EC}_V^h(\\epsilon,\\mathcal{F},T)\\) at step \\(h\\geq 1\\) is the smallest number \\(d\\) so that for any filtered sequence \\(\\{f_t,(x_t,r_t,a_t)\\sim \\pi_{f_t}\\}_{t = 1}^T\\)\n\\[\n\\mathbb{E}\\sum_{t=2}^{T}\\mathcal{E}^{h}(f_{t},x_{t}^{h},a_{t}^{h})\\leq\\sqrt{d\\mathbb{E}\\sum_{t=2}^{T}\\left(\\epsilon+\\sum_{s=1}^{t-1}\\mathbb{E}_{\\tilde{x}\\sim\\pi_{f_{s}}}\\mathcal{E}^{h}(f_{t},\\tilde{x}^{h},\\pi_{f_{t}}(\\tilde{x}^{h}))^{2}\\right)}.\n\\]\nIn order to bound the \\(Q\\)-type Bellman eluder coefficient, we need to assume in Proposition 18.20 that the \\(Q\\) function class can be embedded in a linear vector space. This is not necessary for the \\(V\\)-type Bellman eluder coefficient, which allows general nonlinear \\(Q\\) function class.\nWe note that for linear MDP, from the proof of Proposition 18.18, we know that for any \\( h \\geq 2 \\), there exists \\( u^{h - 1}(f) \\in \\mathcal{H}^{h - 1} \\) so that\n\\[\n\\mathbb{E}_{x^{h}\\mid x^{h-1},a^{h-1}}\\mathcal{E}^{h}(f,x^{h},\\pi_{f}(x^{h}))=\\langle u^{h-1}(f),\\phi^{h-1}(x^{h-1},a^{h-1})\\rangle .\n\\]\nIf moreover, we assume that \\((x^{h - 1},a^{h - 1})\\sim \\pi^{\\prime}\\), where \\(\\pi^\\prime\\) is referred to as a roll-in policy, then we have the following decomposition,\n\\[\n\\mathbb{E}_{x^{h}\\sim\\pi^{\\prime}}\\mathcal{E}^{h}(f,x^{h},\\pi_{f}(x^{h}))=\\langle u^{h-1}(f),\\phi^{h-1}(\\pi^{\\prime})\\rangle,\n\\]\nwhere \\(\\phi^{h - 1}(\\pi^{\\prime}) = \\mathbb{E}_{x^{h - 1},a^{h - 1}\\sim \\pi^{\\prime}}\\phi^{h - 1}(x^{h - 1},a^{h - 1})\\)\nOne may expand this property and define Bellman factorizable MDPs as follows, which includes linear MDP as a special case.\n\ndefinition 18.30. The coverage coefficient \\(\\mathrm{CC}^h(\\epsilon,\\tilde{\\pi},\\mathcal{F})\\) (for \\(\\epsilon >0\\)) of a random policy \\(\\tilde{\\pi}\\) at step \\(h\\) is defined as\n\\[\n\\mathrm{CC}^{h}(\\epsilon,\\tilde{\\pi},\\mathcal{F})=\\sup_{f\\in\\mathcal{F},x^{h}\\in\\mathcal{X}}\\frac{\\mathcal{E}^{h}(f,x^{h},\\pi_{f}(x^{h}))^{2}}{\\epsilon+\\mathbb{E}_{a^{h}\\sim\\tilde{\\pi}^{h}(\\cdot|x^{h})}\\mathcal{E}^{h}(f,x^{h},a^{h})^{2}}.\n\\]\nWe note that Definition 18.30 is consistent with Definition 17.35, which can be regarded as a conditional version of Definition 9.42. Therefore we may employ\na conditional \\( G \\)-optimal design \\( \\tilde{\\pi}^h(a^h|x^h) \\) by conditioning on \\( x^h \\). The following result shows that if Bellman error is linearly embeddable (with possibly unknown feature map), then good coverage coefficient can be obtained using nonlinear \\( G \\)-optimal design. It is a direct consequence of Theorem 9.44.\n\n\\(V\\)-type Bellman Error UCB Algorithm. Algorithm 18.3: \\(V\\)-type Bellman Error UCB Algorithm\nInput: \\(\\lambda, T, \\mathcal{F}, \\mathcal{G}, \\tilde{\\pi}\\)  \n1 Let \\(\\mathcal{F}_0 = \\{f_0\\}\\)  \n2 Let \\(\\beta_0 = 0\\)  \n3 for \\(t = 1, 2, \\ldots, T\\) do  \n4 Observe \\(x_t^1\\)  \n5 Let \\(f_t \\in \\arg \\max_{f \\in \\mathcal{F}_{t-1}} f(x_t^1)\\)  \n6 Let \\(\\pi_t = \\pi_{f_t}\\)  \n7 Draw \\(h_t \\sim [H]\\) uniformly at random  \n8 Play policy \\(\\pi_t\\) and observe trajectory \\((x_t, a_t, r_t)\\) up to \\(x_t^{h_t}\\)  \n9 Play random policy \\(a_t^{h_t} \\sim \\tilde{\\pi}^{h_t}(\\cdot | x_t^{h_t})\\), and observe \\((r_t^{h_t}, x_t^{h_t+1})\\)  \n10 Let  \n\\[ \\mathcal{F}_t = \\left\\{ f \\in \\mathcal{F} : \\sup_{g \\in \\mathcal{G}} \\sum_{s=1}^t L^{h_s}(f, g, x_s^{h_s}, a_s^{h_s}, r_s^{h_s}, x_s^{h_s+1}) \\leq \\beta_t^2 \\right\\} \\] with appropriately chosen \\(\\beta_t\\), where \\(L^h(\\cdot)\\) is defined according to (18.3).\n\nsection 13.1 sequential observation setup. In Section 13.1, data are observed sequentially. At each time \\(t\\), after the history \\(S_{t-1}\\) has been observed, the next observation \\(Z_t\\) is drawn according to a distribution that may depend on that history. The observation is decomposed as \\(Z_t=(Z_t^{(x)},Z_t^{(y)})\\), where \\(Z_t^{(x)}\\) is the component determined by the past history or the chosen action at time \\(t\\), and \\(Z_t^{(y)}\\) is the random response component. The history after time \\(t\\) is \\(S_t=(Z_1,\\ldots,Z_t)\\), and \\(\\mathcal{S}_T\\) denotes the full sequential experiment/history through time \\(T\\). For a function depending on \\(S_t\\), the notation \\(\\mathbb{E}_{Z_t^{(y)}}[\\cdot]\\) means conditional expectation over the response component \\(Z_t^{(y)}\\), conditional on \\(S_{t-1}\\) and on \\(Z_t^{(x)}\\). Stopping times are adapted to the filtration generated by the histories \\(S_t\\).\n\ndefinition 2.11. A sub-Gaussian random variable \\(X\\) has quadratic logarithmic moment generating function for all \\(\\lambda \\in \\mathbb{R}\\):\n\\[\n\\ln\\mathbb{E}e^{\\lambda X}\\leq\\lambda\\mu+\\frac{\\lambda^{2}}{2}b.\\tag{2.10}\n\\]\nUsing (2.10), we can obtain an upper bound of the rate function for sub-Gaussian random variables, which imply the following tail inequality.", "theorems": [{"local_id": "T1", "statement": "In Algorithm 18.3, we let\n\\[\n\\tilde{d}=\\sum_{h=1}^{H}\\operatorname{EC}_{V}^{h}(\\epsilon,\\mathcal{F},T)\\max(1,\\operatorname{CC}^{h}(\\epsilon^{\\prime},\\tilde{\\pi},\\mathcal{F})),\n\\]\nthen\n\\[\n\\mathbb{E}\\sum_{h=1}^{H}\\sum_{t=2}^{T}\\mathcal{E}^{h}(f_{t},x_{t}^{h},a_{t}^{h})\\leq\\sqrt{\\tilde{d}\\mathbb{E}\\sum_{t=2}^{T}\\left(\\epsilon H+\\epsilon^{\\prime}H T+\\sum_{s=1}^{t-1}\\sum_{h=1}^{H}\\bar{\\mathcal{E}}^{h}(f_{t},\\pi_{f_{s}},\\tilde{\\pi})^{2}\\right)},\n\\]\nwhere\n\\[\n\\bar{\\mathcal{E}}^{h}(f,\\pi,\\tilde{\\pi})^{2}=\\mathbb{E}_{\\tilde{x}^{h}\\sim\\pi}\\mathbb{E}_{a^{h}\\sim\\tilde{\\pi}(\\cdot|x^{h})}\\mathcal{E}^{h}(f_{t},x^{h},a^{h})^{2}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\( d_1^h = \\operatorname{EC}_V^h(\\epsilon, \\mathcal{F}, T) \\), and \\( d_2^h = \\operatorname{CC}^h(\\epsilon', \\tilde{\\pi}, \\mathcal{F}) \\)."}, {"n": 2, "claim": "We obtain\n\\[\n\\begin{array}{l}\\mathbb{E}\\sum_{h=1}^{H}\\sum_{t=2}^{T}\\mathcal{E}^{h}\\left(f_{t},x_{t}^{h},a_{t}^{h}\\right)\\leq\\sum_{h=1}^{H}\\sqrt{d_{1}^{h}\\mathbb{E}\\sum_{t=2}^{T}\\left(\\epsilon+\\sum_{s=1}^{t-1}\\mathbb{E}_{\\tilde{x}\\sim\\pi_{f_{s}}}\\mathcal{E}^{h}\\left(f_{t},\\tilde{x}^{h},\\pi_{f_{t}}(\\tilde{x}^{h})\\right)^{2}\\right)}\\\\ \\leq\\sum_{h=1}^{H}\\sqrt{d_{1}^{h}\\mathbb{E}\\sum_{t=2}^{T}\\left(\\epsilon+d_{2}^{h}\\sum_{s=1}^{t-1}\\left(\\epsilon^{\\prime}+\\bar{\\mathcal{E}}^{h}\\left(f_{t},\\pi_{f_{s}},\\tilde{\\pi}\\right)^{2}\\right)\\right)}\\\\ \\leq\\sum_{h=1}^{H}\\sqrt{d_{1}^{h}\\max(1,d_{2}^{h})\\mathbb{E}\\left.\\sum_{t=2}^{T}\\left(\\epsilon+\\epsilon^{\\prime}T+\\sum_{s=1}^{t-1}\\bar{\\mathcal{E}}^{h}(f_{t},\\pi_{f_{s}},\\tilde{\\pi})^{2}\\right)\\right.}.\\\\ \\end{array}\n\\]\nThe first inequality used Definition 18.27."}, {"n": 3, "claim": "The second inequality used Definition 18.30. We then use the Cauchy Schwartz inequality to obtain the result."}]}}, {"local_id": "T2", "statement": "Consider a random variable \\(X \\in [0,1]\\) and \\(\\mathbb{E}X = \\mu\\). We have the following inequality:\n\\[\n\\ln\\mathbb{E}e^{\\lambda X}\\leq\\ln [(1-\\mu)e^{0}+\\mu e^{\\lambda}]\\leq\\lambda\\mu+\\lambda^{2}/8.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\( h_L(\\lambda) = \\mathbb{E}e^{\\lambda X} \\) and \\( h_R(\\lambda) = (1 - \\mu)e^0 + \\mu e^\\lambda \\)."}, {"n": 2, "claim": "We know that \\( h_L(0) = h_R(0) \\). Moreover, when \\( \\lambda \\geq 0 \\):"}, {"n": 3, "claim": "\\[\nh_{L}^{\\prime}(\\lambda)=\\mathbb{E}X e^{\\lambda X}\\leq\\mathbb{E}X e^{\\lambda}=\\mu e^{\\lambda}=h_{R}^{\\prime}(\\lambda),\n\\]\nand similarly \\(h_L'(\\lambda) \\geq h_R'(\\lambda)\\) when \\(\\lambda \\leq 0\\)."}, {"n": 4, "claim": "This proves the first inequality. Now we let\n\\[\nh(\\lambda)=\\ln [(1-\\mu)e^{0}+\\mu e^{\\lambda}].\n\\]\nIt implies that\n\\[\nh^{\\prime}(\\lambda)=\\frac{\\mu e^{\\lambda}}{(1-\\mu)e^{0}+\\mu e^{\\lambda}},\n\\]\nand\n\\[\n\\begin{array}{l}h^{\\prime\\prime}(\\lambda)=\\frac{\\mu e^{\\lambda}}{(1-\\mu)e^{0}+\\mu e^{\\lambda}}-\\frac{(\\mu e^{\\lambda})^{2}}{[(1-\\mu)e^{0}+\\mu e^{\\lambda}]^{2}}\\\\ =\\left|h^{\\prime}(\\lambda)\\right|\\left(1-\\left|h^{\\prime}(\\lambda)\\right|\\right)\\leq 1/4.\\\\ \\end{array}\n\\]\nUsing Taylor expansion, we obtain the inequality \\( h(\\lambda) \\leq h(0) + \\lambda h'(0) + \\lambda^2 / 8 \\), which proves the second inequality."}]}}, {"local_id": "T3", "statement": "Assume that assumption 18.12 holds, \\(\\mathcal{F}\\) is complete with respect to \\(\\mathcal{G}\\), and \\(g^{h}(\\cdot)\\in [0,1]\\) for all \\(g\\in \\mathcal{G}\\). Consider (18.3), and let\n\\[\n\\mathcal{F}_{t}=\\left\\{f\\in\\mathcal{F}:\\sup_{g\\in\\mathcal{G}}\\sum_{h=1}^{H}\\sum_{s=1}^{t}L^{h}(f,g,x_{s}^{h},a_{s}^{h},r_{s}^{h},x_{s}^{h+1})\\leq\\beta_{t}^{2}\\right\\},\n\\]\nwhere\n\\[\n\\beta_{t}^{2}\\geq 4\\epsilon t(4+\\epsilon)H+2\\ln\\left(16 M(\\epsilon,\\mathcal{F},\\|\\cdot\\|_{\\infty})^{2}M(\\epsilon,\\mathcal{G},\\|\\cdot\\|_{\\infty})/\\delta^{2}\\right),\n\\]\nwith \\( M(\\cdot) \\) denotes the \\( \\| \\cdot \\|_{\\infty} \\) packing number, and \\( \\| f \\|_{\\infty} = \\sup_{h,x,a} |f^{h}(x,a)| \\).\nThen with probability at least \\(1 - \\delta\\), for all \\(t \\leq n\\): \\(Q_{*} \\in \\mathcal{F}_{t}\\) and for all \\(f \\in \\mathcal{F}_{t}\\):\n\\[\n\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\mathcal{E}^{h}\\left(f,x_{s}^{h},a_{s}^{h}\\right)^{2}\\leq 4\\beta_{t}^{2}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\mathcal{F}_{\\epsilon}\\) be an \\(\\epsilon \\| \\cdot \\|_{\\infty}\\) packing subset of \\(\\mathcal{F}\\) of size \\(M_1\\), and let \\(\\mathcal{G}_{\\epsilon}\\) be an \\(\\epsilon \\| \\cdot \\|_{\\infty}\\) packing subset of \\(\\mathcal{G}\\) of size \\(M_2\\)."}, {"n": 2, "claim": "For all \\( f \\in \\mathcal{F} \\) and \\( g \\in \\mathcal{G} \\) (or \\( g \\in \\mathcal{F} \\)), we introduce the simplified notations\n\\[\n\\epsilon_{s}^{h}(f)=r_{s}^{h}+f_{s}^{h+1}\\left(x_{s}^{h+1}\\right)-\\left(\\mathcal{T}^{h}f\\right)\\left(x_{s}^{h},a_{s}^{h}\\right),\n\\]\n\\[\n\\Delta_{s}^{h}(f,g)=((\\mathcal{T}^{h}f)(x_{s}^{h},a_{s}^{h})-g^{h}(x_{s}^{h},a_{s}^{h})).\n\\]\nWe have the following sub-Gaussian inequality (see Lemma 2.15) for any fixed \\( f_{\\epsilon} \\in \\mathcal{F}_{\\epsilon} \\), \\( h \\in [H] \\), and \\( \\rho \\in \\mathbb{R} \\):"}, {"n": 3, "claim": "\\[\n\\ln\\mathbb{E}_{r_{s}^{h},x_{s}^{h+1}|x_{s}^{h},a_{s}^{h}}\\exp(\\rho\\epsilon_{s}^{h}(f_{\\epsilon}))\\leq\\frac{\\rho^{2}}{8}.\n\\]\nUsing this logarithmic moment generating function, we can now obtain the following statement from Theorem 13.11, by taking a uniform bound over \\(\\mathcal{G}_{\\epsilon} \\times \\mathcal{F}_{\\epsilon}\\) and consider both under and over estimation (for the absolute value on the left hand side)."}, {"n": 4, "claim": "With probability at least \\(1 - \\delta / 2\\): for all \\(f_{\\epsilon} \\in \\mathcal{F}_{\\epsilon}\\), \\(g_{\\epsilon} \\in \\mathcal{G}_{\\epsilon}\\) and \\(t \\geq 0\\)\n\\[\n\\underbrace{\\left|\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\Delta_{s}^{h}\\left(f_{\\epsilon},g_{\\epsilon}\\right)\\epsilon_{s}^{h}\\left(f_{\\epsilon}\\right)\\right|}_{A}\\leq\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\frac{\\Delta_{s}^{h}\\left(f_{\\epsilon},g_{\\epsilon}\\right)^{2}}{8}+\\ln\\left(4 M_{1}M_{2}/\\delta\\right).\\tag{18.4}\n\\]\nIn the following for each \\(f \\in \\mathcal{F}\\) we choose \\(f_{\\epsilon} \\in \\mathcal{F}_{\\epsilon}\\) so that \\(\\| f - f_{\\epsilon} \\|_{\\infty} \\leq \\epsilon\\)."}, {"n": 5, "claim": "Similarly for each \\(g \\in \\mathcal{G}\\) we choose \\(g_{\\epsilon} \\in \\mathcal{G}_{\\epsilon}\\) so that \\(\\| g - g_{\\epsilon} \\|_{\\infty} \\leq \\epsilon\\)."}, {"n": 6, "claim": "Then\n\\[\n\\begin{array}{l}\\left|\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\Delta_{s}^{h}(f,g)\\epsilon_{s}^{h}(f)\\right|\\\\ \\leq\\left|\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\Delta_{s}^{h}(f,g)\\epsilon_{s}^{h}\\left(f_{\\epsilon}\\right)\\right| +2\\epsilon\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\left|\\Delta_{s}^{h}(f,g)\\right|\\\\ \\leq\\underbrace{\\left|\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\Delta_{s}^{h}(f_{\\epsilon},g_{\\epsilon})\\epsilon_{s}^{h}(f_{\\epsilon})\\right|}_{A}+4H\\epsilon t.\\\\ \\end{array}\n\\]\nThe first inequality used \\( |\\epsilon_s^h (f) - \\epsilon_s^h (f_\\epsilon)|\\leq 2\\epsilon \\)."}, {"n": 7, "claim": "The second inequality used the \\( \\epsilon \\)-covering property \\( |\\Delta_s^h(f,g)-\\Delta_s^h(f_\\epsilon,g_\\epsilon)|\\leq 2\\epsilon \\), \\( |\\epsilon_s^h (f_\\epsilon)|\\leq 1 \\), and \\( |\\Delta_s^h (f,g)|\\leq 1 \\)."}, {"n": 8, "claim": "We also have\n\\[\n\\begin{array}{l}A\\leq\\frac{1}{8}\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\Delta_{s}^{h}\\left(f_{\\epsilon},g_{\\epsilon}\\right)^{2}+\\ln\\left(4 M_{1}M_{2}/\\delta\\right)\\\\ \\leq\\frac{1}{8}\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\left(\\left|\\Delta_{s}^{h}(f,g)\\right| +2\\epsilon\\right)^{2}+\\ln\\left(4 M_{1}M_{2}/\\delta\\right)\\\\ \\leq\\frac{1}{4}\\sum_{s=1}^{t}\\sum_{h=1}^{H}(\\Delta_{s}^{h}(f,g))^{2}+\\epsilon^{2}H t+\\ln(4 M_{1}M_{2}/\\delta).\\\\ \\end{array}\n\\]\nThe first inequality used (18.4)."}, {"n": 9, "claim": "The second inequality used the \\(\\epsilon\\)-covering property \\(|\\Delta_s^h(f,g)-\\Delta_s^h(f_\\epsilon,g_\\epsilon)|\\leq 2\\epsilon\\)."}, {"n": 10, "claim": "The last inequality used standard algebra inequalities \\(|a + 2\\epsilon |^2 /8\\leq (a^2 /4) + \\epsilon^2\\) to simplify the terms."}, {"n": 11, "claim": "That is, we have shown that with probability at least \\(1 - \\delta / 2\\): for all \\(f \\in \\mathcal{F}\\) and \\(g \\in \\mathcal{G}\\) and \\(t \\geq 0\\):"}, {"n": 12, "claim": "\\[\n\\begin{array}{l}\\left|\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\Delta_{s}^{h}(f,g)\\epsilon_{s}^{h}(f)\\right|\\leq\\frac{1}{4}\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\Delta_{s}^{h}(f,g)^{2}\\\\ +\\epsilon tH(4+\\epsilon)+\\ln\\left(4M_{1}M_{2}/\\delta\\right).\\qquad(18. 5)\\\\ \\end{array}\n\\]\nSimilarly, with probability at least \\(1 - \\delta /2\\): for all \\(f\\in \\mathcal{F}\\) and \\(t\\geq 0\\)\n\\[\n\\begin{array}{l}\\left|\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\Delta_{s}^{h}(f,f)\\epsilon_{s}^{h}(f)\\right|\\leq\\frac{1}{4}\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\Delta_{s}^{h}(f,f)^{2}\\\\ +\\epsilon t H(4+\\epsilon)+\\ln\\left(4 M_{1}/\\delta\\right).\\tag{18.6}\\\\ \\end{array}\n\\]\nWe have with probability at least \\(1 - \\delta\\), both (18.5) and (18.6) hold."}, {"n": 13, "claim": "We obtain with probability at least \\(1 - \\delta\\): for all \\(t \\geq 0\\), \\(f \\in \\mathcal{F}\\), \\(g \\in \\mathcal{G}\\):"}, {"n": 14, "claim": "\\[\n\\begin{array}{l}\\left|\\sum_{h=1}^{H}\\sum_{s=1}^{t}\\left[\\Delta_{s}^{h}(f,f)-\\Delta_{s}^{h}(f,g)\\right]\\epsilon_{s}^{h}(f)\\right|\\\\ \\leq\\frac{1}{4}\\sum_{h=1}^{H}\\sum_{s=1}^{t}\\left[\\Delta_{s}^{h}(f,f)^{2}+\\Delta_{s}^{h}(f,g)^{2}\\right]+0. 5\\beta_{t}^{2}.\\tag{18.7}\\\\ \\end{array}\n\\]\nTherefore\n\\[\n\\begin{array}{l}\\sum_{s=1}^{t}\\sum_{h=1}^{H}L^{h}(f,g,x_{s}^{h},a_{s}^{h},r_{s}^{h},x_{s}^{h+1})\\\\ =\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\left[\\Delta_{s}^{h}(f,f)^{2}-\\Delta_{s}^{h}(f,g)^{2}+2[\\Delta_{s}^{h}(f,f)-\\Delta_{s}^{h}(f,g)]\\epsilon_{s}^{h}(f)\\right]\\\\ \\geq\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\left[0. 5\\Delta_{s}^{h}(f,f)^{2}-1. 5\\Delta_{s}^{h}(f,g)^{2}\\right]-\\beta_{t}^{2},\\quad(\\text{b y})\\\\ \\end{array}\n\\]\nand\n\\[\n\\begin{array}{l}\\sum_{s=1}^{t}\\sum_{h=1}^{H}L^{h}\\left(Q_{*},g,x_{s}^{h},a_{s}^{h},r_{s}^{h},x_{s}^{h+1}\\right)\\\\ =\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\left[\\Delta_{s}^{h}(Q_{*},Q_{*})^{2}-\\Delta_{s}^{h}(Q_{*},g)^{2}+2\\left[\\Delta_{s}^{h}(Q_{*},Q_{*})-\\Delta_{s}^{h}(Q_{*},g)\\right]\\epsilon_{s}^{h}(f)\\right]\\\\ \\leq\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\left[1. 5\\Delta_{s}^{h}(Q_{*},Q_{*})^{2}-0. 5\\Delta_{s}^{h}(Q_{*},g)^{2}\\right]+\\beta_{t}^{2}.\\quad \\tag{by(18.7)}\\\\ \\end{array}\n\\]\nTaking sup over \\( g \\in \\mathcal{G} \\), and note that \\( \\Delta_s^h(Q_*, Q_*) = 0 \\), and that \\( \\mathcal{F} \\) is complete with respect to \\( \\mathcal{G} \\), we obtain\n\\[\n\\sup_{g\\in\\mathcal{G}}\\sum_{s=1}^{t}\\sum_{h=1}^{H}L^{h}\\left(Q_{*},g,x_{s}^{h},a_{s}^{h},r_{s}^{h},x_{s}^{h+1}\\right)\\leq\\beta_{t}^{2},\n\\]\n\\[\n\\sup_{g\\in\\mathcal{G}}\\sum_{s=1}^{t}\\sum_{h=1}^{H}L^{h}(f,g,x_{s}^{h},a_{s}^{h},r_{s}^{h},x_{s}^{h+1})\\geq 0. 5\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\underbrace{\\Delta_{s}^{h}(f,f)^{2}}_{\\mathcal{E}^{h}(f,x_{s}^{h},a_{s}^{h})^{2}}-\\beta_{t}^{2}.\n\\]\nIn the two displayed inequalities, the first implies that \\( Q_{*} \\in \\mathcal{F}_{t} \\)."}, {"n": 15, "claim": "The second implies that if \\( f \\in \\mathcal{F}_{t} \\), then\n\\[\n\\beta_{t}^{2}\\geq 0. 5\\sum_{s=1}^{t}\\sum_{h=1}^{H}\\mathcal{E}^{h}(f,x_{s}^{h},a_{s}^{h})^{2}-\\beta_{t}^{2},\n\\]\nwhich implies the second result of the theorem."}]}}, {"local_id": "T4", "statement": "We have for any \\(\\lambda >0\\), with probability at least \\(1 - \\delta\\), for all \\(n\\geq 1\\) and \\(f\\in \\mathcal{F}\\):\n\\[\n\\left[-\\sum_{i=1}^{n}f(Z_{i})-\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda f(Z_{i})}\\right]\\leq\\inf_{\\epsilon>0}\\left[2 n\\epsilon+\\frac{\\ln(M(\\epsilon,\\mathcal{F},\\|\\cdot\\|_{\\infty})/\\delta)}{\\lambda}\\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\mathcal{F}_{\\epsilon} \\subset \\mathcal{F}\\) be an \\(\\epsilon\\) maximal packing of \\(\\mathcal{F}\\), with \\(|\\mathcal{F}_{\\epsilon}| \\leq M(\\epsilon, \\mathcal{F}, \\| \\cdot \\|_{\\infty})\\)."}, {"n": 2, "claim": "We obtain from Theorem 13.2, and the uniform bound over \\(\\mathcal{F}_{\\epsilon}\\) that with probability at least \\(1 - \\delta\\):"}, {"n": 3, "claim": "\\[\n\\sup_{f\\in\\mathcal{F}_{\\epsilon}}\\left[-\\sum_{i=1}^{n}f(Z_{i})-\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda f(Z_{i})}\\right]\\leq\\frac{\\ln(M(\\epsilon,\\mathcal{F},\\|\\cdot\\|_{\\infty})/\\delta)}{\\lambda}.\n\\]\nSince \\(\\mathcal{F}_{\\epsilon}\\) is also an \\(\\epsilon L_{\\infty}\\) cover of \\(\\mathcal{F}\\) (see Theorem 5.2), we obtain\n\\[\n\\begin{array}{l}\\sup_{f\\in\\mathcal{F}}\\left[-\\sum_{i=1}^{n}f(Z_{i})-\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda f(Z_{i})}\\right]\\\\ \\leq 2 n\\epsilon+\\sup_{f\\in\\mathcal{F}_{\\epsilon}}\\left[-\\sum_{i=1}^{n}f(Z_{i})-\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda f(Z_{i})}\\right].\\\\ \\end{array}\n\\]\nThis implies the bound."}]}}, {"local_id": "T5", "statement": "In a linear MDP with feature map \\(\\phi^h (x^h,a^h)\\) on vector spaces \\(\\mathcal{H}^h\\) \\((h\\in [H])\\). Consider the linear candidate \\(Q\\) function class\n\\[\n\\mathcal{F}=\\left\\{\\langle w^{h},\\phi^{h}(x^{h},a^{h})\\rangle:w^{h}\\in\\mathcal{H}^{h},h\\in [H]\\right\\}.\n\\]\nAny function \\( g^{h + 1}(x^{h + 1}) \\) on \\( \\mathcal{X} \\) satisfies\n\\[\n\\left(\\mathcal{T}^{h}g^{h+1}\\right)\\left(x^{h},a^{h}\\right)\\in\\mathcal{F}.\n\\]\nIt implies that \\(\\mathcal{F}\\) is complete, and \\(Q_{*} \\in \\mathcal{F}\\). Moreover, \\(\\forall f \\in \\mathcal{F}\\),\n\\[\n\\mathcal{E}^{h}(f,x^{h},a^{h})\\in\\mathcal{F}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let\n\\[\nu_{g}^{h}=\\int g^{h+1}(x^{h+1})\\nu^{h}(x^{h+1})d\\mu^{h+1}(x^{h+1}).\n\\]\nWe have\n\\[\n\\begin{array}{l}\\mathbb{E}_{x^{h+1}\\mid x^{h},a^{h}}g^{h+1}\\left(x^{h+1}\\right)=\\int g^{h+1}\\left(x^{h+1}\\right)\\langle\\nu^{h}\\left(x^{h+1}\\right),\\phi^{h}\\left(x^{h},a^{h}\\right)\\rangle d\\mu^{h+1}\\left(x^{h+1}\\right)\\\\ =\\langle u_{g}^{h},\\phi^{h}(x^{h},a^{h})\\rangle .\\\\ \\end{array}\n\\]\nThis implies that\n\\[\n\\left(\\mathcal{T}^{h}g\\right)\\left(x^{h},a^{h}\\right)=\\left\\langle\\theta^{h}+u_{g}^{h},\\phi^{h}\\left(x^{h},a^{h}\\right)\\right\\rangle\\in\\mathcal{F}.\n\\]\nSince \\( Q_{*}^{h}(x^{h},a^{h}) = (\\mathcal{T}^{h}Q_{*})(x^{h},a^{h}) \\), we know \\( Q_{*}^{h}(x^{h},a^{h})\\in \\mathcal{F} \\)\nSimilarly, since \\((\\mathcal{T}^h f)(x^h,a^h)\\in \\mathcal{F}\\), we know that \\(f\\in \\mathcal{F}\\) implies\n\\[\n\\mathcal{E}^{h}(f,x^{h},a^{h})=f^{h}\\left(x^{h},a^{h}\\right)-\\left(\\mathcal{T}^{h}f\\right)\\left(x^{h},a^{h}\\right)\\in\\mathcal{F}.\n\\]\nThis proves the desired result."}]}}], "target_local_id": "T3", "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch36_lemma_36_1__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\ndefinition 36.1. \\[d(A_{1},A_{2},x)=\\operatorname*{inf}\\{c a r d\\{i\\leq n:x_{i}\\neq y_{i}^{1}\\ a n d\\ x_{i}\\neq y_{i}^{2}\\},y^{1}\\in A_{1},y^{2}\\in A_{2}\\}\\]\n\nFor \\(f:\\mathcal{X}\\to\\mathbb{R}\\) , the subgraph of \\(f\\) is\n\\[C_{f}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}:0\\leq t\\leq f(x)\\ o r\\ f(x)\\leq t\\leq 0\\}.\\]\nDefine class of sets \\(\\mathcal{C}=\\{C_{f}:f\\in\\mathcal{F}\\}\\).\n\ndefinition 12.1. If \\(\\mathcal{C}\\) is a VC class of sets, then F is VC-subgraph class of functions and, by definition, \\(V C(\\mathcal{F})=V C(\\mathcal{C})\\)\nNote that equivalent definition of \\(C_{f}\\) is\n\\[C_{f}^{\\prime}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}: |f(x)|\\geq|t|\\}.\\]\n\ndefinition 12.3. The ε-packing number, \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) , is the marimal cardinality of an E-separated set.\nNote that \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) is decreasing in ε.\n\ndefinition 12.2. Given \\(\\varepsilon > 0\\) and \\(f_{1},\\ldots,f_{N}\\in\\mathcal{F}\\) ， we say that \\(f_{1},\\ldots,f_{N}\\) are E-separated if \\(d(f_{i},f_{j})>\\varepsilon\\) for any \\(i \\neq j\\)\n\nDenote \\(\\triangle_{n}(\\mathcal{C})=\\operatorname*{sup}_{\\{x_{1},\\ldots,x_{n}\\}}\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})\\leq 2^{n}\\). We will see that for some classes, \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\) for \\(n \\leq V\\) and \\(\\triangle_{n}(\\mathcal{C})<2^{n}\\) for \\(n > V\\) for some constant \\(V\\).\n\ndefinition 8.1. If \\(V < \\infty\\) , then C is called a VC class. V is called VC dimension of C.\n\ndefinition 8.0. For any sample \\(\\{x_{1},\\ldots,x_{n}\\}\\), the ways that \\({\\mathcal{C}}\\) intersects with the sample are \\[\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in{\\mathcal{C}}\\}.\\] Let \\[\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})=\\mathrm{card}~\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in\\mathcal{C}\\},\\] the number of different subsets picked out by \\(C\\in{\\mathcal{C}}\\). If \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\), this means we can find \\(\\{x_{1},\\ldots,x_{n}\\}\\) such that \\({\\mathcal{C}}\\) can pick out any subset of it; in this case \\({\\mathcal{C}}\\) shatters \\(\\{x_{1},\\ldots,x_{n}\\}\\). Equivalently, a set \\(A\\) is shattered by \\({\\mathcal{C}}\\) when \\({\\mathcal{C}}\\) picks out every subset of \\(A\\).", "theorems": [{"local_id": "T1", "statement": "Let \\(\\xi,\\nu\\) - random variables.Assume that\n\\[\\mathbb{P}\\left(\\nu\\geq t\\right)\\leq\\Gamma e^{-\\gamma t}\\]\nwhere \\(\\Gamma\\geq 1,t\\geq 0\\) ，and \\(\\gamma > 0\\) . Furthermore, for all \\(a > 0\\) assume that\n\\[\\mathbb{E}\\phi(\\xi)\\le\\mathbb{E}\\phi(\\nu)\\]\nwhere \\(\\phi(x)=(x-a)_{+}\\) .Then\n\\[\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\Gamma\\cdot e\\cdot e^{-\\gamma t}.\\]", "proof": {"steps": [{"n": 1, "claim": "Since \\(\\phi(x)=(x-a)_{+}\\) ，we have \\(\\phi(\\xi)\\geq\\phi(t)\\) whenever \\(\\xi \\geq t .\\)\n\\[\\begin{array}{ r l }&{\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\mathbb{P}\\left(\\phi(\\xi)\\geq\\phi(t)\\right)}\\\\ &{\\qquad\\leq\\displaystyle\\frac{\\mathbb{E}\\phi(\\xi)}{\\phi(t)}\\leq\\frac{\\mathbb{E}\\phi(\\nu)}{\\phi(t)}=\\frac{\\mathbb{E}\\left(\\nu-a\\right)_{+}}{(t-a)_{+}}}\\end{array}\\]\nFurthermore,\n\\[\\begin{array}{ r l }{\\displaystyle\\mathbb{E}(\\nu-a)_{+}=}&{\\mathbb{E}\\int_{0}^{(\\nu-a)_{+}}\\mathrm{1}d x}\\\\ &{=\\mathbb{E}\\int_{0}^{\\infty}I(x\\leq(\\nu-a)_{+})d x}\\\\ &{=\\int_{0}^{\\infty}\\mathbb{E}I(x\\leq(\\nu-a)_{+})d x}\\\\ &{=\\int_{0}^{\\infty}\\mathbb{P}\\left((\\nu-a)_{+}\\geq x\\right)d x}\\\\ &{=\\int_{0}^{\\infty}\\mathbb{P}\\left(\\nu\\geq a+x\\right)d x}\\\\ &{\\leq\\int_{0}^{\\infty}\\Gamma e^{-\\gamma(a+x)}d x=\\frac{\\Gamma e^{-\\gamma a}}{\\gamma}.}\\end{array}\\]\nHence,\n\\[\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\frac{\\Gamma e^{-\\gamma a}}{\\gamma(t-a)_{+}}=\\frac{\\Gamma\\cdot e\\cdot e^{-\\gamma t}}{1}=\\Gamma\\cdot e\\cdot e^{-\\gamma t}\\]\nwhere we chose optimal \\(\\begin{array}{ r }{a=t-\\frac{1}{\\gamma}}\\end{array}\\) to minimize \\(\\frac{\\Gamma e^{-\\gamma a}}{\\gamma(t-a)_{+}}\\)"}]}}, {"local_id": "T2", "statement": "Setup. Let \\(0<\\varepsilon\\le 1\\).\n\nAssume \\(\\mathcal{F}\\) is a VC-subgraph class and \\(V C(\\mathcal{F})=V\\) Suppose \\(-1\\leq f(x)\\leq 1\\) for all \\(f\\in{\\mathcal{F}}\\) and \\(x\\in\\mathcal{X}\\) .Let \\(x_{1},\\ldots,x_{n}\\in{\\mathcal{X}}\\) and define \\(\\begin{array}{ r }{d(f,g)=\\frac{1}{n}\\sum_{i=1}^{n}|f(x_{i})-g(x_{i})|}\\end{array}\\) . Then\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]\n(which \\(\\begin{array}{ r }{i s\\le\\left(\\frac{K}{\\varepsilon}\\right)^{V+\\delta}}\\end{array}\\) for some \\(\\delta . \\jmath\\)", "proof": {"steps": [{"n": 1, "claim": "Let \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) and \\(f_{1},\\ldots,f_{m}\\) be ε-separated, i.e.\n\\[{\\frac{1}{n}}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})| >\\varepsilon .\\]"}, {"n": 2, "claim": "Let \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) be constructed in the following way:"}, {"n": 3, "claim": "\\(z_{i}\\) is chosen uniformly from \\(x_{1},\\ldots,x_{n}\\) and \\(t_{i}\\) is uniform on \\([-1,1]\\)\nConsider \\(f_{r}\\) and \\(f_{\\ell}\\) from the ε-packing."}, {"n": 4, "claim": "Let \\(C_{f_{\\tau}}\\) and \\(C_{f_{\\ell}}\\) be subgraphs of \\(f_{r}\\) and \\(f_{\\ell}\\) Then\n\\(\\mathbb{P}\\left(C_{f_{\\tau}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n= P(At least one point \\((z_{i},t_{i})\\) is picked by \\(C_{f_{r}}\\) or \\(C_{f_{\\ell}}\\) but not picked by the other)\n\\(=1-\\mathbb{P}\\) (All points \\((z_{i},t_{i})\\) are picked either by both or by none)\n\\(\\mathbf{\\Sigma}=1-\\mathbb{P}\\left(\\left(z_{i},t_{i}\\right)\\right.\\) is picked either by both or by none)k\nSince \\(z_{i}\\) is drawn uniformly from \\(x_{1},\\ldots,x_{n}\\)\nP \\(((z_{1},t_{1})\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)\n\\[\\begin{array}{ l }{\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{P}\\left(\\left(x_{i},t_{1}\\right)\\mathrm{~i s~p i c k e d~b y~b o t h~}C_{f r},C_{f_{\\ell}}\\mathrm{~o r~b y~n e i t h e r}\\right)}\\\\ {\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\left(1-\\frac{1}{2}|f_{r}(x_{i})-f_{\\ell}(x_{i})|\\right)}\\\\ {\\displaystyle=1-\\frac{1}{2}\\frac{1}{n}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})|}\\\\ {\\displaystyle=1-\\frac{1}{2}d(f_{r},f_{\\ell})\\leq 1-\\varepsilon/2\\leq e^{-\\varepsilon/2}}\\end{array}\\]\nSubstituting,\n\\(\\mathbb{P}\\left(C_{f_{r}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n\\(=1-\\mathbb{P}\\left((z_{1},t_{1})\\right.\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)𝑘\n\\[\\begin{array}{ l }{\\geq 1-\\left(e^{-\\varepsilon/2}\\right)^{k}}\\\\ {=1-e^{-k\\varepsilon/2}}\\end{array}\\]\nThere are (m） ways to choose \\(f_{r}\\) and \\(f_{\\ell},\\) s0\nP (All pairs \\(C_{f_{r}}\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\geq 1-{\\binom{m}{2}}e^{-k\\varepsilon/2}.\\)\nWhat k should we choose so that \\(1-{\\binom{m}{2}}e^{-k\\varepsilon/2}>0?\\) Choose\n\\[k>{\\frac{2}{\\varepsilon}}\\log{\\binom{m}{2}}.\\]\nThen there exist \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) such that all \\(C_{f_{\\ell}}\\) pick out different subsets.But \\(\\{C_{f}:f\\in\\mathcal{F}\\}\\) is \\(\\mathrm{VC}\\) so by Sauer's lemma, we can pick out at most \\(\\left({\\frac{e k}{V}}\\right)^{V}\\) out of these k points, provided \\(k \\geq V\\); if instead \\(k < V\\) then \\(m\\leq 2^{k}<2^{V}\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}\\) and the claimed bound already holds, so that case may be set aside."}, {"n": 5, "claim": "Hence, \\(\\begin{array}{ r }{m\\leq\\left(\\frac{e k}{V}\\right)^{V}}\\end{array}\\) as long as \\(k>{\\frac{2}{\\varepsilon}}\\log\\left({{\\binom{m}{2}}}\\right)\\) and \\(k \\geq V\\), the case \\(k < V\\) having been disposed of above."}, {"n": 6, "claim": "The latter holds for \\(\\begin{array}{ r }{k=\\frac{2}{\\varepsilon}\\log m^{2}}\\end{array}\\) ."}, {"n": 7, "claim": "Therefore,\n\\[m\\leq\\left(\\frac{e}{V}\\frac{2}{\\varepsilon}\\log m^{2}\\right)^{V}=\\left(\\frac{4 e}{V\\varepsilon}\\log m\\right)^{V},\\]\nwhere \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) ."}, {"n": 8, "claim": "Hence, we get\n\\[m^{1/V}\\leq\\frac{4 e}{\\varepsilon}\\log m^{1/V}\\]\nand defining \\(m^{1/V}=s .\\)\n\\[s\\leq{\\frac{4 e}{\\varepsilon}}\\log s .\\]\nNote that \\(\\frac{s}{\\log s}\\) is increasing for \\(s \\geq e\\) and so for large enough s, the inequality will be violated."}, {"n": 9, "claim": "We now check that the inequality is violated for \\(\\begin{array}{ r }{s^{\\prime}=\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}}\\end{array}\\) ."}, {"n": 10, "claim": "Indeed, one can show that\n\\[\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{7}{\\varepsilon}\\right)^{2}>\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)\\]\nsince\n\\[\\frac{49}{8e\\varepsilon}>\\log{\\frac{7}{\\epsilon}}.\\]\nHence, \\(m^{1/V}=s\\leq s^{\\prime}\\) and, thus,\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]"}]}}, {"local_id": "T3", "statement": "Setup. Let \\(X_1,\\dots,X_n\\) be i.i.d. random variables with common law \\(P\\) on a measurable space \\((\\mathcal{X},\\mathcal{A})\\), let \\(\\mathcal{C}\\subseteq\\mathcal{A}\\) be a class of measurable sets, and write \\(P(C)=\\mathbb{P}(X_1\\in C)\\). Let \\(X'_1,\\dots,X'_n\\) (the ghost sample) be an independent copy of \\(X_1,\\dots,X_n\\), i.e. i.i.d. \\(P\\) and independent of \\(X_1,\\dots,X_n\\). Suprema over \\(\\mathcal{C}\\) are assumed measurable; otherwise read \\(\\mathbb{P}\\) as outer probability.\n\nIf \\(t\\geq{\\sqrt{\\frac{2}{n}}}.\\) then\n\\[\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\right)\\leq 2\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\geq t/2\\right).\\]", "proof": {"steps": [{"n": 1, "claim": "Suppose the event\n\\[\\operatorname*{sup}_{C\\in{\\mathcal{C}}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\]\noccurs.Let \\(\\begin{array}{ r }{X=(X_{1},\\ldots,X_{n})\\in\\{\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\}}\\end{array}\\) ."}, {"n": 2, "claim": "Then\n\\[\\exists C_{X}{\\mathrm{~s u c h~t h a t~}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nFor a fixed C,\n\\[\\begin{array}{ r }{\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right|\\geq t/2\\right)=\\mathbb{P}\\left(\\left(\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}\\geq t^{2}/4\\right)}\\\\ {\\leq\\mathrm{~(b y~C h e b y s h e v~s~I n e q)~}\\frac{4\\mathbb{E}\\left(\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}}{t^{2}}}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i,j}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))(I(X_{j}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i=1}^{n}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))^{2}=\\frac{4 n\\mathbb{P}\\left(\\mathcal{C}\\right)\\left(1-\\mathbb{P}(\\mathcal{C})\\right)}{n^{2}t^{2}}\\leq\\frac{1}{n t^{2}}\\leq\\frac{1}{2}}\\end{array}\\]\nsince we chose \\(t\\geq\\sqrt{\\textstyle{\\frac{2}{n}}}.\\)\nSo,\n\\[\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\Bigg|\\exists C_{X}\\right)\\geq 1/2\\]\nif \\(t\\geq\\sqrt{2/n}\\) .Assume that the event\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\]\noccurs."}, {"n": 3, "claim": "Recall that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nHence,it must be that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\geq t/2 .\\]\nWe conclude\n\\[\\begin{array}{ r c l }{\\displaystyle\\frac 12}&{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}(C_{X})\\right|\\le t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right).}\\end{array}\\]\nSince indicators are 0,1-valued,\n\\[\\begin{array}{ r l r }&{}&{\\displaystyle\\frac{1}{2}T\\left(\\underbrace{\\operatorname*{sup}_{\\zeta\\in{\\mathcal C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}(C)\\right|\\ge t}_{\\Im C x}\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C x\\right)\\cdot I\\left(\\exists C x\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X,X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]\nNow, take expectation with respect to \\(X_{i}^{,}\\) s to obtain\n\\[\\begin{array}{ r l r }&{}&{\\mathbb{P}_{X}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\ge t\\right)}\\\\ &{}&{\\displaystyle\\le 2\\cdot\\mathbb{P}_{X,X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]"}]}}, {"local_id": "T4", "statement": "Let \\(0\\leq g_{1},g_{2}\\leq 1,g_{i}:\\mathcal{X}\\mapsto [0,1]\\) . Then\n\\[\\int\\operatorname*{min}\\left(2,{\\frac{1}{g_{1}(x)}},{\\frac{1}{g_{2}(x)}}\\right)d P(x)\\cdot\\int g_{1}(x)d P(x)\\cdot\\int g_{2}(x)d P(x)\\leq 1\\]", "proof": {"steps": [{"n": 1, "claim": "Notice that log \\(x \\leq x - 1\\)\nSo enough to show\n\\[\\int\\operatorname*{min}\\left(2,{\\frac{1}{g_{1}}},{\\frac{1}{g_{2}}}\\right)d P+\\int g_{1}d P+\\int g_{2}d P\\leq 3\\]\nwhich is the same as\n\\[\\int\\left[\\operatorname*{min}\\left(2,\\frac{1}{g_{1}},\\frac{1}{g_{2}}\\right)+g_{1}+g_{2}\\right]d P\\leq 3\\]\nIt's enough to show\n\\[\\operatorname*{min}\\left(2,\\frac{1}{g_{1}},\\frac{1}{g_{2}}\\right)+g_{1}+g_{2}\\leq 3 .\\]\nIf min is equal to 2, then \\(\\begin{array}{ r }{g_{1},g_{2}\\le\\frac{1}{2}}\\end{array}\\) and the sum is less than 3."}, {"n": 2, "claim": "If min is equal to \\(\\frac{1}{g_{1}}\\) ,then \\(\\begin{array}{ r }{g_{1}\\ge\\frac{1}{2}}\\end{array}\\) and \\(\\begin{array}{ r }{g_{1}\\geq g_{2},\\mathrm{~s o~}\\operatorname*{min}+g_{1}+g_{2}\\leq\\frac{1}{g_{1}}+2 g_{1}\\leq 3 .}\\end{array}\\)"}]}}], "target_local_id": "T4", "topic_domain": "Others", "topic_subtopic": "Analysis & Geometry", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch41_theorem_41_1__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nFor \\(f:\\mathcal{X}\\to\\mathbb{R}\\) , the subgraph of \\(f\\) is\n\\[C_{f}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}:0\\leq t\\leq f(x)\\ o r\\ f(x)\\leq t\\leq 0\\}.\\]\nDefine class of sets \\(\\mathcal{C}=\\{C_{f}:f\\in\\mathcal{F}\\}\\).\n\ndefinition 12.1. If \\(\\mathcal{C}\\) is a VC class of sets, then F is VC-subgraph class of functions and, by definition, \\(V C(\\mathcal{F})=V C(\\mathcal{C})\\)\nNote that equivalent definition of \\(C_{f}\\) is\n\\[C_{f}^{\\prime}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}: |f(x)|\\geq|t|\\}.\\]\n\ndefinition 12.3. The ε-packing number, \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) , is the marimal cardinality of an E-separated set.\nNote that \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) is decreasing in ε.\n\ndefinition 12.2. Given \\(\\varepsilon > 0\\) and \\(f_{1},\\ldots,f_{N}\\in\\mathcal{F}\\) ， we say that \\(f_{1},\\ldots,f_{N}\\) are E-separated if \\(d(f_{i},f_{j})>\\varepsilon\\) for any \\(i \\neq j\\)\n\nDenote \\(\\triangle_{n}(\\mathcal{C})=\\operatorname*{sup}_{\\{x_{1},\\ldots,x_{n}\\}}\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})\\leq 2^{n}\\). We will see that for some classes, \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\) for \\(n \\leq V\\) and \\(\\triangle_{n}(\\mathcal{C})<2^{n}\\) for \\(n > V\\) for some constant \\(V\\).\n\ndefinition 8.1. If \\(V < \\infty\\) , then C is called a VC class. V is called VC dimension of C.\n\ndefinition 8.0. For any sample \\(\\{x_{1},\\ldots,x_{n}\\}\\), the ways that \\({\\mathcal{C}}\\) intersects with the sample are \\[\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in{\\mathcal{C}}\\}.\\] Let \\[\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})=\\mathrm{card}~\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in\\mathcal{C}\\},\\] the number of different subsets picked out by \\(C\\in{\\mathcal{C}}\\). If \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\), this means we can find \\(\\{x_{1},\\ldots,x_{n}\\}\\) such that \\({\\mathcal{C}}\\) can pick out any subset of it; in this case \\({\\mathcal{C}}\\) shatters \\(\\{x_{1},\\ldots,x_{n}\\}\\). Equivalently, a set \\(A\\) is shattered by \\({\\mathcal{C}}\\) when \\({\\mathcal{C}}\\) picks out every subset of \\(A\\).", "theorems": [{"local_id": "T1", "statement": "Setup. Let \\(X_1,\\dots,X_n\\) be independent random variables with values in a measurable space \\(\\mathcal{X}\\), let \\(\\mathcal{F}\\) be a finite class of measurable functions \\(f:\\mathcal{X}\\to[0,1]\\), write \\(f_i:=f(X_i)\\), and set \\(Z=\\sup_{f\\in\\mathcal{F}}\\sum_{i=1}^{n}f_i\\) with \\(\\mathbb{E}Z>0\\).\n\nLet \\(\\mathcal{F}\\) be a finite set of functions \\(|{\\mathcal{F}}| <\\infty .\\forall f=(f_{1},\\cdots,f_{n})\\in{\\mathcal{F}},0\\leq f_{i}\\leq 1,Z=\\) \\(\\textstyle\\operatorname*{sup}_{f}\\sum_{i=1}^{n}f_{i}\\) ， and define h as \\(h(u)=(1+u)\\log(1+u)-u\\) where \\(u \\geq 0\\) Then \\(\\mathbb{P}(Z\\geq\\mathbb{E}Z+x)\\leq e^{-\\mathbb{E}Z\\cdot h(x/\\mathbb{E}Z)}\\)", "proof": {"steps": [{"n": 1, "claim": "Let\n\\[Z\\quad=\\quad\\operatorname*{sup}_{f\\in{\\mathcal{F}}}\\sum_{i=1}^{n}f_{i}\\ {\\stackrel{\\mathrm{d e f .}}{=}}\\ \\sum_{i=1}^{n}f_{i}^{\\circ}\\]\n\\[Z^{i}\\quad=\\quad\\operatorname*{sup}_{f\\in{\\mathcal{F}}}\\sum_{j\\neq i}f_{j}.\\]\nIt follows that \\(0\\leq Z-Z^{i}\\leq f_{i}^{\\circ}\\leq 1\\) ."}, {"n": 2, "claim": "Since \\(\\phi=e^{x}-x-1\\) is a convex function of \\(x,\\)\n\\[\\phi(-\\lambda(Z-Z^{i}))=\\phi(-\\lambda\\cdot(Z-Z^{i})+0\\cdot(1-(Z-Z^{i})))\\leq(Z-Z^{i})\\phi(-\\lambda)\\]\nand\n\\[\\begin{array}{ r c l }{\\mathbb{E}\\left(\\lambda Z e^{\\lambda Z}\\right)-\\mathbb{E}e^{\\lambda Z}\\log\\mathbb{E}e^{\\lambda Z}}&{\\leq}&{\\mathbb{E}\\left(e^{\\lambda Z}\\displaystyle\\sum_{i=1}^{n}\\phi\\left(-\\lambda(Z-Z^{i})\\right)\\right)}\\\\ &{\\leq}&{\\mathbb{E}\\left(e^{\\lambda Z}\\phi(-\\lambda)\\displaystyle\\sum_{i}\\left(Z-Z^{i}\\right)\\right)}\\\\ &{\\leq}&{\\phi(-\\lambda)\\mathbb{E}\\left(e^{\\lambda Z}\\cdot\\displaystyle\\sum_{i}f_{i}^{\\circ}\\right)}\\\\ &{=}&{\\phi(-\\lambda)\\mathbb{E}\\left(Z\\cdot e^{\\lambda Z}\\right).}\\end{array}\\]\nSet \\(\\tilde{Z}=Z-\\mathbb{E}Z\\ (\\mathrm{i . e .}\\) , center Z),and we get\n\\[\\begin{array}{ r l r }{\\mathbb{E}\\left(\\lambda\\tilde{Z}e^{\\lambda\\tilde{Z}}\\right)-\\mathbb{E}e^{\\lambda\\tilde{Z}}\\log\\mathbb{E}e^{\\lambda\\tilde{Z}}}&{\\leq}&{\\phi(-\\lambda)\\mathbb{E}\\left(Z\\cdot e^{\\lambda\\tilde{Z}}\\right)=\\phi(-\\lambda)\\mathbb{E}\\left(\\left(\\tilde{Z}+\\mathbb{E}Z\\right)\\cdot e^{\\lambda\\tilde{Z}}\\right)}\\\\ {(\\lambda-\\phi(-\\lambda))\\mathbb{E}\\left(\\tilde{Z}e^{\\lambda\\tilde{Z}}\\right)-\\mathbb{E}e^{\\lambda\\tilde{Z}}\\log\\mathbb{E}e^{\\lambda\\tilde{Z}}}&{\\leq}&{\\phi(-\\lambda)\\cdot\\mathbb{E}Z\\cdot\\mathbb{E}e^{\\lambda\\tilde{Z}}.}\\end{array}\\]\nLet \\(v=\\mathbb{E}Z,F(\\lambda)=\\mathbb{E}e^{\\lambda\\tilde{Z}}\\) , 亚 = logF,and we get\n\\[\\begin{array}{ r c l }{{(\\lambda-\\phi(-\\lambda)){\\displaystyle\\frac{F_{\\lambda}^{\\prime}(\\lambda)}{F(\\lambda)}}-\\log F(\\lambda)}}&{{\\le}}&{{v\\phi(-\\lambda)}}\\\\ {{(\\lambda-\\phi(-\\lambda)){(\\log F(\\lambda))}_{\\lambda}^{\\prime}-\\log F(\\lambda)}}&{{\\le}}&{{v\\phi(-\\lambda).}}\\end{array}\\tag{41.1}\\]\nSolving the differential equation\n\\[(\\lambda-\\phi(-\\lambda))\\left(\\underbrace{\\log F(\\lambda)}_{\\Psi_{0}}\\right)_{\\lambda}^{\\prime}-\\underbrace{\\log F(\\lambda)}_{\\Psi_{0}}=v\\phi(-\\lambda),\\tag{41.2}\\]\nyields \\(\\Psi_{0}=v\\cdot\\phi(\\lambda)\\) ."}, {"n": 3, "claim": "We will proceed to show that 亚 satisfying 41.1 has the property \\(\\Psi\\leq\\Psi_{0}\\)\n\\[\\begin{array}{ r l r }{\\mathrm{Substract~41.2~from~41.1,~and~let~\\Psi\\Psi_1=\\Psi-\\Psi_0~}}&{}&\\\\ {(1-e^{-\\lambda})\\Psi_{1}^{\\prime}-\\Psi_{1}}&{\\le}&{0}\\end{array}\\]\n\\[\\begin{array}{ r c l }{\\displaystyle\\big(e^{\\lambda}-1\\big)\\big(1-e^{-\\lambda}\\big)\\frac{1}{e^{\\lambda}-1}=1-e^{-\\lambda}\\mathrm{,a n d}\\big(e^{\\lambda}-1\\big)\\big(1-e^{-\\lambda}\\big)\\frac{e^{\\lambda}}{(e^{\\lambda}-1)^{2}}=1}&&\\\\ {\\displaystyle\\big(e^{\\lambda}-1\\big)\\big(1-e^{-\\lambda}\\big)\\underbrace{\\Bigg(\\frac{1}{e^{\\lambda}-1}\\Psi_{1}^{\\prime}-\\frac{e^{\\lambda}}{\\big(e^{\\lambda}-1\\big)^{2}}\\Psi_{1}\\Bigg)}_{\\big(\\frac{\\Psi_{1}}{e^{\\lambda}-1}\\big)_{\\lambda}^{\\prime}}}&{\\le}&{0}\\end{array}\\]\n\\[{\\frac{\\Psi_{1}(\\lambda)}{e^{\\lambda}-1}}~\\leq~\\operatorname*{lim}_{\\lambda\\to 0}{\\frac{\\Psi_{1}(\\lambda)}{e^{\\lambda}-1}}=0 .\\]\nIt follows that \\(\\Psi\\leq v\\phi(\\lambda)\\) ,and \\(F=\\mathbb{E}e^{\\lambda\\tilde{Z}}\\leq e^{v\\phi(\\lambda)}\\) ."}, {"n": 4, "claim": "By Chebychev's inequality, \\(\\mathbb{P}\\left(Z\\geq\\mathbb{E}Z+t\\right)\\leq e^{-\\lambda t+v\\phi\\left(\\lambda\\right)}\\) Minimizing over all \\(\\lambda > 0\\) ，we get \\(\\mathbb{P}\\left(Z\\geq\\mathbb{E}Z+t\\right)\\leq e^{-v\\cdot h\\left(t/v\\right)}\\) where \\(h(x)=(1+x)\\cdot\\log(1+x)-x.\\)"}]}}, {"local_id": "T2", "statement": "Setup. Let \\(X_1,\\dots,X_n\\) be i.i.d. random variables with common law \\(P\\) on a measurable space \\((\\mathcal{X},\\mathcal{A})\\), let \\(\\mathcal{C}\\subseteq\\mathcal{A}\\) be a class of measurable sets, and write \\(P(C)=\\mathbb{P}(X_1\\in C)\\). Let \\(X'_1,\\dots,X'_n\\) (the ghost sample) be an independent copy of \\(X_1,\\dots,X_n\\), i.e. i.i.d. \\(P\\) and independent of \\(X_1,\\dots,X_n\\). Suprema over \\(\\mathcal{C}\\) are assumed measurable; otherwise read \\(\\mathbb{P}\\) as outer probability.\n\nIf \\(t\\geq{\\sqrt{\\frac{2}{n}}}.\\) then\n\\[\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\right)\\leq 2\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\geq t/2\\right).\\]", "proof": {"steps": [{"n": 1, "claim": "Suppose the event\n\\[\\operatorname*{sup}_{C\\in{\\mathcal{C}}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\]\noccurs.Let \\(\\begin{array}{ r }{X=(X_{1},\\ldots,X_{n})\\in\\{\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\}}\\end{array}\\) ."}, {"n": 2, "claim": "Then\n\\[\\exists C_{X}{\\mathrm{~s u c h~t h a t~}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nFor a fixed C,\n\\[\\begin{array}{ r }{\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right|\\geq t/2\\right)=\\mathbb{P}\\left(\\left(\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}\\geq t^{2}/4\\right)}\\\\ {\\leq\\mathrm{~(b y~C h e b y s h e v~s~I n e q)~}\\frac{4\\mathbb{E}\\left(\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}}{t^{2}}}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i,j}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))(I(X_{j}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i=1}^{n}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))^{2}=\\frac{4 n\\mathbb{P}\\left(\\mathcal{C}\\right)\\left(1-\\mathbb{P}(\\mathcal{C})\\right)}{n^{2}t^{2}}\\leq\\frac{1}{n t^{2}}\\leq\\frac{1}{2}}\\end{array}\\]\nsince we chose \\(t\\geq\\sqrt{\\textstyle{\\frac{2}{n}}}.\\)\nSo,\n\\[\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\Bigg|\\exists C_{X}\\right)\\geq 1/2\\]\nif \\(t\\geq\\sqrt{2/n}\\) .Assume that the event\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\]\noccurs."}, {"n": 3, "claim": "Recall that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nHence,it must be that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\geq t/2 .\\]\nWe conclude\n\\[\\begin{array}{ r c l }{\\displaystyle\\frac 12}&{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}(C_{X})\\right|\\le t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right).}\\end{array}\\]\nSince indicators are 0,1-valued,\n\\[\\begin{array}{ r l r }&{}&{\\displaystyle\\frac{1}{2}T\\left(\\underbrace{\\operatorname*{sup}_{\\zeta\\in{\\mathcal C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}(C)\\right|\\ge t}_{\\Im C x}\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C x\\right)\\cdot I\\left(\\exists C x\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X,X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]\nNow, take expectation with respect to \\(X_{i}^{,}\\) s to obtain\n\\[\\begin{array}{ r l r }&{}&{\\mathbb{P}_{X}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\ge t\\right)}\\\\ &{}&{\\displaystyle\\le 2\\cdot\\mathbb{P}_{X,X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]"}]}}, {"local_id": "T3", "statement": "Setup. Let \\(X_1,\\dots,X_n\\) be independent random variables with values in a measurable space \\(\\mathcal{X}\\), let \\(\\mathcal{F}\\) be a finite class of measurable functions on \\(\\mathcal{X}\\), and write \\(f_i:=f(X_i)\\) for \\(f\\in\\mathcal{F}\\).\n\nLet \\(\\mathcal{F}\\) be a finite set of functions \\(| \\mathcal F | < \\infty\\) .For any \\(f=(f_{1},\\cdot\\cdot\\cdot,f_{n})\\in{\\mathcal{F}},a_{i}\\leq f_{i}\\leq b_{i}\\) \\(\\begin{array}{ r }{L=\\operatorname*{sup}_{f}\\sum_{i=1}^{n}\\left(b_{i}-a_{i}\\right)^{2}}\\end{array}\\) ，and \\(Z=\\operatorname*{sup}_{f}\\sum_{i=1}^{n}f_{i}\\) . Then \\(\\mathbb{P}(Z\\geq\\mathbb{E}Z+\\sqrt{2 L t})\\leq e^{-t}\\)", "proof": {"steps": [{"n": 1, "claim": "Let\n\\[\\begin{array}{ r c l }{Z^{i}}&{=}&{\\displaystyle\\operatorname*{sup}_{f\\in\\mathcal{F}}\\left(a_{i}+\\sum_{j\\neq i}f_{j}\\right)}\\end{array}\\]\n\\[Z=\\operatorname*{sup}_{f\\in{\\mathcal{F}}}\\sum_{i=1}^{n}f_{i}{\\stackrel{\\mathrm{d e f .}}{=}}\\sum_{i=1}^{n}f_{i}^{\\circ}.\\]\nIt follows that\n\\[0\\leq Z-Z^{i}\\leq\\sum_{i}f_{i}^{\\circ}-\\sum_{j\\neq i}f_{j}^{\\circ}-a_{i}=f_{i}^{\\circ}-a_{i}\\leq b_{i}(f^{\\circ})-a_{i}(f^{\\circ}).\\]\nSince \\(\\begin{array}{ r }{\\frac{\\phi(x)}{x^{2}}=\\frac{e^{x}-x-1}{x^{2}}}\\end{array}\\) is increasing in R and lin \\(\\begin{array}{ r }{1_{x\\to 0}\\frac{\\phi(x)}{x^{2}}\\to\\frac{1}{2}}\\end{array}\\) , it follows that \\(\\forall x<0,\\phi(x)\\leq{\\textstyle{\\frac{1}{2}}}x^{2}\\) 2,and\n\\[\\begin{array}{ r c l }{\\displaystyle\\mathbb{E}e^{\\lambda Z}\\lambda Z-\\mathbb{E}e^{\\lambda Z}\\log\\mathbb{E}e^{\\lambda Z}}&{\\leq}&{\\displaystyle\\mathbb{E}e^{\\lambda Z}\\sum_{i}\\phi\\left(-\\lambda(Z-Z^{i})\\right)}\\\\ &&{\\leq}&{\\displaystyle\\frac{1}{2}\\mathbb{E}e^{\\lambda Z}\\sum_{i}\\lambda^{2}(Z-Z^{i})^{2}}\\\\ &&{\\leq}&{\\displaystyle\\frac{1}{2}L\\lambda^{2}\\mathbb{E}e^{\\lambda Z}.}\\end{array}\\]\nCenter \\(Z,\\) and we get\n\\[\\mathbb{E}e^{\\lambda(Z-\\mathbb{E}Z)}\\lambda(Z-\\mathbb{E}Z)-\\mathbb{E}e^{\\lambda(Z-\\mathbb{E}Z)}\\log\\mathbb{E}e^{\\lambda(Z-\\mathbb{E}Z)}\\ \\le\\ \\frac{1}{2}L\\lambda^{2}\\mathbb{E}e^{\\lambda(Z-\\mathbb{E}Z)}.\\]\nLet \\(F(\\lambda)=\\mathbb{E}e^{\\lambda(Z-\\mathbb{E}Z)}\\) ."}, {"n": 2, "claim": "It follows that \\(F_{\\lambda}^{\\prime}(\\lambda)=\\mathbb{E}e^{\\lambda(Z-\\mathbb{E}Z)}(Z-\\mathbb{E}Z)\\) and\n\\[\\begin{array}{ r c l }{\\lambda F_{\\lambda}^{\\prime}(\\lambda)-F(\\lambda)\\log F(\\lambda)}&{\\le}&{\\frac{1}{2}L\\lambda^{2}F(\\lambda)}\\\\ {\\displaystyle\\frac{1}{\\lambda}\\frac{P_{\\lambda}^{\\prime}(\\lambda)}{F(\\lambda)}-\\frac{1}{\\lambda^{2}}\\log F(\\lambda)}&{\\le}&{\\frac{1}{2}L}\\\\ {\\displaystyle(\\frac{1}{\\lambda}\\log F(\\lambda))_{\\lambda}^{\\prime}}&{\\le}&{\\frac{1}{2}L}\\\\ {\\displaystyle\\frac{1}{\\lambda}\\log F(\\lambda)}&{=}&{\\frac{1}{t}\\log F(t)\\vert_{t  0}+\\int_{0}^{\\lambda}(\\frac{1}{t}\\log F(t))_{t}^{\\prime}d t}\\\\ &{\\le}&{\\displaystyle\\frac{1}{2}L\\lambda}\\\\ {F(\\lambda)}&{\\le}&{\\displaystyle\\exp(\\frac{1}{2}L\\lambda^{2}).}\\end{array}\\]\nBy Chebychev inequality, and minimize over \\(\\lambda,\\) we get\n\\[\\begin{array}{ r l r }{\\mathbb{P}(Z\\geq\\mathbb{E}Z+t)}&{{}\\leq}&{e^{-\\lambda t}\\mathbb{E}e^{\\lambda(Z-\\mathbb{E}Z)}}\\end{array}\\]\n\\[\\begin{array}{ r l }{\\leq}&{{}e^{-\\lambda t}e^{\\frac{1}{2}L\\lambda^{2}}}\\end{array}\\]\n\\[\\begin{array}{ r c l }{\\operatorname*{minimize}\\operatorname{over}\\lambda}&{}&{}\\\\ {\\mathbb{P}(Z\\geq\\mathbb{E}Z+t)}&{\\leq}&{e^{-t^{2}/(2L)}}\\end{array}\\]\n"}]}}, {"local_id": "T4", "statement": "Setup. Let \\(0<\\varepsilon\\le 1\\).\n\nAssume \\(\\mathcal{F}\\) is a VC-subgraph class and \\(V C(\\mathcal{F})=V\\) Suppose \\(-1\\leq f(x)\\leq 1\\) for all \\(f\\in{\\mathcal{F}}\\) and \\(x\\in\\mathcal{X}\\) .Let \\(x_{1},\\ldots,x_{n}\\in{\\mathcal{X}}\\) and define \\(\\begin{array}{ r }{d(f,g)=\\frac{1}{n}\\sum_{i=1}^{n}|f(x_{i})-g(x_{i})|}\\end{array}\\) . Then\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]\n(which \\(\\begin{array}{ r }{i s\\le\\left(\\frac{K}{\\varepsilon}\\right)^{V+\\delta}}\\end{array}\\) for some \\(\\delta . \\jmath\\)", "proof": {"steps": [{"n": 1, "claim": "Let \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) and \\(f_{1},\\ldots,f_{m}\\) be ε-separated, i.e.\n\\[{\\frac{1}{n}}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})| >\\varepsilon .\\]"}, {"n": 2, "claim": "Let \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) be constructed in the following way:"}, {"n": 3, "claim": "\\(z_{i}\\) is chosen uniformly from \\(x_{1},\\ldots,x_{n}\\) and \\(t_{i}\\) is uniform on \\([-1,1]\\)\nConsider \\(f_{r}\\) and \\(f_{\\ell}\\) from the ε-packing."}, {"n": 4, "claim": "Let \\(C_{f_{\\tau}}\\) and \\(C_{f_{\\ell}}\\) be subgraphs of \\(f_{r}\\) and \\(f_{\\ell}\\) Then\n\\(\\mathbb{P}\\left(C_{f_{\\tau}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n= P(At least one point \\((z_{i},t_{i})\\) is picked by \\(C_{f_{r}}\\) or \\(C_{f_{\\ell}}\\) but not picked by the other)\n\\(=1-\\mathbb{P}\\) (All points \\((z_{i},t_{i})\\) are picked either by both or by none)\n\\(\\mathbf{\\Sigma}=1-\\mathbb{P}\\left(\\left(z_{i},t_{i}\\right)\\right.\\) is picked either by both or by none)k\nSince \\(z_{i}\\) is drawn uniformly from \\(x_{1},\\ldots,x_{n}\\)\nP \\(((z_{1},t_{1})\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)\n\\[\\begin{array}{ l }{\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{P}\\left(\\left(x_{i},t_{1}\\right)\\mathrm{~i s~p i c k e d~b y~b o t h~}C_{f r},C_{f_{\\ell}}\\mathrm{~o r~b y~n e i t h e r}\\right)}\\\\ {\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\left(1-\\frac{1}{2}|f_{r}(x_{i})-f_{\\ell}(x_{i})|\\right)}\\\\ {\\displaystyle=1-\\frac{1}{2}\\frac{1}{n}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})|}\\\\ {\\displaystyle=1-\\frac{1}{2}d(f_{r},f_{\\ell})\\leq 1-\\varepsilon/2\\leq e^{-\\varepsilon/2}}\\end{array}\\]\nSubstituting,\n\\(\\mathbb{P}\\left(C_{f_{r}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n\\(=1-\\mathbb{P}\\left((z_{1},t_{1})\\right.\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)𝑘\n\\[\\begin{array}{ l }{\\geq 1-\\left(e^{-\\varepsilon/2}\\right)^{k}}\\\\ {=1-e^{-k\\varepsilon/2}}\\end{array}\\]\nThere are (m） ways to choose \\(f_{r}\\) and \\(f_{\\ell},\\) s0\nP (All pairs \\(C_{f_{r}}\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\geq 1-{\\binom{m}{2}}e^{-k\\varepsilon/2}.\\)\nWhat k should we choose so that \\(1-{\\binom{m}{2}}e^{-k\\varepsilon/2}>0?\\) Choose\n\\[k>{\\frac{2}{\\varepsilon}}\\log{\\binom{m}{2}}.\\]\nThen there exist \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) such that all \\(C_{f_{\\ell}}\\) pick out different subsets.But \\(\\{C_{f}:f\\in\\mathcal{F}\\}\\) is \\(\\mathrm{VC}\\) so by Sauer's lemma, we can pick out at most \\(\\left({\\frac{e k}{V}}\\right)^{V}\\) out of these k points, provided \\(k \\geq V\\); if instead \\(k < V\\) then \\(m\\leq 2^{k}<2^{V}\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}\\) and the claimed bound already holds, so that case may be set aside."}, {"n": 5, "claim": "Hence, \\(\\begin{array}{ r }{m\\leq\\left(\\frac{e k}{V}\\right)^{V}}\\end{array}\\) as long as \\(k>{\\frac{2}{\\varepsilon}}\\log\\left({{\\binom{m}{2}}}\\right)\\) and \\(k \\geq V\\), the case \\(k < V\\) having been disposed of above."}, {"n": 6, "claim": "The latter holds for \\(\\begin{array}{ r }{k=\\frac{2}{\\varepsilon}\\log m^{2}}\\end{array}\\) ."}, {"n": 7, "claim": "Therefore,\n\\[m\\leq\\left(\\frac{e}{V}\\frac{2}{\\varepsilon}\\log m^{2}\\right)^{V}=\\left(\\frac{4 e}{V\\varepsilon}\\log m\\right)^{V},\\]\nwhere \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) ."}, {"n": 8, "claim": "Hence, we get\n\\[m^{1/V}\\leq\\frac{4 e}{\\varepsilon}\\log m^{1/V}\\]\nand defining \\(m^{1/V}=s .\\)\n\\[s\\leq{\\frac{4 e}{\\varepsilon}}\\log s .\\]\nNote that \\(\\frac{s}{\\log s}\\) is increasing for \\(s \\geq e\\) and so for large enough s, the inequality will be violated."}, {"n": 9, "claim": "We now check that the inequality is violated for \\(\\begin{array}{ r }{s^{\\prime}=\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}}\\end{array}\\) ."}, {"n": 10, "claim": "Indeed, one can show that\n\\[\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{7}{\\varepsilon}\\right)^{2}>\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)\\]\nsince\n\\[\\frac{49}{8e\\varepsilon}>\\log{\\frac{7}{\\epsilon}}.\\]\nHence, \\(m^{1/V}=s\\leq s^{\\prime}\\) and, thus,\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]"}]}}], "target_local_id": "T3", "topic_domain": "Probability & Concentration", "topic_subtopic": "Functional & Supremum Concentration", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch41_theorem_41_2__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nFor \\(f:\\mathcal{X}\\to\\mathbb{R}\\) , the subgraph of \\(f\\) is\n\\[C_{f}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}:0\\leq t\\leq f(x)\\ o r\\ f(x)\\leq t\\leq 0\\}.\\]\nDefine class of sets \\(\\mathcal{C}=\\{C_{f}:f\\in\\mathcal{F}\\}\\).\n\ndefinition 12.1. If \\(\\mathcal{C}\\) is a VC class of sets, then F is VC-subgraph class of functions and, by definition, \\(V C(\\mathcal{F})=V C(\\mathcal{C})\\)\nNote that equivalent definition of \\(C_{f}\\) is\n\\[C_{f}^{\\prime}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}: |f(x)|\\geq|t|\\}.\\]\n\ndefinition 12.3. The ε-packing number, \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) , is the marimal cardinality of an E-separated set.\nNote that \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) is decreasing in ε.\n\ndefinition 12.2. Given \\(\\varepsilon > 0\\) and \\(f_{1},\\ldots,f_{N}\\in\\mathcal{F}\\) ， we say that \\(f_{1},\\ldots,f_{N}\\) are E-separated if \\(d(f_{i},f_{j})>\\varepsilon\\) for any \\(i \\neq j\\)\n\nDenote \\(\\triangle_{n}(\\mathcal{C})=\\operatorname*{sup}_{\\{x_{1},\\ldots,x_{n}\\}}\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})\\leq 2^{n}\\). We will see that for some classes, \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\) for \\(n \\leq V\\) and \\(\\triangle_{n}(\\mathcal{C})<2^{n}\\) for \\(n > V\\) for some constant \\(V\\).\n\ndefinition 8.1. If \\(V < \\infty\\) , then C is called a VC class. V is called VC dimension of C.\n\ndefinition 8.0. For any sample \\(\\{x_{1},\\ldots,x_{n}\\}\\), the ways that \\({\\mathcal{C}}\\) intersects with the sample are \\[\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in{\\mathcal{C}}\\}.\\] Let \\[\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})=\\mathrm{card}~\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in\\mathcal{C}\\},\\] the number of different subsets picked out by \\(C\\in{\\mathcal{C}}\\). If \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\), this means we can find \\(\\{x_{1},\\ldots,x_{n}\\}\\) such that \\({\\mathcal{C}}\\) can pick out any subset of it; in this case \\({\\mathcal{C}}\\) shatters \\(\\{x_{1},\\ldots,x_{n}\\}\\). Equivalently, a set \\(A\\) is shattered by \\({\\mathcal{C}}\\) when \\({\\mathcal{C}}\\) picks out every subset of \\(A\\).", "theorems": [{"local_id": "T1", "statement": "Setup. Let \\(X_1,\\dots,X_n\\) be i.i.d. random variables with common law \\(P\\) on a measurable space \\((\\mathcal{X},\\mathcal{A})\\), let \\(\\mathcal{C}\\subseteq\\mathcal{A}\\) be a class of measurable sets, and write \\(P(C)=\\mathbb{P}(X_1\\in C)\\). Let \\(X'_1,\\dots,X'_n\\) (the ghost sample) be an independent copy of \\(X_1,\\dots,X_n\\), i.e. i.i.d. \\(P\\) and independent of \\(X_1,\\dots,X_n\\). Suprema over \\(\\mathcal{C}\\) are assumed measurable; otherwise read \\(\\mathbb{P}\\) as outer probability.\n\nIf \\(t\\geq{\\sqrt{\\frac{2}{n}}}.\\) then\n\\[\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\right)\\leq 2\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\geq t/2\\right).\\]", "proof": {"steps": [{"n": 1, "claim": "Suppose the event\n\\[\\operatorname*{sup}_{C\\in{\\mathcal{C}}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\]\noccurs.Let \\(\\begin{array}{ r }{X=(X_{1},\\ldots,X_{n})\\in\\{\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\}}\\end{array}\\) ."}, {"n": 2, "claim": "Then\n\\[\\exists C_{X}{\\mathrm{~s u c h~t h a t~}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nFor a fixed C,\n\\[\\begin{array}{ r }{\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right|\\geq t/2\\right)=\\mathbb{P}\\left(\\left(\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}\\geq t^{2}/4\\right)}\\\\ {\\leq\\mathrm{~(b y~C h e b y s h e v~s~I n e q)~}\\frac{4\\mathbb{E}\\left(\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}}{t^{2}}}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i,j}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))(I(X_{j}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i=1}^{n}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))^{2}=\\frac{4 n\\mathbb{P}\\left(\\mathcal{C}\\right)\\left(1-\\mathbb{P}(\\mathcal{C})\\right)}{n^{2}t^{2}}\\leq\\frac{1}{n t^{2}}\\leq\\frac{1}{2}}\\end{array}\\]\nsince we chose \\(t\\geq\\sqrt{\\textstyle{\\frac{2}{n}}}.\\)\nSo,\n\\[\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\Bigg|\\exists C_{X}\\right)\\geq 1/2\\]\nif \\(t\\geq\\sqrt{2/n}\\) .Assume that the event\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\]\noccurs."}, {"n": 3, "claim": "Recall that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nHence,it must be that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\geq t/2 .\\]\nWe conclude\n\\[\\begin{array}{ r c l }{\\displaystyle\\frac 12}&{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}(C_{X})\\right|\\le t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right).}\\end{array}\\]\nSince indicators are 0,1-valued,\n\\[\\begin{array}{ r l r }&{}&{\\displaystyle\\frac{1}{2}T\\left(\\underbrace{\\operatorname*{sup}_{\\zeta\\in{\\mathcal C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}(C)\\right|\\ge t}_{\\Im C x}\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C x\\right)\\cdot I\\left(\\exists C x\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X,X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]\nNow, take expectation with respect to \\(X_{i}^{,}\\) s to obtain\n\\[\\begin{array}{ r l r }&{}&{\\mathbb{P}_{X}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\ge t\\right)}\\\\ &{}&{\\displaystyle\\le 2\\cdot\\mathbb{P}_{X,X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]"}]}}, {"local_id": "T2", "statement": "Setup. Let \\(X_1,\\dots,X_n\\) be independent random variables with values in a measurable space \\(\\mathcal{X}\\), let \\(\\mathcal{F}\\) be a finite class of measurable functions \\(f:\\mathcal{X}\\to[0,1]\\), write \\(f_i:=f(X_i)\\), and set \\(Z=\\sup_{f\\in\\mathcal{F}}\\sum_{i=1}^{n}f_i\\) with \\(\\mathbb{E}Z>0\\).\n\nLet \\(\\mathcal{F}\\) be a finite set of functions \\(|{\\mathcal{F}}| <\\infty .\\forall f=(f_{1},\\cdots,f_{n})\\in{\\mathcal{F}},0\\leq f_{i}\\leq 1,Z=\\) \\(\\textstyle\\operatorname*{sup}_{f}\\sum_{i=1}^{n}f_{i}\\) ， and define h as \\(h(u)=(1+u)\\log(1+u)-u\\) where \\(u \\geq 0\\) Then \\(\\mathbb{P}(Z\\geq\\mathbb{E}Z+x)\\leq e^{-\\mathbb{E}Z\\cdot h(x/\\mathbb{E}Z)}\\)", "proof": {"steps": [{"n": 1, "claim": "Let\n\\[Z\\quad=\\quad\\operatorname*{sup}_{f\\in{\\mathcal{F}}}\\sum_{i=1}^{n}f_{i}\\ {\\stackrel{\\mathrm{d e f .}}{=}}\\ \\sum_{i=1}^{n}f_{i}^{\\circ}\\]\n\\[Z^{i}\\quad=\\quad\\operatorname*{sup}_{f\\in{\\mathcal{F}}}\\sum_{j\\neq i}f_{j}.\\]\nIt follows that \\(0\\leq Z-Z^{i}\\leq f_{i}^{\\circ}\\leq 1\\) ."}, {"n": 2, "claim": "Since \\(\\phi=e^{x}-x-1\\) is a convex function of \\(x,\\)\n\\[\\phi(-\\lambda(Z-Z^{i}))=\\phi(-\\lambda\\cdot(Z-Z^{i})+0\\cdot(1-(Z-Z^{i})))\\leq(Z-Z^{i})\\phi(-\\lambda)\\]\nand\n\\[\\begin{array}{ r c l }{\\mathbb{E}\\left(\\lambda Z e^{\\lambda Z}\\right)-\\mathbb{E}e^{\\lambda Z}\\log\\mathbb{E}e^{\\lambda Z}}&{\\leq}&{\\mathbb{E}\\left(e^{\\lambda Z}\\displaystyle\\sum_{i=1}^{n}\\phi\\left(-\\lambda(Z-Z^{i})\\right)\\right)}\\\\ &{\\leq}&{\\mathbb{E}\\left(e^{\\lambda Z}\\phi(-\\lambda)\\displaystyle\\sum_{i}\\left(Z-Z^{i}\\right)\\right)}\\\\ &{\\leq}&{\\phi(-\\lambda)\\mathbb{E}\\left(e^{\\lambda Z}\\cdot\\displaystyle\\sum_{i}f_{i}^{\\circ}\\right)}\\\\ &{=}&{\\phi(-\\lambda)\\mathbb{E}\\left(Z\\cdot e^{\\lambda Z}\\right).}\\end{array}\\]\nSet \\(\\tilde{Z}=Z-\\mathbb{E}Z\\ (\\mathrm{i . e .}\\) , center Z),and we get\n\\[\\begin{array}{ r l r }{\\mathbb{E}\\left(\\lambda\\tilde{Z}e^{\\lambda\\tilde{Z}}\\right)-\\mathbb{E}e^{\\lambda\\tilde{Z}}\\log\\mathbb{E}e^{\\lambda\\tilde{Z}}}&{\\leq}&{\\phi(-\\lambda)\\mathbb{E}\\left(Z\\cdot e^{\\lambda\\tilde{Z}}\\right)=\\phi(-\\lambda)\\mathbb{E}\\left(\\left(\\tilde{Z}+\\mathbb{E}Z\\right)\\cdot e^{\\lambda\\tilde{Z}}\\right)}\\\\ {(\\lambda-\\phi(-\\lambda))\\mathbb{E}\\left(\\tilde{Z}e^{\\lambda\\tilde{Z}}\\right)-\\mathbb{E}e^{\\lambda\\tilde{Z}}\\log\\mathbb{E}e^{\\lambda\\tilde{Z}}}&{\\leq}&{\\phi(-\\lambda)\\cdot\\mathbb{E}Z\\cdot\\mathbb{E}e^{\\lambda\\tilde{Z}}.}\\end{array}\\]\nLet \\(v=\\mathbb{E}Z,F(\\lambda)=\\mathbb{E}e^{\\lambda\\tilde{Z}}\\) , 亚 = logF,and we get\n\\[\\begin{array}{ r c l }{{(\\lambda-\\phi(-\\lambda)){\\displaystyle\\frac{F_{\\lambda}^{\\prime}(\\lambda)}{F(\\lambda)}}-\\log F(\\lambda)}}&{{\\le}}&{{v\\phi(-\\lambda)}}\\\\ {{(\\lambda-\\phi(-\\lambda)){(\\log F(\\lambda))}_{\\lambda}^{\\prime}-\\log F(\\lambda)}}&{{\\le}}&{{v\\phi(-\\lambda).}}\\end{array}\\tag{41.1}\\]\nSolving the differential equation\n\\[(\\lambda-\\phi(-\\lambda))\\left(\\underbrace{\\log F(\\lambda)}_{\\Psi_{0}}\\right)_{\\lambda}^{\\prime}-\\underbrace{\\log F(\\lambda)}_{\\Psi_{0}}=v\\phi(-\\lambda),\\tag{41.2}\\]\nyields \\(\\Psi_{0}=v\\cdot\\phi(\\lambda)\\) ."}, {"n": 3, "claim": "We will proceed to show that 亚 satisfying 41.1 has the property \\(\\Psi\\leq\\Psi_{0}\\)\n\\[\\begin{array}{ r l r }{\\mathrm{Substract~41.2~from~41.1,~and~let~\\Psi\\Psi_1=\\Psi-\\Psi_0~}}&{}&\\\\ {(1-e^{-\\lambda})\\Psi_{1}^{\\prime}-\\Psi_{1}}&{\\le}&{0}\\end{array}\\]\n\\[\\begin{array}{ r c l }{\\displaystyle\\big(e^{\\lambda}-1\\big)\\big(1-e^{-\\lambda}\\big)\\frac{1}{e^{\\lambda}-1}=1-e^{-\\lambda}\\mathrm{,a n d}\\big(e^{\\lambda}-1\\big)\\big(1-e^{-\\lambda}\\big)\\frac{e^{\\lambda}}{(e^{\\lambda}-1)^{2}}=1}&&\\\\ {\\displaystyle\\big(e^{\\lambda}-1\\big)\\big(1-e^{-\\lambda}\\big)\\underbrace{\\Bigg(\\frac{1}{e^{\\lambda}-1}\\Psi_{1}^{\\prime}-\\frac{e^{\\lambda}}{\\big(e^{\\lambda}-1\\big)^{2}}\\Psi_{1}\\Bigg)}_{\\big(\\frac{\\Psi_{1}}{e^{\\lambda}-1}\\big)_{\\lambda}^{\\prime}}}&{\\le}&{0}\\end{array}\\]\n\\[{\\frac{\\Psi_{1}(\\lambda)}{e^{\\lambda}-1}}~\\leq~\\operatorname*{lim}_{\\lambda\\to 0}{\\frac{\\Psi_{1}(\\lambda)}{e^{\\lambda}-1}}=0 .\\]\nIt follows that \\(\\Psi\\leq v\\phi(\\lambda)\\) ,and \\(F=\\mathbb{E}e^{\\lambda\\tilde{Z}}\\leq e^{v\\phi(\\lambda)}\\) ."}, {"n": 4, "claim": "By Chebychev's inequality, \\(\\mathbb{P}\\left(Z\\geq\\mathbb{E}Z+t\\right)\\leq e^{-\\lambda t+v\\phi\\left(\\lambda\\right)}\\) Minimizing over all \\(\\lambda > 0\\) ，we get \\(\\mathbb{P}\\left(Z\\geq\\mathbb{E}Z+t\\right)\\leq e^{-v\\cdot h\\left(t/v\\right)}\\) where \\(h(x)=(1+x)\\cdot\\log(1+x)-x.\\)"}]}}, {"local_id": "T3", "statement": "Setup. Let \\(0<\\varepsilon\\le 1\\).\n\nAssume \\(\\mathcal{F}\\) is a VC-subgraph class and \\(V C(\\mathcal{F})=V\\) Suppose \\(-1\\leq f(x)\\leq 1\\) for all \\(f\\in{\\mathcal{F}}\\) and \\(x\\in\\mathcal{X}\\) .Let \\(x_{1},\\ldots,x_{n}\\in{\\mathcal{X}}\\) and define \\(\\begin{array}{ r }{d(f,g)=\\frac{1}{n}\\sum_{i=1}^{n}|f(x_{i})-g(x_{i})|}\\end{array}\\) . Then\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]\n(which \\(\\begin{array}{ r }{i s\\le\\left(\\frac{K}{\\varepsilon}\\right)^{V+\\delta}}\\end{array}\\) for some \\(\\delta . \\jmath\\)", "proof": {"steps": [{"n": 1, "claim": "Let \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) and \\(f_{1},\\ldots,f_{m}\\) be ε-separated, i.e.\n\\[{\\frac{1}{n}}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})| >\\varepsilon .\\]"}, {"n": 2, "claim": "Let \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) be constructed in the following way:"}, {"n": 3, "claim": "\\(z_{i}\\) is chosen uniformly from \\(x_{1},\\ldots,x_{n}\\) and \\(t_{i}\\) is uniform on \\([-1,1]\\)\nConsider \\(f_{r}\\) and \\(f_{\\ell}\\) from the ε-packing."}, {"n": 4, "claim": "Let \\(C_{f_{\\tau}}\\) and \\(C_{f_{\\ell}}\\) be subgraphs of \\(f_{r}\\) and \\(f_{\\ell}\\) Then\n\\(\\mathbb{P}\\left(C_{f_{\\tau}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n= P(At least one point \\((z_{i},t_{i})\\) is picked by \\(C_{f_{r}}\\) or \\(C_{f_{\\ell}}\\) but not picked by the other)\n\\(=1-\\mathbb{P}\\) (All points \\((z_{i},t_{i})\\) are picked either by both or by none)\n\\(\\mathbf{\\Sigma}=1-\\mathbb{P}\\left(\\left(z_{i},t_{i}\\right)\\right.\\) is picked either by both or by none)k\nSince \\(z_{i}\\) is drawn uniformly from \\(x_{1},\\ldots,x_{n}\\)\nP \\(((z_{1},t_{1})\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)\n\\[\\begin{array}{ l }{\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{P}\\left(\\left(x_{i},t_{1}\\right)\\mathrm{~i s~p i c k e d~b y~b o t h~}C_{f r},C_{f_{\\ell}}\\mathrm{~o r~b y~n e i t h e r}\\right)}\\\\ {\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\left(1-\\frac{1}{2}|f_{r}(x_{i})-f_{\\ell}(x_{i})|\\right)}\\\\ {\\displaystyle=1-\\frac{1}{2}\\frac{1}{n}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})|}\\\\ {\\displaystyle=1-\\frac{1}{2}d(f_{r},f_{\\ell})\\leq 1-\\varepsilon/2\\leq e^{-\\varepsilon/2}}\\end{array}\\]\nSubstituting,\n\\(\\mathbb{P}\\left(C_{f_{r}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n\\(=1-\\mathbb{P}\\left((z_{1},t_{1})\\right.\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)𝑘\n\\[\\begin{array}{ l }{\\geq 1-\\left(e^{-\\varepsilon/2}\\right)^{k}}\\\\ {=1-e^{-k\\varepsilon/2}}\\end{array}\\]\nThere are (m） ways to choose \\(f_{r}\\) and \\(f_{\\ell},\\) s0\nP (All pairs \\(C_{f_{r}}\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\geq 1-{\\binom{m}{2}}e^{-k\\varepsilon/2}.\\)\nWhat k should we choose so that \\(1-{\\binom{m}{2}}e^{-k\\varepsilon/2}>0?\\) Choose\n\\[k>{\\frac{2}{\\varepsilon}}\\log{\\binom{m}{2}}.\\]\nThen there exist \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) such that all \\(C_{f_{\\ell}}\\) pick out different subsets.But \\(\\{C_{f}:f\\in\\mathcal{F}\\}\\) is \\(\\mathrm{VC}\\) so by Sauer's lemma, we can pick out at most \\(\\left({\\frac{e k}{V}}\\right)^{V}\\) out of these k points, provided \\(k \\geq V\\); if instead \\(k < V\\) then \\(m\\leq 2^{k}<2^{V}\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}\\) and the claimed bound already holds, so that case may be set aside."}, {"n": 5, "claim": "Hence, \\(\\begin{array}{ r }{m\\leq\\left(\\frac{e k}{V}\\right)^{V}}\\end{array}\\) as long as \\(k>{\\frac{2}{\\varepsilon}}\\log\\left({{\\binom{m}{2}}}\\right)\\) and \\(k \\geq V\\), the case \\(k < V\\) having been disposed of above."}, {"n": 6, "claim": "The latter holds for \\(\\begin{array}{ r }{k=\\frac{2}{\\varepsilon}\\log m^{2}}\\end{array}\\) ."}, {"n": 7, "claim": "Therefore,\n\\[m\\leq\\left(\\frac{e}{V}\\frac{2}{\\varepsilon}\\log m^{2}\\right)^{V}=\\left(\\frac{4 e}{V\\varepsilon}\\log m\\right)^{V},\\]\nwhere \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) ."}, {"n": 8, "claim": "Hence, we get\n\\[m^{1/V}\\leq\\frac{4 e}{\\varepsilon}\\log m^{1/V}\\]\nand defining \\(m^{1/V}=s .\\)\n\\[s\\leq{\\frac{4 e}{\\varepsilon}}\\log s .\\]\nNote that \\(\\frac{s}{\\log s}\\) is increasing for \\(s \\geq e\\) and so for large enough s, the inequality will be violated."}, {"n": 9, "claim": "We now check that the inequality is violated for \\(\\begin{array}{ r }{s^{\\prime}=\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}}\\end{array}\\) ."}, {"n": 10, "claim": "Indeed, one can show that\n\\[\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{7}{\\varepsilon}\\right)^{2}>\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)\\]\nsince\n\\[\\frac{49}{8e\\varepsilon}>\\log{\\frac{7}{\\epsilon}}.\\]\nHence, \\(m^{1/V}=s\\leq s^{\\prime}\\) and, thus,\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]"}]}}, {"local_id": "T4", "statement": "Setup. Let \\(X_1,\\dots,X_n\\) be i.i.d. copies of a random variable \\(X\\) with \\(0\\le X\\le 1\\) and \\(\\mu=\\mathbb{E}X\\), and let \\(0<t\\le 1-\\mu\\).\n\nAssume \\(0\\leq X_{i}\\leq 1\\) and \\(\\mu=\\mathbb{E}X\\) . Then\n\\[\\mathbb{P}\\left(\\frac{1}{n}\\sum_{i=1}^{n}X_{i}-\\mu\\geq t\\right)\\leq e^{-n\\mathcal{D}(\\mu+t,\\mu)}\\]\nwhere the KL-divergence \\(\\begin{array}{ r }{\\mathcal{D}(p,q)=p\\log\\frac{p}{q}+(1-p)\\log\\frac{1-p}{1-q}}\\end{array}\\)", "proof": {"steps": [{"n": 1, "claim": "Note that \\(\\phi(x)=e^{\\lambda x}\\) is convex and so \\(e^{\\lambda x}=e^{\\lambda(x\\cdot 1+(1-x)\\cdot 0)}\\leq x e^{\\lambda}+(1-x)e^{\\lambda\\cdot 0}=1-x+x e^{\\lambda}\\) .Hence,\n\\[\\mathbb{E}e^{\\lambda X}\\leq 1-\\mathbb{E}X+\\mathbb{E}X e^{\\lambda}=1-\\mu+\\mu e^{\\lambda}.\\]\nAgain,we minimize the following bound with respect to \\(\\lambda > 0\\)\n\\[\\begin{array}{ r c l }{\\displaystyle\\mathbb{P}\\left(\\sum_{i=1}^{n}X_{i}\\geq n(\\mu+t)\\right)}&{\\leq}&{e^{-\\lambda n(\\mu+t)}\\mathbb{E}e^{\\lambda\\sum X_{i}}}\\\\ &&{=}&{e^{-\\lambda n(\\mu+t)}\\left(\\mathbb{E}e^{\\lambda X}\\right)^{n}}\\\\ &&{\\leq}&{e^{-\\lambda n(\\mu+t)}\\left(1-\\mu+\\mu e^{\\lambda}\\right)^{n}}\\end{array}\\]\nTake derivative w.r.t. 入:"}, {"n": 2, "claim": "\\[\\begin{array}{ c }{{-n(\\mu+t)e^{-\\lambda n(\\mu+t)}(1-\\mu+\\mu e^{\\lambda})^{n}+n(1-\\mu+\\mu e^{\\lambda})^{n-1}\\mu e^{\\lambda}e^{-\\lambda n(\\mu+t)}=0}}\\\\ {{{}}}\\\\ {{-(\\mu+t)(1-\\mu+\\mu e^{\\lambda})+\\mu e^{\\lambda}=0}}\\\\ {{{}}}\\\\ {{e^{\\lambda}=\\displaystyle\\frac{(1-\\mu)(\\mu+t)}{\\mu(1-\\mu-t)}.}}\\end{array}\\]\nSubstituting,\n\\[\\begin{array}{ r c l }{\\mathbb{P}\\left(\\displaystyle\\sum_{i=1}^{n}X_{i}\\geq n(\\mu+t)\\right)}&{\\leq}&{\\displaystyle\\left(\\left(\\frac{\\mu(1-\\mu-t)}{(1-\\mu)(\\mu+t)}\\right)^{\\mu+t}\\left(1-\\mu+\\frac{(1-\\mu)(\\mu+t)}{1-\\mu-t}\\right)\\right)^{n}}\\\\ &{=}&{\\displaystyle\\left(\\left(\\frac{\\mu}{\\mu+t}\\right)^{\\mu+t}\\left(\\frac{1-\\mu}{1-\\mu-t}\\right)^{1-\\mu-t}\\right)^{n}}\\\\ &{=}&{\\displaystyle\\exp\\left(-n\\left((\\mu+t)\\log\\frac{\\mu+t}{\\mu}+(1-\\mu-t)\\log\\frac{1-\\mu-t}{1-\\mu}\\right)\\right),}\\end{array}\\]\ncompleting the proof."}]}}], "target_local_id": "T2", "topic_domain": "Probability & Concentration", "topic_subtopic": "Functional & Supremum Concentration", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch4_theorem_4_18__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nEmpirical Rademacher complexity. Given a dataset \\(S = \\{z_1, \\dots, z_n\\}\\), the empirical Rademacher complexity is defined as\n\\[\nR_S(\\mathcal{F}) \\stackrel{\\Delta}{=} \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right].\n\\]\n\\(R_S(\\mathcal{F})\\) is a function of both the function class \\(\\mathcal{F}\\) and the dataset \\(S\\).\n\n\\(\\kappa\\)-Lipschitz functions. Let \\(\\kappa \\geq 0\\) and let \\(\\|\\cdot\\|\\) be a norm on the domain \\(D\\). A function \\(L : D \\to \\mathbb{R}\\) is said to be \\(\\kappa\\)-Lipschitz with respect to \\(\\|\\cdot\\|\\) if for all \\(\\theta, \\theta' \\in D\\),\n\\[\n|L(\\theta) - L(\\theta')| \\leq \\kappa\\,\\|\\theta - \\theta'\\|.\n\\]\n\nEpsilon-cover. \\(\\mathcal{C}\\) is an \\(\\epsilon\\)-cover of \\(\\mathcal{Q}\\) with respect to metric \\(\\rho\\) if for all \\(v' \\in \\mathcal{Q}\\), there exists \\(v \\in \\mathcal{C}\\) such that \\(\\rho(v,v')\\le \\epsilon\\).\n\nCovering number. The covering number is defined as the minimum size of an \\(\\epsilon\\)-cover, or explicitly:\n\\[\nN(\\epsilon, \\mathcal{Q}, \\rho) \\stackrel{\\Delta}{=} (\\text{min size of } \\epsilon\\text{-cover of } \\mathcal{Q} \\text{ w.r.t. metric } \\rho).\n\\]\n\n\\(\\epsilon\\)-cover. Let \\(\\epsilon > 0\\). An \\(\\epsilon\\)-cover of a set \\(S\\) with respect to a distance metric \\(\\rho\\) is a subset \\(C \\subseteq S\\) such that for all \\(x \\in S\\) there exists \\(x' \\in C\\) with \\(\\rho(x, x') \\leq \\epsilon\\), or equivalently,\n\\[\nS \\subseteq \\bigcup_{x \\in C} \\mathrm{Ball}(x, \\epsilon, \\rho), \\quad \\text{where} \\quad \\mathrm{Ball}(x, \\epsilon, \\rho) \\triangleq \\{x' : \\rho(x, x') \\leq \\epsilon\\}.\n\\]\n\nRademacher complexity. Let \\(\\mathcal{F}\\) be a family of functions mapping \\(Z \\mapsto \\mathbb{R}\\), and let \\(P\\) be a distribution over \\(Z\\). The (average) Rademacher complexity of \\(\\mathcal{F}\\) is defined as\n\\[\nR_n(F) \\triangleq \\mathbb{E}_{z_1, \\dots, z_n \\overset{\\text{iid}}{\\sim} P} \\left[ \\mathbb{E}_{\\sigma_1, \\dots, \\sigma_n \\overset{\\text{iid}}{\\sim}\\{ \\pm 1 \\}} \\left[ \\sup_{f\\in F} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right] \\right],\n\\]\nwhere \\(\\sigma_1, \\dots, \\sigma_n\\) are independent Rademacher random variables, i.e. each taking on the value of \\(1\\) or \\(-1\\) with probability \\(1/2\\).\n\nSub-Gaussian Random Variables. A random variable \\(X\\) with finite mean \\(\\mu\\) is sub-Gaussian with parameter \\(\\sigma\\) if\n\\[ \\mathbb{E} \\left[ e^{\\lambda(X - \\mu)} \\right] \\leq e^{\\sigma^2\\lambda^2 / 2}, \\quad \\forall\\lambda\\in\\mathbb{R}. \\]\nWe say that \\(X\\) is \\(\\sigma\\)-sub-Gaussian and say it has variance proxy \\(\\sigma^2\\).", "theorems": [{"local_id": "T1", "statement": "Suppose \\(\\phi\\) is \\(\\kappa\\)-Lipschitz, and \\(\\rho = L_2(P_n)\\). Then,\n\\[\n\\log N(\\epsilon, \\phi \\circ \\mathcal{F}, \\rho) \\le \\log N(\\epsilon / \\kappa, \\mathcal{F}, \\rho).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\mathcal{C}\\) denote an \\(\\epsilon/\\kappa\\)-cover for \\(\\mathcal{F}\\). Then \\(\\phi \\circ \\mathcal{C}\\) is an \\(\\epsilon\\)-cover of \\(\\phi \\circ \\mathcal{F}\\):\n\\[\n\\rho(\\phi \\circ f', \\phi \\circ f) = \\sqrt{\\frac{1}{n} \\sum (\\phi(f'(z_i)) - \\phi(f(z_i)))^2} \\le \\sqrt{\\frac{1}{n} \\cdot \\kappa^2 \\sum(f'(z_i) - f(z_i))^2} \\le \\kappa \\cdot \\frac{\\epsilon}{\\kappa} = \\epsilon.\n\\]"}]}}, {"local_id": "T2", "statement": "\\[\n\\mathbb{E}_{z_1, \\dots, z_n \\overset{\\text{iid}}{\\sim} P} \\left[ \\sup_{f\\in F} \\left[ \\frac{1}{n} \\sum^n_{i=1} f(z_i) - \\mathbb{E}_{z\\sim P} [f(z)] \\right]\\right] \\leq 2 R_n(\\mathcal{F}).\n\\]", "proof": {"steps": [{"n": 1, "claim": "We use a technique called symmetrization. We first fix \\(z_1, \\dots, z_n\\) and draw \\(z_1', \\dots, z_n' \\overset{\\text{iid}}{\\sim} P\\)."}, {"n": 2, "claim": "Then we can rewrite the term in the expectation on the LHS:"}, {"n": 3, "claim": "\\[\n\\sup_{f\\in\\mathcal{F}}\\left(\\frac{1}{n}\\sum^n_{i=1}f(z_i)-\\mathbb{E}[f]\\right)=\\sup_{f\\in\\mathcal{F}}\\left(\\frac{1}{n}\\sum^n_{i=1}f(z_i)-\\mathbb{E}_{z_1',\\dots,z_n'}\\left[\\frac{1}{n}\\sum^n_{i=1}f(z_i')\\right]\\right)\n=\\sup_{f\\in\\mathcal{F}}\\left(\\mathbb{E}_{z_1',\\dots,z_n'}\\left[\\frac{1}{n}\\sum^n_{i=1}f(z_i)-\\frac{1}{n}\\sum^n_{i=1}f(z_i')\\right]\\right)\n\\leq\\mathbb{E}_{z_1',\\dots,z_n'}\\left[\\sup_{f\\in\\mathcal{F}}\\left(\\frac{1}{n}\\sum^n_{i=1}f(z_i)-\\frac{1}{n}\\sum^n_{i=1}f(z_i')\\right)\\right].\n\\]\n\nThe last inequality is because in general,\n\\[\n\\sup_u \\left(\\mathbb{E}_v[g(u,v)]\\right) \\leq \\mathbb{E}_v \\left[\\sup_u (g(u,v))\\right]\n\\]\nsince the \\(\\sup\\) over \\(u\\) becomes vacuous after we replace \\(u\\) with \\(u'\\)."}, {"n": 4, "claim": "Taking the expectation over \\(z_1, \\dots, z_n\\): \\[\n\\mathbb{E}_{z_1,\\dots,z_n}\\left[\\sup_{f\\in\\mathcal{F}}\\left(\\frac{1}{n}\\sum^n_{i=1}f(z_i)-\\mathbb{E}[f]\\right)\\right]\n\\leq\\mathbb{E}_{z_i,z_i'}\\left[\\sup_{f\\in\\mathcal{F}}\\left(\\frac{1}{n}\\sum^n_{i=1}(f(z_i)-f(z_i'))\\right)\\right]\n=\\mathbb{E}_{z_i,z_i'}\\left[\\mathbb{E}_{\\sigma_i}\\left[\\sup_{f\\in\\mathcal{F}}\\left(\\frac{1}{n}\\sum^n_{i=1}\\sigma_i\\left(f(z_i)-f(z_i')\\right)\\right)\\right]\\right]\n\\leq\\mathbb{E}_{z_i,z_i',\\sigma_i}\\left[\\sup_{f\\in\\mathcal{F}}\\left(\\frac{1}{n}\\sum^n_{i=1}\\sigma_if(z_i)\\right)+\\sup_{f\\in\\mathcal{F}}\\left(\\frac{1}{n}\\sum^n_{i=1}-\\sigma_if(z_i')\\right)\\right]\n=2R_n(\\mathcal{F}),\n\\]\nwhere the second equality is because \\(\\sigma_i(f(z_i) - f(z_i')) \\stackrel{d}{=} f(z_i) - f(z_i')\\) since \\(f(z_i) - f(z_i')\\) has a symmetric distribution."}, {"n": 5, "claim": "The last equality holds since \\(-\\sigma_i \\overset{d}{=} \\sigma_i\\) and \\(z_i, z_i'\\) are drawn iid from the same distribution."}]}}, {"local_id": "T3", "statement": "Let \\(B, \\epsilon > 0\\), and let \\(S = \\{x \\in \\mathbb{R}^p : \\|x\\|_2 \\leq B\\}\\). Then there exists an \\(\\epsilon\\)-cover of \\(S\\) with respect to the \\(\\ell_2\\)-norm with at most \\(\\max\\!\\left(\\left(\\dfrac{3B\\sqrt{p}}{\\epsilon}\\right)^p,\\, 1\\right)\\) elements.", "proof": {"steps": [{"n": 1, "claim": "If \\(\\epsilon > B\\sqrt{p}\\), then \\(S\\) is trivially contained in the ball of radius \\(\\epsilon\\) centered at the origin, so the \\(\\epsilon\\)-cover has size 1."}, {"n": 2, "claim": "Assume \\(\\epsilon \\leq B\\sqrt{p}\\). Set\n\\[\nC = \\left\\{x \\in S : x_i = k_i \\frac{\\epsilon}{\\sqrt{p}},\\; k_i \\in \\mathbb{Z},\\; |k_i| \\leq \\frac{B\\sqrt{p}}{\\epsilon}\\right\\},\n\\]\ni.e., \\(C\\) is the set of grid points in \\(\\mathbb{R}^p\\) of width \\(\\tfrac{\\epsilon}{\\sqrt{p}}\\) that are contained in \\(S\\)."}, {"n": 3, "claim": "We claim \\(C\\) is an \\(\\epsilon\\)-cover of \\(S\\) with respect to the \\(\\ell_2\\)-norm: for every \\(x \\in S\\) there exists a grid point \\(x' \\in C\\) with \\(|x_i - x_i'| \\leq \\tfrac{\\epsilon}{\\sqrt{p}}\\) for each \\(i\\), so\n\\[\n\\|x - x'\\|_2 = \\sqrt{\\sum_{i=1}^p |x_i - x_i'|^2} \\leq \\sqrt{p \\cdot \\frac{\\epsilon^2}{p}} = \\epsilon.\n\\]\nBounding the size of \\(C\\): each \\(k_i\\) has at most \\(2\\tfrac{B\\sqrt{p}}{\\epsilon} + 1\\) choices, so\n\\[\n|C| \\leq \\left(\\frac{2B\\sqrt{p}}{\\epsilon} + 1\\right)^p \\leq \\left(\\frac{3B\\sqrt{p}}{\\epsilon}\\right)^p.\n\\]"}]}}, {"local_id": "T4", "statement": "Suppose for all \\(f \\in \\mathcal{F}\\), \\(0 \\leq f(z) \\leq 1\\). Then, with probability at least \\(1-\\delta\\),\n\\[\n\\sup_{f\\in \\mathcal{F}} \\left[ \\frac{1}{n} \\sum^n_{i=1} f(z_i) - \\mathbb{E}[f(z)] \\right] \\leq 2 R_S(\\mathcal{F}) + 3\\sqrt{\\frac{\\log{(2/\\delta)}}{2n}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Define\n\\[\ng(z_1, \\dots, z_n) \\triangleq \\sup_{f\\in F} \\left[ \\frac{1}{n} \\sum^{n}_{i=1} f(z_i) - \\mathbb{E}[f(z)]\\right].\n\\]\n\nWe prove the theorem in 4 steps."}, {"n": 2, "claim": "Step 1: We bound \\(g\\) using McDiarmid's Inequality."}, {"n": 3, "claim": "To use McDiarmid's Inequality, we check that the bounded difference condition holds:"}, {"n": 4, "claim": "\\[\ng(z_1,\\dots,z_n)-g(z_1,\\dots,z_i',\\dots,z_n)\n\\leq\\sup_{f\\in\\mathcal{F}}\\left[\\frac{1}{n}\\sum^{n}_{j=1}f(z_j)-\\mathbb{E}[f(z)]\\right]-\\sup_{f\\in\\mathcal{F}}\\left[\\left(\\frac{1}{n}\\sum^{n}_{j=1,j\\neq i}f(z_j)\\right)+\\frac{f(z_i')}{n}-\\mathbb{E}[f(z)]\\right]\n\\leq\\sup_{f\\in\\mathcal{F}}\\left[\\frac{1}{n}\\sum^{n}_{j=1}f(z_j)-\\left(\\frac{1}{n}\\sum^{n}_{j=1,j\\neq i}f(z_j)\\right)-\\frac{f(z_i')}{n}\\right]\n=\\sup_{f\\in\\mathcal{F}}\\left[\\frac{1}{n}\\left(f(z_i)-f(z_i')\\right)\\right]\n\\leq\\frac{1}{n}.\n\\]\nThe first inequality holds because in general \\(\\sup_f A(f) - \\sup_f B(f) \\leq \\sup_f [A(f) - B(f)]\\), and the last inequality holds since \\(f\\) is bounded by \\([0,1]\\)."}, {"n": 5, "claim": "We can thus apply McDiarmid's Inequality with parameters \\(c_1 = \\dots = c_n = 1/n\\):"}, {"n": 6, "claim": "\\[\n\\mathbb{P}\\left[g(z_1,\\dots,z_n)\\geq\\mathbb{E}_{z_1,\\dots,z_n\\overset{\\text{iid}}{\\sim}P}[g]+\\epsilon\\right]\\leq\\exp\\!\\left(\\frac{-2\\epsilon^2}{\\sum^n_{i=1}c_i^2}\\right)=\\exp(-2n\\epsilon^2).\n\\]"}, {"n": 7, "claim": "Step 2: We apply the referenced theorem to get\n\\[\n\\mathbb{E}_{z_1,\\dots, z_n \\overset{\\text{iid}}{\\sim} P}[g] \\leq 2R_n(\\mathcal{F}).\n\\]"}, {"n": 8, "claim": "Step 3: Define\n\\[\n\\tilde{g}(z_1, \\dots, z_n) = R_S(\\mathcal{F}) \\triangleq \\mathbb{E}_{\\sigma_i}\\left[\\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i)\\right].\n\\]\n\nUsing a similar argument to that of Step 1, \\(\\tilde{g}\\) satisfies the bounded difference condition:"}, {"n": 9, "claim": "\\[\n\\tilde{g}(z_1, \\dots, z_n) - \\tilde{g}(z_1, \\dots, z_i', \\dots, z_n)\n\\leq \\mathbb{E}_{\\sigma_i} \\left[\\sup_{f\\in F} \\left[ \\frac{1}{n} \\sum^{n}_{j=1} \\sigma_j f(z_j) \\right] - \\sup_{f\\in F} \\left[ \\left(\\frac{1}{n} \\sum^{n}_{j=1, j \\neq i} \\sigma_j f(z_j)\\right) + \\frac{1}{n} \\sigma_if(z_i')\\right]\\right]\n\\leq \\mathbb{E}_{\\sigma_i}\\left[\\sup_{f\\in F} \\left(\\frac{1}{n} \\sigma_i(f(z_i) - f(z_i'))\\right)\\right]\n\\leq \\frac{1}{n},\n\\]\nsince the term inside the \\(\\sup\\) is always upper bounded by 1."}, {"n": 10, "claim": "We can thus apply McDiarmid's Inequality with parameters \\(c_1 = \\dots = c_n = 1/n\\):"}, {"n": 11, "claim": "\\[\n\\mathbb{P}\\left[ \\tilde{g} - \\mathbb{E}[\\tilde{g}] \\geq \\epsilon \\right] \\leq \\exp(-2n \\epsilon^2), \\quad\\text{and}\\quad \\mathbb{P}\\left[ \\tilde{g} - \\mathbb{E}[\\tilde{g}] \\leq -\\epsilon \\right] \\leq \\exp(-2n \\epsilon^2).\n\\]"}, {"n": 12, "claim": "Step 4: We set \\(\\delta\\) such that \\(\\exp(-2n \\epsilon^2) = \\delta/2\\), implying \\(\\epsilon = \\sqrt{\\frac{\\log(2/\\delta)}{2n}}\\)."}, {"n": 13, "claim": "Then, with probability \\(\\geq 1 - \\delta\\),\n\\[\n\\sup_{f\\in \\mathcal{F}} \\left[ \\frac{1}{n} \\sum^n_{i=1} f(z_i) - \\mathbb{E}[f]\\right] = g \\leq \\mathbb{E}[g] + \\epsilon \\leq 2R_n(\\mathcal{F}) + \\epsilon \\leq 2(R_S(\\mathcal{F}) + \\epsilon) + \\epsilon = 2R_S(\\mathcal{F}) + 3\\epsilon,\n\\]\nas required."}]}}, {"local_id": "T5", "statement": "Suppose \\(f : \\mathbb{R}^n \\to \\mathbb{R}\\) satisfies the bounded difference condition: there exist constants \\(c_1, \\ldots, c_n \\in \\mathbb{R}\\) such that for all real numbers \\(x_1, \\ldots, x_n\\) and \\(x_i'\\),\n\\[\n|f(x_1, \\ldots, x_n) - f(x_1, \\ldots, x_{i-1}, x_i', x_{i+1}, \\ldots, x_n)| \\leq c_i.\n\\]\nThen, for any independent random variables \\(X_1, \\ldots, X_n\\),\n\\[\n\\mathbb{P}\\left[f(X_1, \\ldots, X_n) - \\mathbb{E}[f(X_1, \\ldots, X_n)] \\geq t\\right] \\leq \\exp\\left(-\\frac{2t^2}{\\sum_{i=1}^n c_i^2}\\right).\n\\]\nMoreover, \\(f(X_1, \\ldots, X_n)\\) is \\(O\\!\\left(\\sqrt{\\sum_{i=1}^n c_i^2}\\right)\\)-sub-Gaussian.", "proof": {"steps": [{"n": 1, "claim": "The idea is to take \\(f(X_1,\\dots,X_n) - \\mathbb{E}[f(X_1,\\dots,X_n)]\\) and break it into manageable components by conditioning on portions of the sample."}, {"n": 2, "claim": "Define: \\[\nZ_0 = \\mathbb{E}[f(X_1,\\dots,X_n)] \\quad (\\text{constant}),\n\\]\n\\[\nZ_1 = \\mathbb{E}[f(X_1,\\dots,X_n) \\mid X_1] \\quad (\\text{a function of }X_1),\n\\]\n\\[\n\\vdots\n\\]\n\\[\nZ_i = \\mathbb{E}[f(X_1,\\dots,X_n) \\mid X_1,\\dots,X_i] \\quad (\\text{a function of }X_1,\\dots,X_i),\n\\]\n\\[\n\\vdots\n\\]\n\\[\nZ_n = f(X_1,\\dots,X_n).\n\\]\nBy the law of total expectation, \\(\\mathbb{E}[Z_i] = \\mathbb{E}[f(X_1,\\dots,X_n)] = Z_0\\) for all \\(i\\)."}, {"n": 3, "claim": "Hence \\(\\mathbb{E}[D_i] = 0\\), where \\(D_i = Z_i - Z_{i-1}\\)."}, {"n": 4, "claim": "We can write the quantity of interest as a telescoping sum:"}, {"n": 5, "claim": "\\[\nZ_n - Z_0 = \\sum_{i=1}^n D_i.\n\\]\nConditional on \\(X_1,\\dots,X_{i-1}\\), \\(D_i\\) is a bounded random variable."}, {"n": 6, "claim": "Let\n\\[\nA_i = \\inf_x \\mathbb{E}[f(X_1,\\dots,X_n) \\mid X_1,\\dots,X_{i-1}, X_i = x] - \\mathbb{E}[f(X_1,\\dots,X_n) \\mid X_1,\\dots,X_{i-1}],\n\\]\n\\[\nB_i = \\sup_x \\mathbb{E}[f(X_1,\\dots,X_n) \\mid X_1,\\dots,X_{i-1}, X_i = x] - \\mathbb{E}[f(X_1,\\dots,X_n) \\mid X_1,\\dots,X_{i-1}].\n\\]\nClearly \\(A_i \\leq D_i \\leq B_i\\)."}, {"n": 7, "claim": "By independence,\n\\[\nB_i - A_i \\leq \\sup_{x_{1:i-1}}\\sup_{x,x'} \\int \\left(f(x_1,\\dots,x_{i-1},x,x_{i+1},\\dots,x_n) - f(x_1,\\dots,x_{i-1},x',x_{i+1},\\dots,x_n)\\right)dP(x_{i+1},\\dots,x_n) \\leq c_i.\n\\]\nUsing this bound and the properties of conditional expectation,\n\\[\n\\mathbb{E}\\left[e^{\\lambda(Z_n - Z_0)}\\right] = \\mathbb{E}\\left[e^{\\lambda\\sum_{i=1}^n (Z_i - Z_{i-1})}\\right]\n\\]\n\\[\n= \\mathbb{E}\\left[\\mathbb{E}\\left[e^{\\lambda(Z_n - Z_{n-1})}\\,\\Big|\\,X_1,\\dots,X_{n-1}\\right]e^{\\lambda\\sum_{i=1}^{n-1}(Z_i - Z_{i-1})}\\right]\n\\]\n\\[\n\\leq e^{\\lambda^2 c_n^2/8}\\,\\mathbb{E}\\left[e^{\\lambda\\sum_{i=1}^{n-1}(Z_i - Z_{i-1})}\\right]\n\\]\n\\[\n\\leq \\cdots \\leq e^{\\lambda^2(\\sum_{i=1}^n c_i^2)/8}.\n\\]\nThe tail bound follows by the tail bound for sub-Gaussian random variables."}]}}], "target_local_id": "T4", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch4_theorem_4_25__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nEmpirical Rademacher complexity. Given a dataset \\(S = \\{z_1, \\dots, z_n\\}\\), the empirical Rademacher complexity is defined as\n\\[\nR_S(\\mathcal{F}) \\stackrel{\\Delta}{=} \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right].\n\\]\n\\(R_S(\\mathcal{F})\\) is a function of both the function class \\(\\mathcal{F}\\) and the dataset \\(S\\).\n\nCovering number. The covering number is defined as the minimum size of an \\(\\epsilon\\)-cover, or explicitly:\n\\[\nN(\\epsilon, \\mathcal{Q}, \\rho) \\stackrel{\\Delta}{=} (\\text{min size of } \\epsilon\\text{-cover of } \\mathcal{Q} \\text{ w.r.t. metric } \\rho).\n\\]\n\nEpsilon-cover. \\(\\mathcal{C}\\) is an \\(\\epsilon\\)-cover of \\(\\mathcal{Q}\\) with respect to metric \\(\\rho\\) if for all \\(v' \\in \\mathcal{Q}\\), there exists \\(v \\in \\mathcal{C}\\) such that \\(\\rho(v,v')\\le \\epsilon\\).\n\n\\(\\kappa\\)-Lipschitz functions. Let \\(\\kappa \\geq 0\\) and let \\(\\|\\cdot\\|\\) be a norm on the domain \\(D\\). A function \\(L : D \\to \\mathbb{R}\\) is said to be \\(\\kappa\\)-Lipschitz with respect to \\(\\|\\cdot\\|\\) if for all \\(\\theta, \\theta' \\in D\\),\n\\[\n|L(\\theta) - L(\\theta')| \\leq \\kappa\\,\\|\\theta - \\theta'\\|.\n\\]\n\n\\(\\epsilon\\)-cover. Let \\(\\epsilon > 0\\). An \\(\\epsilon\\)-cover of a set \\(S\\) with respect to a distance metric \\(\\rho\\) is a subset \\(C \\subseteq S\\) such that for all \\(x \\in S\\) there exists \\(x' \\in C\\) with \\(\\rho(x, x') \\leq \\epsilon\\), or equivalently,\n\\[\nS \\subseteq \\bigcup_{x \\in C} \\mathrm{Ball}(x, \\epsilon, \\rho), \\quad \\text{where} \\quad \\mathrm{Ball}(x, \\epsilon, \\rho) \\triangleq \\{x' : \\rho(x, x') \\leq \\epsilon\\}.\n\\]\n\nRademacher complexity. Let \\(\\mathcal{F}\\) be a family of functions mapping \\(Z \\mapsto \\mathbb{R}\\), and let \\(P\\) be a distribution over \\(Z\\). The (average) Rademacher complexity of \\(\\mathcal{F}\\) is defined as\n\\[\nR_n(F) \\triangleq \\mathbb{E}_{z_1, \\dots, z_n \\overset{\\text{iid}}{\\sim} P} \\left[ \\mathbb{E}_{\\sigma_1, \\dots, \\sigma_n \\overset{\\text{iid}}{\\sim}\\{ \\pm 1 \\}} \\left[ \\sup_{f\\in F} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right] \\right],\n\\]\nwhere \\(\\sigma_1, \\dots, \\sigma_n\\) are independent Rademacher random variables, i.e. each taking on the value of \\(1\\) or \\(-1\\) with probability \\(1/2\\).\n\nEvaluation-vector set Q. For a sample \\(S = \\{z_1, \\ldots, z_n\\}\\) and a function class \\(\\mathcal{F}\\) of real-valued functions on \\(Z\\), define\n\\[\n\\mathcal{Q} = \\{(f(z_1), \\ldots, f(z_n)) \\in \\mathbb{R}^n : f \\in \\mathcal{F}\\}.\n\\]\nThus \\(|\\mathcal{Q}|\\), membership \\(v \\in \\mathcal{Q}\\), and the Euclidean norm \\(\\|v\\|_2\\) are taken for these evaluation vectors.", "theorems": [{"local_id": "T1", "statement": "Let \\(\\mathcal{F}\\) be a family of functions \\(Z \\mapsto [-1,1]\\). Then\n\\[\nR_S(\\mathcal{F})\\le\\inf_{\\epsilon>0}\\left(\\epsilon+\\sqrt{\\frac{2\\log N(\\epsilon,\\mathcal{F},L_2(P_n))}{n}}\\right).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Fix any \\(\\epsilon > 0\\). Let \\(\\mathcal{C}\\) be the minimal \\(\\epsilon\\)-cover of \\(\\mathcal{Q}\\) with respect to the metric \\(\\rho(v,v') = \\frac{1}{\\sqrt{n}} \\|v-v'\\|_2\\). Note that \\(|\\mathcal{C}| = N(\\epsilon, \\mathcal{Q}, \\frac{1}{\\sqrt{n}} \\|\\cdot\\|_2) = N(\\epsilon, \\mathcal{F}, L_2(P_n))\\)."}, {"n": 2, "claim": "We aim to bound \\(R_S(\\mathcal{F}) = \\mathbb{E}_\\sigma[\\sup_{v \\in \\mathcal{Q}} \\frac{1}{n} \\langle v, \\sigma \\rangle]\\) by approximating \\(v\\) with \\(v' \\in \\mathcal{C}\\)."}, {"n": 3, "claim": "In particular, for every point \\(v \\in \\mathcal{Q}\\), choose \\(v' \\in \\mathcal{C}\\) such that \\(\\rho(v, v') \\leq \\epsilon\\) and \\(z\\) is small (specifically, \\(\\frac{1}{\\sqrt{n}} \\|z\\|_2 \\le \\epsilon\\))."}, {"n": 4, "claim": "This gives\n\\[\n\\frac{1}{n} \\langle v, \\sigma \\rangle = \\frac{1}{n} \\langle v', \\sigma \\rangle + \\frac{1}{n} \\langle v - v', \\sigma \\rangle \\le \\frac{1}{n} \\langle v', \\sigma \\rangle + \\frac{1}{n} \\|z\\|_2 \\|\\sigma\\|_2 \\le \\frac{1}{n} \\langle v', \\sigma \\rangle + \\epsilon,\n\\]\nwhere \\(z \\stackrel{\\Delta}{=} v - v'\\), using Cauchy-Schwarz and the fact that \\(\\|z\\|_2 \\le \\sqrt{n}\\epsilon\\) and \\(\\|\\sigma\\|_2 \\le \\sqrt{n}\\)."}, {"n": 5, "claim": "Taking the expectation of the supremum on both sides gives\n\\[\nR_S(\\mathcal{F}) = \\mathbb{E}_\\sigma \\left[\\sup_{v\\in \\mathcal{Q}} \\frac{1}{n} \\langle v,\\sigma \\rangle \\right] \\le \\mathbb{E}_\\sigma \\left[\\sup_{v'\\in \\mathcal{C}} \\left(\\frac{1}{n} \\langle v',\\sigma \\rangle + \\epsilon\\right)\\right] = \\epsilon + \\mathbb{E}_\\sigma \\left[\\sup_{v'\\in \\mathcal{C}} \\frac{1}{n} \\langle v',\\sigma \\rangle\\right] \\le \\epsilon + \\sqrt{ \\frac{2\\log |\\mathcal{C}|}{n} },\n\\]\nby Massart's lemma (the referenced result)."}, {"n": 6, "claim": "Since the argument holds for any \\(\\epsilon > 0\\), taking the infimum over all \\(\\epsilon\\) yields \\(R_S(\\mathcal{F})\\le\\inf_{\\epsilon>0}\\left(\\epsilon+\\sqrt{\\frac{2\\log N(\\epsilon,\\mathcal{F},L_2(P_n))}{n}}\\right)\\)."}]}}, {"local_id": "T2", "statement": "Suppose \\(\\phi\\) is \\(\\kappa\\)-Lipschitz, and \\(\\rho = L_2(P_n)\\). Then,\n\\[\n\\log N(\\epsilon, \\phi \\circ \\mathcal{F}, \\rho) \\le \\log N(\\epsilon / \\kappa, \\mathcal{F}, \\rho).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\mathcal{C}\\) denote an \\(\\epsilon/\\kappa\\)-cover for \\(\\mathcal{F}\\). Then \\(\\phi \\circ \\mathcal{C}\\) is an \\(\\epsilon\\)-cover of \\(\\phi \\circ \\mathcal{F}\\):\n\\[\n\\rho(\\phi \\circ f', \\phi \\circ f) = \\sqrt{\\frac{1}{n} \\sum (\\phi(f'(z_i)) - \\phi(f(z_i)))^2} \\le \\sqrt{\\frac{1}{n} \\cdot \\kappa^2 \\sum(f'(z_i) - f(z_i))^2} \\le \\kappa \\cdot \\frac{\\epsilon}{\\kappa} = \\epsilon.\n\\]"}]}}, {"local_id": "T3", "statement": "Let \\(\\mathcal{F}\\) be a collection of functions mapping \\(Z \\mapsto \\mathbb{R}\\) and define \\(\\mathcal{F}' = \\{f'(z) = f(z) + c_0 \\mid f\\in \\mathcal{F}\\}\\) for some \\(c_0\\in\\mathbb{R}\\). Then \\(R_S(\\mathcal{F}) = R_S(\\mathcal{F}')\\) and \\(R_n(\\mathcal{F}) = R_n(\\mathcal{F}')\\).", "proof": {"steps": [{"n": 1, "claim": "We prove that empirical Rademacher complexity is translation invariant."}, {"n": 2, "claim": "\\[\nR_S(\\mathcal{F}') = \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\sup_{f'\\in \\mathcal{F}'} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right]\n= \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i (f(z_i)+c_0) \\right]\n= \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\frac{1}{n} \\sum^n_{i=1} \\sigma_i c_0 + \\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right]\n= \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[\\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right] = R_S(\\mathcal{F}),\n\\]\nwhere the penultimate equality follows because \\(\\mathbb{E}_{\\sigma_1,\\dots,\\sigma_n} \\frac{1}{n}\\sum_{i=1}^n \\sigma_i c_0 = 0\\), since the \\(\\sigma_i\\)'s are Rademacher random variables."}]}}, {"local_id": "T4", "statement": "If \\(\\mathcal{F}\\) is a function class from \\(Z \\mapsto \\mathbb{R}\\), then\n\\[\nR_S(\\mathcal{F}) \\leq 12\\int_{0}^{\\infty}\\sqrt{\\frac{\\log N(\\epsilon, \\mathcal{F}, L_2(P_n))}{n}}\\,d\\epsilon.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\epsilon_0 = \\sup_{f\\in \\mathcal{F}} \\max_i |f(z_i)|\\), so that for all \\(v \\in \\mathcal{Q}\\),\n\\[\n\\epsilon_0 \\ge \\sqrt{\\frac{1}{n} \\sum_{i=1}^n f(z_i)^2} = \\sqrt{\\frac{1}{n} \\|v\\|_2^2}.\n\\]\n\nDefine \\(\\epsilon_j = 2^{-j}\\epsilon_0\\) and let \\(\\mathcal{C}_j\\) be an \\(\\epsilon_j\\)-cover of \\(\\mathcal{Q}\\)."}, {"n": 2, "claim": "Then \\(\\mathcal{C}_0\\) is the coarsest cover of \\(\\mathcal{Q}\\), and as \\(j\\) increases we obtain progressively more fine-grained covers \\(\\mathcal{C}_j\\)."}, {"n": 3, "claim": "We next use this sequence of covers to define a telescoping series that equals \\(v\\)."}, {"n": 4, "claim": "For \\(v \\in \\mathcal{Q}\\), let \\(u_i\\) denote the nearest neighbor of \\(v\\) in \\(\\mathcal{C}_i\\)."}, {"n": 5, "claim": "Note that by definition \\(\\rho(u, v_j) \\leq \\epsilon_j\\)."}, {"n": 6, "claim": "Taking \\(u_0 = 0\\), it follows that as \\(j \\to \\infty\\), \\(\\epsilon_j \\to 0\\) and \\(u_j \\to v\\)."}, {"n": 7, "claim": "Thus,\n\\[\nv = (u_1 - u_0) + (u_2 - u_1) + (u_3 - u_2) + \\cdots = \\sum_{i=1}^\\infty (u_i - u_{i-1}).\n\\]\n\nSubstituting this telescoping expansion into the Rademacher complexity,\n\\[\n\\mathbb{E}\\left[\\sup_{v \\in \\mathcal{Q}} \\frac{1}{n} \\langle v, \\sigma \\rangle\\right] = \\mathbb{E}\\left[\\sup_{v \\in \\mathcal{Q}} \\frac{1}{n} \\sum_{i=1}^\\infty \\langle u_i - u_{i-1}, \\sigma \\rangle\\right] \\le \\sum_{i=1}^\\infty \\mathbb{E}\\left[ \\sup_{u_i \\in \\mathcal{C}_i,\\, u_{i-1} \\in \\mathcal{C}_{i-1}} \\frac{1}{n}\\langle u_i - u_{i-1}, \\sigma \\rangle\\right].\n\\]\n\nEach term is a Rademacher complexity over the finite space \\(\\mathcal{C}_i \\times \\mathcal{C}_{i-1}\\), so we apply Massart's lemma (the referenced result)."}, {"n": 8, "claim": "We first bound \\(\\frac{1}{\\sqrt{n}} \\|u_i - u_{i-1}\\|_2\\):"}, {"n": 9, "claim": "\\[\n\\frac{1}{\\sqrt{n}} \\|u_i - u_{i-1}\\|_2 \\le \\frac{1}{\\sqrt{n}} \\left(\\|u_i - v\\|_2 + \\|u_{i-1} - v\\|_2\\right) \\le \\epsilon_i + \\epsilon_{i-1} = 3\\epsilon_i,\n\\]\nsince \\(\\epsilon_{i-1} = 2\\epsilon_i\\)."}, {"n": 10, "claim": "Applying Massart's lemma with \\(M = 3\\epsilon_i\\) and \\(|\\mathcal{Q}| = |\\mathcal{C}_i \\times \\mathcal{C}_{i-1}| \\leq |\\mathcal{C}_i| \\cdot |\\mathcal{C}_{i-1}|\\),\n\\[\n\\mathbb{E}\\left[\\sup_{u_i \\in \\mathcal{C}_i,\\, u_{i-1} \\in \\mathcal{C}_{i-1}} \\frac{1}{n} \\langle u_i - u_{i-1}, \\sigma \\rangle \\right] \\le \\sqrt{\\frac{2(3\\epsilon_i)^2 \\log(|\\mathcal{C}_i| \\cdot |\\mathcal{C}_{i-1}|)}{n}} \\le \\frac{6\\epsilon_i}{\\sqrt{n}}\\sqrt{\\log |\\mathcal{C}_i|},\n\\]\nusing \\(|\\mathcal{C}_i| \\ge |\\mathcal{C}_{i-1}|\\)."}, {"n": 11, "claim": "Thus,\n\\[\n\\mathbb{E}\\left[\\sup_{v \\in \\mathcal{Q}} \\frac{1}{n} \\langle v, \\sigma \\rangle \\right] \\le \\sum_{i=1}^\\infty \\frac{6\\epsilon_i}{\\sqrt{n}}\\sqrt{\\log N(\\epsilon_i, \\mathcal{F}, L_2(P_n))}.\n\\]\n\nTo relate this sum to the integral, note that \\(\\epsilon_i - \\epsilon_{i+1} = \\frac{\\epsilon_i}{2}\\) and that \\(\\log N(\\epsilon, \\mathcal{F}, L_2(P_n))\\) is monotone decreasing in \\(\\epsilon\\), so the right Riemann sum underestimates the integral:"}, {"n": 12, "claim": "\\[\n(\\epsilon_i - \\epsilon_{i+1}) \\sqrt{\\log |\\mathcal{C}_i|} \\leq \\int_{\\epsilon_{i+1}}^{\\epsilon_i} \\sqrt{\\log N(\\epsilon, \\mathcal{F}, L_2(P_n))}\\,d\\epsilon.\n\\]\nTherefore,\n\\[\n\\sum_{i=1}^\\infty \\frac{6\\epsilon_i}{\\sqrt{n}}\\sqrt{\\log N(\\epsilon_i, \\mathcal{F}, L_2(P_n))} = \\frac{12}{\\sqrt{n}} \\sum_{i=1}^\\infty (\\epsilon_i - \\epsilon_{i+1}) \\sqrt{\\log N(\\epsilon_i, \\mathcal{F}, L_2(P_n))} \\leq \\frac{12}{\\sqrt{n}} \\int_{0}^{\\epsilon_0} \\sqrt{\\log N(\\epsilon, \\mathcal{F}, L_2(P_n))}\\,d\\epsilon.\n\\]\nSince \\(\\log N(\\epsilon, \\mathcal{F}, L_2(P_n)) = 0\\) for all \\(\\epsilon > \\epsilon_0\\), we extend the upper limit to \\(\\infty\\) and obtain\n\\[\n\\mathbb{E}\\left[\\sup_{v \\in \\mathcal{Q}} \\frac{1}{n} \\langle v, \\sigma \\rangle \\right] \\le \\frac{12}{\\sqrt{n}} \\int_{0}^\\infty \\sqrt{\\log N(\\epsilon, \\mathcal{F}, L_2(P_n))}\\,d\\epsilon = 12\\int_{0}^{\\infty}\\sqrt{\\frac{\\log N(\\epsilon, \\mathcal{F}, L_2(P_n))}{n}}\\,d\\epsilon.\n\\]"}]}}, {"local_id": "T5", "statement": "Let \\(B, \\epsilon > 0\\), and let \\(S = \\{x \\in \\mathbb{R}^p : \\|x\\|_2 \\leq B\\}\\). Then there exists an \\(\\epsilon\\)-cover of \\(S\\) with respect to the \\(\\ell_2\\)-norm with at most \\(\\max\\!\\left(\\left(\\dfrac{3B\\sqrt{p}}{\\epsilon}\\right)^p,\\, 1\\right)\\) elements.", "proof": {"steps": [{"n": 1, "claim": "If \\(\\epsilon > B\\sqrt{p}\\), then \\(S\\) is trivially contained in the ball of radius \\(\\epsilon\\) centered at the origin, so the \\(\\epsilon\\)-cover has size 1."}, {"n": 2, "claim": "Assume \\(\\epsilon \\leq B\\sqrt{p}\\). Set\n\\[\nC = \\left\\{x \\in S : x_i = k_i \\frac{\\epsilon}{\\sqrt{p}},\\; k_i \\in \\mathbb{Z},\\; |k_i| \\leq \\frac{B\\sqrt{p}}{\\epsilon}\\right\\},\n\\]\ni.e., \\(C\\) is the set of grid points in \\(\\mathbb{R}^p\\) of width \\(\\tfrac{\\epsilon}{\\sqrt{p}}\\) that are contained in \\(S\\)."}, {"n": 3, "claim": "We claim \\(C\\) is an \\(\\epsilon\\)-cover of \\(S\\) with respect to the \\(\\ell_2\\)-norm: for every \\(x \\in S\\) there exists a grid point \\(x' \\in C\\) with \\(|x_i - x_i'| \\leq \\tfrac{\\epsilon}{\\sqrt{p}}\\) for each \\(i\\), so\n\\[\n\\|x - x'\\|_2 = \\sqrt{\\sum_{i=1}^p |x_i - x_i'|^2} \\leq \\sqrt{p \\cdot \\frac{\\epsilon^2}{p}} = \\epsilon.\n\\]\nBounding the size of \\(C\\): each \\(k_i\\) has at most \\(2\\tfrac{B\\sqrt{p}}{\\epsilon} + 1\\) choices, so\n\\[\n|C| \\leq \\left(\\frac{2B\\sqrt{p}}{\\epsilon} + 1\\right)^p \\leq \\left(\\frac{3B\\sqrt{p}}{\\epsilon}\\right)^p.\n\\]"}]}}], "target_local_id": "T1", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch4_theorem_4_26__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nEmpirical Rademacher complexity. Given a dataset \\(S = \\{z_1, \\dots, z_n\\}\\), the empirical Rademacher complexity is defined as\n\\[\nR_S(\\mathcal{F}) \\stackrel{\\Delta}{=} \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right].\n\\]\n\\(R_S(\\mathcal{F})\\) is a function of both the function class \\(\\mathcal{F}\\) and the dataset \\(S\\).\n\nCovering number. The covering number is defined as the minimum size of an \\(\\epsilon\\)-cover, or explicitly:\n\\[\nN(\\epsilon, \\mathcal{Q}, \\rho) \\stackrel{\\Delta}{=} (\\text{min size of } \\epsilon\\text{-cover of } \\mathcal{Q} \\text{ w.r.t. metric } \\rho).\n\\]\n\n\\(\\kappa\\)-Lipschitz functions. Let \\(\\kappa \\geq 0\\) and let \\(\\|\\cdot\\|\\) be a norm on the domain \\(D\\). A function \\(L : D \\to \\mathbb{R}\\) is said to be \\(\\kappa\\)-Lipschitz with respect to \\(\\|\\cdot\\|\\) if for all \\(\\theta, \\theta' \\in D\\),\n\\[\n|L(\\theta) - L(\\theta')| \\leq \\kappa\\,\\|\\theta - \\theta'\\|.\n\\]\n\nEpsilon-cover. \\(\\mathcal{C}\\) is an \\(\\epsilon\\)-cover of \\(\\mathcal{Q}\\) with respect to metric \\(\\rho\\) if for all \\(v' \\in \\mathcal{Q}\\), there exists \\(v \\in \\mathcal{C}\\) such that \\(\\rho(v,v')\\le \\epsilon\\).\n\n\\(\\epsilon\\)-cover. Let \\(\\epsilon > 0\\). An \\(\\epsilon\\)-cover of a set \\(S\\) with respect to a distance metric \\(\\rho\\) is a subset \\(C \\subseteq S\\) such that for all \\(x \\in S\\) there exists \\(x' \\in C\\) with \\(\\rho(x, x') \\leq \\epsilon\\), or equivalently,\n\\[\nS \\subseteq \\bigcup_{x \\in C} \\mathrm{Ball}(x, \\epsilon, \\rho), \\quad \\text{where} \\quad \\mathrm{Ball}(x, \\epsilon, \\rho) \\triangleq \\{x' : \\rho(x, x') \\leq \\epsilon\\}.\n\\]\n\nRademacher complexity. Let \\(\\mathcal{F}\\) be a family of functions mapping \\(Z \\mapsto \\mathbb{R}\\), and let \\(P\\) be a distribution over \\(Z\\). The (average) Rademacher complexity of \\(\\mathcal{F}\\) is defined as\n\\[\nR_n(F) \\triangleq \\mathbb{E}_{z_1, \\dots, z_n \\overset{\\text{iid}}{\\sim} P} \\left[ \\mathbb{E}_{\\sigma_1, \\dots, \\sigma_n \\overset{\\text{iid}}{\\sim}\\{ \\pm 1 \\}} \\left[ \\sup_{f\\in F} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right] \\right],\n\\]\nwhere \\(\\sigma_1, \\dots, \\sigma_n\\) are independent Rademacher random variables, i.e. each taking on the value of \\(1\\) or \\(-1\\) with probability \\(1/2\\).\n\nEvaluation-vector set Q. For a sample \\(S = \\{z_1, \\ldots, z_n\\}\\) and a function class \\(\\mathcal{F}\\) of real-valued functions on \\(Z\\), define\n\\[\n\\mathcal{Q} = \\{(f(z_1), \\ldots, f(z_n)) \\in \\mathbb{R}^n : f \\in \\mathcal{F}\\}.\n\\]\nThus \\(|\\mathcal{Q}|\\), membership \\(v \\in \\mathcal{Q}\\), and the Euclidean norm \\(\\|v\\|_2\\) are taken for these evaluation vectors.", "theorems": [{"local_id": "T1", "statement": "\\[\n\\mathbb{E}_{z_1, \\dots, z_n \\overset{\\text{iid}}{\\sim} P} \\left[ \\sup_{f\\in F} \\left[ \\frac{1}{n} \\sum^n_{i=1} f(z_i) - \\mathbb{E}_{z\\sim P} [f(z)] \\right]\\right] \\leq 2 R_n(\\mathcal{F}).\n\\]", "proof": {"steps": [{"n": 1, "claim": "We use a technique called symmetrization. We first fix \\(z_1, \\dots, z_n\\) and draw \\(z_1', \\dots, z_n' \\overset{\\text{iid}}{\\sim} P\\)."}, {"n": 2, "claim": "Then we can rewrite the term in the expectation on the LHS:"}, {"n": 3, "claim": "\\[\n\\sup_{f\\in\\mathcal{F}}\\left(\\frac{1}{n}\\sum^n_{i=1}f(z_i)-\\mathbb{E}[f]\\right)=\\sup_{f\\in\\mathcal{F}}\\left(\\frac{1}{n}\\sum^n_{i=1}f(z_i)-\\mathbb{E}_{z_1',\\dots,z_n'}\\left[\\frac{1}{n}\\sum^n_{i=1}f(z_i')\\right]\\right)\n=\\sup_{f\\in\\mathcal{F}}\\left(\\mathbb{E}_{z_1',\\dots,z_n'}\\left[\\frac{1}{n}\\sum^n_{i=1}f(z_i)-\\frac{1}{n}\\sum^n_{i=1}f(z_i')\\right]\\right)\n\\leq\\mathbb{E}_{z_1',\\dots,z_n'}\\left[\\sup_{f\\in\\mathcal{F}}\\left(\\frac{1}{n}\\sum^n_{i=1}f(z_i)-\\frac{1}{n}\\sum^n_{i=1}f(z_i')\\right)\\right].\n\\]\n\nThe last inequality is because in general,\n\\[\n\\sup_u \\left(\\mathbb{E}_v[g(u,v)]\\right) \\leq \\mathbb{E}_v \\left[\\sup_u (g(u,v))\\right]\n\\]\nsince the \\(\\sup\\) over \\(u\\) becomes vacuous after we replace \\(u\\) with \\(u'\\)."}, {"n": 4, "claim": "Taking the expectation over \\(z_1, \\dots, z_n\\): \\[\n\\mathbb{E}_{z_1,\\dots,z_n}\\left[\\sup_{f\\in\\mathcal{F}}\\left(\\frac{1}{n}\\sum^n_{i=1}f(z_i)-\\mathbb{E}[f]\\right)\\right]\n\\leq\\mathbb{E}_{z_i,z_i'}\\left[\\sup_{f\\in\\mathcal{F}}\\left(\\frac{1}{n}\\sum^n_{i=1}(f(z_i)-f(z_i'))\\right)\\right]\n=\\mathbb{E}_{z_i,z_i'}\\left[\\mathbb{E}_{\\sigma_i}\\left[\\sup_{f\\in\\mathcal{F}}\\left(\\frac{1}{n}\\sum^n_{i=1}\\sigma_i\\left(f(z_i)-f(z_i')\\right)\\right)\\right]\\right]\n\\leq\\mathbb{E}_{z_i,z_i',\\sigma_i}\\left[\\sup_{f\\in\\mathcal{F}}\\left(\\frac{1}{n}\\sum^n_{i=1}\\sigma_if(z_i)\\right)+\\sup_{f\\in\\mathcal{F}}\\left(\\frac{1}{n}\\sum^n_{i=1}-\\sigma_if(z_i')\\right)\\right]\n=2R_n(\\mathcal{F}),\n\\]\nwhere the second equality is because \\(\\sigma_i(f(z_i) - f(z_i')) \\stackrel{d}{=} f(z_i) - f(z_i')\\) since \\(f(z_i) - f(z_i')\\) has a symmetric distribution."}, {"n": 5, "claim": "The last equality holds since \\(-\\sigma_i \\overset{d}{=} \\sigma_i\\) and \\(z_i, z_i'\\) are drawn iid from the same distribution."}]}}, {"local_id": "T2", "statement": "Let \\(\\mathcal{F}\\) be a collection of functions mapping \\(Z \\mapsto \\mathbb{R}\\) and define \\(\\mathcal{F}' = \\{f'(z) = f(z) + c_0 \\mid f\\in \\mathcal{F}\\}\\) for some \\(c_0\\in\\mathbb{R}\\). Then \\(R_S(\\mathcal{F}) = R_S(\\mathcal{F}')\\) and \\(R_n(\\mathcal{F}) = R_n(\\mathcal{F}')\\).", "proof": {"steps": [{"n": 1, "claim": "We prove that empirical Rademacher complexity is translation invariant."}, {"n": 2, "claim": "\\[\nR_S(\\mathcal{F}') = \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\sup_{f'\\in \\mathcal{F}'} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right]\n= \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i (f(z_i)+c_0) \\right]\n= \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\frac{1}{n} \\sum^n_{i=1} \\sigma_i c_0 + \\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right]\n= \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[\\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right] = R_S(\\mathcal{F}),\n\\]\nwhere the penultimate equality follows because \\(\\mathbb{E}_{\\sigma_1,\\dots,\\sigma_n} \\frac{1}{n}\\sum_{i=1}^n \\sigma_i c_0 = 0\\), since the \\(\\sigma_i\\)'s are Rademacher random variables."}]}}, {"local_id": "T3", "statement": "Let \\(B, \\epsilon > 0\\), and let \\(S = \\{x \\in \\mathbb{R}^p : \\|x\\|_2 \\leq B\\}\\). Then there exists an \\(\\epsilon\\)-cover of \\(S\\) with respect to the \\(\\ell_2\\)-norm with at most \\(\\max\\!\\left(\\left(\\dfrac{3B\\sqrt{p}}{\\epsilon}\\right)^p,\\, 1\\right)\\) elements.", "proof": {"steps": [{"n": 1, "claim": "If \\(\\epsilon > B\\sqrt{p}\\), then \\(S\\) is trivially contained in the ball of radius \\(\\epsilon\\) centered at the origin, so the \\(\\epsilon\\)-cover has size 1."}, {"n": 2, "claim": "Assume \\(\\epsilon \\leq B\\sqrt{p}\\). Set\n\\[\nC = \\left\\{x \\in S : x_i = k_i \\frac{\\epsilon}{\\sqrt{p}},\\; k_i \\in \\mathbb{Z},\\; |k_i| \\leq \\frac{B\\sqrt{p}}{\\epsilon}\\right\\},\n\\]\ni.e., \\(C\\) is the set of grid points in \\(\\mathbb{R}^p\\) of width \\(\\tfrac{\\epsilon}{\\sqrt{p}}\\) that are contained in \\(S\\)."}, {"n": 3, "claim": "We claim \\(C\\) is an \\(\\epsilon\\)-cover of \\(S\\) with respect to the \\(\\ell_2\\)-norm: for every \\(x \\in S\\) there exists a grid point \\(x' \\in C\\) with \\(|x_i - x_i'| \\leq \\tfrac{\\epsilon}{\\sqrt{p}}\\) for each \\(i\\), so\n\\[\n\\|x - x'\\|_2 = \\sqrt{\\sum_{i=1}^p |x_i - x_i'|^2} \\leq \\sqrt{p \\cdot \\frac{\\epsilon^2}{p}} = \\epsilon.\n\\]\nBounding the size of \\(C\\): each \\(k_i\\) has at most \\(2\\tfrac{B\\sqrt{p}}{\\epsilon} + 1\\) choices, so\n\\[\n|C| \\leq \\left(\\frac{2B\\sqrt{p}}{\\epsilon} + 1\\right)^p \\leq \\left(\\frac{3B\\sqrt{p}}{\\epsilon}\\right)^p.\n\\]"}]}}, {"local_id": "T4", "statement": "Suppose \\(\\phi\\) is \\(\\kappa\\)-Lipschitz, and \\(\\rho = L_2(P_n)\\). Then,\n\\[\n\\log N(\\epsilon, \\phi \\circ \\mathcal{F}, \\rho) \\le \\log N(\\epsilon / \\kappa, \\mathcal{F}, \\rho).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\mathcal{C}\\) denote an \\(\\epsilon/\\kappa\\)-cover for \\(\\mathcal{F}\\). Then \\(\\phi \\circ \\mathcal{C}\\) is an \\(\\epsilon\\)-cover of \\(\\phi \\circ \\mathcal{F}\\):\n\\[\n\\rho(\\phi \\circ f', \\phi \\circ f) = \\sqrt{\\frac{1}{n} \\sum (\\phi(f'(z_i)) - \\phi(f(z_i)))^2} \\le \\sqrt{\\frac{1}{n} \\cdot \\kappa^2 \\sum(f'(z_i) - f(z_i))^2} \\le \\kappa \\cdot \\frac{\\epsilon}{\\kappa} = \\epsilon.\n\\]"}]}}, {"local_id": "T5", "statement": "If \\(\\mathcal{F}\\) is a function class from \\(Z \\mapsto \\mathbb{R}\\), then\n\\[\nR_S(\\mathcal{F}) \\leq 12\\int_{0}^{\\infty}\\sqrt{\\frac{\\log N(\\epsilon, \\mathcal{F}, L_2(P_n))}{n}}\\,d\\epsilon.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\epsilon_0 = \\sup_{f\\in \\mathcal{F}} \\max_i |f(z_i)|\\), so that for all \\(v \\in \\mathcal{Q}\\),\n\\[\n\\epsilon_0 \\ge \\sqrt{\\frac{1}{n} \\sum_{i=1}^n f(z_i)^2} = \\sqrt{\\frac{1}{n} \\|v\\|_2^2}.\n\\]\n\nDefine \\(\\epsilon_j = 2^{-j}\\epsilon_0\\) and let \\(\\mathcal{C}_j\\) be an \\(\\epsilon_j\\)-cover of \\(\\mathcal{Q}\\)."}, {"n": 2, "claim": "Then \\(\\mathcal{C}_0\\) is the coarsest cover of \\(\\mathcal{Q}\\), and as \\(j\\) increases we obtain progressively more fine-grained covers \\(\\mathcal{C}_j\\)."}, {"n": 3, "claim": "We next use this sequence of covers to define a telescoping series that equals \\(v\\)."}, {"n": 4, "claim": "For \\(v \\in \\mathcal{Q}\\), let \\(u_i\\) denote the nearest neighbor of \\(v\\) in \\(\\mathcal{C}_i\\)."}, {"n": 5, "claim": "Note that by definition \\(\\rho(u, v_j) \\leq \\epsilon_j\\)."}, {"n": 6, "claim": "Taking \\(u_0 = 0\\), it follows that as \\(j \\to \\infty\\), \\(\\epsilon_j \\to 0\\) and \\(u_j \\to v\\)."}, {"n": 7, "claim": "Thus,\n\\[\nv = (u_1 - u_0) + (u_2 - u_1) + (u_3 - u_2) + \\cdots = \\sum_{i=1}^\\infty (u_i - u_{i-1}).\n\\]\n\nSubstituting this telescoping expansion into the Rademacher complexity,\n\\[\n\\mathbb{E}\\left[\\sup_{v \\in \\mathcal{Q}} \\frac{1}{n} \\langle v, \\sigma \\rangle\\right] = \\mathbb{E}\\left[\\sup_{v \\in \\mathcal{Q}} \\frac{1}{n} \\sum_{i=1}^\\infty \\langle u_i - u_{i-1}, \\sigma \\rangle\\right] \\le \\sum_{i=1}^\\infty \\mathbb{E}\\left[ \\sup_{u_i \\in \\mathcal{C}_i,\\, u_{i-1} \\in \\mathcal{C}_{i-1}} \\frac{1}{n}\\langle u_i - u_{i-1}, \\sigma \\rangle\\right].\n\\]\n\nEach term is a Rademacher complexity over the finite space \\(\\mathcal{C}_i \\times \\mathcal{C}_{i-1}\\), so we apply Massart's lemma (the referenced result)."}, {"n": 8, "claim": "We first bound \\(\\frac{1}{\\sqrt{n}} \\|u_i - u_{i-1}\\|_2\\):"}, {"n": 9, "claim": "\\[\n\\frac{1}{\\sqrt{n}} \\|u_i - u_{i-1}\\|_2 \\le \\frac{1}{\\sqrt{n}} \\left(\\|u_i - v\\|_2 + \\|u_{i-1} - v\\|_2\\right) \\le \\epsilon_i + \\epsilon_{i-1} = 3\\epsilon_i,\n\\]\nsince \\(\\epsilon_{i-1} = 2\\epsilon_i\\)."}, {"n": 10, "claim": "Applying Massart's lemma with \\(M = 3\\epsilon_i\\) and \\(|\\mathcal{Q}| = |\\mathcal{C}_i \\times \\mathcal{C}_{i-1}| \\leq |\\mathcal{C}_i| \\cdot |\\mathcal{C}_{i-1}|\\),\n\\[\n\\mathbb{E}\\left[\\sup_{u_i \\in \\mathcal{C}_i,\\, u_{i-1} \\in \\mathcal{C}_{i-1}} \\frac{1}{n} \\langle u_i - u_{i-1}, \\sigma \\rangle \\right] \\le \\sqrt{\\frac{2(3\\epsilon_i)^2 \\log(|\\mathcal{C}_i| \\cdot |\\mathcal{C}_{i-1}|)}{n}} \\le \\frac{6\\epsilon_i}{\\sqrt{n}}\\sqrt{\\log |\\mathcal{C}_i|},\n\\]\nusing \\(|\\mathcal{C}_i| \\ge |\\mathcal{C}_{i-1}|\\)."}, {"n": 11, "claim": "Thus,\n\\[\n\\mathbb{E}\\left[\\sup_{v \\in \\mathcal{Q}} \\frac{1}{n} \\langle v, \\sigma \\rangle \\right] \\le \\sum_{i=1}^\\infty \\frac{6\\epsilon_i}{\\sqrt{n}}\\sqrt{\\log N(\\epsilon_i, \\mathcal{F}, L_2(P_n))}.\n\\]\n\nTo relate this sum to the integral, note that \\(\\epsilon_i - \\epsilon_{i+1} = \\frac{\\epsilon_i}{2}\\) and that \\(\\log N(\\epsilon, \\mathcal{F}, L_2(P_n))\\) is monotone decreasing in \\(\\epsilon\\), so the right Riemann sum underestimates the integral:"}, {"n": 12, "claim": "\\[\n(\\epsilon_i - \\epsilon_{i+1}) \\sqrt{\\log |\\mathcal{C}_i|} \\leq \\int_{\\epsilon_{i+1}}^{\\epsilon_i} \\sqrt{\\log N(\\epsilon, \\mathcal{F}, L_2(P_n))}\\,d\\epsilon.\n\\]\nTherefore,\n\\[\n\\sum_{i=1}^\\infty \\frac{6\\epsilon_i}{\\sqrt{n}}\\sqrt{\\log N(\\epsilon_i, \\mathcal{F}, L_2(P_n))} = \\frac{12}{\\sqrt{n}} \\sum_{i=1}^\\infty (\\epsilon_i - \\epsilon_{i+1}) \\sqrt{\\log N(\\epsilon_i, \\mathcal{F}, L_2(P_n))} \\leq \\frac{12}{\\sqrt{n}} \\int_{0}^{\\epsilon_0} \\sqrt{\\log N(\\epsilon, \\mathcal{F}, L_2(P_n))}\\,d\\epsilon.\n\\]\nSince \\(\\log N(\\epsilon, \\mathcal{F}, L_2(P_n)) = 0\\) for all \\(\\epsilon > \\epsilon_0\\), we extend the upper limit to \\(\\infty\\) and obtain\n\\[\n\\mathbb{E}\\left[\\sup_{v \\in \\mathcal{Q}} \\frac{1}{n} \\langle v, \\sigma \\rangle \\right] \\le \\frac{12}{\\sqrt{n}} \\int_{0}^\\infty \\sqrt{\\log N(\\epsilon, \\mathcal{F}, L_2(P_n))}\\,d\\epsilon = 12\\int_{0}^{\\infty}\\sqrt{\\frac{\\log N(\\epsilon, \\mathcal{F}, L_2(P_n))}{n}}\\,d\\epsilon.\n\\]"}]}}], "target_local_id": "T5", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch4_theorem_4_8__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\n\\(\\kappa\\)-Lipschitz functions. Let \\(\\kappa \\geq 0\\) and let \\(\\|\\cdot\\|\\) be a norm on the domain \\(D\\). A function \\(L : D \\to \\mathbb{R}\\) is said to be \\(\\kappa\\)-Lipschitz with respect to \\(\\|\\cdot\\|\\) if for all \\(\\theta, \\theta' \\in D\\),\n\\[\n|L(\\theta) - L(\\theta')| \\leq \\kappa\\,\\|\\theta - \\theta'\\|.\n\\]\n\nRisk and empirical-risk minimizer setup for finite hypothesis classes. Let \\(\\mathcal{H}\\) be a finite hypothesis class and let \\(\\ell\\) be a loss function with \\(0 \\leq \\ell((x,y),h) \\leq 1\\). For \\(h \\in \\mathcal{H}\\), define the population risk\n\\[\nL(h) = \\mathbb{E}_{(x,y) \\sim \\mathcal{D}}[\\ell((x,y),h)].\n\\]\nGiven \\(n\\) iid samples \\((x^{(1)},y^{(1)}), \\dots, (x^{(n)},y^{(n)})\\) from \\(\\mathcal{D}\\), define the empirical risk\n\\[\n\\widehat{L}(h) = \\frac{1}{n}\\sum_{i=1}^n \\ell((x^{(i)},y^{(i)}),h).\n\\]\nLet \\(\\hat{h} \\in \\arg\\min_{h \\in \\mathcal{H}} \\widehat{L}(h)\\) be an empirical-risk minimizer and let \\(h^* \\in \\arg\\min_{h \\in \\mathcal{H}} L(h)\\) be a population-risk minimizer.\n\nEpsilon-cover. \\(\\mathcal{C}\\) is an \\(\\epsilon\\)-cover of \\(\\mathcal{Q}\\) with respect to metric \\(\\rho\\) if for all \\(v' \\in \\mathcal{Q}\\), there exists \\(v \\in \\mathcal{C}\\) such that \\(\\rho(v,v')\\le \\epsilon\\).\n\nCovering number. The covering number is defined as the minimum size of an \\(\\epsilon\\)-cover, or explicitly:\n\\[\nN(\\epsilon, \\mathcal{Q}, \\rho) \\stackrel{\\Delta}{=} (\\text{min size of } \\epsilon\\text{-cover of } \\mathcal{Q} \\text{ w.r.t. metric } \\rho).\n\\]\n\nEmpirical Rademacher complexity. Given a dataset \\(S = \\{z_1, \\dots, z_n\\}\\), the empirical Rademacher complexity is defined as\n\\[\nR_S(\\mathcal{F}) \\stackrel{\\Delta}{=} \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right].\n\\]\n\\(R_S(\\mathcal{F})\\) is a function of both the function class \\(\\mathcal{F}\\) and the dataset \\(S\\).\n\nRademacher complexity. Let \\(\\mathcal{F}\\) be a family of functions mapping \\(Z \\mapsto \\mathbb{R}\\), and let \\(P\\) be a distribution over \\(Z\\). The (average) Rademacher complexity of \\(\\mathcal{F}\\) is defined as\n\\[\nR_n(F) \\triangleq \\mathbb{E}_{z_1, \\dots, z_n \\overset{\\text{iid}}{\\sim} P} \\left[ \\mathbb{E}_{\\sigma_1, \\dots, \\sigma_n \\overset{\\text{iid}}{\\sim}\\{ \\pm 1 \\}} \\left[ \\sup_{f\\in F} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right] \\right],\n\\]\nwhere \\(\\sigma_1, \\dots, \\sigma_n\\) are independent Rademacher random variables, i.e. each taking on the value of \\(1\\) or \\(-1\\) with probability \\(1/2\\).\n\n\\(\\epsilon\\)-cover. Let \\(\\epsilon > 0\\). An \\(\\epsilon\\)-cover of a set \\(S\\) with respect to a distance metric \\(\\rho\\) is a subset \\(C \\subseteq S\\) such that for all \\(x \\in S\\) there exists \\(x' \\in C\\) with \\(\\rho(x, x') \\leq \\epsilon\\), or equivalently,\n\\[\nS \\subseteq \\bigcup_{x \\in C} \\mathrm{Ball}(x, \\epsilon, \\rho), \\quad \\text{where} \\quad \\mathrm{Ball}(x, \\epsilon, \\rho) \\triangleq \\{x' : \\rho(x, x') \\leq \\epsilon\\}.\n\\]", "theorems": [{"local_id": "T1", "statement": "Let \\(B, \\epsilon > 0\\), and let \\(S = \\{x \\in \\mathbb{R}^p : \\|x\\|_2 \\leq B\\}\\). Then there exists an \\(\\epsilon\\)-cover of \\(S\\) with respect to the \\(\\ell_2\\)-norm with at most \\(\\max\\!\\left(\\left(\\dfrac{3B\\sqrt{p}}{\\epsilon}\\right)^p,\\, 1\\right)\\) elements.", "proof": {"steps": [{"n": 1, "claim": "If \\(\\epsilon > B\\sqrt{p}\\), then \\(S\\) is trivially contained in the ball of radius \\(\\epsilon\\) centered at the origin, so the \\(\\epsilon\\)-cover has size 1."}, {"n": 2, "claim": "Assume \\(\\epsilon \\leq B\\sqrt{p}\\). Set\n\\[\nC = \\left\\{x \\in S : x_i = k_i \\frac{\\epsilon}{\\sqrt{p}},\\; k_i \\in \\mathbb{Z},\\; |k_i| \\leq \\frac{B\\sqrt{p}}{\\epsilon}\\right\\},\n\\]\ni.e., \\(C\\) is the set of grid points in \\(\\mathbb{R}^p\\) of width \\(\\tfrac{\\epsilon}{\\sqrt{p}}\\) that are contained in \\(S\\)."}, {"n": 3, "claim": "We claim \\(C\\) is an \\(\\epsilon\\)-cover of \\(S\\) with respect to the \\(\\ell_2\\)-norm: for every \\(x \\in S\\) there exists a grid point \\(x' \\in C\\) with \\(|x_i - x_i'| \\leq \\tfrac{\\epsilon}{\\sqrt{p}}\\) for each \\(i\\), so\n\\[\n\\|x - x'\\|_2 = \\sqrt{\\sum_{i=1}^p |x_i - x_i'|^2} \\leq \\sqrt{p \\cdot \\frac{\\epsilon^2}{p}} = \\epsilon.\n\\]\nBounding the size of \\(C\\): each \\(k_i\\) has at most \\(2\\tfrac{B\\sqrt{p}}{\\epsilon} + 1\\) choices, so\n\\[\n|C| \\leq \\left(\\frac{2B\\sqrt{p}}{\\epsilon} + 1\\right)^p \\leq \\left(\\frac{3B\\sqrt{p}}{\\epsilon}\\right)^p.\n\\]"}]}}, {"local_id": "T2", "statement": "Let \\(\\mathcal{F}\\) be a collection of functions mapping \\(Z \\mapsto \\mathbb{R}\\) and define \\(\\mathcal{F}' = \\{f'(z) = f(z) + c_0 \\mid f\\in \\mathcal{F}\\}\\) for some \\(c_0\\in\\mathbb{R}\\). Then \\(R_S(\\mathcal{F}) = R_S(\\mathcal{F}')\\) and \\(R_n(\\mathcal{F}) = R_n(\\mathcal{F}')\\).", "proof": {"steps": [{"n": 1, "claim": "We prove that empirical Rademacher complexity is translation invariant."}, {"n": 2, "claim": "\\[\nR_S(\\mathcal{F}') = \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\sup_{f'\\in \\mathcal{F}'} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right]\n= \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i (f(z_i)+c_0) \\right]\n= \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\frac{1}{n} \\sum^n_{i=1} \\sigma_i c_0 + \\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right]\n= \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[\\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right] = R_S(\\mathcal{F}),\n\\]\nwhere the penultimate equality follows because \\(\\mathbb{E}_{\\sigma_1,\\dots,\\sigma_n} \\frac{1}{n}\\sum_{i=1}^n \\sigma_i c_0 = 0\\), since the \\(\\sigma_i\\)'s are Rademacher random variables."}]}}, {"local_id": "T3", "statement": "Setup. Throughout, the parameter \\(\\theta\\) ranges over \\(S=\\{\\theta\\in\\mathbb{R}^p:\\|\\theta\\|_2\\le B\\}\\) for some \\(B>0\\).\n\nSuppose \\(\\ell((x,y), \\theta) \\in [0,1]\\), and \\(\\ell((x,y), \\theta)\\) is \\(\\kappa\\)-Lipschitz in \\(\\theta\\) with respect to the \\(\\ell_2\\)-norm for all \\((x, y)\\). Then, with probability at least \\(1-O(\\exp(-\\Omega(p)))\\), we have\n\\[\n\\forall \\theta, \\quad |\\widehat{L}(\\theta)- L(\\theta)| \\leq O\\!\\left(\\sqrt{\\frac{p \\max(\\ln(\\kappa Bn), 1)}{n}}\\right).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Fix parameters \\(\\delta, \\epsilon>0\\) (we will specify their values later). Let \\(C\\) be the \\(\\epsilon\\)-cover of our parameter space \\(S\\) with respect to the \\(\\ell_2\\)-norm constructed in the referenced lemma. Define event \\(E = \\left\\{ \\forall \\theta \\in C, \\; |\\widehat{L}(\\theta) - L(\\theta)| \\le \\delta \\right\\}\\). By Theorem 4.1, we have \\(\\mathbb{P}(E) \\ge 1 - 2|C|\\exp(-2n\\delta^2)\\)."}, {"n": 2, "claim": "Now for any \\(\\theta \\in S\\), we can pick some \\(\\theta_0 \\in C\\) such that \\(\\|\\theta-\\theta_0\\|_2 \\le \\epsilon\\)."}, {"n": 3, "claim": "Since \\(L\\) and \\(\\widehat{L}\\) are \\(\\kappa\\)-Lipschitz functions (this follows from the Lipschitzness of \\(\\ell\\)), we have\n\\[\n|L(\\theta) - L(\\theta_0)| \\le \\kappa \\|\\theta-\\theta_0\\|_2 \\le \\kappa \\epsilon, \\quad\\text{and}\\quad |\\widehat{L}(\\theta) - \\widehat{L}(\\theta_0)| \\le \\kappa \\|\\theta-\\theta_0\\|_2 \\le \\kappa \\epsilon.\n\\]\n\nTherefore, conditional on \\(E\\), we have\n\\[\n|\\widehat{L}(\\theta) - L(\\theta)| \\le |\\widehat{L}(\\theta)-\\widehat{L}(\\theta_0)| + |\\widehat{L}(\\theta_0) - L(\\theta_0)| + |L(\\theta_0) - L(\\theta)| \\le 2 \\kappa\\epsilon+\\delta.\n\\]\n\nSet \\(\\epsilon = \\delta / (2 \\kappa)\\) so that conditional on \\(E\\),\n\\[\n|\\widehat{L}(\\theta) - L(\\theta)| \\le 2\\delta.\n\\]\n\nThe event \\(E\\) happens with probability \\(1 - 2|C|\\exp(-2n\\delta^2) = 1 - 2 \\exp(\\ln{|C|} - 2n\\delta^2)\\)."}, {"n": 4, "claim": "From the referenced remark, \\(\\ln{|C|} \\leq p \\ln{ (3B\\sqrt{p} / (\\delta / (2\\kappa))) }\\)."}, {"n": 5, "claim": "If we ignore the log term and assume \\(\\ln{|C|} \\leq p\\), then:"}, {"n": 6, "claim": "\\[\n2|C| \\exp(-2n\\delta^2) = 2\\exp(\\ln{|C|} - 2n\\delta^2) \\leq 2\\exp(p - 2p) = 2\\exp(-p).\n\\]\n\nSince we cannot drop the log term in \\(\\ln{|C|} \\leq p \\ln{ (3B\\sqrt{p} / (\\delta / (2\\kappa))) }\\), we set \\(\\delta = \\sqrt{\\frac{c_0 p \\max(1, \\ln{(\\kappa Bn)})}{n}}\\) with \\(c_0 = 36\\)."}, {"n": 7, "claim": "Then,\n\\[\n\\ln{|C|} - 2n\\delta^2 \\leq p \\ln\\!\\left(\\frac{6B \\kappa \\sqrt{p}}{\\delta}\\right) - 2n\\delta^2 \\leq p \\ln\\!\\left(\\frac{6B\\kappa \\sqrt{n}}{ \\sqrt{c_0 \\max(1, \\ln{(\\kappa Bn)})} }\\right) - 2n \\frac{c_0p}{n} \\ln(\\kappa Bn)\n\\leq p\\ln\\!\\left(B\\kappa \\sqrt{n}\\right) - 72 p \\ln(\\kappa Bn)\n\\leq p \\ln(B\\kappa n) - 72 p \\ln(B\\kappa n) \\leq -p,\n\\]\nsince \\(\\ln (B\\kappa n) \\geq 1\\) for large enough \\(n\\)."}, {"n": 8, "claim": "Therefore, with probability greater than \\(1 - O(e^{-p})\\), we have\n\\[\n|\\widehat{L}(\\theta) - L(\\theta)| \\leq 2\\delta = O\\!\\left(\\sqrt{\\frac{p}{n}\\max(1,\\ln(\\kappa Bn))}\\right).\n\\]"}]}}, {"local_id": "T4", "statement": "Suppose \\(\\phi\\) is \\(\\kappa\\)-Lipschitz, and \\(\\rho = L_2(P_n)\\). Then,\n\\[\n\\log N(\\epsilon, \\phi \\circ \\mathcal{F}, \\rho) \\le \\log N(\\epsilon / \\kappa, \\mathcal{F}, \\rho).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\mathcal{C}\\) denote an \\(\\epsilon/\\kappa\\)-cover for \\(\\mathcal{F}\\). Then \\(\\phi \\circ \\mathcal{C}\\) is an \\(\\epsilon\\)-cover of \\(\\phi \\circ \\mathcal{F}\\):\n\\[\n\\rho(\\phi \\circ f', \\phi \\circ f) = \\sqrt{\\frac{1}{n} \\sum (\\phi(f'(z_i)) - \\phi(f(z_i)))^2} \\le \\sqrt{\\frac{1}{n} \\cdot \\kappa^2 \\sum(f'(z_i) - f(z_i))^2} \\le \\kappa \\cdot \\frac{\\epsilon}{\\kappa} = \\epsilon.\n\\]"}]}}, {"local_id": "T5", "statement": "Suppose that the hypothesis class \\(\\mathcal{H}\\) is finite and the loss function \\(\\ell\\) is bounded in \\([0,1]\\), i.e. \\(0 \\leq \\ell((x,y),h) \\leq 1\\). Then for all \\(\\delta\\) with \\(0 < \\delta < \\tfrac{1}{2}\\), with probability at least \\(1 - \\delta\\),\n\\[\n|L(h) - \\widehat{L}(h)| \\leq \\sqrt{\\frac{\\ln|\\mathcal{H}| + \\ln(2/\\delta)}{2n}} \\qquad \\forall h \\in \\mathcal{H}.\n\\]\nAs a corollary,\n\\[\nL(\\hat{h}) - L(h^{*}) \\leq \\sqrt{\\frac{2(\\ln|\\mathcal{H}| + \\ln(2/\\delta))}{n}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "The proof proceeds in two steps: (1) use concentration inequalities to prove the bound for a fixed \\(h \\in \\mathcal{H}\\), then (2) apply a union bound across all \\(h \\in \\mathcal{H}\\). (Recall that if \\(E_1, \\dots, E_k\\) are a finite set of events, then \\(\\mathbb{P}(E_1 \\cup \\dots \\cup E_k) \\leq \\sum_{i=1}^k \\mathbb{P}(E_i)\\).)\n\nFix \\(\\epsilon > 0\\)."}, {"n": 2, "claim": "Applying Hoeffding's inequality to \\(\\ell((x^{(i)}, y^{(i)}), h)\\) with \\(a_i = 0\\), \\(b_i = 1\\):"}, {"n": 3, "claim": "\\[\n\\mathbb{P}\\left(|\\widehat{L}(h) - L(h)| \\geq \\epsilon\\right) \\leq 2\\exp\\left(-\\frac{2n^2\\epsilon^2}{n}\\right) = 2\\exp(-2n\\epsilon^2).\n\\]\nTo obtain a bound holding for all \\(h \\in \\mathcal{H}\\), apply the union bound with \\(E_h = \\{|\\widehat{L}(h) - L(h)| \\geq \\epsilon\\}\\):"}, {"n": 4, "claim": "\\[\n\\mathbb{P}\\left(\\exists h \\text{ s.t. } |\\widehat{L}(h) - L(h)| \\geq \\epsilon\\right) \\leq \\sum_{h \\in \\mathcal{H}} \\mathbb{P}\\left(|\\widehat{L}(h) - L(h)| \\geq \\epsilon\\right) \\leq 2|\\mathcal{H}|\\exp(-2n\\epsilon^2).\n\\]\nSetting \\(2|\\mathcal{H}|\\exp(-2n\\epsilon^2) = \\delta\\) gives\n\\[\n\\epsilon = \\sqrt{\\frac{\\ln|\\mathcal{H}| + \\ln(2/\\delta)}{2n}},\n\\]\nwhich proves the uniform bound."}, {"n": 5, "claim": "The excess risk bound follows by taking\n\\[\n\\epsilon = \\sqrt{\\frac{2(\\ln|\\mathcal{H}| + \\ln(2/\\delta))}{n}}\n\\]\nand using\n\\[\n\\mathbb{P}\\left(|L(\\hat{h}) - L(h^*)| \\geq \\epsilon\\right) \\leq \\mathbb{P}\\left(2\\sup_{h \\in \\mathcal{H}}|\\widehat{L}(h) - L(h)| \\geq \\epsilon\\right) \\leq 2|\\mathcal{H}|\\exp\\left(-\\frac{n\\epsilon^2}{2}\\right).\n\\]"}]}}], "target_local_id": "T3", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch6_theorem_6_16__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\ndefinition 6.6. Consider any known data-dependent offset function \\( h(w, S_n) \\). Define the uniform convergence complexity of a function class \\( \\mathcal{G} \\) with offset \\( h \\) as\n\\[\n\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\sup_{w\\in\\Omega}[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})].\\tag{6.2}\n\\]\nThis quantity measures the one-sided expected uniform convergence of function class \\(\\mathcal{G}\\) with offset function \\(h(w, S_n)\\).\nWe note that\n\\[\n\\epsilon_{n}(\\mathcal{G},\\mathcal{D})=\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D}),\\qquad \\text{w i t h} h=0.\n\\]\nWe have the following generalization of Theorem 6.2, with a similar proof.\n\ndefinition 6.11. Consider a function class \\(\\mathcal{G} = \\{\\phi(w, Z) : w \\in \\Omega\\}\\), and let \\(h\\) be an offset function of the following form\n\\[\nh(w,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{i=1}^{n}h(w,Z_{i}),\\quad h(w,z)=h_{0}(w)+h_{1}(w,z).\\tag{6.3}\n\\]\nGiven \\( S_{n}=\\{Z_{1},\\ldots,Z_{n}\\} \\), the (one-sided) empirical Rademacher complexity of \\( \\mathcal{G} \\) with offset \\( h \\) decomposition (6.3) is defined as\n\\[\nR^{h}(\\mathcal{G},\\mathcal{S}_{n})=\\mathbb{E}_{\\sigma}\\sup_{w\\in\\Omega}\\left[\\frac{1}{n}\\sum_{i=1}^{n}\\sigma_{i}[\\phi(w,Z_{i})+0. 5h_{1}(w,Z_{i})]-0. 5h(w,\\mathcal{S}_{n})\\right],\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}^{h}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R^{h}(\\mathcal{G},\\mathcal{S}_{n}).\n\\]\nWe note that the standard Rademacher complexity can be regarded as a special case of the offset Rademacher complexity with \\( h(\\cdot) = h_0(\\cdot) = h_1(\\cdot) = 0 \\):\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n})=R^{0}(\\mathcal{G},\\mathcal{S}_{n}),\\quad R_{n}(\\mathcal{G},\\mathcal{D})=R_{n}^{0}(\\mathcal{G},\\mathcal{D}).\n\\]\nIt should be pointed out that the decomposition of \\( h \\) in (6.3) may not be unique, and the offset Rademacher complexity relies on the specific decomposition used.\nAs shown in Example 6.10, we allow distribution dependent offset in the definition of uniform convergence complexity. We also allow distribution dependent offset in the definition of offset Rademacher complexity.\nThe following result shows that for some cases, offset Rademacher complexity can be obtained easily for some function classes.\n\ndefinition 6.17. Given a function \\( h(w, S_n) \\), we define\n\\[\n\\Delta_{n}h(w)=\\sup\\{n\\cdot|h(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n}^{\\prime})|: |\\mathcal{S}_{n}\\cap\\mathcal{S}_{n}^{\\prime}| =n-1\\}.\n\\]\n\ndefinition 3.8. We define\n\\[\n\\phi(w,z)=L\\left(f(w,x),y\\right)-L_{*}(x,y),\\tag{3.4}\n\\]\nfor \\( w \\in \\Omega \\) and \\( z = (x, y) \\in \\mathcal{Z} = \\mathcal{X} \\times \\mathcal{Y} \\), and a pre-chosen \\( L_{*}(x, y) \\) of \\( z = (x, y) \\) that does not depend on \\( w \\).\nFor training data \\( S_{n}=\\{Z_{i}=(X_{i},Y_{i}):i=1,\\ldots,n\\} \\), we define the training loss for \\( w\\in \\Omega \\) as\n\\[\n\\phi(w,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{i=1}^{n}\\phi(w,Z_{i}).\\tag{3.5}\n\\]\nMoreover, for a distribution \\(\\mathcal{D}\\) on \\(\\mathcal{Z}\\), we define the test loss for \\(w \\in \\Omega\\) as:\n\\[\n\\phi(w,\\mathcal{D})=\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(w,Z).\\tag{3.6}\n\\]\nSince \\( L_{*}(x,y) \\) does not depend on \\( w \\), the ERM solution with respect to the loss \\( L(\\cdot) \\) is equivalent to the ERM solution with respect to \\( \\phi (w,\\cdot) \\). Therefore with the simplified notations, the approximate ERM method (3.3) is a special case of the following method:\n\\[\n\\phi(\\hat{w},\\mathcal{S}_{n})\\leq\\inf_{w\\in\\Omega}\\phi(w,\\mathcal{S}_{n})+\\epsilon^{\\prime}.\\tag{3.7}\n\\]\nIn general, we may simply take \\( L_{*}(x,y) = 0 \\) in (3.4). However, for some applications, we may choose a non-zero \\( L_{*}(x,y) \\) so that\n\\[\nL(f(w,x),y)-L_{*}(x,y)\n\\]\nhas a small variance. For least squares loss, this can be achieved with \\( L_{*}(x,y) = L(f_{*}(x),y) \\) where \\( f_{*}(x) \\) is the optimal prediction function that minimizes the test loss as shown in Example 3.9. The smaller variance, combined with Bernstein's inequality, implies better generalization bound (see Section 3.6 for more details).\n\ndefinition 8.1. Consider a loss function \\(\\phi(f, z): \\mathcal{F} \\times \\mathcal{Z} \\to \\mathbb{R}\\), and a model family \\(\\{\\mathcal{A}(\\theta, S_n): \\Theta \\times \\mathcal{Z}^n \\to \\mathcal{F}, n \\geq 0\\}\\). Consider \\(N \\geq n \\geq 0\\), and iid dataset \\(S_n \\subset S_N \\sim \\mathcal{D}^N\\). A model selection algorithm \\(\\bar{\\mathcal{A}}\\) maps \\(S_N\\) to \\(\\hat{\\theta} = \\hat{\\theta}(S_N) \\in \\Theta\\),\nand then train a model \\(\\hat{f} = \\mathcal{A}(\\hat{\\theta}(\\mathcal{S}_N), \\mathcal{S}_n) = \\bar{\\mathcal{A}}(\\mathcal{S}_N)\\). It satisfies an \\(\\epsilon_{n,N}(\\cdot, \\cdot)\\) oracle inequality if there exists \\(\\epsilon_{n,N}(\\theta, \\delta)\\), such that for all \\(\\delta \\in (0,1)\\), with probability at least \\(1 - \\delta\\) over \\(\\mathcal{S}_N\\):\n\\[\n\\phi(\\mathcal{A}(\\hat{\\theta}(\\mathcal{S}_{N}),\\mathcal{S}_{n}),\\mathcal{D})\\leq\\inf_{\\theta\\in\\Theta}[\\mathbb{E}_{\\mathcal{S}_{n}}\\phi(\\mathcal{A}(\\theta,\\mathcal{S}_{n}),\\mathcal{D})+\\epsilon_{n,N}(\\theta,\\delta)].\n\\]\nMore generally, a learning algorithm \\(\\bar{\\mathcal{A}}: S_N \\to \\mathcal{F}\\) is \\(\\epsilon_{n,N}(\\cdot, \\cdot)\\) adaptive to the model family \\(\\{\\mathcal{A}(\\theta, \\cdot): \\theta \\in \\Theta\\}\\) if there exists \\(\\epsilon_{n,N}(\\theta, \\delta)\\), such that for all \\(\\delta \\in (0,1)\\), with probability at least \\(1 - \\delta\\) over \\(S_N\\):\n\\[\n\\phi(\\bar{\\mathcal{A}}(\\mathcal{S}_{N}),\\mathcal{D})\\leq\\inf_{\\theta\\in\\Theta}\\left[\\mathbb{E}_{\\mathcal{S}_{n}}\\phi(\\mathcal{A}(\\theta,\\mathcal{S}_{n}),\\mathcal{D})+\\epsilon_{n,N}(\\theta,\\delta)\\right].\n\\]\nNote that we have used a slightly different notation \\(\\phi(f, Z)\\), instead of \\(\\phi(w, Z)\\) as in earlier chapters, to emphasize that in model selection, the prediction function \\(f(w, \\cdot)\\) from different models can be compared, but the parameter \\(w\\) of the functions may not be comparable when different models have different functional forms such as decision trees versus neural networks. Without causing confusion, one may consider \\(f\\) to be the function represented by \\(w\\), so that \\(\\phi(f, Z) = L(f(X), Y)\\) and \\(\\phi(w, Z) = L(f(w, X), Y)\\). Similar to Definition 3.8, we use the simplified notation\n\\[\n\\phi(f,\\mathcal{D})=\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(f,Z),\\quad\\phi(f,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{Z\\in\\mathcal{S}_{n}}\\phi(f,Z).\n\\]\nThe concept adaptivity is more general than model selection, as the algorithm does not need to choose a specific model \\(\\mathcal{A}(\\theta,\\cdot)\\) indexed by \\(\\theta\\). For example, a commonly used method to achieve adaptivity is model averaging, where we take the average of several different model outputs \\(\\hat{f}_{\\theta_j} = \\mathcal{A}(\\theta_j,\\mathcal{S}_n)\\) as\n\\[\n\\hat{f}_{\\mathrm{avg}}(x)=\\frac{1}{k}\\sum_{j=1}^{k}\\hat{f}_{\\theta_{j}}(x).\n\\]\nIf different models have the same functional form, with a vector model parameter \\( w \\in \\mathbb{R}^d \\): \\( \\hat{f}_{\\theta_j}(x) = f(\\hat{w}_j, x) \\). Then one may also average the model parameters as follows:\n\\[\n\\hat{w}_{\\mathrm{avg}}=\\frac{1}{k}\\sum_{j=1}^{k}\\hat{w}_{j}.\n\\]\nIn general, the training data \\( S_{n} \\) is a subset of \\( S_{N} \\), and we allow \\( n < N \\) in order to analyze validation methods which are commonly used in practice. In addition to the large probability bounds in Definition 8.1, one may also derive bounds that hold for expectation over \\( S_{N} \\).\n\ndefinition 6.3. Given \\(S_{n}=\\{Z_{1},\\ldots,Z_{n}\\}\\), the (one-sided) empirical Rademacher complexity of \\(\\mathcal{G}\\) is defined as\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n}) = \\mathbb{E}_{\\sigma}\\sup_{w\\in \\Omega}\\frac{1}{n}\\sum_{i = 1}^{n}\\sigma_{i}\\phi (w,Z_{i}),\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R(\\mathcal{G},\\mathcal{S}_{n}).\n\\]\nThe following result shows that the quantity \\(\\epsilon_{n}(\\mathcal{G},\\mathcal{D})\\) can be upper bounded by Rademacher complexity. It follows that an average oracle inequality can be obtained using Rademacher complexity. The proof employs the symmetrization technique, which was also used in Chapter 4 to obtain uniform convergence bounds from empirical covering numbers.\n\nequation (6.1). The approximate regularized ERM method is to find \\( \\hat{w} \\in \\Omega \\) such that, for some \\( \\epsilon' \\geq 0 \\),\n\\[\n\\phi(\\hat{w},\\mathcal{S}_{n})+h(\\hat{w},\\mathcal{S}_{n})\\leq\\inf_{w\\in\\Omega}[\\phi(w,\\mathcal{S}_{n})+h(w,\\mathcal{S}_{n})]+\\epsilon'.\\tag{6.1}\n\\]\nWhen \\(h = 0\\), this reduces to the plain approximate ERM method.\n\nequation (8.1). Let \\(q(\\theta)\\geq 0\\) for \\(\\theta = 1,2,\\ldots\\) satisfy\n\\[\n\\sum_{\\theta=1}^{\\infty} q(\\theta) \\leq 1. \\tag{8.1}\n\\]\n\nequation 8.3. Consider the model selection algorithm that, given \\(\\mathcal{S}_{n}\\), chooses \\(\\hat{\\theta}=\\hat{\\theta}(\\mathcal{S}_{n})\\in\\Theta\\) and \\(\\hat{f}=\\mathcal{A}(\\hat{\\theta},\\mathcal{S}_{n})\\in\\mathcal{F}(\\hat{\\theta})\\) as a \\(\\tilde{\\epsilon}\\)-approximate minimizer of \\(\\phi(f,\\mathcal{S}_{n})+\\tilde{R}(\\theta,f,\\mathcal{S}_{n})\\) over \\(\\theta\\) and \\(f\\in\\mathcal{F}(\\theta)\\):\n\\[\n\\phi(\\hat{f},\\mathcal{S}_{n})+\\tilde{R}(\\hat{\\theta},\\hat{f},\\mathcal{S}_{n})\\leq\\inf_{\\theta,f\\in\\mathcal{F}(\\theta)}\\left[\\phi(f,\\mathcal{S}_{n})+\\tilde{R}(\\theta,f,\\mathcal{S}_{n})\\right]+\\tilde{\\epsilon}.\\tag{8.3}\n\\]\n\nexample 2.14. Consider a bounded random variable: \\( X_{1} \\in [\\alpha, \\beta] \\). Then \\( X_{1} \\) is sub-Gaussian with \\( b = (\\beta - \\alpha)^{2}/4 \\).\nThe tail probability inequality of Theorem 2.12 can also be expressed in a different form. Consider \\( \\delta \\in (0,1) \\) such that \\( \\delta = \\exp(-t) \\), we have \\( t = \\ln(1/\\delta) \\). This means that we can alternatively express the first bound of Theorem 2.12 as follows. With probability at least \\( 1 - \\delta \\), we have\n\\[\n\\bar{X}_{n}<\\mu+\\sqrt{\\frac{2 b\\ln(1/\\delta)}{n}}.\n\\]\nThis form is often preferred in the theoretical analysis of machine learning algorithms.", "theorems": [{"local_id": "T1", "statement": "Consider \\( n \\) independent random variables \\( X_{1},\\ldots,X_{n} \\), and a real-valued function \\( f(X_{1},\\ldots,X_{n}) \\) that satisfies the following inequality\n\\[\n\\sup_{x_{1},\\dots,x_{n},x_{i}^{\\prime}}|f(x_{1},\\dots,x_{n})-f(x_{1},\\dots,x_{i-1},x_{i}^{\\prime},x_{i+1},\\dots,x_{n})|\\leq c_{i}\n\\]\nfor all \\(1 \\leq i \\leq n\\). Then for all \\(\\epsilon > 0\\):\n\\[\n\\Pr\\left[f\\left(X_{1},\\dots,X_{n}\\right)\\geq\\mathbb{E}f\\left(X_{1},\\dots,X_{n}\\right)+\\epsilon\\right]\\leq\\exp\\left(\\frac{-2\\epsilon^{2}}{\\sum_{i=1}^{n}c_{i}^{2}}\\right).\n\\]\nSimilarly:\n\\[\n\\Pr\\left[f\\left(X_{1},\\dots,X_{n}\\right)\\leq\\mathbb{E}f\\left(X_{1},\\dots,X_{n}\\right)-\\epsilon\\right]\\leq\\exp\\left(\\frac{-2\\epsilon^{2}}{\\sum_{i=1}^{n}c_{i}^{2}}\\right).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\( X_{k}^{l}=\\{X_{k},\\ldots,X_{l}\\} \\). Consider \\( X_{1}^{n} \\), and for some \\( 1\\leq k\\leq n \\), we use the simplified notation \\( \\tilde{X}_1^n=\\{X_1,\\dots,X_{k-1},\\tilde{X}_k,X_{k+1},X_n\\} \\)."}, {"n": 2, "claim": "Then we have\n\\[\n\\left|\\mathbb{E}_{X_{k+1}^{n}}f\\left(X_{1}^{n}\\right)-\\mathbb{E}_{X_{k+1}^{n}}f\\left(\\tilde{X}_{1}^{n}\\right)\\right|\\leq c_{k}.\n\\]\nWe now consider \\(\\mathbb{E}_{X_{k + 1}^n}f(X_1^n)\\) as a random variable depending on \\(X_{k}\\), conditioned on \\(X_{1}^{k - 1}\\)."}, {"n": 3, "claim": "It follows from derivation of the Chernoff bound that we have the following logarithmic moment generating function estimate (see Example 2.14):"}, {"n": 4, "claim": "\\[\n\\ln\\mathbb{E}_{X_{k}}\\exp\\left[\\lambda\\mathbb{E}_{X_{k+1}^{n}}f\\left(X_{1}^{n}\\right)\\right]\\leq\\lambda\\mathbb{E}_{X_{k}^{n}}f\\left(X_{1}^{n}\\right)+\\lambda^{2}c_{k}^{2}/8.\n\\]\nNow we may exponentiate the above inequality, and take expectation with respect to \\( X_1^{k-1} \\) to obtain\n\\[\n\\mathbb{E}_{X_{1}^{k}}\\exp [\\lambda\\mathbb{E}_{X_{k+1}^{n}}f(X_{1}^{n})]\\leq\\mathbb{E}_{X_{1}^{k-1}}\\exp [\\lambda\\mathbb{E}_{X_{k}^{n}}f(X_{1}^{n})+\\lambda^{2}c_{k}^{2}/8].\n\\]\nBy taking logarithm, we obtain\n\\[\n\\ln\\mathbb{E}_{X_{1}^{k}}\\exp\\left[\\lambda\\mathbb{E}_{X_{k+1}^{n}}f\\left(X_{1}^{n}\\right)\\right]\\leq\\ln\\mathbb{E}_{X_{1}^{k-1}}\\exp\\left[\\lambda\\mathbb{E}_{X_{k}^{n}}f\\left(X_{1}^{n}\\right)\\right]+\\lambda^{2}c_{k}^{2}/8.\n\\]\nBy summing from \\( k = 1 \\) to \\( k = n \\), and canceling redundant terms, we obtain\n\\[\n\\ln\\mathbb{E}_{X_{1}^{n}}\\exp\\left[\\lambda f\\left(X_{1}^{n}\\right)\\right]\\leq\\lambda\\mathbb{E}_{X_{1}^{n}}f\\left(X_{1}^{n}\\right)+\\lambda^{2}\\sum_{k=1}^{n}c_{k}^{2}/8.\\tag{6.4}\n\\]\nLet\n\\[\n\\delta=\\Pr\\left[f\\left(X_{1}^{n}\\right)\\geq\\mathbb{E}_{X_{1}^{n}}f\\left(X_{1}^{n}\\right)+\\epsilon\\right].\n\\]\nUsing Markov's inequality, we have for all positive \\(\\lambda\\)\n\\[\n\\delta\\leq e^{-\\lambda\\left(\\mathbb{E}_{X_{1}^{n}}f\\left(X_{1}^{n}\\right)+\\epsilon\\right)}\\mathbb{E}_{X_{1}^{n}}e^{\\lambda f\\left(X_{1}^{n}\\right)}\\leq\\exp\\left[-\\lambda\\epsilon+\\frac{\\lambda^{2}}{8}\\sum_{k=1}^{n}c_{k}^{2}\\right].\n\\]\nSince \\(\\lambda > 0\\) is arbitrary, we conclude that\n\\[\n\\ln\\delta\\leq\\inf_{\\lambda\\geq 0}\\left[\\frac{\\lambda^{2}}{8}\\sum_{k=1}^{n}c_{k}^{2}-\\lambda\\epsilon\\right]= -\\frac{2\\epsilon^{2}}{\\sum_{k=1}^{n}c_{k}^{2}}.\n\\]\nThis implies the theorem."}]}}, {"local_id": "T2", "statement": "Assume that for some \\(M \\geq 0\\):\n\\[\n\\sup_{w\\in\\Omega}\\left[\\sup_{z,z^{\\prime}}[\\phi(w,z)-\\phi(w,z^{\\prime})]+\\Delta_{n}h(w)\\right]\\leq M.\n\\]\nThen the approximate ERM method (6.1) satisfies the following oracle inequality. With probability at least \\( 1 - \\delta - \\delta' \\):\n\\[\n\\begin{array}{l}\\phi(\\hat{w},\\mathcal{D})\\leq\\inf_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{D})+\\mathbb{E}_{\\mathcal{S}_{n}}h(w,\\mathcal{S}_{n})+\\Delta_{n}h(w)\\sqrt{\\frac{\\ln\\left(1/\\delta^{\\prime}\\right)}{2 n}}\\right]\\\\ +\\epsilon^{\\prime}+\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})+2 M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}.\\\\ \\end{array}\n\\]\nIf \\( h(\\cdot) \\) has the decomposition (6.3), then\n\\[\n\\begin{array}{l}\\phi(\\hat{w},\\mathcal{D})\\leq\\inf_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{D})+\\mathbb{E}_{\\mathcal{S}_{n}}h(w,\\mathcal{S}_{n})+\\Delta_{n}h(w)\\sqrt{\\frac{\\ln(1/\\delta^{\\prime})}{2 n}}\\right]\\\\ +\\epsilon^{\\prime}+2 R_{n}^{h}(\\mathcal{G},\\mathcal{D})+2 M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}.\\\\ \\end{array}\n\\]", "proof": {"steps": [{"n": 1, "claim": "Given any \\( w \\in \\Omega \\), from the Chernoff bound, we know that with probability \\( 1 - \\delta / 2 \\),\n\\[\n\\phi(w,\\mathcal{S}_{n})\\leq\\phi(w,\\mathcal{D})+M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}.\\tag{6.5}\n\\]\nMoreover, from McDiarmid's inequality, we know that with probability \\(1 - \\delta'\\),\n\\[\nh\\left(w,\\mathcal{S}_{n}\\right)\\leq\\mathbb{E}_{\\mathcal{S}_{n}}h\\left(w,\\mathcal{S}_{n}\\right)+\\Delta_{n}h(w)\\sqrt{\\frac{\\ln\\left(1/\\delta^{\\prime}\\right)}{2 n}}.\\tag{6.6}\n\\]\nTaking the union bound with the inequality of Corollary 6.19 at \\(\\delta /2\\), we obtain at probability \\(1 - \\delta -\\delta^{\\prime}\\)\n\\[\n\\begin{array}{l}\\phi(\\hat{w},\\mathcal{D})\\leq\\phi(\\hat{w},\\mathcal{S}_{n})+h(\\hat{w},\\mathcal{S}_{n})+\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})+M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}\\\\ \\leq\\phi(w,\\mathcal{S}_{n})+h(w,\\mathcal{S}_{n})+\\epsilon^{\\prime}+\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})+M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}\\\\ \\leq\\phi(w,\\mathcal{D})+h(w,\\mathcal{S}_{n})+\\epsilon^{\\prime}+\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})+2 M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}\\\\ \\leq\\phi(w,\\mathcal{D})+\\mathbb{E}_{\\mathcal{S}_{n}}h(w,\\mathcal{S}_{n})+\\Delta_{n}h(w)\\sqrt{\\frac{\\ln(1/\\delta^{\\prime})}{2 n}}\\\\ +\\epsilon^{\\prime}+\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})+2 M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}.\\\\ \\end{array}\n\\]\nIn the above derivation, the first inequality used Corollary 6.19."}, {"n": 2, "claim": "The second inequality used (6.1). The third inequality used (6.5)."}, {"n": 3, "claim": "The last inequality used (6.6). This proves the first desired bound."}, {"n": 4, "claim": "The second desired bound employs Theorem 6.14. \\(\\square\\)"}]}}, {"local_id": "T3", "statement": "Given any non-negative function \\( h(x) \\geq 0 \\), and a set \\( S \\subset \\mathbb{R} \\), we have\n\\[\n\\operatorname*{Pr}(\\bar{X}_{n}\\in S)\\leq\\frac{\\mathbb{E}h(\\bar{X}_{n})}{\\inf_{x\\in S}h(x)}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Since \\( h(x) \\) is non-negative, we have\n\\[\n\\mathbb{E}h\\left(\\bar{X}_{n}\\right)\\geq\\mathbb{E}_{\\bar{X}_{n}\\in S}h\\left(\\bar{X}_{n}\\right)\\geq\\mathbb{E}_{\\bar{X}_{n}\\in S}h_{S}=\\Pr\\left(\\bar{X}_{n}\\in S\\right)h_{S},\n\\]\nwhere \\( h_S = \\inf_{x \\in S} h(x) \\). This leads to the desired bound."}]}}, {"local_id": "T4", "statement": "Assume that for some \\(M \\geq 0\\):\n\\[\n\\sup_{w\\in\\Omega}\\left[\\sup_{z,z^{\\prime}}[\\phi(w,z)-\\phi(w,z^{\\prime})]+\\Delta_{n}h(w)\\right]\\leq M.\n\\]\nThen with probability at least \\(1 - \\delta\\): for all \\(w \\in \\Omega\\)\n\\[\n\\phi(w,\\mathcal{D})\\leq\\phi(w,\\mathcal{S}_{n})+h(w,\\mathcal{S}_{n})+\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})+M\\sqrt{\\frac{\\ln(1/\\delta)}{2 n}}.\n\\]\nMoreover, assume that the decomposition (6.3) holds, then with probability at least \\(1 - \\delta\\): for all \\(w \\in \\Omega\\),\n\\[\n\\phi(w,\\mathcal{D})\\leq\\phi(w,\\mathcal{S}_{n})+h(w,\\mathcal{S}_{n})+2 R_{n}^{h}(\\mathcal{G},\\mathcal{D})+M\\sqrt{\\frac{\\ln(1/\\delta)}{2 n}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Consider \\( \\mathcal{S}_n = \\{Z_1, \\ldots, Z_n\\} \\) and \\( \\mathcal{S}_n' = \\{Z_1, \\ldots, Z_{i-1}, Z_i', Z_{i+1}, \\ldots, Z_n\\} \\)."}, {"n": 2, "claim": "Let \\( f(\\mathcal{S}_n) = \\sup_{w \\in \\Omega} [\\phi(w, \\mathcal{D}) - \\phi(w, \\mathcal{S}_n) - h(w, \\mathcal{S}_n)] \\)."}, {"n": 3, "claim": "For simplicity, we assume that the sup can be achieved at \\( \\hat{w} \\) as\n\\[\n\\hat{w}=\\arg\\max_{w\\in\\Omega}[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})].\n\\]\nThen\n\\[\n\\begin{array}{l}f\\left(\\mathcal{S}_{n}\\right)-f\\left(\\mathcal{S}_{n}^{\\prime}\\right)\\\\ =\\left[\\phi(\\hat{w},\\mathcal{D})-\\phi(\\hat{w},\\mathcal{S}_{n})-h(\\hat{w},\\mathcal{S}_{n})\\right]-\\sup_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n}^{\\prime})-h(w,\\mathcal{S}_{n}^{\\prime})\\right]\\\\ \\leq\\left[\\phi(\\hat{w},\\mathcal{D})-\\phi(\\hat{w},\\mathcal{S}_{n})-h(\\hat{w},\\mathcal{S}_{n})\\right]-\\left[\\phi(\\hat{w},\\mathcal{D})-\\phi(\\hat{w},\\mathcal{S}_{n}^{\\prime})-h(\\hat{w},\\mathcal{S}_{n}^{\\prime})\\right]\\\\ \\leq\\frac{1}{n}[\\phi(\\hat{w},Z_{i}^{\\prime})-\\phi(\\hat{w},Z_{i})+\\Delta_{n}h(\\hat{w})]\\leq M/n.\\\\ \\end{array}\n\\]\nSimilarly, \\( f(\\mathcal{S}_n') - f(\\mathcal{S}_n) \\leq M / n \\)."}, {"n": 4, "claim": "Therefore we may take \\( c_i = M / n \\) in Theorem 6.16, which implies the first desired result."}, {"n": 5, "claim": "The second bound follows from the estimate \\( \\epsilon_n^h (\\mathcal{G},\\mathcal{D}) \\leq 2R_n^h (\\mathcal{G},\\mathcal{D}) \\) of Theorem 6.14."}]}}, {"local_id": "T5", "statement": "Consider offset function of (6.3). We have\n\\[\n\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})\\leq 2 R_{n}^{h}(\\mathcal{G},\\mathcal{D}).\n\\]\nConsequently, the approximate regularized ERM method of (6.1) satisfies\n\\[\n\\mathbb{E}_{\\mathcal{S}_{n}}\\phi(\\hat{w},\\mathcal{D})\\leq\\inf_{w\\in\\Omega}[\\phi(w,\\mathcal{D})+\\mathbb{E}_{\\mathcal{S}_{n}}h(w,\\mathcal{S}_{n})]+\\epsilon^{\\prime}+2 R_{n}^{h}(\\mathcal{G},\\mathcal{D}).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\mathcal{S}_n' = \\{Z_1', \\ldots, Z_n'\\} \\sim \\mathcal{D}^n\\) be \\(n\\) iid samples from \\(\\mathcal{D}\\) that are independent of \\(\\mathcal{S}_n\\)."}, {"n": 2, "claim": "We have\n\\[\n\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\sup_{w\\in\\Omega}[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})]\n\\]\n\\[\n=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\sup_{w\\in\\Omega}\\left[\\mathbb{E}_{\\mathcal{S}_{n}^{\\prime}\\sim\\mathcal{D}^{n}}\\phi(w,\\mathcal{S}_{n}^{\\prime})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})\\right]\n\\]\n\\[\n\\leq\\mathbb{E}_{\\left(\\mathcal{S}_{n},\\mathcal{S}_{n}^{\\prime}\\right)\\sim\\mathcal{D}^{2 n}}\\sup_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{S}_{n}^{\\prime})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})\\right]\n\\]\n\\[\n\\begin{array}{l}=\\mathbb{E}_{\\left(\\mathcal{S}_{n},\\mathcal{S}_{n}^{\\prime}\\right)\\sim\\mathcal{D}^{2n}}\\sup_{w\\in\\Omega}\\left[\\left[\\phi(w,\\mathcal{S}_{n}^{\\prime})+0. 5h_{1}(w,\\mathcal{S}_{n}^{\\prime})\\right]-0. 5h(w,\\mathcal{S}_{n}^{\\prime})\\right.\\\\ -\\left[\\phi(w,\\mathcal{S}_{n})+0. 5h_{1}(w,\\mathcal{S}_{n})\\right]-0. 5h(w,\\mathcal{S}_{n})]\\\\ \\end{array}\n\\]\n\\[\n\\begin{array}{l}\\stackrel{(a)}{=}\\mathbb{E}_{\\left(\\mathcal{S}_{n},\\mathcal{S}_{n}^{\\prime}\\right)\\sim\\mathcal{D}^{2n}}\\mathbb{E}_{\\sigma}\\sup_{w\\in\\Omega}\\frac{1}{n}\\sum_{i=1}^{n}\\left[\\sigma_{i}\\left(\\phi\\left(w,Z_{i}^{\\prime}\\right)+0. 5h_{1}\\left(w,Z_{i}^{\\prime}\\right)\\right)-0. 5h(w,\\mathcal{S}_{n}^{\\prime})\\right.\\\\ \\left.-\\sigma_{i}\\left(\\phi\\left(w,Z_{i}\\right)+0. 5h_{1}\\left(w,Z_{i}\\right)\\right)-0. 5h\\left(w,\\mathcal{S}_{n}\\right)\\right]\\\\ \\end{array}\n\\]\n\\[\n\\begin{array}{l}\\leq\\mathbb{E}_{\\left(\\mathcal{S}_{n},\\mathcal{S}_{n}^{\\prime}\\right)\\sim\\mathcal{D}^{2n}}\\mathbb{E}_{\\sigma}\\sup_{w\\in\\Omega}\\frac{1}{n}\\sum_{i=1}^{n}\\left[\\sigma_{i}\\left(\\phi\\left(w,Z_{i}^{\\prime}\\right)+0. 5h_{1}\\left(w,Z_{i}^{\\prime}\\right)\\right)-0. 5h\\left(w,\\mathcal{S}_{n}^{\\prime}\\right)\\right]\\\\ +\\mathbb{E}_{(\\mathcal{S}_{n},\\mathcal{S}_{n}^{\\prime})\\sim\\mathcal{D}^{2n}}\\mathbb{E}_{\\sigma}\\sup_{w\\in\\Omega}\\frac{1}{n}\\sum_{i=1}^{n}[-\\sigma_{i}(\\phi(w,Z_{i})+0. 5h_{1}(w,Z_{i})-0. 5h(w,\\mathcal{S}_{n})]\\\\ \\end{array}\n\\]\n\\[\n=\\mathbb{E}_{(\\mathcal{S}_{n},\\mathcal{S}_{n}^{\\prime})\\sim\\mathcal{D}^{2 n}}[R^{h}(\\mathcal{G},\\mathcal{S}_{n})+R^{h}(\\mathcal{G},\\mathcal{S}_{n}^{\\prime})]=2 R_{n}^{h}(\\mathcal{G},\\mathcal{D}).\n\\]\nIn the above derivation, \\((a)\\) used the fact that \\(\\sigma_{i}(\\phi (w,Z_{i}^{\\prime}) + 0.5h_{1}(w,Z_{i}^{\\prime})) - \\sigma_{i}(\\phi (w,Z_{i}) + 0.5h_{1}(w,Z_{i}))\\) and \\((\\phi (w,Z_i^{\\prime}) + 0.5h_1(w,Z_i^{\\prime})) - (\\phi (w,Z_i) + 0.5h_1(w,Z_i))\\) have the same distributions."}]}}], "target_local_id": "T1", "topic_domain": "Probability & Concentration", "topic_subtopic": "Functional & Supremum Concentration", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch7_theorem_7_2__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nFor \\(f:\\mathcal{X}\\to\\mathbb{R}\\) , the subgraph of \\(f\\) is\n\\[C_{f}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}:0\\leq t\\leq f(x)\\ o r\\ f(x)\\leq t\\leq 0\\}.\\]\nDefine class of sets \\(\\mathcal{C}=\\{C_{f}:f\\in\\mathcal{F}\\}\\).\n\ndefinition 12.1. If \\(\\mathcal{C}\\) is a VC class of sets, then F is VC-subgraph class of functions and, by definition, \\(V C(\\mathcal{F})=V C(\\mathcal{C})\\)\nNote that equivalent definition of \\(C_{f}\\) is\n\\[C_{f}^{\\prime}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}: |f(x)|\\geq|t|\\}.\\]\n\ndefinition 12.3. The ε-packing number, \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) , is the marimal cardinality of an E-separated set.\nNote that \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) is decreasing in ε.\n\ndefinition 12.2. Given \\(\\varepsilon > 0\\) and \\(f_{1},\\ldots,f_{N}\\in\\mathcal{F}\\) ， we say that \\(f_{1},\\ldots,f_{N}\\) are E-separated if \\(d(f_{i},f_{j})>\\varepsilon\\) for any \\(i \\neq j\\)\n\nDenote \\(\\triangle_{n}(\\mathcal{C})=\\operatorname*{sup}_{\\{x_{1},\\ldots,x_{n}\\}}\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})\\leq 2^{n}\\). We will see that for some classes, \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\) for \\(n \\leq V\\) and \\(\\triangle_{n}(\\mathcal{C})<2^{n}\\) for \\(n > V\\) for some constant \\(V\\).\n\ndefinition 8.1. If \\(V < \\infty\\) , then C is called a VC class. V is called VC dimension of C.\n\ndefinition 8.0. For any sample \\(\\{x_{1},\\ldots,x_{n}\\}\\), the ways that \\({\\mathcal{C}}\\) intersects with the sample are \\[\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in{\\mathcal{C}}\\}.\\] Let \\[\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})=\\mathrm{card}~\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in\\mathcal{C}\\},\\] the number of different subsets picked out by \\(C\\in{\\mathcal{C}}\\). If \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\), this means we can find \\(\\{x_{1},\\ldots,x_{n}\\}\\) such that \\({\\mathcal{C}}\\) can pick out any subset of it; in this case \\({\\mathcal{C}}\\) shatters \\(\\{x_{1},\\ldots,x_{n}\\}\\). Equivalently, a set \\(A\\) is shattered by \\({\\mathcal{C}}\\) when \\({\\mathcal{C}}\\) picks out every subset of \\(A\\).\n\nLemma 15.1. Let \\(\\xi, \\nu\\) - random variables. Assume that\n\\[\\mathbb{P}(\\nu \\geq t) \\leq \\Gamma e^{-\\gamma t}\\]\nwhere \\(\\Gamma \\geq 1\\), \\(t \\geq 0\\), and \\(\\gamma > 0\\). Furthermore, for all \\(a > 0\\) assume that\n\\[\\mathbb{E}\\phi(\\xi) \\leq \\mathbb{E}\\phi(\\nu)\\]\nwhere \\(\\phi(x) = (x - a)_+\\). Then\n\\[\\mathbb{P}(\\xi \\geq t) \\leq \\Gamma \\cdot e \\cdot e^{-\\gamma t}.\\]", "theorems": [{"local_id": "T1", "statement": "Setup. Let \\(X_1,\\dots,X_n\\) be i.i.d. random variables with common law \\(P\\) on a measurable space \\((\\mathcal{X},\\mathcal{A})\\), let \\(\\mathcal{C}\\subseteq\\mathcal{A}\\) be a class of measurable sets, and write \\(P(C)=\\mathbb{P}(X_1\\in C)\\). Let \\(X'_1,\\dots,X'_n\\) (the ghost sample) be an independent copy of \\(X_1,\\dots,X_n\\), i.e. i.i.d. \\(P\\) and independent of \\(X_1,\\dots,X_n\\). Suprema over \\(\\mathcal{C}\\) are assumed measurable; otherwise read \\(\\mathbb{P}\\) as outer probability.\n\nIf \\(t\\geq{\\sqrt{\\frac{2}{n}}}.\\) then\n\\[\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\right)\\leq 2\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\geq t/2\\right).\\]", "proof": {"steps": [{"n": 1, "claim": "Suppose the event\n\\[\\operatorname*{sup}_{C\\in{\\mathcal{C}}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\]\noccurs.Let \\(\\begin{array}{ r }{X=(X_{1},\\ldots,X_{n})\\in\\{\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\}}\\end{array}\\) ."}, {"n": 2, "claim": "Then\n\\[\\exists C_{X}{\\mathrm{~s u c h~t h a t~}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nFor a fixed C,\n\\[\\begin{array}{ r }{\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right|\\geq t/2\\right)=\\mathbb{P}\\left(\\left(\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}\\geq t^{2}/4\\right)}\\\\ {\\leq\\mathrm{~(b y~C h e b y s h e v~s~I n e q)~}\\frac{4\\mathbb{E}\\left(\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}}{t^{2}}}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i,j}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))(I(X_{j}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i=1}^{n}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))^{2}=\\frac{4 n\\mathbb{P}\\left(\\mathcal{C}\\right)\\left(1-\\mathbb{P}(\\mathcal{C})\\right)}{n^{2}t^{2}}\\leq\\frac{1}{n t^{2}}\\leq\\frac{1}{2}}\\end{array}\\]\nsince we chose \\(t\\geq\\sqrt{\\textstyle{\\frac{2}{n}}}.\\)\nSo,\n\\[\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\Bigg|\\exists C_{X}\\right)\\geq 1/2\\]\nif \\(t\\geq\\sqrt{2/n}\\) .Assume that the event\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\]\noccurs."}, {"n": 3, "claim": "Recall that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nHence,it must be that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\geq t/2 .\\]\nWe conclude\n\\[\\begin{array}{ r c l }{\\displaystyle\\frac 12}&{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}(C_{X})\\right|\\le t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right).}\\end{array}\\]\nSince indicators are 0,1-valued,\n\\[\\begin{array}{ r l r }&{}&{\\displaystyle\\frac{1}{2}T\\left(\\underbrace{\\operatorname*{sup}_{\\zeta\\in{\\mathcal C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}(C)\\right|\\ge t}_{\\Im C x}\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C x\\right)\\cdot I\\left(\\exists C x\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X,X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]\nNow, take expectation with respect to \\(X_{i}^{,}\\) s to obtain\n\\[\\begin{array}{ r l r }&{}&{\\mathbb{P}_{X}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\ge t\\right)}\\\\ &{}&{\\displaystyle\\le 2\\cdot\\mathbb{P}_{X,X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]"}]}}, {"local_id": "T2", "statement": "Setup. Let \\(X_1,\\dots,X_n\\) be i.i.d. copies of a random variable \\(X\\) with \\(0\\le X\\le 1\\) and \\(\\mu=\\mathbb{E}X\\), and let \\(0<t\\le 1-\\mu\\).\n\nAssume \\(0\\leq X_{i}\\leq 1\\) and \\(\\mu=\\mathbb{E}X\\) . Then\n\\[\\mathbb{P}\\left(\\frac{1}{n}\\sum_{i=1}^{n}X_{i}-\\mu\\geq t\\right)\\leq e^{-n\\mathcal{D}(\\mu+t,\\mu)}\\]\nwhere the KL-divergence \\(\\begin{array}{ r }{\\mathcal{D}(p,q)=p\\log\\frac{p}{q}+(1-p)\\log\\frac{1-p}{1-q}}\\end{array}\\)", "proof": {"steps": [{"n": 1, "claim": "Note that \\(\\phi(x)=e^{\\lambda x}\\) is convex and so \\(e^{\\lambda x}=e^{\\lambda(x\\cdot 1+(1-x)\\cdot 0)}\\leq x e^{\\lambda}+(1-x)e^{\\lambda\\cdot 0}=1-x+x e^{\\lambda}\\) .Hence,\n\\[\\mathbb{E}e^{\\lambda X}\\leq 1-\\mathbb{E}X+\\mathbb{E}X e^{\\lambda}=1-\\mu+\\mu e^{\\lambda}.\\]\nAgain,we minimize the following bound with respect to \\(\\lambda > 0\\)\n\\[\\begin{array}{ r c l }{\\displaystyle\\mathbb{P}\\left(\\sum_{i=1}^{n}X_{i}\\geq n(\\mu+t)\\right)}&{\\leq}&{e^{-\\lambda n(\\mu+t)}\\mathbb{E}e^{\\lambda\\sum X_{i}}}\\\\ &&{=}&{e^{-\\lambda n(\\mu+t)}\\left(\\mathbb{E}e^{\\lambda X}\\right)^{n}}\\\\ &&{\\leq}&{e^{-\\lambda n(\\mu+t)}\\left(1-\\mu+\\mu e^{\\lambda}\\right)^{n}}\\end{array}\\]\nTake derivative w.r.t. 入:"}, {"n": 2, "claim": "\\[\\begin{array}{ c }{{-n(\\mu+t)e^{-\\lambda n(\\mu+t)}(1-\\mu+\\mu e^{\\lambda})^{n}+n(1-\\mu+\\mu e^{\\lambda})^{n-1}\\mu e^{\\lambda}e^{-\\lambda n(\\mu+t)}=0}}\\\\ {{{}}}\\\\ {{-(\\mu+t)(1-\\mu+\\mu e^{\\lambda})+\\mu e^{\\lambda}=0}}\\\\ {{{}}}\\\\ {{e^{\\lambda}=\\displaystyle\\frac{(1-\\mu)(\\mu+t)}{\\mu(1-\\mu-t)}.}}\\end{array}\\]\nSubstituting,\n\\[\\begin{array}{ r c l }{\\mathbb{P}\\left(\\displaystyle\\sum_{i=1}^{n}X_{i}\\geq n(\\mu+t)\\right)}&{\\leq}&{\\displaystyle\\left(\\left(\\frac{\\mu(1-\\mu-t)}{(1-\\mu)(\\mu+t)}\\right)^{\\mu+t}\\left(1-\\mu+\\frac{(1-\\mu)(\\mu+t)}{1-\\mu-t}\\right)\\right)^{n}}\\\\ &{=}&{\\displaystyle\\left(\\left(\\frac{\\mu}{\\mu+t}\\right)^{\\mu+t}\\left(\\frac{1-\\mu}{1-\\mu-t}\\right)^{1-\\mu-t}\\right)^{n}}\\\\ &{=}&{\\displaystyle\\exp\\left(-n\\left((\\mu+t)\\log\\frac{\\mu+t}{\\mu}+(1-\\mu-t)\\log\\frac{1-\\mu-t}{1-\\mu}\\right)\\right),}\\end{array}\\]\ncompleting the proof."}]}}, {"local_id": "T3", "statement": "Let \\(x=(x_{1},\\ldots,x_{n}),x^{\\prime}=(x_{1}^{\\prime},\\ldots,x_{n}^{\\prime})\\) . If for functions \\(\\varphi_{1}(x,x^{\\prime}),\\varphi_{2}(x,x^{\\prime}),\\varphi_{3}(x,x^{\\prime})\\)\n\\[\\begin{array}{ r }{\\mathbb{P}\\left(\\varphi_{1}(x,x^{\\prime})\\geq\\varphi_{2}(x,x^{\\prime})+\\sqrt{\\varphi_{3}(x,x^{\\prime})\\cdot t}\\right)\\leq\\Gamma e^{-\\gamma t}}\\end{array}\\]\nthen\n\\[\\begin{array}{ r }{\\mathbb{P}\\left(\\mathbb{E}_{x^{\\prime}}\\varphi_{1}(x,x^{\\prime})\\geq\\mathbb{E}_{x^{\\prime}}\\varphi_{2}(x,x^{\\prime})+\\sqrt{\\mathbb{E}_{x^{\\prime}}\\varphi_{3}(x,x^{\\prime})\\cdot t}\\right)\\leq\\Gamma\\cdot e\\cdot e^{-\\gamma t}.}\\end{array}\\]\n(i.e. if the inequality holds, then it holds with averaging over one of the copies)", "proof": {"steps": [{"n": 1, "claim": "First, note that \\(\\begin{array}{ r }{\\sqrt{a b}=\\operatorname*{inf}_{\\delta>0}(\\delta a+\\frac{b}{4\\delta})}\\end{array}\\) with \\(\\begin{array}{ r }{\\delta_{*}=\\sqrt{\\frac{b}{4 a}}}\\end{array}\\) achieving the infima."}, {"n": 2, "claim": "Hence,\n\\[\\begin{array}{ c }{{\\{\\varphi_{1}\\geq\\varphi_{2}+\\sqrt{\\varphi_{3}t}\\}=\\{\\exists\\delta>0,\\varphi_{1}\\geq\\varphi_{2}+\\delta\\varphi_{3}+\\displaystyle\\frac{t}{4\\delta}\\}}}\\\\ {{=\\{\\exists\\delta>0,(\\varphi_{1}-\\varphi_{2}-\\delta\\varphi_{3})4\\delta\\geq t\\}}}\\\\ {{=\\{\\underset{\\underset{\\nu}{\\operatorname*{sup}}}{\\operatorname*{sup}}(\\varphi_{1}-\\varphi_{2}-\\delta\\varphi_{3})4\\delta\\geq t\\}}}\\end{array}\\]\nand similarly\n\\[\\big\\{\\mathbb{E}_{x^{\\prime}}\\varphi_{1}\\geq\\mathbb{E}_{x^{\\prime}}\\varphi_{2}+\\sqrt{\\mathbb{E}_{x^{\\prime}}\\varphi_{3}t}\\big\\}=\\underset{\\xi}{\\underbrace{\\{\\operatorname*{sup}(\\mathbb{E}_{x^{\\prime}}\\varphi_{1}-\\mathbb{E}_{x^{\\prime}}\\varphi_{2}-\\delta\\mathbb{E}_{x^{\\prime}}\\varphi_{3})4\\delta}}\\geq t\\big\\}.\\]\nBy assumption, \\(\\mathbb{P}\\left(\\nu\\geq t\\right)\\leq\\Gamma e^{-\\gamma t}\\) ."}, {"n": 3, "claim": "We want to prove \\(\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\Gamma\\cdot e\\cdot e^{-\\gamma t}\\) ."}, {"n": 4, "claim": "By the previous lemma, we only need to check whether \\(\\mathbb{E}\\phi(\\xi)\\le\\mathbb{E}\\phi(\\nu)\\)\n\\[\\begin{array}{ l }{\\displaystyle\\xi=\\operatorname*{sup}_{\\delta>0}\\mathbb{E}_{x^{\\prime}}(\\varphi_{1}-\\varphi_{2}-\\delta\\varphi_{3})4\\delta}\\\\ {\\displaystyle\\quad\\leq\\mathbb{E}_{x^{\\prime}}\\operatorname*{sup}_{\\delta>0}(\\varphi_{1}-\\varphi_{2}-\\delta\\varphi_{3})4\\delta}\\\\ {\\displaystyle=\\mathbb{E}_{x^{\\prime}}\\nu}\\end{array}\\]\nThus,\n\\[\\phi(\\xi)\\leq\\phi(\\mathbb{E}_{x^{\\prime}}\\nu)\\leq\\mathbb{E}_{x^{\\prime}}\\phi(\\nu)\\]\nby Jensen's inequality (\\(\\phi\\) is convex)."}, {"n": 5, "claim": "Hence,\n\\[\\mathbb{E}\\phi(\\xi)\\le\\mathbb{E}\\mathbb{E}_{x^{\\prime}}\\phi(\\nu)=\\mathbb{E}\\phi(\\nu).\\]"}]}}, {"local_id": "T4", "statement": "Setup. Let \\(0<\\varepsilon\\le 1\\).\n\nAssume \\(\\mathcal{F}\\) is a VC-subgraph class and \\(V C(\\mathcal{F})=V\\) Suppose \\(-1\\leq f(x)\\leq 1\\) for all \\(f\\in{\\mathcal{F}}\\) and \\(x\\in\\mathcal{X}\\) .Let \\(x_{1},\\ldots,x_{n}\\in{\\mathcal{X}}\\) and define \\(\\begin{array}{ r }{d(f,g)=\\frac{1}{n}\\sum_{i=1}^{n}|f(x_{i})-g(x_{i})|}\\end{array}\\) . Then\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]\n(which \\(\\begin{array}{ r }{i s\\le\\left(\\frac{K}{\\varepsilon}\\right)^{V+\\delta}}\\end{array}\\) for some \\(\\delta . \\jmath\\)", "proof": {"steps": [{"n": 1, "claim": "Let \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) and \\(f_{1},\\ldots,f_{m}\\) be ε-separated, i.e.\n\\[{\\frac{1}{n}}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})| >\\varepsilon .\\]"}, {"n": 2, "claim": "Let \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) be constructed in the following way:"}, {"n": 3, "claim": "\\(z_{i}\\) is chosen uniformly from \\(x_{1},\\ldots,x_{n}\\) and \\(t_{i}\\) is uniform on \\([-1,1]\\)\nConsider \\(f_{r}\\) and \\(f_{\\ell}\\) from the ε-packing."}, {"n": 4, "claim": "Let \\(C_{f_{\\tau}}\\) and \\(C_{f_{\\ell}}\\) be subgraphs of \\(f_{r}\\) and \\(f_{\\ell}\\) Then\n\\(\\mathbb{P}\\left(C_{f_{\\tau}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n= P(At least one point \\((z_{i},t_{i})\\) is picked by \\(C_{f_{r}}\\) or \\(C_{f_{\\ell}}\\) but not picked by the other)\n\\(=1-\\mathbb{P}\\) (All points \\((z_{i},t_{i})\\) are picked either by both or by none)\n\\(\\mathbf{\\Sigma}=1-\\mathbb{P}\\left(\\left(z_{i},t_{i}\\right)\\right.\\) is picked either by both or by none)k\nSince \\(z_{i}\\) is drawn uniformly from \\(x_{1},\\ldots,x_{n}\\)\nP \\(((z_{1},t_{1})\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)\n\\[\\begin{array}{ l }{\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{P}\\left(\\left(x_{i},t_{1}\\right)\\mathrm{~i s~p i c k e d~b y~b o t h~}C_{f r},C_{f_{\\ell}}\\mathrm{~o r~b y~n e i t h e r}\\right)}\\\\ {\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\left(1-\\frac{1}{2}|f_{r}(x_{i})-f_{\\ell}(x_{i})|\\right)}\\\\ {\\displaystyle=1-\\frac{1}{2}\\frac{1}{n}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})|}\\\\ {\\displaystyle=1-\\frac{1}{2}d(f_{r},f_{\\ell})\\leq 1-\\varepsilon/2\\leq e^{-\\varepsilon/2}}\\end{array}\\]\nSubstituting,\n\\(\\mathbb{P}\\left(C_{f_{r}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n\\(=1-\\mathbb{P}\\left((z_{1},t_{1})\\right.\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)𝑘\n\\[\\begin{array}{ l }{\\geq 1-\\left(e^{-\\varepsilon/2}\\right)^{k}}\\\\ {=1-e^{-k\\varepsilon/2}}\\end{array}\\]\nThere are (m） ways to choose \\(f_{r}\\) and \\(f_{\\ell},\\) s0\nP (All pairs \\(C_{f_{r}}\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\geq 1-{\\binom{m}{2}}e^{-k\\varepsilon/2}.\\)\nWhat k should we choose so that \\(1-{\\binom{m}{2}}e^{-k\\varepsilon/2}>0?\\) Choose\n\\[k>{\\frac{2}{\\varepsilon}}\\log{\\binom{m}{2}}.\\]\nThen there exist \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) such that all \\(C_{f_{\\ell}}\\) pick out different subsets.But \\(\\{C_{f}:f\\in\\mathcal{F}\\}\\) is \\(\\mathrm{VC}\\) so by Sauer's lemma, we can pick out at most \\(\\left({\\frac{e k}{V}}\\right)^{V}\\) out of these k points, provided \\(k \\geq V\\); if instead \\(k < V\\) then \\(m\\leq 2^{k}<2^{V}\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}\\) and the claimed bound already holds, so that case may be set aside."}, {"n": 5, "claim": "Hence, \\(\\begin{array}{ r }{m\\leq\\left(\\frac{e k}{V}\\right)^{V}}\\end{array}\\) as long as \\(k>{\\frac{2}{\\varepsilon}}\\log\\left({{\\binom{m}{2}}}\\right)\\) and \\(k \\geq V\\), the case \\(k < V\\) having been disposed of above."}, {"n": 6, "claim": "The latter holds for \\(\\begin{array}{ r }{k=\\frac{2}{\\varepsilon}\\log m^{2}}\\end{array}\\) ."}, {"n": 7, "claim": "Therefore,\n\\[m\\leq\\left(\\frac{e}{V}\\frac{2}{\\varepsilon}\\log m^{2}\\right)^{V}=\\left(\\frac{4 e}{V\\varepsilon}\\log m\\right)^{V},\\]\nwhere \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) ."}, {"n": 8, "claim": "Hence, we get\n\\[m^{1/V}\\leq\\frac{4 e}{\\varepsilon}\\log m^{1/V}\\]\nand defining \\(m^{1/V}=s .\\)\n\\[s\\leq{\\frac{4 e}{\\varepsilon}}\\log s .\\]\nNote that \\(\\frac{s}{\\log s}\\) is increasing for \\(s \\geq e\\) and so for large enough s, the inequality will be violated."}, {"n": 9, "claim": "We now check that the inequality is violated for \\(\\begin{array}{ r }{s^{\\prime}=\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}}\\end{array}\\) ."}, {"n": 10, "claim": "Indeed, one can show that\n\\[\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{7}{\\varepsilon}\\right)^{2}>\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)\\]\nsince\n\\[\\frac{49}{8e\\varepsilon}>\\log{\\frac{7}{\\epsilon}}.\\]\nHence, \\(m^{1/V}=s\\leq s^{\\prime}\\) and, thus,\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]"}]}}], "target_local_id": "T2", "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch9_theorem_9_1__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nDenote \\[\\triangle_{n}(\\mathcal{C})=\\sup_{\\{x_{1},\\ldots,x_{n}\\}}\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})\\leq 2^{n}.\\] We will see that for some classes, \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\) for \\(n \\leq V\\) and \\(\\triangle_{n}(\\mathcal{C})<2^{n}\\) for \\(n > V\\) for some constant \\(V\\).\n\ndefinition 8.1. If \\(V < \\infty\\) , then C is called a VC class. V is called VC dimension of C.\n\nLet \\(\\mathcal{F}=\\{f:\\mathcal{X}\\mapsto\\mathbb{R}\\}\\). For \\(f:\\mathcal{X}\\to\\mathbb{R}\\) , the subgraph of \\(f\\) is\n\\[C_{f}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}:0\\leq t\\leq f(x)\\ o r\\ f(x)\\leq t\\leq 0\\}.\\]\nDefine class of sets \\(\\mathcal{C}=\\{C_{f}:f\\in\\mathcal{F}\\}\\).\n\ndefinition 12.1. If \\(\\mathcal{C}\\) is a VC class of sets, then F is VC-subgraph class of functions and, by definition, \\(V C(\\mathcal{F})=V C(\\mathcal{C})\\)\nNote that equivalent definition of \\(C_{f}\\) is\n\\[C_{f}^{\\prime}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}: |f(x)|\\geq|t|\\}.\\]\n\ndefinition 12.3. The ε-packing number, \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) , is the marimal cardinality of an E-separated set.\nNote that \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) is decreasing in ε.\n\ndefinition 12.2. Given \\(\\varepsilon > 0\\) and \\(f_{1},\\ldots,f_{N}\\in\\mathcal{F}\\) ， we say that \\(f_{1},\\ldots,f_{N}\\) are E-separated if \\(d(f_{i},f_{j})>\\varepsilon\\) for any \\(i \\neq j\\)\n\ndefinition 8.0. For any sample \\(\\{x_{1},\\ldots,x_{n}\\}\\), the ways that \\({\\mathcal{C}}\\) intersects with the sample are \\[\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in{\\mathcal{C}}\\}.\\] Let \\[\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})=\\mathrm{card}~\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in\\mathcal{C}\\},\\] the number of different subsets picked out by \\(C\\in{\\mathcal{C}}\\). If \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\), this means we can find \\(\\{x_{1},\\ldots,x_{n}\\}\\) such that \\({\\mathcal{C}}\\) can pick out any subset of it; in this case \\({\\mathcal{C}}\\) shatters \\(\\{x_{1},\\ldots,x_{n}\\}\\). Equivalently, a set \\(A\\) is shattered by \\({\\mathcal{C}}\\) when \\({\\mathcal{C}}\\) picks out every subset of \\(A\\).", "theorems": [{"local_id": "T1", "statement": "Setup. Let \\(X_1,\\dots,X_n\\) be i.i.d. random variables with common law \\(P\\) on a measurable space \\((\\mathcal{X},\\mathcal{A})\\), let \\(\\mathcal{C}\\subseteq\\mathcal{A}\\) be a class of measurable sets, and write \\(P(C)=\\mathbb{P}(X_1\\in C)\\). Let \\(X'_1,\\dots,X'_n\\) (the ghost sample) be an independent copy of \\(X_1,\\dots,X_n\\), i.e. i.i.d. \\(P\\) and independent of \\(X_1,\\dots,X_n\\). Suprema over \\(\\mathcal{C}\\) are assumed measurable; otherwise read \\(\\mathbb{P}\\) as outer probability.\n\nIf \\(t\\geq{\\sqrt{\\frac{2}{n}}}.\\) then\n\\[\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\right)\\leq 2\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\geq t/2\\right).\\]", "proof": {"steps": [{"n": 1, "claim": "Suppose the event\n\\[\\operatorname*{sup}_{C\\in{\\mathcal{C}}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\]\noccurs.Let \\(\\begin{array}{ r }{X=(X_{1},\\ldots,X_{n})\\in\\{\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\}}\\end{array}\\) ."}, {"n": 2, "claim": "Then\n\\[\\exists C_{X}{\\mathrm{~s u c h~t h a t~}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nFor a fixed C,\n\\[\\begin{array}{ r }{\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right|\\geq t/2\\right)=\\mathbb{P}\\left(\\left(\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}\\geq t^{2}/4\\right)}\\\\ {\\leq\\mathrm{~(b y~C h e b y s h e v~s~I n e q)~}\\frac{4\\mathbb{E}\\left(\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}}{t^{2}}}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i,j}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))(I(X_{j}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i=1}^{n}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))^{2}=\\frac{4 n\\mathbb{P}\\left(\\mathcal{C}\\right)\\left(1-\\mathbb{P}(\\mathcal{C})\\right)}{n^{2}t^{2}}\\leq\\frac{1}{n t^{2}}\\leq\\frac{1}{2}}\\end{array}\\]\nsince we chose \\(t\\geq\\sqrt{\\textstyle{\\frac{2}{n}}}.\\)\nSo,\n\\[\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\Bigg|\\exists C_{X}\\right)\\geq 1/2\\]\nif \\(t\\geq\\sqrt{2/n}\\) .Assume that the event\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\]\noccurs."}, {"n": 3, "claim": "Recall that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nHence,it must be that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\geq t/2 .\\]\nWe conclude\n\\[\\begin{array}{ r c l }{\\displaystyle\\frac 12}&{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}(C_{X})\\right|\\le t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right).}\\end{array}\\]\nSince indicators are 0,1-valued,\n\\[\\begin{array}{ r l r }&{}&{\\displaystyle\\frac{1}{2}T\\left(\\underbrace{\\operatorname*{sup}_{\\zeta\\in{\\mathcal C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}(C)\\right|\\ge t}_{\\Im C x}\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C x\\right)\\cdot I\\left(\\exists C x\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X,X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]\nNow, take expectation with respect to \\(X_{i}^{,}\\) s to obtain\n\\[\\begin{array}{ r l r }&{}&{\\mathbb{P}_{X}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\ge t\\right)}\\\\ &{}&{\\displaystyle\\le 2\\cdot\\mathbb{P}_{X,X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]"}]}}, {"local_id": "T2", "statement": "Setup. Fix \\(f_1,\\dots,f_d:\\mathcal{X}\\to\\mathbb{R}\\) and let \\(\\mathcal{C}=\\left\\{\\left\\{x:\\sum_{k=1}^{d}\\alpha_k f_k(x)>0\\right\\}:\\alpha_1,\\dots,\\alpha_d\\in\\mathbb{R}\\right\\}\\).\n\nThen \\(VC(\\mathcal{C})\\le d\\).", "proof": {"steps": [{"n": 1, "claim": "Observation: For any \\(\\{x_{1},\\dots,x_{d+1}\\}\\) if we cannot shatter \\(\\{x_{1},\\dots,x_{d+1}\\}\\longleftrightarrow\\exists I\\subseteq\\{1\\dots d+1\\}\\) s.t. we cannot pick out \\(\\{x_{i},i\\in I\\}\\) ."}, {"n": 2, "claim": "If we can pick out \\(\\{x_{i},i\\in I\\}\\) , then for some \\(C\\in{\\mathcal{C}}\\) there are \\(\\alpha_{1},\\ldots,\\alpha_{d}\\) s.t."}, {"n": 3, "claim": "\\(\\textstyle\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x)>0\\) for \\(i \\in I\\) and \\(\\begin{array}{ r }{\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x)\\le 0}\\end{array}\\) for \\(i \\notin I\\)."}, {"n": 4, "claim": "Denote\n\\[\\left(\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x_{1}),\\ldots,\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x_{d+1})\\right)=F(\\alpha)\\in\\mathbb{R}^{d+1}.\\]\nBy linearity,\n\\[F(\\alpha)=\\sum_{k=1}^{d}\\alpha_{k}\\left(f_{k}(x_{1}),\\ldots,f(x_{d+1})\\right)=\\sum_{k=1}^{d}\\alpha_{k}F_{k}\\subseteq H\\subset\\mathbb{R}^{d+1}\\]\nand H is a d-dim subspace."}, {"n": 5, "claim": "Hence, \\(\\exists\\phi\\neq 0,\\phi\\cdot h=0,\\forall h\\in H\\) (\\(\\phi\\) orthogonal to H)."}, {"n": 6, "claim": "Let \\(I=\\{i:\\phi_{i}>0\\}\\) where \\(\\phi=\\left(\\phi_{1},\\ldots,\\phi_{d+1}\\right)\\) ."}, {"n": 7, "claim": "If \\(I=\\emptyset\\) then take \\(-\\phi\\) instead of \\(\\phi\\) so that \\(\\phi\\) has positive coordinates.\nClaim: We cannot pick out \\(\\{x_{i},i\\in I\\}\\) .Suppose we can: then \\(\\exists\\alpha_{1},\\dotsc,\\alpha_{d}\\) s.t. \\(\\begin{array}{ r }{\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x_{i})>0}\\end{array}\\) for \\(i\\in I\\) and \\(\\begin{array}{ r }{\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x_{i})\\le 0}\\end{array}\\) for \\(i \\not \\in I\\). But \\(\\phi\\cdot F(\\alpha)=0\\) and so\n\\[\\phi_{1}\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x_{1})+\\ldots+\\phi_{d+1}\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x_{d+1})=0 .\\]\nHence,\n\\[\\sum_{i\\in I}\\underbrace{\\phi_{i}\\left(\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x_{i})\\right)}_{>0}=\\sum_{i\\notin I}\\underbrace{(-\\phi_{i})}_{\\geq 0}\\underbrace{\\left(\\sum_{k=1}^{d}\\alpha_{k}f_{k}(x_{i})\\right)}_{\\leq 0}.\\]\nContradiction."}]}}, {"local_id": "T3", "statement": "The number \\(\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})\\) of subsets picked out by C is bounded by the number of subsets shattered by C.", "proof": {"steps": [{"n": 1, "claim": "Without loss of generality, we restrict C to \\({\\mathcal{C}}: =\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in{\\mathcal{C}}\\}\\) , and we have \\(\\operatorname{card}({\\mathcal{C}})=\\) \\(\\Delta_{n}({\\mathcal{C}},x_{1},\\cdots,x_{n})\\)\nWe will say that C is hereditary if and only if whenever \\(B\\subseteq C\\in{\\mathcal{C}},B\\in{\\mathcal{C}}\\) .If C is hereditary, then every \\(C\\in{\\mathcal{C}}\\) is shattered by C,and the lemma is obvious."}, {"n": 2, "claim": "Otherwise,we will transform \\({\\mathcal{C}}\\to{\\mathcal{C}}^{\\prime}\\) ,hereditary, without changing the cardinality of C and without increasing the number of shattered subsets."}, {"n": 3, "claim": "Define the operators \\(T_{i}\\) for \\(i=1,\\cdots,n\\) as the following,\n\\[\\begin{array}{ r c l }{T_{i}(C)}&{=}&{\\left\\{\\begin{array}{ l l }{C-\\{x_{i}\\}}&{\\mathrm{~i f~}C-\\{x_{i}\\}\\mathrm{~i s~n o t~i n~}\\mathcal{C}}\\\\ {C}&{\\mathrm{~o t h e r w i s e}}\\end{array}\\right.}\\\\ {T_{i}(\\mathcal{C})}&{=}&{\\{T_{i}(C):C\\in\\mathcal{C}\\}.}\\end{array}\\]\nIt follows that card \\(T_{i}({\\mathcal{C}})=\\operatorname{card}{\\mathcal{C}}\\) Moreover, every \\(A\\subseteq\\{x_{1},\\cdots,x_{n}\\}\\) that is shattered by \\(T_{i}(\\mathcal{C})\\) is also shattered by \\({\\mathcal{C}}.{\\mathrm{~I f~}}x_{i}\\notin A .\\) ,then \\(\\forall C\\in\\mathcal{C},A\\bigcap C=A\\bigcap T_{i}(C)\\) , thus C and \\(T_{i}(\\mathcal{C})\\) both or neither shatter A."}, {"n": 4, "claim": "On the other hand, if \\(x_{i}\\in A\\) and A is shattered by \\(T_{i}(\\mathcal{C})\\) , then \\(\\forall B\\subseteq A,\\exists C\\in{\\mathcal{C}}\\) ,such that \\(B\\bigcup\\{x_{i}\\}=A\\bigcap T_{i}(C)\\) This means that \\(x_{i}\\in T_{i}(C)\\) ，and that \\(C\\backslash\\{x_{i}\\}\\in\\mathcal{C}\\) Thus both \\(B\\bigcup\\{x_{i}\\}\\) and \\(B\\backslash\\{x_{i}\\}\\) are picked out by C."}, {"n": 5, "claim": "Since either \\(B=B\\cup\\{x_{i}\\}\\) or \\(B=B\\backslash\\{x_{i}\\}\\) , B is picked out by C."}, {"n": 6, "claim": "Thus A is shattered by C. Apply the operator \\(T=T_{1}\\circ\\dots\\circ T_{n}\\) until \\(T^{k+1}({\\mathcal{C}})=T^{k}({\\mathcal{C}})\\) ."}, {"n": 7, "claim": "This will happen for at most \\(\\scriptstyle\\sum_{C\\in{\\mathcal{C}}}\\operatorname{card}(C)\\) times,since \\(\\begin{array}{ r }{\\sum_{C\\in\\mathcal{C}}\\operatorname{card}(T_{i}(C))<\\sum_{C\\in\\mathcal{C}}\\operatorname{card}(C)\\mathrm{~i f~}T_{i}(\\mathcal{C})\\neq\\mathcal{C}}\\end{array}\\) ，The resulting collection \\(\\scriptstyle{\\mathcal{C}}^{\\prime}\\) is hereditary."}, {"n": 8, "claim": "This proves the lemma."}]}}, {"local_id": "T4", "statement": "Setup. Let \\(0<\\varepsilon\\le 1\\).\n\nAssume \\(\\mathcal{F}\\) is a VC-subgraph class and \\(V C(\\mathcal{F})=V\\) Suppose \\(-1\\leq f(x)\\leq 1\\) for all \\(f\\in{\\mathcal{F}}\\) and \\(x\\in\\mathcal{X}\\) .Let \\(x_{1},\\ldots,x_{n}\\in{\\mathcal{X}}\\) and define \\(\\begin{array}{ r }{d(f,g)=\\frac{1}{n}\\sum_{i=1}^{n}|f(x_{i})-g(x_{i})|}\\end{array}\\) . Then\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]\n(which \\(\\begin{array}{ r }{i s\\le\\left(\\frac{K}{\\varepsilon}\\right)^{V+\\delta}}\\end{array}\\) for some \\(\\delta . \\jmath\\)", "proof": {"steps": [{"n": 1, "claim": "Let \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) and \\(f_{1},\\ldots,f_{m}\\) be ε-separated, i.e.\n\\[{\\frac{1}{n}}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})| >\\varepsilon .\\]"}, {"n": 2, "claim": "Let \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) be constructed in the following way:"}, {"n": 3, "claim": "\\(z_{i}\\) is chosen uniformly from \\(x_{1},\\ldots,x_{n}\\) and \\(t_{i}\\) is uniform on \\([-1,1]\\)\nConsider \\(f_{r}\\) and \\(f_{\\ell}\\) from the ε-packing."}, {"n": 4, "claim": "Let \\(C_{f_{\\tau}}\\) and \\(C_{f_{\\ell}}\\) be subgraphs of \\(f_{r}\\) and \\(f_{\\ell}\\) Then\n\\(\\mathbb{P}\\left(C_{f_{\\tau}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n= P(At least one point \\((z_{i},t_{i})\\) is picked by \\(C_{f_{r}}\\) or \\(C_{f_{\\ell}}\\) but not picked by the other)\n\\(=1-\\mathbb{P}\\) (All points \\((z_{i},t_{i})\\) are picked either by both or by none)\n\\(\\mathbf{\\Sigma}=1-\\mathbb{P}\\left(\\left(z_{i},t_{i}\\right)\\right.\\) is picked either by both or by none)k\nSince \\(z_{i}\\) is drawn uniformly from \\(x_{1},\\ldots,x_{n}\\)\nP \\(((z_{1},t_{1})\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)\n\\[\\begin{array}{ l }{\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{P}\\left(\\left(x_{i},t_{1}\\right)\\mathrm{~i s~p i c k e d~b y~b o t h~}C_{f r},C_{f_{\\ell}}\\mathrm{~o r~b y~n e i t h e r}\\right)}\\\\ {\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\left(1-\\frac{1}{2}|f_{r}(x_{i})-f_{\\ell}(x_{i})|\\right)}\\\\ {\\displaystyle=1-\\frac{1}{2}\\frac{1}{n}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})|}\\\\ {\\displaystyle=1-\\frac{1}{2}d(f_{r},f_{\\ell})\\leq 1-\\varepsilon/2\\leq e^{-\\varepsilon/2}}\\end{array}\\]\nSubstituting,\n\\(\\mathbb{P}\\left(C_{f_{r}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n\\(=1-\\mathbb{P}\\left((z_{1},t_{1})\\right.\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)𝑘\n\\[\\begin{array}{ l }{\\geq 1-\\left(e^{-\\varepsilon/2}\\right)^{k}}\\\\ {=1-e^{-k\\varepsilon/2}}\\end{array}\\]\nThere are (m） ways to choose \\(f_{r}\\) and \\(f_{\\ell},\\) s0\nP (All pairs \\(C_{f_{r}}\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\geq 1-{\\binom{m}{2}}e^{-k\\varepsilon/2}.\\)\nWhat k should we choose so that \\(1-{\\binom{m}{2}}e^{-k\\varepsilon/2}>0?\\) Choose\n\\[k>{\\frac{2}{\\varepsilon}}\\log{\\binom{m}{2}}.\\]\nThen there exist \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) such that all \\(C_{f_{\\ell}}\\) pick out different subsets.But \\(\\{C_{f}:f\\in\\mathcal{F}\\}\\) is \\(\\mathrm{VC}\\) so by Sauer's lemma, we can pick out at most \\(\\left({\\frac{e k}{V}}\\right)^{V}\\) out of these k points, provided \\(k \\geq V\\); if instead \\(k < V\\) then \\(m\\leq 2^{k}<2^{V}\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}\\) and the claimed bound already holds, so that case may be set aside."}, {"n": 5, "claim": "Hence, \\(\\begin{array}{ r }{m\\leq\\left(\\frac{e k}{V}\\right)^{V}}\\end{array}\\) as long as \\(k>{\\frac{2}{\\varepsilon}}\\log\\left({{\\binom{m}{2}}}\\right)\\) and \\(k \\geq V\\), the case \\(k < V\\) having been disposed of above."}, {"n": 6, "claim": "The latter holds for \\(\\begin{array}{ r }{k=\\frac{2}{\\varepsilon}\\log m^{2}}\\end{array}\\) ."}, {"n": 7, "claim": "Therefore,\n\\[m\\leq\\left(\\frac{e}{V}\\frac{2}{\\varepsilon}\\log m^{2}\\right)^{V}=\\left(\\frac{4 e}{V\\varepsilon}\\log m\\right)^{V},\\]\nwhere \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) ."}, {"n": 8, "claim": "Hence, we get\n\\[m^{1/V}\\leq\\frac{4 e}{\\varepsilon}\\log m^{1/V}\\]\nand defining \\(m^{1/V}=s .\\)\n\\[s\\leq{\\frac{4 e}{\\varepsilon}}\\log s .\\]\nNote that \\(\\frac{s}{\\log s}\\) is increasing for \\(s \\geq e\\) and so for large enough s, the inequality will be violated."}, {"n": 9, "claim": "We now check that the inequality is violated for \\(\\begin{array}{ r }{s^{\\prime}=\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}}\\end{array}\\) ."}, {"n": 10, "claim": "Indeed, one can show that\n\\[\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{7}{\\varepsilon}\\right)^{2}>\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)\\]\nsince\n\\[\\frac{49}{8e\\varepsilon}>\\log{\\frac{7}{\\epsilon}}.\\]\nHence, \\(m^{1/V}=s\\leq s^{\\prime}\\) and, thus,\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]"}]}}], "target_local_id": "T2", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch11_theorem_11_26__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nequation 11.15. The \\(k\\)-layer neural network function class is defined recursively by\n\\[\n\\mathcal{F}^{(1)} = \\{x \\mapsto \\theta^{\\top}x: \\theta \\in \\Theta\\},\n\\]\nand, for \\(k \\geq 2\\),\n\\[\n\\mathcal{F}^{(k)} = \\left\\{\\frac{1}{m}\\sum_{j=1}^{m}u_j h(f_j(x)): f_j \\in \\mathcal{F}^{(k-1)}, \\|u\\|_1 \\leq mA_k\\right\\}.\n\\]\nHere \\(h \\circ \\mathcal{F}^{(k-1)} = \\{x \\mapsto h(f(x)): f \\in \\mathcal{F}^{(k-1)}\\}\\), so each new layer composes the activation \\(h\\) with functions from the preceding layer and then takes an \\(L_1\\)-regularized linear combination. The parameters \\(A_2,\\ldots,A_k\\) are the layer-wise \\(L_1\\) regularization parameters, and in Theorem 11.26 they enter through \\(A = 2^{k-2}\\prod_{\\ell=2}^{k}A_{\\ell}\\).\n\ndefinition 6.3. Given \\(S_{n}=\\{Z_{1},\\ldots,Z_{n}\\}\\), the (one-sided) empirical Rademacher complexity of \\(\\mathcal{G}\\) is defined as\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n}) = \\mathbb{E}_{\\sigma}\\sup_{w\\in \\Omega}\\frac{1}{n}\\sum_{i = 1}^{n}\\sigma_{i}\\phi (w,Z_{i}),\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R(\\mathcal{G},\\mathcal{S}_{n}).\n\\]\nThe following result shows that the quantity \\(\\epsilon_{n}(\\mathcal{G},\\mathcal{D})\\) can be upper bounded by Rademacher complexity. It follows that an average oracle inequality can be obtained using Rademacher complexity. The proof employs the symmetrization technique, which was also used in Chapter 4 to obtain uniform convergence bounds from empirical covering numbers.\n\ndefinition 10.6. Let \\(\\mathcal{F}_{L_1}(\\Psi)\\) be the point-wise closure of \\(\\cup_{A > 0}\\mathcal{F}_{A,L_1}(\\Psi)\\), then for any \\(f\\in \\mathcal{F}_{L_1}(\\Psi)\\),\n\\[\n\\|f\\|_{1}=\\lim_{\\epsilon\\rightarrow 0}\\inf\\left\\{\\|w\\|_{1}:\\sup_{x}\\left|f(x)-\\sum_{j=1}^{m}w_{j}\\psi(\\theta_{j},x)\\right|\\leq\\epsilon\\right\\}.\n\\]\nFor notational convenience, we write functions in \\(\\mathcal{F}_{L_1}(\\Psi)\\) as\n\\[\nf(x)=w^{\\top}\\psi(x),\n\\]\nwhere \\(\\psi(x)\\) is the infinite dimensional vector \\([\\psi(\\theta, x)]_{\\theta \\in \\Theta}\\), and \\(\\|f\\|_1 = \\|w\\|_1\\).\nFor smooth-loss functions (see Definition A.5), there is a strong relationship between \\( L_{1} \\) regularization and \\( L_{0} \\) regularization, in that functions with small \\( L_{1} \\) norms can be sparsified. Such a sparsification result was first shown for the least squares loss, and referred to as Maurey's lemma in Pisier (1980-1981).\n\nexample 6.26. From Corollary 5.7, we know that if a binary-valued function class \\(\\mathcal{G}\\) (or a VC-subgraph class with values in \\([0,1]\\)) has VC-dimension \\(d\\), then\n\\[\n\\ln N_{2}(\\epsilon,\\mathcal{G},n)\\leq 1+\\ln(d+1)+d\\ln(2 e/\\epsilon^{2}).\n\\]\nSince \\(N_{2}(0.5,\\mathcal{G},n) = 1\\), we have\n\\[\n12\\int_{0}^{\\infty}\\sqrt{\\ln N_{2}(\\epsilon,\\mathcal{G},n)}d\\epsilon\\leq 12\\int_{0}^{0. 5}\\sqrt{1+\\ln(d+1)+d\\ln(2e/\\epsilon^{2})}d\\epsilon\\leq 16\\sqrt{d}.\n\\]\nIt follows that\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n})\\leq\\frac{16\\sqrt{d}}{\\sqrt{n}}.\n\\]\nThe constant isn't optimal, and in fact a better constant can be obtained using packing number (see Example 6.4). The result implies the following uniform convergence result: with probability at least \\(1 - \\delta\\), for all \\(w \\in \\Omega\\),\n\\[\n\\phi(w,\\mathcal{D})\\leq\\phi(w,\\mathcal{S}_{n})+\\frac{32\\sqrt{d}}{\\sqrt{n}}+\\sqrt{\\frac{\\ln(1/\\delta)}{2n}}.\n\\]\nThis bound removes a \\( \\ln n \\) factor from the additive uniform convergence bound in Theorem 4.17.\n\nexample 10.11. In two-layer neural networks, let \\(\\Psi\\) be an \\(L_{2}\\)-regularized ReLU function class:\n\\[\n\\Psi=\\left\\{\\psi(\\theta,x)=\\max(0,\\theta^{\\top}x):\\quad\\|\\theta\\|_{2}\\leq\\alpha,\\|x\\|_{2}\\leq\\beta\\right\\},\n\\]\nand the corresponding \\(L_{1}\\) regularized two-layer NN can be expressed as a function\n\\[\nf(x)=\\sum_{j=1}^{m}w_{j}\\psi\\left(\\theta_{j},x\\right):\\|w\\|_{1}\\leq A,\\|\\theta\\|_{2}\\leq\\alpha,\\|x\\|_{2}\\leq\\beta .\n\\]\nThis function class belongs to \\(\\mathcal{F}_{A,L_1}(\\Psi)\\). We thus obtain the following bound for \\(L_{1}\\) regularized two-layer NN:\n\\[\n\\begin{array}{l}R\\left(\\mathcal{F}_{A,L_{1}}(\\Psi),\\mathcal{S}_{n}\\right)\\leq 2AR(\\Psi,\\mathcal{S}_{n})\\tag{Theorem 10.8}\\\\ \\leq 2A\\alpha\\beta/\\sqrt{n},\\quad(\\text{C o r o l l a r y} 9. 21)\\\\ \\end{array}\n\\]\nwhere we note that \\(\\max (0,f)\\) is 1-Lipschitz in \\(f\\)\nNext, we may consider the following hard-constrained \\( L_{1} \\) regularized learning problem.\n\\[\n\\hat{w}=\\arg\\min_{w}\\frac{1}{n}\\sum_{i=1}^{n}L\\left(w^{\\top}\\psi\\left(X_{i}\\right),Y_{i}\\right),\\quad\\|w\\|_{1}\\leq A.\\tag{10.8}\n\\]\nSimilarly, we may consider the soft-regularized version as:\n\\[\n\\hat{w}=\\arg\\min_{w}\\frac{1}{n}\\sum_{i=1}^{n}L\\left(w^{\\top}\\psi\\left(X_{i}\\right),Y_{i}\\right)+\\lambda\\|w\\|_{1}.\\tag{10.9}\n\\]\nNow by using Theorem 6.31, we obtain the following result.\n\ndefinition 10.2. The sparsity pattern, or support of a weight vector \\( w \\in \\mathbb{R}^m \\) is defined as\n\\[\n\\operatorname{supp}(w)=\\{j:w_{j}\\neq 0\\},\n\\]\nand the \\(L_0\\) norm of \\(w\\) is defined as\n\\[\n\\left\\|w\\right\\|_{0}=\\left|\\operatorname{supp}(w)\\right|.\n\\]\nThe following example compares different regularization conditions with the RKHS functions.\n\nexample 10.3. Consider RBFs of the form\n\\[\n\\psi(\\theta,x)=\\exp(-\\beta\\|x-\\theta\\|_{2}^{2})\n\\]\nfor some \\(\\beta > 0\\). If we treat it as a kernel \\(k(\\cdot, \\cdot) = \\psi(\\cdot, \\cdot)\\) as in Chapter 9, then the corresponding RKHS is given by functions of the form\n\\[\n\\tilde{f}(\\alpha,x)=\\sum_{j=1}^{m}\\alpha_{j}\\exp(-\\beta\\|x-\\theta_{j}\\|_{2}^{2}).\n\\]\nLet \\(\\mathcal{H}\\) be its RKHS, and the corresponding regularization is given by\n\\[\n\\|\\tilde{f}(\\alpha,x)\\|_{\\mathcal{H}}^{2}=\\sum_{i=1}^{m}\\sum_{j=1}^{m}\\alpha_{i}\\alpha_{j}\\exp(-\\beta\\|\\theta_{i}-\\theta_{j}\\|_{2}^{2}).\\tag{10.3}\n\\]\nIf we use RBFs as basis functions in additive models, we have\n\\[\nf([w,\\theta],x)=\\sum_{j=1}^{m}w_{j}\\exp(-\\beta\\|x-\\theta_{j}\\|_{2}^{2}).\n\\]\nIn this case, the kernel associated with the additive model can be defined as\n\\[\nk(x,x^{\\prime})=\\frac{1}{m}\\sum_{i=1}^{m}\\exp(-\\beta\\|x-\\theta_{j}\\|_{2}^{2}-\\beta\\|x^{\\prime}-\\theta_{j}\\|_{2}^{2}).\n\\]\nThe corresponding RKHS norm for additive model is different from (10.3), and can be defined instead as\n\\[\n\\left\\|f([w,\\theta],x)\\right\\|^{2}=m\\sum_{j=1}w_{j}^{2}.\\tag{10.4}\n\\]\nHowever, even for simple 1-dimensional functions such as\n\\[\n\\sum_{j=1}^{m}\\exp(-\\|x-j\\|_{2}^{2})\n\\]\nwith \\(\\beta = 1\\), the complexity measured by the RKHS norm in (10.4) can be rather large. Alternatively, one can also measure the complexity of \\(f([w,\\theta ],x)\\) by the sparsity \\(\\| w\\| _0\\).\nFor each sparsity pattern \\( F \\subset \\{1, \\ldots, m\\} \\), we may define the \\( |F| \\)-dimensional sparse function class\n\\[\n\\mathcal{G}_{F}=\\left\\{\\phi(w,z):\\operatorname{supp}(w)\\subset F\\right\\},\n\\]\nwhere \\(\\phi (w,z) = L(f(w,x),y) = L(w^{\\top}\\psi (x),y)\\). We can now consider the following sparse learning method:\n\\[\n\\hat{w}=\\arg\\min_{w\\in\\mathbb{R}^{d}}\\left[\\frac{1}{n}\\sum_{i=1}^{n}\\phi(w,Z_{i})+r(F)\\right]\\quad \\text{s u b j e c t t o}\\operatorname{supp}(w)\\subset F,\\tag{10.5}\n\\]\nwith a properly defined regularizer \\( r(F) \\) on the sparsity pattern \\( F \\).\nWe note that the exact solution of (10.5) may be difficult to obtain because the sparsity constraint is nonconvex and discontinuous. In practice one often employs approximate solutions. Assume that we can find an approximate solution of (10.5), then we can obtain a generalization error bound that depends only logarithmically on \\( m \\), and linear with respect to the sparsity \\( |F| \\). The following theorem is a direct consequence of Theorem 8.7 which employs an upper bound of the expected Rademacher complexity. Note that similar to Theorem 8.11, one may also replace the expected (sample independent) Rademacher complexity by the sample dependent Rademacher complexity.", "theorems": [{"local_id": "T1", "statement": "Let \\(\\{\\phi_i\\}_{i=1}^n\\) be functions with Lipschitz constants \\(\\{\\gamma_i\\}_{i=1}^n\\) respectively. That is, \\(\\forall i \\in [n]\\):\n\\[\n\\left|\\phi_{i}(\\theta)-\\phi_{i}\\left(\\theta^{\\prime}\\right)\\right|\\leq\\gamma_{i}\\left|\\theta-\\theta^{\\prime}\\right|.\n\\]\nThen for any real valued function \\( h: \\mathcal{F} \\times \\mathcal{Z}^n \\to \\mathbb{R} \\), and \\( S_n = \\{Z_1, \\ldots, Z_n\\} \\subset \\mathcal{Z}^n \\), we have\n\\[\n\\mathbb{E}_{\\sigma}\\sup_{f\\in\\mathcal{F}}\\left[\\sum_{i=1}^{n}\\sigma_{i}\\phi_{i}(f(Z_{i}))-h(f,\\mathcal{S}_{n})\\right]\\leq\\mathbb{E}_{\\sigma}\\sup_{f\\in\\mathcal{F}}\\left[\\sum_{i=1}^{n}\\sigma_{i}\\gamma_{i}f(Z_{i})-h(f,\\mathcal{S}_{n})\\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "The result is a direct consequence of the Lemma 6.29, where we simply set \\( c(w) = -h(w, \\mathcal{S}_n) \\), \\( g_i(w) = \\phi_i(f(Z_i)) \\), and \\( \\tilde{g}_i(w) = \\gamma_i f(Z_i) \\)."}]}}, {"local_id": "T2", "statement": "Assume \\(\\sup_{w,z,z'}[\\phi(w,z) - \\phi(w,z')] \\leq M\\). Let \\(S_n\\) be \\(n\\) iid samples from \\(\\mathcal{D}\\). Then with probability at least \\(1 - \\delta\\), the following bound holds for all \\(w \\in \\mathbb{R}^m\\) and sparsity pattern \\(F\\) such that \\(\\mathrm{supp}(w) \\subset F\\):\n\\[\n\\phi(w,\\mathcal{D})\\leq\\phi(w,\\mathcal{S}_{n})+r(F)+M\\sqrt{\\frac{\\ln(1/\\delta)}{2 n}},\n\\]\nwhere\n\\[\nr(F)\\geq 2 R(\\mathcal{G}_{F},\\mathcal{D})+M\\sqrt{\\frac{|F|\\ln(e m/ |F|)+\\ln(|F| +1)^{2}}{2 n}}\n\\]\nfor all \\( F \\). Consider the sparse learning algorithm in (10.5). We have the following oracle inequality. With probability of at least \\( 1 - \\delta \\):\n\\[\n\\phi(\\hat{w},\\mathcal{D})\\leq\\inf_{w\\in\\mathbb{R}^{m},\\operatorname{supp}(w)\\subset F}[\\phi(w,\\mathcal{D})+r(F)]+2 M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "We note that\n\\[\n\\binom{m}{s}\\leq\\frac{m^{s}}{s!}\\leq\\frac{m^{s}}{s^{s}}\\cdot\\frac{s^{s}}{s!}\\leq\\frac{m^{s}}{s^{s}}\\cdot e^{s}=(m e/s)^{s}.\n\\]\nWe can now consider each \\(\\mathcal{G}_F\\) as a model, with\n\\[\nq_{F}=\\frac{(|F| +1)^{-2}}{(m e/ |F|)^{|F|}}\\leq\\frac{(|F| +1)^{-2}}{\\binom{m}{|F|}}.\n\\]\nTherefore we have\n\\[\n\\sum_{F: |F|\\geq 1}q_{F}\\leq\\sum_{s\\geq 1}\\frac{1}{(s+1)^{2}}\\sum_{F: |F| =s}\\frac{1}{\\binom{m}{|F|}}=\\sum_{s\\geq 1}\\frac{1}{(s+1)^{2}}<1.\n\\]\nFor each index \\( F \\), we may consider \\( \\hat{w} \\) as the ERM solution under the constraint \\( \\operatorname{supp}(w) \\subset F \\)."}, {"n": 2, "claim": "We can thus apply Theorem 8.7 with models indexed by \\( F \\subset \\{1, \\ldots, m\\} \\) to obtain the desired bounds."}]}}, {"local_id": "T3", "statement": "Consider \\(K\\)-layer neural networks defined by (11.15). Assume that \\(h(\\theta^{\\top}x) \\in [0, M]\\) for all \\(\\theta \\in \\Theta\\) and \\(x \\in \\mathcal{X}\\). Assume also that \\(h\\) is 1-Lipschitz, monotone, and satisfies \\(h(0) = 0\\). Then there exists a constant \\(C\\), such that for all distribution \\(D\\) on \\(\\mathcal{X}\\), we have\n\\[\nR(\\mathcal{F}^{(k)},\\mathcal{S}_{n})\\leq A M\\frac{32\\sqrt{d+1}}{\\sqrt{n}},\n\\]\nwhere \\(A = 2^{k - 2}\\prod_{\\ell = 2}^{k}A_{\\ell}\\)", "proof": {"steps": [{"n": 1, "claim": "We prove the statement by induction. The case \\( k = 2 \\) follows from Corollary 11.10."}, {"n": 2, "claim": "Assume the statement holds at layer \\( k - 1 \\), then at layer \\( k \\)."}, {"n": 3, "claim": "We have\n\\[\n\\begin{array}{l}R\\left(\\mathcal{F}^{(k)},\\mathcal{S}_{n}\\right)\\leq A_{k}\\left[R\\left(h\\circ\\mathcal{F}^{(k-1)},\\mathcal{S}_{n}\\right)+R\\left(-h\\circ\\mathcal{F}^{(k-1)},\\mathcal{S}_{n}\\right)\\right]\\\\ \\leq 2 A_{k}R\\left(\\mathcal{F}^{(k-1)},\\mathcal{S}_{n}\\right).\\\\ \\end{array}\n\\]\nThe first inequality used Theorem 10.8."}, {"n": 4, "claim": "The second inequality used Theorem 6.28 with \\(\\gamma_{i} = 1\\) and \\(h = 0\\)."}, {"n": 5, "claim": "Now by using induction, we obtain the desired bound."}]}}, {"local_id": "T4", "statement": "Assume that \\( Y \\in \\{\\pm 1\\} \\), and \\( \\psi(\\theta, x) \\in [-1, 1] \\). Then \\( \\forall \\gamma > 0 \\), with probability \\( 1 - \\delta \\): for all \\( f(x) = \\sum_{j} w_{j} \\psi(\\theta_{j}, x) \\), we have\n\\[\n\\begin{array}{l}\\mathbb{E}_{\\mathcal{D}}\\mathbb{1}(f(X)Y\\leq 0)\\leq\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{1}(f(X_{i})Y_{i}\\leq\\gamma)+4(1+\\gamma^{-1}\\|w\\|_{1})R_{n}(\\Psi_{\\pm},\\mathcal{D})\\\\ +\\left(3+2\\gamma^{-1}\\|w\\|_{1}\\right)\\left[\\sqrt{\\frac{\\ln((2+\\log_{2}(1+\\|w\\|_{1}/\\gamma))^{2})}{2 n}}+\\sqrt{\\frac{\\ln(1/\\delta)}{n}}\\right].\\\\ \\end{array}\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\( L(p, y) = \\min(1, \\max(0, 1 - py / \\gamma)) \\), which is \\( \\gamma^{-1} \\) Lipschitz. Let \\( A_0 = \\gamma \\), \\( B = 1 \\), and \\( M_0 = 1 \\). By using the fact that\n\\[\n\\mathbb{1}(f(X),Y)\\leq L(f(X),Y)\\leq\\mathbb{1}(f(X)Y\\leq\\gamma),\n\\]\nThe result is a directly consequence of Corollary 10.14."}]}}, {"local_id": "T5", "statement": "Let \\(\\mathcal{F} = \\{h(\\theta^\\top x):\\theta \\in \\mathbb{R}^d\\}\\) and \\(\\mathcal{F}_{\\pm} = \\mathcal{F}\\cup -\\mathcal{F}\\). Let\n\\[\n\\mathcal{F}_{A}^{1}=\\{\\mathbb{E}_{\\theta\\sim\\mu}u(\\theta)h(\\theta^{\\top}x):\\mathbb{E}_{\\theta\\sim\\mu}|u(\\theta)|\\leq A\\}.\n\\]\nThen for all monotone function \\( h(\\cdot) \\) with \\( h(\\cdot) \\in [0, M] \\):\n\\[\nR\\left(\\mathcal{F}_{\\pm},\\mathcal{S}_{n}\\right)\\leq M\\frac{32\\sqrt{d+1}}{\\sqrt{n}}\n\\]\n\\[\nR\\left(\\mathcal{F}_{A}^{1},\\mathcal{S}_{n}\\right)\\leq A M\\frac{32\\sqrt{d+1}}{\\sqrt{n}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Since \\( h(\\cdot) \\) is monotone, we know that \\( \\mathcal{F} \\) is a VC-subgraph class with VC dimension \\( d + 1 \\)."}, {"n": 2, "claim": "Therefore from Theorem 5.11 and the calculation in Example 6.26, we obtain\n\\[\nR(\\mathcal{F},\\mathcal{S}_{n})\\leq M\\frac{16\\sqrt{d+1}}{\\sqrt{n}}.\n\\]\nThis leads to the first bound."}, {"n": 3, "claim": "The second bound follows from Theorem 10.8. \\(\\square\\)"}]}}], "target_local_id": "T3", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch2_theorem_2_21__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\ndefinition 2.4. Given a random variable \\( X \\), we may define its logarithmic moment generating function as\n\\[\n\\Lambda_{X}(\\lambda)=\\ln\\mathbb{E}e^{\\lambda X}.\n\\]\nMoreover, given \\(z\\in \\mathbb{R}\\), the rate function \\(I_{X}(z)\\) is defined as\n\\[\nI_{X}(z)=\\left\\{\\begin{array}{l l}\\sup_{\\lambda>0}[\\lambda z-\\Lambda_{X}(\\lambda)]&z>\\mu\\\\ 0&z=\\mu\\\\ \\sup_{\\lambda<0}[\\lambda z-\\Lambda_{X}(\\lambda)]&z<\\mu,\\end{array}\\right.\n\\]\nwhere \\(\\mu = \\mathbb{E}[X]\\)\nThe above definition can be used to obtain exponential tail bounds for sums of independent variables as follows.", "theorems": [{"local_id": "T1", "statement": "If \\( X_{1} \\leq \\mu + b \\), for some \\( b > 0 \\). Let \\( \\psi(z) = (1 + z)\\ln(1 + z) - z \\), then \\( \\forall \\epsilon > 0 \\):\n\\[\n\\Pr\\left[\\bar{X}_{n}\\geq\\mu+\\epsilon\\right]\\leq\\exp\\left[\\frac{-n\\operatorname{Var}\\left(X_{1}\\right)}{b^{2}}\\psi\\left(\\frac{\\epsilon b}{\\operatorname{Var}\\left(X_{1}\\right)}\\right)\\right],\n\\]\n\\[\n\\Pr [\\bar{X}_{n}\\geq\\mu+\\epsilon]\\leq\\exp\\left[\\frac{-n\\epsilon^{2}}{2\\operatorname{Var}(X_{1})+2\\epsilon b/3}\\right].\n\\]\nMoreover, for \\( t > 0 \\):\n\\[\n\\operatorname*{Pr}\\left[\\bar{X}_{n}\\geq\\mu+\\sqrt{\\frac{2\\operatorname{Var}(X_{1})t}{n}}+\\frac{bt}{3n}\\right]\\leq e^{-t}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Lemma 2.20 implies that\n\\[\n-I_{X_{1}}(\\mu+\\epsilon)\\leq\\inf_{\\lambda>0}\\left[-\\lambda\\epsilon+b^{-2}\\left(e^{\\lambda b}-\\lambda b-1\\right)\\operatorname{Var}\\left(X_{1}\\right)\\right].\n\\]\nWe can set the derivative of the objective function on the right hand side with respect to \\(\\lambda\\) to zero at the minimum solution, and obtain the condition for the optimal \\(\\lambda\\) as follows:"}, {"n": 2, "claim": "\\[\n-\\epsilon+b^{-1}(e^{\\lambda b}-1)\\operatorname{Var}\\left(X_{1}\\right)=0.\n\\]\nThis gives the solution \\(\\lambda = b^{-1}\\ln (1 + \\epsilon b / \\mathrm{Var}(X_1))\\)."}, {"n": 3, "claim": "Plugging this solution into the objective function, we obtain\n\\[\n-I_{X_{1}}(\\mu+\\epsilon)\\leq-\\frac{\\operatorname{Var}\\left(X_{1}\\right)}{b^{2}}\\psi\\left(\\frac{\\epsilon b}{\\operatorname{Var}\\left(X_{1}\\right)}\\right).\n\\]\nThe first inequality of the theorem follows from an application of Theorem 2.5."}, {"n": 4, "claim": "Given \\(\\lambda \\in (0,3 / b)\\), it is easy to verify the following inequality using the Taylor expansion of the exponential function\n\\[\n\\begin{array}{l}\\Lambda_{X_{1}}(\\lambda)\\leq\\mu\\lambda+b^{-2}\\left[e^{\\lambda b}-\\lambda b-1\\right]\\operatorname{Var}(X_{1})\\\\ \\leq\\mu\\lambda+\\frac{\\operatorname{Var}\\left(X_{1}\\right)\\lambda^{2}}{2}\\sum_{m=0}^{\\infty}(\\lambda b/3)^{m}=\\mu\\lambda+\\frac{\\operatorname{Var}\\left(X_{1}\\right)\\lambda^{2}}{2(1-\\lambda b/3)}.\\tag{2.13}\\\\ \\end{array}\n\\]\nThe second and the third desired bounds follow from direct applications of Theorem 2.10 with \\(\\alpha = \\operatorname{Var}(X_1)\\) and \\(\\beta = b / 3\\)."}]}}, {"local_id": "T2", "statement": "Assume that \\(X_{1}\\in [0,1]\\).\nThen for all \\(\\epsilon > 0\\):\n\\[\n\\Pr\\left(\\bar{X}_{n}\\geq(1+\\epsilon)\\mu\\right)\\leq\\exp\\left[\\frac{-n\\mu\\epsilon^{2}}{2+\\epsilon}\\right],\n\\]\n\\[\n\\Pr\\left(\\bar{X}_{n}\\leq(1-\\epsilon)\\mu\\right)\\leq\\exp\\left[\\frac{-n\\mu\\epsilon^{2}}{2}\\right].\n\\]\nMoreover, for \\( t > 0 \\), we have:\n\\[\n\\Pr\\left(\\bar{X}_{n}\\geq\\mu+\\sqrt{\\frac{2\\mu t}{n}}+\\frac{t}{3 n}\\right)\\leq e^{-t}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "The first and the second results can be obtained from Theorem 2.17 and the inequality \\(\\mathrm{KL}(z||\\mu) \\geq (z - \\mu)^2 / \\max(2\\mu, \\mu + z)\\) (which is left as an exercise)."}, {"n": 2, "claim": "We then take \\(z = (1 + \\epsilon)\\mu\\) and \\(z = (1 - \\epsilon)\\mu\\) respectively for the first and the second inequalities."}, {"n": 3, "claim": "For the third inequality (which is sharper than the first inequality), we may apply Theorem 2.10."}, {"n": 4, "claim": "Just observe from Lemma 2.15 that when \\(0 < \\lambda < 3\\):"}, {"n": 5, "claim": "\\[\n\\begin{array}{l}\\Lambda_{X_{1}}(\\lambda)\\leq\\ln [(1-\\mu)e^{0}+\\mu e^{\\lambda}]\\\\ \\leq\\mu(e^{\\lambda}-1)=\\mu\\lambda+\\mu\\sum_{k\\geq 2}\\frac{\\lambda^{k}}{k!}\\\\ \\leq\\mu\\lambda+\\frac{\\mu\\lambda^{2}}{2(1-\\lambda/3)}.\\\\ \\end{array}\n\\]\nIn the above derivation, the equality used the Taylor expansion of exponential function."}, {"n": 6, "claim": "The last inequality used \\( k! \\geq 2 \\cdot 3^{k-2} \\) and the sum of infinite geometric series."}, {"n": 7, "claim": "We may take \\( \\alpha = \\mu \\) and \\( \\beta = 1/3 \\) in Theorem 2.10 to obtain the desired bound."}]}}, {"local_id": "T3", "statement": "If \\(X - \\mathbb{E}X \\leq b\\), then \\(\\forall \\lambda \\geq 0\\):\n\\[\n\\ln \\mathbb{E} e^{\\lambda X} \\leq \\lambda \\mathbb{E}X + \\lambda^{2} \\phi(\\lambda b) \\operatorname{Var}(X),\n\\]\nwhere \\(\\phi(z) = (e^{z} - z - 1)/z^{2}\\).", "proof": {"steps": [{"n": 1, "claim": "Let \\(X' = X - \\mathbb{E}X\\). We have\n\\[\n\\begin{array}{l}\n\\ln \\mathbb{E} e^{\\lambda X} = \\lambda \\mathbb{E}X + \\ln \\mathbb{E} e^{\\lambda X'} \\\\\n\\leq \\lambda \\mathbb{E}X + \\mathbb{E} e^{\\lambda X'} - 1 \\\\\n= \\lambda \\mathbb{E}X + \\lambda^{2} \\mathbb{E} \\frac{e^{\\lambda X'} - \\lambda X' - 1}{(\\lambda X')^{2}} (X')^{2} \\\\\n\\leq \\lambda \\mathbb{E}X + \\lambda^{2} \\mathbb{E} \\phi(\\lambda b) (X')^{2},\n\\end{array}\n\\]\nwhere the first inequality used \\(\\ln z \\leq z - 1\\); the second inequality follows from the fact that the function \\(\\phi(z)\\) is non-decreasing (left as an exercise) and \\(\\lambda X' \\leq \\lambda b\\)."}]}}, {"local_id": "T4", "statement": "Let \\(\\sigma_{i} = \\{\\pm 1\\}\\) be independent Bernoulli random variables ( each takes value \\(\\pm 1\\) with equal probability). Let \\(a_{i}\\) be fixed numbers \\((i=1,\\dots,n)\\) Then for all \\(\\epsilon >0\\) ..\n\\[\n\\Pr\\left(n^{-1}\\sum_{i=1}^{n}\\sigma_{i}a_{i}\\geq\\epsilon\\right)\\leq\\exp\\left[-\\frac{n\\epsilon^{2}}{2 n^{-1}\\sum_{i=1}^{n}a_{i}^{2}}\\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "Consider \\( X_{i} = \\sigma_{i}a_{i} \\) in Corollary 2.26. We can take \\( \\mu = 0 \\) and \\( b_{i} = a_{i}^{2} \\) to obtain the desired bound."}]}}, {"local_id": "T5", "statement": "For any \\(n\\) and \\(\\epsilon >0\\):\n\\[\n\\frac{1}{n}\\ln\\Pr(\\bar{X}_{n}\\geq\\mu+\\epsilon)\\leq-I_{X_{1}}(\\mu+\\epsilon)=\\inf_{\\lambda>0}\\left[-\\lambda(\\mu+\\epsilon)+\\ln\\mathbb{E}e^{\\lambda X_{1}}\\right],\n\\]\n\\[\n\\frac{1}{n}\\ln\\Pr(\\bar{X}_{n}\\leq\\mu-\\epsilon)\\leq-I_{X_{1}}(\\mu-\\epsilon)=\\inf_{\\lambda<0}\\left[-\\lambda(\\mu-\\epsilon)+\\ln\\mathbb{E}e^{\\lambda X_{1}}\\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "We choose \\( h(z) = e^{\\lambda nz} \\) in Theorem 2.2 with \\( S = \\{\\bar{X}_n - \\mu \\geq \\epsilon\\} \\)."}, {"n": 2, "claim": "For \\( \\lambda > 0 \\), we have\n\\[\n\\begin{array}{l}\\Pr(\\bar{X}_{n}\\geq\\mu+\\epsilon)\\leq\\frac{\\mathbb{E}e^{\\lambda n\\bar{X}_{n}}}{e^{\\lambda n(\\mu+\\epsilon)}}=\\frac{\\mathbb{E}e^{\\lambda\\sum_{i=1}^{n}X_{i}}}{e^{\\lambda n(\\mu+\\epsilon)}}\\\\ =\\frac{\\mathbb{E}\\prod_{i=1}^{n}e^{\\lambda X_{i}}}{e^{\\lambda n(\\mu+\\epsilon)}}=e^{-\\lambda n(\\mu+\\epsilon)}\\left[\\mathbb{E}e^{\\lambda X_{1}}\\right]^{n}.\\\\ \\end{array}\n\\]\nThe last equation used the independence of \\( X_{i} \\) as well as they are identically distributed."}, {"n": 3, "claim": "Therefore by taking logarithm, we obtain\n\\[\n\\ln\\Pr(\\bar{X}_{n}\\geq\\mu+\\epsilon)\\leq n[-\\lambda(\\mu+\\epsilon)+\\ln\\mathbb{E}e^{\\lambda X_{1}}].\n\\]\nTaking inf over \\(\\lambda > 0\\) on the right hand side, we obtain the first desired bound."}, {"n": 4, "claim": "Similarly, we can obtain the second bound."}]}}], "target_local_id": "T1", "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch7_theorem_7_7__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nPCA rank-1 objective and assumptions. Let \\(M \\in \\mathbb{R}^{d \\times d}\\) be symmetric positive semidefinite. For \\(x \\in \\mathbb{R}^d\\), define \\(g(x) := \\frac{1}{2}\\|M - xx^\\top\\|_F^2\\).\n\nRank-one population objective \\(g\\) and matrix \\(M\\). Let \\(z\\) be the target vector and define \\(M = zz^\\top\\). Consider the rank-one population objective \\[ g(x) = \\frac{1}{2}\\|M - xx^\\top\\|_F^2. \\] For this objective, \\(\\nabla g(x) = (M - xx^\\top)x\\).\n\nRank-one matrix completion objectives. In the rank-one matrix-completion setup, let \\(d\\) be the ambient dimension and let \\(\\epsilon>0\\) be the concentration error parameter. Let \\(z \\in \\mathbb{R}^d\\) be the ground-truth vector, normalized by \\(\\|z\\|_2 = 1\\) and incoherent with \\(\\|z\\|_\\infty \\leq \\mu/\\sqrt d\\), and set \\(M = zz^\\top\\). Let \\(\\Omega\\) be the random observed set of matrix entries, sampled with probability \\(p\\), and let \\(P_\\Omega\\) be the coordinate projection/observation operator onto \\(\\Omega\\). Define the observed empirical objective \\(f(x)=\\frac12\\langle P_\\Omega(M),xx^\\top\\rangle-\\frac14\\langle P_\\Omega(xx^\\top),xx^\\top\\rangle\\), the sampled version over \\(\\Omega\\). Define the population objective \\(g(x)=\\frac12\\langle M,xx^\\top\\rangle-\\frac14\\|xx^\\top\\|_F^2\\), so the proof uses the population-gradient identity \\(\\nabla g(x)=(M-xx^\\top)x\\).\n\nRank-one objective \\(g\\). Let \\(z\\) be the planted unit vector with \\(\\|z\\|_2 = 1\\), and let the rank-one matrix be \\(M = zz^\\top\\). Define the rank-one objective \\(g(x) = \\frac{1}{2}\\|M - xx^\\top\\|_F^2\\). For any vector \\(u\\), its Hessian satisfies the quadratic-form identity \\[u^\\top \\nabla^2 g(x) u = \\|u x^\\top + x u^\\top\\|_F^2 - 2 u^\\top (M - x x^\\top) u.\\] In particular this may be applied with \\(u = z\\).\n\nLocal minimum of a function. We say that \\(x\\) is a local minimum of a function \\(f\\) if there exists an open neighborhood \\(N\\) around \\(x\\) such that in \\(N\\), the function values are at least \\(f(x)\\).\n\nIncoherence. Without loss of generality, assume the ground truth vector \\(z\\in\\mathbb{R}^d\\) satisfies \\(\\|z\\|_2 = 1\\). \\(z\\) satisfies the incoherence condition if \\(\\|z\\|_{\\infty} \\leq \\dfrac{\\mu}{\\sqrt{d}}\\), where \\(\\mu\\) is considered to be a constant or logarithmic in dimension \\(d\\).\n\nProjection onto observed entries. Suppose \\(M \\in \\mathbb{R}^{d \\times d}\\) and \\(\\Omega \\subseteq [d] \\times [d]\\). We define \\(P_\\Omega(M)\\) to be the matrix obtained by zeroing out every entry of \\(M\\) outside \\(\\Omega\\).\n\nMatrix Completion. Suppose \\(M\\in \\mathbb{R}^{d\\times d}\\) and every entry of \\(M\\) is included in \\(\\Omega\\) with probability \\(p\\). The matrix completion task is to recover \\(M\\) (with respect to some loss functions) given the observation \\(P_{\\Omega}(M)\\).\n\nRank-one matrix-completion objectives. Let \\(M = zz^\\top\\). Define the population objective \\(g(x) = \\frac{1}{2}\\|M - xx^\\top\\|_F^2 = \\frac{1}{2}\\|zz^\\top - xx^\\top\\|_F^2\\) and the empirical objective \\(f(x) = \\frac{1}{2p}\\|P_\\Omega(M - xx^\\top)\\|_F^2\\). For the population objective, \\(\\nabla g(x) = (M - xx^\\top)x\\), and the Hessian quadratic form in direction \\(z\\) is \\(\\langle z, \\nabla^2 g(x) z\\rangle = \\|zx^\\top + xz^\\top\\|_F^2 - 2 z^\\top(zz^\\top - xx^\\top)z\\).", "theorems": [{"local_id": "T1", "statement": "Suppose \\(\\|x\\|_\\infty \\leq 2\\mu / \\sqrt{d}\\). If \\(x\\) satisfies \\(\\nabla f(x) = 0\\), then \\(\\langle x,z \\rangle^2 \\geq \\|x\\|_2^4 - \\epsilon\\) with high probability.", "proof": {"steps": [{"n": 1, "claim": "\\[\n\\nabla f(x) = 0 \\implies \\langle x, \\nabla f(x) \\rangle = 0\n\\implies \\langle x, \\nabla g(x) \\rangle \\approx \\langle x, \\nabla f(x) \\rangle/p \\pm \\epsilon \\quad \\text{(by the concentration lemma, i.e., the referenced result)}\n\\implies |\\langle x, (zz^\\top-xx^\\top)x \\rangle| \\leq \\epsilon \\quad \\text{w.h.p.}\n\\implies \\langle x,z \\rangle^2 \\geq \\|x\\|_2^4 - \\epsilon \\quad \\text{w.h.p.}\n\\]"}]}}, {"local_id": "T2", "statement": "Suppose \\(\\|x\\|_\\infty \\leq \\mu / \\sqrt{d}\\). If \\(\\nabla^2 f(x) \\succeq 0\\), then \\(\\|x\\|_2^2 \\geq 1/3 - \\epsilon/3\\) with high probability.", "proof": {"steps": [{"n": 1, "claim": "\\[\n\\nabla^2 f(x) \\succeq 0\n\\implies \\langle z, \\nabla^2 f(x)z \\rangle \\geq 0\n\\implies \\langle z, \\nabla^2g(x)z \\rangle \\geq -\\epsilon \\quad \\text{w.h.p. (by the concentration lemma, i.e., the referenced result)}\n\\implies 3\\|x\\|_2^2 \\geq 1-\\epsilon \\quad \\text{w.h.p.}\n\\implies \\|x\\|_2^2 \\geq 1/3 - \\epsilon/3 \\quad \\text{w.h.p.}\n\\]"}]}}, {"local_id": "T3", "statement": "All local minima of \\(g\\) are global minima (even though \\(g\\) is non-convex).", "proof": {"steps": [{"n": 1, "claim": "Step 1: Show that all stationary points must be eigenvectors. We have \\(\\nabla g(x) = -(M - xx^\\top)x\\), hence\n\\[\n\\nabla g(x) = 0 \\implies Mx = \\|x\\|_2^2 \\cdot x,\n\\]\nwhich implies that \\(x\\) is an eigenvector of \\(M\\) with eigenvalue \\(\\|x\\|_2^2\\). From the Eckart–Young–Mirsky theorem the global minimum (i.e. the best rank-1 approximation) is the eigenvector with the largest eigenvalue."}, {"n": 2, "claim": "Step 2: Show that all local minima must be eigenvectors of the largest eigenvalue."}, {"n": 3, "claim": "We use the second-order condition. For \\(x\\) to be a local minimum we need \\(\\nabla^2 g(x) \\succeq 0\\), i.e. for any \\(v \\in \\mathbb{R}^d\\),\n\\[\n\\langle v, \\nabla^2 g(x)\\, v \\rangle \\geq 0.\n\\]\nExpanding \\(g(x+v)\\) into \\(g(x) + \\text{linear term in }v + \\text{quadratic term in }v\\) (the quadratic term equals \\(\\frac{1}{2}\\langle v, \\nabla^2 g(x)\\,v\\rangle\\)),\n\\[\ng(x+v) = \\tfrac{1}{2}\\|M - (x+v)(x+v)^\\top\\|_F^2 = \\tfrac{1}{2}\\|M - xx^\\top\\|_F^2 - \\langle M - xx^\\top, xv^\\top + vx^\\top\\rangle + \\tfrac{1}{2}\\langle xv^\\top + vx^\\top, xv^\\top + vx^\\top\\rangle - \\langle M - xx^\\top, vv^\\top\\rangle + \\text{higher order terms in }v.\n\\]\nHence,\n\\[\n\\tfrac{1}{2}\\langle v, \\nabla^2 g(x)\\,v\\rangle = \\tfrac{1}{2}\\langle xv^\\top + vx^\\top, xv^\\top + vx^\\top\\rangle - \\langle M - xx^\\top, vv^\\top\\rangle = 2\\langle x, v\\rangle^2 + \\|x\\|_2^2\\|v\\|_2^2 - v^\\top M v.\n\\]\nPicking \\(v = v_1\\), the unit eigenvector with the largest eigenvalue \\(\\lambda_1\\), the local minimum condition requires\n\\[\n\\langle v_1, \\nabla^2 g(x)\\,v_1\\rangle = 2\\langle x, v_1\\rangle^2 - v_1^\\top M v_1 + \\|x\\|_2^2 \\geq 0.\n\\]\nSince all candidates for local minima are eigenvectors of \\(M\\), there are two cases:"}, {"n": 4, "claim": "- Case 1: \\(x\\) has eigenvalue \\(\\lambda_1\\). Then \\(x\\) is the global minimum (by the Eckart–Young–Mirsky theorem)."}, {"n": 5, "claim": "- Case 2: \\(x\\) has eigenvalue \\(\\lambda < \\lambda_1\\). Then \\(x\\) and \\(v_1\\) are orthogonal, so\n\\[\n2\\langle x, v_1\\rangle^2 - v_1^\\top M v_1 + \\|x\\|_2^2 = 0 - \\lambda_1 + \\lambda \\geq 0,\n\\]\nwhich implies \\(\\lambda \\geq \\lambda_1\\), a contradiction."}, {"n": 6, "claim": "In summary, if \\(x\\) is a stationary point and not a global minimum, then moving in the direction of \\(v_1\\) yields a second-order improvement and \\(x\\) cannot be a local minimum."}]}}, {"local_id": "T4", "statement": "If \\(\\nabla^2 g(x) \\succeq 0\\), then \\(\\|x\\|_2^2 \\geq 1/3\\).", "proof": {"steps": [{"n": 1, "claim": "\\[\n\\nabla^2 g(x) \\succeq 0\n\\implies \\langle z, \\nabla^2 g(x)z\\rangle \\geq 0\n\\implies \\|zx^\\top + xz^\\top\\|_F^2 - 2z^\\top(zz^\\top-xx^\\top)z \\geq 0\n\\implies 2 \\|x\\|^2_2 + 2 \\langle x, z\\rangle^2 - 2 + 2 \\langle x, z\\rangle^2 \\geq 0 \\quad\\text{(cyclic trace property)}\n\\implies 3\\|x\\|_2^2 = \\|x\\|_2^2 + 2\\|x\\|_2^2 \\geq \\|x\\|_2^2 + 2\\langle x,z \\rangle^2 \\geq 1 \\quad\\text{(by Cauchy-Schwarz)}\n\\implies \\|x\\|_2^2 \\geq 1/3.\n\\]"}]}}, {"local_id": "T5", "statement": "If \\(x\\) satisfies \\(\\nabla g(x) = 0\\), then \\(\\langle x,z \\rangle^2 = \\|x\\|_2^4\\).", "proof": {"steps": [{"n": 1, "claim": "\\[\n\\nabla g(x) = 0 \\implies \\langle x, \\nabla g(x) \\rangle = 0\n\\implies \\langle x, (zz^\\top-xx^\\top)x \\rangle = 0 \\quad (\\textup{because}~\\nabla g(x) = (M - xx^\\top)x)\n\\implies \\langle x,z \\rangle^2 = \\|x\\|_2^4.\n\\]"}]}}], "target_local_id": "T3", "topic_domain": "Optimization", "topic_subtopic": "Non-convex Geometry", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch15_theorem_15_6__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nequation (15.3). Let \\(\\Lambda_0\\) be a symmetric positive definite regularizer and let \\(\\psi\\) be a feature map. Given \\(\\mathcal{S}_n = \\{(X_1,Y_1),\\ldots,(X_n,Y_n)\\}\\), define\n\\[\n\\hat{\\Lambda}(\\mathcal{S}_{n})=\\Lambda_{0}+\\sum_{i=1}^{n}\\psi(X_{i})\\psi(X_{i})^{\\top}\n\\]\nand the ridge regression estimator\n\\[\n\\hat{w}(\\mathcal{S}_{n})=\\hat{\\Lambda}(\\mathcal{S}_{n})^{-1}\\sum_{i=1}^{n}\\psi(X_{i})Y_{i}.\\tag{15.3}\n\\]\nEquivalently, \\(\\hat{w}(\\mathcal{S}_{n})\\) minimizes\n\\[\n\\sum_{i=1}^{n}(w^{\\top}\\psi(X_{i})-Y_{i})^{2}+\\|w\\|_{\\Lambda_{0}}^{2}.\n\\]\n\ndefinition 10.6. Let \\(\\mathcal{F}_{L_1}(\\Psi)\\) be the point-wise closure of \\(\\cup_{A > 0}\\mathcal{F}_{A,L_1}(\\Psi)\\), then for any \\(f\\in \\mathcal{F}_{L_1}(\\Psi)\\),\n\\[\n\\|f\\|_{1}=\\lim_{\\epsilon\\rightarrow 0}\\inf\\left\\{\\|w\\|_{1}:\\sup_{x}\\left|f(x)-\\sum_{j=1}^{m}w_{j}\\psi(\\theta_{j},x)\\right|\\leq\\epsilon\\right\\}.\n\\]\nFor notational convenience, we write functions in \\(\\mathcal{F}_{L_1}(\\Psi)\\) as\n\\[\nf(x)=w^{\\top}\\psi(x),\n\\]\nwhere \\(\\psi(x)\\) is the infinite dimensional vector \\([\\psi(\\theta, x)]_{\\theta \\in \\Theta}\\), and \\(\\|f\\|_1 = \\|w\\|_1\\).\nFor smooth-loss functions (see Definition A.5), there is a strong relationship between \\( L_{1} \\) regularization and \\( L_{0} \\) regularization, in that functions with small \\( L_{1} \\) norms can be sparsified. Such a sparsification result was first shown for the least squares loss, and referred to as Maurey's lemma in Pisier (1980-1981).\n\ndefinition 6.3. Given \\(S_{n}=\\{Z_{1},\\ldots,Z_{n}\\}\\), the (one-sided) empirical Rademacher complexity of \\(\\mathcal{G}\\) is defined as\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n}) = \\mathbb{E}_{\\sigma}\\sup_{w\\in \\Omega}\\frac{1}{n}\\sum_{i = 1}^{n}\\sigma_{i}\\phi (w,Z_{i}),\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R(\\mathcal{G},\\mathcal{S}_{n}).\n\\]\nThe following result shows that the quantity \\(\\epsilon_{n}(\\mathcal{G},\\mathcal{D})\\) can be upper bounded by Rademacher complexity. It follows that an average oracle inequality can be obtained using Rademacher complexity. The proof employs the symmetrization technique, which was also used in Chapter 4 to obtain uniform convergence bounds from empirical covering numbers.\n\nequation (15.3). The ridge regression method defines, for each \\(n\\),\n\\[\n\\hat{w}(\\mathcal{S}_{n})=\\arg\\min_{w}\\left[\\sum_{i=1}^{n}(Y_{i}-w^{\\top}\\psi(X_{i}))^{2}+\\|w\\|_{\\Lambda_{0}}^{2}\\right].\n\\]\nEquivalently,\n\\[\n\\hat{w}(\\mathcal{S}_{n})=\\Lambda_{n}^{-1}\\sum_{i=1}^{n}\\psi(X_{i})Y_{i},\\quad\\Lambda_{n}=\\Lambda_{0}+\\sum_{s=1}^{n}\\psi(X_{s})\\psi(X_{s})^{\\top}.\n\\]\n\nexample 15.2. For least squares regression with noise variance \\(\\sigma^2\\), we may have\n\\[\np(y|w,x)=\\frac{1}{\\sqrt{2\\pi}\\sigma}\\exp\\left(-\\frac{(y-f(w,x))^{2}}{2\\sigma^{2}}\\right),\n\\]\nand \\((\\operatorname{let} Z = (x, y))\\):\n\\[\n\\phi(w,Z)=\\frac{(y-f(w,x))^{2}}{2\\sigma^{2}}+\\ln(\\sqrt{2\\pi}\\sigma).\n\\]\nWe may also consider the noise as part of the model parameter, and let\n\\[\np(y|[w,\\sigma],x)=\\frac{1}{\\sqrt{2\\pi}\\sigma}\\exp\\left(-\\frac{(y-f(w,x))^{2}}{2\\sigma^{2}}\\right)\n\\]\n\\[\n\\phi([w,\\sigma],Z)=\\frac{(y-f(w,x))^{2}}{2\\sigma^{2}}+\\ln(\\sqrt{2\\pi}\\sigma).\n\\]\nIn the Bayesian setting, the optimal statistical estimation of posterior density for log-loss is the Bayesian posterior averaging estimator. Here we consider a prior \\( p_0(w) \\) on \\( \\Omega \\). Given the training data \\( S_{n}=\\{Z_{1},\\dots,Z_{n}\\} \\), the posterior distribution is\n\\[\np(w|\\mathcal{S}_{n})=\\frac{\\prod_{i=1}^{n}p\\left(Y_{i}\\mid w,X_{i}\\right)p_{0}(w)}{\\int_{\\Omega}\\prod_{i=1}^{n}p\\left(Y_{i}\\mid w^{\\prime},X_{i}\\right)p_{0}\\left(w^{\\prime}\\right)d w^{\\prime}}.\\tag{15.1}\n\\]\nUnlike ERM, the Bayesian posterior averaging method does not use \\( p(y|\\hat{w},x) \\) for a specific \\( \\hat{w}\\in \\Omega \\) that is estimated from the training data, but instead use the averaged probability estimate over the posterior as follows:\n\\[\n\\hat{p}(y|x,\\mathcal{S}_{n})=\\int_{\\Omega}p(y|w,x)p(w|\\mathcal{S}_{n})d w.\\tag{15.2}\n\\]\nAs pointed out in Section 14.1, such a method is referred to as improper online learning.\nIn the online setting, we have the following result, which may be regarded as a generic regret bound for the Bayesian posterior model averaging method.\n\nexample 10.11. In two-layer neural networks, let \\(\\Psi\\) be an \\(L_{2}\\)-regularized ReLU function class:\n\\[\n\\Psi=\\left\\{\\psi(\\theta,x)=\\max(0,\\theta^{\\top}x):\\quad\\|\\theta\\|_{2}\\leq\\alpha,\\|x\\|_{2}\\leq\\beta\\right\\},\n\\]\nand the corresponding \\(L_{1}\\) regularized two-layer NN can be expressed as a function\n\\[\nf(x)=\\sum_{j=1}^{m}w_{j}\\psi\\left(\\theta_{j},x\\right):\\|w\\|_{1}\\leq A,\\|\\theta\\|_{2}\\leq\\alpha,\\|x\\|_{2}\\leq\\beta .\n\\]\nThis function class belongs to \\(\\mathcal{F}_{A,L_1}(\\Psi)\\). We thus obtain the following bound for \\(L_{1}\\) regularized two-layer NN:\n\\[\n\\begin{array}{l}R\\left(\\mathcal{F}_{A,L_{1}}(\\Psi),\\mathcal{S}_{n}\\right)\\leq 2AR(\\Psi,\\mathcal{S}_{n})\\tag{Theorem 10.8}\\\\ \\leq 2A\\alpha\\beta/\\sqrt{n},\\quad(\\text{C o r o l l a r y} 9. 21)\\\\ \\end{array}\n\\]\nwhere we note that \\(\\max (0,f)\\) is 1-Lipschitz in \\(f\\)\nNext, we may consider the following hard-constrained \\( L_{1} \\) regularized learning problem.\n\\[\n\\hat{w}=\\arg\\min_{w}\\frac{1}{n}\\sum_{i=1}^{n}L\\left(w^{\\top}\\psi\\left(X_{i}\\right),Y_{i}\\right),\\quad\\|w\\|_{1}\\leq A.\\tag{10.8}\n\\]\nSimilarly, we may consider the soft-regularized version as:\n\\[\n\\hat{w}=\\arg\\min_{w}\\frac{1}{n}\\sum_{i=1}^{n}L\\left(w^{\\top}\\psi\\left(X_{i}\\right),Y_{i}\\right)+\\lambda\\|w\\|_{1}.\\tag{10.9}\n\\]\nNow by using Theorem 6.31, we obtain the following result.\n\nsection 13.1 sequential observation setup. In Section 13.1, data are observed sequentially. At each time \\(t\\), after the history \\(S_{t-1}\\) has been observed, the next observation \\(Z_t\\) is drawn according to a distribution that may depend on that history. The observation is decomposed as \\(Z_t=(Z_t^{(x)},Z_t^{(y)})\\), where \\(Z_t^{(x)}\\) is the component determined by the past history or the chosen action at time \\(t\\), and \\(Z_t^{(y)}\\) is the random response component. The history after time \\(t\\) is \\(S_t=(Z_1,\\ldots,Z_t)\\), and \\(\\mathcal{S}_T\\) denotes the full sequential experiment/history through time \\(T\\). For a function depending on \\(S_t\\), the notation \\(\\mathbb{E}_{Z_t^{(y)}}[\\cdot]\\) means conditional expectation over the response component \\(Z_t^{(y)}\\), conditional on \\(S_{t-1}\\) and on \\(Z_t^{(x)}\\). Stopping times are adapted to the filtration generated by the histories \\(S_t\\).\n\nequation (15.3). Let \\(\\Lambda_0\\) be a symmetric invertible regularizer and let \\(\\psi\\) be a feature map. Given \\(\\mathcal{S}_n = \\{(X_1,Y_1),\\ldots,(X_n,Y_n)\\}\\), define\n\\[\n\\hat{\\Lambda}(\\mathcal{S}_{n})=\\Lambda_{0}+\\sum_{i=1}^{n}\\psi(X_{i})\\psi(X_{i})^{\\top}\n\\]\nand the ridge regression estimator\n\\[\n\\hat{w}(\\mathcal{S}_{n})=\\hat{\\Lambda}(\\mathcal{S}_{n})^{-1}\\sum_{i=1}^{n}\\psi(X_{i})Y_{i}.\\tag{15.3}\n\\]\nEquivalently, \\(\\hat{w}(\\mathcal{S}_{n})\\) minimizes\n\\[\n\\sum_{i=1}^{n}(w^{\\top}\\psi(X_{i})-Y_{i})^{2}+\\|w\\|_{\\Lambda_{0}}^{2}.\n\\]\n\nexample 2.14. Consider a bounded random variable: \\( X_{1} \\in [\\alpha, \\beta] \\). Then \\( X_{1} \\) is sub-Gaussian with \\( b = (\\beta - \\alpha)^{2}/4 \\).\nThe tail probability inequality of Theorem 2.12 can also be expressed in a different form. Consider \\( \\delta \\in (0,1) \\) such that \\( \\delta = \\exp(-t) \\), we have \\( t = \\ln(1/\\delta) \\). This means that we can alternatively express the first bound of Theorem 2.12 as follows. With probability at least \\( 1 - \\delta \\), we have\n\\[\n\\bar{X}_{n}<\\mu+\\sqrt{\\frac{2 b\\ln(1/\\delta)}{n}}.\n\\]\nThis form is often preferred in the theoretical analysis of machine learning algorithms.", "theorems": [{"local_id": "T1", "statement": "Assume that \\( Y_{t} \\in [0, M] \\) for \\( t \\geq 1 \\). Consider the ridge regression method of (15.3), and let\n\\[\n\\hat{Y}_{t}=\\max(0,\\min(M,\\hat{w}(\\mathcal{S}_{t-1})^{\\top}\\psi(X_{t}))).\n\\]\nWe have\n\\[\n\\sum_{t=1}^{T}(Y_{t}-\\hat{Y}_{t})^{2}\\leq\\inf_{w}\\left[\\sum_{t=1}^{T}(Y_{t}-w^{\\top}\\psi(X_{t}))^{2}+\\|w\\|_{\\Lambda_{0}}^{2}\\right]+M^{2}\\ln\\left|\\Lambda_{0}^{-1}\\Lambda_{T}\\right|,\n\\]\nwhere\n\\[\n\\Lambda_{T}=\\Lambda_{0}+\\sum_{s=1}^{T}\\psi(X_{s})\\psi(X_{s})^{\\top}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "We can apply Theorem 15.6 by taking \\(\\sigma^2 = M^2\\)."}, {"n": 2, "claim": "By using the following inequality\n\\[\n0\\leq\\frac{1-b_{t}}{b_{t}}+\\ln b_{t},\n\\]\nwe obtain\n\\[\n\\begin{array}{l}\\sum_{t=1}^{T}(Y_{t}-\\hat{Y}_{t})^{2}\\leq\\sum_{t=1}^{T}\\left[(Y_{t}-\\hat{Y}_{t})^{2}+M^{2}\\frac{1-b_{t}}{b_{t}}+M^{2}\\ln b_{t}\\right]\\\\ \\leq\\sum_{t=1}^{T}\\left[\\left(Y_{t}-\\hat{Y}_{t}\\right)^{2}+\\left(Y_{t}-\\hat{Y}_{t}\\right)^{2}\\frac{1-b_{t}}{b_{t}}+M^{2}\\ln b_{t}\\right]\\tag{15.5}\\\\ \\leq\\sum_{t=1}^{T}\\left[\\frac{(Y_{t}-\\hat{w}(\\mathcal{S}_{t-1})^{\\top}\\psi(X_{i}))^{2}}{b_{t}}+M^{2}\\ln b_{t}\\right],\\\\ \\end{array}\n\\]\nwhere \\( b_{t} \\) is defined in Theorem 15.6."}, {"n": 3, "claim": "Note that (15.5) used \\( 1 - b_{t} \\leq 0 \\) and \\( (Y_{t} - \\hat{Y}_{t})^{2} \\leq M^{2} \\)."}, {"n": 4, "claim": "The desired result is now a direct application of Theorem 15.6."}]}}, {"local_id": "T2", "statement": "Consider a sequence of random functions \\(\\xi_1(S_1),\\ldots,\\xi_n(S_n)\\) with a fixed number \\(n > 0\\). If for each \\(i\\): \\(\\sup \\xi_i - \\inf \\xi_i\\leq M_i\\) for some constant \\(M_{i}\\), then with probability at least \\(1 - \\delta\\),\n\\[\n\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i}<\\frac{1}{n}\\sum_{i=1}^{n}\\xi_{i}+\\sqrt{\\frac{\\sum_{i=1}^{n}M_{i}^{2}\\ln(1/\\delta)}{2 n^{2}}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Using sub-Gaussian moment bound (see Example 2.14), we can obtain the desired bound by taking \\(\\sigma_{i} = M_{i} / 2\\), and \\(4\\sigma^{2} = \\sum_{i = 1}^{n}M_{i}^{2}\\) in Theorem 13.3. \\(\\square\\)"}]}}, {"local_id": "T3", "statement": "We have\n\\[\n\\begin{array}{l}-\\sum_{t=1}^{T}\\ln\\hat{p}\\left(Y_{t}\\mid X_{t},\\mathcal{S}_{t-1}\\right)= -\\ln\\mathbb{E}_{w\\sim p_{0}}\\prod_{t=1}^{T}p\\left(Y_{t}\\mid w,X_{t}\\right)\\\\ =\\inf_{q\\in\\Delta(\\Omega)}\\left[-\\mathbb{E}_{w\\sim q}\\sum_{t=1}^{T}\\ln p(Y_{t}|w,X_{t})+\\mathbb{E}_{w\\sim q}\\ln\\frac{q(w)}{p_{0}(w)}\\right],\\\\ \\end{array}\n\\]\nwhere \\(\\Delta (\\Omega)\\) is the set of probability distributions over \\(\\Omega\\)", "proof": {"steps": [{"n": 1, "claim": "We have\n\\[\n\\begin{array}{l}\\ln\\hat{p}(Y_{t}|X_{t},\\mathcal{S}_{t-1})=\\ln\\int_{\\Omega}p(Y_{t}|w,X_{t})p(w|\\mathcal{S}_{t-1})d w\\\\ =\\ln\\frac{\\int_{\\Omega}\\prod_{i=1}^{t}p(Y_{i}|w,X_{i})p_{0}(w)d w}{\\int_{\\Omega}\\prod_{i=1}^{t-1}p(Y_{i}|w^{\\prime},X_{i})p_{0}(w^{\\prime})d w^{\\prime}}\\\\ =\\ln\\mathbb{E}_{w\\in p_{0}}\\prod_{i=1}^{t}p(Y_{i}|w,X_{i})-\\ln\\mathbb{E}_{w\\in p_{0}}\\prod_{i=1}^{t-1}p(Y_{i}|w,X_{i}).\\\\ \\end{array}\n\\]\nBy summing over \\( t = 1 \\) to \\( t = T \\), we obtain\n\\[\n\\begin{array}{l}\\sum_{t=1}^{T}\\ln\\hat{p}\\left(Y_{t}\\mid x_{t},\\mathcal{S}_{t-1}\\right)=\\ln\\mathbb{E}_{w\\in p_{0}}\\prod_{t=1}^{T}p\\left(Y_{t}\\mid w,X_{t}\\right)\\\\ =\\sup_{q}\\left[\\mathbb{E}_{w\\sim q}\\ln\\prod_{t=1}^{T}p\\left(Y_{t}|w,X_{t}\\right)-\\mathbb{E}_{w\\sim q}\\ln\\frac{q(w)}{p_{0}(w)}\\right],\\\\ \\end{array}\n\\]\nwhere the second equality used Proposition 7.16."}, {"n": 2, "claim": "This implies the desired result."}]}}, {"local_id": "T4", "statement": "Consider probability model\n\\[\np(y|w,x)=N(w^{\\top}\\psi(x),\\sigma^{2}),\n\\]\nwith prior\n\\[\np_{0}(w)=N(0,\\sigma^{2}\\Lambda_{0}^{-1}).\n\\]\nThen given \\(\\mathcal{S}_n=\\{(X_1,Y_1),\\ldots,(X_n,Y_n)\\}\\) , we have\n\\[\np(w|\\mathcal{S}_{n})=N(\\hat{w}(\\mathcal{S}_{n}),\\sigma^{2}\\hat{\\Lambda}(\\mathcal{S}_{n})^{-1}),\n\\]\nwhere \\(\\hat{w} (S_n)\\) is given by (15.3) and\n\\[\n\\hat{\\Lambda}\\left(\\mathcal{S}_{n}\\right)=\\sum_{i=1}^{n}\\psi\\left(X_{i}\\right)\\psi\\left(X_{i}\\right)^{\\top}+\\Lambda_{0}.\n\\]\nGiven \\( x \\), the posterior distribution of \\( y \\) is\n\\[\n\\hat{p}(y|x,\\mathcal{S}_{n})=N\\left(\\hat{w}(\\mathcal{S}_{n})^{\\top}\\psi(x),\\sigma^{2}+\\sigma^{2}\\psi(x)^{\\top}\\hat{\\Lambda}(\\mathcal{S}_{n})^{-1}\\psi(x)\\right).\n\\]", "proof": {"steps": [{"n": 1, "claim": "It is clear that\n\\[\np(w|\\mathcal{S}_{n})\\propto\\exp\\left(-\\sum_{i=1}^{n}\\frac{(w^{\\top}\\psi(X_{i})-Y_{i})^{2}}{2\\sigma^{2}}-\\frac{\\|w\\|_{\\Lambda_{0}}^{2}}{2\\sigma^{2}}\\right).\n\\]\nNote that (15.3) implies that\n\\[\n\\begin{array}{l}\\sum_{i=1}^{n}\\frac{\\left(w^{\\top}\\psi\\left(X_{i}\\right)-Y_{i}\\right)^{2}}{2\\sigma^{2}}+\\frac{\\|w\\|_{\\Lambda_{0}}^{2}}{2\\sigma^{2}}\\\\ =\\sum_{i=1}^{n}\\frac{\\left(\\hat{w}^{\\top}\\psi\\left(X_{i}\\right)-Y_{i}\\right)^{2}}{2\\sigma^{2}}+\\frac{\\|\\hat{w}\\|_{\\Lambda_{0}}^{2}}{2\\sigma^{2}}+\\frac{1}{2\\sigma^{2}}(w-\\hat{w})^{\\top}\\hat{\\Lambda}\\left(S_{n}\\right)(w-\\hat{w}).\\tag{15.4}\\\\ \\end{array}\n\\]\nThis implies the first desired result."}, {"n": 2, "claim": "Moreover, given \\( x \\), and let the random variable \\( u = w^{\\top}\\psi (x) \\) with \\( w\\sim p(w|S_n) \\), we have\n\\[\nu|x,\\mathcal{S}_{n}\\sim N\\left(\\hat{w}^{\\top}\\psi(x),\\sigma^{2}\\psi(x)^{\\top}\\hat{\\Lambda}(\\mathcal{S}_{n})^{-1}\\psi(x)\\right).\n\\]\nSince in posterior distribution, the observation \\( y|u \\sim u + \\epsilon \\) with \\( \\epsilon \\sim N(0, \\sigma^2) \\), we know that\n\\[\ny|x,\\mathcal{S}_{n}\\sim N(\\hat{w}^{\\top}\\psi(x),\\sigma^{2}\\psi(x)^{\\top}\\hat{\\Lambda}^{-1}\\psi(x)+\\sigma^{2}).\n\\]\nThis implies the desired result."}]}}, {"local_id": "T5", "statement": "Consider the ridge regression method of (15.3). We have the following result for any \\(\\sigma \\geq 0\\) and for all observed sequence \\(S_T\\):\n\\[\n\\begin{array}{l}\\sum_{t=1}^{T}\\left[\\frac{\\left(Y_{t}-\\hat{w}\\left(\\mathcal{S}_{t-1}\\right)^{\\top}\\psi\\left(X_{t}\\right)\\right)^{2}}{b_{t}}+\\sigma^{2}\\ln b_{t}\\right]\\\\ =\\inf_{w}\\left[\\sum_{t=1}^{T}\\left(Y_{t}-w^{\\top}\\psi\\left(X_{t}\\right)\\right)^{2}+\\|w\\|_{\\Lambda_{0}}^{2}\\right]+\\sigma^{2}\\ln\\left|\\Lambda_{0}^{-1}\\Lambda_{T}\\right|,\\\\ \\end{array}\n\\]\nwhere\n\\[\n\\Lambda_{t}=\\Lambda_{0}+\\sum_{s=1}^{t}\\psi(X_{s})\\psi(X_{s})^{\\top},\n\\]\nand \\(b_{t} = 1 + \\psi (X_{t})^{\\top}\\Lambda_{t - 1}^{-1}\\psi (X_{t}).\\)", "proof": {"steps": [{"n": 1, "claim": "Assume \\( w \\in \\mathbb{R}^d \\). We note that from Gaussian integration that\n\\[\n\\begin{array}{l}\\mathbb{E}_{w\\sim p_{0}}\\prod_{i=1}^{T}p\\left(Y_{i}|w,X_{i}\\right)\\\\ =\\int\\frac{\\left|\\Lambda_{0}^{-1}\\right|^{-1/2}}{(2\\pi)^{(T+d)/2}\\sigma^{T}\\sigma^{d}}\\exp\\left(-\\frac{1}{2\\sigma^{2}}\\sum_{i=1}^{T}\\left(Y_{i}-w^{\\top}\\psi\\left(X_{i}\\right)\\right)^{2}-\\frac{\\left\\|w\\right\\|_{\\Lambda_{0}}^{2}}{2\\sigma^{2}}\\right)d w\\\\ =\\frac{\\left|\\Lambda_{0}^{-1}\\Lambda_{T}\\right|^{-1/2}}{\\left(2\\pi\\right)^{T/2}\\sigma^{T}}\\exp\\left(-\\frac{1}{2\\sigma^{2}}\\sum_{i=1}^{T}\\left(Y_{i}-\\hat{w}\\left(\\mathcal{S}_{T}\\right)^{\\top}\\psi(X_{i})\\right)^{2}-\\frac{\\left\\|\\hat{w}\\left(\\mathcal{S}_{T}\\right)\\right\\|_{\\Lambda_{0}}^{2}}{2\\sigma^{2}}\\right),\\\\ \\end{array}\n\\]\nwhere the last equation used Gaussian integration with decomposition (15.4)."}, {"n": 2, "claim": "That is,\n\\[\n\\begin{array}{l}-\\ln\\mathbb{E}_{w\\sim p_{0}}\\prod_{i=1}^{T}p(Y_{i}|w,X_{i})=T\\ln(\\sqrt{2\\pi}\\sigma)+\\frac{1}{2}\\ln\\left|\\Lambda_{0}^{-1}\\Lambda_{T}\\right|\\\\ +\\frac{1}{2\\sigma^{2}}\\sum_{i=1}^{T}(Y_{i}-\\hat{w}(\\mathcal{S}_{T})^{\\top}\\psi(X_{i}))^{2}+\\frac{\\|\\hat{w}(\\mathcal{S}_{T})\\|_{\\Lambda_{0}}^{2}}{2\\sigma^{2}}.\\\\ \\end{array}\n\\]\nMoreover, from Proposition 15.5, we have\n\\[\n\\begin{array}{l}\\sum_{t=1}^{T}-\\ln\\hat{p}(Y_{t}|X_{t},\\mathcal{S}_{t-1})\\\\ =\\sum_{t=1}^{T}\\left[\\frac{\\left(Y_{t}-\\hat{w}\\left(\\mathcal{S}_{t-1}\\right)^{\\top}\\psi\\left(X_{t}\\right)\\right)^{2}}{2\\sigma^{2}b_{t}}+\\frac{1}{2}\\ln(b_{t})+\\ln(\\sqrt{2\\pi}\\sigma)\\right].\\\\ \\end{array}\n\\]\nThe desired result now follows from Theorem 15.3."}]}}], "target_local_id": "T5", "topic_domain": "Online Learning & RL", "topic_subtopic": "Online Learning & Regret", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch5_theorem_5_10__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nEmpirical Rademacher complexity. Given a dataset \\(S = \\{z_1, \\dots, z_n\\}\\), the empirical Rademacher complexity is defined as\n\\[\nR_S(\\mathcal{F}) \\stackrel{\\Delta}{=} \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right].\n\\]\n\\(R_S(\\mathcal{F})\\) is a function of both the function class \\(\\mathcal{F}\\) and the dataset \\(S\\).\n\nTwo-layer ReLU network. Given a width \\(m\\), let \\(w_1,\\dots,w_m \\in \\mathbb{R}\\), \\(u_1,\\dots,u_m \\in \\mathbb{R}^d\\), and let \\(\\phi\\) denote the ReLU activation. For \\(\\theta=(w_1,\\dots,w_m,u_1,\\dots,u_m)\\), define the two-layer ReLU network\n\\[\nf_{\\theta}(x)=\\sum_{j=1}^m w_j \\phi\\left(u_j^\\top x\\right).\n\\]\n\n\\(\\epsilon\\)-cover. Let \\(\\epsilon > 0\\). An \\(\\epsilon\\)-cover of a set \\(S\\) with respect to a distance metric \\(\\rho\\) is a subset \\(C \\subseteq S\\) such that for all \\(x \\in S\\) there exists \\(x' \\in C\\) with \\(\\rho(x, x') \\leq \\epsilon\\), or equivalently,\n\\[\nS \\subseteq \\bigcup_{x \\in C} \\mathrm{Ball}(x, \\epsilon, \\rho), \\quad \\text{where} \\quad \\mathrm{Ball}(x, \\epsilon, \\rho) \\triangleq \\{x' : \\rho(x, x') \\leq \\epsilon\\}.\n\\]\n\nGeneralized margin. Let \\(f:\\mathbb{R}^d \\to \\mathbb{R}\\) be a classification model. A function \\(g_f(x,y)\\) is a generalized margin for \\(f\\) if \\(g_f(x,y)=0\\) whenever \\(f(x)y \\leq 0\\), and \\(g_f(x,y)>0\\) whenever \\(f(x)y > 0\\).\n\nInfinity-covering number. For a class \\(\\mathcal{F}\\) of functions on an input domain \\(\\mathcal{X}\\), \\(N_\\infty(\\epsilon,\\mathcal{F})\\) is the minimum cardinality of an \\(\\epsilon\\)-cover with respect to the metric \\(\\rho(f,f') = \\sup_{x \\in \\mathcal{X}} |f(x)-f'(x)| = \\|f-f'\\|_\\infty\\). In the multi-output case, the Euclidean norm is used inside the supremum.\n\nRademacher complexity. Let \\(\\mathcal{F}\\) be a family of functions mapping \\(Z \\mapsto \\mathbb{R}\\), and let \\(P\\) be a distribution over \\(Z\\). The (average) Rademacher complexity of \\(\\mathcal{F}\\) is defined as\n\\[\nR_n(F) \\triangleq \\mathbb{E}_{z_1, \\dots, z_n \\overset{\\text{iid}}{\\sim} P} \\left[ \\mathbb{E}_{\\sigma_1, \\dots, \\sigma_n \\overset{\\text{iid}}{\\sim}\\{ \\pm 1 \\}} \\left[ \\sup_{f\\in F} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right] \\right],\n\\]\nwhere \\(\\sigma_1, \\dots, \\sigma_n\\) are independent Rademacher random variables, i.e. each taking on the value of \\(1\\) or \\(-1\\) with probability \\(1/2\\).", "theorems": [{"local_id": "T1", "statement": "Let \\(C(\\theta)=\\sum_{j=1}^{m}\\left|w_{j}\\right|\\left\\|u_{j}\\right\\|_{2}\\), and for some constant \\(B > 0\\) consider the hypothesis class\n\\[\n\\mathcal{H}=\\left\\{f_{\\theta} \\mid C(\\theta) \\leq B\\right\\}.\n\\]\nIf \\(\\left\\|x^{(i)}\\right\\|_{2} \\leq C\\) for all \\(i \\in\\{1, \\ldots, n\\}\\), then\n\\[\nR_{S}(\\mathcal{H}) \\leq \\frac{2 B C}{\\sqrt{n}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Due to the positive homogeneity of the ReLU function \\(\\phi\\), it will be useful to define the \\(\\ell_2\\)-normalized weight vector \\(\\bar{u}_j \\stackrel{\\Delta}{=} u_j / \\|u_j\\|_2\\) so that \\(\\phi\\left(u_j^\\top x\\right) = \\|u_j\\|_2 \\cdot \\phi(\\bar{u}_j^\\top x)\\)."}, {"n": 2, "claim": "The empirical Rademacher complexity satisfies\n\\[\n\\begin{aligned}\nR_S(\\mathcal{H}) &= \\frac{1}{n}\\mathbb{E}_{\\sigma}\\left[ \\sup_{\\theta} \\sum_{i=1}^n \\sigma_i f_{\\theta}\\left(x^{(i)}\\right) \\right] \\\\\n&= \\frac{1}{n}\\mathbb{E}_{\\sigma}\\left[ \\sup_{\\theta} \\sum_{i=1}^n \\sigma_i \\left[\\sum_{j=1}^m w_j \\phi\\left(u_j^\\top x^{(i)}\\right) \\right] \\right] \\quad \\text{(by definition of }f_\\theta\\text{)} \\\\\n&= \\frac{1}{n}\\mathbb{E}_{\\sigma}\\left[ \\sup_{\\theta} \\sum_{i=1}^n \\sigma_i \\left[\\sum_{j=1}^m w_j \\|u_j\\|_2 \\phi\\left(\\bar{u}_j^\\top x^{(i)}\\right) \\right] \\right] \\quad \\text{(by positive homogeneity of }\\phi\\text{)} \\\\\n&= \\frac{1}{n}\\mathbb{E}_{\\sigma}\\left[ \\sup_{\\theta} \\sum_{j=1}^m w_j \\|u_j\\|_2 \\left[ \\sum_{i=1}^n \\sigma_i \\phi\\left(\\bar{u}_j^\\top x^{(i)}\\right) \\right] \\right] \\\\\n&\\leq \\frac{1}{n}\\mathbb{E}_{\\sigma}\\left[ \\sup_{\\theta} \\sum_{j=1}^m |w_j| \\|u_j\\|_2 \\max_{k \\in [m]}\\left| \\sum_{i=1}^n \\sigma_i \\phi\\left(\\bar{u}_k^\\top x^{(i)}\\right) \\right| \\right] \\quad \\left(\\text{because }\\sum_j \\alpha_j \\beta_j \\leq \\sum_j |\\alpha_j| \\max_{k} |\\beta_k|\\right) \\\\\n&\\leq \\frac{B}{n} \\mathbb{E}_{\\sigma}\\left[ \\sup_{\\theta = (w, U)} \\max_{k \\in [m]} \\left| \\sum_{i=1}^n \\sigma_i \\phi\\left(\\bar{u}_k^\\top x^{(i)}\\right) \\right| \\right] \\quad \\text{(because }C(\\theta) \\leq B\\text{)} \\\\\n&= \\frac{B}{n} \\mathbb{E}_{\\sigma}\\left[ \\sup_{\\bar{u}: \\|\\bar{u}\\|_2 = 1} \\left| \\sum_{i=1}^n \\sigma_i \\phi\\left(\\bar{u}^\\top x^{(i)}\\right) \\right| \\right] \\\\\n&\\le \\frac{B}{n} \\mathbb{E}_{\\sigma}\\left[ \\sup_{\\bar{u}: \\|\\bar{u}\\|_2 \\le 1} \\left| \\sum_{i=1}^n \\sigma_i \\phi\\left(\\bar{u}^\\top x^{(i)}\\right) \\right| \\right] \\\\\n&\\le \\frac{2B}{n} \\mathbb{E}_{\\sigma}\\left[ \\sup_{\\bar{u}: \\|\\bar{u}\\|_2 \\le 1} \\sum_{i=1}^n \\sigma_i \\phi\\left(\\bar{u}^\\top x^{(i)}\\right) \\right] \\quad \\text{(see the referenced result)} \\\\\n&= 2B \\cdot R_S(\\mathcal{H}'),\n\\end{aligned}\n\\]\nwhere \\(\\mathcal{H}' = \\left\\{x \\mapsto \\phi(\\bar{u}^\\top x) : \\bar{u} \\in \\mathbb{R}^d, \\|\\bar{u}\\|_2 \\leq 1 \\right\\}\\)."}, {"n": 3, "claim": "By Talagrand's lemma, since \\(\\phi\\) is \\(1\\)-Lipschitz, \\(R_S(\\mathcal{H}') \\leq R_S(\\mathcal{H}'')\\) where \\(\\mathcal{H}'' = \\left\\{x \\mapsto \\bar{u}^\\top x : \\bar{u} \\in \\mathbb{R}^d, \\|\\bar{u}\\|_2 \\leq 1 \\right\\}\\) is a linear hypothesis space."}, {"n": 4, "claim": "Using \\(R_S(\\mathcal{H}'') \\leq \\frac{C}{\\sqrt{n}}\\) by the referenced \\(\\ell_2\\) linear-model theorem then concludes the proof."}]}}, {"local_id": "T2", "statement": "Let \\(\\mathcal{H} = \\{x \\mapsto \\langle w,x \\rangle \\mid w \\in \\mathbb{R}^d,\\, \\|w\\|_1 \\le B\\}\\) for some constant \\(B > 0\\). Moreover, assume \\(\\|x^{(i)}\\|_\\infty \\leq C\\) for some constant \\(C > 0\\) and all points in \\(S = \\{x^{(i)}\\}_{i=1}^n \\subset \\mathbb{R}^d\\). Then\n\\[\nR_S(\\mathcal{H}) \\leq BC\\sqrt{\\frac{2\\log(2d)}{n}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "By definition,\n\\[\nR_S(\\mathcal{H}) = \\mathbb{E}_\\sigma\\left[ \\sup_{\\|w\\|_1 \\le B} \\frac{1}{n} \\sum_{i=1}^n \\sigma_i \\langle w, x^{(i)} \\rangle \\right] = \\frac{B}{n} \\mathbb{E}_\\sigma\\left[ \\left\\|\\sum_{i=1}^n \\sigma_i x^{(i)}\\right\\|_\\infty \\right],\n\\]\nusing \\(\\sup_{\\|w\\|_1 \\leq B}\\langle w,v\\rangle = B\\|v\\|_\\infty\\) (since the \\(\\ell_\\infty\\)-norm is dual to the \\(\\ell_1\\)-norm by Holder's inequality)."}, {"n": 2, "claim": "Since the supremum over \\(\\|w\\|_1 \\leq 1\\) is attained at one of the vertices \\(\\mathcal{W} = \\bigcup_{i=1}^d \\{-e_i, e_i\\}\\), defining the restricted class \\(\\bar{\\mathcal{H}} = \\{x \\mapsto \\langle w, x \\rangle \\mid w \\in \\mathcal{W}\\}\\) gives\n\\[\nR_S(\\mathcal{H}) = B R_S(\\bar{\\mathcal{H}}).\n\\]\nThe class \\(\\bar{\\mathcal{H}}\\) is finite with \\(|\\bar{\\mathcal{H}}| = 2d\\)."}, {"n": 3, "claim": "For any \\(w \\in \\mathcal{W}\\),\n\\[\n\\frac{1}{n}\\sum_{i=1}^n \\langle w, x^{(i)} \\rangle^2 \\leq \\frac{1}{n}\\sum_{i=1}^n \\|x^{(i)}\\|^2_\\infty \\leq C^2,\n\\]\nso \\(\\bar{\\mathcal{H}}\\) is bounded in the \\(\\ell_2\\)-metric."}, {"n": 4, "claim": "Applying Massart's lemma,\n\\[\nR_S(\\mathcal{H}) = B R_S(\\bar{\\mathcal{H}}) \\leq BC\\sqrt{\\frac{2\\log|\\bar{\\mathcal{H}}|}{n}} = BC\\sqrt{\\frac{2\\log(2d)}{n}}.\n\\]"}]}}, {"local_id": "T3", "statement": "Let \\(\\sigma = (\\sigma_1, \\dots, \\sigma_n)\\) and \\(f_{\\theta}(x) = \\left(f_{\\theta}\\left(x^{(1)}\\right), \\dots, f_{\\theta}\\left(x^{(n)}\\right)\\right)\\). Suppose that for any \\(\\sigma \\in \\{\\pm 1\\}^n\\), \\(\\sup_{\\theta} \\langle \\sigma, f_{\\theta}(x) \\rangle \\geq 0\\). Then,\n\\[\n\\mathbb{E}_{\\sigma}\\left[ \\sup_{\\theta} \\left| \\langle \\sigma, f_{\\theta}(x) \\rangle \\right| \\right] \\leq 2\\, \\mathbb{E}_{\\sigma}\\left[ \\sup_{\\theta} \\langle \\sigma, f_{\\theta}(x) \\rangle \\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "Letting \\(\\phi\\) be the ReLU function, the lemma's assumption implies that \\(\\sup_{\\theta} \\phi\\left(\\langle \\sigma, f_{\\theta}(x) \\rangle\\right) = \\sup_{\\theta}\\langle \\sigma, f_{\\theta}(x) \\rangle\\) for any \\(\\sigma \\in \\{\\pm 1\\}^n\\)."}, {"n": 2, "claim": "Observing that \\(|z| = \\phi(z) + \\phi(-z)\\),\n\\[\n\\begin{aligned}\n\\sup_{\\theta} \\left|\\langle \\sigma, f_{\\theta}(x) \\rangle\\right|\n&= \\sup_{\\theta} \\left[ \\phi \\left(\\langle \\sigma, f_{\\theta}(x) \\rangle \\right) + \\phi \\left(\\langle -\\sigma, f_{\\theta}(x) \\rangle \\right)\\right] \\\\\n&\\le \\sup_{\\theta} \\phi \\left(\\langle \\sigma, f_{\\theta}(x) \\rangle \\right) + \\sup_{\\theta} \\phi \\left(\\langle -\\sigma, f_{\\theta}(x) \\rangle \\right) \\\\\n&= \\sup_{\\theta} \\langle \\sigma, f_{\\theta}(x) \\rangle + \\sup_{\\theta} \\langle -\\sigma, f_{\\theta}(x) \\rangle.\n\\end{aligned}\n\\]\nTaking the expectation over \\(\\sigma\\) (and noting that \\(\\sigma \\overset{d}{=} -\\sigma\\)), we get the desired conclusion."}]}}, {"local_id": "T4", "statement": "Let \\(F\\) and \\(f\\) be functions taking \\(\\Omega\\) into \\(\\mathbb{R}\\), and assume that \\(F\\) is \\(\\alpha\\)-strongly-convex with respect to the norm \\(\\|\\cdot\\|\\) and that \\(f\\) is convex. Let \\(w = \\textup{argmin}_{z \\in \\Omega} F(z)\\) and \\(w' = \\textup{argmin}_{z \\in \\Omega} [f(z) + F(z)]\\). Then\n\\[\n0 \\leq f(w) - f(w') \\leq \\frac{1}{\\alpha} \\|\\nabla f(w)\\|_*^2,\n\\]\nwhere \\(\\|\\cdot\\|_*\\) is the dual norm of \\(\\|\\cdot\\|\\).", "proof": {"steps": [{"n": 1, "claim": "By strong convexity,\n\\[\nF(w') - F(w) \\geq \\langle \\nabla F(w), w' - w \\rangle + \\frac{\\alpha}{2} \\|w - w'\\|^2 \\geq \\frac{\\alpha}{2} \\|w - w'\\|^2,\n\\]\nwhere in the second step we used the fact that the KKT optimality conditions for \\(w\\) imply \\(\\langle \\nabla F(w), w' - w \\rangle \\geq 0\\)."}, {"n": 2, "claim": "(Informally, if \\(\\Omega = \\mathbb{R}^d\\), then \\(\\nabla F(w) = 0\\) as \\(w\\) minimizes \\(F\\). If \\(\\Omega\\) is a convex subset of \\(\\mathbb{R}^d\\), then the gradient \\(\\nabla F(w)\\) must be perpendicular to the tangent to \\(\\Omega\\) at \\(w\\); otherwise, we could move in the direction of the negative gradient and project back to the set \\(\\Omega\\) to lower the value of \\(F\\).) Since \\(F + f\\) is also \\(\\alpha\\)-strongly convex, exactly the same argument implies:"}, {"n": 3, "claim": "\\[\n[F(w) + f(w)] - [F(w') + f(w')] \\geq \\frac{\\alpha}{2} \\|w - w'\\|^2.\n\\]\nAdding these two inequalities gives\n\\[\nf(w) - f(w') \\geq \\alpha \\|w - w'\\|^2.\n\\]\nSince this lower bound is clearly positive, this shows \\(0 \\leq f(w) - f(w')\\)."}, {"n": 4, "claim": "Next, we prove the upper bound on \\(f(w) - f(w')\\)."}, {"n": 5, "claim": "Rearranging the lower bound inequality, we obtain\n\\[\n\\|w - w'\\| \\leq \\sqrt{\\frac{1}{\\alpha} [f(w) - f(w')]}.\n\\]\nSince \\(f\\) is convex, we have \\(f(w') \\geq f(w) + \\langle \\nabla f(w), w' - w \\rangle\\)."}, {"n": 6, "claim": "Rearranging gives\n\\[\nf(w) - f(w') \\leq \\langle \\nabla f(w), w - w' \\rangle \\leq \\|\\nabla f(w)\\|_* \\cdot \\|w - w'\\| \\leq \\|\\nabla f(w)\\|_* \\sqrt{\\frac{1}{\\alpha} [f(w) - f(w')]},\n\\]\nwhere the second inequality is by Cauchy-Schwarz."}, {"n": 7, "claim": "Since \\(f(w) - f(w') \\geq 0\\), we can square both sides to conclude\n\\[\n[f(w) - f(w')]^2 \\leq \\|\\nabla f(w)\\|_*^2 \\frac{1}{\\alpha} [f(w) - f(w')].\n\\]\nDividing both sides by \\(f(w) - f(w')\\) gives the desired upper bound."}]}}, {"local_id": "T5", "statement": "Let \\(\\mathcal{H} = \\{x \\mapsto \\langle w,x \\rangle \\mid w \\in \\mathbb{R}^d,\\, \\|w\\|_2 \\le B\\}\\) for some constant \\(B > 0\\). Moreover, assume \\(\\mathbb{E}_{x \\sim P}[\\|x\\|_2^2] \\leq C^2\\), where \\(P\\) is some distribution and \\(C > 0\\) is a constant. Then\n\\[\nR_S(\\mathcal{H}) \\le \\frac{B}{n} \\sqrt{\\sum_{i=1}^n \\|x^{(i)}\\|_2^2},\n\\]\nand\n\\[\nR_n(\\mathcal{H}) \\le \\frac{BC}{\\sqrt{n}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "By definition,\n\\[\nR_S(\\mathcal{H}) = \\mathbb{E}_\\sigma\\left[ \\sup_{\\|w\\|_2 \\le B} \\frac{1}{n} \\sum_{i=1}^n \\sigma_i \\langle w, x^{(i)} \\rangle \\right] = \\frac{1}{n} \\mathbb{E}_\\sigma\\left[ \\sup_{\\|w\\|_2 \\le B} \\left\\langle w, \\sum_{i=1}^n \\sigma_i x^{(i)} \\right\\rangle \\right] = \\frac{B}{n} \\mathbb{E}_\\sigma\\left[ \\left\\|\\sum_{i=1}^n \\sigma_i x^{(i)}\\right\\|_2 \\right],\n\\]\nusing \\(\\sup_{\\|w\\|_2 \\le B} \\langle w,v\\rangle = B\\|v\\|_2\\)."}, {"n": 2, "claim": "By Jensen's inequality applied to \\(\\alpha \\mapsto \\alpha^2\\),\n\\[\n\\frac{B}{n} \\mathbb{E}_\\sigma\\left[ \\left\\|\\sum_{i=1}^n \\sigma_i x^{(i)}\\right\\|_2 \\right] \\leq \\frac{B}{n} \\sqrt{ \\mathbb{E}_\\sigma\\left[\\left\\|\\sum_{i=1}^n \\sigma_i x^{(i)}\\right\\|_2^2\\right] } = \\frac{B}{n} \\sqrt{\\sum_{i=1}^n \\|x^{(i)}\\|_2^2},\n\\]\nwhere the last equality uses independence of the \\(\\sigma_i\\) and \\(\\mathbb{E}[\\sigma_i] = 0\\)."}, {"n": 3, "claim": "This proves the bound on \\(R_S(\\mathcal{H})\\). For the average Rademacher complexity,\n\\[\nR_n(\\mathcal{H}) = \\mathbb{E}[R_S(\\mathcal{H})] \\le \\frac{B}{n} \\mathbb{E}\\left[\\sqrt{\\sum_{i=1}^n \\|x^{(i)}\\|_2^2}\\right] \\le \\frac{B}{n} \\sqrt{\\sum_{i=1}^n \\mathbb{E}\\left[\\|x^{(i)}\\|_2^2\\right]} \\le \\frac{BC}{\\sqrt{n}},\n\\]\nwhere the first inequality is Jensen's inequality and the second uses \\(\\mathbb{E}_{x \\sim P}[\\|x\\|_2^2] \\leq C^2\\)."}]}}], "target_local_id": "T1", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch6_theorem_6_25__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nEmpirical and Uniform Covering Number. Given an empirical distribution \\( S_{n}=\\{Z_{1},\\ldots,Z_{n}\\} \\), we define the pseudometric \\( d = L_{p}(S_{n}) \\) as\n\\[\nd(\\phi,\\phi^{\\prime})=\\left[\\frac{1}{n}\\sum_{i=1}^{n}|\\phi(Z_{i})-\\phi^{\\prime}(Z_{i})|^{p}\\right]^{1/p}.\n\\]\nThe corresponding metric covering number \\(N(\\epsilon, \\mathcal{G}, L_p(\\mathcal{S}_n))\\) is referred to as the\nempirical \\( L_{p} \\) covering number. Given \\( n \\), the largest \\( L_{p} \\) covering number over empirical distribution \\( S_{n} \\) is referred to as the uniform \\( L_{p} \\) covering number\n\\[\nN_{p}(\\epsilon,\\mathcal{G},n)=\\sup_{\\mathcal{S}_{n}}N(\\epsilon,\\mathcal{G},L_{p}(\\mathcal{S}_{n})).\n\\]\nSince the \\(L_{p}(\\mathcal{S}_{n})\\) pseudometric increases with \\(p\\), we have the following simple result.\n\ntheorem 6.23. If \\(\\mathcal{G}\\) is a finite function class with \\(|\\mathcal{G}| = N\\), then\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n})\\leq\\sup_{g\\in\\mathcal{G}}\\|g\\|_{L_{2}(\\mathcal{S}_{n})}\\cdot\\sqrt{\\frac{2\\ln N}{n}}.\n\\]\n\ndefinition 6.3. Given \\(S_{n}=\\{Z_{1},\\ldots,Z_{n}\\}\\), the (one-sided) empirical Rademacher complexity of \\(\\mathcal{G}\\) is defined as\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n}) = \\mathbb{E}_{\\sigma}\\sup_{w\\in \\Omega}\\frac{1}{n}\\sum_{i = 1}^{n}\\sigma_{i}\\phi (w,Z_{i}),\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R(\\mathcal{G},\\mathcal{S}_{n}).\n\\]\nThe following result shows that the quantity \\(\\epsilon_{n}(\\mathcal{G},\\mathcal{D})\\) can be upper bounded by Rademacher complexity. It follows that an average oracle inequality can be obtained using Rademacher complexity. The proof employs the symmetrization technique, which was also used in Chapter 4 to obtain uniform convergence bounds from empirical covering numbers.\n\ndefinition 6.11. Consider a function class \\(\\mathcal{G} = \\{\\phi(w, Z) : w \\in \\Omega\\}\\), and let \\(h\\) be an offset function of the following form\n\\[\nh(w,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{i=1}^{n}h(w,Z_{i}),\\quad h(w,z)=h_{0}(w)+h_{1}(w,z).\\tag{6.3}\n\\]\nGiven \\( S_{n}=\\{Z_{1},\\ldots,Z_{n}\\} \\), the (one-sided) empirical Rademacher complexity of \\( \\mathcal{G} \\) with offset \\( h \\) decomposition (6.3) is defined as\n\\[\nR^{h}(\\mathcal{G},\\mathcal{S}_{n})=\\mathbb{E}_{\\sigma}\\sup_{w\\in\\Omega}\\left[\\frac{1}{n}\\sum_{i=1}^{n}\\sigma_{i}[\\phi(w,Z_{i})+0. 5h_{1}(w,Z_{i})]-0. 5h(w,\\mathcal{S}_{n})\\right],\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}^{h}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R^{h}(\\mathcal{G},\\mathcal{S}_{n}).\n\\]\nWe note that the standard Rademacher complexity can be regarded as a special case of the offset Rademacher complexity with \\( h(\\cdot) = h_0(\\cdot) = h_1(\\cdot) = 0 \\):\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n})=R^{0}(\\mathcal{G},\\mathcal{S}_{n}),\\quad R_{n}(\\mathcal{G},\\mathcal{D})=R_{n}^{0}(\\mathcal{G},\\mathcal{D}).\n\\]\nIt should be pointed out that the decomposition of \\( h \\) in (6.3) may not be unique, and the offset Rademacher complexity relies on the specific decomposition used.\nAs shown in Example 6.10, we allow distribution dependent offset in the definition of uniform convergence complexity. We also allow distribution dependent offset in the definition of offset Rademacher complexity.\nThe following result shows that for some cases, offset Rademacher complexity can be obtained easily for some function classes.\n\ndefinition 6.6. Consider any known data-dependent offset function \\( h(w, S_n) \\). Define the uniform convergence complexity of a function class \\( \\mathcal{G} \\) with offset \\( h \\) as\n\\[\n\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\sup_{w\\in\\Omega}[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})].\\tag{6.2}\n\\]\nThis quantity measures the one-sided expected uniform convergence of function class \\(\\mathcal{G}\\) with offset function \\(h(w, S_n)\\).\nWe note that\n\\[\n\\epsilon_{n}(\\mathcal{G},\\mathcal{D})=\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D}),\\qquad \\text{w i t h} h=0.\n\\]\nWe have the following generalization of Theorem 6.2, with a similar proof.\n\ndefinition 6.17. Given a function \\( h(w, S_n) \\), we define\n\\[\n\\Delta_{n}h(w)=\\sup\\{n\\cdot|h(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n}^{\\prime})|: |\\mathcal{S}_{n}\\cap\\mathcal{S}_{n}^{\\prime}| =n-1\\}.\n\\]\n\ndefinition 6.1. Given an empirical process \\(\\{\\phi(w, S_n) : w \\in \\Omega\\}\\), with \\(S_n \\sim \\mathcal{D}^n\\). Define the expected supremum of this empirical process as\n\\[\n\\epsilon_{n}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}}\\sup_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})\\right],\n\\]\nwhich will be referred to as the uniform convergence complexity of the function class \\(\\mathcal{G}\\).\nThe smaller this quantity is, the closer the gap between the training error and the test error is, which implies that we have less overfitting. In Chapter 3, we\nobtained large probability uniform convergence results for empirical processes, and then derived oracle inequalities in large probabilities. In the following, we show that if average convergence can be obtained, then we can derive oracle inequalities in expectation directly.\n\nequation (6.1). The approximate regularized ERM method is to find \\( \\hat{w} \\in \\Omega \\) such that, for some \\( \\epsilon' \\geq 0 \\),\n\\[\n\\phi(\\hat{w},\\mathcal{S}_{n})+h(\\hat{w},\\mathcal{S}_{n})\\leq\\inf_{w\\in\\Omega}[\\phi(w,\\mathcal{S}_{n})+h(w,\\mathcal{S}_{n})]+\\epsilon'.\\tag{6.1}\n\\]\nWhen \\(h = 0\\), this reduces to the plain approximate ERM method.\n\nexample 2.14. Consider a bounded random variable: \\( X_{1} \\in [\\alpha, \\beta] \\). Then \\( X_{1} \\) is sub-Gaussian with \\( b = (\\beta - \\alpha)^{2}/4 \\).\nThe tail probability inequality of Theorem 2.12 can also be expressed in a different form. Consider \\( \\delta \\in (0,1) \\) such that \\( \\delta = \\exp(-t) \\), we have \\( t = \\ln(1/\\delta) \\). This means that we can alternatively express the first bound of Theorem 2.12 as follows. With probability at least \\( 1 - \\delta \\), we have\n\\[\n\\bar{X}_{n}<\\mu+\\sqrt{\\frac{2 b\\ln(1/\\delta)}{n}}.\n\\]\nThis form is often preferred in the theoretical analysis of machine learning algorithms.", "theorems": [{"local_id": "T1", "statement": "Consider a real valued function class \\(\\mathcal{F} = \\{f(z):\\mathcal{Z}\\to \\mathbb{R}\\}\\). Let \\(\\mathcal{D}\\) be a distribution on \\(\\mathcal{Z}\\). Assume that there exists \\(M,\\sigma >0\\) so that \\(\\forall f\\in \\mathcal{F}\\),\n\\(\\sigma^2 \\geq \\operatorname{Var}_{Z \\sim \\mathcal{D}}[f(Z)]\\), and \\(\\sup_{z' \\in \\mathcal{Z}} [\\mathbb{E}_{Z \\sim \\mathcal{D}} f(Z) - f(z')] \\leq M\\). Let \\(\\mathcal{S}_n = \\{Z_1, \\ldots, Z_n\\}\\) be \\(n\\) independent random variables from \\(\\mathcal{D}\\). Then with probability at least \\(1 - \\delta\\) over \\(\\mathcal{S}_n\\), for all \\(f \\in \\mathcal{F}\\),\n\\[\n\\begin{array}{l}\\mathbb{E}_{Z\\sim\\mathcal{D}}f(Z)-\\frac{1}{n}\\sum_{i=1}^{n}f\\left(Z_{i}\\right)\\\\ \\leq\\epsilon_{n}(\\mathcal{F},\\mathcal{D})+\\sqrt{\\frac{(4 M\\epsilon_{n}(\\mathcal{F},\\mathcal{D})+2\\sigma^{2})\\ln(1/\\delta)}{n}}+\\frac{M\\ln(1/\\delta)}{3 n}\\\\ \\leq 2\\epsilon_{n}(\\mathcal{F},\\mathcal{D})+\\sqrt{\\frac{2\\sigma^{2}\\ln(1/\\delta)}{n}}+\\frac{4 M\\ln(1/\\delta)}{3 n},\\\\ \\end{array}\n\\]\nwhere \\(\\epsilon_{n}(\\mathcal{F},\\mathcal{D})\\) is Definition 6.1.", "proof": {"steps": [{"n": 1, "claim": "Let\n\\[\n\\zeta=\\sup_{f\\in\\mathcal{F}}\\left[n\\mathbb{E}_{Z\\sim\\mathcal{D}}f(Z)-\\sum_{i=1}^{n}f(Z_{i})\\right],\n\\]\nand\n\\[\n\\zeta_{k}=\\sup_{f\\in\\mathcal{F}}\\left[(n-1)\\mathbb{E}_{Z\\sim\\mathcal{D}}f(Z)-\\sum_{i\\neq k}f(Z_{i})\\right].\n\\]\nAssume that \\(\\zeta_{k}\\) is achieved at \\(f_{k}\\), and \\(\\zeta\\) achieved at \\(f_{0}\\)."}, {"n": 2, "claim": "Then\n\\[\n\\sum_{k=1}^{n}\\left[\\zeta-\\zeta_{k}\\right]\\leq\\sum_{k=1}^{n}\\left[\\mathbb{E}_{Z\\sim\\mathcal{D}}f_{0}(Z)-f_{0}\\left(Z_{k}\\right)\\right]=\\zeta .\n\\]\nLet\n\\[\n\\zeta_{k}^{\\prime}=\\left[n\\mathbb{E}_{Z\\sim\\mathcal{D}}f_{k}(Z)-\\sum_{i=1}^{n}f_{k}(Z_{i})\\right]-\\zeta_{k},\n\\]\nthen\n\\[\n\\zeta_{k}^{\\prime}\\leq\\zeta-\\zeta_{k}\\leq\\left[\\mathbb{E}_{Z}f_{0}(Z)-f_{0}(Z_{k})\\right]\\leq M.\n\\]\nMoreover, since \\(Z_{k}\\) is independent of \\(f_{k}\\), we have\n\\[\n\\mathbb{E}_{Z_{k}}\\zeta_{k}^{\\prime}=\\mathbb{E}_{Z_{k}}\\left[\\mathbb{E}_{Z}f_{k}(Z)-f_{k}\\left(Z_{k}\\right)\\right]=0\n\\]\n\\[\n\\mathbb{E}_{Z_{k}}\\left(\\zeta_{k}^{\\prime}\\right)^{2}=\\mathbb{E}_{Z_{k}}\\left[\\mathbb{E}_{Z}f_{k}(Z)-f_{k}(Z_{k})\\right]^{2}\\leq\\sigma^{2}.\n\\]\nBy taking \\( u = 1 \\) in Theorem 6.33, we obtain the first desired bound."}, {"n": 3, "claim": "The second bound is a consequence of\n\\[\n\\begin{array}{l}\\sqrt{\\left(4 M\\epsilon_{n}(\\mathcal{F},\\mathcal{D})+2\\sigma^{2}\\right)\\ln(1/\\delta)/n}\\\\ \\leq\\sqrt{4 M\\epsilon_{n}(\\mathcal{F},\\mathcal{D})\\ln(1/\\delta)/n}+\\sqrt{2\\sigma^{2}\\ln(1/\\delta)/n}\\\\ \\leq\\epsilon_{n}(\\mathcal{F},\\mathcal{D})+M\\ln(1/\\delta)/n+\\sqrt{2\\sigma^{2}\\ln(1/\\delta)/n}.\\\\ \\end{array}\n\\]\nThe first inequality used \\(\\sqrt{a + b} \\leq \\sqrt{a} + \\sqrt{b}\\), and the second inequality used \\(\\sqrt{4ab} \\leq a + b\\)."}]}}, {"local_id": "T2", "statement": "We have\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n})\\leq\\inf_{\\epsilon\\geq 0}\\left[4\\epsilon+12\\int_{\\epsilon}^{\\infty}\\sqrt{\\frac{\\ln N(\\epsilon^{\\prime},\\mathcal{G},L_{2}(\\mathcal{S}_{n}))}{n}}d\\epsilon^{\\prime}\\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(B = \\sup_{g\\in \\mathcal{G}}\\| g\\|_{L_2(S_n)}\\), and let \\(\\epsilon_{\\ell} = 2^{-\\ell}B\\) for \\(\\ell = 0,1,\\ldots\\)."}, {"n": 2, "claim": "Let \\(\\mathcal{G}_{\\ell}\\) be an \\(\\epsilon_{\\ell}\\)-cover of \\(\\mathcal{G}\\) with metric \\(L_{2}(S_{n})\\), and \\(N_{\\ell} = |\\mathcal{G}_{\\ell}| = N(\\epsilon_{\\ell},\\mathcal{G},L_{2}(S_{n}))\\)."}, {"n": 3, "claim": "We may let \\(\\mathcal{G}_0 = \\{0\\}\\) at scale \\(\\epsilon_0 = B\\)."}, {"n": 4, "claim": "For each \\( g \\in \\mathcal{G} \\), we consider \\( g_{\\ell}(g) \\in \\mathcal{G}_{\\ell} \\) so that \\( \\| g - g_{\\ell}(g)\\|_{L_2(\\mathcal{S}_n)} \\leq \\epsilon_{\\ell} \\)."}, {"n": 5, "claim": "The key idea in chaining is to rewrite \\( g \\in \\mathcal{G} \\) using the following multi-scale decomposition:"}, {"n": 6, "claim": "\\[\ng=(g-g_{L}(g))+\\sum_{\\ell=1}^{L}(g_{\\ell}(g)-g_{\\ell-1}(g)).\n\\]\nWe also have\n\\[\n\\left\\|g_{\\ell}(g)-g_{\\ell-1}(g)\\right\\|_{L_{2}\\left(\\mathcal{S}_{n}\\right)}\\leq\\left\\|g_{\\ell}(g)-g\\right\\|_{L_{2}\\left(\\mathcal{S}_{n}\\right)}+\\left\\|g_{\\ell-1}(g)-g\\right\\|_{L_{2}\\left(\\mathcal{S}_{n}\\right)}\\leq 3\\epsilon_{\\ell}.\\tag{6.7}\n\\]\nThe number of distinct \\( g_{\\ell}(g) - g_{\\ell - 1}(g) \\) is no more than \\( N_{\\ell}N_{\\ell - 1} \\)."}, {"n": 7, "claim": "It implies that\n\\[\n\\begin{array}{l}R(\\mathcal{G},\\mathcal{S}_{n})=\\mathbb{E}_{\\sigma}\\sup_{g\\in\\mathcal{G}}\\frac{1}{n}\\sum_{i=1}^{n}\\sigma_{i}\\left[(g-g_{L}(g))(Z_{i})+\\sum_{\\ell=1}^{L}(g_{\\ell}(g)-g_{\\ell-1}(g))(Z_{i})\\right]\\\\ \\leq\\mathbb{E}_{\\sigma}\\sup_{g\\in\\mathcal{G}}\\frac{1}{n}\\sum_{i=1}^{n}\\sigma_{i}(g-g_{L}(g))(Z_{i})+\\sum_{\\ell=1}^{L}\\mathbb{E}_{\\sigma}\\sup_{g\\in\\mathcal{G}}\\frac{1}{n}\\sum_{i=1}^{n}\\sigma_{i}(g_{\\ell}(g)-g_{\\ell-1}(g))(Z_{i})\\\\ \\stackrel{(a)}{\\leq}\\epsilon_{L}+\\sum_{\\ell=1}^{L}\\sup_{g\\in\\mathcal{G}}\\|g_{\\ell}(g)-g_{\\ell-1}(g)\\|_{L_{2}(\\mathcal{S}_{n})}\\sqrt{\\frac{2\\ln [N_{\\ell}N_{\\ell-1}]}{n}}\\\\ \\stackrel{(b)}{\\leq}\\epsilon_{L}+3\\sum_{\\ell=1}^{L}\\epsilon_{\\ell}\\sqrt{\\frac{2\\ln\\left[N_{\\ell}N_{\\ell-1}\\right]}{n}}\\\\ \\leq\\epsilon_{L}+12\\sum_{\\ell=1}^{L}\\left(\\epsilon_{\\ell}-\\epsilon_{\\ell+1}\\right)\\sqrt{\\frac{\\ln\\left[N_{\\ell}\\right]}{n}}\\\\ \\leq\\epsilon_{L}+12\\int_{\\epsilon_{L}/2}^{\\infty}\\sqrt{\\frac{\\ln N(\\epsilon^{\\prime},\\mathcal{G},L_{2}(\\mathcal{S}_{n}))}{n}}d\\epsilon^{\\prime}.\\\\ \\end{array}\n\\]\nThe derivation of (a) used Theorem 6.23, and the derivation of (b) used (6.7)."}, {"n": 8, "claim": "The next two inequalities used \\( N(\\epsilon, \\mathcal{G}, L_2(S_n)) \\) is a non-increasing function of \\( \\epsilon \\)."}, {"n": 9, "claim": "Now given any \\( \\epsilon > 0 \\), we can choose \\( \\epsilon_L \\) so that \\( \\epsilon \\in [\\epsilon_L / 4, \\epsilon_L / 2] \\)."}, {"n": 10, "claim": "This leads to the desired result."}]}}, {"local_id": "T3", "statement": "Consider (6.1), and define\n\\[\n\\tilde{h}(w,\\mathcal{S}_{n})=h(w,\\mathcal{S}_{n})+h^{\\prime}(w),\n\\]\nwhere \\( h'(w) \\) is an arbitrary function of \\( w \\). Then\n\\[\n\\mathbb{E}_{\\mathcal{S}_{n}}[\\phi(\\hat{w},\\mathcal{D})-h^{\\prime}(\\hat{w})]\\leq\\inf_{w\\in\\Omega}[\\phi(w,\\mathcal{D})+\\mathbb{E}_{\\mathcal{S}_{n}}h(w,\\mathcal{S}_{n})]+\\epsilon^{\\prime}+\\epsilon_{n}^{\\tilde{h}}(\\mathcal{G},\\mathcal{D}).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let\n\\[\n\\tilde{\\phi}(w,z)=\\phi(w,z)-h^{\\prime}(w),\n\\]\nthen (6.1) remains the same with \\(\\phi(w, z)\\) replaced by \\(\\tilde{\\phi}(w, z)\\), and \\(h(w, S_n)\\) replaced by \\(\\tilde{h}(w, S_n)\\)."}, {"n": 2, "claim": "We can now apply Theorem 6.7 with \\(\\epsilon_n^h(\\mathcal{G}, \\mathcal{D})\\) replaced by\n\\[\n\\epsilon_{n}^{\\bar{h}}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}}\\sup_{w\\in\\Omega}[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})-h^{\\prime}(w)]\n\\]\nto obtain\n\\[\n\\mathbb{E}_{\\mathcal{S}_{n}}\\tilde{\\phi}(\\hat{w},\\mathcal{D})\\leq\\inf_{w\\in\\Omega}\\left[\\tilde{\\phi}(w,\\mathcal{D})+\\mathbb{E}_{\\mathcal{S}_{n}}\\tilde{h}(w,\\mathcal{S}_{n})\\right]+\\epsilon^{\\prime}+\\epsilon_{n}^{\\tilde{h}}(\\mathcal{G},\\mathcal{D}).\n\\]\nThis implies the desired bound."}]}}, {"local_id": "T4", "statement": "Assume that for some \\(M \\geq 0\\):\n\\[\n\\sup_{w\\in\\Omega}\\left[\\sup_{z,z^{\\prime}}[\\phi(w,z)-\\phi(w,z^{\\prime})]+\\Delta_{n}h(w)\\right]\\leq M.\n\\]\nThen with probability at least \\(1 - \\delta\\): for all \\(w \\in \\Omega\\)\n\\[\n\\phi(w,\\mathcal{D})\\leq\\phi(w,\\mathcal{S}_{n})+h(w,\\mathcal{S}_{n})+\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})+M\\sqrt{\\frac{\\ln(1/\\delta)}{2 n}}.\n\\]\nMoreover, assume that the decomposition (6.3) holds, then with probability at least \\(1 - \\delta\\): for all \\(w \\in \\Omega\\),\n\\[\n\\phi(w,\\mathcal{D})\\leq\\phi(w,\\mathcal{S}_{n})+h(w,\\mathcal{S}_{n})+2 R_{n}^{h}(\\mathcal{G},\\mathcal{D})+M\\sqrt{\\frac{\\ln(1/\\delta)}{2 n}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Consider \\( \\mathcal{S}_n = \\{Z_1, \\ldots, Z_n\\} \\) and \\( \\mathcal{S}_n' = \\{Z_1, \\ldots, Z_{i-1}, Z_i', Z_{i+1}, \\ldots, Z_n\\} \\)."}, {"n": 2, "claim": "Let \\( f(\\mathcal{S}_n) = \\sup_{w \\in \\Omega} [\\phi(w, \\mathcal{D}) - \\phi(w, \\mathcal{S}_n) - h(w, \\mathcal{S}_n)] \\)."}, {"n": 3, "claim": "For simplicity, we assume that the sup can be achieved at \\( \\hat{w} \\) as\n\\[\n\\hat{w}=\\arg\\max_{w\\in\\Omega}[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})].\n\\]\nThen\n\\[\n\\begin{array}{l}f\\left(\\mathcal{S}_{n}\\right)-f\\left(\\mathcal{S}_{n}^{\\prime}\\right)\\\\ =\\left[\\phi(\\hat{w},\\mathcal{D})-\\phi(\\hat{w},\\mathcal{S}_{n})-h(\\hat{w},\\mathcal{S}_{n})\\right]-\\sup_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n}^{\\prime})-h(w,\\mathcal{S}_{n}^{\\prime})\\right]\\\\ \\leq\\left[\\phi(\\hat{w},\\mathcal{D})-\\phi(\\hat{w},\\mathcal{S}_{n})-h(\\hat{w},\\mathcal{S}_{n})\\right]-\\left[\\phi(\\hat{w},\\mathcal{D})-\\phi(\\hat{w},\\mathcal{S}_{n}^{\\prime})-h(\\hat{w},\\mathcal{S}_{n}^{\\prime})\\right]\\\\ \\leq\\frac{1}{n}[\\phi(\\hat{w},Z_{i}^{\\prime})-\\phi(\\hat{w},Z_{i})+\\Delta_{n}h(\\hat{w})]\\leq M/n.\\\\ \\end{array}\n\\]\nSimilarly, \\( f(\\mathcal{S}_n') - f(\\mathcal{S}_n) \\leq M / n \\)."}, {"n": 4, "claim": "Therefore we may take \\( c_i = M / n \\) in Theorem 6.16, which implies the first desired result."}, {"n": 5, "claim": "The second bound follows from the estimate \\( \\epsilon_n^h (\\mathcal{G},\\mathcal{D}) \\leq 2R_n^h (\\mathcal{G},\\mathcal{D}) \\) of Theorem 6.14."}]}}, {"local_id": "T5", "statement": "Setup. Throughout, \\(\\gamma > 0\\).\n\nIf \\(\\mathcal{G}\\) is a finite function class with \\(|\\mathcal{G}| = N\\), then\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n})\\leq\\sup_{g\\in\\mathcal{G}}\\|g\\|_{L_{2}(\\mathcal{S}_{n})}\\cdot\\sqrt{\\frac{2\\ln N}{n}}.\n\\]\nIf moreover, for all \\( g \\in \\mathcal{G} \\), \\( g(z) \\in [0,1] \\). Consider the offset decomposition (6.3), and let \\( h_0(g) = 0 \\), and \\( h_1(g, S_n) = (\\gamma / n)\\sum_{i=1}^{n} g(Z_i) \\). Then we have\n\\[\nR^{h}(\\mathcal{G},\\mathcal{S}_{n})\\leq\\frac{(1+0.5\\gamma)^{2}\\ln N}{\\gamma n}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(B = \\sup_{g\\in \\mathcal{G}}\\| g\\|_{L_2(\\mathcal{S}_n)}\\)."}, {"n": 2, "claim": "Then we have for all \\(\\lambda >0\\)\n\\[\n\\begin{array}{l}R(\\mathcal{G},\\mathcal{S}_{n})=\\mathbb{E}_{\\sigma}\\sup_{g\\in\\mathcal{G}}\\frac{1}{n}\\sum_{i=1}^{n}\\sigma_{i}g(Z_{i})\\\\ \\stackrel{(a)}{\\leq}\\mathbb{E}_{\\sigma}\\frac{1}{\\lambda n}\\ln\\sum_{g\\in\\mathcal{G}}\\exp\\left[\\lambda\\sum_{i=1}^{n}\\sigma_{i}g(Z_{i})\\right]\\\\ \\stackrel{(b)}{\\leq}\\frac{1}{\\lambda n}\\ln\\mathbb{E}_{\\sigma}\\sum_{g\\in\\mathcal{G}}\\exp\\left[\\lambda\\sum_{i=1}^{n}\\sigma_{i}g(Z_{i})\\right]\\\\ =\\frac{1}{\\lambda n}\\ln\\sum_{g\\in\\mathcal{G}}\\prod_{i=1}^{n}\\mathbb{E}_{\\sigma_{i}}\\exp\\left[\\lambda\\sigma_{i}g(Z_{i})\\right]\\\\ \\stackrel{(c)}{\\leq}\\frac{1}{\\lambda n}\\ln\\sum_{g\\in\\mathcal{G}}\\prod_{i=1}^{n}\\exp\\left[\\lambda^{2}g\\left(Z_{i}\\right)^{2}/2\\right]\\leq\\frac{1}{\\lambda n}\\ln N\\exp\\left[\\lambda^{2}n B^{2}/2\\right].\\\\ \\end{array}\n\\]\nIn \\((a)\\), we used soft-max to bound the max operator."}, {"n": 3, "claim": "In \\((b)\\), we used Jensen's inequality and the concavity of logarithm."}, {"n": 4, "claim": "In \\((c)\\), we used the moment generating function for bounded random variables (see Example 2.14)."}, {"n": 5, "claim": "Now we can obtain the first desired bound by optimizing over \\(\\lambda > 0\\)."}, {"n": 6, "claim": "For the second desired bound, we can obtain by duplicating the previous steps up to step \\((c)\\) to obtain\n\\[\n\\begin{array}{l}R^{h}(\\mathcal{G},\\mathcal{S}_{n})=\\mathbb{E}_{\\sigma}\\sup_{g\\in\\mathcal{G}}\\frac{1}{n}\\sum_{i=1}^{n}[\\sigma_{i}(1+0. 5\\gamma)g(Z_{i})-0. 5\\gamma g(Z_{i})]\\\\ \\leq\\frac{1}{\\lambda n}\\ln\\sum_{g\\in\\mathcal{G}}\\exp\\left[\\sum_{i=1}^{n}[\\lambda^{2}(1+0. 5\\gamma)^{2}g(Z_{i})^{2}/2-0. 5\\lambda\\gamma g(Z_{i})]\\right].\\\\ \\end{array}\n\\]\nNow set \\(\\lambda = \\gamma /(1 + 0.5\\gamma)^2\\), we obtain \\(R^{h}(\\mathcal{G},S_{n})\\leq \\frac{1}{\\lambda n}\\ln \\sum_{g\\in \\mathcal{G}}1\\), which implies the second bound."}]}}], "target_local_id": "T2", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch9_theorem_9_2__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nBlack-Box Procedure. A \\textit{black-box procedure} generates a sequence of points \\(\\{x_t\\}\\) such that \\[ x_{t+1} \\in x_0 + \\Span\\{g_1, \\dots, g_t\\}, \\] and \\(g_s \\in \\partial f (x_s)\\).\n\nStrong convexity. A function \\(f\\colon\\Omega\\to\\mathbb{R}\\) is \\emph{\\(\\alpha\\)-strongly convex} if, for all \\(x,y\\in\\Omega\\), the following inequality holds for some \\(\\alpha>0\\): \\[ f(y) \\geqslant f(x) + g(x)^\\top(y-x) + \\frac{\\alpha}{2}\\left\\lVert x-y\\right\\rVert^2 \\]\n\nUniform stability. The uniform stability of an algorithm \\(A\\) is defined as\n\\[\n\\Delta_{\\sup}(A) = \\sup_{S, S' \\in (X\\times Y)^m } \\sup_{i \\in [m]} |\\ell(A(S), z_i') - \\ell(A(S^{(i)}, z_i')|\n\\]\n\nConvex set. A set \\(K\\subseteq\\mathbb{R}^n\\) is convex if the line segment between any two points in \\(K\\) is also contained in \\(K\\). Formally, for all \\(x,y\\in K\\) and all scalars \\(\\gamma\\in[0,1]\\) we have \\(\\gamma x+(1-\\gamma)y\\in K\\).", "theorems": [{"local_id": "T1", "statement": "If \\(f\\) is in addition \\(\\alpha\\)-strongly convex w.r.t to \\(\\|\\cdot\\|_{[1-\\gamma]}\\), then we get\n\\[\n\\mathbb{E}[f(x_{t+1}) - \\arg\\min_{x \\in \\mathbb{R}^{n}} f(x)] \\leqslant \\left(1 - \\frac{\\alpha}{\\sum_{i=1}^n \\beta_i^\\gamma}\\right)^t (f(x_1) - f(x^*)) \\ .\n\\]", "proof": {"steps": [{"n": 1, "claim": "We need the following lemma:\n\nLemma 12.4. Let \\(f\\) be an \\(\\alpha\\)-strongly convex function w.r.t. a norm \\(\\|\\cdot\\|\\). Then, \\(f(x) - f(x^{*}) \\leqslant \\frac{1}{2\\alpha}\\|\\nabla f(x)\\|_{*}^{2}\\)."}, {"n": 2, "claim": "Lemma 12.4 shows that\n\\[\n\\|\\nabla f(x_s)\\|^{*2}_{[1-\\gamma]} \\geqslant 2 \\alpha \\delta_s \\ .\n\\]\nOn the other hand, Theorem 12.2 showed that\n\\[\n\\delta_{t+1} - \\delta_t \\leqslant -\\frac{\\|\\nabla f(x_t)\\|^{*2}_{[1-\\gamma]}}{2\\sum_{i=1}^n \\beta_i^{\\gamma}}\n\\]\nCombining these two, we get\n\\begin{align*}\n\\delta_{t+1} - \\delta_t &\\leqslant -\\frac{\\alpha \\delta_t}{\\sum_{i=1}^n \\beta_i^{\\gamma}} \\\\\n\\delta_{t+1} &\\leqslant \\delta_t \\left(1 -\\frac{\\alpha}{\\sum_{i=1}^n \\beta_i^{\\gamma}} \\right)~.\n\\end{align*}\nApplying the above inequality recursively and recalling that \\(\\delta_t = \\mathbb{E}[f(x_t) - f(x^*)]\\) gives the result."}]}}, {"local_id": "T2", "statement": "For any function \\(\\varphi(x,z)\\), \\[\\min_x \\max_z \\varphi(x,z) \\geqslant \\max_z \\min_x \\varphi(x,z)\\,.\\]", "proof": {"steps": [{"n": 1, "claim": "The proof is essentially tautological. Note that we always have \\[\\varphi(x,z) \\geqslant \\min_x \\varphi(x,z).\\] Taking the maximization with respect to the second argument verifies that \\[\\max_z \\varphi(x,z) \\geqslant \\max_z \\min_x \\varphi(x,z) \\quad \\forall x\\,.\\] Now, minimizing the left hand side with respect to \\(x\\) proves \\[\\min_x \\max_z \\varphi(x,z) \\geqslant \\max_z \\min_x \\varphi(x,z)\\] which is precisely our assertion."}]}}, {"local_id": "T3", "statement": "Setup. A black-box procedure generates points with \\(x_{t+1}\\in x_0+\\mathrm{Span}\\{g_1,\\dots,g_t\\}\\), \\(g_s\\in\\partial f(x_s)\\); throughout we normalize the starting point to \\(x_0=0\\), which is without loss of generality since the hard instance may be translated.\n\nLet \\(t \\leqslant n\\), \\(L, R > 0\\). There exists a convex \\(L\\)-Lipschitz function \\(f\\) such that any black-box procedure satisfies \\[ \\min_{1 \\leqslant s \\leqslant t} f(x_s) - \\min_{x \\in B_2(R)} f(x) \\geqslant \\frac{RL}{2(1+\\sqrt{t})}. \\] Furthermore, there is an \\(\\alpha\\)-strongly convex, \\(L\\)-Lipschitz function \\(f\\) such that \\[ \\min_{1 \\leqslant s \\leqslant t} f(x_s) - \\min_{x \\in B_2(\\frac{L}{2\\alpha})} f(x) \\geqslant \\frac{L^2}{8\\alpha t}. \\]", "proof": {"steps": [{"n": 1, "claim": "Consider the function \\[ f(x) = \\gamma \\max_{1 \\leqslant i \\leqslant t} x[i] + \\frac{\\alpha}{2} \\|x\\|^2, \\] for some \\(\\gamma, \\alpha \\in \\mathbb{R}\\)."}, {"n": 2, "claim": "In the strongly convex case, \\(\\gamma\\) is a free parameter, and in the Lipschitz case both \\(\\alpha\\) and \\(\\gamma\\) are free parameters."}, {"n": 3, "claim": "By the subdifferential calculus, \\[ \\partial f(x) = \\alpha x + \\gamma \\conv\\{e_i \\colon i \\in \\argmax_{1 \\leqslant j \\leqslant t} x(j)\\}. \\] The function \\(f\\) is evidently \\(\\alpha\\)-strongly convex."}, {"n": 4, "claim": "Furthermore, if \\(\\|x\\| \\leqslant R\\) and \\(g \\in \\partial f(x)\\), then \\(\\|g\\| \\leqslant \\alpha R + \\gamma\\), so \\(f\\) is \\((\\alpha R + \\gamma)\\)-Lipschitz on \\(B_2(R)\\)."}, {"n": 5, "claim": "Suppose the gradient oracle returns \\(g_i = \\alpha x + \\gamma e_i\\), where \\(i\\) is the first coordinate such that \\(x[i] = \\max_{1 \\leqslant j \\leqslant t} x[j]\\)."}, {"n": 6, "claim": "An inductive argument then shows \\[ x_s \\in \\Span\\{e_1, \\dots, e_{s-1}\\} \\] Consequently, for \\(s \\leqslant t\\), \\(f(x_s) \\geqslant 0\\)."}, {"n": 7, "claim": "However, consider \\(y\\in \\mathbb{R}^n\\) such that \\[ y[i] = \\begin{cases} -\\frac{\\gamma}{\\alpha t} &\\text{if } 1 \\leqslant i \\leqslant t\\\\ 0 &\\text{otherwise}. \\end{cases} \\] Since \\(0 \\in \\partial f(y)\\), \\(y\\) is a minimizer of \\(f\\) with objective value \\[ f(y) = \\frac{-\\gamma^2}{\\alpha t} + \\frac{\\alpha}{2}\\frac{\\gamma^2}{\\alpha^2 t} = -\\frac{\\gamma^2}{2\\alpha t}, \\] and hence \\(f(x_s) - f(y) \\geqslant \\frac{\\gamma^2}{2 \\alpha t}\\)."}, {"n": 8, "claim": "We conclude the proof by appropriately choosing \\(\\alpha\\) and \\(\\gamma\\)."}, {"n": 9, "claim": "In the convex, Lipschitz case, set \\[ \\alpha = \\frac{L}{R} \\frac{1}{1 + \\sqrt{t}} \\quad \\text{and} \\quad \\gamma = L \\frac{\\sqrt{t}}{1 + \\sqrt{t}}. \\] Then, \\(f\\) is \\(L\\)-Lipschitz, \\[ \\|y\\| = \\sqrt{t \\left(\\frac{-\\gamma}{\\alpha t} \\right)^2} = \\frac{\\gamma}{\\alpha \\sqrt{t}} = R \\] and hence \\[ f(x_s) - \\min_{x \\in B_2(R)} f(x) = f(x_s) - f(y) \\geqslant \\frac{\\gamma^2}{2\\alpha t} = \\frac{RL}{2(1+\\sqrt{t})}. \\] In the strongly-convex case, set \\(\\gamma = \\frac{L}{2}\\) and take \\(R = \\frac{L}{2\\alpha}\\)."}, {"n": 10, "claim": "Then, \\(f\\) is \\(L\\)-Lipschitz, \\[ \\|y\\| = \\frac{\\gamma}{\\alpha \\sqrt{t}} = \\frac{L}{2\\alpha \\sqrt{t}} = \\frac{R}{\\sqrt{t}} \\leqslant R, \\] and therefore \\[ f(x_s) - \\min_{x \\in B_2(L/2\\alpha)} f(x) = f(x_s) - f(y) \\geqslant \\frac{LR}{4t} = \\frac{L^2}{8\\alpha t}. \\]"}]}}, {"local_id": "T4", "statement": "Let \\(\\Omega \\subseteq \\mathbb{R}^d\\) be a closed convex set. Let \\(\\mathcal{A}\\) denote the affine set \\(\\{x : Ax = b\\}\\) for some \\(A \\in \\mathbb{R}^{n \\times d}\\) and \\(b \\in \\mathbb{R}^n\\). Suppose that the set \\(\\operatorname{ri}(\\Omega) \\cap \\mathcal{A}\\) is non-empty. Then for any \\(x \\in \\Omega \\cap \\mathcal{A}\\), \\[\\mathcal{N}_{\\Omega \\cap \\mathcal{A}}(x) = \\mathcal{N}_{\\Omega}(x) + \\{A^\\top \\lambda : \\lambda \\in \\mathbb{R}^n\\}\\,.\\]", "proof": {"steps": [{"n": 1, "claim": "The ``\\(\\supseteq\\)'' assertion is straightforward."}, {"n": 2, "claim": "To see this, suppose \\(z \\in \\Omega \\cap \\mathcal{A}\\) and note that \\(z-x \\in \\operatorname{null}(A)\\) so that \\((z-x)^\\top A^\\top \\lambda = \\lambda^\\top A(z-x) = 0\\) for all \\(\\lambda \\in \\mathbb{R}^n\\)."}, {"n": 3, "claim": "If \\(u \\in \\mathcal{N}_{\\Omega}(x)\\), then \\((z-x)^\\top u \\leqslant 0\\), so for \\(\\lambda \\in \\mathbb{R}^n\\), we have \\[\\langle z-x, u + A^\\top \\lambda \\rangle = \\langle z-x, u \\rangle \\leqslant 0\\] implying that \\(u + A^\\top \\lambda \\in \\mathcal{N}_{\\Omega \\cap \\mathcal{A}}(x)\\)."}, {"n": 4, "claim": "For the reverse inclusion, let \\(v \\in \\mathcal{N}_{\\Omega \\cap \\mathcal{A}}(x)\\)."}, {"n": 5, "claim": "Then we have \\[v^\\top (z-x) \\leqslant 0 \\text{ for all } z \\in \\Omega \\cap \\mathcal{A}.\\] Now define the sets \\[\\begin{aligned} C_1 &= \\left\\{ (y,\\mu) \\in \\mathbb{R}^{d+1} : y = z-x,\\,\\, z \\in \\Omega,\\,\\, \\mu \\leqslant v^\\top y \\right\\}\\\\ C_2 &= \\left\\{ (y,\\mu) \\in \\mathbb{R}^{d+1} : y \\in \\operatorname{ker}(A),\\,\\, \\mu = 0 \\right\\}\\,. \\end{aligned}\\] Note that \\(\\operatorname{ri}(C_1) \\cap C_2 = \\emptyset\\) because otherwise there would exist a \\((\\hat{y},\\hat{\\mu})\\) such that \\[v^\\top \\hat{y} > \\hat{\\mu} = 0\\] and \\(\\hat{y} \\in \\mathcal{T}_{\\Omega \\cap \\mathcal{A}}(x)\\)."}, {"n": 6, "claim": "This would contradict our assumption that \\(v \\in \\mathcal{N}_{\\Omega \\cap \\mathcal{A}}(x)\\)."}, {"n": 7, "claim": "Since their intersection is empty, we can properly separate \\(\\operatorname{ri}(C_1)\\) from \\(C_2\\)."}, {"n": 8, "claim": "Indeed, since \\(C_2\\) is a subspace and \\(C_1\\) has nonempty relative interior, there must be a \\((w,\\gamma)\\) such that \\[\\inf_{(y,\\mu)\\in C_1} \\{ w^\\top y + \\gamma \\mu\\} < \\sup_{(y,\\mu)\\in C_1} \\{ w^\\top y + \\gamma \\mu\\} \\leqslant 0\\] while \\[w^\\top u = 0 \\text{ for all } u \\in \\ker(A).\\] In particular, this means that \\(w = A^\\top \\lambda\\) for some \\(\\lambda \\in \\mathbb{R}^n\\)."}, {"n": 9, "claim": "Now, \\(\\gamma\\) must be nonnegative, as otherwise, \\[\\sup_{(y,\\mu)\\in C_1} \\{ w^\\top y + \\gamma \\mu\\} = \\infty\\] (which can be seen by letting \\(\\mu\\) tend to negative infinity)."}, {"n": 10, "claim": "If \\(\\gamma = 0\\), then \\[\\sup_{y \\in C_1} w^\\top y \\leqslant 0\\] but the set \\(\\{y : w^\\top y = 0\\}\\) does not contain the set \\(\\{z-x : z \\in \\Omega\\}\\) as the infimum is strictly negative."}, {"n": 11, "claim": "This means that the relative interior of \\(\\Omega - \\{x\\}\\) cannot intersect the kernel of \\(A\\) which contradicts our assumptions."}, {"n": 12, "claim": "Thus, we can conclude that \\(\\gamma\\) is strictly positive."}, {"n": 13, "claim": "By homogeneity, we may as well assume that \\(\\gamma = 1\\)."}, {"n": 14, "claim": "To complete the argument, note that we now have \\[(w+v)^\\top (z-x) \\leqslant 0 \\text{ for all } z \\in \\Omega.\\] This means that \\(v+w \\in \\mathcal{N}_{\\Omega}(x)\\) and we have already shown that \\(w = A^\\top \\lambda\\)."}, {"n": 15, "claim": "Thus, \\[v = (v+w)-w \\in \\mathcal{N}_{\\Omega}(x) + \\mathcal{N}_{\\mathcal{A}}(x)\\,.\\]"}]}}, {"local_id": "T5", "statement": "Assume \\(\\ell(w, z)\\) is \\(\\alpha\\)-strongly convex over the domain \\(\\Omega\\) and \\(L\\)-Lipschitz. Let \\(\\hat w_S = \\arg \\min_{w\\in \\Omega} \\frac{1}{m} \\sum_{i=1}^m \\ell(w, z_i)\\) denote the empirical risk minimizer (ERM). Then, ERM satisfies\n\\[\n\\Delta_{\\sup} (\\text{ERM}) \\leqslant \\frac{4L^2}{\\alpha m}\\,.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Denote by \\(\\hat w_S\\) the empirical risk minimizer on a sample \\(S.\\) Fix arbitrary samples \\(S,S'\\) of size \\(m\\) and an index \\(i\\in[m].\\) We need to show that\n\\[\n|(\\ell(\\hat w_{S^{(i)}},z_i')-\\ell(\\hat w_{S},z_i'))|\\leqslant\\frac{4L^2}{\\alpha m}\\,.\n\\]\nOn one hand, by strong convexity we know that\n\\[\nR_S(\\hat w_{S^{(i)}}) - R_S(\\hat w_{S}) \\geqslant \\frac{\\alpha}{2} \\| \\hat w_{S} - \\hat w_{S^{(i)}} \\|^2 \\,.\n\\]\nOn the other hand,\n\\begin{align*}\n&R_S(\\hat w_{S^{(i)}})-R_S(\\hat w_{S})\\\\\n&=\\frac{1}{m}(\\ell(\\hat w_{S^{(i)}},z_i)-\\ell(\\hat w_{S},z_i))+\\frac{1}{m}\\sum_{i\\neq j}(\\ell(\\hat w_{S^{(i)}},z_j)-\\ell(\\hat w_{S},z_j))\\\\\n&=\\frac{1}{m}(\\ell(\\hat w_{S^{(i)}},z_i)-\\ell(\\hat w_{S},z_i))+\\frac{1}{m}(\\ell(\\hat w_{S},z_i')-\\ell(\\hat w_{S^{(i)}},z_i'))+\\left(R_{S^{(i)}}(\\hat w_{S^{(i)}})-R_{S^{(i)}}(\\hat w_{S})\\right)\\\\\n&\\leqslant\\frac{1}{m}|\\ell(\\hat w_{S^{(i)}},z_i)-\\ell(\\hat w_{S},z_i)| +\\frac{1}{m}|(\\ell(\\hat w_{S},z_i')-\\ell(\\hat w_{S^{(i)}},z_i'))|\\\\\n&\\leqslant\\frac{2L}{m}\\|\\hat w_{S^{(i)}}-\\hat w_{S}\\|\\,.\n\\end{align*}\nHere, we used that\n\\[\nR_{S^{(i)}}(\\hat w_{S^{(i)}}) - R_{S^{(i)}}(\\hat w_{S})) \\leqslant 0\n\\]\nand the fact that \\(\\ell\\) is \\(L\\)-Lipschitz."}, {"n": 2, "claim": "Putting it all together \\(\\| \\hat w_{S^{(i)}} - \\hat w_{S} \\| \\leqslant \\frac{4 L}{\\alpha m}\\). Then by the Lipschitz condition we have that\n\\[\n|(\\ell(\\hat w_{S^{(i)}},z_i')-\\ell(\\hat w_{S},z_i'))|\\leqslant L\\|\\hat w_{S^{(i)}}-\\hat w_{S}\\|\\leqslant\\frac{4L^2}{\\alpha m}\\,.\n\\]\nHence, \\(\\Delta_{\\sup}(\\text{ERM}) \\leqslant \\frac{4 L ^ 2}{\\alpha m}\\)."}]}}], "target_local_id": "T3", "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch12_proposition_12_21__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nPacking Number. Let \\((\\mathcal{V},d)\\) be a pseudometric space with metric \\(d(\\cdot,\\cdot)\\). A finite subset \\(\\mathcal{G}(\\epsilon)\\subset \\mathcal{G}\\) is an \\(\\epsilon\\)-packing of \\(\\mathcal{G}\\) if \\(d(\\phi,\\phi^{\\prime})>\\epsilon\\) for all \\(\\phi,\\phi^{\\prime}\\in\\mathcal{G}(\\epsilon)\\). The \\(\\epsilon\\)-packing number of \\(\\mathcal{G}\\), denoted by \\(M(\\epsilon,\\mathcal{G},d)\\), is the largest cardinality of \\(\\epsilon\\)-packing of \\(\\mathcal{G}\\).\nThe following results illustrate the equivalence between covering number and packing number. One advantage of using an \\(\\epsilon\\) packing of \\(\\mathcal{G}\\) instead of an \\(\\epsilon\\) cover of \\(\\mathcal{G}\\) is that all members in the \\(\\epsilon\\) packing also belong to \\(\\mathcal{G}\\). There if members in \\(\\mathcal{G}\\) satisfy certain assumptions such as the variance condition, then members of its \\(\\epsilon\\) packing also satisfy such assumptions. For this reason, we will use packing numbers instead of covering numbers in some of the theoretical analysis in later chapters.\n\ndefinition 12.9. Consider a distribution family \\(\\mathcal{P}\\) on sample space \\(\\mathcal{Z}\\), a parameter space \\(\\Theta\\). A learning algorithm \\(\\mathcal{A}:\\mathcal{Z}^n\\to \\Theta\\), a loss function \\(Q:\\Theta \\times \\mathcal{P}\\rightarrow \\mathbb{R}\\). Then the worst case expected risk of a learning algorithm (i.e., a statistical estimator) \\(\\mathcal{A}\\) with respect to \\(\\mathcal{P}\\) is given by\n\\[\nr_{n}(\\mathcal{A},\\mathcal{P},Q)=\\sup_{\\mathcal{D}\\in\\mathcal{P}}\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\mathbb{E}_{\\mathcal{A}}Q(\\mathcal{A}(\\mathcal{S}_{n}),\\mathcal{D}),\n\\]\nwhere \\(\\mathbb{E}_{\\mathcal{A}}\\) is the expectation over any internal randomization of \\(\\mathcal{A}\\). Moreover, the minimax risk is defined as:\n\\[\nr_{n}(\\mathcal{P},Q)=\\inf_{\\mathcal{A}}r_{n}(\\mathcal{A},\\mathcal{P},Q).\n\\]\nThe standard statistical framework for optimality is to find an algorithm with the smallest worse case risk \\( r_n(\\mathcal{A}, \\mathcal{P}, Q) \\). This type of analysis is referred to as minimax analysis. The minimax risk \\( r_n(\\mathcal{P}, Q) \\) depends only on the sample size \\( n \\), distribution family \\( \\mathcal{P} \\), and the loss function \\( Q \\). In nonparametric statistics, it is difficult to find the exact expression of minimax risk. Therefore we often consider rate optimal algorithms \\( \\mathcal{A} \\), which achieves the optimal minimax risk \\( r_n(\\mathcal{P}, Q) \\) up to a constant factor as \\( n \\to \\infty \\).\nIn minimax analysis, we are interested in estimating the minimax risk (up to a constant factor). To establish an upper bound of \\( r_n(\\mathcal{P}, Q) \\), we consider specific learning algorithms and analyze their convergence rates. For example, if we consider the ERM method \\( \\mathcal{A}_{\\mathrm{erm}} \\) for the least squares regression problem, then we may obtain an upper bound of\n\\[\nr_{n}(\\mathcal{P},Q)\\leq r_{n}(\\mathcal{A}_{\\mathrm{erm}},\\mathcal{P},Q)=O(n^{-r}),\n\\]\nfor some \\(r > 0\\), based on the analysis of Example 6.49.\nOn the other hand, if we can show a lower bound \\(r_n(\\mathcal{P}, Q) \\geq cn^{-r}\\) for some constant \\(c\\) that may depend on \\(\\mathcal{P}\\) but independent of \\(n\\), then we know that the\nERM method achieves the optimal minimax lower bound. That is, no statistical estimator can perform much better than ERM up to a constant factor for this distribution family. If the lower bound is \\( O(n^{-r^{\\prime}}) \\) for \\( r^{\\prime} > r \\), then ERM may not be optimal. As we will see, both situations can happen depending on the entropy of the function class \\( \\mathcal{F} \\) to be estimated.\n\nexample 12.6. For least squares problem, \\( f(x) \\) is a real valued regression function. Let \\( f_{\\mathcal{D}}(x) = \\mathbb{E}_{\\mathcal{D}}[Y|X = x] \\). We may define\n\\[\nQ(f,\\mathcal{D})=\\mathbb{E}_{X\\sim\\mathcal{D}}(f(X)-f_{\\mathcal{D}}(X))^{2}.\n\\]\nFor the least squares regression problem in Example 12.6, we consider a function class \\(\\mathcal{F}\\) that contains the optimal prediction rule \\(f_{\\mathcal{D}}(X) = \\mathbb{E}[Y|X]\\), and\n\\[\nQ_{\\mathrm{LS}}(f,\\mathcal{D})=\\mathbb{E}_{X\\sim\\mathcal{D}}(f(X)-f_{\\mathcal{D}}(X))^{2}.\n\\]\nGiven a function class \\(\\mathcal{F}\\), instead of running least squares on \\(\\mathcal{F}\\) with\n\\[\n\\hat{f}=\\arg\\min_{f\\in\\mathcal{F}}\\sum_{i=1}^{n}(f(X_{i})-Y_{i})^{2},\n\\]\nthe sieve method considers a subset \\(\\mathcal{F}_n \\subset \\mathcal{F}\\), and then perform least squares regression restricted to this subset:\n\\[\n\\hat{f}_{\\mathcal{F}_{n}}=\\arg\\min_{f\\in\\mathcal{F}_{n}}\\sum_{i=1}^{n}(f(X_{i})-Y_{i})^{2}.\n\\]\n\nexample 6.49. Consider a function class \\(\\mathcal{F}\\) and the ERM method for least squares regression:\n\\[\n\\hat{f}=\\arg\\min_{f\\in\\mathcal{F}}\\frac{1}{n}\\sum_{i=1}^{n}(f(X_{i})-Y_{i})^{2},\n\\]\nwhere \\( Z_{i} = (X_{i},Y_{i}) \\) are iid samples from \\( \\mathcal{D} \\). Assume that \\( |f(X) - Y| \\in [0,1] \\) for all \\( X \\) and \\( Y \\). Example 3.18 implies that the loss function \\( \\phi (f,Z) = [(f(X) - Y)^2 -(f_*(X) - Y)^2 ] \\) satisfies the variance condition if the true regression function \\( f_{*} \\in \\mathcal{F} \\). Assume also that the empirical covering number of \\( \\mathcal{F} \\) satisfies:\n\\[\n\\ln N_{2}(\\epsilon,\\mathcal{F},n)\\leq\\frac{c}{\\epsilon^{p}}\\tag{6.9}\n\\]\nfor some constant \\( c > 0 \\) and \\( p > 0 \\). We consider the following two situations: \\( p \\in (0,2) \\) and \\( p \\geq 2 \\). Let \\( h(f) = \\phi(f,\\mathcal{D}) \\) with \\( h_0(f) = 0 \\).\n- \\( p \\in (0, 2) \\). The conditions of Corollary 6.44 hold with \\( c_0 = 0 \\). This implies the following bound (see Exercises 6.9) on the rate function with constant \\( \\alpha \\):\n\\[\n\\bar{r}^{h}(\\alpha,\\mathcal{F},\\mathcal{D})=O\\left(n^{-2/(2+p)}\\right).\n\\]\nWe thus have with probability at least \\(1 - \\delta\\):\n\\[\n\\mathbb{E}_{\\mathcal{D}}L(\\hat{f}(X),Y)\\leq\\mathbb{E}_{\\mathcal{D}}L(f_{*}(X),Y)+O\\left(n^{-2/(2+p)}+\\frac{\\ln(1/\\delta)}{n}\\right).\n\\]\n- \\( p > 2 \\). The entropy integral of Theorem 6.25 implies that\n\\[\nR_{n}(\\mathcal{F}^{h}(b),\\mathcal{D})\\leq\\frac{\\tilde{c}_{1}}{n^{1/p}}\n\\]\nfor some constant \\(\\tilde{c}_1\\). We thus obtain a rate of convergence of\n\\[\n\\bar{r}^{h}(\\alpha,\\mathcal{F},\\mathcal{D})=O\\left(n^{-1/p}\\right)\n\\]\nfor local Rademacher complexity. It can be shown that this is the same rate as what we can obtain from the standard non-localized Rademacher complexity (see Exercise 6.7).", "theorems": [{"local_id": "T1", "statement": "Assume that the distribution of \\( X \\) is \\( \\mathcal{D}_X \\). Let \\( \\mathcal{F}_n \\) be an \\( \\epsilon \\) packing subset of \\( \\mathcal{F} \\) in the \\( L_2(\\mathcal{D}_X) \\) metric with \\( M(\\epsilon, \\mathcal{F}, L_2(\\mathcal{D}_X)) \\) members. Assume there exists \\( b > 0 \\) such that \\( [f(X) - f'(X)] \\leq 2b \\) for all \\( f, f' \\in \\mathcal{F} \\). Assume that \\( f_{\\mathcal{D}} \\in \\mathcal{F} \\) and \\( Y \\) is sub-Gaussian:\n\\[\n\\ln\\mathbb{E}_{Y|X}\\exp(\\lambda(Y-f_{\\mathcal{D}}(X)))\\leq\\frac{\\lambda^{2}b^{2}}{2}.\n\\]\nThen\n\\[\n\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\mathbb{E}_{X\\sim\\mathcal{D}_{X}}(\\hat{f}_{\\mathcal{F}_{n}}(X)-f_{\\mathcal{D}}(X))^{2}\\leq\\left[4\\epsilon^{2}+\\frac{14 b^{2}}{n}\\ln M(\\epsilon,\\mathcal{F},L_{2}(\\mathcal{D}_{X}))\\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\phi(f,Z) = (f(X) - Y)^2 - (f_{\\mathcal{D}}(X) - Y)^2\\) with \\(Z = (X,Y)\\)."}, {"n": 2, "claim": "Let \\(\\rho = b^{-2} / 4\\). Then we have\n\\[\n\\begin{array}{l}\\Lambda(\\rho,f)= -\\frac{1}{\\rho}\\ln\\mathbb{E}_{Z}e^{-\\rho\\phi(f,Z)}\\quad \\text{(d e f i n i t i o n o f}\\Lambda \\text{i n T h e o r e m 1 0 . 1 8)}\\\\ = -\\frac{1}{\\rho}\\ln\\mathbb{E}_{Z}e^{-\\rho(f(X)-f_{\\mathcal{D}}(X))^{2}+2\\rho(f(X)-f_{\\mathcal{D}}(X))(Y-f_{\\mathcal{D}}(X))}\\quad(\\text{d e f i n i t i o n o f}\\phi)\\\\ \\geq-\\frac{1}{\\rho}\\ln\\mathbb{E}_{X}e^{(-\\rho+2\\rho^{2}b^{2})(f(X)-f_{\\mathcal{D}}(X))^{2}}\\quad(\\mathrm{sub-Gaussiannoiseassumption})\\\\ \\geq\\frac{1}{\\rho}\\left[1-\\mathbb{E}_{X}e^{-0. 5\\rho(f(X)-f_{\\mathcal{D}}(X))^{2}}\\right]\\quad(-\\ln z\\geq 1-z)\\\\ \\geq\\frac{1}{\\rho}\\mathbb{E}_{X}e^{-2\\rho b^{2}}0. 5\\rho(f(X)-f_{\\mathcal{D}}(X))^{2}\\quad(1-e^{-z}\\geq e^{-a}z \\text{f o r} z\\in [0,a])\\\\ \\geq 0. 3\\mathbb{E}_{X}(f(X)-f_{\\mathcal{D}}(X))^{2}.\\quad\\left(2\\rho b^{2}=0. 5\\right)\\\\ \\end{array}\n\\]\nIt follows from Theorem 10.18 by setting \\( 1 / (\\lambda n) = \\rho = b^{-2} / 4 \\) (which implies that \\( \\lambda = 4b^{2} / n \\)) that for any random estimator \\( \\hat{q} \\) on \\( \\mathcal{F}_n \\):"}, {"n": 3, "claim": "\\[\n\\begin{array}{l}0. 3\\mathbb{E}_{\\mathcal{S}_{n}}\\mathbb{E}_{f\\sim\\hat{q}(\\cdot|\\mathcal{S}_{n})}\\mathbb{E}_{X}(f(X)-f_{\\mathcal{D}}(X))^{2}\\\\ \\leq\\mathbb{E}_{\\mathcal{S}_{n}}\\mathbb{E}_{f\\sim\\hat{q}(\\cdot|\\mathcal{S}_{n})}\\frac{1}{n}\\sum_{i=1}^{n}\\phi(f,Z_{i})+\\frac{4b^{2}}{n}\\mathbb{E}_{\\mathcal{S}_{n}}\\mathrm{KL}(\\hat{q}\\|q_{0}).\\tag{12.1}\\\\ \\end{array}\n\\]\nLet \\(\\hat{q}\\) corresponding to the deterministic estimator \\(\\hat{f}_{\\mathcal{F}_n}\\), and \\(q_0\\) be the uniform distribution on \\(\\mathcal{F}_n\\)."}, {"n": 4, "claim": "Then \\(\\mathrm{KL}(\\hat{q} || q_0) \\leq \\ln |\\mathcal{F}_n|\\)."}, {"n": 5, "claim": "Therefore we obtain\n\\[\n\\begin{array}{l}0. 3\\mathbb{E}_{\\mathcal{S}_{n}}Q_{\\mathrm{LS}}(\\hat{f}_{\\mathcal{F}_{n}},\\mathcal{D})\\leq\\mathbb{E}_{\\mathcal{S}_{n}}\\frac{1}{n}\\sum_{i=1}^{n}\\phi(\\hat{f}_{\\mathcal{F}_{n}},Z_{i})+\\frac{4b^{2}}{n}\\ln|\\mathcal{F}_{n}|\\\\ \\leq\\inf_{f\\in\\mathcal{F}_{n}}\\mathbb{E}_{\\mathcal{S}_{n}}\\frac{1}{n}\\sum_{i=1}^{n}\\phi(f,Z_{i})+\\frac{4b^{2}}{n}\\ln|\\mathcal{F}_{n}|\\\\ \\left(\\hat{f}_{\\mathcal{F}_{n}}\\text{i s e m p i r i c a l r i k m i n i m i z e r o f}\\mathcal{F}_{n}\\right)\\\\ \\leq\\inf_{f\\in\\mathcal{F}_{n}}Q_{\\mathrm{LS}}(f,\\mathcal{D})+\\frac{4b^{2}}{n}\\ln|\\mathcal{F}_{n}|\\leq\\epsilon^{2}+\\frac{4b^{2}}{n}\\ln|\\mathcal{F}_{n}|.\\\\ \\end{array}\n\\]\nThe last inequality used the fact that \\(\\mathcal{F}_n\\) is an \\(\\epsilon\\) packing of \\(\\mathcal{F}\\) and \\(f_{\\mathcal{D}} \\in \\mathcal{F}\\)."}, {"n": 6, "claim": "This implies the desired bound."}]}}, {"local_id": "T2", "statement": "The mutual information \\( I(j,Z) \\) in Theorem 12.11 satisfies the inequality\n\\[\nI(j,Z)\\leq\\frac{1}{N^{2}}\\sum_{j=1}^{N}\\sum_{k=1}^{N}\\mathrm{KL}(\\mathcal{D}_{j}| |\\mathcal{D}_{k})\\leq\\sup_{j,k}\\mathrm{KL}(\\mathcal{D}_{j}| |\\mathcal{D}_{k}).\n\\]", "proof": {"steps": [{"n": 1, "claim": "We have\n\\[\n\\begin{array}{l}I(j,Z)=\\frac{1}{N}\\sum_{j=1}^{N}\\mathbb{E}_{Z\\sim\\mathcal{D}_{j}}\\ln\\frac{p_{\\mathcal{D}_{j}}(Z)}{\\frac{1}{N}\\sum_{k=1}^{N}p_{\\mathcal{D}_{k}}(Z)}\\\\ \\leq\\frac{1}{N^{2}}\\sum_{j=1}^{N}\\sum_{k=1}^{N}\\mathbb{E}_{Z\\sim\\mathcal{D}_{j}}\\ln\\frac{p_{\\mathcal{D}_{j}}(Z)}{p_{\\mathcal{D}_{k}}(Z)},\\\\ \\end{array}\n\\]\nwhere the inequality used Jensen's inequality and the convexity of \\(-\\ln z\\)"}]}}, {"local_id": "T3", "statement": "Consider a finite family of distributions \\(\\mathcal{P}\\). Let \\(d \\geq 1\\) be an integer, and \\(Q\\) can be decomposed as\n\\[\nQ(\\theta,\\mathcal{D})=\\sum_{j=1}^{d}Q_{j}(\\theta,\\mathcal{D}),\n\\]\nwhere \\( Q_{j} \\geq 0 \\) are all non-negative. Assume for all \\( j \\), there exists a partition \\( M_{j} \\) of \\( \\mathcal{P} \\). We use notation \\( \\mathcal{D}' \\sim_{j} \\mathcal{D} \\) to indicate that \\( \\mathcal{D}' \\) and \\( \\mathcal{D} \\) belong to the same partition in \\( M_{j} \\). Let \\( m_{j}(\\mathcal{D}) \\) be the number of elements in the partition containing \\( \\mathcal{D} \\). Assume there exist \\( \\epsilon, \\beta \\geq 0 \\) such that\n\\[\n\\begin{array}{l}\\forall\\mathcal{D}^{\\prime}\\sim_{j}\\mathcal{D},\\mathcal{D}^{\\prime}\\neq\\mathcal{D}:\\quad\\inf_{\\theta}[Q_{j}(\\theta,\\mathcal{D}^{\\prime})+Q_{j}(\\theta,\\mathcal{D})]\\geq\\epsilon,\\\\ \\forall\\mathcal{D}\\in\\mathcal{P}:\\frac{1}{d(\\mathcal{P})}\\sum_{j=1}^{d}\\sum_{\\mathcal{D}\\in\\mathcal{P}_{j}}\\frac{1}{m_{j}(\\mathcal{D})-1}\\sum_{\\mathcal{D}^{\\prime}\\sim_{j}\\mathcal{D}}\\|\\mathcal{D}^{\\prime}-\\mathcal{D}\\|_{\\mathrm{TV}}\\leq\\beta,\\\\ \\end{array}\n\\]\nwhere \\(\\mathcal{P}_j = \\{\\mathcal{D}\\in \\mathcal{P}:m_j(\\mathcal{D}) > 1\\}\\) and \\(d(\\mathcal{P}) = \\sum_{j = 1}^{d}|\\mathcal{P}_j|\\). Let \\(\\mathcal{A}(Z)\\) be any estimator, we have\n\\[\n\\frac{1}{|\\mathcal{P}|}\\sum_{\\mathcal{D}\\in\\mathcal{P}}\\mathbb{E}_{Z\\sim\\mathcal{D}}\\mathbb{E}_{\\mathcal{A}}Q(\\mathcal{A}(Z),\\mathcal{D})\\geq\\frac{\\epsilon d(\\mathcal{P})}{2|\\mathcal{P}|}[1-\\beta],\n\\]\nwhere \\(\\mathbb{E}_A\\) is with respect to the internal randomization in \\(\\mathcal{A}\\).", "proof": {"steps": [{"n": 1, "claim": "For notation convenient, assume there exists \\(d\\mu (z)\\) so that for all \\(\\mathcal{D}\\in \\mathcal{P}\\), \\(p_{\\mathcal{D}}(z)d\\mu (z)\\) is the distribution of \\(\\mathcal{D}\\)."}, {"n": 2, "claim": "Then\n\\[\n\\begin{array}{l}\\frac{2}{|\\mathcal{P}|}\\sum_{j=1}^{d}\\sum_{\\mathcal{D}\\in\\mathcal{P}}\\mathbb{E}_{Z\\sim\\mathcal{D}}Q_{j}(\\mathcal{A}(Z),\\mathcal{D})\\geq\\frac{2}{|\\mathcal{P}|}\\sum_{j=1}^{d}\\sum_{\\mathcal{D}\\in\\mathcal{P}_{j}}\\mathbb{E}_{Z\\sim\\mathcal{D}}Q_{j}(\\mathcal{A}(Z),\\mathcal{D})\\\\ =\\frac{1}{|\\mathcal{P}|}\\sum_{j=1}^{d}\\sum_{\\mathcal{D}\\in\\mathcal{P}_{j}}\\frac{1}{m_{j}(\\mathcal{D})-1}\\sum_{\\substack{\\mathcal{D}^{\\prime}\\sim_{j}\\mathcal{D}\\\\ \\mathcal{D}^{\\prime}\\neq\\mathcal{D}}}\\left[\\mathbb{E}_{Z\\sim\\mathcal{D}^{\\prime}}Q_{j}(\\mathcal{A}(Z),\\mathcal{D}^{\\prime})+\\mathbb{E}_{Z\\sim\\mathcal{D}}Q_{j}(\\mathcal{A}(Z),\\mathcal{D})\\right]\\\\ =\\frac{1}{|\\mathcal{P}|}\\sum_{j=1}^{d}\\sum_{\\mathcal{D}\\in\\mathcal{P}_{j}}\\frac{1}{m_{j}(\\mathcal{D})-1}\\sum_{\\substack{\\mathcal{D}^{\\prime}\\sim_{j}\\mathcal{D}\\\\ \\mathcal{D}^{\\prime}\\neq\\mathcal{D}}}\\int [Q_{j}(\\mathcal{A}(z),\\mathcal{D})p_{\\mathcal{D}}(z)+Q_{j}(\\mathcal{A}(z),\\mathcal{D}^{\\prime})p_{\\mathcal{D}^{\\prime}}(z)]d\\mu(z)\\\\ \\stackrel{(a)}{\\geq}\\frac{1}{|\\mathcal{P}|}\\sum_{j=1}^{d}\\sum_{\\mathcal{D}\\in\\mathcal{P}_{j}}\\frac{1}{m_{j}(\\mathcal{D})-1}\\sum_{\\substack{\\mathcal{D}^{\\prime}\\sim_{j}\\mathcal{D}\\\\ \\mathcal{D}^{\\prime}\\neq\\mathcal{D}}}\\int\\epsilon\\min(p_{\\mathcal{D}}(z),p_{\\mathcal{D}^{\\prime}}(z))d\\mu(z)\\\\ \\stackrel{(b)}{=}\\frac{1}{|\\mathcal{P}|}\\sum_{j=1}^{d}\\sum_{\\mathcal{D}\\in\\mathcal{P}_{j}}\\frac{1}{m_{j}(\\mathcal{D})-1}\\sum_{\\substack{\\mathcal{D}^{\\prime}\\sim_{j}\\mathcal{D}\\\\ \\mathcal{D}^{\\prime}\\neq\\mathcal{D}}}\\epsilon(1-\\|\\mathcal{D}-\\mathcal{D}^{\\prime}\\|_{\\mathrm{TV}}).\\\\ \\end{array}\n\\]\nIn the above derivation, the first inequality used \\( Q_{j}(\\cdot) \\geq 0 \\); (a) used the assumption of the lemma, and (b) used the definition of TV-norm (see Appendix B)."}, {"n": 3, "claim": "By rearranging the last term and using the definition of \\( \\beta \\), we obtain the desired bound."}]}}, {"local_id": "T4", "statement": "If for some \\(C > 0\\) and \\(\\epsilon > 0\\):\n\\[\nC^{-1}\\epsilon^{-q}\\leq\\ln M(\\epsilon,\\mathcal{F},L_{2}(\\mathcal{D}_{X}))\\leq C\\epsilon^{-q}.\n\\]\nFor noise model that satisfies the condition of Theorem 12.17, we have\n\\[\nr_{n}(\\mathcal{P},Q_{\\mathrm{LS}})\\geq C^{\\prime}n^{-2/(2+q)}\n\\]\nfor some \\(C' > 0\\).", "proof": {"steps": [{"n": 1, "claim": "We consider an \\(\\epsilon\\) packing subset \\(\\mathcal{F}'\\) of \\(\\mathcal{F}\\) with size of at least \\(\\exp(C^{-1}\\epsilon^{-q})\\)."}, {"n": 2, "claim": "Since for some \\(C_0 > 0\\),\n\\[\n\\ln N\\left(0. 5C_{0}\\epsilon,\\mathcal{F},L_{2}\\left(\\mathcal{D}_{X}\\right)\\right)\\leq 0. 5C^{-1}\\epsilon^{-q},\n\\]\nit implies that there exists a ball of size \\(0.5C_0\\epsilon\\), which contains at least\n\\[\n\\frac{\\exp(C^{-1}\\epsilon^{-q})}{\\exp(0.5C^{-1}\\epsilon^{-q})}=\\exp(0. 5C^{-1}\\epsilon^{-q})\n\\]\nmembers of \\(\\mathcal{F}'\\)."}, {"n": 3, "claim": "This means we can find \\(N \\geq \\exp(0.5C^{-1}\\epsilon^{-q})\\) functions \\(\\{f_1, \\ldots, f_N\\}\\) such that\n\\[\n\\begin{array}{l}\\sup_{j\\neq k}Q(f_{j},f_{k})\\leq C_{0}^{2}\\epsilon^{2},\\\\ \\inf_{j\\neq k}Q(f_{j},f_{k})\\geq\\epsilon^{2},\\\\ \\end{array}\n\\]\nwhere \\( Q(f, f') = \\mathbb{E}_{X \\sim \\mathcal{D}_X}(f(X) - f'(X))^2 \\)."}, {"n": 4, "claim": "Now let \\( n = \\lceil (C' / \\epsilon^2)^{(q + 2) / 2} \\rceil \\) for a sufficiently small constant \\( C' \\), then we have\n\\[\n\\ln N\\geq 0.5 C^{-1}\\epsilon^{-q}\\geq\\ln 4+n\\sigma^{-2}C_{0}^{2}\\epsilon^{2}\\geq\\ln 4+n\\sigma^{-2}\\sup_{j\\neq k}Q(f_{j},f_{k}).\n\\]"}, {"n": 5, "claim": "Theorem 12.17 implies that \\(r_{n}(\\mathcal{P},Q_{\\mathrm{LS}})\\geq 0.125\\epsilon^{2}\\). Since \\(\\epsilon^{2}\\geq C^{\\prime}n^{-2/(q+2)}\\), we obtain the desired bound."}]}}, {"local_id": "T5", "statement": "Consider a randomized algorithm \\(\\mathcal{A}\\) that returns a distribution \\(\\hat{q}(w|S_n)\\) on the parameter space \\(\\Omega\\) for each training data \\(S_n \\in \\mathcal{Z}^n\\). Let\n\\[\n\\Lambda(\\lambda,w)= -\\frac{1}{\\lambda}\\ln\\mathbb{E}_{Z\\sim\\mathcal{D}}\\exp(-\\lambda\\phi(w,Z)).\n\\]\nThen for any data independent distribution \\(q_0\\) on \\(\\Omega\\) and \\(\\lambda > 0\\), we have\n\\[\n\\mathbb{E}_{\\mathcal{S}_{n}}\\mathbb{E}_{w\\sim\\hat{q}(\\cdot|\\mathcal{S}_{n})}\\Lambda(1/(\\lambda n),w)\\leq\\mathbb{E}_{\\mathcal{S}_{n}}\\mathbb{E}_{w\\sim\\hat{q}(\\cdot|\\mathcal{S}_{n})}\\frac{1}{n}\\sum_{i=1}^{n}\\phi(w,Z_{i})+\\lambda\\mathbb{E}_{\\mathcal{S}_{n}}\\mathrm{KL}(\\hat{q}| |q_{0}).\n\\]\nMoreover, for any \\(\\lambda > 0\\), with probability at least \\(1 - \\delta\\) over \\(S_{n}\\):\n\\[\n\\mathbb{E}_{w\\sim\\hat{q}(\\cdot|\\mathcal{S}_{n})}\\Lambda(1/(\\lambda n),w)\\leq\\mathbb{E}_{w\\sim\\hat{q}(\\cdot|\\mathcal{S}_{n})}\\frac{1}{n}\\sum_{i=1}^{n}\\phi(w,Z_{i})+[\\lambda\\mathrm{KL}(\\hat{q}| |q_{0})+\\lambda\\ln(1/\\delta)].\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let\n\\[\n\\Delta(\\mathcal{S}_{n})=\\sup_{\\hat{q}}\\left[\\mathbb{E}_{w\\sim\\hat{q}(\\cdot|\\mathcal{S}_{n})}\\left(\\Lambda(1/(\\lambda n),w)-\\frac{1}{n}\\sum_{i=1}^{n}\\phi(w,Z_{i})\\right)-\\lambda\\mathrm{KL}(\\hat{q}| |q_{0})\\right].\n\\]\nWe have\n\\[\n\\begin{array}{l}\\ln\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\exp(\\lambda^{-1}\\Delta(\\mathcal{S}_{n}))\\\\ =\\ln\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\exp\\left[\\ln\\mathbb{E}_{w\\sim q_{0}}\\exp\\left(\\lambda^{-1}\\Lambda(1/(\\lambda n),w)-\\frac{\\lambda^{-1}}{n}\\sum_{i=1}^{n}\\phi(w,Z_{i})\\right)\\right]\\\\ =\\ln\\mathbb{E}_{w\\sim q_{0}}\\left[\\exp\\left(\\lambda^{-1}\\Lambda(1/(\\lambda n),w)\\right)\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\exp\\left(-\\frac{\\lambda^{-1}}{n}\\sum_{i=1}^{n}\\phi(w,Z_{i})\\right)\\right]\\\\ =\\ln\\mathbb{E}_{w\\sim q_{0}}\\left[\\exp\\left(\\lambda^{-1}\\Lambda(1/(\\lambda n),w)\\right)\\left(\\mathbb{E}_{Z\\sim\\mathcal{D}}\\exp\\left(-\\frac{\\lambda^{-1}}{n}\\phi(w,Z)\\right)\\right)^{n}\\right]\\\\ =\\ln\\mathbb{E}_{w\\sim q_{0}}\\left[\\exp\\left(\\lambda^{-1}\\Lambda(1/(\\lambda n),w)\\right)\\left(\\exp\\left(-\\frac{1}{\\lambda n}\\Lambda(1/(\\lambda n),w)\\right)\\right)^{n}\\right]\\\\ =0.\\\\ \\end{array}\n\\]\nThe first equation used Proposition 7.16."}, {"n": 2, "claim": "The second equation used algebraic manipulation. The third equation used the independence of \\( Z_{i} \\)."}, {"n": 3, "claim": "The fourth equation used the definition of \\( \\Lambda(\\lambda,w) \\)."}, {"n": 4, "claim": "This implies that\n\\[\n\\lambda^{-1}\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\Delta(\\mathcal{S}_{n})\\leq\\ln\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\exp(\\lambda^{-1}\\Delta(\\mathcal{S}_{n}))=0.\n\\]\nThis implies the first bound."}, {"n": 5, "claim": "For the second inequality, we know that\n\\[\n\\operatorname*{Pr}\\left(\\lambda^{-1}\\Delta(\\mathcal{S}_{n})\\geq\\ln(1/\\delta)\\right)\\leq\\mathbb{E}_{\\mathcal{S}_{n}}\\exp\\left(\\lambda^{-1}\\Delta(\\mathcal{S}_{n})-\\ln(1/\\delta)\\right)=\\delta .\n\\]\nThis proves the second result."}]}}], "target_local_id": "T1", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch13_theorem_13_24__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nsection 13.1 sequential observation setup. In Section 13.1, data are observed sequentially. At each time \\(t\\), after the history \\(S_{t-1}\\) has been observed, the next observation \\(Z_t\\) is drawn according to a distribution that may depend on that history. The observation is decomposed as \\(Z_t=(Z_t^{(x)},Z_t^{(y)})\\), where \\(Z_t^{(x)}\\) is the component determined by the past history or the chosen action at time \\(t\\), and \\(Z_t^{(y)}\\) is the random response component. The history after time \\(t\\) is \\(S_t=(Z_1,\\ldots,Z_t)\\), and \\(\\mathcal{S}_T\\) denotes the full sequential experiment/history through time \\(T\\). For a function depending on \\(S_t\\), the notation \\(\\mathbb{E}_{Z_t^{(y)}}[\\cdot]\\) means conditional expectation over the response component \\(Z_t^{(y)}\\), conditional on \\(S_{t-1}\\) and on \\(Z_t^{(x)}\\). Stopping times are adapted to the filtration generated by the histories \\(S_t\\).\n\nSequential Statistical Estimation. Consider a family of environment distributions \\(\\mathcal{P}_{\\mathcal{Z}}\\), where each \\(q \\in \\mathcal{P}_{\\mathcal{Z}}\\) determines the probability for generating \\(Z_{t}\\) based on policy \\(\\pi_{t}\\) as \\(q(Z_{t}|\\pi_{t}, S_{t-1})\\). Consider also a family of learning algorithms, represented by \\(\\mathcal{P}_{\\mathcal{A}}\\). Each learning algorithm \\(\\hat{q} \\in \\mathcal{P}_{\\mathcal{A}}\\) maps the history \\(S_{t-1}\\) deterministically to the next policy \\(\\pi_{t} \\in \\Pi\\) as \\(\\pi_{t} = \\hat{q}(S_{t-1})\\). Given \\(q \\in \\mathcal{P}_{\\mathcal{Z}}\\), and \\(\\hat{q} \\in \\mathcal{P}_{\\mathcal{A}}\\), the data generation probability is fully determined as\n\\[\np(\\mathcal{S}_{n}|\\hat{q},q)=\\prod_{t=1}^{n}q(Z_{t}|\\hat{q}(\\mathcal{S}_{t-1}),\\mathcal{S}_{t-1}).\n\\]\nAfter observing \\( S_{n} \\) for some \\( n \\), the learning algorithm \\( \\hat{q} \\) determines a distribution \\( \\hat{q}(\\theta | S_{n}) \\), and draws an estimator \\( \\theta \\in \\Theta \\) according to \\( \\hat{q}(\\theta | S_{n}) \\). The learning algorithm suffers a loss (also referred to as regret) \\( Q(\\theta, q) \\). The overall probability of \\( \\theta \\) and \\( S_{n} \\) is\n\\[\np(\\theta,\\mathcal{S}_{n}|\\hat{q},q)=\\hat{q}(\\theta|\\mathcal{S}_{n})\\prod_{t=1}^{n}q\\left(Z_{t}|\\hat{q}\\left(\\mathcal{S}_{t-1}\\right),\\mathcal{S}_{t-1}\\right).\\tag{13.8}\n\\]\nNote that although the notation in Definition 13.16 assumes that \\(\\pi_t\\) is a deterministic function of \\(S_{t-1}\\), it can also incorporate randomized algorithms. In such case, a policy is simply a distribution over the random choices. Since a random choice can always be characterized by a distribution (over the random choices) that is a deterministic function of \\(S_{t-1}\\), when we consider random policies in subsequent chapters, it should be understood that such policies can be handled by Definition 13.16. An example is given below.\n\nPacking Number. Let \\((\\mathcal{V},d)\\) be a pseudometric space with metric \\(d(\\cdot,\\cdot)\\). A finite subset \\(\\mathcal{G}(\\epsilon)\\subset \\mathcal{G}\\) is an \\(\\epsilon\\)-packing of \\(\\mathcal{G}\\) if \\(d(\\phi,\\phi^{\\prime})>\\epsilon\\) for all \\(\\phi,\\phi^{\\prime}\\in\\mathcal{G}(\\epsilon)\\). The \\(\\epsilon\\)-packing number of \\(\\mathcal{G}\\), denoted by \\(M(\\epsilon,\\mathcal{G},d)\\), is the largest cardinality of \\(\\epsilon\\)-packing of \\(\\mathcal{G}\\).\nThe following results illustrate the equivalence between covering number and packing number. One advantage of using an \\(\\epsilon\\) packing of \\(\\mathcal{G}\\) instead of an \\(\\epsilon\\) cover of \\(\\mathcal{G}\\) is that all members in the \\(\\epsilon\\) packing also belong to \\(\\mathcal{G}\\). There if members in \\(\\mathcal{G}\\) satisfy certain assumptions such as the variance condition, then members of its \\(\\epsilon\\) packing also satisfy such assumptions. For this reason, we will use packing numbers instead of covering numbers in some of the theoretical analysis in later chapters.\n\ndefinition 4.1. Let \\((\\mathcal{V},d)\\) be a pseudometric space with metric \\(d(\\cdot,\\cdot)\\). A finite\nset \\(\\mathcal{G}(\\epsilon) \\subset \\mathcal{V}\\) is an \\(\\epsilon\\)-cover (or \\(\\epsilon\\)-net) of \\(\\mathcal{G} \\subset \\mathcal{V}\\) if, for all \\(\\phi \\in \\mathcal{G}\\), there exists \\(\\phi' \\in \\mathcal{G}(\\epsilon)\\) so that \\(d(\\phi', \\phi) \\leq \\epsilon\\). The \\(\\epsilon\\)-covering number of \\(\\mathcal{G}\\) with metric \\(d\\) is the smallest cardinality \\(N(\\epsilon, \\mathcal{G}, d)\\) of such \\(\\mathcal{G}(\\epsilon)\\). The number \\(\\ln N(\\epsilon, \\mathcal{G}, d)\\) is called the \\(\\epsilon\\)-entropy.\nFor a function class \\(\\mathcal{G}\\) with seminorm \\(L_{p}(\\mathcal{D})\\) in (3.19), we denote the corresponding \\(L_{p}(\\mathcal{D})\\)-covering number as \\(N(\\epsilon, \\mathcal{G}, L_{p}(\\mathcal{D}))\\). When \\(1 \\leq p \\leq q\\), we have\n\\[\nN(\\epsilon,\\mathcal{G},L_{p}(\\mathcal{D}))\\leq N(\\epsilon,\\mathcal{G},L_{q}(\\mathcal{D})).\n\\]\nIt is easy to verify the following relationship, which implies that \\( L_{p}(\\mathcal{D}) \\) bracketing cover is a stronger requirement than \\( L_{p}(\\mathcal{D}) \\) cover.\n\ntheorem 13.11: We have for any \\(\\lambda>0\\), with probability at least \\(1-\\delta\\), for all \\(n\\geq 1\\) and \\(f\\in\\mathcal{F}\\): \\[\\left[-\\sum_{i=1}^{n} f(Z_i)-\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_i^{(y)}}e^{-\\lambda f(Z_i)}\\right]\\leq\\inf_{\\epsilon>0}\\left[2n\\epsilon+\\frac{\\ln(M(\\epsilon,\\mathcal{F},\\|\\cdot\\|_\\infty)/\\delta)}{\\lambda}\\right].\\]", "theorems": [{"local_id": "T1", "statement": "Assume that for all \\( f \\in \\mathcal{F} \\) and \\( Z_{t}: -f(Z_{t}) + \\mathbb{E}_{Z_{t}^{(y)}}f(Z_{t}) \\leq b \\), and the following variance condition holds for all \\( f \\in \\mathcal{F} \\) and \\( S_{t} \\):\n\\[\n\\operatorname{Var}_{Z_{t}^{(y)}}[f(Z_{t})]\\leq c_{0}^{2}+c_{1}\\mathbb{E}_{Z_{t}^{(y)}}[f(Z_{t})],\n\\]\nthen with probability at least \\(1 - \\delta\\), for all \\(f \\in \\mathcal{F}\\):\n\\[\n\\begin{array}{l}\\left(1-\\lambda c_{1}\\phi(\\lambda b)\\right)\\sum_{t=1}^{n}\\mathbb{E}_{Z_{t}^{(y)}}[f(Z_{t})]\\leq\\sum_{t=1}^{n}f(Z_{t})+\\lambda n c_{0}^{2}\\phi(\\lambda b)\\\\ +\\inf_{\\epsilon>0}\\left[2 n\\epsilon+\\frac{\\ln(M(\\epsilon,\\mathcal{F},\\|\\cdot\\|_{\\infty})/\\delta)}{\\lambda}\\right],\\\\ \\end{array}\n\\]\nwhere \\(\\phi (z) = (e^{z} - z - 1) / z^{2}\\)", "proof": {"steps": [{"n": 1, "claim": "Note that by employing the moment estimation of Bennett's inequality in Lemma 2.20, we obtain\n\\[\n\\begin{array}{l}\\frac{1}{\\lambda}\\ln\\mathbb{E}_{Z_{t}^{(y)}}e^{-\\lambda f(Z_{t})}\\\\ \\leq-\\mathbb{E}_{Z_{t}^{(y)}}[f(Z_{t})]+\\lambda\\phi(\\lambda b)\\operatorname{Var}_{Z_{t}^{(y)}}[f(Z_{t})]\\\\ \\leq-\\mathbb{E}_{Z_{t}^{(y)}}[f(Z_{t})]+\\lambda\\phi(\\lambda b)[c_{0}^{2}+c_{1}\\mathbb{E}_{Z_{t}^{(y)}}[f(Z_{t})]].\\\\ \\end{array}\n\\]\nWe can now obtain the desired bound from Theorem 13.11."}]}}, {"local_id": "T2", "statement": "Let \\(\\Sigma_0\\) be a \\(d\\times d\\) symmetric positive definite matrix, and \\(\\{\\psi (X_t)\\}\\) be a sequence of vectors in \\(\\mathbb{R}^d\\). Let \\(\\Sigma_{t} = \\Sigma_{0} + \\sum_{s = 1}^{t}\\psi (X_{s})\\psi (X_{s})^{\\top}\\), then\n\\[\n\\sum_{s=1}^{t}\\frac{\\psi(X_{s})^{\\top}\\Sigma_{s-1}^{-1}\\psi(X_{s})}{1+\\psi(X_{s})^{\\top}\\Sigma_{s-1}^{-1}\\psi(X_{s})}\\leq\\ln|\\Sigma_{0}^{-1}\\Sigma_{t}|.\n\\]", "proof": {"steps": [{"n": 1, "claim": "We have\n\\[\n\\begin{array}{l}\\frac{\\psi(X_{s})^{\\top}\\Sigma_{s-1}^{-1}\\psi(X_{s})}{1+\\psi(X_{s})^{\\top}\\Sigma_{s-1}^{-1}\\psi(X_{s})}=\\psi(X_{s})^{\\top}\\Sigma_{s}^{-1}\\psi(X_{s})\\\\ =\\operatorname{trace}\\left(\\Sigma_{s}^{-1}\\left(\\Sigma_{s}-\\Sigma_{s-1}\\right)\\right)\\\\ =\\operatorname{trace}\\left(\\nabla\\ln\\left|\\Sigma_{s}\\right|\\left(\\Sigma_{s}-\\Sigma_{s-1}\\right)\\right)\\\\ \\leq\\ln|\\Sigma_{s}| -\\ln|\\Sigma_{s-1}|.\\\\ \\end{array}\n\\]\nThe first equality is due to the Sherman-Morrison formula."}, {"n": 2, "claim": "The last inequality used the fact that \\(\\ln |Z|\\) is a concave function of \\(Z\\) (see Theorem A.18)."}, {"n": 3, "claim": "Now by summing over \\(s = 1\\) to \\(s = t\\), we obtain the desired bound."}]}}, {"local_id": "T3", "statement": "Let \\(\\{(X_t, \\epsilon_t)\\}\\) be a filtered sequence in \\(\\mathcal{X} \\times \\mathbb{R}\\) so that \\(\\epsilon_t\\) is conditional zero-mean sub-Gaussian noise: for all \\(\\lambda \\in \\mathbb{R}\\),\n\\[\n\\ln\\mathbb{E}\\left[e^{\\lambda\\epsilon_{t}}\\mid X_{t},\\mathcal{S}_{t-1}\\right]\\leq\\frac{\\lambda^{2}}{2}\\sigma^{2},\n\\]\nwhere \\( S_{t-1} \\) denotes the history data. Assume that \\( Y_t = f_*(X_t) + \\epsilon_t \\), with \\( f_*(x) \\in \\mathcal{F}: \\mathcal{X} \\to \\mathbb{R} \\). Let \\( \\hat{f}_t \\) be an approximate ERM solution for some \\( \\epsilon' > 0 \\):\n\\[\n\\left(\\sum_{s=1}^{t}\\left(\\hat{f}_{t}\\left(X_{s}\\right)-Y_{s}\\right)^{2}\\right)^{1/2}\\leq\\min_{f\\in\\mathcal{F}}\\left(\\sum_{s=1}^{t}\\left(f\\left(X_{s}\\right)-Y_{s}\\right)^{2}\\right)^{1/2}+\\sqrt{t}\\epsilon^{\\prime}.\n\\]\nThen with probability at least \\(1 - \\delta\\), for all \\(t \\geq 0\\):\n\\[\n\\sum_{s=1}^{t}(\\hat{f}_{t}(X_{t})-f_{*}(X_{t}))^{2}\\leq\\inf_{\\epsilon>0}\\left[8 t(\\epsilon+\\epsilon^{\\prime})(\\sigma+2\\epsilon+2\\epsilon^{\\prime})+12\\sigma^{2}\\ln\\frac{2 N(\\epsilon,\\mathcal{F},\\|\\cdot\\|_{\\infty})}{\\delta}\\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\eta = \\sigma^{-2} / 4\\), and for \\(f \\in \\mathcal{F}\\), let\n\\[\n\\phi(f,Z_{t})=\\left[\\left(f(X_{t})-Y_{t}\\right)^{2}-\\left(f_{*}(X_{t})-Y_{t}\\right)^{2}\\right].\n\\]\nLet \\(\\mathcal{F}_{\\epsilon}\\) be an \\(\\epsilon \\| \\cdot \\|_{\\infty}\\) cover of \\(\\mathcal{F}\\) with \\(N = N(\\epsilon, \\mathcal{F}, \\| \\cdot \\|_{\\infty})\\) members."}, {"n": 2, "claim": "Given any \\(f \\in \\mathcal{F}_{\\epsilon}\\),\n\\[\n\\begin{array}{l}\\ln\\mathbb{E}_{Y_{t}|X_{t},S_{t-1}}\\exp(-\\eta\\phi(f,Z_{t}))\\\\ = -\\eta\\left(f\\left(X_{t}\\right)-f_{*}\\left(X_{t}\\right)\\right)^{2}\\\\ +\\ln\\mathbb{E}_{Y_{t}\\mid X_{t},S_{t-1}}\\exp(-2\\eta(f(X_{t})-f_{*}(X_{t}))(f_{*}(X_{t})-Y_{t}))\\\\ \\leq-\\eta(f(X_{t})-f_{*}(X_{t}))^{2}+\\frac{4\\eta^{2}\\sigma^{2}}{2}(f(X_{t})-f_{*}(X_{t}))^{2}\\\\ = -\\frac{\\left(f(X_{t})-f_{*}(X_{t})\\right)^{2}}{8\\sigma^{2}}.\\\\ \\end{array}\n\\]\nThe inequality used the sub-Gaussian noise assumption."}, {"n": 3, "claim": "It follows from Theorem 13.11 that with probability at least \\( 1 - \\delta / 2 \\), for all \\( f \\in \\mathcal{F}_{\\epsilon} \\) and \\( t \\geq 0 \\):"}, {"n": 4, "claim": "\\[\n\\sum_{i=1}^{t}-\\eta\\phi(f,Z_{i})\\leq-\\sum_{i=1}^{t}\\frac{\\left(f\\left(X_{i}\\right)-f_{*}\\left(X_{i}\\right)\\right)^{2}}{8\\sigma^{2}}+\\ln(2 N/\\delta).\\tag{13.5}\n\\]\nUsing Exercise 13.3, we have with probability at least \\(1 - \\delta /2\\), for all \\(t\\geq 0\\)\n\\[\n\\underbrace{\\sum_{i=1}^{t}\\left(f_{*}\\left(X_{i}\\right)-Y_{i}\\right)^{2}}_{A}\\leq 2 t\\sigma^{2}+3\\sigma^{2}\\ln(2/\\delta).\\tag{13.6}\n\\]\nIt follows that with probability at least \\(1 - \\delta\\), for all \\(t \\geq 0\\) and \\(f \\in \\mathcal{F}_{\\epsilon}\\), both (13.5)\nand (13.6) hold."}, {"n": 5, "claim": "Now, let \\( f \\in \\mathcal{F}_{\\epsilon} \\) so that \\( \\| \\hat{f}_t - f \\|_{\\infty} \\leq \\epsilon \\)."}, {"n": 6, "claim": "This implies that\n\\[\n\\begin{array}{l}\\sum_{i=1}^{t}\\left[(f(X_{i})-Y_{i})^{2}-(f_{*}(X_{i})-Y_{i})^{2}\\right]\\leq\\left(\\sqrt{\\sum_{i=1}^{t}(\\hat{f}_{t}(X_{i})-Y_{i})^{2}}+\\sqrt{t}\\epsilon\\right)^{2}-A\\\\ \\leq\\left(\\sqrt{A}+\\sqrt{t}(\\epsilon+\\epsilon^{\\prime})\\right)^{2}-A\\\\ \\stackrel{(a)}{\\leq}(\\epsilon^{\\prime}+\\epsilon)^{2}t+2\\sqrt{t}(\\epsilon^{\\prime}+\\epsilon)\\sigma\\sqrt{2 t+3\\ln(2/\\delta)}\\\\ \\leq 4\\left(\\epsilon^{\\prime}+\\epsilon\\right)^{2}t+3\\sigma t\\left(\\epsilon^{\\prime}+\\epsilon\\right)+\\sigma^{2}\\ln(2/\\delta).\\tag{13.7}\\\\ \\end{array}\n\\]\nIn the above derivations, the first inequality used \\( |f(X_i) - Y_i| \\leq |\\hat{f}_t(X_i) - Y_i| + \\epsilon \\) for all \\( i \\)."}, {"n": 7, "claim": "The second inequality used the fact that \\( \\hat{f}_t \\) is approximate ERM. (a) follows from (13.6)."}, {"n": 8, "claim": "The last inequality used \\( 2a\\sigma \\sqrt{2b + 3c} \\leq 3a\\sigma \\sqrt{b} + 3a^2 + \\sigma^2 c \\)."}, {"n": 9, "claim": "We thus obtain\n\\[\n\\begin{array}{l}\\left(\\sum_{i=1}^{t}(\\hat{f}(X_{i})-f_{*}(X_{i}))^{2}\\right)^{1/2}\\\\ \\leq\\sqrt{\\epsilon^{2}t}+\\left(\\sum_{i=1}^{t}\\left(f\\left(X_{i}\\right)-f_{*}\\left(X_{i}\\right)\\right)^{2}\\right)^{1/2}\\quad(\\text{t r i a n g l e i n e q u a l i t y})\\\\ \\leq\\sqrt{\\epsilon^{2}t}+\\left(8\\sigma^{2}\\ln(2N/\\delta)+8\\sigma^{2}\\sum_{i=1}^{t}\\eta\\phi(f,Z_{i})\\right)^{1/2}\\quad(\\text{f r o m}(13. 5))\\\\ =\\sqrt{\\epsilon^{2}t}+\\left(8\\sigma^{2}\\ln(2N/\\delta)+2\\sum_{i=1}^{t}\\left[\\left(f\\left(X_{i}\\right)-Y_{i}\\right)^{2}-\\left(f_{*}\\left(X_{i}\\right)-Y_{i}\\right)^{2}\\right]\\right)^{1/2}\\\\ \\leq\\sqrt{\\epsilon^{2}t}+\\left(10\\sigma^{2}\\ln(2N/\\delta)+8t\\left(\\epsilon^{\\prime}+\\epsilon\\right)^{2}+6\\sigma t\\left(\\epsilon^{\\prime}+\\epsilon\\right)\\right)^{1/2}\\quad(\\text{f r o m}(13. 7))\\\\ \\leq\\left(12\\sigma^{2}\\ln(2N/\\delta)+16t(\\epsilon+\\epsilon^{\\prime})^{2}+8\\sigma t(\\epsilon+\\epsilon^{\\prime})\\right)^{1/2},\\\\ \\end{array}\n\\]\nwhere the last inequality used \\(\\sqrt{a} +\\sqrt{b}\\leq \\sqrt{6a + 1.2b}\\) ."}, {"n": 10, "claim": "This implies the bound."}]}}, {"local_id": "T4", "statement": "Consider a sequence of random functions \\(\\xi_1(S_1),\\ldots,\\xi_t(S_t),\\ldots\\) Assume each \\(\\xi_{i}\\) is sub-Gaussian with respect to \\(Z_{i}^{(y)}\\):\n\\[\n\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{\\lambda\\xi_{i}}\\leq\\lambda\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i}+\\frac{\\lambda^{2}\\sigma_{i}^{2}}{2}\n\\]\nfor some \\(\\sigma_{i}\\) that may depend on \\(S_{i - 1}\\) and \\(Z_{i}^{(x)}\\). Then for all \\(\\sigma >0\\), with probability at least \\(1 - \\delta\\)\n\\[\n\\forall n>0:\\sum_{i=1}^{n}\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i}<\\sum_{i=1}^{n}\\xi_{i}+\\left(\\sigma+\\frac{\\sum_{i=1}^{n}\\sigma_{i}^{2}}{\\sigma}\\right)\\sqrt{\\frac{\\ln(1/\\delta)}{2}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "We can let \\(\\lambda = \\sqrt{2\\ln(1 / \\delta)} /\\sigma\\), and then apply the first inequality of Theorem 13.2."}]}}, {"local_id": "T5", "statement": "Let \\(d \\geq 1\\) and \\(m \\geq 2\\) be integers, and let \\(\\mathcal{P}_{\\mathcal{Z}} = \\{q^{\\tau} : \\tau \\in \\{1, \\ldots, m\\}^{d}\\}\\) contain \\(m^{d}\\) probability measures. Suppose that the loss function \\(Q\\) can be decomposed as\n\\[\nQ(\\theta,q)=\\sum_{j=1}^{d}Q_{j}(\\theta,q),\n\\]\nwhere \\( Q_{j} \\geq 0 \\) are all non-negative. For each \\( j \\), \\( \\tau \\sim_{j} \\tau' \\) if \\( \\tau = \\tau' \\) or if \\( \\tau \\) and \\( \\tau' \\) differ by only one component \\( j \\). Assume that there exists \\( \\epsilon, \\beta \\geq 0 \\) such that\n\\[\n\\forall\\tau^{\\prime}\\sim_{j}\\tau,\\tau^{\\prime}\\neq\\tau:[Q_{j}(\\theta,q^{\\tau})+Q_{j}(\\theta,q^{\\tau^{\\prime}})]\\geq\\epsilon,\n\\]\nand there exists \\( q_j^\\tau \\) such that all \\( \\tau' \\sim_j \\tau \\) map to the same value: \\( q_j^{\\tau'} = q_j^\\tau \\). Given any learning algorithm \\( \\hat{q} \\). If for all \\( \\tau \\), \\( j \\in [d] \\), time step \\( t \\), and \\( S_{t-1} \\):\n\\[\n\\frac{1}{m}\\sum_{\\tau^{\\prime}\\sim_{j}\\tau}\\mathrm{KL}(q_{j}^{\\tau}(\\cdot|\\hat{q}(\\mathcal{S}_{t-1}),\\mathcal{S}_{t-1})| |q^{\\tau^{\\prime}}(\\cdot|\\hat{q}(\\mathcal{S}_{t-1}),\\mathcal{S}_{t-1})))\\leq\\beta_{j,t}^{2},\n\\]\nthen\n\\[\n\\frac{1}{m^{d}}\\sum_{\\tau}\\mathbb{E}_{\\theta,S_{n}\\sim p(\\cdot|\\hat{q},q^{\\tau})}Q(\\theta,q^{\\tau})\\geq 0.5 d\\epsilon\\left(1-\\sqrt{\\frac{2}{d}\\sum_{j=1}^{d}\\sum_{t=1}^{n}\\beta_{j,t}^{2}}\\right).\n\\]", "proof": {"steps": [{"n": 1, "claim": "We have\n\\[\n\\begin{array}{l}\\frac{1}{dm^{d}}\\sum_{\\tau}\\sum_{j=1}^{d}\\frac{1}{m-1}\\sum_{\\tau^{\\prime}\\sim_{j}\\tau}\\|p(\\cdot|\\hat{q},q^{\\tau})-p(\\cdot|\\hat{q},q^{\\tau^{\\prime}})\\|_{\\mathrm{TV}}\\\\ \\leq\\frac{1}{dm^{d}}\\sum_{\\tau}\\sum_{j=1}^{d}\\frac{1}{m-1}\\sum_{\\tau^{\\prime}\\sim_{j}\\tau,\\tau^{\\prime}\\neq\\tau}\\left[\\|p(\\cdot|\\hat{q},q_{j}^{\\tau})-p(\\cdot|\\hat{q},q^{\\tau^{\\prime}})\\|_{\\mathrm{TV}}\\right.\\\\ +\\|p(\\cdot|\\hat{q},q_{j}^{\\tau})-p(\\cdot|\\hat{q},q^{\\tau})\\|_{\\mathrm{TV}}\\big]\\\\ =\\frac{2}{dm^{d}}\\sum_{\\tau}\\sum_{j=1}^{d}\\frac{1}{m}\\sum_{\\tau^{\\prime}\\sim_{j}\\tau}\\|p(\\cdot|\\hat{q},q_{j}^{\\tau})-p(\\cdot|\\hat{q},q^{\\tau^{\\prime}})\\|_{\\mathrm{TV}}\\\\ \\leq\\frac{2}{m^{d}}\\sum_{\\tau}\\sqrt{\\frac{1}{dm}\\sum_{j=1}^{d}\\sum_{\\tau^{\\prime}\\sim_{j}\\tau}\\|p(\\cdot|\\hat{q},q_{j}^{\\tau})-p(\\cdot|\\hat{q},q^{\\tau^{\\prime}})\\|_{\\mathrm{TV}}^{2}}\\\\ \\leq\\frac{2}{m^{d}}\\sum_{\\tau}\\sqrt{\\frac{1}{2dm}\\sum_{j=1}^{d}\\sum_{\\tau^{\\prime}\\sim_{j}\\tau}\\mathrm{KL}(p(\\cdot|\\hat{q},q_{j}^{\\tau})| |p(\\cdot|\\hat{q},q^{\\tau^{\\prime}}))}\\\\ \\leq\\sqrt{\\frac{2}{d}\\sum_{j=1}^{d}\\sum_{t=1}^{n}\\beta_{t}^{2}}.\\\\ \\end{array}\n\\]\nThe first inequality is triangle inequality for TV-norm."}, {"n": 2, "claim": "The first equality used \\(\\hat{q}_j^\\tau = \\hat{q}_j^{\\tau '}\\) when \\(\\tau \\sim_{j}\\tau^{\\prime}\\)."}, {"n": 3, "claim": "The second inequality used Jensen's inequality and the\nconcavity of \\(\\sqrt{\\cdot}\\)."}, {"n": 4, "claim": "The third inequality used Theorem B.9. The last inequality used Lemma 13.21."}, {"n": 5, "claim": "Now in Lemma 12.27, we let \\(M_{j}(q^{\\tau}) = \\{q^{\\tau^{\\prime}}:\\tau^{\\prime}\\sim_{j}\\tau \\}\\) be the partitions."}, {"n": 6, "claim": "The result is a simple application of Lemma 12.27 with \\(m_j(q^\\tau) = m\\), \\(|\\mathcal{P}| = m^{d}\\), and \\(d(\\mathcal{P}) = dm^{d}\\)."}]}}], "target_local_id": "T5", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch13_theorem_13_6__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nsection 13.1 sequential observation setup. In Section 13.1, data are observed sequentially. At each time \\(t\\), after the history \\(S_{t-1}\\) has been observed, the next observation \\(Z_t\\) is drawn according to a distribution that may depend on that history. The observation is decomposed as \\(Z_t=(Z_t^{(x)},Z_t^{(y)})\\), where \\(Z_t^{(x)}\\) is the component determined by the past history or the chosen action at time \\(t\\), and \\(Z_t^{(y)}\\) is the random response component. The history after time \\(t\\) is \\(S_t=(Z_1,\\ldots,Z_t)\\), and \\(\\mathcal{S}_T\\) denotes the full sequential experiment/history through time \\(T\\). For a function depending on \\(S_t\\), the notation \\(\\mathbb{E}_{Z_t^{(y)}}[\\cdot]\\) means conditional expectation over the response component \\(Z_t^{(y)}\\), conditional on \\(S_{t-1}\\) and on \\(Z_t^{(x)}\\). Stopping times are adapted to the filtration generated by the histories \\(S_t\\).\n\nPacking Number. Let \\((\\mathcal{V},d)\\) be a pseudometric space with metric \\(d(\\cdot,\\cdot)\\). A finite subset \\(\\mathcal{G}(\\epsilon)\\subset \\mathcal{G}\\) is an \\(\\epsilon\\)-packing of \\(\\mathcal{G}\\) if \\(d(\\phi,\\phi^{\\prime})>\\epsilon\\) for all \\(\\phi,\\phi^{\\prime}\\in\\mathcal{G}(\\epsilon)\\). The \\(\\epsilon\\)-packing number of \\(\\mathcal{G}\\), denoted by \\(M(\\epsilon,\\mathcal{G},d)\\), is the largest cardinality of \\(\\epsilon\\)-packing of \\(\\mathcal{G}\\).\nThe following results illustrate the equivalence between covering number and packing number. One advantage of using an \\(\\epsilon\\) packing of \\(\\mathcal{G}\\) instead of an \\(\\epsilon\\) cover of \\(\\mathcal{G}\\) is that all members in the \\(\\epsilon\\) packing also belong to \\(\\mathcal{G}\\). There if members in \\(\\mathcal{G}\\) satisfy certain assumptions such as the variance condition, then members of its \\(\\epsilon\\) packing also satisfy such assumptions. For this reason, we will use packing numbers instead of covering numbers in some of the theoretical analysis in later chapters.\n\ndefinition 4.1. Let \\((\\mathcal{V},d)\\) be a pseudometric space with metric \\(d(\\cdot,\\cdot)\\). A finite\nset \\(\\mathcal{G}(\\epsilon) \\subset \\mathcal{V}\\) is an \\(\\epsilon\\)-cover (or \\(\\epsilon\\)-net) of \\(\\mathcal{G} \\subset \\mathcal{V}\\) if, for all \\(\\phi \\in \\mathcal{G}\\), there exists \\(\\phi' \\in \\mathcal{G}(\\epsilon)\\) so that \\(d(\\phi', \\phi) \\leq \\epsilon\\). The \\(\\epsilon\\)-covering number of \\(\\mathcal{G}\\) with metric \\(d\\) is the smallest cardinality \\(N(\\epsilon, \\mathcal{G}, d)\\) of such \\(\\mathcal{G}(\\epsilon)\\). The number \\(\\ln N(\\epsilon, \\mathcal{G}, d)\\) is called the \\(\\epsilon\\)-entropy.\nFor a function class \\(\\mathcal{G}\\) with seminorm \\(L_{p}(\\mathcal{D})\\) in (3.19), we denote the corresponding \\(L_{p}(\\mathcal{D})\\)-covering number as \\(N(\\epsilon, \\mathcal{G}, L_{p}(\\mathcal{D}))\\). When \\(1 \\leq p \\leq q\\), we have\n\\[\nN(\\epsilon,\\mathcal{G},L_{p}(\\mathcal{D}))\\leq N(\\epsilon,\\mathcal{G},L_{q}(\\mathcal{D})).\n\\]\nIt is easy to verify the following relationship, which implies that \\( L_{p}(\\mathcal{D}) \\) bracketing cover is a stronger requirement than \\( L_{p}(\\mathcal{D}) \\) cover.\n\nSequential Statistical Estimation. Consider a family of environment distributions \\(\\mathcal{P}_{\\mathcal{Z}}\\), where each \\(q \\in \\mathcal{P}_{\\mathcal{Z}}\\) determines the probability for generating \\(Z_{t}\\) based on policy \\(\\pi_{t}\\) as \\(q(Z_{t}|\\pi_{t}, S_{t-1})\\). Consider also a family of learning algorithms, represented by \\(\\mathcal{P}_{\\mathcal{A}}\\). Each learning algorithm \\(\\hat{q} \\in \\mathcal{P}_{\\mathcal{A}}\\) maps the history \\(S_{t-1}\\) deterministically to the next policy \\(\\pi_{t} \\in \\Pi\\) as \\(\\pi_{t} = \\hat{q}(S_{t-1})\\). Given \\(q \\in \\mathcal{P}_{\\mathcal{Z}}\\), and \\(\\hat{q} \\in \\mathcal{P}_{\\mathcal{A}}\\), the data generation probability is fully determined as\n\\[\np(\\mathcal{S}_{n}|\\hat{q},q)=\\prod_{t=1}^{n}q(Z_{t}|\\hat{q}(\\mathcal{S}_{t-1}),\\mathcal{S}_{t-1}).\n\\]\nAfter observing \\( S_{n} \\) for some \\( n \\), the learning algorithm \\( \\hat{q} \\) determines a distribution \\( \\hat{q}(\\theta | S_{n}) \\), and draws an estimator \\( \\theta \\in \\Theta \\) according to \\( \\hat{q}(\\theta | S_{n}) \\). The learning algorithm suffers a loss (also referred to as regret) \\( Q(\\theta, q) \\). The overall probability of \\( \\theta \\) and \\( S_{n} \\) is\n\\[\np(\\theta,\\mathcal{S}_{n}|\\hat{q},q)=\\hat{q}(\\theta|\\mathcal{S}_{n})\\prod_{t=1}^{n}q\\left(Z_{t}|\\hat{q}\\left(\\mathcal{S}_{t-1}\\right),\\mathcal{S}_{t-1}\\right).\\tag{13.8}\n\\]\nNote that although the notation in Definition 13.16 assumes that \\(\\pi_t\\) is a deterministic function of \\(S_{t-1}\\), it can also incorporate randomized algorithms. In such case, a policy is simply a distribution over the random choices. Since a random choice can always be characterized by a distribution (over the random choices) that is a deterministic function of \\(S_{t-1}\\), when we consider random policies in subsequent chapters, it should be understood that such policies can be handled by Definition 13.16. An example is given below.\n\ntheorem 13.11: We have for any \\(\\lambda>0\\), with probability at least \\(1-\\delta\\), for all \\(n\\geq 1\\) and \\(f\\in\\mathcal{F}\\): \\[\\left[-\\sum_{i=1}^{n} f(Z_i)-\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_i^{(y)}}e^{-\\lambda f(Z_i)}\\right]\\leq\\inf_{\\epsilon>0}\\left[2n\\epsilon+\\frac{\\ln(M(\\epsilon,\\mathcal{F},\\|\\cdot\\|_\\infty)/\\delta)}{\\lambda}\\right].\\]", "theorems": [{"local_id": "T1", "statement": "Let \\(d \\geq 1\\) and \\(m \\geq 2\\) be integers, and let \\(\\mathcal{P}_{\\mathcal{Z}} = \\{q^{\\tau} : \\tau \\in \\{1, \\ldots, m\\}^{d}\\}\\) contain \\(m^{d}\\) probability measures. Suppose that the loss function \\(Q\\) can be decomposed as\n\\[\nQ(\\theta,q)=\\sum_{j=1}^{d}Q_{j}(\\theta,q),\n\\]\nwhere \\( Q_{j} \\geq 0 \\) are all non-negative. For each \\( j \\), \\( \\tau \\sim_{j} \\tau' \\) if \\( \\tau = \\tau' \\) or if \\( \\tau \\) and \\( \\tau' \\) differ by only one component \\( j \\). Assume that there exists \\( \\epsilon, \\beta \\geq 0 \\) such that\n\\[\n\\forall\\tau^{\\prime}\\sim_{j}\\tau,\\tau^{\\prime}\\neq\\tau:[Q_{j}(\\theta,q^{\\tau})+Q_{j}(\\theta,q^{\\tau^{\\prime}})]\\geq\\epsilon,\n\\]\nand there exists \\( q_j^\\tau \\) such that all \\( \\tau' \\sim_j \\tau \\) map to the same value: \\( q_j^{\\tau'} = q_j^\\tau \\). Given any learning algorithm \\( \\hat{q} \\). If for all \\( \\tau \\), \\( j \\in [d] \\), time step \\( t \\), and \\( S_{t-1} \\):\n\\[\n\\frac{1}{m}\\sum_{\\tau^{\\prime}\\sim_{j}\\tau}\\mathrm{KL}(q_{j}^{\\tau}(\\cdot|\\hat{q}(\\mathcal{S}_{t-1}),\\mathcal{S}_{t-1})| |q^{\\tau^{\\prime}}(\\cdot|\\hat{q}(\\mathcal{S}_{t-1}),\\mathcal{S}_{t-1})))\\leq\\beta_{j,t}^{2},\n\\]\nthen\n\\[\n\\frac{1}{m^{d}}\\sum_{\\tau}\\mathbb{E}_{\\theta,S_{n}\\sim p(\\cdot|\\hat{q},q^{\\tau})}Q(\\theta,q^{\\tau})\\geq 0.5 d\\epsilon\\left(1-\\sqrt{\\frac{2}{d}\\sum_{j=1}^{d}\\sum_{t=1}^{n}\\beta_{j,t}^{2}}\\right).\n\\]", "proof": {"steps": [{"n": 1, "claim": "We have\n\\[\n\\begin{array}{l}\\frac{1}{dm^{d}}\\sum_{\\tau}\\sum_{j=1}^{d}\\frac{1}{m-1}\\sum_{\\tau^{\\prime}\\sim_{j}\\tau}\\|p(\\cdot|\\hat{q},q^{\\tau})-p(\\cdot|\\hat{q},q^{\\tau^{\\prime}})\\|_{\\mathrm{TV}}\\\\ \\leq\\frac{1}{dm^{d}}\\sum_{\\tau}\\sum_{j=1}^{d}\\frac{1}{m-1}\\sum_{\\tau^{\\prime}\\sim_{j}\\tau,\\tau^{\\prime}\\neq\\tau}\\left[\\|p(\\cdot|\\hat{q},q_{j}^{\\tau})-p(\\cdot|\\hat{q},q^{\\tau^{\\prime}})\\|_{\\mathrm{TV}}\\right.\\\\ +\\|p(\\cdot|\\hat{q},q_{j}^{\\tau})-p(\\cdot|\\hat{q},q^{\\tau})\\|_{\\mathrm{TV}}\\big]\\\\ =\\frac{2}{dm^{d}}\\sum_{\\tau}\\sum_{j=1}^{d}\\frac{1}{m}\\sum_{\\tau^{\\prime}\\sim_{j}\\tau}\\|p(\\cdot|\\hat{q},q_{j}^{\\tau})-p(\\cdot|\\hat{q},q^{\\tau^{\\prime}})\\|_{\\mathrm{TV}}\\\\ \\leq\\frac{2}{m^{d}}\\sum_{\\tau}\\sqrt{\\frac{1}{dm}\\sum_{j=1}^{d}\\sum_{\\tau^{\\prime}\\sim_{j}\\tau}\\|p(\\cdot|\\hat{q},q_{j}^{\\tau})-p(\\cdot|\\hat{q},q^{\\tau^{\\prime}})\\|_{\\mathrm{TV}}^{2}}\\\\ \\leq\\frac{2}{m^{d}}\\sum_{\\tau}\\sqrt{\\frac{1}{2dm}\\sum_{j=1}^{d}\\sum_{\\tau^{\\prime}\\sim_{j}\\tau}\\mathrm{KL}(p(\\cdot|\\hat{q},q_{j}^{\\tau})| |p(\\cdot|\\hat{q},q^{\\tau^{\\prime}}))}\\\\ \\leq\\sqrt{\\frac{2}{d}\\sum_{j=1}^{d}\\sum_{t=1}^{n}\\beta_{t}^{2}}.\\\\ \\end{array}\n\\]\nThe first inequality is triangle inequality for TV-norm."}, {"n": 2, "claim": "The first equality used \\(\\hat{q}_j^\\tau = \\hat{q}_j^{\\tau '}\\) when \\(\\tau \\sim_{j}\\tau^{\\prime}\\)."}, {"n": 3, "claim": "The second inequality used Jensen's inequality and the\nconcavity of \\(\\sqrt{\\cdot}\\)."}, {"n": 4, "claim": "The third inequality used Theorem B.9. The last inequality used Lemma 13.21."}, {"n": 5, "claim": "Now in Lemma 12.27, we let \\(M_{j}(q^{\\tau}) = \\{q^{\\tau^{\\prime}}:\\tau^{\\prime}\\sim_{j}\\tau \\}\\) be the partitions."}, {"n": 6, "claim": "The result is a simple application of Lemma 12.27 with \\(m_j(q^\\tau) = m\\), \\(|\\mathcal{P}| = m^{d}\\), and \\(d(\\mathcal{P}) = dm^{d}\\)."}]}}, {"local_id": "T2", "statement": "Let \\(\\Sigma_0\\) be a \\(d\\times d\\) symmetric positive definite matrix, and \\(\\{\\psi (X_t)\\}\\) be a sequence of vectors in \\(\\mathbb{R}^d\\). Let \\(\\Sigma_{t} = \\Sigma_{0} + \\sum_{s = 1}^{t}\\psi (X_{s})\\psi (X_{s})^{\\top}\\), then\n\\[\n\\sum_{s=1}^{t}\\frac{\\psi(X_{s})^{\\top}\\Sigma_{s-1}^{-1}\\psi(X_{s})}{1+\\psi(X_{s})^{\\top}\\Sigma_{s-1}^{-1}\\psi(X_{s})}\\leq\\ln|\\Sigma_{0}^{-1}\\Sigma_{t}|.\n\\]", "proof": {"steps": [{"n": 1, "claim": "We have\n\\[\n\\begin{array}{l}\\frac{\\psi(X_{s})^{\\top}\\Sigma_{s-1}^{-1}\\psi(X_{s})}{1+\\psi(X_{s})^{\\top}\\Sigma_{s-1}^{-1}\\psi(X_{s})}=\\psi(X_{s})^{\\top}\\Sigma_{s}^{-1}\\psi(X_{s})\\\\ =\\operatorname{trace}\\left(\\Sigma_{s}^{-1}\\left(\\Sigma_{s}-\\Sigma_{s-1}\\right)\\right)\\\\ =\\operatorname{trace}\\left(\\nabla\\ln\\left|\\Sigma_{s}\\right|\\left(\\Sigma_{s}-\\Sigma_{s-1}\\right)\\right)\\\\ \\leq\\ln|\\Sigma_{s}| -\\ln|\\Sigma_{s-1}|.\\\\ \\end{array}\n\\]\nThe first equality is due to the Sherman-Morrison formula."}, {"n": 2, "claim": "The last inequality used the fact that \\(\\ln |Z|\\) is a concave function of \\(Z\\) (see Theorem A.18)."}, {"n": 3, "claim": "Now by summing over \\(s = 1\\) to \\(s = t\\), we obtain the desired bound."}]}}, {"local_id": "T3", "statement": "Assume that for all \\( f \\in \\mathcal{F} \\) and \\( Z_{t}: -f(Z_{t}) + \\mathbb{E}_{Z_{t}^{(y)}}f(Z_{t}) \\leq b \\), and the following variance condition holds for all \\( f \\in \\mathcal{F} \\) and \\( S_{t} \\):\n\\[\n\\operatorname{Var}_{Z_{t}^{(y)}}[f(Z_{t})]\\leq c_{0}^{2}+c_{1}\\mathbb{E}_{Z_{t}^{(y)}}[f(Z_{t})],\n\\]\nthen with probability at least \\(1 - \\delta\\), for all \\(f \\in \\mathcal{F}\\):\n\\[\n\\begin{array}{l}\\left(1-\\lambda c_{1}\\phi(\\lambda b)\\right)\\sum_{t=1}^{n}\\mathbb{E}_{Z_{t}^{(y)}}[f(Z_{t})]\\leq\\sum_{t=1}^{n}f(Z_{t})+\\lambda n c_{0}^{2}\\phi(\\lambda b)\\\\ +\\inf_{\\epsilon>0}\\left[2 n\\epsilon+\\frac{\\ln(M(\\epsilon,\\mathcal{F},\\|\\cdot\\|_{\\infty})/\\delta)}{\\lambda}\\right],\\\\ \\end{array}\n\\]\nwhere \\(\\phi (z) = (e^{z} - z - 1) / z^{2}\\)", "proof": {"steps": [{"n": 1, "claim": "Note that by employing the moment estimation of Bennett's inequality in Lemma 2.20, we obtain\n\\[\n\\begin{array}{l}\\frac{1}{\\lambda}\\ln\\mathbb{E}_{Z_{t}^{(y)}}e^{-\\lambda f(Z_{t})}\\\\ \\leq-\\mathbb{E}_{Z_{t}^{(y)}}[f(Z_{t})]+\\lambda\\phi(\\lambda b)\\operatorname{Var}_{Z_{t}^{(y)}}[f(Z_{t})]\\\\ \\leq-\\mathbb{E}_{Z_{t}^{(y)}}[f(Z_{t})]+\\lambda\\phi(\\lambda b)[c_{0}^{2}+c_{1}\\mathbb{E}_{Z_{t}^{(y)}}[f(Z_{t})]].\\\\ \\end{array}\n\\]\nWe can now obtain the desired bound from Theorem 13.11."}]}}, {"local_id": "T4", "statement": "Let \\(\\{(X_t, \\epsilon_t)\\}\\) be a filtered sequence in \\(\\mathcal{X} \\times \\mathbb{R}\\) so that \\(\\epsilon_t\\) is conditional zero-mean sub-Gaussian noise: for all \\(\\lambda \\in \\mathbb{R}\\),\n\\[\n\\ln\\mathbb{E}\\left[e^{\\lambda\\epsilon_{t}}\\mid X_{t},\\mathcal{S}_{t-1}\\right]\\leq\\frac{\\lambda^{2}}{2}\\sigma^{2},\n\\]\nwhere \\( S_{t-1} \\) denotes the history data. Assume that \\( Y_t = f_*(X_t) + \\epsilon_t \\), with \\( f_*(x) \\in \\mathcal{F}: \\mathcal{X} \\to \\mathbb{R} \\). Let \\( \\hat{f}_t \\) be an approximate ERM solution for some \\( \\epsilon' > 0 \\):\n\\[\n\\left(\\sum_{s=1}^{t}\\left(\\hat{f}_{t}\\left(X_{s}\\right)-Y_{s}\\right)^{2}\\right)^{1/2}\\leq\\min_{f\\in\\mathcal{F}}\\left(\\sum_{s=1}^{t}\\left(f\\left(X_{s}\\right)-Y_{s}\\right)^{2}\\right)^{1/2}+\\sqrt{t}\\epsilon^{\\prime}.\n\\]\nThen with probability at least \\(1 - \\delta\\), for all \\(t \\geq 0\\):\n\\[\n\\sum_{s=1}^{t}(\\hat{f}_{t}(X_{t})-f_{*}(X_{t}))^{2}\\leq\\inf_{\\epsilon>0}\\left[8 t(\\epsilon+\\epsilon^{\\prime})(\\sigma+2\\epsilon+2\\epsilon^{\\prime})+12\\sigma^{2}\\ln\\frac{2 N(\\epsilon,\\mathcal{F},\\|\\cdot\\|_{\\infty})}{\\delta}\\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\eta = \\sigma^{-2} / 4\\), and for \\(f \\in \\mathcal{F}\\), let\n\\[\n\\phi(f,Z_{t})=\\left[\\left(f(X_{t})-Y_{t}\\right)^{2}-\\left(f_{*}(X_{t})-Y_{t}\\right)^{2}\\right].\n\\]\nLet \\(\\mathcal{F}_{\\epsilon}\\) be an \\(\\epsilon \\| \\cdot \\|_{\\infty}\\) cover of \\(\\mathcal{F}\\) with \\(N = N(\\epsilon, \\mathcal{F}, \\| \\cdot \\|_{\\infty})\\) members."}, {"n": 2, "claim": "Given any \\(f \\in \\mathcal{F}_{\\epsilon}\\),\n\\[\n\\begin{array}{l}\\ln\\mathbb{E}_{Y_{t}|X_{t},S_{t-1}}\\exp(-\\eta\\phi(f,Z_{t}))\\\\ = -\\eta\\left(f\\left(X_{t}\\right)-f_{*}\\left(X_{t}\\right)\\right)^{2}\\\\ +\\ln\\mathbb{E}_{Y_{t}\\mid X_{t},S_{t-1}}\\exp(-2\\eta(f(X_{t})-f_{*}(X_{t}))(f_{*}(X_{t})-Y_{t}))\\\\ \\leq-\\eta(f(X_{t})-f_{*}(X_{t}))^{2}+\\frac{4\\eta^{2}\\sigma^{2}}{2}(f(X_{t})-f_{*}(X_{t}))^{2}\\\\ = -\\frac{\\left(f(X_{t})-f_{*}(X_{t})\\right)^{2}}{8\\sigma^{2}}.\\\\ \\end{array}\n\\]\nThe inequality used the sub-Gaussian noise assumption."}, {"n": 3, "claim": "It follows from Theorem 13.11 that with probability at least \\( 1 - \\delta / 2 \\), for all \\( f \\in \\mathcal{F}_{\\epsilon} \\) and \\( t \\geq 0 \\):"}, {"n": 4, "claim": "\\[\n\\sum_{i=1}^{t}-\\eta\\phi(f,Z_{i})\\leq-\\sum_{i=1}^{t}\\frac{\\left(f\\left(X_{i}\\right)-f_{*}\\left(X_{i}\\right)\\right)^{2}}{8\\sigma^{2}}+\\ln(2 N/\\delta).\\tag{13.5}\n\\]\nUsing Exercise 13.3, we have with probability at least \\(1 - \\delta /2\\), for all \\(t\\geq 0\\)\n\\[\n\\underbrace{\\sum_{i=1}^{t}\\left(f_{*}\\left(X_{i}\\right)-Y_{i}\\right)^{2}}_{A}\\leq 2 t\\sigma^{2}+3\\sigma^{2}\\ln(2/\\delta).\\tag{13.6}\n\\]\nIt follows that with probability at least \\(1 - \\delta\\), for all \\(t \\geq 0\\) and \\(f \\in \\mathcal{F}_{\\epsilon}\\), both (13.5)\nand (13.6) hold."}, {"n": 5, "claim": "Now, let \\( f \\in \\mathcal{F}_{\\epsilon} \\) so that \\( \\| \\hat{f}_t - f \\|_{\\infty} \\leq \\epsilon \\)."}, {"n": 6, "claim": "This implies that\n\\[\n\\begin{array}{l}\\sum_{i=1}^{t}\\left[(f(X_{i})-Y_{i})^{2}-(f_{*}(X_{i})-Y_{i})^{2}\\right]\\leq\\left(\\sqrt{\\sum_{i=1}^{t}(\\hat{f}_{t}(X_{i})-Y_{i})^{2}}+\\sqrt{t}\\epsilon\\right)^{2}-A\\\\ \\leq\\left(\\sqrt{A}+\\sqrt{t}(\\epsilon+\\epsilon^{\\prime})\\right)^{2}-A\\\\ \\stackrel{(a)}{\\leq}(\\epsilon^{\\prime}+\\epsilon)^{2}t+2\\sqrt{t}(\\epsilon^{\\prime}+\\epsilon)\\sigma\\sqrt{2 t+3\\ln(2/\\delta)}\\\\ \\leq 4\\left(\\epsilon^{\\prime}+\\epsilon\\right)^{2}t+3\\sigma t\\left(\\epsilon^{\\prime}+\\epsilon\\right)+\\sigma^{2}\\ln(2/\\delta).\\tag{13.7}\\\\ \\end{array}\n\\]\nIn the above derivations, the first inequality used \\( |f(X_i) - Y_i| \\leq |\\hat{f}_t(X_i) - Y_i| + \\epsilon \\) for all \\( i \\)."}, {"n": 7, "claim": "The second inequality used the fact that \\( \\hat{f}_t \\) is approximate ERM. (a) follows from (13.6)."}, {"n": 8, "claim": "The last inequality used \\( 2a\\sigma \\sqrt{2b + 3c} \\leq 3a\\sigma \\sqrt{b} + 3a^2 + \\sigma^2 c \\)."}, {"n": 9, "claim": "We thus obtain\n\\[\n\\begin{array}{l}\\left(\\sum_{i=1}^{t}(\\hat{f}(X_{i})-f_{*}(X_{i}))^{2}\\right)^{1/2}\\\\ \\leq\\sqrt{\\epsilon^{2}t}+\\left(\\sum_{i=1}^{t}\\left(f\\left(X_{i}\\right)-f_{*}\\left(X_{i}\\right)\\right)^{2}\\right)^{1/2}\\quad(\\text{t r i a n g l e i n e q u a l i t y})\\\\ \\leq\\sqrt{\\epsilon^{2}t}+\\left(8\\sigma^{2}\\ln(2N/\\delta)+8\\sigma^{2}\\sum_{i=1}^{t}\\eta\\phi(f,Z_{i})\\right)^{1/2}\\quad(\\text{f r o m}(13. 5))\\\\ =\\sqrt{\\epsilon^{2}t}+\\left(8\\sigma^{2}\\ln(2N/\\delta)+2\\sum_{i=1}^{t}\\left[\\left(f\\left(X_{i}\\right)-Y_{i}\\right)^{2}-\\left(f_{*}\\left(X_{i}\\right)-Y_{i}\\right)^{2}\\right]\\right)^{1/2}\\\\ \\leq\\sqrt{\\epsilon^{2}t}+\\left(10\\sigma^{2}\\ln(2N/\\delta)+8t\\left(\\epsilon^{\\prime}+\\epsilon\\right)^{2}+6\\sigma t\\left(\\epsilon^{\\prime}+\\epsilon\\right)\\right)^{1/2}\\quad(\\text{f r o m}(13. 7))\\\\ \\leq\\left(12\\sigma^{2}\\ln(2N/\\delta)+16t(\\epsilon+\\epsilon^{\\prime})^{2}+8\\sigma t(\\epsilon+\\epsilon^{\\prime})\\right)^{1/2},\\\\ \\end{array}\n\\]\nwhere the last inequality used \\(\\sqrt{a} +\\sqrt{b}\\leq \\sqrt{6a + 1.2b}\\) ."}, {"n": 10, "claim": "This implies the bound."}]}}, {"local_id": "T5", "statement": "Consider a sequence of random functions \\(\\xi_1(S_1),\\ldots,\\xi_n(S_n)\\). Assume that \\(\\xi_{i}\\geq \\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i} - b\\) for some constant \\(b > 0\\). Then for any \\(\\lambda \\in (0,3 / b)\\), with probability at least \\(1 - \\delta\\):\n\\[\n\\forall n>0:\\sum_{i=1}^{n}\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i}<\\sum_{i=1}^{n}\\xi_{i}+\\frac{\\lambda\\sum_{i=1}^{n}\\mathrm{Var}_{Z_{i}^{(y)}}(\\xi_{i})}{2(1-\\lambda b/3)}+\\frac{\\ln(1/\\delta)}{\\lambda}.\n\\]\nThis implies that for all \\(\\sigma > 0\\), with probability at least \\(1 - \\delta\\):\n\\[\n\\forall n>0:\\sum_{i=1}^{n}\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i}<\\sum_{i=1}^{n}\\xi_{i}+\\sigma\\sqrt{2\\ln(1/\\delta)}+\\frac{b\\ln(1/\\delta)}{3}or\\sum_{i=1}^{n}\\mathrm{Var}_{Z_{i}^{(y)}}(\\xi_{i})>\\sigma^{2}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Using the logarithmic moment generating function (2.13), we obtain the first inequality directly from the first inequality of Theorem 13.2."}, {"n": 2, "claim": "Moreover, we can obtain from the second inequality of Theorem 13.2 with \\( A_{n} = \\{S_{n}:\\sum_{i = 1}^{n}\\mathrm{Var}_{Z_{i}^{(y)}}(\\xi_{i})\\leq \\sigma^{2}\\} \\), \\( z_{n} = \\sum_{i = 1}^{n}\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i} - \\epsilon -\\epsilon^{2}b / (6\\sigma^{2}) \\), and the rate function estimate corresponding to the third inequality of Lemma 2.9:"}, {"n": 3, "claim": "\\[\n\\Pr\\left[\\exists n>0:\\sum_{i=1}^{n}\\xi_{i}\\leq z_{n}\\text{a n d}\\mathcal{S}_{n}\\in A_{n}\\right]\\leq\\exp\\left(-\\frac{\\epsilon^{2}}{2\\sigma^{2}}\\right).\n\\]\nThis implies the second desired inequality with \\(\\epsilon = \\sigma \\sqrt{2\\ln(1 / \\delta)}\\)"}]}}], "target_local_id": "T5", "topic_domain": "Probability & Concentration", "topic_subtopic": "Martingales & Dependent Data", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch15_proposition_15_5__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nequation (15.3). Let \\(\\Lambda_0\\) be a symmetric positive definite regularizer and let \\(\\psi\\) be a feature map. Given \\(\\mathcal{S}_n = \\{(X_1,Y_1),\\ldots,(X_n,Y_n)\\}\\), define\n\\[\n\\hat{\\Lambda}(\\mathcal{S}_{n})=\\Lambda_{0}+\\sum_{i=1}^{n}\\psi(X_{i})\\psi(X_{i})^{\\top}\n\\]\nand the ridge regression estimator\n\\[\n\\hat{w}(\\mathcal{S}_{n})=\\hat{\\Lambda}(\\mathcal{S}_{n})^{-1}\\sum_{i=1}^{n}\\psi(X_{i})Y_{i}.\\tag{15.3}\n\\]\nEquivalently, \\(\\hat{w}(\\mathcal{S}_{n})\\) minimizes\n\\[\n\\sum_{i=1}^{n}(w^{\\top}\\psi(X_{i})-Y_{i})^{2}+\\|w\\|_{\\Lambda_{0}}^{2}.\n\\]\n\ndefinition 10.6. Let \\(\\mathcal{F}_{L_1}(\\Psi)\\) be the point-wise closure of \\(\\cup_{A > 0}\\mathcal{F}_{A,L_1}(\\Psi)\\), then for any \\(f\\in \\mathcal{F}_{L_1}(\\Psi)\\),\n\\[\n\\|f\\|_{1}=\\lim_{\\epsilon\\rightarrow 0}\\inf\\left\\{\\|w\\|_{1}:\\sup_{x}\\left|f(x)-\\sum_{j=1}^{m}w_{j}\\psi(\\theta_{j},x)\\right|\\leq\\epsilon\\right\\}.\n\\]\nFor notational convenience, we write functions in \\(\\mathcal{F}_{L_1}(\\Psi)\\) as\n\\[\nf(x)=w^{\\top}\\psi(x),\n\\]\nwhere \\(\\psi(x)\\) is the infinite dimensional vector \\([\\psi(\\theta, x)]_{\\theta \\in \\Theta}\\), and \\(\\|f\\|_1 = \\|w\\|_1\\).\nFor smooth-loss functions (see Definition A.5), there is a strong relationship between \\( L_{1} \\) regularization and \\( L_{0} \\) regularization, in that functions with small \\( L_{1} \\) norms can be sparsified. Such a sparsification result was first shown for the least squares loss, and referred to as Maurey's lemma in Pisier (1980-1981).\n\ndefinition 6.3. Given \\(S_{n}=\\{Z_{1},\\ldots,Z_{n}\\}\\), the (one-sided) empirical Rademacher complexity of \\(\\mathcal{G}\\) is defined as\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n}) = \\mathbb{E}_{\\sigma}\\sup_{w\\in \\Omega}\\frac{1}{n}\\sum_{i = 1}^{n}\\sigma_{i}\\phi (w,Z_{i}),\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R(\\mathcal{G},\\mathcal{S}_{n}).\n\\]\nThe following result shows that the quantity \\(\\epsilon_{n}(\\mathcal{G},\\mathcal{D})\\) can be upper bounded by Rademacher complexity. It follows that an average oracle inequality can be obtained using Rademacher complexity. The proof employs the symmetrization technique, which was also used in Chapter 4 to obtain uniform convergence bounds from empirical covering numbers.\n\nequation (15.3). The ridge regression method defines, for each \\(n\\),\n\\[\n\\hat{w}(\\mathcal{S}_{n})=\\arg\\min_{w}\\left[\\sum_{i=1}^{n}(Y_{i}-w^{\\top}\\psi(X_{i}))^{2}+\\|w\\|_{\\Lambda_{0}}^{2}\\right].\n\\]\nEquivalently,\n\\[\n\\hat{w}(\\mathcal{S}_{n})=\\Lambda_{n}^{-1}\\sum_{i=1}^{n}\\psi(X_{i})Y_{i},\\quad\\Lambda_{n}=\\Lambda_{0}+\\sum_{s=1}^{n}\\psi(X_{s})\\psi(X_{s})^{\\top}.\n\\]\n\nexample 15.2. For least squares regression with noise variance \\(\\sigma^2\\), we may have\n\\[\np(y|w,x)=\\frac{1}{\\sqrt{2\\pi}\\sigma}\\exp\\left(-\\frac{(y-f(w,x))^{2}}{2\\sigma^{2}}\\right),\n\\]\nand \\((\\operatorname{let} Z = (x, y))\\):\n\\[\n\\phi(w,Z)=\\frac{(y-f(w,x))^{2}}{2\\sigma^{2}}+\\ln(\\sqrt{2\\pi}\\sigma).\n\\]\nWe may also consider the noise as part of the model parameter, and let\n\\[\np(y|[w,\\sigma],x)=\\frac{1}{\\sqrt{2\\pi}\\sigma}\\exp\\left(-\\frac{(y-f(w,x))^{2}}{2\\sigma^{2}}\\right)\n\\]\n\\[\n\\phi([w,\\sigma],Z)=\\frac{(y-f(w,x))^{2}}{2\\sigma^{2}}+\\ln(\\sqrt{2\\pi}\\sigma).\n\\]\nIn the Bayesian setting, the optimal statistical estimation of posterior density for log-loss is the Bayesian posterior averaging estimator. Here we consider a prior \\( p_0(w) \\) on \\( \\Omega \\). Given the training data \\( S_{n}=\\{Z_{1},\\dots,Z_{n}\\} \\), the posterior distribution is\n\\[\np(w|\\mathcal{S}_{n})=\\frac{\\prod_{i=1}^{n}p\\left(Y_{i}\\mid w,X_{i}\\right)p_{0}(w)}{\\int_{\\Omega}\\prod_{i=1}^{n}p\\left(Y_{i}\\mid w^{\\prime},X_{i}\\right)p_{0}\\left(w^{\\prime}\\right)d w^{\\prime}}.\\tag{15.1}\n\\]\nUnlike ERM, the Bayesian posterior averaging method does not use \\( p(y|\\hat{w},x) \\) for a specific \\( \\hat{w}\\in \\Omega \\) that is estimated from the training data, but instead use the averaged probability estimate over the posterior as follows:\n\\[\n\\hat{p}(y|x,\\mathcal{S}_{n})=\\int_{\\Omega}p(y|w,x)p(w|\\mathcal{S}_{n})d w.\\tag{15.2}\n\\]\nAs pointed out in Section 14.1, such a method is referred to as improper online learning.\nIn the online setting, we have the following result, which may be regarded as a generic regret bound for the Bayesian posterior model averaging method.\n\nexample 10.11. In two-layer neural networks, let \\(\\Psi\\) be an \\(L_{2}\\)-regularized ReLU function class:\n\\[\n\\Psi=\\left\\{\\psi(\\theta,x)=\\max(0,\\theta^{\\top}x):\\quad\\|\\theta\\|_{2}\\leq\\alpha,\\|x\\|_{2}\\leq\\beta\\right\\},\n\\]\nand the corresponding \\(L_{1}\\) regularized two-layer NN can be expressed as a function\n\\[\nf(x)=\\sum_{j=1}^{m}w_{j}\\psi\\left(\\theta_{j},x\\right):\\|w\\|_{1}\\leq A,\\|\\theta\\|_{2}\\leq\\alpha,\\|x\\|_{2}\\leq\\beta .\n\\]\nThis function class belongs to \\(\\mathcal{F}_{A,L_1}(\\Psi)\\). We thus obtain the following bound for \\(L_{1}\\) regularized two-layer NN:\n\\[\n\\begin{array}{l}R\\left(\\mathcal{F}_{A,L_{1}}(\\Psi),\\mathcal{S}_{n}\\right)\\leq 2AR(\\Psi,\\mathcal{S}_{n})\\tag{Theorem 10.8}\\\\ \\leq 2A\\alpha\\beta/\\sqrt{n},\\quad(\\text{C o r o l l a r y} 9. 21)\\\\ \\end{array}\n\\]\nwhere we note that \\(\\max (0,f)\\) is 1-Lipschitz in \\(f\\)\nNext, we may consider the following hard-constrained \\( L_{1} \\) regularized learning problem.\n\\[\n\\hat{w}=\\arg\\min_{w}\\frac{1}{n}\\sum_{i=1}^{n}L\\left(w^{\\top}\\psi\\left(X_{i}\\right),Y_{i}\\right),\\quad\\|w\\|_{1}\\leq A.\\tag{10.8}\n\\]\nSimilarly, we may consider the soft-regularized version as:\n\\[\n\\hat{w}=\\arg\\min_{w}\\frac{1}{n}\\sum_{i=1}^{n}L\\left(w^{\\top}\\psi\\left(X_{i}\\right),Y_{i}\\right)+\\lambda\\|w\\|_{1}.\\tag{10.9}\n\\]\nNow by using Theorem 6.31, we obtain the following result.\n\nequation (15.3). Let \\(\\Lambda_0\\) be a symmetric invertible regularizer and let \\(\\psi\\) be a feature map. Given \\(\\mathcal{S}_n = \\{(X_1,Y_1),\\ldots,(X_n,Y_n)\\}\\), define\n\\[\n\\hat{\\Lambda}(\\mathcal{S}_{n})=\\Lambda_{0}+\\sum_{i=1}^{n}\\psi(X_{i})\\psi(X_{i})^{\\top}\n\\]\nand the ridge regression estimator\n\\[\n\\hat{w}(\\mathcal{S}_{n})=\\hat{\\Lambda}(\\mathcal{S}_{n})^{-1}\\sum_{i=1}^{n}\\psi(X_{i})Y_{i}.\\tag{15.3}\n\\]\nEquivalently, \\(\\hat{w}(\\mathcal{S}_{n})\\) minimizes\n\\[\n\\sum_{i=1}^{n}(w^{\\top}\\psi(X_{i})-Y_{i})^{2}+\\|w\\|_{\\Lambda_{0}}^{2}.\n\\]", "theorems": [{"local_id": "T1", "statement": "Consider probability model\n\\[\np(y|w,x)=N(w^{\\top}\\psi(x),\\sigma^{2}),\n\\]\nwith prior\n\\[\np_{0}(w)=N(0,\\sigma^{2}\\Lambda_{0}^{-1}).\n\\]\nThen given \\(\\mathcal{S}_n=\\{(X_1,Y_1),\\ldots,(X_n,Y_n)\\}\\) , we have\n\\[\np(w|\\mathcal{S}_{n})=N(\\hat{w}(\\mathcal{S}_{n}),\\sigma^{2}\\hat{\\Lambda}(\\mathcal{S}_{n})^{-1}),\n\\]\nwhere \\(\\hat{w} (S_n)\\) is given by (15.3) and\n\\[\n\\hat{\\Lambda}\\left(\\mathcal{S}_{n}\\right)=\\sum_{i=1}^{n}\\psi\\left(X_{i}\\right)\\psi\\left(X_{i}\\right)^{\\top}+\\Lambda_{0}.\n\\]\nGiven \\( x \\), the posterior distribution of \\( y \\) is\n\\[\n\\hat{p}(y|x,\\mathcal{S}_{n})=N\\left(\\hat{w}(\\mathcal{S}_{n})^{\\top}\\psi(x),\\sigma^{2}+\\sigma^{2}\\psi(x)^{\\top}\\hat{\\Lambda}(\\mathcal{S}_{n})^{-1}\\psi(x)\\right).\n\\]", "proof": {"steps": [{"n": 1, "claim": "It is clear that\n\\[\np(w|\\mathcal{S}_{n})\\propto\\exp\\left(-\\sum_{i=1}^{n}\\frac{(w^{\\top}\\psi(X_{i})-Y_{i})^{2}}{2\\sigma^{2}}-\\frac{\\|w\\|_{\\Lambda_{0}}^{2}}{2\\sigma^{2}}\\right).\n\\]\nNote that (15.3) implies that\n\\[\n\\begin{array}{l}\\sum_{i=1}^{n}\\frac{\\left(w^{\\top}\\psi\\left(X_{i}\\right)-Y_{i}\\right)^{2}}{2\\sigma^{2}}+\\frac{\\|w\\|_{\\Lambda_{0}}^{2}}{2\\sigma^{2}}\\\\ =\\sum_{i=1}^{n}\\frac{\\left(\\hat{w}^{\\top}\\psi\\left(X_{i}\\right)-Y_{i}\\right)^{2}}{2\\sigma^{2}}+\\frac{\\|\\hat{w}\\|_{\\Lambda_{0}}^{2}}{2\\sigma^{2}}+\\frac{1}{2\\sigma^{2}}(w-\\hat{w})^{\\top}\\hat{\\Lambda}\\left(S_{n}\\right)(w-\\hat{w}).\\tag{15.4}\\\\ \\end{array}\n\\]\nThis implies the first desired result."}, {"n": 2, "claim": "Moreover, given \\( x \\), and let the random variable \\( u = w^{\\top}\\psi (x) \\) with \\( w\\sim p(w|S_n) \\), we have\n\\[\nu|x,\\mathcal{S}_{n}\\sim N\\left(\\hat{w}^{\\top}\\psi(x),\\sigma^{2}\\psi(x)^{\\top}\\hat{\\Lambda}(\\mathcal{S}_{n})^{-1}\\psi(x)\\right).\n\\]\nSince in posterior distribution, the observation \\( y|u \\sim u + \\epsilon \\) with \\( \\epsilon \\sim N(0, \\sigma^2) \\), we know that\n\\[\ny|x,\\mathcal{S}_{n}\\sim N(\\hat{w}^{\\top}\\psi(x),\\sigma^{2}\\psi(x)^{\\top}\\hat{\\Lambda}^{-1}\\psi(x)+\\sigma^{2}).\n\\]\nThis implies the desired result."}]}}, {"local_id": "T2", "statement": "Assume that \\( Y_{t} \\in [0, M] \\) for \\( t \\geq 1 \\). Consider the ridge regression method of (15.3), and let\n\\[\n\\hat{Y}_{t}=\\max(0,\\min(M,\\hat{w}(\\mathcal{S}_{t-1})^{\\top}\\psi(X_{t}))).\n\\]\nWe have\n\\[\n\\sum_{t=1}^{T}(Y_{t}-\\hat{Y}_{t})^{2}\\leq\\inf_{w}\\left[\\sum_{t=1}^{T}(Y_{t}-w^{\\top}\\psi(X_{t}))^{2}+\\|w\\|_{\\Lambda_{0}}^{2}\\right]+M^{2}\\ln\\left|\\Lambda_{0}^{-1}\\Lambda_{T}\\right|,\n\\]\nwhere\n\\[\n\\Lambda_{T}=\\Lambda_{0}+\\sum_{s=1}^{T}\\psi(X_{s})\\psi(X_{s})^{\\top}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "We can apply Theorem 15.6 by taking \\(\\sigma^2 = M^2\\)."}, {"n": 2, "claim": "By using the following inequality\n\\[\n0\\leq\\frac{1-b_{t}}{b_{t}}+\\ln b_{t},\n\\]\nwe obtain\n\\[\n\\begin{array}{l}\\sum_{t=1}^{T}(Y_{t}-\\hat{Y}_{t})^{2}\\leq\\sum_{t=1}^{T}\\left[(Y_{t}-\\hat{Y}_{t})^{2}+M^{2}\\frac{1-b_{t}}{b_{t}}+M^{2}\\ln b_{t}\\right]\\\\ \\leq\\sum_{t=1}^{T}\\left[\\left(Y_{t}-\\hat{Y}_{t}\\right)^{2}+\\left(Y_{t}-\\hat{Y}_{t}\\right)^{2}\\frac{1-b_{t}}{b_{t}}+M^{2}\\ln b_{t}\\right]\\tag{15.5}\\\\ \\leq\\sum_{t=1}^{T}\\left[\\frac{(Y_{t}-\\hat{w}(\\mathcal{S}_{t-1})^{\\top}\\psi(X_{i}))^{2}}{b_{t}}+M^{2}\\ln b_{t}\\right],\\\\ \\end{array}\n\\]\nwhere \\( b_{t} \\) is defined in Theorem 15.6."}, {"n": 3, "claim": "Note that (15.5) used \\( 1 - b_{t} \\leq 0 \\) and \\( (Y_{t} - \\hat{Y}_{t})^{2} \\leq M^{2} \\)."}, {"n": 4, "claim": "The desired result is now a direct application of Theorem 15.6."}]}}, {"local_id": "T3", "statement": "Consider \\(\\mathcal{D}_f(X,Y)\\) so that \\(X\\sim \\mathcal{D}_X\\) is identical for all \\(f\\in \\mathcal{F}\\), and \\(Y\\sim N(f(X),\\sigma^2)\\) for some constant \\(\\sigma >0\\). Then\n\\[\n\\mathbb{E}_{X\\sim\\mathcal{D}_{X}}(f(X)-f^{\\prime}(X))^{2}=2\\sigma^{2}\\mathrm{KL}(\\mathcal{D}_{f}| |\\mathcal{D}_{f^{\\prime}}).\n\\]", "proof": {"steps": [{"n": 1, "claim": "We consider a distribution \\(\\mathcal{D}\\) over \\(Z = (X,Y)\\) associated with each \\(f\\in \\mathcal{F}\\), with density defined as\n\\[\np_{\\mathcal{D}}(Z)=\\sigma^{-1}p_{\\mathcal{D}_{X}}(X)p_{N(0,1)}((Y-f(X))/\\sigma),\n\\]\nwhere \\(p_{\\mathcal{D}_X}(X)\\) is the density of \\(X \\sim \\mathcal{D}_X\\) and \\(p_{N(0,1)}\\) is the density of standard normal distribution."}, {"n": 2, "claim": "Using this notation, we have\n\\[\n\\mathrm{KL}(\\mathcal{D}_{\\hat{f}}| |\\mathcal{D}_{f})=\\frac{\\mathbb{E}_{X}(\\hat{f}(X)-f(X))^{2}}{2\\sigma^{2}}.\n\\]\nThis proves the desired result."}]}}, {"local_id": "T4", "statement": "Consider the ridge regression method of (15.3). We have the following result for any \\(\\sigma \\geq 0\\) and for all observed sequence \\(S_T\\):\n\\[\n\\begin{array}{l}\\sum_{t=1}^{T}\\left[\\frac{\\left(Y_{t}-\\hat{w}\\left(\\mathcal{S}_{t-1}\\right)^{\\top}\\psi\\left(X_{t}\\right)\\right)^{2}}{b_{t}}+\\sigma^{2}\\ln b_{t}\\right]\\\\ =\\inf_{w}\\left[\\sum_{t=1}^{T}\\left(Y_{t}-w^{\\top}\\psi\\left(X_{t}\\right)\\right)^{2}+\\|w\\|_{\\Lambda_{0}}^{2}\\right]+\\sigma^{2}\\ln\\left|\\Lambda_{0}^{-1}\\Lambda_{T}\\right|,\\\\ \\end{array}\n\\]\nwhere\n\\[\n\\Lambda_{t}=\\Lambda_{0}+\\sum_{s=1}^{t}\\psi(X_{s})\\psi(X_{s})^{\\top},\n\\]\nand \\(b_{t} = 1 + \\psi (X_{t})^{\\top}\\Lambda_{t - 1}^{-1}\\psi (X_{t}).\\)", "proof": {"steps": [{"n": 1, "claim": "Assume \\( w \\in \\mathbb{R}^d \\). We note that from Gaussian integration that\n\\[\n\\begin{array}{l}\\mathbb{E}_{w\\sim p_{0}}\\prod_{i=1}^{T}p\\left(Y_{i}|w,X_{i}\\right)\\\\ =\\int\\frac{\\left|\\Lambda_{0}^{-1}\\right|^{-1/2}}{(2\\pi)^{(T+d)/2}\\sigma^{T}\\sigma^{d}}\\exp\\left(-\\frac{1}{2\\sigma^{2}}\\sum_{i=1}^{T}\\left(Y_{i}-w^{\\top}\\psi\\left(X_{i}\\right)\\right)^{2}-\\frac{\\left\\|w\\right\\|_{\\Lambda_{0}}^{2}}{2\\sigma^{2}}\\right)d w\\\\ =\\frac{\\left|\\Lambda_{0}^{-1}\\Lambda_{T}\\right|^{-1/2}}{\\left(2\\pi\\right)^{T/2}\\sigma^{T}}\\exp\\left(-\\frac{1}{2\\sigma^{2}}\\sum_{i=1}^{T}\\left(Y_{i}-\\hat{w}\\left(\\mathcal{S}_{T}\\right)^{\\top}\\psi(X_{i})\\right)^{2}-\\frac{\\left\\|\\hat{w}\\left(\\mathcal{S}_{T}\\right)\\right\\|_{\\Lambda_{0}}^{2}}{2\\sigma^{2}}\\right),\\\\ \\end{array}\n\\]\nwhere the last equation used Gaussian integration with decomposition (15.4)."}, {"n": 2, "claim": "That is,\n\\[\n\\begin{array}{l}-\\ln\\mathbb{E}_{w\\sim p_{0}}\\prod_{i=1}^{T}p(Y_{i}|w,X_{i})=T\\ln(\\sqrt{2\\pi}\\sigma)+\\frac{1}{2}\\ln\\left|\\Lambda_{0}^{-1}\\Lambda_{T}\\right|\\\\ +\\frac{1}{2\\sigma^{2}}\\sum_{i=1}^{T}(Y_{i}-\\hat{w}(\\mathcal{S}_{T})^{\\top}\\psi(X_{i}))^{2}+\\frac{\\|\\hat{w}(\\mathcal{S}_{T})\\|_{\\Lambda_{0}}^{2}}{2\\sigma^{2}}.\\\\ \\end{array}\n\\]\nMoreover, from Proposition 15.5, we have\n\\[\n\\begin{array}{l}\\sum_{t=1}^{T}-\\ln\\hat{p}(Y_{t}|X_{t},\\mathcal{S}_{t-1})\\\\ =\\sum_{t=1}^{T}\\left[\\frac{\\left(Y_{t}-\\hat{w}\\left(\\mathcal{S}_{t-1}\\right)^{\\top}\\psi\\left(X_{t}\\right)\\right)^{2}}{2\\sigma^{2}b_{t}}+\\frac{1}{2}\\ln(b_{t})+\\ln(\\sqrt{2\\pi}\\sigma)\\right].\\\\ \\end{array}\n\\]\nThe desired result now follows from Theorem 15.3."}]}}, {"local_id": "T5", "statement": "We have\n\\[\n\\begin{array}{l}-\\sum_{t=1}^{T}\\ln\\hat{p}\\left(Y_{t}\\mid X_{t},\\mathcal{S}_{t-1}\\right)= -\\ln\\mathbb{E}_{w\\sim p_{0}}\\prod_{t=1}^{T}p\\left(Y_{t}\\mid w,X_{t}\\right)\\\\ =\\inf_{q\\in\\Delta(\\Omega)}\\left[-\\mathbb{E}_{w\\sim q}\\sum_{t=1}^{T}\\ln p(Y_{t}|w,X_{t})+\\mathbb{E}_{w\\sim q}\\ln\\frac{q(w)}{p_{0}(w)}\\right],\\\\ \\end{array}\n\\]\nwhere \\(\\Delta (\\Omega)\\) is the set of probability distributions over \\(\\Omega\\)", "proof": {"steps": [{"n": 1, "claim": "We have\n\\[\n\\begin{array}{l}\\ln\\hat{p}(Y_{t}|X_{t},\\mathcal{S}_{t-1})=\\ln\\int_{\\Omega}p(Y_{t}|w,X_{t})p(w|\\mathcal{S}_{t-1})d w\\\\ =\\ln\\frac{\\int_{\\Omega}\\prod_{i=1}^{t}p(Y_{i}|w,X_{i})p_{0}(w)d w}{\\int_{\\Omega}\\prod_{i=1}^{t-1}p(Y_{i}|w^{\\prime},X_{i})p_{0}(w^{\\prime})d w^{\\prime}}\\\\ =\\ln\\mathbb{E}_{w\\in p_{0}}\\prod_{i=1}^{t}p(Y_{i}|w,X_{i})-\\ln\\mathbb{E}_{w\\in p_{0}}\\prod_{i=1}^{t-1}p(Y_{i}|w,X_{i}).\\\\ \\end{array}\n\\]\nBy summing over \\( t = 1 \\) to \\( t = T \\), we obtain\n\\[\n\\begin{array}{l}\\sum_{t=1}^{T}\\ln\\hat{p}\\left(Y_{t}\\mid x_{t},\\mathcal{S}_{t-1}\\right)=\\ln\\mathbb{E}_{w\\in p_{0}}\\prod_{t=1}^{T}p\\left(Y_{t}\\mid w,X_{t}\\right)\\\\ =\\sup_{q}\\left[\\mathbb{E}_{w\\sim q}\\ln\\prod_{t=1}^{T}p\\left(Y_{t}|w,X_{t}\\right)-\\mathbb{E}_{w\\sim q}\\ln\\frac{q(w)}{p_{0}(w)}\\right],\\\\ \\end{array}\n\\]\nwhere the second equality used Proposition 7.16."}, {"n": 2, "claim": "This implies the desired result."}]}}], "target_local_id": "T1", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch4_theorem_4_12__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\ndefinition 4.1. Let \\((\\mathcal{V},d)\\) be a pseudometric space with metric \\(d(\\cdot,\\cdot)\\). A finite\nset \\(\\mathcal{G}(\\epsilon) \\subset \\mathcal{V}\\) is an \\(\\epsilon\\)-cover (or \\(\\epsilon\\)-net) of \\(\\mathcal{G} \\subset \\mathcal{V}\\) if, for all \\(\\phi \\in \\mathcal{G}\\), there exists \\(\\phi' \\in \\mathcal{G}(\\epsilon)\\) so that \\(d(\\phi', \\phi) \\leq \\epsilon\\). The \\(\\epsilon\\)-covering number of \\(\\mathcal{G}\\) with metric \\(d\\) is the smallest cardinality \\(N(\\epsilon, \\mathcal{G}, d)\\) of such \\(\\mathcal{G}(\\epsilon)\\). The number \\(\\ln N(\\epsilon, \\mathcal{G}, d)\\) is called the \\(\\epsilon\\)-entropy.\nFor a function class \\(\\mathcal{G}\\) with seminorm \\(L_{p}(\\mathcal{D})\\) in (3.19), we denote the corresponding \\(L_{p}(\\mathcal{D})\\)-covering number as \\(N(\\epsilon, \\mathcal{G}, L_{p}(\\mathcal{D}))\\). When \\(1 \\leq p \\leq q\\), we have\n\\[\nN(\\epsilon,\\mathcal{G},L_{p}(\\mathcal{D}))\\leq N(\\epsilon,\\mathcal{G},L_{q}(\\mathcal{D})).\n\\]\nIt is easy to verify the following relationship, which implies that \\( L_{p}(\\mathcal{D}) \\) bracketing cover is a stronger requirement than \\( L_{p}(\\mathcal{D}) \\) cover.\n\ndefinition 3.8. We define\n\\[\n\\phi(w,z)=L\\left(f(w,x),y\\right)-L_{*}(x,y),\\tag{3.4}\n\\]\nfor \\( w \\in \\Omega \\) and \\( z = (x, y) \\in \\mathcal{Z} = \\mathcal{X} \\times \\mathcal{Y} \\), and a pre-chosen \\( L_{*}(x, y) \\) of \\( z = (x, y) \\) that does not depend on \\( w \\).\nFor training data \\( S_{n}=\\{Z_{i}=(X_{i},Y_{i}):i=1,\\ldots,n\\} \\), we define the training loss for \\( w\\in \\Omega \\) as\n\\[\n\\phi(w,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{i=1}^{n}\\phi(w,Z_{i}).\\tag{3.5}\n\\]\nMoreover, for a distribution \\(\\mathcal{D}\\) on \\(\\mathcal{Z}\\), we define the test loss for \\(w \\in \\Omega\\) as:\n\\[\n\\phi(w,\\mathcal{D})=\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(w,Z).\\tag{3.6}\n\\]\nSince \\( L_{*}(x,y) \\) does not depend on \\( w \\), the ERM solution with respect to the loss \\( L(\\cdot) \\) is equivalent to the ERM solution with respect to \\( \\phi (w,\\cdot) \\). Therefore with the simplified notations, the approximate ERM method (3.3) is a special case of the following method:\n\\[\n\\phi(\\hat{w},\\mathcal{S}_{n})\\leq\\inf_{w\\in\\Omega}\\phi(w,\\mathcal{S}_{n})+\\epsilon^{\\prime}.\\tag{3.7}\n\\]\nIn general, we may simply take \\( L_{*}(x,y) = 0 \\) in (3.4). However, for some applications, we may choose a non-zero \\( L_{*}(x,y) \\) so that\n\\[\nL(f(w,x),y)-L_{*}(x,y)\n\\]\nhas a small variance. For least squares loss, this can be achieved with \\( L_{*}(x,y) = L(f_{*}(x),y) \\) where \\( f_{*}(x) \\) is the optimal prediction function that minimizes the test loss as shown in Example 3.9. The smaller variance, combined with Bernstein's inequality, implies better generalization bound (see Section 3.6 for more details).\n\nEmpirical and Uniform Covering Number. Given an empirical distribution \\( S_{n}=\\{Z_{1},\\ldots,Z_{n}\\} \\), we define the pseudometric \\( d = L_{p}(S_{n}) \\) as\n\\[\nd(\\phi,\\phi^{\\prime})=\\left[\\frac{1}{n}\\sum_{i=1}^{n}|\\phi(Z_{i})-\\phi^{\\prime}(Z_{i})|^{p}\\right]^{1/p}.\n\\]\nThe corresponding metric covering number \\(N(\\epsilon, \\mathcal{G}, L_p(\\mathcal{S}_n))\\) is referred to as the\nempirical \\( L_{p} \\) covering number. Given \\( n \\), the largest \\( L_{p} \\) covering number over empirical distribution \\( S_{n} \\) is referred to as the uniform \\( L_{p} \\) covering number\n\\[\nN_{p}(\\epsilon,\\mathcal{G},n)=\\sup_{\\mathcal{S}_{n}}N(\\epsilon,\\mathcal{G},L_{p}(\\mathcal{S}_{n})).\n\\]\nSince the \\(L_{p}(\\mathcal{S}_{n})\\) pseudometric increases with \\(p\\), we have the following simple result.\n\nBracketing Number. Let \\(\\mathcal{G} = \\{\\phi(w, \\cdot) : w \\in \\Omega\\}\\) be a real-valued function class, equipped with a pseudometric \\(d\\). We say\n\\[\n\\mathcal{G}(\\epsilon)=\\left\\{\\left[\\phi_{1}^{L}(z),\\phi_{1}^{U}(z)\\right],\\dots,\\left[\\phi_{N}^{L}(z),\\phi_{N}^{U}(z)\\right]\\right\\}\n\\]\nis an \\(\\epsilon\\)-bracket of \\(\\mathcal{G}\\) under metric \\(d\\) if for all \\(w \\in \\Omega\\), there exists \\(j = j(w)\\) such that \\(\\forall z\\):\n\\[\n\\phi_{j}^{L}(z)\\leq\\phi(w,z)\\leq\\phi_{j}^{U}(z),\\quad d(\\phi_{j}^{L},\\phi_{j}^{U})\\leq\\epsilon .\n\\]\nThe \\(\\epsilon\\)-bracketing number is the smallest cardinality \\(N_{\\mathbb{I}}(\\epsilon, \\mathcal{G}, d)\\) of such \\(\\mathcal{G}(\\epsilon)\\). The quantity \\(\\ln N_{\\mathbb{I}}(\\epsilon, \\mathcal{G}, d)\\) is called \\(\\epsilon\\) bracketing entropy.\nIn particular, given a distribution \\(\\mathcal{D}\\) and \\(p \\geq 1\\), we define \\(L_{p}\\)-seminorm in function space as\n\\[\n\\left\\|f-f^{\\prime}\\right\\|_{L_{p}(\\mathcal{D})}=\\left[\\mathbb{E}_{Z\\sim\\mathcal{D}}|f(Z)-f^{\\prime}(Z)|^{p}\\right]^{1/p}.\\tag{3.19}\n\\]\nIt induces a pseudometric, denoted as \\( d = L_{p}(\\mathcal{D}) \\), and the corresponding bracketing number is \\( N_{\\parallel}(\\epsilon, \\mathcal{G}, L_{p}(\\mathcal{D})) \\).\nFor \\(p = \\infty\\), the \\(L_{\\infty}(\\mathcal{D})\\)-seminorm is defined as the essential supremum seminorm, which leads to the pseudometric\n\\[\nd(f,f^{\\prime})=\\inf\\left\\{\\omega:\\Pr_{\\mathcal{D}}[|f(Z)-f^{\\prime}(Z)|\\leq\\omega]=1\\right\\}.\n\\]\nHowever, when \\( p = \\infty \\), instead of using the \\( L_{\\infty} \\) bracketing number, it is more conventional to use the equivalent notion of \\( L_{\\infty} \\) covering number (see Proposition 4.3).\nWe have the following relationship of lower bracketing number and bracketing numbers with different norms.\n\nVC-dimension. We say that \\(\\mathcal{G}\\) shatters \\(S_{n}\\) if the number of elements \\(|\\mathcal{G}(S_n)|\\) is \\(2^{n}\\). That is, we can always find \\(w\\in \\Omega\\) so that \\(\\phi (w,z)\\) matches any arbitrary possible choice of \\(\\{0,1\\} ^n\\) values at the \\(n\\) points. The maximum \\(n\\) such that \\(\\mathcal{G}\\) shatters at least one instance of \\(S_{n}\\in \\mathcal{Z}^{n}\\), denoted by \\(\\mathrm{vc}(\\mathcal{G})\\), is called the VC-dimension of \\(\\mathcal{G}\\).\nNote that the maximum number of functions in \\(\\mathcal{G}(\\mathcal{S}_n)\\) is \\(2^n\\). If \\(n > d\\), then for any \\(n\\) samples \\(\\mathcal{S}_n\\), \\(\\mathcal{G}(\\mathcal{S}_n)\\) contains fewer than \\(2^n\\) elements. Surprisingly, if a binary-valued function class \\(\\mathcal{G}\\) has VC dimension \\(d\\), then when \\(n > d\\), the size of set \\(\\mathcal{G}(\\mathcal{S}_n)\\) can grow only polynomially in \\(n\\). This gives an \\(O(d\\ln n)\\) upper bound on the uniform entropy of the function class \\(\\mathcal{G}\\) with a finite VC-dimension (see Vapnik and Chervonenkis, 1968, 1971; Sauer, 1972).\n\ndefinition 4.7. Consider a real-valued function family \\(\\mathcal{F}:\\mathcal{Z}\\to \\mathbb{R}\\). Consider \\(n\\) iid Bernoulli random variables \\(\\sigma_{i}\\in \\{\\pm 1\\}\\), where \\(\\operatorname*{Pr}(\\sigma_i=1)=\\operatorname*{Pr}(\\sigma_i= -1)=0.5\\). The symmetrized empirical process is\n\\[\nf(\\sigma,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{i=1}^{n}\\sigma_{i}f(Z_{i})\\qquad f\\in\\mathcal{F},\n\\]\nwhere the randomness is with respect to both \\(S_{n} = \\{Z_{i}\\}\\) and \\(\\sigma = \\{\\sigma_{i}\\}\\).\nNote that in order to bound the symmetrized empirical process, we only need a covering number result on \\( S_{n} \\). Therefore the symmetrization analysis allows us to work with empirical covering numbers. Next we will show that a bound on the symmetrized empirical process can be used to obtain uniform convergence of the empirical process \\( \\{f(S_{n}): f \\in \\mathcal{F}\\} \\) to the corresponding result on the validation data. Since one can relate the validation loss of a learning algorithm to its generalization error on the test data, we can use this result to obtain generalization bounds.\nThe following lemma shows that if we can obtain an upper bound on the symmetrized empirical process \\(\\{f(\\sigma, S_n) : f \\in \\mathcal{F}\\}\\), and the upper bound satisfies a superadditive property, then we can obtain the uniform convergence of the empirical process \\(\\{f(S_n) : f \\in \\mathcal{F}\\}\\) on the training data \\(S_n\\) to the corresponding result on the validation data \\(S_n'\\).", "theorems": [{"local_id": "T1", "statement": "Let \\(\\psi_{\\mathrm{trn}}: \\mathcal{F} \\times \\mathcal{Z}^n \\to \\mathbb{R}\\), \\(\\psi_{\\mathrm{val}}: \\mathcal{F} \\times \\mathcal{Z}^n \\to \\mathbb{R}\\), \\(\\psi_{\\mathrm{tst}}: \\mathcal{F} \\times \\mathcal{D} \\to \\mathbb{R}\\) be appropriately training (where \\(\\mathcal{D}\\) denotes probability distributions on \\(\\mathcal{Z}\\)), validation, and test statistics. Assume that for any \\(\\delta_1 \\in (0,1)\\), the following uniform convergence result holds. With probability at least \\(1 - \\delta_1\\) over randomly drawn training and validation sets \\((S_n, S_n') \\sim \\mathcal{D}^{2n}\\):\n\\[\n\\forall f\\in\\mathcal{F}:\\psi_{\\mathrm{val}}(f,\\mathcal{S}_{n}^{\\prime})\\leq\\psi_{\\mathrm{trn}}(f,\\mathcal{S}_{n})+\\epsilon_{n}^{1}(\\delta_{1}).\n\\]\nMoreover, assume \\(\\forall f\\in \\mathcal{F}\\), we have with probability \\(1 - \\delta_{2}\\) over randomly drawn \\(\\mathcal{S}_n^\\prime \\sim \\mathcal{D}\\):\n\\[\n\\psi_{\\mathrm{tst}}(f,\\mathcal{D})\\leq\\psi_{\\mathrm{val}}(f,\\mathcal{S}_{n}^{\\prime})+\\epsilon_{n}^{2}(\\delta_{2}).\n\\]\nThen the following uniform convergence statement holds. With probability at least \\(1 - \\delta_1 - \\delta_2\\),\n\\[\n\\forall f\\in\\mathcal{F}:\\psi_{\\mathrm{tst}}(f,\\mathcal{D})\\leq\\psi_{\\mathrm{trn}}(f,\\mathcal{S}_{n})+\\epsilon_{n}^{1}(\\delta_{1})+\\epsilon_{n}^{2}(\\delta_{2}).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\( Q(f, \\mathcal{S}_n) = \\psi_{\\mathrm{tst}}(f, \\mathcal{D}) - \\psi_{\\mathrm{trn}}(f, \\mathcal{S}_n) - (\\epsilon_n^1(\\delta_1) + \\epsilon_n^2(\\delta_2)) \\), and let \\( E \\) be the event that \\( \\sup_{f \\in \\mathcal{F}} Q(f, \\mathcal{S}_n) \\leq 0 \\)."}, {"n": 2, "claim": "We pick \\( \\hat{f}(\\mathcal{S}_n) \\in \\mathcal{F} \\) so that if \\( E \\) holds, then we choose an arbitrary \\( Q(\\hat{f}(\\mathcal{S}_n), \\mathcal{S}_n) \\leq 0 \\), and if \\( E \\) does not hold, we choose \\( \\hat{f}(\\mathcal{S}_n) \\) so that \\( Q(\\hat{f}(\\mathcal{S}_n), \\mathcal{S}_n) > 0 \\)."}, {"n": 3, "claim": "We consider sample \\( (\\mathcal{S}_n, \\mathcal{S}_n') \\sim \\mathcal{D}^{2n} \\)."}, {"n": 4, "claim": "For simplicity, in the following, we let \\( \\hat{f} = \\hat{f}(\\mathcal{S}_n) \\)."}, {"n": 5, "claim": "The uniform convergence condition of the theorem implies that with probability at least \\( 1 - \\delta_1 \\), the following event holds:"}, {"n": 6, "claim": "\\[\n{E_{1}:}{\\psi_{\\mathrm{val}}(\\hat{f},\\mathcal{S}_{n}^{\\prime})\\leq\\psi_{\\mathrm{trn}}(\\hat{f},\\mathcal{S}_{n})+\\epsilon_{n}^{1}(\\delta_{1}).}\n\\]\nNote that the validation data \\( S_{n}^{\\prime} \\) is independent of the training data \\( S_{n} \\)."}, {"n": 7, "claim": "Therefore \\( S_{n}^{\\prime} \\) is also independent of \\( \\hat{f} \\)."}, {"n": 8, "claim": "Therefore the condition of the theorem implies that with probability at least \\( 1 - \\delta_2 \\), the following event holds:"}, {"n": 9, "claim": "\\[\nE_{2}:\\psi_{\\mathrm{tst}}(\\hat{f},\\mathcal{D})\\leq\\psi_{\\mathrm{val}}(\\hat{f},\\mathcal{S}_{n}^{\\prime})+\\epsilon_{n}^{2}(\\delta_{2}).\n\\]\nIf both events \\(E_{1}\\) and \\(E_{2}\\) hold, then\n\\[\n\\begin{array}{l}\\psi_{\\mathrm{tst}}(\\hat{f},\\mathcal{D})\\leq\\psi_{\\mathrm{val}}(\\hat{f},\\mathcal{S}_{n}^{\\prime})+\\epsilon_{n}^{2}(\\delta_{2})\\\\ \\leq\\psi_{\\mathrm{trn}}(\\hat{f},\\mathcal{S}_{n})+\\epsilon_{n}^{1}(\\delta_{1})+\\epsilon_{n}^{2}(\\delta_{2}).\\\\ \\end{array}\n\\]\nThe definition of \\(\\hat{f}\\) implies that \\(E\\) holds."}, {"n": 10, "claim": "Therefore \\(\\operatorname*{Pr}(E)\\geq \\operatorname*{Pr}(E_1\\& E_2)\\geq 1 - \\delta_1 - \\delta_2\\)."}, {"n": 11, "claim": "This implies the desired bound."}]}}, {"local_id": "T2", "statement": "Let \\(\\sigma_{i} = \\{\\pm 1\\}\\) be independent Bernoulli random variables ( each takes value \\(\\pm 1\\) with equal probability). Let \\(a_{i}\\) be fixed numbers \\((i=1,\\dots,n)\\) Then for all \\(\\epsilon >0\\) ..\n\\[\n\\Pr\\left(n^{-1}\\sum_{i=1}^{n}\\sigma_{i}a_{i}\\geq\\epsilon\\right)\\leq\\exp\\left[-\\frac{n\\epsilon^{2}}{2 n^{-1}\\sum_{i=1}^{n}a_{i}^{2}}\\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "Consider \\( X_{i} = \\sigma_{i}a_{i} \\) in Corollary 2.26. We can take \\( \\mu = 0 \\) and \\( b_{i} = a_{i}^{2} \\) to obtain the desired bound."}]}}, {"local_id": "T3", "statement": "(Symmetrization) Consider a real valued function family \\(F = \\{f : Z \\rightarrow \\mathbb{R}\\}\\). Assume there exists a function \\(\\psi : F \\times Z^{n} \\rightarrow \\mathbb{R}\\) and \\(\\epsilon_{n} : (0,1) \\rightarrow \\mathbb{R}\\) so that with probability at least \\(1-\\delta\\):\n\\[\n\\forall f \\in F, \\qquad f(\\sigma, S_{n}) \\leq \\psi(f, S_{n}) + \\epsilon_{n}(\\delta),\n\\]\nwhere the randomness is over both \\(S_{n} \\sim D^{n}\\) and \\(\\sigma\\). If there exists \\(\\tilde{\\psi}(f, S_{n} \\cup S_{n}^{\\prime})\\) so that the following superadditive inequality holds for all \\((S_{n}, S_{n}^{\\prime})\\)\n\\[\n\\psi(f, S_{n}) + \\psi(f, S_{n}^{\\prime}) \\leq \\tilde{\\psi}(f, S_{n} \\cup S_{n}^{\\prime}),\n\\]\nthen with probability at least \\(1-\\delta\\) over independent random data \\((S_{n}, S_{n}^{\\prime}) \\sim D^{2n}\\):\n\\[\n\\forall f \\in F, \\qquad f(S_{n}^{\\prime}) \\leq f(S_{n}) + \\tilde{\\psi}(f, S_{n} \\cup S_{n}^{\\prime}) + 2\\epsilon_{n}(\\delta/2).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Consider independent random samples \\((S_{n}, S_{n}^{\\prime}) \\sim D^{2n}\\)."}, {"n": 2, "claim": "The distribution of \\(f(S_{n}) - f(S_{n}^{\\prime})\\) is the same as that of \\(f(\\sigma, S_{n}) - f(\\sigma, S_{n}^{\\prime})\\), and the latter contains additional randomness from Bernoulli random variables \\(\\sigma\\), drawn independently of \\((S_{n}, S_{n}^{\\prime})\\), as in Definition 4.7."}, {"n": 3, "claim": "It follows that\n\\[\n\\begin{array}{l}\n\\Pr\\Bigl(\\exists f \\in F,\\ f(S_{n}^{\\prime}) > f(S_{n}) + \\tilde{\\psi}(f, S_{n} \\cup S_{n}^{\\prime}) + 2\\epsilon_{n}(\\delta/2)\\Bigr) \\\\\n= \\Pr\\Bigl(\\exists f \\in F,\\ f(\\sigma, S_{n}^{\\prime}) > f(\\sigma, S_{n}) + \\tilde{\\psi}(f, S_{n} \\cup S_{n}^{\\prime}) + 2\\epsilon_{n}(\\delta/2)\\Bigr) \\\\\n\\stackrel{(a)}{\\leq} \\Pr\\Bigl(\\underbrace{\\exists f \\in F,\\ f(\\sigma, S_{n}^{\\prime}) > f(\\sigma, S_{n}) + (\\psi(f, S_{n}) + \\psi(f, S_{n}^{\\prime})) + 2\\epsilon_{n}(\\delta/2)}_{E_{0}}\\Bigr)\n\\end{array}\n\\]\n\\[\n\\begin{array}{l}\n\\stackrel{(b)}{\\leq} \\Pr\\Bigl(\\underbrace{\\exists f \\in F,\\ f(\\sigma, S_{n}^{\\prime}) > \\psi(f, S_{n}^{\\prime}) + \\epsilon_{n}(\\delta/2)}_{E_{1}}\\Bigr) \\\\\n\\quad + \\Pr\\Bigl(\\underbrace{\\exists f \\in F,\\ -f(\\sigma, S_{n}) > \\psi(f, S_{n}) + \\epsilon_{n}(\\delta/2)}_{E_{2}}\\Bigr)\n\\end{array}\n\\]\n\\[\n= 2\\Pr\\Bigl(\\exists f \\in F,\\ f(\\sigma, S_{n}) > \\psi(f, S_{n}) + \\epsilon_{n}(\\delta/2)\\Bigr) \\leq 2(\\delta/2).\n\\]\nIn the above derivation, the first equation used the fact that \\(f(S_{n}) - f(S_{n}^{\\prime})\\) and \\(f(\\sigma, S_{n}) - f(\\sigma, S_{n}^{\\prime})\\) have the same distributions. (a) used the assumption \\(\\psi(S_{n}) + \\psi(S_{n}^{\\prime}) \\leq \\widetilde{\\psi}(S_{n} \\cup S_{n}^{\\prime})\\). (b) used the union bound, and the fact that if event \\(E_{0}\\) holds, then either event \\(E_{1}\\) holds or event \\(E_{2}\\) holds."}, {"n": 4, "claim": "The next equation used the symmetry of \\(-f(\\sigma, S_{n})\\) and \\(f(\\sigma, S_{n})\\), and the last inequality used the assumption of the lemma."}, {"n": 5, "claim": "The result implies the desired bound."}]}}, {"local_id": "T4", "statement": "If \\(\\operatorname{vc}(\\mathcal{G}) = d\\), then we have for all \\(n > 0\\) and empirical samples \\(\\mathcal{S}_n = \\{Z_1, \\ldots, Z_n\\} \\in \\mathcal{Z}^n\\):\n\\[\n|\\mathcal{G}(\\mathcal{S}_{n})|\\leq\\sum_{\\ell=0}^{d}\\binom{n}{\\ell}\\leq\\max(2,e n/d)^{d}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "First, we prove the statement under the assumption that \\(|\\mathcal{G}(\\mathcal{S}_n)|\\) is upper bounded by the number of subsets of \\(\\mathcal{S}_n\\) (including the empty set) that are shattered by \\(\\mathcal{G}\\)."}, {"n": 2, "claim": "Under this assumption, since any subset shattered by \\(\\mathcal{G}\\) cannot be larger than \\(d\\) by the definition of VC-dimension, and the number of subsets of size \\(\\ell\\) is \\(\\binom{n}{\\ell}\\), we know that the number of subsets shattered by \\(\\mathcal{G}\\) cannot be more than \\(\\sum_{\\ell=0}^{d} \\binom{n}{\\ell}\\)."}, {"n": 3, "claim": "When \\(n \\geq d\\), we have (see Exercise 4.1)\n\\[\n\\sum_{\\ell=0}^{d}\\binom{n}{\\ell}\\leq(e n/d)^{d}.\\tag{4.2}\n\\]\nWhen \\(n \\leq d\\), we have \\(\\sum_{\\ell=0}^{d} \\binom{n}{\\ell} \\leq 2^d\\)."}, {"n": 4, "claim": "This implies the desired result. In the following, we only need to prove the statement that \\(|\\mathcal{G}(\\mathcal{S}_n)|\\) is upper bounded by the number of subsets of \\(\\mathcal{S}_n\\) that are shattered by \\(\\mathcal{G}\\)."}, {"n": 5, "claim": "This can be proved by induction on \\(n\\). When \\(n = 1\\), one can check that the claim holds trivially."}, {"n": 6, "claim": "Now assume that the claim holds for all empirical samples of size no more than \\( n - 1 \\)."}, {"n": 7, "claim": "Consider \\( n \\) samples \\( \\{Z_1, \\ldots, Z_n\\} \\)."}, {"n": 8, "claim": "We define\n\\[\n\\phi(w,\\mathcal{S}_{k})=[\\phi(w,Z_{1}),\\dots,\\phi(w,Z_{k})],\n\\]\n\\[\n\\mathcal{G}_{n-1}(\\mathcal{S}_{n})=\\left\\{\\left[\\phi(w,\\mathcal{S}_{n-1}),1\\right]:\\left[\\phi(w,\\mathcal{S}_{n-1}),0\\right],\\left[\\phi(w,\\mathcal{S}_{n-1}),1\\right]\\in\\mathcal{G}(\\mathcal{S}_{n})\\right\\}.\n\\]\nUsing the induction hypothesis, we know that \\(|\\mathcal{G}_{n - 1}(\\mathcal{S}_n)|\\) is bounded by the number of shattered subset \\(\\mathcal{S}\\subset \\mathcal{S}_{n - 1}\\); for each shattered \\(\\mathcal{S}\\subset \\mathcal{S}_{n - 1}\\), \\(\\mathcal{S}\\cup \\{Z_n\\}\\) is shattered by \\(\\mathcal{G}\\) because both \\([\\phi (w,\\mathcal{S}_{n - 1}),1]\\) and \\([\\phi (w,S_{n - 1}),0]\\) belong to \\(\\mathcal{G}(\\mathcal{S}_n)\\)."}, {"n": 9, "claim": "Therefore \\(|\\mathcal{G}_{n - 1}(\\mathcal{S}_n)|\\) is no more than the number of shattered subsets of \\(\\mathcal{S}_n\\) that contains \\(Z_{n}\\)."}, {"n": 10, "claim": "Moreover, since for \\(\\phi(w,\\cdot) \\in \\mathcal{G}(\\mathcal{S}_n) - \\mathcal{G}_{n-1}(\\mathcal{S}_n)\\), \\(\\phi(w,Z_n)\\) is uniquely determined by its values at \\(\\mathcal{S}_{n-1}\\) (if not, then both \\([\\phi(w,\\mathcal{S}_{n-1}),0]\\) and \\([\\phi(w,\\mathcal{S}_{n-1}),1]\\) can be achieved in \\(\\mathcal{G}(\\mathcal{S}_n) - \\mathcal{G}_{n-1}(\\mathcal{S}_n)\\), which is impossible because by definition, we should have put \\([\\phi(w,\\mathcal{S}_{n-1}),1]\\) in \\(\\mathcal{G}_{n-1}(\\mathcal{S}_n)\\)), it follows that \\(|\\mathcal{G}(\\mathcal{S}_n) - \\mathcal{G}_{n-1}(\\mathcal{S}_n)|\\) is no more than \\(|\\mathcal{G}(\\mathcal{S}_{n-1})|\\)."}, {"n": 11, "claim": "By induction hypothesis, \\(|\\mathcal{G}(\\mathcal{S}_{n-1})|\\) is no more than the number of shattered subsets of \\(\\mathcal{S}_n\\) that does not contain \\(Z_n\\)."}, {"n": 12, "claim": "By combining the above two facts, \\(|\\mathcal{G}(\\mathcal{S}_n)|\\) is no more than the number of shattered subsets of \\(\\mathcal{S}_n\\)."}]}}, {"local_id": "T5", "statement": "Assume that \\(\\phi(w, z) \\in [0,1]\\) for all \\(w\\) and \\(z\\). Then given \\(\\delta \\in (0,1)\\), with probability at least \\(1 - \\delta\\), the following inequality holds:\n\\[\n\\forall w\\in\\Omega:\\phi(w,\\mathcal{D})\\leq\\phi(w,\\mathcal{S}_{n})+\\epsilon_{n}(\\delta),\n\\]\nwhere\n\\[\n\\epsilon_{n}(\\delta)=\\inf_{\\epsilon>0}\\left[2\\epsilon+3\\sqrt{\\frac{\\ln(3 N_{1}(\\epsilon,\\mathcal{G},2 n)/\\delta)}{2 n}}\\right].\n\\]\nMoreover, for any \\(\\gamma \\in (0,1)\\), with probability at least \\(1 - \\delta\\), the following inequality holds:\n\\[\n\\forall w\\in\\Omega:(1-\\gamma)^{2}\\phi(w,\\mathcal{D})\\leq\\phi(w,\\mathcal{S}_{n})+\\epsilon_{n}(\\delta),\n\\]\nwhere\n\\[\n\\epsilon_{n}(\\delta)=\\inf_{\\epsilon>0}\\left[2\\epsilon+\\frac{(5-4\\gamma)\\ln(3 N_{1}(\\epsilon,\\mathcal{G},n)/\\delta)}{2\\gamma n}\\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\mathcal{F} = \\{f(z) = \\phi(w,z) - 0.5 : w \\in \\Omega\\}\\)."}, {"n": 2, "claim": "Given \\(S_{n}\\), we consider an \\(\\epsilon-L_{1}(S_{n})\\) cover \\(\\mathcal{F}_{\\epsilon}(S_{n})\\) of \\(\\mathcal{F}\\), of size no more \\(N = N_{1}(\\epsilon, \\mathcal{G}, n)\\)."}, {"n": 3, "claim": "We may assume that \\(f(Z_{i}) \\in [-0.5, 0.5]\\) for \\(f \\in \\mathcal{F}_{\\epsilon}(S_{n})\\)."}, {"n": 4, "claim": "From Corollary 2.27 (with \\(a_{i} = 0.5\\)) and the union bound, we obtain the following uniform convergence result over \\(\\mathcal{F}_{\\epsilon}(S_{n})\\)."}, {"n": 5, "claim": "With probability \\(1-\\delta\\): \\[\n\\forall f\\in\\mathcal{F}_{\\epsilon}(\\mathcal{S}_{n}):f(\\sigma,\\mathcal{S}_{n})\\leq\\sqrt{\\frac{\\ln(N/\\delta)}{2 n}}.\n\\]\nSince for all \\( f \\in \\mathcal{F} \\), we can find \\( f' \\in \\mathcal{F}_{\\epsilon}(\\mathcal{S}_n) \\) so that \\( n^{-1} \\sum_{Z \\in \\mathcal{S}_n} |f(Z) - f'(Z)| \\leq \\epsilon \\) for all \\( Z \\in \\mathcal{S}_n \\)."}, {"n": 6, "claim": "It follows that\n\\[\nf(\\sigma,\\mathcal{S}_{n})\\leq f^{\\prime}(\\sigma,\\mathcal{S}_{n})+\\epsilon\\leq\\epsilon+\\sqrt{\\frac{\\ln(N/\\delta)}{2 n}}.\n\\]\nUsing Lemma 4.8 with \\(\\psi = 0\\), this uniform convergence result for the symmetrized\nempirical process implies the following uniform convergence result."}, {"n": 7, "claim": "With probability at least \\(1 - \\delta_{1}\\) over \\((S_n, S'_n) \\sim \\mathcal{D}^{2n}\\):"}, {"n": 8, "claim": "\\[\n\\forall w\\in\\Omega:\\underbrace{\\phi(w,\\mathcal{S}_{n}^{\\prime})}_{\\psi_{\\mathrm{val}}}\\leq\\underbrace{\\phi(w,\\mathcal{S}_{n})}_{\\psi_{\\mathrm{trn}}}+\\underbrace{2\\epsilon+\\sqrt{\\frac{2\\ln(2 N/\\delta_{1})}{n}}}_{\\epsilon_{n}^{1}(\\delta_{1})}.\n\\]\nThe standard additive Chernoff bound implies that for all \\( w \\in \\Omega \\), with probability at least \\( 1 - \\delta_2 \\):"}, {"n": 9, "claim": "\\[\n\\underbrace{\\phi(w,\\mathcal{D})}_{\\psi_{\\mathrm{tst}}}\\leq\\underbrace{\\phi(w,\\mathcal{S}_{n}^{\\prime})}_{\\psi_{\\mathrm{val}}}+\\underbrace{\\sqrt{\\frac{\\ln(1/\\delta_{2})}{2 n}}}_{\\epsilon_{n}^{2}(\\delta_{2})}.\n\\]\nTherefore in Lemma 4.11, we can take symbols as defined above, together with \\(\\delta_1 = 2\\delta /3\\) and \\(\\delta_{2} = \\delta /3\\) to obtain the desired bound."}, {"n": 10, "claim": "Similarly, we consider \\(\\mathcal{F} = \\{f(z) = \\phi (w,z):w\\in \\Omega \\}\\)."}, {"n": 11, "claim": "Given \\(S_{n}\\), we consider an \\(\\epsilon -L_{1}(S_{n})\\) cover \\(\\mathcal{F}_{\\epsilon}(\\mathcal{S}_n)\\) of \\(\\mathcal{F}\\), of size no more \\(N=N_{1}(\\epsilon,\\mathcal{G},n)\\)."}, {"n": 12, "claim": "We assume that \\(f(Z_{i})\\in [0,1]\\) for all \\(f\\in \\mathcal{F}_{\\epsilon}(\\mathcal{S}_n)\\)."}, {"n": 13, "claim": "From Corollary 2.27 and the union bound, we obtain the following uniform convergence result over \\(\\mathcal{F}_{\\epsilon}(\\mathcal{S}_n)\\)."}, {"n": 14, "claim": "With probability at least \\(1 - \\delta\\): \\[\n\\begin{array}{l}\\forall f\\in\\mathcal{F}_{\\epsilon}(\\mathcal{S}_{n}):f(\\sigma,\\mathcal{S}_{n})\\leq\\sqrt{\\frac{2\\sum_{Z\\in\\mathcal{S}_{n}}f(Z)^{2}\\ln(N/\\delta)}{n^{2}}}\\\\ \\leq\\gamma^{\\prime}\\frac{1}{n}\\sum_{Z\\in\\mathcal{S}_{n}}f(Z)+\\frac{\\ln(N/\\delta)}{2\\gamma^{\\prime}n}.\\\\ \\end{array}\n\\]\nThe first inequality used Corollary 2.27."}, {"n": 15, "claim": "The second inequality used \\(\\sqrt{2ab} \\leq \\gamma' a + b / (2\\gamma')\\) and \\(f(Z)^2 \\leq f(Z)\\)."}, {"n": 16, "claim": "Since for all \\(f \\in \\mathcal{F}\\), we can find \\(f' \\in \\mathcal{F}_{\\epsilon}(\\mathcal{S}_n)\\) so that \\(\\frac{1}{n} \\sum_{Z \\in \\mathcal{S}_n} |f(Z) - f'(Z)| \\leq \\epsilon\\)."}, {"n": 17, "claim": "It follows that\n\\[\n\\begin{array}{l}f(\\sigma,\\mathcal{S}_{n})\\leq f^{\\prime}(\\sigma,\\mathcal{S}_{n})+\\epsilon\\\\ \\leq\\gamma^{\\prime}\\frac{1}{n}\\sum_{Z\\in\\mathcal{S}_{n}}f^{\\prime}(Z)+\\frac{\\ln(N/\\delta)}{2\\gamma^{\\prime}n}+\\epsilon\\\\ \\leq\\gamma^{\\prime}\\frac{1}{n}\\sum_{Z\\in\\mathcal{S}_{n}}f(Z)+\\frac{\\ln(N/\\delta)}{2\\gamma^{\\prime}n}+(1+\\gamma^{\\prime})\\epsilon .\\\\ \\end{array}\n\\]\nNow, with \\(\\psi(f, \\mathcal{S}_n) = \\widetilde{\\psi}(f, \\mathcal{S}_n) = \\gamma' \\frac{1}{n} \\sum_{Z \\in \\mathcal{S}_n} f(Z)\\), we obtain from Lemma 4.8 the following uniform convergence result."}, {"n": 18, "claim": "With probability at least \\(1 - \\delta_1\\) over \\((\\mathcal{S}_n, \\mathcal{S}_n') \\sim \\mathcal{D}^{2n}\\):"}, {"n": 19, "claim": "\\[\n\\forall w\\in\\Omega:(1-\\gamma^{\\prime})\\phi(w,\\mathcal{S}_{n}^{\\prime})\\leq(1+\\gamma^{\\prime})\\phi(w,\\mathcal{S}_{n})+2(1+\\gamma^{\\prime})\\epsilon+\\frac{\\ln(2 N/\\delta_{1})}{\\gamma^{\\prime}n}.\n\\]\nLet \\(\\gamma' = \\gamma / (2 - \\gamma)\\), then it is easy to check algebraically that \\((1 - \\gamma') / (1 + \\gamma') = 1 - \\gamma\\)\nand \\(1 / (\\gamma^{\\prime}(1 + \\gamma^{\\prime})) = (2 - \\gamma)^{2} / (2\\gamma)\\)."}, {"n": 20, "claim": "We thus obtain: \\[\n\\forall w\\in\\Omega:\\underbrace{(1-\\gamma)\\phi(w,\\mathcal{S}_{n}^{\\prime})}_{\\psi_{\\mathrm{val}}}\\leq\\underbrace{\\phi(w,\\mathcal{S}_{n})}_{\\psi_{\\mathrm{trn}}}+\\underbrace{2\\epsilon+\\frac{(2-\\gamma)^{2}\\ln(2 N/\\delta_{1})}{2\\gamma n}}_{\\epsilon_{n}^{1}(\\delta_{1})}.\\tag{4.1}\n\\]\nThe standard multiplicative Chernoff bound in (2.11) implies that with probability \\( 1 - \\delta_2 \\):"}, {"n": 21, "claim": "\\[\n\\underbrace{(1-\\gamma)(1-\\gamma)\\phi(w,\\mathcal{D})}_{\\psi_{\\mathrm{tst}}}\\leq\\underbrace{(1-\\gamma)\\phi(w,\\mathcal{S}_{n}^{\\prime})}_{\\psi_{\\mathrm{val}}}+\\underbrace{(1-\\gamma)\\frac{\\ln(1/\\delta_{2})}{2\\gamma n}}_{\\epsilon_{n}^{2}(\\delta_{2})}.\n\\]\nTherefore in Lemma 4.11, we can use symbols as displayed above, together with \\(\\delta_1 = 2\\delta /3\\) and \\(\\delta_{2} = \\delta /3\\) to obtain the desired bound."}]}}], "target_local_id": "T5", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch16_theorem_16_3__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\ndefinition 16.1. In the multi-armed bandit (or MAB) problem, we consider \\(K\\) arms. The environment generates a sequence of reward vectors for time steps \\(t \\geq 1\\) as \\(r_t = [r_t(1), \\ldots, r_t(K)]\\). Each \\(r_t(a)\\) is associated with an arm \\(a \\in \\{1, \\ldots, K\\}\\). In the literature, an arm \\(a\\) is also referred to as an action. At each time step \\(t = 1, 2, \\ldots, T\\),\n- The player pulls one of the arms \\( a_{t} \\in \\{1, \\dots, K\\} \\).\n- The environment returns the reward \\( r_t(a_t) \\), but does not reveal information on any other arm \\( a \\neq a_t \\).\nAt each time \\( t \\), a (randomized) bandit algorithm takes the historic observations observed so far, and maps it to a distribution \\( \\hat{\\pi}_{t-1} \\) over actions \\( a \\in \\{1, \\ldots, K\\} \\). We then draw a random action \\( a_t \\) (arm) from \\( \\hat{\\pi}_{t-1} \\). Here \\( \\hat{\\pi}_{t-1} \\) is referred to as policy.\nIn the adversarial bandit problem, we are given an arbitrary reward sequence \\(\\{[r_1(1),\\ldots,r_t(K)]:t\\geq 1\\}\\) before hand. The adversary in this setting is called oblivious, who decides the rewards of all the rounds before the game starts (but may take advantage of the bandit algorithm's weakness). For this reward se\nquence, the expected cumulative reward of a randomized bandit algorithm is\n\\[\n\\mathbb{E}\\sum_{t=1}^{T}r_{t}(a_{t}),\n\\]\nwhere the randomization is over the internal randomization of the bandit algorithm. The expected regret is defined as\n\\[\n\\operatorname{REG}_{T}=\\max_{a}\\sum_{t=1}^{T}r_{t}(a)-\\mathbb{E}\\sum_{t=1}^{T}r_{t}\\left(a_{t}\\right).\\tag{16.1}\n\\]\nIf the bandit algorithm is deterministic: i.e. pulling a single arm \\(a_{t}\\) only at any time step, then the regret is\n\\[\n\\mathrm{REG}_{T}=\\max_{a}\\sum_{t=1}^{T}r_{t}(a)-\\sum_{t=1}^{T}r_{t}(a_{t}).\n\\]\nIn the adversarial bandit problem, our goal is to minimize the expected regret \\(\\mathrm{REG}_T\\) (and the expectation is with respect to the internal randomization of the algorithm instead of the data).\nAnother form of bandit problem, called stochastic bandit, assumes that the reward \\( r_t(a) \\) is drawn independently from a distribution \\( \\mathcal{D}_a \\), with mean \\( \\mu(a) = \\mathbb{E}_{r_t(a) \\sim \\mathcal{D}_a} [r_t(a)] \\). In this setting, the goal of bandit is to find \\( a \\) that maximizes the expected reward \\( \\mu(a) \\). We are interested in the expected regret defined as:\n\\[\n\\operatorname{REG}_{T}=T\\max_{a}\\mu(a)-\\mathbb{E}\\sum_{t=1}^{T}\\mu\\left(a_{t}\\right).\\tag{16.2}\n\\]\nAgain, this regret is defined for each realization of the stochastic rewards, and the expectation is with respect to the randomness of the algorithm. If algorithm is deterministic and outputs \\( a_{t} \\) at each time \\( t \\), then\n\\[\n\\mathrm{REG}_{T}=T\\max_{a}\\mu(a)-\\sum_{t=1}^{T}\\mu(a_{t}).\n\\]\nWe can further include randomization over data into the regret of stochastic bandit, by considering expected regret as follows:\n\\[\n\\mathbb{E}\\operatorname{REG}_{T}=T\\max_{a}\\mu(a)-\\mathbb{E}\\sum_{t=1}^{T}\\mu(a_{t}),\n\\]\nwhere the expectation is with respect to both the data and the internal randomization of the learning algorithm.\n\nequation 16.3. For each arm \\(a\\), define the number of pulls up to time \\(t\\) and the corresponding empirical mean by\n\\[\n\\hat n_t(a)=\\sum_{s=1}^{t}\\mathbb{1}(a_s=a), \\qquad\n\\hat\\mu_t(a)=\\frac{1}{\\hat n_t(a)}\\sum_{s=1}^{t} r_s(a_s)\\mathbb{1}(a_s=a),\n\\]\nwith the initialization \\(\\hat n_0(a)=0\\) and \\(\\hat\\mu_0(a)=0\\). In the UCB algorithm, after the initialization rounds, the arm is selected according to\n\\[\na_t\\in\\arg\\max_a\\left[\\hat\\mu_{t-1}(a)+\\hat c_{t-1}(a)\\right]. \\tag{16.3}\n\\]\nHere \\(\\hat c_t(a)\\) denotes the confidence width used by the algorithm.\n\nUCB Algorithm. Algorithm 16.1: UCB Algorithm\nInput: \\(K\\) and \\(T\\geq K\\)   \n1 for \\(a=1,\\dots,K\\) do   \n2 Let \\(\\hat{n}_0(a) = 0\\)   \n3 Let \\(\\hat{\\mu}_0(a) = 0\\)   \n4 for \\(t=1,2,\\ldots,T\\) do   \n5 if \\(t\\leq K\\) then   \n6 Let \\(a_{t} = t\\)   \n7 else   \n8 Let \\(a_{t}\\in \\arg \\max_{a}[\\hat{\\mu}_{t - 1}(a) + \\hat{c}_{t - 1}(a)]\\) according to (16.3)   \n9 Pull arm \\(a_{t}\\) and observe reward \\(r_t(a_t)\\)\n\nsection 13.1 sequential observation setup. In Section 13.1, data are observed sequentially. At each time \\(t\\), after the history \\(S_{t-1}\\) has been observed, the next observation \\(Z_t\\) is drawn according to a distribution that may depend on that history. The observation is decomposed as \\(Z_t=(Z_t^{(x)},Z_t^{(y)})\\), where \\(Z_t^{(x)}\\) is the component determined by the past history or the chosen action at time \\(t\\), and \\(Z_t^{(y)}\\) is the random response component. The history after time \\(t\\) is \\(S_t=(Z_1,\\ldots,Z_t)\\), and \\(\\mathcal{S}_T\\) denotes the full sequential experiment/history through time \\(T\\). For a function depending on \\(S_t\\), the notation \\(\\mathbb{E}_{Z_t^{(y)}}[\\cdot]\\) means conditional expectation over the response component \\(Z_t^{(y)}\\), conditional on \\(S_{t-1}\\) and on \\(Z_t^{(x)}\\). Stopping times are adapted to the filtration generated by the histories \\(S_t\\).", "theorems": [{"local_id": "T1", "statement": "Let \\(a_* \\in \\arg \\max_a \\mu(a)\\). Let\n\\[\n\\delta_{1}=\\Pr [\\exists t>K:\\mu(a_{*})>\\hat{\\mu}_{t-1}(a_{*})+\\hat{c}_{t-1}(a_{*})]\n\\]\nbe the probability that the upper confident bound fails on \\( a_* \\). Let\n\\[\n\\delta_{2}=\\Pr\\left[\\exists t>K\\&a\\in\\{1,\\dots,K\\}\\backslash\\left\\{a_{*}\\right\\}:\\mu(a)<\\hat{\\mu}_{t-1}(a)-\\hat{c}_{t-1}^{\\prime}(a)\\right]\n\\]\nbe the probability that the lower confident bound fails on \\( a \\neq a_* \\). Then for Algorithm 16.1, we have with probability at least \\( 1 - \\delta_1 - \\delta_2 \\):\n\\[\n\\operatorname{REG}_{T}\\leq\\sum_{a=1}^{K}[\\mu(a_{*})-\\mu(a)]+\\sum_{t=K+1}^{T}[\\hat{c}_{t-1}(a_{t})+\\hat{c}_{t-1}^{\\prime}(a_{t})]\\mathbb{1}(a_{t}\\neq a_{*}).\n\\]", "proof": {"steps": [{"n": 1, "claim": "We have with probability \\(1 - \\delta_1 - \\delta_2\\), the following hold for all \\(t > K\\):"}, {"n": 2, "claim": "\\[\n\\mu\\left(a_{*}\\right)\\leq\\hat{\\mu}_{t-1}\\left(a_{*}\\right)+\\hat{c}_{t-1}\\left(a_{*}\\right),\\quad \\text{(u p p e r c o n f i d e n c e b o u n d)}\n\\]\n\\[\n\\hat{\\mu}_{t-1}\\left(a_{t}\\right)\\mathbb{1}\\left(a_{t}\\neq a_{*}\\right)\\leq\\left[\\mu\\left(a_{t}\\right)+\\hat{c}_{t-1}^{\\prime}\\left(a_{t}\\right)\\right]\\mathbb{1}\\left(a_{t}\\neq a_{*}\\right).\\quad(\\text{l o w e r c o n f i d e n c e b o u n d})\n\\]\nIt follows that for all \\(t > K\\):"}, {"n": 3, "claim": "\\[\n\\begin{array}{l}\\mu\\left(a_{*}\\right)\\mathbb{1}\\left(a_{t}\\neq a_{*}\\right)\\\\ \\leq\\left[\\hat{\\mu}_{t-1}\\left(a_{*}\\right)+\\hat{c}_{t-1}\\left(a_{*}\\right)\\right]\\mathbb{1}\\left(a_{t}\\neq a_{*}\\right)\\quad(\\text{u p p e r c o n f i d e n c e b o u n d})\\\\ \\leq\\left[\\hat{\\mu}_{t-1}\\left(a_{t}\\right)+\\hat{c}_{t-1}\\left(a_{t}\\right)\\right]\\mathbb{1}\\left(a_{t}\\neq a_{*}\\right)\\quad(\\text{U C B a l g o r i t h m})\\\\ \\leq\\left[\\mu\\left(a_{t}\\right)+\\hat{c}_{t-1}^{\\prime}\\left(a_{t}\\right)+\\hat{c}_{t-1}\\left(a_{t}\\right)\\right]\\mathbb{1}\\left(a_{t}\\neq a_{*}\\right).\\quad \\text{(l o w e r c o n f i d e n c e b o u n d)}\\\\ \\end{array}\n\\]\nFor \\(t \\leq K\\), we have\n\\[\n\\mu(a_{*})=\\mu(a_{t})+[\\mu(a_{*})-\\mu(a_{t}=t)].\n\\]\nWe obtain the bound by summing over \\( t = 1 \\) to \\( t = T \\)."}]}}, {"local_id": "T2", "statement": "Given \\( K \\geq 2 \\) and \\( T \\geq 1 \\). Then there exists a distribution over the assignment of rewards \\( r_t(a) \\in [0,1] \\) such that the expected regret of any algorithm (where the expectation is taken with respect to both the randomization over rewards and the algorithm's internal randomization) is at least\n\\[\n\\min (0.02 T, 0.06 \\sqrt{K T}).\n\\]", "proof": {"steps": [{"n": 1, "claim": "We would like to apply Theorem 13.24 with \\(d = 1\\) and \\(m = K\\)."}, {"n": 2, "claim": "We consider a family of \\(K\\) distributions \\(\\mathcal{P}_{\\mathcal{Z}} = \\{q^{\\tau}, \\tau = 1, \\ldots, K\\}\\), and for each arm \\(a\\), the distribution \\(q^{\\tau}(a)\\) is a Bernoulli distribution \\(r \\in \\{0, 1\\}\\) with mean \\(\\mathbb{E}[r] = 0.5 + \\epsilon \\mathbb{1}(a = \\tau)\\) with \\(\\epsilon \\in (0, 0.1]\\) to be determined later."}, {"n": 3, "claim": "We also define \\(q_0(a)\\) as a Bernoulli distribution \\(r \\in \\{0, 1\\}\\) with mean \\(\\mathbb{E}[r] = 0.5\\)."}, {"n": 4, "claim": "Since by construction, \\(q^{\\tau} \\sim_{1} q^{\\tau'}\\) for all \\(\\tau, \\tau' \\in [K]\\), we can map them all to \\(q_1^\\tau = q_0\\)."}, {"n": 5, "claim": "If we pull an arm \\(\\theta\\), we have\n\\[\nQ(\\theta,q^{\\tau})=\\epsilon\\mathbb{1}(\\theta\\neq\\tau).\n\\]\nIt is clear that \\([Q(\\theta, q^{\\tau}) + Q(\\theta, q^{\\tau'})] \\geq \\epsilon\\) for \\(\\tau \\neq \\tau'\\)."}, {"n": 6, "claim": "Consider \\(n \\leq T\\) samples generated sequentially by an arbitrary bandit algorithm and \\(q^{\\tau}(a)\\), and let the resulting distribution that generates \\(\\mathcal{S}_n\\) as \\(p^{\\tau}(\\mathcal{S}_n)\\), where \\(\\mathcal{S}_n = \\{a_1, r_1, \\ldots, a_n, r_n\\}\\)."}, {"n": 7, "claim": "Let \\(\\epsilon = \\min (0.1, 0.24\\sqrt{K / T})\\). We have for all \\(a_{t}\\):"}, {"n": 8, "claim": "\\[\n\\begin{array}{l}\\frac{1}{K}\\sum_{\\tau=1}^{K}\\mathrm{KL}\\left(q_{0}\\left(a_{t}\\right)| |q^{\\tau}\\left(a_{t}\\right)\\right)=\\frac{1}{K}\\sum_{\\tau=1}^{K}\\mathrm{KL}\\left(q_{0}\\left(a_{t}\\right)| |q^{\\tau}\\left(a_{t}\\right)\\right)\\mathbb{1}\\left(a_{t}=\\tau\\right)\\\\ =\\frac{1}{K}\\mathrm{KL}(0. 5| |0. 5+\\epsilon)\\leq\\frac{0.5\\epsilon^{2}}{K(0.5+\\epsilon)(0.5-\\epsilon)}\\leq\\frac{2.1}{K}\\epsilon^{2}.\\\\ \\end{array}\n\\]\nThe first inequality is left as Exercise 16.4."}, {"n": 9, "claim": "The second inequality used \\(\\epsilon \\leq 0.1\\). Theorem 13.24 (with \\(d = 1\\), \\(m = K\\), and \\(\\beta_{1,t}^2 = \\frac{2.1}{K}\\epsilon^2\\)) implies that at the end of the \\(n\\)-th iteration, \\(\\hat{\\theta}\\) of any learning algorithm satisfies\n\\[\n\\begin{array}{l}\\frac{1}{K}\\sum_{\\tau=1}^{K}\\mathbb{E}_{q^{\\tau}}\\mathbb{E}_{\\hat{\\theta}}Q(\\hat{\\theta},q^{\\tau})\\geq 0. 5\\epsilon\\left(1-\\sqrt{2\\times 2.1(n/K)\\epsilon^{2}}\\right)\\\\ \\geq 0. 25\\epsilon=0. 25\\min\\left(0. 1,0. 24\\sqrt{K/T}\\right).\\\\ \\end{array}\n\\]\nThe second inequality used \\((n / K)\\epsilon^2\\leq 0.24^2\\)."}, {"n": 10, "claim": "Since this holds for all steps \\(n\\leq T\\) we obtain the desired bound."}]}}, {"local_id": "T3", "statement": "Let \\(a_* \\in \\arg \\max_a \\mu(a)\\). Let\n\\[\n\\delta_{1}=\\Pr\\left[\\exists t>K:\\mu\\left(a_{*}\\right)>\\hat{\\mu}_{t-1}\\left(a_{*}\\right)+\\hat{c}_{t-1}\\left(a_{*}\\right)\\right],\n\\]\n\\[\n\\delta_{2}=\\Pr\\left[\\exists t>K,a\\in\\{1,\\dots,K\\}\\backslash\\{a_{*}\\}:\\mu(a)<\\hat{\\mu}_{t-1}(a)-\\hat{c}_{t-1}^{\\prime}(a)\\right].\n\\]\nDefine \\(\\Delta(a) = \\mu(a_{*}) - \\mu(a)\\), and\n\\[\nT(a)=\\max\\left\\{m:\\Delta(a)\\leq\\hat{c}_{t-1}(a)+\\hat{c}_{t-1}^{\\prime}(a),\\hat{n}_{t-1}(a)=m,K<t\\leq T\\right\\}\\cup\\{0\\}.\n\\]\nThen for Algorithm 16.1, we have with probability at least \\(1 - \\delta_{1} - \\delta_{2}\\):\n\\[\n\\operatorname{REG}_{T}\\leq\\inf_{\\Delta_{0}>0}\\left[\\sum_{a=1}^{K}T(a)\\Delta(a)\\mathbb{1}(\\Delta(a)>\\Delta_{0})+(T-K)\\Delta_{0}\\right]+\\sum_{a=1}^{K}\\Delta(a).\n\\]", "proof": {"steps": [{"n": 1, "claim": "The proof of Lemma 16.2 shows that with probability at least \\(1 - \\delta\\), for all \\(t \\geq K + 1\\):"}, {"n": 2, "claim": "\\[\n\\mu(a_{*})\\leq\\mu(a_{t})+[\\hat{c}_{t-1}^{\\prime}(a_{t})+\\hat{c}_{t-1}(a_{t})]\\mathbb{1}(a_{t}\\neq a_{*}),\n\\]\nwhich implies that\n\\[\n\\Delta(a_{t})\\leq [\\hat{c}_{t-1}^{\\prime}(a_{t})+\\hat{c}_{t-1}(a_{t})]\\mathbb{1}(a_{t}\\neq a_{*}).\n\\]\nUsing the assumption of the lemma, we obtain for all \\(a \\neq a_*\\) and \\(1 \\leq t \\leq T\\), \\(\\hat{n}_{t-1}(a) \\mathbb{1}(a_t = a) \\leq T(a)\\)."}, {"n": 3, "claim": "It follows that for all \\( a \\neq a_* \\), let \\( t \\) be the last time such that \\( a_t = a \\), then \\( t \\leq T \\) and \\( \\hat{n}_T(a) = \\hat{n}_{t-1}(a_t) + 1 \\leq T(a) + 1 \\)."}, {"n": 4, "claim": "We thus obtain the following regret bound for any \\( \\Delta_0 \\geq 0 \\):"}, {"n": 5, "claim": "\\[\n\\begin{array}{l}\\sum_{t=1}^{T}\\left[\\mu\\left(a_{*}\\right)-\\mu\\left(a_{t}\\right)\\right]=\\sum_{a=1}^{K}\\hat{n}_{T}(a)\\Delta(a)\\\\ \\leq\\sum_{a=1}^{K}\\Delta(a)+\\sum_{a=1}^{K}(\\hat{n}_{T}(a)-1)\\Delta_{0}+\\sum_{a=1}^{K}(\\hat{n}_{T}(a)-1)\\Delta(a)\\mathbb{1}(\\Delta(a)>\\Delta_{0})\\\\ \\leq\\sum_{a=1}^{K}\\Delta(a)+(T-K)\\Delta_{0}+\\sum_{a=1}^{K}T(a)\\Delta(a)\\mathbb{1}(\\Delta(a)>\\Delta_{0}).\\\\ \\end{array}\n\\]\nThe second inequality used \\(\\sum_{a=1}^{K} (\\hat{n}_T(a) - 1) = T - K\\) and \\(\\hat{n}_T(a) - 1 \\leq T(a)\\)."}, {"n": 6, "claim": "This implies the bound."}]}}, {"local_id": "T4", "statement": "Assume that rewards \\(r_t(a) \\in [0,1]\\). Let \\(a_* \\in \\arg \\max_a \\mu(a)\\). With a choice of\n\\[\n\\hat{c}_{t}(a)=\\sqrt{\\frac{\\ln(2(\\hat{n}_{t}(a)+1)^{2}/\\delta)}{2\\hat{n}_{t}(a)}},\n\\]\nwe have with probability at least \\(1 - \\delta\\):\n\\[\n\\mathrm{REG}_{T}\\leq(K-1)+\\sqrt{8\\ln(2 K T^{2}/\\delta)(T-K)K}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Given any integer \\( m \\geq 1 \\) and \\( a \\in \\{1, \\ldots, K\\} \\)."}, {"n": 2, "claim": "We know that the sequence \\( r(a_{t})\\mathbb{1}(a_{t} = a) \\) satisfies the sub-Gaussian bound in Theorem 13.3 with \\( \\sigma_{i} = 0.5\\mathbb{1}(a_{i} = a) \\)."}, {"n": 3, "claim": "This means that \\( \\sum_{i=1}^{t}\\sigma_{i}^{2} = 0.25\\hat{n}_{t}(a) \\)."}, {"n": 4, "claim": "By letting \\( \\sigma = 0.5\\sqrt{m} \\) for a constant \\( m \\), and considering the event \\( \\hat{n}_{t}(a) = m \\), we obtain with probability at most \\( 0.5\\delta / (m + 1)^{2} \\):"}, {"n": 5, "claim": "\\[\n\\exists t\\geq 1:\\mu(a)>\\hat{\\mu}_{t}(a)+\\hat{c}_{t}(a)\\&\\hat{n}_{t}(a)=m.\n\\]\nSimilarly, with probability at most \\((0.5 / K)\\delta /(m + 1)^2\\)\n\\[\n\\exists t\\geq 1:\\mu(a)<\\hat{\\mu}_{t}(a)-\\hat{c}_{t}^{\\prime}(a)\\&\\hat{n}_{t}(a)=m,\n\\]\nwhere the lower confidence interval size is defined as\n\\[\n\\hat{c}_{t}^{\\prime}(a)=\\sqrt{\\frac{\\ln(2 K(\\hat{n}_{t}(a)+1)^{2}/\\delta)}{2\\hat{n}_{t}(a)}}.\n\\]\nIt follows that the failure probability of upper confidence bound is given by\n\\[\n\\begin{array}{l}\\Pr\\left[\\exists t>K:\\mu\\left(a_{*}\\right)>\\hat{\\mu}_{t}\\left(a_{*}\\right)+\\hat{c}_{t}\\left(a_{*}\\right)\\right]\\\\ \\leq\\sum_{m=1}^{\\infty}\\Pr\\left[\\exists t>K:\\hat{n}_{t}\\left(a_{*}\\right)=m\\&\\mu\\left(a_{*}\\right)>\\hat{\\mu}_{t}\\left(a_{*}\\right)+\\hat{c}_{t}\\left(a_{*}\\right)\\right]\\\\ \\leq\\sum_{m=1}^{\\infty}0. 5\\delta/(m+1)^{2}\\leq 0. 5\\delta .\\\\ \\end{array}\n\\]\nMoreover, the failure probability of lower confidence bound is given by\n\\[\n\\begin{array}{l}\\Pr\\left[\\exists t>K\\&a\\in\\{1,\\dots,K\\}:\\mu(a)<\\hat{\\mu}_{t}(a)-\\hat{c}_{t}^{\\prime}(a)\\right]\\\\ \\leq\\sum_{a=1}^{K}\\Pr\\left[\\exists t>K:\\mu(a)<\\hat{\\mu}_{t}(a)-\\hat{c}_{t}^{\\prime}(a)\\right]\\\\ \\leq\\sum_{a=1}^{K}\\sum_{m=1}^{\\infty}\\Pr\\left[\\exists t>K:\\hat{n}_{t}(a)=m\\&\\mu(a)<\\hat{\\mu}_{t}(a)-\\hat{c}_{t}^{\\prime}(a)\\right]\\\\ \\leq\\sum_{a=1}^{K}\\sum_{m=1}^{\\infty}\\frac{\\delta}{2 K(m+1)^{2}}\\leq\\delta/2.\\\\ \\end{array}\n\\]\nThe following bound follows from Lemma 16.2."}, {"n": 6, "claim": "With probability at least \\(1 - \\delta\\): \\[\n\\begin{array}{l}\\sum_{t=1}^{T}[\\mu(a_{*})-\\mu(a_{t})]-\\sum_{a=1}^{K}[\\mu(a_{*})-\\mu(a)]\\leq\\sum_{t=K+1}^{T}[\\hat{c}_{t-1}(a_{t})+\\hat{c}_{t-1}^{\\prime}(a_{t})]\\\\ \\leq 2\\sum_{t=K+1}^{T}\\sqrt{\\frac{\\ln(2 K(\\hat{n}_{t-1}(a_{t})+1)^{2}/\\delta)}{2\\hat{n}_{t-1}(a_{t})}}\\\\ =2\\sum_{a=1}^{K}\\sum_{t=K+1}^{T}\\sqrt{\\frac{\\ln\\left(2 K\\left(\\hat{n}_{t-1}\\left(a_{t}\\right)+1\\right)^{2}/\\delta\\right)}{2\\hat{n}_{t-1}\\left(a_{t}\\right)}}\\mathbb{1}\\left(a_{t}=a\\right)\\\\ \\stackrel{(a)}{=}2\\sum_{a=1}^{K}\\sum_{m=1}^{\\hat{n}_{T}(a)-1}\\sqrt{\\frac{\\ln(2 K(m+1)^{2}/\\delta)}{2 m}}\\\\ \\leq 2\\sum_{a=1}^{K}\\int_{0}^{\\hat{n}_{T}(a)-1}\\sqrt{\\frac{\\ln\\left(2 K\\hat{n}_{T}(a)^{2}/\\delta\\right)}{2 t}}d t\\\\ =4\\sum_{a=1}^{K}\\sqrt{\\frac{\\ln(2 K\\hat{n}_{T}(a)^{2}/\\delta)(\\hat{n}_{T}(a)-1)}{2}}\\\\ \\leq 4\\sqrt{\\frac{\\ln(2 K T^{2}/\\delta)(T-K)K}{2}}.\\\\ \\end{array}\n\\]\nEquation (a) used the fact that for each arm \\( a \\), with \\( a_{t} = a \\), \\( \\hat{n}_{t-1}(a_{t}) \\) takes values from \\( m = 1 \\) to \\( m = \\hat{n}_T(a) - 1 \\), once for each value."}, {"n": 7, "claim": "In the derivation of the last inequality, we used Jensen's inequality and the concavity of \\( \\sqrt{z} \\), which implies that\n\\[\n\\sum_{a=1}^{K}\\sqrt{\\hat{n}_{T}(a)-1}\\leq\\sqrt{K\\sum_{a=1}^{K}(\\hat{n}_{T}(a)-1)}=\\sqrt{K(T-K)}.\n\\]\nThis implies the desired bound."}]}}, {"local_id": "T5", "statement": "Consider a sequence of random functions \\(\\xi_1(S_1),\\ldots,\\xi_t(S_t),\\ldots\\) Assume each \\(\\xi_{i}\\) is sub-Gaussian with respect to \\(Z_{i}^{(y)}\\):\n\\[\n\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{\\lambda\\xi_{i}}\\leq\\lambda\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i}+\\frac{\\lambda^{2}\\sigma_{i}^{2}}{2}\n\\]\nfor some \\(\\sigma_{i}\\) that may depend on \\(S_{i - 1}\\) and \\(Z_{i}^{(x)}\\). Then for all \\(\\sigma >0\\), with probability at least \\(1 - \\delta\\)\n\\[\n\\forall n>0:\\sum_{i=1}^{n}\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i}<\\sum_{i=1}^{n}\\xi_{i}+\\left(\\sigma+\\frac{\\sum_{i=1}^{n}\\sigma_{i}^{2}}{\\sigma}\\right)\\sqrt{\\frac{\\ln(1/\\delta)}{2}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "We can let \\(\\lambda = \\sqrt{2\\ln(1 / \\delta)} /\\sigma\\), and then apply the first inequality of Theorem 13.2."}]}}], "target_local_id": "T4", "topic_domain": "Online Learning & RL", "topic_subtopic": "Bandits", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch3_theorem_3_3__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nStrong convexity. A function \\(f\\colon\\Omega\\to\\mathbb{R}\\) is \\emph{\\(\\alpha\\)-strongly convex} if, for all \\(x,y\\in\\Omega\\), the following inequality holds for some \\(\\alpha>0\\): \\[ f(y) \\geqslant f(x) + g(x)^\\top(y-x) + \\frac{\\alpha}{2}\\left\\lVert x-y\\right\\rVert^2 \\]\n\nSmoothness. A continuously differentiable function \\(f\\) is \\(\\beta\\) smooth if the gradient map \\(\\nabla f\\colon\\mathbb{R}^n\\to\\mathbb{R}^n\\) is \\(\\beta\\)-Lipschitz, i.e., \\[ \\|\\nabla f(x) - \\nabla f(y)\\| \\leqslant \\beta\\|x-y\\|\\,. \\]\n\nUniform stability. The uniform stability of an algorithm \\(A\\) is defined as\n\\[\n\\Delta_{\\sup}(A) = \\sup_{S, S' \\in (X\\times Y)^m } \\sup_{i \\in [m]} |\\ell(A(S), z_i') - \\ell(A(S^{(i)}, z_i')|\n\\]", "theorems": [{"local_id": "T1", "statement": "Assume \\(f\\colon\\mathbb{R}^n\\to\\mathbb{R}\\) is \\(\\alpha\\)-strongly convex and \\(\\beta\\)-smooth. Let \\(x^{*}\\) be an optimizer of \\(f\\), and let \\(x_{t}\\) be the updated point at step \\(t\\) using gradient descent with a constant step size \\(\\frac{1}{\\beta}\\), i.e. using the update rule \\(x_{t+1} = x_t - \\frac{1}{\\beta}\\nabla f(x_t)\\). Then, \\[ \\|x_{t+1} - x^*\\|^2 \\leq \\exp\\left(-t \\frac{\\alpha}{\\beta}\\right)\\|x_1 - x^*\\|^2 \\]", "proof": {"steps": [{"n": 1, "claim": "\\begin{align*} \\|x_{t+1} -x^*\\|^2 &= \\|x_t - \\frac{1}{\\beta}\\nabla f(x_t) - x^*\\|^2 \\\\ &= \\|x_t-x^*\\|^2 - \\frac{2}{\\beta} \\nabla f(x_t)^\\top(x_t - x^*) + \\frac{1}{\\beta^2}\\|\\nabla f(x_t)\\|^2 \\\\ &\\leq \\big(1-\\frac{\\alpha}{\\beta}\\big)\\|x_t-x^*\\|^2 \\tag{Use of Lemma with $y=x^*$, $x=x_t$}\\\\ &\\leq \\big(1-\\frac{\\alpha}{\\beta}\\big)^t \\|x_1-x^*\\|^2 \\\\ &\\leq \\exp\\left(-t \\frac{\\alpha}{\\beta}\\right) \\|x_1-x^*\\|^2 \\end{align*}"}]}}, {"local_id": "T2", "statement": "Assume \\(f\\) as in Theorem 3.5. Then \\(\\forall x,y \\in \\mathbb{R}^n\\) and an update of the form \\(x^+ = x - \\frac{1}{\\beta}\\nabla f(x)\\), \\[ f(x^+) - f(y) \\leq \\nabla f(x)^\\top (x-y) - \\frac{1}{2\\beta}\\|\\nabla f(x)\\|^2 - \\frac{\\alpha}{2}\\|x-y\\|^2 \\]", "proof": {"steps": [{"n": 1, "claim": "\\begin{align*} f(x^+) - f(x) + f(x) - f(y) &\\leq \\nabla f(x)^\\top (x^+ - x) + \\frac{\\beta}{2}\\|x^+ - x\\|^2 \\tag{Smoothness} \\\\ & \\quad \\ + \\nabla f(x)^\\top (x-y) - \\frac{\\alpha}{2}\\|x-y\\|^2 \\tag{Strong convexity} \\\\ &= \\nabla f(x)^\\top(x^+ - y) + \\frac{1}{2\\beta}\\|\\nabla f(x)\\|^2 - \\frac{\\alpha}{2}\\|x-y\\|^2 \\tag{Definition of $x^+$} \\\\ &= \\nabla f(x)^\\top (x - y) - \\frac{1}{2\\beta}\\|\\nabla f(x)\\|^2 - \\frac{\\alpha}{2}\\|x-y\\|^2 \\tag{Definition of $x^+$} \\end{align*}"}]}}, {"local_id": "T3", "statement": "Assume \\(\\ell(w, z)\\) is \\(\\alpha\\)-strongly convex over the domain \\(\\Omega\\) and \\(L\\)-Lipschitz. Let \\(\\hat w_S = \\arg \\min_{w\\in \\Omega} \\frac{1}{m} \\sum_{i=1}^m \\ell(w, z_i)\\) denote the empirical risk minimizer (ERM). Then, ERM satisfies\n\\[\n\\Delta_{\\sup} (\\text{ERM}) \\leqslant \\frac{4L^2}{\\alpha m}\\,.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Denote by \\(\\hat w_S\\) the empirical risk minimizer on a sample \\(S.\\) Fix arbitrary samples \\(S,S'\\) of size \\(m\\) and an index \\(i\\in[m].\\) We need to show that\n\\[\n|(\\ell(\\hat w_{S^{(i)}},z_i')-\\ell(\\hat w_{S},z_i'))|\\leqslant\\frac{4L^2}{\\alpha m}\\,.\n\\]\nOn one hand, by strong convexity we know that\n\\[\nR_S(\\hat w_{S^{(i)}}) - R_S(\\hat w_{S}) \\geqslant \\frac{\\alpha}{2} \\| \\hat w_{S} - \\hat w_{S^{(i)}} \\|^2 \\,.\n\\]\nOn the other hand,\n\\begin{align*}\n&R_S(\\hat w_{S^{(i)}})-R_S(\\hat w_{S})\\\\\n&=\\frac{1}{m}(\\ell(\\hat w_{S^{(i)}},z_i)-\\ell(\\hat w_{S},z_i))+\\frac{1}{m}\\sum_{i\\neq j}(\\ell(\\hat w_{S^{(i)}},z_j)-\\ell(\\hat w_{S},z_j))\\\\\n&=\\frac{1}{m}(\\ell(\\hat w_{S^{(i)}},z_i)-\\ell(\\hat w_{S},z_i))+\\frac{1}{m}(\\ell(\\hat w_{S},z_i')-\\ell(\\hat w_{S^{(i)}},z_i'))+\\left(R_{S^{(i)}}(\\hat w_{S^{(i)}})-R_{S^{(i)}}(\\hat w_{S})\\right)\\\\\n&\\leqslant\\frac{1}{m}|\\ell(\\hat w_{S^{(i)}},z_i)-\\ell(\\hat w_{S},z_i)| +\\frac{1}{m}|(\\ell(\\hat w_{S},z_i')-\\ell(\\hat w_{S^{(i)}},z_i'))|\\\\\n&\\leqslant\\frac{2L}{m}\\|\\hat w_{S^{(i)}}-\\hat w_{S}\\|\\,.\n\\end{align*}\nHere, we used that\n\\[\nR_{S^{(i)}}(\\hat w_{S^{(i)}}) - R_{S^{(i)}}(\\hat w_{S})) \\leqslant 0\n\\]\nand the fact that \\(\\ell\\) is \\(L\\)-Lipschitz."}, {"n": 2, "claim": "Putting it all together \\(\\| \\hat w_{S^{(i)}} - \\hat w_{S} \\| \\leqslant \\frac{4 L}{\\alpha m}\\). Then by the Lipschitz condition we have that\n\\[\n|(\\ell(\\hat w_{S^{(i)}},z_i')-\\ell(\\hat w_{S},z_i'))|\\leqslant L\\|\\hat w_{S^{(i)}}-\\hat w_{S}\\|\\leqslant\\frac{4L^2}{\\alpha m}\\,.\n\\]\nHence, \\(\\Delta_{\\sup}(\\text{ERM}) \\leqslant \\frac{4 L ^ 2}{\\alpha m}\\)."}]}}, {"local_id": "T4", "statement": "Assume \\(f\\colon\\Omega\\to\\mathbb{R}\\) is \\emph{\\(\\alpha\\)-strongly convex} and \\emph{\\(L\\)-Lipschitz}. Let \\(x^{*}\\) be an optimizer of \\(f\\), and let \\(x_{s}\\) be the updated point at step \\(s\\) using projected gradient descent. Let the max number of iterations be \\(t\\) with an adaptive step size \\(\\eta_{s} = \\frac{2}{\\alpha(s+1)}\\), then \\[ f\\left(\\sum_{s=1}^{t}\\frac{2s}{t(t+1)}x_{s}\\right) - f(x^{*})\\leqslant\\frac{2 L^{2}}{\\alpha(t+1)} \\]", "proof": {"steps": [{"n": 1, "claim": "Recall the two steps update rule of projected gradient descent\n\\begin{align*}\ny_{s+1} &= x_{s} - \\eta_{s}\\nabla f(x_{s})\\\\\nx_{s+1} &= \\Pi_{\\Omega}(y_{s+1})\n\\end{align*}\nFirst, the proof begins by exploring an upper bound of difference between function values \\(f(x_{s})\\) and \\(f(x^{*})\\)."}, {"n": 2, "claim": "\\begin{align*}\nf(x_{s}) - f(x^{*}) &\\leqslant \\nabla f(x_{s})^{\\top}(x_{s}-x^{*}) - \\frac{\\alpha}{2}\\|x_{s}-x^{*}\\|^{2}\\\\\n\t\t\t &=\\frac{1}{\\eta_{s}} (x_{s} - y_{s+1})^{\\top}(x_{s}-x^{*}) - \\frac{\\alpha}{2}\\|x_{s}-x^{*}\\|^{2} \\tag{by update rule}\\\\\n\t\t\t &=\\frac{1}{2\\eta_{s}} (\\|x_{s} - x^{*}\\|^{2} + \\|x_{s} - y_{s+1}\\|^{2} - \\|y_{s+1}-x^{*}\\|^{2}) - \\frac{\\alpha}{2}\\|x_{s}-x^{*}\\|^{2} \\tag{by \"Fundamental Theorem of Optimization\"}\\\\\n\t\t\t &=\\frac{1}{2\\eta_{s}} (\\|x_{s} - x^{*}\\|^{2} - \\|y_{s+1}-x^{*}\\|^{2}) +\\frac{\\eta_{s}}{2}\\|\\nabla f(x_{s})\\|^{2} - \\frac{\\alpha}{2}\\|x_{s}-x^{*}\\|^{2} \\tag{by update rule}\\\\\n\t\t\t &\\leqslant \\frac{1}{2\\eta_{s}} (\\|x_{s} - x^{*}\\|^{2} - \\|x_{s+1}-x^{*}\\|^{2}) +\\frac{\\eta_{s}}{2}\\|\\nabla f(x_{s})\\|^{2} - \\frac{\\alpha}{2}\\|x_{s}-x^{*}\\|^{2} \\tag{by Pythagorean lemma}\\\\\n\t\t\t &\\leqslant (\\frac{1}{2\\eta_{s}} - \\frac{\\alpha}{2})\\|x_{s}-x^{*}\\|^{2} -\\frac{1}{2\\eta_{s}}\\|x_{s+1}-x^{*}\\|^{2} + \\frac{\\eta_{s}L^{2}}{2} \\tag{by Lipschitzness}\n\\end{align*}\nBy multiplying \\(s\\) on both sides and substituting the step size \\(\\eta_{s}\\) by \\(\\frac{2}{\\alpha(s+1)}\\), we get\n\\[\ns(f(x_{s}) - f(x^{*})) \\leqslant \\frac{L^{2}}{\\alpha} + \\frac{\\alpha}{4}(s(s-1)\\|x_{s}-x^{*}\\|^{2} - s(s+1)\\|x_{s+1} - x^{*}\\|^{2})\n\\]\nFinally, we can find the upper bound of the function value obtained using \\(t\\) steps projected gradient descent\n\\begin{align*}\nf\\left(\\sum_{s=1}^{t}\\frac{2s}{t(t+1)}x_{s}\\right)\n&\\leqslant \\sum_{s=1}^{t}\\frac{2s}{t(t+1)}f(x_{s})\n\\tag{by Jensen's inequality}\\\\\n&\\leqslant \\frac{2}{t(t+1)}\\sum_{s=1}^{t}\\left(sf(x^{*}) + \\frac{L^{2}}{\\alpha}\n+  \\frac{\\alpha}{4}(s(s-1)\\|x_{s}-x^{*}\\|^{2} - s(s+1)\\|x_{s+1} - x^{*}\\|^{2}) \\right)\\\\\n&= \\frac{2}{t(t+1)}\\sum_{s=1}^{t}sf(x^{*}) + \\frac{2L^{2}}{\\alpha(t+1)} - \\frac{\\alpha}{2}\\|x_{t+1} - x^{*}\\|^{2} \\tag{by telescoping sum}\\\\\n&\\leqslant f(x^{*}) + \\frac{2L^{2}}{\\alpha(t+1)}\n\\end{align*}\nThis concludes that solving an optimization problem with a strongly convex objective function with projected gradient descent has a convergence rate of the order \\(\\frac{1}{t+1}\\), which is faster compared to the case purely with Lipschitzness."}]}}, {"local_id": "T5", "statement": "If \\(f\\) is in addition \\(\\alpha\\)-strongly convex w.r.t to \\(\\|\\cdot\\|_{[1-\\gamma]}\\), then we get\n\\[\n\\mathbb{E}[f(x_{t+1}) - \\arg\\min_{x \\in \\mathbb{R}^{n}} f(x)] \\leqslant \\left(1 - \\frac{\\alpha}{\\sum_{i=1}^n \\beta_i^\\gamma}\\right)^t (f(x_1) - f(x^*)) \\ .\n\\]", "proof": {"steps": [{"n": 1, "claim": "We need the following lemma:\n\nLemma 12.4. Let \\(f\\) be an \\(\\alpha\\)-strongly convex function w.r.t. a norm \\(\\|\\cdot\\|\\). Then, \\(f(x) - f(x^{*}) \\leqslant \\frac{1}{2\\alpha}\\|\\nabla f(x)\\|_{*}^{2}\\)."}, {"n": 2, "claim": "Lemma 12.4 shows that\n\\[\n\\|\\nabla f(x_s)\\|^{*2}_{[1-\\gamma]} \\geqslant 2 \\alpha \\delta_s \\ .\n\\]\nOn the other hand, Theorem 12.2 showed that\n\\[\n\\delta_{t+1} - \\delta_t \\leqslant -\\frac{\\|\\nabla f(x_t)\\|^{*2}_{[1-\\gamma]}}{2\\sum_{i=1}^n \\beta_i^{\\gamma}}\n\\]\nCombining these two, we get\n\\begin{align*}\n\\delta_{t+1} - \\delta_t &\\leqslant -\\frac{\\alpha \\delta_t}{\\sum_{i=1}^n \\beta_i^{\\gamma}} \\\\\n\\delta_{t+1} &\\leqslant \\delta_t \\left(1 -\\frac{\\alpha}{\\sum_{i=1}^n \\beta_i^{\\gamma}} \\right)~.\n\\end{align*}\nApplying the above inequality recursively and recalling that \\(\\delta_t = \\mathbb{E}[f(x_t) - f(x^*)]\\) gives the result."}]}}], "target_local_id": "T4", "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch8_lemma_8_3__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\ndefinition 8.0. For any sample \\(\\{x_{1},\\ldots,x_{n}\\}\\), the ways that \\({\\mathcal{C}}\\) intersects with the sample are \\[\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in{\\mathcal{C}}\\}.\\] Let \\[\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})=\\mathrm{card}~\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in\\mathcal{C}\\},\\] the number of different subsets picked out by \\(C\\in{\\mathcal{C}}\\). If \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\), this means we can find \\(\\{x_{1},\\ldots,x_{n}\\}\\) such that \\({\\mathcal{C}}\\) can pick out any subset of it; in this case \\({\\mathcal{C}}\\) shatters \\(\\{x_{1},\\ldots,x_{n}\\}\\). Equivalently, a set \\(A\\) is shattered by \\({\\mathcal{C}}\\) when \\({\\mathcal{C}}\\) picks out every subset of \\(A\\).\n\nDenote \\(\\triangle_{n}(\\mathcal{C})=\\operatorname*{sup}_{\\{x_{1},\\ldots,x_{n}\\}}\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})\\leq 2^{n}\\). We will see that for some classes, \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\) for \\(n \\leq V\\) and \\(\\triangle_{n}(\\mathcal{C})<2^{n}\\) for \\(n > V\\) for some constant \\(V\\).\n\ndefinition 8.1. If \\(V < \\infty\\) , then C is called a VC class. V is called VC dimension of C.\n\nFor \\(f:\\mathcal{X}\\to\\mathbb{R}\\) , the subgraph of \\(f\\) is\n\\[C_{f}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}:0\\leq t\\leq f(x)\\ o r\\ f(x)\\leq t\\leq 0\\}.\\]\nDefine class of sets \\(\\mathcal{C}=\\{C_{f}:f\\in\\mathcal{F}\\}\\).\n\ndefinition 12.1. If \\(\\mathcal{C}\\) is a VC class of sets, then F is VC-subgraph class of functions and, by definition, \\(V C(\\mathcal{F})=V C(\\mathcal{C})\\)\nNote that equivalent definition of \\(C_{f}\\) is\n\\[C_{f}^{\\prime}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}: |f(x)|\\geq|t|\\}.\\]\n\ndefinition 12.3. The ε-packing number, \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) , is the marimal cardinality of an E-separated set.\nNote that \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) is decreasing in ε.\n\ndefinition 12.2. Given \\(\\varepsilon > 0\\) and \\(f_{1},\\ldots,f_{N}\\in\\mathcal{F}\\) ， we say that \\(f_{1},\\ldots,f_{N}\\) are E-separated if \\(d(f_{i},f_{j})>\\varepsilon\\) for any \\(i \\neq j\\)", "theorems": [{"local_id": "T1", "statement": "Setup. Let \\(0<\\varepsilon\\le 1\\).\n\nAssume \\(\\mathcal{F}\\) is a VC-subgraph class and \\(V C(\\mathcal{F})=V\\) Suppose \\(-1\\leq f(x)\\leq 1\\) for all \\(f\\in{\\mathcal{F}}\\) and \\(x\\in\\mathcal{X}\\) .Let \\(x_{1},\\ldots,x_{n}\\in{\\mathcal{X}}\\) and define \\(\\begin{array}{ r }{d(f,g)=\\frac{1}{n}\\sum_{i=1}^{n}|f(x_{i})-g(x_{i})|}\\end{array}\\) . Then\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]\n(which \\(\\begin{array}{ r }{i s\\le\\left(\\frac{K}{\\varepsilon}\\right)^{V+\\delta}}\\end{array}\\) for some \\(\\delta . \\jmath\\)", "proof": {"steps": [{"n": 1, "claim": "Let \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) and \\(f_{1},\\ldots,f_{m}\\) be ε-separated, i.e.\n\\[{\\frac{1}{n}}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})| >\\varepsilon .\\]"}, {"n": 2, "claim": "Let \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) be constructed in the following way:"}, {"n": 3, "claim": "\\(z_{i}\\) is chosen uniformly from \\(x_{1},\\ldots,x_{n}\\) and \\(t_{i}\\) is uniform on \\([-1,1]\\)\nConsider \\(f_{r}\\) and \\(f_{\\ell}\\) from the ε-packing."}, {"n": 4, "claim": "Let \\(C_{f_{\\tau}}\\) and \\(C_{f_{\\ell}}\\) be subgraphs of \\(f_{r}\\) and \\(f_{\\ell}\\) Then\n\\(\\mathbb{P}\\left(C_{f_{\\tau}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n= P(At least one point \\((z_{i},t_{i})\\) is picked by \\(C_{f_{r}}\\) or \\(C_{f_{\\ell}}\\) but not picked by the other)\n\\(=1-\\mathbb{P}\\) (All points \\((z_{i},t_{i})\\) are picked either by both or by none)\n\\(\\mathbf{\\Sigma}=1-\\mathbb{P}\\left(\\left(z_{i},t_{i}\\right)\\right.\\) is picked either by both or by none)k\nSince \\(z_{i}\\) is drawn uniformly from \\(x_{1},\\ldots,x_{n}\\)\nP \\(((z_{1},t_{1})\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)\n\\[\\begin{array}{ l }{\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{P}\\left(\\left(x_{i},t_{1}\\right)\\mathrm{~i s~p i c k e d~b y~b o t h~}C_{f r},C_{f_{\\ell}}\\mathrm{~o r~b y~n e i t h e r}\\right)}\\\\ {\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\left(1-\\frac{1}{2}|f_{r}(x_{i})-f_{\\ell}(x_{i})|\\right)}\\\\ {\\displaystyle=1-\\frac{1}{2}\\frac{1}{n}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})|}\\\\ {\\displaystyle=1-\\frac{1}{2}d(f_{r},f_{\\ell})\\leq 1-\\varepsilon/2\\leq e^{-\\varepsilon/2}}\\end{array}\\]\nSubstituting,\n\\(\\mathbb{P}\\left(C_{f_{r}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n\\(=1-\\mathbb{P}\\left((z_{1},t_{1})\\right.\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)𝑘\n\\[\\begin{array}{ l }{\\geq 1-\\left(e^{-\\varepsilon/2}\\right)^{k}}\\\\ {=1-e^{-k\\varepsilon/2}}\\end{array}\\]\nThere are (m） ways to choose \\(f_{r}\\) and \\(f_{\\ell},\\) s0\nP (All pairs \\(C_{f_{r}}\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\geq 1-{\\binom{m}{2}}e^{-k\\varepsilon/2}.\\)\nWhat k should we choose so that \\(1-{\\binom{m}{2}}e^{-k\\varepsilon/2}>0?\\) Choose\n\\[k>{\\frac{2}{\\varepsilon}}\\log{\\binom{m}{2}}.\\]\nThen there exist \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) such that all \\(C_{f_{\\ell}}\\) pick out different subsets.But \\(\\{C_{f}:f\\in\\mathcal{F}\\}\\) is \\(\\mathrm{VC}\\) so by Sauer's lemma, we can pick out at most \\(\\left({\\frac{e k}{V}}\\right)^{V}\\) out of these k points, provided \\(k \\geq V\\); if instead \\(k < V\\) then \\(m\\leq 2^{k}<2^{V}\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}\\) and the claimed bound already holds, so that case may be set aside."}, {"n": 5, "claim": "Hence, \\(\\begin{array}{ r }{m\\leq\\left(\\frac{e k}{V}\\right)^{V}}\\end{array}\\) as long as \\(k>{\\frac{2}{\\varepsilon}}\\log\\left({{\\binom{m}{2}}}\\right)\\) and \\(k \\geq V\\), the case \\(k < V\\) having been disposed of above."}, {"n": 6, "claim": "The latter holds for \\(\\begin{array}{ r }{k=\\frac{2}{\\varepsilon}\\log m^{2}}\\end{array}\\) ."}, {"n": 7, "claim": "Therefore,\n\\[m\\leq\\left(\\frac{e}{V}\\frac{2}{\\varepsilon}\\log m^{2}\\right)^{V}=\\left(\\frac{4 e}{V\\varepsilon}\\log m\\right)^{V},\\]\nwhere \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) ."}, {"n": 8, "claim": "Hence, we get\n\\[m^{1/V}\\leq\\frac{4 e}{\\varepsilon}\\log m^{1/V}\\]\nand defining \\(m^{1/V}=s .\\)\n\\[s\\leq{\\frac{4 e}{\\varepsilon}}\\log s .\\]\nNote that \\(\\frac{s}{\\log s}\\) is increasing for \\(s \\geq e\\) and so for large enough s, the inequality will be violated."}, {"n": 9, "claim": "We now check that the inequality is violated for \\(\\begin{array}{ r }{s^{\\prime}=\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}}\\end{array}\\) ."}, {"n": 10, "claim": "Indeed, one can show that\n\\[\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{7}{\\varepsilon}\\right)^{2}>\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)\\]\nsince\n\\[\\frac{49}{8e\\varepsilon}>\\log{\\frac{7}{\\epsilon}}.\\]\nHence, \\(m^{1/V}=s\\leq s^{\\prime}\\) and, thus,\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]"}]}}, {"local_id": "T2", "statement": "Let \\(0\\leq g_{1},g_{2}\\leq 1,g_{i}:\\mathcal{X}\\mapsto [0,1]\\) . Then\n\\[\\int\\operatorname*{min}\\left(2,{\\frac{1}{g_{1}(x)}},{\\frac{1}{g_{2}(x)}}\\right)d P(x)\\cdot\\int g_{1}(x)d P(x)\\cdot\\int g_{2}(x)d P(x)\\leq 1\\]", "proof": {"steps": [{"n": 1, "claim": "Notice that log \\(x \\leq x - 1\\)\nSo enough to show\n\\[\\int\\operatorname*{min}\\left(2,{\\frac{1}{g_{1}}},{\\frac{1}{g_{2}}}\\right)d P+\\int g_{1}d P+\\int g_{2}d P\\leq 3\\]\nwhich is the same as\n\\[\\int\\left[\\operatorname*{min}\\left(2,\\frac{1}{g_{1}},\\frac{1}{g_{2}}\\right)+g_{1}+g_{2}\\right]d P\\leq 3\\]\nIt's enough to show\n\\[\\operatorname*{min}\\left(2,\\frac{1}{g_{1}},\\frac{1}{g_{2}}\\right)+g_{1}+g_{2}\\leq 3 .\\]\nIf min is equal to 2, then \\(\\begin{array}{ r }{g_{1},g_{2}\\le\\frac{1}{2}}\\end{array}\\) and the sum is less than 3."}, {"n": 2, "claim": "If min is equal to \\(\\frac{1}{g_{1}}\\) ,then \\(\\begin{array}{ r }{g_{1}\\ge\\frac{1}{2}}\\end{array}\\) and \\(\\begin{array}{ r }{g_{1}\\geq g_{2},\\mathrm{~s o~}\\operatorname*{min}+g_{1}+g_{2}\\leq\\frac{1}{g_{1}}+2 g_{1}\\leq 3 .}\\end{array}\\)"}]}}, {"local_id": "T3", "statement": "Setup. Let \\(X_1,\\dots,X_n\\) be i.i.d. random variables with common law \\(P\\) on a measurable space \\((\\mathcal{X},\\mathcal{A})\\), let \\(\\mathcal{C}\\subseteq\\mathcal{A}\\) be a class of measurable sets, and write \\(P(C)=\\mathbb{P}(X_1\\in C)\\). Let \\(X'_1,\\dots,X'_n\\) (the ghost sample) be an independent copy of \\(X_1,\\dots,X_n\\), i.e. i.i.d. \\(P\\) and independent of \\(X_1,\\dots,X_n\\). Suprema over \\(\\mathcal{C}\\) are assumed measurable; otherwise read \\(\\mathbb{P}\\) as outer probability.\n\nIf \\(t\\geq{\\sqrt{\\frac{2}{n}}}.\\) then\n\\[\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\right)\\leq 2\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\geq t/2\\right).\\]", "proof": {"steps": [{"n": 1, "claim": "Suppose the event\n\\[\\operatorname*{sup}_{C\\in{\\mathcal{C}}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\]\noccurs.Let \\(\\begin{array}{ r }{X=(X_{1},\\ldots,X_{n})\\in\\{\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\}}\\end{array}\\) ."}, {"n": 2, "claim": "Then\n\\[\\exists C_{X}{\\mathrm{~s u c h~t h a t~}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nFor a fixed C,\n\\[\\begin{array}{ r }{\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right|\\geq t/2\\right)=\\mathbb{P}\\left(\\left(\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}\\geq t^{2}/4\\right)}\\\\ {\\leq\\mathrm{~(b y~C h e b y s h e v~s~I n e q)~}\\frac{4\\mathbb{E}\\left(\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}}{t^{2}}}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i,j}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))(I(X_{j}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i=1}^{n}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))^{2}=\\frac{4 n\\mathbb{P}\\left(\\mathcal{C}\\right)\\left(1-\\mathbb{P}(\\mathcal{C})\\right)}{n^{2}t^{2}}\\leq\\frac{1}{n t^{2}}\\leq\\frac{1}{2}}\\end{array}\\]\nsince we chose \\(t\\geq\\sqrt{\\textstyle{\\frac{2}{n}}}.\\)\nSo,\n\\[\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\Bigg|\\exists C_{X}\\right)\\geq 1/2\\]\nif \\(t\\geq\\sqrt{2/n}\\) .Assume that the event\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\]\noccurs."}, {"n": 3, "claim": "Recall that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nHence,it must be that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\geq t/2 .\\]\nWe conclude\n\\[\\begin{array}{ r c l }{\\displaystyle\\frac 12}&{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}(C_{X})\\right|\\le t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right).}\\end{array}\\]\nSince indicators are 0,1-valued,\n\\[\\begin{array}{ r l r }&{}&{\\displaystyle\\frac{1}{2}T\\left(\\underbrace{\\operatorname*{sup}_{\\zeta\\in{\\mathcal C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}(C)\\right|\\ge t}_{\\Im C x}\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C x\\right)\\cdot I\\left(\\exists C x\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X,X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]\nNow, take expectation with respect to \\(X_{i}^{,}\\) s to obtain\n\\[\\begin{array}{ r l r }&{}&{\\mathbb{P}_{X}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\ge t\\right)}\\\\ &{}&{\\displaystyle\\le 2\\cdot\\mathbb{P}_{X,X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]"}]}}, {"local_id": "T4", "statement": "The number \\(\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})\\) of subsets picked out by C is bounded by the number of subsets shattered by C.", "proof": {"steps": [{"n": 1, "claim": "Without loss of generality, we restrict C to \\({\\mathcal{C}}: =\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in{\\mathcal{C}}\\}\\) , and we have \\(\\operatorname{card}({\\mathcal{C}})=\\) \\(\\Delta_{n}({\\mathcal{C}},x_{1},\\cdots,x_{n})\\)\nWe will say that C is hereditary if and only if whenever \\(B\\subseteq C\\in{\\mathcal{C}},B\\in{\\mathcal{C}}\\) .If C is hereditary, then every \\(C\\in{\\mathcal{C}}\\) is shattered by C,and the lemma is obvious."}, {"n": 2, "claim": "Otherwise,we will transform \\({\\mathcal{C}}\\to{\\mathcal{C}}^{\\prime}\\) ,hereditary, without changing the cardinality of C and without increasing the number of shattered subsets."}, {"n": 3, "claim": "Define the operators \\(T_{i}\\) for \\(i=1,\\cdots,n\\) as the following,\n\\[\\begin{array}{ r c l }{T_{i}(C)}&{=}&{\\left\\{\\begin{array}{ l l }{C-\\{x_{i}\\}}&{\\mathrm{~i f~}C-\\{x_{i}\\}\\mathrm{~i s~n o t~i n~}\\mathcal{C}}\\\\ {C}&{\\mathrm{~o t h e r w i s e}}\\end{array}\\right.}\\\\ {T_{i}(\\mathcal{C})}&{=}&{\\{T_{i}(C):C\\in\\mathcal{C}\\}.}\\end{array}\\]\nIt follows that card \\(T_{i}({\\mathcal{C}})=\\operatorname{card}{\\mathcal{C}}\\) Moreover, every \\(A\\subseteq\\{x_{1},\\cdots,x_{n}\\}\\) that is shattered by \\(T_{i}(\\mathcal{C})\\) is also shattered by \\({\\mathcal{C}}.{\\mathrm{~I f~}}x_{i}\\notin A .\\) ,then \\(\\forall C\\in\\mathcal{C},A\\bigcap C=A\\bigcap T_{i}(C)\\) , thus C and \\(T_{i}(\\mathcal{C})\\) both or neither shatter A."}, {"n": 4, "claim": "On the other hand, if \\(x_{i}\\in A\\) and A is shattered by \\(T_{i}(\\mathcal{C})\\) , then \\(\\forall B\\subseteq A,\\exists C\\in{\\mathcal{C}}\\) ,such that \\(B\\bigcup\\{x_{i}\\}=A\\bigcap T_{i}(C)\\) This means that \\(x_{i}\\in T_{i}(C)\\) ，and that \\(C\\backslash\\{x_{i}\\}\\in\\mathcal{C}\\) Thus both \\(B\\bigcup\\{x_{i}\\}\\) and \\(B\\backslash\\{x_{i}\\}\\) are picked out by C."}, {"n": 5, "claim": "Since either \\(B=B\\cup\\{x_{i}\\}\\) or \\(B=B\\backslash\\{x_{i}\\}\\) , B is picked out by C."}, {"n": 6, "claim": "Thus A is shattered by C. Apply the operator \\(T=T_{1}\\circ\\dots\\circ T_{n}\\) until \\(T^{k+1}({\\mathcal{C}})=T^{k}({\\mathcal{C}})\\) ."}, {"n": 7, "claim": "This will happen for at most \\(\\scriptstyle\\sum_{C\\in{\\mathcal{C}}}\\operatorname{card}(C)\\) times,since \\(\\begin{array}{ r }{\\sum_{C\\in\\mathcal{C}}\\operatorname{card}(T_{i}(C))<\\sum_{C\\in\\mathcal{C}}\\operatorname{card}(C)\\mathrm{~i f~}T_{i}(\\mathcal{C})\\neq\\mathcal{C}}\\end{array}\\) ，The resulting collection \\(\\scriptstyle{\\mathcal{C}}^{\\prime}\\) is hereditary."}, {"n": 8, "claim": "This proves the lemma."}]}}], "target_local_id": "T4", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "ch16_lemma_16_1__correct", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nLet \\(x=(x_{1},\\ldots,x_{n})\\), \\(x^{\\prime}=(x_{1}^{\\prime},\\ldots,x_{n}^{\\prime})\\), where \\(x_{1},\\ldots,x_{n},x_{1}^{\\prime},\\ldots,x_{n}^{\\prime}\\) are i.i.d. random variables, and\n\\[d(f,g)=\\left(\\frac{1}{n}\\sum_{i=1}^{n}\\left(f(x_{i})-f(x_{i}^{\\prime})-g(x_{i})+g(x_{i}^{\\prime})\\right)^{2}\\right)^{1/2}.\\]\n\ndefinition 16.1. We say that F satisfies uniform entropy condition if\n\\[\\forall n,\\ \\forall(x_{1},\\ldots,x_{n}),\\ D({\\mathcal{F}},\\varepsilon,d_{x})\\leq D({\\mathcal{F}},\\varepsilon)\\]\nwhere \\(\\begin{array}{ r }{d_{x}(f,g)=\\left(\\frac{1}{n}\\sum_{i=1}^{n}\\left(f(x_{i})-g(x_{i})\\right)^{2}\\right)^{1/2}}\\end{array}\\)\n\ndefinition 12.3. The ε-packing number, \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) , is the marimal cardinality of an E-separated set.\nNote that \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) is decreasing in ε.\n\ndefinition 12.1. If \\(\\mathcal{C}\\) is a VC class of sets, then F is VC-subgraph class of functions and, by definition, \\(V C(\\mathcal{F})=V C(\\mathcal{C})\\)\nFor \\(f:\\mathcal{X}\\to\\mathbb{R}\\) , the subgraph of \\(f\\) is\n\\[C_{f}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}:0\\leq t\\leq f(x)\\ o r\\ f(x)\\leq t\\leq 0\\}.\\]\nDefine class of sets \\(\\mathcal{C}=\\{C_{f}:f\\in\\mathcal{F}\\}\\).\nNote that equivalent definition of \\(C_{f}\\) is\n\\[C_{f}^{\\prime}=\\{(x,t)\\in\\mathcal{X}\\times\\mathbb{R}: |f(x)|\\geq|t|\\}.\\]\n\ndefinition 12.2. Given \\(\\varepsilon > 0\\) and \\(f_{1},\\ldots,f_{N}\\in\\mathcal{F}\\) ， we say that \\(f_{1},\\ldots,f_{N}\\) are E-separated if \\(d(f_{i},f_{j})>\\varepsilon\\) for any \\(i \\neq j\\)\n\nDenote \\[\\triangle_{n}(\\mathcal{C})=\\sup_{\\{x_{1},\\ldots,x_{n}\\}}\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})\\leq 2^{n}.\\] We will see that for some classes, \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\) for \\(n \\leq V\\) and \\(\\triangle_{n}(\\mathcal{C})<2^{n}\\) for \\(n > V\\) for some constant \\(V\\).\n\ndefinition 8.1. If \\(V < \\infty\\) , then C is called a VC class. V is called VC dimension of C.\n\ndefinition 8.0. For any sample \\(\\{x_{1},\\ldots,x_{n}\\}\\), the ways that \\({\\mathcal{C}}\\) intersects with the sample are \\[\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in{\\mathcal{C}}\\}.\\] Let \\[\\triangle_{n}(\\mathcal{C},x_{1},\\ldots,x_{n})=\\mathrm{card}~\\{C\\cap\\{x_{1},\\ldots,x_{n}\\}:C\\in\\mathcal{C}\\},\\] the number of different subsets picked out by \\(C\\in{\\mathcal{C}}\\). If \\(\\triangle_{n}(\\mathcal{C})=2^{n}\\), this means we can find \\(\\{x_{1},\\ldots,x_{n}\\}\\) such that \\({\\mathcal{C}}\\) can pick out any subset of it; in this case \\({\\mathcal{C}}\\) shatters \\(\\{x_{1},\\ldots,x_{n}\\}\\). Equivalently, a set \\(A\\) is shattered by \\({\\mathcal{C}}\\) when \\({\\mathcal{C}}\\) picks out every subset of \\(A\\).", "theorems": [{"local_id": "T1", "statement": "Setup. Let \\(X_1,\\dots,X_n\\) be i.i.d. random variables with common law \\(P\\) on a measurable space \\((\\mathcal{X},\\mathcal{A})\\), let \\(\\mathcal{C}\\subseteq\\mathcal{A}\\) be a class of measurable sets, and write \\(P(C)=\\mathbb{P}(X_1\\in C)\\). Let \\(X'_1,\\dots,X'_n\\) (the ghost sample) be an independent copy of \\(X_1,\\dots,X_n\\), i.e. i.i.d. \\(P\\) and independent of \\(X_1,\\dots,X_n\\). Suprema over \\(\\mathcal{C}\\) are assumed measurable; otherwise read \\(\\mathbb{P}\\) as outer probability.\n\nIf \\(t\\geq{\\sqrt{\\frac{2}{n}}}.\\) then\n\\[\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\right)\\leq 2\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\geq t/2\\right).\\]", "proof": {"steps": [{"n": 1, "claim": "Suppose the event\n\\[\\operatorname*{sup}_{C\\in{\\mathcal{C}}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\]\noccurs.Let \\(\\begin{array}{ r }{X=(X_{1},\\ldots,X_{n})\\in\\{\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\}}\\end{array}\\) ."}, {"n": 2, "claim": "Then\n\\[\\exists C_{X}{\\mathrm{~s u c h~t h a t~}}\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nFor a fixed C,\n\\[\\begin{array}{ r }{\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right|\\geq t/2\\right)=\\mathbb{P}\\left(\\left(\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}\\geq t^{2}/4\\right)}\\\\ {\\leq\\mathrm{~(b y~C h e b y s h e v~s~I n e q)~}\\frac{4\\mathbb{E}\\left(\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C})\\right)^{2}}{t^{2}}}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i,j}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))(I(X_{j}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))}\\\\ {=\\frac{4}{n^{2}t^{2}}\\displaystyle\\sum_{i=1}^{n}\\mathbb{E}(I(X_{i}^{\\prime}\\in\\mathcal{C})-\\mathbb{P}(\\mathcal{C}))^{2}=\\frac{4 n\\mathbb{P}\\left(\\mathcal{C}\\right)\\left(1-\\mathbb{P}(\\mathcal{C})\\right)}{n^{2}t^{2}}\\leq\\frac{1}{n t^{2}}\\leq\\frac{1}{2}}\\end{array}\\]\nsince we chose \\(t\\geq\\sqrt{\\textstyle{\\frac{2}{n}}}.\\)\nSo,\n\\[\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\Bigg|\\exists C_{X}\\right)\\geq 1/2\\]\nif \\(t\\geq\\sqrt{2/n}\\) .Assume that the event\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\leq t/2\\]\noccurs."}, {"n": 3, "claim": "Recall that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\mathbb{P}\\left(C_{X}\\right)\\right|\\geq t .\\]\nHence,it must be that\n\\[\\left|{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-{\\frac{1}{n}}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\geq t/2 .\\]\nWe conclude\n\\[\\begin{array}{ r c l }{\\displaystyle\\frac 12}&{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})-\\mathbb{P}(C_{X})\\right|\\le t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &{\\le}&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C_{X})-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C_{X})\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right)}\\\\ &&{\\displaystyle\\mathbb{P}_{X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac 1n\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac 1n\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C_{X}\\right).}\\end{array}\\]\nSince indicators are 0,1-valued,\n\\[\\begin{array}{ r l r }&{}&{\\displaystyle\\frac{1}{2}T\\left(\\underbrace{\\operatorname*{sup}_{\\zeta\\in{\\mathcal C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}(C)\\right|\\ge t}_{\\Im C x}\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\Bigg|\\exists C x\\right)\\cdot I\\left(\\exists C x\\right)}\\\\ &{}&{\\displaystyle\\le\\mathbb{P}_{X,X^{\\prime}}\\left(\\underset{C\\in{\\mathcal C}}{\\operatorname*{sup}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]\nNow, take expectation with respect to \\(X_{i}^{,}\\) s to obtain\n\\[\\begin{array}{ r l r }&{}&{\\mathbb{P}_{X}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\ge t\\right)}\\\\ &{}&{\\displaystyle\\le 2\\cdot\\mathbb{P}_{X,X^{\\prime}}\\left(\\displaystyle\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\ge t/2\\right).}\\end{array}\\]"}]}}, {"local_id": "T2", "statement": "Let \\(\\xi,\\nu\\) - random variables.Assume that\n\\[\\mathbb{P}\\left(\\nu\\geq t\\right)\\leq\\Gamma e^{-\\gamma t}\\]\nwhere \\(\\Gamma\\geq 1,t\\geq 0\\) ，and \\(\\gamma > 0\\) . Furthermore, for all \\(a > 0\\) assume that\n\\[\\mathbb{E}\\phi(\\xi)\\le\\mathbb{E}\\phi(\\nu)\\]\nwhere \\(\\phi(x)=(x-a)_{+}\\) .Then\n\\[\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\Gamma\\cdot e\\cdot e^{-\\gamma t}.\\]", "proof": {"steps": [{"n": 1, "claim": "Since \\(\\phi(x)=(x-a)_{+}\\) ，we have \\(\\phi(\\xi)\\geq\\phi(t)\\) whenever \\(\\xi \\geq t .\\)\n\\[\\begin{array}{ r l }&{\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\mathbb{P}\\left(\\phi(\\xi)\\geq\\phi(t)\\right)}\\\\ &{\\qquad\\leq\\displaystyle\\frac{\\mathbb{E}\\phi(\\xi)}{\\phi(t)}\\leq\\frac{\\mathbb{E}\\phi(\\nu)}{\\phi(t)}=\\frac{\\mathbb{E}\\left(\\nu-a\\right)_{+}}{(t-a)_{+}}}\\end{array}\\]\nFurthermore,\n\\[\\begin{array}{ r l }{\\displaystyle\\mathbb{E}(\\nu-a)_{+}=}&{\\mathbb{E}\\int_{0}^{(\\nu-a)_{+}}\\mathrm{1}d x}\\\\ &{=\\mathbb{E}\\int_{0}^{\\infty}I(x\\leq(\\nu-a)_{+})d x}\\\\ &{=\\int_{0}^{\\infty}\\mathbb{E}I(x\\leq(\\nu-a)_{+})d x}\\\\ &{=\\int_{0}^{\\infty}\\mathbb{P}\\left((\\nu-a)_{+}\\geq x\\right)d x}\\\\ &{=\\int_{0}^{\\infty}\\mathbb{P}\\left(\\nu\\geq a+x\\right)d x}\\\\ &{\\leq\\int_{0}^{\\infty}\\Gamma e^{-\\gamma(a+x)}d x=\\frac{\\Gamma e^{-\\gamma a}}{\\gamma}.}\\end{array}\\]\nHence,\n\\[\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\frac{\\Gamma e^{-\\gamma a}}{\\gamma(t-a)_{+}}=\\frac{\\Gamma\\cdot e\\cdot e^{-\\gamma t}}{1}=\\Gamma\\cdot e\\cdot e^{-\\gamma t}\\]\nwhere we chose optimal \\(\\begin{array}{ r }{a=t-\\frac{1}{\\gamma}}\\end{array}\\) to minimize \\(\\frac{\\Gamma e^{-\\gamma a}}{\\gamma(t-a)_{+}}\\)"}]}}, {"local_id": "T3", "statement": "If \\({\\mathcal{F}}\\) satisfies uniform entropy condition, then\n\\[\\mathbb{E}_{x^{\\prime}}\\int_{0}^{d(0,f)}\\log^{1/2}\\mathcal{D}(\\mathcal{F},\\varepsilon,d)d\\varepsilon\\leq\\int_{0}^{\\sqrt{\\mathbb{E}_{x^{\\prime}}d(0,f)^{2}}}\\log^{1/2}\\mathcal{D}(\\mathcal{F},\\varepsilon/2)d\\varepsilon\\]", "proof": {"steps": [{"n": 1, "claim": "Using inequality \\((a+b)^{2}\\leq 2(a^{2}+b^{2})\\)\n\\[\\begin{array}{ r c l }{d(f,g)}&{=}&{\\displaystyle\\left(\\frac{1}{n}\\sum_{i=1}^{n}\\left(f(x_{i})-g(x_{i})+g(x_{i}^{\\prime})-f(x_{i}^{\\prime})\\right)^{2}\\right)^{1/2}}\\\\ &{\\le}&{\\displaystyle\\left(\\frac{2}{n}\\sum_{i=1}^{n}\\left(\\left(f(x_{i})-g(x_{i})\\right)^{2}+\\left(g(x_{i}^{\\prime})-f(x_{i}^{\\prime})\\right)^{2}\\right)\\right)^{1/2}}\\\\ &{=}&{\\displaystyle 2\\left(\\frac{1}{2 n}\\sum_{i=1}^{n}\\left(\\left(f(x_{i})-g(x_{i})\\right)^{2}+\\left(g(x_{i}^{\\prime})-f(x_{i}^{\\prime})\\right)^{2}\\right)\\right)^{1/2}}\\\\ &{=}&{\\displaystyle 2 d_{x,x^{\\prime}}(f,g)}\\end{array}\\]\nSince \\(d(f,g)\\leq 2 d_{x,x^{\\prime}}(f,g)\\) ， we also have\n\\[\\begin{array}{ r }{\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\mathcal{D}(\\mathcal{F},\\varepsilon/2,d_{x,x^{\\prime}}).}\\end{array}\\]\nIndeed, let \\(f_{1},...,f_{N}\\) be optimal ε-packing w.r.t. distance d."}, {"n": 2, "claim": "Then\n\\[\\varepsilon\\le d(f_{i},f_{j})\\le 2 d_{x,x^{\\prime}}(f_{i},f_{j})\\]\nand, hence,\n\\[\\varepsilon/2\\leq d_{x,x^{\\prime}}(f_{i},f_{j}).\\]\nSo, \\(f_{1},...,f_{N}\\) is ε/2-packing w.r.t. \\(d_{x,x^{\\prime}}\\) ."}, {"n": 3, "claim": "Therefore, can pack at least N and so \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\mathcal{D}(\\mathcal{F},\\varepsilon/2,d_{x,x^{\\prime}})\\)\n\\[\\begin{array}{ r l }&{\\mathbb{E}_{x^{\\prime}}\\int_{0}^{d(0,f)}\\log^{1/2}\\mathcal{D}(\\mathcal{F},\\varepsilon,d)d\\varepsilon\\le\\mathbb{E}_{x^{\\prime}}\\int_{0}^{d(0,f)}\\log^{1/2}\\mathcal{D}(\\mathcal{F},\\varepsilon/2,d_{x,x^{\\prime}})d\\varepsilon}\\\\ &{\\qquad\\le\\displaystyle\\int_{0}^{d(0,f)}\\log^{1/2}\\mathcal{D}(\\mathcal{F},\\varepsilon/2)d\\varepsilon}\\end{array}\\]\nLet \\(\\begin{array}{ r }{\\phi(x)=\\int_{0}^{x}\\log^{1/2}\\mathcal{D}(\\mathcal{F},\\varepsilon)d\\varepsilon}\\end{array}\\) .It is concave because \\(\\phi^{\\prime}(x)=\\log^{1/2}\\mathcal{D}(\\mathcal{F},\\varepsilon/2)\\) is decreasing when x is increasing (can pack less with larger balls)."}, {"n": 4, "claim": "Hence, by Jensen's inequality,\n\\[\\begin{array}{ r }{\\mathbb{E}_{x^{\\prime}}\\phi(d(0,f))\\leq\\phi(\\mathbb{E}_{x^{\\prime}}d(0,f))=\\phi(\\mathbb{E}_{x^{\\prime}}\\sqrt{d(0,f)^{2}})\\leq\\phi(\\sqrt{\\mathbb{E}_{x^{\\prime}}d(0,f)^{2}}).}\\end{array}\\]"}]}}, {"local_id": "T4", "statement": "Setup. Let \\(0<\\varepsilon\\le 1\\).\n\nAssume \\(\\mathcal{F}\\) is a VC-subgraph class and \\(V C(\\mathcal{F})=V\\) Suppose \\(-1\\leq f(x)\\leq 1\\) for all \\(f\\in{\\mathcal{F}}\\) and \\(x\\in\\mathcal{X}\\) .Let \\(x_{1},\\ldots,x_{n}\\in{\\mathcal{X}}\\) and define \\(\\begin{array}{ r }{d(f,g)=\\frac{1}{n}\\sum_{i=1}^{n}|f(x_{i})-g(x_{i})|}\\end{array}\\) . Then\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]\n(which \\(\\begin{array}{ r }{i s\\le\\left(\\frac{K}{\\varepsilon}\\right)^{V+\\delta}}\\end{array}\\) for some \\(\\delta . \\jmath\\)", "proof": {"steps": [{"n": 1, "claim": "Let \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) and \\(f_{1},\\ldots,f_{m}\\) be ε-separated, i.e.\n\\[{\\frac{1}{n}}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})| >\\varepsilon .\\]"}, {"n": 2, "claim": "Let \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) be constructed in the following way:"}, {"n": 3, "claim": "\\(z_{i}\\) is chosen uniformly from \\(x_{1},\\ldots,x_{n}\\) and \\(t_{i}\\) is uniform on \\([-1,1]\\)\nConsider \\(f_{r}\\) and \\(f_{\\ell}\\) from the ε-packing."}, {"n": 4, "claim": "Let \\(C_{f_{\\tau}}\\) and \\(C_{f_{\\ell}}\\) be subgraphs of \\(f_{r}\\) and \\(f_{\\ell}\\) Then\n\\(\\mathbb{P}\\left(C_{f_{\\tau}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n= P(At least one point \\((z_{i},t_{i})\\) is picked by \\(C_{f_{r}}\\) or \\(C_{f_{\\ell}}\\) but not picked by the other)\n\\(=1-\\mathbb{P}\\) (All points \\((z_{i},t_{i})\\) are picked either by both or by none)\n\\(\\mathbf{\\Sigma}=1-\\mathbb{P}\\left(\\left(z_{i},t_{i}\\right)\\right.\\) is picked either by both or by none)k\nSince \\(z_{i}\\) is drawn uniformly from \\(x_{1},\\ldots,x_{n}\\)\nP \\(((z_{1},t_{1})\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)\n\\[\\begin{array}{ l }{\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{P}\\left(\\left(x_{i},t_{1}\\right)\\mathrm{~i s~p i c k e d~b y~b o t h~}C_{f r},C_{f_{\\ell}}\\mathrm{~o r~b y~n e i t h e r}\\right)}\\\\ {\\displaystyle=\\frac{1}{n}\\sum_{i=1}^{n}\\left(1-\\frac{1}{2}|f_{r}(x_{i})-f_{\\ell}(x_{i})|\\right)}\\\\ {\\displaystyle=1-\\frac{1}{2}\\frac{1}{n}\\sum_{i=1}^{n}|f_{r}(x_{i})-f_{\\ell}(x_{i})|}\\\\ {\\displaystyle=1-\\frac{1}{2}d(f_{r},f_{\\ell})\\leq 1-\\varepsilon/2\\leq e^{-\\varepsilon/2}}\\end{array}\\]\nSubstituting,\n\\(\\mathbb{P}\\left(C_{f_{r}}\\right.\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\)\n\\(=1-\\mathbb{P}\\left((z_{1},t_{1})\\right.\\) is picked by both \\(C_{f_{r}},C_{f_{\\ell}}\\) or by neither)𝑘\n\\[\\begin{array}{ l }{\\geq 1-\\left(e^{-\\varepsilon/2}\\right)^{k}}\\\\ {=1-e^{-k\\varepsilon/2}}\\end{array}\\]\nThere are (m） ways to choose \\(f_{r}\\) and \\(f_{\\ell},\\) s0\nP (All pairs \\(C_{f_{r}}\\) and \\(C_{f_{\\ell}}\\) pick out different subsets of \\((z_{1},t_{1}),\\ldots,(z_{k},t_{k}))\\geq 1-{\\binom{m}{2}}e^{-k\\varepsilon/2}.\\)\nWhat k should we choose so that \\(1-{\\binom{m}{2}}e^{-k\\varepsilon/2}>0?\\) Choose\n\\[k>{\\frac{2}{\\varepsilon}}\\log{\\binom{m}{2}}.\\]\nThen there exist \\((z_{1},t_{1}),\\dots,(z_{k},t_{k})\\) such that all \\(C_{f_{\\ell}}\\) pick out different subsets.But \\(\\{C_{f}:f\\in\\mathcal{F}\\}\\) is \\(\\mathrm{VC}\\) so by Sauer's lemma, we can pick out at most \\(\\left({\\frac{e k}{V}}\\right)^{V}\\) out of these k points, provided \\(k \\geq V\\); if instead \\(k < V\\) then \\(m\\leq 2^{k}<2^{V}\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}\\) and the claimed bound already holds, so that case may be set aside."}, {"n": 5, "claim": "Hence, \\(\\begin{array}{ r }{m\\leq\\left(\\frac{e k}{V}\\right)^{V}}\\end{array}\\) as long as \\(k>{\\frac{2}{\\varepsilon}}\\log\\left({{\\binom{m}{2}}}\\right)\\) and \\(k \\geq V\\), the case \\(k < V\\) having been disposed of above."}, {"n": 6, "claim": "The latter holds for \\(\\begin{array}{ r }{k=\\frac{2}{\\varepsilon}\\log m^{2}}\\end{array}\\) ."}, {"n": 7, "claim": "Therefore,\n\\[m\\leq\\left(\\frac{e}{V}\\frac{2}{\\varepsilon}\\log m^{2}\\right)^{V}=\\left(\\frac{4 e}{V\\varepsilon}\\log m\\right)^{V},\\]\nwhere \\(m=\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) ."}, {"n": 8, "claim": "Hence, we get\n\\[m^{1/V}\\leq\\frac{4 e}{\\varepsilon}\\log m^{1/V}\\]\nand defining \\(m^{1/V}=s .\\)\n\\[s\\leq{\\frac{4 e}{\\varepsilon}}\\log s .\\]\nNote that \\(\\frac{s}{\\log s}\\) is increasing for \\(s \\geq e\\) and so for large enough s, the inequality will be violated."}, {"n": 9, "claim": "We now check that the inequality is violated for \\(\\begin{array}{ r }{s^{\\prime}=\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}}\\end{array}\\) ."}, {"n": 10, "claim": "Indeed, one can show that\n\\[\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{7}{\\varepsilon}\\right)^{2}>\\frac{4 e}{\\varepsilon}\\log\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)\\]\nsince\n\\[\\frac{49}{8e\\varepsilon}>\\log{\\frac{7}{\\epsilon}}.\\]\nHence, \\(m^{1/V}=s\\leq s^{\\prime}\\) and, thus,\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]"}]}}], "target_local_id": "T3", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2025-23-0832", "context": "In this section, we introduce our notation, data generation process, neural network architecture and the optimization algorithm.\n\nNotations. We use lower case letters to denote scalars and boldface letters and symbols (e.g. x) to denote vectors and matrices. We use $\\odot$ to denote element-wise product. For an integer n, we use [n] to denote the set of integers $\\{1,2,\\ldots,n\\}$ . We use $x=O(y),x=$ $\\Omega(y),x=\\Theta(y)$ to denote that there exists a constant C such that $x\\leq Cy,\\ x\\geq Cy,x=Cy$ respectively. We use $\\widetilde{O},\\widetilde{\\Omega}$ and Θe to hide polylogarithmic factor in these notations. Finally, we use $x=\\mathrm{poly}(y)$ if $x=O(y^{C})$ for some positive constant $C_{i}$ , and $x={\\mathrm{poly}}\\log y$ if $x=\\operatorname{poly}(\\log y)$\n\nDefinition 1 (Data distribution of K classes) Consider we are given the set of signal vectors $\\{\\mu\\mathbf{e}_{i}\\}_{i=1}^{K}$ , where $\\mu > 0$ denotes the strength of the signal, and $\\mathbf{e}_{i}\\in\\mathbb{R}^{d}$ (with $d>K)$ denotes the $_{i-t h}$ standard basis vector with its i-th entry being 1 and all other coordinates being 0. Each data point $\\left(\\mathbf{x},y\\right)$ with $\\mathbf{x}=[\\mathbf{x}_{1}^{\\top},\\mathbf{x}_{2}^{\\top}]^{\\top}\\in\\mathbb{R}^{2\\bar{d}}$ and $y \\in [ K ]$ is generated from the following distribution $\\mathcal{D}.$ :\n\n1. The label y is generated from a uniform distribution over $[ K ]$  \n2. A noise vector $\\boldsymbol{\\xi}$ is generated from the Gaussian distribution ${\\mathcal{N}}(\\mathbf{0},\\sigma_{n}^{2}\\mathbf{I})$  \n3. With probability $1/2_{;}$ , assign $\\mathbf{x}_{1}=\\pmb{\\mu}_{y},\\ \\mathbf{x}_{2}=\\pmb{\\xi}$ ; with probability $1/2_{;}$ , assign ${\\bf x}_{2}=$ $\\mu_{y},~\\mathbf{x}_{1}=\\pmb{\\xi}$ where $\\mu_{y}=\\mu\\mathbf{e}_{y}$\n\nNetwork architecture and random pruning. We consider a two-layer convolutional neural network model with polynomial ReLU activation $\\sigma(z)=(\\operatorname*{max}\\{0,z\\})^{q}$ , where we focus on the case when $q=3\\AA^{1}$ The network is pruned at the initialization by mask M where each entry in the mask M is generated i.i.d. from Bernoulli(p). Given the data $\\left(\\mathbf{x},y\\right)$ , the output of the neural network can be written as $F(\\mathbf{W}\\odot\\mathbf{M},\\mathbf{x})=(F_{1}(\\mathbf{W}_{1}\\odot\\mathbf{M}_{1},\\mathbf{x}),F_{2}(\\mathbf{W}_{2}\\odot$ $\\mathbf{M}_{2},\\mathbf{x}),\\ldots,F_{K}(\\mathbf{W}_{K}\\odot\\mathbf{M}_{K},\\mathbf{x}))$ where the j-th output is given by\n\n$$\n\\begin{array}{l}F_{j}(\\mathbf{W}_{j}\\odot\\mathbf{M}_{j},\\mathbf{x})=\\sum_{r=1}^{m}[\\sigma(\\langle\\mathbf{w}_{j,r}\\odot\\mathbf{m}_{j,r},\\mathbf{x}_{1}\\rangle)+\\sigma(\\langle\\mathbf{w}_{j,r}\\odot\\mathbf{m}_{j,r},\\mathbf{x}_{2}\\rangle)]\\\\ =\\sum_{r=1}^{m}[\\sigma(\\langle\\mathbf{w}_{j,r}\\odot\\mathbf{m}_{j,r},\\boldsymbol{\\mu}_{y}\\rangle)+\\sigma(\\langle\\mathbf{w}_{j,r}\\odot\\mathbf{m}_{j,r},\\boldsymbol{\\xi}\\rangle)],\\\\ \\end{array}\n$$\n\nwhere $\\mathbf{m}_{j,r}$ denotes the r-th row of $\\mathbf{M}_{j}$ . The mask M is only sampled once at the initialization and remains fixed through the entire training process. From now on, we use tilde over a symbol to denote its masked version, e.g., $\\widetilde{\\mathbf{W}}=\\mathbf{W}\\odot\\mathbf{M}$ and $\\widetilde{\\mathbf{w}}_{j,r}=\\mathbf{w}_{j,r}\\odot\\mathbf{m}_{j,r}$\n\nSince we have $\\pmb{\\mu}_{j}\\odot\\mathbf{m}_{j,r}=\\mathbf{0}$ , with probability $1-p,$ some neurons will not receive the corresponding signal at all and will only learn noise. Therefore, for each class $j \\in [ K ]$ , we split the neurons into two sets based on whether it receives its corresponding signal or not:\n\n$$\n\\mathcal{S}_{\\text{signal}}^{j}=\\{r\\in [m]:\\boldsymbol{\\mu}_{j}\\odot\\mathbf{m}_{j,r}\\neq\\mathbf{0}\\},\n$$\n\n$$\n\\mathcal{S}_{\\mathrm{noise}}^{j}=\\{r\\in [m]:\\boldsymbol{\\mu}_{j}\\odot\\mathbf{m}_{j,r}=\\mathbf{0}\\}.\n$$\n\nGradient descent algorithm. We consider the network is trained by cross-entropy loss with softmax. We denote by $\\begin{array}{ r }{\\mathrm{logit}_{i}(F,\\mathbf{x}): =\\frac{e^{F_{i}(\\mathbf{x})}}{\\sum_{j\\in [K]}e^{F_{j}(\\mathbf{x})}}}\\end{array}$ and the cross-entropy loss can be written as $\\ell(F(\\mathbf{x},y))= -\\log\\log\\mathrm{i}\\mathbf{\\xi}_{y}(F,\\mathbf{x})$ . The convolutional neural network is trained by minimizing the empirical cross-entropy loss given by\n\n$$\nL_{S}(\\mathbf{W})=\\underset{S}{\\mathbb{E}}\\ell [F(\\mathbf{W}\\odot\\mathbf{M};\\mathbf{x}_{i},y_{i})]=\\frac{1}{n}\\sum_{i=1}^{n}\\ell [F(\\mathbf{W}\\odot\\mathbf{M};\\mathbf{x}_{i},y_{i})],\n$$\n\nwhere $S=\\{(\\mathbf{x}_{i},y_{i})\\}_{i=1}^{n}$ is the training data set. Similarly, we define the generalization loss as\n\n$$\nL_{\\mathcal{D}}:=\\underset{(\\mathbf{x},y)}{\\mathbb{E}}[\\ell(F(\\mathbf{W}\\odot\\mathbf{M};\\mathbf{x},y))].\n$$\n\nThe model weights are initialized from a i.i.d. Gaussian $\\mathcal{N}(0,\\sigma_{0}^{2})$ . The gradient of the cross-entropy loss is given by $\\ell_{j,i}^{\\prime}: =\\ell_{j}^{\\prime}(\\mathbf{x}_{i},y_{i})=\\mathrm{logit}_{j}(F,\\mathbf{x}_{i})-\\mathbb{I}(j=y_{i})$\n\nSince\n\n$$\n\\begin{array}{l}\\nabla_{\\mathbf{w}_{j,r}}L_{S}(\\mathbf{W}\\odot\\mathbf{M})=\\nabla_{\\mathbf{w}_{j,r}\\odot\\mathbf{m}_{j,r}}L_{S}(\\mathbf{W}\\odot\\mathbf{M})\\odot\\mathbf{m}_{j,r}\\\\ =\\nabla_{\\widetilde{\\mathbf{w}}_{j,r}}L_{S}(\\widetilde{\\mathbf{W}})\\odot\\mathbf{m}_{j,r},\\\\ \\end{array}\n$$\n\nwe can write the full-batch gradient descent update of the weights as\n\n$$\n\\begin{array}{l}\\widetilde{\\mathbf{w}}_{j,r}^{(t+1)}=\\widetilde{\\mathbf{w}}_{j,r}^{(t)}-\\eta\\nabla_{\\widetilde{\\mathbf{w}}_{j,r}}L_{S}(\\widetilde{\\mathbf{W}})\\odot\\mathbf{m}_{j,r}\\\\ =\\widetilde{\\mathbf{w}}_{j,r}^{(t)}-\\frac{\\eta}{n}\\sum_{i=1}^{n}\\ell_{j,i}^{\\prime(t)}\\cdot\\sigma^{\\prime}\\left(\\left\\langle\\widetilde{\\mathbf{w}}_{j,r}^{(t)},\\boldsymbol{\\xi}_{i}\\right\\rangle\\right)\\cdot\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}\\\\ -\\frac{\\eta}{n}\\sum_{i=1}^{n}\\ell_{j,i}^{\\prime(t)}\\sigma^{\\prime}\\left(\\left\\langle\\widetilde{\\mathbf{w}}_{j,r}^{(t)},\\boldsymbol{\\mu}_{y_{i}}\\right\\rangle\\right)\\boldsymbol{\\mu}_{y_{i}}\\odot\\mathbf{m}_{j,r},\\\\ \\end{array}\n$$\n\nfor $j \\in [ K ]$ and $r \\in [ m ]$ , where $\\widetilde{\\pmb{\\xi}}_{j,r,i}=\\pmb{\\xi}_{i}\\odot\\mathbf{m}_{j,r}$\n\nCondition 2 We consider the parameter regime described as follows: (1) Number of classes $K=O(\\log d).\\quad(2)$ Total number of training samples n = poly log d. (3) Dimension $d\\geq C_{d}$ for some suficiently large constant $\\it C_{d}.\\it(4)$ Relationship between signal strength and noise strength: $\\mu=\\Theta(\\sigma_{n}\\sqrt{d}\\log d)=\\Theta(1)$ . (5) The number of neurons in the network $m=\\Omega({\\mathrm{poly}}\\log d).\\ (6)$ Initialization variance: $\\begin{array}{ r }{\\sigma_{0}=\\widetilde{\\Theta}(m^{-4}n^{-1}\\mu^{-1})}\\end{array}$ . (7) Learning rate: $\\Omega(1/\\operatorname{poly}(d))\\leq\\eta\\leq\\widetilde{O}(1/\\mu^{2})$ . (8) Target training loss: $\\epsilon=\\Theta(1/\\operatorname{poly}(d))$ ).\n\nConditions (1) ensures that there are not too many classes which is a mild assumption and can be satisfied by real world dataset like MNIST. Condition (2) ensure that there are enough samples in each class with high probability and at the same time not too many samples such that the noise will interfere with the neural network learning the signals. Condition (3) ensures that our setting is in high-dimensional regime. Condition (4) ensures that the full model can be trained to exhibit good generalization. Condition (5), (6) and (7) ensures that the neural network is suficiently overparameterized and can be optimized eficiently by gradient descent. Condition (7) and (8) further ensures that training time is polynomial in d. We further discuss the practical consideration of η and ε to justify their condition in Theorem 23.\n\nOur second result shows that there exists a relatively large pruning fraction (i.e., small p) such that the learned model yields poor generalization, although gradient descent is still able to drive the training error toward zero. The full proof is defered to Appendix D.\n\nTheorem 9 (Main theorem for over pruning, informal version of Theorem 41 ) Under Condition 2 if $\\begin{array}{ r }{p\\ =\\ \\Theta(\\frac{1}{K m\\log d})}\\end{array}$ , then with probability at least $1 - 1 /$ poly log d over the randomness in the data, network initialization and pruning, there exists $T =$ $O(\\eta^{-1}n\\sigma_{0}^{q-2}\\sigma_{n}^{-q}(p d)^{-q/2}+\\eta^{-1}\\epsilon^{-1}m^{4}n\\sigma_{n}^{-2}(p d)^{-1})$ such that\n\n1. The training loss is below $\\epsilon\\colon L_{S}(\\widetilde{\\mathbf{W}}^{(T)})\\le\\epsilon$  \n2. The generalization loss is large: $L_{\\mathcal{D}}(\\widetilde{\\mathbf{W}}^{(T)})\\ge\\Omega(\\log K)$\n\nIn this section, we introduce the following signal-noise decomposition of each neuron weight from Cao et al. (2022), and some useful properties for the terms in such a decomposition, which are useful in our analysis.\n\nDefinition 14 (signal-noise decomposition) For each neuron weight $j\\in [K],\\ r\\in [m]$ • there exist coeficients $\\gamma_{j,r,k}^{(t)},\\zeta_{j,r,i}^{(t)},\\omega_{j,r,i}^{(t)}$ such that\n\n$$\n\\begin{array}{l}\\widetilde{\\mathbf{w}}_{j,r}^{(t)}\\\\ =\\widetilde{\\mathbf{w}}_{j,r}^{(0)}+\\sum_{k=1}^{K}\\gamma_{j,r,k}^{(t)}\\cdot\\|\\boldsymbol{\\mu}_{k}\\|_{2}^{-2}\\cdot\\boldsymbol{\\mu}_{k}\\odot\\mathbf{m}_{j,r}+\\sum_{i=1}^{n}\\zeta_{j,r,i}^{(t)}\\cdot\\left\\|\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}\\right\\|_{2}^{-2}\\cdot\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}+\\sum_{i=1}^{n}\\omega_{j,r,i}^{(t)}\\left\\|\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}\\right\\|_{2}^{-2}\\cdot\\widetilde{\\boldsymbol{\\xi}}_{j,r,i},\\\\ \\end{array}\n$$\n\nwhere $\\gamma_{j,r,j}^{(t)}\\geq 0,\\gamma_{j,r,k}^{(t)}\\leq 0,\\zeta_{j,r,i}^{(t)}\\geq 0,\\omega_{j,r,i}^{(t)}\\leq 0 .$\n\nIt is straightforward to see the following:\n\n$$\n\\begin{array}{l}\\gamma_{j,r,k}^{(0)},\\zeta_{j,r,i}^{(0)},\\omega_{j,r,i}^{(0)}=0,\\\\ \\gamma_{j,r,j}^{(t+1)}=\\gamma_{j,r,j}^{(t)}-\\mathbb{I}(r\\in\\mathcal{S}_{\\mathrm{signal}}^{j})\\frac{\\eta}{n}\\sum_{i=1}^{n}\\ell_{j,i}^{\\prime(t)}\\cdot\\sigma^{\\prime}\\left(\\left\\langle\\widetilde{\\mathbf{w}}_{j,r}^{(t)},\\boldsymbol{\\mu}_{y_{i}}\\right\\rangle\\right)\\|\\boldsymbol{\\mu}_{y_{i}}\\|_{2}^{2}\\mathbb{I}(y_{i}=j),\\\\ \\gamma_{j,r,k}^{(t+1)}=\\gamma_{j,r,k}^{(t)}-\\mathbb{I}((\\mathbf{m}_{j,r})_{k}=1)\\frac{\\eta}{n}\\sum_{i=1}^{n}\\ell_{j,i}^{\\prime(t)}\\cdot\\sigma^{\\prime}\\left(\\left\\langle\\widetilde{\\mathbf{w}}_{j,r}^{(t)},\\boldsymbol{\\mu}_{y_{i}}\\right\\rangle\\right)\\left\\|\\boldsymbol{\\mu}_{y_{i}}\\right\\|_{2}^{2}\\mathbb{I}(y_{i}=k),\\forall j\\neq k,\\\\ \\end{array}\n$$\n\n$$\n\\zeta_{j,r,i}^{(t+1)}=\\zeta_{j,r,i}^{(t)}-\\frac{\\eta}{n}\\cdot\\ell_{j,i}^{\\prime(t)}\\cdot\\sigma^{\\prime}\\left(\\left\\langle\\widetilde{\\mathbf{w}}_{j,r}^{(t)},\\boldsymbol{\\xi}_{i}\\right\\rangle\\right)\\left\\|\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}\\right\\|_{2}^{2}\\mathbb{I}(j=y_{i}),\n$$\n\n$$\n\\omega_{j,r,i}^{(t+1)}=\\omega_{j,r,i}^{(t)}-\\frac{\\eta}{n}\\cdot\\ell_{j,i}^{\\prime(t)}\\cdot\\sigma^{\\prime}\\left(\\left\\langle\\widetilde{\\mathbf{w}}_{j,r}^{(t)},\\boldsymbol{\\xi}_{i}\\right\\rangle\\right)\\left\\|\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}\\right\\|_{2}^{2}\\mathbb{I}(j\\neq y_{i}),\n$$\n\nwhere $\\{\\gamma_{j,r,j}^{(t)}\\}_{t=1}^{T},\\{\\zeta_{j,r,i}^{(t)}\\}_{t=1}^{T}$ are increasing sequences and $\\{\\gamma_{j,r,k}^{(t)}\\}_{t=1}^{T},\\{\\omega_{j,r,i}^{(t)}\\}_{t=1}^{T}$ are decreasing sequences, because $-\\ell_{j,i}^{\\prime(t)}\\geq 0$ when $j=y_{i},$ and $-\\ell_{j,i}^{\\prime(t)}\\leq 0$ when $j\\neq y_{i}$ . By Lemma 18, we have $p d > n + K$ , and hence the set of vectors $\\{\\pmb{\\mu}_{k}\\}_{k=1}^{K}\\cup\\{\\widetilde{\\pmb{\\xi}}_{i}\\}_{i=1}^{n}$ is linearly independent with probability measure 1 over the Gaussian distribution for each $j\\in [K],r\\in [m]$ . Therefore the decomposition is unique.\n\nWe first formally restate Theorem 3.\n\nTheorem 15 (Formal Restatement of Theorem 3) Under Condition 2, choose initialization variance $\\sigma_{0}=\\widetilde\\Theta(m^{-4}n^{-1}\\mu^{-1})$ and learning rate $\\eta\\le\\widetilde{\\cal O}(1/\\mu^{2})$ . For $\\epsilon > 0$ , if $p\\ge C_{1}\\frac{\\log d}{m}$ for some suficiently large constant $C_{1}$ , then with probability at least $1-O(d^{-1})$ over the randomness in the data, network initialization and pruning, there exists $T =$ $\\widetilde O(K\\eta^{-1}\\sigma_{0}^{2-q}\\mu^{-q}+K^{2}m^{4}\\mu^{-2}\\eta^{-1}\\epsilon^{-1})$ such that the following holds:\n\n1. The training loss is below ε: $L_{S}(\\widetilde{\\mathbf{W}}^{(T)})\\le\\epsilon .$  \n2. The weights of the CNN highly correlate with its corresponding class signal: ma $\\mathfrak{c}_{r}\\gamma_{j,r,j}^{(T)}\\geq$ $\\Omega(m^{-1/q})$ for all $j \\in [ K ]$  \n3. The weights of the CNN doesn’t have high correlation with the signal from diferent classes: ma $\\mathfrak{c}_{j\\neq k,r\\in [m]}|\\gamma_{j,r,k}^{(T)}|\\leq\\widetilde{O}(\\sigma_{0}\\mu)$  \n4. None of the weights is highly correlated with the noise: max $_{:j,r,i}\\zeta_{j,r,i}^{(T)}=\\widetilde{O}(\\sigma_{0}\\sigma_{n}\\sqrt{p d})$ 2 and max $_{j,r,i}|\\omega_{j,r,i}^{(T)}| =\\widetilde{O}(\\sigma_{0}\\sigma_{n}\\sqrt{p d})$\n\nMoreover, the testing loss is upper-bounded by\n\n$$\nL_{\\mathcal{D}}(\\widetilde{\\mathbf{W}}^{(T)})\\leq O(K\\epsilon)+\\exp(-n^{2}/p).\n$$\n\nThe proof of Theorem 3 consists of the analysis of the pruning on the signal and noise for three stages of gradient descent: initialization, feature growing phase, and converging phase, and the establishment of the generalization property. We present these analysis in detail in the following subsections. A special note is that the constant C showing up in the following proof of each subsequent Lemmas is defined locally instead of globally, which means the constant C within each Lemma is the same but may be diferent across diferent Lemma.\n\nLemma 19 Suppose $p=\\Omega(1/\\mathrm{poly}\\log d)$ , and m, $n={\\mathrm{poly}}\\log d .$ . With probability at least $1-1/d,$ we have\n\n$$\n\\begin{array}{l}\\left\\|\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}\\right\\|_{2}^{2}=\\Theta(\\sigma_{n}^{2}p d),\\\\ \\left|\\left\\langle\\widetilde{\\pmb{\\xi}}_{j,r,i},\\pmb{\\xi}_{i^{\\prime}}\\right\\rangle\\right|\\leq O(\\sigma_{n}^{2}\\sqrt{p d\\log d}),\\\\ \\left|\\left\\langle\\boldsymbol{\\mu}_{k},\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}\\right\\rangle\\right|\\leq|\\left\\langle\\boldsymbol{\\mu},\\boldsymbol{\\xi}_{i}\\right\\rangle|\\leq O(\\sigma_{n}\\mu\\sqrt{\\log d}),\\\\ \\end{array}\n$$\n\nfor all $j\\in\\{-1,1\\},\\ r\\in [m],\\ i,i^{\\prime}\\in [n]$ and $i\\neq i^{\\prime}$\n\nThis subsection establishes a few properties (summarized in Theorem 24) that will be used in the analysis of feature growing phase and converging phase of gradient descent presented in the next two subsections. Define $T^{\\star}=\\eta^{-1}\\mathrm{poly}(1/\\epsilon,\\mu,d^{-1},\\sigma_{n}^{-2},\\sigma_{0}^{-1}n,m,d)$ . Denote $\\alpha=\\Theta(\\log^{1/q}(T^{\\star})),~\\beta=2\\operatorname*{max}_{i,j,r,k}\\left\\{\\left|\\left.\\widetilde{\\mathbf{w}}_{j,r}^{(0)},\\pmb{\\mu}_{k}\\right.\\right|,\\left|\\left.\\widetilde{\\mathbf{w}}_{j,r}^{(0)},\\pmb{\\xi}_{i}\\right.\\right|\\right\\}$ . We need the following bound holds for our subsequent analysis.\n\n$$\n4 m^{1/q}\\max_{j,r,i}\\left\\{\\left\\langle\\widetilde{\\mathbf{w}}_{j,r}^{(0)},\\boldsymbol{\\mu}_{y_{i}}\\right\\rangle,C n\\alpha\\frac{\\mu\\sqrt{\\log d}}{\\sigma_{n}p d},\\left\\langle\\widetilde{\\mathbf{w}}_{j,r}^{(0)},\\boldsymbol{\\xi}_{i}\\right\\rangle,3 C n\\alpha\\sqrt{\\frac{\\log d}{p d}}\\right\\}\\leq 1 \\tag{1}\n$$\n\nProposition 24 Under Theorem 2, during the training time $t<T^{\\star}$ , we have\n\n1. $\\gamma_{j,r,j}^{(t)}, \\zeta_{j,r,i}^{(t)} \\le \\alpha$,  \n2. $\\omega_{j,r,i}^{(t)} \\ge -\\beta - 6Cn\\alpha\\sqrt{\\frac{\\log d}{pd}}$.  \n3. $\\gamma_{j,r,k}^{(t)} \\ge -\\beta - 2Cn\\alpha\\frac{\\mu\\sqrt{\\log d}}{\\sigma_n pd}$.\n\nNotice that the lower bound has absolute value smaller than the upper bound.\n\nLemma 28 Under Theorem $\\mathrm{{}}^{\\mathcal{Q}},f o r t\\leq T$ , we have that\n\n$$\n\\begin{array}{l}1.\\omega_{j,r,i}^{(t)}\\geq-\\beta-6 C n\\alpha\\sqrt{\\frac{\\log d}{p d}};\\\\ \\gamma_{j,r,k}^{(t)}\\geq-\\beta-2 C n\\alpha\\frac{\\mu\\sqrt{\\log d}}{\\sigma_{n}p d}.\\\\ \\end{array}\n$$\n\nLemma 29 Under Theorem ${\\mathcal{Q}},$ for $t \\leq T$ , we have $\\gamma_{j,r,j}^{(t)},\\zeta_{j,r,i}^{(t)}\\leq\\alpha$\n\nLemma 31 (Formal Restatement of Theorem 5) Under the same assumption as $T h e -$ orem 15, there exists time $\\begin{array}{ r }{T_{1}=\\frac{\\log 2 m^{-1/q}}{\\log(1+\\Theta(\\frac{\\eta}{K})\\mu^{q}\\sigma_{0}^{q-2})}=O(K\\eta^{-1}\\sigma_{0}^{2-q}\\mu^{-q}\\log 2 m^{-1/q})}\\end{array}$ such that\n\n1. max<sub>r</sub> $\\cdot\\gamma_{j,r,j}^{(T_{1})}\\geq m^{-1/q}\\ f o r\\ j\\in [K]$  \n2. $|\\zeta_{j,r,i}^{(t)}|,|\\omega_{j,r,i}^{(t)}|\\le{\\cal O}(\\sigma_{0}\\sigma_{n}\\sqrt{p d})$ for all j ∈ [K], r ∈ [m], i ∈ [n] and t ≤ T<sub>1</sub>.  \n3. $|\\gamma_{j,r,k}^{(t)}|\\leq O(\\sigma_{0}\\mu\\operatorname{poly}\\log d)\\ f o r\\ a l l\\ j,k\\in [K],\\ j\\neq k,\\ r\\in [m]\\ a n d\\ t\\leq T_{1}.$\n\nIn this subsection, we show that gradient descent can drive the training loss toward zero while the signal in the feature is still large. An important intermediate step in our argument is the development of the following gradient upper bound for multi-class cross-entropy loss.\n\nIn this phase, we are going to show that\n\n• max<sub>r</sub> $\\gamma_{j,r,j}^{(t)}\\geq m^{1/q}$ for all $j \\in [ K ]$  \n• ma $\\mathrm{x}_{j\\neq k,r\\in [m]}|\\gamma_{j,r,k}^{(t)}|\\leq\\beta_{1}$ where $\\beta_{1}=\\widetilde{O}(\\sigma_{0}\\mu)$  \n• max $\\cdot_{j,r,i}\\{\\zeta_{j,r,i}^{(t)},|\\omega_{j,r,i}^{(t)}|\\}\\le\\beta_{2}$ where $\\beta_{2}=O(\\sigma_{0}\\sigma_{n}\\sqrt{p d})$\n\nDefine $\\mathbf{W}^{\\star}$ as follows:\n\n$$\n\\mathbf{w}_{j,r}^{\\star}=\\mathbf{w}_{j,r}^{(0)}+\\Theta(m\\log(1/\\epsilon))\\frac{\\boldsymbol{\\mu}_{j}}{\\mu^{2}}.\n$$\n\nLemma 33 (Gradient upper bound, formal version of Theorem 6) Under Theorem 2, $f o r t\\leq T^{\\star}$ , there exists constant $C=O(K m^{2/q}$ max $\\{\\mu^{2},\\sigma_{n}^{2}p d\\})$ ) such that\n\n$$\n\\left\\|\\nabla L_{S}(\\widetilde{\\mathbf{W}}^{(t)})\\odot\\mathbf{M}\\right\\|_{F}^{2}\\leq C L_{S}(\\widetilde{\\mathbf{W}}^{(t)}).\n$$\n\nLemma 34 For $T_{1}\\leq t\\leq T^{\\star}$ , we have for all $j\\neq y_{i}$\n\n$$\n\\left\\langle\\nabla F_{y_{i}}(\\widetilde{\\mathbf{W}}_{y_{i}}^{(t)},\\mathbf{x}_{i}),\\widetilde{\\mathbf{W}}_{y_{i}}^{\\star}\\right\\rangle-\\left\\langle\\nabla F_{j}(\\widetilde{\\mathbf{W}}_{j}^{(t)},\\mathbf{x}_{i}),\\widetilde{\\mathbf{W}}_{j}^{\\star}\\right\\rangle\\geq q\\log\\frac{2 q K}{\\epsilon}.\n$$\n\nLemma 37 Under the same assumption as Theorem 15, we have\n\n$$\n\\left\\|\\widetilde{\\mathbf{W}}^{(t)}-\\widetilde{\\mathbf{W}}^{\\star}\\right\\|_{F}^{2}-\\left\\|\\widetilde{\\mathbf{W}}^{(t+1)}-\\widetilde{\\mathbf{W}}^{\\star}\\right\\|_{F}^{2}\\geq 5\\eta L_{S}(\\widetilde{\\mathbf{W}}^{(t)})-\\eta\\epsilon .\n$$\n\nIn this section, we show that there exists a relatively large pruning fraction (i.e., small p) such that while gradient descent is still able to drive the training error toward zero, the learned model yields poor generalization. We first provide a formal restatement of Theorem 9.\n\nTheorem 41 (Formal Version of Theorem 9) Under Condition 2, choose initialization variance $\\sigma_{0}=\\widetilde\\Theta(m^{-4}n^{-1}\\mu^{-1})$ and learning rate $\\eta\\overset{\\cdot}{\\le}\\widetilde{O}(1/\\mu^{2})$ . For $\\begin{array}{ r }{\\epsilon>0,\\ i f p=\\Theta\\big(\\frac{1}{K m\\log d}\\big)}\\end{array}$ then with probability at least $1-1/\\log(d)$ , there exists $T={\\cal O}(\\eta^{-1}n\\sigma_{0}^{q-2}\\sigma_{n}^{-q}(p d)^{-q/2}+$ $\\eta^{-1}\\epsilon^{-1}m^{4}n\\sigma_{n}^{-2}(p d)^{-1})$ such that the following holds:\n\n1. The training loss is below ε: $L_{S}(\\widetilde{\\mathbf{W}}^{(T)})\\le\\epsilon .$  \n2. The model weight doesn’t learn any of its corresponding signal at all: $\\gamma_{j,r,j}^{(t)}=0$ for all $j\\in [K],\\ r\\in [m]$  \n3. The model weights is highly correlated with the noise: $\\begin{array}{ r }{\\operatorname*{max}_{r\\in [m]}\\zeta_{j,r,i}^{(T)}\\geq\\Omega(m^{-1/q})}\\end{array}$ if $y_{i}=j$\n\nMoreover, the testing loss is large:\n\n$$\nL_{\\mathcal{D}}(\\widetilde{\\mathbf{W}}^{(T)})\\geq\\Omega(\\log K).\n$$\n\nThe proof of Theorem 9 consists of the analysis of the over-pruning for three stages of gradient descent: initialization, feature growing phase, and converging phase, and the establishment of the generalization property. We present these analysis in detail in the following subsections.\n\nLemma 43 (Formal Restatement of Theorem 12) Under the same assumption as $T h e \\textmd{ - }$ orem $\\not 4 .$ , there exists $T_{1}<T^{\\star}$ such that $T_{1}=O(\\eta^{-1}n\\sigma_{0}^{q-2}\\sigma_{n}^{-q}(p d)^{-q/2})$ and we have\n\n• max<sub>r</sub> $\\cdot\\zeta_{y_{i},r,i}\\geq m^{-1/q}$ for all $i \\in [ n ]$  \n• max $\\mathfrak{c}_{j,r,i}|\\omega_{j,r,i}^{(t)}| =\\widetilde O(\\sigma_{0}\\sigma_{n}\\sqrt{p d})$  \n• max $\\mathsf{\\tilde{\\rho}}_{j,r,k}|\\gamma_{j,r,k}^{(t)}|\\le\\widetilde O(\\sigma_{0}\\mu)$\n\nFrom the first stage we know that\n\n$$\n\\widetilde{\\mathbf{w}}_{j,r}^{(T_{1})}=\\widetilde{\\mathbf{w}}_{j,r}^{(0)}+ +\\sum_{k\\neq j}\\gamma_{j,r,k}^{(t)}\\frac{\\boldsymbol{\\mu}_{k}\\odot\\mathbf{m}_{j,r}}{\\mu^{2}}+\\sum_{i=1}^{n}\\zeta_{j,r,i}^{(T_{1})}\\frac{\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}}{\\left\\|\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}\\right\\|_{2}^{2}}+\\sum_{i=1}^{n}\\omega_{j,r,i}^{(T_{1})}\\frac{\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}}{\\left\\|\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}\\right\\|_{2}^{2}}.\n$$\n\nNow we define $\\widetilde{\\mathbf{W}}^{\\star}$ as follows:\n\n$$\n\\widetilde{\\mathbf{w}}_{j,r}^{\\star}=\\widetilde{\\mathbf{w}}_{j,r}^{(0)}+\\Theta(m\\log(1/\\epsilon))\\left[\\sum_{i=1}^{n}\\mathbb{I}(j=y_{i})\\frac{\\widetilde{\\pmb{\\xi}}_{j,r,i}}{\\left\\|\\widetilde{\\pmb{\\xi}}_{j,r,i}\\right\\|_{2}^{2}}\\right].\n$$\n\nLemma 44 Based on the result from feature growing phase,\n\n$$\n\\left\\|\\widetilde{\\mathbf{W}}^{(T_{1})}-\\widetilde{\\mathbf{W}}^{\\star}\\right\\|_{F}\\leq O(m^{2}n^{1/2}\\log(1/\\epsilon)\\sigma_{n}^{-1}(p d)^{-1/2}).\n$$", "theorems": [{"local_id": "T1", "statement": "Lemma 42 When m = poly log d and $\\begin{array}{ r }{p=\\Theta(\\frac{1}{K m\\log d})}\\end{array}$ , with probability $1-O(1/\\log d)$ , for all class $j \\in [ K ]$ we have $|S_{\\mathrm{signal}}^{j}| =0$", "proof": {"steps": [{"n": 1, "claim": "Proof First, the probability that a given class j receives no signal is $(1-p)^{m}$ . We use the inequality that\n\n$$\n1 + t \\geq \\exp \\left\\{O (t) \\right\\} \\quad \\forall t \\in (- 1 / 4, 1 / 4).\n$$"}, {"n": 2, "claim": "Then the probability that $|S_{\\mathrm{signal}}^{j}| =0,\\ \\forall j\\in [K]$ is given by\n\n$$\n(1-p)^{K m}\\geq\\exp\\left\\{-O(p K m)\\right\\}\\geq 1-O\\left(\\frac{1}{\\log d}\\right).\n$$"}]}}, {"local_id": "T2", "statement": "Lemma 48 Under the same assumption as Theorem $\\mathit{41,}$ we have\n\n$$\n\\left\\|\\mathbf{W}^{(t)}-\\mathbf{W}^{\\star}\\right\\|_{F}^{2}-\\left\\|\\mathbf{W}^{(t+1)}-\\mathbf{W}^{\\star}\\right\\|_{F}^{2}\\geq C\\eta L_{S}(\\widetilde{\\mathbf{W}}^{(t)})-\\eta\\epsilon .\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof To simplify our notation, we define $\\widehat{F}_{j}^{(t)}(\\mathbf{x}_{i})=\\left.\\nabla F_{j}(\\widetilde{\\mathbf{W}}_{j}^{(t)},\\mathbf{x}_{i}),\\widetilde{\\mathbf{W}}_{j}^{\\star}\\right.$ . The proof is exactly the same as the proof of Lemma 37.\n\n$$\n\\left\\|\\widetilde{\\mathbf{W}}^{(t)}-\\widetilde{\\mathbf{W}}^{\\star}\\right\\|_{F}^{2}-\\left\\|\\widetilde{\\mathbf{W}}^{(t+1)}-\\widetilde{\\mathbf{W}}^{\\star}\\right\\|_{F}^{2}\n$$\n\n$$\n\\begin{array}{l}=2\\eta\\left\\langle\\nabla L_{S}(\\widetilde{\\mathbf{W}}^{(t)})\\odot\\mathbf{M},\\widetilde{\\mathbf{W}}^{(t)}-\\widetilde{\\mathbf{W}}^{\\star}\\right\\rangle-\\eta^{2}\\left\\|\\nabla L_{S}(\\widetilde{\\mathbf{W}}^{(t)})\\odot\\mathbf{M}\\right\\|_{F}^{2}\\\\ =\\frac{2\\eta}{n}\\sum_{i=1}^{n}\\sum_{j=1}^{K}\\ell_{j,i}^{\\prime(t)}\\left[q F_{j}(\\widetilde{\\mathbf{W}}_{j}^{(t)};\\mathbf{x}_{i},y_{i})-\\left\\langle\\nabla F_{j}(\\widetilde{\\mathbf{W}}_{j}^{(t)},\\mathbf{x}_{i}),\\widetilde{\\mathbf{W}}_{j}^{\\star}\\right\\rangle\\right]-\\eta^{2}\\left\\|\\nabla L_{S}(\\widetilde{\\mathbf{W}}^{(t)})\\odot\\mathbf{M}\\right\\|_{F}^{2}\\\\ \\geq\\frac{2 q\\eta}{n}\\sum_{i=1}^{n}\\left[\\log(1+\\sum_{j\\neq y_{i}}e^{F_{j}-F_{y_{i}}})-\\log(1+\\sum_{j\\neq y_{i}}e^{(\\hat{F}_{j}-\\hat{F}_{y_{i}})/q})\\right]-\\eta^{2}\\left\\|\\nabla L_{S}(\\widetilde{\\mathbf{W}}^{(t)})\\odot\\mathbf{M}\\right\\|_{F}^{2}\\\\ \\geq\\frac{2 q\\eta}{n}\\sum_{i=1}^{n}\\left[\\ell\\left(\\widetilde{\\mathbf{W}}^{(t)};\\mathbf{x}_{i},y_{i}\\right)-\\log\\left(1+K e^{-\\log(2 q K/\\epsilon)}\\right)\\right]-\\eta^{2}\\left\\|\\nabla L_{S}\\left(\\widetilde{\\mathbf{W}}^{(t)}\\right)\\odot\\mathbf{M}\\right\\|_{F}^{2}\\\\ \\geq\\frac{2 q\\eta}{n}\\sum_{i=1}^{n}\\left[\\ell(\\widetilde{\\mathbf{W}}^{(t)};\\mathbf{x}_{i},y_{i})-\\frac{\\epsilon}{2 q}\\right]-\\eta^{2}\\left\\|\\nabla L_{S}(\\widetilde{\\mathbf{W}}^{(t)})\\odot\\mathbf{M}\\right\\|_{F}^{2}\\\\ \\geq C\\eta L_{S}(\\widetilde{\\mathbf{W}}^{(t)})-\\eta\\epsilon,\\\\ \\end{array}\n$$\n\nwhere the first inequality follows from the convexity of the cross-entropy loss with softmax, the second inequality follows from Lemma 34, the third inequality follows because $\\log(1+x)\\leq x$ , and the last inequality follows from Lemma 33 for some constant $C>0 .\\ \\mathsf{I}$"}]}}, {"local_id": "T3", "statement": "Lemma 49 (Formal Restatement of Theorem 13) Under the same assumption as $T h e \\textmd{ - }$ orem $\\mathit{41,}$ choose $T_{2}=T_{1}+\\left\\lceil\\frac{\\left\\|\\widetilde{\\mathbf{W}}^{(T_{1})}-\\widetilde{\\mathbf{W}}^{\\star}\\right\\|_{F}^{2}}{2\\eta\\epsilon}\\right\\rceil=T_{1}+\\widetilde{O}(\\eta^{-1}\\epsilon^{-1}m^{4}n\\sigma_{n}^{-2}(p d)^{-1})$ . Then for any time t during this stage we have max $_{j,r}\\left|\\omega_{j,r,i}^{(t)}\\right| =O(\\sigma_{0}\\sqrt{p d})$ and\n\n$$\n\\frac{1}{t-T_{1}}\\sum_{s=T_{1}}^{t}L_{S}(\\widetilde{\\mathbf{W}}^{(s)})\\leq\\frac{\\left\\|\\widetilde{\\mathbf{W}}^{(T_{1})}-\\widetilde{\\mathbf{W}}^{\\star}\\right\\|_{F}^{2}}{C\\eta(t-T_{1})}+\\frac{\\epsilon}{C}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof We have\n\n$$\n\\left\\|\\widetilde{\\mathbf{W}}^{(s)}-\\widetilde{\\mathbf{W}}^{\\star}\\right\\|_{F}^{2}-\\left\\|\\widetilde{\\mathbf{W}}^{(s+1)}-\\widetilde{\\mathbf{W}}^{\\star}\\right\\|_{F}^{2}\\geq C\\eta L_{S}(\\widetilde{\\mathbf{W}}^{(s)})-\\eta\\epsilon .\n$$"}, {"n": 2, "claim": "Taking a telescopic sum from $T_{1}$ to t yields\n\n$$\n\\sum_{s=T_{1}}^{t}L_{S}(\\widetilde{\\mathbf{W}}^{(s)})\\leq\\frac{\\left\\|\\widetilde{\\mathbf{W}}^{(T_{1})}-\\widetilde{\\mathbf{W}}^{\\star}\\right\\|_{F}^{2}+\\eta\\epsilon(t-T_{1}+1)}{C\\eta}.\n$$"}, {"n": 3, "claim": "Combining Lemma 44, we have\n\n$$\n\\sum_{s=T_{1}}^{t}L_{S}(\\widetilde{\\mathbf{W}}^{(s)})\\leq O(\\eta^{-1}\\left\\|\\widetilde{\\mathbf{W}}^{(T_{1})}-\\widetilde{\\mathbf{W}}^{\\star}\\right\\|_{F}^{2})=\\widetilde{O}(\\eta^{-1}m^{4}n\\sigma_{n}^{-2}(p d)^{-1}).\n$$"}]}}, {"local_id": "T4", "statement": "Theorem 50 (Formal Version of the Generalization Result of Theorem 9) Under the same assumption as Theorem $\\not 4.1\\ :$ , within $O(\\eta^{-1}n\\sigma_{0}^{q-2}\\sigma_{n}^{-q}(p d)^{-q/2}+\\eta^{-1}\\epsilon^{-1}m^{4}n\\sigma_{n}^{-2}(p d)^{-1})$ iterations, we can find $\\widetilde{{\\mathbf W}}^{(T)}$ such that $L_{S}(\\widetilde{\\mathbf{W}}^{(T)})\\le\\epsilon_{:}$ , and $L_{\\mathcal{D}}(\\widetilde{\\mathbf{W}}^{(t)})\\geq\\Omega(\\log K)$", "proof": {"steps": [{"n": 1, "claim": "Proof First of all, from Theorem 49 we know there exists $t\\in [T_{1},T_{2}]$ such that $L_{S}(\\widetilde{\\mathbf{W}}^{(T)})\\le\\epsilon$ Then, we can bound\n\n$$\n\\begin{array}{l}\\left\\|\\widetilde{\\mathbf{w}}_{j,r}^{(t)}\\right\\|_{2}=\\left\\|\\widetilde{\\mathbf{w}}_{j,r}^{(0)}+\\sum_{k\\neq j}\\gamma_{j,r,k}^{(t)}\\frac{\\boldsymbol{\\mu}_{k}}{\\mu^{2}}+\\sum_{i=1}^{n}\\zeta_{j,r,i}^{(t)}\\frac{\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}}{\\left\\|\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}\\right\\|_{2}^{2}}+\\sum_{i=1}^{n}\\omega_{j,r,i}^{(t)}\\frac{\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}}{\\left\\|\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}\\right\\|_{2}^{2}}\\right\\|_{2}\\\\ \\leq\\left\\|\\widetilde{\\mathbf{w}}_{j,r}^{(0)}\\right\\|_{2}+\\sum_{k\\neq j}|\\gamma_{j,r,k}^{(t)}|\\frac{1}{\\mu}+\\sum_{i=1}^{n}\\zeta_{j,r,i}^{(t)}\\frac{1}{\\left\\|\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}\\right\\|_{2}}+\\sum_{i=1}^{n}|\\omega_{j,r,i}^{(t)}|\\frac{1}{\\left\\|\\widetilde{\\boldsymbol{\\xi}}_{j,r,i}\\right\\|_{2}}\\\\ \\leq O(\\sigma_{0}\\sqrt{d})+\\widetilde{O}(n\\sigma_{n}^{-1}(p d)^{-1/2}).\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "Consider a new example $\\left(\\mathbf{x},y\\right)$ . Taking a union bound over r, with probability at least $1-d^{-1}$ , we have\n\n$$\n\\left|\\left\\langle\\mathbf{w}_{y,r}^{(t)},\\pmb{\\xi}\\right\\rangle\\right| =\\widetilde{O}(\\sigma_{0}\\sigma_{n}\\sqrt{d}+n(p d)^{-1/2}),\n$$\n\nfor all $r \\in [ m ]$ . Then,\n\n$$\n\\begin{array}{l}F_{y}(\\mathbf{x})=\\sum_{r=1}^{m}\\sigma\\left(\\left\\langle\\widetilde{\\mathbf{w}}_{j,r}^{(t)},\\boldsymbol{\\mu}_{y}\\right\\rangle\\right)+\\sigma\\left(\\left\\langle\\widetilde{\\mathbf{w}}_{j,r}^{(t)},\\boldsymbol{\\xi}\\right\\rangle\\right)\\\\ \\leq m\\max_{r}\\left|\\left\\langle\\mathbf{w}_{y,r}^{(t)},\\boldsymbol{\\xi}\\right\\rangle\\right|^{q}\\\\ \\leq m\\widetilde{O}(\\sigma_{0}^{q}\\sigma_{n}^{q}d^{q/2}+n^{q}(p d)^{-q/2})\\\\ \\leq 1,\\\\ \\end{array}\n$$\n\nwhere the last inequality follows because $\\sigma_{0}\\le\\widetilde{{\\cal O}}(m^{-1/q}\\mu^{-1})$ and $d\\geq\\widetilde\\Omega(m^{2/q}n^{2})$ . Thus, with probability at least $1 - 1 / d .$\n\n$$\n\\ell(F(\\widetilde{\\mathbf{W}}^{(t)};\\mathbf{x}))\\geq\\log(1+(K-1)e^{-1}).\n$$"}]}}], "target_local_id": null, "topic_domain": "Deep Learning Theory", "topic_subtopic": "Training Dynamics & Implicit Bias", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2025-23-1519", "context": "We begin by introducing some fundamental notations that will frequently appear. Suppose that the data $D_{n}: =\\{X_{1},\\ldots,X_{n}\\}$ are independent and identically distributed (i.i.d.) and drawn from an unknown distribution P that is absolutely continuous with respect to the Lebesgue measure $\\mu$ and admits a unique density function $f .$ . In this paper, we assume that $f$ is supported on $[0,1]^{d}$ , which we denote as X . Recall that for $1 \\leq p < \\infty$ and a vector $x\\in\\mathbb{R}^{d}$ , the $\\ell_{p}{\\mathrm{-norm}}$ is defined as $\\|x\\|_{p}: =(x_{1}^{p}+\\cdot\\cdot\\cdot+x_{d}^{p})^{1/p}$ , and the $\\ell_{\\infty}{\\mathrm{-norm}}$ is defined as $\\|x\\|_{\\infty}: =\\operatorname*{max}_{i=1,\\ldots,d}|x_{i}|$ . For a measurable function $g:\\mathcal{X}\\mathbb{R}$ , we define the $L_{p}{\\mathrm{-norm}}$ as $\\begin{array}{ r }{\\|g\\|_{p}: =\\big(\\int_{\\mathcal X}|g(x)|^{p}dx\\big)^{1/p}}\\end{array}$ . Let $B(x,r): =\\{x^{\\prime}\\in\\mathbb{R}^{d}:\\|x^{\\prime}-x\\|_{2}\\leq r\\}$ denote a ball in Euclidean space $\\mathbb{R}^{d}$ centered at $x\\in\\mathbb{R}^{d}$ with radius $r\\in(0,+\\infty)$ . We use $V_{d}$ to denote the volume of the d-dimensional closed unit ball. In addition, for $n\\in\\mathbb{N}_{+}$ , we write $[n]: =\\{1,\\dots,n\\}$ as the set containing integers from 1 to n and $\\mathcal{W}_{n}: =\\{(w_{1},\\ldots,w_{n})\\in\\mathbb{R}^{n}$ $\\begin{array}{ r }{\\sum_{i=1}^{n}w_{i}=1,~w_{i}\\ge 0,~i\\in [n]\\}}\\end{array}$ . For any $x\\in\\mathbb{R}$ , let bxc be the largest integer less than or equal to x and dxe be the smallest ingeter larger than or equal to x.\n\nThroughout this paper, we use $a\\vee b=\\operatorname*{max}\\{a,b\\}$ and $a\\wedge b=\\operatorname*{min}\\{a,b\\}$ . Moreover, we use the following notations to compare the magnitudes of quantities: $a_{n}\\lesssim b_{n}$ or $a_{n}={\\mathcal{O}}(b_{n})$ indicates that there exists a positive constant $c > 0$ that is independent of n such that $a_{n}\\leq cb_{n};a_{n}\\gtrsim b_{n}$ implies that there exists a positive constant $c > 0$ such that $a_{n}\\geq cb_{n};$ and $a_{n}\\asymp b_{n}$ means that $a_{n}\\lesssim b_{n}$ and $b_{n}\\lesssim a_{n}$ hold simultaneously. In this paper, we focus on the case of a fixed dimension, allowing the constant c to depend on the dimension $d .$ Finally, we use $C,C^{\\prime},c,$ and $c^{\\prime}$ to represent positive constants.\n\nAssumption 1 The data $D_{n}$ consists ofi.i.d. samples drawn from a distribution P satisfying the contamination model:\n\n$$\n\\mathrm{P}=(1-\\Pi)\\cdot\\mathrm{P}_{0}+\\Pi\\cdot\\mathrm{P}_{1},\\tag{1}\n$$\n\nwhere $\\mathrm{P_{0}}$ and $\\mathrm{P_{1}}$ denote the distributions of normal and anomalies, respectively, and $\\Pi \\in$ (0, 1) is the contamination proportion. Additionally, we assume that $\\mathrm{P_{0}}$ has a probability density function $f_{0}$ , while $\\mathrm{P_{1}}$ is uniformly distributed over $[0,1]^{d}$ with density function $f_{1}$\n\nBuilding on this framework, we explore the distance-based method for unsupervised anomaly detection, motivated by the connection between distance functions and k-nearest neighbor (kNN) density estimations highlighted by Biau et al. (2011). This method leverages the distances between data points and their nearest neighbors to assess anomalies. For any $\\boldsymbol{x}\\in\\mathbb{R}^{d}$ and a dataset $D_{n}.$ , we denote $X_{(k)}(x;D_{n})$ as the k-th nearest neighbor of x in $D_{n}$ We then define $R_{n,(k)}(x): =\\left\\|x-X_{(k)}(x;D_{n})\\right\\|_{2}$ as the distance between x and $X_{(k)}(x;D_{n})$ referred to as the k-nearest neighbor distance, or k-distance of x in $D_{n}$\n\nDistance-based methods are important and efective for anomaly detection. However, when handling large datasets, the number of nearest neighbors that need to be searched grows substantially, leading to significant computational overhead. To mitigate this issue, we incorporate the bagging technique by averaging the weighted k-distances computed on multiple disjoint sub-datasets randomly drawn from the original dataset $D_{n}$ without replacement. Let B be the number of bagging rounds pre-specified by the user, and $\\{D_{s}^{b}\\}_{b=1}^{B}$ be B disjoint subsets of $D_{n}$ , each of size s. Since the sub-samples are disjoint and the data $D_{n}$ is supposed i.i.d., this procedure is mathematically equivalent to taking the first s samples for bag 1, the following s samples for bag 2, etc. In each subset $D_{s}^{b},\\ b\\in [B]$ ， let $R_{s,(k)}^{b}(x): =\\left. x-X_{(k)}(x;D_{s}^{b})\\right._{2}$ be the k-distance of x in $D_{s}^{b}$ for any integer $k\\leq s,$ , and let the weighted k-distance be defined as $\\begin{array}{ r }{R_{s}^{w,b}(x): =\\sum_{i=1}^{s}w_{i}^{b}R_{s,(i)}^{b}(x)}\\end{array}$ , with $w^{b}\\in\\mathcal{W}_{s}$ . The bagged weighted k-distances are obtained by averaging these weighted k-distances across the B sub-datasets:\n\n$$\nR_{n}^{B}(x):=\\frac{1}{B}\\sum_{b=1}^{B}R_{s}^{w,b}(x).\n$$\n\nThen, we introduce the bagged weighted k-distances for density estimation (BWDDE) as\n\n$$\nf_{n}^{B}(x):=\\frac{1}{V_{d}R_{n}^{B}(x)^{d}}\\left(\\frac{1}{B}\\sum_{b=1}^{B}\\sum_{i=1}^{s}w_{i}^{b}\\gamma_{s,i}\\right)^{d},\\tag{2}\n$$\n\nwhere $V_{d}={\\pi}^{d/2}/\\Gamma(d/2+1)$ is the volume of the unit ball and\n\n$$\n\\gamma_{s,i}:=\\frac{\\Gamma(i+1/d)\\Gamma(s+1)}{\\Gamma(i)\\Gamma(s+1+1/d)}.\\tag{3}\n$$\n\nSince the underlying density function f in (4) is unknown, standard optimization techniques for parameter selection cannot be directly applied to weight selection in density estimation. To address this, we seek a surrogate for the empirical risk in (4) and minimize it to determine the nearest neighbor weights. To proceed, we introduce the following regularity assumptions on the underlying probability distribution P.\n\nAssumption 2 Assume that P has a Lebesgue density f with support $\\mathcal{X}=[0,1]^{d}$\n\n(i) [Lipschitz Continuity] The density f is Lipschitz continuous on $[0,1]^{d}$ , i.e., for all x, $,y\\in [0,1]^{d}$ , there exists a constant $c_{L}>0$ such that $|f(x)-f(y)|\\leq c_{L}\\|x-y\\|_{2}$\n\n(ii) [Boundness] There exist constants $\\overline{{c}}\\geq\\underline{{c}}>0$ such that $\\underline{{c}}\\le f(x)\\le\\overline{{c}}$ for all $x\\in\\mathcal{X}$\n\nUnder these assumptions, along with additional conditions on the weights, the next proposition presents a surrogate for the empirical risk (4).\n\nProposition 1 (Surrogate Risk) Let Assumption 2 hold, and let L denote the absolute value loss. Let $\\{D_{s}^{\\bar{b}}\\}_{b=1}^{B}$ be B disjoint subsets randomly drawn from $D_{n},$ with $D_{s}^{b}= $ $\\{X_{1}^{b},\\ldots,X_{s}^{b}\\}$ , and define $\\begin{array}{ r }{\\overline{{R}}_{s,(i)}^{b}: =\\sum_{j=1}^{s}R_{s,(i)}^{b}(X_{j}^{b})/s}\\end{array}$ as the average i-distances for any integer $i \\leq s$ on the subset $D_{s}^{b}$ . Furthermore, let f be the true density function and $f_{n}^{B}$ be the BWDDE as in (2). Moreover, let $k^{b}: =k(w^{b}): =\\operatorname*{sup}\\{i\\in [s]:w_{i}^{b}\\neq 0\\},\\ :\\underline{{k}}: =\\operatorname*{min}_{b\\in [B]}k^{b}$ and $\\overline{{k}}: =\\operatorname*{max}_{b\\in [B]}k^{b}$ . Finally, suppose that the following four conditions hold:\n\n(i) There exists a sequence $c_{n}\\asymp$ log n such that $\\sum_{i=1}^{cn}w_{i}^{b}\\lesssim(\\log n)/k^{b}$ for all $b \\in \\left[ B \\right]$  \n(ii) $\\begin{array}{ r }{\\underline{{k}}\\gtrsim(\\log n)^{2},\\underline{{k}}\\asymp\\overline{{k}},B\\geq 2(d^{2}+4)(\\log n)/3,B\\lesssim(\\overline{{k}}/(\\log n))^{1+2/d}}\\end{array}$ , log $s \\asymp \\log n$ , and $s\\geq\\operatorname*{max}\\{c_{1}^{\\prime},2\\overline{{k}}\\}$ , where $c_{1}^{\\prime}$ is a constant defined in Lemma ${\\mathit{12}};$  \n(iii) $\\|w^b\\|_2 \\gtrsim (k^b)^{-1/2}$ and $\\sum_{i=1}^s i^{1/d} w_i^b \\asymp (k^b)^{1/d}$ for $b \\in [B]$;  \n(iv) There exist constants $C_{n,i}$ such that $\\max_{b\\in[B]} w_i^b \\lesssim C_{n,i}$ for $i\\in[s]$ and $\\sum_{i=1}^s i^{1/d-1/2} C_{n,i} \\lesssim \\overline{k}^{1/d-1/2}$.\n\nThen, there exists $N_{1}^{*}\\in\\mathbb{N}$ , specified in the proof, such that for all $n\\geq N_{1}^{*}$ and $X_{i}^{b}$ satisfying $B(X_{i}^{b},R_{s,(k^{b^{\\prime}})}^{b^{\\prime}}(X_{i}^{b}))\\stackrel{^{\\prime}}{\\subset}[0,1]^{d}$ for all $b^{\\prime}\\in [B]$ , there holds\n\n$$\nL(X_{i}^{b},f_{n}^{B})\\lesssim\\sqrt{(\\log s)/B}\\cdot\\|w^{b}\\|_{2}+R_{s}^{w,b}(X_{i}^{b}),\\qquad i\\in [s],b\\in [B],\\tag{5}\n$$\n\nwith probability $\\mathrm{P}^{Bs}$ at least $1-4/n^{2}$ . Furthermore, we have\n\n$$\n\\begin{array}{l}\\mathcal{R}_{L,D_{s}^{B}}(f_{n}^{B})\\lesssim\\mathcal{R}_{L,D_{s}^{B}}^{\\text{sur}}(f_{n}^{B}):=\\frac{1}{B}\\sum_{b=1}^{B}\\left(\\sqrt{(\\log s)/B}\\cdot\\|w^{b}\\|_{2}+\\frac{1}{s}\\sum_{i=1}^{s}R_{s}^{w,b}(X_{i}^{b})\\right)\\\\ =\\frac{1}{B}\\sum_{b=1}^{B}\\left(\\sqrt{(\\log s)/B}\\cdot\\|w^{b}\\|_{2}+\\sum_{i=1}^{s}w_{i}^{b}\\overline{{R}}_{s,(i)}^{b}\\right).\\tag{6}\\\\ \\end{array}\n$$\n\nSurrogate risk minimization (SRM). From the expression of the surrogate risk in (6), minimizing the surrogate risk is equivalent to solving the following optimization problems:\n\n$$\nw^{b,*}:=\\underset{w^{b}\\in\\mathcal{W}_{s}}{\\arg\\min}\\sqrt{(\\log s)/B}\\cdot\\|w^{b}\\|_{2}+\\sum_{i=1}^{s}w_{i}^{b}\\overline{{R}}_{s,(i)}^{b},\\quad b\\in [B].\\tag{7}\n$$\n\nDensity estimation. The discussions above indicate that the minimization problem (7) provides a practical method for determining the weights of nearest neighbors in density estimation.\n\nHowever, these weights depend on the data, introducing statistical dependence between the weights $w_{i}^{b,*}$ and the i-distance $R_{s,(i)}^{b}(x)$ for $i \\in [ s ]$ in each bag $b \\in [ B ]$ . Such dependence complicates the theoretical analysis of weighted k-distances and density estimation within the framework of statistical learning theory.\n\nTo address this issue, we employ distinct subsets of the data for computing the kdistances, ensuring their independence from the optimized weights. For simplicity, we assume that $n = 2 B s$ , which facilitates a more straightforward theoretical derivation without loss of generality. This assumption ensures that each subset is of equal size; if n is not exactly divisible by 2B, minor modifications to the partitioning scheme can be applied without afecting the theoretical conclusions. We randomly partition $D_{n}$ into B disjoint pairs of subsets $\\{(D_{s}^{b},\\widetilde{D}_{s}^{b})\\}_{b=1}^{B}$ , where each subset has size s and $D_{s}^{b}\\cap\\widetilde{D}_{s}^{b}=\\varnothing$ for each b. Let $w^{b,*}$ represent the weights obtained by solving the optimization problem (7) using $\\{D_{s}^{b}\\}_{b=1}^{B},$ and let $\\widetilde{R}_{s,(i)}^{b}(x)$ denote the i-distance of x computed using $\\{\\widetilde{D}_{s}^{b}\\}_{b=1}^{B}$ for $i \\in [ s ]$ . Then, we define the regularized k-distances as follows:\n\n$$\nR_{s}^{b,*}(x):=R_{s}^{w^{b,*},b}(x)=\\sum_{i=1}^{s}w_{i}^{b,*}\\widetilde{R}_{s,(i)}^{b}(x).\\tag{9}\n$$\n\nWe refer to the weighted average of these k-distances as the bagged regularized k-distance\n\n$$\nR_{n}^{B,*}(x):=\\frac{1}{B}\\sum_{b=1}^{B}R_{s}^{b,*}(x).\\tag{10}\n$$\n\nBy incorporating the $w^{b,*}$ and $R_{n}^{B,*}(x)$ into the BWDDE formula (2), we are able to obtain a new nearest-neighbor-based density estimator called bagged regularized k-distances for density estimation (BRDDE), expressed as\n\n$$\nf_{n}^{B,*}(x):=\\frac{1}{V_{d}R_{n}^{B,*}(x)^{d}}\\left(\\frac{1}{B}\\sum_{b=1}^{B}\\sum_{i=1}^{s}w_{i}^{b,*}\\gamma_{s,i}\\right)^{d}.\\tag{11}\n$$\n\n<div class=\"mineru-algorithm\" style=\"white-space: pre-wrap; font-family:monospace;\">\nAlgorithm 2: Bagged Regularized $k$-Distances for Anomaly Detection (BRDAD)\nInput: Data $D = \\{X_1, \\cdots, X_n\\}$; Number of anomalies $m$;\nBagging rounds $B$; Subsampling size $s$.\nRandomly partition $D_n$ into $B$ disjoint pairs of subsets, denoted as $\\{(D_s^b, \\widetilde{D}_s^b)\\}_{b=1}^B$, such that $D_s^b \\cap \\widetilde{D}_s^b = \\emptyset$ for each $b$.\nfor $b \\in [B]$ do\n    Compute weights $w^{b,*}$ using $D_s^b$ by (7);\n    Compute the regularized $k$-distances $R_s^{b,*}(X_i)$ for $1 \\leq i \\leq n$ using (9), based on $w^{b,*}$ and $\\widetilde{D}_s^b$.\nend\nCompute the bagged regularized $k$-distances $R_n^{B,*}(X_i)$ by (10) for $1 \\leq i \\leq n$.\nSort the data $D_n = \\{X_1, \\ldots, X_n\\}$ as $\\{X_1', \\ldots, X_n'\\}$ in descending order according to their bagged regularized $k$-distances, i.e., $R_n^{B,*}(X_1') \\geq \\cdots \\geq R_n^{B,*}(X_n')$.\nOutput: Anomalies $\\{X_i'\\}_{i=1}^m$.\n</div>\n\nAnomaly Detection. By applying BRDDE to all samples, anomalies can be identified as instances with lower BRDDE values. However, explicit density estimation is not necessary for anomaly detection. Since density estimates serve as anomaly scores, any monotone transformation preserves their ranking (possibly in reverse) and maintains the same family of level sets, leading to identical AUC values. Thus, bagged regularized k-distances provide a suficient and practical alternative for anomaly detection. They can be accurately computed in high-dimensional spaces, and their associated weights can be eficiently optimized from (7), making them an efective approach for density-based anomaly detection.\n\nWe now introduce our anomaly detection algorithm, bagged regularized k-distances for anomaly detection (BRDAD). The dataset $D_{n}$ is sorted into the sequence $\\{X_{1}^{\\prime},\\ldots,X_{n}^{\\prime}\\}$ based on their bagged regularized k-distances in descending order, i.e. $R_{n}^{B,*}(X_{1}^{\\prime})\\geq\\cdots\\geq$ $R_{n}^{B,*}(X_{n}^{\\prime})$ . Given the pre-specified number of anomalies $m,$ the first m instances $\\{X_{i}^{\\prime}\\}_{i=1}^{m},$ are considered as the m anomalies. The complete procedure of our BRDAD algorithm is presented in Algorithm 2. As illustrated above, SRM mitigates the challenge of hyperparameter selection in density estimation. Consequently, BRDAD retains the advantages of BRDDE by using the same weights to address the sensitivity of hyperparameter selection in nearest-neighbor-based methods for unsupervised anomaly detection.\n\nIn this section, we present theoretical results related to our BRDAD algorithm. We first investigate the Huber contamination model in Section 3.1, in which we can analyze the performance of the bagged regularized k-distances from a learning theory perspective. Then, we present the convergence rates of BRDDE and BRDAD in Section 3.2 and 3.3, respectively. Finally, we provide comments and discussions on our algorithms and theoretical results in Section 3.4. We also compare our theoretical findings on the convergences of both BRDDE and BRDAD with other nearest-neighbor-based methods in this section.\n\nIn the Huber contamination model (HCM) in Assumption 1, for every instance X from $\\mathrm{P}{\\mathrm{,}}$ we can use a latent variable $Y\\in\\{0,1\\}$ that indicates which distribution it is from. More specifically, $Y = 0$ and $Y = 1$ indicate that the instance is from the normal and the anomalous distribution, respectively. As a result, the anomaly detection problem can be converted into a bipartite ranking problem where instances are labeled positive or negative implicitly according to whether it is normal or not. Let $\\widetilde{\\mathrm{P}}$ represent the joint probability distribution of $\\mathcal{X}\\times\\mathcal{Y}$ In this case, our learning goal is to learn a score function that minimizes the probability of mis-ranking a pair of normal and anomalous instances, i.e. that maximizes the area under the ROC curve (AUC). Therefore, we can study regret bounds for the AUC of the bagged regularized k-distances to evaluate its performance from the learning theory perspective. Let $r:\\mathcal{X}\\mathbb{R}$ be a score function, then the AUC of r can be written as\n\n$$\n\\mathrm{AUC}(r)=\\mathbb{E}\\big [\\mathbf{1}\\{(Y-Y^{\\prime})(r(X)-r(X^{\\prime})>0)\\}+\\mathbf{1}\\{r(X)=r(X^{\\prime})\\}/2|Y\\neq Y^{\\prime}\\big],\n$$\n\nwhere $(X,Y),(X^{\\prime},Y^{\\prime})$ are assumed to be drawn i.i.d. from $\\widetilde{\\mathrm{P}}$ . In other words, the AUC of $r$ is the probability that a randomly drawn anomaly is ranked higher than a randomly drawn normal instance by the score function $r .$ . Given the HCM in (1) and the assumption that $\\mathrm{P_{1}}$ is uniformly distributed over $[0,1]^{d}$ in Assumption 1, the posterior probability function with respect to $\\widetilde{\\mathrm{P}}$ is given by\n\n$$\n\\eta(x):=\\widetilde{\\mathrm{P}}(Y=1|X=x)=\\frac{\\Pi f_{1}(x)}{(1-\\Pi)f_{0}(x)+\\Pi f_{1}(x)}=\\Pi f(x)^{-1}.\\tag{12}\n$$\n\nThen, the optimal AUC is defined as\n\n$$\n\\mathrm{AUC}^{*}:=\\sup_{r:\\mathcal{X}\\to\\mathbb{R}}\\mathrm{AUC}(r)=1-\\frac{1}{2\\Pi(1-\\Pi)}\\mathbb{E}_{X,X^{\\prime}}\\big [\\min\\big(\\eta(X)(1-\\eta(X^{\\prime})),\\eta(X^{\\prime})(1-\\eta(X))\\big)\\big].\n$$\n\nFinally, the AUC regret of a score function r is defined as\n\n$$\n\\mathrm{Reg}^{\\mathrm{AUC}}(r):=\\mathrm{AUC}^{*}-\\mathrm{AUC}(r).\n$$\n\nAs discussed in Section 2.2, BRDAD is a density-based anomaly detection method. To establish its convergence rates under the Huber contamination model, we first derive the theoretical convergence rates of BRDDE in (11), which are presented in the next subsection.\n\nIn this section, we first provide the error decomposition for the BWDDE $f_{n}^{B}(x)$ in (2). Then, we present the upper bounds for these error terms.\n\nLet the term (I) be defined as\n\n$$\n(I):=\\frac{1}{V_{d}R_{n}^{B}(x)^{d}}\\sum_{j=0}^{d-1}\\left(\\frac{1}{B}\\sum_{b=1}^{B}\\sum_{i=1}^{s}w_{i}^{b}\\gamma_{s,i}\\right)^{j}\\left(V_{d}^{1/d}f(x)^{1/d}R_{n}^{B}(x)\\right)^{d-1-j}.\\tag{14}\n$$\n\nIf the terms (II) and (III) are defined respectively as\n\n$$\n(II):=\\sum_{i=1}^{s}\\left|\\frac{1}{B}\\sum_{b=1}^{B}w_{i}^{b}\\left(\\gamma_{s,i}-\\mathrm{P}\\left(B\\left(x,R_{s,(i)}^{b}(x)\\right)\\right)^{1/d}\\right)\\right|,\\tag{17}\n$$\n\n$$\n(III):=\\sum_{i=1}^{s}\\left|\\frac{1}{B}\\sum_{b=1}^{B}w_{i}^{b}\\big(\\mathrm{P}(B(x,R_{s,(i)}^{b}(x)))^{1/d}-V_{d}^{1/d}f(x)^{1/d}R_{s,(i)}^{b}(x)\\big)\\right|,\\tag{18}\n$$\n\nthen applying the triangle inequality to (16) yields the error decomposition\n\n$$\n\\left|f_{n}^{B}(x)-f(x)\\right|\\leq(I)\\cdot(II)+(I)\\cdot(III).\\tag{19}\n$$\n\nWe emphasize that $\\gamma_{s,i}=\\mathbb{E}[\\mathrm{P}(B(x,R_{s,(i)}^{b}(x)))^{1/d}]$ for a fixed $x\\in\\mathcal{X},b\\in [B]$ , and $i \\in [ s ]$ Therefore, standard concentration inequalities can be applied to establish the convergence rates for term (II). The derivation of this equality is explained as follows: Since P has a density over $[0,1]^{d}$ with respect to the Lebesgue measure, the random variable $\\|X-x\\|_{2}$ is continuous for a fixed $x\\in\\mathcal{X}$ . By the probability integral transform, $\\operatorname{P}(B(x,\\|X-x\\|_{2}))$ is uniformly distributed over $[0,1]$ . For any $b \\in [ B ]$ , note that $X_{1}^{b},\\ldots,X_{s}^{b}$ are i.i.d. with the same distribution P. Let $U_{1}^{b},\\ldots,U_{s}^{b}$ be i.i.d. uniform [0, 1] random variables. Then\n\n$$\n\\bigl(\\mathrm{P}(x,\\|X_{1}^{b}-x\\|_{2}),\\dots,\\mathrm{P}(x,\\|X_{s}^{b}-x\\|_{2})\\bigr)\\stackrel{{\\mathcal{D}}}{{=}}\\bigl(U_{1}^{b},\\dots,U_{s}^{b}\\bigr).\n$$\n\nReordering the samples such that $\\|X_{(1)}^{b}(x)-x\\|_{2}\\leq\\cdot\\cdot\\cdot\\leq\\|X_{(s)}^{b}(x)-x\\|_{2}$ , we obtain\n\n$$\n\\left(\\mathrm{P}(B(x,R_{s,(1)}^{b}(x))),\\dots,\\mathrm{P}(B(x,R_{s,(s)}^{b}(x)))\\right)\\stackrel{{\\mathcal{D}}}{{=}}\\left(U_{(1)}^{b},\\dots,U_{(s)}^{b}\\right),\\tag{20}\n$$\n\nwhere $U_{(i)}^{b}$ is the i-th order statistic of $U_{1}^{b},\\ldots,U_{s}^{b}$ . This reduces the study of $\\mathrm{P}(B(x,R_{s,(i)}^{b}(x)))$ to the study of $U_{(i)}^{b}$ . By Corollary 1.2 in Biau and Devroye (2015), $U_{(i)}^{b}\\sim\\mathrm{Beta}(i,s+1-i)$ Hence, $\\mathbb{E}[\\mathrm{P}(B(x,R_{s,(i)}^{b}(x)))^{1/d}]=\\mathbb{E}[(U_{(i)}^{b})^{1/d}]=\\gamma_{s,i}.$\n\nThe following proposition provides the upper bounds for the error terms $(I),(II)$ , and (III), respectively.\n\nProposition 4 Let Assumption 2 hold. Furthermore, let $(I),(II)$ , and (III) be defined as in $(14),(17)$ , and (18), respectively. Let $k^{b},\\underline{{k}},$ and k be defined as in Proposition 1. Suppose that the conditions $(i)-(iv)$ in Proposition 1 hold. Then there exists $N_{1}\\in\\mathbb{N}$ , which will be specified in the proof, such that for all $n>N_{1}$ and x satisfying $B(x,R_{s,(k^{b})}^{b}(x))\\subset [0,1]^{d}$ for all $b \\in [ B ]$ , the following statements hold with probability $\\mathrm{P}^{Bs}$ at least $1-2/n^{2}$\n\n(I) $\\lesssim (\\overline{k}/s)^{-1/d}$;\n\n(II) $\\lesssim (\\overline{k}/s)^{1/d}((\\log n)/(\\overline{k}B))^{1/2}$;\n\n(III) $\\lesssim (\\log n)^{1+1/d}/(s^{1/d}\\overline{k}) + (\\overline{k}/s)^{2/d}$.\n\nThe following Proposition ensures that the solution to the surrogate risk minimization satisfies the conditions in Proposition 1 with high probability, thereby addressing Problem (i) and validating the efectiveness of our algorithm.\n\nProposition 5 Let Assumption 2 hold. Let $\\{D_{s}^{b}\\}_{b=1}^{B}$ be B disjoint subsets of size s randomly drawn from the data set $D_{n}$ . Moreover, let $w^{b,*}$ be defined as in (7), and $k^{b,*}: =k(w^{b,*}): =$ sup $\\{i\\in [n]:w_{i}^{b,*}\\neq 0\\}$ . If we choose s and B as in (13) in Theorem 2, then there exists $N_{2}\\in\\mathbb{N}$ , which will be specified in the proof, such that for all $n>N_{2}$ , the following two statements hold with probability $\\mathrm{P}^{Bs}$ at least $1-1/n^{2}$\n\n1. The conditions $(i)-(iv)$ in Proposition 1 hold for $w^{b,*}$ and $k^{b,*}$  \n2. Furthermore, we have $k^{b,*}\\asymp(n/\\log n)^{1/(d+2)}for\\ b\\in [B]$\n\nThe following lemma, which will be used several times in the sequel, provides the uniform bound on the distance between any point and its k-th nearest neighbor with a high probability when the distribution has bounded support.\n\nLemma 12 Let Assumption 2 hold. Furthermore, let $\\{D_{s}^{b}\\}_{b=1}^{B}$ be B disjoint subsets of size s randomly drawn from the data set $D_{n}$ , where $D_{s}^{b}=\\{X_{1}^{b},\\ldots,\\bar{X}_{s}^{b}\\}$ and $R_{s,(i)}^{b}(x)$ denotes the i-distance of x in the subset $D_{s}^{b}$ for $i \\in [ s ]$ . Additionally, let the sequence $c_{n}: =\\lceil 48(2d+$ 9) log $n+48\\log 2+1447$ . Suppose that $s\\geq c_{1}^{\\prime}: =\\operatorname*{max}\\{4e,d+2,C\\}$ , where C is the constant specified in Lemma 11. Then, there exist constants $0<c_{2}^{\\prime}<c_{3}^{\\prime}$ such that, with probability $\\mathrm{P}^{Bs}$ at least $1-1/n^{2}$ , for all $x\\in\\mathcal{X},b\\in [B]$ , and $c_{n}\\leq i\\leq s,$ , the following inequalities hold:\n\n$$\nc_{2}^{\\prime}(i/s)^{1/d}\\leq R_{s,(i)}^{b}(x)\\leq c_{3}^{\\prime}(i/s)^{1/d}\\tag{21}\n$$\n\nand\n\n$$\n\\mathrm{P}(B(x,R_{s,(i)}^{b}(x)))\\asymp i/s.\\tag{22}\n$$\n\nBy the definition of $\\gamma_{s,i}$ in (3), we conclude that\n\n$$\n\\left(\\frac{i+1/d-1}{s+1+1/d}\\right)^{1/d}<\\gamma_{s,i}<\\left(\\frac{i+1/d}{s+1/d}\\right)^{1/d}.\\tag{38}\n$$\n\nLemma 17 Let Assumption 2 hold. Additionally, let $\\{D_{s}^{b}\\}_{b=1}^{B}$ be B disjoint subsets of size s randomly drawn from data $D_{n}$ . Let $R_{s,(i)}^{b}(x)$ be the i-distance of x in the subset $D_{s}^{b}$ . Define k as in Proposition 1. Suppose that $B\\geq 2(d^{2}+4)(\\log n)/3,s\\geq 2\\overline{{k}}$ . Furthermore, assume that there exists constants $C_{n,i}^{\\prime}$ such that $w_{i}^{b}\\leq C_{n,i}^{\\prime}for$ all $b \\in [ B ]$ and $i \\in [ s ]$ . Then, there exists $n_{1}: =2d^{d}+1$ , such that for all $x\\in{\\mathcal{X}},1\\leq i\\leq{\\overline{{k}}}$ , and all $n>n_{1}$ , the following statement holds with probability $\\dot{\\mathrm{P}}^{Bs}$ at least $1-1/n^{2}.$\n\n$$\n\\left|\\frac{1}{B}\\sum_{b=1}^{B}w_{i}^{b}\\big(\\mathrm{P}(B(x,R_{s,(i)}^{b}(x)))^{1/d}-\\gamma_{s,i}\\big)\\right|\\lesssim C_{n,i}^{\\prime}\\left(\\frac{i}{s}\\right)^{1/d}\\sqrt{\\frac{\\log n}{iB}}.\\tag{41}\n$$\n\nfor all $x\\in\\mathcal{X}$ . This implies\n\n$$\nR_{n}^{B}(x)=\\frac{1}{B}\\sum_{b=1}^{B}R_{s}^{w,b}(x)\\gtrsim(\\underline{{k}}/s)^{1/d}\\tag{47}\n$$\n\nUsing this bound with the choices of B and s in (61), and the inequality (72), we derive\n\n$$\n\\begin{array}{l}k^{b,*}\\leq\\frac{2}{c_{4}^{\\prime}(c_{3}^{\\prime})^{2d/(2+d)}}\\left(\\int_{1/2}^{1}(1-t^{1/d})^{2}dt\\right)^{-\\frac{d}{d+2}}\\left(\\frac{\\log s}{B}\\right)^{\\frac{d}{d+2}}s^{\\frac{2}{d+2}}\\\\ \\leq\\frac{2(c_{6}^{\\prime})^{\\frac{2}{d+2}}}{c_{4}^{\\prime}((c_{3}^{\\prime})^{2}c_{7}^{\\prime})^{\\frac{d}{d+2}}}\\left(\\int_{1/2}^{1}(1-t^{1/d})^{2}dt\\right)^{-\\frac{d}{d+2}}\\left(\\frac{n}{\\log n}\\right)^{\\frac{1}{d+2}}=\\frac{c_{8}^{\\prime}}{2}\\left(\\frac{n}{\\log n}\\right)^{\\frac{1}{d+2}}.\\tag{73}\\\\ \\end{array}\n$$\n\nIn this section, we present the proof related to BRDDE. The weights $w^{b,*}$ are derived using SRM based on the data for BRDDE, whereas Proposition 4 assumes that the weights are fixed and independent of the data. As a result, Proposition 4 cannot be directly applied to establish the convergence rate of our density estimator. However, Proposition 5 ensures that the weights returned by SRM satisfy the required conditions in Proposition 1 with high probability. Therefore, by making slight modifications to the proof of Proposition 4, we can establish the error decomposition of BRDDE as stated in Proposition 21.\n\nBefore proceeding, we introduce additional notation. Regarding the expression of $f^{B,*}(x)$ in (11), we define the error term $(I^{\\prime}),(II^{\\prime})$ , and $\\left(III^{\\prime}\\right)$ as follows, corresponding to $(I),(II)$ ， and (III) in (14), (17), and (18):\n\n$$\n(I^{\\prime}):=\\frac{1}{V_{d}R_{n}^{B,*}(x)^{d}}\\sum_{j=0}^{d-1}\\left(\\frac{1}{B}\\sum_{b=1}^{B}\\sum_{i=1}^{s}w_{i}^{b,*}\\gamma_{s,i}\\right)^{j}\\left(V_{d}^{1/d}f(x)^{1/d}R_{n}^{B,*}(x)\\right)^{d-1-j}.\\tag{79}\n$$\n\n$$\n(II^{\\prime}):=\\sum_{i=1}^{s}\\left|\\frac{1}{B}\\sum_{b=1}^{B}w_{i}^{b,*}\\big(\\gamma_{s,i}-\\mathrm{P}(B(x,\\widetilde{R}_{s,(i)}^{b}(x)))^{1/d}\\big)\\right|,\\tag{80}\n$$\n\n$$\n(III^{\\prime}):=\\sum_{i=1}^{s}\\left|\\frac{1}{B}\\sum_{b=1}^{B}w_{i}^{b,*}\\big(\\mathrm{P}(B(x,\\widetilde{R}_{s,(i)}^{b}(x)))^{1/d}-V_{d}^{1/d}f(x)^{1/d}\\widetilde{R}_{s,(i)}^{b}(x)\\big)\\right|.\\tag{81}\n$$\n\nFollowing a similar derivation as in (19), we obtain the error decomposition:\n\n$$\n\\left|f_{n}^{B,*}(x)-f(x)\\right|\\leq\\left(I^{\\prime}\\right)\\cdot\\left(II^{\\prime}\\right)+\\left(I^{\\prime}\\right)\\cdot\\left(III^{\\prime}\\right).\\tag{82}\n$$\n\nProposition 21 Let Assumption 2 hold. Let $(I^{\\prime}),(II^{\\prime})$ , and $\\left(III^{\\prime}\\right)$ be defined in (79), (80), and (81), respectively. Then, there exists an integer $N_{4}\\in\\mathbb{N}$ such that for all $n>N_{4}$ and any x satisfying $\\bar{B}(x,\\widetilde{R}_{s,(k^{b,*})}^{b}(x))\\subset [0,1]^{d}$ for all $b \\in [ B ]$ , with probability $\\mathrm{P}^{n}$ at least $1-2/n^{2}$ $(I^{\\prime}),\\ (II^{\\prime})$ , and $(II\\dot{I}^{\\prime})$ have upper bounds of the same asymptotic order as (I), (II), and (III) in Proposition $\\it 4$\n\nProof [of Proposition 21] Let $\\Omega_{5}$ denote the event defined by the statements in Proposition $5 .$ Applying Lemma 12 to the subset $\\{\\widetilde{D}_{s}^{b}\\}_{b=1}^{B}$ , we obtain that, with probability at least $1-1/n^{2}$ the following holds: $c_{2}^{\\prime}(i/s)^{1/d}\\leq\\widetilde{R}_{s,(i)}^{b}(x)\\leq c_{3}^{\\prime}(i/s)^{1/d}$ and $\\mathrm{P}(B(x,\\widetilde{R}_{s,(i)}^{b}(x)))\\asymp i/s$ for all $x\\in\\mathcal{X},b\\in [B]$ , and $c_{n}\\leq i\\leq s.$ . We define this event as $\\Omega_{6}$\n\nSince the datasets $D_{s}^{b}$ and $\\widetilde{D}_{s}^{b}$ are independent for $b \\in [ B ]$ and $w^{b,*}$ is the solution to the SRM in (7), we have\n\n$$\n\\begin{array}{l}\\mathbb{E}\\big [w_{i}^{b,*}((\\mathrm{P}(B(x,\\widetilde{R}_{s,(i)}^{b}(x))))^{1/d}-\\gamma_{s,i})\\big|\\Omega_{5}\\big]\\\\ =\\mathbb{E}\\big [w_{i}^{b,*}\\big|\\Omega_{5}\\big]\\cdot\\mathbb{E}\\big [(\\mathrm{P}(B(x,\\widetilde{R}_{s,(i)}^{b}(x))))^{1/d}-\\gamma_{s,i}\\big]=0\\\\ \\end{array}\n$$\n\nfor a fixed $x\\in\\mathcal{X}$ , where the last equality follows from (20) and the expectation is taken with respect to the empirical measure $\\widetilde{D}_{s}^{b}\\mathrm{.}$ , conditional on the event $\\Omega_{5}$ . Following the proof of Lemma 17 and conditioning on the event $\\Omega_{5}$ , for all $n>n_{1}=2d^{d}+1$ , we have\n\n$$\n\\mathrm{P}\\left(\\sup_{x\\in\\mathcal{X},i\\in [\\bar{k}]}\\left|\\frac{1}{B}\\sum_{b=1}^{B}w_{i}^{b,*}\\big(\\mathrm{P}(B(x,\\widetilde{R}_{s,(i)}^{b}(x)))^{1/d}-\\gamma_{s,i}\\big)\\right|\\lesssim C_{n,i}\\left(\\frac{i}{s}\\right)^{1/d}\\sqrt{\\frac{\\log n}{iB}}\\Bigg|\\Omega_{5}\\right)\\geq 1-\\frac{1}{n^{2}}.\n$$\n\nSince Proposition 5 ensures that $\\mathrm{P}(\\Omega_{5})\\geq 1-1/n^{2}$ , applying the conditional probability formula yields\n\n$$\n\\mathrm{P}^{n}\\Bigg(\\sup_{x\\in\\mathcal{X},i\\in [\\overline{{k}}]}\\bigg|\\frac{1}{B}\\sum_{b=1}^{B}w_{i}^{b,*}\\big(\\mathrm{P}(B(x,\\widetilde{R}_{s,(i)}^{b}(x)))^{1/d}-\\gamma_{s,i}\\big)\\bigg|\\lesssim C_{n,i}\\bigg(\\frac{i}{s}\\bigg)^{1/d}\\sqrt{\\frac{\\log n}{iB}}\\Bigg)\\geq 1-\\frac{2}{n^{2}}\n$$\n\nfor all $n>n_{1}\\lor N_{2}$ , where $N_{2}$ is the integer specified in Proposition 5. Denote this event as $\\Omega_{7}$ . By the union bound, the event $\\Omega_{5}\\cap\\Omega_{6}\\cap\\Omega_{7}$ holds with probability $\\mathrm{P}^{n}$ at least $1-4/n^{2}$ for all $n>n_{1}\\lor N_{2}$ . Since the events $\\Omega_{6}$ and $\\Omega_{7}$ correspond to the events $\\Omega_{1}$ and $\\Omega_{2}$ in the proof of Proposition 4, respectively, and given that conditions $(i)-(iv)$ in Proposition 1 hold for $w^{b,*}$ and $k^{b,*}$ on $\\Omega_{5}$ , there exists $N_{4}\\in\\mathbb{N}$ such that the upper bound for $(I^{\\prime}),(II^{\\prime})$ and $\\left(III^{\\prime}\\right)$ follow for all $n>N_{4}$ by similar arguments as in the proof of Proposition 4. (Note that $N_{4}$ may difer $N_{1}$ in that proposition.) The details are omitted. 7", "theorems": [{"local_id": "T1", "statement": "Theorem 2 Let Assumption 2 hold. Suppose that the dataset $D_{n}$ is randomly partitioned into B disjoint pairs of subsets, denoted as $\\{(D_{s}^{b},\\widetilde{D}_{s}^{b})\\}_{b=1}^{B}$ , such that $D_{s}^{b}\\cap\\widetilde{D}_{s}^{b}=\\varnothing$ for each b. Let f be the true density function, and let $f_{n}^{B,*}$ be the BRDDE defined in (11). If we choose\n\n$$\ns\\asymp(n/\\log n)^{(d+1)/(d+2)}\\quad a n d\\quad B\\asymp n^{1/(d+2)}(\\log n)^{(d+1)/(d+2)},\\tag{13}\n$$\n\nthen there exists $N_{2}^{*}\\in\\mathbb{N}$ , which will be specified in the proof, such that for all $n>N_{2}^{*}$ , with probability $\\mathrm{P}^{n}$ at least $1-4/n^{2}$ , we have\n\n$$\n\\int_{\\mathcal{X}}|f_{n}^{B,*}(x)-f(x)|d x\\lesssim n^{-1/(2+d)}(\\log n)^{(d+3)/(d+2)}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof [of Theorem 2] Let $\\Omega_{5},\\Omega_{6}$ , and $\\Omega_{7}$ be the events defined in the proof of Proposition 21. Following the arguments therein, the event $\\Omega_{5}\\cap\\Omega_{6}\\cap\\Omega_{7}$ holds with probability $\\mathrm{P}^{n}$ at least $1-4/n^{2}$ for all $n\\geq N_{4}$ . For the remainder of the proof, we assume that the event $\\Omega_{5}\\cap\\Omega_{6}\\cap\\Omega_{7}$ holds and that $n>N_{2}^{*}: =N_{4}$\n\nFrom (73) in the proof of Proposition 5, we have $\\overline{{k}}\\le c_{8}^{\\prime}(n/\\log n)^{1/(d+2)}/2$ . Define\n\n$$\n\\Delta_{n}:=[c_{3}^{\\prime}(c_{8}^{\\prime}(n/\\log n)^{1/(d+2)}/(2 s))^{1/d},1-c_{3}^{\\prime}(c_{8}^{\\prime}(n/\\log n)^{1/(d+2)}/(2 s))^{1/d}]^{d}.\n$$"}, {"n": 2, "claim": "On the event $\\Omega_{6}$ , for all $x\\in\\Delta_{n}$ and any $b \\in [ B ]$ , we have\n\n$$\n\\widetilde{R}_{s,(k^{b,*})}^{b}(x)\\leq c_{3}^{\\prime}(k^{b,*}/s)^{1/d}\\leq c_{3}^{\\prime}(c_{8}^{\\prime}(n/\\log n)^{1/(d+2)}/(2 s))^{1/d}.\n$$"}, {"n": 3, "claim": "Thus, for any $y\\in B\\big(x,\\widetilde{R}_{s,(k^{b,*})}^{b}(x)\\big)$ , we have\n\n$$\nd(y,\\mathbb{R}^{d}\\setminus [0,1]^{d})\\geq c_{3}^{\\prime}(c_{8}^{\\prime}(n/\\log n)^{1/(d+2)}/(2 s))^{1/d}-\\widetilde{R}_{s,(k^{b,*})}^{b}(x)\\geq 0.\n$$"}, {"n": 4, "claim": "This implies that $B\\left(x,\\widetilde{R}_{s,(k^{b,*})}^{b}(x)\\right)\\subset [0,1]^{d}$ for all $x\\in\\Delta_{n}$ and $b \\in [ B ]$ . Therefore, from Proposition 21 and inequality (82), we obtain\n\n$$\n\\left|f_{n}^{B,*}(x)-f(x)\\right|\\lesssim((\\log n)/\\overline{{k}})^{1+1/d}+\\big((\\log n)/(\\overline{{k}}B)\\big)^{1/2}+(\\overline{{k}}/s)^{1/d},\\quad x\\in\\Delta_{n}.\n$$"}, {"n": 5, "claim": "Using $k^{b,*}\\asymp(n/\\log n)^{1/(d+2)}$ from Proposition 5 and substituting the choices of B and $s,$ we obtain\n\n$$\n\\big|f_{n}^{B,*}(x)-f(x)\\big|\\lesssim n^{-1/(d+2)}(\\log n)^{(d+3)/(d+2)},\\qquad x\\in\\Delta_{n}\n$$"}, {"n": 6, "claim": "Integrating over $\\Delta_{n}$ , we get\n\n$$\n\\int_{\\Delta_{n}}|f_{n}^{B,*}(x)-f(x)|d x\\lesssim n^{-1/(d+2)}(\\log n)^{(d+3)/(d+2)}.\\tag{83}\n$$"}, {"n": 7, "claim": "On the other hand, on the event $\\Omega_{5}$ , condition (iii) in Proposition 1 holds for $w^{b,*}$ and $k^{b,*}$ , which implies that $\\textstyle\\sum_{i=1}^{s}w_{i}^{b,*}i^{1/d}\\lesssim(k^{b,*})^{1/d}$ for all $b \\in [ B ]$ . Since $\\gamma_{s,i}<((i+1/d)/(s+$ $1/d))^{1/d}\\leq(2 i\\bar{/}s)^{1/d}$ by (38), we have\n\n$$\n\\frac{1}{B}\\sum_{b=1}^{B}\\sum_{i=1}^{s}w_{i}^{b,*}\\gamma_{s,i}\\lesssim\\frac{1}{B}\\sum_{b=1}^{B}\\sum_{i=1}^{s}w_{i}^{b,*}(i/s)^{1/d}\\lesssim\\frac{1}{B}\\sum_{b=1}^{B}(k^{b,*}/s)^{1/d}\\lesssim(\\overline{{k}}/s)^{1/d}.\n$$"}, {"n": 8, "claim": "Following similar arguments as in (47) from Proposition 4, we obtain $R_{n}^{B,*}(x)\\gtrsim(\\underline{{k}}/s)^{1/d}$ for all $x\\in\\mathcal{X}$ on the event $\\Omega_{5}\\cap\\Omega_{6}$ . This implies\n\n$$\nf_{n}^{B,*}(x)=\\frac{1}{V_{d}R_{n}^{B,*}(x)^{d}}\\bigg(\\frac{1}{B}\\sum_{b=1}^{B}\\sum_{i=1}^{s}w_{i}^{b,*}\\gamma_{s,i}\\bigg)^{d}\\lesssim\\overline{{k}}/\\underline{{k}}\\lesssim 1,\\qquad x\\in\\mathcal{X}.\n$$"}, {"n": 9, "claim": "Since $\\|f\\|_{\\infty}\\leq\\overline{{c}}$ from Assumption 2, we conclude that $\\|f_{n}^{B,*}-f\\|_{\\infty}$ is bounded by a constant. Therefore, we have\n\n$$\n\\begin{array}{l}\\int_{\\mathcal{X}\\backslash\\Delta_{n}}|f_{n}^{B,*}(x)-f(x)|d x\\lesssim\\mu(\\mathcal{X}\\setminus\\Delta_{n})\\lesssim((n/\\log n)^{1/(d+2)}/s)^{1/d}\\\\ \\lesssim n^{-1/(d+2)}(\\log n)^{(d+3)/(d+2)}.\\\\ \\end{array}\n$$"}, {"n": 10, "claim": "Finally, combining this with (83), we have\n\n$$\n\\int_{\\mathcal{X}}\\left|f_{n}^{B,*}(x)-f(x)\\right|d x=\\left(\\int_{\\Delta_{n}}+\\int_{\\mathcal{X}\\backslash\\Delta_{n}}\\right)\\left|f_{n}^{B,*}(x)-f(x)\\right|d x\\lesssim n^{-1/(d+2)}(\\log n)^{(d+3)/(d+2)},\n$$\n\nwhich completes the proof."}]}}, {"local_id": "T2", "statement": "Theorem 3 Let Assumptions 1 and 2 hold. Suppose the conditions in Theorem 2 hold, including the dataset partitioning and choice of parameters s and B. Let $R_{n}^{B,*}$ be the bagged regularized k-distances returned by Algorithm 2. Then there exists $N_{2}^{*}\\in\\mathbb{N}$ , as specified in Theorem ${\\mathcal{Q}},$ such that for all $n>N_{2}^{*}$ , with probability $\\mathrm{P}^{n}$ at least $1-4/n^{2}$ , we have\n\n$$\n\\mathrm{Reg}^{\\mathrm{AUC}}(R_{n}^{B,*})\\lesssim n^{-1/(2+d)}(\\log n)^{(d+3)/(d+2)}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof [of Theorem 3] Let $\\Omega_{5},\\Omega_{6}$ , and $\\Omega_{7}$ be the events defined in the proof of Proposition 21. Following the arguments therein, the event $\\Omega_{5}\\cap\\Omega_{6}\\cap\\Omega_{7}$ holds with probability $\\mathrm{P}^{n}$ at least $1-4/n^{2}$ for all $n\\geq N_{4}$ . For the subsequent arguments, we assume that $\\Omega_{5}\\cap\\Omega_{6}\\cap\\Omega_{7}$ holds and that $n>N_{2}^{*}=N_{4}$\n\nLet $\\left\\{c_{n}\\right\\}$ denote the sequence from Lemma 12. On the event $\\Omega_{6},$ , for all $x\\in\\mathcal{X}$ , we have\n\n$$\n\\begin{array}{l}R_{s}^{b,*}(x)=\\sum_{i=1}^{s}w_{i}^{b,*}\\widetilde{R}_{s,(i)}^{b}(x)=\\sum_{i=1}^{c_{n}}w_{i}^{b,*}\\widetilde{R}_{s,(i)}^{b}(x)+\\sum_{i=c_{n}+1}^{s}w_{i}^{b,*}\\widetilde{R}_{s,(i)}^{b}(x)\\\\ \\leq\\widetilde{R}_{s,(c_{n})}^{b,*}(x)+\\sum_{i=c_{n}+1}^{s}w_{i}^{b,*}\\widetilde{R}_{s,(i)}^{b}(x)\\lesssim(c_{n}/s)^{1/d}+\\sum_{i=1}^{s}w_{i}^{b,*}(i/s)^{1/d},\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "By Proposition 5, on the event $\\Omega_{5}$ , we have $\\textstyle\\sum_{i=1}^{s}w_{i}^{b,*}i^{1/d}\\lesssim\\overline{{k}}^{1/d}$ and $\\underline{{k}}\\gtrsim(\\log n)^{2}$ . This implies that $R_{s}^{b,*}(x)\\lesssim(c_{n}/s)^{1/d}+(\\overline{{k}}/s)^{1/d}\\lesssim(\\overline{{k}}/s)^{1/d}$ . Averaging over b in $[ B ]$ , we obtain\n\n$$\nR_{n}^{B,*}(x)=\\frac{1}{B}\\sum_{b=1}^{B}R_{s}^{b,*}(x)\\lesssim(\\bar{k}/s)^{1/d}\\tag{85}\n$$"}, {"n": 3, "claim": "On the other hand, using (38), we have\n\n$$\n\\sum_{i=1}^{s}w_{i}^{b,*}\\gamma_{s,i}>\\sum_{i=1}^{s}w_{i}^{b,*}\\bigg(\\frac{i+1/d-1}{s+1+1/d}\\bigg)^{1/d}\\gtrsim\\sum_{i=1}^{s}w_{i}^{b,*}(i/s)^{1/d},\n$$\n\nwhere we use the inequality $(i+1/d-1)/(s+1+1/d)\\geq(i-1)/(s+2)\\geq(i-1)/(2s)$ Applying Proposition 5 again, we get $\\textstyle\\sum_{i=1}^{s}w_{i}^{b,*}i^{1/d}\\gtrsim\\overline{{k}}^{1/d}$ . Averaging over b in [B], we have\n\n$$\n\\frac{1}{B}\\sum_{b=1}^{B}\\sum_{i=1}^{s}w_{i}^{b,*}\\gamma_{s,i}\\gtrsim(\\overline{{k}}/s)^{1/d}.\n$$"}, {"n": 4, "claim": "Combining this with (85), we conclude that $f_{n}^{B,*}(x)$ is lower bounded by a constant for $x\\in\\mathcal{X}$ . Consequently, by Theorem 2 and Proposition 6, we obtain the desired assertion."}]}}, {"local_id": "T3", "statement": "Proposition 6 Let Assumptions 1 and 2 hold, and let f be the true density function. Let $R_{n}^{B,*}$ be the bagged regularized k-distances as in (10) and $f_{n}^{B,*}(x)$ be the BRDDE as in (11). Suppose that there exists a constant $c > 0$ such that $\\|f_{n}^{B,*}\\|_{\\infty}\\overset{\\cdot}{\\geq}c$ . Then, we have\n\n$$\n\\mathrm{Reg}^{\\mathrm{AUC}}(R_{n}^{B,*})\\lesssim\\int_{\\mathcal{X}}\\big|f_{n}^{B,*}(x)-f(x)\\big|d x.\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof [of Proposition 6] Under the Huber contamination model in Assumption 1, let $\\eta(x)$ be defined as in (12), and define $\\widehat{\\eta}(x)=\\Pi\\cdot f_{n}^{B,*}(x)^{-1}$ . By the expression of $f_{n}^{^{\\prime}{},*}(x)$ in (11), it follows that ${\\bf 1}\\{R_{n}^{B,*}(X)-R_{n}^{B,*}(X^{\\prime})>0\\}={\\bf 1}\\{\\widehat{\\eta}(X)-\\widehat{\\eta}(X^{\\prime})>0\\}$ and $\\mathbf{1}\\{R_{n}^{B,*}(X)=$ ${\\cal{R}}_{n}^{B,*}(X^{\\prime})\\}=\\mathbf{1}\\{{\\widehat\\eta}(X){=}{\\widehat\\eta}(X^{\\prime})\\}$ . Consequently, we obtain\n\n$$\n\\begin{array}{l}\\mathrm{AUC}(R_{n}^{B,*})\\\\ =\\mathbb{E}\\left[\\mathbf{1}\\left\\{\\left(Y-Y^{\\prime}\\right)\\left(R_{n}^{B,*}(X)-R_{n}^{B,*}\\left(X^{\\prime}\\right)>0\\right)\\right\\}+\\mathbf{1}\\left\\{R_{n}^{B,*}(X)=R_{n}^{B,*}\\left(X^{\\prime}\\right)\\right\\}/2\\mid Y\\neq Y^{\\prime}\\right]\\\\ =\\mathbb{E}\\big [\\mathbf{1}\\{(Y-Y^{\\prime})(\\widehat{\\eta}(X)-\\widehat{\\eta}(X^{\\prime})>0)\\}+\\mathbf{1}\\{\\widehat{\\eta}(X)=\\widehat{\\eta}(X^{\\prime})\\}/2|Y\\neq Y^{\\prime}\\big]=\\mathrm{AUC}(\\widehat{\\eta}).\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "Therefore, we have $\\mathrm{Reg}^{\\mathrm{AUC}}(R_{n}^{B,*})=\\mathrm{Reg}^{\\mathrm{AUC}}(\\widehat\\eta)$ . Applying Agarwal (2013, Corollary 11), we obtain\n\n$$\n\\mathrm{Reg}^{\\mathrm{AUC}}(R_{n}^{B,*})=\\mathrm{Reg}^{\\mathrm{AUC}}(\\widehat{\\eta})\\leq\\frac{1}{\\Pi(1-\\Pi)}\\int_{\\mathcal{X}}|\\widehat{\\eta}(x)-\\eta(x)|d\\mathrm{P}_{X}(x).\\tag{84}\n$$"}, {"n": 3, "claim": "From Assumption 2, we have $\\|f\\|_{\\infty}\\geq\\underline{{c}},$ and given that $\\|f_{n}^{B,*}\\|_{\\infty}\\geq c$ , it follows that\n\n$$\n|\\widehat{\\eta}(x)-\\eta(x)| =\\frac{\\Pi\\big|f_{n}^{B,*}(x)-f(x)\\big|}{f_{n}^{B,*}(x)f(x)}\\lesssim\\big|f_{n}^{B,*}(x)-f(x)\\big|.\n$$"}, {"n": 4, "claim": "Combining this with (84) and the condition $\\|f\\|_{\\infty}\\leq\\overline{{c}}$ from Assumption 2, we establish the desired result. ■"}]}}], "target_local_id": null, "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2025-24-1178", "context": "In this work, we tackle feature learning and function estimation jointly through the paradigm of empirical risk minimisation. We consider a classical supervised learning problem. We have i.i.d. samples $(x_{i},y_{i})_{i\\in [n]}$ from a random variable $(X,Y)\\in\\mathcal{X}\\times\\mathcal{Y}\\subset\\mathbb{R}^{d}\\times\\mathbb{R}$ Our goal is to minimise the expected risk, which is defined as $\\mathcal{R}(f): =\\mathbb{E}_{X,Y}[\\ell(Y,f(X))]$ over some class of functions ${\\mathcal{F}}_{:}$ where \\` is a loss function mapping from $\\mathbb{R}\\times\\mathbb{R}$ to R. This can be achieved through the framework of regularised empirical risk minimisation, where the empirical risk is defined as $\\begin{array}{ r }{\\widehat{\\mathcal{R}}(f): =\\frac{1}{n}\\sum_{i=1}^{n}\\ell(y_{i},f(x_{i}))}\\end{array}$ . Our interest in regularised empirical risk minimisation stems from its flexibility, allowing it to be applied to a wide range of problems as long as the objective can be defined as the optimisation of an expected loss. Our primary objective is to achieve the lowest possible risk, which we study in theory and in practice, while we explore the recovery of underlying features in numerical experiments. Our method draws inspiration from several lines of work, namely positive definite kernels and neural networks with their mean field limit, which we briefly review, together with the main limitations we aim to alleviate.\n\nMean-field limit. To apply a similar framework to our future estimator, we introduce the mean-field limit of an over-parameterised one-hidden layer neural network (Nitanda and Suzuki, 2017; Mei et al., 2019; Chizat and Bach, 2022; Sirignano and Spiliopoulos, 2020). When the number of neurons m is very large, the network can be rescaled as follows\n\n$$\nf(x)=\\frac{1}{m}\\sum_{j=1}^{m}\\eta_{j}\\sigma(w_{j}^{\\top}x+b_{j}),\\text{which approximates}\\int\\eta\\sigma(w^{\\top}x+b)\\mathrm{d}\\mu(\\eta,w,b),\\tag{1}\n$$\n\nwhere $\\mu$ is a probability distribution, and we can take the weights and constant terms $(w,b)$ to be constrained when the activation is 1-homogeneous,<sup>1</sup> such as the ReLU or step func tion. This approach is valuable because, as noted by Chizat and Bach (2022), under certain conditions (convexity of the loss and penalty functions, homogeneity of the activation function), the regularised empirical risk problem optimised via gradient descent in the infinitely small step-size limit converges to the minimiser of the corresponding problem with infinitely many particles. This allows us to use a finite number of particles m in practice while still leveraging the theoretical benefits derived from the continuous framework.\n\nWe use the following notations. For a positive integer m, we define $[m]: =\\{1,\\dots,m\\}$ For a d-dimensional vector α and $i\\in [d],\\alpha_{i}$ denotes its i-th element. For a matrix A, tr A denotes its trace when A is square, $A^{-1}$ its inverse when well defined, while $A_{i,j}$ the element in its i-th row and j-th column, and $A^{\\top}$ its transpose. $I_{d}$ is the $d \\times d$ identity matrix. We use $S^{d-1}$ to denote the unit sphere in $\\mathbb{R}^{d}$ for $\\| \\cdot \\|$ a generic norm and $\\| \\cdot \\| .$ <sub>∗</sub> its dual norm. The $\\ell_{2},\\ell_{1}$ , and $\\ell_{\\infty}$ norms are denoted as $\\|\\cdot\\|_{2},\\|\\cdot\\|_{1}$ , and $\\|\\cdot\\|_{\\infty}$ respectively. We use $O(\\cdot)$ to denote the asymptotic behaviour of functions, indicating the order of growth. The set of probability measures on a given space S is denoted by ${\\mathcal{P}}(S)$ . A normal random variable is denoted as following the law N(mean, variance). 1 is the indicator function. For two spaces $S_{1},S_{2},S_{1}^{S_{2}}$ is the set of functions from $S_{2}$ to $S_{1}$\n\nBuilding on the limitations of current methods discussed in the introduction, we propose a novel architecture that integrates neural networks with kernel methods. This approach can be interpreted in two ways: as learning with a kernel that is itself learned during training, or as employing a one-hidden layer neural network where the weights from the input layer to the hidden layer are learned through gradient descent, while the weights and non-linearity from the hidden layer to the output are optimised explicitly. In this section, we introduce the custom function space we propose, revisit key properties of reproducing kernel Hilbert spaces (RKHS), and explore the connections between BKerNN model, kernel methods, and neural networks. Additionally, we present the various regularisation penalties we consider throughout our analysis.\n\nWe begin by considering the continuous setting, which mirrors the mean-field limit of overparameterised one-hidden layer neural networks discussed in Section 1.\n\nDefinition 1 (Infinite-Width Function Space) Let\n\n$$\n\\mathcal{F}_{\\infty}:=\\left\\{f\\mid f(\\cdot)=c+\\int_{\\mathcal{H}\\times\\mathcal{S}^{d-1}}g(w^{\\top}\\cdot)\\mathrm{d}\\mu(g,w),\\quad\\Omega_{0}(f)<\\infty\\right\\},\n$$\n\nwhere $c$ is a constant in $\\mathbb{R},S^{d-1}$ is the unit sphere for some norm $\\| \\cdot \\|$ on $\\mathbb{R}^{d}$ (typically either $\\ell_{2}$ or $\\ell_{1}\\ j$ , H is a space of functions, and µ is a probability measure on $\\mathcal{H}\\times S^{d-1}$ . We define H as $\\{g:\\mathbb{R}\\to\\mathbb{R}\\mid g(0)=0$ , g has a weak derivative $g^{\\prime}$ $\\textstyle\\int_{\\mathbb{R}}(g^{\\prime})^{2}<\\infty\\}$ . H is a Hilbert space and a Sobolev space, with the inner product defined as $\\begin{array}{ r }{\\langle\\tilde{g},\\bar{g}\\rangle_{\\mathcal{H}}=\\int\\tilde{g}^{\\prime}g^{\\prime}}\\end{array}$ . We define $\\Omega_{0}$ as the infimum over all measures which can be used to define f, i.e.\n\n$$\n\\Omega_{0}(f):=\\inf_{\\mu\\in\\mathcal{P}(\\mathcal{H}\\times\\mathcal{S}^{d-1})}\\int_{\\mathcal{H}\\times\\mathcal{S}^{d-1}}\\|g\\|_{\\mathcal{H}}\\mathrm{d}\\mu(g,w),\\tag{2}\n$$\n\nwith the infimum over $\\mu$ such that $\\begin{array}{ r }{f=c+\\int_{\\mathcal{H}\\times S^{d-1}}g(w^{\\top}\\cdot)\\mathrm{d}\\mu(g,w)}\\end{array}$ , where $\\begin{array}{ r }{\\|g\\|_{\\mathcal{H}}^{2}: =\\int_{-\\infty}^{+\\infty}(g^{\\prime})^{2}}\\end{array}$\n\nNote that c corresponds to the value of the function f evaluated at the null vector and is therefore unique, which is not necessarily the case of the measure $\\mu .\\_F_{\\infty}$ is well-defined using a “variation norm” on couples $(g,w)$ integrated w.r.t a Borel measure on $\\mathcal{H}\\times S^{d-1}$ This is the variation norm associated to the map $(g,w)\\in\\mathcal{H}\\times{\\mathcal{S}}^{d-1}\\to g(w^{\\top}.)$ (which is a function from $\\mathbb{R}^{d}$ to $\\mathbb{R},$ , see Kurkova and Sanguineti (2001) and Bach (2024, Section 9.3.2).\n\nThe function space $\\mathcal{F}_{\\infty}$ is inspired by infinite-width single hidden layer neural networks: with the addition of the intercept c, each function in this space can be seen as the integral of a linear part w and a non-linearity g over some probability distribution, as in Equation (1) where the non-linearity is $\\eta\\sigma(\\cdot)$ . Thus here the activation functions are learnt.\n\nThe approximation of $\\mathcal{F}_{\\infty}$ with m particles can then be obtained as follows.\n\nDefinition 2 (Finite-Width Function Space) Let\n\n$$\n\\mathcal{F}_{m}:=\\left\\{f\\mid f(\\cdot)=c+\\frac{1}{m}\\sum_{j=1}^{m}g_{j}(w_{j}^{\\top}\\cdot),w_{j}\\in\\mathcal{S}^{d-1},g_{j}\\in\\mathcal{H},c\\in\\mathbb{R}\\right\\}.\n$$\n\nRemark that $\\forall m\\in\\mathbb{N}^{*},\\mathcal{F}_{m}\\subset\\mathcal{F}_{\\infty}.$ , by taking the discrete probability measure uniformly supported by the particles $w_{1},\\ldots,w_{m}$\n\nWe now consider regularised empirical risk minimisation starting with the basic penalty $\\Omega_{0}$ This penalty enforces the regularity of the function and, because we use penalisation with non-squared norms, limits the cardinality of the support of $\\mu$ on the space H. While this penalty is not specifically aimed at feature learning, by limiting the number of non-zero particles, it indirectly promotes feature learning to some extent. This serves as a starting point, and we introduce more targeted penalties in Section 2.5 with a stronger feature learning behaviour. For $f\\in\\mathcal{F}_{m}$ written as in Definition 2, the penalty simplifies to $\\begin{array}{ r }{\\Omega_{0}(f)=\\frac{1}{m}\\sum_{j=1}^{m}\\|g_{j}\\|_{\\mathcal{H}}}\\end{array}$ . The learning objective is thus defined as\n\n$$\n\\hat{f}_{\\lambda}:=\\underset{f\\in\\mathcal{F}}{\\arg\\min}\\widehat{\\mathcal{R}}(f)+\\lambda\\Omega(f),\\tag{3}\n$$\n\nwhere $\\lambda > 0$ is a regularisation parameter and Ω is currently $\\Omega_{0}$ from Equation (2). The function space $\\mathcal{F}$ is either $\\mathcal{F}_{\\infty}$ or $\\mathcal{F}_{m}$ . For statistical analysis in Section 4, we consider $\\mathcal{F}_{\\infty}$ while in practice, we compute the estimator using $\\mathcal{F}_{m}$ as discussed in Section 3. The rationale for using $\\mathcal{F}_{m}$ and expecting the statistical properties of $\\mathcal{F}_{\\infty}$ is elaborated in Section 3.2.\n\nIn the continuous setting, Equation (3) corresponds to\n\n$$\n\\min_{c\\in\\mathbb{R},\\mu\\in\\mathcal{P}(\\mathcal{H}\\times\\mathcal{S}^{d-1})}\\frac{1}{n}\\sum_{i=1}^{n}\\ell\\left(y_{i},c+\\int_{\\mathcal{H}\\times\\mathcal{S}^{d-1}}g(w^{\\top}x_{i})\\mathrm{d}\\mu(g,w)\\right)+\\lambda\\int_{\\mathcal{H}\\times\\mathcal{S}^{d-1}}\\|g\\|_{\\mathcal{H}}\\mathrm{d}\\mu(g,w),\\tag{4}\n$$\n\nwhile in the m-particles setting, Equation (3) becomes\n\n$$\n\\min_{c\\in\\mathbb{R},w_{1},\\dots,w_{m}\\in\\mathcal{S}^{d-1},g_{1},\\dots,g_{m}\\in\\mathcal{H}}\\frac{1}{n}\\sum_{i=1}^{n}\\ell\\left(y_{i},c+\\frac{1}{m}\\sum_{j=1}^{m}g_{j}\\left(w_{j}^{\\top}x_{i}\\right)\\right)+\\lambda\\frac{1}{m}\\sum_{j=1}^{m}\\|g_{j}\\|_{\\mathcal{H}},\\tag{5}\n$$\n\nwhere we clearly see a sparsity-inducing “grouped” penalty (Yuan and Lin, 2006).\n\nIn this subsection, we succinctly present some properties of reproducing kernel Hilbert spaces (RKHS) that are essential for our analysis. See Aronszajn (1950); Berlinet and Thomas-Agnan (2011) for an introduction to RKHS. Recall that we defined the Hilbert space H as\n\n$$\n\\mathcal{H}:=\\left\\{g:\\mathbb{R}\\to\\mathbb{R}\\mid g(0)=0,\\int_{\\mathbb{R}}(g^{\\prime})^{2}<   +\\infty\\right\\},\n$$\n\nwith the inner product $\\begin{array}{ r }{\\langle\\tilde{g},g\\rangle_{\\mathcal{H}}=\\int_{\\mathbb{R}}\\tilde{g}^{\\prime}g^{\\prime}}\\end{array}$ . This space is a reproducing kernel Hilbert space with the reproducing kernel $k^{(B)}(a,b)=(|a| + |b| - |a-b|)/2=\\mathrm{min}(|a|,|b|)\\mathbb{1}_{a b>0}$ . This kernel, which can be referred to as the “Brownian” kernel, corresponds to the covariance of the Brownian motion at times a and b (Mishura and Shevchenko, 2017). To see that $k^{(B)}$ is the reproducing kernel of H, it sufices to check that we have the reproducing property, i.e. that\n\n$$\n\\forall g\\in\\mathcal{H},\\forall a\\in\\mathbb{R},g(a)=\\langle g,k_{a}^{(B)}\\rangle_{\\mathcal{H}},\n$$\n\nwhere $k_{a}^{(B)}:b\\in\\mathbb{R}\\to k^{(B)}(a,b)\\in\\mathbb{R}^{2}$ As a reproducing kernel, it is positive definite, meaning that for any $n\\in\\mathbb{N},\\alpha\\in\\mathbb{R}^{n}$ , and $a\\in\\mathbb{R}^{n}$ , we have $\\begin{array}{ r }{\\sum_{i,j=1}^{n}\\alpha_{i}k^{(B)}(a_{i},a_{j})\\alpha_{j}\\geq 0}\\end{array}$\n\nAdditionally, we observe that $\\lVert\\boldsymbol{k}_{a}^{(B)}\\rVert_{\\mathcal{H}}^{2}=\\langle\\boldsymbol{k}_{a}^{(B)},\\boldsymbol{k}_{a}^{(B)}\\rangle=k^{(B)}(a,a)= |a|$ and we can obtain $\\|\\boldsymbol{k}_{a}^{(B)}-\\boldsymbol{k}_{b}^{(B)}\\|_{\\mathcal{H}}^{2}= |a-b|$ in a similar manner. It is also noteworthy that by definition, the functions in H are necessarily continuous, in fact even $1 / 2 \\cdot$ -Hölder continuous as we see in Lemma 3.\n\nWe first transform the optimisation problem before considering our setup from two diferent perspectives: one through kernel learning and the other through neural networks. To transform the optimisation problem, we use the representer theorem, a well-known result in RKHS theory that allows us to replace the optimisation over functions in the RKHS with optimisation over a finite weighted sum of the kernel at the data points.\n\nIn summary, we aim to perform multiple kernel learning with a kernel parametrized by a probability measure (Lemma 6), and approximate it using particles (Lemma 5).\n\nLemma 5 shows that we are solving a regularised kernel ridge regression problem where the kernel $\\begin{array}{ r }{\\frac{1}{m}\\sum_{j=1}^{m}(|w_{j}^{\\top}x| + |w_{j}^{\\top}x^{\\prime}| - |w_{j}^{\\top}(x-x^{\\prime})|)/2}\\end{array}$ is also learnt through the weights $(w_{j})_{j\\in [m]}$ and the third term $\\begin{array}{ r }{\\frac{\\lambda}{2}\\frac{1}{m}\\sum_{j=1}^{m}\\|w_{j}\\|\\quad}\\end{array}$ serves as a penalty to improve kernel learning.\n\nThe homogeneity of the kernel $k^{(B)}$ leads to well-behaved optimisation, as we discuss in Section 3.2 and see in Experiment 1 in Section 5.2. The kernel matrix K is indeed positively 1-homogeneous in the particles $(w_{j})_{j\\in [m]}$ . If we had chosen H to be the RKHS corresponding to the exponential kernel (or the Gaussian kernel), we would have faced the challenge of learning the kernel $\\begin{array}{ r }{\\sum_{j=1}^{m}e^{- |w_{j}^{\\top}(x-x^{\\prime})|}}\\end{array}$ , which exhibits a complex and nonhomogeneous dependency on the weights $(w_{j})_{j\\in [m]}$ . By using the Brownian kernel instead of the exponential kernel, we only slightly change the regularisation, regularising with $\\int_{\\mathbb{R}}(g^{\\prime})^{2}$ instead of $\\begin{array}{ r }{\\int_{\\mathbb{R}}g^{2}+\\int_{\\mathbb{R}}(g^{\\prime})^{2}}\\end{array}$ while making the optimisation more tractable.\n\nWe now present other penalties designed to achieve diferent efects. The three terms in Equation (8) correspond to the empirical risk, the standard penalty from KRR on the RKHS norm of the function, and an extra regularisation term on the learnt kernel weights. This additional term, $\\begin{array}{ r }{\\frac{\\lambda}{2 m}\\sum_{j=1}^{m}\\|w_{j}\\|}\\end{array}$ , originates from the penalty $\\Omega_{0}(f)$ in Equation (2). However, we can explore other penalties on $w_{1},\\ldots,w_{m}$ that induce various additional sparsity efects, even if they do not directly correspond to penalties on $f\\in\\mathcal{F}_{m}$ . Let $W\\in\\mathbb{R}^{d\\times m}$ be the matrix with $(w_{1},\\ldots,w_{m})$ as columns, denote by $W^{(a)}$ the a-th row of $W$ , and let $W=U S V^{\\top}$ be its singular value decomposition, with S a diagonal matrix composed of $S_{1},\\ldots,S_{\\operatorname*{min}(m,d)}$ Recall that ν is a probability distribution on $\\mathbb{R}^{d}$\n\n1. Basic penalty: $\\begin{array}{ r }{\\Omega_{\\mathrm{basic}}(w_{1},\\dots,w_{m})=\\frac{1}{2 m}\\sum_{j=1}^{m}\\left| |w_{j}|\\right|}\\end{array}$ , which we discussed in Section 2.1. In the continuous setting, it corresponds to $\\begin{array}{ r }{\\frac{1}{2}\\int_{\\mathbb{R}^{d}}\\|w\\|\\mathrm{d}\\nu(w)}\\end{array}$ . This penalty, which does not target any specific pattern in the data-generating mechanism, is the one for which we provide theoretical results in Section 4. However, it does not work as well in practice as the following penalties.\n\n2. Variable penalty: $\\begin{array}{ r }{\\Omega_{\\mathrm{variable}}(w_{1},\\dots,w_{m})=\\frac{1}{2}\\sum_{a=1}^{d}\\big(\\frac{1}{m}\\sum_{j=1}^{m}(w_{j})_{a}^{2}\\big)^{1/2}}\\end{array}$ , which is also equal to $\\begin{array}{ r }{\\frac{1}{2\\sqrt{m}}\\sum_{a=1}^{\\operatorname*{min}(m,d)}\\|W^{(a)}\\|_{2}}\\end{array}$ . This penalty, inspired by the group Lasso (Yuan and Lin, 2006), is designed for variable selection, pushing quantities $\\lVert W^{(a)}\\rVert_{2}$ towards zero, thus encouraging dependence on a few variables. In the continuous setting, it corresponds to $\\begin{array}{ r }{\\frac{1}{2}\\sum_{a=1}^{\\operatorname*{min}(m,d)}\\big(\\int_{\\mathbb{R}^{d}}|w_{a}|^{2}{\\mathrm{d}}\\nu(w)\\big)^{1/2}}\\end{array}$\n\n3. Feature penalty: $\\begin{array}{ r }{\\Omega_{\\mathrm{feature}}(w_{1},\\ldots,w_{m})=\\frac{1}{2}\\mathrm{tr}\\left(\\left(\\frac{1}{m}\\sum_{j=1}^{m}w_{j}w_{j}^{\\top}\\right)^{1/2}\\right)}\\end{array}$ , which is also equal to ${\\frac{1}{2}}\\sum_{a=1}^{\\operatorname*{min}(m,d)}{\\frac{S_{a}}{\\sqrt{m}}}$ and to the nuclear norm of W divided by $2\\sqrt{m}$ . It is used for feature learning as it is a convex relaxation of the rank, encouraging W to have low rank and thus dependence on only a few linear transformations of the data. Regularisation using the nuclear norm in the context of feature learning is wellestablished in the literature, as demonstrated by Argyriou et al. (2008). It corresponds to $\\begin{array}{ r }{\\frac 12\\operatorname{tr}\\big(\\big(\\int_{\\mathbb R^{d}}ww^{\\top}\\mathrm{d}\\nu(w)\\big)^{1/2}\\big)}\\end{array}$ in the continuous setting.\n\n4. Concave variable penalty: The concave version of the penalty for variable selection, Ω<sub>concave</sub> <sub>variable</sub> $\\begin{array}{ r }{(w_{1},\\ldots,w_{m})=\\frac{1}{2 s}\\sum_{a=1}^{d}\\log\\left(1+\\frac{s}{\\sqrt{m}}\\|W^{(a)}\\|_{2}\\right)}\\end{array}$ , with $s \\geq 0$ . The appeal of the added concavity is discussed below. In the continuous setting, it corresponds to $\\begin{array}{ r }{\\frac{1}{2 s}\\sum_{a=1}^{d}\\log\\left(1+s\\int_{\\mathbb{R}^{d}}(w_{a})^{2}{\\mathrm{d}\\nu}(w)\\right)^{1/2})}\\end{array}$\n\n5. Concave feature penalty: The concave version of the penalty intended for feature learning, $\\Omega_{\\mathrm{concave}}$ feature $\\begin{array}{ r }{(w_{1},\\ldots,w_{m})=\\frac{1}{2 s}\\sum_{a=1}^{\\operatorname*{min}(m,d)}\\log\\left(1+\\frac{s}{\\sqrt{m}}S_{a}\\right)}\\end{array}$ for feature selection, with $s \\geq 0$ . The appeal of the added concavity is discussed below. In the continuous setting it corresponds to $\\begin{array}{ r }{\\frac{1}{2 s}\\sum_{a=1}^{d}\\log\\big(1+s\\big(\\big(\\int_{\\mathbb{R}^{d}}w w^{\\top}\\mathrm{d}\\nu(w)\\big)^{1/2}\\big)_{a,a}\\big)}\\end{array}$\n\nThe first penalty is convex in both ν and W, making it straightforward to optimise. The second and third penalties, while not convex in $\\nu,$ are convex in W due to the presence of squared and square root terms on the components of W, easing optimisation in the m particles setting. The fourth and fifth penalties are neither convex in ν nor $W .$ , instead, they are concave in W. As s approaches zero, these penalties revert to their non-concave versions. Convex penalties, while easier to handle, can be detrimental by diminishing relevant variables or features to achieve sparsity. Mitigating this efect can involve retraining with the selected variables/features or employing concave penalties, which is the choice we made here. Although concave penalties are more complex to analyse, they often yield better performance because they drive the solution towards the boundary, promoting sparsity (Fan and Li, 2001; Bach et al., 2012). We discuss the impact of the choice of regularisation in Experiment 3 in Section 5.3.\n\nNote that the link with infinite-dimensional normed function spaces and the theoretical analysis of Section 4. only applies to the first “basic” penalty. Extending the analysis of concave penalties from finite dimension (Fan and Li, 2001; Zhang, 2010) to infinite dimension remains an open problem.\n\nIn this section, we detail the process of computing the estimator for each of the penalties presented in Section 2.5. We then discuss the importance of the homogeneity of the Brownian kernel and how the optimisation with particles relates to the continuous setting.\n\nIn this section, we focus on the square loss $\\ell(y,y^{\\prime})=\\textstyle{\\frac{1}{2}}(y-y^{\\prime})^{2}$ , which allows for explicit computations. However, the method can be extended to other loss functions using gradientbased techniques, (see Bach, 2024, Chapter 5). Recalling Equation (8) and the penalties described in Section 2.5, the optimisation problem we aim to solve is\n\n$$\n\\min_{w_{1},\\dots,w_{m}\\in\\mathbb{R}^{d},c\\in\\mathbb{R},\\alpha\\in\\mathbb{R}^{n}}\\frac{1}{2 n}\\|Y-K\\alpha-c\\mathbb{1}_{n}\\|_{2}^{2}+\\frac{\\lambda}{2}\\alpha^{\\top}K\\alpha+\\lambda\\Omega_{\\text{weights}}\\left(w_{1},\\dots,w_{m}\\right),\\tag{10}\n$$\n\nwhere $\\begin{array}{ r }{K=\\frac{1}{m}\\sum_{j=1}^{m}K^{(w_{j})}}\\end{array}$ and $\\Omega_{\\mathrm{weights}}$ represents any of the penalties from Section 2.5.\n\nTo solve this problem, we alternate between exact minimisation with respect to α and $c,$ which is done in closed-form, and one improvement step for the minimisation with respect to $w_{1},\\ldots,w_{m}$ which is done using one step of proximal gradient descent. This corresponds to minimizing with respect to the $w_{1},\\ldots,w_{m}$ the infimum with respect to c and $\\alpha .$\n\nWhen the weights $w_{1},\\ldots,w_{m}$ are fixed, the kernel matrix K is also fixed, allowing us to find the solution for the constant c and the coeficients α in closed-form. By centring both the kernel matrix and the response $Y$ , we transform the problem into a classical kernel ridge regression problem, for which explicit solutions are well-known.\n\nNext, we focus on optimising $w_{1},\\ldots,w_{m}$ while keeping c and α fixed. The goal is to solve\n\n$$\n\\min_{w_{1},\\dots,w_{m}\\in\\mathbb{R}^{d}}G(w_{1},\\dots,w_{m})+\\lambda\\Omega_{\\text{weights}}(w_{1},\\dots,w_{m}),\\tag{11}\n$$\n\nwhere the dependence on $(w_{j})_{j\\in [m]}$ in the first term is through the kernel matrix K. Note that G is convex in K but not in $w_{1},\\ldots,w_{m}$ . Additionally, G is diferentiable almost everywhere, except where $w_{j}^{\\top}(x_{i}-x_{i^{\\prime}})$ for some $j\\in [m],i\\neq i^{\\prime}\\in [n]$ . However, standard practice assumes that these non-diferentiabilities average out with many data points. Meanwhile, the penalties $\\Omega_{\\mathrm{weights}}$ are not diferentiable at certain fixed points, independently of the data, similarly to the Lasso penalty. Therefore, we use proximal gradient descent to solve Equation (11). With a step-size $\\gamma > 0$ , this involves minimising\n\n$$\n\\sum_{j=1}^{m}\\frac{\\partial G}{\\partial w_{j}}(w^{\\mathrm{old}})^{\\top}(w_{j}-w_{j}^{\\mathrm{old}})+\\frac{1}{2\\gamma}\\sum_{j=1}^{m}\\|w_{j}-w_{j}^{\\mathrm{old}}\\|_{2}^{2}+\\lambda\\Omega_{\\mathrm{weights}}(w_{1},\\ldots,w_{m}),\n$$\n\nover $w_{1},\\ldots,w_{m}\\in\\mathbb{R}^{d}$ . This corresponds to the simultaneous proximal gradient descent steps $\\begin{array}{ r }{w_{j}\\mathrm{prox}_{\\lambda\\gamma\\Omega}(w_{j}-\\gamma\\frac{\\partial G}{\\partial w_{i}})}\\end{array}$ . We therefore compute the gradient and the proximal operator. For the gradient, we have the following lemma.", "theorems": [{"local_id": "T1", "statement": "Lemma 5 (Kernel Formulation of Finite-Width) Equation (5) is equivalent to\n\n$$\n\\min_{w_{1},\\dots,w_{m}\\in\\mathbb{R}^{d},c\\in\\mathbb{R},\\alpha\\in\\mathbb{R}^{n}}\\frac{1}{n}\\sum_{i=1}^{n}\\ell(y_{i},(K\\alpha)_{i}+c)+\\frac{\\lambda}{2}\\alpha^{\\top}K\\alpha+\\frac{\\lambda}{2}\\frac{1}{m}\\sum_{j=1}^{m}\\|w_{j}\\|,\\tag{8}\n$$\n\nwhere $\\begin{array}{ r }{K=\\frac{1}{m}\\sum_{j=1}^{m}K^{(w_{j})}}\\end{array}$ , and $K^{(w_{j})}\\in\\mathbb{R}^{n\\times n}$ is the kernel matrix $f o r$ kernel $k^{(B)}$ and projected data $(w_{j}^{\\top}x_{1},\\ldots,w_{j}^{\\top}x_{n}),\\ i . e .,K_{i,i^{\\prime}}^{(w_{j})}=(|w_{j}^{\\top}x_{i}| + |w_{j}^{\\top}x_{i^{\\prime}}| - |w_{j}^{\\top}(x_{i}-x_{i^{\\prime}})|)/2$ . Notice that there are no constraints on the particles $(w_{j})_{j\\in [m]}$ to belong to the unit sphere anymore.", "proof": {"steps": [{"n": 1, "claim": "Proof [Proof of Lemma 5] Our goal is to transform Equation (5). We begin with the following trick for the m particles setting\n\n$$\n\\frac{1}{m}\\sum_{j=1}^{m}\\|g_{j}\\|_{\\mathcal{H}}=\\inf_{\\beta\\in\\mathbb{R}_{+}^{m}}\\frac{1}{2 m}\\sum_{j=1}^{m}\\left(\\frac{\\|g_{j}\\|_{\\mathcal{H}}^{2}}{\\beta_{j}}+\\beta_{j}\\right).\n$$"}, {"n": 2, "claim": "This allows us to rewrite Equation 5 in the following way\n\n$$\n\\min_{c\\in\\mathbb{R},w_{1},\\dots,w_{m}\\in\\mathcal{S}^{d-1},g_{1},\\dots,g_{m}\\in\\mathcal{H},\\beta\\in\\mathbb{R}_{+}^{m}}\\frac{1}{n}\\sum_{i=1}^{n}\\ell\\left(y_{i},c+\\frac{1}{m}\\sum_{j=1}^{m}g_{j}(w_{j}^{\\top}x_{i})\\right)+\\lambda\\frac{1}{2 m}\\sum_{j=1}^{m}\\left(\\frac{\\|g_{j}\\|_{\\mathcal{H}}^{2}}{\\beta_{j}}+\\beta_{j}\\right).\n$$"}, {"n": 3, "claim": "Now if we fix $(w_{j})_{j\\in [m]}$ and $(\\beta_{j})_{j\\in [m]}$ in the equation above, the minimisation problem is equivalent to\n\n$$\n\\min_{c\\in\\mathbb{R},g_{1},\\dots,g_{m}\\in\\mathcal{H}}\\frac{1}{n}\\sum_{i=1}^{n}\\ell(y_{i},c+\\frac{1}{m}\\sum_{j=1}^{m}g_{j}(w_{j}^{\\top}x_{i}))+\\frac{\\lambda}{2}\\frac{1}{m}\\sum_{j=1}^{m}\\frac{\\|g_{j}\\|_{\\mathcal{H}}^{2}}{\\beta_{j}}.\\tag{17}\n$$"}, {"n": 4, "claim": "Using the representer theorem (Schölkopf et al., 2001), we express each $x  g_{j}(w_{j}^{\\top}x)$ as\n\n$$\nx\\to\\sum_{i=1}^{n}\\alpha_{i}^{(j)}k^{(B)}(w_{j}^{\\top}x_{i},w_{j}^{\\top}x),\n$$\n\nwhich leads to\n\n$$\n\\|g_{j}\\|_{\\mathcal{H}}^{2}=\\sum_{i,i^{\\prime}=1}^{n}\\alpha_{i}^{(j)}\\alpha_{i^{\\prime}}^{(j)}k^{(B)}(w_{j}^{\\top}x_{i},w_{j}^{\\top}x_{i^{\\prime}}).\n$$"}, {"n": 5, "claim": "Rewriting the norm and evaluation in kernel form with $K_{i,i^{\\prime}}^{(w_{j})}=k^{(B)}(w_{j}^{\\top}x_{i},w_{j}^{\\top}x_{i^{\\prime}})$ , we obtain\n\n$$\n\\|g_{j}\\|_{\\mathcal{H}}^{2}=(\\alpha^{(j)})^{\\top}K^{(w_{j})}\\alpha^{(j)},\n$$\n\nand\n\n$$\ng_{j}(w_{j}^{\\top}x_{i})=(K^{(w_{j})}\\alpha^{(j)})_{i}.\n$$"}, {"n": 6, "claim": "Thus, we transform Equation (17) into\n\n$$\n\\min_{c\\in\\mathbb{R},\\alpha^{(1)},\\dots,\\alpha^{(m)}\\in\\mathbb{R}^{n}}\\frac{1}{n}\\sum_{i=1}^{n}\\ell(y_{i},\\frac{1}{m}\\sum_{j=1}^{m}(K^{(w_{j})}\\alpha^{(j)})_{i}+c)+\\frac{\\lambda}{2}\\frac{1}{m}\\sum_{j=1}^{m}\\frac{(\\alpha^{(j)})^{\\top}K^{(w_{j})}\\alpha^{(j)}}{\\beta_{j}}.\n$$"}, {"n": 7, "claim": "We show that minimisation is attained for vectors $\\alpha^{(j)}$ equal to $\\beta_{j}\\alpha$ for a single vector $\\alpha .$ Consider the convex problem\n\n$$\n\\min_{\\alpha^{(1)},\\dots,\\alpha^{(m)}\\in\\mathbb{R}^{n}}\\frac{1}{2}\\frac{1}{m}\\sum_{j=1}^{m}\\frac{(\\alpha^{(j)})^{\\top}K^{(w_{j})}\\alpha^{(j)}}{\\beta_{j}},\n$$\n\nsubject to $\\begin{array}{ r }{\\frac{1}{m}\\sum_{j=1}^{m}K^{(w_{j})}\\alpha^{(j)}=z}\\end{array}$ where $z\\in\\mathbb{R}^{d}$ . We define the Lagrangian\n\n$$\n\\mathcal{L}(\\alpha^{(1)},\\dots,\\alpha^{(m)},\\alpha)=\\frac{1}{2}\\frac{1}{m}\\sum_{j=1}^{m}\\frac{(\\alpha^{(j)})^{\\top}K^{(w_{j})}\\alpha^{(j)}}{\\beta_{j}}+\\alpha^{\\top}\\left(z-\\frac{1}{m}\\sum_{j}K^{(w_{j})}\\alpha^{(j)}\\right).\n$$"}, {"n": 8, "claim": "By taking the diferential of L with respect to $\\alpha^{(j)}$ at the optimum, we get\n\n$$\n\\frac{\\partial\\mathcal{L}}{\\partial\\alpha^{(j)}}=\\frac{1}{m}K^{(w_{j})}\\left(\\frac{\\alpha^{(j)}}{\\beta_{j}}-\\alpha\\right)=0.\n$$"}, {"n": 9, "claim": "The diferential with respect to α yields that at the optimum, the constraint is verified, $\\begin{array}{ r }{\\mathrm{i . e .,}z=\\frac{1}{m}\\sum_{i}K^{(w_{j})}\\alpha^{(j)}}\\end{array}$ . We note that for $\\alpha^{(j)}=\\beta_{j}\\alpha$ , all equations are satisfied, yielding the desired result."}, {"n": 10, "claim": "We can then write Equation (5) as\n\n$$\n\\min_{w_{1},\\dots,w_{m}\\in\\mathbb{R}^{d},c\\in\\mathbb{R},\\beta\\in\\mathbb{R}_{+}^{m},\\alpha\\in\\mathbb{R}^{n}}\\frac{1}{n}\\sum_{i=1}^{n}\\ell(y_{i},(K\\alpha)_{i}+c)+\\frac{\\lambda}{2}\\alpha^{\\top}K\\alpha+\\frac{\\lambda}{2}\\frac{1}{m}\\sum_{j=1}^{m}\\beta_{j},\n$$\n\nwith the constraints $\\forall j\\in [m],w_{j}\\in S^{d-1}$ , and $\\begin{array}{ r }{K=\\frac{1}{m}\\sum_{j=1}^{m}\\beta_{j}K^{(w_{j})}}\\end{array}$\n\nWe notice that $\\beta_{j}K^{(w_{j})}=K^{(\\beta_{j}w_{j})}$ due to the positive homogeneity of the Brownian kernel. We therefore introduce the change of variable $\\beta_{j}w_{j}=\\tilde{w}_{j}$\n\n$$\n\\min_{\\tilde{w}_{1},\\dots,\\tilde{w}_{m}\\in\\mathbb{R}^{d},c\\in\\mathbb{R},\\alpha\\in\\mathbb{R}^{n}}\\frac{1}{n}\\sum_{i=1}^{n}\\ell(y_{i},(K\\alpha)_{i}+c)+\\frac{\\lambda}{2}\\alpha^{\\top}K\\alpha+\\frac{\\lambda}{2}\\frac{1}{m}\\sum_{j=1}^{m}\\|\\tilde{w}_{j}\\|,\n$$\n\nwith $\\begin{array}{ r }{K=\\frac{1}{m}\\sum_{j=1}^{m}K^{(\\tilde{w}_{j})}}\\end{array}$ and no constraint on the norm of $\\tilde{w}_{j}$ . For ease of exposition in the main text, we replace w˜ by w. ■"}]}}, {"local_id": "T2", "statement": "Lemma 6 (Kernel Formulation of Infinite-Width) Equation (4) is equivalent to:\n\n$$\n\\min_{\\nu\\in\\mathcal{P}(\\mathbb{R}^{d}),c\\in\\mathbb{R},\\alpha\\in\\mathbb{R}^{n}}\\frac{1}{n}\\sum_{i=1}^{n}\\ell(y_{i},(K\\alpha)_{i}+c)+\\frac{\\lambda}{2}\\alpha^{\\top}K\\alpha+\\frac{\\lambda}{2}\\int_{\\mathbb{R}^{d}}\\|w\\|\\mathrm{d}\\nu(w),\\tag{9}\n$$\n\nwith $\\begin{array}{ r }{K=\\int_{\\mathbb{R}^{d}}K^{(w)}\\mathrm{d}\\nu(w)}\\end{array}$ and and $K^{(w)}\\in\\mathbb{R}^{n\\times n}$ is the kernel matrix for kernel $k^{(B)}$ and data $(w^{\\top}x_{1},\\ldots,w^{\\top}x_{n})$", "proof": {"steps": [{"n": 1, "claim": "Proof [Proof of Lemma 6] The proof follows the same steps as the proof of Lemma $5,$ systematically replacing any $\\textstyle{\\frac{1}{m}}\\sum_{j=1}^{m}$ with the appropriate integral over $\\mathcal{H}\\times\\mathcal{S}^{d-1}$ with respect to measure $\\mu .$ . Before the change of variables, the problem is\n\n$$\n\\min_{\\mu\\in\\mathcal{P}(\\mathcal{H}\\times\\mathcal{S}^{d-1}),c\\in\\mathbb{R},(\\beta_{g})_{g}\\in\\mathbb{R}_{+}^{\\mathcal{H}},\\alpha\\in\\mathbb{R}^{n}}\\frac{1}{n}\\sum_{i=1}^{n}\\ell(y_{i},(K\\alpha)_{i}+c)+\\frac{\\lambda}{2}\\alpha^{\\top}K\\alpha+\\frac{\\lambda}{2}\\int_{\\mathcal{H}\\times\\mathcal{S}^{d-1}}\\beta_{g}\\mathrm{d}\\mu(g,w),\n$$\n\nwhere $\\begin{array}{ r }{K=\\int_{\\mathcal{H}\\times S^{d-1}}\\beta_{g}K^{(w)}d\\mu(g,w)=\\int_{\\mathcal{H}\\times S^{d-1}}K^{(\\beta_{g}w)}d\\mu(g,w)}\\end{array}$ ."}, {"n": 2, "claim": "The change of variables $\\beta_{g}w=\\tilde{w}$ transforms the problem into\n\n$$\n\\min_{(\\beta_{g})_{g}\\in\\mathbb{R}_{+}^{\\mathcal{H}},\\nu\\in\\mathcal{P}(\\{\\beta_{g}w,g\\in\\mathcal{H},w\\in\\mathcal{S}^{d-1}\\}),c\\in\\mathbb{R},\\alpha\\in\\mathbb{R}^{n}}\\frac{1}{n}\\sum_{i=1}^{n}\\ell(y_{i},(K\\alpha)_{i}+c)+\\frac{\\lambda}{2}\\alpha^{\\top}K\\alpha+\\frac{\\lambda}{2}\\int_{\\mathbb{R}^{d}}\\|\\tilde{w}\\|\\mathrm{d}\\nu(\\tilde{w}),\n$$\n\nwith $\\begin{array}{ r }{K=\\int_{\\mathbb{R}^{d}}K^{(\\tilde{w})}\\mathrm{d}\\nu(\\tilde{w})}\\end{array}$ . We can consider the integral over R<sup>d</sup> instead of $\\{\\beta_{g}w,g\\in\\mathcal{H},w\\in$ $S^{d-1}\\bigr\\}$ by extending ν with $\\nu(\\mathbb{R}^{d}\\setminus\\{\\beta_{g}w,g\\in\\mathcal{H},w\\in\\mathcal{S}^{d-1}\\})=0$ ."}, {"n": 3, "claim": "This is equivalent to considering the minimum over $\\nu\\in\\mathcal{P}(\\mathbb{R}^{d})$ instead of the minimum over $(\\beta_{g})_{g\\in\\mathcal{H}}\\in\\mathbb{R}_{+}^{\\mathcal{H}}$ and $\\nu\\in\\mathcal{P}(\\{\\beta_{g}w,g\\in\\mathcal{H},w\\in\\mathcal{S}^{d-1}\\})$ , as we discuss now."}, {"n": 4, "claim": "Indeed, the first minimum is smaller as it is considered over a larger space, but they are equal because both the norm $\\| \\cdot \\|$ and the kernel K are positively homogeneous. Hence, the problem finally becomes\n\n$$\n\\min_{\\nu\\in\\mathcal{P}(\\mathbb{R}^{d}),c\\in\\mathbb{R},\\alpha\\in\\mathbb{R}^{n}}\\frac{1}{n}\\sum_{i=1}^{n}\\ell(y_{i},(K\\alpha)_{i}+c)+\\frac{\\lambda}{2}\\alpha^{\\top}K\\alpha+\\frac{\\lambda}{2}\\int_{\\mathbb{R}^{d}}\\|\\tilde{w}\\|\\mathrm{d}\\nu(\\tilde{w}),\n$$\n\nwith $\\begin{array}{ r }{K=\\int_{\\mathbb{R}^{d}}K^{(\\tilde{w})}\\mathrm{d}\\nu(\\tilde{w})}\\end{array}$ . For ease of exposition in the main text, we replace $\\tilde{w}$ by w."}]}}, {"local_id": "T3", "statement": "Lemma 7 (Optimisation for Fixed Particles) For fixed $w_{1},\\ldots,w_{m}$ and hence a fixed $K$ , define\n\n$$\nG(w_{1},\\dots,w_{m}):=\\min_{\\alpha\\in\\mathbb{R}^{n},c\\in\\mathbb{R}}\\frac{1}{2 n}\\|Y-K\\alpha-c\\mathbb{1}_{n}\\|_{2}^{2}+\\frac{\\lambda}{2}\\alpha^{\\top}K\\alpha .\n$$\n\nThe optimisation problem defining G is solved by\n\n$$\n\\alpha=(\\tilde{K}+n\\lambda I)^{-1}\\tilde{Y}\\quad\\mathrm{and}\\quad c=\\frac{\\mathbb{1}^{\\top}Y}{n}-\\frac{\\mathbb{1}^{\\top}K\\alpha}{n},\n$$\n\nwhere $\\tilde{K}: =\\Pi K\\Pi$ and $\\begin{array}{ r }{\\tilde{Y}: =Y-\\frac{\\mathbb{1}\\mathbb{1}^{\\top}Y}{n}}\\end{array}$ , with $\\begin{array}{ r }{\\Pi=I-\\frac{\\mathbb{1}\\mathbb{1}^{\\top}}{n}}\\end{array}$ being the centring matrix. The objective value is then\n\n$$\nG(w_{1},\\dots,w_{m})=\\frac{\\lambda}{2}\\tilde{Y}^{\\top}(\\tilde{K}+\\lambda n I)^{-1}\\tilde{Y}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof [Proof of Lemma 7] For a fixed $\\alpha,$ , the optimal c is given by $\\begin{array}{ r }{c=\\frac{\\mathbb{1}^{\\top}Y}{n}-\\frac{\\mathbb{1}^{\\top}K\\alpha}{n}}\\end{array}$ Substituting this back into the objective function, we obtain\n\n$$\n\\min_{\\alpha\\in\\mathbb{R}^{n}}\\frac{1}{2 n}\\|\\Pi Y-\\Pi K\\alpha\\|_{2}^{2}+\\frac{\\lambda}{2}\\alpha^{\\top}K\\alpha,\n$$\n\nwhich is minimised for α satisfying $(K\\Pi K+n\\lambda K)\\alpha=K\\Pi Y$ . We can further simplify this by observing that if $(\\Pi K+n\\lambda I)\\alpha=\\Pi Y$ , then the previous condition is satisfied."}, {"n": 2, "claim": "From the equation nλα $= \\Pi Y - \\Pi K \\alpha$ , we can deduce that $\\Pi \\alpha = \\alpha$ because $\\Pi^{2}=\\Pi$ Therefore, we can express α as $\\alpha=\\Pi\\tilde{\\alpha}$ . Substituting this change of variable into the original problem, we define $\\tilde{K}: =\\Pi K\\Pi$ and $\\tilde{Y}: =\\Pi Y$ , transforming the problem into\n\n$$\n\\min_{\\tilde{\\alpha}\\in\\mathbb{R}^{n}}\\frac{1}{2 n}\\|\\tilde{Y}-\\tilde{K}\\tilde{\\alpha}\\|_{2}^{2}+\\frac{\\lambda}{2}\\tilde{\\alpha}^{\\top}\\tilde{K}\\tilde{\\alpha}.\n$$"}, {"n": 3, "claim": "This is a standard kernel ridge regression problem (noting that $\\tilde{K}$ is still a valid kernel matrix), for which the solution is known to be $\\tilde{\\alpha}=(\\tilde{K}+n\\lambda I)^{-1}\\tilde{Y}$ . We also have $\\Pi\\tilde{\\alpha}=\\tilde{\\alpha}.$ implying $\\alpha=\\tilde{\\alpha}$ because one can show that $\\mathbb{1}^{\\top}\\tilde{\\alpha}=0$ . To see why, note that $\\mathbb{1}^{\\top}\\tilde{\\alpha}=\\langle\\mathbb{1},\\tilde{\\alpha}\\rangle=$ $\\langle(\\tilde{K}+n\\lambda I)^{-1}\\mathbf{1},\\tilde{Y}\\rangle$ . Since $(\\tilde{K}+n\\lambda I)^{-1}\\mathbf{1}$ is proportional to 1 (as 1 is an eigenvector of $\\tilde{K}+n\\lambda I$ and its inverse), and $\\langle\\tilde{Y},\\mathbb{1}\\rangle=0$ , we obtain the desired result."}, {"n": 4, "claim": "Finally, we verify the optimal condition $(K\\Pi K+n\\lambda K)\\alpha=K\\Pi Y$ . Given $(\\Pi K\\Pi+$ $n\\lambda I)\\tilde{\\alpha}=\\Pi Y$ by definition, multiplying by K yields $(K\\Pi K\\Pi+n\\lambda K)\\tilde{\\alpha}=K\\Pi Y$ . Since $\\tilde{\\alpha}=\\alpha=\\Pi\\alpha$ , the desired result follows. ■"}]}}, {"local_id": "T4", "statement": "Lemma 8 (Gradient of G) Let $j \\in [ m ]$ , then\n\n$$\n\\frac{\\partial G}{\\partial w_{j}}=\\frac{\\lambda}{4}\\frac{1}{m}\\sum_{i,i^{\\prime}=1}^{n}z_{i}z_{i^{\\prime}}\\mathrm{sign}(w_{j}^{\\top}(x_{i}-x_{i^{\\prime}}))(x_{i}-x_{i^{\\prime}}),\n$$\n\nwhere $z=(\\tilde{K}+n\\lambda I)^{-1}\\tilde{Y}$", "proof": {"steps": [{"n": 1, "claim": "Proof [Proof of Lemma 8] First, we compute the derivative of $\\begin{array}{ r }{G=\\frac{\\lambda}{2}\\tilde{Y}^{\\top}(\\tilde{K}+\\lambda n I)^{-1}\\tilde{Y}}\\end{array}$ with respect to $w_{j}$\n\n$$\n\\begin{array}{l}\\frac{\\partial G}{\\partial w_{j}}=\\sum_{i,i^{\\prime}=1}^{n}\\frac{\\partial G}{\\partial K_{i,i^{\\prime}}}\\frac{\\partial K_{i,i^{\\prime}}}{\\partial w_{j}}\\\\ =\\frac{1}{m}\\sum_{i,i^{\\prime}=1}^{n}\\frac{\\partial G}{\\partial K_{i,i^{\\prime}}}\\frac{\\left(\\operatorname{sign}\\left(w_{j}^{\\top}x_{i}\\right)x_{i}+\\operatorname{sign}\\left(w_{j}^{\\top}x_{i^{\\prime}}\\right)x_{i^{\\prime}}-\\operatorname{sign}\\left(w_{j}^{\\top}\\left(x_{i}-x_{i^{\\prime}}\\right)\\right)\\left(x_{i}-x_{i^{\\prime}}\\right)\\right)}{2}.\\tag{18}\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "We know that\n\n$$\n\\frac{\\partial G}{\\partial(\\tilde{K}+\\lambda n I)}= -\\frac{\\lambda}{2}(\\tilde{K}+\\lambda n I)^{-1}\\tilde{Y}\\tilde{Y}^{\\top}(\\tilde{K}+\\lambda n I)^{-1},\n$$\n\nthus\n\n$$\n\\begin{array}{l}\\frac{\\partial G}{\\partial K_{i,i^{\\prime}}}=\\sum_{l,k}\\frac{\\partial G}{\\partial(\\tilde{K}+\\lambda n I)_{l,k}}\\frac{\\partial(\\Pi K\\Pi+\\lambda n I)_{l,k}}{\\partial K_{i,i^{\\prime}}}\\\\ =\\sum_{l,k}-\\frac{\\lambda}{2}((\\tilde{K}+\\lambda n I)^{-1}\\tilde{Y}\\tilde{Y}^{\\top}(\\tilde{K}+\\lambda n I)^{-1})_{l,k}\\Pi_{l,i}\\Pi_{i^{\\prime},k}\\\\ = -\\frac{\\lambda}{2}\\left(\\Pi(\\tilde{K}+\\lambda n I)^{-1}\\tilde{Y}\\tilde{Y}^{\\top}(\\tilde{K}+\\lambda n I)^{-1}\\Pi\\right)_{i,i^{\\prime}}\\\\ = -\\frac{\\lambda}{2}(\\Pi(\\tilde{K}+\\lambda n I)^{-1}\\tilde{Y})_{i}(\\Pi(\\tilde{K}+\\lambda n I)^{-1}\\tilde{Y})_{i^{\\prime}}.\\\\ \\end{array}\n$$"}, {"n": 3, "claim": "Substituting this back into Equation (18) and introducing $S_{j}\\in\\mathbb{R}^{n\\times n}$ with $(S_{j})_{i,i^{\\prime}}=$ $(\\mathrm{sign}(w_{j}^{\\top}x_{i})x_{i}+\\mathrm{sign}(w_{j}^{\\top}x_{i^{\\prime}})x_{i^{\\prime}}-\\mathrm{sign}(w_{j}^{\\top}(x_{i}-x_{i^{\\prime}}))(x_{i}-x_{i^{\\prime}}))/2,$ , we get\n\n$$\n\\begin{array}{l}\\frac{\\partial G}{\\partial w_{j}}= -\\frac{\\lambda}{2 m}\\sum_{i,i^{\\prime}=1}^{n}(\\Pi(\\tilde{K}+\\lambda n I)^{-1}\\tilde{Y})_{i}(\\Pi(\\tilde{K}+\\lambda n I)^{-1}\\tilde{Y})_{i^{\\prime}}(S_{j})_{i,i^{\\prime}}\\\\ = -\\frac{\\lambda}{2 m}\\operatorname{tr}\\left((\\Pi(\\tilde{K}+\\lambda n I)^{-1}\\tilde{Y})^{\\top}S_{j}(\\Pi(\\tilde{K}+\\lambda n I)^{-1}\\tilde{Y})\\right)\\\\ = -\\frac{\\lambda}{2 m}\\operatorname{tr}\\left(((\\tilde{K}+\\lambda n I)^{-1}\\tilde{Y})^{\\top}\\Pi S_{j}\\Pi((\\tilde{K}+\\lambda n I)^{-1}\\tilde{Y})\\right).\\\\ \\end{array}\n$$"}, {"n": 4, "claim": "This implies that we can replace $(S_{j})_{i,i^{\\prime}}$ with the i-th, i<sup>0</sup>-th component of any matrix with the same centred version, such as ${\\tilde{S}}_{j}$ where $(\\tilde{S}_{j})_{i,i^{\\prime}}= -\\mathrm{sign}(w_{j}^{\\top}(x_{i}-x_{i}^{\\prime}))(x_{i}-x_{i}^{\\prime})$ 2 yielding the desired result."}]}}], "target_local_id": null, "topic_domain": "Deep Learning Theory", "topic_subtopic": "Representation & Expressivity", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2025-24-2016", "context": "Throughout this paper, real numbers are represented by lowercase, non-bold letters, such as a, while vectors in $\\mathbb{R}^{d}$ for d $\\geq 2$ are written in boldface lowercase, such as a. Constant vectors are denoted using bold numerals, such as 1 and 0. The ith element of a is denoted by $a_{i}.$ For indexed vector such as $\\mathbf{a}_{j}$ , its ith element is denoted as $a_{j i}$ . Unless otherwise specified, random variables are written in uppercase non-bold (for example, X), while random vectors use bold uppercase (for example, X). The ith element of X and $\\mathbf{X}_{j}$ are denoted as $X_{i}$ and $X_{j i}$ , respectively. The set of non-negative integers is denoted by $\\mathbb{N}_{0}: =\\{0,1,2,...\\}$ . For positive integers u and v, [u] represents $\\{1,\\ldots,u\\}$ , and $[u]^{v}$ denotes its Cartesian product taken v times. A set of u elements indexed by i is written as $\\{a_{i}\\}_{i\\in [u]}: =\\{a_{1},...,a_{u}\\}$ . For any real $s > 0$ , s denotes the largest integer strictly smaller than s. For $a,b\\in\\mathbb{R}$ , we define $a\\vee b: =\\operatorname*{max}\\{a,b\\}$ and $a\\wedge b: =\\operatorname*{min}\\{a,b\\}$ . Given $\\mathbf{w}\\in\\mathbb{R}^{d}$ and $p \\geq 1$ , its $\\ell_{p}.$ -norm and $\\ell_{\\infty}{\\mathrm{-n o r m}}$ are defined by $\\begin{array}{ r }{\\|\\mathbf{w}\\|_{p}: =\\left(\\sum_{i=1}^{d}|w_{i}|^{p}\\right)^{1/p}}\\end{array}$ and $\\|\\mathbf{w}\\|_{\\infty}: =\\operatorname*{max}\\{|w_{1}|,\\ldots,|w_{d}|\\}$ respectively. For a function supported on $[0,1]^{d}$ , its $\\mathbb{L}_{p}$ -norm and sup-norm are defined as follows:\n\n$$\n\\|f\\|_{\\mathbb{L}_{p}}:=\\left(\\int_{[0,1]^{d}}|f(\\mathbf{x})|^{p}d\\mathbf{x}\\right)^{1/p},\\quad 1\\leq p<\\infty,\\quad\\|f\\|_{\\infty}:=\\sup\\left\\{|f(\\mathbf{x})|:\\mathbf{x}\\in [0,1]^{d}\\right\\}.\n$$\n\nFor any u, $\\mathbf{v}\\in\\mathbb{R}^{d}$ , with $u_{j}\\leq v_{j}$ for $j=1,\\ldots,d,$ a hyperrectangle $[\\mathbf{u},\\mathbf{v}]$ is defined as follows:\n\n$$\n[\\mathbf{u},\\mathbf{v}]:=\\prod_{j=1}^{d}[u_{j},v_{j}].\n$$\n\nGiven the privacy parameter $\\alpha > 0$ , we write $z_{\\alpha}: =e^{2\\alpha}-e^{-2\\alpha}=2\\sinh(2\\alpha)$ . A constant that only depends on parameters $\\theta_{1},\\theta_{2},\\ldots$ . is denoted as $C(\\theta_{1},\\theta_{2},\\ldots)$ . The indicator function $\\mathbb{1}(\\mathcal{A})$ takes value 1 if the event $\\mathcal{A}$ is true and 0 otherwise.\n\nLet $\\mathcal{P}$ denote the space of pairs of distributions of interest: multinomial distributions taking values in [k] or continuous distributions taking values in $[0,1]^{d}.$ . Given a pair of distributions $(P_{Y},P_{Z})\\in\\mathcal{P}$ , for each $i\\in [n_{1}]$ , the ith data owner draws $Y_{i}$ from $P_{Y}$ independently from the others. Similarly, for each $j\\in [n_{2}]$ , the jth data owner independently draws $Z_{j}$ from $P_{Z}$ We allow the sample sizes $n_{1}$ and $n_{2}$ to difer, and assume $n_{1}~\\leq~n_{2}$ , without loss of generality, throughout this paper. We denote the pooled sample size as $n: =n_{1}+n_{2}$ Under the LDP constraint, each owner releases only a randomized transformation of their raw sample, formally referred to as an LDP view:\n\nDefinition 1 (Local diferential privacy) Given a privacy level $\\alpha > 0$ , let $X_{i}$ and ${\\tilde{X}}_{i}$ be random elements mapped to measurable spaces $(\\mathcal{X},\\mathcal{F})$ and $(\\tilde{\\mathcal{X}}_{i},\\tilde{\\mathcal{F}}_{i})$ , respectively, for each $i \\in [ n ]$ . Then ${\\tilde{X}}_{i}$ is an α-local diferentially private $\\left(\\alpha-LDP\\right)$ view of $X_{i}$ if there exists a bivariate function $Q_{i}({\\cdot}\\mid\\cdot):\\tilde{\\mathcal{F}}_{i}\\times\\mathcal{X}\\mapsto [0,1]$ such that:\n\n1. For any $x\\in\\mathcal{X},Q_{i}(\\cdot|x)$ is a conditional distribution of ${\\tilde{X}}_{i}$ given $X_{i}=x_{\\mathrm{~}}$  \n2. For any $A\\in\\tilde{\\mathcal{F}}_{i},x\\mapsto Q_{i}(A\\vert x)$ is a measurable function on $\\mathcal{X}_{i}$ , and  \n3. For any $x,x^{\\prime}\\in\\mathcal{X}\\times\\mathcal{X}$ and $A\\in\\tilde{\\mathcal{F}}_{i}$ , the inequality $Q_{i}(A\\mid x)\\le e^{\\alpha}Q_{i}(A\\mid x^{\\prime})$ holds.\n\nLet ${\\mathcal{Q}}_{\\alpha}$ denote the collection of function sets $\\{Q_{i}\\}_{i=1}^{n}$ satisfying the above properties. Then $Q\\in\\mathcal{Q}_{\\alpha}$ is called an α-LDP mechanism (or channel) associated with $\\{X_{i}\\}_{i=1}^{n}$\n\nLet $\\mathcal{P}_{0}$ and $\\mathcal{P}_{1}$ denote the collections of null and alternative distributions, respectively, corresponding to $H_{0}$ and $H_{1}$ introduced in (1). In the minimax framework, we focus on a subset of $\\mathcal{P}_{1}$ , denoted as $\\mathcal{P}_{1}(\\rho_{n_{1},n_{2}})$ , where $\\rho_{n_{1},n_{2}}$ indicates a minimum separation between $\\mathcal{P}_{1}(\\rho_{n_{1},n_{2}})$ and $\\mathcal{P}_{0}$\n\nGiven a privacy mechanism $Q,$ the curator evaluates a test function $\\begin{array}{ r }{\\Delta_{Q}:\\prod_{i=1}^{n}\\tilde{\\mathcal{X}}_{i}\\mapsto}\\end{array}$ $\\{0,1\\}$ on the α-LDP views and rejects $H_{0}$ if $\\Delta_{Q}\\ =\\ 1$ and accepts $H_{0}$ otherwise. Our objective is to design the private test $\\Delta_{Q}$ that controls the type I and II errors uniformly over distributions in $\\mathcal{P}_{0}$ and $\\mathcal{P}_{1}(\\rho_{n_{1},n_{2}})$ , respectively. In particular, for fixed $\\gamma,\\beta\\in(0,1)$ ， we aim for the private test $\\Delta_{Q}$ to satisfy the following conditions:\n\n$$\n\\text{Type I error:}\\sup_{(P_{Y},P_{Z})\\in\\mathcal{P}_{0}}\\mathbb{E}[\\Delta_{Q}]\\leq\\gamma\\quad \\text{and}\\quad \\text{Type II error:}\\sup_{(P_{Y},P_{Z})\\in\\mathcal{P}_{1}(\\rho_{n_{1},n_{2}})}\\mathbb{E}[1-\\Delta_{Q}]\\leq\\beta .\\tag{2}\n$$\n\nLet $\\Phi_{\\gamma}^{\\alpha}$ be the set of α-LDP level-γ tests, which take α-LDP views as inputs and control the type I error as in (2). The quality of a level-γ test $\\Delta_{Q,\\gamma}\\in\\Phi_{\\gamma}^{\\alpha}$ is measured by its uniform separation, which quantifies how close two hypotheses can be while still being reliably distinguished by $\\Delta_{Q,\\gamma}$ . In more technical terms, the uniform separation $\\tilde{\\rho}_{n_{1},n_{2}}(\\Delta_{Q,\\gamma})$ is the smallest separation $\\rho_{n_{1},n_{2}}$ which accomplishes the type II error control as in (2), namely,\n\n$$\n\\tilde{\\rho}_{n_{1},n_{2}}\\left(\\Delta_{Q,\\gamma}\\right):=\\inf\\left\\{\\rho_{n_{1},n_{2}}>0:\\sup_{\\left(P_{Y},P_{Z}\\right)\\in\\mathcal{P}_{1}\\left(\\rho_{n_{1},n_{2}}\\right)}\\mathbb{E}\\left[1-\\Delta_{Q,\\gamma}\\right]\\leq\\beta\\right\\}.\\tag{3}\n$$\n\nSince diferent α-LDP mechanism $Q$ may assume diferent $\\tilde{\\mathcal{X}}_{i}^{\\flat}\\mathrm{s}$ , a test $\\Delta_{Q,\\gamma}$ depends on a particular LDP mechanism $Q .$ . Therefore, an optimal private level-γ test can be described as a pair of an α-LDP mechanism and a test function, designed to achieve the minimal uniform separation (3). We define this minimal uniform separation as the optimal testing separation under LDP, which we refer to as the minimax testing separation:\n\nDefinition 2 (α-LDP non-asymptotic minimax testing separation) For a fixed privacy level $\\alpha > 0$ , the α-LDP non-asymptotic minimax separation of testing is defined as\n\n$$\n\\rho_{n_{1},n_{2},\\alpha}^{*}:=\\inf_{Q\\in\\mathcal{Q}_{\\alpha}}\\inf_{\\Delta_{Q,\\gamma}\\in\\Phi_{\\gamma}^{\\alpha}}\\tilde{\\rho}_{n_{1},n_{2}}(\\Delta_{Q,\\gamma}).\\tag{4}\n$$\n\nIn this section, we switch gears to testing for equality between two multivariate densities under the LDP constraint. To this end, we consider two classes of smooth densities, namely the Hölder ball and Besov ball, and establish the minimax testing separation in terms of the $\\mathbb{L}_{2}$ distance for each class. Especially, we derive the upper bound for the minimax testing separation by building on the multinomial permutation test introduced in Section 3.2 with a careful discretization scheme. We also introduce an aggregated test, which is adaptive to the unknown smoothness parameter.\n\nWe start by formally defining the Hölder ball and Besov ball. The Hölder ball generalizes Lipschitz continuity and can be thought of as functions with bounded fractional derivatives. The following definition of the Hölder ball rephrases the one stated in Section 2.1 of Arias-Castro et al. (2018).\n\nDefinition 8 (Hölder ball) The Hölder ball with smoothness parameter $s > 0$ and radius $R > 0$ , denoted as $B_{d,s}^{\\mathrm{H}}(R)$ , is the class of functions $f:[0,1]^{d}\\mapsto\\mathbb{R}$ satisfying the following conditions:\n\n1. $\\left|f^{(\\lfloor s\\rfloor)}(\\mathbf{x})-f^{(\\lfloor s\\rfloor)}(\\mathbf{x}^{\\prime})\\right|\\leq R\\left\\|\\mathbf{x}-\\mathbf{x}^{\\prime}\\right\\|^{s-\\lfloor s\\rfloor}$ , for all $\\mathbf{x},\\mathbf{x}^{\\prime}\\in [0,1]^{d}$ .  \n2. $\\|f^{(s^{\\prime})}\\|_{\\infty}\\leq R$ , for each $s^{\\prime}\\in [\\left\\lfloor s\\right\\rfloor]$\n\nwhere $f^{(\\lfloor s\\rfloor)}$ denotes the $\\lfloor s \\rfloor$ -order derivative of $f .$\n\nThe Besov ball, on the other hand, measures the smoothness of a function by capturing its abrupt oscillations through wavelets. In this respect, it can address spatially inhomogeneous functions whose smoothness can vary across their domain. To elaborate, we consider an orthonormal wavelet basis of $\\mathbb{L}_{2}([0,1]^{d})$ at a fixed prime resolution level $J\\in{\\mathbb{N}}_{0}$ . We denote this basis as ${\\overline{{\\Phi}}}_{J}\\cup(\\cup_{j\\geq J}{\\overline{{\\Psi}}}_{j})$ , which are classified into two distinct categories. For $\\phi\\in\\overline{{\\Phi}}_{J}$ the scaling coeficient $\\begin{array}{ r }{\\mathbf{\\bar{\\theta}}_{\\phi}(\\mathbf{\\bar{f}}): =\\int_{[0,1]^{d}}f(\\mathbf{x})\\phi(\\mathbf{x})}\\end{array}$ dx detects an overall trend of $f .$ On the other hand, for $\\psi\\in\\overline{{\\Psi}}_{j}$ with its resolution level $j \\geq J$ , the wavelet coeficient $\\theta_{\\psi}(f): =$ $\\begin{array}{ r }{\\int_{[0,1]^{d}}f(\\mathbf{x})\\psi(\\mathbf{x})}\\end{array}$ dx captures abrupt oscillations from the general trend. As J and $j$ increase, the coeficients capture more detailed behaviors. Among many existing types of wavelet basis, this paper focuses on Haar multivariate wavelet basis. It is useful because projecting densities onto a subset of this basis is equivalent to applying equal-sized binning, as used in our test proposed in Section 4.3. Consequently, the discretization error inherent in the test can be characterized by the corresponding wavelet coeficients. This basis is constructed by taking tensor products of many rescaled and shifted versions of two basic univariate functions (see Appendix B.3 for the details).\n\nThe Besov ball can be defined through the magnitudes of wavelet coeficients. The following definition paraphrases the one presented in Section 3 of Tang and Yang (2023).\n\nDefinition 9 (Besov seminorm and Besov ball) Fix a smoothness parameter $s > 0$ , a microscopic parameter $1 \\leq q \\leq \\infty$ , and a wavelet basis $\\overline{{\\Phi}}_{0}\\cup(\\bigcup_{j\\geq 0}\\overline{{\\Psi}}_{j})$ . The Besov seminorm of $f\\in\\mathbb{L}_{2}([0,1]^{d})$ is defined using the sequences of its wavelet coeficients $\\left(\\theta_{\\psi}(f)\\right)_{\\psi\\in\\overline{{\\Psi}}_{j}}$ as follows:\n\n$$\n\\|f\\|_{s,2,q}:=\\left\\{\\begin{array}{l l}\\left[\\sum_{j=0}^{\\infty}2^{j s q}\\big(\\sum_{\\psi\\in\\overline{{\\Psi}}_{j}}|\\theta_{\\psi}(f)|^{2}\\big)^{\\frac{q}{2}}\\right]^{\\frac{1}{q}},&1\\leq q<\\infty,\\\\ \\sup_{j\\in\\mathbb{N}_{0}}2^{j s}\\big(\\sum_{\\psi\\in\\overline{{\\Psi}}_{j}}|\\theta_{\\psi}(f)|^{2}\\big)^{\\frac{1}{2}},&q=\\infty .\\end{array}\\right.\n$$\n\nFor a radius $R > 0$ , we define the Besov ball $B_{d,s,q}^{\\mathrm{B}}(R)$ as\n\n$$\n\\mathcal{B}_{d,s,q}^{\\mathrm{B}}(R):=\\{f\\in\\mathbb{L}_{2}([0,1]^{d}):\\|f\\|_{s,2,q}\\leq R\\}.\n$$\n\nBuilding on the smooth distribution classes defined in Definition 10, we formally define the two density testing problems of interest and present the minimax testing rate applicable to both. In this subsection, we assume that the smoothness parameter s for Hölder ball and Besov ball is known, addressing the case of unknown s in Section 4.4.\n\nAssume that the pair of data-generating distributions $(P_{\\mathbf{Y}},P_{\\mathbf{Z}})$ is contained in a class $\\mathcal{P}.$ The curator uses two sets of α-LDP views $\\{\\tilde{\\mathbf{Y}}_{i}\\}_{i\\in [n_{1}]}$ and $\\{\\tilde{\\mathbf{Z}}_{j}\\}_{j\\in [n_{2}]}$ , privatized as described in Section 2.1, to decide whether $(P_{\\mathbf{Y}},P_{\\mathbf{Z}})$ came from\n\n$$\n\\mathcal{P}_{0}:=\\left\\{\\left(P_{\\mathbf{Y}},P_{\\mathbf{Z}}\\right)\\in\\mathcal{P}:f_{\\mathbf{Y}}=f_{\\mathbf{Z}}\\right\\}\\text{or}\\mathcal{P}_{1}\\left(\\rho_{n_{1},n_{2}}\\right):=\\left\\{\\left(P_{\\mathbf{Y}},P_{\\mathbf{Z}}\\right)\\in\\mathcal{P}:\\|f_{\\mathbf{Y}}-f_{\\mathbf{Z}}\\|_{\\mathbb{L}_{2}}\\geq\\rho_{n_{1},n_{2}}\\right\\},\\tag{12}\n$$\n\nwhere $f_{\\mathbf{Y}}$ and $f_{\\mathbf{Z}}$ are densities of $P_{\\mathbf{Y}}$ and $P_{\\mathbf{Z}}.$ , respectively. We consider the problems of $\\mathcal{P}=\\mathcal{P}_{d,s}^{\\mathrm{H,2}}(R)$ and $\\mathcal{P}=\\mathcal{P}_{d,s,q}^{\\mathrm{B},2}(R)$ . Notably, while we focus on the class of either being Hölder or Besov smooth distributions, the permutation-based test presented in Section 4.3 guarantees type I error control over a broader class of null distributions beyond those defined over Hölder or Besov balls.\n\nLet us fix the type I error $\\gamma\\in(0,1)$ and the type II error $\\beta\\in(0,1)$ such that $2 \\gamma + \\beta < 1$ as in Section 3.1, and assume further that $n_{\\mathrm{1}}\\alpha^{2}\\geq 1$ , similarly considered in Lam-Weil et al. (2022). The main result of this section, stated below, establishes a lower bound as well as an upper bound for the minimax separation for two-sample density testing under the LDP constraint.\n\nWe outline the construction of multivariate Haar wavelets, following Giné and Nickl (2015), Section 4.3.6, and Autin et al. (2010), Section 2. For $f:[0,1]\\to\\mathbb{R}$ and integers $u,v,$ define the re-scaled and shifted version as $f_{u,v}(x): =2^{u/2}f(2^{u}x-v)$ . Define the univariate Haar scaling and wavelet functions from [0, 1] to $\\mathbb{R}$ as\n\n$$\n\\phi(x):=\\mathbb{1}(0\\leq x<1)\\quad \\text{and}\\quad\\psi(x):=\\mathbb{1}(0\\leq x<1/2)-\\mathbb{1}(1/2\\leq x<1),\n$$\n\nrespectively. A multivariate Haar wavelet basis is indexed by its prime resolution level $J\\in{\\mathbb{N}}_{0}$ , with each element characterized by up to three (multi) indices that range over:\n\n1. Re-scaling: $\\mathbb{N}\\setminus [J-1]$ 2  \n2. Shifting: $\\Lambda(j): =\\{0,1,\\ldots,(2^{j}-1)\\}^{d}$ , for an integer $j \\geq J$  \n3. On-of: $\\mathcal{T}: =\\{0,1\\}^{d}\\backslash\\{0$\n\nrespectively. For any $\\pmb{k}=(k_{1},\\dots,k_{d})\\in\\Lambda(J)$ , let $\\phi_{J,k}:[0,1]^{d}\\to\\mathbb{R}$ be a tensor product of the re-scaled and shifted $\\phi^{\\prime}\\mathrm{s},$ , evaluated at $\\mathbf{x}=(x_{1},\\ldots,x_{d})\\in [0,1]^{d}\\mathrm{.}$\n\n$$\n\\phi_{J,\\boldsymbol{k}}(\\mathbf{x}):=\\prod_{p=1}^{d}\\phi_{J,k_{p}}(x_{p}).\n$$\n\nFor a resolution level $j\\geq J,$ , and for any $\\ell=(\\ell_{1},\\dots,\\ell_{d})\\in\\Lambda(j)$ , let $\\psi_{i,\\ell}^{\\epsilon}:[0,1]^{d}\\to\\mathbb{R}$ be a mixed tensor product of re-scaled and shifted $\\phi^{\\prime}\\mathrm{s}$ and $\\psi\\mathrm{s}$ , evaluated at $\\mathbf{x}=(x_{1},\\ldots,x_{d})\\in$ $[0,1]^{d}$ :\n\n$$\n\\psi_{j,\\ell}^{\\epsilon}(\\mathbf{x}):=\\prod_{p=1}^{d}\\left\\{\\phi_{j,\\ell_{p}}(x_{p})\\right\\}^{1-\\epsilon_{p}}\\left\\{\\psi_{j,\\ell_{p}}(x_{p})\\right\\}^{\\epsilon_{p}}.\n$$\n\nThe Haar multivariate wavelet basis at prime resolution level J is defined as\n\n$$\n\\Phi_{J}\\cup\\left(\\bigcup_{j\\geq J}\\Psi_{j}\\right),\\text{where}\\Phi_{J}:=\\left\\{\\phi_{J,\\boldsymbol{k}}\\right\\}_{\\boldsymbol{k}\\in\\Lambda(J)}\\text{and}\\Psi_{j}:=\\left\\{\\psi_{j,\\ell}^{\\epsilon}\\right\\}_{\\ell\\in\\Lambda(j),\\epsilon\\in\\mathcal{I}}.\\tag{26}\n$$\n\nThis section analyzes the error arising from comparing discretized multinomial probability vectors instead of the original multivariate densities. Given the number of bins $\\kappa,$ let $B_{1},\\ldots,B_{\\kappa^{d}}$ enumerate d-dimensional hypercubes with side length $1 / \\kappa$ . For a density $f_{\\mathbf{Y}}.$ its step function approximation is defined as:\n\n$$\n\\hat{f}_{\\mathbf{Y}}(\\mathbf{y}):=\\sum_{m\\in [\\kappa^{d}]}\\mathbb{1}(\\mathbf{y}\\in B_{m})\\kappa^{d}\\int_{B_{m}}f_{\\mathbf{Y}}(\\mathbf{t})d\\mathbf{t}.\\tag{27}\n$$\n\nBy the argument in Arias-Castro et al. (2018), the lower bound for the minimax testing rate in the one-sample problem, denoted as $\\rho_{n_{1},\\alpha}^{*}$ , also provides a lower bound for the minimax separation in two-sample case, denoted as $\\rho_{n_{1},n_{2},\\alpha}^{*}.$ . Thus, we focus on bounding $\\rho_{n_{1},\\alpha}^{*}$ in the following proof.\n\nProof In the one-sample setting, let $f_{0}$ denote the known density $f_{\\mathbf{Z}}$ , and assume that the data-generating distribution $P\\mathbf{\\hat{Y}}$ lies within the following class:\n\nDefinition 23 (Smooth distribution classes) Let $\\mathcal{P}_{d,s}^{\\mathrm{H},1}(R)$ denote the set of distributions $P\\mathbf{\\hat{Y}}$ whose density function $f_{\\mathbf{Y}}$ satisfies $(f_{\\mathbf{Y}}-f_{0})\\in B_{d,s}^{\\mathrm{H}}(R)$ and $\\|f_{\\mathbf{Y}}-f_{0}\\|_{\\mathbb{L}_{2}}\\leq R$ Similarly, define $\\mathcal{P}_{d,s,q}^{\\mathrm{B},1}(R)$ by replacing $B_{d,s}^{\\mathrm{H}}(R)$ with $B_{d,s,q}^{\\mathrm{B}}(R)$\n\nThe one-sample density testing problem is defined as follows: Given $\\{\\tilde{\\mathbf{Y}}_{i}\\}_{i\\in [n_{1}]}$ generated from $P\\mathbf{\\hat{Y}}$ and privatized via α-LDP mechanism $Q,$ , decide whether $P\\mathbf{\\hat{Y}}$ came from\n\n$$\n\\mathcal{P}_{0}=\\{P_{\\mathbf{Y}}\\in\\mathcal{P}:f_{\\mathbf{Y}}=f_{0}\\}\\quad \\text{or}\\quad\\mathcal{P}_{1}(\\rho_{n_{1}})=\\{P_{\\mathbf{Y}}\\in\\mathcal{P}:\\|f_{\\mathbf{Y}}-f_{0}\\|_{\\mathbb{L}_{2}}\\geq\\rho_{n_{1}}\\}.\\tag{72}\n$$\n\nWe consider two problems: (i) $\\mathcal{P}=\\mathcal{P}_{d,s}^{\\mathrm{H},1}(R)$ , and (ii) $\\mathcal{P}=\\mathcal{P}_{d,s,q}^{\\mathrm{B},1}(R)$ , with significant overlap in most of the proof steps.\n\nWe derive the lower bound by testing the uniform null hypothesis against a carefully constructed mixture alternative. This alternative balances two objectives regarding the distances between the distributions: achieving suficient separation in $\\mathbb{L}_{2}$ distance without the $\\alpha{\\mathrm{-}}\\mathrm{LDP}$ constraint, and ensuring indistinguishability in the total variation distance under the $\\alpha{\\mathrm{-}}\\mathrm{LDP}$ constraint. The mixture construction follows the approach in Appendix A.2.2 of Lam-Weil et al. (2022), where uniform densities are perturbed by eigenfunctions of the integral operator created from the $\\alpha{\\mathrm{-}}\\mathrm{LDP}$ mechanism. We generalize this construction to multivariate Besov and Hölder smoothness classes.\n\nAs defined in Definition 1, let $Q\\in\\mathcal{Q}_{\\alpha}$ be a non-interactive $\\alpha{\\mathrm{-}}\\mathrm{LDP}$ mechanism, and $\\{Q_{i}\\}_{i\\in [n_{1}]}$ be its marginals. By Lemma B.1 of Lam-Weil et al. (2022), for each $i\\in [n_{1}]$ , there exists a probability measure $\\mu_{i}$ such that $Q_{i}(\\cdot|\\mathbf{y}_{i})$ is absolutely continuous with respect to $\\mu_{i}$ for all $\\mathbf{y}_{i}\\in [0,1]^{d}$ . Denote its density with respect to $\\mu_{i}$ by $q_{i}(\\tilde{\\mathbf{y}}_{i}|\\mathbf{y}_{i})$ . Using this, we define the private counterpart of a density function on $[0,1]^{d};$\n\nDefinition 24 (Private counterpart of density) Let f be a probability density on $[0,1]^{d}$ and Q be an $\\alpha{-}L D P$ mechanism with marginal conditional densities $q_{i}(\\tilde{\\bf y}_{i}\\mid\\bf y)$ with respect to $\\mu_{i}$ . The private counterpart of f is defined as\n\n$$\n\\tilde{f}_{i}(\\tilde{\\mathbf{y}}_{i}):=\\int_{[0,1]^{d}}q_{i}(\\tilde{\\mathbf{y}}_{i}\\mid\\mathbf{y})f(\\mathbf{y})d\\mathbf{y}.\n$$\n\nLet $f_{0}$ denote the uniform density supported on $[0,1]^{d}$ . Using its private counterpart, we introduce the following integral operator whose kernel resembles the likelihood ratio of the privatized densities:\n\nDefinition 25 (Privacy mechanism integral operator) For each $i\\in [n_{1}]$ , define an operator $L_{i}:[0,1]^{d}\\times\\tilde{\\mathcal{X}}\\mathbb{R}$ as:\n\n$$\nL_{i}(\\mathbf{s},\\mathbf{t}):=\\frac{q_{i}(\\mathbf{t}|\\mathbf{s})}{\\sqrt{\\tilde{f}_{0,i}(\\mathbf{t})}}.\n$$\n\nUsing $L_{i}$ as a kernel function, we define an integral operator $K_{i}:\\mathbb{L}_{2}([0,1]^{d})\\to\\mathbb{L}_{2}(\\tilde{\\mathcal{X}},d\\mu_{i})$ as:\n\n$$\n(K_{i}f)(\\cdot):=\\int_{[0,1]^{d}}L_{i}(\\mathbf{y},\\cdot)f(\\mathbf{y})d\\mathbf{y}.\n$$\n\nLet $K_{i}^{*}$ denote the adjoint of $K_{i}$ . Finally, by aggregating all the operators for $i\\in [n_{1}]$ , we define a symmetric and positive semidefinite integral operator:\n\n$$\nK:=\\sum_{i=1}^{n_{1}}\\frac{1}{n_{1}}K_{i}^{*}K_{i}.\n$$\n\nFor $f\\in\\mathbb{L}_{2}\\big([0,1]^{d}\\big)$ , the operator K yields the expected squared privatized likelihood ratio:\n\n$$\n\\begin{array}{l}\\langle K f,f\\rangle=\\frac{1}{n_{1}}\\sum_{i=1}^{n}\\int_{\\tilde{\\mathcal{X}}_{i}}\\frac{\\left(\\int_{[0,1]^{d}}q_{i}(\\tilde{\\mathbf{y}}_{i}|\\mathbf{y})f(\\mathbf{y})d\\mathbf{y}\\right)^{2}}{\\tilde{f}_{0,i}(\\tilde{\\mathbf{y}}_{i})}d\\mu_{i}(\\tilde{\\mathbf{y}}_{i})\\\\ =\\frac{1}{n_{1}}\\sum_{i=1}^{n}\\int_{\\tilde{\\mathcal{X}}_{i}}\\frac{\\tilde{f}_{i}(\\tilde{\\mathbf{y}}_{i})^{2}}{\\tilde{f}_{0,i}(\\tilde{\\mathbf{y}}_{i})}\\frac{d Q_{f_{0}}}{\\tilde{f}_{0,i}(\\tilde{\\mathbf{y}}_{i})}=\\frac{1}{n_{1}}\\sum_{i=1}^{n}\\mathbb{E}_{Q_{f_{0}}}\\left[\\frac{\\tilde{f}_{i}^{2}(\\tilde{\\mathbf{y}}_{i})}{\\tilde{f}_{0,i}^{2}(\\tilde{\\mathbf{y}}_{i})}\\right].\\tag{73}\\\\ \\end{array}\n$$\n\nWe construct each mixture element by adding carefully chosen eigenfunctions of K as bumps to $f_{0},$ , ensuring the resulting densities belong to the Hölder or Besov classes. Specifically, we define two orthonormal sets of smoothness-inducing functions—one for Besov smoothness and one for Hölder smoothness. For a positive integer J and $\\omega{>}0$ , we set $\\kappa=2^{J}$ and determine the number of functions as $\\kappa^{d}$ . The parameter $\\omega$ controls the perturbation height. The value of $J$ and $\\omega$ will be optimized later to achieve a tight lower bound. Both orthonormal sets have cardinality $\\kappa^{d}$ , the supports of their elements partition $[0,1]^{d}$ without overlap, each element has the $\\mathbb{L}_{\\infty}$ norm bounded by $\\kappa^{d/2}$ , and is orthogonal to $f_{0}$ . We start by defining the orthonormal set for the Besov ball, consisting of bounded step functions from the multivariate Haar basis.\n\nDefinition 26 (Orthonormal set for the Besov ball) Consider the Haar wavelet basis on $[0,1]^{d}$ with a prime resolution level $J\\in\\mathbb{N}$ , as defined in Appendix B.3. Set $\\kappa=2^{J}$ and define $\\Lambda(\\bar{J}): =\\{0,1,\\ldots,(\\kappa{-}1)\\}^{d}.$ . Fix a d-dimensional on-of multi-index $\\epsilon^{*}: =\\{1,0,0,\\dots,0\\}$ Let B $B$ represent the elements with this multi-index, drawn from the mixed tensor product set at the lowest resolution level J. Specifically, we define:\n\n$$\n\\mathbb{B}_{B}:=\\{\\psi_{J,\\ell}^{\\epsilon^{*}}\\}_{\\ell\\in\\Lambda(J)}.\n$$\n\nSince all results ahead hold for any non-zero $\\epsilon^{*}$ , the specific choice in Definition 26 is arbitrary. Each element of $\\mathbb{B}_{B}$ is orthogonal to $f_{0}$ because its multi-index has at least one non-zero entry. In that dimension of non-zero entry, the scaled and shifted Haar wavelet function $\\psi(x)=\\mathbb{1}_{[0,1/2)}(x)-\\mathbb{1}_{[1/2,1)}(x)$ , which integrates to zero, contributes to the integral. By Fubini’s theorem, this ensures that the entire integral equals zero. Next we define the orthonormal set for Hölder ball:\n\nDefinition 27 (Orthonormal set for the Hölder ball) For a smoothness parameter $s >$ 0, let $\\bar{\\varphi}:\\mathbb{R}^{d}\\to$ R be an infinitely diferentiable function supported on $[0,1]^{d}$ satisfying $\\begin{array}{ r }{\\|\\bar{\\varphi}\\|_{\\mathbb{L}_{2}}=1,\\ \\|\\bar{\\varphi}\\|_{\\mathbb{L}_{\\infty}}<\\infty,\\int_{[0,1]^{d}}\\bar{\\varphi}=0}\\end{array}$ , and $\\|\\bar{\\varphi}^{(s^{\\prime})}\\|_{\\infty}<\\infty$ for $s^{\\prime}\\in\\{1,\\ldots,(\\lfloor s\\rfloor+1)\\}$ . For $J\\in\\mathbb{N}_{.}$ , set $\\kappa=2^{J}$ and define $\\Lambda(J): =\\{0,1,...,(\\kappa-1)\\}^{d},\\varphi_{J,\\ell}(\\mathbf{y}): =\\kappa^{d/2}\\bar{\\varphi}(\\kappa\\mathbf{y}-\\ell)$ for $\\ell\\in\\Lambda(J)$ , and $\\mathbb{B}_{H}: =\\{\\varphi_{J,\\ell}\\}_{\\ell\\in\\Lambda(J)}$ , meaning scaled and shifted copies $o f\\bar{\\varphi}$\n\nNow we are ready to define the mixtures, one for Besov and one for Hölder ball.\n\nDefinition 28 (Mixture alternatives) For the Hölder ball, let $V=\\operatorname{Span}(\\{f_{0}\\}\\cup\\mathbb{B}_{H})$ Construct an orthonormal basis of V as $\\{f_{0}\\}\\cup\\{u_{t}\\}_{t=1}^{\\kappa^{d}}$ , where each $u_{t}$ is an eigenfunction of K with eigenvalue $\\lambda_{t}$ . For each $t\\in [\\kappa^{d}]$ , since u<sub>t</sub> is orthogonal to $f_{0}$ , it integrates to zero. Recall that $z_{\\alpha}=e^{2\\alpha}-e^{-2\\alpha}~f o r~\\alpha>0$ , and set $\\widetilde{\\lambda}_{t}=(\\lambda_{t}/z_{\\alpha}^{2})\\vee\\kappa^{-d}$ . Fix $\\omega > 0$ . For any $\\pmb{\\eta}=(\\eta_{1},\\dots,\\eta_{\\kappa^{d}})\\in\\{-1,1\\}^{\\kappa^{d}}$ , define the perturbed density as:\n\n$$\nf_{\\omega,\\kappa}^{\\eta}:=f_{0}+\\omega\\sum_{t=1}^{\\kappa^{d}}\\eta_{t}\\tilde{\\lambda}_{t}^{-1/2}u_{t}.\\tag{74}\n$$\n\nSince $u_{t}\\in V$ for each $t\\in [\\kappa^{d}]$ , we can also write:\n\n$$\nf_{\\omega,\\kappa}^{\\eta}=f_{0}+\\omega\\sum_{t=1}^{\\kappa^{d}}\\eta_{t}\\tilde{\\lambda}_{t}^{-1/2}\\sum_{\\psi\\in\\mathbb{B}_{H}}\\theta_{\\psi}(u_{t})\\psi,\\tag{75}\n$$\n\nwhere $\\theta_{\\psi}(u_{t})$ is the Fourier coeficient of $u_{t}$ with respect to $\\psi$ . The mixture for Hölder ball, denoted as $\\nu_{\\rho}^{H}$ , is defined as the uniform probability measure over $\\{f_{\\omega,\\kappa}^{\\eta}:\\eta\\in\\{-1,1\\}^{\\kappa^{d}}\\}$ $T h e$ mixture for the Besov ball, denoted as $\\nu_{\\rho}^{B}$ , is defined through the same procedure as above, with B $H$ replaced by $\\mathbb{B}_{B}$\n\nWe next establish a series of conditions on κ and $\\omega$ to ensure that the mixture elements belong to the smooth density class with high probability.\n\nLemma 30 (Nonnegativity) For any $f_{\\omega,\\kappa}^{\\eta}$ drawn from either the distribution $\\nu_{\\rho}^{H}$ or $\\nu_{\\rho}^{B}$ 2 we always have $\\begin{array}{ r }{\\int_{[0,1]^{d}}f_{\\omega,\\kappa}^{\\eta}(\\mathbf{y})d\\mathbf{y}=1}\\end{array}$ . Also, $f_{\\omega,\\kappa}^{\\eta}\\geq 0$ with a probability greater than $1 - \\gamma$ if\n\n$$\n\\omega\\leq\\frac{\\kappa^{-d}}{\\sqrt{2\\log(2\\kappa^{d}/\\gamma)}}.\\tag{77}\n$$\n\nLemma 32 (Besov class) With probability at least $1 - \\gamma$ , a function drawn from $\\nu_{\\rho}^{\\mathrm{B}}$ lies in $B_{d,s,q}^{\\mathrm{B}}(R)$ if\n\n$$\n\\omega\\leq\\frac{R\\kappa^{-(s+d)}}{\\sqrt{2\\log(\\kappa^{d}/\\gamma)}}.\\tag{78}\n$$\n\nFinally, we characterize the condition for κ and $\\omega$ that ensures any valid private test fails.\n\nVerification of nonnegativeness with high probability. The proof follows the approach of Lemma B.5 in Lam-Weil et al. (2022), with an adjustment for the change of basis. Since the proof is the same for $\\nu_{\\rho}^{H}$ and $\\nu_{\\rho}^{B}$ , we state if only for $\\nu_{\\rho}^{B}$ . For each $\\boldsymbol{\\psi}\\in\\mathbb{B}_{B}$ , define an\n\nevent $A_{\\psi}$ as:\n\n$$\nA_{\\psi}:=\\left[\\left|\\omega\\sum_{t=1}^{\\kappa^{d}}\\eta_{t}\\tilde{\\lambda}_{t}^{-1/2}\\theta_{\\psi}(u_{t})\\right|\\geq\\kappa^{-d/2}\\right],\n$$\n\nwhich belongs to σ-algebra generated by $\\eta .$ . Since the sum of independent scaled Rademacher variables inside $A_{\\psi}$ is sub-Gaussian with variance proxy $\\begin{array}{ r }{\\omega^{2}\\tilde{\\lambda}_{t}^{-1}\\sum_{t=1}^{\\kappa^{d}}\\theta_{\\psi}^{2}(u_{t})}\\end{array}$ , by the union bound, we have:\n\n$$\n\\begin{array}{l}\\mathbb{P}_{\\nu_{\\rho}^{B}}\\left(\\bigcup_{\\psi\\in\\mathbb{B}_{B}}A_{\\psi}\\right)\\leq\\sum_{\\psi\\in\\mathbb{B}_{B}}2\\exp\\left(\\frac{-\\kappa^{-d}}{2\\omega^{2}\\tilde{\\lambda}_{t}^{-1}\\sum_{t=1}^{\\kappa^{d}}\\theta_{\\psi}^{2}(u_{t})}\\right)\\\\ \\stackrel{(a)}{\\leq}2\\sum_{\\psi\\in\\mathbb{B}_{B}}\\exp\\left(\\frac{-\\kappa^{-2 d}}{2\\omega^{2}\\sum_{t=1}^{\\kappa^{d}}\\theta_{\\psi}^{2}(u_{t})}\\right)\\\\ \\stackrel{(b)}{=}2\\kappa^{d}\\exp\\left(\\frac{-\\kappa^{-2 d}}{2\\omega^{2}}\\right),\\tag{81}\\\\ \\end{array}\n$$\n\nwhere step (a) uses $\\tilde{\\lambda_{t}}\\leq\\kappa^{-d}$ and step (b) uses the following equality that stems from the fact that $u_{t}\\mathrm{'s}$ are orthonormal eigenbasis of V and the orthonormality of $\\mathbb{B}_{B}$ :\n\n$$\n\\sum_{t=1}^{\\kappa^{d}}\\theta_{\\psi}^{2}(u_{t})=\\langle\\psi,\\sum_{t=1}^{\\kappa^{d}}\\langle\\psi,u_{t}\\rangle_{\\mathbb{L}_{2}}u_{t}\\rangle_{\\mathbb{L}_{2}}=\\langle\\psi,\\psi\\rangle_{\\mathbb{L}_{2}}=1.\n$$", "theorems": [{"local_id": "T1", "statement": "Lemma 29 $(\\mathbb{L}_{2}$ separation) For any density $f_{\\omega,\\kappa}^{\\eta}$ drawn from $\\nu_{\\rho},$ the $\\mathbb{L}_{2}$ distance from $f_{0}$ is bounded as:\n\n$$\n\\left\\|\\left\\|f_{\\omega,\\kappa}^{\\eta}-f_{0}\\right\\|\\right\\|_{\\mathbb{L}_{2}}\\geq\\kappa^{d}\\omega\\sqrt{3/4}.\\tag{76}\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof Since $f_{\\omega,\\kappa}^{\\eta}-f_{0}$ is a weighted sum of orthonormal singular vectors of $Q,$ its $\\mathbb{L}_{2}$ norm $\\|f_{\\eta}-f_{0}\\|_{\\mathbb{L}_{2}}$ is bounded below by a quantity involving the sum of the corresponding singular values of $Q .$ . Bounding this sum by $z_{\\alpha}^{2}=e^{2\\alpha}-e^{-2\\alpha}$ , using the definition of the α-LDP constraint, completes the proof. 7"}]}}, {"local_id": "T2", "statement": "Lemma 31 (Hölder class) With probability at least $1 - \\gamma$ , a function drawn from $\\nu_{\\rho}^{H}$ lies in $B_{d,s}^{\\mathrm{H}}(R)$ if\n\n$$\n\\omega\\leq\\frac{R\\kappa^{-(d+s)}}{C(s,\\bar{\\varphi})\\sqrt{2\\log(2\\kappa^{d}/\\gamma)}},\n$$\n\nwhere $\\begin{array}{ r }{C(s,\\bar{\\varphi}): =\\operatorname*{max}_{s^{\\prime}\\in\\{1,\\dots,(\\lfloor s\\rfloor+1)\\}}4\\|\\bar{\\varphi}^{(s^{\\prime})}\\|_{\\infty}}\\end{array}$", "proof": {"steps": [{"n": 1, "claim": "Proof We verify the two steps mentioned above in order."}, {"n": 2, "claim": "Hölder continuity of derivatives with high probability. Fix arbitrary $\\mathbf{y}_{1},\\mathbf{y}_{2}\\in [0,1]^{d}$ . Since each $\\varphi\\in\\mathbb{B}_{H}$ has disjoint support, we can specify the element of $\\mathbb{B}_{H}$ whose $\\lfloor s \\rfloor -$ derivative has support containing $\\mathbf{y}_{1}$ and $\\mathbf{y}_{2}$ , say $\\varphi_{\\mathbf{y}_{1}}$ and $\\varphi_{\\mathbf{y}_{2}}$ . Then using the fact that $\\varphi_{\\mathbf{y}_{1}}^{(\\lfloor s\\rfloor)}(\\mathbf{y}_{2})-$ $\\varphi_{\\mathbf{y}_{2}}^{(\\lfloor s\\rfloor)}(\\mathbf{y}_{1})=0$ , we can write the diference of the s -derivative of $f_{\\omega,\\kappa}^{\\eta}$ as\n\n$$\n\\begin{array}{l}|(f_{\\omega,\\kappa}^{\\eta})^{(\\lfloor s\\rfloor)}(\\mathbf{y}_{1})-(f_{\\omega,\\kappa}^{\\eta})^{(\\lfloor s\\rfloor)}(\\mathbf{y}_{2})|\\\\ =\\omega\\left|\\sum_{t=1}^{\\kappa^{d}}\\eta_{t}\\tilde{\\lambda}_{t}^{-1/2}\\left\\{\\theta_{\\varphi_{\\mathbf{y}_{1}}}(u_{t})\\varphi_{\\mathbf{y}_{1}}^{(\\lfloor s\\rfloor)}(\\mathbf{y}_{1})-\\theta_{\\varphi_{\\mathbf{y}_{2}}}(u_{t})\\varphi_{\\mathbf{y}_{2}}^{(\\lfloor s\\rfloor)}(\\mathbf{y}_{2})\\right\\}\\right|\\\\ \\leq\\omega\\left|\\sum_{t=1}^{\\kappa^{d}}\\theta_{\\varphi_{\\mathbf{y}_{1}}}\\left(u_{t}\\right)\\eta_{t}\\tilde{\\lambda}_{t}^{-1/2}\\left(\\varphi_{\\mathbf{y}_{1}}^{(\\lfloor s\\rfloor)}\\left(\\mathbf{y}_{1}\\right)-\\varphi_{\\mathbf{y}_{1}}^{(\\lfloor s\\rfloor)}\\left(\\mathbf{y}_{2}\\right)\\right)\\right|(82)\\\\ +\\omega\\left|\\sum_{t=1}^{\\kappa^{d}}\\theta_{\\varphi_{\\mathbf{y}_{2}}}\\left(u_{t}\\right)\\eta_{t}\\tilde{\\lambda}_{t}^{-1/2}\\left(\\varphi_{\\mathbf{y}_{2}}^{(\\lfloor s\\rfloor)}\\left(\\mathbf{y}_{1}\\right)-\\varphi_{\\mathbf{y}_{2}}^{(\\lfloor s\\rfloor)}\\left(\\mathbf{y}_{2}\\right)\\right)\\right|,(83)\\\\ \\end{array}\n$$\n\nwhere the last inequality uses the triangle inequality."}, {"n": 3, "claim": "We separately bound the terms (82) and (83), starting with (82). For $\\mathbf{y}~\\in~[0,1]^{d}$ , recall from (27) that $\\varphi_{\\mathbf{y}_{1}}$ has the form $\\kappa^{d/2}\\varphi(\\kappa\\mathbf{y}-\\pmb{\\ell})$ . By the chain rule, its $\\lfloor s \\rfloor$ -mixed partial derivatives are written as\n\n$$\n\\varphi_{\\mathbf{y}_{1}}^{(\\lfloor s\\rfloor)}(\\mathbf{y})=\\kappa^{(d/2)+\\lfloor s\\rfloor}\\bar{\\varphi}^{(\\lfloor s\\rfloor)}(\\kappa\\mathbf{y}-\\boldsymbol{\\ell}).\\tag{84}\n$$"}, {"n": 4, "claim": "Using the above equality, the mean value theorem and the boundedness of all partial mixed derivatives specified in Definition 27, the term (82) is upper bounded by\n\n$$\n\\omega\\kappa^{(d/2)+\\lfloor s\\rfloor}\\bigg|\\sum_{t=1}^{\\kappa^{d}}\\theta_{\\varphi_{\\mathbf{y}_{1}}}(u_{t})\\eta_{t}\\tilde{\\lambda}_{t}^{-1/2}\\bigg|\\left(2\\|\\bar{\\varphi}^{(\\lfloor s\\rfloor)}\\|_{\\infty}\\wedge\\|\\bar{\\varphi}^{(\\lfloor s+1\\rfloor)}\\|_{\\infty}\\kappa\\|\\mathbf{y}_{1}-\\mathbf{y}_{2}\\|\\right),\n$$\n\nBounding the term (83) using the same argument, we obtain the following bound:\n\n$$\n\\begin{array}{l}\\left|\\left(f_{\\omega,\\kappa}^{\\boldsymbol{\\eta}}\\right)^{(\\lfloor s\\rfloor)}\\left(\\mathbf{y}_{1}\\right)-\\left(f_{\\omega,\\kappa}^{\\boldsymbol{\\eta}}\\right)^{(\\lfloor s\\rfloor)}\\left(\\mathbf{y}_{2}\\right)\\right|\\\\ \\leq\\omega\\kappa^{(d/2)+\\lfloor s\\rfloor}\\left(\\left|\\sum_{t=1}^{\\kappa^{d}}\\theta_{\\varphi_{\\mathbf{y}_{1}}}\\left(u_{t}\\right)\\eta_{t}\\tilde{\\lambda}_{t}^{-1/2}\\right| +\\left|\\sum_{t=1}^{\\kappa^{d}}\\theta_{\\varphi_{\\mathbf{y}_{2}}}\\left(u_{t}\\right)\\eta_{t}\\tilde{\\lambda}_{t}^{-1/2}\\right|\\right)\\left(2\\|\\bar{\\varphi}^{(\\lfloor s\\rfloor)}\\|_{\\infty}\\wedge\\|\\bar{\\varphi}^{(\\lfloor s+1\\rfloor)}\\|_{\\infty}\\kappa\\|\\mathbf{y}_{1}-\\mathbf{y}_{2}\\|\\right).\\tag{85}\\\\ \\end{array}\n$$"}, {"n": 5, "claim": "For each $\\varphi\\in\\mathbb{B}_{H}$ , let $A_{\\varphi}$ be an event within the σ-algebra generated by $\\eta_{\\mathrm{:}}$ , defined as follows:\n\n$$\nA_{\\varphi}:=\\left[\\left|\\sum_{t=1}^{\\kappa^{d}}\\eta_{t}\\tilde{\\lambda}_{t}^{-1/2}\\theta_{\\varphi}(u_{t})\\right|\\geq\\sqrt{2\\kappa^{d}\\log(2\\kappa^{d}/\\gamma)}\\right].\\tag{86}\n$$"}, {"n": 6, "claim": "Since the sum of independent scaled Rademacher variables inside $A_{\\varphi}$ is sub-Gaussian with variance proxy $\\begin{array}{ r }{\\tilde{\\lambda}_{t}^{-1}\\sum_{t=1}^{\\kappa^{d}}\\theta_{\\varphi}^{2}(u_{t})}\\end{array}$ , by the same technique as (81), we have $\\begin{array}{ r }{\\mathbb{P}_{\\nu_{\\rho}^{H}}\\bigl(\\bigcap_{\\varphi\\in\\mathbb{B}_{H}}A_{\\varphi}^{c}\\bigr)\\geq}\\end{array}$\n\n$1 - \\gamma / 2$ . Under the event $\\cap_{\\varphi\\in\\mathbb{B}_{H}}A_{\\varphi}^{c}$ , the bound (85) continues to\n\n$$\n\\begin{array}{l}|(f_{\\omega,\\kappa}^{\\boldsymbol{\\eta}})^{(\\lfloor s\\rfloor)}(\\mathbf{y}_{1})-(f_{\\omega,\\kappa}^{\\boldsymbol{\\eta}})^{(\\lfloor s\\rfloor)}(\\mathbf{y}_{2})|\\stackrel{(a)}{\\leq}\\omega\\kappa^{\\frac{d}{2}+\\lfloor s\\rfloor}2\\sqrt{2\\kappa^{d}\\log(2\\kappa^{d}/\\gamma)}\\left(2\\|\\bar{\\varphi}^{(\\lfloor s\\rfloor)}\\|_{\\infty}\\wedge\\|\\bar{\\varphi}^{(\\lfloor s+1\\rfloor)}\\|_{\\infty}\\kappa\\|\\mathbf{y}_{1}-\\mathbf{y}_{2}\\|\\right)\\\\ \\stackrel{(b)}{\\leq}\\omega\\kappa^{\\frac{d}{2}+\\lfloor s\\rfloor}\\sqrt{2\\kappa^{d}\\log(2\\kappa^{d}/\\gamma)}C_{2}(s,\\bar{\\varphi})(1\\wedge\\kappa\\|\\mathbf{y}_{1}-\\mathbf{y}_{2}\\|)\\\\ \\stackrel{(c)}{\\leq}\\omega\\kappa^{\\frac{d}{2}+\\lfloor s\\rfloor}\\sqrt{2\\kappa^{d}\\log(2\\kappa^{d}/\\gamma)}C_{2}(s,\\bar{\\varphi})\\kappa^{(s-\\lfloor s\\rfloor)}\\|\\mathbf{y}_{1}-\\mathbf{y}_{2}\\|^{s-\\lfloor s\\rfloor}\\\\ =\\omega\\kappa^{d+s}\\sqrt{2\\log(2\\kappa^{d}/\\gamma)}C_{2}(s,\\bar{\\varphi})\\|\\mathbf{y}_{1}-\\mathbf{y}_{2}\\|^{s-\\lfloor s\\rfloor},\\tag{87}\\\\ \\end{array}\n$$\n\nwhere $C_{2}(s,\\bar{\\varphi}): =\\big(4\\||\\bar{\\varphi}^{(\\lfloor s\\rfloor)}\\|_{L_{\\infty}}\\vee 2\\||\\bar{\\varphi}^{(\\lfloor s+1\\rfloor)}\\|_{L_{\\infty}}\\big)$ , step (a) uses the event $\\cap_{\\varphi\\in{\\mathbb B}_{H}}A_{\\varphi}^{c}$ , step (b) uses the basic inequality that\n\n$$\n(a\\times 1)\\land(b\\times c)\\leq(a\\lor b)\\times(1\\land c)\\quad \\text{for} a,b,c\\geq 0,\n$$\n\nand step (c) holds by the following fact: When $(1\\land c)\\leq 1$ and $0 \\leq s - \\lfloor s \\rfloor < 1$ , it holds that\n\n$$\n(1\\land c)\\leq(1\\land c)^{s-\\lfloor s\\rfloor}\\leq c^{s-\\lfloor s\\rfloor}.\n$$"}, {"n": 7, "claim": "Starting from inequality (87), we solve for $\\omega$ in $\\omega\\kappa^{d+s}\\sqrt{2\\log(2\\kappa^{d}/\\gamma)}C_{2}(s,\\bar{\\varphi})\\leq R .$ Solving for $\\omega,$ we obtain\n\n$$\n\\omega\\leq\\frac{R\\kappa^{-(d+s)}}{C_{2}(s,\\bar{\\varphi})\\sqrt{2\\log(2\\kappa^{d}/\\gamma)}}.\\tag{88}\n$$"}, {"n": 8, "claim": "Therefore, if $\\omega$ satisfies this inequality, then with probability at least $1 - \\gamma / 2$ , the Lipschitz property from Definition 8 holds for a function $f_{\\omega,\\kappa}^{\\eta}$ drawn from $\\nu_{\\rho}^{H}$\n\nBounded derivatives with high probability. Fix an integer $s^{\\prime}\\in\\left[\\left\\lfloor s\\right\\rfloor\\right]$ and an arbitrary ${\\textbf{ y }}\\in$ $[0,1]^{d}$ . Since each $\\varphi\\in\\mathbb{B}_{H}$ has disjoint support, we can specify the element of $\\mathbb{B}_{H}$ whose s<sup>0</sup>-derivative has support containing y, say $\\varphi_{\\mathbf{y}}$ . Using (84) to the basis expansion of $f_{\\omega,\\kappa}^{\\eta}$ with respect to $\\mathbb{B}_{H}$ , we compute the $s^{\\prime}.$ -derivative of $f_{\\omega,\\kappa}^{\\eta}$ as:\n\n$$\n(f_{\\omega,\\kappa}^{\\pmb{\\eta}})^{(s^{\\prime})}(\\mathbf{y})=\\omega\\kappa^{\\frac{d}{2}+s^{\\prime}}\\biggl(\\sum_{t=1}^{\\kappa^{d}}\\eta_{t}\\tilde{\\lambda}_{t}^{-1/2}\\theta_{\\varphi_{\\mathbf{y}}}(u_{t})\\biggr)\\bar{\\varphi}^{(s^{\\prime})}(\\kappa\\mathbf{y}-\\pmb{\\ell}).\n$$"}, {"n": 9, "claim": "Under the event $\\cap_{\\varphi\\in{\\mathbb{B}}_{H}}A_{\\varphi}^{c}$ as defined in (86), the derivative above is bounded by\n\n$$\n(f_{\\omega,\\kappa}^{\\eta})^{(s^{\\prime})}(\\mathbf{y})\\leq\\kappa^{d+s^{\\prime}}\\||\\varphi^{(s^{\\prime})}\\||_{\\infty}\\omega\\sqrt{2\\log(2\\kappa^{d}/\\gamma)}.\n$$"}, {"n": 10, "claim": "We solve for $\\omega$ in $\\kappa^{d+s^{\\prime}}\\|\\varphi^{(s^{\\prime})}\\|_{\\infty}\\omega\\sqrt{2\\log(2\\kappa^{d}/\\gamma)}\\leq R$ . Solving for $\\omega,$ , we obtain\n\n$$\n\\omega\\leq\\frac{R\\kappa^{-(d+s^{\\prime})}}{\\|\\|\\varphi^{(s^{\\prime})}\\|\\|_{\\infty}\\sqrt{2\\log(2\\kappa^{d}/\\gamma)}}.\\tag{89}\n$$"}, {"n": 11, "claim": "Therefore, if ω satisfies this inequality, then with probability at least $1 - \\gamma / 2$ , the s<sup>0</sup>-derivative of $f_{\\omega,\\kappa}^{\\eta}$ drawn from $\\nu_{\\rho}^{H}$ is bounded by R. Combining the above condition for $s^{\\prime}\\in [\\left\\lfloor s\\right\\rfloor]$ , we conclude that if\n\n$$\n\\omega\\leq\\frac{R\\kappa^{-(d+s)}}{C_{3}(s,\\bar{\\varphi})\\sqrt{2\\log(2\\kappa^{d}/\\gamma)}},\\tag{90}\n$$\n\nwhere $\\begin{array}{ r }{C_{3}(s,\\bar{\\varphi}): =\\operatorname*{max}_{s^{\\prime}\\in\\{1,\\dots,\\lfloor s\\rfloor\\}}\\||\\bar{\\varphi}^{(s^{\\prime})}\\||_{\\infty}}\\end{array}$ , then with probability at least $1 - \\gamma / 2$ , the bounded derivatives property of Definition 8 holds for a function $f_{\\omega,\\kappa}^{\\eta}$ drawn from $\\nu_{\\rho}^{H}$"}, {"n": 12, "claim": "Conclusion.\n\nCombining the conditions (88) and (90), we conclude that if\n\n$$\n\\omega\\leq\\frac{R\\kappa^{-(d+s)}}{C_{4}(s,\\bar{\\varphi})\\sqrt{2\\log(2\\kappa^{d}/\\gamma)}},\\tag{91}\n$$\n\nwhere $\\begin{array}{ r }{C_{4}(s,\\bar{\\varphi}): =\\operatorname*{max}_{s^{\\prime}\\in\\{1,\\dots,(\\lfloor s\\rfloor+1)\\}}4\\||\\bar{\\varphi}^{(s^{\\prime})}\\|_{\\infty}}\\end{array}$ , then with probability at least $1 - \\gamma$ , for a function $f_{\\omega,\\kappa}^{\\eta}$ drawn from $\\nu_{\\rho}^{H}$ , we have $f_{\\omega,\\kappa}^{\\eta}\\in\\mathcal{B}_{d,s}^{\\mathrm{H}}(R)$ . This completes the proof Lemma 31."}]}}, {"local_id": "T3", "statement": "Lemma 33 (Indistinguishability) Under the conditions of Lemmas 30 and $^{31,}$ if\n\n$$\n\\omega\\leq(n z_{\\alpha}^{2})^{-1/2}\\left(\\frac{\\log [1+4(1-2\\gamma-\\beta)^{2}]}{\\kappa^{d}}\\right)^{1/4},\\tag{79}\n$$\n\nthen no valid $\\alpha{-}L D P$ test of level γ can distinguish between the uniform density $f_{0}$ and the alternatives in (72) (with $\\dot{\\mathcal{P}}=\\mathcal{P}_{d,s}^{\\mathrm{H},\\mathrm{i}}(R))$ with type II error less than $\\beta$ . The same conclusion holds when $\\mathcal{P}_{d,s}^{\\mathrm{H},1}(R)$ and the condition in 31 is replaced by $\\mathcal{P}_{d,s,q}^{\\mathrm{B},1}(R)$ and the condition in 32, respectively.", "proof": {"steps": [{"n": 1, "claim": "Proof The argument applies to both $(\\mathcal{P}_{d,s}^{\\mathrm{H},1}(R),\\nu_{\\rho}^{H})$ and $(\\mathcal{P}_{d,s,q}^{\\mathrm{B},1}(R),\\nu_{\\rho}^{B})$ . For simplicity, denote the distribution class and mixture as $\\mathcal{P}$ and $\\nu_{\\rho},$ , respectively. Let $L_{Q_{\\nu_{o}}^{n}}(\\tilde{\\mathbf{Y}}_{1},\\ldots,\\tilde{\\mathbf{Y}}_{n_{1}})$ be the likelihood ratio between $\\boldsymbol{Q}_{\\nu_{\\rho}}^{n}$ and $Q_{f_{0}}^{n}$ . Define the total variation distance as $\\|\\mathbb{P}_{Q_{\\nu_{\\rho}}^{n}}-$ $\\begin{array}{ r }{\\mathbb{P}_{Q_{f_{0}}^{n}}\\|_{\\mathrm{TV}}: =\\frac{1}{2}\\int|LQ_{\\nu_{\\rho}}^{n}-1|\\ d\\mathbb{P}_{Q_{f_{0}}^{n}}}\\end{array}$ , and chi-square divergence as $\\begin{array}{ r }{\\chi^{2}(Q_{\\nu_{\\rho}}^{n}\\Vert Q_{f_{0}}^{n}): =\\frac 12\\big(\\mathbb{E}_{Q_{f_{0}}^{n}}[L_{Q_{\\nu_{\\rho}}^{n}}(\\tilde{\\mathbf{Y}}_{1},\\ldots,\\tilde{\\mathbf{Y}}_{n_{1}})^{2}-}\\end{array}$ $1\\bar{\\jmath})^{1/2}$ ."}, {"n": 2, "claim": "The minimax type II error is lower bounded as:\n\n$$\n\\begin{array}{l}\\inf_{\\Delta_{\\gamma,Q}}\\sup_{f\\in\\mathcal{P}}\\mathbb{P}_{Q_{f}^{n}}\\left(\\Delta_{\\gamma,Q}(\\tilde{\\mathbf{Y}}_{1},\\dots,\\tilde{\\mathbf{Y}}_{n_{1}})=0\\right)\\geq\\inf_{\\Delta_{\\gamma,Q}}\\mathbb{P}_{Q_{\\nu_{\\rho}}^{n}}\\left(\\Delta_{\\gamma,Q}(\\tilde{\\mathbf{Y}}_{1},\\dots,\\tilde{\\mathbf{Y}}_{n_{1}})=0\\right)-\\gamma\\\\ \\geq 1-2\\gamma-\\|Q_{\\nu_{\\rho}^{H}}^{n}-Q_{f_{0}}^{n}\\|_{\\mathrm{TV}}\\\\ \\geq 1-2\\gamma-\\frac{1}{2}\\sqrt{\\chi^{2}(Q_{\\nu_{\\rho}}^{n}\\|Q_{f_{0}}^{n})},\\tag{80}\\\\ \\end{array}\n$$\n\nwhere the last inequality uses the inequality between chi-square divergence and total variation distance $\\chi^{2}(Q_{\\nu_{\\rho}}^{n}\\|Q_{f_{0}}^{n})\\geq 4\\|Q_{\\nu_{\\rho}}^{n}-Q_{f_{0}}^{n}\\|_{\\mathrm{TV}}^{2}$ (see, for example, Lemma B.2 of Lam-Weil et al., 2022)."}, {"n": 3, "claim": "Thus it sufices to verify that the condition in (79) implies that the last righthand side term in (80) is lower bounded by $\\beta .$ . Thus the proof is completed by bounding the chi-square divergence as $\\chi^{2}(Q_{\\nu_{\\rho}}^{n}\\Vert Q_{f_{0}}^{n})\\leq-1+$ $\\exp(n_{1}^{2}\\omega^{4}z_{\\alpha}^{4}\\kappa^{d})$, by using the equation (73) and the definition in (74) involving the orthonormal singular vectors of $Q$"}]}}], "target_local_id": null, "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2026-24-0771", "context": "Notation. Scalars are denoted in lower-case and vectors in lower-case bold, e.g. $x\\in\\mathbb{R}$ and $\\mathbf{x}\\in\\mathbb{R}^{n}$ . Note that these will sometimes denote random variables depending on the context. We maintain an analogous notation for scalar-valued and vector-valued functions, e.g., $f$ and $f .$ . The ith coordinate of the vector x is denoted by $\\mathbf{\\mathcal{x}}_{i}$ . The set containing the first n integers excluding 0 is denoted by [n]. Given a subset of indices $S \\subseteq [ n ]$ , x denotes the subvector consisting of entries $\\mathbf{\\nabla}_{\\mathbf{x}_{i}}$ for $i \\in S$ . Given a sequence of T random vectors $(\\pmb{x}^{1},\\dots,\\pmb{x}^{T})$ , the subsequence consisting of the first t elements is denoted by $\\pmb{x}^{\\le t}: =(\\pmb{x}^{1},...,\\pmb{x}^{t})$ , and analogously for $\\scriptstyle{\\pmb x}^{<t}$ . We will sometimes combine these notation to get $\\pmb{x}_{S}^{\\le t}: =(\\pmb{x}_{S}^{1},\\ldots,\\pmb{x}_{S}^{t})$ . Given a function $f:\\mathbb{R}^{n}\\mathbb{R}^{m}$ , its Jacobian matrix evaluated at $\\mathbf{\\boldsymbol{x}}\\in\\mathbb{R}^{n}$ is denoted by $D f(\\pmb{x})\\in\\mathbb{R}^{m\\times n}$ . See Table 5 in the Appendix for more.\n\nIn this section, we introduce the latent variable model under consideration (Section 2.1), entanglement graphs (Section 2.2), identifiability and observational equivalence (Section 2.3), equivalence up to diffeomorphism (Section 2.4) as well as permutation equivalence (Section 2.5).\n\nWe now specify the setting under consideration. Assume that we observe the realization of a sequence of $d_{x}$ -dimensional random vectors $\\{x^{t}\\}_{t=1}^{T}$ and a sequence of $d_{a}$ -dimensional auxiliary vectors $\\{a^{t}\\}_{t=0}^{T-1}$ . The coordinates of $\\pmb{a}^{t}$ are discrete or continuous and can potentially represent, for example, an action taken by an agent or the index of an intervention or environment (see Section 3.4.1). Observations $\\{x^{t}\\}$ are assumed to be explained by a sequence of continuous random vectors $d_{z}$ of hidden dimensions $\\{z^{t}\\}_{t=1}^{T}$ using a ground-truth decoder function $f$\n\nAssumption 1 (Observation model) For all $t \\in [ T ]$ , the observations ${\\boldsymbol{x}}^{t}$ are given by\n\n$$\n\\boldsymbol{x}^{t}=\\boldsymbol{f}(\\boldsymbol{z}^{t})+\\boldsymbol{n}^{t},\n$$\n\nwhere $\\pmb{n}^{t}\\sim\\mathcal{N}(0,\\sigma^{2}\\pmb{I})$ are mutually independent across time and independent ofall $z^{t}$ and $\\pmb{a}^{t}$ with $\\sigma^{2}\\geq 0 .$ . Moreover, $d_{z}\\ \\leq\\ d_{x}$ and $\\pmb{f}:\\mathbb{R}^{d_{z}}\\mathbb{R}^{d_{x}}$ is a diffeomorphism onto its image<sup>2</sup>. Lastly, assume that $f(\\mathbb{R}^{d_{z}})$ is closed in $\\mathbb{R}^{d_{x}}$\n\nImportantly, we suppose that each factor $\\boldsymbol{z}_{i}^{t}$ contains interpretable information about the observation, e.g., for high-dimensional images, the coordinates $\\boldsymbol{z}_{i}^{t}$ might be the position of an object, its color, or its orientation in space. This idea that there exists a ground-truth decoder $f$ that captures the relationship between the so-called “natural factors of variations” and the observations x is of capital importance, since it is the very basis for a mathematical definition of disentanglement (Definition 7). Appendix D.1 discusses the implications of the diffeomorphism assumption (see also Mansouri et al. (2022)). We denote $\\boldsymbol{z}^{\\le t}: =\\left[\\boldsymbol{\\bar{z}}^{1}\\ \\cdot\\ \\cdot\\ \\boldsymbol{z}^{t}\\right]\\in\\mathbb{R}^{d_{z}\\times t}$ and analogously for $z^{<t}$ and other random vectors.\n\nIn a similar spirit to previous works on nonlinear ICA (Hyvarinen et al.¨ , 2019; Khemakhem et al., 2020a), we assume that the latent factors $\\boldsymbol{z}_{i}^{t}$ are conditionally independent given the past.\n\nAssumption 2 (Conditionally independent latent factors) The latent factors $\\boldsymbol{z}_{i}^{t}$ are conditionally mutually independent given $z^{<t}$ and $\\mathbf{\\pmb{a}}^{<t}$ :\n\n$$\np(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})=\\prod_{i=1}^{d_{z}}p(\\boldsymbol{z}_{i}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t}),\\tag{1}\n$$\n\nwhere $p(z^{t}\\mid z^{<t},\\pmb{a}^{<t})$ is a density function w.r.t. the Lebesgue measure on $\\mathbb{R}^{d_{z}}$ . We assume that the support $o f p(z_{i}^{t}\\mid z^{<t},\\pmb{a}^{<t})$ is R for all $z^{<t}$ and $\\pmb{a}^{<t}$ . The support of $p(z^{t}\\mid z^{<t},\\pmb{a}^{<t})$ is thus given by $\\mathbb{R}^{d_{z}}$ . At $t = 1$ , we set $\\begin{array}{ r }{p(z^{1}\\mid z^{<1},{\\pmb a}^{<1})=p(z^{1}\\mid{\\pmb a}^{0})=\\prod_{i=1}^{d_{z}}p(z_{i}^{1}\\mid{\\pmb a}^{0})}\\end{array}$\n\nWe will refer to the l.h.s. of (1) as the transition model and to the factors $p(\\pmb{z}_{i}^{t}\\ |\\ \\pmb{z}^{<t},\\pmb{a}^{<t})$ as mechanisms. In a causal setting, this conditional independence assumption states that latent variables cannot exert influence on other latent variables instantaneously. We believe this to be a reasonable assumption when the time steps are sampled at a sufficiently high rate since, at a fundamental level, causal effects cannot travel faster than the speed of light. Of course in many practical scenarios, a low sampling rate can entail instantaneous dependencies which, if too strong, might result in model misspecification. In Section 7, we discuss and contrast with causal representation learning methods which allow for instantaneous influence.\n\nNotice that we do not assume the dynamical system to be $M a r k o\\nu i a n^{3}.$ , i.e. the distribution over future states can depend on the whole history of latents and auxiliary variables $(z^{<t},a^{<t})$ ). In addition, this model can represent non-homogeneous processes by taking the auxiliary variable a to be a time index (Hyvarinen et al.¨ , 2019).\n\nWe are going to describe the dependency structure of the latent and auxiliary variables over time using a probabilistic directed graphical model composed of two bipartite graphs, $G^{z}\\in\\{0,1\\}^{d_{z}\\times d_{z}}$ which relates $z^{<t}$ to $z^{t},$ , and $G^{a}\\in\\{0,1\\}^{d_{z}\\times d_{a}}$ , which relates $\\pmb{a}^{<t}$ to $z^{t}$ . A directed edge points from $z_{i}^{<t}$ to $\\boldsymbol{z}_{i}^{t}$ if and only if $G_{i,j}^{z}=1$ . Analogously, a directed edge points from $\\pmb{a}_{\\ell}^{<t}$ to $z_{i}^{t}.$ , if and only $\\mathrm{if}^{\\prime}G_{i,\\ell}^{a}=1$ . Figure 1 shows an example of such graphs together with its adjacency matrix $G: =[G^{z},G^{a}]$ . The following assumption specifies the relationship between these graphs and the transition model.\n\nAssumption 3 (Transition model $p$ follows G) For every mechanism $i\\in [d_{z}]$\n\n$$\np(\\boldsymbol{z}_{i}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})=p(\\boldsymbol{z}_{i}^{t}\\mid\\boldsymbol{z}_{\\mathbf{Pa}_{i}^{z}}^{<t},\\boldsymbol{a}_{\\mathbf{Pa}_{i}^{a}}^{<t}),\\tag{2}\n$$\n\nwhere $\\mathbf{Pa}_{i}^{z}\\subseteq [d_{z}]$ and $\\mathbf{Pa}_{i}^{a}\\subseteq [d_{a}]$ are the sets of parents of $\\cdot_{z_{i}^{t}}$ in $G^{z}$ and $G^{a}$ , respectively.\n\nThe graph G thus encodes a set of conditional independence statements about the latent and auxiliary variables.<sup>4</sup> We will say that mechanisms are sparse when the graphs $G^{a}$ and $G^{z}$ are sparse.\n\nThis model has three components that need to be learned: (i) the decoder function f, (ii) the transition model over latent variables $p,$ and (iii) the dependency graph $G$ . We gather all these components into $\\pmb\\theta: =(\\pmb f,p,\\pmb G)$ . Everything else in the model, i.e. $d_{z}$ and $\\sigma^{2}$ , is assumed to be known. We assume that $\\sigma^{2}$ is known here mainly for simplicity, since, when it is not, it can be identified as shown by Lachapelle et al. (2022, Appendix A.4.1), as long as $d_{x}>d_{z}$\n\nNotice how we have not specified any model for the auxiliary variable $\\pmb{a}^{t}$ . We do not intend to do so in this work, as we are solely interested in modeling the conditional distribution of $\\mathbf{\\boldsymbol{x}}^{\\le T}$ and $z^{\\le T}$ given $\\mathbf{\\pmb{a}}^{<T}$ . We denote by $\\mathcal{A}\\subseteq\\mathbb{R}^{d_{a}}$ the set of possible values for the auxiliary variable $\\pmb{a}^{t}$ . We thus have that, for all values of $\\pmb{a}^{<T}\\in\\mathcal{A}^{T}$ , our model induces a conditional distribution\n\n$$\np(\\boldsymbol{x}^{\\leq T}\\mid\\boldsymbol{a}^{<T})=\\int\\prod_{t=1}^{T}p(\\boldsymbol{x}^{t}\\mid\\boldsymbol{z}^{t})p(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})d\\boldsymbol{z}^{\\leq T},\n$$\n\nwhere $p(\\pmb{x}^{t}\\mid z^{t})=\\mathcal{N}(\\pmb{x}^{t};\\pmb{f}(z^{t}),\\sigma^{2}\\pmb{I})$ . We note that if $\\sigma^{2}=0;$ , the conditional distribution of ${\\boldsymbol{x}}^{t}$ given $z^{t}$ is a Dirac centered at $\\pmb{f}(\\pmb{z}^{t})$ and thus has no density w.r.t. the Lebesgue measure. Even if, in that case, the above integral does not make sense, the conditional distribution of $\\mathbf{\\boldsymbol{x}}^{\\le T}$ given $\\pmb{a}^{<T}$ is still well-defined and all the results of this work still hold since none of the proofs requires $\\sigma^{2}>0$\n\nIn this section, we define entanglement maps, which describe the functional relationship between the learned and ground-truth representations, and entanglement graphs, which describe their entanglement structure.\n\nDefinition 1 (Entanglement maps) Let f and $\\tilde{f}$ be two diffeomorphisms from $\\mathbb{R}^{d_{z}}$ to their images such that $\\pmb{f}(\\mathbb{R}^{d_{z}})=\\tilde{\\pmb{f}}(\\mathbb{R}^{d_{z}})$ . The entanglement map of the pair $(f,{\\tilde{f}})$ is given by\n\n$$\n\\boldsymbol{v}:=\\boldsymbol{f}^{-1}\\circ\\tilde{\\boldsymbol{f}}.\\tag{3}\n$$\n\nWe now define the dependency graph of some function h so that each edge indicates that some input i influences some output j:\n\nDefinition 2 (Functional dependency graph) Let h be a function from $\\mathbb{R}^{n}$ to $\\mathbb{R}^{m}$ . The dependency graph of h is a bipartite directed graph from [n] to [m] with adjacency matrix $\\pmb{H}\\in\\{0,1\\}^{m\\times n}$ such that\n\n$$\n\\boldsymbol{H}_{i,j}=0\\iff \\text{There is a function}\\bar{\\boldsymbol{h}}\\text{such that, for all}\\boldsymbol{a}\\in\\mathbb{R}^{n},\\boldsymbol{h}_{i}(\\boldsymbol{a})=\\bar{\\boldsymbol{h}}_{i}(\\boldsymbol{a}_{-j}),\n$$\n\nwhere $\\pmb{a}_{-j}$ is a with its jth coordinate removed.\n\nWe will be particularly interested in the dependency graph of the entanglement map $\\pmb{v}: =\\pmb{f}^{-1}$ ◦ ${\\hat{f}},$ denoted by V.\n\nDefinition 3 (Entanglement graphs) Let f and $\\tilde{f}$ be two diffeomorphisms from $\\mathbb{R}^{d_{z}}$ to their images such that $\\pmb{f}(\\mathbb{R}^{d_{z}})=\\tilde{\\pmb{f}}(\\mathbb{R}^{d_{z}})$ . The entanglement graph of the pair $(f,{\\tilde{f}})$ is the dependency graph (Definition 2) of their entanglement map $\\pmb{v}: =\\pmb{f}^{-1}\\circ\\tilde{\\pmb{f}},$ , which we denote $V\\in\\{0,1\\}^{d_{z}\\times d_{z}}$\n\nTo analyze formally whether a specific algorithm is expected to yield a disentangled representation, we will rely on the notion of identifiability. Before we define what we mean by identifiability, we need the notion of observationally equivalent models. Two models are observationally equivalent if both models represent the same distribution over observations. The following formalizes this definition.\n\nDefinition 4 (Observational equivalence) We say that two models $\\pmb\\theta: =(\\pmb f,p,\\pmb G)$ and $\\tilde{\\pmb{\\theta}}: =(\\tilde{f},\\tilde{p},\\tilde{G})$ satisfying Assumption 1 are observationally equivalent, denoted $\\theta\\sim_{\\mathrm{obs}}\\theta,$ , if and only if, for all $\\mathbf{\\Psi}_{\\mathbf{\\lambda}}\\mathbf{a}^{<T}\\in\\mathcal{\\bar{A}}^{T}$ and all $\\pmb{x}^{\\le T}\\in\\mathbb{R}^{d_{x}\\times T}$\n\n$$\np(\\boldsymbol{x}^{\\leq T}\\mid\\boldsymbol{a}^{<T})=\\tilde{p}(\\boldsymbol{x}^{\\leq T}\\mid\\boldsymbol{a}^{<T}).\n$$\n\nWe start by defining equivalence up to diffeomorphism. This equivalence relation is important since we will show later on that it is actually the same as observational equivalence and will thus be our first step in all our identifiability results. In what follows, we overload the notation and write $\\pmb{v}(z^{<t}): =[\\pmb{v}(z^{1}),\\dots,\\pmb{v}(z^{t-1})]$ , and similarly for other functions.\n\nDefinition 5 (Equivalence up to diffeomorphism) We say that two models $\\pmb\\theta: =(\\pmb f,p,\\pmb G)$ and $\\tilde{\\pmb{\\theta}}: =(\\tilde{\\pmb{f}},\\tilde{p},\\tilde{G})$ satisfying Assumption 1 are equivalent up to diffeomorphism, denoted $\\theta\\sim_{\\mathrm{diff}}\\tilde{\\theta},$ if and only if $\\cdot f(\\mathbb{R}^{d_{z}})=\\tilde{\\pmb{f}}(\\mathbb{R}^{d_{z}})$ and, for all $t \\in [ T ]$ , all $\\mathbf{\\boldsymbol{a}}^{<t}\\in\\mathcal{A}^{t}$ and all $\\pmb{z}^{\\leq t}\\in\\mathbb{R}^{d_{z}\\times t}$\n\n$$\n\\tilde{p}(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})=p(\\boldsymbol{v}(\\boldsymbol{z}^{t})\\mid\\boldsymbol{v}(\\boldsymbol{z}^{<t}),\\boldsymbol{a}^{<t})|\\det D\\boldsymbol{v}(\\boldsymbol{z}^{t})|,\\tag{6}\n$$\n\nwhere $\\pmb{v}: =\\pmb{f}^{-1}\\circ\\tilde{\\pmb{f}}$ (entanglement map) is a diffeomorphism and Dv denotes its Jacobian matrix.\n\nThe following proposition establishes the converse, i.e. that $\\theta\\sim_{\\mathrm{obs}}\\tilde{\\theta}$ implies $\\theta\\sim_{\\mathrm{diff}}\\tilde{\\theta}.$ . Since its proof is more involved, we present it in the Appendix A.3. Note that this first identifiability result is relatively weak and should be seen as a first step towards stronger guarantees. A very similar result was shown by Ahuja et al. (2022a, Theorem 3.1) to highlight the fact that the representation $f$ is identifiable up to the equivariances v of the transition model $p .$\n\nProposition 2 (Identifiability up to diffeomorphism) Let $\\pmb\\theta: =(\\pmb f,p,\\pmb G)$ and $\\hat{\\pmb\\theta}: =(\\hat{f},\\hat{p},\\hat{G})$ be two models satisfying Assumption 1. $H f\\pmb\\theta\\sim_{\\mathrm{obs}}\\hat{\\pmb\\theta}(D e f . 4),$ , then $\\pmb\\theta\\sim_{\\mathrm{diff}}\\hat{\\pmb\\theta}(D e f . 5)$\n\nIn this section, we show that without further assumptions the model introduced so far is unidentifiable (Section 3.1), provide a first insight as to why mechanism sparsity can make the model identifi able and thus lead to disentanglement (Section 3.2), introduce the machinery of G-preserving maps (Section 3.3) which leads up to theorems showing identifiability up to a-consistency (Section 3.4) and z-consistency (Section 3.5), which corresponds to partial disentanglement. We also relate these results to interventions (Section 3.4.1), show how to combine both regularization on $\\hat{G}^{a}$ and $\\hat{G}^{z}$ to obtain stronger guarantees (Section 3.6) and introduce a graphical criterion guaranteeing complete disentanglement (Section 3.7). Finally, we introduce the sufficient influence assumptions and prove the identifiability results (Section 3.8), and provide multiple examples to build intuition (Sec tion 3.9).\n\nDefinition 9 (Vectors & matrices as index sets) Let $\\mathbf{\\pmb{a}}\\in\\mathbb{R}^{n}$ and $\\pmb{A}\\in\\mathbb{R}^{m\\times n}$ . We will sometimes use a to denote the set of indices corresponding to the support of the vector a, i.e.\n\n$$\n\\boldsymbol{a}\\sim\\{i\\in [n]\\mid\\boldsymbol{a}_{i}\\neq 0\\}.\n$$\n\nThis will allow us to write things like $i\\in a o r\\mathbf{a}\\subseteq b,$ where $\\ b{b}\\in\\mathbb{R}^{n}$ . We will use an analogous convention for matrices, $i . e .$\n\n$$\n\\boldsymbol{A}\\sim\\{(i,j)\\in [m]\\times [n]\\mid\\boldsymbol{A}_{i,j}\\neq 0\\},\n$$\n\nThis will allow us to write things like $(i,j)\\in A$ and ${\\mathbf{}{\\mathbf{}{\\mathbf{}{A}}}\\subseteq{\\mathbf{}{\\mathbf{}{B}}}}$ , where $\\pmb{{B}}\\in\\mathbb{R}^{m\\times n}$\n\nRecall that we would like to show that $\\theta\\sim_{\\mathrm{obs}}\\hat{\\theta}$ implies $\\theta\\sim_{\\mathrm{perm}}\\hat{\\pmb\\theta}$ , i.e., disentanglement (or partial disentanglement). Our approach will be to start from (6), which is guaranteed by Proposition 2, and perform a series of algebraic manipulations to gain mathematical insight into how regularizing $\\hat{G}$ to be sparse (mechanism sparsity) can induce disentanglement. A key manipulation will be taking first and second order derivatives. For this to be possible, we require a certain level of smoothness for the transition models:\n\nAssumption 4 (Smoothness of transition model) When a is continuous, the transition densities $p(z_{i}^{t}\\mid z^{<t},\\pmb{a}^{<t})$ are $C^{2}$ functions from $\\mathbb{R}\\times\\mathbb{R}^{d_{z}\\times(t-1)}\\times\\mathcal{A}^{t}$ to R and $\\mathcal{A}\\subseteq\\mathbb{R}^{\\ell}$ is regular closed<sup>5</sup>. When a is discrete (e.g. Section $3.4.l),$ , for all $a^{<t},p(z_{i}^{t}\\mid z^{<t},{}a^{<t})$ are $C^{2}$ functions from R $\\times$ $\\mathbb{R}^{d_{z}\\times(t-1)}t o\\mathbb{R}.$\n\nWe start by taking the log on both sides of (6) and let $q : =$ log p and $\\hat{q}: =$ log ˆp:\n\n$$\n\\hat{q}(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})=q(\\boldsymbol{v}(\\boldsymbol{z}^{t})\\mid\\boldsymbol{v}(\\boldsymbol{z}^{<t}),\\boldsymbol{a}^{<t})+\\log|\\det D\\boldsymbol{v}(\\boldsymbol{z}^{t})|.\n$$\n\nWe then take the derivative w.r.t. $z^{t}$ on both sides:\n\n$$\nD_{z}^{t}\\hat{q}(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})=D_{z}^{t}q(\\boldsymbol{v}(\\boldsymbol{z}^{t})\\mid\\boldsymbol{v}(\\boldsymbol{z}^{<t}),\\boldsymbol{a}^{<t})D\\boldsymbol{v}(\\boldsymbol{z}^{t})+\\eta(\\boldsymbol{z}^{t})\\in\\mathbb{R}^{1\\times d_{z}},\\tag{8}\n$$\n\nwhere $D_{z}^{t}q$ denotes the Jacobian of $q(\\pmb{z}^{t}\\ |\\ \\pmb{z}^{<t},\\pmb{a}^{<t})$ w.r.t. $z^{t}$ and analogously for $D_{z}^{t}\\hat{q}$ . The term $\\eta(\\boldsymbol{z}^{t})$ is the derivative of log | det $D{\\pmb v}({\\pmb z}^{t})\\vert$ | w.r.t. $z^{t}$\n\nSparse temporal dependencies. In what precedes, we made use of the sparsity of the graph $\\hat{G}^{a}$ to argue that Dv must also be sparse. We now show a similar intuition based on the sparsity of $\\hat{G}^{z}$ . Starting from (8), instead of differentiating w.r.t. $\\pmb{a}^{\\tau}$ , we will differentiate w.r.t. $z^{\\tau}$ , for some $\\tau < t .$ , which yields:\n\n$$\nH_{z,z}^{t,\\tau}\\hat{q}(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})=D\\boldsymbol{v}(\\boldsymbol{z}^{t})^{\\top}H_{z,z}^{t,\\tau}q(\\boldsymbol{v}(\\boldsymbol{z}^{t})\\mid\\boldsymbol{v}(\\boldsymbol{z}^{<t}),\\boldsymbol{a}^{<t})D\\boldsymbol{v}(\\boldsymbol{z}^{\\tau})\\in\\mathbb{R}^{d_{z}\\times d_{z}},\\tag{14}\n$$\n\nwhere $H_{z,z}^{t,\\tau}q$ is the Hessian matrix of second derivatives of $q$ w.r.t. $z^{t}$ and $z^{\\tau}$ , and analogously for $H_{z,z}^{t,\\tau}\\hat{q}$ . Using an argument perfectly analogous to Equations (10) to (12), we can show that whenever $G_{i,j}^{z}=0$ , we also have $(H_{z,z}^{t,\\tau}q)_{i,j}=0$ , and similarly for $\\hat{G}^{z}$ and $H_{z,z}^{t,\\tau}\\hat{q}$ . In other words, $H_{z,z}^{t,\\tau}q\\subseteq G^{z}$ and $H_{z,z}^{t,\\tau}\\hat{q}\\subseteq\\hat{G}^{z}$ . Therefore, analogously to (13), regularizing $\\hat{G}^{z}$ to be sparse intuitively should force Dv to be sparse as well, i.e., bringing us closer to disentanglement:\n\n$$\n\\underbrace{H_{z,z}^{t,\\tau}\\hat{q}\\left(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t}\\right)}_{\\subseteq\\hat{\\boldsymbol{G}}^{z}}=\\underbrace{D\\boldsymbol{v}\\left(\\boldsymbol{z}^{t}\\right)^{\\top}}_{\\text{forced to be sparse}}\\underbrace{H_{z,z}^{t,\\tau}q\\left(\\boldsymbol{v}\\left(\\boldsymbol{z}^{t}\\right)\\mid\\boldsymbol{v}\\left(\\boldsymbol{z}^{<t}\\right),\\boldsymbol{a}^{<t}\\right)}_{\\subseteq\\boldsymbol{G}^{z}}\\underbrace{D\\boldsymbol{v}\\left(\\boldsymbol{z}^{\\tau}\\right)}_{\\text{forced to be sparse}}.\\tag{15}\n$$\n\nTheorems 1, 2, 3 & 5 will show how regularizing $\\hat{G}$ to be sparse can force the dependency graph of the entanglement map v to be sparse as well. These results characterize the functional dependency structure of the entanglement map v as a function of the ground-truth graph $G .$ . This link will be made precise thanks to the notion of graph preserving maps, which we define next. Before going further, we need to set up the following notation.\n\nDefinition 10 (Aligned subspaces of $\\mathbb{R}^{m}$ and $\\mathbb{R}^{m\\times n})$ Given a binary vector $\\pmb{b}\\in\\{0,1\\}^{m}$ , let\n\n$$\n\\mathbb{R}_{\\boldsymbol{b}}^{m}:=\\left\\{\\boldsymbol{x}\\in\\mathbb{R}^{m}\\mid\\boldsymbol{b}_{i}=0\\implies\\boldsymbol{x}_{i}=0\\right\\}\n$$\n\nGiven a binary matrix $\\pmb{{\\cal B}}\\in\\{0,1\\}^{m\\times n}$ , let\n\n$$\n\\mathbb{R}_{\\boldsymbol{B}}^{m\\times n}:=\\left\\{\\boldsymbol{M}\\in\\mathbb{R}^{m\\times n}\\mid\\boldsymbol{B}_{i,j}=0\\implies\\boldsymbol{M}_{i,j}=0\\right\\}.\n$$\n\nTo start formally reasoning about what will happen if $\\hat{G}^{a}$ is regularized sparse, we temporarily assume that $\\hat{G}^{a}=G^{a}$ . With this assumption, we can interpret (13) as implying that $D v(z^{t})^{\\top}$ must preserve the “sparsity structure” of the matrix $H_{z,a}^{t,\\tau}q$ . This observation motivates the following definitions, which will be central to our contribution.\n\nDefinition 11 (G-preserving matrix) Given $\\pmb{G}\\in\\{0,1\\}^{m\\times n}$ , a matrix $C\\in\\mathbb{R}^{m\\times m}$ is G-preserving when\n\n$$\n\\boldsymbol{C}^{\\top}\\mathbb{R}_{\\boldsymbol{G}}^{m\\times n}\\subseteq\\mathbb{R}_{\\boldsymbol{G}}^{m\\times n}.\n$$\n\nDefinition 12 (G-preserving functions) Given $G\\in\\{0,1\\}^{m\\times n}$ , a function $c:\\mathbb{R}^{m}\\mathbb{R}^{m}$ is $G\\mathrm{-}$ preserving when its dependency graph C (Definition 2) is G-preserving.\n\nWe now show that G-preserving functions can be defined alternatively in terms of a simple condition on their dependency graph. This characterization of G-preserving functions is key to understanding how (partial) disentanglement results from sparsity regularization.\n\nProposition 3 A function c with dependency graph C (Definition 2) is G-preserving if and only\n\n$$\n\\boldsymbol{G}_{i,\\cdot}\\not\\subseteq\\boldsymbol{G}_{j,\\cdot}\\implies\\boldsymbol{C}_{i,j}=0,\\text{   for   all   } i,j.\n$$\n\nWe now characterize differentiable G-preserving functions in terms of their Jacobian matrices.\n\nLemma 1 A differentiable function $c:\\mathbb{R}^{m}\\mathbb{R}^{m}$ is G-preserving if and only if, for all $z\\in\\mathbb{R}^{m}$ $Dc(z)$ is G-preserving.\n\nIn this section, we introduce our first identifiability results based on the sparsity of the graph $G^{a}$ which describes the structure of the dependencies between $\\pmb{a}^{<t}$ and $z^{t}$ . We will see that, under some assumptions, regularizing the learned graph $\\hat{G}^{a}$ to be sparse will allow identifiability up to the following equivalence class:\n\nDefinition 13 (a-consistency equivalence) We say two models $\\pmb\\theta: =(\\pmb f,p,\\pmb G)$ and $\\tilde{\\pmb{\\theta}}: =(\\tilde{f},\\tilde{p},\\tilde{G})$ satisfying Assumptions 1, 2 & 3 are a-consistent, denoted $\\theta\\sim_{\\mathrm{con}}^{a}\\tilde{\\theta},$ , if and only if there exists a permutation matrix $_{r}$ such that\n\n1. $\\pmb\\theta\\sim_{\\mathrm{diff}}\\tilde{\\pmb\\theta}(D e f . 5)$ , and $\\tilde{G}^{a}=P G^{a}$ ; and  \n2. the entanglement map $\\pmb{v}: =\\pmb{f}^{-1}\\circ\\tilde{\\pmb{f}}$ can be written as $\\pmb{v}=\\pmb{c}\\circ\\pmb{P}^{\\top}$ where c is a $G^{a}$ -preserving diffeomorphism (Def. 12).\n\nThe first result provides conditions under which regularizing the learned graph $\\hat{G}^{a}$ to be as sparse as the ground-truth graph $G^{a}$ will induce the learned model to be a-consistent with the ground-truth one.\n\nTheorem 1 (Nonparametric disentanglement from continuous a with sparse influence) Let the parameters $\\pmb\\theta: =(\\pmb f,p,\\pmb G)$ and $\\hat{\\pmb\\theta}: =(\\hat{f},\\hat{p},\\hat{G})$ correspond to two models satisfying Assumptions 1, 2, 3, & 4. Further assume that\n\n1. [Observational equivalence] $\\pmb{\\theta}\\sim_{\\mathrm{obs}}\\hat{\\pmb{\\theta}}(D e f . 4),$  \n2. [Sufficient influence of a] The Hessian matrix $H_{z,a}^{t,\\tau}$ log $p(z^{t}\\mid z^{<t},\\pmb{a}^{<t})$ varies “sufficiently”, as formalized in Assumption ${ \\it 6 ; }$\n\nThen, there exists a permutation matrix P such that $P G^{a}\\subseteq{\\hat{G}}^{a}$ . Further assume that\n\n3. [Sparsity regularization] $| |\\hat{G}^{a}| |_{0}\\leq| |G^{a}| |_{0};$\n\nThen, $\\pmb\\theta\\sim_{\\mathrm{con}}^{\\pmb a}\\hat{\\pmb\\theta}(D e f . I 3)$\n\nThis section is analogous to the previous one, but instead of leveraging the sparsity of $G^{a}$ to show identifiability, it leverages the sparsity of $G^{z}$ , which describes the structure of the dependencies between the latents from one time step to another. We will see that, under some assumptions, regularizing the learned graph $\\hat{G}^{z}$ to be sparse will allow identifiability up to the following equivalence class:\n\nDefinition 14 (z-consistency equivalence) We say two models $\\pmb\\theta: =(\\pmb f,p,\\pmb G)$ and $\\tilde{\\pmb{\\theta}}: =(\\tilde{\\pmb{f}},\\tilde{p},\\tilde{G})$ satisfying Assumptions 1, 2 & 3 are z-consistent, denoted $\\theta\\sim_{\\mathrm{con}}^{z}\\tilde{\\theta},$ , if and only if there exists a permutation matrix P such that\n\n1. $\\pmb\\theta\\sim_{\\mathrm{diff}}\\tilde{\\pmb\\theta}(D e f . 5)$ and $\\tilde{G}^{z}=P G^{z}P^{\\top}$ ; and  \n2. the entanglement map $\\pmb{v}: =\\pmb{f}^{-1}\\circ\\tilde{\\pmb{f}}$ can be written as $\\pmb{v}=\\pmb{c}\\circ\\pmb{P}^{\\top}$ where c is a $G^{z}$ -preserving and $\\left(G^{z}\\right)^{\\top}$ -preserving diffeomorphism (Definition 12).\n\nAssumption 5 (Graphical criterion, Lachapelle et al. (2022)) Let $G=[G^{z}\\ G^{a}]$ be a graph. For all $i\\in\\{1,\\ldots,d_{z}\\}$ ,\n\n$$\n\\left(\\bigcap_{j\\in\\mathbf{Ch}_{i}^{z}}\\mathbf{Pa}_{j}^{z}\\right)\\cap\\left(\\bigcap_{j\\in\\mathbf{Pa}_{i}^{z}}\\mathbf{Ch}_{j}^{z}\\right)\\cap\\left(\\bigcap_{\\ell\\in\\mathbf{Pa}_{i}^{a}}\\mathbf{Ch}_{\\ell}^{a}\\right)=\\{i\\},\n$$\n\nwhere $\\mathbf{Pa}_{i}^{z}$ and $\\mathbf{Ch}_{i}^{z}$ are the sets of parents and children of node $z_{i}$ in $G^{z}$ , respectively, while Ch<sup>a</sup> is the set of children of $\\mathbf{a}_{\\ell}$ in $G^{a}$\n\nIn this section, we introduce the sufficient influence assumptions and use them to prove Theorems 1, 2 & 3. In the next section (Section 3.9), we provide multiple examples to gain intuition about the sufficient influence assumptions. Throughout, the following lemma will come in handy.\n\nLemma 2 (Invertible matrix contains a permutation) Let $\\pmb{L}\\in\\mathbb{R}^{m\\times m}$ be an invertible matrix. Then, there exists a permutation σ such that $\\begin{array}{ r }{{\\cal L}_{i,\\sigma(i)}\\neq 0 f o r a l l i,}\\end{array}$ or, in other words, $P^{\\top}\\subseteq L$ where P is the permutation matrix associated with σ, i.e., $P e_{i}=e_{\\sigma(i)}$ . Note that this implies that PL and LP have no zero on their diagonals.\n\nProof [of Theorem 1] Recall equation (13), which we derived in Section 3.2:\n\n$$\n\\underbrace{H_{z,a}^{t,\\tau}\\hat{q}\\left(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t}\\right)}_{\\subseteq\\hat{\\boldsymbol{G}}^{a}}=D\\boldsymbol{v}\\left(\\boldsymbol{z}^{t}\\right)^{\\top}\\underbrace{H_{z,a}^{t,\\tau}q\\left(\\boldsymbol{v}\\left(\\boldsymbol{z}^{t}\\right)\\mid\\boldsymbol{v}\\left(\\boldsymbol{z}^{<t}\\right),\\boldsymbol{a}^{<t}\\right)}_{\\subseteq\\boldsymbol{G}^{a}}.\\tag{16}\n$$\n\nNotice that Assumption 6 holds only “almost everywhere”, i.e. on a set $\\mathbb{R}^{d_{z}}\\setminus E_{0}$ where $E_{0}$ has zero Lebesgue measure. Fix an arbitrary $\\boldsymbol{z}\\in\\mathbb{R}^{d_{z}}\\setminus E_{0}$ . For notational convenience, define\n\n$$\n\\Lambda(\\boldsymbol{z},\\gamma):=H_{z,a}^{t,\\tau}q(\\boldsymbol{v}(\\boldsymbol{z})\\mid\\boldsymbol{v}(\\boldsymbol{z}^{<t}),\\boldsymbol{a}^{<t})\\qquad\\hat{\\Lambda}(\\boldsymbol{z},\\gamma):=H_{z,a}^{t,\\tau}\\hat{q}(\\boldsymbol{z}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t}),\n$$\n\nwhere $\\gamma:=(t,\\tau,\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})$ . This allows us to rewrite (16) with a much lighter notation:\n\n$$\n\\hat{\\Lambda}(\\boldsymbol{z},\\gamma)=D\\boldsymbol{v}(\\boldsymbol{z})^{\\top}\\Lambda(\\boldsymbol{z},\\gamma).\\tag{17}\n$$\n\nNow, notice that the sufficient influence assumption (Assumption 6) requires that, for all $\\ell\\in[d_{a}]$ there exists $\\{\\gamma_{(r)}\\}_{r=1}^{|\\mathbf{Ch}_{\\ell}^{a}|}$ such that $\\mathrm{span}\\{\\Lambda(\\boldsymbol{z},\\gamma_{(r)})_{\\cdot,\\ell}\\}_{r=1}^{|\\mathbf{Ch}_{\\ell}^{a}|}=\\mathbb{R}_{\\mathbf{Ch}_{\\ell}^{a}}^{d_{z}}$ . We can thus write\n\n$$\nD\\boldsymbol{v}(\\boldsymbol{z})^{\\top}\\mathbb{R}_{\\boldsymbol{G}_{\\cdot,\\ell}^{a}}^{d_{z}}=D\\boldsymbol{v}(\\boldsymbol{z})^{\\top}\\mathrm{span}\\{\\Lambda(\\boldsymbol{z},\\gamma_{(r)})_{\\cdot,\\ell}\\}_{r=1}^{|\\mathbf{Ch}_{\\ell}^{a}|}=\\mathrm{span}\\{\\hat{\\Lambda}(\\boldsymbol{z},\\gamma_{(r)})_{\\cdot,\\ell}\\}_{r=1}^{|\\mathbf{Ch}_{\\ell}^{a}|}\\subseteq\\mathbb{R}_{\\hat{\\boldsymbol{G}}_{\\cdot,\\ell}^{a}}^{d_{z}}\\tag{18}\n$$\n\nSince $D\\boldsymbol{v}(\\boldsymbol{z})$ is invertible, there exists a permutation $\\boldsymbol{P}(\\boldsymbol{z})$ such that $D\\boldsymbol{v}(\\boldsymbol{z})\\boldsymbol{P}(\\boldsymbol{z})$ has no zero on its diagonal (Lemma 2). Let $\\boldsymbol{C}(\\boldsymbol{z}):=D\\boldsymbol{v}(\\boldsymbol{z})\\boldsymbol{P}(\\boldsymbol{z})$ . By left-multiplying (18) by $\\boldsymbol{P}(\\boldsymbol{z})^{\\top}$ , we get\n\n$$\n\\boldsymbol{C}(\\boldsymbol{z})^{\\top}\\mathbb{R}_{\\boldsymbol{G}_{\\cdot,\\ell}^{a}}^{d_{z}}\\subseteq\\mathbb{R}_{\\boldsymbol{P}(\\boldsymbol{z})^{\\top}\\hat{\\boldsymbol{G}}_{\\cdot,\\ell}^{a}}^{d_{z}}.\\tag{19}\n$$\n\nWe would like to show that $\\boldsymbol{C}(\\boldsymbol{z})$ is $\\boldsymbol{G}^{a}$ -preserving. Notice how the above equation is almost exactly the definition of $\\boldsymbol{G}^{a}$ -preserving. All that is left to prove is that $\\boldsymbol{P}(\\boldsymbol{z})^{\\top}\\hat{\\boldsymbol{G}}^{a}=\\boldsymbol{G}^{a}$ .\n\nWe start by showing $\\boldsymbol{P}(\\boldsymbol{z})^{\\top}\\hat{\\boldsymbol{G}}^{a}\\supseteq\\boldsymbol{G}^{a}$ . Take $(i,\\ell)\\in\\boldsymbol{G}^{a}$ . Since $\\boldsymbol{e}_{i}\\in\\mathbb{R}_{\\boldsymbol{G}_{\\cdot,\\ell}^{a}}^{d_{z}}$ , equation (19) implies\n\n$$\n\\boldsymbol{C}(\\boldsymbol{z})^{\\top}\\boldsymbol{e}_{i}=\\boldsymbol{C}(\\boldsymbol{z})_{i,\\cdot}\\in\\mathbb{R}_{\\boldsymbol{P}(\\boldsymbol{z})^{\\top}\\hat{\\boldsymbol{G}}_{\\cdot,\\ell}^{a}}^{d_{z}}.\n$$\n\nSince $\\boldsymbol{C}(\\boldsymbol{z})_{i,i}\\neq0$ (all elements on its diagonal are nonzero), we must have that $(i,\\ell)\\in\\boldsymbol{P}(\\boldsymbol{z})^{\\top}\\hat{\\boldsymbol{G}}^{a}$ .\n\nNow, since $||\\boldsymbol{P}(\\boldsymbol{z})^{\\top}\\hat{\\boldsymbol{G}}^{a}||_{0}=||\\hat{\\boldsymbol{G}}^{a}||_{0}\\leq||\\boldsymbol{G}^{a}||_{0}$ , we have $\\boldsymbol{P}(\\boldsymbol{z})^{\\top}\\hat{\\boldsymbol{G}}^{a}=\\boldsymbol{G}^{a}$ . This implies\n\n$$\n\\boldsymbol{C}(\\boldsymbol{z})^{\\top}\\mathbb{R}_{\\boldsymbol{G}^{a}}^{d_{z}\\times d_{a}}\\subseteq\\mathbb{R}_{\\boldsymbol{G}^{a}}^{d_{z}\\times d_{a}},\n$$\n\ni.e. $\\boldsymbol{C}(\\boldsymbol{z})$ is a $\\boldsymbol{G}^{a}$ -preserving matrix, as desired.\n\nTo recap, we now have that, for all $\\boldsymbol{z}\\in\\mathbb{R}^{d_{z}}\\setminus E_{0}$ , there exists a permutation $\\boldsymbol{P}(\\boldsymbol{z})$ s.t. $D\\boldsymbol{v}(\\boldsymbol{z})\\boldsymbol{P}(\\boldsymbol{z})$ is $\\boldsymbol{G}^{a}$ -preserving. We are not done yet, since, a priori, the permutation $\\boldsymbol{P}(\\boldsymbol{z})$ can be different for different values of $\\boldsymbol{z}$ , and we do not know what happens on the measure-zero set $E_{0}$ . What we need to show is that there exists a permutation $\\boldsymbol{P}$ such that, for all $\\boldsymbol{z}$ , $D\\boldsymbol{v}(\\boldsymbol{z})\\boldsymbol{P}$ is $\\boldsymbol{G}^{a}$ -preserving. Lemma 12 in Appendix A.5 shows precisely this, by leveraging the continuity of $D\\boldsymbol{v}(\\boldsymbol{z})$ ( $\\boldsymbol{v}$ is a diffeomorphism and thus $C^{1}$ ).\n\nNotice that $D(\\boldsymbol{v}\\circ\\boldsymbol{P})(\\boldsymbol{z})=D\\boldsymbol{v}(\\boldsymbol{P}\\boldsymbol{z})\\boldsymbol{P}$ , which is $\\boldsymbol{G}^{a}$ -preserving everywhere. Using Lemma 1, we conclude that the function $\\boldsymbol{c}:=\\boldsymbol{v}\\circ\\boldsymbol{P}$ is $\\boldsymbol{G}^{a}$ -preserving. This concludes the proof.\n\nOne can see that, if a is discrete, Theorem 1 cannot be applied because its sufficient influence assumption (Assumption 6) refers to the cross derivative of log p w.r.t. $z^{t}$ and $\\pmb{a}^{\\tau}$ , which, of course, is not well defined when a is discrete. The discrete case is important to discuss interventions with unknown-targets as we did in Section 3.4.1, which is why we have a specialized result (Theorem 2) which has an analogous sufficient influence assumption based on partial differences.\n\nDefinition 16 (Partial difference) Let us define\n\n$$\n\\Delta_{a,\\ell}^{\\tau,\\epsilon}D_{z}^{t}\\log p(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t}):=D_{z}^{t}\\log p(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t}+\\epsilon\\boldsymbol{E}^{(\\ell,\\tau)})-D_{z}^{t}\\log p(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t}),\n$$\n\nwhere $\\epsilon\\in\\mathbb{R}$ and $\\pmb{{\\cal E}^{(\\ell,\\tau)}}$ is a matrix with a one at entry $(\\ell,\\tau)$ and zeros everywhere else.\n\nOne can see that $\\Delta_{a,\\ell}^{\\tau,\\epsilon}D_{z}^{t}$ log p is essentially the discrete analog of $(H_{z,a}^{t,\\tau}\\log p)., $ . Apart from this difference, the sufficient influence assumption for discrete a is the same as for continuous a.\n\nAssumption 7 (Sufficient influence of a (nonparametric/discrete)) For almost all $\\boldsymbol{z}\\in\\mathbb{R}^{d_{\\boldsymbol{z}}}\\left(i . e .\\right.$ except on a set with zero Lebesgue measure) and all $\\ell\\in [d_{a}];$ , there exists\n\n$$\n\\left\\{\\left(t_{(r)},\\tau_{(r)},\\boldsymbol{z}_{(r)},\\boldsymbol{a}_{(r)}^{<t},\\epsilon_{(r)}\\right)\\right\\}_{r=1}^{\\left|\\mathbf{Ch}_{\\ell}^{\\boldsymbol{a}}\\right|},\n$$\n\nsuch that $t_{(r)}\\in [T],\\tau_{(r)}<t_{(r)},z_{(r)}\\in\\mathbb{R}^{d_{z}\\times(t_{(r)}-1)},{\\pmb a}_{(r)}\\in\\mathcal{A}^{t_{(r)}},\\epsilon_{(r)}\\in\\mathbb{R},({\\pmb a}_{(r)})..._{\\tau_{(r)}}+\\epsilon_{(r)}e_{\\ell}\\in\\mathcal{A}$ and\n\n$$\n\\operatorname{span}\\left\\{\\Delta_{a,\\ell}^{\\tau_{(r)},\\epsilon_{(r)}}D_{z}^{t_{(r)}}\\log p(\\boldsymbol{z}\\mid\\boldsymbol{z}_{(r)},\\boldsymbol{a}_{(r)})\\right\\}_{r=1}^{|\\mathbf{Ch}_{\\ell}^{a}|}=\\mathbb{R}_{\\mathbf{Ch}_{\\ell}^{a}}^{d_{z}}.\n$$\n\nWe now introduce the sufficient influence assumption of Theorem 3, which showed how regularizing the temporal dependency graph $\\hat{G}^{z}$ to be sparse can result in disentanglement. Again, it is very similar to other sufficient influence assumptions we have seen so far.\n\nAssumption 8 (Sufficient influence of z (nonparametric)) For almost all $z\\in\\mathbb{R}^{d_{z}}$ (i.e. except on a set with zero Lebesgue measure), there exists\n\n$$\n\\{(\\boldsymbol{t}_{(r)},\\boldsymbol{\\tau}_{(r)},\\boldsymbol{z}_{(r)},\\boldsymbol{a}_{(r)})\\}_{r=1}^{| |\\boldsymbol{G}^{z}| |_{0}},\n$$\n\nsuch that $t_{(r)}\\in [T],\\tau_{(r)}<t_{(r)},z_{(r)}\\in\\mathbb{R}^{d_{z}\\times\\left(t_{(r)}-1\\right)},a_{(r)}\\in\\mathcal{A}^{t_{(r)}},z=z_{(r)}^{\\tau_{(r)}}$ and\n\n$$\n\\operatorname{span}\\left\\{H_{z,z}^{t_{(r)},\\tau_{(r)}}q(\\boldsymbol{z}\\mid\\boldsymbol{z}_{(r)},\\boldsymbol{a}_{(r)})\\right\\}_{r=1}^{| |\\boldsymbol{G}^{z}| |_{0}}=\\mathbb{R}_{\\boldsymbol{G}^{z}}^{d_{z}}.\n$$\n\nWe will assume that the conditional densities $p(z_{i}^{t}\\mid z^{<t},\\pmb{a}^{<t})$ are from an exponentialfamily (Wainwright and Jordan, 2008):\n\nAssumption 9 (Exponential family transition model) For all $i\\in [d_{z}]$ , we have\n\n$$\np(\\boldsymbol{z}_{i}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})=h_{i}(\\boldsymbol{z}_{i}^{t})\\exp\\{\\boldsymbol{s}_{i}(\\boldsymbol{z}_{i}^{t})^{\\top}\\boldsymbol{\\lambda}_{i}(\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})-\\psi_{i}(\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})\\}.\\tag{30}\n$$\n\nWell-known distributions which belong to this family include the Gaussian and beta distributions. In the Gaussian case, the sufficient statistic is $\\pmb{s}_{i}(z): =(z,z^{2})$ and the base measure is $\\begin{array}{ r }{h_{i}(z): =\\frac{1}{\\sqrt{2\\pi}}}\\end{array}$ . The function $\\lambda_{i}(z^{<t},\\pmb{a}^{<t})$ outputs the vectors of natural parameters for the conditional distribution and can be itself parametrized, for instance, by a multi-layer perceptron (MLP) or a recurrent neural network (RNN). We will refer to the functions $\\lambda_{i}$ as the mechanisms or the transition functions. In the Gaussian case, the natural parameter is two-dimensional and is related to the usual parameters $\\mu$ and $\\sigma^{2}$ via the equation $\\begin{array}{ r }{(\\lambda_{1},\\lambda_{2})=(\\frac{\\mu}{\\sigma^{2}},-\\frac{1}{2\\sigma^{2}})}\\end{array}$ . We will denote by k the dimensionality of the natural parameter and that of the sufficient statistic (which are equal). Thus, $k = 2$ in the Gaussian case. The remaining term $\\psi_{i}(z^{<t},\\pmb{a}^{<t})$ acts as a normalization constant.\n\nWe define $\\pmb{\\lambda}(\\pmb{z}^{<t},\\pmb{a}^{<t})~\\in~\\mathbb{R}^{k d_{z}}$ as the concatenation of all $\\lambda_{i}(z^{<t},\\pmb{a}^{<t})$ and similarly for $\\pmb{s}(z^{t})\\in\\mathbb{R}^{k d_{z}}$ . Similarly to the nonparameteric case, the learnable parameters are $\\pmb\\theta: =(f,\\lambda,G)$ Note that throughout, we assume that the sufficient statistic s is not learned and known in advance. With this notation, we can write the full transition model as\n\n$$\np(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})=h(\\boldsymbol{z}^{t})\\exp\\{\\boldsymbol{s}(\\boldsymbol{z}^{t})^{\\top}\\boldsymbol{\\lambda}(\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})-\\psi(\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})\\},\n$$\n\nwhere $\\begin{array}{ r }{h: =\\prod_{i=1}^{d_{z}}h_{i}}\\end{array}$ and $\\begin{array}{ r }{\\psi=\\sum_{i=1}^{d_{z}}\\psi_{i}}\\end{array}$\n\nIn this section, we follow Khemakhem et al. (2020a) and show that the exponential family assumption combined with an additional sufficient variability assumption allows us to go from identifiability up to diffeomorphism (Definition 5) to identifiability up to quasi-linearity, which we define next:\n\nDefinition 17 (Quasi-linear equivalence) We say that two models $\\pmb\\theta: =(f,\\lambda,G)$ and $\\tilde{\\pmb{\\theta}}: =(\\tilde{f},\\tilde{\\lambda},\\tilde{G})$ satisfying Assumptions $I,2\\&9$ are equivalent up to quasi-linearity, denoted $\\theta\\sim_{\\mathrm{lin}}\\tilde{\\theta},$ if and only $i f\\theta\\sim_{\\mathrm{diff}}\\tilde{\\theta}$ and there exist an invertible matrix $\\pmb{L}\\in\\mathbb{R}^{k d_{z}\\times k d_{z}}$ and a vector $\\pmb{b}\\in\\mathbb{R}^{k d_{z}}$ such that the map $\\pmb{v}: =\\pmb{f}^{-1}\\circ\\tilde{\\pmb{f}}$ satisfies\n\n$$\n\\boldsymbol{s}(\\boldsymbol{v}(\\boldsymbol{z}))=\\boldsymbol{L}\\boldsymbol{s}(\\boldsymbol{z})+\\boldsymbol{b},\\forall\\boldsymbol{z}\\in\\mathbb{R}^{d_{z}}.\n$$\n\nIfthe sufficient statistic s is invertible, one obtains\n\n$$\n\\boldsymbol{v}(\\boldsymbol{z})=\\boldsymbol{s}^{-1}(\\boldsymbol{L}\\boldsymbol{s}(\\boldsymbol{z})+\\boldsymbol{b}),\\forall\\boldsymbol{z}\\in\\mathbb{R}^{d_{z}}.\\tag{31}\n$$\n\nThe following theorem provides conditions to guarantee identifiability up to quasi-linearity. This is an adaptation and minor extension of Theorem 1 from Khemakhem et al. (2020a). For completeness, we provide a proof in Appendix B.2.\n\nTheorem 4 (Conditions for linear identifiability - Adapted from Khemakhem et al. (2020a)) Let $\\pmb\\theta: =(f,\\lambda,G)$ and $\\hat{\\pmb\\theta}: =(\\hat{f},\\hat{\\lambda},\\hat{G})$ be two models satisfying Assumptions 1, 2 & 9. Further assume that\n\n1. [Observational equivalence] $\\theta\\sim_{\\mathrm{obs}}\\hat{\\theta}(D e f i n i t i o n 4).$  \n2. [Minimal sufficient statistics] For all i, the sufficient statistic $\\mathbf{\\delta}_{s_{i}}$ is minimal (see below).  \n3. [Sufficient variability] The natural parameter λ varies “sufficiently” as formalized by Assumption 10 (see below).\n\nThen, $\\pmb\\theta\\sim_{\\mathrm{lin}}\\hat{\\pmb\\theta}(D e f . I 7).$\n\nAssumption 10 (Sufficient variability in exponential families) There exist $(z_{(r)},\\mathbf{a}_{(r)})_{p=0}^{k d_{z}}$ in their respective supports such that the $k d_{z}$ -dimensional vectors $(\\lambda(z_{(r)},\\pmb{a}_{(r)})-\\lambda(z_{(0)},\\pmb{a}_{(0)}))_{r=1}^{k d_{z}}$ are linearly independent.\n\nNotice that $z_{(r)}$ represents values of $z^{<t}$ for potentially different values of t and therefore can vary in shape.\n\nWe now provide a (partial) disentanglement guarantee that takes advantage of sparsity regularization of $\\hat{G}^{z}$ and is specialized for exponential families with a one-dimensional sufficient statistic $(k=1)$ . We will see that this extra parametric assumption on the transition model allows us to weaken the sufficient influence assumption of Theorem 3 (Assumption 8). In particular, this is going to allow for Markovian transitions without auxiliary variables, which was not allowed by the nonparametric result (Remark 7).\n\nThe sufficient influence assumption for z specialized to exponential families with $k = 1$ is directly taken from Lachapelle et al. (2022):\n\nAssumption 11 (Sufficient influence of z (Lachapelle et al., 2022)) Assume $k = 1$ and $Ds(z)$ is invertible everywhere. There exist $\\{(z_{(r)},\\pmb{a}_{(r)},\\tau_{(r)})\\}_{r=1}^{| |\\pmb{G}^{z}| |_{0}}$ belonging to their respective support such that\n\n$$\n\\mathrm{span}\\left\\{D_{z}^{\\tau_{(r)}}\\pmb{\\lambda}(\\pmb{z}_{(r)},\\pmb{a}_{(r)})D\\pmb{s}(\\pmb{z}_{(r)}^{\\tau_{(r)}})^{-1}\\right\\}_{r=1}^{| |\\pmb{G}^{z}| |_{0}}=\\mathbb{R}_{\\pmb{G}^{z}}^{d_{z}\\times d_{z}},\n$$\n\nwhere $D_{z}^{\\tau_{(r)}}\\lambda$ and Ds are Jacobians with respect to $z^{\\tau_{(r)}}$ and $z,$ respectively\n\nThe goal of the next result is to relax the conditions of Lemma 11 so that $\\pmb{L}(z)\\pmb{P}(z)$ is $\\mathbf{G}\\mathrm{-}$ preserving for almost all $\\boldsymbol{z}\\in\\mathbb{R}^{d_{z}}$\n\nLemma 12 Let $G\\in\\{0,1\\}^{m\\times n}$ and let $\\pmb{L}:\\mathbb{R}^{d_{z}}\\mathbb{R}^{m\\times m}$ be a continuous function such that $\\pmb{L}(z)$ is invertible for all $z\\in\\mathbb{R}^{d_{z}}$ . Suppose that, for almost all $z\\in\\mathbb{R}^{d_{z}}$ (i.e. except on a set E of Lebesgue measure zero), there exists a permutation matrix $P(z)$ such that $\\pmb{L}(z)\\pmb{P}(z)$ is G-preserving. Then, there exists a permutation matrix P such that, for all $z\\in\\mathbb{R}^{d_{z}},\\pmb{L}(z)\\pmb{P}$ is G-preserving.\n\nWe recall the definition of a minimal sufficient statistic in an exponential family, which can be found in Wainwright and Jordan (2008, p. 40).\n\nDefinition 20 (Minimal sufficient statistic) Given a parameterized distribution in the exponential family, as in (30), we say that its sufficient statistic $\\mathbf{\\delta}_{s_{i}}$ is minimal when there is no $v \\neq 0$ such that $v^{\\top}s_{i}(z)$ is constantfor all $z\\in{\\mathcal{Z}}$\n\nLemma 15 Let $\\pmb\\theta: =(\\pmb f,p,\\pmb G)$ satisfy Assumptions 1, 2, 3, 4 & 9 and let $q : = \\log p .$ . Then\n\n$$\n\\begin{array}{l}D_{z}^{t}q(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})=\\boldsymbol{\\lambda}(\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})^{\\top}D\\boldsymbol{s}(\\boldsymbol{z}^{t})+D(\\log h)(\\boldsymbol{z}^{t})\\\\ H_{z,a}^{t,\\tau}q(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})=D\\boldsymbol{s}(\\boldsymbol{z}^{t})^{\\top}D_{a}^{\\tau}\\boldsymbol{\\lambda}(\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})\\\\ H_{z,z}^{t,\\tau}q(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})=D\\boldsymbol{s}(\\boldsymbol{z}^{t})^{\\top}D_{z}^{\\tau}\\boldsymbol{\\lambda}(\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t}).\\\\ \\end{array}\n$$", "theorems": [{"local_id": "T1", "statement": "Theorem 2 (Nonparametric disentanglement via discrete a with sparse influence) Let the parameters $\\pmb\\theta: =(\\pmb f,p,\\pmb G)$ and $\\hat{\\pmb\\theta}: =(\\hat{f},\\hat{p},\\hat{G})$ correspond to two models satisfying Assumptions 1, 2, 3 & 4. Further assume that\n\n1. [Observational equivalence] $\\pmb{\\theta}\\sim_{\\mathrm{obs}}\\hat{\\pmb{\\theta}}(D e f . 4),$  \n2. [Sufficient influence of a] The vector of derivatives $D_{z}^{t}$ log $p(z^{t}\\mid z^{<t},\\pmb{a}^{<t})$ depends “sufficiently strongly” on each component $^{a\\ell}$ as formalized in Assumption $7 ;$\n\nThen, there exists a permutation matrix P such that $P G^{a}\\subseteq{\\hat{G}}^{a}$ . Further assume that\n\n3. [Sparsity regularization] $| |\\hat{G}^{a}| |_{0}\\leq| |G^{a}| |_{0};$\n\nThen, $\\pmb\\theta\\sim_{\\mathrm{con}}^{\\pmb a}\\hat{\\pmb\\theta}(D e f . I 3)$", "proof": {"steps": [{"n": 1, "claim": "Proof [of Theorem 2] We recall equation (8) derived in Section 3.2:\n\n$$\nD_{z}^{t}\\hat{q}(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})=D_{z}^{t}q(\\boldsymbol{v}(\\boldsymbol{z}^{t})\\mid\\boldsymbol{v}(\\boldsymbol{z}^{<t}),\\boldsymbol{a}^{<t})D\\boldsymbol{v}(\\boldsymbol{z}^{t})+\\eta(\\boldsymbol{z}^{t})\\in\\mathbb{R}^{1\\times d_{z}}.\\tag{20}\n$$"}, {"n": 2, "claim": "Now, instead of differentiating w.r.t. $\\pmb{a}_{\\ell}^{\\tau}$ for some $\\tau < t$ and $\\ell\\in [d_{a}]$ , we are going to take a partial difference. That is, we evaluate the above equation on at $\\pmb{a}^{<t}$ and $\\pmb{a}^{<t}+\\epsilon\\pmb{E}^{(\\ell,\\tau)}$ and $\\epsilon\\in\\mathbb{R}$ , where $\\pmb{E}^{(\\ell,\\tau)}$ is a “one-hot matrix”, while keeping everything else constant, and take the difference."}, {"n": 3, "claim": "This yields:\n\n$$\n\\begin{array}{l}[D_{z}^{t}\\hat{q}(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t}+\\epsilon\\boldsymbol{E}^{(\\ell,\\tau)})-D_{z}^{t}\\hat{q}(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})]^{\\top}\\\\ =D\\boldsymbol{v}(\\boldsymbol{z}^{t})^{\\top}[D_{z}^{t}q(\\boldsymbol{v}(\\boldsymbol{z}^{t})\\mid\\boldsymbol{v}(\\boldsymbol{z}^{<t}),\\boldsymbol{a}^{<t}+\\epsilon\\boldsymbol{E}^{(\\ell,\\tau)})-D_{z}^{t}q(\\boldsymbol{v}(\\boldsymbol{z}^{t})\\mid\\boldsymbol{v}(\\boldsymbol{z}^{<t}),\\boldsymbol{a}^{<t})]^{\\top}\\\\ \\Delta_{a,\\ell}^{\\tau,\\epsilon}D_{z}^{t}\\hat{q}\\left(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t}\\right)^{\\top}=D\\boldsymbol{v}\\left(\\boldsymbol{z}^{t}\\right)^{\\top}\\Delta_{a,\\ell}^{\\tau,\\epsilon}D_{z}^{t}q\\left(\\boldsymbol{v}\\left(\\boldsymbol{z}^{t}\\right)\\mid\\boldsymbol{v}\\left(\\boldsymbol{z}^{<t}\\right),\\boldsymbol{a}^{<t}\\right)^{\\top},\\tag{21}\\\\ \\end{array}\n$$\n\nwhere we used the notation for partial differences introduced in Definition 16."}, {"n": 4, "claim": "Notice that the difference on the left is $\\subseteq\\hat{G}_{\\cdot,\\ell}^{a}$ and the difference on the right $\\mathrm{is}\\subseteq G_{\\cdot,\\ell}^{a}.$ This equation is thus analogous to (16) from the continuous case. For that reason, we can employ a completely analogous strategy. Hence, we define\n\n$$\n\\hat{\\Lambda}(\\boldsymbol{z}^{t},\\gamma)_{\\cdot,\\ell}:=\\Delta_{a,\\ell}^{\\tau,\\epsilon}D_{z}^{t}\\hat{q}(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})^{\\top}\\quad\\Lambda(\\boldsymbol{z}^{t},\\gamma)_{\\cdot,\\ell}:=\\Delta_{a,\\ell}^{\\tau,\\epsilon}D_{z}^{t}q(\\boldsymbol{v}(\\boldsymbol{z}^{t})\\mid\\boldsymbol{v}(\\boldsymbol{z}^{<t}),\\boldsymbol{a}^{<t})^{\\top},\n$$\n\nwhere $\\gamma=(t,\\tau,z^{<t},\\pmb{a}^{<t},\\vec{\\epsilon})$ ."}, {"n": 5, "claim": "This notation allows us to rewrite (21) more compactly as\n\n$$\n\\underbrace{\\hat{\\Lambda}(\\boldsymbol{z}^{t},\\gamma)}_{\\subseteq\\hat{\\boldsymbol{G}}^{a}}=\\boldsymbol{L}(\\boldsymbol{z}^{t})^{\\top}\\underbrace{\\Lambda(\\boldsymbol{z}^{t},\\gamma)}_{\\subseteq\\boldsymbol{G}^{a}}.\n$$"}, {"n": 6, "claim": "From here, the rest of the argument is exactly analogous to the proof of Theorem 1."}]}}, {"local_id": "T2", "statement": "Theorem 3 (Nonparametric disentanglement via sparse temporal dependencies) Let the parameters $\\pmb\\theta: =(\\pmb f,p,\\pmb G)$ and $\\hat{\\pmb\\theta}: =(\\hat{f},\\hat{p},\\hat{G})$ correspond to two models satisfying Assumptions $I,2,3\\ \\&$ 4. Further assume that\n\n1. [Observational equivalence] $\\pmb{\\theta}\\sim_{\\mathrm{obs}}\\hat{\\pmb{\\theta}}(D e f . 4),$  \n2. [Sufficient influence of z] The Hessian matrix $H_{z,z}^{t,\\tau}$ log $p(z^{t}\\mid z^{<t},\\pmb{a}^{<t})$ varies “sufficiently”, asformalized in Assumption $\\mathbf{\\nabla}_{\\delta,\\cdot}$\n\nThen, there exists a permutation matrix P such that $P G^{z}P^{\\top}\\subseteq\\hat{G}^{z}$ . Further assume that\n\n3. [Sparsity regularization] $| |\\hat{G}^{z}| |_{0}\\leq| |G^{z}| |_{0};$\n\nThen, $\\theta\\sim_{\\mathrm{con}}^{z}\\hat{\\pmb{\\theta}}(D e f . I 4)$", "proof": {"steps": [{"n": 1, "claim": "Proof [of Theorem 3] We recall equation (15) derived in Section 3.2:\n\n$$\n\\underbrace{H_{z,z}^{t,\\tau}\\hat{q}(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})}_{\\subseteq\\hat{\\boldsymbol{G}}^{z}}=D\\boldsymbol{v}(\\boldsymbol{z}^{t})^{\\top}\\underbrace{H_{z,z}^{t,\\tau}q(\\boldsymbol{v}(\\boldsymbol{z}^{t})\\mid\\boldsymbol{v}(\\boldsymbol{z}^{<t}),\\boldsymbol{a}^{<t})}_{\\subseteq\\boldsymbol{G}^{z}}D\\boldsymbol{v}(\\boldsymbol{z}^{\\tau}).\n$$"}, {"n": 2, "claim": "This equation holds for all pairs of $z^{t}$ and $z^{\\tau}$ in $\\mathbb{R}^{d_{z}}$ . We can thus evaluate it at a point such that $z^{t}=z^{\\tau}$ , which yields\n\n$$\nH_{z,z}^{t,\\tau}\\hat{q}(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})=D\\boldsymbol{v}(\\boldsymbol{z}^{t})^{\\top}H_{z,z}^{t,\\tau}q(\\boldsymbol{v}(\\boldsymbol{z}^{t})\\mid\\boldsymbol{v}(\\boldsymbol{z}^{<t}),\\boldsymbol{a}^{<t})D\\boldsymbol{v}(\\boldsymbol{z}^{t}).\\tag{22}\n$$"}, {"n": 3, "claim": "Recall that Assumption 8 holds for all $\\boldsymbol{z}^{t}\\in\\mathbb{R}^{d_{\\boldsymbol{z}}}\\setminus E_{0}$ where $E_{0}$ has Lebesgue measure zero. Fix an arbitrary $z^{t}\\in\\mathbb{R}^{d_{z}}\\setminus E_{0}$ and set $z^{\\tau}=z^{t}$ . Let us define\n\n$$\n\\Lambda(\\pmb{z}^{t},\\gamma):=H_{z,z}^{t,\\tau}q(\\pmb{v}(\\pmb{z}^{t})\\mid\\pmb{v}(\\pmb{z}^{<t}),\\pmb{a}^{<t})\\qquad\\hat{\\Lambda}(\\pmb{z}^{t},\\gamma):=H_{z,z}^{t,\\tau}\\hat{q}(\\pmb{z}^{t}\\mid\\pmb{z}^{<t},\\pmb{a}^{<t}),\n$$\n\nwhere $\\gamma: =(t,\\tau,z_{-\\tau}^{<t},\\pmb{a}^{<t})$ and $ {\\boldsymbol{z}}_{-\\tau}^{<t}$ is $z^{<t}$ but without $z^{\\tau}$ . We can now rewrite (22) compactly as\n\n$$\n\\hat{\\Lambda}(\\boldsymbol{z}^{t},\\gamma)=D\\boldsymbol{v}(\\boldsymbol{z}^{t})^{\\top}\\Lambda(\\boldsymbol{z}^{t},\\gamma)D\\boldsymbol{v}(\\boldsymbol{z}^{t}).\n$$"}, {"n": 4, "claim": "Now, notice that the sufficient influence assumption (Assumption 8) requires that, there exists $\\{\\gamma_{(r)}\\}_{r=1}^{| |G^{z}| |_{0}}$ such that span $\\{\\Lambda(z^{t},\\gamma_{(r)})\\}_{r=1}^{| |G^{z}| |_{0}}=\\bar{\\mathbb{R}}_{G^{z}}^{d_{z}\\times d_{z}}$ . We can thus write\n\n$$\nD\\boldsymbol{v}(\\boldsymbol{z}^{t})^{\\top}\\mathrm{span}\\{\\Lambda(\\boldsymbol{z}^{t},\\gamma_{(r)})\\}_{r=1}^{| |\\boldsymbol{G}^{z}| |_{0}}D\\boldsymbol{v}(\\boldsymbol{z}^{t})=\\mathrm{span}\\{\\hat{\\Lambda}(\\boldsymbol{z}^{t},\\gamma_{(r)})\\}_{r=1}^{| |\\boldsymbol{G}^{z}| |_{0}}\\subseteq\\mathbb{R}_{\\hat{\\boldsymbol{G}}^{z}}^{d_{z}\\times d_{z}}\n$$\n\n$$\n\\Longrightarrow D\\boldsymbol{v}\\left(\\boldsymbol{z}^{t}\\right)^{\\top}\\mathbb{R}_{\\boldsymbol{G}^{z}}^{d_{z}\\times d_{z}}D\\boldsymbol{v}\\left(\\boldsymbol{z}^{t}\\right)\\subseteq\\mathbb{R}_{\\hat{\\boldsymbol{G}}^{z}}^{d_{z}\\times d_{z}}\\tag{23}\n$$"}, {"n": 5, "claim": "Since $Dv(z)$ is invertible, there exists a permutation $P(z)$ such that $D{\\pmb v}({\\pmb z})P({\\pmb z})$ has no zero on its diagonal (Lemma 2). Let $C(z): =Dv(z)P(z)$ . If we left and right-multiply (23) by $P(z)^{\\top}$ and $P(z)$ , respectively, we obtain\n\n$$\n\\boldsymbol{C}(\\boldsymbol{z}^{t})^{\\top}\\mathbb{R}_{\\boldsymbol{G}^{z}}^{d_{z}\\times d_{z}}\\boldsymbol{C}(\\boldsymbol{z}^{t})\\subseteq\\mathbb{R}_{\\boldsymbol{P}(\\boldsymbol{z})^{\\top}\\hat{\\boldsymbol{G}}^{z}\\boldsymbol{P}(\\boldsymbol{z})}^{d_{z}\\times d_{z}}.\\tag{24}\n$$"}, {"n": 6, "claim": "We now show that $\\pmb{G}^{z}\\subseteq\\pmb{P}(\\pmb{z})^{\\top}\\hat{G}^{z}\\pmb{P}(\\pmb{z})$ . Take $(i,j)\\in G^{z}$ . Since $e_{i}e_{j}^{\\top}\\in\\mathbb{R}_{G^{z}}^{d_{z}\\times d_{z}}$ , equation (24) implies\n\n$$\n\\boldsymbol{C}(\\boldsymbol{z}^{t})^{\\top}\\boldsymbol{e}_{i}\\boldsymbol{e}_{j}^{\\top}\\boldsymbol{C}(\\boldsymbol{z}^{t})=\\left(\\boldsymbol{C}(\\boldsymbol{z}^{t})_{i,\\cdot}\\right)^{\\top}\\boldsymbol{C}(\\boldsymbol{z}^{t})_{j,\\cdot}\\subseteq\\mathbb{R}_{\\boldsymbol{P}(\\boldsymbol{z})^{\\top}\\hat{\\boldsymbol{G}}^{z}\\boldsymbol{P}(\\boldsymbol{z})}^{d_{z}\\times d_{z}}\\tag{25}\n$$"}, {"n": 7, "claim": "Since $C(z^{t})_{i,i}C(z^{t})_{j,j}~\\ne~0$ (recall the diagonal of $C(\\boldsymbol{z}^{t})$ has no zero), we must have $(i,j)\\in$ $P(z)^{\\top}\\hat{G}^{z}P(z)$ . This shows that $\\pmb{G}^{z}\\subseteq\\pmb{P}(\\pmb{z})^{\\top}\\hat{G}^{z}\\pmb{P}(\\pmb{z})$\n\nSince $| |P(z)^{\\top}\\hat{G}^{z}P(z)| |_{0}= | |\\hat{G}^{z}| |_{0}\\leq| |G^{z}| |_{0}$ , we must have $\\pmb{G}^{z}=\\pmb{P}(\\pmb{z})^{\\top}\\hat{G}^{z}\\pmb{P}(\\pmb{z})$ , which yields\n\n$$\n\\boldsymbol{C}(\\boldsymbol{z}^{t})^{\\top}\\mathbb{R}_{\\boldsymbol{G}^{z}}^{d_{z}\\times d_{z}}\\boldsymbol{C}(\\boldsymbol{z}^{t})\\subseteq\\mathbb{R}_{\\boldsymbol{G}^{z}}^{d_{z}\\times d_{z}}.\\tag{26}\n$$"}, {"n": 8, "claim": "We are now going to show that the above implies that $C(\\boldsymbol{z}^{t})$ is both $G^{z}$ -preserving and $(G^{z})^{\\top}$ preserving. Start by rewriting (25) as follows:\n\n$$\n\\text{for all}(i,j)\\in\\boldsymbol{G}^{z},\\left(\\boldsymbol{C}(\\boldsymbol{z}^{t})_{i,\\cdot}\\right)^{\\top}\\boldsymbol{C}(\\boldsymbol{z}^{t})_{j,\\cdot}\\subseteq\\mathbb{R}_{\\boldsymbol{G}^{z}}^{d_{z}\\times d_{z}}.\\tag{27}\n$$"}, {"n": 9, "claim": "We start by showing $G^{z}$ -preservation. To do so, we leverage the characterization of Proposition 3. We must show that $G_{i,\\cdot}^{z}\\not\\subseteq G_{j,\\cdot}^{z}$ implies $C(z^{t})_{i,j}=0$ . Because $G_{i,\\cdot}^{z}\\not\\subseteq G_{j,\\cdot}^{z}$ , there must exists k s.t. $G_{i,k}^{z}~=~1$ and $G_{j,k}^{z}=0$ . We thus have, by (27), that $(\\pmb{C}(\\pmb{z}^{t})_{i,\\cdot})^{\\top}\\pmb{C}(\\pmb{z}^{t})_{k,\\cdot}\\ \\subseteq\\ \\mathbb{R}_{\\pmb{G}^{z}}^{d_{z}\\times d_{z}}$ Because $G_{j,k}^{z}=0$ , we have $C(z^{t})_{i,j}C(z^{t})_{k,k}~=~0$ . But since $C(z^{t})_{k,k}\\ \\ne\\ 0$ , we must have that $C(z^{t})_{i,j}=0$ , as desired. To show $\\left(G^{z}\\right)^{\\top}$ -preservation, one can use a completely analogous argument."}, {"n": 10, "claim": "We showed that $C(\\boldsymbol{z}^{t})$ is G<sup>z</sup>-preserving and $\\left(G^{z}\\right)^{\\top}$ -preserving. It is easy to verify that this is equivalent to being $[G^{z}\\ (G^{z})^{\\top}]$ -preserving (where [· ·] stands for column concatenation). This remark will be useful below."}, {"n": 11, "claim": "Similarly to the proof of Theorem 1, we must now show that there exists a single permutation that works for all $\\boldsymbol{z}^{t}\\in\\mathbb{R}^{d_{z}}$ . To achieve this, we use Lemma 12 with $G: =[\\bar{G}^{z}\\bar{(}G^{z})^{\\top}]$ and $\\pmb{L}(z): =D\\pmb{v}(z)$ . This allows us to say that there exists a permutation P such that $D{\\pmb v}({\\pmb z})P$ is $[G^{z}\\left(G^{z}\\right)^{\\top}]$ ]-preserving for all z (not “almost all”)."}, {"n": 12, "claim": "Notice that $D(\\pmb{v}\\circ\\pmb{P})(z)=D\\pmb{v}(\\pmb{P}z)\\pmb{P}$ , which is $[G^{z}\\ (G^{z})^{\\top}]$ ]-preserving everywhere. Using Lemma 1, we conclude that the function $c: =v\\circ P{\\mathrm{~i s~}}[G^{z}\\left(G^{z}\\right)^{\\top}]$ -preserving."}]}}, {"local_id": "T3", "statement": "Theorem 5 (Disentanglement via sparse temporal dependencies in exponential families) Let $\\pmb\\theta: =(\\pmb f,\\lambda,\\pmb G)$ and $\\hat{\\pmb\\theta}: =(\\hat{f},\\hat{\\lambda},\\hat{G})$ be two models satisfying Assumptions 1, 2, 3, 4, 9 as well as all assumptions ofTheorem 4. Further suppose that\n\n1. The sufficient statistic s is $d_{z}$ -dimensional $(k=1)$ and is a diffeomorphism from $\\mathbb{R}^{d_{z}}$ to $\\boldsymbol{s}(\\mathbb{R}^{d_{z}})$ ;  \n2. [Sufficient influence of z] The Jacobian of the ground-truth transition function λ with respect to z varies “sufficiently”, asformalized in Assumption 11;\n\nThen, there exists a permutation matrix P such that $P G^{z}P^{\\top}\\subseteq\\hat{G}^{z}$ . Further assume that\n\n3. [Sparsity regularization] $| |\\hat{G}^{z}| |_{0}\\leq| |G^{z}| |_{0},$\n\nThen, $\\theta\\sim_{\\mathrm{con}}^{z}\\hat{\\theta}$ (Def. 14) & $\\pmb{\\theta}\\sim_{\\mathrm{lin}}\\hat{\\pmb{\\theta}}(D e f . I 7)$ , which together implies that\n\n$$\n\\boldsymbol{v}(\\boldsymbol{z})=\\boldsymbol{s}^{-1}(\\boldsymbol{CP}^{\\top}\\boldsymbol{s}(\\boldsymbol{z})+\\boldsymbol{b}),\n$$\n\nwhere $\\pmb{b}\\in\\mathbb{R}^{d_{z}}$ and $C\\in\\mathbb{R}^{d_{z}\\times d_{z}}$ is invertible, $G^{z}.$ and $(G^{z})^{\\top}$ -preserving (Definition 11).", "proof": {"steps": [{"n": 1, "claim": "Proof Recall the equation we derived in Section 3.2:\n\n$$\nH_{z,z}^{t,\\tau}\\hat{q}(\\boldsymbol{z}^{t}\\mid\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})=D\\boldsymbol{v}(\\boldsymbol{z}^{t})^{\\top}H_{z,z}^{t,\\tau}q(\\boldsymbol{v}(\\boldsymbol{z}^{t})\\mid\\boldsymbol{v}(\\boldsymbol{z}^{<t}),\\boldsymbol{a}^{<t})D\\boldsymbol{v}(\\boldsymbol{z}^{\\tau}).\n$$"}, {"n": 2, "claim": "Using Lemma 15, we get\n\n$$\nD\\boldsymbol{s}(\\boldsymbol{z}^{t})^{\\top}D_{z}^{\\tau}\\hat{\\boldsymbol{\\lambda}}(\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})=D\\boldsymbol{v}(\\boldsymbol{z}^{t})^{\\top}D\\boldsymbol{s}(\\boldsymbol{v}(\\boldsymbol{z}^{t}))^{\\top}D_{z}^{\\tau}\\boldsymbol{\\lambda}(\\boldsymbol{v}(\\boldsymbol{z}^{<t}),\\boldsymbol{a}^{<t})D\\boldsymbol{v}(\\boldsymbol{z}^{\\tau}).\\tag{65}\n$$"}, {"n": 3, "claim": "Note that Assumption 3 requires that $D_{z}^{\\tau}\\lambda(z^{<t},\\pmb{a}^{<t})\\subseteq\\pmb{G}^{z}$ and $D_{z}^{\\tau}\\hat{\\lambda}(z^{<t},{\\pmb a}^{<t})\\subseteq\\hat{\\pmb{G}}^{z}$ . Theorem 4 implies that there exist an invertible matrix $\\pmb{L}\\in\\mathbb{R}^{d_{z}\\times d_{z}}$ and a vector $\\pmb{b}\\in\\mathbb{R}^{d_{z}}$ such that\n\n$$\n\\boldsymbol{v}(\\boldsymbol{z})=\\boldsymbol{s}^{-1}(\\boldsymbol{L}\\boldsymbol{s}(\\boldsymbol{z})+\\boldsymbol{b}).\n$$"}, {"n": 4, "claim": "Taking the derivative of the above w.r.t. z, we obtain\n\n$$\n\\begin{array}{l}D\\boldsymbol{v}(\\boldsymbol{z})=D\\boldsymbol{s}^{-1}(\\boldsymbol{L}\\boldsymbol{s}(\\boldsymbol{z})+\\boldsymbol{b})\\boldsymbol{L}D\\boldsymbol{s}(\\boldsymbol{z})(66)\\\\ =D\\boldsymbol{s}^{-1}(\\boldsymbol{s}(\\boldsymbol{v}(\\boldsymbol{z})))\\boldsymbol{L}D\\boldsymbol{s}(\\boldsymbol{z})(67)\\\\ =D\\boldsymbol{s}(\\boldsymbol{v}(\\boldsymbol{z}))^{-1}\\boldsymbol{L}D\\boldsymbol{s}(\\boldsymbol{z}),(68)\\\\ \\end{array}\n$$\n\nwhere we used $\\pmb{\\mathscr{s}}(\\pmb{v}(z))=\\pmb{L\\mathscr{s}}(z)+\\pmb{b}$ to go from the first to the second line and used the inverse function theorem to go from the second to the third line."}, {"n": 5, "claim": "Plugging (68) into (65) yields\n\n$$\n\\begin{array}{l}D\\boldsymbol{s}(\\boldsymbol{z}^{t})^{\\top}D_{z}^{\\tau}\\hat{\\boldsymbol{\\lambda}}(\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})\\\\ =D\\boldsymbol{s}(\\boldsymbol{z}^{t})^{\\top}\\boldsymbol{L}^{\\top}D\\boldsymbol{s}(\\boldsymbol{v}(\\boldsymbol{z}^{t}))^{-\\top}D\\boldsymbol{s}(\\boldsymbol{v}(\\boldsymbol{z}^{t}))^{\\top}D_{z}^{\\tau}\\boldsymbol{\\lambda}(\\boldsymbol{v}(\\boldsymbol{z}^{<t}),\\boldsymbol{a}^{<t})D\\boldsymbol{s}(\\boldsymbol{v}(\\boldsymbol{z}^{\\tau}))^{-1}\\boldsymbol{L}D\\boldsymbol{s}(\\boldsymbol{z}^{\\tau})\\\\ =D\\boldsymbol{s}(\\boldsymbol{z}^{t})^{\\top}\\boldsymbol{L}^{\\top}D_{z}^{\\tau}\\boldsymbol{\\lambda}(\\boldsymbol{v}(\\boldsymbol{z}^{<t}),\\boldsymbol{a}^{<t})D\\boldsymbol{s}(\\boldsymbol{v}(\\boldsymbol{z}^{\\tau}))^{-1}\\boldsymbol{L}D\\boldsymbol{s}(\\boldsymbol{z}^{\\tau}),\\\\ \\end{array}\n$$\n\nwhich implies\n\n$$\nD\\boldsymbol{s}\\left(\\boldsymbol{z}^{t}\\right)^{\\top}D_{z}^{\\tau}\\hat{\\boldsymbol{\\lambda}}\\left(\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t}\\right)=D\\boldsymbol{s}\\left(\\boldsymbol{z}^{t}\\right)^{\\top}\\boldsymbol{L}^{\\top}D_{z}^{\\tau}\\boldsymbol{\\lambda}\\left(\\boldsymbol{v}\\left(\\boldsymbol{z}^{<t}\\right),\\boldsymbol{a}^{<t}\\right)D\\boldsymbol{s}\\left(\\boldsymbol{v}\\left(\\boldsymbol{z}^{\\tau}\\right)\\right)^{-1}\\boldsymbol{L}D\\boldsymbol{s}\\left(\\boldsymbol{z}^{\\tau}\\right)\\tag{69}\n$$\n\n$$\nD_{z}^{\\tau}\\hat{\\boldsymbol{\\lambda}}(\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})D\\boldsymbol{s}(\\boldsymbol{z}^{\\tau})^{-1}=\\boldsymbol{L}^{\\top}D_{z}^{\\tau}\\boldsymbol{\\lambda}(\\boldsymbol{v}(\\boldsymbol{z}^{<t}),\\boldsymbol{a}^{<t})D\\boldsymbol{s}(\\boldsymbol{v}(\\boldsymbol{z}^{\\tau}))^{-1}\\boldsymbol{L},\\tag{70}\n$$\n\nwhere we left- and right-multiplied by $D s(z^{t})^{-\\top}$ and $D s(z^{\\tau})^{-1}$ , respectively."}, {"n": 6, "claim": "Let us define\n\n$$\n\\Lambda(\\gamma):=D_{z}^{\\tau}\\boldsymbol{\\lambda}(\\boldsymbol{v}(\\boldsymbol{z}^{<t}),\\boldsymbol{a}^{<t})D\\boldsymbol{s}(\\boldsymbol{v}(\\boldsymbol{z}^{\\tau}))^{-1}\\quad\\hat{\\Lambda}(\\gamma):=D_{z}^{\\tau}\\hat{\\boldsymbol{\\lambda}}(\\boldsymbol{z}^{<t},\\boldsymbol{a}^{<t})D\\boldsymbol{s}(\\boldsymbol{z}^{\\tau})^{-1},\n$$\n\nwhere $\\gamma=(t,\\tau,z^{<t},\\pmb{a}^{<t})$ . Note that because Ds is diagonal, we have that $\\Lambda(\\gamma)\\subseteq G^{z}$ and $\\hat{\\boldsymbol{\\Lambda}}(\\gamma)\\subseteq\\hat{\\boldsymbol{G}}^{z}$ . Using this notation, we can rewrite (70) as\n\n$$\n\\underbrace{\\hat{\\Lambda}(\\gamma)}_{\\subseteq\\hat{G}^{z}}=\\boldsymbol{L}^{\\top}\\underbrace{\\Lambda(\\gamma)}_{\\subseteq\\boldsymbol{G}^{z}}\\boldsymbol{L}.\n$$"}, {"n": 7, "claim": "Thanks to Assumption 11, we can apply the same argument as in Theorem 3 to show that $\\pmb{L}=\\pmb{C P}^{\\top}$ where C is a matrix that is both $G^{z}$ -preserving and $\\left(G^{z}\\right)^{\\top}$ -preserving, as desired."}]}}], "target_local_id": null, "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2026-24-1748", "context": "Learning a model using several machines over their collective data is appealing. The motivation behind this distributed machine learning (ML) scheme (a.k.a. federated learning (Kairouz et al., 2021)) is usually eficiency. Another motivation is data protection, as each machine retains control over its local data. The distributed ML problem can be precisely stated as follows in a standard server-based system comprising n machines (referred as workers), represented by set $[n]: =\\{1,\\dots,n\\}$ , and a server. Each worker i has access to a common data space $\\mathcal{X}$ through a local distribution $\\mathcal{D}^{(i)}$ . A model parameterized by $\\boldsymbol{\\theta}\\in\\mathbb{R}^{d}$ incurs a loss for each data point $x\\in\\mathcal{X}$ measured by a real-valued loss function $q:\\mathbb{R}^{d}\\times\\mathcal{X}\\mathbb{R}$ Then, for each worker $i \\in [ n ]$ , the local loss function is given by\n\n$$\nQ^{(i)}(\\theta):=\\mathbb{E}_{x\\sim\\mathcal{D}^{(i)}}[q(\\theta,x)].\\tag{1}\n$$\n\nThe server aims to compute a model parameter $\\theta^{*}\\in\\mathbb{R}^{d}$ minimizing the global loss function\n\n$$\nQ(\\theta):=\\frac{1}{n}\\sum_{i=1}^{n}Q^{(i)}(\\theta).\\tag{2}\n$$\n\nWe assume that the gradient of the loss function $q(\\theta,x)$ with respect to $\\theta,$ denoted by $\\nabla q(\\theta,x)$ , exists and is continuous at all $\\theta\\in\\mathbb{R}^{d}$ and $x\\in\\mathcal{X}$ , which is standard in machine learning (Bottou et al., 2018).\n\nSolution to Byzantine failure is tight with respect to data poisoning. We consider the class of ML problems that can be solved by optimizing L-Lipschitz smooth loss functions satisfying the $\\mu{\\mathrm{-}}\\mathrm{P}\\mathrm{L}$ inequality, where the local gradients (of honest workers) have bounded covariance trace of $\\sigma^{2}$ . These conditions are satisfied in many cases (Bottou et al., 2018). We further assume that the global gradient dissimilarity that characterizes data heterogeneity is bounded by $\\zeta^{2}$ , which is essential to tackling misbehaving workers of either type (Karimireddy et al., 2022; Allouah et al., 2023a). We assume that the total number of fully corrupted workers is bounded by f. Note that the case of $f \\geq n / 2$ is trivial as the learning error can be arbitrarily large (Lemma 1 in (Liu et al., 2021)): we thus assume $f < n / 2$ in all our results.\n\nWe consider a server-based system architecture with n workers and a central server. The workers only communicate with the server and there is no communication between workers. We assume that at most f out of n workers may be faulty, either as per Byzantine failure or fully-poisonous local data. We denote by H the set of $n - f$ honest workers, and let $Q^{(\\mathcal{H})}(\\theta)$ denote their average loss, i.e.,\n\n$$\nQ^{(\\mathcal{H})}(\\theta)=\\frac{1}{|\\mathcal{H}|}\\sum_{i\\in\\mathcal{H}}Q^{(i)}(\\theta),\\quad\\forall\\theta\\in\\mathbb{R}^{d}.\\tag{5}\n$$\n\nWe assume that ${Q}^{(\\mathcal{H})}$ admits a minimum, i.e., $\\exists\\theta^{*}\\in\\mathbb{R}^{d}$ such that for all $\\theta\\in\\mathbb{R}^{d},Q^{(\\mathcal{H})}(\\theta)\\geq$ $Q^{(\\mathcal{H})}(\\theta^{\\ast})$ . We let $Q^{*}: =Q^{(\\mathcal{H})}(\\theta^{*})$ . Furthermore, we consider the class of smooth loss functions satisfying the Polyak-Łojasiewicz (PL) inequality, which is more general than strong convexity (Bottou et al., 2018) and can indeed be satisfied by some non-convex functions (Karimi et al., 2016).\n\nAssumption 1 (Smoothness) There exists $L < \\infty$ such that for all $i \\in [ n ]$ and θ, $,\\theta^{\\prime}\\in\\mathbb{R}^{d}$\n\n$$\n\\left\\|\\nabla Q^{(i)}(\\theta)-\\nabla Q^{(i)}(\\theta^{\\prime})\\right\\|\\leq L\\left\\|\\theta^{\\prime}-\\theta\\right\\|.\n$$\n\nAssumption 2 (PL-condition) There exists $\\mu \\geq 0$ such that for all $\\boldsymbol{\\theta}\\in\\mathbb{R}^{d}$\n\n$$\n\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta)\\right\\|^{2}\\geq 2\\mu\\left(Q^{(\\mathcal{H})}(\\theta)-Q^{*}\\right).\n$$\n\nAs stated below, we also assume that the stochastic gradients computed by the honest workers have a bounded local covariance trace. This assumption is standard for analyzing the convergence of stochastic first-order methods (Tang et al., 2018). For all $i\\in\\mathcal{H}$ , by definition of $Q^{(i)}$ , and the assumption that $\\nabla q\\left(\\theta,x\\right)$ is continuous in θ and x, we have $\\mathbb{E}_{\\boldsymbol{x}\\sim\\mathcal{D}^{(i)}}\\left[\\nabla q\\left(\\theta,\\boldsymbol{x}\\right)\\right]=\\nabla Q^{(i)}(\\theta)$\n\nAssumption 3 (Stochasticity) There exists $\\sigma < \\infty$ such that for all $i\\in\\mathcal{H}$ and $\\boldsymbol{\\theta}\\in\\mathbb{R}^{d}$\n\n$$\n\\mathbb{E}_{x\\sim\\mathcal{D}^{(i)}}\\left[\\left\\|\\nabla q(\\theta,x)-\\nabla Q^{(i)}(\\theta)\\right\\|^{2}\\right]\\leq\\sigma^{2}.\n$$\n\nLastly, as stated below, we assume the local gradients of the honest workers to have bounded diversity (or heterogeneity) over the parameter space. Without this assumption we cannot obtain meaningful guarantees in the threat models we consider, as shown by Karimireddy et al. (2022).\n\nAssumption 4 (Heterogeneity) There exists $\\zeta < \\infty$ such that for all $\\theta\\in\\mathbb{R}^{d}$\n\n$$\n\\frac{1}{|\\mathcal{H}|}\\sum_{i\\in\\mathcal{H}}\\left\\|\\nabla Q^{(i)}(\\theta)-\\nabla Q^{(\\mathcal{H})}(\\theta)\\right\\|^{2}\\leq\\zeta^{2}.\n$$\n\nWe present here a matching upper bound (up to a multiplicative factor of condition number) for Theorem 1, considering a Byzantine adversary. We first describe the algorithm we consider, and then present its convergence guarantee.\n\nThe algorithm follows the skeleton of DSGD and imparts robustness to the learning procedure by applying a momentum operation at the workers’ level and a trimmed mean operation at the server (instead of averaging), as described in Algorithm 1. Essentially, in each iteration $t \\geq 0$ , each honest worker i computes a stochastic gradient\n\n$$\ng_{t}^{(i)}:=\\nabla q(\\theta_{t},x^{(i)}),\\quad \\text{where} x^{(i)}\\sim\\mathcal{D}^{(i)},\\tag{6}\n$$\n\nand returns a Polyak’s momentum of its stochastic gradients, denoted by ${m}_{t}^{\\left(i\\right)}$ and defined:\n\n$$\nm_{t}^{(i)}=\\beta_{t}m_{t-1}^{(i)}+(1-\\beta_{t})g_{t}^{(i)},\\tag{7}\n$$\n\n<div class=\"mineru-algorithm\" style=\"white-space: pre-wrap; font-family:monospace;\">\nAlgorithm 1: DSGD with distributed momentum and trimmed mean aggregation\nInput : $T \\geq 2$, $(\\gamma_0, \\ldots, \\gamma_{T-1})$ and $(\\beta_0, \\ldots, \\beta_{T-1})$.\nServer chooses arbitrarily $\\theta_0 \\in \\mathbb{R}^d$. Each honest worker $i$ sets $m_{-1}^{(i)} = 0$.\nfor $t = 0$ to $T - 1$ do\n    Server broadcasts $\\theta_t$ to all workers;\n    for each honest worker $i$ (in parallel) do\n        Compute a stochastic gradient $g_t^{(i)}$, as defined in (6);\n        Send to the server the momentum $m_t^{(i)}$, as defined in (7);\n    end\n    % A corrupted worker $i$ may send an arbitrary value for $m_t^{(i)}$ to the server.\n    Server updates the parameter vector $\\theta_{t+1}=\\theta_t-\\gamma_t \\text{TM}^{(f)}\\left(m_t^{(1)},\\ldots,m_t^{(n)}\\right)$;\nend\nOutput: $\\hat{\\theta} = \\theta_T$\n</div>\n\nwhere $\\beta_{t}\\in [0,1)$ is the momentum coeficient, and $m_{-1}^{(i)}=0$ by convention. The server updates its current parameter vector $\\theta_{t}$ by aggregating the workers’ momentums using coordinate-wise trimmed mean (TM), defined below. Hereafter, for any $z\\in\\mathbb{R}^{d}$ and $k \\in [ d ] .$ we denote by $[z]_{k}$ the k-th coordinate of $z .$ . Then, given n input vectors $z_{1},...,z_{n}\\in\\mathbb{R}^{d}$ 2 for all $k \\in [ d ]$ , we denote by $\\tau_{k}$ the permutation on [n] that sorts the k-th coordinates of the input vectors in non-decreasing order, i.e., $[z_{\\tau_{k}(1)}]_{k}\\leq [z_{\\tau_{k}(2)}]_{k}\\leq ...\\leq [z_{\\tau_{k}(n)}]_{k}$ . Then, the trimmed mean of $z_{1},\\ldots,z_{n},$ with trimming parameter $f$ is a vector in $\\mathbb{R}^{d}$ whose k-th coordinate is defined as follows,\n\n$$\n\\left[\\mathrm{TM}^{(f)}(z_{1},\\dots,z_{n})\\right]_{k}:=\\frac{1}{n-2 f}\\sum_{j\\in [f+1,n-f]}[z_{\\tau_{k}(j)}]_{k}.\n$$\n\nTheorem 3 below establishes the convergence of Algorithm 1, with a Byzantine adversary, assuming a scheduled decreasing step sizes and increasing momentum coeficients. Note that the algorithm is oblivious to the identity of faulty workers that may send arbitrary values to the server. We denote by E [·] the expectation on the randomness of the algorithm, formally defined in Appendix B.\n\nWe present here the key steps involved in proving Theorem 3. Our proof is based on a new Lyapunov function, denoted by $V_{t}$ . We first motivate the design of $V_{t}.$ , and define it formally. We then analyze the growth of $V_{t}$ along the trajectory of Algorithm 1. Lastly, we show the convergence of the sequence $(V_{t})_{t=0}^{\\bar{T}-1}$ for the specified diminishing step sizes, thereby proving our result.\n\nAnalyzing the growth of the loss function. We analyze the growth of the loss function ${Q^{(\\mathcal{H})}(\\boldsymbol{\\theta}_{t})}$ along the trajectory of Algorithm 1. For any $t \\geq 0$ we denote the average momentum of the honest workers as $\\begin{array}{ r }{\\overline{{m}}_{t}: =\\frac{1}{(n-f)}\\sum_{i\\in\\mathcal{H}}m_{t}^{(i)}}\\end{array}$ . Combining the result of Allouah et al. (2023a) on the robustness of TM with the standard decomposition of the loss function under smoothness assumption (Bottou et al., 2018), we get the following bound on the growth of the loss function.\n\nIncorporating the drift and deviation in the Lyapunov function. In the remaining, for any $t \\geq 0$ , we denote the deviation and the drift of each worker i as\n\n$$\n\\delta_{t}:=\\overline{{m}}_{t}-\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\quad \\text{and}\\quad\\Delta m_{t}^{(i)}:=m_{t}^{(i)}-\\overline{{m}}_{t},\\forall i\\in\\mathcal{H}.\\tag{8}\n$$\n\nDue to the time-varying step size and momentum coeficient in Algorithm 1, it is dificult to derive a uniform bound (i.e., a bound that holds true for any $t\\geq 0)$ on the second and third terms in the right hand side of Lemma 5. Accordingly, we cannot simply and directly analyze the variation of E $\\left[Q^{(\\mathcal{H})}\\left(\\theta_{t}\\right)-Q^{\\ast}\\right]$ with t. Instead, we have to incorporate the drift and the deviation in the analysis. Specifically, we define the following Lyapunov function for our problem.\n\n$$\nV_{t}:=\\mathbb{E}\\left[Q^{(\\mathcal{H})}(\\theta_{t})-Q^{*}+\\rho\\|\\delta_{t}\\|^{2}+\\rho\\frac{\\lambda}{n-f}\\sum_{i\\in\\mathcal{H}}\\left\\|\\Delta m_{t}^{(i)}\\right\\|^{2}\\right],\\tag{9}\n$$\n\nwhere $\\begin{array}{ r }{\\rho=\\frac{1}{12L}}\\end{array}$ . Then, by definition of $V_{t}.$ , we have E $\\left[Q^{(\\mathcal{H})}(\\theta_{T})-Q^{*}\\right]\\leq V_{T}$ . Hence an upper bound on $V_{T}$ gives us an upper bound on E $\\left[Q^{(\\mathcal{H})}(\\theta_{T}^{\\mathsf{^{*}}})-Q^{*}\\right]$ . With this Lyapunov function at hand, we can construct the proof by following three critical steps: (i) determining a recursive bound on the Lyapunov function V<sub>t</sub>, (ii) choosing a desirable sequence $(\\gamma_{0},\\dots,\\gamma_{T-1})$ to obtain tight convergence rate, and (iii) combining (i) and (ii) to derive the final bound on E $\\left[Q^{(\\mathcal{H})}(\\theta_{T})-Q^{\\ast}\\right]$\n\nBefore proving a few simple lemmas that will be used in the subsequent proofs, let us introduce some useful notations.\n\nNotation: We denote by $\\mathcal{P}_{t}$ the history of nodes from steps 0 to t. Specifically, we define\n\n$$\n\\mathcal{P}_{t}:=\\left\\{\\theta_{0},\\dots,\\theta_{t};m_{1}^{(i)},\\dots,m_{t-1}^{(i)};i=1,\\dots,n\\right\\}.\n$$\n\nBy convention, ${\\mathcal P}_{0}=\\{\\theta_{0}\\}$ . Furthermore, we denote by $\\mathbb{E}_{t}\\left[\\cdot\\right]: =\\mathbb{E}\\left[\\cdot\\mid\\mathcal{P}_{t}\\right]$ the conditional expectation given the history $\\mathcal{P}_{t}$ , and by E [·] the total expectation over the randomness of the algorithm; thus, $\\mathbb{E}\\left[\\cdot\\right]: =\\mathbb{E}_{0}\\left[\\cdot\\cdot\\cdot\\mathbb{E}_{T}\\left[\\cdot\\right]\\right]$ . Also denote by\n\n$$\nR_{t}:=\\mathrm{TM}\\left(m_{t}^{(1)},\\dots,m_{t}^{(n)}\\right),\\tag{27}\n$$\n\nthe output of trimmed mean operation.\n\nNote that by decomposing the update rule computed by the server at step t, we can treat Algorithm 1 as DSGD with a momentum term and a bias $\\gamma_{t}\\left(R_{t}-{\\overline{{m}}}_{t}\\right)$ . Specifically, we have\n\n$$\n\\theta_{t+1}=\\theta_{t}-\\gamma_{t}R_{t}=\\theta_{t}-\\gamma_{t}\\overline{{m}}_{t}-\\gamma_{t}(R_{t}-\\overline{{m}}_{t}).\\tag{28}\n$$\n\nThe key to better understand the bias term in (28) is the analysis of TM, that attempts to robustly estimate the average of the honest momentums at every step. Using a recent result in (Allouah et al., 2023a), we can actually bound the bias $\\left(R_{t}-{\\overline{{m}}}_{t}\\right)$ from above by the spread of honest nodes’ momentums. Specifically, we have the following lemma.\n\nLemma 15 (Proposition 2 in (Allouah et al., 2023a)) Let $n > 2 f$ . Consider $A l g o -$ rithm 1 and $R_{t}$ as defined in (27). For any $t \\geq 0$ , we have\n\n$$\n\\|R_{t}-\\overline{{m}}_{t}\\|^{2}\\leq\\frac{\\lambda}{n-f}\\sum_{i\\in\\mathcal{H}}\\left\\|m_{t}^{(i)}-\\overline{{m}}_{t}\\right\\|^{2},\\quad w i t h\\quad\\lambda=\\frac{6 f}{n-2 f}\\left(1+\\frac{f}{n-2 f}\\right).\n$$\n\nLemma 17 Consider an arbitrary non-empty set $S\\subseteq\\{1,\\dots,n\\}$ . For any set $o f \\left| S \\right|$ realvalued vectors $\\{x^{(i)}\\}_{i\\in S}$ , we obtain that\n\n$$\n\\frac{1}{|S|}\\sum_{i\\in S}\\left\\|x^{(i)}-\\bar{x}\\right\\|^{2}=\\frac{1}{2|S|^{2}}\\sum_{i,j\\in S}\\left\\|x^{(i)}-x^{(j)}\\right\\|^{2},\\quad w h e r e\\quad\\bar{x}=\\frac{1}{|S|}\\sum_{i\\in S}x^{(i)}.\n$$", "theorems": [{"local_id": "T1", "statement": "Lemma 5. Suppose Assumption 1 holds true. Consider Algorithm 1 with $T \\geq 2$ , and $\\gamma_{t}\\leq 1/L$ for all $t\\in\\{0,\\ldots,T-1\\}$ . Let λ be as defined in Lemma 15. Then, for all $t,$ the following holds true\n\n$$\n\\begin{array}{l}\\mathbb{E}\\left[Q^{(\\mathcal{H})}(\\theta_{t+1})-Q^{(\\mathcal{H})}(\\theta_{t})\\right]\\leq-\\frac{\\gamma_{t}}{2}\\mathbb{E}\\left[\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}\\right]+\\gamma_{t}\\frac{\\lambda}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|m_{t}^{(i)}-\\overline{{m}}_{t}\\right\\|^{2}\\right]\\\\ +\\gamma_{t}\\mathbb{E}\\left[\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})-\\overline{{m}}_{t}\\right\\|^{2}\\right].\\\\ \\end{array}\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof Consider an arbitrary step t. Note that Assumption 1 implies the Lipschitz continuity of $\\nabla Q^{(\\mathcal{H})}({\\boldsymbol{\\theta}})$ with coeficient L. Thus, we have\n\n$$\nQ^{(\\mathcal{H})}(\\theta_{t+1})-Q^{(\\mathcal{H})}(\\theta_{t})\\leq\\left\\langle\\theta_{t+1}-\\theta_{t},\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\rangle+\\frac{L}{2}\\left\\|\\theta_{t+1}-\\theta_{t}\\right\\|^{2}.\n$$"}, {"n": 2, "claim": "Substituting from Algorithm 1, $\\theta_{t+1}=\\theta_{t}-\\gamma_{t}R_{t}$ , we obtain that\n\n$$\nQ^{(\\mathcal{H})}(\\theta_{t+1})-Q^{(\\mathcal{H})}(\\theta_{t})\\leq-\\gamma_{t}\\left\\langle R_{t},\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\rangle+\\frac{L\\gamma_{t}^{2}}{2}\\left\\|R_{t}\\right\\|^{2}.\n$$"}, {"n": 3, "claim": "Using the fact that $2\\left. a,b\\right.=\\left\\|a\\right\\|^{2}+\\left\\|b\\right\\|^{2}-\\left\\|a-b\\right\\|^{2}$ , we obtain that\n\n$$\n\\begin{array}{l}Q^{(\\mathcal{H})}(\\theta_{t+1})-Q^{(\\mathcal{H})}(\\theta_{t})\\leq-\\frac{\\gamma_{t}}{2}\\left\\|R_{t}\\right\\|^{2}-\\frac{\\gamma_{t}}{2}\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}+\\frac{\\gamma_{t}}{2}\\left\\|R_{t}-\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}+\\frac{L\\gamma_{t}^{2}}{2}\\left\\|R_{t}\\right\\|^{2}\\\\ =\\left(\\frac{L\\gamma_{t}^{2}}{2}-\\frac{\\gamma_{t}}{2}\\right)\\|R_{t}\\|^{2}-\\frac{\\gamma_{t}}{2}\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}+\\frac{\\gamma_{t}}{2}\\left\\|R_{t}-\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}.\\\\ \\end{array}\n$$"}, {"n": 4, "claim": "As $\\gamma_{t}\\leq 1/L$ , we obtain that\n\n$$\n\\begin{array}{l}Q^{(\\mathcal{H})}(\\theta_{t+1})-Q^{(\\mathcal{H})}(\\theta_{t})\\leq-\\frac{\\gamma_{t}}{2}\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}+\\frac{\\gamma_{t}}{2}\\left\\|R_{t}-\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}\\\\ \\leq-\\frac{\\gamma_{t}}{2}\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}+\\gamma_{t}\\left\\|R_{t}-\\overline{{m}}_{t}\\right\\|^{2}+\\gamma_{t}\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})-\\overline{{m}}_{t}\\right\\|^{2}.\\\\ \\end{array}\n$$"}, {"n": 5, "claim": "Using Lemma 15, we then obtain that\n\n$$\nQ^{(\\mathcal{H})}(\\theta_{t+1})-Q^{(\\mathcal{H})}(\\theta_{t})\\leq-\\frac{\\gamma_{t}}{2}\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}+\\gamma_{t}\\frac{\\lambda}{n-f}\\sum_{i\\in\\mathcal{H}}\\left\\|m_{t}^{(i)}-\\overline{{m}}_{t}\\right\\|^{2}+\\gamma_{t}\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})-\\overline{{m}}_{t}\\right\\|^{2}.\n$$"}, {"n": 6, "claim": "Taking the total expectation from both sides we then have\n\n$$\n\\begin{array}{l}\\mathbb{E}\\left[Q^{(\\mathcal{H})}(\\theta_{t+1})-Q^{(\\mathcal{H})}(\\theta_{t})\\right]\\leq-\\frac{\\gamma_{t}}{2}\\mathbb{E}\\left[\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}\\right]+\\gamma_{t}\\frac{\\lambda}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|m_{t}^{(i)}-\\overline{{m}}_{t}\\right\\|^{2}\\right]\\\\ +\\gamma_{t}\\mathbb{E}\\left[\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})-\\overline{{m}}_{t}\\right\\|^{2}\\right],\\\\ \\end{array}\n$$\n\nwhich is the desired result."}]}}, {"local_id": "T2", "statement": "Lemma 6. Suppose assumptions 3, and 4 hold true, and consider Algorithm 1 with $T \\geq 2$ Then, for any $t\\in\\{0,\\ldots,T-1\\}$ , the following holds true\n\n$$\n\\frac{1}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|\\Delta m_{t}^{(i)}\\right\\|^{2}\\right]\\leq\\beta_{t}\\frac{1}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|\\Delta m_{t-1}^{(i)}\\right\\|^{2}\\right]+(1-\\beta_{t})\\zeta^{2}+(1-\\beta_{t})^{2}\\sigma^{2}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof Consider two arbitrary correct nodes i and j. By the definition of the momentum vector from (7), we obtain that\n\n$$\n\\begin{array}{l}m_{t}^{(i)}-m_{t}^{(j)}=\\beta_{t}(m_{t-1}^{(i)}-m_{t-1}^{(j)})+(1-\\beta_{t})(g_{t}^{(i)}-g_{t}^{(j)})\\\\ =\\beta_{t}(m_{t-1}^{(i)}-m_{t-1}^{(j)})+(1-\\beta_{t})\\left(\\nabla Q^{(i)}(\\theta_{t})-\\nabla Q^{(j)}(\\theta_{t})\\right)\\\\ +\\left(1-\\beta_{t}\\right)\\left(g_{t}^{(i)}-\\nabla Q^{(i)}(\\theta_{t})-g_{t}^{(j)}+\\nabla Q^{(j)}(\\theta_{t})\\right).\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "Taking the squared norm from both sides, we obtain that\n\n$$\n\\begin{array}{l}\\left\\|m_{t}^{(i)}-m_{t}^{(j)}\\right\\|^{2}=\\left\\|\\beta_{t}(m_{t-1}^{(i)}-m_{t-1}^{(j)})+(1-\\beta_{t})(\\nabla Q^{(i)}(\\theta_{t})-\\nabla Q^{(j)}(\\theta_{t}))\\right\\|^{2}\\\\ +\\left\\|(1-\\beta_{t})\\left(g_{t}^{(i)}-\\nabla Q^{(i)}(\\theta_{t})-g_{t}^{(j)}+\\nabla Q^{(j)}(\\theta_{t})\\right)\\right\\|^{2}\\\\ +2\\left\\langle\\beta_{t}(m_{t-1}^{(i)}-m_{t-1}^{(j)})+(1-\\beta_{t})(\\nabla Q^{(i)}(\\theta_{t})-\\nabla Q^{(j)}(\\theta_{t}))\\right.,(1-\\beta_{t})(g_{t}^{(i)}-\\nabla Q^{(i)}(\\theta_{t})-g_{t}^{(j)}+\\nabla Q^{(j)}(\\theta_{t}))\\Biggr>.\\\\ \\end{array}\n$$"}, {"n": 3, "claim": "Taking the conditional expectation $\\mathbb{E}_{t}\\left[\\cdot\\right]$ from both sides and noting that $\\mathbb{E}_{t}\\left[g_{t}^{(i)}\\right]=\\nabla Q^{(i)}(\\theta_{t})$ and $\\mathbb{E}_{t}\\left[g_{t}^{(j)}\\right]=\\nabla Q^{(j)}(\\theta_{t})$ , we obtain that\n\n$$\n\\begin{array}{l}\\mathbb{E}_{t}\\left[\\left\\|m_{t}^{(i)}-m_{t}^{(j)}\\right\\|^{2}\\right]=\\left\\|\\beta_{t}(m_{t-1}^{(i)}-m_{t-1}^{(j)})+(1-\\beta_{t})\\left(\\nabla Q^{(i)}(\\theta_{t})-\\nabla Q^{(j)}(\\theta_{t})\\right)\\right\\|^{2}\\\\ +(1-\\beta_{t})^{2}\\mathbb{E}_{t}\\left[\\left\\|g_{t}^{(i)}-\\nabla Q^{(i)}(\\theta_{t})\\right\\|^{2}\\right]+(1-\\beta_{t})^{2}\\mathbb{E}_{t}\\left[\\left\\|g_{t}^{(j)}-\\nabla Q^{(j)}(\\theta_{t})\\right\\|^{2}\\right].\\\\ \\end{array}\n$$"}, {"n": 4, "claim": "Using Assumption 3, we then obtain that\n\n$$\n\\mathbb{E}_{t}\\left[\\left\\|m_{t}^{(i)}-m_{t}^{(j)}\\right\\|^{2}\\right]\\leq\\left\\|\\beta_{t}(m_{t-1}^{(i)}-m_{t-1}^{(j)})+(1-\\beta_{t})\\left(\\nabla Q^{(i)}(\\theta_{t})-\\nabla Q^{(j)}(\\theta_{t})\\right)\\right\\|^{2}+2(1-\\beta_{t})^{2}\\sigma^{2}.\n$$"}, {"n": 5, "claim": "By Jensen’s inequality, we then have\n\n$$\n\\mathbb{E}_{t}\\left[\\left\\|m_{t}^{(i)}-m_{t}^{(j)}\\right\\|^{2}\\right]\\leq\\beta_{t}\\left\\|m_{t-1}^{(i)}-m_{t-1}^{(j)}\\right\\|^{2}+(1-\\beta_{t})\\left\\|\\nabla Q^{(i)}(\\theta_{t})-\\nabla Q^{(j)}(\\theta_{t})\\right\\|^{2}+2(1-\\beta_{t})^{2}\\sigma^{2}.\n$$"}, {"n": 6, "claim": "Taking total expectation and averaging over all possible $i,j\\in{\\mathcal{H}}$ , we then obtain that\n\n$$\n\\begin{array}{l}\\frac{1}{(n-f)^{2}}\\sum_{i,j\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|m_{t}^{(i)}-m_{t}^{(j)}\\right\\|^{2}\\right]\\leq\\beta_{t}\\frac{1}{(n-f)^{2}}\\sum_{i,j\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|m_{t-1}^{(i)}-m_{t-1}^{(j)}\\right\\|^{2}\\right]\\\\ +(1-\\beta_{t})\\frac{1}{(n-f)^{2}}\\sum_{i,j\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|\\nabla Q^{(i)}(\\theta_{t})-\\nabla Q^{(j)}(\\theta_{t})\\right\\|^{2}\\right]+2(1-\\beta_{t})^{2}\\sigma^{2}.\\\\ \\end{array}\n$$"}, {"n": 7, "claim": "Using Lemma 17, we then obtain that\n\n$$\n\\begin{array}{l}\\frac{1}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|m_{t}^{(i)}-\\overline{{m}}_{t}\\right\\|^{2}\\right]\\leq\\beta_{t}\\frac{1}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|m_{t-1}^{(i)}-\\overline{{m}}_{t-1}\\right\\|^{2}\\right]\\\\ +(1-\\beta_{t})\\frac{1}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|\\nabla Q^{(i)}(\\theta_{t})-\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}\\right]+(1-\\beta_{t})^{2}\\sigma^{2}.\\\\ \\end{array}\n$$"}, {"n": 8, "claim": "By Assumption 4, we then obtain that\n\n$$\n\\frac{1}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|m_{t}^{(i)}-\\overline{{m}}_{t}\\right\\|^{2}\\right]\\leq\\beta_{t}\\frac{1}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|m_{t-1}^{(i)}-\\overline{{m}}_{t-1}\\right\\|^{2}\\right]+(1-\\beta_{t})\\zeta^{2}+(1-\\beta_{t})^{2}\\sigma^{2}.\n$$\n\nThis is the desired result."}]}}, {"local_id": "T3", "statement": "Lemma 7. Suppose assumptions $\\begin{array}{ r }{I,~\\mathcal{3},}\\end{array}$ and 4 hold true, consider Algorithm 1 with $T \\geq 2$ 2 and λ as defined in Lemma 5. Then for any $t\\in\\{0,\\ldots,T-1\\}$ , the following holds true\n\n$$\n\\begin{array}{l}\\mathbb{E}\\left[\\left\\|\\delta_{t+1}\\right\\|^{2}\\right]\\leq\\beta_{t+1}^{2}\\left(1+4\\gamma_{t}L+3\\gamma_{t}^{2}L^{2}\\right)\\mathbb{E}\\left[\\left\\|\\delta_{t}\\right\\|^{2}\\right]+(1-\\beta_{t+1})^{2}\\frac{\\sigma^{2}}{n-f}\\\\ \\left.+3\\beta_{t+1}^{2}\\left(\\gamma_{t}^{2}L^{2}+\\gamma_{t}L\\right)\\left(\\frac{\\lambda}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|\\Delta m_{t}^{(i)}\\right\\|^{2}\\right]+\\mathbb{E}\\left[\\left\\|\\nabla Q^{(\\mathcal{H})}\\left(\\theta_{t}\\right)\\right\\|^{2}\\right]\\right)\\right).\\\\ \\end{array}\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof We recall that at any round $t\\in\\{0,\\ldots,T-1\\}$ and any worker $i\\in\\mathcal{H}$ , the momentum ${m}_{t}^{\\left(i\\right)}$ is computed as follows\n\n$$\nm_{t}^{(i)}=\\beta_{t}m_{t-1}^{(i)}+(1-\\beta_{t})g_{t}^{(i)}.\n$$"}, {"n": 2, "claim": "Hence, we have\n\n$$\n\\delta_{t+1}=\\overline{{m}}_{t+1}-\\nabla Q^{(\\mathcal{H})}(\\theta_{t+1})=\\beta_{t+1}\\overline{{m}}_{t}+(1-\\beta_{t+1})\\overline{{g}}_{t+1}-\\nabla Q^{(\\mathcal{H})}(\\theta_{t+1}).\n$$\n\nwhere for any $\\begin{array}{ r }{t\\in\\{0,\\ldots,T-1\\},\\ :\\overline{{m}}_{t}: =\\frac{1}{(n-f)}\\sum_{i\\in\\mathcal{H}}m_{t}^{(i)}}\\end{array}$ and $\\begin{array}{ r }{\\overline{{g}}_{t}: =\\frac{1}{(n-f)}\\sum_{i\\in\\mathcal{H}}g_{t}^{(i)}}\\end{array}$\n\nAdding and subtracting $\\beta_{t+1}\\nabla Q^{(\\mathcal{H})}(\\theta_{t})$ , we obtain that\n\n$$\n\\begin{array}{l}\\delta_{t+1}=\\beta_{t+1}\\left(\\overline{{m}}_{t}-\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right)+(1-\\beta_{t+1})\\overline{{g}}_{t+1}-\\nabla Q^{(\\mathcal{H})}(\\theta_{t+1})+\\beta_{t+1}\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\\\ =\\beta_{t+1}\\left(\\overline{{m}}_{t}-\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right)+(1-\\beta_{t+1})\\left(\\overline{{g}}_{t+1}-\\nabla Q^{(\\mathcal{H})}(\\theta_{t+1})\\right)+\\beta_{t+1}\\left(\\nabla Q^{(\\mathcal{H})}(\\theta_{t})-\\nabla Q^{(\\mathcal{H})}(\\theta_{t+1})\\right).\\\\ \\end{array}\n$$"}, {"n": 3, "claim": "Now by Assumption 3, we have $\\begin{array}{ r }{\\mathbb{E}_{t+1}\\left[\\overline{{g}}_{t+1}\\right]=\\nabla Q^{(\\mathcal{H})}(\\theta_{t+1})\\mathrm{and}\\mathbb{E}_{t+1}\\left[\\left.\\overline{{g}}_{t+1}-\\nabla Q^{(\\mathcal{H})}(\\theta_{t+1})\\right.^{2}\\right]\\leq}\\end{array}$ $\\frac{\\sigma^{2}}{n-f}$ . Therefore,\n\n$$\n\\mathbb{E}_{t+1}\\left[\\left\\|\\delta_{t+1}\\right\\|^{2}\\right]\\leq\\beta_{t+1}^{2}\\left\\|\\delta_{t}+\\nabla Q^{(\\mathcal{H})}(\\theta_{t})-\\nabla Q^{(\\mathcal{H})}(\\theta_{t+1})\\right\\|^{2}+(1-\\beta_{t+1})^{2}\\frac{\\sigma^{2}}{n-f}.\n$$"}, {"n": 4, "claim": "Now as $(a+b)^{2}\\leq(1+c)a^{2}+(1+1/c)b^{2}$ for any c, we obtain that\n\n$$\n\\begin{array}{l}\\mathbb{E}_{t+1}\\left[\\left\\|\\delta_{t+1}\\right\\|^{2}\\right]\\leq\\beta_{t+1}^{2}(1+\\gamma_{t}L)\\left\\|\\delta_{t}\\right\\|^{2}+\\beta_{t+1}^{2}(1+\\frac{1}{\\gamma_{t}L})\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})-\\nabla Q^{(\\mathcal{H})}(\\theta_{t+1})\\right\\|^{2}\\\\ +(1-\\beta_{t+1})^{2}\\frac{\\sigma^{2}}{n-f}.\\\\ \\end{array}\n$$"}, {"n": 5, "claim": "From Assumption 1, we have $\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})-\\nabla Q^{(\\mathcal{H})}(\\theta_{t+1})\\right\\|\\le L\\left\\|\\theta_{t}-\\theta_{t+1}\\right\\|$ . Using this above, we obtain that\n\n$$\n\\begin{array}{l}\\mathbb{E}_{t+1}\\left[\\left\\|\\delta_{t+1}\\right\\|^{2}\\right]\\leq\\beta_{t+1}^{2}(1+\\gamma_{t}L)\\left\\|\\delta_{t}\\right\\|^{2}+\\beta_{t+1}^{2}\\left(1+\\frac{1}{\\gamma_{t}L}\\right)L^{2}\\left\\|\\theta_{t}-\\theta_{t+1}\\right\\|^{2}\\\\ +(1-\\beta_{t+1})^{2}\\frac{\\sigma^{2}}{n-f}.\\tag{29}\\\\ \\end{array}\n$$"}, {"n": 6, "claim": "Now recall that $\\theta_{t}-\\theta_{t+1}=\\gamma_{t}R_{t}$ . Therefore,\n\n$$\n\\begin{array}{l}\\left\\|\\theta_{t}-\\theta_{t+1}\\right\\|^{2}=\\gamma_{t}^{2}\\left\\|R_{t}\\right\\|^{2}\\\\ =\\gamma_{t}^{2}\\left\\|R_{t}-\\overline{{m}}_{t}+\\overline{{m}}_{t}-\\nabla Q^{(\\mathcal{H})}(\\theta_{t})+\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}\\\\ \\leq 3\\gamma_{t}^{2}\\left\\|R_{t}-\\overline{{m}}_{t}\\right\\|^{2}+3\\gamma_{t}^{2}\\left\\|\\overline{{m}}_{t}-\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}+3\\gamma_{t}^{2}\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}\\\\ \\leq 3\\gamma_{t}^{2}\\frac{\\lambda}{n-f}\\sum_{i\\in\\mathcal{H}}\\left\\|m_{t}^{(i)}-\\overline{{m}}_{t}\\right\\|^{2}+3\\gamma_{t}^{2}\\left\\|\\overline{{m}}_{t}-\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}+3\\gamma_{t}^{2}\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2},\\\\ \\end{array}\n$$\n\nwhere in the last inequality we used 15."}, {"n": 7, "claim": "Combining this with (29), we obtain that\n\n$$\n\\begin{array}{l}\\mathbb{E}_{t+1}\\left[\\|\\delta_{t+1}\\|^{2}\\right]\\leq\\beta_{t+1}^{2}(1+\\gamma_{t}L)\\|\\delta_{t}\\|^{2}+(1-\\beta_{t+1})^{2}\\frac{\\sigma^{2}}{n-f}\\\\ +\\beta_{t+1}^{2}(1+\\frac{1}{\\gamma_{t}L})L^{2}\\left(3\\gamma_{t}^{2}\\frac{\\lambda}{n-f}\\sum_{i\\in\\mathcal{H}}\\left\\|m_{t}^{(i)}-\\overline{{m}}_{t}\\right\\|^{2}+3\\gamma_{t}^{2}\\left\\|\\delta_{t}\\right\\|^{2}+3\\gamma_{t}^{2}\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}\\right).\\\\ \\end{array}\n$$"}, {"n": 8, "claim": "Rearranging the terms and taking the total expectation, we obtain that\n\n$$\n\\begin{array}{l}\\mathbb{E}\\left[\\|\\delta_{t+1}\\|^{2}\\right]\\leq\\beta_{t+1}^{2}(1+4\\gamma_{t}L+3\\gamma_{t}^{2}L^{2})\\mathbb{E}\\left[\\|\\delta_{t}\\|^{2}\\right]+(1-\\beta_{t+1})^{2}\\frac{\\sigma^{2}}{n-f}\\\\ +3\\beta_{t+1}^{2}(\\gamma_{t}^{2}L^{2}+\\gamma_{t}L)\\left(\\frac{\\lambda}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|m_{t}^{(i)}-\\overline{{m}}_{t}\\right\\|^{2}\\right]+\\mathbb{E}\\left[\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}\\right]\\right).\\\\ \\end{array}\n$$\n\nThis is the desired result."}]}}, {"local_id": "T4", "statement": "Lemma 8. Suppose assumptions $\\begin{array}{ r }{\\boldsymbol{{1}},\\boldsymbol{{\\mathcal{Q}}},\\boldsymbol{{\\mathcal{B}}},}\\end{array}$ and 4 hold true. Consider Algorithm 1 with $T \\geq 2$ and a set of parameters such that $t\\in\\{0,\\cdots,T\\},\\gamma_{t}\\leq\\frac{1}{18L}$ , and $1-\\beta_{t+1}=18\\gamma_{t}L$ . Finally, let $(V_{t})_{t\\geq 0}$ be as defined in (9) and λ as defined in Lemma 15. Then the following holds true\n\n$$\nV_{t+1}\\leq\\left(1-\\frac{\\mu\\gamma_{t}}{3}\\right)V_{t}+27L\\left(\\lambda+\\frac{1}{n-f}\\right)\\sigma^{2}\\gamma_{t}^{2}+\\frac{3}{2}\\lambda\\zeta^{2}\\gamma_{t}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof Consider an arbitrary $t\\in\\{0,\\ldots,T-1\\}$ . Combining Lemmas 6, 7, and 5, we obtain that\n\n$$\n\\begin{array}{l}V_{t+1}=\\mathbb{E}\\left[Q^{(\\mathcal{H})}(\\theta_{t+1})\\right]-Q^{*}+\\rho\\mathbb{E}\\left[\\|\\delta_{t+1}\\|^{2}\\right]+\\rho\\frac{\\lambda}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|m_{t+1}^{(i)}-\\overline{{m}}_{t+1}\\right\\|^{2}\\right]\\\\ \\leq\\mathbb{E}\\left[Q^{(\\mathcal{H})}(\\theta_{t})\\right]-Q^{*}-\\frac{\\gamma_{t}}{2}\\mathbb{E}\\left[\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}\\right]+\\gamma_{t}\\frac{\\lambda}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|m_{t}^{(i)}-\\overline{{m}}_{t}\\right\\|^{2}\\right]\\\\ +\\gamma_{t}\\mathbb{E}\\left[\\|\\delta_{t}\\|^{2}\\right]+\\rho\\beta_{t+1}^{2}(1+4\\gamma_{t}L+3\\gamma_{t}^{2}L^{2})\\mathbb{E}\\left[\\|\\delta_{t}\\|^{2}\\right]+\\rho(1-\\beta_{t+1})^{2}\\frac{\\sigma^{2}}{n-f}\\\\ +3\\rho\\beta_{t+1}^{2}(\\gamma_{t}^{2}L^{2}+\\gamma_{t}L)\\left(\\frac{\\lambda}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|m_{t}^{(i)}-\\overline{{m}}_{t}\\right\\|^{2}\\right]+\\mathbb{E}\\left[\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}\\right]\\right)\\\\ +\\rho\\lambda\\beta_{t+1}\\frac{1}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|m_{t}^{(i)}-\\overline{{m}}_{t}\\right\\|^{2}\\right]+\\rho\\lambda(1-\\beta_{t+1})\\zeta^{2}+\\rho\\lambda(1-\\beta_{t+1})^{2}\\sigma^{2}.\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "Re-arranging the terms, we obtain that\n\n$$\n\\begin{array}{l}V_{t+1}\\leq\\mathbb{E}\\left[Q^{(\\mathcal{H})}\\left(\\theta_{t}\\right)\\right]-Q^{*}+\\left(-\\frac{\\gamma_{t}}{2}+3\\rho\\beta_{t+1}^{2}(\\gamma_{t}^{2}L^{2}+\\gamma_{t}L)\\right)\\mathbb{E}\\left[\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}\\right]\\\\ +\\left(\\gamma_{t}+3\\rho\\beta_{t+1}^{2}(\\gamma_{t}^{2}L^{2}+\\gamma_{t}L)+\\rho\\beta_{t+1}\\right)\\frac{\\lambda}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|m_{t}^{(i)}-\\overline{{m}}_{t}\\right\\|^{2}\\right]\\\\ +\\left(\\gamma_{t}+\\rho\\beta_{t+1}^{2}(1+4\\gamma_{t}L+3\\gamma_{t}^{2}L^{2})\\right)\\mathbb{E}\\left[\\|\\delta_{t}\\|^{2}\\right]\\\\ +\\rho\\lambda(1-\\beta_{t+1})\\zeta^{2}+\\rho\\lambda(1-\\beta_{t+1})^{2}\\sigma^{2}+\\rho(1-\\beta_{t+1})^{2}\\frac{\\sigma^{2}}{n-f}.\\tag{30}\\\\ \\end{array}\n$$"}, {"n": 3, "claim": "We denote,\n\n$$\n\\begin{array}{l}A:= -\\frac{\\gamma_{t}}{2}+3\\rho\\beta_{t+1}^{2}(\\gamma_{t}^{2}L^{2}+\\gamma_{t}L),\\\\ B:=\\gamma_{t}+3\\rho\\beta_{t+1}^{2}(\\gamma_{t}^{2}L^{2}+\\gamma_{t}L)+\\rho\\beta_{t+1},\\\\ C:=\\gamma_{t}+\\rho\\beta_{t+1}^{2}(1+4\\gamma_{t}L+3\\gamma_{t}^{2}L^{2})\\\\ D:=\\rho\\lambda(1-\\beta_{t+1})\\zeta^{2}+\\rho\\lambda(1-\\beta_{t+1})^{2}\\sigma^{2}+\\rho(1-\\beta_{t+1})^{2}\\frac{\\sigma^{2}}{n-f}.\\\\ \\end{array}\n$$"}, {"n": 4, "claim": "Substituting from above in (30) we obtain that\n\n$$\n\\begin{array}{l}V_{t+1}\\leq\\mathbb{E}\\left[Q^{(\\mathcal{H})}\\left(\\theta_{t}\\right)\\right]-Q^{*}+A\\mathbb{E}\\left[\\left\\|\\nabla Q^{(\\mathcal{H})}\\left(\\theta_{t}\\right)\\right\\|^{2}\\right]\\\\ +B\\frac{\\lambda}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|m_{t}^{(i)}-\\overline{{m}}_{t}\\right\\|^{2}\\right]+C\\mathbb{E}\\left[\\left\\|\\delta_{t}\\right\\|^{2}\\right]+D.\\\\ \\end{array}\n$$"}, {"n": 5, "claim": "Now, we separately analyse the terms A, B, C and D below by using the following,\n\n$$\n\\rho=\\frac{1}{12L},\\gamma_{t}\\leq\\frac{1}{18L},\\text{and} 1-\\beta_{t+1}=18\\gamma_{t}L.\\tag{31}\n$$"}, {"n": 6, "claim": "Note that the condition on $\\gamma_{t}$ above follows\n\nTerm A. Using the facts that $\\rho=1/12L,\\gamma_{t}\\leq 1/18L\\leq 1/3L$ and that $\\beta_{t+1}^{2}<1$ , we obtain that\n\n$$\n\\begin{array}{l}A= -\\frac{\\gamma_{t}}{2}+3\\rho\\beta_{t+1}^{2}(\\gamma_{t}^{2}L^{2}+\\gamma_{t}L)\\leq-\\frac{\\gamma_{t}}{2}+3\\rho(\\gamma_{t}^{2}L^{2}+\\gamma_{t}L)\\\\ \\leq-\\frac{\\gamma_{t}}{2}+\\frac{1}{4 L}\\left(\\frac{\\gamma_{t}L}{3}+\\gamma_{t}L\\right)= -\\frac{\\gamma_{t}}{6}.\\tag{32}\\\\ \\end{array}\n$$"}, {"n": 7, "claim": "Term B. We obtain that\n\n$$\nB=\\gamma_{t}+3\\rho\\beta_{t+1}^{2}(\\gamma_{t}^{2}L^{2}+\\gamma_{t}L)+\\rho\\beta_{t+1}=\\rho(12\\gamma_{t}L+3\\beta_{t+1}^{2}(\\gamma_{t}^{2}L^{2}+\\gamma_{t}L)+\\beta_{t+1}).\n$$"}, {"n": 8, "claim": "Noting that $\\beta_{t+1}\\leq 1,\\beta_{t+1}=1-18\\gamma_{t}L$ and $\\gamma_{t}\\leq 1/18L\\leq 1/12L$ we obtain that\n\n$$\nB\\leq\\rho\\left(12\\gamma_{t}L+3\\gamma_{t}L+\\frac{\\gamma_{t}L}{4}+(1-18\\gamma_{t}L)\\right)\\leq\\rho\\left(1-\\frac{11\\gamma_{t}L}{4}\\right)\\leq\\rho\\left(1-\\frac{\\gamma_{t}L}{3}\\right)\\leq\\rho\\left(1-\\frac{\\mu\\gamma_{t}}{3}\\right),\n$$\n\nwhere in the last inequality we used $\\mu \\leq L$\n\nTerm C. Using the facts that $\\beta_{t+1}<1$ and $\\rho=1/12L$ , we obtain that\n\n$$\n\\begin{array}{l}C:=\\gamma_{t}+\\rho\\beta_{t+1}^{2}(1+4\\gamma_{t}L+3\\gamma_{t}^{2}L^{2})\\leq\\rho\\left(\\frac{\\gamma_{t}}{\\rho}+\\beta_{t+1}+4\\gamma_{t}L+3\\gamma_{t}^{2}L^{2}\\right)\\\\ =\\rho\\left(12\\gamma_{t}L+\\beta_{t+1}+4\\gamma_{t}L+3\\gamma_{t}^{2}L^{2}\\right).\\\\ \\end{array}\n$$"}, {"n": 9, "claim": "Using the fact $\\gamma_{t}\\leq 1/18L\\leq 1/12L$ we then have\n\n$$\nC\\leq\\rho\\left(16\\gamma_{t}L+\\frac{\\gamma_{t}L}{4}+(1-18\\gamma_{t}L)\\right)\\leq\\rho\\left(1-\\frac{7\\gamma_{t}L}{4}\\right)\\leq\\rho\\left(1-\\frac{\\gamma_{t}L}{3}\\right)\\leq\\rho\\left(1-\\frac{\\mu\\gamma_{t}}{3}\\right),\\tag{33}\n$$"}, {"n": 10, "claim": "Term D.\n\n$$\n\\begin{array}{l}D=\\rho\\lambda(1-\\beta_{t+1})\\zeta^{2}+\\rho\\lambda(1-\\beta_{t+1})^{2}\\sigma^{2}+\\rho(1-\\beta_{t+1})^{2}\\frac{\\sigma^{2}}{n-f}\\\\ =\\frac{3}{2}\\gamma_{t}\\lambda\\zeta^{2}+27\\gamma_{t}^{2}L\\left(\\lambda+\\frac{1}{n-f}\\right)\\sigma^{2}.\\\\ \\end{array}\n$$"}, {"n": 11, "claim": "Combining all, we obtain that\n\n$$\n\\begin{array}{l}V_{t+1}\\leq\\mathbb{E}\\left[Q^{(\\mathcal{H})}(\\theta_{t})\\right]-Q^{*}-\\frac{\\gamma_{t}}{6}\\mathbb{E}\\left[\\left\\|\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right\\|^{2}\\right]+\\left(1-\\frac{\\mu\\gamma_{t}}{3}\\right)\\rho\\mathbb{E}\\left[\\|\\delta_{t}\\|^{2}\\right]\\\\ +\\left(1-\\frac{\\mu\\gamma_{t}}{3}\\right)\\rho\\frac{\\lambda}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|m_{t}^{(i)}-\\overline{{m}}_{t}\\right\\|^{2}\\right]+\\frac{3}{2}\\gamma_{t}\\lambda\\zeta^{2}+27\\gamma_{t}^{2}L\\left(\\lambda+\\frac{1}{n-f}\\right)\\sigma^{2}.\\\\ \\end{array}\n$$"}, {"n": 12, "claim": "Recall from Assumption 2 that $\\left.\\nabla Q^{(\\mathcal{H})}(\\theta_{t})\\right.^{2}\\geq 2\\mu\\left(Q^{(\\mathcal{H})}\\left(\\theta_{t}\\right)-Q^{*}\\right)$ . Therefore,\n\n$$\n\\begin{array}{l}V_{t+1}\\leq\\left(1-\\frac{\\mu\\gamma_{t}}{3}\\right)\\left(\\mathbb{E}\\left[Q^{(\\mathcal{H})}\\left(\\theta_{t}\\right)\\right]-Q^{*}+\\rho\\mathbb{E}\\left[\\|\\delta_{t}\\|^{2}\\right]+\\rho\\frac{\\lambda}{n-f}\\sum_{i\\in\\mathcal{H}}\\mathbb{E}\\left[\\left\\|m_{t}^{(i)}-\\overline{{m}}_{t}\\right\\|^{2}\\right]\\right)\\\\ +\\frac{3}{2}\\gamma_{t}\\lambda\\zeta^{2}+27\\gamma_{t}^{2}L\\left(\\lambda+\\frac{1}{n-f}\\right)\\sigma^{2}\\\\ =\\left(1-\\frac{\\mu\\gamma_{t}}{3}\\right)V_{t}+27L\\left(\\lambda+\\frac{1}{n-f}\\right)\\sigma^{2}\\gamma_{t}^{2}+\\frac{3}{2}\\lambda\\zeta^{2}\\gamma_{t}.\\\\ \\end{array}\n$$"}]}}], "target_local_id": null, "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2025-24-0579", "context": "Notation. Throughout this paper, we denote by $\\Delta(\\boldsymbol{S})$ the probability simplex over a set , and $[K]: =\\{1,\\cdots,K\\}$ for any positive integer $K > 0$ . In addition, $f(\\cdot)=\\widetilde O(g(\\cdot))$ or $f \\lesssim g$ (resp. $f(\\cdot)=\\widetilde\\Omega(g(\\cdot))$ or $f\\gtrsim g)$ means that $f(\\cdot)$ is orderwise no larger than (resp. no smaller than) $g(\\cdot)$ modulo some logarithmic factors. The notation $f \\asymp g$ means $f \\lesssim g$ and $f \\gtrsim g$ hold simultaneously.\n\nIn this section, we introduce the mathematical model and background of Markov decision processes.\n\nInfinite-horizon Markov decision process. We consider an infinite-horizon Markov decision process (MDP), which is represented by $\\boldsymbol{\\mathcal{M}}=(\\boldsymbol{\\mathcal{S}},\\boldsymbol{\\mathcal{A}},\\boldsymbol{P},\\boldsymbol{r},\\boldsymbol{\\gamma})$ . Here, and $\\mathcal{A}$ denote the state space and the action space, respectively, $P:\\mathcal{S}\\times\\mathcal{A}\\times\\mathcal{S}[0,1]$ indicates the transition kernel such that $P(s^{\\prime}\\mid s,a)$ denotes the probability that action a in state s leads to state $s^{\\prime},r:\\mathcal{S}\\times\\mathcal{A}\\to [0,1]$ denotes a deterministic reward function, where $r(s,a)$ is the immediate reward for action a in state s, and $\\gamma\\in [0,1)$ is the discount factor.\n\nPolicy, value function, and Q-function. A policy is an action-selection rule denoted by the mapping $\\pi:S\\Delta({\\mathcal{A}})$ , such that $\\pi(a|s)$ is the probability of taking action a in state s. For a given policy $\\pi_{\\ i}$ the value function $V^{\\pi}:{\\mathcal{S}}\\mathbb{R}$ , which measures the expected discounted cumulative reward from an initial state $s,$ is defined as\n\n$$\n\\forall s\\in\\mathcal{S}:\\quad V^{\\pi}(s):=\\mathbb{E}\\left[\\sum_{t=0}^{\\infty}\\gamma^{t}r(s_{t},a_{t})\\mid s_{0}=s\\right].\\tag{6}\n$$\n\nHere, the expectation is taken with respect to the randomness of the trajectory $\\{s_{t},a_{t},r_{t}\\}_{t=0}^{\\infty},$ sampled based on the transition kernel $(\\mathrm{i . e .,~}s_{t+1}\\sim P(\\cdot|s_{t},a_{t}))$ and the policy $\\pi\\ \\left(\\mathrm{i . e .}\\right.$ 2 $a_{t}\\sim\\pi(\\cdot|s_{t}))$ for any $t \\geq 0$ . Similarly, the state-action value function $(\\mathrm{i . e .,~}Q\\mathrm{-}f u n c t i o n)$ $Q^{\\pi}:\\mathcal{S}\\times\\mathcal{A}\\mathbb{R}$ , which measures the expected discounted cumulative reward from an initial state-action pair $(s,a)$ , is defined as\n\n$$\n\\forall(s,a)\\in\\mathcal{S}\\times\\mathcal{A}:\\qquad Q^{\\pi}(s,a):=r(s,a)+\\mathbb{E}\\left[\\sum_{t=1}^{\\infty}\\gamma^{t}r(s_{t},a_{t})\\mid s_{0}=s,a_{0}=a\\right].\n$$\n\nAgain here, the expectation is taken with respect to the randomness of the trajectory $\\{s_{t},a_{t},r_{t}\\}_{t=1}^{\\infty}$ generated similarly as above. Since the rewards lie within $[0,1]$ ], it follows that for any policy $\\pi .$\n\n$$\n0\\leq V^{\\pi}\\leq\\frac{1}{1-\\gamma},\\quad 0\\leq Q^{\\pi}\\leq\\frac{1}{1-\\gamma}.\\tag{7}\n$$\n\nOptimal policy and Bellman’s principle of optimality. A policy that maximizes the value function uniformly over all states is called an optimal policy and denoted by $\\pi^{\\star}$ . Note that the existence of such an optimal policy is always guaranteed (Puterman, 2014), which also maximizes the Q-function simultaneously. The corresponding optimal value function and Q-function are denoted by $V^{\\star}: =V^{\\pi^{\\star}}$ and $Q^{\\star}: =Q^{\\pi^{\\star}}$ , respectively. It is well-known that the optimal Q-function $Q^{\\star}$ can be determined as the unique fixed point of the Bellman operator $\\tau$ , given by\n\n$$\n\\mathcal{T}(Q)(s,a):=r(s,a)+\\gamma\\underset{s^{\\prime}\\sim P(\\cdot|s,a)}{\\mathbb{E}}\\left[\\max_{a^{\\prime}\\in\\mathcal{A}}Q(s^{\\prime},a^{\\prime})\\right].\\tag{8}\n$$\n\nIn this section, we begin with understanding federated synchronous Q-learning, where all the state-action pairs are updated simultaneously assuming access to a generative model or simulator at all the agents.\n\nIn the synchronous setting, each agent $k \\in [ K ]$ has access to a generative model, and generates a new sample\n\n$$\ns_{t}^{k}(s,a)\\sim P(\\cdot|s,a)\\tag{9}\n$$\n\nfor every state-action pair $(s,a)\\in\\mathcal{S}\\times\\mathcal{A}$ independently at every iteration t. Our goal is to learn the optimal Q-function $Q^{\\star}$ collaboratively by aggregating the local Q-learning estimates periodically.\n\nWe propose a natural federated synchronous Q-learning algorithm called FedSynQ that alternates between local updates at agents and periodic averaging at a central server. The complete description is summarized in Algorithm 1. FedSynQ initializes a local Q-function as $Q_{0}^{k}=Q_{0}$ at each agent $k \\in [ K ]$ . Suppose at the beginning of each iteration $t \\geq 1$ , each agent maintains a local Q-function estimate $Q_{t-1}^{k}$ and a local value function estimate $V_{t-1}^{k}$ 2 which are related via\n\n$$\n\\forall s\\in\\mathcal{S}:\\quad V_{t}^{k}(s):=\\max_{a\\in\\mathcal{A}}Q_{t}^{k}(s,a).\\tag{11}\n$$\n\nFedSynQ proceeds according to the following steps in the rest of the t-th iteration.\n\n1. Local updates: Each agent first independently updates all entries of its Q-estimate $Q_{t-1}^{k}$ to reach some intermediate estimate following the update rule:\n\n$$\n\\forall(s,a)\\in\\mathcal{S}\\times\\mathcal{A}:\\quad Q_{t-\\frac{1}{2}}^{k}(s,a)=(1-\\eta)Q_{t-1}^{k}(s,a)+\\eta\\left(r(s,a)+\\gamma V_{t-1}^{k}(s_{t}^{k}(s,a))\\right),\\tag{12}\n$$\n\nwhere $s_{t}^{k}(s,a)$ is drawn according to (9), and $\\eta \\geq 0$ is the learning rate.\n\n2. Periodic averaging: These intermediate estimates will be periodically averaged by the server to form the updated estimate $Q_{t}^{k}$ at the end of the t-th iteration. Formally, denoting $\\tau \\geq 1$ as the synchronization period, it follows\n\n$$\n\\forall(s,a)\\in\\mathcal{S}\\times\\mathcal{A}:\\quad Q_{t}^{k}(s,a)=\\left\\{\\begin{array}{l l}\\frac{1}{K}\\sum_{k=1}^{K}Q_{t-\\frac{1}{2}}^{k}(s,a)&\\text{if} t\\equiv 0(\\bmod\\tau)\\\\ Q_{t-\\frac{1}{2}}^{k}(s,a)&\\text{otherwise}\\end{array}.\\right.\\tag{13}\n$$\n\nDenoting the number of total iterations by $T_{i}$ , the algorithm outputs the final Q-estimate as the average of all local estimates, i.e. $\\begin{array}{ r }{Q_{T}=\\frac{1}{K}\\sum_{k}Q_{T}^{k}}\\end{array}$ . Without loss of generality, we assume the total number of iterations T is divisible by $\\tau_{\\ast}$ where $C_{\\sf r o u n d}=T/\\tau$ is the rounds of communication.\n\nAlgorithm 1: Federated Synchronous Q-learning (FedSynQ)\n\n<div class=\"mineru-algorithm\" style=\"white-space: pre-wrap; font-family:monospace;\">\ninputs: learning rate $\\eta$, discount factor $\\gamma$, number of agents $K$, synchronization period $\\tau$, number of iterations $T$.\ninitialization: $Q_0^k = Q_0$ for all $k$.\nfor $t = 1, \\cdots, T$ do\n    for $k \\in [K]$ do\n        Draw $s_t^k(s, a) \\sim P(\\cdot | s, a)$ for all $(s, a) \\in \\mathcal{S} \\times \\mathcal{A}$.\n        Compute $Q_{t-\\frac{1}{2}}^k$ according to (12).\n        Compute $Q_t^k$ according to (13).\n    end for\nend for\nreturn: $Q_T = \\frac{1}{K} \\sum_k Q_T^k$.\n</div>\n\nWe are ready to provide the finite-time convergence analysis of Algorithm 1.\n\nTheorem 1 (Finite-time convergence of FedSynQ) Consider any given $\\delta\\in(0,1)$ and $\\varepsilon\\in(0,\\frac{1}{1-\\gamma}]$ . Suppose that the initialization of Algorithm 1 satisfies $\\begin{array}{ r }{0\\leq Q_{0}\\leq\\frac{1}{1-\\gamma}}\\end{array}$ , and the synchronization period τ obeys\n\n$$\n\\tau\\leq 1+\\frac{1}{\\eta}\\min\\left\\{\\frac{1-\\gamma}{8\\gamma},\\frac{1}{K}\\right\\}.\\tag{14a}\n$$\n\nThere exist some suficiently large constant $c_{T}>0$ and suficiently small constant $c_{\\eta}>0$ 2 both independent of problem parameters, such that with probability at least $1 - \\delta$ , the output of Algorithm 1 satisfies $\\|Q_{T}-Q^{\\star}\\|_{\\infty}\\leq\\varepsilon$ , provided that the sample size per agent T and the learning rate η satisfy\n\n$$\nT\\geq\\frac{c_{T}}{K(1-\\gamma)^{5}\\varepsilon^{2}}(\\log((1-\\gamma)^{2}\\varepsilon))^{2}\\log\\frac{|\\mathcal{S}| |\\mathcal{A}|K T}{\\delta},\\tag{14b}\n$$\n\n$$\n\\eta=c_{\\eta}K(1-\\gamma)^{4}\\varepsilon^{2}\\frac{1}{\\log\\frac{|\\mathcal{S}| |\\mathcal{A}|K T}{\\delta}}.\\tag{14c}\n$$\n\nIn this section, we study the sample complexity of federated asynchronous Q-learning, where K agents sample local trajectories using diferent behavior policies. In particular, we propose a novel aggregation algorithm FedAsynQ-ImAvg that leverages the heterogeneity of these policies and dramatically improves the sample complexity.\n\nIn the asynchronous setting, each agent $k \\in [ K ]$ independently collects a sample trajectory $\\{s_{t}^{k},a_{t}^{k},r_{t}^{k}\\}_{t=0}^{\\infty}$ from the same underlying MDP  following some stationary local behavior policy $\\pi_{\\mathfrak{b}}^{k}$ such that\n\n$$\na_{t}^{k}\\sim\\pi_{\\mathsf{b}}^{k}(\\cdot|s_{t}^{k}),\\quad r_{t}^{k}=r(s_{t}^{k},a_{t}^{k}),\\quad s_{t+1}^{k}\\sim P(\\cdot|s_{t}^{k},a_{t}^{k})\\tag{15}\n$$\n\nfor all $t \\geq 0 .$ , where the initial state is initialized as $s_{0}^{k}$ for each agent k. Note that the behavior policies $\\{\\pi_{\\mathfrak{b}}^{k}\\}_{k\\in [K]}$ are heterogeneous across agents and can be diferent from the optimal policy $\\pi^{\\star}$ . Contrary to the generative model considered in the synchronous setting, the samples collected under the asynchronous setting are no longer independent across time but are Markovian, making the analysis significantly more challenging. The sample trajectory at each agent can be viewed as sampling a time-homogeneous Markov chain over the set of state-action pairs. Throughout this paper, we make the following standard uniform ergodicity assumption (Paulin, 2015; Li et al., 2021b).\n\nAssumption 1 (Uniform ergodicity) For every agent $k \\in [ K ]$ , the Markov chain induced by the stationary behavior policy $\\pi_{\\mathfrak{b}}^{k}$ is uniformly ergodic over the entire state-action space $\\mathcal{S}\\times\\mathcal{A}$\n\nUniform ergodicity guarantees that the distribution of the state-action pair $(s_{t},a_{t})$ of a trajectory converges to the stationary distribution of the Markov chain geometrically fast regardless of the initial state-action pair, and eventually, each state-action pair is visited in proportion to the stationary distribution.\n\nKey parameters. Two important quantities concerning the resulting Markov chains will govern the performance guarantees. The first one is the stationary state-action distribution $\\mu_{\\mathrm{b}}^{k}$ , which is the stationary distribution of the Markov chain induced by $\\pi_{\\mathfrak{b}}^{k}$ over all stateaction pairs; the second one is $t_{\\mathrm{mix}}^{k}$ , which is the mixing time of the same Markov chain given by\n\n$$\nt_{\\text{mix}}^{k}:=\\min\\left\\{t\\mid\\max_{(s_{0},a_{0})\\in\\mathcal{S}\\times\\mathcal{A}}d_{\\mathrm{TV}}\\left(P_{t}^{k}(\\cdot\\mid s_{0},a_{0}),\\mu_{\\mathrm{b}}^{k}\\right)\\leq\\frac{1}{4}\\right\\},\\tag{16}\n$$\n\nwhere $P_{t}^{k}(\\cdot|s_{0},a_{0})$ denote the distribution of $(s_{t},a_{t})$ conditioned on $(s_{0},a_{0})$ for agent $k,$ and $d_{\\mathsf{TV}}(\\cdot,\\cdot)$ is the total variation distance. Further, let the largest mixing time of all the Markov chains induced by local behavior policies be\n\n$$\nt_{\\text{mix}}^{\\max}:=\\max_{k\\in [K]}t_{\\text{mix}}^{k}.\\tag{17}\n$$\n\nLet us further define a few key parameters that measure the coverage and heterogeneity of the stationary state-action distribution $\\mu_{\\mathrm{b}}^{k}$ across agents. First, define\n\n$$\n\\mu_{\\min}:=\\min_{k\\in [K]}\\mu_{\\min}^{k},\\quad \\text{where}\\quad\\mu_{\\min}^{k}:=\\min_{(s,a)\\in\\mathcal{S}\\times\\mathcal{A}}\\mu_{\\mathrm{b}}^{k}(s,a).\\tag{18}\n$$\n\nState-action pairs with small stationary probabilities are visited less frequently, and therefore can become bottlenecks in improving the quality of Q-function estimates. Clearly, $\\mu_{\\mathrm{{m i n}}}\\leq{\\frac{1}{|S| |A|}}$ . In addition, denote\n\n$$\n\\mu_{\\mathrm{avg}}:=\\min_{(s,a)\\in\\mathcal{S}\\times\\mathcal{A}}\\frac{1}{K}\\sum_{k=1}^{K}\\mu_{\\mathrm{b}}^{k}(s,a).\\tag{19}\n$$\n\nIn words, $\\mu_{\\mathsf{avg}}$ is the minimum entry of the average stationary state-action distribution of all agents. The diference between $\\mu_{\\sf a v g}$ and $\\mu_{\\mathrm{{m i n}}}$ stands out when an individual agent fails to cover the entire state-action space. While $\\mu_{\\mathrm{{m i n}}}=0$ in such a case, $\\mu_{\\mathsf{avg}}$ can still be positive as long as each state-action pair is explored by at least one of the agents, i.e., $\\textstyle\\sum_{k=1}^{K}\\mu_{\\mathsf{b}}^{k}(s,a)>0$ . Note that $\\mu_{\\sf a v g}$ is always greater than or equal to $\\mu_{\\mathrm{{m i n}}}$ since\n\n$$\n\\mu_{\\mathrm{avg}}=\\min_{(s,a)\\in\\mathcal{S}\\times\\mathcal{A}}\\frac{1}{K}\\sum_{k=1}^{K}\\mu_{\\mathrm{b}}^{k}(s,a)\\geq\\min_{(s,a)\\in\\mathcal{S}\\times\\mathcal{A},k\\in [K]}\\mu_{\\mathrm{b}}^{k}(s,a)=\\mu_{\\min}.\\tag{20}\n$$\n\nLast but not least, we measure the heterogeneity of the stationary state-action distributions across agents by\n\n$$\nC_{\\mathrm{het}}:=\\max_{k\\in [K]}\\max_{(s,a)\\in\\mathcal{S}\\times\\mathcal{A}}\\frac{\\mu_{\\mathrm{b}}^{k}(s,a)}{\\frac{1}{K}\\sum_{k=1}^{K}\\mu_{\\mathrm{b}}^{k}(s,a)},\\tag{21}\n$$\n\nwhich satisfies $1\\leq C_{\\mathsf{het}}\\leq\\operatorname*{min}\\{K,1/\\mu_{\\mathsf{avg}}\\}$ , and in particular, $C_{\\mathsf{het}}=1$ when $\\mu_{b}^{k}=\\mu_{b}$ are all equal.\n\nSimilar to the synchronous setting, we describe a federated asynchronous Q-learning algorithm, called FedAsynQ (see Algorithm 2), that learns the optimal Q-function by periodically averaging the local Q-estimates with the aid of a central server. See Figure 1 for an illustration. Inheriting the notation of $Q_{t}^{k}$ and $V_{t}^{k}$ from the synchronous setting (cf. (11)), FedAsynQ proceeds as follows in the rest of the t-th iteration.\n\n1. Local updates: Each agent k samples a transition $(s_{t-1}^{k},a_{t-1}^{k},r_{t-1}^{k},s_{t}^{k})$ from its Markovian trajectory generated by the behavior policy $\\pi_{\\mathfrak{b}}^{k}$ according to (15) and updates a single\n\nFigure 1: Federated asynchronous Q-learning with K agents and a parameter server. Each agent k performs $\\tau$ local updates on its local Q-table along a Markovian trajectory induced by behavior policy $\\pi_{\\mathfrak{b}}^{k}$ and sends the Q-table to the server. The server averages and synchronizes the local Q-tables every τ iterations. For importance averaging, the agents additionally send the number of visits over all the state-action pairs within each synchronization period, which is not pictured.\n\nentry of its local Q-estimate $Q_{t-1}^{k}$\n\n$$\nQ_{t-\\frac{1}{2}}^{k}(s,a)=\\left\\{\\begin{array}{l l}(1-\\eta)Q_{t-1}^{k}(s,a)+\\eta\\big(r_{t-1}^{k}+\\gamma V_{t-1}^{k}(s_{t}^{k})\\big)&\\text{if}(s,a)=(s_{t-1}^{k},a_{t-1}^{k})\\\\ Q_{t-1}^{k}(s,a),&\\text{otherwise}\\end{array},\\right.\\tag{23}\n$$\n\nwhere η denotes the learning rate.\n\n2. Periodic averaging: The intermediate local estimates will be averaged every $\\tau$ iterations, where $\\tau \\geq 1$ is the synchronization period. Here, we consider a more general weighted averaging scheme, where the updated estimate $Q_{t}^{k}$ is:\n\n$$\n\\forall(s,a)\\in\\mathcal{S}\\times\\mathcal{A}:\\qquad Q_{t}^{k}(s,a)=\\left\\{\\begin{array}{l l}\\sum_{k=1}^{K}\\alpha_{t}^{k}(s,a)Q_{t-\\frac{1}{2}}^{k}(s,a)&\\text{if} t\\equiv 0(\\bmod\\tau)\\\\ Q_{t-\\frac{1}{2}}^{k}(s,a)&\\text{otherwise}\\end{array},\\right.\\tag{24}\n$$\n\nwhere $\\alpha_{t}^{k}=[\\alpha_{t}^{k}(s,a)]_{s\\in{\\cal S},a\\in{\\cal A}}\\in [0,1]^{|{\\cal S}| |{\\cal A}|}$ is an entry-wise weight assigned to agent k such that\n\n$$\n\\forall(s,a)\\in\\mathcal{S}\\times\\mathcal{A}:\\quad\\sum_{k=1}^{K}\\alpha_{t}^{k}(s,a)=1.\n$$\n\nAfter a total of T iterations, for all $(s,a)\\in\\mathcal{S}\\times\\mathcal{A}$ , FedAsynQ outputs a global Q-estimate $\\begin{array}{ r }{Q_{T}(s,a)=\\sum_{k=1}^{K}\\alpha_{T}^{k}(s,a)Q_{T}^{k}(s,a)}\\end{array}$ . In the subsections below, we provide two possible ways (equal and importance weighting) to choose $\\alpha_{t}^{k}$ and their corresponding sample complexity analyses.\n\nAlgorithm 2: Federated Asynchronous Q-learning (FedAsynQ)\n\n<div class=\"mineru-algorithm\" style=\"white-space: pre-wrap; font-family:monospace;\">\ninputs: learning rate $\\{\\eta\\}$, discount factor $\\gamma$, number of agents $K$, synchronization period $\\tau$, total number of iterations $T$.\ninitialization: $Q_0^k = Q_0$ for all $k \\in [K]$.\nfor $t = 1, \\cdots, T$ do\n    for $k \\in [K]$ do\n        Draw action $a_{t-1}^k \\sim \\pi_b^k(s_{t-1}^k)$, observe reward $r_{t-1}^k = r(s_{t-1}^k, a_{t-1}^k)$, and draw next state $s_t^k \\sim P(\\cdot | s_{t-1}^k, a_{t-1}^k)$.\n        Compute $Q_{t-\\frac{1}{2}}^k$ according to (23).\n        Compute $Q_t^k$ according to (24).\n    end for\nend for\nreturn: $Q_T(s, a) = \\sum_{k=1}^{K} \\alpha_T^k(s, a) Q_T^k(s, a)$, for all $(s, a) \\in \\mathcal{S} \\times \\mathcal{A}$.\n</div>\n\nWe begin with the most natural choice, which equally weights the local Q-estimates, that is,\n\n$$\n\\alpha_{t}^{k}(s,a)=\\frac{1}{K}.\\tag{25}\n$$\n\nWe call the resulting scheme FedAsynQ-EqAvg, which is also analyzed in Khodadadian et al. (2022). We have the following improved performance guarantee in the next theorem.\n\nTheorem 2 (Finite-time convergence of FedAsynQ-EqAvg) Consider any given $\\delta\\in(0,1)$ and $\\varepsilon\\in(0,\\frac{1}{1-\\gamma}]$ . Suppose that the initialization of FedAsynQ-EqAvg satisfies $\\begin{array}{ r }{0\\leq Q_{0}\\leq\\frac{1}{1-\\gamma}.}\\end{array}$ There exist some suficiently large constant $c_{T}>0$ and suficiently small constant $c_{\\eta}>0$ both independent of problem parameters, such that with probability at least $1-\\delta_{\\mathrm{i}}$ , the output of FedAsynQ-EqAvg satisfies $\\|Q_{T}-Q^{\\star}\\|_{\\infty}\\leq\\varepsilon$ , provided that the synchronization period $\\tau,$ the sample size per agent T, and the learning rate η satisfy\n\n$$\n\\tau_{0}\\leq\\tau\\leq\\frac{1}{4\\eta}\\min\\left\\{\\frac{1-\\gamma}{4},\\frac{1}{K}\\right\\},\\tag{26a}\n$$\n\n$$\nT\\geq c_{T}\\left(\\frac{C_{\\mathrm{het}}}{K\\mu_{\\mathrm{avg}}(1-\\gamma)^{5}\\varepsilon^{2}}+T_{0}\\right)(\\log((1-\\gamma)^{2}\\varepsilon))^{2}\\log(TK)\\log\\frac{|\\mathcal{S}| |\\mathcal{A}|T^{2}K}{\\delta},\\quad(26\\mathrm{b})\n$$\n\n$$\n\\eta=c_{\\eta}\\min\\left\\{\\frac{K(1-\\gamma)^{4}\\varepsilon^{2}}{C_{\\mathrm{het}}},\\eta_{0}\\right\\}\\frac{1}{\\log(T K)\\log\\frac{|\\mathcal{S}| |\\mathcal{A}|T^{2}K}{\\delta}},\\tag{26c}\n$$\n\nwhere $\\begin{array}{ r }{\\tau_{0}=\\frac{2176t_{\\mathrm{mix}}^{\\mathrm{max}}}{\\mu_{\\mathrm{avg}}}}\\end{array}$ log 8K log <sup>4|S||A|T2</sup><sub>δ</sub> $\\begin{array}{ r }{T_{0}=\\frac{1}{\\mu_{\\mathsf{avg}}(1-\\gamma)\\eta_{0}}}\\end{array}$ , and $\\begin{array}{ r }{\\eta_{0}=\\frac{\\mu_{\\mathrm{avg}}\\operatorname*{min}\\left\\{1-\\gamma,K^{-1}\\right\\}}{t_{\\mathrm{mix}}^{m a\\times}}}\\end{array}$ , independent $o f \\varepsilon .$\n\nLet the matrix $P\\in\\mathbb{R}^{|S| |\\mathcal{A}|\\times|\\mathcal{A}|}$ represent the transition kernel of the underlying MDP, where $P(s,a)=P(\\cdot|s,a)$ is the probability vector corresponding to the state transition at the state-action pair $(s,a)$ . For any vector $V\\in\\mathbb{R}^{|S|}$ , we define the variance parameter $\\mathsf{Var}_{s,a}(V)$ with respect to the probability vector $P(s,a)$ as\n\n$$\n\\mathsf{Var}_{s,a}(V):=\\mathbb{E}_{s^{\\prime}\\sim P(\\cdot|s,a)}\\left[V(s^{\\prime})-P(s,a)V\\right]^{2}=P(s,a)(V\\circ V)-[P(s,a)V]\\circ [P(s,a)V].\\tag{29}\n$$\n\nHere, denotes the Hadamard product such that $a\\circ b=[a_{i}b_{i}]_{i=1}^{n}$ for any vector $a =$ $[a_{i}]_{i=1}^{n},b=[b_{i}]_{i=1}^{n}\\in\\mathbb{R}^{n}$ . With slight abuse of notation, we shall also assume $V^{\\star}\\in\\mathbb{R}^{|{\\cal S}|}$ $V_{t}^{k}\\bar{\\in}\\mathbb{R}^{|S|},~Q^{\\star}~\\bar{\\in}~\\mathbb{R}^{|S| |\\mathcal{A}|}$ 2 $Q_{t}^{k}\\ \\in\\ \\mathbb{R}^{|S| |A|},\\ Q_{t+\\frac{1}{2}}^{k}\\ \\in\\ \\mathbb{R}^{|S| |A|}$ and $r~\\in~\\mathbb{R}^{|S| |A|}$ represent the corresponding functions in the matrix/vector form.\n\nWe first state a few basic facts that hold both for the synchronous and the asynchronous settings. It is easy to establish, by induction, that all iterates satisfy for all $1 \\leq k \\leq K$ and $t \\geq 0$ that\n\n$$\n0\\leq Q_{t}^{k}\\leq\\frac{1}{1-\\gamma},\\quad 0\\leq V_{t}^{k}\\leq\\frac{1}{1-\\gamma},\\tag{30}\n$$\n\nas long as $\\textstyle 0\\leq Q_{0}=Q_{0}^{k}\\leq{\\frac{1}{1-\\gamma}}$ ; see a similar argument, $\\mathrm{e . g .}$ , in Li et al. (2023, Lemma 4). In addition, observe that\n\n$$\n\\|V_{t}^{k}-V^{\\star}\\|_{\\infty}\\leq\\|Q_{t}^{k}-Q^{\\star}\\|_{\\infty}\\tag{31}\n$$\n\nsince\n\n$$\n\\|V_{t}^{k}-V^{\\star}\\|_{\\infty}=\\max_{s\\in\\mathcal{S}}\\Big|\\max_{a\\in\\mathcal{A}}Q_{t}^{k}(s,a)-\\max_{a\\in\\mathcal{A}}Q^{\\star}(s,a)\\Big|\\leq\\max_{s\\in\\mathcal{S},a\\in\\mathcal{A}}\\big|Q_{t}^{k}(s,a)-Q^{\\star}(s,a)\\big|\\leq\\|Q_{t}^{k}-Q^{\\star}\\|_{\\infty}.\n$$\n\nLetting $Q_{t}$ be the average of the local Q-estimates at the end of the t-th iteration, i.e., $\\begin{array}{ r }{Q_{t}=\\frac{1}{K}\\dot{\\sum_{k=1}^{K}Q_{t}^{k}}}\\end{array}$ , it follows from (13) and (24) that for all $t \\geq 0$ that\n\n$$\nQ_{t}=\\frac{1}{K}\\sum_{k=1}^{K}Q_{t}^{k}=\\frac{1}{K}\\sum_{k=1}^{K}Q_{t-\\frac{1}{2}}^{k}.\\tag{32}\n$$\n\nDenote the error between $Q_{t}$ and $Q^{\\star}$ by\n\n$$\n\\Delta_{t}=Q^{\\star}-Q_{t},\n$$\n\nwhich is the quantity we aim to control. From (30), it holds immediately that for all $t \\geq 0$\n\n$$\n\\|\\Delta_{t}\\|_{\\infty}\\leq\\frac{1}{1-\\gamma}.\\tag{33}\n$$\n\nNext, we also introduce the following functions pertaining to periodic averaging. For any $t,$\n\ndefine $\\begin{array}{ r }{\\iota(t): =\\tau\\lfloor\\frac{t}{\\tau}\\rfloor}\\end{array}$ as the most recent synchronization step until $t ;$  \ndefine $\\begin{array}{ r }{\\phi(t): =\\lfloor\\frac{t}{\\tau}\\rfloor}\\end{array}$ as the number of synchronization steps until t.\n\nDefine the local empirical transition matrix at the t-th iteration $P_{t}^{k}\\in\\{0,1\\}^{|S| |A|\\times|S|}$ as\n\n$$\nP_{t}^{k}((s,a),s^{\\prime}):=\\left\\{\\begin{array}{l l}1,&\\text{if} s^{\\prime}=s_{t}^{k}(s,a)\\\\ 0,&\\text{otherwise}\\end{array},\\right.\\tag{34}\n$$\n\nthen the local update rule (12) can be rewritten as\n\n$$\nQ_{t-\\frac{1}{2}}^{k}=(1-\\eta)Q_{t-1}^{k}+\\eta\\left(r+\\gamma P_{t}^{k}V_{t-1}^{k}\\right).\\tag{35}\n$$\n\nStep 1: error decomposition. To analyze the error $\\Delta_{t}.$ , we first decompose the error into three terms, each of which can be bounded in a simple form. From (32), it follows that\n\n$$\n\\begin{array}{l}\\Delta_{t}=\\frac{1}{K}\\sum_{k=1}^{K}\\left(Q^{\\star}-Q_{t-\\frac{1}{2}}^{k}\\right)\\stackrel{{\\text{(i)}}}{{=}}\\frac{1}{K}\\sum_{k=1}^{K}\\left((1-\\eta)(Q^{\\star}-Q_{t-1}^{k})+\\eta(Q^{\\star}-r-\\gamma P_{t}^{k}V_{t-1}^{k})\\right)\\\\ \\stackrel{\\text{(ii)}}{=}(1-\\eta)\\Delta_{t-1}+\\eta\\frac{\\gamma}{K}\\sum_{k=1}^{K}\\left(P V^{\\star}-P_{t}^{k}V_{t-1}^{k}\\right)\\\\ =(1-\\eta)\\Delta_{t-1}+\\eta\\frac{\\gamma}{K}\\sum_{k=1}^{K}\\bigl(P-P_{t}^{k}\\bigr)V_{t-1}^{k}+\\eta\\frac{\\gamma}{K}\\sum_{k=1}^{K}P\\bigl(V^{\\star}-V_{t-1}^{k}\\bigr),\\\\ \\end{array}\n$$\n\nwhere (i) follows from (35), and (ii) follows from Bellman’s optimality equation $Q^{\\star}=$ $r+\\gamma P V^{\\star}$ . By recursion over the above relation, we obtain\n\n$$\n\\Delta_{t}=\\underbrace{(1-\\eta)^{t}\\Delta_{0}}_{=:E_{t}^{1}}+\\underbrace{\\eta\\frac{\\gamma}{K}\\sum_{i=1}^{t}(1-\\eta)^{t-i}\\sum_{k=1}^{K}\\left(P-P_{i}^{k}\\right)V_{i-1}^{k}}_{=:E_{t}^{2}}+\\underbrace{\\eta\\frac{\\gamma}{K}\\sum_{i=1}^{t}(1-\\eta)^{t-i}\\sum_{k=1}^{K}P\\left(V^{\\star}-V_{i-1}^{k}\\right)}_{=:E_{t}^{3}}.\\tag{36}\n$$\n\nFor simplicity, we introduce the following notation. Let $\\mathcal{U}_{v_{1},v_{2}}^{k}(s,a)$ represent a set of iteration indices between $[v_{1},v_{2})$ for some $0\\leq v_{1}\\leq v_{2}\\leq T$ where agent k visits $(s,a)$ i.e.,\n\n$$\n\\mathcal{U}_{v_{1},v_{2}}^{k}(s,a):=\\left\\{u\\in [v_{1},v_{2}):(s_{u}^{k},a_{u}^{k})=(s,a)\\right\\},\n$$\n\nand $N_{v_{1},v_{2}}^{k}(s,a)$ denotes the number of visits of agent k on $(s,a)$ during iterations between $[v_{1},v_{2}),{\\mathrm{i . e .}}$\n\n$$\nN_{v_{1},v_{2}}^{k}(s,a)= |\\mathcal{U}_{v_{1},v_{2}}^{k}(s,a)|.\n$$\n\nDefine the local empirical transition matrix at the t-th iteration $P_{t}^{k}\\in\\{0,1\\}^{|S| |A|\\times|S|}$ as\n\n$$\nP_{t}^{k}\\left(\\left(s,a\\right),s^{\\prime}\\right):=\\left\\{\\begin{array}{l l}1&\\text{if}\\left(s,a,s^{\\prime}\\right)=\\left(s_{t-1}^{k},a_{t-1}^{k},s_{t}^{k}\\right)\\\\ 0&\\text{otherwise}\\end{array}\\quad .\\right.\\tag{46}\n$$\n\nThen the local update rule (23) can be rewritten as\n\n$$\nQ_{t-\\frac{1}{2}}^{k}(s,a)=\\left\\{\\begin{array}{l l}(1-\\eta)Q_{t-1}^{k}(s,a)+\\eta(r_{t-1}^{k}+\\gamma P_{t}^{k}(s,a)V_{t-1}^{k})&\\text{if}(s,a)=(s_{t-1}^{k},a_{t-1}^{k})\\\\ Q_{t-1}^{k}(s,a),&\\text{otherwise}\\end{array}.\\right.\\tag{47}\n$$\n\nStep 1: error decomposition. Consider any $0 \\leq t \\leq T$ such that t 0 (mod τ), i.e., t is a synchronization step. To analyze $\\Delta_{t}$ , we first decompose the error for each $(s,a)\\in\\mathcal S\\times\\mathcal A$ as follows:\n\n$$\n\\begin{array}{l}\\Delta_{t}(s,a)=\\frac{1}{K}\\sum_{k=1}^{K}(Q^{\\star}(s,a)-Q_{t-\\frac{1}{2}}^{k}(s,a))\\\\ =\\left(\\frac{1}{K}\\sum_{k=1}^{K}(1-\\eta)^{N_{t-\\tau,t}^{k}(s,a)}\\right)\\Delta_{t-\\tau}(s,a)\\\\ +\\frac{\\gamma}{K}\\sum_{k=1}^{K}\\sum_{u\\in\\mathcal{U}_{t-\\tau,t}^{k}(s,a)}\\eta(1-\\eta)^{N_{u+1,t}^{k}(s,a)}(P(s,a)-P_{u+1}^{k}(s,a))V_{u}^{k}\\\\ +\\frac{\\gamma}{K}\\sum_{k=1}^{K}\\sum_{u\\in\\mathcal{U}_{t-\\tau,t}^{k}(s,a)}\\eta(1-\\eta)^{N_{u+1,t}^{k}(s,a)}P(s,a)(V^{\\star}-V_{u}^{k}),\\tag{48}\\\\ \\end{array}\n$$\n\nwhere we invoke the following recursive relation of the local error at iteration u such that $(s_{u-1},a_{u-1})=(s,a)$ :\n\n$$\n\\begin{array}{l}Q^{\\star}(s,a)-Q_{u-\\frac{1}{2}}^{k}(s,a)\\\\ =(1-\\eta)(Q^{\\star}(s,a)-Q_{u-1}^{k}(s,a))+\\eta(Q^{\\star}(s,a)-r_{u-1}^{k}-\\gamma P_{u}^{k}(s,a)V_{u-1}^{k})\\\\ =(1-\\eta)(Q^{\\star}(s,a)-Q_{u-1}^{k}(s,a))+\\eta(\\gamma P(s,a)V^{\\star}-\\gamma P_{u}^{k}(s,a)V_{u-1}^{k})\\\\ =(1-\\eta)(Q^{\\star}(s,a)-Q_{u-1}^{k}(s,a))+\\gamma\\eta(P(s,a)-P_{u}^{k}(s,a))V_{u-1}^{k}+\\gamma P(s,a)(V^{\\star}-V_{u-1}^{k})49)\\\\ \\end{array}\n$$\n\nHere, the second equality follows from Bellman’s optimality equation. Denoting\n\n$$\n\\lambda_{v_{1},v_{2}}(s,a):=\\frac{1}{K}\\sum_{k=1}^{K}(1-\\eta)^{N_{v_{1},v_{2}}^{k}(s,a)}\\tag{50}\n$$\n\nfor any integer $0\\leq v_{1}\\leq v_{2}\\leq T$ , we apply recursion to the relation (48) over the synchronization periods, and obtain\n\n$$\n\\begin{array}{l}\\Delta_{t}(s,a)=\\left(\\prod_{h=0}^{\\phi(t)-1}\\lambda_{h\\tau,(h+1)\\tau}(s,a)\\right)\\Delta_{0}(s,a)\\\\ +\\frac{\\gamma}{K}\\sum_{h=0}^{\\phi(t)-1}\\left(\\prod_{l=(h+1)}^{\\phi(t)-1}\\lambda_{l\\tau,(l+1)\\tau}(s,a)\\right)\\\\ \\times\\sum_{k=1}^{K}\\sum_{u\\in\\mathcal{U}_{h\\tau,(h+1)\\tau}^{k}(s,a)}\\eta(1-\\eta)^{N_{u+1,(h+1)\\tau}^{k}(s,a)}(P(s,a)-P_{u+1}^{k}(s,a))V_{u}^{k}\\\\ +\\frac{\\gamma}{K}\\sum_{h=0}^{\\phi(t)-1}\\left(\\prod_{l=(h+1)}^{\\phi(t)-1}\\lambda_{l\\tau,(l+1)\\tau}(s,a)\\right)\\\\ \\times\\sum_{k=1}^{K}\\sum_{u\\in\\mathcal{U}_{h\\tau,(h+1)\\tau}^{k}(s,a)}\\eta(1-\\eta)^{N_{u+1,(h+1)\\tau}^{k}(s,a)}P(s,a)(V^{\\star}-V_{u}^{k})\\\\ =\\underbrace{\\omega_{0,t}(s,a)\\Delta_{0}(s,a)}_{=:E_{t}^{1}(s,a)}+\\underbrace{\\gamma\\sum_{k=1}^{K}\\sum_{u\\in\\mathcal{U}_{0,t}^{k}(s,a)}\\omega_{u,t}^{k}(s,a)(P(s,a)-P_{u+1}^{k}(s,a))V_{u}^{k}}_{=:E_{t}^{2}(s,a)}\\\\ +\\underbrace{\\gamma\\sum_{k=1}^{K}\\sum_{u\\in\\mathcal{U}_{0,t}^{k}(s,a)}\\omega_{u,t}^{k}(s,a)P(s,a)(V^{\\star}-V_{u}^{k})}_{=:E_{t}^{3}(s,a)},\\tag{51}\\\\ \\end{array}\n$$\n\nwhich is decomposed in a similar manner as (36). Here, we define\n\n$$\n\\omega_{0,t}(s,a):=\\prod_{h=0}^{\\phi(t)-1}\\lambda_{h\\tau,(h+1)\\tau}(s,a),\\tag{52a}\n$$\n\n$$\n\\omega_{u,t}^{k}(s,a):=\\frac{1}{K}\\eta(1-\\eta)^{N_{u+1,(\\phi(u)+1)\\tau}^{k}(s,a)}\\prod_{l=\\phi(u)+1}^{\\phi(t)-1}\\lambda_{l\\tau,(l+1)\\tau}(s,a).\\tag{52b}\n$$\n\nWe record the following useful lemma whose proof is provided in Appendix C.2.\n\nLemma 6 Consider integers $v_{1}$ and $v_{2}$ such that $0\\leq v_{1}\\leq v_{2}\\leq t\\leq T$ , where $t\\equiv 0\\left(mod\\tau\\right)$ and a state-action pair $(s,a)\\in\\mathcal{S}\\times\\mathcal{A}$ . Suppose that $\\eta \\tau \\leq 1$ . The parameters defined in (52) satisfy\n\n$$\n\\lambda_{v_{1},v_{2}}(s,a)\\leq\\exp\\left(-\\frac{\\eta}{2 K}\\sum_{k=1}^{K}N_{v_{1},v_{2}}^{k}(s,a)\\right),\\tag{53a}\n$$\n\n$$\n\\omega_{0,t}(s,a)+\\sum_{k=1}^{K}\\sum_{u\\in\\mathcal{U}_{0,t}^{k}(s,a)}\\omega_{u,t}^{k}(s,a)=1,\\tag{53b}\n$$\n\n$$\n\\sum_{k=1}^{K}\\sum_{u\\in\\mathcal{U}_{0,h^{\\prime}\\tau}^{k}(s,a)}\\omega_{u,t}^{k}(s,a)\\leq\\exp\\left(-\\frac{\\eta}{2K}\\sum_{k=1}^{K}N_{h^{\\prime}\\tau,t}^{k}(s,a)\\right),\\quad\\forall 0\\leq h^{\\prime}\\leq\\phi(t),(53c)\n$$\n\n$$\n\\sum_{k=1}^{K}\\sum_{u\\in\\mathcal{U}_{0,t}^{k}(s,a)}(\\omega_{u,t}^{k}(s,a))^{2}\\leq\\frac{2\\eta}{K}.\\tag{53d}\n$$\n\nIn addition, we provide the concentration bound of the total number of visits of multiple agents agents with independent uniformly ergodic Markov chains, whose proof is provided in Appendix C.1. Denote\n\n$$\nt_{\\mathrm{th}}(s,a):=\\frac{2176t_{\\mathrm{mix}}^{\\mathrm{max}}\\log 8K\\log\\frac{4|\\mathcal{S}| |\\mathcal{A}|T^{2}}{\\delta}}{\\mu_{\\mathrm{avg}}(s,a)}\\text{and} t_{\\mathrm{th}}:=\\frac{2176t_{\\mathrm{mix}}^{\\mathrm{max}}\\log 8K\\log\\frac{4|\\mathcal{S}| |\\mathcal{A}|T^{2}}{\\delta}}{\\mu_{\\mathrm{avg}}}.\\tag{78}\n$$\n\nHere, $\\begin{array}{ r }{\\mu_{\\mathsf{avg}}(s,a): =\\frac{1}{K}\\sum_{k=1}^{K}\\mu_{\\mathsf{b}}^{k}(s,a)}\\end{array}$ is the average behavior policy over all agents.\n\nLemma 15 Consider any $\\delta\\in(0,1)$ . Under the asynchronous sampling, for any $(s,a)\\in$ $\\mathcal{S}\\times\\mathcal{A}$ and $0 \\leq u < v \\leq T$ such that $v-u\\geq t_{\\mathsf{th}}(s,a)$ , the following holds :\n\n$$\n\\frac{1}{4}(v-u)K\\mu_{\\mathrm{avg}}(s,a)\\leq\\sum_{k=1}^{K}N_{u,v}^{k}(s,a)\\leq 2(v-u)K\\mu_{\\mathrm{avg}}(s,a)\\tag{79}\n$$\n\nwith probability at least $1-{\\frac{\\delta}{|S| |{\\mathcal{A}}|T^{2}}}$\n\nNext, we have\n\n$$\n\\begin{array}{l}W_{t}(s,a):=\\sum_{i=1}^{t}\\sum_{k=1}^{K}\\mathbb{E}\\big[(z_{i}^{k}(s,a))^{2}|V_{i-1}^{K},\\ldots,V_{i-1}^{1},\\ldots,V_{0}^{K},\\ldots,V_{0}^{1}\\big]\\\\ =\\sum_{i=1}^{t}\\sum_{k=1}^{K}\\mathsf{Var}\\big(z_{i}^{k}(s,a)|V_{i-1}^{K},\\ldots,V_{i-1}^{1},\\ldots,V_{0}^{K},\\ldots,V_{0}^{1}\\big)\\\\ =\\sum_{i=1}^{t}\\sum_{k=1}^{K}\\eta^{2}(1-\\eta)^{2(t-i)}\\mathsf{Var}_{s,a}(V_{i-1}^{k})\\\\ \\leq\\frac{2K}{(1-\\gamma)^{2}}\\sum_{i=1}^{t}\\eta^{2}(1-\\eta)^{2(t-i)}\\leq\\frac{2\\eta K}{(1-\\gamma)^{2}}:=\\sigma^{2},\\end{array}\\tag{84}\n$$\n\nwhere we recall the definition of $\\mathsf{Var}_{s,a}$ in (29). Here, the first inequality holds since\n\n$$\n\\mathsf{Var}_{s,a}(V_{i-1}^{k})\\leq\\|P(s,a)\\|_{1}(\\|V_{i-1}^{k}\\|_{\\infty})^{2}+(\\|P(s,a)\\|_{1}\\|V_{i-1}^{k}\\|_{\\infty})^{2}\\leq\\frac{2}{(1-\\gamma)^{2}}\n$$\n\nand the last inequality follows from\n\n$$\n\\sum_{i=1}^{t}\\eta^{2}(1-\\eta)^{2(t-i)}\\leq\\frac{\\eta^{2}(1-(1-\\eta)^{2 t})}{1-(1-\\eta)^{2}}\\leq\\eta .\\tag{85}\n$$\n\nFor any $(s,a)\\in\\mathcal{S}\\times\\mathcal{A}$ , $k\\in[K]$ , and $0\\leq i<T$ , by the definition (90), it follows that\n\n$$\n\\left|d_{\\iota(i),i}^{k}(s,a)\\right|=\\left|\\sum_{j=\\iota(i)}^{i-1}d_{j,j+1}^{k}(s,a)\\right|\\leq\\underbrace{2\\eta\\sum_{j=\\iota(i)}^{i-1}\\|\\Delta_{j}^{k}\\|_{\\infty}}_{:=B_{1}}+\\underbrace{\\gamma\\eta\\left|\\sum_{j=\\iota(i)}^{i-1}(P_{j+1}^{k}(s,a)-P(s,a))V^{\\star}\\right|}_{:=B_{2}},\\tag{92}\n$$\n\nwhere\n\n$$\n\\Delta_{j}^{k}=Q^{\\star}-Q_{j}^{k}.\\tag{93}\n$$\n\nBounding $B_{2}$ . Using the fact that the empirical transitions are independent and centered on the true transition probability, by invoking Hoefding’s inequality and the union bound, we can claim that the following holds for all $(s,a,k,t)\\in\\mathcal{S}\\times\\mathcal{A}\\times [K]\\times [T]$ ，\n\n$$\n\\begin{array}{l}\\gamma\\eta\\left|\\sum_{j=\\iota(i)}^{i-1}(P_{j+1}^{k}(s,a)-P(s,a))V^{\\star}\\right|\\leq\\frac{\\gamma\\eta}{1-\\gamma}\\sqrt{\\frac{1}{2}\\sum_{j=\\iota(i)}^{i-1}\\log\\frac{|\\mathcal{S}| |\\mathcal{A}|K T}{\\delta}}\\\\ \\leq\\frac{\\gamma\\eta}{1-\\gamma}\\sqrt{(\\tau-1)\\log\\frac{|\\mathcal{S}| |\\mathcal{A}|K T}{\\delta}}\\tag{97}\\\\ \\end{array}\n$$\n\nwith probability at least $1 - \\delta$ for any given $\\delta\\in(0,1)$ , where $\\tau$ is the synchronization period.", "theorems": [{"local_id": "T1", "statement": "Lemma 8 Let β be any integer that satisfies $0<\\beta\\leq\\phi(T)$ . For any given $\\delta\\in(0,1)$ ), the following holds\n\n$$\n\\begin{array}{l}|E_{t}^{3}(s,a)|\\leq\\frac{2\\gamma}{1-\\gamma}\\exp\\left(-\\frac{\\eta\\mu_{\\mathrm{avg}}\\beta\\tau}{8}\\right)+\\frac{8\\gamma\\eta\\sqrt{\\tau-1}}{1-\\gamma}\\sqrt{\\log\\frac{2|\\mathcal{S}| |\\mathcal{A}|T K}{\\delta}}\\\\ +\\frac{1+\\gamma}{2}\\max_{\\phi(t)-\\beta\\leq h\\leq\\phi(t)-1}\\|\\Delta_{h\\tau}\\|_{\\infty},\\tag{57}\\\\ \\end{array}\n$$\n\nfor all $\\beta \\tau \\leq t \\leq T$ with probability at least $1-\\delta_{i}$ , as long as $\\beta\\tau\\geq t_{\\mathrm{th}}$ and $\\eta \\le$ min $\\textstyle{\\left\\{\\frac{1-\\gamma}{4\\gamma\\tau},\\frac{1}{2\\tau}\\right\\}}$", "proof": {"steps": [{"n": 1, "claim": "For any $t \\ge \\beta \\tau$ , the error term can be decomposed as follows:\n\n$$\n\\begin{array}{l}E_{t}^{3}(s,a)=\\gamma\\sum_{k=1}^{K}\\sum_{u\\in\\mathcal{U}_{0,t}^{k}(s,a)}\\omega_{u,t}^{k}(s,a)P(s,a)(V^{\\star}-V_{u}^{k})\\\\ =\\underbrace{\\gamma\\sum_{k=1}^{K}\\sum_{u\\in\\mathcal{U}_{0,(\\phi(t)-\\beta)\\tau}^{k}(s,a)}\\omega_{u,t}^{k}(s,a)P(s,a)(V^{\\star}-V_{u}^{k})}_{=:E_{t}^{3 a}(s,a)}\\\\ +\\underbrace{\\gamma\\sum_{k=1}^{K}\\sum_{u\\in\\mathcal{U}_{(\\phi(t)-\\beta)\\tau,t}^{k}(s,a)}\\omega_{u,t}^{k}(s,a)P(s,a)(V^{\\star}-V_{u}^{k})}_{=:E_{t}^{3 b}(s,a)}.\\tag{154}\\\\ \\end{array}\n$$\n\nWe shall these two terms separately."}, {"n": 2, "claim": "Bounding $E_{t}^{3 a}(s,a)$ . First, the bound on $E_{t}^{3 a}(s,a)$ is derived as follows:\n\n$$\n\\begin{array}{l}|E_{t}^{3a}(s,a)|\\leq\\gamma\\sum_{k=1}^{K}\\sum_{u\\in\\mathcal{U}_{0,(\\phi(t)-\\beta)\\tau}^{k}(s,a)}\\omega_{u,t}^{k}(s,a)\\|P(s,a)\\|_{1}\\|(V^{\\star}-V_{u}^{k})\\|_{\\infty}\\\\ \\stackrel{\\text{(i)}}{\\leq}\\frac{2\\gamma}{1-\\gamma}\\sum_{k=1}^{K}\\sum_{u\\in\\mathcal{U}_{0,(\\phi(t)-\\beta)\\tau}^{k}(s,a)}\\omega_{u,t}^{k}(s,a)\\\\ \\stackrel{\\text{(ii)}}{\\leq}\\frac{2\\gamma}{1-\\gamma}\\exp\\left(-\\frac{\\eta}{2 K}\\sum_{k=1}^{K}N_{(\\phi(t)-\\beta)\\tau,t}^{k}(s,a)\\right)\\\\ \\stackrel{\\text{(iii)}}{\\leq}\\frac{2\\gamma}{1-\\gamma}\\exp\\left(-\\frac{\\eta\\mu_{\\mathrm{avg}}\\beta\\tau}{8}\\right),\\tag{155}\\\\ \\end{array}\n$$\n\nwhere (i) holds because $\\begin{array}{ r }{\\|V_{u}^{k}\\|_{\\infty},\\|V^{\\star}\\|_{\\infty}\\leq\\frac{1}{1-\\gamma}}\\end{array}$ (cf. (30)) and $\\|P(s,a)\\|_{1}\\leq 1$ , (ii) holds due to (53c) in Lemma 6, and (iii) follows from the fact that $\\begin{array}{ r }{\\sum_{k=1}^{K}N_{(\\phi(t)-\\beta)\\tau,t}^{k}(s,a)\\geq}\\end{array}$ $\\frac{K\\mu_{\\mathsf{avg}}\\beta\\tau}{4}$ according to Lemma 15 as long as $\\beta\\tau\\geq t_{\\mathrm{th}}$"}, {"n": 3, "claim": "Bounding $E_{t}^{3 b}(s,a)$ .\n\nNext, we bound $E_{t}^{3 b}(s,a)$ as follows:\n\n$$\n\\begin{array}{l}|E_{t}^{3b}(s,a)|\\leq\\gamma\\sum_{k=1}^{K}\\sum_{u\\in\\mathcal{U}_{(\\phi(t)-\\beta)\\tau,t}^{k}(s,a)}\\omega_{u,t}^{k}(s,a)\\left\\|V^{\\star}-V_{u}^{k}\\right\\|_{\\infty}\\\\ \\stackrel{\\text{(i)}}{\\leq}\\gamma\\sum_{k=1}^{K}\\sum_{h=\\phi(t)-\\beta}^{\\phi(t)-1}\\sum_{u\\in\\mathcal{U}_{h\\tau,(h+1)\\tau}^{k}(s,a)}\\omega_{u,t}^{k}(s,a)(\\|\\Delta_{h\\tau}\\|_{\\infty}+\\|Q_{u}^{k}-Q_{h\\tau}^{k}\\|_{\\infty})\\\\ \\stackrel{\\text{(ii)}}{\\leq}\\gamma\\sum_{k=1}^{K}\\sum_{h=\\phi(t)-\\beta}^{\\phi(t)-1}\\sum_{u\\in\\mathcal{U}_{h\\tau,(h+1)\\tau}^{k}(s,a)}\\omega_{u,t}^{k}(s,a)((1+2\\eta\\tau)\\|\\Delta_{h\\tau}\\|_{\\infty}+\\sigma_{\\mathrm{loca}})156)\\\\ \\end{array}\n$$\n\nwhere (i) follows from the following bound, which will be shown in Appendix C.4.1,\n\n$$\n\\|V^{\\star}-V_{u}^{k}\\|_{\\infty}\\leq\\|\\Delta_{\\iota(u)}^{k}\\|_{\\infty}+\\|Q_{u}^{k}-Q_{\\iota(u)}^{k}\\|_{\\infty},\\tag{157}\n$$\n\nand (ii) holds due to the following lemma."}, {"n": 4, "claim": "Lemma 19 Assume $\\eta\\tau\\leq\\frac{1}{2}$ . For any given $\\delta\\in(0,1)$ , the following holds for any $k \\in [ K ]$ and $0 \\leq u < T$ :\n\n$$\n\\|Q_{u}^{k}-Q_{\\iota(u)}^{k}\\|_{\\infty}\\leq 2\\eta\\tau\\|\\Delta_{\\iota(u)}^{k}\\|_{\\infty}+\\frac{8\\gamma\\eta\\sqrt{\\tau-1}}{1-\\gamma}\\sqrt{\\log\\frac{2|\\mathcal{S}| |\\mathcal{A}|T K}{\\delta}}\\tag{158}\n$$\n\nwith probability at least $1 - \\delta$\n\nHere, for notation simplicity, we denote $\\begin{array}{ r }{\\sigma_{\\mathsf{local}}: =\\frac{8\\gamma\\eta\\sqrt{\\tau-1}}{1-\\gamma}\\sqrt{\\log\\frac{2|\\mathcal{S}| |\\mathcal{A}|T K}{\\delta}}}\\end{array}$\n\nThen, with some algebraic calculations, we can obtain the bound on $E_{t}^{3 b}(s,a)$ as follows:\n\n$$\n\\begin{array}{l}|E_{t}^{3 b}(s,a)|\\stackrel{\\mathrm{(i)}}{\\leq}\\sigma_{\\mathrm{local}}+\\gamma\\sum_{h=\\phi(t)-\\beta}^{\\phi(t)-1}(1+2\\eta\\tau)\\|\\Delta_{h\\tau}\\|_{\\infty}\\sum_{k=1}^{K}\\sum_{u\\in\\mathcal{U}_{h\\tau,(h+1)\\tau}^{k}(s,a)}\\omega_{u,t}^{k}(s,a)\\\\ \\stackrel{\\text{(ii)}}{\\leq}\\sigma_{\\text{local}}+\\frac{1+\\gamma}{2}\\max_{\\phi(t)-\\beta\\leq h<\\phi(t)}\\|\\Delta_{h\\tau}\\|_{\\infty}\\sum_{k=1}^{K}\\sum_{h=\\phi(t)-\\beta}^{\\phi(t)-1}\\sum_{u\\in\\mathcal{U}_{h\\tau,(h+1)\\tau}^{k}(s,a)}\\omega_{u,t}^{k}(s,a)\\\\ \\stackrel{\\text{(iii)}}{\\leq}\\sigma_{\\text{local}}+\\frac{1+\\gamma}{2}\\max_{\\phi(t)-\\beta\\leq h<\\phi(t)}\\|\\Delta_{h\\tau}\\|_{\\infty},\\tag{159}\\\\ \\end{array}\n$$\n\nwhere (i) holds according to (53b) of Lemma 6, (ii) holds when $\\eta$ is small enough that $\\begin{array}{ r }{\\eta\\le\\frac{1-\\gamma}{4\\gamma\\tau}}\\end{array}$ , and (iii) follows from (53b) of Lemma 6."}, {"n": 5, "claim": "Now we have the bounds of $E_{t}^{3 a}(s,a)$ and $E_{t}^{3 b}(s,a)$ separately obtained above. By combining the bounds in (154), we can claim the advertised bound, which completes the proof."}]}}, {"local_id": "T2", "statement": "Lemma 16 Assume $\\begin{array}{ r }{\\tau\\eta\\leq\\frac{1}{2}}\\end{array}$ . For any given $\\delta\\in(0,1)$ , the following bound holds for any $1 \\leq i \\leq T$ and $k \\in [ K ]$\n\n$$\n\\|\\Delta_{i}^{k}\\|_{\\infty}\\leq\\|\\Delta_{\\iota(i)}\\|_{\\infty}+\\frac{2}{1-\\gamma}\\sqrt{\\eta\\log\\frac{|\\mathcal{S}| |\\mathcal{A}|K T}{\\delta}}\\tag{95}\n$$\n\nwith at least probability $1-\\delta,$ where $\\iota(i)$ is the most recent synchronization step until $i .$", "proof": {"steps": [{"n": 1, "claim": "By applying the decomposition in (36) to the local error for agent $k,$ we decompose $\\Delta_{i}^{k}$ as follows:\n\n$$\n\\begin{array}{l}\\Delta_{i}^{k}(s,a)=\\underbrace{(1-\\eta)^{i-\\iota(i)}\\Delta_{\\iota(i)}^{k}(s,a)}_{:=D_{1}}+\\underbrace{\\gamma\\sum_{j=\\iota(i)+1}^{i}\\eta(1-\\eta)^{i-j}(P(s,a)-P_{j}^{k}(s,a))V^{\\star}}_{:=D_{2}}\\\\ +\\underbrace{\\gamma\\sum_{j=\\iota(i)+1}^{i}\\eta(1-\\eta)^{i-j}P_{j}^{k}(s,a)(V^{\\star}-V_{j-1}^{k})}_{:=D_{3}}.\\tag{103}\\\\ \\end{array}\n$$\n\nWe shall bound each term separately."}, {"n": 2, "claim": "Bounding $D_{1}$ . Since $\\Delta_{\\iota(i)}^{k}=\\Delta_{\\iota(i)}$ for every agent k at the synchronization step $\\iota(i)$\n\n$$\n|D_{1}|\\leq(1-\\eta)^{i-\\iota(i)}\\|\\Delta_{\\iota(i)}\\|_{\\infty}.\\tag{104}\n$$"}, {"n": 3, "claim": "Bounding $D_{2}$ . In a similar manner to (97), by invoking Hoefding inequality and using the fact that $\\begin{array}{ r }{\\sum_{i=\\iota(i)+1}^{i}(\\eta(1-\\eta)^{i-j})^{2}\\leq\\eta}\\end{array}$ (cf. (85)), we can claim that the following holds for all $(\\overleftarrow{s},\\overrightarrow{a},\\overrightarrow{k},t)\\in\\mathcal{S}\\times\\mathcal{A}\\times [K]\\times [T]$\n\n$$\n\\left|D_{2}\\right|\\leq\\gamma\\sqrt{\\sum_{j=\\iota(i)+1}^{i}\\left(\\eta(1-\\eta)^{i-j}\\right)^{2}\\|V^{\\star}\\|_{\\infty}^{2}\\log\\frac{|\\mathcal{S}| |\\mathcal{A}|K T}{\\delta}}\\leq\\frac{\\gamma}{1-\\gamma}\\sqrt{\\eta\\log\\frac{|\\mathcal{S}| |\\mathcal{A}|K T}{\\delta}}\\tag{105}\n$$\n\nwith probability at least $1 - \\delta$ for any given $\\delta\\in(0,1)$\n\nBounding $D_{3}$ . By bounding $\\|V^{\\star}-V_{j-1}^{k}\\|_{\\infty}$ with the local error $\\|\\Delta_{j-1}^{k}\\|_{\\infty}$ (cf. (31)) and using $\\|P_{j}^{k}(s,a)\\|_{1}\\leq 1$ , we have\n\n$$\n\\left|D_{3}\\right|\\leq\\gamma\\sum_{j=\\iota(i)+1}^{i}\\eta(1-\\eta)^{i-j}\\|P_{j}^{k}(s,a)\\|_{1}\\|V^{\\star}-V_{j-1}^{k}\\|_{\\infty}\\leq\\gamma\\sum_{j=\\iota(i)+1}^{i}\\eta(1-\\eta)^{i-j}\\|\\Delta_{j-1}^{k}\\|_{\\infty}.\\tag{106}\n$$"}, {"n": 4, "claim": "By combining the bounds obtained above in (103), we obtain the following recursive relation\n\n$$\n\\|\\Delta_{i}^{k}\\|_{\\infty}\\leq(1-\\eta)^{i-\\iota(i)}\\|\\Delta_{\\iota(i)}\\|_{\\infty}+\\underbrace{\\frac{\\gamma}{1-\\gamma}\\sqrt{\\eta\\log\\frac{|\\mathcal{S}| |\\mathcal{A}|K T}{\\delta}}}_{:=\\rho}+\\gamma\\sum_{j=\\iota(i)+1}^{i}\\eta(1-\\eta)^{i-j}\\|\\Delta_{j-1}^{k}\\|_{\\infty}.\\tag{107}\n$$"}, {"n": 5, "claim": "By invoking the recursive relation with some algebraic calculations, we obtain the following bound\n\n$$\n\\begin{array}{l}\\|\\Delta_{i}^{k}\\|_{\\infty}\\\\ \\leq(1-\\eta)^{i-\\iota(i)}\\|\\Delta_{\\iota(i)}\\|_{\\infty}+\\rho\\\\ +\\gamma\\sum_{j_{1}=\\iota(i)+1}^{i}\\eta(1-\\eta)^{i-j_{1}}\\left((1-\\eta)^{j_{1}-1-\\iota(i)}\\|\\Delta_{\\iota(i)}\\|_{\\infty}+\\rho+\\gamma\\sum_{j_{2}=\\iota(i)+1}^{j_{1}-1}\\eta(1-\\eta)^{j_{1}-1-j_{2}}\\|\\Delta_{j_{2}-1}^{k}\\|_{\\infty}\\right)\\\\ =\\left((1-\\eta)^{i-\\iota(i)}+\\gamma\\sum_{j_{1}=\\iota(i)+1}^{i}\\eta(1-\\eta)^{i-1-\\iota(i)}\\right)\\|\\Delta_{\\iota(i)}\\|_{\\infty}+\\left(1+\\gamma\\sum_{j_{1}=\\iota(i)+1}^{i}\\eta(1-\\eta)^{i-j_{1}}\\right)\\rho\\\\ +\\gamma^{2}\\sum_{j_{1}=\\iota(i)+1}^{i}\\sum_{j_{2}=\\iota(i)+1}^{j_{1}-1}\\eta^{2}(1-\\eta)^{i-1-j_{2}}\\|\\Delta_{j_{2}-1}^{k}\\|_{\\infty}\\\\ \\leq\\left((1-\\eta)^{i-\\iota(i)}+\\gamma\\sum_{j_{1}=\\iota(i)+1}^{i}\\eta(1-\\eta)^{i-1-\\iota(i)}\\right)\\|\\Delta_{\\iota(i)}\\|_{\\infty}+\\left(1+\\gamma\\sum_{j_{1}=\\iota(i)+1}^{i}\\eta(1-\\eta)^{i-j_{1}}\\right)\\rho\\\\ +\\gamma^{2}\\sum_{j_{1}=\\iota(i)+1}^{i}\\sum_{j_{2}=\\iota(i)+1}^{j_{1}-1}\\eta^{2}(1-\\eta)^{i-1-j_{2}}\\left((1-\\eta)^{j_{2}-1-\\iota(i)}\\|\\Delta_{\\iota(i)}\\|_{\\infty}+\\rho+\\dots\\right)\\\\ \\leq\\|\\Delta_{\\iota(i)}\\|_{\\infty}\\\\ \\times\\left((1-\\eta)^{i-\\iota(i)}+\\gamma\\sum_{j_{1}=\\iota(i)+1}^{i}\\eta(1-\\eta)^{i-1-\\iota(i)}+\\dots+\\gamma^{l}\\sum_{j_{1}=\\iota(i)+1}^{i}\\dots\\sum_{j_{l}=\\iota(i)+1}^{j_{l-1}-1}\\eta^{l}(1-\\eta)^{i-l-\\iota(i)}\\right)\\\\ \\left.+\\left(1+\\gamma\\sum_{j_{1}=\\iota(i)+1}^{i}\\eta(1-\\eta)^{i-j_{1}}+\\dots\\gamma^{l}\\sum_{j_{1}=\\iota(i)+1}^{i}\\dots\\sum_{j_{l}=\\iota(i)+1}^{j_{l-1}-1}\\eta^{l}(1-\\eta)^{i-l+1-j_{l}}\\right)\\rho\\right.\\\\ +\\gamma^{l+1}\\sum_{j_{1}=\\iota(i)+1}^{i}\\dots\\sum_{j_{l+1}=\\iota(i)+1}^{j_{l}-1}\\eta^{l+1}(1-\\eta)^{i-l-j_{l+1}}\\left(\\|\\Delta_{j_{l+1}-1}^{k}\\|\\right)\\\\ \\stackrel{\\text{(i)}}{\\leq}\\sum_{l=0}^{i-\\iota(i)}\\gamma^{l}\\binom{i-\\iota(i)}{l}\\eta^{l}(1-\\eta)^{i-\\iota(i)-l}\\|\\Delta_{\\iota(i)}^{k}\\|_{\\infty}+\\sum_{l=0}^{i-\\iota(i)-1}\\gamma^{l}\\binom{i-\\iota(i)}{l}\\eta^{l}\\rho\\\\ \\leq((1-\\eta)+\\gamma\\eta)^{i-\\iota(i)}\\|\\Delta_{\\iota(i)}^{k}\\|_{\\infty}+(1+\\gamma\\eta)^{i-\\iota(i)}\\rho\\\\ \\stackrel{\\text{(ii)}}{\\leq}\\|\\Delta_{\\iota(i)}^{k}\\|_{\\infty}+2\\rho,\\tag{108}\\\\ \\end{array}\n$$\n\n$$\n\\begin{array}{l}\\times\\left((1-\\eta)^{i-\\iota(i)}+\\gamma\\sum_{j_{1}=\\iota(i)+1}^{i}\\eta(1-\\eta)^{i-1-\\iota(i)}+\\dots+\\gamma^{l}\\sum_{j_{1}=\\iota(i)+1}^{i}\\dots\\sum_{j_{l}=\\iota(i)+1}^{j_{l-1}-1}\\eta^{l}(1-\\eta)^{i-l-\\iota(i)}\\right)\\\\ \\left.+\\left(1+\\gamma\\sum_{j_{1}=\\iota(i)+1}^{i}\\eta(1-\\eta)^{i-j_{1}}+\\dots\\gamma^{l}\\sum_{j_{1}=\\iota(i)+1}^{i}\\dots\\sum_{j_{l}=\\iota(i)+1}^{j_{l-1}-1}\\eta^{l}(1-\\eta)^{i-l+1-j_{l}}\\right)\\rho\\right.\\\\ +\\gamma^{l+1}\\sum_{j_{1}=\\iota(i)+1}^{i}\\dots\\sum_{j_{l+1}=\\iota(i)+1}^{j_{l}-1}\\eta^{l+1}(1-\\eta)^{i-l-j_{l+1}}\\left(\\|\\Delta_{j_{l+1}-1}^{k}\\|\\right)\\\\ \\end{array}\n$$\n\nwhere (i) follows from $\\Delta_{j_{i-\\iota(i)}-1}^{k}=\\Delta_{\\iota(i)}^{k}$ since $j_{l}\\leq i-l+1$ 2\n\n$$\n\\sum_{j_{1}=\\iota(i)+1}^{i}\\sum_{j_{2}=\\iota(i)+1}^{j_{1}-1}\\dots\\sum_{j_{l}=\\iota(i)+1}^{j_{l-1}-1}\\eta^{l}(1-\\eta)^{i-l-\\iota(i)}=\\binom{i-\\iota(i)}{l}\\eta^{l}(1-\\eta)^{i-l-\\iota(i)},\n$$\n\n$$\n\\sum_{j_{1}=\\iota(i)+1}^{i}\\dots\\sum_{j_{l}=\\iota(i)+1}^{j_{l-1}-1}\\eta^{l}(1-\\eta)^{i-l+1-j_{l}}\\leq\\sum_{j_{1}=\\iota(i)+1}^{i}\\dots\\sum_{j_{l}=\\iota(i)+1}^{j_{l-1}-1}\\eta^{l}\\leq\\binom{i-\\iota(i)}{l}\\eta^{l},\n$$\n\nand (ii) follows from $(1+\\gamma\\eta)^{i-\\iota(i)}\\leq(1+\\gamma\\eta)^{\\tau}\\leq e^{\\tau\\eta}\\leq 2$ since $i-\\iota(i)\\leq\\tau$ and $\\begin{array}{ r }{\\tau\\eta\\leq\\frac{1}{2}}\\end{array}$ This completes the proof."}]}}, {"local_id": "T3", "statement": "Lemma 20 Assume $\\eta\\tau\\leq\\frac{1}{2}$ . For any given $\\delta\\in(0,1)$ , the following holds for any $k \\in [ K ]$ and $0 \\leq u < T$ :\n\n$$\n\\|\\Delta_{u}^{k}\\|_{\\infty}\\leq\\|\\Delta_{\\iota(u)}^{k}\\|_{\\infty}+\\frac{2\\gamma}{1-\\gamma}\\sqrt{\\eta\\log\\frac{|\\mathcal{S}| |\\mathcal{A}|T K}{\\delta}}\\tag{166}\n$$\n\nwith probability at least $1 - \\delta$", "proof": {"steps": [{"n": 1, "claim": "For each state-action $(s,a)\\in\\mathcal{S}\\times\\mathcal{A}$ and agent $k,$ by invoking the recursive relation (49) derived from the local Q-learning update in (23), $\\Delta_{u}^{k}$ is decomposed as follows:\n\n$$\n\\begin{array}{l}\\Delta_{u}^{k}(s,a)=\\underbrace{(1-\\eta)^{N_{\\iota(u),u}^{k}(s,a)}\\Delta_{\\iota(u)}^{k}(s,a)}_{=:D_{1}}+\\underbrace{\\gamma\\sum_{v\\in\\mathcal{U}_{\\iota(u),u}^{k}(s,a)}\\eta(1-\\eta)^{N_{v+1,u}^{k}(s,a)}(P(s,a)-P_{v+1}^{k}(s,a))V^{\\star}}_{=:D_{2}}\\\\ +\\underbrace{\\gamma\\sum_{v\\in\\mathcal{U}_{\\iota(u),u}^{k}(s,a)}\\eta\\left(1-\\eta\\right)^{N_{v+1,u}^{k}(s,a)}P_{v+1}^{k}(s,a)\\left(V^{\\star}-V_{v}^{k}\\right)}_{=:D_{3}}.\\tag{169}\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "Now, we obtain the bound on the three decomposed terms separately."}, {"n": 3, "claim": "Bounding $D_{1}$ . The term $D_{1}$ can be bounded by\n\n$$\n|D_{1}|\\leq(1-\\eta)^{N_{\\iota(u),u}^{k}(s,a)}\\|\\Delta_{\\iota(u)}^{k}\\|_{\\infty}.\\tag{170}\n$$"}, {"n": 4, "claim": "Bounding $D_{2}$ . By applying the Hoefding bound using the independence of transitions, the second term is bounded as follows:\n\n$$\n|D_{2}|\\leq\\gamma\\sqrt{\\sum_{v\\in\\mathcal{U}_{\\iota(u),u}^{k}(s,a)}(\\eta(1-\\eta)^{N_{v+1,u}^{k}(s,a)})^{2}(\\|V^{\\star}\\|_{\\infty})^{2}\\log\\frac{|\\mathcal{S}| |\\mathcal{A}|T K}{\\delta}}\n$$\n\n$$\n\\leq\\frac{\\gamma}{1-\\gamma}\\sqrt{\\eta\\log\\frac{|\\mathcal{S}| |\\mathcal{A}|T K}{\\delta}}:=\\rho \\tag{171}\n$$\n\nwith probability at least $1-\\delta,$ where the last inequality holds due to the fact that $\\begin{array}{ r }{\\|V^{\\star}\\|_{\\infty}\\leq\\frac{1}{1-\\gamma}}\\end{array}$ and\n\n$$\n\\sum_{v\\in\\mathcal{U}_{\\iota(u),u}^{k}(s,a)}(\\eta(1-\\eta)^{N_{v+1,u}^{k}(s,a)})^{2}\\leq\\eta^{2}(1+(1-\\eta)^{2}+(1-\\eta)^{4}+\\dots)\\leq\\eta .\n$$"}, {"n": 5, "claim": "See (Li et al., 2021b, Lemma 1) for the detailed explanation of the bound."}, {"n": 6, "claim": "Bounding $D_{3}$ . Lastly, we bound the third term as follows:\n\n$$\n\\begin{array}{l}|D_{3}|\\leq\\gamma\\sum_{v\\in\\mathcal{U}_{\\iota(u),u}^{k}(s,a)}\\eta(1-\\eta)^{N_{v+1,u}^{k}(s,a)}\\|P_{v+1}^{k}(s,a)\\|_{1}\\|V^{\\star}-V_{v}^{k}\\|_{\\infty}\\\\ \\leq\\gamma\\sum_{v\\in\\mathcal{U}_{\\iota(u),u}^{k}(s,a)}\\eta(1-\\eta)^{N_{v+1,u}^{k}(s,a)}\\|\\Delta_{v}^{k}\\|_{\\infty},\\tag{172}\\\\ \\end{array}\n$$\n\nwhere the last inequality follows from the fact that $\\|P_{v+1}^{k}(s,a)\\|_{1}=1$ and\n\n$$\nQ_{v}^{k}(s,a^{\\star}(s))-Q^{\\star}(s,a^{\\star}(s))\\leq V_{v}^{k}(s)-V^{\\star}(s)\\leq Q_{v}^{k}(s,a_{v}^{k}(s))-Q^{\\star}(s,a_{v}^{k}(s))\n$$\n\nfor any $s \\in S$ , where we denote $\\begin{array}{ r }{a^{\\star}(s)=\\arg\\operatorname*{max}_{a}Q^{\\star}(s,a),\\ a_{v}^{k}(s)=\\arg\\operatorname*{max}_{a}Q_{v}^{k}(s,a)}\\end{array}$\n\nBy combining the bounds of the above three terms, we obtain the following recursive relation:\n\n$$\n\\left|\\Delta_{u}^{k}(s,a)\\right|\\leq(1-\\eta)^{N_{\\iota(u),u}^{k}(s,a)}\\|\\Delta_{\\iota(u)}^{k}\\|_{\\infty}+\\rho+\\gamma\\sum_{v\\in\\mathcal{U}_{\\iota(u),u}^{k}(s,a)}\\eta(1-\\eta)^{N_{v+1,u}^{k}(s,a)}\\|\\Delta_{v}^{k}\\|_{\\infty}.\\tag{173}\n$$"}, {"n": 7, "claim": "Using the recursive relation, we will prove that the following claim holds for any $0 \\leq m < \\tau$ by induction:\n\n$$\n\\|\\Delta_{\\iota(u)+m}^{k}\\|_{\\infty}\\leq\\|\\Delta_{\\iota(u)}^{k}\\|_{\\infty}+2\\rho,\\tag{174}\n$$\n\nwhich completes the proof of Lemma 20. First, if $m = 0$ , the claim is obviously true. Suppose the claim holds for $\\iota(u),\\iota(u)+1,\\cdot\\cdot\\cdot,\\iota(u)+m-1$ . Then, for $u=\\iota(u)+m$ , by invoking the recursive relation (173), we can show that the claim (174) holds for m as follows:\n\n$$\n\\begin{array}{l}|\\Delta_{\\iota(u)+m}^{k}(s,a)|\\\\ \\leq(1-\\eta)^{N_{\\iota(u),u}^{k}(s,a)}\\|\\Delta_{\\iota(u)}^{k}\\|_{\\infty}+\\rho+\\gamma\\sum_{v\\in\\mathcal{U}_{\\iota(u),u}^{k}(s,a)}\\eta(1-\\eta)^{N_{v+1,u}^{k}(s,a)}(\\|\\Delta_{\\iota(u)}^{k}\\|_{\\infty}+2\\rho)\\\\ =((1-\\eta)^{N_{\\iota(u),u}^{k}(s,a)}+\\gamma\\sum_{v\\in\\mathcal{U}_{\\iota(u),u}^{k}(s,a)}\\eta(1-\\eta)^{N_{v+1,u}^{k}(s,a)})\\|\\Delta_{\\iota(u)}^{k}\\|_{\\infty}\\\\ \\end{array}\n$$\n\n$$\n\\begin{array}{l}+(1+2\\gamma\\sum_{v\\in\\mathcal{U}_{\\iota(u),u}^{k}(s,a)}\\eta(1-\\eta)^{N_{v+1,u}^{k}(s,a)})\\rho\\\\ =((1-\\eta)^{N_{\\iota(u),u}^{k}(s,a)}+\\gamma(1-(1-\\eta)^{N_{\\iota(u),u}^{k}(s,a)})\\|\\Delta_{\\iota(u)}^{k}\\|_{\\infty}+(1+2\\gamma(1-(1-\\eta)^{N_{\\iota(u),u}^{k}(s,a)}))\\rho\\\\ \\leq\\|\\Delta_{\\iota(u)}^{k}\\|_{\\infty}+2\\rho,\\tag{175}\\\\ \\end{array}\n$$\n\nwhere the last inequality holds since\n\n$$\n(1-\\eta)^{N_{\\iota(u),u}^{k}(s,a)}\\geq(1-\\eta)^{\\tau}\\geq(\\frac{1}{4})^{\\eta\\tau}\\geq\\frac{1}{2}\n$$\n\nprovided that $\\begin{array}{ r }{\\eta\\tau\\leq\\frac{1}{2}}\\end{array}$"}]}}], "target_local_id": null, "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2025-25-0183", "context": "Throughout the paper we limit the discussion to estimation of $\\mathcal{M}\\in\\mathcal{C}^{k}$ , k-times smooth, compact submanifolds of $\\mathbb{R}^{D}$ without boundary. This limitation is important for the analysis, but the algorithm we present is local and thus from a practical perspective, a local version of these assumptions should sufice. The smoothness requirement is also crucial for the algorithm. However, it is shown in Kouˇrimsk´a et al. (2024) that a non-smooth manifold can be approximated arbitrarily well by a smooth manifold without a significant change to the reach. This implies that the method presented in this paper is useful for non-smooth manifolds as well. A key concept in the analysis of manifold estimates is the reach of a manifold (e.g., see the analyses at Aamari and Levrard (2018, 2019); Feferman et al. (2018); Niyogi et al. (2008)), which was introduced by Federer in Federer (1959) and is defined as the maximal distance for which there exists a unique projection onto M.\n\nDefinition 1 (Reach Federer (1959)) The reach of a subset A $o f\\mathbb{R}^{D}$ , is the largest τ (possibly ∞) such that for any $\\boldsymbol{x}\\in\\mathbb{R}^{D}$ that maintains dis $t(A,x)\\leq\\tau$ , there exists a unique point $P r o j_{A}(x)\\in A$ , nearest to x.\n\nOur noise model in the analysis is as follows: We assume that we are given a finite set of samples $\\{r_{i}\\}_{i=1}^{n}$ drawn independently from\n\n$$\n\\mathcal{M}_{\\sigma}\\triangleq\\{q\\in\\mathbb{R}^{D}\\mid d i s t(q,\\mathcal{M})<\\sigma\\},\\tag{1}\n$$\n\na tubular neighborhood of M. Explicitly, we assume that $r_{i}\\sim\\mathrm{Unif}(\\mathcal{M}_{\\sigma})$ , which is the uniform distribution on $\\mathcal{M}_{\\sigma};$ i.e., the normalized Lebesgue measure with respect to $\\mathbb{R}^{D}$\n\nFinally, let $p\\in\\mathcal{M}$ , we wish to describe $W_{p}\\subset{\\mathcal{M}}$ a neighborhood of p as a graph of a function\n\n$$\n\\varphi_{p}:W_{T_{p}\\mathcal{M}}\\to\\mathcal{M}\\tag{2}\n$$\n\nwhere $W_{T_{p},M}=P r o j_{T_{p},M}(W_{p})$ , the projection of $W_{p}$ onto the tangent, and $\\varphi_{p}$ is defined by\n\n$$\n\\varphi_{p}(x)=p+(x,\\phi_{p}(x))_{T_{p}\\mathcal{M}},\\tag{3}\n$$\n\nwhere $\\boldsymbol{x}\\in\\mathbb{R}^{d}\\simeq T_{p}\\boldsymbol{\\mathcal{M}},\\phi_{p}(\\boldsymbol{x})\\in\\mathbb{R}^{D-d}\\simeq T_{p}\\boldsymbol{\\mathcal{M}}^{\\perp}$ , and $(x,y)_{T_{p}\\mathcal{M}}\\in\\mathbb{R}^{D}$ denotes that $\\boldsymbol{x}\\in\\mathbb{R}^{d}$ and $\\boldsymbol{y}\\in\\mathbb{R}^{D-d}$ represent a point in some basis of $T_{p}{\\mathcal{M}}$ and $T_{p}\\mathcal{M}^{\\perp}$ correspondingly. Then, we define the graph of $\\phi_{p}$ to be\n\n$$\n\\Gamma_{\\phi_{p},W_{p}}\\triangleq\\{p+(x,\\phi_{p}(x))_{T_{p}\\mathcal{M}}|x\\in W_{T_{p}\\mathcal{M}}\\}.\\tag{4}\n$$\n\nFor simplicity, throughout the paper we identify the graph of $\\phi_{p}$ with $\\Gamma_{\\phi_{p},W_{p}}$ . That is, we refer to M as locally a graph of $\\phi_{p}$ (see Figure 1).\n\nWe would like to stress that throughout the paper there is a slight misuse of notation. Explicitly, when we refer to $T_{p}{\\mathcal{M}}$ and $T_{p}{\\mathcal{M}}^{\\perp}$ , we sometimes look at it as elements in the Grassmannian $Gr(d,D)$ and $Gr(D-d,D)$ ; i.e., subspaces of $\\mathbb{R}^{D}$ with dimensions d and $D - d$ correspondingly. On the other hand, in some other occasions (as in (4)) we neglect the fact that these are subsets of ${\\mathbb R}^{D}$ which is equivalent to choosing some basis and working in it.\n\nThroughout this paper we assume that the (unknown) manifold M and the samples $\\{r_{i}\\}_{i=1}^{n}$ satisfy the following requirements:\n\n1. $M\\in{\\mathcal{C}}^{k}$ is a compact d-dimensional sub-manifold of $\\mathbb{R}^{D}$ without boundary  \n2. $\\begin{array}{ r }{M=\\frac{\\tau}{\\sigma}}\\end{array}$ is large enough, where τ is the reach of M and $\\sigma > 0$ is the noise level.  \n3. $\\{r_{i}\\}_{i=1}^{n}$ are samples drawn independently and uniformly from $\\mathcal{M}_{\\sigma}\\left(\\mathrm{i . e .,}r_{i}\\sim\\mathrm{Unif}(\\mathcal{M}_{\\sigma})\\right)$\n\nGiven $(\\boldsymbol{q},H)\\in\\mathbb{R}^{D}\\times G r(\\boldsymbol{d},D)$ we define the following minimization scheme, known as local polynomial regression $\\left(\\mathrm{e . g .}\\right.$ , Cleveland (1979); McLain (1976)): Find $\\pi\\in\\Pi_{k-1}^{d\\mapsto D}$ a polynomial of total degree $deg(\\pi)\\leq k-1$ from $\\mathbb{R}^{d}$ to R $_{\\cdot}^{D-d}$ which minimizes\n\n$$\nJ_{2}(\\pi\\mid q,H)=\\frac{1}{N_{q,H}}\\sum_{r_{i}\\in U_{\\mathrm{ROI}}^{n}}\\|r_{i}-(x_{i},\\pi(x_{i}))_{H}\\|^{2},\\tag{8}\n$$\n\nwhere $x_{i}\\in\\mathbb{R}^{d}$ are the projections of $r_{i}-q$ onto $H,(x,y)_{H}\\in\\mathbb{R}^{d}\\times\\mathbb{R}^{D-d}$ are coeficients in a basis of $H\\times H^{\\perp},U_{\\mathrm{ROI}}^{n}(q,H)$ is defined through a bandwidth $\\epsilon_{n}$\n\n$$\nU_{\\mathrm{ROI}}^{n}(q,H)=\\left\\{r_{i}\\in U_{\\mathrm{ROI}}\\mid\\|x_{i}\\|<\\epsilon_{n}\\right\\},\\tag{9}\n$$\n\nand $N_{q,H}$ denotes the number of samples in $U_{\\mathrm{ROI}}^{n}(q,H)$ . Explicitly, the local polynomial regression is defined through\n\n$$\n\\pi_{q,H}^{*}=\\underset{\\pi\\in\\Pi_{k-1}^{d\\mapsto D}}{\\operatorname{argmin}}J_{2}(\\pi\\mid q,H).\\tag{10}\n$$\n\nAs required to ensure convergence in probability for local polynomial regression Aizenbud and Sober (2021a); Stone (1980), we demand that the bandwidth $\\epsilon_{n}\\to 0$ as n → ∞ is such that\n\n$$\n0<\\lim_{n\\to\\infty}n^{1/(2 k+d)}\\cdot\\epsilon_{n}<\\infty .\\tag{11}\n$$\n\nUnfortunately, the minimization problem stated in Equation (10) could not be used as-is, since we need an explicit relation between the failure probability, the error bound, and the number of samples. The analysis of convergence of local polynomial regression of vector-valued functions in Theorem 3.2 of Aizenbud and Sober (2021a), uses a variant of local polynomial regression, namely some sort of “median trick” Alon et al. (1999) on (10)\n\nto unveil such an explicit connection. For simplicity of notations, we abuse the definition of $\\pi_{q,H}^{*}$ in (10), and define\n\n$$\n\\pi_{q,H}^{*}=\\text{Algorithm 2 of Aizenbud and Sober (2021a)}(x_{i},y_{i}),\\tag{12}\n$$\n\nwhere $y_{i}$ are the projection of $r_{i}-q$ onto $H^{\\perp}$ . Any derivative of $\\pi_{q,H}^{*}$ can also be estimated by means of Algorithm 2 of Aizenbud and Sober (2021a). For simplicity of presentation, throughout the paper when we write $\\partial\\pi_{q,H}^{*}$ or $\\mathcal{D}_{\\pi_{q_{\\ell},H_{\\ell}}^{*}}$ what we actually mean is the estimate of the derivative rather than the derivative of $\\pi_{q,H}^{*}.$\n\nWe begin with $q^{*}$ (a coarse approximation of a point on the manifold) and $H_{0}=H^{*}$ (the initial tangent estimate) resulting from Algorithm 1. Then, we update iteratively the directions of H with respect to a tangent of M. Explicitly, in iteration $\\ell,$ we define $H_{\\ell+1}$ to be the subspace coinciding with Im $(\\mathcal{D}_{\\pi_{r,H_{\\ell}}^{*}}[0])$ the image of the diferential of $(\\mathrm{I}_{d},\\pi_{r,H_{\\ell}}^{*})$ $\\mathbb{R}^{d}\\mathbb{R}^{D}$ at 0 (i.e., the tangent to the graph of $\\pi_{r,H_{\\ell}}^{*})$ . In other words, we look at the manifold as a local graph of a function\n\n$$\nf_{\\ell}:\\mathbb{R}^{d}\\simeq H_{\\ell}\\to\\mathbb{R}^{D-d}\\simeq H_{\\ell}^{\\perp}.\\tag{13}\n$$\n\nThat is, we define a local patch of the manifold through the graph\n\n$$\n\\Gamma_{f_{\\ell},r,H_{\\ell}}=\\{r+(x,f_{\\ell}(x))_{H_{\\ell}}|x\\in B_{H_{\\ell}}(0,\\rho)\\}\\subset\\mathcal{M},\\tag{14}\n$$\n\nwhere $\\rho$ is some radius where this function is defined (see Lemma 17 for more details regarding the existence of such $\\rho>0)$ . Then, we estimate the first order diferential of $f_{\\ell}$ through taking the diferential of the local polynomial regression estimate $\\pi_{r,H_{\\ell}}^{*}$ . The image of the diferential determines d-directions in $\\mathbb{R}^{D}$ (i.e., an element in the Grassmannian $Gr(d,D))$ , by which we define $H_{\\ell+1}$ . Following this, we define\n\n$$\nf_{\\ell+1}:(r,H_{\\ell+1})\\rightarrow H_{\\ell+1}^{\\perp}\\simeq\\mathbb{R}^{D-d}\\tag{15}\n$$\n\nTheorem 2 Let $(q^{*}(r),H^{*}(r))$ , the output of Algorithm 1, and let $p=P r o j_{\\mathcal{M}}(q^{*})$ . Denote $\\alpha=\\sqrt{C_{M}/M}$ for some constant $C_{M}$ (independent of α and $M)$ . Then, for any $\\delta > 0$ arbitrarily small, there exists $N_{\\delta}$ such that for all $n>N_{\\delta}$\n\n$$\n\\angle_{\\max}(H^{*},T_{p}\\mathcal{M})\\leq\\alpha\n$$\n\nwith probability of at least $1 - \\delta$ . Furthermore, we have\n\n$$\n\\|p-q^{*}\\|\\leq 3\\sigma\n$$\n\nTheorem 3 Assume that $M>C_{\\tau}{\\sqrt{D\\log D}}$ . Denote $\\mathbf{p}=P r o j_{\\mathcal{M}}(r)$ where r is the input point for Algorithm 2. Let $(q,H)$ be an initial coordinate system for which $\\| q - p \\| \\leq 3 \\sigma$ and $\\angle_{\\mathrm{max}}(H,T_{p}\\mathcal{M})\\leq\\sqrt{c_{M}/M}$ , where $p=P r o j_{\\mathcal{M}}(q)\\in\\mathcal{M}$ . For any $\\delta > 0$ arbitrarily small, denote by $\\hat{p}_{n},\\mathcal{T}_{\\hat{p}_{n}}\\mathcal{\\hat{M}}$ the estimates derived from Algorithm 2 initialized with $(q,H)$ with the number of iterations κ specified in Lemma 27. Then, there is $N_{\\delta}$ such that for all $n>N_{\\delta}$ we have\n\n$$\nd i s t(\\hat{p}_{n},\\mathcal{M})\\leq\\frac{C_{1}\\ln\\left(\\frac{1}{\\delta}\\right)}{n^{r_{0}}}=\\mathcal{O}(n^{-r_{0}}),\\tag{17}\n$$\n\n$$\n\\|\\hat{p}_{n}-\\mathbf{p}\\|\\leq C_{2}\\ln\\left(\\frac{1}{\\delta}\\right)\\left(\\frac{n}{\\ln(\\ln(n))^{2}}\\right)^{-r_{1}}=\\widetilde{\\mathcal{O}}\\left(n^{-r_{1}}\\right),\\tag{18}\n$$\n\nand\n\n$$\n\\angle_{\\max}(\\widehat{T_{\\hat{p}_{n}}\\mathcal{M}},T_{\\mathbf{p}}\\mathcal{M})\\leq C_{3}\\ln\\left(\\frac{1}{\\delta}\\right)\\left(\\frac{n}{\\ln(\\ln(n))^{2}}\\right)^{-r_{1}}=\\widetilde{\\mathcal{O}}(n^{-r_{1}})\\tag{19}\n$$\n\nwith probability of at least $1 - \\delta$ , where $\\begin{array}{ r }{r_{0}=\\frac{k}{2 k+d},r_{1}=\\frac{k-1}{2 k+d}}\\end{array}$ , and $C_{1},C_{2},C_{3}$ are constants independent of d, δ.\n\nBefore delving into the details of the proof, we wish to reiterate the steps of Algorithm 2 while introducing some useful notations. According to the assumptions of Theorem 3 we have a local coordinate system $(\\boldsymbol{q},H)\\in\\mathbb{R}^{D}\\times G r(\\boldsymbol{d},D)$ , such that $\\| q - p \\| \\leq 3 \\sigma$ and $\\angle\\operatorname*{max}(T_{p}\\mathcal{M},H)\\le\\alpha .$ , where $p\\in{\\mathcal{M}}$ . As Algorithm 2 involves an iterative process, we denote the initial directions of the tangent estimation as $H_{0}: =H$ . Then, at each iteration we update the local coordinates’ directions $H_{\\ell}$ (for $\\ell=1,\\ldots,\\kappa)$\n\nFigure 3: Road-map for proof of Theorem 3\n\nSimilar to (4) and using the result of Lemma 17, we begin by looking at the manifold patch $\\mathcal{M}\\cap\\mathrm{Cyl}_{H_{0}}(p,c_{\\pi/4}\\tau,\\tau/2)$ as the graph of a function $f_{0}:(r,H_{0})\\simeq\\mathbb{R}^{d}\\mathbb{R}^{D-d};$ i.e.,\n\n$$\n\\Gamma_{f_{0},r,H_{0}}\\triangleq\\{r+(x,f_{0}(x))_{H_{0}}\\mid x\\in B_{H_{0}}(0,c_{\\pi/4}\\tau)\\},\\tag{21}\n$$\n\nwhere $\\mathrm{Cyl}_{H_{0}}(p,c_{\\pi/4}\\tau,\\tau/2)$ is the D-dimensional cylinder with the base $B_{H}(p,c_{\\pi/4}\\tau)\\subset H_{0}$ and height $\\tau / 2$ in any direction on $H_{0}^{\\perp}$ . For the remainder of this section, we assume that $\\alpha=\\sqrt{C_{M}/M}$ is small enough (see the sampling assumptions in Section 2.1). Then, at each iteration of Algorithm 2 we update $H_{\\ell}$ to $H_{\\ell+1}$ through taking the linear space coinciding with the directions of the estimated tangent to $f_{\\ell}$ at zero. Explicitly, given $(r,H_{\\ell})$ , we look at the manifold as $\\Gamma_{f_{\\ell},r,H_{\\ell}}$ the local graph of a function $f_{\\ell}$ and estimate $T_{0}f_{\\ell}\\in G r(d,D)$ the tangent to the graph of $f_{\\ell}$ at 0 through taking the image of $\\mathcal{D}_{\\pi_{r,H_{\\ell}}^{*}}[0]$ , the first order diferential of $\\pi_{r,H_{\\ell}}^{*}$ at zero. In other words, we “rotate” the coordinate system to the point where $H_{\\ell+1}$ aligns with the former tangent estimation to get $f_{\\ell+1}$\n\nHowever, if we want to use the well-known convergence rates of local polynomial regression (i.e., the minimization of (10)), a key assumption in the analysis is that the noise has a zero mean (see Stone (1980)). However, in our case, for any $x\\in H_{\\ell}.$ the samples above x are uniformly distributed in\n\n$$\n\\Omega_{\\ell}(x)=(x+H_{\\ell}^{\\perp})\\cap\\mathcal{M}_{\\sigma},\\tag{22}\n$$\n\nwhere x $\\vdash H_{\\ell}^{\\perp}=\\{x+y\\mid y\\in H_{\\ell}^{\\perp}\\}$ . That is, $\\Omega_{\\ell}(x):H_{\\ell}\\simeq\\mathbb{R}^{d}\\to 2^{\\mathbb{R}^{D-d}}$ . Thus, denoting $\\eta_{\\ell}(x)\\sim\\mathrm{Unif}(\\Omega(x))$ ) and defining\n\n$$\n\\widetilde{f}_{\\ell}(x)=\\mathbb{E}[\\eta_{\\ell}(y|x)]\\neq f_{\\ell},\\tag{23}\n$$\n\nthe result of the regression will estimate $\\widetilde{f}_{\\ell}$ rather than $f_{\\ell}$ itself (see Figure 2). Thus, when we wish to estimate $f_{\\ell}$ or its derivatives using local polynomial regression, we estimate a diferent function, for which the noise has zero mean (see Figure 4).\n\nThe concept of Principal Angles between flats were first introduced by Jordan in 1875 Jordan (1875). Below, we use the definition of Principal Angles between subspaces as described in Bj¨orck and Golub (1973).\n\nDefinition 4 (Principal Angles) Let V be an inner product space. Given two sub-spaces U, W of dimensions dim $(\\mathcal{U})=k,d i m(\\mathcal{W})=l$ , where $k \\leq l$ there exists a sequence of k angles $0\\le\\beta_{1}\\le ...\\le\\beta_{k}\\le\\pi/2$ called the principal angles and their corresponding principal pairs of vectors $(u_{i},w_{i})\\in\\mathcal{U}\\times\\mathcal{W}\\ f o r\\ i=1,\\dots,k$ such that $\\angle(u_{i},w_{i})=\\beta_{i}$ are defined by:\n\n$$\nu_{1},w_{1}\\triangleq\\operatorname*{argmin}_{\\substack{u\\in\\mathcal{U},w\\in\\mathcal{W}\\\\ \\|u\\|=\\|w\\|=1}}\\arccos\\left(|\\langle u,w\\rangle|\\right),\\beta_{1}\\triangleq\\angle(u_{1},w_{1}),\n$$\n\nand $f o r i > 1$\n\n$$\nu_{i},w_{i}\\triangleq\\operatorname*{argmin}_{\\substack{u\\perp\\mathcal{U}_{i-1},w\\perp\\mathcal{W}_{i-1}\\\\ \\|u\\|=\\|w\\|=1}}\\arccos\\left(|\\langle u,w\\rangle|\\right),\\beta_{i}\\triangleq\\angle(u_{i},w_{i}),\n$$\n\nwhere\n\n$$\n\\mathcal{U}_{i}\\triangleq S p a n\\{u_{j}\\}_{j=1}^{i},\\mathcal{W}_{i}\\triangleq S p a n\\{w_{j}\\}_{j=1}^{i}.\n$$\n\nIn other words, given two linear subspaces of $\\mathbb{R}^{D}$ of the same dimension d we can measure the distance between them based upon the principal angles. In our case, we measure the distance between two subspaces by taking the maximal principal angle (maximal angle) , and denote it as\n\n$$\n\\angle_{\\max}(\\mathcal{U},\\mathcal{W})\\triangleq\\max_{1\\leq i\\leq d}\\beta_{i}.\\tag{36}\n$$\n\nLemmas 8 and 9 are reformulation of results proven in Knyazev and Argentati (2007) and will be also used later on.\n\nLemma 8 Let F and G be two linear spaces ofdimension d in $\\mathbb{R}^{D}$ . Assume that $\\angle_{\\mathrm{max}}(F,G)\\leq$ α. Then for any vector $v\\in F^{\\rfloor}$\n\n$$\n\\min_{w\\in G^{\\perp}}\\angle(v,w)\\leq\\alpha\n$$\n\nIt is well known that, locally, a sub-manifold of $\\mathbb{R}^{D}$ can be described as a graph of a function defined from the tangent space to its orthogonal complement. In this section, we deal with expressing a manifold as a local function graph with respect to some tilted coordinate system and bounding the size of the neighborhood for which this definition still holds. The results reported below are general and relate closely to the concept of the Reach (see Definition 1), which was introduced by Federer (1959) and further studied by Boissonat, Lieutier, and Wintraecken Boissonnat et al. (2019).\n\nLemma 17 (M is locally a function graph over a tilted plane) Let M be a d-dimensional sub-manifold of $\\mathbb{R}^{D}$ with reach τ. For any $p\\in\\mathcal{M}$ , let $T_{p}{\\mathcal{M}}$ be the tangent of M at $p .$ . Let $H\\in Gr(d,D)$ , such that $\\angle\\mathrm{max}(H,T_{p}\\mathcal{M})=\\alpha\\leq\\pi/4$ Then $\\mathcal{M}\\cap\\mathrm{Cyl}_{H}(p,\\rho,\\tau/2)$ is locally a function over H, where $\\mathrm{Cyl}_{H}(p,\\rho,\\tau/2)$ is the D-dimensional cylinder with the base\n\n$B_{H}(p,\\rho)\\subset H$ and height $\\tau / 2$ in any direction on $H^{\\perp}$ , for $\\rho=c_{\\pi/4}\\tau$ where $c_{\\pi/4}$ is some general constant.\n\nExplicitly, there exists a function\n\n$$\nf:B_{H}(p,\\rho)\\to H^{\\perp}\n$$\n\nsuch that the graph of f defined as\n\n$$\n\\Gamma_{f}=\\{p+(x,f(x))|x\\in B_{H}(p,\\rho)\\}\n$$\n\nidentifies with $\\mathcal{M}\\cap\\mathrm{Cyl}_{H}(p,\\rho,\\tau/2)$\n\nLemma 19 (Function version of Lemma 13) Let M be a d-dimensional sub-manifold $o f\\mathbb{R}^{D}$ with reach $\\tau .$ . For any $p\\in\\mathcal{M}$ , let $T_{p}{\\mathcal{M}}$ be the tangent of M at p and let $H\\in Gr(d,D)$ such that $\\angle_{\\mathrm{max}}(T_{p}\\mathcal{M},H)=\\alpha\\leq\\pi/4$ . Let $f_{p}:\\mathbb{R}^{d}\\mathbb{R}^{D-d}$ be such that the neighborhood $W_{p}\\subset{\\mathcal{M}}$ can be descried as the the graph of $f_{p}$\n\n$$\n\\Gamma_{f_{p},W_{p}}=\\{p+(x,f_{p}(x))|x\\in P r o j_{H}(W_{p})\\}\n$$\n\nThen, for any $v\\in H^{\\perp}$ , such that $\\lVert\\boldsymbol{v}\\rVert=1$\n\n$$\n-\\tau\\cos\\alpha+\\sqrt{\\tau^{2}-(\\|x\\|-\\tau\\sin\\alpha)^{2}}\\leq\\langle v,f_{p}(x)\\rangle\\leq\\tau\\cos\\alpha-\\sqrt{\\tau^{2}-(\\|x\\|+\\tau\\sin\\alpha)^{2}}\n$$\n\nLemma 29 Let the assumptions of Theorem 3 hold, and let $f:H\\simeq\\mathbb{R}^{d}\\to\\mathbb{R}^{D-d}$ be a function such that its graph coincides with a neighborhood on the manifold M (see Lemma $\\boldsymbol{\\mathit{17}})$ . Let $\\widetilde{f}:\\mathbb{R}^{d}\\to\\mathbb{R}^{D-d}$ be the regression function defined in (23). For any $\\varepsilon > 0$ , denoting $\\alpha=\\angle_{\\mathrm{max}}(H,T_{0}f)$ , there is a constant $C_{\\tau}$ large enough such that for $\\begin{array}{ r }{M={\\frac{\\tau}{\\sigma}}>C_{\\tau}\\sqrt{D\\log D}}\\end{array}$ and $\\alpha\\leq\\sqrt{1/D}$ , where $C_{M}$ is from Theorem ${\\mathit{2}},$ we have\n\n$$\n\\|\\mathcal{D}\\widetilde{f}(0)-\\mathcal{D}f(0)\\|<\\varepsilon\\alpha\n$$\n\nProof We reiterate the definition of $\\Omega(x):H\\simeq\\mathbb{R}^{d}\\to 2^{\\mathbb{R}^{D-d}}$ from (22)\n\n$$\n\\Omega(x)=(x+H^{\\perp})\\cap\\mathcal{M}_{\\sigma},\n$$\n\nwhere $x+H^{\\perp}=\\{x+y\\mid y\\in H^{\\perp}\\}$ . Next, denoting $S_{D-d}$ to be the $D - d$ dimensional unit sphere, we define $g(x,\\theta):H\\times S_{D-d}\\to\\mathbb{R}$ the maximal length from $f(x)$ in the direction θ that is inside $\\Omega(x)$ . Explicitly,\n\n$$\ng(x,\\theta)=\\max\\{y\\in\\mathbb{R}\\mid(x,f(x)+y\\cdot\\theta)_{H}\\in\\Omega(x)\\}.\\tag{87}\n$$\n\nNote, that the farthest point from $f(x)$ in $\\Omega(x)$ at each direction $\\theta,$ by which we define $g(x,\\theta)$ , belongs to $\\partial\\mathcal{M}_{\\sigma}$ (the boundary of the domain $\\boldsymbol{\\mathcal{M}}_{\\sigma})$ , and is therefore exactly σ away from some point on the manifold itself. Since we are viewing the manifold locally as the graph of the function $f:H  H^{\\bot}$ we denote this point by $(\\widetilde{x}_{\\theta},f(\\widetilde{x}_{\\theta}(x)))_{H}$ . Explicitly, $\\widetilde{x}_{\\theta}:H  H$ is such that\n\n$$\n(x,f(x)+g(x,\\theta)\\cdot\\theta)_{H}^{T}=(\\widetilde{x}_{\\theta}(x),f(\\widetilde{x}_{\\theta}(x)))_{H}^{T}+\\sigma\\vec{N}_{\\theta}(x),\\tag{88}\n$$\n\nwhere $\\vec{N}_{\\theta}\\in\\mathbb{R}^{D}$ is perpendicular to $T_{\\widetilde{x}_{\\theta}}f$ . We introduced the definition of $\\scriptstyle{\\widetilde{x}}_{\\theta}$ here as it will be pivotal in the proofs of Lemmas 32 and 33 upon which the the current proof relies. Furthermore, we wish to stress here that by Lemma 40 $\\widetilde{x}_{\\theta}(0)\\leq\\sigma\\sin(\\alpha+c\\alpha\\sigma/\\tau)$ for some general constant c. Therefore, for suficiently large $\\begin{array}{ r }{M=\\frac{\\tau}{\\sigma},\\ :\\widetilde{x}_{\\theta}(0)}\\end{array}$ is within the domain of definition of the function $f$ which by Lemma 17 is of radius of at least $c_{\\pi/4}\\tau$\n\nNext, by the definitions and Eq. (87) and (22) we have that\n\n$$\n\\widetilde{f}(x)-f(x)=\\frac{\\int_{y\\in\\Omega(x)}y d y}{\\int_{y\\in\\Omega(x)}d y}.\n$$\n\nSince $\\Omega\\subset\\mathbb{R}^{D}$ is perpendicular to $H\\simeq\\mathbb{R}^{d}$ we get that $\\Omega\\simeq\\mathbb{R}^{D-d}$ . Thus, by change of variables we can breakdown the integrals over $\\Omega(x)$ to a radial component r and directions on the $(D-d-1)$ )-dimensional sphere. Explicitly,\n\n$$\n\\widetilde{f}(x)-f(x)=\\frac{\\int_{S_{D-d-1}}\\int_{0}^{g(x,\\theta)}\\theta r r^{D-d-1}d r d\\theta}{\\int_{S_{D-d-1}}\\int_{0}^{g(x,\\theta)}r^{D-d-1}d r d\\theta}=\\frac{(D-d)\\int_{S_{D-d-1}}\\theta g(x,\\theta)^{D-d+1}d\\theta}{(D-d+1)\\int_{S_{D-d-1}}g(x,\\theta)^{D-d}d\\theta},\n$$\n\nwhere $d r$ is the measure over the radial component, $r^{D-d-1}$ is the Jacobian introduced by the change of variables and $d \\theta$ is the measure over the $(D-d-1)$ -dimensional sphere. For brevity we introduce the notation $\\widetilde{D}=D-d$ and get\n\n$$\n\\widetilde{f}(x)-f(x)=\\frac{\\int_{\\mathbb{S}_{\\widetilde{D}-1}}\\int_{0}^{g(x,\\theta)}\\theta r r^{\\widetilde{D}-1}d r d\\theta}{\\int_{\\mathbb{S}_{\\widetilde{D}-1}}\\int_{0}^{g(x,\\theta)}r^{\\widetilde{D}-1}d r d\\theta}=\\frac{\\widetilde{D}\\int_{\\mathbb{S}_{\\widetilde{D}-1}}\\theta g(x,\\theta)^{\\widetilde{D}+1}d\\theta}{(\\widetilde{D}+1)\\int_{\\mathbb{S}_{\\widetilde{D}-1}}g(x,\\theta)^{\\widetilde{D}}d\\theta}.\\tag{89}\n$$\n\nLemma 32 Let the conditions ofLemma 29 hold. Let $g(x,\\theta)$ be as defined in Equation (87). Then, for α smaller than some constant and M larger then some constant,\n\n$$\n\\sigma\\leq g(0,\\theta)\\leq\\sigma+4\\sigma\\alpha^{2}\n$$\n\nLemma 47 Let $G_{0}$ be a d-dimensional linear space in $\\mathbb{R}^{D}$ , and let $g_{0}:G_{0}\\to\\mathbb{R}^{D-d}$ , such that the graph of g is a manifold with reach bounded by $\\tau .$ Assume that $\\begin{array}{ r }{\\angle_{\\mathrm{max}}(T_{0}g_{0},G_{0})\\le}\\end{array}$ $\\alpha$ . Let $G_{1}$ be a d dimensional linear space in $\\mathbb{R}^{D}$ , such that $\\angle\\operatorname*{max}(G_{0},G_{1})\\leq\\beta$ . Define $g_{1}:G_{1}\\to\\mathbb{R}^{D-d}$ as the function who’s graph coincides with the graph of g .\n\nThen, for $\\alpha\\le\\pi/16,\\beta\\le\\beta_{c}$ where $\\beta_{c}$ is some constant dependent only on $c_{\\pi/4}$ of Lemma $^{17,}$ and $\\|g_{0}(0)\\|\\le c_{\\pi/4}\\cdot\\tau_{;}$ , for the constant $c_{\\pi/4}$ defined in Lemma $^{17,}$ we have\n\n$$\n\\angle_{\\max}(G_{1},T_{0}g_{1})\\leq\\angle_{\\max}(T_{0}g_{0},G_{1})+\\frac{4\\|g_{0}(0)\\|(2+\\|g_{0}(0)\\|/\\tau)\\beta}{\\tau}\n$$\n\nMoreover,\n\n$$\n\\|g_{1}(0)\\|\\leq\\|g_{0}(0)\\|(1+(\\alpha+\\beta)^{2}(3+2\\|g_{0}(0)\\|/\\tau))\n$$\n\nNow, we are ready to bound the distance $\\left\\|\\tilde{o}_{1}-\\bar{o}_{1}\\right\\|$\n\n$$\n\\begin{array}{l}\\|\\tilde{o}_{1}-\\bar{o}_{1}\\|^{2}=\\|\\tilde{o}_{1}-x_{o}\\|^{2}+\\|x_{o}-\\bar{o}_{1}\\|^{2}=(x+\\Delta x)^{2}+y^{2}\\\\ =x^{2}\\left(1+\\sin^{2}(\\bar{\\alpha}+\\bar{\\beta})\\left(\\|g_{0}(0)\\|/\\tau+1\\right)\\right)^{2}\\\\ \\left.+\\left(\\cos^{2}(\\bar{\\alpha}+\\bar{\\beta})\\left(x^{2}/\\tau+x\\tan(\\bar{\\alpha}+\\bar{\\beta})\\right)\\right)^{2}\\right.\\\\ =x^{2}\\left(1+\\sin^{2}(\\bar{\\alpha}+\\bar{\\beta})\\left(\\|g_{0}(0)\\|/\\tau+1\\right)\\right)^{2}\\\\ \\left.+x^{2}\\left(\\cos^{2}(\\bar{\\alpha}+\\bar{\\beta})\\left(x/\\tau+\\tan(\\bar{\\alpha}+\\bar{\\beta})\\right)\\right)^{2}\\right.\\\\ =x^{2}\\left(1+\\sin^{2}(\\bar{\\alpha}+\\bar{\\beta})\\left(\\|g_{0}(0)\\|/\\tau+1\\right)\\right)^{2}\\\\ \\left.+x^{2}\\left(\\cos(\\bar{\\alpha}+\\bar{\\beta})\\left(\\|g_{0}(0)\\|\\sin\\bar{\\beta}/\\tau+\\sin(\\bar{\\alpha}+\\bar{\\beta})\\right)\\right)^{2}\\right.\\\\ \\leq x^{2}\\left(1+\\sin^{2}(\\bar{\\alpha}+\\bar{\\beta})\\left(\\|g_{0}(0)\\|/\\tau+1\\right)\\right)^{2}\\tag{154}\\\\ \\left.+x^{2}\\left(\\cos(\\bar{\\alpha}+\\bar{\\beta})\\sin(\\bar{\\alpha}+\\bar{\\beta})\\left(\\|g_{0}(0)\\|/\\tau+1\\right)\\right)^{2}\\right.\\\\ \\leq x^{2}\\left(1+\\sin^{2}(\\bar{\\alpha}+\\bar{\\beta})\\left(\\|g_{0}(0)\\|/\\tau+1\\right)\\right)^{2}\\\\ \\left.+x^{2}\\left(\\sin(\\bar{\\alpha}+\\bar{\\beta})\\left(\\|g_{0}(0)\\|/\\tau+1\\right)\\right)^{2}\\right.\\\\ \\leq 2 x^{2}\\left(2+\\sin(\\bar{\\alpha}+\\bar{\\beta})\\|g_{0}(0)\\|/\\tau\\right)^{2}\\\\ \\leq\\frac{2(\\|g_{0}(0)\\|\\sin\\bar{\\beta})^{2}\\left(2+\\sin(\\bar{\\alpha}+\\bar{\\beta})\\|g_{0}(0)\\|/\\tau\\right)^{2}}{\\cos(\\bar{\\alpha}+\\bar{\\beta})}\\\\ \\leq 4(\\|g_{0}(0)\\|\\sin\\bar{\\beta})^{2}\\left(2+\\sin(\\bar{\\alpha}+\\bar{\\beta})\\|g_{0}(0)\\|/\\tau\\right)^{2}\\\\ \\end{array}\n$$\n\nwhere the last inequality is correct under the assumption that $\\cos(\\bar{\\alpha}+\\bar{\\beta})\\geq 1/2$", "theorems": [{"local_id": "T1", "statement": "Lemma 46 For $f_{\\ell}$ defined in (15). Denote $\\alpha_{\\ell}=\\angle_{\\mathrm{max}}(T_{0}f_{\\ell},H_{\\ell})$ and assume $\\alpha_{\\ell}<1/D$ Then, for any δ there is N such that for any number of samples $n > N$ , we have\n\n$$\n\\|\\pi_{r,H_{\\ell}}^{*}(0)-f_{\\ell}(0)\\|\\leq 8\\sigma D\\alpha_{\\ell}^{2}+\\frac{c\\ln\\left(\\frac{1}{\\delta}\\right)}{n^{r_{0}}}.\n$$\n\nwith probability of at least $1 - \\delta$ , where $\\textstyle r_{0}={\\frac{k}{2 k+d}}$ and $\\pi_{r,H_{\\ell}}^{*}(0)$ is the polynomial defined in Equation (10), and c is some general constant.", "proof": {"steps": [{"n": 1, "claim": "Proof Using the triangle inequality, we have\n\n$$\n\\|\\pi_{r,H_{\\ell}}^{*}(0)-f_{\\ell}(0)\\|\\leq\\|\\pi_{r,H_{\\ell}}^{*}(0)-\\widetilde{f}_{\\ell}(0)\\|+\\|\\widetilde{f}_{\\ell}(0)-f_{\\ell}(0)\\|.\\tag{138}\n$$"}, {"n": 2, "claim": "Applying Theorem 2.1 of Aizenbud and Sober (2021a) on $\\lVert\\pi_{r,H_{\\ell}}^{*}(0)-\\widetilde{f}_{\\ell}(0)\\rVert$ we have that\n\n$$\n\\|\\pi_{r,H_{\\ell}}^{*}(0)-\\widetilde{f}_{\\ell}(0)\\|\\leq\\frac{c\\ln\\left(\\frac{1}{\\delta}\\right)}{n^{r_{0}}}\\tag{139}\n$$\n\nwith probability of at least $1 - \\delta$\n\nNow we focus on bounding $\\|\\widetilde{f}_{\\ell}(0)-f_{\\ell}(0)\\|$ . From (89) we have that\n\n$$\n\\widetilde{f}_{\\ell}(x)-f_{\\ell}(x)=\\frac{\\widetilde{D}\\int_{\\mathbb{S}_{\\widetilde{D}-1}}\\theta g(x,\\theta)^{\\widetilde{D}+1}d\\theta}{(\\widetilde{D}+1)\\int_{\\mathbb{S}_{\\widetilde{D}-1}}g(x,\\theta)^{\\widetilde{D}}d r d\\theta}=\\frac{\\widetilde{D}\\int_{\\mathbb{S}_{\\widetilde{D}-1}}\\theta(g(x,\\theta)^{\\widetilde{D}+1}-\\sigma^{\\widetilde{D}+1})d\\theta}{(\\widetilde{D}+1)\\int_{\\mathbb{S}_{\\widetilde{D}-1}}g(x,\\theta)^{\\widetilde{D}}d r d\\theta}\n$$\n\nor, looking at some direction $\\vec{z}$ we have\n\n$$\nz^{T}\\cdot(\\widetilde{f}_{\\ell}(x)-f_{\\ell}(x))=\\frac{\\widetilde{D}\\int_{\\mathbb{S}_{\\widetilde{D}-1}}z(g(x,\\theta)^{\\widetilde{D}+1}-\\sigma^{\\widetilde{D}+1})d\\theta}{(\\widetilde{D}+1)\\int_{\\mathbb{S}_{\\widetilde{D}-1}}g(x,\\theta)^{\\widetilde{D}}d\\theta}\n$$\n\n$$\n\\leq\\frac{\\widetilde{D}}{(\\widetilde{D}+1)}\\sigma((1+4\\alpha_{\\ell}^{2})^{\\widetilde{D}+1}-1)\n$$"}, {"n": 3, "claim": "We note that for $x < 1 / n$ the following holds\n\n$$\n(1+x)^{n}-1<x((1+1/n)^{n}-1)/(1/n)<n x(e-1)<2 n x.\n$$"}, {"n": 4, "claim": "Using the above observation, and the fact that $4\\alpha_{\\ell}^{2}<\\frac{1}{\\widetilde{D}+1}$ we have that\n\n$$\nz^{T}\\cdot(\\widetilde{f_{\\ell}}(x)-f_{\\ell}(x))\\leq\\frac{8\\sigma\\widetilde{D}}{(\\widetilde{D}+1)}(\\widetilde{D}+1)\\alpha_{\\ell}^{2}\\leq 8\\sigma D\\alpha_{\\ell}^{2}\n$$"}, {"n": 5, "claim": "Combining this with (138) and (139), we have that for any $\\delta > 0$ , for $\\begin{array}{ r }{\\alpha_{\\ell}<\\frac{1}{D}}\\end{array}$ , and for number of samples $n > N$ large enough,\n\n$$\n\\|\\pi_{r,H_{\\ell}}^{*}(0)-f_{\\ell}(0)\\|\\leq 8\\sigma D\\alpha_{\\ell}^{2}+\\frac{C_{2}\\ln\\left(\\frac{1}{\\delta}\\right)}{n^{r_{0}}},\n$$\n\nwith probability of at least $1 - \\delta$"}]}}, {"local_id": "T2", "statement": "Lemma 48 Under the conditions of Lemma $ 47,g_{1}(0)$ exists and\n\n$$\n\\|P r o j_{T_{0}g_{0}}(o_{1}+(0,g_{1}(0))_{G_{1}}-\\tilde{o}_{1})\\|\\leq\\tau/2,\n$$\n\nwhere $o_{1}=(0,0)_{G_{0}}$ is the origin and $\\tilde{o}_{1}=(0,g_{0}(0))_{G_{0}}$", "proof": {"steps": [{"n": 1, "claim": "Proof We begin with defining the coordinate system $(\\tilde{o}_{1},G_{1})$ with $\\tilde{o}_{1}=(0,g_{0}(0))_{G_{0}}$ . Let $\\widetilde{g}_{1}:(\\widetilde{o}_{1},G_{1})\\simeq\\mathbb{R}^{d}\\mathbb{R}^{D-d}$ be the function defined in Lemma $17$ , such that\n\n$$\n\\Gamma_{\\widetilde{g}_{1}}=\\mathcal{M}\\cap\\mathrm{Cyl}(\\widetilde{o}_{1},c_{\\pi/4}\\tau,\\tau/2)\n$$"}, {"n": 2, "claim": "From Lemma 17 we know that $\\widetilde{g}_{1}$ is defined for any $\\boldsymbol{x}\\in\\mathbb{R}^{d}$ such that $\\|x\\|\\leq c_{\\pi/4}\\tau$ . Now, we denote $x_{o}=P r o j_{G_{1}}(o_{1}-\\tilde{o}_{1})$ , the projection of $o_{1}$ onto the afine space defined by $(\\tilde{o}_{1},G_{1})$ From the assumptions we know that $\\|\\tilde{o}_{1}-o_{1}\\|\\leq c_{\\pi/4}\\tau$ . Since $\\angle_{\\mathrm{max}}(G_{1},G_{0})\\leq\\beta$ from Lemma 8 we have that $\\|x_{o}\\|\\le c_{\\pi/4}\\tau$ sin $\\beta .$ . Thus, for any $\\beta$ we have $\\|x_{o}\\|\\le c_{\\pi/4}\\tau$ , and $\\widetilde{g}_{1}(\\boldsymbol{x}_{o})$ is therefore defined (by Lemma 17). Since $\\widetilde{g}_{1}$ identifies with $g_{1}$ up to some shift in the domain and target, it follows that $g_{1}(0)$ is well defined."}, {"n": 3, "claim": "Next we bound $\\left\\|P r o j_{T_{0}g_{0}}(o_{1}+(0,g_{1}(0))_{G_{1}}-\\tilde{o}_{1})\\right\\|$ . Since $\\widetilde{g}_{1}(0)=0$ , from Lemma 19 and the triangle inequality for maximal angles between flats we have that\n\n$$\n\\|\\widetilde{g}_{1}(x_{o})\\|\\leq\\tau\\cos(\\alpha+\\beta)-\\sqrt{\\tau^{2}-(\\|x_{o}\\|+\\tau\\sin(\\alpha+\\beta))^{2}}\n$$"}, {"n": 4, "claim": "Substituting $\\|x_{o}\\|$ in the right hand side we set\n\n$$\n\\begin{array}{l}\\|\\widetilde{g}_{1}(x_{o})\\|\\leq\\tau\\cos(\\alpha+\\beta)-\\sqrt{\\tau^{2}-(\\tau c_{\\pi/4}\\sin\\beta+\\tau\\sin(\\alpha+\\beta))^{2}}\\\\ =\\tau\\left(\\cos(\\alpha+\\beta)-\\sqrt{1-(c_{\\pi/4}\\sin\\beta+\\sin(\\alpha+\\beta))^{2}}\\right)\\tag{140}\\\\ \\end{array}\n$$"}, {"n": 5, "claim": "Since\n\n$$\n\\begin{array}{l}\\|P r o j_{T_{0}g_{0}}(o_{1}+(0,g_{1}(0))_{G_{1}}-\\tilde{o}_{1})\\|\\leq\\|o_{1}+(0,g_{1}(0))_{G_{1}}-\\tilde{o}_{1})\\|=\\sqrt{\\|\\widetilde{g}_{1}(x_{o})\\|^{2}+\\|x_{o}\\|^{2}}\\\\ =\\tau\\sqrt{c_{\\pi/4}^{2}\\sin^{2}\\beta+\\left(\\cos(\\alpha+\\beta)-\\sqrt{1-\\left(c_{\\pi/4}\\sin\\beta+\\sin(\\alpha+\\beta)\\right)^{2}}\\right)^{2}}\\tag{141}\\\\ \\end{array}\n$$\n\nwhich, for small enough $\\beta\\ (\\mathrm{i . e .}$ , smaller than some constant $\\beta_{c}$ depending only on $c_{\\pi/4})$ and fixed α is smaller than 0.5τ."}]}}, {"local_id": "T3", "statement": "Lemma 49 Let $G,H\\in Gr(d,D)$ be two d-dimensional linear subspaces of $\\mathbb{R}^{D}$ . Let $_{\\textit{ g }:}$ $G\\to\\mathbb{R}^{D-d}$ be such that the graph of g is a manifold M with a reach bounded by τ and let $h:H\\to\\mathbb{R}^{D-d}$ denote the function over H whose graph coincides with M in some neighborhood (see Figure $\\boldsymbol{{17}}$ . Assume that $T_{0}g$ , the tanget of g at $0,$ , is parallel to $G,$ and that $\\Vert g(0)\\Vert\\leq\\sigma .\\ I f\\ L_{\\operatorname*{max}}(H,T_{0}h)\\leq a$ is small enough, then\n\n$$\n\\|p_{g}-p_{h}\\|\\leq\\frac{a}{\\left(\\frac{1}{2\\sigma(2+\\sigma/\\tau)}-\\frac{c}{\\tau}\\right)},\n$$\n\nwhere $p_{g}=(0,g(0))_{0,G}\\in\\mathcal{M}\\ a n d\\ p_{h}=(0,h(0))_{0,H}\\in\\mathcal{M}.$", "proof": {"steps": [{"n": 1, "claim": "Proof\n\nFigure 17: Supporting illustration for the proof of Lemma 49 where we bound the diference between $p_{h}$ and $p_{g}.$ , the evaluation of r under the functions $h$ and $g$ correspondingly, given that the maximal angle between $T_{p_{h}}{\\mathcal{M}}$ and $T_{p_{q}}{\\mathcal{M}}$ is bounded by $a .$\n\nFrom the triangle inequality and Boissonnat et al. (2019) we bound from above the angle between $T_{0}g$ and $H$ ."}, {"n": 2, "claim": "$$\n\\angle_{\\max}(H,T_{0}g)\\leq\\angle_{\\max}(H,T_{0}h)+\\angle_{\\max}(T_{0}h,T_{0}g)\\leq a+c\\|p_{g}-p_{h}\\|/\\tau .\\tag{161}\n$$"}, {"n": 3, "claim": "We now wish to give another bound tying between $\\angle\\operatorname*{max}(H,T_{0}g)$ and $\\|p_{g}-p_{h}\\|$ just from the other direction so that we could create an inequality limiting the values of $\\|p_{g}-p_{h}\\|$ . Using the notation in Lemma $47.$ , we can follow the derivations with $\\alpha = 0$ , and with $\\|g(0)\\|\\le\\sigma$ 2 we get from (154) that\n\n$$\n\\|p_{g}-p_{h}\\|^{2}\\leq 4(\\sigma\\sin\\angle_{\\max}(H,T_{0}g))^{2}(2+\\sigma\\sin\\angle_{\\max}(H,T_{0}g)/\\tau)^{2}\\leq 4(\\sigma\\sin\\angle_{\\max}(H,T_{0}g))^{2}(2+\\sigma/\\tau)^{2}.\n$$"}, {"n": 4, "claim": "Thus,\n\n$$\n\\|p_{g}-p_{h}\\|\\leq 2\\sigma(2+\\sigma/\\tau)\\sin\\angle_{\\max}(H,T_{0}g)\\leq 2\\sigma(2+\\sigma)\\angle_{\\max}(H,T_{0}g),\n$$\n\nor,\n\n$$\n\\angle_{\\max}(H,T_{0}g)\\geq\\frac{\\|p_{g}-p_{h}\\|}{2\\sigma(2+\\sigma/\\tau)}.\n$$"}, {"n": 5, "claim": "Combining this with (161), we have\n\n$$\n\\frac{\\|p_{g}-p_{h}\\|}{2\\sigma(2+\\sigma/\\tau)}\\leq a+c\\|p_{g}-p_{h}\\|/\\tau\n$$\n\nor,\n\n$$\n\\|p_{g}-p_{h}\\|\\left(\\frac{1}{2\\sigma(2+\\sigma/\\tau)}-\\frac{c}{\\tau}\\right)\\leq a.\n$$"}, {"n": 6, "claim": "Finally, we have,\n\n$$\n\\|p_{g}-p_{h}\\|\\leq\\frac{a}{\\left(\\frac{1}{2\\sigma(2+\\sigma/\\tau)}-\\frac{c}{\\tau}\\right)}.\n$$"}]}}], "target_local_id": null, "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2026-25-0012", "context": "Some norms of vector, matrix and tensor are used. The $\\ell_{1},\\ell_{\\infty},\\ell_{2,\\infty}$ and Frobenius norms of A are defined as $\\begin{array}{ r }{\\|\\pmb{A}\\|_{1}=\\sum_{i,j,k}|\\pmb{A}_{i,j,k}|,\\|\\pmb{A}\\|_{\\infty}=\\operatorname*{max}_{i,j,k}|\\pmb{A}_{i,j,k}|,\\|\\pmb{A}\\|_{2,\\infty}=}\\end{array}$ $\\operatorname*{max}_{i}\\|\\pmb{\\mathcal{A}}(i,:,:)\\|_{F}$ and $\\begin{array}{ r }{\\|\\pmb{\\mathscr{A}}\\|_{F}=\\sqrt{\\sum_{i,j,k}|\\pmb{\\mathscr{A}}_{i,j,k}|^{2}}}\\end{array}$ , respectively. For $\\pmb{v}\\in\\mathbb{C}^{n}$ , its $\\ell_{2}$ norm is denoted as $\\begin{array}{ r }{\\|\\pmb{v}\\|_{2}=\\sqrt{\\sum_{i}|v_{i}|^{2}}}\\end{array}$ and we use $\\ell_{0}$ norm to denote the number of nonzero elements in v. The spectral norm of a matrix A is denoted as $\\|A\\|=\\operatorname*{max}_{i}\\sigma_{i}(A)$ , where $\\sigma_{i}(A)\\ '_{\\mathrm{i}}$ s are the singular values of A. The matrix nuclear norm of A is $\\begin{array}{ r }{\\|\\pmb{A}\\|_{*}=\\sum_{i}\\sigma_{i}(\\pmb{A})}\\end{array}$\n\nThe framework of tensor singular value decomposition (t-SVD) is based on the t-product under an invertible linear transform L (Kernfeld et al., 2015). In this paper, the transformation matrix Φ defining the transform L is restricted to be orthogonal, i.e., $\\Phi\\in\\mathbb{C}^{n_{3}\\times n_{3}}$ satisfying\n\n$$\n\\boldsymbol{\\Phi}\\boldsymbol{\\Phi}^{H}=\\boldsymbol{\\Phi}^{H}\\boldsymbol{\\Phi}=\\ell\\boldsymbol{I}_{n_{3}},\\tag{5}\n$$\n\nwhere $\\ell > 0$ is a constant. We define the associated linear transform $L(\\cdot)$ with its inverse mapping $L^{-1}(\\cdot)$ on any $\\pmb{\\mathcal{A}}\\in\\mathbb{R}^{n_{1}\\times n_{2}\\times n_{3}}$ as\n\n$$\n\\bar{\\boldsymbol{\\mathcal{A}}}:=L(\\boldsymbol{\\mathcal{A}})=\\boldsymbol{\\mathcal{A}}\\times_{3}\\boldsymbol{\\Phi}\\quad \\text{and}\\quad L^{-1}(\\boldsymbol{\\mathcal{A}}):=\\boldsymbol{\\mathcal{A}}\\times_{3}\\boldsymbol{\\Phi}^{-1},\\tag{6}\n$$\n\nwhere $\\times{}_{3}$ denotes the mode-3 tensor-matrix product (Kolda and Bader, 2009), i.e., for any $\\pmb{\\mathcal{A}}\\in\\mathbb{C}^{n_{1}\\times n_{2}\\times n_{3}}$ and $\\begin{array}{ r }{M\\in\\mathbb{C}^{n_{4}\\times n_{3}},\\mathcal{B}=\\pmb{\\mathcal{A}}\\times_{3}\\pmb{M}\\in\\mathbb{C}^{n_{1}\\times n_{2}\\times n_{4}}}\\end{array}$ such that\n\n$$\n\\boldsymbol{\\mathcal{B}}_{i,j,l}=\\sum_{k=1}^{n_{3}}\\boldsymbol{\\mathcal{A}}_{i,j,k}\\boldsymbol{M}_{l,k},\\quad i\\in [n_{1}],j\\in [n_{2}],l\\in [n_{4}].\n$$\n\nDefinition 1 (t-product (Kernfeld et al., 2015)) The t-product of any $\\pmb{\\mathcal{A}}\\in\\mathbb{C}^{n_{1}\\times n_{2}\\times n_{3}}$ and $\\pmb{{\\cal B}}\\in\\mathbb{C}^{n_{2}\\times n_{4}\\times n_{3}}$ under transform L in (6), is a unique tensor $\\pmb{\\mathcal{C}}=\\pmb{\\mathcal{A}}*_{\\Phi}\\pmb{\\mathcal{B}}\\in\\mathbb{C}^{n_{1}\\times n_{4}\\times n_{3}}$ such that $L(\\pmb{C})=L(\\pmb{A})\\triangle L(\\pmb{B})$\n\nWe denote $\\ b{\\overline{{A}}}\\in\\mathbb{C}^{n_{1}n_{3}\\times n_{2}n_{3}}$ as the block diagonal matrix with its i-th diagonal block corresponding to the i-th frontal slice $\\bar{A}^{(i)}$ of A in (6), i.e.,\n\n$$\n\\overline{{\\boldsymbol{A}}}=\\operatorname{bdiag}(\\overline{{\\boldsymbol{A}}})=\\left[\\begin{array}{c c c}\\overline{{\\boldsymbol{A}}}^{(1)}&&\\\\ &\\ddots&\\\\ &&\\overline{{\\boldsymbol{A}}}^{(n_{3})}\\end{array}\\right],\\tag{7}\n$$\n\nwhere bdiag is an operator that maps $\\bar{A}$ to $\\bar{A}$ . Using (5), we have the following properties:\n\n$$\n\\|\\boldsymbol{\\mathcal{A}}\\|_{F}=\\frac{1}{\\sqrt{\\ell}}\\|\\bar{\\boldsymbol{\\mathcal{A}}}\\|_{F}=\\frac{1}{\\sqrt{\\ell}}\\|\\bar{\\boldsymbol{A}}\\|_{F}\\quad \\text{and}\\quad\\langle\\boldsymbol{\\mathcal{A}},\\boldsymbol{\\mathcal{B}}\\rangle=\\frac{1}{\\ell}\\langle\\bar{\\boldsymbol{\\mathcal{A}}},\\bar{\\boldsymbol{\\mathcal{B}}}\\rangle=\\frac{1}{\\ell}\\langle\\bar{\\boldsymbol{A}},\\bar{\\boldsymbol{B}}\\rangle .\\tag{8}\n$$\n\nDefinition 2 (Conjugate transpose (Song et al., 2020)) The conjugate transpose of a tensor $\\pmb{\\mathcal{A}}\\in\\mathbb{C}^{n_{1}\\times n_{2}\\times n_{3}}$ under L is the tensor $\\pmb{\\mathcal{A}}^{H}\\in\\mathbb{C}^{n_{2}\\times n_{1}\\times n_{3}}$ satisfying $L(\\mathcal{A}^{\\bar{H}})^{(i)}\\stackrel{.}{=}$ $(L(\\pmb{A})^{(i)})^{H},i=1,2,...,n_{3}$\n\nDefinition 3 (Identity tensor (Song et al., 2020)) The identity tensor $\\pmb{\\mathcal{I}}_{n}\\in\\mathbb{C}^{n\\times n\\times n_{3}}$ under L is a tensor such that each frontal slice of $L(\\pmb{\\mathcal{I}}_{n})=\\bar{\\pmb{\\mathcal{I}}}_{n}$ is the identity matrix $\\boldsymbol{I}_{n}$ Then $\\pmb{\\mathcal{I}}_{n}=L^{-1}(\\bar{\\pmb{\\mathcal{I}}}_{n})$ gives the identity tensor under L.\n\nDefinition 4 (Orthogonal tensor (Song et al., 2020)) A tensor $\\pmb{\\mathcal{Q}}\\in\\mathbb{C}^{n\\times n\\times n_{3}}$ is orthogonal if it satisfies $\\pmb{\\mathcal{Q}}*_{\\Phi}\\pmb{\\mathcal{Q}}^{H}=\\pmb{\\mathcal{Q}}^{H}*_{\\Phi}\\pmb{\\mathcal{Q}}=\\pmb{\\mathcal{I}}_{n}$\n\nDefinition 5 (Invertible tensor) A tensor $\\pmb{\\mathcal{A}}\\in\\mathbb{R}^{r\\times r\\times n_{3}}$ is invertible if there exists a tensor $\\pmb{\\mathcal{B}}\\in\\mathbb{R}^{r\\times r\\times n_{3}}$ such that $\\pmb{\\mathcal{A}}*_{\\Phi}\\pmb{\\mathcal{B}}=\\pmb{\\mathcal{B}}*_{\\Phi}\\pmb{\\mathcal{A}}=\\pmb{\\mathcal{I}}_{r}$ . The set of invertible tensors in $\\mathbb{R}^{r\\times r\\times n_{3}}$ is denoted by GL(r).\n\nDefinition 6 (t-SVD (Song et al., 2020)) Let L be any invertible linear transform in (6). For any $\\pmb{\\mathcal{A}}\\in\\mathbb{C}^{n_{1}\\times n_{2}\\times n_{3}}$ , it can be factorized as $\\pmb{\\mathcal{A}}=\\pmb{\\mathcal{U}}*_{\\Phi}\\pmb{\\mathcal{G}}*_{\\Phi}\\pmb{\\mathcal{V}}^{H}$ , where $\\pmb{\\mathcal{U}}\\in\\mathbb{C}^{n_{1}\\times n_{1}\\times n_{3}}$ and $\\pmb{\\mathcal{V}}\\in\\mathbb{C}^{n_{2}\\times n_{2}\\times n_{3}}$ are orthogonal, and $\\pmb{\\mathcal{G}}\\in\\mathbb{C}^{n_{1}\\times n_{2}\\times n_{3}}$ is an f-diagonal tensor, whose frontal slices are diagonal matrices.\n\nFor any A, we have the following relationship between its t-SVD and the matrix SVD of its block-diagonal matrix (Kernfeld et al., 2015):\n\n$$\n\\mathcal{A}=\\mathcal{U}*_{\\Phi}\\mathcal{G}*_{\\Phi}\\mathcal{V}^{H}\\Leftrightarrow\\bar{A}=\\bar{U}\\bar{G}\\bar{V}^{H}.\n$$\n\nIn words, the t-product in the spatial domain corresponds to matrix multiplication of the frontal slices in the transformed domain. Note that when $n_{3}=1$ , the operator ∗<sub>Φ</sub> reduces to matrix multiplication.\n\nDefinition 8 (Tensor nuclear norm and spectral norm (Lu et al., 2019)) The tensor nuclear norm of $\\pmb{\\mathcal{A}}\\in\\mathbb{C}^{n_{1}\\times n_{2}\\times n_{3}}$ under L is defined as $\\begin{array}{ r }{\\|\\pmb{A}\\|_{*}=\\frac{1}{\\ell}\\sum_{i=1}^{n_{3}}\\|\\overline{{\\pmb{A}}}^{(i)}\\|_{*}=\\frac{1}{\\ell}\\|\\overline{{\\pmb{A}}}\\|_{*}}\\end{array}$ The spectral norm of A is defined as $\\|{\\bf\\cal A}\\|=\\|\\overline{{{\\cal A}}}\\|$\n\nSuppose that the ground truth tubal rank-r tensor $\\scriptstyle x_{\\star}$ with multi-rank r admits the following compact t-SVD $\\pmb{\\mathcal{X}}_{\\star}=\\pmb{\\mathcal{U}}_{\\star}*_{\\Phi}\\pmb{\\mathcal{G}}_{\\star}*_{\\Phi}\\pmb{\\mathcal{V}}_{\\star}^{H}$ , where $\\mathcal{U}_{\\star}\\in\\mathbb{R}^{n_{1}\\times r\\times n_{3}},\\mathcal{V}_{\\star}\\in\\mathbb{R}^{n_{2}\\times r\\times n_{3}}$ , and $\\pmb{\\mathcal{G}}_{\\star}\\in\\mathbb{R}^{r\\times r\\times n_{3}}$ . Define the ground truth low-rank factors as\n\n$$\n\\mathcal{L}_{\\star}=\\mathcal{U}_{\\star}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\quad \\text{and}\\quad\\mathcal{R}_{\\star}=\\mathcal{V}_{\\star}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\n$$\n\nso that $\\pmb{\\mathcal{X}}_{\\star}=\\pmb{\\mathcal{L}}_{\\star}\\ast_{\\Phi}\\pmb{\\mathcal{R}}_{\\star}^{H}$ . Here, the “square root” of a tensor $\\pmb{\\mathcal{A}}$, denoted by $\\pmb{\\mathcal{A}}^{\\frac{1}{2}}$ , is obtained by setting $\\pmb{\\mathcal{A}}^{\\frac{1}{2}}:=\\pmb{\\mathcal{B}}=L^{-1}(\\bar{\\pmb{\\mathcal{B}}})$ , where the i-th frontal slice of $\\bar{\\pmb{\\mathcal{B}}}$ is $\\bar{\\boldsymbol{B}}^{(i)}=(\\overline{\\boldsymbol{A}}^{(i)})^{\\frac{1}{2}}$ . The factored representation is not unique in that for any invertible tensor $\\pmb{\\mathcal{Q}}\\in\\mathrm{GL}(r)$ , one has $\\mathcal{L}_{\\star}*_{\\Phi}\\mathcal{R}_{\\star}^{H}=(\\mathcal{L}_{\\star}*_{\\Phi}\\mathcal{Q})*_{\\Phi}(\\mathcal{R}_{\\star}*_{\\Phi}\\mathcal{Q}^{-H})^{H}$ . We define the following two important singular values of tensor $x,$ <sub>⋆</sub> as\n\n$$\n\\bar{\\sigma}_{1}(\\boldsymbol{\\mathcal{X}}_{\\star})=\\max\\{\\bar{\\boldsymbol{\\mathcal{G}}}_{i,i,k}|\\bar{\\boldsymbol{\\mathcal{G}}}_{i,i,k}>0,i\\leq r_{k},k\\in [n_{3}]\\}\n$$\n\nand $\\bar{\\sigma}_{s_{r}}(\\boldsymbol{\\mathcal{X}}_{\\star})=\\min\\{\\bar{\\boldsymbol{\\mathcal{G}}}_{i,i,k}|\\bar{\\boldsymbol{\\mathcal{G}}}_{i,i,k}>0,i\\leq r_{k},k\\in [n_{3}]\\}$.\n\nDefinition 10 (Incoherence) For a tubal rank-r tensor $\\pmb{\\chi}_{\\star}\\in\\mathbb{R}^{n_{1}\\times n_{2}\\times n_{3}}$ , assume that the multi-rank of X <sub>⋆</sub> is r and it has the t-SVD $\\pmb{\\mathcal{X}}_{\\star}=\\pmb{\\mathcal{U}}_{\\star}\\ast_{\\Phi}\\pmb{\\mathcal{G}}_{\\star}\\ast_{\\Phi}\\pmb{\\mathcal{V}}_{\\star}^{H}$ . Then $\\pmb{\\chi}_{\\star}$ is said to satisfy the tensor incoherence conditions with parameter $\\mu$ if\n\n$$\n\\|\\boldsymbol{\\mathcal{U}}_{\\star}\\|_{2,\\infty}=\\max_{i\\in [n_{1}]}\\|\\boldsymbol{\\mathcal{U}}_{\\star}^{H}*_{\\boldsymbol{\\Phi}}\\mathring{\\mathfrak{e}}_{i}\\|_{F}\\leq\\sqrt{\\frac{\\mu s_{r}}{n_{1}n_{3}\\ell}}\\quad \\text{and}\\quad\\|\\boldsymbol{\\mathcal{V}}_{\\star}\\|_{2,\\infty}=\\max_{j\\in [n_{2}]}\\|\\boldsymbol{\\mathcal{V}}_{\\star}^{H}*_{\\boldsymbol{\\Phi}}\\mathring{\\mathfrak{e}}_{j}\\|_{F}\\leq\\sqrt{\\frac{\\mu s_{r}}{n_{2}n_{3}\\ell}}.\\tag{10}\n$$\n\nHere, $\\overset{\\circ}{\\boldsymbol{\\mathfrak{e}}}_{i}$ is a tensor of size $n_{1}\\times 1\\times n_{3}$ with the entries of the (i, 1)-th mode-3 tube of $L(\\mathbf{\\vec{e}}_{i})$ equaling 1 and the rest equaling $\\theta,$ and $\\begin{array}{ r }{s_{r}=\\sum_{k=1}^{n_{3}}r_{k}}\\end{array}$\n\nNotice that when all the $r_{k}\\mathrm{{^{*}s}}$ are equal to the tubal rank r, i.e., $s_{r}~=~n_{3}r$ , the above conditions will be equivalent to the ones in Lu (2021, Proposition 1). Roughly speaking, a small incoherence parameter ensures that the tensor columns are not correlated with the standard tensor basis, i.e., the energy of the tensor is evenly distributed across its entries. To track the performance of ScaledGD iterates $\\pmb{\\mathcal{F}}_{t}=[\\pmb{\\mathcal{L}}_{t}^{\\bar{H}},\\pmb{\\mathcal{R}}_{t}^{H}]^{H}$ , one needs a distance metric that properly takes account of the factor ambiguity due to invertible transforms. Motivated by the analysis in Tong et al. (2021a), we consider the following distance metric that resolves the ambiguity in the t-SVD.\n\nDefinition 11 (Distance metric) Let $\\pmb{\\mathcal{F}}=\\left[\\pmb{\\mathcal{L}}\\right]\\in\\mathbb{R}^{(n_{1}+n_{2})\\times r\\times n_{3}}$ and $\\mathcal{F}_{\\star}=\\left[\\mathcal{L}_{\\star}\\right]\\in$ $\\mathbb{R}^{(n_{1}+n_{2})\\times r\\times n_{3}}$ , denote\n\n$$\n\\mathrm{dist}(\\boldsymbol{\\mathcal{F}},\\boldsymbol{\\mathcal{F}}_{\\star})=\\sqrt{\\inf_{\\boldsymbol{\\mathcal{Q}}\\in\\mathrm{GL}(r)}\\|(\\boldsymbol{\\mathcal{L}}*_{\\Phi}\\boldsymbol{\\mathcal{Q}}-\\boldsymbol{\\mathcal{L}}_{\\star})*_{\\Phi}\\boldsymbol{\\mathcal{G}}_{\\star}^{\\frac{1}{2}}\\|_{F}^{2}+\\|(\\boldsymbol{\\mathcal{R}}*_{\\Phi}\\boldsymbol{\\mathcal{Q}}^{-H}-\\boldsymbol{\\mathcal{R}}_{\\star})*_{\\Phi}\\boldsymbol{\\mathcal{G}}_{\\star}^{\\frac{1}{2}}\\|_{F}^{2}}.\\tag{11}\n$$\n\nIf the infimum is attained at the argument $\\scriptstyle{\\mathfrak{Q}},$ , it is called the optimal alignment tensor between $\\mathcal{F}$ and $\\mathcal{F}_{\\star}$\n\nAs indicated in Appendix A, for the ScaledGD iterates $\\{\\mathcal{F}_{t}\\}$ , the optimal alignment tensors $\\left\\{\\pmb{\\mathcal{Q}}_{t}\\right\\}$ always exist and hence are well-defined.\n\nIn the following, we use $\\mathscr{Q}_{t}$ to denote the optimal alignment tensor between $\\mathcal{F}_{t}=\\bigg [\\mathcal{L}_{t}\\bigg]$ and $\\mathcal{F}_{\\star}=\\left\\lceil\\frac{\\mathcal{L}_{\\star}}{\\mathcal{R}_{\\star}}\\right\\rceil$ . For notational convenience, we denote $\\pmb{\\mathcal{L}}_{\\sharp}: =\\pmb{\\mathcal{L}}_{t}\\ast\\pmb{\\Phi}\\pmb{\\mathcal{Q}}_{t},\\pmb{\\mathcal{R}}_{\\sharp}: =\\pmb{\\mathcal{R}}_{t}\\ast\\pmb{\\Phi}\\pmb{\\mathcal{Q}}_{t}^{-H}$ $\\pmb{\\mathcal{L}}_{\\triangle}: =\\pmb{\\mathcal{L}}_{\\sharp}-\\pmb{\\mathcal{L}}_{\\star},\\pmb{\\mathcal{R}}_{\\triangle}: =\\pmb{\\mathcal{R}}_{\\sharp}-\\pmb{\\mathcal{R}}_{\\star}$ , and $\\pmb{S}_{\\triangle}: =\\pmb{S}_{t+1}-\\pmb{S}_{\\star}$ .\n\nLemma 35 Let $\\pmb{\\mathcal{A}}\\in\\mathbb{R}^{n_{1}\\times n_{2}\\times n_{3}}$ and $\\pmb{\\mathcal{B}}\\in\\mathbb{R}^{n_{4}\\times n_{2}\\times n_{3}}$ be two tensors, then\n\n$$\n\\|\\boldsymbol{\\mathcal{A}}*_{\\Phi}\\boldsymbol{\\mathcal{B}}^{H}\\|_{\\infty}\\leq\\sqrt{\\ell}\\|\\boldsymbol{\\mathcal{A}}\\|_{2,\\infty}\\|\\boldsymbol{\\mathcal{B}}\\|_{2,\\infty}.\n$$\n\nLemma 44 For any $\\pmb{\\mathcal{L}}_{\\sharp}\\in\\mathbb{R}^{n_{1}\\times r\\times n_{3}},\\ \\pmb{\\mathcal{R}}_{\\sharp}\\in\\mathbb{R}^{n_{2}\\times r\\times n_{3}}$ , denote $\\begin{array}{ r }{\\pmb{\\mathscr{L}}_{\\triangle}: =\\pmb{\\mathscr{L}}_{\\sharp}-\\pmb{\\mathscr{L}},}\\end{array}$ and $\\mathcal{R}_{\\triangle}: =$ $\\mathcal{R}_{\\sharp}-\\mathcal{R}_{\\star}$ . Suppose that $\\|\\pmb{\\mathscr{L}}_{\\triangle}*_{\\Phi}\\pmb{\\mathscr{G}}_{\\star}^{-\\frac{1}{2}}\\|\\vee\\|\\pmb{\\mathscr{R}}_{\\triangle}*_{\\Phi}\\pmb{\\mathscr{G}}_{\\star}^{-\\frac{1}{2}}\\|<1$ , then\n\n$$\n\\left\\|\\mathcal{L}_{\\sharp}*_{\\Phi}\\left(\\mathcal{L}_{\\sharp}^{H}*_{\\Phi}\\mathcal{L}_{\\sharp}\\right)^{-1}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\right\\|\\leq\\frac{1}{1-\\left\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\right\\|};\\tag{41}\n$$\n\n$$\n\\left\\|\\mathcal{R}_{\\sharp}*_{\\Phi}\\left(\\mathcal{R}_{\\sharp}^{H}*_{\\Phi}\\mathcal{R}_{\\sharp}\\right)^{-1}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\right\\|\\leq\\frac{1}{1-\\left\\|\\mathcal{R}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\right\\|};\\tag{42}\n$$\n\n$$\n\\left\\|\\mathcal{L}_{\\sharp}*_{\\Phi}\\left(\\mathcal{L}_{\\sharp}^{H}*_{\\Phi}\\mathcal{L}_{\\sharp}\\right)^{-1}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}-\\mathcal{U}_{\\star}\\right\\|\\leq\\frac{\\sqrt{2}\\left\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\right\\|}{1-\\left\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\right\\|};\\tag{43}\n$$\n\n$$\n\\|\\mathcal{R}_{\\sharp}*_{\\Phi}\\left(\\mathcal{R}_{\\sharp}^{H}*_{\\Phi}\\mathcal{R}_{\\sharp}\\right)^{-1}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}-\\mathcal{V}_{\\star}\\|\\leq\\frac{\\sqrt{2}\\|\\mathcal{R}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\|}{1-\\|\\mathcal{R}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\|}.\\tag{44}\n$$\n\nInduction step. At the $t$ -th iteration, we assume the conditions\n\n$$\n\\mathrm{dist}(\\boldsymbol{\\mathcal{F}}_{t},\\boldsymbol{\\mathcal{F}}_{\\star})\\leq\\frac{\\epsilon}{\\sqrt{\\ell}}\\rho^{t}\\bar{\\sigma}_{s_{r}}(\\boldsymbol{\\mathcal{X}}_{\\star}),\n$$\n\n$$\n\\sqrt{n_{1}}\\|(\\mathcal{L}_{t}*_{\\Phi}\\mathcal{Q}_{t}-\\mathcal{L}_{\\star})*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{2,\\infty}\\vee\\sqrt{n_{2}}\\|(\\mathcal{R}_{t}*_{\\Phi}\\mathcal{Q}_{t}^{-H}-\\mathcal{R}_{\\star})*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{2,\\infty}\\leq\\sqrt{\\frac{\\mu s_{r}}{n_{3}\\ell}}\\rho^{t}\\bar{\\sigma}_{s_{r}}(\\boldsymbol{\\chi}_{\\star})\n$$\n\nhold.\n\nThe last step is to switch the alignment tensor from $\\mathscr{Q}_{t}$ to $\\pmb{\\mathcal{Q}}_{t+1}$ . Note that (54) together with Lemma 41 confirms the existence of $\\pmb{\\mathcal{Q}}_{t+1}$ . Applying the triangle inequality and Lemma 49, we have\n\n$$\n\\begin{array}{l}\\|(\\mathcal{L}_{t+1}*_{\\Phi}\\mathcal{Q}_{t+1}-\\mathcal{L}_{\\star})*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{2,\\infty}\\\\ \\leq\\|(\\mathcal{L}_{t+1}*_{\\Phi}\\mathcal{Q}_{t}-\\mathcal{L}_{\\star})*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{2,\\infty}+\\|(\\mathcal{L}_{t+1}*_{\\Phi}(\\mathcal{Q}_{t+1}-\\mathcal{Q}_{t})*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{2,\\infty}\\\\ =\\|(\\mathcal{L}_{t+1}*_{\\Phi}\\mathcal{Q}_{t}-\\mathcal{L}_{\\star})*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{2,\\infty}\\\\ \\qquad+\\|(\\mathcal{L}_{t+1}*_{\\Phi}\\mathcal{Q}_{t}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}*_{\\Phi}\\mathcal{Q}_{t}^{-1}*_{\\Phi}(\\mathcal{Q}_{t+1}-\\mathcal{Q}_{t})*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{2,\\infty}\\\\ \\leq\\|(\\mathcal{L}_{t+1}*_{\\Phi}\\mathcal{Q}_{t}-\\mathcal{L}_{\\star})*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{2,\\infty}\\\\ \\qquad+\\|(\\mathcal{L}_{t+1}*_{\\Phi}\\mathcal{Q}_{t}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\|_{2,\\infty}\\|\\mathcal{G}_{\\star}^{\\frac{1}{2}}*_{\\Phi}\\mathcal{Q}_{t}^{-1}*_{\\Phi}(\\mathcal{Q}_{t+1}-\\mathcal{Q}_{t})*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|\\\\ \\leq\\|(\\mathcal{L}_{t+1}*_{\\Phi}\\mathcal{Q}_{t}-\\mathcal{L}_{\\star})*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{2,\\infty}+\\Big(\\|(\\mathcal{L}_{t+1}*_{\\Phi}\\mathcal{Q}_{t}-\\mathcal{L}_{\\star})*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\|_{2,\\infty}\\\\ \\qquad+\\|\\mathcal{L}_{\\star}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\|_{2,\\infty}\\Big)\\|\\mathcal{G}_{\\star}^{\\frac{1}{2}}*_{\\Phi}\\mathcal{Q}_{t}^{-1}*_{\\Phi}(\\mathcal{Q}_{t+1}-\\mathcal{Q}_{t})*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|\\\\ \\leq\\Big((1-\\eta)+\\eta\\frac{\\epsilon}{1-\\epsilon}+6\\eta\\frac{\\sqrt{\\alpha\\mu s_{r}}}{1-\\epsilon}\\Big)\\sqrt{\\frac{\\mu s_{r}}{n_{1}n_{3}\\ell}}\\rho^{t}\\bar{\\sigma}_{s_{r}}(\\boldsymbol{\\mathcal{X}}_{\\star})\\\\ \\qquad+\\Big(\\big((1-\\eta)+\\eta\\frac{\\epsilon}{1-\\epsilon}+6\\eta\\frac{\\sqrt{\\alpha\\mu s_{r}}}{1-\\epsilon}\\big)\\sqrt{\\frac{\\mu s_{r}}{n_{1}n_{3}\\ell}}\\rho^{t}+\\sqrt{\\frac{\\mu s_{r}}{n_{1}n_{3}\\ell}}\\Big)\\frac{2\\epsilon}{1-\\epsilon}\\rho^{t+1}\\bar{\\sigma}_{s_{r}}(\\boldsymbol{\\mathcal{X}}_{\\star})\\\\ \\leq\\Big((1-\\eta)+\\eta\\frac{\\epsilon}{1-\\epsilon}+6\\eta\\frac{\\sqrt{\\alpha\\mu s_{r}}}{1-\\epsilon}+\\frac{2\\epsilon}{1-\\epsilon}\\big((2-\\eta)+\\eta\\frac{\\epsilon}{1-\\epsilon}+6\\eta\\frac{\\sqrt{\\alpha\\mu s_{r}}}{1-\\epsilon}\\big)\\Big)\\sqrt{\\frac{\\mu s_{r}}{n_{1}n_{3}\\ell}}\\rho^{t}\\bar{\\sigma}_{s_{r}}(\\boldsymbol{\\mathcal{X}}_{\\star})\\\\ \\leq(1-0.6\\eta)\\sqrt{\\frac{\\mu s_{r}}{n_{1}n_{3}\\ell}}\\rho^{t}\\bar{\\sigma}_{s_{r}}(\\boldsymbol{\\mathcal{X}}_{\\star}),\\end{array}\n$$\n\nwhere we use $\\epsilon=0.02$ , $\\alpha\\mu s_{r}<\\alpha\\mu s_{r}^{1.5}\\frac{n_{1}+n_{2}n_{3}}{n_{3}\\sqrt{n_{(2)}n_{2}}}\\leq 10^{-4}$ , and $\\frac{1}{4}\\leq\\eta\\leq\\frac{8}{9}$ in the last step.", "theorems": [{"local_id": "T1", "statement": "Lemma 45 For any $\\pmb{\\mathcal{L}}_{\\sharp}\\in\\mathbb{R}^{n_{1}\\times r\\times n_{3}},\\ \\pmb{\\mathcal{R}}_{\\sharp}\\in\\mathbb{R}^{n_{2}\\times r\\times n_{3}}$ , denote $\\pmb{\\mathscr{L}}_{\\triangle}: =\\pmb{\\mathscr{L}}_{\\sharp}-\\pmb{\\mathscr{L}}_{\\star}$ and $\\mathcal{R}_{\\triangle}: =$ $\\mathcal{R}_{\\sharp}-\\mathcal{R}_{\\star}$ , then\n\n$$\n\\begin{array}{l}\\|\\mathcal{L}_{\\sharp}*_{\\Phi}\\mathcal{R}_{\\sharp}^{H}-\\mathcal{X}_{\\star}\\|_{F}\\leq\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{R}_{\\star}^{H}\\|_{F}+\\|\\mathcal{L}_{\\star}*_{\\Phi}\\mathcal{R}_{\\triangle}^{H}\\|_{F}+\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{R}_{\\triangle}^{H}\\|_{F}\\\\ \\leq\\left(1+\\frac{1}{2}(\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\|\\vee\\|\\mathcal{R}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\|)\\right)\\\\ \\Big(\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{F}+\\|\\mathcal{R}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{F}\\Big).\\\\ \\end{array}\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof Using the decomposition that $\\mathcal{L}_{\\sharp}\\ast_{\\Phi}\\mathcal{R}_{\\sharp}^{H}-\\pmb{\\mathcal{X}}_{\\star}=\\mathcal{L}_{\\triangle}\\ast_{\\Phi}\\pmb{\\mathcal{R}}_{\\star}^{H}+\\mathcal{L}_{\\star}\\ast_{\\Phi}\\pmb{\\mathcal{R}}_{\\triangle}^{H}+\\mathcal{L}_{\\triangle}\\ast_{\\Phi}\\pmb{\\mathcal{R}}_{\\triangle}^{H}$ and the facts that\n\n$$\n\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{R}_{\\star}^{H}\\|_{F}=\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}*_{\\Phi}\\mathcal{V}_{\\star}^{H}\\|_{F}=\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{F},\n$$\n\nand ∥L<sub>⋆</sub> ∗<sub>Φ</sub> R<sup>H</sup><sub>△</sub>∥<sub>F</sub> = ∥U<sub>⋆</sub> ∗<sub>Φ</sub> G <sup>1/2</sup><sub>⋆</sub> ∗<sub>Φ</sub> R<sup>H</sup><sub>△</sub>∥<sub>F</sub> = ∥R<sub>△</sub> ∗<sub>Φ</sub> G <sup>1/2</sup><sub>⋆</sub> ∥<sub>F</sub> ,\n\nas well as the triangle inequality, we have\n\n$$\n\\begin{array}{l}\\|\\mathcal{L}_{\\sharp}*_{\\Phi}\\mathcal{R}_{\\sharp}^{H}-\\mathcal{X}_{\\star}\\|_{F}\\leq\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{R}_{\\star}^{H}\\|_{F}+\\|\\mathcal{L}_{\\star}*_{\\Phi}\\mathcal{R}_{\\triangle}^{H}\\|_{F}+\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{R}_{\\triangle}^{H}\\|_{F}\\\\ =\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{F}+\\|\\mathcal{R}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{F}+\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{R}_{\\triangle}^{H}\\|_{F}.\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "This together with the following upper bound\n\n$$\n\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{R}_{\\triangle}^{H}\\|_{F}=\\frac{1}{2}\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}*_{\\Phi}(\\mathcal{R}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}})^{H}\\|_{F}+\\frac{1}{2}\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}*_{\\Phi}(\\mathcal{R}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}})^{H}\\|_{F}\n$$\n\n$$\n=\\frac{1}{2\\sqrt{\\ell}}\\|\\bar{\\boldsymbol{L}}_{\\triangle}\\bar{\\boldsymbol{G}}_{\\star}^{\\frac{1}{2}}(\\bar{\\boldsymbol{R}}_{\\triangle}\\bar{\\boldsymbol{G}}_{\\star}^{-\\frac{1}{2}})^{H}\\|_{F}+\\frac{1}{2\\sqrt{\\ell}}\\|\\bar{\\boldsymbol{L}}_{\\triangle}\\bar{\\boldsymbol{G}}_{\\star}^{-\\frac{1}{2}}(\\bar{\\boldsymbol{R}}_{\\triangle}\\bar{\\boldsymbol{G}}_{\\star}^{\\frac{1}{2}})^{H}\\|_{F}\n$$\n\n$$\n\\leq\\frac{1}{2\\sqrt{\\ell}}\\|\\bar{\\boldsymbol{L}}_{\\triangle}\\bar{\\boldsymbol{G}}_{\\star}^{\\frac{1}{2}}\\|_{F}\\|\\bar{\\boldsymbol{R}}_{\\triangle}\\bar{\\boldsymbol{G}}_{\\star}^{-\\frac{1}{2}}\\|+\\frac{1}{2\\sqrt{\\ell}}\\|\\bar{\\boldsymbol{L}}_{\\triangle}\\bar{\\boldsymbol{G}}_{\\star}^{-\\frac{1}{2}}\\|\\|\\bar{\\boldsymbol{R}}_{\\triangle}\\bar{\\boldsymbol{G}}_{\\star}^{\\frac{1}{2}}\\|_{F}\n$$\n\n$$\n\\leq\\frac{1}{2\\sqrt{\\ell}}(\\|\\overline{{\\boldsymbol{L}}}_{\\triangle}\\overline{{\\boldsymbol{G}}}_{\\star}^{-\\frac{1}{2}}\\|\\vee\\|\\overline{{\\boldsymbol{R}}}_{\\triangle}\\overline{{\\boldsymbol{G}}}_{\\star}^{-\\frac{1}{2}}\\|)\\Big(\\|\\overline{{\\boldsymbol{L}}}_{\\triangle}\\overline{{\\boldsymbol{G}}}_{\\star}^{\\frac{1}{2}}\\|_{F}+\\|\\overline{{\\boldsymbol{R}}}_{\\triangle}\\overline{{\\boldsymbol{G}}}_{\\star}^{\\frac{1}{2}}\\|_{F}\\Big)\n$$\n\n$$\n=\\frac{1}{2}(\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\|\\vee\\|\\mathcal{R}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\|)\\Big(\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{F}+\\|\\mathcal{R}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{F}\\Big)\n$$\n\nfinishes the proof."}]}}, {"local_id": "T2", "statement": "Lemma 48 If\n\n$$\n\\mathrm{dist}(\\boldsymbol{\\mathcal{F}}_{t},\\boldsymbol{\\mathcal{F}}_{\\star})\\leq\\frac{\\epsilon}{\\sqrt{\\ell}}\\rho^{t}\\bar{\\sigma}_{s_{r}}(\\boldsymbol{\\mathcal{X}}_{\\star})\n$$\n\nfor some $0 < \\epsilon < 1$ , then the following inequalities hold\n\n$$\n\\|\\boldsymbol{\\mathcal{L}}_{\\triangle}*_{\\Phi}\\boldsymbol{\\mathcal{G}}_{\\star}^{\\frac{1}{2}}\\|_{F}\\vee\\|\\boldsymbol{\\mathcal{R}}_{\\triangle}*_{\\Phi}\\boldsymbol{\\mathcal{G}}_{\\star}^{\\frac{1}{2}}\\|_{F}\\leq\\frac{\\epsilon}{\\sqrt{\\ell}}\\rho^{t}\\bar{\\sigma}_{s_{r}}(\\boldsymbol{\\mathcal{X}}_{\\star});\n$$\n\n$$\n\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|\\vee\\|\\mathcal{R}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|\\leq\\epsilon\\rho^{t}\\bar{\\sigma}_{s_{r}}(\\boldsymbol{\\chi}_{\\star});\n$$\n\n$$\n\\|\\mathcal{L}_{\\sharp}*_{\\Phi}\\left(\\mathcal{L}_{\\sharp}^{H}*_{\\Phi}\\mathcal{L}_{\\sharp}\\right)^{-1}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|\\vee\\|\\mathcal{R}_{\\sharp}*_{\\Phi}\\left(\\mathcal{R}_{\\sharp}^{H}*_{\\Phi}\\mathcal{R}_{\\sharp}\\right)^{-1}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|\\leq\\frac{1}{1-\\epsilon}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof Notice that dist $\\begin{array}{ r }{(\\mathcal{F}_{t},\\mathcal{F}_{\\star})\\geq\\sqrt{\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{F}^{2}\\vee\\|\\mathcal{R}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{F}^{2}}.}\\end{array}$ . The first claim is directly followed by the definition of dist(·, ·). By the fact that $\\|\\pmb{\\mathscr{A}}\\|=\\|\\overline{{\\pmb{A}}}\\|\\leq\\|\\overline{{\\pmb{A}}}\\|_{F}=$ $\\|\\bar{\\pmb{A}}\\|_{F}=\\sqrt{\\ell}\\|\\pmb{A}\\|_{F}$ for any tensor, the second claim can be deduced from the first claim. As a consequence, we have $\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\|\\vee\\|\\mathcal{R}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\|\\le\\epsilon\\rho^{t}\\le\\epsilon .$ , given $\\rho=1-0.6\\eta<1$ 2 then the third claim can be obtained by applying (41) and (42)."}]}}, {"local_id": "T3", "statement": "Lemma 50 If\n\n$$\n\\mathrm{dist}(\\boldsymbol{\\mathcal{F}}_{t},\\boldsymbol{\\mathcal{F}}_{\\star})\\leq\\frac{\\epsilon}{\\sqrt{\\ell}}\\rho^{t}\\bar{\\sigma}_{s_{r}}(\\boldsymbol{\\mathcal{X}}_{\\star})\\quad \\text{and}\n$$\n\n$$\n\\sqrt{n_{1}}\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{2,\\infty}\\vee\\sqrt{n_{2}}\\|\\mathcal{R}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{2,\\infty}\\leq\\sqrt{\\frac{\\mu s_{r}}{n_{3}\\ell}}\\rho^{t}\\bar{\\sigma}_{s_{r}}(\\pmb{\\mathscr{X}}_{\\star}),\n$$\n\nthen\n\n$$\n\\|\\pmb{\\mathscr{X}}_{\\star}-\\pmb{\\mathscr{X}}_{t}\\|_{\\infty}\\leq 3\\frac{\\mu s_{r}}{n_{3}\\sqrt{n_{1}n_{2}\\ell}}\\rho^{t}\\bar{\\sigma}_{s_{r}}(\\pmb{\\mathscr{X}}_{\\star}).\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof First, based on Definition 10 and the assumption of this lemma, we have\n\n$$\n\\begin{array}{l}\\|\\mathcal{R}_{\\sharp}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\|_{2,\\infty}\\leq\\|\\mathcal{R}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{2,\\infty}\\|\\mathcal{G}_{\\star}^{-1}\\|+\\|\\mathcal{R}_{\\star}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\|_{2,\\infty}\\\\ \\leq(\\rho^{t}+1)\\sqrt{\\frac{\\mu s_{r}}{n_{2}n_{3}\\ell}}\\leq 2\\sqrt{\\frac{\\mu s_{r}}{n_{2}n_{3}\\ell}}.\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "Furthermore, using Lemma 35, we obtain\n\n$$\n\\begin{array}{l}\\|\\boldsymbol{\\mathcal{X}}_{\\star}-\\boldsymbol{\\mathcal{X}}_{t}\\|_{\\infty}=\\|\\boldsymbol{\\mathcal{L}}_{\\triangle}*_{\\Phi}\\boldsymbol{\\mathcal{R}}_{\\sharp}^{H}+\\boldsymbol{\\mathcal{L}}_{\\star}*_{\\Phi}\\boldsymbol{\\mathcal{R}}_{\\triangle}^{H}\\|_{\\infty}\\leq\\|\\boldsymbol{\\mathcal{L}}_{\\triangle}*_{\\Phi}\\boldsymbol{\\mathcal{R}}_{\\sharp}^{H}\\|_{\\infty}+\\|\\boldsymbol{\\mathcal{L}}_{\\star}*_{\\Phi}\\boldsymbol{\\mathcal{R}}_{\\triangle}^{H}\\|_{\\infty}\\\\ \\leq\\sqrt{\\ell}\\|\\mathcal{L}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{2,\\infty}\\|\\mathcal{R}_{\\sharp}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\|_{2,\\infty}+\\sqrt{\\ell}\\|\\mathcal{L}_{\\star}*_{\\Phi}\\mathcal{G}_{\\star}^{-\\frac{1}{2}}\\|_{2,\\infty}\\|\\mathcal{R}_{\\triangle}*_{\\Phi}\\mathcal{G}_{\\star}^{\\frac{1}{2}}\\|_{2,\\infty}\\\\ \\leq\\sqrt{\\ell}(2\\sqrt{\\frac{\\mu s_{r}}{n_{2}n_{3}\\ell}}+\\sqrt{\\frac{\\mu s_{r}}{n_{2}n_{3}\\ell}})\\sqrt{\\frac{\\mu s_{r}}{n_{1}n_{3}\\ell}}\\rho^{t}\\bar{\\sigma}_{s_{r}}(\\pmb{\\mathscr{X}}_{\\star})\\\\ =3\\frac{\\mu s_{r}}{n_{3}\\sqrt{n_{1}n_{2}\\ell}}\\rho^{t}\\bar{\\sigma}_{s_{r}}(\\pmb{\\mathscr{X}}_{\\star}).\\\\ \\end{array}\n$$\n\nThis finishes the proof."}]}}], "target_local_id": null, "topic_domain": "Others", "topic_subtopic": "Matrix & Tensor Analysis", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2026-25-1449", "context": "In this paper, we study the two-time-scale SA (Borkar, 1997), a variant of the classical SA algorithm, designed to identify the roots of systems comprising two coupled, potentially nonlinear equations. We focus on two unknown Lipschitz operators, $F:\\mathbb{R}^{d_{x}}\\times\\mathbb{R}^{d_{y}}\\mathbb{R}^{d_{x}}$ and $G:\\mathbb{R}^{d_{x}}\\times\\mathbb{R}^{d_{y}}\\mathbb{R}^{d_{y}}$ , with the aim of finding the root pair $(x^{\\star},y^{\\star})$ satisfying\n\n$$\n\\left\\{\\begin{array}{l}F(x^{\\star},y^{\\star})=0,\\\\ G(x^{\\star},y^{\\star})=0.\\end{array}\\right.\\tag{1}\n$$\n\nGiven that $F$ and $G$ are unknown, we assume access to a stochastic oracle that provides noisy evaluations of $F(x,y)$ and $G(x,y)$ at any input pair $(x,y)$ . Specifically, for any x and $y,$ the oracle returns $F(x,y)+\\xi$ and $G(x,y)~+~\\psi$ where $\\xi$ and $\\psi$ represent noise components. Utilizing this stochastic oracle, we apply the nonlinear two-time-scale $\\mathrm{SA}$ to solve the problem defined in (1) by iteratively updating estimates $x_{t}$ and $y_{t}$ for $x^{\\star}$ and $y^{\\star}$ , respectively, as follows\n\n$$\nx_{t+1}=x_{t}-\\alpha_{t}\\left(F(x_{t},y_{t})+\\xi_{t}\\right),\\tag{2a}\n$$\n\n$$\ny_{t+1}=y_{t}-\\beta_{t}\\left(G(x_{t},y_{t})+\\psi_{t}\\right),\\tag{2b}\n$$\n\nwhere $x_{0}$ and $y_{0}$ are initialized arbitrarily in $\\mathbb{R}^{d_{x}}$ and $\\mathbb{R}^{d_{y}}$ . The noise components $\\xi_{t}$ and $\\psi_{t}$ are modeled as martingale diference sequences. The step sizes $\\alpha_{t}$ and $\\beta_{t}$ are chosen such that $\\beta_{t}\\ll\\alpha_{t}$ , making $y_{t}$ “quasi-static” relative to $x_{t}$ . This diference in step sizes simulates a setting where $y_{t}$ remains nearly fixed while $x_{t}$ undergoes rapid updates, as discussed in Konda and Tsitsiklis (2004).\n\nAssuming that for a given y, the equation $F(x,y)=0 $ has a unique solution $H(y)$ , where H is a Lipschitz operator. Under this assumption, we can regard two-time-scale SA as a single-loop approximation of the following double-loop process\n\nInner loop: compute $x=H(y)$ , or equivalently, solve $F(x,y)=0 $ for a fixed $y,$ (3)\n\nOuter loop: iteratively find the root of $G(H(y),y)=0$\n\nThis formulation ensures that the solution to (1) satisfies $x^{\\star}=H(y^{\\star})$ ; in other words, for $y=y^{\\star},x^{\\star}$ is precisely the root of the inner loop. We designate the update of $x_{t}$ as the “fast-time-scale” update and that of $y_{t}$ as the “slow-time-scale” update, referring $x_{t}$ and $y_{t}$ as the fast iterate and slow iterate, respectively.\n\nNotation. For a vector $x,\\ \\|x\\|$ denotes the Euclidean norm; for a matrix $A,\\parallel A\\parallel$ to denote the spectral norm. We use $o(\\cdot),\\mathcal{O}(\\cdot),\\Omega(\\cdot)$ , and $\\Theta(\\cdot)$ to hide universal constants and $\\widetilde{\\mathcal{O}}(\\cdot)$ to hide both universal constants and log factors. We denote max $\\{a,b\\}$ as $a \\vee b$ and min{a, b} as $a \\wedge b .$ . The ceiling function ⌈·⌉ denotes the smallest integer greater than or equal to the input number. For two non-negative numbers a and $b,a\\lesssim b(a\\propto b)$ indicates the existence of a positive number C such that $a\\leq Cb(a=Cb)$ with $C$ depending on parameters of no interest. For two positive sequence $\\left\\{a_{n}\\right\\}$ and $\\{b_{n}\\},a_{n}\\sim b_{n}$ signifies lim $_{1}{_{n\\to\\infty}}a_{n}/b_{n}=1$ $\\operatorname{By}{\\stackrel{a.s.}{\\to}}$ we denote almost sure convergence; by $\\xrightarrow{d}$ we denote the convergence in distribution. We abbreviate $\\{1,2,\\ldots,n\\}$ as [n].\n\nAssumption 1 (Lipschitz conditions of F and H) For any fixed $\\boldsymbol{y}\\in\\mathbb{R}^{d_{y}}$ , there exists an operator $H:\\mathbb{R}^{d_{y}}\\mathbb{R}^{d_{x}}$ such that $x=H(y)$ is the unique solution of $F(x,y)=0$ . H and F satisfy that for $\\forall x\\in\\mathbb{R}^{d_{x}},y_{1},y_{2}\\in\\mathbb{R}^{d_{y}}$\n\n$$\n\\|H(y_{1})-H(y_{2})\\|\\leq L_{H}\\|y_{1}-y_{2}\\|,\\tag{11}\n$$\n\n$$\n\\|F(x,y_{1})-F(H(y_{1}),y_{1})\\|\\leq L_{F}\\|x-H(y_{1})\\|.\\tag{12}\n$$\n\nCondition (12) introduces a star-type Lipschitz condition, which is less restrictive and allows for greater flexibility in modeling and analysis.\n\nAssumption 2 (Nested Lipschitz condition of G) The operator G satisfies that ∀ $\\boldsymbol{\\mathscr{r}}_{\\mathcal{X}}\\in$ $\\mathbb{R}^{d_{x}},y\\in\\mathbb{R}^{d_{y}}$\n\n$$\n\\|G(x,y)-G(H(y),y)\\|\\leq L_{G,x}\\|x-H(y)\\|,\\tag{13}\n$$\n\n$$\n\\|G(H(y),y)-G(H(y^{\\star}),y^{\\star})\\|\\leq L_{G,y}\\|y-y^{\\star}\\|.\\tag{14}\n$$\n\nHere $y^{\\star}$ is the unique solution to $G(H(y),y)=0$\n\nAssumption 3 (Star-type strong monotonicity) For any fixed $y~\\in~\\mathbb{R}^{d_{y}},~F(\\cdot,y)$ is strongly monotone at $H(y),i.e.,$ there exists a constant $\\mu_{F}>0$ such that\n\n$$\n\\langle x-H(y),F(x,y)-F(H(y),y)\\rangle\\geq\\mu_{F}\\|x-H(y)\\|^{2},\\quad\\forall x\\in\\mathbb{R}^{d_{x}}.\\tag{15}\n$$\n\n$G(H(\\cdot),\\cdot)$ is strongly monotone at $y^{\\star},i . e .$ , there exists a constant $\\mu_{G}>0$ such that\n\n$$\n\\langle y-y^{\\star},G(H(y),y)-G(H(y^{\\star}),y^{\\star})\\rangle\\geq\\mu_{G}\\|y-y^{\\star}\\|^{2},\\quad\\forall y\\in\\mathbb{R}^{d_{y}}.\\tag{16}\n$$\n\nThe star-type strong monotonicity assumption of F and G also appears in previous work (Doan, 2022; Shen and Chen, 2022). A direct consequence of the above assumptions is $L_{F}\\geq\\mu_{F}$ and $L_{G,y}\\geq\\mu_{G}$\n\nAssumption 4 (Uniform local linearity of H) Assume that H is diferentiable and there exist constants $S_{H}\\geq 0$ and $\\delta_{H}\\in [0.5,1]$ such that $\\forall y_{1},y_{2}\\in\\mathbb{R}^{d_{y}}$ ，\n\n$$\n\\|H(y_{1})-H(y_{2})-\\nabla H(y_{2})(y_{1}-y_{2})\\|\\leq S_{H}\\|y_{1}-y_{2}\\|^{1+\\delta_{H}}.\n$$\n\nAlthough a finite-time decoupled convergence rate for nonlinear two-time-scale SA is not established in the literature, asymptotic decoupled convergence has been explored under a local linearity assumption (Mokkadem and Pelletier, 2006). Inspired by this work, we consider the following nested local linearity assumption.\n\nAssumption 5 (Nested local linearity up to order $(1+\\delta_{F},1+\\delta_{G}))$ There exist matrices $B_{1},B_{2},B_{3}$ with compatible dimensions, constants $S_{B,F},S_{B,G}\\geq 0$ and $\\delta_{F},\\delta_{G}\\in(0,1]$ such that\n\n$$\n\\|F(x,y)-B_{1}(x-H(y))\\|\\leq S_{B,F}\\left(\\|x-H(y)\\|^{1+\\delta_{F}}+\\|y-y^{\\star}\\|^{1+\\delta_{F}}\\right),\\tag{19}\n$$\n\n$$\n\\|G(x,y)-B_{2}(x-H(y))-B_{3}(y-y^{\\star})\\|\\leq S_{B,G}\\left(\\|x-H(y)\\|^{1+\\delta_{G}}+\\|y-y^{\\star}\\|^{1+\\delta_{G}}\\right).\\tag{20}\n$$\n\nProposition 2 Suppose that Assumptions $1\\mathrm{-}\\mathit{4}$ hold.\n\n(i) If Assumption 5† holds, then $\\begin{array}{ r }{\\|A_{11}\\|\\le L_{F},\\ \\|A_{21}\\|\\le L_{G,x},\\ \\frac{A_{11}+A_{11}^{\\top}}{2}\\ \\succeq\\ \\mu_{F}I}\\end{array}$ , and $\\nabla H(y^{\\star})= -A_{11}^{-1}A_{12}$ . If we further assume $\\delta_{H}\\geq\\delta_{F}\\vee\\delta_{G}.$ , then Assumption 5 holds with parameters\n\n$$\nB_{1}=A_{11},\\quad B_{2}=A_{21},\\quad B_{3}=A_{22}-A_{21}A_{11}^{-1}A_{12},\n$$\n\n$$\nS_{B,F}=S_{A,F}+L_{F}(S_{H}\\vee 2 L_{H}),S_{B,G}=S_{A,G}+L_{G,x}(S_{H}\\vee 2 L_{H}).\n$$\n\n(ii) If Assumption 5 holds, then $\\begin{array}{ r }{\\|B_{1}\\|\\le L_{F},\\|B_{2}\\|\\le L_{G,x},\\|B_{3}\\|\\le L_{G,y},\\frac{B_{1}+B_{1}^{\\top}}{2}\\succeq\\mu_{F}I,}\\end{array}$ and $\\frac{B_{3}+B_{3}^{\\top}}{2}\\succeq\\mu_{G}I .$\n\nNext, we focus on noise models. We denote by $\\mathcal{F}_{t}$ the filtration containing all the history generated by (2) before time t, i.e., $\\mathcal{F}_{t}=\\sigma\\{x_{0},y_{0},\\xi_{0},\\psi_{0},\\xi_{1},\\psi_{1},\\hdots,\\xi_{t-1},\\psi_{t-1}\\}$\n\nAssumption 6 (Conditions on the noise) The sequences of random variables $\\{\\xi_{t}\\}_{t=0}^{\\infty}$ and $\\{\\psi_{t}\\}_{t=0}^{\\infty}$ are martingale diference sequences satisfying\n\n$$\n\\begin{array}{l}\\mathbb{E}\\left[\\xi_{t}\\mid\\mathcal{F}_{t}\\right]=0,\\quad\\mathbb{E}\\left[\\psi_{t}\\mid\\mathcal{F}_{t}\\right]=0,\\quad\\mathbb{E}\\left[\\|\\xi_{t}\\|^{4}\\mid\\mathcal{F}_{t}\\right]\\leq\\Gamma_{11}^{2},\\quad\\mathbb{E}\\left[\\|\\psi_{t}\\|^{4}\\mid\\mathcal{F}_{t}\\right]\\leq\\Gamma_{22}^{2},\\end{array}\\tag{23}\n$$\n\n$$\n\\|\\mathbb{E}[\\xi_{t}\\xi_{t}^{\\top}\\mid\\mathcal{F}_{t}]\\|\\leq\\Sigma_{11},\\quad\\|\\mathbb{E}[\\psi_{t}\\psi_{t}^{\\top}\\mid\\mathcal{F}_{t}]\\|\\leq\\Sigma_{22},\\quad\\|\\mathbb{E}[\\xi_{t}\\psi_{t}^{\\top}\\mid\\mathcal{F}_{t}]\\|\\leq\\Sigma_{12}.\n$$\n\nIn Assumption 6, the boundedness of the fourth-order moments is imposed to control the higher-order error terms introduced by Assumption 5.\n\nThe final assumption is the requirements for the step sizes.\n\nAssumption 7 (Conditions on step sizes) With the constants $\\mu_{F},\\mu_{G},\\delta_{F}$ and $\\delta_{G}$ defined in Assumptions 3 and 5, the step sizes $\\{\\alpha_{t}\\}_{t=0}^{\\infty}$ and $\\{\\beta_{t}\\}_{t=0}^{\\infty}$ satisfy the following conditions:\n\n(i) Constant bounds: $\\begin{array}{ r }{\\alpha_{t}\\le\\iota_{1},\\beta_{t}\\le\\iota_{2},\\frac{\\beta_{t}}{\\alpha_{t}}\\le\\kappa,\\frac{\\beta_{t}^{2}}{\\alpha_{t}}\\le\\rho,}\\end{array}$ , where $\\iota_{1},\\iota_{2},\\kappa$ and ρ are problemdependent constants with specific forms defined in (49) in Appendix A.\n\n(ii) Growth conditions: $\\begin{array}{ r }{1\\leq\\frac{\\alpha_{t-1}}{\\alpha_{t}}\\leq 1+(\\frac{\\delta_{F}\\mu_{F}}{16}\\alpha_{t})\\wedge(\\frac{\\delta_{F}\\mu_{G}}{16}\\beta_{t})\\wedge(\\frac{\\delta_{G}\\mu_{G}}{8}\\beta_{t})\\ and\\ 1\\leq\\frac{\\beta_{t-1}}{\\beta_{t}}\\ \\leq}\\end{array}$ $1+{\\frac{\\mu_{G}}{64}}\\beta_{t}$ for any $t \\geq 1$  \n(iii) $\\frac{\\beta_{t}}{\\alpha_{t}}$ is non-increasing in t, and $\\begin{array}{ r }{\\prod_{\\tau=0}^{t}\\Big(1-\\frac{\\mu_{G}\\beta_{\\tau}}{4}\\Big)=\\mathcal{O}(\\alpha_{t}^{2})}\\end{array}$\n\nIn Section 4.1, we establish upper bounds for $\\mathbb{E}\\Vert y_{t}-y^{\\star}\\Vert^{2}$ and $\\mathbb{E}\\Vert x_{t}-H(y_{t})\\Vert^{2}$ , and prove finite-time decoupled convergence under the local linearity condition. To demonstrate the necessity of local linearity for decoupled convergence, we also construct an example in which local linearity fails and derive a corresponding lower bound in Section 4.2.\n\nTo analyze the convergence rate of $(x_{t},y_{t})$ to $(x^{\\star},y^{\\star})$ , it is common to examine the mean square errors $\\mathbb{E}[\\|x_{t}-x^{\\star}\\|^{2}]$ and $\\mathbb{E}[\\|y_{t}-y^{\\star}\\|^{2}]$ . Recall that two-time-scale SA can be viewed as an approximation of the two-loop procedure in (3). It is thus more fundamental to consider the following residual variables (Doan, 2022):\n\n$$\n\\hat{x}_{t}=x_{t}-H(y_{t}),\\quad\\hat{y}_{t}=y_{t}-y^{\\star}.\\tag{24}\n$$\n\nHere, $\\hat{x}_{t}$ and $\\hat{y}_{t}$ represent the errors of the inner and outer loops in (3), respectively. Furthermore, given the Lipschitz continuity of H in Assumption 1, we can bound $\\|x_{t}-x^{\\star}\\|$ as $\\|x_{t}-x^{\\star}\\|\\leq\\|\\hat{x}_{t}\\|+L_{H}\\|\\hat{y}_{t}\\|$ . Thus, it sufices to focus on $(\\hat{x}_{t},\\hat{y}_{t})$\n\nOur main results, based on the assumptions in Section 3, are stated as follows.\n\nIn this section, we outline the framework of our proof for Theorem 3. We first present the high-level idea and comparison with prior works, with the detailed procedure developed in Sections 5.1–5.4.\n\nHigh-level idea. Our main technical contribution is a systematic framework for handling the cross term $\\|\\mathbb{E}\\hat{x}_{t}\\hat{y}_{t}^{\\top}\\|$ in the nonlinear setting. It’s crucial for the sharp convergence characterization of the interacting sequences $\\{x_{t}\\}_{t=0}^{\\infty}$ and $\\{y_{t}\\}_{t=0}^{\\infty}$ . While similar analysis exists for the linear case (e.g., Kaledin et al. 2020), the nonlinear case is harder due to the nonlinearity of F, G, and H. We approximate these mappings by their linear parts, which introduces residual errors. The key challenge is that the cross term’s dynamics intertwine with these (higher-order) residuals. Our framework systematically tracks this interaction and controls the error accumulation via a fourth-moment analysis, showing the residuals are indeed higher-order.\n\nThe whole proof is organized into the following four steps:\n\nStep 1: Derive Convergence Rates without Local Linearity of F and G\n\nStep 2: Introduce the Matrix Cross Term and Derive Refined One-Step Descent Lemmas\n\nStep 3: Analyze the Convergence Rates of Fourth-Order Moments\n\nStep 4: Integrate the Above Ingredients and Derive Decoupled Convergence Rates\n\nComparison with prior works. Overall, the nonlinear setting requires several additional components beyond those in Doan (2022) and Kaledin et al. (2020), even though a few individual steps are similar in spirit to these earlier analyses. The first step is close in spirit to Doan (2022), but it only provides a preliminary convergence rate and serves as a starting point for our later analysis. The second step is partly inspired by Kaledin et al. (2020), where we introduce the matrix cross term $\\|\\mathbb{E}\\hat{x}_{t}\\hat{y}_{t}^{\\top}\\|$ to refine the one-step descent analysis for the slow iterate; however, in the nonlinear setting, the local linear approximations of $F .$ , $G,$ and H also produce higher-order error terms, which require additional control arguments. The third step, namely the convergence analysis of fourth-order moments, does not appear in either Doan (2022) or Kaledin et al. (2020), and is needed to handle the higher-order terms caused by nonlinearity. Finally, the last step integrates all these ingredients to establish decoupled convergence, while simultaneously handling the nonlinear remainder terms and the matrix cross term.\n\nWe first establish a coarse convergence rate without local linearity as a starting point. To this end, we first present the one-step descent lemmas for the squared errors The analysis in this step does not require the local linearity of $F$ and G in Assumption 5, nor does it require the bounded fourth-order moments in Assumption 6. The requirement on the step sizes in Assumption $7$ can also be weakened. Thus, the results in this step is based on the weaker version of Assumptions 6 and $^{7,}$ denoted as Assumptions 8 and 9, with the details deferred to Appendix B.\n\nLemma 6 (One-step descent of the fast iterate $\\hat{x}_{t})$ Suppose that Assumptions $1\\mathrm{-}\\mathit{4}$ and 8–9 hold. For any $t \\geq 0$ , we have\n\n$$\n\\begin{array}{l}\\mathbb{E}\\left[\\|\\hat{x}_{t+1}\\|^{2}\\mid\\mathcal{F}_{t}\\right]\\leq(1-\\mu_{F}\\alpha_{t})\\|\\hat{x}_{t}\\|^{2}+c_{x,1}\\beta_{t}^{2}\\|\\hat{y}_{t}\\|^{2}+2\\Gamma_{11}\\alpha_{t}^{2}\\\\ +c_{x,2}\\beta_{t}\\sqrt{1-\\alpha_{t}\\mu_{F}}\\|\\hat{x}_{t}\\|\\|\\hat{y}_{t}\\|+c_{x,3}\\beta_{t}^{2}+c_{x,4}\\frac{\\beta_{t}^{2+2\\delta_{H}}}{\\alpha_{t}},\\tag{33}\\\\ \\end{array}\n$$\n\nwhere $\\{c_{x,i}\\}_{i\\in [4]}$ are problem-dependent constants defined in (59).\n\nLemma $\\mathbf{7}$ (One-step descent of the slow iterate $\\hat{y}_{t})$ Suppose that Assumptions $1\\mathrm{-}\\mathit{4}$ and 8–9 hold. For any $t \\geq 0$ , we have\n\n$$\n\\begin{array}{l}\\mathbb{E}\\left[\\|\\hat{y}_{t+1}\\|^{2}\\mid\\mathcal{F}_{t}\\right]\\leq(1-\\mu_{G}\\beta_{t})\\|\\hat{y}_{t}\\|^{2}+L_{G,x}^{2}\\beta_{t}^{2}\\|\\hat{x}_{t}\\|^{2}\\\\ +2L_{G,x}\\beta_{t}\\sqrt{1-\\mu_{G}\\beta_{t}}\\|\\hat{x}_{t}\\|\\|\\hat{y}_{t}\\|+\\Gamma_{22}\\beta_{t}^{2}.\\\\ \\end{array}\n$$\n\nWith Assumption 5, we could replace $\\mathbb{E}{\\lVert\\hat{x}_{t}\\rVert\\lVert\\hat{y}_{t}\\rVert}$ in Lemma 6 and 7 with the matrix cross term $\\|\\mathbb{E}\\hat{x}_{t}\\hat{y}_{t}^{\\top}\\|$ at the cost of introducing the higher-order residual terms. We also need to derive the one-step descent lemma of the matrix cross term $\\|\\mathbb{E}\\hat{x}_{t}\\hat{y}_{t}^{\\top}\\|$ . To simplify the notation, we define\n\n$$\n\\mathcal{Z}_{t,\\delta}:=\\mathbb{E}\\|\\hat{x}_{t}\\|^{\\delta}+\\mathbb{E}\\|\\hat{y}_{t}\\|^{\\delta}.\n$$\n\nTo analyze the residual terms $\\Delta_{x,t},\\Delta_{y,t}$ and $\\Delta_{x y,t}$ , we focus on a single quantity $\\mathcal{Z}_{t,4}$ due to the observation from Jensen’s inequality: $\\mathcal{Z}_{t,\\delta}\\leq(\\mathcal{Z}_{t,4})^{\\delta/4}$ if $\\delta\\leq 4 .^{4}$ It motivates us to analyze fourth-order moments of errors, i.e., $\\mathbb{E}\\Vert\\hat{x}_{t+1}\\Vert^{4}$ and $\\mathbb{E}\\Vert\\hat{y}_{t+1}\\Vert^{4}$ . To that end, we derive one-step recursions for the conditional fourth-order moments. The derivation process closely parallels that of Lemmas 9 and 10. Moreover, we emphasize that the analysis of fourth-order moments does not require the local linearity of F and G in Assumption 5. Because Assumption 7 involves the parameter $\\delta_{F}$ and $\\delta_{G}$ in Assumption 5. The analysis relies on the following weaker version of Assumption 7 instead.\n\nAssumption 7† (Conditions on step sizes) The conditions in Assumption 7 holds with $\\mu_{F}=\\mu_{G}=d_{x}=d_{y}=1$ , where $d_{x}$ and $d_{y}$ appear in in (49).\n\nIn this section, we present the omitted details in Section 3.\n\nFirst, we give the detailed definition of the constants in Assumption 7.\n\n$$\n\\iota_{1}=\\frac{\\mu_{F}}{4L_{F}^{2}}\\wedge\\frac{1}{12\\mu_{F}},\\iota_{2}=\\frac{\\mu_{G}}{L_{G,x}^{2}}\\wedge\\frac{1}{14\\mu_{G}},\\kappa=\\frac{\\mu_{F}\\mu_{G}}{\\left(28d_{x}\\vee 200L_{G,y}\\right)L_{H}L_{G,x}}\\wedge\\frac{\\mu_{F}}{5\\mu_{G}},\\tag{49}\n$$\n\n$$\n\\rho=\\frac{\\mu_{F}}{(16d_{x}\\vee 200)L_{H}^{2}L_{G,x}^{2}}.\n$$\n\nIn this section, we focus on the convergence rates of nonlinear two-time-scale SA without imposing local linearity on F and G, i.e., without Assumption 5. For Assumption 4, we consider the more general case $\\delta_{H}\\in [0,1]$ . Moreover, we only need the following weaker version of Assumptions 6 and 7 throughout this section.\n\nAssumption 8 (Martingale diference noise with bounded variance) The sequences of random variables $\\{\\xi_{t}\\}_{t=0}^{\\infty}$ and $\\{\\psi_{t}\\}_{t=0}^{\\infty}$ are martingale diference sequences satisfying\n\n$$\n\\mathbb{E}\\left[\\xi_{t}\\mid\\mathcal{F}_{t}\\right]=0,\\quad\\mathbb{E}\\left[\\psi_{t}\\mid\\mathcal{F}_{t}\\right]=0,\\quad\\mathbb{E}\\left[\\|\\xi_{t}\\|^{2}\\mid\\mathcal{F}_{t}\\right]\\leq\\Gamma_{11},\\quad\\mathbb{E}\\left[\\|\\psi_{t}\\|^{2}\\mid\\mathcal{F}_{t}\\right]\\leq\\Gamma_{22}.\\tag{58}\n$$\n\nAssumption 9 (Conditions on step sizes) The step sizes $\\{\\alpha_{t}\\}_{t=0}^{\\infty}$ and $\\{\\beta_{t}\\}_{t=0}^{\\infty}$ satisfy the following conditions that $f o r \\forall t \\geq 1$ :\n\n• Constant bounds: $\\begin{array}{ r }{\\alpha_{t}\\le\\iota_{1},\\beta_{t}\\le\\iota_{2},\\frac{\\beta_{t}}{\\alpha_{t}}\\le\\kappa,\\frac{\\beta_{t}^{2}}{\\alpha_{t}}\\le\\rho}\\end{array}$ with\n\n$$\n\\iota_{1}=\\frac{\\mu_{F}}{4L_{F}^{2}},\\iota_{2}=\\frac{\\mu_{G}}{L_{G,y}^{2}},\\kappa=\\frac{\\mu_{F}\\mu_{G}}{28L_{H}L_{G,x}(1\\vee L_{G,y})}\\wedge\\frac{\\mu_{F}}{\\mu_{G}},\\rho=\\frac{\\mu_{F}}{16L_{H}^{2}L_{G,x}^{2}}.\n$$\n\n• Growth conditions: $\\begin{array}{ r }{1\\leq\\frac{\\alpha_{t-1}}{\\alpha_{t}}\\leq 1+(\\frac{\\mu_{F}}{4}\\alpha_{t})\\wedge(\\frac{\\mu_{G}}{8}\\beta_{t})}\\end{array}$ and $\\begin{array}{ r }{1\\leq\\frac{\\beta_{t-1}}{\\beta_{t}}\\leq 1+\\frac{\\mu_{G}}{32}\\beta_{t}}\\end{array}$\n\n$\\frac{\\beta_{t}}{\\alpha_{t}}$ is non-increasing in t, and $\\begin{array}{ r }{\\prod_{\\tau=0}^{t}\\left(1-\\frac{\\mu_{G}\\beta_{\\tau}}{4}\\right)=\\mathcal{O}(\\alpha_{t})}\\end{array}$\n\nRecall that $\\hat{x}_{t}=x_{t}-H(y_{t})$ . We then consider\n\n$$\n\\hat{x}_{t+1}=x_{t+1}-H(y_{t+1})=\\hat{x}_{t}-\\alpha_{t}F(x_{t},y_{t})-\\alpha_{t}\\xi_{t}+H(y_{t})-H(y_{t+1}),\\tag{60}\n$$\n\nwhich implies\n\n$$\n\\begin{array}{l}\\|\\hat{x}_{t+1}\\|^{2}=\\|\\hat{x}_{t}-\\alpha_{t}F(x_{t},y_{t})\\|^{2}+\\|H(y_{t})-H(y_{t+1})-\\alpha_{t}\\xi_{t}\\|^{2}\\\\ +2\\left\\langle\\hat{x}_{t}-\\alpha_{t}F\\left(x_{t},y_{t}\\right),H\\left(y_{t}\\right)-H\\left(y_{t+1}\\right)\\right\\rangle-2\\alpha_{t}\\left\\langle\\hat{x}_{t}-\\alpha_{t}F\\left(x_{t},y_{t}\\right),\\xi_{t}\\right\\rangle .\\tag{61}\\\\ \\end{array}\n$$\n\nFor the first term, noting that $F(H(y_{t}),y_{t})=0$ , we have\n\n$$\n\\begin{array}{l}\\|\\hat{x}_{t}-\\alpha_{t}F(x_{t},y_{t})\\|^{2}\\\\ =\\|\\hat{x}_{t}\\|^{2}-2\\alpha_{t}\\langle\\hat{x}_{t},F(x_{t},y_{t})-F(H(y_{t}),y_{t})\\rangle+\\alpha_{t}^{2}\\|F(x_{t},y_{t})-F(H(y_{t}),y_{t})\\|^{2}\\\\ \\leq\\|\\hat{x}_{t}\\|^{2}-2\\mu_{F}\\alpha_{t}\\|\\hat{x}_{t}\\|^{2}+L_{F}^{2}\\alpha_{t}^{2}\\|\\hat{x}_{t}\\|^{2}\\leq\\left(1-\\frac{7\\mu_{F}\\alpha_{t}}{4}\\right)\\|\\hat{x}_{t}\\|^{2},\\tag{62}\\\\ \\end{array}\n$$\n\nwhere the inequality uses strong monotone and Lispchitz continuity of F in (15) and (12) respectively.\n\nFor the second term, recall that $\\mathcal{F}_{t}=\\sigma\\{x_{0},y_{0},\\xi_{0},\\psi_{0},\\xi_{1},\\psi_{1},\\hdots,\\xi_{t-1},\\psi_{t-1}\\}$ . We then take the conditional expectation of the second term on the right-hand side of (61) w.r.t. $\\mathcal{F}_{t}$ and using Assumption 8 to have\n\n$$\n\\begin{array}{l}\\mathbb{E}\\left[\\|H(y_{t})-H(y_{t+1})-\\alpha_{t}\\xi_{t}\\|^{2}\\mid\\mathcal{F}_{t}\\right]\\\\ \\leq 2L_{H}^{2}\\beta_{t}^{2}\\left\\|G(x_{t},y_{t})\\right\\|^{2}+2L_{H}^{2}\\beta_{t}^{2}\\mathbb{E}\\left[\\left\\|\\psi_{t}\\right\\|^{2}\\mid\\mathcal{F}_{t}\\right]+2\\alpha_{t}^{2}\\mathbb{E}\\left[\\left\\|\\xi_{t}\\right\\|^{2}\\mid\\mathcal{F}_{t}\\right]\\\\ \\leq 4L_{H}^{2}L_{G,x}^{2}\\beta_{t}^{2}\\|\\hat{x}_{t}\\|^{2}+4L_{H}^{2}L_{G,y}^{2}\\beta_{t}^{2}\\|\\hat{y}_{t}\\|^{2}+2\\beta_{t}^{2}L_{H}^{2}\\Gamma_{22}+2\\alpha_{t}^{2}\\Gamma_{11},\\tag{63}\\\\ \\end{array}\n$$\n\nwhere the last inequality uses the following inequality that depends on the fact $G(H(y^{\\star}),y^{\\star})=$ 0 and Assumption 2\n\n$$\n\\|G(x_{t},y_{t})\\|\\leq\\|G(x_{t},y_{t})-G(H(y_{t}),y_{t})\\|+\\|G(H(y_{t}),y_{t})-G(H(y^{\\star}),y^{\\star})\\|\n$$\n\n$$\n\\leq L_{G,x}\\|\\hat{x}_{t}\\|+L_{G,y}\\|\\hat{y}_{t}\\|,\\tag{64}\n$$\n\nand thus $\\Vert G(x_{t},y_{t})\\Vert^{2}\\leq 2 L_{G,x}^{2}\\Vert\\hat{x}_{t}\\Vert^{2}+2 L_{G,y}^{2}\\Vert\\hat{y}_{t}\\Vert^{2}$\n\nCombing these two inequalities, we then obtain the preceding relation\n\n$$\n\\mathbb{E}\\left[\\langle\\hat{x}_{t}-\\alpha_{t}F(x_{t},y_{t}),H(y_{t})-H(y_{t+1})\\rangle|\\mathcal{F}_{t}\\right]\n$$\n\n$$\n\\begin{array}{l}\\leq 7L_{H}\\beta_{t}\\sqrt{1-\\alpha_{t}\\mu_{F}}\\|\\hat{x}_{t}\\|\\|G(x_{t},y_{t})\\|+4S_{H}\\sqrt{1-\\alpha_{t}\\mu_{F}}\\beta_{t}^{1+\\delta_{H}}\\|\\hat{x}_{t}\\|\\Gamma_{22}^{\\frac{1+\\delta_{H}}{2}}\\\\ \\stackrel{(64)}{\\leq}7L_{H}\\beta_{t}\\left(L_{G,x}\\sqrt{1-\\alpha_{t}\\mu_{F}}\\|\\hat{x}_{t}\\|^{2}+L_{G,y}\\sqrt{1-\\alpha_{t}\\mu_{F}}\\|\\hat{x}_{t}\\|\\|\\hat{y}_{t}\\|\\right)+4S_{H}\\sqrt{1-\\alpha_{t}\\mu_{F}}\\beta_{t}^{1+\\delta_{H}}\\|\\hat{x}_{t}\\|\\Gamma_{22}^{\\frac{1+\\delta_{H}}{2}}\\\\ \\leq\\frac{\\mu_{F}\\alpha_{t}}{2}\\|\\hat{x}_{t}\\|^{2}+7L_{H}L_{G,y}\\beta_{t}\\sqrt{1-\\alpha_{t}\\mu_{F}}\\|\\hat{x}_{t}\\|\\|\\hat{y}_{t}\\|+\\frac{16S_{H}^{2}\\Gamma_{22}^{1+\\delta_{H}}}{\\mu_{F}}\\frac{\\beta_{t}^{2+2\\delta_{H}}}{\\alpha_{t}}\\tag{67}\\\\ \\end{array}\n$$\n\nwhere the last inequality uses the relation obtained from the choices of step sizes $7 L_{H}L_{G,x}\\beta_{t}\\leq$ $\\frac{\\mu_{F}\\alpha_{t}}{4}$ and Cauchy–Schwarz inequality.\n\nProof [Proof of Lemma 7] Recall that $\\hat{y}=y-y^{\\star}$ . Using (2) we consider\n\n$$\n\\hat{y}_{t+1}=\\hat{y}_{t}-\\beta_{t}G(H(y_{t}),y_{t})+\\beta_{t}\\left(G(H(y_{t}),y_{t})-G(x_{t},y_{t})\\right)-\\beta_{t}\\psi_{t},\\tag{68}\n$$\n\nwhich implies that\n\n$$\n\\begin{array}{l}\\|\\hat{y}_{t+1}\\|^{2}=\\|\\hat{y}_{t}-\\beta_{t}G(H(y_{t}),y_{t})\\|^{2}+\\|\\beta_{t}\\left(G(H(y_{t}),y_{t})-G(x_{t},y_{t})\\right)-\\beta_{t}\\psi_{t}\\|^{2}\\\\ +2\\beta_{t}\\langle\\hat{y}_{t}-\\beta_{t}G(H(y_{t}),y_{t}),G(H(y_{t}),y_{t})-G(x_{t},y_{t})\\rangle-2\\beta_{t}\\langle\\hat{y}_{t}-\\beta_{t}G(H(y_{t}),y_{t}),\\psi_{t}\\rangle .\\tag{69}\\\\ \\end{array}\n$$\n\nWe next analyze each term on the right-hand side of (69).\n\nFor the first term, we have that\n\n$$\n\\begin{array}{l}\\|\\hat{y}_{t}-\\beta_{t}G(H(y_{t}),y_{t})\\|^{2}\\stackrel{(16)}{\\leq}\\|\\hat{y}_{t}\\|^{2}-2\\mu_{G}\\beta_{t}\\|\\hat{y}_{t}\\|^{2}+\\beta_{t}^{2}\\|G(H(y_{t}),y_{t})-G(H(y^{\\star}),y^{\\star})\\|^{2}\\\\ \\leq(1-2\\mu_{G}\\beta_{t})\\|\\hat{y}_{t}\\|^{2}+L_{G,y}^{2}\\beta_{t}^{2}\\|\\hat{y}_{t}\\|^{2}\\leq(1-\\mu_{G}\\beta_{t})\\|\\hat{y}_{t}\\|^{2},\\tag{70}\\\\ \\end{array}\n$$\n\nwhere the first inequality also uses $G(H(y^{\\star}),y^{\\star})=0$ and the last inequality uses the relation $L_{G,y}^{2}\\beta_{t}^{2}\\le\\mu_{G}\\beta_{t}$\n\nFor the second term, taking the conditional expectation on its both sides w.r.t $\\mathcal{F}_{t}$ and using Assumptions 8 and 2, we have\n\n$$\n\\mathbb{E}\\left[\\|\\beta_{t}\\left(G(H(y_{t}),y_{t})-G(x_{t},y_{t})\\right)-\\beta_{t}\\psi_{t}\\|^{2}\\mid\\mathcal{F}_{t}\\right]\\leq L_{G,x}^{2}\\beta_{t}^{2}\\|\\hat{x}_{t}\\|^{2}+\\beta_{t}^{2}\\Gamma_{22}.\\tag{71}\n$$\n\nFor the third term, it follows that\n\n$$\n\\begin{array}{l}2\\beta_{t}\\langle\\hat{y}_{t}-\\beta_{t}G(H(y_{t}),y_{t}),G(H(y_{t}),y_{t})-G(x_{t},y_{t})\\rangle\\\\ \\stackrel{(13)}{\\leq}2\\beta_{t}\\|\\hat{y}_{t}-\\beta_{t}G(H(y_{t}),y_{t})\\|\\cdot L_{G,x}\\|\\hat{x}_{t}\\|\\stackrel{(70)}{\\leq}2L_{G,x}\\beta_{t}\\sqrt{1-\\mu_{G}\\beta_{t}}\\|\\hat{x}_{t}\\|\\|\\hat{y}_{t}\\|.\\tag{72}\\\\ \\end{array}\n$$\n\nFinally, taking the conditional expectation of (69) w.r.t $\\mathcal{F}_{t}$ and using (70)–(72) yields\n\n$$\n\\mathbb{E}\\left[\\|\\hat{y}_{t+1}\\|^{2}\\mid\\mathcal{F}_{t}\\right]\\leq(1-\\mu_{G}\\beta_{t})\\|\\hat{y}_{t}\\|^{2}+L_{G,x}^{2}\\beta_{t}^{2}\\|\\hat{x}_{t}\\|^{2}+\\beta_{t}^{2}\\Gamma_{22}+2L_{G,x}\\beta_{t}\\sqrt{1-\\mu_{G}\\beta_{t}}\\|\\hat{x}_{t}\\|\\|\\hat{y}_{t}\\|,\n$$\n\nwhich concludes our proof.\n\nBefore present the detailed proof, we first give an operator decomposition lemma, which states that with Assumption 5, we could decompose the nonlinear operators F and G as the linear parts plus higher-order error terms.\n\nLemma 19 (Operator decomposition) Suppose that Assumption 5 holds. With $\\hat{x}_{t}$ and $\\hat{y}_{t}$ defined in (24), we have the following results.\n\n(i) $F(x_{t},y_{t})=B_{1}\\hat{x}_{t}+R_{t}^{F}$ with $\\lvert|R_{t}^{F}\\rvert|\\leq S_{B,F}(\\lvert|\\hat{x}_{t}\\rvert|^{1+\\delta_{F}}+\\lvert|\\hat{y}_{t}\\rvert|^{1+\\delta_{F}})$ . It follows that $\\hat{x}_{t}-\\alpha_{t}F(x_{t},y_{t})=(I-\\alpha_{t}B_{1})\\hat{x}_{t}-\\alpha_{t}R_{t}^{F}$ . We further have $\\|I-\\alpha_{t}B_{1}\\|\\le 1-\\mu_{F}\\alpha_{t}$ if $\\begin{array}{ r }{0\\leq\\alpha_{t}\\leq\\frac{\\mu_{F}}{L_{F}^{2}}.}\\end{array}$  \n(ii) G $(H(y_{t}),y_{t})~=~B_{3}\\hat{y}_{t}+R_{t}^{G H}$ with $\\|R_{t}^{G H}\\|~\\le~S_{B,G}\\|\\hat{y}_{t}\\|^{1+\\delta_{G}}$ . It follows that $\\hat{y}_{t}-$ $\\beta_{t}G(H(y_{t}),y_{t})=(I-\\beta_{t}B_{3})\\hat{y}_{t}-\\beta_{t}R_{t}^{G H}$ . We further have $\\|I-\\beta_{t}B_{3}\\|\\le 1-\\mu_{G}\\beta_{t}$ $\\begin{array}{ r }{i f 0\\le\\beta_{t}\\le\\frac{\\mu_{G}}{L_{G,y}^{2}}.}\\end{array}$  \n(iii) G(x<sub>t</sub>, y<sub>t</sub>) = B<sub>2</sub>xˆ<sub>t</sub> + B<sub>3</sub>yˆ<sub>t</sub> + R<sup>G</sup> with ∥R<sup>G</sup>∥ ≤ S<sub>B,G</sub>(∥xˆ<sub>t</sub>∥<sup>1+δ</sup>G + ∥yˆ<sub>t</sub>∥<sup>1+δ</sup>G ).", "theorems": [{"local_id": "T1", "statement": "Lemma 10 (Refined one-step descent of the slow iterate $\\hat{y}_{t})$ Suppose that Assumptions 2–3 and 5–7 hold. We have for any $t \\geq 0$\n\n$$\n\\begin{array}{l}\\mathbb{E}\\|\\hat{y}_{t+1}\\|^{2}\\leq\\left(1-\\frac{2\\mu_{G}\\beta_{t}}{3}\\right)\\mathbb{E}\\|\\hat{y}_{t}\\|^{2}+2L_{G,x}^{2}\\beta_{t}^{2}\\mathbb{E}\\|\\hat{x}_{t}\\|^{2}+2d_{y}L_{G,x}\\beta_{t}\\|\\mathbb{E}\\hat{x}_{t}\\hat{y}_{t}^{\\top}\\|\\tag{38}\\\\ +\\Gamma_{22}\\beta_{t}^{2}+\\Delta_{y,t},\\\\ \\end{array}\n$$\n\nwhere $\\Delta_{y,t}$ is a higher-order residual given in the following\n\n$$\n\\Delta_{y,t}=S_{B,G}^{2}\\beta_{t}\\left(15d_{y}^{2}/\\mu_{G}+d_{y}^{2}\\beta_{t}+8d_{y}\\beta_{t}\\right)\\mathcal{Z}_{t,2+2\\delta_{G}}.\\tag{39}\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof [Proof of Lemma 10] The decomposition in (69) implies that\n\n$$\n\\begin{array}{l}\\mathbb{E}\\|\\hat{y}_{t+1}\\|^{2}\\stackrel{(70)+(71)}{\\leq}(1-\\mu_{G}\\beta_{t})\\mathbb{E}\\|\\hat{y}_{t}\\|^{2}+L_{G,x}^{2}\\beta_{t}^{2}\\mathbb{E}\\|\\hat{x}_{t}\\|^{2}+\\beta_{t}^{2}\\Gamma_{22}\\tag{97}\\\\ +2\\beta_{t}\\mathbb{E}\\langle\\hat{y}_{t}-\\beta_{t}G(H(y_{t}),y_{t}),G(H(y_{t}),y_{t})-G(x_{t},y_{t})\\rangle\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "The proof is almost identical to that of Lemma $7$ except that we take additional care on the last term."}, {"n": 3, "claim": "By Lemma 19, we have\n\n$$\n\\hat{y}_{t}-\\beta_{t}G(H(y_{t}),y_{t})=(I-\\beta_{t}B_{3})\\hat{y}_{t}-\\beta_{t}R_{t}^{G H}\n$$\n\n$$\nG(H(y_{t}),y_{t})=B_{3}\\hat{y}_{t}+R_{t}^{G H}\n$$\n\n$$\nG(x_{t},y_{t})=B_{2}\\hat{x}_{t}+B_{3}\\hat{y}_{t}+R_{t}^{G}\n$$\n\nwith $\\lVert R_{t}^{G H}\\rVert\\leq S_{B,G}\\rVert\\hat{y}_{t}\\rVert^{1+\\delta_{G}}$ and $\\lVert R_{t}^{G}\\rVert\\leq S_{B,G}(\\lVert\\hat{x}_{t}\\rVert^{1+\\delta_{G}}+\\lVert\\hat{y}_{t}\\rVert^{1+\\delta_{G}})$ . Therefore, we have\n\n$$\n\\begin{array}{l}\\mathbb{E}\\langle\\hat{y}_{t}-\\beta_{t}G(H(y_{t}),y_{t}),G(H(y_{t}),y_{t})-G(x_{t},y_{t})\\rangle\\\\ \\leq d_{y}\\left\\|\\mathbb{E}((I-\\beta_{t}B_{3})\\hat{y}_{t}-\\beta_{t}R_{t}^{GH})(-B_{2}\\hat{x}_{t}+R_{t}^{GH}-R_{t}^{G})^{\\top}\\right\\|\\\\ \\leq d_{y}\\Big [(\\sqrt{1-\\mu_{G}\\beta_{t}})\\|B_{2}\\|\\|\\mathbb{E}\\hat{y}_{t}\\hat{x}_{t}^{\\top}\\|+(\\sqrt{1-\\mu_{G}\\beta_{t}})\\mathbb{E}\\|\\hat{y}_{t}\\|(\\|R_{t}^{GH}\\|+\\|R_{t}^{G}\\|)\\\\ \\left.+\\beta_{t}\\|B_{2}\\|\\mathbb{E}\\|R_{t}^{GH}\\|\\|\\hat{x}_{t}\\|+\\beta_{t}\\mathbb{E}\\|R_{t}^{GH}\\|(\\|R_{t}^{GH}\\|+\\|R_{t}^{G}\\|)\\right]\\\\ \\leq d_{y}L_{G,x}\\left\\|\\mathbb{E}\\hat{x}_{t}\\hat{y}_{t}^{\\top}\\right\\|+\\frac{\\mu_{G}}{6}\\mathbb{E}\\left\\|\\hat{y}_{t}\\right\\|^{2}+\\frac{L_{G,x}^{2}}{2}\\beta_{t}\\mathbb{E}\\|\\hat{x}_{t}\\|^{2}\\tag{98}\\\\ +S_{B,G}^{2}\\left(\\frac{15d_{y}^{2}}{2\\mu_{G}}+\\frac{d_{y}^{2}}{2}\\beta_{t}+4d_{y}\\beta_{t}\\right)\\left(\\mathbb{E}\\|\\hat{x}_{t}\\|^{2+2\\delta_{G}}+\\mathbb{E}\\|\\hat{y}_{t}\\|^{2+2\\delta_{G}}\\right),\\\\ \\end{array}\n$$\n\nwhere the second inequality uses $\\|I-\\beta_{t}B_{3}\\|\\le\\sqrt{1-\\mu_{G}\\beta_{t}}$ and the last inequality uses Cauchy–Schwarz inequality."}, {"n": 4, "claim": "Plugging (98) into (97) yields\n\n$$\n\\begin{array}{l}\\mathbb{E}\\|\\hat{y}_{t+1}\\|^{2}\\stackrel{(97)}{\\leq}(1-\\mu_{G}\\beta_{t})\\mathbb{E}\\|\\hat{y}_{t}\\|^{2}+L_{G,x}^{2}\\beta_{t}^{2}\\mathbb{E}\\|\\hat{x}_{t}\\|^{2}+\\beta_{t}^{2}\\Gamma_{22}\\\\ +2\\beta_{t}\\mathbb{E}\\langle\\hat{y}_{t}-\\beta_{t}G(H(y_{t}),y_{t}),G(H(y_{t}),y_{t})-G(x_{t},y_{t})\\rangle\\\\ \\stackrel{(98)}{\\leq}\\left(1-\\frac{2\\mu_{G}\\beta_{t}}{3}\\right)\\mathbb{E}\\|\\hat{y}_{t}\\|^{2}+2L_{G,x}^{2}\\beta_{t}^{2}\\mathbb{E}\\|\\hat{x}_{t}\\|^{2}+2d_{y}L_{G,x}\\beta_{t}\\|\\mathbb{E}\\hat{x}_{t}\\hat{y}_{t}^{\\top}\\|+\\Gamma_{22}\\beta_{t}^{2}+\\Delta_{y,t},\\\\ \\end{array}\n$$\n\nwhere we use $\\Delta_{y,t}$ defined in (39) to collect the remaining terms."}]}}, {"local_id": "T2", "statement": "Lemma 12 (One-step descent of the fourth-order moment of the fast iterate $\\hat{x}_{t})$ Suppose that Assumptions $1-4,\\ 6,$ , and 7† hold. We have for any $t \\geq 0$ ，\n\n$$\n\\begin{array}{l}\\mathbb{E}\\left[\\|\\hat{x}_{t+1}\\|^{4}\\mid\\mathcal{F}_{t}\\right]\\leq(1-\\mu_{F}\\alpha_{t})\\|\\hat{x}_{t}\\|^{4}+c_{xx,1}^{de}\\beta_{t}\\|\\hat{x}_{t}\\|^{3}\\|\\hat{y}_{t}\\|+c_{xx,2}^{de}\\beta_{t}^{2}\\|\\hat{x}_{t}\\|^{2}\\|\\hat{y}_{t}\\|^{2}\\\\ +c_{xx,3}^{de}\\beta_{t}^{4}\\|\\hat{y}_{t}\\|^{4}+32\\alpha_{t}^{4}\\Gamma_{11}^{2}+224L_{H}^{4}\\beta_{t}^{4}\\Gamma_{22}^{2}\\tag{42}\\\\ +\\left(20\\Gamma_{11}\\alpha_{t}^{2}+20L_{H}^{2}\\Gamma_{22}\\beta_{t}^{2}+c_{xx,4}^{de}\\frac{\\beta_{t}^{2+2\\delta_{H}}}{\\alpha_{t}}\\right)\\|\\hat{x}_{t}\\|^{2},\\\\ \\end{array}\n$$\n\nwhere $\\{c_{x x,i}\\}_{i\\in [5]}$ are problem-dependent constants defined in (107).", "proof": {"steps": [{"n": 1, "claim": "Proof [Proof of Lemma 12] We first present the specific forms of the constants:\n\n$$\nc_{xx,1}^{de}=28L_{H}L_{G,y},c_{xx,2}^{de}=40L_{H}^{2}L_{G,y}^{2},c_{xx,3}^{de}=1280L_{H}^{4}L_{G,y}^{4},c_{xx,4}^{de}=\\frac{64S_{H}^{2}\\Gamma_{22}^{1+\\delta_{H}}}{\\mu_{F}}.\\tag{107}\n$$"}, {"n": 2, "claim": "From the proof of Lemma 6, we have\n\n$$\n\\begin{array}{l}\\|\\hat{x}_{t+1}\\|^{2}\\stackrel{(61)}{=}\\underbrace{\\|\\hat{x}_{t}-\\alpha_{t}F(x_{t},y_{t})\\|^{2}}_{\\mathcal{C}_{1}}+\\underbrace{\\|H(y_{t})-H(y_{t+1})-\\alpha_{t}\\xi_{t}\\|^{2}}_{\\mathcal{C}_{2}}\\\\ +\\underbrace{2\\langle\\hat{x}_{t}-\\alpha_{t}F(x_{t},y_{t}),H(y_{t})-H(y_{t+1})\\rangle}_{\\mathcal{C}_{3}}+\\underbrace{2\\alpha_{t}\\langle\\hat{x}_{t}-\\alpha_{t}F(x_{t},y_{t}),-\\xi_{t}\\rangle}_{\\mathcal{C}_{4}}.\\\\ \\end{array}\n$$"}, {"n": 3, "claim": "Taking the square of both sides yields\n\n$$\n\\|\\hat{x}_{t+1}\\|^{4}\\leq\\mathcal{C}_{1}^{2}+4\\mathcal{C}_{2}^{2}+4\\mathcal{C}_{3}^{2}+4\\mathcal{C}_{4}^{2}+2\\mathcal{C}_{1}(\\mathcal{C}_{2}+\\mathcal{C}_{3}+\\mathcal{C}_{4}),\\tag{108}\n$$\n\nwhere the last inequality is due to $(a+b+c)^{2}\\leq 3(a^{2}+b^{2}+c^{2})$ for any $a,b\\in\\mathbb{R}$ . We then analyze these terms respectively in the following."}, {"n": 4, "claim": "• For $\\mathcal{C}_{1}^{2}$ , we have\n\n$$\n\\mathcal{C}_{1}^{2}\\stackrel{(62)}{\\leq}\\left(1-\\frac{7\\mu_{F}\\alpha_{t}}{2}+\\frac{49\\mu_{F}^{2}\\alpha_{t}^{2}}{16}\\right)\\|\\hat{x}_{t}\\|^{4}.\\tag{109}\n$$"}, {"n": 5, "claim": "• For $\\mathcal{C}_{2}^{2}$ , taking the expectation w.r.t. $\\mathcal{F}_{t}$ , we have\n\n$$\n\\begin{array}{l}\\mathbb{E}\\left[\\mathcal{C}_{2}^{2}\\mid\\mathcal{F}_{t}\\right]\\stackrel{(11)}{\\leq}8\\left(L_{H}^{4}\\beta_{t}^{4}\\mathbb{E}\\left[\\|G(x_{t},y_{t})+\\psi_{t}\\|^{4}\\mid\\mathcal{F}_{t}\\right]+\\alpha_{t}^{4}\\Gamma_{11}^{2}\\right)\\\\ \\stackrel{(111)}{\\leq}8\\left(5L_{H}^{4}\\beta_{t}^{4}\\|G(x_{t},y_{t})\\|^{4}+7L_{H}^{4}\\beta_{t}^{4}\\Gamma_{22}^{2}+\\alpha_{t}^{4}\\Gamma_{11}^{2}\\right),\\tag{110}\\\\ \\end{array}\n$$\n\nwhere the first inequality also uses $(a+b)^{4}\\leq 8(a^{4}+b^{4})$ for any $a,b\\in\\mathbb{R}$ and the last inequality uses (111) below\n\n$$\n\\mathbb{E}\\left[\\|G(x_{t},y_{t})+\\psi_{t}\\|^{4}\\right]\\leq 5\\|G(x_{t},y_{t})\\|^{4}+7\\Gamma_{22}^{2}.\\tag{111}\n$$"}, {"n": 6, "claim": "To derive (111), we first notice that\n\n$$\n\\begin{array}{l}\\|G(x_{t},y_{t})+\\psi_{t}\\|^{4}\\leq\\|G(x_{t},y_{t})\\|^{4}+6\\|G(x_{t},y_{t})\\|^{2}\\|\\psi_{t}\\|^{2}+4\\|G(x_{t},y_{t})\\|\\|\\psi_{t}\\|^{3}+\\|\\psi_{t}\\|^{4}\\\\ +4\\|G(x_{t},y_{t})\\|^{2}\\langle G(x_{t},y_{t}),\\psi_{t}\\rangle\\\\ \\leq 5\\|G(x_{t},y_{t})\\|^{4}+7\\|\\psi_{t}\\|^{4}+4\\|G(x_{t},y_{t})\\|^{2}\\langle G(x_{t},y_{t}),\\psi_{t}\\rangle,\\\\ \\end{array}\n$$\n\nwhere the last inequality follows from Young’s inequality. Taking the conditional expectation gives (111)."}, {"n": 7, "claim": "Then we plug (64) into (110) and obtain\n\n$$\n\\mathbb{E}\\left[\\mathcal{C}_{2}^{2}\\mid\\mathcal{F}_{t}\\right]\\leq 320L_{H}^{4}L_{G,x}^{4}\\beta_{t}^{4}\\|\\hat{x}_{t}\\|^{4}+320L_{H}^{4}L_{G,y}^{4}\\beta_{t}^{4}\\|\\hat{y}_{t}\\|^{4}+56L_{H}^{4}\\beta_{t}^{4}\\Gamma_{22}^{2}+8\\alpha_{t}^{4}\\Gamma_{11}^{2},\\tag{112}\n$$\n\nwhere the inequality also uses $(a+b)^{4}\\leq 8(a^{4}+b^{4})$ for any $a,b\\in\\mathbb{R}$\n\n• For $\\mathcal{C}_{3}^{2}$ , taking the expectation w.r.t. $\\mathcal{F}_{t}$ , we have\n\n$$\n\\begin{array}{l}\\mathbb{E}\\left[\\mathcal{C}_{3}^{2}\\mid\\mathcal{F}_{t}\\right]\\stackrel{(11)+(62)}{\\leq}4L_{H}^{2}\\|\\hat{x}_{t}\\|^{2}\\mathbb{E}\\left[\\|y_{t}-y_{t+1}\\|^{2}\\mid\\mathcal{F}_{t}\\right]\\\\ \\stackrel{(64)}{\\leq}4L_{H}^{2}\\beta_{t}^{2}\\|\\hat{x}_{t}\\|^{2}\\left(2L_{G,x}^{2}\\|\\hat{x}_{t}\\|^{2}+2L_{G,y}^{2}\\|\\hat{y}_{t}\\|^{2}+\\Gamma_{22}\\right)\\\\ =8L_{H}^{2}L_{G,x}^{2}\\beta_{t}^{2}\\|\\hat{x}_{t}\\|^{4}+8L_{H}^{2}L_{G,y}^{2}\\beta_{t}^{2}\\|\\hat{x}_{t}\\|^{2}\\|\\hat{y}_{t}\\|^{2}+4L_{H}^{2}\\Gamma_{22}\\beta_{t}^{2}\\|\\hat{x}_{t}\\|^{2}.\\tag{113}\\\\ \\end{array}\n$$"}, {"n": 8, "claim": "• For $\\mathcal{C}_{4}^{2}$ , taking the expectation w.r.t. $\\mathcal{F}_{t}$ , we have\n\n$$\n\\mathbb{E}\\left[\\mathcal{C}_{4}^{2}\\mid\\mathcal{F}_{t}\\right]\\leq 4\\alpha_{t}^{2}\\|\\hat{x}_{t}-\\alpha_{t}F(x_{t},y_{t})\\|^{2}\\mathbb{E}\\left[\\|\\xi_{t}\\|^{2}\\mid\\mathcal{F}_{t}\\right]\\stackrel{(62)}{\\leq}4\\alpha_{t}^{2}\\Gamma_{11}\\|\\hat{x}_{t}\\|^{2}.\\tag{114}\n$$"}, {"n": 9, "claim": "• For $\\mathcal{C}_{1}\\mathcal{C}_{2}$ , taking the expectation w.r.t. $\\mathcal{F}_{t}$ , we have\n\n$$\n\\begin{array}{l}\\mathbb{E}\\left[\\mathcal{C}_{1}\\mathcal{C}_{2}\\mid\\mathcal{F}_{t}\\right]=\\mathcal{C}_{1}\\mathbb{E}\\left[\\mathcal{C}_{2}\\mid\\mathcal{F}_{t}\\right]\\\\ \\stackrel{(62)+(63)}{\\leq}\\|\\hat{x}_{t}\\|^{2}\\left(4L_{H}^{2}L_{G,x}^{2}\\beta_{t}^{2}\\|\\hat{x}_{t}\\|^{2}+4L_{H}^{2}L_{G,y}^{2}\\beta_{t}^{2}\\|\\hat{y}_{t}\\|^{2}+2\\beta_{t}^{2}L_{H}^{2}\\Gamma_{22}+2\\alpha_{t}^{2}\\Gamma_{11}\\right)\\\\ \\leq 4L_{H}^{2}L_{G,x}^{2}\\beta_{t}^{2}\\|\\hat{x}_{t}\\|^{4}+4L_{H}^{2}L_{G,y}^{2}\\beta_{t}^{2}\\|\\hat{x}_{t}\\|^{2}\\|\\hat{y}_{t}\\|^{2}+2L_{H}^{2}\\Gamma_{22}\\beta_{t}^{2}\\|\\hat{x}_{t}\\|^{2}+2\\Gamma_{11}\\alpha_{t}^{2}\\|\\hat{x}_{t}\\|^{2}.\\tag{115}\\\\ \\end{array}\n$$"}, {"n": 10, "claim": "• For $\\mathcal{C}_{1}\\mathcal{C}_{3}$ , taking the expectation w.r.t. $\\mathcal{F}_{t}$ , we have\n\n$$\n\\begin{array}{l}\\mathbb{E}\\left[\\mathcal{C}_{1}\\mathcal{C}_{3}\\mid\\mathcal{F}_{t}\\right]=\\mathcal{C}_{1}\\mathbb{E}\\left[\\mathcal{C}_{3}\\mid\\mathcal{F}_{t}\\right]\\\\ \\stackrel{(62)+(67)}{\\leq}\\|\\hat{x}_{t}\\|^{2}\\left(\\mu_{F}\\alpha_{t}\\|\\hat{x}_{t}\\|^{2}+14L_{H}L_{G,y}\\beta_{t}\\|\\hat{x}_{t}\\|\\|\\hat{y}_{t}\\|+\\frac{32S_{H}^{2}\\Gamma_{22}^{1+\\delta_{H}}}{\\mu_{F}}\\frac{\\beta_{t}^{2+\\delta_{H}}}{\\alpha_{t}}\\right)\\\\ \\leq\\mu_{F}\\alpha_{t}\\|\\hat{x}_{t}\\|^{4}+14L_{H}L_{G,y}\\beta_{t}\\|\\hat{x}_{t}\\|^{3}\\|\\hat{y}_{t}\\|+\\frac{32S_{H}^{2}\\Gamma_{22}^{1+\\delta_{H}}}{\\mu_{F}}\\frac{\\beta_{t}^{2+2\\delta_{H}}}{\\alpha_{t}}\\|\\hat{x}_{t}\\|^{2}.\\tag{116}\\\\ \\end{array}\n$$"}, {"n": 11, "claim": "• For $\\mathcal{C}_{1}\\mathcal{C}_{4}$ , taking the expectation w.r.t. $\\mathcal{F}_{t}$ , we have\n\n$$\n\\mathbb{E}\\left[\\mathcal{C}_{1}\\mathcal{C}_{4}\\mid\\mathcal{F}_{t}\\right]=\\mathcal{C}_{1}\\mathbb{E}\\left[\\mathcal{C}_{4}\\mid\\mathcal{F}_{t}\\right]=0.\\tag{117}\n$$"}, {"n": 12, "claim": "Moreover, since $\\begin{array}{ r }{\\alpha_{t}\\leq\\frac{1}{12\\mu_{F}}}\\end{array}$ and $\\begin{array}{ r }{\\frac{\\beta_{t}^{2}}{\\alpha_{t}}\\leq\\frac{\\mu_{F}}{200L_{H}^{2}L_{G,x}^{2}}}\\end{array}$ , we have\n\n$$\n1-\\frac{3\\mu_{F}\\alpha_{t}}{2}+\\frac{49\\mu_{F}^{2}\\alpha_{t}^{2}}{16}+40L_{H}^{2}L_{G,x}^{2}\\beta_{t}^{2}+1280L_{H}^{4}L_{G,x}^{4}\\beta_{t}^{4}\\leq 1-\\mu_{F}\\alpha_{t}.\n$$"}, {"n": 13, "claim": "Taking the expectation on both sides of (108) w.r.t. $\\mathcal{F}_{t}$ and plugging (109), (112) to (117) into it, together with the above inequality and the definition of $\\{c_{x x,i}\\}_{i\\in [4]}$ in (107), we obtain (42)."}]}}, {"local_id": "T3", "statement": "Lemma 13 (One-step descent of the fourth-order moment of the slow iterate $\\hat{y}_{t})$ Suppose that Assumptions 2, 3, 6, and 7† hold. We have for any $t \\geq 0$ ，\n\n$$\n\\begin{array}{l}\\mathbb{E}\\left[\\|\\hat{y}_{t+1}\\|^{4}\\mid\\mathcal{F}_{t}\\right]\\leq\\left(1-\\frac{3\\mu_{G}\\beta_{t}}{2}\\right)\\|\\hat{y}_{t}\\|^{4}+4L_{G,x}\\beta_{t}\\|\\hat{x}_{t}\\|\\|\\hat{y}_{t}\\|^{3}+18L_{G,x}^{2}\\beta_{t}^{2}\\|\\hat{x}_{t}\\|^{2}\\|\\hat{y}_{t}\\|^{2}\\tag{43}\\\\ +20L_{G,x}^{4}\\beta_{t}^{4}\\|\\hat{x}_{t}\\|^{4}+18\\Gamma_{22}\\beta_{t}^{2}\\|\\hat{y}_{t}\\|^{2}+28\\beta_{t}^{4}\\Gamma_{22}^{2}.\\\\ \\end{array}\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof [Proof of Lemma 13] From the proof of Lemma $^{7,}$ we have\n\n$$\n\\begin{array}{l}\\|\\hat{y}_{t+1}\\|^{2}\\stackrel{(69)}{=}\\underbrace{\\|\\hat{y}_{t}-\\beta_{t}G(H(y_{t}),y_{t})\\|^{2}}_{\\mathcal{D}_{1}}+\\underbrace{\\|\\beta_{t}\\left(G(H(y_{t}),y_{t})-G(x_{t},y_{t})\\right)-\\beta_{t}\\psi_{t}\\|^{2}}_{\\mathcal{D}_{2}}\\\\ +\\underbrace{2\\beta_{t}\\langle\\hat{y}_{t}-\\beta_{t}G(H(y_{t}),y_{t}),G(H(y_{t}),y_{t})-G(x_{t},y_{t})\\rangle}_{\\mathcal{D}_{3}}+\\underbrace{2\\beta_{t}\\langle\\hat{y}_{t}-\\beta_{t}G(H(y_{t}),y_{t}),-\\psi_{t}\\rangle}_{\\mathcal{D}_{4}}.\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "Taking the square of both sides yields\n\n$$\n\\|\\hat{y}_{t+1}\\|^{4}\\leq\\mathcal{D}_{1}^{2}+4\\mathcal{D}_{2}^{2}+4\\mathcal{D}_{3}^{2}+4\\mathcal{D}_{4}^{2}+2\\mathcal{D}_{1}(\\mathcal{D}_{2}+\\mathcal{D}_{3}+\\mathcal{D}_{4}),\\tag{118}\n$$\n\nwhere the last inequality is due to $(a+b+c)^{2}\\leq 3(a^{2}+b^{2}+c^{2})$ for any $a,b\\in\\mathbb{R}$ . We then analyze these terms respectively in the following."}, {"n": 3, "claim": "• For $\\mathcal{D}_{1}^{2}$ , we have\n\n$$\n\\mathcal{D}_{1}^{2}\\stackrel{(70)}{\\leq}(1-2\\mu_{G}\\beta_{t}+\\mu_{G}^{2}\\beta_{t}^{2})\\|\\hat{y}_{t}\\|^{4}\\leq\\left(1-\\frac{3\\mu_{G}\\beta_{t}}{2}\\right)\\|\\hat{y}_{t}\\|^{4},\\tag{119}\n$$\n\nwhere the last inequality is due to $\\begin{array}{ r }{\\beta_{t}\\le\\frac{1}{2\\mu_{G}}}\\end{array}$\n\n• For $\\mathcal{D}_{2}^{2}$ , taking the expectation w.r.t. $\\mathcal{F}_{t}$ , we have\n\n$$\n\\mathbb{E}\\left[\\mathcal{D}_{2}^{2}\\mid\\mathcal{F}_{t}\\right]=\\beta_{t}^{4}\\mathbb{E}\\left[\\left\\|G(H(y_{t}),y_{t})-G(x_{t},y_{t})-\\psi_{t}\\right\\|^{4}\\mid\\mathcal{F}_{t}\\right].\n$$"}, {"n": 4, "claim": "Similar to the derivation of (111), we can obtain\n\n$$\n\\mathbb{E}\\left[\\|G(H(y_{t}),y_{t})-G(x_{t},y_{t})-\\psi_{t}\\|^{4}\\right]\\leq 5\\|G(x_{t},y_{t})-G(H(y_{t}),y_{t})\\|^{4}+7\\Gamma_{22}^{2}.\n$$"}, {"n": 5, "claim": "It follows that\n\n$$\n\\mathbb{E}\\left[\\mathcal{D}_{2}^{2}\\mid\\mathcal{F}_{t}\\right]\\leq 5\\beta_{t}^{4}\\|G(x_{t},y_{t})-G(H(y_{t}),y_{t})\\|^{4}+7\\beta_{t}^{4}\\Gamma_{22}^{2}\\stackrel{(13)}{\\leq}5L_{G,x}^{4}\\beta_{t}^{4}\\|\\hat{x}_{t}\\|^{4}+7\\beta_{t}^{4}\\Gamma_{22}^{2}.\\tag{120}\n$$"}, {"n": 6, "claim": "• For $\\mathcal{D}_{3}^{2}$ , we have\n\n$$\n\\mathcal{D}_{3}^{2}\\leq 4\\beta_{t}^{2}\\|\\hat{y}_{t}-\\beta_{t}G(H(y_{t}),y_{t})\\|^{2}\\|G(H(y_{t}),y_{t})-G(x_{t},y_{t})\\|^{2}\\stackrel{(13)+(70)}{\\leq}4L_{G,x}^{2}\\beta_{t}^{2}\\|\\hat{y}_{t}\\|^{2}\\|\\hat{x}_{t}\\|^{2}.\\tag{121}\n$$"}, {"n": 7, "claim": "• For $\\mathcal{D}_{4}^{2}$ , taking the expectation w.r.t. $\\mathcal{F}_{t}$ , we have\n\n$$\n\\mathbb{E}\\left[\\mathcal{D}_{4}^{2}\\mid\\mathcal{F}_{t}\\right]\\leq 4\\beta_{t}^{2}\\|\\hat{y}_{t}-\\beta_{t}G(H(y_{t}),y_{t})\\|^{2}\\mathbb{E}\\left[\\|\\psi_{t}\\|^{2}\\mid\\mathcal{F}_{t}\\right]\\stackrel{(70)}{\\leq}4\\Gamma_{22}\\beta_{t}^{2}\\|\\hat{y}_{t}\\|^{2}.\\tag{122}\n$$"}, {"n": 8, "claim": "• For $\\mathcal{D}_{1}\\mathcal{D}_{2}$ , taking the expectation w.r.t. $\\mathcal{F}_{t}$ , we have\n\n$$\n\\mathbb{E}\\left[\\mathcal{D}_{1}\\mathcal{D}_{2}\\mid\\mathcal{F}_{t}\\right]=\\mathcal{D}_{1}\\mathbb{E}\\left[\\mathcal{D}_{2}\\mid\\mathcal{F}_{t}\\right]\\stackrel{(70)+(71)}{\\leq}L_{G,x}^{2}\\beta_{t}^{2}\\|\\hat{x}_{t}\\|^{2}\\|\\hat{y}_{t}\\|^{2}+\\Gamma_{22}\\beta_{t}^{2}\\|\\hat{y}_{t}\\|^{2}.\\tag{123}\n$$"}, {"n": 9, "claim": "• For $\\mathcal{D}_{1}\\mathcal{D}_{3}$ , taking the expectation w.r.t. $\\mathcal{F}_{t}$ , we have\n\n$$\n\\mathbb{E}\\left[\\mathcal{D}_{1}\\mathcal{D}_{3}\\mid\\mathcal{F}_{t}\\right]=\\mathcal{D}_{1}\\mathbb{E}\\left[\\mathcal{D}_{3}\\mid\\mathcal{F}_{t}\\right]\\stackrel{(70)+(72)}{\\leq}2L_{G,x}\\beta_{t}\\|\\hat{x}_{t}\\|\\|\\hat{y}_{t}\\|^{3}.\\tag{124}\n$$"}, {"n": 10, "claim": "• For $\\mathcal{D}_{1}\\mathcal{D}_{4}$ , taking the expectation w.r.t. $\\mathcal{F}_{t}$ , we have\n\n$$\n\\mathbb{E}\\left[\\mathcal{D}_{1}\\mathcal{D}_{4}\\mid\\mathcal{F}_{t}\\right]=\\mathcal{D}_{1}\\mathbb{E}\\left[\\mathcal{D}_{4}\\mid\\mathcal{F}_{t}\\right]=0.\\tag{125}\n$$"}, {"n": 11, "claim": "Taking the expectation on both sides of (118) w.r.t. $\\mathcal{F}_{t}$ and plugging (119) to (125) into it yields (43). ■"}]}}], "target_local_id": null, "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2025-21-0039", "context": "The limitations of dimension reduction and the potential applications of manifold fitting underline the value of formulating the manifold fitting problem, as follows. Suppose the observed data samples $X=\\{x_{i}\\in\\mathbb{R}^{D}\\}_{i=1}^{N}$ are in the form\n\n$$\nx_{i}=y_{i}+\\xi_{i},\n$$\n\nwhere $y_{1},\\cdots,y_{N}$ are unobserved variables drawn from the uniform distribution supported on the latent manifold M with dimension $d < D$ . Generally, M is assumed to be a compact and smooth sub-manifold embedded in the ambient space $\\mathbb{R}^{D}$ . The precise conditions on M will be detailed in Section 2.1. The uniform distribution assumption of $y_{i}$ sampled from M is the same as those used in the related works (Genovese et al., 2012c, 2014; Mohammed and\n\nNarayanan, 2017). Here, $\\xi_{1},\\cdots\\xi_{N}$ are drawn from a distribution G. The assumptions about the noisy distribution G difer among the related work. The simplest assumption is that the observed samples are noiseless (Feferman et al., 2016; Mohammed and Narayanan, 2017). However, some literature assumes that the noise is distributed in a bounded region centered at the origin, which means that the observed samples are located in a tube centered at M (Genovese et al., 2012b). Other literature, such as Genovese et al. (2012c, 2014); Feferman et al. (2018), assumes $G$ to be a Gaussian distribution supported on $\\mathbb{R}^{D}$ , whose density at ξ is\n\n$$\n(\\frac{1}{2\\pi\\sigma^{2}})^{\\frac{D}{2}}\\exp(-\\frac{\\|\\xi\\|_{2}^{2}}{2\\sigma^{2}}).\\tag{1.1}\n$$\n\nThe tail of the Gaussian distribution might make the theoretical analysis more challenging than in the previous two cases. Strictly speaking, previous manifold fitting methods have not directly addressed this problem, nor have they proved the convergence of the fitted manifold under this assumption. In this paper, we are concerned with the Gaussian assumption of noisy distribution, denoting it as $G_{\\sigma}$ to stress the deviation parameter σ hereafter. Under the above settings, the goal is to produce a smooth manifold $\\mathcal{M}_{\\mathrm{out}}$ convergent to M. Specifically, if $\\sigma$ is suficiently small, one could derive $M_{\\mathrm{out}}$ such that $d(x,{\\mathcal{M}})\\leq O(\\sigma)$ holds for any arbitrary $x\\in\\mathcal{M}_{\\mathrm{out}}$ . In particular, $\\mathcal{M}_{\\mathrm{out}}$ converges to M when $\\sigma  0$ The convergence with respect to $\\sigma$ is the domain that Feferman et al. (2018) is built on, although its final theoretical result is expressed through sample complexity.\n\nThroughout this paper, the latent manifold is denoted as M and our approximation to M is denoted as $\\mathcal{M}_{\\mathrm{out}}$ . For a set $A\\subset\\mathbb{R}^{D}$ and a point $\\boldsymbol{x}\\in\\mathbb{R}^{D}$ 2 $\\Pi_{A}x$ denotes the projection of x onto $A,$ namely the nearest point in A to x. Hence $\\Pi_{\\mathcal{M}}x$ is the projection of x onto the latent manifold. If there is no ambiguity, we might use $x^{*}$ instead of $\\Pi_{\\mathcal{M}}x$ for simplicity. The distance between x and A, denoted by $d(x,A)$ , is the Euclidean distance between $\\Pi_{A}{x}$ and x. For any $x^{*}\\in\\mathcal{M},T_{x^{*}}\\mathcal{M}$ denotes the tangent space of M at $x^{*}$ and $\\Pi_{x^{*}}$ denotes the orthogonal projection onto the normal space of M at $x^{*}$ . We will make frequent use of the lower-cases $c,c_{0},c_{1}$ , etc. and upper-cases $C,C_{0},C_{1}$ etc., in the rest of this paper with the lower-cases denoting generic constants less than 1, while the upper-cases denote generic constants greater than 1. Values of the generic constants may change from line to line. By constants, we mean they are independent of the radius r, the standard deviation σ or x, while the constants may depend on some other constants used to characterize the manifold, such as the reach of M.\n\nWe denote $B_{D}(x,r)$ as the Euclidean ball in $\\mathbb{R}^{D}$ centered at x of radius r, which defines a neighborhood of x. The index set $I_{x,r}$ is defined as the indices of the sample points in $B_{D}(x,r)$ , and $\\left|I_{x,r}\\right|$ denotes the cardinality of $I_{x,r}$ . As given in (1.1), σ represents the standard deviation of noise. Throughout this paper, we assume\n\n$$\nr=O(\\sqrt{\\sigma}),\\quad\\sigma<1 \\tag{2.1}\n$$\n\nwithout loss of generality, otherwise the data could be rescaled so that $\\sigma < 1$ holds. Here $r=O({\\sqrt{\\sigma}})$ means that there exist constants c and C such that $c{\\sqrt{\\sigma}}\\leq r\\leq C{\\sqrt{\\sigma}}$ . Noticing $\\sigma < 1$ , we obtain\n\n$$\nr\\leq C\\sqrt{\\sigma}<C.\\tag{2.2}\n$$\n\nThis means r can be bounded above by certain constant. In subsequent proofs, under the premise that it does not afect the final precision of conclusive upper bound, we will relax some r to C in order to simplify the proof.\n\nThe latent manifold M is supposed to be boundaryless, compact, d-dimensional, and twice diferentiable, with a reach bounded by $\\tau > 0$ . The concept reach is a measure of the regularity of the manifold, first introduced by Federer (Federer, 1959) as follows:\n\nDefinition 1 (Reach). Let M be a closed subset of $\\mathbb{R}^{D}$ . The reach of M, denoted by reach(M), is the largest number τ to have the property that any point at a distance $r < \\tau$ from M has a unique nearest point in M.\n\nThis section introduces a novel method for estimating the bias $f(x)$ from a point x to the latent manifold M, and defines the approximated manifold $\\mathcal{M}_{\\mathrm{out}}$ as the points satisfying $f(x)=\\mathbf{0}$ . Unlike the aforementioned methods, which rely on discs centered at sample points to estimate the bias $f(x)$ , here we build upon the fact that a Riemannian manifold can be locally treated as an afine space and calculate the bias from x to $T_{x^{*}}{\\mathcal{M}}$ as an equivalent measure of the bias $f(x)$ from x to $\\mathcal{M}$ . Thus, the key to addressing such a bias is to find an afine space $\\{x^{\\prime}:\\Psi_{x}^{\\alpha}(x^{\\prime}-\\mathbf{b})\\}$ approximating $T_{x^{*}}{\\mathcal{M}}$ , where $\\Psi_{x}^{\\alpha}$ estimates the orthogonal projection onto the normal space at $x^{*}$ and b estimates one points in $T_{x^{*}}{\\mathcal{M}}$\n\nIn order to approximate the orthogonal projection onto the normal space of $\\mathcal{M}$ at $x^{*}$ $\\Psi_{x}^{\\alpha}$ is defined as the weighted average of $\\{P_{x_{i}}\\}_{i\\in I_{x,r}}$ , where $P_{x_{i}}$ is the orthogonal projection perpendicular to the first d principal components in $B_{D}(x_{i},r^{\\prime})$ . Mathematically, $P_{x_{i}}=$ $V_{\\perp}V_{\\perp}^{T}$ , where $V_{\\perp}$ is the orthogonal component of V and V is the $D \\times d$ matrix whose columns are the eigenvectors corresponding to the largest d eigenvalues of $\\begin{array}{ r }{\\sum_{j\\in I_{x_{i},r^{\\prime}}}(x_{j}-x_{i})(x_{j}-x_{i})^{T}}\\end{array}$ The radius $r^{\\prime}$ should be suficiently large, so that the intersection of $B_{D}(x_{i},r^{\\prime})$ and M is nonempty. Further analysis in Section 3.1 explains that we need $r^{\\prime}\\geq 2 r$\n\nAs the weighted average of $\\{P_{x_{i}}\\}_{i\\in I_{x,r}}$\n\n$$\n\\Psi_{x}^{\\alpha}=\\Pi_{\\mathrm{hi}}(A_{x}),\\quad A_{x}=\\sum_{i\\in I_{x,r}}\\alpha_{i}(x)P_{x_{i}}.\\tag{2.4}\n$$\n\nHere, $\\Pi_{\\mathrm{hi}}(A)$ denotes the projection onto the span of the eigenvectors corresponding to the largest $D - d$ eigenvalues of A. Specifically, $\\Pi_{\\mathrm{hi}}(A)=V{\\bar{V}}^{T}$ , V is a $D\\times(D-d)$ matrix whose columns are the eigenvectors corresponding to the largest $D - d$ eigenvalues of A. Further, the weights $\\alpha_{i}:\\mathbb{R}^{D}\\mathbb{R}$ in (2.4) are defined as follows:\n\n$$\n\\tilde{\\alpha}_{i}(x)=\\left\\{\\begin{array}{l l}\\left(1-\\frac{\\|x-x_{i}\\|_{2}^{2}}{r^{2}}\\right)^{\\beta},&x\\in B_{D}(x_{i},r)\\\\ 0,&\\text{otherwise}\\end{array},\\quad\\tilde{\\alpha}(x)=\\sum_{i}\\tilde{\\alpha}_{i}(x),\\quad\\alpha_{i}(x)=\\frac{\\tilde{\\alpha}_{i}(x)}{\\tilde{\\alpha}(x)},\\right.\\tag{2.5}\n$$\n\nwith $\\beta \\geq 2$ a fixed integer guaranteeing $f(x)$ in (2.6) to be twice diferentiable.\n\nUnder the assumption that a manifold can be approximated well by an afine space locally, samples in the neighborhood of x lie close to $T_{x^{*}}{\\mathcal{M}}$ , with the exception of noise. Therefore, a convex combination of these samples also lies close to $T_{x^{*}}{\\mathcal{M}}$ . Thus, we can estimate b using the average of sample points in the neighborhood of x. Recalling the weights in (2.5), we formulate b $\\textstyle=\\sum_{i\\in I_{x,r}}\\alpha_{i}(x)x_{i}$ as the weighted average of sample points in the neighborhood of $x .$ Then the bias from x to the space $\\{x^{\\prime}:\\Psi_{x}^{\\alpha}(x^{\\prime}-\\mathbf{b})\\}$ is\n\n$$\nf(x):\\mathbb{R}^{D}\\rightarrow\\mathbb{R}^{D},\\quad f(x)=\\Psi_{x}^{\\alpha}\\bigg(x-\\sum_{i\\in I_{x,r}}\\alpha_{i}(x)x_{i}\\bigg).\\tag{2.6}\n$$\n\nFinally, the approximation is defined as\n\n$$\n\\mathcal{M}_{\\text{out}}=\\{x:d(x,\\mathcal{M})\\leq c r,f(x)=\\mathbf{0},c<1\\},\\tag{2.7}\n$$\n\nthat is, the points with zero bias.\n\nIn Theorem 4, we prove any arbitrary neighborhood of $\\mathcal{M}_{\\mathrm{out}}$ is a d-dimensional manifold in high probability. In Theorem $5,$ we characterize the convergence of $\\mathcal{M}_{\\mathrm{out}}$ in the probability $\\delta_{0}(1-\\delta)^{2}$ , where we denote\n\n$$\n\\delta_{0}=1-d\\exp\\{\\frac{-c r^{d+2}N}{2\\ln 2}\\}\\tag{2.8}\n$$\n\nTheorem 11. Suppose $r^{\\prime}=O({\\sqrt{\\sigma}})$ and $d(z,\\mathcal{M})\\leq r^{\\prime}$ . For any given $\\delta$ ,there exists $C$ such that the diference between $P_{z}$ and $\\Pi_{z}$ ∗ is bounded by\n\n$$\n\\begin{array}{l}\\|P_{z}-\\Pi_{z^{*}}\\|_{F}\\leq\\frac{C}{r^{\\prime 2}}\\frac{1}{|I_{z,r^{\\prime}}|}\\sum_{i\\in I_{z,r^{\\prime}}}\\left(\\|\\xi_{i}\\|_{2}^{4}+\\|\\xi_{i}\\|_{2}^{3}+\\|\\xi_{i}\\|_{2}^{2}+r^{\\prime}\\|\\xi_{i}\\|_{2}\\right)\\\\ +C\\bigg(r^{\\prime}+\\frac{\\|z-z^{*}\\|_{2}}{r^{\\prime}}+\\frac{\\|z-z^{*}\\|_{2}^{2}}{r^{2}}\\bigg),\\quad \\text{in probability}\\delta_{0}.\\\\ \\end{array}\n$$\n\nProposition 12. For a point x satisfying $d(x,\\mathcal{M})\\leq c r$ , there exist constants $c_{0}$ and $c_{0}^{\\prime}$ such that\n\n(i) $\\tilde{\\alpha}(x)$ is bounded below by $c_{0}|I_{x,r}|$ , with probability $1-C_{0}/\\sqrt{|I_{x,r}|}$  \n(ii) $\\tilde{\\alpha}(x)$ is bounded below by a constant $c_{0}^{\\prime}$ with probability $1-(1-c r^{d})^{N}=O(N r^{d})$\n\nLemma 13. Suppose $d(x,\\mathcal{M})\\leq c r$ with some constant $c < 1$ and $r=O({\\sqrt{\\sigma}})$ . For any given δ, there exist constants $C,c_{0}$ and $n_{0}$ such that if $N\\geq n_{0}r^{-d}$ , then $\\tilde{\\alpha}(x)\\geq c_{0}|I_{x,r}|$ with probability at least $(1-\\delta)$ and\n\n$$\n\\sum_{i\\in I_{x,r}}\\alpha_{i}(x)\\|\\xi_{i}\\|_{2}^{k}\\leq C\\sigma^{k}\\quad \\text{and}\\quad\\frac{1}{|I_{x,r}|^{2}}\\sum_{i,j\\in I_{x,r}}\\|\\xi_{i}\\|_{2}^{s}\\|\\xi_{i}\\|_{2}^{t}\\leq C\\sigma^{s+t}\\tag{3.2}\n$$\n\nhold for $k,s,t\\leq 4$ with probability at least $(1-\\delta)^{2}$\n\nTheorem 15. Suppose $d(x,\\mathcal{M})\\leq c r$ with some constant $c < 1$ and $r=O({\\sqrt{\\sigma}})$ . For any given δ, there exist constants C and $n_{0}$ such that if $N\\geq n_{0}r^{-d}$ , then\n\n$$\n\\|\\Psi_{x}^{\\alpha}-\\Pi_{x^{*}}\\|_{2}\\leq\\|\\Psi_{x}^{\\alpha}-\\Pi_{x^{*}}\\|_{F}\\leq C r \\tag{3.4}\n$$\n\nholds with probability $\\delta_{0}(1-\\delta)^{2}$\n\nTheorem 16. Suppose x $\\in\\mathcal{M}$ and $r=O({\\sqrt{\\sigma}})$ . For any given δ, there exist constants C and $n_{0}$ such that if $N\\geq n_{0}r^{-d}$ , then $\\|f(x)\\|_{2}\\leq C r^{2}$ with probability $\\delta_{0}(1-\\delta)^{2}$\n\nTheorem 17. Suppose $d(x,\\mathcal{M})\\leq c r$ and $r=O({\\sqrt{\\sigma}})$ . For any given $\\delta_{i}$ , there exist constants $C$ and $n_{0}$ such that if $N\\geq n_{0}r^{-d}$\n\n$$\n\\|\\partial_{v}f(x)-\\Psi_{x}^{\\alpha}v\\|_{2}\\leq C r,\\tag{3.7}\n$$\n\nwith probability $\\delta_{0}(1-\\delta)^{2}$\n\nCorollary 18. Suppose $d(x,{\\mathcal{M}})\\leq c r$ and $r~=~O({\\sqrt{\\sigma}})$ . For any given $\\delta,$ , there exist constants $C$ and $n_{0}$ such that $i f N\\geq n_{0}r^{-d}$\n\n$$\n\\|J_{f}(x)-\\Psi_{x}^{\\alpha}\\|_{F}\\leq C r \\tag{3.8}\n$$\n\nwith probability $\\delta_{0}(1-\\delta)^{2}$\n\nTheorem 19. Suppose $d(x,\\mathcal{M})\\leq c r$ with some constant $c < 1$ and $r=O({\\sqrt{\\sigma}})$ . For any given $\\delta,$ there exist constants C and $n_{0}$ such that if $N\\geq n_{0}r^{-d}$ , then $\\|\\partial_{v}\\partial_{u}f(x)\\|_{2}\\leq C$ with probability $\\delta_{0}(1-\\delta)^{2}$\n\nProposition 20. Let $\\begin{array}{ r }{\\epsilon=\\operatorname*{min}\\{\\sqrt{\\frac{\\alpha(x)}{|I_{x,2 r}|^{2}}\\frac{r^{3}}{\\beta}},r\\}}\\end{array}$ for given x, then\n\n$$\n\\|\\Psi_{x}^{\\alpha}-\\Psi_{z}^{\\alpha}\\|_{2}\\leq C r,\\quad\\forall z\\in B_{D}(x,\\epsilon)\n$$\n\nwith probability $\\delta_{0}\\big(1-\\delta\\big)^{2}\\big(1-(1-c r^{d})^{N}\\big)$\n\nProposition 21. For any fixed point $x\\in\\mathcal{M}_{\\mathrm{out}}$ , set $W_{x}$ to be the basis of the spanning space of $J_{f}(x)^{T}$ and\n\n$$\ng(z)=W_{x}^{T}f(z).\\tag{4.3}\n$$\n\nThe following two statements\n\n(i) $g$ is a function from $\\mathbb{R}^{D}\\ t o\\ \\mathbb{R}^{D\\times(D-d)}$  \n(ii) Given $z\\in B_{D}(x,r\\tau),g(z)=\\mathbf{0}$ if and only if $f(z)=\\mathbf{0}$\n\nhold simultaneously with probability at least $\\delta_{0}^{2}(1-\\delta)^{4}\\big(1-(1-c r^{d})^{N}\\big)$\n\nLemma 22. Suppose function $g$ is defined as $(4.3)$ . The implicit function $\\phi:\\mathbb{R}^{d}\\mathbb{R}^{D-}$ −d satisfying $g(\\cdot,\\phi(\\cdot))=\\mathbf{0}$ exists, and its first and second derivatives are bounded above by\n\n$$\n\\partial_{s}\\phi(\\zeta)\\leq C\\|(\\zeta;\\phi(\\zeta))-x\\|_{2},\\quad\\partial_{t}\\partial_{s}\\phi(\\zeta)\\leq C,\n$$\n\nwith probability at least $\\delta_{0}\\big(1-\\delta\\big)^{2}\\big(1-(1-c r^{d})^{N}\\big)$ , for any $\\|s\\|_{2}=\\|t\\|_{2}=1$", "theorems": [{"local_id": "T1", "statement": "Corollary 6. For any arbitrary $x\\in\\mathcal{M}_{\\mathrm{out}},d(x,\\mathcal{M})\\to 0$ as $\\sigma  0$ with probability at least $\\delta_{0}(1-\\delta)^{2}$", "proof": {"steps": [{"n": 1, "claim": "Proof Given $r=O({\\sqrt{\\sigma}})$ , there exists $C_{0}$ such that $r=C_{0}\\sqrt{\\sigma}$ . For any $\\varepsilon > 0$ , let $\\begin{array}{ r }{\\sigma=\\frac{\\varepsilon}{C C_{0}^{2}}}\\end{array}$ and then $d(x,\\mathcal{M})\\leq C r^{2}=C C_{0}^{2}\\sigma=\\varepsilon$"}]}}, {"local_id": "T2", "statement": "Lemma 14. Suppose x and y are two points on $\\mathcal{M}$ , then\n\n$$\n\\|\\Pi_{x}^{*}-\\Pi_{y}^{*}\\|_{2}\\leq\\|\\Pi_{x}^{*}-\\Pi_{y}^{*}\\|_{F}\\leq C\\frac{\\|x-y\\|_{2}}{\\tau}.\\tag{3.3}\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof of Lemma 14 Corollary 12 of Boissonnat et al. (2019) shows that\n\n$$\n\\left\\|\\sin\\frac{\\theta(U_{x},U_{y})}{2}\\right\\|_{2}\\leq\\frac{\\|x-y\\|_{2}}{2\\mathrm{reach}(\\mathcal{M})},\n$$\n\nwhere $U_{x}$ and $U_{y}$ are the basis of $T_{x}{\\mathcal{M}}$ and $T_{y}{\\mathcal{M}}_{:}$ respectively. Letting the orthogonal complements of $U_{x}$ and $U_{y}$ be denoted by $V_{x}$ and $V_{y}$ , respectively, we obtain $\\Pi_{x}=V_{x}V_{x}^{T}$ and $\\Pi_{y}=V_{y}V_{y}^{T}$ . Then, in accordance with (ii) of Lemma $8 .$ ,\n\n$$\n\\begin{array}{l}\\|\\Pi_{x}-\\Pi_{y}\\|_{F}=\\|V_{x}V_{x}^{T}-V_{y}V_{y}^{T}\\|_{F}=\\|U_{x}U_{x}-U_{y}U_{y}\\|_{F}\\leq C\\Big\\|\\sin\\theta(U_{x},U_{y})\\Big\\|_{2}\\\\ \\leq 2 C\\Big\\|\\sin\\frac{\\theta(U_{x},U_{y})}{2}\\Big\\|_{2}\\leq C\\frac{\\|x-y\\|_{2}}{\\tau}.\\\\ \\end{array}\n$$"}]}}, {"local_id": "T3", "statement": "Lemma 28. Suppose $d(x,\\mathcal{M})\\leq c r$ with some constant $c<1,r=O({\\sqrt{\\sigma}})$ and $\\beta \\geq 2$ . For any given δ, there exist constants C and n<sub>0</sub> such that if $N\\geq n_{0}r^{-d}$ , then\n\n$$\n\\|\\big(\\partial_{v}\\alpha_{i}(x)\\big)_{i\\in I_{x,r}}\\|_{2}\\leq\\frac{C}{r}|I_{x,r}|^{-\\frac{1}{2}}\\text{with probability} 1-\\delta .\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof By Lemma 13, $\\tilde{\\alpha}(x)\\geq c_{0}|I_{x,r}|$ with probability at least $1 - \\delta$ . Based on this, we obtain the following inequalities given $0\\leq\\tilde{\\alpha}_{i}(x)\\leq 1$ :\n\n$$\n\\begin{array}{l}\\left\\|\\left(\\partial_{v}\\alpha_{i}(x)\\right)_{i\\in I_{x,r}}\\right\\|_{2}\\leq\\left\\|\\left(\\frac{\\partial_{v}\\tilde{\\alpha}_{i}(x)}{\\tilde{\\alpha}(x)}\\right)_{i\\in I_{x,r}}\\right\\|_{2}+\\left\\|\\left(\\frac{\\left(\\partial_{v}\\tilde{\\alpha}(x)\\right)\\tilde{\\alpha}_{i}(x)}{\\tilde{\\alpha}^{2}(x)}\\right)_{i\\in I_{x,r}}\\right\\|_{2}\\\\ \\leq\\frac{C}{r}\\left\\|\\left(\\frac{\\tilde{\\alpha}_{i}(x)^{\\frac{\\beta-1}{\\beta}}}{\\tilde{\\alpha}(x)}\\right)_{i\\in I_{x,r}}\\right\\|_{2}+\\left|\\frac{\\partial_{v}\\tilde{\\alpha}(x)}{\\tilde{\\alpha}^{2}(x)}\\right|\\left\\|\\left(\\tilde{\\alpha}_{i}(x)\\right)_{i\\in I_{x,r}}\\right\\|_{2}\\\\ \\leq\\frac{C}{r}\\left\\|\\left(\\frac{1}{\\tilde{\\alpha}(x)}\\right)_{i\\in I_{x,r}}\\right\\|_{2}+ |\\frac{\\partial_{v}\\tilde{\\alpha}(x)}{\\tilde{\\alpha}^{2}(x)}|\\|(1)_{i\\in I_{x,r}}\\|_{2}\\\\ \\leq\\frac{C}{r}\\left|I_{x,r}\\right|^{-\\frac{1}{2}}+\\frac{C}{r}\\left|I_{x,r}\\right|^{\\frac{1}{2}}\\frac{\\sum_{i\\in I_{x,r}}\\tilde{\\alpha}_{i}(x)^{\\frac{\\beta-1}{\\beta}}}{\\tilde{\\alpha}^{2}(x)}\\\\ \\leq\\frac{C}{r}|I_{x,r}|^{-\\frac{1}{2}}+\\frac{C}{r}|I_{x,r}|^{\\frac{1}{2}}\\frac{|I_{x,r}|}{|I_{x,r}|^{2}}\\leq\\frac{C}{r}|I_{x,r}|^{-\\frac{1}{2}}.\\\\ \\end{array}\n$$"}]}}], "target_local_id": null, "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2025-23-0455", "context": "In this section we establish notation for reference in the rest of the text.\n\nWe let S be a countably infinite set with the discrete topology. In sections 4 and 5 where we study guarantees for kernels on arbitrary infinite discrete spaces, S can be any countable infinite set; in the rest of the paper we will specifically be interested in the case when S is the space of sequences (defined below). For any finite set B we will define |B| as its cardinality. We let N be the set of natural numbers $\\{0,1,2,\\ldots\\}$ . We also define $\\mathbb{1}(P)$ to be the indicator function, which is 1 if P is true and 0 if it is false. If $y,z\\in\\mathbb{R}$ , we define $y \\vee z$ as their maximum and $y\\wedge z{\\mathrm{~a s}}$ their minimum.\n\nFor $X \\ \\in \\ S$ , by $\\delta_{X}$ we either mean the function or measure that is 1 on X and 0 everywhere else. We call $C_{0}(S)$ the space of functions on S that vanish at infinity and $\\|\\cdot\\|_{\\infty}$ the infinity norm on $C_{0}(S)$ . We define $C_{C}(S)$ to be the set of functions on S that are non-zero at only finitely many points. We define ${\\mathcal{P}}(S)$ to be the space of probability distributions on S.\n\nA kernel is a function $k:S\\times S\\to\\mathbb{R}$ such that for any finite set of $Y_{1},\\dots,Y_{N}\\ \\in$ S and $\\begin{array}{ r }{\\alpha_{1},\\dots,\\alpha_{N}\\in\\mathbb{R},\\sum_{n,m}\\alpha_{n}\\alpha_{m}k(Y_{n},Y_{m})\\geq 0}\\end{array}$ We define the inner product $(\\cdot|\\cdot)_{k}$ on linear combinations of functions $k_{Y}=k(Y,\\cdot)$ so that $\\begin{array}{ r l }{{\\bigl(\\sum_{n=1}^{N}\\alpha_{n}k_{X_{n}}\\big|\\sum_{m=1}^{M}\\beta_{m}k_{Y_{m}}\\bigr)_{k}=}\\qquad}&{{}}\\end{array}$ $\\begin{array}{ r }{\\sum_{n,m}\\alpha_{n}\\beta_{m}k(X_{n},Y_{m})}\\end{array}$ for $X_{n},Y_{m}\\in S$ and $\\alpha_{n},\\beta_{m}\\in\\mathbb{R}$ . We denote the associated norm as $\\|\\cdot\\|_{k}$ . We say that a kernel is strictly positive definite if $\\textstyle\\sum_{n=1}^{N}\\alpha_{n}k_{X_{n}}\\neq 0$ when the $X_{n}$ are distinct and $\\alpha_{n}$ are non-zero. We define the reproducing kernel Hilbert space (RKHS) of the kernel $\\mathcal{H}_{k}$ as the Hilbert space completion using the inner product $(\\cdot|\\cdot)_{k}$ . We can write every $f\\in\\mathcal{H}_{k}$ as a function on S given by $f(X)=(f|k_{X})_{k}$ . For a set of vectors $\\{v_{\\lambda}\\}_{\\lambda}\\subset{\\mathcal{H}}_{k}$ we define span $\\{v_{\\lambda}\\}_{\\lambda}$ as the set of finite linear combinations of elements of $\\{v_{\\lambda}\\}_{\\lambda}$\n\nSay µ is a signed measure on S such that $\\begin{array}{ r }{\\int d|\\mu|(X)\\sqrt{k(X,X)}<\\infty}\\end{array}$ . Then there is a $\\textit{ f }\\in\\mathcal{H}_{k}$ such that for all $\\textit{ g }\\in\\mathcal{H}_{k}$ we have $(f|g)_{k}\\ =\\ \\textstyle\\int d\\mu(X)g(X)$ and $\\|f\\|_{k}^{2}=$ $\\begin{array}{ r }{\\int d\\mu(X)\\int d\\mu(Y)k(X,Y)}\\end{array}$ . This element of $\\mathcal{H}_{k}$ is called the “kernel embedding” of the measure $\\mu,$ denoted $\\textstyle{\\int d\\mu(X)k_{X}}$\n\nLet B be a finite set, the “alphabet” (for example, this would be the four nucleotides for DNA, or the twenty amino acids for proteins). We define the set of sequences as $\\cup_{L=0}^{\\infty}B^{L}$ where $B^{0}$ is defined to be the set containing just the sequence of length zero, ∅. For a sequence X, and numbers ${\\mathit{l}},{{\\mathit{l}}^{\\prime}}.$ , we define $X_{(:l)}$ as the first l letters in X, $X_{(l:l^{\\prime})}$ as the l<sup>0</sup> − l letters after the first l letters and $X_{(l:)}$ as all letters after the first l letters, which is potentially the empty sequence ∅. We call $X_{(l)}$ the l-th letter of X, starting counting at 0. The concatenation of sequences X, Y is denoted $X + Y$ . For any number l, the sequence X concatenated to itself l times is denoted $l \\times X$ . We call $d_{H}(X,Y)$ the Hamming distance between the sequences X and Y , that is the total number of mismatches between the two sequences after they have been padded with an infinite tail of stop symbols \\$.\n\nIn this section we introduce the discrete mass property and describe its implications for kernel flexibility. We focus on properties necessary for procedures useful in biological sequence analysis. In particular, we show that if a kernel has discrete masses, it is (1) universal (an important property for consistent regression), (2) characteristic (important for consistent two-sample testing (Gretton et al., 2012; Fukumizu et al., 2007)) and (3) metrizes the space of distributions (important for optimizing sequence distributions (Li et al., 2017; Pronzato and Zhigljavsky, 2020; Husz´ar and Duvenaud, 2012; Futami et al., 2019)). Later, we will design biological sequence kernels that have discrete masses to satisfy these three properties.\n\nWe say a kernel has discrete masses if its Hilbert space includes delta functions at all points in the space S. Note that in this section we take S to be an arbitrary discrete space; later (starting in Section 6) we will specialize to the setting where S is sequence space.\n\nDefinition 1 (The discrete mass property) We say k has discrete masses if $C_{C}(S)\\subset$ $\\mathcal{H}_{k}$ , where $C_{C}(S)$ is the set of all functions on S that are non-zero at only finitely many points. Since $\\mathcal{H}_{k}$ is a linear space, this is equivalent to $\\delta_{X}\\in\\mathcal{H}_{k}$ for all $X \\in S$\n\nNext we consider re-weighting kernels to emphasize or de-emphasize certain areas of sequence space. More precisely, we consider tilting a kernel k by some function $A:S\\to(0,\\infty)$ to obtain a new kernel $k^{A}(X,Y)=A(X)k(X,Y)A(Y)$ for $X,Y\\in S$ . The discrete mass property is preserved after tilting.\n\nProposition 16 If k is a kernel on S that has discrete masses, then $k^{A}$ has discrete masses.\n\nFinally, we look at a novel transformation of kernels, specific to biological sequences, that involves “reparameterizing” the alphabet B. The basic idea starts from representing letters in the alphabet B as one-hot encoded vectors, i.e. we treat each $b \\in B$ as a vector of length |B| with zeros everywhere except at one position. The alphabet B thus forms a basis of $\\mathbb{R}^{B}$ However, we may also consider an alternative basis $\\tilde{B}$ of $\\mathbb{R}^{B}$ . This alternative basis gives rise to an alternative set of sequences $\\tilde{S}=\\cup_{L=0}^{\\infty}\\tilde{B}^{L}$ . By treating S and $\\tilde{S}$ as sets of vectors, there will be a natural way to extend a kernel on S to $\\tilde{S}.$ . We will see that the property of discrete masses is invariant to this change in basis: if the kernel has discrete masses over $S,$ it also has discrete masses over ${\\tilde{S}}_{;}$ , and vice versa. (Note in this section and all following sections, S will be the set of sequences, rather than an arbitrary infinite discrete space.)\n\nTo be more precise, we must define what it means to apply a sequence kernel to vector encoding of a sequence. Any sequence $Y$ in $S=\\cup_{{\\cal L}=0}^{\\infty}B^{L}$ can be represented as a one-hot encoding, which consists of a vector $V\\in\\mathbb{R}^{|Y|\\times B}$ such that $V_{(l)}\\in\\mathbb{R}^{B}$ is the one-hot encoding of the letter $Y_{(l)}$ . If the vector $V\\in\\mathbb{R}^{L\\times B}$ is a one-hot encoding of a sequence then we define its embedding into $\\mathcal{H}_{k}$ as the embedding of the sequence it encodes. We can write this using the formula\n\n$$\nk_{V}=\\sum_{X\\in S: |X| =L}\\left(\\prod_{l=0}^{L-1}V_{(l),X_{(l)}}\\right)k_{X}.\\tag{3}\n$$\n\nThus if V is a one-hot encoding of $Y$ , we recover $k_{V}=k_{Y}$ . The kernel has not changed; all we have done is rewrite it to embed vector encoded sequences.\n\nWe now apply the same kernel to diferent sequences that use diferent encodings, which are not one-hot. In particular, let $\\tilde{B}$ be an alternative basis of $\\mathbb{R}^{B}$ . Each sequence $\\tilde{Y}$ in $\\tilde{S}=\\cup_{L=0}^{\\infty}\\tilde{B}^{L}$ can be represented by a vector encoding $\\tilde{V}\\in\\mathbb{R}^{|\\tilde{Y}|\\times B}$ , where $\\tilde{V}_{(l)}$ is the encoding of the letter $\\tilde{Y}_{(l)}$ . We define the kernel embedding of $\\tilde{Y}$ by plugging $\\tilde{V}$ into Equation $s ;$ it consists of a linear combination of kernel embeddings $k_{X}$ from each sequence $X \\in S$ with $|X| = |{\\tilde{V}}|$ . We will show that if the kernel k has discrete masses over $S,$ and we apply k to ${\\tilde{S}}_{:}$ it will still have discrete masses. We call this shift from the alphabet B to $\\tilde{B}$ a “reparameterization” of the kernel’s alphabet.\n\nProposition 17 Say k is a strictly positive definite kernel on $S_{i}$ , and $\\tilde{B}$ is a basis $o f\\mathbb{R}^{B}$ Then k has discrete masses as a kernel on $\\tilde{S}$ if and only if it has discrete masses as a kernel on $S$\n\nThroughout the rest of the paper, for any kernel on S and any $V,W\\in\\cup_{L=0}^{\\infty}\\mathbb{R}^{L\\times B}$ we will write $k(V,W)=(k_{V}|k_{W})_{k}$ where $k_{V}$ and k<sub>W</sub> are defined by Eqn 3.\n\nIn this section, we design kernels with discrete masses that compare sequences based on pairwise alignments. Whereas position-wise comparison kernels judge two sequences X and Y to be similar only if there are a small number of substitutions that can transform X into Y, alignment kernels judge two sequences to be similar if there are a small number of substitutions, insertions and/or deletions that can transform X into Y. Biologically, sequences that are similar in this way often share similar phenotypes, and are closely related evolutionarily. However, this notion of similarity is poorly captured by position-wise comparison kernels. For example, the sequence $X = A T G C$ difers from the sequence $Y = T G C$ only by the insertion of a single A at the start of the sequence, but kernels based on the Hamming distance will judge the two sequences to be completely diferent, with distance $d_{H}(X,Y)=4= |X|$\n\nOne popular, but problematic, technique for dealing with insertions and deletions is to pre-process the data set using a multiple sequence alignment algorithm, which adds gap symbols to each sequence to make it the same length. In essence, the idea behind multiple sequence alignment algorithms is to construct a point estimate of which letters in each sequence are evolutionarily related to one another via substitutions, and place each related letter in the same position, i.e. in the same column of the pre-processed data matrix. Once sequences have been aligned, it is more reasonable to apply a position-wise comparison kernel; insertions and deletions are compared by including the gap symbol in the alphabet B. The problem with multiple sequence alignment pre-processing is that (a) it does not take into account uncertainty in which positions are related to one another, instead relying on a point estimate, and (b) it prevents downstream machine learning methods from generalizing to unseen sequences, since adding new sequences to the data set can change the multiple sequence alignment, for instance, if the new sequence is longer than those previously observed (Weinstein and Marks, 2021).\n\nAlignment kernels ofer an alternative approach to accounting for insertions and deletions (Haussler, 1999). Rather than transform the entire data set, they consider two sequences X and Y to be similar if they difer by a small number of insertions and deletions, as well as substitutions. Alignment kernels are of special relevance to problems involving sequences with high length variation, such as in the analysis of human antibodies or of distantly related evolutionary homologs.\n\nAlignment kernels are typically computed with a dynamic programming procedure – comparing two sequences of length L costs $\\mathcal{O}(L^{2})$ (Watkins, 2000). However, the algorithm can be made much more eficient in practice with modest assumptions and the use of GPUs (Rush, 2020).\n\nIn this section we define the alignment kernel and show that it has discrete masses if and only if its hyperparameters are set to values in a certain range.\n\nIn biological sequence analysis, a pairwise alignment between two sequences is a matching between a subset of positions in each sequence, with the restrictions that (a) each position in each sequence can be matched to at most one position in the other sequence, and (b) matchings must be ordered, such that if site $l_{X}$ in sequence $X$ is matched to $l_{Y}$ in sequence $Y,$ site $l_{X}^{\\prime}>l_{X}$ in $X$ cannot match to a site $l_{Y}^{\\prime}<l_{Y}$ in Y (Durbin et al., 1998, chap. 2). For example, one alignment between the sequences $X=\\mathtt{ATGC}$ and $Y=\\mathtt{ACC}$ is,\n\n<table><tr><td>ATGC</td></tr><tr><td>|||</td></tr><tr><td>AC-C</td></tr></table>\n\nwhere vertical lines | denote a matching. Here, - is a gap symbol, denoting the fact that the nucleotide G in X is unmatched; we can interpret the G as an insertion in X relative to $Y,\\mathrm{or},$ conversely, interpret the gap - as a deletion in Y relative to X. The alignment kernel $k(X,Y)$ considers all possible pairwise alignments between X and Y, scores each alignment according to a position-wise comparison kernel applied to the aligned sequences, and sums up those scores to produce its value.\n\nMathematically, to define the alignment kernel, we start with two simpler kernels and then convolve them, following the construction of Haussler (1999). The first kernel, $k_{s}.$ evaluates matches between aligned letters. The second kernel, $k_{I}$ , evaluates insertions and deletions; it applies an “afine” gap penalty, which separately penalizes the presence of a gap and the gap’s total length. For any two kernels $k_{1}$ and $k_{2}$ on $S_{;}$ , the convolution kernel $k_{1}\\star k_{2}$ is,\n\n$$\n(k_{1}\\star k_{2})(X,Y) = \\sum_{X^{(1)} + X^{(2)} = X}\\sum_{Y^{(1)} + Y^{(2)} = Y}k_{1}(X^{(1)},Y^{(1)})k_{2}(X^{(2)},Y^{(2)})\n$$\n\nwhere the sum is over all sequences $X^{(1)},X^{(2)},Y^{(1)},Y^{(2)}\\in S$ such that $X^{(1)}+X^{(2)}=X$ and $Y^{(1)}+Y^{(2)}=Y$ . Recall that S includes the sequence of length zero, so the sum includes such terms as $X^{(1)}=X,X^{(2)}=\\varnothing$\n\nDefinition 20 (Alignment kernel) Let $k_{s}^{\\prime}$ be a strictly positive definite kernel on B. Define $k_{s}\\ b y$ extending $k_{s}^{\\prime}$ to S with $k_{s}(X,Y)=0\\ i f\\ |X|\\neq 1\\ o r\\ |Y|\\neq 1$\n\nLet $\\Delta \\mu > 0$ be the penalty for starting an insertion and let $\\mu > 0$ be the penalty for the insertion’s length. The insertion penalty function g is then defined as $g(X)=1$ if $| X | = 0$ and $g(X)=\\exp\\left(-\\Delta\\mu- |X|\\mu\\right)$ otherwise. Define the non-negative-definite kernel $k_{I}(X,Y)=g(X)g(Y)~f o r~X,Y\\in S$\n\nThe alignment kernel k is,\n\n$$\nk=\\sum_{l=0}^{\\infty}k_{I}\\star(k_{s}\\star k_{I})^{\\star l},\n$$\n\nwhere the exponent ?l denotes the convolution of l kernels $k_{s}\\star k_{I}$\n\nWe start by connecting alignment kernels to kmer spectrum kernels. The basic connection between the two kernels has led to similar dynamic programming methods to compute each (Lodhi et al., 2002) and has been noted in a special case to motivate neural network architectures (Chen et al., 2019). Here we formally describe the connection in terms of an orthonormal basis of the RKHS. We show that, for a particular tilting, the features of the alignment kernel are a weighted sum of kmer counts, allowing for some kmer mismatches. More precisely, we show that the tilted alignment kernel is equivalent to an infinite-dimensional gapped kmer spectrum kernel. Since tilting preserves discrete masses (Proposition 16), this gapped kmer spectrum kernel has discrete masses.\n\nIntuitively, a “gapped kmer” is a kmer that includes gaps; it matches another sequence if each of the non-gap characters match, regardless of the letters at the gap positions. For example, the gapped kmer sequence $\\mathtt{AT}\\mathtt{-G}$ matches ATCG, ATGG, ATAG and ATTG. Formally, a gapped kmer of length L, with M non-gap letters, is defined by an ordered set of indices $J=\\left(j_{0},\\ldots,j_{M+1}\\right)$ , with $-1=j_{-1}<j_{0}<j_{2}<\\cdots<j_{M}<j_{M}=L$ , and a sequence of M non-gap letters, $Z\\in B^{M}$ . Each $j_{m}$ for $m\\in\\{0,\\ldots,M-1\\}$ indexes the location of the non-gap letter $Z_{(m)}$ . There is a gap after the mth letter if $j_{m}+1\\ne j_{m+1}$ . Let $G(L,M,g)$ be the set of all gapped kmer indices $J,$ with length L, M non-gap letters, and g gaps. Let $G(L,g)=\\cup_{M}G(L,M,g)$ be the union over all possible M. For any sequence $X \\in S$ , and $J=\\left(j_{0},\\ldots,j_{M+1}\\right)\\in G(|X|,g)$ , define $X_{(J)}=X_{(j_{0})}+\\cdot\\cdot\\cdot+X_{(j_{M-1})}$\n\nThe following theorem equates alignment kernels to gapped kmer spectrum kernels. In particular, it shows that we can represent alignment kernels in terms of features $u_{V}(X)$ that depend on the number of times a sequence $V \\in S$ appears in X, allowing for gaps. In other words, each feature $u_{V}(X)$ depends on a sum over the counts of all gapped kmers with non-gap letters V.\n\nTheorem 24 (The features of alignment kernels are gapped kmer counts) Consider an alignment kernel k with $k_{s}(b,b^{\\prime})=\\sigma^{-1}|B|\\delta_{b^{\\prime}}(b)$ for $b,b^{\\prime}\\in\\ B$ . Tilting the kernel by $\\boldsymbol{A}(\\boldsymbol{X})\\stackrel{-}{=}\\boldsymbol{e}^{\\mu|\\boldsymbol{X}|}$ gives $k^{A}(X,Y)=A(X)A(Y)k(X,Y)$ . For all sequences $X \\in S$ , define the\n\nfeatures $\\{u_{V}(X)\\}_{V\\in S}$ , given by\n\n$$\nu_{V}(X)=e^{\\frac{1}{2}\\zeta|V|}\\sum_{g=0}^{\\infty}e^{-\\Delta\\mu g}\\sum_{J\\in G(|X|,g)}\\mathbb{1}(X_{(J)}=V),\\tag{7}\n$$\n\nwhere $\\zeta = 2 \\mu -$ log σ + log |B|. Now, $\\{u_{V}(X)\\}_{V\\in S}$ is an orthonormal basis for $\\mathcal{H}_{k^{A}}$ . We call the kernel $k^{A}(X,Y)$ the “gapped kmer spectrum kernel”.\n\nIn this section, we derive a representation of the alignment kernel in terms of features. These features are gapped kmer counts.\n\nWe now consider the general case, allowing $k_{s}$ to be any strictly positive definite kernel on letters, instead of requiring that it is diagonal. The first step is to show that any alignment kernel can be written as a reparameterization of an alignment kernel with diagonal $k_{s}$\n\nProposition 35 Consider an alignment kernel k, with parameters $\\mu,~\\Delta\\mu$ and $k_{s}$ . Let $K_{b,b^{\\prime}}=k_{s}(b,b^{\\prime})$ for $b,b^{\\prime}\\in B$ and $\\sigma=\\mathbf{1}_{B}^{T}K^{-1}\\mathbf{1}_{B}$ . There is an alphabet B<sup>˜</sup>, inducing the set of sequences $\\tilde{S}$ , such that k applied to S<sup>˜</sup> is an alignment kernel with parameters $\\mu,\\Delta\\mu$ and $k_{s}(\\tilde{b},\\tilde{b}^{\\prime})=\\sigma^{-1}|B|\\delta_{\\tilde{b}}(\\tilde{b}^{\\prime})$ for all $\\tilde{b},\\tilde{b}^{\\prime}\\in\\tilde{B}$ . As well, for any $\\begin{array}{ r }{b\\in\\mathcal{B},\\sum_{\\tilde{b}\\in\\tilde{\\mathcal{B}}}k_{s}(\\tilde{b},b)=\\sigma^{-1}|\\mathcal{B}|}\\end{array}$ . We call $\\tilde{B}$ a rectified alphabet for k and $\\tilde{S}$ a rectified set of sequences for k.\n\nIn this section we show that alignment kernels have discrete masses under certain parameter settings when the alphabet size is one. This result is crucial for proving the more general result, covering arbitrary alphabet sizes, in Theorem 21 and Appendix C. It builds on the feature representation described in Theorem 24 and Appendix E.\n\nThe result will depend on the theory of the Riordan group, for which we will need to introduce some notation. If $\\textstyle f(x)=\\sum_{i=0}^{\\infty}C_{i}x^{i}$ is a formal power series, define $[x^{n}]f=C_{n}$ For two formal power series $f(y),g(y)$ we define the Riordan array of $f$ and $g$ as the infinite upper triangular matrix with entries\n\n$$\nQ_{L^{\\prime},L}=[x^{L^{\\prime}}y^{L}]\\frac{g(y)}{1-x y f(y)}=[x^{L^{\\prime}}y^{L}]\\sum_{l=0}^{\\infty}g(y)x^{l}y^{l}f(y)^{l}=[y^{L}]y^{L^{\\prime}}g(y)f(y)^{L^{\\prime}}\n$$\n\nfor numbers $L,L^{\\prime}\\ge 0$ . We say that $Q$ is in the Bell subgroup if $f = g$ . The result we will need is that one can invert a Riordan array by inverting the power series $f$ and $g$ (Shapiro et al., 1991).\n\nProposition 38 (Shapiro et al., 1991) Let\n\n$$\nQ_{L^{\\prime},L}=[x^{L^{\\prime}}y^{L}]\\frac{g(y)}{1-x y f(y)}\n$$\n\nand let $\\bar{f}$ be the unique power series such that the equation $y\\bar{f}(y)f(y\\bar{f}(y))=y~h o l d s .^{\\mathrm{\\tiny~8}}$ Then, letting $\\bar{g}(y)=\\left(g(y\\bar{f}(y))\\right)^{-1}$ ，\n\n$$\nQ_{L^{\\prime},L}^{-1}=[x^{L^{\\prime}}y^{L}]\\frac{\\bar{g}(y)}{1-x y\\bar{f}(y)}.\n$$\n\nNote $i f Q$ is in the Bell subgroup, $g(y)=f(y)$ and $\\bar{f}(y)f(y\\bar{f}(y))=1\\ s o\\ \\bar{g}(y)=\\bar{f}(y)$ Thus $Q^{-1}$ is also in the Bell subgroup.", "theorems": [{"local_id": "T1", "statement": "Theorem 34 (proofofTheorem $\\it{24})$ Consider an alignment kernel k with $k_{s}(b,b^{\\prime})=\\sigma^{-1}|B|\\delta_{b^{\\prime}}(b)$ $f o r\\ b,b^{\\prime}\\in B$ . Tilting the kernel by ${\\dot{A}}(X)=e^{\\mu|X|}$ gives $k^{A}(X,Y)=A(X)A(Y)k(X,Y)$ . For all sequences $X \\in S$ , define the features $\\{u_{V}(X)\\}_{V\\in S},g i v e n\\ b y$\n\n$$\nu_{V}(X)=e^{\\frac{1}{2}\\zeta|V|}\\sum_{g=0}^{\\infty}e^{-\\Delta\\mu g}\\sum_{J\\in G(|X|,g)}\\mathbb{1}(X_{(J)}=V),\\tag{11}\n$$\n\nwhere $\\zeta=2\\mu-\\log\\sigma+\\log|\\boldsymbol{\\mathcal{B}}|$ . Now we show that $\\{u_{V}(X)\\}_{V\\in S}$ is an orthonormal basis for $\\mathcal{H}_{k^{A}}$", "proof": {"steps": [{"n": 1, "claim": "Proof We can represent alignments between two sequences $X \\in S$ and $Y \\in S$ by pairs of indices $J\\in G(|X|,M),J^{\\prime}\\in G(|Y|,M)$ that describe which letters in X and $Y$ are aligned. In particular, $k(X,Y)$ is equal to\n\n$$\n\\begin{array}{l}\\sum_{M=0}^{\\infty}\\sum_{(j_{l})_{l= -1}^{M}\\in G(|X|,M)}\\sum_{(j_{l}^{\\prime})_{l= -1}^{M}\\in G(|Y|,M)}k_{i}(X_{(:j_{0})},Y_{(:j_{0}^{\\prime})})\\\\ \\times\\prod_{m=0}^{M-1}k_{s}(X_{(j_{m})},Y_{(j_{m}^{\\prime})})k_{i}(X_{(j_{m+1:j_{m+1}})},Y_{(j_{m}^{\\prime}+1:j_{m+1}^{\\prime})})\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "By our assumption on $k_{s}$ , for each alignment $J\\in G(|X|,M,g),J^{\\prime}\\in G(|Y|,M,g^{\\prime})$ , the contribution from the aligned letters, $\\begin{array}{ r l }{\\prod_{m=0}^{M-1}k_{s}(X_{(j_{m})},\\ddot{Y}_{(j_{m}^{\\prime})})}&{{}}\\end{array}$ , is 0 if $X_{(J)}\\ne Y_{(J^{\\prime})}$ and is $\\sigma^{-M}|B|^{M}$ otherwise. On the other hand, the contribution from the insertions and deletions is $\\begin{array}{ r }{\\prod_{m=0}^{\\underline{{M}}^{\\underline{{\\cdot}}}1}k_{I}(X_{(j_{m};j_{m+1})},Y_{(j_{m}^{\\prime}+1;j_{m+1}^{\\prime})})\\ =\\ e^{-\\mu(|X| -M+ |Y| -M)}e^{-\\Delta\\mu(g+g^{\\prime})}}\\end{array}$ ."}, {"n": 3, "claim": "Thus, $k^{A}(X,Y)$ is equal to\n\n$$\n\\begin{array}{l}\\sum_{M=0}^{\\infty}\\left(e^{2\\mu}\\sigma^{-1}|\\mathcal{B}|\\right)^{M}\\sum_{g,g^{\\prime}}e^{-\\Delta\\mu(g+g^{\\prime})}\\sum_{J\\in G(|X|,M,g)}\\sum_{J^{\\prime}\\in G(|Y|,M,g^{\\prime})}\\mathbb{1}(X_{(J)}=Y_{(J^{\\prime})})\\\\ =\\sum_{M=0}^{\\infty}e^{\\zeta M}\\sum_{V\\in\\mathcal{B}^{M}}\\sum_{g,g^{\\prime}}e^{-\\Delta\\mu(g+g^{\\prime})}\\sum_{J\\in G(|X|,M,g)}\\sum_{J^{\\prime}\\in G(|Y|,M,g^{\\prime})}\\mathbb{1}(X_{(J)}=V)\\mathbb{1}(Y_{(J^{\\prime})}=V)\\tag{12}\\\\ =\\sum_{M=0}^{\\infty}\\sum_{V\\in\\mathcal{B}^{M}}u_{V}(X)u_{V}(Y)\\\\ \\end{array}\n$$"}, {"n": 4, "claim": "Thus the features of $k^{A}$ are $X\\mapsto(u_{V}(X))_{V\\in S}$\n\nNow we will show that $u_{V}\\in\\mathcal{H}_{k}$ for all $V \\in S$ and that $(u_{V})_{V}$ make up an orthonormal basis. Define the infinite matrix Q as $Q_{V,Z}=e^{-\\frac{1}{2}\\zeta|V|}u_{V}(Z)$ for all $V,Z\\in S$ . Pick some total ordering, $\\leq,$ of $S$ such that $V\\leq Z{\\mathrm{~i f~}}|V|\\leq|Z|$ . In this case, $Q$ is an infinite upper triangular matrix with 1’s along the diagonal. Thus $Q$ has an infinite upper triangular matrix $Q^{-1}$ as its inverse with 1s along its diagonal. In the remainder of the proof, we will represent any $\\textstyle\\sum_{Z}\\alpha_{Z}k_{Z}^{A}\\in\\operatorname{span}\\{k_{Z}^{A}\\}_{Z\\in S}$ as an infinite vector, indexed by elements of $S_{i}$ with $\\alpha_{Z}$ in position $Z .\\mathrm{~\\tiny~\\textnormal~{~S o~}~}$ , for any function $f\\in\\operatorname{span}\\{k_{Z}^{A}\\}_{Z\\in S}$ , we can think of the matrix-vector product $Q f$ as the featurization of f."}, {"n": 5, "claim": "Now, for any $f,g\\in\\operatorname{span}\\{k_{Z}^{A}\\}_{Z\\in S}$ , we have from Equation 12,\n\n$$\n(f|g)_{k^{A}}=f^{T}Q^{T}\\mathrm{diag}(e^{\\zeta|V|})_{V\\in S}Q g,\n$$\n\nwhere dia $\\mathfrak{v}(e^{\\zeta|V|})_{V\\in S}$ denotes the infinite diagonal matrix with entries $e^{\\zeta|V|}$\n\nWe will next show that $u_{X}\\in\\mathcal{H}_{k^{A}}$ , with the representation $f_{X}=e^{-\\frac{1}{2}\\zeta\\left|X\\right|}\\sum_{Z}Q_{Z,X}^{-1}k_{Z}\\in$ span $\\{k_{Z}^{A}\\}_{Z\\in S}$ (note the sum has only finitely many non-zero terms as $Q_{Z,X}^{-1}\\neq 0$ for only finitely many $Z$ with $Z\\leq X)$ ."}, {"n": 6, "claim": "We have,\n\n$$\n\\mathrm{diag}(e^{\\zeta|V|})_{V\\in S}Q f_{X}=e^{\\frac{1}{2}\\zeta|X|}d_{X},\n$$\n\nwhere $d_{X}$ is the infinite-dimensional vector indexed by $S$ with 0’s in every position except 1 in position $X$ . For any $Y\\in S,(f_{X}|k_{Y}^{A})_{k^{A}}=e^{\\frac 12\\zeta|X|}\\dot{Q}_{X,Y}=u_{X}(Y)$ . Thus, $u_{X}\\in\\mathcal{H}_{k^{A}}$ . We also have,\n\n$$\n(u_{X}|u_{Y})_{k^{A}}=f_{X}^{T}Q^{T}\\mathrm{diag}(e^{\\zeta|V|})_{V\\in S}Q f_{Y}=d_{X}^{T}d_{Y}=d_{X}(Y).\n$$"}, {"n": 7, "claim": "Thus $\\{u_{X}\\}_{X\\in S}$ is a set of orthogonal non-zero vectors, normalized such that $\\|u_{X}\\|_{k^{A}}=1$ for all $X \\in S$ . Finally, by Equation 12,\n\n$$\nk^{A}(X,Y)=\\sum_{V\\in S}^{\\infty}(u_{V}|k_{X}^{A})_{k^{A}}(u_{V}|k_{Y}^{A})_{k^{A}}\n$$\n\nfor all $X,Y\\in S$ so that $\\{u_{X}\\}_{X\\in S}$ is also a basis."}]}}, {"local_id": "T2", "statement": "Corollary 36 Consider an alignment kernel k and let $\\tilde{S}$ be a rectified set $o f$ sequences. For $Y\\in S,V\\in\\tilde{S}$ with $L= |\\boldsymbol{Y}| = |\\boldsymbol{V}|$ , define the pairwise comparison using $k_{s}$ as,\n\n$$\nc_{p w}(Y,V)=\\prod_{l=0}^{L-1}\\sigma|\\mathcal{B}|^{-1}k_{s}(Y_{(l)},V_{(l)}).\n$$\n\nTilting the kernel by $A(X)=e^{\\mu|X|}$ gives $k^{A}(X,Y)=A(X)A(Y)k(X,Y)$ . For all sequences $X \\in S$ , define the features $\\{u_{V}(X)\\}_{V\\in\\tilde{S}}$ , given by\n\n$$\nu_{V}(X)=e^{\\frac{1}{2}\\zeta|V|}\\sum_{g=0}^{\\infty}e^{-\\Delta\\mu g}\\sum_{J\\in G(|X|,g)}c_{p w}(X_{(J)},V),\\tag{13}\n$$\n\nwhere $\\zeta=2\\mu-\\log\\sigma+\\log|\\boldsymbol{\\mathcal{B}}|$ . Now, $\\{u_{V}(X)\\}_{V\\in\\tilde{S}}$ is an orthonormal basis for $\\mathcal{H}_{k^{A}}$", "proof": {"steps": [{"n": 1, "claim": "Proof Let $\\tilde{B},\\tilde{S}$ a rectified alphabet and a rectified set of sequences for k. By Theorem 24, there is an orthonormal basis of $\\mathcal{H}_{k^{A}},\\left\\{u_{V}\\right\\}_{V\\in\\tilde{S}}$ such that for any $V,X\\in\\tilde{S}$\n\n$$\n(u_{V}|k_{X}^{A})_{k^{A}}=e^{\\frac{1}{2}\\zeta|V|}\\sum_{g=0}^{\\infty}e^{-\\Delta\\mu g}\\sum_{J\\in G(|X|,g)}\\mathbb{1}(X_{(J)}=V).\n$$"}, {"n": 2, "claim": "Note that reparameterization does not afect the RKHS $\\mathcal{H}_{k^{A}}$ , so that $\\{u_{V}\\}_{V\\in\\tilde{S}}$ is an orthonormal basis for the kernel $k^{A}$ applied to the space $S$ as well."}, {"n": 3, "claim": "Now, consider a $Y \\in S$ , with $L = | Y |$ . Since $\\tilde{B}$ is a basis for $\\mathbb{R}^{B}$ , we can write each letter $Y_{(l)}$ as $\\begin{array}{ r }{Y_{(l)}=\\sum_{\\tilde{b}\\in\\tilde{\\mathcal{B}}}Y_{(l),\\tilde{b}}\\tilde{b}}\\end{array}$ for some scalar coeficients $\\{Y_{(l),\\tilde{b}}\\}_{\\tilde{b}\\in\\tilde{B}}$ (to be clear, we are treating $Y_{(l)}$ here as a one-hot encoding of the lth letter of Y). To derive the coeficients $Y_{(l),\\tilde{b}},$ recall that $\\{\\sqrt{\\sigma|B|^{-1}}\\tilde{b}\\}_{\\tilde{b}\\in\\tilde{B}}$ is an orthonormal basis of $\\mathbb{R}^{B}$ , with inner product defined by $k_{s}$ Thus, we have $Y_{(l),\\tilde{b}}=k_{s}(Y_{(l)},\\sqrt{\\sigma|\\mathcal{B}|^{-1}}\\tilde{b})\\sqrt{\\sigma|\\mathcal{B}|^{-1}}=\\sigma|\\mathcal{B}|^{-1}k_{s}(Y_{(l)},\\tilde{b})$ for all $\\tilde{b}\\in\\tilde{\\cal B}$ ."}, {"n": 4, "claim": "We also have, applying the properties of the rectified alphabet in Proposition 35, $\\textstyle\\sum_{\\tilde{b}\\in\\tilde{B}}Y_{(l),\\tilde{b}}=$\n\n$$\n\\sigma|\\mathcal{B}|^{-1}\\sum_{\\tilde{b}\\in\\tilde{\\mathcal{B}}}k_{s}(Y_{(l)},\\tilde{b})=1.\\mathrm{Thus},\n$$\n\n$$\n\\begin{array}{l}(u_{V}|k_{Y}^{A})_{k}^{A}=\\sum_{X\\in\\tilde{S}: |X| =L}\\left(\\prod_{l=0}^{L-1}Y_{(l),X_{(l)}}\\right)e^{\\frac{1}{2}\\zeta|V|}\\sum_{g=0}^{\\infty}e^{-\\Delta\\mu g}\\sum_{J\\in G(|X|,g)}\\mathbb{1}(X_{(J)}=V)\\\\ =e^{\\frac{1}{2}\\zeta|V|}\\sum_{g=0}^{\\infty}e^{-\\Delta\\mu g}\\sum_{J\\in G(|X|,g)}\\sum_{\\substack{X\\in\\tilde{S}:|X| =L}}\\left(\\prod_{l=0}^{L-1}Y_{(l),X_{(l)}}\\right)\\prod_{m=0}^{M-1}\\mathbb{1}(X_{(j_{m})}=V_{(m)})\\\\ =e^{\\frac{1}{2}\\zeta|V|}\\sum_{g=0}^{\\infty}e^{-\\Delta\\mu g}\\sum_{J\\in G(|X|,g)}\\left(\\prod_{l\\notin J}\\sum_{b\\in\\tilde{\\mathcal{B}}}Y_{(l),b}\\right)\\prod_{m=0}^{M-1}Y_{(j_{m}),V_{(m)}}\\\\ =e^{\\frac{1}{2}\\zeta|V|}\\sum_{g=0}^{\\infty}e^{-\\Delta\\mu g}\\sum_{J\\in G(|X|,g)}1\\times\\prod_{m=0}^{M-1}\\sigma|\\mathcal{B}|^{-1}k_{s}(Y_{(j_{m})},V_{(m)})\\\\ =e^{\\frac{1}{2}\\zeta|V|}\\sum_{g=0}^{\\infty}e^{-\\Delta\\mu g}\\sum_{J\\in G(|X|,g)}c_{p w}(Y_{(J)},V)\\\\ \\end{array}\n$$"}]}}, {"local_id": "T3", "statement": "Theorem 39 Say $| \\boldsymbol B | = 1$ and call A the only letter in B. If $\\Delta \\mu > 0$ , k has discrete masses if and only if $2\\mu+\\log k_{s}(A,A)\\geq 0 .\\ I f\\Delta\\mu=0$ , k has discrete masses $i f$ and only if $2\\mu+\\log k_{s}(A,A)>0$", "proof": {"steps": [{"n": 1, "claim": "Proof Note in this case $\\zeta=2\\mu+\\log k_{s}(A,A)$ We will replace k with $k^{A}(X,Y)=$ $e^{\\mu|X|}e^{\\mu|Y|}k(X,Y)$ throughout. Clearly, this does not afect whether or not the kernel has discrete masses. Index ${\\cal S}=\\{0,1,\\dots\\}$ Call $Q$ the infinite upper triangular matrix with $\\begin{array}{ r }{e^{-\\frac{1}{2}\\zeta L^{\\prime}}u_{L^{\\prime}}(L)=\\sum_{q=0}^{\\infty}e^{-g\\Delta\\mu}|G(L,L^{\\prime},g)|}\\end{array}$ in position $(L^{\\prime},L)$ . Note that the Gram matrix of 9 k on $\\{0,1,\\ldots L\\}$ is $Q_{:L+1,:L+1}^{T}\\mathrm{diag}(e^{\\zeta L^{\\prime}})_{L^{\\prime}=0}^{L}Q_{:L+1,:L+1}$ ."}, {"n": 2, "claim": "Since $Q$ is infinite upper triangular with no zeros along its diagonal it has an infinite upper triangular inverse $Q^{-1}$ with the property that $(Q_{:L,:L})^{-1}=(Q^{-1})_{:L,:L}$ . Thus the inverse of the Gram matrix of k on $\\{0,1,\\ldots L\\}$ is $Q_{:L+1,:L+1}^{-1}\\mathrm{diag}(e^{-\\zeta L^{\\prime}})_{L^{\\prime}=0}^{L}Q_{:L+1,:L+1}^{-1 T}$ which has, in position $\\begin{array}{ r }{(L^{\\prime},L^{\\prime}),\\sum_{l=L^{\\prime}}^{L}e^{-\\zeta l}(Q_{L^{\\prime},l}^{-1})^{2}}\\end{array}$ Thus, by Proposition 11, $\\delta_{L^{\\prime}}\\in\\mathcal{H}_{k}$ if and only if\n\n$$\n\\sum_{l=L^{\\prime}}^{\\infty}e^{-\\zeta l}(Q_{L^{\\prime},l}^{-1})^{2}<\\infty .\\tag{14}\n$$"}, {"n": 3, "claim": "The proof will now proceed in two parts using the technique of generating functions and the theory of the Riordan group. In part 1 we will equate $u_{L}(L^{\\prime})$ with the coeficients of a formal power series. We will then explicitly invert the matrix Q by inverting this power series. In part 2 we will equate Equation 14 with the convergence of a power series, which we will use to prove the result."}, {"n": 4, "claim": "Part 1: Say g is an integer greater than or equal to 1, and $L\\geq L^{\\prime}$ . We want to find the size of $G(L,L^{\\prime},g)$ . Each $J=(j_{-1},\\dots,j_{L^{\\prime}})\\in G(L,L^{\\prime},g)$ is uniquely characterized by the numbers $j_{m_{1}},\\ldots,j_{m_{g}},$ which indicate the positions at which each of the g gaps start, and $j_{m_{1}+1},\\ldots,j_{m_{g}+1}$ , which indicate the position at which each of the g gaps end. In particular,\n\n$$\nJ=(-1,0,1,\\dots,j_{m_{1}}-1,j_{m_{1}},j_{m_{1}+1},j_{m_{1}+1}+1,\\dots,j_{m_{2}}-1,j_{m_{2}},j_{m_{2}+1},\\dots,L).\n$$"}, {"n": 5, "claim": "Thus each J is also uniquely characterized by the sizes of its g gaps $l_{1}=j_{m_{1}+1}-j_{m_{1}}-1,l_{2}=$ $j_{m_{2}+1}-j_{m_{2}}-1,...$ . as well as the sizes of its $g + 1$ contiguous regions $c_{0}=j_{m_{1}}+1,c_{1}=$ $j_{m_{2}}-j_{m_{1}+1}+1,c_{2}=j_{m_{3}}-j_{m_{2}+1}+1,\\ldots,c_{g}=L-j_{m_{g}+1}.$ . On the other hand, given numbers $l_{1},\\ldots,l_{g},c_{0},\\ldots,c_{g}$ , one gets a unique valid $J\\in G(L,L^{\\prime},g)$ if and only if $\\textstyle\\sum_{i=0}^{g}c_{i}=$ $L^{\\prime},\\sum_{i=1}^{g}l_{i}=L-L^{\\prime},l_{i}\\geq 1$ for all $i,$ and $c_{i}\\geq 1$ for all $i=1,2,\\ldots,g-1\\ :(c_{0}=0$ implies a gap at the beginning of the sequence while $c_{g}=0$ implies a gap at the end)."}, {"n": 6, "claim": "The number of $g + 1$ ordered numbers $(c_{0},\\ldots,c_{g})$ with $\\textstyle\\sum_{i=0}^{\\bar{g}}c_{i}=L^{\\prime}$ and $c_{i}\\geq 1$ for all $i=1,2,\\ldots,g-1$ is the L<sup>0</sup>-th coeficient of the power series\n\n$$\n\\left(1+x+x^{2}+\\dots\\right)\\left(x+x^{2}+\\dots\\right)^{g-1}\\left(1+x+x^{2}+\\dots\\right)=\\frac{x^{g-1}}{(1-x)^{g+1}}.\n$$"}, {"n": 7, "claim": "Similarly, the number of sets of number of g ordered numbers $(l_{1},\\ldots,l_{g})$ with $\\textstyle\\sum_{i=1}^{g}l_{i}=$ $L-L^{\\prime}$ and $l_{i}\\geq 1$ for all i is the $L-L^{\\prime}–\\mathrm{th}$ coeficient of the power series $\\frac{y^{g}}{(1-y)^{g}}$ . On the other hand if $g=0,G(L,L^{\\prime},g)$ is empty if $L\\neq L^{\\prime}$ otherwise it has one element. In particular, $G(L,L^{\\prime},0)$ is (trivially) the coeficient in front of $x^{L^{\\prime}}y^{L-L^{\\prime}}$ of $1/(1-x)$ . Thus, the size of $G(L,L^{\\prime},g)$ is the coeficient in front of $x^{L^{\\prime}}y^{L-L^{\\prime}}z^{g}$ of the power series\n\n$$\n\\begin{array}{l}\\sum_{g=1}^{\\infty}z^{g}\\frac{x^{g-1}}{(1-x)^{g+1}}\\frac{y^{g}}{(1-y)^{g}}+\\frac{1}{1-x}=\\frac{z y}{(1-x)^{2}(1-y)}\\sum_{g=0}^{\\infty}\\left(z\\frac{x}{1-x}\\frac{y}{1-y}\\right)^{g}+\\frac{1}{1-x}\\\\ =\\frac{z y}{(1-x)^{2}(1-y)}\\frac{(1-x)(1-y)}{(1-x)(1-y)-z x y}+\\frac{1}{1-x}\\\\ =\\frac{1}{1-x}\\left(\\frac{z y}{(1-x)(1-y)-z x y}+1\\right)\\\\ =\\frac{1}{1-x}\\left(\\frac{z y+1-x-y+x y-z x y}{(1-x)(1-y)-z x y}\\right)\\\\ =\\frac{1}{1-x}\\left(\\frac{(z y+1-y)(1-x)}{(1-x)(1-y)-z x y}\\right)\\\\ =\\frac{1-(1-z)y}{(1-x)(1-y)-z x y}\\\\ =\\frac{\\frac{1-(1-z)y}{1-y}}{1-x\\frac{1-(1-z)y}{1-y}}.\\\\ \\end{array}\n$$"}, {"n": 8, "claim": "Call $\\begin{array}{ r }{f_{z}(y)=\\frac{1-(1-z)y}{1-y}}\\end{array}$ . Next we replace x with xy so that we get that $|G(L,L^{\\prime},g)|$ is $\\begin{array}{ r }{[(x y)^{L^{\\prime}}y^{L-L^{\\prime}}z^{g}]\\frac{f_{z}(y)}{1-(x y)f_{z}(y)}=[x^{L^{\\prime}}y^{L}z^{g}]\\frac{f_{z}(y)}{1-x y f_{z}(y)}}\\end{array}$ . Thus, substituting $z=e^{-\\Delta\\mu}$ and calling $f=f_{e^{-\\Delta\\mu}}$\n\n$$\nQ_{L^{\\prime},L}=\\sum_{g=0}^{\\infty}e^{-g\\Delta\\mu}|G(L,L^{\\prime},g)| =[x^{L^{\\prime}}y^{L}]\\sum_{g=0}^{\\infty}e^{-g\\Delta\\mu}[z^{g}]\\frac{f_{z}(y)}{1-x y f_{z}(y)}=[x^{L^{\\prime}}y^{L}]\\frac{f(y)}{1-x y f(y)}.\n$$"}, {"n": 9, "claim": "$Q$ is in the Bell subgroup of Riordan arrays (Shapiro et al. (1991) and chapter 3 of Shapiro et al. $\\left(2022\\right))$ . This means that by Proposition 38, defining $\\bar{f}$ as the unique power series with $y{\\bar{f}}(y)f(y{\\bar{f}}(y))=y$ 2\n\n$$\nQ_{L^{\\prime},L}^{-1}=[x^{L^{\\prime}}y^{L}]\\frac{\\bar{f}(y)}{1-x y\\bar{f}(y)}.\n$$"}, {"n": 10, "claim": "Part 2: Since\n\n$$\nQ_{L^{\\prime},L}^{-1}=[x^{L^{\\prime}}y^{L}]\\frac{\\bar{f}(y)}{1-x y\\bar{f}(y)}=[y^{L}]\\bar{f}(y)(y\\bar{f}(y))^{L^{\\prime}}=[y^{L-L^{\\prime}}]\\bar{f}(y)^{L^{\\prime}+1},\n$$\n\nwe have,\n\n$$\n\\sum_{l=L}^{\\infty}e^{-\\zeta l}(Q_{L,l}^{-1})^{2}=e^{-\\zeta L}\\sum_{l=0}^{\\infty}(e^{-\\frac{1}{2}\\zeta l}[y^{l}]\\bar{f}(y)^{L+1})^{2}.\n$$"}, {"n": 11, "claim": "This sum is finite if $e^{-{\\frac{1}{2}}\\zeta}$ is smaller than the radius of convergence of $\\bar{f}(y)^{L+1}$ (as the terms in the series decay exponentially) and infinite if it is larger (as the terms do not approach 0). We will now show that the radius of convergence of $\\bar{f}$ is exactly 1, so that if $\\zeta < 0$ then $\\delta_{0}\\notin\\mathcal{H}_{k}$ and if $\\zeta > 0$ then, since the radius of convergence of $\\bar{f}(y)^{L}$ is also at least 1 for all $L,$ k has discrete masses. We will afterwards carefully consider the case of $\\zeta = 0$\n\nLet us first consider the case $\\Delta \\mu = 0$ . In this case we have, from the definition of $f,$ $\\begin{array}{ r }{y f(y)=\\frac{y}{1-y}.}\\end{array}$ , so $\\textstyle y{\\bar{f}}(y)={\\frac{y}{1+y}}$ . In this case $\\bar{f}$ clearly has radius of convergence 1. We can also see explicitly that $[y^{l}]\\bar{f}(y)^{L+1}=(-1)^{l}{\\binom{L+l}{l}}$ . Thus Equation 14 also does not hold for any L when $\\zeta = 0$\n\nWe now assume $\\Delta \\mu > 0$ . We will derive the form of ${\\bar{f}}.$ . Call $\\xi=1-e^{-\\Delta\\mu}\\in(0,1)$ , and $t=y\\bar{f}(y)$\n\n$$\nt f(t)=y\\iff t(1-\\xi t)=(1-t)y\\iff 0=\\xi t^{2}-(y+1)t+y.\n$$"}, {"n": 12, "claim": "Thus, picking the root that is 0 when $y = 0$ , we get\n\n$$\ny\\bar{f}(y)=\\frac{1}{2\\xi}\\left(1+y-\\sqrt{(1+y)^{2}-4\\xi y}\\right).\n$$"}, {"n": 13, "claim": "The roots of the polynomial $(1+y)^{2}-4\\xi y$ are\n\n$$\n(2\\xi-1)\\pm\\sqrt{(2\\xi-1)^{2}-1}=(2\\xi-1)\\pm i\\sqrt{1-(2\\xi-1)^{2}}.\n$$"}, {"n": 14, "claim": "Note both of these roots have norm 1 and can thus be written $e^{i\\theta},e^{-i\\theta}$ for some $\\pi > \\theta > 0$ Thus, we may write\n\n$$\ny\\bar{f}(y)=\\frac{1}{2\\xi}\\left(1+y-\\sqrt{(1-e^{i\\theta}y)(1-e^{-i\\theta}y)}\\right).\n$$"}, {"n": 15, "claim": "${\\sqrt{1-e^{i\\theta}y}},{\\sqrt{1-e^{-i\\theta}y}}$ each have radius of convergence $1,\\ \\mathrm{so},\\ y\\bar{f}$ has radius of convergence at least 1. On the other hand, the derivative of $y{\\bar{f}}$ diverges when y approaches $e^{i\\theta}\\mathrm{or}e^{-i\\theta}$ ， so that $y{\\bar{f}}$ , and thus ${\\bar{f}},$ , has radius of convergence exactly 1."}, {"n": 16, "claim": "The case when $\\zeta ~ = ~ 0$ is more delicate. First call $g(y)~=~\\sqrt{(1+y)^{2}-4\\xi y}~=$ $\\sqrt{(1-e^{i\\theta}y)(1-e^{-i\\theta}y)}$ . For real $0 < r < 1$ , define\n\n$$\n\\begin{array}{l}q_{L}(r)=\\int_{0}^{1}d t g(\\sqrt{r}e^{2\\pi i t})^{L}g(\\sqrt{r}e^{-2\\pi i t})^{L}\\\\ =\\int_{0}^{1}d t\\sum_{n}r^{n/2}e^{2\\pi n i t}[y^{n}]g(y)^{L}\\sum_{m}r^{m/2}e^{-2\\pi m i t}[z^{m}]g(z)^{L}\\\\ =\\sum_{n,m}\\left([y^{n}]g(y)^{L}[z^{m}]g(z)^{L}\\right)r^{(n+m)/2}\\int_{0}^{1}d t e^{2\\pi i t(n-m)}\\\\ =\\sum_{n}([y^{n}]g(y)^{L})^{2}r^{n}.\\\\ \\end{array}\n$$"}, {"n": 17, "claim": "$g$ is analytic with real coeficients at 0 so $([y^{n}]g(y)^{L})^{2}\\ge 0$ for all $L,n.$ . Thus, by the monotone convergence theorem, $\\begin{array}{ r }{\\sum_{n}([y^{n}]g(y)^{L})^{2}=\\operatorname*{lim}_{r\\to 1^{-}}q_{L}(r)}\\end{array}$ . However,\n\n$$\ng(\\sqrt{r}e^{2\\pi i z})^{L}g(\\sqrt{r}e^{-2\\pi i z})^{L}=\\left|1-e^{i(\\theta+2\\pi z)}\\sqrt{r}\\right|^{L}\\left|1-e^{i(\\theta-2\\pi z)}\\sqrt{r}\\right|^{L}\\leq 4^{L},\n$$\n\nso $\\begin{array}{ r }{\\sum_{n}([y^{n}]g(y)^{L})^{2}<\\infty}\\end{array}$ for all L. Now note\n\n$$\n\\begin{array}{l}\\sum_{n}([y^{n}](y\\bar{f}(y))^{L})^{2}=\\sum_{n}\\frac{1}{(2\\xi)^{2 L}}\\left(\\sum_{l=0}^{L}\\binom{L}{l}[y^{n}](1+y)^{l}(-1)^{L-l}g(y)^{L-l}\\right)^{2}\\\\ =\\sum_{n}\\frac{1}{(2\\xi)^{2 L}}\\left(\\sum_{l=0}^{L}(-1)^{L-l}\\sum_{l^{\\prime}=0}^{l}\\binom{l}{l^{\\prime}}[y^{n-l^{\\prime}}]g(y)^{L-l}\\right)^{2}\\\\ \\leq\\sum_{n}\\frac{1}{(2\\xi)^{2 L}}\\left(L^{2}\\max_{l,l^{\\prime}}\\left|\\binom{l}{l^{\\prime}}[y^{n-l^{\\prime}}]g(y)^{L-l}\\right|\\right)^{2}\\\\ \\leq\\frac{1}{(2\\xi)^{2 L}}(L^{2})^{2}\\sum_{n}\\max_{l,l^{\\prime}}\\left|\\binom{l}{l^{\\prime}}^{2}\\left([y^{n-l^{\\prime}}]g(y)^{L-l}\\right)^{2}\\right|\\\\ \\leq\\frac{1}{(2\\xi)^{2 L}}(L^{2})^{2}\\sum_{l=0}^{L}\\sum_{l^{\\prime}=0}^{l}\\binom{l}{l^{\\prime}}^{2}\\sum_{n}\\left([y^{n-l^{\\prime}}]g(y)^{L-l}\\right)^{2}<\\infty .\\\\ \\end{array}\n$$"}, {"n": 18, "claim": "Thus k has discrete masses when $\\zeta = 0$ as well."}]}}], "target_local_id": null, "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2025-24-0194", "context": "Notations. For a vector $\\mathbf{a}\\in\\mathbb{R}^{m}$ , the i-the entry is denoted by $a_{i}$ . Random vectors are denoted by bold upper-case letters and their realizations are denoted by bold lower-case letters, e.g., X and x. Matrices are denoted by bold upper-case letters, e.g., A, where $\\mathbf{A}_{i}$ denotes the i-th row of A and $\\mathbf{A}_{i,j}$ denotes the entry at row i and column $j$ . For matrices A and B with the same shapes, $\\mathbf A \\preccurlyeq \\mathbf B$ denotes component-wise inequality. We denote the indicator function by 1, and for a matrix $\\mathbf{A}\\in\\mathbb{R}^{m\\times n}$ , we use the convention that $\\mathbb{1}\\{\\mathbf{A}\\}\\in\\{0,1\\}^{m\\times n}$ , where the entries are specified by $[\\mathbb{1}\\{\\mathbf{A}\\}]_{i,j}=\\mathbb{1}\\{\\mathbf{A}_{i,j}\\neq 0\\}$ . For a positive integer $n\\in\\mathbb{N}$ , we define $[n]\\triangleq\\{1,\\dots,n\\}$ . The permutation matrix associated with any permutation π of [n] is denoted by $\\mathbf{P}_{\\pi},\\mathrm{i . e .}$ , [π<sub>1</sub> $\\pi_{2}\\ldots\\pi_{n}]^{\\top}=\\mathbf{P}_{\\pi}\\cdot [12\\ldots n]^{\\top}$ . The n-dimensional identity matrix is denoted by ${\\mathbf{I}}_{n\\times n}$ , and the Hadamard product is denoted by . We use im(f) to denote the image of the function $f .$ We let $\\mathbf{e}_{i}$ denote the i-th standard basis vector of R<sup>n</sup>. Given a function $f:\\mathbb{R}^{s}\\mathbb{R}^{r}$ that has first-order partial derivatives on $\\mathbb{R}^{s}$ we denote the Jacobian of $f$ at $\\mathbf{z}\\in\\mathbb{R}^{s}$ by $J_{f}(\\mathbf{z})\\in\\mathbb{R}^{r\\times s}$ with entries $[J_{f}(\\mathbf{z})]_{i,j}=\\partial f(\\mathbf{z})_{i}/\\partial\\mathbf{z}_{j}$\n\nConsider latent causal random variables $\\mathbf{Z}\\triangleq [Z_{1},\\ldots,Z_{n}]^{\\intercal}$ . An unknown transformation $g:\\mathbb{R}^{n}\\mathbb{R}^{d}$ generates the observable random variables $\\mathbf{X}\\triangleq [X_{1},\\ldots,X_{d}]^{\\intercal}$ from the latent variables Z according to:\n\n$$\n\\mathbf{X}=g(\\mathbf{Z}).\\tag{3}\n$$\n\nWe assume that $d \\geq n .$ , and the transformation $g$ is continuously diferentiable and a difeomorphism onto its image (otherwise, identifiability is ill-posed). We denote the image of g by $\\mathcal{X}\\triangleq\\operatorname{im}(g)\\subseteq\\mathbb{R}^{d}$ . The probability density functions (pdfs) of Z and X are denoted by p and $p\\mathbf{x}$ , respectively. We assume that $p$ is absolutely continuous with respect to the n-dimensional Lebesgue measure. Subsequently, $p\\mathbf{x}$ , which is defined on the image manifold $\\operatorname{im}(g)$ , is absolutely continuous with respect to the n-dimensional Hausdorf measure rather than d-dimensional Lebesgue measure<sup>1</sup>. The distribution of latent variables Z factorizes with respect to a DAG that consists of $n$ nodes and is denoted by $\\mathcal{G}$ . Node $i \\in [ n ]$ of $\\mathcal{G}$ represents $Z_{i}$ and $p$ factorizes according to:\n\n$$\np(\\mathbf{z})=\\prod_{i=1}^{n}p_{i}(z_{i}\\mid\\mathbf{z}_{\\mathrm{pa}(i)}),\\tag{4}\n$$\n\nwhere $\\mathrm{pa}(i)$ denotes the set of parents of node i and $p_{i}\\big(z_{i}\\mid\\mathbf{z}_{\\mathrm{pa}(i)}\\big)$ is the conditional pdf of $z_{i}$ given the variables of its parents. We use ch(i), an(i), and $\\mathrm{de}(i)$ to denote the children, ancestors, and descendants of node $i,$ respectively. Accordingly, for each node $i \\in [ n ]$ we also define\n\n$$\n\\overline{{\\mathrm{pa}}}(i)\\triangleq\\mathrm{pa}(i)\\cup\\{i\\},\\overline{{\\mathrm{ch}}}(i)\\triangleq\\mathrm{ch}(i)\\cup\\{i\\},\\overline{{\\mathrm{an}}}(i)\\triangleq\\mathrm{an}(i)\\cup\\{i\\},\\text{and}\\overline{{\\mathrm{de}}}(i)\\triangleq\\mathrm{de}(i)\\cup\\{i\\}.\\tag{5}\n$$\n\nWe denote the transitive closure and transitive reduction of $\\mathcal{G}$ by $\\mathcal{G}_{\\mathrm{tc}}$ and $\\mathcal{G}_{\\mathrm{tr}}$ , respectively<sup>2</sup>. The parental relationships in these graphs are denoted by $\\mathrm{pa}_{\\mathrm{tc}}$ and $\\mathrm{pa}_{\\mathrm{tr}}.$ , and other graphical relationships are denoted similarly. Based on the modularity property, a change in the causal mechanism of node i does not afect those of the other nodes. We also assume that all conditional pdfs $\\{p_{i}(z_{i}\\mid\\mathbf{z}_{\\mathrm{pa}(i)}):i\\in [n]\\}$ are continuously diferentiable with respect to all variables and $p(\\mathbf{z})\\neq 0$ for all $\\mathbf{z}\\in\\mathbb{R}^{n}$ . We consider the general structural causal models (SCMs) based on which for each $i \\in [ n ]$ ,\n\n$$\nZ_{i}=f_{i}(Z_{\\mathrm{pa}(i)},N_{i}),\\tag{6}\n$$\n\nwhere $\\{f_{i}:i\\in [n]\\}$ are general functions that capture the dependence of node i on its parents and $\\{N_{i}:i\\in [n]\\}$ account for the exogenous noise terms that we assume to have pdfs with full support.\n\nDefinition 1 (Valid Causal Order) We refer to a permutation $(\\pi_{1},\\ldots,\\pi_{n})$ of [n] as a valid causal order<sup>3</sup> if $\\pi_{i}\\in\\mathrm{pa}(\\pi_{j})$ indicates that $i < j$\n\nIn this paper, without loss of generality, we assume that $(1,\\ldots,n)$ is a valid causal order. We also define a graphical notion that will be useful for presenting our results and analysis on CRL under a linear transformation.\n\nThe score function associated with a pdf is defined as the gradient of its logarithm. The score function associated with $p$ is denoted by\n\n$$\n\\boldsymbol{s}(\\mathbf{z})\\triangleq\\nabla_{\\mathbf{z}}\\log p(\\mathbf{z}).\\tag{9}\n$$\n\nNoting the connection $\\mathbf{X}=g(\\mathbf{Z})$ , the density of X under $\\mathcal{E}^{0}$ , denoted by $p\\mathbf{x}$ , is supported on an n-dimensional manifold  embedded in $\\mathbb{R}^{d}$ . Hence, specifying the score function of X requires notions from diferential geometry. For this purpose, we denote the tangent space of the manifold  at point $\\mathbf{x}\\in\\mathcal{X}$ by $T_{\\mathbf{x}}\\mathcal{X}$ . Tangent vectors $\\mathbf{v}\\in T_{x}\\mathcal{X}$ are equivalence classes of continuously diferentiable curves $\\gamma\\colon(-1,1)\\to\\mathcal{X}\\subseteq\\mathbb{R}^{d}$ with $\\gamma(0)=\\mathbf{x}$ and $\\gamma^{\\prime}(0)=\\mathbf{v}$ Furthermore, given a function $f\\colon\\mathcal{X}\\mathbb{R}$ , denote its directional derivative at point $\\mathbf{x}\\in\\mathcal{X}$ along a tangent vector $\\mathbf{v}\\in T_{\\mathbf{x}}\\mathcal{X}$ by $D_{\\mathbf{v}}f(\\mathbf{x})$ , which is defined as\n\n$$\nD_{\\mathbf{v}}f(\\mathbf{x})\\triangleq\\left.\\frac{\\mathrm{d}}{\\mathrm{d}t}(f\\circ\\gamma)(t)\\right|_{t=0},\\tag{10}\n$$\n\nfor any curve $\\gamma$ in equivalence class v. The diferential of $f$ at point $\\mathbf{x}\\in\\mathcal{X}$ , denoted by $\\mathrm{d}f_{\\mathbf{x}}$ is the linear operator mapping tangent vector $\\mathbf{v}\\in T_{\\mathbf{x}}\\mathcal{X}$ to $D_{\\mathbf{v}}f_{\\mathbf{x}}$ (Simon, 2014, p. 57), i.e.,\n\n$$\n\\mathrm{d}f_{x}\\colon T_{\\mathbf{x}}\\mathcal{X}\\ni v\\mapsto D_{\\mathbf{v}}f(\\mathbf{x})\\in\\mathbb{R}.\\tag{11}\n$$\n\nLet $\\mathbf{B}\\in\\mathbb{R}^{d\\times n}$ be a matrix for which the columns of B form an orthonormal basis for $T_{\\mathbf{x}}\\mathcal{X}$ Denote the directional derivative of $f$ along the i-th column of B by $D_{i}f$ for all $i \\in [ n ]$ Then, the diferential operator can be expressed by the vector\n\n$$\nD f_{\\mathbf{x}}\\triangleq\\mathbf{B}\\cdot\\left[D_{1}f_{\\mathbf{x}}\\dots D_{n}f_{\\mathbf{x}}\\right]^{\\top}\\in\\mathbb{R}^{d},\\tag{12}\n$$\n\nsuch that\n\n$$\n\\mathrm{d}f_{\\mathbf{x}}(\\mathbf{v})=\\mathbf{v}^{\\top}\\cdot D f_{\\mathbf{x}},\\quad\\forall\\mathbf{x}\\in\\mathcal{X},\\quad\\forall\\mathbf{v}\\in T_{\\mathbf{x}}\\mathcal{X}.\\tag{13}\n$$\n\nNote that the diferential operator $\\mathrm{d}f$ is a generalization of the gradient. Hence, we can generalize the definition of the score function using the diferential operator by setting f to the logarithm of pdf. Therefore, the score function of $\\mathbf{X}$ under $\\mathcal{E}^{0}$ is specified as follows:\n\n$$\n\\boldsymbol{s}_{\\mathbf{X}}(\\mathbf{x})\\triangleq D\\log p_{\\mathbf{X}}(\\mathbf{x}),\\quad\\forall\\mathbf{x}\\in\\mathcal{X}.\\tag{14}\n$$\n\nWe consider two types of interventions. A soft intervention on node i (also referred to as imperfect intervention in literature), changes the conditional distribution $p_{i}\\big(z_{i}\\mid\\mathbf{z}_{\\mathrm{pa}(i)}\\big)$ to a distinct conditional distribution, which we denote by $q_{i}\\big(z_{i}\\mid\\mathbf{z}_{\\mathrm{pa}(i)}\\big)$ . A soft intervention does not necessarily remove the functional dependence of an intervened node on its parents but rather alters it to a diferent mechanism. A stochastic hard intervention on node i (also referred to as perfect intervention) is stricter than a soft intervention and removes the edges incident on i. A hard intervention on node changes $p_{i}\\big(z_{i}\\mid\\mathbf{z}_{\\mathrm{pa}(i)}\\big)$ to $q_{i}(z_{i})$ that emphasizes the lack of dependence of $z_{i}$ on $\\mathbf{z}_{\\mathrm{pa}(i)}$ . Finally, we note that in some settings, we assume two hard interventions per node, in which case the two hard interventional mechanisms for node i are denoted by two distinct pdfs $q_{i}(z_{i})$ and $\\tilde{q}_{i}(z_{i})$\n\nInterventional environments. We consider atomic interventional environments in which each environment one node is intervened in, as is customary in the closely related CRL literature (Squires et al., 2023; Ahuja et al., 2023; Buchholz et al., 2023). In some settings (linear transformation), we will have one interventional environment per node and denote the interventional environments by ${\\mathcal{E}}\\triangleq\\{\\mathcal{E}^{1},\\ldots,\\mathcal{E}^{n}\\}$ , where we call the atomic environment set. We denote the node intervened in environment $\\mathcal{E}^{m}$ by $I^{m}\\in [n]$ . For other settings (general transformation), we will have two interventional environments per node and denote the second atomic environment set by $\\tilde{\\mathcal{E}}=\\{\\tilde{\\mathcal{E}}^{1},\\ldots,\\tilde{\\mathcal{E}}^{n}\\}$ . Similarly, we denote the intervened node in $\\tilde{\\mathcal{E}}^{m}$ by $\\tilde{I}^{m}$ for each $m \\in [ n ]$ . We assume that node-environment pairs are unspecified, $\\mathrm{i . e .}$ , the ordered intervention sets ${\\mathcal{T}}\\triangleq(I^{1},\\ldots,I^{n})$ and $\\tilde{\\mathcal{T}}\\triangleq(\\tilde{I}^{1},\\dots,\\tilde{I}^{\\bar{n}})$ are two unknown permutations of [n]. We also adopt the convention that $\\mathcal{E}^{0}$ is the observational environment and $I^{0}\\triangleq\\varnothing$ . Next, we define the notion of coupling between the environment sets $\\mathcal{E}$ and $\\tilde{\\mathcal{E}}.$\n\nNext, we define $p^{m}$ as the pdf of Z in environment $\\mathcal{E}^{m}$ . Hence, under soft and hard intervention for each $m\\in [n],p^{m}$ can be factorized as follows.\n\n$$\n\\text{soft int. in}\\mathcal{E}^{m}:p^{m}(\\mathbf{z})=q_{\\ell}(z_{\\ell}\\mid\\mathbf{z}_{\\mathrm{pa}(\\ell)})\\prod_{i\\neq\\ell}p_{i}(z_{i}\\mid\\mathbf{z}_{\\mathrm{pa}(i)}),\\quad \\text{where}\\ell=I^{m},\\tag{15}\n$$\n\n$$\n\\text{hard int. in}\\mathcal{E}^{m}:p^{m}(\\mathbf{z})=q_{\\ell}(z_{\\ell})\\prod_{i\\neq\\ell}p_{i}(z_{i}\\mid\\mathbf{z}_{\\mathrm{pa}(i)}),\\quad \\text{where}\\ell=I^{m}.\\tag{16}\n$$\n\nSimilarly, we define $\\tilde{p}^{m}$ as the pdf of Z in $\\tilde{\\mathcal{E}}^{m}$ , which can be factorized similarly to (16) with $\\tilde{q}_{\\ell}$ replaced with $q \\ell$ . Hence, the score functions associated with $p^{m}$ and $\\tilde{p}^{m}$ are specified as follows.\n\n$$\n\\boldsymbol{s}^{m}(\\mathbf{z})\\triangleq\\nabla_{\\mathbf{z}}\\log p^{m}(\\mathbf{z}),\\quad \\text{and}\\quad\\tilde{\\boldsymbol{s}}^{m}(\\mathbf{z})\\triangleq\\nabla_{\\mathbf{z}}\\log\\tilde{p}^{m}(\\mathbf{z}).\\tag{17}\n$$\n\nWe denote the score functions of the observed variables X under ${\\mathcal{E}}^{m}$ and $\\tilde{\\mathcal{E}}^{m}$ by $s_{\\mathbf{X}}^{m}$ and $\\tilde{\\pmb{s}}_{\\mathbf{X}}^{m}$ respectively. Note that the score functions change across diferent environments, which is induced by the changes in the distribution of Z. Specifically, following (4) and (15), the latent scores $\\pmb{s}(\\mathbf{z})$ and $\\pmb{s}^{m}(\\mathbf{z})$ are decomposed as\n\n$$\n\\boldsymbol{s}(\\mathbf{z})=\\nabla_{\\mathbf{z}}\\log p_{\\ell}(z_{\\ell}\\mid\\mathbf{z}_{\\mathrm{pa}(\\ell)})+\\sum_{i\\neq\\ell}\\nabla_{\\mathbf{z}}\\log p_{i}(z_{i}\\mid\\mathbf{z}_{\\mathrm{pa}(i)}),\\tag{18}\n$$\n\n$$\n\\mathrm{and}\\quad\\boldsymbol{s}^{m}(\\mathbf{z})=\\nabla_{\\mathbf{z}}\\log q_{\\ell}(z_{\\ell}\\mid\\mathbf{z}_{\\mathrm{pa}(\\ell)})+\\sum_{i\\neq\\ell}\\nabla_{\\mathbf{z}}\\log p_{i}(z_{i}\\mid\\mathbf{z}_{\\mathrm{pa}(i)}).\\tag{19}\n$$\n\nwhere $\\ell=I^{m}$ . Hence, $\\pmb{s}(\\mathbf{z})$ and $\\pmb{s}^{m}(\\mathbf{z})$ difer in only the causal mechanism of the intervened node in environment $\\mathcal{E}^{m}$ . In Section 4, we investigate these discrepancies between s and $\\pmb{s}^{m}\\left(\\mathrm{or}\\ \\tilde{\\pmb{s}}^{m}\\right)$ and characterize the relationship between the scores in the observational and interventional environments.\n\nDefinition 4 (Latent Graph Identifiability) For identifiability of the latent graph, we define:\n\n1. Perfect DAG recovery: DAG recovery is said to be perfect $i f{\\hat{\\mathcal{G}}}$ is isomorphic to $\\mathcal{G}$  \n2. Transitive closure recovery: DAG recovery is said to maintain transitive closure if $\\hat{\\mathcal G}$ and have the same ancestral relationships, $i.e.,\\hat{\\mathcal{G}}_{\\mathrm{tr}}$ is isomorphic to $\\mathcal{G}_{\\mathrm{tr}}$\n\n3. Consistency up to mixing with parents: The estimator $\\hat{\\mathbf{Z}}(\\mathbf{X})$ satisfies consistency up to mixing with parents if there exists a permutation π of [n] and a constant matrix $\\mathbf{C}_{\\mathrm{pa}}\\in\\mathbb{R}^{n\\times n}$ such that\n\n$$\n\\hat{\\mathbf{Z}}(\\mathbf{X})=\\mathbf{P}_{\\pi}\\cdot\\mathbf{C}_{\\mathrm{pa}}\\cdot\\mathbf{Z},\\quad\\forall\\mathbf{z}\\in\\mathbb{R}^{n},\\tag{22}\n$$\n\nwhere $\\mathbf{C}_{\\mathrm{{p a}}}$ has nonzero diagonal entries and for all $j\\notin\\overline{{\\mathrm{pa}}}(i),[\\mathbf{C}_{\\mathrm{pa}}]_{i,j}=0$ . Equivalently, $\\hat{Z}_{\\pi_{i}}$ is a linear function of $\\{Z_{j}:j\\in\\overline{{\\mathrm{pa}}}(i)\\}$ for all $i \\in [ n ]$\n\nDefinition 6 (Node-level Partial Identifiability) For identifiability of a single latent variable, we define:\n\n1. Partial latent recovery: The estimator $\\hat{\\mathbf{Z}}(\\mathbf{X})$ satisfies partial latent recovery for node i if\n\n$$\n\\left[\\hat{\\mathbf{Z}}(\\mathbf{X})\\right]_{i}=\\phi_{k}(Z_{k}),\\tag{24}\n$$\n\nfor some $k \\in [ n ]$ where $\\phi_{k}:\\mathbb{R}\\mathbb{R}$ is a difeomorphism.\n\n2. Partial consistency up to mixing with parents: The estimator $\\hat{\\mathbf{Z}}(\\mathbf{X})$ satisfies partial consistency up to mixing with parents for node i if\n\n$$\n[\\hat{\\mathbf{Z}}(\\mathbf{X})]_{i}=\\sum_{j\\in\\overline{{\\mathrm{pa}}}(k)}c_{k}\\cdot Z_{k},\\tag{25}\n$$\n\nfor some $k \\in [ n ]$ and constants $\\{c_{j}:j\\in\\overline{{\\mathrm{pa}}}(k)\\}$\n\nFor formalizing the achievability results and designing the associated algorithms, generating the estimates $\\hat{\\mathbf{Z}}(\\mathbf{X})$ and $\\hat{\\mathcal G}$ is facilitated by estimating the inverse of $g$ based on the observed data X. Specifically, an estimate of $g^{-1}$ , where $g^{-1}$ denotes the inverse of $g,$ facilitates recovering Z via ${\\bf Z}=g^{-1}({\\bf X})$ . Throughout the rest of this paper, we refer to $g^{-1}$ as the true encoder. To formalize the procedures of estimating $g^{-1}$ , we define $\\mathcal{H}$ as the set of possible valid encoders, i.e., candidates for $g^{-1}$ . A function $h \\in \\mathcal H$ can be such a candidate if it is invertible, that is, there exists an associated decoder $h^{-1}$ such that $(h^{-1}\\circ h)(\\mathbf{X})=\\mathbf{X}$ . Hence, the set of valid encoders is specified by\n\n$$\n\\mathcal{H}\\triangleq\\{h:\\mathcal{X}\\to\\mathbb{R}^{n}:\\exists h^{-1}:\\mathbb{R}^{n}\\to\\mathbb{R}^{d}\\text{such that}(h^{-1}\\circ h)(\\mathbf{x})=\\mathbf{x},\\forall\\mathbf{x}\\in\\mathcal{X}\\}.\\tag{26}\n$$\n\nNext, corresponding to any pair of observation X and valid encoder $h \\in \\mathcal H$ , we define $\\hat{\\mathbf{Z}}(\\mathbf{X};h)$ as an auxiliary estimate of $\\mathbf{Z}$ generated by applying the valid encoder h on X, i.e.,\n\n$$\n\\hat{\\mathbf{Z}}(\\mathbf{X};h)\\triangleq h(\\mathbf{X})=(h\\circ g)(\\mathbf{Z}),\\quad\\forall h\\in\\mathcal{H}.\\tag{27}\n$$\n\nThe estimate $\\hat{\\mathbf{Z}}(\\mathbf{X};h)$ inherits its randomness from X, and its statistical model is governed by that of Z under the chosen h. To emphasize the dependence on $h,$ we denote the score functions associated with the pdfs of $\\hat{\\mathbf{Z}}(\\mathbf{X};h)$ under environments $\\mathcal{E}^{0},\\mathcal{E}^{m}$ , and $\\tilde{\\mathcal{E}}^{m}$ respectively, by\n\n$$\n\\boldsymbol{s}_{\\hat{\\mathbf{Z}}}(\\cdot;h),\\quad\\boldsymbol{s}_{\\hat{\\mathbf{Z}}}^{m}(\\cdot;h),\\quad\\mathrm{and}\\quad\\tilde{\\boldsymbol{s}}_{\\hat{\\mathbf{Z}}}^{m}(\\cdot;h).\\tag{28}\n$$\n\nWe will be addressing both general and linear transformations $g .$ In the linear transformation setting, the true linear transformation $g$ is denoted by the matrix $\\mathbf{G}\\in\\mathbb{R}^{d\\times n}$ . Accordingly, we denote a valid linear encoder by $\\mathbf{H}\\in\\mathbb{R}^{n\\times d}$ . For a given valid encoder H, the associated valid decoder is given by its Moore-Penrose inverse, i.e., $\\mathbf H^{\\dagger}\\triangleq\\mathbf H^{\\top}\\cdot(\\mathbf H\\cdot\\mathbf H^{\\top})^{-1}$\n\nScore functions and their variations across diferent interventional environments play pivotal roles in our approach to identifying latent representations. In this section, we present the key properties of the score functions that will be leveraged in Section 5 and Section 6 to construct identifiability results, along with the corresponding algorithms.\n\nWe first investigate score variations across pairs of environments, such as the observational environment and an interventional one (under both soft and hard atomic interventions) or two interventional environments, either coupled or uncoupled. The key insight is that an intervention causes changes in only certain coordinates of the score function, as indicated by the sparse changes in the decompositions of the score functions in (18) and (19). These sparse changes further reflect the graph structure in the score diferences. For instance, for a single-node intervention $I^{m}=\\{i\\}$ in m-th environment, j-th coordinate of the score diference $s-s^{m}$ becomes\n\n$$\n\\left[\\boldsymbol{s}(\\mathbf{z})-\\boldsymbol{s}^{m}(\\mathbf{z})\\right]_{j}=\\frac{\\partial}{\\partial z_{j}}\\log p_{i}(z_{i}\\mid\\mathbf{z}_{\\mathrm{pa}(i)})-\\frac{\\partial}{\\partial z_{j}}\\log q_{i}(z_{i}\\mid\\mathbf{z}_{\\mathrm{pa}(i)}),\\tag{29}\n$$\n\nwhich is zero if $j\\not\\in\\overline{{\\mathrm{pa}}}(i)$ . The following lemma formalizes this property for all relevant cases.\n\nLemma 7 (Score Changes under Interventions) Consider the observational environment $\\mathcal{E}^{0}$ and an interventional environment $\\mathcal{E}^{m}$ with unknown intervention target $I^{m}$ . Then, for any causal model and intervention type, $\\mathbb{E}\\Big [\\big|\\pmb{s}(\\mathbf{Z})-\\pmb{s}^{m}(\\mathbf{Z})\\big|_{i}\\Big]\\neq 0$ implies that $i\\in\\overline{{\\mathrm{pa}}}(I^{m})$ . For the reverse direction, we have the following results specified for each relevant case.\n\n(i) Hard interventions: If the intervention in $\\mathcal{E}^{m}~(o r~\\tilde{\\mathcal{E}}^{m})$ is hard, then score functions s and $\\pmb{s}^{m}\\left(o r\\tilde{\\pmb{s}}^{m}\\right)$ difer in their i-th coordinate if and only if node i or one of its children is intervened in $\\mathcal{E}^{m}$ (or in $\\tilde{\\mathcal{E}}^{m})$\n\n$$\n\\mathbb{E}\\left[\\left|\\boldsymbol{s}(\\mathbf{Z})-\\boldsymbol{s}^{m}(\\mathbf{Z})\\right|_{i}\\right]\\neq 0\\quad\\Longleftrightarrow\\quad i\\in\\overline{{\\mathrm{pa}}}(I^{m}),\\tag{30}\n$$\n\n$$\na n d\\quad\\mathbb{E}\\Big [\\big|\\boldsymbol{s}(\\mathbf{Z})-\\tilde{\\boldsymbol{s}}^{m}(\\mathbf{Z})\\big|_{i}\\Big]\\neq 0\\quad\\Longleftrightarrow\\quad i\\in\\overline{{\\mathrm{pa}}}(\\tilde{I}^{m}).\\tag{31}\n$$\n\n(ii) Soft interventions: If the intervention in $\\mathcal{E}^{m}$ is soft and the latent causal model is an additive noise model, then score functions s and $s^{m}$ difer in their i-th coordinate if and only if node i or one of its children is intervened in $\\mathcal{E}^{m}$\n\n$$\n\\mathbb{E}\\left[\\left|\\boldsymbol{s}(\\mathbf{Z})-\\boldsymbol{s}^{m}(\\mathbf{Z})\\right|_{i}\\right]\\neq 0\\quad\\Longleftrightarrow\\quad i\\in\\overline{{\\mathrm{pa}}}(I^{m}).\\tag{32}\n$$\n\n(iii) Coupled environments $I^{m}=\\tilde{I}^{m}$ : In the coupled environment setting, $s^{m}$ and $\\tilde{\\pmb{s}}^{m}$ difer in their i-th coordinate if and only $i f i$ is intervened.\n\n$$\n\\mathbb{E}\\left[\\left|\\boldsymbol{s}^{m}(\\mathbf{Z})-\\tilde{\\boldsymbol{s}}^{m}(\\mathbf{Z})\\right|_{i}\\right]\\neq 0\\quad\\Longleftrightarrow\\quad i=I^{m}.\\tag{33}\n$$\n\n(iv) Uncoupled environments $I^{m}\\neq\\tilde{I}^{m}$ : Consider two interventional environments $\\mathcal{E}^{m}$ and $\\tilde{\\mathcal{E}}^{m}$ with diferent intervention targets ${\\cal I}^{m}\\ne{\\tilde{\\cal I}}^{m}$ , and consider additive noise models specified in (7). Given that $p(\\mathbf{Z})$ is twice diferentiable, the score functions $s^{m}$ and $\\tilde{\\pmb{s}}^{m}$ difer in their i-th coordinate if and only if node i or one of its children is intervened.\n\n$$\n\\mathbb{E}\\left[\\left|\\boldsymbol{s}^{m}(\\mathbf{Z})-\\tilde{\\boldsymbol{s}}^{m}(\\mathbf{Z})\\right|_{i}\\right]\\neq 0\\quad\\Longleftrightarrow\\quad i\\in\\overline{{\\mathrm{pa}}}(I^{m},\\tilde{I}^{m}).\\tag{34}\n$$\n\nTheorem 7 provides the necessary and suficient conditions for the invariance of the coordinates of the score functions of latent variables. The core idea of the score-based framework is that tracing these sparse changes in the score functions of the latent variables guides finding reliable estimates for the inverse of the transformation g, which in turn facilitates estimating $\\mathbf{Z}.$ Intuitively, we will look for the encoders $h \\in \\mathcal H$ such that the variations between the score estimates $\\begin{array}{ r }{{\\pmb{s}}_{\\hat{\\mathbf{Z}}}(\\hat{\\bf{z}};h),{\\pmb{s}}_{\\hat{\\mathbf{Z}}}^{m}(\\hat{\\bf{z}};h)}\\end{array}$ , and $\\tilde{\\pmb{s}}_{\\hat{\\mathbf{z}}}^{m}(\\hat{\\mathbf{z}};h)$ will be similar to the true score variations given by Theorem 7. However, the scores of the latent variables are not directly accessible. To circumvent this, we need to understand the connection between score functions of X and Z. In the following lemma, we leverage the change of variables formula for injective mappings and establish this relationship for any injective mapping f from latent to observed space.\n\nLemma 8 (Score Diference Transformation) Consider random vectors $\\mathbf{Y}_{1},\\mathbf{Y}_{2}\\in\\mathbb{R}^{r}$ and $\\mathbf{W}_{1},\\mathbf{W}_{2}\\in\\mathbb{R}^{s}$ that are related through\n\n$$\n\\mathbf{Y}_{1}=f(\\mathbf{W}_{1}),\\quad a n d\\quad\\mathbf{Y}_{2}=f(\\mathbf{W}_{2}),\\tag{35}\n$$\n\nsuch that $r\\geq s,$ probability measures of $\\mathbf{W}_{1},\\mathbf{W}_{2}$ are absolutely continuous with respect to the s-dimensional Lebesgue measure, and $f:\\mathbb{R}^{s}\\mathbb{R}^{r}$ is an injective and continuously diferentiable function. The diference of the score functions of Y<sub>1</sub> and $\\mathbf{Y}_{2}$ , and that of W<sub>1</sub> and $\\mathbf{W}_{2}$ are related as\n\n$$\ns_{\\mathbf{W}_{1}}(\\mathbf{w})-s_{\\mathbf{W}_{2}}(\\mathbf{w})=\\left[J_{f}(\\mathbf{w})\\right]^{\\top}\\cdot\\left[s_{\\mathbf{Y}_{1}}(\\mathbf{y})-s_{\\mathbf{Y}_{2}}(\\mathbf{y})\\right],\\quad \\text{where}\\mathbf{y}=f(\\mathbf{w}),\\tag{36}\n$$\n\nwhere $J_{f}(\\mathbf{w})$ denotes the Jacobian of f at point w. Furthermore, for the reverse direction, we have\n\n$$\ns_{\\mathbf{Y}_{1}}(\\mathbf{y})-s_{\\mathbf{Y}_{2}}(\\mathbf{y})=\\left[\\left[J_{f}(\\mathbf{w})\\right]^{\\dagger}\\right]^{\\top}\\cdot\\left[s_{\\mathbf{W}_{1}}(\\mathbf{w})-s_{\\mathbf{W}_{2}}(\\mathbf{w})\\right],\\quad \\text{where}\\mathbf{y}=f(\\mathbf{w}).\\tag{37}\n$$\n\nFor CRL under linear transformations in Section 5, we use the following corollary of Theorem 8.\n\nCorollary 9 In Theorem 8, if f is a linear transform, that is Y = F W for a full-rank matrix $\\mathbf{F}\\in\\mathbb{R}^{r\\times s}$ , then the score functions of Y and W are related through $\\begin{array}{ r }{s_{\\mathbf{W}}(\\mathbf{w})=\\mathbf{F}^{\\top}\\cdot s_{\\mathbf{Y}}(\\mathbf{y})}\\end{array}$ and $s_{\\mathbf{Y}}(\\mathbf{y})=\\left[\\mathbf{F}^{\\dagger}\\right]^{\\top}\\cdot s_{\\mathbf{W}}(\\mathbf{w})$ , where $\\mathbf{y}=\\mathbf{F}\\cdot\\mathbf{w}$\n\nIn this section, we consider CRL under linear transformation, in which the general transformation model in (3) becomes:\n\n$$\n\\mathbf{X}=\\mathbf{G}\\cdot\\mathbf{Z},\\tag{41}\n$$\n\nwhere $\\mathbf{G}\\in\\mathbb{R}^{d\\times n}$ is an unknown full-rank matrix mapping the latent variables to the observed ones. We will present steps for leveraging the properties of the score functions presented in Section 4 to design an algorithm that identifies the true encoder and recovers the true causal representations. The algorithm is referred to as Linear Score-based Causal Latent Estimation via Interventions (LSCALE-I). The theoretical guarantees associated with the algorithm steps will also be presented, serving as constructive proof of identifiability.\n\nLSCALE-I consists of three stages outlined in Algorithm 1. In the first stage, we will learn an encoder $\\begin{array}{ r }{\\hat{\\bf H},}\\end{array}$ row by row, where each row is an estimate of a row of the true encoder $\\mathbf{G}^{\\dagger}$ . This encoder estimate will recover each $Z_{i}$ up to mixing with $\\mathbf{Z}_{\\mathrm{pa}(i)}$ . In the second <sub>stage, we will use the estimated latent variables H</sub>ˆ <sub>to obtain the transitive closure of the</sub> latent graph . Finally, as an optional third stage for hard interventions, we will leverage the independence statements implied by the hard interventions to refine our encoder estimate. We will demonstrate that this refinement yields scaling consistency and perfect DAG recovery. We will also discuss the relevance and distinctions of our results vis-à-vis the results in the existing literature.\n\nStatistical diversity. Similarly to the existing identifiability results, it is necessary to have some regularity conditions <sup>5</sup> on the probability distributions of observational and interventional environments. We adopt the following assumption for the case of linear transformations.\n\nAssumption 1 For every possible pair $(i,k)$ where $i\\in [n],k\\in\\mathrm{pa}(i)$ , the following term cannot be a constant function in z,\n\n$$\n\\frac{\\partial}{\\partial z_{k}}\\left(\\log\\frac{p_{i}(z_{i}\\mid\\mathbf{z}_{\\mathrm{pa}(i)})}{q_{i}(z_{i}\\mid\\mathbf{z}_{\\mathrm{pa}(i)})}\\right)\\left[\\frac{\\partial}{\\partial z_{i}}\\log\\frac{p_{i}(z_{i}\\mid\\mathbf{z}_{\\mathrm{pa}(i)})}{q_{i}(z_{i}\\mid\\mathbf{z}_{\\mathrm{pa}(i)})}\\right]^{-1}.\\tag{42}\n$$\n\nEssentially, $\\begin{array}{ r }{\\frac{\\partial}{\\partial z_{k}}\\big(\\log\\frac{p_{i}\\left(z_{i}\\left|\\mathbf{z}_{\\mathrm{pa}(i)}\\right.\\right)}{q_{i}\\left(z_{i}\\left|\\mathbf{z}_{\\mathrm{pa}(i)}\\right.\\right)}\\big)}\\end{array}$ captures the efect of intervening on node i on the score associated with node k. Therefore, Assumption 1 ensures that an intervention suficiently diferentiates the target variable and its parents in the score function. We note that this is a very mild assumption and holds for a wide range of commonly used models (including additive noise models under hard interventions) and is discussed in more detail in Section 5.5.\n\nBefore we explain the details of LSCALE-I and establish identifiability results in the following subsections, we first present the rationale and derive the partial identifiability result, which will serve as the building block. In the rest of the paper, for brevity, we denote the score diference functions for each $m \\in [ n ]$ by\n\n$$\n\\boldsymbol{d}_{\\mathbf{Z}}^{m}(\\mathbf{z})\\triangleq\\boldsymbol{s}^{m}(\\mathbf{z})-\\boldsymbol{s}(\\mathbf{z})\\quad \\text{and}\\quad\\boldsymbol{d}_{\\mathbf{X}}^{m}(\\mathbf{x})\\triangleq\\boldsymbol{s}_{\\mathbf{X}}^{m}(\\mathbf{x})-\\boldsymbol{s}_{\\mathbf{X}}(\\mathbf{x}).\\tag{43}\n$$\n\nTo construct an algorithm that achieves identifiability under linear transformations, we first analyze the score diferences for the linear transformation. Note that for linear transformation ${\\mathbf{X}}={\\mathbf{G}}\\cdot{\\mathbf{Z}}.$ Jacobian $J_{\\mathbf{G}}(\\mathbf{z})$ is independent of z and equal to matrix G. Also, we have $\\mathbf{z}=\\mathbf{G}^{\\dagger}\\cdot\\mathbf{x}$ for all $\\mathbf{x}\\in\\mathrm{col}(\\mathbf{G})$ . Then, using Theorem 9 we have\n\n$$\n\\boldsymbol{d}_{\\mathbf{X}}^{m}(\\mathbf{x})=[\\mathbf{G}^{\\dagger}]^{\\top}\\cdot\\boldsymbol{d}_{\\mathbf{Z}}^{m}(\\mathbf{z}).\\tag{44}\n$$\n\nLet us denote the correlation matrices of $d_{\\mathbf{X}}^{m}(\\mathbf{x})$ and $d_{\\mathbf{Z}}^{m}(\\mathbf{z})$ , respectively, by\n\n$$\n\\mathbf{R}_{\\mathbf{X}}^{m}\\triangleq\\mathbb{E}\\big [\\pmb{d}_{\\mathbf{X}}^{m}(\\mathbf{x})\\cdot [\\pmb{d}_{\\mathbf{X}}^{m}(\\mathbf{x})]^{\\top}\\big]\\quad\\mathrm{and}\\quad\\mathbf{R}_{\\mathbf{Z}}^{m}\\triangleq\\mathbb{E}\\big [\\pmb{d}_{\\mathbf{Z}}^{m}(\\mathbf{z})\\cdot [\\pmb{d}_{\\mathbf{Z}}^{m}(\\mathbf{z})]^{\\top}\\big].(45)\n$$\n\nNote that (44) implies that $\\mathbf{R}_{\\mathbf{X}}^{m}=[\\mathbf{G}^{\\dagger}]^{\\top}\\cdot\\mathbf{R}_{\\mathbf{Z}}^{m}\\cdot\\mathbf{G}^{\\dagger}$ . The next lemma specifies that the structure of the column space of correlation matrix R<sup>m</sup> is heavily constrained by the true encoder $\\mathbf{G}^{\\dagger}$ , the graph $\\mathcal{G}_{:}$ and the intervention target $I^{m}$ in m-th environment.\n\nAfter obtaining the partial identifiability result from a single interventional environment, our goal is to construct an algorithm that uses the complete set of interventional environments $\\mathcal{E}=\\{\\mathcal{E}^{1},\\ldots,\\mathcal{E}^{n}\\}$ with targets $\\cup_{m\\in [n]}I^{m}=[n]$ to form estimates of the true encoder $\\mathbf{G}^{\\dagger}$ and the latent graph $\\mathcal{G}$ .\n\nEncoder estimation. In Stage L1 of Algorithm 1, we form an encoder estimate $\\hat{\\bf H}$ via using the column spaces of $\\{\\mathbf{R}_{\\mathbf{X}}^{m}:m\\in [n]\\}$ . Specifically, for each $m \\in [ n ]$ we select a vector randomly from $\\mathrm{col}(\\mathbf{R}_{\\mathbf{X}}^{m})$ and assign its transpose to m-th row of Hˆ .\n\nAlgorithm 1 Linear Score-based Causal Latent Estimation via Interventions (LSCALE-I)\n\n<div class=\"mineru-algorithm\" style=\"white-space: pre-wrap; font-family:monospace;\">\nInput: $\\mathbf{R}_{\\mathbf{X}}^{m}$ for all $m \\in [n]$ $\\triangleright$ compute using (45)\n\nStage L1: Encoder estimation\n$\\hat{\\mathbf{H}} \\leftarrow \\mathbf{0}_{n \\times d}$\nfor $m \\in (1, \\ldots, n)$ do\n    Select any $\\mathbf{h} \\in \\text{col}(\\mathbf{R}_{\\mathbf{X}}^{m})$ $\\hat{\\mathbf{H}}_{m} \\leftarrow \\mathbf{h}^{\\top}$\n\nStage L2: Latent graph estimation\nConstruct latent DAG estimate $\\hat{\\mathcal{G}}$ with parent sets\n$\\hat{\\text{pa}}(m) \\leftarrow \\left\\{ i \\neq m : \\mathbb{E} \\left[ \\left| \\boldsymbol{d}_{\\hat{\\mathbf{Z}}}^{m}(\\hat{\\mathbf{Z}}; \\hat{\\mathbf{H}}) \\right|_{i} \\right] \\neq 0 \\right\\}, \\quad \\forall m \\in [n]$\n\nif the interventions are hard then\n    Stage L3: Unmixing procedure\n    $\\pi \\leftarrow$ a topological order of $\\hat{\\mathcal{G}}$\n    for $m \\in (\\pi_1, \\ldots, \\pi_n)$ do\n        $\\hat{\\mathbf{Z}} \\leftarrow \\hat{\\mathbf{Z}}^{m}(\\mathbf{X}; \\hat{\\mathbf{H}})$ $\\mathbf{u} \\leftarrow \\text{Cov}(\\hat{Z}_{m}, \\hat{\\mathbf{Z}}_{\\hat{\\text{pa}}(m)}) \\cdot [\\text{Cov}(\\hat{\\mathbf{Z}}_{\\hat{\\text{pa}}(m)})]^{-1}$ $\\hat{\\mathbf{H}}_{m} \\leftarrow \\hat{\\mathbf{H}}_{m} - \\mathbf{u} \\cdot \\hat{\\mathbf{H}}_{\\hat{\\text{pa}}(m)}$\n    Refine $\\hat{\\mathcal{G}}$ according to\n    $\\hat{\\text{pa}}(m) \\leftarrow \\left\\{ i \\neq m : \\mathbb{E} \\left[ \\left| \\boldsymbol{d}_{\\hat{\\mathbf{Z}}}^{m}(\\hat{\\mathbf{Z}}; \\hat{\\mathbf{H}}) \\right|_{i} \\right] \\neq 0 \\right\\}, \\quad \\forall m \\in [n]$\n\nReturn $\\hat{\\mathcal{G}}, \\hat{\\mathbf{H}},$ and $\\hat{\\mathbf{Z}}$\n</div>\n\nLatent graph estimation. Next, we use the encoder estimate $\\hat{\\bf H}$ to recover the transitive closure of the latent graph $\\mathcal{G}$ . Motivated by Theorem $7(\\mathrm{ii})$ , we form $\\hat{\\mathcal G}$ with parent sets\n\n$$\n\\hat{\\mathrm{pa}}(m)\\triangleq\\left\\{i\\neq m:\\mathbb{E}\\left[|\\boldsymbol{s}_{\\hat{\\mathbf{Z}}}(\\hat{\\mathbf{Z}};\\hat{\\mathbf{H}})-\\boldsymbol{s}_{\\hat{\\mathbf{Z}}}^{m}(\\hat{\\mathbf{Z}};\\hat{\\mathbf{H}})|_{i}\\right]\\neq 0\\right\\}.\\tag{53}\n$$\n\nUsing the result that $\\hat{\\mathbf{Z}}(\\mathbf{X};\\hat{\\mathbf{H}})$ recovers Z up to mixing with parents, we show that the transitive closures of the estimate $\\hat{\\mathcal G}$ and the true graph $\\mathcal{G}$ are the same under a graph isomorphism.\n\nFirst, we provide a linear algebraic property, which will be used in the proofs.\n\nLemma 33 Consider the latent causal graph , and a matrix $\\mathbf{L}\\in\\mathbb{R}^{n\\times n}$\n\n1. Let $\\mathbf{L}_{\\mathrm{pa}}$ and $\\mathbf{L}_{\\mathrm{an}}$ be binary matrices that denote the parental and ancestral relationships in $\\mathcal{G}$ , respectively, i.e.,\n\n$$\n[\\mathbf{L}_{\\mathrm{pa}}]_{i,j}\\triangleq\\left\\{\\begin{array}{l l}1,&\\text{if}\\quad j\\in\\overline{{\\mathrm{pa}}}(i)\\\\ 0,&\\text{otherwise}\\end{array},\\text{and}\\quad [\\mathbf{L}_{\\mathrm{an}}]_{i,j}\\triangleq\\left\\{\\begin{array}{l l}1,&\\text{if}\\quad j\\in\\overline{{\\mathrm{an}}}(i)\\\\ 0,&\\text{otherwise}\\end{array}.\\right.\\right.\\tag{170}\n$$\n\nThen, $if\\mathbf{I}_{n\\times n}\\preccurlyeq\\mathbf{1}\\{\\mathbf{L}\\}\\preccurlyeq\\mathbf{L}_{\\mathrm{pa}}$ , we also have ${\\bf\\cal I}_{n\\times n}\\preccurlyeq 1\\{{\\bf L}^{-1}\\}\\preccurlyeq{{\\bf L}}_{\\mathrm{an}}$\n\n2. Let $\\mathbf{L}_{\\mathrm{sur}}$ be a binary matrix that denotes the surrounding relationships in ${\\mathcal{G}},$ , i.e.,\n\n$$\n[\\mathbf{L}_{\\text{sur}}]_{i,j}\\triangleq\\left\\{\\begin{array}{l l}1,&\\text{if}\\overline{{\\operatorname{ch}}}(i)\\subseteq\\overline{{\\operatorname{ch}}}(j),\\\\ 0,&\\text{otherwise}\\end{array}.\\right.\\tag{171}\n$$\n\nThen, $i f\\mathbf{I}_{n\\times n}\\preccurlyeq\\mathbb{1}\\{\\mathbf{L}\\}\\preccurlyeq\\mathbf{L}_{\\operatorname{sur}}$ , we also have ${\\bf I}_{n\\times n}\\preccurlyeq\\mathbb{1}\\{{\\bf L}^{-1}\\}\\preccurlyeq{\\bf L}_{\\mathrm{sur}}$", "theorems": [{"local_id": "T1", "statement": "Lemma 10 For any interventional environment $\\mathcal{E}^{m}$ , co $|\\big(\\mathbf{R}_{\\mathbf{X}}^{m}\\big)\\subseteq\\mathrm{span}\\big\\{ [\\mathbf{G}_{i}^{\\dagger}]^{\\top}:i\\in\\overline{{\\mathrm{pa}}}(I^{m})\\big\\}$", "proof": {"steps": [{"n": 1, "claim": "Proof: Using the sparse score changes property in (29), we know that $\\left[d_{\\mathbf{Z}}^{m}(\\mathbf{z})\\right]_{i}=0$ for all $i\\not\\in\\overline{{\\mathrm{pa}}}(I^{m})$ . Then, using (44), for any $\\mathbf{x}=\\mathbf{G}\\cdot\\mathbf{z}$ we have\n\n$$\n\\boldsymbol{d}_{\\mathbf{X}}^{m}(\\mathbf{x})=\\sum_{i\\in\\overline{{\\mathrm{pa}}}(I^{m})}\\left[\\mathbf{G}_{i}^{\\dagger}\\right]^{\\top}\\cdot\\left[\\boldsymbol{d}_{\\mathbf{Z}}^{m}(\\mathbf{z})\\right]_{i}\\in\\operatorname{span}\\left\\{\\left[\\mathbf{G}_{i}^{\\dagger}\\right]^{\\top}:i\\in\\overline{{\\mathrm{pa}}}(I^{m})\\right\\}.\\tag{46}\n$$"}, {"n": 2, "claim": "Since this holds for any $\\mathbf{x}\\in\\mathrm{col}(\\mathbf{G})$ , we have\n\n$$\n\\operatorname{span}\\left\\{\\boldsymbol{d}_{\\mathbf{X}}^{m}(\\mathbf{x}):\\mathbf{x}\\in\\operatorname{col}(\\mathbf{G})\\right\\}\\subseteq\\operatorname{span}\\left\\{\\left[\\mathbf{G}_{i}^{\\dagger}\\right]^{\\top}:i\\in\\overline{{\\mathrm{pa}}}\\left(I^{m}\\right)\\right\\}.\\tag{47}\n$$"}, {"n": 3, "claim": "By definition of $\\mathbf{R}_{\\mathbf{X}}^{m}$ , we have $\\mathrm{col}(\\mathbf{R}_{\\mathbf{X}}^{m})=\\operatorname{span}\\{\\boldsymbol{d}_{\\mathbf{X}}^{m}(\\mathbf{x}):\\mathbf{x}\\in\\mathrm{col}(\\mathbf{G})\\}$ which concludes the proof."}]}}, {"local_id": "T2", "statement": "Theorem 11 (Linear – Node-level partial identifiability) A single-node soft interventional environment $\\mathcal{E}^{m}$ with intervention target $\\ell=I^{m}$ is suficient to recover $Z_{\\ell}$ up to mixing with its parents.", "proof": {"steps": [{"n": 1, "claim": "Proof: Let us pick y uniformly at random from unit sphere $\\mathbb{S}^{d-1}\\subset\\mathbb{R}^{d}$ and let $\\mathbf{h}=\\mathbf{R}_{\\mathbf{X}}^{m}\\cdot\\mathbf{y}$ Note that $\\left[d_{\\mathbf{Z}}^{m}(\\mathbf{z})\\right]_{i}=0$ for all $i\\not\\in{\\overline{{\\mathrm{pa}}}}(\\ell)$ implies that $\\left[{\\bf R}_{\\bf Z}^{m}\\right]_{i}={\\bf 0}$ for all $i\\not\\in{\\overline{{\\mathrm{pa}}}}(\\ell)$ . Then, using $\\mathbf{R}_{\\mathbf{X}}^{m}=[\\mathbf{G}^{\\dagger}]^{\\top}\\cdot\\mathbf{R}_{\\mathbf{Z}}^{m}\\cdot\\mathbf{G}^{\\dagger}$ , we have\n\n$$\n\\mathbf{h}=\\mathbf{R}_{\\mathbf{X}}^{m}\\cdot\\mathbf{y}=[\\mathbf{G}^{\\dagger}]^{\\top}\\cdot\\mathbf{R}_{\\mathbf{Z}}^{m}\\cdot\\mathbf{G}^{\\dagger}\\cdot\\mathbf{y}\\tag{48}\n$$\n\n$$\n=\\sum_{i\\in\\overline{{\\mathrm{pa}}}(\\ell)}\\left[\\mathbf{R}_{\\mathbf{Z}}^{m}\\right]_{i}\\cdot\\mathbf{G}^{\\dagger}\\cdot\\mathbf{y}\\cdot [\\mathbf{G}_{i}^{\\dagger}]^{\\top}\\tag{49}\n$$"}, {"n": 2, "claim": "Denote $c_{i}=\\left[\\mathbf{R}_{\\mathbf{Z}}^{m}\\right]_{i}\\cdot\\mathbf{G}^{\\dagger}\\cdot\\mathbf{y}$ for each $i\\in{\\overline{{\\mathrm{pa}}}}(\\ell)$ . Then, using $\\mathbf{G}_{i}^{\\dagger}\\cdot\\mathbf{G}=\\mathbf{e}_{i}$ , i-th standard basis vector, we have $[{\\bf h}^{\\top}\\cdot{\\bf G}]_{i}=c_{i}$ for $i\\in{\\overline{{\\mathrm{pa}}}}(\\ell)$ . As such, we can use h to obtain\n\n$$\n\\mathbf{h}^{\\top}\\cdot\\mathbf{X}=\\mathbf{h}^{\\top}\\cdot\\mathbf{G}\\cdot\\mathbf{Z}=c_{\\ell}\\cdot Z_{\\ell}+\\sum_{i\\in\\mathrm{pa}(\\ell)}c_{i}\\cdot Z_{i}.\\tag{50}\n$$"}, {"n": 3, "claim": "Finally, note that $\\left[\\mathbf{R}_{\\mathbf{Z}}^{m}\\right]_{\\ell}$ is not a zero vector since $[d_{\\mathbf{Z}}^{m}]_{\\ell}\\neq 0$ . Therefore, since rows of $\\mathbf{G}^{\\dagger}$ are linearly independent, we know that $\\left[{\\bf R}_{{\\bf Z}}^{m}\\right]_{i}\\cdot{\\bf G}^{\\dagger}$ is a nonzero vector. Thus, $\\mathbf{y}\\in\\mathbb{S}^{d-1}$ ensures that $c_{\\ell}$ is nonzero with probability 1. Then, $\\mathbf{h}^{\\top}\\cdot\\mathbf{X}$ is an estimate of $Z_{\\ell}$ that satisfies consistency up to mixing with parents."}]}}, {"local_id": "T3", "statement": "Lemma 12 (Linear – Encoder via Soft Interventions) Under soft interventions, output H<sup>ˆ</sup> ofAlgorithm 1 achieves identifiability up to mixing with parents. Specifically, $\\hat{\\bf Z}({\\bf X};\\hat{\\bf H})=$ ${\\bf P}_{\\mathcal{T}}\\cdot{\\bf C}_{\\mathrm{pa}}\\cdot{\\bf Z}$ such that diagonal entries of $\\mathbf{C}_{\\mathrm{{p a}}}$ are nonzero and $[\\mathbf{C}_{\\mathrm{pa}}]_{i,j}=0$ for all $j\\not\\in\\overline{{\\mathrm{pa}}}(i)$", "proof": {"steps": [{"n": 1, "claim": "Proof: Following Theorem 11 for all $m \\in [ n ]$ immediately implies the desired result. Specifically, according to (50), $[\\hat{\\mathbf{H}}_{m}]^{\\top}\\in\\mathrm{col}(\\mathbf{R}_{\\mathbf{X}}^{m})$ satisfies\n\n$$\n[\\hat{\\mathbf{Z}}(\\mathbf{X};\\hat{\\mathbf{H}})]_{m}=\\hat{\\mathbf{H}}_{m}\\cdot\\mathbf{X}=c_{\\ell}\\cdot Z_{\\ell}+\\sum_{i\\in\\mathrm{pa}(\\ell)}c_{i}\\cdot Z_{i},\\quad \\text{where}\\ell=I^{m},\\tag{51}\n$$\n\nfor some constants $\\{c_{i}:i\\in\\overline{{\\mathrm{pa}}}(\\ell)\\}$ where $c_{\\ell}\\neq 0$ with probability 1. Then, combining all n identities, we have\n\n$$\n\\hat{\\mathbf{Z}}(\\mathbf{X};\\hat{\\mathbf{H}})=\\mathbf{P}_{\\mathcal{I}}\\cdot\\mathbf{C}_{\\mathrm{pa}}\\cdot\\mathbf{Z},\\tag{52}\n$$\n\nwhere $\\mathbf{P}_{\\mathcal{T}}$ is the permutation matrix of the intervention order $(I^{1},\\ldots,I^{n})$ , and $\\mathbf{C}_{\\mathrm{{p a}}}$ has nonzero diagonal entries and satisfies $[\\mathbf{C}_{\\mathrm{pa}}]_{i,j}=0$ for all $j\\not\\in\\overline{{\\mathrm{pa}}}(i)$"}]}}, {"local_id": "T4", "statement": "Lemma 13 (Linear – Graph via Soft Interventions) Under Assumption 1 and soft interventions, output $\\hat{\\mathcal G}$ of Algorithm 1 recovers the transitive closure of $\\mathcal{G}$", "proof": {"steps": [{"n": 1, "claim": "Proof: Let $\\hat{\\mathbf{Z}}$ denote $\\hat{\\bf Z}({\\bf X};\\hat{\\bf H})={\\bf P}_{\\mathcal{T}}\\cdot{\\bf C}_{\\mathrm{pa}}\\cdot{\\bf Z}$ . Recall that $\\left[{\\bf C}_{\\mathrm{pa}}\\right]_{i,j}=0$ for all $j\\not\\in\\overline{{\\mathrm{pa}}}(i)$ , and since nodes $(1,\\ldots,n)$ are topologically ordered, $\\mathbf{C}_{\\mathrm{{p a}}}$ is a lower triangular matrix. By using standard linear algebra for the expansion of an inverse of a lower triangular matrix, we can show that $\\left[{\\bf C}_{\\mathrm{pa}}^{-1}\\right]_{i,j}=0$ for all $j\\not\\in\\overline{{\\mathrm{an}}}(i)$ (see Theorem 33 in Appendix B.1 for the detailed steps). Then, using Theorem 9 for the scores of $\\mathbf{Z},$ we have\n\n$$\n\\boldsymbol{d}_{\\hat{\\mathbf{Z}}}^{m}(\\hat{\\mathbf{Z}})=\\mathbf{P}_{\\mathcal{I}}\\cdot [\\mathbf{C}_{\\mathrm{pa}}^{-1}]^{\\top}\\cdot\\boldsymbol{d}_{\\mathbf{Z}}^{m}(\\mathbf{Z}).\\tag{54}\n$$"}, {"n": 2, "claim": "Next, taking the transpose of $\\mathbf{C}_{\\mathrm{{p a}}}^{-1}$ , we have $\\big [\\mathbf{C}_{\\mathrm{pa}}^{-\\top}\\big]_{i,j}=0$ for all $j\\not\\in\\overline{{\\mathrm{de}}}(i)$ . Combining this observation with the fact that $[d_{\\mathbf{Z}}^{m}]_{j}=0$ for all $j\\not\\in\\widetilde{{\\mathrm{pa}}}(I^{m})$ , we obtain\n\n$$\n\\left[\\boldsymbol{d}_{\\hat{\\mathbf{Z}}}^{m}(\\hat{\\mathbf{Z}})\\right]_{k}=\\left[\\mathbf{C}_{\\mathrm{pa}}^{-\\top}\\right]_{I^{k}}\\cdot\\boldsymbol{d}_{\\mathbf{Z}}^{m}(\\mathbf{Z})=\\sum_{j\\in\\overline{{\\mathrm{de}}}(I^{k})\\cap\\overline{{\\mathrm{pa}}}(I^{m})}\\left[\\mathbf{C}_{\\mathrm{pa}}^{-\\top}\\right]_{I^{k},j}\\cdot\\left[\\boldsymbol{d}_{\\mathbf{Z}}^{m}\\right]_{j}\\tag{55}\n$$"}, {"n": 3, "claim": "Subsequently, $\\mathbb{E}\\big [|{d}_{\\hat{\\mathbf{Z}}}^{m}(\\hat{\\mathbf{Z}})|_{k}\\big]\\neq 0$ implies that $\\overline{{\\mathrm{de}}}(I^{k})\\cap\\overline{{\\mathrm{pa}}}(I^{m})\\neq\\emptyset$ , and $I^{k}\\in\\overline{{\\mathrm{an}}}(I^{m})$ . Therefore, the estimated parent set in (53) satisfies ${\\hat{\\mathrm{pa}}}(m)\\subseteq{\\overline{{\\mathrm{an}}}}(I^{m})$ , and the transitive closure of $\\mathcal{G}$ is a supergraph of $\\hat{\\mathcal G}$ under relabeling of the nodes with permutation . Next, we show that $\\hat{\\mathcal G}$ contains all the edges in the transitive reduction of ${\\mathcal{G}}.$ Let $I^{k}\\to I^{m}$ be an edge in the transitive reduction of , i.e., there is no other directed path between nodes $I^{k}$ and $I^{m}$ in $\\mathcal{G}.$ This implies that $\\overline{{\\mathrm{de}}}(I^{k})\\cap\\overline{{\\mathrm{pa}}}(I^{m})=\\{I^{k},I^{m}\\}$ , and (54) becomes\n\n$$\n\\left[\\boldsymbol{d}_{\\hat{\\mathbf{Z}}}^{m}(\\hat{\\mathbf{Z}})\\right]_{k}=\\left[\\mathbf{C}_{\\mathrm{pa}}^{-\\top}\\right]_{I^{k},I^{k}}\\cdot\\left[\\boldsymbol{d}_{\\mathbf{Z}}^{m}\\right]_{I^{k}}+\\left[\\mathbf{C}_{\\mathrm{pa}}^{-\\top}\\right]_{I^{k},I^{m}}\\cdot\\left[\\boldsymbol{d}_{\\mathbf{Z}}^{m}\\right]_{I^{m}}.\\tag{56}\n$$"}, {"n": 4, "claim": "Assumption 1 ensures that $\\big [d_{\\hat{\\mathbf{Z}}}^{m}(\\hat{\\mathbf{Z}})\\big]_{k}$ is not constantly zero, which means that $\\hat{\\mathcal G}$ contains $k  m$ edge, preserving $I^{k}\\to I^{m}$ in $\\mathcal{G}$ . Therefore, $\\hat{\\mathcal G}$ preserves all the edges in the transitive reduction of ${\\mathcal{G}},$ and taking its transitive closure gives $\\mathcal{G}_{\\mathrm{tc}}$ , under relabeling of the nodes with permutation $\\mathcal{Z}.$"}]}}], "target_local_id": null, "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2025-25-0657", "context": "Consider the following finite-sum convex minimization problem:\n\n$$\nf(\\mathbf{x})=\\frac{1}{n}\\sum_{i=1}^{n}\\psi_{i}(\\mathbf{x}),\\qquad \\text{with}\\qquad\\mathbf{x}^{*}\\in\\operatorname{argmin}f(\\mathbf{x}).\\tag{1}\n$$\n\nAssumption 1 Consider a twice continuously diferentiablefunction $f:\\mathbb{R}^{d} $ ℝ defined as in (1), where $\\psi_{i}:\\mathbb{R}^{d}\\mathbb{R}$ and $i\\in\\{1,2,...,n\\}$ . We make standard 𝜆-smoothness, 𝜇-strong convexity, and 𝐿-Lipschitz Hessian assumptions, with $\\kappa = \\lambda / \\mu$ as the condition number.\n\n1. For each $i\\in\\{1,2,...,n\\},\\psi_{i}$ is a 𝜆-smooth convexfunction:\n\n$$\n\\psi_{i}(\\mathbf{y})\\leq\\psi_{i}(\\mathbf{x})+\\nabla\\psi_{i}(\\mathbf{x})^{\\top}(\\mathbf{y}-\\mathbf{x})+\\frac{\\lambda}{2}\\|\\mathbf{y}-\\mathbf{x}\\|^{2},\\forall\\mathbf{x},\\mathbf{y}\\in\\mathbb{R}^{d},\n$$\n\nand 𝑓 is a 𝜇-strongly convexfunction:\n\n$$\nf(\\mathbf{y})\\geq f(\\mathbf{x})+\\nabla f(\\mathbf{x})^{\\top}(\\mathbf{y}-\\mathbf{x})+\\frac{\\mu}{2}\\|\\mathbf{y}-\\mathbf{x}\\|^{2},\\forall\\mathbf{x},\\mathbf{y}\\in\\mathbb{R}^{d}.\n$$\n\n2. The Hessian $\\nabla f^{2}(\\mathbf{x})$ of the objective function is 𝐿-Lipschitz continuous:\n\n$$\n\\left\\|\\nabla f^{2}(\\mathbf{x})-\\nabla f^{2}(\\mathbf{y})\\right\\|\\leq L\\left\\|\\mathbf{x}-\\mathbf{y}\\right\\|,\\forall\\mathbf{x},\\mathbf{y}\\in\\mathbb{R}^{d}.\n$$\n\nDefinition 1 (Gradient oracle) Given mini-batch size $1 \\leq b \\leq n$ and indices $i_{1},...,i_{b}\\in\\{1,...,n\\}$ the gradient oracle $\\mathcal{G}$ returns $\\hat{\\mathbf{g}}\\sim\\mathcal{G}(\\{i_{1},...,i_{b}\\})$ such that:\n\n$$\n\\hat{\\mathbf{g}}(\\mathbf{x})=\\frac{1}{b}\\sum_{j=1}^{b}\\nabla\\psi_{i_{j}}(\\mathbf{x}),\\qquad f o r a n y\\mathbf{x}\\in\\mathbb{R}^{d}.\n$$\n\nDefinition 3 (Hessian oracle) Given fixed $0~<~\\beta_{1}~\\le~\\beta_{2}$ , we say that $\\mathcal{H}\\ =\\ \\mathcal{H}_{\\beta_{1},\\beta_{2}}$ is a $\\beta_{2}/\\beta_{1}\\cdot$ approximate Hessian oracle, and use $\\alpha: =\\beta_{2}/\\beta_{1}$ to denote the Hessian approximation factor, if for any $\\mathbf{x}\\in\\mathbb{R}^{d}$ , $\\mathcal{H}$ returns a possibly random $\\bar{\\mathbf{H}}\\sim\\mathcal{H}(\\mathbf{x})$ such that:\n\n$$\n\\beta_{1}\\cdot\\nabla^{2}f(\\mathbf{x})\\preceq\\bar{\\mathbf{H}}\\preceq\\beta_{2}\\cdot\\nabla^{2}f(\\mathbf{x}).\n$$\n\nAlgorithm 1 Mini-batch Stochastic Variance-Reduced Newton (Mb-SVRN)\n\n<div class=\"mineru-algorithm\" style=\"white-space: pre-wrap; font-family:monospace;\">\nRequire: $\\tilde{\\mathbf{x}}_0$, gradient mini-batch size $b$, gradient/Hessian oracles $\\mathcal{G}, \\mathcal{H}$, inner iterations $t_{\\max}$ for $s = 0, 1, 2, \\cdots$, do\n    Compute the Hessian estimate $\\bar{\\mathbf{H}}_s \\sim \\mathcal{H}(\\tilde{\\mathbf{x}}_s)$\n    Compute the full gradient $\\mathbf{g}_s = \\mathbf{g}(\\tilde{\\mathbf{x}}_s)$ for $\\mathbf{g} \\sim \\mathcal{G}(\\{1, ..., n\\})$\n    Set $\\mathbf{x}_{0,s} = \\tilde{\\mathbf{x}}_s$\n    for $t = 0, 1, 2, \\cdots, t_{\\max} - 1$ do\n        Compute $\\hat{\\mathbf{g}} \\sim \\mathcal{G}(\\{i_1, ..., i_b\\})$ for $i_1, ..., i_b \\sim \\{1, ..., n\\}$ uniformly random\n        Compute $\\hat{\\mathbf{g}}_{t,s} = \\hat{\\mathbf{g}}(\\mathbf{x}_{t,s})$ and $\\hat{\\mathbf{g}}_{0,s} = \\hat{\\mathbf{g}}(\\tilde{\\mathbf{x}}_s)$\n        Compute variance-reduced gradient $\\bar{\\mathbf{g}}_{t,s} = \\hat{\\mathbf{g}}_{t,s} - \\hat{\\mathbf{g}}_{0,s} + \\mathbf{g}_s$\n        Update $\\mathbf{x}_{t+1,s} = \\mathbf{x}_{t,s} - \\eta\\bar{\\mathbf{H}}_s^{-1}\\bar{\\mathbf{g}}_{t,s}$\n    end for\n    $\\tilde{\\mathbf{x}}_{s+1} = \\mathbf{x}_{t_{max},s}$\nend for\n</div>\n\nNotation. Let $\\tilde{\\mathbf{x}}_{0}$ denote the starting outer iterate and $\\tilde{\\mathbf{X}}_{s}$ denote the outer iterate after 𝑠 outer iterations. Within the $(s+1)^{t h}$ outer iteration, the inner iterates are indexed as $\\mathbf{x}_{t,s}.$ . Since our results require working only within one outer iteration, we drop the subscript 𝑠 throughout the analysis, and denote $\\tilde{\\mathbf{x}}_{0}$ as 𝐱̃ and corresponding $t^{t h}$ inner iterate as $\\mathbf{x}_{t}.$ . Similarly let 𝐇<sup>̃</sup> and $\\mathbf{H}_{t}$ denote the Hessian at $\\tilde{\\mathbf{x}}$ and $\\mathbf{x}_{t},$ respectively. For gradient mini-batch size $^{b,}$ , let $\\hat{\\bf g}$ denote $\\begin{array}{ r }{\\frac{1}{b}\\sum_{j=1}^{b}\\dot{\\nabla}\\psi_{i_{j}}(\\tilde{\\mathbf{x}})}\\end{array}$ , let $\\hat{\\bf g}_{t}$ denote $\\begin{array}{ r }{\\frac{1}{b}\\sum_{j=1}^{b}\\nabla\\psi_{i_{j}}(\\mathbf{x}_{t})}\\end{array}$ and let $\\bar{\\mathbf{g}}_{t}=\\hat{\\mathbf{g}}_{t}-\\hat{\\mathbf{g}}+\\tilde{\\mathbf{g}}$ denote the variance-reduced gradient at $\\mathbf{x}_{t}.$ , where 𝐠̃ represents exact gradient at 𝐱̃. Moreover, 𝐠 represents exact gradient at $\\mathbf{x}_{t}$ , 𝐇 represents Hessian at $\\mathbf{X}^{*}$ , and let $\\tilde{\\Delta}=\\tilde{\\mathbf{x}}-\\mathbf{x}^{*},\\Delta_{t}=\\mathbf{x}_{t}-\\mathbf{x}^{*}$ . Also, we use $\\eta$ to denote the fixed step size, and $\\alpha=\\beta_{2}/\\beta_{1}$ denotes the Hessian approximation factor. Finally, we refer to $\\mathbb{E}_{t}$ as the conditional expectation given the event that the algorithm has reached iterate $\\mathbf{x}_{t}$ starting with the outer iterate 𝐱̃ .\n\nNext, we define the local convergence neighborhood, by using the notion of a Mahalanobis norm: $\\|\\mathbf{x}\\|_{\\mathbf{M}}={\\sqrt{\\mathbf{x}^{\\top}\\mathbf{M}\\mathbf{x}}}$ , for any positive semi-definite matrix 𝐌 and vector 𝐱.\n\nDefinition 6 Given $\\epsilon_{0}>0 $ , we define a local neighborhood $\\boldsymbol{\\nu}_{f}(\\epsilon_{0})$ around $\\mathbf{X}^{*}$ as follows:\n\n$$\n\\mathcal{V}_{f}(\\epsilon_{0}):=\\left\\{\\mathbf{x}:\\left\\|\\mathbf{x}-\\mathbf{x}^{*}\\right\\|_{\\mathbf{H}}^{2}<\\frac{\\mu^{3/2}}{L}\\epsilon_{0}\\right\\},\n$$\n\nwhere $L > 0$ is the Lipschitz constant for Hessians of 𝑓, and $\\mu > 0$ is the strong convexity parameter $f o r f .$\n\nLemma 7 (Upper bound on variance of stochastic gradient) Let $\\tilde{\\mathbf{x}}\\in\\mathcal{V}_{f}(\\epsilon_{0}\\eta)$ and $\\mathbf{x}_{t}\\in\\mathcal{V}_{f}(c\\epsilon_{0}\\eta)$ for some $c \\geq 1$ . Then:\n\n$$\n\\mathbb{E}_{t}[\\|\\bar{\\bf g}_{t}-{\\bf g}_{t}\\|^{2}]\\leq(1+c\\epsilon_{0}\\eta)\\frac{\\lambda}{b}\\|\\tilde{\\bf x}-{\\bf x}_{t}\\|_{\\bf H}^{2}.\n$$\n\nAssuming the access to Hessian oracle returning 𝐇<sup>̄</sup> according to Definition 3, we define $\\hat{\\textbf{ H }}: =$ $\\bar{\\bf H}/\\sqrt{\\beta_{1}\\beta_{2}}$ . It immediately follows that,\n\n$$\n\\frac{1}{\\sqrt{\\alpha}}\\nabla^{2}f(\\tilde{\\mathbf{x}})\\preceq\\hat{\\mathbf{H}}\\preceq\\sqrt{\\alpha}\\nabla^{2}f(\\tilde{\\mathbf{x}}),\n$$\n\nfor $\\alpha=\\beta_{2}/\\beta_{1}$ . We refer to the above property as 𝛼-Hessian approximation and say $\\hat{\\bf H}$ is an $\\alpha \\cdot$ approximate Hessian at $\\tilde{\\mathbf{X}},$ denoted as $\\hat{\\textbf{ H }}\\approx_{\\sqrt{\\alpha}}\\tilde{\\textbf{ H }}$ . Thus, without loss of generality, in the rest of the proof we will use 𝐇<sup>̂</sup> instead of $\\bar{\\bf H}.$ In the following result, we use standard approximate Newton analysis to establish that, using an 𝛼-approximate Hessian at 𝐱̃, a Newton step with a suficiently small step size 𝜂 reduces the distance to $\\mathbf{X}^{*}$ (in 𝐇 norm), at least by a factor of $\\left(1-\\frac{\\eta}{8\\sqrt{\\alpha}}\\right)$ . We call this an approximate Newton step since the gradient is exact and only the second-order information is stochastic.\n\nLemma 9 (High-probability bound on stochastic gradient noise) Let $\\begin{array}{ r }{\\epsilon_{0}<\\frac{1}{8 c\\sqrt{\\alpha}},\\eta<\\frac{1}{4\\sqrt{\\alpha}},\\tilde{\\mathbf{x}}\\in}\\end{array}$\n\n$\\mathcal{V}_{f}(\\epsilon_{0}\\eta),\\mathbf{x}_{t}\\in\\mathcal{V}_{f}(c\\epsilon_{0}\\eta)$ for some $c \\geq 1$ . Then, for any $\\delta>0,$ , with probability at least $1-\\delta\\frac{b^{2}}{n^{2}}$ (depending on the gradient mini-batch size 𝑏):\n\n$$\n\\left\\|\\mathbf{g}_{t}-\\bar{\\mathbf{g}}_{t}\\right\\|\\leq\\left\\{\\begin{array}{l l}(16\\sqrt{\\kappa\\lambda}/3b)\\ln(n/b\\delta)\\left\\|\\mathbf{x}_{t}-\\tilde{\\mathbf{x}}\\right\\|_{\\mathbf{H}}&forb<\\frac{8}{9}\\kappa,\\\\ 4\\sqrt{\\lambda/b}\\ln(n/b\\delta)\\left\\|\\mathbf{x}_{t}-\\tilde{\\mathbf{x}}\\right\\|_{\\mathbf{H}}&forb\\geq\\frac{8}{9}\\kappa .\\end{array}\\right.\n$$\n\nLemma 10 (Mean Value Theorem) Let $\\tilde{\\mathbf{x}}\\in\\mathcal{V}_{f}(\\epsilon_{0}\\eta),\\mathbf{x}_{t}\\in\\mathcal{V}_{f}(c\\epsilon_{0}\\eta)$ for $\\begin{array}{ r }{\\epsilon_{0}<\\frac{1}{8 c\\sqrt{\\alpha}}}\\end{array}$ and for some $c \\geq 1$ . If the Hessian estimate satisfies 𝐇<sup>̂</sup> ${\\approx}\\sqrt{\\alpha}^{\\tilde{\\mathbf{H}},}$ , then\n\n$$\n\\left\\|\\hat{\\mathbf{H}}^{-1}\\mathbf{g}_{t}\\right\\|_{\\mathbf{H}}\\leq 2\\sqrt{\\alpha}\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}.\n$$\n\nTheorem 11 (One inner iteration conditional expectation result) Let $\\begin{array}{ r }{\\epsilon_{0}<\\frac{1}{8 c\\sqrt{\\alpha}},\\tilde{\\mathbf{x}}\\in\\mathcal{V}_{f}(\\epsilon_{0}\\eta)}\\end{array}$ $\\mathbf{x}_{t}\\in\\mathcal{V}_{f}(c\\epsilon_{0}\\eta)$ for some $c \\geq 1$ . Consider Mb-SVRN with step size $\\begin{array}{ r }{0<\\eta<\\operatorname*{min}\\{\\frac{1}{4\\sqrt{\\alpha}},\\frac{b}{48\\alpha^{3/2}\\kappa}\\}}\\end{array}$ and gradient mini-batch size 𝑏. Then:\n\n$$\n\\mathbb{E}_{t}[\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}^{2}]\\leq\\left(1-\\frac{\\eta}{8\\sqrt{\\alpha}}\\right)\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}+3\\eta^{2}\\frac{\\alpha\\kappa}{b}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}.\n$$\n\nWe proceed to formally define the random stopping and resume times. Consider a fixed $\\gamma > 0$ Define random stopping times $T_{i}$ and random resume times $L_{i}$ , with $L_{0}=0$ and for $i \\geq 0$ , as\n\n$$\nT_{i}=\\min\\left(\\left\\{t>L_{i}\\mid\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}>e^{2}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}\\text{or}\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}<\\frac{\\kappa\\alpha^{2}\\gamma}{n}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}\\right\\}\\cup\\left\\{\\frac{n}{b}\\right\\}\\right),\n$$\n\n$$\nL_{i+1}=\\min\\left(\\left\\{t>T_{i}\\mid\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}\\leq e^{2}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}\\text{and}\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}\\geq\\frac{\\kappa\\alpha^{2}\\gamma}{n}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}\\right\\}\\cup\\left\\{\\frac{n}{b}\\right\\}\\right).\\tag{3}\n$$\n\nThe random stopping time $T_{i}$ denotes the $(i+1)^{t h}$ iteration index when the random sequence $\\left(\\mathbf{x}_{t}\\right)$ leaves the local neighborhood $\\mathcal{V}_{f}(e^{2}\\epsilon_{0}\\eta)$ or satisfies $\\begin{array}{ r }{\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}<\\frac{\\kappa\\alpha^{2}\\gamma}{n}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}}\\end{array}$ . The random resume time $L_{i}$ denotes the $(i+1)^{t h}$ iteration index when the random sequence $\\left(\\mathbf{x}_{t}\\right)$ returns back to the local neighborhood and also satisfies $\\begin{array}{ r }{\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}\\ge\\frac{\\kappa\\alpha^{2}\\gamma}{n}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}}\\end{array}$ . The condition $\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}\\leq e^{2}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}$ , ensures that $\\mathbf{x}_{t}$ lies in a small local neighborhood around $\\mathbf{X}^{*}$ such that the previous results (Lemma 7, 8, 9) hold with $c=e^{2}$ , whereas the condition $\\begin{array}{ r }{\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}\\geq\\frac{\\kappa\\alpha^{2}\\gamma}{n}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}}\\end{array}$ intuitively means that $\\mathbf{x}_{t}$ hasn’t reached too close to $\\mathbf{X}^{*}$ (necessarily far from $ {\\widetilde{\\mathbf{X}}})$ such that behaviour of $\\mathbb{E}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}$ is determined by noise in the stochastic gradient. The latter condition also captures the fact that the optimal convergence rate has not been achieved at $\\mathbf{X}_{t}.$ . These two diferent conditions are required to design a submartingale from random time $L_{i}$ to $T_{i}$ . Note that, in hindsight, we have defined stopping time in a way similar to our convergence guarantee, $\\begin{array}{ r }{\\left\\|\\Delta_{\\frac{n}{b}}\\right\\|_{\\mathbf{H}}^{2}\\lesssim\\frac{\\kappa\\alpha^{2}}{n}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}}\\end{array}$ . We add a factor $\\gamma$ to capture constants and log factors that can appear in the convergence guarantee. One can show that for $T_{i}\\leq t<L_{i+1}$ , the behavior of $\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}$ is dictated by the variance in the stochastic gradient and not the progress made by the approximate Newton step.\n\nFinally, as the last building block of the submartingale framework, we establish a high probability upper and lower bound on $\\|\\Delta_{t+1}\\|_{\\mathbf{H}}$ in terms of $\\|\\Delta_{t}\\|_{\\mathbf{H}}$ . Here, due to the noise in the stochastic gradient, we cannot prove almost sure bounds. However, by Lemmas 9 and 16, we can get the upper and lower bounds, holding with probability at least $1-\\delta\\frac{b^{2}}{n^{2}}$ , for any $\\delta > 0$ . For notational convenience, we use 𝑀 to denote the high probability upper bound on ${\\left\\|\\bar{\\bf g}_{t}-{\\bf g}_{t}\\right\\|}$ , guaranteed in Lemma 9,\n\n$$\nM=\\left\\{\\begin{array}{l}\\frac{\\kappa}{b}\\quad \\text{if} b<\\frac{8}{9}\\kappa\\\\ \\sqrt{\\frac{\\kappa}{b}}\\text{if} b\\geq\\frac{8}{9}\\kappa .\\end{array}\\right.\n$$", "theorems": [{"local_id": "T1", "statement": "Lemma 14 (Submartingale property till stopping time) Let the gradient mini-batch size be $1 \\leq$ $b \\ \\leq \\ n$ and $\\begin{array}{ r }{{\\epsilon_{0}}<\\frac{1}{8e^{2}\\sqrt{\\alpha}}.\\ Let{\\tilde{\\textbf{ x }}}\\in\\ {\\mathcal{V}}_{f}(\\epsilon_{0}\\eta),\\ \\mathbf{x}_{t}\\in\\ {\\mathcal{V}}_{f}(e^{2}\\epsilon_{0}\\eta)\\ ,\\ \\eta=\\ \\frac{b\\sqrt{\\alpha}\\beta}{n}\\ <\\ \\operatorname*{min}\\{\\frac{1}{4\\sqrt{\\alpha}},\\frac{b}{48\\alpha^{3/2}k}\\}}\\end{array}$ $\\begin{array}{ r }{\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}\\leq\\frac{n}{\\kappa\\alpha^{2}\\gamma}\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2},}\\end{array}$ , and $\\begin{array}{ r }{\\frac{3\\beta}{\\gamma}<\\frac{1}{16}}\\end{array}$ . Then:\n\n$$\n\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}\\leq\\left(1-\\frac{\\eta}{32\\sqrt{\\alpha}}\\right)\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof By Theorem 11 and using $\\begin{array}{ r }{\\frac{3\\beta}{\\gamma}<\\frac{1}{16}}\\end{array}$ , it follows that,\n\n$$\n\\begin{array}{l}\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}^{2}\\leq\\left(1-\\frac{\\eta}{8\\sqrt{\\alpha}}\\right)\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}+3\\eta^{2}\\frac{\\alpha\\kappa}{b}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}\\\\ \\leq\\left(1-\\frac{\\eta}{8\\sqrt{\\alpha}}\\right)\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}+3\\eta\\cdot\\frac{b\\sqrt{\\alpha}\\beta}{n}\\cdot\\frac{\\alpha\\kappa}{b}\\cdot\\frac{n}{\\kappa\\alpha^{2}\\gamma}\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}\\\\ =\\left(1-\\frac{\\eta}{8\\sqrt{\\alpha}}+\\frac{3\\eta}{\\sqrt{\\alpha}}\\cdot\\frac{\\beta}{\\gamma}\\right)\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}\\\\ \\leq\\left(1-\\frac{\\eta}{16\\sqrt{\\alpha}}\\right)\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2},\\\\ \\end{array}\n$$\n\nimplying that\n\n$$\n\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}\\leq\\left(1-\\frac{\\eta}{32\\sqrt{\\alpha}}\\right)\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}},\n$$\n\nwhich concludes the proof."}]}}, {"local_id": "T2", "statement": "Lemma 16 (Bounded variation) Let $\\tilde{\\mathbf{x}}\\in\\mathcal{V}_{f}(\\epsilon_{0}\\eta)$ with $\\epsilon_{0}~<~{\\frac{1}{8 e^{2}\\sqrt{\\alpha}}}$ and $\\begin{array}{ r }{\\eta\\ \\leq\\ \\frac{1}{4\\sqrt{\\alpha}}}\\end{array}$ . Also, let $\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}\\leq e^{2}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}$ and $\\begin{array}{ r }{\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}\\geq\\frac{\\kappa\\alpha^{2}\\gamma}{n}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}}\\end{array}$ for some $\\gamma > 0 .$ . Then, with $\\begin{array}{ r }{\\omega=\\sqrt{\\frac{n}{\\kappa\\alpha^{2}\\gamma}}.}\\end{array}$\n\n$$\n\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}\\leq 2\\omega\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof We apply Jensen’s inequality to $\\mathbb{E}_{t}\\left.\\Delta_{t+1}\\right._{\\mathbf{H}}$ to get,\n\n$$\n\\begin{array}{l}\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}\\geq\\left\\|\\mathbb{E}_{t}\\Delta_{t+1}\\right\\|_{\\mathbf{H}}=\\left\\|\\Delta_{t}-\\eta\\hat{\\mathbf{H}}^{-1}\\mathbf{g}_{t}\\right\\|_{\\mathbf{H}}\\\\ \\geq\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}-\\eta\\left\\|\\hat{\\mathbf{H}}^{-1}\\mathbf{g}_{t}\\right\\|_{\\mathbf{H}}.\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "Since $\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}\\leq e^{2}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}$ , we have $\\mathbf{x}_{t}\\in\\mathcal{V}_{f}(e^{2}\\epsilon_{0}\\eta)$ . By using Lemma 10 on the term $\\left\\|\\hat{\\mathbf{H}}^{-1}\\mathbf{g}_{t}\\right\\|_{\\mathbf{H}}$\n\n$$\n\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}\\geq\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}-2\\eta\\sqrt{\\alpha}\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}\\geq\\frac{1}{2}\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}.\n$$"}, {"n": 3, "claim": "The last inequality holds because $\\begin{array}{ r }{\\eta\\leq\\frac{1}{4\\sqrt{\\alpha}}}\\end{array}$ . Finally, we use the condition that $\\begin{array}{ r }{\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}\\geq\\frac{\\kappa\\alpha^{2}\\gamma}{n}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}}\\end{array}$ concluding the proof."}]}}, {"local_id": "T3", "statement": "Lemma 17 (Predictable quadratic variation bound) Let $\\tilde{\\mathbf{x}}\\in\\mathcal{V}_{f}(\\epsilon_{0}\\eta)$ with $\\begin{array}{ r }{\\epsilon_{0}<\\frac{1}{8 e^{2}\\sqrt{\\alpha}}}\\end{array}$ and $\\eta =$ $\\begin{array}{ r }{\\frac{b\\sqrt{\\alpha}\\beta}{n}\\leq\\operatorname*{min}\\left\\{\\frac{1}{4\\sqrt{\\alpha}},\\frac{b}{48\\alpha^{3/2}\\kappa}\\right\\}}\\end{array}$ for some $\\beta > 0 .$ . Also, let $\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}\\leq e^{2}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}$ and $\\begin{array}{ r }{\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}\\ge\\frac{\\kappa\\alpha^{2}\\gamma}{n}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}}\\end{array}$ for some $\\gamma > 0 .$ . Then, with $\\begin{array}{ r }{\\omega=\\sqrt{\\frac{n}{\\kappa\\alpha^{2}\\gamma}}.}\\end{array}$\n\n$$\n\\mathbb{E}_{t}\\left(\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}-\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}\\right)^{2}<80\\eta\\frac{\\beta}{\\gamma\\sqrt{\\alpha}}\\left(\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}\\right)^{2}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof Consider $\\mathbb{E}_{t}\\left(\\left.\\Delta_{t+1}\\right._{\\mathbf{H}}-\\mathbb{E}_{t}\\left.\\Delta_{t+1}\\right._{\\mathbf{H}}\\right)^{2}$ . We have,\n\n$$\n\\begin{array}{l}\\mathbb{E}_{t}\\left(\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}-\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}\\right)^{2}=\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}^{2}-(\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}})^{2}\\\\ =\\mathbb{E}_{t}\\left\\|\\Delta_{t}-\\eta\\hat{\\mathbf{H}}^{-1}\\bar{\\mathbf{g}}_{t}\\right\\|_{\\mathbf{H}}^{2}-(\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}})^{2}\\\\ =\\left\\|\\Delta_{t}-\\eta\\hat{\\mathbf{H}}^{-1}\\mathbf{g}_{t}\\right\\|_{\\mathbf{H}}^{2}+\\eta^{2}\\mathbb{E}_{t}\\left\\|\\hat{\\mathbf{H}}^{-1}(\\bar{\\mathbf{g}}_{t}-\\mathbf{g}_{t})\\right\\|_{\\mathbf{H}}^{2}\\\\ +2\\eta\\mathbb{E}_{t}\\left(\\boldsymbol{\\Delta}_{t}-\\eta\\hat{\\mathbf{H}}^{-1}\\mathbf{g}_{t}\\right)^{\\top}\\left(\\hat{\\mathbf{H}}^{-1}(\\bar{\\mathbf{g}}_{t}-\\mathbf{g}_{t})\\right)-(\\mathbb{E}_{t}\\left\\|\\boldsymbol{\\Delta}_{t+1}\\right\\|_{\\mathbf{H}})^{2}.\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "Using 𝔼<sub>𝑡</sub> $\\left(\\hat{\\mathbf{H}}^{-1}(\\bar{\\mathbf{g}}_{t}-\\mathbf{g}_{t})\\right)=\\hat{\\mathbf{H}}^{-1}\\mathbb{E}_{t}\\left(\\bar{\\mathbf{g}}_{t}-\\mathbf{g}_{t}\\right)=0\\mathrm{~a n d~}\\Delta_{t}-\\eta\\hat{\\mathbf{H}}^{-1}\\mathbf{g}_{t}=\\mathbb{E}_{t}\\Delta_{t+1},\\mathrm{~w e~g e t},$\n\n$$\n\\mathbb{E}_{t}\\left(\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}-\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}\\right)^{2}=\\left\\|\\mathbb{E}_{t}\\Delta_{t+1}\\right\\|_{\\mathbf{H}}^{2}+\\eta^{2}\\mathbb{E}_{t}\\left\\|\\hat{\\mathbf{H}}^{-1}(\\bar{\\mathbf{g}}_{t}-\\mathbf{g}_{t})\\right\\|_{\\mathbf{H}}^{2}-(\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}})^{2}.\n$$"}, {"n": 3, "claim": "Using Jensen’s inequality on the first term, we have $\\left\\|\\mathbb{E}_{t}\\Delta_{t+1}\\right\\|_{\\mathbf{H}}\\leq\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}$ , from which it follows that,\n\n$$\n\\begin{array}{l}\\mathbb{E}_{t}\\left(\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}-\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}\\right)^{2}\\leq\\left(\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}\\right)^{2}+\\eta^{2}\\mathbb{E}_{t}\\left\\|\\hat{\\mathbf{H}}^{-1}(\\bar{\\mathbf{g}}_{t}-\\mathbf{g}_{t})\\right\\|_{\\mathbf{H}}^{2}-(\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}})^{2}\\\\ =\\eta^{2}\\mathbb{E}_{t}\\left\\|\\hat{\\mathbf{H}}^{-1}(\\bar{\\mathbf{g}}_{t}-\\mathbf{g}_{t})\\right\\|_{\\mathbf{H}}^{2}\\\\ =\\eta^{2}\\mathbb{E}_{t}\\|\\mathbf{H}^{1/2}\\hat{\\mathbf{H}}^{-1/2}\\hat{\\mathbf{H}}^{-1/2}(\\bar{\\mathbf{g}}_{t}-\\mathbf{g}_{t})\\|^{2}\\\\ \\leq\\eta^{2}\\cdot\\|\\mathbf{H}^{1/2}\\hat{\\mathbf{H}}^{-1/2}\\|^{2}\\cdot\\mathbb{E}_{t}\\|\\hat{\\mathbf{H}}^{-1/2}(\\bar{\\mathbf{g}}_{t}-\\mathbf{g}_{t})\\|^{2}.\\tag{6}\\\\ \\end{array}\n$$"}, {"n": 4, "claim": "Using the fact that 𝐇<sup>̂</sup> $\\approx_{\\sqrt{\\alpha}}$ 𝐇<sup>̃</sup> and 𝐇<sup>̃</sup> $\\approx_{(1+\\epsilon_{0}\\eta)}$ 𝐇, we have 𝐇<sup>̂</sup> $\\approx\\sqrt{\\alpha}(1{+}\\epsilon_{0}\\eta)$ 𝐇. Therefore, $\\|\\mathbf{H}^{1/2}\\hat{\\mathbf{H}}^{-1/2}\\|^{2}=$ $\\left\\|\\mathbf{H}^{1/2}\\hat{\\mathbf{H}}^{-1}\\mathbf{H}^{1/2}\\right\\|\\leq\\sqrt{\\alpha}(1+\\epsilon_{0}\\eta)$ . Combining this with the inequality (6),\n\n$$\n\\mathbb{E}_{t}\\left(\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}-\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}\\right)^{2}\\leq\\eta^{2}\\sqrt{\\alpha}(1+\\epsilon_{0}\\eta)\\cdot\\mathbb{E}_{t}\\|\\hat{\\mathbf{H}}^{-1/2}(\\mathbf{g}_{t}-\\bar{\\mathbf{g}}_{t})\\|^{2}.\n$$"}, {"n": 5, "claim": "Upper bounding $\\begin{array}{ r }{\\|\\hat{\\mathbf{H}}^{-1/2}\\|^{2}=\\left\\|\\hat{\\mathbf{H}}^{-1}\\right\\|\\leq\\frac{\\sqrt{\\alpha}}{\\mu}}\\end{array}$ , we get,\n\n$$\n\\mathbb{E}_{t}\\left(\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}-\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}\\right)^{2}\\leq\\eta^{2}\\frac{\\alpha}{\\mu}(1+\\epsilon_{0}\\eta)\\cdot\\mathbb{E}_{t}[\\|(\\mathbf{g}_{t}-\\bar{\\mathbf{g}}_{t})\\|^{2}].\\tag{7}\n$$"}, {"n": 6, "claim": "Since $\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}<e^{2}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}$ , we have $\\mathbf{x}_{t}\\in\\mathcal{V}_{f}(e^{2}\\epsilon_{0}\\eta)$ . By using Lemma 7 on the last term of the inequality (7), it follows that,\n\n$$\n\\mathbb{E}_{t}\\left[\\left\\|\\mathbf{g}_{t}-\\bar{\\mathbf{g}}_{t}\\right\\|^{2}\\right]\\leq\\frac{(1+c\\epsilon_{0}\\eta)\\lambda}{b}\\left\\|\\Delta_{t}-\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}\\leq\\frac{2(1+c\\epsilon_{0}\\eta)\\lambda}{b}\\left(\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}+\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}\\right).\\tag{8}\n$$"}, {"n": 7, "claim": "Substituting (8) in the inequality (7), we get,\n\n$$\n\\mathbb{E}_{t}\\left(\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}-\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}\\right)^{2}\\leq 2(1+\\epsilon_{0}\\eta)(1+c\\epsilon_{0}\\eta)\\eta^{2}\\frac{\\kappa\\alpha}{b}\\left(\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}+\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}\\right).\n$$"}, {"n": 8, "claim": "Again, using $\\left\\|\\Delta_{t}\\right\\|_{\\mathbf{H}}^{2}<e^{2}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}$\n\n$$\n\\mathbb{E}_{t}\\left(\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}-\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}\\right)^{2}<20\\eta^{2}\\frac{\\kappa\\alpha}{b}\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}.\n$$"}, {"n": 9, "claim": "Since $\\begin{array}{ r }{\\eta\\leq\\frac{1}{4\\sqrt{\\alpha}}}\\end{array}$ , we use Lemma 16 on $\\left\\|\\tilde{\\Delta}\\right\\|_{\\mathbf{H}}^{2}$ to obtain,\n\n$$\n\\mathbb{E}_{t}\\left(\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}-\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}\\right)^{2}<80\\eta^{2}\\frac{\\kappa\\alpha}{b}\\omega^{2}\\left(\\mathbb{E}_{t}\\left\\|\\Delta_{t+1}\\right\\|_{\\mathbf{H}}\\right)^{2}.\n$$"}, {"n": 10, "claim": "Substituting one of the 𝜂 factors as $\\frac{b{\\sqrt{\\alpha}}\\beta}{n}$ , and $\\begin{array}{ r }{\\omega^{2}=\\frac{n}{\\kappa\\alpha^{2}\\gamma}}\\end{array}$ , concludes the proof."}]}}], "target_local_id": null, "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2026-24-0456", "context": "Statistical model. We consider the classical nonparametric regression setting with fixed regular grid design, observing $n=N^{d}$ points,\n\n$$\nX_{i}=f(x_{i})+\\sigma\\varepsilon_{i}\n$$\n\nwith $(x_{i})_{1\\leq i\\leq n}$ the points of the regular grid $G_{1/N}$ over $[0,1]^{d}$ of step size $1 / N$ , σ the level of noise and $(\\varepsilon_{i})_{1\\leq i\\leq n}$ independent standard Gaussian variables $(\\mathrm{i . e .,}\\mathcal{N}(0,1))$ . This provides a standard and general framework for our analysis. In terms of applications, it is particularly well-suited for modeling noisy images, where the observed points correspond to pixels (or more generally, voxels) observed with additive noise.\n\nEstimator. In this context, our goal is to estimate $\\operatorname{dgm}(f)$ , the collection of persistence diagrams associated with the sublevel sets of $f_{i}$ , by computing the diagram associated with the sublevel sets of a histogram estimator of the signal. More formally, let $h > 0$ such that Nh is an integer (and so is $n h^{d})$ , consider $G_{h}$ the regular orthogonal grid over $[0,1]^{d}$ of step h and $C_{h}$ the collection of all the closed hypercubes of side h composing $G_{h}$ . For all $\\lambda\\in\\mathbb{R}$ 2 we define the estimator of $\\mathcal{F}_{\\lambda}=f^{-1}(]-\\infty,\\lambda])$ as follows:\n\n$$\n\\widehat{\\mathcal{F}}_{\\lambda}=\\bigcup_{H\\in C_{h,\\lambda}}H,\\text{with} C_{h,\\lambda}=\\left\\{H\\in C_{h}\\text{such that}\\frac{1}{n h^{d}}\\sum_{x_{i}\\in H}X_{i}\\leq\\lambda\\right\\}.\n$$\n\nthe set ${\\widehat{\\mathcal{F}}}_{\\lambda}$ represents the sublevel set indexed by λ of the histogram estimator of $f$ and $n h^{d}$ represents the cardinality of the set $H\\cap G_{h}$ for all $H\\in C_{h}$ . Next, we consider, for all $s\\in\\{0,\\bar{\\ldots},d\\},\\widehat{\\mathbb{V}}_{f,s}$ the persistence module induced by the collection of homology groups:\n\n$$\n\\left(H_{s}\\left(\\widehat{\\mathcal{F}}_{\\lambda}\\right)\\right)_{\\lambda\\in\\mathbb{R}}\n$$\n\nequipped with the linear maps $\\widehat{v}_{\\lambda}^{\\lambda^{\\prime}}$ induced by the inclusion $\\widehat{\\mathcal{F}}_{\\lambda}\\subset\\widehat{\\mathcal{F}}_{\\lambda^{\\prime}},\\lambda^{\\prime}\\geq\\lambda$ . We define $\\widehat{\\mathrm{dgm}(f)}$ as the associated collection of persistence diagrams. A natural question is how to calibrate the window size h for signals. From the proof of Lemma 7 (Appendix B.1), a good choice is taking h such that\n\n$$\nh^{\\alpha}>\\sqrt{\\frac{\\log(1/h^{d})}{n h^{d}}}\\tag{1}\n$$\n\nthus, we can choose:\n\n$$\nh\\asymp\\left(\\frac{\\log(n)}{n}\\right)^{\\frac{1}{d+2\\alpha}}.\n$$\n\nDefinition 5 A subspace A of X is called a deformation retract of X if there is a continuous $F:X\\times [0,1]\\to X$ such that for all $x \\in X$ and $a \\in A$ •\n\n$F(x,0)=x$  \n$F(x,1)\\in A$  \n$F(a,1)=a.$\n\nThe function F is then called a (deformation) retraction from X to A.\n\nA deformation retraction F from X to A ensures that X and A are “homotopy equivalent” (see Hatcher, 2000, pages 2–3), and thus are topologically similar. In particular, F induces isomorphisms between the homology groups of X and A. More precisely, for all $t\\in [0,1]$ $F(.,t)$ induces a group morphism $F^{\\#}(.,t):C_{s}(X)\\to C_{S}(X)$ between s−chains of X defined by composing each singular s-simplex $\\sigma:\\Delta^{s}X$ with $F(.,t)$ to get a singular s-simplex $F^{\\#}(\\sigma,t)=F(.,t)\\circ\\sigma:\\Delta^{s}X$ , then extending $F^{\\#}(.,t)$ linearly via $\\begin{array}{ r }{F^{\\#}\\left(\\sum_{i}n_{i}\\sigma_{i},t\\right)=}\\end{array}$ $\\begin{array}{ r }{\\sum_{i}n_{i}F^{\\#}\\left(\\sigma_{i},t\\right)=\\sum_{i}n_{i}F(.,t)\\circ\\sigma_{i}}\\end{array}$ . The group morphism $F^{*}(.,t):H_{s}(X)H_{s}(X)$ defined by $[C]\\mapsto [F^{\\#}(C,t)]$ can be shown to be an isomorphism for all $t\\in [0,1]$ (see Hatcher, 2000, pages 110-113). In particular, $F^{*}(.,1):H_{s}(X)\\to H_{s}(A)$ is an isomorphism.\n\nThe idea behind persistent homology is to encode the evolution of the topology (in the homology sense) of a nested family of topological spaces, called a filtration. As we move along indices, topological features (connected components, cycles, cavities, ...) can appear or disappear (existing connected components merge, cycles or cavities are filled, ...). Two keys to formalize this idea, which we use throughout this paper, are the notions of filtration and of persistence module.\n\nDefinition 6 Let $\\Lambda\\subset\\mathbb{R}$ be a set of indices. A filtration indexed by Λ is a family $(\\kappa_{\\lambda})_{\\lambda\\in\\Lambda}$ of topological spaces satisfying, for all $\\lambda,\\lambda^{\\prime}\\in\\Lambda,\\lambda\\leqslant\\lambda^{\\prime}$\n\n$$\n\\mathcal{K}_{\\lambda}\\subset\\mathcal{K}_{\\lambda^{\\prime}}.\n$$\n\nThe typical filtration that we will consider in this paper is, for a function $f:\\mathbb{R}^{d}\\mathbb{R}$ , the family of sublevel sets $({\\mathcal{F}}_{\\lambda})_{\\lambda\\in\\mathbb{R}}=\\left(f^{-1}(]-\\infty,\\lambda])\\right)_{\\lambda\\in\\mathbb{R}}$\n\nDefinition 7 Let $\\Lambda\\subset\\mathbb{R}$ be a set of indices. A persistence module over Λ is a family $\\mathbb{V}=(\\mathbb{V}_{\\lambda})_{\\lambda\\in\\Lambda}$ of vector spaces equipped with linear maps $v_{\\lambda}^{\\lambda^{\\prime}}:\\mathbb{V}_{\\lambda}\\to\\mathbb{V}_{\\lambda^{\\prime}}$ such that, for all $\\lambda\\leqslant\\lambda^{\\prime}\\leqslant\\lambda^{\\prime\\prime}\\in\\Lambda$\n\n$$\nv_{\\lambda}^{\\lambda}=i d\n$$\n\nand\n\n$$\nv_{\\lambda^{\\prime}}^{\\lambda^{\\prime\\prime}}\\circ v_{\\lambda}^{\\lambda^{\\prime}}=v_{\\lambda}^{\\lambda^{\\prime\\prime}}.\n$$\n\nThe typical persistence module that we will consider in this paper is, for a function $f:\\mathbb{R}^{d} $ R and $s \\in \\mathbb N$ , the family of homology groups $\\mathbb{V}_{f,s}=(H_{s}\\left(\\mathcal{F}_{\\lambda}\\right))_{\\lambda\\in\\mathbb{R}}$ equipped with $v_{\\lambda}^{\\lambda^{\\prime}}$ the linear maps induced by the inclusion $\\mathcal{F}_{\\lambda}\\subset\\mathcal{F}_{\\lambda^{\\prime}}$ . To be more precise, in this paper, $H_{s}(.)$ is the singular homology functor in degree s with coeficients in a field (typically $\\mathbb{Z}/2\\mathbb{Z})$ Hence, $H_{s}\\left(\\mathcal{F}_{\\lambda}\\right)$ is a vector space.\n\nUnder the q-tameness assumption (see Definition 10 in Appendix A), a persistence module V can be fully described by a multi-set in $\\mathbb{R}^{2}$ , known as its persistence diagram:\n\n$$\n\\mathrm{dgm}(\\mathbb{V})=\\{(b_{j},d_{j})\\mid j\\in J\\}\\subset\\overline{{\\mathbb{R}}}^{2}.\n$$\n\nIn this representation, each $b_{j}$ represents the birth time of a topological feature $(\\mathrm{e . g .}$ , the emergence of a connected component or the formation of a cycle), while $d_{j}$ represents its death time $(\\mathrm{e . g .}$ , the merging of two components or the filling of a cycle). The diference $d_{j}-b_{j}$ corresponds to the feature’s lifetime. For a detailed construction of persistence diagrams, we refer the reader to Chazal et al. (2016). In cases where we consider the collection of modules $(\\mathbb{V}_{f,s})_{s\\in\\mathbb{N}}$ associated with the sublevel sets filtration of $f,$ we denote by dgm(f) the collection of associated persistence diagrams.\n\nWe here detail and discuss the regularity assumptions on f mentioned in the Introduction. For a set $A\\subset [0,1]^{d}$ , we denote $\\overline{{A}}$ its closure, $A^{\\circ}$ its interior, ∂A its boundary and $A^{c}$ its complement. Let $f:[0,1]^{d}\\to\\mathbb{R}$ , we make the following assumptions about $f \\colon$\n\nA0. f is bounded by M, i.e., for all $x\\in [0,1]^{d},|f(x)|\\leq M$\n\nA1. f is a piecewise $(L,\\alpha)$ −Hölder-continuous function, i.e., there exist $M_{1},...,M_{l}$ disjoint open sets of $[0,1]^{d}$ such that\n\n$$\n\\bigcup_{i=1}^{l}\\overline{{M_{i}}}=[0,1]^{d}\n$$\n\nand for all $i\\in\\{1,...,l\\}$ and $x,y\\in M_{i}$ 2\n\n$$\n|f(x)-f(y)|\\leq L\\|x-y\\|_{2}^{\\alpha}.\n$$\n\nA2. f satisfies, for all $x_{0}\\in [0,1]^{d}$ 2\n\n$$\n\\operatorname*{liminf}_{\\substack{x\\in\\bigcup_{i=1}^{l}M_{i}\\to x_{0}}}f(x)=f(x_{0})\n$$\n\nIn this context, two signals, difering only on a null set, are statistically indistinguishable. Persistent homology is sensitive to point-wise irregularity. As a result, two signals difering only on a null set can have very diferent persistence diagrams. Assumption A2 prevents such scenarios. Furthermore, note that for any piecewise Hölder-continuous function $f,$ there exists a modification $\\tilde{f}$ satisfying Assumption A2 such that $f$ and $\\tilde{f}$ coincide except on a set of measure zero. Following this remark, note that, although this assumption is necessary for the proofs of our main results, it is more of a convention to avoid statistical indistinguishability than a true limitation, at least from a practical perspective.\n\nA3. $\\cup_{i=1}^{l}\\partial M_{i}\\cap\\ l]0,1[^{d}$ is a $C^{1,1}$ hypersurface, and, for $R>0,$ , it satisfies:\n\n$$\n\\text{reach}\\left(\\left]0,1\\right[^{d}\\cap\\bigcup_{i=1}^{l}\\partial M_{i}\\right)\\geq R \\text{and} d_{2}\\left(\\bigcup_{i=1}^{l}\\partial M_{i}\\cap]0,1\\left[^{d},\\partial [0,1]^{d}\\right.\\right)\\geq R\n$$\n\nwhere, for a set $A\\subset{\\mathbb{R}}^{d}$ , reach $(A)=$\n\n$$\n\\sup\\left\\{r\\in\\mathbb{R}:\\forall x\\in\\mathbb{R}^{d}\\backslash A \\text{with} d_{2}(\\{x\\},A)<r,\\exists!y\\in A \\text{s.t.}| |x-y| |_{2}=d_{2}(\\{x\\},A)\\right\\}\n$$\n\nand,\n\n$$\nd_{2}(A,B)=\\max\\left(\\sup_{x\\in B}\\inf_{y\\in A}| |x-y| |_{2},\\sup_{x\\in A}\\inf_{y\\in B}| |x-y| |_{2}\\right).\n$$\n\nis the Hausdorf distance between two sets A and B. The reach is a curvature measure introduced by Federer (1959). An intuitive way to understand it is that if A has a reach $R,$ we can roll a ball of radius R along the boundary of A. Positive reach assumptions are fairly common in statistical TDA (Balakrishnan et al., 2012; Niyogi et al., 2008) and geometric inference (Genovese et al., 2012; Kim et al., 2016; Aamari and Levrard, 2017; Aamari et al., 2019; Berenfeld et al., 2021). Here, the first part of Assumption A3 ensures geometric control over the union of the boundary of the $M_{i},i\\in\\{1,...,l\\}$ , in the interior of $[0,1]^{d}$ , for example, it prevents the appearance of cusps, corners, and multiple points. The second part ensures that discontinuities do not appear too close to the boundary of the cube $[0,1]^{d}$\n\nWe denote $S_{d}(M,L,\\alpha,R)$ the set of such functions. The combination of Assumptions A1, A2 and A3 ensures that the persistence diagrams of $f\\in S_{d}(M,L,\\alpha,R)$ are well-defined (see Appendix A, Proposition 1).\n\nThis section is devoted to the proof of Theorem 2. First, we introduce some notations. For a point $x\\in [0,1]^{d}$ and $b > 0$ , we denote the closed Euclidean ball centered at x of radius b:\n\n$$\nB_{2}(x,b)=\\{y\\in [0,1]^{d},| |x-y| |_{2}\\leq b\\}\n$$\n\nSimilarly, for a set $A\\subset{\\mathbb{R}}^{d}$ and $b \\geq 0$ , we denote,\n\n$$\nA^{b}=B_{2}(A,b)=\\left\\{x\\in [0,1]^{d}\\text{s.t.} d_{2}(\\{x\\},A)\\leq b\\right\\}\n$$\n\nand\n\n$$\nA^{-b}=\\left((A^{c})^{b}\\right)^{c}.\n$$\n\nWe also define, for all $h > 0 \\colon$\n\n$$\nN_{h}=\\frac{\\max_{H\\in C_{h}}\\left|\\frac{\\sigma}{n h^{d}}\\sum_{x_{i}\\in H}\\varepsilon_{i}\\right|}{\\sqrt{2\\sigma^{2}\\frac{\\log(1/h^{d})}{n h^{d}}}}.\n$$\n\nTo formalize this issue, we define the set of problematic points as\n\n$$\nS_{\\lambda,h}=\\left(\\left(\\bigcup_{i=1}^{l}\\partial M_{i}\\cap]0,1[^{d}\\right)^{h}\\setminus\\mathcal{F}_{\\lambda+L h^{\\alpha}}\\right)\\cap\\mathcal{F}_{\\lambda}^{h}.\n$$\n\nIntuitively, $S_{\\lambda,h}$ consists of the regions where $\\mathcal{F}_{\\lambda}^{h}$ extends beyond $\\mathcal{F}_{\\lambda+L h^{\\alpha}}$ , preventing a controlled inclusion. Our goal is to smoothly \"push\" these points into $\\mathcal{F}_{\\lambda+\\varepsilon}$ for some $\\varepsilon\\asymp h^{\\alpha}$ To achieve this, we will construct a deformation retraction using Theorem 4.8 of Federer (1959). Under Assumption A3, this theorem guarantees that for every $x\\in [0,1]^{d}$ at a (Euclidean) distance strictly smaller than R from $\\begin{array}{ r }{\\left]0,1[^{d}\\cap\\bigcup_{i=1}^{l}\\partial M_{i}\\right.}\\end{array}$ , there exists a unique closest point in $\\begin{array}{ r }{\\left]0,1[{}^{d}\\cap\\bigcup_{i=1}^{l}\\partial M_{i}\\right.}\\end{array}$ , denoted by $\\xi(x)$ , i.e., there exists a well-defined projection map. Moreover, the function $\\xi$ is continuous.\n\nUsing Assumptions A1 and A2, we can show (see the proof of Lemma 2) that for all $x\\in S_{\\lambda,h}$ , the projection $\\xi(x)$ belongs to $\\mathcal{F}_{\\lambda+\\varepsilon}$ for some $\\varepsilon\\asymp h^{\\alpha}$ . The idea is then to map each point $x\\in S_{\\lambda,h}$ to $\\xi(x)$ , moving along the line segment $[x,\\xi(x)]$ . However, turning this idea into a proper deformation retraction requires several technical adjustments.\n\nThe first issue is that the segments $[x,\\xi(x)]$ for $x\\in S_{\\lambda,h}$ may not be contained entirely in $\\mathcal{F}_{\\lambda}^{h}$ . To resolve this, we consider a slightly larger set, adding the segments $[x,\\xi(x)]$ for all $x\\in S_{\\lambda,h}$ :\n\n$$\n\\mathcal{K}_{\\lambda,h}:=\\mathcal{F}_{\\lambda}^{h}\\cup\\left(\\bigcup_{x\\in S_{\\lambda,h}}[x,\\xi(x)]\\right)\\subset\\mathcal{F}_{\\lambda}^{2 h}.\n$$\n\nAdditionally, to ensure the continuity of the deformation retraction, we must avoid moving excessively points of $\\begin{array}{ r }{\\bigcup_{x\\in S_{\\lambda,h}}[x,\\xi(x)]\\cap M_{i}}\\end{array}$ that are close to $\\mathcal{F}_{\\lambda+L h^{\\alpha}}\\cap M_{i}$ for all $i\\in\\{1,\\ldots,l\\}$ To achieve this, we introduce the condition (C1): there exists some $i\\in\\{1,\\ldots,l\\}$ such that\n\n$$\nx\\in\\bigcup_{x\\in S_{\\lambda,h}}[x,\\xi(x)]\\cap M_{i}\\tag{2}\n$$\n\nand\n\n$$\nd_{2}(\\xi(x),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}})\\geq 2 h- | |x-\\xi(x)| |_{2}.\\tag{3}\n$$\n\nWe then define the function $F_{\\lambda,h}:{\\cal K}_{\\lambda,h}\\times [0,1]{\\cal K}_{\\lambda,h}$ , which will serve as our deformation retraction, by:\n\n$$\nF_{\\lambda,h}(x)=\\left\\{\\begin{array}{l l}(1-t)x+t\\left(\\xi(x)+(2 h-d_{2}(\\xi(x),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}}))_{+}\\frac{x-\\xi(x)}{| |x-\\xi(x)| |_{2}}\\right),&\\text{under}(\\mathbf{C 1}),\\\\ x,&\\text{otherwise}.\\end{array}\\right.\n$$\n\nwhere $(.)_{+}=\\operatorname*{max}(.,0)$ denotes the positive part. For $\\begin{array}{ r }{x\\in\\bigcup_{z\\in S_{\\lambda,h}}[z,\\xi(z)]}\\end{array}$ , the function $F_{\\lambda,h}$ pushes x along the segment\n\n$$\n\\left[x,\\xi(x)+(2 h-d_{2}(\\xi(x),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}}))_{+}\\frac{x-\\xi(x)}{| |x-\\xi(x)| |_{2}}\\right]\\subset [x,\\xi(x)].\n$$\n\nFrom a geometric point of view, $F_{\\lambda,h}(x)$ pushes in the direction of the normalized vector $-{\\frac{x-\\xi(x)}{\\|x-\\xi(x)\\|_{2}}}$ , which corresponds to the opposite of the gradient of the distance function to the set $\\cup_{i=1}^{l}\\partial M_{i}\\cap(0,1)^{d}$ at the point x. This perspective links naturally to existing results that use the (generalized) gradient of the distance function to construct deformation retracts (see for instance Kim et al., 2020, Theorem 12). Moreover, as $\\cup_{i=1}^{l}\\partial M_{i}\\cap(0,1)^{d}$ is an $C^{1,1}$ hypersurface, it is worth noting that the vector $-{\\frac{x-\\xi(x)}{\\|x-\\xi(x)\\|_{2}}}$ generates the normal cone of the hypersurface at $\\xi(x)$ (see Federer, 1959, Theorem 4.8.).\n\nNote that the term $(2 h-d_{2}\\left(\\xi(x),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}}\\right))_{+}$ and the condition (C1) will be instrumental in establishing the continuity of $F_{\\lambda,h}$ , as they ensure that, for any point $x\\in\\bigcup_{x\\in S_{\\lambda . h}}[x,\\xi(x)]\\cap M_{i}$ , the closer x is to $\\mathcal{F}_{\\lambda+L h^{\\alpha}}\\cap M_{i}$ , the less it will be displaced by the deformation $F_{\\lambda,h}$\n\nFinally, we define\n\n$$\n\\mathcal{G}_{\\lambda,h}=\\mathrm{Im}\\left(x\\longmapsto F_{\\lambda,h}(x,1)\\right),\n$$\n\nwhich will serve as our deformation retract. With this, we have all the necessary formalism to state Lemma 2.\n\nLemma 2 For all $\\textstyle 0<h<{\\frac{R}{2}}$ and $\\lambda\\in\\mathbb{R},F_{\\lambda,h}$ is a deformation retraction of $\\kappa_{\\lambda,h}$ onto $\\mathcal{G}_{\\lambda,h}$ . Furthermore, we have $\\mathcal{F}_{\\lambda}^{\\tilde{h}}\\subset\\mathcal{K}_{\\lambda,h}\\subset\\mathcal{F}_{\\lambda}^{2 h}$ and $\\mathcal{F}_{\\lambda}\\subset\\mathcal{G}_{\\lambda,h}\\subset\\mathcal{F}_{\\lambda+L(1+3^{\\alpha})h^{\\alpha}}$\n\nThis section is devoted to proving the claim that the persistence diagrams we considered in this work are well-defined. Following the construction of Chazal et al. (2016), it sufices to prove that the associated persistence modules are $q\\mathrm{-t a m e}$\n\nDefinition 10 A persistence module V is said to be q-tame if for all $\\lambda<\\lambda^{\\prime}\\in\\Lambda$ , rank $\\left(v_{\\lambda}^{\\lambda^{\\prime}}\\right)$ is finite.\n\nThis section is dedicated to the proof of Lemma 1, from Section 3, which relies on the following lemma.\n\nThis section is devoted to the proof of $F_{\\lambda,h;}$ , claimed in the proof of Lemma 2.", "theorems": [{"local_id": "T1", "statement": "Lemma 1 Let $f:[0,1]^{d}\\to\\mathbb{R}$ . For all $\\lambda\\in\\mathbb{R}$ and $h > 0$ satisfying $(1)$ ,\n\n$$\n\\mathcal{F}_{\\lambda-\\sqrt{2}\\sigma N_{h}h^{\\alpha}}^{-\\sqrt{d}h}\\subset\\widehat{\\mathcal{F}}_{\\lambda}\\subset\\mathcal{F}_{\\lambda+\\sqrt{2}\\sigma N_{h}h^{\\alpha}}^{\\sqrt{d}h}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof [Proof of Lemma 1] Let $x\\in\\mathcal{F}_{\\lambda-\\sqrt{2}\\sigma N_{h}h^{\\alpha}}^{-\\sqrt{d}h}$ and H be the hypercube of $C_{h}$ containing x. We then have:\n\n$$\nH\\subset\\mathcal{F}_{\\lambda-\\sqrt{2}\\sigma N_{h}h^{\\alpha}}.\n$$"}, {"n": 2, "claim": "Hence, by Lemma 7, $\\begin{array}{ r }{\\sum_{x_{i}\\in H}X_{i}-n h^{d}\\lambda<0}\\end{array}$ , thus,\n\n$$\nH\\subset\\widehat{\\mathcal{F}}_{\\lambda}.\n$$"}, {"n": 3, "claim": "Now, let $x\\in\\left(\\mathcal{F}_{\\lambda+\\sqrt{2}\\sigma N_{h}h^{\\alpha}}^{\\sqrt{d}h}\\right)^{c}$ , and H be the hypercube of $C_{h}$ containing x. We then have:\n\n$$\nH\\subset\\mathcal{F}_{\\lambda+\\sqrt{2}\\sigma N_{h}h^{\\alpha}}^{c}.\n$$"}, {"n": 4, "claim": "Hence, by Lemma 7, $\\begin{array}{ r }{\\sum_{x_{i}\\in H}X_{i}-n h^{d}\\lambda>0}\\end{array}$ , thus,\n\n$$\nH\\subset\\widehat{\\mathcal{F}}_{\\lambda}^{c}\n$$\n\nand Lemma 1 is proved."}]}}, {"local_id": "T2", "statement": "Lemma 6 Let $f\\in S_{d}(M,L,\\alpha,R)$ . For all $s\\in\\mathbb{N}$ and $\\begin{array}{ r }{h<\\frac{R}{2}}\\end{array}$ , for all $\\lambda\\in\\mathbb{R}$ , there exists a morphism $\\phi_{\\lambda}$ such that\n\n$$\nH_{s}\\left(\\mathcal{F}_{\\lambda}\\right)\\xrightarrow{}H_{s}\\left(\\mathcal{F}_{\\lambda}^{h}\\right)\\xrightarrow [\\phi_{\\lambda}]{}H_{s}\\left(\\mathcal{F}_{\\lambda+L(1+3^{\\alpha})h^{\\alpha}}\\right)\\tag{16}\n$$\n\nis a commutative diagram (unspecified maps come from set inclusions).", "proof": {"steps": [{"n": 1, "claim": "Proof Let $\\tilde{\\phi}_{\\lambda}:H_{s}(\\mathcal{K}_{\\lambda,h})\\to H_{s}(\\mathcal{G}_{\\lambda,h})$ be the linear map induced by the deformation retraction from Lemma 2. We also denote $i_{1,\\lambda}:H_{s}\\left(\\mathcal{F}_{\\lambda}^{h}\\right)\\to H_{s}\\left(\\mathcal{K}_{\\lambda,h}\\right)$ the linear map induced by the inclusion $\\mathcal{F}_{\\lambda}^{h}\\subset\\mathcal{K}_{\\lambda,h}$ and $i_{2,\\lambda}:H_{s}(\\mathcal{G}_{\\lambda,h})\\to H_{s}(\\mathcal{F}_{\\lambda+L(1+3^{\\alpha})h^{\\alpha}})$ the linear map induced by the inclusion $\\mathcal G_{\\lambda,h}\\subset\\mathcal F_{\\lambda+L(1+3^{\\alpha})h^{\\alpha}}$ , also provided by Lemma 2. We take $\\phi_{\\lambda}=i_{2,\\lambda}\\circ\\tilde{\\phi}_{\\lambda}\\circ i_{1,\\lambda}$ . Diagram 16 then is (unspecified maps are induced by set inclusion),\n\n$$\n\\begin{array}{ccccc}\n & & H_{s}\\left(\\mathcal{F}_{\\lambda}\\right) & \\xrightarrow{\\qquad\\qquad\\qquad} & H_{s}\\left(\\mathcal{F}_{\\lambda+L(1+3^{\\alpha})h^{\\alpha}}\\right) \\\\\n & \\swarrow\\;\\;(F1) & \\Big\\downarrow & (F2)\\;\\;\\searrow\\;\\;(F3) & \\Big\\uparrow{\\scriptstyle i_{2,\\lambda}} \\\\\nH_{s}\\left(\\mathcal{F}_{\\lambda}^{h}\\right) & \\xrightarrow[i_{1,\\lambda}]{\\quad} & H_{s}\\left(\\mathcal{K}_{\\lambda,h}\\right) & \\underset{\\tilde{\\phi}_{\\lambda}}{\\longleftrightarrow} & H_{s}\\left(\\mathcal{G}_{\\lambda,h}\\right)\n\\end{array}\\tag{17}\n$$"}, {"n": 2, "claim": "Faces (F1) and $(F3)$ simply commutes by inclusion. Face $(F2)$ commutes as $\\tilde{\\phi}_{\\lambda}$ is induced by a deformation retraction. All faces of Diagram 17 commute, hence Diagram 17 (and equivalently Diagram 16) is commutative. ■"}]}}, {"local_id": "T3", "statement": "Lemma 7 Let $f:[0,1]^{d}\\ \\to\\ \\mathbb{R}$ and h satisfying (1). Let $H\\subset\\mathcal{F}_{\\lambda+\\sqrt{2}\\sigma N_{h}h^{\\alpha}}^{c}\\cap C_{h}$ and $H^{\\prime}\\subset\\mathcal{F}_{\\lambda-\\sqrt{2}\\sigma N_{h}h^{\\alpha}}\\cap C_{h}$ . We have that:\n\n$$\n\\frac{1}{n h^{d}}\\sum_{x_{i}\\in H}X_{i}>\\lambda \\text{and}\\frac{1}{n h^{d}}\\sum_{x_{i}\\in H^{\\prime}}X_{i}<\\lambda .\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof Let us consider here the case where in ${H}^{\\prime}\\subset\\mathcal{F}_{\\lambda-\\sqrt{2}\\sigma N_{h}h^{\\alpha}}$ (the proof being the same in both cases). We have:\n\n$$\n\\begin{array}{l}\\frac{1}{n h^{d}}\\sum_{x_{i}\\in H^{\\prime}}X_{i}\\\\ =\\frac{1}{n h^{d}}\\sum_{x_{i}\\in H^{\\prime}}f(x_{i})+\\sigma\\varepsilon_{i}\\\\ \\leq\\lambda-\\sqrt{2}\\sigma N_{h}h^{\\alpha}+N_{h}\\sqrt{2}\\sigma\\sqrt{\\frac{\\log(1/h^{d})}{n h^{d}}}\\\\ <\\lambda\\\\ \\end{array}\n$$\n\nBy the choice made for h."}]}}, {"local_id": "T4", "statement": "Lemma 8 Let $R / 2 > h > 0$ and $\\lambda\\in\\mathbb{R},F_{\\lambda,h}$ is continuous.", "proof": {"steps": [{"n": 1, "claim": "Proof Let $\\delta,\\delta^{\\prime}>0,x,y\\in\\mathcal{K}_{\\lambda,h}$ such that $| |x-y| |_{2}\\leq\\delta$ and $t,s\\in [0,1]$ such that $|t-s|\\leq\\delta^{\\prime}$ Let’s check the diferent cases."}, {"n": 2, "claim": "We begin with the cases where $x\\in\\overline{{M}}_{i}$ <sub>i</sub> and $y\\in\\overline{{M}}_{j},i\\neq j$ . Let $\\delta < h$ , then $| |x-\\xi(x)| |_{2}\\leq$ $\\delta,| |y-\\xi(y)| |_{2}\\leq\\delta$ , and thus $F_{\\lambda,h}(x,t),F_{\\lambda,h}(y,s)\\in [x,\\xi(x)]\\cup [y,\\xi(y)]\\subset B_{2}(x,2\\delta)$ . Then,\n\n$$\n| |F_{\\lambda,h}(x,t)-F_{\\lambda,h}(y,s)| |_{2}\\leq 4\\delta\n$$"}, {"n": 3, "claim": "And the conclusion follows in this case. From now on, we suppose that $x,y\\in M_{i}$\n\n• If x does not satisfy (2) or (3) and y does not satisfy (2) or (3), then, directly,\n\n$$\n| |F_{\\lambda,h}(x,t)-F_{\\lambda,h}(y,s)| |_{2}= | |x-y| |_{2}\\leq\\delta .\n$$"}, {"n": 4, "claim": "• If x satisfies (2) and (3), and y does not satisfy (2), then, $y\\in\\mathcal{F}_{\\lambda+L h^{\\alpha}}$ . Thus,\n\n$$\nd_{2}\\left(\\xi(x),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}}\\right)\\leq| |x-\\xi(x)| | + | |x-y| |_{2}\n$$\n\nand thus, by (3),\n\n$$\n| |x-\\xi(x)| |\\geq 2 h-d_{2}\\left(\\xi(x),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}}\\right)\\geq| |x-\\xi(x)| | -\\delta .\n$$"}, {"n": 5, "claim": "Consequently,\n\n$$\n\\left\\|\\xi(x)+(2 h-d_{2}(\\xi(x),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}}))_{+}\\frac{x-\\xi(x)}{| |x-\\xi(x)| |_{2}}-x\\right\\|_{2}\\leq\\delta\n$$"}, {"n": 6, "claim": "Then,\n\n$$\n\\begin{array}{l}\\|F_{\\lambda,h}(x,t)-F_{\\lambda,h}(y,s)\\|_{2}\\\\ =\\left\\|(1-t)x+t\\left(\\xi(x)+\\left(2 h-d_{2}(\\xi(x),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}})\\right)_{+}\\frac{x-\\xi(x)}{| |x-\\xi(x)| |_{2}}\\right)-y\\right\\|_{2}\\\\ \\leq| |x-y| |_{2}\\quad+\\left\\|\\xi(x)+\\left(2 h-d_{2}\\left(\\xi(x),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}}\\right)\\right)_{+}\\frac{x-\\xi(x)}{| |x-\\xi(x)| |_{2}}-x\\right\\|_{2}\\\\ \\leq 2\\delta .\\\\ \\end{array}\n$$"}, {"n": 7, "claim": "• If x satisfies (2) and (3), and y satisfies (2) but not (3). Then,\n\n$$\n\\begin{array}{l}\\|x-\\xi(x)\\|_{2}\\geq 2 h-d_{2}(\\xi(x),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}})\\\\ =2 h-d_{2}\\left(\\xi(y),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}}\\right)\\\\ +d_{2}\\left(\\xi(y),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}}\\right)-d_{2}\\left(\\xi(x),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}}\\right)\\\\ \\geq 2 h-d_{2}\\left(\\xi(y),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}}\\right)- | |\\xi(x)-\\xi(y)| |_{2}\\\\ \\geq| |y-\\xi(y)| |_{2}- | |\\xi(x)-\\xi(y)| |_{2}\\\\ \\geq| |x-\\xi(x)| |_{2}-2| |\\xi(x)-\\xi(y)| |_{2}- | |x-y| |_{2}.\\\\ \\end{array}\n$$"}, {"n": 8, "claim": "Hence,\n\n$$\n\\begin{array}{l}\\left\\|\\xi(x)+\\left(2 h-d_{2}\\left(\\xi(x),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}}\\right)\\right)_{+}\\frac{x-\\xi(x)}{| |x-\\xi(x)| |_{2}}-x\\right\\|_{2}\\\\ \\leq| |x-y| |_{2}+2| |\\xi(x)-\\xi(y)| |_{2}.\\\\ \\end{array}\n$$"}, {"n": 9, "claim": "And thus, we have:\n\n$$\n\\begin{array}{l}\\|F_{\\lambda,h}(x,t)-F_{\\lambda,h}(y,s)\\|_{2}\\\\ =\\left\\|(1-t)x+t\\left(\\xi(x)+\\left(2 h-d_{2}(\\xi(x),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}})\\right)_{+}\\frac{x-\\xi(x)}{| |x-\\xi(x)| |_{2}}\\right)-y\\right\\|_{2}\\\\ \\end{array}\n$$\n\n$$\n\\begin{array}{l}\\leq| |x-y| |_{2}\\quad+\\left\\|\\xi(x)+\\left(2 h-d_{2}\\left(\\xi(x),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}}\\right)\\right)_{+}\\frac{x-\\xi(x)}{| |x-\\xi(x)| |_{2}}-x\\right\\|_{2}\\\\ \\leq 2| |x-y| |_{2}+2| |\\xi(x)-\\xi(y)| |_{2}\\\\ \\leq 2\\delta+2| |\\xi(x)-\\xi(y)| |_{2}\\\\ \\end{array}\n$$\n\nand we conclude by continuity of $\\xi .$ ."}, {"n": 10, "claim": "• Finally, if x and y both satisfy (2) and (3), then,\n\n$$\n\\begin{array}{l}\\|F_{\\lambda,h}(x,t)-F_{\\lambda,h}(y,s)\\|_{2}\\\\ =\\left\\|(1-t)x+t\\left(\\xi(x)+\\left(2 h-d_{2}\\left(\\xi(x),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}}\\right)\\right)_{+}\\frac{x-\\xi(x)}{\\|x-\\xi(x)\\|_{2}}\\right)\\right.\\\\ \\left.-(1-s)y-s\\left(\\xi(y)+\\left(2 h-d_{2}\\left(\\xi(y),M_{i}\\cap\\mathcal{F}_{\\lambda+L h^{\\alpha}}\\right)\\right)_{+}\\frac{y-\\xi(y)}{\\|y-\\xi(y)\\|_{2}}\\right)\\right\\|_{2}\\\\ \\end{array}\n$$\n\nand the conclusion follows again, in this case by continuity of $\\xi .$\n\nAll possible cases have been checked, the proof is complete."}]}}], "target_local_id": null, "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2026-24-1484", "context": "We denote column vectors in $\\mathbb{R}^{d}$ by lowercase bold letters, that is, $\\pmb{x}: =(x_{1},\\ldots,x_{d})^{\\top}$ and write $\\|\\pmb{x}\\|_{2}: =(\\pmb{x}^{\\top}\\pmb{x})^{1/2}$ for the Euclidean norm. The expectation and covariance of random vectors are respectively denoted by $\\mathbb{E}[\\cdot]$ and $\\operatorname{Cov}(\\cdot)$ . For two positive number sequences $\\left(a_{n}\\right)$ and $\\left(b_{n}\\right)$ , we say $a_{n}=O(b_{n})$ (resp. $a_{n}\\asymp b_{n})$ if there exists $c > 0$ such that $a_{n}/b_{n}\\leq c$ (resp. $1/c\\le a_{n}/b_{n}\\le c)$ for all large n, and say $a_{n}=o(b_{n})$ if $a_{n}/b_{n}\\to 0$ as $n \\to \\infty$ . Let $(x_{n})$ and $(y_{n})$ be two sequences of random variables. Write $x_{n}=O_{\\mathbb{P}}(y_{n})$ if for $\\forall \\epsilon > 0$ , there exists $c > 0$ such that $\\mathbb{P}(|x_{n}/y_{n}|\\le c)>1-\\epsilon$ for all large n, and say $x_{n}=o_{\\mathbb{P}}(y_{n}){\\mathrm{~i f~}}x_{n}/y_{n}\\to 0$ in probability as $n \\to \\infty$\n\nWe denote matrices by uppercase letters. The $d \\times d$ identity matrix is symbolized by $I_{d}.$ . Given matrices A and B of compatible dimension, their matrix product is denoted by juxtaposition. Write $A^{\\top}$ for the transpose of A and define $\\mathbb{A}: =A^{\\top}A$ . When A and B are of the same dimension, the Hadamard product $A \\odot B$ is given by element-wise multiplication $(A\\odot B)_{i j}=A_{i j}B_{i j}$ . For any $A\\in\\mathbb{R}^{d\\times d}$ , let Dia $\\mathrm{g}(A): =I_{d}\\odot A$ denote the diagonal matrix with the same main diagonal as A. Given $p\\in(0,1)$ , define the matrices\n\n$$\n\\overline{{A}}:=A-\\operatorname{Diag}(A),\n$$\n\n$$\nA_{p}:=p A+(1-p)\\mathrm{Diag}(A).\n$$\n\nIn particular, $A_{p}=p\\overline{{A}}+\\mathrm{Diag}(A)$ , so $A_{p}$ results from re-scaling the of-diagonal entries of A by p. For a matrix A, the operator norm induced by the Euclidean norm $\\|\\cdot\\|_{2}$ is the spectral norm and will always be written without sub-script, that is, $\\|A\\|: =\\|A\\|_{\\mathrm{op}}$\n\nWe consider the least-squares criterion $\\begin{array}{ r }{\\frac 12\\|\\pmb{y}-\\pmb{X}\\pmb{\\beta}\\|_{2}^{2}}\\end{array}$ for the estimation of $\\beta^{*}$ . For the minimization, we adopt a constant learning-rate gradient descent algorithm with random dropouts in each iteration. Following the seminal work on dropout by Srivastava et al. (2014), we call a $d \\times d$ random diagonal matrix D a p-dropout matrix if its diagonal entries satisfy $D_{i i}\\stackrel{\\mathrm{i . i . d .}}{\\sim}$ Bernoulli(p), with some retaining probability $p\\in(0,1)$ . On average, D has pd diagonal entries equal to 1 and $(1-p)d$ diagonal entries equal to 0. For simplicity, the dependence of D on p will only be stated if unclear from the context. For a sequence of independent and identically distributed (i.i.d.) dropout matrices $D_{k},k=1,2,\\ldots,$ and some constant learning rate $\\alpha > 0$ , the k-th step gradient descent iterate with dropout takes the form\n\n$$\n\\begin{array}{l}\\tilde{\\boldsymbol{\\beta}}_{k}(\\alpha)=\\tilde{\\boldsymbol{\\beta}}_{k-1}(\\alpha)-\\alpha\\nabla_{\\tilde{\\boldsymbol{\\beta}}_{k-1}}\\frac{1}{2}\\big\\|\\boldsymbol{y}-X D_{k}\\tilde{\\boldsymbol{\\beta}}_{k-1}(\\alpha)\\big\\|_{2}^{2}\\\\ =\\tilde{\\boldsymbol{\\beta}}_{k-1}(\\alpha)+\\alpha D_{k}X^{\\top}\\big(\\boldsymbol{y}-X D_{k}\\tilde{\\boldsymbol{\\beta}}_{k-1}(\\alpha)\\big),\\end{array}\\tag{2}\n$$\n\nwith $\\tilde{\\boldsymbol{\\beta}}_{0}$ a given initial vector.\n\nIn the previous section, we considered a fixed design matrix and (full) gradient descent with dropout. Computing the gradient over the entire dataset can be computationally expensive, especially with large datasets. We now investigate stochastic gradient descent with dropout regularization.\n\nConsider i.i.d. covariate vectors $\\pmb{x}_{k}\\in\\mathbb{R}^{d},k=1,2,....$ from some distribution Π, and the realizations $y_{k}|\\pmb{x}_{k}$ from a linear regression model\n\n$$\ny_{k}=\\boldsymbol{x}_{k}^{\\top}\\boldsymbol{\\beta}^{*}+\\epsilon_{k},\\tag{31}\n$$\n\nwith unknown regression vector $\\beta^{*}\\in\\mathbb{R}^{d}$ . We assume that the model is in reduced form, which here means that min $_i(\\mathbb{E}[\\pmb{x}_{1}\\pmb{x}_{1}^{\\top}])_{i i}>0$ . In addition, we assume that the i.i.d. random noises $\\epsilon_{k}$ satisfy $\\mathbb{E}[\\epsilon_{k}]=0$ and $\\mathrm{Var}(\\epsilon_{k})=1$ . In this paper, we focus on the classical case where the SGD computes the gradient based on an individual observation $(y_{k},\\pmb{x}_{k})$ and a dropout matrix $D_{k}$ . For constant learning rate α and initialization $\\breve{\\beta}_{0}$ , the k-th step SGD iterate with Bernoulli dropout is\n\n$$\n\\begin{array}{l}\\breve{\\boldsymbol{\\beta}}_{k}(\\alpha)=\\breve{\\boldsymbol{\\beta}}_{k-1}(\\alpha)-\\alpha\\nabla_{\\breve{\\boldsymbol{\\beta}}_{k-1}}\\frac{1}{2}\\big(y_{k}-\\boldsymbol{x}_{k}^{\\top}D_{k}\\breve{\\boldsymbol{\\beta}}_{k-1}(\\alpha)\\big)^{2}\\\\ =\\breve{\\boldsymbol{\\beta}}_{k-1}(\\alpha)+\\alpha D_{k}\\boldsymbol{x}_{k}\\left(y_{k}-\\boldsymbol{x}_{k}^{\\top}D_{k}\\breve{\\boldsymbol{\\beta}}_{k-1}(\\alpha)\\right).\\tag{32}\\\\ \\end{array}\n$$\n\nThis is a sequential estimation, or online learning scheme, as computing $\\breve{\\beta}_{k}(\\alpha)$ from $\\breve{\\beta}_{k-1}(\\alpha)$ only requires the k-th sample $(y_{k},\\pmb{x}_{k})$ and the dropout matrix $D_{k}$ . To study the contraction property of the SGD dropout iterates $\\breve{\\beta}_{k}(\\alpha)$ , we express the recursion (32) by an iterated random function $\\breve{f}:\\mathbb{R}^{d\\times\\hat{d}}\\times\\mathbb{R}\\times\\mathbb{R}^{d}\\times\\mathbb{R}^{d}\\mapsto\\mathbb{R}^{d}$ with\n\n$$\nf_{D,(y,\\boldsymbol{x})}(\\boldsymbol{u})=\\boldsymbol{u}+\\alpha D\\boldsymbol{x}(y-\\boldsymbol{x}D\\boldsymbol{u}),\n$$\n\nthat is,\n\n$$\n\\begin{array}{l}\\breve{\\boldsymbol{\\beta}}_{k}(\\alpha)=\\breve{\\boldsymbol{\\beta}}_{k-1}(\\alpha)+\\alpha D_{k}\\boldsymbol{x}_{k}\\big(y_{k}-\\boldsymbol{x}_{k}^{\\top}D_{k}\\breve{\\boldsymbol{\\beta}}_{k-1}(\\alpha)\\big)\\\\ =:\\check{f}_{D_{k},(y_{k},\\boldsymbol{x}_{k})}(\\check{\\boldsymbol{\\beta}}_{k-1}(\\alpha)).\\tag{33}\\\\ \\end{array}\n$$\n\nWe shall show that this iterated random function $\\breve{f}$ is geometrically contracting under suitable conditions, and therefore, there exists a unique stationary distribution $\\breve{\\pi}_{\\alpha}$ such that $\\breve{\\beta}_{k}(\\alpha)\\Rightarrow\\breve{\\pi}_{\\alpha}$ , where ⇒ denotes the convergence in distribution.\n\nFrom now on, let $(y,x)$ be a sample with the same distribution as $(y_{k},\\pmb{x}_{k})$ . By marginalizing over all randomness, we can view the SGD dropout in (32) as a minimizer of the $\\ell^{2}.$ -regularized least-squares loss\n\n$$\n\\breve{\\boldsymbol{\\beta}}:=\\arg\\min_{\\boldsymbol{\\beta}\\in\\mathbb{R}^{d}}\\mathbb{E}_{(y,\\boldsymbol{x})}\\mathbb{E}_{D}\\left[\\frac{1}{2}\\left(y-\\boldsymbol{x}^{\\top}D\\boldsymbol{\\beta}\\right)^{2}\\right].\\tag{34}\n$$\n\nHere, the expectation is take over both the random sample $(y,x)$ and the dropout matrix $D$ . Throughout the rest of the paper, we shall write $\\mathbb{E}[\\cdot]=\\mathbb{E}_{(y,\\pmb{x})}\\mathbb{E}_{D}[\\cdot]$ when no confusion should be caused.\n\nDenote the $d \\times d$ Gram matrix by $\\mathbb{X}_{k}=\\pmb{x}_{k}\\pmb{x}_{k}^{\\top}$ , and define\n\n$$\n\\overline{{\\mathbb{X}}}_{k}=\\mathbb{X}_{k}-\\operatorname{Diag}(\\mathbb{X}_{k}),\\quad\\mathbb{X}_{k,p}=p\\mathbb{X}_{k}+(1-p)\\operatorname{Diag}(\\mathbb{X}_{k}).\\tag{35}\n$$\n\nBy Lemma 28 in the Appendix, we have a closed form solution for $\\breve{\\beta}$ as follows\n\n$$\n\\breve{\\boldsymbol{\\beta}}=p\\Big(p^{2}\\mathbb{E}[\\boldsymbol{x}_{1}\\boldsymbol{x}_{1}^{\\top}]+p(1-p)\\mathrm{Diag}(\\mathbb{E}[\\boldsymbol{x}_{1}\\boldsymbol{x}_{1}^{\\top}])\\Big)^{-1}\\mathbb{E}[y_{1}\\boldsymbol{x}_{1}]=(\\mathbb{E}[\\mathbb{X}_{1,p}])^{-1}\\mathbb{E}[y_{1}\\boldsymbol{x}_{1}],\n$$\n\nand thus, we obtain the relationship $\\mathbb{E}[\\mathbb{X}_{1,p}]\\breve{\\beta}=\\mathbb{E}[y_{1}\\pmb{x}_{1}]$ . To study the SGD with dropout, we now focus on the diference process $\\{\\breve{\\beta}_{k}(\\alpha)-\\breve{\\beta}\\}_{k\\in\\mathbb{N}}$ . As in the case of gradient descent, this process can be written in autoregressive form,\n\n$$\n\\breve{\\boldsymbol{\\beta}}_{k}(\\alpha)-\\breve{\\boldsymbol{\\beta}}=\\underbrace{\\left(I_{d}-\\alpha D_{k}\\mathbb{X}_{k}D_{k}\\right)}_{=:\\breve{A}_{k}(\\alpha)}\\left(\\breve{\\boldsymbol{\\beta}}_{k-1}(\\alpha)-\\breve{\\boldsymbol{\\beta}}\\right)+\\underbrace{\\alpha D_{k}\\boldsymbol{x}_{k}\\left(y_{k}-\\boldsymbol{x}_{k}^{\\top}D_{k}\\breve{\\boldsymbol{\\beta}}\\right)}_{=:\\breve{\\boldsymbol{b}}_{k}(\\alpha)}.\\tag{36}\n$$\n\nEstablishing the geometric-moment contraction (GMC) property to the stochastic gradient descent iterates with dropout is non-trivial as the randomness of $\\breve{\\beta}_{k}(\\alpha)$ not only comes from the dropout matrix $D_{k}$ , but also the random sample $(y_{k},\\pmb{x}_{k})$ . Recall that ${\\mathbb X}_{k,p}$ is defined in (35) and by Lemma 21(ii), $\\mathbb{E}[D\\mathbb{X}_{k}D]=p\\mathbb{E}[\\mathbb{X}_{k,p}]$\n\nLemma 10 (Learning-rate range in SGD dropout) Assume that for some $q \\geq 2$ , the q-th moment $\\mu_{q}(\\pmb{v})=(\\mathbb{E}\\|D_{k}\\mathbb{X}_{k}D_{k}\\pmb{v}\\|_{2}^{q})^{1/q}<\\infty$ for all unit vectors $\\pmb{v}\\in\\mathbb{R}^{d}$ . If the learning rate $\\alpha > 0$ satisfies\n\n$$\n\\frac{\\alpha(q-1)}{2}\\sup_{\\boldsymbol{v}\\in\\mathbb{R}^{d},\\|\\boldsymbol{v}\\|_{2}=1}\\frac{(1+\\alpha\\mu_{q}(\\boldsymbol{v}))^{q-2}\\mu_{q}(\\boldsymbol{v})^{2}}{p\\boldsymbol{v}^{\\top}\\mathbb{E}[\\mathbb{X}_{k,p}]\\boldsymbol{v}}<1,\\tag{37}\n$$\n\nthen, for a dropout matrix $D$\n\n$$\n\\check{r}_{\\alpha,q}^{q}:=\\sup_{\\boldsymbol{v}\\in\\mathbb{R}^{d},\\|\\boldsymbol{v}\\|_{2}=1}\\mathbb{E}\\left\\|\\left(I_{d}-\\alpha D\\mathbb{X}_{k}D\\right)\\boldsymbol{v}\\right\\|_{2}^{q}<1.\\tag{38}\n$$\n\nFor the geometric-moment contraction for the SGD dropout sequence, we impose the following moment conditions.\n\nAssumption 1 (Finite moment) Assume that for some $q \\geq 2$ , the random noises ϵ and the random sample x in model (31) have finite 2q-th moment $\\mathbb{E}[|\\epsilon|^{2 q}]+\\|\\pmb{x}\\|_{2}^{2 q}]<\\infty$\n\nLemma 30 in the Appendix shows that this assumption ensures the finite q-th moment of the stochastic gradient in (32) evaluated at the true parameter $\\beta^{*}$ and the $\\ell^{2}.$ -minimizer $\\breve{\\beta}$ in model (31), that is,\n\n$$\n\\left(\\mathbb{E}\\Big\\|\\nabla_{\\boldsymbol{\\beta}^{*}}\\frac{1}{2}\\big(y-\\boldsymbol{x}^{\\top}D\\boldsymbol{\\beta}^{*}\\big)^{2}\\Big\\|_{2}^{q}\\right)^{1/q}=\\left(\\mathbb{E}\\Big\\|D\\boldsymbol{x}\\big(y-\\boldsymbol{x}^{\\top}D\\boldsymbol{\\beta}^{*}\\big)\\Big\\|_{2}^{q}\\right)^{1/q}<\\infty,\n$$\n\nand $(\\mathbb{E}\\|\\nabla_{\\check{\\beta}}\\frac{1}{2}\\big(y-\\pmb{x}^{\\top}D\\check{\\beta}\\big)^{2}\\|_{2}^{q})^{1/q}<\\infty$ . Assumption 1 moreover guarantees $(\\mathbb{E}\\Vert\\nabla_{\\beta}\\frac{1}{2}(y- $ $\\pmb{x}^{\\top}D\\beta)^{2}\\|_{2}^{q})^{1/q}<\\infty$ for all $\\beta\\in\\mathbb{R}^{d}$\n\nTheorem 12 (Geometric-moment contraction of SGD dropout) Let $q \\geq 2$ . Suppose that Assumption 1 holds and the learning rate α satisfies (37). For two dropout sequences $\\breve{\\beta}_{k}(\\alpha)$ and $\\breve{\\beta}_{k}^{\\prime}(\\alpha),k=0,1,...,$ that are generated by the recursion (32) with the same sequence of dropout matrices $\\{D_{k}\\}_{k\\in\\mathbb{N}}$ but possibly diferent initializations $\\breve{\\beta}_{0},\\breve{\\beta}_{0}^{\\prime}$ , we have\n\n$$\n\\left(\\mathbb{E}\\left\\|\\check{\\boldsymbol{\\beta}}_{k}(\\alpha)-\\check{\\boldsymbol{\\beta}}_{k}^{\\prime}(\\alpha)\\right\\|_{2}^{q}\\right)^{1/q}\\leq\\check{r}_{\\alpha,q}^{k}\\|\\check{\\boldsymbol{\\beta}}_{0}-\\check{\\boldsymbol{\\beta}}_{0}^{\\prime}\\|_{2},\\quad \\text{for all} k=1,2,\\dots,\\tag{41}\n$$\n\nwith\n\n$$\n\\check{r}_{\\alpha,q}=\\Big(\\sup_{\\boldsymbol{v}\\in\\mathbb{R}^{d}:\\|\\boldsymbol{v}\\|_{2}=1}\\mathbb{E}\\big\\|\\breve{A}_{1}(\\alpha)\\boldsymbol{v}\\big\\|_{2}^{q}\\Big)^{1/q}<1.\n$$\n\nMoreover, for any initial vector $\\breve{\\beta}_{0}\\in\\mathbb{R}^{d}$ , there exists a unique stationary distribution $\\breve{\\pi}_{\\alpha}$ which does not depend on $\\breve{\\beta}_{0}$ , such that $\\breve{\\beta}_{k}(\\alpha)\\Rightarrow\\breve{\\pi}_{\\alpha}$ as $k \\to \\infty$\n\nBy Theorem 12, initializing $\\breve{\\beta}_{0}^{\\circ}\\sim\\breve{\\pi}_{o}$ leads to the stationary SGD dropout sequence $\\{\\breve{\\beta}_{k}^{\\circ}(\\alpha)\\}_{k\\in\\mathbb{N}}$ by following the recursion\n\n$$\n\\breve{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\breve{\\boldsymbol{\\beta}}=\\breve{A}_{k}(\\alpha)\\left(\\breve{\\boldsymbol{\\beta}}_{k-1}^{\\circ}(\\alpha)-\\breve{\\boldsymbol{\\beta}}\\right)+\\breve{\\boldsymbol{b}}_{k}(\\alpha),\\quad k=1,2,\\dots,\\tag{42}\n$$\n\nwhere the ℓ<sup>2</sup>-regularized minimizer $\\breve{\\beta}$ is defined in (34), and the random coeficients $\\breve{A}_{k}(\\alpha)=$ $I_{d}{-}\\alpha D_{k}\\mathbb{X}_{k}D_{k}$ and $\\breve{\\pmb{b}}_{k}(\\alpha)=\\alpha D_{k}\\pmb{x}_{k}(y_{k}-\\pmb{x}_{k}^{\\top}D_{k}\\breve{\\beta})$ are defined in (36). Furthermore, recall the iterated random function $\\breve{f}_{D,(y,{\\pmb x})}(\\beta)$ defined in (33). As a direct consequence of Theorem 12, we have\n\n$$\n\\breve{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)=\\breve{f}_{D_{k},(y_{k},\\boldsymbol{x}_{k})}\\left(\\breve{\\boldsymbol{\\beta}}_{k-1}^{\\circ}(\\alpha)\\right),\\tag{43}\n$$\n\nwhich holds for all $k\\in\\mathbb{Z}$ . To see the case with $k \\leq 0$ , we only need to notice that, for any $\\beta\\in\\mathbb{R}^{d}$ , we have the limit\n\n$$\n\\breve{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha):=\\lim_{m\\rightarrow\\infty}\\check{f}_{\\boldsymbol{\\xi}_{k}}\\circ\\dots\\check{f}_{\\boldsymbol{\\xi}_{k-m}}(\\boldsymbol{\\beta})=:\\check{h}_{\\alpha}\\left(\\boldsymbol{\\xi}_{k},\\boldsymbol{\\xi}_{k-1},\\dots\\right),\\tag{44}\n$$\n\nwhere $\\breve{h}_{\\alpha}$ is a measurable function that depends on $\\alpha,$ and we use ${\\boldsymbol{\\xi}}_{k}$ to denote all the new-coming random parts in the k-th iteration, that is,\n\n$$\n\\boldsymbol{\\xi}_{k}=(D_{k},(y_{k},\\boldsymbol{x}_{k})),\\quad k\\in\\mathbb{Z}.\\tag{45}\n$$\n\nFor $k\\le 0,\\xi_{k}$ can be viewed as an i.i.d. copy of $\\xi_{j}$ for some $j \\geq 1$ . The limit $\\breve{h}_{\\alpha}(\\xi_{k},\\xi_{k-1},\\dots)$ exists almost surely and does not depend on $\\beta$ . Therefore, the SGD dropout iteration $\\breve{\\beta}_{k}^{\\circ}(\\alpha)=\\breve{f}_{D_{k},(y_{k},{\\pmb x}_{k})}(\\breve{\\beta}_{k-1}^{\\circ}(\\alpha))$ in (43) holds for all $k\\in\\mathbb{Z}$\n\nIn this section, we provide the asymptotics for the k-th iterate of SGD dropout and the Ruppert-Polyak averaged version.\n\nLemma 14 (Moment convergence of iterative SGD dropout) Let $q \\geq 2$ and suppose that Assumption 1 holds. For the stationary SGD dropout sequence $\\{\\breve{\\beta}_{k}^{\\circ}(\\alpha)\\}_{k\\in\\mathbb{N}}$ defined in $(42)$ with learning rate α satisfying (37), we have\n\n$$\n\\max_{k}\\left(\\mathbb{E}\\|\\breve{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\breve{\\boldsymbol{\\beta}}\\|_{2}^{q}\\right)^{1/q}=O(\\sqrt{\\alpha}).\\tag{47}\n$$\n\nBesides the stochastic order of the last iterate of SGD dropout $\\breve{\\beta}_{k}(\\alpha)$ , we are also interested in the limiting distribution of the Ruppert-Polyak averaged SGD dropout, which can efectively reduce the variance and keep the online computing scheme. In particular, we define\n\n$$\n\\bar{\\boldsymbol{\\beta}}_{k}^{\\mathrm{sgd}}(\\alpha)=\\frac{1}{k}\\sum_{i=1}^{k}\\breve{\\boldsymbol{\\beta}}_{i}(\\alpha).\\tag{48}\n$$\n\nLemma 21 (Clara et al., 2024) For any matrices A and B in $\\mathbb{R}^{d\\times d},p\\in(0,1)$ , and a diagonal matrix $D\\in\\mathbb{R}^{d\\times d}$ , the following results hold:\n\n(i) AD = AD, DA = DA, and ${\\overline{{A}}}_{p}=p{\\overline{{A}}}={\\overline{{A_{p}}}},$\n\nIf in addition, the diagonal matrix D is random and independent of A and $B,$ with the diagonal entries satisfying $D_{i i}\\stackrel{\\mathrm{i . i . d .}}{\\sim}$ Bernoulli(p), $1\\leq i\\leq d,$ , then,\n\n(ii) $\\mathbb{E}[D A D]=p A_{p},$ , where $A_{p}=p A+(1-p)\\mathrm{Diag}(A).$ ;  \n(iii) $\\begin{array}{ r }{\\mathbb{E}[D A D B D]=p A_{p}B_{p}+p^{2}(1-p)\\mathrm{Diag}(\\overline{A}B),\\ w h e r e\\ \\overline{A}=A-\\mathrm{Diag}(A);}\\end{array}$  \n(iv) E[DADBDCD] = pA<sub>p</sub>B<sub>p</sub>C<sub>p</sub>+p<sup>2</sup>(1−p)-Diag(AB<sub>p</sub>C)+A<sub>p</sub>Diag(BC)+Diag(AB)C<sub>p</sub>+ $(1-p)A\\odot{\\overline{{B}}}^{\\top}\\odot C]$ , where ⊙ denotes the Hadamard product.\n\nSpecifically, for all $q \\geq 2$ , Theorem 1 in Wu (2005) pointed out a useful inequality for the functional dependence measure as follows,\n\n$$\n\\sum_{k=0}^{\\infty}\\left(\\mathbb{E}\\|\\mathcal{P}_{0}[\\tilde{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)]\\|_{2}^{q}\\right)^{1/q}\\leq\\sum_{k=0}^{\\infty}\\theta_{k,q}(\\alpha)=\\Theta_{0,q}(\\alpha).\\tag{119}\n$$\n\nHere, we introduce Theorem 3 in Wu (2011) and Theorem 2 in Karmakar and Wu (2020), which are the fundamental tools for the proofs of Theorems 7 and 9, respectively.\n\nLemma 25 (Asymptotic normality (Wu, 2011)) Consider a sequence of stationary mean-zero random variables $x_{k}=g(\\epsilon_{k},\\epsilon_{k-1},\\ldots)\\in\\mathbb{R}$, for $k=1,\\ldots,n$ , where $\\epsilon_{k}\\mathit{^{\\prime}s}$ are $i . i . d .$ random variables, and $g(\\cdot)$ is a measurable function such that each $x_{k}$ is a proper random variable. Recall the projection operator $\\mathcal{P}_{0}[\\cdot]$ defined in (115). Let $\\theta_{q}(i)=(\\mathbb{E}\\lVert\\mathcal{P}_{0}[x_{i}]\\rVert_{2}^{q})^{1/q}$, $q > 1$ . Assume $\\mathbb{E}[x_{i}]=0$ and\n\n$$\n\\Theta_{q}:=\\sum_{i=0}^{\\infty}\\theta_{q}(i)<\\infty .\\tag{121}\n$$\n\nLet $\\textstyle S_{n}=\\sum_{k=1}^{n}x_{k}$ and define the process $S_{t}=S_{\\lfloor t\\rfloor}+(t-\\lfloor t\\rfloor)x_{\\lfloor t\\rfloor+1}$, for $t\\ge 0$ and the floor function $\\lfloor t\\rfloor=\\operatorname*{max}\\{k\\in\\mathbb{Z}:k\\leq t\\}$ . Then,\n\n(i) we have the moment inequality\n\n$$\n(\\mathbb{E}\\|S_{n}\\|_{2}^{q})^{1/q}\\leq\\left\\{\\begin{array}{l l}(q-1)^{1/2}n^{1/2}\\Theta_{q},&q>2,\\\\ (q-1)^{-1}n^{1/q}\\Theta_{q},&1<q\\leq 2.\\end{array}\\right.\\tag{122}\n$$\n\n(ii) If moreover (121) holds with $q = 2$ , the invariance principle\n\n$$\n\\left\\{\\frac{1}{\\sqrt{n}}S_{n u},0\\leq u\\leq 1\\right\\}\\Rightarrow\\{\\sigma\\mathbb{B}(u),0\\leq u\\leq 1\\},\\tag{123}\n$$\n\nholds with long-run variance $\\begin{array}{ r }{\\sigma^{2}=\\mathbb{E}\\|\\sum_{i=0}^{\\infty}\\mathcal{P}_{0}[x_{i}]\\|_{2}^{2}}\\end{array}$\n\nLemma 26 (Gaussian approximation (Karmakar and Wu, 2020)) Suppose that we have a sequence of nonstationary mean-zero random vectors $\\pmb{x}_{k}=g_{k}(\\epsilon_{k},\\epsilon_{k-1},...)\\in\\mathbb{R}^{d}$ , for $k=1,\\ldots,n$ , where the $\\epsilon_{k}$ ’s are i.i.d. random variables, and $g_{k}(\\cdot)$ is a measurable function such that each $\\scriptstyle{\\mathbf{{\\mathit{x}}}}_{k}$ is a proper random vector. Let $\\begin{array}{ r }{S_{j}=\\sum_{k=1}^{j}\\pmb{x}_{k}}\\end{array}$ . Assume the following conditions hold for some $q > 2$\n\n(i) The series $(\\|\\pmb{x}_{k}\\|_{2}^{q})_{k\\ge 1}$ is uniformly integrable: $\\begin{array}{ r }{\\operatorname*{sup}_{k\\geq 1}\\mathbb{E}\\big [\\|\\pmb{x}_{k}\\|_{2}^{q}\\mathbf{1}_{\\|\\pmb{x}_{k}\\|_{2}\\geq u}\\big]0}\\end{array}$ as $u $ $\\infty_{i}$  \n(ii) The eigenvalues of covariance matrices of increment processes are lower-bounded, that is, there exists $\\lambda_{*}>0$ and $l_{*}\\in\\mathbb{N}$ , such that for all $t\\geq 1,l\\geq l_{*}$ ，\n\n$$\n\\lambda_{\\mathrm{min}}\\big(\\mathrm{Cov}(S_{t+l}-S_{t})\\big)\\geq\\lambda_{*}l;\n$$\n\n(iii) There exist constants $\\chi>\\chi_{0}$ and $\\kappa > 0$ , where\n\n$$\n\\chi_{0}=\\frac{q^{2}-4+(q-2)\\sqrt{q^{2}+20q+4}}{8q},\n$$\n\nsuch that the tail cumulative dependence measure\n\n$$\n\\Theta_{m,q}(\\alpha)=\\sum_{k=m}^{\\infty}\\theta_{k,q}(\\alpha)=O\\big\\{m^{-\\chi}\\big(\\log(m)\\big)^{-\\kappa}\\big\\}.\\tag{124}\n$$\n\nThen, for all $q > 2$ , there exists a probability space $(\\Omega^{\\star},{\\mathcal{A}}^{\\star},\\mathbb{P}^{\\star})$ on which we can define random vectors $\\boldsymbol{x}_{k}^{\\star},$ , with the partial sum process $\\begin{array}{ r }{S_{i}^{\\star}=\\sum_{k=1}^{i}{\\pmb x}_{k}^{\\star}}\\end{array}$ and a Gaussian process $\\begin{array}{ r }{G_{i}^{\\star}=\\sum_{k=1}^{i}z_{k}^{\\star}}\\end{array}$ . Here $z_{k}^{\\star}$ is a mean-zero independent Gaussian vector, such that $(S_{i}^{\\star})_{1\\leq i\\leq n}\\stackrel{\\mathcal{D}}{=}$ $(S_{i})_{1\\leq i\\leq n}$ and\n\n$$\n\\max_{i\\leq n}|S_{i}^{\\star}-G_{i}^{\\star}| =o_{\\mathbb{P}}(n^{1/q})\\quad i n(\\Omega^{\\star},\\mathcal{A}^{\\star},\\mathbb{P}^{\\star}).\n$$\n\nWe notice that condition (ii) in Lemma 26 on the non-singularity is required when the sequence $\\{\\pmb{x}_{k}\\}_{k\\in\\mathbb{N}}$ is non-stationary. However, if the function $g_{k}(\\cdot)\\equiv g(\\cdot)$ , that is, the sequence $\\{\\pmb{x}_{k}\\}_{k\\in\\mathbb{N}}$ is stationary, then the covariance matrix of the increments is allowed to be singular. To see this, consider a stationary partial sum $\\boldsymbol{S_{l}}=(S_{l,1},\\ldots,S_{l,d})^{\\top}$ with a singular covariance matrix $\\Sigma\\in\\mathbb{R}^{d\\times d}$ and assume $\\operatorname{rank}(\\Sigma)=d-1$ . Then, there exists a unit vector $\\pmb{v}\\in\\mathbb{R}^{d}$ such that $\\Sigma{\\boldsymbol{v}}=0$ , which indicates that $S_{l,1}$ can be written into a linear combination of $S_{l,2},\\ldots,S_{l,d};$ and the covariance matrix of this linear combination is non-singular. Hence, condition (ii) in Lemma 26 is not required for stationary processes.\n\nThat is, for all $\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ},\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ^{\\prime}}\\in\\mathbb{R}^{d}$ , such that $\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ}\\neq\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ^{\\prime}}$ , we have\n\n$$\n\\begin{array}{l}\\big(\\mathbb{E}\\|f_{D_{k}}\\circ\\cdots\\circ f_{D_{1}}(\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ})-f_{D_{k}}\\circ\\cdots\\circ f_{D_{1}}(\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ^{\\prime}})\\|_{2}^{q}\\big)^{1/q}\\\\ =\\big(\\mathbb{E}\\|h_{\\alpha}(D_{k},\\ldots,D_{1},D_{0},D_{-1},\\ldots)-h_{\\alpha}(D_{k},\\ldots,D_{1},D_{0}^{\\prime},D_{-1}^{\\prime},\\ldots)\\|_{2}^{q}\\big)^{1/q}\\\\ =\\big(\\mathbb{E}\\|h_{\\alpha}(\\mathcal{F}_{k})-h_{\\alpha}(\\mathcal{F}_{k,\\{0,-1,\\ldots\\}})\\|_{2}^{q}\\big)^{1/q}\\\\ \\leq c_{q}r_{\\alpha,q}^{k},\\end{array}\\tag{126}\n$$\n\nwhere we recall the filtration $\\mathcal{F}_{i,\\{j\\}}=\\sigma(D_{i},\\ldots,D_{j+1},D_{j}^{\\prime},D_{j-1},\\ldots)$ , and $c_{q}>0$ is some constant independent of k. Moreover, since $h_{\\alpha}(\\mathcal{F}_{k})$ is stationary over k and $D_{i}$ and $D_{j}^{\\prime}$ are i.i.d. random matrices, for all $i,j\\in\\mathbb{Z},$ , it follows that\n\n$$\n\\begin{array}{l}\\mathbb{E}\\|h_{\\alpha}(\\mathcal{F}_{k,\\{0\\}})-h_{\\alpha}(\\mathcal{F}_{k,\\{0,-1,\\dots\\}})\\|_{2}^{q}\\\\ =\\mathbb{E}\\|h_{\\alpha}(\\mathcal{F}_{k})-h_{\\alpha}(\\mathcal{F}_{k,\\{-1,-2\\dots\\}})\\|_{2}^{q}\\\\ =\\mathbb{E}\\|h_{\\alpha}(\\mathcal{F}_{k+1})-h_{\\alpha}(\\mathcal{F}_{k+1,\\{0,-1,\\dots\\}})\\|_{2}^{q}\\leq c_{q}^{\\prime}r_{\\alpha,q}^{k},\\tag{127}\\\\ \\end{array}\n$$\n\nwhere the constant $c_{q}^{\\prime} > 0$ is also independent of k.\n\nNext, we bound the diference between $\\tilde{S}_{t}^{\\circ}(\\alpha)$ and $\\tilde{S}_{t}^{\\tilde{\\beta}_{0}}(\\alpha)$ for any arbitrarily fixed $\\tilde{\\beta}_{0}\\in$ $\\mathbb{R}^{d}$ in the q-th moment, for all $q \\geq 2$ . For the constant learning rate $\\alpha > 0$ satisfying $\\alpha\\|\\mathbb{X}\\|<2$ , applying Theorem 3 yields\n\n$$\n\\begin{array}{l}\\left(\\mathbb{E}\\big\\|\\tilde{S}_{t}^{\\circ}(\\alpha)-\\tilde{S}_{t}^{\\tilde{\\boldsymbol{\\beta}}_{0}}(\\alpha)\\big\\|_{2}^{q}\\right)^{1/q}\\\\ =\\left(\\mathbb{E}\\big\\|\\left[f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ}(\\alpha)\\right)+f_{D_{2}}\\circ f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ}(\\alpha)\\right)+\\dots+f_{D_{t}}\\circ\\dots\\circ f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ}(\\alpha)\\right)\\right]\\right.\\\\ \\left.-\\left[f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}(\\alpha)\\right)+f_{D_{2}}\\circ f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}(\\alpha)\\right)+\\dots+f_{D_{t}}\\circ\\dots\\circ f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}(\\alpha)\\right)\\right]\\|_{2}^{q}\\right)^{1/q}\\\\ \\leq\\left(\\sum_{k=1}^{t}r_{\\alpha,q}^{k}\\right)\\|\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ}-\\tilde{\\boldsymbol{\\beta}}_{0}\\|_{2}.\\tag{131}\\\\ \\end{array}\n$$\n\nSince the contraction constant $r_{\\alpha,q}\\in(0,1)$ , we can derive the limit for the sum of the geometric series $\\{r_{\\alpha,q}^{k}\\}_{k=1}^{t}$ as follows\n\n$$\n\\lim_{t\\rightarrow\\infty}\\sum_{k=1}^{t}r_{\\alpha,q}^{k}=\\lim_{t\\rightarrow\\infty}\\frac{r_{\\alpha,q}\\left(1-r_{\\alpha,q}^{t}\\right)}{1-r_{\\alpha,q}}=\\frac{r_{\\alpha,q}}{1-r_{\\alpha,q}}.\\tag{132}\n$$\n\nThis, together with (131) gives\n\n$$\n\\left(\\mathbb{E}\\big\\|\\tilde{S}_{t}^{\\circ}(\\alpha)-\\tilde{S}_{t}^{\\tilde{\\beta}_{0}}(\\alpha)\\big\\|_{2}^{q}\\right)^{1/q}=O(1)=o(\\sqrt{t}),\\tag{133}\n$$\n\nwhich yields the quenched central limit theorem for the partial sum $\\tilde{S}_{t}^{\\tilde{\\beta}_{0}}(\\alpha)$ defined in (114), that is, for any fixed initial point $\\tilde{\\beta}_{0}\\in\\mathbb{R}^{d}$ 2\n\n$$\nt^{-1/2}\\tilde{S}_{t}^{\\tilde{\\beta}_{0}}(\\alpha)\\Rightarrow\\mathcal{N}(0,\\Sigma(\\alpha)).\\tag{134}\n$$\n\nMoreover, we notice that by Markov’s inequality and (143), for any $u\\in\\mathbb{R}$ and $\\delta > 0$ , we have\n\n$$\n\\begin{array}{l}\\sup_{k\\geq 1}\\mathbb{E}\\big [\\|\\tilde{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\tilde{\\boldsymbol{\\beta}}_{\\text{mean}}(\\alpha)\\|_{2}^{q}\\mathbf{1}_{\\|\\tilde{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\tilde{\\boldsymbol{\\beta}}_{\\text{mean}}(\\alpha)\\|_{2}\\geq u}\\big]\\\\ \\leq\\sup_{k\\geq 1}\\mathbb{E}\\Big [\\|\\tilde{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\tilde{\\boldsymbol{\\beta}}_{\\text{mean}}(\\alpha)\\|_{2}^{q}\\cdot\\|\\tilde{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\tilde{\\boldsymbol{\\beta}}_{\\text{mean}}(\\alpha)\\|_{2}^{\\delta}/u^{\\delta}\\Big]\\\\ =\\sup_{k\\geq 1}\\mathbb{E}\\big [\\|\\tilde{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\tilde{\\boldsymbol{\\beta}}_{\\text{mean}}(\\alpha)\\|_{2}^{q+\\delta}\\big]/u^{\\delta}\\\\ =O\\bigl\\{\\alpha^{(q+\\delta)/2}/u^{\\delta}\\bigr\\},\\tag{144}\\\\ \\end{array}\n$$\n\nwhich converges to 0 as $u  \\infty$ . Therefore, condition (i) in Lemma 26 is satisfied. Since $\\{\\tilde{\\beta}_{k}^{\\circ}(\\alpha)\\}_{k\\in\\mathbb{N}}$ is stationary, following the arguments below Lemma 26, condition (ii) is not required. Regarding condition (iii), for the constant learning rate $\\alpha > 0$ satisfying $\\alpha\\|\\mathbb{X}\\|<2$ it follows from (128) that the functional dependence measure $\\begin{array}{ r }{\\theta_{k,q}(\\alpha)\\leq c\\cdot r_{\\alpha,q}^{k},}\\end{array}$ for all $q > 2$ and $k \\in \\mathbb N$ , where the constant $c > 0$ is independent of $k .$ . Consequently, there exists a constant $\\kappa > 0$ such that the tail cumulative dependence measure of $\\{\\tilde{\\beta}_{k}^{\\circ}(\\alpha)\\}_{k\\in\\mathbb{N}}$ can be bounded by\n\n$$\n\\Theta_{m,q}(\\alpha)=\\sum_{k=m}^{\\infty}\\theta_{k,q}(\\alpha)=O\\big\\{m^{-\\chi}\\big(\\log(m)\\big)^{-\\kappa}\\big\\},\\tag{145}\n$$\n\nwhere $\\chi>0$ is some constant that can be taken to be arbitrarily large.\n\nNext, recall the partial sum $\\begin{array}{ r }{\\tilde{S}_{i}^{\\tilde{\\beta}_{0}}(\\alpha)=\\sum_{k=1}^{i}[\\tilde{\\beta}_{k}(\\alpha)-\\tilde{\\beta}_{\\mathrm{mean}}(\\alpha)]}\\end{array}$ as defined in (114), given an arbitrarily fixed initial point $\\tilde{\\beta}_{0}\\in\\mathbb{R}^{d}$ . It follows from the triangle inequality that\n\n$$\n\\begin{array}{l}\\left(\\mathbb{E}\\Big [\\max_{1\\leq i\\leq t}\\big\\|\\tilde{S}_{i}^{\\tilde{\\boldsymbol{\\beta}}_{0}}(\\alpha)-\\Sigma^{1/2}(\\alpha)\\tilde{G}_{i}^{\\star}\\big\\|_{2}\\Big]^{q}\\right)^{1/q}\\\\ =\\left(\\mathbb{E}\\left[\\max_{1\\leq i\\leq t}\\left\\|\\tilde{S}_{i}^{\\tilde{\\beta}_{0}}(\\alpha)-\\tilde{S}_{i}^{\\circ}(\\alpha)+\\tilde{S}_{i}^{\\circ}(\\alpha)-\\Sigma^{1/2}(\\alpha)\\tilde{G}_{i}^{\\star}\\right\\|_{2}\\right]^{q}\\right)^{1/q}\\\\ \\leq\\left(\\mathbb{E}\\left[\\max_{1\\leq i\\leq t}\\left\\|\\tilde{S}_{i}^{\\tilde{\\beta}_{0}}(\\alpha)-\\tilde{S}_{i}^{\\circ}(\\alpha)\\right\\|_{2}+\\max_{1\\leq i\\leq t}\\left\\|\\tilde{S}_{i}^{\\circ}(\\alpha)-\\Sigma^{1/2}(\\alpha)\\tilde{G}_{i}^{\\star}\\right\\|_{2}\\right]^{q}\\right)^{1/q}\\\\ \\leq\\left(\\mathbb{E}\\left[\\max_{1\\leq i\\leq t}\\left\\|\\tilde{S}_{i}^{\\tilde{\\beta}_{0}}(\\alpha)-\\tilde{S}_{i}^{\\circ}(\\alpha)\\right\\|_{2}\\right]^{q}\\right)^{1/q}+\\left(\\mathbb{E}\\left[\\max_{1\\leq i\\leq t}\\left\\|\\tilde{S}_{i}^{\\circ}(\\alpha)-\\Sigma^{1/2}(\\alpha)\\tilde{G}_{i}^{\\star}\\right\\|_{2}\\right]^{q}\\right)^{1/q}.\\tag{148}\\\\ \\end{array}\n$$\n\nTherefore, to show the invariance principle for $\\big(\\tilde{S}_{i}^{\\tilde{\\beta}_{0}}(\\alpha)\\big)_{1\\leq i\\leq t},$ it sufices to bound the difference part max $_{1\\le i\\le t}\\|\\tilde{S}_{i}^{\\tilde{\\beta}_{0}}(\\alpha)-\\tilde{S}_{i}^{\\scriptscriptstyle\\mathrm{o}}(\\alpha)\\|_{2}$ in terms of the $q\\mathrm{-}$ -th moment. To this end, recall the iterative function $f_{D}(\\beta)=\\beta+\\alpha D X^{\\top}({\\pmb y}-X D\\beta)$ in (9) that rewrites the GD dropout recursion (2). We note that\n\n$$\n\\begin{array}{l}\\max_{1\\leq i\\leq t}\\left\\|\\tilde{S}_{i}^{\\tilde{\\beta}_{0}}(\\alpha)-\\tilde{S}_{i}^{\\circ}(\\alpha)\\right\\|_{2}\\\\ =\\max_{1\\leq i\\leq t}\\left\\|\\left[f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}\\right)-f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ}\\right)\\right]+\\dots+\\left[f_{D_{i}}\\circ\\dots\\circ f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}\\right)-f_{D_{i}}\\circ\\dots\\circ f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ}\\right)\\right]\\right\\|_{2}\\\\ \\leq\\max_{1\\leq i\\leq t}\\left(\\left\\|f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}\\right)-f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ}\\right)\\right\\|_{2}+\\dots+\\left\\|f_{D_{i}}\\circ\\dots\\circ f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}\\right)-f_{D_{i}}\\circ\\dots\\circ f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ}\\right)\\right\\|_{2}\\right)\\\\ =\\left\\|f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}\\right)-f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ}\\right)\\right\\|_{2}+\\dots+\\left\\|f_{D_{t}}\\circ\\dots\\circ f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}\\right)-f_{D_{t}}\\circ\\dots\\circ f_{D_{1}}\\left(\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ}\\right)\\right\\|_{2}.\\tag{149}\\\\ \\end{array}\n$$\n\nThis, along with the triangle inequality and Theorem 3 yields\n\n$$\n\\begin{array}{l}\\left(\\mathbb{E}\\left[\\max_{1\\leq i\\leq t}\\left\\|\\tilde{S}_{i}^{\\tilde{\\boldsymbol{\\beta}}_{0}}(\\alpha)-\\tilde{S}_{i}^{\\circ}(\\alpha)\\right\\|_{2}\\right]^{q}\\right)^{1/q}\\\\ \\leq\\left(\\mathbb{E}\\big\\|f_{D_{1}}(\\tilde{\\boldsymbol{\\beta}}_{0})-f_{D_{1}}(\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ})\\big\\|_{2}^{q}\\right)^{1/q}+\\dots+\\left(\\mathbb{E}\\big\\|f_{D_{t}}\\circ\\dots\\circ f_{D_{1}}(\\tilde{\\boldsymbol{\\beta}}_{0})-f_{D_{t}}\\circ\\dots\\circ f_{D_{1}}(\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ})\\big\\|_{2}^{q}\\right)^{1/q}\\\\ \\leq\\frac{r_{\\alpha,q}\\left(1-r_{\\alpha,q}^{t}\\right)}{1-r_{\\alpha,q}}\\|\\tilde{\\boldsymbol{\\beta}}_{0}-\\tilde{\\boldsymbol{\\beta}}_{0}^{\\circ}\\|_{2}=o\\left(t^{1/q}\\right).\\tag{150}\\\\ \\end{array}\n$$\n\nWe insert this result back into (149), which together with (148) gives the invariance principle for the partial sum $\\begin{array}{ r }{\\tilde{S}_{i}^{\\tilde{\\beta}_{0}}(\\alpha)=\\sum_{k=1}^{i}[\\tilde{\\beta}_{k}(\\alpha)-\\tilde{\\beta}_{\\mathrm{mean}}(\\alpha)]}\\end{array}$\n\nFinally, let the partial sum $\\begin{array}{ r }{S_{i}^{\\beta_{0}}(\\alpha)=\\sum_{k=1}^{i}[\\tilde{\\beta}_{k}(\\alpha)-\\tilde{\\beta}]}\\end{array}$ be as defined in Theorem 9. We shall bound the diference between $S_{i}^{\\tilde{\\beta}_{0}}(\\alpha)$ and $\\tilde{S}_{i}^{\\tilde{\\beta}_{0}}(\\alpha)$ . Since $\\widetilde{\\beta}_{\\mathrm{mean}}(\\alpha)=\\mathbb{E}_{D}[\\widetilde{\\beta}_{1}^{\\circ}(\\alpha)]$ as defined in (111) and $\\tilde{\\beta}_{\\mathrm{mean}}(\\alpha)-\\tilde{\\beta}=\\mathbb{E}_{D}\\widetilde [\\tilde{\\beta}_{1}^{\\circ}(\\alpha)-\\tilde{\\beta}]\\stackrel{\\circ}{=}0$ by (137), it follows that\n\n$$\n\\left(\\mathbb{E}\\Big [\\max_{1\\leq i\\leq t}\\big\\|\\tilde{S}_{i}^{\\tilde{\\boldsymbol{\\beta}}_{0}}(\\alpha)-S_{i}^{\\tilde{\\boldsymbol{\\beta}}_{0}}(\\alpha)\\big\\|_{2}\\Big]^{q}\\right)^{1/q}=\\left(\\mathbb{E}\\Big [\\max_{1\\leq i\\leq t}\\big\\|\\sum_{k=1}^{i}\\tilde{\\boldsymbol{\\beta}}_{\\mathrm{mean}}(\\alpha)-\\tilde{\\boldsymbol{\\beta}}\\big\\|_{2}\\Big]^{q}\\right)^{1/q}=0.\\tag{151}\n$$\n\nProof The recursion in (36) is $\\breve{\\beta}_{k}(\\alpha)-\\breve{\\beta}=\\breve{A}_{k}(\\alpha)(\\breve{\\beta}_{k-1}(\\alpha)-\\breve{\\beta})+\\breve{b}_{k}(\\alpha)$ , with random matrix $\\breve{A}_{k}(\\alpha)=I_{d}-\\alpha D_{k}\\mathbb{X}_{k}D_{k}$ , and random vector $\\breve{\\pmb{b}}_{k}(\\alpha)=\\alpha D_{k}\\pmb{x}_{k}(y_{k}-\\pmb{x}_{k}^{\\top}D_{k}\\breve{\\beta})$ . Recall $\\begin{array}{ r }{\\breve{\\beta}=\\arg\\operatorname*{min}_{\\beta\\in\\mathbb{R}^{d}}\\mathbb E[(y-\\pmb{x}^{\\top}D\\beta)^{2}/2]}\\end{array}$ in (34), where the expectation is taken over both $(y,x)$ and D. By Lemma 21 (ii),\n\n$$\n\\begin{array}{l}\\mathbb{E}_{(y,\\pmb{x})}\\mathbb{E}_{D}[\\breve{\\pmb{b}}_{k}(\\alpha)]=\\mathbb{E}_{(y,\\pmb{x})}\\mathbb{E}_{D}[\\alpha D_{k}\\pmb{x}_{k}(y_{k}-\\pmb{x}_{k}^{\\top}D_{k}\\breve{\\pmb{\\beta}})]\\\\ =\\mathbb{E}_{(y,\\boldsymbol{x})}[\\alpha p I_{d}y_{k}\\boldsymbol{x}_{k}-\\alpha p\\mathbb{X}_{k,p}\\breve{\\boldsymbol{\\beta}}]\\\\ =\\alpha p\\mathbb{E}[\\mathbb{X}_{1,p}]\\breve{\\boldsymbol{\\beta}}-\\alpha p\\mathbb{E}[\\mathbb{X}_{1,p}]\\breve{\\boldsymbol{\\beta}}\\\\ =0.\\tag{193}\\\\ \\end{array}\n$$\n\nThe proofs of the quenched CLT and the invariance principle for averaged SGD dropout follow similar arguments as the ones for averaged GD dropout. The key diferences lie in the functional dependence measures, because for SGD settings, both the dropout matrix $D_{k}$ and the sequential observation $\\mathbf{{x}}_{k}$ are random. We shall first introduce some necessary definitions and then proceed with the rigorous proofs.\n\nRecall the generic dropout matrix $D\\in\\mathbb{R}^{d\\times d}$ and random sample $(y,\\pmb{x})\\in\\mathbb{R}\\times\\mathbb{R}^{d}$ . For the SGD dropout sequence $\\{\\breve{\\beta}_{k}(\\alpha)\\}_{k\\in\\mathbb{N}}$ , we define the centering term\n\n$$\n\\breve{\\boldsymbol{\\beta}}_{\\text{mean}}(\\alpha)=\\lim_{k\\rightarrow\\infty}\\mathbb{E}[\\breve{\\boldsymbol{\\beta}}_{k}(\\alpha)]=\\mathbb{E}[\\breve{\\boldsymbol{\\beta}}_{1}^{\\circ}(\\alpha)],\\tag{204}\n$$\n\nwhere the expectation is taken over both $(y,x)$ and D, and $\\breve{\\beta}_{1}^{\\circ}(\\alpha)$ defined in (44) follows the unique stationary distribution $\\breve{\\pi}_{\\alpha}$ . We first use Lemma 25 to prove the CLT for the partial sum of the stationary sequence $\\{\\breve{\\beta}_{k}^{\\circ}(\\alpha)-\\breve{\\beta}_{\\mathrm{mean}}(\\alpha)\\}_{k\\in\\mathbb{N}}$ , and then apply the geometricmoment contraction in Theorem 12 to show the quenched CLT for the partial sum of the non-stationary one $\\{\\breve{\\beta}_{k}(\\alpha)-\\breve{\\beta}_{\\mathrm{mean}}(\\alpha)\\}_{k\\in\\mathbb{N}}$ . Finally, we extend the quenched CLT to the partial sum of $\\{\\breve{\\beta}_{k}(\\alpha)-\\breve{\\beta}_{\\mathrm{mean}}(\\alpha)\\}_{k\\in\\mathbb{N}}$ by providing the upper bound of $\\breve{\\beta}_{\\mathrm{mean}}(\\alpha)-\\breve{\\beta}$ in terms of the q-th moment for some $q \\geq 2$\n\nSimilar to Section D.1, we introduce the functional dependence measure in Wu (2005) for the stationary SGD dropout sequence $\\{\\breve{\\beta}_{k}^{\\circ}(\\alpha)\\}_{k\\in\\mathbb{N}}$ . However, the randomness in $\\breve{\\beta}_{k}^{\\circ}(\\alpha)$ is induced from both the dropout matrix $D_{k}$ and the random sample $(y_{k},\\pmb{x}_{k})$ . Therefore, we define a new filtration $\\breve{\\mathcal{F}}_{k}$ by\n\n$$\n\\breve{\\mathcal{F}}_{k}=\\sigma(\\boldsymbol{\\xi}_{k},\\boldsymbol{\\xi}_{k-1},\\dots),\\quad k\\in\\mathbb{Z},\\tag{205}\n$$\n\nwhere the i.i.d. random elements $\\pmb{\\xi}_{k}=(D_{k},(y_{k},\\pmb{x}_{k})),k\\in\\mathbb{Z},$ are defined in (45). For any random vector $\\zeta\\in\\mathbb{R}^{d}$ satisfying $\\mathbb{E}\\|\\zeta\\|_{2}<\\infty$ , define projection operators\n\n$$\n\\breve{\\mathcal{P}}_{k}[\\boldsymbol{\\zeta}]=\\mathbb{E}[\\boldsymbol{\\zeta}\\mid\\breve{\\mathcal{F}}_{k}]-\\mathbb{E}[\\boldsymbol{\\zeta}\\mid\\breve{\\mathcal{F}}_{k-1}],\\quad k\\in\\mathbb{Z}.\\tag{206}\n$$\n\nBy Theorem 12 and (44), there exists a measurable function $\\breve{h}_{\\alpha}(\\cdot)$ such that the stationary SGD dropout sequence $\\{\\breve{\\beta}_{k}^{\\circ}(\\alpha)\\}_{k\\in\\mathbb{N}}$ can be represented by a causal process\n\n$$\n\\breve{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)=\\breve{h}_{\\alpha}\\left(\\boldsymbol{\\xi}_{k},\\boldsymbol{\\xi}_{k-1},\\dots\\right)=\\breve{h}_{\\alpha}\\left(\\breve{\\mathcal{F}}_{k}\\right).\\tag{207}\n$$\n\nWe denote the coupled version of $\\breve{\\mathscr{F}}_{i}$ by\n\n$$\n\\breve{\\mathcal{F}}_{i,\\{j\\}}=\\sigma(\\boldsymbol{\\xi}_{i},\\dots,\\boldsymbol{\\xi}_{j+1},\\boldsymbol{\\xi}_{j}^{\\prime},\\boldsymbol{\\xi}_{j-1}\\dots),\\tag{208}\n$$\n\nand let $\\breve{\\mathcal{F}}_{i,\\{j\\}}=\\breve{\\mathcal{F}}_{i}\\mathrm{~i f~}j>i$ , where $\\xi_{j}^{\\prime}$ is an i.i.d. copy of $\\xi_{i}$ . For $q > 1$ , define the functional dependence measure of $\\breve{\\beta}_{k}(\\alpha)$ as\n\n$$\n\\check{\\theta}_{k,q}(\\alpha)=\\left(\\mathbb{E}\\|\\check{\\beta}_{k}^{\\circ}(\\alpha)-\\check{\\beta}_{k,\\{0\\}}^{\\circ}(\\alpha)\\|_{2}^{q}\\right)^{1/q},\\quad \\text{where}\\check{\\beta}_{k,\\{0\\}}^{\\circ}(\\alpha)=\\check{h}_{\\alpha}(\\check{\\mathcal{F}}_{k,\\{0\\}}).\\tag{209}\n$$\n\nIn addition, if $\\begin{array}{ r }{\\sum_{k=0}^{\\infty}\\breve{\\theta}_{k,q}(\\alpha)<\\infty .}\\end{array}$ , we define the tail of cumulative dependence measure as\n\n$$\n\\check{\\Theta}_{m,q}(\\alpha)=\\sum_{k=m}^{\\infty}\\check{\\theta}_{k,q}(\\alpha),\\quad m\\in\\mathbb{N}.\\tag{210}\n$$\n\nBoth $\\breve{\\theta}_{k,q}(\\alpha)$ and $\\check{\\Theta}_{k,q}(\\alpha)$ are useful to study the dependence structure of the stationary SGD dropout iteration $\\breve{\\beta}_{k}^{\\circ}(\\alpha)=f_{D_{k},(y_{k},{\\pmb x}_{k})}(\\breve{\\beta}_{k-1}^{\\circ}(\\alpha))$ . To apply Lemma 25, we only need to show that the stationary SGD dropout sequence $\\{\\breve{\\beta}_{k}^{\\circ}(\\alpha)\\}_{k\\in\\mathbb{N}}$ is short-range dependent in the sense that $\\check{\\Theta}_{0,q}(\\alpha)<\\infty$ , for some $q \\geq 2$", "theorems": [{"local_id": "T1", "statement": "Lemma 13 (Dependence on d in SGD with dropout) If the covariate vectors $\\mathbf{{x}}_{k}$ are independently drawn from a standard multivariate Gaussian, then,\n\n$$\n\\breve{r}_{\\alpha,2}^{2}=1-2\\alpha p+\\alpha^{2}[(d-1)p^{2}+3 p].\\tag{46}\n$$\n\nHence, $\\alpha<2/((d-1)p+3)$ ensures $\\breve{r}_{\\alpha,2}<1$", "proof": {"steps": [{"n": 1, "claim": "Proof For any fixed unit vector $\\pmb{u}\\in\\mathbb{R}^{d}$ , since $\\breve{A}_{k}(\\alpha)=I_{d}-\\alpha D_{k}\\mathbb{X}_{k}D_{k}$ as defined in (36) with $\\mathbb{X}_{k}=\\pmb{x}_{k}\\pmb{x}_{k}^{\\top}$ , it follows that\n\n$$\n\\begin{array}{l}\\|\\breve{A}_{k}(\\alpha)\\boldsymbol{v}\\|_{2}^{2}=\\|(I_{d}-\\alpha D_{k}\\boldsymbol{x}_{k}\\boldsymbol{x}_{k}^{\\top}D_{k})\\boldsymbol{v}\\|_{2}^{2}\\\\ =\\|\\boldsymbol{v}\\|_{2}^{2}-2\\alpha\\boldsymbol{v}^{\\top}D_{k}\\boldsymbol{x}_{k}\\boldsymbol{x}_{k}^{\\top}D_{k}\\boldsymbol{v}+\\alpha^{2}\\boldsymbol{v}^{\\top}D_{k}\\boldsymbol{x}_{k}\\boldsymbol{x}_{k}^{\\top}D_{k}^{2}\\boldsymbol{x}_{k}\\boldsymbol{x}_{k}^{\\top}D_{k}\\boldsymbol{v}\\\\ =1-2\\alpha(\\boldsymbol{x}_{k}^{\\top}D_{k}\\boldsymbol{v})^{2}+\\alpha^{2}\\|D_{k}\\boldsymbol{x}_{k}\\|_{2}^{2}(\\boldsymbol{x}_{k}^{\\top}D_{k}\\boldsymbol{v})^{2}.\\tag{180}\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "Write $\\pmb{v}=(v_{1},\\ldots,v_{d})^{\\top}$ and $\\pmb{x}_{k}=(x_{k 1},\\ldots,x_{k d})^{\\top}$ . Let $D_{k,j j}$ be the j-th diagonal element in $D_{k}$ . Since $\\pmb{x}_{k}\\sim\\mathcal{N}(0,I_{d})$ and $D_{k}$ is independent of $\\scriptstyle{\\mathbf{{\\mathit{x}}}}_{k}$ , it follows that\n\n$$\n\\mathbb{E}(\\boldsymbol{x}_{k}^{\\top}D_{k}\\boldsymbol{v})^{2}=\\sum_{j=1}^{d}v_{j}^{2}\\mathbb{E}[D_{k,j j}x_{k j}^{2}]=\\sum_{j=1}^{d}v_{j}^{2}p=p.\\tag{181}\n$$"}, {"n": 3, "claim": "Moreover,\n\n$$\n\\begin{array}{l}\\|D_{k}\\boldsymbol{x}_{k}\\|_{2}^{2}(\\boldsymbol{x}_{k}^{\\top}D_{k}\\boldsymbol{v})^{2}=\\Big(\\sum_{i=1}^{d}D_{k,i i}x_{k i}^{2}\\Big)\\Big(\\sum_{j,l=1}^{d}D_{k,j j}D_{k,l l}v_{j}v_{l}x_{k j}x_{k l}\\Big)\\\\ =\\sum_{i,j,l=1}^{d}D_{k,i i}D_{k,j j}D_{k,l l}v_{j}v_{l}x_{k i}^{2}x_{k j}x_{k l}.\\tag{182}\\\\ \\end{array}\n$$"}, {"n": 4, "claim": "For $j\\neq l,\\mathbb{E}[x_{k i}^{2}x_{k j}x_{k l}]=0$ . For $j = l$ and $i\\neq j,\\mathbb{E}[x_{k i}^{2}x_{k j}^{2}]=1$ . For $j = l$ and $i = j$ , $\\mathbb{E}[x_{k i}^{4}]=3$ . Therefore, by the independence of the diagonal elements in $D_{k}$ , we have\n\n$$\n\\begin{array}{l}\\mathbb{E}\\big [\\|D_{k}\\boldsymbol{x}_{k}\\|_{2}^{2}(\\boldsymbol{x}_{k}^{\\top}D_{k}\\boldsymbol{v})^{2}\\big]=\\sum_{j=1}^{d}v_{j}^{2}\\mathbb{E}\\Big(3 D_{k,j j}+\\sum_{i\\neq j}D_{k,i i}D_{k,j j}\\Big)\\\\ =\\sum_{j=1}^{d}v_{j}^{2}\\big(3 p+p^{2}(d-1)\\big)\\\\ =3 p+p^{2}(d-1).\\tag{183}\\\\ \\end{array}\n$$"}, {"n": 5, "claim": "Finally, we obtain\n\n$$\n\\mathbb{E}\\|\\breve{A}_{k}(\\alpha)\\pmb{v}\\|_{2}^{2}=1-2\\alpha p+\\alpha^{2}[3 p+(d-1)p^{2}],\n$$\n\nfor all unit vectors $\\pmb{v}\\in\\mathbb{R}^{d}$ . This completes the proof."}]}}, {"local_id": "T2", "statement": "Theorem 15 (Quenched CLT of averaged SGD dropout) Under Assumption 1, if the learning rate α satisfies (37), then,\n\n$$\n\\sqrt{k}\\big(\\bar{\\beta}_{k}^{\\mathrm{sgd}}(\\alpha)-\\breve{\\beta}\\big)\\Rightarrow\\mathcal{N}\\big(0,\\breve{\\Sigma}(\\alpha)\\big),\\quad a s k\\rightarrow\\infty,\\tag{49}\n$$\n\nwith $\\begin{array}{ r }{\\breve{\\Sigma}(\\alpha): =\\sum_{i= -\\infty}^{\\infty}\\mathbb{E}[(\\breve{\\beta}_{0}^{\\circ}(\\alpha)-\\breve{\\beta})(\\breve{\\beta}_{i}^{\\circ}(\\alpha)-\\breve{\\beta})^{\\top}]}\\end{array}$ the long-run covariance matrix of the stationary process $\\breve{\\beta}_{i}^{\\circ}(\\alpha)\\sim\\breve{\\pi}_{\\alpha}$", "proof": {"steps": [{"n": 1, "claim": "Proof [Theorem 15] Consider two initial vectors $\\breve{\\beta}_{0}^{\\circ},\\breve{\\beta}_{0}^{\\circ^{\\prime}}$ following the stationary distribution $\\breve{\\pi}_{\\alpha}$ defined in Theorem 12. By the recursion in (36), we obtain two stationary SGD dropout sequences $\\{\\breve{\\beta}_{k}^{\\circ}(\\alpha)\\}_{k\\in\\mathbb{N}}$ and $\\{\\breve{\\beta}_{k}^{\\circ^{\\prime}}(\\alpha)\\}_{k\\in\\mathbb{N}}$ . It follows from Theorem 12 that for all $q \\geq 2$ 2\n\n$$\n\\sup_{\\check{\\boldsymbol{\\beta}}_{0}^{\\circ},\\check{\\boldsymbol{\\beta}}_{0}^{\\circ^{\\prime}}\\in\\mathbb{R}^{d},\\check{\\boldsymbol{\\beta}}_{0}^{\\circ}\\neq\\check{\\boldsymbol{\\beta}}_{0}^{\\circ^{\\prime}}}\\frac{\\left(\\mathbb{E}\\|\\check{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\check{\\boldsymbol{\\beta}}_{k}^{\\circ^{\\prime}}(\\alpha)\\|_{2}^{q}\\right)^{1/q}}{\\|\\check{\\boldsymbol{\\beta}}_{0}^{\\circ}-\\check{\\boldsymbol{\\beta}}_{0}^{\\circ^{\\prime}}\\|_{2}}\\leq\\check{r}_{\\alpha,q}^{k},\\quad k\\in\\mathbb{N},\\tag{211}\n$$\n\nwith $\\begin{array}{ r }{\\breve{r}_{\\alpha,q}=\\left(\\operatorname*{sup}_{{\\pmb v}\\in\\mathbb{R}^{d}:\\|{\\pmb v}\\|_{2}=1}\\mathbb{E}\\|\\breve{A}_{1}(\\alpha){\\pmb v}\\|_{2}^{q}\\right)^{1/q}}\\end{array}$ as defined in (38) and random matrix $\\breve{A}_{1}(\\alpha)=$ $I_{d}-\\alpha D_{1}\\mathbb{X}_{1}D_{1}$ ."}, {"n": 2, "claim": "When the constant learning rate $\\alpha > 0$ satisfies the condition in (37), we have $\\breve{r}_{\\alpha,q}~\\in~(0,1)$ as shown in Theorem 12. Recall the coupled filtration $\\breve{\\mathcal{F}}_{i,\\{j\\}}=$ $\\sigma(\\pmb{\\xi}_{i},\\dots,\\pmb{\\xi}_{j+1},\\pmb{\\xi}_{j}^{\\prime},\\pmb{\\xi}_{j-1}\\dots)$ as defined in (208). Then, (44) and (211) show that\n\n$$\n\\begin{array}{l}\\left(\\mathbb{E}\\|\\breve{f}_{\\boldsymbol{\\xi}_{k}}\\circ\\dots\\circ\\breve{f}_{\\boldsymbol{\\xi}_{1}}\\left(\\breve{\\boldsymbol{\\beta}}_{0}^{\\circ}\\right)-\\breve{f}_{\\boldsymbol{\\xi}_{k}}\\circ\\dots\\circ\\breve{f}_{\\boldsymbol{\\xi}_{1}}\\left(\\breve{\\boldsymbol{\\beta}}_{0}^{\\circ^{\\prime}}\\right)\\|_{2}^{q}\\right)^{1/q}\\\\ =\\left(\\mathbb{E}\\|\\breve{h}_{\\alpha}\\left(\\boldsymbol{\\xi}_{k},\\dots,\\boldsymbol{\\xi}_{1},\\boldsymbol{\\xi}_{0},\\boldsymbol{\\xi}_{-1},\\dots\\right)-\\breve{h}_{\\alpha}\\left(\\boldsymbol{\\xi}_{k},\\dots,\\boldsymbol{\\xi}_{1},\\boldsymbol{\\xi}_{0}^{\\prime},\\boldsymbol{\\xi}_{-1}^{\\prime},\\dots\\right)\\|_{2}^{q}\\right)^{1/q}\\\\ =\\left(\\mathbb{E}\\|\\breve{h}_{\\alpha}(\\breve{\\mathcal{F}}_{k})-\\breve{h}_{\\alpha}(\\breve{\\mathcal{F}}_{k,\\{0,-1,\\dots\\}})\\|_{2}^{q}\\right)^{1/q}\\\\ \\leq\\check{c}_{q}\\check{r}_{\\alpha,q}^{k},\\tag{212}\\\\ \\end{array}\n$$\n\nfor some constant $\\breve{c}_{q}>0$ that is independent of k."}, {"n": 3, "claim": "Following a similar argument in (127), for all $q \\geq 2$ and $k \\in \\mathbb N$ , we can bound the functional dependence measure $\\cup_{k,q}(\\alpha)$ in (209) as follows\n\n$$\n\\begin{array}{l}\\breve{\\theta}_{k,q}(\\alpha)=\\left(\\mathbb{E}\\|\\breve{h}_{\\alpha}(\\breve{\\mathcal{F}}_{k})-\\breve{h}_{\\alpha}(\\breve{\\mathcal{F}}_{k,\\{0\\}})\\|_{2}^{q}\\right)^{1/q}\\\\ \\leq\\left(\\mathbb{E}\\|\\breve{h}_{\\alpha}(\\breve{\\mathcal{F}}_{k})-\\breve{h}_{\\alpha}(\\breve{\\mathcal{F}}_{k,\\{0,-1,\\dots\\}})\\|_{2}^{q}\\right)^{1/q}+\\left(\\mathbb{E}\\|\\breve{h}_{\\alpha}(\\breve{\\mathcal{F}}_{k,\\{0,-1,\\dots\\}})-\\breve{h}_{\\alpha}(\\breve{\\mathcal{F}}_{k,\\{0\\}})\\|_{2}^{q}\\right)^{1/q}\\\\ \\leq\\check{c}_{q}^{\\prime}\\check{r}_{\\alpha,q}^{k},\\tag{213}\\\\ \\end{array}\n$$\n\nfor some constant $\\breve{c}_{q}^{\\prime}>0$ that is independent of k."}, {"n": 4, "claim": "Consequently, the cumulative dependence measure $\\breve{\\Theta}_{m,q}(\\alpha)$ in (210) is also bounded for all $q \\geq 2$ and $m\\in\\mathbb{N}$ , that is,\n\n$$\n\\breve{\\Theta}_{m,q}(\\alpha)=\\sum_{k=m}^{\\infty}\\breve{\\theta}_{k,q}(\\alpha)=O\\left(\\check{r}_{\\alpha,q}^{m}\\right)<\\infty .\\tag{214}\n$$"}, {"n": 5, "claim": "The inequality in (119) derived by Wu (2005) holds for a general class of functional dependence measures, as long as the inputs of the functional system (i.e., the measurable function $\\breve{h}(\\xi_{k},\\xi_{k-1},...))$ are i.i.d. elements. Thus, we can apply (119) to the projection operator $\\breve{\\mathcal{P}}_{k}[\\cdot]$ in (206) and obtain\n\n$$\n\\sum_{k=0}^{\\infty}\\left(\\mathbb{E}\\|\\check{\\mathcal{P}}_{0}[\\check{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)]\\|_{2}^{q}\\right)^{1/q}\\leq\\sum_{k=0}^{\\infty}\\check{\\theta}_{k,q}(\\alpha)=\\check{\\Theta}_{0,q}(\\alpha)<\\infty,\\tag{215}\n$$\n\nimplying the short-range dependence of the stationary SGD dropout sequence $\\{\\breve{\\beta}_{k}^{\\circ}(\\alpha)\\}_{k\\in\\mathbb{N}}$ Then, it follows from Lemma 25 that\n\n$$\nt^{-1/2}\\sum_{k=1}^{t}\\left(\\breve{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\breve{\\boldsymbol{\\beta}}_{\\text{mean}}(\\alpha)\\right)\\Rightarrow\\mathcal{N}(0,\\breve{\\Sigma}(\\alpha)),\\tag{216}\n$$\n\nwhere the long-run covariance matrix $\\breve{\\Sigma}(\\alpha)$ is defined in Theorem 15."}, {"n": 6, "claim": "Following similar arguments as in (131)–(133), for any initial vector $\\breve{\\beta}_{0}\\in\\mathbb{R}^{d}$ , we can leverage the geometricmoment contraction in Theorem 12 and achieve the quenched CLT for the corresponding SGD dropout sequence $\\{\\breve{\\beta}_{k}(\\alpha)\\}_{k\\in\\mathbb{N}}$ , that is,\n\n$$\nt^{-1/2}\\sum_{k=1}^{t}\\left(\\breve{\\boldsymbol{\\beta}}_{k}(\\alpha)-\\breve{\\boldsymbol{\\beta}}_{\\text{mean}}(\\alpha)\\right)\\Rightarrow\\mathcal{N}(0,\\breve{\\Sigma}(\\alpha)).\\tag{217}\n$$"}, {"n": 7, "claim": "Recall the $\\ell^{2}.$ -minimizer $\\breve{\\beta}$ in (34) and the centering term $\\begin{array}{ r }{\\breve{\\beta}_{\\mathrm{mean}}(\\alpha)=\\operatorname*{lim}_{k\\to\\infty}\\mathbb{E}(\\breve{\\beta}_{k}(\\alpha))=}\\end{array}$ $\\mathbb{E}(\\breve{\\beta}_{1}^{\\circ}(\\alpha))$ in (204). We shall prove $\\begin{array}{ r }{\\|\\sum_{k=1}^{t}\\mathbb{E}[\\breve{\\beta}_{k}(\\alpha)-\\breve{\\beta}]\\|_{2}=o(\\sqrt{t})}\\end{array}$ . For any two initial vectors $\\breve{\\beta}_{0}$ and $\\breve{\\beta}_{0}^{\\circ}$ , where $\\breve{\\beta}_{0}^{\\circ}$ follows the stationary distribution $\\breve{\\pi}_{\\alpha}$ in Theorem 12, while $\\breve{\\beta}_{0}$ is an arbitrary initial vector in $\\mathbb{R}^{d}$ , it follows from the triangle inequality that\n\n$$\n\\begin{array}{l}\\left\\|\\sum_{k=1}^{t}\\mathbb{E}[\\breve{\\boldsymbol{\\beta}}_{k}(\\alpha)-\\breve{\\boldsymbol{\\beta}}]\\right\\|_{2}=\\left\\|\\mathbb{E}\\left[\\sum_{k=1}^{t}\\left(\\breve{\\boldsymbol{\\beta}}_{k}(\\alpha)-\\breve{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)+\\breve{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\breve{\\boldsymbol{\\beta}}\\right)\\right]\\right\\|_{2}\\\\ \\leq\\left\\|\\mathbb{E}\\left[\\sum_{k=1}^{t}\\left(\\breve{\\boldsymbol{\\beta}}_{k}(\\alpha)-\\breve{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)\\right)\\right]\\right\\|_{2}+\\left\\|\\mathbb{E}\\left[\\sum_{k=1}^{t}\\left(\\breve{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\breve{\\boldsymbol{\\beta}}\\right)\\right]\\right\\|_{2}\\\\ =:\\check{\\mathrm{I}}_{1}+\\check{\\mathrm{I}}_{2}.\\tag{218}\\\\ \\end{array}\n$$"}, {"n": 8, "claim": "For the part $\\breve{\\mathrm{I}}_{1}$ , Jensen’s inequality and a similar argument as for (131) yield\n\n$$\n\\begin{array}{l}\\breve{\\mathrm{I}}_{1}=\\left\\|\\mathbb{E}\\left[\\sum_{k=1}^{t}\\left(\\breve{\\boldsymbol{\\beta}}_{k}(\\alpha)-\\breve{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)\\right)\\right]\\right\\|_{2}\\\\ \\leq\\left(\\mathbb{E}\\left\\|\\sum_{k=1}^{t}\\left(\\breve{\\boldsymbol{\\beta}}_{k}(\\alpha)-\\breve{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)\\right)\\right\\|_{2}^{2}\\right)^{1/2}\\\\ \\leq\\left(\\sum_{k=1}^{t}\\check{r}_{\\alpha,2}^{k}\\right)\\left\\|\\breve{\\boldsymbol{\\beta}}_{0}-\\breve{\\boldsymbol{\\beta}}_{0}^{\\circ}\\right\\|_{2}.\\tag{219}\\\\ \\end{array}\n$$"}, {"n": 9, "claim": "For the part $\\breve{\\mathrm{I}}_{2}$ , we recall the random matrix $\\breve{A}_{1}(\\alpha)=I_{d}-\\alpha D_{1}\\mathbb{X}_{1}D_{1}$ in (36) with $\\mathbb{X}_{1}=\\pmb{x}_{1}\\pmb{x}_{1}^{\\top}$ Recall the notation $\\mathbb{X}_{1,p}=p\\mathbb{X}_{1}+(1-p)\\mathrm{Diag}(\\mathbb{X}_{1})$ in (35). Notice that by Lemma 21 (ii), we have $\\begin{array}{ r }{\\mathbb{E}_{(y,x)}\\mathbb{E}_{D}[\\check{A}_{1}(\\boldsymbol{\\hat{\\alpha}})]=\\mathbb{E}_{(y,x)}\\mathbb{E}_{D}[I_{d}-\\alpha D_{1}\\mathbb{X}_{1}D_{1}]=\\mathbb{E}_{(y,x)}[I_{d}-\\alpha p\\mathbb{X}_{1,p}]=I_{d}-\\alpha p\\mathbb{E}[\\mathbb{X}_{1,p}]}\\end{array}$ which along with E $[\\breve{b}_{k}]=0$ in (193) gives $\\mathbb{E}[\\breve{\\beta}_{k}^{\\circ}(\\alpha)-\\breve{\\beta}]=\\bigl(I_{d}-\\alpha p\\mathbb{E}[\\mathbb{X}_{1,p}]\\bigr)\\mathbb{E}[\\breve{\\beta}_{k-1}^{\\circ}(\\alpha)-\\breve{\\beta}]$ Since $\\{\\breve{\\beta}_{k}^{\\circ}(\\alpha)\\}_{k\\in\\mathbb{N}}$ is stationary and $\\mathbb{E}[\\mathbb{X}_{1,p}]$ is non-singular by the reduced-form condition mi $\\mathbf{\\sigma}_{\\mathfrak{i}}(\\mathbb{E}[\\pmb{x}_{1}\\pmb{x}_{1}^{\\top}])_{i i}>0$ imposed on model (31), it follows that\n\n$$\n\\mathbb{E}[\\breve{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\breve{\\boldsymbol{\\beta}}]=0,\\quad \\text{for all} k\\in\\mathbb{N}.\\tag{220}\n$$"}, {"n": 10, "claim": "This further yields\n\n$$\n\\check{\\mathrm{I}}_{2}=\\left\\|\\mathbb{E}\\left[\\sum_{k=1}^{t}\\left(\\check{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\check{\\boldsymbol{\\beta}}\\right)\\right]\\right\\|_{2}=\\left\\|\\sum_{k=1}^{t}\\mathbb{E}\\left[\\check{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\check{\\boldsymbol{\\beta}}\\right]\\right\\|_{2}=0.\\tag{221}\n$$"}, {"n": 11, "claim": "By applying the results for $\\breve{\\mathrm{I}}_{1}$ and $\\breve{\\mathrm{I}}_{2}$ to (218), we obtain\n\n$$\n\\left\\|\\sum_{k=1}^{t}\\mathbb{E}\\left[\\check{\\boldsymbol{\\beta}}_{k}(\\alpha)-\\check{\\boldsymbol{\\beta}}\\right]\\right\\|_{2}\\leq\\left(\\sum_{k=1}^{t}\\check{r}_{\\alpha,2}^{k}\\right)\\left\\|\\check{\\boldsymbol{\\beta}}_{0}-\\check{\\boldsymbol{\\beta}}_{0}^{\\circ}\\right\\|_{2}.\\tag{222}\n$$"}, {"n": 12, "claim": "When the constant learning rate $\\alpha > 0$ satisfies the condition in (37), $\\breve{r}_{\\alpha,2}\\in(0,1)$ by Theorem 12, and hence (222) remains bounded as $t \\to \\infty$ . By this result and (217), the desired quenched CLT for the partial sum $\\scriptstyle\\sum_{k=1}^{t}({\\check{\\beta}}_{k}^{\\circ}(\\alpha)-{\\check{\\beta}})$ follows."}]}}, {"local_id": "T3", "statement": "Theorem 17 (Quenched invariance principle of averaged SGD dropout) Let Assumption 1 hold for some $q > 2$ and consider the learning rate α satisfying (37). Define a partial sum process $(\\breve{S}_{i}^{\\circ}(\\alpha))_{1\\leq i\\leq t}$ for $t\\in\\mathbb{N}_{+}$ with\n\n$$\n\\breve{S}_{i}^{\\circ}(\\alpha)=\\sum_{k=1}^{i}\\left(\\breve{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\breve{\\boldsymbol{\\beta}}\\right).\\tag{51}\n$$\n\nThen, there exists a (richer) probability space $(\\breve{\\Omega}^{\\star},\\breve{A}^{\\star},\\breve{\\mathbb{P}}^{\\star})$ on which one can define ddimensional random vectors $\\breve{\\beta}_{k}^{\\star}$ , the associated partial sum process $\\begin{array}{ r }{\\breve{S}_{i}^{\\star}(\\alpha)=\\sum_{k=1}^{i}(\\breve{\\beta}_{k}^{\\star}(\\alpha)-}\\end{array}$ $\\breve{\\beta})$ , and a Gaussian process $\\begin{array}{ r }{\\breve{G}_{i}^{\\star}=\\sum_{k=1}^{i}\\breve{z}_{k}^{\\star}}\\end{array}$ , with independent Gaussian random vectors $\\breve{z}_{k}\\sim\\mathcal N(0,I_{d})$ , such that $(\\breve{S}_{i}^{\\circ})_{1\\leq i\\leq t}\\underline{{\\stackrel{\\mathcal{D}}{=}(\\breve{S}_{i}^{\\star})_{1\\leq i\\leq t}}}$ and\n\n$$\n\\max_{1\\leq i\\leq t}\\left\\|\\breve{S}_{i}^{\\star}-\\breve{\\Sigma}^{1/2}(\\alpha)\\breve{G}_{i}^{\\star}\\right\\|_{2}=o_{\\mathbb{P}}(t^{1/q}),\\quad i n(\\breve{\\Omega}^{\\star},\\breve{\\mathcal{A}}^{\\star},\\breve{\\mathbb{P}}^{\\star}),\\tag{52}\n$$\n\nwhere $\\breve{\\Sigma}(\\alpha)$ is the long-run covariance matrix defined in Theorem 15. In addition, this approximation holds for all $(\\breve{S}_{i}^{\\breve{\\beta}_{0}}(\\alpha))_{1\\leq i\\leq t}$ given any arbitrary initialization $\\breve{\\beta}_{0}\\in\\mathbb{R}^{d}$ , where\n\n$$\n\\breve{S}_{i}^{\\breve{\\boldsymbol{\\beta}}_{0}}(\\alpha)=\\sum_{k=1}^{i}\\left(\\breve{\\boldsymbol{\\beta}}_{k}(\\alpha)-\\breve{\\boldsymbol{\\beta}}\\right).\\tag{53}\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof Consider a stationary SGD dropout sequence $\\{\\breve{\\beta}_{k}^{\\circ}(\\alpha)\\}_{k\\in\\mathbb{N}}$ following the stationary distribution $\\breve{\\pi}_{\\alpha}$ in Theorem 12. Define the mean-zero stationary partial sum\n\n$$\n\\breve{S}_{i}^{\\circ}(\\alpha)=\\sum_{k=1}^{i}\\left(\\breve{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\breve{\\boldsymbol{\\beta}}_{\\text{mean}}\\right),\\quad i\\in\\mathbb{N}.\\tag{223}\n$$"}, {"n": 2, "claim": "Recall that $\\breve{\\beta}_{\\mathrm{mean}}(\\alpha)=\\mathbb{E}[\\breve{\\beta}_{1}^{\\circ}(\\alpha)]$ as defined in (204). Due to the stationarity, we have $\\mathbb{E}[\\breve{\\beta}_{k}^{\\circ}(\\alpha)-\\breve{\\beta}]=0$ for all $k \\in \\mathbb N$ by (220). This gives\n\n$$\n\\left(\\mathbb{E}\\|\\breve{\\boldsymbol{\\beta}}_{\\text{mean}}(\\alpha)-\\breve{\\boldsymbol{\\beta}}\\|_{2}^{q}\\right)^{1/q}=\\left(\\mathbb{E}\\|\\mathbb{E}[\\breve{\\boldsymbol{\\beta}}_{1}^{\\circ}(\\alpha)-\\breve{\\boldsymbol{\\beta}}]\\|_{2}^{q}\\right)^{1/q}=0.\\tag{224}\n$$"}, {"n": 3, "claim": "Since we supposed in Theorem 17 that Assumption 1 holds for some $q > 2$ , it follows from Lemma 14 that, for all $q > 2$ 2\n\n$$\n\\left(\\mathbb{E}\\|\\breve{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\breve{\\boldsymbol{\\beta}}_{\\text{mean}}(\\alpha)\\|_{2}^{q}\\right)^{1/q}\\leq\\left(\\mathbb{E}\\|\\breve{\\boldsymbol{\\beta}}_{k}^{\\circ}(\\alpha)-\\breve{\\boldsymbol{\\beta}}\\|_{2}^{q}\\right)^{1/q}+\\left(\\mathbb{E}\\|\\breve{\\boldsymbol{\\beta}}_{\\text{mean}}(\\alpha)-\\breve{\\boldsymbol{\\beta}}\\|_{2}^{q}\\right)^{1/q}=O(\\sqrt{\\alpha}).\\tag{225}\n$$"}, {"n": 4, "claim": "Following a similar argument as for (144), we can show that $\\breve{\\beta}_{k}^{\\circ}(\\alpha)-\\breve{\\beta}_{\\mathrm{mean}}(\\alpha)$ satisfies condition (i) on the uniform integrability in Lemma 26. Due to the stationarity of $\\{\\breve{\\beta}_{k}^{\\circ}(\\alpha)\\}_{k\\in\\mathbb{N}}.$ condition (ii) in Lemma 26 is not required (see the discussion below Lemma 26 for details). Condition (iii) is also satisfied, since when the constant learning rate $\\alpha > 0$ satisfies condition (37), the stationary SGD dropout sequence $\\{\\breve{\\beta}_{k}^{\\circ}(\\alpha)\\}_{k\\in\\mathbb{N}}$ is shown to be short-range dependent by (215), i.e., the tail of cumulative dependence measure $\\begin{array}{ r }{\\breve{\\Theta}_{m,q}(\\alpha)=\\sum_{k=m}^{\\infty}\\breve{\\theta}_{k,q}(\\alpha)<}\\end{array}$ ∞."}, {"n": 5, "claim": "Hence, by Lemma 26, there exists a (richer) probability space $(\\breve{\\Omega}^{\\star},\\breve{A}^{\\star},\\breve{\\mathbb{P}}^{\\star})$ on which we can define random vectors $\\breve{\\beta}_{k}^{\\star}\\mathrm{s}$ with the partial sum process $\\begin{array}{ r }{\\breve{S}_{i}^{\\star}=\\sum_{k=1}^{i}(\\breve{\\beta}_{k}^{\\star}-\\breve{\\beta}_{\\mathrm{mean}})}\\end{array}$ , and a Gaussian process $\\begin{array}{ r }{\\breve{G}_{i}^{\\star}=\\sum_{k=1}^{i}\\breve{z}_{k}^{\\star}}\\end{array}$ , where $\\breve{z}_{k}^\\star\\}\\mathrm{s}$ are independent Gaussian random vectors in $\\mathbb{R}^{d}$ following $\\mathcal{N}(0,I_{d})$ , such that\n\n$$\n(\\breve{S}_{i}^{\\star})_{1\\leq i\\leq t}\\stackrel{{\\mathcal{D}}}{{=}}(\\breve{S}_{i}^{\\circ})_{1\\leq i\\leq t},\\tag{226}\n$$\n\nand\n\n$$\n\\max_{1\\leq i\\leq t}\\left\\|\\check{S}_{i}^{\\star}-\\check{\\Sigma}^{1/2}(\\alpha)\\check{G}_{i}^{\\star}\\right\\|_{2}=o_{\\mathbb{P}}(t^{1/q}),\\quad \\text{in}(\\check{\\Omega}^{\\star},\\check{\\mathcal{A}}^{\\star},\\check{\\mathbb{P}}^{\\star}),\\tag{227}\n$$\n\nwhere the long-run covariance matrix $\\breve{\\Sigma}(\\alpha)$ is defined in Theorem 15."}, {"n": 6, "claim": "Following similar arguments as for (148)–(150), we can leverage the geometric-moment contraction in Theorem 12 to show the same Gaussian approximation rate, i.e., $O_{\\mathbb{P}}(t^{1/q})$ , for the partial sum sequence $\\begin{array}{ r }{\\big(\\sum_{k=1}^{i}(\\breve{\\beta}_{k}(\\alpha)-\\breve{\\beta}_{\\mathrm{mean}}(\\alpha))\\big)_{1<i<t},}\\end{array}$ for any arbitrarily fixed initial vector $\\breve{\\beta}_{0}\\in\\mathbb{R}^{d}$ . Finally, recall the partial sum process $\\begin{array}{ r }{\\breve{S}_{i}^{\\breve{\\beta}_{0}}(\\alpha)=\\sum_{k=1}^{i}(\\breve{\\beta}_{k}(\\alpha)-\\breve{\\beta})}\\end{array}$ . By a similar argument in (151), the desired Gaussian approximation result for the partial sum process $(\\breve{S}_{i}^{\\breve{\\beta}_{0}}(\\alpha))_{1\\leq i\\leq t}$ follows. ■"}]}}], "target_local_id": null, "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2026-25-2383", "context": "Throughout, we assume $f(x; S)$ is convex in $x$ where $S$ is a random variable with sample space $\\mathcal{S}$ . For conciseness, we denote $f(x; S_i)$ by $f_i(x)$ where $S_i$ represents the $i$ th sample. Define $F(x) := \\mathbb{E}[f(x; S)]$ , $F^{\\star} := \\inf_{x \\in \\mathcal{X}} F(x) > -\\infty$ , $\\bar{F}(x) := \\frac{1}{n} \\sum_{i=1}^{n} f_i(x)$ . Let $\\mathcal{X}$ be a closed convex set with $\\mathcal{X} \\subseteq \\mathbb{R}^d$ . In most practical cases, $\\mathcal{X} = \\mathbb{R}^d$ . Let $\\mathcal{X}^{\\star} := \\arg \\min_{x \\in \\mathcal{X}} F(x)$ and $R^{\\star} := \\inf_{x \\in \\mathcal{X}^{\\star}} \\|x\\|$ . If $\\mathcal{X}^{\\star}$ is empty, then $R^{\\star} = \\infty$ . The dual norm is $\\|z\\|_* = \\sup\\{z \\cdot x : \\|x\\| \\leq 1\\}$ . Let $\\nabla h(x)$ be any subgradient of a function $h(x)$ and $\\nabla_j h(x)$ be the $j$ th coordinate of $\\nabla h(x)$ . Define $a \\wedge b := \\min\\{a, b\\}$ and $a \\vee b := \\max\\{a, b\\}$ . For a scalar $K$ , we let $[K] := \\{0, 1, \\ldots, K\\}$ and, with slight abuse of notation, for a vector $v$ we let $[v]_j$ be the $j$ th coordinate of $v$ . Let $\\mathbf{0}$ be a vector of zeros. We say the function $h(x)$ is $L$ -Lipschitz if and only if for all $x, x' \\in \\mathcal{X}$ , we have $|h(x) - h(x')| \\leq L \\|x - x' \\|$ . An equivalent definition is that $\\| \\nabla h(x)\\|_* \\leq L$ for all $x \\in \\mathcal{X}$ . We say the function $f$ is $L$ -Lipschitz if and only if $h(x) = f(x; S)$ is $L$ -Lipschitz almost surely. We say the function $h: \\mathcal{X} \\to \\mathbb{R}$ is $\\mu$ -strongly convex if and only if for all $u, v \\in \\mathcal{X}$ we have $h(u) - h(v) \\geq \\nabla h(v) \\cdot (u - v) + \\frac{\\mu}{2} \\|u - v\\|^2$ . We say the function $f$ is $\\mu$ -strongly convex if and only if $h(x) = f(x; S)$ is $\\mu$ -strongly convex almost surely. We assume there exists some reference model $x_{0}$ , a model the user identifies as a reasonable baseline. In the theory (Section 2 and Section 3), for simplicity we set the reference model to be the origin, i.e., $x_{0} = 0$ . Nonetheless, our results hold for an arbitrary reference model by shifting the origin.\n\nIn this section, we consider the case where there is a finite set of models $x_{0}, x_{1}, \\ldots, x_{K}$ . In this setting, our goal is to find the best model among these $K + 1$ models. We will also explore applications of these techniques to selecting hyperparameters for stochastic optimization methods.\n\nSection 2.1 studies the performance of standard model selection which chooses the model that minimizes the validation error. In particular, we show that it can effectively tune hyperparameters when strong convexity is present but otherwise may perform poorly compared to parameter-free optimization methods. Section 2.2 studies the performance of a new algorithm called RELIABLEMODELSELECTION, which obtains good worst-case performance both with and without strong convexity.\n\nConversely, without strong convexity, standard model selection can perform poorly at tuning learning rates. For a concrete example, consider adaptive SGD (also known as isotropic AdaGrad (Gupta et al., 2017)), which is a scalar variant of ADAGRAD (Duchi et al., 2011). While we chose adaptive SGD as an example, similar issues occur for tuning other standard stochastic optimization methods such as regularized ERM or SGD with a fixed step size. The formula for adaptive SGD is\n\n$$\nu_{t+1}\\leftarrow\\mathbf{proj}_{R}\\left(u_{t}-R\\frac{\\boldsymbol{\\nabla}f_{n+t}\\left(u_{t}\\right)}{\\sqrt{\\sum_{j=1}^{t}\\left\\|\\boldsymbol{\\nabla}f_{n+j}\\left(u_{j}\\right)\\right\\|_{2}^{2}}}\\right),\\bar{u}_{t}\\leftarrow\\frac{1}{t}\\sum_{j=1}^{t}u_{j},\\text{for} t=1,\\dots,n \\tag{4}\n$$\n\nwhere $u_1 = \\mathbf{0}$ , $\\text{proj}_R(\\hat{u}) := \\arg\\min_{u \\in \\mathcal{X}: \\|u\\|_2 \\leq R} \\|u - \\hat{u}\\|_2$ and $R > 0$ is the learning rate which represents our estimated distance to optimality. We add the constraint $\\|u\\|_2 \\leq R$ to ensure the domain is bounded proportionally to the learning rate. To clearly separate the validation set and training set, Equation (4) uses gradient $\\nabla f_{n+t}(u_t)$ instead of $\\nabla f_t(u_t)$ . Thus, we can think of $f_1, \\ldots, f_n$ as the validation set and $f_{n+1}, \\ldots, f_{2n}$ as the training set. Denote ADASGD(R) = $\\bar{u}_n$ . It is well-known, e.g., Orabona (2019, Theorem 3.9 & 4.14), that assuming $f$ is $L$ -Lipschitz, adaptive SGD starting from $u_1 = \\mathbf{0}$ , with probability $1 - \\delta$ obtains\n\n$$\nF(\\mathrm{ADASGD}(R))-\\min_{x\\in\\mathcal{X}:\\|x\\|_{2}\\leq R}F(x)\\leq O\\left(\\frac{L R}{\\sqrt{n}}\\sqrt{\\ln 2/\\delta}\\right)\\tag{5}\n$$\n\nSetting $R = R^{\\star}$ yields the optimal $O(LR^{\\star}\\sqrt{\\ln 2 / \\delta} /\\sqrt{n})$ suboptimality guarantee. However, this approach relies on $R^{\\star}$ being known.\n\nNext, we consider how to reliably select high-quality models and avoid the issues that plagued standard model selection for tuning learning rates in Section 2.1. Our method, described in Algorithm 1, requires a list of confidence interval widths $\\tau_{1},\\ldots,\\tau_{K}$ that satisfy Condition 1. The algorithm identifies a set of safe models $\\mathcal{F}$ over which we minimize the validation error. The algorithmic parameter $\\gamma \\in [1,\\infty)$ controls the size of $\\mathcal{F}$ , with larger values of $\\gamma$ behaving more like standard model selection. When $\\gamma = 1$ , the algorithm reduces to minimizing the upper confidence bound on the function value, but this can be overly conservative, heavily favoring models that make predictions similar to the reference model.\n\nCondition 1 Let $\\tau_1, \\ldots, \\tau_K$ be nonnegative scalars such that for some $\\delta \\in (0,1)$ , $\\mathbb{P}(\\exists k \\in \\{1, \\ldots, K\\}) : |F(x_k) - \\bar{F}(x_k) - (F(\\mathbf{0}) - \\bar{F}(\\mathbf{0}))| > \\tau_k) \\leq \\delta$ .\n\nCondition 1 supposes that we have confidence intervals on the closeness of our validation loss to the population loss relative to the reference model, $x_0 = \\mathbf{0}$ . Depending on the problem at hand, there are multiple ways to generate such confidence intervals. For example, if $f$ is $L$ -Lipschitz and some upper bound $\\hat{L} \\geq L$ is known, then a result of Maurer and Pontil (2009) (Theorem 14 in Appendix A) with $Z_i = f_i(x_k) - f_i(\\mathbf{0})$ gives that with probability at most $\\delta_k \\in (0,1)$ , $|F(x_k) - F(\\mathbf{0}) - (\\bar{F}(x_k) - \\bar{F}(\\mathbf{0}))| > \\sqrt{\\frac{2\\mathbb{V}_k}{n}\\ln\\frac{4}{\\delta_k}} + \\frac{14\\hat{L}\\|x_k\\|}{3(n-1)}\\ln\\frac{4}{\\delta_k}$ where\n\n$$\n\\mathbb{V}_{k}:=\\frac{1}{n-1}\\sum_{i=1}^{n}(f_{i}(x_{k})-f_{i}(\\mathbf{0})-(\\bar{F}(x_{k})-\\bar{F}(\\mathbf{0})))^{2}\n$$\n\nis the sample variance of the paired differences $f_{i}(x_{k}) - f_{i}(\\mathbf{0})$ . Setting $\\delta_{k} = \\frac{\\delta}{K}$ and applying a union bound shows Condition 1 holds with\n\n$$\n\\tau_{k}=\\sqrt{\\frac{2\\tilde{c}\\mathbb{V}_{k}}{n}}+\\tilde{c}\\frac{14\\hat{L}\\|x_{k}\\|}{3(n-1)}\\text{where}\\tilde{c}:=\\ln\\frac{4K}{\\delta}.\\tag{6}\n$$\n\nBlack dots show the validation errors of candidate models $x_{0}$ to $x_{4}$ . Orange and red intervals depict $\\tau_{k}$ and $\\gamma\\tau_{k}$ , respectively, which are smaller for models with predictions more similar to the reference model, $x_{0}$ . The threshold $\\theta$ (red dashed line) is the smallest extended upper bound $\\bar{F}(x_{k}) + \\gamma\\tau_{k}$ . Models with $\\bar{F}(x_{k}) + \\tau_{k} \\leq \\theta$ form the candidate set F (in this example $F = \\{1, 2\\}$ ). Among these, $x_{2}$ has the lowest error and thus $k_{rely} = 2$ . Standard model selection chooses the lowest validation error across all models, $k_{std} = 4$ .\n\ninput A scalar $\\gamma \\in [1,\\infty)$\n\ninput Candidate solutions $x_0, \\ldots, x_K$\n\ninput Sample functions $f_{1},\\ldots,f_{n}$\n\ninput $\\tau_{1},\\ldots,\\tau_{K}$ satisfying Condition 1.\n\n1: $\\tau_{0} := 0$  \n2: $\\theta \\leftarrow \\min_{k\\in [K]}\\bar{F} (x_k) + \\gamma \\tau_k$  \n3: $\\mathcal{F} \\leftarrow \\{k \\in [K] : \\bar{F}(x_k) + \\tau_k \\leq \\theta\\}$  \n4: $k_{\\mathrm{rely}} := \\arg \\min_{k \\in \\mathcal{F}} \\bar{F}(x_k)$\n\noutput The chosen index: $k_{\\mathrm{rely}}$ .\n\nTheorem 4 Suppose that $f$ is $L$ -Lipschitz and $\\tau_{k}$ is defined as per Equation (6). Let $\\delta \\in (0,1)$ and $\\tilde{c} := \\ln \\frac{4K}{\\delta}$ . Then, with probability at least $1 - \\delta$ ,\n\n$$\nF(x_{k_{\\mathrm{rely}}})\\leq\\min_{k\\in [K]}F(x_{k})+(1+\\gamma)\\left(L\\sqrt{\\frac{2\\tilde{c}}{n-1}}+\\frac{14\\hat{L}\\tilde{c}}{3(n-1)}\\right)\\|x_{k}\\|.\\tag{7}\n$$\n\nAdditionally, if $F$ is $\\mu$ -strongly convex and $\\gamma \\geq 3$ , then, with probability at least $1 - 2\\delta$ ,\n\n$$\nF(x_{k_{\\mathrm{rely}}})-F^{\\star}\\leq 4\\Big(\\min_{k\\in [K]}F(x_{k})-F^{\\star}\\Big)+\\frac{128}{\\mu}\\left(2L\\sqrt{\\frac{2\\tilde{c}}{n-1}}+\\frac{14\\hat{L}\\tilde{c}}{3(n-1)}\\right)^{2}.\\tag{8}\n$$\n\nWe can use Equation (7) to generically obtain sample complexity bounds for parameter-free stochastic optimization. For example, Corollary 5 shows how to obtain such bounds by applying RELIABLEMODELSELECTION to tune the learning rate for adaptive SGD. To interpret Corollary 5, recall that $f_1, \\ldots, f_{2n}$ are i.i.d. samples where $f_1, \\ldots, f_n$ is the validation set (i.e., used by RELIABLEMODELSELECTION) and $f_{n+1}, \\ldots, f_{2n}$ is the training set, i.e., used by adaptive SGD, see Equation (4).\n\nThis section develops a method with perfect adaptivity to unknown distance to optimality, matching optimal known-parameter high-probability bounds. Our results hold for p-norms with three values of p: 1, 2, and $\\infty$ . While our method requires an estimate of the Lipschitz constant (Assumption 1), we show that for a sample of size n, an estimate within $\\tilde{O}(\\sqrt{n})$ of the true Lipschitz constant suffices to match optimal known-parameter high-probability bounds.\n\nAssumption 1 (Upper bound on Lipschitz constants) Suppose that $f$ is differentiable, convex in $x$ , and satisfies $\\| \\nabla f(x; S) \\|_2 \\leq L \\leq \\hat{L}$ and $|\\nabla_j f(x; S)| \\leq \\mathbf{L}_j \\leq \\hat{\\mathbf{L}}_j$ almost surely for all $j \\leq d$ and $x \\in \\mathcal{X}$ , where $\\hat{L}$ and $\\hat{\\mathbf{L}}$ are known.\n\nAssumption 1 treats $f$ as differentiable only for the sake of simplicity: our results extend to nondifferentiable functions via a smoothing argument, e.g., using the Moreau envelope (Moreau, 1965). Our optimal adaptive method consists of two stages. First, using half our sample, we apply regularized empirical risk minimization (ERM) to find a radius $R$ such that, with high probability, the minimum of $F$ in a ball of radius $R$ is close to its global minimum, and also $R = O(R^{\\star})$ . Second, using the other half of our sample, we apply an off-the-shelf optimal known-parameter stochastic optimization method to approximately minimize $F$ in a ball of radius $R$ . Since $R$ is given to the method, it need not be adaptive to the distance to optimality. Our approach is summarized in Algorithm 2.\n\nSTAGE 1: LOCALIZATION VIA ERM. The key component of our approach is (non-squared) norm regularization. For a given regularization parameter $\\lambda_{n}$ , consider the following regularized ERM and regularized population risk minimizers:\n\n$$\n\\hat{x}_{\\lambda_{n}}\\in\\underset{x\\in\\mathcal{X}}{\\arg\\min}\\left\\{\\bar{F}(x)+\\lambda_{n}\\|x\\|\\right\\}\\text{and} x_{\\lambda_{n}}^{\\star}\\in\\underset{x\\in\\mathcal{X}}{\\arg\\min}\\left\\{F(x)+\\lambda_{n}\\|x\\|\\right\\}.\n$$\n\nTo obtain our results, it will suffice to choose $\\lambda_{n}$ such that the following condition holds.\n\nCondition 2 Let $f$ be differentiable. Suppose $\\delta \\in (0,1)$ and $\\lambda_n > 0$ are (known) constants such that $\\mathbb{P}\\big(\\big\\| \\nabla \\bar{F}(x_{\\zeta}^{\\star}) - \\nabla F(x_{\\zeta}^{\\star})\\big\\|_{*} > \\lambda_{n} / 2\\big) \\leq \\delta$ for $\\zeta \\in \\{\\frac{\\lambda_n}{3}, 3\\lambda_n\\}$ .", "theorems": [{"local_id": "T1", "statement": "Lemma 3 With probability at least $1 - \\delta$ , we have $F(x_{k_{\\mathrm{rely}}}) \\leq F(x_k) + (1 + \\gamma)\\tau_k$ for all $k \\in [K]$ .", "proof": {"steps": [{"n": 1, "claim": "Proof For all $k \\in [K]$ we have\n\n$$\n\\begin{array}{l}0\\stackrel{(i)}{\\leq}\\theta-\\bar{F}(x_{k_{\\mathrm{rely}}})-\\tau_{k_{\\mathrm{rely}}}\\stackrel{(i i)}{\\leq}\\bar{F}(x_{k})+\\gamma\\tau_{k}-\\bar{F}(x_{k_{\\mathrm{rely}}})-\\tau_{k_{\\mathrm{rely}}}\\\\ =\\gamma\\tau_{k}+\\bar{F}(x_{k})-\\bar{F}(\\mathbf{0})+\\bar{F}(\\mathbf{0})-\\bar{F}(x_{k_{\\text{rely}}})-\\tau_{k_{\\text{rely}}}\\stackrel{(i i i)}{\\leq}(1+\\gamma)\\tau_{k}+F(x_{k})-F(x_{k_{\\text{rely}}})\\\\ \\end{array}\n$$\n\ndue to (i) $k_{rely} \\in F$ , (ii) the definition of $\\theta$ , and (iii) Condition 1."}]}}, {"local_id": "T2", "statement": "Corollary 5 Suppose $f$ is $L$ -Lipschitz with known upper bound $\\hat{L} \\geq L$ , and that $R^{\\star} \\in [1, \\rho]$ for some known $\\rho > 1$ . For $k = 1, 2, \\ldots, \\lceil \\ln \\rho \\rceil$ , let $x_k = \\mathrm{ADASGD}(e^k)$ be the output of adaptive SGD, i.e., Equation (4), with learning rate $R_k = e^k$ . Then, for any $\\delta \\in (0, 1)$ , Algorithm 1 applied to $x_1, \\ldots, x_K$ with $K = \\lceil \\ln \\rho \\rceil$ and $\\tau_1, \\ldots, \\tau_K$ defined in Equation (6) achieves, with probability at least $1 - 2\\delta$ ,\n\n$$\nF(x_{k_{\\mathrm{rely}}})-F^{\\star}\\leq O\\left(\\sqrt{\\ln\\frac{\\lceil\\ln\\rho\\rceil}{\\delta}}\\cdot\\frac{L R^{\\star}}{\\sqrt{n}}+\\ln\\frac{\\lceil\\ln\\rho\\rceil}{\\delta}\\cdot\\frac{\\hat{L}R^{\\star}}{n}\\right).\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof Since the learning rates $R_{k}=e^{k}$ form a geometric grid over $[1,e^{\\lceil\\ln\\rho\\rceil}]$ , there exists some $k^{*}\\in\\{1,\\ldots,\\lceil\\ln\\rho\\rceil\\}$ with $R^{\\star}\\leq R_{k^{*}}\\leq e\\cdot R^{\\star}$ . By Equation (5), for this $k^{*}$ , with probability $1-\\delta$ ,\n\n$$\nF(x_{k^{*}})-F^{\\star}\\leq O\\left(\\frac{L R_{k^{*}}\\sqrt{\\ln(2/\\delta)}}{\\sqrt{n}}\\right)=O\\left(\\frac{L R^{\\star}\\sqrt{\\ln(2/\\delta)}}{\\sqrt{n}}\\right).\n$$"}, {"n": 2, "claim": "Applying Equation (7) of Theorem 4 with $k = k^{*}$ and $\\| x_{k^*}\\| \\leq R_{k^*} \\leq e \\cdot R^{\\star}$ , and then union-bounding gives that with probability at least $1 - 2\\delta$\n\n$$\nF(x_{k_{\\mathrm{rely}}})-F^{\\star}\\leq F(x_{k^{*}})-F^{\\star}+(1+\\gamma)\\left(\\sqrt{2\\tilde{c}}\\frac{L\\|x_{k^{*}}\\|}{\\sqrt{n-1}}+\\frac{14\\tilde{c}}{3}\\cdot\\frac{\\hat{L}\\|x_{k^{*}}\\|}{n-1}\\right)\\leq O\\left(\\sqrt{\\tilde{c}}\\frac{LR^{\\star}}{\\sqrt{n}}+\\tilde{c}\\frac{\\hat{L}R^{\\star}}{n-1}\\right).\n$$"}, {"n": 3, "claim": "Since our number of candidates $K = \\lceil \\ln \\rho \\rceil$ , we have $\\tilde{c} = \\ln \\frac{4\\lceil \\ln \\rho \\rceil}{\\delta}$ ."}]}}, {"local_id": "T3", "statement": "Lemma 6 If Condition 2 holds then $\\mathbb{P}(\\| x_{3\\lambda_n}^\\star \\| \\leq 2\\|\\hat{x}_{\\lambda_n}\\| \\leq 14\\| x_{\\lambda_n / 3}^\\star \\|)\\geq 1 - 2\\delta$ .", "proof": {"steps": [{"n": 1, "claim": "Proof The remainder of this proof will assume that\n\n$$\n\\left\\|\\boldsymbol{\\nabla}\\bar{F}(x_{\\lambda_{n}/3}^{\\star})-\\boldsymbol{\\nabla}F(x_{\\lambda_{n}/3}^{\\star})\\right\\|_{*}\\leq\\lambda_{n}/2 \\tag{17a}\n$$\n\n$$\n\\left\\|\\boldsymbol{\\nabla}\\bar{F}(x_{3\\lambda_{n}}^{\\star})-\\boldsymbol{\\nabla}F(x_{3\\lambda_{n}}^{\\star})\\right\\|_{*}\\leq\\lambda_{n}/2 \\tag{17b}\n$$\n\nand under this assumption we will show that $\\|x_{3\\lambda_{n}}^{\\star}\\|\\leq2\\|\\hat{x}_{\\lambda_{n}}\\|\\leq14\\|x_{\\lambda_{n}/3}^{\\star}\\|$ . Proving this implication suffices to prove the lemma because Condition 2 and a union bound implies Equation (17) holds with probability $1-2\\delta$ ."}, {"n": 2, "claim": "We will find the following well-known fact useful (Oden and Kikuchi, 1980, Theorem 1.5.1, Section 1.5): if $H$ is convex and differentiable, $\\alpha > 0$ , and $z \\in \\arg \\min_{x \\in \\mathcal{X}} \\{H(x) + \\alpha \\| x\\| \\}$ , then for any $y \\in \\mathcal{X}$ ,\n\n$$\n\\nabla H(z)\\cdot(y-z)\\geq\\alpha(\\|z\\|-\\|y\\|).\\tag{18}\n$$"}, {"n": 3, "claim": "First, we will show that Equation (17a) implies\n\n$$\n\\|\\hat{x}_{\\lambda_{n}}\\|\\leq 7\\|x_{\\lambda_{n}/3}^{\\star}\\|.\\tag{19}\n$$"}, {"n": 4, "claim": "Let $q := \\nabla \\bar{F}(x_{\\lambda_n/3}^\\star) - \\nabla F(x_{\\lambda_n/3}^\\star)$ . By (i) definition of $\\hat{x}_{\\lambda_n}$ , (ii) convexity of $\\bar{F}$ , (iii) Equation (18) applied with $H = F$ , $z = x_{\\lambda_n/3}^\\star$ , $y = \\hat{x}_{\\lambda_n}$ , and $\\alpha = \\lambda_n/3$ , and (iv) $\\|q\\|_* \\leq \\lambda_n/2$ from Equation (17a), Hölder's inequality, and the triangle inequality, we have\n\n$$\n\\begin{array}{l}\\bar{F}(x_{\\lambda_{n}/3}^{\\star})+\\lambda_{n}\\|x_{\\lambda_{n}/3}^{\\star}\\|\\underset{(i)}{\\geq}\\bar{F}(\\hat{x}_{\\lambda_{n}})+\\lambda_{n}\\|\\hat{x}_{\\lambda_{n}}\\|\\\\ \\underset{(i i)}{\\geq}\\bar{F}(x_{\\lambda_{n}/3}^{\\star})+\\nabla\\bar{F}(x_{\\lambda_{n}/3}^{\\star})\\cdot(\\hat{x}_{\\lambda_{n}}-x_{\\lambda_{n}/3}^{\\star})+\\lambda_{n}\\|\\hat{x}_{\\lambda_{n}}\\|\\\\ =\\bar{F}(x_{\\lambda_{n}/3}^{\\star})+\\nabla F(x_{\\lambda_{n}/3}^{\\star})\\cdot(\\hat{x}_{\\lambda_{n}}-x_{\\lambda_{n}/3}^{\\star})+q\\cdot(\\hat{x}_{\\lambda_{n}}-x_{\\lambda_{n}/3}^{\\star})+\\lambda_{n}\\|\\hat{x}_{\\lambda_{n}}\\|\\\\ \\underset{(i i i)}{\\geq}\\bar{F}(x_{\\lambda_{n}/3}^{\\star})+\\frac{\\lambda_{n}}{3}\\left(\\|x_{\\lambda_{n}/3}^{\\star}\\|-\\|\\hat{x}_{\\lambda_{n}}\\|\\right)+q\\cdot(\\hat{x}_{\\lambda_{n}}-x_{\\lambda_{n}/3}^{\\star})+\\lambda_{n}\\|\\hat{x}_{\\lambda_{n}}\\|\\left.\\right.\\\\ \\underset{(i v)}{\\geq}\\bar{F}(x_{\\lambda_{n}/3}^{\\star})+\\frac{\\lambda_{n}}{3}(\\|x_{\\lambda_{n}/3}^{\\star}\\|-\\|\\hat{x}_{\\lambda_{n}}\\|)-\\frac{\\lambda_{n}}{2}(\\|\\hat{x}_{\\lambda_{n}}\\|+\\|x_{\\lambda_{n}/3}^{\\star}\\|)+\\lambda_{n}\\|\\hat{x}_{\\lambda_{n}}\\|.\\\\ \\end{array}\n$$"}, {"n": 5, "claim": "Rearranging gives $\\|\\hat{x}_{\\lambda_n}\\|\\leq 7\\|x_{\\lambda_n/3}^\\star\\|,$ which implies Equation (19)."}, {"n": 6, "claim": "The remainder of the proof will show that Equation (17b) implies\n\n$$\n\\|x_{3\\lambda_{n}}^{\\star}\\|\\leq 2\\|\\hat{x}_{\\lambda_{n}}\\|.\\tag{20}\n$$"}, {"n": 7, "claim": "Let $s := \\nabla \\bar{F}(x_{3\\lambda_n}^\\star) - \\nabla F(x_{3\\lambda_n}^\\star)$ . By (i) convexity of $\\bar{F}$ , (ii) Equation (18) applied with $H = F$ , $z = x_{3\\lambda_n}^\\star$ , $y = \\hat{x}_{\\lambda_n}$ , and $\\alpha = 3\\lambda_n$ , and (iii) convexity of $\\bar{F}$ , we get\n\n$$\n\\begin{array}{l}\\bar{F}(\\hat{x}_{\\lambda_{n}})+3\\lambda_{n}\\|\\hat{x}_{\\lambda_{n}}\\|\\underset{(i)}{\\geq}\\bar{F}(x_{3\\lambda_{n}}^{\\star})+\\boldsymbol{\\nabla}\\bar{F}(x_{3\\lambda_{n}}^{\\star})\\cdot(\\hat{x}_{\\lambda_{n}}-x_{3\\lambda_{n}}^{\\star})+3\\lambda_{n}\\|\\hat{x}_{\\lambda_{n}}\\|\\\\ =\\bar{F}(x_{3\\lambda_{n}}^{\\star})+\\nabla F(x_{3\\lambda_{n}}^{\\star})\\cdot(\\hat{x}_{\\lambda_{n}}-x_{3\\lambda_{n}}^{\\star})+s\\cdot(\\hat{x}_{\\lambda_{n}}-x_{3\\lambda_{n}}^{\\star})+3\\lambda_{n}\\|\\hat{x}_{\\lambda_{n}}\\|\\\\ \\geq_{(i i)}\\bar{F}(x_{3\\lambda_{n}}^{\\star})+3\\lambda_{n}\\|x_{3\\lambda_{n}}^{\\star}\\|+s\\cdot(\\hat{x}_{\\lambda_{n}}-x_{3\\lambda_{n}}^{\\star})\\\\ \\underset{(i i i)}{\\geq}\\bar{F}(\\hat{x}_{\\lambda_{n}})+(\\nabla\\bar{F}(\\hat{x}_{\\lambda_{n}})-s)\\cdot(x_{3\\lambda_{n}}^{\\star}-\\hat{x}_{\\lambda_{n}})+3\\lambda_{n}\\|x_{3\\lambda_{n}}^{\\star}\\|\\\\ \\end{array}\n$$"}, {"n": 8, "claim": "Next, by (i) rearranging the previous display, (ii) Equation (18) applied with $H = \\bar{F}$ , $z = \\hat{x}_{\\lambda_n}$ , $y = x_{3\\lambda_n}^\\star$ , and $\\alpha = \\lambda_n$ , and (iii) Hölder's inequality, the triangle inequality, and $\\| s \\|_* \\leq \\lambda_n / 2$ by Equation (17b), we get\n\n$$\n\\begin{array}{l}\\|\\hat{x}_{\\lambda_{n}}\\|\\geq_{(i)}\\|x_{3\\lambda_{n}}^{\\star}\\|+\\frac{(\\nabla\\bar{F}(\\hat{x}_{\\lambda_{n}})-s)\\cdot(x_{3\\lambda_{n}}^{\\star}-\\hat{x}_{\\lambda_{n}})}{3\\lambda_{n}}\\\\ \\geq_{(i i)}\\|x_{3\\lambda_{n}}^{\\star}\\|+\\frac{\\|\\hat{x}_{\\lambda_{n}}\\|-\\|x_{3\\lambda_{n}}^{\\star}\\|}{3}-\\frac{s\\cdot(x_{3\\lambda_{n}}^{\\star}-\\hat{x}_{\\lambda_{n}})}{3\\lambda_{n}}\\\\ \\underset{(i i i)}{\\geq}\\|x_{3\\lambda_{n}}^{\\star}\\|+\\frac{\\|\\hat{x}_{\\lambda_{n}}\\|-\\|x_{3\\lambda_{n}}^{\\star}\\|}{3}-\\frac{\\|\\hat{x}_{\\lambda_{n}}\\|+\\|x_{3\\lambda_{n}}^{\\star}\\|}{6}.\\\\ \\end{array}\n$$"}, {"n": 9, "claim": "Rearranging yields $\\|x_{3\\lambda_{n}}^{\\star}\\|\\leq\\frac{5}{3}\\|\\hat{x}_{\\lambda_{n}}\\|\\leq2\\|\\hat{x}_{\\lambda_{n}}\\|$ , which proves Equation (20). Combining this with Equation (19) proves the claim."}]}}], "target_local_id": null, "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2025-22-1262", "context": "Let $(\\mathcal{X},d)$ be a finite metric space with finite cardinality $M: = |{\\mathcal{X}}|$ and denote by $\\mathcal{M}_{+}(\\mathcal{X}): =$ $\\{\\mu\\in\\mathbb{R}^{M}\\mid\\mu(x)\\geq 0\\forall x\\in\\mathcal{X}\\}$ the set of non-negative measures<sup>1</sup> on $\\mathcal{X}$ . The total mass of a measure $\\mu\\in\\mathcal{M}_{+}(\\mathcal{X})$ is defined as $\\begin{array}{ r }{\\mathbb{M}(\\mu): =\\sum_{x\\in\\mathcal{X}}\\mu(x)}\\end{array}$ and the subset $\\mathcal{P}(\\mathcal{X})\\subset\\mathcal{M}_{+}(\\mathcal{X})$\n\nof measures with total mass one is the set of probability measures. If $\\pi\\in\\mathcal{M}_{+}(\\mathcal{X}\\times\\mathcal{X})$ is a measure on the product space $\\mathcal{X}\\times\\mathcal{X}$ its marginals are defined as $\\textstyle\\pi(x,\\mathcal{X}): =\\sum_{x^{\\prime}}\\pi(x,x^{\\prime})$ and $\\begin{array}{ r }{\\pi(\\boldsymbol{\\mathcal{X}},\\boldsymbol{x}^{\\prime}): =\\sum_{\\boldsymbol{x}^{\\prime}\\in\\mathcal{X}}\\pi(\\boldsymbol{x},\\boldsymbol{x}^{\\prime})}\\end{array}$ , respectively. For two measures $\\mu,\\nu\\in\\mathcal{M}_{+}(\\mathcal{X})$ define the set of non-negative sub-couplings as\n\n$$\n\\Pi_{\\leq}(\\mu,\\nu):=\\{\\pi\\in\\mathcal{M}_{+}(\\mathcal{X}\\times\\mathcal{X})\\mid\\pi(x,\\mathcal{X})\\leq\\mu(x),\\pi(\\mathcal{X},x^{\\prime})\\leq\\nu(x^{\\prime})\\forall x,x^{\\prime}\\in\\mathcal{X}\\}.\\tag{1}\n$$\n\nFollowing Heinemann et al. (2023), for $p \\geq 1$ and a parameter $C > 0$ , the $(p,C){-}Kanto-$ rovich-Rubinstein distance (KRD) between two measures $\\mu,\\nu\\in\\mathcal{M}_{+}(\\mathcal{X})$ is defined as\n\n$$\n\\mathrm{KR}_{p,C}(\\mu,\\nu):=\\left(\\min_{\\pi\\in\\Pi_{\\leq}(\\mu,\\nu)}\\sum_{x,x^{\\prime}\\in\\mathcal{X}}d^{p}(x,x^{\\prime})\\pi(x,x^{\\prime})+C^{p}\\left(\\frac{\\mathbb{M}(\\mu)+\\mathbb{M}(\\nu)}{2}-\\mathbb{M}(\\pi)\\right)\\right)^{\\frac{1}{p}}.\\tag{2}\n$$\n\nFor $p \\geq 1$ , it defines a distance on $\\mathcal{M}_{+}(\\mathcal{X})$ robust to spatial outliers<sup>2</sup> and naturally extends the well-known $p { - } t h$ order OT distance defined for measures of equal total mass (Heinemann et al., 2023). Notably, by compactness of $\\Pi_{\\le}(\\mu,\\nu)$ and continuity of the objective, an optimizer $\\pi\\in\\Pi_{\\leq}(\\mu,\\nu)$ of (2) always exists and is called unbalanced optimal transport plan (UOT plan).\n\nFor the Poisson intensity model we fix a parameter $t > 0$ and a success probability $s\\in(0,1]$ Consider a collection of $|\\mathcal{X}|$ independent Poisson random variables $P_{x}{\\sim}\\mathrm{Poi}(t\\mu(x))$ with intensity $t\\mu(x)$ at each location $x\\in\\mathcal{X}$ and independently of them Bernoulli random variables $B_{x}\\sim\\mathrm{Ber}(s)$ for each $x \\in X$ . A suitable unbiased estimator for $\\mu$ is defined by\n\n$$\n\\widehat{\\mu}_{t,s}:=\\frac{1}{s t}\\sum_{x\\in\\mathcal{X}}B_{x}P_{x}\\delta_{x}.\\tag{8}\n$$\n\nIndeed, by reverse triangle inequality and our stability bound in Section 2, Lemma 3 it holds\n\n$$\n\\begin{array}{l}\\mathbb{E}\\left[|\\mathrm{KR}_{p,C}(\\widehat{\\mu},\\widehat{\\nu})-\\mathrm{KR}_{p,C}(\\mu,\\nu)|\\right]\\leq\\min\\Big(\\mathbb{E}\\left[\\mathrm{KR}_{p,C}(\\widehat{\\mu},\\mu)\\right]+\\mathbb{E}\\left[\\mathrm{KR}_{p,C}(\\widehat{\\nu},\\nu)\\right],\\\\ \\qquad 2C^{p}\\mathrm{KR}_{p,C}^{1-p}(\\mu,\\nu)\\left(\\mathbb{E}\\left[\\mathrm{TV}(\\widehat{\\mu},\\mu)\\right]+\\mathbb{E}\\left[\\mathrm{TV}(\\widehat{\\nu},\\nu)\\right]\\right)\\Big),\\end{array}\\tag{10}\n$$\n\nwhere $\\begin{array}{ r }{\\mathrm{TV}(\\mu,\\nu)=\\sum_{x\\in\\mathcal{X}}\\lvert\\mu(x)-\\nu(x)\\rvert}\\end{array}$ is the total variation distance. This asserts that the $\\mathrm{KR}_{p,C}(\\mu,\\nu)$ is well-approximated by $\\mathrm{KR}_{p,C}(\\widehat{\\mu},\\widehat{\\nu})$ as soon as the empirical estimators $\\widehat{\\mu}$ and νb approximate the population measures well (Corollary 7 and Theorem 8).\n\nIn this section we investigate the Poisson model and analyze how fast the empirical measure estimator tends to its population counterpart in terms of the $(p,C)$ -Kantorovich-Rubinstein distance and the total variation norm. These bounds allow us to quantify how fast the plug-in estimator tends to the population $(p,C)$ -Kantorovich-Rubinstein distance. Results for the multinomial and Bernoulli model follow along the same reasoning. Corresponding deviation bounds and proofs are provided in Appendices A and B.\n\nThe convergence results of this section are based on novel stability bounds for the UOT distance, which we detail in the following. The first is based on a tree approximation of the space X (Lemma 1), whereas the second relies on the dual formulation of the UOT cost (Lemma 3). The proofs for this section are detailed in Appendix D.1.\n\nLet $\\tau=(V,E)$ be a rooted, ultrametric tree with height function $h:V\\longrightarrow\\mathbb{R}_{+}$ and root r. For two nodes $\\mathsf{u},\\mathsf{v}\\in V$ , denote the unique path between u and v in T by $\\mathcal{P}(\\mathsf{u},\\mathsf{v})$ . For a node $\\textsf{ v }\\in\\ V$ its children are the elements of the set ${\\mathcal{C}}({\\boldsymbol{\\mathsf{v}}})=\\{{\\boldsymbol{\\mathsf{w}}}\\in V~|~{\\boldsymbol{\\mathsf{v}}}\\in{\\mathcal{P}}({\\boldsymbol{\\mathsf{w}}},{\\boldsymbol{\\mathsf{r}}})\\}$ The parent par(v) of a node v is the unique node with $(p a r(\\mathsf{v}),\\mathsf{v})\\in E$ and $h(\\mathsf{v})<h(p a r(\\mathsf{v}))$ ). For any $C > 0$ , define the set\n\n$$\n\\mathcal{R}(C):=\\{\\mathrm{v}\\in V\\mid h(\\mathrm{v})\\leq C/2<h(\\operatorname{par}(\\mathrm{v}))\\}\\tag{12}\n$$\n\nwith the convention that $\\mathcal{R}(C)=\\{\\mathfrak{r}\\}\\ \\mathrm{if}\\ \\frac{C}{2}\\geq h(\\mathfrak{r})$ . The goal is to control the $(p,C)\\mathrm{-K R D}$ on the finite metric space $(\\mathcal{X},d)$ by bounding it from above by a dominating distance $d_{T}$ induced<sup>7</sup> from a tree T with the elements of X as vertices and a height function h such that $d(x,x^{\\prime})\\leq d\\tau(x,x^{\\prime})$ . In this case and by the definition of the Kantorovich-Rubinstein distance it holds for all measures $\\mu,\\nu\\in\\mathcal{M}_{+}(\\mathcal{X})$ that\n\n$$\n\\mathrm{KR}_{p,C}(\\mu,\\nu)\\leq\\mathrm{KR}_{d_{\\mathcal{T}}^{p},C}(\\mu,\\nu),\\tag{13}\n$$\n\nwhere $\\mathrm{KR}_{d_{T}^{p},C}(\\mu,\\nu)$ denotes the $(p,C)\\mathrm{-K R D}$ w.r.t. the ground space $(\\mathcal{X},d\\tau)$ . Moreover, if $\\tau$ is an ultrametric tree with leaf nodes L and height function $h\\colon V\\to\\mathbb{R}_{+}$ inducing<sup>7</sup> the tree metric $d_{T}$ and the two measures $\\mu^{L},\\nu^{L}\\in\\mathcal{M}_{+}(L)$ supported on the leaf nodes of $\\tau$ then it holds (Heinemann et al., 2023, Theorem 2.3) that\n\n$$\n\\begin{array}{l}\\mathrm{KR}_{d_{\\mathcal{T},C}^{p}}^{p}\\left(\\mu^{L},\\nu^{L}\\right)=\\\\ \\sum_{\\mathrm{v}\\in\\mathcal{R}(\\mathcal{C})}\\left(2^{p-1}\\sum_{\\mathrm{w}\\in\\mathcal{C}(\\mathrm{v})\\backslash\\{\\mathrm{v}\\}}\\left((h(\\operatorname{par}(\\mathrm{w}))^{p}-h(\\mathrm{w})^{p})\\left|\\mu^{L}(\\mathcal{C}(\\mathrm{w}))-\\nu^{L}(\\mathcal{C}(\\mathrm{w}))\\right|\\right)\\right.\\tag{14}\\\\ \\left.+\\left(\\frac{C^{p}}{2}-2^{p-1}h(\\mathrm{v})^{p}\\right)\\left|\\mu^{L}(\\mathcal{C}(\\mathrm{v}))-\\nu^{L}(\\mathcal{C}(\\mathrm{v}))\\right|\\right).\\\\ \\end{array}\n$$\n\nThe construction of $\\tau,$ such that (13) holds, is as follows. Fix some depth level $L \\in$ N. For some resolution $q > 1$ and level $j=0,\\ldots,L$ define the covering $\\quad\\sec^{8}\\ Q_{j}\\ : =$ $\\mathcal{N}(\\mathcal{X},q^{-j}\\mathrm{diam}(\\mathcal{X}))\\subset\\mathcal{X}$ and let $Q_{L+1}: =\\mathcal{X}$ . Any point $x\\in Q_{j}$ is considered as a node at level $j$ of a tree T and denoted as $(x,j)$ to emphasize its level position. An illustration of this approximation is given in Figure 4. For level $j = 0$ this yields a single element in $Q_{0}$ which serves as the root of the tree. For $j=0,\\ldots,L$ a node $(x,j)$ at level $j$ is connected to one node $(x^{\\prime},j+1)$ at level $j + 1$ if their distance satisfies $d(x,x^{\\prime})\\leq q^{-j}\\dim(\\chi)$ (ties are broken arbitrarily). The edge weight of the corresponding edge is set equal to $q^{-j}\\mathrm{diam}(\\mathcal{X})$ Consequently, the height of each node only depends on its assigned level $0 \\leq l \\leq L + 1$ and is defined as $h_{q,L}\\colon\\{0,\\ldots,L+1\\}\\to\\mathbb{R}\\mathrm{~b y~}^{\\mathfrak{g}}$\n\n$$\nh_{q,L}(l)=\\sum_{j=l}^{L}q^{-j}\\mathrm{diam}(\\mathcal{X})=\\frac{q^{1-l}-q^{-L}}{q-1}\\mathrm{diam}(\\mathcal{X}).\\tag{15}\n$$\n\nBy definition the space $\\mathcal{X}$ is embedded in level $L + 1$ as the leaf nodes of $\\tau$ with height $h_{q,L}(L+1)=0$ . By a straightforward computation it holds for two points $x,x^{\\prime}\\in{\\mathcal{X}}$ considered as embedded in $\\tau$ as $(x,L+1)$ and $(x^{\\prime},L+1)$ that\n\n$$\nd^{p}(x,x^{\\prime})\\leq d_{\\mathcal{T}}^{p}((x,L+1),(x^{\\prime},L+1)).\\tag{16}\n$$\n\nThe measures $\\mu,\\nu$ are embedded into $\\tau$ as measures $\\mu^{L},\\nu^{L}$ supported only on leaf nodes of $\\tau .$ , and thus it follows from (16) that\n\n$$\n\\mathrm{KR}_{p,C}(\\mu,\\nu)\\leq\\mathrm{KR}_{d_{\\mathcal{T},C}^{p}}\\left(\\mu^{L},\\nu^{L}\\right).\n$$\n\nIn combination with the closed formula from (14) this yields an upper bound on the $(p,C)$ KRD. Whenever clear from the context the notation is alleviated by writing $\\mathsf{v}\\in Q_{l}$ instead of $(\\mathsf{v},l)\\in Q_{l}$\n\nLemma 1 Let $(\\mathcal{X},d)$ be a finite metric space and let $\\mu,\\nu\\in\\mathcal{M}_{+}(\\mathcal{X})$ with total mass $\\mathbb{M}(\\mu)$ and $\\mathbb{M}(\\nu)$ , respectively. Let $p \\geq 1$ and $C > 0$ . Then for any resolution $q > 1$ , depth $L\\in\\mathbb{N}$ and height function (15) with\n\n$$\nh_{q,L}(k)=\\frac{q^{1-k}-q^{-L}}{q-1}\\mathrm{diam}(\\mathcal{X})\n$$\n\nit holds that\n\n$$\n\\mathrm{KR}_{p,C}^{p}(\\mu,\\nu)\\leq\\left\\{\\begin{array}{l}\\left(\\frac{C^{p}}{2}-2^{p-1}h_{q,L}(0)^{p}\\right)|\\mathbb{M}(\\mu)-\\mathbb{M}(\\nu)|\\\\ +B_{q,p,L,\\mathcal{X}}(1),\\\\ \\quad \\text{if} C\\geq 2 h_{q,L}(0),\\\\ B_{q,p,L,\\mathcal{X}}(l-1),\\\\ \\quad \\text{if} 2 h_{q,L}(l)\\leq C<2 h_{q,L}(l-1),\\\\ \\frac{C^{p}}{2}\\mathrm{TV}(\\mu,\\nu),\\\\ \\quad \\text{if} C\\leq\\left(2 h_{q,L}(L)\\vee\\min_{x\\neq x^{\\prime}}d(x,x^{\\prime})\\right),\\end{array}\\right.\n$$\n\nwhere we define\n\n$$\nB_{q,p,L,\\mathcal{X}}(l):=2^{p-1}\\sum_{j=l}^{L+1}\\sum_{x\\in Q_{j}}(h_{q,L}(j-1)^{p}-h_{q,L}(j)^{p})\\left|\\mu^{L}(\\mathcal{C}(x))-\\nu^{L}(\\mathcal{C}(x))\\right|.\n$$\n\nIn this section we first derive statistical deviation bounds for approximation error empirical estimators for their population counterpart with respect to the TV norm and $(p,C)\\mathrm{-K R D}$ We then continue by analyzing the performance of plug-in estimators. The omitted proofs of the following Theorems 4 and 5 can be found in Appendix D.2.\n\nTheorem 4 (Expected deviation of estimator in TV and KRD) Let $(\\mathcal{X},d)$ be $a\\ f_{-} $ nite metric space and $\\mu\\in\\mathcal{M}_{+}(\\mathcal{X})$ with total mass $\\mathbb{M}(\\mu)$ . Let $\\widehat{\\mu}_{t,s}$ be the estimator from (8). Then, for any $p \\geq 1$ and $C > 0$ it holds that\n\n$$\n\\mathbb{E}\\left[\\mathrm{KR}_{p,C}^{p}\\left(\\widehat{\\mu}_{t,s},\\mu\\right)\\right]\\leq C^{p}\\mathbb{E}\\left[\\mathrm{TV}\\left(\\widehat{\\mu}_{t,s},\\mu\\right)\\right]\\leq C^{p}\\left(2(1-s)\\mathbb{M}(\\mu)+\\frac{s}{\\sqrt{t}}\\sum_{x\\in\\mathcal{X}}\\sqrt{\\mu(x)}\\right).\n$$\n\nIn this section we derive novel quantitative convergence statements for empirical UOT plans. We rely on a novel stability result for the balanced setting, which we also include as Theorem 14 as it might be interesting on its own; the proof is stated in Appendix D.3. The rest of the omitted proofs can be found in Appendix D.4.\n\nTo set notation, we denote the collection of UOT plans between measures $\\mu,\\nu\\in\\mathcal{M}_{+}(\\mathcal{X})$ for parameters $p \\geq 1$ and $C \\geq 0$ by\n\n$$\n\\overline{{\\mathbf{P}}}_{p,C}^{*}(\\mu,\\nu):=\\underset{\\pi\\in\\Pi_{\\leq}(\\mu,\\nu)}{\\operatorname{argmin}}\\sum_{x,x^{\\prime}\\in\\mathcal{X}}d^{p}(x,x^{\\prime})\\pi(x,x^{\\prime})+C^{p}\\left(\\frac{\\mathbb{M}(\\mu)+\\mathbb{M}(\\nu)}{2}-\\mathbb{M}(\\pi)\\right).\\tag{20}\n$$\n\nMoreover, we additionally define the domain $\\mathcal{D}(C): =\\{(x,x^{\\prime})\\in\\mathcal{X}\\times\\mathcal{X}|d(x,x^{\\prime})<C\\}$ and consider the set of restricted UOT plans\n\n$$\n\\mathbf{P}_{p,C}^{*}(\\mu,\\nu):=\\left\\{\\pi\\cdot\\mathbb{1}(\\cdot\\in\\mathcal{D}(C))\\mid\\pi\\in\\overline{{\\mathbf{P}}}_{p,C}^{*}(\\mu,\\nu)\\right\\}.\\tag{21}\n$$\n\nLemma 36 in Appendix D.4 shows that $\\mathbf{P}_{p,C}^{*}(\\mu,\\nu)$ is a subset of $\\overline{{\\mathbf{P}}}_{p,C}^{*}(\\mu,\\nu)$ , and that every sub-coupling in $\\Pi_{\\le}(\\mu,\\nu)$ which is the sum of an element in $\\overline{{\\mathbf{P}}}_{p,C}^{*}(\\mu,\\nu)$ and a non-negative measure supported on $\\{(x,x^{\\prime})\\in\\mathcal{X}\\times\\mathcal{X}|d(x,x^{\\prime})=C\\}$ is contained in $\\overline{{\\mathbf{P}}}_{p,C}^{*}(\\mu,\\nu)$ . Based on this insight, we restrict our attention to the set of restricted UOT plans for empirical and population measures,\n\n$$\n\\mathbf{P}_{p,C}^{*}(\\widehat{\\mu}_{t,s},\\widehat{\\nu}_{t,s})\\quad\\mathrm{and}\\quad\\mathbf{P}_{p,C}^{*}(\\mu,\\nu),\n$$\n\nand quantify the accuracy in estimating the latter in terms of the former via the Hausdorf distance induced by the total variation norm. Crucial for our analysis is the following novel stability bound of the UOT plans.\n\nFor our stability bound of the UOT plan we establish a novel stability bound for the vanilla OT plan between measures with equal mass which might be of independent interest. To formalize these results, we employ the following notation. Let $\\mu,\\nu\\in\\mathcal{M}_{+}(\\mathcal{X})$ be measures with identical mass $\\mathbb{M}(\\mu)=\\mathbb{M}(\\nu)$ , let c : ${\\mathcal{X}}\\times{\\mathcal{X}}\\to [0,\\infty)$ be a cost function and denote the respective collection of OT plans as\n\n$$\n\\widetilde{\\mathbf{P}}_{c}^{*}(\\mu,\\nu):=\\underset{\\pi\\in\\Pi=(\\mu,\\nu)}{\\operatorname{argmin}}\\sum_{x,x^{\\prime}\\in\\mathcal{X}}c(x,x^{\\prime})\\pi(x,x^{\\prime}).\\tag{22}\n$$\n\nNote that the assumption of identical masses ensures that the collection of transport plans is always non-empty and compact, hence $\\widetilde{\\mathbf{P}}_{c}^{*}(\\mu,\\nu)$ is always non-empty.\n\nTheorem 14 (Stability bound for balanced OT plans) Let X be a finite discrete space and $\\mu_{1},\\nu_{1},\\mu_{2},\\nu_{2}\\in\\mathcal{M}_{+}(\\mathcal{X})$ such that $\\mathbb{M}(\\mu_{1})=\\mathbb{M}(\\nu_{1})$ and $\\mathbb{M}(\\mu_{2})=\\mathbb{M}(\\nu_{2})$ . Then, for every cost function c: ${\\mathcal{X}}\\times{\\mathcal{X}}\\to [0,\\infty)$ it holds that\n\n$$\n\\mathcal{H}_{\\mathrm{TV}}\\left(\\widetilde{\\mathbf{P}}_{c}^{*}(\\mu_{1},\\nu_{1}),\\widetilde{\\mathbf{P}}_{c}^{*}(\\mu_{2},\\nu_{2})\\right)\\leq 4|\\mathcal{X}|\\left(\\mathrm{TV}(\\mu_{1},\\mu_{2})+\\mathrm{TV}(\\nu_{1},\\nu_{2})\\right).\n$$\n\nA key tool in establishing properties of the $(p,C)\\mathrm{-K R D}$ and the $(p,C)$ -barycenter is the lift of these problems to the space of probability measures by augmenting the space X with a dummy point having a fixed distance to all points in $\\mathcal{X}_{:}$ , see (Heinemann et al., 2023, Section 3.1) for details. For a fixed parameter $C > 0$ , consider a dummy point d and define the augmented space ${\\tilde{x}}: ={\\mathcal{X}}\\cup\\{{\\mathfrak{o}}\\}$ with metric cost\n\n$$\n\\widetilde{d}_{C}^{p}(x,x^{\\prime})=\\left\\{\\begin{array}{l l}d^{p}(x,x^{\\prime})\\wedge C^{p},&\\text{if} x,x^{\\prime}\\in\\mathcal{X},\\\\ \\frac{C^{p}}{2},&\\text{if} x\\in\\mathcal{X},x^{\\prime}=\\mathfrak{d},\\\\ \\frac{C^{p}}{2},&\\text{if} x=\\mathfrak{d},x^{\\prime}\\in\\mathcal{X},\\\\ 0,&\\text{if} x=x^{\\prime}=\\mathfrak{d}.\\end{array}\\right.\\tag{27}\n$$\n\nConsider the subset $\\mathcal{M}_{+}^{B}(\\mathcal{X})\\ : =\\ \\{\\mu\\in\\mathcal{M}_{+}(\\mathcal{X})\\ |\\ \\mathbb{M}(\\mu)\\leq B\\}\\ \\subset\\ \\mathcal{M}_{+}(\\mathcal{X})$ of non-negative measures whose total mass is bounded by $B .$ Setting $\\widetilde{\\mu}: =\\mu+(B-\\mathbb{M}(\\mu))\\delta_{\\mathfrak{d}}$ , any measure $\\mu\\in\\mathcal{M}_{+}^{B}(\\mathcal{X})$ defines an augmented measure $\\widetilde{\\mu}$ on $\\mathcal{X}$ such that $\\mathbb{M}(\\widetilde{\\mu})=B$ . For any $\\mu,\\nu\\in$ $\\mathcal{M}_{+}(\\mathcal{X})$ and their augmented versions $\\widetilde{\\mu},\\widetilde{\\nu}\\in\\mathcal{M}_{+}(\\mathcal{X})$ it holds\n\n$$\n\\mathrm{KR}_{C,p}^{p}(\\mu,\\nu)=\\widetilde{\\mathrm{OT}}_{\\widetilde{d}_{C}^{p}}(\\widetilde{\\mu},\\widetilde{\\nu}).\\tag{28}\n$$\n\nHere, $\\widetilde{\\mathrm{OT}}_{\\widetilde{d}_{C}^{p}}$ denotes the OT cost defined for measures $\\mu,\\nu$ on $(\\widetilde{\\mathcal{X}},\\widetilde{d})$ with $\\mathbb{M}(\\mu)=\\mathbb{M}(\\nu)$ as\n\n$$\n\\widetilde{\\mathrm{OT}}_{\\widetilde{d}_{C}^{p}}(\\mu,\\nu):=\\min_{\\pi\\in\\Pi_{=}(\\mu,\\nu)}\\sum_{x,x^{\\prime}\\in\\mathcal{X}}\\widetilde{d}_{C}^{p}(x,x^{\\prime})\\pi(x,x^{\\prime}),\n$$\n\nwhere the set of couplings $\\scriptstyle\\Pi=(\\mu,\\nu)$ is the set $\\Pi_{\\le}(\\mu,\\nu)$ with inequalities replaced by equalities.\n\nLemma 36 (Relation between restricted and unrestricted UOT plans) Let $(\\mathcal{X},d)$ be a finite space, $\\mu,\\nu\\in\\mathcal{M}_{+}(\\mathcal{X})$ . Then, for $p \\geq 1$ and $C \\geq 0$ the sets $\\overline{{\\mathbf{P}}}_{p,C}^{*}(\\mu,\\nu)$ defined in (20) and $\\mathbf{P}_{p,C}^{*}(\\mu,\\nu)$ defined in (21) are related as follows.\n\n1. It holds that $\\mathbf{P}_{p,C}^{*}(\\mu,\\nu)\\subseteq\\overline{{\\mathbf{P}}}_{p,C}^{*}(\\mu,\\nu)$ , i.e., the set $\\mathbf{P}_{p,C}^{*}(\\mu,\\nu)$ consists of unbalanced optimal transport plans.  \n2. Every unbalanced optimal transport plan $\\pi\\in\\overline{{\\mathbf{P}}}_{p,C}^{*}(\\mu,\\nu)$ can be represented as the sum of a restricted transport plan $\\mathbf{P}_{p,C}^{*}(\\mu,\\nu)$ and a non-negative measure supported on $\\{(x,x^{\\prime})\\in\\mathcal{X}\\times\\mathcal{X}|d(x,x^{\\prime})=C\\}$  \n3. Every sub-coupling $\\pi~\\in~\\Pi_{\\leq}(\\mu,\\nu)$ which is the sum of a restricted transport plan $\\mathbf{P}_{p,C}^{*}(\\mu,\\nu)$ and a non-negative measure supported on $\\{(x,x^{\\prime})\\in\\mathcal{X}\\times\\mathcal{X}|d(x,x^{\\prime})=C\\}$ is also an unbalanced optimal transport plan, i.e., $\\pi\\in\\overline{{\\mathbf{P}}}_{p,C}^{*}(\\mu,\\nu)$\n\nLemma 38 Let $\\mu^{i},\\nu^{i}\\in\\mathcal{M}_{+}(\\mathcal{X})$ for $i\\in\\{1,2\\}$ be two non-negative measures and consider their augmented measures $\\widetilde{\\mu}^{i},\\widetilde{\\nu}^{i}\\in\\mathcal{M}_{+}^{B}(\\widetilde{\\mathcal{X}})$ to $\\widetilde{\\chi}$ for $B=\\operatorname*{max}(\\mathbb{M}(\\mu^{i}),\\mathbb{M}(\\nu^{i}))$ , then\n\n$$\n\\begin{array}{l}\\mathrm{TV}(\\widetilde{\\mu}^{1},\\widetilde{\\mu}^{2})+\\mathrm{TV}(\\widetilde{\\nu}^{1},\\widetilde{\\nu}^{2})\\\\ \\leq\\mathrm{TV}(\\mu^{1},\\mu^{2})+\\mathrm{TV}(\\nu^{1},\\nu^{2})+ |\\mathbb{M}(\\mu^{1})-\\mathbb{M}(\\mu^{2})| + |\\mathbb{M}(\\nu^{1})-\\mathbb{M}(\\nu^{2})|\\\\ \\leq 2\\left(\\mathrm{TV}(\\mu^{1},\\mu^{2})+\\mathrm{TV}(\\nu^{1},\\nu^{2})\\right).\\\\ \\end{array}\n$$", "theorems": [{"local_id": "T1", "statement": "Theorem 5 (Expected deviation of estimator in KRD) Let $(\\mathcal{X},d)$ be a finite metric space and $\\mu\\in\\mathcal{M}_{+}(\\mathcal{X})$ . Let $\\widehat{\\mu}_{t,s}$ be the estimator from (8). Then, for any $p\\geq 1,C>0$ resolution $q > 1$ and depth $L\\in\\mathbb{N}$ it holds that\n\n$$\n\\mathbb{E}\\left[\\mathrm{KR}_{p,C}\\left(\\widehat{\\mu}_{t,s},\\mu\\right)\\right]\\leq\\mathcal{E}_{p,\\mathcal{X},\\mu}^{\\mathrm{Poi}}(C,q,L)^{1/p}\\left\\{\\begin{array}{c}\\left(2(1-s)\\mathbb{M}(\\mu)+\\frac{s}{\\sqrt{t}}\\sum_{x\\in\\mathcal{X}}\\sqrt{\\mu(x)}\\right)^{\\frac{1}{p}},\\\\ i f C\\leq\\left(2 h_{q,L}(L)\\vee\\min_{x\\neq x^{\\prime}}d(x,x^{\\prime})\\right),\\\\ \\left(\\frac{1}{s t}\\mathbb{M}(\\mu)+\\frac{1-s}{s}\\sum_{x\\in\\mathcal{X}}\\mu(x)^{2}\\right)^{\\frac{1}{2 p}},\\\\ e l s e.\\end{array}\\right.\n$$\n\nFor\n\n$$\nA_{q,p,L,\\mathcal{X}}(l):=\\mathrm{diam}(\\mathcal{X})^{p}2^{p-1}\\left(q^{-L p}|\\mathcal{X}|^{\\frac{1}{2}}+\\left(\\frac{q}{q-1}\\right)^{p}\\sum_{j=l}^{L}q^{p-j p}|Q_{j}|^{\\frac{1}{2}}\\right),\n$$\n\nthe constant is equal to\n\n$$\n\\mathcal{E}_{p,\\mathcal{X},\\mu}^{\\mathrm{Poi}}(C,q,L)=\\left\\{\\begin{array}{l}\\left(\\frac{C^{p}}{2}-2^{p-1}\\left(\\frac{q-q^{-L}}{q-1}\\mathrm{diam}(\\mathcal{X})\\right)^{p}\\right)+A_{q,p,L,\\mathcal{X}}(1),\\\\ \\quad \\text{if} C\\geq 2 h_{q,L}(0),\\\\ A_{q,p,L,\\mathcal{X}}(l),\\\\ \\quad \\text{if} 2 h_{q,L}(l)\\leq C<2 h_{q,L}(l-1),\\\\ \\frac{C^{p}}{2},\\\\ \\quad \\text{if} C\\leq\\left(2 h_{q,L}(L)\\vee\\min_{x\\neq x^{\\prime}}d(x,x^{\\prime})\\right),\\end{array}\\right.\n$$\n\nFurthermore, for $p = 1$ the factor $\\frac{q}{(q-1)}$ in $A_{q,1,L,\\mathcal{X}}(l)$ can be removed for all $l=1,\\ldots,L$", "proof": {"steps": [{"n": 1, "claim": "Proof [Proof of Theorem 5] Let $\\widehat{\\mu}_{t,s}$ be the estimator from (8). We fix $p = 1$ and detail the case $p > 1$ at the end of the proof. Suppose first that $C\\leq\\mathrm{min}_{x\\neq x^{\\prime}}d(x,x^{\\prime})$ . According to (Heinemann et al., 2023, Theorem 2.2 (ii)) it holds that\n\n$$\n\\mathbb{E}\\left[\\mathrm{KR}_{1,C}(\\widehat{\\mu}_{t,s},\\mu)\\right]=\\frac{C}{2}\\mathbb{E}\\left[\\sum_{x\\in\\mathcal{X}}|\\widehat{\\mu}_{t,s}-\\mu(x)|\\right]=\\frac{C}{2}\\mathbb{E}\\left[\\mathrm{TV}(\\widehat{\\mu}_{t,s},\\mu)\\right].\n$$"}, {"n": 2, "claim": "This yields the total variation bounds (see Theorem 4). Next, consider the tree approximation as outlined in Section 2.1.1 and construct an ultrametric tree T such that $\\begin{array}{ r }{\\mathrm{KR}_{1,C}(\\widehat{\\mu}_{t,s},\\mu)\\leq\\mathrm{KR}_{d_{T},C}\\left(\\widehat{\\mu}_{t,s}^{L},\\mu^{L}\\right)}\\end{array}$ . Applying Lemma 1 for $p = 1$ where by definition the diference of height function is equal to\n\n$$\nh_{q,L}(j-1)-h_{q,L}(j)=\\frac{\\mathrm{diam}(\\mathcal{X})}{q-1}\\left(q^{2-j}-q^{1-j}\\right)=\\mathrm{diam}(\\mathcal{X})q^{1-j}\n$$\n\nand yields the upper bound\n\n$$\n\\mathbb{E}\\left[\\mathrm{KR}_{1,C}(\\widehat{\\mu}_{t,s},\\mu)\\right]\\leq\\mathbb{E}\\left[\\mathrm{KR}_{d_{\\mathcal{T}},C}\\left(\\widehat{\\mu}_{t,s}^{L},\\mu^{L}\\right)\\right]\n$$\n\n$$\n\\leq\\left\\{\\begin{array}{l l}\\left(\\frac{C}{2}-h_{q,L}(0)\\right)\\mathbb{E}\\left[|\\mathbb{M}(\\widehat{\\mu}_{t,s})-\\mathbb{M}(\\mu)|\\right]\\\\ +\\mathrm{diam}(\\mathcal{X})\\sum_{j=1}^{L+1}q^{1-j}\\sum_{x\\in Q_{j}}\\mathbb{E}\\left[\\left|\\widehat{\\mu}_{t,s}^{L}(\\mathcal{C}(x))-\\mu^{L}(\\mathcal{C}(x))\\right|\\right],&\\text{if} C\\geq 2 h_{q,L}(0)\\\\ \\mathrm{diam}(\\mathcal{X})\\sum_{j=l}^{L+1}q^{1-j}\\sum_{x^{\\prime}\\in Q_{j}}\\mathbb{E}\\left[\\left|\\widehat{\\mu}_{t,s}^{L}(\\mathcal{C}(x^{\\prime}))-\\mu^{L}(\\mathcal{C}(x^{\\prime}))\\right|\\right],&\\text{if} 2 h_{q,L}(l)\\leq C<2 h_{q,L}(l-1),\\\\ \\frac{C^{p}}{2}\\mathbb{E}\\left[\\mathrm{TV}(\\widehat{\\mu}_{t,s},\\mu)\\right],&\\text{if} C\\leq\\left(2 h_{q,L}(L)\\vee\\min_{x\\neq x^{\\prime}}d(x,x^{\\prime})\\right).\\end{array}\\right.\n$$"}, {"n": 3, "claim": "For the estimator from (8) with $B_{x}\\sim\\mathrm{Ber}(s)$ and $P_{x}\\sim\\mathrm{Poi}(t\\mu(x))$ for all $x\\in\\mathcal{X}$ , it holds\n\n$$\n\\begin{array}{l}\\sum_{x\\in Q_{l}}\\mathbb{E}\\left[\\left|\\widehat{\\mu}_{t,s}^{L}(\\mathcal{C}(x))-\\mu^{L}(\\mathcal{C}(x))\\right|\\right]\\\\ =\\sum_{x\\in Q_{l}}\\frac{1}{s t}\\mathbb{E}\\left[\\left|\\sum_{y\\in\\mathcal{C}(x)}P_{y}B_{y}-s t\\sum_{y\\in\\mathcal{C}(x)}\\mu(y)\\right|\\right]\\\\ \\leq\\sum_{x\\in Q_{l}}\\frac{1}{s t}\\sqrt{\\operatorname{Var}\\left(\\sum_{y\\in\\mathcal{C}(x)}P_{y}B_{y}\\right)}=\\sum_{x\\in Q_{l}}\\frac{1}{s t}\\sqrt{\\sum_{y\\in\\mathcal{C}(x)}\\operatorname{Var}\\left(P_{y}B_{y}\\right)}\\\\ =\\sum_{x\\in Q_{l}}\\frac{1}{s t}\\sqrt{\\sum_{y\\in\\mathcal{C}(x)}s(1-s)t\\mu(y)+s(1-s)t^{2}\\mu(y)^{2}+t\\mu(y)s^{2}}\\\\ =\\sum_{x\\in Q_{l}}\\sqrt{\\frac{1-s}{s t}\\sum_{y\\in\\mathcal{C}(x)}\\mu(y)+\\frac{1-s}{s}\\sum_{y\\in\\mathcal{C}(x)}\\mu(y)^{2}+\\frac{1}{t}\\sum_{y\\in\\mathcal{C}(x)}\\mu(y)}\\\\ =\\sum_{x\\in Q_{l}}\\sqrt{\\frac{1}{s t}\\mu(\\mathcal{C}(x))+\\frac{1-s}{s}\\sum_{y\\in\\mathcal{C}(x)}\\mu(y)^{2}}\\\\ \\leq\\sqrt{|Q_{l}|}\\sqrt{\\frac{1}{s t}\\sum_{x\\in Q_{l}}\\mu(\\mathcal{C}(x))+\\frac{1-s}{s}\\sum_{x\\in Q_{l}}\\sum_{y\\in\\mathcal{C}(x)}\\mu(y)^{2}}\\\\ =\\sqrt{|Q_{l}|}\\sqrt{\\frac{1}{s t}\\mathbb{M}(\\mu)+\\frac{1-s}{s}\\sum_{x\\in\\mathcal{X}}\\mu(x)^{2}}\\\\ \\end{array}\n$$"}, {"n": 4, "claim": "Following an analogous computation one bounds the estimation error for the total mass intensity as\n\n$$\n\\mathbb{E}\\left[|\\mathbb{M}(\\widehat{\\mu}_{t,s})-\\mathbb{M}(\\mu)|\\right]\\leq\\sqrt{\\operatorname{Var}(\\mathbb{M}(\\widehat{\\mu}_{t,s}))}\\leq\\sqrt{\\frac{1}{s t}\\mathbb{M}(\\mu)+\\frac{1-s}{s}\\sum_{x\\in\\mathcal{X}}\\mu(x)^{2}}.\\tag{31}\n$$"}, {"n": 5, "claim": "Applying both of these bounds to the previous upper bound on the $(p,C)\\mathrm{-K R D}$ in Lemma 1 yields the claim."}, {"n": 6, "claim": "For $p > 1$ , we first observe again that if $C\\leq\\mathrm{min}_{x\\neq x^{\\prime}}d(x,x^{\\prime})$ then according to (Heinemann et al., 2023, Theorem 2.2) it holds that\n\n$$\n\\mathbb{E}\\left[\\mathrm{KR}_{p,C}^{p}(\\widehat{\\mu}_{t,s},\\mu)\\right]=\\frac{C^{p}}{2}\\mathbb{E}\\left[\\mathrm{TV}(\\widehat{\\mu}_{t,s},\\mu)\\right]\n$$\n\nwhich yields the total variation bounds. For more general $C,$ we repeat the previous calculations with the upper bounds on the diference of height function $h_{q,L}(j-1)^{p}-h_{q,L}(j)^{p}\\leq$ diam $(\\mathcal{X})^{{p}}\\left(\\textstyle{\\frac{q}{q-1}}\\right)^{{p}}q^{{p}-{j p}}$ . Since $h_{q,L}(L+1)=0$ we also have\n\n$$\nh_{q,L}(L)^{p}-h_{q,L}(L+1)^{p}=\\mathrm{diam}(\\mathcal{X})^{p}q^{-L p}.\n$$"}, {"n": 7, "claim": "The expectations are bounded identically as before. Finally, using Jensen’s inequality,\n\n$$\n\\mathbb{E}\\left[\\mathrm{KR}_{p,C}(\\mu,\\widehat{\\mu}_{t,s})\\right]\\leq\\left(\\mathbb{E}\\left[\\mathrm{KR}_{p,C}^{p}(\\mu,\\widehat{\\mu}_{t,s})\\right]\\right)^{\\frac{1}{p}},\n$$\n\nfinishes the proof."}]}}, {"local_id": "T2", "statement": "Theorem 11 (Stability bound for UOT plans) Let $(\\mathcal{X},d)$ be a finite metric space and $\\mu^{1},\\mu^{2},\\nu^{1},\\nu^{2}\\in\\dot{\\mathcal{M}}_{+}(\\mathcal{X})$ . Then, for any $p \\geq 1$ and $C \\geq 0$ it holds that\n\n$$\n\\begin{array}{l}\\mathcal{H}_{\\mathrm{TV}}\\left(\\mathbf{P}_{p,C}^{*}(\\mu^{1},\\nu^{1}),\\mathbf{P}_{p,C}^{*}(\\mu^{2},\\nu^{2})\\right)\\\\ \\leq 4\\left(|\\mathcal{X}| +1\\right)\\left(\\mathrm{TV}(\\mu^{1},\\mu^{2})+ |\\mathbb{M}(\\mu^{1})-\\mathbb{M}(\\mu^{2})| +\\mathrm{TV}(\\nu^{1},\\nu^{2})+ |\\mathbb{M}(\\nu^{1})-\\mathbb{M}(\\nu^{2})|\\right),\\\\ \\end{array}\n$$\n\nwhere $\\mathcal{H}_{\\mathrm{TV}}$ denotes the Hausdorf distance induced by the total variation norm.", "proof": {"steps": [{"n": 1, "claim": "Proof [Proof of Theorem 11] For the proof we again rely on the lift to optimal transport as detailed in Appendix C. To this end, consider the augmented space ${\\widetilde{\\mathcal{X}}}=\\mathcal{X}\\cup\\{\\mathfrak{d}\\}$ for some dummy element d, let B be a majorant for the masses of the measures $\\mu^{1},\\mu^{2},\\nu^{1},\\nu^{2}$ , and define for $i\\in\\{1,2\\}$ the augmented measures\n\n$$\n\\widetilde{\\mu}^{i}:=\\mu^{i}+\\delta_{\\mathfrak{d}}(\\max(\\mathbb{M}(\\mu^{i}),\\mathbb{M}(\\nu^{i}))-\\mathbb{M}(\\mu^{i}))\\quad \\text{and}\n$$\n\n$$\n\\widetilde{\\nu}^{i}:=\\nu^{i}+\\delta_{\\mathfrak{d}}(\\max(\\mathbb{M}(\\mu^{i}),\\mathbb{M}(\\nu^{i}))-\\mathbb{M}(\\nu^{i})).\n$$"}, {"n": 2, "claim": "Then, upon denoting the collection of OT plans between $\\widetilde{\\mu}^{i}$ and $\\widetilde{\\nu}^{i}$ with respect to the cost function $\\widetilde{d}_{C}^{p}$ by $\\widetilde{\\mathbf{P}}_{p,C}^{*}(\\widetilde{\\mu}^{i},\\widetilde{\\nu}^{i})$ , it follows by Lemma 37 and Theorem 14 that\n\n$$\n\\begin{array}{l}\\mathcal{H}_{\\mathrm{TV}}\\left(\\mathbf{P}_{p,C}^{*}(\\mu^{1},\\nu^{1}),\\mathbf{P}_{p,C}^{*}(\\mu^{2},\\nu^{2})\\right)\\leq\\mathcal{H}_{\\mathrm{TV}}\\left(\\widetilde{\\mathbf{P}}_{p,C}^{*}(\\widetilde{\\mu}^{1},\\widetilde{\\nu}^{1}),\\widetilde{\\mathbf{P}}_{p,C}^{*}(\\widetilde{\\mu}^{2},\\widetilde{\\nu}^{2})\\right)\\\\ \\leq 4\\left|\\widetilde{\\mathcal{X}}\\right|\\left(\\mathrm{TV}(\\widetilde{\\mu}^{1},\\widetilde{\\mu}^{2})+\\mathrm{TV}(\\widetilde{\\nu}^{1},\\widetilde{\\nu}^{2})\\right).\\\\ \\end{array}\n$$"}, {"n": 3, "claim": "The assertion now follows by $|\\widetilde{\\mathcal{X}}| = |\\mathcal{X}| +1$ and the subsequent Lemma 38."}]}}, {"local_id": "T3", "statement": "Theorem 12 (Expected deviation of empirical UOT plans) Let $(\\mathcal{X},d)$ be a finite metric space and $\\mu,\\nu\\in\\mathcal{M}_{+}(\\mathcal{X})$ . Let $\\widehat{\\mu}_{t,s},\\widehat{\\nu}_{t,s}$ be the estimator from (8). Then, for any $p \\geq 1$ and $C \\geq 0$ it follows that\n\n$$\n\\begin{array}{l}\\mathbb{E}\\left[\\sup_{p\\geq 1,C>0}\\mathcal{H}_{\\mathrm{TV}}\\left(\\mathbf{P}_{p,C}^{*}(\\widehat{\\mu}_{t,s},\\widehat{\\nu}_{t,s}),\\mathbf{P}_{p,C}^{*}(\\mu,\\nu)\\right)\\right]\\\\ \\leq 4\\left(|\\mathcal{X}| +1\\right)\\Bigg(2(1-s)\\mathbb{M}(\\mu+\\nu)+\\frac{s}{\\sqrt{t}}\\sum_{x\\in\\mathcal{X}}\\left[\\sqrt{\\mu(x)}+\\sqrt{\\nu(x)}\\right]\\\\ \\left.+\\sqrt{\\frac{1}{s t}\\mathbb{M}(\\mu)+\\frac{1-s}{s}\\sum_{x\\in\\mathcal{X}}\\mu(x)^{2}}+\\sqrt{\\frac{1}{s t}\\mathbb{M}(\\nu)+\\frac{1-s}{s}\\sum_{x\\in\\mathcal{X}}\\nu(x)^{2}}\\right).\\\\ \\end{array}\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof The assertion follows by combining the stability bound from Theorem 11 above with the bound on the total variation distance between the estimators $\\widehat{\\mu}_{t,s},\\widehat{\\nu}_{t,s}$ and $\\mu,\\nu$ (Theorem 4) and on the absolute deviation between their masses (31)."}]}}, {"local_id": "T4", "statement": "Lemma 37 Let $(\\mathcal{X},d)$ be a finite metric space and let $p \\geq 1$ and $C \\geq 0$ . Then, the following assertions hold.\n\n1. For every pair of measures $\\mu,\\nu\\in\\mathcal{M}_{+}(\\mathcal{X})$ and their augmented counterparts $\\widetilde\\mu,\\widetilde\\nu\\in$ $\\mathcal{M}_{+}^{B}(\\widetilde{\\mathcal{X}})=\\{\\mu\\in\\mathcal{M}_{+}(\\mathcal{X})\\mid\\mathbb{M}(\\mu)\\leq B\\}\\ f o r\\ B=\\operatorname*{max}(\\mathbb{M}(\\mu),\\mathbb{M}(\\nu))$ it holds that\n\n$$\n\\mathbf{P}_{p,C}^{*}(\\mu,\\nu)=\\left\\{\\widetilde{\\pi}\\cdot\\mathbb{1}(\\cdot\\in\\mathcal{D}(C))\\Big|\\widetilde{\\pi}\\in\\widetilde{\\mathbf{P}}_{p,C}^{*}(\\widetilde{\\mu},\\widetilde{\\nu})\\right\\}.\\tag{36}\n$$\n\n2. For every two pairs of measures $\\mu^{1},\\nu^{1},\\mu^{2},\\nu^{2}\\in\\mathcal{M}_{+}(\\mathcal{X})$ and corresponding augmented measures µe<sup>i</sup>, $\\tilde{\\nu^{i}}\\in\\mathcal{M}_{+}^{\\tilde{B}_{i}}(\\tilde{\\mathcal{X}})$ for $B_{i}=\\operatorname*{max}(\\mathbb{M}(\\mu^{i}),\\mathbb{M}(\\nu^{i}))$ and $i\\in\\{1,2\\}$ it holds that\n\n$$\n\\mathcal{H}_{\\mathrm{TV}}\\left(\\mathbf{P}_{p,C}^{*}(\\mu^{1},\\nu^{1}),\\mathbf{P}_{p,C}^{*}(\\mu^{2},\\nu^{2})\\right)\\leq\\mathcal{H}_{\\mathrm{TV}}\\left(\\widetilde{\\mathbf{P}}_{p,C}^{*}(\\widetilde{\\mu}^{1},\\widetilde{\\nu}^{1}),\\widetilde{\\mathbf{P}}_{p,C}^{*}(\\widetilde{\\mu}^{2},\\widetilde{\\nu}^{2})\\right).\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof For the first assertion assume without loss of generality that $\\mathbb{M}(\\mu)\\geq\\mathbb{M}(\\nu)$ which implies $\\widetilde{\\mu}=\\mu$ . Further, throughout the proof we will make use of the following equality which has been shown by Heinemann et al. (2023, p. 17) and is discussed in Appendix C of this manuscript,\n\n$$\n\\mathrm{KR}_{p,C}^{p}(\\mu,\\nu)=\\mathrm{UOT}_{d^{p}\\wedge C^{p},C}(\\mu,\\nu)=\\widetilde{\\mathrm{OT}}_{\\widetilde{d}_{C}^{p}}(\\widetilde{\\mu},\\widetilde{\\nu}).\\tag{37}\n$$"}, {"n": 2, "claim": "To show that the right-hand side of (36) is included in the left-hand side, let $\\widetilde{\\pi}\\in\\widetilde{\\mathbf{P}}_{p,C}^{*}(\\widetilde{\\mu},\\widetilde{\\nu})$ and define $\\pi: =\\widetilde{\\pi}\\cdot\\mathbb{1}(\\cdot\\in\\mathcal{D}(C))$ and $\\pi^{*}: =\\widetilde{\\pi}\\cdot\\mathbb{1}(\\cdot\\in\\mathcal{X}\\times\\mathcal{X})$ . Note that $\\widetilde{\\pi}(\\{\\mathfrak{d}\\}\\times\\widetilde{\\mathcal{X}})=$ $\\widetilde{\\mu}(\\{\\mathfrak{d}\\})=0$ . As a result, by the marginal constraints on $\\widetilde{\\pi},\\widetilde{\\pi}(\\widetilde{\\mathcal{X}}\\times\\mathcal{X})=\\widetilde{\\nu}(\\mathcal{X})=\\nu(\\mathcal{X})$ , it follows that $\\mathbb{M}(\\pi^{*})=\\mathbb{M}(\\nu)$ ."}, {"n": 3, "claim": "From this it is evident that\n\n$$\n\\begin{array}{l}\\sum_{x,x^{\\prime}\\in\\mathcal{X}}d^{p}(x,x^{\\prime})\\pi(x,x^{\\prime})+C^{p}\\left(\\frac{\\mathbb{M}(\\mu)+\\mathbb{M}(\\nu)}{2}-\\mathbb{M}(\\pi)\\right)\\\\ =\\sum_{x,x^{\\prime}\\in\\mathcal{X}}d^{p}(x,x^{\\prime})\\pi(x,x^{\\prime})+C^{p}\\mathbb{M}(\\pi^{*}-\\pi)+C^{p}\\left(\\frac{\\mathbb{M}(\\mu)+\\mathbb{M}(\\nu)}{2}-\\mathbb{M}(\\pi^{*})\\right)\\\\ =\\sum_{x,x^{\\prime}\\in\\mathcal{X}}\\widetilde{d}_{C}^{p}(x,x^{\\prime})\\pi^{*}(x,x^{\\prime})+\\frac{C^{p}}{2}\\left(\\mathbb{M}(\\mu)-\\mathbb{M}(\\nu)\\right)\\\\ =\\sum_{x,x^{\\prime}\\in\\mathcal{X}}\\widetilde{d}_{C}^{p}(x,x^{\\prime})\\widetilde{\\pi}(x,x^{\\prime})+\\sum_{x\\in\\mathcal{X}}\\widetilde{d}_{C}^{p}(x,\\mathfrak{d})\\widetilde{\\pi}(x,\\mathfrak{d})\\\\ =\\sum_{x,x^{\\prime}\\in\\widetilde{\\mathcal{X}}}\\widetilde{d}_{C}^{p}(x,x^{\\prime})\\widetilde{\\pi}(x,x^{\\prime})=\\widetilde{\\mathrm{OT}}_{\\widetilde{d}_{C}^{p}}(\\widetilde{\\mu},\\widetilde{\\nu})=\\mathrm{KR}_{p,C}^{p}(\\mu,\\nu),\\\\ \\end{array}\n$$\n\nwhere the equality third to last is a consequence of $\\widetilde{\\pi}(\\mathfrak{d}\\times\\widetilde{\\mathcal{X}})=\\widetilde{\\mu}(\\mathfrak{d})=0 .$ , the second to last equality follows by optimality of ${\\widetilde{\\pi}},$ , and the final equality is a consequence of (37)."}, {"n": 4, "claim": "In conjunction with Lemma 36 this asserts that $\\pi\\in\\mathbf{P}_{p,C}^{*}(\\mu,\\nu)$\n\nConversely, to show that the left-hand side of (36) is contained in the right-hand side, let $\\pi\\in\\mathbf{P}_{p,C}^{*}(\\mu,\\nu)$ . Further, define the measure $\\check{\\pi}\\in\\mathcal{M}_{+}(\\mathcal{X}\\times\\mathcal{X})$ for $x,x^{\\prime}\\in\\mathcal{X}$ by\n\n$$\n\\check{\\pi}(x,x^{\\prime}):=\\left\\{\\begin{array}{l l}0,&\\text{if}\\mathbb{M}(\\nu)=\\mathbb{M}(\\pi),\\\\ \\frac{\\mu(x)-\\pi(x,\\mathcal{X})}{\\mathbb{M}(\\mu)-\\mathbb{M}(\\pi)}\\left(\\nu(x^{\\prime})-\\pi(\\mathcal{X},x^{\\prime})\\right)&,\\text{if}\\mathbb{M}(\\nu)>\\mathbb{M}(\\pi),\\end{array}\\right.\n$$\n\nand further set $\\pi^{*}: =\\pi+{\\check{\\pi}}$ ."}, {"n": 5, "claim": "Then, it follows that $\\pi^{*}\\in\\Pi_{\\leq}(\\mu,\\nu)$ and $\\mathbb{M}(\\pi^{*})=\\mathbb{M}(\\nu)$ Further, from optimality of π we infer that\n\n$$\n\\begin{array}{l}\\mathrm{KR}_{p,C}^{p}(\\mu,\\nu)=\\sum_{x,x^{\\prime}\\in\\mathcal{X}}d^{p}(x,x^{\\prime})\\pi(x,x^{\\prime})+C^{p}\\left(\\frac{\\mathbb{M}(\\mu)+\\mathbb{M}(\\nu)}{2}-\\mathbb{M}(\\pi)\\right)\\\\ =\\sum_{x,x^{\\prime}\\in\\mathcal{X}}(d^{p}(x,x^{\\prime})\\wedge C^{p})\\pi(x,x^{\\prime})+C^{p}\\mathbb{M}(\\check{\\pi})+C^{p}\\left(\\frac{\\mathbb{M}(\\mu)+\\mathbb{M}(\\nu)}{2}-\\mathbb{M}(\\pi+\\check{\\pi})\\right)\\\\ \\geq\\sum_{x,x^{\\prime}\\in\\mathcal{X}}(d^{p}(x,x^{\\prime})\\land C^{p})\\pi^{*}(x,x^{\\prime})+C^{p}\\left(\\frac{\\mathbb{M}(\\mu)+\\mathbb{M}(\\nu)}{2}-\\mathbb{M}(\\pi^{*})\\right)\\\\ \\geq\\mathrm{UOT}_{d^{p}\\wedge C^{p},C}(\\mu,\\nu)=\\mathrm{KR}_{p,C}^{p}(\\mu,\\nu),\\tag{38}\\\\ \\end{array}\n$$\n\nwhere the last equality follows (37) and thus asserts that all inequalities are qualities."}, {"n": 6, "claim": "This implies that ˇπ is concentrated on $\\{(x,x^{\\prime})\\in\\mathcal{X}\\times\\mathcal{X}\\mid d(x,x^{\\prime})\\geq C\\}$ and, thus, $\\pi^{*}\\cdot\\mathbb{1}(\\cdot\\in$ ${\\mathcal{D}}(C))=\\pi$ . Next, define $\\begin{array}{ r }{\\widetilde{\\pi}: =\\pi^{*}+\\sum_{x\\in{\\mathcal K}}\\delta_{(x,0)}\\left(\\mu(x)-\\pi^{*}(x,{\\mathcal X})\\right)}\\end{array}$ which fulfills $\\widetilde{\\pi}\\in\\Pi_{=}(\\widetilde{\\mu},\\widetilde{\\nu})$ and $\\widetilde{\\pi}\\cdot\\mathbb{1}(\\cdot\\in\\mathcal{D}(C))=\\pi$ . In particular, from the above derivation in (38) we infer from $\\mathbb{M}(\\pi^{*})=\\mathbb{M}(\\nu),\\mathbb{M}(\\widetilde{\\pi})=\\mathbb{M}(\\widetilde{\\mu})=\\mathbb{M}(\\mu)$ and since $\\widetilde{\\pi}(\\mathfrak{d},\\widetilde{\\mathcal{X}})=\\widetilde{\\mu}(\\mathfrak{d})=0$ that\n\n$$\n\\begin{array}{l}\\mathrm{KR}_{p,C}^{p}(\\mu,\\nu)=\\sum_{x,x^{\\prime}\\in\\mathcal{X}}(d^{p}(x,x^{\\prime})\\wedge C^{p})\\pi^{*}(x,x^{\\prime})+C^{p}\\left(\\frac{\\mathbb{M}(\\mu)+\\mathbb{M}(\\nu)}{2}-\\mathbb{M}(\\pi^{*})\\right)\\\\ =\\sum_{x,x^{\\prime}\\in\\mathcal{X}}(d^{p}(x,x^{\\prime})\\land C^{p})\\pi^{*}(x,x^{\\prime})+\\frac{C^{p}}{2}\\left(\\mathbb{M}(\\mu)-\\mathbb{M}(\\nu)\\right)\\\\ =\\sum_{x,x^{\\prime}\\in\\mathcal{X}^{\\prime}}\\widetilde{d}_{C}^{p}(x,x^{\\prime})\\pi^{*}(x,x^{\\prime})\\geq\\widetilde{\\mathrm{OT}}_{\\widetilde{d}_{C}^{p}}(\\widetilde{\\mu},\\widetilde{\\nu})=\\mathrm{KR}_{p,C}^{p}(\\mu,\\nu).\\\\ \\end{array}\n$$"}, {"n": 7, "claim": "Here, the final equality is a consequence of (37), and altogether confirms that $\\widetilde{\\pi}\\in\\widetilde{\\mathbf{P}}_{p,C}^{*}(\\widetilde{\\mu},\\widetilde{\\nu})$ This concludes the proof of the first assertion."}, {"n": 8, "claim": "For the proof of the second assertion note from the considerations above that for every pair of UOT plans $\\pi^{i}\\in\\mathbf{P}^{*}(\\mu^{i},\\nu^{i})$ with $i\\in\\{1,2\\}$ , every pair of corresponding OT plans $\\bar{\\boldsymbol{\\pi}}^{i}\\in\\widetilde{\\mathbf{P}}^{*}(\\mu^{i},\\nu^{i})$ with $\\widetilde{\\pi}^{i}\\cdot\\mathbb{1}(\\cdot\\in{\\mathcal{D}}(C))=\\pi^{i}$ fulfills the inequality\n\n$$\n\\mathrm{TV}(\\pi^{1},\\pi^{2})\\leq\\mathrm{TV}(\\widetilde{\\pi}^{1},\\widetilde{\\pi}^{2}).\n$$"}, {"n": 9, "claim": "Based on this, the claim follows from the definition of the Hausdorf distance induced by the total variation norm. ■"}]}}], "target_local_id": null, "topic_domain": "Others", "topic_subtopic": "Optimal Transport & Sampling", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2025-23-1679", "context": "Let a compact set $\\mathcal{X}\\subseteq\\mathbb{R}^{d}$ denote the input space and $\\mathcal{V}\\subseteq\\mathbb{R}$ denote the output space. Let $\\rho=\\rho_{d}$ be an unknown probability distribution on $\\mathcal{X}\\times\\mathcal{V}$ and denote the corresponding marginal distribution on X by $\\mu=\\mu_{d}$ . We use $L^{p}(\\mathcal{X},\\mu)$ (in short $L^{p})$ to represent the $L^{p}{\\mathrm{-s p a c e s}}$ . Denote the conditional mean as\n\n$$\nf_{\\rho}^{*}(\\boldsymbol{x})=f_{\\rho_{d}}^{*}(\\boldsymbol{x}):=\\mathbb{E}_{\\rho_{d}}[\\boldsymbol{y}\\mid\\boldsymbol{x}]=\\int_{\\mathcal{Y}}y\\mathrm{d}\\rho_{d}(y|\\boldsymbol{x}).\n$$\n\nThroughout the paper, we make the following assumption:\n\nAssumption 1 Suppose that $\\mathcal{H}~=~\\mathcal{H}_{d}$ is a separable reproducing kernel Hilbert space (RKHS) on $\\boldsymbol{\\mathcal{X}}\\subset\\mathbb{R}^{d}$ with respect to a continuous kernel function $k=k_{d}$ satisfying\n\n$$\n\\sup_{\\boldsymbol{x}\\in\\mathcal{X}}k_{d}(\\boldsymbol{x},\\boldsymbol{x})\\leq\\kappa^{2},\n$$\n\nwhere κ is an absolute constant.\n\nFor example, in Section 3.2, we consider $k_{d}$ as the inner product kernel on the unit sphere, which takes the form $k_{d}({\\pmb x},{\\pmb x}^{\\prime})=\\Phi(\\langle{\\pmb x},{\\pmb x}^{\\prime}\\rangle)$ , where $\\pmb{x},\\pmb{x}^{\\prime}\\in\\mathbb{S}^{d}$ and $\\Phi(\\cdot)$ is a function independent of the dimension d (see Assumption 4).\n\nBy convention in statistical learning, we consider the large-sample-size limit, i.e., the case where the sample size n is suficiently large, although our results are non-asymptotic. This paper allows the dimension d to grow as n grows, meaning that we consider $d=d(n)$ and a sequence of estimation problems:\n\nEstimating {f<sup>∗</sup><sub>ρ</sub> }<sub>d≥1</sub> using the RKHS {H<sub>d</sub>}<sub>d≥1</sub> or kernel {k<sub>d</sub>}<sub>d≥1</sub>.\n\nIn Section 3.1, we will consider the general case $d=d(n)$ , where d can be fixed or vary with n. In Section 3.2 and Section 3.3, we consider a specific large-dimensional setting, where $n\\asymp d^{\\gamma}$ for some $\\gamma > 0$ . In both cases, we suppose that Assumption 1 holds uniformly for all $d \\geq 1$ . For brevity, in the remainder of this paper, we frequently omit the subscript d in $\\mu_{d},\\rho_{d},f_{\\rho_{d}}^{*},k_{d},\\mathcal{H}_{d}.$ and other quantities that depend on d.\n\nSuppose that the samples $\\{(x_{i},y_{i})\\}_{i=1}^{n}$ are i.i.d. sampled from $\\rho .$ Kernel ridge regression (KRR) constructs an estimator ${\\ddot{f}}_{\\lambda}$ by solving the penalized least squares problem\n\n$$\n\\hat{f}_{\\lambda}=\\underset{f\\in\\mathcal{H}}{\\arg\\min}\\left(\\frac{1}{n}\\sum_{i=1}^{n}\\left(y_{i}-f(\\boldsymbol{x}_{i})\\right)^{2}+\\lambda\\|f\\|_{\\mathcal{H}}^{2}\\right),\n$$\n\nwhere $\\lambda > 0$ is referred to as the regularization parameter.\n\nDenote the samples as $\\pmb{X}=(\\pmb{x}_{1},\\cdots,\\pmb{x}_{n})$ and $\\mathbf{y}=\\left(y_{1},\\ldots,y_{n}\\right)^{\\top}$ . The representer theorem (see, e.g., Steinwart and Christmann 2008) provides an explicit formula for the KRR estimator:\n\n$$\n\\hat{f}_{\\lambda}(\\boldsymbol{x})=\\mathbb{K}(\\boldsymbol{x},\\boldsymbol{X})(\\mathbb{K}(\\boldsymbol{X},\\boldsymbol{X})+n\\lambda\\mathbf{I})^{-1}\\mathbf{y},\\tag{1}\n$$\n\nwhere\n\n$$\n\\mathbb{K}(\\pmb{X},\\pmb{X})=(k(\\pmb{x}_{i},\\pmb{x}_{j}))_{n\\times n},\\mathbb{K}(\\pmb{x},\\pmb{X})=(k(\\pmb{x},\\pmb{x}_{1}),\\dots,k(\\pmb{x},\\pmb{x}_{n})).\n$$\n\nWe aim to analyze the convergence rate of the generalization error (excess risk) of $\\hat{f}_{\\lambda};$\n\n$$\n\\mathbb{E}_{\\pmb{x}\\sim\\mu}\\left[\\left(\\hat{f}_{\\lambda}(\\pmb{x})-f_{\\rho}^{*}(\\pmb{x})\\right)^{2}\\right]=\\left\\|\\hat{f}_{\\lambda}-f_{\\rho}^{*}\\right\\|_{L^{2}(\\mathcal{X},\\mu)}^{2}.\n$$\n\nNotations. We use the standard asymptotic notations $O(\\cdot),\\ o(\\cdot),\\ \\Omega(\\cdot)$ , and $\\Theta(\\cdot)$ . We also write $a_{n}\\asymp b_{n}$ for $a_{n}=\\Theta(b_{n});a_{n}\\lesssim b_{n}$ for $a_{n}=O(b_{n});a_{n}\\gtrsim b_{n}$ for $a_{n}=\\Omega(b_{n});a_{n}\\ll b_{n}$ for $a_{n}=o(b_{n})$ . We will also use the probability versions of the asymptotic notations such as $O_{\\mathbb{P}}(\\cdot),o_{\\mathbb{P}}(\\cdot),\\Omega_{\\mathbb{P}}(\\cdot)$ , and $\\Theta_{\\mathbb{P}}(\\cdot)$ . For instance, we say the random variables $X_{n},Y_{n}$ satisfying $X_{n}={\\cal O}_{\\mathbb{P}}(Y_{n})$ if and only if for any $\\varepsilon > 0$ , there exist a constant $C_{\\varepsilon}$ and $N_{\\varepsilon}$ such that $P\\left(|X_{n}|\\geq C_{\\varepsilon}|Y_{n}|\\right)\\leq\\varepsilon,\\forall n>N_{\\varepsilon}$\n\nLet $S_{k}:\\mathcal{H}\\to L^{2}(\\mathcal{X},\\mu)$ denote the natural embedding operator. Its adjoint operator $S_{k}^{*}:L^{2}(\\mathcal{X},\\mu)\\to\\mathcal{H}$ is an integral operator. Specifically, for $f\\in L^{2}(\\mathcal{X},\\mu)$ and $\\mathbf{\\boldsymbol{x}}\\in\\mathcal{X}$ ，\n\n$$\n(S_{k}^{*}f)(\\pmb{x})=\\int_{\\mathcal{X}}k(\\pmb{x},\\pmb{x}^{\\prime})f(\\pmb{x}^{\\prime})\\mathrm{d}\\mu(\\pmb{x}^{\\prime}).\n$$\n\nUnder Assumption $1,S_{k}$ and $S_{k}^{*}$ are Hilbert-Schmidt (HS) operators and therefore compact. Their HS norms (denoted as $\\|{\\cdot}\\|_{2})$ satisfy\n\n$$\n\\|S_{k}^{*}\\|_{2}=\\|S_{k}\\|_{2}=\\|k\\|_{L^{2}(\\mathcal{X},\\mu)}:=\\left(\\int_{\\mathcal{X}}k(\\boldsymbol{x},\\boldsymbol{x})\\mathrm{d}\\mu(\\boldsymbol{x})\\right)^{1/2}\\leq\\kappa .\n$$\n\nNext, we define two integral operators:\n\n$$\nL_{k}:=S_{k}S_{k}^{*}:L^{2}(\\mathcal{X},\\mu)\\to L^{2}(\\mathcal{X},\\mu),\\quad T:=S_{k}^{*}S_{k}:\\mathcal{H}\\to\\mathcal{H}.\\tag{2}\n$$\n\n$L_{k}$ and $T$ are self-adjoint, positive-definite and trace class (hence Hilbert-Schmidt and compact). Their trace norms (denoted as $\\left.\\cdot\\right._{1})$ satisfy\n\n$$\n\\|L_{k}\\|_{1}=\\|T\\|_{1}=\\|S_{k}\\|_{2}^{2}=\\|S_{k}^{*}\\|_{2}^{2}.\n$$\n\nThe spectral theorem for self-adjoint compact operators yields that there is an at most countable index set $N .$ , a non-increasing summable sequence $\\{\\lambda_{i}\\}_{i\\in N}\\subseteq(0,\\infty)$ and a family $\\{e_{i}\\}_{i\\in N}\\subseteq\\mathcal{H}$ , such that $\\{e_{i}\\}_{i\\in N}$ is an orthonormal basis (ONB) of Ran ${\\overline{{S_{k}}}}\\subseteq L^{2}({\\mathcal{X}},{\\mu})$ and $\\{\\lambda_{i}^{1/2}e_{i}\\}_{i\\in N}$ is an ONB of H. Further, the integral operators can be written as\n\n$$\nL_{k}=\\sum_{i\\in N}\\lambda_{i}\\left\\langle\\cdot,e_{i}\\right\\rangle_{L^{2}}e_{i}\\quad \\text{and}\\quad T=\\sum_{i\\in N}\\lambda_{i}\\left\\langle\\cdot,\\lambda_{i}^{1/2}e_{i}\\right\\rangle_{\\mathcal{H}}\\lambda_{i}^{1/2}e_{i}.\n$$\n\nWe refer to $\\{e_{i}\\}_{i\\in N}$ and $\\{\\lambda_{i}\\}_{i\\in N}$ as the eigenfunctions and eigenvalues. Mercer’s theorem $(\\mathrm{see},\\mathrm{e . g .}$ , Steinwart and Christmann 2008, Theorem 4.49) states that\n\n$$\nk\\left(\\boldsymbol{x},\\boldsymbol{x}^{\\prime}\\right)=\\sum_{i\\in N}\\lambda_{i}e_{i}(\\boldsymbol{x})e_{i}\\left(\\boldsymbol{x}^{\\prime}\\right),\\quad\\boldsymbol{x},\\boldsymbol{x}^{\\prime}\\in\\mathcal{X},\n$$\n\nwhere the convergence is absolute and uniform in $\\boldsymbol{x},\\boldsymbol{x}^{\\prime}$.\n\nSince we are going to consider the source condition in subsequent sections, we now introduce the interpolation spaces (power spaces) of RKHS. For any $s \\geq 0$ , the fractional power operator $L_{k}^{s}:L^{2}(\\mathcal X,\\mu)\\to L^{2}(\\mathcal X,\\mu)$ is defined as\n\n$$\nL_{k}^{s}(f)=\\sum_{i\\in N}\\lambda_{i}^{s}\\left\\langle f,e_{i}\\right\\rangle_{L^{2}}e_{i}.\n$$\n\nThen the interpolation space (power space) $[\\mathcal{H}]^{s}$ is given by\n\n$$\n\\left[\\mathcal{H}\\right]^{s}:=\\operatorname{Ran}L_{k}^{s/2}=\\left\\{\\sum_{i\\in N}a_{i}\\lambda_{i}^{s/2}e_{i}:(a_{i})_{i\\in N}\\in\\ell_{2}(N)\\right\\}\\subseteq L^{2}(\\mathcal{X},\\mu),\\tag{3}\n$$\n\nequipped with the inner product\n\n$$\n\\langle f,g\\rangle_{[\\mathcal{H}]^{s}}=\\left\\langle L_{k}^{-s/2}f,L_{k}^{-s/2}g\\right\\rangle_{L^{2}}.\n$$\n\nIt follows that $[{\\mathcal{H}}]^{s}$ is also a separable Hilbert space with orthogonal basis $\\{\\lambda_{i}^{s/2}e_{i}\\}_{i\\in N}$ . In particular, we have $[\\mathcal{H}]^{1}=\\mathcal{H}$ . For $0<s_{1}<s_{2}$ , the embeddings $[\\mathcal{H}]^{s_{2}}\\hookrightarrow [\\mathcal{\\dot{H}}]^{\\check{s}_{1}}\\hookrightarrow [\\mathcal{H}]^{0}\\hookrightarrow$ $L^{2}(\\mathcal{X},\\mu)$ are well-defined and compact (Fischer and Steinwart, 2020). For the functions in $[{\\mathcal{H}}]^{s}$ with larger $s,$ we say they have higher regularity (smoothness) with respect to the RKHS. In the remainder of this paper, we assume $| N | = \\infty$ . Also note that $\\{\\lambda_{i}\\}_{i=1}^{\\infty}$ and $\\{e_{i}\\}_{i=1}^{\\infty}$ are dependent on ${\\mathcal{H}},$ and hence depend on d.\n\nGiven the RKHS H and the corresponding eigenvalues and eigenfunctions, the true function can be decomposed as $\\begin{array}{ r }{f_{\\rho}^{*}=\\sum_{i=1}^{\\infty}f_{i}e_{i}({\\pmb x})\\in L^{2}({\\pmb\\chi},{\\mu})}\\end{array}$ for some sequence $\\{f_{i}\\}_{i=1}^{\\infty}$ . We define the following important quantities, which are determined by $\\{\\lambda_{i}\\}_{i=1}^{\\infty},\\{e_{i}\\}_{i=1}^{\\infty},\\{f_{i}\\}_{i=1}^{\\infty}$ and regularization parameter λ:\n\n$$\n\\mathcal{N}_{1}(\\lambda)=\\sum_{i=1}^{\\infty}\\left(\\frac{\\lambda_{i}}{\\lambda_{i}+\\lambda}\\right);\\quad\\mathcal{N}_{2}(\\lambda)=\\sum_{i=1}^{\\infty}\\left(\\frac{\\lambda_{i}}{\\lambda_{i}+\\lambda}\\right)^{2};\\tag{4}\n$$\n\n$$\n\\mathcal{M}_{1}(\\lambda)=\\operatorname*{esssup}_{\\boldsymbol{x}\\in\\mathcal{X}}\\left|\\sum_{i=1}^{\\infty}\\left(\\frac{\\lambda}{\\lambda_{i}+\\lambda}f_{i}e_{i}(\\boldsymbol{x})\\right)\\right|;\\mathcal{M}_{2}(\\lambda)=\\sum_{i=1}^{\\infty}\\left(\\frac{\\lambda}{\\lambda_{i}+\\lambda}f_{i}\\right)^{2}.\n$$\n\nAssumption 2 Suppose that for some absolute constant $\\sigma > 0$ ，\n\n$$\n\\mathbb{E}_{(\\boldsymbol{x},y)\\sim\\rho}\\left[\\left(y-f_{\\rho}^{*}(\\boldsymbol{x})\\right)^{2}\\Big|\\boldsymbol{x}\\right]=\\sigma^{2},\\quad\\mu \\text{-a.e.}\\boldsymbol{x}\\in\\mathcal{X}.\n$$\n\nAssumption 2 ensures that the noise is non-vanishing, a condition satisfied in the standard nonparametric regression model $y=f_{\\rho}^{*}(\\pmb{x})+\\epsilon$ where ε is an independent nonzero noise.\n\nAssumption 3 Suppose that\n\n$$\n\\operatorname*{esssup}_{\\boldsymbol{x}\\in\\mathcal{X}}\\sum_{i=1}^{\\infty}\\left(\\frac{\\lambda_{i}}{\\lambda_{i}+\\lambda}\\right)^{2}e_{i}^{2}(\\boldsymbol{x})\\leq\\mathcal{N}_{2}(\\lambda),\\tag{5}\n$$\n\nand\n\n$$\n\\operatorname*{esssup}_{\\boldsymbol{x}\\in\\mathcal{X}}\\sum_{i=1}^{\\infty}\\frac{\\lambda_{i}}{\\lambda_{i}+\\lambda}e_{i}^{2}(\\boldsymbol{x})\\leq\\mathcal{N}_{1}(\\lambda).\\tag{6}\n$$\n\nSimilar to Assumption 1, we suppose that Assumption 2 and 3 hold uniformly for all $d \\geq 1$ . Assumption 3 naturally holds for RKHSs with uniformly bounded eigenfunctions, i.e., $\\begin{array}{ r }{\\operatorname*{sup}_{i\\geq 1}\\operatorname*{sup}_{{\\pmb x}\\in{\\mathcal{X}}}|e_{i}({\\pmb x})|\\leq 1}\\end{array}$ . Additionally, RKHSs associated with the inner product kernel on the unit sphere under the uniform distribution satisfy Assumption 3 (see Lemma 22).\n\nIn this subsection, we consider the inner product kernel on the unit sphere $\\mathbb{S}^{d}$ with uniform distribution. In the large-dimensional setting $n\\asymp d^{\\gamma},\\gamma\\searrow 0$ , and under an additional source condition assumption, we apply Theorem 1 to establish the convergence rates of the generalization error of the KRR estimator. We then derive the corresponding minimax lower bound, allowing us to discuss the minimax optimality and the saturation efect of KRR.\n\nSuppose that $\\mathcal{X}=\\mathbb{S}^{d}$ and $\\mu$ is the uniform distribution on $\\mathbb{S}^{d}$ . We consider the inner product kernel, i.e., there exists a function $\\Phi(t):[-1,1]\\ \\to\\ \\mathbb{R}$ such that $k_{d}({\\pmb x},{\\pmb x}^{\\prime})=$ Φ $\\ L\\left(\\left.\\pmb{x},\\pmb{x}^{\\prime}\\right.\\right),\\forall\\pmb{x},\\pmb{x}^{\\prime}\\in\\mathbb{S}^{d}$ . Then Mercer’s decomposition for the inner product kernel is then expressed in the basis of spherical harmonics:\n\n$$\nk_{d}(\\boldsymbol{x},\\boldsymbol{x}^{\\prime})=\\sum_{k=0}^{\\infty}\\mu_{k}\\sum_{l=1}^{N(d,k)}Y_{k,l}(\\boldsymbol{x})Y_{k,l}(\\boldsymbol{x}^{\\prime}),\n$$\n\nwhere $\\{Y_{k,l}\\}_{l=1}^{N(d,k)}$ are spherical harmonic polynomials of degree $k;\\mu_{k}$ are the eigenvalues (also depending on d) with multiplicity $\\begin{array}{ r }{N(d,0)=1;N(d,k)=\\frac{2 k+d-1}{k}\\frac{(k+d-2)!}{(d-1)!(k-1)!},k=1,2,\\cdot\\cdot\\cdot}\\end{array}$\n\nAssumption 4 (Inner product kernel) Suppose that $k=\\{k_{d}\\}_{d=1}^{\\infty}$ satisfies\n\n$$\nk_{d}(\\pmb{x},\\pmb{x}^{\\prime})=\\Phi\\left(\\langle\\pmb{x},\\pmb{x}^{\\prime}\\rangle\\right),\\forall\\pmb{x},\\pmb{x}^{\\prime}\\in\\mathbb{S}^{d},\n$$\n\nwhere $\\Phi(t)\\in\\mathcal{C}^{\\infty}\\left([-1,1]\\right)$ is a fixed function independent of d and\n\n$$\n\\Phi(t)=\\sum_{j=0}^{\\infty}a_{j}t^{j},a_{j}>0,\\forall j=0,1,2,\\dots\n$$\n\nAssumption 4 formally defines the kernel considered in this subsection. We assume all coeficients $\\{a_{j}\\}_{j=0}^{\\infty}$ are positive to simplify the main results and proofs. In fact, the proof remains similar for other inner product kernels, provided that the positive coeficients can be identified, for example, the neural tangent kernel in Section 3.3. We assume $\\Phi(t)$ to be fixed (i.e., assume $\\{a_{j}\\}_{j=0}^{\\infty}$ to be independent of $d)$ and ignore the dependence of constants on $\\{a_{j}\\}_{j=0}^{\\infty}$ in the rest of our paper.\n\nWe next introduce the source condition, which characterizes the relative smoothness of $f_{\\rho}^{\\ast}$ with respect to $\\mathcal{H}$\n\nAssumption 5 (Source condition)\n\n(a) Suppose that $f_{\\rho}^{*}(\\pmb{x})=f_{\\rho_{d}}^{*}(\\pmb{x})=\\sum_{i=1}^{\\infty}f_{i}e_{i}(\\pmb{x})\\in [\\mathcal{H}]^{s}$ for some $s > 0$ and satisfies that,\n\n$$\n\\left\\|f_{\\rho}^{*}\\right\\|_{[\\mathcal{H}]^{s}}\\leq R_{\\gamma},\n$$\n\nwhere $R_{\\gamma}$ is a constant only depending on $\\gamma .$\n\n(b) Denote q as the smallest integer such that $q > \\gamma$ and $\\mu_{q}\\neq 0$ . For $k \\in \\mathbb N$ , define $\\mathcal{T}_{k}$ as the index set satisfying $\\lambda_{i}\\equiv\\mu_{k},i\\in\\mathcal{I}_{k}$ . Further suppose that there exists an absolute constant $c_{0}>0$ such that for any d and $k\\in\\{0,1,\\cdots,q\\}$ with $\\mu_{k}\\neq 0$ , we have\n\n$$\n\\sum_{i\\in\\mathcal{I}_{k}}\\mu_{k}^{-s}f_{i}^{2}\\geq c_{0}.\\tag{10}\n$$\n\nNote that $f_{\\rho}^{\\ast}$ and H vary with $d,$ thus $\\{f_{i}\\}_{i=1}^{\\infty},\\{\\mu_{k}\\}_{k=0}^{\\infty},\\{\\mathcal{T}_{k}\\}_{k=0}^{\\infty}$ also depend on d. We omit the subscript d and assume Assumption 5 to hold uniformly for all $d \\geq 1$\n\nThe proof of Theorem 1 is based on the traditional bias-variance decomposition. The contribution in this paper is that we refine the tools in Li et al. (2024) and Li et al. (2023b) to extend the applicability to the large-dimensional case. Throughout the proof, we denote\n\n$$\nT_{\\lambda}=T+\\lambda;\\quad T_{\\boldsymbol{X}\\lambda}=T_{\\boldsymbol{X}}+\\lambda,\\tag{14}\n$$\n\nwhere λ is the regularization parameter. $T + \\lambda$ actually means $T + \\lambda I$ , where I is the identity operator. We use $\\|\\cdot\\|_{\\mathcal{B}(B_{1},B_{2})}$ to denote the operator norm of a bounded linear operator from a Banach space $B_{1}$ to $B_{2}$ , i.e., $\\|A\\|_{\\mathcal{B}(B_{1},B_{2})}=\\operatorname*{sup}_{\\|f\\|_{B_{1}}=1}\\|A f\\|_{B_{2}}$ . For brevity, we denote it simply as $\\| \\cdot \\|$ where no ambiguity arises. In addition, we use trA and $\\|A\\|_{1}$ to denote the trace and the trace norm of an operator. We denote $\\|A\\|_{2}$ as the Hilbert-Schmidt norm. In addition, we use $L^{2}(\\mathcal{X},\\mu)$ and $L^{\\infty}(\\mathcal{X},\\mu)$ simply as $L^{2}$ and $L^{\\infty}$ throughout the proof for brevity.\n\nWe also need the following essential notations, which are frequently used in related literature. Denote the samples $\\pmb{Z}=\\{(\\pmb{x}_{i},y_{i})\\}_{i=1}^{n}$ . Define the sampling operator $K_{x}:\\mathbb{R} $ $\\mathcal{H},\\ y\\mapsto\\ y k(\\pmb{x},\\cdot)$ and its adjoint operator $K_{\\pmb{x}}^{*}:\\mathcal{H}\\mathbb{R},\\ f\\mapsto f(\\pmb{x})$ . Then we can define $T_{x}=K_{x}K_{x}^{*}$ . Furthermore, we define the sample covariance operator $T_{X}:{\\mathcal{H}}\\to{\\mathcal{H}}$ as\n\n$$\nT_{\\boldsymbol{X}}:=\\frac{1}{n}\\sum_{i=1}^{n}K_{\\boldsymbol{x}_{i}}K_{\\boldsymbol{x}_{i}}^{*}.\\tag{15}\n$$\n\nThen we know that $\\|T_{\\mathbf{\\mathbf{X}}}\\|\\le\\|T_{\\mathbf{\\mathbf{X}}}\\|_{1}\\le\\kappa^{2}$ and $T_{X}$ is a trace class thus compact operator. Furthermore, define the sample basis function\n\n$$\ng_{\\boldsymbol{Z}}:=\\frac{1}{n}\\sum_{i=1}^{n}K_{\\boldsymbol{x}_{i}}y_{i}\\in\\mathcal{H}.\n$$\n\nFollowing Caponnetto and de Vito (2007), the KRR estimator (1) can be expressed in operator form as\n\n$$\n\\hat{f}_{\\lambda}=(T_{\\boldsymbol{X}}+\\lambda)^{-1}g_{\\boldsymbol{Z}},\n$$\n\nIn order to derive the bias term, we define\n\n$$\n\\tilde{g}_{\\boldsymbol{Z}}:=\\mathbb{E}\\left(g_{\\boldsymbol{Z}}|\\boldsymbol{X}\\right)=\\frac{1}{n}\\sum_{i=1}^{n}K_{\\boldsymbol{x}_{i}}f_{\\rho}^{*}(\\boldsymbol{x}_{i})\\in\\mathcal{H};\n$$\n\nand\n\n$$\n\\tilde{f}_{\\lambda}:=\\mathbb{E}\\left(\\hat{f}_{\\lambda}|\\boldsymbol{X}\\right)=(T_{\\boldsymbol{X}}+\\lambda)^{-1}\\tilde{g}_{\\boldsymbol{Z}}\\in\\mathcal{H}.\\tag{16}\n$$\n\nWe also need to define the expectation of $g z$ as\n\n$$\ng=\\mathbb{E}g_{\\boldsymbol{Z}}=\\int_{\\mathcal{X}}k(\\boldsymbol{x},\\cdot)f_{\\rho}^{*}(\\boldsymbol{x})d\\mu(\\boldsymbol{x})=S_{k}^{*}f_{\\rho}^{*}\\in\\mathcal{H},\n$$\n\nand\n\n$$\nf_{\\lambda}=(T+\\lambda)^{-1}g=(T+\\lambda)^{-1}S_{k}^{*}f_{\\rho}^{*}.\\tag{17}\n$$\n\nOur next goal is to prove that second term in (41) is higher order infinitesimal, i.e., $\\Big\\|\\tilde{f}_{\\lambda}-f_{\\lambda}\\Big\\|_{L^{2}}=o_{\\mathbb{P}}(\\mathcal{M}_{2}(\\lambda)^{\\frac{1}{2}})$\n\nLemma 17 Suppose that Assumption 1, 2 and 3 hold. If the following approximation conditions hold for some $\\lambda=\\lambda(d,n)\\to 0$ :\n\n$$\n\\frac{\\mathcal{N}_{1}(\\lambda)}{n}\\ln n=o(1);\\quad n^{-1}\\mathcal{N}_{1}(\\lambda)^{\\frac{1}{2}}\\mathcal{M}_{1}(\\lambda)=o\\left(\\mathcal{M}_{2}(\\lambda)^{\\frac{1}{2}}\\right),\\tag{43}\n$$\n\nthen we have\n\n$$\n\\left\\|\\tilde{f}_{\\lambda}-f_{\\lambda}\\right\\|_{L^{2}}=o_{\\mathbb{P}}(\\mathcal{M}_{2}(\\lambda)^{\\frac{1}{2}}).\n$$\n\nProof To begin with, by definition, we rewrite $\\left\\|\\tilde{f}_{\\lambda}-f_{\\lambda}\\right\\|_{L_{2}}$ as follows\n\n$$\n\\begin{array}{l}\\left\\|\\tilde{f}_{\\lambda}-f_{\\lambda}\\right\\|_{L_{2}}=\\left\\|S_{k}\\left(\\tilde{f}_{\\lambda}-f_{\\lambda}\\right)\\right\\|_{L^{2}}\\\\ =\\left\\|S_{k}T_{\\lambda}^{-\\frac{1}{2}}\\cdot T_{\\lambda}^{\\frac{1}{2}}T_{\\boldsymbol{X}\\lambda}^{-1}T_{\\lambda}^{\\frac{1}{2}}\\cdot T_{\\lambda}^{-\\frac{1}{2}}T_{\\boldsymbol{X}\\lambda}\\left(\\tilde{f}_{\\lambda}-f_{\\lambda}\\right)\\right\\|_{L^{2}}\\\\ \\leq\\left\\|S_{k}T_{\\lambda}^{-\\frac{1}{2}}\\right\\|_{\\mathscr{B}(\\mathcal{H},L^{2})}\\cdot\\left\\|T_{\\lambda}^{\\frac{1}{2}}T_{\\boldsymbol{X}\\lambda}^{-1}T_{\\lambda}^{\\frac{1}{2}}\\right\\|_{\\mathscr{B}(\\mathcal{H},\\mathcal{H})}\\cdot\\left\\|T_{\\lambda}^{-\\frac{1}{2}}\\left(\\tilde{g}_{\\boldsymbol{Z}}-T_{\\boldsymbol{X}\\lambda}f_{\\lambda}\\right)\\right\\|_{\\mathcal{H}}.\\tag{44}\\\\ \\end{array}\n$$\n\n(iii) For the third term in (44), it can be rewritten as\n\n$$\n\\begin{array}{l}\\left\\|T_{\\lambda}^{-\\frac{1}{2}}\\left(\\tilde{g}_{\\boldsymbol{Z}}-T_{\\boldsymbol{X}\\lambda}f_{\\lambda}\\right)\\right\\|_{\\mathcal{H}}=\\left\\|T_{\\lambda}^{-\\frac{1}{2}}\\left[\\left(\\tilde{g}_{\\boldsymbol{Z}}-\\left(T_{\\boldsymbol{X}}+\\lambda+T-T\\right)f_{\\lambda}\\right)\\right]\\right\\|_{\\mathcal{H}}\\\\ =\\left\\|T_{\\lambda}^{-\\frac{1}{2}}\\left[\\left(\\tilde{g}_{\\boldsymbol{Z}}-T_{\\boldsymbol{X}}f_{\\lambda}\\right)-(T+\\lambda)f_{\\lambda}+T f_{\\lambda}\\right]\\right\\|_{\\mathcal{H}}\\\\ =\\left\\|T_{\\lambda}^{-\\frac{1}{2}}\\left[\\left(\\tilde{g}_{\\boldsymbol{Z}}-T_{\\boldsymbol{X}}f_{\\lambda}\\right)-(g-T f_{\\lambda})\\right]\\right\\|_{\\mathcal{H}}.\\tag{47}\\\\ \\end{array}\n$$\n\nIn this section, we aim to apply Theorem 1 to prove the results in Section 3.2. We will see that the application is nontrivial and is an important contribution of this paper.\n\nWe first introduce more necessary preliminaries in Appendix B.1, which is a preparation for subsequent calculations. Next, in order to apply Theorem 1 to get specific convergence rates, we calculate the exact convergence rates of the key quantities therein in Appendix B.2. Finally, we state the proof of Theorem 4 and Theorem 5 in turn in Appendix B.3 and B.4. We will see that there are essential diferences in the proof of these two theorems.\n\nSuppose that $\\mathcal{X}=\\mathbb{S}^{d}$ and $\\mu$ is the uniform distribution on $\\mathbb{S}^{d}$ . Recall that in Section 3.2, we consider the inner product kernel, i.e., there exists a function $\\Phi(t):[-1,1]\\to $ R such that $k({\\pmb x},{\\pmb x}^{\\prime})=\\Phi\\left(\\langle{\\pmb x},{\\pmb x}^{\\prime}\\rangle\\right),\\forall{\\pmb x},{\\pmb x}^{\\prime}\\in\\mathbb{S}^{d}$ . Then Mercer’s decomposition for the inner product kernel is given in the basis of spherical harmonics:\n\n$$\nk(\\boldsymbol{x},\\boldsymbol{y})=\\sum_{k=0}^{\\infty}\\mu_{k}\\sum_{l=1}^{N(d,k)}Y_{k,l}(\\boldsymbol{x})Y_{k,l}(\\boldsymbol{y}),\\tag{54}\n$$\n\nwhere $\\{Y_{k,l}\\}_{l=1}^{N(d,k)}$ are spherical harmonic polynomials of degree $k;\\mu_{k}$ are the eigenvalues with multiplicity $\\begin{array}{ r }{N(d,0)=1;N(d,k)=\\frac{2 k+d-1}{k}\\cdot\\frac{(k+d-2)!}{(d-1)!(k-1)!},k=1,2,\\cdot\\cdot\\cdot}\\end{array}$\n\nLemma 19 Let $k=k_{d}$ be the inner product kernel on $\\mathbb{S}^{d}$ satisfying Assumption 1 and $\\it 4 .$ For any fixed integer $p \\geq 0$ , there exist constants ${\\mathfrak{C}},{\\mathfrak{C}}_{1}$ and ${\\mathfrak{C}}_{2}$ only depending on $p$ and $\\{a_{j}\\}_{j\\le p+1}$ , such that for any $d\\geq\\mathfrak{C}$ , we have\n\n$$\n\\mathfrak{C}_{1}d^{-k}\\leq\\mu_{k}\\leq\\mathfrak{C}_{2}d^{-k},k=0,1,\\dots,p+1.\n$$\n\nLemma 20 Let $k=k_{d}$ be the inner product kernel on $\\mathbb{S}^{d}$ satisfying Assumption 1 and $\\it 4 .$ For any fixed integer $p \\geq 0$ , there exist constants C only depending on p and $\\{a_{j}\\}_{j\\le p+1}$ , such that for any $d\\geq\\mathfrak{C}$ , we have\n\n$$\n\\mu_{k}\\leq\\frac{\\mathfrak{C}_{2}}{\\mathfrak{C}_{1}}d^{-1}\\mu_{p},\\quad k=p+1,p+2,\\dots\n$$\n\nwhere ${\\mathfrak{C}}_{1}$ and ${\\mathfrak{C}}_{2}$ are constants given in Lemma 19.\n\nLemma 21 For an integer $k\\geq 0,$ , denote $N(d,k)$ as the multiplicity of the eigenspace corresponding to $\\mu_{k}$ in the Mercer’s decomposition. For any fixed integer $p \\geq 0$ , there exist constants ${\\mathfrak{C}}_{3},{\\mathfrak{C}}_{4}$ and C only depending on $p,$ such that for any $d\\geq\\mathfrak{C}$ , we have\n\n$$\n\\mathfrak{C}_{3}d^{k}\\leq N(d,k)\\leq\\mathfrak{C}_{4}d^{k},\\quad k=0,1,\\dots,p+1.\\tag{56}\n$$\n\nThe following lemma verifies that if we consider the inner product kernel on the unit sphere, then Assumption 3 naturally holds.\n\nLemma 22 Suppose that $\\mathcal{X}=\\mathbb{S}^{d}$ and $\\mu$ is the uniform distribution on $\\mathbb{S}^{d}$ . Suppose that k is an inner product kernel, then Assumption 3 holds.\n\nBased on the information of the eigenvalues in the last subsection, this subsection determines the exact convergence rates of the quantities appeared in Theorem 1. These rates will finally determine the convergence rates in Theorem 4 and Theorem 5. Note that we assume that d grows with n in Theorem 4 and Theorem 5 .\n\nLemma 24 Consider $\\mathcal{X}=\\mathbb{S}^{d}$ and the marginal distribution µ to be the uniform distribution. Let $k=k_{d}$ be the inner product kernel on $\\mathbb{S}^{d}$ satisfying Assumption 1 and 4. Further suppose that Assumption 5 holds for some $s > 0$ . Suppose that $\\alpha, \\beta$ are two real numbers such that\n\n$$\ns+\\alpha-\\beta\\leq 0;\\quad s+\\alpha\\geq 0.\\tag{63}\n$$\n\nThen by choosing $\\lambda=d^{-l}$ for some $0 < l < \\gamma$ , if $p \\leq l \\leq p + 1$ for some $p\\in\\{0,1,2\\cdots\\}$ , we have\n\n$$\n\\mathcal{M}_{\\alpha,\\beta}(\\lambda):=\\sum_{i=1}^{\\infty}\\frac{\\lambda_{i}^{\\alpha}}{(\\lambda_{i}+\\lambda)^{\\beta}}f_{i}^{2}=\\Theta\\left(d^{-(s+\\alpha-\\beta)p}+\\lambda^{-\\beta}d^{-(p+1)(s+\\alpha)}\\right).\n$$\n\nThe notation Θ involves constants only depending on $s,p,c_{0}$ and $R_{\\gamma}$ , where $c_{0}$ and $R_{\\gamma}$ are the constants from Assumption 5.\n\nLemma 39 Given the definition of $\\mathcal{N}_{1}(\\lambda)$ as in (4). If the condition (6) in Assumption 3 holds, we have\n\n$$\n\\|T_{\\lambda}^{-\\frac{1}{2}}k(\\boldsymbol{x},\\cdot)\\|_{\\mathcal{H}}^{2}\\leq\\mathcal{N}_{1}(\\lambda),\\mu \\text{-a.e.}\\boldsymbol{x}\\in\\mathcal{X}.\n$$", "theorems": [{"local_id": "T1", "statement": "Lemma 23 Consider $\\mathcal{X}=\\mathbb{S}^{d}$ and the marginal distribution µ to be the uniform distribution. Let $k=k_{d}$ be the inner product kernel on $\\mathbb{S}^{d}$ satisfying Assumption 1 and 4. By choosing $\\lambda=d^{-l}$ for some $l > 0$ , we have:\n\n$$\n\\mathcal{N}_{1}(\\lambda)=\\Theta\\left(\\lambda^{-1}\\right);\\tag{57}\n$$\n\nI ${}^{f}p\\leq l\\leq p+1$ for some $p\\in\\{0,1,2\\cdot\\cdot\\cdot\\}$ , we have\n\n$$\n\\mathcal{N}_{2}(\\lambda)=\\Theta\\left(d^{p}+\\lambda^{-2}d^{-(p+1)}\\right).\\tag{58}\n$$\n\nThe notation Θ involves constants only depending on κ and $p$ .", "proof": {"steps": [{"n": 1, "claim": "Proof If $p \\leq l \\leq p + 1$ for some $p\\in\\{0,1,2\\cdot\\cdot\\cdot\\}$ , Lemma 19 and Lemma 21 show that there exist constants ${\\mathfrak{C}},{\\mathfrak{C}}_{1},{\\mathfrak{C}}_{2},{\\mathfrak{C}}_{3}$ and ${\\mathfrak{C}}_{4}$ only depending on $p$ (recall that we ignore the dependence on $\\{a_{j}\\}_{j=0}^{\\infty})$ , such that for any $d\\geq\\mathfrak{C}$ , we have\n\n$$\n\\mathfrak{C}_{2}^{-1}\\mu_{p+1}\\leq\\lambda\\leq\\mathfrak{C}_{1}^{-1}\\mu_{p};\n$$\n\nand for $k=0,1,\\cdots,p+1$ 2\n\n$$\n\\mathfrak{C}_{1}d^{-k}\\leq\\mu_{k}\\leq\\mathfrak{C}_{2}d^{-k};\\quad\\mathfrak{C}_{3}d^{k}\\leq N(d,k)\\leq\\mathfrak{C}_{4}d^{k}.\n$$"}, {"n": 2, "claim": "We first prove (57). On the one hand, for any $d\\geq\\mathfrak{C}$ , we have\n\n$$\n\\begin{array}{l}\\mathcal{N}_{1}(\\lambda)=\\sum_{k=0}^{p}\\frac{\\mu_{k}}{\\mu_{k}+\\lambda}N(d,k)+\\sum_{k=p+1}^{\\infty}\\frac{\\mu_{k}}{\\mu_{k}+\\lambda}N(d,k)\\\\ \\leq\\sum_{k=0}^{p}\\frac{\\mu_{k}}{\\mu_{k}}N(d,k)+\\sum_{k=p+1}^{\\infty}\\frac{\\mu_{k}}{\\lambda}N(d,k)\\\\ \\leq(p+1)\\mathfrak{C}_{4}d^{p}+\\lambda^{-1}\\sum_{k=p+1}^{\\infty}\\mu_{k}N(d,k)\\\\ \\leq(p+1)\\mathfrak{C}_{4}d^{p}+\\lambda^{-1}\\kappa^{2}\\\\ \\lesssim\\lambda^{-1},\\tag{59}\\\\ \\end{array}\n$$\n\nwhere we use the fact that $\\begin{array}{ r }{\\sum_{k=p+1}^{\\infty}\\mu_{k}N(d,k)\\le\\sum_{i=1}^{\\infty}\\lambda_{i}\\le\\operatorname*{sup}_{x\\in\\mathcal{X}}k(x,x)\\le\\kappa^{2}}\\end{array}$ for the third inequality."}, {"n": 3, "claim": "On the other hand, for any $d\\geq\\mathfrak{C}$ , we have\n\n$$\n\\begin{array}{l}\\mathcal{N}_{1}(\\lambda)=\\sum_{k=0}^{p}\\frac{\\mu_{k}}{\\mu_{k}+\\lambda}N(d,k)+\\sum_{k=p+1}^{\\infty}\\frac{\\mu_{k}}{\\mu_{k}+\\lambda}N(d,k)\\\\ \\geq\\frac{\\mu_{p}}{\\mu_{p}+\\lambda}N(d,p)+\\frac{\\mu_{p+1}}{\\mu_{p+1}+\\lambda}N(d,p+1)\\\\ \\geq\\frac{\\mu_{p}}{\\mu_{p}+\\mathfrak{C}_{1}^{-1}\\mu_{p}}N(d,p)+\\frac{\\mu_{p+1}}{\\mathfrak{C}_{2}\\lambda+\\lambda}N(d,p+1)\\\\ \\geq\\frac{\\mathfrak{C}_{3}}{1+\\mathfrak{C}_{1}^{-1}}d^{p}+\\frac{\\mathfrak{C}_{1}\\mathfrak{C}_{3}}{\\mathfrak{C}_{2}+1}\\lambda^{-1}\\\\ \\gtrsim\\lambda^{-1}.\\tag{60}\\\\ \\end{array}\n$$"}, {"n": 4, "claim": "Combining (59) and (60), we finish the proof of (57)."}, {"n": 5, "claim": "Now we begin to prove (58). On the one hand, for any $d\\geq\\mathfrak{C}$ we have\n\n$$\n\\begin{array}{l}\\mathcal{N}_{2}(\\lambda)=\\sum_{k=0}^{p}\\left(\\frac{\\mu_{k}}{\\mu_{k}+\\lambda}\\right)^{2}N(d,k)+\\sum_{k=p+1}^{\\infty}\\left(\\frac{\\mu_{k}}{\\mu_{k}+\\lambda}\\right)^{2}N(d,k)\\\\ \\leq\\sum_{k=0}^{p}\\left(\\frac{\\mu_{k}}{\\mu_{k}}\\right)^{2}N(d,k)+\\sum_{k=p+1}^{\\infty}\\left(\\frac{\\mu_{k}}{\\lambda}\\right)^{2}N(d,k)\\\\ \\leq(p+1)\\mathfrak{C}_{4}d^{p}+\\lambda^{-2}\\sum_{k=p+1}^{\\infty}\\mu_{k}^{2}N(d,k)\\\\ \\leq(p+1)\\mathfrak{C}_{4}d^{p}+\\lambda^{-2}\\mu_{p+1}\\sum_{k=p+1}^{\\infty}\\mu_{k}N(d,k)\\\\ \\leq(p+1)\\mathfrak{C}_{4}d^{p}+\\lambda^{-2}\\mathfrak{C}_{2}d^{-(p+1)}\\kappa^{2}.\\tag{61}\\\\ \\end{array}\n$$"}, {"n": 6, "claim": "Note that for the forth equation, we use the fact that $\\mu_{k}\\leq\\mu_{p+1},\\forall k\\geq p+1$ (when d is suficiently large), which can be proved by Lemma 20. On the other hand, for any $d\\geq\\mathfrak{C}.$ , we have\n\n$$\n\\begin{array}{l}\\mathcal{N}_{2}(\\lambda)=\\sum_{k=0}^{p}\\left(\\frac{\\mu_{k}}{\\mu_{k}+\\lambda}\\right)^{2}N(d,k)+\\sum_{k=p+1}^{\\infty}\\left(\\frac{\\mu_{k}}{\\mu_{k}+\\lambda}\\right)^{2}N(d,k)\\\\ \\geq\\left(\\frac{\\mu_{p}}{\\mu_{p}+\\mathfrak{C}_{1}^{-1}\\mu_{p}}\\right)^{2}N(d,p)+\\left(\\frac{\\mu_{p+1}}{\\mathfrak{C}_{2}\\lambda+\\lambda}\\right)^{2}N(d,p+1)\\\\ \\geq\\frac{\\mathfrak{C}_{3}}{\\left(1+\\mathfrak{C}_{1}^{-1}\\right)^{2}}d^{p}+\\lambda^{-2}\\left(\\frac{\\mathfrak{C}_{1}}{\\mathfrak{C}_{2}+1}\\right)^{2}\\mathfrak{C}_{3}d^{-(p+1)}.\\tag{62}\\\\ \\end{array}\n$$"}, {"n": 7, "claim": "Combining (61) and (62), we prove that $\\displaystyle{\\mathcal{N}}_{2}(\\lambda)=\\Theta\\left(d^{p}+\\lambda^{-2}d^{-(p+1)}\\right)$"}]}}, {"local_id": "T2", "statement": "Lemma 25 Consider $\\mathcal{X}=\\mathbb{S}^{d}$ and the marginal distribution µ to be the uniform distribution. Let $k=k_{d}$ be the inner product kernel on $\\mathbb{S}^{d}$ satisfying Assumption 1 and 4. Further suppose that Assumption 5 holds for some $s > 0$ . Define $\\tilde{s}=\\operatorname*{min}\\{s,2\\}$ . By choosing $\\lambda=d^{-l}$ for some $0 < l < \\gamma$ , we have: if $p \\leq l \\leq p + 1$ for some $p\\in\\{0,1,2,\\cdots\\}$ , we have\n\n$$\n\\mathcal{M}_{2}(\\lambda)=\\Theta\\left(\\lambda^{2}d^{(2-\\tilde{s})p}+d^{-(p+1)\\tilde{s}}\\right).\\tag{66}\n$$\n\nThe notation Θ involves constants only depending on $s,p,c_{0}$ and $R_{\\gamma}$ , where $c_{0}$ and $R_{\\gamma}$ are the constants from Assumption 5.", "proof": {"steps": [{"n": 1, "claim": "Proof When $0<s\\leq 2,\\mathcal{M}_{2}(\\lambda)$ can be viewed as $\\lambda^{2}{\\mathcal{M}}_{\\alpha,\\beta}(\\lambda)$ in Lemma 24 with $\\alpha=0,\\beta=2$ The conditions (63) are satisfied and Lemma 24 shows that\n\n$$\n\\mathcal{M}_{2}(\\lambda)=\\Theta\\left(\\lambda^{2}d^{(2-s)p}+d^{-(p+1)s}\\right).\\tag{67}\n$$"}, {"n": 2, "claim": "When $s > 2$ , without loss of generality, we can assume $\\lambda_{i}\\leq 1,\\forall i$ and $\\lambda \\leq 1$ . Recall we have proved in Lemma 19 that $\\mu_{0}$ remains as a constant as $d \\to \\infty$ . On the one hand, Assumption 5 (b) also implies $f_{1}^{2}\\geq c_{0}$ , which further implies\n\n$$\n\\mathcal{M}_{2}(\\lambda)=\\lambda^{2}\\sum_{i=1}^{\\infty}\\frac{f_{i}^{2}}{(\\lambda_{i}+\\lambda)^{2}}\\geq\\frac{1}{4}\\lambda^{2}\\sum_{i=1}^{\\infty}f_{i}^{2}\\geq\\frac{1}{4}\\lambda^{2}c_{0}.\\tag{68}\n$$"}, {"n": 3, "claim": "On the other hand, since $s > 2$ , we have\n\n$$\n\\begin{array}{l}\\mathcal{M}_{2}(\\lambda)=\\lambda^{2}\\sum_{k=0}^{\\infty}\\left(\\frac{\\mu_{k}^{s}}{(\\mu_{k}+\\lambda)^{2}}\\sum_{i\\in\\mathcal{I}_{k}}\\mu_{k}^{-s}f_{i}^{2}\\right)\\\\ \\leq\\lambda^{2}\\left(\\sup_{k\\geq 0}\\frac{\\mu_{k}^{s}}{(\\mu_{k}+\\lambda)^{2}}\\cdot\\sum_{k=0}^{\\infty}\\sum_{i\\in\\mathcal{I}_{k}}\\mu_{k}^{-s}f_{i}^{2}\\right)\\\\ \\leq\\lambda^{2}\\cdot\\sup_{k\\geq 0}\\frac{\\mu_{k}^{s}}{(\\mu_{k}+\\lambda)^{2}}\\cdot R_{\\gamma}^{2}\\\\ \\leq\\lambda^{2}R_{\\gamma}^{2}.\\tag{69}\\\\ \\end{array}\n$$"}, {"n": 4, "claim": "Further note that since $s > 2$ and $p \\leq l \\leq p + 1$ , (68) and (69) implies\n\n$$\n\\mathcal{M}_{2}(\\lambda)=\\Theta(\\lambda^{2})=\\Theta(\\lambda^{2}d^{(2-\\tilde{s})p}+d^{-(p+1)\\tilde{s}}).\n$$\n\nWe finish the proof."}]}}, {"local_id": "T3", "statement": "Lemma 26 Consider $\\mathcal{X}=\\mathbb{S}^{d}$ and the marginal distribution µ to be the uniform distribution. Let $k=k_{d}$ be the inner product kernel on $\\mathbb{S}^{d}$ satisfying Assumption 1 and $\\it 4 .$ . Further suppose that Assumption 5 holds for some $s \\geq 1$ . Define $\\tilde{s}=\\operatorname*{min}\\{s,2\\}$ . By choosing $\\lambda=d^{-l}$ for some $0 < l < \\gamma$ , we have: if $p \\leq l \\leq p + 1$ for some $p\\in\\{0,1,2\\cdot\\cdot\\cdot\\}$ , we have\n\n$$\n\\mathcal{M}_{1}(\\lambda)=O\\left(\\lambda^{\\frac{1}{2}}d^{\\frac{(2-\\tilde{s})p}{2}}+d^{-\\frac{(\\tilde{s}-1)(p+1)}{2}}\\right).\\tag{70}\n$$\n\nThe notation O involves constants only depending on $s,\\kappa,p$ and $R_{\\gamma}$ , where $R_{\\gamma}$ is the constant from Assumption 5.", "proof": {"steps": [{"n": 1, "claim": "Proof First, Cauchy-Schwarz inequality shows that\n\n$$\n\\begin{array}{l}\\mathcal{M}_{1}(\\lambda)^{2}=\\underset{\\boldsymbol{x}\\in\\mathcal{X}}{\\text{ess sup}}\\left|\\sum_{i=1}^{\\infty}\\left(\\frac{\\lambda}{\\lambda_{i}+\\lambda}f_{i}e_{i}(\\boldsymbol{x})\\right)\\right|^{2}\\\\ \\leq\\left(\\sum_{i=1}^{\\infty}\\frac{\\lambda^{2}\\lambda_{i}^{-1}}{\\lambda_{i}+\\lambda}f_{i}^{2}\\right)\\cdot\\operatorname{ess}\\sup_{\\boldsymbol{x}\\in\\mathcal{X}}\\sum_{i=1}^{\\infty}\\left(\\frac{\\lambda_{i}}{\\lambda_{i}+\\lambda}e_{i}(\\boldsymbol{x})^{2}\\right)\\\\ \\leq\\left(\\sum_{i=1}^{\\infty}\\frac{\\lambda^{2}\\lambda_{i}^{-1}}{\\lambda_{i}+\\lambda}f_{i}^{2}\\right)\\cdot\\sum_{i=1}^{\\infty}\\left(\\frac{\\lambda_{i}}{\\lambda_{i}+\\lambda}\\right)\\\\ :=\\mathcal{Q}_{1}(\\lambda)\\cdot\\mathcal{N}_{1}(\\lambda),\\tag{71}\\\\ \\end{array}\n$$\n\nwhere we use (6) in Assumption 3 for the second inequality."}, {"n": 2, "claim": "When $1\\leq s\\leq 2,\\mathcal{Q}_{1}(\\lambda)$ defined above can be viewed as $\\lambda^{2}{\\mathcal{M}}_{\\alpha,\\beta}(\\lambda)$ in Lemma 24 with $\\alpha= -1,\\beta=1$ . In addition, the conditions (63) are satisfied, thus Lemma 24 shows that\n\n$$\n\\mathcal{Q}_{1}(\\lambda)=\\Theta\\left(\\lambda^{2}d^{(2-s)p}+\\lambda d^{-(s-1)(p+1)}\\right).\\tag{72}\n$$"}, {"n": 3, "claim": "Since we assume the kernel to be bounded in Assumption 1, we can assume $\\lambda_{i}\\leq 1$ , ∀i and $\\lambda \\leq 1$ without loss of generality. When $s > 2$ , on the one hand, Assumption $5\\ (\\mathrm{b})$ also implies\n\n$$\n\\mathcal{Q}_{1}(\\lambda)=\\lambda^{2}\\sum_{i=1}^{\\infty}\\frac{f_{i}^{2}}{(\\lambda_{i}+\\lambda)\\lambda_{i}}\\geq\\frac{1}{2}\\lambda^{2}\\sum_{i=1}^{\\infty}f_{i}^{2}\\geq\\frac{1}{2}\\lambda^{2}c_{0}.\\tag{73}\n$$"}, {"n": 4, "claim": "On the other hand, since $s > 2$ , we have\n\n$$\n\\begin{array}{l}\\mathcal{Q}_{1}(\\lambda)=\\lambda^{2}\\sum_{k=0}^{\\infty}\\left(\\frac{\\mu_{k}^{s-1}}{\\mu_{k}+\\lambda}\\sum_{i\\in\\mathcal{I}_{k}}\\mu_{k}^{-s}f_{i}^{2}\\right)\\\\ \\leq\\lambda^{2}\\left(\\sup_{k\\geq 0}\\frac{\\mu_{k}^{s-1}}{\\mu_{k}+\\lambda}\\cdot\\sum_{k=0}^{\\infty}\\sum_{i\\in\\mathcal{I}_{k}}\\mu_{k}^{-s}f_{i}^{2}\\right)\\\\ \\leq\\lambda^{2}\\cdot\\frac{\\mu_{k}^{s-1}}{\\mu_{k}+\\lambda}\\cdot R_{\\gamma}^{2}\\\\ \\leq\\lambda^{2}R_{\\gamma}^{2}.\\tag{74}\\\\ \\end{array}\n$$"}, {"n": 5, "claim": "Further note that since $s > 2$ and $p \\leq l \\leq p + 1$ , (73) and (74) implies\n\n$$\n\\mathcal{Q}_{1}(\\lambda)=\\Theta(\\lambda^{2})=\\Theta(\\lambda^{2}d^{(2-\\tilde{s})p}+\\lambda d^{-(\\tilde{s}-1)(p+1)}).\n$$"}, {"n": 6, "claim": "Therefore, we have $\\mathcal{Q}_{1}(\\lambda)=\\Theta\\left(\\lambda^{2}d^{(2-\\tilde{s})p}+\\lambda d^{-(\\tilde{s}-1)(p+1)}\\right)$ for any $s > 0$\n\nFurther recalling that Lemma 23 proves $\\mathcal{N}_{1}(\\lambda)=\\Theta\\left(\\lambda^{-1}\\right)$ , use (71) and we finish the proof."}]}}, {"local_id": "T4", "statement": "Lemma 28 Suppose that Assumption 1, 2 and 3 hold. Further suppose that Assumption 5 holds for some $0 < s < 1$ . If the following approximation conditions hold for some $\\lambda=\\lambda(d,n)\\to 0$:\n\n$$\n\\frac{\\mathcal{N}_{1}(\\lambda)}{n}\\ln n=o(1);\\quad n^{-1}\\mathcal{N}_{1}(\\lambda)^{\\frac{1}{2}}\\left\\|f_{\\lambda}\\right\\|_{L^{\\infty}}=o\\left(\\mathcal{M}_{2}(\\lambda)^{\\frac{1}{2}}\\right);\\tag{80}\n$$\n\nand there exists $\\varepsilon > 0$ , such that\n\n$$\nn^{-1}\\mathcal{N}_{1}(\\lambda)^{\\frac{1}{2}}n^{\\frac{1-s}{2}+\\varepsilon}=o\\left(\\mathcal{M}_{2}(\\lambda)^{\\frac{1}{2}}\\right),\\tag{81}\n$$\n\nthen we have\n\n$$\n\\left\\|\\tilde{f}_{\\lambda}-f_{\\lambda}\\right\\|_{L^{2}}=o_{\\mathbb{P}}(\\mathcal{M}_{2}(\\lambda)^{\\frac{1}{2}}),\n$$\n\nwhere the notation $o_{\\mathbb{P}}$ involves constants only depending on s and $\\kappa .$", "proof": {"steps": [{"n": 1, "claim": "Proof Recall the decomposition (44) in the proof of Lemma 17. The first two terms in (44) can be handled without any diference. Our goal here is to prove the following equation and we will finish the proof:\n\n$$\n\\left\\|T_{\\lambda}^{-\\frac{1}{2}}\\left(\\tilde{g}_{\\boldsymbol{Z}}-T_{\\boldsymbol{X}\\lambda}f_{\\lambda}\\right)\\right\\|_{\\mathcal{H}}=o_{\\mathbb{P}}\\left(\\mathcal{M}_{2}(\\lambda)^{\\frac{1}{2}}\\right).\\tag{82}\n$$"}, {"n": 2, "claim": "Similarly as (47), we rewrite the left hand side of (82) as\n\n$$\n\\left\\|T_{\\lambda}^{-\\frac{1}{2}}\\left(\\tilde{g}_{\\mathbf{Z}}-T_{\\mathbf{X}\\lambda}f_{\\lambda}\\right)\\right\\|_{\\mathcal{H}}=\\left\\|T_{\\lambda}^{-\\frac{1}{2}}\\left[\\left(\\tilde{g}_{\\mathbf{Z}}-T_{\\mathbf{X}}f_{\\lambda}\\right)-(g-T f_{\\lambda})\\right]\\right\\|_{\\mathcal{H}}.\\tag{83}\n$$"}, {"n": 3, "claim": "Denote $\\xi_{i}=\\xi({\\pmb x}_{i})=T_{\\lambda}^{-\\frac 12}(K_{{\\pmb x}_{i}}f_{\\rho}^{*}({\\pmb x}_{i})-T_{{\\pmb x}_{i}}f_{\\lambda})$ . Further consider the subset $\\Omega_{1}=\\{{\\pmb x}\\in\\mathcal{X}$ $|f_{\\rho}^{*}({\\pmb x})|\\leq t\\}$ and $\\Omega_{2}=\\mathcal{X}\\backslash\\Omega_{1}$ , where t will be chosen appropriately later. Decompose $\\xi_{i}$ as $\\xi_{i}I_{{\\pmb x}_{i}\\in\\Omega_{1}}+\\xi_{i}I_{{\\pmb x}_{i}\\in\\Omega_{2}}$ and we have the following decomposition of (83):\n\n$$\n\\begin{array}{l}\\left\\|\\frac{1}{n}\\sum_{i=1}^{n}\\xi_{i}-\\mathbb{E}\\xi_{x}\\right\\|_{\\mathcal{H}}\\leq\\left\\|\\frac{1}{n}\\sum_{i=1}^{n}\\xi_{i}I_{\\boldsymbol{x}_{i}\\in\\Omega_{1}}-\\mathbb{E}\\xi_{\\boldsymbol{x}}I_{\\boldsymbol{x}\\in\\Omega_{1}}\\right\\|_{\\mathcal{H}}+\\|\\frac{1}{n}\\sum_{i=1}^{n}\\xi_{i}I_{\\boldsymbol{x}_{i}\\in\\Omega_{2}}\\|_{\\mathcal{H}}+\\|\\mathbb{E}\\xi_{\\boldsymbol{x}}I_{\\boldsymbol{x}\\in\\Omega_{2}}\\|_{\\mathcal{H}}\\\\ :=\\mathrm{I}+\\mathrm{II}+\\mathrm{III}.\\tag{84}\\\\ \\end{array}\n$$"}, {"n": 4, "claim": "Next we choose $\\begin{array}{ r }{t=n^{\\frac{1-s}{2}+\\varepsilon_{t}},q=\\frac{2}{1-s}-\\varepsilon_{q}}\\end{array}$ such that\n\n$$\n\\varepsilon_{t}<\\varepsilon;\\text{and}\\frac{1-s}{2}+\\varepsilon_{t}>1/\\left(\\frac{2}{1-s}-\\varepsilon_{q}\\right),\\tag{85}\n$$\n\nwhere $\\varepsilon$ is given in (81). Then we can bound the three terms in (84) as follows:"}, {"n": 5, "claim": "(i) For the first term in (84), denoted as I, notice that\n\n$$\n\\left\\|\\left(f_{\\lambda}-f_{\\rho}^{*}\\right)I_{\\pmb{x}_{i}\\in\\Omega_{1}}\\right\\|_{L^{\\infty}}\\leq\\|f_{\\lambda}\\|_{L^{\\infty}}+n^{\\frac{1-s}{2}+\\varepsilon_{t}}.\n$$"}, {"n": 6, "claim": "Imitating the procedure (iii) in the proof of Lemma 17 and using (80), (81), we have\n\n$$\n\\mathrm{I}=o_{\\mathbb{P}}\\left(\\mathcal{M}_{2}(\\lambda)^{\\frac{1}{2}}\\right).\\tag{86}\n$$"}, {"n": 7, "claim": "(ii) For the second term in (84), denoted as II. Since $\\begin{array}{ r }{q=\\frac{2}{1-s}-\\varepsilon_{q}<\\frac{2}{1-s}}\\end{array}$ , Lemma 44 shows that,\n\n$$\n[\\mathcal{H}]^{s}\\hookrightarrow L^{q}(\\mathcal{X},\\mu),\n$$\n\nwith embedding norm less than a constant $C_{s,\\kappa}$ . Then Assumption 5 (a) implies that there exists $0<C_{q}<\\infty$ only depending on $\\gamma,s$ and κ such that $\\|f_{\\rho}^{*}\\|_{L^{q}(\\mathcal{X},\\mu)}\\leq C_{q}$ . Using the Markov inequality, we have\n\n$$\nP(\\pmb{x}\\in\\Omega_{2})=P\\Big(|f_{\\rho}^{*}(\\pmb{x})| >t\\Big)\\leq\\frac{\\mathbb{E}|f_{\\rho}^{*}(\\pmb{x})|^{q}}{t^{q}}\\leq\\frac{(C_{q})^{q}}{t^{q}}.\n$$"}, {"n": 8, "claim": "Further, since (85) guarantees $t^{q}\\gg n$ , we have\n\n$$\n\\begin{array}{l}\\tau_{n}:=P\\left(\\mathrm{II}\\geq\\mathcal{M}_{2}(\\lambda)^{\\frac{1}{2}}\\right)\\leq P\\Big(\\exists\\boldsymbol{x}_{i}\\text{s.t.}\\boldsymbol{x}_{i}\\in\\Omega_{2},\\Big)=1-P\\Big(\\boldsymbol{x}_{i}\\notin\\Omega_{2},\\forall\\boldsymbol{x}_{i},i=1,2,\\dots,n\\Big)\\\\ =1-P\\left(\\boldsymbol{x}\\notin\\Omega_{2}\\right)^{n}\\\\ =1-P\\Big(|f_{\\rho}^{*}(\\boldsymbol{x})|\\leq t\\Big)^{n}\\\\ \\leq 1-\\left(1-\\frac{\\left(C_{q}\\right)^{q}}{t^{q}}\\right)^{n}\\rightarrow 0.\\tag{87}\\\\ \\end{array}\n$$"}, {"n": 9, "claim": "(iii) For the third term in (84), denoted as III. Since Lemma 39 implies that $\\|T_{\\lambda}^{-\\frac{1}{2}}k(\\pmb{x},\\cdot)\\|_{\\mathcal{H}}\\leq$ $\\mathcal{N}_{1}(\\lambda)^{\\frac{1}{2}},\\mu\\mathrm{{\\mathrm{-}}a . e .}~\\pmb{x}\\in\\mathcal{X}$ , so\n\n$$\n\\begin{array}{l}\\mathrm{III}\\leq\\mathbb{E}\\|\\xi_{\\boldsymbol{x}}I_{\\boldsymbol{x}\\in\\Omega_{2}}\\|_{\\mathcal{H}}\\leq\\mathbb{E}\\left[\\|T_{\\lambda}^{-\\frac{1}{2}}k(\\boldsymbol{x},\\cdot)\\|_{\\mathcal{H}}\\cdot\\left|\\left(f_{\\rho}^{*}-f_{\\lambda}(\\boldsymbol{x})\\right)I_{\\boldsymbol{x}\\in\\Omega_{2}}\\right|\\right]\\\\ \\leq\\mathcal{N}_{1}(\\lambda)^{\\frac{1}{2}}\\mathbb{E}\\big|\\big(f_{\\rho}^{*}-f_{\\lambda}(\\boldsymbol{x})\\big)I_{\\boldsymbol{x}\\in\\Omega_{2}}\\big|\\\\ \\leq\\mathcal{N}_{1}(\\lambda)^{\\frac{1}{2}}\\left\\|f_{\\rho}^{*}-f_{\\lambda}\\right\\|_{L^{2}}^{\\frac{1}{2}}\\cdot P\\left(\\boldsymbol{x}\\in\\Omega_{2}\\right)^{\\frac{1}{2}}\\\\ \\leq\\mathcal{N}_{1}(\\lambda)^{\\frac{1}{2}}\\mathcal{M}_{2}(\\lambda)^{\\frac{1}{2}}t^{-\\frac{q}{2}},\\tag{88}\\\\ \\end{array}\n$$\n\nwhere we use Cauchy-Schwarz inequality for the third inequality and (16) for the forth inequality."}, {"n": 10, "claim": "Recalling that the choices of t, q satisfy $t^{-q}\\ll n^{-1}$ and we have assumed $\\mathcal{N}_{1}(\\lambda)$ ln $n/n=o(1)$ , we have\n\n$$\n\\mathrm{III}=o\\left(\\mathcal{M}_{2}(\\lambda)^{\\frac{1}{2}}\\right).\\tag{89}\n$$"}, {"n": 11, "claim": "Plugging (86), (87) and (89) into (84), we finish the proof."}]}}], "target_local_id": null, "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2025-24-2230", "context": "We consider a two-state HMM with real-valued emissions, in which we observe the first n entries of a sequence $\\pmb{Y}=(Y_{1},Y_{2},\\dots)\\in [0,1]^{\\mathbb{N}}$ which, under a parameter $\\theta=(p,q,f_{0},f_{1})$ satisfies\n\n$$\n\\mathbb{P}_{\\theta}(Y_{n}\\in A\\mid\\boldsymbol{X})=\\int_{A}f_{X_{n}}(y)\\mathrm{d}y,\\tag{1}\n$$\n\n$$\n\\boldsymbol{X}=(X_{n})_{n\\in\\mathbb{N}}\\sim\\operatorname{Markov}(\\pi,Q_{\\theta}),\n$$\n\nwith the $Y_{n},\\ n\\in\\mathbb{N}$ conditionally independent given X. The vector X of ‘hidden states’, which we assume is started from its invariant distribution $X_{1}\\sim\\pi$ , takes values in $\\{0,1\\}^{\\mathbb{N}}$ The transition matrix of the chain is given by\n\n$$\nQ=Q_{\\theta}:=\\left(\\begin{array}{c c}1-p&p\\\\ q&1-q\\end{array}\\right),\\tag{2}\n$$\n\nwith the convention that for $j\\geq 1,\\mathbb{P}_{\\theta}(X_{j+1}=0\\mid X_{j}=0)=1-p<1$ and $\\mathbb{P}_{\\theta}(X_{j+1}=0\\ |$ $X_{i}=1)=q>0$ . The functions $f_{0},f_{1}\\in L^{2}([0,1])$ are density functions. Thus all $Y_{k},k\\geq 1$ follow the mixture distribution $\\pi_{0}f_{0}+\\pi_{1}f_{1}$\n\nThe goal is to estimate the parameter θ. This is a nonlinear inverse problem known to be solvable, up to a label-switching issue, even without any modelling assumptions on $f_{0}$ and $f_{1}$ (Gassiat et al., 2016; Alexandrovich et al., 2016): specifically, given that the highly non-identifiable i.i.d. nonparametric mixture is a degenerate submodel of a HMM, under conditions which rule out independence. There are three ways in which the data $(Y_{n})_{n\\in\\mathbb{N}}$ can fail to exhibit dependence: when the hidden states themselves are in reality independently distributed; when the emission distributions are identical; or when only one population is observed. We adopt the minimax paradigm and we analyse the smallest maximum risk attainable over the following class of parameters. We define for some $\\delta,\\epsilon\\in(0,1)$ and some $\\zeta,s_{0},s_{1},R>0$\n\n$$\n\\Theta_{\\delta,\\epsilon,\\zeta}^{s_{0},s_{1}}(R):=\\{\\theta:p,q\\geq\\delta,|1-p-q|\\geq\\epsilon,\\|f_{0}-f_{1}\\|_{L^{2}}\\geq\\zeta,\\|f_{i}\\|_{B_{2,\\infty}^{s_{i}}}\\leq R\\}.\\tag{3}\n$$\n\nHere $\\|\\cdot\\|_{B_{2,\\infty}^{s}}$ denotes a Besov norm whose precise definition as used in this paper is delayed to equation (15) below. The space $B_{2,\\infty}^{s}$ can be thought to be similar to the subspace of s-times diferentiable functions with continuous s-derivative that are square-integrable, but it allows for slightly more general functions with comparable smoothness. We refer to Triebel (1983) for a thorough introduction to Besov spaces and their history. The quantities $\\delta,\\epsilon$ and ζ lower bound the “distance” to the i.i.d. submodel. Indeed if $\\delta = 0$ , we may be unable to estimate both $f_{0}$ and $f_{1}$ since we may see data from one of these alone; if $\\zeta = 0$ we may be unable to estimate $p$ and $q ;$ and if $\\epsilon = 0$ then we may be unable to identify the contributions of $f_{0}$ and $f_{1}$ to the mixture $\\pi_{0}f_{0}+\\pi_{1}f_{1}$ . We use concentration inequalities for Markov chains (Paulin, 2015) to build our estimators. This requires us to slightly shrink the set $\\Theta_{\\delta,\\epsilon,\\zeta}^{s_{0},s_{1}}(R)$ ) and restrict our attention to parameters that are also in\n\n$$\n\\Sigma_{\\gamma^{*}}(L):=\\{\\theta:1- |1-p-q|\\geq\\gamma^{*},\\max_{j=0,1}\\|f_{j}\\|_{\\infty}\\leq L\\},\\tag{4}\n$$\n\ni.e. parameters with uniformly bounded emission densities (here $\\|\\cdot\\|_{\\infty}$ denotes the usual supremum norm) and having an absolute spectral gap. The assumption that the Markov chain starts from its stationary distribution could be relaxed as explained in (Paulin, 2015, Section 3.3), at the price of increasing the constants in the upper bounds, and longer proofs. We throughout use $\\mathbb{P}_{\\theta}$ to denote the law of $(X,Y)$ , and all induced marginal and conditional laws.\n\nIn this section we construct estimators whose maximum risk over $\\Theta_{\\delta,\\epsilon,\\zeta}^{s_{0},s_{1}}(R)\\cap\\Sigma_{\\gamma^{*}}(L)$ match those established in the lower bounds of Theorems 2 and 3 in most cases.\n\nHere we describe the heuristic we use to build a near minimax optimal estimator of $\\theta =$ $(p,q,f_{0},f_{1})$ . As noted previously (Gassiat et al., 2016), understanding the law of three consecutive observations is key to recovering the model parameters. A reparametrisation simplifies the expression for said law, and allows the dependence on the parameters $\\delta,\\epsilon$ and ζ to appear more naturally. Set\n\n$$\n\\phi(\\theta)=\\big(\\frac{q-p}{p+q},1-p-q,\\|f_{0}-f_{1}\\|_{L^{2}}\\big),\\quad\\psi(\\theta)=\\big(\\frac{q f_{0}+p f_{1}}{p+q},\\frac{f_{0}-f_{1}}{\\|f_{0}-f_{1}\\|_{L^{2}}}\\big).\\tag{7}\n$$\n\nFor $m \\geq 1$ , let $P_{\\phi,\\psi}^{(m)}$ denote the law of $(Y_{1},\\ldots,Y_{m})$ under parameter $(\\phi,\\psi)$ , and let $p_{\\phi,\\psi}^{(m)}$ denote the corresponding density with respect to Lebesgue measure on $[0,1]^{m}$ . In the parametrisation (7), defining for $\\phi=\\left(\\phi_{1},\\phi_{2},\\phi_{3}\\right)$\n\n$$\nr(\\phi)=\\frac{1}{4}(1-\\phi_{1}^{2})\\phi_{2}\\phi_{3}^{2},\\tag{8}\n$$\n\none computes, with $f \\otimes g$ defined by $(f\\otimes g)(x,y)=f(x)g(y)$ 2\n\n$$\n\\begin{array}{l}p_{\\phi,\\psi}^{(3)}=\\psi_{1}\\otimes\\psi_{1}\\otimes\\psi_{1}+r(\\phi)\\big(\\psi_{2}\\otimes\\psi_{2}\\otimes\\psi_{1}+\\psi_{1}\\otimes\\psi_{2}\\otimes\\psi_{2}\\big)\\\\ +\\phi_{2}r(\\phi)\\psi_{2}\\otimes\\psi_{1}\\otimes\\psi_{2}-\\phi_{1}\\phi_{2}\\phi_{3}r(\\phi)\\psi_{2}\\otimes\\psi_{2}\\otimes\\psi_{2}.\\tag{9}\\\\ \\end{array}\n$$\n\nWe propose to estimate $(p,q)$ and the wavelet coeficients of $f_{0}$ and $f_{1}$ using the method of moments. In the inversion procedure described above we replace the moments by their empirical versions computed using\n\n$$\n\\mathbb{P}_{n}^{(s)}(H):=\\frac{1}{n-s+1}\\sum_{i=1}^{n-s+1}H(Y_{i},\\dots,Y_{i+s-1}),\\quad H:[0,1]^{s}\\rightarrow\\mathbb{R},s\\geq 1.\\tag{13}\n$$\n\nAs suggested by equation (12), the formula for computing (φ, ψ) given the moments is unstable if the function h is chosen poorly, so that the estimates may be far from the true values if $\\langle\\psi_{2},h\\rangle$ is too small even if empirical moments are close to their means. No fixed choice of h works uniformly over the parameter space: given $h,$ there exists a parameter $(\\phi,\\psi)$ such that $\\langle\\psi_{2},h\\rangle$ is arbitrarily small, resulting in an arbitrarily large maximum risk over $\\Theta_{\\delta,\\epsilon,\\zeta}^{s_{0},s_{1}}(R)$ . To avoid a deteriorated maximum risk, it is therefore necessary to estimate h from the data. The oracle choice for h would maximize $\\begin{array}{ r }{h\\mapsto\\frac{|\\langle\\psi_{2},h\\rangle|}{\\|h\\|}}\\end{array}$ and hence be given by $h=\\psi_{2}$ . Thus, a crucial step in our estimation procedure is to provide an initial (crude) estimator $\\tilde{\\psi}_{2}$ of $\\psi_{2}$ such that $\\|\\tilde{\\psi}_{2}\\|_{L^{2}}=1$ and such that\n\n$$\n\\tilde{\\mathcal{I}}:=\\left\\langle\\psi_{2},\\tilde{\\psi}_{2}\\right\\rangle \\tag{14}\n$$\n\nis suficiently bounded away from zero with high probability under each parameter $(\\phi,\\psi)\\in$ $\\Theta_{\\delta,\\epsilon,\\zeta}^{s_{0},s_{1}}(R)\\cap\\Sigma_{\\gamma^{*}}(L)$ . For this reason we describe $\\tilde{\\psi}_{2}$ as a separating function: since $\\psi_{2}= $ $(f_{0}-f_{1})/\\|f_{0}-f_{1}\\|_{L^{2}}$ , finding $\\tilde{\\psi}_{2}$ is tantamount to finding an hyperplane in $L^{2}[0,1]$ which separates $f_{0}$ and $f_{1}$ suficiently well. The estimator $\\tilde{\\psi}_{2}$ is built in Section 3.3.\n\nAs explained in Section 3.1, our estimation procedure is based on computing empirical averages of the type $\\begin{array}{ r }{\\mathbb{P}_{n}^{(2)}(\\tilde{\\psi}_{2}\\otimes f)=\\frac{1}{n-1}\\sum_{i=1}^{n-1}\\tilde{\\psi}_{2}(Y_{i})f(Y_{i+1})}\\end{array}$ where $\\tilde{\\psi}_{2}$ is a crude estimator of $\\psi_{2}$ . If $\\tilde{\\psi}_{2}$ is also estimated from $(Y_{1},\\ldots,Y_{n})$ , it is not clear at all that these empirical average approach $\\mathbb{E}_{\\theta}(\\psi_{2}(Y_{1})f(Y_{2}))$ , as they are sum of somewhat complex dependent random variables, each term of which depends on the whole sample $(Y_{1},\\ldots,Y_{n})$ . A classical trick is to estimate $\\tilde{\\psi}_{2}$ using a sample $({\\tilde{Y}}_{1},\\dots,{\\tilde{Y}}_{n})$ that is independent from the sample $(Y_{1},\\ldots,Y_{n})$ used to compute the average $\\begin{array}{ r }{\\frac{1}{n-1}\\sum_{i=1}^{n-1}\\tilde{\\psi}_{2}(Y_{i})f(Y_{i+1})}\\end{array}$ . In the context of HMM, however, the sample cannot be split into two independent parts. Fortunately this is not too worrisome. As explained in Section $\\mathrm{A},$ it is possible to split the sample $(Y_{1},\\ldots,Y_{n})$ into three parts, and then use the first third to estimate $\\tilde{\\psi}_{2}$ and the last third for empirical averages. This way, deviation inequalities for the empirical averages of functions involving $\\tilde{\\psi}_{2}$ can be achieved as if $\\tilde{\\psi}_{2}$ were independent of the observations used in the empirical averages, up to a term $C e^{-c\\gamma^{*}n}$ for $C$ and c universal constants. Thus, to facilitate reading, we will throughout assume that $\\tilde{\\psi}_{2}$ is estimated using a sample $(\\tilde{Y}_{1},\\ldots,\\tilde{Y}_{n})\\sim P_{\\phi,\\psi}^{(n)}$ independent of $(Y_{1},\\ldots,Y_{n})$\n\nFor notational convenience, we define the set of wavelet indices\n\n$$\n\\Lambda(M):=\\{0,1,\\dots,2^{J-1}\\}\\cup\\{(j,k):j=J,\\dots,M,k=0,\\dots,2^{j}-1\\}\n$$\n\nincluding all father indices and mother indices of levels $J \\le j \\le M$ , and for all $\\lambda\\in\\Lambda(M)$ we set $e_{\\lambda}=\\Phi_{J k}{\\mathrm{~i f~}}\\lambda=k$ and $e_{\\lambda}=\\Psi_{j k}{\\mathrm{~i f~}}\\lambda=(j,k)$\n\nFor M large enough (see Theorem 4 below) compute the $2^{M}\\times 2^{M}$ matrix $\\tilde{\\mathcal{G}}$ with entries\n\n$$\n\\tilde{\\mathcal{G}}_{\\lambda,\\lambda^{\\prime}}=\\frac{1}{2}\\tilde{\\mathbb{P}}_{n}^{(2)}(e_{\\lambda}\\otimes e_{\\lambda^{\\prime}}+e_{\\lambda^{\\prime}}\\otimes e_{\\lambda})-\\tilde{\\mathbb{P}}_{n}^{(1)}(e_{\\lambda})\\tilde{\\mathbb{P}}_{n}^{(1)}(e_{\\lambda^{\\prime}}).\n$$\n\nThe matrix $\\tilde{\\mathcal{G}}$ is an estimator of the matrix $\\mathcal{G}$ with entries\n\n$$\n\\mathcal{G}_{\\lambda,\\lambda^{\\prime}}=\\frac{1}{2}\\mathbb{E}_{\\theta}(e_{\\lambda}\\otimes e_{\\lambda^{\\prime}}+e_{\\lambda^{\\prime}}\\otimes e_{\\lambda})-\\mathbb{E}_{\\theta}(e_{\\lambda})\\mathbb{E}_{\\theta}(e_{\\lambda})=r(\\phi)\\langle\\psi_{2},e_{\\lambda}\\rangle\\langle\\psi_{2},e_{\\lambda^{\\prime}}\\rangle\n$$\n\nwhere the second equality follows from equation (9). Hence, G is proportional to the Gram matrix of the vector $V_{\\theta}\\propto(\\langle\\psi_{2},e_{\\lambda}\\rangle:\\lambda\\in\\Lambda(M))$ . The matrices $\\tilde{\\mathcal{G}}$ and $\\mathcal{G}$ are real symmetric, and thus by the spectral theorem are always diagonalizable. By concentration arguments, we expect that $\\tilde{\\mathcal{G}}$ will have an eigenvalue approximately equal to $r(\\phi)$ (which can be positive or negative) and the rest of eigenvalues will be smaller in absolute value. The eigenvector $\\tilde{V}$ (chosen such that $\\|\\tilde{V}\\|=1)$ corresponding to the leading eigenvalue is then an estimator of $\\pm V_{\\theta}/\\|V_{\\theta}\\|$ . We suggest to set\n\n$$\n\\tilde{\\psi}_{2}(x):=\\frac{\\max\\big(-\\tau,\\min\\big(\\tau,\\sum_{\\lambda\\in\\Lambda(M)}\\tilde{V}_{\\lambda}e_{\\lambda}(x)\\big)\\big)}{\\Big(\\int_{0}^{1}\\max\\big(-\\tau,\\min\\big(\\tau,\\sum_{\\lambda\\in\\Lambda(M)}\\tilde{V}_{\\lambda}e_{\\lambda}(y)\\big)\\big)^{2}d y\\Big)^{1/2}}\n$$\n\nwhere the truncation $\\tau \\geq 1$ is intended to prevent technicalities within the proofs. The next theorem shows that $\\tilde{\\psi}_{2}$ is well aligned with $\\psi_{2}$ with high probability under $\\theta\\in\\Theta_{\\delta,\\epsilon,\\zeta}^{s_{0},s_{1}}(R)\\cap$ $\\Sigma_{\\gamma^{*}}(L)$ . The proof of Theorem 4 can be found in Section C.7.\n\nEasy computations lead to (recall $\\tilde{\\mathcal{I}}: =\\langle\\psi_{2},\\tilde{\\psi}_{2}\\rangle$ in (14) and $r(\\phi)=(1/4)(1-\\phi_{1}^{2})\\phi_{2}\\phi_{3}^{2}$ in (8))\n\n$$\nm(\\phi)\\equiv\\left(r(\\phi)\\tilde{\\mathcal{I}}^{2},r(\\phi)\\phi_{2}\\tilde{\\mathcal{I}}^{2},r(\\phi)\\phi_{1}\\phi_{2}\\phi_{3}\\tilde{\\mathcal{I}}^{3}\\right),\\tag{16}\n$$\n\nsee Lemma 13 in $\\mathrm{Appendix}\\mathrm{C}$ . The moments in the previous display can be inverted modulo label-switching. Namely, it is possible to express $\\phi_{1}\\mathrm{sgn}(\\tilde{\\mathcal{I}}),\\phi_{2}$ , and $\\phi_{3}|\\tilde{\\mathcal{I}}|$ as functions of $m(\\phi)$ . The inversion formulas for m are given in Lemma 14. By replacing $m(\\phi)$ with the empirical estimates in the inversion formula we define\n\n$$\n\\hat{\\phi}_{1}:=\\frac{\\hat{m}_{3}}{[4\\hat{m}_{1}^{2}(\\hat{m}_{2})_{+}+\\hat{m}_{3}^{2}]^{1/2}},\\quad\\hat{\\phi}_{2}:=\\max\\Big(-1,\\min\\Big(\\frac{\\hat{m}_{2}}{\\hat{m}_{1}},1\\Big)\\Big).\n$$\n\nNotice that since m $\\nu(\\phi)_{2}\\geq 0$ , we replaced $\\hat{m}_{2}$ by $(\\hat{m}_{2})_{+}=\\operatorname*{max}(\\hat{m}_{2},0)$ . We then build an estimator of $p$ and q justified by (10) by letting\n\n$$\n\\begin{array}{l}\\hat{p}=\\frac{1}{2}(1-\\hat{\\phi}_{1})(1-\\hat{\\phi}_{2}),\\\\ \\hat{q}=\\frac{1}{2}(1+\\hat{\\phi}_{1})(1-\\hat{\\phi}_{2}).\\end{array}\n$$\n\nRecalling the definition (16) of m, define\n\n$$\ng:=\\phi_{3}|\\tilde{\\mathcal{I}}| =\\frac{\\sqrt{4 m_{1}^{2}m_{2}+m_{3}^{2}}}{m_{2}},\n$$\n\nand define\n\n$$\nf_{\\pm}:=\\psi_{1}\\pm\\frac{g(1\\mp\\tilde{s}\\phi_{1})}{2m_{1}}G,\\qquad G:=\\frac{m_{1}\\psi_{2}}{\\tilde{\\mathcal{I}}}.\n$$\n\nLemma 14 (Inversion formulas for m) Let $m(\\phi)=(r(\\phi)\\tilde{\\mathbb{Z}}^{2},r(\\phi)\\phi_{2}\\tilde{\\mathbb{Z}}^{2},r(\\phi)\\phi_{1}\\phi_{2}\\phi_{3}\\tilde{\\mathbb{Z}}^{3})$ with $\\tilde{\\mathcal{T}}\\neq 0$ . Then,\n\n$$\n\\begin{array}{l}\\mathrm{sgn}(\\tilde{\\mathcal{I}})\\phi_{1}=\\frac{m_{3}(\\phi)}{\\sqrt{4 m_{1}(\\phi)^{2}m_{2}(\\phi)+m_{3}(\\phi)^{2}}},\\\\ \\phi_{2}=\\frac{m_{2}(\\phi)}{m_{1}(\\phi)},\\\\ \\phi_{3}|\\tilde{\\mathcal{I}}| =\\frac{\\sqrt{4 m_{1}(\\phi)^{2}m_{2}(\\phi)+m_{3}(\\phi)^{2}}}{m_{2}(\\phi)}.\\\\ \\end{array}\n$$\n\nLemma 16 Let $m_{1},m_{2},m_{3}$ be defined as in (16) and let $v: =4 m_{1}^{2}m_{2}+m_{3}^{2}$ . Then $0 \\leq$ $m_{2}\\leq|m_{1}|$ and $\\sqrt{v}=\\tilde{\\mathcal{T}}^{3}r(\\phi)\\phi_{2}\\phi_{3}=\\tilde{\\mathcal{T}}m_{2}\\phi_{3}$ . Furthermore, for every $\\theta\\in\\Theta_{\\delta,\\epsilon,\\zeta}^{s_{0},s_{1}}(R)$ and $0<\\delta\\leq 1,0<\\epsilon\\leq 1$ , and $0 < \\zeta \\leq 1$ :\n\n$$\n\\left|\\frac{g}{m_{1}}\\right|\\leq\\frac{4}{\\delta\\epsilon\\zeta|\\tilde{\\mathcal{I}}|},\\quad\\frac{\\max(1,g)}{m_{2}}\\leq\\frac{4}{\\delta\\epsilon^{2}\\zeta^{2}|\\tilde{\\mathcal{I}}|^{2}},\\quad\\frac{\\max(1,g)}{g m_{2}}\\leq\\frac{4}{\\delta\\epsilon^{2}\\zeta^{3}|\\tilde{\\mathcal{I}}|^{3}}.\n$$\n\nLemma 17 For any $k \\geq 1$\n\n$$\n\\|p_{\\theta}^{(k)}\\|_{\\infty}\\leq\\max\\big(\\|f_{0}\\|_{\\infty},\\|f_{1}\\|_{\\infty}\\big)^{k}.\n$$\n\nConsequently, for any $\\theta\\in\\Sigma_{\\gamma^{*}}(L)$ and any measurable function $h:\\mathbb{R}^{k}\\mathbb{R}$ , we have\n\n$$\n\\mathbb{E}_{\\theta}[h(Y_{1},\\dots,Y_{k})^{2}]\\leq L^{k}\\|h\\|_{L^{2}}^{2}.\n$$\n\nRemark 18 The proof adapts to yield $E_{\\theta}[h(Y_{1},Y_{3})^{2}]\\leq L^{2}\\|h\\|_{L^{2}}^{2}$ rather than the weaker bound $L^{3}\\|h\\|_{L^{2}}^{2}$ directly obtainable using the lemma. Indeed, we have\n\n$$\n\\sup_{y_{1},y_{3}}\\Big|\\int p_{\\theta}^{(3)}(y_{1},y_{2},y_{3})\\mathrm{d}y_{2}\\Big| =\\sum_{x_{1},x_{2},x_{3}}\\mathbb{P}_{\\theta}(X_{1}=x_{1},X_{2}=x_{2},X_{3}=x_{3})f_{x_{1}}(y_{1})f_{x_{3}}(y_{3})\\leq L^{2},\n$$\n\nand the rest of the proof is the same.\n\nWe now recall the following result, which is adapted from (Paulin, 2015) and will be key to getting deviation inequalities of empirical ingredients in our procedures.\n\nLemma 20 Let $1 \\leq k \\leq 3$ and let $h:\\mathbb{R}^{k}\\mathbb{R}$ be measurable. There is a universal constant $C > 0$ such that for all $\\theta_{\\perp}$ , all $n \\geq 4$ such that $n\\gamma^{*}\\geq 1/99$ , and all $t \\geq 0$\n\n$$\n\\mathbb{P}_{\\theta}\\Big(|\\mathbb{P}_{n}^{(k)}(h)-\\mathbb{E}_{\\theta}(h)|\\geq t\\Big)\\leq\\exp\\Big(-\\frac{C n t^{2}\\gamma^{*}}{\\mathbb{E}_{\\theta}(h^{2})+\\|h\\|_{\\infty}t}\\Big).\n$$\n\nThis in particular implies that there is a is a universal constant $C > 0$ such that for all $\\theta_{i}$ , all $n \\geq 4$ such that $n\\gamma^{*}\\geq 1/99$ , and all $x \\geq 0$\n\n$$\n\\mathbb{P}_{\\theta}\\Bigg(|\\mathbb{P}_{n}^{(k)}(h)-\\mathbb{E}_{\\theta}(h)|\\geq C\\sqrt{\\frac{\\mathbb{E}_{\\theta}[h^{2}]x}{n\\gamma^{*}}}+\\frac{C\\|h\\|_{\\infty}x}{n\\gamma^{*}}\\Bigg)\\leq e^{-x}.\n$$\n\nRecall the definitions\n\n$$\n\\begin{array}{l}\\hat{m}_{1}:=\\mathbb{P}_{n}^{(2)}(\\tilde{\\psi}_{2}\\otimes\\tilde{\\psi}_{2})-\\mathbb{P}_{n}^{(1)}(\\tilde{\\psi}_{2})^{2},\\\\ \\hat{m}_{2}:=\\mathbb{P}_{n}^{(3)}(\\tilde{\\psi}_{2}\\otimes 1\\otimes\\tilde{\\psi}_{2})-\\mathbb{P}_{n}^{(1)}(\\tilde{\\psi}_{2})^{2}\\\\ \\hat{m}_{3}= -\\mathbb{P}_{n}^{(3)}(\\tilde{\\psi}_{2}\\otimes\\tilde{\\psi}_{2}\\otimes\\tilde{\\psi}_{2})+\\mathbb{P}_{n}^{(1)}(\\tilde{\\psi}_{2})^{3}+(2\\hat{m}_{1}+\\hat{m}_{2})\\mathbb{P}_{n}^{(1)}(\\tilde{\\psi}_{2}),\\\\ \\end{array}\n$$\n\nestimators of the functional m defined in (16) as $m=(r(\\phi)\\tilde{\\cal{T}}^{2},r(\\phi)\\phi_{2}\\tilde{\\cal{T}}^{2},r(\\phi)\\phi_{1}\\phi_{2}\\phi_{3}\\tilde{\\cal{T}}^{3})$ with $\\tilde{\\mathcal{I}}=\\langle\\psi_{2},\\tilde{\\psi}_{2}\\rangle$ , and deduced from Lemma 13 to be equal to what is obtained in the expressions for ˆm on replacing every instance of an empirical estimator by the expectation operator. [This does not mean that $\\mathbb{E}_{\\theta}\\hat{m}=m,$ , since there are powers and products in the expressions.] In this section, we prove deviation inequalities for the estimators of $m,$ from which we deduce bounds in expectation. The results of this section will be used to prove Theorem 5 and Theorem 6.\n\nWe remark that the results are mostly uniform over the whole class $\\Sigma_{\\gamma^{*}}(L)$ , not our final parameter set $\\Theta_{\\delta,\\epsilon,\\zeta}^{s_{0},s_{1}}(R)\\cap\\Sigma_{\\gamma^{*}}(L)$ . The need to intersect with $\\Theta_{\\delta,\\epsilon,\\zeta}^{s_{0},s_{1}}(R)$ arises for ensuring the parameters θ are identifiable from $m$\n\nProposition 23 Let $n\\gamma^{*}\\geq 1/99$ . Then there exists a universal constant $C > 0$ such that for all $x \\geq 0$\n\n$$\n\\sup_{\\theta\\in\\Sigma_{\\gamma^{*}}(L)}\\mathbb{P}_{\\theta}\\left(\\max_{j=1,2,3}|\\hat{m}_{j}-m_{j}|\\geq C L^{3/2}\\sqrt{\\frac{x}{n\\gamma^{*}}}+C\\max(\\tau,\\sqrt{L})^{3}\\frac{x}{n\\gamma^{*}}\\right)\\leq 4 e^{-x}.\n$$\n\nProposition 33 Let $n\\gamma^{*}\\geq 1/99$ and $\\|\\tilde{\\psi}_{2}\\|_{\\infty}\\leq\\tau$ . Then, there is a universal constant $C > 0$ such that for all $\\theta\\in\\Sigma_{\\gamma^{*}}(L)$ and all $x \\geq 0$\n\n$$\n\\mathbb{P}_{\\theta}\\left(\\|\\hat{G}^{J_{n}}-G^{J_{n}}\\|_{L^{2}}\\geq C L\\sqrt{\\frac{x}{n\\gamma^{*}}}+C\\max(\\tau 2^{J_{n}/2},\\sqrt{L}2^{J_{n}/2},\\tau\\sqrt{L})\\frac{x}{n\\gamma^{*}}\\right)\\leq 4\\cdot 24^{2^{J_{n}}}e^{-x}.\n$$", "theorems": [{"local_id": "T1", "statement": "Lemma 26 Assume $\\theta\\in\\Sigma_{\\gamma^{*}}(L)$ and $n\\gamma^{*}\\geq 1/99$ . Write $Z_{1}=\\mathbb{P}_{n}^{(1)}(\\tilde{\\psi}_{2})-\\mathbb{E}_{\\theta}(\\tilde{\\psi}_{2}),Z_{2}=$ $\\begin{array}{ r }{\\mathbb{P}_{n}^{(2)}\\big(\\tilde{\\psi}_{2}\\otimes\\tilde{\\psi}_{2}\\big)-\\mathbb{E}_{\\theta}\\big(\\tilde{\\psi}_{2}\\otimes\\tilde{\\psi}_{2}\\big),Z_{3}=\\mathbb{P}_{n}^{(3)}\\big(\\tilde{\\psi}_{2}\\otimes 1\\otimes\\tilde{\\psi}_{2}\\big)-\\mathbb{E}_{\\theta}\\big(\\tilde{\\psi}_{2}\\otimes 1\\otimes\\tilde{\\psi}_{2}\\big)}\\end{array}$ , and $Z_{4}=\\mathbb{P}_{n}^{(3)}(\\tilde{\\psi}_{2}$ ⊗ $\\tilde{\\psi}_{2}\\otimes\\tilde{\\psi}_{2})-\\mathbb{E}_{\\theta}(\\tilde{\\psi}_{2}\\otimes\\tilde{\\psi}_{2}\\otimes\\tilde{\\psi}_{2})$ . Then\n\n$$\n\\mathbb{P}_{\\theta}\\Big(|Z_{1}|\\geq C\\sqrt{\\frac{L x}{n\\gamma^{*}}}+C\\tau\\frac{x}{n\\gamma^{*}}\\Big)\\leq e^{-x},\n$$\n\n$$\n\\mathbb{P}_{\\theta}\\Big(|Z_{j}|\\geq C L\\sqrt{\\frac{x}{n\\gamma^{*}}}+C\\tau^{2}\\frac{x}{n\\gamma^{*}}\\Big)\\leq e^{-x},\\quad j=2,3,\n$$\n\n$$\n\\mathbb{P}_{\\theta}\\Big(|Z_{4}|\\geq C L^{3/2}\\sqrt{\\frac{x}{n\\gamma^{*}}}+C\\tau^{3}\\frac{x}{n\\gamma^{*}}\\Big)\\leq e^{-x}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof For $Z_{4},$ , use Lemma 20 together with the facts that $\\|\\tilde{\\psi}_{2}\\otimes\\tilde{\\psi}_{2}\\otimes\\tilde{\\psi}_{2}\\|_{\\infty}=\\|\\tilde{\\psi}_{2}\\|_{\\infty}^{3}\\leq\\tau^{3}$ and that $\\mathbb{E}_{\\theta}[(\\tilde{\\psi}_{2}\\otimes\\tilde{\\psi}_{2}\\otimes\\tilde{\\psi}_{2})^{2}]\\le\\breve{L}^{3}\\|\\tilde{\\psi}_{2}\\|_{I^{2}}^{6}=L^{3}$ by Lemma 17. The arguments are similar for $j=1,2,3$ , though note for $j = 3$ we use Remark 18 rather than Lemma 17 itself."}]}}, {"local_id": "T2", "statement": "Lemma 28 Let $n\\gamma^{*}\\geq 1/99$ . Then, there exists a universal constant $C > 0$ such that for all $\\theta\\in\\Sigma_{\\gamma^{*}}(L)$\n\n$$\n\\mathbb{P}_{\\theta}\\Big(\\max_{j=1,2,3}|\\hat{m}_{j}-m_{j}|\\geq\\frac{gm_{2}}{44\\max(1,g)}\\Big)\\leq 4\\exp\\left(-\\frac{Cn\\gamma^{*}g^{2}m_{2}^{2}/\\max(1,g)^{2}}{L^{3}+\\max(\\tau,\\sqrt{L})^{3}gm_{2}/\\max(1,g)}\\right).\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof By Proposition 23, applied with $x \\geq 0$ such that\n\n$$\nCL^{3/2}\\sqrt{\\frac{x}{n\\gamma^{*}}}+C\\max(\\tau,\\sqrt{L})^{3}\\frac{x}{n\\gamma^{*}}=\\frac{gm_{2}}{44\\max(1,g)}\n$$\n\nie,\n\n$$\n\\begin{array}{l}\\sqrt{\\frac{x}{n\\gamma^{*}}}=\\frac{L^{3/2}}{2\\max(\\tau,\\sqrt{L})^{3}}\\left(\\sqrt{1+\\frac{4\\max(\\tau,\\sqrt{L})^{3}gm_{2}}{44CL^{3}\\max(1,g)}}-1\\right)\\\\ \\geq\\frac{1}{44CL^{3/2}}\\frac{gm_{2}/\\max(1,g)}{\\sqrt{1+\\frac{4\\max(\\tau,\\sqrt{L})^{3}gm_{2}}{44CL^{3}\\max(1,g)}}},\\\\ \\end{array}\n$$\n\nwe obtain the result."}]}}, {"local_id": "T3", "statement": "Lemma 31 The following bounds holds true.\n\n$$\n|\\hat{\\phi}_{2}-\\phi_{2}|\\leq 2\\min\\left(1,\\frac{2\\max_{j=1,2}|\\hat{m}_{j}-m_{j}|}{|m_{1}|}\\right).\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof We let $\\Delta_{1}: =\\hat{m}_{1}-m_{1}$ and $\\Delta_{2}: =\\hat{m}_{2}-m_{2}$ . We also let $f(x): =\\operatorname*{max}(-1,\\operatorname*{min}(x,1))$ . It is easily seen that $|f(x)-f(y)|\\leq\\operatorname*{min}(2,|x-y|)$ . Suppose first that $|\\Delta_{1}| > |m_{1}| /2$ . Then, $\\begin{array}{ r }{|\\hat{\\phi}_{2}-\\phi_{2}|\\leq 2\\leq\\operatorname*{min}(2,\\frac{4|\\Delta_{1}|}{|m_{1}|})}\\end{array}$ . On the other hand, if $|\\Delta_{1}|\\leq|m_{1}| /2$ , then, recalling that $m_{2}\\leq|m_{1}|$ we have from Lemma 14\n\n$$\n\\begin{array}{l}|\\hat{\\phi}_{2}-\\phi_{2}| = |f(\\hat{m}_{2}/\\hat{m}_{1})-f(m_{2}/m_{1})|\\\\ \\leq\\min\\left(2,\\left|\\frac{m_{2}+\\Delta_{2}}{m_{1}+\\Delta_{1}}-\\frac{m_{2}}{m_{1}}\\right|\\right)\\\\ =\\min\\left(2,\\left|\\frac{m_{1}\\Delta_{2}-m_{2}\\Delta_{1}}{m_{1}(m_{1}+\\Delta_{1})}\\right|\\right)\\\\ \\leq\\min\\Big(2,\\frac{2|\\Delta_{1}| +2|\\Delta_{2}|}{|m_{1}|}\\Big).\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "The conclusion follows since $x\\mapsto(x)_{+}$ is 1-Lipschitz and thus $|\\Delta_{2}| = |\\hat{m}_{2}-m_{2}|$."}]}}], "target_local_id": null, "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2026-22-1194", "context": "Notations. For any positive integer n we let [n] denote the set $\\{1,\\ldots,n\\}$ . For any set A we let $\\Delta(A)$ denote the set of probability measures over A. For sets $A,B,$ we let $A \\times B$ be the Cartesian product of the two. During the whole paper, we use $k \\in [ K ]$ to refer to an episode and $h \\in [ H ]$ to refer to a horizon. In addition, we use $\\tilde{k}$ to refer to a bufer period associated with the k-th episode.\n\nWe consider a repeated (lazy) multi-phase second-price auction with personalized reserve prices. Particularly, we assume that there are N rational bidders, indexed by [N], and one seller participating in the auction. For ease of presentation, we use “he” to refer to a specific bidder and “she” to the seller.\n\nSecond Price Auction with Personalized Reserve Prices. We begin by describing a single round of the auction. Each bidder $i \\in [ N ]$ submits some bid $b_{i}\\in\\mathbb{R}_{\\geq 0}$ and the seller determines the personalized reserve prices for the bidders in the form of reserve price vector $\\boldsymbol{\\rho}\\in\\mathbb{R}_{>0}^{N}$ , with $\\rho_{i}$ denoting bidder i’s reserve price. The bidder with the highest bid only wins if he also clears his personal reserve price, i.e., $b_{i}\\geq\\rho_{i}$ . If the bidder i receives the item, he pays the seller the maximum of his personalized reserve and the second highest bid, namely max $\\{\\rho_{i},\\operatorname*{max}_{j\\neq i}b_{j}\\}$ , which we dub $m_{i}$ for simplicity. When the bidder with the highest bid fails to clear his personalized reserve price, the auction fails, the seller gains zero, and the item remains unsold. In summary, bidder i receives the item if and only if $b_{i}\\geq m_{i}$ and the price he pays is $m_{i}$ . For any round of auction, we let $q_{i}=11(\\mathrm{bidder}\\ i$ receives the item) indicate whether bidder i received the item or not. For the sake of convenience, throughout the paper, we assume that there are no ties in the submitted bids.\n\nA Multi-Phase Second Price Auction. We now characterize the dynamics of the multiphase auction setting we study. Assume that the transition dynamic between rounds can be modeled as an episodic Markov Decision Process $(\\mathrm{MDP})^{1}$ . A multi-phase second price auction with personalized reserves is parameterized as $(\\mathcal{S},\\Upsilon,H,\\mathbb{P},\\{r_{i}\\}_{i=1}^{N})$ , with the state space denoted by S, seller’s item choice space $\\Upsilon^{2}$ , horizon H, transition kernel $\\mathbb{P}=\\{\\mathbb{P}_{h}\\}_{h=1}^{H}$ where $\\mathbb{P}_{h}:\\mathcal{S}\\times\\Upsilon\\Delta(\\mathcal{S})$ , and the individual bidders’ reward functions $r_{i}=\\{r_{i h}\\}_{h=1}^{H}$ for all $i \\in [ N ]$ . The choice of item $\\upsilon \\in \\Upsilon$ afects the bidders’ rewards as well as the transition.\n\nThe interaction between the bidders and the seller is then defined as follows. We assume without loss of generality that the state at the initial step is fixed at some $x_{1}\\in S$ . For each $h \\in [ H ]$ , the seller and the bidders engage in a single round of the second-price auction. Given the seller’s item choice at step $h,v_{h}$ , nature transitions to the next state according to the transition kernel $\\mathbb{P}_{h}$\n\nBidder Rewards. We assume that for each bidder $i \\in [ N ]$ at time $h \\in [ H ]$ , his reward<sup>3</sup> depends on both the state x and item being auctioned of at that round $\\upsilon \\in \\Upsilon$ , which we formalize as\n\n$$\nr_{i h}(x,\\upsilon)=1+\\mu_{i h}(x,\\upsilon)+z_{i h},\\text{where} z_{i h}\\stackrel{\\mathrm{i.i.d.}}{\\sim}F.\n$$\n\nHere, $z_{i h}$ denotes the randomness within bidders’ rewards and is drawn i.i.d. from the market noise distribution $F(\\cdot)$ . We assume that $F(\\cdot)$ is supported on [−1, 1] and has mean 0. Let $\\mu_{i,h}:S\\times\\Upsilon  [0,1]$ denote the conditional expectation of the reward less one, where the constant is added to ensure $r_{i h}(x,v)\\in [0,3]$\n\nPolicies and Value Functions. Before we describe the seller’s policy, we first discuss the action space $\\mathcal{A}=\\Upsilon\\times\\mathbb{R}_{\\ge 0}^{N}$ . At each $h \\in [ H ]$ , the seller chooses some action $a_{h}=(v_{h},\\rho_{h})$ ， comprising of item choice $\\upsilon \\in \\Upsilon$ and reserve price vector $\\rho\\in\\mathbb{R}_{\\geq 0}^{N}$ . The seller’s policy is then $\\pi=\\{\\pi_{h}\\}_{h=1}^{H}$ , where $\\pi_{h}:S\\to\\Delta({\\mathcal{A}})$ . We let $\\pi^{v}$ and $\\pi^{\\rho}$ denote the marginal item choice and reserve price policies, respectively. Recall that the seller garners revenue only when the item is sold to a bidder. At each $h \\in [ H ]$ , her per-step expected revenue is then\n\n$$\nR_{h}=\\mathbb{E}_{\\{z_{i h}\\}_{i=1}^{N}}\\left[\\sum_{i=1}^{N}m_{i h}\\mathbb{1}(m_{i h}\\leq b_{i h})\\right]\\tag{1}\n$$\n\nas we recall that $m_{i h}=\\operatorname*{max}\\{\\rho_{i h},\\operatorname*{max}_{j\\neq i}b_{j h}\\}$ and bidder i pays the seller $m_{i h}$ if and only if $b_{i h}\\geq m_{i h}$ . The value function (V-function) of the seller’s revenue for any policy π and the action-value function (Q-function) is $Q_{h}^{\\pi}:\\mathcal{S}\\times\\mathcal{A}\\mathbb{R}$ are then\n\n$$\nV_{h}^{\\pi}(x)=\\mathbb{E}_{\\pi}\\left[\\sum_{h^{\\prime}=h}^{H}R_{h^{\\prime}}(x_{h^{\\prime}},a_{h^{\\prime}})\\mid x_{h}=x\\right]\n$$\n\nand\n\n$$\nQ_{h}^{\\pi}(x,a)=\\mathbb{E}_{\\pi}\\left[\\sum_{h^{\\prime}=h}^{H}R_{h^{\\prime}}(x_{h^{\\prime}},a_{h^{\\prime}})\\mid x_{h}=x,a_{h}=a\\right],\n$$\n\nrespectively.\n\nSince the bidder reward only depends on state x and the choice of item υ instead of reserve $\\rho,$ we have a family of mappings from $\\mathcal{S}\\times\\Upsilon$ to $\\mathbb{R}_{\\geq 0}^{N}$ that determines $\\rho .$ Therefore, with a slight abuse of notation, we can rewrite our Q-function as $Q(x,a)=Q(x,(v,\\rho(x,v)))$ , restricting the role of setting reserve prices using such mappings without loss of generality. From now on, we use $Q(x,v)$ to denote the Q-function for simplicity. For any function $f:{\\mathcal{S}}\\mathbb{R}$ , we define the transition operator $\\mathcal{P}$ and the Bellman operator $\\boldsymbol{B}$ as\n\n$$\n(\\mathcal{P}_{h}f)(x,a)=\\mathbb{E}[f(x_{h+1})\\mid x_{h}=x,a_{h}=a],(\\mathcal{B}_{h}f)(x,a)=\\mathbb{E}[R_{h}(x_{h},a_{h})]+(\\mathbb{P}_{h}f)(x,a),\n$$\n\nrespectively. Finally, we let $\\pi^{\\star}$ denote the optimal policy when the bidders’ reward functions, the MDP’s underlying transition, and the market noise distribution are all known to the seller. We remark that when these parameters are known, second price auctions with personalized reserve prices are inherently incentive compatible and rational bidders will bid truthfully.\n\nImpatient Utility-Maximizing Bidders. We assume the bidders are equipped with some discount rate $\\gamma\\in(0,1)$ while the seller’s reward is not discounted. For the sake of simplicity, we assume $\\gamma$ is common knowledge. Drutsa (2020) consider a scenario where $\\gamma$ is unknown but with a strictly less than one upper bound. We highlight that it also works with our CLUB algorithm as long as we replace $\\gamma$ with its upper bound. We can have regret bounds with the same order because we adopt more conservative estimators, and buyers won’t violate as much as the corresponding results of $\\gamma$ . Then all results in our paper hold up to some changes of absolute constants. Rational bidder i’s utility at step h is given by $(r_{i h}(x_{h},v_{h})-m_{i h})\\mathbb{1}(b_{i h}\\geq m_{i h})$ , as we note that he only receives nonzero utility upon winning the auction. His objective is to maximize his discounted cumulative utility\n\n$$\n\\mathrm{Utility}_{i}=\\sum_{k=1}^{K}\\gamma^{k}\\mathbb{E}_{\\pi_{k}}\\left[\\sum_{h=1}^{H}(r_{i h}(x_{h}^{k},v_{h}^{k})-m_{i h}^{k})\\mathbb{1}(b_{i h}^{k}\\geq m_{i h}^{k})\\mid x_{1}^{k}=x_{1}\\right].\n$$\n\nLinear Markov Decision Process. As a concrete setting, we study linear function approximation.\n\nAssumption 1. Assume that there exists known feature mapping $\\phi:\\mathcal{S}\\times\\Upsilon\\mathbb{R}^{d}$ such that there exist d-dimension unknown (signed) measures $\\mathcal{M}_{h}$ over $\\boldsymbol{s}$ and unknown vectors $\\{\\theta_{i h}\\}_{i=1}^{N}\\in\\mathbb{R}^{d}$ that satisfy\n\n$$\n\\mathbb{P}_{h}(x^{\\prime}|x,v)=\\langle\\phi(x,v),\\mathcal{M}_{h}(x^{\\prime})\\rangle,\\mu_{i h}(x,v)=\\langle\\phi(x,v),\\theta_{i h}\\rangle\n$$\n\nfor all $(x,v,x^{\\prime})\\in\\mathcal{S}\\times\\Upsilon\\times\\mathcal{S},i\\in [N]$ , and $h \\in [ H ]$ . Without loss of generality, we assume that $\\|\\phi(x,v)\\|\\leq 1for$ all $(x,v)\\in{S\\times\\Upsilon}$ ， $\\|{\\mathcal{M}}_{h}(S)\\|\\leq\\sqrt{d},$ , and $\\lVert\\theta_{i h}\\rVert\\leq\\sqrt{d}$ for all $h \\in [ H ]$ and $i \\in [ N ]$\n\nAlgorithm 2 Bufer Period with Known $F(\\cdot)$\n\n<div class=\"mineru-algorithm\" style=\"white-space: pre-wrap; font-family:monospace;\">\nReceives buffer start buffer.s( $\\tilde{k}+1$ ) = k and end buffer.e( $\\tilde{k}+1$ ) =  $k + \\frac{3\\log K}{\\log(1/\\gamma)}$ .\nDo nothing for all episodes buffer.s( $\\tilde{k}+1$ ) ≤ k &lt; buffer.e( $\\tilde{k}+1$ ), i.e., do nothing during the buffer period before the end.\nAt the end of the buffer period, update policy estimate  $\\pi_{\\tilde{k}+1}$  and Q-function estimate  $\\hat{Q}_{h}^{\\pi_{\\tilde{k}+1}}(\\cdot,\\cdot)$  using Algorithm 4, and then increment buffer period counter  $\\tilde{k} \\leftarrow \\tilde{k} + 1$ .\n</div>\n\nWith bufer periods defined, we summarize CLUB’s update schedule in Algorithm 3 and include Figure 1 for visual representation. Let $\\frac{1}{H K}\\circ\\pi_{\\mathrm{rand}}+\\left(1-\\frac{1}{H K}\\right)\\circ\\pi_{\\tilde{k}}$ represent a mixture policy combining $\\pi_{\\mathrm{rand}}$ and $\\pi_{\\tilde{k}}$ where for each $h,$ with probability $\\frac{1}{H K}$ we act according to $\\pi_{\\mathrm{rand}}$ and with probability $1-{\\frac{1}{H K}}$ according to $\\pi_{\\tilde{k}}$ . For convenience, we assume buffer. ${\\tt e}(\\tilde{k})$ is an integer, as rounding up buffer. $\\boldsymbol{\\mathbf{\\rho}}_{\\boldsymbol{\\mathbf{e}}}(\\tilde{k})$ does not afect asymptotic regret. Unlike a typical low switching cost RL algorithm, Algorithm 4 further delays updating for $\\frac{3\\log K}{\\log(1/\\gamma)}$ episodes after the switching criterion in Line 4 is satisfied.\n\nAlgorithm 3 Contextual-LSVI-UCB-Bufer (CLUB) with Known F\n\n<div class=\"mineru-algorithm\" style=\"white-space: pre-wrap; font-family:monospace;\">\nInitialize policy estimate $\\pi_0$, buffer period counter $\\tilde{k} = 0$, buffer period starting points buffer.s(0) = 1, and buffer period end points buffer.e(0) = 1.\nfor episodes $k = 1, \\ldots, K$ do\n    Execute mixture policy $\\frac{1}{HK} \\circ \\pi_{\\text{rand}} + (1 - \\frac{1}{HK}) \\circ \\pi_{\\tilde{k}}$, collecting outcomes $q_{ih}^\\tau$ and updating matrices $\\Lambda_h^k \\leftarrow \\sum_{\\tau=1}^k \\phi(x_h^\\tau, v_h^\\tau) \\phi(x_h^\\tau, v_h^\\tau)^T + I$ for all $h \\in [H]$.\n    If there exists $h \\in [H]$ such that $(\\Lambda_h^{\\text{buffer.e}(\\tilde{k})})^{-1} \\not\\leq 2(\\Lambda_h^k)^{-1}$, schedule a new buffer period starting at buffer.s($\\tilde{k} + 1$) = $k$ and ending at buffer.e($\\tilde{k} + 1$) = $k + \\frac{3\\log K}{\\log(1/\\gamma)}$ using Algorithm 2, and set $k \\leftarrow \\text{buffer.e}(\\tilde{k} + 1)$.\n</div>\n\nAddressing Challenges 2 and 3: Regret Minimization and Nonlinear Revenue. Having shown that our algorithm punishes untruthful behavior, we begin by showing that the resulting reports are suficiently truthful for obtaining accurate parameter estimates. It’s still quite intricate as regret depends on both the state, action related to the transition kernel, and reserve prices. Traditional point estimation with uncertainty quantity is not enough since we need to not only combine the structure of the underlying MDP and coordinate with bufer periods, the so-called lazy updates, but also consider the small proportion of untruthful bids. Whereas LSVI-UCB directly learns from empirical rewards, here we use indicators $q_{i h}^{k}$ , which we recall is one if bidder i receives the item at episode k step h and zero otherwise. As we cannot guarantee that the empirical covariance matrix is positive definite, existing techniques in Amin et al. (2014); Golrezaei et al. (2019) cannot be applied. We instead have\n\n$$\n\\hat{\\theta}_{i h}=\\underset{\\|\\theta\\|\\leq 2\\sqrt{d}}{\\arg\\min}\\sum_{\\tau=1}^{\\text{buffer.e}(\\tilde{k}+1)}(q_{i h}^{\\tau}-1+F(m_{i h}^{\\tau}-1-\\langle\\phi(x_{h}^{\\tau},v_{h}^{\\tau}),\\theta\\rangle))^{2},\\tag{3}\n$$\n\nwhere $\\rho_{i h}^{\\tau}$ is agent i’s reserve price and $m_{i h}^{\\tau}=\\operatorname*{max}\\{\\operatorname*{max}_{j\\neq i}b_{i h}^{\\tau},\\rho_{i h}^{\\tau}\\}$ . Equation (3) is justified by the observation that, assuming that he bids truthfully, bidder i wins the auction with probability $1-F(m_{i h}^{\\tau}-1-\\langle\\phi(x_{h}^{\\tau},v_{h}^{\\tau}),\\theta\\rangle)$ , conditioned on $x_{h}^{\\tau},v_{h}^{\\tau}.$ , and $m_{i h}^{\\tau}$ . Controlling the uncertainty around $\\dot{\\theta}_{i h}$ then resembles controlling the uncertainty of a generalized linear model with $F(\\cdot)$ being the link function. As bidders need to overbid or underbid significantly to alter the outcome of the auction, $\\hat{\\theta}_{i h}$ is less susceptible to untruthfulness.\n\nWhile we use a typical linear function approximation assumption, the seller’s revenue function $R_{h}$ is not linear, and we cannot directly apply existing approaches. We instead directly estimate $R_{h}$ and link our uncertainty on the seller’s revenue to the typical linear MDP uncertainty quantifier, summarized Algorithm 4.\n\nAlgorithm 4 Estimation of $\\hat{Q}_{h}^{\\pi_{\\tilde{k}+1}}(\\cdot,\\cdot)$\n\n<div class=\"mineru-algorithm\" style=\"white-space: pre-wrap; font-family:monospace;\">\nEstimate $\\hat{\\theta}_{ih}$ using Equation (3) and set $\\hat{\\mu}_{ih}(\\cdot,\\cdot)\\leftarrow\\langle\\phi(\\cdot,\\cdot),\\hat{\\theta}_{ih}\\rangle$ for all $i,h$.\nEstimate reserve price $\\hat{\\rho}_{ih}(\\cdot,\\cdot)=\\arg\\max_y y(1-F(y-1-\\hat{\\mu}_{ih}(\\cdot,\\cdot)))$ for all $i,h$.\nEstimate revenue $\\hat{R}_h(\\cdot,\\cdot)\\leftarrow\\mathbb{E}[\\max\\{\\tilde{b}_h^-(\\cdot,\\cdot),\\hat{\\rho}_h^+(\\cdot,\\cdot)\\}\\,1(\\tilde{b}_h^+(\\cdot,\\cdot)\\geq\\hat{\\rho}_h^+(\\cdot,\\cdot))]$.\nfor $h=H,\\ldots,1$ do\n    $\\triangleright$ Estimate $Q$-function and optimal policy.\n    $\\Lambda_h \\leftarrow \\sum_{\\tau=1}^{\\text{buffer.e}(\\tilde{k}+1)} \\phi(x_h^\\tau,v_h^\\tau)\\phi(x_h^\\tau,v_h^\\tau)^T + \\lambda I.$ $\\triangleright$ We set $\\lambda=1$ in this paper.\n    $\\omega_h \\leftarrow \\Lambda_h^{-1} \\sum_{\\tau=1}^{\\text{buffer.e}(\\tilde{k}+1)} \\phi(x_h^\\tau,v_h^\\tau)[\\max_v \\hat{Q}_{h+1}(x_{h+1}^\\tau,v)]$.\n    $\\hat{Q}_h^{\\pi_{\\tilde{k}+1}}(\\cdot,\\cdot)\\leftarrow\\min\\{\\omega_h^T\\phi(\\cdot,\\cdot)+\\hat{R}(\\cdot,\\cdot)+\\text{poly}(\\log K)\\|\\phi(\\cdot,\\cdot)\\|_{\\Lambda_h^{-1}},3H\\}.$ $\\pi_{\\tilde{k}+1,h}^v(\\cdot)\\leftarrow\\arg\\max_v \\hat{Q}_h^{\\pi_{\\tilde{k}+1}}(\\cdot,v).$ $\\pi_{\\tilde{k}+1,h}^{\\rho_i}(\\cdot)\\leftarrow\\hat{\\rho}_{ih}(\\cdot,\\pi_{\\tilde{k}+1,h}^v(\\cdot)).$\nend for\nReturn $\\{\\hat{Q}_h^{\\pi_{\\tilde{k}+1}}(\\cdot,\\cdot)\\}_{h=1}^H$ and $\\{\\pi_{\\tilde{k}+1,h}(\\cdot)\\}_{h=1}^H$.\n</div>\n\nWe let $\\tilde{b}^{+}$ and $\\rho^{+}$ denote the highest truthful bid and the highest reserve price, respectively. Similarly, let $\\tilde{b}^{-}$ <sup>−</sup> and $\\rho^{-}$ <sup>−</sup> denote the second-highest. Algorithm 4 estimates the Q-function optimistically by dividing the problem into two halves: per-step revenue estimation (Lines 1 to 3) and transition estimation (Lines 4 to 10). In the first half, we use Equation (3) to estimate all $\\theta_{i h}$ , which in turn gives estimates for bidders’ rewards in the form of $\\hat{\\mu}_{i h}$ . We then feed the reward function estimates to Line $^{2,}$ yielding an estimate for the optimal reserve price. With Algorithms 1, 2, and 3, the efects of untruthful reports are controlled, and we can ensure that the revenue estimate is suficiently close to the ground truth. With $\\rho_{i h}$ estimated, we then obtain revenue estimates for all states and item choices via Line 3. Consequently, we decide both nearly optimal reserve prices and the order of items, addressing the second challenge of regret minimization.\n\nWe introduce the following assumptions before we bound the regret. These regularity assumptions are commonly found in economics literature (Kleiber and Kotz, 2003; Bagnoli and Bergstrom, 2006).\n\nAssumption 2. Market noise pdf f is bounded, i.e. there exist constants $c_{1},C_{1}$ such that $c_{1}\\leq f\\leq C_{1}$\n\nAssumption 3. Market noise pdf f is diferentiable and its derivative is bounded. That is, there exists a constant L such that $|f^{\\prime}|\\le L$\n\nAssumption 4. Market noise cdf $F(\\cdot)$ and $1-F(\\cdot)$ are log-concave.\n\nWe are now ready to state our results. If we set poly(log $K)=C_{7}+C_{6}H\\log^{2}K$ in Algorithm 4, where constant $C_{6}$ is determined in Lemma 16 and constant $C_{7}=B_{8}H^{\\frac{3}{2}}$ log K with constant $B_{8}$ determined in Lemma 39, then we have Theorem 6.\n\nTheorem 6. Under Assumption ${\\mathit{1}},{\\mathit{2}},{\\mathit{3}}$ and 4, for any fixed failure probability $\\delta\\in(0,1)$ with probability at least $1 - \\delta$ , Algorithm 3 achieves at most $\\tilde{\\mathcal O}(\\sqrt{H^{5}K})$ revenue regret, where $\\tilde{\\mathcal{O}}(\\cdot)$ hides only absolute constants and logarithmic terms.\n\nProof See Section B for a detailed proof.\n\nWe now discuss when the market noise distribution is unknown. Recall from previous discussions that our second challenge lies in minimizing revenue regret when the market noise distribution is unknown. Existing techniques, similar to the one in Golrezaei et al. (2019), incorporate pure exploration rounds to address the challenge, yet necessitate a $\\tilde{\\mathcal{O}}(K^{2/3})$ revenue regret. In this section, we instead introduce a novel technique dubbed “simulation”, which eliminates the need for pure exploration rounds and achieves instead a $\\tilde{\\mathcal{O}}(\\sqrt{K})$ regret. While the first and third challenges have been previously addressed, the approaches in Section 3 also require careful adjustments, as the unknown market noise distribution makes a direct application of these approaches impossible. We detail our techniques and procedures in the rest of this section.\n\nSimilarly, there are three steps to do auction design when $F(\\cdot)$ is unknown. First, we leverage Algorithm 1 and Algorithm 5 to motivate an approximately truthful mechanism. Second, we utilize Algorithm 8 in coordination with the newly proposed Algorithm 7 to estimate the underlying MDP and set reserve prices. We motivate truthfulness through bufer periods and quantify the uncertainty by constructing corresponding ellipsoid bounds. Finally, we add up all these uncertainties and minimize regret with high probability.\n\nAddressing Challenge 1: Untruthfulness. When the market noise distribution is unknown, the techniques used in Section 3 cannot be applied directly, necessitating careful adaptations. We summarize the changes to these techniques, beginning by introducing Algorithm 5, the counterpart to Algorithm 2, for when $F(\\cdot)$ is unknown. The key diference lies in the optimization subroutine called in Line 3, which is required for addressing the third challenge when the market noise distribution $F(\\cdot)$ is unknown.\n\nWe then discuss Algorithm 6, a close variant of Algorithm 3, whose biggest change lies in the update schedule in Line 4. Algorithm 3 maintains only an accurate estimate of the underlying MDP, achieved with a low switching cost style update schedule, which in turn deters untruthful bidding. On the other hand, Algorithm 6 needs accurate estimates of both the MDP and the market noise distribution $F(\\cdot)$ . We force additional updates whenever k is a power of 2, also ensuring that $\\hat{F}(\\cdot)$ is close to $F(\\cdot)$ . As the number of updates remains in ${\\mathcal{O}}(\\log K)$ , the extraneous updates do not afect the regret asymptotically.\n\nAlgorithm 5 Bufer Period with Unknown $F(\\cdot)$\n\n<div class=\"mineru-algorithm\" style=\"white-space: pre-wrap; font-family:monospace;\">\n1: Receives buffer start buffer.s( $\\tilde{k}+1$ ) = k and end buffer.e( $\\tilde{k}+1$ ) =  $k + \\frac{3\\log K}{\\log(1/\\gamma)}$ .\n2: Do nothing for all episodes buffer.s( $\\tilde{k}+1$ ) ≤ k &lt; buffer.e( $\\tilde{k}+1$ ), i.e., do nothing during the buffer period before the end.\n3: At the end of the buffer period, update policy estimate  $\\pi_{\\tilde{k}+1}$  and Q-function estimate  $\\hat{Q}_{h}^{\\pi_{\\tilde{k}+1}}(\\cdot,\\cdot)$  using Algorithm 8, and then increment buffer period counter  $\\tilde{k} \\leftarrow \\tilde{k} + 1$ .\n</div>\n\nAlgorithm 6 Contextual-LSVI-UCB-Bufer (CLUB) with Unknown F\n\n<div class=\"mineru-algorithm\" style=\"white-space: pre-wrap; font-family:monospace;\">\nInitialize policy estimate $\\pi_0$, buffer period counter $\\tilde{k} = 0$, buffer period starting points buffer.s(0) = 1, and buffer period end points buffer.e(0) = 1.\nfor episodes $k = 1, \\ldots, K$ do\n    Execute mixture policy $\\frac{1}{HK} \\circ \\pi_{\\text{rand}} + (1 - \\frac{1}{HK}) \\circ \\pi_{\\tilde{k}}$, collecting outcomes $q_{ih}^\\tau$ and updating matrices $\\Lambda_h^k \\leftarrow \\sum_{\\tau=1}^k \\phi(x_h^\\tau, v_h^\\tau) \\phi(x_h^\\tau, v_h^\\tau)^T + I$ for all $h \\in [H]$.\n    If there exists $h \\in [H]$ such that $(\\Lambda_h^{\\text{buffer.e}(\\tilde{k})})^{-1} \\not\\leq 2(\\Lambda_h^k)^{-1}$ or $\\log_2(k)$ is an integer, schedule a new buffer period starting at buffer.s($\\tilde{k} + 1$) = $k$ and ending at buffer.e($\\tilde{k} + 1$) = $k + \\frac{3\\log K}{\\log(1/\\gamma)}$ using Algorithm 5, and set $k \\leftarrow \\text{buffer.e}(\\tilde{k} + 1)$.\nend for\n</div>\n\nHere we introduce a new random variable $\\tilde{q}_{i h}^{\\tau}=1(b_{i h}^{\\tau}\\geq\\tilde{\\rho}_{i h}^{\\tau})$ , where for each $h,\\tau$ we select one $i \\in [ N ]$ uniformly at random and then draw $\\tilde{\\rho}_{i h}^{\\tau}$ from $\\mathrm{{U n i f}}([0,3])$ . For all $j \\neq i$ we set $\\tilde{\\rho}_{i h}^{\\tau}$ to ∞. At a high level, $\\tilde{q}_{i h}^{\\tau}$ “simulates” executing $\\pi_{\\mathrm{rand}}.$ holding $x_{h}^{\\tau}$ and $v_{h}^{\\tau}$ constant, what would be the outcome if we were to act according to $\\pi_{\\mathrm{rand}}$ instead? As we do not need to execute $\\pi_{\\mathrm{rand}}.$ , revenue regret can be decreased. Furthermore, $\\tilde{q}_{i h}^{\\tau}$ still enjoys the same resilience towards untruthful reporting that $q_{i h}^{\\tau}$ does.\n\n1: for $h=1,\\ldots,H$ and $\\tau=1,\\dots,K$ do  \n2: Generate virtual reserve prices $\\tilde{\\rho}_{i h}^{\\tau}$ by selecting one bidder $i \\in [ N ]$ uniformly at random. Let $\\tilde{\\rho}_{i h}^{\\tau}\\sim\\mathrm{Unif}([0,3])$ and set all other reserve prices to infinity, i.e. $\\tilde{\\rho}_{j h}^{\\tau}=\\infty$ for all $j \\neq i .$  \nUse real bidding data $b_{i h}^{\\tau}$ and simulated reserve prices $\\tilde{\\rho}_{i h}^{\\tau}$ to simulate outcome $\\tilde{q}_{i h}^{\\tau}$ for all $i \\in [ N ]$ , namely set $b_{i h}^{\\tau}=\\mathbb{1}(b_{i h}^{\\tau}\\geq\\tilde{\\rho}_{i h}^{\\tau})$ for all $i \\in [ N ]$  \n4: end for  \n5: Return the simulated outcomes $\\{\\tilde{q}_{i h}^{k}\\}$\n\nAddressing Challenge 3: Nonlinear Revenue. With the first challenge addressed by carefully adjusting techniques in Section 3 and the second by the simulation technique detailed in Algorithm 7, we now discuss the third challenge: provably eficient reinforcement learning when the revenue is nonlinear and $F(\\cdot)$ is unknown. We start with summarizing how we simultaneously estimate $\\theta_{i h}$ and $F(\\cdot)$ in the form of Equation (4).\n\n$$\n\\hat{\\theta}_{i h}=\\underset{\\|\\theta\\|\\leq 2\\sqrt{d}}{\\arg\\min}\\sum_{\\tau=1}^{\\text{buffer.e}(\\tilde{k}+1)}\\left(3 N\\tilde{q}_{i h}^{\\tau}-\\left(1+\\left\\langle\\phi\\left(x_{h}^{\\tau},v_{h}^{\\tau}\\right),\\theta\\right\\rangle\\right)\\right)^{2},\\tag{4}\n$$\n\n$$\n\\hat{F} (z) = \\frac{1}{N\\text{\\rm{buffer.e}}(\\tilde{k} + 1)H}\\sum_{i = 1}^{N}\\sum_{\\tau = 1}^{\\text{\\rm{buffer.e}}(\\tilde{k} +1)}\\sum_{h = 1}^{H}\\mathbb{1}(b_{i\\tau h} - 1 - \\langle \\phi_{h}^{\\tau},\\hat{\\theta}_{ih}\\rangle \\leq z).\n$$\n\nWe note that we are simply using a histogram to estimate $F(\\cdot)$ and, as we have successfully decoupled the estimation error of $F(\\cdot)$ from that of $\\theta_{i h}$ , using histogram is suficient for achieving $\\tilde{\\mathcal{O}}(\\sqrt{K})$ revenue regret. We then introduce Algorithm 8, whose key diference with Algorithm 4 lies in the added uncertainty due to $\\hat{F}(\\cdot)$ and the inclusion of the simulation subroutine. Similar to Section 3, the procedure then provides us with suficiently accurate policy and Q-function estimates, resolving our third and final challenge.\n\nIn summary, we have addressed all three challenges when the market noise distribution is unknown. The first challenge is resolved by carefully adjusting the techniques introduced in Section 3, ensuring that they are still valid when $F(\\cdot)$ is unknown. For the second challenge we feature a novel technique dubbed “simulation” that allows us to “simulate” pure exploration rounds without actually executing them, reducing revenue regret. For the third challenge, we build of of the simulation technique and introduce a new estimation procedure for jointly estimating $F(\\cdot)$ and θ.\n\nAlgorithm 8 Estimation of $\\hat{Q}_{h}^{\\pi_{\\tilde{k}+1}}(\\cdot,\\cdot)$ with Unknown $F(\\cdot)$\n\n<div class=\"mineru-algorithm\" style=\"white-space: pre-wrap; font-family:monospace;\">\nCollect simulation outcome $\\tilde{q}$ using Algorithm 7.\nEstimate $\\hat{\\theta}_{ih}, \\hat{F}(\\cdot)$ using Equation (4).\nEstimate $\\hat{\\mu}_{ih}(\\cdot, \\cdot) \\leftarrow \\langle \\phi(\\cdot, \\cdot), \\hat{\\theta}_{ih} \\rangle$.\nSet reserve price $\\hat{\\rho}_{ih}(\\cdot, \\cdot) = \\arg\\max_y y(1 - \\hat{F}(y - 1 - \\hat{\\mu}(\\cdot, \\cdot)))$.\nEstimate revenue $\\hat{R}_h(\\cdot, \\cdot) \\leftarrow \\mathbb{E}[\\max\\{\\tilde{b}_h^-(\\cdot, \\cdot), \\hat{\\rho}_h^+(\\cdot, \\cdot)\\} \\mathbb{1}(\\tilde{b}_h^+(\\cdot, \\cdot) \\geq \\hat{\\rho}_h^+(\\cdot, \\cdot))]$.\nfor $h = H, \\ldots, 1$ do\n    ▷ Estimate $Q$-function and optimal policy.\n    $\\Lambda_h \\leftarrow \\sum_{\\tau=1}^{\\text{buffer.e}(\\tilde{k}+1)} \\phi(x_h^\\tau, v_h^\\tau) \\phi(x_h^\\tau, v_h^\\tau)^T + \\lambda I.$\n    ▷ We set $\\lambda = 1$ in this paper.\n    $\\omega_h \\leftarrow \\Lambda_h^{-1} \\sum_{\\tau=1}^{\\text{buffer.e}(\\tilde{k}+1)} \\phi(x_h^\\tau, v_h^\\tau)[\\max_a \\hat{Q}_{h+1}(x_{h+1}^\\tau, a)]$.\n    $\\hat{Q}_h^{\\pi_{\\tilde{k}+1}}(\\cdot, \\cdot) \\leftarrow \\min\\{\\omega_h^T \\phi(\\cdot, \\cdot) + \\hat{R}(\\cdot, \\cdot) + \\text{poly}_1(\\log K) \\| \\phi(\\cdot, \\cdot) \\|_{\\Lambda_h^{-1}} + \\frac{\\text{poly}_2(\\log K)}{\\sqrt{\\text{buffer.e}(\\tilde{k}+1)}}, 3H\\}$ $\\pi_{\\tilde{k}+1,h}^v(\\cdot) \\leftarrow \\arg\\max_v \\hat{Q}_h^{\\pi_{\\tilde{k}+1}}(\\cdot, v)$.\n    $\\pi_{\\tilde{k}+1,h}^{\\rho_i}(\\cdot) \\leftarrow \\hat{\\rho}_{ih}(\\cdot, \\pi_{\\tilde{k}+1,h}^a(\\cdot)).$\nend for\nReturn $\\{\\hat{Q}_h^{\\pi_{\\tilde{k}+1}}(\\cdot, \\cdot)\\}_{h=1}^H$ and $\\{\\pi_{\\tilde{k}+1,h}(\\cdot)\\}_{h=1}^H$.\n</div>\n\nWe now argue that Algorithm 6 achieves $\\tilde{\\mathcal{O}}(\\sqrt{K})$ regret. We begin with a slight detour, making a basic assumption on the hypothesis class for $F(\\cdot)$\n\nAssumption 7. The market noise distribution $F(\\cdot)$ belongs to a distribution family ${ \\mathcal F } .$\n\nWe further let $\\mathcal{N}_{\\epsilon}(\\mathcal{F})$ be the ϵ-covering number of $\\mathcal{F}$ with respect to the metric that dist $\\begin{array}{ r }{(F,G)=\\operatorname*{sup}_{x}|F(x)-G(x)|}\\end{array}$ . We now have our main theorem when the noise distribution is unknown. If we let poly (log $K)=C_{15}+C_{13}H\\log^{2}K$ and $\\mathrm{poly_{2}}(\\log K)=C_{14}H^{2}\\log^{4}K$ in Algorithm 8, where $C_{15}=D_{7}H^{\\frac{3}{2}}$ and the constant $D_{7}$ is determined in Lemma 47, constants $C_{13}$ and $C_{14}$ are determined in Lemma 24, we would attain the following regret guarantee.\n\nTheorem 8. Under Assumptions 1, 2, 3, 4 and $^{7,}$ when $F(\\cdot)$ is unknown, for any fixed failure probability $\\delta\\in(0,1)$ , Algorithm 6 achieves at most $\\tilde{\\mathcal{O}}(H^{3}\\sqrt{K}+H^{2.5}\\sqrt{K\\log\\mathcal{N}_{1/K}(\\mathcal{F})})$ regret with probability at least $1 - \\delta$ in the worst case, where $\\tilde{\\mathcal{O}}(\\cdot)$ hides only absolute constants and logarithmic terms.\n\nProof See Section C for a detailed proof.\n\nIn this section, we show some useful lemmas in order to prove theorems in Section 3. We organize the section as follows. Firstly, we introduce lemmas to bound the efect of untruthful bidding. Then, we will show that we are able to estimate unknown parameters accurately. Finally, combining them leads to bounded regret with high probability.\n\nNow, we focus on the gap between $R(\\cdot,\\cdot)$ and the estimate $\\hat{R}(\\cdot,\\cdot)$ . We are ready to show the following lemma.\n\nFirst of all, we introduce some notations. $\\begin{array}{ r }{R_{h}^{k}(\\cdot,\\cdot)\\ : =\\ :\\sum_{i=1}^{N}}\\end{array}$ E[max $\\{r_{i h}^{k-},\\alpha_{i h}^{k*}\\}\\mathbb{1}(r_{i h}^{k}\\ge$ max $\\{r_{i h}^{k-},\\alpha_{i h}^{k*}\\}\\}$ and $\\begin{array}{ r }{\\hat{R}_{h}^{k}(\\cdot,\\cdot)=\\sum_{i=1}^{N}}\\end{array}$ E[max $\\{\\hat{r}_{i h}^{k-},\\alpha_{i h}^{k}\\}\\mathbb{1}(\\hat{r}_{i h}^{k}\\geq\\operatorname*{max}\\{\\hat{r}_{i h}^{k-},\\alpha_{i h}^{k}\\})]$ . In short, $R(\\cdot,\\cdot)$ is the expectation of revenue if we choose the optimal reserve price $\\alpha_{i h}^{k*}$ for every bidder based on the knowledge of $\\mu_{i h}^{k}(\\cdot,\\cdot)$ and everyone bids truthfully based on his valuation. Respectively, $\\hat{R}(\\cdot,\\cdot)$ is the one we choose reserve price $\\alpha_{i h}^{k}$ with the estimation of $\\mu_{i h}^{k}(\\cdot,\\cdot)$ i.e., $\\hat{\\mu}_{i h}^{k}(\\cdot,\\cdot)$\n\nLemma 16. When Lemma 15 holds, we have\n\n$$\n|R_{h}^{k}(\\cdot,\\cdot)-\\hat{R}_{h}^{k}(\\cdot,\\cdot)|\\leq C_{6}H\\log^{2}K\\|\\phi(\\cdot,\\cdot)\\|_{(\\Lambda_{h}^{\\mathrm{buffer.e}(\\tilde{k})})^{-1}},\n$$\n\nwhere $C_{6}$ is a constant independent of K and $H$ .\n\nCompared to Section B, this section introduces a well-performed estimator to estimate the underlying distribution. With its help, we prove corresponding theorems when the market noise distribution is unknown.\n\nIn order to estimate noise distribution, we have the following lemma (Dvoretzky et al., 1956) to bound the gap between the true distribution and the empirical distribution. We assume that $\\hat{F}(\\cdot)$ and ${\\hat{f}}(\\cdot)$ inherit all the properties of $F(\\cdot)$ and $f(\\cdot)$ , because we can easily use some smooth kernels<sup>1</sup> to achieve this goal. However, in order to make the paper easy to understand, we do not explicitly write down the choice of a smooth kernel.\n\nNow, we introduce a lemma bounding the gap between the noise distribution $F(\\cdot)$ and $\\hat{F}(\\cdot)$\n\nLemma 23. Conditional on Good Event ${\\mathcal{E}},$ , it holds with probability at least $1 - \\delta$ that for any x in episode buffer. $.{\\tt e}(\\tilde{k})$\n\n$$\n\\begin{array}{l}|F(x)-\\hat{F}(x)|\\leq\\sqrt{\\frac{1}{2}\\log\\frac{2K}{\\delta}}(NH \\text{buffer.e}(\\tilde{k}))^{-\\frac{1}{2}}+\\frac{C_{1}C_{3}H}{K}+\\frac{C_{9}H\\log^{2}K}{\\text{buffer.e}(\\tilde{k})}\\\\ +C_{1}C_{11}\\sqrt{H}\\log K\\overline{{\\|\\phi(x_{h}^{\\tau},v_{h}^{\\tau})\\|}}_{(\\Lambda_{h}^{\\text{buffer.e}(\\tilde{k})})^{-1}}\\\\ \\leq C_{12}\\frac{H\\log^{2}K}{\\sqrt{\\mathrm{buffer.e}(\\tilde{k})}},\\\\ \\end{array}\n$$\n\nwhere $C_{12}$ is a constant.\n\nNow, we begin to bound the wedge of $R(\\cdot,\\cdot)$ and $\\hat{R}(\\cdot,\\cdot)$ corresponding to $\\hat{F}(\\cdot)$ . It holds the following lemma.\n\nLemma 24. Conditional on Good Event ${\\mathcal{E}},$ we have\n\n$$\n|R_{h}^{k}(\\cdot,\\cdot)-\\hat{R}_{h}^{k}(\\cdot,\\cdot)|\\leq C_{13}H\\log^{2}K\\|\\phi(\\cdot,\\cdot)\\|_{(\\Lambda_{h}^{\\mathrm{buffer.e}(\\tilde{k})})^{-1}}+C_{14}\\frac{H^{2}\\log^{4}K}{\\sqrt{\\mathrm{buffer.e}(\\tilde{k})}},\n$$\n\nwhere $C_{13}$ and $C_{14}$ are constants independent of K and $H$ .\n\nIn order to prove Lemma 17, we have the following lemmas for help.\n\nLemma 33. For any fixed policy π, let $\\{\\omega_{h}^{\\pi}\\}_{h\\in [H]}$ be the corresponding vectors such that $Q_{h}^{\\pi}(\\cdot,\\cdot)=R(\\cdot,\\cdot)+\\langle\\phi(\\cdot,\\cdot),\\omega_{h}^{\\pi}\\rangle$ for any h. Then, it holds that\n\n$$\n\\|\\omega_{h}^{\\pi}\\|\\leq 3 H\\sqrt{d},\n$$\n\nfor any h.\n\nLemma 37. Let V denote a class of functions mapping from S to R with the following parametric form\n\n$$\nV(\\cdot)=\\min\\{\\max_{a}\\omega^{T}\\phi(\\cdot,v)+\\hat{R}(\\cdot,v)+\\beta\\|\\phi(\\cdot,v)\\|_{\\Lambda^{-1}},3 H\\},\n$$\n\nwhere $\\|\\omega\\|\\le L,\\beta\\in [0,B]$ and the minimum eigenvalue satisfies $\\lambda_{\\operatorname*{min}}(\\Lambda)\\ \\geq\\ \\lambda$ $S u p \\ –$ pose $\\|\\phi(\\cdot,\\cdot)\\|\\leq 1$ and let $\\mathcal{N}_{\\epsilon}$ be the ϵ-covering number of $\\nu$ with respect to the distance dist $\\begin{array}{ r }{(V,V^{\\prime})=\\operatorname*{sup}_{x}|V(x)-V^{\\prime}(x)|}\\end{array}$ . Then, it holds\n\n$$\n\\log\\mathcal{N}_{\\epsilon}\\leq d\\log(1+\\frac{8L}{\\epsilon})+d^{2}\\log(1+\\frac{32\\sqrt{d}B^{2}}{\\lambda\\epsilon^{2}})+dN\\log(1+\\frac{8NB_{5}\\sqrt{d}}{\\epsilon}),\n$$\n\nwhere $B_{5}$ is a constant.\n\nNow, with lemmas prepared, we have the following lemma.\n\nLemma 38. For any δ, with probability at least $1 - \\delta$ , there exists constants $B_{6}$ and $B_{7}$ independent of K and H so that\n\n$$\n\\begin{array}{l}\\forall(k,h)\\in [K]\\times [H]:\\|\\sum_{\\tau=1}^{k}\\phi_{h}^{\\tau}\\left[\\hat{V}_{h+1}^{k}\\left(x_{h+1}^{\\tau}\\right)-\\mathbb{P}\\hat{V}_{h+1}^{k}\\left(x_{h}^{\\tau},v_{h}^{\\tau}\\right)\\right]\\|_{\\left(\\Lambda_{h}^{\\text{buffer.e}(\\tilde{k})}\\right)^{-1}}^{2}\\leq B_{6}H^{3}\\log^{2}K\\\\ +B_{7}H^{2}\\log C_{7}.\\\\ \\end{array}\n$$\n\nIn order to prove Lemma 24, we introduce the following lemma first.\n\nNow, we introduce some lemmas in parallel in order to prove Lemma $25$\n\nThen, it holds the following lemma.\n\nLemma 46. For any δ, with probability at least $1 - \\delta$ , there exists constants $B_{6}$ and $B_{7}$ independent of K and H so that\n\n$$\n\\begin{array}{l}\\forall(k,h)\\in [K]\\times [H]:\\|\\sum_{\\tau=1}^{k}\\phi_{h}^{\\tau}\\left[\\hat{V}_{h+1}^{k}\\left(x_{h+1}^{\\tau}\\right)-\\mathbb{P}\\hat{V}_{h+1}^{k}\\left(x_{h}^{\\tau},v_{h}^{\\tau}\\right)\\right]\\|_{\\left(\\Lambda_{h}^{\\text{buffer.e}(\\tilde{k})}\\right)^{-1}}^{2}\\leq D_{5}H^{3}\\\\ +D_{6}H^{2}\\log C_{15},\\\\ \\end{array}\n$$\n\nwhere $D_{5}\\sim\\tilde{\\mathcal{O}}(1)$ omitting log K and $D_{6}$ is a constant.", "theorems": [{"local_id": "T1", "statement": "Lemma 39. There exist a constant $B_{8}$ so that $C_{7}=B_{8}H^{\\frac{3}{2}}$ log $K_{\\cdot}$ , and for any fixed policy $\\pi,$ on Good Event ${\\mathcal{E}}^{\\mathcal{C}},i . e .,$ all inequalities hold, we have for all $(x,v,h,k)\\in\\mathcal{S}\\times\\Upsilon\\times [H]\\times [K]$ that:\n\n$$\n\\langle\\phi(\\cdot,\\cdot),\\omega_{h}^{k}\\rangle+\\hat{R}_{h}^{k}(\\cdot,\\cdot)-Q_{h}^{\\pi}(\\cdot,\\cdot)=\\mathbb{P}_{h}(\\hat{V}_{h+1}^{k}-V_{h+1}^{\\pi})(\\cdot,\\cdot)+\\Delta_{h}^{k}(\\cdot,\\cdot),\n$$\n\nwhere $\\Delta_{h}^{k}(\\cdot,\\cdot)\\leq(C_{7}+C_{6}H\\log^{2}K)\\lVert\\phi(\\cdot,\\cdot)\\rVert_{(\\Lambda_{h}^{\\mathrm{b u f f e r . e}(\\tilde{k})})^{-1}}.$", "proof": {"steps": [{"n": 1, "claim": "Proof Due to Bellman equation, we know that for any $(x,v,h)\\in\\mathcal S\\times\\Upsilon\\times [H]$ , it holds\n\n$$\nQ_{h}^{\\pi}(\\cdot,\\cdot)=R_{h}(\\cdot,\\cdot)+\\langle\\phi(\\cdot,\\cdot),\\omega_{h}^{\\pi}\\rangle=(R_{h}+\\mathbb{P}_{h}V_{h+1}^{\\pi})(\\cdot,\\cdot).\n$$"}, {"n": 2, "claim": "Therefore, it gives\n\n$$\n\\langle\\phi(\\cdot,\\cdot),\\omega_{h}^{k}\\rangle+\\hat{R}_{h}^{k}(\\cdot,\\cdot)-Q_{h}^{\\pi}(\\cdot,\\cdot)=\\langle\\phi(\\cdot,\\cdot),\\omega_{h}^{k}-\\omega_{h}^{\\pi}\\rangle+(\\hat{R}_{h}^{k}-R_{h})(\\cdot,\\cdot).\n$$"}, {"n": 3, "claim": "Then, since $\\omega_{h}^{k}=\\omega_{h}^{\\tt b u f f e r . e(\\tilde{k})}$ , it holds that\n\n$$\n\\begin{array}{l}\\omega_{h}^{k}-\\omega_{h}^{\\pi}=(\\Lambda_{h}^{\\mathrm{buffer.e}(\\tilde{k})})^{-1}\\sum_{\\tau=1}^{\\mathrm{buffer.e}(\\tilde{k})}\\phi_{h}^{\\tau}\\hat{V}_{h+1}^{k}(x_{h+1}^{\\tau})-\\omega_{h}^{\\pi}\\\\ =(\\Lambda_{h}^{\\text{buffer.e}(\\tilde{k})})^{-1}\\{-\\lambda\\omega_{h}^{\\pi}+\\sum_{\\tau=1}^{\\text{buffer.e}(\\tilde{k})}\\phi_{h}^{\\tau}[\\hat{V}_{h+1}^{k}(x_{h+1}^{\\tau})-\\mathbb{P}_{h}V_{h+1}^{\\pi}(x_{h}^{\\tau},v_{h}^{\\tau})]\\}\\\\ =\\delta_{1}+\\delta_{2}+\\delta_{3},\\\\ \\end{array}\n$$\n\nwhere\n\n$$\n\\delta_{1}= -\\lambda(\\Lambda_{h}^{\\mathrm{buffer.e}(\\tilde{k})})^{-1}w_{h}^{\\pi},\n$$\n\n$$\n\\delta_{2}=(\\Lambda_{h}^{\\mathrm{buffer.e}(\\tilde{k})})^{-1}\\sum_{\\tau=1}^{\\mathrm{buffer.e}(\\tilde{k})}\\phi_{h}^{\\tau}[\\hat{V}_{h+1}^{k}(x_{h+1}^{\\tau})-\\mathbb{P}_{h}\\hat{V}_{h+1}^{k}(x_{h}^{\\tau},v_{h}^{\\tau})],\n$$\n\n$$\n\\delta_{3}=(\\Lambda_{h}^{\\mathrm{buffer.e}(\\tilde{k})})^{-1}\\sum_{\\tau=1}^{\\mathrm{buffer.e}(\\tilde{k})}\\phi_{h}^{\\tau}\\mathbb{P}_{h}(\\hat{V}_{h+1}^{k}-V_{h+1}^{\\pi})(x_{h}^{\\tau},v_{h}^{\\tau}).\n$$"}, {"n": 4, "claim": "Then, we begin to bound items corresponding to $\\delta_{1},\\delta_{2}$ , and $\\delta_{3}$ individually."}, {"n": 5, "claim": "Firstly, it holds\n\n$$\n\\begin{array}{l}|\\langle\\phi(\\cdot,\\cdot),\\delta_{1}\\rangle|\\leq\\sqrt{\\lambda}\\|w_{h}^{\\pi}\\|\\|\\phi(\\cdot,\\cdot)\\|_{(\\Lambda_{h}^{\\text{buffer.e}(\\tilde{k})})^{-1}}\\\\ \\leq 3 H\\sqrt{d\\lambda}\\|\\phi(\\cdot,\\cdot)\\|_{(\\Lambda_{h}^{\\text{buffer.e}(\\tilde{k})})^{-1}}.\\\\ \\end{array}\n$$"}, {"n": 6, "claim": "The first inequality holds due to Cauchy inequality and $\\Lambda_{\\mathsf{b u f f e r . e}(\\tilde{k})}\\ \\succeq\\ \\lambda I$ . The second inequality holds due to Lemma 33."}, {"n": 7, "claim": "Secondly, it holds that\n\n$$\n|\\langle\\phi(\\cdot,\\cdot),\\delta_{2}\\rangle|\\leq\\sqrt{B_{6}H^{3}\\log^{2}K+B_{7}H^{2}\\log C_{7}}\\|\\phi(\\cdot,\\cdot)\\|_{(\\Lambda_{h}^{\\text{buffer.e}(\\tilde{k})})^{-1}}.\n$$\n\nIt holds because of Lemma 38."}, {"n": 8, "claim": "Lastly, we have\n\n$$\n\\begin{array}{l}\\langle\\phi(\\cdot,\\cdot),\\delta_{3}\\rangle=\\langle\\phi(\\cdot,\\cdot),(\\Lambda_{h}^{\\text{buffer.e}(\\tilde{k})})^{-1}\\sum_{\\tau=1}^{\\text{buffer.e}(\\tilde{k})}\\phi_{h}^{\\tau}\\mathbb{P}_{h}(\\hat{V}_{h+1}^{k}-V_{h+1}^{\\pi})(x_{h}^{\\tau},v_{h}^{\\tau})\\rangle\\\\ =\\langle\\phi(\\cdot,\\cdot),(\\Lambda_{h}^{\\text{buffer.e}(\\tilde{k})})^{-1}\\sum_{1}^{\\text{buffer.e}(\\tilde{k})}\\phi_{h}^{\\tau}(\\phi_{h}^{\\tau})^{T}\\int(\\hat{V}_{h+1}^{k}-V_{h+1}^{\\pi})(x^{\\prime})d\\mathcal{M}_{h}(x^{\\prime})\\rangle\\\\ =\\langle\\phi(\\cdot,\\cdot),\\int(\\hat{V}_{h+1}^{k}-V_{h+1}^{\\pi})(x^{\\prime})d\\mathcal{M}_{h}(x^{\\prime})\\rangle-\\lambda\\langle\\phi(\\cdot,\\cdot),(\\Lambda_{h}^{\\mathrm{buffer.e}(\\tilde{k})})^{-1}\\int(\\hat{V}_{h+1}^{k}-V_{h+1}^{\\pi})d\\mathcal{M}_{h}\\rangle\\\\ =\\mathbb{P}_{h}(\\hat{V}_{h+1}^{k}-V_{h+1}^{\\pi})(\\cdot,\\cdot)-\\lambda\\langle\\phi(\\cdot,\\cdot),(\\Lambda_{h}^{\\mathrm{buffer.e}(\\tilde{k})})^{-1}\\int(\\hat{V}_{h+1}^{k}-V_{h+1}^{\\pi})(x^{\\prime})d\\mathcal{M}_{h}(x^{\\prime})\\rangle\\\\ \\leq\\mathbb{P}_{h}(\\hat{V}_{h+1}^{k}-V_{h+1}^{\\pi})(\\cdot,\\cdot)+3 H\\sqrt{d\\lambda}\\|\\phi(\\cdot,\\cdot)\\|_{(\\Lambda_{h}^{\\text{buffer.e}(\\tilde{k})})^{-1}}.\\\\ \\end{array}\n$$"}, {"n": 9, "claim": "The second and fourth equations hold due to the definition of the operator $\\mathbb{P}_{h}$ . The third equation holds due to simple algebraic arrangement. The inequality holds due to Cauchy inequality, $V(\\cdot)\\leq 3H$ and $\\Lambda_{\\mathsf{b u f f e r . e}(\\tilde{k})}\\succeq\\lambda I$\n\nWith the bounds in hand, we have $\\Delta_{k}^{h}(\\cdot,\\cdot)\\leq(3 H\\sqrt{d\\lambda}+\\sqrt{B_{6}H^{3}\\log^{2}K+B_{7}H^{2}\\log C_{7}}+$ $3 H\\sqrt{d\\lambda}+C_{6}H\\log^{2}K)\\|\\phi(\\cdot,\\cdot)\\|_{(\\Lambda_{h}^{\\mathrm{b u f f e r . e}(\\tilde{k})})^{-1}}.$ Then, it is obviously that there exists a constant $B_{8},$ so that $B_{8}H^{\\frac{3}{2}}$ log $K\\geq 3 H\\sqrt{d\\lambda}+\\sqrt{B_{6}H^{3}\\log^{2}K+B_{7}H^{2}\\log C_{7}}+3 H\\sqrt{d\\lambda}$ and it finishes the proof. ■"}]}}, {"local_id": "T2", "statement": "Lemma 42. Under assumption Assumption 3, when Lemma 23 holds, using histogram method to estimate $p.d.ff(\\cdot)$ leads to the following bound that for any x\n\n$$\n|f(x)-\\hat{f}(x)|\\leq D_{1}\\frac{\\sqrt{H}\\log K}{\\textbf{b u f f e r . e}(\\tilde{k})^{\\frac{1}{4}}},\n$$\n\nwhere $D_{1}$ is a constant.", "proof": {"steps": [{"n": 1, "claim": "With Lemma 23 in hand, we divide [−1, 1] into 2M parts denoted by $\\{-M,\\dotsc,0,\\dotsc,M-1\\}$ uniformly, then we have\n\n$$\n\\hat{f}(x)=M[\\hat{F}(\\frac{i+1}{M})-\\hat{F}(\\frac{i}{M})],\n$$\n\nwhere $\\textstyle x\\in{\\bigl(}{\\frac{i}{M}},{\\frac{i+1}{M}}{\\bigr]}$\n\nUnder assumption Assumption 3, it holds that\n\n$$\n|f(x)-M[F(\\frac{i+1}{M})-F(\\frac{i}{M})]|\\leq\\frac{L}{M}.\n$$"}, {"n": 2, "claim": "Therefore, it holds that\n\n$$\n|f(x)-\\hat{f}(x)|\\leq 2MC_{12}\\frac{H\\log^{2}K}{\\sqrt{\\mathrm{buffer.e}(\\tilde{k})}}+\\frac{L}{M}.\n$$"}, {"n": 3, "claim": "By setting $\\begin{array}{ r }{M=\\frac{\\mathtt{b u f f e r . e}(\\tilde{k})^{\\frac{1}{4}}}{\\sqrt{H}\\log K}}\\end{array}$ , we finish our proof."}]}}, {"local_id": "T3", "statement": "Lemma 45. Let V denote a class of functions mapping from S to R with the following parametric form\n\n$$\nV(\\cdot)=\\min\\{\\max_{a}\\omega^{T}\\phi(\\cdot,v)+\\hat{R}(\\cdot,v)+\\beta\\|\\phi(\\cdot,v)\\|_{\\Lambda^{-1}}+A,3 H\\},\n$$\n\nwhere $\\begin{array}{ r }{\\|\\omega\\|\\le L,\\beta\\in [0,B],A=\\frac{C_{14}H^{2}\\log^{4}K}{\\sqrt{\\mathsf{buffer}.\\mathsf{e}(\\tilde{k})}}}\\end{array}$ in episode k and the minimum eigenvalue satisfies $\\lambda_{\\operatorname*{min}}(\\Lambda)\\geq\\lambda$ . Suppose $\\|\\phi(\\cdot,\\cdot)\\|\\leq 1$ and let $\\mathcal{N}_{\\epsilon}$ be the ϵ-covering number of V with respect to the distance dist $\\begin{array}{ r }{(V,V^{\\prime})=\\operatorname*{sup}_{x}|V(x)-V^{\\prime}(x)|}\\end{array}$ |. Then, it holds\n\n$$\n\\log\\mathcal{N}_{\\epsilon}\\leq d\\log(1+\\frac{8L}{\\epsilon})+d^{2}\\log(1+\\frac{32\\sqrt{d}B^{2}}{\\lambda\\epsilon^{2}})+dN\\log(1+\\frac{16NB_{5}\\sqrt{d}}{\\epsilon})+\\log\\mathcal{N}_{\\frac{\\epsilon}{12N^{2}}}(\\mathcal{F}),\n$$\n\nwhere $B_{5}$ is a constant.", "proof": {"steps": [{"n": 1, "claim": "When $F(\\cdot)$ is unknown, it holds that\n\n$$\n\\begin{array}{l}\\sup_{x,v}|\\hat{R}_{1}(x,v)-\\hat{R}_{2}(x,v)| =\\sup_{x,v}|\\hat{R}_{1}(x,v,\\hat{F}_{1})-\\hat{R}_{2}(x,v,\\hat{F}_{2})|\\\\ \\leq\\sup_{x,v}|\\hat{R}_{1}(x,v,\\hat{F}_{1})-\\hat{R}_{2}(x,v,\\hat{F}_{1})|\\\\ +\\sup_{x,v}|\\hat{R}_{2}(x,v,\\hat{F}_{1})-\\hat{R}_{2}(x,v,\\hat{F}_{2})|.\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "Then, we use $C_{\\hat{\\theta}}$ to denote the cardinality of the balls corresponding to $\\hat{\\theta}$ and $C_{\\mathcal{F}}$ to denote the cardinality of the balls corresponding to ${ \\mathcal F } .$\n\nLike the proof of Lemma 37, we simply use $\\frac{\\epsilon}{4 N B_{5}}–\\mathrm{ball}$ to cover $\\hat{\\theta}_{i}$ , and it holds that\n\n$$\n|C_{\\hat{\\theta}}|\\leq(1+\\frac{16NB_{5}\\sqrt{d}}{\\epsilon})^{dN}.\n$$"}, {"n": 3, "claim": "Conditional on $\\omega,$ A and $\\{\\hat{\\theta}_{i}\\}_{i=1}^{N}$ , with Lemma 24, we know that in order to satisfy $\\begin{array}{ r }{\\operatorname*{sup}_{x,v}|\\hat{R}(x,v,\\hat{F})-\\hat{R}(x,v,F)|\\le\\frac{\\dot{\\epsilon}}{4}}\\end{array}$ , what we need is $\\begin{array}{ r }{\\|\\hat{F}-F\\|_{\\infty}\\leq\\frac{\\epsilon}{12N^{2}}}\\end{array}$ . Then, it ends the proof. □"}]}}, {"local_id": "T4", "statement": "Lemma 47. There exist $D_{7}\\sim\\tilde{\\mathcal{O}}(1)$ so that $C_{15}=D_{7}H^{\\frac{3}{2}}$ , and for any fixed policy π, on Good Event E, i.e., all inequalities hold, we have for all $(x,v,h,k)\\in\\mathcal{S}\\times\\Upsilon\\times [H]\\times [K]$ that:\n\n$$\n\\langle\\phi(\\cdot,\\cdot),\\omega_{h}^{k}\\rangle+\\hat{R}_{h}^{k}(\\cdot,\\cdot)-Q_{h}^{\\pi}(\\cdot,\\cdot)=\\mathbb{P}_{h}(\\hat{V}_{h+1}^{k}-V_{h+1}^{\\pi})(\\cdot,\\cdot)+\\Delta_{h}^{k}(\\cdot,\\cdot),\n$$\n\nwhere $\\begin{array}{ r }{\\Delta_{h}^{k}(\\cdot,\\cdot)\\leq(C_{15}+C_{13}H\\log^{2}K)\\|\\phi(\\cdot,\\cdot)\\|_{(\\Lambda_{h}^{\\mathrm{buffer.e}(\\tilde{k})})^{-1}}+C_{14}\\frac{H^{2}\\log^{4}K}{\\sqrt{\\mathrm{buffer.e}(\\tilde{k})}}.}\\end{array}$", "proof": {"steps": [{"n": 1, "claim": "Proof The proof of Lemma 47 is the same as proof of Lemma 39. Let’s show the determination of $D_{7}$ in parallel. With Lemma 46 in hand, it holds that\n\n$$\nD_{7}H^{\\frac{3}{2}}\\geq 3H\\sqrt{d\\lambda}+\\sqrt{D_{5}H^{3}+D_{6}H^{2}\\log C_{15}}+3H\\sqrt{d\\lambda}.\n$$"}, {"n": 2, "claim": "Then, it is easy to see the existence of $D_{7}$ where $D_{7}\\sim\\tilde{\\mathcal{O}}(1)$"}]}}], "target_local_id": null, "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2026-24-1538", "context": "We consider a Markov Reward Process $(\\mathrm{M R P^{1}}$ , Bellman (1957); Puterman (2014)) with a state space ${\\mathcal{S}}\\subseteq\\mathbb{R}^{K}$ for some $K\\in\\mathbb{N}$ . The MRP employs an initial distribution $p_{0}:B(S)[0,1]$ , where B denotes the Borel algebra. The dynamics of the $\\mathrm{MRP}$ are characterized by a transition kernel $p:B({\\mathcal{S}})\\times{\\mathcal{S}}[0,1]^{2}$ The $\\mathrm{MRP}$ also adopts a measurable and bounded reward function $r:{\\mathcal{S}}\\mathbb{R}$ . At time step $t,$ suppose the agent is at state $S_{t}.$ . It transitions to the next state $S_{t+1}\\sim p(\\cdot|S_{t})$ following the transition kernel. At the same time, the environment emits a reward $R_{t+1}\\doteq r(S_{t})$ to the agent for transitioning out of $S_{t}$ . Note that $(p,p_{0})$ defines a Markov chain $\\{S_{t}\\}$ . We further define a p-induced operator $P_{\\pi}:{\\mathcal{F}}{\\mathcal{F}}$ , where ${\\mathcal{F}}\\doteq\\{f:S\\to\\mathbb{R}^{n}\\mid f$ is measurable and integrable}, as $(P_{\\pi}f)(s)\\doteq\\int_{S}f(s^{\\prime})p(\\mathrm{d}s^{\\prime}\\mid s)$\n\nThe state-value function $v_{\\pi}:S\\mathbb{R}$ is defined as $\\begin{array}{ r }{v_{\\pi}(s)\\doteq\\mathbb{E}\\bigl [\\sum_{t=0}^{\\infty}\\gamma^{t}R_{t+1}|S_{0}=s\\bigr]}\\end{array}$ , where $\\gamma\\in [0,1)$ is a discount factor. The state-value function maps each state to the expected cumulative reward the agent gets starting from that state. We consider a linear function approximation of $v_{\\pi}$ with a feature mapping $x:S\\mathbb{R}^{d}$ and a weight vector $w\\in\\mathbb{R}^{d}$ . The function x maps each state $s\\in{\\mathcal{S}}$ to a d-dimensional real vector $x(s)$ . One simply takes $x(s)^{\\top}$ w to compute the approximated state value for s. The goal is thus to adjust w such that $x(s)^{\\top}w\\approx v_{\\pi}(s)$ for all $s\\in{\\mathcal{S}}$ . Linear TD updates the weight w iteratively as\n\n$$\nw_{t+1}=w_{t}+\\alpha_{t}\\Big(R_{t+1}+\\gamma x(S_{t+1})^{\\top}w_{t}-x(S_{t})^{\\top}w_{t}\\Big)x(S_{t}).\\qquad(\\text{Linear TD})\n$$\n\nwhere $\\alpha_{t}$ is the learning rate and we recall that $\\{S_{t}\\}$ is a Markov chain evolving as $S_{t+1}\\sim$ $p(\\cdot|S_{t})$ and $S_{0}$ is sampled from an arbitrary initial distribution. The following assumptions are commonly made in analyzing linear TD (Tsitsiklis and Roy, 1996).\n\nAssumption 2.1 The learning rates $\\left\\{\\alpha_{t}\\right\\}$ is a decreasing sequence of positive real numbers such that $\\begin{array}{ r }{\\sum_{t=0}^{\\infty}\\alpha_{t}=\\infty,\\sum_{t=0}^{\\infty}\\alpha_{t}^{2}<\\infty}\\end{array}$ , and lim $\\begin{array}{ r }{t\\to\\infty\\left(\\frac{1}{\\alpha_{t+1}}-\\frac{1}{\\alpha_{t}}\\right)<\\infty}\\end{array}$\n\nAssumption 2.2 The Markov chain $\\{S_{t}\\}$ admits a well-defined and unique stationary distribution $\\mu:B(S)\\to [0,1]$ , such that $\\textstyle\\mu(Z)=\\int_{S}p(Z\\mid s)\\mu(\\mathrm{d}s)$ for all $Z\\in B({\\cal S})$ and $\\mu(U)>0$ for all non-empty open sets $U \\subseteq S$\n\nThe sequence $\\textstyle\\left\\{{\\frac{1}{(t+1)^{k}}}\\right\\}$ where $k\\in(0.5,1]$ satisfies Assumption 2.1 as a valid example. In light of Assumption 2.2, we define an inner product $\\langle\\cdot,\\cdot\\rangle_{\\mu}$ induced by $\\mu$ as $\\langle f,g\\rangle_{\\mu}\\doteq$ $\\begin{array}{ r }{\\int_{S}\\langle f(s),g(s)\\rangle\\mu(\\mathrm{d}s)}\\end{array}$ , where $f:S\\mathbb{R}^{n}$ and $g:S\\mathbb{R}^{n}$ are measurable functions. The inne product $\\langle\\cdot,\\cdot\\rangle_{\\mu}$ further induces a semi-norm $\\|f\\|_{\\mu}^{2}\\doteq\\langle f,f\\rangle_{\\mu}$\n\nAssumption 2.3 The feature function and reward function are bounded, $i . e .$ sup ∥x(s)∥ < ∞, sup |r(s)| < ∞.\n\nWe use $L_{p}(S,\\mu)$ to denote the set of functions defined on $s$ that are p-integrable with respect to $\\mu .\\mathrm{~A~}$ direct consequence of Assumption 2.3 is that $x\\in L_{p}(S,\\mu)$ for $p\\in [1,\\infty)$ because $\\begin{array}{ r }{\\big(\\int_{S}\\|x(s)\\|^{p}\\mu(\\mathrm{d}s)\\big)^{1/p}\\leq\\big((\\operatorname*{sup}_{s\\in S}\\|x(s)\\|)^{p}\\int_{S}\\mu(\\mathrm{d}s)\\big)^{1/p}=\\operatorname*{sup}_{s\\in S}\\|x(s)\\|<\\infty .}\\end{array}$\n\nAssumption 2.4 Given any $f:{\\mathcal{S}}\\mathbb{R}^{n}$ and $f = 0 \\ a . e$ . with respect to µ, it holds that $P_{\\pi}f=0~a . e$ . with respect to $\\mu$ as well.\n\nIn the rest of the paper, we use $\\because_{\\mathrm{a . e .}}\\ d s^{\\prime}$ to denote “a.e. with respect to $\\mu^{\\dag}$ for simplifying notations. Notably, Assumptions 2.2 - 2.4 trivially hold when S is finite and {S<sub>t</sub>} is irreducible.\n\nIntuitively, when {S<sub>t</sub>} reaches the stationary distribution and the learning rate α is “small”, linear TD behaves like a deterministic algorithm as $w_{t+1}=w_{t}+\\alpha_{t}(A w_{t}+b)$ , where\n\n$$\n\\begin{array}{l}A\\doteq\\mathbb{E}_{S\\sim\\mu,S^{\\prime}\\sim p(\\cdot|S)}\\left[x(S)\\big(\\gamma x(S^{\\prime})^{\\top}-x(S)^{\\top}\\big)\\right]\\\\ =\\int_{\\mathcal{S}}x(s)\\left(\\gamma\\left(P_{\\pi}x\\right)(s)^{\\top}-x(s)^{\\top}\\right)\\mu(\\mathrm{d}s)\\in\\mathbb{R}^{d\\times d}\\tag{1}\\\\ \\end{array}\n$$\n\n$$\nb\\doteq\\mathbb{E}_{S\\sim\\mu}[x(S)r(S)]=\\int_{\\mathcal{S}}x(s)r(s)\\mu(\\mathrm{d}s)\\in\\mathbb{R}^{d}.\\tag{2}\n$$\n\nWe can then further relate the stochastic and discrete iterative updates (Linear TD) with deterministic and continuous trajectories of the following ordinary diferential equation (ODE)\n\n$$\n\\frac{\\mathrm{d}w(t)}{\\mathrm{d}t}=A w(t)+b,\\tag{3}\n$$\n\nwhich is known as the ODE method in stochastic approximation (Benveniste et al., 1990; Kushner and Yin, 2003; Borkar, 2009; Borkar et al., 2025; Liu et al., 2025a).\n\nLemma 2 Let Assumptions 2.2 & 2.3 hold. Then A is negative semi-definite.\n\nAs previously discussed, linear TD is approximately a stochastic discretization of ODE (3). The fixed points of linear TD, commonly referred to as TD fixed points (Tsitsiklis and Roy, 1996; Sutton and Barto, 2018), are linked to the equilibria of this ODE. We hence consider the linear system\n\n$$\nA w+b=0,\\tag{4}\n$$\n\nwhere A is defined in (1) and b is defined in (2). With linearly independent features, (4) adopts a unique solution — the unique TD fixed point. Without assuming linear independence, matrix A is merely negative semi-definite (Lemma 2), so (4) can potentially adopt infinitely many solutions. In light of this, we refer to all solutions to the linear system as TD fixed points. Namely, we define $\\mathcal{W}_{*}\\doteq\\{w_{*}|A w_{*}+b=0\\}$ and refer to W as the set of TD fixed points. A few questions arise naturally from this definition.\n\nWe shall give afirmative answers to all the questions above in the rest of the paper. We use\n\n$$\nv_{w}(s)\\doteq x(s)^{\\top}w \\tag{5}\n$$\n\nto denote the value estimate for a state $s$ given a weight $w$ .\n\nLemma 4 Let Assumptions 2.2, 2.3, & 2.4 hold. Then $\\mathcal{W}_{*}$ is non-empty.\n\nLemma 5 Let Assumptions 2.2 & 2.3 hold. Given any $w\\in\\mathcal{W}_{*}$ and any $\\boldsymbol{w^{\\prime}}\\in\\mathbb{R}^{d}$ , it holds that $v_{w}=v_{w^{\\prime}}\\ a . e .\\iff w^{\\prime}\\in\\mathcal{W}_{*}$\n\nWe use $w(t;w_{0})$ to denote the solution to (3) with the initial condition $w(0;w_{0})=w_{0}$ .\n\nTheorem 13 Let Assumptions 2.2, 2.3, & 2.4 hold. Let $w(t)$ be a bounded solution to ODE (3) on $(-\\infty,+\\infty),\\ i . e .,\\ \\operatorname*{sup}_{t\\in(-\\infty,+\\infty)}\\|w(t)\\|<\\infty$ . It then holds that $w(t)$ is constant and is in $\\psi_{*},\\ i . e .$ , there exists some $w_{*}\\in\\mathcal{W}_{*}$ such that $w(t)=w_{*}$ holds for any $t \\in$ $(-\\infty,+\\infty)$\n\nCorollary 14 Let Assumptions 2.2, 2.3, & 2.4 hold. If W is a bounded invariant set of ODE (3), then $\\boldsymbol{\\mathcal{W}}\\subseteq\\boldsymbol{\\mathcal{W}}_{*}$ .\n\nHaving fully characterized the mean ODE (3), we are now ready to connect the linear TD update (4) with the mean ODE. To this end, we consider the joint process $Y_{t}=(S_{t},S_{t+1})$ which is also a Markov chain. Recall that $S_{t}\\in\\mathbb{R}^{K}$ . We then regard $Y_{t}$ as a vector in $\\mathbb{R}^{2 K}$ As a result, the Markov chain {Y<sub>t</sub>} evolves in a subset of $\\mathbb{R}^{2 K}$ , denoted as Y. Since {Y<sub>t</sub>} is essentially the same chain as {S<sub>t</sub>} but viewed diferently, Assumption 2.2 implies that $\\{Y_{t}\\}$ adopts a unique stationary distribution, referred as η. We additionally refer to the transition kernel of $\\{Y_{t}\\}$ as $P_{\\mathcal{Y}}$ . Given a measurable and integrable function $f:\\mathcal{V}\\to\\mathbb{R}^{d}$ we define $P_{\\mathcal{V}}f$ as $(P y f)(y)\\doteq\\int_{\\mathcal{V}}f(y^{\\prime})P y(\\mathrm{d}y^{\\prime}\\mid y)$ . With this joint process, the linear TD update (Linear TD) can be written as $w_{t+1}=w_{t}+\\alpha_{t}H(w_{t},Y_{t+1})$ , where $H(w,y)=$ $\\big(r(s)+\\gamma w^{\\top}x(s^{\\prime})-w^{\\top}x(s)\\big)x(s)$ . Here, we have used shorthand $y\\doteq(s,s^{\\prime})$ . The expected update is then $h(w)=\\mathbb{E}_{y\\sim\\eta}[H(w,y)]=A w+b$ . We now make a few standard assumptions on the behavior of $\\{Y_{t}\\}$\n\nAssumption 5.1 (Poisson Equation) There exists a function $\\nu:\\mathcal{y}\\mathbb{R}^{d}$ , such that $\\nu_{w}(y)-(P y\\nu_{w})(y)=H(w,y)-h(w)$ holds for all $w,y$ . Furthermore, there exists a constant $C_{5,1}$ such that ∀w, $y,\\|\\nu_{w}(y)\\|\\leq C_{\\mathcal{S}. I}(1+\\|w\\|)(1+\\|y\\|);\\quad\\|(P_{\\mathcal{V}}\\nu_{w})(y)-(P_{\\mathcal{V}}\\nu_{w^{\\prime}})(y)\\|\\leq$ $C_{5.4}\\|w-w^{\\prime}\\|(1+\\|y\\|)$\n\nAssumption 5.2 (Law of Iterated Logarithm) For any w, there exists a sample-pathdependent finite constant $(i.e.$ , a random variable that is finite $a . s .{\\big)}\\ \\zeta_{5 .\\mathcal{Q}},$ such that $\\begin{array}{ r }{\\|\\sum_{t=1}^{n}\\left(H(w,Y_{t})-h(w)\\right)\\|\\leq\\zeta_{5.2}\\sqrt{n\\log\\log n}\\quad a.s.}\\end{array}$\n\nWe additionally make the following assumption regarding the state space.\n\nAssumption 5.3 $\\mathcal{S}$ is compact.\n\nHere, $m(t,T)$ is defined as $m(t,T)\\doteq\\operatorname*{max}\\left\\{n|\\sum_{i=t}^{n}\\alpha_{i}\\le T\\right\\}$ . Intuitively, the magnitude of the updates of linear TD in (Linear TD) is controlled by the learning rate $\\alpha_{i}$ . Then, $\\{t,t+1,\\ldots,m(t,T)\\}$ denotes a period during which the total magnitude of updates is no more than $T$ . The proof is in E.3. Corollary 19 essentially confirms that $\\{w_{t}\\}$ will visit (arbitrarily small) neighbors of $w_{*}$ infinitely many times. The number of update steps during each visit $(\\mathrm{i . e .,}m(t_{k},T)-t_{k})$ diverges to $\\infty$\n\nAssumption A.1 $\\left\\{\\alpha_{t}\\right\\}$ is a decreasing sequence of positive real numbers such that\n\n$$\n\\sum_{t=0}^{\\infty}\\alpha_{t}=\\infty,\\quad\\sum_{t=0}^{\\infty}\\alpha_{t}^{2}<\\infty,\\quad\\lim_{t\\to\\infty}\\left(\\frac{1}{\\alpha_{t+1}}-\\frac{1}{\\alpha_{t}}\\right)<\\infty .\n$$\n\nAssumption A.2 The Markov chain {Y } admits a well-defined and unique stationary distribution $\\eta:B(\\mathbb{R}^{k})\\to [0,1]$ , such that $\\begin{array}{ r }{\\eta(Z)=\\int_{\\mathbb{R}^{k}}P_{\\mathcal{Y}}(Z\\mid\\boldsymbol{y})\\eta(\\mathrm{d}\\boldsymbol{y})}\\end{array}$ for all $Z\\in B(\\mathbb{R}^{k})$ and $\\eta(U)>0$ for all non-empty open sets $U\\subseteq\\mathbb{R}^{k}$\n\nAssumption A.3 There exists a constant $K_{A . 3}\\in\\mathbb{R}$ , such that for all $w\\in\\mathbb{R}^{d}$ and $\\boldsymbol{y}\\in\\mathbb{R}^{k}$ we have $\\|H(w,y)\\|\\le K_{A . 3}(1+\\|w\\|)(1+\\|y\\|)$\n\nFor any function $f(w,y)$ on $\\mathbb{R}^{d}\\times\\mathbb{R}^{k}$ , we shall denote the partial mapping $y\\to f(w,y)$ by $f_{w}$ . We shall also denote the function $\\begin{array}{ r }{y\\int_{\\mathbb{R}^{k}}f(w,y^{\\prime})P_{\\mathcal{V}}(\\mathrm{d}y^{\\prime}\\mid y)}\\end{array}$ as $P_{\\mathcal{Y}}f_{w}$\n\nAssumption A.4 There exists a function $g:\\mathbb{R}^{d}\\mathbb{R}^{d}$ , and for each $w\\in\\mathbb{R}^{d}$ a function $\\nu_{w}:\\mathbb{R}^{k}\\to\\mathbb{R}^{d}$ , such that\n\n(i) g is locally Lipschitz on $\\mathbb{R}^{d}.$ ;  \n(ii) $\\nu_{w}-P y\\nu_{w}=H_{w}-g(w)\\ f o r\\ a l l\\ w\\in\\mathbb{R}^{d};$  \n(iii) there exists a constant $K_{A . 4}\\in\\mathbb{R}$ , such that for all $w\\in\\mathbb{R}^{d}$ and $y\\in\\mathbb{R}^{k},~\\|\\nu_{w}(y)\\|~\\leq$ $K_{A . 4}(1+\\|w\\|)(1+\\|y\\|)$ ;  \n(iv) there exists a constant $K_{A,\\mathcal{A}}^{\\prime}\\in\\mathbb{R}$ , such that for all $w,w^{\\prime}\\in\\mathbb{R}^{d}$ and $\\boldsymbol{y}\\in\\mathbb{R}^{k}$ ∥(P<sub>Y</sub>ν<sub>w</sub>)(y) − (P<sub>Y</sub>ν<sub>w</sub>′)(y)∥ ≤ K<sup>′</sup><sub>A.4</sub>∥w − w<sup>′</sup>∥(1 + ∥y∥).\n\nAssumption A.5 For all $a\\in\\mathbb{R}^{d},q>0$ , and $n \\geq 0$ , there exists $K_{A . 5}(q)<\\infty$ , such that $\\begin{array}{ r }{\\mathbb{E}[1+\\|Y_{n+1}\\|^{q}\\mid w_{0}=a,Y_{0}=y]\\le K_{A . 5}(q)(1+\\|y\\|^{q})}\\end{array}$\n\nDefine $h(w)\\doteq\\mathbb{E}_{Y\\sim\\eta}[H(w,Y)]$ . Then, we have the following theorem in Benveniste et al. (1990).\n\nTheorem A.3 (Theorem $\\boldsymbol{\\mathit{17}}(\\boldsymbol{a})$ of Benveniste et al. (1990)) Let Assumptions A.1 and $A . 3 ~ \\cdot$ A.5 hold. Assume there exists a function $U:\\mathbb{R}^{d}[0,\\infty)$ of class $C^{2}$ with bounded second derivatives<sup>15</sup> and a constant $c > 0$ , such that for all $w\\in\\mathbb{R}^{d}$\n\ni. $\\left.\\nabla_{w}U(w),h(w)\\right.\\leq 0$ , and\n\nii. $U(w)\\geq c\\Vert w\\Vert^{2},$\n\nthen for all $w_{0}\\in\\mathbb{R}^{d}$ , the iterates $\\{w_{t}\\}$ generated by (23) is stable, $i . e .,\\operatorname*{sup}_{t}\\|w_{t}\\|<\\infty$ a.s.\n\nAssumption A.6 There exists a function $L:\\mathcal{V}\\to\\mathbb{R}$ such that for any $w,w^{\\prime},y,$ $\\|H(w,y)-H(w^{\\prime},y)\\|\\leq L(y)\\|w-w^{\\prime}\\|$ . Moreover, the following expectations are well-defined and finite for any $w\\in\\mathbb{R}^{d}\\colon h(w)\\doteq\\mathbb{E}_{Y\\sim\\eta}[H(w,Y)],L\\doteq\\mathbb{E}_{Y\\sim\\eta}[L(Y)]$\n\nAssumption A.7 For all $w_{0}~\\in~\\mathbb{R}^{d}$ , the iterates $\\{w_{t}\\}$ generated by (23) is stable, i.e., $\\operatorname*{sup}_{t}\\|w_{t}\\|<\\infty\\ a . s$\n\nTheorem A.4 (Corollary 8 of Liu et al. $(2025a))$ Let Assumptions A.1, A.2, A.6 and A.7 hold. Then the iterates $\\{w_{t}\\}$ generated by (23) converge almost surely to a (possibly sample path dependent) bounded invariant set of the ODE $\\begin{array}{ r }{\\frac{\\mathrm{d}w(t)}{\\mathrm{d}t}=h(w(t))}\\end{array}$\n\nLemma E.2 Let Assumptions 2.2 and 2.3 hold. Then there exists some $K_{E . 2}>0 $ , such that for any w, $w^{\\prime}\\in\\mathbb{R}^{d}$ , it holds that $\\|h(w)-h(w^{\\prime})\\|\\leq K_{E . 2}\\|w-w^{\\prime}\\|$\n\nLemma E.3 Let Assumption 2.3 hold. Then there exists a constant $K_{E . 3}>0$ , such that for all $y\\in\\mathcal{V}$ , it holds that $\\|H(w,y)\\|\\le K_{E . 3}(1+\\|w\\|)$", "theorems": [{"local_id": "T1", "statement": "Theorem 17 Let Assumptions 2.1, 2.2, 2.3, 2.4, 5.1, & 5.3 hold. Then, the iterates $\\{w_{t}\\}$ generated by (Linear TD) is stable, $\\therefore e .,\\operatorname*{sup}_{t}\\|w_{t}\\|<\\infty\\quad a . s .$", "proof": {"steps": [{"n": 1, "claim": "Proof We shall show that Assumptions A.1 and A.3 – A.5 hold. Invoking Theorem A.3 will then complete the proof. First, Assumption A.1 is satisfied by Assumption 2.1. Assumption A.3 holds by Lemma E.3. We then proceed to show Assumption A.4 can be satisfied. Choosing $g = h$ , where we recall that $\\begin{array}{ r }{h(w)\\doteq\\int_{\\mathcal{V}}H(w,y)\\eta(\\mathrm{d}y)}\\end{array}$ , Assumption A.4 (i) holds by Lemma E.2. Assumption 5.1 automatically satisfies (ii), (iii), and (iv) of Assumption A.4. Lastly, we prove Assumption A.5 holds. Since S is compact by Assumption 5.3, then Y is also compact. Thus, $\\operatorname{sup}_{y\\in{\\mathcal{Y}}}\\left\\|y\\right\\|$ is finite. We denote this quantity as $C_{y}$ . Then, for all $a\\in\\mathbb{R}^{d},q>0$ and $n \\geq 0$ , it holds that $\\mathbb{E}[1+\\lVert Y_{n+1}\\rVert^{q}\\ |\\ w_{0}=a,Y_{0}=y]\\le 1+C_{y}^{q}$ . Thus, by setting $K_{\\mathrm{A . 5}}(q)\\doteq(1+C_{y}^{q})$ , Assumption A.5 is satisfied."}, {"n": 2, "claim": "Having verified the assumptions, we now show the existence of the function U required by Theorem A.3. Fixing an arbitrary $w_{*}\\in\\mathcal{W}_{*}$ , we define U as $U(w)\\doteq\\lVert w-w_{*}\\rVert^{2}+\\lVert w_{*}\\rVert^{2}$ . The second derivative of U is $2 I_{d}$ . Therefore, U is of class $C^{2}$ with bounded second derivatives. We now show that it satisfies Condition (i) of Theorem A.3. Taking the gradient, we have\n\n$$\n\\langle\\nabla_{w}U(w),h(w)\\rangle=\\langle 2(w-w_{*}),A w+b\\rangle=2\\langle(w-w_{*}),A w+b-(A w_{*}+b)\\rangle\n$$\n\n$$\n=2\\left\\langle w-w_{*},A(w-w_{*})\\right\\rangle\\leq 0.\\tag{Lemma2}\n$$"}, {"n": 3, "claim": "Condition (i) holds for our selected U. Next, we prove that Condition (ii) also holds by showing that $\\begin{array}{ r }{U(w)\\geq\\frac{1}{2}\\Vert w\\Vert^{2}}\\end{array}$ for all $w\\in\\mathbb{R}^{d}$ . We have\n\n$$\n\\begin{array}{l}U(w)-\\frac{1}{2}\\|w\\|^{2}=\\|w-w_{*}\\|^{2}+\\|w_{*}\\|^{2}-\\frac{1}{2}\\|w\\|^{2}=\\frac{1}{2}\\|w\\|^{2}-2\\left\\langle w,w_{*}\\right\\rangle+2\\|w_{*}\\|^{2}\\\\ =\\frac{1}{2}\\Big(\\|w\\|^{2}-4\\langle w,w_{*}\\rangle+4\\|w_{*}\\|^{2}\\Big)=\\frac{1}{2}\\|w-2 w_{*}\\|^{2}\\geq 0.\\\\ \\end{array}\n$$"}, {"n": 4, "claim": "Hence, Condition (ii) holds by setting $\\begin{array}{ r }{c=\\frac{1}{2}}\\end{array}$ . The proof is completed."}]}}, {"local_id": "T2", "statement": "Lemma D.1 Under Assumptions 2.2 and 2.3, it holds that $\\begin{array}{ r }{\\operatorname*{sup}_{t\\in [0,\\infty)}\\|w(t;w_{0})\\|<\\infty}\\end{array}$ for all $\\boldsymbol w_{0}\\in\\mathbb{R}^{d}$", "proof": {"steps": [{"n": 1, "claim": "Proof Fix an arbitrary $w_{*}\\in\\mathcal{W}_{*}$ . We first show that $\\frac{\\mathrm{d}\\|w(t;w_{0})-w_{*}\\|^{2}}{\\mathrm{d}t}\\leq 0$ for any $\\boldsymbol w_{0}\\in\\mathbb{R}^{d}$\n\n$$\n\\begin{array}{l}\\frac{\\mathrm{d}\\|w(t;w_{0})-w_{*}\\|^{2}}{\\mathrm{d}t}=2\\left(w\\left(t;w_{0}\\right)-w_{*}\\right)^{\\top}\\left(A w\\left(t;w_{0}\\right)+b\\right)\\\\ =2\\left(w\\left(t;w_{0}\\right)-w_{*}\\right)^{\\top}\\left(A w\\left(t;w_{0}\\right)+b-\\left(A w_{*}+b\\right)\\right)\\quad\\left(A w_{*}+b=0\\right)\\\\ =2\\left(w\\left(t;w_{0}\\right)-w_{*}\\right)^{\\top}A\\left(w\\left(t;w_{0}\\right)-w_{*}\\right)\\leq 0.\\tag{Lemma2}\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "Thus, $\\|w(t;w_{0})-w_{*}\\|$ is monotonically decreasing given any $\\boldsymbol{w}_{0}\\in\\mathbb{R}^{d}$ . Hence, $\\|w(t;w_{0})-w_{*}\\|\\leq\\|w_{0}-w_{*}\\|$ for $t \\geq 0 .$ , and by triangle inequality we get $\\begin{array}{ r }{\\operatorname*{sup}_{t\\in [0,\\infty)}\\|w(t;w_{0})\\|\\leq\\|w_{*}\\|+\\|w_{0}-w_{*}\\|}\\end{array}$ , which completes the proof."}]}}, {"local_id": "T3", "statement": "Lemma E.1 Let Assumption 2.3 hold. Then there exists some $K_{E,1}>0$ , such that for any w, $w^{\\prime}\\in\\mathbb{R}^{d}$ and $y\\in\\mathcal{V}$ , it holds that $\\|H(w,y)-H(w^{\\prime},y)\\|\\leq K_{E . l}\\|w-w^{\\prime}\\|$", "proof": {"steps": [{"n": 1, "claim": "Proof Given $w,w^{\\prime}\\in\\mathbb{R}^{d}$ , we have\n\n$$\n\\begin{array}{l}\\|H(w,y)-H(w^{\\prime},y)\\|=\\left\\|\\left(\\gamma(w-w^{\\prime})^{\\top}x(s^{\\prime})-(w-w^{\\prime})^{\\top}x(s)\\right)x(s)\\right\\|\\quad\\left(Y\\doteq\\left[\\begin{array}{c c}s&s^{\\prime}\\end{array}\\right]^{\\top}\\right)\\\\ = |\\gamma\\langle w-w^{\\prime},x(s^{\\prime})\\rangle-\\langle w-w^{\\prime},x(s)\\rangle|\\|x(s)\\|\\\\ \\leq C_{x}(|\\gamma\\langle w-w^{\\prime},x(s^{\\prime})\\rangle| + |\\langle w-w^{\\prime},x(s)\\rangle|)\\\\ (C_{x}\\doteq\\sup_{s\\in\\mathcal{S}}\\|x(s)\\|<\\infty)\\\\ \\leq C_{x}(\\gamma C_{x}\\|w-w^{\\prime}\\|+C_{x}\\|w-w^{\\prime}\\|)\\\\ =(1+\\gamma)C_{x}^{2}\\|w-w^{\\prime}\\|.\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "Setting $K_{\\mathrm{E . 1}}=(1+\\gamma)C_{x}^{2}$ , the lemma is proved."}]}}], "target_local_id": null, "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2026-25-0376", "context": "Let $X_{1},\\ldots,X_{n}$ be n independent and identically distributed random vectors from an unknown distribution with probability density function $f_{0}$ on $\\Omega\\subseteq\\mathbb{R}^{d}$ . The task is to estimate $f_{0}$ nonparametrically based on the observed data $\\{X_{i}\\}_{i=1}^{n}$\n\nOur proposed data-augmented nonparametric noise contrastive density estimation is to first convert the unsupervised density estimation problem into a supervised task of classification, building on the ideas from Hastie et al. (2009) and Gutmann and Hyv¨arinen (2012). The key diference is that we estimate a mixture density $f_{\\mathcal{M}}$ by classification, instead of estimating the target density $f_{0}$ directly. To achieve this, we need to introduce external data points sampled from some known density $f_{\\mathcal{R}}$ . Specifically, our proposed procedure consists of three main steps:\n\nLet $f_{\\mathcal{R}}$ be a known density function on $\\Omega_{;}$ termed as the reference density. We generate a random sample $\\{\\tilde{Z}_{1},\\dots,\\tilde{Z}_{\\rho n}\\}$ from the reference density $f_{\\mathcal{R}}$ , independent of the observed data $\\{X_{i}\\}_{i=1}^{n}$ , where $\\rho$ is a positive constant such that $\\rho n\\in\\mathbb{Z}_{+}$ and $\\mathbb{Z}_{+}$ denotes the set of positive integers. Given the observed data $\\{X_{i}\\}_{i=1}^{n}\\sim f_{0}$ and the reference data $\\{\\tilde{Z}_{i}\\}_{i=1}^{\\rho n}\\sim$ $f_{\\mathcal{R}}$ , we term $\\{Y_{i}\\}_{i=1}^{n+\\rho n}: -\\{X_{i}\\}_{i=1}^{n}\\cup\\{\\tilde{Z}_{i}\\}_{i=1}^{\\rho n}$ as the augmented data.\n\nWe assign a binary label ${\\tilde{C}}_{i}$ to each $Y_{i}$ according to whether it is the observed data. That is, we assign $\\tilde{C}_{i}=1\\mathrm{if}Y_{i}\\in\\{X_{j}\\}_{j=1}^{n}$ and $\\tilde{C}_{i}=0\\mathrm{if}Y_{i}\\in\\{\\tilde{Z}_{j}\\}_{j=1}^{\\rho n}$ . Then, the conditional densities of $Y$ are\n\n$$\nf(y\\mid\\tilde{C}=1)=f_{0}(y),\\quad f(y\\mid\\tilde{C}=0)=f_{\\mathcal{R}}(y).\n$$\n\nNote that the probability mass of label $\\tilde{C}$ is $\\operatorname*{Pr}(\\tilde{C}=1)=1/(1+\\rho)$ and $\\mathrm{Pr}(\\tilde{C}=0)=$ $\\rho/(1+\\rho)$ . Then, the marginal density of the augmented data $Y$ is\n\n$$\n\\begin{array}{l}f(y)=f(y\\mid\\tilde{C}=1)\\mathrm{Pr}(\\tilde{C}=1)+f(y\\mid\\tilde{C}=0)\\mathrm{Pr}(\\tilde{C}=0)\\\\ =\\frac{1}{1+\\rho}f_{0}(y)+\\frac{\\rho}{1+\\rho}f_{\\mathcal{R}}(y): -f_{\\mathcal{M}}(y).\\tag{1}\\\\ \\end{array}\n$$\n\nHere $f_{\\mathcal{M}}$ is referred to as the mixture density. It can be easily derived from (1) that $f_{0}=(1+\\rho)f_{\\mathcal{M}}-\\rho f_{\\mathcal{R}}$ . We can obtain an estimator for $f_{0}$ once the estimator for the mixture density $f_{\\mathcal{M}}$ is available.\n\nStep 2. Estimating the mixture density $f_{\\mathcal{M}}$ with a noise contrastive estimation.\n\nWe generate an additional set of reference data $\\{Z_{i}\\}_{i=1}^{\\nu(n+\\rho n)}$ from $f_{\\mathcal{R}}$ independent of $\\{\\tilde{Z}_{i}\\}_{i=1}^{\\rho n}$ , termed as the contrastive noise. Here ν is the ratio of sample sizes between the contrastive noise and the augmented data. We let $\\nu\\in\\mathbb{Z}_{+}$ for convenience. Henceforth, we write the union of the augmented data and the contrastive noise as $S=\\{Y_{1},\\ldots,Y_{n+\\rho n}\\}$ $Z_{1},\\dots,Z_{\\nu(n+\\rho n)}\\}$ , the sample used for density estimation.\n\nSimilar to Step 1, we rewrite S as $\\{U_{t}\\}_{t=1}^{(1+\\nu)(n+\\rho n)}$ and assign a binary label $C_{t}$ to each $U_{t}$ according to whether it is the augmented data. That is, we assign $C_{t}=1\\mathrm{~i f~}U_{t}\\in\\{Y_{i}\\}_{i=1}^{n+\\rho n}$ 2 and $C_{t}=0{\\mathrm{~i f~}}U_{t}\\in\\{Z_{i}\\}_{i=1}^{\\nu(n+\\rho n)}$ . Then, the conditional densities of U are:\n\n$$\nf(u\\mid C=1)=f_{\\mathcal{M}}(u),\\quad f(u\\mid C=0)=f_{\\mathcal{R}}(u).\n$$\n\nThe prior marginal probability mass of label C in the combined data S is $\\operatorname*{Pr}(C=1)=$ $1/(1{+}\\nu)$ and $\\operatorname*{Pr}(C=0)=\\nu/(1{+}\\nu)$ . Then, by the Bayes’ formula, the posterior probabilities for the labels are\n\n$$\n\\operatorname*{Pr}(C=1\\mid u)=\\frac{f_{\\mathcal{M}}(u)}{f_{\\mathcal{M}}(u)+\\nu f_{\\mathcal{R}}(u)},\\quad\\operatorname*{Pr}(C=0\\mid u)=\\frac{\\nu f_{\\mathcal{R}}(u)}{f_{\\mathcal{M}}(u)+\\nu f_{\\mathcal{R}}(u)}.\n$$\n\nThen, the log-likelihood of $f$ for the sample $S$ is\n\n$$\n\\begin{array}{l}\\ell(f;S)=\\sum_{t=1}^{(1+\\nu)(n+\\rho n)}\\left\\{C_{t}\\log\\Pr(C_{t}=1\\mid U_{t};f)+(1-C_{t})\\log\\Pr(C_{t}=0\\mid U_{t};f)\\right\\}\\\\ =\\sum_{i=1}^{n+\\rho n}\\log\\Pr(C_{i}=1\\mid Y_{i};f)+\\sum_{i=1}^{\\nu(n+\\rho n)}\\log\\Pr(C_{i}=0\\mid Z_{i};f),\\\\ \\end{array}\n$$\n\nwhere $\\operatorname*{Pr}(C=1\\mid u;f): -f(u)/\\{f(u)+\\nu f_{\\mathcal{R}}(u)\\}$ and $\\operatorname*{Pr}(C=0\\mid u;f): -\\nu f_{\\mathcal{R}}(u)/\\{f(u)+$ $\\nu f_{\\mathcal{R}}(u)\\}$ . Intuitively, an estimator for $f_{\\mathcal{M}}$ can be obtained by maximizing the log-likelihood function $\\ell(f;S)$ over $f\\in\\mathcal{F}_{n}$ , a prespecified function class. Equivalently, we minimize\n\n$$\n\\begin{array}{l}\\hat{R}_{S}(f): - -\\frac{1}{(1+\\rho)n}\\left\\{\\sum_{i=1}^{(1+\\rho)n}\\log\\Pr\\left(C_{i}=1\\mid Y_{i};f\\right)+\\sum_{i=1}^{\\nu(1+\\rho)n}\\log\\Pr\\left(C_{i}=0\\mid Z_{i};f\\right)\\right\\}\\\\ = -\\frac{1}{(1+\\rho)n}\\Bigl\\{\\sum_{i=1}^{(1+\\rho)n}\\log\\frac{f(Y_{i})}{f(Y_{i})+\\nu f_{\\mathcal{R}}(Y_{i})}+\\sum_{i=1}^{\\nu(1+\\rho)n}\\log\\frac{\\nu f_{\\mathcal{R}}(Z_{i})}{f(Z_{i})+\\nu f_{\\mathcal{R}}(Z_{i})}\\Bigr\\},\\\\ \\end{array}\n$$\n\nwhich equals to $-\\ell(f;S)/\\{(1+\\rho)n\\}$ . We refer to $\\hat{R}_{S}(f)$ as the empirical noise contrastive risk given sample S, which is also written as $\\hat{R}(f)$ to suppress its dependence on sample S when there is no confusion.\n\nAs a result, we define the estimator for the mixture density as the empirical risk minimizer:\n\n$$\n\\hat{f}_{\\mathcal{M}}\\in\\underset{f\\in\\mathcal{F}_{n}}{\\arg\\min}\\hat{R}(f).\\tag{3}\n$$\n\nStep 3. Deriving the estimator for the target density $f_{0}$ .\n\nIn view of (1) and the estimator $\\hat{f}_{\\mathcal{M}}$ defined in (3), our proposed estimator for $f_{0}$ is given by\n\n$$\n\\tilde{f}_{n}=(1+\\rho)\\hat{f}_{\\mathcal{M}}-\\rho f_{\\mathcal{R}},\\tag{4}\n$$\n\nwhich is referred to as data-augmented nonparametric noise contrastive estimation.\n\nRemark 1 For any $f\\in\\mathcal{F}_{n}$ , the population-level noise contrastive risk corresponding to (2) is\n\n$$\n\\begin{array}{l}R(f)= -\\bigl\\{\\mathbb{E}_{Y\\sim f_{\\mathcal{M}}}\\log P r(C=1\\mid Y;f)+\\nu\\mathbb{E}_{Z\\sim f_{\\mathcal{R}}}\\log P r(C=0\\mid Z;f)\\bigr\\}\\\\ = -\\Bigl\\{\\mathbb{E}_{Y\\sim f_{\\mathcal{M}}}\\log\\frac{f(Y)}{f(Y)+\\nu f_{\\mathcal{R}}(Y)}+\\nu\\mathbb{E}_{Z\\sim f_{\\mathcal{R}}}\\log\\frac{\\nu f_{\\mathcal{R}}(Z)}{f(Z)+\\nu f_{\\mathcal{R}}(Z)}\\Bigr\\}.\\\\ \\end{array}\n$$\n\nWe now introduce the class of neural networks used for density function approximation. In recent years, advancements on the approximation theory of the feedforward neural networks with Rectified Linear Unit (ReLU) activation function are reported by Yarotsky (2017), Chen et al. (2019), Shen et al. (2020), Farrell et al. (2021), and Lu et al. (2021). In this paper, we consider multi-layer perceptron, an important and widely-used subclass of feedforward neural networks. The following expression characterizes the architecture of multi-layer perceptron:\n\n$$\nf_{\\theta}(x)=\\mathcal{L}_{\\mathcal{D}}\\circ\\pmb{\\sigma}\\circ\\mathcal{L}_{\\mathcal{D}-1}\\circ\\pmb{\\sigma}\\circ\\dots\\circ\\mathcal{L}_{1}\\circ\\pmb{\\sigma}\\circ\\mathcal{L}_{0}(x),x\\in\\mathbb{R}^{p_{0}},\n$$\n\nwhere $\\sigma$ is the component-wise version of ReLU activation function $\\sigma(x)=\\operatorname*{max}\\{0,x\\}$ , and $\\mathcal{L}_{i}(x)=W_{i}x+b_{i},i=0,\\ldots,D$ are linear transformations with weight matrices $W_{i}\\in\\mathbb{R}^{p_{i+1}\\times p_{i}}$ and bias vectors $b_{i}~\\in~\\mathbb{R}^{p_{i+1}}$ . Here, D is the depth of the neural network, that is, the number of hidden layers. The (D + 2)-tuple $(p_{0},\\ldots,p_{\\mathcal{D}+1})$ denotes the number of neurons in each layer, and specifically, $p_{0}=d,p_{D+1}=1$ . The width of the network is defined as $\\mathcal{W}=\\operatorname*{max}\\{p_{1},...,p_{D}\\}$ . The total number of parameters $\\begin{array}{ r }{S=\\sum_{i=0}^{\\mathcal{D}}p_{i+1}(p_{i}+1)}\\end{array}$ is referred to as the size of the network. We denote the collection of networks $f_{\\theta}$ described above as $\\mathcal{N N}(\\mathcal{D},\\mathcal{W},\\mathcal{S})$ . Note that the architecture parameters of the network class satisfies max $\\{\\mathcal{W},\\mathcal{D}\\}\\le\\mathcal{S}\\le\\mathcal{W}(d+1)+(\\mathcal{W}^{2}+\\mathcal{W})(\\mathcal{D}-1)+\\mathcal{W}+1=O(\\mathcal{W}^{2}\\mathcal{D})$\n\nWe take ${\\mathcal{F}}_{n}$ to be the following class of neural networks:\n\n$$\n\\mathcal{F}(\\mathcal{D},\\mathcal{W},\\mathcal{S},\\gamma,\\Gamma): -\\{f_{\\theta}\\in\\mathcal{N N}(\\mathcal{D},\\mathcal{W},\\mathcal{S}):\\gamma\\leq f_{\\theta}\\leq\\Gamma\\},\n$$\n\nfor some positive constants $\\gamma,\\Gamma$ . Note that the architecture parameters $\\mathcal{D},\\mathcal{W}_{:}$ , and $\\boldsymbol{\\mathcal{S}}$ can depend on n, and we suppress the subscript n for simplicity. Furthermore, the output range constraint $\\gamma\\leq f_{\\theta}\\leq\\Gamma$ can be satisfied by adding an additional layer $\\phi(x)=\\sigma(x-\\gamma)-\\sigma(x-$ $\\Gamma)+\\gamma$ at the end of the network. Then, the network depth will increase by 1 accordingly. Such adjustment will only afect the constant prefactors in our non-asymptotic error bounds and will not afect the convergence rate, and thus the details are omitted for simplicity.\n\nIn this section, we provide a theoretical analysis of the proposed data-augmented nonparametric noise contrastive estimator. We will study the convergence properties of the mixture density estimator $\\hat{f}_{\\mathcal{M}}$ , which can imply the convergence properties of $\\tilde{f_{n}}$ , the estimator for the target density $f_{0}$\n\nTo evaluate the quality of an estimator $\\hat{f}$ for a target function $f_{*},$ we use the expected $L_{2}.$ -risk $\\mathbb{E}\\Vert\\hat{f}-f_{*}\\Vert_{L_{2}(f_{*})}^{2}$ , where the expectation is taken with respect to training sample and\n\n$$\n\\|f-f_{*}\\|_{L_{2}(f_{*})}^{2}: -\\mathbb{E}_{X\\sim f_{*}}|f(X)-f_{*}(X)|^{2}=\\int_{\\Omega}|f(x)-f_{*}(x)|^{2}f_{*}(x)d x.\n$$\n\nThe main procedure for the error analysis is to decompose the expected $L_{\\mathrm{{2^{-}}}}\\mathrm{risk}$ of $\\hat{f}_{\\mathcal{M}}$ into stochastic error and approximation error, and bound them separately. Thereafter, an upper bound of $\\mathbb{E}_{S}\\Vert\\tilde{f}_{n}-f_{0}\\Vert_{L_{2}(f_{0})}^{2}$ can be established based on that of $\\mathbb{E}_{S}\\Vert\\hat{f}_{\\mathcal{M}}-f_{\\mathcal{M}}\\Vert_{L_{2}(f_{\\mathcal{M}})}^{2}$\n\nWe first present some conditions imposed on the target density $f_{0}$ and reference density $f_{\\mathcal{R}}$ . We take the domain Ω of $f_{0}$ and $f_{\\mathcal{R}}$ to be the d-dimensional hypercube $[0,1]^{d}$ for simplicity. Extension to accommodate density with unbounded domain will be discussed in Section 3.5.\n\nAssumption 3 The reference density $f_{\\mathcal{R}}$ is positive and continuous on $[0,1]^{d}$ , and there exists a positive constant $L_{0}$ such that $f_{0}\\leq L_{0}$\n\nLemma 4 For any $\\Omega\\subseteq\\mathbb{R}^{d}$ , the data-augmented nonparametric noise contrastive estimator ${\\tilde{f}}_{n}$ defined in $(4)$ satisfies\n\n$$\n\\mathbb{E}_{S}\\|\\tilde{f}_{n}-f_{0}\\|_{L_{2}(f_{0})}^{2}\\leq(1+\\rho)^{3}\\mathbb{E}_{S}\\|\\hat{f}_{\\mathcal{M}}-f_{\\mathcal{M}}\\|_{L_{2}(f_{\\mathcal{M}})}^{2},\n$$\n\nwhere $S$ is the training sample.\n\nLemma 5 (Calibration) Under Assumption 3, there exist positive constants $c_{1}$ and $c_{2}$ such that, for all $f\\in\\mathcal{F}_{n}$ 2\n\n$$\nc_{1}\\|f-f_{\\mathcal{M}}\\|_{L_{2}(f_{\\mathcal{M}})}^{2}\\leq R(f)-R(f_{\\mathcal{M}})\\leq c_{2}\\|f-f_{\\mathcal{M}}\\|_{L_{2}(f_{\\mathcal{M}})}^{2}.\\tag{5}\n$$\n\nMoreover, $c_{1}$ and $c_{2}$ only depend on $\\nu,\\rho,L_{0},\\gamma,\\Gamma$ , and $f_{\\mathcal{R}}$ .\n\nLemma 6 (Error decomposition) Under Assumption 3, we have\n\n$$\n\\mathbb{E}_{S}\\|\\hat{f}_{\\mathcal{M}}-f_{\\mathcal{M}}\\|_{L_{2}(f_{\\mathcal{M}})}^{2}\\leq\\frac{1}{c_{1}}\\mathbb{E}_{S}[R(\\hat{f}_{\\mathcal{M}})-\\hat{R}(\\hat{f}_{\\mathcal{M}})]+\\frac{c_{2}}{c_{1}}\\inf_{f\\in\\mathcal{F}_{n}}\\|f-f_{\\mathcal{M}}\\|_{L_{2}(f_{\\mathcal{M}})}^{2},\\tag{6}\n$$\n\nwhere $c_{1},c_{2}$ are the same constants as in Lemma ${ 5 . }$\n\nThe stochastic error of the density estimator $\\hat{f}_{\\mathcal{M}}$ will be controlled by the complexity measures of the neural network class ${\\mathcal{F}}_{n}$ . The definitions of uniform covering number and pseudo-dimension are given in Appendix B.\n\nLemma 7 (Stochastic error) Let $\\mathcal{F}_{n}=\\mathcal{F}(\\mathcal{D},\\mathcal{W},\\mathcal{S},\\gamma,\\Gamma)$ be the class of feedforward neural networks activated by continuous piecewise linear activation function with finitely many inflection points. Let $\\begin{array}{ r }{\\hat{f}_{\\mathcal{M}}\\in\\arg\\operatorname*{min}_{f\\in\\mathcal{F}_{n}}\\hat{R}(f)}\\end{array}$ be the empirical risk minimizer over $\\mathcal{F}_{n}$ Suppose that Assumption 3 holds. Then $f o r n\\ge P d i m(\\mathcal{F}_{n})/\\{(1+\\nu)(1+\\rho)\\}$\n\n$$\n\\mathbb{E}_{S}[R(\\hat{f}_{\\mathcal{M}})-\\hat{R}(\\hat{f}_{\\mathcal{M}})]\\leq c_{3}\\mathcal{S D}\\log\\mathcal{S}\\frac{\\log n}{n},\n$$\n\nwhere $c_{3}$ is a constant depending only on $\\nu,\\rho,L_{0},\\gamma,\\Gamma$ , and $f_{\\mathcal{R}}$ .\n\nTo bound the approximation error inf ${\\ L}_{f\\in\\mathcal{F}_{n}}\\parallel f-f_{\\mathcal{M}}\\parallel_{L_{2}(f_{\\mathcal{M}})}^{2};$ , we impose some smoothness conditions on the target density $f_{0}$ and the reference density $f_{\\mathcal{R}}$ . In this paper, we assume that both $f_{0}$ and $f_{\\mathcal{R}}$ belong to some H¨older class (Definition 28, Appendix B).\n\nAssumption 8 (H¨older smoothness) The densities $f_{0},f_{\\mathcal{R}}\\in\\mathcal{H}^{\\beta}([0,1]^{d},B_{0})$ for a given $\\beta > 0$ and some finite constant $B_{0}>0$\n\nThe assumption on $f_{\\mathcal{R}}$ in Assumption 8 is not restrictive, as many popular distributions satisfy this assumption, such as the uniform distribution on $[0,1]^{d}$ . Notably, Assumption 8 implies that the mixture density $f_{\\mathcal{M}}\\in\\mathcal{H}^{\\beta}([0,1]^{d},B_{0})$\n\nIn this subsection, we establish non-asymptotic error bounds for our proposed density estimator under the total variation distance. Specifically, we consider the expected total variation distance $\\mathbb{E}_{S}\\mathrm{TV}(\\tilde{f}_{n},f_{0})$ between our proposed estimator ${\\tilde{f}}_{n}$ and the target density $f_{0}$ , where\n\n$$\n\\mathrm{TV}(f,f_{*})=\\frac{1}{2}\\int_{\\Omega}|f(x)-f_{*}(x)|d x.\n$$\n\nIn this subsection, we extend our method to handle densities with unbounded supports. Specifically, we let the target density $f_{0}$ and the reference density $f_{\\mathcal{R}}$ be functions on $\\Omega=\\mathbb{R}^{d}$ , the entire d-dimensional Euclidean space. To tackle the challenges associated with unbounded support, we consider a truncated version of noise contrastive risk.\n\nWe define the truncated domain $\\Omega_{t}$ as a d-dimensional hypercube $[-t,t]^{d}$ , where t is a positive constant (the truncation level) and can depend on the sample size n. Then, the truncated noise contrastive risk is defined as\n\n$$\nR_{t}(f)= -\\Big\\{\\mathbb{E}_{Y\\sim f_{\\mathcal{M}}}\\Big [\\log\\frac{f(Y)}{f(Y)+\\nu f_{\\mathcal{R}}(Y)}\\mathbb{1}_{\\Omega_{t}}(Y)\\Big]+\\nu\\mathbb{E}_{Z\\sim f_{\\mathcal{R}}}\\Big [\\log\\frac{\\nu f_{\\mathcal{R}}(Z)}{f(Z)+\\nu f_{\\mathcal{R}}(Z)}\\mathbb{1}_{\\Omega_{t}}(Z)\\Big]\\Big\\},\n$$\n\nwhere $\\mathbb{1}_{A}$ denotes the indicator function on a set $A,$ , i.e $\\mathbb{1}_{A}$ is equal to 1 on A and 0 outside A. Given the training sample ${\\cal S}=\\{Y_{1},\\dots,Y_{n+\\rho n},Z_{1},\\dots,Z_{\\nu(n+\\rho n)}\\}$ , the truncated empirical risk is defined by replacing the expectations in $R_{t}$ by sample averages:\n\n$$\n\\hat{R}_{t}(f)= -\\frac{1}{(1+\\rho)n}\\left\\{\\sum_{i=1}^{(1+\\rho)n}\\log\\frac{f(Y_{i})}{f(Y_{i})+\\nu f_{\\mathcal{R}}(Y_{i})}\\mathbb{1}_{\\Omega_{t}}(Y_{i})+\\sum_{i=1}^{\\nu(1+\\rho)n}\\log\\frac{\\nu f_{\\mathcal{R}}(Z_{i})}{f(Z_{i})+\\nu f_{\\mathcal{R}}(Z_{i})}\\mathbb{1}_{\\Omega_{t}}(Z_{i})\\right\\}.\n$$\n\nWe define the estimator for the mixture density $f_{\\mathcal{M}}$ as the minimizer of the truncated empirical risk, that is, $\\begin{array}{ r }{\\check{f}_{\\mathcal{M}}\\in\\arg\\operatorname*{min}_{f\\in\\mathcal{F}_{n}}\\hat{R}_{t}(f)}\\end{array}$ . Then, similar to (4), the resulting estimator for the target density $f_{0}$ is given by\n\n$$\n\\bar{f}_{n}=(1+\\rho)\\check{f}_{\\mathcal{M}}-\\rho f_{\\mathcal{R}}.\\tag{9}\n$$\n\nLemma 19 (Approximation error) Suppose that Assumptions 3 and 8 hold, and $\\gamma \\leq$ $\\rho l/(1+\\rho),\\Gamma\\geq L_{0}\\vee L$ where l, L are positive constants such that $l\\le f_{\\mathcal{R}}\\le L$ . Then for any $P,Q\\in\\mathbb{Z}_{+}$ , there exists a function $f_{*}\\in\\mathcal{F}_{n}=\\mathcal{F}(\\mathcal{D},\\mathcal{W},\\mathcal{S},\\gamma,\\Gamma)$ with width $\\mathcal{W}=$ $38(\\lfloor\\beta\\rfloor+1)^{2}d^{\\lfloor\\beta\\rfloor+1}P\\lceil\\log_{2}(8P)\\rceil$ and depth $\\mathcal{D}=21(\\lfloor\\beta\\rfloor+1)^{2}Q\\lceil\\log_{2}(8Q)\\rceil$ such that\n\n$$\n\\|f_{*}-f_{\\mathcal{M}}\\|_{L_{2}(f_{\\mathcal{M}})}^{2}\\leq 324B_{0}^{2}(\\lfloor\\beta\\rfloor+1)^{4}d^{2\\lfloor\\beta\\rfloor+\\beta\\vee 1}(PQ)^{-4\\beta/d}.\n$$\n\nIn this subsection, we present an error analysis of our estimator for estimating densities with unbounded support, with a focus on the diferences between the theoretical analysis in this setting and the bounded case. First, a positive and continuous reference density supported on a compact domain is lower bounded, while this property does not hold for unbounded domains. Second, conventional neural network approximation theory for approximating function with compact domain and is not directly applicable to the unbounded support case. To address the complications associated with unbounded support, we will leverage the truncation technique and assume that the target distribution has a sub-exponentially decaying tail.\n\nIn the following, we denote the target distribution (probability measure) as $\\mu_{0};$ that is, $\\mu_{0}(d x)=f_{0}(x)d x$ . We make the following assumptions on the target density $f_{0}$ and the reference density $f_{\\mathcal{R}}{}^{\\mathrm{~}}$\n\nAssumption 20 The target distribution $\\mu_{0}$ is sub-exponential in the sense that\n\n$$\n\\mu_{0}(\\mathbb{R}^{d}\\setminus [-x,x]^{d})\\leq C d\\exp(-a x),\\forall x>0,\\tag{22}\n$$\n\nfor some positive constants a and $C$ . Furthermore, there exist positive constants $L_{0}$ and $L_{\\mathcal{R}}$ such that $f_{0}\\leq L_{0}$ and $0<f_{\\mathcal{R}}\\le L_{\\mathcal{R}}$ on $\\mathbb{R}^{d}$ . In addition, there exists a positive number $l_{\\mathcal{R}}$ that depends on t and $d,$ such that $f_{\\mathcal{R}}\\geq l_{\\mathcal{R}}$ on $\\Omega_{t}$\n\nAssumption 21 (H¨older smoothness) The densities $f_{0},f_{\\mathcal{R}}\\in\\mathcal{H}^{\\beta}(\\mathbb{R}^{d},B_{0})$ for a given $\\beta > 0$ and some finite constant $B_{0}>0$\n\nIn Assumption 20, we require the reference distribution to satisfy $f_{\\mathcal{R}}\\geq l_{\\mathcal{R}}$ on $\\Omega_{t}$ . Since the integral of $f_{\\mathcal{R}}$ over $\\Omega_{t}$ is upper bounded by 1, it follows that $l_{\\mathcal{R}}\\leq(2 t)^{-d}$ . Therefore, $l_{\\mathcal{R}}$ depends on both t and d. We also select the lower bound of the neural network, $\\gamma,$ to be a function of t and $d .$ In our theoretical results, we establish the dependence of the error bounds on $l_{\\mathcal{R}}$ and $\\gamma$ explicitly.\n\nNext, we decompose the expected $L_{2}.$ -risk of ${\\bar{f}}_{n}$ into the error on the truncated domain and the error due to truncation. Before proceeding, similar to the definition of $\\|f-f_{*}\\|_{L_{2}(f_{*})}^{2}$ we let\n\n$$\n\\|g\\|_{L_{2}(f_{*};A)}^{2}: -\\mathbb{E}_{X\\sim f_{*}}\\left[|g(X)|^{2}\\mathbb{1}_{A}(X)\\right]=\\int_{A}|g(x)|^{2}f_{*}(x)d x\n$$\n\nfor any probability density $f_{*}$ on $\\mathbb{R}^{d}$ , function $g:\\mathbb{R}^{d}\\mathbb{R}$ , and subset $A\\subseteq\\mathbb{R}^{d}$\n\nLemma 22 (Error decomposition) Given any $t \\geq 1 / 2$ , suppose that Assumption 20 holds, and $\\gamma \\leq 1$ . Then, the expected $\\scriptstyle{L_{\\mathrm{2}}-r i s k}$ of the data-augmented nonparametric noise contrastive estimator ${\\bar{f}}_{n}$ defined in $(9)$ satisfies\n\n$$\n\\begin{array}{l}\\mathbb{E}_{S}\\|\\bar{f}_{n}-f_{0}\\|_{L_{2}(f_{0})}^{2}\\leq c_{13}\\gamma^{-1}l_{\\mathcal{R}}^{-2}\\mathbb{E}_{S}[R_{t}(\\check{f}_{\\mathcal{M}})-\\hat{R}_{t}(\\check{f}_{\\mathcal{M}})]+c_{14}\\gamma^{-3}l_{\\mathcal{R}}^{-2}\\inf_{f\\in\\mathcal{F}_{n}}\\|f-f_{\\mathcal{M}}\\|_{L_{2}(f_{\\mathcal{M}};\\Omega_{t})}^{2}\\\\ +c_{15}\\mathbb{E}_{S}\\|\\check{f}_{\\mathcal{M}}-f_{\\mathcal{M}}\\|_{L_{2}\\left(f_{0};\\Omega_{t}^{\\complement}\\right)}^{2},\\tag{23}\\\\ \\end{array}\n$$\n\nwhere $c_{13}$ and $c_{14}$ are constants only depending on $\\nu,\\rho,L_{0},\\Gamma$ , and $L_{\\mathcal{R}}$ , and $c_{15}$ is a constant only depending on $\\rho .$\n\nLemma 23 (Stochastic error) Let $\\mathcal{F}_{n}=\\mathcal{F}(\\mathcal{D},\\mathcal{W},\\mathcal{S},\\gamma,\\Gamma)$ be the class of feedforward neural networks activated by continuous piecewise linear activation function with finitely many inflection points, and let $\\gamma \\leq 1$ . Given any $t \\geq 1 / 2$ , let $\\begin{array}{ r }{\\check{f}_{\\mathcal{M}}\\in\\arg\\operatorname*{min}_{f\\in\\mathcal{F}_{n}}\\hat{R}_{t}(f)}\\end{array}$ be a minimizer of the truncated empirical risk over $\\mathcal{F}_{n}$ , and suppose that Assumption 20 holds. Then, for $n\\geq P d i m(\\mathcal{F}_{n})/\\{(1+\\nu)(1+\\rho)\\}$ ，\n\n$$\n\\mathbb{E}_{S}[R_{t}(\\check{f}_{\\mathcal{M}})-\\hat{R}_{t}(\\check{f}_{\\mathcal{M}})]\\leq c_{1}^{\\prime}\\gamma^{-2}l_{\\mathcal{R}}^{-1}\\mathcal{S D}\\log\\mathcal{S}\\frac{\\log n}{n},\n$$\n\nwhere $c_{1}^{\\prime}$ is a constant depending only on $\\nu,\\rho,L_{0},\\Gamma $ , and $L_{\\mathcal{R}}$ .\n\nLemma 25 (Approximation error) Given any $t > 0$ , suppose that Assumptions ${\\it 20}$ and 21 hold, $\\gamma\\leq\\rho l_{\\mathcal{R}}/(1+\\rho)$ , and $\\Gamma\\geq L_{0}\\vee L_{\\mathcal{R}}$ . Then for any $P,Q\\in\\mathbb{Z}_{+}$ , there exists a function $f_{*}\\in\\mathcal{F}_{n}=\\mathcal{F}(\\mathcal{D},\\mathcal{W},\\mathcal{S},\\gamma,\\Gamma)$ with width $\\mathcal{W}=38(\\lfloor\\beta\\rfloor+1)^{2}d^{\\lfloor\\beta\\rfloor+1}P\\lceil\\log_{2}(8P)\\rceil$ and depth $\\mathcal{D}=21(\\lfloor\\beta\\rfloor+1)^{2}Q\\lceil\\log_{2}(8Q)\\rceil+1$ such that\n\n$$\n\\|f_{*}-f_{\\mathcal{M}}\\|_{L_{2}(f_{\\mathcal{M}};\\Omega_{t})}^{2}\\leq 324(2t)^{2\\beta}B_{0}^{2}(s+1)^{4}d^{2s+\\beta\\vee 1}(PQ)^{-4\\beta/d}.\n$$\n\nLemma 26 (Truncation error) Given any $t > 0$ , suppose that Assumption 20 holds. Then, the truncation error satisfies\n\n$$\n\\mathbb{E}_{S}\\|\\check{f}_{\\mathcal{M}}-f_{\\mathcal{M}}\\|_{L_{2}(f_{0};\\Omega_{t}^{\\complement})}^{2}\\leq c_{2}^{\\prime}C d\\exp(-a t),\n$$\n\nwhere $c_{2}^{\\prime}$ is a constant only depending on $L_{0},\\Gamma_{}$ , and $L_{\\mathcal{R}}$ .\n\nDefinition 27 Suppose that $\\mathcal{G}$ is a class of functions from $\\mathcal{X}$ to $\\mathbb{R}$ , given a sequence $x_{1:k}:=(x_{1},\\ldots,x_{k})\\in\\mathcal{X}^{k}$ , let $\\mathcal{N}_{p}(\\epsilon,\\mathcal{G},x_{1:k})$ be the $\\epsilon$ -covering number of $\\mathcal{G}|_{x_{1:k}}:=\\{(g(x_{1}),\\ldots,g(x_{k})):g\\in\\mathcal{G}\\}\\subset\\mathbb{R}^{k}$ with respect to the $\\|\\cdot\\|_{p}$ $(1\\leq p\\leq\\infty)$ norm. The uniform covering number $\\mathcal{N}_{p}(\\epsilon,\\mathcal{G},k)$ is defined as the maximum of $\\mathcal{N}_{p}(\\epsilon,\\mathcal{G},x_{1:k})$ over all $x_{1:k}\\in\\mathcal{X}^{k}$ , i.e.\n\n$$\n\\mathcal{N}_{p}(\\epsilon,\\mathcal{G},k)=\\max\\{\\mathcal{N}_{p}(\\epsilon,\\mathcal{G},x_{1:k}):x_{1:k}\\in\\mathcal{X}^{k}\\}.\n$$\n\nThe pseudo-dimension $Pdim(g)$ is the largest interger k for which there exists $(x_{1},\\ldots,x_{k}$ 2 $y_{1},\\ b ...,y_{k})\\in\\mathcal{X}^{k}\\times\\mathbb{R}^{k}$ such that, for any $(a_{1},\\dotsc,a_{k})\\in\\{0,1\\}^{k}$ there exists $g\\in{\\mathcal{G}}$ such that $g(x_{i})>y_{i}\\iff a_{i}=1$ for all i.\n\nThe pseudo-dimension is closely related with VC-dimension in the sense that $\\operatorname{Pdim}({\\mathcal{F}})=$ VCdim(F) if F is a class of functions generated by a neural network with fixed architecture and fixed activation function (Theorem 14.1 in Anthony and Bartlett (1999)).\n\nDefinition 28 (H¨older class) Let $\\beta = s + r > 0$ with $s=\\lfloor\\beta\\rfloor\\in\\mathbb{N}$ and $r\\in(0,1]$ , where bβc denotes the largest integer strictly less than $\\beta$ and N is the set of nonnegative integers. For a finite $B_{0}>0$ , the H¨older class $\\mathcal{H}^{\\beta}(\\mathbb{R}^{d},B_{0})$ is defined as\n\n$$\n\\begin{array}{l}\\mathcal{H}^{\\beta}(\\mathbb{R}^{d},B_{0})\\\\ =\\{f:\\mathbb{R}^{d}\\to\\mathbb{R},\\max_{\\|\\alpha\\|_{1}\\leq s}\\|\\partial^{\\alpha}f\\|_{\\infty}\\leq B_{0},\\max_{\\|\\alpha\\|_{1}=s}\\sup_{x\\neq y}\\frac{|\\partial^{\\alpha}f(x)-\\partial^{\\alpha}f(y)|}{\\|x-y\\|_{2}^{r}}\\leq B_{0}\\},\\\\ \\end{array}\n$$\n\nwhere $\\alpha=(\\alpha_{1},\\ldots,\\alpha_{d})\\in\\mathbb{N}^{d},\\ \\partial^{\\alpha}=\\partial^{\\alpha_{1}}\\ldots\\partial^{\\alpha_{d}}$ and $\\begin{array}{ r }{\\|\\alpha\\|_{1}=\\sum_{i=1}^{d}\\alpha_{i}}\\end{array}$ . For any subset $\\mathcal{X}\\subseteq\\mathbb{R}^{d}$ , we denote $\\mathcal{H}^{\\beta}(\\mathcal{X},B_{0})=\\{f:\\mathcal{X}\\to\\mathbb{R},f\\in\\mathcal{H}^{\\beta}(\\mathbb{R}^{d},B_{0})\\}$", "theorems": [{"local_id": "T1", "statement": "Theorem 10 (Non-asymptotic error bound) Suppose that Assumptions 3 and 8 hold, and $\\gamma\\leq\\rho l/(1+\\rho),\\Gamma\\geq L_{0}\\vee L$ where l, L are positive constants such that $l\\le f_{\\mathcal{R}}\\le L$ Then, for any $P,Q\\in\\mathbb{Z}_{+}$ , the function class of ReLU-activated multi-layer perceptrons $\\mathcal{F}_{n}=$ $\\mathcal{F}(\\mathcal{D},\\mathcal{W},\\mathcal{S},\\gamma,\\Gamma)$ with width $\\mathcal{W}=38(\\lfloor\\beta\\rfloor+1)^{2}d^{\\lfloor\\beta\\rfloor+1}P\\lceil\\log_{2}(8P)\\rceil$ and depth $\\mathcal{D}=21(\\lfloor\\beta\\rfloor+$ $1)^{2}Q\\lceil\\log_{2}(8 Q)\\rceil$ , for $n\\geq P d i m(\\mathcal{F}_{n})/\\{(1+\\rho)(1+\\nu)\\}$ , the data-augmented nonparametric noise contrastive estimation ${\\tilde{f}}_{n}$ defined in $(4)$ satisfies\n\n$$\n\\mathbb{E}_{S}\\|\\tilde{f}_{n}-f_{0}\\|_{L_{2}(f_{0})}^{2}\\leq c_{4}\\mathcal{S D}\\log\\mathcal{S}\\frac{\\log n}{n}+c_{5}B_{0}^{2}(\\lfloor\\beta\\rfloor+1)^{4}d^{2\\lfloor\\beta\\rfloor+\\beta\\vee 1}(P Q)^{-4\\beta/d},\n$$\n\nwhere $c_{4}$ and $c_{5}$ are constants only depending on $\\nu,\\rho,L_{0},\\gamma,\\Gamma$ , and $f_{\\mathcal{R}}$ . Furthermore, if we set\n\n$$\n\\begin{array}{l}\\mathcal{W}=114(\\lfloor\\beta\\rfloor+1)^{2}d^{\\lfloor\\beta\\rfloor+1},\\\\ \\mathcal{D}=21(\\lfloor\\beta\\rfloor+1)^{2}\\lceil n^{d/\\{2(d+2\\beta)\\}}\\log_{2}(8n^{d/\\{2(d+2\\beta)\\}})\\rceil,\\\\ \\mathcal{S}=O((\\lfloor\\beta\\rfloor+1)^{6}d^{2\\lfloor\\beta\\rfloor+2}\\lceil n^{d/\\{2(d+2\\beta)\\}}\\log_{2}n\\rceil),\\\\ \\end{array}\n$$\n\nthen $f o r n\\geq P d i m(\\mathcal{F}_{n})/\\{(1+\\rho)(1+\\nu)\\},\\ :\\tilde{f}_{n}$ satisfies\n\n$$\n\\mathbb{E}_{S}\\|\\tilde{f}_{n}-f_{0}\\|_{L_{2}(f_{0})}^{2}\\leq c_{6}(B_{0}\\vee 1)^{2}(\\lfloor\\beta\\rfloor+1)^{9}d^{2\\lfloor\\beta\\rfloor+\\beta\\vee 3}n^{-\\frac{2\\beta}{d+2\\beta}}(\\log_{2}n)^{4},\\tag{7}\n$$\n\nwhere $c_{6}$ is a constant only depending on $\\nu,\\rho,L_{0},\\gamma,\\Gamma$ , and $f_{\\mathcal{R}}$ .", "proof": {"steps": [{"n": 1, "claim": "Proof Due to Lemma 6, the expected $L_{\\mathrm{{2^{-r i s k}}}}$ for estimating $f_{\\mathcal{M}}$ can be decomposed into the sum of stochastic error and approximation error:\n\n$$\n\\mathbb{E}_{S}\\|\\hat{f}_{\\mathcal{M}}-f_{\\mathcal{M}}\\|_{L_{2}(f_{\\mathcal{M}})}^{2}\\leq\\frac{1}{c_{1}}\\mathbb{E}_{S}[R(\\hat{f}_{\\mathcal{M}})-\\hat{R}(\\hat{f}_{\\mathcal{M}})]+\\frac{c_{2}}{c_{1}}\\inf_{f\\in\\mathcal{F}_{n}}\\|f-f_{\\mathcal{M}}\\|_{L_{2}(f_{\\mathcal{M}})}^{2}.\\tag{18}\n$$"}, {"n": 2, "claim": "By Lemma 7, we bound the stochastic error by\n\n$$\n\\mathbb{E}_{S}\\left[R\\left(\\hat{f}_{\\mathcal{M}}\\right)-\\hat{R}\\left(\\hat{f}_{\\mathcal{M}}\\right)\\right]\\leq c_{3}\\mathcal{S D}\\log\\mathcal{S}\\frac{\\log n}{n}.\\tag{19}\n$$"}, {"n": 3, "claim": "By Lemma 19 in Appendix A.5, we bound the approximation error by\n\n$$\n\\inf_{f\\in\\mathcal{F}_{n}}\\|f-f_{\\mathcal{M}}\\|_{L_{2}(f_{\\mathcal{M}})}^{2}\\leq 324B_{0}^{2}(\\lfloor\\beta\\rfloor+1)^{4}d^{2\\lfloor\\beta\\rfloor+\\beta\\vee 1}(PQ)^{-4\\beta/d}.\\tag{20}\n$$"}, {"n": 4, "claim": "Combining (18), (19), and (20), we have\n\n$$\n\\mathbb{E}_{S}\\|\\hat{f}_{\\mathcal{M}}-f_{\\mathcal{M}}\\|_{L_{2}(f_{\\mathcal{M}})}^{2}\\leq c_{4}^{\\prime}\\mathcal{S D}\\log\\mathcal{S}\\frac{\\log n}{n}+c_{5}^{\\prime}B_{0}^{2}(\\lfloor\\beta\\rfloor+1)^{4}d^{2\\lfloor\\beta\\rfloor+\\beta\\vee 1}(P Q)^{-4\\beta/d}.\\tag{21}\n$$"}, {"n": 5, "claim": "Since $c_{1},c_{2}$ and $c_{3}$ only depend on $\\nu,\\rho,L_{0},\\gamma,\\Gamma$ , and $f_{\\mathcal{R}}$ , it is clear that $c_{4}^{\\prime},c_{5}^{\\prime}$ are constants only depending on $\\nu,\\rho,L_{0},\\gamma,\\Gamma $ , and $f_{\\mathcal{R}}$ . Finally, Lemma 4 implies that\n\n$$\n\\mathbb{E}_{S}\\|\\tilde{f}_{n}-f_{0}\\|_{L_{2}(f_{0})}^{2}\\leq c_{4}\\mathcal{S D}\\log\\mathcal{S}\\frac{\\log n}{n}+c_{5}B_{0}^{2}(\\lfloor\\beta\\rfloor+1)^{4}d^{2\\lfloor\\beta\\rfloor+\\beta\\vee 1}(P Q)^{-4\\beta/d},\n$$\n\nwhere $c_{4}=(1+\\rho)^{3}c_{4}^{\\prime}$ and $c_{5}=(1+\\rho)^{3}c_{5}^{\\prime}$ . Setting $P = 1$ and $Q=\\lceil n^{d/\\{2(d+2\\beta)\\}}\\rceil$ , we obtain the error bound (7) in Theorem 10. This concludes the proof of Theorem 10. ■"}]}}, {"local_id": "T2", "statement": "Theorem 13 (Non-asymptotic error bound in total variation distance) Suppose that Assumptions 3 and 8 hold, and $\\gamma\\le\\rho l/(1+\\rho),\\Gamma\\ge L_{0}\\vee L$ where l, L are positive constants such that $l\\ \\leq\\ f_{\\mathcal{R}}\\ \\leq\\ L$ . Then, for any $P,Q\\in\\mathbb{Z}_{+}$ , the function class of ReLU-activated multi-layer perceptrons $\\mathcal{F}_{n}=\\mathcal{F}(\\mathcal{D},\\mathcal{W},\\mathcal{S},\\gamma,\\Gamma)$ with width $\\mathcal{W}=38(\\lfloor\\beta\\rfloor+$ $1)^{2}d^{\\lfloor\\beta\\rfloor+1}P\\lceil\\log_{2}(8 P)\\rceil$ and depth $\\mathcal{D}=21(\\lfloor\\beta\\rfloor+1)^{2}Q\\lceil\\log_{2}(8Q)^{-}$ , for $n\\geq P d i m(\\mathcal{F}_{n})/\\{(1+$ $\\rho)(1+\\nu)\\}$ , the data-augmented nonparametric noise contrastive estimation ${\\tilde{f}}_{n}$ defined in (4) satisfies\n\n$$\n\\mathbb{E}_{S}T V(\\tilde{f}_{n},f_{0})\\leq c_{7}\\Big(\\mathcal{S D}\\log\\mathcal{S}\\frac{\\log n}{n}\\Big)^{1/2}+c_{8}B_{0}(\\lfloor\\beta\\rfloor+1)^{2}d^{\\lfloor\\beta\\rfloor+(\\beta\\vee 1)/2}(P Q)^{-2\\beta/d},\n$$\n\nwhere $c_{7}$ and $c_{8}$ are constants only depending on $\\nu,\\rho,L_{0},\\gamma,\\Gamma$ , and $f_{\\mathcal{R}}$ . Furthermore, if we set\n\n$$\n\\mathcal{W}=114(\\lfloor\\beta\\rfloor+1)^{2}d^{\\lfloor\\beta\\rfloor+1},\n$$\n\n$$\n\\mathcal{D}=21(\\lfloor\\beta\\rfloor+1)^{2}\\lceil n^{d/2(d+2\\beta)}\\log_{2}(8n^{d/\\{2(d+2\\beta)\\}})\\rceil,\n$$\n\n$$\n\\mathcal{S}=O((\\lfloor\\beta\\rfloor+1)^{6}d^{2\\lfloor\\beta\\rfloor+2}\\lceil n^{d/\\{2(d+2\\beta)\\}}\\log_{2}n\\rceil),\n$$\n\nthen $f o r n\\geq P d i m(\\mathcal{F}_{n})/\\{(1+\\rho)(1+\\nu)\\},\\ :\\tilde{f}_{n}$ satisfies\n\n$$\n\\mathbb{E}_{S}T V(\\tilde{f}_{n},f_{0})\\leq c_{9}(B_{0}\\vee 1)(\\lfloor\\beta\\rfloor+1)^{9/2}d^{\\lfloor\\beta\\rfloor+(\\beta\\vee 3)/2}n^{-\\frac{\\beta}{d+2\\beta}}(\\log_{2}n)^{2},\\tag{8}\n$$\n\nwhere $c_{9}$ is a constant only depending on $\\nu,\\rho,L_{0},\\gamma,\\Gamma$ , and $f_{\\mathcal{R}}$ .", "proof": {"steps": [{"n": 1, "claim": "Proof Suppose that $f_{\\mathcal{R}}\\geq l,$ , then $f_{\\mathcal{M}}\\geq\\rho l/(1+\\rho)$ . It follows from the definition of total variation distance that\n\n$$\n\\begin{array}{l}\\mathbb{E}_{S}\\mathrm{TV}(\\hat{f}_{\\mathcal{M}},f_{\\mathcal{M}})=\\frac{1}{2}\\mathbb{E}_{S}\\int|\\hat{f}_{\\mathcal{M}}(y)-f_{\\mathcal{M}}(y)|d y\\\\ \\leq\\frac{1}{2}\\mathbb{E}_{S}\\int|\\hat{f}_{\\mathcal{M}}(y)-f_{\\mathcal{M}}(y)|f_{\\mathcal{M}}(x)\\frac{1+\\rho}{\\rho l}d y\\\\ =\\frac{1+\\rho}{2\\rho l}\\mathbb{E}_{S}[\\mathbb{E}_{Y\\sim f_{\\mathcal{M}}}|\\hat{f}_{\\mathcal{M}}(Y)-f_{\\mathcal{M}}(Y)|]\\\\ \\leq\\frac{1+\\rho}{2\\rho l}\\bigl\\{\\mathbb{E}_{S}[\\mathbb{E}_{Y\\sim f_{\\mathcal{M}}}|\\hat{f}_{\\mathcal{M}}(Y)-f_{\\mathcal{M}}(Y)|^{2}]\\bigr\\}^{1/2}\\\\ =\\frac{1+\\rho}{2\\rho l}\\bigl\\{\\mathbb{E}_{S}\\|\\hat{f}_{\\mathcal{M}}-f_{\\mathcal{M}}\\|_{L_{2}(f_{\\mathcal{M}})}^{2}\\bigr\\}^{1/2},\\\\ \\end{array}\n$$\n\nwhere the penultimate line follows from the Jensen’s inequality."}, {"n": 2, "claim": "By inequality (21) in Appendix $\\mathrm{A . 6}$ ,\n\n$$\n\\mathbb{E}_{S}\\|\\hat{f}_{\\mathcal{M}}-f_{\\mathcal{M}}\\|_{L_{2}(f_{\\mathcal{M}})}^{2}\\leq c_{4}^{\\prime}\\mathcal{S D}\\log\\mathcal{S}\\frac{\\log n}{n}+c_{5}^{\\prime}B_{0}^{2}(\\lfloor\\beta\\rfloor+1)^{4}d^{2\\lfloor\\beta\\rfloor+\\beta\\vee 1}(P Q)^{-4\\beta/d},\n$$\n\nwhere $c_{4}^{\\prime}$ and $c_{5}^{\\prime}$ are constants only depending on $\\nu,\\rho,L_{0},\\gamma,\\Gamma$ and, $f_{\\mathcal{R}}$ . Therefore,\n\n$$\n\\begin{array}{l}\\mathbb{E}_{S}\\mathrm{TV}(\\hat{f}_{\\mathcal{M}},f_{\\mathcal{M}})\\leq\\frac{1+\\rho}{2\\rho l}\\bigl\\{\\mathbb{E}_{S}\\|\\hat{f}_{\\mathcal{M}}-f_{\\mathcal{M}}\\|_{L_{2}(f_{\\mathcal{M}})}^{2}\\bigr\\}^{1/2}\\\\ \\leq\\frac{1+\\rho}{2\\rho l}\\Bigl\\{c_{4}^{\\prime}\\mathcal{S D}\\log\\mathcal{S}\\frac{\\log n}{n}+c_{5}^{\\prime}B_{0}^{2}(\\lfloor\\beta\\rfloor+1)^{4}d^{2\\lfloor\\beta\\rfloor+\\beta\\vee 1}(P Q)^{-4\\beta/d}\\Bigr\\}^{1/2}\\\\ \\leq\\frac{(1+\\rho)\\sqrt{c_{4}^{\\prime}}}{2\\rho l}\\Big(\\mathcal{S D}\\log\\mathcal{S}\\frac{\\log n}{n}\\Big)^{1/2}\\\\ +\\frac{(1+\\rho)\\sqrt{c_{5}^{\\prime}}}{2\\rho l}B_{0}(\\lfloor\\beta\\rfloor+1)^{2}d^{\\lfloor\\beta\\rfloor+(\\beta\\vee 1)/2}(P Q)^{-2\\beta/d}.\\\\ \\end{array}\n$$"}, {"n": 3, "claim": "Note that $\\mathrm{TV}(\\tilde{f}_{n},f_{0})=(1+\\rho)\\mathrm{TV}(\\tilde{f}_{\\mathcal{M}},f_{\\mathcal{M}})$ . Letting $c_{7}=(1+\\rho)^{2}\\sqrt{c_{4}^{\\prime}}/(2\\rho l),c_{8}=(1+$ $\\rho)^{2}\\sqrt{c_{5}^{\\prime}}/(2\\rho l)$ , we have\n\n$$\n\\mathbb{E}_{S}\\mathrm{TV}(\\tilde{f}_{n},f_{0})\\leq c_{7}\\Big(\\mathcal{S D}\\log\\mathcal{S}\\frac{\\log n}{n}\\Big)^{1/2}+c_{8}B_{0}(\\lfloor\\beta\\rfloor+1)^{2}d^{\\lfloor\\beta\\rfloor+(\\beta\\vee 1)/2}(P Q)^{-2\\beta/d},\n$$\n\nand $c_{7},\\ :c_{8}$ only depend on $\\nu,\\rho,L_{0},\\gamma,\\Gamma$ , and $f_{\\mathcal{R}}$ . Setting $P = 1$ and $Q=\\lceil n^{d/\\{2(d+2\\beta)\\}}\\rceil$ , we obtain the error bound (8) in Theorem $^{13.}$ . This concludes the proof of Theorem 13."}]}}, {"local_id": "T3", "statement": "Theorem 16 (Non-asymptotic error bound) Suppose that Assumptions 20 and 21 hold, $t=a^{-1}\\log n,\\gamma\\leq\\rho l_{\\mathcal{R}}/(1+\\rho)$ , and $\\Gamma\\geq L_{0}\\vee L_{\\mathcal{R}}$ . Then, for any $P,Q\\in\\mathbb{Z}_{+}$ , the function class of ReLU-activated multi-layer perceptrons $\\mathcal{F}_{n}~=~\\mathcal{F}(\\mathcal{D},\\mathcal{W},\\mathcal{S},\\gamma,\\Gamma)$ with width $\\mathcal{W}=38(\\lfloor\\beta\\rfloor+1)^{2}d^{\\lfloor\\beta\\rfloor+1}P\\lceil\\log_{2}(8P)\\rceil$ and depth $\\mathcal{D}=21(\\lfloor\\beta\\rfloor+1)^{2}Q\\lceil\\log_{2}(8Q)\\rceil+1$ , for $n\\geq [P d i m(\\mathcal{F}_{n})/\\{(1+\\rho)(1+\\nu)\\}]\\vee\\exp(a/2)$ , the data-augmented nonparametric noise contrastive estimator ${\\bar{f}}_{n}$ defined in (9) satisfies\n\n$$\n\\begin{array}{l}\\mathbb{E}_{S}\\|\\bar{f}_{n}-f_{0}\\|_{L_{2}(f_{0})}^{2}\\leq c_{16}\\gamma^{-3}l_{\\mathcal{R}}^{-3}\\mathcal{SD}\\log\\mathcal{S}\\frac{\\log n}{n}\\\\ +c_{17}\\gamma^{-3}l_{\\mathcal{R}}^{-2}(2/a)^{2\\beta}(\\lfloor\\beta\\rfloor+1)^{4}d^{2\\lfloor\\beta\\rfloor+\\beta\\vee 1}(PQ)^{-4\\beta/d}(\\log n)^{2\\beta}\\\\ +c_{18}dn^{-1},\\\\ \\end{array}\n$$\n\nwhere $c_{16},\\ c_{17},$ , and $c_{18}$ are constants only depending on $\\nu,\\ \\rho,\\ C,\\ L_{0},\\ \\Gamma,\\ L_{\\mathcal{R}}$ , and $B_{0}$ . Furthermore, $i f$ we set\n\n$$\n\\mathcal{W}=114(\\lfloor\\beta\\rfloor+1)^{2}d^{\\lfloor\\beta\\rfloor+1},\n$$\n\n$$\n\\mathcal{D}=21(\\lfloor\\beta\\rfloor+1)^{2}\\lceil n^{d/\\{2(d+2\\beta)\\}}\\log_{2}(8n^{d/\\{2(d+2\\beta)\\}})\\rceil+1,\n$$\n\n$$\n\\mathcal{S}=O((\\lfloor\\beta\\rfloor+1)^{6}d^{2\\lfloor\\beta\\rfloor+2}\\lceil n^{d/\\{2(d+2\\beta)\\}}\\log_{2}n\\rceil),\n$$\n\nthen for n ≥ [Pdim(F<sub>n</sub>)/{(1 + ρ)(1 + ν)}] ∨ exp(a/2), <sup>¯</sup>f<sub>n</sub> satisfies\n\n$$\n\\mathbb{E}_{S}\\|\\bar{f}_{n}-f_{0}\\|_{L_{2}(f_{0})}^{2}\\leq c_{19}\\gamma^{-3}l_{\\mathcal{R}}^{-3}\\{(2/a)\\vee 1\\}^{2\\beta}(\\lfloor\\beta\\rfloor+1)^{9}d^{2\\lfloor\\beta\\rfloor+3(\\beta\\vee 1)}n^{-\\frac{2\\beta}{d+2\\beta}}(\\log_{2}n)^{2(\\beta\\vee 2)},\\tag{10}\n$$\n\nwhere $c_{19}$ is a constant only depending on $\\nu,\\rho,C,L_{0},\\Gamma,L_{\\mathcal{R}}$ , and $B_{0}$ .", "proof": {"steps": [{"n": 1, "claim": "Proof of Theorem 16 $\\mathrm{By}$ Lemma 22, the expected L<sub>2</sub>-risk of ${\\bar{f}}_{n}$ can be decomposed into the sum of stochastic error, approximation error, and truncation error:\n\n$$\n\\begin{array}{l}\\mathbb{E}_{S}\\|\\bar{f}_{n}-f_{0}\\|_{L_{2}(f_{0})}^{2}\\leq c_{13}\\gamma^{-1}l_{\\mathcal{R}}^{-2}\\mathbb{E}_{S}[R_{t}(\\check{f}_{\\mathcal{M}})-\\hat{R}_{t}(\\check{f}_{\\mathcal{M}})]+c_{14}\\gamma^{-3}l_{\\mathcal{R}}^{-2}\\inf_{f\\in\\mathcal{F}_{n}}\\|f-f_{\\mathcal{M}}\\|_{L_{2}(f_{\\mathcal{M}};\\Omega_{t})}^{2}\\\\ +c_{15}\\mathbb{E}_{S}\\|\\check{f}_{\\mathcal{M}}-f_{\\mathcal{M}}\\|_{L_{2}\\left(f_{0};\\Omega_{t}^{\\complement}\\right)}^{2}.\\tag{30}\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "By Lemma 23, we bound the stochastic error by\n\n$$\n\\mathbb{E}_{S}[R_{t}(\\check{f}_{\\mathcal{M}})-\\hat{R}_{t}(\\check{f}_{\\mathcal{M}})]\\leq c_{1}^{\\prime}\\gamma^{-2}l_{\\mathcal{R}}^{-1}\\mathcal{S D}\\log\\mathcal{S}\\frac{\\log n}{n}.\\tag{31}\n$$"}, {"n": 3, "claim": "By Lemma 25, we bound the approximation error by\n\n$$\n\\inf_{f\\in\\mathcal{F}_{n}}\\|f-f_{\\mathcal{M}}\\|_{L_{2}(f_{\\mathcal{M}};\\Omega_{t})}^{2}\\leq 324(2t)^{2\\beta}B_{0}^{2}(s+1)^{4}d^{2s+\\beta\\vee 1}(PQ)^{-4\\beta/d}.\\tag{32}\n$$"}, {"n": 4, "claim": "By Lemma 26, we bound the truncation error by\n\n$$\n\\mathbb{E}_{S}\\|\\check{f}_{\\mathcal{M}}-f_{\\mathcal{M}}\\|_{L_{2}(f_{0};\\Omega_{t}^{\\complement})}^{2}\\leq c_{2}^{\\prime}C d\\exp(-a t)=c_{2}^{\\prime}C d n^{-1}.\\tag{33}\n$$"}, {"n": 5, "claim": "Combining (30), (31), (32) and (33), we have\n\n$$\n\\begin{array}{l}\\mathbb{E}_{S}\\|\\bar{f}_{n}-f_{0}\\|_{L_{2}(f_{0})}^{2}\\leq c_{16}\\gamma^{-3}l_{\\mathcal{R}}^{-3}\\mathcal{SD}\\log\\mathcal{S}\\frac{\\log n}{n}+c_{17}\\gamma^{-3}l_{\\mathcal{R}}^{-2}(2t)^{2\\beta}(s+1)^{4}d^{2s+\\beta\\vee 1}(PQ)^{-4\\beta/d}\\\\ +c_{18}dn^{-1}\\\\ =c_{16}\\gamma^{-3}l_{\\mathcal{R}}^{-3}\\mathcal{SD}\\log\\mathcal{S}\\frac{\\log n}{n}\\\\ +c_{17}\\gamma^{-3}l_{\\mathcal{R}}^{-2}(2/a)^{2\\beta}(s+1)^{4}d^{2s+\\beta\\vee 1}(PQ)^{-4\\beta/d}(\\log n)^{2\\beta}\\\\ +c_{18}dn^{-1}.\\\\ \\end{array}\n$$"}, {"n": 6, "claim": "Notice that $c_{13},\\ :c_{14}$ and $c_{1}^{\\prime}$ are constants only depending on $\\nu,\\rho,L_{0},\\Gamma .$ , and $L_{\\mathcal{R}},c_{15}$ only depends on $\\rho,$ and $c_{2}^{\\prime}$ only depends on $L_{0},\\Gamma .$ , and $L_{\\mathcal{R}}$ . Then, it is clear that\n\n$c_{16}$ is a constant only depending on $\\nu,\\rho,L_{0},\\Gamma$ , and $L_{\\mathcal{R}}$ ;  \n$c_{17}$ is a constant only depending on $\\nu,\\rho,L_{0},\\Gamma,L_{\\mathcal{R}}$ , and $B_{0}$ ;  \n$c_{18}$ is a constant only depending on $\\rho,C,L_{0},\\Gamma,L_{\\mathcal{R}}$\n\nSetting $P = 1$ and $Q=\\lceil n^{d/\\{2(d+2\\beta)\\}}\\rceil$ , we obtain the error bound (10) in Theorem 16. This concludes the proof of Theorem 16."}]}}], "target_local_id": null, "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2026-25-0989", "context": "With $\\Pi_{0}=\\Pi$ , the Bayesian online learning framework considered in this paper consists of the following inductive steps. For $t \\geq 1$ , let $\\widetilde{\\Pi}_{t}(\\cdot\\mathrm{~\\vert~}\\mathbf{D}_{t})$ be the posterior distribution obtained by updating the prior $\\Pi_{t-1}$ with the data $\\mathbf{D}_{t}$ using Bayes’ formula:\n\n$$\n\\widetilde{\\Pi}_{t}(\\mathcal{A}\\mid\\mathbf{D}_{t})=\\frac{\\int_{\\mathcal{A}}\\exp\\left\\{L_{t}(\\theta)\\right\\}\\mathrm{d}\\Pi_{t-1}(\\theta)}{\\int_{\\Theta}\\exp\\left\\{L_{t}(\\theta)\\right\\}\\mathrm{d}\\Pi_{t-1}(\\theta)},\\quad \\text{for any measurable}\\mathcal{A}\\subset\\Theta,\\tag{1}\n$$\n\nwhere $L_{t}(\\cdot)$ is the log-likelihood corresponding to the t-th mini-batch $\\mathbf{D}_{t}$ . Next, we approximate the updated posterior $\\widetilde{\\Pi}_{t}(\\cdot\\mathrm{~\\bf~|~D}_{t})$ by projecting it onto Q, the space of prior distributions. Since we use a variational approximation, the approximated posterior is given by\n\n$$\n\\Pi_{t}=\\underset{Q\\in\\mathcal{Q}}{\\operatorname{argmin}}K\\left(Q;\\widetilde{\\Pi}_{t}(\\cdot\\mid\\mathbf{D}_{t})\\right),\\tag{2}\n$$\n\nwhere $K(P;Q)$ denotes the Kullback–Leibler (KL) divergence, defined as\n\n$$\nK(P;Q)=\\int\\log\\left(\\frac{\\mathrm{d}P}{\\mathrm{d}Q}\\right)\\mathrm{d}P.\\tag{3}\n$$\n\nFor two real numbers a and $b,a\\lor b$ and $a \\wedge b$ denote the maximum and minimum of a and $b,$ respectively. For two positive sequences $\\left(a_{n}\\right)$ and $(b_{n}),a_{n}\\lesssim b_{n}(\\mathrm{or}a_{n}=O(b_{n}))$ means that $a_{n}\\leq C b_{n}$ for some constant $C\\in(0,\\infty)$ . Also, $a_{n}\\asymp b_{n}$ indicates that $a_{n}\\lesssim b_{n}$ and $b_{n}\\lesssim a_{n}$ . The notation $a_{n}\\ll b_{n}{\\mathrm{~}}({\\mathrm{o r~}}a_{n}=o(b_{n}){\\mathrm{,}}$ ) implies that $a_{n}/b_{n}\\to 0$ as $n \\to \\infty$ . We use the standard $O_{P}$ and $o_{p}$ notation for stochastic order symbols; see van der Vaart (2000). With a slight abuse of notation, for $\\nu_{n}\\in\\mathbb{R}^{p}$ satisfying $\\|\\nu_{n}\\|_{2}=o(1)$ , we write $\\nu_{n}=o(1)$ . For $1\\leq q\\leq\\infty,\\|\\cdot\\|_{q}$ indicates the $\\ell_{q}{\\mathrm{-n o r m}}$ of a vector. For $m\\in\\mathbb{N}$ , let $[m]=\\{1,2,...,m\\}$\n\nTable 1: Important notations\n\n<table><tr><td>Notation</td><td>Location</td><td>Notation</td><td>Location</td></tr><tr><td> $\\Theta(\\theta, \\mathbf{F}, r), \\ \\Theta(\\mathbf{F}, r)$ </td><td>(6), (7)</td><td> $\\mathbf{F}_{t,\\theta}, \\ \\widetilde{\\mathbf{F}}_{t,\\theta}$ </td><td>(5)</td></tr><tr><td> $r_{\\text{eff},t}, \\ \\widetilde{r}_{\\text{eff},1:t}, \\ r_{\\text{eff},1:t}$ </td><td>(28), (43)</td><td> $\\mathcal{E}_{\\text{est},1}, \\ \\mathcal{E}_{\\text{est},2}$ </td><td>(32), (44)</td></tr><tr><td> $p_{\\text{eff},t}, \\ \\lambda_t$ </td><td>(28)</td><td> $r_{\\text{LA}}, \\ \\widehat{\\tau}_{3,t}, \\ \\widehat{\\tau}_{4,t}$ </td><td>(19)</td></tr><tr><td> $M_n$ </td><td>(27), (42)</td><td> $\\widehat{\\tau}_{3,t,r}$ </td><td>(24)</td></tr><tr><td> $p_*$ </td><td>(EX)</td><td> $\\tau_{3,t}^*$ </td><td>(29)</td></tr><tr><td> $d_V(\\cdot, \\cdot), \\ K(\\cdot; \\cdot)$ </td><td>(20), (3)</td><td> $\\epsilon_{n,t,\\text{TV}}, \\ \\epsilon_{n,t,\\text{KL}}$ </td><td>Theorem 2, 3</td></tr><tr><td> $\\widehat{\\theta}_t, \\ \\theta_t^*$ </td><td>(9)</td><td> $K_{\\text{min}}, \\ K_{\\text{max}}$ </td><td>(34), (35), (36)</td></tr><tr><td> $\\widehat{\\theta}_{1:t}, \\ \\widehat{\\theta}_{1:t}^{\\text{ML}}, \\ \\theta_{1:t}^*$ </td><td>(40)</td><td> $K_{\\text{low}}, \\ K_{\\text{up}}$ </td><td>Proposition 6</td></tr></table>\n\nFor $z=(z_{j})_{j\\in [p]}\\in\\mathbb{R}^{p}$ and $k\\geq 2,$ let\n\n$$\nz^{\\otimes k}=(z_{i_{1}}\\times ...\\times z_{i_{k}})_{i_{1},\\dots,i_{k}\\in [p]}\\in\\mathbb{R}^{p^{k}}.\n$$\n\nFor two k-order tensors $\\mathbf{A}=(A_{i_{1},\\ldots,i_{k}})_{i_{1},\\ldots,i_{k}\\in [p]}\\in\\mathbb{R}^{p^{k}}$ and $\\mathbf{B}=(B_{i_{1},\\ldots,i_{k}})_{i_{1},\\ldots,i_{k}\\in [p]}\\in\\mathbb{R}^{p^{k}}$ ， let\n\n$$\n\\langle\\mathbf{A},\\mathbf{B}\\rangle=\\sum_{i_{1},\\dots,i_{k}\\in [p]}A_{i_{1},\\dots,i_{k}}B_{i_{1},\\dots,i_{k}}.\n$$\n\nLet $S_{+ +}^{p}$ denote the set of all $p\\times p{\\mathrm{-d i m e n s i o n a l}}$ symmetric positive definite matrices. Let $\\mathbf{I}_{p}\\in\\mathbb{R}^{p\\times p}$ denote the identity matrix. For a matrix $\\mathbf{A}=(a_{i j})\\in\\mathbb{R}^{n\\times p}$ , let $\\lambda_{\\mathrm{min}}(\\mathbf{A})$ and $\\lambda_{\\operatorname*{max}}(\\mathbf{A})$ denote the smallest and largest singular values of A, respectively. For simplicity, $\\lVert\\mathbf{A}\\rVert_{2}$ will often be used interchangeably with $\\lambda_{\\operatorname*{max}}(\\mathbf{A})$ . Let $\\|\\mathbf{A}\\|_{\\mathrm{F}}=(\\sum_{i j}a_{i j}^{2})^{1/2}$ be the Frobenius norm. For two distinct matrices A, $\\mathbf{B}\\in\\mathbb{R}^{n\\times n},\\mathbf{A}\\succeq\\mathbf{B}$ means $\\mathbf{A}-\\mathbf{B}$ is positive semi-definite matrix. For a k-th order tensor $\\mathbf{A}=(A_{i_{1},\\ldots,i_{k}})_{i_{1},\\ldots,i_{k}\\in [p]}\\in\\mathbb{R}^{p^{k}}$ , define the operator norm of A by\n\n$$\n\\|\\mathbf{A}\\|_{\\mathrm{op}}=\\sup_{u_{1},\\dots,u_{k}\\in\\mathcal{U}}|\\langle\\mathbf{A},u_{1}\\otimes\\dots\\otimes u_{k}\\rangle|,\n$$\n\nwhere $\\mathcal{U}=\\{u\\in\\mathbb{R}^{p}:\\|u\\|_{2}=1\\}$\n\nIn this subsection, we precisely formulate the Bayesian online learning procedure briefly introduced in the introduction. Suppose the entire dataset $\\mathbf{D}=(Y_{i})_{i\\in [N]}$ consists of independent (not necessarily identically distributed) observations $Y_{1},\\dots,Y_{N}$ . The dataset is partitioned into $T$ mini-batches of equal size $n,$ so that the total sample size up to the t-th mini-batch is given by $N_{t}=n t$ for all $t\\in\\{0,1,2,\\ldots,T\\}$ . The t-th mini-batch and the collection of all samples up to the t-th mini-batch are denoted as\n\n$$\n\\mathbf{D}_{t}=(Y_{N_{t-1}+1},Y_{N_{t-1}+2},...,Y_{N_{t}}),\\quad \\text{and}\\quad\\mathbf{D}_{1:t}=(\\mathbf{D}_{1},\\mathbf{D}_{2},...,\\mathbf{D}_{t})=(Y_{i})_{i\\in [N_{t}]}.\n$$\n\nFor $i \\in [ N ]$ , let $p_{\\theta,i}(\\cdot)$ be the probability density function for $Y_{i}$ parametrized by $\\theta \\in$ $\\Theta\\subset\\mathbb{R}^{p}$ , and let $\\ell_{\\theta,i}(y)=\\log p_{\\theta,i}(y)$ be the log density. For $t \\in [ T ]$ ], let\n\n$$\nL_{t}(\\theta)=L_{t}(\\theta;\\mathbf{D}_{t})=\\sum_{i=N_{t-1}+1}^{N_{t}}\\ell_{\\theta,i}(Y_{i}).\n$$\n\nLet $\\mathbb{P}_{\\theta}^{(N)}$ denote the joint probability measure corresponding to the product density function $(y_{1},y_{2},...,y_{N})\\mapsto\\prod_{i=1}^{N}p_{\\theta,i}(y_{i})$ . We assume that the model is well-specified; that is, D is generated from $\\mathbb{P}_{\\theta_{0}}^{(N)}$ for some true parameter $\\theta_{0}\\in\\Theta$ . Let $\\mathbb{P}_{0,t}(\\cdot)$ denote the joint probability measure corresponding to the product density\n\n$$\n(y_{N_{t-1}+1},y_{N_{t-1}+2},\\dots,y_{N_{t}})\\mapsto\\prod_{i=N_{t-1}+1}^{N_{t}}p_{\\theta_{0},i}(y_{i}),\n$$\n\nand let $\\mathbb{E}_{t}$ denote the expectation under $\\mathbb{P}_{0,t}$ . Since the model is assumed to be well-specified, we have $\\theta_{0}=\\operatorname{argmax}_{\\theta\\in\\Theta}\\mathbb{E}_{t}L_{t}(\\theta)$ for all $t \\in [ T ]$\n\nNow, we introduce our online learning procedure. Let Q be the collection of all Gaussian measures with nonsingular covariance matrices. Given an initial prior $\\Pi_{0}$ and the log-likelihood $L_{t}(\\theta)$ for the t-th mini-batch data $\\mathbf{D}_{t}.$ , we iteratively define the posterior distribution $\\widetilde{\\Pi}_{t}(\\cdot\\mid\\mathbf{D}_{t})$ and the corresponding variational approximation $\\Pi_{t}(\\cdot)$ as described in (1) and (2). We denote by $\\widetilde{\\pi}_{t}(\\cdot\\mid\\mathbf{D}_{t})$ and $\\pi_{t}(\\cdot)$ the density functions of $\\widetilde{\\Pi}_{t}(\\cdot\\mid\\mathbf{D}_{t})$ and $\\Pi_{t}(\\cdot)$ respectively.\n\nIn particular, we consider a normal distribution $\\mathcal{N}(\\mu_{0},\\Omega_{0}^{-1})$ as the initial prior $\\Pi_{0}$ • where $\\mu_{0}\\in\\mathbb{R}^{p}$ and $\\Omega_{0}\\in{S}_{+ +}^{p}$ . For $t\\in\\{0,1,...,T\\}$ , we denote\n\n$$\n\\Pi_{t}=\\mathcal{N}(\\mu_{t},\\boldsymbol{\\Omega}_{t}^{-1})\\tag{4}\n$$\n\nfor $\\mu_{t}\\in\\mathbb{R}^{p}$ and $\\Omega_{t}\\in S_{+ +}^{p}$\n\nFor a four times diferentiable function $f:\\mathbb{R}^{p} $ R with $\\theta\\mapsto f(\\theta)$ , let\n\n$$\n\\nabla f(\\theta)=\\left(\\frac{\\partial}{\\partial\\theta_{i_{1}}}f(\\theta)\\right)_{i_{1}\\in [p]}\\in\\mathbb{R}^{p},\\quad\\nabla^{2}f(\\theta)=\\left(\\frac{\\partial^{2}}{\\partial\\theta_{i_{1}}\\partial\\theta_{i_{2}}}f(\\theta)\\right)_{i_{1},i_{2}\\in [p]}\\in\\mathbb{R}^{p\\times p},\n$$\n\n$$\n\\nabla^{3}f(\\theta)=\\left(\\frac{\\partial^{3}}{\\partial\\theta_{i_{1}}\\partial\\theta_{i_{2}}\\partial\\theta_{i_{3}}}f(\\theta)\\right)_{i_{1},i_{2},i_{3}\\in [p]}\\in\\mathbb{R}^{p\\times p\\times p},\n$$\n\n$$\n\\nabla^{4}f(\\theta)=\\left(\\frac{\\partial^{4}}{\\partial\\theta_{i_{1}}\\partial\\theta_{i_{2}}\\partial\\theta_{i_{3}}\\partial\\theta_{i_{4}}}f(\\theta)\\right)_{i_{1},i_{2},i_{3},i_{4}\\in [p]}\\in\\mathbb{R}^{p\\times p\\times p\\times p}.\n$$\n\nTo ensure an accurate approximation of the posterior, we will impose several smoothness conditions on the log-likelihood $L_{t},$ , requiring that it is at least four times continuously diferentiable with probability 1. For such a diferentiable $L_{t}(\\cdot)$ , we introduce some notations used for posterior analysis. For $t \\in [ T ]$ and $\\theta \\in \\Theta$ , define\n\n$$\n\\mathbf{F}_{t,\\theta}= -\\nabla^{2}L_{t}(\\theta)\\in\\mathbb{R}^{p\\times p},\\quad\\widetilde{\\mathbf{F}}_{t,\\theta}=\\boldsymbol{\\Omega}_{t-1}+\\mathbf{F}_{t,\\theta},\\tag{5}\n$$\n\nwhere $\\Omega_{t-1}$ is defined in (4). For $\\theta_{\\mathrm{c}}\\in\\mathbb{R}^{p},r\\geq 0$ and $\\mathbf{F}\\in S_{+ +}^{p}$ , let\n\n$$\n\\Theta\\left(\\theta_{\\mathrm{c}},\\mathbf{F},r\\right)=\\left\\{\\theta\\in\\Theta:\\left\\|\\mathbf{F}^{1/2}\\left(\\theta-\\theta_{\\mathrm{c}}\\right)\\right\\|_{2}\\leq r\\right\\}\\tag{6}\n$$\n\ndenote the local elliptical vicinity of $\\theta_{\\mathrm{c}}$ . For notational simplicity, let\n\n$$\n\\Theta(\\mathbf{F},r)=\\Theta(0,\\mathbf{F},r).\\tag{7}\n$$\n\nGiven the prior distribution $\\Pi_{t-1}$ , which reflects the information in $\\mathbf{D}_{1:t-1}$ , we define the penalized log-likelihood function as\n\n$$\n\\widetilde{L}_{t}(\\theta)=L_{t}(\\theta)-\\frac{1}{2}\\left\\|\\boldsymbol{\\Omega}_{t-1}^{1/2}\\left(\\theta-\\mu_{t-1}\\right)\\right\\|_{2}^{2}.\\tag{8}\n$$\n\nFor each $t \\in [ T ]$ , the penalized maximum likelihood estimator (pMLE) and its population version are defined as\n\n$$\n\\widehat{\\theta}_{t}=\\underset{\\theta\\in\\Theta}{\\operatorname{argmax}}\\widetilde{L}_{t}(\\theta),\\quad\\theta_{t}^{*}=\\underset{\\theta\\in\\Theta}{\\operatorname{argmax}}\\mathbb{E}_{t}\\widetilde{L}_{t}(\\theta).\\tag{9}\n$$\n\nNow, we formalize the notion of smoothness. Let $f:\\Theta\\to\\mathbb{R}$ be a four-times diferentiable function. For some $\\mathbf{F}\\in S_{+ +}^{p}$ and $\\tau_{3},r\\geq 0$ , we say that $f$ satisfies the third order smoothness at $\\theta \\in \\Theta$ with parameters $(\\tau_{3},\\mathbf{F},r)$ if\n\n$$\n\\sup_{u\\in\\Theta(\\mathbf{F},r)}\\sup_{z\\in\\mathbb{R}^{p}}\\frac{\\left|\\langle\\nabla^{3}f(\\theta+u),z^{\\otimes 3}\\rangle\\right|}{\\left\\|\\mathbf{F}^{1/2}z\\right\\|_{2}^{3}}\\leq\\tau_{3}.\\tag{10}\n$$\n\nIn this section, we demonstrate that the posterior distribution $\\widetilde{\\Pi}_{t}\\left(\\cdot\\mid\\mathbf{D}_{t}\\right)$ is well approximated by its variational approximation $\\Pi_{t}$ . Before establishing the precise relation between $\\Pi_{t}$ and $\\widetilde{\\Pi}_{t}\\left(\\cdot\\mid\\mathbf{D}_{t}\\right)$ , we first show that the posterior can be accurately approximated by its Laplace approximation, defined as\n\n$$\n\\Pi_{t}^{\\mathsf{LA}}=\\mathcal{N}\\left(\\widehat{\\theta}_{t},\\widetilde{\\mathbf{F}}_{t,\\widehat{\\theta}_{t}}^{-1}\\right),\\quad\\forall t\\in [T],\n$$\n\nwhich admits a density function $\\pi_{t}^{\\tt L A}(\\cdot)$ . Although our primary focus is on $\\Pi_{t}$ rather than $\\Pi_{t}^{\\mathtt{LA}}$ , understanding the concentration behavior of the pair $(\\widehat{\\theta}_{t},\\widetilde{\\mathbf{F}}_{t,\\widehat{\\theta}_{t}})$ allows us to facilitate the theoretical analysis needed to establish $\\Pi_{T}\\approx\\Pi(\\cdot\\mid\\mathbf{D})$ . We will revisit this issue in Section 8.\n\nOur main results and the corresponding proofs in this section are largely inspired by those in Spokoiny (2023). Notably, for the total variation (TV) metric, we refine the dimension dependency of Theorem 2.4 in Spokoiny (2023) by leveraging advanced analysis in Spokoiny (2024). Moreover, for the KL divergence, we significantly improve the convergence rate presented in Theorem 2.6 of Spokoiny (2023); see the discussion following Theorem 3. Since the variational posterior $\\Pi_{t}$ minimizes the KL divergence, obtaining a sharp upper bound for the KL divergence between $\\Pi_{t}^{\\mathtt{LA}}$ and $\\widetilde{\\Pi}_{t}(\\cdot\\mathrm{~\\bf~|~D}_{t})$ is a crucial step in bounding $K(\\Pi_{t};\\widetilde{\\Pi}_{t}(\\cdot{|\\textbf{ D }_{t})})$ ). A detailed proof of this section is provided in Appendix B, where we present self-contained non-asymptotic proofs adapted to the online learning framework.\n\nAs discussed in Section 2.3, to accurately approximate $\\widetilde{\\Pi}_{t}(\\cdot\\mathrm{~\\bf~|~}\\mathbf{D}_{t})$ , we adopt selfconcordance-type smoothness conditions. To this end, we impose the following assumption:\n\n(A0) For every $t \\in [ T ]$ , assume that the map $\\theta\\mapsto L_{t}(\\theta)$ is concave on $\\Theta$ and at least four times continuously diferentiable with probability 1.\n\nFor the remainder of this paper, every event is treated as the intersection with the event under which assumption (A0) holds, without further explicit restatement.\n\nLet\n\n$$\n\\begin{array}{l}r_{\\mathrm{LA}}=2\\sqrt{p}+\\sqrt{2\\log N}\\\\ \\widehat{\\tau}_{3,t}=\\inf\\left\\{\\tau\\in\\mathbb{R}_{+}:\\sup_{u\\in\\Theta\\left(\\widetilde{\\mathbf{F}}_{t,\\widehat{\\theta}_{t}},4 r_{\\mathrm{LA}}\\right)}\\sup_{z\\in\\mathbb{R}^{p}}\\frac{\\left|\\langle\\nabla^{3}\\widetilde{L}_{t}\\left(\\widehat{\\theta}_{t}+u\\right),z^{\\otimes 3}\\rangle\\right|}{\\left\\|\\widetilde{\\mathbf{F}}_{t,\\widehat{\\theta}_{t}}^{1/2}z\\right\\|_{2}^{3}}\\leq\\tau\\right\\},\\tag{19}\\\\ \\end{array}\n$$\n\n$$\n\\widehat{\\tau}_{4,t} = \\inf \\left\\{\\tau \\in \\mathbb{R}_{+}:\\sup_{u\\in \\Theta (\\widetilde{\\mathbf{F}}_{t,\\widehat{\\theta}_{t}},4r_{\\mathrm{LA}})}\\sup_{z\\in \\mathbb{R}^{p}}\\frac{\\left|\\langle\\nabla^{4}\\widetilde{L}_{t}(\\widehat{\\theta}_{t} + u),z^{\\otimes 4}\\rangle\\right|}{\\left\\| \\widetilde{\\mathbf{F}}_{t,\\widehat{\\theta}_{t}}^{1 / 2}z\\right\\|_{2}^{4}}\\leq \\tau \\right\\}.\n$$\n\nThen, with probability 1, $\\widetilde L_{t}(\\cdot)$ satisfies the third and fourth order smoothness at $\\widehat{\\theta}_{t}$ with parameters $(\\widehat{\\tau}_{3,t},\\widetilde{\\mathbf{F}}_{t,\\widehat{\\theta}_{t}},4r_{\\mathrm{LA}})$ and $(\\widehat{\\tau}_{4,t},\\widetilde{\\mathbf{F}}_{t,\\widehat{\\theta}_{t}},4r_{\\mathrm{LA}})$ , respectively. Note that $\\widehat{\\tau}_{3,t}$ and $\\widehat{\\tau}_{4,t}$ are finite almost surely under (A0) because $\\widetilde{L}_{t}$ is four times continuously diferentiable and $\\widetilde{\\mathbf{F}}_{t,\\widehat{\\theta_{t}}}\\in S_{+ +}^{p}$\n\nFor two probability distributions $P$ and $Q,$ , the total variation (TV) distance is defined as\n\n$$\nd_{V}(P,Q)=\\sup_{\\mathcal{A}}|P(\\mathcal{A})-Q(\\mathcal{A})|,\\tag{20}\n$$\n\nwhere the supremum is taken over all measurable sets $\\mathcal{A}$ .\n\nTheorem 2 (Laplace approximation: TV distance) Suppose that (A0) holds. $Then,$ with probability 1, we have\n\n$$\nd_{V}\\left(\\Pi_{t}^{\\mathrm{LA}}(\\cdot),\\widetilde{\\Pi}_{t}(\\cdot\\mid\\mathbf{D}_{t})\\right)\\leq K\\epsilon_{n,t,\\mathrm{TV}},\\quad\\forall t\\in [T],\\tag{21}\n$$\n\nwhere\n\n$$\n\\epsilon_{n,t,\\mathbf{TV}}=\\left(\\widehat{\\tau}_{4,t}+\\widehat{\\tau}_{3,t}^{2}\\right)p^{2}+\\widehat{\\tau}_{3,t}p+\\widehat{\\tau}_{3,t}^{3}\\log^{3}N+e^{-8\\log N-8p}\n$$\n\nand $K > 0$ is a universal constant.\n\nTo approximate the posterior in terms of the KL divergence, an additional step is required to carefully control the behavior of the posterior density in the tail region. In particular, the log-likelihood ratio θ 7→ log $(\\pi_{t}^{\\mathrm{LA}}(\\theta)/\\widetilde{\\pi}_{t}(\\theta\\mid\\mathbf{D}_{t}))$ on $\\Theta^{\\mathrm{c}}(\\widehat{\\theta_{t}},\\widetilde{\\mathbf{F}}_{t,\\widehat{\\theta_{t}}},4r_{\\mathrm{LA}})$ should not grow too rapidly. To quantify this growth rate, we introduce the following quantity:\n\n$$\n\\widehat{\\tau}_{3,t,r}=\\inf\\left\\{\\tau\\in\\mathbb{R}_{+}:\\sup_{u\\in\\Theta(\\widetilde{\\mathbf{F}}_{t,\\widehat{\\theta}_{t}},r)}\\sup_{z\\in\\mathbb{R}^{p}}\\frac{\\left|\\langle\\nabla^{3}\\widetilde{L}_{t}(\\widehat{\\theta}_{t}+u),z^{\\otimes 3}\\rangle\\right|}{\\left\\|\\widetilde{\\mathbf{F}}_{t,\\widehat{\\theta}_{t}}^{1/2}z\\right\\|_{2}^{3}}\\leq\\tau\\right\\},\\quad r>4 r_{\\mathrm{LA}}.\\tag{24}\n$$\n\nThis quantity plays a key role in bounding the KL divergence $K(\\Pi_{t}^{\\mathtt{LA}}(\\cdot);\\ \\widetilde{\\Pi}_{t}\\left(\\cdot\\mid\\mathbf{D}_{t}\\right))$ . To ensure that this divergence remains suficiently small, we impose the following regularity condition.\n\n(KL) Suppose that $N \\geq 2$ . Assume also that, on an event $\\mathcal{E}_{1}$ the following inequalities hold uniformly for all $t \\in [ T ]$\n\n$$\n\\left(\\widehat{\\tau}_{3,t}r_{\\mathrm{LA}}^{2}\\right)\\vee\\left(\\widehat{\\tau}_{4,t}p^{2}\\right)\\leq\\frac{1}{8},\\tag{25}\n$$\n\n$$\n\\widehat{\\tau}_{3,t,r}\\leq N e^{8 p}\\exp\\left(\\left[\\sqrt{p}+\\sqrt{2\\log N}-3\\right]r\\right),\\quad\\forall r>4 r_{\\mathrm{LA}}.\n$$\n\nTheorem 3 (Laplace approximation: KL divergence) Suppose that $(AO)$ and (KL) hold on some event $\\mathcal{E}_{1}$ . Then, on $\\mathcal{E}_{1}$ , the following inequality holds uniformly for all $t \\in [ T ]$\n\n$$\nK\\left(\\Pi_{t}^{\\mathsf{LA}}(\\cdot);\\widetilde{\\Pi}_{t}\\left(\\cdot\\mid\\mathbf{D}_{t}\\right)\\right)\\leq K\\epsilon_{n,t,\\mathrm{KL}}^{2},\n$$\n\nwhere $K > 0$ is a universal constant and\n\n$$\n\\epsilon_{n,t,\\mathrm{KL}}=\\left(\\left[\\widehat{\\tau}_{4,t}+\\widehat{\\tau}_{3,t}^{2}\\right]p^{2}+\\widehat{\\tau}_{3,t}^{3}\\log^{3}N+e^{-7\\log N}\\right)^{1/2}.\n$$\n\nThis section presents theoretical results for the penalized M-estimator $\\widehat{\\theta}_{t}$ , which has been well-studied in the literature. Recently, Spokoiny (2012, 2017, 2024) developed advanced techniques for analyzing statistical behaviors of M-estimators with finite-sample guarantees. Accordingly, we adopt these techniques with modifications to adapt them to our framework.\n\nWe say the model is stochastically linear if the map $\\theta\\mapsto\\zeta_{t}(\\theta)=L_{t}(\\theta)-\\mathbb{E}_{t}L_{t}(\\theta)$ is linear for every $t ;$ that is, $\\zeta_{t}(\\theta)=a^{\\top}\\theta+b$ for some (random) quantities $a\\in\\mathbb{R}^{p}$ and $b\\in\\mathbb{R}$ . Note that stochastic linearity implies that $\\nabla^{k}L_{t}(\\theta)$ is non-random for all $t \\in [ T ]$ and $k\\in\\{2,3,4\\}$ Also, since $\\nabla\\zeta_{t}({\\boldsymbol{\\theta}})$ does not depend on $\\theta,$ we hereafter denote this random vector by $\\nabla{\\zeta}_{t}$ The stochastically linear framework encompasses many important statistical models, such as the logistic regression, Poisson regression, nonparametric regression (Spokoiny and Panov, 2025), nonlinear inverse problem (Spokoiny, 2019), and covariance estimation (Puchkin et al., 2025). See Section 1.3 in Spokoiny (2024) for a detailed discussion.\n\nWe now introduce the assumptions and notations used in the estimation procedure.\n\n(A1) The model is stochastically linear. Also, $\\mathbf{F}_{t,\\theta}$ is nonsingular for every $t \\in [ T ]$ and $\\theta\\in\\Theta(\\theta_{0},\\mathbf{I}_{p},1/2)$ . Furthermore, there exist $\\{\\mathbf{V}_{t}:t\\in [T]\\}\\subset S_{+ +}^{p}$ and $M_{n}>1$ such that\n\n$$\n\\mathbb{P}_{0,t}\\left(\\left\\|\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}}^{-1/2}\\nabla\\zeta_{t}\\right\\|_{2}\\geq r_{\\text{eff},t}\\right)\\leq e^{-(\\log n+\\log T)},\\tag{26}\n$$\n\n$$\n\\max_{t\\in [T]}\\sup_{\\theta\\in\\Theta\\left(\\theta_{0},\\mathbf{I}_{p},1/2\\right)}\\left\\|\\mathbf{F}_{t,\\theta}^{-1}\\mathbf{V}_{t}\\right\\|_{2}\\leq\\frac{M_{n}^{2}}{9},\\tag{27}\n$$\n\nwhere\n\n$$\nr_{\\text{eff},t}=p_{\\text{eff},t}^{1/2}+\\sqrt{2\\lambda_{t}(\\log n+\\log T)},\\quad\\left(\\simeq_{1,\\dots}\\right)\\|\\simeq_{1,\\dots}\\|\\tag{28}\n$$\n\n$$\np_{\\mathrm{eff},t}=\\mathrm{tr}\\left(\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}}^{-1}\\mathbf{V}_{t}\\right),\\quad\\lambda_{t}=\\left\\|\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}}^{-1}\\mathbf{V}_{t}\\right\\|_{2}.\n$$\n\nAs discussed in Section 2.3, we employ the self-concordance condition on $\\mathbb{E}_{t}{\\widetilde{L}}_{t}(\\cdot)$ over the local set $\\Theta(\\theta_{t}^{*},\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}},4 r_{\\mathrm{eff},t})$ . For this purpose, we define the following quantity:\n\n$$\n\\tau_{3,t}^{*}=\\inf\\left\\{\\tau_{3}\\in\\mathbb{R}_{+}:\\sup_{u\\in\\Theta(\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}},4 r_{\\mathrm{eff},t})}\\sup_{z\\in\\mathbb{R}^{p}}\\frac{\\left|\\langle\\nabla^{3}\\mathbb{E}_{t}\\widetilde{L}_{t}(\\theta_{t}^{*}+u),z^{\\otimes 3}\\rangle\\right|}{\\left\\|\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}}^{1/2}z\\right\\|_{2}^{3}}\\leq\\tau_{3}\\right\\}.\\tag{29}\n$$\n\nNote that $\\tau_{3,t}^{*}$ is random because both $\\theta_{t}^{*}$ and $\\Omega_{t}$ <sub>−1</sub> depend on $(\\mathbf{D}_{1},...,\\mathbf{D}_{t-1})$ . To prove the convergence of the pMLE $\\widehat{\\theta}_{t}$ , we further impose the following assumption.\n\n(Est) On an event ${\\mathcal{E}}_{2}$ , the following inequality holds uniformly for all $t \\in [ T ]$\n\n$$\n\\tau_{3,t}^{*}r_{\\mathrm{eff},t}\\leq 1/16.\\tag{30}\n$$\n\nTo state the main theorem in this section, let $\\mathcal{E}_{\\mathrm{{e s t},1}}^{\\mathcal{O}}$ be the event on which the following inequality holds:\n\n$$\n\\left\\|\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}}^{-1/2}\\nabla\\zeta_{t}\\right\\|_{2}\\leq r_{\\text{eff},t}\\quad \\text{for all} t\\in [T].\\tag{32}\n$$\n\nThen, $\\mathbb{P}_{0}^{(N)}(\\mathcal{E}_{\\mathrm{est},1})\\geq 1-n^{-1}$ under (A1). The event $\\mathcal{E}_{\\mathrm{{e s t},1}}^{\\mathcal{o}}$ plays a crucial role throughout this paper, as it ensures the statistical validity of the online estimation procedure.\n\nFor $t \\in [ T ]$ , let\n\n$$\n\\begin{array}{l}\\Delta_{t}=\\left\\|\\boldsymbol{\\Omega}_{t}^{1/2}\\left(\\mu_{t}-\\widehat{\\theta}_{t}\\right)\\right\\|_{2}\\vee\\left\\|\\widetilde{\\mathbf{F}}_{t,\\widehat{\\theta}_{t}}^{1/2}\\left(\\mu_{t}-\\widehat{\\theta}_{t}\\right)\\right\\|_{2}\\tag{100}\\\\ \\vee\\left\\|\\boldsymbol{\\Omega}_{t}^{-1/2}\\widetilde{\\mathbf{F}}_{t,\\widehat{\\theta}_{t}}\\boldsymbol{\\Omega}_{t}^{-1/2}-\\mathbf{I}_{p}\\right\\|_{\\mathrm{F}}\\vee\\left\\|\\widetilde{\\mathbf{F}}_{t,\\widehat{\\theta}_{t}}^{-1/2}\\boldsymbol{\\Omega}_{t}\\widetilde{\\mathbf{F}}_{t,\\widehat{\\theta}_{t}}^{-1/2}-\\mathbf{I}_{p}\\right\\|_{\\mathrm{F}}.\\\\ \\end{array}\n$$\n\nIn this subsection, assume we are given a function $f:\\Theta\\to\\mathbb{R}$ that is four times continuously diferentiable. For $\\theta \\in \\Theta$ , let $\\mathbf{F}_{\\theta}= -\\nabla^{2}f(\\theta)$ , and assume $\\mathbf{F}_{\\theta}$ is nonsingular in this subsection.\n\nLemma 40 For a given $\\theta \\in \\Theta$ , suppose that f satisfies the third order smoothness at θ with parameters $(\\tau_{3},\\mathbf{F}_{\\theta},r)$ . Then,\n\n$$\n\\sup_{\\theta^{\\prime}\\in\\Theta(\\theta,\\mathbf{F}_{\\theta},r)}\\left\\|\\mathbf{F}_{\\theta}^{-1/2}\\mathbf{F}_{\\theta^{\\prime}}\\mathbf{F}_{\\theta}^{-1/2}-\\mathbf{I}_{p}\\right\\|_{2}\\leq\\tau_{3}r.\n$$\n\nConsequently, for every $u\\in\\mathbb{R}^{p}$ and $\\theta^{\\prime}\\in\\Theta(\\theta,\\mathbf{F}_{\\theta},r)$ , we have\n\n$$\n(1-\\tau_{3}r)\\left\\|\\mathbf{F}_{\\theta}^{1/2}u\\right\\|_{2}^{2}\\leq\\left\\|\\mathbf{F}_{\\theta^{\\prime}}^{1/2}u\\right\\|_{2}^{2}\\leq(1+\\tau_{3}r)\\left\\|\\mathbf{F}_{\\theta}^{1/2}u\\right\\|_{2}^{2}.\n$$\n\nLemma 44 For $\\mu_{1},\\mu_{2}\\in\\mathbb{R}^{p}$ and $\\Omega_{1},\\Omega_{2}\\in{\\mathcal{S}}_{+ +}^{p}$ , let $Q_{1}={\\mathcal{N}}(\\mu_{1},\\Omega_{1}^{-1}),Q_{2}={\\mathcal{N}}(\\mu_{2},\\Omega_{2}^{-1})$ Suppose that\n\n$$\nd_{V}\\left(Q_{1},Q_{2}\\right)\\leq\\frac{1}{600}.\n$$\n\nLet\n\n$$\n\\Delta=\\left\\|\\boldsymbol{\\Omega}_{1}^{1/2}\\left(\\mu_{1}-\\mu_{2}\\right)\\right\\|_{2}\\vee\\left\\|\\boldsymbol{\\Omega}_{2}^{-1/2}\\boldsymbol{\\Omega}_{1}\\boldsymbol{\\Omega}_{2}^{-1/2}-\\mathbf{I}_{p}\\right\\|_{\\mathrm{F}}.\n$$\n\nThen,\n\n$$\n\\frac{\\Delta}{200}\\leq d_{V}\\left(Q_{1},Q_{2}\\right)\\leq\\frac{\\Delta}{\\sqrt{2}}.\n$$", "theorems": [{"local_id": "T1", "statement": "Theorem 4 (Variational approximation) Suppose that (A0) and (KL) hold on some event $\\mathcal{E}_{1}$ . Then, on $\\mathcal{E}_{1}$ , the following inequalities hold uniformly for all $t \\in [ T ]$\n\n$$\nK\\left(\\Pi_{t}(\\cdot);\\widetilde{\\Pi}_{t}(\\cdot|\\mathbf{D}_{t})\\right)\\leq K\\epsilon_{n,t,\\mathrm{KL}}^{2},d_{V}\\left(\\Pi_{t}(\\cdot),\\widetilde{\\Pi}_{t}(\\cdot|\\mathbf{D}_{t})\\right)\\leq K\\epsilon_{n,t,\\mathrm{KL}},\n$$\n\nwhere $K > 0$ is a universal constant.", "proof": {"steps": [{"n": 1, "claim": "Proof [Proof of Theorem 4] In this proof, we will work on the event $\\mathcal{E}_{1}$ without explicitly referring to it. By Theorems 2 and 3, we have\n\n$$\nd_{V}\\left(\\Pi_{t}^{\\mathsf{LA}}(\\cdot),\\widetilde{\\Pi}_{t}\\left(\\cdot\\mid\\mathbf{D}_{t}\\right)\\right)\\leq K_{1}\\epsilon_{n,t,\\mathtt{TV}},\\quad K\\left(\\Pi_{t}^{\\mathsf{LA}}(\\cdot),\\widetilde{\\Pi}_{t}\\left(\\cdot\\mid\\mathbf{D}_{t}\\right)\\right)\\leq K_{2}\\epsilon_{n,t,\\mathtt{KL}}^{2},\n$$\n\nwhere $K_{1}$ and $K_{2}$ denote the constant K in Theorems 2 and 3, respectively. By the definition of $\\Pi_{t}(\\cdot)$ , it follows that\n\n$$\nK\\left(\\Pi_{t}(\\cdot);\\widetilde{\\Pi}_{t}(\\cdot\\mid\\mathbf{D}_{t})\\right)\\leq K\\left(\\Pi_{t}^{\\mathrm{LA}}(\\cdot);\\widetilde{\\Pi}_{t}(\\cdot\\mid\\mathbf{D}_{t})\\right)\\leq K_{2}\\epsilon_{n,t,\\mathrm{KL}}^{2}.\n$$"}, {"n": 2, "claim": "By Pinsker’s inequality, we have\n\n$$\n\\begin{array}{l}d_{V}\\left(\\Pi_{t}(\\cdot),\\widetilde{\\Pi}_{t}\\left(\\cdot\\mid\\mathbf{D}_{t}\\right)\\right)\\leq\\sqrt{\\frac{1}{2}K\\left(\\Pi_{t}(\\cdot),\\widetilde{\\Pi}_{t}\\left(\\cdot\\mid\\mathbf{D}_{t}\\right)\\right)}\\leq\\sqrt{\\frac{K_{2}}{2}}\\epsilon_{n,t,\\mathrm{KL}},\\\\ d_{V}\\left(\\Pi_{t}^{\\mathrm{LA}}(\\cdot),\\widetilde{\\Pi}_{t}\\left(\\cdot\\mid\\mathbf{D}_{t}\\right)\\right)\\leq\\sqrt{\\frac{1}{2}K\\left(\\Pi_{t}^{\\mathrm{LA}}(\\cdot),\\widetilde{\\Pi}_{t}\\left(\\cdot\\mid\\mathbf{D}_{t}\\right)\\right)}\\leq\\sqrt{\\frac{K_{2}}{2}}\\epsilon_{n,t,\\mathrm{KL}}.\\\\ \\end{array}\n$$"}, {"n": 3, "claim": "By taking the constant K in Theorem 4 as $K=K_{2}\\vee\\sqrt{K_{2}/2}$ , we complete the proof."}]}}, {"local_id": "T2", "statement": "Theorem 5 Suppose that (A0), (A1) and (Est) hold. Then, on $\\mathcal{E}_{\\mathrm{est},1}\\cap\\mathcal{E}_{2}$ , the following inequality holds uniformly for all $t \\in [ T ]$\n\n$$\n\\left\\|\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}}^{1/2}\\left(\\widehat{\\theta}_{t}-\\theta_{t}^{*}\\right)\\right\\|_{2}\\leq 4 r_{\\mathrm{eff},t},\\quad\\forall t\\in [T].\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof [Proof of Theorem 5] In this proof, we work on the event $\\mathcal{E}_{2}\\cap\\mathcal{E}_{\\mathrm{est},1}$ without explicitly referring to it. Let $t \\in [ T ]$ ]. For simplicity in notations, let $\\Theta_{n,t}=\\Theta(\\theta_{t}^{*},\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}},4 r_{\\mathrm{eff},t})$ ， $\\overline{{\\Theta}}_{n,t}=\\Theta_{n,t}(\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}},4 r_{\\mathrm{eff},t})$ in this proof, and\n\n$$\n\\partial\\Theta_{n,t}=\\left\\{\\theta\\in\\Theta:\\left\\|\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}}^{1/2}\\left(\\theta-\\theta_{t}^{*}\\right)\\right\\|_{2}=4 r_{\\text{eff},t}\\right\\}.\n$$"}, {"n": 2, "claim": "For any $\\theta \\in \\Theta$ , let\n\n$$\ng_{t}(\\theta)=\\mathbb{E}_{t}\\widetilde{L}_{t}(\\theta)+\\langle\\nabla\\zeta_{t},\\theta\\rangle=\\mathbb{E}_{t}\\widetilde{L}_{t}(\\theta)+\\langle\\nabla\\widetilde{L}_{t}(\\widehat{\\theta}_{t})-\\nabla\\mathbb{E}_{t}\\widetilde{L}_{t}(\\widehat{\\theta}_{t}),\\theta\\rangle .\\tag{101}\n$$"}, {"n": 3, "claim": "By the right-hand side of the last display and the strong concavity of $\\theta\\mapsto\\mathbb{E}_{t}\\widetilde{L}_{t}(\\theta)$ , note that $\\nabla g_{t}(\\widehat{\\theta}_{t})=0$ and $\\theta\\mapsto g_{t}(\\theta)$ is concave. Hence, we only need to prove that the first-order stationary point $\\widehat{\\theta}_{t}$ of $g_{t}(\\cdot)$ is located in $\\Theta_{n,t}$\n\nBy the concavity of $g_{t}(\\cdot)$ , for any $\\theta\\in\\Theta_{n,t}^{\\mathrm{c}},$ we have\n\n$$\ng_{t}(\\bar{\\theta})\\geq\\omega g_{t}(\\theta)+(1-\\omega)g_{t}(\\theta_{t}^{*}),\\tag{102}\n$$\n\nwhere $\\overline{{\\theta}}=\\omega\\theta+(1-\\omega)\\theta_{t}^{*}$ and $\\omega=4 r_{\\mathrm{eff},t}\\Vert\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}}^{1/2}(\\theta-\\theta_{t}^{*})\\Vert_{2}^{-1}\\in(0,1)$ . At the end of this proof, we will show that\n\n$$\n\\sup_{\\theta^{\\circ}\\in\\partial\\Theta_{n,t}}g_{t}(\\theta^{\\circ})-g_{t}(\\theta_{t}^{*})\\leq-2 r_{\\text{eff},t}^{2}<0.\\tag{103}\n$$"}, {"n": 4, "claim": "It follows that, for any $\\theta\\in\\Theta_{n,t}^{\\mathrm{c}}.$ 7\n\n$$\n0> -2r_{\\mathrm{eff},t}^{2}\\geq\\sup_{\\theta^{\\circ}\\in\\partial\\Theta_{n,t}}g_{t}(\\theta^{\\circ})-g_{t}(\\theta_{t}^{*})\\stackrel{(102)}{\\geq}\\omega\\bigg [g_{t}(\\theta)-g_{t}(\\theta_{t}^{*})\\bigg]\\geq g_{t}(\\theta)-g_{t}(\\theta_{t}^{*}),\n$$\n\nwhich implies that $\\widehat{\\theta}_{t}\\in\\Theta_{n,t}$\n\nTo complete the proof, we only need to prove (103). Let $\\theta^{\\circ}\\in\\partial\\Theta_{n,t}$ and $u=\\theta^{\\circ}-\\theta_{t}^{*}$ By Taylor’s theorem, there exists some $\\widetilde{u}\\in\\overline{{\\Theta}}_{n,t}$ such that, on $\\mathcal{E}_{\\mathrm{{e s t},1}}^{\\mathcal{o}}$\n\n$$\n\\begin{array}{l}g_{t}(\\theta^{\\circ})-g_{t}(\\theta_{t}^{*})=\\nabla g_{t}(\\theta_{t}^{*})^{\\top}u+\\frac{1}{2}\\langle\\nabla^{2}g_{t}(\\theta_{t}^{*}+\\widetilde{u}),u^{\\otimes 2}\\rangle\\\\ =\\left[\\nabla\\mathbb{E}_{t}\\widetilde{L}_{t}(\\theta_{t}^{*})+\\nabla\\zeta_{t}\\right]^{\\top}u+\\frac{1}{2}\\langle\\nabla^{2}\\mathbb{E}_{t}\\widetilde{L}_{t}(\\theta_{t}^{*}+\\widetilde{u}),u^{\\otimes 2}\\rangle\\\\ =\\nabla\\zeta_{t}^{\\top}u+\\frac{1}{2}\\langle\\nabla^{2}\\mathbb{E}_{t}\\widetilde{L}_{t}(\\theta_{t}^{*}+\\widetilde{u}),u^{\\otimes 2}\\rangle\\\\ =\\left[\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}}^{-1/2}\\nabla\\zeta_{t}\\right]^{\\top}\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}}^{1/2}u-\\frac{1}{2}\\langle\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}+\\widetilde{u}},u^{\\otimes 2}\\rangle\\\\ \\stackrel{\\mathrm{Lemma40}}{\\leq}\\left(\\left\\|\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}}^{-1/2}\\nabla\\zeta_{t}\\right\\|_{2}-\\frac{1}{2}\\left(1-4\\tau_{3,t}^{*}r_{\\mathrm{eff},t}\\right)\\left\\|\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}}^{1/2}u\\right\\|_{2}\\right)\\left\\|\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}}^{1/2}u\\right\\|_{2}\\\\ \\stackrel{(\\mathbf{A 1})}{\\leq}\\left[r_{\\text{eff},t}-2\\left(1-4\\tau_{3,t}^{*}r_{\\text{eff},t}\\right)r_{\\text{eff},t}\\right]\\times 4 r_{\\text{eff},t},\\quad(\\because\\left\\|\\widetilde{\\mathbf{F}}_{t,\\theta_{t}^{*}}^{1/2}u\\right\\|_{2}=4 r_{\\text{eff},t})\\\\ \\leq-2 r_{\\mathrm{eff},t}^{2},\\\\ \\end{array}\n$$\n\nwhere the last inequality holds by $\\tau_{3,t}^{*}r_{\\mathrm{eff},t}\\leq 1/16$ ."}, {"n": 5, "claim": "This completes the proof."}]}}, {"local_id": "T3", "statement": "Corollary 17 Suppose that conditions in Theorem 4 hold. Also, assume that\n\n$$\n\\epsilon_{n,t,\\mathrm{KL}}\\leq(1200K)^{-1},\\quad\\forall t\\in [T]\n$$\n\non $\\mathcal{E}_{1}$ defined in Theorem $^{4,}$ where K is the universal constant specified in Theorem $\\it 4 .$ Then, on $\\mathcal{E}_{1}$\n\n$$\n\\Delta_{t}\\leq 400K\\epsilon_{n,t,\\mathrm{KL}},\\quad\\forall t\\in [T].\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof From the proof of Theorem 4, we have\n\n$$\nd_{V}\\left(\\Pi_{t}(\\cdot),\\widetilde{\\Pi}_{t}\\left(\\cdot\\mid\\mathbf{D}_{t}\\right)\\right)\\leq K_{1}\\epsilon_{n,t,\\mathrm{KL}},\n$$\n\n$$\nd_{V}\\left(\\Pi_{t}^{\\mathsf{LA}}(\\cdot),\\widetilde{\\Pi}_{t}\\left(\\cdot\\mid\\mathbf{D}_{t}\\right)\\right)\\leq K_{1}\\epsilon_{n,t,\\mathrm{KL}},\n$$\n\nwhere $K_{1}$ denotes the constant K in Theorem 4. It follows that\n\n$$\nd_{V}\\left(\\Pi_{t}(\\cdot),\\Pi_{t}^{\\mathsf{LA}}(\\cdot)\\right)\\leq d_{V}\\left(\\Pi_{t}(\\cdot),\\widetilde{\\Pi}_{t}\\left(\\cdot\\mid\\mathbf{D}_{t}\\right)\\right)+d_{V}\\left(\\widetilde{\\Pi}_{t}\\left(\\cdot\\mid\\mathbf{D}_{t}\\right),\\Pi_{t}^{\\mathsf{LA}}(\\cdot)\\right)\n$$\n\n$$\n\\leq K_{1}\\epsilon_{n,t,\\mathrm{KL}}+K_{1}\\epsilon_{n,t,\\mathrm{KL}}\\leq 2 K_{1}\\epsilon_{n,t,\\mathrm{KL}}.\n$$"}, {"n": 2, "claim": "Since $2K_{1}\\epsilon_{n,t,\\mathrm{KL}}\\leq 1/600$ , we can apply Lemma 44. By Lemma 44 and symmetry of $d_{V}(\\cdot,\\cdot)$ therefore, we have\n\n$$\n\\frac{\\Delta_{t}}{200}\\leq d_{V}\\left(\\Pi_{t}(\\cdot),\\Pi_{t}^{\\mathrm{LA}}(\\cdot)\\right)\\leq 2K_{1}\\epsilon_{n,t,\\mathrm{KL}},\n$$\n\nwhich completes the proof."}]}}], "target_local_id": null, "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2025-23-0102", "context": "We consider error-in-variables regression in a high-dimensional setting with fixed design. Formally, we observe a labeled dataset of size $n,$ denoted as $\\{(y_{i},z_{i}):i\\le n\\}$ . Here, $y_{i}\\in$ R is the response variable and $z_{i}\\in\\mathbb{R}^{p}$ is the observed covariate. For any $i \\geq 1$ , we posit that\n\n$$\ny_{i}=\\left\\langle\\boldsymbol{x}_{i},\\boldsymbol{\\beta}^{*}\\right\\rangle+\\varepsilon_{i},\\tag{1}\n$$\n\nwhere $\\beta^{*}\\in\\mathbb{R}^{p}$ is the unknown model parameter, $\\pmb{x}_{i}\\in\\mathbb{R}^{p}$ is a fixed covariate, and $\\varepsilon_{i}\\in\\mathbb{R}$ is the response noise. Unlike traditional settings where $z_{i}={\\pmb x}_{i}$ , the error-in-variables (EiV) setting reveals a corrupted version of the covariate $\\mathbf{\\delta}_{\\mathbf{\\mathcal{X}}_{i}}$ . Precisely, for any $i \\geq 1$ , let\n\n$$\n\\boldsymbol{z}_{i}=(\\boldsymbol{x}_{i}+\\boldsymbol{w}_{i})\\circ\\boldsymbol{\\pi}_{i},\\tag{2}\n$$\n\nwhere $\\pmb{w}_{i}\\in\\mathbb{R}^{p}$ is the covariate measurement noise, $\\pi_{i}\\in\\{1,\\mathsf{NA}\\}^{p}$ is a binary mask with NA denoting a missing value, and ◦ is the Hadamard (entrywise) product. Further, we consider a high-dimensional setting where n and p are growing with n possibly smaller than p.\n\nFor a matrix $\\pmb{A}\\in\\mathbb{R}^{a\\times b}$ , we denote its operator (spectral), Frobenius, and max element-wise norms as $\\|A\\|_{2},\\|A\\|_{F}$ , and $\\|A\\|_{\\operatorname*{max}}$ , respectively. By rowspan(A), we denote the subspace of $\\mathbb{R}^{b}$ spanned by the rows of A. Let $A^{\\dagger}$ denote the pseudoinverse of A. For a vector $\\pmb{v}\\in\\mathbb{R}^{a}$ , let $\\|\\pmb{v}\\|_{p}$ denote its $\\ell_{p}{\\mathrm{-n o r m}}$ for $p\\in [1,\\infty]$ . We define the sub-gaussian (Orlicz) norm as $\\lVert\\pmb{v}\\rVert_{\\psi_{2}}$ . Let $\\langle\\cdot,\\cdot\\rangle$ and $\\otimes$ denote the inner and outer products, respectively. For any two numbers $a,b\\in\\mathbb{R}$ , we use $a \\wedge b$ to denote min(a, b) and $a \\lor b$ to denote $\\operatorname*{max}(a,b)$ . Let $[a]=\\{1,\\dots,a\\}$ for any positive integer a.\n\nLet f and g be two functions defined on the same space. We say that $f(n)=O(g(n))$ if and only if there exists a positive real number M and a real number $n_{0}$ such that for all $n\\geq n_{0},|f(n)|\\leq M|g(n)|$ |. Analogously we say $f(n)\\ : =\\ :\\Theta(g(n))$ if and only if there exists positive real numbers m, M such that for all $n\\geq n_{0},\\ m|g(n)|\\leq|f(n)|\\leq M|g(n)|$ ; $f(n)=o(g(n))$ if for any $m > 0$ , there exists $n_{0}$ such that for all $n\\geq n_{0},|f(n)|\\leq m|g(n)|$ ; $f(n)=\\omega(g(n))$ if for any $m > 0$ , there exists $n_{0}$ such that for all $n\\geq n_{0},|f(n)|\\geq m|g(n)|$ |. ${\\bar{O}}(\\cdot)$ is defined analogously to $O(\\cdot)$ , but ignores log dependencies.\n\nAs described in Section 1, our in-sample (train) data consists of n labeled observations $\\{(y_{i},z_{i}):i\\leq n\\}$ . By contrast, our out-of-sample (test) data consists of $m \\geq 1$ unlabeled observations. That is, for $i ~ > ~ n$ , we observe the covariates $z_{i}$ but do not observe the associated response variables $y_{i}$ . Let $\\pmb{Z}=[\\pmb{z}_{i}^{\\top}:i\\leq n]\\in\\mathbb{R}^{n\\times p}$ and $Z^{\\prime}=[z_{i}^{\\top}:i>n]\\in\\mathbb{R}^{m\\times p}$ denote the matrices of in- and out-of-sample covariates, respectively.\n\nWe describe PCR, as introduced in Jollife (1982), with a variation to handle missing data.\n\nI: Parameter estimation. Let $\\widehat{\\rho}$ denote the fraction of observed entries in $Z .$ Replace all missing values (NA) in the covariate matrices with zero. Let $\\begin{array}{ r }{\\widetilde{\\pmb{Z}}=(1/\\widehat{\\rho})\\pmb{Z}=\\sum_{i=1}^{n\\wedge p}\\widehat{s}_{i}\\widehat{\\pmb{u}}_{i}\\otimes\\widehat{\\pmb{v}}_{i}}\\end{array}$ 2 where $\\widehat{s}_{i}\\in\\mathbb{R}$ are the singular values (in decreasing order) and $\\widehat{\\pmb{u}}_{i}\\in\\mathbb{R}^{n},\\widehat{\\pmb{v}}_{i}\\in\\mathbb{R}^{p}$ are the corresponding left and right singular vectors, respectively. For a hyperparameter $k \\in [ n \\land p ]$ ， let $\\begin{array}{ r }{\\widetilde{Z}^{\\bar{k}}=\\sum_{i=1}^{k^{-}}\\widehat{s}_{i}\\widehat{\\pmb{u}}_{i}\\otimes\\widehat{\\pmb{v}}_{i}}\\end{array}$ and define the estimated model parameter as\n\n$$\n\\widehat{\\boldsymbol{\\beta}}=(\\widetilde{\\boldsymbol{Z}}^{k})^{\\dagger}\\boldsymbol{y}=\\Big(\\sum_{i=1}^{k}(1/\\widehat{s}_{i})\\widehat{\\boldsymbol{v}}_{i}\\otimes\\widehat{\\boldsymbol{u}}_{i}\\Big)\\boldsymbol{y}.\\tag{3}\n$$\n\nProperty 2.1 The PCR solution, ${\\widehat{\\beta}},$ , as given in (3), is\n\n1. the unique solution to the following program:\n\n$$\n\\text{minimize}\\quad\\|\\boldsymbol{\\beta}\\|_{2}\\quad \\text{over}\\quad\\boldsymbol{\\beta}\\in\\mathbb{R}^{p}\\quad \\text{such that}\\quad\\boldsymbol{\\beta}\\in\\underset{\\boldsymbol{b}\\in\\mathbb{R}^{p}}{\\arg\\min}\\|\\boldsymbol{y}-\\widetilde{\\boldsymbol{Z}}^{k}\\boldsymbol{b}\\|_{2}^{2}.\n$$\n\n2. embedded within the rowspan(Ze<sup>k</sup>).\n\nThis section formalizes our problem setup. Let $\\pmb{X}=[\\pmb{x}_{i}^{\\top}:i\\leq n]\\in\\mathbb{R}^{n\\times p}$ and $\\pmb{X}^{\\prime}=[\\pmb{x}_{i}^{\\top}$ $i>n]\\in\\mathbb{R}^{m\\times p}$ represent the underlying in- and out-of-sample covariates, respectively.\n\nCollectively, we assume (1) and (2) are satisfied. We make the additional assumptions.\n\nAssumption 3.1 (Response noise) Let $\\{\\varepsilon_{i}:i\\leq n\\}$ be a sequence of independent mean zero subgaussian random variables with $\\|\\varepsilon_{i}\\|_{\\psi_{2}}\\leq\\sigma$\n\nAssumption 3.1 is a standard assumption in the regression literature that posits the idiosyncratic response noise to be independent draws from a subgaussian distribution.\n\nAssumption 3.2 (Covariate noise and missing values) Let $\\{w_{i}:i\\leq n+m\\}$ be a sequence of independent mean zero subgaussian random vectors with $\\|\\pmb{w}_{i}\\|_{\\psi_{2}}\\ \\leq\\ K$ and $\\|\\mathbb{E}[\\pmb{w}_{i}\\otimes\\pmb{w}_{i}]\\|_{2}\\le\\gamma^{2}$ . Let $\\pi_{i}\\in\\{1,\\mathsf{NA}\\}^{p}$ , where NA denotes a missing value, be a vector of independent Bernoulli variables with parameter $\\rho\\in\\left(0,1\\right]$ . Further, let $\\varepsilon_{i},\\ w_{i},\\ \\pi_{i}$ be mutually independent.\n\nConsistent with standard assumptions in the error-in-variables (EiV) regression literature, Assumption 3.2 posits the idiosyncratic EiV vector-valued noise ${\\pmb w}_{i}$ to be subgaussian and independent across measurements; note, however, that the noise is allowed to be dependent within a measurement, i.e., the coordinates of ${\\pmb w}_{i}$ can be correlated. Finally, we require missing entries in the observed covariate vector to be missing completely at random (MCAR). In Section 4.3.1, we discuss ways to allow for more heterogeneous missingness patterns.\n\nAssumption 3.3 (Bounded covariates) Let $\\|X\\|_{\\operatorname*{max}}\\leq 1\\ a n d\\ \\|X^{\\prime}\\|_{\\operatorname*{max}}\\leq 1$\n\nThis section addresses Q1–Q2. For ease of notation, let $C,c>0$ be absolute constants whose values may change from line to line or even within a line. Let $H=X^{\\dagger}X\\in\\mathbb{R}^{p\\times p}$ and ${\\pmb{H}}_{\\perp}={\\pmb{I}}-{\\pmb{H}}$ denote the projection matrices onto the rowspace and nullspace of $\\boldsymbol{X}$ respectively. Let ${\\pmb H}^{\\prime},{\\pmb H}_{\\perp}^{\\prime}\\in\\mathbb{R}^{p\\times p}$ be defined analogously with respect to $X^{\\prime}$ . We define $\\tilde{\\beta}^{*}=H\\beta^{*}$ as the projection of $\\beta^{*}$ onto the linear space spanned by the rows of $\\boldsymbol{X}$\n\nQ1: “When $p > n$ , is there a model parameter that PCR consistently identifies?”\n\nThe model parameter $\\beta^{*}$ is not identifiable in the high-dimensional regime as infinitely many solutions satisfy (1). Among all feasible parameters, we show that PCR recovers $\\tilde{\\beta}^{*}$ the unique parameter with minimum $\\ell_{2}{\\mathrm{-n o r m}}$ that is entirely embedded in the rowspace of $\\boldsymbol{X}$ , provided the number of principal components k is aptly chosen.\n\nFrom Property 2.1, recall that PCR enforces $\\widehat{\\beta}\\in\\mathrm{rowspan}(\\widetilde{Z}^{k})$ . Hence, if $k = r =$ rank(X) and the rowspace of $\\widetilde{\\pmb{Z}}^{r}$ is “close” to the rowspace of X, then ${\\widehat{\\beta}}\\approx{\\widetilde{\\beta}}^{*}$ . The “noise” in $z$ arises from the missingness pattern induced by $\\pi$ and the measurement error $W ;$ meanwhile, the “signal” in $z$ arises from $\\boldsymbol{X}$ , where its strength is captured by the magnitude of its singular values. Accordingly, we define the snr as\n\n$$\n\\mathrm{snr}:=\\frac{\\rho s_{r}}{\\sqrt{n}+\\sqrt{p}}.\\tag{5}\n$$\n\nHere, $s_{r}$ is the smallest nonzero singular value of $X,\\rho$ determines the fraction of observed entries, and ${\\sqrt{n}}+{\\sqrt{p}}$ is induced by the perturbation in the singular values from W. As one would expect, the signal strength snr scales linearly with $\\rho .$ From standard concentration results for sub-gaussian matrices, it follows that $\\|\\boldsymbol{W}\\|_{2}=\\widetilde{O}(\\sqrt{n}+\\sqrt{p})$ (see Lemma 9). With this notation, we state the main result on model identification.\n\nTheorem 4.1 Let Assumptions 3.1–3.3 hold. Consider $(i)$ PCR with $k=r=\\mathrm{rank}(X)$ )， $(i i)\\rho\\geq c(n p)^{-1}\\log^{2}(n p)$ , and $(iii)$ snr $\\ge C(K+1)(\\gamma+1)$ . Then w.p. at least $1-O((np)^{-10})$ ,\n\n$$\n\\|\\widehat{\\boldsymbol{\\beta}}-\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{2}^{2}\\leq C_{\\text{noise}}\\log(n p)\\cdot\\left\\{\\frac{\\|\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{2}^{2}}{\\mathsf{snr}^{2}}+\\frac{\\sqrt{n}\\|\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{1}}{(n\\vee p)\\mathsf{snr}^{2}}+\\frac{r(1\\vee\\|\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{1}^{2})}{(n\\vee p)\\mathsf{snr}^{2}}+\\frac{\\|\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{1}^{2}}{\\mathsf{snr}^{4}}\\right\\},\\tag{6}\n$$\n\nwhere $C_{\\mathrm{noise}}=C(K+1)^{4}(\\gamma+1)^{2}(\\sigma^{2}+1)$ . Further, if $\\langle\\pmb{x}_{i},\\beta^{*}\\rangle\\in [-d,d]$ for all $i \\leq n$ , then\n\n$$\n\\|\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{2}\\leq s_{r}^{-1}\\cdot d\\sqrt{n}\\quad a n d\\quad\\|\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{1}\\leq s_{r}^{-1}\\cdot d\\sqrt{n p}.\\tag{7}\n$$\n\nWe start with some useful notation. Note $\\pmb{X}\\beta^{*}=\\pmb{X}\\tilde{\\beta}^{*}$ . Let $\\pmb{y}=\\pmb{X}\\tilde{\\beta}^{*}+\\varepsilon$ be the vector notation of (1) with $\\pmb{y}=[y_{i}:i\\le n]\\in\\mathbb{R}^{n},\\varepsilon=[\\varepsilon_{i}:i\\le n]\\in\\mathbb{R}^{n}$ . Throughout, let $\\pmb{X}=\\pmb{U}\\pmb{S}\\pmb{V}^{\\top}$ denote the singular value decomposition (SVD) of X. Recall that we write $\\widetilde{Z}=\\widehat{\\rho}^{-1}Z=\\widehat{U}\\widehat{S}\\widehat{V}^{\\top}$ for the SVD of $\\widetilde{\\boldsymbol{z}}$ Its truncation using the top k singular components is denoted as $\\widetilde{Z}^{k}=\\widehat{U}_{k}\\widehat{S}_{k}\\widehat{V}_{k}^{\\top}$\n\nFurther, we will often use the following bound: for any $\\pmb{A}\\in\\mathbb{R}^{a\\times b},\\pmb{v}\\in\\mathbb{R}^{b}$ ，\n\n$$\n\\|\\boldsymbol{A}\\boldsymbol{v}\\|_{2}=\\|\\sum_{j=1}^{b}\\boldsymbol{A}._{j}v_{j}\\|_{2}\\leq\\left(\\max_{j\\leq b}\\|\\boldsymbol{A}._{j}\\|_{2}\\right)\\left(\\sum_{j=1}^{b}|v_{j}|\\right)=\\|\\boldsymbol{A}\\|_{2,\\infty}\\|\\boldsymbol{v}\\|_{1},\\tag{S1}\n$$\n\nwhere $\\|A\\|_{2,\\infty}=\\operatorname*{max}_{j}\\|A_{\\cdot j}\\|_{2}$ with $A ._{j}$ representing the $j\\mathrm{-t h}$ column of A.\n\nLemma 16 (Modified Hoefding Inequality) Let $\\ b{X}\\in\\mathbb{R}^{n}$ be random vector with independent mean-zero sub-Gaussian random coordinates with $\\|X_{i}\\|_{\\psi_{2}}\\leq K$ . Let $\\mathbf{\\pmb{a}}\\in\\mathbb{R}^{n}$ be another random vector that satisfies $\\|\\mathbf{a}\\|_{2}\\leq b$ almost surely for some constant $b \\geq 0$ . Then for all $t \\geq 0$ ，\n\n$$\n\\mathbb{P}\\Big(\\Big|\\sum_{i=1}^{n}a_{i}X_{i}\\Big|\\geq t\\Big)\\leq 2\\exp\\Big(-\\frac{c t^{2}}{K^{2}b^{2}}\\Big),\n$$\n\nwhere $c > 0$ is a universal constant.\n\nLemma 17 (Modified Hanson-Wright Inequality) Let $\\ b{X}\\in\\mathbb{R}^{n}$ be a random vector with independent mean-zero sub-Gaussian coordinates with $\\|X_{i}\\|_{\\psi_{2}}\\leq K$ . Let $\\ b{A}\\in\\mathbb{R}^{n\\times n}$ be a random matrix satisfying $\\|A\\|_{2}\\leq a$ and $\\|A\\|_{F}^{2}\\le b$ almost surely for some $a,b\\geq 0$ . Then for any $t \\geq 0$\n\n$$\n\\mathbb{P}\\left(\\left|\\boldsymbol{X}^{\\top}\\boldsymbol{A}\\boldsymbol{X}-\\mathbb{E}[\\boldsymbol{X}^{\\top}\\boldsymbol{A}\\boldsymbol{X}]\\right|\\geq t\\right)\\leq 2\\cdot\\exp\\Big(-c\\min\\left(\\frac{t^{2}}{K^{4}b},\\frac{t}{K^{2}a}\\right)\\Big).\n$$", "theorems": [{"local_id": "T1", "statement": "Lemma 2 Consider the setup of Theorem 4.1, and PCR with parameter $k = r$ . Then, for any $t > 0$ , the following holds w.p. at least $1-\\exp\\bigl(-t^{2}\\bigr)$\n\n$$\n\\|\\boldsymbol{U}\\boldsymbol{U}^{\\top}-\\widehat{\\boldsymbol{U}}_{r}\\widehat{\\boldsymbol{U}}_{r}^{\\top}\\|_{2}\\leq C(K+1)(\\gamma+1)\\frac{\\sqrt{n}+\\sqrt{p}+t}{\\rho s_{r}},\n$$\n\n$$\n\\|\\boldsymbol{V}\\boldsymbol{V}^{\\top}-\\widehat{\\boldsymbol{V}}_{r}\\widehat{\\boldsymbol{V}}_{r}^{\\top}\\|_{2}\\leq C(K+1)(\\gamma+1)\\frac{\\sqrt{n}+\\sqrt{p}+t}{\\rho s_{r}}.\n$$\n\nHere, $s_{r}>0$ represents the r-th largest singular value of X.", "proof": {"steps": [{"n": 1, "claim": "Recall that U, V denote the left and right singular vectors of X (equivalently, $\\rho X)$ , respectively; meanwhile, $\\widehat{U}_{k},\\widehat{V}_{k}$ denote the top k left and right singular vectors of Ze (equivalently, $z)$ , respectively. Further, observe that $\\mathbb{E}[Z]=\\rho X$ and let $\\tilde{W}=Z-\\rho X$ . To arrive at our result, we recall Wedin’s Theorem (Wedin, 1972)."}, {"n": 2, "claim": "Theorem C.1 (Wedin’s Theorem) Given $\\boldsymbol{A},\\boldsymbol{B}\\in\\mathbb{R}^{n\\times p}$ , let $\\boldsymbol{A}=\\boldsymbol{U}\\boldsymbol{S}\\boldsymbol{V}^{\\top}$ and $\\boldsymbol{B}=\\widehat{\\boldsymbol{U}}\\widehat{\\boldsymbol{S}}\\widehat{\\boldsymbol{V}}^{\\top}$ be their respective SVDs . Let $U_{k},V_{k}$ (respectively, $\\widehat{U}_{k},\\widehat{V}_{k})$ correspond to the truncation of U, V (respectively, ${\\widehat{U}},{\\widehat{V}})$ that retains the columns corresponding to the top k singular values of A (respectively, B). Let $s_{k}$ denote the k-th singular value of A. Then,\n\n$$\n\\max\\left(\\|\\boldsymbol{U}_{k}\\boldsymbol{U}_{k}^{\\top}-\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\|_{2},\\|\\boldsymbol{V}_{k}\\boldsymbol{V}_{k}^{\\top}-\\widehat{\\boldsymbol{V}}_{k}\\widehat{\\boldsymbol{V}}_{k}^{\\top}\\|_{2}\\right)\\leq\\frac{2\\|\\boldsymbol{A}-\\boldsymbol{B}\\|_{2}}{s_{k}-s_{k+1}}.\n$$"}, {"n": 3, "claim": "Using Theorem C.1 for $k = r$ , it follows that\n\n$$\n\\max\\left(\\|\\boldsymbol{U}\\boldsymbol{U}^{\\top}-\\widehat{\\boldsymbol{U}}_{r}\\widehat{\\boldsymbol{U}}_{r}^{\\top}\\|_{2},\\|\\boldsymbol{V}\\boldsymbol{V}^{\\top}-\\widehat{\\boldsymbol{V}}_{r}\\widehat{\\boldsymbol{V}}_{r}^{\\top}\\|_{2}\\right)\\leq\\frac{2\\|\\tilde{\\boldsymbol{W}}\\|_{2}}{\\rho s_{r}},\\tag{S24}\n$$\n\nwhere $s_{r}$ is the smallest nonzero singular value of X. Next, we obtain a high probability bound on $\\|\\tilde{W}\\|_{2}$ . To that end,\n\n$$\n\\frac{1}{n}\\|\\tilde{\\boldsymbol{W}}\\|_{2}^{2}=\\frac{1}{n}\\|\\tilde{\\boldsymbol{W}}^{\\top}\\tilde{\\boldsymbol{W}}\\|_{2}\\leq\\frac{1}{n}\\|\\tilde{\\boldsymbol{W}}^{\\top}\\tilde{\\boldsymbol{W}}-\\mathbb{E}[\\tilde{\\boldsymbol{W}}^{\\top}\\tilde{\\boldsymbol{W}}]\\|_{2}+\\frac{1}{n}\\|\\mathbb{E}[\\tilde{\\boldsymbol{W}}^{\\top}\\tilde{\\boldsymbol{W}}]\\|_{2}.\\tag{S25}\n$$"}, {"n": 4, "claim": "We bound the two terms in (S25) separately. We recall the following lemma, which is a direct extension of Theorem 4.6.1 of Vershynin (2018) for the non-isotropic setting, and we present its proof for completeness in Section C.5."}, {"n": 5, "claim": "Lemma 6 (Independent sub-gaussian rows) Let A be an $n \\times p$ matrix whose rows $A_{i}$ are independent, mean zero, sub-gaussian random vectors in $\\mathbb{R}^{p}$ with second moment matrix $\\Sigma=(1/n)\\mathbb{E}[A^{\\top}A]$ . Then for any $t \\geq 0$ , the following holds w.p. at least $1-\\exp(-t^{2})$\n\n$$\n\\|\\frac{1}{n}\\boldsymbol{A}^{\\top}\\boldsymbol{A}-\\boldsymbol{\\Sigma}\\|_{2}\\leq K^{2}\\max(\\delta,\\delta^{2}),\\quad \\text{where}\\delta=C\\sqrt{\\frac{p}{n}}+\\frac{t}{\\sqrt{n}};\\tag{S26}\n$$\n\nhere, $K=\\operatorname*{max}_{i}\\|A_{i}\\|_{\\psi_{2}}$\n\nThe matrix $\\tilde{W}=Z-\\rho X$ has independent rows by Assumption 3.2. We state the following Lemma about the distribution property of the rows of $\\tilde{W}$ , the proof of which can be found in Section C.6."}, {"n": 6, "claim": "Lemma 7 Let Assumption 3.2 hold. Then, $z_{i}-\\rho x_{i}$ is a sequence of independent, mean zero, sub-gaussian random vectors satisfying $\\|\\mathbf{\\cdot}\\mathbf{-}\\rho\\mathbf{x}_{i}\\|_{\\psi_{2}}\\leq C(K+1)$\n\nFrom Lemmas 6 and 7, w.p. at least $1-\\exp\\bigl(-t^{2}\\bigr)$\n\n$$\n\\frac{1}{n}\\|\\tilde{\\boldsymbol{W}}^{\\top}\\tilde{\\boldsymbol{W}}-\\mathbb{E}[\\tilde{\\boldsymbol{W}}^{\\top}\\tilde{\\boldsymbol{W}}]\\|_{2}\\leq C(K+1)^{2}\\left(1+\\frac{p}{n}+\\frac{t^{2}}{n}\\right).\\tag{S27}\n$$"}, {"n": 7, "claim": "Finally, we claim the following bound on $\\lVert\\mathbb{E}[\\tilde{\\mathbf{W}}^{\\top}\\tilde{\\mathbf{W}}]\\rVert_{2}$ , the proof of which is in Section C.7."}, {"n": 8, "claim": "Lemma 8 Let Assumption 3.2 hold. Then, we have\n\n$$\n\\|\\mathbb{E}[\\tilde{\\boldsymbol{W}}^{\\top}\\tilde{\\boldsymbol{W}}]\\|_{2}\\leq C(K+1)^{2}n(\\rho-\\rho^{2})+n\\rho^{2}\\gamma^{2}.\n$$"}, {"n": 9, "claim": "From (S25), (S27) and Lemma 8, we have w.p. at least $1-\\exp(-t^{2})$ for any $t > 0$\n\n$$\n\\|\\tilde{\\boldsymbol{W}}\\|_{2}^{2}\\leq C(K+1)^{2}(n+p+t^{2})+n(\\rho(1-\\rho)(K+1)^{2}+\\rho^{2}\\gamma^{2}).\n$$"}, {"n": 10, "claim": "For this, we conclude the following lemma."}, {"n": 11, "claim": "Lemma 9 For any $t>0_{i}$ , the following holds w.p. at least $1-\\exp\\bigl(-t^{2}\\bigr)$\n\n$$\n\\|\\boldsymbol{Z}-\\rho\\boldsymbol{X}\\|_{2}\\leq C(K+1)(\\gamma+1)(\\sqrt{n}+\\sqrt{p}+t).\n$$"}, {"n": 12, "claim": "Using the above and (S24), we conclude the proof of Lemma 2."}]}}, {"local_id": "T2", "statement": "Lemma 3 Consider PCR with parameter $k = r$ and $\\rho\\geq c(n p)^{-1}\\log^{2}(n p)$ . Then w.p. at least $1-O(1/(np)^{10})$ ，\n\n$$\n\\begin{array}{l}\\|\\boldsymbol{X}-\\widetilde{\\boldsymbol{Z}}^{r}\\|_{2,\\infty}^{2}\\\\ \\leq C(K+1)^{4}(\\gamma+1)^{2}\\left(\\frac{(n+p)(n+\\sqrt{n}\\log(n p))}{\\rho^{4}s_{r}^{2}}+\\frac{r+\\sqrt{r}\\log(n p)}{\\rho^{2}}\\right)+C\\frac{\\log(n p)}{\\rho p}.\\\\ \\end{array}\n$$", "proof": {"steps": [{"n": 1, "claim": "We want to bound $\\|X-\\widetilde{Z}^{k}\\|_{2,\\infty}^{2}$ . To that end, let $\\Delta_{j}=X_{\\cdot j}-\\widetilde{Z}_{\\cdot j}^{k}$ for any $j \\in [ p ]$ . Our interest is in bounding $\\|\\Delta_{j}\\|_{2}^{2}$ for all $j \\in [ p ]$ . Consider,\n\n$$\n\\widetilde{\\boldsymbol{Z}}_{. j}^{k}-\\boldsymbol{X}._{j}=(\\widetilde{\\boldsymbol{Z}}_{. j}^{k}-\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}._{j})+(\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}._{j}-\\boldsymbol{X}._{j}).\n$$"}, {"n": 2, "claim": "Now, note that $\\widetilde{Z}_{. j}^{k}-\\widehat{U}_{k}\\widehat{U}_{k}^{\\top}X ._{j}$ belongs to the subspace spanned by column vectors of $\\widehat{U}_{k}$ while $\\widehat{U}_{k}\\widehat{U}_{k}^{\\top}X ._{j}-\\mathbf{\\widehat{X}}._{j}$ belongs to its orthogonal complement with respect to $\\mathbb{R}^{n}$ . As a result,\n\n$$\n\\|\\widetilde{\\boldsymbol{Z}}_{. j}^{k}-\\boldsymbol{X}._{j}\\|_{2}^{2}=\\|\\widetilde{\\boldsymbol{Z}}_{. j}^{k}-\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}._{j}\\|_{2}^{2}+\\|\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}._{j}-\\boldsymbol{X}._{j}\\|_{2}^{2}.\\tag{S28}\n$$"}, {"n": 3, "claim": "Bounding $\\|\\widetilde{\\pmb{Z}}_{\\cdot j}^{k}-\\widehat{U}_{k}\\widehat{\\pmb{U}}_{k}^{\\top}\\pmb{X}._{j}\\|_{2}^{2}$ Recall that $\\widetilde{Z}\\ =\\ (1/\\widehat{\\rho})Z\\ =\\ \\widehat{U}\\widehat{S}\\widehat{V}^{\\intercal}$ , and hence $z =$ $\\widehat{\\rho}\\widehat{\\pmb{U}}\\widehat{\\pmb{S}}\\widehat{V}^{\\top}$ . Consequently,\n\n$$\n\\frac{1}{\\widehat{\\rho}}\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{Z}_{\\cdot j}=\\frac{1}{\\widehat{\\rho}}\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{Z}\\boldsymbol{e}_{j}=\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\widehat{\\boldsymbol{U}}\\widehat{\\boldsymbol{S}}\\widehat{\\boldsymbol{V}}^{\\top}\\boldsymbol{e}_{j}\n$$\n\n$$\n=\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{S}}_{k}\\widehat{\\boldsymbol{V}}_{k}^{\\top}\\boldsymbol{e}_{j}=\\widetilde{\\boldsymbol{Z}}_{. j}^{k}.\n$$"}, {"n": 4, "claim": "Therefore, we have\n\n$$\n\\begin{array}{l}\\widetilde{\\boldsymbol{Z}}_{. j}^{k}-\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}._{j}=\\frac{1}{\\widehat{\\rho}}\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{Z}._{j}-\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}._{j}\\\\ =\\frac{1}{\\widehat{\\rho}}\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}(\\boldsymbol{Z}._{j}-\\rho\\boldsymbol{X}._{j})+\\Big(\\frac{\\rho-\\widehat{\\rho}}{\\widehat{\\rho}}\\Big)\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}._{j}.\\\\ \\end{array}\n$$"}, {"n": 5, "claim": "Therefore,\n\n$$\n\\begin{array}{l}\\|\\widetilde{\\boldsymbol{Z}}_{. j}^{k}-\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}_{. j}\\|_{2}^{2}\\leq\\frac{2}{\\widehat{\\rho}^{2}}\\|\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}(\\boldsymbol{Z}_{. j}-\\rho\\boldsymbol{X}_{. j})\\|_{2}^{2}+2\\left(\\frac{\\rho-\\widehat{\\rho}}{\\widehat{\\rho}}\\right)^{2}\\|\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}_{. j}\\|_{2}^{2}\\\\ \\leq\\frac{2}{\\widehat{\\rho}^{2}}\\|\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}(\\boldsymbol{Z}._{j}-\\rho\\boldsymbol{X}._{j})\\|_{2}^{2}+2\\left(\\frac{\\rho-\\widehat{\\rho}}{\\widehat{\\rho}}\\right)^{2}\\|\\boldsymbol{X}._{j}\\|_{2}^{2},\\\\ \\end{array}\n$$\n\nwhere we have used the fact that $\\|\\widehat{U}_{k}\\widehat{U}_{k}^{\\top}\\|_{2}=1$ ."}, {"n": 6, "claim": "Recall that $\\pmb{U}\\in\\mathbb{R}^{n\\times r}$ represents the left singular vectors of X. Thus,\n\n$$\n\\begin{array}{l}\\|\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}(\\boldsymbol{Z}. j-\\rho\\boldsymbol{X}. j)\\|_{2}^{2}\\leq 2\\|(\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}-\\boldsymbol{U}\\boldsymbol{U}^{\\top})(\\boldsymbol{Z}. j-\\rho\\boldsymbol{X}. j)\\|_{2}^{2}+2\\|\\boldsymbol{U}\\boldsymbol{U}^{\\top}(\\boldsymbol{Z}. j-\\rho\\boldsymbol{X}. j)\\|_{2}^{2}\\\\ \\leq 2\\|\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}-\\boldsymbol{U}\\boldsymbol{U}^{\\top}\\|_{2}^{2}\\|\\boldsymbol{Z}._{j}-\\rho\\boldsymbol{X}._{j}\\|_{2}^{2}+2\\|\\boldsymbol{U}\\boldsymbol{U}^{\\top}(\\boldsymbol{Z}._{j}-\\rho\\boldsymbol{X}._{j})\\|_{2}^{2}.\\\\ \\end{array}\n$$"}, {"n": 7, "claim": "By Assumption 3.3, we have that $\\|{\\boldsymbol{X}}_{\\cdot j}\\|_{2}^{2}\\leq n$ . This yields\n\n$$\n\\begin{array}{l}\\|\\widetilde{\\boldsymbol{Z}}_{. j}^{k}-\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}_{. j}\\|_{2}^{2}\\leq\\frac{4}{\\widehat{\\rho}^{2}}\\|\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}-\\boldsymbol{U}\\boldsymbol{U}^{\\top}\\|_{2}^{2}\\|\\boldsymbol{Z}_{. j}-\\rho\\boldsymbol{X}_{. j}\\|_{2}^{2}\\\\ +\\frac{4}{\\widehat{\\rho}^{2}}\\|\\boldsymbol{U}\\boldsymbol{U}^{\\top}\\left(\\boldsymbol{Z}_{\\cdot j}-\\rho\\boldsymbol{X}_{\\cdot j}\\right)\\|_{2}^{2}+2 n\\left(\\frac{\\rho-\\widehat{\\rho}}{\\widehat{\\rho}}\\right)^{2}.\\tag{S29}\\\\ \\end{array}\n$$"}, {"n": 8, "claim": "We now state Lemmas 10 and 11. Their proofs are in Sections C.8 and C.9, respectively."}, {"n": 9, "claim": "Lemma 10 For any $\\alpha > 1$ ，\n\n$$\n\\mathbb{P}\\left(\\rho/\\alpha\\leq\\widehat{\\rho}\\leq\\alpha\\rho\\right)\\geq 1-2\\exp\\Big(-\\frac{(\\alpha-1)^{2}n p\\rho}{2\\alpha^{2}}\\Big).\n$$"}, {"n": 10, "claim": "Therefore, for $\\rho\\geq c{\\frac{\\log^{2}n p}{n p}}$ , we have w.p. $1-O(1/(np)^{10})$\n\n$$\n\\frac{\\rho}{2}\\leq\\widehat{\\rho}\\leq 2\\rho\\quad a n d\\quad\\left(\\frac{\\rho-\\widehat{\\rho}}{\\widehat{\\rho}}\\right)^{2}\\leq C\\frac{\\log(n p)}{\\rho n p}.\n$$"}, {"n": 11, "claim": "Lemma 11 Consider any matrix $\\ b{Q}\\in\\mathbb{R}^{n\\times\\ell}$ with $1 \\leq \\ell \\leq n$ such that its columns $Q_{\\cdot j}~f o r$ $j \\in [ \\ell ]$ are orthonormal vectors. Then for any $t > 0$ ，\n\n$$\n\\begin{array}{l}\\mathbb{P}\\Big(\\max_{j\\in [p]}\\left\\|\\boldsymbol{Q}\\boldsymbol{Q}^{\\top}(\\boldsymbol{Z}._{j}-\\rho\\boldsymbol{X}._{j})\\right\\|_{2}^{2}\\geq\\ell C(K+1)^{2}+t\\Big)\\\\ \\leq p\\cdot\\exp\\Big(-c\\min\\Big(\\frac{t^{2}}{C(K+1)^{4}\\ell},\\frac{t}{C(K+1)^{2}}\\Big)\\Big).\\\\ \\end{array}\n$$"}, {"n": 12, "claim": "Subsequently, w.p. $1-O(1/(np)^{10})$ ，\n\n$$\n\\max_{j\\in [p]}\\left\\|\\boldsymbol{Q}\\boldsymbol{Q}^{\\top}(\\boldsymbol{Z}_{. j}-\\rho\\boldsymbol{X}_{. j})\\right\\|_{2}^{2}\\leq C(K+1)^{2}(\\ell+\\sqrt{\\ell}\\log(n p)).\n$$"}, {"n": 13, "claim": "Both terms $\\|Z_{\\cdot j}-\\rho X_{\\cdot j}\\|_{2}^{2}$ and $\\|{U U}^{\\top}(Z . j-\\rho X ._{j})\\|_{2}^{2}$ can be bounded by Lemma 11: for the first term $Q=\\mathbf{I}$ , and for the second term $Q = U$ . In summary, w.p. $1-O(1/(np)^{10})$ , we have\n\n$$\n\\max_{j\\in [p]}\\|\\boldsymbol{Z}_{\\cdot j}-\\rho\\boldsymbol{X}_{\\cdot j}\\|_{2}^{2}\\leq C(K+1)^{2}(n+\\sqrt{n}\\log(n p)),\\tag{S30}\n$$\n\nand\n\n$$\n\\max_{j\\in [p]}\\|\\boldsymbol{U}\\boldsymbol{U}^{\\top}(\\boldsymbol{Z}._{j}-\\rho\\boldsymbol{X}._{j})\\|_{2}^{2}\\leq C(K+1)^{2}(r+\\sqrt{r}\\log(n p)).\\tag{S31}\n$$"}, {"n": 14, "claim": "Using (S29), (S30), (S31), and Lemmas 2 and 10 with $k = r .$ , we conclude that w.p. 1 − ${O}(1/(np)^{10})$ ,\n\n$$\n\\begin{array}{l}\\max_{j\\in [p]}\\|\\widetilde{\\boldsymbol{Z}}_{\\cdot j}^{k}-\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}_{\\cdot j}\\|_{2}^{2}\\\\ \\leq C(K+1)^{4}(\\gamma+1)^{2}\\left(\\frac{(n+p)(n+\\sqrt{n}\\log(n p))}{\\rho^{4}s_{r}^{2}}+\\frac{r+\\sqrt{r}\\log(n p)}{\\rho^{2}}\\right)+C\\frac{\\log(n p)}{\\rho p}(\\mathrm{S32})\\\\ \\end{array}\n$$"}, {"n": 15, "claim": "Bounding $\\|\\widehat{U}_{k}\\widehat{U}_{k}^{\\top}X ...-X ._{j}\\|_{2}^{2}$ . Recalling $\\pmb{X}=\\pmb{U}\\pmb{S}\\pmb{V}^{\\top}$ , we obtain $U U^{\\top}X_{\\cdot j}=X_{\\cdot j}$ since $U U^{\\top}$ is the projection onto the column space of X. Therefore,\n\n$$\n\\begin{array}{l}\\|\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}._{j}-\\boldsymbol{X}._{j}\\|_{2}^{2}=\\|\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}._{j}-\\boldsymbol{U}\\boldsymbol{U}^{\\top}\\boldsymbol{X}._{j}\\|_{2}^{2}\\\\ \\leq\\|\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}-\\boldsymbol{U}\\boldsymbol{U}^{\\top}\\|_{2}^{2}\\|\\boldsymbol{X}_{\\cdot j}\\|_{2}^{2}.\\\\ \\end{array}\n$$"}, {"n": 16, "claim": "Using Property 3.3, note that $\\|X_{\\cdot j}\\|_{2}^{2}\\leq n$ . Thus using Lemma 2 with $k = r$ , we have that w.p. at least $1-O(1/(np)^{10})$ , we have\n\n$$\n\\|\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}._{j}-\\boldsymbol{X}._{j}\\|_{2}^{2}\\leq C(K+1)^{2}(\\gamma+1)^{2}\\frac{n(n+p)}{\\rho^{2}s_{r}^{2}}.\\tag{S33}\n$$"}, {"n": 17, "claim": "Concluding. From (S28), (S32), and (S33), we claim w.p. at least $1-O(1/(np)^{10})$\n\n$$\n\\begin{array}{l}\\|\\boldsymbol{X}-\\widetilde{\\boldsymbol{Z}}^{k}\\|_{2,\\infty}^{2}\\\\ \\leq C(K+1)^{4}(\\gamma+1)^{2}\\left(\\frac{(n+p)(n+\\sqrt{n}\\log(n p))}{\\rho^{4}s_{r}^{2}}+\\frac{r+\\sqrt{r}\\log(n p)}{\\rho^{2}}\\right)+C\\frac{\\log(n p)}{\\rho p}.\\\\ \\end{array}\n$$\n\nThis completes the proof of Lemma 3."}]}}, {"local_id": "T3", "statement": "Lemma 4 $I f\\rho\\geq c(n p)^{-1}\\log^{2}(n p)$ , then for any k, we have $w . p .$ at least $1-O(1/(np)^{10})$ ,\n\n$$\n|\\widehat{s}_{k}-s_{k}|\\leq C(K+1)(\\gamma+1)\\frac{\\sqrt{n}+\\sqrt{p}}{\\rho}+C\\frac{\\sqrt{\\log(n p)}}{\\sqrt{\\rho n p}}s_{k}.\n$$", "proof": {"steps": [{"n": 1, "claim": "To bound $\\widehat{s}_{k}$ , we recall Weyl’s inequality."}, {"n": 2, "claim": "Lemma 12 (Weyl’s inequality) Given A, $\\pmb{{B}}\\in\\mathbb{R}^{m\\times n}$ , let $\\sigma_{i}$ and $\\widehat{\\sigma}_{i}$ be the i-th singular values of A and B, respectively, in decreasing order and repeated by multiplicities. Then for all $i \\in [ m \\land n ]$ 2\n\n$$\n\\left|\\sigma_{i}-\\widehat{\\sigma}_{i}\\right|\\leq\\left\\|\\boldsymbol{A}-\\boldsymbol{B}\\right\\|_{2}.\n$$"}, {"n": 3, "claim": "Let $\\tilde{s}_{k}$ be the k-th singular value of Z. Then, $\\widehat{s}_{k}=(1/\\widehat{\\rho})\\widetilde{s}_{k}$ since it is the k-th singular value of $\\widetilde{Z}=(1/\\widehat{\\rho})Z$ . By Lemma 12, we have\n\n$$\n\\left|\\tilde{s}_{k}-\\rho s_{k}\\right|\\leq\\|\\mathbf{Z}-\\rho\\mathbf{X}\\|_{2};\n$$\n\nrecall that $s_{k}$ is the k-th singular value of X. As a result,\n\n$$\n\\begin{array}{l}|\\widehat{s}_{k}-s_{k}| =\\frac{1}{\\widehat{\\rho}}|\\tilde{s}_{k}-\\widehat{\\rho}s_{k}|\\\\ \\leq\\frac{1}{\\widehat{\\rho}}|\\tilde{s}_{k}-\\rho s_{k}| +\\frac{|\\rho-\\widehat{\\rho}|}{\\widehat{\\rho}}s_{k}\\\\ \\leq\\frac{\\|\\boldsymbol{Z}-\\rho\\boldsymbol{X}\\|_{2}}{\\widehat{\\rho}}+\\frac{|\\rho-\\widehat{\\rho}|}{\\widehat{\\rho}}s_{k}.\\\\ \\end{array}\n$$"}, {"n": 4, "claim": "From Lemma 9 and Lemma 10, it follows that w.p. at least $1-O(1/(np)^{10})$ ,\n\n$$\n|\\widehat{s}_{k}-s_{k}|\\leq C(K+1)(\\gamma+1)\\frac{\\sqrt{n}+\\sqrt{p}}{\\rho}+C\\frac{\\sqrt{\\log(n p)}}{\\sqrt{\\rho n p}}s_{k}.\n$$\n\nThis completes the proof of Lemma 4."}]}}, {"local_id": "T4", "statement": "Lemma 5 Given $\\widetilde{\\boldsymbol{Z}}^{r}$ , the following holds w.p. at least $1-O(1/(np)^{10})$ with respect to the randomness in ε:\n\n$$\n\\langle\\widetilde{\\boldsymbol{Z}}^{r}(\\widehat{\\boldsymbol{\\beta}}-\\widetilde{\\boldsymbol{\\beta}}^{*}),\\boldsymbol{\\varepsilon}\\rangle\\leq\\sigma^{2}r+C\\sigma\\sqrt{\\log(n p)}\\left(\\sigma\\sqrt{r}+\\sigma\\sqrt{\\log(n p)}+\\|\\widetilde{\\boldsymbol{\\beta}}^{*}\\|_{1}(\\sqrt{n}+\\|\\widetilde{\\boldsymbol{Z}}^{r}-\\boldsymbol{X}\\|_{2,\\infty})\\right).\n$$", "proof": {"steps": [{"n": 1, "claim": "We need to bound $\\langle\\widetilde{Z}^{k}(\\widehat{\\beta}-\\widetilde{\\beta}^{*}),\\varepsilon\\rangle$ . To that end, we recall that $\\widehat{\\beta}=\\widehat{V}_{k}\\widehat{S}_{k}^{-1}\\widehat{U}_{k}^{\\top}y,\\widetilde{Z}^{k}=$ $\\widehat{U}_{k}\\widehat{S}_{k}\\widehat{V}_{k}^{\\top}$ , and $\\pmb{y}=\\pmb{X}\\tilde{\\beta}^{*}+\\varepsilon$ . Thus,\n\n$$\n\\widetilde{\\boldsymbol{Z}}^{k}\\widehat{\\boldsymbol{\\beta}}=\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{S}}_{k}\\widehat{\\boldsymbol{V}}_{k}^{\\top}\\widehat{\\boldsymbol{V}}_{k}\\widehat{\\boldsymbol{S}}_{k}^{-1}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{y}=\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}\\tilde{\\boldsymbol{\\beta}}^{*}+\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{\\varepsilon}.\n$$"}, {"n": 2, "claim": "Therefore,\n\n$$\n\\langle\\widetilde{\\boldsymbol{Z}}^{k}(\\widehat{\\boldsymbol{\\beta}}-\\widetilde{\\boldsymbol{\\beta}}^{*}),\\boldsymbol{\\varepsilon}\\rangle=\\langle\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}\\widetilde{\\boldsymbol{\\beta}}^{*},\\boldsymbol{\\varepsilon}\\rangle+\\langle\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{\\varepsilon},\\boldsymbol{\\varepsilon}\\rangle-\\langle\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{S}}_{k}\\widehat{\\boldsymbol{V}}_{k}^{\\top}\\widetilde{\\boldsymbol{\\beta}}^{*},\\boldsymbol{\\varepsilon}\\rangle .\\tag{S34}\n$$"}, {"n": 3, "claim": "Now, ε is independent of $\\widehat{U}_{k},\\widehat{S}_{k},\\widehat{V}_{k}$ since $\\widetilde{Z}^{k}$ is determined by Z, which is independent of ε. As a result,\n\n$$\n\\begin{array}{l}\\mathbb{E}\\big [\\langle\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{\\varepsilon},\\boldsymbol{\\varepsilon}\\rangle\\big]=\\mathbb{E}\\big [\\boldsymbol{\\varepsilon}^{\\top}\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{\\varepsilon}\\big]\\\\ =\\mathbb{E}\\big [\\operatorname{tr}\\Big(\\boldsymbol{\\varepsilon}^{\\top}\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{\\varepsilon}\\Big)\\big]=\\mathbb{E}\\big [\\operatorname{tr}\\Big(\\boldsymbol{\\varepsilon}\\boldsymbol{\\varepsilon}^{\\top}\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\Big)\\big]\\\\ =\\operatorname{tr}\\left(\\mathbb{E}\\left[\\boldsymbol{\\varepsilon}\\boldsymbol{\\varepsilon}^{\\top}\\right]\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\right)\\leq C\\operatorname{tr}\\left(\\sigma^{2}\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\right)\\\\ =C\\sigma^{2}\\|\\widehat{\\boldsymbol{U}}_{k}\\|_{F}^{2}=C\\sigma^{2}k.\\tag{S35}\\\\ \\end{array}\n$$"}, {"n": 4, "claim": "Therefore, it follows that\n\n$$\n\\mathbb{E}[\\langle\\widetilde{\\boldsymbol{Z}}^{k}(\\widehat{\\boldsymbol{\\beta}}-\\tilde{\\boldsymbol{\\beta}}^{*}),\\boldsymbol{\\varepsilon}\\rangle]\\leq C\\sigma^{2}k,\\tag{S36}\n$$\n\nwhere we used the fact $\\mathbb{E}[\\varepsilon]=\\mathbf{0}$ . To obtain a high probability bound, using Lemma 16 it follows that for any $t > 0$\n\n$$\n\\mathbb{P}\\left(\\langle\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}\\tilde{\\boldsymbol{\\beta}}^{*},\\varepsilon\\rangle\\geq t\\right)\\leq\\exp\\left(-\\frac{c t^{2}}{n\\|\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{1}^{2}\\sigma^{2}}\\right)\\tag{S37}\n$$\n\ndue to Assumption 3.1, and\n\n$$\n\\|\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{X}\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{2}\\leq\\|\\boldsymbol{X}\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{2}\\leq\\|\\boldsymbol{X}\\|_{2,\\infty}\\|\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{1}\\leq\\sqrt{n}\\|\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{1};\n$$\n\nnote that we have used the fact that $\\widehat{U}_{k}\\widehat{U}_{k}^{\\top}$ is a projection matrix and $\\|X\\|_{2,\\infty}\\leq{\\sqrt{n}}$ due to Assumption 3.3."}, {"n": 5, "claim": "Similarly, for any $t > 0$\n\n$$\n\\mathbb{P}\\left(\\langle\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{S}}_{k}\\widehat{\\boldsymbol{V}}_{k}^{\\top}\\tilde{\\boldsymbol{\\beta}}^{*},\\boldsymbol{\\varepsilon}\\rangle\\geq t\\right)\\leq\\exp\\Big(-\\frac{c t^{2}}{\\sigma^{2}(n+\\|\\widetilde{\\boldsymbol{Z}}^{k}-\\boldsymbol{X}\\|_{2,\\infty}^{2})\\|\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{1}^{2}}\\Big),\\tag{S38}\n$$\n\ndue to Assumption 3.1, and\n\n$$\n\\begin{array}{l}\\|\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{S}}_{k}\\widehat{\\boldsymbol{V}}_{k}^{\\top}\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{2}=\\|(\\widetilde{\\boldsymbol{Z}}^{k}-\\boldsymbol{X})\\tilde{\\boldsymbol{\\beta}}^{*}+\\boldsymbol{X}\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{2}\\leq\\|(\\widetilde{\\boldsymbol{Z}}^{k}-\\boldsymbol{X})\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{2}+\\|\\boldsymbol{X}\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{2}\\\\ \\leq\\left(\\|\\widetilde{\\boldsymbol{Z}}^{k}-\\boldsymbol{X}\\|_{2,\\infty}+\\|\\boldsymbol{X}\\|_{2,\\infty}\\right)\\|\\tilde{\\boldsymbol{\\beta}}^{*}\\|_{1}.\\\\ \\end{array}\n$$"}, {"n": 6, "claim": "Finally, using Lemma 17 and (S36), it follows that for any $t > 0$\n\n$$\n\\mathbb{P}\\left(\\langle\\widehat{\\boldsymbol{U}}_{k}\\widehat{\\boldsymbol{U}}_{k}^{\\top}\\boldsymbol{\\varepsilon},\\boldsymbol{\\varepsilon}\\rangle\\geq\\sigma^{2}k+t\\right)\\leq\\exp\\Big(-c\\min\\left(\\frac{t^{2}}{k\\sigma^{4}},\\frac{t}{\\sigma^{2}}\\right)\\Big),\\tag{S39}\n$$\n\nsince $\\widehat{U}_{k}\\widehat{U}_{k}^{\\top}$ is a projection matrix and by Assumption 3.1."}, {"n": 7, "claim": "From (S34), (S37), (S38), and (S39), we conclude that w.p. at least $1-O(1/(np)^{10})$\n\n$$\n\\langle\\widetilde{\\boldsymbol{Z}}^{k}(\\widehat{\\boldsymbol{\\beta}}-\\widetilde{\\boldsymbol{\\beta}}^{*}),\\boldsymbol{\\varepsilon}\\rangle\\leq\\sigma^{2}k+C\\sigma\\sqrt{\\log(n p)}\\left(\\sigma\\sqrt{k}+\\sigma\\sqrt{\\log(n p)}+\\|\\widetilde{\\boldsymbol{\\beta}}^{*}\\|_{1}(\\sqrt{n}+\\|\\widetilde{\\boldsymbol{Z}}^{k}-\\boldsymbol{X}\\|_{2,\\infty})\\right).\n$$\n\nThis completes the proof of Lemma 5."}]}}], "target_local_id": null, "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "jmlr-2025-23-1104", "context": "In this paper, we consider the following bilevel optimization problem:\n\n$$\n\\min_{x\\in\\mathbb{R}^{d_{x}}}\\varphi(x):=f(x,y^{*}(x)),\\quad \\text{where} y^{*}(x)=\\arg\\min_{y\\in\\mathbb{R}^{d_{y}}}g(x,y).\\tag{1}\n$$\n\nWe call f the upper-level problem, $g$ the lower-level problem, and $\\varphi$ the hyper-objective. This formulation covers many applications, including but not limited to hyperparameter tuning (Franceschi et al., 2018; Pedregosa, 2016; Bae and Grosse, 2020; Mackay et al., 2018), neural architecture search (Liu et al., 2019; Wang et al., 2022b; Zoph and Le, 2016; Zhang et al., 2021), meta-learning (Finn et al., 2017; Fallah et al., 2020; Andrychowicz et al., 2016; Mishchenko et al., 2023; Rajeswaran et al., 2019; Chandra et al., 2022; Ji et al., 2022; Zhou et al., 2019), adversarial training (Zhang et al., 2022; Br¨uckner and Schefer, 2011; Wang et al., 2021; Bishop et al., 2020; Wang et al., 2022a; Robey et al., 2023) and data hyper-cleaning (Gao et al., 2022; Zhou et al., 2022; Ren et al., 2018; Yong et al., 2022; Shu et al., 2019; Li et al., 2022).\n\nHere $g^{*}(x)=g(x,y^{*}(x))$ is usually called the value-function. The Lagrangian with a multiplier $\\lambda > 0$ for this inequality-constrained problem takes the form of:\n\n$$\n\\mathcal{L}_{\\lambda}(x,y):=f(x,y)+\\lambda\\left(g(x,y)-g^{*}(x)\\right).\\tag{4}\n$$\n\nKwon et al. (2023) further defines the following auxiliary function:\n\n$$\n\\mathcal{L}_{\\lambda}^{*}(x):=\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x)),\\text{where} y_{\\lambda}^{*}(x)=\\arg\\min_{y\\in\\mathbb{R}^{d_{y}}}\\mathcal{L}_{\\lambda}(x,y),\\tag{5}\n$$\n\nand showed that $\\mathcal{L}_{\\lambda}^{*}(x)$ is a good proxy of $\\varphi(x)$ with\n\n$$\n\\|\\nabla\\mathcal{L}_{\\lambda}^{*}(x)-\\nabla\\varphi(x)\\|=\\mathcal{O}\\left(\\kappa^{3}/\\lambda\\right).\n$$\n\nNotations. We use notations $\\mathcal{O}(\\cdot),\\tilde{\\mathcal{O}}(\\cdot),\\Omega(\\cdot),\\mathit{\\Theta}\\asymp\\mathrm{as}$ follows: given two functions $p :$ : $\\mathbb{R}^{+}\\to\\mathbb{R}^{+}$ and $q:\\mathbb{R}^{+}\\mathbb{R}^{+},p(x)=\\mathcal{O}(q(x))$ means lim $\\begin{array}{ r }{\\operatorname*{sup}_{x\\to+\\infty}p(x)/q(x)< +\\infty;p(x)=}\\end{array}$ $\\tilde{\\mathcal{O}}(\\boldsymbol{p}(\\boldsymbol{x}))$ means there exists some positive integer $k \\geq 0$ such that $p(x)=\\mathcal{O}(q(x)\\log^{k}(q(x)))$ $p(x)=\\Omega(q(x))$ means lim $\\textstyle\\operatorname*{sup}_{x\\to+\\infty}p(x)/q(x)>0$ , and $p(x)\\asymp q(x)$ means we both have $p(x)=\\mathcal{O}(q(x))$ and $p(x)=\\Omega(q(x))$ . We use $I_{d}\\in\\mathbb{R}^{d\\times d}$ to denote a d=dimensional identity matrix. For two symmetric matrices A and $B,$ , we use $A \\succeq B$ to indicate that $A - B$ is positive semidefinite. We use $\\mathbb{B}(r)$ to denote the Euclidean ball centered at the origin and radius r. For a function $h:\\mathbb{R}^{d}\\mathbb{R}$ , we use $\\nabla h\\in\\mathbb{R}^{d},\\nabla^{2}h\\in\\mathbb{R}^{d\\times d},\\nabla^{3}h\\in\\mathbb{R}^{d\\times d\\times d}$ to denote its gradient, Hessian, and third-order derivative, and use $h^{*}$ to denote the global minimum of $h(\\cdot)$ . We denote $\\| \\cdot \\|$ to be the operator norm of a tensor and more details about the notations of tensors can be found in Appendix A.\n\nAssumption 3.1 Suppose that\n\na. $g(x,y)$ is µ-strongly convex in $y ;$  \nb. $g(x,y)$ is $L_{g}$ -gradient Lipschitz;  \nc. $g(x,y)$ is $\\rho_{g}$ -Hessian Lipschitz;  \nd. $f(x,y)$ is C<sub>f</sub>-Lipschitz in $y ;$\n\ne. $f(x,y)$ is L<sub>f</sub>-gradient Lipschitz;  \n$f.~f(x,y)$ is two-times continuous diferentiable;  \ng. $\\varphi(x)$ is lower bounded, i.e. i $\\operatorname{if}_{\\substack{x\\in\\mathbb{R}^{d_{x}}}}\\varphi(x)> -\\infty;$\n\nDefinition 3.1 Under Assumption 3.1, we define the largest smoothness constant $\\ell : =$ max $\\{C_{f},L_{f},L_{g},\\rho_{g}\\}$ and the condition number $\\kappa : = \\ell / \\mu$\n\nAssumption 3.2 Suppose that\n\na. $f(x,y)$ is three-times continuous diferentiable;  \nb. $f(x,y)$ is ρ<sub>f</sub>-Hessian Lipschitz;\n\nc. $g(x,y)$ is $\\nu_{g}$ -third-order derivative Lipschitz.\n\nDefinition 3.3 Under Assumption 3.1 and 3.2, we define the largest smoothness constant $\\ell: =\\operatorname*{max}\\{C_{f},L_{f},L_{g},\\rho_{g},\\rho_{f},\\nu_{g}\\}$ and the condition number $\\kappa : = \\ell / \\mu$\n\nWe follow the notations of tensors used in Kolda and Bader (2009). For a three-way tensor $\\mathcal{X}\\in\\mathbb{R}^{d_{1}\\times d_{2}\\times d_{3}}$ , we use $[\\mathcal{X}]_{i_{1},i_{2},i_{3}}$ to represent its $(i_{1},i_{2},i_{3})$ -th element. The inner product of two three-way tensors $x,y$ is defined by $\\begin{array}{ r }{\\langle\\mathcal{X},\\mathcal{Y}\\rangle: =\\sum_{i_{1},i_{2},i_{3}}[\\mathcal{X}]_{i_{1},i_{2},i_{3}}\\cdot [\\mathcal{V}]_{i_{1},i_{2},i_{3}}}\\end{array}$ . The operator norm of three-way tensor $\\mathcal{X}\\in\\mathbb{R}^{d_{1}\\times d_{2}\\times d_{3}}$ is defined by $\\|{\\mathcal{X}}\\|: =\\operatorname*{sup}_{\\|x_{i}\\|=1}\\langle{\\mathcal{X}},x_{1}\\circ x_{2}\\circ x_{3}\\rangle$ ， where the elements of $x_{1}\\circ x_{2}\\circ x_{3}\\in\\mathbb{R}^{d_{1}\\times d_{2}\\times d_{3}}$ is $[x_{1}\\circ x_{2}\\circ x_{3}]_{i_{1},i_{2},i_{3}}: =[x_{1}]_{i_{1}}\\cdot [x_{2}]_{i_{2}}\\cdot [x_{3}]_{i_{3}}$ It can be verified that such a definition generalizes the Euclidean norm of a vector and the spectral norm of a matrix to tensors. For a three-way tensor $\\mathcal{X}\\in\\mathbb{R}^{d_{1}\\times d_{2}\\times d_{3}}$ and a vector $v\\in\\mathbb{R}^{d_{1}}$ , their mode-1 product, denoted by $\\mathcal{X}\\bar{\\times}_{1}v$ , gives a matrix in $\\mathbb{R}^{d_{2}\\times d_{3}}$ with elements $\\begin{array}{ r }{[\\mathcal{X}\\breve{\\times}_{1}v]_{i_{2},i_{3}}: =\\sum_{i_{1}}[\\mathcal{X}]_{i_{1},i_{2},i_{3}}\\cdot [v]_{i_{1}}}\\end{array}$ . We define $\\bar{\\mathsf{x}}_{2}$ and ${\\bar{\\mathsf{x}}}_{3}$ in a similar way, and it can be verified that $\\|{\\mathcal{X}}{\\bar{\\times}}_{i}v\\|\\leq\\|{\\mathcal{X}}\\|\\|v\\|$ . For a three-way tensor $\\mathcal{X}\\in\\mathbb{R}^{d_{1}\\times d_{2}\\times d_{3}}$ and a matrix $A\\in\\mathbb{R}^{d_{1}^{\\prime}\\times d_{1}}$ , their mode-1 product, denoted by ${\\mathcal{X}}\\times_{1}A$ , gives a tensor in $\\mathbb{R}^{d_{1}^{\\prime}\\times d_{2}\\times d_{3}}$ with elements $\\begin{array}{ r }{[\\mathcal{X}\\times_{1}A]_{i_{1}^{\\prime},i_{2},i_{3}}: =\\sum_{i_{1}}[\\mathcal{X}]_{i_{1},i_{2},i_{3}}\\cdot [A]_{i_{1}^{\\prime},i_{1}}}\\end{array}$ . We define $\\times_{2}$ and $\\times_{3}$ in a similar way, and it can also be verified that $\\left\\|{\\mathcal{X}}\\times_{i}A\\right\\|\\leq\\left\\|{\\dot{\\mathcal{X}}}\\right\\|\\left\\|A\\right\\|$\n\nFor a function $h(x,y):\\mathbb{R}^{d_{x}}\\times\\mathbb{R}^{d_{y}}\\mathbb{R}$ , we denote $\\nabla_{x}h(x,y)\\in\\mathbb{R}^{d_{x}}$ to be the partial gradient with respect to $x,$ with elements given by $[\\nabla_{x}h(x,y)]_{i}~: =~\\partial f(x,y)/\\partial [x]_{i}$ And we define $\\nabla_{y}h(x,y)~\\in~\\mathbb{R}^{d_{y}}$ in a similar way. We denote $\\nabla_{x x}^{2}h(x,y)\\in\\mathbb R^{d_{x}}\\times\\mathbb R^{d_{x}}$ to be the partial Hessian with respect to $x,$ , with elements given by $[\\nabla_{x x}^{2}h(x,y)]_{i_{1},i_{2}}: =$ $\\partial^{2}h(x,y)/(\\partial [x]_{i_{1}}\\partial [x]_{i_{2}})$ . And we define $\\nabla_{x y}^{2}h(x,y)\\in\\mathbb{R}^{d_{x}}\\times\\mathbb{R}^{d_{y}},\\nabla_{y x}^{2}h(x,y)\\overset{\\mathrm{\\scriptsize~\\cdot~}}{\\in}\\mathbb{R}^{d_{y}}\\times\\mathbb{R}^{\\bar{d}_{x}}$ and $\\nabla_{y y}^{2}h(x,y)\\in\\mathbb{R}^{d_{y}}\\times\\mathbb{R}^{d_{y}}$ in a similar way. We denote $\\nabla_{x x x}^{3}h(x,y)\\in\\bar{\\mathbb{R}}^{d_{x}}\\times\\mathbb{R}^{d_{x}}\\times\\mathbb{R}^{d_{x}}$ to be the partial third-order derivative with respect to $x,$ , with elements given by $[\\nabla_{x x x}^{3}h(x,y)]_{i_{1},i_{2},i_{3}}: =$ $\\partial^{3}h(x,y)/(\\partial [x]_{i_{1}}\\partial [x]_{i_{2}}\\partial [x]_{i_{3}})$ . And we define $\\nabla_{x x y}^{3}h(x,y)\\in\\mathbb{R}^{d_{x}}\\times\\mathbb{R}^{d_{x}}\\times\\mathbb{R}^{d_{y}},\\nabla_{x y x}^{3}h(x,y)\\in$ $\\begin{array}{ r }{\\mathbb{R}^{d_{x}}\\times\\mathbb{R}^{d_{y}}\\times\\mathbb{R}^{d_{x}},\\nabla_{x y u}^{3}h(x,y)\\in\\mathbb{R}^{d_{x}}\\times\\mathbb{R}^{d_{y}}\\times\\mathbb{R}^{d_{y}},\\nabla_{y y u}^{3}h(x,y)\\in\\mathbb{R}^{d_{y}}\\times\\mathbb{R}^{d_{y}}\\times\\mathbb{R}^{d_{y}},\\nabla_{y y x}^{3}h(x,y)\\in\\mathbb{R}^{d_{z}}.}\\end{array}$ $\\mathbb{R}^{d_{y}}\\times\\mathbb{R}^{d_{y}}\\times\\mathbb{R}^{d_{x}},\\mathring{\\nabla}_{u x u}^{3}h(x,y)\\in\\mathbb{R}^{d_{y}}\\times\\mathbb{R}^{d_{x}}\\times\\mathbb{R}^{\\tilde{d_{y}}}$ and $\\nabla_{y x x}^{3}h(x,y)\\in\\mathbb{R}^{d_{y}}\\times\\mathbb{R}^{d_{x}}\\times\\mathbb{R}^{d_{x}}$ in a similar way. We denote $\\nabla y^{*}(x)\\in\\mathbb{R}^{d_{x}}\\times\\mathbb{R}^{d_{y}}$ to be the matrix with elements given by $[\\nabla y^{*}(x)]_{i_{1},i_{2}}: =\\partial [y^{*}(x)]_{i_{2}}/\\partial [x]_{i_{1}}$ . We denote $\\nabla^{2}y^{*}(x)\\in\\mathbb{R}^{d_{x}}\\times\\mathbb{R}^{d_{x}}\\times\\mathbb{R}^{d_{y}}$ to be the threeway tensor with elements given by $[\\nabla^{2}y^{*}(x)]_{i_{1},i_{2},i_{3}}: =\\partial^{2}[y^{*}(x)]_{i_{3}}/(\\partial [x]_{i_{1}}\\partial [x]_{i_{2}})$ . And we define $\\nabla y_{\\lambda}^{\\ast}(x)$ and $\\nabla^{2}y_{\\lambda}^{*}(x)$ in a similar way.\n\nLemma B.2 Under Assumption 3.1, for $\\lambda\\ge 2 L_{f}/\\mu$ , it holds that\n\n$$\n\\|y_{\\lambda}^{*}(x)-y^{*}(x)\\|\\leq\\frac{C_{f}}{\\lambda\\mu}\n$$\n\nLemma B.5 Under Assumption 3.1, for $\\lambda\\geq 2 L_{f}/\\mu$ , it holds that $\\|\\nabla y^{*}(x)-\\nabla y_{\\lambda}^{*}(x)\\|\\leq$ $D_{2}/\\lambda$ , where\n\n$$\nD_{2}:=\\left(\\frac{1}{\\mu}+\\frac{2 L_{g}}{\\mu^{2}}\\right)\\left(L_{f}+\\frac{C_{f}\\rho_{g}}{\\mu}\\right)=\\mathcal{O}(\\kappa^{3}).\n$$\n\nRearranging, we can obtain\n\n$$\n\\nabla y^{*}(x)= -\\nabla_{x y}^{2}g(x,y^{*}(x))[\\nabla_{y y}^{2}g(x,y^{*}(x))]^{-1}.\\tag{16}\n$$\n\nSimilarly, we also have\n\n$$\n\\nabla y_{\\lambda}^{*}(x)= -\\nabla_{x y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))[\\nabla_{y y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))]^{-1}.\\tag{17}\n$$\n\nUsing the matrix identity $A^{-1}-B^{-1}=A^{-1}(B-A)B^{-1}$ , we have\n\n$$\n\\begin{array}{l}\\left\\|\\left[\\nabla_{y y}^{2}g(x,y^{*}(x))\\right]^{-1}-\\left[\\frac{\\nabla_{y y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))}{\\lambda}\\right]^{-1}\\right\\|\\\\ \\leq\\left\\|\\left[\\nabla_{y y}^{2}g(x,y_{\\lambda}^{*}(x))\\right]^{-1}\\right\\|\\left\\|\\frac{\\nabla_{y y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))}{\\lambda}-\\nabla_{y y}^{2}g(x,y^{*}(x))\\right\\|\\left\\|\\left[\\frac{\\nabla_{y y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))}{\\lambda}\\right]^{-1}\\right\\|\\\\ \\leq\\frac{2}{\\mu^{2}}\\left\\|\\frac{\\nabla_{y y}^{2}f(x,y_{\\lambda}^{*}(x))}{\\lambda}+\\nabla_{y y}^{2}g(x,y_{\\lambda}^{*}(x))-\\nabla_{y y}^{2}g(x,y^{*}(x))\\right\\|\\\\ \\leq\\frac{2}{\\mu^{2}}\\left(\\frac{L_{f}}{\\lambda}+\\rho_{g}\\|y_{\\lambda}^{*}(x)-y^{*}(x)\\|\\right)\\\\ \\leq\\frac{2}{\\lambda\\mu^{2}}\\left(L_{f}+\\frac{C_{f}\\rho_{g}}{\\mu}\\right).\\tag{18}\\\\ \\end{array}\n$$\n\nLemma B.6 Under Assumption ${\\it 3.1,}$ for $\\lambda\\ge 2 L_{f}/\\mu$ , it holds that $\\|\\nabla y_{\\lambda}^{*}(x)\\|\\leq 4 L_{g}/\\mu$\n\nLemma B.7 Under Assumption 3.1, for $\\lambda\\ge 2 L_{f}/\\mu,\\nabla\\mathcal{L}_{\\lambda}^{\\ast}(x)$ is $D_{3}{-}L i p s c h i t z$ , where\n\n$$\nD_{3}:=L_{f}+\\frac{4 L_{f}L_{g}}{\\mu}+\\frac{C_{f}\\rho_{g}}{\\mu}+\\frac{C_{f}L_{g}\\rho_{g}}{\\mu^{2}}+L_{g}D_{2}=\\mathcal{O}(\\kappa^{3}).\n$$", "theorems": [{"local_id": "T1", "statement": "Lemma C.1 Under Assumption 3.1 and 3.2, for $\\lambda\\ge 2 L_{f}/\\mu$ , we have $\\|\\nabla^{2}y^{*}(x)-\\nabla^{2}y_{\\lambda}^{*}(x)\\|\\le$ $D_{4}/\\lambda$ , where\n\n$$\nD_{4}:=\\frac{2\\rho_{g}}{\\mu^{2}}\\left(1+\\frac{L_{g}}{\\mu}\\right)^{2}\\left(L_{f}+\\frac{C_{f}\\rho_{g}}{\\mu}\\right)+\\frac{14L_{g}\\rho_{g}D_{2}}{\\mu^{2}}+\\frac{50L_{g}^{2}}{\\mu^{3}}\\left(\\frac{C_{f}\\nu_{g}}{\\mu}+\\rho_{f}\\right)=\\mathcal{O}(\\kappa^{5}).\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof First of all, we calculate the explicit form of $\\nabla^{2}y^{*}(x)$ and $\\nabla^{2}y_{\\lambda}^{*}(x)$\n\nBy taking the derivative with respect to x on\n\n$$\n\\nabla_{x y}^{2}g(x,y^{*}(x))+\\nabla y^{*}(x)\\nabla_{y y}^{2}g(x,y^{*}(x))=0,\n$$\n\nwe obtain\n\n$$\n\\begin{array}{l}\\nabla_{x x y}^{3}g(x,y^{*}(x))+\\nabla_{y x y}^{3}g(x,y^{*}(x))\\times_{1}\\nabla y^{*}(x)+\\nabla^{2}y^{*}(x)\\times_{3}\\nabla_{y y}^{2}g(x,y^{*}(x))\\\\ +\\nabla_{x y y}^{3}g(x,y^{*}(x))\\times_{2}\\nabla y^{*}(x)+\\nabla_{y y y}^{3}g(x,y^{*}(x))\\times_{1}\\nabla y^{*}(x)\\times_{2}\\nabla y^{*}(x)=0.\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "Rearranging to get\n\n$$\n\\begin{array}{l}\\nabla^{2}y^{*}(x)= -\\left(\\nabla_{x x y}^{3}g(x,y^{*}(x))+\\nabla_{y x y}^{3}g(x,y^{*}(x))\\times_{1}\\nabla y^{*}(x)\\right)\\times_{3}\\left[\\nabla_{y y}^{2}g(x,y^{*}(x))\\right]^{-1}\\\\ -\\nabla_{x y y}^{3}g\\left(x,y^{*}(x)\\right)\\times_{2}\\nabla y^{*}(x)\\times_{3}\\left[\\nabla_{y y}^{2}g\\left(x,y^{*}(x)\\right)\\right]^{-1}\\tag{21}\\\\ -\\nabla_{y y y}^{3}g(x,y^{*}(x))\\times_{1}\\nabla y^{*}(x)\\times_{2}\\nabla y^{*}(x)\\times_{3}[\\nabla_{y y}^{2}g(x,y^{*}(x))]^{-1}.\\\\ \\end{array}\n$$"}, {"n": 3, "claim": "Similarly,\n\n$$\n\\begin{array}{l}\\nabla^{2}y_{\\lambda}^{*}(x)= -\\left(\\nabla_{x x y}^{3}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))+\\nabla_{y x y}^{3}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))\\times_{1}\\nabla y_{\\lambda}^{*}(x)\\right)\\times_{3}\\left[\\nabla_{y y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))\\right]^{-1}\\\\ -\\nabla_{x y y}^{3}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))\\times_{2}\\nabla y_{\\lambda}^{*}(x)\\times_{3}[\\nabla_{y y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))]^{-1}\\\\ -\\nabla_{y y y}^{3}\\mathcal{L}_{\\lambda}\\left(x,y_{\\lambda}^{*}(x)\\right)\\times_{1}\\nabla y_{\\lambda}^{*}(x)\\times_{2}\\nabla y_{\\lambda}^{*}(x)\\times_{3}\\left[\\nabla_{y y}^{2}\\mathcal{L}_{\\lambda}\\left(x,y_{\\lambda}^{*}(x)\\right)\\right]^{-1}.\\tag{22}\\\\ \\end{array}\n$$"}, {"n": 4, "claim": "We then prove $\\|\\nabla^{2}y^{*}(x)-\\nabla^{2}y_{\\lambda}^{*}(x)\\|=\\mathcal{O}(1/\\lambda)$ . We prove this by showing that the diference between each corresponding term of $\\nabla^{2}y^{*}(x)$ and $\\nabla^{2}y_{\\lambda}^{*}(x)$ is $\\mathcal{O}(1/\\lambda)$ . Note that\n\n$$\n\\left\\|\\nabla_{x x y}^{3}g(x,y^{*}(x))-\\frac{\\nabla_{x x y}^{3}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))}{\\lambda}\\right\\|\\leq\\nu_{g}\\|y_{\\lambda}^{*}(x)-y^{*}(x)\\|+\\frac{\\rho_{f}}{\\lambda}\\leq\\frac{1}{\\lambda}\\left(\\frac{C_{f}\\nu_{g}}{\\mu}+\\rho_{f}\\right),\n$$\n\nand\n\n$$\n\\begin{array}{l}\\left\\|\\nabla_{y x y}^{3}g(x,y^{*}(x))\\times_{1}\\nabla y^{*}(x)-\\frac{\\nabla_{y x y}^{3}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))\\times_{1}\\nabla y_{\\lambda}^{*}(x)}{\\lambda}\\right\\|\\\\ \\leq\\|\\nabla y^{*}(x)-\\nabla y_{\\lambda}^{*}(x)\\|\\|\\nabla_{y x y}^{3}g(x,y^{*}(x))\\|+\\|\\nabla y_{\\lambda}^{*}(x)\\|\\left\\|\\nabla_{y x y}^{3}g(x,y^{*}(x))-\\frac{\\nabla_{y x y}^{3}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))}{\\lambda}\\right\\|\\\\ \\leq\\frac{\\rho_{g}D_{2}}{\\lambda}+\\frac{4 L_{g}}{\\lambda\\mu}\\left(\\frac{C_{f}\\nu_{g}}{\\mu}+\\rho_{f}\\right),\\\\ \\end{array}\n$$\n\nand\n\n$$\n\\begin{array}{l}\\left\\|\\nabla_{yyy}^{3}g(x,y^{*}(x))\\times_{1}\\nabla y^{*}(x)\\times_{2}\\nabla y^{*}(x)-\\frac{\\nabla_{yyy}^{3}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))\\times_{1}\\nabla y_{\\lambda}^{*}(x)\\times_{2}\\nabla y_{\\lambda}^{*}(x)}{\\lambda}\\right\\|\\\\ \\leq\\|\\nabla y^{*}(x)\\|\\|\\nabla_{yyy}^{3}g(x,y^{*}(x))\\|\\|\\nabla y^{*}(x)-\\nabla y_{\\lambda}^{*}(x)\\|\\\\ +\\|\\nabla y_{\\lambda}^{*}(x)\\|\\|\\nabla_{yyy}^{3}g(x,y^{*}(x))\\|\\|\\nabla y^{*}(x)-\\nabla y_{\\lambda}^{*}(x)\\|\\\\ +\\left\\|\\nabla y_{\\lambda}^{*}(x)\\right\\|^{2}\\left\\|\\nabla_{yyy}^{3}g(x,y^{*}(x))-\\frac{\\nabla_{yyy}^{3}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))}{\\lambda}\\right\\|\\\\ \\leq\\frac{5L_{g}\\rho_{g}D_{2}}{\\lambda\\mu}+\\frac{16L_{g}^{2}}{\\lambda\\mu^{2}}\\left(\\frac{C_{f}\\nu_{g}}{\\mu}+\\rho_{f}\\right),\\\\ \\end{array}\n$$\n\nwe can obtain that\n\n$$\n\\begin{array}{l}\\|\\nabla^{2}y^{*}(x)-\\nabla^{2}y_{\\lambda}^{*}(x)\\|\\\\ \\leq\\rho_{g}\\left(1+\\frac{L_{g}}{\\mu}\\right)^{2}\\left\\|\\left[\\nabla_{yy}^{2}g\\left(x,y^{*}(x)\\right)\\right]^{-1}-\\left[\\frac{\\nabla_{yy}^{2}\\mathcal{L}_{\\lambda}\\left(x,y_{\\lambda}^{*}(x)\\right)}{\\lambda}\\right]^{-1}\\right\\|\\\\ +\\left(\\frac{7L_{g}\\rho_{g}D_{2}}{\\lambda\\mu}+\\frac{25L_{g}^{2}}{\\lambda\\mu^{2}}\\left(\\frac{C_{f}\\nu_{g}}{\\mu}+\\rho_{f}\\right)\\right)\\left\\|\\left[\\frac{\\nabla_{yy}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))}{\\lambda}\\right]^{-1}\\right\\|\\\\ \\leq\\frac{2\\rho_{g}}{\\lambda\\mu^{2}}\\left(1+\\frac{L_{g}}{\\mu}\\right)^{2}\\left(L_{f}+\\frac{C_{f}\\rho_{g}}{\\mu}\\right)+\\frac{14L_{g}\\rho_{g}D_{2}}{\\lambda\\mu^{2}}+\\frac{50L_{g}^{2}}{\\lambda\\mu^{3}}\\left(\\frac{C_{f}\\nu_{g}}{\\mu}+\\rho_{f}\\right),\\\\ \\end{array}\n$$\n\nwhere we use Equation 18 in the second inequality."}]}}, {"local_id": "T2", "statement": "Lemma C.2 Under Assumption 3.1 and 3.2, for $\\lambda\\ge 2 L_{f}/\\mu,\\nabla^{2}{\\mathcal{L}}_{\\lambda}^{*}(x)$ is $D_{5}– L i p s c h i t z .$ where\n\n$$\nD_{5}:=\\left(1+\\frac{4 L_{g}}{\\mu}\\right)^{2}\\left(3\\rho_{f}+\\frac{2 L_{f}\\rho_{g}}{\\mu}\\right)+\\left(1+\\frac{L_{g}}{\\mu}\\right)^{2}\\frac{C_{f}\\nu_{g}}{\\mu}\n$$\n\n$$\n+\\left(2+\\frac{5 L_{g}}{\\mu}\\right)D_{2}\\rho_{g}+\\left(1+\\frac{L_{g}}{\\mu}\\right)^{2}\\frac{C_{f}\\rho_{g}^{2}}{\\mu^{2}}+L_{g}D_{4}=\\mathcal{O}(\\ell\\kappa^{5}).\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof Similar to the proof of Lemma B.7, we split $\\nabla^{2}\\mathcal{L}_{\\lambda}^{*}(x)$ into two terms:\n\n$$\n\\nabla^{2}\\mathcal{L}_{\\lambda}^{*}(x)=\\nabla A(x)+\\nabla B(x),\n$$\n\nwhere both the mappings $A(x)$ and $B(x)$ both follow the definitions in Equation 19. rTaking total derivative on $A(x)$ , we obtain\n\n$$\n\\nabla A(x)=\\nabla_{x x}^{2}f(x,y_{\\lambda}^{*}(x))+\\nabla y_{\\lambda}^{*}(x)\\nabla_{y x}^{2}f(x,y_{\\lambda}^{*}(x)).\n$$"}, {"n": 2, "claim": "And recall Equation 20 that\n\n$$\n\\begin{array}{l}\\nabla B(x)=\\lambda(\\nabla_{x x}^{2}g(x,y_{\\lambda}^{*}(x))-\\nabla_{x x}^{2}g(x,y^{*}(x)))\\\\ +\\lambda\\left(\\nabla y_{\\lambda}^{*}(x)\\nabla_{y x}^{2}g(x,y_{\\lambda}^{*}(x))-\\nabla y^{*}(x)\\nabla_{y x}^{2}g(x,y^{*}(x))\\right).\\\\ \\end{array}\n$$"}, {"n": 3, "claim": "Then we bound the Lipschitz coeficient of $\\nabla A(x)$ and $\\nabla B(x)$ , respectively."}, {"n": 4, "claim": "From Equation 21 we can calculate that\n\n$$\n\\|\\nabla^{2}y^{*}(x)\\|\\leq\\left(1+\\frac{L_{g}}{\\mu}\\right)^{2}\\frac{\\rho_{g}}{\\mu}.\\tag{23}\n$$"}, {"n": 5, "claim": "Similarly, from Equation 22 we can calculate that\n\n$$\n\\|\\nabla^{2}y_{\\lambda}^{*}(x)\\|\\leq\\left(1+\\frac{4 L_{g}}{\\mu}\\right)^{2}\\left(\\frac{\\rho_{f}}{\\lambda}+\\rho_{g}\\right)\\frac{2}{\\mu}\\leq\\left(1+\\frac{4 L_{g}}{\\mu}\\right)^{2}\\left(\\frac{2\\rho_{f}}{L_{f}}+\\frac{2\\rho_{g}}{\\mu}\\right).\\tag{24}\n$$"}, {"n": 6, "claim": "From Lemma B.6 we know that $y_{\\lambda}^{*}(x)$ is $(4 L_{g}/\\mu)$ -Lipschitz. This further implies that both $\\nabla_{x x}^{2}f(x,y_{\\lambda}^{*}(x))$ and $\\nabla_{y x}^{2}f(x,y_{\\lambda}^{*}(x))$ are $(1+4 L_{g}/\\mu)\\rho_{f}\\mathrm{-L i p s c h i t z}$ . Then, for any $x_{1},x_{2}\\in\\mathbb{R}^{d_{x}}$ we have\n\n$$\n\\begin{array}{l}\\|\\nabla A(x_{1})-\\nabla A(x_{2})\\|\\\\ \\leq\\left\\|\\nabla_{x x}^{2}f(x_{1},y_{\\lambda}^{*}(x_{1}))-\\nabla_{x x}^{2}f(x_{2},y_{\\lambda}^{*}(x_{2}))\\right\\|\\\\ +\\left\\|\\nabla y_{\\lambda}^{*}(x_{1})\\nabla_{y x}^{2}f(x_{1},y_{\\lambda}^{*}(x_{1}))-\\nabla y_{\\lambda}^{*}(x_{2})\\nabla_{y x}^{2}f(x_{1},y_{\\lambda}^{*}(x_{1}))\\right\\|\\\\ +\\left\\|\\nabla y_{\\lambda}^{*}(x_{2})\\nabla_{y x}^{2}f(x_{1},y_{\\lambda}^{*}(x_{1}))-\\nabla y_{\\lambda}^{*}(x_{2})\\nabla_{y x}^{2}f(x_{2},y_{\\lambda}^{*}(x_{2}))\\right\\|\\\\ \\leq\\left\\|\\nabla_{x x}^{2}f(x_{1},y_{\\lambda}^{*}(x_{1}))-\\nabla_{x x}^{2}f(x_{2},y_{\\lambda}^{*}(x_{2}))\\right\\|\\\\ +\\left\\|\\nabla y_{\\lambda}^{*}(x_{1})-\\nabla y_{\\lambda}^{*}(x_{2})\\right\\|\\left\\|\\nabla_{y x}^{2}f(x_{1},y_{\\lambda}^{*}(x_{1}))\\right\\|\\\\ +\\|\\nabla y_{\\lambda}^{*}(x_{2})\\|\\|\\nabla_{y x}^{2}f(x_{1},y_{\\lambda}^{*}(x_{1}))-\\nabla_{y x}^{2}f(x_{2},y_{\\lambda}^{*}(x_{2}))\\|\\\\ \\leq\\underbrace{\\left(\\left(1+\\frac{4 L_{g}}{\\mu}\\right)^{2}\\rho_{f}+\\left(1+\\frac{4 L_{g}}{\\mu}\\right)^{2}\\left(\\frac{2\\rho_{f}}{L_{f}}+\\frac{2\\rho_{g}}{\\mu}\\right)L_{f}\\right)}_{C_{1}}\\|x_{1}-x_{2}\\|,\\\\ \\end{array}\n$$\n\nwhere $C_{1}$ gives the upper bound of the Lipschitz coeficient of the mapping $\\nabla A(x)$\n\nTo bound the Lipschitz coeficient of $\\nabla B(x)$ , we first derive the explicit form of the mapping $\\nabla^{2}B(x):\\mathbb{R}^{d_{x}}\\mathbb{R}^{d_{x}}\\times\\mathbb{R}^{d_{x}}\\times\\mathbb{R}^{d_{x}}$ by:\n\n$$\n\\begin{array}{l}\\nabla^{2}B(x)=\\lambda(\\nabla_{x x x}^{3}g(x,y_{\\lambda}^{*}(x))-\\nabla_{x x x}^{3}g(x,y^{*}(x)))\\\\ +\\lambda(\\nabla_{y x x}^{3}g(x,y_{\\lambda}^{*}(x))\\times_{1}\\nabla y_{\\lambda}^{*}(x)-\\nabla_{y x x}^{3}g(x,y^{*}(x))\\times_{1}\\nabla y^{*}(x))\\\\ +\\lambda\\left(\\nabla_{x y x}^{3}g(x,y_{\\lambda}^{*}(x))\\times_{2}\\nabla y_{\\lambda}^{*}(x)-\\nabla_{x y x}^{3}g(x,y^{*}(x))\\times_{2}\\nabla y^{*}(x)\\right)\\\\ +\\lambda\\left(\\nabla_{y y x}^{3}g(x,y_{\\lambda}^{*}(x))\\times_{1}\\nabla y_{\\lambda}^{*}(x)\\times_{2}\\nabla y_{\\lambda}^{*}(x)-\\nabla_{y y x}^{3}g(x,y^{*}(x))\\times_{1}\\nabla y^{*}(x)\\times_{2}\\nabla y^{*}(x)\\right)\\\\ +\\lambda\\left(\\nabla^{2}y_{\\lambda}^{*}(x)\\times_{3}[\\nabla_{y x}^{2}g(x,y_{\\lambda}^{*}(x))]^{\\top}-\\nabla^{2}y^{*}(x)\\times_{3}[\\nabla_{y x}^{2}g(x,y^{*}(x))]^{\\top}\\right).\\\\ \\end{array}\n$$"}, {"n": 7, "claim": "Then we can bound the Lipschitz coeficient of $\\nabla B(x)$ via the operator norm of $\\nabla^{2}B(x)$\n\n$$\n\\begin{array}{l}C_{2}:=\\|\\nabla^{2}B(x)\\|\\\\ \\leq\\lambda\\|\\nabla_{x x x}^{3}g(x,y^{*}(x))-\\nabla_{x x x}^{3}g(x,y_{\\lambda}^{*}(x))\\|\\\\ +\\lambda\\|\\nabla y^{*}(x)\\|\\|\\nabla_{y x x}^{3}g(x,y^{*}(x))-\\nabla_{y x x}^{3}g(x,y_{\\lambda}^{*}(x))\\|\\\\ +\\lambda\\|\\nabla y_{\\lambda}^{*}(x)-\\nabla y^{*}(x)\\|\\|\\nabla_{y x x}^{3}g(x,y_{\\lambda}^{*}(x))\\|\\\\ +\\lambda\\|\\nabla y^{*}(x)\\|\\|\\nabla_{x y x}^{3}g(x,y^{*}(x))-\\nabla_{x y x}^{3}g(x,y_{\\lambda}^{*}(x))\\|\\\\ +\\lambda\\|\\nabla y^{*}(x)-\\nabla y_{\\lambda}^{*}(x)\\|\\|\\nabla_{x y x}^{3}g(x,y_{\\lambda}^{*}(x))\\|\\\\ +\\lambda\\|\\nabla y^{*}(x)\\|\\|\\nabla_{y y x}^{3}g(x,y^{*}(x))\\|\\|\\nabla y_{\\lambda}^{*}(x)-\\nabla y^{*}(x)\\|\\\\ +\\lambda\\|\\nabla y_{\\lambda}^{*}(x)\\|\\|\\nabla_{y y x}^{3}g(x,y^{*}(x))\\|\\|\\nabla y_{\\lambda}^{*}(x)-\\nabla y^{*}(x)\\|\\\\ +\\lambda\\|\\nabla y^{*}(x)\\|^{2}\\|\\nabla_{y y x}^{3}g(x,y^{*}(x))-\\nabla_{y y x}^{3}g(x,y_{\\lambda}^{*}(x))\\|\\\\ +\\lambda\\|\\nabla^{2}y^{*}(x)\\|\\|\\nabla_{y x}^{2}g(x,y^{*}(x))-\\nabla_{y x}^{2}g(x,y_{\\lambda}^{*}(x))\\|\\\\ +\\lambda\\|\\nabla^{2}y^{*}(x)-\\nabla^{2}y_{\\lambda}^{*}(x)\\|\\|\\nabla_{y x}^{2}g(x,y_{\\lambda}^{*}(x))\\|,\\\\ \\end{array}\n$$\n\nwhich only requires using triangle inequality multiple times."}, {"n": 8, "claim": "Then we plug\n\n$$\n\\|y_{\\lambda}^{*}(x)-y^{*}(x)\\|=\\mathcal{O}(1/\\lambda),\\text{By Lemma B.2}\n$$\n\n$$\n\\|\\nabla y_{\\lambda}^{*}(x)-\\nabla y^{*}(x)\\|=\\mathcal{O}(1/\\lambda),\\text{By Lemma B.5}.\\tag{25}\n$$\n\n$$\n\\|\\nabla^{2}y_{\\lambda}^{*}(x)-\\nabla^{2}y^{*}(x)\\|=\\mathcal{O}(1/\\lambda),\\text{By Lemma C.1.}\n$$\n\nand\n\n$$\n\\|\\nabla y^{*}(x)\\|=\\mathcal{O}(1),\\text{By Equation 16.}\n$$\n\n$$\n\\|\\nabla y_{\\lambda}^{*}(x)\\|=\\mathcal{O}(1),\\text{By Lemma B.6}.\\tag{26}\n$$\n\n$$\n\\|\\nabla^{2}y^{*}(x)\\|=\\mathcal{O}(1),\\text{By Equation 23.}\n$$\n\ninto the bound for $C_{2}$ to obtain that\n\n$$\n\\begin{array}{l}C_{1}+C_{2}\\leq\\left(1+\\frac{4 L_{g}}{\\mu}\\right)^{2}\\left(3\\rho_{f}+\\frac{2 L_{f}\\rho_{g}}{\\mu}\\right)+\\left(1+\\frac{L_{g}}{\\mu}\\right)^{2}\\frac{C_{f}\\nu_{g}}{\\mu}\\\\ +\\left(2+\\frac{5 L_{g}}{\\mu}\\right)D_{2}\\rho_{g}+\\left(1+\\frac{L_{g}}{\\mu}\\right)^{2}\\frac{C_{f}\\rho_{g}^{2}}{\\mu^{2}}+L_{g}D_{4}.\\\\ \\end{array}\n$$"}]}}, {"local_id": "T3", "statement": "Lemma C.3 Under Assumption 3.1 and 3.2, for $\\lambda\\ge 2 L_{f}/\\mu_{\\ i}$ , we have $\\|\\nabla^{2}\\mathcal{L}_{\\lambda}^{*}(x)-\\nabla^{2}\\varphi(x)\\|\\le$ $D_{6}/\\lambda$ , where\n\n$$\nD_{6}:=2 L_{g}D_{2}^{2}+\\left(1+\\frac{L_{g}}{\\mu}\\right)^{2}\\left(\\frac{C_{f}\\rho_{f}}{\\mu}+\\frac{C_{f}L_{f}\\rho_{g}}{\\mu^{2}}+\\frac{C_{f}^{2}\\nu_{g}}{2\\mu^{2}}+\\frac{C_{f}^{2}\\rho_{g}^{2}}{2\\mu^{3}}\\right)=\\mathcal{O}(\\ell\\kappa^{6}).\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof Taking total derivative on $\\nabla\\varphi(x)=\\nabla_{x}f(x,y^{*}(x))+\\nabla y^{*}(x)\\nabla_{y}f(x,y^{*}(x))$ yields\n\n$$\n\\begin{array}{l}\\begin{array}{l}\\nabla^{2}\\varphi(x)=\\nabla_{x x}^{2}f\\left(x,y^{*}(x)\\right)+\\nabla y^{*}(x)\\nabla_{y x}^{2}f\\left(x,y^{*}(x)\\right)+\\nabla^{2}y^{*}(x)\\bar{\\times}_{3}\\nabla_{y}f\\left(x,y^{*}(x)\\right)\\end{array}\\tag{27}\\\\ +\\nabla_{x y}^{2}f(x,y^{*}(x))[\\nabla y^{*}(x)]^{\\top}+\\nabla y^{*}(x)\\nabla_{y y}^{2}f(x,y^{*}(x))[\\nabla y^{*}(x)]^{\\top}.\\\\ \\end{array}\n$$"}, {"n": 2, "claim": "Plug into the close form of $\\nabla^{2}y^{*}(x)$ given by Equation 21, we arrive at\n\n$$\n\\begin{array}{l}\\nabla^{2}\\varphi(x)\\\\ =\\underbrace{\\nabla_{x x}^{2}f(x,y^{*}(x))-\\nabla_{x x y}^{3}g(x,y^{*}(x))\\times_{3}\\left[\\nabla_{y y}^{2}g(x,y^{*}(x))\\right]^{-1}\\bar{\\times}_{3}\\nabla_{y}f(x,y^{*}(x))}_{\\text{(I)}}\\\\ +\\underbrace{\\left(\\nabla_{y x}^{2}f\\left(x,y^{*}(x)\\right)-\\nabla_{y x y}^{3}g\\left(x,y^{*}(x)\\right)\\times_{3}\\left[\\nabla_{y y}^{2}g\\left(x,y^{*}(x)\\right]^{-1}\\bar{\\times}_{3}\\nabla_{y}f\\left(x,y^{*}(x)\\right)\\right)\\right)\\times_{1}\\nabla y^{*}(x)}_{\\text{(II)}}\\\\ +\\underbrace{\\left(\\nabla_{x y}^{2}f\\left(x,y^{*}(x)\\right)-\\nabla_{x y y}^{3}g\\left(x,y^{*}(x)\\right)\\times_{3}\\left[\\nabla_{y y}^{2}g\\left(x,y^{*}(x)\\right]^{-1}\\bar{\\times}_{3}\\nabla_{y}f\\left(x,y^{*}(x)\\right)\\right)\\right)\\times_{2}\\nabla y^{*}(x)}_{\\text{(III)}}\\\\ +\\underbrace{\\left(\\nabla_{y y}^{2}f(x,y^{*}(x))-\\nabla_{y y y}^{3}g(x,y^{*}(x))\\times_{3}\\left[\\nabla_{y y}^{2}g(x,y^{*}(x)\\right]^{-1}\\bar{\\times}_{3}\\nabla_{y}f(x,y^{*}(x))\\right)\\times_{1}\\nabla y^{*}(x)\\times_{2}\\nabla y^{*}(x)}_{\\text{(IV)}}.\\\\ \\end{array}\n$$"}, {"n": 3, "claim": "Recall that\n\n$$\n\\begin{array}{l}\\nabla^{2}\\mathcal{L}_{\\lambda}^{*}(x)=\\nabla_{x x}^{2}f(x,y_{\\lambda}^{*}(x))+\\lambda(\\nabla_{x x}^{2}g(x,y_{\\lambda}^{*}(x))-\\nabla_{x x}^{2}g(x,y^{*}(x)))\\\\ +\\nabla y_{\\lambda}^{*}(x)\\nabla_{y x}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))-\\lambda\\nabla y^{*}(x)\\nabla_{y x}^{2}g(x,y^{*}(x)).\\\\ \\end{array}\n$$"}, {"n": 4, "claim": "Our goal is show that $\\nabla\\mathcal{L}_{\\lambda}^{*}(x)\\approx\\nabla^{2}\\varphi(x)$ . At first glance, these two quantities are very diferent and we can not directly bound their diference: $\\nabla^{2}\\varphi({\\boldsymbol{x}})$ takes the form of $A +$ $B C+C^{\\top}B^{\\top}+B D B^{\\top}$ , while $\\nabla^{2}\\mathcal{L}_{\\lambda}^{*}(x)$ looks diferent. Below, we introduce an intermediary quantity $\\tilde{\\nabla}^{2}\\mathcal{L}_{\\lambda}^{*}(x)$ which takes a similar form as $\\nabla^{2}\\varphi({\\boldsymbol{x}})$ to serves as a bridge:\n\n$$\n\\begin{array}{l}\\tilde{\\nabla}^{2}\\mathcal{L}_{\\lambda}^{*}(x)=\\underbrace{\\nabla_{x x}^{2}f(x,y_{\\lambda}^{*}(x))+\\lambda(\\nabla_{x x}^{2}g(x,y_{\\lambda}^{*}(x))-\\nabla_{x x}^{2}g(x,y^{*}(x)))}_{(I^{\\prime})}\\\\ +\\underbrace{\\nabla y^{*}(x)\\left(\\nabla_{y x}^{2}\\mathcal{L}_{\\lambda}\\left(x,y_{\\lambda}^{*}(x)\\right)-\\lambda\\nabla_{y x}^{2}g\\left(x,y^{*}(x)\\right)\\right)}_{\\left(\\Pi^{\\prime}\\right)}\\left.\\right)\\\\ +\\underbrace{\\left(\\nabla_{x y}^{2}\\mathcal{L}_{\\lambda}\\left(x,y_{\\lambda}^{*}(x)\\right)-\\lambda\\nabla_{x y}^{2}g\\left(x,y^{*}(x)\\right)\\right)\\left[\\nabla y^{*}(x)\\right]^{\\top}}_{\\text{(III')}}\\tag{28}\\\\ +\\underbrace{\\nabla y^{*}(x)\\left(\\nabla_{y y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))-\\lambda\\nabla_{y y}^{2}g(x,y^{*}(x))\\right)[\\nabla y^{*}(x)]^{\\top}}_{(\\mathrm{IV}^{\\prime})}.\\\\ \\end{array}\n$$"}, {"n": 5, "claim": "Now we show ∇<sup>2</sup>L<sup>∗</sup><sub>λ</sub>(x) ≈ ∇<sup>˜</sup> <sup>2</sup>L<sup>∗</sup><sub>λ</sub>(x) ≈ ∇<sup>2</sup>ϕ(x).\n\n$$\n\\begin{array}{l}\\tilde{\\nabla}^{2}\\mathcal{L}_{\\lambda}^{*}(x)-\\nabla^{2}\\mathcal{L}_{\\lambda}^{*}(x)\\\\ =(\\nabla y^{*}(x)-\\nabla y_{\\lambda}^{*}(x))\\nabla_{y x}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))+\\\\ +\\nabla_{x y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))[\\nabla y^{*}(x)]^{\\top}+\\nabla y^{*}(x)\\nabla_{y y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))[\\nabla y^{*}(x)]^{\\top}\\\\ =(\\nabla y^{*}(x)-\\nabla y_{\\lambda}^{*}(x))\\nabla_{y x}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))+\\nabla_{x y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))[\\nabla y^{*}(x)-\\nabla y_{\\lambda}^{*}(x)]^{\\top}\\\\ +\\nabla y^{*}(x)\\nabla_{y y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))[\\nabla y^{*}(x)]^{\\top}-\\nabla y_{\\lambda}^{*}(x)\\nabla_{y y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))[\\nabla y_{\\lambda}^{*}(x)]^{\\top}\\\\ =(\\nabla y^{*}(x)-\\nabla y_{\\lambda}^{*}(x))\\nabla_{y x}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))+\\nabla_{x y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))[\\nabla y^{*}(x)-\\nabla y_{\\lambda}^{*}(x)]^{\\top}\\\\ +(\\nabla y^{*}(x)-\\nabla y_{\\lambda}^{*}(x))\\nabla_{y y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))[\\nabla y_{\\lambda}^{*}(x)]^{\\top}\\\\ +\\nabla y_{\\lambda}^{*}(x)\\nabla_{y y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))(\\nabla y^{*}(x)-\\nabla y_{\\lambda}^{*}(x))^{\\top}\\\\ +(\\nabla y^{*}(x)-\\nabla y_{\\lambda}^{*}(x))\\nabla_{y y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))[\\nabla y^{*}(x)-\\nabla y_{\\lambda}^{*}(x)]^{\\top}\\\\ =(\\nabla y^{*}(x)-\\nabla y_{\\lambda}^{*}(x))\\nabla_{y y}^{2}\\mathcal{L}_{\\lambda}(x,y_{\\lambda}^{*}(x))[\\nabla y^{*}(x)-\\nabla y_{\\lambda}^{*}(x)]^{\\top},\\\\ \\end{array}\n$$\n\nwhere we use the identity\n\n$$\nU A U^{\\top}-V A V^{\\top}=(U-V)A V^{\\top}+V A(U-V)^{\\top}+(U-V)A(U-V)^{\\top}\n$$\n\nin the second last step, and we cancel the first four terms by Equation 17 in the final step."}, {"n": 6, "claim": "Noting that $\\nabla_{y y}^{2}\\mathcal{L}_{\\lambda}(\\cdot,\\cdot)\\preceq 2\\lambda L_{g},$ we have\n\n$$\n\\|\\tilde{\\nabla}^{2}\\mathcal{L}_{\\lambda}^{*}(x)-\\nabla^{2}\\mathcal{L}_{\\lambda}^{*}(x)\\|\\leq 2\\lambda L_{g}\\|\\nabla y^{*}(x)-\\nabla y_{\\lambda}^{*}(x)\\|^{2}\\leq\\frac{2 L_{g}D_{2}^{2}}{\\lambda}.\n$$"}, {"n": 7, "claim": "Now, we have successfully simplified our goal to showing $\\tilde{\\nabla}^{2}\\mathcal{L}_{\\lambda}^{*}(x)\\approx\\nabla^{2}\\varphi(x)$ , which can be done by showing $(\\mathrm{I})\\approx(\\mathrm{I^{\\prime}}),(\\mathrm{II})\\approx(\\mathrm{II^{\\prime}}),(\\mathrm{III})\\approx(\\mathrm{III^{\\prime}})$ , and $\\mathrm{(I V)\\approx(I V^{\\prime})}$ , separately. Firstly,\n\n$$\n\\begin{array}{l}(I)-(I^{\\prime})\\\\ =\\nabla_{x x}^{2}f(x,y^{*}(x))-\\nabla_{x x}^{2}f(x,y_{\\lambda}^{*}(x))\\\\ +\\nabla_{x x y}^{3}g(x,y^{*}(x))\\times_{3}[\\nabla_{y y}^{2}g(x,y^{*}(x))]^{-1}\\bar{\\times}_{3}(\\nabla_{y}f(x,y_{\\lambda}^{*}(x))-\\nabla_{y}f(x,y^{*}(x)))\\\\ \\left.+\\lambda\\left(\\nabla_{x x}^{2}g(x,y^{*}(x))-\\nabla_{x x}^{2}g(x,y_{\\lambda}^{*}(x))+\\nabla_{x x y}^{3}g(x,y^{*}(x))\\bar{\\times}_{3}(y_{\\lambda}^{*}(x)-y^{*}(x))\\right)\\right)\\\\ +\\lambda\\nabla_{x x y}^{3}g(x,y^{*}(x))\\times_{3}[\\nabla_{y y}^{2}g(x,y^{*}(x))]^{-1}\\\\ \\bar{\\times}_{3}\\left(\\nabla_{y}g(x,y_{\\lambda}^{*}(x))-\\nabla_{y}g(x,y^{*}(x))-\\nabla_{y y}^{2}g(x,y^{*}(x))(y_{\\lambda}^{*}(x)-y^{*}(x))\\right).\\\\ \\end{array}\n$$"}, {"n": 8, "claim": "Therefore,\n\n$$\n\\begin{array}{l}\\|(\\mathrm{I})-(\\mathrm{I}^{\\prime})\\|\\leq\\rho_{f}\\|y^{*}(x)-y_{\\lambda}^{*}(x)\\|+\\frac{L_{f}\\rho_{g}}{\\mu}\\|y^{*}(x)-y_{\\lambda}^{*}(x)\\|\\\\ +\\frac{\\lambda\\nu_{g}}{2}\\|y^{*}(x)-y_{\\lambda}^{*}(x)\\|^{2}+\\frac{\\lambda\\rho_{g}^{2}}{2\\mu}\\|y^{*}(x)-y_{\\lambda}^{*}(x)\\|^{2}\\\\ \\leq\\frac{C_{f}\\rho_{f}}{\\lambda\\mu}+\\frac{C_{f}L_{f}\\rho_{g}}{\\lambda\\mu^{2}}+\\frac{C_{f}^{2}\\nu_{g}}{2\\lambda\\mu^{2}}+\\frac{C_{f}^{2}\\rho_{g}^{2}}{2\\lambda\\mu^{3}}.\\\\ \\end{array}\n$$"}, {"n": 9, "claim": "Using $\\|\\nabla y^{*}(x)\\|\\le L_{g}/\\mu$ , we can similarly bound the diference between (II) and (II<sup>0</sup>) by\n\n$$\n\\begin{array}{l}\\|(\\mathrm{II})-(\\mathrm{II}^{\\prime})\\|\\\\ \\leq\\|\\nabla y^{*}(x)\\|\\Big\\|\\nabla_{y x}^{2}f(x,y^{*}(x))-\\nabla_{y x y}^{3}g(x,y^{*}(x))\\times_{3}[\\nabla_{y y}^{2}g(x,y^{*}(x)]^{-1}\\bar{\\times}_{3}\\nabla_{y}f(x,y^{*}(x))\\\\ \\left.-\\nabla_{y x}^{2}f(x,y_{\\lambda}^{*}(x))-\\lambda\\left(\\nabla_{y x}^{2}g(x,y_{\\lambda}^{*}(x))-\\nabla_{y x}^{2}g(x,y^{*}(x))\\right)\\right\\|,\\\\ \\leq\\frac{L_{g}}{\\mu}\\left(\\frac{C_{f}\\rho_{f}}{\\lambda\\mu}+\\frac{C_{f}L_{f}\\rho_{g}}{\\lambda\\mu^{2}}+\\frac{C_{f}^{2}\\nu_{g}}{2\\lambda\\mu^{2}}+\\frac{C_{f}^{2}\\rho_{g}^{2}}{2\\lambda\\mu^{3}}\\right),\\\\ \\end{array}\n$$"}, {"n": 10, "claim": "And the bound for (III) and (III<sup>0</sup>) is the same. Finally, we know that\n\n$$\n\\begin{array}{l}\\left\\|(\\mathrm{IV})-(\\mathrm{IV}^{\\prime})\\right\\|\\\\ \\leq\\|\\nabla y^{*}(x)\\|^{2}\\left\\|\\nabla_{y y}^{2}f(x,y^{*}(x))-\\nabla_{y y y}^{3}g(x,y^{*}(x))\\times_{3}[\\nabla_{y y}^{2}g(x,y^{*}(x))]^{-1}\\bar{\\times}_{3}\\nabla_{y}f(x,y^{*}(x))\\right.\\\\ \\left.-\\nabla_{y y}^{2}f(x,y_{\\lambda}^{*}(x))-\\lambda\\left(\\nabla_{y y}^{2}g(x,y_{\\lambda}^{*}(x))-\\nabla_{y y}^{2}g(x,y^{*}(x))\\right)\\right\\|,\\\\ \\leq\\frac{L_{g}^{2}}{\\mu^{2}}\\left(\\frac{C_{f}\\rho_{f}}{\\lambda\\mu}+\\frac{C_{f}L_{f}\\rho_{g}}{\\lambda\\mu^{2}}+\\frac{C_{f}^{2}\\nu_{g}}{2\\lambda\\mu^{2}}+\\frac{C_{f}^{2}\\rho_{g}^{2}}{2\\lambda\\mu^{3}}\\right).\\\\ \\end{array}\n$$"}, {"n": 11, "claim": "Combing the above bounds completes our proof."}]}}], "target_local_id": null, "topic_domain": "Optimization", "topic_subtopic": "Non-convex Geometry", "label": "correct", "error_step": null, "error_class6": [], "error_explanation": null}
{"id": "SySpa-Z0Z", "context": "Neural networks trained with dropout can be viewed as approximating a Gaussian Process (Gal & Ghahramani, 2015). We sample $\\mathbf{w}_k \\sim P(\\mathbf{w})$ and $b_k \\sim P(b)$ for $K$ hidden units in a single layer. Define $\\phi(\\mathbf{x},\\mathbf{W},\\mathbf{b}) = \\sqrt{\\frac{1}{K}}\\sigma(\\mathbf{W}\\mathbf{x}+\\mathbf{b})$, this is the activation output of a single network layer, and $\\sigma$ is a non-linear function. The training dataset contains $N$ samples, we can define $\\Phi = [\\phi(\\mathbf{x},\\mathbf{W},\\mathbf{b})]_{n=1}^{N}$, with $\\Phi \\in \\mathbb{R}^{N\\times K}$. We can then form a valid covariance matrix by computing the outer product of our layer output $\\hat{K}(\\mathbf{X},\\mathbf{X}) = \\Phi\\Phi^{T}$. We define the following Gaussian Process: $\\mathbf{T}|\\mathbf{X},\\mathbf{W},\\mathbf{b} \\sim \\mathcal{N}(0,\\hat{K}(\\mathbf{X},\\mathbf{X}))$; $\\mathbf{Y}|\\mathbf{T} \\sim \\mathcal{N}(\\mathbf{T}, 0 \\cdot I_N)$; $c_n|\\mathbf{Y} \\sim \\mathrm{Categorical}(e^{y_{nd}} / \\sum_{d'} e^{y_{nd'}})$. (Eq. 2)\n\nNatural actor-critic (setting of Proposition 1). We consider a discounted Markov Decision Process (MDP) given by the tuple $\\mathcal{M}:=(\\mathcal{S},\\mathcal{A},P,R,\\gamma)$, where $\\mathcal{S}$ is a bounded measurable state space, $\\mathcal{A}$ is the finite set of actions. $P:\\mathcal{S}\\times\\mathcal{A}\\to\\mathcal{P}(\\mathcal{S})$ is the probability transition kernel, $R:\\mathcal{S}\\times\\mathcal{A}\\to\\mathcal{P}([0,R_{\\max}])$ is the reward kernel on the state action space with $R_{\\max}$ being the absolute value of the maximum reward, and $0<\\gamma<1$ is the discount factor. A policy $\\pi:\\mathcal{S}\\to\\mathcal{P}(\\mathcal{A})$ maps a state to a probability distribution over the action space. The action value function for a given policy $\\pi$ is given by $Q^{\\pi}(s,a)=\\mathbb{E}\\left[\\sum_{t=0}^{\\infty}\\gamma^{t}r(s_t,a_t)|s_0=s,a_0=a\\right]$, where $r(s_t,a_t)\\sim R(\\cdot|s_t,a_t)$, $a_t\\sim\\pi(\\cdot|s_t)$ and $s_{t+1}\\sim P(\\cdot|s_t,a_t)$ for $t=\\{0,\\cdots,\\infty\\}$. A policy that achieves the optimal action-value functions $Q^{*}(s,a)=\\sup_{\\pi}Q^{\\pi}(s,a)$ is known as the optimal policy and is denoted as $\\pi^{*}$. From the definition of $Q^{\\pi}(s,a)$, the value function satisfies $V^{\\pi}(s)=\\mathbb{E}_{a\\sim\\pi}[Q^{\\pi}(s,a)]$; the optimal value function is $V^{*}(s)=\\sup_{\\pi}V^{\\pi}(s)$, $\\forall s\\in\\mathcal{S}$. We further define $V^{\\pi}(\\nu)=\\mathbb{E}_{s_0\\sim\\nu}[V^{\\pi}(s_0)]$, where $\\nu$ is an initial state distribution. We can define the visitation distribution as $d^{\\pi}_{s_0}(s)=(1-\\gamma)\\sum_{t=0}^{\\infty}\\gamma^{t}Pr^{\\pi}(s_t=s|s_0)$. Here $Pr^{\\pi}(s_t=s|s_0)$ denotes the probability the state at time $t$ is $s$ given a starting state of $s_0$. Hence, we can write $d^{\\pi}_{\\rho}(s)=\\mathbb{E}_{s_0\\sim\\rho}[d^{\\pi}_{s_0}(s)]$. In a natural policy gradient algorithm, the policy is parameterized as $\\{\\pi_\\lambda,\\lambda\\in\\Lambda\\}$ and $\\Lambda\\subset\\mathbb{R}^d$ where $d$ is a positive integer. We have $K$ total iterations of the Algorithm. Here $A^{\\pi_{\\lambda_k}}(s,a)=Q^{\\pi_{\\lambda_k}}(s,a)-V^{\\pi_{\\lambda_k}}(s)$. The smoothness property: for any $\\lambda_1,\\lambda_2\\in\\Lambda$ and $(s,a)\\in(\\mathcal{S}\\times\\mathcal{A})$ we have $\\|\\nabla\\log(\\pi_{\\lambda_1})(a|s)-\\nabla\\log(\\pi_{\\lambda_2})(a|s)\\|_2\\le\\beta\\|\\lambda_1-\\lambda_2\\|_2$ where $\\beta>0$. We summarize the Natural Actor-Critic approach in Algorithm 1. It has one main for loop indexed by the iteration counter $k$. The first inner for loop indexed by $j$ is the loop where the critic step is performed.\nAlgorithm 1 (Natural Actor Critic with Neural Parametrization). Input: $\\mathcal{S},\\mathcal{A},\\gamma$, Time Horizon $K\\in\\mathcal{Z}$, Updates per time step $J\\in\\mathcal{Z}$, starting state sampling distribution $\\nu$, Actor step sizes $\\beta_{i,k},\\forall k\\in\\{1,\\cdots,K\\},i\\in\\{1,\\cdots,n.J\\}$, Critic step size $\\alpha$, policy gradient step size $\\eta$.\n1: Initialize: $\\lambda_0=\\{0\\}^d$.\n2: for $k\\in\\{1,\\cdots,K\\}$ do\n3: Initialize $X=\\varnothing$, $Q_k(s,a)=0\\ \\forall(s,a)\\in\\mathcal{S}\\times\\mathcal{A}$\n4: for $j\\in\\{1,\\cdots,J\\}$ do\n5: Sample $s_1$ from $\\nu$ and $a_1$ by following $\\pi_{\\lambda_k}$\n6: Initialize $\\theta_0$ using a standard Gaussian.\n7: for $i\\in\\{1,\\cdots,n\\}$ do\n8: Sample the tuple $s_{i+1},a_{i+1}$ by following the policy $\\pi_{\\lambda_k}$\n9: Set $y_i=r'(s_i,a_i)+\\gamma Q_k(s_{i+1},a_{i+1})$,\n10: $\\theta_i=\\theta_{i-1}+\\alpha(y_i-Q_{\\theta_i}(s_i,a_i))\\nabla Q_{\\theta_i}(s_i,a_i)$\n11: end for\n12: $Q_k=Q_{\\theta_n}$\n13: Append the $n$ $(s_i,a_i)$ pairs to the data-set $X$\n14: end for\n15: Initialize $w_0=0^d$\n16: for $i\\in\\{1,\\cdots,|X|\\}$ do\n17: $A_k(s_i,a_i)=Q_k(s_i,a_i)-\\sum_{a\\in\\mathcal{A}}\\pi_{\\lambda_k}(a|s_i)Q_k(s_i,a)$\n18: $w_i=w_{i-1}-\\beta_{i,k}\\left(w_i\\cdot\\nabla_\\lambda\\log\\pi_{\\lambda_k}(a_i|s_i)-A_k(s_i,a_i)\\right)\\nabla_\\lambda\\log\\pi_{\\lambda_k}(a_i|s_i)$\n19: end for\n20: Update $\\lambda_{k+1}=\\lambda_k+\\eta w_{|X|}$\n21: end for\nOutput: $\\pi_{\\lambda_{K+1}}$\nUsing the smoothness property one obtains a bound on the expectation of the difference between our estimated value function and the optimal value function:\n$$\\min_{k\\in\\{1,\\cdots,K\\}}V^{*}(\\nu)-V^{\\pi_{\\lambda_k}}(\\nu)\\le\\frac{\\log(|\\mathcal{A}|)}{K\\eta(1-\\gamma)}+\\frac{\\eta\\beta W^{2}}{2(1-\\gamma)}+\\frac{1}{K}\\sum_{k=1}^{K}\\frac{err_k}{1-\\gamma},\\tag{3}$$\nwhere\n$$err_k=\\mathbb{E}_{s\\sim d_{\\nu}^{\\pi^{*}},a\\sim\\pi^{*}(\\cdot|s)}(|A^{\\pi_{\\lambda_k}}-w^{k}(s,a)\\nabla\\log(\\pi_{\\lambda_k}(a|s))|),\\tag{4}$$\nwhere $W$ is a constant such that $\\|w^{k}\\|_2\\le W\\ \\forall k$, where $k$ denotes the iteration of the outer for loop of Algorithm 1. $A_{k,J}$ is our estimate of $A^{\\pi_{\\lambda_k}}$; this estimation is carried out in the first inner for loop of Algorithm 1.\n\nApproximate gradient flow (setting of Lemma 2). Let $\\pi$ be a target distribution on $\\mathcal{X}$ and $\\mathcal{P}_2(\\mathcal{X})$ the set of probability measures with finite second moment.\nAssumption 1. We say that the target distribution $\\pi$ satisfies the LSI, if, for any $\\mu \\in \\mathcal{P}_2(\\mathcal{X})$, there exists a positive constant $C_{\\mathrm{LS}}$ such that $\\mathrm{KL}(\\mu|\\pi) \\le \\frac{1}{C_{\\mathrm{LS}}}\\left\\|\\nabla \\log \\frac{\\mu}{\\pi}\\right\\|_{L^2(\\mu)}^2$.\nLet us assume that a gradient flow on the Wasserstein space exists, which is induced by some velocity $v_t = g_{\\mu_t}(x) \\in L^2(\\mu_t)$ for $x \\in \\mathcal{X}$. Here, $g_{\\mu_t}(x)$ represents a function of $x$ only depending on $\\mu_t$. When considering discrete time, we assume that the recursion $\\mu_{t+1} = (I - \\gamma_t g_{\\mu_t})\\#\\mu_t$ exists. As for the discrete-time setting, we assume the following inequality with a kind of descent property:\n$$\n\\mathrm{KL}(\\mu_{t+1}|\\pi) \\le \\mathrm{KL}(\\mu_t|\\pi) - \\eta_t \\left\\langle \\nabla \\log \\frac{\\mu_t}{\\pi}, g_{\\mu_t} \\right\\rangle_{L^2(\\mu_t)}, \\tag{5}\n$$\nwhere $\\eta_t$ is some positive constant.\nDefinition ($(\\epsilon,\\delta)$-AGF). Suppose that $\\nabla \\log \\frac{\\mu_t}{\\pi}(x) < \\infty$ (a.e.) and $\\|\\nabla \\log \\frac{\\mu_t}{\\pi}\\|_{L^2(\\mu_t)} < \\infty$ for all $t$. Then, we say a function $g_{\\mu_t}(x) \\in L^2(\\mu_t)$ is $(\\epsilon_t, \\delta_t)$-AGF if the following condition holds:\n$$\n-\\left\\langle \\nabla \\log \\frac{\\mu_t}{\\pi}, g_{\\mu_t} \\right\\rangle_{L^2(\\mu_t)} \\le -\\epsilon_t \\left\\| \\nabla \\log \\frac{\\mu_t}{\\pi} \\right\\|_{L^2(\\mu_t)}^2 + \\delta_t, \\tag{6}\n$$\nwhere $\\epsilon_t, \\delta_t \\ge 0$.", "theorems": [{"local_id": "T1", "statement": "For each outer iteration $k$ of Algorithm 1, the error term $err_k=\\mathbb{E}_{s\\sim d_{\\nu}^{\\pi^{*}},a\\sim\\pi^{*}(\\cdot|s)}\\big|A^{\\pi_{\\lambda_k}}(s,a)-w^{k}(s,a)\\nabla\\log\\pi_{\\lambda_k}(a|s)\\big|$ appearing in the value-suboptimality bound (Eqn 3) admits the upper bound $err_k\\le \\underbrace{\\mathbb{E}_{s,a}|A^{\\pi_{\\lambda_k}}-A_{k,J}|}_{I}+\\underbrace{\\mathbb{E}_{s,a}|A_{k,J}-w^{k}\\nabla\\log\\pi_{\\lambda_k}(a|s)|}_{II}$, separating the critic estimation error $I$ from the actor estimation error $II$.", "proof": {"steps": [{"n": 1, "claim": "The value-suboptimality sketch yields $\\min_{k}V^{*}(\\nu)-V^{\\pi_{\\lambda_k}}(\\nu)\\le\\frac{\\log|\\mathcal{A}|}{K\\eta(1-\\gamma)}+\\frac{\\eta\\beta W^{2}}{2(1-\\gamma)}+\\frac{1}{K}\\sum_{k=1}^{K}\\frac{err_k}{1-\\gamma}$, with $err_k$ as defined in Eqn 4 and $\\|w^{k}\\|_2\\le W$."}, {"n": 2, "claim": "Inside the absolute value defining $err_k$, insert the critic advantage estimate $A_{k,J}$ produced by the first inner for loop: $A^{\\pi_{\\lambda_k}}-w^{k}\\nabla\\log\\pi_{\\lambda_k}=(A^{\\pi_{\\lambda_k}}-A_{k,J})+(A_{k,J}-w^{k}\\nabla\\log\\pi_{\\lambda_k})$."}, {"n": 3, "claim": "Applying the triangle inequality pointwise and then monotonicity of the expectation gives $err_k\\le\\mathbb{E}_{s,a}|A^{\\pi_{\\lambda_k}}-A_{k,J}|+\\mathbb{E}_{s,a}|A_{k,J}-w^{k}\\nabla\\log\\pi_{\\lambda_k}(a|s)|=I+II$."}, {"n": 4, "claim": "Term $I$ is the critic error (the gap between the true advantage $A^{\\pi_{\\lambda_k}}$ and its estimate $A_{k,J}$ from the first inner loop) and term $II$ is the actor error (the linear-regression residual against $\\nabla\\log\\pi_{\\lambda_k}$ optimized in the second inner loop)."}]}}, {"local_id": "T2", "statement": "Suppose that Assumption 1 is satisfied. Then, under Eq. (5), for any $T \\in \\mathbb{N}$, we obtain $\\mathrm{KL}(\\mu_T|\\pi) \\le \\prod_{t=0}^{T-1}(1-\\eta_t\\epsilon_t C_{\\mathrm{LS}})\\mathrm{KL}(\\mu_0|\\pi) + \\sum_{t=0}^{T-1}\\delta_t\\eta_t\\prod_{j=t+1}^{T-1}(1-\\eta_j\\epsilon_j C_{\\mathrm{LS}})$.", "proof": {"steps": [{"n": 1, "claim": "By substituting Eq. (5) into Eq. (6) and applying the LSI, we obtain $\\mathrm{KL}(\\mu_{t+1}|\\pi) \\le (1-\\eta_t\\epsilon_t C_{\\mathrm{LS}})\\mathrm{KL}(\\mu_t|\\pi) + \\eta_t\\delta_t$. By induction in the above, we obtain the claim."}]}}, {"local_id": "T3", "statement": "In the Variational Dropout / Gaussian Process setting where $\\phi(\\mathbf{x},\\mathbf{W},\\mathbf{b}) = \\sqrt{\\frac{1}{K}}\\sigma(\\mathbf{W}x+\\mathbf{b})$ is a single-layer activation output, $\\Phi = [\\phi(\\mathbf{x},\\mathbf{W},\\mathbf{b})]_{n=1}^{N} \\in \\mathbb{R}^{N\\times K}$, and $\\hat{K}(X,X) = \\Phi\\Phi^{T}$ is the induced covariance, with the GP $\\mathbf{T}|\\mathbf{X},\\mathbf{W},b \\sim \\mathcal{N}(0,\\hat{K}(\\mathbf{X},\\mathbf{X}))$ and prior $r(\\mathbf{T}) \\sim \\mathcal{N}(0,I)$, so that $\\Sigma_1 = I$, $\\mu_0=\\mu_1=0$, $\\Sigma_0 = \\hat{K}(X,X)$: the Information Bottleneck penalty $I(X;T)$ (the term $D_{\\mathrm{KL}}(p(\\mathbf{T}|\\mathbf{X},\\mathbf{W},\\mathbf{b})||r(\\mathbf{T}))$) is claimed to be minimized by, i.e. equivalent to, the $L_2$ activation norm penalty $\\eta\\frac{N}{K}\\sum_i^N[\\phi(x)]^2$, giving the objective $\\mathcal{L}_{\\mathrm{GP-ANP}}(\\theta) \\approx -\\mathcal{L}_{\\mathrm{GP-VI}}(\\theta) + \\eta\\frac{N}{K}\\sum_i^N[\\phi(x)]^2$ (Eq. 7).", "proof": {"steps": [{"n": 1, "claim": "$D_{\\mathrm{KL}}(p(\\mathbf{T}|\\mathbf{X},\\mathbf{W},\\mathbf{b})||r(\\mathbf{T})) = \\frac{1}{2}\\{\\mathrm{tr}(\\Sigma_1^{-1}\\Sigma_0) + (\\mu_1-\\mu_0)^{T}\\Sigma_1^{-1}(\\mu_1-\\mu_0) - k + \\ln\\frac{|\\Sigma_1|}{|\\Sigma_0|}\\}$."}, {"n": 2, "claim": "$D_{\\mathrm{KL}}(p(\\mathbf{T}|\\mathbf{X},\\mathbf{W},\\mathbf{b})||r(\\mathbf{T})) = \\frac{1}{2}\\{\\mathrm{tr}(\\hat{K}(\\mathbf{X},\\mathbf{X})) - \\ln|\\hat{K}(\\mathbf{X},\\mathbf{X})|\\}$."}, {"n": 3, "claim": "$\\mathrm{tr}(\\hat{K}(\\mathbf{X},\\mathbf{X})) = \\frac{N}{K}\\sum_{i}^{N}[\\sigma(\\mathbf{W}^{T}x_i+b)]^2$."}, {"n": 4, "claim": "The trace term $\\mathrm{tr}(\\hat{K}(\\mathbf{X},\\mathbf{X})) = \\frac{N}{K}\\sum_i^N[\\sigma(\\mathbf{W}^Tx_i+b)]^2$ is equivalent to an $L_2$ norm penalty over the output of the final non-linear layer (the activation norm penalty, ANP)."}, {"n": 5, "claim": "$\\frac{1}{2}\\{\\mathrm{tr}(\\hat{K}(\\mathbf{X},\\mathbf{X})) - \\ln|\\hat{K}(\\mathbf{X},\\mathbf{X})|\\} \\longrightarrow \\eta\\frac{N}{K}\\sum_i^N[\\phi(x)]^2$, i.e. the KL divergence (hence $I(X;T)$) is equivalent to the $L_2$ activation norm penalty, giving $\\mathcal{L}_{\\mathrm{GP-ANP}}(\\theta) \\approx -\\mathcal{L}_{\\mathrm{GP-VI}}(\\theta) + \\eta\\frac{N}{K}\\sum_i^N[\\phi(x)]^2$; regularizing on the activations of the final layer is equivalent to minimizing $I(X;T)$ on the corresponding Gaussian Process."}]}}], "target_local_id": "T3", "topic_domain": "Deep Learning Theory", "topic_subtopic": "Bayesian Neural Networks", "label": "flawed", "error_step": 2, "error_class6": ["computation_or_constant", "invalid_bound_or_inequality", "others"], "error_explanation": "Step 2 substitutes the theorem's own values into the Gaussian KL identity of step 1 -- Sigma_1 = I, Sigma_0 = Khat(X,X) = Phi Phi^T with Phi in R^{N x K} so Khat in R^{N x N} and k = N, mu_0 = mu_1 = 0 -- and drops the -k term. The substitution yields (1/2){tr(Khat) - N - ln|Khat|}; the step prints (1/2){tr(Khat) - ln|Khat|}. As a stated EQUALITY it is false for every N >= 1 and every Khat, overstating the KL by exactly N/2. Sharpest witness: at Khat = I_N the two distributions p and r are identical, so the KL is exactly 0, while the printed expression returns N/2 (0.5, 1.0, 2.0, 4.0 at N = 1, 2, 4, 8). A 2,000,000-draw Monte-Carlo estimate of E_p[log p - log r], assuming no closed form, matches the -N version to MC noise and sits N/2 below the printed one at N = 1, 2, 3. Step 1 itself carries the -k explicitly, so the omission is visible within the proof. Two further defects sit LATER in the same proof and are recorded rather than graded, since the anchor is the FIRST invalid step: step 3's trace identity prints the prefactor N/K where tr(Phi Phi^T) = (1/K) sum_n ||sigma(W x_n + b)||^2 gives 1/K, a spurious factor N; and step 5 discards the log-determinant term -ln|Khat| from the KL, which is the flaw ICLR reviewers raised about this paper."}
{"id": "nEMiSX_ipXr", "context": "In a nonempty clean input set $\\mathcal{X}\\subset\\mathbb{R}^d$, let every sample x exclusively belong to one of the classes $\\mathcal{Y}$, and their classes will be denoted as $c_x$. A classifier f assigns a class label from $\\mathcal{Y}$ for each sample $x\\in\\mathbb{R}^d$. One-dimensional toy example: there are only two classes $-1$ and $1$, i.e., $\\mathcal{Y}=\\{-1,1\\}$, and one-dimensional clean input set $\\mathcal{X}=[-2,-1)\\cup[1,2)\\subseteq\\mathbb{R}$. $c_x=-1$ when $x\\in[-2,-1)$ and $c_x=1$ when $x\\in[1,2)$. Let us define three classifiers $f_1$, $f_2$ and $f_3$ for this toy example. When step function $\\mathrm{step}(x)$ is defined as $\\mathrm{step}(x)=1$ if $x\\ge 0$ and $\\mathrm{step}(x)=-1$ if $x<0$, let $f_1(x)=\\mathrm{step}(x-1)$, $f_2(x)=1-\\mathrm{step}(x+4)+\\mathrm{step}(x)$, and $f_3(x)=\\mathrm{step}(x)$. Notice that the accuracy for all three classifiers is 1. The oracle classifier will be $f_3$. Definition 1 (Genuine adversarial accuracy). We define genuine adversarial accuracy that uses exact perturbation norm. The complement set of a set S will be denoted as $S^{\\mathrm{c}}$, $\\bar{\\mathcal{X}}$ is the topological closure of X, i.e., union of X and its boundary. Voronoi boundary $VB(\\mathcal{X})$ is defined as $\\{x'\\in\\mathbb{R}^d\\mid\\exists x_1,x_2\\in\\bar{\\mathcal{X}}:x_1\\neq x_2,\\ \\|x'-x_1\\|=\\|x'-x_2\\|\\}$. Previously allowed perturbation region $\\mathcal{X}_\\epsilon$ is defined as $\\{x'\\in VB(\\mathcal{X})^{\\mathsf{c}}\\mid \\|x'-x\\|<\\epsilon, x\\in\\bar{\\mathcal{X}}\\}$. Note that $\\mathbb{1}()$ is an indicator function that has value 1 if the condition in the bracket holds and value 0 if the condition in the bracket does not hold. Then, genuine adversarial accuracy (by exact perturbation norm) $a_{gen;exact}(\\epsilon)$ is defined as follows. (i) $a_{gen;exact}(\\epsilon)=\\mathbb{E}_{x\\in S_{exact}(\\epsilon)}[\\mathbb{1}(f(x^*)=c_x)]$ where $S_{exact}(\\epsilon)=\\{x\\in\\bar{\\mathcal{X}}\\mid\\exists x'\\in\\mathcal{X}_\\epsilon^{\\mathrm{c}}\\cap VB(\\mathcal{X})^{\\mathrm{c}}:\\|x'-x\\|=\\epsilon\\}$ and $x^*=\\arg\\max_{x'\\in\\mathcal{X}_\\epsilon^{\\mathrm{c}}\\cap VB(\\mathcal{X})^{\\mathrm{c}}:\\|x'-x\\|=\\epsilon}L(\\theta,x',c_x)$ when $\\epsilon>0$. (ii) $a_{gen;exact}(0)=\\mathbb{E}_{x\\in\\mathcal{X}}[\\mathbb{1}(f(x)=c_x)]$ when $\\epsilon=0$. Theorem 2. Genuine adversarial accuracy only use each point only once to calculate adversarial accuracy. In other words, it does not allow overlaps. The proof is in Section C of the appendix. Theorem 3. A single nearest neighbor (1-NN) classifier maximizes genuine adversarial accuracy, and it is the almost everywhere unique classifier that satisfies this except for $\\bar{\\mathcal{X}}\\setminus\\mathcal{X}$ and Voronoi boundary $VB(\\mathcal{X})$. Proof of the theorem is in Section D of the appendix.\n\nMulti-modal variational bounds (setting of Lemma 1). We consider a set of $M$ random variables $\\{X_1,\\ldots,X_M\\}$ with empirical density $p_d$, where each random variable $X_s$, $s\\in\\mathcal{M}=\\{1,\\ldots,M\\}$, can be used to model a different data modality taking values in $\\mathsf{X}_s$. With some abuse of notation, we write $X=\\{X_1,\\ldots,X_M\\}$ and for any subset $\\mathcal{S}\\subset\\mathcal{M}$, we set $X=(X_{\\mathcal{S}},X_{\\backslash\\mathcal{S}})$ for two partitions of the random variables into $X_{\\mathcal{S}}=\\{X_s\\}_{s\\in\\mathcal{S}}$ and $X_{\\backslash\\mathcal{S}}=\\{X_s\\}_{s\\in\\mathcal{M}\\backslash\\mathcal{S}}$. For a latent variable $Z\\in\\mathsf{Z}$ with prior density $p_\\theta(z)$, we posit a joint generative model $p_\\theta(z,x)=p_\\theta(z)\\prod_{s=1}^{M}p_\\theta(x_s|z)$, where $p_\\theta(x_s|z)$ is commonly referred to as the decoding distribution for modality $s$. $\\mathrm{I}_q(X,Y)=\\int q(x,y)\\log\\frac{q(x,y)}{q(x)q(y)}$ is the mutual information of random variables $X$ and $Y$ having marginal and joint densities $q$, whilst $\\mathcal{H}(X|Y)=-\\int q(x,y)\\log q(x|y)\\,\\mathrm{d}x\\mathrm{d}y$ is the conditional entropy of $X$ given $Y$. We need to learn encoding distributions $q_\\phi(z|x_{\\mathcal{S}})$ for any $\\mathcal{S}\\in\\mathcal{P}(\\mathcal{M})$. For $\\mathcal{S}\\subset\\mathcal{M}$ and $\\beta>0$, we introduce an (approximate) conditional lower bound $\\mathcal{L}_{\\backslash\\mathcal{S}}(x,\\theta,\\phi,\\beta)=\\int q_\\phi(z|x)\\left[\\log p_\\theta(x_{\\backslash\\mathcal{S}}|z)\\right]\\mathrm{d}z-\\beta\\mathsf{KL}(q_\\phi(z|x)|q_\\phi(z|x_{\\mathcal{S}}))$. We set $R_{\\backslash\\mathcal{S}}=\\int p_d(x)\\mathsf{KL}(q_\\phi(z|x)|q_\\phi(z|x_{\\mathcal{S}}))\\mathrm{d}x$ for a conditional or cross rate. Similarly, set $D_{\\backslash\\mathcal{S}}=-\\int p_d(x)q_\\phi(z|x)\\log p_\\theta(x_{\\backslash\\mathcal{S}}|z)\\mathrm{d}z\\mathrm{d}x$.", "theorems": [{"local_id": "T1", "statement": "Lemma 1 (Variational bounds on the conditional mutual information). It holds that $\\begin{array}{ r }{-\\int\\mathcal{L}_{\\backslash\\mathcal{S}}(x,\\theta,\\phi,\\beta)p_{d}(\\mathrm{d}x)=D_{\\backslash\\mathcal{S}}+\\beta R_{\\backslash\\mathcal{S}}}\\end{array}$ and for $\\Delta_{\\backslash{S,1}},\\Delta_{\\backslash{S,2}}\\geq 0\\mathrm{~}$\n\n$$\n\\mathcal{H}_{\\backslash\\mathcal{S}}-D_{\\backslash\\mathcal{S}}+\\Delta_{\\backslash\\mathcal{S},1}=\\mathrm{I}_{q_{\\phi}}(X_{\\backslash\\mathcal{S}},Z_{\\mathcal{M}}|X_{\\mathcal{S}})=R_{\\backslash\\mathcal{S}}-\\Delta_{\\backslash\\mathcal{S},2}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof of Lemma 1. The proof follows by adapting the arguments in Alemi et al. (2018). The law of $X_{\\backslash s}$ and $Z$ conditional on $X_{\\mathcal{S}}$ on the encoder path can be written as\n\n$$\nq_{\\phi}(z,x\\backslash s|x_{S})=p_{d}(x\\backslash s|x_{S})q_{\\phi}(z|x)=q_{\\phi,\\backslash S}^{\\mathrm{agg}}(z|x_{S})q^{\\star}(x\\backslash s|z,x_{S})\n$$\n\nwith $q^{\\star}(x_{\\backslash\\mathcal{S}}|z,x_{\\mathcal{S}})=q_{\\phi}(z,x_{\\backslash\\mathcal{S}}|x_{\\mathcal{S}})/q_{\\phi,\\backslash\\mathcal{S}}^{\\mathrm{agg}}(z|x_{\\mathcal{S}})$ . To prove a lower bound on the conditional mutual information, note that\n\n$$\n\\begin{array}{ r l }&{\\quad\\mathrm{I}_{q_{\\phi}}(X_{\\backslash\\mathcal{S}},Z_{\\mathcal{M}}|X_{\\mathcal{S}})}\\\\ &{=\\displaystyle\\int p_{d}(x_{\\mathcal{S}})\\int q_{\\phi,\\backslash\\mathcal{S}}^{\\mathrm{agg}}(z|x_{\\mathcal{S}})\\int q^{\\star}(x_{\\backslash\\mathcal{S}}|z,x_{\\mathcal{S}})\\log\\frac{q_{\\phi,\\backslash\\mathcal{S}}^{\\mathrm{agg}}(z|x_{\\mathcal{S}})q^{\\star}(x_{\\backslash\\mathcal{S}}|z,x_{\\mathcal{S}})}{q_{\\phi,\\backslash\\mathcal{S}}^{\\mathrm{agg}}(z|x_{\\mathcal{S}})p_{d}(x_{\\backslash\\mathcal{S}}|x_{\\backslash\\mathcal{S}})}\\mathrm{d}z\\mathrm{d}x_{\\backslash\\mathcal{S}}\\mathrm{d}x_{\\mathcal{S}}}\\\\ &{=\\displaystyle\\int p_{d}(x_{\\mathcal{S}})\\int q_{\\phi,\\backslash\\mathcal{S}}^{\\mathrm{agg}}(z|x_{\\mathcal{S}})\\left[q^{\\star}(x_{\\backslash\\mathcal{S}}|z,x_{\\mathcal{S}})\\log p_{\\theta}(x_{\\backslash\\mathcal{S}}|z))+\\mathsf{KL}(q^{\\star}(x_{\\backslash\\mathcal{S}}|z,x_{\\mathcal{S}})|p_{\\theta}(x_{\\backslash\\mathcal{S}}|z))\\right]\\mathrm{d}z\\mathrm{d}x_{\\mathcal{S}}}\\\\ &{\\quad-\\displaystyle\\int p_{d}(x_{\\mathcal{S}})\\int p_{d}(x_{\\backslash\\mathcal{S}}|x_{\\mathcal{S}})\\log p_{d}(x_{\\backslash\\mathcal{S}}|x_{\\mathcal{S}})\\mathrm{d}x}\\\\ &{=\\displaystyle\\int p_{d}(x)\\int q_{\\phi}(z|x)\\log p_{\\theta}(x_{\\backslash\\mathcal{S}}|z)\\mathrm{d}z\\mathrm{d}x-\\underbrace{\\int p_{d}(x_{\\mathcal{S}})\\int p_{d}(x_{\\backslash\\mathcal{S}}|x_{\\mathcal{S}})\\log p_{d}(x_{\\backslash\\mathcal{S}}|x_{\\mathcal{S}})\\mathrm{d}x}_{= -\\mathcal{H}_{\\backslash\\mathcal{S}}= -\\mathcal{H}(X_{\\backslash\\mathcal{S}}|X_{\\mathcal{S}})}}\\\\ &{\\quad+\\underbrace{\\int p_{d}(x_{\\mathcal{S}})\\int q_{\\phi,\\backslash\\mathcal{S}}^{\\mathrm{agg}}(z|x_{\\mathcal{S}})\\mathsf{KL}(q^{\\star}(x_{\\backslash\\mathcal{S}}|z,x_{\\mathcal{S}})|p_{\\theta}(x_{\\backslash\\mathcal{S}}|z))\\mathrm{d}x_{\\mathcal{S}}}_{=\\Delta_{\\backslash\\mathcal{S},1}\\geq 0}}\\\\ &{=\\Delta_{\\backslash\\mathcal{S},1}+D_{\\backslash\\mathcal{S}}+\\mathcal{H}_{\\backslash\\mathcal{S}}.}\\end{array}\n$$"}, {"n": 2, "claim": "The upper bound follows by observing that\n\n$$\n\\begin{array}{ r l }&{\\quad\\mathrm{I}_{q_{\\phi}}(X_{\\backslash S},Z_{\\mathcal{M}}|\\dot{X}_{S})}\\\\ &{=\\displaystyle\\int p_{d}(x_{S})\\int p_{d}(x_{\\backslash S}|x_{\\backslash})\\log\\frac{q_{\\phi}(z|x)p_{d}(x_{\\backslash S}|x_{S})}{q_{\\phi,\\backslash S}^{\\mathrm{agg}}(z|x_{S})p_{d}(x_{\\backslash S}|x_{S})}\\mathrm{d}z\\mathrm{d}x}\\\\ &{=\\displaystyle\\int p_{d}(x){\\mathsf{KL}}(q_{\\phi}(z|x)|q_{\\phi}(z|x_{S}))\\mathrm{d}x-\\underbrace{\\int p_{d}(x_{S}){\\mathsf{KL}}(q_{\\phi,\\backslash S}^{\\mathrm{agg}}(z|x_{S})|q_{\\phi}(z|x_{S}))\\mathrm{d}x_{S}}_{=\\Delta_{\\backslash S,2}\\geq 0}}\\\\ &{=R_{\\backslash S}-\\Delta_{\\backslash S,2}.}\\end{array}\n$$"}]}}, {"local_id": "T2", "statement": "Proposition 1. Correctness of Algorithm 1, Line 13 : see Appendix B.2 for the proof. [Algorithm 1 (Reverse-mode differentiation of Adam), lines 1–13:]\n1: Input: $\\mathbf{w}_{T},\\mathbf{m}_{T},\\mathbf{v}_{T},\\gamma,\\alpha,\\epsilon,L(w,x)$ , meta-objective $f(w)$\n2: Initialize: $d\\mathbf{m}\\leftarrow 0,d\\mathbf{x}\\leftarrow 0,d\\mathbf{w}\\leftarrow\\nabla_{\\mathbf{w}}f(\\mathbf{w}_{T})$\n3: for $t = T$ to $1$ do\n4: $\\hat{\\mathbf{m}}_{t}\\triangleq\\mathbf{m}_{t}/(1-\\beta_{1}^{t})$\n5: $\\hat{\\mathbf{v}}_{t}\\triangleq\\mathbf{v}_{t}/(1-\\beta_{2}^{t})$\n6: $\\mathbf{w}_{t-1}=\\mathbf{w}_{t}+\\alpha\\cdot\\hat{\\mathbf{m}}_{t}/(\\hat{\\mathbf{v}}_{t}+\\epsilon)$\n7: $\\mathbf{g}_{t}\\triangleq\\nabla_{\\mathbf{w}}L(\\mathbf{w}_{t-1},\\mathbf{x})$\n8: $\\mathbf{m}_{t-1}=[\\mathbf{m}_{t}-(1-\\beta_{1})\\cdot\\mathbf{g}_{t}]/\\beta_{1}$\n9: $\\mathbf{v}_{t-1}=[\\mathbf{v}_{t}-(1-\\beta_{2})\\cdot\\mathbf{g}_{t}^{2}]/\\beta_{2}$\n10: $\\epsilon^{\\prime}\\triangleq\\epsilon\\cdot\\sqrt{1-\\beta_{2}^{t}}$\n11: $\\alpha^{\\prime}\\triangleq\\alpha\\cdot\\sqrt{1-\\beta_{2}^{t}}/(1-\\beta_{1}^{t})$\n12: $\\beta^{\\prime}\\triangleq(1-\\beta_{2})/(1-\\beta_{1})$\n13: $\\begin{array}{ r }{d\\mathbf{m}=d\\mathbf{m}+\\alpha^{\\prime}\\cdot\\left(\\frac{\\boldsymbol{\\beta}^{\\prime}\\cdot\\mathbf{m}_{t}\\cdot\\mathbf{g}_{t}}{\\sqrt{\\mathbf{v}_{t}}\\cdot(\\sqrt{\\mathbf{v}_{t}}+\\epsilon^{\\prime})^{2}}-\\frac{1}{\\sqrt{\\mathbf{v}_{t}}+\\epsilon^{\\prime}}\\right)\\cdot d\\mathbf{w}}\\end{array}$", "proof": {"steps": [{"n": 1, "claim": "Proof. Using the chain rule of derivatives: $$\\begin{array}{ l }{\\displaystyle\\frac{d f\\big(\\mathbf{w}_{T}\\big)}{d\\mathbf{m}_{0}}\\triangleq d\\mathbf{m}=d\\mathbf{m}+\\frac{\\partial w_{t}}{\\partial m_{t}}\\cdot d\\mathbf{w}}\\\\ {=d\\mathbf{m}-\\displaystyle\\frac{\\alpha}{1-\\beta_{1}^{t}}\\left(\\frac{\\big [\\sqrt{\\hat{\\mathbf{v}_{t}}}+\\epsilon\\big]-\\Big [\\mathbf{m}_{t}\\cdot\\Big(\\frac{\\partial\\mathbf{v}_{t}/\\partial\\mathbf{m}_{t}}{2\\cdot\\sqrt{\\hat{\\mathbf{v}_{t}}}\\cdot(1-\\beta_{2}^{t})}\\Big)\\Big]}{(\\sqrt{\\hat{\\mathbf{v}_{t}}}+\\epsilon)^{2}}\\right)\\cdot d\\mathbf{w}}\\\\ {=d\\mathbf{m}+\\alpha^{\\prime}\\cdot\\displaystyle\\left(\\frac{\\Big [\\mathbf{m}_{t}\\cdot\\Big(\\frac{\\partial\\mathbf{v}_{t}/\\partial\\mathbf{m}_{t}}{2\\cdot\\sqrt{\\mathbf{v}_{t}}}\\Big)\\Big]}{(\\sqrt{\\mathbf{v}_{t}}+\\epsilon^{\\prime})^{2}}-\\frac{1}{\\sqrt{\\mathbf{v}_{t}}+\\epsilon^{\\prime}}\\right)\\cdot d\\mathbf{w}~,}\\end{array}$$ where, $\\epsilon^{\\prime}\\triangleq\\epsilon\\cdot\\sqrt{1-\\beta_{2}^{t}}$ and $\\begin{array}{ r }{\\alpha^{\\prime}\\triangleq\\frac{\\alpha\\cdot\\sqrt{1-\\beta_{2}^{t}}}{1-\\beta_{1}^{t}}}\\end{array}$"}, {"n": 2, "claim": "Using the chain rule again: $$\\frac{\\partial\\mathbf{v}_{t}}{\\partial\\mathbf{m}_{t}}=\\frac{\\partial\\mathbf{v}_{t}/\\partial\\mathbf{g}_{t}}{\\partial\\mathbf{m}_{t}/\\partial\\mathbf{g}_{t}}=\\frac{2\\cdot(1-\\beta_{2})\\cdot\\mathbf{g}_{t}}{(1-\\beta_{1})}=2\\cdot\\beta^{\\prime}\\cdot\\mathbf{g}_{t},$$ where, $\\begin{array}{ r }{\\beta^{\\prime}\\triangleq\\frac{1-\\beta_{2}}{1-\\beta_{1}}}\\end{array}$ , leading to finally:"}, {"n": 3, "claim": "$$\\begin{array}{ r }{\\frac{d f(\\mathbf{w}_{T})}{d\\mathbf{m}_{0}}\\triangleq d\\mathbf{m}=d\\mathbf{m}+\\alpha^{\\prime}\\cdot\\left(\\frac{\\left[\\mathbf{m}_{t}\\cdot\\left(\\frac{\\partial\\mathbf{v}_{t}/\\partial\\mathbf{m}_{t}}{2\\cdot\\sqrt{\\mathbf{v}_{t}}}\\right)\\right]}{(\\sqrt{\\mathbf{v}_{t}}+\\epsilon^{\\prime})^{2}}-\\frac{1}{\\sqrt{\\mathbf{v}_{t}}+\\epsilon^{\\prime}}\\right)\\cdot d\\mathbf{w}}\\\\ {=d\\mathbf{m}+\\alpha^{\\prime}\\cdot\\left(\\frac{\\boldsymbol{\\beta}^{\\prime}\\cdot\\mathbf{m}_{t}\\cdot\\mathbf{g}_{t}}{\\sqrt{\\mathbf{v}_{t}}\\cdot(\\sqrt{\\mathbf{v}_{t}}+\\epsilon^{\\prime})^{2}}-\\frac{1}{\\sqrt{\\mathbf{v}_{t}}+\\epsilon^{\\prime}}\\right)\\cdot d\\mathbf{w}}\\end{array}$$"}]}}, {"local_id": "T3", "statement": "In the one-dimensional toy example with two classes $\\mathcal{Y}=\\{-1,1\\}$ and clean input set $\\mathcal{X}=[-2,-1)\\cup[1,2)\\subseteq\\mathbb{R}$ (with closure $\\bar{\\mathcal{X}}=[-2,-1]\\cup[1,2]$), where $c_x=-1$ for $x\\in[-2,-1)$ and $c_x=1$ for $x\\in[1,2)$, define the Voronoi boundary as $VB(\\mathcal{X})=\\left\\{ x'\\in\\mathbb{R}^d\\ \\middle|\\ \\exists x_1,x_2\\in\\bar{\\mathcal{X}}:x_1\\neq x_2,\\ \\|x'-x_1\\|=\\|x'-x_2\\|\\right\\}$. Using this Voronoi boundary in the genuine adversarial accuracy $a_{gen;exact}(\\epsilon)=\\mathbb{E}_{x\\in S_{exact}(\\epsilon)}[\\mathbb{1}(f(x^*)=c_x)]$ (with $S_{exact}(\\epsilon)$ and $x^*$ restricted to $\\mathcal{X}_\\epsilon^{\\mathrm{c}}\\cap VB(\\mathcal{X})^{\\mathrm{c}}$), the paper's Figure 6 / Appendix A calculation asserts $VB(\\mathcal{X})=\\{0\\}$ and concludes that $f_3(x)=\\mathrm{step}(x)$ (the 1-NN classifier) is the most robust classifier for this data, consistent with the oracle, and that the same $VB(\\mathcal{X})$ construction validates Theorem 3's claim of 1-NN optimality almost everywhere except for $\\bar{\\mathcal{X}}\\setminus\\mathcal{X}$ and $VB(\\mathcal{X})$.", "proof": {"steps": [{"n": 1, "claim": "The Voronoi boundary is $VB(\\mathcal{X})=\\left\\{ x'\\in\\mathbb{R}^d\\mid\\exists x_1,x_2\\in\\bar{\\mathcal{X}}:x_1\\neq x_2,\\ \\|x'-x_1\\|=\\|x'-x_2\\|\\right\\}$, i.e. the set of points that have two distinct clean points (in the closure $\\bar{\\mathcal{X}}$) at equal distance."}, {"n": 2, "claim": "For the one-dimensional example with $\\bar{\\mathcal{X}}=[-2,-1]\\cup[1,2]$, this Voronoi boundary evaluates to $VB(\\mathcal{X})=\\{0\\}$ (equivalently, for the unit-sphere/ball example $VB(\\mathcal{X})=\\{0\\}$)."}, {"n": 3, "claim": "Hence $VB(\\mathcal{X})^{\\mathrm{c}}=\\mathbb{R}\\setminus\\{0\\}$ is essentially the whole input space, so the perturbation constraint $x'\\in \\mathcal{X}_\\epsilon^{\\mathrm{c}}\\cap VB(\\mathcal{X})^{\\mathrm{c}}$ admits (almost) all candidate perturbations."}, {"n": 4, "claim": "Evaluating $a_{gen;exact}(\\epsilon)=\\mathbb{E}_{x\\in S_{exact}(\\epsilon)}[\\mathbb{1}(f(x^*)=c_x)]$ over these admissible perturbations yields the reported curves for $f_1,f_2,f_3$, from which $f_3(x)=\\mathrm{step}(x)$ (the 1-NN classifier) is the most robust classifier, consistent with the oracle classifier."}, {"n": 5, "claim": "The identical $VB(\\mathcal{X})$ construction underlies Theorem 3's claim that the 1-NN classifier maximizes genuine adversarial accuracy and is the almost-everywhere unique such classifier, except for $\\bar{\\mathcal{X}}\\setminus\\mathcal{X}$ and the Voronoi boundary $VB(\\mathcal{X})$."}]}}], "target_local_id": "T3", "topic_domain": "Others", "topic_subtopic": "Robustness, Privacy & Security", "label": "flawed", "error_step": 2, "error_class6": ["logical_gap_or_missing_case", "others", "computation_or_constant"], "error_explanation": "The set-builder $VB(\\mathcal{X})=\\{x'\\in\\mathbb{R}^d\\mid\\exists x_1,x_2\\in\\bar{\\mathcal{X}}:x_1\\neq x_2,\\ \\|x'-x_1\\|=\\|x'-x_2\\|\\}$ marks $x'$ as a boundary point whenever ANY two distinct clean points are equidistant from it. Since $\\bar{\\mathcal{X}}=[-2,-1]\\cup[1,2]$ has uncountably many points, almost every $x'$ admits two equidistant clean points, so $VB(\\mathcal{X})$ balloons to (nearly) the whole space rather than the isolated locus $\\{0\\}$. Concretely, for any $x'\\in(0,0.5)$, taking $x_1=-1\\in\\bar{\\mathcal{X}}$ gives $\\|x'-x_1\\|=x'+1$, and $x_2=2x'+1\\in\\bar{\\mathcal{X}}$ also gives $\\|x'-x_2\\|=x'+1$; hence $(0,0.5)\\subset VB(\\mathcal{X})$, and symmetrically $(-0.5,0)\\subset VB(\\mathcal{X})$, so $VB(\\mathcal{X})\\supset(-0.5,0.5)$, NOT $\\{0\\}$. Consequently $VB(\\mathcal{X})^{\\mathrm{c}}$ (and thus $S_{exact}(\\epsilon)$) is empty or degenerate for such supports (for a unit sphere $VB(\\mathcal{X})=\\mathbb{R}^d$, so $VB(\\mathcal{X})^{\\mathrm{c}}=\\varnothing$), making genuine adversarial accuracy and every downstream result ill-defined. The correct notion uses OPEN Voronoi cells $Vor(x)=\\{x':\\|x-x'\\|<\\|x_{clean}-x'\\|\\ \\forall x_{clean}\\in\\bar{\\mathcal{X}}\\setminus\\{x\\}\\}$ with $VB(\\mathcal{X})=(\\bigcup_x Vor(x))^{\\mathrm{c}}$, under which the sphere example gives $VB(\\mathcal{X})=\\varnothing$ and the 1-D example gives $VB(\\mathcal{X})=\\{0\\}$."}
{"id": "pJBSzGmb9a", "context": "We consider a discounted MDP $\\mathcal{M}:=(\\mathcal{S},\\mathcal{A},P,R,\\gamma)$, where $\\mathcal{S}$ is a bounded measurable state space, $\\mathcal{A}$ is a finite action set, $P:\\mathcal{S}\\times\\mathcal{A}\\to\\mathcal{P}(\\mathcal{S})$ is the transition kernel, $R:\\mathcal{S}\\times\\mathcal{A}\\to\\mathcal{P}([0,R_{\\max}])$ is the reward kernel ($R_{\\max}$ the maximum absolute reward), and $0<\\gamma<1$ is the discount factor. A policy $\\pi:\\mathcal{S}\\to\\mathcal{P}(\\mathcal{A})$ maps a state to a distribution over actions. The action value function is $Q^{\\pi}(s,a)=\\mathbb{E}\\big[\\sum_{t=0}^{\\infty}\\gamma^{t}r(s_t,a_t)\\mid s_0=s,a_0=a\\big]$ (Eqn 3), with $r(s_t,a_t)\\sim R(\\cdot|s_t,a_t)$, $a_t\\sim\\pi(\\cdot|s_t)$, $s_{t+1}\\sim P(\\cdot|s_t,a_t)$; the optimal value is $Q^{*}(s,a)=\\sup_{\\pi}Q^{\\pi}(s,a)$ (Eqn 4). The value function is $V^{\\pi}(s)=\\mathbb{E}[\\sum_{t=0}^{\\infty}\\gamma^{t}r(s_t,a_t)\\mid s_0=s]$ with $V^{\\pi}(s)=\\mathbb{E}_{a\\sim\\pi}[Q^{\\pi}(s,a)]$, and the advantage is $A^{\\pi}(s,a)=Q^{\\pi}(s,a)-V^{\\pi}(s)$. We write $\\rho_{\\nu}^{\\pi}$ for the stationary state distribution under $\\pi$ from initial distribution $\\nu$, $\\zeta_{\\nu}^{\\pi}(s,a)=\\rho_{\\nu}^{\\pi}(s)\\pi(a|s)$ for the stationary state-action distribution, and $d_{\\nu}^{\\pi}(s)=(1-\\gamma)\\sum_{t=0}^{\\infty}\\gamma^{t}\\Pr^{\\pi}(s_t=s)$ (averaged over $s_0\\sim\\nu$) for the discounted visitation distribution. For a measurable $f$ and measure $\\nu$, $\\mathbb{E}(f)_{\\nu}=\\int_{\\mathcal{S}\\times\\mathcal{A}}f\\,d\\nu$. The one-step Markov transition operator is $P^{\\pi}Q(s,a)=\\mathbb{E}[Q(s',a')\\mid s'\\sim P(\\cdot|s,a),a'\\sim\\pi(\\cdot|s')]$ (Eqn 6). NAC (Algorithm 1): the policy is parametrized as $\\{\\pi_{\\lambda},\\lambda\\in\\Lambda\\subset\\mathbb{R}^{d}\\}$; at iteration $k$ the natural gradient step is $\\lambda_{k+1}=\\lambda_k+\\eta F_{\\nu}^{\\dagger}(\\lambda_k)\\nabla_{\\lambda}V^{\\pi_{\\lambda_k}}(\\nu)$ (Eqn 7), with $\\nabla_{\\lambda_k}V^{\\pi_{\\lambda_k}}(\\nu)=\\mathbb{E}_{s,a}[\\nabla\\log\\pi_{\\lambda_k}(a|s)\\,Q^{\\pi_{\\lambda_k}}(s,a)]$ (Eqn 8), $F_{\\nu}(\\lambda_k)=\\mathbb{E}_{s,a}[\\nabla\\log\\pi_{\\lambda_k}(a|s)(\\nabla\\log\\pi_{\\lambda_k}(a|s))^{\\top}]$ (Eqn 9), $s\\sim d_{\\nu}^{\\pi_{\\lambda_k}},a\\sim\\pi_{\\lambda_k}$; compatible function approximation gives $F_{\\nu}^{\\dagger}(\\lambda_k)\\nabla_{\\lambda_k}V^{\\pi_{\\lambda_k}}(\\nu)=\\tfrac{1}{1-\\gamma}w_k^{*}$ (Eqn 10) with $w_k^{*}=\\arg\\min_{w}\\mathbb{E}_{s,a}(A^{\\pi_{\\lambda_k}}(s,a)-w\\nabla_{\\lambda}\\log\\pi_{\\lambda_k}(a|s))^{2}$ (Eqn 11). The critic maintains a neural-network estimate $Q_{\\theta}$; at outer iteration $k$, inner iteration $j$ the critic solves $\\arg\\min_{\\theta}\\mathbb{E}_{s,a}(T^{\\pi_{\\lambda_k}}Q_{k,j-1}(s,a)-Q_{\\theta}(s,a))^{2}$ (Eqn 12), with target $T^{\\pi_{\\lambda_k}}Q_{k,j-1}$ fixed (DQN-style target network) and update $y_i=r'(s_i,a_i)+\\gamma Q_k(s_{i+1},a_{i+1})$. Intermediate critic minimizers: $Q_{k,j}^{1}$ is the best neural-network approximation of $T^{\\pi_{\\lambda_k}}Q_{k,j-1}$ (Def 2, Eqn 13); $Q_{k,j}^{2}$ minimizes the expected one-sample-target loss $\\mathbb{E}(Q_{\\theta}(s,a)-(r'(s,a)+\\gamma Q_{k,j-1}(s',a')))^{2}$ (Def 3, Eqn 14); $Q_{k,j}^{3}$ minimizes the empirical version over $n$ sampled tuples (Def 4, Eqn 15). Assumptions: (2) $\\|\\nabla\\log\\pi_{\\lambda_1}(a|s)-\\nabla\\log\\pi_{\\lambda_2}(a|s)\\|_2\\le\\beta\\|\\lambda_1-\\lambda_2\\|_2$; (3) geometric mixing $d_{TV}(\\Pr((s_\\tau,a_\\tau)\\in\\cdot),\\zeta_{\\nu}^{\\pi_\\lambda})\\le p\\rho^{\\tau}$; (4) $\\min_{w}\\mathbb{E}_{s,a\\sim\\zeta_{\\nu}^{\\pi_{\\lambda_k}}}(A_{\\theta}(s,a)-w^{\\top}\\nabla\\log\\pi_{\\lambda}(a|s))^{2}\\le\\epsilon_{bias}$; (5) $\\min_{\\theta_1\\in\\Theta}\\mathbb{E}_{s,a\\sim\\zeta_{\\nu}^{\\pi_{\\lambda_k}}}(Q_{\\theta_1}(s,a)-T^{\\pi_{\\lambda}}Q_{\\theta}(s,a))^{2}\\le\\epsilon_{approx}$. The value-suboptimality sketch gives $\\min_{k}V^{*}(\\nu)-V^{\\pi_{\\lambda_k}}(\\nu)\\le\\tfrac{\\log|\\mathcal{A}|}{K\\eta(1-\\gamma)}+\\tfrac{\\eta\\beta W^{2}}{2(1-\\gamma)}+\\tfrac{1}{K}\\sum_k\\tfrac{err_k}{1-\\gamma}$ (Eqn 16) with $err_k=\\mathbb{E}_{s\\sim d_{\\nu}^{\\pi^{*}},a\\sim\\pi^{*}(\\cdot|s)}|A^{\\pi_{\\lambda_k}}-w^{k}(s,a)\\nabla\\log\\pi_{\\lambda_k}(a|s)|$ (Eqn 17), and the critic bound $\\mathbb{E}_{s,a}|Q^{\\pi_{\\lambda_k}}-Q_{k,J}|\\le\\sum_{j=1}^{J-1}\\gamma^{J-j-1}(P^{\\pi_{\\lambda_k}})^{J-j-1}\\mathbb{E}|\\epsilon_{k,j}|+\\gamma^{J}\\tfrac{R_{\\max}}{1-\\gamma}$ (Eqn 18), where $\\epsilon_{k,j}=T^{\\pi_{\\lambda_k}}Q_{k,j-1}-Q_{k,j}$.", "theorems": [{"local_id": "T1", "statement": "For a discounted MDP $\\mathcal{M} := (\\mathcal{S}, \\mathcal{A}, P, R, \\gamma)$ with bounded measurable state space $\\mathcal{S}$, finite action set $\\mathcal{A}$, transition kernel $P : \\mathcal{S} \\times \\mathcal{A} \\to \\mathcal{P}(\\mathcal{S})$, reward kernel $R : \\mathcal{S} \\times \\mathcal{A} \\to \\mathcal{P}([0, R_{\\max}])$, discount factor $0 < \\gamma < 1$, and a policy $\\pi : \\mathcal{S} \\to \\mathcal{P}(\\mathcal{A})$, the Bellman operator for policy $\\pi$ acting on a function $Q : \\mathcal{S} \\times \\mathcal{A} \\to \\mathbb{R}$ is well-defined as a deterministic operator, and it satisfies the fixed-point identity $T^{\\pi} Q^{\\pi}(s,a) = Q^{\\pi}(s,a)$ (where $Q^{\\pi}(s,a) = \\mathbb{E}[\\sum_{t=0}^{\\infty} \\gamma^t r(s_t,a_t) \\mid s_0=s, a_0=a]$), so that the resulting Bellman errors and the approximation error $\\epsilon_{approx}$ of Assumption 5 control the critic estimation error used in the convergence analysis.", "proof": {"steps": [{"n": 1, "claim": "$(T^{\\pi} Q)(s, a) = r'(s,a) + \\gamma \\int Q(s', \\pi(s')) P(ds' | s, a)$, where $r'(s,a)$ is a single realization of the random reward $R(\\cdot|s,a)$."}, {"n": 2, "claim": "The fixed-point identity $T^{\\pi} Q^{\\pi}(s,a) = Q^{\\pi}(s,a)$ holds for the action value function $Q^{\\pi}$."}, {"n": 3, "claim": "The Bellman error at inner iteration $j$, outer iteration $k$ is $\\epsilon_{k,j} = T^{\\pi_{\\lambda_k}} Q_{k,j-1} - Q_{k,j}$, and the critic step minimizes $\\mathbb{E}_{s,a}(T^{\\pi_{\\lambda_k}} Q_{k,j-1}(s,a) - Q_\\theta(s,a))^2$ (Eqn 12)."}, {"n": 4, "claim": "The approximation error $\\epsilon_{approx}$ of Assumption 5, $\\min_{\\theta_1\\in\\Theta}\\mathbb{E}_{s,a\\sim\\zeta_\\nu^{\\pi_{\\lambda_k}}}(Q_{\\theta_1}(s,a) - T^{\\pi_\\lambda}Q_\\theta(s,a))^2 \\le \\epsilon_{approx}$, together with the fixed-point identity, bounds the critic estimation error $\\mathbb{E}_{s,a}|Q^{\\pi_{\\lambda_k}} - Q_{k,J}|$ via the error recursion $\\sum_{j=1}^{J-1}\\gamma^{J-j-1}(P^{\\pi_{\\lambda_k}})^{J-j-1}\\mathbb{E}|\\epsilon_{k,j}| + \\gamma^J\\frac{R_{max}}{1-\\gamma}$ (Eqn 18, Eqns 19-22)."}]}}, {"local_id": "T2", "statement": "For outer iteration $k$ and inner iteration $j$ of Algorithm 1, the Bellman error $\\epsilon_{k,j}=T^{\\pi_{\\lambda_k}}Q_{k,j-1}-Q_{k,j}$ admits the exact decomposition $\\epsilon_{k,j}=\\epsilon_{k,j}^{1}+\\epsilon_{k,j}^{2}+\\epsilon_{k,j}^{3}+\\epsilon_{k,j}^{4}$ into the approximation, estimation, sampling, and optimization errors, defined through the intermediate minimizers $Q_{k,j}^{1},Q_{k,j}^{2},Q_{k,j}^{3}$.", "proof": {"steps": [{"n": 1, "claim": "The Bellman error is $\\epsilon_{k,j}=T^{\\pi_{\\lambda_k}}Q_{k,j-1}-Q_{k,j}$, the gap between the fixed critic target $T^{\\pi_{\\lambda_k}}Q_{k,j-1}$ and the critic estimate $Q_{k,j}$ produced at inner iteration $j$."}, {"n": 2, "claim": "Insert the three intermediate Q-functions and telescope: $\\epsilon_{k,j}=(T^{\\pi_{\\lambda_k}}Q_{k,j-1}-Q_{k,j}^{1})+(Q_{k,j}^{1}-Q_{k,j}^{2})+(Q_{k,j}^{2}-Q_{k,j}^{3})+(Q_{k,j}^{3}-Q_{k,j})$."}, {"n": 3, "claim": "Name the four brackets: $\\epsilon_{k,j}^{1}=T^{\\pi_{\\lambda_k}}Q_{k,j-1}-Q_{k,j}^{1}$ (approximation error), $\\epsilon_{k,j}^{2}=Q_{k,j}^{1}-Q_{k,j}^{2}$ (estimation error), $\\epsilon_{k,j}^{3}=Q_{k,j}^{2}-Q_{k,j}^{3}$ (sampling error), $\\epsilon_{k,j}^{4}=Q_{k,j}^{3}-Q_{k,j}$ (optimization error)."}, {"n": 4, "claim": "Hence $\\epsilon_{k,j}=\\epsilon_{k,j}^{1}+\\epsilon_{k,j}^{2}+\\epsilon_{k,j}^{3}+\\epsilon_{k,j}^{4}$, and each component is bounded separately (Lemmas 1-4) before being propagated through the critic bound (Eqn 18)."}]}}, {"local_id": "T3", "statement": "Under a smooth (e.g. softmax) policy parametrization satisfying Assumption 2, the natural-policy-gradient direction of Algorithm 1 satisfies the compatible-function-approximation identity $F_{\\nu}^{\\dagger}(\\lambda_k)\\nabla_{\\lambda_k}V^{\\pi_{\\lambda_k}}(\\nu)=\\frac{1}{1-\\gamma}w_k^{*}$, where $w_k^{*}=\\arg\\min_{w}\\mathbb{E}_{s,a}\\big(A^{\\pi_{\\lambda_k}}(s,a)-w\\nabla_{\\lambda}\\log\\pi_{\\lambda_k}(a|s)\\big)^{2}$ with $s\\sim d_{\\nu}^{\\pi_{\\lambda_k}},a\\sim\\pi_{\\lambda_k}(\\cdot|s)$.", "proof": {"steps": [{"n": 1, "claim": "The policy gradient theorem gives $\\nabla_{\\lambda_k}V^{\\pi_{\\lambda_k}}(\\nu)=\\mathbb{E}_{s,a}[\\nabla\\log\\pi_{\\lambda_k}(a|s)\\,Q^{\\pi_{\\lambda_k}}(s,a)]$ with $s\\sim d_{\\nu}^{\\pi_{\\lambda_k}},a\\sim\\pi_{\\lambda_k}$."}, {"n": 2, "claim": "The natural-gradient preconditioner is the Fisher information matrix $F_{\\nu}(\\lambda_k)=\\mathbb{E}_{s,a}[\\nabla\\log\\pi_{\\lambda_k}(a|s)(\\nabla\\log\\pi_{\\lambda_k}(a|s))^{\\top}]$."}, {"n": 3, "claim": "Subtracting the state-dependent baseline $V^{\\pi_{\\lambda_k}}$ (which leaves the gradient unchanged) replaces $Q^{\\pi_{\\lambda_k}}$ by $A^{\\pi_{\\lambda_k}}=Q^{\\pi_{\\lambda_k}}-V^{\\pi_{\\lambda_k}}$, and projecting onto the compatible features $\\nabla\\log\\pi_{\\lambda_k}$ gives $F_{\\nu}^{\\dagger}(\\lambda_k)\\nabla_{\\lambda_k}V^{\\pi_{\\lambda_k}}(\\nu)=\\frac{1}{1-\\gamma}w_k^{*}$ with $w_k^{*}$ solving Eqn 11."}, {"n": 4, "claim": "Therefore the NPG update $\\lambda_{k+1}=\\lambda_k+\\eta F_{\\nu}^{\\dagger}(\\lambda_k)\\nabla_{\\lambda_k}V^{\\pi_{\\lambda_k}}(\\nu)$ reduces to advantage regression: estimate $w_k^{*}$ by SGD on the loss in Eqn 11 and set $\\lambda_{k+1}=\\lambda_k+\\eta w_k$."}]}}, {"local_id": "T4", "statement": "For each outer iteration $k$ of Algorithm 1, the error term $err_k=\\mathbb{E}_{s\\sim d_{\\nu}^{\\pi^{*}},a\\sim\\pi^{*}(\\cdot|s)}\\big|A^{\\pi_{\\lambda_k}}(s,a)-w^{k}(s,a)\\nabla\\log\\pi_{\\lambda_k}(a|s)\\big|$ appearing in the value-suboptimality bound (Eqn 16) admits the upper bound $err_k\\le \\underbrace{\\mathbb{E}_{s,a}|A^{\\pi_{\\lambda_k}}-A_{k,J}|}_{I}+\\underbrace{\\mathbb{E}_{s,a}|A_{k,J}-w^{k}\\nabla\\log\\pi_{\\lambda_k}(a|s)|}_{II}$, separating the critic estimation error $I$ from the actor estimation error $II$.", "proof": {"steps": [{"n": 1, "claim": "The value-suboptimality sketch yields $\\min_{k}V^{*}(\\nu)-V^{\\pi_{\\lambda_k}}(\\nu)\\le\\frac{\\log|\\mathcal{A}|}{K\\eta(1-\\gamma)}+\\frac{\\eta\\beta W^{2}}{2(1-\\gamma)}+\\frac{1}{K}\\sum_{k=1}^{K}\\frac{err_k}{1-\\gamma}$, with $err_k$ as defined in Eqn 17 and $\\|w^{k}\\|_2\\le W$."}, {"n": 2, "claim": "Inside the absolute value defining $err_k$, insert the critic advantage estimate $A_{k,J}$ produced by the first inner for loop: $A^{\\pi_{\\lambda_k}}-w^{k}\\nabla\\log\\pi_{\\lambda_k}=(A^{\\pi_{\\lambda_k}}-A_{k,J})+(A_{k,J}-w^{k}\\nabla\\log\\pi_{\\lambda_k})$."}, {"n": 3, "claim": "Applying the triangle inequality pointwise and then monotonicity of the expectation gives $err_k\\le\\mathbb{E}_{s,a}|A^{\\pi_{\\lambda_k}}-A_{k,J}|+\\mathbb{E}_{s,a}|A_{k,J}-w^{k}\\nabla\\log\\pi_{\\lambda_k}(a|s)|=I+II$."}, {"n": 4, "claim": "Term $I$ is the critic error (the gap between the true advantage $A^{\\pi_{\\lambda_k}}$ and its estimate $A_{k,J}$ from the first inner loop) and term $II$ is the actor error (the linear-regression residual against $\\nabla\\log\\pi_{\\lambda_k}$ optimized in the second inner loop)."}]}}], "target_local_id": "T1", "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "label": "flawed", "error_step": 1, "error_class6": ["missing_assumption", "others", "computation_or_constant"], "error_explanation": "The Bellman operator is defined using a single realization $r'(s,a)$ of the random reward $R(\\cdot|s,a)$ as its first term. Because $r'(s,a)$ is a random draw, $(T^{\\pi}Q)(s,a)$ is itself a random variable rather than a deterministic function of $(s,a)$. Consequently the deterministic fixed-point identity $T^{\\pi}Q^{\\pi} = Q^{\\pi}$ — on which every subsequent error bound depends — fails whenever the reward is not deterministic, and $\\epsilon_{approx}$ in Assumption 4 can be as large as $O(1)$ (the reward variance), rendering the convergence theorem meaningless. The first term should have been the conditional expected reward $\\mathbb{E}(r(s,a))$, which keeps $T^{\\pi}Q$ deterministic and makes $T^{\\pi}Q^{\\pi}=Q^{\\pi}$ hold even for random rewards."}
{"id": "ch10_proposition_10_1__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nDefinition 6.3. Given \\(S_{n}=\\{Z_{1},\\ldots,Z_{n}\\}\\), the (one-sided) empirical Rademacher complexity of \\(\\mathcal{G}\\) is defined as\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n}) = \\mathbb{E}_{\\sigma}\\sup_{w\\in \\Omega}\\frac{1}{n}\\sum_{i = 1}^{n}\\sigma_{i}\\phi (w,Z_{i}),\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R(\\mathcal{G},\\mathcal{S}_{n}).\n\\]\nThe following result shows that the quantity \\(\\epsilon_{n}(\\mathcal{G},\\mathcal{D})\\) can be upper bounded by Rademacher complexity. It follows that an average oracle inequality can be obtained using Rademacher complexity. The proof employs the symmetrization technique, which was also used in Chapter 4 to obtain uniform convergence bounds from empirical covering numbers.\n\nDefinition 10.6. Let \\(\\mathcal{F}_{L_1}(\\Psi)\\) be the point-wise closure of \\(\\cup_{A > 0}\\mathcal{F}_{A,L_1}(\\Psi)\\), then for any \\(f\\in \\mathcal{F}_{L_1}(\\Psi)\\),\n\\[\n\\|f\\|_{1}=\\lim_{\\epsilon\\rightarrow 0}\\inf\\left\\{\\|w\\|_{1}:\\sup_{x}\\left|f(x)-\\sum_{j=1}^{m}w_{j}\\psi(\\theta_{j},x)\\right|\\leq\\epsilon\\right\\}.\n\\]\nFor notational convenience, we write functions in \\(\\mathcal{F}_{L_1}(\\Psi)\\) as\n\\[\nf(x)=w^{\\top}\\psi(x),\n\\]\nwhere \\(\\psi(x)\\) is the infinite dimensional vector \\([\\psi(\\theta, x)]_{\\theta \\in \\Theta}\\), and \\(\\|f\\|_1 = \\|w\\|_1\\).\nFor smooth-loss functions (see Definition A.5), there is a strong relationship between \\( L_{1} \\) regularization and \\( L_{0} \\) regularization, in that functions with small \\( L_{1} \\) norms can be sparsified. Such a sparsification result was first shown for the least squares loss, and referred to as Maurey's lemma in Pisier (1980-1981).\n\nPacking Number. Let \\((\\mathcal{V},d)\\) be a pseudometric space with metric \\(d(\\cdot,\\cdot)\\). A finite subset \\(\\mathcal{G}(\\epsilon)\\subset \\mathcal{G}\\) is an \\(\\epsilon\\)-packing of \\(\\mathcal{G}\\) if \\(d(\\phi,\\phi^{\\prime})>\\epsilon\\) for all \\(\\phi,\\phi^{\\prime}\\in\\mathcal{G}(\\epsilon)\\). The \\(\\epsilon\\)-packing number of \\(\\mathcal{G}\\), denoted by \\(M(\\epsilon,\\mathcal{G},d)\\), is the largest cardinality of \\(\\epsilon\\)-packing of \\(\\mathcal{G}\\).\nThe following results illustrate the equivalence between covering number and packing number. One advantage of using an \\(\\epsilon\\) packing of \\(\\mathcal{G}\\) instead of an \\(\\epsilon\\) cover of \\(\\mathcal{G}\\) is that all members in the \\(\\epsilon\\) packing also belong to \\(\\mathcal{G}\\). There if members in \\(\\mathcal{G}\\) satisfy certain assumptions such as the variance condition, then members of its \\(\\epsilon\\) packing also satisfy such assumptions. For this reason, we will use packing numbers instead of covering numbers in some of the theoretical analysis in later chapters.\n\nDefinition 4.1. Let \\((\\mathcal{V},d)\\) be a pseudometric space with metric \\(d(\\cdot,\\cdot)\\). A finite\nset \\(\\mathcal{G}(\\epsilon) \\subset \\mathcal{V}\\) is an \\(\\epsilon\\)-cover (or \\(\\epsilon\\)-net) of \\(\\mathcal{G} \\subset \\mathcal{V}\\) if, for all \\(\\phi \\in \\mathcal{G}\\), there exists \\(\\phi' \\in \\mathcal{G}(\\epsilon)\\) so that \\(d(\\phi', \\phi) \\leq \\epsilon\\). The \\(\\epsilon\\)-covering number of \\(\\mathcal{G}\\) with metric \\(d\\) is the smallest cardinality \\(N(\\epsilon, \\mathcal{G}, d)\\) of such \\(\\mathcal{G}(\\epsilon)\\). The number \\(\\ln N(\\epsilon, \\mathcal{G}, d)\\) is called the \\(\\epsilon\\)-entropy.\nFor a function class \\(\\mathcal{G}\\) with seminorm \\(L_{p}(\\mathcal{D})\\) in (3.19), we denote the corresponding \\(L_{p}(\\mathcal{D})\\)-covering number as \\(N(\\epsilon, \\mathcal{G}, L_{p}(\\mathcal{D}))\\). When \\(1 \\leq p \\leq q\\), we have\n\\[\nN(\\epsilon,\\mathcal{G},L_{p}(\\mathcal{D}))\\leq N(\\epsilon,\\mathcal{G},L_{q}(\\mathcal{D})).\n\\]\nIt is easy to verify the following relationship, which implies that \\( L_{p}(\\mathcal{D}) \\) bracketing cover is a stronger requirement than \\( L_{p}(\\mathcal{D}) \\) cover.\n\nEmpirical and Uniform Covering Number. Given an empirical distribution \\( S_{n}=\\{Z_{1},\\ldots,Z_{n}\\} \\), we define the pseudometric \\( d = L_{p}(S_{n}) \\) as\n\\[\nd(\\phi,\\phi^{\\prime})=\\left[\\frac{1}{n}\\sum_{i=1}^{n}|\\phi(Z_{i})-\\phi^{\\prime}(Z_{i})|^{p}\\right]^{1/p}.\n\\]\nThe corresponding metric covering number \\(N(\\epsilon, \\mathcal{G}, L_p(\\mathcal{S}_n))\\) is referred to as the\nempirical \\( L_{p} \\) covering number. Given \\( n \\), the largest \\( L_{p} \\) covering number over empirical distribution \\( S_{n} \\) is referred to as the uniform \\( L_{p} \\) covering number\n\\[\nN_{p}(\\epsilon,\\mathcal{G},n)=\\sup_{\\mathcal{S}_{n}}N(\\epsilon,\\mathcal{G},L_{p}(\\mathcal{S}_{n})).\n\\]\nSince the \\(L_{p}(\\mathcal{S}_{n})\\) pseudometric increases with \\(p\\), we have the following simple result.\n\nL1-regularized function classes. The \\(L_{1}\\)-regularized nonnegative function class induced by \\(\\Psi = \\{\\psi(\\theta,x):\\theta\\in\\Theta\\}\\) at radius \\(A\\) is defined as\n\\[\n\\mathcal{F}_{A,L_{1}}^{+}(\\Psi)=\\left\\{\\sum_{j=1}^{m}w_{j}\\psi\\left(\\theta_{j},x\\right):m>0,\\|w\\|_{1}\\leq A,w_{j}\\geq 0,\\theta_{j}\\in\\Theta\\right\\}.\n\\]\nThe signed \\(L_{1}\\)-regularized function class is defined as\n\\[\n\\mathcal{F}_{A,L_{1}}(\\Psi)=\\left\\{\\sum_{j=1}^{m}w_{j}\\psi\\left(\\theta_{j},x\\right):m>0,\\|w\\|_{1}\\leq A,w_{j}\\in\\mathbb{R},\\theta_{j}\\in\\Theta\\right\\}.\n\\]\n\nEquation 10.2. \\[\n\\mathcal{F}^{\\prime}=\\{f(w,x)=w^\\top\\psi(x):\\|w\\|_2\\leq A\\}.\n\\]\n\nExample 10.11. In two-layer neural networks, let \\(\\Psi\\) be an \\(L_{2}\\)-regularized ReLU function class:\n\\[\n\\Psi=\\left\\{\\psi(\\theta,x)=\\max(0,\\theta^{\\top}x):\\quad\\|\\theta\\|_{2}\\leq\\alpha,\\|x\\|_{2}\\leq\\beta\\right\\},\n\\]\nand the corresponding \\(L_{1}\\) regularized two-layer NN can be expressed as a function\n\\[\nf(x)=\\sum_{j=1}^{m}w_{j}\\psi\\left(\\theta_{j},x\\right):\\|w\\|_{1}\\leq A,\\|\\theta\\|_{2}\\leq\\alpha,\\|x\\|_{2}\\leq\\beta .\n\\]\nThis function class belongs to \\(\\mathcal{F}_{A,L_1}(\\Psi)\\). We thus obtain the following bound for \\(L_{1}\\) regularized two-layer NN:\n\\[\n\\begin{array}{l}R\\left(\\mathcal{F}_{A,L_{1}}(\\Psi),\\mathcal{S}_{n}\\right)\\leq 2AR(\\Psi,\\mathcal{S}_{n})\\tag{Theorem 10.8}\\\\ \\leq 2A\\alpha\\beta/\\sqrt{n},\\quad(\\text{C o r o l l a r y} 9. 21)\\\\ \\end{array}\n\\]\nwhere we note that \\(\\max (0,f)\\) is 1-Lipschitz in \\(f\\)\nNext, we may consider the following hard-constrained \\( L_{1} \\) regularized learning problem.\n\\[\n\\hat{w}=\\arg\\min_{w}\\frac{1}{n}\\sum_{i=1}^{n}L\\left(w^{\\top}\\psi\\left(X_{i}\\right),Y_{i}\\right),\\quad\\|w\\|_{1}\\leq A.\\tag{10.8}\n\\]\nSimilarly, we may consider the soft-regularized version as:\n\\[\n\\hat{w}=\\arg\\min_{w}\\frac{1}{n}\\sum_{i=1}^{n}L\\left(w^{\\top}\\psi\\left(X_{i}\\right),Y_{i}\\right)+\\lambda\\|w\\|_{1}.\\tag{10.9}\n\\]\nNow by using Theorem 6.31, we obtain the following result.", "theorems": [{"local_id": "T1", "statement": "Let \\(\\| \\cdot \\|\\) be a seminorm on \\(\\mathbb{R}^k\\). Let \\(B(r) = \\{z \\in \\mathbb{R}^k : \\| z \\| \\leq r\\}\\) be the \\(\\| \\cdot \\|\\)-ball with radius \\(r\\). Then\n\\[\nM(\\epsilon,B(r),\\|\\cdot\\|)\\leq\\left(1+2 r/\\epsilon\\right)^{k}.\n\\]\nMoreover, if \\(\\| \\cdot \\|\\) is a norm, then\n\\[\nN(\\epsilon,B(r),\\|\\cdot\\|)\\geq(r/\\epsilon)^{k}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "For simplicity, we only consider the case \\( \\| \\cdot \\| \\) is a norm."}, {"n": 2, "claim": "Let \\( \\{z_1,\\ldots,z_M\\}\\subset B(r) \\) be a maximal \\( \\epsilon \\) packing of \\( B(r) \\)."}, {"n": 3, "claim": "Let \\( B_{j} = \\{z\\in \\mathbb{R}^{k}:||z - z_{j}||\\leq \\epsilon /2\\} \\), then \\( B_{j}\\cap B_{k} = \\emptyset \\) for \\( j\\neq k \\) and \\( B_{j}\\subset B(r + \\epsilon /2) \\) for all \\( j \\)."}, {"n": 4, "claim": "It follows that\n\\[\n\\sum_{j=1}^{M}\\operatorname{volume}(B_{j})=\\operatorname{volume}(\\cup_{j=1}^{M}B_{j})\\leq\\operatorname{volume}(B(r+\\epsilon/2)).\n\\]\nLet \\( v = \\mathrm{volume}(B(1)) \\)."}, {"n": 5, "claim": "Since \\( \\mathrm{volume}(B_j) = (\\epsilon / 2)^k v \\) and \\( \\mathrm{volume}(B(r + \\epsilon / 2)) = (r + \\epsilon / 2)^k v \\), we have\n\\[\nM(\\epsilon/2)^{k}v\\leq(r+\\epsilon/2)^{k}v.\n\\]\nThis implies the first bound."}, {"n": 6, "claim": "Let \\(\\{z_1, \\ldots, z_N\\} \\subset \\mathbb{R}^k\\) be a cover of \\(B(r)\\)."}, {"n": 7, "claim": "If we define \\(B_j = \\{z \\in \\mathbb{R}^k : \\|z - z_j\\| \\leq \\epsilon\\}\\), then \\(B(r) \\subset \\cup_j B_j\\)."}, {"n": 8, "claim": "Therefore\n\\[\n\\operatorname{volume}(B(r))\\leq\\operatorname{volume}\\left(\\cup_{j=1}^{N}B_{j}\\right)\\leq\\sum_{j=1}^{N}\\operatorname{volume}\\left(B_{j}\\right).\n\\]\nLet \\( v = \\mathrm{volume}(B(1)) \\)."}, {"n": 9, "claim": "Since \\( \\mathrm{volume}(B_j) = (\\epsilon)^k v \\) and \\( \\mathrm{volume}(B(r)) = r^k v \\), we have\n\\[\nr^{k}v\\leq N\\epsilon^{k}v.\n\\]\nThis implies the second bound."}]}}, {"local_id": "T2", "statement": "Assume that \\( L(p, y) \\geq 0 \\) is \\( \\gamma \\) Lipschitz, \\( M_0 = \\sup_y L(0, y) \\), and \\( B = \\sup_{x, \\psi \\in \\Psi} |\\psi(x)| \\). Consider \\( A_0 > 0 \\), then with probability at least \\( 1 - \\delta \\): the following inequality holds for all \\( w \\):\n\\[\n\\begin{array}{l}\\mathbb{E}_{\\mathcal{D}}L(w^{\\top}\\psi(X),Y)\\leq\\frac{1}{n}\\sum_{i=1}^{n}L(w^{\\top}\\psi(X_{i}),Y_{i})+4\\gamma(A_{0}+\\|w\\|_{1})R_{n}(\\Psi_{\\pm},\\mathcal{D})\\\\ +\\left(M_{0}+2\\gamma B\\left(A_{0}+\\|w\\|_{1}\\right)\\right)\\left[\\sqrt{\\frac{\\ln\\left(2+\\log_{2}(1+\\|w\\|_{1}/A_{0})\\right)}{n}}+\\sqrt{\\frac{\\ln(1/\\delta)}{2 n}}\\right].\\\\ \\end{array}\n\\]\nConsider (10.9) with\n\\[\n\\lambda\\geq 4\\gamma R_{n}(\\Psi_{\\pm},\\mathcal{D})+2\\gamma B\\sqrt{\\frac{\\ln(2+\\log_{2}(1+M_{0}/(\\lambda A_{0})))}{n}}.\n\\]\nWe have the following oracle inequality. With probability at least \\(1 - \\delta\\):\n\\[\n\\begin{array}{l}\\mathbb{E}_{\\mathcal{D}}L\\left(\\hat{w}^{\\top}\\psi(X),Y\\right)\\leq\\inf_{w}\\left[\\mathbb{E}_{\\mathcal{D}}L\\left(w^{\\top}\\psi(X),Y\\right)+\\left(\\lambda+4\\gamma B\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}\\right)\\|w\\|_{1}\\right]\\\\ +\\epsilon_{n}(\\delta),\\\\ \\end{array}\n\\]\nwhere\n\\[\n\\begin{array}{l}\\epsilon_{n}(\\delta)=4\\gamma A_{0}R_{n}(\\Psi_{\\pm},\\mathcal{D})+(M_{0}+2\\gamma A_{0}B)\\sqrt{\\frac{\\ln((2+\\log_{2}(1+M_{0}/(\\lambda A_{0}))))}{n}}\\\\ +(2 M_{0}+4\\gamma A_{0}B)\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}.\\\\ \\end{array}\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\( A_{\\theta} = 2^{\\theta}A_{0} \\), with \\( q(\\theta) = (1 + \\theta)^{-2} \\) for \\( \\theta = 1,2,\\ldots \\), and let \\( f(x) = w^{\\top}\\psi(x) \\)."}, {"n": 2, "claim": "Consider \\( \\mathcal{F}(1) = \\{w^{\\top}\\psi(x):\\|w\\|_{1}\\leq A_{1}\\} \\), and \\( \\mathcal{F}(\\theta) = \\{w^{\\top}\\psi(x):A_{\\theta-1}\\leq \\|w\\|_{1}\\leq A_{\\theta}\\} \\) for \\( \\theta > 1 \\)."}, {"n": 3, "claim": "We have\n\\[\nR_{n}(\\mathcal{F}(\\theta),\\mathcal{D})\\leq\\gamma A_{\\theta}R_{n}(\\Psi_{\\pm},\\mathcal{D}).\n\\]\nGiven any \\(w\\), let \\(\\theta\\) be the smallest number such that \\(f(w, x) = w^\\top \\Psi(x) \\in \\mathcal{F}(\\theta)\\), then \\(A_\\theta \\leq 2(A_0 + \\|w\\|_1)\\)."}, {"n": 4, "claim": "Therefore\n\\[\nL(f(x),y)\\leq L(0,y)+\\gamma|f(x)|\\leq M_{0}+\\gamma A_{\\theta}B\\leq M_{0}+2\\gamma(A_{0}+\\|w\\|_{1})B.\n\\]\nWe can take \\( h = 0 \\) in Theorem 8.7, \\( M(\\theta) \\leq M_0 + 2\\gamma (A_0 + \\| w\\| _1)B \\), and \\( 1 / q(\\theta) \\leq (2 + \\log_2(1 + \\| w\\| _1 / A_0))^2 \\)."}, {"n": 5, "claim": "Let\n\\[\n\\begin{array}{l}\\tilde{R}(\\theta,f,\\mathcal{S}_{n})=4\\gamma\\left(A_{0}+\\|w\\|_{1}\\right)R_{n}\\left(\\Psi_{\\pm},\\mathcal{D}\\right)\\\\ +\\left(M_{0}+2\\gamma B\\left(A_{0}+\\|w\\|_{1}\\right)\\right)\\sqrt{\\frac{\\ln\\left(2+\\log_{2}\\left(1+\\|w\\|_{1}/A_{0}\\right)\\right)}{n}}.\\tag{10.10}\\\\ \\end{array}\n\\]\nThis implies the desired uniform convergence result."}, {"n": 6, "claim": "Now we would like to show the second desired oracle inequality."}, {"n": 7, "claim": "With the condition of \\(\\lambda\\), we have \\(\\| \\hat{w} \\|_1 \\leq M_0 / \\lambda\\), and by considering \\(\\| w \\|_1 \\leq M_0 / \\lambda\\), we can redefine\n\\[\n\\begin{array}{l}\\tilde{R}(\\theta)=\\tilde{R}(\\theta,f,S_{n})=\\lambda\\|w\\|_{1}+4\\gamma A_{0}R_{n}\\left(\\Psi_{\\pm},\\mathcal{D}\\right)\\\\ +\\left(M_{0}+2\\gamma A_{0}B\\right)\\sqrt{\\frac{\\ln\\left(2+\\log_{2}\\left(1+M_{0}/\\left(\\lambda A_{0}\\right)\\right)\\right)}{n}}.\\\\ \\end{array}\n\\]\nThis definition of \\(\\tilde{R}(\\theta)\\) is an upper bound of (10.10)."}, {"n": 8, "claim": "We can thus apply Theorem 8.7 again to obtain the desired oracle inequality, where we also use \\(2M_0 + 4\\gamma A_0B + 4\\gamma B\\|w\\|_1\\) as an upper bound for \\(2M(\\theta)\\)."}]}}, {"local_id": "T3", "statement": "Assume that the \\(m\\) feature functions \\(\\{\\psi_j(X)\\}\\) are orthonormal when \\(X\\sim \\mathcal{D}\\). Then there exists an absolute constant \\(c > 0\\) such that for sufficiently large \\(n\\),\n\\[\nR_{n}(\\mathcal{F}^{\\prime},\\mathcal{D})\\geq c\\sqrt{\\frac{m}{n}}A,\n\\]\nwhere \\(\\mathcal{F}'\\) is given by (10.2).", "proof": {"steps": [{"n": 1, "claim": "For sufficiently large \\(n\\), the feature vectors \\(\\{\\psi_j\\}\\) are near orthonormal with respect to the empirical distribution \\(S_n\\)."}, {"n": 2, "claim": "With constant probability, \\(\\mathcal{F}'\\) is nearly an \\(m\\)-dimensional ball of radius \\(A\\) with respect to \\(L_2(S_n)\\)."}, {"n": 3, "claim": "The packing number satisfies \\(M(\\epsilon,\\mathcal{F},L_2(S_n)) \\leq 2^{m}\\) at \\(\\epsilon = c_0 A\\) for some \\(c_0>0\\)."}, {"n": 4, "claim": "Sudakov minoration gives \\(R_n(\\mathcal{F}',\\mathcal{D}) \\geq c\\sqrt{m/n}\\,A\\) for an absolute constant \\(c>0\\)."}]}}], "target_local_id": "T3", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "label": "flawed", "error_step": 3, "error_class6": ["invalid_bound_or_inequality", "logical_gap_or_missing_case", "computation_or_constant"], "error_explanation": "The inequality direction is flipped: Theorem 5.3 yields a packing LOWER bound \\(M(\\epsilon,\\mathcal{F},L_2(S_n)) \\geq 2^m\\), not \\(\\leq 2^m\\). Sudakov minoration requires a large packing number (lower bound) to force a large Rademacher complexity; an upper bound \\(\\leq 2^m\\) gives no such conclusion, so the relational claim is genuinely false."}
{"id": "ch13_proposition_13_2__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nConvex set. A set \\(K\\subseteq\\mathbb{R}^n\\) is convex if the line segment between any two points in \\(K\\) is also contained in \\(K\\). Formally, for all \\(x,y\\in K\\) and all scalars \\(\\gamma\\in[0,1]\\) we have \\(\\gamma x+(1-\\gamma)y\\in K\\).\n\nStrong convexity. A function \\(f\\colon\\Omega\\to\\mathbb{R}\\) is \\emph{\\(\\alpha\\)-strongly convex} if, for all \\(x,y\\in\\Omega\\), the following inequality holds for some \\(\\alpha>0\\): \\[ f(y) \\geqslant f(x) + g(x)^\\top(y-x) + \\frac{\\alpha}{2}\\left\\lVert x-y\\right\\rVert^2 \\]\n\nUniform stability. The uniform stability of an algorithm \\(A\\) is defined as\n\\[\n\\Delta_{\\sup}(A) = \\sup_{S, S' \\in (X\\times Y)^m } \\sup_{i \\in [m]} |\\ell(A(S), z_i') - \\ell(A(S^{(i)}, z_i')|\n\\]", "theorems": [{"local_id": "T1", "statement": "Let \\(\\Omega \\subseteq \\mathbb{R}^d\\) be a closed convex set. Let \\(\\mathcal{A}\\) denote the affine set \\(\\{x : Ax = b\\}\\) for some \\(A \\in \\mathbb{R}^{n \\times d}\\) and \\(b \\in \\mathbb{R}^n\\). Suppose that the set \\(\\operatorname{ri}(\\Omega) \\cap \\mathcal{A}\\) is non-empty. Then for any \\(x \\in \\Omega \\cap \\mathcal{A}\\), \\[\\mathcal{N}_{\\Omega \\cap \\mathcal{A}}(x) = \\mathcal{N}_{\\Omega}(x) + \\{A^\\top \\lambda : \\lambda \\in \\mathbb{R}^n\\}\\,.\\]", "proof": {"steps": [{"n": 1, "claim": "\\(\\supseteq\\): for \\(z\\in\\Omega\\cap\\mathcal{A}\\), \\(z-x\\in\\operatorname{null}(A)\\), \\((z-x)^\\top A^\\top\\lambda=0\\), and \\(u\\in\\mathcal{N}_\\Omega(x)\\) gives \\(u+A^\\top\\lambda\\in\\mathcal{N}_{\\Omega\\cap\\mathcal{A}}(x)\\)."}, {"n": 2, "claim": "\\(\\subseteq\\): let \\(v\\in\\mathcal{N}_{\\Omega\\cap\\mathcal{A}}(x)\\), so \\(v^\\top(z-x)\\leqslant 0\\) on \\(\\Omega\\cap\\mathcal{A}\\)."}, {"n": 3, "claim": "Set \\(C_1=\\{(y,\\mu):y=z-x,\\ z\\in\\Omega,\\ \\mu\\leqslant v^\\top y\\}\\), \\(C_2=\\{(y,\\mu):y\\in\\ker(A),\\ \\mu=0\\}\\), with \\(C_1\\cap C_2=\\emptyset\\)."}, {"n": 4, "claim": "By the Hahn–Banach separation theorem, the disjoint convex sets \\(C_1\\) and \\(C_2\\) admit a hyperplane \\((w,\\gamma)\\) with \\(\\sup_{C_1}\\{w^\\top y+\\gamma\\mu\\}\\leqslant 0<\\inf_{C_2}\\) and \\(w\\perp\\ker(A)\\)."}, {"n": 5, "claim": "Thus \\(w=A^\\top\\lambda\\) for some \\(\\lambda\\)."}, {"n": 6, "claim": "\\(\\gamma>0\\) after excluding \\(\\gamma<0\\) (unbounded sup) and \\(\\gamma=0\\) (relative-interior hypothesis); normalize \\(\\gamma=1\\)."}, {"n": 7, "claim": "Then \\((w+v)^\\top(z-x)\\leqslant 0\\) for all \\(z\\in\\Omega\\), so \\(v+w\\in\\mathcal{N}_\\Omega(x)\\)."}, {"n": 8, "claim": "Hence \\(v=(v+w)-w\\in\\mathcal{N}_\\Omega(x)+\\{A^\\top\\lambda\\}\\)."}]}}, {"local_id": "T2", "statement": "For any function \\(\\varphi(x,z)\\), \\[\\min_x \\max_z \\varphi(x,z) \\geqslant \\max_z \\min_x \\varphi(x,z)\\,.\\]", "proof": {"steps": [{"n": 1, "claim": "The proof is essentially tautological. Note that we always have \\[\\varphi(x,z) \\geqslant \\min_x \\varphi(x,z).\\] Taking the maximization with respect to the second argument verifies that \\[\\max_z \\varphi(x,z) \\geqslant \\max_z \\min_x \\varphi(x,z) \\quad \\forall x\\,.\\] Now, minimizing the left hand side with respect to \\(x\\) proves \\[\\min_x \\max_z \\varphi(x,z) \\geqslant \\max_z \\min_x \\varphi(x,z)\\] which is precisely our assertion."}]}}, {"local_id": "T3", "statement": "Assume \\(\\ell(w, z)\\) is \\(\\alpha\\)-strongly convex over the domain \\(\\Omega\\) and \\(L\\)-Lipschitz. Let \\(\\hat w_S = \\arg \\min_{w\\in \\Omega} \\frac{1}{m} \\sum_{i=1}^m \\ell(w, z_i)\\) denote the empirical risk minimizer (ERM). Then, ERM satisfies\n\\[\n\\Delta_{\\sup} (\\text{ERM}) \\leqslant \\frac{4L^2}{\\alpha m}\\,.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Denote by \\(\\hat w_S\\) the empirical risk minimizer on a sample \\(S.\\) Fix arbitrary samples \\(S,S'\\) of size \\(m\\) and an index \\(i\\in[m].\\) We need to show that\n\\[\n|(\\ell(\\hat w_{S^{(i)}},z_i')-\\ell(\\hat w_{S},z_i'))|\\leqslant\\frac{4L^2}{\\alpha m}\\,.\n\\]\nOn one hand, by strong convexity we know that\n\\[\nR_S(\\hat w_{S^{(i)}}) - R_S(\\hat w_{S}) \\geqslant \\frac{\\alpha}{2} \\| \\hat w_{S} - \\hat w_{S^{(i)}} \\|^2 \\,.\n\\]\nOn the other hand,\n\\begin{align*}\n&R_S(\\hat w_{S^{(i)}})-R_S(\\hat w_{S})\\\\\n&=\\frac{1}{m}(\\ell(\\hat w_{S^{(i)}},z_i)-\\ell(\\hat w_{S},z_i))+\\frac{1}{m}\\sum_{i\\neq j}(\\ell(\\hat w_{S^{(i)}},z_j)-\\ell(\\hat w_{S},z_j))\\\\\n&=\\frac{1}{m}(\\ell(\\hat w_{S^{(i)}},z_i)-\\ell(\\hat w_{S},z_i))+\\frac{1}{m}(\\ell(\\hat w_{S},z_i')-\\ell(\\hat w_{S^{(i)}},z_i'))+\\left(R_{S^{(i)}}(\\hat w_{S^{(i)}})-R_{S^{(i)}}(\\hat w_{S})\\right)\\\\\n&\\leqslant\\frac{1}{m}|\\ell(\\hat w_{S^{(i)}},z_i)-\\ell(\\hat w_{S},z_i)| +\\frac{1}{m}|(\\ell(\\hat w_{S},z_i')-\\ell(\\hat w_{S^{(i)}},z_i'))|\\\\\n&\\leqslant\\frac{2L}{m}\\|\\hat w_{S^{(i)}}-\\hat w_{S}\\|\\,.\n\\end{align*}\nHere, we used that\n\\[\nR_{S^{(i)}}(\\hat w_{S^{(i)}}) - R_{S^{(i)}}(\\hat w_{S})) \\leqslant 0\n\\]\nand the fact that \\(\\ell\\) is \\(L\\)-Lipschitz."}, {"n": 2, "claim": "Putting it all together \\(\\| \\hat w_{S^{(i)}} - \\hat w_{S} \\| \\leqslant \\frac{4 L}{\\alpha m}\\). Then by the Lipschitz condition we have that\n\\[\n\\frac{1}{m}|(\\ell(\\hat w_{S^{(i)}},z_i')-\\ell(\\hat w_{S},z_i'))|\\leqslant L\\|\\hat w_{S^{(i)}}-\\hat w_{S}\\|\\leqslant\\frac{4L^2}{\\alpha m}\\,.\n\\]\nHence, \\(\\Delta_{\\sup}(\\text{ERM}) \\leqslant \\frac{4 L ^ 2}{\\alpha m}\\)."}]}}], "target_local_id": "T1", "topic_domain": "Optimization", "topic_subtopic": "Convex Analysis & Duality", "label": "flawed", "error_step": 3, "error_class6": ["logical_gap_or_missing_case", "others", "computation_or_constant"], "error_explanation": "Step 3 wrongly asserts C_1 ∩ C_2 = empty. The two sets in fact intersect: (0,0) belongs to both (take z=x, giving y=0 in ker(A) and mu=0 <= v^T y = 0), and so does any y in ker(A)∩(Omega-x) with v^T y = 0 (since v in N_{Omega∩A}(x) forces v^T y <= 0 there while mu=0 <= v^T y forces v^T y >= 0, hence equality). The correct construction, as in the gold proof, requires only that the RELATIVE INTERIOR of C_1 be disjoint from C_2, ri(C_1) ∩ C_2 = empty, which supports PROPER separation of ri(C_1) from the subspace C_2 (inf < sup <= 0). The error is thus a false disjointness / wrong separation setup, not a missing hypothesis of the Hahn-Banach theorem."}
{"id": "ch13_theorem_13_10__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nsection 13.1 sequential observation setup. In Section 13.1, data are observed sequentially. At each time \\(t\\), after the history \\(S_{t-1}\\) has been observed, the next observation \\(Z_t\\) is drawn according to a distribution that may depend on that history. The observation is decomposed as \\(Z_t=(Z_t^{(x)},Z_t^{(y)})\\), where \\(Z_t^{(x)}\\) is the component determined by the past history or the chosen action at time \\(t\\), and \\(Z_t^{(y)}\\) is the random response component. The history after time \\(t\\) is \\(S_t=(Z_1,\\ldots,Z_t)\\), and \\(\\mathcal{S}_T\\) denotes the full sequential experiment/history through time \\(T\\). For a function depending on \\(S_t\\), the notation \\(\\mathbb{E}_{Z_t^{(y)}}[\\cdot]\\) means conditional expectation over the response component \\(Z_t^{(y)}\\), conditional on \\(S_{t-1}\\) and on \\(Z_t^{(x)}\\). Stopping times are adapted to the filtration generated by the histories \\(S_t\\).\n\nDefinition 2.11. A sub-Gaussian random variable \\(X\\) has quadratic logarithmic moment generating function for all \\(\\lambda \\in \\mathbb{R}\\):\n\\[\n\\ln\\mathbb{E}e^{\\lambda X}\\leq\\lambda\\mu+\\frac{\\lambda^{2}}{2}b.\\tag{2.10}\n\\]\nUsing (2.10), we can obtain an upper bound of the rate function for sub-Gaussian random variables, which imply the following tail inequality.", "theorems": [{"local_id": "T1", "statement": "Let \\(\\{(X_t, \\epsilon_t)\\}\\) be a sequence in \\(\\mathbb{R}^d \\times \\mathbb{R}\\) with respect to a filtration \\(\\{\\mathcal{F}_t\\}\\) so that\n\\[\n\\mathbb{E}\\left[\\epsilon_{t}\\mid X_{t},\\mathcal{F}_{t-1}\\right]=0,\\quad\\operatorname{Var}\\left[\\epsilon_{t}\\mid X_{t},\\mathcal{F}_{t-1}\\right]\\leq\\sigma^{2}.\n\\]\nLet \\(\\Lambda_0\\) be a positive definite matrix, and\n\\[\n\\Lambda_{t}=\\Lambda_{0}+\\sum_{s=1}^{t}X_{s}X_{s}^{\\top}.\n\\]\nIf there exists \\(\\alpha > 0\\) such that \\(|\\epsilon_t| \\leq \\alpha\\), then for any \\(\\delta > 0\\), with probability at least \\(1 - \\delta\\), for all \\(t \\geq 0\\):\n\\[\n\\left\\|\\sum_{s=1}^{t}\\epsilon_{s}X_{s}\\right\\|_{\\Lambda_{t}^{-1}}^{2}\\leq 1.3\\sigma^{2}\\ln|\\Lambda_{0}^{-1}\\Lambda_{t}| +4\\alpha^{2}\\ln(2/\\delta).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Define \\(\\hat{w}_t = \\Lambda_t^{-1}\\sum_{s=1}^t \\epsilon_s X_s\\), \\(f_t = \\hat{w}_{t-1}^\\top X_t\\), and \\(b_t = 1 + X_t^\\top \\Lambda_{t-1}^{-1} X_t\\)."}, {"n": 2, "claim": "\\(\\left\\|\\sum_{s=1}^t \\epsilon_s X_s\\right\\|_{\\Lambda_t^{-1}}^2 = \\sum_{s=1}^t \\frac{(b_s-1)\\epsilon_s^2 + 2\\epsilon_s f_s - f_s^2}{b_s}\\)."}, {"n": 3, "claim": "\\(\\left|2\\epsilon_s f_s / b_s\\right| \\le 2\\alpha f_s/\\sqrt{b_s}\\)."}, {"n": 4, "claim": "By the sub-Gaussian exponential inequality of Lemma 2.15, with probability \\(\\ge 1-\\delta/2\\): \\(\\sum_{s=1}^t \\frac{2\\epsilon_s f_s - f_s^2}{b_s} \\le \\sum_{s=1}^t \\frac{2\\lambda\\alpha^2 f_s^2 - f_s^2}{b_s} + \\frac{\\ln(2/\\delta)}{\\lambda}\\) for \\(\\lambda>0\\)."}, {"n": 5, "claim": "Setting \\(\\lambda = 0.5\\alpha^{-2}\\) gives \\(\\sum_{s=1}^t \\frac{2\\epsilon_s f_s - f_s^2}{b_s} \\le 4\\alpha^2\\ln(2/\\delta)\\) (13.3)."}, {"n": 6, "claim": "\\((1-b_s^{-1})\\epsilon_s^2 \\le \\alpha^2\\)."}, {"n": 7, "claim": "With probability \\(\\ge 1-\\delta/2\\): \\(\\sum_{s=1}^t (1-b_s^{-1})\\epsilon_s^2 \\le \\frac{1}{\\lambda\\alpha^2}\\sum_{s=1}^t (e^{\\lambda\\alpha^2}-1)(1-b_s^{-1})\\mathbb{E}_{\\epsilon_s}\\epsilon_s^2 + \\frac{\\ln(2/\\delta)}{\\lambda}\\)."}, {"n": 8, "claim": "Setting \\(\\lambda = 0.5\\alpha^{-2}\\): \\(\\sum_{s=1}^t (1-b_s^{-1})\\epsilon_s^2 \\le 1.3\\sigma^2\\ln|\\Lambda_0^{-1}\\Lambda_t| + 2\\alpha^2\\ln(2/\\delta)\\) (13.4)."}, {"n": 9, "claim": "Combining (13.2), (13.3), (13.4) by a union bound gives the stated result with probability \\(\\ge 1-\\delta\\)."}]}}, {"local_id": "T2", "statement": "Let \\(\\Sigma_0\\) be a \\(d\\times d\\) symmetric positive definite matrix, and \\(\\{\\psi (X_t)\\}\\) be a sequence of vectors in \\(\\mathbb{R}^d\\). Let \\(\\Sigma_{t} = \\Sigma_{0} + \\sum_{s = 1}^{t}\\psi (X_{s})\\psi (X_{s})^{\\top}\\), then\n\\[\n\\sum_{s=1}^{t}\\frac{\\psi(X_{s})^{\\top}\\Sigma_{s-1}^{-1}\\psi(X_{s})}{1+\\psi(X_{s})^{\\top}\\Sigma_{s-1}^{-1}\\psi(X_{s})}\\leq\\ln|\\Sigma_{0}^{-1}\\Sigma_{t}|.\n\\]", "proof": {"steps": [{"n": 1, "claim": "We have\n\\[\n\\begin{array}{l}\\frac{\\psi(X_{s})^{\\top}\\Sigma_{s-1}^{-1}\\psi(X_{s})}{1+\\psi(X_{s})^{\\top}\\Sigma_{s-1}^{-1}\\psi(X_{s})}=\\psi(X_{s})^{\\top}\\Sigma_{s}^{-1}\\psi(X_{s})\\\\ =\\operatorname{trace}\\left(\\Sigma_{s}^{-1}\\left(\\Sigma_{s}-\\Sigma_{s-1}\\right)\\right)\\\\ =\\operatorname{trace}\\left(\\nabla\\ln\\left|\\Sigma_{s}\\right|\\left(\\Sigma_{s}-\\Sigma_{s-1}\\right)\\right)\\\\ \\leq\\ln|\\Sigma_{s}| -\\ln|\\Sigma_{s-1}|.\\\\ \\end{array}\n\\]\nThe first equality is due to the Sherman-Morrison formula."}, {"n": 2, "claim": "The last inequality used the fact that \\(\\ln |Z|\\) is a concave function of \\(Z\\) (see Theorem A.18)."}, {"n": 3, "claim": "Now by summing over \\(s = 1\\) to \\(s = t\\), we obtain the desired bound."}]}}, {"local_id": "T3", "statement": "Consider a random variable \\(X \\in [0,1]\\) and \\(\\mathbb{E}X = \\mu\\). We have the following inequality:\n\\[\n\\ln\\mathbb{E}e^{\\lambda X}\\leq\\ln [(1-\\mu)e^{0}+\\mu e^{\\lambda}]\\leq\\lambda\\mu+\\lambda^{2}/8.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\( h_L(\\lambda) = \\mathbb{E}e^{\\lambda X} \\) and \\( h_R(\\lambda) = (1 - \\mu)e^0 + \\mu e^\\lambda \\)."}, {"n": 2, "claim": "We know that \\( h_L(0) = h_R(0) \\). Moreover, when \\( \\lambda \\geq 0 \\):"}, {"n": 3, "claim": "\\[\nh_{L}^{\\prime}(\\lambda)=\\mathbb{E}X e^{\\lambda X}\\leq\\mathbb{E}X e^{\\lambda}=\\mu e^{\\lambda}=h_{R}^{\\prime}(\\lambda),\n\\]\nand similarly \\(h_L'(\\lambda) \\geq h_R'(\\lambda)\\) when \\(\\lambda \\leq 0\\)."}, {"n": 4, "claim": "This proves the first inequality. Now we let\n\\[\nh(\\lambda)=\\ln [(1-\\mu)e^{0}+\\mu e^{\\lambda}].\n\\]\nIt implies that\n\\[\nh^{\\prime}(\\lambda)=\\frac{\\mu e^{\\lambda}}{(1-\\mu)e^{0}+\\mu e^{\\lambda}},\n\\]\nand\n\\[\n\\begin{array}{l}h^{\\prime\\prime}(\\lambda)=\\frac{\\mu e^{\\lambda}}{(1-\\mu)e^{0}+\\mu e^{\\lambda}}-\\frac{(\\mu e^{\\lambda})^{2}}{[(1-\\mu)e^{0}+\\mu e^{\\lambda}]^{2}}\\\\ =\\left|h^{\\prime}(\\lambda)\\right|\\left(1-\\left|h^{\\prime}(\\lambda)\\right|\\right)\\leq 1/4.\\\\ \\end{array}\n\\]\nUsing Taylor expansion, we obtain the inequality \\( h(\\lambda) \\leq h(0) + \\lambda h'(0) + \\lambda^2 / 8 \\), which proves the second inequality."}]}}, {"local_id": "T4", "statement": "Consider a sequence of random functions \\(\\xi_1(S_1),\\ldots,\\xi_t(S_t),\\ldots\\) such that \\(\\xi_{i}\\in [0,1]\\) for all \\(i\\). We have for \\(\\lambda >0\\), with probability at least \\(1 - \\delta\\):\n\\[\n\\forall n>0:\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i}<\\frac{\\lambda}{1-e^{-\\lambda}}\\frac{1}{n}\\sum_{i=1}^{n}\\xi_{i}+\\frac{\\ln(1/\\delta)}{\\left(1-e^{-\\lambda}\\right)n}.\n\\]\nSimilarly, for \\(\\lambda > 0\\), with probability at least \\(1 - \\delta\\):\n\\[\n\\forall n>0:\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i}>\\frac{\\lambda}{e^{\\lambda}-1}\\frac{1}{n}\\sum_{i=1}^{n}\\xi_{i}-\\frac{\\ln(1/\\delta)}{(e^{\\lambda}-1)n}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "We obtain from Lemma 2.15 and Theorem 13.2 that with probability at least \\(1 - \\delta\\)\n\\[\n\\begin{array}{l}-\\sum_{i=1}^{n}\\xi_{i}<\\frac{\\ln(1/\\delta)}{\\lambda}+\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln(1+(e^{-\\lambda}-1)\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i})\\\\ \\leq\\frac{\\ln(1/\\delta)}{\\lambda}+\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\left(e^{-\\lambda}-1\\right)\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i}.\\\\ \\end{array}\n\\]\nThis implies the first bound."}, {"n": 2, "claim": "Moreover, with probability at least \\(1 - \\delta\\):"}, {"n": 3, "claim": "\\[\n-\\sum_{i=1}^{n}[1-\\xi_{i}]<\\frac{\\ln(1/\\delta)}{\\lambda}+\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln(1+(e^{-\\lambda}-1)\\mathbb{E}_{Z_{i}^{(y)}}[1-\\xi_{i}])\n\\]\nimplies that\n\\[\n\\begin{array}{l}\\sum_{i=1}^{n}\\xi_{i}<\\frac{\\ln(1/\\delta)}{\\lambda}+\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln(1+(e^{\\lambda}-1)\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i})\\\\ \\leq\\frac{\\ln(1/\\delta)}{\\lambda}+\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\left(e^{\\lambda}-1\\right)\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i}.\\\\ \\end{array}\n\\]\nThis implies the second bound."}]}}, {"local_id": "T5", "statement": "Consider a sequence of random functions \\(\\xi_1(S_1),\\ldots,\\xi_t(S_t),\\ldots\\) with filtration \\(\\{\\mathcal{F}_t\\}\\). We have for any \\(\\delta \\in (0,1)\\) and \\(\\lambda >0\\):\n\\[\n\\operatorname*{Pr}\\left[\\exists n>0: -\\sum_{i=1}^{n}\\xi_{i}\\geq\\frac{\\ln(1/\\delta)}{\\lambda}+\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda\\xi_{i}}\\right]\\leq\\delta .\n\\]\nMoreover, consider a sequence of \\(\\{z_t\\in \\mathbb{R}\\}\\) adapted to \\(\\{\\mathcal{F}_t\\}\\), and events \\(A_{t}\\) on \\(\\mathcal{F}_t\\):\n\\[\n\\ln\\Pr\\left[\\exists n>0:\\sum_{i=1}^{n}\\xi_{i}\\leq z_{n}\\&\\mathcal{S}_{n}\\in A_{n}\\right]\\leq\\inf_{\\lambda>0}\\sup_{n>0}\\sup_{\\mathcal{S}_{n}\\in A_{n}}\\left[\\lambda z_{n}+\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda\\xi_{i}}\\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "We will prove the result for a finite time sequence \\(\\xi_1(S_1),\\ldots,\\xi_T(S_T)\\)."}, {"n": 2, "claim": "It implies the desired result by letting \\(T\\to \\infty\\)."}, {"n": 3, "claim": "Let\n\\[\n\\xi_{\\tau}(\\lambda)= -\\sum_{i=1}^{\\tau}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda\\xi_{i}}-\\lambda\\sum_{i=1}^{\\tau}\\xi_{i},\n\\]\nwhere \\(\\tau\\) is a stopping time, then we have from Lemma 13.1:"}, {"n": 4, "claim": "\\(\\mathbb{E}e^{\\xi_{\\tau}(\\lambda)} = 1\\). Now for any given sequence of \\(\\tilde{z}_n(S_n)\\) and \\(A_{n}\\), define the stopping time \\(\\tau\\) as either \\(T\\), or the first time step \\(n\\) so that\n\\[\n\\xi_{n}(\\lambda)\\geq-\\tilde{z}_{n}\\left(\\mathcal{S}_{n}\\right)\\&\\mathcal{S}_{n}\\in A_{n}\n\\]\nfor each sequence \\(S_T\\)."}, {"n": 5, "claim": "It follows that\n\\[\n\\begin{array}{l}\\Pr\\left(\\exists n:\\xi_{n}(\\lambda)\\geq-\\tilde{z}_{n}\\left(\\mathcal{S}_{n}\\right)\\&\\mathcal{S}_{n}\\in A_{n}\\right)\\inf_{n>0,\\mathcal{S}_{n}\\in A_{n}}e^{-\\tilde{z}_{n}\\left(\\mathcal{S}_{n}\\right)}\\\\ \\leq\\mathbb{E}\\left[e^{\\xi_{\\tau}(\\lambda)+\\tilde{z}_{\\tau}\\left(\\mathcal{S}_{\\tau}\\right)}\\mathbb{1}\\left(\\mathcal{S}_{\\tau}\\in A_{\\tau}\\right)\\right]\\inf_{n>0,\\mathcal{S}_{n}\\in A_{n}}e^{-\\tilde{z}_{n}\\left(\\mathcal{S}_{n}\\right)}\\\\ \\leq\\mathbb{E}\\left[e^{\\xi_{\\tau}(\\lambda)+\\tilde{z}_{\\tau}\\left(\\mathcal{S}_{\\tau}\\right)}\\mathbb{1}\\left(\\mathcal{S}_{\\tau}\\in A_{\\tau}\\right)e^{-\\tilde{z}_{\\tau}\\left(\\mathcal{S}_{\\tau}\\right)}\\right]\\\\ \\leq\\mathbb{E}e^{\\xi_{\\tau}(\\lambda)}=1.\\\\ \\end{array}\n\\]\nTherefore we obtain\n\\[\n\\begin{array}{l}\\ln\\operatorname*{Pr}\\left[\\exists n>0: -\\lambda\\sum_{i=1}^{n}\\xi_{i}\\geq-\\tilde{z}_{n}(\\mathcal{S}_{n})+\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda\\xi_{i}}\\&\\mathcal{S}_{n}\\in A_{n}\\right]\\\\ \\leq\\sup_{n>0:\\mathcal{S}_{n}\\in A_{n}}\\tilde{z}_{n}(\\mathcal{S}_{n}).\\\\ \\end{array}\n\\]\nLet \\(\\tilde{z}(\\mathcal{S}_n) = \\ln \\delta\\), we obtain the first inequality."}, {"n": 6, "claim": "Let\n\\[\n\\tilde{z}_{n}(\\mathcal{S}_{n})=\\lambda z_{n}+\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda\\xi_{i}},\n\\]\nwe obtain the second inequality."}]}}], "target_local_id": "T1", "topic_domain": "Probability & Concentration", "topic_subtopic": "Martingales & Dependent Data", "label": "flawed", "error_step": 3, "error_class6": ["invalid_bound_or_inequality", "computation_or_constant", "others"], "error_explanation": "Step 3 asserts $\\left|2\\epsilon_s f_s/b_s\\right| \\le 2\\alpha f_s/\\sqrt{b_s}$, but the right-hand side has no absolute value on $f_s$: since $f_s=\\hat w_{s-1}^\\top X_s$ carries no sign restriction, whenever $f_s<0$ the left side is nonnegative while $2\\alpha f_s/\\sqrt{b_s}$ is negative, so the stated inequality is false as written (it needs $2\\alpha|f_s|/\\sqrt{b_s}$ on the right). Counterexample verified in Lean: $\\alpha=1,\\epsilon_s=1,f_s=-1,b_s=1$ satisfies $|\\epsilon_s|\\le\\alpha$ and $b_s\\ge1$, yet $|2\\epsilon_s f_s/b_s|=2$ while $2\\alpha f_s/\\sqrt{b_s}=-2$. All four independent verifiers run when this item was built (two reasoning over the informal text, two producing machine-checked counterexamples in Lean) unanimously placed the first error at step 3, for exactly this reason."}
{"id": "ch14_proposition_14_14__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nEquation 14.9. The posterior distribution \\(p_t(w)\\) used by the Hedge algorithm is given by\n\\[\np_t(w) \\propto p_0(w) \\exp \\left(-\\eta \\sum_{s=1}^{t} \\ell_s(w)\\right), \\tag{14.9}\n\\]\nequivalently, recursively,\n\\[\np_t(w) \\propto p_{t-1}(w) \\exp(-\\eta \\ell_t(w)).\n\\]\n\nOnline Gradient Descent. Algorithm 14.3: Online Gradient Descent\nInput: Sequence of loss functions \\(\\ell_1,\\dots,\\ell_T\\) defined on \\(\\Omega\\) Output: \\(\\hat{w}\\)   \n1 Let \\(w_{0} = 0\\)   \n2 for \\(t=1,2,\\ldots,T\\) do   \n3 Observe loss \\(\\ell_t(w_{t - 1})\\)   \n4 Let \\(\\tilde{w}_t = w_{t - 1} - \\eta_t\\nabla \\ell_t(w_{t - 1})\\)   \n5 Let \\(w_{t} = \\arg \\min_{w\\in \\Omega}\\| w - \\tilde{w}_{t}\\|_{2}^{2}\\)   \n6 Let \\(\\hat{w} = T^{-1}\\sum_{t = 1}^{T}w_{t - 1}\\) or \\(\\hat{w} = w_{s}\\) for a random \\(s\\) from O to \\(T - 1\\) Return: \\(\\hat{w}\\)\n\nHedge Algorithm. Hedge Algorithm\nInput: \\(T\\), prior \\(p_{0}(w)\\) on \\(\\Omega\\), learning rate \\(\\eta > 0\\)\n1 Randomly draw \\(w_{0} \\sim p_{0}(w)\\)\n2 for \\(t = 1, 2, \\ldots, T\\) do\n3 \\quad Observe loss \\(\\ell_{t}(w_{t-1})\\)\n4 \\quad Randomly draw \\(w_{t} \\sim p_{t}(w)\\) according to \\((14.9)\\)\n\nWe have the following result for the Hedge algorithm, using conditions similar to those of Theorem 7.17.\n\nExample 14.10. Consider the structured-SVM loss of Example 9.32, where\n\\[\n\\phi(w,z)=\\max_{\\ell}[\\gamma(y,\\ell)-w^{\\top}(\\psi(x,y)-\\psi(x,\\ell))].\n\\]\nIf \\( \\| \\psi(x, y) - \\psi(, \\ell) \\|_2 \\leq B \\), then we can take \\( G = B \\). Using Theorem 14.5, we obtain\n\\[\n\\frac{1}{T}\\sum_{t=1}^{T}\\phi(w_{t-1},Z_{t})\\leq\\frac{1}{T}\\sum_{t=1}^{T}\\phi(w,Z_{t})+\\frac{\\|w_{0}-w\\|_{2}^{2}}{2\\eta T}+\\frac{\\eta}{2}B^{2}.\n\\]\nBy taking expectation, and using Jensen's inequality with \\(\\lambda = 1 / (\\eta T)\\), we obtain\n\\[\n\\mathbb{E}_{\\mathcal{S}_{T}}\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(\\hat{w},Z)\\leq\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(w,Z)+\\frac{\\lambda}{2}\\|w_{0}-w\\|_{2}^{2}+\\frac{1}{2\\lambda T}B^{2}.\n\\]\nFor \\( L_{2} \\) regularization (or kernel methods), one can obtain a better bound using strong convexity. Observe that for regularized loss, we take\n\\[\n\\ell_{t}(w)=\\phi\\left(w,Z_{t}\\right)+\\frac{\\lambda}{2}\\|w-w_{0}\\|_{2}^{2}.\\tag{14.8}\n\\]\nIf \\(\\phi (w,z)\\) is convex in \\(w\\), then \\(\\ell_t(w)\\) is \\(\\lambda\\) strongly convex.\nThe following result holds for strongly convex loss functions, and the specific learning rate schedule was proposed by (Shalev-Shwartz et al., 2011) to solve SVMs.", "theorems": [{"local_id": "T1", "statement": "Consider \\(\\Omega = \\{w_0, w_1, \\ldots, w_T\\}\\) that contains \\(T+1\\) members. Given any deterministic online learning algorithm \\(\\mathcal{A}\\) that returns a model parameter \\(w_{t-1}\\) based on \\(\\{\\ell_s : s = 1, \\ldots, t-1\\}\\), there exists a loss sequence \\(\\ell_t(w) \\in \\{0, 1\\}\\) so that\n\\[\n\\inf_{w}\\sum_{t=1}^{T}\\ell_{t}(w)=0,\\qquad\\sum_{t=1}^{T}\\ell_{t}(w_{t-1})=T.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Define the adversarial loss at each time \\(t\\) by \\(\\ell_t(w)=\\mathbb{1}(w=w_{t-1})\\), where \\(w_{t-1}\\) is the parameter returned by \\(\\mathcal{A}\\)."}, {"n": 2, "claim": "With this choice, \\(\\sum_{t=1}^T\\ell_t(w_{t-1})=T\\)."}, {"n": 3, "claim": "\\(\\Omega=\\{w_0,\\dots,w_T\\}\\) contains \\(T+1\\) elements."}, {"n": 4, "claim": "Take \\(w=w_0\\), the initial parameter; it differs from \\(w_1,\\dots,w_T\\), so it is never penalized."}, {"n": 5, "claim": "Therefore \\(\\sum_{t=1}^T\\ell_t(w_0)=0\\)."}, {"n": 6, "claim": "Consequently \\(\\inf_w\\sum_{t=1}^T\\ell_t(w)=0\\)."}]}}, {"local_id": "T2", "statement": "Consider the regularized loss function (14.8) with \\( w_0 = 0 \\), where \\( \\phi(w, z) \\) is convex in \\( w \\), and \\( G \\) Lipschitz in \\( w \\). Moreover assume that \\( d(\\Omega) = \\sup \\{ \\| w \\|_2 : w \\in \\Omega \\} \\). If \\( Z_1, \\ldots, Z_T \\sim \\mathcal{D} \\) are independent samples, then we can obtain the following expected oracle inequality for Algorithm 14.3 if we take learning rate in Theorem 14.11:\n\\[\n\\mathbb{E}_{\\mathcal{S}_{T}}\\mathbb{E}_{\\mathcal{D}}\\phi(\\hat{w},Z)+\\frac{\\lambda}{2}\\|\\hat{w}\\|_{2}^{2}\\leq\\inf_{w\\in\\Omega}\\left[\\mathbb{E}_{\\mathcal{D}}\\phi(w,Z)+\\frac{\\lambda}{2}\\|w\\|_{2}^{2}\\right]+\\frac{\\ln(e T)}{2\\lambda T}[G+\\lambda d(\\Omega)]^{2}.\n\\]\nWe can also obtain the following expected oracle inequality for Algorithm 14.3 if we take learning rate in Theorem 14.12 with\n\\[\n\\hat{w}^{\\prime}=\\sum_{t=1}^{T}\\frac{2(t+1)}{T^{2}+3 T}w_{t-1},\n\\]\nthen\n\\[\n\\mathbb{E}_{\\mathcal{S}_{T}}\\mathbb{E}_{\\mathcal{D}}\\phi(\\hat{w}^{\\prime},Z)+\\frac{\\lambda}{2}\\|\\hat{w}^{\\prime}\\|_{2}^{2}\\leq\\inf_{w\\in\\Omega}\\left[\\mathbb{E}_{\\mathcal{D}}\\phi(w,Z)+\\frac{\\lambda}{2}\\|w\\|_{2}^{2}\\right]+\\frac{2[G+\\lambda d(\\Omega)]^{2}}{\\lambda(T+3)}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Note that \\(\\ell_t(w)\\) is \\(\\lambda\\)-strongly convex in \\(w\\)."}, {"n": 2, "claim": "Moreover, it is \\(G + \\lambda d(\\Omega)\\) Lipschitz."}, {"n": 3, "claim": "We simply take expectation and apply Jensen's inequality to obtain the desired bounds."}]}}, {"local_id": "T3", "statement": "Assume that for all \\(t\\):\n\\[\n\\sup_{w\\in\\Omega}\\ell_{t}(w)-\\inf_{w\\in\\Omega}\\ell_{t}(w)\\leq M,\n\\]\nthen Algorithm 14.4 has regret\n\\[\n\\sum_{t=1}^{T}\\mathbb{E}_{w_{t-1}\\sim p_{t-1}(\\cdot)}\\ell_{t}(w_{t-1})\\leq\\inf_{p\\in\\Delta(\\Omega)}\\left[\\mathbb{E}_{w\\sim p}\\sum_{t=1}^{T}\\ell_{t}(w)+\\frac{1}{\\eta}\\mathrm{KL}(p| |p_{0})\\right]+\\frac{\\eta TM^{2}}{8},\n\\]\nwhere \\(\\Delta (\\Omega)\\) denotes the set of probability distributions on \\(\\Omega\\)", "proof": {"steps": [{"n": 1, "claim": "Let\n\\[\nZ_{t}= -\\ln\\mathbb{E}_{w\\sim p_{0}}\\exp\\left(-\\eta\\sum_{s=1}^{t}\\ell_{s}(w)\\right)\n\\]\nbe the log-partition function for observations up to time \\( t \\)."}, {"n": 2, "claim": "We have\n\\[\n\\begin{array}{l}Z_{t-1}-Z_{t}=\\ln\\mathbb{E}_{w\\sim p_{t-1}}\\exp(-\\eta\\ell_{t}(w))\\\\ \\leq-\\eta\\mathbb{E}_{w\\sim p_{t-1}}\\ell_{t}(w)+\\frac{\\eta^{2}M^{2}}{8},\\\\ \\end{array}\n\\]\nwhere the first equation is simple algebra, and the inequality follows from the estimate of logarithmic moment generation function in Lemma 2.15."}, {"n": 3, "claim": "By summing\nover \\(t = 1\\) to \\(T\\), and noticing that \\(Z_{0} = 0\\), we obtain\n\\[\n\\sum_{t=1}^{T}\\mathbb{E}_{w_{t-1}\\sim p_{t-1}(\\cdot)}\\ell_{t}(w_{t-1})\\leq\\frac{1}{\\eta}Z_{T}+\\frac{\\eta T M^{2}}{8}.\n\\]\nThe desired bound follows by applying Proposition 7.16 to reformulate the log-partition function \\( Z_{T} \\)."}]}}], "target_local_id": "T1", "topic_domain": "Online Learning & RL", "topic_subtopic": "Online Learning & Regret", "label": "flawed", "error_step": 4, "error_class6": ["logical_gap_or_missing_case", "computation_or_constant", "invalid_bound_or_inequality"], "error_explanation": "The penalized set is actually $\\{w_0,\\dots,w_{T-1}\\}$ (the loss at time $t$ penalizes $w_{t-1}$), so $w_0$ is penalized at $t=1$, giving $\\ell_1(w_0)=1$. Choosing $w=w_0$ omits the $t=1$ case and the claim that $w_0$ is never penalized is false; the correct witness must avoid $w_0,\\dots,w_{T-1}$."}
{"id": "ch15_proposition_15_16__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nDefinition 15.12. A convex function \\( g(u) \\) is \\( \\alpha \\)-exponential concave for some \\( \\alpha > 0 \\) if\n\\[\ne^{-\\alpha g(u)}\n\\]\nis concave in \\(u\\).\nThe following property is convenient to show exponential concavity.\n\nEquation 15.3 ridge regression estimator. The ridge regression estimator for the sample \\(\\mathcal{S}_{n}\\) is defined by\n\\[\n\\hat{w}(\\mathcal{S}_{n})=\\arg\\min_{w}\\left[\\sum_{i=1}^{n}(w^{\\top}\\psi(X_{i})-Y_{i})^{2}+\\|w\\|_{\\Lambda_{0}}^{2}\\right].\n\\]\nEquivalently,\n\\[\n\\hat{w}(\\mathcal{S}_{n})=\\hat{\\Lambda}(\\mathcal{S}_{n})^{-1}\\sum_{i=1}^{n}\\psi(X_{i})Y_{i},\n\\]\nwhere\n\\[\n\\hat{\\Lambda}(\\mathcal{S}_{n})=\\Lambda_{0}+\\sum_{i=1}^{n}\\psi(X_{i})\\psi(X_{i})^{\\top}.\n\\]\n\nTheorem 15.6. Consider the ridge regression method of (15.3). We have the following result for any \\(\\sigma \\geq 0\\) and for all observed sequence \\(\\mathcal{S}_T\\):\n\\[\n\\sum_{t=1}^{T}\\left[\\frac{(Y_t-\\hat{w}(\\mathcal{S}_{t-1})^\\top\\psi(X_t))^2}{b_t}+\\sigma^2\\ln b_t\\right]\n=\\inf_{w}\\left[\\sum_{t=1}^{T}(Y_t-w^\\top\\psi(X_t))^2+\\|w\\|_{\\Lambda_0}^2\\right]+\\sigma^2\\ln\\left|\\Lambda_0^{-1}\\Lambda_T\\right|,\n\\]\nwhere\n\\[\n\\Lambda_t=\\Lambda_0+\\sum_{s=1}^{t}\\psi(X_s)\\psi(X_s)^\\top,\n\\]\nand \\(b_t=1+\\psi(X_t)^\\top\\Lambda_{t-1}^{-1}\\psi(X_t)\\).\n\nExample 15.2. For least squares regression with noise variance \\(\\sigma^2\\), we may have\n\\[\np(y|w,x)=\\frac{1}{\\sqrt{2\\pi}\\sigma}\\exp\\left(-\\frac{(y-f(w,x))^{2}}{2\\sigma^{2}}\\right),\n\\]\nand \\((\\operatorname{let} Z = (x, y))\\):\n\\[\n\\phi(w,Z)=\\frac{(y-f(w,x))^{2}}{2\\sigma^{2}}+\\ln(\\sqrt{2\\pi}\\sigma).\n\\]\nWe may also consider the noise as part of the model parameter, and let\n\\[\np(y|[w,\\sigma],x)=\\frac{1}{\\sqrt{2\\pi}\\sigma}\\exp\\left(-\\frac{(y-f(w,x))^{2}}{2\\sigma^{2}}\\right)\n\\]\n\\[\n\\phi([w,\\sigma],Z)=\\frac{(y-f(w,x))^{2}}{2\\sigma^{2}}+\\ln(\\sqrt{2\\pi}\\sigma).\n\\]\nIn the Bayesian setting, the optimal statistical estimation of posterior density for log-loss is the Bayesian posterior averaging estimator. Here we consider a prior \\( p_0(w) \\) on \\( \\Omega \\). Given the training data \\( S_{n}=\\{Z_{1},\\dots,Z_{n}\\} \\), the posterior distribution is\n\\[\np(w|\\mathcal{S}_{n})=\\frac{\\prod_{i=1}^{n}p\\left(Y_{i}\\mid w,X_{i}\\right)p_{0}(w)}{\\int_{\\Omega}\\prod_{i=1}^{n}p\\left(Y_{i}\\mid w^{\\prime},X_{i}\\right)p_{0}\\left(w^{\\prime}\\right)d w^{\\prime}}.\\tag{15.1}\n\\]\nUnlike ERM, the Bayesian posterior averaging method does not use \\( p(y|\\hat{w},x) \\) for a specific \\( \\hat{w}\\in \\Omega \\) that is estimated from the training data, but instead use the averaged probability estimate over the posterior as follows:\n\\[\n\\hat{p}(y|x,\\mathcal{S}_{n})=\\int_{\\Omega}p(y|w,x)p(w|\\mathcal{S}_{n})d w.\\tag{15.2}\n\\]\nAs pointed out in Section 14.1, such a method is referred to as improper online learning.\nIn the online setting, we have the following result, which may be regarded as a generic regret bound for the Bayesian posterior model averaging method.", "theorems": [{"local_id": "T1", "statement": "If \\(\\Omega = \\{w_{1},\\ldots \\}\\) is discrete, then\n\\[\n-\\sum_{t=1}^{T}\\ln\\hat{p}\\left(Y_{t}\\mid X_{t},\\mathcal{S}_{t-1}\\right)\\leq\\inf_{w\\in\\Omega}\\left[-\\sum_{t=1}^{T}\\ln p\\left(Y_{t}|w,X_{t}\\right)-\\ln p_{0}(w)\\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "Given any \\( w' \\in \\Omega \\), if we choose \\( q(w) = 1 \\) when \\( w = w' \\), and \\( q(w) = 0 \\) when \\( w \\neq w' \\), then\n\\[\n-\\mathbb{E}_{w\\sim q}\\sum_{t=1}^{T}\\ln p(Y_{t}|w,X_{t})+\\mathbb{E}_{w\\sim q}\\ln\\frac{q(w)}{p_{0}(w)}= -\\sum_{t=1}^{T}\\ln p(Y_{t}|w^{\\prime},X_{t})-\\ln p_{0}(w^{\\prime}).\n\\]\nThis implies the result."}]}}, {"local_id": "T2", "statement": "Consider an arbitrary sequence \\(\\{(X_t,Y_t)\\}\\), where \\(Y_{t}|X_{t}\\) is generated from a distribution \\(\\mathcal{D}\\). Assume that \\(f_{*}(X_{t}) = \\mathbb{E}[Y_{t}|X_{t}]\\in [0,M]\\) and \\(\\operatorname{Var}[Y_t|X_t]\\leq \\sigma^2\\). Consider the ridge regression method of (15.3), and let\n\\[\n\\hat{f}(w,x)=\\max(0,\\min(M,w^{\\top}\\psi(X))).\n\\]\nWe have\n\\[\n\\begin{array}{l}\\mathbb{E}_{\\mathcal{S}_{T}}\\sum_{t=1}^{T}\\left(f_{*}\\left(X_{t}\\right)-\\hat{f}\\left(\\hat{w}\\left(\\mathcal{S}_{t-1},X_{t}\\right)\\right)^{2}\\right.\\\\ \\leq\\mathbb{E}_{\\mathcal{S}_{T}}\\inf_{w}\\sum_{t=1}^{T}\\left[\\left(f_{*}(X_{t})-w^{\\top}\\psi(X_{t})\\right)^{2}+\\|w\\|_{\\Lambda_{0}}^{2}\\right]+\\left(M^{2}+\\sigma^{2}\\right)\\mathbb{E}_{\\mathcal{S}_{T}}\\ln\\left|\\Lambda_{0}^{-1}\\Lambda_{T}\\right|,\\\\ \\end{array}\n\\]\nwhere \\(\\Lambda_T = \\Lambda_0 + \\sum_{s=1}^T \\psi(X_s) \\psi(X_s)^\\top\\).", "proof": {"steps": [{"n": 1, "claim": "We have the following derivation, which is similar to that of Corollary 15.7:"}, {"n": 2, "claim": "\\[\n\\begin{array}{l}\\mathbb{E}_{\\mathcal{S}_{T}}\\sum_{t=1}^{T}\\left[\\operatorname{Var}[Y_t|X_t]+\\left(f_{*}(X_{t})-\\hat{f}\\left(\\hat{w}_{t-1}(\\mathcal{S}_{t-1}),X_{t}\\right)\\right)^{2}\\right]\\\\ \\leq\\mathbb{E}_{\\mathcal{S}_{T}}\\sum_{t=1}^{T}\\left[\\left[\\operatorname{Var}[Y_t|X_t]+\\left(f_{*}(X_{t})-\\hat{f}\\left(\\hat{w}_{t-1}(\\mathcal{S}_{t-1}),X_{t}\\right)\\right)^{2}\\right]\\right.\\\\ \\left.+\\left[\\operatorname{Var}[Y_t|X_t]+\\left(f_{*}\\left(X_{t}\\right)-\\hat{f}\\left(\\hat{w}_{t-1}\\left(\\mathcal{S}_{t-1}\\right),X_{t}\\right)\\right)^{2}\\right]\\frac{b_{t}-1}{b_{t}}+\\left[M^{2}+\\sigma^{2}\\right]\\ln b_{t}\\right]\\\\ \\leq\\mathbb{E}_{\\mathcal{S}_{T}}\\sum_{t=1}^{T}\\left[\\frac{\\left(Y_{t}-\\hat{w}_{t-1}\\left(\\mathcal{S}_{t-1}\\right)^{\\top}\\psi\\left(X_{t}\\right)\\right)^{2}}{b_{t}}+\\left[M^{2}+\\sigma^{2}\\right]\\ln b_{t}\\right].\\\\ \\end{array}\n\\]\nIn the above derivation, the proof of the first inequality is the same as that of (15.5)."}, {"n": 3, "claim": "The last inequality used (by noticing the truncation definition of \\(\\hat{f}(w,x)\\))\n\\[\n\\left(f_{*}\\left(X_{t}\\right)-\\hat{f}\\left(\\hat{w}_{t-1}\\left(\\mathcal{S}_{t-1}\\right),X_{t}\\right)\\right)^{2}\\leq\\left(f_{*}\\left(X_{t}\\right)-\\hat{w}_{t-1}\\left(\\mathcal{S}_{t-1}\\right)^{\\top}\\psi\\left(X_{t}\\right)\\right)^{2}.\n\\]\nNow using Theorem 15.6, we obtain\n\\[\n\\begin{array}{l}\\mathbb{E}_{\\mathcal{S}_{T}}\\sum_{t=1}^{T}\\left[\\operatorname{Var}[Y_t|X_t]+\\left(f_{*}(X_{t})-\\hat{f}(\\hat{w}(\\mathcal{S}_{t-1}),X_{t})\\right)^{2}\\right]\\\\ \\leq\\inf_{w}\\left[\\mathbb{E}_{\\mathcal{S}_{T}}\\sum_{t=1}^{T}(Y_{t}-w^{\\top}\\psi(X_{t}))^{2}+\\|w\\|_{\\Lambda_{0}}^{2}\\right]+[M^{2}+\\sigma^{2}]\\mathbb{E}_{\\mathcal{S}_{T}}\\ln|\\Lambda_{0}^{-1}\\Lambda_{T}|\\\\ \\leq\\inf_{w}\\left[\\mathbb{E}_{\\mathcal{S}_{T}}\\sum_{t=1}^{T}[\\operatorname{Var}[Y_t|X_t]+(f_{*}(X_{t})-w^{\\top}\\psi(X_{t}))^{2}]+\\|w\\|_{\\Lambda_{0}}^{2}\\right]+[M^{2}+\\sigma^{2}]\\mathbb{E}_{\\mathcal{S}_{T}}\\ln|\\Lambda_{0}^{-1}\\Lambda_{T}|.\\\\ \\end{array}\n\\]\nThis implies the result."}]}}, {"local_id": "T3", "statement": "Assume \\(\\phi(u)\\) is \\(\\alpha\\) exponentially concave in \\(u\\). Let \\(\\tilde{\\phi}(w) = \\phi(w^\\top X)\\), then \\(\\tilde{\\phi}\\) is also \\(\\alpha\\) exponentially concave in \\(w\\).", "proof": {"steps": [{"n": 1, "claim": "\\(\\phi\\) \\(\\alpha\\) exponentially concave means \\(g(u)=e^{-\\alpha\\phi(u)}\\) is concave."}, {"n": 2, "claim": "\\(w\\mapsto w^\\top X\\) is affine, so it maps line segments to line segments lying in the domain, and therefore \\(g(w^\\top X)\\) is concave."}, {"n": 3, "claim": "Hence \\(e^{-\\alpha\\tilde{\\phi}(w)}=g(w^\\top X)\\) is concave in \\(w\\)."}, {"n": 4, "claim": "Therefore \\(\\tilde{\\phi}\\) is \\(\\alpha\\) exponentially concave in \\(w\\)."}]}}], "target_local_id": "T2", "topic_domain": "Online Learning & RL", "topic_subtopic": "Online Learning & Regret", "label": "flawed", "error_step": 2, "error_class6": ["computation_or_constant", "invalid_bound_or_inequality", "others"], "error_explanation": "Step 2's middle display carries the correction factor (b_t - 1)/b_t on the second bracketed term. With the leading term it collapses to 1 + (b_t - 1)/b_t = 2 - 1/b_t, whereas the next line requires the factor 1/b_t. Since b_t = |Lambda_t|/|Lambda_{t-1}| with Lambda_t = Lambda_{t-1} + psi(X_t)psi(X_t)^T, we always have b_t >= 1, so 2 - 1/b_t >= 1 >= 1/b_t with strict inequality whenever b_t > 1: the middle line is LARGER than what the final line provides, and the second inequality of the chain fails. Numerically the overshoot is 1.40x at b_t = 1.2, 3.00x at b_t = 2 and 19.0x at b_t = 10. The correct factor is (1 - b_t)/b_t, a non-positive correction, which collapses to exactly 1/b_t. The first inequality of the chain is unaffected - the inflated middle line still dominates the leading term - so the defect is precisely the step-2 link, and the theorem itself remains true."}
{"id": "ch16_theorem_16_3__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nsection 13.1 sequential observation setup. In Section 13.1, data are observed sequentially. At each time \\(t\\), after the history \\(S_{t-1}\\) has been observed, the next observation \\(Z_t\\) is drawn according to a distribution that may depend on that history. The observation is decomposed as \\(Z_t=(Z_t^{(x)},Z_t^{(y)})\\), where \\(Z_t^{(x)}\\) is the component determined by the past history or the chosen action at time \\(t\\), and \\(Z_t^{(y)}\\) is the random response component. The history after time \\(t\\) is \\(S_t=(Z_1,\\ldots,Z_t)\\), and \\(\\mathcal{S}_T\\) denotes the full sequential experiment/history through time \\(T\\). For a function depending on \\(S_t\\), the notation \\(\\mathbb{E}_{Z_t^{(y)}}[\\cdot]\\) means conditional expectation over the response component \\(Z_t^{(y)}\\), conditional on \\(S_{t-1}\\) and on \\(Z_t^{(x)}\\). Stopping times are adapted to the filtration generated by the histories \\(S_t\\).\n\nDefinition 16.1. In the multi-armed bandit (or MAB) problem, we consider \\(K\\) arms. The environment generates a sequence of reward vectors for time steps \\(t \\geq 1\\) as \\(r_t = [r_t(1), \\ldots, r_t(K)]\\). Each \\(r_t(a)\\) is associated with an arm \\(a \\in \\{1, \\ldots, K\\}\\). In the literature, an arm \\(a\\) is also referred to as an action. At each time step \\(t = 1, 2, \\ldots, T\\),\n- The player pulls one of the arms \\( a_{t} \\in \\{1, \\dots, K\\} \\).\n- The environment returns the reward \\( r_t(a_t) \\), but does not reveal information on any other arm \\( a \\neq a_t \\).\nAt each time \\( t \\), a (randomized) bandit algorithm takes the historic observations observed so far, and maps it to a distribution \\( \\hat{\\pi}_{t-1} \\) over actions \\( a \\in \\{1, \\ldots, K\\} \\). We then draw a random action \\( a_t \\) (arm) from \\( \\hat{\\pi}_{t-1} \\). Here \\( \\hat{\\pi}_{t-1} \\) is referred to as policy.\nIn the adversarial bandit problem, we are given an arbitrary reward sequence \\(\\{[r_1(1),\\ldots,r_t(K)]:t\\geq 1\\}\\) before hand. The adversary in this setting is called oblivious, who decides the rewards of all the rounds before the game starts (but may take advantage of the bandit algorithm's weakness). For this reward se\nquence, the expected cumulative reward of a randomized bandit algorithm is\n\\[\n\\mathbb{E}\\sum_{t=1}^{T}r_{t}(a_{t}),\n\\]\nwhere the randomization is over the internal randomization of the bandit algorithm. The expected regret is defined as\n\\[\n\\operatorname{REG}_{T}=\\max_{a}\\sum_{t=1}^{T}r_{t}(a)-\\mathbb{E}\\sum_{t=1}^{T}r_{t}\\left(a_{t}\\right).\\tag{16.1}\n\\]\nIf the bandit algorithm is deterministic: i.e. pulling a single arm \\(a_{t}\\) only at any time step, then the regret is\n\\[\n\\mathrm{REG}_{T}=\\max_{a}\\sum_{t=1}^{T}r_{t}(a)-\\sum_{t=1}^{T}r_{t}(a_{t}).\n\\]\nIn the adversarial bandit problem, our goal is to minimize the expected regret \\(\\mathrm{REG}_T\\) (and the expectation is with respect to the internal randomization of the algorithm instead of the data).\nAnother form of bandit problem, called stochastic bandit, assumes that the reward \\( r_t(a) \\) is drawn independently from a distribution \\( \\mathcal{D}_a \\), with mean \\( \\mu(a) = \\mathbb{E}_{r_t(a) \\sim \\mathcal{D}_a} [r_t(a)] \\). In this setting, the goal of bandit is to find \\( a \\) that maximizes the expected reward \\( \\mu(a) \\). We are interested in the expected regret defined as:\n\\[\n\\operatorname{REG}_{T}=T\\max_{a}\\mu(a)-\\mathbb{E}\\sum_{t=1}^{T}\\mu\\left(a_{t}\\right).\\tag{16.2}\n\\]\nAgain, this regret is defined for each realization of the stochastic rewards, and the expectation is with respect to the randomness of the algorithm. If algorithm is deterministic and outputs \\( a_{t} \\) at each time \\( t \\), then\n\\[\n\\mathrm{REG}_{T}=T\\max_{a}\\mu(a)-\\sum_{t=1}^{T}\\mu(a_{t}).\n\\]\nWe can further include randomization over data into the regret of stochastic bandit, by considering expected regret as follows:\n\\[\n\\mathbb{E}\\operatorname{REG}_{T}=T\\max_{a}\\mu(a)-\\mathbb{E}\\sum_{t=1}^{T}\\mu(a_{t}),\n\\]\nwhere the expectation is with respect to both the data and the internal randomization of the learning algorithm.\n\nEquation 16.3. For each arm \\(a\\), define the number of pulls up to time \\(t\\) and the corresponding empirical mean by\n\\[\n\\hat n_t(a)=\\sum_{s=1}^{t}\\mathbb{1}(a_s=a), \\qquad\n\\hat\\mu_t(a)=\\frac{1}{\\hat n_t(a)}\\sum_{s=1}^{t} r_s(a_s)\\mathbb{1}(a_s=a),\n\\]\nwith the initialization \\(\\hat n_0(a)=0\\) and \\(\\hat\\mu_0(a)=0\\). In the UCB algorithm, after the initialization rounds, the arm is selected according to\n\\[\na_t\\in\\arg\\max_a\\left[\\hat\\mu_{t-1}(a)+\\hat c_{t-1}(a)\\right]. \\tag{16.3}\n\\]\nHere \\(\\hat c_t(a)\\) denotes the confidence width used by the algorithm.\n\nUCB Algorithm. Algorithm 16.1: UCB Algorithm\nInput: \\(K\\) and \\(T\\geq K\\)   \n1 for \\(a=1,\\dots,K\\) do   \n2 Let \\(\\hat{n}_0(a) = 0\\)   \n3 Let \\(\\hat{\\mu}_0(a) = 0\\)   \n4 for \\(t=1,2,\\ldots,T\\) do   \n5 if \\(t\\leq K\\) then   \n6 Let \\(a_{t} = t\\)   \n7 else   \n8 Let \\(a_{t}\\in \\arg \\max_{a}[\\hat{\\mu}_{t - 1}(a) + \\hat{c}_{t - 1}(a)]\\) according to (16.3)   \n9 Pull arm \\(a_{t}\\) and observe reward \\(r_t(a_t)\\)", "theorems": [{"local_id": "T1", "statement": "Let \\(a_* \\in \\arg \\max_a \\mu(a)\\). Let\n\\[\n\\delta_{1}=\\Pr [\\exists t>K:\\mu(a_{*})>\\hat{\\mu}_{t-1}(a_{*})+\\hat{c}_{t-1}(a_{*})]\n\\]\nbe the probability that the upper confident bound fails on \\( a_* \\). Let\n\\[\n\\delta_{2}=\\Pr\\left[\\exists t>K\\&a\\in\\{1,\\dots,K\\}\\backslash\\left\\{a_{*}\\right\\}:\\mu(a)<\\hat{\\mu}_{t-1}(a)-\\hat{c}_{t-1}^{\\prime}(a)\\right]\n\\]\nbe the probability that the lower confident bound fails on \\( a \\neq a_* \\). Then for Algorithm 16.1, we have with probability at least \\( 1 - \\delta_1 - \\delta_2 \\):\n\\[\n\\operatorname{REG}_{T}\\leq\\sum_{a=1}^{K}[\\mu(a_{*})-\\mu(a)]+\\sum_{t=K+1}^{T}[\\hat{c}_{t-1}(a_{t})+\\hat{c}_{t-1}^{\\prime}(a_{t})]\\mathbb{1}(a_{t}\\neq a_{*}).\n\\]", "proof": {"steps": [{"n": 1, "claim": "We have with probability \\(1 - \\delta_1 - \\delta_2\\), the following hold for all \\(t > K\\):"}, {"n": 2, "claim": "\\[\n\\mu\\left(a_{*}\\right)\\leq\\hat{\\mu}_{t-1}\\left(a_{*}\\right)+\\hat{c}_{t-1}\\left(a_{*}\\right),\\quad \\text{(u p p e r c o n f i d e n c e b o u n d)}\n\\]\n\\[\n\\hat{\\mu}_{t-1}\\left(a_{t}\\right)\\mathbb{1}\\left(a_{t}\\neq a_{*}\\right)\\leq\\left[\\mu\\left(a_{t}\\right)+\\hat{c}_{t-1}^{\\prime}\\left(a_{t}\\right)\\right]\\mathbb{1}\\left(a_{t}\\neq a_{*}\\right).\\quad(\\text{l o w e r c o n f i d e n c e b o u n d})\n\\]\nIt follows that for all \\(t > K\\):"}, {"n": 3, "claim": "\\[\n\\begin{array}{l}\\mu\\left(a_{*}\\right)\\mathbb{1}\\left(a_{t}\\neq a_{*}\\right)\\\\ \\leq\\left[\\hat{\\mu}_{t-1}\\left(a_{*}\\right)+\\hat{c}_{t-1}\\left(a_{*}\\right)\\right]\\mathbb{1}\\left(a_{t}\\neq a_{*}\\right)\\quad(\\text{u p p e r c o n f i d e n c e b o u n d})\\\\ \\leq\\left[\\hat{\\mu}_{t-1}\\left(a_{t}\\right)+\\hat{c}_{t-1}\\left(a_{t}\\right)\\right]\\mathbb{1}\\left(a_{t}\\neq a_{*}\\right)\\quad(\\text{U C B a l g o r i t h m})\\\\ \\leq\\left[\\mu\\left(a_{t}\\right)+\\hat{c}_{t-1}^{\\prime}\\left(a_{t}\\right)+\\hat{c}_{t-1}\\left(a_{t}\\right)\\right]\\mathbb{1}\\left(a_{t}\\neq a_{*}\\right).\\quad \\text{(l o w e r c o n f i d e n c e b o u n d)}\\\\ \\end{array}\n\\]\nFor \\(t \\leq K\\), we have\n\\[\n\\mu(a_{*})=\\mu(a_{t})+[\\mu(a_{*})-\\mu(a_{t}=t)].\n\\]\nWe obtain the bound by summing over \\( t = 1 \\) to \\( t = T \\)."}]}}, {"local_id": "T2", "statement": "Assume that rewards \\(r_t(a) \\in [0,1]\\). Let \\(a_* \\in \\arg \\max_a \\mu(a)\\). With a choice of\n\\[\n\\hat{c}_{t}(a)=\\sqrt{\\frac{\\ln(2(\\hat{n}_{t}(a)+1)^{2}/\\delta)}{2\\hat{n}_{t}(a)}},\n\\]\nwe have with probability at least \\(1 - \\delta\\):\n\\[\n\\mathrm{REG}_{T}\\leq(K-1)+\\sqrt{8\\ln(2 K T^{2}/\\delta)(T-K)K}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "For any integer \\(m\\ge 0\\) and arm \\(a\\), the sequence \\(r(a_t)\\mathbb{1}(a_t=a)\\) is sub-Gaussian with \\(\\sigma_i=0.5\\,\\mathbb{1}(a_i=a)\\), so \\(\\sum_{i=1}^t\\sigma_i^2=0.25\\,\\hat n_t(a)\\)."}, {"n": 2, "claim": "\\(\\Pr[\\exists t\\ge1:\\mu(a)>\\hat\\mu_t(a)+\\hat c_t(a)\\ \\&\\ \\hat n_t(a)=m]\\le 0.5\\delta/(m+1)^2\\) for all \\(m\\ge0\\)."}, {"n": 3, "claim": "\\(\\Pr[\\exists t\\ge1:\\mu(a)<\\hat\\mu_t(a)-\\hat c'_t(a)\\ \\&\\ \\hat n_t(a)=m]\\le (0.5/K)\\delta/(m+1)^2\\), with \\(\\hat c'_t(a)=\\sqrt{\\ln(2K(\\hat n_t(a)+1)^2/\\delta)/(2\\hat n_t(a))}\\)."}, {"n": 4, "claim": "\\(\\Pr[\\exists t>K:\\mu(a_*)>\\hat\\mu_t(a_*)+\\hat c_t(a_*)]\\le \\sum_{m=0}^\\infty 0.5\\delta/(m+1)^2 \\le 0.5\\delta\\)."}, {"n": 5, "claim": "\\(\\Pr[\\exists t>K,\\,a:\\mu(a)<\\hat\\mu_t(a)-\\hat c'_t(a)]\\le \\sum_{a=1}^K\\sum_{m=1}^\\infty \\delta/(2K(m+1)^2)\\le \\delta/2\\)."}, {"n": 6, "claim": "On the good event (probability \\(\\ge1-\\delta\\)), Lemma 16.2 gives \\(\\sum_{t=1}^T[\\mu(a_*)-\\mu(a_t)]-\\sum_{a=1}^K[\\mu(a_*)-\\mu(a)]\\le \\sum_{t=K+1}^T[\\hat c_{t-1}(a_t)+\\hat c'_{t-1}(a_t)]\\)."}, {"n": 7, "claim": "\\(\\sum_{t=K+1}^T[\\hat c_{t-1}(a_t)+\\hat c'_{t-1}(a_t)]\\le 2\\sum_{t=K+1}^T\\sqrt{\\frac{\\ln(2K(\\hat n_{t-1}(a_t)+1)^2/\\delta)}{2\\hat n_{t-1}(a_t)}}\\)."}, {"n": 8, "claim": "\\(=2\\sum_{a=1}^K\\sum_{m=1}^{\\hat n_T(a)-1}\\sqrt{\\frac{\\ln(2K(m+1)^2/\\delta)}{2m}}\\)."}, {"n": 9, "claim": "\\(\\le 2\\sum_{a=1}^K\\int_0^{\\hat n_T(a)-1}\\sqrt{\\frac{\\ln(2K\\hat n_T(a)^2/\\delta)}{2t}}\\,dt\\)."}, {"n": 10, "claim": "\\(=4\\sum_{a=1}^K\\sqrt{\\frac{\\ln(2K\\hat n_T(a)^2/\\delta)(\\hat n_T(a)-1)}{2}}\\)."}, {"n": 11, "claim": "\\(\\le 4\\sqrt{\\frac{\\ln(2KT^2/\\delta)(T-K)K}{2}}\\)."}, {"n": 12, "claim": "\\(\\mathrm{REG}_T\\le (K-1)+\\sqrt{8\\ln(2KT^2/\\delta)(T-K)K}\\)."}]}}, {"local_id": "T3", "statement": "Let \\(a_* \\in \\arg \\max_a \\mu(a)\\). Let\n\\[\n\\delta_{1}=\\Pr\\left[\\exists t>K:\\mu\\left(a_{*}\\right)>\\hat{\\mu}_{t-1}\\left(a_{*}\\right)+\\hat{c}_{t-1}\\left(a_{*}\\right)\\right],\n\\]\n\\[\n\\delta_{2}=\\Pr\\left[\\exists t>K,a\\in\\{1,\\dots,K\\}\\backslash\\{a_{*}\\}:\\mu(a)<\\hat{\\mu}_{t-1}(a)-\\hat{c}_{t-1}^{\\prime}(a)\\right].\n\\]\nDefine \\(\\Delta(a) = \\mu(a_{*}) - \\mu(a)\\), and\n\\[\nT(a)=\\max\\left\\{m:\\Delta(a)\\leq\\hat{c}_{t-1}(a)+\\hat{c}_{t-1}^{\\prime}(a),\\hat{n}_{t-1}(a)=m,K<t\\leq T\\right\\}\\cup\\{0\\}.\n\\]\nThen for Algorithm 16.1, we have with probability at least \\(1 - \\delta_{1} - \\delta_{2}\\):\n\\[\n\\operatorname{REG}_{T}\\leq\\inf_{\\Delta_{0}>0}\\left[\\sum_{a=1}^{K}T(a)\\Delta(a)\\mathbb{1}(\\Delta(a)>\\Delta_{0})+(T-K)\\Delta_{0}\\right]+\\sum_{a=1}^{K}\\Delta(a).\n\\]", "proof": {"steps": [{"n": 1, "claim": "The proof of Lemma 16.2 shows that with probability at least \\(1 - \\delta\\), for all \\(t \\geq K + 1\\):"}, {"n": 2, "claim": "\\[\n\\mu(a_{*})\\leq\\mu(a_{t})+[\\hat{c}_{t-1}^{\\prime}(a_{t})+\\hat{c}_{t-1}(a_{t})]\\mathbb{1}(a_{t}\\neq a_{*}),\n\\]\nwhich implies that\n\\[\n\\Delta(a_{t})\\leq [\\hat{c}_{t-1}^{\\prime}(a_{t})+\\hat{c}_{t-1}(a_{t})]\\mathbb{1}(a_{t}\\neq a_{*}).\n\\]\nUsing the assumption of the lemma, we obtain for all \\(a \\neq a_*\\) and \\(1 \\leq t \\leq T\\), \\(\\hat{n}_{t-1}(a) \\mathbb{1}(a_t = a) \\leq T(a)\\)."}, {"n": 3, "claim": "It follows that for all \\( a \\neq a_* \\), let \\( t \\) be the last time such that \\( a_t = a \\), then \\( t \\leq T \\) and \\( \\hat{n}_T(a) = \\hat{n}_{t-1}(a_t) + 1 \\leq T(a) + 1 \\)."}, {"n": 4, "claim": "We thus obtain the following regret bound for any \\( \\Delta_0 \\geq 0 \\):"}, {"n": 5, "claim": "\\[\n\\begin{array}{l}\\sum_{t=1}^{T}\\left[\\mu\\left(a_{*}\\right)-\\mu\\left(a_{t}\\right)\\right]=\\sum_{a=1}^{K}\\hat{n}_{T}(a)\\Delta(a)\\\\ \\leq\\sum_{a=1}^{K}\\Delta(a)+\\sum_{a=1}^{K}(\\hat{n}_{T}(a)-1)\\Delta_{0}+\\sum_{a=1}^{K}(\\hat{n}_{T}(a)-1)\\Delta(a)\\mathbb{1}(\\Delta(a)>\\Delta_{0})\\\\ \\leq\\sum_{a=1}^{K}\\Delta(a)+(T-K)\\Delta_{0}+\\sum_{a=1}^{K}T(a)\\Delta(a)\\mathbb{1}(\\Delta(a)>\\Delta_{0}).\\\\ \\end{array}\n\\]\nThe second inequality used \\(\\sum_{a=1}^{K} (\\hat{n}_T(a) - 1) = T - K\\) and \\(\\hat{n}_T(a) - 1 \\leq T(a)\\)."}, {"n": 6, "claim": "This implies the bound."}]}}, {"local_id": "T4", "statement": "Consider a sequence of random functions \\(\\xi_1(S_1),\\ldots,\\xi_t(S_t),\\ldots\\) Assume each \\(\\xi_{i}\\) is sub-Gaussian with respect to \\(Z_{i}^{(y)}\\):\n\\[\n\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{\\lambda\\xi_{i}}\\leq\\lambda\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i}+\\frac{\\lambda^{2}\\sigma_{i}^{2}}{2}\n\\]\nfor some \\(\\sigma_{i}\\) that may depend on \\(S_{i - 1}\\) and \\(Z_{i}^{(x)}\\). Then for all \\(\\sigma >0\\), with probability at least \\(1 - \\delta\\)\n\\[\n\\forall n>0:\\sum_{i=1}^{n}\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i}<\\sum_{i=1}^{n}\\xi_{i}+\\left(\\sigma+\\frac{\\sum_{i=1}^{n}\\sigma_{i}^{2}}{\\sigma}\\right)\\sqrt{\\frac{\\ln(1/\\delta)}{2}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "We can let \\(\\lambda = \\sqrt{2\\ln(1 / \\delta)} /\\sigma\\), and then apply the first inequality of Theorem 13.2."}]}}], "target_local_id": "T2", "topic_domain": "Online Learning & RL", "topic_subtopic": "Bandits", "label": "flawed", "error_step": 4, "error_class6": ["invalid_bound_or_inequality", "others", "computation_or_constant"], "error_explanation": "Summing the per-count bound from $m=0$ is illegitimate: at $m=0$ we have $\\hat n_t(a_*)=0$, so $\\hat c_t(a_*)=\\sqrt{\\ln(\\cdot)/(2\\cdot0)}$ is undefined (division by zero), and the deviation bound does not apply for an arm with zero pulls. Moreover including the $m=0$ term inflates the series to $\\sum_{m=0}^\\infty 1/(m+1)^2=\\pi^2/6$, giving $\\approx0.82\\delta>0.5\\delta$, so the stated $\\le0.5\\delta$ actually fails. The sum must start at $m=1$."}
{"id": "ch1_proposition_1_11__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nConvex function. A function \\(f\\colon\\Omega\\to\\mathbb{R}\\) is convex if for all \\(x,y\\in\\Omega\\) and all scalars \\(\\gamma\\in[0,1]\\) we have \\(f(\\gamma x+(1-\\gamma)y)\\leqslant \\gamma f(x)+(1-\\gamma)f(y)\\).\n\nSmoothness. A continuously differentiable function \\(f\\) is \\(\\beta\\) smooth if the gradient map \\(\\nabla f\\colon\\mathbb{R}^n\\to\\mathbb{R}^n\\) is \\(\\beta\\)-Lipschitz, i.e., \\[ \\|\\nabla f(x) - \\nabla f(y)\\| \\leqslant \\beta\\|x-y\\|\\,. \\]", "theorems": [{"local_id": "T1", "statement": "If \\(f\\) is twice continuously differentiable on its domain \\(\\Omega\\), then \\(f\\) is convex if and only if \\(\\nabla^2 f(x)\\succeq 0\\) for all \\(x\\in\\Omega\\).", "proof": {"steps": [{"n": 1, "claim": "Assume first that \\(f\\) is convex. Fix \\(x\\in\\Omega\\), an arbitrary vector \\(u\\), and a scalar \\(\\alpha\\), and define \\(y=x+\\alpha u\\)."}, {"n": 2, "claim": "\\(f(y)-f(x)-\\nabla f(x)^\\top(y-x)\\ge 0\\)."}, {"n": 3, "claim": "\\(0\\le \\int_0^1 (1-\\gamma)\\dfrac{\\partial^2 f(x+\\gamma(y-x))}{\\partial\\gamma^2}\\,d\\gamma\\)."}, {"n": 4, "claim": "There is some \\(\\gamma\\in(0,1)\\) such that \\(0\\le (1-\\gamma)\\dfrac{\\partial^2 f(x+\\gamma(y-x))}{\\partial\\gamma^2}\\)."}, {"n": 5, "claim": "\\(\\dfrac{\\partial^2 f(x+\\gamma(y-x))}{\\partial\\gamma^2}=\\|y-x\\|^2\\,\\lambda_{\\max}\\!\\big(\\nabla^2 f(x+\\gamma(y-x))\\big)\\), the curvature of \\(f\\) at \\(x+\\gamma(y-x)\\) along the segment."}, {"n": 6, "claim": "For every \\(\\alpha\\ne 0\\), \\(0\\le u^\\top\\nabla^2 f(x+\\alpha\\gamma u)u\\)."}, {"n": 7, "claim": "\\(\\nabla^2 f(x)\\succeq 0\\)."}, {"n": 8, "claim": "Conversely, assume \\(\\nabla^2 f(z)\\succeq 0\\) for all \\(z\\in\\Omega\\). Then \\(\\int_0^1 (1-\\gamma)(y-x)^\\top\\nabla^2 f(x+\\gamma(y-x))(y-x)\\,d\\gamma\\ge 0\\)."}, {"n": 9, "claim": "Hence \\(f(y)-f(x)-\\nabla f(x)^\\top(y-x)\\ge 0\\) for all \\(x,y\\), and therefore \\(f\\) is convex."}]}}, {"local_id": "T2", "statement": "Assume \\(f\\colon\\Omega\\to\\mathbb{R}\\) is differentiable. Then, \\(f\\) is convex if and only if for all \\(x,y\\in\\Omega\\) we have\n\\[\nf(y) \\geqslant f(x) + \\nabla f(x)^\\top (y-x)\\,.\n\\]", "proof": {"steps": [{"n": 1, "claim": "First, suppose \\(f\\) is convex, then by definition\n\\begin{align*}\nf(y) &\\geqslant \\frac{f((1-\\gamma)x+\\gamma y)-(1-\\gamma)f(x)}{\\gamma}\\\\\n&\\geqslant f(x) +\\frac{f(x+\\gamma(y-x))-f(x)}\\gamma\\\\\n&\\rightarrow f(x) +\\nabla f(x)^\\top (y-x) \\quad\\text{as $\\gamma\\to 0$} \\tag{by Fact 1d-nd-gradient.}\n\\end{align*}\nOn the other hand, fix two points \\(x, y\\in\\Omega\\) and \\(\\gamma\\in[0,1]\\)."}, {"n": 2, "claim": "Putting \\(z=\\gamma x + (1-\\gamma)y\\) we get from applying the first-order lower bound twice,\n\\[\nf(x) \\geqslant f(z) + \\nabla f(z)^\\top (x-z)\n\\quad\\text{and}\\quad\nf(y) \\geqslant f(z) + \\nabla f(z)^\\top (y-z)\n\\]\nAdding these inequalities scaled by \\(\\gamma\\) and \\((1-\\gamma)\\), respectively, we get \\(\\gamma f(x)+(1-\\gamma)f(y)\\geqslant f(z),\\) which establishes convexity."}]}}, {"local_id": "T3", "statement": "Let \\(f\\) be a \\(\\beta\\)-smooth function on \\(\\mathbb{R}^n\\). Then, for every \\(x,y \\in \\mathbb{R}^n\\), \\[ \\left|f(y) - f(x) - \\nabla f(x)^\\top(y-x)\\right| \\leqslant \\frac{\\beta}{2}\\|y-x\\|^2\\,. \\]", "proof": {"steps": [{"n": 1, "claim": "Express \\(f(x) - f(y)\\) as an integral, then apply Cauchy-Schwarz and \\(\\beta\\)-smoothness as follows:\n\\begin{align*}\n|f(y) - f(x) - \\nabla f(x)^\\top(y-x)|\n&=\\left|\\int\\limits_{0}^1 \\nabla f(x + t(y-x))^\\top(y-x)dt - \\nabla\nf(x)^\\top(y-x)\\right|\\\\\n&\\leqslant  \\int\\limits_{0}^1 \\|\\nabla f(x + t(y-x)) - \\nabla f(x)\\|\\cdot \\|y-x\\|dt\\\\\n&\\leqslant \\int\\limits_{0}^1 \\beta t\\|y-x\\|^2dt\\\\\n&= \\frac{\\beta}{2}\\|y-x\\|^2\n\\end{align*}"}]}}], "target_local_id": "T1", "topic_domain": "Optimization", "topic_subtopic": "Convex Analysis & Duality", "label": "flawed", "error_step": 5, "error_class6": ["computation_or_constant", "logical_gap_or_missing_case"], "error_explanation": "Step 5 asserts d^2/dgamma^2 f(x+gamma(y-x)) = ||y-x||^2 * lambda_max(Hess f(x+gamma(y-x))). The chain rule gives the quadratic form (y-x)^T Hess f(x+gamma(y-x)) (y-x), which depends on the direction y-x; lambda_max is a single direction-independent scalar attached to the matrix. The two agree only when y-x happens to be a top eigenvector. Numerically, for f(z)=e^{z1}+z1*z2^2 at x=(0.3,-0.4), y=(1.1,0.7), gamma=0.37: g''=2.6285 while ||y-x||^2 * lambda_max = 3.3580, so the displayed identity is false, not merely under-justified. It propagates: substituting into step 4 and dividing by the positive (1-gamma) yields only lambda_max(Hess f(x+alpha*gamma*u)) >= 0, from which step 6's 0 <= u^T Hess f(x+alpha*gamma*u) u does not follow for arbitrary u -- diag(1,-1) has nonnegative largest eigenvalue but is not PSD. Letting alpha -> 0 delivers at most lambda_max(Hess f(x)) >= 0, strictly weaker than step 7's Hess f(x) >= 0, so the 'only if' direction genuinely fails. Step 8 still displays the correct quadratic form, so the proof also contradicts itself on the same identity."}
{"id": "ch1_theorem_1_2__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nConvex set. A set \\(K\\subseteq\\mathbb{R}^n\\) is convex if the line segment between any two points in \\(K\\) is also contained in \\(K\\). Formally, for all \\(x,y\\in K\\) and all scalars \\(\\gamma\\in[0,1]\\) we have \\(\\gamma x+(1-\\gamma)y\\in K\\).\n\nSmoothness. A continuously differentiable function \\(f\\) is \\(\\beta\\) smooth if the gradient map \\(\\nabla f\\colon\\mathbb{R}^n\\to\\mathbb{R}^n\\) is \\(\\beta\\)-Lipschitz, i.e., \\[ \\|\\nabla f(x) - \\nabla f(y)\\| \\leqslant \\beta\\|x-y\\|\\,. \\]\n\nConvex function. A function \\(f\\colon\\Omega\\to\\mathbb{R}\\) is convex if for all \\(x,y\\in\\Omega\\) and all scalars \\(\\gamma\\in[0,1]\\) we have \\(f(\\gamma x+(1-\\gamma)y)\\leqslant \\gamma f(x)+(1-\\gamma)f(y)\\).", "theorems": [{"local_id": "T1", "statement": "Let \\(f\\) be a \\(\\beta\\)-smooth convex function, then for every \\(x,y \\in \\mathbb{R}^n\\), we have \\[ f(x) - f(y)\\leqslant \\nabla f(x)^\\top(x-y) - \\frac{1}{2\\beta}\\|\\nabla f(x) - \\nabla f(y)\\|^2\\,. \\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(z = y - \\frac{1}{\\beta}(\\nabla f(y) - \\nabla f(x))\\). Then,\n\\begin{align*}\nf(x) - f(y)\n    &= f(x) - f(z) + f(z) - f(y) \\\\\n    &\\leqslant \\nabla f(x)^\\top(x-z) + \\nabla f(y)^\\top(z-y) + \\frac{\\beta}{2}\\|z-y\\|^2 \\\\\n    &= \\nabla f(x)^\\top(x-y) + (\\nabla f(x) - \\nabla f(y))^\\top(y-z) + \\frac{1}{2\\beta}\\|\\nabla f(x) - \\nabla f(y)\\|^2 \\\\\n    &= \\nabla f(x)^\\top(x-y) - \\frac{1}{2\\beta} \\|\\nabla f(x) - \\nabla f(y)\\|^2\n\\end{align*}\nHere, the inequality follows from convexity and smoothness."}]}}, {"local_id": "T2", "statement": "If \\(f\\) is twice continuously differentiable on its domain \\(\\Omega\\), then \\(f\\) is convex if and only if \\(\\nabla^2 f(x)\\succeq 0\\) for all \\(x\\in\\Omega\\).", "proof": {"steps": [{"n": 1, "claim": "Suppose \\(f\\) is convex and our goal is to show that the Hessian is positive semidefinite."}, {"n": 2, "claim": "Let \\(y=x + \\alpha u\\) for some arbitrary vector \\(u\\) and scalar \\(\\alpha\\)."}, {"n": 3, "claim": "Proposition first-order-lower-bound shows\n\\[\nf(y) - f(x) -\\nabla f(x)^\\top (y-x) \\geqslant 0\n\\]\nHence, by Proposition first-order-taylor,\n\\begin{align*}\n0&\\leqslant  \\int_0^1 (1-\\gamma)\\frac{\\partial^2 f(x+\\gamma(y-x))}{\\partial\\gamma^2}\\,\\mathrm{d}\\gamma\\\\\n&=  (1-\\gamma)\\frac{\\partial^2 f(x+\\gamma(y-x))}{\\partial\\gamma^2}\n\\quad\\text{for some $\\gamma\\in(0,1)$} \\tag{by the mean value theorem}\\\\\n&= (1-\\gamma)(y-x)^\\top \\nabla^2 f(x+\\gamma (y-x))(y-x)\n\\tag{by Fact 1d-nd-hessian}\\,.\n\\end{align*}\nPlugging in our choice of \\(y\\), this shows \\(0\\leqslant u^\\top \\nabla^2 f(x+\\alpha\\gamma u)u\\)."}, {"n": 4, "claim": "Letting \\(\\alpha\\) tend to zero establishes that \\(\\nabla^2 f(x)\\succeq 0\\). (Note that \\(\\gamma\\) generally depends on \\(\\alpha\\) but is always bounded by \\(1\\).)\n\nNow, suppose the Hessian is positive semidefinite everywhere in \\(\\Omega\\) and our goal is to show that the function \\(f\\) is convex."}, {"n": 5, "claim": "Using the same derivation as above, we can see that the second-order error term in Taylor's theorem must be non-negative."}, {"n": 6, "claim": "Hence, the first-order approximation is a global lower bound and so the function \\(f\\) is convex by Proposition first-order-lower-bound."}]}}, {"local_id": "T3", "statement": "Let \\(C, K\\subseteq\\mathbb{R}^n\\) be convex sets with empty intersection \\(C\\cap K=\\emptyset\\). Then there exists a point \\(a\\in\\mathbb{R}^n\\) and a number \\(b\\in\\mathbb{R}\\) such that\n\\begin{enumerate}\n\\item for all \\(x\\in C\\), we have \\(\\langle a, x\\rangle \\geqslant b\\).\n\\item for all \\(x\\in K\\), we have \\(\\langle a, x\\rangle \\leqslant b\\).\n\\end{enumerate}\nIf \\(C\\) and \\(K\\) are closed and at least one of them is bounded, then we can replace the inequalities by strict inequalities.", "proof": {"steps": [{"n": 1, "claim": "We prove the case where both sets are compact (i.e., closed and bounded). In this case, the Cartesian product \\(C\\times K\\) is also compact. Therefore, \\(d(x,y)=\\|x-y\\|\\) attains its minimum over \\(C\\times K\\)."}, {"n": 2, "claim": "Let \\(p\\in C\\), \\(q\\in K\\) attain the minimum with \\(\\delta=\\|p-q\\|>0\\)."}, {"n": 3, "claim": "Set \\(a=p-q\\neq0\\), \\(m=(p+q)/2\\), \\(b=\\langle a,m\\rangle=(\\langle a,p\\rangle+\\langle a,q\\rangle)/2\\)."}, {"n": 4, "claim": "Since \\(\\langle a,p\\rangle-\\langle a,q\\rangle=\\|a\\|^2>0\\), we have \\(\\langle a,p\\rangle<b<\\langle a,q\\rangle\\)."}, {"n": 5, "claim": "If \\(r\\in C\\) lies on \\(H=\\{x:\\langle a,x\\rangle=b\\}\\), then \\(z(t)=p+t(r-p)\\in C\\) for \\(t\\in[0,1]\\)."}, {"n": 6, "claim": "\\(f(t)=\\|z(t)-q\\|^2\\) has \\(f'(0)=2(b-\\langle a,p\\rangle)=-\\|a\\|^2<0\\)."}, {"n": 7, "claim": "Hence a nearer point exists, contradicting minimality; symmetry handles \\(K\\)."}]}}], "target_local_id": "T3", "topic_domain": "Optimization", "topic_subtopic": "Convex Analysis & Duality", "label": "flawed", "error_step": 4, "error_class6": ["computation_or_constant", "invalid_bound_or_inequality", "logical_gap_or_missing_case"], "error_explanation": "From $\\langle a,p\\rangle-\\langle a,q\\rangle=\\|a\\|^2>0$ it follows that $\\langle a,p\\rangle>\\langle a,q\\rangle$, hence $\\langle a,p\\rangle>b>\\langle a,q\\rangle$. The stated chain $\\langle a,p\\rangle<b<\\langle a,q\\rangle$ reverses the inequality direction and is false."}
{"id": "ch2_corollary_2_18__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nLet \\(X_1,\\ldots,X_n\\) be \\(n\\) real-valued independent and identically distributed (iid) random variables, with expectation \\(\\mu=\\mathbb{E}X_i\\). Let\n\\[\n\\bar{X}_n=\\frac{1}{n}\\sum_{i=1}^{n}X_i. \\tag{2.1}\n\\]\n\nDefinition 2.4. Given a random variable \\( X \\), we may define its logarithmic moment generating function as\n\\[\n\\Lambda_{X}(\\lambda)=\\ln\\mathbb{E}e^{\\lambda X}.\n\\]\nMoreover, given \\(z\\in \\mathbb{R}\\), the rate function \\(I_{X}(z)\\) is defined as\n\\[\nI_{X}(z)=\\left\\{\\begin{array}{l l}\\sup_{\\lambda>0}[\\lambda z-\\Lambda_{X}(\\lambda)]&z>\\mu\\\\ 0&z=\\mu\\\\ \\sup_{\\lambda<0}[\\lambda z-\\Lambda_{X}(\\lambda)]&z<\\mu,\\end{array}\\right.\n\\]\nwhere \\(\\mu = \\mathbb{E}[X]\\)\nThe above definition can be used to obtain exponential tail bounds for sums of independent variables as follows.", "theorems": [{"local_id": "T1", "statement": "Assume that \\(X_{1}\\in [0,1]\\).\nThen for all \\(\\epsilon > 0\\):\n\\[\n\\Pr\\left(\\bar{X}_{n}\\geq(1+\\epsilon)\\mu\\right)\\leq\\exp\\left[\\frac{-n\\mu\\epsilon^{2}}{2+\\epsilon}\\right],\n\\]\n\\[\n\\Pr\\left(\\bar{X}_{n}\\leq(1-\\epsilon)\\mu\\right)\\leq\\exp\\left[\\frac{-n\\mu\\epsilon^{2}}{2}\\right].\n\\]\nMoreover, for \\( t > 0 \\), we have:\n\\[\n\\Pr\\left(\\bar{X}_{n}\\geq\\mu+\\sqrt{\\frac{2\\mu t}{n}}+\\frac{t}{3 n}\\right)\\leq e^{-t}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "The first and the second results can be obtained from Theorem 2.17 and the inequality \\(\\mathrm{KL}(z||\\mu) \\geq (z - \\mu)^2 / (2\\mu)\\) (which is left as an exercise)."}, {"n": 2, "claim": "We then take \\(z = (1 + \\epsilon)\\mu\\) and \\(z = (1 - \\epsilon)\\mu\\) respectively for the first and the second inequalities."}, {"n": 3, "claim": "For the third inequality (which is sharper than the first inequality), we may apply Theorem 2.10. Just observe from Lemma 2.15 that when \\(0 < \\lambda < 3\\):\n\\[\n\\begin{array}{l}\\Lambda_{X_{1}}(\\lambda)\\leq\\ln [(1-\\mu)e^{0}+\\mu e^{\\lambda}]\\\\ \\leq\\mu(e^{\\lambda}-1)=\\mu\\lambda+\\mu\\sum_{k\\geq 2}\\frac{\\lambda^{k}}{k!}\\\\ \\leq\\mu\\lambda+\\frac{\\mu\\lambda^{2}}{2(1-\\lambda/3)}.\\\\ \\end{array}\n\\]"}, {"n": 4, "claim": "In the above derivation, the equality used the Taylor expansion of exponential function. The last inequality used \\( k! \\geq 2 \\cdot 3^{k-2} \\) and the sum of infinite geometric series."}, {"n": 5, "claim": "We may take \\( \\alpha = \\mu \\) and \\( \\beta = 1/3 \\) in Theorem 2.10 to obtain the desired bound."}]}}, {"local_id": "T2", "statement": "Assume that \\(X_{1}\\in [0,1]\\) . Then for all \\(\\epsilon >0\\) , we have\n\\[\n\\begin{array}{l}\\Pr(\\bar{X}_{n}\\geq\\mu+\\epsilon)\\leq e^{-n\\mathrm{KL}(\\mu+\\epsilon| |\\mu)},\\\\ \\Pr(\\bar{X}_{n}\\leq\\mu-\\epsilon)\\leq e^{-n\\mathrm{KL}(\\mu-\\epsilon| |\\mu)},\\\\ \\end{array}\n\\]\nwhere \\(\\mathrm{KL}(z||\\mu)\\) is the Kullback-Leibler divergence (KL divergence) defined as\n\\[\n\\mathrm{KL}(z| |\\mu)=z\\ln{\\frac{z}{\\mu}}+(1-z)\\ln{\\frac{1-z}{1-\\mu}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Consider the case \\(z = \\mu +\\epsilon\\) .We have\n\\[\n-I_{X_{1}}(z)\\leq\\inf_{\\lambda>0}[-\\lambda z+\\ln((1-\\mu)e^{0}+\\mu e^{\\lambda})].\n\\]\nAssume that the optimal value of \\(\\lambda\\) on the right hand side is achieved at \\(\\lambda_{*}\\)."}, {"n": 2, "claim": "By setting the derivative to zero, we obtain the expression:"}, {"n": 3, "claim": "\\[\nz=\\frac{\\mu e^{\\lambda_{*}}}{(1-\\mu)e^{0}+\\mu e^{\\lambda_{*}}},\n\\]\nwhich implies that\n\\[\ne^{\\lambda_{*}}=\\frac{z(1-\\mu)}{\\mu(1-z)}.\n\\]\nThis implies that \\(-I_{X_1}(z) \\leq -\\mathrm{KL}(z||\\mu)\\)."}, {"n": 4, "claim": "The case of \\(z = \\mu - \\epsilon\\) is similar. We can thus obtain the desired bound from Theorem 2.5."}]}}, {"local_id": "T3", "statement": "We have\n\\[\n\\Pr\\left(\\left|\\bar{X}_{n}-\\mu\\right|\\geq\\epsilon\\right)\\leq\\frac{\\operatorname{Var}\\left(X_{1}\\right)}{n\\epsilon^{2}}.\\tag{2.2}\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\( h(x) = (x - \\mu)^2 \\), then\n\\[\n\\mathbb{E}h(\\bar{X}_{n})=\\mathbb{E}(\\bar{X}_{n}-\\mu)^{2}=\\frac{1}{n}\\mathrm{Var}(X_{1}).\n\\]\nThe desired bound follows from the Markov inequality with \\( S = \\{|\\bar{X}_n - \\mu| \\geq \\epsilon\\} \\)."}]}}], "target_local_id": "T1", "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "label": "flawed", "error_step": 1, "error_class6": ["invalid_bound_or_inequality", "computation_or_constant"], "error_explanation": "Step 1 asserts the displayed inequality KL(z||mu) >= (z-mu)^2/(2mu). The correct auxiliary inequality has denominator max(2mu, mu+z); the mu+z branch has been dropped. The claim as displayed is false whenever z is appreciably larger than mu: for mu=0.1, z=0.9 one has KL(0.9||0.1) ~= 1.758 while (z-mu)^2/(2mu) = 0.64/0.2 = 3.2, so the asserted lower bound exceeds the actual divergence. The dropped branch is exactly the one active in the upper tail, since for z=(1+eps)mu we have max(2mu, mu+z) = (2+eps)mu. Feeding the corrupted inequality into Theorem 2.17 at Step 2 with z=(1+eps)mu yields exponent n*mu*eps^2/2 instead of the stated n*mu*eps^2/(2+eps) -- a bound that is itself false (Bernoulli(0.1), eps=8, n=1: Pr(Xbar>=0.9)=0.1 > e^{-3.2}~=0.041, while the corollary's true bound e^{-0.64}~=0.527 holds comfortably). The corollary's first displayed inequality is therefore never derived, and nothing downstream repairs it: Steps 3-5 handle only the third inequality by a separate Theorem 2.10 argument."}
{"id": "ch2_lemma_2_22__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nLet \\(X_{1},\\ldots,X_{n}\\) be \\(n\\) real-valued independent and identically distributed (iid) random variables, with expectation \\(\\mu = \\mathbb{E}X_{i}\\). Let\n\\[\n\\bar{X}_{n}=\\frac{1}{n}\\sum_{i=1}^{n}X_{i}.\\tag{2.1}\n\\]\n\nDefinition 2.4. Given a random variable \\( X \\), we may define its logarithmic moment generating function as\n\\[\n\\Lambda_{X}(\\lambda)=\\ln\\mathbb{E}e^{\\lambda X}.\n\\]\nMoreover, given \\(z\\in \\mathbb{R}\\), the rate function \\(I_{X}(z)\\) is defined as\n\\[\nI_{X}(z)=\\left\\{\\begin{array}{l l}\\sup_{\\lambda>0}[\\lambda z-\\Lambda_{X}(\\lambda)]&z>\\mu\\\\ 0&z=\\mu\\\\ \\sup_{\\lambda<0}[\\lambda z-\\Lambda_{X}(\\lambda)]&z<\\mu,\\end{array}\\right.\n\\]\nwhere \\(\\mu = \\mathbb{E}[X]\\)\nThe above definition can be used to obtain exponential tail bounds for sums of independent variables as follows.\n\nConsider a random variable \\(X\\) so that \\(\\mathbb{E}[X] = \\mu\\). Assume that there exists \\(\\alpha > 0\\) and \\(\\beta \\geq 0\\) such that for \\(\\lambda \\in [0, \\beta^{-1})\\):\n\\[\n\\Lambda_{X}(\\lambda)\\leq\\lambda\\mu+\\frac{\\alpha\\lambda^{2}}{2(1-\\beta\\lambda)},\\tag{2.9}\n\\]\nthen for \\(\\epsilon > 0\\):\n\\[\n\\begin{array}{l}-I_{X}(\\mu+\\epsilon)\\leq-\\frac{\\epsilon^{2}}{2(\\alpha+\\beta\\epsilon)},\\\\ -I_{X}\\left(\\mu+\\epsilon+\\frac{\\beta\\epsilon^{2}}{2\\alpha}\\right)\\leq-\\frac{\\epsilon^{2}}{2\\alpha}.\\end{array}\n\\]", "theorems": [{"local_id": "T1", "statement": "If \\( X_{1} \\) has a logarithmic moment generating function that satisfies (2.9) for \\( \\lambda > 0 \\), then all \\( \\epsilon > 0 \\):\n\\[\n\\operatorname*{Pr}(\\bar{X}_{n}\\geq\\mu+\\epsilon)\\leq\\exp\\left[\\frac{-n\\epsilon^{2}}{2(\\alpha+\\beta\\epsilon)}\\right].\n\\]\nMoreover, for \\(t > 0\\), we have\n\\[\n\\operatorname*{Pr}\\left(\\bar{X}_{n}\\geq\\mu+\\sqrt{\\frac{2\\alpha t}{n}}+\\frac{\\beta t}{n}\\right)\\leq e^{-t}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "The first inequality of the theorem follows from the first inequality of Lemma 2.9 and Theorem 2.5. The second inequality of the theorem follows from the second inequality of Lemma 2.9 and Theorem 2.5, with \\(\\epsilon = \\sqrt{2\\alpha t / n}\\)."}]}}, {"local_id": "T2", "statement": "Assume that \\( X_{1} \\) satisfies the moment condition in Lemma 2.22. Then for all \\( \\epsilon > 0 \\):\n\\[\n\\Pr [\\bar{X}_{n}\\geq\\mu+\\epsilon]\\leq\\exp\\left[\\frac{-n\\epsilon^{2}}{2 V+2\\epsilon b/3}\\right],\n\\]\nand for all \\(t > 0\\):\n\\[\n\\operatorname*{Pr}\\left[\\bar{X}_{n}\\geq\\mu+\\sqrt{\\frac{2Vt}{n}}+\\frac{bt}{3n}\\right]\\leq e^{-t}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "We simply set \\(\\alpha = V\\) and \\(\\beta = b / 3\\) in Theorem 2.10."}]}}, {"local_id": "T3", "statement": "If \\( X \\) satisfies the following moment condition with \\( b, V > 0 \\) for integers \\( m \\geq 2 \\):\n\\[\n\\mathbb{E}[X-c]^{m}\\leq m!(b/3)^{m-2}V/2,\n\\]\nwhere \\(c\\) is arbitrary. Then when \\(\\lambda \\in (0,3 / b)\\):\n\\[\n\\ln\\mathbb{E}e^{\\lambda X}\\leq\\lambda\\mathbb{E}X+\\frac{\\lambda^{2}V}{2(1-\\lambda b/3)}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "\\(\\ln\\mathbb{E}e^{\\lambda X}=\\lambda c+\\ln\\mathbb{E}e^{\\lambda(X-c)}\\)."}, {"n": 2, "claim": "\\(\\ln\\mathbb{E}e^{\\lambda(X-c)}\\le\\mathbb{E}e^{\\lambda(X-c)}-1\\)."}, {"n": 3, "claim": "\\(\\mathbb{E}e^{\\lambda(X-c)}-1=\\sum_{m\\ge1}\\frac{\\lambda^m\\,\\mathbb{E}(X-c)^m}{m!}\\)."}, {"n": 4, "claim": "\\(\\mathbb{E}e^{\\lambda(X-c)}-1=\\lambda\\mathbb{E}(X-c)+\\sum_{m\\ge2}\\frac{\\lambda^m\\,\\mathbb{E}(X-c)^m}{m!}\\)."}, {"n": 5, "claim": "\\(\\ln\\mathbb{E}e^{\\lambda X}\\le\\lambda\\mathbb{E}X+\\sum_{m\\ge2}\\frac{\\lambda^m\\,\\mathbb{E}(X-c)^m}{m!}\\)."}, {"n": 6, "claim": "For \\(m\\ge2\\), \\(\\dfrac{\\lambda^m\\,\\mathbb{E}(X-c)^m}{m!}\\le\\lambda^m(b/6)^{m-2}\\dfrac{V}{2}\\)."}, {"n": 7, "claim": "\\(\\sum_{m\\ge2}\\lambda^m(b/3)^{m-2}\\dfrac{V}{2}=\\dfrac{\\lambda^2 V}{2(1-\\lambda b/3)}\\)."}, {"n": 8, "claim": "\\(\\ln\\mathbb{E}e^{\\lambda X}\\le\\lambda\\mathbb{E}X+\\dfrac{\\lambda^2 V}{2(1-\\lambda b/3)}\\)."}]}}], "target_local_id": "T3", "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "label": "flawed", "error_step": 6, "error_class6": ["invalid_bound_or_inequality", "computation_or_constant", "logical_gap_or_missing_case"], "error_explanation": "The moment hypothesis only guarantees $\\mathbb{E}(X-c)^m\\le m!(b/3)^{m-2}V/2$, but the step replaces the constant $b/3$ with the strictly smaller $b/6$, asserting a tighter bound $(b/6)^{m-2}\\le(b/3)^{m-2}$ that is not implied and is false in general (the moment can attain the $(b/3)$ value)."}
{"id": "ch2_lemma_2_8__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nSmoothness. A continuously differentiable function \\(f\\) is \\(\\beta\\) smooth if the gradient map \\(\\nabla f\\colon\\mathbb{R}^n\\to\\mathbb{R}^n\\) is \\(\\beta\\)-Lipschitz, i.e., \\[ \\|\\nabla f(x) - \\nabla f(y)\\| \\leqslant \\beta\\|x-y\\|\\,. \\]\n\nConvex function. A function \\(f\\colon\\Omega\\to\\mathbb{R}\\) is convex if for all \\(x,y\\in\\Omega\\) and all scalars \\(\\gamma\\in[0,1]\\) we have \\(f(\\gamma x+(1-\\gamma)y)\\leqslant \\gamma f(x)+(1-\\gamma)f(y)\\).", "theorems": [{"local_id": "T1", "statement": "Let \\(f\\) be convex and \\(\\beta\\)-smooth on \\(\\mathbb{R}^n\\) then gradient descent with \\(\\eta = \\frac{1}{\\beta}\\) satisfies \\[ f(x_t) - f(x^*) \\leqslant \\frac{2\\beta\\|x_1 - x^*\\|^2}{t-1} \\]", "proof": {"steps": [{"n": 1, "claim": "By the update rule and the smoothness lemma we have\n\\[f(x_{s+1}) - f(x_s) \\leqslant -\\frac{1}{2\\beta}\\|\\nabla f(x_s)\\|^2 \\]\nIn particular, denoting \\(\\delta_s = f(x_s) - f(x^*)\\) this shows\n\\[\\delta_{s+1} \\leqslant \\delta_s - \\frac{1}{2\\beta}\\|\\nabla f(x_s)\\|^2 \\]\nOne also has by convexity\n\\[\\delta_s \\leqslant \\nabla f(x_s)^\\top(x_s - x^*) \\leqslant \\|x_s - x^*\\| \\cdot \\|\\nabla f(x_s)\\|\\]\nWe will prove that \\(\\|x_s - x^*\\|\\) is decreasing with \\(s\\), which with the two above displays will imply\n\\[\\delta_{s+1}\\leqslant \\delta_s - \\frac{1}{2\\beta\\|x_1 - x^*\\|^2}\\delta_s^2\\]\nWe solve the recurrence as follows."}, {"n": 2, "claim": "Let \\(w = \\frac{1}{2\\beta\\|x_1 - x^*\\|^2}\\), then\n\\[w\\delta_s^2 + \\delta_{s+1} \\leqslant \\delta_s \\iff w\\frac{\\delta_s}{\\delta_{s+1}} + \\frac\n{1}{\\delta_s} \\leqslant \\frac{1}{\\delta_{s+1}} \\implies \\frac{1}{\\delta_{s+1}} - \\frac{1}{\\delta_s} \\geqslant w \\implies \\frac{1}{\\delta_t} \\geqslant w(t-1)\\]\nTo finish the proof it remains to show that \\(\\|x_s - x^*\\|\\) is decreasing with \\(s\\)."}, {"n": 3, "claim": "Using the cocoercivity lemma, we get\n\\[\n(\\nabla f(x) - \\nabla f(y))^\\top(x - y) \\geqslant \\frac{1}{\\beta} \\|\\nabla f(x)\n- \\nabla f(y)\\|^2\\,.\n\\]\nWe use this and the fact that \\(\\nabla f(x^*) = 0\\), to show\n\\begin{align*}\n    \\|x_{s+1} - x^*\\|^2 &= \\|x_s - \\frac{1}{\\beta}\\nabla f(x_s) - x^*\\|^2 \\\\\n    &= \\|x_s - x^*\\|^2 - \\frac{2}{\\beta}\\nabla f(x_s)^\\top(x_s - x^*) + \\frac{1}{\\beta^2}\\|\\nabla f(x_s)\\|^2 \\\\\n    &\\leqslant \\|x_s - x^*\\|^2 - \\frac{1}{\\beta^2} \\|\\nabla f(x_s)\\|^2 \\\\\n    &\\leqslant \\|x_s - x^*\\|^2\\,.\n\\end{align*}"}]}}, {"local_id": "T2", "statement": "Let \\(f\\) be a \\(\\beta\\)-smooth function on \\(\\mathbb{R}^n\\). Then, for every \\(x,y \\in \\mathbb{R}^n\\), \\[ \\left|f(y) - f(x) - \\nabla f(x)^\\top(y-x)\\right| \\leqslant \\frac{\\beta}{2}\\|y-x\\|^2\\,. \\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(D:=f(y)-f(x)-\\nabla f(x)^\\top(y-x)\\). It suffices to bound \\(D\\) from above by \\(\\frac{\\beta}{2}\\|y-x\\|^2\\)."}, {"n": 2, "claim": "\\(D=\\int_0^1\\big(\\nabla f(x+t(y-x))-\\nabla f(x)\\big)^\\top(y-x)\\,dt\\)."}, {"n": 3, "claim": "\\(D\\leqslant\\int_0^1\\|\\nabla f(x+t(y-x))-\\nabla f(x)\\|\\cdot\\|y-x\\|\\,dt\\)."}, {"n": 4, "claim": "\\(\\int_0^1\\|\\nabla f(x+t(y-x))-\\nabla f(x)\\|\\cdot\\|y-x\\|\\,dt\\leqslant\\int_0^1\\beta t\\|y-x\\|^2\\,dt=\\frac{\\beta}{2}\\|y-x\\|^2\\)."}, {"n": 5, "claim": "Therefore \\(|D|\\leqslant\\frac{\\beta}{2}\\|y-x\\|^2\\)."}]}}, {"local_id": "T3", "statement": "Let \\(f\\) be a \\(\\beta\\)-smooth convex function, then for every \\(x,y \\in \\mathbb{R}^n\\), we have \\[ f(x) - f(y)\\leqslant \\nabla f(x)^\\top(x-y) - \\frac{1}{2\\beta}\\|\\nabla f(x) - \\nabla f(y)\\|^2\\,. \\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(z = y - \\frac{1}{\\beta}(\\nabla f(y) - \\nabla f(x))\\). Then,\n\\begin{align*}\nf(x) - f(y)\n    &= f(x) - f(z) + f(z) - f(y) \\\\\n    &\\leqslant \\nabla f(x)^\\top(x-z) + \\nabla f(y)^\\top(z-y) + \\frac{\\beta}{2}\\|z-y\\|^2 \\\\\n    &= \\nabla f(x)^\\top(x-y) + (\\nabla f(x) - \\nabla f(y))^\\top(y-z) + \\frac{1}{2\\beta}\\|\\nabla f(x) - \\nabla f(y)\\|^2 \\\\\n    &= \\nabla f(x)^\\top(x-y) - \\frac{1}{2\\beta} \\|\\nabla f(x) - \\nabla f(y)\\|^2\n\\end{align*}\nHere, the inequality follows from convexity and smoothness."}]}}], "target_local_id": "T2", "topic_domain": "Optimization", "topic_subtopic": "Convex Analysis & Duality", "label": "flawed", "error_step": 1, "error_class6": ["logical_gap_or_missing_case", "invalid_bound_or_inequality", "others"], "error_explanation": "Only the upper bound $D\\leqslant\\frac{\\beta}{2}\\|y-x\\|^2$ was proved; bounding $|D|$ also requires $-D\\leqslant\\frac{\\beta}{2}\\|y-x\\|^2$ (the symmetric lower case). Step 1's claim that one direction controls $|D|=\\max(D,-D)$ is false, so the missing branch makes the conclusion unjustified."}
{"id": "ch4_lemma_4_5__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\n\\(\\epsilon\\)-cover. Let \\(\\epsilon > 0\\). An \\(\\epsilon\\)-cover of a set \\(S\\) with respect to a distance metric \\(\\rho\\) is a subset \\(C \\subseteq S\\) such that for all \\(x \\in S\\) there exists \\(x' \\in C\\) with \\(\\rho(x, x') \\leq \\epsilon\\), or equivalently,\n\\[\nS \\subseteq \\bigcup_{x \\in C} \\mathrm{Ball}(x, \\epsilon, \\rho), \\quad \\text{where} \\quad \\mathrm{Ball}(x, \\epsilon, \\rho) \\triangleq \\{x' : \\rho(x, x') \\leq \\epsilon\\}.\n\\]\n\n\\(\\kappa\\)-Lipschitz functions. Let \\(\\kappa \\geq 0\\) and let \\(\\|\\cdot\\|\\) be a norm on the domain \\(D\\). A function \\(L : D \\to \\mathbb{R}\\) is said to be \\(\\kappa\\)-Lipschitz with respect to \\(\\|\\cdot\\|\\) if for all \\(\\theta, \\theta' \\in D\\),\n\\[\n|L(\\theta) - L(\\theta')| \\leq \\kappa\\,\\|\\theta - \\theta'\\|.\n\\]\n\nRisk and empirical-risk minimizer setup for finite hypothesis classes. Let \\(\\mathcal{H}\\) be a finite hypothesis class and let \\(\\ell\\) be a loss function with \\(0 \\leq \\ell((x,y),h) \\leq 1\\). For \\(h \\in \\mathcal{H}\\), define the population risk\n\\[\nL(h) = \\mathbb{E}_{(x,y) \\sim \\mathcal{D}}[\\ell((x,y),h)].\n\\]\nGiven \\(n\\) iid samples \\((x^{(1)},y^{(1)}), \\dots, (x^{(n)},y^{(n)})\\) from \\(\\mathcal{D}\\), define the empirical risk\n\\[\n\\widehat{L}(h) = \\frac{1}{n}\\sum_{i=1}^n \\ell((x^{(i)},y^{(i)}),h).\n\\]\nLet \\(\\hat{h} \\in \\arg\\min_{h \\in \\mathcal{H}} \\widehat{L}(h)\\) be an empirical-risk minimizer and let \\(h^* \\in \\arg\\min_{h \\in \\mathcal{H}} L(h)\\) be a population-risk minimizer.\n\nEpsilon-cover. \\(\\mathcal{C}\\) is an \\(\\epsilon\\)-cover of \\(\\mathcal{Q}\\) with respect to metric \\(\\rho\\) if for all \\(v' \\in \\mathcal{Q}\\), there exists \\(v \\in \\mathcal{C}\\) such that \\(\\rho(v,v')\\le \\epsilon\\).\n\nCovering number. The covering number is defined as the minimum size of an \\(\\epsilon\\)-cover, or explicitly:\n\\[\nN(\\epsilon, \\mathcal{Q}, \\rho) \\stackrel{\\Delta}{=} (\\text{min size of } \\epsilon\\text{-cover of } \\mathcal{Q} \\text{ w.r.t. metric } \\rho).\n\\]", "theorems": [{"local_id": "T1", "statement": "Setup. Throughout, the parameter \\(\\theta\\) ranges over \\(S=\\{\\theta\\in\\mathbb{R}^p:\\|\\theta\\|_2\\le B\\}\\) for some \\(B>0\\).\n\nSuppose \\(\\ell((x,y), \\theta) \\in [0,1]\\), and \\(\\ell((x,y), \\theta)\\) is \\(\\kappa\\)-Lipschitz in \\(\\theta\\) with respect to the \\(\\ell_2\\)-norm for all \\((x, y)\\). Then, with probability at least \\(1-O(\\exp(-\\Omega(p)))\\), we have\n\\[\n\\forall \\theta, \\quad |\\widehat{L}(\\theta)- L(\\theta)| \\leq O\\!\\left(\\sqrt{\\frac{p \\max(\\ln(\\kappa Bn), 1)}{n}}\\right).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Fix parameters \\(\\delta, \\epsilon>0\\) (we will specify their values later). Let \\(C\\) be the \\(\\epsilon\\)-cover of our parameter space \\(S\\) with respect to the \\(\\ell_2\\)-norm constructed in the referenced lemma. Define event \\(E = \\left\\{ \\forall \\theta \\in C, \\; |\\widehat{L}(\\theta) - L(\\theta)| \\le \\delta \\right\\}\\). By Theorem 4.1, we have \\(\\mathbb{P}(E) \\ge 1 - 2|C|\\exp(-2n\\delta^2)\\)."}, {"n": 2, "claim": "Now for any \\(\\theta \\in S\\), we can pick some \\(\\theta_0 \\in C\\) such that \\(\\|\\theta-\\theta_0\\|_2 \\le \\epsilon\\)."}, {"n": 3, "claim": "Since \\(L\\) and \\(\\widehat{L}\\) are \\(\\kappa\\)-Lipschitz functions (this follows from the Lipschitzness of \\(\\ell\\)), we have\n\\[\n|L(\\theta) - L(\\theta_0)| \\le \\kappa \\|\\theta-\\theta_0\\|_2 \\le \\kappa \\epsilon, \\quad\\text{and}\\quad |\\widehat{L}(\\theta) - \\widehat{L}(\\theta_0)| \\le \\kappa \\|\\theta-\\theta_0\\|_2 \\le \\kappa \\epsilon.\n\\]\n\nTherefore, conditional on \\(E\\), we have\n\\[\n|\\widehat{L}(\\theta) - L(\\theta)| \\le |\\widehat{L}(\\theta)-\\widehat{L}(\\theta_0)| + |\\widehat{L}(\\theta_0) - L(\\theta_0)| + |L(\\theta_0) - L(\\theta)| \\le 2 \\kappa\\epsilon+\\delta.\n\\]\n\nSet \\(\\epsilon = \\delta / (2 \\kappa)\\) so that conditional on \\(E\\),\n\\[\n|\\widehat{L}(\\theta) - L(\\theta)| \\le 2\\delta.\n\\]\n\nThe event \\(E\\) happens with probability \\(1 - 2|C|\\exp(-2n\\delta^2) = 1 - 2 \\exp(\\ln{|C|} - 2n\\delta^2)\\)."}, {"n": 4, "claim": "From the referenced remark, \\(\\ln{|C|} \\leq p \\ln{ (3B\\sqrt{p} / (\\delta / (2\\kappa))) }\\)."}, {"n": 5, "claim": "If we ignore the log term and assume \\(\\ln{|C|} \\leq p\\), then:"}, {"n": 6, "claim": "\\[\n2|C| \\exp(-2n\\delta^2) = 2\\exp(\\ln{|C|} - 2n\\delta^2) \\leq 2\\exp(p - 2p) = 2\\exp(-p).\n\\]\n\nSince we cannot drop the log term in \\(\\ln{|C|} \\leq p \\ln{ (3B\\sqrt{p} / (\\delta / (2\\kappa))) }\\), we set \\(\\delta = \\sqrt{\\frac{c_0 p \\max(1, \\ln{(\\kappa Bn)})}{n}}\\) with \\(c_0 = 36\\)."}, {"n": 7, "claim": "Then,\n\\[\n\\ln{|C|} - 2n\\delta^2 \\leq p \\ln\\!\\left(\\frac{6B \\kappa \\sqrt{p}}{\\delta}\\right) - 2n\\delta^2 \\leq p \\ln\\!\\left(\\frac{6B\\kappa \\sqrt{n}}{ \\sqrt{c_0 \\max(1, \\ln{(\\kappa Bn)})} }\\right) - 2n \\frac{c_0p}{n} \\ln(\\kappa Bn)\n\\leq p\\ln\\!\\left(B\\kappa \\sqrt{n}\\right) - 72 p \\ln(\\kappa Bn)\n\\leq p \\ln(B\\kappa n) - 72 p \\ln(B\\kappa n) \\leq -p,\n\\]\nsince \\(\\ln (B\\kappa n) \\geq 1\\) for large enough \\(n\\)."}, {"n": 8, "claim": "Therefore, with probability greater than \\(1 - O(e^{-p})\\), we have\n\\[\n|\\widehat{L}(\\theta) - L(\\theta)| \\leq 2\\delta = O\\!\\left(\\sqrt{\\frac{p}{n}\\max(1,\\ln(\\kappa Bn))}\\right).\n\\]"}]}}, {"local_id": "T2", "statement": "Let \\(B, \\epsilon > 0\\), and let \\(S = \\{x \\in \\mathbb{R}^p : \\|x\\|_2 \\leq B\\}\\). Then there exists an \\(\\epsilon\\)-cover of \\(S\\) with respect to the \\(\\ell_2\\)-norm with at most \\(\\max\\!\\left(\\left(\\dfrac{3B\\sqrt{p}}{\\epsilon}\\right)^p,\\, 1\\right)\\) elements.", "proof": {"steps": [{"n": 1, "claim": "Define the grid \\(C=\\{x\\in S: x_i=k_i\\tfrac{\\epsilon}{\\sqrt p},\\,k_i\\in\\mathbb{Z},\\,|k_i|\\le \\tfrac{B\\sqrt p}{\\epsilon}\\}\\)."}, {"n": 2, "claim": "For each \\(x\\in S\\) there is \\(x'\\in C\\) with \\(|x_i-x'_i|\\le \\tfrac{\\epsilon}{\\sqrt p}\\) for all \\(i\\)."}, {"n": 3, "claim": "Hence \\(\\|x-x'\\|_2=\\sqrt{\\sum_{i=1}^p|x_i-x'_i|^2}\\le\\sqrt{p\\cdot\\tfrac{\\epsilon^2}{p}}=\\epsilon\\), so \\(C\\) is an \\(\\epsilon\\)-cover."}, {"n": 4, "claim": "Each coordinate index \\(k_i\\) has at most \\(\\tfrac{2B\\sqrt p}{\\epsilon}+1\\le\\tfrac{3B\\sqrt p}{\\epsilon}\\) choices, so \\(|C|\\le\\big(\\tfrac{3B\\sqrt p}{\\epsilon}\\big)^p\\)."}, {"n": 5, "claim": "Therefore for all \\(B,\\epsilon>0\\) the set \\(S\\) has an \\(\\epsilon\\)-cover of size at most \\(\\max((3B\\sqrt p/\\epsilon)^p,1)\\)."}]}}, {"local_id": "T3", "statement": "Suppose \\(\\phi\\) is \\(\\kappa\\)-Lipschitz, and \\(\\rho = L_2(P_n)\\). Then,\n\\[\n\\log N(\\epsilon, \\phi \\circ \\mathcal{F}, \\rho) \\le \\log N(\\epsilon / \\kappa, \\mathcal{F}, \\rho).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\mathcal{C}\\) denote an \\(\\epsilon/\\kappa\\)-cover for \\(\\mathcal{F}\\). Then \\(\\phi \\circ \\mathcal{C}\\) is an \\(\\epsilon\\)-cover of \\(\\phi \\circ \\mathcal{F}\\):\n\\[\n\\rho(\\phi \\circ f', \\phi \\circ f) = \\sqrt{\\frac{1}{n} \\sum (\\phi(f'(z_i)) - \\phi(f(z_i)))^2} \\le \\sqrt{\\frac{1}{n} \\cdot \\kappa^2 \\sum(f'(z_i) - f(z_i))^2} \\le \\kappa \\cdot \\frac{\\epsilon}{\\kappa} = \\epsilon.\n\\]"}]}}], "target_local_id": "T2", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "label": "flawed", "error_step": 4, "error_class6": ["invalid_bound_or_inequality", "logical_gap_or_missing_case", "others"], "error_explanation": "Step 4 asserts 2B sqrt(p)/eps + 1 <= 3B sqrt(p)/eps, which is equivalent to B sqrt(p)/eps >= 1, i.e. eps <= B sqrt(p). The proof never establishes that. This is exactly what the perturbation removed: the unmodified lemma (present in this bench as ch4_theorem_4_18__correct T3, with a byte-identical statement) opens with 'If eps > B sqrt(p), then S is trivially contained in the ball of radius eps centered at the origin, so the eps-cover has size 1' and only then says 'Assume eps <= B sqrt(p)'. Both steps are gone here, so step 4's inequality is applied unconditionally and is false whenever eps > B sqrt(p) -- precisely the regime step 5 then quantifies over ('for all B, eps > 0'). Concretely, p = 1, B = 1, eps = 2: C = {0}, so |C| = 1, while (3B sqrt(p)/eps)^p = 3/2 happens to hold, but at eps = 4 the claimed bound (3/4)^p = 3/4 < 1 = |C| fails. NOT step 2: step 2 only claims that SOME x' in C lies within eps/sqrt(p) of x coordinatewise, and that is true -- round each coordinate TOWARD ZERO, x'_i = sign(x_i) floor(|x_i| sqrt(p)/eps) eps/sqrt(p). Then |x'_i| <= |x_i|, so ||x'||_2 <= ||x||_2 <= B and x' stays in S; |x_i - x'_i| < eps/sqrt(p); and |k_i| <= |x_i| sqrt(p)/eps <= B sqrt(p)/eps. Hence x' is in C. Only NEAREST-point rounding can leave the ball, and the step does not claim nearest. Gold was anchored at step 2 until 2026-08-06 on that mistaken counterargument. All four configurations of the pv137b run had independently localised the defect at step 4 (recorded in adjudication/out_M as a post-gold claim and dismissed there); in pv127, three of four again pointed at step 4."}
{"id": "ch4_theorem_4_12__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nDefinition 4.1. Let \\((\\mathcal{V},d)\\) be a pseudometric space with metric \\(d(\\cdot,\\cdot)\\). A finite\nset \\(\\mathcal{G}(\\epsilon) \\subset \\mathcal{V}\\) is an \\(\\epsilon\\)-cover (or \\(\\epsilon\\)-net) of \\(\\mathcal{G} \\subset \\mathcal{V}\\) if, for all \\(\\phi \\in \\mathcal{G}\\), there exists \\(\\phi' \\in \\mathcal{G}(\\epsilon)\\) so that \\(d(\\phi', \\phi) \\leq \\epsilon\\). The \\(\\epsilon\\)-covering number of \\(\\mathcal{G}\\) with metric \\(d\\) is the smallest cardinality \\(N(\\epsilon, \\mathcal{G}, d)\\) of such \\(\\mathcal{G}(\\epsilon)\\). The number \\(\\ln N(\\epsilon, \\mathcal{G}, d)\\) is called the \\(\\epsilon\\)-entropy.\nFor a function class \\(\\mathcal{G}\\) with seminorm \\(L_{p}(\\mathcal{D})\\) in (3.19), we denote the corresponding \\(L_{p}(\\mathcal{D})\\)-covering number as \\(N(\\epsilon, \\mathcal{G}, L_{p}(\\mathcal{D}))\\). When \\(1 \\leq p \\leq q\\), we have\n\\[\nN(\\epsilon,\\mathcal{G},L_{p}(\\mathcal{D}))\\leq N(\\epsilon,\\mathcal{G},L_{q}(\\mathcal{D})).\n\\]\nIt is easy to verify the following relationship, which implies that \\( L_{p}(\\mathcal{D}) \\) bracketing cover is a stronger requirement than \\( L_{p}(\\mathcal{D}) \\) cover.\n\nDefinition 3.8. We define\n\\[\n\\phi(w,z)=L\\left(f(w,x),y\\right)-L_{*}(x,y),\\tag{3.4}\n\\]\nfor \\( w \\in \\Omega \\) and \\( z = (x, y) \\in \\mathcal{Z} = \\mathcal{X} \\times \\mathcal{Y} \\), and a pre-chosen \\( L_{*}(x, y) \\) of \\( z = (x, y) \\) that does not depend on \\( w \\).\nFor training data \\( S_{n}=\\{Z_{i}=(X_{i},Y_{i}):i=1,\\ldots,n\\} \\), we define the training loss for \\( w\\in \\Omega \\) as\n\\[\n\\phi(w,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{i=1}^{n}\\phi(w,Z_{i}).\\tag{3.5}\n\\]\nMoreover, for a distribution \\(\\mathcal{D}\\) on \\(\\mathcal{Z}\\), we define the test loss for \\(w \\in \\Omega\\) as:\n\\[\n\\phi(w,\\mathcal{D})=\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(w,Z).\\tag{3.6}\n\\]\nSince \\( L_{*}(x,y) \\) does not depend on \\( w \\), the ERM solution with respect to the loss \\( L(\\cdot) \\) is equivalent to the ERM solution with respect to \\( \\phi (w,\\cdot) \\). Therefore with the simplified notations, the approximate ERM method (3.3) is a special case of the following method:\n\\[\n\\phi(\\hat{w},\\mathcal{S}_{n})\\leq\\inf_{w\\in\\Omega}\\phi(w,\\mathcal{S}_{n})+\\epsilon^{\\prime}.\\tag{3.7}\n\\]\nIn general, we may simply take \\( L_{*}(x,y) = 0 \\) in (3.4). However, for some applications, we may choose a non-zero \\( L_{*}(x,y) \\) so that\n\\[\nL(f(w,x),y)-L_{*}(x,y)\n\\]\nhas a small variance. For least squares loss, this can be achieved with \\( L_{*}(x,y) = L(f_{*}(x),y) \\) where \\( f_{*}(x) \\) is the optimal prediction function that minimizes the test loss as shown in Example 3.9. The smaller variance, combined with Bernstein's inequality, implies better generalization bound (see Section 3.6 for more details).\n\nEmpirical and Uniform Covering Number. Given an empirical distribution \\( S_{n}=\\{Z_{1},\\ldots,Z_{n}\\} \\), we define the pseudometric \\( d = L_{p}(S_{n}) \\) as\n\\[\nd(\\phi,\\phi^{\\prime})=\\left[\\frac{1}{n}\\sum_{i=1}^{n}|\\phi(Z_{i})-\\phi^{\\prime}(Z_{i})|^{p}\\right]^{1/p}.\n\\]\nThe corresponding metric covering number \\(N(\\epsilon, \\mathcal{G}, L_p(\\mathcal{S}_n))\\) is referred to as the\nempirical \\( L_{p} \\) covering number. Given \\( n \\), the largest \\( L_{p} \\) covering number over empirical distribution \\( S_{n} \\) is referred to as the uniform \\( L_{p} \\) covering number\n\\[\nN_{p}(\\epsilon,\\mathcal{G},n)=\\sup_{\\mathcal{S}_{n}}N(\\epsilon,\\mathcal{G},L_{p}(\\mathcal{S}_{n})).\n\\]\nSince the \\(L_{p}(\\mathcal{S}_{n})\\) pseudometric increases with \\(p\\), we have the following simple result.\n\nBracketing Number. Let \\(\\mathcal{G} = \\{\\phi(w, \\cdot) : w \\in \\Omega\\}\\) be a real-valued function class, equipped with a pseudometric \\(d\\). We say\n\\[\n\\mathcal{G}(\\epsilon)=\\left\\{\\left[\\phi_{1}^{L}(z),\\phi_{1}^{U}(z)\\right],\\dots,\\left[\\phi_{N}^{L}(z),\\phi_{N}^{U}(z)\\right]\\right\\}\n\\]\nis an \\(\\epsilon\\)-bracket of \\(\\mathcal{G}\\) under metric \\(d\\) if for all \\(w \\in \\Omega\\), there exists \\(j = j(w)\\) such that \\(\\forall z\\):\n\\[\n\\phi_{j}^{L}(z)\\leq\\phi(w,z)\\leq\\phi_{j}^{U}(z),\\quad d(\\phi_{j}^{L},\\phi_{j}^{U})\\leq\\epsilon .\n\\]\nThe \\(\\epsilon\\)-bracketing number is the smallest cardinality \\(N_{\\mathbb{I}}(\\epsilon, \\mathcal{G}, d)\\) of such \\(\\mathcal{G}(\\epsilon)\\). The quantity \\(\\ln N_{\\mathbb{I}}(\\epsilon, \\mathcal{G}, d)\\) is called \\(\\epsilon\\) bracketing entropy.\nIn particular, given a distribution \\(\\mathcal{D}\\) and \\(p \\geq 1\\), we define \\(L_{p}\\)-seminorm in function space as\n\\[\n\\left\\|f-f^{\\prime}\\right\\|_{L_{p}(\\mathcal{D})}=\\left[\\mathbb{E}_{Z\\sim\\mathcal{D}}|f(Z)-f^{\\prime}(Z)|^{p}\\right]^{1/p}.\\tag{3.19}\n\\]\nIt induces a pseudometric, denoted as \\( d = L_{p}(\\mathcal{D}) \\), and the corresponding bracketing number is \\( N_{\\parallel}(\\epsilon, \\mathcal{G}, L_{p}(\\mathcal{D})) \\).\nFor \\(p = \\infty\\), the \\(L_{\\infty}(\\mathcal{D})\\)-seminorm is defined as the essential supremum seminorm, which leads to the pseudometric\n\\[\nd(f,f^{\\prime})=\\inf\\left\\{\\omega:\\Pr_{\\mathcal{D}}[|f(Z)-f^{\\prime}(Z)|\\leq\\omega]=1\\right\\}.\n\\]\nHowever, when \\( p = \\infty \\), instead of using the \\( L_{\\infty} \\) bracketing number, it is more conventional to use the equivalent notion of \\( L_{\\infty} \\) covering number (see Proposition 4.3).\nWe have the following relationship of lower bracketing number and bracketing numbers with different norms.\n\nDefinition 4.7. Consider a real-valued function family \\(\\mathcal{F}:\\mathcal{Z}\\to \\mathbb{R}\\). Consider \\(n\\) iid Bernoulli random variables \\(\\sigma_{i}\\in \\{\\pm 1\\}\\), where \\(\\operatorname*{Pr}(\\sigma_i=1)=\\operatorname*{Pr}(\\sigma_i= -1)=0.5\\). The symmetrized empirical process is\n\\[\nf(\\sigma,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{i=1}^{n}\\sigma_{i}f(Z_{i})\\qquad f\\in\\mathcal{F},\n\\]\nwhere the randomness is with respect to both \\(S_{n} = \\{Z_{i}\\}\\) and \\(\\sigma = \\{\\sigma_{i}\\}\\).\nNote that in order to bound the symmetrized empirical process, we only need a covering number result on \\( S_{n} \\). Therefore the symmetrization analysis allows us to work with empirical covering numbers. Next we will show that a bound on the symmetrized empirical process can be used to obtain uniform convergence of the empirical process \\( \\{f(S_{n}): f \\in \\mathcal{F}\\} \\) to the corresponding result on the validation data. Since one can relate the validation loss of a learning algorithm to its generalization error on the test data, we can use this result to obtain generalization bounds.\nThe following lemma shows that if we can obtain an upper bound on the symmetrized empirical process \\(\\{f(\\sigma, S_n) : f \\in \\mathcal{F}\\}\\), and the upper bound satisfies a superadditive property, then we can obtain the uniform convergence of the empirical process \\(\\{f(S_n) : f \\in \\mathcal{F}\\}\\) on the training data \\(S_n\\) to the corresponding result on the validation data \\(S_n'\\).", "theorems": [{"local_id": "T1", "statement": "Let \\(\\psi_{\\mathrm{trn}}: \\mathcal{F} \\times \\mathcal{Z}^n \\to \\mathbb{R}\\), \\(\\psi_{\\mathrm{val}}: \\mathcal{F} \\times \\mathcal{Z}^n \\to \\mathbb{R}\\), \\(\\psi_{\\mathrm{tst}}: \\mathcal{F} \\times \\mathcal{D} \\to \\mathbb{R}\\) be appropriately training (where \\(\\mathcal{D}\\) denotes probability distributions on \\(\\mathcal{Z}\\)), validation, and test statistics. Assume that for any \\(\\delta_1 \\in (0,1)\\), the following uniform convergence result holds. With probability at least \\(1 - \\delta_1\\) over randomly drawn training and validation sets \\((S_n, S_n') \\sim \\mathcal{D}^{2n}\\):\n\\[\n\\forall f\\in\\mathcal{F}:\\psi_{\\mathrm{val}}(f,\\mathcal{S}_{n}^{\\prime})\\leq\\psi_{\\mathrm{trn}}(f,\\mathcal{S}_{n})+\\epsilon_{n}^{1}(\\delta_{1}).\n\\]\nMoreover, assume \\(\\forall f\\in \\mathcal{F}\\), we have with probability \\(1 - \\delta_{2}\\) over randomly drawn \\(\\mathcal{S}_n^\\prime \\sim \\mathcal{D}\\):\n\\[\n\\psi_{\\mathrm{tst}}(f,\\mathcal{D})\\leq\\psi_{\\mathrm{val}}(f,\\mathcal{S}_{n}^{\\prime})+\\epsilon_{n}^{2}(\\delta_{2}).\n\\]\nThen the following uniform convergence statement holds. With probability at least \\(1 - \\delta_1 - \\delta_2\\),\n\\[\n\\forall f\\in\\mathcal{F}:\\psi_{\\mathrm{tst}}(f,\\mathcal{D})\\leq\\psi_{\\mathrm{trn}}(f,\\mathcal{S}_{n})+\\epsilon_{n}^{1}(\\delta_{1})+\\epsilon_{n}^{2}(\\delta_{2}).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\( Q(f, \\mathcal{S}_n) = \\psi_{\\mathrm{tst}}(f, \\mathcal{D}) - \\psi_{\\mathrm{trn}}(f, \\mathcal{S}_n) - (\\epsilon_n^1(\\delta_1) + \\epsilon_n^2(\\delta_2)) \\), and let \\( E \\) be the event that \\( \\sup_{f \\in \\mathcal{F}} Q(f, \\mathcal{S}_n) \\leq 0 \\)."}, {"n": 2, "claim": "We pick \\( \\hat{f}(\\mathcal{S}_n) \\in \\mathcal{F} \\) so that if \\( E \\) holds, then we choose an arbitrary \\( Q(\\hat{f}(\\mathcal{S}_n), \\mathcal{S}_n) \\leq 0 \\), and if \\( E \\) does not hold, we choose \\( \\hat{f}(\\mathcal{S}_n) \\) so that \\( Q(\\hat{f}(\\mathcal{S}_n), \\mathcal{S}_n) > 0 \\)."}, {"n": 3, "claim": "We consider sample \\( (\\mathcal{S}_n, \\mathcal{S}_n') \\sim \\mathcal{D}^{2n} \\)."}, {"n": 4, "claim": "For simplicity, in the following, we let \\( \\hat{f} = \\hat{f}(\\mathcal{S}_n) \\)."}, {"n": 5, "claim": "The uniform convergence condition of the theorem implies that with probability at least \\( 1 - \\delta_1 \\), the following event holds:"}, {"n": 6, "claim": "\\[\n{E_{1}:}{\\psi_{\\mathrm{val}}(\\hat{f},\\mathcal{S}_{n}^{\\prime})\\leq\\psi_{\\mathrm{trn}}(\\hat{f},\\mathcal{S}_{n})+\\epsilon_{n}^{1}(\\delta_{1}).}\n\\]\nNote that the validation data \\( S_{n}^{\\prime} \\) is independent of the training data \\( S_{n} \\)."}, {"n": 7, "claim": "Therefore \\( S_{n}^{\\prime} \\) is also independent of \\( \\hat{f} \\)."}, {"n": 8, "claim": "Therefore the condition of the theorem implies that with probability at least \\( 1 - \\delta_2 \\), the following event holds:"}, {"n": 9, "claim": "\\[\nE_{2}:\\psi_{\\mathrm{tst}}(\\hat{f},\\mathcal{D})\\leq\\psi_{\\mathrm{val}}(\\hat{f},\\mathcal{S}_{n}^{\\prime})+\\epsilon_{n}^{2}(\\delta_{2}).\n\\]\nIf both events \\(E_{1}\\) and \\(E_{2}\\) hold, then\n\\[\n\\begin{array}{l}\\psi_{\\mathrm{tst}}(\\hat{f},\\mathcal{D})\\leq\\psi_{\\mathrm{val}}(\\hat{f},\\mathcal{S}_{n}^{\\prime})+\\epsilon_{n}^{2}(\\delta_{2})\\\\ \\leq\\psi_{\\mathrm{trn}}(\\hat{f},\\mathcal{S}_{n})+\\epsilon_{n}^{1}(\\delta_{1})+\\epsilon_{n}^{2}(\\delta_{2}).\\\\ \\end{array}\n\\]\nThe definition of \\(\\hat{f}\\) implies that \\(E\\) holds."}, {"n": 10, "claim": "Therefore \\(\\operatorname*{Pr}(E)\\geq \\operatorname*{Pr}(E_1\\& E_2)\\geq 1 - \\delta_1 - \\delta_2\\)."}, {"n": 11, "claim": "This implies the desired bound."}]}}, {"local_id": "T2", "statement": "(Symmetrization) Consider a real valued function family \\(F = \\{f : Z \\rightarrow \\mathbb{R}\\}\\). Assume there exists a function \\(\\psi : F \\times Z^{n} \\rightarrow \\mathbb{R}\\) and \\(\\epsilon_{n} : (0,1) \\rightarrow \\mathbb{R}\\) so that with probability at least \\(1-\\delta\\):\n\\[\n\\forall f \\in F, \\qquad f(\\sigma, S_{n}) \\leq \\psi(f, S_{n}) + \\epsilon_{n}(\\delta),\n\\]\nwhere the randomness is over both \\(S_{n} \\sim D^{n}\\) and \\(\\sigma\\). If there exists \\(\\tilde{\\psi}(f, S_{n} \\cup S_{n}^{\\prime})\\) so that the following superadditive inequality holds for all \\((S_{n}, S_{n}^{\\prime})\\)\n\\[\n\\psi(f, S_{n}) + \\psi(f, S_{n}^{\\prime}) \\leq \\tilde{\\psi}(f, S_{n} \\cup S_{n}^{\\prime}),\n\\]\nthen with probability at least \\(1-\\delta\\) over independent random data \\((S_{n}, S_{n}^{\\prime}) \\sim D^{2n}\\):\n\\[\n\\forall f \\in F, \\qquad f(S_{n}^{\\prime}) \\leq f(S_{n}) + \\tilde{\\psi}(f, S_{n} \\cup S_{n}^{\\prime}) + 2\\epsilon_{n}(\\delta/2).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Consider independent random samples \\((S_{n}, S_{n}^{\\prime}) \\sim D^{2n}\\)."}, {"n": 2, "claim": "The distribution of \\(f(S_{n}) - f(S_{n}^{\\prime})\\) is the same as that of \\(f(\\sigma, S_{n}) - f(\\sigma, S_{n}^{\\prime})\\), and the latter contains additional randomness from Bernoulli random variables \\(\\sigma\\), drawn independently of \\((S_{n}, S_{n}^{\\prime})\\), as in Definition 4.7."}, {"n": 3, "claim": "It follows that\n\\[\n\\begin{array}{l}\n\\Pr\\Bigl(\\exists f \\in F,\\ f(S_{n}^{\\prime}) > f(S_{n}) + \\tilde{\\psi}(f, S_{n} \\cup S_{n}^{\\prime}) + 2\\epsilon_{n}(\\delta/2)\\Bigr) \\\\\n= \\Pr\\Bigl(\\exists f \\in F,\\ f(\\sigma, S_{n}^{\\prime}) > f(\\sigma, S_{n}) + \\tilde{\\psi}(f, S_{n} \\cup S_{n}^{\\prime}) + 2\\epsilon_{n}(\\delta/2)\\Bigr) \\\\\n\\stackrel{(a)}{\\leq} \\Pr\\Bigl(\\underbrace{\\exists f \\in F,\\ f(\\sigma, S_{n}^{\\prime}) > f(\\sigma, S_{n}) + (\\psi(f, S_{n}) + \\psi(f, S_{n}^{\\prime})) + 2\\epsilon_{n}(\\delta/2)}_{E_{0}}\\Bigr)\n\\end{array}\n\\]\n\\[\n\\begin{array}{l}\n\\stackrel{(b)}{\\leq} \\Pr\\Bigl(\\underbrace{\\exists f \\in F,\\ f(\\sigma, S_{n}^{\\prime}) > \\psi(f, S_{n}^{\\prime}) + \\epsilon_{n}(\\delta/2)}_{E_{1}}\\Bigr) \\\\\n\\quad + \\Pr\\Bigl(\\underbrace{\\exists f \\in F,\\ -f(\\sigma, S_{n}) > \\psi(f, S_{n}) + \\epsilon_{n}(\\delta/2)}_{E_{2}}\\Bigr)\n\\end{array}\n\\]\n\\[\n= 2\\Pr\\Bigl(\\exists f \\in F,\\ f(\\sigma, S_{n}) > \\psi(f, S_{n}) + \\epsilon_{n}(\\delta/2)\\Bigr) \\leq 2(\\delta/2).\n\\]\nIn the above derivation, the first equation used the fact that \\(f(S_{n}) - f(S_{n}^{\\prime})\\) and \\(f(\\sigma, S_{n}) - f(\\sigma, S_{n}^{\\prime})\\) have the same distributions. (a) used the assumption \\(\\psi(S_{n}) + \\psi(S_{n}^{\\prime}) \\leq \\widetilde{\\psi}(S_{n} \\cup S_{n}^{\\prime})\\). (b) used the union bound, and the fact that if event \\(E_{0}\\) holds, then either event \\(E_{1}\\) holds or event \\(E_{2}\\) holds."}, {"n": 4, "claim": "The next equation used the symmetry of \\(-f(\\sigma, S_{n})\\) and \\(f(\\sigma, S_{n})\\), and the last inequality used the assumption of the lemma."}, {"n": 5, "claim": "The result implies the desired bound."}]}}, {"local_id": "T3", "statement": "Let \\(\\sigma_{i} = \\{\\pm 1\\}\\) be independent Bernoulli random variables ( each takes value \\(\\pm 1\\) with equal probability). Let \\(a_{i}\\) be fixed numbers \\((i=1,\\dots,n)\\) Then for all \\(\\epsilon >0\\) ..\n\\[\n\\Pr\\left(n^{-1}\\sum_{i=1}^{n}\\sigma_{i}a_{i}\\geq\\epsilon\\right)\\leq\\exp\\left[-\\frac{n\\epsilon^{2}}{2 n^{-1}\\sum_{i=1}^{n}a_{i}^{2}}\\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "Consider \\( X_{i} = \\sigma_{i}a_{i} \\) in Corollary 2.26. We can take \\( \\mu = 0 \\) and \\( b_{i} = a_{i}^{2} \\) to obtain the desired bound."}]}}, {"local_id": "T4", "statement": "Assume that \\(\\phi(w, z) \\in [0,1]\\) for all \\(w\\) and \\(z\\). Then given \\(\\delta \\in (0,1)\\), with probability at least \\(1 - \\delta\\), the following inequality holds:\n\\[\n\\forall w\\in\\Omega:\\phi(w,\\mathcal{D})\\leq\\phi(w,\\mathcal{S}_{n})+\\epsilon_{n}(\\delta),\n\\]\nwhere\n\\[\n\\epsilon_{n}(\\delta)=\\inf_{\\epsilon>0}\\left[2\\epsilon+3\\sqrt{\\frac{\\ln(3 N_{1}(\\epsilon,\\mathcal{G},2 n)/\\delta)}{2 n}}\\right].\n\\]\nMoreover, for any \\(\\gamma \\in (0,1)\\), with probability at least \\(1 - \\delta\\), the following inequality holds:\n\\[\n\\forall w\\in\\Omega:(1-\\gamma)^{2}\\phi(w,\\mathcal{D})\\leq\\phi(w,\\mathcal{S}_{n})+\\epsilon_{n}(\\delta),\n\\]\nwhere\n\\[\n\\epsilon_{n}(\\delta)=\\inf_{\\epsilon>0}\\left[2\\epsilon+\\frac{(5-4\\gamma)\\ln(3 N_{1}(\\epsilon,\\mathcal{G},n)/\\delta)}{2\\gamma n}\\right].\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\mathcal{F}=\\{f(z)=\\phi(w,z)-0.5:w\\in\\Omega\\}\\)."}, {"n": 2, "claim": "Pick an \\(\\epsilon\\)-\\(L_1(\\mathcal{S}_n)\\) cover \\(\\mathcal{F}_\\epsilon(\\mathcal{S}_n)\\) of \\(\\mathcal{F}\\) with \\(|\\mathcal{F}_\\epsilon(\\mathcal{S}_n)|\\le N=N_1(\\epsilon,\\mathcal{G},n)\\), assuming \\(f(Z_i)\\in[-0.5,0.5]\\)."}, {"n": 3, "claim": "By Corollary 2.27 (with \\(a_i=0.5\\)) and the union bound, w.p. \\(1-\\delta\\): \\(\\forall f\\in\\mathcal{F}_\\epsilon(\\mathcal{S}_n):\\ f(\\sigma,\\mathcal{S}_n)\\le\\sqrt{\\tfrac{\\ln(N/\\delta)}{2n}}\\)."}, {"n": 4, "claim": "For any \\(f\\in\\mathcal{F}\\) choose \\(f'\\) in the cover with \\(n^{-1}\\sum_{Z}|f-f'|\\le\\epsilon\\), giving \\(f(\\sigma,\\mathcal{S}_n)\\le\\epsilon+\\sqrt{\\tfrac{\\ln(N/\\delta)}{2n}}\\)."}, {"n": 5, "claim": "By Lemma 4.8 (\\(\\psi=0\\)), w.p. \\(\\ge1-\\delta_1\\) over \\((\\mathcal{S}_n,\\mathcal{S}_n')\\): \\(\\forall w,\\ \\phi(w,\\mathcal{S}_n')\\le\\phi(w,\\mathcal{S}_n)+2\\epsilon+\\sqrt{\\tfrac{2\\ln(2N/\\delta_1)}{n}}\\)."}, {"n": 6, "claim": "Additive Chernoff: for each \\(w\\), w.p. \\(\\ge1-\\delta_2\\), \\(\\phi(w,\\mathcal{D})\\le\\phi(w,\\mathcal{S}_n')+\\sqrt{\\tfrac{\\ln(1/\\delta_2)}{2n}}\\)."}, {"n": 7, "claim": "Combining via Lemma 4.11 with \\(\\delta_1=2\\delta/3,\\ \\delta_2=\\delta/3\\) yields the first bound with \\(\\epsilon_n(\\delta)=\\inf_{\\epsilon>0}[2\\epsilon+3\\sqrt{\\tfrac{\\ln(3N_1(\\epsilon,\\mathcal{G},2n)/\\delta)}{2n}}]\\)."}, {"n": 8, "claim": "For the second claim let \\(\\mathcal{F}=\\{f(z)=\\phi(w,z)\\}\\), cover size \\(\\le N=N_1(\\epsilon,\\mathcal{G},n)\\), \\(f(Z_i)\\in[0,1]\\)."}, {"n": 9, "claim": "By Corollary 2.27, \\(f(\\sigma,\\mathcal{S}_n)\\le\\sqrt{\\tfrac{2\\sum_Z f(Z)^2\\ln(N/\\delta)}{n^2}}\\le\\gamma'\\tfrac1n\\sum_Z f(Z)+\\tfrac{\\ln(N/\\delta)}{4\\gamma' n}\\)."}, {"n": 10, "claim": "Cover approximation: \\(f(\\sigma,\\mathcal{S}_n)\\le\\gamma'\\tfrac1n\\sum_Z f(Z)+\\tfrac{\\ln(N/\\delta)}{2\\gamma' n}+(1+\\gamma')\\epsilon\\)."}, {"n": 11, "claim": "By Lemma 4.8: w.p. \\(\\ge1-\\delta_1\\), \\(\\forall w,\\ (1-\\gamma')\\phi(w,\\mathcal{S}_n')\\le(1+\\gamma')\\phi(w,\\mathcal{S}_n)+2(1+\\gamma')\\epsilon+\\tfrac{\\ln(2N/\\delta_1)}{\\gamma' n}\\)."}, {"n": 12, "claim": "With \\(\\gamma'=\\gamma/(2-\\gamma)\\), \\(\\tfrac{1-\\gamma'}{1+\\gamma'}=1-\\gamma\\) and \\(\\tfrac{1}{\\gamma'(1+\\gamma')}=\\tfrac{(2-\\gamma)^2}{2\\gamma}\\), giving (4.1)."}, {"n": 13, "claim": "Multiplicative Chernoff: w.p. \\(\\ge1-\\delta_2\\), \\((1-\\gamma)^2\\phi(w,\\mathcal{D})\\le(1-\\gamma)\\phi(w,\\mathcal{S}_n')+(1-\\gamma)\\tfrac{\\ln(1/\\delta_2)}{2\\gamma n}\\)."}, {"n": 14, "claim": "Lemma 4.11 with \\(\\delta_1=2\\delta/3,\\ \\delta_2=\\delta/3\\) yields the second bound."}]}}], "target_local_id": "T4", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "label": "flawed", "error_step": 9, "error_class6": ["computation_or_constant", "invalid_bound_or_inequality", "others"], "error_explanation": "The AM-GM step is stated as $\\sqrt{2ab}\\le\\gamma'a+\\tfrac{b}{4\\gamma'}$ (the constant $2\\gamma'$ was replaced by $4\\gamma'$). The true bound is $\\gamma'a+\\tfrac{b}{2\\gamma'}\\ge2\\sqrt{ab/2}=\\sqrt{2ab}$; with $b/(4\\gamma')$ the right side is strictly smaller and the inequality is false (e.g. $a=b=\\gamma'=1$ gives $\\sqrt2\\le1.25$, false)."}
{"id": "ch5_theorem_5_20__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nLayerwise deep-net composition setup. Consider an \\(r\\)-layer composed class \\(\\mathcal{F}_r \\circ \\cdots \\circ \\mathcal{F}_1\\), where layer \\(i\\) has function class \\(\\mathcal{F}_i\\). For \\(f_i \\in \\mathcal{F}_i\\), assume that \\(f_i\\) is \\(\\kappa_i\\)-Lipschitz with respect to the relevant norm. Let \\(\\rho = L_2(P_n)\\) be the empirical \\(L_2\\) metric. If every input \\(z^{(j)}\\) to \\(f_i\\) satisfies \\(\\|z^{(j)}\\|_2 \\leq c_{i-1}\\), write \\(g(\\epsilon_i, c_{i-1})\\) for the per-layer covering-bound function such that \\(\\log N(\\epsilon_i, \\mathcal{F}_i, L_2(P_n)) \\leq g(\\epsilon_i, c_{i-1})\\).\n\nEmpirical Rademacher complexity. Given a function class \\(\\mathcal{F}\\) and a sample \\(S = (x^{(1)}, \\ldots, x^{(n)})\\), the empirical Rademacher complexity is\n\\[\nR_S(\\mathcal{F}) = \\frac{1}{n}\\mathbb{E}_{\\sigma}\\left[\\sup_{f \\in \\mathcal{F}} \\sum_{i=1}^n \\sigma_i f(x^{(i)})\\right],\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent Rademacher random variables.\n\nEmpirical Rademacher complexity. Given a dataset \\(S = \\{z_1, \\dots, z_n\\}\\), the empirical Rademacher complexity is defined as\n\\[\nR_S(\\mathcal{F}) \\stackrel{\\Delta}{=} \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right].\n\\]\n\\(R_S(\\mathcal{F})\\) is a function of both the function class \\(\\mathcal{F}\\) and the dataset \\(S\\).\n\nDense neural network function class. Let \\(r\\) be the number of layers and let \\(d_0, d_1, \\ldots, d_r\\) be layer dimensions. For each layer \\(i \\in \\{1,\\ldots,r\\}\\), let \\(W_i \\in \\mathbb{R}^{d_i \\times d_{i-1}}\\) and let \\(\\phi_i : \\mathbb{R}^{d_i} \\to \\mathbb{R}^{d_i}\\) be a 1-Lipschitz activation, applied coordinatewise, with \\(\\phi_i(0)=0\\). Define the layer map\n\\[\nF_i(z) = \\phi_i(W_i z)\n\\]\nand the dense neural network\n\\[\nf_\\theta = F_r \\circ F_{r-1} \\circ \\cdots \\circ F_1.\n\\]\nFor a sample \\(S=(x^{(1)},\\ldots,x^{(n)})\\), assume \\(\\|x^{(j)}\\|_2 \\leq c\\) for all examples \\(j\\). Let \\(c_0=c\\), and for each layer input let \\(c_{i-1}\\) denote a bound such that every input to layer \\(i\\) satisfies \\(\\|z^{(j)}\\|_2 \\leq c_{i-1}\\). With \\(\\rho = L_2(P_n)\\), define the norm-bounded dense network class\n\\[\n\\mathcal{F} = \\left\\{f_\\theta : \\|W_i\\|_{\\textup{op}} \\leq \\kappa_i,\\ \\|W_i^\\top\\|_{2,1} \\leq b_i\\ \\text{ for all } i=1,\\ldots,r\\right\\}.\n\\]\n\n\\(\\epsilon\\)-cover. Let \\(\\epsilon > 0\\). An \\(\\epsilon\\)-cover of a set \\(S\\) with respect to a distance metric \\(\\rho\\) is a subset \\(C \\subseteq S\\) such that for all \\(x \\in S\\) there exists \\(x' \\in C\\) with \\(\\rho(x, x') \\leq \\epsilon\\), or equivalently,\n\\[\nS \\subseteq \\bigcup_{x \\in C} \\mathrm{Ball}(x, \\epsilon, \\rho), \\quad \\text{where} \\quad \\mathrm{Ball}(x, \\epsilon, \\rho) \\triangleq \\{x' : \\rho(x, x') \\leq \\epsilon\\}.\n\\]\n\n\\(\\kappa\\)-Lipschitz functions. Let \\(\\kappa \\geq 0\\) and let \\(\\|\\cdot\\|\\) be a norm on the domain \\(D\\). A function \\(L : D \\to \\mathbb{R}\\) is said to be \\(\\kappa\\)-Lipschitz with respect to \\(\\|\\cdot\\|\\) if for all \\(\\theta, \\theta' \\in D\\),\n\\[\n|L(\\theta) - L(\\theta')| \\leq \\kappa\\,\\|\\theta - \\theta'\\|.\n\\]\n\nCovering number. The covering number is defined as the minimum size of an \\(\\epsilon\\)-cover, or explicitly:\n\\[\nN(\\epsilon, \\mathcal{Q}, \\rho) \\stackrel{\\Delta}{=} (\\text{min size of } \\epsilon\\text{-cover of } \\mathcal{Q} \\text{ w.r.t. metric } \\rho).\n\\]\n\nEpsilon-cover. \\(\\mathcal{C}\\) is an \\(\\epsilon\\)-cover of \\(\\mathcal{Q}\\) with respect to metric \\(\\rho\\) if for all \\(v' \\in \\mathcal{Q}\\), there exists \\(v \\in \\mathcal{C}\\) such that \\(\\rho(v,v')\\le \\epsilon\\).", "theorems": [{"local_id": "T1", "statement": "Under the setup given above, if every input to \\(f_i\\) satisfies \\(\\|z^{(j)}\\|_2 \\leq c_{i-1}\\), we assume that\n\\[\n\\log N(\\epsilon_i, \\mathcal{F}_i, L_2(P_n)) \\leq g(\\epsilon_i, c_{i-1}).\n\\]\n(If \\(\\mathcal{F}_i\\) defines a collection of linear models, then \\(\\log N(\\epsilon_i, \\mathcal{F}_i, L_2(P_n)) \\leq \\left\\lceil \\frac{c_{i-1}^2}{\\epsilon_i^2} \\right\\rceil\\).)\nThen, there exists an \\(\\epsilon\\)-cover \\(\\mathcal{C}\\) of \\(\\mathcal{F}_r \\circ \\cdots \\circ \\mathcal{F}_1\\) for \\(\\epsilon = \\epsilon_r + \\kappa_r\\epsilon_{r-1} + \\cdots + \\kappa_r\\kappa_{r-1}\\cdots\\kappa_2\\epsilon_1\\) such that\n\\[\n\\log |\\mathcal{C}| \\leq \\sum_{i=1}^{r} g\\left(\\epsilon_i, c_{i-1}\\right).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\epsilon_1,\\dots,\\epsilon_r\\) be the radius for each layer."}, {"n": 2, "claim": "Let \\(\\mathcal{C}_1\\) be an \\(\\epsilon_1\\)-cover of \\(\\mathcal{F}_1\\)."}, {"n": 3, "claim": "Then, for all \\(f_1' \\in \\mathcal{C}_1\\), we define \\(\\mathcal{C}_{2,f_1'}\\) as an \\(\\epsilon_2\\)-covering of the set\n\\[\n\\mathcal{F}_2 \\circ f_1' = \\left\\{f_2\\left(f_1'\\left(X\\right)\\right) : f_2 \\in \\mathcal{F}_2 \\right\\}.\n\\]\nTaking a union of this covering over all \\(f_1' \\in \\mathcal{C}_1\\) yields an \\(\\epsilon\\)-covering for \\(\\mathcal{F}_2 \\circ \\mathcal{F}_1\\)."}, {"n": 4, "claim": "In particular, if\n\\[\n\\mathcal{C}_2 = \\bigcup_{f_1'\\in \\mathcal{C}_1}\\mathcal{C}_{2,f_1'},\n\\]\nthen \\(\\mathcal{C}_2\\) is an \\(\\epsilon\\)-cover of \\(\\mathcal{F}_2 \\circ \\mathcal{F}_1\\) with \\(\\epsilon = \\epsilon_1 \\cdot \\kappa_2 + \\epsilon_2\\)."}, {"n": 5, "claim": "Next, we bound the sizes of these covers. Directly applying the assumption, we conclude that\n\\[\n\\log |\\mathcal{C}_{2,f_1'}| \\leq g\\left(\\epsilon_2, c_1\\right).\n\\]\nThen, because \\(\\mathcal{C}_2 = \\bigcup_{f_1'\\in \\mathcal{C}_1}\\mathcal{C}_{2,f_1'}\\), it immediately follows that\n\\[\n\\begin{aligned}\n|\\mathcal{C}_{2}| &\\leq |\\mathcal{C}_{1}| \\exp\\left(g\\left(\\epsilon_2, c_1\\right)\\right) \\\\\n\\log|\\mathcal{C}_{2}| &\\leq \\log|\\mathcal{C}_{1}| + g\\left(\\epsilon_2, c_1\\right) \\\\\n&\\leq g\\left(\\epsilon_1, c_0\\right) + g\\left(\\epsilon_2, c_1\\right).\n\\end{aligned}\n\\]\nSimilarly, given \\(\\mathcal{C}_k\\), for any \\(f_k' \\circ f_{k-1}' \\circ \\cdots \\circ f_1' \\in \\mathcal{C}_k\\), we construct a \\(\\mathcal{C}_{k+1,f_k',\\dots,f_1'}\\) that is an \\(\\epsilon_{k+1}\\)-covering of \\(\\mathcal{F}_{k+1} \\circ f_k' \\circ \\cdots \\circ f_1'\\)."}, {"n": 6, "claim": "We similarly define\n\\[\n\\mathcal{C}_{k+1} = \\bigcup_{\\substack{f_i \\in \\mathcal{C}_i \\\\ i \\leq k}} \\mathcal{C}_{k+1,f_k',\\dots,f_1'}.\n\\]\nThen, inducting on the argument, we conclude that\n\\[\n\\log |\\mathcal{C}_{k+1}| \\leq g\\left(\\epsilon_{k+1}, c_k\\right) + \\cdots + g\\left(\\epsilon_1, c_0\\right).\n\\]\nNext, we show that for the above construction, the radius of the cover for \\(\\mathcal{F}\\) is\n\\[\n\\epsilon = \\sum_{i=1}^{r} \\left(\\epsilon_i \\prod_{j=i+1}^{r}\\kappa_{j}\\right).\n\\]\nFor any choice of \\(f_r \\circ \\cdots \\circ f_1 \\in \\mathcal{F}_r \\circ \\mathcal{F}_{r-1} \\circ \\cdots \\circ \\mathcal{F}_1\\), by definition of \\(\\mathcal{C}_1\\), there exists \\(f_1' \\in \\mathcal{C}_1\\) such that\n\\[\n\\rho(f_1, f_1') \\leq \\epsilon_1.\n\\]\nSimilarly, there exists \\(f_2' \\circ f_1' \\in \\mathcal{C}_{2,f_1'}\\) such that\n\\[\n\\rho\\left(f_2' \\circ f_1', f_2\\circ f_1' \\right) \\leq \\epsilon_2.\n\\]\nWe can leverage these two facts and the triangle inequality to prove that \\(f_2' \\circ f_1'\\) is close to \\(f_2 \\circ f_1\\):"}, {"n": 7, "claim": "\\[\n\\begin{aligned}\n\\rho\\left(f_2' \\circ f_1', f_2 \\circ f_1\\right)\n&\\leq \\rho\\left(f_2' \\circ f_1', f_2 \\circ f_1'\\right) + \\rho\\left(f_2 \\circ f_1', f_2 \\circ f_1\\right) \\quad \\text{(triangle ineq.)} \\\\\n&\\leq \\epsilon_2 + \\rho\\left(f_2 \\circ f_1', f_2 \\circ f_1\\right) \\quad \\text{(def. of }\\mathcal{C}_{2,f_1'}\\text{)} \\\\\n&\\leq \\epsilon_2 + \\kappa_2 \\rho\\left(f_1', f_1\\right) \\quad \\text{(Lipschitz condition)} \\\\\n&\\leq \\epsilon_2 + \\kappa_2\\epsilon_1 \\quad \\text{(def. of }\\mathcal{C}_{1}\\text{)}.\n\\end{aligned}\n\\]\nInducting to prove this for arbitrary \\(k\\), we similarly apply the definition of \\(\\mathcal{C}_{k,f'_{k-1},\\dots,f'_1}\\) to conclude that there exists \\(f'_k \\circ f'_{k-1} \\circ \\cdots \\circ f'_1 \\in \\mathcal{C}_k\\) such that\n\\[\n\\rho(f'_k \\circ f'_{k-1} \\circ \\cdots \\circ f'_1, f_k \\circ f'_{k-1} \\circ \\cdots \\circ f'_1) \\leq \\epsilon_k.\n\\]\nThen, expanding using the triangle inequality and peeling off terms by applying the definition of each \\(\\epsilon_i\\)-cover,\n\\[\n\\begin{aligned}\n&\\rho\\left(f_k' \\circ f_{k-1}' \\circ \\cdots \\circ f_1', f_k \\circ \\cdots \\circ f_1\\right) \\\\\n&\\leq \\rho\\left(f_k' \\circ f_{k-1}'\\circ \\cdots \\circ f_1', f_k \\circ f_{k-1}'\\circ \\cdots \\circ f_1' \\right) \\\\\n&\\quad + \\rho\\left(f_k \\circ f_{k-1}'\\circ f_{k-2}' \\circ \\cdots \\circ f_1', f_k \\circ f_{k-1}\\circ f_{k-2}' \\circ \\cdots \\circ f_1'\\right) \\\\\n&\\quad + \\cdots + \\rho\\left(f_k \\circ f_{k-1}\\circ \\cdots \\circ f_2 \\circ f_1', f_k \\circ f_{k-1}\\circ \\cdots \\circ f_1\\right) \\\\\n&\\leq \\rho\\left(f_k' \\circ f_{k-1}'\\circ \\cdots \\circ f_1', f_k \\circ f_{k-1}'\\circ \\cdots \\circ f_1' \\right) \\\\\n&\\quad + \\kappa_{k} \\cdot \\rho(f'_{k-1} \\circ \\cdots \\circ f'_1, f_{k-1} \\circ f'_{k-2} \\circ \\cdots \\circ f'_1) \\\\\n&\\quad + \\cdots + \\left(\\prod_{j=2}^k \\kappa_j\\right) \\rho(f'_1, f_1) \\\\\n&\\leq \\sum_{i=1}^{k} \\left(\\epsilon_i\\prod_{j=i+1}^{k}\\kappa_{j}\\right).\n\\end{aligned}\n\\]\nThe first inequality follows by the triangle inequality, the second by the \\(\\kappa_i\\)-Lipschitz continuity of \\(f_i\\), and the third by applying the definition of each of the \\(\\epsilon_i\\)-covers."}]}}, {"local_id": "T2", "statement": "Suppose \\(\\phi\\) is \\(\\kappa\\)-Lipschitz, and \\(\\rho = L_2(P_n)\\). Then,\n\\[\n\\log N(\\epsilon, \\phi \\circ \\mathcal{F}, \\rho) \\le \\log N(\\epsilon / \\kappa, \\mathcal{F}, \\rho).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\mathcal{C}\\) denote an \\(\\epsilon/\\kappa\\)-cover for \\(\\mathcal{F}\\). Then \\(\\phi \\circ \\mathcal{C}\\) is an \\(\\epsilon\\)-cover of \\(\\phi \\circ \\mathcal{F}\\):\n\\[\n\\rho(\\phi \\circ f', \\phi \\circ f) = \\sqrt{\\frac{1}{n} \\sum (\\phi(f'(z_i)) - \\phi(f(z_i)))^2} \\le \\sqrt{\\frac{1}{n} \\cdot \\kappa^2 \\sum(f'(z_i) - f(z_i))^2} \\le \\kappa \\cdot \\frac{\\epsilon}{\\kappa} = \\epsilon.\n\\]"}]}}, {"local_id": "T3", "statement": "Suppose that for all \\(i\\), \\(\\|x^{(i)}\\|_2 \\leq c\\) and let\n\\[\n\\mathcal{F} = \\{f_\\theta : \\|W_i\\|_{\\textup{op}} \\leq \\kappa_i,\\ \\|W_i^\\top\\|_{2,1} \\leq b_i\\}.\n\\]\nThen\n\\[\nR_S(\\mathcal{F})\\leq\\frac{c}{\\sqrt{n}}\\cdot\\underbrace{\\left(\\prod_{i=1}^r\\kappa_i\\right)}_{\\textup{(I)}}\\cdot\\underbrace{\\left(\\sum_{i=1}^r\\frac{b_i^{2/3}}{\\kappa_i^{2/3}}\\right)^{3/2}}_{\\textup{(II)}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Each layer has covering number \\(g(\\epsilon_i,c_{i-1}) = \\widetilde{O}\\!\\left(\\frac{c_{i-1}^2 b_i^2}{\\epsilon_i^2}\\right)\\)."}, {"n": 2, "claim": "\\(\\log N(\\epsilon,\\mathcal{F},\\rho) = \\widetilde{O}\\!\\left(\\sum_{i=1}^r \\frac{c_{i-1}^2 b_i^2}{\\epsilon_i^2}\\right)\\) with \\(\\epsilon = \\sum_{i=1}^r \\epsilon_i \\prod_{j=i+1}^r \\kappa_j\\)."}, {"n": 3, "claim": "Holding \\(\\epsilon\\) fixed, minimize \\(\\sum_i c_{i-1}^2 b_i^2/\\epsilon_i^2\\) over \\(\\{\\epsilon_i\\}\\)."}, {"n": 4, "claim": "With \\(\\alpha_i^2 = c_{i-1}^2 b_i^2\\), \\(\\beta_i=\\prod_{j>i}\\kappa_j\\), Hölder \\((3,\\tfrac32)\\) gives \\(\\sum_i \\frac{\\alpha_i^2}{\\epsilon_i^2} \\geq \\frac{R}{\\epsilon^2}\\), \\(R=(\\sum_i (\\alpha_i\\beta_i)^{2/3})^{3/2}\\)."}, {"n": 5, "claim": "Equality holds at \\(\\epsilon_i = \\left(\\frac{c_{i-1}^2 b_i^2}{\\prod_{j>i}\\kappa_j}\\right)^{1/3}\\cdot \\frac{\\epsilon}{(\\sum_i b_i^{2/3}/\\kappa_i^{2/3})\\prod_i \\kappa_i^{2/3}}\\)."}, {"n": 6, "claim": "Substituting, \\(\\sum_i \\frac{c_{i-1}^2 b_i^2}{\\epsilon_i^2} = \\big(c^{2/3} S \\prod_i\\kappa_i^{2/3}\\big)^3 \\frac{1}{\\epsilon^2} = c^2 \\prod_i \\kappa_i^3 \\, S^3\\, \\frac{1}{\\epsilon^2}\\), where \\(S=\\sum_i (b_i/\\kappa_i)^{2/3}\\)."}, {"n": 7, "claim": "Since \\(\\log N \\le R/\\epsilon^2\\), the Rademacher bound gives \\(R_S(\\mathcal{F}) \\lesssim \\sqrt{R/n}\\)."}, {"n": 8, "claim": "Therefore \\(R_S(\\mathcal{F}) \\lesssim \\frac{c}{\\sqrt{n}}\\prod_i \\kappa_i \\left(\\sum_i (b_i/\\kappa_i)^{2/3}\\right)^{3/2}\\)."}]}}], "target_local_id": "T3", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "label": "flawed", "error_step": 4, "error_class6": ["computation_or_constant", "invalid_bound_or_inequality", "logical_gap_or_missing_case"], "error_explanation": "Step 4's Hölder bound is false: minimizing sum_i alpha_i^2/epsilon_i^2 subject to sum_i beta_i*epsilon_i = epsilon has minimum value (sum_i (alpha_i*beta_i)^(2/3))^3 / epsilon^2 (exponent 3), not (sum_i (alpha_i*beta_i)^(2/3))^(3/2) / epsilon^2 (exponent 3/2) as Step 4 states. This was independently confirmed two ways: (1) Lagrange multipliers on the constrained minimization directly give optimal epsilon_i proportional to (alpha_i^2/beta_i)^(1/3), substituting back yields the cubed value; (2) direct Hölder application with conjugate exponents (3, 3/2) on sum_i (alpha_i*beta_i)^(2/3) = sum_i (alpha_i^2/epsilon_i^2)^(1/3) * (beta_i*epsilon_i)^(2/3) gives the same cubed bound. Concrete counterexample to the stated 3/2-power claim: alpha=beta=1/2, epsilon_1=1 (so epsilon=1/2); Step 4 as written asserts 1/4 >= 1, which is false. This is corroborated by all four independent verifiers run when this item was built (two reasoning over the informal text, two producing machine-checked numerical counterexamples in Lean), and is internally consistent with Step 5's own equality condition and the originally-marked Step 6's own cubed substitution (which silently abandons Step 4's stated 3/2 power). Step 4 is therefore the first genuine error, strictly before the originally-marked Step 6 (which contains a separate, independent error: mis-cubing kappa_i^(2/3) to kappa_i^3 instead of kappa_i^2)."}
{"id": "ch6_corollary_6_19__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nDefinition 6.6. Consider any known data-dependent offset function \\( h(w, S_n) \\). Define the uniform convergence complexity of a function class \\( \\mathcal{G} \\) with offset \\( h \\) as\n\\[\n\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\sup_{w\\in\\Omega}[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})].\\tag{6.2}\n\\]\nThis quantity measures the one-sided expected uniform convergence of function class \\(\\mathcal{G}\\) with offset function \\(h(w, S_n)\\).\nWe note that\n\\[\n\\epsilon_{n}(\\mathcal{G},\\mathcal{D})=\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D}),\\qquad \\text{w i t h} h=0.\n\\]\nWe have the following generalization of Theorem 6.2, with a similar proof.\n\nDefinition 6.11. Consider a function class \\(\\mathcal{G} = \\{\\phi(w, Z) : w \\in \\Omega\\}\\), and let \\(h\\) be an offset function of the following form\n\\[\nh(w,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{i=1}^{n}h(w,Z_{i}),\\quad h(w,z)=h_{0}(w)+h_{1}(w,z).\\tag{6.3}\n\\]\nGiven \\( S_{n}=\\{Z_{1},\\ldots,Z_{n}\\} \\), the (one-sided) empirical Rademacher complexity of \\( \\mathcal{G} \\) with offset \\( h \\) decomposition (6.3) is defined as\n\\[\nR^{h}(\\mathcal{G},\\mathcal{S}_{n})=\\mathbb{E}_{\\sigma}\\sup_{w\\in\\Omega}\\left[\\frac{1}{n}\\sum_{i=1}^{n}\\sigma_{i}[\\phi(w,Z_{i})+0. 5h_{1}(w,Z_{i})]-0. 5h(w,\\mathcal{S}_{n})\\right],\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}^{h}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R^{h}(\\mathcal{G},\\mathcal{S}_{n}).\n\\]\nWe note that the standard Rademacher complexity can be regarded as a special case of the offset Rademacher complexity with \\( h(\\cdot) = h_0(\\cdot) = h_1(\\cdot) = 0 \\):\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n})=R^{0}(\\mathcal{G},\\mathcal{S}_{n}),\\quad R_{n}(\\mathcal{G},\\mathcal{D})=R_{n}^{0}(\\mathcal{G},\\mathcal{D}).\n\\]\nIt should be pointed out that the decomposition of \\( h \\) in (6.3) may not be unique, and the offset Rademacher complexity relies on the specific decomposition used.\nAs shown in Example 6.10, we allow distribution dependent offset in the definition of uniform convergence complexity. We also allow distribution dependent offset in the definition of offset Rademacher complexity.\nThe following result shows that for some cases, offset Rademacher complexity can be obtained easily for some function classes.\n\nDefinition 6.17. Given a function \\( h(w, S_n) \\), we define\n\\[\n\\Delta_{n}h(w)=\\sup\\{n\\cdot|h(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n}^{\\prime})|: |\\mathcal{S}_{n}\\cap\\mathcal{S}_{n}^{\\prime}| =n-1\\}.\n\\]\n\nDefinition 3.8. We define\n\\[\n\\phi(w,z)=L\\left(f(w,x),y\\right)-L_{*}(x,y),\\tag{3.4}\n\\]\nfor \\( w \\in \\Omega \\) and \\( z = (x, y) \\in \\mathcal{Z} = \\mathcal{X} \\times \\mathcal{Y} \\), and a pre-chosen \\( L_{*}(x, y) \\) of \\( z = (x, y) \\) that does not depend on \\( w \\).\nFor training data \\( S_{n}=\\{Z_{i}=(X_{i},Y_{i}):i=1,\\ldots,n\\} \\), we define the training loss for \\( w\\in \\Omega \\) as\n\\[\n\\phi(w,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{i=1}^{n}\\phi(w,Z_{i}).\\tag{3.5}\n\\]\nMoreover, for a distribution \\(\\mathcal{D}\\) on \\(\\mathcal{Z}\\), we define the test loss for \\(w \\in \\Omega\\) as:\n\\[\n\\phi(w,\\mathcal{D})=\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(w,Z).\\tag{3.6}\n\\]\nSince \\( L_{*}(x,y) \\) does not depend on \\( w \\), the ERM solution with respect to the loss \\( L(\\cdot) \\) is equivalent to the ERM solution with respect to \\( \\phi (w,\\cdot) \\). Therefore with the simplified notations, the approximate ERM method (3.3) is a special case of the following method:\n\\[\n\\phi(\\hat{w},\\mathcal{S}_{n})\\leq\\inf_{w\\in\\Omega}\\phi(w,\\mathcal{S}_{n})+\\epsilon^{\\prime}.\\tag{3.7}\n\\]\nIn general, we may simply take \\( L_{*}(x,y) = 0 \\) in (3.4). However, for some applications, we may choose a non-zero \\( L_{*}(x,y) \\) so that\n\\[\nL(f(w,x),y)-L_{*}(x,y)\n\\]\nhas a small variance. For least squares loss, this can be achieved with \\( L_{*}(x,y) = L(f_{*}(x),y) \\) where \\( f_{*}(x) \\) is the optimal prediction function that minimizes the test loss as shown in Example 3.9. The smaller variance, combined with Bernstein's inequality, implies better generalization bound (see Section 3.6 for more details).\n\nEquation (6.1). The approximate regularized ERM method is to find \\( \\hat{w} \\in \\Omega \\) such that, for some \\( \\epsilon' \\geq 0 \\),\n\\[\n\\phi(\\hat{w},\\mathcal{S}_{n})+h(\\hat{w},\\mathcal{S}_{n})\\leq\\inf_{w\\in\\Omega}[\\phi(w,\\mathcal{S}_{n})+h(w,\\mathcal{S}_{n})]+\\epsilon'.\\tag{6.1}\n\\]\nWhen \\(h = 0\\), this reduces to the plain approximate ERM method.\n\nExample 2.14. Consider a bounded random variable: \\( X_{1} \\in [\\alpha, \\beta] \\). Then \\( X_{1} \\) is sub-Gaussian with \\( b = (\\beta - \\alpha)^{2}/4 \\).\nThe tail probability inequality of Theorem 2.12 can also be expressed in a different form. Consider \\( \\delta \\in (0,1) \\) such that \\( \\delta = \\exp(-t) \\), we have \\( t = \\ln(1/\\delta) \\). This means that we can alternatively express the first bound of Theorem 2.12 as follows. With probability at least \\( 1 - \\delta \\), we have\n\\[\n\\bar{X}_{n}<\\mu+\\sqrt{\\frac{2 b\\ln(1/\\delta)}{n}}.\n\\]\nThis form is often preferred in the theoretical analysis of machine learning algorithms.", "theorems": [{"local_id": "T1", "statement": "Assume that for some \\(M \\geq 0\\):\n\\[\n\\sup_{w\\in\\Omega}\\left[\\sup_{z,z^{\\prime}}[\\phi(w,z)-\\phi(w,z^{\\prime})]+\\Delta_{n}h(w)\\right]\\leq M.\n\\]\nThen the approximate ERM method (6.1) satisfies the following oracle inequality. With probability at least \\( 1 - \\delta - \\delta' \\):\n\\[\n\\begin{array}{l}\\phi(\\hat{w},\\mathcal{D})\\leq\\inf_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{D})+\\mathbb{E}_{\\mathcal{S}_{n}}h(w,\\mathcal{S}_{n})+\\Delta_{n}h(w)\\sqrt{\\frac{\\ln\\left(1/\\delta^{\\prime}\\right)}{2 n}}\\right]\\\\ +\\epsilon^{\\prime}+\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})+2 M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}.\\\\ \\end{array}\n\\]\nIf \\( h(\\cdot) \\) has the decomposition (6.3), then\n\\[\n\\begin{array}{l}\\phi(\\hat{w},\\mathcal{D})\\leq\\inf_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{D})+\\mathbb{E}_{\\mathcal{S}_{n}}h(w,\\mathcal{S}_{n})+\\Delta_{n}h(w)\\sqrt{\\frac{\\ln(1/\\delta^{\\prime})}{2 n}}\\right]\\\\ +\\epsilon^{\\prime}+2 R_{n}^{h}(\\mathcal{G},\\mathcal{D})+2 M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}.\\\\ \\end{array}\n\\]", "proof": {"steps": [{"n": 1, "claim": "Given any \\( w \\in \\Omega \\), from the Chernoff bound, we know that with probability \\( 1 - \\delta / 2 \\),\n\\[\n\\phi(w,\\mathcal{S}_{n})\\leq\\phi(w,\\mathcal{D})+M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}.\\tag{6.5}\n\\]\nMoreover, from McDiarmid's inequality, we know that with probability \\(1 - \\delta'\\),\n\\[\nh\\left(w,\\mathcal{S}_{n}\\right)\\leq\\mathbb{E}_{\\mathcal{S}_{n}}h\\left(w,\\mathcal{S}_{n}\\right)+\\Delta_{n}h(w)\\sqrt{\\frac{\\ln\\left(1/\\delta^{\\prime}\\right)}{2 n}}.\\tag{6.6}\n\\]\nTaking the union bound with the inequality of Corollary 6.19 at \\(\\delta /2\\), we obtain at probability \\(1 - \\delta -\\delta^{\\prime}\\)\n\\[\n\\begin{array}{l}\\phi(\\hat{w},\\mathcal{D})\\leq\\phi(\\hat{w},\\mathcal{S}_{n})+h(\\hat{w},\\mathcal{S}_{n})+\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})+M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}\\\\ \\leq\\phi(w,\\mathcal{S}_{n})+h(w,\\mathcal{S}_{n})+\\epsilon^{\\prime}+\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})+M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}\\\\ \\leq\\phi(w,\\mathcal{D})+h(w,\\mathcal{S}_{n})+\\epsilon^{\\prime}+\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})+2 M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}\\\\ \\leq\\phi(w,\\mathcal{D})+\\mathbb{E}_{\\mathcal{S}_{n}}h(w,\\mathcal{S}_{n})+\\Delta_{n}h(w)\\sqrt{\\frac{\\ln(1/\\delta^{\\prime})}{2 n}}\\\\ +\\epsilon^{\\prime}+\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})+2 M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}.\\\\ \\end{array}\n\\]\nIn the above derivation, the first inequality used Corollary 6.19."}, {"n": 2, "claim": "The second inequality used (6.1). The third inequality used (6.5)."}, {"n": 3, "claim": "The last inequality used (6.6). This proves the first desired bound."}, {"n": 4, "claim": "The second desired bound employs Theorem 6.14. \\(\\square\\)"}]}}, {"local_id": "T2", "statement": "Consider offset function of (6.3). We have\n\\[\n\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})\\leq 2 R_{n}^{h}(\\mathcal{G},\\mathcal{D}).\n\\]\nConsequently, the approximate regularized ERM method of (6.1) satisfies\n\\[\n\\mathbb{E}_{\\mathcal{S}_{n}}\\phi(\\hat{w},\\mathcal{D})\\leq\\inf_{w\\in\\Omega}[\\phi(w,\\mathcal{D})+\\mathbb{E}_{\\mathcal{S}_{n}}h(w,\\mathcal{S}_{n})]+\\epsilon^{\\prime}+2 R_{n}^{h}(\\mathcal{G},\\mathcal{D}).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\mathcal{S}_n' = \\{Z_1', \\ldots, Z_n'\\} \\sim \\mathcal{D}^n\\) be \\(n\\) iid samples from \\(\\mathcal{D}\\) that are independent of \\(\\mathcal{S}_n\\)."}, {"n": 2, "claim": "We have\n\\[\n\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\sup_{w\\in\\Omega}[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})]\n\\]\n\\[\n=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\sup_{w\\in\\Omega}\\left[\\mathbb{E}_{\\mathcal{S}_{n}^{\\prime}\\sim\\mathcal{D}^{n}}\\phi(w,\\mathcal{S}_{n}^{\\prime})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})\\right]\n\\]\n\\[\n\\leq\\mathbb{E}_{\\left(\\mathcal{S}_{n},\\mathcal{S}_{n}^{\\prime}\\right)\\sim\\mathcal{D}^{2 n}}\\sup_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{S}_{n}^{\\prime})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})\\right]\n\\]\n\\[\n\\begin{array}{l}=\\mathbb{E}_{\\left(\\mathcal{S}_{n},\\mathcal{S}_{n}^{\\prime}\\right)\\sim\\mathcal{D}^{2n}}\\sup_{w\\in\\Omega}\\left[\\left[\\phi(w,\\mathcal{S}_{n}^{\\prime})+0. 5h_{1}(w,\\mathcal{S}_{n}^{\\prime})\\right]-0. 5h(w,\\mathcal{S}_{n}^{\\prime})\\right.\\\\ -\\left[\\phi(w,\\mathcal{S}_{n})+0. 5h_{1}(w,\\mathcal{S}_{n})\\right]-0. 5h(w,\\mathcal{S}_{n})]\\\\ \\end{array}\n\\]\n\\[\n\\begin{array}{l}\\stackrel{(a)}{=}\\mathbb{E}_{\\left(\\mathcal{S}_{n},\\mathcal{S}_{n}^{\\prime}\\right)\\sim\\mathcal{D}^{2n}}\\mathbb{E}_{\\sigma}\\sup_{w\\in\\Omega}\\frac{1}{n}\\sum_{i=1}^{n}\\left[\\sigma_{i}\\left(\\phi\\left(w,Z_{i}^{\\prime}\\right)+0. 5h_{1}\\left(w,Z_{i}^{\\prime}\\right)\\right)-0. 5h(w,\\mathcal{S}_{n}^{\\prime})\\right.\\\\ \\left.-\\sigma_{i}\\left(\\phi\\left(w,Z_{i}\\right)+0. 5h_{1}\\left(w,Z_{i}\\right)\\right)-0. 5h\\left(w,\\mathcal{S}_{n}\\right)\\right]\\\\ \\end{array}\n\\]\n\\[\n\\begin{array}{l}\\leq\\mathbb{E}_{\\left(\\mathcal{S}_{n},\\mathcal{S}_{n}^{\\prime}\\right)\\sim\\mathcal{D}^{2n}}\\mathbb{E}_{\\sigma}\\sup_{w\\in\\Omega}\\frac{1}{n}\\sum_{i=1}^{n}\\left[\\sigma_{i}\\left(\\phi\\left(w,Z_{i}^{\\prime}\\right)+0. 5h_{1}\\left(w,Z_{i}^{\\prime}\\right)\\right)-0. 5h\\left(w,\\mathcal{S}_{n}^{\\prime}\\right)\\right]\\\\ +\\mathbb{E}_{(\\mathcal{S}_{n},\\mathcal{S}_{n}^{\\prime})\\sim\\mathcal{D}^{2n}}\\mathbb{E}_{\\sigma}\\sup_{w\\in\\Omega}\\frac{1}{n}\\sum_{i=1}^{n}[-\\sigma_{i}(\\phi(w,Z_{i})+0. 5h_{1}(w,Z_{i})-0. 5h(w,\\mathcal{S}_{n})]\\\\ \\end{array}\n\\]\n\\[\n=\\mathbb{E}_{(\\mathcal{S}_{n},\\mathcal{S}_{n}^{\\prime})\\sim\\mathcal{D}^{2 n}}[R^{h}(\\mathcal{G},\\mathcal{S}_{n})+R^{h}(\\mathcal{G},\\mathcal{S}_{n}^{\\prime})]=2 R_{n}^{h}(\\mathcal{G},\\mathcal{D}).\n\\]\nIn the above derivation, \\((a)\\) used the fact that \\(\\sigma_{i}(\\phi (w,Z_{i}^{\\prime}) + 0.5h_{1}(w,Z_{i}^{\\prime})) - \\sigma_{i}(\\phi (w,Z_{i}) + 0.5h_{1}(w,Z_{i}))\\) and \\((\\phi (w,Z_i^{\\prime}) + 0.5h_1(w,Z_i^{\\prime})) - (\\phi (w,Z_i) + 0.5h_1(w,Z_i))\\) have the same distributions."}]}}, {"local_id": "T3", "statement": "Consider \\( n \\) independent random variables \\( X_{1},\\ldots,X_{n} \\), and a real-valued function \\( f(X_{1},\\ldots,X_{n}) \\) that satisfies the following inequality\n\\[\n\\sup_{x_{1},\\dots,x_{n},x_{i}^{\\prime}}|f(x_{1},\\dots,x_{n})-f(x_{1},\\dots,x_{i-1},x_{i}^{\\prime},x_{i+1},\\dots,x_{n})|\\leq c_{i}\n\\]\nfor all \\(1 \\leq i \\leq n\\). Then for all \\(\\epsilon > 0\\):\n\\[\n\\Pr\\left[f\\left(X_{1},\\dots,X_{n}\\right)\\geq\\mathbb{E}f\\left(X_{1},\\dots,X_{n}\\right)+\\epsilon\\right]\\leq\\exp\\left(\\frac{-2\\epsilon^{2}}{\\sum_{i=1}^{n}c_{i}^{2}}\\right).\n\\]\nSimilarly:\n\\[\n\\Pr\\left[f\\left(X_{1},\\dots,X_{n}\\right)\\leq\\mathbb{E}f\\left(X_{1},\\dots,X_{n}\\right)-\\epsilon\\right]\\leq\\exp\\left(\\frac{-2\\epsilon^{2}}{\\sum_{i=1}^{n}c_{i}^{2}}\\right).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\( X_{k}^{l}=\\{X_{k},\\ldots,X_{l}\\} \\). Consider \\( X_{1}^{n} \\), and for some \\( 1\\leq k\\leq n \\), we use the simplified notation \\( \\tilde{X}_1^n=\\{X_1,\\dots,X_{k-1},\\tilde{X}_k,X_{k+1},X_n\\} \\)."}, {"n": 2, "claim": "Then we have\n\\[\n\\left|\\mathbb{E}_{X_{k+1}^{n}}f\\left(X_{1}^{n}\\right)-\\mathbb{E}_{X_{k+1}^{n}}f\\left(\\tilde{X}_{1}^{n}\\right)\\right|\\leq c_{k}.\n\\]\nWe now consider \\(\\mathbb{E}_{X_{k + 1}^n}f(X_1^n)\\) as a random variable depending on \\(X_{k}\\), conditioned on \\(X_{1}^{k - 1}\\)."}, {"n": 3, "claim": "It follows from derivation of the Chernoff bound that we have the following logarithmic moment generating function estimate (see Example 2.14):"}, {"n": 4, "claim": "\\[\n\\ln\\mathbb{E}_{X_{k}}\\exp\\left[\\lambda\\mathbb{E}_{X_{k+1}^{n}}f\\left(X_{1}^{n}\\right)\\right]\\leq\\lambda\\mathbb{E}_{X_{k}^{n}}f\\left(X_{1}^{n}\\right)+\\lambda^{2}c_{k}^{2}/8.\n\\]\nNow we may exponentiate the above inequality, and take expectation with respect to \\( X_1^{k-1} \\) to obtain\n\\[\n\\mathbb{E}_{X_{1}^{k}}\\exp [\\lambda\\mathbb{E}_{X_{k+1}^{n}}f(X_{1}^{n})]\\leq\\mathbb{E}_{X_{1}^{k-1}}\\exp [\\lambda\\mathbb{E}_{X_{k}^{n}}f(X_{1}^{n})+\\lambda^{2}c_{k}^{2}/8].\n\\]\nBy taking logarithm, we obtain\n\\[\n\\ln\\mathbb{E}_{X_{1}^{k}}\\exp\\left[\\lambda\\mathbb{E}_{X_{k+1}^{n}}f\\left(X_{1}^{n}\\right)\\right]\\leq\\ln\\mathbb{E}_{X_{1}^{k-1}}\\exp\\left[\\lambda\\mathbb{E}_{X_{k}^{n}}f\\left(X_{1}^{n}\\right)\\right]+\\lambda^{2}c_{k}^{2}/8.\n\\]\nBy summing from \\( k = 1 \\) to \\( k = n \\), and canceling redundant terms, we obtain\n\\[\n\\ln\\mathbb{E}_{X_{1}^{n}}\\exp\\left[\\lambda f\\left(X_{1}^{n}\\right)\\right]\\leq\\lambda\\mathbb{E}_{X_{1}^{n}}f\\left(X_{1}^{n}\\right)+\\lambda^{2}\\sum_{k=1}^{n}c_{k}^{2}/8.\\tag{6.4}\n\\]\nLet\n\\[\n\\delta=\\Pr\\left[f\\left(X_{1}^{n}\\right)\\geq\\mathbb{E}_{X_{1}^{n}}f\\left(X_{1}^{n}\\right)+\\epsilon\\right].\n\\]\nUsing Markov's inequality, we have for all positive \\(\\lambda\\)\n\\[\n\\delta\\leq e^{-\\lambda\\left(\\mathbb{E}_{X_{1}^{n}}f\\left(X_{1}^{n}\\right)+\\epsilon\\right)}\\mathbb{E}_{X_{1}^{n}}e^{\\lambda f\\left(X_{1}^{n}\\right)}\\leq\\exp\\left[-\\lambda\\epsilon+\\frac{\\lambda^{2}}{8}\\sum_{k=1}^{n}c_{k}^{2}\\right].\n\\]\nSince \\(\\lambda > 0\\) is arbitrary, we conclude that\n\\[\n\\ln\\delta\\leq\\inf_{\\lambda\\geq 0}\\left[\\frac{\\lambda^{2}}{8}\\sum_{k=1}^{n}c_{k}^{2}-\\lambda\\epsilon\\right]= -\\frac{2\\epsilon^{2}}{\\sum_{k=1}^{n}c_{k}^{2}}.\n\\]\nThis implies the theorem."}]}}, {"local_id": "T4", "statement": "Assume that for some \\(M \\geq 0\\):\n\\[\n\\sup_{w\\in\\Omega}\\left[\\sup_{z,z^{\\prime}}[\\phi(w,z)-\\phi(w,z^{\\prime})]+\\Delta_{n}h(w)\\right]\\leq M.\n\\]\nThen with probability at least \\(1 - \\delta\\): for all \\(w \\in \\Omega\\)\n\\[\n\\phi(w,\\mathcal{D})\\leq\\phi(w,\\mathcal{S}_{n})+h(w,\\mathcal{S}_{n})+\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})+M\\sqrt{\\frac{\\ln(1/\\delta)}{2 n}}.\n\\]\nMoreover, assume that the decomposition (6.3) holds, then with probability at least \\(1 - \\delta\\): for all \\(w \\in \\Omega\\),\n\\[\n\\phi(w,\\mathcal{D})\\leq\\phi(w,\\mathcal{S}_{n})+h(w,\\mathcal{S}_{n})+2 R_{n}^{h}(\\mathcal{G},\\mathcal{D})+M\\sqrt{\\frac{\\ln(1/\\delta)}{2 n}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\mathcal S_n=\\{Z_1,\\dots,Z_n\\}\\) and \\(\\mathcal S_n'=\\{Z_1,\\dots,Z_{i-1},Z_i',Z_{i+1},\\dots,Z_n\\}\\), and define \\(f(\\mathcal S_n)=\\sup_{w\\in\\Omega}[\\phi(w,\\mathcal D)-\\phi(w,\\mathcal S_n)-h(w,\\mathcal S_n)]\\)."}, {"n": 2, "claim": "The supremum is attained at \\(\\hat w=\\arg\\max_{w\\in\\Omega}[\\phi(w,\\mathcal D)-\\phi(w,\\mathcal S_n)-h(w,\\mathcal S_n)]\\)."}, {"n": 3, "claim": "\\(f(\\mathcal S_n)-f(\\mathcal S_n')=[\\phi(\\hat w,\\mathcal D)-\\phi(\\hat w,\\mathcal S_n)-h(\\hat w,\\mathcal S_n)]-\\sup_{w}[\\phi(w,\\mathcal D)-\\phi(w,\\mathcal S_n')-h(w,\\mathcal S_n')]\\)."}, {"n": 4, "claim": "\\(f(\\mathcal S_n)-f(\\mathcal S_n')\\le[\\phi(\\hat w,\\mathcal D)-\\phi(\\hat w,\\mathcal S_n)-h(\\hat w,\\mathcal S_n)]-[\\phi(\\hat w,\\mathcal D)-\\phi(\\hat w,\\mathcal S_n')-h(\\hat w,\\mathcal S_n')]\\)."}, {"n": 5, "claim": "\\(f(\\mathcal S_n)-f(\\mathcal S_n')\\le\\tfrac1n[\\phi(\\hat w,Z_i')-\\phi(\\hat w,Z_i)+\\Delta_n h(\\hat w)]\\le M/n\\)."}, {"n": 6, "claim": "Symmetrically \\(f(\\mathcal S_n')-f(\\mathcal S_n)\\le M/n\\), so \\(f\\) has bounded differences with \\(c_i=M/n\\)."}, {"n": 7, "claim": "By McDiarmid's inequality, with probability at least \\(1-\\delta\\), \\(f(\\mathcal S_n)\\le \\mathbb E[f(\\mathcal S_n)]+M\\sqrt{\\frac{\\ln(1/\\delta)}{n}}\\)."}, {"n": 8, "claim": "Since \\(\\mathbb E[f(\\mathcal S_n)]=\\epsilon_n^h(\\mathcal G,\\mathcal D)\\), rearranging \\(f(\\mathcal S_n)\\ge\\phi(w,\\mathcal D)-\\phi(w,\\mathcal S_n)-h(w,\\mathcal S_n)\\) gives the first bound for all \\(w\\)."}, {"n": 9, "claim": "The second bound follows since \\(\\epsilon_n^h(\\mathcal G,\\mathcal D)\\le 2R_n^h(\\mathcal G,\\mathcal D)\\)."}]}}], "target_local_id": "T4", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "label": "flawed", "error_step": 8, "error_class6": ["logical_gap_or_missing_case", "invalid_bound_or_inequality", "computation_or_constant"], "error_explanation": "The originally-marked Step 7 is not actually a false inequality: with sum_i c_i^2 = M^2/n, solving exp(-2t^2/(M^2/n)) = delta for t gives t = M*sqrt(ln(1/delta)/(2n)), and Step 7's stated t = M*sqrt(ln(1/delta)/n) equals sqrt(2) times that tight threshold -- i.e. Step 7 uses a LARGER, more conservative t, so its McDiarmid probability bound P[f - E[f] >= t] <= exp(-2t^2/sum c_i^2) still holds (in fact more strongly). Step 7 as literally written is therefore a true but looser bound, not a false relational claim. The genuine gap is at Step 8, which invokes Step 7 to conclude the theorem's stated bound using the theorem's TIGHTER constant M*sqrt(ln(1/delta)/(2n)) -- but that tighter conclusion does not follow from Step 7's looser M*sqrt(ln(1/delta)/n) bound. This is an unjustified inference at Step 8 (silently swapping in a smaller deviation term than Step 7 actually established), not a false inequality at Step 7."}
{"id": "ch6_theorem_6_14__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nDefinition 6.6. Consider any known data-dependent offset function \\( h(w, S_n) \\). Define the uniform convergence complexity of a function class \\( \\mathcal{G} \\) with offset \\( h \\) as\n\\[\n\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\sup_{w\\in\\Omega}[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})].\\tag{6.2}\n\\]\nThis quantity measures the one-sided expected uniform convergence of function class \\(\\mathcal{G}\\) with offset function \\(h(w, S_n)\\).\nWe note that\n\\[\n\\epsilon_{n}(\\mathcal{G},\\mathcal{D})=\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D}),\\qquad \\text{w i t h} h=0.\n\\]\nWe have the following generalization of Theorem 6.2, with a similar proof.\n\nDefinition 6.11. Consider a function class \\(\\mathcal{G} = \\{\\phi(w, Z) : w \\in \\Omega\\}\\), and let \\(h\\) be an offset function of the following form\n\\[\nh(w,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{i=1}^{n}h(w,Z_{i}),\\quad h(w,z)=h_{0}(w)+h_{1}(w,z).\\tag{6.3}\n\\]\nGiven \\( S_{n}=\\{Z_{1},\\ldots,Z_{n}\\} \\), the (one-sided) empirical Rademacher complexity of \\( \\mathcal{G} \\) with offset \\( h \\) decomposition (6.3) is defined as\n\\[\nR^{h}(\\mathcal{G},\\mathcal{S}_{n})=\\mathbb{E}_{\\sigma}\\sup_{w\\in\\Omega}\\left[\\frac{1}{n}\\sum_{i=1}^{n}\\sigma_{i}[\\phi(w,Z_{i})+0. 5h_{1}(w,Z_{i})]-0. 5h(w,\\mathcal{S}_{n})\\right],\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}^{h}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R^{h}(\\mathcal{G},\\mathcal{S}_{n}).\n\\]\nWe note that the standard Rademacher complexity can be regarded as a special case of the offset Rademacher complexity with \\( h(\\cdot) = h_0(\\cdot) = h_1(\\cdot) = 0 \\):\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n})=R^{0}(\\mathcal{G},\\mathcal{S}_{n}),\\quad R_{n}(\\mathcal{G},\\mathcal{D})=R_{n}^{0}(\\mathcal{G},\\mathcal{D}).\n\\]\nIt should be pointed out that the decomposition of \\( h \\) in (6.3) may not be unique, and the offset Rademacher complexity relies on the specific decomposition used.\nAs shown in Example 6.10, we allow distribution dependent offset in the definition of uniform convergence complexity. We also allow distribution dependent offset in the definition of offset Rademacher complexity.\nThe following result shows that for some cases, offset Rademacher complexity can be obtained easily for some function classes.\n\nDefinition 3.8. We define\n\\[\n\\phi(w,z)=L\\left(f(w,x),y\\right)-L_{*}(x,y),\\tag{3.4}\n\\]\nfor \\( w \\in \\Omega \\) and \\( z = (x, y) \\in \\mathcal{Z} = \\mathcal{X} \\times \\mathcal{Y} \\), and a pre-chosen \\( L_{*}(x, y) \\) of \\( z = (x, y) \\) that does not depend on \\( w \\).\nFor training data \\( S_{n}=\\{Z_{i}=(X_{i},Y_{i}):i=1,\\ldots,n\\} \\), we define the training loss for \\( w\\in \\Omega \\) as\n\\[\n\\phi(w,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{i=1}^{n}\\phi(w,Z_{i}).\\tag{3.5}\n\\]\nMoreover, for a distribution \\(\\mathcal{D}\\) on \\(\\mathcal{Z}\\), we define the test loss for \\(w \\in \\Omega\\) as:\n\\[\n\\phi(w,\\mathcal{D})=\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(w,Z).\\tag{3.6}\n\\]\nSince \\( L_{*}(x,y) \\) does not depend on \\( w \\), the ERM solution with respect to the loss \\( L(\\cdot) \\) is equivalent to the ERM solution with respect to \\( \\phi (w,\\cdot) \\). Therefore with the simplified notations, the approximate ERM method (3.3) is a special case of the following method:\n\\[\n\\phi(\\hat{w},\\mathcal{S}_{n})\\leq\\inf_{w\\in\\Omega}\\phi(w,\\mathcal{S}_{n})+\\epsilon^{\\prime}.\\tag{3.7}\n\\]\nIn general, we may simply take \\( L_{*}(x,y) = 0 \\) in (3.4). However, for some applications, we may choose a non-zero \\( L_{*}(x,y) \\) so that\n\\[\nL(f(w,x),y)-L_{*}(x,y)\n\\]\nhas a small variance. For least squares loss, this can be achieved with \\( L_{*}(x,y) = L(f_{*}(x),y) \\) where \\( f_{*}(x) \\) is the optimal prediction function that minimizes the test loss as shown in Example 3.9. The smaller variance, combined with Bernstein's inequality, implies better generalization bound (see Section 3.6 for more details).\n\nDefinition 6.17. Given a function \\( h(w, S_n) \\), we define\n\\[\n\\Delta_{n}h(w)=\\sup\\{n\\cdot|h(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n}^{\\prime})|: |\\mathcal{S}_{n}\\cap\\mathcal{S}_{n}^{\\prime}| =n-1\\}.\n\\]\n\nEquation (6.1). The approximate regularized ERM method is to find \\( \\hat{w} \\in \\Omega \\) such that, for some \\( \\epsilon' \\geq 0 \\),\n\\[\n\\phi(\\hat{w},\\mathcal{S}_{n})+h(\\hat{w},\\mathcal{S}_{n})\\leq\\inf_{w\\in\\Omega}[\\phi(w,\\mathcal{S}_{n})+h(w,\\mathcal{S}_{n})]+\\epsilon'.\\tag{6.1}\n\\]\nWhen \\(h = 0\\), this reduces to the plain approximate ERM method.", "theorems": [{"local_id": "T1", "statement": "Assume that for some \\(M \\geq 0\\):\n\\[\n\\sup_{w\\in\\Omega}\\left[\\sup_{z,z^{\\prime}}[\\phi(w,z)-\\phi(w,z^{\\prime})]+\\Delta_{n}h(w)\\right]\\leq M.\n\\]\nThen with probability at least \\(1 - \\delta\\): for all \\(w \\in \\Omega\\)\n\\[\n\\phi(w,\\mathcal{D})\\leq\\phi(w,\\mathcal{S}_{n})+h(w,\\mathcal{S}_{n})+\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})+M\\sqrt{\\frac{\\ln(1/\\delta)}{2 n}}.\n\\]\nMoreover, assume that the decomposition (6.3) holds, then with probability at least \\(1 - \\delta\\): for all \\(w \\in \\Omega\\),\n\\[\n\\phi(w,\\mathcal{D})\\leq\\phi(w,\\mathcal{S}_{n})+h(w,\\mathcal{S}_{n})+2 R_{n}^{h}(\\mathcal{G},\\mathcal{D})+M\\sqrt{\\frac{\\ln(1/\\delta)}{2 n}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "Consider \\( \\mathcal{S}_n = \\{Z_1, \\ldots, Z_n\\} \\) and \\( \\mathcal{S}_n' = \\{Z_1, \\ldots, Z_{i-1}, Z_i', Z_{i+1}, \\ldots, Z_n\\} \\)."}, {"n": 2, "claim": "Let \\( f(\\mathcal{S}_n) = \\sup_{w \\in \\Omega} [\\phi(w, \\mathcal{D}) - \\phi(w, \\mathcal{S}_n) - h(w, \\mathcal{S}_n)] \\)."}, {"n": 3, "claim": "For simplicity, we assume that the sup can be achieved at \\( \\hat{w} \\) as\n\\[\n\\hat{w}=\\arg\\max_{w\\in\\Omega}[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})].\n\\]\nThen\n\\[\n\\begin{array}{l}f\\left(\\mathcal{S}_{n}\\right)-f\\left(\\mathcal{S}_{n}^{\\prime}\\right)\\\\ =\\left[\\phi(\\hat{w},\\mathcal{D})-\\phi(\\hat{w},\\mathcal{S}_{n})-h(\\hat{w},\\mathcal{S}_{n})\\right]-\\sup_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n}^{\\prime})-h(w,\\mathcal{S}_{n}^{\\prime})\\right]\\\\ \\leq\\left[\\phi(\\hat{w},\\mathcal{D})-\\phi(\\hat{w},\\mathcal{S}_{n})-h(\\hat{w},\\mathcal{S}_{n})\\right]-\\left[\\phi(\\hat{w},\\mathcal{D})-\\phi(\\hat{w},\\mathcal{S}_{n}^{\\prime})-h(\\hat{w},\\mathcal{S}_{n}^{\\prime})\\right]\\\\ \\leq\\frac{1}{n}[\\phi(\\hat{w},Z_{i}^{\\prime})-\\phi(\\hat{w},Z_{i})+\\Delta_{n}h(\\hat{w})]\\leq M/n.\\\\ \\end{array}\n\\]\nSimilarly, \\( f(\\mathcal{S}_n') - f(\\mathcal{S}_n) \\leq M / n \\)."}, {"n": 4, "claim": "Therefore we may take \\( c_i = M / n \\) in Theorem 6.16, which implies the first desired result."}, {"n": 5, "claim": "The second bound follows from the estimate \\( \\epsilon_n^h (\\mathcal{G},\\mathcal{D}) \\leq 2R_n^h (\\mathcal{G},\\mathcal{D}) \\) of Theorem 6.14."}]}}, {"local_id": "T2", "statement": "Let \\( S_{n} \\) be \\( n \\) iid samples from \\( \\mathcal{D} \\). Then the approximate ERM method of (6.1) satisfies\n\\[\n\\mathbb{E}_{\\mathcal{S}_{n}}\\phi(\\hat{w},\\mathcal{D})\\leq\\inf_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{D})+\\mathbb{E}_{\\mathcal{S}_{n}}h(w,\\mathcal{S}_{n})\\right]+\\epsilon^{\\prime}+\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D}).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Given any \\(w \\in \\Omega\\), we have for each training data \\(\\mathcal{S}_n\\)\n\\[\n\\begin{array}{l}\\phi(\\hat{w},\\mathcal{D})\\\\ =\\left[\\phi(\\hat{w},\\mathcal{S}_{n})+h(\\hat{w},\\mathcal{S}_{n})\\right]+\\left[\\phi(\\hat{w},\\mathcal{D})-\\phi(\\hat{w},\\mathcal{S}_{n})-h(\\hat{w},\\mathcal{S}_{n})\\right]\\\\ \\leq\\left[\\phi(\\hat{w},\\mathcal{S}_{n})+h(\\hat{w},\\mathcal{S}_{n})\\right]+\\sup_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})\\right]\\\\ \\leq\\left[\\phi(w,\\mathcal{S}_{n})+h(w,\\mathcal{S}_{n})\\right]+\\epsilon^{\\prime}+\\sup_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})\\right].\\\\ \\end{array}\n\\]\nIn the derivation of the last inequality, we used (6.1)."}, {"n": 2, "claim": "Taking expectation with respect to \\( S_{n} \\), and note that \\( w \\) does not depend on \\( S_{n} \\), we obtain\n\\[\n\\begin{array}{l}\\mathbb{E}_{\\mathcal{S}_{n}}\\phi(\\hat{w},\\mathcal{D})\\leq\\phi(w,\\mathcal{D})+\\mathbb{E}_{\\mathcal{S}_{n}}h(w,\\mathcal{S}_{n})+\\epsilon^{\\prime}\\\\ +\\mathbb{E}_{\\mathcal{S}_{n}}\\sup_{w\\in\\Omega}[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})].\\\\ \\end{array}\n\\]\nThis implies the desired bound."}]}}, {"local_id": "T3", "statement": "Consider offset function of (6.3). We have\n\\[\n\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})\\leq 2 R_{n}^{h}(\\mathcal{G},\\mathcal{D}).\n\\]\nConsequently, the approximate regularized ERM method of (6.1) satisfies\n\\[\n\\mathbb{E}_{\\mathcal{S}_{n}}\\phi(\\hat{w},\\mathcal{D})\\leq\\inf_{w\\in\\Omega}[\\phi(w,\\mathcal{D})+\\mathbb{E}_{\\mathcal{S}_{n}}h(w,\\mathcal{S}_{n})]+\\epsilon^{\\prime}+2 R_{n}^{h}(\\mathcal{G},\\mathcal{D}).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\(\\mathcal{S}_n'=\\{Z_1',\\ldots,Z_n'\\}\\sim\\mathcal{D}^n\\) be iid samples independent of \\(\\mathcal{S}_n\\)."}, {"n": 2, "claim": "\\(\\epsilon_n(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_n}\\sup_{w\\in\\Omega}[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_n)-h(w,\\mathcal{S}_n)]\\)."}, {"n": 3, "claim": "\\(\\phi(w,\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_n'}\\phi(w,\\mathcal{S}_n')\\)."}, {"n": 4, "claim": "\\(\\epsilon_n\\le\\mathbb{E}_{(\\mathcal{S}_n,\\mathcal{S}_n')}\\sup_{w}[\\phi(w,\\mathcal{S}_n')-\\phi(w,\\mathcal{S}_n)-h(w,\\mathcal{S}_n)]\\)."}, {"n": 5, "claim": "\\(=\\mathbb{E}\\sup_w\\big[[\\phi(w,\\mathcal{S}_n')+0.5h_1(w,\\mathcal{S}_n')]-0.5h(w,\\mathcal{S}_n')-[\\phi(w,\\mathcal{S}_n)+0.5h_1(w,\\mathcal{S}_n)]-0.5h(w,\\mathcal{S}_n)\\big]\\)."}, {"n": 6, "claim": "\\(\\stackrel{(a)}{=}\\mathbb{E}\\,\\mathbb{E}_\\sigma\\sup_w\\frac1n\\sum_i[\\sigma_i(\\phi(w,Z_i')+0.5h_1(w,Z_i'))-0.5h(w,\\mathcal{S}_n')-\\sigma_i(\\phi(w,Z_i)+0.5h_1(w,Z_i))-0.5h(w,\\mathcal{S}_n)]\\)."}, {"n": 7, "claim": "\\(\\le\\mathbb{E}\\,\\mathbb{E}_\\sigma\\sup_w\\frac1n\\sum_i[\\sigma_i(\\phi(w,Z_i')+0.5h_1(w,Z_i'))-0.5h(w,\\mathcal{S}_n')]+\\mathbb{E}\\,\\mathbb{E}_\\sigma\\sup_w\\frac1n\\sum_i[-\\sigma_i(\\phi(w,Z_i)+0.5h_1(w,Z_i))-0.5h(w,\\mathcal{S}_n)]\\)."}, {"n": 8, "claim": "\\(=\\mathbb{E}[R^h(\\mathcal{G},\\mathcal{S}_n)+R^h(\\mathcal{G},\\mathcal{S}_n')]=R_n^h(\\mathcal{G},\\mathcal{D})\\)."}, {"n": 9, "claim": "Substituting \\(\\epsilon_n^h\\le2R_n^h\\) into the ERM bound yields the claim."}]}}], "target_local_id": "T3", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "label": "flawed", "error_step": 8, "error_class6": ["computation_or_constant", "invalid_bound_or_inequality", "logical_gap_or_missing_case"], "error_explanation": "The sum of two equal expectations $\\mathbb{E}R^h(\\mathcal{G},\\mathcal{S}_n)+\\mathbb{E}R^h(\\mathcal{G},\\mathcal{S}_n')=R_n^h+R_n^h$ equals $2R_n^h$, not $R_n^h$. Stating it as $R_n^h$ drops the factor of two, an arithmetic error."}
{"id": "ch7_lemma_7_18__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nRank-one population objective \\(g\\) and matrix \\(M\\). Let \\(z\\) be the target vector and define \\(M = zz^\\top\\). Consider the rank-one population objective \\[ g(x) = \\frac{1}{2}\\|M - xx^\\top\\|_F^2. \\] For this objective, \\(\\nabla g(x) = (M - xx^\\top)x\\).\n\nRank-one matrix-factorization objective \\(g\\). Let \\(x\\) and \\(z\\) be vectors in a finite-dimensional real Euclidean space, assume \\(\\|z\\|_2 = 1\\), and set \\(M = zz^\\top\\). Define \\(g(x) = \\frac{1}{2}\\|M - xx^\\top\\|_F^2\\). Then \\(\\nabla g(x) = (zz^\\top - xx^\\top)x = (M - xx^\\top)x\\), and the Hessian quadratic form used below is \\(\\langle z, \\nabla^2 g(x)z\\rangle = \\|zx^\\top + xz^\\top\\|_F^2 - 2z^\\top(zz^\\top-xx^\\top)z\\).\n\nRank-one matrix completion objectives. In the rank-one matrix-completion setup, let \\(d\\) be the ambient dimension and let \\(\\epsilon>0\\) be the concentration error parameter. Let \\(z \\in \\mathbb{R}^d\\) be the ground-truth vector, normalized by \\(\\|z\\|_2 = 1\\) and incoherent with \\(\\|z\\|_\\infty \\leq \\mu/\\sqrt d\\), and set \\(M = zz^\\top\\). Let \\(\\Omega\\) be the random observed set of matrix entries, sampled with probability \\(p\\), and let \\(P_\\Omega\\) be the coordinate projection/observation operator onto \\(\\Omega\\). Define the observed empirical objective \\(f(x)=\\frac12\\langle P_\\Omega(M),xx^\\top\\rangle-\\frac14\\langle P_\\Omega(xx^\\top),xx^\\top\\rangle\\), the sampled version over \\(\\Omega\\). Define the population objective \\(g(x)=\\frac12\\langle M,xx^\\top\\rangle-\\frac14\\|xx^\\top\\|_F^2\\), so the proof uses the population-gradient identity \\(\\nabla g(x)=(M-xx^\\top)x\\).\n\nLocal minimum of a function. We say that \\(x\\) is a local minimum of a function \\(f\\) if there exists an open neighborhood \\(N\\) around \\(x\\) such that in \\(N\\), the function values are at least \\(f(x)\\).", "theorems": [{"local_id": "T1", "statement": "If \\(x\\) satisfies \\(\\nabla g(x) = 0\\), then \\(\\langle x,z \\rangle^2 = \\|x\\|_2^4\\).", "proof": {"steps": [{"n": 1, "claim": "\\(\\nabla g(x)=0\\)."}, {"n": 2, "claim": "\\(\\langle x,(zz^\\top-xx^\\top)x\\rangle=0\\)."}, {"n": 3, "claim": "\\(\\langle x,zz^\\top x\\rangle=(z^\\top x)^2=\\langle x,z\\rangle^2\\)."}, {"n": 4, "claim": "By the Cauchy–Schwarz equality, \\(\\langle x,z\\rangle^2=\\|x\\|_2^2\\,\\|z\\|_2^2\\)."}, {"n": 5, "claim": "\\(\\langle x,xx^\\top x\\rangle=\\|x\\|_2^4\\) and \\(\\|z\\|_2^2=\\|x\\|_2^2\\)."}, {"n": 6, "claim": "\\(\\langle x,z\\rangle^2=\\|x\\|_2^2\\|z\\|_2^2=\\|x\\|_2^4\\)."}]}}, {"local_id": "T2", "statement": "All local minima of \\(g\\) are global minima.", "proof": {"steps": [{"n": 1, "claim": "\\[\n\\nabla g(x) = 0\n\\implies \\langle z, \\nabla g(x) \\rangle = 0\n\\implies \\langle z, (zz^\\top-xx^\\top)x \\rangle = 0\n\\implies \\langle x,z\\rangle (1-\\|x\\|_2^2) = 0.\n\\]\nSince \\(|\\langle x,z \\rangle| \\geq 1/3 \\neq 0\\) (by the bound-norm-for-\\(g\\) lemma, i.e., the referenced result), we must have \\(\\|x\\|_2^2 = 1\\). But then the connecting-inner-product-and-norm-for-\\(g\\) lemma implies \\(\\langle x, z\\rangle^2 = \\|x\\|_2^4 = 1\\), so \\(x = \\pm z\\) by Cauchy-Schwarz."}]}}, {"local_id": "T3", "statement": "Suppose \\(\\|x\\|_\\infty \\leq 2\\mu / \\sqrt{d}\\). If \\(x\\) satisfies \\(\\nabla f(x) = 0\\), then \\(\\langle x,z \\rangle^2 \\geq \\|x\\|_2^4 - \\epsilon\\) with high probability.", "proof": {"steps": [{"n": 1, "claim": "\\[\n\\nabla f(x) = 0 \\implies \\langle x, \\nabla f(x) \\rangle = 0\n\\implies \\langle x, \\nabla g(x) \\rangle \\approx \\langle x, \\nabla f(x) \\rangle/p \\pm \\epsilon \\quad \\text{(by the concentration lemma, i.e., the referenced result)}\n\\implies |\\langle x, (zz^\\top-xx^\\top)x \\rangle| \\leq \\epsilon \\quad \\text{w.h.p.}\n\\implies \\langle x,z \\rangle^2 \\geq \\|x\\|_2^4 - \\epsilon \\quad \\text{w.h.p.}\n\\]"}]}}], "target_local_id": "T1", "topic_domain": "Optimization", "topic_subtopic": "Non-convex Geometry", "label": "flawed", "error_step": 4, "error_class6": ["invalid_bound_or_inequality", "missing_assumption", "others"], "error_explanation": "Cauchy–Schwarz only gives the inequality \\(\\langle x,z\\rangle^2\\le\\|x\\|_2^2\\|z\\|_2^2\\); equality holds solely when \\(x\\) and \\(z\\) are colinear, a precondition not established here, so asserting the equality is a misapplication of the named result."}
{"id": "ch7_lemma_7_19__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nRank-one matrix completion objectives. In the rank-one matrix-completion setup, let \\(d\\) be the ambient dimension and let \\(\\epsilon>0\\) be the concentration error parameter. Let \\(z \\in \\mathbb{R}^d\\) be the ground-truth vector, normalized by \\(\\|z\\|_2 = 1\\) and incoherent with \\(\\|z\\|_\\infty \\leq \\mu/\\sqrt d\\), and set \\(M = zz^\\top\\). Let \\(\\Omega\\) be the random observed set of matrix entries, sampled with probability \\(p\\), and let \\(P_\\Omega\\) be the coordinate projection/observation operator onto \\(\\Omega\\). Define the observed empirical objective \\(f(x)=\\frac12\\langle P_\\Omega(M),xx^\\top\\rangle-\\frac14\\langle P_\\Omega(xx^\\top),xx^\\top\\rangle\\), the sampled version over \\(\\Omega\\). Define the population objective \\(g(x)=\\frac12\\langle M,xx^\\top\\rangle-\\frac14\\|xx^\\top\\|_F^2\\), so the proof uses the population-gradient identity \\(\\nabla g(x)=(M-xx^\\top)x\\).\n\nRank-one population objective \\(g\\) and matrix \\(M\\). Let \\(z\\) be the target vector and define \\(M = zz^\\top\\). Consider the rank-one population objective \\[ g(x) = \\frac{1}{2}\\|M - xx^\\top\\|_F^2. \\] For this objective, \\(\\nabla g(x) = (M - xx^\\top)x\\).\n\nLocal minimum of a function. We say that \\(x\\) is a local minimum of a function \\(f\\) if there exists an open neighborhood \\(N\\) around \\(x\\) such that in \\(N\\), the function values are at least \\(f(x)\\).", "theorems": [{"local_id": "T1", "statement": "Assume \\(p = \\dfrac{\\operatorname{poly}(\\mu, \\log d)}{d\\epsilon^2}\\). Then with high probability, all local minima of \\(f\\) are \\(O(\\sqrt{\\epsilon})\\)-close to \\(+z\\) or \\(-z\\).", "proof": {"steps": [{"n": 1, "claim": "Observe that \\(\\|x-z\\|_2^2 = \\|x\\|_2^2 + \\|z\\|_2^2 - 2\\langle x,z \\rangle \\leq \\|x\\|_2^2 + 1 - 2\\langle x,z \\rangle\\). The goal is to show this quantity is small with high probability; the following bounds hold with high probability throughout."}, {"n": 2, "claim": "Let \\(x\\) be such that \\(\\nabla f(x) = 0\\). For \\(\\epsilon \\leq 1/16\\),\n\\[\n\\langle x,z \\rangle^2 \\geq \\|x\\|_2^4 - \\epsilon \\quad\\text{(by the connecting-inner-product-and-norm-for-}f\\text{ lemma)}\n\\geq (1/3-\\epsilon/3)^2 - \\epsilon \\quad\\text{(by the bound-norm-for-}f\\text{ lemma)}\n\\geq 1/32.\n\\]\n\nWith this, we bound \\(\\|x\\|_2^2\\):"}, {"n": 3, "claim": "\\[\n\\nabla f(x) = 0\n\\implies \\langle x, \\nabla f(x) \\rangle = 0\n\\implies |\\langle z, \\nabla g(x) \\rangle| \\leq \\epsilon \\quad\\text{(by the concentration lemma)}\n\\implies |\\langle x,z\\rangle| \\cdot |1-\\|x\\|_2^2| \\leq \\epsilon \\quad\\text{(by definition of }g\\text{)}\n\\implies |1-\\|x\\|_2^2| \\leq 32\\epsilon = O(\\epsilon) \\quad\\text{(using }\\langle x,z\\rangle^2 \\geq 1/32\\text{)}\n\\implies \\|x\\|_2^2 = 1 \\pm O(\\epsilon).\n\\]\n\nNext, we bound \\(\\langle x,z \\rangle\\):"}, {"n": 4, "claim": "\\[\n\\langle x, z \\rangle^2 \\geq \\|x\\|_2^4 - \\epsilon \\quad\\text{(by the connecting-inner-product-and-norm-for-}f\\text{ lemma)}\n\\geq (1-O(\\epsilon))^2 - \\epsilon\n= 1 - O(\\epsilon).\n\\]\n\nFinally, we put these together to bound \\(\\|x-z\\|_2^2\\)."}, {"n": 5, "claim": "There are two cases."}, {"n": 6, "claim": "Case 1: \\(\\langle x,z\\rangle \\geq 1 - O(\\epsilon)\\). Then\n\\[\n\\|x-z\\|_2^2 = \\|x\\|_2^2 + \\|z\\|_2^2 - 2\\langle x,z \\rangle\n\\leq \\|x\\|_2^2 + 1 - 2\\langle x,z \\rangle\n\\leq 1 + O(\\epsilon) + 1 - 2(1-O(\\epsilon))\n\\leq O(\\epsilon).\n\\]\nHence \\(x\\) is \\(O(\\sqrt{\\epsilon})\\)-close to \\(z\\)."}, {"n": 7, "claim": "Case 2: \\(\\langle x,z\\rangle \\leq -(1 - O(\\epsilon))\\). Then by an analogous argument, \\(x\\) is \\(O(\\sqrt{\\epsilon})\\)-close to \\(-z\\)."}]}}, {"local_id": "T2", "statement": "Suppose \\(\\|x\\|_\\infty \\leq 2\\mu / \\sqrt{d}\\). If \\(x\\) satisfies \\(\\nabla f(x) = 0\\), then \\(\\langle x,z \\rangle^2 \\geq \\|x\\|_2^4 - \\epsilon\\) with high probability.", "proof": {"steps": [{"n": 1, "claim": "\\(\\nabla f(x)=0\\implies\\langle x,\\nabla f(x)\\rangle=0\\)."}, {"n": 2, "claim": "\\(\\langle x,\\nabla g(x)\\rangle\\le\\langle x,\\nabla f(x)\\rangle/p+\\epsilon\\) w.h.p."}, {"n": 3, "claim": "\\(\\langle x,(zz^\\top-xx^\\top)x\\rangle=\\langle x,z\\rangle^2-\\|x\\|_2^4\\), hence \\(\\langle x,z\\rangle^2-\\|x\\|_2^4\\le\\epsilon\\) w.h.p."}, {"n": 4, "claim": "\\(\\langle x,z\\rangle^2\\ge\\|x\\|_2^4-\\epsilon\\) w.h.p."}]}}, {"local_id": "T3", "statement": "If \\(x\\) satisfies \\(\\nabla g(x) = 0\\), then \\(\\langle x,z \\rangle^2 = \\|x\\|_2^4\\).", "proof": {"steps": [{"n": 1, "claim": "\\[\n\\nabla g(x) = 0 \\implies \\langle x, \\nabla g(x) \\rangle = 0\n\\implies \\langle x, (zz^\\top-xx^\\top)x \\rangle = 0 \\quad (\\textup{because}~\\nabla g(x) = (M - xx^\\top)x)\n\\implies \\langle x,z \\rangle^2 = \\|x\\|_2^4.\n\\]"}]}}], "target_local_id": "T2", "topic_domain": "Optimization", "topic_subtopic": "Non-convex Geometry", "label": "flawed", "error_step": 4, "error_class6": ["invalid_bound_or_inequality", "logical_gap_or_missing_case", "others"], "error_explanation": "The decisive error is at step 4. Every earlier step is true as printed: step 2 gives only the one-sided estimate <x,grad g(x)> <= <x,grad f(x)>/p + epsilon, which with step 1 yields <x,grad g(x)> <= epsilon and nothing below; step 3's identity is exact and its consequence <x,z>^2 - ||x||^4 <= epsilon is a valid one-sided conclusion. Step 4 then asserts the LOWER bound <x,z>^2 >= ||x||^4 - epsilon, i.e. <x,z>^2 - ||x||^4 >= -epsilon, which does not follow from an upper bound alone -- nothing in the retained chain bounds the quantity from below, so <x,z>^2 could be arbitrarily smaller than ||x||^4. The correct proof carries the two-sided estimate |<x,z>^2 - ||x||^4| <= epsilon throughout, whose lower half is exactly what step 4 needs."}
{"id": "ch8_theorem_8_3__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nUnderdetermined least-squares setup. In the underdetermined least-squares setting, let \\(X := [x^{(1)}, \\ldots, x^{(n)}]^\\top \\in \\mathbb{R}^{n \\times d}\\) be the design matrix and let \\(\\vec{y} := [y^{(1)}, \\ldots, y^{(n)}]^\\top \\in \\mathbb{R}^n\\) be the response vector. The regime is \\(n < d\\), and \\(X\\) is assumed to have full row rank. The rows of \\(X\\) are denoted \\(x_1, \\ldots, x_n\\), so \\(\\zeta \\perp x_1, \\ldots, x_n\\) is equivalent to \\(X\\zeta = 0\\). The empirical squared loss is \\(\\hat L(\\beta) := \\frac{1}{2}\\|\\vec{y} - X\\beta\\|_2^2\\).\n\nEmpirical least-squares loss. In the underdetermined least-squares setting of Chapter 8, let \\(X \\in \\mathbb{R}^{n \\times d}\\) have rows \\(x^{(1)}, \\dots, x^{(n)}\\) and let \\(\\vec{y} \\in \\mathbb{R}^n\\) be the response vector. Define the empirical least-squares loss by \\(\\widehat{L}(\\beta) = \\tfrac12\\|X\\beta - \\vec{y}\\|_2^2\\). Its gradient is \\(\\nabla \\widehat{L}(\\beta) = X^\\top(X\\beta - \\vec{y})\\), and \\(\\widehat{L}(\\beta) = 0\\) if and only if \\(X\\beta = \\vec{y}\\).\n\nOverparametrized linear regression setup. Let \\(X \\defeq [x^{(1)}, \\dots, x^{(n)}]^\\top \\in \\mathbb{R}^{n \\times d}\\) denote the data matrix, where \\(n < d\\) and \\(X\\) is full rank, and let \\(\\vec{y} \\defeq [y^{(1)}, \\dots, y^{(n)}]^\\top \\in \\mathbb{R}^n\\) denote the label vector. Define the empirical loss \\(\\widehat{L}(\\beta) \\defeq \\frac{1}{2}\\|\\vec{y}-X\\beta\\|_2^2\\), and define the minimum-norm interpolant \\(\\beta^\\star \\defeq \\argmin_{\\beta:\\widehat{L}(\\beta)=0}\\|\\beta\\|_2\\).\n\nGradient descent. Suppose we learn \\(\\beta\\) using gradient descent with initialization \\(\\beta^0\\), where at iteration \\(t\\) we set \\(\\beta^t=\\beta^{t-1}-\\eta\\nabla\\widehat{L}(\\beta^{t-1})\\) for some learning rate \\(\\eta\\).", "theorems": [{"local_id": "T1", "statement": "Suppose gradient descent on \\(\\widehat{L}(\\beta)\\) with initialization \\(\\beta^0 = 0\\) converges to a solution \\(\\hat{\\beta}\\) such that \\(\\widehat{L}(\\hat{\\beta}) = 0\\). Then \\(\\hat{\\beta} = \\beta^\\star\\).", "proof": {"steps": [{"n": 1, "claim": "\\(\\beta^0=0\\in\\operatorname{span}\\{x^{(1)},\\dots,x^{(n)}\\}\\)."}, {"n": 2, "claim": "If \\(\\beta^{t-1}\\in\\operatorname{span}\\{x^{(1)},\\dots,x^{(n)}\\}\\), then \\(\\beta^t\\in\\operatorname{span}\\{x^{(1)},\\dots,x^{(n)}\\}\\)."}, {"n": 3, "claim": "Hence \\(\\hat\\beta\\in\\operatorname{span}\\{x^{(1)},\\dots,x^{(n)}\\}\\), so \\(\\hat\\beta=X^\\top v\\) for some \\(v\\in\\mathbb{R}^n\\)."}, {"n": 4, "claim": "From \\(\\widehat{L}(\\hat\\beta)=0\\) we get \\(0=X\\hat\\beta-\\vec y=XX^\\top v-\\vec y\\), so \\(XX^\\top v < \\vec y\\) componentwise, giving \\(v=(XX^\\top)^{-1}\\vec y\\)."}, {"n": 5, "claim": "Therefore \\(\\hat\\beta=X^\\top v=X^\\top(XX^\\top)^{-1}\\vec y=X^+\\vec y=\\beta^\\star\\)."}]}}, {"local_id": "T2", "statement": "\\(\\beta^\\star = X^+\\vec{y}\\).", "proof": {"steps": [{"n": 1, "claim": "Take any \\(\\beta\\) such that \\(\\widehat{L}(\\beta) = 0\\), and write \\(\\beta = X^+\\vec{y} + \\zeta\\)."}, {"n": 2, "claim": "Then from the definition of \\(X^+\\) and the fact that \\(X\\zeta = 0\\) (see the proof of the referenced lemma), we have\n\\begin{align} \\|\\beta\\|_2^2 &= \\|X^+\\vec{y}\\|_2^2 + \\|\\zeta\\|_2^2 + 2\\langle X^+\\vec{y},\\, \\zeta \\rangle \\\\ &= \\|X^+\\vec{y}\\|_2^2 + \\|\\zeta\\|_2^2 + 2\\langle X^\\top(XX^\\top)^{-1}\\vec{y},\\, \\zeta \\rangle \\\\ &= \\|X^+\\vec{y}\\|_2^2 + \\|\\zeta\\|_2^2 + 2\\langle (XX^\\top)^{-1}y,\\, X\\zeta \\rangle \\\\ &= \\|X^+\\vec{y}\\|_2^2 + \\|\\zeta\\|_2^2 & \\text{(because }X\\zeta = 0\\text{)} \\\\ &\\geq \\|X^+\\vec{y}\\|_2^2, \\end{align}\nwith equality if and only if \\(\\zeta = 0\\)."}]}}, {"local_id": "T3", "statement": "Let \\(X^+\\) denote the pseudoinverse of \\(X\\) (since \\(X\\) is full rank, \\(XX^\\top\\) is invertible and \\(X^+ = X^\\top(XX^\\top)^{-1}\\); note that \\(XX^+X = X\\)). Then \\(\\beta\\) is a global minimizer if and only if \\(\\beta = X^+\\vec{y} + \\zeta\\) for some \\(\\zeta\\) such that \\(\\zeta \\perp x_1, \\dots, x_n\\).", "proof": {"steps": [{"n": 1, "claim": "For any \\(\\beta \\in \\mathbb{R}^d\\), we can decompose it as \\(\\beta = X^+\\vec{y} + \\zeta\\) for some \\(\\zeta \\in \\mathbb{R}^d\\). Since\n\\[ X\\beta = X(X^+\\vec{y} + \\zeta) = \\vec{y} + X\\zeta, \\]\n\\(\\beta\\) is a global minimizer if and only if \\(X\\zeta = 0\\), which happens if and only if \\(\\zeta \\perp x_1, \\dots, x_n\\)."}]}}], "target_local_id": "T1", "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "label": "flawed", "error_step": 4, "error_class6": ["invalid_bound_or_inequality", "computation_or_constant", "logical_gap_or_missing_case"], "error_explanation": "The inserted relational claim \\(XX^\\top v<\\vec y\\) is false: the zero-loss condition gives the exact equation \\(XX^\\top v=\\vec y\\), not a strict inequality. A strict inequality contradicts the very equality used to solve for \\(v\\)."}
{"id": "ch8_theorem_8_9__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nDefinition 6.11. Consider a function class \\(\\mathcal{G} = \\{\\phi(w, Z) : w \\in \\Omega\\}\\), and let \\(h\\) be an offset function of the following form\n\\[\nh(w,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{i=1}^{n}h(w,Z_{i}),\\quad h(w,z)=h_{0}(w)+h_{1}(w,z).\\tag{6.3}\n\\]\nGiven \\( S_{n}=\\{Z_{1},\\ldots,Z_{n}\\} \\), the (one-sided) empirical Rademacher complexity of \\( \\mathcal{G} \\) with offset \\( h \\) decomposition (6.3) is defined as\n\\[\nR^{h}(\\mathcal{G},\\mathcal{S}_{n})=\\mathbb{E}_{\\sigma}\\sup_{w\\in\\Omega}\\left[\\frac{1}{n}\\sum_{i=1}^{n}\\sigma_{i}[\\phi(w,Z_{i})+0. 5h_{1}(w,Z_{i})]-0. 5h(w,\\mathcal{S}_{n})\\right],\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}^{h}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R^{h}(\\mathcal{G},\\mathcal{S}_{n}).\n\\]\nWe note that the standard Rademacher complexity can be regarded as a special case of the offset Rademacher complexity with \\( h(\\cdot) = h_0(\\cdot) = h_1(\\cdot) = 0 \\):\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n})=R^{0}(\\mathcal{G},\\mathcal{S}_{n}),\\quad R_{n}(\\mathcal{G},\\mathcal{D})=R_{n}^{0}(\\mathcal{G},\\mathcal{D}).\n\\]\nIt should be pointed out that the decomposition of \\( h \\) in (6.3) may not be unique, and the offset Rademacher complexity relies on the specific decomposition used.\nAs shown in Example 6.10, we allow distribution dependent offset in the definition of uniform convergence complexity. We also allow distribution dependent offset in the definition of offset Rademacher complexity.\nThe following result shows that for some cases, offset Rademacher complexity can be obtained easily for some function classes.\n\nDefinition 8.1. Consider a loss function \\(\\phi(f, z): \\mathcal{F} \\times \\mathcal{Z} \\to \\mathbb{R}\\), and a model family \\(\\{\\mathcal{A}(\\theta, S_n): \\Theta \\times \\mathcal{Z}^n \\to \\mathcal{F}, n \\geq 0\\}\\). Consider \\(N \\geq n \\geq 0\\), and iid dataset \\(S_n \\subset S_N \\sim \\mathcal{D}^N\\). A model selection algorithm \\(\\bar{\\mathcal{A}}\\) maps \\(S_N\\) to \\(\\hat{\\theta} = \\hat{\\theta}(S_N) \\in \\Theta\\),\nand then train a model \\(\\hat{f} = \\mathcal{A}(\\hat{\\theta}(\\mathcal{S}_N), \\mathcal{S}_n) = \\bar{\\mathcal{A}}(\\mathcal{S}_N)\\). It satisfies an \\(\\epsilon_{n,N}(\\cdot, \\cdot)\\) oracle inequality if there exists \\(\\epsilon_{n,N}(\\theta, \\delta)\\), such that for all \\(\\delta \\in (0,1)\\), with probability at least \\(1 - \\delta\\) over \\(\\mathcal{S}_N\\):\n\\[\n\\phi(\\mathcal{A}(\\hat{\\theta}(\\mathcal{S}_{N}),\\mathcal{S}_{n}),\\mathcal{D})\\leq\\inf_{\\theta\\in\\Theta}[\\mathbb{E}_{\\mathcal{S}_{n}}\\phi(\\mathcal{A}(\\theta,\\mathcal{S}_{n}),\\mathcal{D})+\\epsilon_{n,N}(\\theta,\\delta)].\n\\]\nMore generally, a learning algorithm \\(\\bar{\\mathcal{A}}: S_N \\to \\mathcal{F}\\) is \\(\\epsilon_{n,N}(\\cdot, \\cdot)\\) adaptive to the model family \\(\\{\\mathcal{A}(\\theta, \\cdot): \\theta \\in \\Theta\\}\\) if there exists \\(\\epsilon_{n,N}(\\theta, \\delta)\\), such that for all \\(\\delta \\in (0,1)\\), with probability at least \\(1 - \\delta\\) over \\(S_N\\):\n\\[\n\\phi(\\bar{\\mathcal{A}}(\\mathcal{S}_{N}),\\mathcal{D})\\leq\\inf_{\\theta\\in\\Theta}\\left[\\mathbb{E}_{\\mathcal{S}_{n}}\\phi(\\mathcal{A}(\\theta,\\mathcal{S}_{n}),\\mathcal{D})+\\epsilon_{n,N}(\\theta,\\delta)\\right].\n\\]\nNote that we have used a slightly different notation \\(\\phi(f, Z)\\), instead of \\(\\phi(w, Z)\\) as in earlier chapters, to emphasize that in model selection, the prediction function \\(f(w, \\cdot)\\) from different models can be compared, but the parameter \\(w\\) of the functions may not be comparable when different models have different functional forms such as decision trees versus neural networks. Without causing confusion, one may consider \\(f\\) to be the function represented by \\(w\\), so that \\(\\phi(f, Z) = L(f(X), Y)\\) and \\(\\phi(w, Z) = L(f(w, X), Y)\\). Similar to Definition 3.8, we use the simplified notation\n\\[\n\\phi(f,\\mathcal{D})=\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(f,Z),\\quad\\phi(f,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{Z\\in\\mathcal{S}_{n}}\\phi(f,Z).\n\\]\nThe concept adaptivity is more general than model selection, as the algorithm does not need to choose a specific model \\(\\mathcal{A}(\\theta,\\cdot)\\) indexed by \\(\\theta\\). For example, a commonly used method to achieve adaptivity is model averaging, where we take the average of several different model outputs \\(\\hat{f}_{\\theta_j} = \\mathcal{A}(\\theta_j,\\mathcal{S}_n)\\) as\n\\[\n\\hat{f}_{\\mathrm{avg}}(x)=\\frac{1}{k}\\sum_{j=1}^{k}\\hat{f}_{\\theta_{j}}(x).\n\\]\nIf different models have the same functional form, with a vector model parameter \\( w \\in \\mathbb{R}^d \\): \\( \\hat{f}_{\\theta_j}(x) = f(\\hat{w}_j, x) \\). Then one may also average the model parameters as follows:\n\\[\n\\hat{w}_{\\mathrm{avg}}=\\frac{1}{k}\\sum_{j=1}^{k}\\hat{w}_{j}.\n\\]\nIn general, the training data \\( S_{n} \\) is a subset of \\( S_{N} \\), and we allow \\( n < N \\) in order to analyze validation methods which are commonly used in practice. In addition to the large probability bounds in Definition 8.1, one may also derive bounds that hold for expectation over \\( S_{N} \\).\n\nExample 2.14. Consider a bounded random variable: \\( X_{1} \\in [\\alpha, \\beta] \\). Then \\( X_{1} \\) is sub-Gaussian with \\( b = (\\beta - \\alpha)^{2}/4 \\).\nThe tail probability inequality of Theorem 2.12 can also be expressed in a different form. Consider \\( \\delta \\in (0,1) \\) such that \\( \\delta = \\exp(-t) \\), we have \\( t = \\ln(1/\\delta) \\). This means that we can alternatively express the first bound of Theorem 2.12 as follows. With probability at least \\( 1 - \\delta \\), we have\n\\[\n\\bar{X}_{n}<\\mu+\\sqrt{\\frac{2 b\\ln(1/\\delta)}{n}}.\n\\]\nThis form is often preferred in the theoretical analysis of machine learning algorithms.", "theorems": [{"local_id": "T1", "statement": "Consider \\( h \\) with decomposition (6.3) as \\( h(f,z) = h_0(f) + h_1(f,z) \\). Assume that for some \\( M \\geq 0 \\):\n\\[\n\\sup_{f\\in\\mathcal{F}}\\sup_{z,z^{\\prime}}\\left[|\\phi(f,z)-\\phi(f,z^{\\prime})| + |h_{1}(f,z)-h_{1}(f,z^{\\prime})|\\right]\\leq M.\n\\]\nThen with probability at least \\(1 - \\delta\\), for all \\(f \\in \\mathcal{F}\\):\n\\[\n\\phi(f,\\mathcal{D})\\leq\\phi(f,\\mathcal{S}_{n})+h(f,\\mathcal{S}_{n})+2 R^{h}(\\mathcal{F},\\mathcal{S}_{n})+3 M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "With probability at least \\( 1 - \\delta / 2 \\), Corollary 6.19 holds."}, {"n": 2, "claim": "With probability at least \\( 1 - \\delta / 2 \\), Theorem 8.9 holds."}, {"n": 3, "claim": "The desired inequality follows by taking the union bound of the two events."}]}}, {"local_id": "T2", "statement": "Consider \\( n \\) independent random variables \\( X_{1},\\ldots,X_{n} \\), and a real-valued function \\( f(X_{1},\\ldots,X_{n}) \\) that satisfies the following inequality\n\\[\n\\sup_{x_{1},\\dots,x_{n},x_{i}^{\\prime}}|f(x_{1},\\dots,x_{n})-f(x_{1},\\dots,x_{i-1},x_{i}^{\\prime},x_{i+1},\\dots,x_{n})|\\leq c_{i}\n\\]\nfor all \\(1 \\leq i \\leq n\\). Then for all \\(\\epsilon > 0\\):\n\\[\n\\Pr\\left[f\\left(X_{1},\\dots,X_{n}\\right)\\geq\\mathbb{E}f\\left(X_{1},\\dots,X_{n}\\right)+\\epsilon\\right]\\leq\\exp\\left(\\frac{-2\\epsilon^{2}}{\\sum_{i=1}^{n}c_{i}^{2}}\\right).\n\\]\nSimilarly:\n\\[\n\\Pr\\left[f\\left(X_{1},\\dots,X_{n}\\right)\\leq\\mathbb{E}f\\left(X_{1},\\dots,X_{n}\\right)-\\epsilon\\right]\\leq\\exp\\left(\\frac{-2\\epsilon^{2}}{\\sum_{i=1}^{n}c_{i}^{2}}\\right).\n\\]", "proof": {"steps": [{"n": 1, "claim": "Let \\( X_{k}^{l}=\\{X_{k},\\ldots,X_{l}\\} \\). Consider \\( X_{1}^{n} \\), and for some \\( 1\\leq k\\leq n \\), we use the simplified notation \\( \\tilde{X}_1^n=\\{X_1,\\dots,X_{k-1},\\tilde{X}_k,X_{k+1},X_n\\} \\)."}, {"n": 2, "claim": "Then we have\n\\[\n\\left|\\mathbb{E}_{X_{k+1}^{n}}f\\left(X_{1}^{n}\\right)-\\mathbb{E}_{X_{k+1}^{n}}f\\left(\\tilde{X}_{1}^{n}\\right)\\right|\\leq c_{k}.\n\\]\nWe now consider \\(\\mathbb{E}_{X_{k + 1}^n}f(X_1^n)\\) as a random variable depending on \\(X_{k}\\), conditioned on \\(X_{1}^{k - 1}\\)."}, {"n": 3, "claim": "It follows from derivation of the Chernoff bound that we have the following logarithmic moment generating function estimate (see Example 2.14):"}, {"n": 4, "claim": "\\[\n\\ln\\mathbb{E}_{X_{k}}\\exp\\left[\\lambda\\mathbb{E}_{X_{k+1}^{n}}f\\left(X_{1}^{n}\\right)\\right]\\leq\\lambda\\mathbb{E}_{X_{k}^{n}}f\\left(X_{1}^{n}\\right)+\\lambda^{2}c_{k}^{2}/8.\n\\]\nNow we may exponentiate the above inequality, and take expectation with respect to \\( X_1^{k-1} \\) to obtain\n\\[\n\\mathbb{E}_{X_{1}^{k}}\\exp [\\lambda\\mathbb{E}_{X_{k+1}^{n}}f(X_{1}^{n})]\\leq\\mathbb{E}_{X_{1}^{k-1}}\\exp [\\lambda\\mathbb{E}_{X_{k}^{n}}f(X_{1}^{n})+\\lambda^{2}c_{k}^{2}/8].\n\\]\nBy taking logarithm, we obtain\n\\[\n\\ln\\mathbb{E}_{X_{1}^{k}}\\exp\\left[\\lambda\\mathbb{E}_{X_{k+1}^{n}}f\\left(X_{1}^{n}\\right)\\right]\\leq\\ln\\mathbb{E}_{X_{1}^{k-1}}\\exp\\left[\\lambda\\mathbb{E}_{X_{k}^{n}}f\\left(X_{1}^{n}\\right)\\right]+\\lambda^{2}c_{k}^{2}/8.\n\\]\nBy summing from \\( k = 1 \\) to \\( k = n \\), and canceling redundant terms, we obtain\n\\[\n\\ln\\mathbb{E}_{X_{1}^{n}}\\exp\\left[\\lambda f\\left(X_{1}^{n}\\right)\\right]\\leq\\lambda\\mathbb{E}_{X_{1}^{n}}f\\left(X_{1}^{n}\\right)+\\lambda^{2}\\sum_{k=1}^{n}c_{k}^{2}/8.\\tag{6.4}\n\\]\nLet\n\\[\n\\delta=\\Pr\\left[f\\left(X_{1}^{n}\\right)\\geq\\mathbb{E}_{X_{1}^{n}}f\\left(X_{1}^{n}\\right)+\\epsilon\\right].\n\\]\nUsing Markov's inequality, we have for all positive \\(\\lambda\\)\n\\[\n\\delta\\leq e^{-\\lambda\\left(\\mathbb{E}_{X_{1}^{n}}f\\left(X_{1}^{n}\\right)+\\epsilon\\right)}\\mathbb{E}_{X_{1}^{n}}e^{\\lambda f\\left(X_{1}^{n}\\right)}\\leq\\exp\\left[-\\lambda\\epsilon+\\frac{\\lambda^{2}}{8}\\sum_{k=1}^{n}c_{k}^{2}\\right].\n\\]\nSince \\(\\lambda > 0\\) is arbitrary, we conclude that\n\\[\n\\ln\\delta\\leq\\inf_{\\lambda\\geq 0}\\left[\\frac{\\lambda^{2}}{8}\\sum_{k=1}^{n}c_{k}^{2}-\\lambda\\epsilon\\right]= -\\frac{2\\epsilon^{2}}{\\sum_{k=1}^{n}c_{k}^{2}}.\n\\]\nThis implies the theorem."}]}}, {"local_id": "T3", "statement": "Consider \\( h \\) with decomposition 6.3 as \\( h(f,z) = h_0(f) + h_1(f,z) \\). Assume that for some \\( M \\geq 0 \\):\n\\[\n\\sup_{f\\in\\mathcal{F}}\\sup_{z,z^{\\prime}}[|\\phi(f,z)-\\phi(f,z^{\\prime})| + |h_{1}(f,z)-h_{1}(f,z^{\\prime})|]\\leq M.\n\\]\nWe have with probability at least \\(1 - \\delta\\)\n\\[\nR_{n}^{h}(\\mathcal{F},\\mathcal{D})\\leq R^{h}(\\mathcal{F},\\mathcal{S}_{n})+M\\sqrt{\\frac{\\ln(1/\\delta)}{2 n}}.\n\\]\nSimilarly, with probability at least \\(1 - \\delta\\), we have\n\\[\nR_{n}^{h}(\\mathcal{F},\\mathcal{D})\\geq R^{h}(\\mathcal{F},\\mathcal{S}_{n})-M\\sqrt{\\frac{\\ln(1/\\delta)}{2 n}}.\n\\]", "proof": {"steps": [{"n": 1, "claim": "\\(S_n\\) and \\(S_n'\\) are identical except at index \\(k\\)."}, {"n": 2, "claim": "\\(\\hat f(\\sigma,\\mathcal{S}_n)=\\arg\\max_f\\big[\\tfrac1n\\sum_i\\sigma_i(\\phi(f,Z_i)+0.5h_1(f,Z_i))-0.5h(f,\\mathcal{S}_n)\\big]\\), with \\(\\hat f,\\hat f'\\) the maximizers on \\(\\mathcal{S}_n,\\mathcal{S}_n'\\)."}, {"n": 3, "claim": "\\(\\phi'(f,Z)=\\phi(f,Z)+0.5h_1(f,Z)\\)."}, {"n": 4, "claim": "\\(\\tfrac1n\\sum_i\\sigma_i\\phi'(\\hat f,Z_i')-0.5h(\\hat f,\\mathcal{S}_n')\\le\\tfrac1n\\sum_i\\sigma_i\\phi'(\\hat f',Z_i')-0.5h(\\hat f',\\mathcal{S}_n')\\). (8.4)"}, {"n": 5, "claim": "\\(R^h(\\mathcal{F},\\mathcal{S}_n)-R^h(\\mathcal{F},\\mathcal{S}_n')\\le\\mathbb{E}_\\sigma\\big[\\tfrac1n\\sigma_k\\phi'(\\hat f,Z_k)-\\tfrac1n\\sigma_k\\phi'(\\hat f,Z_k')+\\tfrac1{2n}h_1(\\hat f,Z_k')-\\tfrac1{2n}h_1(\\hat f,Z_k)\\big]\\le M/n\\)."}, {"n": 6, "claim": "Therefore \\(|R^h(\\mathcal{F},\\mathcal{S}_n)-R^h(\\mathcal{F},\\mathcal{S}_n')|\\le M/n\\)."}, {"n": 7, "claim": "McDiarmid with \\(c_i=M/n\\) gives \\(\\Pr[\\,\\cdot\\ge t\\,]\\le\\exp(-2t^2 n/M^2)\\)."}, {"n": 8, "claim": "Setting the bound to \\(\\delta\\) yields \\(t=M\\sqrt{\\ln(1/\\delta)/(2n)}\\)."}, {"n": 9, "claim": "Hence both \\(R_n^h\\le R^h(\\mathcal{F},\\mathcal{S}_n)+M\\sqrt{\\ln(1/\\delta)/(2n)}\\) and \\(R_n^h\\ge R^h(\\mathcal{F},\\mathcal{S}_n)-M\\sqrt{\\ln(1/\\delta)/(2n)}\\) hold, each w.p. \\(1-\\delta\\)."}, {"n": 10, "claim": "This completes the proof."}]}}], "target_local_id": "T3", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "label": "flawed", "error_step": 6, "error_class6": ["logical_gap_or_missing_case", "invalid_bound_or_inequality", "circular_or_unproven"], "error_explanation": "The absolute-value bound is asserted from only one direction. Step 5 establishes only $R^h(\\mathcal{F},\\mathcal{S}_n)-R^h(\\mathcal{F},\\mathcal{S}_n')\\le M/n$; bounding $|R^h(\\mathcal{F},\\mathcal{S}_n)-R^h(\\mathcal{F},\\mathcal{S}_n')|$ also requires the symmetric inequality $R^h(\\mathcal{F},\\mathcal{S}_n')-R^h(\\mathcal{F},\\mathcal{S}_n)\\le M/n$, which the gold proof obtains by exchanging the roles of $\\mathcal{S}_n$ and $\\mathcal{S}_n'$. That branch is silently dropped, so the two-sided bound does not follow as stated."}
{"id": "ch9_lemma_9_9__perturbation", "context": "Shared setting for the results below. All definitions, standing assumptions and numbered equations that the proofs rely on are collected here.\n\nContrastive affinity setup. Let \\(X = \\{x_1,\\ldots,x_N\\}\\) be a finite domain. Let \\(p(x,z)\\) be the joint distribution of positive pairs on \\(X\\times X\\), and write its marginal as \\(p(x)=\\sum_{z\\in X} p(x,z)\\). A positive pair \\((x,x^+)\\) is sampled from this joint distribution \\(p(x,z)\\), while a random pair \\((x,x')\\) is sampled independently from the product distribution \\(p(x)p(z)\\). The normalized affinity matrix \\(\\bar{A}\\in\\mathbb{R}^{N\\times N}\\) is defined entrywise by \\(\\bar{A}_{xz}=p(x,z)/(\\sqrt{p(x)}\\sqrt{p(z)})\\).\n\nStochastic block model. Consider the stochastic block model on the vertex set \\([n] = \\{1, \\ldots, n\\}\\). Partition \\([n] = S \\cup \\bar{S}\\) with \\(|S| = n/2\\). For vertices \\(i,j\\) in the same community, meaning either both are in \\(S\\) or both are in \\(\\bar{S}\\), sample \\(G_{ij} \\sim \\operatorname{Bernoulli}(p)\\); for vertices \\(i,j\\) in distinct communities, sample \\(G_{ij} \\sim \\operatorname{Bernoulli}(q)\\), with \\(p > q\\). Let \\(\\bar{G} = \\mathbb{E}[G]\\). Then \\(\\bar{G}_{ij} = p\\) when \\(i,j\\) are in the same community, and \\(\\bar{G}_{ij} = q\\) otherwise.", "theorems": [{"local_id": "T1", "statement": "Let \\(\\bar{G} = \\mathbb{E}[G]\\) for the stochastic block model. Then, letting \\(u_i(A)\\) denote the \\(i\\)-th eigenvector of the matrix \\(A\\),\n\\[\nu_1(\\bar{G}) = \\vec{1},\n\\]\n\\[\nu_2(\\bar{G}) = [\\underbrace{1, \\dots, 1}_{|S|}, \\underbrace{-1, \\dots, -1}_{|\\bar{S}|}]^\\top,\n\\]\nwhere \\(u_2(\\bar{G})\\) has \\(|S|\\) entries of \\(1\\) and \\(|\\bar{S}|\\) entries of \\(-1\\).", "proof": {"steps": [{"n": 1, "claim": "We begin by directly proving that \\(u_1(\\bar{G}) = \\vec{1}\\).\n\\[\n\\bar{G} \\cdot \\vec{1} = \\begin{bmatrix} \\frac{pn}{2} + \\frac{qn}{2} \\\\ \\vdots \\\\ \\frac{pn}{2} + \\frac{qn}{2} \\end{bmatrix} = \\frac{p+q}{2} \\cdot n \\cdot \\vec{1}.\n\\]\nMore generally, \\(\\vec{1}\\) is the top eigenvector for any matrix with fixed row sum or any graph with uniform degree (i.e., a regular graph)."}, {"n": 2, "claim": "Next, we prove that \\(u_2(\\bar{G})\\) has the stated form."}, {"n": 3, "claim": "Let\n\\[\nG' = \\left[\\begin{array}{c|c} r \\cdots r & \\\\ \\vdots & \\text{\\huge 0} \\\\ r \\cdots r & \\\\ \\hline & r \\cdots r \\\\ \\text{\\huge 0} & \\vdots \\\\ & r \\cdots r \\end{array}\\right]\n\\]\nfor \\(r = p - q\\)."}, {"n": 4, "claim": "To precisely define \\(G'\\), note that \\(G'\\) is block diagonal with two blocks of size \\(|S|\\) and \\(|\\bar{S}|\\), respectively."}, {"n": 5, "claim": "Then,\n\\[\n\\bar{G} = \\vec{1}\\vec{1}^\\top q + G'.\n\\]\nLet \\(u = [\\underbrace{1,\\dots,1}_{|S|}, \\underbrace{-1,\\dots,-1}_{|\\bar{S}|}]^\\top\\)."}, {"n": 6, "claim": "Then,\n\\[\nG' \\cdot u = r \\cdot \\frac{n}{2} \\cdot u.\n\\]\nBecause \\(u \\perp \\vec{1}\\), we obtain\n\\[\n\\bar{G} \\cdot u = G' \\cdot u = r \\cdot \\frac{n}{2} \\cdot u = \\frac{p-q}{2} \\cdot n \\cdot u.\n\\]\nThus, \\(u\\) has eigenvalue \\(\\frac{p-q}{2} \\cdot n\\)."}]}}, {"local_id": "T2", "statement": "Let \\(L(f) = -2\\,\\mathbb{E}_{(x,x^+)}\\, f(x)^\\top f(x^+) + \\mathbb{E}_{(x,x')}\\left(f(x)^\\top f(x')\\right)^2\\). Suppose \\(X = \\{x_1,\\ldots,x_N\\}\\) and let the matrix\n\\[\nF = \\begin{bmatrix} p(x_1)^{1/2}\\, f(x_1)^\\top \\\\ \\vdots \\\\ p(x_N)^{1/2}\\, f(x_N)^\\top \\end{bmatrix}.\n\\]\nThen,\n\\[\nL(f) = \\|\\bar{A} - FF^\\top\\|_F^2 + \\text{const}.\n\\]\nHence, minimizing \\(L(f)\\) with respect to \\(f\\) is equivalent to eigendecomposition of \\(\\bar{A}\\).", "proof": {"steps": [{"n": 1, "claim": "\\(\\|\\bar{A}-FF^\\top\\|_F^2 = \\sum_{x,z}\\left(\\frac{p(x,z)}{\\sqrt{p(x)p(z)}} - f(x)^\\top f(z)\\sqrt{p(x)p(z)}\\right)^2\\)."}, {"n": 2, "claim": "\\(= \\text{const} - 2\\sum_{x,z}p(x,z)f(x)^\\top f(z) + \\sum_{x,z}p(x)p(z)(f(x)^\\top f(z))^2\\)."}, {"n": 3, "claim": "\\(= \\text{const} - 2\\mathbb{E}_{(x,x^+)}f(x)^\\top f(x^+) + \\mathbb{E}_{(x,x')}(f(x)^\\top f(x'))^2 = L(f)+\\text{const}.\\)"}, {"n": 4, "claim": "Thus \\(L(f)=\\|\\bar{A}-FF^\\top\\|_F^2+\\text{const}\\), so minimizing \\(L\\) minimizes \\(\\|\\bar{A}-FF^\\top\\|_F^2\\)."}, {"n": 5, "claim": "By the Eckart–Young–Mirsky theorem, the rank-\\(k\\) matrix \\(M\\) minimizing \\(\\|\\bar{A}-M\\|_F^2\\) is \\(M^\\star = \\sum_{i=1}^k \\sigma_i u_i v_i^\\top\\), the truncated SVD of \\(\\bar{A}\\)."}, {"n": 6, "claim": "Since \\(M^\\star = FF^\\top\\) at the optimum, the optimal \\(F\\) is given by the top-\\(k\\) eigenvectors of \\(\\bar{A}\\), i.e. minimizing \\(L\\) is eigendecomposition of \\(\\bar{A}\\)."}, {"n": 7, "claim": "This establishes the stated equivalence."}]}}, {"local_id": "T3", "statement": "Suppose \\(z \\sim \\mathcal{N}(v, I)\\). Then,\n\\[\n\\mathbb{E}[z \\otimes z \\otimes z] = v \\otimes v \\otimes v + \\sum_{l=1}^d \\mathbb{E}[z] \\otimes e_l \\otimes e_l + \\sum_{l=1}^d e_l \\otimes \\mathbb{E}[z] \\otimes e_l + \\sum_{l=1}^d e_l \\otimes e_l \\otimes \\mathbb{E}[z],\n\\]\nwhere \\(e_1,\\dots,e_d\\) denote the canonical basis vectors.", "proof": {"steps": [{"n": 1, "claim": "We compute the third moment element-wise. That is,\n\\[\n(\\mathbb{E}[z \\otimes z \\otimes z])_{ijk} = \\mathbb{E}[z_i z_j z_k] = \\mathbb{E}[(v_i + \\xi_i)(v_j + \\xi_j)(v_k + \\xi_k)] \\quad (z = v + \\xi,\\ \\xi \\sim \\mathcal{N}(0, I))\n\\]\n\\[\n= v_i v_j v_k + \\underbrace{\\mathbb{E}[v_i v_j \\xi_k] + \\mathbb{E}[v_i \\xi_j v_k] + \\mathbb{E}[\\xi_i v_j v_k]}_{=0} + \\mathbb{E}[v_i \\xi_j \\xi_k] + \\mathbb{E}[v_j \\xi_i \\xi_k] + \\mathbb{E}[v_k \\xi_i \\xi_j] + \\mathbb{E}[\\xi_i \\xi_j \\xi_k].\n\\]\nTo explicitly compute the last four terms, we note that:"}, {"n": 2, "claim": "\\[\n\\mathbb{E}[\\xi_i \\xi_k] = \\begin{cases} 0 & \\text{if } i \\neq k \\\\ \\mathbb{E}[\\xi_i^2] = 1 & \\text{if } i = k, \\end{cases}\n\\]\nand that for any choice of \\(i, j, k\\),\n\\[\n\\mathbb{E}[\\xi_i \\xi_j \\xi_k] = 0.\n\\]\nTherefore,\n\\[\n(\\mathbb{E}[z \\otimes z \\otimes z])_{ijk} = v_i v_j v_k + v_i \\mathbf{1}[j=k] + v_j \\mathbf{1}[i=k] + v_k \\mathbf{1}[i=j].\n\\]\nSince \\(\\left(\\sum_{l=1}^d v \\otimes e_l \\otimes e_l\\right)_{ijk} = \\sum_{l=1}^d v_i e_{lj} e_{lk} = v_i \\mathbb{I}(j=k)\\), we have proven that:"}, {"n": 3, "claim": "\\[\n\\mathbb{E}[z \\otimes z \\otimes z] = v \\otimes v \\otimes v + \\sum_{l=1}^d v \\otimes e_l \\otimes e_l + \\sum_{l=1}^d e_l \\otimes v \\otimes e_l + \\sum_{l=1}^d e_l \\otimes e_l \\otimes v.\n\\]"}]}}], "target_local_id": "T2", "topic_domain": "Deep Learning Theory", "topic_subtopic": "Representation & Expressivity", "label": "flawed", "error_step": 6, "error_class6": ["logical_gap_or_missing_case", "missing_assumption", "others"], "error_explanation": "The first decisive error is Step 6, not Step 5. Step 5 correctly states unconstrained Eckart-Young (truncated SVD minimizes ||Abar-M||_F^2 over arbitrary rank-k M). The error is Step 6's claim that this unconstrained minimizer M* equals FF^T at the optimum: the feasible set is M=FF^T, i.e. symmetric PSD of rank <= k, and when Abar is not PSD the truncated SVD is generally not symmetric PSD and thus not realizable as FF^T. The correct constrained minimizer keeps only the positive-eigenvalue part of the eigendecomposition of Abar, so equating the raw SVD factors with FF^T (and hence with 'the top-k eigenvectors of Abar') is invalid."}
{"id": "H1xaJn05FQ", "context": "Setting (Sliced-Wasserstein Auto-Encoders). Let $X$ be a compact domain of a manifold in Euclidean space, and let $\\rho_X$ be a Borel probability measure on $X$ with density $p_X$, i.e. $d\\rho_X(x)=p_X(x)\\,dx$. Let $\\phi:X\\to Z$ be a deterministic (parametric) encoder and $\\psi:Z\\to X$ a deterministic decoder. Using Random Variable Transformation, the pushforward $\\rho_Z=\\phi_*(\\rho_X)$ has density $p_Z(z)=\\int_X p_X(x)\\,\\delta(z-\\phi(x))\\,dx$ (Eq. 2) and the decoded (auto-encoded) distribution has density $p_Y(y)=\\int_X p_X(x)\\,\\delta(y-\\psi(\\phi(x)))\\,dx$ (Eq. 3); when $\\psi=\\phi^{-1}$ one has $\\psi(\\phi(\\cdot))=\\mathrm{id}$ and $p_Y=p_X$. Let $c:X\\times Y\\to\\mathbb{R}^+$ be a transportation cost. The Kantorovich Wasserstein distance is $W_c(p_X,p_Y)=\\inf_{\\gamma\\in\\Gamma(\\rho_X,\\rho_Y)}\\int_{X\\times Y} c(x,y)\\,d\\gamma(x,y)$ (Eq. 4), where $\\Gamma(\\rho_X,\\rho_Y)$ is the set of all transport plans (couplings) with marginals $p_X,p_Y$. When a transport map $f:X\\to Y$ satisfies $p_Y(y)=\\int_X p_X(x)\\,\\delta(y-f(x))\\,dx$ (Eq. 5), the Monge formulation gives $W_c(p_X,p_Y)=\\min_{f\\in MP}\\int_X c(x,f(x))\\,d\\rho_X(x)$ (Eq. 6), the min taken over all admissible maps $MP$. In one dimension, with CDFs $P_X,P_Y$, the Wasserstein distance has the closed form $W_c(p_X,p_Y)=\\int_0^1 c(P_X^{-1}(\\tau),P_Y^{-1}(\\tau))\\,d\\tau$ (Eq. 7). The one-dimensional Radon slices of $p_X$ are $\\mathcal{R}p_X(t;\\theta)=\\int_X p_X(x)\\,\\delta(t-\\theta\\cdot x)\\,dx$, $\\theta\\in\\mathbb{S}^{d-1}$ (Eq. 8), and the sliced-Wasserstein distance is $SW_c(p_X,p_Y)=\\int_{\\mathbb{S}^{d-1}} W_c(\\mathcal{R}p_X(\\cdot;\\theta),\\mathcal{R}p_Y(\\cdot;\\theta))\\,d\\theta = \\mathbb{E}_{\\theta\\sim U(\\mathbb{S}^{d-1})}[\\,W_c(\\text{slices})\\,]$ (Eq. 9), where $d\\theta$ is the uniform probability measure on the unit sphere. For the cost $c(x,y)=\\|x-y\\|_p^p$ with $p\\ge 2$ one writes $\\alpha_{d,p}=\\tfrac1d\\int_{\\mathbb{S}^{d-1}}\\|\\theta\\|_p^p\\,d\\theta$ (Eq. 10). Given i.i.d. samples $\\{x_m\\}_{m=1}^M\\sim p_X$ (resp. $\\{y_m\\}\\sim p_Y$), the empirical measures are $p_{X,M}=\\tfrac1M\\sum_{m=1}^M\\delta_{x_m}$ and $p_{Y,M}=\\tfrac1M\\sum_{m=1}^M\\delta_{y_m}$, with empirical CDF $P_{X,M}(t)=\\tfrac1M\\sum_{m=1}^M u(t-x_m)$ ($u$ the unit step). The $p$-Wasserstein distance is a true metric on measures with finite $p$-th moment.", "theorems": [{"local_id": "T1", "statement": "Let $p_{X,M},p_{Y,M}$ be the empirical measures of $M$ i.i.d. samples drawn from $p_X,p_Y$ respectively. For the $1$-Wasserstein distance, the empirical estimate underestimates the true distance by at most $O(1/\\sqrt M)$ in expectation: $\\mathbb{E}\\big(W_1(p_X,p_Y)-W_1(p_{X,M},p_{Y,M})\\big)\\le \\tfrac{C}{\\sqrt M}$ for an absolute constant $C$.", "proof": {"steps": [{"n": 1, "claim": "$W_1$ is a true metric on probability measures with finite first moment, hence obeys the triangle inequality."}, {"n": 2, "claim": "$W_1(p_X,p_Y)\\le W_1(p_X,p_{X,M})+W_1(p_{X,M},p_{Y,M})+W_1(p_{Y,M},p_Y)$."}, {"n": 3, "claim": "Rearranging, $W_1(p_X,p_Y)-W_1(p_{X,M},p_{Y,M})\\le W_1(p_X,p_{X,M})+W_1(p_Y,p_{Y,M})$."}, {"n": 4, "claim": "Taking expectations and applying the empirical convergence bound $\\mathbb{E}(W_1(p_X,p_{X,M}))\\le \\tfrac{C'}{\\sqrt M}$ (and likewise for $Y$)."}, {"n": 5, "claim": "Therefore $\\mathbb{E}\\big(W_1(p_X,p_Y)-W_1(p_{X,M},p_{Y,M})\\big)\\le \\tfrac{C}{\\sqrt M}$ for an absolute constant $C$."}]}}, {"local_id": "T2", "statement": "For the transportation cost $c(x,y)=\\|x-y\\|_2^p$ with $p\\ge 2$, the sliced-Wasserstein distance is dominated by the Wasserstein distance: $SW_p^p(p_X,p_Y)\\le \\alpha_{d,p}\\,W_p^p(p_X,p_Y)$ with $\\alpha_{d,p}=\\tfrac1d\\int_{\\mathbb{S}^{d-1}}\\|\\theta\\|_p^p\\,d\\theta\\le 1$. In particular, for $p=2$ one has $\\alpha_{2,d}=\\tfrac1d$ and hence $SW_2(p_X,p_Y)\\le \\tfrac{1}{\\sqrt d}\\,W_2(p_X,p_Y)$.", "proof": {"steps": [{"n": 1, "claim": "The sliced-Wasserstein distance is the average, over directions $\\theta\\in\\mathbb{S}^{d-1}$, of the one-dimensional Wasserstein distances between the projected marginals: $SW_c(p_X,p_Y)=\\int_{\\mathbb{S}^{d-1}} W_c(\\mathcal{R}p_X(\\cdot;\\theta),\\mathcal{R}p_Y(\\cdot;\\theta))\\,d\\theta$."}, {"n": 2, "claim": "For $c(x,y)=\\|x-y\\|_2^p$ with $p\\ge 2$, $SW_p^p(p_X,p_Y)\\le \\alpha_{d,p}\\,W_p^p(p_X,p_Y)$ with $\\alpha_{d,p}=\\tfrac1d\\int_{\\mathbb{S}^{d-1}}\\|\\theta\\|_p^p\\,d\\theta$."}, {"n": 3, "claim": "The constant satisfies $\\alpha_{d,p}\\le 1$."}, {"n": 4, "claim": "For $p=2$, $\\alpha_{2,d}=\\tfrac1d$."}, {"n": 5, "claim": "Therefore $SW_2(p_X,p_Y)\\le \\tfrac{1}{\\sqrt d}\\,W_2(p_X,p_Y)$."}]}}, {"local_id": "T3", "statement": "Let $X$ be a compact domain of a manifold in Euclidean space with Borel probability measure $\\rho_X$ and density $p_X$ ($d\\rho_X(x)=p_X(x)dx$). Let $\\phi:X\\to Z$ be a deterministic encoder and $\\psi:Z\\to X$ a deterministic decoder, and let $p_Y(y)=\\int_X p_X(x)\\,\\delta(y-\\psi(\\phi(x)))\\,dx$ be the decoded distribution. Let $c:X\\times Y\\to\\mathbb{R}^+$ be a transportation cost and $W_c(p_X,p_Y)=\\inf_{\\gamma\\in\\Gamma(\\rho_X,\\rho_Y)}\\int_{X\\times Y} c(x,y)\\,d\\gamma(x,y)$ the (Kantorovich) Wasserstein distance. Claim: for deterministic encoder-decoder pairs $(\\phi,\\psi)$, minimizing the Wasserstein distance between $p_X$ and $p_Y$ is EQUIVALENT to minimizing the pointwise reconstruction cost, i.e. $\\inf_{\\phi,\\psi} W_c(p_X,p_Y)=\\inf_{\\phi,\\psi}\\mathbb{E}_{X\\sim p_X}[c(X,\\psi(\\phi(X)))]=\\inf_{\\phi,\\psi}\\int_X c(x,\\psi(\\phi(x)))\\,p_X(x)\\,dx$.", "proof": {"steps": [{"n": 1, "claim": "The encoded distribution is $p_Z(z)=\\int_X p_X(x)\\,\\delta(z-\\phi(x))\\,dx$ (the pushforward $\\rho_Z=\\phi_*(\\rho_X)$), and the decoded distribution is $p_Y(y)=\\int_X p_X(x)\\,\\delta(y-\\psi(\\phi(x)))\\,dx$ (Eq. 2 and Eq. 3)."}, {"n": 2, "claim": "The auto-encoder objective reduces to learning $\\phi,\\psi$ that minimize a dissimilarity measure between $p_Y$ and $p_X$ (together with matching $p_Z$ to the prior $q_Z$)."}, {"n": 3, "claim": "Choose the Wasserstein distance $W_c(p_X,p_Y)=\\inf_{\\gamma\\in\\Gamma(\\rho_X,\\rho_Y)}\\int_{X\\times Y} c(x,y)\\,d\\gamma(x,y)$ as the dissimilarity measure between $p_X$ and $p_Y$ (Kantorovich formulation, Eq. 4)."}, {"n": 4, "claim": "The deterministic composition $f(\\cdot)=\\psi(\\phi(\\cdot))$ is a transport map between $p_X$ and $p_Y$, i.e. $y=f(x)$ pushes $p_X$ to $p_Y$, and evaluating the transport cost along this map gives $\\mathbb{E}_{X\\sim p_X}[c(X,\\psi(\\phi(X)))]=\\int_X c(x,\\psi(\\phi(x)))\\,p_X(x)\\,dx$."}, {"n": 5, "claim": "Therefore, for deterministic encoder-decoder pairs, $\\inf_{\\phi,\\psi} W_c(p_X,p_Y)=\\inf_{\\phi,\\psi}\\mathbb{E}_{X\\sim p_X}[c(X,\\psi(\\phi(X)))]=\\inf_{\\phi,\\psi}\\int_X c(x,\\psi(\\phi(x)))\\,p_X(x)\\,dx$; this equivalence is Theorem 1 of Bousquet et al. (2017) for deterministic encoder-decoder pairs."}, {"n": 6, "claim": "Consequently, one may minimize $W_c(p_X,p_Y)$ over $(\\phi,\\psi)$ by directly minimizing the unconstrained reconstruction objective $\\mathbb{E}_{X\\sim p_X}[c(X,\\psi(\\phi(X)))]$, empirically approximated by $\\frac{1}{N}\\sum_{n=1}^{N} c(x_n,\\psi(\\phi(x_n)))$ (Eq. 11)."}]}}], "target_local_id": "T3", "topic_domain": "Others", "topic_subtopic": "Optimal Transport & Sampling", "label": "flawed", "error_step": 5, "error_class6": ["logical_gap_or_missing_case", "invalid_bound_or_inequality", "circular_or_unproven"], "error_explanation": "The claimed equivalence $\\inf_{\\phi,\\psi} W_c(p_X,p_Y) = \\inf_{\\phi,\\psi}\\mathbb{E}_{X\\sim p_X}[c(X,\\psi(\\phi(X)))]$ is false. Computing $W_c(p_X,p_Y)$ requires an infimum over all transport plans $\\gamma\\in\\Gamma(\\rho_X,\\rho_Y)$ (Kantorovich) or all valid transport maps (Monge). The deterministic composition $f=\\psi\\circ\\phi$ induces only the specific plan $\\gamma(x,y)=\\delta(y-\\psi(\\phi(x)))\\,p_X(x)$, which need not be optimal. Theorem 1 of Bousquet et al. gives $W_c(p_X,\\psi_*p_Z)=\\inf_{f:\\,f_*p_X=p_Z}\\mathbb{E}_{p_X}[c(X,\\psi(f(X)))]$, whose RHS is itself a CONSTRAINED infimum over encoders $f$; fixing $f=\\phi$ drops that infimum and can only increase the value. Hence the reconstruction cost is only an UPPER BOUND, $W_c(p_X,p_Y)\\le\\mathbb{E}_{x\\sim p_X}[c(x,\\psi(\\phi(x)))]$, with equality only in the special case where the induced plan happens to be optimal. Replacing $\\le$ with $=$ (equivalence) is the flaw, admitted by the authors as an oversight."}
{"id": "0f-0I6RFAch", "context": "We introduce our main contribution, namely the Indirection Layer (InLay). InLay takes a sequence of objects as input and transforms the sequence into a new indirect graph-structured representation. Concretely, let $X=(x_{1},x_{2},\\ldots,x_{k})^{\\top}\\in\\mathbb{R}^{k\\times n}$ be the input sequence for InLay, where k is the number of objects and each $x_{i}\\in\\mathbb{R}^{n}$ represents an object. To better exploit data internal relationships, we treat each input sequence as a directed complete weighted graph (with no self-loop) whose vertices represent the objects and edges represent relationships as scalars in [−1, 1]. Specifically, for each sequence X, we denote $G_{X}$ as its corresponding graph. We define $\\mathcal{G}_{k}$ to be the space of all directed weighted complete graphs G with k vertices and edge weights in [−1, 1]. From now on, we will only write G instead of $G_{X}$ when it is not necessary to specify X, and we denote $A_{G}$ as the adjacency matrix of G. This adjacency matrix captures the internal relationships of the corresponding data sequence.\n\nRemark 1. (Canonical indexing assumption) As the set of graph vertices may permute, a graph G with k vertices may not have an unique adjacency matrix. To assure the well-definedness of $A_{G}$ , we assume that (when computing the adjacency matrix) the i-th vertex represents the i-th element of the input sequence. We show in Appendix C that the indirection representations are still maintained if the canonical indexing assumption is not obeyed.\n\nIn indirection, each object is bound to a symbol. We denote by $V^{\\mathrm{ind}}=\\left(v_{1}^{\\mathrm{ind}},v_{2}^{\\mathrm{ind}},\\ldots,v_{k}^{\\mathrm{ind}}\\right)^{\\top}\\in\\mathbb{R}^{k\\times n}$ to be the set of symbols where each $v_{i}^{\\mathrm{ind}}\\in\\mathbb{R}^{n}$ is data-independent and trainable. Let $\\mathcal{G}_{k}^{\\mathrm{ind}}$ be the subset of $\\mathcal{G}_{k}$ that consists of all graphs whose set of vertices is $V^{\\mathrm{ind}}$ (i.e., each vertex represents some $v_{i}^{\\mathrm{ind}}$ and no two vertices represent the same $v_{i}^{\\mathrm{ind}})$ . The space $\\mathcal{G}_{k}^{\\mathrm{ind}}$ can be interpreted as the space of symbolic graphs with fixed vertices. We define the indirection operator $\\mathcal{T}$ as follows.\n\nDefinition 1. Given an input sequence $X=(x_{1},x_{2},\\dots,x_{k})$ and its corresponding graph $G_{X}\\in$ $\\mathcal{G}_{k}.$ ,the indirection operator I is a mapping from $\\mathcal{G}_{k}$ to $\\mathcal{G}_{k}^{\\mathrm{ind}}$ that maps $G_{X}$ to $\\bar{\\mathcal{T}}(\\bar{G}_{X})$ so that $A_{G_{X}}=A_{\\mathcal{T}(G_{X})}$ and the i-th vertex of $\\mathcal{T}(G_{X})$ represents $v_{i}^{\\mathrm{ind}}$\n\nRemark 2. Definition 1 is introduced in the case when the canonical indexing assumption (see Remark 1) is obeyed. The vertex order emerges when computing the adjacency matrix. A more general definition is given in Appendix C.\n\nAfter indirection, each data graph G is mapped to a symbolic graph ${\\mathcal{T}}(G)$ . This operation can be interpreted as follows: at first, edge weights of ${\\mathcal{T}}(G)$ are unspecified; then the indirection operator $\\mathcal{T}$ assigns edge weights from the data to ${\\mathcal{T}}(G)$ . Once receiving this information from data, ${\\mathcal{T}}(G)$ is propagated and the updated vertex features are indirection representations of the input sequence. Formally, for an input sequence $X,$ ,if we denote $r_{X}$ to be the indirection representations of $X$ then $r_{X}=A_{G_{X}}V^{\\mathrm{ini}}$ . This symbolic $r_{X}$ is used as a new representation for X for prediction steps afterward.\n\nDefinition 2. Given two graphs $G=(V,E)\\in\\mathcal{G}_{k}$ and $G^{\\prime}=(V^{\\prime},E^{\\prime})\\in\\mathcal{G}_{k}$ with associated adjacency matrices $A_{G}=(a_{i j})_{i,j=\\overline{{1,k}}}$ and $A_{G^{\\prime}}=(a_{i j}^{\\prime})_{i,j=\\overline{{1,k}}}$ We say G and $G^{\\prime}$ are isomorphic, denote by $G\\cong G^{\\prime}$ , if there exists a bijection $\\phi:V\\to{\\bar{V}}^{\\prime}$ so that $a_{i j}=a_{\\phi(i)\\phi(j)}^{\\prime}$ for every $i,j\\in V$\n\nTwo isomorphic graphs can be interpreted as being identical up to isomorphism, and thus a graph distance defined on $\\mathcal{G}_{k}$ should respect this property, i.e., the distance between two isomorphic graphs is 0. One such distance is the cut distance $\\hat{\\delta}_{\\square}$ (Borgs et al., 2008), which is a useful tool to compare similarities betwen structures (Liu et al., 2018), and also for studying the convergence of sequence of graphs (Borgs et al., 2008). A formal definition for $\\hat{\\delta}_{\\square}$ is given in Appendix A.\n\nIt follows from the definition of $\\hat{\\delta}_{\\square}$ that $\\hat{\\delta}_{\\square}(G,G^{\\prime})=0$ if and only if G is isomorphic with $G^{\\prime}$ Moreover, if $G_{1}\\cong G_{2}$ then ${\\hat{\\delta}}_{\\square}(G_{1},G^{\\prime})={\\hat{\\delta}}_{\\square}(G_{2},G^{\\prime})$ for any $G^{\\prime}$ Since $G\\cong{\\mathcal{T}}(G)$ , the indirection operator I preserves $\\hat{\\delta}_{\\square}$ distance, $\\begin{array}{ r }{\\mathrm{i . e .,}\\hat{\\delta}_{\\square}(G,G^{\\prime})=\\hat{\\delta}_{\\square}({\\mathcal T}(G),{\\mathcal T}(G^{\\prime}))}\\end{array}$ for every $G,G^{\\prime}\\in\\mathcal{G}_{k}$\n\nWe have shown that the indirection operator I admits invariant properties with respect to the graph spectrum, Laplacian and the cut graph distance $\\hat{\\delta}_{\\square}$ . The following result shows that the distances between indirection representations are bounded by cut distances between corresponding graphs. For each $G\\in\\mathcal{G}_{k}$ , we denote $r_{G}=A_{G}V^{\\mathrm{ind}}$ to be its associated indirection representation.\n\nDefinition 3. (Borgs et al., 2008) Given two graphs $G=(V,E)\\in\\mathcal{G}_{k}$ and $G^{\\prime}=(V,E^{\\prime})\\in\\mathcal{G}_{k}$ with associated adjacency matrices $A_{G}=(a_{i j})_{i,j=\\overline{{1,k}}}$ and $A_{G^{\\prime}}=(a_{i j}^{\\prime})_{i,j=\\overline{{1,k}}}.$ The cut distance $d_{\\square}$ between G and $G^{\\prime}$ is\n\n$$\nd_{\\square}(G,G^{\\prime})=\\operatorname*{max}_{S,T\\subset V}\\frac{1}{k^{2}}|e_{G}(S,T)-e_{G^{\\prime}}(S,T)|,\\tag{1}\n$$\n\nwhere\n\n$$\ne_{G}(S,T)=\\sum_{i\\in S,j\\in T}a_{i j}\\quad\\mathrm{and}\\quad e_{G^{\\prime}}(S,T)=\\sum_{i\\in S,j\\in T}a_{i j}^{\\prime}.\n$$\n\nDefinition 4. (Borgs et al., 2008) Given two graphs $G=(V,E)\\in\\mathcal{G}_{k}$ and $G^{\\prime}=(V,E^{\\prime})\\in\\mathcal{G}_{k}$ The generalized cut distance $\\hat{\\delta}_{\\square}$ between $G$ and $G^{\\prime}$ is computed as $\\begin{array}{ r }{\\hat{\\delta}_{\\square}(G,G^{\\prime})=\\operatorname*{min}_{\\tilde{G}\\cong G}d_{\\square}(\\tilde{G},G^{\\prime})}\\end{array}$ , where $\\tilde{G}$ shares the same set of vertices with $G^{\\prime}$\n\nDefinition 5. For a given matrix $A=\\left(a_{i j}\\right)_{i=\\overline{{1,k}},j=\\overline{{1,n}}}.$ its infinity norm is computed as $\\|A\\|_{\\infty}=$ $\\operatorname*{max}_{1\\leq i\\leq k}\\sum_{j=1}^{n}\\vert a_{i j}\\vert$ . In words, the matrix infinity norm is the max row sum.\n\nProposition 1. The matrix infinity norm is sub-multiplicative, $i . e .,\\|A B\\|_{\\infty}\\leq\\|A\\|_{\\infty}\\|B\\|_{\\infty}$\n\nFor each $G\\in\\mathcal{G}_{k}$ , denote $R_{G}=\\{r_{\\tilde{G}}:\\tilde{G}\\in\\mathcal{G}_{k},\\tilde{G}\\cong G\\}$ . We now focus on the distance between $R_{G}$ and $R_{G^{\\prime}}$ , and we use the popular Hausdorff metric to measure this distance. The distance between $R_{G}$ and $R_{G^{\\prime}}$ is meaningful in the sense that for any $r^{\\prime}\\in R_{G^{\\prime}}$ , we can find $r\\in R_{G}$ so that $\\|r-r^{\\prime}\\|_{\\infty}\\leq d_{H}(R_{G},R_{G^{\\prime}})$ . If $d_{H}(R_{G},R_{G^{\\prime}})$ is small and $r$ is observed, then $r^{\\prime}$ is likely to be treated similarly as $r$ . The following theorem shows that the Hausdorff distance $d_{H}$ with respect to the $\\|\\cdot\\|_{\\infty}$ norm between $R_{G}$ and $R_{G^{\\prime}}$ also depends on the distance between $G$ and $G^{\\prime}$ .", "theorems": [{"local_id": "T1", "statement": "Theorem 1. Given two graphs $G\\in\\mathcal G_{k}$ and $G^{\\prime}\\in\\mathcal{G}_{k}$ . The following inequality holds: $$ d_{H}(R_{G},R_{G^{\\prime}})\\leq k^{3}\\|V^{i n d}\\|_{\\infty}\\hat{\\delta}(G,G^{\\prime}).\\tag{2} $$", "proof": {"steps": [{"n": 1, "claim": "Proof of Theorem 1. Denote $[G]=\\{\\tilde{G}\\in\\mathcal{G}_{k}:\\tilde{G}\\cong G\\}$ ."}, {"n": 2, "claim": "We will prove that for every $G_{1}\\in [G]$ there exits $G_{2}\\in [G^{\\prime}]$ so that $\\|r_{G_{1}}-r_{G_{2}}\\|_{\\infty}\\leq k^{3}\\|V^{\\mathrm{ind}}\\|_{\\infty}\\hat{\\delta}(G,G^{\\prime})$"}, {"n": 3, "claim": "Following the definition of $\\hat{\\delta}_{\\square}$ , for $G_{1}~\\in~[G]$ , there exists $G_{2}~\\in~[G^{\\prime}]$ so that $\\hat{\\delta}_{\\square}(G_{1},G^{\\prime})=$ $d_{\\square}(G_{1},G_{2})$ ."}, {"n": 4, "claim": "On the other hand, since $G_{1}\\cong G$ , it follows that ${\\hat{\\delta}}_{\\square}(G_{1},G^{\\prime})={\\hat{\\delta}}_{\\square}(G,G^{\\prime})$ , and hence $d_{\\square}(G_{1},G_{2})=\\hat{\\delta}_{\\square}(G,G^{\\prime})$ ."}, {"n": 5, "claim": "\\|r_{G_{1}}-r_{G_{2}}\\|_{\\infty}=\\|A_{G_{1}}V^{\\mathrm{ind}}-A_{G_{2}}V^{\\mathrm{ind}}\\|_{\\infty}"}, {"n": 6, "claim": "\\qquad\\leq\\|A_{G_{1}}-A_{G_{2}}\\|_{\\infty}\\|V^{\\mathrm{ind}}\\|_{\\infty}"}, {"n": 7, "claim": "\\qquad\\leq k^{3}d_{\\square}(G_{1},G_{2})\\|V^{\\mathrm{ind}}\\|_{\\infty}"}, {"n": 8, "claim": "\\qquad=k^{3}\\hat{\\delta}_{\\square}(G,G^{\\prime})\\|V^{\\mathrm{ind}}\\|_{\\infty}."}, {"n": 9, "claim": "This means"}, {"n": 10, "claim": "d(r_{G_{1}},R_{G^{\\prime}})=\\operatorname*{inf}_{\\tilde{G}\\in [G^{\\prime}]}\\|r_{G_{1}}-r_{\\tilde{G}}\\|\\leq k^{3}\\hat{\\delta}_{\\square}(G,G^{\\prime})\\|V^{\\mathrm{ind}}\\|_{\\infty}"}, {"n": 11, "claim": "for every $G_{1}\\in [G]$ .Similary, $d(\\boldsymbol{r}_{G_{2}},\\boldsymbol{R}_{G})=\\operatorname*{inf}_{\\tilde{G}\\in [G]}\\|\\boldsymbol{r}_{G_{2}}-\\boldsymbol{r}_{\\tilde{G}}\\|\\leq k^{3}\\hat{\\delta}_{\\square}(G,G^{\\prime})\\|V^{\\mathrm{ind}}\\|_{\\infty}$ for every $G_{2}\\in [G^{\\prime}]$ . This leads to"}, {"n": 12, "claim": "d_{H}(R_{G},R_{G^{\\prime}})=\\operatorname*{max}\\left\\{\\operatorname*{sup}_{G_{1}\\in [G]}d(r_{G_{1}},R_{G^{\\prime}}),\\operatorname*{sup}_{G_{2}\\in [G^{\\prime}]}d(r_{G_{2}},R_{G})\\right\\}\\leq k^{3}\\Vert V^{\\mathrm{ind}}\\Vert_{\\infty}\\hat{\\delta}(G,G^{\\prime})."}]}}, {"local_id": "T2", "statement": "Let $\\mathcal G_k$ be the space of directed weighted complete graphs on $k$ vertices with edge weights in $[-1,1]$, and for $G\\in\\mathcal G_k$ let $A_G$ be its adjacency matrix. Let $V^{\\mathrm{ind}}=(v_1^{\\mathrm{ind}},\\dots,v_k^{\\mathrm{ind}})^\\top\\in\\mathbb R^{k\\times n}$ be the trainable symbols, and define the associated indirection representation $r_G=A_G V^{\\mathrm{ind}}$. Let $\\hat\\delta_\\square$ denote the generalized cut distance on $\\mathcal G_k$ and let $\\|\\cdot\\|_\\infty$ be the entrywise max norm ($\\|A\\|_\\infty=\\max_{ij}|a_{ij}|$). Then for any two graphs $G\\in\\mathcal G_k$ and $G'\\in\\mathcal G_k$, $\\|r_G-r_{G'}\\|_\\infty\\le\\big(2+k^2\\,\\hat\\delta_\\square(G,G')\\big)\\,\\|V^{\\mathrm{ind}}\\|_\\infty$.", "proof": {"steps": [{"n": 1, "claim": "$\\varepsilon:=\\hat\\delta_\\square(G,G')$."}, {"n": 2, "claim": "$\\hat\\delta_\\square(\\mathcal I(G),\\mathcal I(G'))=\\hat\\delta_\\square(G,G')=\\varepsilon$."}, {"n": 3, "claim": "There exists $G^{\\mathrm{ind}}\\in\\mathcal G_k^{\\mathrm{ind}}$ with $G^{\\mathrm{ind}}\\cong\\mathcal I(G)$ and $d_\\square(G^{\\mathrm{ind}},\\mathcal I(G'))=\\varepsilon$."}, {"n": 4, "claim": "Setting $\\mathcal E:=A_{G^{\\mathrm{ind}}}-A_{\\mathcal I(G')}$, each entry of $\\mathcal E$ has absolute value at most $k^2\\varepsilon$."}, {"n": 5, "claim": "$\\|\\mathcal E\\|_\\infty\\le k^2\\varepsilon$."}, {"n": 6, "claim": "$A_{\\mathcal I(G')}=A_{G'}$."}, {"n": 7, "claim": "$\\|r_G-r_{G'}\\|_\\infty=\\|A_G V^{\\mathrm{ind}}-A_{G'}V^{\\mathrm{ind}}\\|_\\infty$."}, {"n": 8, "claim": "$\\|A_G V^{\\mathrm{ind}}-A_{G'}V^{\\mathrm{ind}}\\|_\\infty\\le\\|A_G-A_{G^{\\mathrm{ind}}}+\\mathcal E\\|_\\infty\\,\\|V^{\\mathrm{ind}}\\|_\\infty$."}, {"n": 9, "claim": "$\\|A_G-A_{G^{\\mathrm{ind}}}+\\mathcal E\\|_\\infty\\,\\|V^{\\mathrm{ind}}\\|_\\infty\\le\\big(\\|A_G\\|_\\infty+\\|A_{G^{\\mathrm{ind}}}\\|_\\infty+\\|\\mathcal E\\|_\\infty\\big)\\|V^{\\mathrm{ind}}\\|_\\infty$."}, {"n": 10, "claim": "$\\big(\\|A_G\\|_\\infty+\\|A_{G^{\\mathrm{ind}}}\\|_\\infty+\\|\\mathcal E\\|_\\infty\\big)\\|V^{\\mathrm{ind}}\\|_\\infty\\le(2+k^2\\varepsilon)\\|V^{\\mathrm{ind}}\\|_\\infty$."}]}}, {"local_id": "T3", "statement": "This shows that the indirection representations of $x_{i}$ 's are unchanged when the vertices of $G_{X}$ permute.", "proof": {"steps": [{"n": 1, "claim": "Definition 6. Given an input sequence $X=(x_{1},x_{2},\\dots,x_{k})$ and its corresponding graph $G_{X}\\in$ $\\mathcal{G}_{k}.$ , the indirection operator I is a mapping from $\\mathcal{G}_{k}$ to $\\mathcal{G}_{k}^{\\mathrm{ind}}$ that maps $G_{X}$ to $\\mathcal{T}(G_{X})$ so that 1. $A_{G_{X}}=A_{\\mathcal{T}(G_{X})}$ and 2. if the i-th vertex of $G_{X}$ represents for $x_{j}$ , then the i-th vertex of $\\mathcal{T}(G_{X})$ represents for $v_{j}^{\\mathrm{ind}}$"}, {"n": 2, "claim": "Consider an input sequence $X=(x_{1},x_{2},\\dots,x_{k})$ and its corresponding graph $G_{X}$ with adjacency matrix $A_{G_{X}}$ , which is computed based on the assumption in Remark 1."}, {"n": 3, "claim": "The associated indirection representation computed by Eq. (3) is rx, i.e., the i-th element of $r_{X}$ is the indirection representation for $x_{i}$"}, {"n": 4, "claim": "Now consider an arbitrary graph $G_{X}^{\\prime}$ that also represents X with adjacency matrix $A_{G_{X}^{\\prime}}$ and set of vertices $\\{v_{1}^{\\prime},v_{2}^{\\prime},\\ldots,v_{k}^{\\prime}\\}$ ."}, {"n": 5, "claim": "This means there exists a permutation σ (with an associated permutation matrix $P)$ so that $v_{\\sigma(i)}^{\\prime}$ represents for $x_{i}$ for all i, and $A_{G_{\\mathrm{X}}^{\\prime}}=P A_{G_{X}}P^{\\intercal}$ ."}, {"n": 6, "claim": "Suppose that $G_{X}^{\\prime}$ is mapped to $\\mathcal{T}(G_{X}^{\\prime})$ with set of vertices $\\{u_{1}^{\\prime},u_{2}^{\\prime},\\ldots,u_{k}^{\\prime}\\}$ ."}, {"n": 7, "claim": "By the definition of indirection operator (Definition (1)), $\\boldsymbol{v}_{i}^{\\prime}$ is mapped to $u_{i}^{\\prime}$ (so that $A_{G_{X}^{\\prime}}=A_{\\mathcal{T}(G_{X}^{\\prime})})$ and $u_{\\sigma(i)}^{\\prime}$ represents $v_{i}^{\\mathrm{ind}}$ (since $v_{\\sigma(i)}^{\\prime}$ represents for $x_{i})$ ."}, {"n": 8, "claim": "It follows that the associated indirection representation $r_{X}^{\\prime}$ with respect to $G_{X}^{\\prime}$ is computed as"}, {"n": 9, "claim": "r_{X}^{\\prime}=A_{G_{X}^{\\prime}}P V^{\\mathrm{ind}}=P A_{G_{X}}P^{\\top}P V^{\\mathrm{ind}}=P A_{G_{X}}V^{\\mathrm{ind}}=P r_{X}."}, {"n": 10, "claim": "This means the $\\sigma(i)$ -th element of $r_{X}^{\\prime}$ is the i-th element of $r_{X}$ ."}, {"n": 11, "claim": "On the other hand, after graph propagation, $u_{\\sigma(i)}^{\\prime}$ represents for the $\\sigma(i)$ -th element of $r_{X}^{\\prime}$ , which is the i-th element of $r_{X}$ ."}, {"n": 12, "claim": "Since $v_{\\sigma(i)}^{\\prime}$ represents for $x_{i}$ and $v_{\\sigma(i)}^{\\prime}$ is mapped to $u_{\\sigma(i)}^{\\prime}$ by the indirection operator I, it follows that the i-th element of $r_{X}$ is the indirection representation for $x_{i}$ ."}]}}], "target_local_id": "T2", "topic_domain": "Deep Learning Theory", "topic_subtopic": "Representation & Expressivity", "label": "flawed", "error_step": 8, "error_class6": ["invalid_bound_or_inequality", "others", "computation_or_constant"], "error_explanation": "The step invokes $\\|AB\\|_\\infty\\le\\|A\\|_\\infty\\|B\\|_\\infty$ for the ENTRYWISE max norm $\\|A\\|_\\infty=\\max_{ij}|a_{ij}|$ to pull $\\|V^{\\mathrm{ind}}\\|_\\infty$ out of $\\|(A_G-A_{G'})V^{\\mathrm{ind}}\\|_\\infty$. But the entrywise max norm is NOT sub-multiplicative, so this inequality is false. Counterexample: $A=B=\\begin{bmatrix}2&2\\\\2&2\\end{bmatrix}$ gives $AB=\\begin{bmatrix}8&8\\\\8&8\\end{bmatrix}$, so $\\|AB\\|_\\infty=8>4=\\|A\\|_\\infty\\|B\\|_\\infty$. This nonexistent property is what factors out $V^{\\mathrm{ind}}$ and (together with the max-norm bounds $\\|A_G\\|_\\infty\\le1$, $\\|\\mathcal E\\|_\\infty\\le k^2\\varepsilon$) yields the claimed prefactor $2+k^2\\varepsilon$. The correct argument requires the induced matrix infinity norm (max row sum), which IS sub-multiplicative and instead gives the larger factor $k(2+k^2\\varepsilon)$."}
{"id": "5ZkuWAbxzT", "context": "In this section, we formally state the problem settings of RL from human feedback (RLHF), where we consider two types of feedback: absolute rating and preference. In the contextual bandit setting, at each round $t$ , the agent observes a context $x_{t}\\in\\mathcal{X}$ generated from a distribution $d_{0}$ and chooses an action $a_{t}\\in\\mathcal{A}$ . The agent receives a stochastic reward $r_{t}\\in\\mathbb{R}$ depending on the context $x_{t}$ and the action $a_{t}$ . The goal is to maximize the expected cumulative reward over $T$ rounds. The learner has access to a family of reward functions $R(\\theta,x,a)$ parameterized by $\\theta \\in \\Theta$ , such that there exists $\\theta_{*}\\in\\Theta$ satisfying $\\mathbb{E}[r_{t}|x_{1:t},a_{1:t}]=R(\\theta_{*},x_{t},a_{t})$ . WLOG, we assume that the reward feedback $r_{t}$ at all rounds is a non-negative real number bounded by $B$ . We consider a KL-regularized objective as follows: $$ Q(\\pi)=\\mathbb{E}_{x\\sim d_{0}}\\mathbb{E}_{a\\sim\\pi(\\cdot|x)}\\bigg [R(\\theta_{*},x,a)-\\eta^{-1}\\ln\\frac{\\pi(a|x)}{\\pi_{0}(a|x)}\\bigg],\\tag{1} $$ where $\\pi_{0}$ is a known fixed policy, and $\\eta > 0$ is a hyperparameter that controls the trade-off between maximizing rewards and staying close to the reference policy π0.\n\nReward function class. We consider a function class $\\mathcal{R}=\\{R(\\theta,\\cdot,\\cdot)|\\theta\\in\\Theta\\}$ and for the realizability, we assume that the ground truth reward function $R(\\theta_{*},x,a)$ is in the function class $\\mathcal{R}$ . Then, we define the covering number of $\\mathcal{R}$ as follows.\n\nDefinition 1 (e-cover and covering number). Given a function class ${\\mathcal F},$ for each $\\epsilon > 0$ , an e-cover $o f{\\mathcal{F}}$ with respect to $| |\\cdot| |_{\\infty}$ , denoted by $\\mathcal{C}(\\mathcal{F},\\epsilon)$ , satisfies that for any $f\\in{\\mathcal{F}},$ ,we can find $f^{\\prime}\\in\\mathcal{C}(\\mathcal{F},\\epsilon)$ such that $| |\\boldsymbol{\\hat{f}}^{\\prime}-\\boldsymbol{f^{\\prime}}| |_{\\infty}^{\\cdots}\\leq\\epsilon .$ The e-covering number, denoted as $\\dot{N_{\\mathscr{F}}(\\epsilon)}$ , is the smallest cardinality of such $\\mathcal{C}(\\mathcal{F},\\epsilon)$\n\nDefinition 2 (Policy Improvement Oracle). For a reward function $R(\\theta,\\cdot,\\cdot)\\in\\mathcal{R}$ and a reference policy π0, for any prompt ${\\bar{\\mathbf{\\alpha}}}_{x}\\sim d_{0},$ , we can compute: $$ \\pi_{\\theta}^{\\eta}(\\cdot|x): =\\operatorname*{argmax}_{\\pi(\\cdot|x)\\in\\Delta(A)}\\mathbb{E}_{a\\sim\\pi(\\cdot|x)}\\left[R(\\theta,x,a)-\\eta^{-1}\\log\\frac{\\pi(a|x)}{\\pi_{0}(a|x)}\\right]\\propto\\pi_{0}(\\cdot|x)\\cdot\\exp\\left(\\eta R(\\theta,x,\\cdot)\\right). $$ Hence, the comparator policy is the solution of the oracle given the true reward function $R(\\theta^{*},\\cdot,\\cdot);$ $\\pi^{*}(\\cdot|\\cdot)\\propto\\pi_{0}(\\cdot|\\cdot)\\cdot\\exp(\\dot{\\eta}R(\\theta^{\\dagger},\\cdot,\\cdot))$ . The goal is to minimize the sub-optimality of our learned policy πwith $\\pi^{*}\\colon Q(\\pi^{*})-Q(\\widehat{\\pi})$\n\nCoverage conditions. It is crucial to assume that our data-collector policy $\\pi_{0}$ possesses good coverage, which can ensure that the learned reward function can generalize well to unseen contexts (prompts) and actions (responses), and thus can enable us to approximate the optimal policy. The global coverage is the uniform cover over all the policies in the considered class II, which is standard in offline RL (Munos & Szepesvári, 2008; Song et al., 2024) and online RL (Xie et al., 2022; Rosset et al., 2024). Essentially, Song et al. (2024) demonstrated that global coverage is necessary for offline framework and Direct Preference Optimization (DPO) fails without global coverage. Hence, we introduce two types of global-coverage conditions.\n\nDefinition 3 (Data Coverage). Given a reference policy $\\pi_{0},\\ D^{2}$ is the minimum positive real number satisfying $\\forall\\left(x,a\\right)\\in\\mathbf{\\mathcal{X}}\\times\\mathbf{\\mathcal{A}}$ ,s.t. $\\pi(a|x)>0$ and $\\dot{\\boldsymbol{b}}:\\mathcal{X}[-\\boldsymbol{B},\\boldsymbol{B}],$ we have $$ \\operatorname*{sup}_{\\theta,\\theta^{\\prime}\\in\\Theta}\\frac{|R(\\theta^{\\prime},x,a)-R(\\theta,x,a)-b(x)|^{2}}{\\mathbb{E}_{x^{\\prime}\\sim d_{0}}\\mathbb{E}_{a^{\\prime}\\sim\\pi_{0}(\\cdot\\vert x^{\\prime})}|R(\\theta^{\\prime},x^{\\prime},a^{\\prime})-R(\\theta,x^{\\prime},a^{\\prime})-b(x^{\\prime})|^{2}}\\le D^{2}. $$ The coverage coefficient D measures how well the in-sample error induced by distribution $d_{0}\\times$ $\\pi_{0}$ can cover the out-of-sample error, identifically speaking, it depicts the ability of $\\pi_{0}$ to cover the action space. This concept is adapted from the F-design for online RL under general function approximation (Agarwal et al., 2024), and resembles the coverage coefficient for offline RL (Ye et al., 2024c;a), and the eluder dimension (Wang et al., 2020; Ye et al., 2023; Agarwal et al., 2023; Zhao et al., 2023a) for online RL.\n\nDefinition 4 (Global-Policy Coverage). Given a reference policy $\\pi_{0},C_{\\mathrm{GL}}$ is the minimum positive real number satisfying that for any $\\pi:\\dot{\\mathcal{X}}A$ $$ \\operatorname*{sup}_{x\\sim d_{0},a\\in\\mathcal{A}}\\frac{\\pi(a|x)}{\\pi_{0}(a|x)}\\leq C_{\\mathrm{GL}}. $$ The two conditions require the reference policy to cover all possible policy distributions, which is standard and common in RL literature. Additionally, although the two conditions defined above are both global, it is obvious that $D^{2}\\leq C_{\\mathrm{GL}}$ , indicating that it is more general to assume a finite D.\n\nBecause of the KL-regularization for RLHF, the learned policy will not move too far from the reference policy. Hence, it is natural to relax the global coverage to local coverage inside the KL-ball (Song et al., 2024).\n\nDefinition 5 (Local KL-ball Coverage). Given a reference policy $\\pi_{0},$ for a positive constant ρKL $<$ $\\infty,$ and all policy satisfying that $\\begin{array}{ r }{\\mathbb{E}_{x\\sim d_{0}}[\\mathrm{KL}(\\pi,\\pi_{0})]\\stackrel{}{\\le}\\rho_{\\mathrm{KL}}}\\end{array}$ , we define $$ \\operatorname*{sup}_{x\\sim d_{0},a\\in\\mathcal{A}}\\frac{\\pi(a|x)}{\\pi_{0}(a|x)}: =C_{\\rho_{\\mathrm{KL}}}. $$ \n\nRemark 1 (Relation between Local and Global Coverage Conditions). The local-coverage condition (Definition 5) is more precise because compared to the global conditions targeting all possible policies, it only constrains the coverage to a KL-ball. In Song et al. (2024), because of the specific form of the oracle (Definition 2), the considered policy class is $\\Pi=\\{\\pi(\\cdot|\\cdot)\\propto $ $\\pi_{0}(\\cdot|\\cdot)\\exp(\\eta R(\\theta,\\cdot,\\cdot)):R(\\theta,\\cdot,\\cdot)\\in\\mathscr{R}\\}$ . Thus, they only need to assume that the condition hold for $\\rho=2\\eta B,$ indicating that $\\dot{C}_{\\rho_{\\mathrm{KL}}}\\leq C_{\\mathrm{GL}}$ On the other hand, the data coverage condition (Definition $3)$ is measured on the level of reward functions instead of policies. In this sense, the data coverage condition and local-coverage condition do not encompass each other.", "theorems": [{"local_id": "T1", "statement": "Lemma 1 (Generalization error of reward function). For an arbitrary policy π, a set of contextaction pairs $\\{(x_{i},a_{i})\\}_{i=1}^{n}$ generated i.i.d. from π, and a distance threshold $0<\\epsilon_{c}\\le B$ , we have with probability at least $1-\\delta,$ for any pair of parameters $\\theta_{1}$ and $\\theta_{2},$ $$ \\begin{array}{ l l }&{\\displaystyle\\mathbb{E}_{\\pi}|R(\\theta_{1},x,a)-R(\\theta_{2},x,a)|^{2}}\\\\ &{\\displaystyle\\leq\\frac{2}{n}\\sum_{i=1}^{n}|R(\\theta_{1},x_{i},a_{i})-R(\\theta_{2},x_{i},a_{i})|^{2}+\\frac{32B^{2}}{3n}\\log(2N_{\\Re}(\\epsilon_{c})/\\delta)+10\\epsilon_{c}B.}\\end{array} $$ ", "proof": {"steps": [{"n": 1, "claim": "For any $R(\\theta,\\cdot,\\cdot)\\in\\mathcal{R}$ , there exists $\\theta^{c}$ such that $\\|R(\\theta,\\cdot,\\cdot)-R(\\theta^{c},\\cdot,\\cdot)\\|_{\\infty}\\leq$ $\\epsilon_{c}.$"}, {"n": 2, "claim": "By Lemma 2, for each pair of $\\theta_{1}^{c},\\theta_{2}^{c}\\in\\Theta^{c}$ (corresponding to $\\theta_{1},\\theta_{2})$ , we have with probability at least $1 - \\delta$ , $$ \\begin{array}{ r l }&{\\displaystyle\\left.\\frac{1}{n}\\sum_{i=1}^{n}(R(\\theta_{1}^{c},x_{i},a_{i})-R(\\theta_{2}^{c},x_{i},a_{i}))^{2}-\\mathbb{E}_{\\pi}\\lvert R(\\theta_{1}^{c},x,a)-R(\\theta_{2}^{c},x,a)\\rvert^{2}\\right.}\\\\ &{\\le\\sqrt{\\frac{2\\mathrm{Var}_{\\pi}\\lvert R(\\theta_{1}^{c},x,a)-R(\\theta_{2}^{c},x,a)\\rvert^{2}}{n}\\log(2/\\delta)}+\\frac{2}{3 n}B^{2}\\log(2/\\delta)}\\\\ &{\\le\\sqrt{\\frac{2 B^{2}\\mathbb{E}_{\\pi}\\lvert R(\\theta_{1}^{c},x,a)-R(\\theta_{2}^{c},x,a)\\rvert^{2}}{n}\\log(2/\\delta)}+\\frac{2}{3 n}B^{2}\\log(2/\\delta)}\\end{array} $$ "}, {"n": 3, "claim": "where the second inequality follows from the fact that $R(\\theta_{1}^{c},x,a),R(\\theta_{2}^{c},x,a)\\leq B$"}, {"n": 4, "claim": "Using union bound over all $\\theta_{1}^{c},\\theta_{2}^{c}\\in\\Theta^{c}$ , we have with probability at least $1-\\delta,$ for all $\\theta_{1}^{c},\\theta_{2}^{c}\\in\\Theta^{c}$ $$ \\begin{array}{ r l }&{\\displaystyle\\mathbb{E}_{\\pi}|R(\\theta_{1}^{c},x,a)-R(\\theta_{2}^{c},x,a)|^{2}-\\frac{1}{n}\\sum_{i=1}^{n}(R(\\theta_{1}^{c},x_{i},a_{i})-R(\\theta_{2}^{c},x_{i},a_{i}))^{2}}\\\\ &{\\displaystyle\\leq\\sqrt{\\frac{4 B^{2}\\mathbb{E}_{\\pi}|R(\\theta_{1}^{c},x,a)-R(\\theta_{2}^{c},x,a)|^{2}}{n}\\log(2 N_{\\mathcal{R}}(\\epsilon_{c})/\\delta)}+\\frac{4 B^{2}}{3 n}\\log(2 N_{\\mathcal{R}}(\\epsilon_{c})/\\delta),}\\end{array} $$ "}, {"n": 5, "claim": "from which we further obtain the following inequality by Lemma 3, $$ \\mathbb{E}_{\\pi}|R(\\theta_{1}^{c},x,a)-R(\\theta_{2}^{c},x,a)|^{2}\\leq\\frac{2}{n}\\sum_{i=1}^{n}(R(\\theta_{1}^{c},x_{i},a_{i})-R(\\theta_{2}^{c},x_{i},a_{i}))^{2}+\\frac{32B^{2}}{3n}\\log(2N_{\\Re}(\\epsilon_{c})/\\delta).\\tag{2} $$ "}, {"n": 6, "claim": "Then we can complete the proof by the definition of e-net."}]}}, {"local_id": "T2", "statement": "Consider the KL-regularized contextual bandit / RLHF setting with reward class $\\mathcal{R}=\\{R(\\theta,\\cdot,\\cdot)\\mid\\theta\\in\\Theta\\}$, ground truth $R(\\theta_*,\\cdot,\\cdot)\\in\\mathcal{R}$, reference policy $\\pi_0$, regularization $\\eta>0$, objective $Q(\\pi)=\\mathbb{E}_{x\\sim d_0}\\mathbb{E}_{a\\sim\\pi(\\cdot|x)}[R(\\theta_*,x,a)-\\eta^{-1}\\ln(\\pi(a|x)/\\pi_0(a|x))]$. Let $D^2$ (Definition 3) be minimum positive real s.t. for all $(x,a)$ with $\\pi(a|x)>0$ and all $b:\\mathcal{X}\\to[-B,B]$, $\\sup_{\\theta,\\theta'\\in\\Theta}\\frac{|R(\\theta',x,a)-R(\\theta,x,a)-b(x)|^2}{\\mathbb{E}_{x'\\sim d_0}\\mathbb{E}_{a'\\sim\\pi_0(\\cdot|x')}|R(\\theta',x',a')-R(\\theta,x',a')-b(x')|^2}\\le D^2$; and $C_{\\mathrm{GL}}$ (Definition 4) minimum positive real s.t. for any policy $\\pi$, $\\sup_{x\\sim d_0,a\\in\\mathcal{A}}\\frac{\\pi(a|x)}{\\pi_0(a|x)}\\le C_{\\mathrm{GL}}$. Claim: (i) $D^2\\le C_{\\mathrm{GL}}$, so finite $D$ is more general than finite $C_{\\mathrm{GL}}$; (ii) (Theorems 1, 2) for any $\\epsilon\\in(0,1)$, $\\eta>0$, any algorithm $A$, there is a KL-regularized contextual bandit (resp. preference learning) instance with $O(1)$ coverage coefficient and class $\\mathcal{R}$ s.t. $A$ needs at least $\\Omega\\big(\\min(\\frac{\\eta\\log N_{\\mathcal{R}}(\\epsilon)}{\\epsilon},\\frac{\\log N_{\\mathcal{R}}(\\epsilon)}{\\epsilon^2})\\big)$ rounds (resp. samples) for suboptimality $\\epsilon$.", "proof": {"steps": [{"n": 1, "claim": "$D^2$ (Def 3) and $C_{\\mathrm{GL}}$ (Def 4) are both global coverage conditions on $\\pi_0$."}, {"n": 2, "claim": "Both being global, they can be compared on the same footing."}, {"n": 3, "claim": "It is obvious that $D^2\\le C_{\\mathrm{GL}}$."}, {"n": 4, "claim": "Hence finite $D$ is a more general assumption than finite $C_{\\mathrm{GL}}$."}, {"n": 5, "claim": "Construction (Thm 1 proof, Sec. C; Thm 2 proof, Sec. D.1): $|\\mathcal{X}|=M$, $|\\mathcal{A}|=2$, $R(\\theta,x,a)=c$ if $a=\\theta(x)$, else $0$."}, {"n": 6, "claim": "This instance has $O(1)$ coverage coefficient."}, {"n": 7, "claim": "For any $\\epsilon\\in(0,1)$, $\\eta>0$, any algorithm $A$ needs at least $\\Omega\\big(\\min(\\frac{\\eta\\log N_{\\mathcal{R}}(\\epsilon)}{\\epsilon},\\frac{\\log N_{\\mathcal{R}}(\\epsilon)}{\\epsilon^2})\\big)$ rounds/samples for gap $\\epsilon$."}]}}, {"local_id": "T3", "statement": "Corollary 1. Let $C_{\\rho_{\\mathrm{KL}}}$ be in Definition 5 where $\\rho = 2 \\eta B .$ For any $\\delta\\in(0,1/6)$ and $\\epsilon>0,$ if we set $n=c_{m,n}m=\\Theta(C_{\\rho_{\\mathrm{KL}}}\\eta/\\epsilon\\cdot e^{B}\\log(N_{\\mathcal{R}}(\\epsilon_{c})/\\delta))$ (where constant $\\begin{array}{ r }{c_{m,n}>0,\\epsilon_{c}=\\frac{\\epsilon}{2(1+c_{m,n}^{-1})e^{B}})}\\end{array}$ then with probability at least $1 - 6 \\delta$ the output policy of Algorithm $2~\\pi_{\\widehat{\\theta}}^{\\eta}$ is $O(\\epsilon)$ optimal.", "proof": {"steps": [{"n": 1, "claim": "It still holds that $$ Q(\\pi^{*})-Q(\\pi_{\\widehat{\\theta_{0}}}^{\\eta})\\leq\\eta\\cdot\\mathbb{E}_{\\pi_{f}^{\\eta}}\\big [\\big(R(\\widehat{\\theta}_{0},x,a)-R(\\theta_{*},x,a)-b(x)\\big)^{2}\\big] $$ "}, {"n": 2, "claim": "where $\\pi_{f}^{\\eta}(a|x)\\propto\\pi_{0}(a|x)\\cdot\\exp(\\eta\\cdot f(x,a))$ and $f(\\cdot,\\cdot)=\\gamma [R(\\widehat{\\theta}_{0},\\cdot,\\cdot)-b(\\cdot)]+(1-\\gamma)R(\\theta_{*},\\cdot,\\cdot)$ for some $\\gamma\\in(0,1)$ ."}, {"n": 3, "claim": "Thus, We have $\\mathrm{KL}(\\pi_{f}^{\\eta}(a|x)\\|\\pi_{0})\\leq 2\\eta B$ , which further implies that $$ Q(\\pi^{*})-Q(\\pi_{\\widehat{\\theta}}^{\\eta})\\leq\\eta\\cdot C_{\\rho_{\\mathrm{KL}}}\\cdot O\\big(\\frac{1}{n}e^{B}\\log(N_{\\mathcal{R}}(\\epsilon_{c})/\\delta)+e^{B}(1+c_{m,n}^{-1})\\epsilon_{c}\\big) $$ "}, {"n": 4, "claim": "by Lemma 4. Then we can conclude by substituting the value of m into the suboptimality gap. □"}]}}, {"local_id": "T4", "statement": "Lemma 3. Suppose $a,b\\geq 0 .\\ I f x^{2}\\leq a+b\\cdot x,$ then $x^{2}\\leq 2 b^{2}+$ 2a.", "proof": {"steps": [{"n": 1, "claim": ",we obtain max $\\{x_{1},x_{2}\\}=$ $(b+\\sqrt{b^{2}+4 a})/2$"}, {"n": 2, "claim": "Hence, we have $x\\leq(b+\\sqrt{b^{2}+4 a})/2$ provided that $q(x)\\leq 0$"}, {"n": 3, "claim": "Then we further have $$ x^{2}\\leq{\\frac{1}{4}}\\left(b+{\\sqrt{b^{2}+4 a}}\\right)^{2}\\leq{\\frac{1}{4}}\\cdot 2\\left(b^{2}+b^{2}+4 a\\right)\\leq 2 b^{2}+2 a .\\tag{3} $$ "}]}}], "target_local_id": "T2", "topic_domain": "Online Learning & RL", "topic_subtopic": "Bandits", "label": "flawed", "error_step": 3, "error_class6": ["invalid_bound_or_inequality", "missing_assumption"], "error_explanation": "The claim $D^2\\le C_{\\mathrm{GL}}$ is false in general, so the downstream assertion that the constructed hard instances have an $O(1)$ coverage coefficient is also wrong. The data-coverage coefficient $D$ (Definition 2.6) is NOT $O(1)$: taking $\\theta=\\theta'$ and $b(x)=B\\cdot\\mathbb{1}[x=x_0]$ at the minimum-probability context $x_0$, the ratio in Definition 2.6 scales as $1/d_0(x_0)\\ge M$, i.e. at least linearly in the number of contexts $M$. By contrast the global-policy coverage $C_{\\mathrm{GL}}$ (Definition 2.7) is a likelihood ratio $\\pi/\\pi_0$ that does NOT scale with the context-space size. Hence $D^2$ can grow with $M$ while $C_{\\mathrm{GL}}$ stays bounded, so $D^2\\le C_{\\mathrm{GL}}$ fails, and the coverage coefficient of the hard instance is not constant — the coverage factor should carry a $\\log N_{\\mathcal{R}}(\\epsilon)$ (context-size) dependence rather than being $O(1)$. (The authors admitted $D^2\\le C_{\\mathrm{GL}}$ 'does not always hold' and agreed to remove both this claim and the '$O(1)$ coverage' qualifier, leaving the coverage coefficient as an explicit term in the final bounds.)"}
{"id": "8rvqpiTTFv", "context": "Assumption 1 (Expected Residual Condition). Let $f^{\\operatorname*{inf}}=\\operatorname*{inf}_{x\\in\\mathbb{R}^{n}}f(x)$ . We say the Expected Residual condition holds if there exist parameters $A,B,C\\geq 0$ such that for an unbiased estimator $g(x)$ of $\\nabla f(x)$ , we have that for all $\\boldsymbol{x}\\in\\mathbb{R}^{d}$\n\n$$\n\\begin{array}{ r }{\\mathbb{E}_{\\mathcal{D}}\\|g(x)\\|^{2}\\leq 2 A[f(x)-f^{\\mathrm{inf}}]+B\\|\\nabla f(x)\\|^{2}+C .}\\end{array}\\tag{ER}\n$$\n\nUsing the stochastic reformulation, the update rule of Unified SAM can be rewritten as follows:\n\n$$\nx^{t+1}=x^{t}-\\gamma_{t}g\\left(x^{t}+\\rho_{t}\\left(1-\\lambda_{t}+\\frac{\\lambda_{t}}{\\|g(x^{t})\\|}\\right)g(x^{t})\\right),\n$$\n\n(Unified SAM)\n\nwhere $g(x^{t})\\sim\\mathcal{D}$ is sampled i.i.d at each iteration and $\\rho_{t}\\geq 0,\\gamma_{t}>0$ and $\\lambda_{t}\\in [0,1]$ . The name unified stems from the fact that the update rule indeed unifies both USAM and SAM, however, we acknowledge that there exist other SAM-like variants that our approach does not cover.\n\nDefinition 1 (L-smooth). A differentiable function $f:\\mathbb{R}^{d}\\mathbb{R}$ is L-smooth if there exists a constant $L > 0$ such that\n\n$$\n\\|\\nabla f(x)-\\nabla f(y)\\|\\leq L\\|x-y\\|,\n$$\n\nor equivalently\n\n$$\nf(x)-f(y)\\leq\\langle\\nabla f(y),x-y\\rangle+{\\frac{L}{2}}\\|x-y\\|^{2}\n$$\n\nfor all $x,y\\in\\mathbb{R}^{d}.$\n\nDefinition $2\\ ({\\mu}{-}\\mathrm{PL})$ . We say that a differentiable function $f:{\\mathbb{R}^{d}}\\to{\\mathbb{R}}$ satisfies the Polyak-Lojasiewicz condition if\n\n$$\n\\|\\nabla f(x)\\|^{2}\\geq 2\\mu(f(x)-f(x^{*})),\n$$\n\nfor all $x\\in\\mathbb{R}^{d}.$\n\nLemma 1. Let f be an L-smooth function. Then for all $x\\in\\mathbb{R}^{n}$ we have\n\n$$\n\\begin{array}{ r }{\\|\\nabla f(x)\\|^{2}\\leq 2 L[f(x)-f^{\\mathrm{inf}}].}\\end{array}\n$$\n\nLemma 2 (Young's Inequality). For any $a,b\\in\\mathbb{R}^{n}$ and $\\beta \\neq 0$ we have\n\n$$\n\\langle a,b\\rangle\\leq\\frac{1}{2\\beta}\\|a\\|^{2}+\\frac{\\beta}{2}\\|b\\|^{2}.\n$$\n\nTheorem 1. Assume that each $f_{i}$ is $L_{i}$ -smooth, $f$ is $\\mu$ -PL and the ER is satisfied. Set $L_{\\mathrm{max}}=\\operatorname*{max}_{i\\in [n]}L_{i}$ and let\n\n$$\n\\rho_{t}=\\rho\\leq\\frac{\\mu}{L_{\\operatorname*{max}}\\left(\\mu+2[B\\mu+A](1-\\lambda)^{2}\\right)},\\gamma_{t}=\\gamma\\leq\\frac{\\mu-L_{\\operatorname*{max}}\\rho\\left(\\mu+2[B\\mu+A](1-\\lambda)^{2}\\right)}{2 L_{\\operatorname*{max}}(B\\mu+A)\\left[2 L_{\\operatorname*{max}}^{2}\\rho^{2}(1-\\lambda)^{2}+1\\right]}.\n$$\n\nThen for Unified SAM it holds\n\n$$\n\\mathbb{E}[f(x^{t})-f(x^{*})]\\leq(1-\\gamma\\mu)^{t}\\left[f(x^{0})-f(x^{*})\\right]+N,\n$$\n\n$$\n\\begin{array}{ r }{N=\\frac{L_{\\mathrm{max}}}{\\mu}\\left(C\\gamma+\\rho(1+2\\gamma L_{\\mathrm{max}}^{2}\\rho)\\left[\\lambda^{2}+C(1-\\lambda)^{2}\\right]\\right).}\\end{array}\n$$\n\nProposition 1 (Prop. 2, (Khaled & Richtárik, 2020)). Assume that each $f_{i}$ is $L_{i}$ -smooth and that $\\mathbb{E}_{\\mathcal{D}}[v_{i}^{2}]<\\infty$ for all $i \\in [ n ]$ . Let $\\sigma^{*}=\\frac{1}{n}\\sum_{i=1}^{n}f_{i}(x^{*})-f_{i}^{*}\\geq 0$ , where $f_{i}^{*}=\\operatorname*{inf}_{x\\in\\mathbb{R}^{n}}f_{i}(x)$ . Then ER holds with $A=\\operatorname*{max}_{i}L_{i}\\mathbb{E}_{\\mathcal{D}}[v_{i}^{2}]$ , $B = 0$ and $C=2 A\\sigma^{*}$ .\n\nProposition 2. Assume that each $f_{i}$ is $L_{i}$ -smooth and the ER is satisfied. Then the iterates of Unified SAM with $\\rho \\leq \\min \\left\\{ \\frac{1}{4 L_{\\max}}, \\frac{1}{8 B L_{\\max} (1-\\lambda)^2} \\right\\}$ and $\\gamma \\leq \\frac{1}{8 B L_{\\max}}$, satisfy:\n\n$$\n\\min_{t=0,\\ldots,T-1} \\mathbb{E}\\|\\nabla f(x^t)\\|^2 \\leq \\frac{2\\left(1+2A\\gamma L_{\\max}\\left[\\rho(1-\\lambda)^2(1+2\\gamma\\rho L_{\\max}^2)+\\gamma\\right]\\right)^T}{T\\gamma}[f(x^0)-f^{\\mathrm{inf}}] + 2L_{\\max}\\left[C\\gamma+\\rho(1+2\\gamma\\rho L_{\\max}^2)(\\lambda^2+C(1-\\lambda)^2)\\right] .\n$$", "theorems": [{"local_id": "T1", "statement": "Lemma 3. Let $(r_{t})_{t\\geq 0}$ and $(\\delta_{t})_{t\\geq 0}$ be sequences of non-negative real numbers and let $g > 1$ and $N \\geq 0$ Assume that the following recursive relationship holds:\n\n$$\nr_{t}\\le g\\delta_{t}-\\delta_{t+1}+N\\tag{1}\n$$\n\nThen it holds\n\n$$\n\\operatorname*{min}_{0\\leq t\\leq T-1}r_{t}\\leq\\frac{g^{T}}{T}\\delta_{0}+N .\n$$", "proof": {"steps": [{"n": 1, "claim": "Set $\\gamma_{t}=g^{-t}$ for any $t\\in\\mathbb{Z}$ Then multiply both sides of (1) with $g_{t}$ .This yields"}, {"n": 2, "claim": "g_{t}r_{t}\\leq g_{t-1}\\delta_{t}-g_{t}\\delta_{t+1}+N g_{t}."}, {"n": 3, "claim": "Summing for $t=0,\\ldots,T$ and telescoping we get"}, {"n": 4, "claim": "\\begin{array}{ c }{{\\displaystyle\\sum_{t=0}^{T-1}g_{t}r_{t}\\le g_{-1}\\delta_{0}-g_{T-1}\\delta_{T}+N\\displaystyle\\sum_{t=0}^{T-1}g_{t}}}\\\\ {{\\le g_{-1}\\delta_{0}+N\\displaystyle\\sum_{t=0}^{T-1}g_{t}.}}\\end{array}\\tag{2}"}, {"n": 5, "claim": "Now let $\\begin{array}{ r }{G=\\sum_{t=0}^{T-1}g_{t}}\\end{array}$ Note that the sequence $\\left(g_{t}\\right)$ is decreasing for $t \\geq 0$ , thus $G\\ge T g_{T-1}$ . Now dividing (2) by G we get"}, {"n": 6, "claim": "\\begin{array}{ r l r }{{\\operatorname*{min}_{0\\leq t\\leq T-1}r_{t}\\leq\\frac{1}{G}\\sum_{t=0}^{T-1}g_{t}r_{t}}}\\\\ &{}&\\\\ &{}&{\\leq\\frac{g_{-1}\\delta_{0}}{G}+N}\\\\ &{}&{\\leq\\frac{g_{-1}\\delta_{0}}{T g_{T-1}}+N}\\\\ &{}&\\\\ &{}&{=\\frac{g^{T}\\delta_{0}}{T}+N,}\\end{array}"}, {"n": 7, "claim": "as wanted."}]}}, {"local_id": "T2", "statement": "Theorem 2. Assume that each $f_i$ is $L_i$-smooth, lower bounded, the ER condition holds, and $X^* \\neq \\emptyset$. Let $\\varepsilon > 0$, $\\delta_0 = f(x_0) - f^{\\inf} \\geq 0$. Then for any $\\rho \\leq \\bar\\rho$ and $\\gamma \\leq \\bar\\gamma$, with $\\rho = O(1)$ and $\\gamma = O(1/T)$, the iterates of Unified SAM $x^{t+1} = x^t - \\gamma_t g(x^t + \\rho_t(1-\\lambda_t + \\frac{\\lambda_t}{\\|g(x^t)\\|})g(x^t))$ satisfy $\\min_{t=0,\\ldots,T-1} \\mathbb{E}\\|\\nabla f(x^t)\\| \\leq \\varepsilon$ in at most $O(\\varepsilon^{-4})$ iterations.", "proof": {"steps": [{"n": 1, "claim": "$\\min_{t} \\mathbb{E}\\|\\nabla f(x^t)\\|^2 \\leq \\frac{2(1 + 2A\\gamma L_{\\max}[\\rho(1-\\lambda)^2 + 2\\gamma L_{\\max}^2\\rho^2(1-\\lambda)^2 + \\gamma])^T}{T\\gamma}[f(x^0)-f^{\\inf}] + 2L_{\\max}[C\\gamma+\\rho(1+2\\gamma\\rho L_{\\max}^2)(\\lambda^2+C(1-\\lambda)^2)]$ by Prop. 2."}, {"n": 2, "claim": "The factor $(1 + 2A\\gamma L_{\\max}[\\rho(1-\\lambda)^2 + 2\\gamma L_{\\max}^2\\rho^2(1-\\lambda)^2 + \\gamma])^T \\leq \\exp(1) < 3$."}, {"n": 3, "claim": "Hence $\\min_{t} \\mathbb{E}\\|\\nabla f(x^t)\\|^2 \\leq \\frac{6\\delta_0}{T\\gamma} + 2L_{\\max}[C\\rho(1-\\lambda)^2 + 2C\\gamma L_{\\max}^2\\rho^2(1-\\lambda)^2 + C\\gamma + \\rho\\lambda^2 + 2\\gamma L_{\\max}^2\\rho^2\\lambda^2]$."}, {"n": 4, "claim": "Require $\\frac{6\\delta_0}{T\\gamma} \\leq \\frac{\\varepsilon^2}{2}$, i.e. $T \\geq \\frac{12\\delta_0}{\\gamma\\varepsilon^2}$, and bound the remaining constant term by $\\varepsilon^2/2$ via $\\rho,\\gamma$."}, {"n": 5, "claim": "Thus for any $\\rho \\leq \\bar\\rho$ and $\\gamma \\leq \\bar\\gamma$, with $\\rho = O(1)$, $\\gamma = O(1/T)$, both terms are $\\leq \\varepsilon^2/2$, so $\\min_{t} \\mathbb{E}\\|\\nabla f(x^t)\\|^2 \\leq \\varepsilon^2$."}, {"n": 6, "claim": "Hence $\\min_{t=0,\\ldots,T-1} \\mathbb{E}\\|\\nabla f(x^t)\\| \\leq \\varepsilon$ for $T \\geq \\frac{\\delta_0 L_{\\max}}{\\varepsilon^2}\\max\\{96B, 24(1-\\lambda)\\sqrt{3L_{\\max}A}, \\frac{5184 L_{\\max}A^2(1-\\lambda)^4\\delta_0}{\\varepsilon^2}, \\frac{864\\delta_0 A}{\\varepsilon^2}, \\frac{144C}{\\varepsilon^2}, \\frac{288 L_{\\max}^2(1-\\lambda)^2}{\\varepsilon^2}\\}$, i.e. $O(\\varepsilon^{-4})$ iterations."}]}}, {"local_id": "T3", "statement": "Theorem 3. Assume that each $f_{i}$ is $L_{i}$ -smooth, f is $\\mu{-}P L$ and the ER is satisfied. Let $\\rho_{t}=$ min $\\textstyle\\left\\{{\\frac{1}{2 t+1}},\\rho^{*}\\right\\}$ and $\\gamma_{t}=$ min $\\textstyle\\left\\{{\\frac{2 t+1}{(t+1)^{2}\\mu}},\\gamma^{*}\\right\\}$ , where $\\rho^{*}$ and $\\gamma^{*}$ are the upper bounds of ρ and $\\gamma,$ respectively, in Theorem 1. Then for Unified SAM it holds\n\n$$\nE[f(x^{t})-f(x^{*})]\\leq O\\left({\\frac{1}{t}}\\right).\n$$", "proof": {"steps": [{"n": 1, "claim": "Since γt = min $\\textstyle\\left\\{{\\frac{2 t+1}{(t+1)^{2}\\mu}},\\gamma^{*}\\right\\}\\leq\\gamma^{*}$ and $\\begin{array}{ r }{\\rho_{t}=m i n\\left\\{\\frac{1}{2 t+1},\\rho^{*}\\right\\}\\le\\rho^{*}}\\end{array}$ we get that inequality (3) holds for any $t \\geq 0$ hence we have:"}, {"n": 2, "claim": "\\begin{array}{ r l }&{\\mathbb{E}[f({x}^{t+1})-f({x}^{*})]\\le(1-\\gamma_{t}\\mu)\\mathbb{E}[f({x}^{t})-f({x}^{*})]+\\gamma_{t}L_{\\operatorname*{max}}\\left(C\\gamma_{t}+\\rho_{t}(1+2\\gamma_{t}\\rho_{t}L_{\\operatorname*{max}}^{2})\\left[\\lambda^{2}+C(1-\\lambda)^{2}\\right]\\right)}\\\\ &{\\qquad=(1-\\gamma_{t}\\mu)\\mathbb{E}[f({x}^{t})-f({x}^{*})]+C L_{\\operatorname*{max}}\\gamma_{t}^{2}+\\gamma_{t}\\rho_{t}L_{\\operatorname*{max}}^{2}(1+2\\gamma_{t}\\rho_{t}L_{\\operatorname*{max}}^{2})\\left[\\lambda^{2}+C(1-\\lambda)^{2}\\right]}\\\\ &{\\qquad\\le(1-\\gamma_{t}\\mu)\\mathbb{E}[f({x}^{t})-f({x}^{*})]+C L_{\\operatorname*{max}}\\gamma_{t}^{2}+\\gamma_{t}\\rho_{t}L_{\\operatorname*{max}}^{2}\\left(1+\\frac{2 L_{\\operatorname*{max}}^{2}}{\\mu}\\right)\\left[\\lambda^{2}+C(1-\\lambda)^{2}\\right],}\\end{array}\\tag{4}"}, {"n": 3, "claim": "Now set $\\delta_{t}=\\mathbb{E}[f(x^{t})-f(x^{*})],R=C L_{\\operatorname*{max}}$ and $\\begin{array}{ r }{Q=L_{\\mathrm{max}}^{2}\\left(1+\\frac{2 L_{\\mathrm{max}}^{2}}{\\mu}\\right)\\left[\\lambda^{2}+C(1-\\lambda)^{2}\\right]}\\end{array}$ Then the inequality (4) takes the following form:"}, {"n": 4, "claim": "\\delta_{t+1}\\leq(1-\\gamma_{t}\\mu)\\delta_{t}+Q\\gamma_{t}\\rho_{t}+R\\gamma_{t}^{2}.\\tag{5}"}, {"n": 5, "claim": "Now since the sequences $\\frac{2 t+1}{(t+1)^{2}\\mu}$ and $\\frac{1}{2 t+1}$ are clearly decreasing there exists an integer $t^{*}\\in\\mathbb{N}$ such that for any $t\\geq t^{*}$ we have $\\begin{array}{ r }{\\gamma_{t}=\\frac{2 t+1}{(t+1)^{2}\\mu}}\\end{array}$ and $\\rho_{t}\\frac{1}{2 t+1}$ . Substituting in (5) we get that for any $t\\geq t^{*}$ we have"}, {"n": 6, "claim": "\\begin{array}{ l }{\\displaystyle\\delta_{t+1}\\leq\\frac{t^{2}}{(t+1)^{2}}\\delta_{t}+\\frac{Q}{\\mu(t+1)^{2}}+\\frac{R(2 t+1)^{2}}{\\mu^{2}(t+1)^{4}}}\\\\ {\\displaystyle\\leq\\frac{t^{2}}{(t+1)^{2}}\\delta_{t}+\\frac{Q\\mu+4 R}{\\mu^{2}(t+1)^{2}},}\\end{array}"}, {"n": 7, "claim": "Multiplying both sides with $(t+1)^{2}$ and rearranging we have"}, {"n": 8, "claim": "(t+1)^{2}\\delta_{t+1}-t^{2}\\delta_{t}\\leq\\frac{Q\\mu+4 R}{\\mu^{2}}."}, {"n": 9, "claim": "Summing for $t=t^{*},\\dots,T-1$ and telescoping we have"}, {"n": 10, "claim": "T^{2}\\delta_{T}\\leq(t^{*})^{2}\\delta_{t^{*}}+\\frac{Q\\mu+4 R}{\\mu^{2}}(T-t^{*})."}, {"n": 11, "claim": "Changing notation from T to t we get"}, {"n": 12, "claim": "\\begin{array}{ l }{\\displaystyle\\mathbb{E}[f(x^{t})-f(x^{*})]\\leq\\frac{(t^{*})^{2}\\delta_{t^{*}}}{t^{2}}+\\frac{Q\\mu+4 R}{\\mu^{2}}\\frac{t-t^{*}}{t^{2}}}\\\\ {\\displaystyle\\leq\\frac{(t^{*})^{2}\\delta_{t^{*}}}{t^{2}}+\\frac{Q\\mu+4 R}{\\mu^{2}}\\frac{1}{t}}\\\\ {\\displaystyle=O\\left(\\frac{1}{t}\\right),}\\end{array}"}, {"n": 13, "claim": "as wanted."}]}}, {"local_id": "T4", "statement": "Lemma 4. Assume that each $f_{i}$ .1 $L_{i}$ -smooth and consider the iterates of Unified SAM. Then for any $\\beta > 0$ it holds\n\n$$\n\\begin{array}{ r l }&{\\mathbb{E}_{\\mathcal{D}}\\left. g\\left(x^{t}+\\rho_{t}\\left(1-\\lambda_{t}+\\frac{\\lambda_{t}}{\\|g(x^{t})\\|}\\right)g(x^{t})\\right),\\nabla f(x^{t})\\right.}\\\\ &{\\ge\\left(1-\\displaystyle\\frac{\\beta}{2}\\right)\\|\\nabla f(x^{t})\\|^{2}-\\frac{L_{\\operatorname*{max}}^{2}\\rho_{t}^{2}\\lambda_{t}^{2}}{\\beta}-\\frac{L_{\\operatorname*{max}}^{2}\\rho_{t}^{2}(1-\\lambda_{t})^{2}}{\\beta}\\mathbb{E}_{\\mathcal{D}}\\|g(x^{t})\\|^{2}.}\\end{array}\n$$\n\nIn particular, for $\\beta=L_{\\operatorname*{max}}\\rho_{t}$ we get\n\n$$\n\\begin{array}{ r l }&{\\mathbb{E}_{\\mathcal{D}}\\left. g\\left(x^{t}+\\rho_{t}\\left(1-\\lambda_{t}+\\frac{\\lambda_{t}}{\\|g(x^{t})\\|}\\right)g(x^{t})\\right),\\nabla f(x^{t})\\right.}\\\\ &{\\ge\\left(1-\\frac{L_{\\operatorname*{max}}\\rho_{t}}{2}\\right)\\|\\nabla f(x^{t})\\|^{2}-L_{\\operatorname*{max}}\\rho_{t}\\lambda_{t}^{2}-L_{\\operatorname*{max}}\\rho_{t}(1-\\lambda_{t})^{2}\\mathbb{E}_{\\mathcal{D}}\\|g(x^{t})\\|^{2}.}\\end{array}\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof. We have"}, {"n": 2, "claim": "\\begin{array}{ r l }&{\\mathbb{E}_{\\mathcal{D}}\\left. g\\left(x^{t}+\\rho_{t}\\left(1-\\lambda_{t}+\\frac{\\lambda_{t}}{\\|g(x^{t})\\|}\\right)g(x^{t})\\right),\\nabla f(x^{t})\\right.}\\\\ &{=\\mathbb{E}_{\\mathcal{D}}\\left. g\\left(x^{t}+\\rho_{t}\\left(1-\\lambda_{t}+\\frac{\\lambda_{t}}{\\|g(x^{t})\\|}\\right)g(x^{t})\\right)-g(x^{t}),\\nabla f(x^{t})\\right.}\\\\ &{\\quad+\\mathbb{E}_{\\mathcal{D}}\\langle g(x^{t}),\\nabla f(x^{t})\\rangle}\\\\ &{=\\mathbb{E}_{\\mathcal{D}}\\left. g\\left(x^{t}+\\rho_{t}\\left(1-\\lambda_{t}+\\frac{\\lambda_{t}}{\\|g(x^{t})\\|}\\right)g(x^{t})\\right)-g(x^{t}),\\nabla f(x^{t})\\right.+\\|\\nabla f(x^{t})\\|^{2}.}\\end{array}"}, {"n": 3, "claim": "Now"}, {"n": 4, "claim": "\\begin{array}{ l }{\\displaystyle-\\mathbb{E}_{\\mathcal{D}}\\left. g\\left(x^{t}+\\rho_{t}\\left(1-\\lambda_{t}+\\frac{\\lambda_{t}}{\\|g(x^{t})\\|}\\right)g(x^{t})\\right)-g(x^{t}),\\nabla f(x^{t})\\right.}\\\\ {\\displaystyle\\underset{\\leq}{L e m m a~2}\\frac{1}{2\\beta}\\mathbb{E}_{\\mathcal{D}}\\left\\|g\\left(x^{t}+\\rho_{t}\\left(1-\\lambda_{t}+\\frac{\\lambda_{t}}{\\|g(x^{t})\\|}\\right)g(x^{t})\\right)-g(x^{t})\\right\\|^{2}}\\\\ {\\displaystyle+\\frac{\\beta}{2}\\mathbb{E}_{\\mathcal{D}}\\|\\nabla f(x^{t})\\|^{2}}\\\\ {\\displaystyle\\overset{s m o o t h n e s s}{\\leq}\\frac{L_{\\operatorname*{max}}^{2}\\rho_{t}^{2}}{2\\beta}\\mathbb{E}_{\\mathcal{D}}\\left\\|(1-\\lambda_{t})g(x^{t})+\\frac{\\lambda_{t}}{\\|g(x^{t})\\|}g(x^{t})\\right\\|^{2}+\\frac{\\beta}{2}\\|\\nabla f(x^{t})\\|^{2}}\\end{array}"}, {"n": 5, "claim": "\\begin{array}{ r l }&{\\overset{L e m m a~2}{\\leq}\\frac{L_{\\operatorname*{max}}^{2}\\rho_{t}^{2}}{2\\beta}2(1-\\lambda_{t})^{2}\\mathbb{E}_{\\mathcal{D}}\\|g(x^{t})\\|^{2}+\\frac{L_{\\operatorname*{max}}^{2}\\rho_{t}^{2}}{2\\beta}2\\lambda_{t}^{2}+\\frac{\\beta}{2}\\|\\nabla f(x^{t})\\|^{2}}\\\\ &{=\\frac{L_{\\operatorname*{max}}^{2}\\rho_{t}^{2}\\lambda_{t}^{2}}{\\beta}+\\frac{L_{\\operatorname*{max}}^{2}\\rho_{t}^{2}(1-\\lambda_{t})^{2}}{\\beta}\\mathbb{E}_{\\mathcal{D}}\\|g(x^{t})\\|^{2}+\\frac{\\beta}{2}\\|\\nabla f(x^{t})\\|^{2},}\\end{array}"}, {"n": 6, "claim": "thus"}, {"n": 7, "claim": "\\begin{array}{ r l }&{\\mathbb{E}_{\\mathcal{D}}\\langle g(x^{t}+\\rho_{t}g(x^{t})),\\nabla f(x^{t})\\rangle}\\\\ &{\\ge-\\frac{L_{\\operatorname*{max}}^{2}\\rho_{t}^{2}\\lambda_{t}^{2}}{\\beta}-\\frac{L_{\\operatorname*{max}}^{2}\\rho_{t}^{2}(1-\\lambda_{t})^{2}}{\\beta}\\mathbb{E}_{\\mathcal{D}}\\|g(x^{t})\\|^{2}-\\frac{\\beta}{2}\\|\\nabla f(x^{t})\\|^{2}+\\|\\nabla f(x^{t})\\|^{2}}\\\\ &{=\\left(1-\\displaystyle\\frac{\\beta}{2}\\right)\\|\\nabla f(x^{t})\\|^{2}-\\frac{L_{\\operatorname*{max}}^{2}\\rho_{t}^{2}\\lambda_{t}^{2}}{\\beta}-\\frac{L_{\\operatorname*{max}}^{2}\\rho_{t}^{2}(1-\\lambda_{t})^{2}}{\\beta}\\mathbb{E}_{\\mathcal{D}}\\|g(x^{t})\\|^{2}.}\\end{array}"}]}}], "target_local_id": "T2", "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "label": "flawed", "error_step": 5, "error_class6": ["logical_gap_or_missing_case", "invalid_bound_or_inequality", "missing_assumption"], "error_explanation": "Step 5 defines the set of admissible step-sizes as 'any $\\rho \\leq \\bar\\rho$ and $\\gamma \\leq \\bar\\gamma$' (with the original scaling $\\rho=O(1)$, $\\gamma=O(1/T)$) but fails to carry the lower-bound restriction $T \\geq \\frac{12\\delta_0}{\\gamma\\varepsilon^2}$ that Step 4 derived. That restriction is exactly the hypothesis needed to make the first term $\\frac{6\\delta_0}{T\\gamma}$ of the Eq. (11) bound fall below $\\varepsilon^2/2$: it is equivalent to a strictly positive lower bound on $\\gamma$. Because the theorem admits 'any $\\gamma \\leq \\bar\\gamma$', it does not exclude $\\gamma \\to 0$ (or $\\gamma=0$), for which $\\frac{6\\delta_0}{T\\gamma}$ remains a positive constant and no progress is made. Equivalently, with the stated $\\gamma=O(1/T)$ scaling the coefficient of $[f(x^0)-f^{\\inf}]$ behaves as $\\frac{(1+\\Theta(\\gamma))^T}{T\\gamma} \\geq \\frac{1+\\Theta(T\\gamma)}{T\\gamma} = \\Theta(1)$, so the RHS is at least a constant and the claimed $O(\\varepsilon^{-4})$ convergence does not hold. The correct version keeps the restriction $T \\geq \\frac{12\\delta_0}{\\gamma\\varepsilon^2}$ and fixes $\\rho=O(1/\\sqrt{T})$, $\\gamma=O(1/\\sqrt{T})$ as a single choice, restoring the guarantee."}
{"id": "BgZzJISvpY", "context": "In statistics, the Extreme Value Theorem is also known as the FisherTippett theorem, and it tells us that the maximum of i.i.d. samples from exponentially tailed distributions will asymptotically converge to the Gumbel distribution, whose probability density function $(PDF)$ is $g\\left(x\\right)=$ $\\exp{(-(z+e^{-z}))}/\\beta$ where $z=\\left(x-\\mu\\right)/\\beta,\\mu$ is the location parameter and $\\beta$ is the scale parameter.\n\nTheorem 2 (Extreme Value Theorem) (Pranklin, 1974; Fisher & Tippett, 1928). For i.i.d. random variables $X_{1},...,X_{n}\\sim f_{X}$ with exponential tails, lim $\\scriptstyle{\\mathfrak{n}}\\to\\infty$ maxi (Xi) follows the Gumbel distribution. Furthermore, G is max-stable, i.e. $i f X_{i}\\sim\\mathcal{G}$ , then $m a x_{i}(X_{i})\\sim\\mathcal G$ holds.\n\nWe use the Extreme Value Theorem to reconsider the practice of truncating the tail of a distribution (Kuznetsov et al., 2020). In continuous control domains in DRL, the existence of unavoidable noise in function approximations leads to the emergence of overestimated outliers, which can be effectively modeled using the Gumbel distribution. As the Extreme Value Theorem tells us, the maximum value of i.i.d. samples with an exponential tail distribution converges asymptotically to the Gumbel distribution. Therefore, we can reasonably assume that the topmost atoms of the approximated reward distribution are anomalies with large estimation bias caused by the uncertainty of the function approximators. Truncating these anomalies is a good choice when there is no good way to use them. Although this may sacrifice some information on the right tail, it can make the model updates more stable and not be impacted by extreme values.\n\nNow we further explore the application of the Extreme Value Theorem in DRL. We directly model the distribution after truncation to reduce the consumption of model representation.\n\nTheorem 3 (Distribution of Truncate DRL). For a distribution of any shape, there exists a truncated number N of truncations to the right tail of times, $\\forall n>N,$ such that the remaining distribution after n truncations follows the extreme minimum distribution. In RL, truncating the maximum output of the critic network will cause the remaining portion to follow the extreme minimum distribution.\n\nWe give the proof of Theorem 3 in Appendix A.1. There are three forms of Extreme Value Theorem. We introduce our method in the main text using the type—I generalized extreme value distribution as an example. We will discuss the other two types of distributions in Appendix D. Despite sharing a similar generative process with the Gumbel distribution, the extreme minimum distribution has received little attention in the literature. Before further elaborating our approach, we provide here the theorem of the extreme minimum distribution, which is modeled after the definition of Gumbel distribution:\n\nDue to the fact that the information related to extreme minimum distribution is always passed in literature related to Extreme Value Theorem, we provide a simple proof of Theorem 4, Theorem 5 and Theorem 6 in Appendix A.1.\n\nTheorem 5 and Theorem 6 ensure that the minimum distribution maintains its shape and does not get converted to other distributions during the RL update process. Therefore, the distribution can be used to continuously model the truncated cumulative discounted reward.\n\nDue to the symmetrical structure of the extreme minimum distribution and Gumbel distribution, they have very symmetrical properties. But they are rarely mentioned in previous papers. Here, we provide a simple proof using the properties related to Gumbel distribution. The relevant properties of these extreme minimum distributions are not our work.\n\nWe use $F_{0}\\left(x\\right)$ to represent the initial distribution, $\\mathcal{G}_{i}\\left(x\\right)(i>0)$ to represent the distribution of extreme atoms truncated at the i-th iteration, and $F_{i}\\left(x\\right)\\left(i>0\\right)$ to represent the remaining distribution after the i-th truncation.\n\nProof: Let us consider an arbitrary distribution function, whose cumulative density function we denote by $F(x)$ . When truncating, we mark the cumulative density function of the distribution before truncation as $F_{b e f o r e},$ and the cumulative density function of the distribution after truncation as $F_{a f t e r} $ By the property of truncated distribution, we can write the distribution of the tail atoms after truncating the T as:\n\nF_{a f t e r}\\left(x\\right)=F_{b e f o r e}\\left(x\\right)/F_{b e f o r e}\\left(\\tau\\right)\\tag{1}\n\nTherefore, we perform the same operation during each truncation process (assuming that the proportion of truncation is the same each time for ease of computation, truncature the parts of $(\\tau_{t r u},1]$ which has no effect on the final result). After any number of truncations, the remaining distribution $F_{n}\\left(x\\right)$ can be expressed as:\n\n\\begin{array}{ r l }&{F_{n}\\left(x\\right)=F_{n-1}\\left(x\\right)/F_{n-1}\\left(\\tau_{d e l}\\right)}\\\\ &{\\qquad=F_{n-2}\\left(x\\right)/\\left(F_{n-1}\\left(\\tau_{d e l}\\right)F_{n-2}\\left(\\tau_{d e l}\\right)\\right)}\\\\ &{\\qquad\\quad\\vdots}\\\\ &{\\qquad=F_{0}\\left(x\\right)/\\left(F_{n-1}\\left(\\tau_{d e l}\\right)F_{n-2}\\left(\\tau_{d e l}\\right)\\cdot\\cdot\\cdot F_{0}\\left(\\tau_{d e l}\\right)\\right)}\\end{array}\\tag{2}", "theorems": [{"local_id": "T1", "statement": "Theorem 4 (Extreme Minimum Distribution) (Fisher & Tippett, 1928). For i.i.d. random variables $X_{1},\\dots,X_{n}\\sim f_{X}$ , with left exponential tails, lin $\\scriptstyle 1_{n\\to\\infty}$ mini $(X_{i})$ follows the extreme minimum distribution $\\mathcal{F}\\left(\\mu,\\beta\\right)$ , which has PDF $f\\left(x\\right)=e x p\\left(z-e^{z}\\right)/\\beta$ and CDF $F(x)=1-\\exp\\left(-e^{z}\\right)$ , where $z=\\left(x-\\mu\\right)/\\beta$ with location parameter µ and scale parameter $\\beta .$", "proof": {"steps": [{"n": 1, "claim": "Proof: For n sets of i.i.d. random variables, $X_{1},\\dots,X_{n}\\sim f_{X}$ , the location parameter of the distribution of lim $\\scriptstyle{\\mathrm{i}n\\to\\infty}$ mini $(X_{i})\\ i s\\ \\mu .$ We consider the opposite of these sets of random variables $-X_{1},\\cdot\\cdot\\cdot,-X_{n}\\sim f_{-X}$ , according to the definition of the Gumbel distribution, we can get the distribution of the maximum value:"}, {"n": 2, "claim": "\\operatorname*{lim}_{n\\to\\infty}\\operatorname*{max}_{i}\\left(-X_{i}\\right)\\sim\\mathcal{G}\\left(-\\mu,\\beta\\right)\\tag{4}"}, {"n": 3, "claim": "where $- \\mu$ is location parameter and $\\beta$ is scale parameter of the distribution ${\\mathcal{G}},$ and the $C D F$ is $G\\left(-x\\right)=\\exp{\\left(-e^{-\\frac{-x-(-\\mu)}{\\beta}}\\right)}$ . Since lin $\\begin{array}{ r }{\\mathrm{n}_{n\\infty}\\mathrm{max}_{i}(-X_{i})= -\\operatorname*{lim}_{n\\infty}\\mathrm{min}_{i}(X_{i})}\\end{array}$ , the CDF of the extreme minimum distribution can be written by:"}, {"n": 4, "claim": "\\begin{array}{ r l }&{F\\left(x\\right)=P\\left(X\\leq x\\right)}\\\\ &{\\qquad=1-P\\left(X>x\\right)}\\\\ &{\\qquad=1-P\\left(-X< -x\\right)}\\\\ &{\\qquad=1-G\\left(-x\\right)}\\\\ &{\\qquad=1-\\exp\\left(-e^{\\frac{x-\\mu}{\\beta}}\\right).}\\end{array}\\tag{5}"}, {"n": 5, "claim": "Taking the derivative of Eq. 5 yields the PDF of the distribution:"}, {"n": 6, "claim": "f\\left(x\\right)=\\frac{1}{\\beta}\\exp\\left(\\frac{x-\\mu}{\\beta}-e^{\\frac{x-\\mu}{\\beta}}\\right).\\tag{3}"}]}}, {"local_id": "T2", "statement": "For a distribution of any shape, there exists a number $N$ of right-tail truncations such that for all $n > N$ the remaining distribution after $n$ truncations follows the extreme minimum distribution $\\mathcal{F}(\\mu,\\beta)$ (PDF $f(x)=\\exp(z-e^{z})/\\beta$, CDF $F(x)=1-\\exp(-e^{z})$, where $z=(x-\\mu)/\\beta$). In RL, truncating the maximum output of the critic network will cause the remaining portion to follow the extreme minimum distribution. Here truncation removes the right tail beyond a point $\\tau$, the same proportion is truncated at each step (truncating the parts of $(\\tau_{tru},1]$), and $F_i$ denotes the CDF of the distribution remaining after the $i$-th truncation.", "proof": {"steps": [{"n": 1, "claim": "For a piece of string from which we cut off the $1/10$-th of the very tail end each time, after enough truncation operations the remaining string must be a small part of the very front end of the original string."}, {"n": 2, "claim": "For an arbitrary distribution with CDF $F(x)$, when truncating at $\\tau$ the post-truncation CDF satisfies $F_{after}(x) = F_{before}(x)/F_{before}(\\tau)$ for $x \\le \\tau$."}, {"n": 3, "claim": "Performing the same truncation each step (same proportion each time, truncating the parts of $(\\tau_{tru},1]$, which has no effect on the final result), the remaining distribution after $n$ truncations is $F_n(x) = F_{n-1}(x)/F_{n-1}(\\tau_{del}) = F_{n-2}(x)/(F_{n-1}(\\tau_{del})F_{n-2}(\\tau_{del})) = \\cdots = F_0(x)/(F_{n-1}(\\tau_{del})F_{n-2}(\\tau_{del})\\cdots F_0(\\tau_{del}))$."}, {"n": 4, "claim": "The normalizing quantiles satisfy $0 < F_{n-1}(\\tau_{del}) \\le \\cdots \\le F_0(\\tau_{del}) < 1$."}, {"n": 5, "claim": "Setting $K(n) = F_{n-1}(\\tau_{del})F_{n-2}(\\tau_{del})\\cdots F_0(\\tau_{del})$, we have $\\lim_{n\\to+\\infty} K(n) = 0^{+}$."}, {"n": 6, "claim": "Using $\\tau$ for the truncation point, for all $\\tau$ we have $F_0(\\tau) > 0$, and there exists an integer $N$ such that for all $n > N$, $K(n) < F_0(\\tau)$; so as the distribution is continuously truncated, there will always be $K(n) < F_0(\\tau)$."}, {"n": 7, "claim": "The distribution of the remaining (left) portion after truncation can therefore be considered as the extreme minimum distribution $\\mathcal{F}(\\mu,\\beta)$."}]}}, {"local_id": "T3", "statement": "Theorem 5 (Min-stable) (Mood, 1963). F is min-stable, i.e. if $X_{i}\\sim{\\mathcal{F}}$ , then mini $(X_{i})\\sim{\\mathcal{F}}$ holds.", "proof": {"steps": [{"n": 1, "claim": "Proof: Consider $X_{i}\\in\\mathcal{F}(\\mu,\\beta)$ , mini $\\begin{array}{ r }{\\left(X_{i}\\right)= -\\operatorname*{max}_{i}\\left(-X_{i}\\right)\\sim-\\mathcal{G}\\left(-\\mu,\\beta\\right)}\\end{array}$ ."}, {"n": 2, "claim": "Same as Eq. 5, we can get mini $(X_{i})\\sim{\\mathcal{F}}$"}]}}, {"local_id": "T4", "statement": "Theorem 6 (Hazan & Jaakkola, 2012). For i.i.d. $\\epsilon_{i}\\sim\\mathcal{F}(0,\\beta)$ added to a set $\\{x_{1},\\ldots,x_{n}\\}\\in\\mathbb{R}$ mini $\\begin{array}{ r }{(x_{i}+\\epsilon_{i})\\sim\\mathcal{F}\\left(-\\beta\\log\\sum_{i}\\exp{(-x_{i}/\\beta)},\\beta\\right)}\\end{array}$", "proof": {"steps": [{"n": 1, "claim": "Before giving the proof of this theorem, we give Lemma 2 to help the proof of Theorem 6."}, {"n": 2, "claim": "Lemma 2 (Gumbel-Max Trick) (Hazan & Jaakkola, 2012). For i.i.d. $\\epsilon_{i}\\sim\\mathcal{G}\\left(0,\\beta\\right)$ added to a set $\\begin{array}{ r }{\\{x_{1},\\cdots,x_{n}\\}\\in\\mathbb{R},\\operatorname*{max}_{i}\\left(x_{i}+\\epsilon_{i}\\right)\\sim\\mathcal{G}\\left(\\beta\\log\\sum_{i}\\exp\\left(x_{i}/\\beta\\right),\\beta\\right)}\\end{array}$"}, {"n": 3, "claim": "We then prove Theorem 6."}, {"n": 4, "claim": "Proof: For i.i.d. $\\epsilon_{i}~\\sim~\\mathcal{F}\\left(0,\\beta\\right)$ and a set $\\{x_{1},\\ldots,x_{n}\\}\\ \\in\\ \\mathbb{R},$ by comparing the expressions of $\\mathcal{F}(0,\\beta)$ and $\\mathcal{G}(0,\\beta)$ ,there are $-\\epsilon_{i}~\\sim~\\mathcal{G}(0,\\beta)$ ,"}, {"n": 5, "claim": "then we can get mini $\\begin{array}{ r l }{(x_{i}+\\epsilon_{i})}&{{}=}\\end{array}$ $\\begin{array}{ r }{-\\operatorname*{max}_{i}\\left(-x_{i}+(-\\epsilon_{i})\\right)\\sim-\\mathcal{G}\\left(\\beta\\log\\sum_{i}\\exp\\left(-x_{i}/\\beta\\right),\\beta\\right)}\\end{array}$"}, {"n": 6, "claim": "Same as Eq. 5, according to Gumbel-Max Trick, we have mini $(x_{i}+\\epsilon_{i})\\sim\\mathcal{F}\\left(-\\beta\\log\\sum_{i}\\exp{(-x_{i}/\\beta)},\\beta\\right)$"}]}}], "target_local_id": "T2", "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "label": "flawed", "error_step": 7, "error_class6": ["logical_gap_or_missing_case", "circular_or_unproven", "missing_assumption"], "error_explanation": "The step is a non-sequitur: it establishes only that the cumulative normalizing constant $K(n)=\\prod_i F_i(\\tau_{del})$ shrinks to $0^{+}$ (the surviving probability mass concentrates on the front/left portion), and then asserts without any argument that the surviving distribution therefore equals the extreme minimum (Gumbel-type) distribution $\\mathcal{F}(\\mu,\\beta)$. But $K(n)\\to 0$ says nothing about the SHAPE of $F_n$ converging to $\\mathcal{F}(\\mu,\\beta)$. A valid claim would require an actual convergence-in-distribution argument (Fisher–Tippett/Extreme Value Theorem for the minimum): the limiting shape of the surviving left portion depends on the LEFT-tail behavior of the original distribution, and only under specific domain-of-attraction conditions does $\\min_i(X_i)$ converge, after affine renormalization, to a generalized extreme-value / extreme-minimum law. (The reliance on Theorem 3, mis-stated with 'left exponential tails' while the method truncates the RIGHT tail, is an error the authors admitted after review.)"}
{"id": "EnrY5TOrbQ", "context": "More formally, let $\\tilde{D}$ be a distribution over $\\mathbb{R}^{d}\\times\\mathbb{R}.$ A ReLU function is parameterized by $w=(\\tilde{w},b_{w})$ where $\\tilde{w}\\in\\mathbb{R}^{d}$ and $b_{w}\\in\\mathbb{R}$ For notational convenience, we will consider the points to be in $\\mathbb{R}^{d+1}$ by appending x with a fixed coordinate 1 as $x=(\\widetilde{x},1)$ . Let $D$ be the distribution over $\\mathbb{R}^{d+1}\\times\\mathbb{R}$ induced by D. We define the loss incurred at $w=(\\tilde{w},b_{w})$ to be\n\nL(w)=\\frac{1}{2}\\operatorname{\\mathbb{E}}_{(\\tilde{x},y)\\sim\\tilde{D}}\\left[(\\sigma(\\tilde{w}^{\\top}\\tilde{x}+b_{w})-y)^{2}\\right]=\\frac{1}{2}\\operatorname{\\mathbb{E}}_{(x,y)\\sim D}\\left[(\\sigma(w^{\\top}x)-y)^{2}\\right].\n\nHere $\\sigma(x)=\\operatorname*{max}(x,0)$ is the standard rectified linear unit popularly used in deep learning. The goal in agnostic learning of a ReLU function (or agnostic ReLU regression) is to design a polynomial time learning algorithm that takes as input i.i.d. samples from $D$ and outputs $w=(\\tilde{w},b_{w})$ such that $L(w)$ compares favorably with $O P T$ that is given by\n\nO P T: =\\operatorname*{min}_{w=(\\tilde{w},b_{w})\\in H}\\frac{1}{2}\\operatorname*{\\mathbb{E}}_{(x,y)\\sim D}[(\\sigma(w^{\\top}x)-y)^{2}].\n\nO(1)-regular marginals: Assumptions about the marginals over $\\widetilde{x}$ We make the following assumptions about the marginal distribution $\\widetilde{\\mathcal{D}}_{x}$ over $\\widetilde{\\boldsymbol{x}}\\in\\mathbb{R}^{d}$ there exists absolute constants $\\beta_{1},\\beta_{2}^{\\prime},\\beta_{2},\\beta_{3},\\beta_{4},\\beta_{5}>0$ and $\\beta_{0}:\\mathbb{R}_{+}\\to\\mathbb{R}_{+}$ , such that\n\n(i) Approximate isotropicity and bounded fourth moments: for every unit vector $u\\in\\mathbb{R}^{d}$ $\\mathbb{E}_{\\widetilde{x}\\sim\\widetilde{D}_{x}}[\\langle u,\\widetilde{x}\\rangle^{2}]\\in [\\bar{1}/\\beta_{2}^{\\prime},\\beta_{2}]$ , and $\\mathbb{E}_{\\widetilde{x}\\sim\\widetilde{D}_{r}}[\\langle u,\\widetilde{x}\\rangle^{4}]\\le\\beta_{4}$\n\n(ii) Anti-concentration: there exists an absolute constant $\\beta_{3}>0$ such that for every unit vector $\\widetilde u\\in\\mathbb{R}^{d}$ and $\\delta > 0$ ,\n\n\\operatorname*{sup}_{t\\in\\mathbb{R}}\\operatorname*{\\mathbb{P}}_{\\tilde{x}\\sim\\widetilde{D}_{x}}\\left[\\langle\\tilde{u},\\widetilde{x}\\rangle\\in(t-\\delta,t+\\delta)\\right]\\leq\\operatorname*{min}\\{\\beta_{3}\\delta,1\\}.\n\n(iii) Spread out: there exists $\\beta_{0}:\\mathbb{R}_{+}\\to\\mathbb{R}_{+}$ such that $\\beta_{0}(|b_{v}|)>0$ is a constant when $\\left|b_{v}\\right|$ is a constant, and\n\n\\forall\\widetilde{v}\\in{\\mathbf{S}}^{d-1},\\quad\\underset{\\widetilde{x}\\sim\\widetilde{\\mathcal{D}}_{x}}{\\mathbb{E}}\\left[\\sigma(\\widetilde{v}^{\\top}\\widetilde{x}+b_{v})\\right]\\geq\\beta_{0}(|b_{v}|).\n\n(iv) 2-D projections: In every 2-dimensional subspace of $\\mathbb{R}^{d}$ spanned by orthonormal unit vectors $\\bar{\\tilde{u}}_{1},\\tilde{u}_{2}\\in\\mathbb{R}^{d}$ , we have a set $G_{\\tilde{u}_{1},\\tilde{u}_{2}}\\subset\\mathbb{R}$ such that ,\n\n\\mathbb{P}_{\\widetilde{x}\\sim\\widetilde{D}_{x}}[\\widetilde{u}_{2}^{\\top}\\widetilde{x}\\in G_{\\widetilde{u}_{1},\\widetilde{u}_{2}}]=1-o(1),\\ \\mathrm{and}\\tag{2}\n\n\\forall t\\in G_{\\tilde{u}_{1},\\tilde{u}_{2}},\\quad\\underset{\\tilde{x}\\sim\\widetilde{\\mathcal{D}}_{x}}{\\mathbb{E}}\\left[\\sigma(\\tilde{u}_{1}^{\\top}\\widetilde{x})\\ |\\ \\tilde{u}_{2}^{\\top}\\widetilde{x}=t\\right]\\geq\\beta_{5}\\cdot\\underset{\\tilde{x}\\sim\\widetilde{\\mathcal{D}}_{x}}{\\mathbb{E}}\\left[\\sigma(\\tilde{u}_{1}^{\\top}\\widetilde{x})\\right].\\tag{3}\n\nIn other words, the conditional expectation of $\\sigma(\\widetilde{u}_{1}^{\\top}\\widetilde{x})$ is not much smaller after conditioning on the projection in an orthogonal direction $\\tilde{u}_{2}$ , for most values of $\\widetilde{u}_{2}^{\\top}\\widetilde{x}$ . Note that for a Gaussian $\\bar{N}(0,I)$ , the r.v.s $\\tilde{u}_{1}^{\\top}\\widetilde{x},\\tilde{u}_{2}^{\\top}\\widetilde{x}$ are independent, so this condition holds with $\\beta_{5}=1$ and $G_{\\tilde{u}_{1},\\tilde{u}_{2}}=\\mathbb{R}$\n\nWe remark that Gaussian distribution $\\mathcal{N}(0,I)$ is O(1)-regular i.e., all the constants $\\beta_{1},\\beta_{2},\\beta_{2}^{\\prime},\\beta_{5}=$ $1,\\beta_{3}\\leq 2$ , and $\\begin{array}{ r }{\\beta_{0}(b_{v})=\\mathbb{E}_{q\\sim N(0,1)}[\\sigma(g+b_{v})]>0}\\end{array}$ for all $b_{v}\\in(-\\infty,\\infty)$ ;in fact $\\beta_{0}$ is an increasing function that is 0 only at $- \\infty .$ We also note that assumptions of this flavor have also been used in prior works includingVardi et al. (2021), which inspired parts of our analysis. In particular, Vardi et al. (2021) assume a lower-bound on the density for any 2-dimensional marginal; our assumption (4) on the 2-dimensional marginals is qualitatively weaker (it is potentially satisfied by even discrete distributions), and moreover we only need the condition to be satisfied for a large fraction of values of $\\widetilde{u}_{2}^{\\top}\\widetilde{x}$ (and not all). See SectionB for the generalized version of our main theorem.\n\nWe consider agnostically learning a single ReLU neuron with bias through gradient descent under the supervised learning setting. We assume we are given data $(x,y)$ , where $\\bar{\\boldsymbol{x}}\\in\\mathbb{R}^{d+1}$ follows the standard Gaussian distribution $\\mathcal{N}(0,I)$ in the first d dimensions and the $d + 1 \\ '$ th dimension being a constant 1. We also assume the labels $y \\in \\mathbb R$ are arbitrarily correlated with x and $\\sigma(w^{\\top}x)$\n\nNote that throughout the paper, we will use $\\widetilde{w},\\widetilde{v},\\widetilde{x}$ to denote the first d dimensions of w, v, x respectively, with the last dimension of w being $b_{w}\\in\\mathbb{R}$ (similarly for $b_{v}\\in\\mathbb{R})$ . Therefore, $w^{\\top}$ x is in fact $\\widetilde{w}^{\\top}\\widetilde{x}+b_{w}$\n\nIn the analysis, we will compare the current iterate w to any optimizer of the loss $L(w)$\n\nv: =\\arg\\operatorname*{min}_{w\\in H}L(w),{\\mathrm{~w h e r e~}}L(w)=\\frac{1}{2}\\operatorname*{\\mathbb{E}}_{(x,y)\\sim D}\\Big [(\\sigma(w^{\\top}x)-y)^{2}\\Big],\\tag{4}\n\nand the hypothesis set $H=\\{w=(\\tilde{w},b_{w}):\\|\\tilde{w}\\|\\in [\\frac{1}{C_{1}},C_{1}],|b_{w}|\\le C_{2})\\}$ , where $C_{1}$ and $C_{2}$ are absolute constants. This is to ensure that the relative bias $\\lceil b_{w}| /\\|\\tilde{w}\\|_{2}$ is bounded; as described earlier AppendixCallows us to assume $\\|\\tilde{w}\\|\\in [\\frac{1}{C_{1}},C_{1}]$ without loss of generality.\n\nAs we are in the agnostic setting, there may be no w that achieves zero loss. We can split the loss function $L(w)$ into two components, one of which is $F(w)$ defined by\n\n\\begin{array}{ r }{F(w): =\\frac{1}{2}\\mathbb E\\left[(\\sigma(w^{\\top}x)-\\sigma(v^{\\top}x))^{2}\\right],\\nabla F(w): =\\mathbb E\\left[(\\sigma(w^{\\top}x)-\\sigma(v^{\\top}x))\\sigma^{\\prime}(w^{\\top}x)x\\right].}\\end{array}\\tag{5}\n\nWe will often refer to $F(w)$ as the realizable loss, since it captures the difference between w and $v ;$ in the realizable setting $\\dot{L}(w)=F(w)$ . Note that $F(v)=0$\n\nGradient of the Loss. The gradient of $L(w)$ with respect to w is\n\n\\nabla L(w)=\\mathbb{E}\\left[(\\sigma(w^{\\top}x)-y)\\sigma^{\\prime}(w^{\\top}x)x\\right]\\tag{6}\n\nwhere $\\sigma^{\\prime}(\\cdot)$ is the derivative of $\\sigma(\\cdot)$ , defined as $\\sigma^{\\prime}(z)=\\mathbb{1}\\{z\\geq 0\\}$ . Note that the ReLU function $\\sigma(z)$ is differentiable everywhere except at $z = 0$ Following standard convention in this literature, we define $\\sigma^{\\prime}(0)=1$ . Note that the exact value of $\\sigma^{\\prime}(0)$ will have no effect on our results.\n\nWe can also decompose $\\nabla L(w)$ as\n\n\\nabla L(w)=\\mathbb{E}\\left[(\\sigma(w^{\\top}x)-\\sigma(v^{\\top}x))\\sigma^{\\prime}(w^{\\top}x)x\\right]+\\mathbb{E}\\left[(\\sigma(v^{\\top}x)-y)\\sigma^{\\prime}(w^{\\top}x)x\\right]\\tag{7}\n\n\\mathrm{Therefore},\\nabla L(w)=\\nabla F(w)+\\mathbb{E}\\left[(\\sigma(v^{\\top}x)-y)\\sigma^{\\prime}(w^{\\top}x)x\\right]\\tag{8}\n\nGradient Descent. Finally, our paper focuses on the standard gradient descent algorithm with a fixed learning rate $\\eta > 0$ We initialize at some point $w_{0}\\in\\mathbb{R}^{d+\\bar{1}}$ , and at each iteration $t\\in\\mathbb{N}$ we have $w_{t+1}=w_{t}-\\eta\\nabla F(w_{t})$ . We do not optimize the iteration count in this paper; hence it will be instructive to think of  as a non-negligible parameter that can be set to be sufficiently small (e.., an inverse polynomial for polynomial time guarantees).\n\nSimplification. For sake of exposition we will assume that $\\|\\tilde{v}\\|_{2}=1$ ; the same analysis goes through when $\\lVert\\tilde{v}\\rVert_{2}\\in [1/C_{1},C_{1}]$ as well. Moreover Proposition1shows that assuming that $\\|\\tilde{v}\\|_{2}$ is normalized is without loss of generality. Note that we cannot make such a simplifying assumption about the vectors $\\boldsymbol{w}_{t}=\\left(\\boldsymbol{\\tilde{w}_{t}},\\boldsymbol{b_{w}}\\right)$ in the intermediate iterations.\n\nFinally, please see Section Bfor the weaker distributional guarantees and guarantees.\n\n4 Overview of the Analysis (Proof of Theorem 1)\n\nWe now provide an overview of our analysis. For complete proofs of the lemmas and propositions, please refer to the supplementary material (Appendix A). Recall that our goal throughout the learning process is to find a $\\mathcal{w}\\in\\mathbb{R}^{d+1}$ such that $L(w)$ achieves a comparable performance to $OPT=L(v)$ In order to accomplish this, we aim to find w such that it is close to $v,\\mathrm{i . e .}\\parallel w-v\\parallel$ is small. Note that approximating v suffices to achieve an error close to $O P T$ , since we can upper-bound $L(w)$ as\n\n\\begin{array}{ c }{\\displaystyle\\boldsymbol{L}(\\boldsymbol{w})=\\frac 1 2\\mathbb E\\left[\\big(\\sigma(\\boldsymbol{w}^{\\top}\\boldsymbol{x})-y)^{2}\\right]=\\frac 1 2\\mathbb E\\left[\\big(\\sigma(\\boldsymbol{w}^{\\top}\\boldsymbol{x})-\\sigma(\\boldsymbol{v}^{\\top}\\boldsymbol{x})+\\sigma(\\boldsymbol{v}^{\\top}\\boldsymbol{x})-y)^{2}\\right]}\\\\ {\\le 2\\cdot\\frac 1 2\\mathbb E\\left[\\big(\\sigma(\\boldsymbol{w}^{\\top}\\boldsymbol{x})-\\sigma(\\boldsymbol{v}^{\\top}\\boldsymbol{x}))^{2}\\right]+2\\cdot\\frac 1 2\\mathbb E\\left[\\big(\\sigma(\\boldsymbol{v}^{\\top}\\boldsymbol{x})-y)^{2}\\right]=~2 F(\\boldsymbol{w})+2 O P T}\\end{array}\n\nthrough Young's inequality. The realizable portion of the loss $F(w)$ becomes $O(OPT)$ when $\\|w-v\\|\\leq O({\\sqrt{O P T}})$ (see Lemma1for a proof), and as a consequence we will get $O(OPT)$ ) error in total.\n\nTo formalize our intuition above, we adopt a similar proof strategy used inFrei et al. (2020). Namely, we argue that when optimizing with respect to the agnostic loss $L(w_{t})$ , we are always making some non-trivial progress due to a decrease in $\\lVert\\boldsymbol{w}_{t}-\\boldsymbol{v}\\rVert$ and due to a decrease in $F(w_{t})$ (which is just the realizable portion of the loss). Moreover, whenever we stop making progress, we will argue that at this point either $\\lVert w_{t}-v\\rVert\\leq O({\\sqrt{O P T}})$ or $\\begin{array}{ r }{\\|\\nabla F(w_{t})\\|\\leq O(\\sqrt{O P T});}\\end{array}$ in both cases, this iterate already achieves an error of $O(OPT)$ due to Lemma 1and Lemma 2\n\nChallenges in arguing progress. At a high-level the analysis of gradient descent follows a similar approach toFrei et al. (2020) which only handles zero bias. Yet there are several new ideas needed to obtain the stronger $\\overline{{O(O P T)}}$ guarantee even for the zero-bias case. Moreover, allowing non-zero bias terms imposes extra technical challenges. For example, the probability measure of $\\{w^{\\top}x\\geq 0,v^{\\top}x\\geq 0\\}$ under Gaussian distributions, which is vital to deriving the gain in each gradient descent step, does not have a closed-form expression when bias is present. Furthermore we cannot afford to lose any dimension dependent factors or assume boundedness. Thus, to address these difficulties, more detailed analyses (e.g. Lemma3 4) are needed to facilitate our argument.\n\nMoreover tackling non-zero bias terms requires additional assumptions when initializing $w_{0}$ as well. The initializer finds a $w_{0}$ such that $F(w_{0})$ is strictly less than $F(0)$ by a constant amount $\\delta > 0$ (this is inspired by Vardi et al.(2021), however δ in their case can have an inverse-polynomial dependence on the dimension ). In fact our multiscale random initialization and the improved analysis is crucial to obtaining a dimension-independent bound on the error. The high-level intuition behind why this property is useful is that it ensures that gradient descent does not get trapped around a highly non-smooth region (e.g. when $w=0)$ by making it start at somewhere better than it, so that w keeps moving closer to v. Moreover, in our case the analysis is more challenging to implement compared to Vardi et al. (2021)because of the agnostic setting. This is becauseVardi et al. (2021heavily relies on the realizability assumption to simplify its analysis.\n\nWe also highlight our improvements on the dependency of the dimension d. In previous works, the guarantees of the algorithm has a dependence on d either explicitly or implicitly. For instance, inFrei et al.(2020the $O({\\sqrt{O P T}})$ guarantee for ReLU neurons includes a coefficient in terms of $B_{X}$ (the upper-bound for $\\lVert x\\rVert)$ , which for Gaussian inputs is in fact ${\\sqrt{d}};$ or for example inVardi et al. the gain for each gradient descent iteration γ comes with a dependency on c (the upper-bound for $\\|{\\boldsymbol{x}}\\|)\\operatorname{of}c^{-8}$ , which for Gaussian is $d^{-4}$ . In contrast, we avoid such dependencies on the dimension d in order to obtain our guarantees.\n\nWe first establish two important lemmas we will later utilize in proving progress in each iteration. As stated in the preliminaries, we assume in the rest of the section that $\\bar{| |\\tilde{v}| |_{2}}=1$ . The first lemma gives a lower bound on the measure of the region where both $\\sigma(v^{\\top}x)$ and $\\dot{\\sigma}(w_{t}^{\\top}x)$ are non-zero. Our inductive hypotheses will ensure that this lower bound is a constant (if $\\left|b_{v}\\right|$ is a constant).\n\nLemma 3 (Lower bound on the measure of the intersection). Suppose the marginal distribution $\\widetilde{\\mathcal{D}}_{x}o\\nu e r\\widetilde{x}i s O(1)$ -regular. There exists an absolute constant $c > 0$ such that for all $\\delta>0_{:}$ if $F(w)\\leq F(0)-\\delta$ then\n\n\\mathbb P[w^{\\top}x\\geq 0,v^{\\top}x\\geq 0]\\geq\\frac{\\delta^{2}}{c\\|w\\|_{2}^{4}\\|v\\|_{2}^{4}}=\\frac{\\delta^{2}}{c\\|w\\|_{2}^{4}(1+ |b_{v}|^{2})^{2}}.\\tag{9}\n\nWith Lemma 3the following lemma allows us to get an improvement on the realizable portion of the loss function as long as the gradient is non-negligible. We state and prove this lemma for the general case of $O(1)$ -regular marginal distributions.\n\nLemma 4 (Improvement from the first order term). Suppose the marginal over x is $O(1)$ -regular. There exists absolute constants $c_{1},c_{2}\\ >\\ 0$ such that for any $\\delta>\\bar{0},i f\\|v\\|_{2},\\|w\\|_{2}\\leq B$ and $F(w)\\leq F(0)-\\delta,$ then $\\langle\\nabla F(w),w-v\\rangle\\geq\\gamma\\Vert w-v\\Vert^{2}$ , where $\\begin{array}{ r }{\\gamma=\\frac{c_{1}\\delta^{9}}{B^{28}}}\\end{array}$\n\nThe constants $c_{1},c_{2}$ depend on the constants $\\beta_{1},\\beta_{2}^{\\prime},\\beta_{2},\\beta_{4}$ etc. in the regularity assumption of $\\widetilde{\\mathcal{D}}_{x}$ We remark that for our setting of parameters $\\delta=\\Omega(1)$ and $B=O(1)$ , and hence we will conclude that $\\langle\\nabla F,w-v\\rangle\\geq\\Omega(\\|w-\\bar{v}\\|_{2}^{2})$ . Please refer to AppendixAfor all the complete proofs.\n\n4.1 Main Proof Strategy\n\nWith these two key lemmas, we are now ready to discuss the proof overview of the main theorem (Theorem[1). We inductively maintain two invariants in every iteration of the algorithm:\n\n\\begin{array}{ r }{\\mathrm{(A)}\\quad\\|w_{t}-v\\|_{2}\\leq O(1),\\quad\\mathrm{and}\\quad\\mathrm{(B)}\\ F(0)-F(w_{t})=\\Omega(1).}\\end{array}\n\nThese two invariants are true at $t = 0$ due to our initialization $w_{0}$ Lemma 5guarantees with at least constant probability $\\Omega(1)$ , both the invariants hold for $w_{0}$ . The proof that both the invariants continue to hold follows from the progress made by the algorithm due to a decrease in both $\\lVert\\boldsymbol{w}_{t}-\\boldsymbol{v}\\rVert_{2}$ and $F(w_{t})$ (note that we only need to show they do not increase to maintain the invariant).\n\nThe argument consists of two parts. First, assuming $F(w_{t})\\le F(0)-\\delta$ holds (for some constant $\\delta>0)$ , we establish that whenever $\\|w_{t}-v\\|^{2}>\\gamma\\breve{O}P T$ for some constant $\\gamma > 0 .$ gradient descent always makes progress i.e. $\\|w_{t}-v\\|^{2}-\\|w_{t+1}^{\\cdot}-\\dot{v}\\|^{2}$ is lower bounded. Next, we argue that if $w_{0}$ is initialized such that $F(w_{0})\\leq F(0)-\\delta$ for some constant $\\delta > 0$ , then throughout gradient descent $F(w_{t})$ always decreases, i.e. the inequality $F(w_{t})\\leq F(w_{0})\\leq F(0)-\\delta$ always holds.\n\nHowever, unlikeVardi et al. (2021) where they focus on the realizable setting, analyzing gradient descent on the agnostic loss $\\bar{\\boldsymbol{L}}(\\boldsymbol{w})$ is more challenging, since the update depends on $\\dot{\\nabla}L(\\bar{w})$ and not $\\nabla F(w)$ . In fact, the additional term from the \"non-realizable\" portion of the loss $L(w)$ can overwhelm the contribution from the realizable loss when either $\\lVert\\nabla F\\rVert_{2}~\\le~O(\\sqrt{O P T})$ or $\\|w_{t}\\mathrm{~-~}v\\|_{2}\\ \\leq$ $O({\\sqrt{O P T}})$ . The following two lemmas argue that in both of these cases, the current iterate already achieves $\\mathop{O}(O P T)$ error (and this iterate will be the T that satisfies the guarantee of Theorem[1)", "theorems": [{"local_id": "T1", "statement": "Assume $\\widetilde{\\mathcal{D}}_{x}i s O(1)$ -regular with parameters defined above. $I f\\|w_{t}-v\\|_{2}\\leq O({\\sqrt{O P T+\\varepsilon}})f,$ or some $\\varepsilon > 0$ , then $F(w_{t})\\leq O(O P T+\\varepsilon)$ •", "proof": {"steps": [{"n": 1, "claim": "Since ReLU function is 1-Lipschitz $(\\mathrm{i . e .}|\\sigma(z)-\\sigma(z^{\\prime})|\\le|z-z^{\\prime}|)$"}, {"n": 2, "claim": "F(w_{t})=\\frac{1}{2}\\operatorname{\\mathbb{E}}\\left[(\\sigma(w_{t}^{\\top}x)-\\sigma(v^{\\top}x))^{2}\\right]\\leq\\frac{1}{2}\\operatorname{\\mathbb{E}}\\left[(w_{t}^{\\top}x-v^{\\top}x)^{2}\\right]=\\frac{\\|w_{t}-v\\|^{2}}{2}\\operatorname{\\mathbb{E}}\\left[(u^{\\top}x)^{2}\\right]"}, {"n": 3, "claim": "where we defined $\\begin{array}{ r }{u=\\frac{w_{t}-v}{\\lVert w_{t}-v\\rVert}}\\end{array}$ , hence the last equation. Now, notice by using Young's inequality, we get"}, {"n": 4, "claim": "\\mathbb{E}\\left[(u^{\\top}x)^{2}\\right]=\\ \\mathbb{E}\\left[(\\widetilde{u}^{\\top}\\widetilde{x}+b_{u})^{2}\\right]\\leq\\ 2\\mathbb{E}\\left[(\\widetilde{u}^{\\top}\\widetilde{x})^{2}\\right]+2 b_{u}^{2}\\leq\\ 2\\beta_{2}+2 b_{u}^{2}\\leq\\ O(1)"}, {"n": 5, "claim": "due to the regularity assumption on $\\widetilde{\\mathcal{D}}_{x}$ . Hence"}, {"n": 6, "claim": "F(w_{t})\\leq\\frac{\\|w_{t}-v\\|^{2}}{2}\\cdot O(1)\\leq O(\\|w_{t}-v\\|_{2}^{2})\\leq O(O P T+\\varepsilon)"}, {"n": 7, "claim": "which concludes the proof."}]}}, {"local_id": "T2", "statement": "Suppose the marginal distribution $\\widetilde{\\mathcal{D}}_x$ over $\\tilde{x} \\in \\mathbb{R}^d$ is $O(1)$-regular (in particular there exists an absolute constant $\\beta_4 > 0$ with $\\mathbb{E}_{\\tilde{x}\\sim\\widetilde{\\mathcal{D}}_x}[\\langle u,\\tilde{x}\\rangle^4] \\le \\beta_4$ for every unit vector $u \\in \\mathbb{R}^d$), where points are lifted to $\\mathbb{R}^{d+1}$ via $x = (\\tilde{x}, 1)$ so that $w^\\top x = \\tilde{w}^\\top \\tilde{x} + b_w$, $\\sigma(z) = \\max(z,0)$ is the ReLU with $\\sigma'(z) = \\mathbb{1}\\{z \\ge 0\\}$, and $F(w) := \\frac{1}{2} \\mathbb{E}[(\\sigma(w^\\top x) - \\sigma(v^\\top x))^2]$ with $\\nabla F(w) := \\mathbb{E}[(\\sigma(w^\\top x) - \\sigma(v^\\top x)) \\sigma'(w^\\top x) x]$ and $F(0)$ the realizable loss at $w = 0$. Then (Lemma 4) there exist absolute constants $c_1, c_2 > 0$ such that for any $\\delta > 0$, if $\\|v\\|_2, \\|w\\|_2 \\le B$ and $F(w) \\le F(0) - \\delta$, then $\\langle \\nabla F(w), w - v\\rangle \\ge \\gamma \\|w - v\\|^2$, where $\\gamma = c_1 \\delta^9 / B^{28}$.", "proof": {"steps": [{"n": 1, "claim": "It suffices to upper bound $\\mathbb{P}_{\\tilde{x}\\sim\\widetilde{\\mathcal{D}}_x}[|\\tilde{u}^\\top \\tilde{x} + b_u| < \\tau]$ for the relevant unit vector $u = (\\tilde{u}, b_u)$ with $\\|u\\|_2 = 1$."}, {"n": 2, "claim": "Suppose $\\beta = \\|\\tilde{u}\\|_2 \\le \\frac{1}{4\\beta_4} (q/2)^{1/4}$. Then $|b_u| > 1/2$."}, {"n": 3, "claim": "$\\mathbb{P}_{\\tilde{x}\\sim\\widetilde{\\mathcal{D}}_x}[|\\tilde{u}^\\top \\tilde{x} + b_u| < \\tau] \\le \\mathbb{P}[|\\tilde{u}^\\top \\tilde{x}| > 1/4]$."}, {"n": 4, "claim": "$\\mathbb{P}[|\\tilde{u}^\\top \\tilde{x}| > 1/4] \\le \\mathbb{E}[\\langle \\tilde{u}, \\tilde{x}\\rangle^4] / (1/4)^4$."}, {"n": 5, "claim": "$\\mathbb{E}[\\langle \\tilde{u}, \\tilde{x}\\rangle^4] / (1/4)^4 \\le \\beta_4 (4\\|\\tilde{u}\\|_2)^4 \\le q/2$."}, {"n": 6, "claim": "$\\mathbb{E}[\\sigma(w^\\top x) \\sigma(v^\\top x)] \\le \\sqrt{\\mathbb{P}[\\cdot]} \\cdot \\sqrt{2 \\mathbb{E}[(w^\\top x)^4] + 2 \\mathbb{E}[(v^\\top x)^4]}$."}, {"n": 7, "claim": "$\\mathbb{E}[\\sigma(w^\\top x) \\sigma(v^\\top x)] \\le 8 \\sqrt{\\mathbb{P}[\\cdot]} \\cdot \\sqrt{\\mathbb{E}_{\\tilde{x}\\sim\\mathcal{N}(0,I)}[(\\tilde{w}^\\top \\tilde{x})^4] + b_w^4} \\cdot \\sqrt{\\mathbb{E}_{\\tilde{x}\\sim\\mathcal{N}(0,I)}[(\\tilde{v}^\\top \\tilde{x})^4] + b_v^4}$."}, {"n": 8, "claim": "$\\le 8 \\sqrt{\\mathbb{P}[\\cdot]} \\cdot \\sqrt{(\\beta_4 \\|\\tilde{w}\\|_2^4 + b_w^4)(\\beta_4 \\|\\tilde{v}\\|_2^4 + b_v^4)}$."}, {"n": 9, "claim": "$\\le c' \\sqrt{\\mathbb{P}[\\cdot]} \\cdot \\|w\\|_2^2 \\|v\\|_2^2$."}]}}, {"local_id": "T3", "statement": "Suppose $B,\\delta\\ >\\ 0$ are constants such that $\\|v\\|_{2},\\|w\\|_{2}\\ \\leq\\ B$ and $F(w)\\leq F(0)-\\delta .$ Then there exists a constant $C_{G}>0$ ,such that if $\\lVert\\nabla F(w)\\rVert\\leq C_{G}\\sqrt{O P T}$ then $\\|w-v\\|_{2}\\leq O({\\sqrt{O P T}})$ .", "proof": {"steps": [{"n": 1, "claim": "We can first apply Lemma $4$ to conclude that $\\langle\\nabla F(w),w-v\\rangle\\geq\\gamma\\Vert w-v\\Vert^{2}$ for some constant $\\gamma > 0$ (since $B,\\delta>0$ are constants), hence we have $\\begin{array}{ r }{\\|\\nabla F(w)\\|\\|w-v\\|\\geq\\langle\\nabla F(w),w-}\\end{array}$ $v\\rangle\\geq\\gamma\\Vert w-v\\Vert^{2}$ ."}, {"n": 2, "claim": "Thus $\\|w-v\\|_{2}=O({\\sqrt{O P T}})$ which implies the lemma."}]}}], "target_local_id": "T2", "topic_domain": "Optimization", "topic_subtopic": "Non-convex Geometry", "label": "flawed", "error_step": 2, "error_class6": ["missing_assumption", "invalid_bound_or_inequality", "logical_gap_or_missing_case"], "error_explanation": "The step concludes |b_u| > 1/2 from the case assumption β = ‖ũ‖2 ≤ (1/(4β4))(q/2)^{1/4} together with ‖u‖2 = 1. This is unjustified: the O(1)-regularity assumption only requires the fourth-moment constant β4 > 0, and β4 may be arbitrarily small. When β4 is small the threshold (1/(4β4))(q/2)^{1/4} becomes arbitrarily large, so ‖ũ‖2 can be large while still satisfying the bound, and from ‖ũ‖2^2 + b_u^2 = 1 one cannot conclude |b_u| > 1/2. The correct argument must keep the expectation under the general O(1)-regular distribution D̃x and use its fourth-moment bound explicitly, folding the β4 dependence into the case threshold so that |b_u| > 1/2 still follows when β4 is small (the authors later patch this by adding O(1)-regularity to Lemma 4.3). A closely related instance of the same defect appears at Eq.(20)/the Eq.(11) fourth-moment step, where the expectation over the general O(1)-regular D̃x is silently replaced by an expectation over the Gaussian N(0,I)."}
{"id": "FUdBF49WRV1", "context": "Let $G=(V,E)$ be a graph with V the set of vertices and $E \\subset V \\times V$ the set of edges. The graph is undirected meaning that $(i,j)\\in E\\mathrm{iff}(j,i)\\in E$ . Define the vector spaces $L^{2}(V)$ and $L^{2}({\\bar{E}})$ as the set of maps $V\\mathbb{R}$ and $E\\to\\mathbb{R}$ with ${\\bf{\\bar{x}}},{\\bf{\\bar{y}}}\\in L^{2}(V)$ and $F,H\\in L^{\\hat{2}}(E)$ and scalar products\n\n$$\n\\langle{\\pmb x},{\\pmb y}\\rangle_{L^{2}(V)}: =\\sum_{i\\in V}x_{i}{\\pmb y}_{i}\\quad,\\qquad\\langle{\\pmb F},{\\pmb H}\\rangle_{L^{2}(E)}: =\\sum_{(i,j)\\in E}{\\pmb F}_{(i,j)}{\\pmb H}_{(i,j)}\\tag{3}\n$$\n\nThink of E as the \"tangent space\" to V and of $L^{2}(E)$ as the set of \"vector fields\" on the space V with each row $F_{i,:}$ representing a vector at the i-th node. Define the pointwise scalar product as the map $L^{2}(E)\\times L^{2}(E){\\overset{\\cdot}{\\to}}L^{2}(V)$ taking 2 vector fields and returning their inner product at each point of V , at the node i is defined by equation 4.\n\n$$\n\\langle F,H\\rangle_{i}: =\\sum_{j:(i,j)\\in E}F_{i,j}H_{i,j}\\tag{4}\n$$\n\nIn equation 5, we define the gradient  as a mapping $L^{2}(V)\\to L^{2}(E)$ and the divergence div as a mapping $L^{2}(E)\\to L^{2}(V)$ , thus leading to an analogue of the directional derivative in equation 6.\n\n$$\n(\\nabla\\pmb{x})_{(i,j)}: =\\pmb{x}(j)-\\pmb{x}(i)\\quad,\\qquad(\\mathrm{div}\\pmb{F})_{i}: =\\sum_{j:(i,j)\\in E}\\pmb{F}_{(i,j)}\\tag{5}\n$$\n\nDefinition 2. The directional derivative of the function x on the graph G in the direction of the vector field  where each vector is of unit-norm is\n\n$$\nD_{\\hat{\\cal F}}{\\pmb x}(i): =\\langle\\nabla{\\pmb x},\\hat{\\pmb F}\\rangle_{i}=\\sum_{j:(i,j)\\in E}({\\pmb x}(j)-{\\pmb x}(i))\\hat{\\pmb F}_{i,j}\\tag{6}\n$$\n\n|F| will denote the absolute value of F and $| |F_{i,:}| |_{L^{p}}$ the $L^{p}$ -norm of the i-th row of F. We also define the forward/backward directions as the positive/negative parts of the field $F^{\\pm}$\n\nThe directional average matrix $B_{a v}$ is the weighted aggregation matrix such that all weights are positives and all rows have an L1-norm equal to 1, as shown in equation 7 and theorem 1, with a proof in the appendix C.1.\n\n$$\nB_{a v}(\\boldsymbol{F})_{i,:}=\\frac{|F_{i,:}|}{| |F_{i,:}| |_{L^{1}}+\\epsilon}\\tag{7}\n$$\n\nThe variable  is an arbitrarily small positive number used to avoid floating-point errors. The $L^{1}.$ norm denominator is a local row-wise normalization. The aggregator works by assigning a large weight to the elements in the forward or backward direction of the field, while assigning a small weight to the other elements, with a total weight of 1.\n\nThe directional derivative matrix $\\mathbf{\\delta}_{B_{d x}}$ is defined in (8) and theorem 2, with the proof in appendix C.2. Again, the denominator is a local row-wise normalization but can be replaced by a global normalization. diag(a) is a square, diagonal matrix with diagonal entries given by a. The aggregator works by subtracting the projected forward message by the backward message (similar to a center derivative), with an additional diagonal term to balance both directions.\n\n$$\nB_{d x}(F)_{i,:}=\\hat{F}_{i,:}-\\mathrm{diag}\\Big(\\sum_{j}\\hat{F}_{:,j}\\Big)_{i,:}\\qquad\\hat{F}_{i,:}=\\left(\\frac{F_{i,:}}{| |F_{i,:}| |_{L^{1}}+\\epsilon}\\right)\\tag{8}\n$$\n\nWe also compare the expressiveness of the Directional Graph Networks with the classical WL graph isomorphism test which is often used to classify the expressivity of graph neural networks (Xu et al., 2018a). In theorem . (prove in appendix C.8) we prove that DGNs are capable of distinguishing pairs of graphs that the 1-WL test (and so ordinary GNNs) cannot differentiate.\n\nFigure 8: Illustration of an example pair of graphs which the 1-WL test cannot distinguish but DGNs can. The table shows the node feature updates done at every layer. MPNN with mean/sum aggregators and the 1-WL test only use the updates in the first row and therefore cannot distinguish between the nodes in the two graphs. DGNs also use directional aggregators that, with the vector field given by the first eigenvector of the Laplacian matrix, provides different updates to the nodes in the two graphs.\n\n[Figure content: two graphs titled \"Graph 1\" and \"Graph 2\", each with six nodes, two labelled $a$ and four labelled $b$. The figure shows only the labels; the node names $u_1,\\dots,u_6$ and $v_1,\\dots,v_6$ are ours.]\n\n$$\n\\begin{array}{c|c|l}\n\\textbf{Graph 1} & \\text{label} & \\text{neighbours} \\\\\n\\hline\nu_1 & b & u_2,\\ u_4 \\\\\nu_2 & a & u_1,\\ u_3,\\ u_5 \\\\\nu_3 & b & u_2,\\ u_6 \\\\\nu_4 & b & u_1,\\ u_5 \\\\\nu_5 & a & u_2,\\ u_4,\\ u_6 \\\\\nu_6 & b & u_3,\\ u_5\n\\end{array}\n\\qquad\n\\begin{array}{c|c|l}\n\\textbf{Graph 2} & \\text{label} & \\text{neighbours} \\\\\n\\hline\nv_1 & b & v_2,\\ v_3 \\\\\nv_2 & b & v_1,\\ v_3 \\\\\nv_3 & a & v_1,\\ v_2,\\ v_4 \\\\\nv_4 & a & v_3,\\ v_5,\\ v_6 \\\\\nv_5 & b & v_4,\\ v_6 \\\\\nv_6 & b & v_4,\\ v_5\n\\end{array}\n$$\n\nGraph 1 edges: $\\{u_1,u_2\\},\\{u_2,u_3\\},\\{u_4,u_5\\},\\{u_5,u_6\\},\\{u_1,u_4\\},\\{u_2,u_5\\},\\{u_3,u_6\\}$. Graph 2 edges: $\\{v_1,v_2\\},\\{v_1,v_3\\},\\{v_2,v_3\\},\\{v_3,v_4\\},\\{v_4,v_5\\},\\{v_4,v_6\\},\\{v_5,v_6\\}$.\n\n$$\n\\begin{array}{c|c|c}\n\\textbf{Aggregation matrix} & \\textbf{Graph 1} & \\textbf{Graph 2} \\\\\n\\hline\n\\boldsymbol{A} & \\begin{array}{c} 1a + 1b \\to b \\\\ 1a + 2b \\to a \\end{array} & \\begin{array}{c} 1a + 1b \\to b \\\\ 1a + 2b \\to a \\end{array} \\\\\n\\hline\n\\boldsymbol{B}^{1}_{dx} & \\begin{array}{c} |1a - 1b| \\to b \\\\ 0 \\to a \\end{array} & \\begin{array}{c} |1a - 1b| \\to b \\\\ |0.44b - 0.44a| \\to a \\end{array} \\\\\n\\hline\n\\boldsymbol{B}^{1}_{av} & \\begin{array}{c} 1a \\to b \\\\ 1b \\to a \\end{array} & \\begin{array}{c} 1a \\to b \\\\ 0.44b + 0.56a \\to a \\end{array}\n\\end{array}\n$$\n\nAnother important result is that the directions in the graph allow to replicate some of the most common data augmentation techniques used in computer vision, namely reflection, rotation and distortion. The main difference is that, instead of modifying the image (such as a $5^{\\circ}$ rotation), the proposed transformation is applied on the vector field defining the aggregation kernel (thus rotating the kernel by $-5^{\\circ}$ without changing the image). This offers the advantage of avoiding to pre-process the data since the augmentation is done directly on the kernel at each iteration of the training.\n\nThe simplest augmentation is the vector field flipping, which is done changing the sign of the field F, as stated in definition 3. This changes the sign of $\\mathbf{\\delta}_{B_{d x}}$ , but leaves $B_{a v}$ unchanged.\n\nDefinition 3 (Reflection of the vector field). For a vector field F, the reflected field $i s - { \\pmb F }$\n\nLet $F_{1},F_{2}$ be vector fields in a graph, with $\\hat{F}_{1}$ and $\\hat{F}_{2}$ being the field normalized such that each row has a unitary $L^{2}\\mathrm{-n o r m}$ Define the angle vector α by $\\langle(\\hat{F}_{1})_{i,:},(\\hat{F}_{2})_{i,:}\\rangle=\\cos(\\alpha_{i})$ . The vector field $\\hat{F}_{2}^{\\perp}$ is the normalized component of $\\hat{F}_{2}$ perpendicular to $\\hat{F}_{1}$ . The equation below defines $\\hat{F}_{2}^{\\perp}$ . The next equation defines the angle\n\n$$\n(\\hat{F}_{2}^{\\perp})_{i,:}=\\frac{(\\hat{F}_{2}-\\langle\\hat{F}_{1},\\hat{F}_{2}\\rangle\\hat{F}_{1})_{i,:}}{| |(\\hat{F}_{2}-\\langle\\hat{F}_{1},\\hat{F}_{2}\\rangle\\hat{F}_{1})_{i,:}| |}\n$$\n\nNotice that we then have the decomposition $(\\hat{F}_{2})_{i,:}=\\cos(\\alpha_{i})(\\hat{F}_{1})_{i,:}+\\sin(\\alpha_{i})(\\hat{F}_{2}^{\\perp})_{i,:}.$\n\nDefinition 4 (Rotation of the vector fields). For $\\hat{F}_{1}$ and $\\hat{F}_{2}$ non-colinear vector fields with each vector of unitary length, their rotation by the angle θ in the plane formed by $\\{\\hat{F}_{1},\\hat{F}_{2}\\}$ is\n\n$$\n\\hat{F}_{1}^{\\theta}=\\hat{F}_{1}\\mathrm{diag}(\\cos\\theta)+\\hat{F}_{2}^{\\bot}\\mathrm{diag}(\\sin\\theta)\\quad,\\quad\\hat{F}_{2}^{\\theta}=\\hat{F}_{1}\\mathrm{diag}(\\cos(\\theta+\\alpha))+\\hat{F}_{2}^{\\bot}\\mathrm{diag}(\\sin(\\theta+\\alpha))\\tag{9}\n$$", "theorems": [{"local_id": "T1", "statement": "The operation $\\pmb{y}=\\pmb{B}_{a v}\\pmb{x}$ is the directional average of x, in the sense that $\\mathbf{\\nabla}_{\\mathbf{{\\boldsymbol{y}}}_{u}}$ is the mean of $\\scriptstyle{\\mathbf{x}}_{v},$ weighted by the direction and amplitude of F.", "proof": {"steps": [{"n": 1, "claim": "This should be a simple proof, that if we want a weighted average of our neighbours, we simply need to multiply the weights by each neighbour, and divide by the sum of the weights."}, {"n": 2, "claim": "Of course, the weights should be positive."}]}}, {"local_id": "T2", "statement": "Let $F_1$ and $F_2$ be non-colinear vector fields on a graph $G=(V,E)$, let $\\alpha$ be defined by $\\langle (\\hat{F}_1)_{i,:}, (\\hat{F}_2)_{i,:}\\rangle = \\cos(\\alpha_i)$, and let $F_2^{\\perp}$ be the normalized component of $F_2$ perpendicular to $F_1$, $(\\hat{F}_2^{\\perp})_{i,:} = \\frac{(\\hat{F}_2 - \\langle \\hat{F}_1,\\hat{F}_2\\rangle \\hat{F}_1)_{i,:}}{\\|(\\hat{F}_2 - \\langle \\hat{F}_1,\\hat{F}_2\\rangle \\hat{F}_1)_{i,:}\\|}$. The rotation by angle $\\theta$ in the plane $\\{F_1,F_2\\}$, $F_1^{\\theta} = F_1\\,\\mathrm{diag}(\\cos\\theta) + F_2^{\\perp}\\,\\mathrm{diag}(\\sin\\theta)$ and $F_2^{\\theta} = F_1\\,\\mathrm{diag}(\\cos(\\theta+\\alpha)) + F_2^{\\perp}\\,\\mathrm{diag}(\\sin(\\theta+\\alpha))$, reduces to the identity at $\\theta=0$: $F_1^{\\theta=0}=F_1$ and $F_2^{\\theta=0}=F_2$.", "proof": {"steps": [{"n": 1, "claim": "The property to check is that the rotation reduces to the identity at $\\theta=0$: $F_1^{\\theta=0}=F_1$ and $F_2^{\\theta=0}=F_2$."}, {"n": 2, "claim": "At $\\theta=0$, $F_1^{\\theta=0} = F_1\\,\\mathrm{diag}(\\cos 0) + F_2^{\\perp}\\,\\mathrm{diag}(\\sin 0) = F_1$."}, {"n": 3, "claim": "$F_2 = \\cos(\\alpha)F_1 + \\sin(\\alpha)F_2^{\\perp}$."}, {"n": 4, "claim": "At $\\theta=0$, $F_2^{\\theta=0} = F_1\\,\\mathrm{diag}(\\cos\\alpha) + F_2^{\\perp}\\,\\mathrm{diag}(\\sin\\alpha) = \\cos(\\alpha)F_1 + \\sin(\\alpha)F_2^{\\perp}$."}, {"n": 5, "claim": "Hence $F_2^{\\theta=0} = F_2$, so the formulas give a rotation of $(F_1,F_2)$ by $\\theta$."}]}}, {"local_id": "T3", "statement": "DGNs using the mean aggregator, any directional aggregator of the first eigenvector and injective degree-scalers are strictly more powerful than the 1-WL test.", "proof": {"steps": [{"n": 1, "claim": "We will show that (1) DGNs are at least as powerful as the 1-WL test and (2) there is a pair of graphs which are not distinguishable by the 1-WL test which DGNs can discriminate."}, {"n": 2, "claim": "Since the DGNs include the mean aggregator combined with at least an injective degree-scaler, Corso et al. (2020) show that the resulting architecture is at least as powerful as the 1-WL test."}, {"n": 3, "claim": "Then, to show that the DGNs are strictly more powerful than the 1-WL test it suffices to provide an example of a pair of graphs that DGNs can differentiate and 1-WL cannot. Such a pair of graphs is illustrated in figure 8."}, {"n": 4, "claim": "The 1-WL test (as any MPNN with, for example, sum aggregator) will always have the same features for all the nodes labelled with a and for all the nodes labelled with b and, therefore, will classify the graphs as isomorphic."}, {"n": 5, "claim": "DGNs, via the directional smoothing or directional derivative aggregators based on the first eigenvector of the Laplacian matrix, will update the features of the a nodes differently in the two graphs (figure 8 presents also the aggregation functions) and will, therefore, be capable of distinguishing them."}]}}, {"local_id": "T4", "statement": "Suppose $\\hat{F}$ have rows of unit $L^{1}$ norm. The operation ${\\boldsymbol{\\mathsf{y}}}=$ $B_{d x}({\\hat{F}}){:}$ x is the centered directional derivative of x in the direction of F, in the sense of equation 6, i.e. $$ {\\pmb y}=D_{\\hat{\\pmb F}}{\\pmb x}=\\bigg(\\hat{\\pmb F}-\\mathrm{diag}\\bigg(\\sum_{j}\\hat{\\pmb F}_{:,j}\\bigg)\\bigg){\\pmb x} $$", "proof": {"steps": [{"n": 1, "claim": "Since F rows have unit $L^{1}$ norm, $\\begin{array}{ r l r }{\\hat{\\bf F}}&{{}=}&{F}\\end{array}$"}, {"n": 2, "claim": "The i-th coordinate of the vector $\\begin{array}{ r }{\\bigg(\\boldsymbol{F}-\\mathrm{diag}\\left(\\sum_{j}\\boldsymbol{F}_{:,j}\\right)\\bigg)}\\end{array}$ x is"}, {"n": 3, "claim": "\\begin{array}{ l }{\\left(F x-\\mathrm{diag}\\left(\\displaystyle\\sum_{j}F\\right)x\\right)_{i}=\\displaystyle\\sum_{j}F_{i,j}x(j)-\\left(\\displaystyle\\sum_{j}F_{i,j}\\right)x(i)}\\\\ {\\displaystyle=\\displaystyle\\sum_{j:(i,j)\\in E}(x(j)-x(i))F_{i,j}}\\\\ {=D_{F}x(i)}\\end{array}"}]}}], "target_local_id": "T2", "topic_domain": "Others", "topic_subtopic": "Graph Signal Processing", "label": "flawed", "error_step": 3, "error_class6": ["missing_assumption", "computation_or_constant", "invalid_bound_or_inequality"], "error_explanation": "Step 3 asserts the decomposition $F_2 = \\cos(\\alpha)F_1 + \\sin(\\alpha)F_2^{\\perp}$ for arbitrary (non-unit-norm) vector fields $F_1,F_2$. This identity holds only when each row vector has unit $L^2$-norm. Since $F_2^{\\perp}$ is defined as the NORMALIZED perpendicular component, the expression $\\cos(\\alpha)F_1 + \\sin(\\alpha)F_2^{\\perp}$ reproduces the parallel part $F_2^{\\parallel}$ correctly but fixes the perpendicular magnitude to $\\sin(\\alpha)$ instead of the true magnitude $k=\\|F_2 - F_2^{\\parallel}\\|$. For a general vector these differ ($\\|F_2\\|\\neq 1$), so the decomposition — and hence the identity $F_2^{\\theta=0}=F_2$ derived from it in step 5 — fails. The rotation therefore does not preserve the field even at zero angle. The fix is to require unit $L^2$-norm rows (use $\\hat{F}_1,\\hat{F}_2$), under which $\\hat{F}_2=\\cos(\\alpha)\\hat{F}_1+\\sin(\\alpha)\\hat{F}_2^{\\perp}$ holds exactly."}
{"id": "H1lNb0NtPH", "context": "2 Designing a Dataset Difficulty Measure\n\nFor supervised learning across data modalities such as images and text, data samples can usually be viewed as feature-label pairs (x, y) where $x\\in\\mathcal{X}\\subset\\mathbb{R}^{d_{x}}$ , and $y\\in\\mathcal{V}$ . We focus on classification problems where the labels y are discrete, i.e., $\\mathcal{V}\\subset\\mathbb{Z}^{+}$ . We denote the joint distribution of the feature-label pairs as $\\mathcal{P}_{\\mathcal{X}\\mathcal{Y}}$ . The marginal distributions of the features and labels are denoted as $\\mathcal{P}_{\\mathcal{X}}$ and $\\mathcal{P}_{\\mathcal{Y}}$ respectively. We make the following widely adopted assumption from learning theory literature about how samples are generated:\n\nAssumption 2. The feature-label pairs $(x,y)$ in the datasets, both training and testing, are sampled i.i.d. from a static distribution $(x,y)\\sim{\\mathcal{P}}_{\\chi y}$\n\nIntuitively, there are many possible indicators for the potential difficulty of a dataset: the number of features, the number of classes, the number of samples, the distinguishability of samples across classes, as well as the difference between the data distributions of the training set and the testing set. However, none of these indicators alone can fully describe the relative difficulty of a dataset.\n\nFrom Assumption 2, if the samples $(x,y)$ are sampled i.i.d. from $\\mathcal{P}_{\\mathcal{X}\\mathcal{Y}}$ , where Y is discrete, a natural measure that characterizes the difficulty of the data distribution is the best probability of the 0-1 error that can be achieved by any estimator.\n\nDefinition 2 (Model-Agnostic Error). $\\begin{array}{ r }{\\mathcal{P}_{e}=\\operatorname*{inf}_{f}\\mathcal{P}_{x,y\\sim\\mathcal{P}x y}\\left[f(x)\\neq y\\right]}\\end{array}$\n\nThe measure 1 is straightforward but unfortunately it is hard to compute since it involves evaluations against all possible estimators. However, with mild assumptions, it can be lower bounded by terms related to the conditional entropy, which is much easier to evaluate.\n\n2.1 Discrete Features\n\nIn the case where the features $x\\in\\mathcal{X}$ are discrete, according to Fano's theorem, $\\mathcal{P}_{e}$ is lower bounded: Fano's inequality. If both X and Y are discrete random variables, then $\\begin{array}{ r }{\\mathcal{P}_{e}\\ge\\frac{\\mathbb{H}(Y|X)-1}{\\log|\\mathcal{V}|}}\\end{array}$ , where |V| is the cardinality of the label set Y, and $\\mathbb{H}(Y|X)$ is the conditional entropy.\n\nHowever, even though data can be represented using discrete integers, treating the features as discrete random variables leads to the following difficulties:\n\n1.The cardinality of the feature space becomes extremely large if discrete features are used. For image data, since each pixel is represented as an integer, the (raw) feature dimension would bee he mber  pixel in tage, whic can b xtey rSmilarly, or n data, when the sequence length is long, the feature dimension becomes large very quickly.\n\n2Given the large feature space, finding a matching set of features between training and testing data from the limited number of training and test samples would be unlikely. As a result, probability mass estimation on each discrete value would be impractical, since we may only see at most one sample for each discrete configuration.\n\n2.2 Continuous Features\n\nAs opposed to treating the features x as discrete random variables, if we view them as i.i.d. samples from a continuous distribution with probability density $p_{x},$ we can estimate the conditional entropy $\\mathbb{H}(Y|X)$ under some smoothness assumptions and also infer the model-agnostic error $\\mathcal{P}_{e}$\n\nHowever, classical Fano's inequality only holds for discrete random variables. For the case with the continuous features and discrete labels, it has not been shown how $\\mathcal{P}_{e}$ can be controlled. In this paper, we prove a generalized version of Fano's inequality that holds for the continuous-featurediscrete-label scenario. Formally, for the continuous-X-discrete-Y case:\n\nDefinition 3. Hybrid Conditional Entropy\n\n$$\n\\mathbb{H}(Y|X): =\\mathbb{E}_{X}\\left[-\\sum_{y=1}^{|\\mathcal{V}|}\\mathbb{P}(Y=y|X)\\log\\mathbb{P}(Y=y|X)\\right],\\tag{2}\n$$\n\nwhere $\\mathbb{P}(Y=y|X): =\\mathbb{E}[\\mathbf{1}(Y=y)|X]$\n\nThis definition of $\\mathbb{H}(Y|X)$ is consistent with the classical definition, in the sense that both of them give the intuition that how much information or uncertainty is left for Y given X.\n\nNext, to connect $\\mathcal{P}_{e}$ and the hybrid $\\mathbb{H}(Y|X)$ , we introduce an assumption on the function f\n\nDefinition 4. Smooth Discretization Property: The function $f:\\mathcal{X}\\mathcal{Y}$ satisfies the smooth discretization property if for every $y\\in\\mathcal{V},$ almost every $x\\in\\mathcal{X}$ (x a.e. in $\\mathcal{X})$\n\n$$\nf(x)=y\\iff\\exists\\delta>0\\quad s . t .\\quad\\forall\\tilde{x}\\in B_{\\delta}(x),\\quad f(\\tilde{x})=y,\n$$\n\nwhere $B_{\\delta}(x): =\\{\\tilde{x}\\in\\Omega:\\|\\tilde{x}-x\\|_{2}<\\delta\\}$ is a δ—neighborhood of x in X.\n\nThis assumption on the classifier function f is not unnatural considering f maps a continuous variable to a discrete variable. Without this assumption, it would be extremely hard to quantify the population error probability $P(f(X)\\neq Y)$ since the behavior of f may be erratic. Furthermore, in the real data setting, this assumption is always satisfied for every classifier $f,$ as we can always construct a small enough neighborhood of each data point such that they are disjoint and assume f is a constant in each neighborhood. In this sense the assumption on $f$ is pretty minimal. Now we are ready to extend Fano's inequality:\n\nTheorem 2. Fano's Inequality for Continuous Features: Let $\\mathcal{P}_{e}$ be the minimum error probability, i.e.,\n\n$$\n{\\mathcal{P}}_{e}=\\operatorname*{inf}_{f}P(f(X)\\neq Y)\n$$\n\nwhere f is any estimator of Y based on the observation X that satisfies the smooth discretization property. Then we have\n\n$$\n\\mathbb{H}(\\mathcal{P}_{e})+\\mathcal{P}_{e}\\log(m-1)\\geq\\mathbb{H}(Y|X),\\tag{3}\n$$\n\nwhere $\\mathbb H(\\mathcal P_{e}): = -\\mathcal P_{e}\\log\\mathcal P_{e}-(1-\\mathcal P_{e})\\log(1-\\mathcal P_{e}).$\n\nProof. See appendix B.\n\n3 Estimating the Lower Bound\n\nIt is natural to consider $\\mathcal{P}_{e}$ defined in Theorem 2 as a measure of dataset difficulty. Unfortunately, direct estimation of $\\mathcal{P}_{e}$ is impractical since one has to evaluate the estimation error against all possible estimators. However, theorem 2 provides an alternative towards estimating a lower bound on $\\mathcal{P}_{\\epsilon}$ through estimating the hybrid conditional entropy $\\mathbb{H}(Y|X)$ defined in Equation (2).\n\n3.1 Conditional Entropy Estimation\n\nIn real applications, direct calculation of hybrid conditional entropy $\\mathbb{H}(Y|X)$ according to Definition 3 is impossible since $\\mathbb{P}(Y=y|X)$ is unknown. However, similar to the conditional entropy for discrete random variables, the hybrid conditional entropy H $[(Y|X)$ can also be written as\n\n$$\n\\mathbb{H}(Y|X)=\\mathbb{H}(Y)-\\sum_{y=1}^{|\\mathcal{D}|}P(Y=y)\\mathbb{K}\\mathbb{L}(X|Y=y| |X).\\tag{4}\n$$\n\nPlease refer to Appendix A for a detailed proof of Equation (4). We also define the hybrid mutual information $\\mathbb{I}(X;{\\bar{Y}})$ , which is compatible with H(Y |X):\n\n$$\n\\mathbb{I}(X;Y)=\\sum_{y=1}^{|\\mathcal{V}|}P(Y=y)\\mathbb{K L}(X|Y=y| |X)\\tag{5}\n$$\n\nIn some benchmark datasets with balanced classes (e.g., CIFAR-10 and MNIST), $\\mathbb{H}(Y|X)\\ =$ log $\\begin{array}{ r }{|\\mathcal{V}| -\\frac{1}{|\\mathcal{V}|}\\sum_{y=1}^{|\\mathcal{V}|}\\mathbb{K}\\mathbb{L}(X|Y=y| |X)}\\end{array}$ . This indicates that if a dataset has more classes and the features for different classes are closer to each other on average, then $\\mathbb{H}(Y|X)$ would be larger.\n\n3.2 A Variational KL Divergence Estimator using Neural Networks\n\nWe now discuss the practical design of an estimator for $\\mathbb{K L}(X|Y)$ . Even though there are nonparametric KL divergence estimators such as kNN based techniques (Noshad et al., 2017; Perez-Cruz, 2008; Wang et al., 2009), they rely on some knowledge of the internal dimension of the data manifold. However, the internal dimension of the data manifold is usually much smaller than the dimension of the raw features and is hard to estimate. As an alternative, we design a neural networkbased estimator for KL divergence estimation, inspired by Belghazi et al. (2018). The estimator is based on the following theorem:\n\nDonsker-Varadhan representation.\n\n$$\n\\mathbb{K L}(\\mathcal{P}| |\\mathcal{Q})=\\operatorname*{sup}_{T:\\mathcal{X}\\mathbb{R}}\\mathbb{E}_{\\mathcal{P}}[T]-\\log\\mathbb{E}_{\\mathcal{Q}}[e^{T}]\\tag{6}\n$$\n\nIf we parameterize the function T using a neural network, we obtain:\n\n$$\n\\mathbb{K L}(\\mathcal{P}| |\\mathcal{Q})\\geq\\operatorname*{sup}_{\\theta\\in\\Theta}\\mathbb{E}_{\\mathcal{P}}[T_{\\theta}]-\\log\\mathbb{E}_{\\mathcal{Q}}[e^{T_{\\theta}}],\\tag{7}\n$$\n\nwhere θ represents the neural network parameters.\n\nIn our empirical evaluation, we optimize the empirical average instead of the expectation. However this may cause issues such as overftting. To mitigate this issue, we split the data into separate training and validation sets. The neural network models are trained on the training set and the estimation is made using the validation set. See Algorithm 1 for details.\n\n```latex\nAlgorithm 1 DIME\nInitialize $|\\mathcal{V}|$ neural networks: $T_{\\theta_{1}},T_{\\theta_{2}},\\ldots,T_{\\theta_{|y|}}.$ ,one per class.\nInitialize class counters $k_{1}=0,\\ldots,k_{c}=0$ , one per class. Initialize the sample counter $k = 0 .$\nfor t in $0,\\ldots,T$ do\nDraw a batch of b training examples $\\{(x_{i},y_{i})\\}$ , and b evaluation examples $\\{(x_{i}^{e},y_{i}^{e})\\}$\nfor each class $c\\in\\{1,\\ldots,|\\mathcal{V}|\\}$ do\nFind samples of the c-th class: $S_{c}=\\{i|y_{i}=c\\},S_{c}^{e}=\\{i|y_{i}^{e}=c\\}$\nTrain $\\begin{array}{ r }{\\theta_{c}=\\arg\\operatorname*{max}_{\\theta_{c}}\\frac{1}{|S_{c}|}\\sum_{S_{c}}T_{\\theta_{c}}(x_{i})-\\log [\\frac{1}{b}\\sum_{j}\\exp T_{\\theta_{c}}(x_{j})]}\\end{array}$\nEvaluate $\\begin{array}{ r }{\\widehat{\\mathbb{K}\\mathbb{L}}_{\\theta_{c}}=\\frac{1}{|S_{c}^{e}|}\\sum_{S_{c}^{e}}T_{\\theta_{c}}(x_{i}^{e})-\\log [\\frac{1}{b}\\sum_{j}}\\end{array}$ exp Tθc(xe)]\nIf $\\widehat{\\mathbb{K L}}_{\\theta_{c}}$ stops increasing, pause the training for $T_{\\theta_{c}}$\nUpdate counters: $k_{c}+\\doteq\\lvert S_{c}^{e}\\rvert .$\nend for\nUpdate the sample counter: $k + = b$\nIf all neural networks stop updating, break.\nend for\nEstimate the class probability $\\begin{array}{ r }{\\widehat{p_{c}}=\\frac{k_{c}}{k},c=\\{1,\\dots,|\\mathcal{V}|\\}}\\end{array}$\nCalculate ${\\widehat{\\mathbb{H}}}(Y|X)$ using $(4),\\widehat{p}_{c},$ ,and $\\widehat{\\mathbb{K}\\mathbb{L}}_{\\theta_{c}},c=\\{1,\\dots,|\\mathcal{V}|\\}$\nSolve (3) and output the solution.\n```\n\n3.3 DIME as a Difficulty Measure\n\nGiven that we are using a neural network model to estimate the KL divergence between $\\mathcal{P}(X|y)$ and ${\\mathcal{P}}(X)$ , according to Equation $(7),{\\widehat{\\mathbb{K L}}}(X|y| |X)\\leq{\\mathbb{K L}}(X|y| |X)$ . As a consequence, the estimated mutual information $\\widehat{\\mathbb{I}}(X;Y)\\leq\\mathbb{I}(X;Y)$ , which leads to a larger ${\\widehat{\\mathbb{H}}}(Y|X)$ . In the end, DIME could be larger as compared to true lower bound on $\\mathcal{P}_{e}$\n\nThe caveat is that $\\mathcal{P}_{e}$ is defined as a model-agnostic measure, but we use a neural network model to estimate its lower bound instead. The hope is, if the function class of the neural network is large enough, the gap between ${\\widehat{\\mathbb{K L}}}(X|y| |X)$ and $\\mathbb{K L}(X|y| |X)$ is small enough so that DIME won't deviate too much from the true lower bound of $\\mathcal{P}_{e}$ . We now describe our experiments for DIME using vision and language benchmark datasets.\n\nA Hybrid Conditional Entropy\n\nConsider the usual classification setting, X, representing the feature, is a random vector in $\\mathbb{R}^{p};Y$ representing the label, is a random variable(or vector) in $\\{1,\\ldots,|\\mathcal{V}|\\}$ . Assume X is a continuous variable with density $p_{X}(x)$ for x in some compact domain $\\Omega\\subset\\mathcal{X}\\subset\\mathbb{R}^{p}$ Can we define a hybrid conditional entropy $\\operatorname{\\mathbb{H}}(Y|X)$ while X is continuous and Y is discrete? We give a formal definition for this mixed-pair case and demonstrate that $\\mathbb{H}(Y|X)$ defined in our way can be a good indicator for the hardness of classification on the data $(X,Y)$ , which is consistent with the classical definition in the sense that both of them give the intuition that how much information or uncertainty is left for Y given X.\n\nOur definition is\n\n$$\n\\mathbb{H}(Y|X): =\\mathbb{E}_{X}\\left[-\\sum_{y=1}^{|\\mathcal{V}|}\\mathbb{P}(Y=y|X)\\log\\mathbb{P}(Y=y|X)\\right],\\tag{8}\n$$\n\nwhere $\\mathbb{P}(Y=y|X): =\\mathbb{E}[\\mathbf{1}(Y=y)|X]$ . By the definition of conditional expectation (or probability), it is easy to check that\n\n$$\n\\mathbb{E}[{\\bf 1}(Y=y)|{X}]=\\frac{P(Y=y)p(X|{Y=y})}{p_{X}(X)},\\tag{9}\n$$\n\nwhere $p(x|Y=y),x\\in\\mathbb{R}^{p}$ is the conditional density function of X given $Y = y$ .Denote $p_{y}: =$ $P(Y=y)$ , we have\n\n$$\n\\begin{array}{ l }{\\displaystyle\\mathbb{H}(Y|X)=\\int_{\\Omega}-\\sum_{y=1}^{|\\mathcal{V}|}p_{y}p(x|Y=y)\\log\\frac{p_{y}p(x|Y=y)}{p_{X}(x)}d x}\\\\ {\\displaystyle= -\\sum_{y=1}^{|\\mathcal{V}|}p_{y}\\log p_{y}-\\sum_{y=1}^{|\\mathcal{V}|}p_{y}\\int_{\\Omega}p(x|Y=y)\\log\\frac{p(x|Y=y)}{p_{X}(x)}d x}\\\\ {\\displaystyle=\\mathbb{H}(Y)-\\sum_{y=1}^{|\\mathcal{V}|}p_{y}\\mathbb{E}\\mathbb{L}(X|Y=y| |X).}\\end{array}\\tag{10}\n$$\n\nThis proves (4).\n\nWe summarize some basic properties of $\\mathbb{H}(Y|X)$ under our definition (8):\n\nB Proof of Fano's Inequality for Continuous Features\n\nWe see the above essential properties for $\\mathbb{H}(Y|X)$ have been preserved for the mixture case. Furthermore, we prove Fano's inequality still holds for our case. Let us restate Theorem 2 here:", "theorems": [{"local_id": "T1", "statement": "Lemma 2. (i) $0\\leq\\mathbb{H}(Y|X)\\leq\\mathbb{H}(Y)\\leq\\log|\\mathcal{V}|.$\n\n(ii) $\\mathbb{H}(Y|X)=\\mathbb{H}(Y)$ if and only if X and Y are independent.\n\n(ii) $\\mathbb{H}(Y|X)~=~0~i f$ and only $i f Y$ is a function of X. That is, there exist |y| subsets of Ω: $\\Omega_{1},\\dots,\\Omega_{|\\mathcal{V}|}$ such that $\\mathsf{P}(X\\in\\mathsf{\\Omega}\\partial_{y}\\cap\\Omega_{y^{\\prime}})\\ =\\ 0$ for any $y\\ne y^{\\prime}\\in\\{1,...,|\\mathcal{V}|\\}$ $P(X\\in\\cup_{y=1}^{|\\mathcal{V}|}\\Omega_{y})=1 a n d P(Y=y|X\\in\\Omega_{y})=1 for any y\\in\\{1,\\ldots,|\\mathcal{V}|\\}$", "proof": {"steps": [{"n": 1, "claim": "Proof. (iThis is trivial by (8), (10) and the fact that KL divergence is always nonnegative."}, {"n": 2, "claim": "(ii) By (10), it is easy to get\n\n$$\n{\\begin{array}{ r l }{\\mathbb{H}(Y|X)=\\mathbb{H}(Y)\\iff X|Y=y{\\overset{d}{\\operatorname{d}}}X,\\quad\\forall y\\in\\{1,\\dots,|\\mathcal{V}|\\}.}&{}\\\\ {\\iff P(X\\in A,Y=y)=P(X\\in A)P(Y=y),{\\mathrm{f o r~a n y}}}&{}\\\\ {{\\mathrm{measurable set}}A{\\mathrm{i n~}}\\mathcal{X},{\\mathrm{a n y~}}y\\in\\{1,\\dots,|\\mathcal{V}|\\}.}&{}\\\\ {\\iff X,Y{\\mathrm{~are independent .}}}&{}\\end{array}}\n$$"}, {"n": 3, "claim": "(iii) Sufficiency: If Y is a function of X, then by assumptions, for any $y_{1}\\neq y_{2}$ , we have\n\n$$\nP(X\\in\\Omega_{y_{1}},Y=y_{2})\\leq P(X\\in\\Omega_{y_{1}})-P(X\\in\\Omega_{y_{1}},Y=y_{1})=0 .\n$$"}, {"n": 4, "claim": "Therefore, $p(x|Y=y_{2})=0$ for $x\\in\\Omega_{y_{1}}(\\mathrm{a . e .})$ .Then for any $y\\in\\{1,\\ldots,|y|\\}$ , we have\n\n$$\n\\mathbb{K L}(X|Y=y)=\\int_{\\Omega_{y}}p(x|Y=y)\\log\\frac{p(x|Y=y)}{p_{X}(x)}d x\\tag{11}\n$$\n\nNotice that for $\\begin{array}{ r }{x\\in\\Omega_{y}(\\mathrm{a . e .}),p_{X}(x)=\\sum_{y^{\\prime}=1}^{|\\mathcal{V}|}p_{y^{\\prime}}p(x|Y=y^{\\prime})=p_{y}p(x|Y=y)}\\end{array}$ , and thus\n\n$$\n\\mathbb{K L}(X|Y=y)= -\\log p_{y}.\\tag{12}\n$$\n\nThen by (10), $\\mathbb{H}(Y|X)=0$"}, {"n": 5, "claim": "Necessity: Assume $\\mathbb{H}(Y|X)=0$ .Notice that $\\begin{array}{ r }{p_{X}(x)=\\sum_{y=1}^{|\\mathcal{V}|}p_{y}p(x|Y=y),\\forall x\\in\\Omega}\\end{array}$ Therefore, for any $y\\in\\{1,\\ldots,|\\mathcal{V}|\\}$ ,\n\n$$\n\\begin{array}{ r l }{{\\mathbb{K L}(X|Y=y| |X)=\\int_{\\Omega}p(x|Y=y)\\log\\frac{p(x|Y=y)}{p_{X}(x)}d x}}\\\\ &{\\leq\\int_{\\Omega}p(x|Y=y)\\log\\frac{p(x|Y=y)}{p_{y}p(x|Y=y)}d x}\\\\ &{= -\\log p_{y}.}\\end{array}\\tag{13}\n$$"}, {"n": 6, "claim": "Combining (10) and (13), we know if $\\mathbb{H}(Y|X)=0$ , then\n\n$$\np_{X}(x)=p_{y}p(x|Y=y),\\forall x\\in\\{x\\in\\Omega:p(x|Y=y)>0\\}(a . e .),\\forall 1\\leq y\\leq|\\mathcal{Y}|.\\tag{14}\n$$\n\nDefine $\\Omega_{y}: =\\{x\\in\\Omega:p(x|Y=y)>0\\}$ , for every $y\\in\\{1,\\ldots,|\\mathcal{V}|\\}$ . Without loss of generality, we assume $p_{X}(x)>0$ for every $x \\in \\Omega$ (Otherwise, we can replace Ω by $\\Omega\\cap\\left\\{p_{X}(x)>0\\right\\}$ and the following argument is still true.) Then we have\n\n$$\nP(X\\in\\cup_{y=1}^{|\\mathcal{V}|}\\Omega_{y})=P(X\\in\\Omega)=1 .\n$$"}, {"n": 7, "claim": "By (14), we have\n\n$$\n\\begin{array}{ l }{P(X\\in\\Omega_{y})=\\displaystyle\\int_{\\Omega_{y}}p_{X}(x)d x=\\int_{\\Omega_{y}}p_{y}p(x|Y=y)d x}\\\\ {\\quad=p_{y}P(X\\in\\Omega_{y}|Y=y)=P(X\\in\\Omega_{y},Y=y).}\\end{array}\\tag{15}\n$$\n\nTherefore, $P(Y=y|X\\in\\Omega_{y})=1$ . Lastly, we show $P(X\\in\\Omega_{u}\\cap\\Omega_{u^{\\prime}})=0$ for any $y\\ne y^{\\prime}$ This is immediate if we notice that by replacing $\\Omega_{y}$ in (15) with $\\Omega_{y}\\cap\\Omega_{y^{\\prime}}$ , we can have\n\n$$\nP(X\\in\\Omega_{y}\\cap\\Omega_{y^{\\prime}})=P(X\\in\\Omega_{y}\\cap\\Omega_{y^{\\prime}},Y=y).\n$$\n\nChanging the position of y and $y^{\\prime}.$ , we have\n\n$$\nP(X\\in\\Omega_{y}\\cap\\Omega_{y^{\\prime}})=P(X\\in\\Omega_{y}\\cap\\Omega_{y^{\\prime}},Y=y^{\\prime}).\n$$\n\nThus, $P(X\\in\\Omega_{y}\\cap\\Omega_{y^{\\prime}})=0 .$\n\nBefore the end of the proof, we want to add a remark that the condition $P(X\\in\\Omega_{y}\\cap\\Omega_{y^{\\prime}})=$ 0 is actually just a consequence of $P(Y=y|X\\in\\Omega_{y})=1,\\forall y$ We add it explicitly in the conditions to make it clear that $\\Omega_{1},\\ldots,\\Omega_{|\\mathcal{V}|}$ is a partition of Ω according to the value of Y."}]}}, {"local_id": "T2", "statement": "Setting: feature-label pairs $(x,y)$ with $x \\in \\mathcal{X} \\subset \\mathbb{R}^{d_x}$ and discrete labels $y \\in \\mathcal{V} \\subset \\mathbb{Z}^{+}$, sampled i.i.d. from a static joint distribution $\\mathcal{P}_{\\mathcal{X}\\mathcal{Y}}$ (Assumption 2). Let $\\mathcal{P}_e = \\inf_f P(f(X) \\neq Y)$ be the model-agnostic minimum error probability, and let the hybrid conditional entropy $\\mathbb{H}(Y|X)$, hybrid mutual information $\\mathbb{I}(X;Y) = \\sum_{y=1}^{|\\mathcal{V}|} P(Y=y)\\,\\mathbb{KL}(X|Y=y \\,||\\, X)$, and the decomposition $\\mathbb{H}(Y|X) = \\mathbb{H}(Y) - \\mathbb{I}(X;Y)$ be as defined in Eqs. (2),(4),(5), with Fano's inequality (Theorem 2) $\\mathbb{H}(\\mathcal{P}_e) + \\mathcal{P}_e \\log(m-1) \\geq \\mathbb{H}(Y|X)$. Claim (Section 3.3): since a neural network is used to estimate each KL divergence via the Donsker-Varadhan / MINE bound of Eq. (7), the estimate satisfies $\\widehat{\\mathbb{KL}}(X|y \\,||\\, X) \\leq \\mathbb{KL}(X|y \\,||\\, X)$; consequently $\\widehat{\\mathbb{I}}(X;Y) \\leq \\mathbb{I}(X;Y)$, giving a larger $\\widehat{\\mathbb{H}}(Y|X)$, so that DIME (obtained by solving Eq. (3) with $\\widehat{\\mathbb{H}}(Y|X)$) is $\\geq$ and hence a valid conservative estimate of the true lower bound on $\\mathcal{P}_e$.", "proof": {"steps": [{"n": 1, "claim": "For each class $c$, a network $T_{\\theta_c}$ is trained by $\\theta_c = \\arg\\max_{\\theta_c} \\frac{1}{|S_c|}\\sum_{S_c} T_{\\theta_c}(x_i) - \\log[\\frac{1}{b}\\sum_j \\exp T_{\\theta_c}(x_j)]$, and the KL is then evaluated on the held-out set as $\\widehat{\\mathbb{KL}}_{\\theta_c} = \\frac{1}{|S_c^e|}\\sum_{S_c^e} T_{\\theta_c}(x_i^e) - \\log[\\frac{1}{b}\\sum_j \\exp T_{\\theta_c}(x_j^e)]$."}, {"n": 2, "claim": "This estimator is the finite-sample version of the neural-network KL bound of Eq. (7): the population term $\\log \\mathbb{E}_{\\mathcal{Q}}[e^{T}]$ is replaced by the empirical Monte-Carlo quantity $\\log[\\frac{1}{b}\\sum_j \\exp T_{\\theta_c}(x_j)]$."}, {"n": 3, "claim": "The population Donsker-Varadhan / MINE bound (Eq. 7) gives $\\mathbb{KL}(\\mathcal{P}||\\mathcal{Q}) \\geq \\sup_{\\theta \\in \\Theta} \\mathbb{E}_{\\mathcal{P}}[T_\\theta] - \\log \\mathbb{E}_{\\mathcal{Q}}[e^{T_\\theta}]$."}, {"n": 4, "claim": "According to Equation (7), the neural-network estimate satisfies $\\widehat{\\mathbb{KL}}(X|y \\,||\\, X) \\leq \\mathbb{KL}(X|y \\,||\\, X)$."}, {"n": 5, "claim": "Consequently the estimated hybrid mutual information satisfies $\\widehat{\\mathbb{I}}(X;Y) \\leq \\mathbb{I}(X;Y)$."}, {"n": 6, "claim": "Therefore the estimated conditional entropy satisfies $\\widehat{\\mathbb{H}}(Y|X) \\geq \\mathbb{H}(Y|X)$ (DIME yields a larger $\\widehat{\\mathbb{H}}(Y|X)$)."}, {"n": 7, "claim": "Hence DIME, obtained by solving Fano's inequality (Eq. 3) with $\\widehat{\\mathbb{H}}(Y|X)$, is $\\geq$ and thus a valid conservative estimate of the true lower bound on $\\mathcal{P}_e$."}]}}, {"local_id": "T3", "statement": "Theorem 3. Let $\\mathcal{P}_{e}$ be the minimum error probability, i.e.\n\n$$\n{\\mathcal{P}}_{e}=\\operatorname*{inf}_{f}P(f(X)\\neq Y)\n$$\n\n![](images/82453a7027947d731d7310ea0d52fec4214c36a7cc3f96f3cdc1bc773f103df1.jpg)  \nFigure 4: An example of f with Smooth Discretization Property given the data and classifier. Red points represent $\\{x_{i},f_{0}(x_{i})\\}_{i=1}^{10},x_{i}\\sim U[0,1]$ , fo is some given classifier in practice.\n\nwhere f is any \"continuous\" estimator of Y based on the observation X. That is, f is any function satisfying that for any $y=1,\\ldots,|\\mathcal{Y}|$ , almost every $x\\in\\Omega,$ (x a.e. in Ω)\n\n$$\nf(x)=y\\iff\\exists\\delta>0\\quad s . t .\\quad\\forall\\tilde{x}\\in B_{\\delta}(x),\\quad f(\\tilde{x})=y,\n$$\n\nwhere $B_{\\delta}(x): =\\{\\tilde{x}\\in\\Omega:\\|\\tilde{x}-x\\|_{2}<\\delta\\}$ is a δ—neighborhood of x in X. Then we have\n\n$$\n\\mathbb{H}(\\mathcal{P}_{e})+\\mathcal{P}_{e}\\log(|\\mathcal{V}| -1)\\geq\\mathbb{H}(Y|X).\\tag{16}\n$$\n\n$$\nH e r e,\\mathbb{H}(\\mathcal{P}_{e}): = -\\mathcal{P}_{e}\\log\\mathcal{P}_{e}-(1-\\mathcal{P}_{e})\\log(1-\\mathcal{P}_{e}).\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof. We basically follow the idea in the classical proof in Cover & Thomas (2012), although many details need to be dealt with carefully under our definition. Assume there exists a function f achieves the minimum error probability, i.e. $P(f(X)\\neq Y)={\\mathcal{P}}_{\\epsilon}$ (If such f doesn't exist, we can always find some f such that $P(f(X)\\neq Y)\\leq\\mathcal{P}_{e}+\\epsilon$ for some little $\\epsilon>0,$ then let $\\epsilon  0$ we can get the same result). Define an error random variable,\n\n$$\nE={\\left\\{\\begin{array}{ l l }{1,}&{{\\mathrm{if}}\\quad f(X)\\not=Y}\\\\ {0,}&{{\\mathrm{if}}\\quad f(X)=Y.}\\end{array}\\right.}\\tag{17}\n$$"}, {"n": 2, "claim": "Under our definition (8), we have (in what follows, we use the notation $Y = i$ for $i=1,\\dots,|\\mathcal{V}|$ instead of $Y = y$ to remind the reader that Y is discrete.)\n\n$$\nH(E,Y|X): =\\mathbb{E}_{X}\\left[-\\sum_{i=1}^{|\\mathcal{V}|}\\sum_{j=0}^{1}\\mathbb{P}(Y=i,E=j|X)\\log\\mathbb{P}(Y=i,E=j|X)\\right].\\tag{18}\n$$\n\nIt is easy to check that\n\n$$\n\\mathbb{P}(Y=i,E=j|X)=\\frac{P(Y=i,E=j)p(X|Y=i,E=j)}{p_{X}(X)},\\tag{19}\n$$\n\nwhere $p(x|Y=i,E=j)$ is the conditional density function of X given $Y=i,E=j$ Similarly, one may check that (Footnote 3: Strictly speaking, by the definition of conditional expectation or probability, we should have $\\mathbb{P}(E=j|Y=i,X)=\\frac{P(Y=i,E=j)p(X|Y=i,E=j)}{P(Y=i)p(X|Y=i)}\\mathbf{1}(Y=i)+\\frac{P(Y=i,E=j)p(X|Y=i,E=j)}{P(Y\\neq i)p(X|Y\\neq i)}\\mathbf{1}(Y\\neq i)$ . Footnote 4: For the consistency of the notations, we still use $\\mathbb{P}(E=j|Y=i,X)$ to denote the right side of (20). So actually, one should understand $\\mathbb{P}(E=j|Y=i,X)$ as $\\mathbb{P}(E=j|Y=i,X)\\mathbf{1}(Y=i)$ in this context. Similar for $\\mathbb{P}(Y=i|E=j,X)$ below.)\n\n$$\n\\mathbb{P}(E=j|Y=i,X)=\\frac{P(Y=i,E=j)p(X|Y=i,E=j)}{P(Y=i)p(X|Y=i)}\\tag{20}\n$$\n\nCombining the above two equations and (9), we have\n\n$$\n\\mathbb{P}(Y=i,E=j|X)=\\mathbb{P}(Y=i|X)\\mathbb{P}(E=j|Y=i,X).\\tag{21}\n$$"}, {"n": 3, "claim": "Plug it in (18), and use the notation\n\n$$\n\\mathbb{H}(E|Y,X): =\\mathbb{E}_{X}\\left[-\\sum_{i=1}^{|\\mathcal{V}|}\\sum_{j=0}^{1}\\mathbb{P}(Y=i,E=j|X)\\log\\mathbb{P}(E=j|Y=i,X)\\right],\\tag{22}\n$$\n\nwe get\n\n$$\n\\mathbb{H}(E,Y|X)=\\mathbb{H}(Y|X)+\\mathbb{H}(E|Y,X)\\tag{23}\n$$"}, {"n": 4, "claim": "For any $x\\in\\Omega,i=1,\\dots,|\\mathcal{V}|$ , we know $\\exists \\delta > 0$ ,such that\n\n$$\nP(X\\in B_{\\delta}(x),Y=i,E=0)=\\left\\{{\\begin{array}{ l l l }{P(X\\in B_{\\delta}(x),Y=i),}&{{\\mathrm{if}}}&{f(x)=i}\\\\ {0,}&{{\\mathrm{if}}}&{f(x)\\neq i.}\\end{array}}\\right.\\tag{24}\n$$\n\nLet $\\delta  0 .$ we have $p(x|Y=i,E=0)P(Y=i,E=0)=p(x|Y=i)P(Y=i)$ or 0 for any $x \\in \\Omega$ and $1 \\leq i \\leq m$ Therefore, by (20), we know $\\mathbb{P}(E=0|Y=i,X)$ is always 0 or 1 for any $1\\leq i\\leq|\\mathcal{V}|$ . So is $\\mathbb{P}(E=1|Y=i,\\dot{X})$ . Then by (22), we have\n\n$$\n\\mathbb{H}(E|Y,X)=0 .\\tag{25}\n$$"}, {"n": 5, "claim": "Similar to (23), we have\n\n$$\n\\mathbb{H}(E,Y|X)=\\mathbb{H}(E|X)+\\mathbb{H}(Y|E,X),\\tag{26}\n$$\n\nwhere\n\n$$\n\\mathbb{H}(E|X): =\\mathbb{E}_{X}\\left[-\\sum_{j=0}^{1}\\mathbb{P}(E=j|X)\\log\\mathbb{P}(E=j|X)\\right]\\tag{27}\n$$\n\nand\n\n$$\n\\mathbb{H}(Y|E,X): =\\mathbb{E}_{X}\\left[-\\sum_{i=1}^{|\\mathcal{V}|}\\sum_{j=0}^{1}\\mathbb{P}(Y=i,E=j|X)\\log\\mathbb{P}(Y=i|E=j,X)\\right].\\tag{28}\n$$\n\nNotice that $\\mathbb{H}(E|X)\\le\\mathbb{H}(E)=\\mathbb{H}(\\mathcal{P}_{e})$ , it suffices to show\n\n$$\n\\mathbb{H}(Y|E,X)\\leq\\mathcal{P}_{e}\\log(|\\mathcal{Y}| -1).\\tag{29}\n$$"}, {"n": 6, "claim": "Similar to (24), we know for any $x\\in\\Omega,i=1,\\ldots,|y|,\\exists\\delta>0,$ ,such that\n\n$$\nP(X\\in B_{\\delta}(x),Y=i,E=0)=\\left\\{{\\begin{array}{ l l l }{P(X\\in B_{\\delta}(x),E=0),}&{{\\mathrm{if}}}&{f(x)=i}\\\\ {0,}&{{\\mathrm{if}}}&{f(x)\\neq i.}\\end{array}}\\right.\\tag{30}\n$$\n\nThen we get $\\mathbb{P}(Y=i|E=0,X)$ is always 0 or 1 for any $i=1,\\ldots,|\\mathcal{V}|$ .Therefore,\n\n$$\n\\mathbb H(Y|E=0,X): =\\mathbb E_{X}\\left[-\\sum_{i=1}^{|\\mathcal V|}\\mathbb P(Y=i,E=0|X)\\log\\mathbb P(Y=i|E=0,X)\\right]=0 .\\tag{31}\n$$\n\nFor the case $E = 1$ , we have\n\n$$\nP(X\\in B_{\\delta}(x),Y=i,E=1)=\\left\\{{\\begin{array}{ l l l }{P(X\\in B_{\\delta}(x),Y=i),}&{{\\mathrm{if}}}&{f(x)\\neq i}\\\\ {0,}&{{\\mathrm{if}}}&{f(x)=i.}\\end{array}}\\right.\\tag{32}\n$$\n\nThus, we have\n\n$$\n\\begin{array}{ r }{\\mathbb{P}(Y=i|E=1,X)=\\frac{P(Y=i,E=1)p(X|Y=i,E=1)}{P(E=1)p(X|E=1)}}\\\\ {=\\left\\{\\begin{array}{ l l }{\\frac{P(Y=i)p(X|Y=i)}{P(E=1)p(X|E=1)},}&{\\mathrm{if}\\quad f(X)\\neq i}\\\\ {0,}&{\\mathrm{if}\\quad f(X)=i.}\\end{array}\\right.}\\end{array}\\tag{33}\n$$\n\n(34)\n\nSimilar to (21), we have\n\n$$\n\\begin{array}{ r }{\\mathbb{P}(Y=i,E=1|X)=\\mathbb{P}(E=1|X)\\mathbb{P}(Y=i|E=1,X).}\\end{array}\\tag{35}\n$$\n\nPlug in the above two equations to $\\mathbb{H}(Y|E=1,X)$ , we get\n\n$$\n\\begin{array}{ r l }&{\\mathbb{H}(Y|E=1,X)=\\mathbb{E}_{X}\\left[-\\displaystyle\\sum_{i=1}^{|\\mathcal{Y}|}\\mathbb{P}(E=1|X)\\mathbb{P}(Y=i|E=1,X)\\log\\mathbb{P}(Y=i|E=1,X)\\right]}\\\\ &{\\quad\\quad=\\mathbb{E}_{X}\\left[\\mathbb{P}(E=1|X)\\left(-\\displaystyle\\sum_{i\\neq f(X)}\\mathbb{P}(Y=i|E=1,X)\\log\\mathbb{P}(Y=i|E=1,X)\\right)\\right]}\\\\ &{\\quad\\le\\mathbb{E}_{X}\\left[\\mathbb{P}(E=1|X)\\log(|\\mathcal{Y}| -1)\\right]}\\\\ &{\\quad=\\mathcal{P}_{e}\\log(|\\mathcal{Y}| -1).}\\end{array}\n$$"}, {"n": 7, "claim": "Therefore,\n\n$$\n\\mathbb H(Y|E,X)=\\mathbb H(Y|E=0,X)+\\mathbb H(Y|E=1,X)\\le\\mathcal P_{e}\\log(|\\mathcal V| -1).\n$$"}]}}], "target_local_id": "T2", "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "label": "flawed", "error_step": 4, "error_class6": ["invalid_bound_or_inequality", "others", "computation_or_constant"], "error_explanation": "Step 4 asserts $\\widehat{\\mathbb{KL}} \\leq \\mathbb{KL}$ \"according to Equation (6)\", but Eq. (6) is the POPULATION Donsker-Varadhan bound, in which $\\log\\mathbb{E}_{\\mathcal{Q}}[e^{T}]$ is an exact expectation. Algorithm 1's estimator replaces this with a Monte-Carlo average inside the log, $\\log[\\frac{1}{b}\\sum_j \\exp T_{\\theta_c}(x_j)]$. By Jensen's inequality, a Monte-Carlo estimate of a log-expectation is a stochastic LOWER bound of the true log-expectation; subtracting it therefore makes $\\widehat{\\mathbb{KL}}$ a stochastic UPPER bound of the proposed KL lower bound, so $\\widehat{\\mathbb{KL}} \\leq \\mathbb{KL}$ is NOT guaranteed. This severs the entire downstream chain ($\\widehat{\\mathbb{I}} \\leq \\mathbb{I} \\Rightarrow \\widehat{\\mathbb{H}}(Y|X) \\geq \\mathbb{H}(Y|X) \\Rightarrow$ DIME lower-bounds $\\mathcal{P}_e$): the procedure computes an upper bound of a lower bound, which does not preserve the inequality direction. The empirical MINE estimator is biased and, even de-biased, cannot in general be claimed to bound $\\mathcal{P}_e$."}
{"id": "HJNMYceCW", "context": "Formally, we consider a (possibly virtual) learning agent that interacts directly with its environment. The interaction between the agent and the environment is governed by a restricted class of finitehorizon Markov Decision Processes (MDP), defined as a tuple $\\{\\mathcal{S},s_{0},\\dot{\\mathcal{A}},\\mathcal{P},\\mathcal{L},H\\}$ where: S is a large but finite state space, typically $S\\subset\\mathbb{R}^{d};s_{0}\\in\\mathcal{S}$ is a start state; A is a finite action $\\mathrm{space}^{2}$ of size K; $\\mathcal{P}=\\{\\mathcal{P}\\overset{\\vartriangle}{\\mathcal{P}}(\\ v s^{\\prime}|\\boldsymbol s,\\boldsymbol{a}):\\boldsymbol s,\\boldsymbol{s}^{\\prime}\\in\\mathcal{S},\\boldsymbol{a}\\in\\mathcal{A}\\}$ is the set of Markovian transition probabilities; $\\mathcal{L}\\in\\mathbb{R}^{|S|}$ is the state dependent loss function, defined only at terminal states $s \\in S .$ H is the horizon (maximum length of an episode).\n\nThe goal is to learn a policy π, which defines the behavior of the agent in the environment. We consider policies that are potentially functions of entire trajectories3, and potentially produce distributions over actions: $\\pi(s{\\bar{)}}\\in\\Delta^{\\mathcal{A}}$ , where $\\Delta^{\\mathcal{A}}$ is the A-dimensional probability simplex. However, to ease exposition, we will present the background in terms of policies that depend only on states; this can be accomplished by simply blowing up the state space.\n\nLet $d_{h}^{\\pi}$ denote the distribution of states visited at time step h when starting at state $s_{0}$ and operating according to $\\pi\\colon d_{h+1}^{\\pi}(s^{\\prime})\\ : =\\ :\\mathbb{E}_{s_{h}\\sim d_{h}^{\\pi},a_{h}\\sim\\pi(s_{h})}{\\mathcal P}(s^{\\prime}\\mid s\\ =\\ s_{h},a\\ =a_{h})$ The quality of the policy $\\pi$ is quantified by its value function or q-value function: $V^{\\pi}(s)\\in$ R associates each state with the expected future loss for starting at this state and following π afterwards; $Q^{\\pi}(s,a)\\in\\mathbb{R}$ associates each state/action pair with the same expected future loss: $V^{\\pi}(s_{h})=\\mathbb{E}_{s_{H}\\sim d_{H}^{\\pi}\\mid s_{h}}{\\mathcal{L}}(s_{H})$ and $Q^{\\pi}(s_{h},a_{h})=\\mathbb{E}_{s_{H}\\sim d_{H}^{\\pi}\\mid s_{h},a_{h}}{\\mathcal{L}}(s_{H})$ The learning goal is to estimate a policy  from a hypothesis class of policies II with minimal expected loss: $J(\\pi)=V^{\\pi}(s_{0})$\n\nAkin to the coffee shop example, REsLopE learns a decomposition of the episodic loss (i.e total time spent from work to the coffee shop) into a sum of per-time-step losses (i.e. timing activities along the route). REsLOPE operates as a reduction from reinforcement learning with episodic loss to contextual bandits. In thi way, Re solves he ce assmet proble by prei residual losses, and relies on the underlying contextual bandit oracle to solve explore/exploit. REs-LOPE operates online, incrementally updating a policy $\\pi^{\\mathrm{learn}}$ once per episode. In the structured contextual bandit setting, we assume access to a reference policy, $\\pi^{\\mathrm{ref}}.$ , that was perhaps pretrained on supervised data, and which we wish to improve; a hyperparameter $\\beta$ controls how much we trust $\\pi^{\\mathrm{ref}}.{\\dot{\\mathrm{As}}}\\pi^{\\mathrm{learn}}$ improves, we replace $\\pi^{\\mathrm{ref}}$ with $\\cdot\\pi^{\\mathrm{{l e a r n}}}$ . In the RL setting, we set $\\beta = 0$\n\n1. CB. $\\mathbf{ACT}(\\pi^{\\mathrm{learn}},\\pmb{x})$ , where x is the input example; this returns a tuple $(a,p)$ , where a is the selected action, and p is the probability with which that action was selected.\n\n2. CB $\\boldsymbol{\\cdot}\\cos\\operatorname{T}\\big(\\pi^{\\mathrm{learn}},\\pmb{x},a\\big)$ returns the estimated cost of taking action a in the context.\n\n3. CB. $\\mathrm{UPDATE}\\left(\\pi^{\\mathrm{learn}},x,a,p,c\\right)$ , where x is the input example, $a \\in [ K ]$ is the selected action, $p\\in(0,1]$ is the probability of that action, and $c\\in\\mathbb{R}$ is the target cost.\n\nFor simplicity, we first consider the case where we have access to a good reference policy $\\pi^{\\mathrm{ref}}$ but do not have access to good Q-value estimates under $\\pi^{\\mathrm{ref}}$ . The only way one can obtain a Q-value estimate is to do a roll-out, but in a non-resettable environment, we can only do this once. We will subsequently consider the case of suboptimal (or missing) reference policies, in which the goal of the analysis will change from competing with $\\pi^{\\mathrm{ref}}$ to competing with both $\\pi^{\\mathrm{ref}}$ and a local optimality guarantee.", "theorems": [{"local_id": "T1", "statement": "Lemma 2 (Bagnell et al. (2004); Kakade et al. (2003)). For all policies π and $\\pi^{\\prime}{:}$\n\n$$\nJ(\\pi)-J(\\pi^{\\prime})=\\sum_{h=1}^{H}\\mathbb{E}_{s_{h}\\sim d_{\\pi}^{h}}\\left[Q^{\\pi^{\\prime}}(s_{h},\\pi)-V^{\\pi^{\\prime}}(s_{h})\\right]\\tag{1}\n$$", "proof": {"steps": [{"n": 1, "claim": "In a now-classic lemma, Kakade et al. (2003) and Bagnell et al. (2004) show that the difference in total loss between two policies can be computed exactly as a sum of per-time-step advantages of one over the other:"}, {"n": 2, "claim": "$$\nJ(\\pi)-J(\\pi^{\\prime})=\\sum_{h=1}^{H}\\mathbb{E}_{s_{h}\\sim d_{\\pi}^{h}}\\left[Q^{\\pi^{\\prime}}(s_{h},\\pi)-V^{\\pi^{\\prime}}(s_{h})\\right]\\tag{1}\n$$"}]}}, {"local_id": "T2", "statement": "Consider a finite-horizon MDP $\\{\\mathcal{S}, s_0, \\mathcal{A}, \\mathcal{P}, \\mathcal{L}, H\\}$ with state-dependent loss $\\mathcal{L}$ defined at terminal states, objective $J(\\pi) = V^\\pi(s_0)$, and a structured contextual bandit setting with access to a reference policy $\\pi^{ref}$ and a contextual bandit oracle CB (supporting CB.ACT, CB.COST, CB.UPDATE). Setting $\\beta = 1$, running RESLOPE for $N$ episodes with a contextual bandit algorithm, the average returned policy $\\bar\\pi = \\mathbb{E}_n \\pi_n$ has regret equal to the suboptimality of $\\pi^{ref}$, namely: $Regret(\\bar\\pi) \\leq Regret(\\pi^{ref}) + \\frac{1}{N}\\epsilon_{CB}(N)$, where $\\epsilon_{CB}(N)$ is the cumulative regret of the underlying contextual bandit algorithm after $N$ steps. In particular, when the contextual bandit algorithm is no-regret, RESLOPE is also no-regret.", "proof": {"steps": [{"n": 1, "claim": "$J(\\bar\\pi)-J(\\pi^{ref}) = \\mathbb{E}_n \\sum_h \\mathbb{E}_{s\\sim d_{\\pi_n}^h}[Q^{\\pi^{ref}}(s,\\pi_n) - Q^{\\pi^{ref}}(s,\\pi^{ref})]$ (Eq. 2)."}, {"n": 2, "claim": "Fix $n$, $h^{dev}$: $\\mathbb{E}_{s\\sim d_{\\pi_n}^{h^{dev}}}[Q^{\\pi^{ref}}(s,\\pi_n) - Q^{\\pi^{ref}}(s,\\pi^{ref})] + \\sum_{h\\neq h^{dev}} \\mathbb{E}_{s\\sim d_{\\pi_n}^h}[Q^{\\pi^{ref}}(s,\\pi_n) - Q^{\\pi^{ref}}(s,\\pi^{ref})]$ (Eq. 3)."}, {"n": 3, "claim": "$= \\mathbb{E}_{s\\sim d_{\\pi_n}^{h^{dev}}}\\left[Q^{\\pi^{ref}}(s,\\pi_n) - \\left(Q^{\\pi^{ref}}(s,\\pi^{ref}) - \\sum_{h\\neq h^{dev}} \\mathbb{E}_{s'\\sim d_{\\pi_n}^h}[Q^{\\pi^{ref}}(s',\\pi_n) - Q^{\\pi^{ref}}(s',\\pi^{ref})]\\right)\\right]$ (Eq. 4)."}, {"n": 4, "claim": "$= \\mathbb{E}_{s\\sim d_{\\pi_n}^{h^{dev}}}\\left[Q^{\\pi^{ref}}(s,\\pi_n) - \\left(\\mathbb{E}_{s_H\\sim\\pi^{ref}\\mid s_{h^{dev}}=s}\\,\\ell(s_H) - \\sum_{h\\neq h^{dev}} \\mathbb{E}_{s'\\sim d_{\\pi_n}^h}\\,\\mathbf{CB.cost}(\\pi_n,s',\\pi_n(s'))\\right)\\right] = \\mathbb{E}_{s\\sim d_{\\pi_n}^{h^{dev}}}[Q^{\\pi^{ref}}(s,\\pi_n) - \\mathrm{Residual}(\\pi_n,h^{dev},s)]$ (Eq. 5)."}, {"n": 5, "claim": "$J(\\bar\\pi)-J(\\pi^{ref}) = \\frac{1}{N}\\epsilon_{CB}(N)$, so $Regret(\\bar\\pi) \\leq Regret(\\pi^{ref}) + \\frac{1}{N}\\epsilon_{CB}(N)$."}, {"n": 6, "claim": "If CB is no-regret, $\\frac{1}{N}\\epsilon_{CB}(N)\\to 0$ as $N\\to\\infty$; RESLOPE is no-regret."}]}}, {"local_id": "T3", "statement": "Theorem 3. For arbitrary β, define the combined regret of π as: $R e g r e t_{\\beta}(\\bar{\\pi})=\\beta [J(\\bar{\\pi})-J(\\pi^{r e f})]+$ $\\begin{array}{ r }{(1-\\beta)\\sum_{h}[J(\\bar{\\pi})-\\operatorname*{min}_{\\pi\\in\\Pi}\\mathbb{E}_{s\\sim d_{\\bar{\\pi}}^{h}}Q^{\\bar{\\pi}}(s,\\pi)]}\\end{array}$ . The first term is suboptimality to $\\pi^{r e f};$ the second term is suboptimality to the policy's own realizable one-step deviations. Given a contextual bandit learning algorithm, and under a realizability assumption, the combined regret of π satisfies: $R e g r e t_{\\beta}(\\bar{\\pi})\\le\\frac{1}{N}\\epsilon_{C B}(N)+\\epsilon_{a p p r o x}$", "proof": {"steps": [{"n": 1, "claim": "Proof of Theorem 3. The proof follows a combination of the proof of Theorem 2 with the LOLS analysis. Using the same notation as before, additionally let $\\pi_{n}^{\\mathrm{{o u t}}}$ be the mixture of $\\pi_{n}$ with $\\pi^{\\mathrm{ref}}$ for rollout."}, {"n": 2, "claim": "First, we observe (LOLS Eq 6):\n\n$$\nJ(\\bar{\\pi})-J(\\pi^{\\mathrm{ref}})=\\mathbb{E}_{n}\\sum_{h}\\mathbb{E}_{s\\sim d_{\\pi_{n}}^{h}}[Q^{\\pi^{\\mathrm{ref}}}(s,\\pi_{n})-Q^{\\pi^{\\mathrm{ref}}}(s,\\pi^{\\mathrm{ref}})]\\tag{6}\n$$"}, {"n": 3, "claim": "Then (LOLS Eq 7):\n\n$$\n\\sum_{h}\\left[J(\\bar{\\pi})-\\operatorname*{min}_{\\pi\\in\\Pi}\\mathbb{E}_{s\\sim d_{\\bar{\\pi}}^{h}}Q^{\\bar{\\pi}}(s,\\pi)\\right]\\leq\\mathbb{E}_{n}\\sum_{h}\\mathbb{E}_{s\\sim d_{\\pi_{n}}^{h}}\\left[Q^{\\pi_{n}}(s,\\pi_{n})-\\operatorname*{min}_{a}Q^{\\pi_{n}}(s,a)\\right]\\tag{7}\n$$"}, {"n": 4, "claim": "So far nothing has changed. It will be convenient to define $\\begin{array}{ r }{Q_{\\beta}^{\\pi_{n}}(s)=\\beta\\operatorname*{min}_{a}Q^{\\pi^{\\mathrm{ref}}}(s,a)+(1-}\\end{array}$ $\\beta)$ mina $Q^{\\pi_{n}}(s,a)$ . For each n fix the deviation time step $h_{n}^{\\mathrm{dev}}$ . We plug these together ala LOLS and get:"}, {"n": 5, "claim": "$$\n\\beta\\Big(J(\\bar{\\pi})-J(\\pi^{\\mathrm{ref}})\\Big)+(1-\\beta)\\Big(J(\\bar{\\pi})-\\operatorname*{min}_{\\pi\\in\\Pi}\\mathbb{E}_{s\\sim d_{\\bar{\\pi}}^{h}}Q^{\\bar{\\pi}}(s,\\pi)\\Big)\\tag{8}\n$$\n\n$$\n\\leq\\mathbb{E}_{n}\\sum_{h}\\mathbb{E}_{s\\sim d_{\\pi_{n}}^{h}}\\left[Q^{\\pi_{n}^{\\mathrm{out}}}(s,\\pi_{n})-\\beta\\operatorname*{min}_{a}Q^{\\pi^{\\mathrm{ref}}}(s,a)-(1-\\beta)\\operatorname*{min}_{a}Q^{\\pi_{n}}(s,a)\\right]\\tag{9}\n$$\n\n$$\n\\displaystyle\\mathbb{=}\\mathbb{E}_{n}\\sum_{h}\\mathbb{E}_{s\\sim d_{\\pi_{n}}^{h}}\\left[Q^{\\pi_{n}^{\\mathrm{out}}}(s,\\pi_{n})-Q_{\\beta}^{\\pi_{n}}(s)\\right]\\tag{10}\n$$"}, {"n": 6, "claim": "$$\n=\\mathbb{E}_{n}\\mathbb{E}_{s^{\\mathrm{dev}}\\sim d_{\\pi_{n}}^{h_{n}^{\\mathrm{dev}}}}\\left[Q^{\\pi_{n}^{\\mathrm{out}}}(s^{\\mathrm{dev}},\\pi_{n})-\\left(Q_{\\beta}^{\\pi_{n}}(s^{\\mathrm{dev}})-\\sum_{h\\neq h_{n}^{\\mathrm{dev}}}\\mathbb{E}_{s_{h}\\sim d_{\\pi_{n}}^{h}}\\left(Q^{\\pi_{n}^{\\mathrm{out}}}(s_{h},\\pi_{n})-Q_{\\beta}^{\\pi_{n}}(s_{h})\\right)\\right)\\right]\\tag{11}\n$$\n\n$$\n=\\mathbb{E}_{n}\\mathbb{E}_{s^{\\mathrm{dev}}\\sim d_{\\pi_{n}}^{h_{n}^{\\mathrm{dev}}}}\\left[Q^{\\pi_{n}^{\\mathrm{out}}}(s^{\\mathrm{dev}},\\pi_{n})-\\left(\\mathbb{E}_{s_{H}\\sim d_{\\pi_{n}}^{H}\\mid s_{h_{n}^{\\mathrm{dev}}}=s^{\\mathrm{dev}}}\\mathcal{L}_{n}(s_{H})-\\sum_{h\\neq h^{\\mathrm{dev}}}\\mathrm{C B . C O S T}(\\pi_{n},s_{h})\\right)\\right]\\tag{12}\n$$"}, {"n": 7, "claim": "The final step follows because the inner-most expectation is exactly what the contextual bandit algorithm is estimating, and $Q_{\\beta}^{\\pi_{n}}\\left(s^{\\mathrm{dev}}\\right)$ is exactly the expectation of the observed loss. At this point the rest of the proof follows that of Theorem 2, relying on the same internal-to-external regret transformation, and the joint no-regret minimization of all \"players.\""}]}}], "target_local_id": "T2", "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "label": "flawed", "error_step": 4, "error_class6": ["missing_assumption", "invalid_bound_or_inequality", "circular_or_unproven"], "error_explanation": "This step replaces the sum of TRUE per-step Q-advantages $\\sum_{h\\neq h^{dev}} \\mathbb{E}_{s'}[Q^{\\pi^{ref}}(s',\\pi_n) - Q^{\\pi^{ref}}(s',\\pi^{ref})]$ by the estimated contextual-bandit residual $\\sum_{h\\neq h^{dev}} \\mathbb{E}_{s'}\\,\\mathbf{CB.cost}(\\pi_n,s',\\pi_n(s'))$ as an EXACT equality, with no error term. But CB.cost is only an ESTIMATE of the per-step (dis)advantage produced by a cost-sensitive learner, not the true $Q$-difference. Equating them silently assumes access to exact quantities and drops the regret/approximation error of the cost-sensitive learner. A missing additive term $\\epsilon_{CS}$ (the cost-sensitive learner's regret, called $\\epsilon_{approx}$ in the corrected paper) must appear additively in Eq. (8)/(9), and Theorem 1 requires an added realizability assumption for that term to vanish as $N\\to\\infty$. Without it the final bound $Regret(\\bar\\pi)\\le Regret(\\pi^{ref})+\\frac{1}{N}\\epsilon_{CB}(N)$ is unjustified."}
{"id": "HOjLHrlZhmx", "context": "Q-learning and Deep Q-Networks (DQNs). Markov decision processes (MDPs) are at the core of RL. Our focus is on discounted discrete-time MDPs, which are defined by tuple $(S,\\mathcal{A},R,P,\\gamma,d_{0})$ where S is a set of states (each with dimensionality N), A represents a set of discrete actions, $R:S\\times\\mathcal{A}\\mathbb{R}$ is the reward function, and $P:\\bar{S\\times A}\\bar{\\mathcal{P}(S)}$ is the transition function with P(·) defining the set of probability measures, $\\gamma\\in [0,1]$ is the discount factor, and $d_{0}\\in{\\mathcal{P}}(S)$ is the distribution over the initial state. At time step t, the agent is in the state $s_{t}\\in S$ After choosing action $a_{t}\\in A .$ , the agent transitions to the next state $s_{t+1}\\sim P(s_{t},a_{t})$ and receives reward $R(s_{t},a_{t})$ . The goal is to learn ${\\mathrm{a~p o l i c y~}}\\pi:S{\\mathcal{P}}(A)$ that maximizes the expected cumulative reward $\\mathbb{E}[\\sum_{t}\\gamma^{t}r_{t}]$ .\n\nDefinition 2 (Certification for per-state action). Given a trained network $Q^{\\pi}$ with policy π, we define the robustness certification for per-state action as the maximum perturbation magnitude , such that for any perturbation $\\delta\\in B^{\\bar{\\varepsilon}}$ , the predicted action under the perturbed state will be the same as the action taken in the clean environment, $i . e .,\\pi(s+\\delta)=\\pi(s),\\mathcal{\\overline{{\\forall}}}\\delta\\in\\mathcal{B}^{\\bar{\\varepsilon}}$\n\nDefinition 3 (Cumulative reward). Let $P:\\mathcal{S}\\times\\mathcal{A}\\mathcal{P}(\\mathcal{S})$ be the transition function of the environment with $\\mathcal{P}(\\cdot)$ defining the set of probability measures. Let $R,d_{0},\\gamma,Q^{\\pi}$ , π be the reward function, initial state distribution, discount factor, a given trained Q-network, and the corresponding greedy policy as introduced in Section $2,J(\\pi)$ represents the cumulative reward and $J_{\\varepsilon}(\\pi)$ represents the perturbed cumulative reward under perturbations $\\delta_{t}\\in B^{\\varepsilon}$ at each time step t:\n\n$$\nJ(\\pi): =\\sum_{t=0}^{\\infty}\\gamma^{t}R(s_{t},\\pi(s_{t})),\n$$\n\n$$\n\\mathrm{where}\\ s_{t+1}\\sim P(s_{t},a_{t}),s_{0}\\sim d_{0},\n$$\n\nand\n\n$$\nJ_{\\varepsilon}(\\pi): =\\sum_{t=0}^{\\infty}\\gamma^{t}R(s_{t},\\pi(s_{t}+\\delta_{t})),\\tag{4}\n$$\n\n$$\n\\mathrm{where}\\ s_{t+1}\\sim P(s_{t},\\pi(s_{t}+\\delta_{t})),s_{0}\\sim d_{0}.\n$$\n\nThe randomness of $J(\\pi)$ arises from the environment dynamics, while that of $J_{\\varepsilon}(\\pi)$ includes additional randomness from the perturbations $\\{\\delta_{t}\\}$ . We focus on a finite horizon H in this paper, where a sufficiently large H can approximate $J(\\pi)$ and $J_{\\varepsilon}(\\pi)$ to arbitrary precision when $\\gamma < 1$\n\nDefinition 4 (Robustness certification for cumulative reward). The robustness certification for cumulative reward is the lower bound of perturbed cumulative reward J such that $\\underline{{J}}~\\leq~J_{\\varepsilon}(\\pi)$ under perturbation in $B^{\\varepsilon}=\\{\\delta\\in\\mathbb{R}^{n}\\mid\\|\\delta\\|_{2}\\leq\\varepsilon\\}$ applied to all time steps.\n\nLet $Q^{\\pi}$ be the action-value function given by the trained network Q with policy π. We derive a smoothed function ${\\widetilde{Q}}^{\\pi}$ through per-state local smoothing. Specifically, at each time step t, for each action $a\\in{\\mathcal{A}}$ , we draw random noise from a Gaussian distribution $\\mathcal{N}(\\mathrm{~0~},\\sigma^{2}I_{N})$ to smooth $Q^{\\pi}(\\cdot,a)$\n\n$$\n\\widetilde{Q}^{\\pi}(s_{t},a): =\\mathbb{E}_{\\Delta_{t}\\sim\\mathcal{N}(0,\\sigma^{2}I_{N})}Q^{\\pi}(s_{t}+\\Delta_{t},a)\\quad\\forall s_{t}\\in\\mathcal{S},a\\in\\mathcal{A},\\mathrm{~a n d~}\\tilde{\\pi}(s_{t}): =\\operatorname*{argmax}_{a}\\widetilde{Q}^{\\pi}(s_{t},a)\\quad\\forall s_{t}\\in\\mathcal{S}.\\tag{5}\n$$\n\nDefinition 5 (σ-randomized trajectory and σ-randomized policy). Given a state trajectory $\\left(s_{0},s_{1},\\ldots,s_{H-1}\\right)$ of length H where $s_{t+1}\\sim P(s_{t},\\pi(s_{t})),s_{0}\\sim d_{0}.$ with π the greedy policy of the action-value function $Q^{\\pi}$ , we derive a σ-randomized trajectory as $(s_{0}^{\\prime},s_{1}^{\\prime},\\ldots,s_{H-1}^{\\prime})$ , where $s_{t+1}^{\\prime}\\sim$ $P(s_{t}^{\\prime},\\pi(s_{t}^{\\prime}+\\Delta_{t})),\\Delta_{t}\\sim\\mathcal{N}(0,\\sigma^{2}I_{N})$ , and $s_{0}^{\\prime}=s_{0}\\sim d_{0}$ We correspondingly define a σ-randomized policy π' based on π in the following form: $\\overline{{\\pi}}^{\\prime}(s_{t}): =\\pi(s_{t}+\\Delta_{t})$ where $\\bar{\\Delta_{t}}\\sim\\mathcal N(0,\\sigma^{2}I_{N})$ .\n\nLet the operator $\\oplus$ concatenates given input states or noise that are added to each state. The sampled noise sequence is denoted by $\\Delta\\bar{=}\\oplus_{t=0}^{H-\\bar{1}}\\Delta_{t}$ , where $\\Delta_{t}\\sim\\mathcal{N}(0,\\sigma^{2}I_{N})$\n\nDefinition 6 (Perturbed return function). Let $R,P,\\gamma,d_{0}$ be the reward function, transition function, discount factor, and initial state distribution in Definition 3. We define a bounded perturbed return function $F_{\\pi}:\\mathbb{R}^{H\\times N}[J_{\\operatorname*{min}},J_{\\operatorname*{max}}]$ representing cumulative reward with potential perturbation $\\delta \\colon$\n\n$$\nF_{\\pi}\\left(\\oplus_{t=0}^{H-1}\\delta_{t}\\right): =\\sum_{t=0}^{H}\\gamma^{t}R(s_{t},\\pi(s_{t}+\\delta_{t})),\\quad{\\mathrm{w h e r e~}}s_{t+1}\\sim P(s_{t},\\pi(s_{t}+\\delta_{t})),s_{0}\\sim d_{0}.\\tag{6}\n$$\n\nWe can see when there is no perturbation $(\\delta_{t}=\\mathbf{0}),F_{\\pi}(\\oplus_{t=0}^{H-1}\\mathbf{0})=J(\\pi)$ ; when there are adversarial perturbations $\\delta_{t}\\in B^{\\varepsilon}$ at each time step, $F_{\\pi}(\\oplus_{t=0}^{H-1}\\delta_{t})=J_{\\varepsilon}(\\pi)$ ,i.e., perturbed cumulative reward.\n\nMean Smoothing: Expectation bound. Here we propose to sample noise sequences $\\Delta$ to perform global smoothing for the entire state trajectory, and calculate the lower bound of the expected perturbed cumulative reward $\\mathbb{E}_{\\Delta}\\left[J_{\\varepsilon}(\\pi^{\\prime})\\right]$ under all possible $\\ell_{2}$ -bounded perturbations within magnitude ε. The expectation is over the noise sequence $\\Delta$ involved in the σ-randomized policy $\\pi^{\\prime}$ in Definition 5.", "theorems": [{"local_id": "T1", "statement": "Lemma 2 (Lipschitz continuity of the smoothed value function). Given the action-value function $Q^{\\pi}:S\\times{\\mathcal{A}}[V_{\\operatorname*{min}},V_{\\operatorname*{max}}]$ , the smoothed function ${\\widetilde{Q}}^{\\pi}$ with smoothing parameter σ is L-Lipschitz continuous with $\\begin{array}{ r }{L=\\frac{V_{\\mathrm{max}}-V_{\\mathrm{min}}}{\\sigma}\\sqrt{2/\\pi}}\\end{array}$ w.r.t. the state input.", "proof": {"steps": [{"n": 1, "claim": "Proof. To prove Lemma 2, we leverage the technique in the proof for Lemma 1 of Salman et al.   \n(2019) in their Appendix A."}, {"n": 2, "claim": "For each action $a\\in{\\mathcal{A}}.$ ,our smoothed value function is\n\n$$\n\\widetilde{Q}^{\\pi}(s,a): =\\underset{\\Delta\\sim\\mathcal{N}(0,\\sigma^{2}I_{N})}{\\mathbb{E}}Q^{\\pi}(s+\\Delta,a)=\\frac{1}{(2\\pi)^{N/2}\\sigma^{N}}\\int_{\\mathbb{R}^{n}}Q^{\\pi}(t,a)\\exp\\left(-\\frac{1}{2\\sigma^{2}}\\|s-t\\|^{2}\\right)d t .\n$$"}, {"n": 3, "claim": "Taking the gradient w.r.t. s, we obtain\n\n$$\n\\nabla_{s}\\widetilde{Q}^{\\pi}(s,a)=\\frac{1}{(2\\pi)^{N/2}\\sigma^{N}}\\int_{\\mathbb{R}^{n}}Q^{\\pi}(t,a)\\frac{1}{\\sigma^{2}}(t-s)\\exp\\left(-\\frac{1}{2\\sigma^{2}}\\|s-t\\|^{2}\\right)d t .\n$$"}, {"n": 4, "claim": "For any unit direction u, we have\n\n$$\n\\begin{array}{ r l }&{u\\cdot\\nabla_{s}\\widetilde{Q}^{\\pi}(s,a)\\leq\\displaystyle\\frac{1}{(2\\pi)^{N/2}\\sigma^{N}}\\int_{\\mathbb R^{n}}\\frac{V_{\\operatorname*{max}}-V_{\\operatorname*{min}}}{\\sigma^{2}}\\displaystyle|u(s-t)|\\exp\\left(-\\frac{1}{2\\sigma^{2}}\\|s-t\\|^{2}\\right)d t}\\\\ &{\\quad\\quad\\quad=\\displaystyle\\frac{V_{\\operatorname*{max}}-V_{\\operatorname*{min}}}{\\sigma^{2}}\\cdot\\int_{\\mathbb R^{n}}\\frac{1}{(2\\pi)^{1/2}\\sigma}|s_{i}-t|\\exp\\left(-\\frac{1}{2\\sigma^{2}}|s_{i}-t|^{2}\\right)d t}\\\\ &{\\quad\\quad\\quad\\cdot\\displaystyle\\prod_{j\\neq i}\\int_{\\mathbb R^{n}}\\frac{1}{(2\\pi)^{1/2}\\sigma}\\exp\\left(-\\frac{1}{2\\sigma^{2}}|s_{j}-t|^{2}\\right)d t}\\\\ &{\\quad\\quad\\quad=\\displaystyle\\frac{V_{\\operatorname*{max}}-V_{\\operatorname*{min}}}{\\sigma}\\sqrt{\\frac{2}{\\pi}}}\\end{array}\n$$"}, {"n": 5, "claim": "Thus, ${\\widetilde{Q}}^{\\pi}$ is L-Lipschitz continuous with $\\begin{array}{ r }{L=\\frac{V_{\\mathrm{max}}-V_{\\mathrm{min}}}{\\sigma}\\sqrt{2/\\pi}}\\end{array}$ w.r.t. the state input."}]}}, {"local_id": "T2", "statement": "Theorem 2. Let $Q^{\\pi}:S\\times{\\mathcal{A}}[V_{\\operatorname*{min}},V_{\\operatorname*{max}}]$ be a trained value network, ${\\widetilde{Q}}^{\\pi}$ be the smoothed function with (5). At time step t with state $s_{t},$ we can compute the lower bound $r_{t}$ of maximum perturbation magnitude $\\bar{\\varepsilon}(s_{t})\\ :(i . e .,\\bar{r_{t}}\\leq\\bar{\\varepsilon}(s_{t})$ , ε defined in Definition 2) for locally smoothed policy π:\n\n$$\nr_{t}=\\frac{\\sigma}{2}\\left(\\Phi^{-1}\\left(\\frac{\\widetilde{Q}^{\\pi}(s_{t},a_{1})-V_{\\operatorname*{min}}}{V_{\\operatorname*{max}}-V_{\\operatorname*{min}}}\\right)-\\Phi^{-1}\\left(\\frac{\\widetilde{Q}^{\\pi}(s_{t},a_{2})-V_{\\operatorname*{min}}}{V_{\\operatorname*{max}}-V_{\\operatorname*{min}}}\\right)\\right),\\tag{7}\n$$\n\nwhere $\\Phi^{-1}$ is the inverse CDF function, $a_{1}$ is the action with the highest ${\\widetilde{Q}}^{\\pi}$ value at state $s_{t},$ ,and $a_{2}$ is the runner-up action. We name the lower bound $r_{t}$ as certified radius for the state $s_{t}$", "proof": {"steps": [{"n": 1, "claim": "Proof for Theorem. Let the perturbation be $\\delta_{t}$ , based on the Lipschitz continuity of the mapping $\\eta,$ we have\n\n$$\n\\eta_{a_{1}}(s_{t})-\\eta_{a_{1}}(s_{t}+\\delta_{t})\\leq\\|\\delta_{t}\\|_{2},\\tag{8}\n$$"}, {"n": 2, "claim": "$$\n\\eta_{a_{2}}(s_{t}+\\delta_{t})-\\eta_{a_{2}}(s_{t})\\leq\\|\\delta_{t}\\|_{2}.\\tag{9}\n$$"}, {"n": 3, "claim": "Suppose that under perturbation $\\delta_{t}$ , the action selection would be misled in the sense that the smoothed value for the original action $a_{1}$ is lower than that of another action $a_{2}$ ,i.e., $\\widetilde Q^{\\pi}(s_{t}+\\delta_{t},a_{1})\\ :\\leq$ $\\widetilde{Q}^{\\pi}(s_{t}+\\delta_{t},a_{2})$ . Then, based on the monotonicity of $\\eta,$ we have\n\n$$\n\\eta_{a_{1}}(s_{t}+\\delta)\\leq\\eta_{a_{2}}(s_{t}+\\delta).\\tag{10}\n$$"}, {"n": 4, "claim": "Summing up (8), (9), and (10), we obtain\n\n$$\n\\begin{array}{ r l r }{{\\|\\delta_{t}\\|_{2}\\geq\\frac{1}{2}(\\eta_{a_{1}}(s_{t})-\\eta_{a_{2}}(s_{t}))}}\\\\ &{}&{=\\frac{\\sigma}{2}(\\Phi^{-1}(\\frac{\\widetilde{Q}^{\\pi}(s_{t},a_{1})-V_{\\operatorname*{min}}}{V_{\\operatorname*{max}}-V_{\\operatorname*{min}}})-\\Phi^{-1}(\\frac{\\widetilde{Q}^{\\pi}(s_{t},a_{2})-V_{\\operatorname*{min}}}{V_{\\operatorname*{max}}-V_{\\operatorname*{min}}}))}\\end{array}\n$$"}, {"n": 5, "claim": "which is a lower bound of the maximum perturbation magnitude $\\bar{\\varepsilon}(s_{t})$ that can be tolerated at state $s_{t}$ Hence, when $r_{t}$ takes the value of the computed lower bound, it satisfies the condition that $r_{t}\\le\\bar{\\varepsilon}(s_{t})$"}]}}, {"local_id": "T3", "statement": "Let $R,P,\\gamma,d_0$ be the reward function, transition function, discount factor, and initial state distribution. Let $F_\\pi : \\mathbb{R}^{H\\times N} \\to [J_{\\min},J_{\\max}]$ be the bounded perturbed return function defined by $F_\\pi(\\oplus_{t=0}^{H-1}\\delta_t) := \\sum_{t=0}^{H}\\gamma^t R(s_t,\\pi(s_t+\\delta_t))$, where $s_{t+1}\\sim P(s_t,\\pi(s_t+\\delta_t))$, $s_0\\sim d_0$, and $\\oplus$ concatenates the per-step perturbations. Then the smoothed perturbed return function $\\widetilde{F}_\\pi$, defined by $\\widetilde{F}_\\pi(\\oplus_{t=0}^{H-1}\\delta_t) := \\mathbb{E}_{\\Delta\\sim\\mathcal{N}(0,\\sigma^2 I_{H\\times N})} F_\\pi(\\oplus_{t=0}^{H-1}(\\delta_t+\\Delta_t))$ with smoothing parameter $\\sigma$, is $\\frac{(J_{\\max}-J_{\\min})}{\\sigma}\\sqrt{2/\\pi}$-Lipschitz continuous w.r.t. the concatenated input. Writing $L=\\frac{(J_{\\max}-J_{\\min})}{\\sigma}\\sqrt{2/\\pi}$ and letting $\\pi'$ denote the $\\sigma$-randomized policy, it follows that for every adversarial perturbation sequence $\\oplus_{t=0}^{H-1}\\delta_t$ with per-step budget $\\|\\delta_t\\|\\le\\varepsilon$, the perturbed cumulative reward admits the certified lower bound $\\mathbb{E}[J_\\varepsilon^H(\\pi')] \\ge \\widetilde{F}_\\pi(\\oplus_{t=0}^{H-1}\\mathbf{0}) - L\\varepsilon\\sqrt{H}$.", "proof": {"steps": [{"n": 1, "claim": "$\\widetilde{F}_\\pi$ is obtained by taking the expectation of $F_\\pi$ over sampled $\\sigma$-randomized trajectories, exactly analogous to how the smoothed value function $\\widetilde{Q}^\\pi$ in Lemma 2 is obtained by taking the expectation over sampled states."}, {"n": 2, "claim": "The only difference from the Lemma 2 setting is the output range: the Q-network outputs lie in $[V_{\\min},V_{\\max}]$, whereas the perturbed return function outputs lie in $[J_{\\min},J_{\\max}]$."}, {"n": 3, "claim": "Therefore $\\widetilde{F}_\\pi$ is $\\frac{(J_{\\max}-J_{\\min})}{\\sigma}\\sqrt{2/\\pi}$-Lipschitz continuous, obtained by substituting the range $J_{\\max}-J_{\\min}$ for $V_{\\max}-V_{\\min}$ in the Lemma 2 Lipschitz constant."}, {"n": 4, "claim": "$\\widetilde{F}_\\pi(\\oplus_{t=0}^{H-1}\\delta_t) = \\mathbb{E}[F_\\pi(\\oplus_{t=0}^{H-1}(\\delta_t+\\zeta_t))]$."}, {"n": 5, "claim": "$\\mathbb{E}[F_\\pi(\\oplus_{t=0}^{H-1}(\\delta_t+\\zeta_t))] = \\mathbb{E}[F_{\\pi'}(\\oplus_{t=0}^{H-1}\\delta_t)]$."}, {"n": 6, "claim": "$\\mathbb{E}[F_{\\pi'}(\\oplus_{t=0}^{H-1}\\delta_t)] = \\mathbb{E}[J_\\varepsilon^H(\\pi')]$."}, {"n": 7, "claim": "The lower bound $\\widetilde{F}_\\pi(\\oplus_{t=0}^{H-1}\\mathbf{0}) - L\\varepsilon\\sqrt{H}$ on $\\mathbb{E}[J_\\varepsilon^H(\\pi')]$ is obtained."}]}}, {"local_id": "T4", "statement": "Lemma 4. Let  be the CDF of a standard normal distribution, the mapping $\\eta_{a}(s): =\\sigma$ $\\Phi^{-1}\\left(\\frac{\\widetilde Q^{\\pi}(s,a)-V_{\\mathrm{min}}}{V_{\\mathrm{max}}-V_{\\mathrm{min}}}\\right)$ is 1-Lipschitz continuous.", "proof": {"steps": [{"n": 1, "claim": "The lemma can be proved following the same technique as the proof for Lemma 2 in Appendix A.1. The detailed proof can be referred to in the proof for Lemma 2 of Salman et al. (2019) in their Appendix A. We next show how to leverage Lemma 4 to prove Theorem 2."}]}}, {"local_id": "T5", "statement": "Theorem 3. Let $(r_{t}^{1},\\ldots,r_{t}^{|\\mathcal{A}| -1})$ be a sequence of certified radii for state $s_{t}$ at time step t, where $r_{t}^{k}$ denotes the radius such that $i f\\varepsilon<r_{t}^{k};$ , the possible action at time step t will belong to the actions corresponding to top k action values of $\\widetilde{Q}$ at state $s_{t}.$ The definition of $r_{t}$ in Theorem 2 is equivalent to $\\boldsymbol{r}_{t}^{1}$ here. The radii can be computed similarly as follows:\n\n$$\nr_{t}^{k}=\\frac{\\sigma}{2}\\left(\\Phi^{-1}\\left(\\frac{\\widetilde{Q}^{\\pi}(s_{t},a_{1})-V_{\\operatorname*{min}}}{V_{\\operatorname*{max}}-V_{\\operatorname*{min}}}\\right)-\\Phi^{-1}\\left(\\frac{\\widetilde{Q}^{\\pi}(s_{t},a_{k+1})-V_{\\operatorname*{min}}}{V_{\\operatorname*{max}}-V_{\\operatorname*{min}}}\\right)\\right),\\quad 1\\leq k< |A|,\n$$\n\nwhere $a_{1}$ is the action of the highest $\\widetilde{Q}$ value at state $s_{t}$ and $a_{k+1}$ is the $(k+1)–th$ best action. We additionally define $r_{t}^{0}(s_{t})=0;$ which is also compatible with the definition above.", "proof": {"steps": [{"n": 1, "claim": "Proof. Replacing $a_{2}$ with $\\boldsymbol a_{k+1}$ in the proof for Theorem 2 in Appendix A.2 directly leads to Theorem 3."}]}}], "target_local_id": "T3", "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "label": "flawed", "error_step": 7, "error_class6": ["missing_assumption", "logical_gap_or_missing_case", "others"], "error_explanation": "The certified lower bound is asserted for every perturbation sequence, including one chosen ADAPTIVELY, i.e. where the adversary picks $\\delta_t$ as a function of the smoothed actions/states already observed at steps $0,\\ldots,t$. However, the proof of this bound at Step 7 lifts the single-shot Lipschitz / randomized-smoothing argument, whose validity requires generating the $m$ randomized trajectories from a KNOWN, pre-committed noise sequence $\\{\\delta_t\\}_{t=0}^{H-1}$. This silently assumes a NON-ADAPTIVE adversary. As the reviewer put it, 'the presented Lemma 2 only holds if the attacker commits to the values of $\\zeta_t$ beforehand,' and the authors concede it 'cannot be applied against an adaptive attacker' (a stated limitation of CROP-GRe). The underlying Lipschitz constant established earlier (Step 3) is itself correct as a property of a fixed input; the missing-assumption error is that Step 7 uses it to certify against an ARBITRARY adaptive adversary without assuming the perturbation/noise sequence is fixed and known in advance."}
{"id": "HyWG0H5ge", "context": "An algorithm has no-regret if lim $_{N\\to\\infty}\\mathrm{Regret}(N)/N\\to 0$ for any sequence of losses with bounded gradients. For example, Kingma & Ba (2015) prove that Adam has no-regret:\n\nTheorem 2 (Adam has no-regret, (Kingma & Ba, 2015)).\n\nSuppose the the losses $\\ell^{n}$ have bounded gradients $\\|\\nabla_{\\mathbf{W}}\\ell^{n}(\\mathbf{W})\\|_{2}\\leq G a n d\\|\\nabla_{\\mathbf{W}}\\ell^{n}(\\mathbf{W})\\|_{\\infty}\\leq G$ for all $\\mathbf{W}\\in\\mathcal{H}$ and suppose that the weights chosen by the algorithm $\\ L\\ L_{\\{a t i s f y}\\|\\mathbf{W}^{m}-\\mathbf{W}^{n}\\|_{2}\\leq D$ and $\\|\\mathbf{W}^{m}-\\mathbf{W}^{n}\\|_{\\infty}\\leq D$ for all $m,n\\in\\{1,\\ldots,N\\}$ . Then Adam satisfies\n\n$$\n\\mathsf{Regret}(N)/N\\le O\\left(1/\\sqrt{N}\\right)\\quad f o r a l l N\\ge 1 .\\tag{3}\n$$\n\nThe regret of gradient descent, AdaGrad (Duchi et al., 2011), mirror descent and a variety of related algorithms satisfy (3), albeit with different constant terms that are hidden in the big-O notation. Finally, the $1/\\sqrt{N}$ rate is also lower-bound. It cannot be improved without additional assumptions.\n\n---\n\nConsider a network with $L - 1$ hidden layers and weight matrices $\\mathbf{W}: =\\{\\mathbf{W}_{1},\\dots,\\mathbf{W}_{L}\\}$ . Let $\\mathbf{x}_{\\mathrm{0}}$ denote the input. For hidden layers $l\\in\\{\\bar{1},\\ldots,L-1\\bar\\}$ }, set $\\mathbf{a}_{l}=\\mathbf{W}_{l}\\cdot\\mathbf{x}_{l-1}$ and ${\\mathbf x}_{l}=s({\\mathbf a}_{l})$ where $s(\\cdot)$ is applied coordinatewise. The last layer outputs $\\mathbf{x}_{L}=\\mathbf{a}_{L}=\\mathbf{W}_{L}\\cdot\\mathbf{x}_{L-1}$ . Let $p_{l}$ denote the size of the $l^{\\mathrm{th}}$ layer; $p_{0}$ is the size of the input and $p_{L}$ is the size of the output. Suppose the loss $\\ell(\\mathbf{f},y)$ is smooth and convex in the first argument. The training data is $(\\mathbf{x}^{d},y^{d})_{d=1}^{D}.$ The network is trained on stochastic samples from the training data on a series of rounds $n=1,\\ldots,N$ For simplicity we assume minibatch size $1 ;$ the results generalize without difficulty.\n\nWe recall backprop using notation from Martens et al. (2012). Let $\\mathbf{J_{b}^{a}}$ denote the Jacobian matrix of the vector a with respect to the vector b. By the chain rule the gradient decomposes as\n\n$$\n\\nabla_{\\mathbf{W}_{l}}\\ell\\big(\\mathbf{f}_{\\mathbf{W}}(\\mathbf{x}_{0}),y\\big)=\\underbrace{\\mathbf{J}_{L}^{\\mathcal{E}}\\cdot\\mathbf{J}_{L-1}^{L}\\cdot\\cdot\\cdot\\mathbf{J}_{l}^{l+1}}_{\\delta_{l}}\\otimes\\mathbf{x}_{l-1}=\\underbrace{\\mathbf{J}_{L}^{\\mathcal{E}}}_{\\nabla_{\\mathbf{f}}\\ell(\\mathbf{f},y)}\\cdot\\underbrace{\\mathbf{J}_{l}^{L}\\otimes\\mathbf{x}_{l-1}}_{\\nabla_{\\mathbf{W}_{l}}\\mathbf{f}_{\\mathbf{W}}(\\mathbf{x}_{0})= :\\mathbf{G}_{l}}\\tag{4}\n$$\n\nwhere $\\delta_{l}=\\mathbf{J}_{l}^{\\varepsilon}$ is the backpropagated error computed recursively via $\\delta_{l}=\\delta_{l+1}\\cdot\\mathbf{J}_{l}^{l+1}.^{2}$ The middle expression in (4) is the standard representation of backpropagated gradients. The expression on the right factorizes the backpropagated error $\\delta_{l}=\\mathbf{J}_{L}^{\\varepsilon}\\cdot\\mathbf{\\hat{J}}_{l}^{L}$ into the gradient of the loss $\\mathbf{J}_{L}^{\\mathcal{E}}$ and the Jacobian $\\mathbf{J}_{l}^{L}$ between layers, which describes gradient flow within the network.\n\nThe first-order Taylor approximation to a differentiable function $f:\\mathbb{R}\\mathbb{R}$ near a is $T_{a}(x)=$ $f(a)+f^{\\prime}(a)\\cdot(x-a)$ . The neural Taylor approximation for a fully connected network is as follows. Definition 2. The Jacobian tensor of layer $l,{\\bf G}_{l}: ={\\bf J}_{{l}}^{L}\\otimes{\\bf x}_{l-1}$ , is the gradient of the output of the neural network with respect to the weights of layer l. It is the outer product of a $\\left(p_{L}\\times p_{l}\\right)$ -matrix with a pl−1-vector, and so is a $(p_{L},p_{l},p_{l-1})– t e n s o r$\n\nGiven $\\mathbf{G}_{l}$ and $\\left(p_{l}\\times p_{l-1}\\right)$ -matrix V, the expression $\\langle\\mathbf{G}_{l},\\mathbf{V}\\rangle: =\\mathbf{J}_{l}^{L}\\cdot\\mathbf{V}\\cdot\\mathbf{x}_{l-1}$ is the pL-vector computed via matrix-matrix-vector multiplication. The neural Taylor approximation to f in a neighborhood of $\\mathbf{W}^{n}$ , given input $\\mathbf{x}_{0}^{n}$ , is the first-order Taylor expansion\n\n$$\n\\mathbf{T}^{n}(\\mathbf{V}): =\\mathbf{f}_{\\mathbf{W}^{n}}(\\mathbf{x}_{0}^{n})+\\sum_{l=1}^{L}\\left.\\mathbf{G}_{l},\\mathbf{V}_{l}-\\mathbf{W}_{l}^{n}\\right.\\approx\\mathbf{f}_{\\mathbf{V}}(\\mathbf{x}_{0}^{n}).\n$$\n\nFinally, the Taylor loss of the network on round n is $\\mathcal{T}^{n}(\\mathbf{V})=\\ell(\\mathbf{T}^{n}(\\mathbf{V}),y^{n})$\n\nThe Taylor approximation to layer l is $\\mathbf{T}_{l}^{n}(\\mathbf{V}_{l}): =\\mathbf{f}_{\\mathbf{W}^{n}}\\big(\\mathbf{x}_{0}^{n}\\big)+\\big\\langle\\mathbf{G}_{l},\\mathbf{V}_{l}-\\mathbf{W}_{l}^{n}\\big\\rangle$ . We can also construct the Taylor approximation to neuron α in layer l. Let the pL-vector $\\mathbf{J}_{\\alpha}^{L}: =\\mathbf{\\dot{J}}_{l}^{L}[:,\\alpha]$ denote the Jacobian with respect to neuron α and let the $\\left(p_{L}\\times p_{l-1}\\right)$ -matrix $\\mathbf{G}_{\\alpha}: =\\mathbf{J}_{\\alpha}^{L}\\otimes\\mathbf{x}_{l-1}$ denote the Jacobian with respect to the weights of neuron α. The Taylor approximation to neuron α is $\\mathbf{T}_{\\alpha}^{n}(\\mathbf{V}_{\\alpha}): =\\mathbf{f}_{\\mathbf{W}^{n}}(\\mathbf{x}_{0}^{n})+\\left.\\mathbf{G}_{\\alpha},\\mathbf{V}_{\\alpha}-\\mathbf{W}_{\\alpha}^{n}\\right.$\n\nThe Taylor losses are the simplest non-trivial (i.e. non-affine) convex functions encoding the information generated by backprop, see section A.4.\n\n---\n\n2.3 Empirical Analysis of Online Neural Optimization and Regret\n\nThis section empirically investigates the Taylor optimum and regret terms in theorem 3 on two tasks:\n\nAutoencoder trained on MNIST. Dense layers with architecture $784\\to 50\\to 30\\to 20\\to 30\\to$ 50 → 784 and ReLU non-linearities. Trained with MSE loss using minibatches of 64.\n\nConvnet trained on CIFAR-10. Three convolutional layers with stack size 64 and $5 \\times 5$ receptive fields, ReLU non-linearities and $2 \\times 2$ max-pooling. Followed by a 192 unit fully-connected layer with ReLU before a ten-dimensional fully-connected output layer. Trained with cross-entropy loss using minibatches of 128.\n\nFor both tasks we compare the optimization performance of Adam, RMSProp and SGD (figure 6). Learning rates were tuned for optimal performance. Additional parameters for Adam and RMSProp were left at default. For the convnet all three methods perform equally well: achieving a small loss and an accuracy $\\mathrm{of}\\geq 99\\%$ on the training set. However, SGD exhibits slightly more variance. For the autoencoder, although it is an extremely simple model, SGD with the best (fixed) learning rate performs significantly worse than the adaptive optimizers.\n\nThe neuronal and layer-wise regret are evaluated for each model. At every iteration we record the training error  the left-hand-side of eq. (5). To evaluate the Taylor loss, we record the input to the neuron/layer, it's weights, the output of the network and the gradient tensor $\\mathbf{G}_{l}$ After training, we minimize the Taylor loss with respect to V to find the Taylor optimum at each round. The regret is the difference between the observed training loss and the optimal Taylor loss.\n\n---\n\n3.1 The Role of Activation Configurations in Optimization\n\nWe describe how configurations of active neurons relate to smooth regions of rectifier networks and to neural Taylor approximations. Recall that the loss of a neural net on its training data is $\\begin{array}{ r }{\\hat{\\ell}(\\mathbf{W})=\\frac{1}{D}\\sum_{d=1}^{D}\\ell\\big(f_{\\mathbf{W}}(\\mathbf{x}^{d}),y^{d}\\big)}\\end{array}$\n\nDefinition 3. Enumerate the data as $[D]=\\{1,\\dots,D\\}$ and neurons as $[ M ]$ .The activation configuration A(W) is a $(D\\times M)$ binary matrix representing the active neurons for each input. The set of all possible activation configurations corresponds to the set of all $(D\\times M)$ binary matrices.\n\nObservation 3 (activation configurations correspond to smooth regions in rectifier networks).\n\nA parameter value exhibits a kink in iff an infinitesimal change alters the of activation configuration, i.e.  is not smooth at W iff there is a V s.t. $\\pmb{\\mathcal{A}}(\\mathbf{W})\\neq\\pmb{\\mathcal{A}}(\\mathbf{W}+\\delta\\mathbf{V})$ for all $\\delta > 0 .$\n\n---\n\nA.1 Background on Convex Optimization\n\nA continuous function $f$ is smooth if there exists a $\\beta > 0$ such that $\\|\\nabla f(\\mathbf{x})-\\nabla f(\\mathbf{y})\\|_{2}\\ \\leq$ $\\beta\\cdot\\|\\mathbf{x}-\\mathbf{y}\\|_{2}$ for all x and y in the domain. Rectifiers are not smooth for any value of β.\n\nNonsmooth convex functions. Let $X\\subset\\mathbb{R}^{p}$ be a convex set contained in a ball of radius R. Let $\\ell : X \\to$ R be a convex function. Section 3.1 of (Bubeck, 2015) shows that projected gradient descent has convergence guarantee\n\n$$\n\\ell\\left({\\frac{1}{N}}\\sum_{n=1}^{N}\\mathbf{w}^{n}\\right)-\\ell(\\mathbf{w}^{*})\\leq O\\left({\\frac{1}{\\sqrt{N}}}\\right)\n$$\n\nwhere $\\mathbf{w}^{n}$ are generated by gradient descent and $\\mathbf{w}^{*}: =\\mathrm{argmin}_{\\mathbf{w}\\in X}\\ell(\\mathbf{w})$ is the minimizer of l. It is also shown, section 3.5, that\n\n$$\n\\operatorname*{min}_{1\\le n\\le N}\\ell(\\mathbf{w}^{n})-\\ell(\\mathbf{w}^{*})\\geq\\Omega\\left(\\frac{1}{\\sqrt{N}}\\right)\n$$\n\nwhere the weights are in the span of the previously observed gradients: $\\mathbf{w}^{n}\\in$ span $\\left\\{\\nabla\\ell(\\mathbf{w}^{1}),\\ldots,\\nabla\\ell(\\mathbf{w}^{n-1})\\right\\}$ for all $n\\in\\{1,\\ldots,\\bar{N}\\}$ •\n\nThe gradient of a convex function increases monotonically. That is\n\n$$\n\\left.\\nabla\\ell(\\mathbf{w})-\\nabla\\ell(\\mathbf{v}),\\mathbf{w}-\\mathbf{v}\\right.\\geq 0\n$$\n\nfor all points w, v where the gradient exists. Gradients at one point of a nonsmooth convex function therefore do contain information about other points, although not as much information as in the smooth case. In contrast, the gradients of nonsmooth nonconvex functions can vary wildly as shown in Fig. 1.\n\nSmooth convex functions. In the smooth setting, gradient descent converges at rate $\\textstyle{\\frac{1}{N}}$ . The lower bound for convergence is even better, $\\scriptstyle{\\frac{1}{N^{2}}}$ . The lower bound is achieved by Nesterov's accelerated gradient descent method.", "theorems": [{"local_id": "T1", "statement": "Observation 3 (activation configurations correspond to smooth regions in rectifier networks). Assume the per-example loss $\\ell(\\mathbf{f},y)$ is smooth in its first argument. If the empirical loss $\\hat{\\ell}(\\mathbf{W})=\\frac{1}{D}\\sum_d \\ell(f_{\\mathbf{W}}(\\mathbf{x}^d),y^d)$ is not smooth at $\\mathbf{W}$ (i.e. $\\mathbf{W}$ exhibits a kink), then an infinitesimal change of the parameters alters the activation configuration: there exist a direction $\\mathbf{V}$ and $\\delta_0>0$ such that $\\boldsymbol{\\mathcal{A}}(\\mathbf{W}) \\neq \\boldsymbol{\\mathcal{A}}(\\mathbf{W}+\\delta\\mathbf{V})$ for all $0<\\delta<\\delta_0$.", "proof": {"steps": [{"n": 1, "claim": "Wherever the activation configuration is locally constant, ℓ̂ is smooth at W."}, {"n": 2, "claim": "Therefore a kink at W forces some direction V to change the activation configuration for all sufficiently small δ>0."}]}}, {"local_id": "T2", "statement": "Theorem 3 (no-regret relative to Taylor optimum).\n\nSuppose, as in Theorem 2, the Taylor losses have bounded gradients and the weights of the neural network have bounded diameter during training. Suppose the neural net is optimized by an algorithm with Regret $(N)\\leq O({\\sqrt{N}})$ such as gradient descent, AdaGrad, Adam or mirror descent.\n\nNetwork guarantee: The running average of the training error of the neural network satisfies\n\n$$\n\\underbrace{\\frac{1}{N}\\sum_{n=1}^{N}\\ell\\left(f_{\\mathbf{W}^{n}}\\left(\\mathbf{x}_{0}^{n}\\right),y^{n}\\right)}_{\\text{ running average of training errors }}\\leq\\underbrace{\\min_{\\mathbf{V}}\\left\\{\\frac{1}{N}\\sum_{n=1}^{N}\\mathcal{T}^{n}(\\mathbf{V})\\right\\}}_{\\text{ Taylor optimum }}+\\underbrace{O\\left(\\frac{1}{\\sqrt{N}}\\right)}_{\\mathrm{Regret}(N)/N}.\\tag{6}\n$$\n\n• Layer-wise / Neuron-wise guarantee: The Taylor loss of [layer-l / neuron-α] is $\\mathcal{T}_{l/\\alpha}^{\\bar{n}}(\\mathbf{V}_{l/\\alpha}): =\\ell(\\mathbf{T}_{l/\\alpha}^{n}(\\mathbf{V}_{l/\\alpha}),y^{n})$ . Then,\n\n$$\n\\underbrace{\\frac{1}{N}\\sum_{n=1}^{N}\\ell\\left(f_{\\mathbf{W}^{n}}\\left(\\mathbf{x}_{0}^{n}\\right),y^{n}\\right)}_{\\text{ running average of training errors }}\\leq\\underbrace{\\min_{\\mathbf{V}_{l/\\alpha}}\\left\\{\\frac{1}{N}\\sum_{n=1}^{N}\\mathcal{T}_{l/\\alpha}^{n}(\\mathbf{V}_{l/\\alpha})\\right\\}}_{\\text{ layer-wise/neuronal Taylor optimum }}+\\underbrace{O\\left(\\frac{1}{\\sqrt{N}}\\right)}_{\\mathrm{Regret}(N)/N}.\\tag{5}\n$$", "proof": {"steps": [{"n": 1, "claim": "The Taylor loss has three key properties by construction: T1. The Taylor loss $\\mathcal{T}^{n}$ coincides with the loss at $\\mathbf{W}^{n}$\n\n$$\n\\ell(\\mathbf{f}_{\\mathbf{W}^{n}}(\\mathbf{x}_{0}^{n}),y^{n})=\\mathcal{T}^{n}(\\mathbf{V})_{|\\mathbf{V}=\\mathbf{W}^{n}}\n$$"}, {"n": 2, "claim": "T2. The Taylor loss gradient ${\\mathcal{T}}^{n}$ coincides with the loss gradient at $\\mathbf{W}^{n}$\n\n$$\n\\nabla_{\\mathbf{W}}\\ell(\\mathbf{f}_{\\mathbf{W}^{n}}(\\mathbf{x}_{0}^{n}),y^{n})=\\nabla_{\\mathbf{V}}\\mathcal{T}^{n}(\\mathbf{V})_{|\\mathbf{V}=\\mathbf{W}^{n}}\n$$"}, {"n": 3, "claim": "T3. The Taylor losses are convex functions of V because $\\ell(\\mathbf{f},y)$ is convex in its first argument and convexity is invariant under affine maps. If  is a convex function, then so is $g(x)=$ $\\ell(Ax+b)$ , where $A\\in\\mathbb{R}^{m\\times n}$ and $b\\in\\mathbb{R}^{m}$ •"}, {"n": 4, "claim": "By T1, the training loss, i.e. the left-hand side of (6), exactly coincides with the Taylor losses. By T2, the gradients of the Taylor losses exactly coincide with the errors computed by backpropagation on the training losses. That is, the training loss over n rounds is indistinguishable from the Taylor losses to the first order:\n\n$$\n\\begin{array}{ c }{\\displaystyle\\mathrm{losses:}\\ \\frac{1}{N}\\sum_{n=1}^{N}\\ell\\left(f_{\\mathbf{W}^{n}}\\left(\\mathbf{x}_{0}^{n}\\right),y^{n}\\right)=\\frac{1}{N}\\sum_{n=1}^{N}\\mathcal{T}^{n}\\left(\\mathbf{W}^{n}\\right)}\\\\ {\\displaystyle\\mathrm{gradients:}\\ \\nabla_{\\mathbf{W}}\\left(\\frac{1}{N}\\sum_{n=1}^{N}\\ell\\left(f_{\\mathbf{W}^{n}}\\left(\\mathbf{x}_{0}^{n}\\right),y^{n}\\right)\\right)=\\nabla_{\\mathbf{W}}\\left(\\frac{1}{N}\\sum_{n=1}^{N}\\mathcal{T}^{n}\\left(\\mathbf{W}^{n}\\right)\\right)}\\end{array}\n$$"}, {"n": 5, "claim": "We can therefore substitute the Taylor losses in place of the training loss $(f_{\\mathbf{W}}(\\mathbf{x}_{0}),y)$ without altering either the losses incurred during training or the dynamics of backpropagation (or any first-order method)."}]}}, {"local_id": "T3", "statement": "Consider a network with $L-1$ hidden layers and weight matrices $\\mathbf{W}:=\\{\\mathbf{W}_1,\\dots,\\mathbf{W}_L\\}$, trained by online convex optimization on Taylor losses $\\mathcal{T}^n(\\mathbf{V})=\\ell(\\mathbf{T}^n(\\mathbf{V}),y^n)$, where $\\mathbf{T}^n(\\mathbf{V})$ is the first-order (neural Taylor) expansion of the network output around $\\mathbf{W}^n$. Suppose the loss $\\ell(\\mathbf{f},y)$ is smooth and convex in the first argument, the Taylor losses have bounded gradients, and the weights of the neural network have bounded diameter during training. Let $a_n$ denote the Taylor optimum on round $n$ (the minimizer of the round-$n$ Taylor loss, i.e. the path-dependent Taylor optimum appearing in Theorem 3). Then the sequence of Taylor optima $\\{a_n\\}_{n\\ge 1}$ converges.", "proof": {"steps": [{"n": 1, "claim": "Suffices to show $\\lVert a_n - a_{n+1}\\rVert \\to 0$."}, {"n": 2, "claim": "$\\lVert a_n - a_{n+1}\\rVert \\to 0$ as $n\\to\\infty$."}, {"n": 3, "claim": "Hence $\\{a_n\\}$ converges."}]}}, {"local_id": "T4", "statement": "Observation 4 (signed-gradient is a maximizer).\n\nSuppose none of the coordinates in  are zero. The signed-gradient satisfies\n\n$$\n\\mathbf{sign}(\\overline{{\\nabla}}\\ell)=\\underset{\\mathbf{x}\\in B_{\\infty}^{p}}{\\operatorname{argmax}}\\left.\\|\\mathbf{x}\\|_{1}:\\langle\\mathbf{x},\\overline{{\\nabla}}\\ell\\rangle\\geq 0\\right.,\\quad where\\quad B_{\\infty}^{p}=\\{\\mathbf{x}\\in\\mathbb{R}^{p}:\\operatorname*{max}_{i=1,\\ldots,p}|x_{i}|\\leq 1\\}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof. Immediate."}]}}], "target_local_id": "T3", "topic_domain": "Deep Learning Theory", "topic_subtopic": "Training Dynamics & Implicit Bias", "label": "flawed", "error_step": 1, "error_class6": ["logical_gap_or_missing_case", "circular_or_unproven", "others"], "error_explanation": "Step 1 claims that to prove $\\{a_n\\}$ converges it suffices to show the successive differences vanish, $\\lVert a_n - a_{n+1}\\rVert \\to 0$. This sufficiency claim is invalid: successive differences tending to zero does NOT imply the sequence is Cauchy and hence does NOT imply convergence. Being Cauchy requires the strictly stronger condition $\\lVert a_n - a_m\\rVert \\to 0$ for ALL $m>n$ (equivalently $\\sup_{m>n}\\lVert a_n-a_m\\rVert\\to 0$ in a complete space). Classic counterexamples where $\\lVert a_n-a_{n+1}\\rVert\\to 0$ yet the sequence diverges include the partial sums of the harmonic series and $a_n=\\sqrt{n}$. The false reduction is introduced here at step 1; steps 2-3 merely establish the successive-difference condition and draw the unwarranted convergence conclusion from it."}
{"id": "HygegyrYwH", "context": "The dataset is denoted by $\\{(x_i,y_i)\\}_{i=1}^n$ where $x_i\\in\\mathbb R^d$ and $y_i\\in\\{-1,+1\\}$. For simplicity, we assume that $\\|x_i\\|_2=1$ for any $1\\le i\\le n$.\n\nThe two-layer network has weight matrices $W\\in\\mathbb R^{m\\times d}$ and $a\\in\\mathbb R^m$. We use the following parameterization:\n\n$$\nf(x;W,a):=\\frac{1}{\\sqrt m}\\sum_{s=1}^m a_s\\sigma\\big(\\langle w_s,x\\rangle\\big),\n$$\n\nwith initialization\n\n$$\nw_{s,0}\\sim\\mathcal N(0,I_d),\\quad\\text{and}\\quad a_s\\sim\\mathrm{unif}\\big(\\{-1,+1\\}\\big).\n$$\n\nNote that $w_{s,t}$ denotes the $s$-th row of $W$ at step $t$. We fix $a$ and only train $W$. We consider the ReLU activation $\\sigma(z):=\\max\\{0,z\\}$.\n\nWe use the logistic (binary cross entropy) loss $\\ell(z):=\\ln\\big(1+\\exp(-z)\\big)$ and gradient descent. For any $1\\le i\\le n$ and any $W$, let $f_i(W):=f(x_i;W,a)$. The empirical risk and its gradient are given by\n\n$$\n\\widehat{\\mathcal R}(W):=\\frac1n\\sum_{i=1}^n\\ell\\big(y_if_i(W)\\big),\\quad\\text{and}\\quad\\nabla\\widehat{\\mathcal R}(W)=\\frac1n\\sum_{i=1}^n\\ell'\\big(y_if_i(W)\\big)y_i\\nabla f_i(W).\n$$\n\nFor any $t\\ge0$, the gradient descent step is given by $W_{t+1}:=W_t-\\eta_t\\nabla\\widehat{\\mathcal R}(W_t)$. Also define\n\n$$\nf_i^{(t)}(W):=\\big\\langle\\nabla f_i(W_t),W\\big\\rangle,\\quad\\text{and}\\quad\\widehat{\\mathcal R}^{(t)}(W):=\\frac1n\\sum_{i=1}^n\\ell\\Big(y_if_i^{(t)}(W)\\Big).\n$$\n\nNote that $f_i^{(t)}(W_t)=f_i(W_t)$. This property generally holds due to homogeneity: for any $W$ and any $1\\le s\\le m$,\n\n$$\n\\frac{\\partial f_i}{\\partial w_s}=\\frac{1}{\\sqrt m}a_s\\mathbb 1\\big[\\langle w_s,x_i\\rangle>0\\big]x_i,\\quad\\text{and}\\quad\\Big\\langle\\frac{\\partial f_i}{\\partial w_s},w_s\\Big\\rangle=\\frac{1}{\\sqrt m}a_s\\sigma\\big(\\langle w_s,x_i\\rangle\\big),\n$$\n\nand thus $\\big\\langle\\nabla f_i(W),W\\big\\rangle=f_i(W)$.\n\nIn this section, we consider a fixed training set and empirical risk minimization. We first state our assumption on the separability of the NTK, and then give our main result and a proof sketch.\n\nThe key idea of the NTK is to do the first-order Taylor approximation:\n\n$$\nf(x;W,a)\\approx f(x;W_{0},a)+\\left.\\nabla_{W}f(x;W_{0},a),W-W_{0}\\right..\n$$\n\nIn other words, we want to do learning using the features given by $\\nabla f_{i}(W_{0})\\in\\mathbb{R}^{m\\times d}$ A natural assumption is that there exists $\\overline{{U}}\\in\\mathbb{R}^{m\\times d}$ which can separate $\\{(\\nabla f_{i}(W_{0}),y_{i})\\}_{i=1}^{n}$ with a positive margin:\n\n$$\n\\operatorname*{min}_{1\\leq i\\leq n}\\left(y_{i}\\left.\\overline{{U}},\\nabla f_{i}(W_{0})\\right.\\right)=\\operatorname*{min}_{1\\leq i\\leq n}\\left(y_{i}\\frac{1}{\\sqrt{m}}\\sum_{s=1}^{m}a_{s}\\langle\\bar{u}_{s},x_{i}\\rangle\\mathbb{1}\\left[\\langle w_{s,0},x_{i}\\rangle>0\\right]\\right)>0 .\\tag{1}\n$$\n\nThe infinite-width limit of eq. (1) is formalized as Assumption 1, with an additional bound on the $(2,\\infty)$ norm of the separator. A concrete construction of $\\overline{U}$ using Assumption 1 is given in eq. (2).\n\nLet $\\mu_{\\mathcal{N}}$ denote the Gaussian measure on $\\mathbb{R}^{d}$ , given by the Gaussian density with respect to the Lebesgue measure on $\\mathbb{R}^{d}$ . We consider the following Hilbert space\n\n$$\n\\mathcal{H}: =\\left\\{w:\\mathbb{R}^{d}\\to\\mathbb{R}^{d}\\bigg|\\int\\|w(z)\\|_{2}^{2}\\mathrm{d}\\mu_{\\mathcal{N}}(z)<\\infty\\right\\}.\n$$\n\nFor any $x\\in\\mathbb{R}^{d}$ , define $\\phi_{x}\\in\\mathcal{H}$ by\n\n$$\n\\phi_{x}(z): =x\\mathbb{1}\\left[\\langle z,x\\rangle>0\\right],\n$$\n\nand particularly define $\\phi_{i}: =\\phi_{x_{i}}$ for the training input $x_{i}$\n\nAssumption 1. There exists $\\bar{v}\\in\\mathcal H$ and $\\gamma > 0$ , such that $\\left.\\bar{v}(z)\\right._{2}\\leq 1$ for any $z\\in\\mathbb{R}^{d}$ , and for any $1 \\leq i \\leq n$\n\n$$\ny_{i}\\left.\\bar{v},\\phi_{i}\\right._{\\mathcal{H}}: =y_{i}\\int\\left.\\bar{v}(z),\\phi_{i}(z)\\right.\\mathrm{d}\\mu_{N}(z)\\geq\\gamma .\\tag{}\n$$\n\nAs discussed in Section 5, the space H is the reproducing kernel Hilbert space (RKHS) induced by the infinite-width NTK with respect to $W$ , and $\\phi_{x}$ maps x into H. Assumption 1 supposes that the induced training set $\\{(\\phi_{i},y_{i})\\}_{i=1}^{n}$ can be separated by some $\\bar{v}\\in\\mathcal{H}.$ with an additional bound on $\\left.\\bar{v}(z)\\right._{2}$ which is crucial in our analysis. It is also possible to give a dual characterization of the separation margin (cf. eq. (3)), which also allows us to show that Assumption 1 always holds when there are no parallel inputs (cf. Proposition 1). However, it is often more convenient to construct v directly; see Section 5 for some examples.\n\nGiven an initialization $(W_{0},a)$ , for any $1 \\leq s \\leq m$ , define\n\n$$\n\\bar{u}_{s}: =\\frac{1}{\\sqrt{m}}a_{s}\\bar{v}(w_{s,0}),\\tag{2}\n$$\n\nwhere v is given by Assumption 1. Collect $\\bar{u}_{s}$ into a matrix $\\overline{{U}}\\in\\mathbb{R}^{m\\times d}$ . It holds that $\\Vert\\bar{u}_{s}\\Vert_{2}\\leq$ $1/{\\sqrt{m}}$ ,and $\\|\\overline{{U}}\\|_{F}\\le 1$\n\nFor any W, any $\\epsilon_{2}>0$ , and any $1 \\leq i \\leq n$ , define\n\n$$\n\\alpha_{i}(W,\\epsilon_{2})=\\frac{1}{m}\\sum_{s=1}^{m}\\mathbb{1}\\left[\\left|\\left. w_{s},x_{i}\\right.\\right|\\leq\\epsilon_{2}\\right].\n$$\n\nLemma 3 controls the output of the network at initialization.\n\nLemma 3. Given any $\\delta\\in(0,1)$, if $m\\ge25\\ln(2n/\\delta)$, then with probability $1-\\delta$, it holds simultaneously for all $1\\le i\\le n$ that\n\n$$\n\\big|f(x_i;W_0,a)\\big|\\le\\sqrt{2\\ln\\big(4n/\\delta\\big)}.\n$$\n\nWe analyze gradient descent in this subsection. First, define\n\n$$\n{\\widehat{\\mathcal{Q}}}(W): ={\\frac{1}{n}}\\sum_{i=1}^{n}-\\ell^{\\prime}\\left(y_{i}f_{i}(W)\\right).\n$$\n\nWe have the following observations.\n\n•For any W and any $1\\leq s\\leq m,\\left\\|\\partial f_{i}/\\partial w_{s}\\right\\|_{2}\\leq 1/\\sqrt{m},$ and thus $\\left\\|\\nabla f_{i}(W)\\right\\|_{F}\\leq 1$ Therefore by the triangle inequality, $\\left\\|\\nabla\\widehat{\\mathcal{R}}(W)\\right\\|_{F}\\leq\\widehat{\\mathcal{Q}}(W)$\n\nThe logistic loss satisfies $0\\leq-\\ell^{\\prime}\\leq 1$ , and thus $0\\leq\\widehat{\\mathcal{Q}}(W)\\leq 1$\n\nThe logistic loss satisfies $-\\ell^{\\prime}\\leq\\ell,$ and thus ${\\widehat{\\mathcal{Q}}}(W)\\leq{\\widehat{\\mathcal{R}}}(W)$", "theorems": [{"local_id": "T1", "statement": "Under the condition of Lemma 2, for any $\\epsilon_{2}>0,$ with probability $1-\\delta,$ it holds simultaneously for all $1 \\leq i \\leq n$ that\n\n$$\n\\alpha_{i}\\left(W_{0},\\epsilon_{2}\\right)\\leq\\sqrt{\\frac{2}{\\pi}}\\epsilon_{2}+\\sqrt{\\frac{\\ln(n/\\delta)}{2 m}}\\leq\\epsilon_{2}+\\frac{\\epsilon_{1}}{2}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Given any fixed $\\epsilon_{2}$ and $1 \\leq i \\leq n$\n\n$$\n\\mathbb{E}\\left[\\alpha_{i}(W_{0},\\epsilon_{2})\\right]=\\mathbb{P}\\left(\\left|\\langle w,x_{i}\\rangle\\right|\\leq\\epsilon_{2}\\right)\\leq\\frac{2\\epsilon_{2}}{\\sqrt{2\\pi}}=\\sqrt{\\frac{2}{\\pi}}\\epsilon_{2},\n$$"}, {"n": 2, "claim": "because $\\langle w,x_{i}\\rangle$ is a standard Gaussian r.v. and the density of standard Gaussian has maximum $1/{\\sqrt{2\\pi}}$ ."}, {"n": 3, "claim": "Since $\\alpha_{i}(W_{0},\\epsilon_{2})$ is the empirical mean of Bernoulli r.v.'s, by Hoeffding's inequality, with probability $1 - \\delta / n$\n\n$$\n\\alpha_{i}(W_{0},\\epsilon_{2})\\leq\\mathbb{E}\\left[\\alpha_{i}(W_{0},\\epsilon_{2})\\right]+\\sqrt{\\frac{\\ln(n/\\delta)}{2 m}}\\leq\\sqrt{\\frac{2}{\\pi}}\\epsilon_{2}+\\sqrt{\\frac{\\ln(n/\\delta)}{2 m}}.\n$$"}, {"n": 4, "claim": "Applying a union bound finishes the proof."}]}}, {"local_id": "T2", "statement": "Under Assumption 1, for the two-layer ReLU network $f(x;W,a)=\\frac{1}{\\sqrt m}\\sum_{s=1}^m a_s\\sigma(\\langle w_s,x\\rangle)$ with $w_{s,0}\\sim\\mathcal N(0,I_d)$, $a_s\\sim\\mathrm{unif}(\\{-1,+1\\})$ fixed, $W$ trained by gradient descent $W_{t+1}:=W_t-\\eta_t\\nabla\\widehat{\\mathcal R}(W_t)$ on $\\widehat{\\mathcal R}(W):=\\frac1n\\sum_{i=1}^n\\ell(y_i f_i(W))$, $\\ell(z):=\\ln(1+\\exp(-z))$, and inputs with $\\|x_i\\|_2=1$, $y_i\\in\\{-1,+1\\}$: given $\\epsilon\\in(0,1)$ and $\\delta\\in(0,1/3)$, let $\\lambda:=\\frac{\\sqrt{2\\ln(4n/\\delta)}+\\ln(4/\\epsilon)}{\\gamma/4}$, $M:=\\frac{4096\\lambda^2}{\\gamma^6}$. For any $m\\ge M$ and constant step size $\\eta\\le1$, with probability $1-3\\delta$, $\\frac1T\\sum_{t<T}\\widehat{\\mathcal R}(W_t)\\le\\epsilon$ where $T:=\\lceil 2\\lambda^2/(\\eta\\epsilon)\\rceil$, and $\\|w_{s,t}-w_{s,0}\\|_2\\le\\frac{4\\lambda}{\\gamma\\sqrt m}$ for all $0\\le t<T$, $1\\le s\\le m$.", "proof": {"steps": [{"n": 1, "claim": "It suffices to show $y_i\\langle\\nabla f_i(W_t),\\overline W\\rangle\\ge\\ln(4/\\epsilon)$ for all $1\\le i\\le n$ and $0\\le t<t_1$, since $\\ell(z)\\le\\exp(-z)$ gives $\\widehat{\\mathcal R}^{(t)}(\\overline W)\\le\\epsilon/4$, where $\\overline W:=W_0+\\lambda\\overline U$ and $t_1$ is the first step some neuron leaves the ball of radius $4\\lambda/(\\gamma\\sqrt m)$."}, {"n": 2, "claim": "Write $\\overline W=W_0+\\lambda\\overline U$ and split\n\n$$\ny_i\\langle\\nabla f_i(W_t),\\overline W\\rangle = y_i\\langle\\nabla f_i(W_0),W_0\\rangle + \\lambda\\,y_i\\langle\\nabla f_i(W_t),\\overline U\\rangle.\n$$"}, {"n": 3, "claim": "By Lemma 3, $|y_i\\langle\\nabla f_i(W_0),W_0\\rangle|\\le\\sqrt{2\\ln(4n/\\delta)}$."}, {"n": 4, "claim": "By Lemma 2, $\\lambda y_i\\langle\\nabla f_i(W_t),\\overline U\\rangle\\ge\\lambda(\\gamma-\\epsilon_1)\\ge3\\lambda\\gamma/4$ with $\\epsilon_1=\\gamma^2/8$."}, {"n": 5, "claim": "Combining, with $\\lambda=\\frac{\\sqrt{2\\ln(4n/\\delta)}+\\ln(4/\\epsilon)}{\\gamma/4}$,\n\n$$\ny_i\\langle\\nabla f_i(W_t),\\overline W\\rangle\\ge\\frac{\\lambda\\gamma}{4}-\\sqrt{2\\ln(4n/\\delta)}=\\ln(4/\\epsilon),\n$$\n\nso $\\widehat{\\mathcal R}^{(t)}(\\overline W)\\le\\epsilon/4$ for all $0\\le t<t_1$."}, {"n": 6, "claim": "By Lemma 4,\n\n$$\n\\frac1T\\sum_{t<T}\\widehat{\\mathcal R}(W_t)\\le\\epsilon,\\quad T:=\\lceil2\\lambda^2/(\\eta\\epsilon)\\rceil,\n$$\n\nand $\\|w_{s,t}-w_{s,0}\\|_2\\le4\\lambda/(\\gamma\\sqrt m)$ for all $0\\le t<T$."}]}}, {"local_id": "T3", "statement": "Under Assumption 1, given any $\\delta\\in(0,1)$ and any $\\epsilon_{1}~\\in~(0,\\gamma),~i f~m~\\geq$ $\\left(2\\ln(n/\\delta)\\right)/\\epsilon_{1}^{2},$ then with probability $1-\\delta,$ it holds simultaneously for all $1 \\leq i \\leq n$ that\n\n$$\ny_{i}f_{i}^{(0)}\\left(\\overline{{U}}\\right)=y_{i}\\left.\\nabla f_{i}(W_{0}),\\overline{{U}}\\right.\\geq\\gamma-\\sqrt{\\frac{2\\ln(n/\\delta)}{m}}\\geq\\gamma-\\epsilon_{1}.\n$$", "proof": {"steps": [{"n": 1, "claim": "By Assumption 1, given any $1 \\leq i \\leq n$\n\n$$\n\\mu: =\\mathbb{E}_{w\\sim\\mathcal{N}(0,I_{d})}\\left[y_{i}\\left.\\bar{v}(w),x_{i}\\right.\\mathbb{1}\\left[\\left. w,x_{i}\\right.>0\\right]\\right]\\geq\\gamma .\n$$"}, {"n": 2, "claim": "On the other hand,\n\n$$\ny_{i}f_{i}^{(0)}\\left(\\overline{{U}}\\right)=\\frac{1}{m}\\sum_{s=1}^{m}y_{i}\\left.\\bar{v}(w_{s,0}),x_{i}\\right.\\mathbb{1}\\left[\\left. w_{s,0},x_{i}\\right.>0\\right]\n$$"}, {"n": 3, "claim": "is the empirical mean of i.i.d. r.v.'s supported on $[-1,+1]$ with mean $\\mu .$ Therefore by Hoeffding's inequality, with probability $1-\\delta/n,$\n\n$$\ny_{i}f_{i}^{(0)}\\left(\\overline{{U}}\\right)-\\gamma\\geq y_{i}f_{i}^{(0)}\\left(\\overline{{U}}\\right)-\\mu\\geq-\\sqrt{\\frac{2\\ln(n/\\delta)}{m}}.\n$$"}, {"n": 4, "claim": "Applying a union bound finishes the proof."}]}}, {"local_id": "T4", "statement": "For any $t \\geq 0$ and any $\\overline{{W}},i f\\eta_{t}\\leq 1$ , then\n\n$$\n\\eta_{t}\\widehat{\\mathcal{R}}(W_{t})\\leq\\left\\|W_{t}-\\overline{{W}}\\right\\|_{F}^{2}-\\left\\|W_{t+1}-\\overline{{W}}\\right\\|_{F}^{2}+2\\eta_{t}\\widehat{\\mathcal{R}}^{(t)}\\left(\\overline{{W}}\\right).\n$$\n\nConsequently, if we use a constant step size $\\eta\\leq 1for0\\leq\\tau<t,$ then\n\n$$\n\\eta\\left(\\sum_{\\tau<t}{\\widehat{\\mathcal{R}}(W_{\\tau})}\\right)+\\Big\\|W_{t}-\\overline{{W}}\\Big\\|_{F}^{2}\\leq\\Big\\|W_{0}-\\overline{{W}}\\Big\\|_{F}^{2}+2\\eta\\left(\\sum_{\\tau<t}{\\widehat{\\mathcal{R}}^{(\\tau)}\\left(\\overline{{W}}\\right)}\\right).\n$$", "proof": {"steps": [{"n": 1, "claim": "We have\n\n$$\n\\Big\\|{W}_{t+1}-\\overline{{W}}\\Big\\|_{F}^{2}=\\Big\\|{W}_{t}-\\overline{{W}}\\Big\\|_{F}^{2}-2\\eta_{t}\\Big\\langle\\nabla\\widehat{\\mathcal{R}}(W_{t}),W_{t}-\\overline{{W}}\\Big\\rangle+\\eta_{t}^{2}\\Big\\|\\nabla\\widehat{\\mathcal{R}}(W_{t})\\Big\\|_{F}^{2}.\\tag{4}\n$$"}, {"n": 2, "claim": "The first order term of eq. (4) can be handled using the convexity of  and homogeneity of ReLU:\n\n$$\n\\begin{array}{ r l }{\\Bigl\\langle\\nabla\\widehat{\\mathcal{R}}(W_{t}),W_{t}-\\overline{{W}}\\Bigr\\rangle=\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}\\ell^{\\prime}\\left(y_{i}f_{i}(W_{t})\\right)y_{i}\\Bigl\\langle\\nabla f_{i}(W_{t}),W_{t}-\\overline{{W}}\\Bigr\\rangle}&{}\\\\ {=\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}\\ell^{\\prime}\\left(y_{i}f_{i}(W_{t})\\right)\\left(y_{i}f_{i}(W_{t})-y_{i}f_{i}^{(t)}\\left(\\overline{{W}}\\right)\\right)}&{}\\\\ {\\geq\\displaystyle\\frac{1}{n}\\sum_{i=1}^{n}\\left(\\ell\\left(y_{i}f_{i}(W_{t})\\right)-\\ell\\left(y_{i}f_{i}^{(t)}\\left(\\overline{{W}}\\right)\\right)\\right)=\\widehat{\\mathcal{R}}(W_{t})-\\widehat{\\mathcal{R}}^{(t)}\\left(\\overline{{W}}\\right).}\\end{array}\\tag{5}\n$$"}, {"n": 3, "claim": "The second-order term of eq. (4) can be bounded as follows\n\n$$\n\\eta_{t}^{2}\\Big\\lVert\\nabla\\widehat{\\mathcal{R}}(W_{t})\\Big\\rVert_{F}^{2}\\leq\\eta_{t}^{2}\\widehat{\\mathcal{Q}}(W_{t})^{2}\\leq\\eta_{t}\\widehat{\\mathcal{Q}}(W_{t})\\leq\\eta_{t}\\widehat{\\mathcal{R}}(W_{t}),\\tag{6}\n$$"}, {"n": 4, "claim": "because $\\left\\|\\nabla\\widehat{\\mathcal{R}}(W_{t})\\right\\|_{F}\\leq\\widehat{\\mathcal{Q}}(W_{t})$ , and $\\eta_{t},{\\widehat{\\mathcal{Q}}}(W_{t})\\leq 1$ , and $\\widehat{\\mathcal{Q}}(W_{t})\\leq\\widehat{\\mathcal{R}}(W_{t})$ . Combining eqs. (4) to (6) gives\n\n$$\n\\eta_{t}\\widehat{\\mathcal{R}}(W_{t})\\leq\\left\\|W_{t}-\\overline{{W}}\\right\\|_{F}^{2}-\\left\\|W_{t+1}-\\overline{{W}}\\right\\|_{F}^{2}+2\\eta_{t}\\widehat{\\mathcal{R}}^{(t)}\\left(\\overline{{W}}\\right).\n$$"}, {"n": 5, "claim": "Telescoping gives the other claim."}]}}], "target_local_id": "T2", "topic_domain": "Deep Learning Theory", "topic_subtopic": "Training Dynamics & Implicit Bias", "label": "flawed", "error_step": 2, "error_class6": ["logical_gap_or_missing_case", "invalid_bound_or_inequality", "computation_or_constant"], "error_explanation": "The two-term split silently replaces $y_i\\langle\\nabla f_i(W_t),W_0\\rangle$ with $y_i\\langle\\nabla f_i(W_0),W_0\\rangle$, thereby dropping the cross-term $y_i\\langle\\nabla f_i(W_t)-\\nabla f_i(W_0),W_0\\rangle = \\frac{y_i}{\\sqrt m}\\sum_{s=1}^m a_s(\\mathbb 1[\\langle w_{s,t},x_i\\rangle>0]-\\mathbb 1[\\langle w_{s,0},x_i\\rangle>0])\\langle w_{s,0},x_i\\rangle$. Because $\\nabla f_i$ is evaluated at $W_t$ and not $W_0$, the ReLU indicator patterns differ between $W_t$ and $W_0$ on the set $S_c$ of neurons that flipped sign, so $\\langle\\nabla f_i(W_t),W_0\\rangle\\ne\\langle\\nabla f_i(W_0),W_0\\rangle$ and Lemma 2.5 alone does NOT control $\\langle\\nabla f_i(W_t),W_0\\rangle$. The omitted term is nonzero, so the derived bound $y_i\\langle\\nabla f_i(W_t),\\overline W\\rangle\\ge\\ln(4/\\epsilon)$ and hence $\\widehat{\\mathcal R}^{(t)}(\\overline W)\\le\\epsilon/4$ are unjustified. Correctly the cross-term must be added and bounded by $\\le\\lambda\\gamma/2$ (via $|S_c|\\le m(4\\lambda/(\\gamma\\sqrt m)+\\gamma^2/16)$ from Lemma 2.4 and $m\\ge4096\\lambda^2/\\gamma^6$), which forces a larger width and worsens the $1/\\gamma$ dependence to $1/\\gamma^8$."}
{"id": "J7FaSJw-xCM", "context": "While quantum theory encompasses a broad scope of subjects, quantum information theory or quantum Shannon theory is a sub-field that focuses on the quantum equivalent of Shannon information theory (Wilde, 2013).A brief introduction is provided in Appendix A.Among the extensive results, we utilize only the basic concepts of von Neumann entropy, conditional entropy, and mutual information in this work. While Shannon entropy is calculated for a classical probability distribution, von Neumann entropy (also called quantum entropy) is calculated for a density operator ρ (Nielsen & Chuang, 2002), a positive semidefinite hermitian matrix in a Hilbert space H with the trace value of one. More details can be found in Appendix A. Similar to Shannon information theory, it measures the uncertainty associated with a quantum system. Definition 1 (von Neumann entropy (Nielsen & Chuang, 2002)). The von Neumann entropy (quantum entropy) of a quantum state with density operator p is defined as below. $$ S(\\rho)\\equiv-t r(\\rho\\log\\rho)= -\\sum_{x}\\lambda_{x}\\log\\lambda_{x},w h e r e\\lambda_{x}\\ a r e\\ t h e\\ e i g e n\\nu a l u e s\\ o f\\rho . $$ In case some of the singular values are zero, we exclude the corresponding dimensions with $\\overline{{0\\cdot log(0)=0.5(\\rho)}}$ ranges from zero (when $\\rho$ is a pure state) to log d (when $\\rho$ is uniformly mixed), where ρ is in a d-dimensional space. The von Neumann entropy agrees with the Shannon entropy if and only if the states $|\\psi_{i}\\rangle$ are orthogonal (Nielsen & Chuang, 2002). When we have two quantum systems A and $B,$ their composite system is called AB. The corresponding density operators are denoted as $\\rho_{A},\\rho_{B}$ , and $\\rho_{A B}$ , respectively. Then, conditional entropy and mutual information are defined as below, where the quantum system name instead of the density operator is used as the argument's notation. Definition 2 (Conditional entropy and mutual information (Nielsen & Chuang, 2002)). The quantum conditional entropy and mutual information of a composite system AB with components A and B are defined as below. $$ \\begin{array}{ c }{{S(A|B)\\equiv S(A B)-S(B).}}\\\\ {{S(A;B)\\equiv S(A)-S(A|B).}}\\end{array} $$ The value of conditional entropy S(A|B) is reduced as the two systems become entangled. In fact, it can be even negative (Cerf & Adami, 1997). In general, measuring quantum entanglement is computationally challenging (Huang, 2014). The quantum mutual information $S(A;B)$ can be understood in a similar way as in Shannon information theory, and it is a measure of shared information between the two systems. In this section, we provide the derivation of von Neumann entropy lower bound for $H(\\mathbf{Z})$ and derive the negative cosine similarity upper bound for $H(\\mathbf{Z}_{1}|\\mathbf{Z}_{2})$ . In Section 5, the two bounds are combined to form a new lower bound of mutual information.non-contrastive loss for unsupervised learning. Consider two random variables $\\mathbf{Z}_{1}$ and $\\mathbf{Z}_{2}$ , where $\\mathbf{Z}_{2}$ is assumed to be on a unit hypersphere in $\\mathbb{R}^{d}$ If $\\mathbf{Z}_{1}$ is an augmented version of $\\mathbf{Z}_{2}$ with a weak Gaussian perturbation, the two random variables are related as $\\mathbf{\\bar{Z}}_{1}\\sim\\mathcal{N}(\\mathbf{Z}_{2},\\mathbf{\\Sigma}\\mathbf{)}$ . For this special augmentation, the main result of Theorem 1 can be proved. For the two lemmas, it is assumed that Σ is a symmetric positive definite matrix whose eigendecomposition is $\\pmb{\\Sigma}=\\pmb{Q}\\pmb{\\Lambda}\\pmb{Q}^{T},\\pmb{\\Lambda}=\\mathrm{diag}(\\lambda_{1},...,\\lambda_{d}),$ ,and $0<\\dot{\\lambda}_{d}\\leq ...\\leq\\lambda_{1}\\ll 1$ Also, note T that $| |z_{1}| |^{2}\\approx| |z_{2}| |^{2}=1$ and $\\begin{array}{ r }{z_{1}\\cdot z_{2}\\approx\\frac{z_{1}^{\\prime}z_{2}}{| |z_{1}| |\\cdot| |z_{2}| |}=\\cos{(z_{1},z_{2})}>0}\\end{array}$ due to the weak Gaussian noise perturbation condition. Lemma 2. $-\\frac{1}{2}(z_{1}-z_{2})^{T}\\Sigma^{-1}(z_{1}-z_{2})\\geq-1/\\lambda_{d}+z_{1}\\cdot z_{2}/\\lambda_{1}$ . The Theorem 2 was derived for random variables $\\mathbf{Z}_{1}$ and $\\mathbf{Z}_{2}$ . For unsupervised representation learning, however, augmentation is applied to input random variables $\\mathbf{X}_{1}$ and $\\mathbf{X}_{2}$ . Therefore, we assume the commonly used ReLU as the activation vector and utilize its piecewise linearity to prove the following theorem.", "theorems": [{"local_id": "T1", "statement": "Theorem 3. Let $z_{i}\\in\\mathbf{Z}$ be a unit column vector with probability p(i). For $\\begin{array}{ r }{\\rho\\equiv\\sum_{i}p(i)z_{i}(z_{i})^{T}}\\end{array}$ $\\approx$ $\\begin{array}{ r }{\\sum_{i=1}^{n}\\frac{1}{n}z_{i}(z_{i})^{T}=\\frac{1}{n}{\\cal U}^{T}{\\cal U}=\\tilde{\\rho},}\\end{array}$ where $U=[z_{1},z_{2},...,z_{n}]^{T}$ , $$ H(\\mathbf{Z})\\geq S(\\rho)\\approx S(\\tilde{\\rho})= -\\sum_{j}\\lambda_{j}\\log\\lambda_{j},where\\lambda_{j}denote the eigenvalues of U^{T}U/n . $$", "proof": {"steps": [{"n": 1, "claim": "By the quantum data-processing inequality, $H(\\mathbf{Z})\\geq S(\\rho)$ holds for the Shannon entropy $H(\\mathbf{Z})$ and the von Neumann entropy $S(\\rho)$ (Nielsen & Chuang, 2002; Wilde, 2013)."}, {"n": 2, "claim": "For sufficiently large n, the density operator $\\rho$ can be approximated as the empirical density operator $\\tilde{\\rho}$ that satisfies $\\tilde{\\rho}\\geq 0$ and $t\\dot{r}(\\tilde{\\rho})=1$ ."}, {"n": 3, "claim": "Then, $S(\\tilde{\\rho})$ can be evaluated with the eigenvalues of $\\tilde{\\rho}$ following the Definition 1."}]}}, {"local_id": "T2", "statement": "Let $\\mathbf{X}_2$ be an input distribution and $\\mathbf{X}_1 (= \\{x_1 \\mid x_1 = x_2 + \\mathcal{N}(0, c\\cdot I),\\ x_2 \\in \\mathbf{X}_2\\})$ be a Gaussian noise perturbation of $\\mathbf{X}_2$ with $c \\to 0$. For a deep feedforward neural network with piecewise linear (e.g. ReLU) activations $f(\\cdot)$ whose output vectors are L2-normalized, and with $\\mathbf{Z}_1 = f(\\mathbf{X}_1)$, $\\mathbf{Z}_2 = f(\\mathbf{X}_2)$, there exist constants $\\alpha$ and $\\beta>0$ such that $H(\\mathbf{Z}_1\\mid\\mathbf{Z}_2) \\leq \\alpha - \\beta\\,\\mathbb{E}[\\cos(\\mathbf{Z}_1,\\mathbf{Z}_2)]$.", "proof": {"steps": [{"n": 1, "claim": "A deep feedforward neural network with piecewise linear activations $f(\\cdot)$ is piecewise linear."}, {"n": 2, "claim": "For each $x_2\\in\\mathbf{X}_2$ lying in the interior of some linear region, since $f$ is piecewise linear, $\\exists\\epsilon>0$, $A$, $b$ such that $f$ agrees with the affine map $x\\mapsto Ax+b$ on the ball $B_\\epsilon(x_2)=\\{x\\mid \\|x-x_2\\|<\\epsilon\\}$."}, {"n": 3, "claim": "Hence $z_1\\sim\\mathcal{N}(z_2,\\Sigma)$, $\\Sigma=\\mathbb{E}[(z_1-z_2)(z_1-z_2)^T]=A\\,\\mathbb{E}[(x_1-x_2)(x_1-x_2)^T]\\,A^T=cAA^T$, and $\\|\\Sigma\\|\\ll1$ as $c\\to0$."}, {"n": 4, "claim": "By Theorem 1, $H(\\mathbf{Z}_1\\mid\\mathbf{Z}_2)\\leq\\frac{d}{2}\\log2\\pi\\lambda_1+\\frac{1}{\\lambda_d}-\\frac{1}{\\lambda_1}\\mathbb{E}[\\cos(\\mathbf{Z}_1,\\mathbf{Z}_2)]$, where $0<\\lambda_d\\leq\\dots\\leq\\lambda_1$ are the eigenvalues of $\\Sigma$."}, {"n": 5, "claim": "Hence $\\exists\\alpha,\\exists\\beta>0$ with $H(\\mathbf{Z}_1\\mid\\mathbf{Z}_2)\\leq\\alpha-\\beta\\,\\mathbb{E}[\\cos(\\mathbf{Z}_1,\\mathbf{Z}_2)]$."}]}}, {"local_id": "T3", "statement": "$-\\log p(z_{1}|z_{2})\\leq\\frac{d}{2}\\log 2\\pi\\lambda_{1}+\\frac{1}{\\lambda_{d}}-\\frac{1}{\\lambda_{1}}z_{1}\\cdot z_{2}$ .", "proof": {"steps": [{"n": 1, "claim": "Proof. It suffices to prove that $p(z_{1}|z_{2})\\geq(2\\pi\\lambda_{1})^{-\\frac{d}{2}}\\exp\\{-1/\\lambda_{d}+z_{1}\\cdot z_{2}/\\lambda_{1}\\}$"}, {"n": 2, "claim": "p(z_{1}|z_{2})=\\displaystyle\\frac{1}{(2\\pi)^{d/2}}\\displaystyle\\frac{1}{\\operatorname*{det}({\\bf{\\vec{\\Sigma}}})^{1/2}}\\exp\\biggl\\{-\\displaystyle\\frac{1}{2}(z_{1}-z_{2})^{T}{\\Sigma}^{-1}(z_{1}-z_{2})\\biggr\\}"}, {"n": 3, "claim": "\\geq\\displaystyle\\frac{1}{(2\\pi)^{d/2}}\\displaystyle\\frac{1}{\\lambda_{1}^{d/2}}\\exp\\{-1/\\lambda_{d}+z_{1}\\cdot z_{2}/\\lambda_{1}\\}."}, {"n": 4, "claim": "Where the inequality follows from det $\\mathbf{\\tau}(\\Sigma)=\\Pi_{i=1}^{d}\\lambda_{i}\\leq\\lambda_{1}^{d}$ and by Lemma 2."}]}}], "target_local_id": "T2", "topic_domain": "Deep Learning Theory", "topic_subtopic": "Representation & Expressivity", "label": "flawed", "error_step": 3, "error_class6": ["missing_assumption", "others", "logical_gap_or_missing_case"], "error_explanation": "The step concludes $z_1 \\sim \\mathcal{N}(z_2, \\Sigma)$ with $\\Sigma = cAA^T$ by treating the piecewise-linear map $f$ as a single fixed affine map $Ax+b$ on the whole noise neighborhood. This is invalid: a piecewise-linear $f(x)=\\sum_i (A_i x + b_i)\\mathbb{1}(x\\in\\mathcal{B}_i)$ does not preserve Gaussianity, because conditional on $x_1$ the region indicators $\\mathbb{1}(x_1+e\\in\\mathcal{B}_i)$ are random (Bernoulli). No matter how small the noise variance $c$, the support of the perturbation $e$ is all of $\\mathbb{R}^d$, so there is nonzero probability of crossing a linear-region boundary; hence $e_f = f(x_2)-f(x_1)$ is not Gaussian and $\\Sigma$ is not a single fixed matrix $cAA^T$. Consequently the Gaussian conditional-entropy machinery of Theorem 4.4 cannot legitimately be applied here. (Compounding this, $\\alpha=\\tfrac{d}{2}\\log 2\\pi\\lambda_1+\\tfrac{1}{\\lambda_d}$ and $\\beta=\\tfrac{1}{\\lambda_1}$ are built from the eigenvalues of the local $\\Sigma$ at one $x_2$, so they cannot serve as global constants; by Proposition A, $h(z_2|z_1)=\\tfrac{1}{2}\\log((2\\pi e)^d\\det\\Sigma)$ is unbounded over $\\Sigma$, so no fixed $\\alpha,\\beta$ can bound $H(\\mathbf{Z}_1|\\mathbf{Z}_2)$ for all augmentations.)"}
{"id": "J7hbPeOZ39", "context": "Notation: We use bold lowercase font for vectors x and uppercase font for matrices X. For a vector x, we denote its i-th entry by $x_{i}$ and we use $\\lVert\\boldsymbol{x}\\rVert$ to denote its $\\ell^{2}{\\mathrm{-norm}}$ For two vectors x and y, we use $({\\pmb x};{\\pmb y})$ to denote their concatenation and use $\\langle{\\pmb x},{\\pmb y}\\rangle$ to denote their inner product. For a vector x and a positive semi-definite matrix $W .$ ,we use $\\|\\pmb{x}\\|_{W}$ to denote the weighted $\\ell^{2}$ -norm. For any positive integer $n,$ we use $[ n ]$ to denote the set $\\{1,2,\\ldots,n\\}$\n\nAssumption 2 (Multinomial logit choice under linear utility). The utility of the buyer at time t for item i is given by the linear model\n\n$$\nu_{ti}(p)=\\langle\\psi^{*},{\\pmb x}_{ti}\\rangle-\\langle\\phi^{*},{\\pmb x}_{ti}\\rangle\\cdot p\n$$\n\nwhere $\\boldsymbol{\\psi}^{*}\\in\\mathbb{R}^{d}$ and $\\phi^{*}\\in\\mathbb{R}^{d}$ are time-invariant parameter vectors unknown to the seller agent. In this model, the $\\alpha_{ti}: =\\langle\\psi^{*},\\pmb{x}_{ti}\\rangle$ term represents the buyer's base valuation of the item while the $\\beta_{ti}: =\\langle\\phi^{*},\\pmb{x}_{ti}\\rangle$ term represents the buyer's price sensitivity.\n\nThen, given an assortment $S_t$ with prices $\\mathbf{p}_t$, the probability that the buyer selects item $i \\in S_t$ is\n\n$$\nq_t(i|S_t,\\mathbf{p}_t) := \\frac{\\exp\\{u_{ti}(p_{ti})\\}}{1+\\sum_{j\\in S_t}\\exp\\{u_{tj}(p_{tj})\\}}.\n$$\n\nConsequently, the probability of no purchase is\n\n$$\nq_t(0|S_t,\\mathbf{p}_t) := \\frac{1}{1+\\sum_{j\\in S_t}\\exp\\{u_{tj}(p_{tj})\\}}.\n$$\n\nAssumption 3. There exists a constant $L_{0}>0$ such that price sensitivity $\\beta_{ti}=\\langle\\phi^{*},\\pmb{x}_{ti}\\rangle$ satisfies $\\beta_{ti}\\geq L_{0}$ for all $t \\in [ T ]$ and $i \\in [ n ]$ , almost surely.\n\nFor convenience, we let $\\pmb\\theta=(\\psi;\\phi)$ and $\\widetilde{\\pmb{x}}_{ti}=\\left(\\pmb{x}_{ti};-p_{ti}\\pmb{x}_{ti}\\right)$ denote the extended parameter and feature vectors such that\n\n$$\n\\left.\\pmb{\\theta},\\widetilde{\\pmb{x}}_{ti}\\right.=\\left.\\pmb{\\psi},\\pmb{x}_{ti}\\right.-\\left.\\pmb{\\phi},\\pmb{x}_{ti}\\right.\\cdot\\boldsymbol{p}_{ti}.\n$$\n\nThen, we write the MNL choice probabilities under some parameter $\\boldsymbol{\\theta} = (\\boldsymbol{\\psi}, \\boldsymbol{\\phi})$ using the notation\n\n$$\nq_{ti}(\\boldsymbol{\\theta}) = \\frac{e^{\\langle \\boldsymbol{\\theta}, \\widetilde{\\mathbf{x}}_{ti}\\rangle}}{1+\\sum_{j\\in S_t} e^{\\langle \\boldsymbol{\\theta}, \\widetilde{\\mathbf{x}}_{tj}\\rangle}}.\n$$\n\nWith this notation, the negative log-likelihood function for the observations up to time $t$ is given by\n\n$$\n\\ell_t(\\boldsymbol{\\theta}) := -\\sum_{s=1}^{t-1} \\log q_{si}(\\boldsymbol{\\theta}).\n$$\n\nThe maximum likelihood estimator is the parameter $\\widehat{\\boldsymbol{\\theta}}_t$ that minimizes $\\ell_t(\\boldsymbol{\\theta})$ over the parameter space.\n\nFurthermore, if the Gram matrix $\\begin{array}{ r }{V_{t-1}=\\sum_{\\tau=1}^{t-1}\\sum_{i\\in S_{\\tau}}\\widetilde{\\pmb{x}}_{\\tau i}\\widetilde{\\pmb{x}}_{\\tau i}^{\\top}}\\end{array}$ is positive definite, then $\\ell_{t}(\\pmb\\theta)$ is strongly convex and thus admits a unique minimizer.\n\n$V_{t}\\gets V_{t-1}+\\sum_{i\\in S_{t}}\\widetilde{\\pmb{x}}_{ti}\\widetilde{\\pmb{x}}_{ti}^{\\top}$\n\nCompute $g_{ti}: =\\alpha_{t}\\lVert(\\pmb{x}_{ti},\\pmb{x}_{ti})\\rVert_{V_{t}^{-1}}$ for all $i \\in [ n ]$ Confidence bonus\n\nAssumption 4 (Stochastic and bounded features). Each feature vector $\\mathbf{\\nabla}_{\\mathbf{\\boldsymbol{x}}_{ti}}$ is an independent random variable with unknown distribution; they satisfy $\\|\\pmb{x}_{ti}\\|\\le 1$ , and there exists a constant $\\sigma_{0}>0$ such that $\\mathbb{E}[\\pmb{x}_{ti}\\pmb{x}_{ti}^{\\top}]\\succcurlyeq\\sigma_{0}\\pmb{I}.$ Furthermore, parameter vectors satisfy $\\begin{array}{ r l r }{{\\|({\\pmb{\\theta}}^{*},{\\pmb{\\phi}}^{*})\\|\\leq 1}}\\end{array}$\n\nLemma 2 (Bounded optimum prices). Consider that the utility function for each item $i \\in [ n ]$ is given by $\\ \\cdot\\ u_{ti}(p)=\\alpha_{ti}-\\beta_{ti}\\cdot p$ for some $\\alpha_{ti}\\ \\leq\\ 1$ and $L_{0}\\le\\dot{\\beta}_{ti}\\le 1$ Then, $B_{t}\\in [0,P_{0}]$ and $p_{ti}^{*}\\in [1,P]fo$ or constants $P_{0}$ and P that only depend on K and $L_{0}$\n\nLemma 3 (Initialization). Define $\\begin{array}{ r }{V_{T_{0}}=\\sum_{t=1}^{T_{0}}\\sum_{i\\in S_{t}}\\widetilde{\\pmb{x}}_{ti}\\widetilde{\\pmb{x}}_{ti}^{\\top}}\\end{array}$ .\n\nLemma 3 implies that we can have $\\lambda_{\\operatorname*{min}}(V_{T_{0}})\\ :\\geq\\ :K(1+P^{2})$ with a high probability if we run the random initialization for $\\mathcal{O}(\\sigma_{0}^{-2}(d+\\log T+P^{2}))$ rounds.\n\nLemma 4. Let θ be some parameter such that $\\|\\pmb\\theta-\\pmb\\theta^{*}\\|\\leq 1$ . Then, there exists some constant $\\kappa > 0$ that depends only on K and $L_{0}$ such that\n\n$$\nq_{t}(i|S_{t},\\pmb{p}_{t};\\pmb{\\theta})q_{t}(0|S_{t},\\pmb{p}_{t};\\pmb{\\theta})\\geq\\kappa\n$$\n\nfor all $i\\in S_{t}$ at any round $t \\in [ T ]$\n\nLemma 5 (Consistency of MLE). Let $T_{0}$ be any round such that\n\n$$\n\\lambda_{\\operatorname*{min}}(V_{T_{0}})\\geq\\operatorname*{max}\\left\\{\\frac{16}{\\kappa^{2}}(4d+2\\log(T)),K(1+P^{2})\\right\\}.\n$$\n\nThen for any $\\begin{array}{ r }{t\\geq T_{0},\\mathbb{P}(\\|\\widehat{\\pmb\\theta}_{t}-{\\pmb\\theta}^{*}\\|>1)\\leq 1-\\frac{1}{T^{2}}}\\end{array}$\n\nCombining the results of Lemma 3 and Lemma 5, we can show that the condition $\\lVert\\widehat{{\\pmb\\theta}_{t}}-{\\pmb\\theta}^{*}\\rVert\\leq 1$ is satisfied for any $t\\geq T_{0}$ with probability $1-\\mathcal{O}(T^{-2})$ if we select\n\n$$\nT_{0}=\\Theta\\left(\\frac{d+\\log T+P^{2}}{\\sigma_{0}^{2}\\kappa^{2}}\\right).\\tag{1}\n$$\n\nThus, we can define a good event\n\n$$\n{\\mathcal{E}}_{0}=\\left\\{\\|{\\widehat{\\pmb{\\theta}}}_{t}-{\\pmb{\\theta}}^{*}\\|\\leq 1,\\forall t\\geq T_{0}\\right\\}\n$$\n\nand show that it holds with probability $1-\\mathcal{O}(T^{-1})$\n\nFor the selection of $T_{0}$ given in equation 1, we already showed that ${\\mathcal{E}}_{0}$ holds with probability $1-\\mathcal{O}(T^{-1})$ . Therefore, conditioned on ${\\mathcal{E}}_{0}$ happens, we can further ensure that $\\lVert\\widehat{{\\pmb\\theta}_{t}}-{\\pmb\\theta}^{*}\\rVert_{V_{t}}\\leq\\alpha_{t}$ holds with probability at least $1-t^{-2}$ if we choose the confidence radius as\n\n$$\n\\alpha_{t}=\\frac{2}{\\kappa}\\sqrt{d\\log\\left(\\frac{t}{2d}\\right)+2\\log t}.\\tag{2}\n$$\n\nIn other words, for any $t\\geq T_{0}$ , we can define another good event $\\mathcal{E}_{t}=\\{\\|\\widehat{\\pmb{\\theta}}_{t}-\\pmb{\\theta}^{*}\\|_{V_{t}}\\leq\\alpha_{t}\\}$ that holds with probability at least $\\dot{1}-t^{-2}$ conditioned on ${\\mathcal{E}}_{0}$\n\nLemma 6. Suppose $\\|x_{ti}\\|\\leq 1$ and $p_{ti}\\leq P$ for all i and t. $If\\lambda_{\\operatorname*{min}}(V_{T_{0}})\\geq 2PK$ , then\n\n$$\n\\sum_{t=T_{0}+1}^{n}\\sum_{i\\in S_{t}}\\|\\widetilde{\\pmb{x}}_{ti}\\|_{V_{t}^{-1}}^{2}\\leq 2\\log\\bigg(\\frac{\\operatorname*{det}(V_{n})}{\\operatorname*{det}(V_{T_{0}})}\\bigg)\n$$", "theorems": [{"local_id": "T1", "statement": "Lemma 7. Suppose $\\|x_{t i}\\|\\leq 1$ and $1\\leq p_{t i}\\leq P$ for all i and t. Then det(Vt) is increasing with respect to t and\n\n$$\n\\operatorname*{det}(V_{t})\\leq\\left({\\frac{tK(1+P^{2})}{2d}}\\right)^{2d}\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof. Let $\\lambda_{1},\\ldots,\\lambda_{2 d}$ be the eigenvalues of $V_{t}.$ Then, using the AM-GM inequality we can write"}, {"n": 2, "claim": "\n\n$$\n\\begin{array}{ r l }{\\operatorname*{det}(V_{i})=}&{\\displaystyle\\prod_{i=1}^{2 d}\\lambda_{i}}\\\\ &{\\le\\left(\\frac{\\sum_{i=1}^{2 d}\\lambda_{i}}{2 d}\\right)^{2 d}}\\\\ &{=\\left(\\frac{\\operatorname{trace}(V_{i})}{2 d}\\right)^{2 d}}\\\\ &{=\\left(\\frac{\\sum_{i=1}^{t}\\sum_{i\\in S_{i}}\\vert\\vert\\tilde{x}_{i}-\\vert\\vert_{2}^{2}}{2 d}\\right)^{2 d}}\\\\ &{\\le\\left(\\frac{t K(1+P^{2})}{2 d}\\right)^{2 d}.}\\end{array}\n$$"}]}}, {"local_id": "T2", "statement": "Under the MNL linear-utility model with $u_{ti}(p) = \\langle \\psi^*, x_{ti}\\rangle - \\langle \\phi^*, x_{ti}\\rangle \\cdot p$ (Assumption 2), price sensitivity $\\beta_{ti} = \\langle \\phi^*, x_{ti}\\rangle \\geq L_0 > 0$ (Assumption 3), and using the extended feature $\\widetilde{x}_{ti} = (x_{ti}, -p\\, x_{ti})$ with Gram matrix $V_t = \\sum_{\\tau=1}^{t-1}\\sum_{i\\in S_\\tau} \\widetilde{x}_{\\tau i}\\widetilde{x}_{\\tau i}^\\top$: assume that $\\|\\widehat{\\theta}_t - \\theta^*\\|_{V_t} \\leq \\alpha_t$ holds for some $\\alpha_t$, and let $h_{ti} : \\mathbb{R} \\to \\mathbb{R}$ be the linear function $h_{ti}(p) = \\min\\{\\langle \\widehat{\\psi}_t, x_{ti}\\rangle + g_{ti}, 1\\} - \\max\\{\\langle \\widehat{\\phi}_t, x_{ti}\\rangle - g_{ti}, L_0\\}\\cdot p$, where $\\widehat{\\theta}_t = (\\widehat{\\psi}_t, \\widehat{\\phi}_t)$ and $g_{ti} := \\alpha_t \\|(x_{ti}, x_{ti})\\|_{V_t^{-1}}$. Then $h_{ti}(p)$ satisfies $h_{ti}(p) \\geq u_{ti}(p)$ for all $p \\geq 0$ (3), and $h_{ti}(p) - u_{ti}(p) \\leq 3P g_{ti}$ for all $p \\in [1, P]$ (4).", "proof": {"steps": [{"n": 1, "claim": "Recall $u_{ti}(p) = \\langle \\psi^*, x_{ti}\\rangle - \\langle \\phi^*, x_{ti}\\rangle \\cdot p$, and define the auxiliary linear function $\\widetilde{h}_{ti}(p) = \\langle \\widehat{\\psi}_t, x_{ti}\\rangle + g_{ti} - (\\langle \\widehat{\\phi}_t, x_{ti}\\rangle - g_{ti})\\cdot p$."}, {"n": 2, "claim": "$|\\langle \\widehat{\\theta}_t, \\widetilde{x}_{ti}\\rangle - \\langle \\theta^*, \\widetilde{x}_{ti}\\rangle| = |\\langle V_t^{1/2}(\\widehat{\\theta}_t - \\theta^*), V_t^{-1/2}\\widetilde{x}_{ti}\\rangle|$."}, {"n": 3, "claim": "$|\\langle V_t^{1/2}(\\widehat{\\theta}_t - \\theta^*), V_t^{-1/2}\\widetilde{x}_{ti}\\rangle| \\leq \\|V_t^{1/2}(\\widehat{\\theta}_t - \\theta^*)\\|\\,\\|V_t^{-1/2}\\widetilde{x}_{ti}\\| = \\|\\widehat{\\theta}_t - \\theta^*\\|_{V_t}\\,\\|\\widetilde{x}_{ti}\\|_{V_t^{-1}}.$"}, {"n": 4, "claim": "$\\|\\widehat{\\theta}_t - \\theta^*\\|_{V_t}\\,\\|\\widetilde{x}_{ti}\\|_{V_t^{-1}} \\leq \\|\\widehat{\\theta}_t - \\theta^*\\|_{V_t}\\,\\|(x_{ti}, -p x_{ti})\\|_{V_t^{-1}}.$"}, {"n": 5, "claim": "$|\\langle \\widehat{\\theta}_t, \\widetilde{x}_{ti}\\rangle - \\langle \\theta^*, \\widetilde{x}_{ti}\\rangle| \\leq (1+p) g_{ti}$ for all $p \\geq 0$."}, {"n": 6, "claim": "$|\\langle \\widehat{\\theta}_t, \\widetilde{x}_{ti}\\rangle - \\langle \\theta^*, \\widetilde{x}_{ti}\\rangle| \\leq P g_{ti}$ for all $p \\in [1, P]$."}, {"n": 7, "claim": "For all $p \\in [1, P]$, $h_{ti}(p) - u_{ti}(p) \\leq \\widetilde{h}_{ti}(p) - u_{ti}(p) = \\langle \\widehat{\\theta}_t, \\widetilde{x}_{ti}\\rangle + (1+p) g_{ti} - \\langle \\theta^*, \\widetilde{x}_{ti}\\rangle \\leq 3P g_{ti}.$"}, {"n": 8, "claim": "For all $p \\geq 0$, $u_{ti}(p) - \\widetilde{h}_{ti}(p) = \\langle \\psi^*, x_{ti}\\rangle - \\langle \\widehat{\\psi}_t, x_{ti}\\rangle - g_{ti} - (\\langle \\phi^*, x_{ti}\\rangle - \\langle \\widehat{\\phi}_t, x_{ti}\\rangle + g_{ti})\\, p \\leq 0.$"}, {"n": 9, "claim": "$\\langle \\psi^*, x_{ti}\\rangle - \\langle \\widehat{\\psi}_t, x_{ti}\\rangle - g_{ti} \\leq 0$ and $\\langle \\phi^*, x_{ti}\\rangle - \\langle \\widehat{\\phi}_t, x_{ti}\\rangle + g_{ti} \\geq 0.$"}, {"n": 10, "claim": "$u_{ti}(p) - h_{ti}(p) \\leq 0$ for all $p \\geq 0$, i.e. $h_{ti}(p) \\geq u_{ti}(p)$."}]}}, {"local_id": "T3", "statement": "Lemma 9 (Normality of MLE). Suppose $\\|\\widehat{\\pmb{\\theta}}_{t}-\\pmb{\\theta}^{*}\\|\\leq 1 f o r t\\geq T_{0}$ Then\n\n$$\n\\lVert\\widehat{{\\boldsymbol{\\theta}}}_{t}-{\\boldsymbol{\\theta}}^{*}\\rVert_{V_{t}}\\leq\\frac{2}{\\kappa}\\sqrt{d\\log\\left(\\frac{t}{2 d}\\right)+2\\log t}\\tag{5}\n$$\n\nwith probability at least $1-t^{-2}$", "proof": {"steps": [{"n": 1, "claim": "Proof. Following the proof of Lemma 5, we obtain\n\n$$\n\\lVert G(\\widehat{\\pmb{\\theta}}_{t})\\rVert_{V_{t}^{-1}}^{2}\\geq\\kappa^{2}\\lVert\\widehat{\\pmb{\\theta}}_{t}-\\pmb{\\theta}^{*}\\rVert_{V_{t}}^{2}\n$$\n\nfor any $\\widehat{\\pmb{\\theta}}_{t}\\in\\{\\pmb{\\theta}:\\lVert\\pmb{\\theta}-\\pmb{\\theta}^{*}\\rVert\\leq 1\\}$ ."}, {"n": 2, "claim": "Then, we use Theorem 1 in Abbasi-Yadkori (2011), which states if the noise $\\epsilon_{t i}$ is sub-Gaussian with parameter 1, then\n\n$$\n\\|G(\\widehat{\\theta}_{t})\\|_{V_{t}^{-1}}^{2}\\leq 2\\log\\left(\\frac{\\operatorname*{det}(V_{t})^{1/2}\\operatorname*{det}(V_{T_{0}})^{-1/2}}{\\delta}\\right)\n$$\n\nwith probability at least $1 - \\delta$"}, {"n": 3, "claim": "Then we combine with Lemma 7 to obtain\n\n$$\n\\begin{array}{ r l r }{{\\|G(\\widehat{\\theta}_{t})\\|_{V_{t}^{-1}}^{2}\\leq 2[d\\log(\\frac{t K(1+P^{2})}{2 d})-\\frac{1}{2}\\log\\operatorname*{det}(V_{T_{0}})+\\log\\frac{1}{\\delta}]}}\\\\ &{}&{\\leq 2[d\\log(\\frac{t}{2 d})+\\log\\frac{1}{\\delta}]~}\\end{array}\n$$"}, {"n": 4, "claim": "\n\nwhere the second inequality is from $\\lambda_{\\operatorname*{min}}(V_{T_{0}})\\geq K(1+P^{2})$ . Combining these inequalities and setting $\\delta=t^{-2}$ gives the result. L"}]}}, {"local_id": "T4", "statement": "Lemma 10. For any positive definite matrix $V,$\n\n$$\n\\|g_{t}(\\pmb\\theta)\\|_{V}^{2}\\leq 4\\operatorname*{max}_{i\\in S_{t}}\\|\\widetilde{\\pmb x}_{t i}\\|_{V}^{2}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof. Let $y_{t i}$ be a binary variable such that $y_{t i}=1$ if $i_{t}\\ =\\ i$ and $y_{t i}=0$ otherwise. For convenience also denote $q_{t i}=q_{t}(i|S_{t},\\pmb{p}_{t};\\pmb{\\theta})$ . Then, we note that $\\textstyle\\sum_{i\\in S_{t}}q_{t i}\\leq 1$ and $\\textstyle\\sum_{i\\in S_{t}}y_{t i}\\leq 1$"}, {"n": 2, "claim": "Consequently, we can write\n\n$$\n\\begin{array}{ r l }{{\\|g_{t}(\\theta)\\|_{V}^{2}=\\sum_{i\\in S_{t}}\\sum_{j\\in S_{t}}(q_{t i}-y_{t i})(q_{t j}-y_{t j})\\widetilde{x}_{t i}^{\\top}V\\widetilde{x}_{t j}}}\\\\ &{\\le\\sum_{i\\in S_{t}}\\sum_{j\\in S_{t}}(q_{t i}q_{t j}+y_{t i}y_{t j}+q_{t i}y_{t j}+q_{t j}y_{t i})|\\widetilde{x}_{t i}^{\\top}V\\widetilde{x}_{t j}|}\\\\ &{\\le 4\\operatorname*{max}_{i,j\\in S_{t}}|\\widetilde{x}_{t i}^{\\top}V\\widetilde{x}_{t j}|}\\\\ &{\\le 4\\operatorname*{max}_{i\\in S_{t}}|\\widetilde{x}_{t i}^{\\top}V\\widetilde{x}_{t i}|}\\\\ &{=4\\operatorname*{max}_{i\\in S_{t}}\\|\\widetilde{x}_{t i}\\|_{V}^{2}.}\\end{array}\n$$"}]}}], "target_local_id": "T2", "topic_domain": "Online Learning & RL", "topic_subtopic": "Bandits", "label": "flawed", "error_step": 5, "error_class6": ["invalid_bound_or_inequality", "others", "computation_or_constant"], "error_explanation": "The step concludes $|\\langle \\widehat{\\theta}_t, \\widetilde{x}_{ti}\\rangle - \\langle \\theta^*, \\widetilde{x}_{ti}\\rangle| \\leq (1+p) g_{ti}$ from the Cauchy-Schwarz line $|\\langle \\widehat{\\theta}_t, \\widetilde{x}_{ti}\\rangle - \\langle \\theta^*, \\widetilde{x}_{ti}\\rangle| \\leq \\|\\widehat{\\theta}_t - \\theta^*\\|_{V_t}\\,\\|(x_{ti}, -p x_{ti})\\|_{V_t^{-1}} \\leq \\alpha_t\\,\\|(x_{ti}, -p x_{ti})\\|_{V_t^{-1}}$. Doing so silently requires $\\|(x_{ti}, -p x_{ti})\\|_{V_t^{-1}} \\leq (1+p)\\,\\|(x_{ti}, x_{ti})\\|_{V_t^{-1}}$, i.e. $\\|(x, -p x)\\|_M \\leq (1+p)\\|(x, x)\\|_M$ for the PSD matrix $M = V_t^{-1}$. This inequality is false in general: under a general PSD weighting, the sign flip on the second block changes the cross-term interactions, so $(x_{ti}, -p x_{ti})$ can have a strictly larger $M$-norm than $(1+p)$ times the $M$-norm of $(x_{ti}, x_{ti})$. Hence $g_{ti}$ as defined does not dominate the estimation error of the price-dependent extended feature, invalidating bounds (13)-(14). The fix (conceded by the authors) is to strengthen the bonus to $g_{ti} = \\alpha_t \\max_{1 \\leq p \\leq P} \\|(x_{ti}, -p x_{ti})\\|_{V_t^{-1}}$, which directly yields $|\\langle \\widehat{\\theta}_t, \\widetilde{x}_{ti}\\rangle - \\langle \\theta^*, \\widetilde{x}_{ti}\\rangle| \\leq g_{ti}$."}
{"id": "K98byXpOpU", "context": "Notations. Here we give several important notations used in this paper. $\\| \\cdot \\|$ denotes the $\\ell_{2}$ norm for vectors and spectral norm for matrices. $I_{d}$ denotes a d-dimensional identity matrix. $A^{\\top}$ denotes transpose of matrix A. Given a convex set $\\mathcal{X}_{\\mathrm{{:}}}$ ,we define a projection operation to $\\mathcal{X}$ as $\\mathcal{P}_{\\mathcal{X}}(x^{\\prime})=\\arg\\operatorname*{min}_{x\\in\\mathcal{X}}{1}/{2}\\|x-x^{\\prime}\\|^{2}$\n\nIn this paper, we consider the following BO problems where both upper- and lower-level problems have the convex constraints,\n\n$$\n\\operatorname*{min}_{x\\in\\mathcal{X}\\subseteq\\mathbb{R}^{d_{1}}}F(x)=f(x,y^{*}(x))\\quad s . t .\\quad y^{*}(x)=\\underset{y\\in\\mathcal{Y}\\subseteq\\mathbb{R}^{d_{2}}}{\\arg\\operatorname*{min}}g(x,y).\\tag{1}\n$$\n\nThen, we introduce several mild assumptions on the Problem (1).\n\nAssumption 3. The upper-level function $f(x,y)$ satisfies the following conditions:\n\n1. $\\nabla_{x}f(x,y)$ is $L_{f}. L$ ipschitz continuous w.r.t. $(x,y)\\in\\mathcal{X}\\times\\mathcal{Y}$ and $\\nabla_{y}f(x,y)$ is L f-Lipschitz continuous w.r.t $(x,y)\\in\\mathcal{X}\\times\\mathcal{Y}_{:}$ ,where $L_{f}\\geq 0$\n\n2. For any $x\\in\\mathcal{X}$ and $y\\in\\mathcal{V},$ we have $\\|\\nabla_{y}f(x,y)\\|\\le C_{f y}$\n\nAssumption 4. The lower-level function $g(x,y)$ satisfies the following conditions:\n\n1. For any $x\\in\\mathcal{X}$ and $y\\in\\mathcal{y},g(x,y)$ is twice continuously differentiable in $(x,y)$\n\n2. Fix $x\\in\\mathcal{X},\\nabla_{y}g(x,y)$ is Lg-Lipschitz continuous w.r.t $y\\in\\mathcal{V}$ for some $L_{g}\\ge 1$\n\n3. Fix $x\\in{\\mathcal{X}},$ for any $y\\in\\mathcal{y},g(x,y)$ is $\\mu_{g}$ -strongly-convex in y for some $\\mu_{g}>0$\n\n4. $\\nabla_{x y}^{2}g(x,y)$ is $L_{g x y}– L i_{I}$ oschitz continuous w.r.t $(x,y)\\in\\mathcal{X}\\times\\mathcal{Y}$ and $\\nabla_{y y}^{2}g(x,y)$ is $L_{g y y}$ Lipschitz continuous w.r.t $(x,y)\\in\\mathcal{X}\\times\\mathcal{Y}$ , where $L_{g x y}\\ge 0$ and $L_{g y y}\\ge\\dot{0}.$\n\n5. For any $x\\in\\mathcal{X}$ and $y\\in\\mathcal{V}$ ,we have $\\begin{array}{ r }{\\|\\nabla_{x y}^{2}g(x,y)\\|\\leq C_{g x y}.}\\end{array}$\n\nGiven a non-expansive projection operator $\\mathcal{P}_{\\mathcal{Y}}(z)$ and a distribution $\\mathbb{P} = \\mathcal{N}(0, I_{d_2})$, we define the smoothing function as $\\mathcal{P}_{\\mathcal{Y}\\delta}(z) = \\mathbb{E}_{u\\sim\\mathbb{P}}[\\mathcal{P}_{\\mathcal{Y}}(z+\\delta u)]$.\n\nUsing this Gaussian smoothing function, we can approximate the hypergradient as follows,\n\n$$\n\\nabla F_\\delta(x) = -\\eta\\nabla^2_{xy} g(x,y^*(x))\\nabla\\mathcal{P}_{\\mathcal{Y}\\delta}(z^*)^\\top\\left[I_{d_2} - (I_{d_2}-\\eta\\nabla^2_{yy} g(x,y^*(x)))\\nabla\\mathcal{P}_{\\mathcal{Y}\\delta}(z^*)^\\top\\right]^{-1}\\nabla_y f(x,y^*(x)) + \\nabla_x f(x,y^*(x)),\n$$\n\nwhere $z^* = y^*(x) - \\eta\\nabla_y g(x,y^*(x))$.\n\nTo calculate the hypergradient $\\nabla F_\\delta(x)$, we need the optimal solution $y^*(x)$, which is usually difficult. One proper method is to use $y$ to approximate $y^*(x)$. Using this method, we can approximate the hypergradient at $(x, y)$ as follows,\n\n$$\n\\nabla f_\\delta(x,y) = \\nabla_x f(x,y) - \\eta\\nabla^2_{xy} g(x,y)\\nabla\\mathcal{P}_{\\mathcal{Y}\\delta}(z)^\\top\\left[I_{d_2} - (I_{d_2}-\\eta\\nabla^2_{yy} g(x,y))\\nabla\\mathcal{P}_{\\mathcal{Y}\\delta}(z)^\\top\\right]^{-1}\\nabla_y f(x,y),\n$$\n\nwhere $z = y - \\eta\\nabla_y g(x,y)$.\n\nIn addition, we can use the following unbiased estimator of the gradient $\\nabla\\mathcal{P}_{\\mathcal{Y}\\delta}(z)$ to replace $\\nabla\\mathcal{P}_{\\mathcal{Y}\\delta}(z)$,\n\n$$\n\\bar{H}(z;u) = \\frac{1}{\\delta}\\left(\\mathcal{P}_{\\mathcal{Y}}(z+\\delta u) - \\mathcal{P}_{\\mathcal{Y}}(z)\\right)u^\\top.\n$$\n\nFor $\\bar{H}(z;u)$, we have the following conclusion.\n\nLemma 2. We have $\\mathbb{E}_u\\left[\\bar{H}(z;u)\\right] = \\nabla\\mathcal{P}_{\\mathcal{Y}\\delta}(z)$ and $\\mathbb{E}_u\\left[\\left\\|\\bar{H}(z;u) - \\nabla\\mathcal{P}_{\\mathcal{Y}\\delta}(z)\\right\\|^2\\right] \\le (d_2+4)^2 + 2$.\n\nTo further reduce the complexity caused by calculating multiple Jacobian-vector products, we can introduce an additional stochastic layer on the finite sum. Specifically, assume we have a parameter $Q > 0$ and a collection of $Q + 2$ independent samples $\\bar{\\xi} := \\{u^0, \\cdots, u^{Q-1}, c(Q)\\}$, where $c(Q) \\sim \\mathcal{U}\\{0, \\cdots, Q-1\\}$. Then, we can approximate the gradient as follows,\n\n$$\n\\bar{\\nabla} f_\\delta(x,y;\\bar{\\xi}) = \\nabla_x f(x,y) - \\eta Q\\nabla^2_{xy} g(x,y)\\bar{H}(z;u^0)^\\top\\prod_{i=1}^{c(Q)}\\left((I_{d_2}-\\eta\\nabla^2_{yy} g(x,y))\\bar{H}(z;u^i)^\\top\\right)\\nabla_y f(x,y),\n$$\n\nwhere we have used the convention $\\prod_{i=1}^{0} A = I$.\n\n3.3 Double-Momentum Method for Lower-Level Constrained Bilevel Optimization\n\nEquipped with the hypergradient $\\bar{\\nabla}f_{\\delta}(x,y;\\bar{\\xi})$ , our next endeavor is to design a single-loop singletimescale algorithm to solve the constrained bilevel optimization problem (1). Our main idea is to adopt the double-momentum-based method and adaptive step-size method developed in (Huang & Huang, 2021; Khanduri et al., 2021; Shi et al., 2022). Our algorithm is summarized in Algorithm 1. Since we use the double-momentum method to solve the lower-level constrained bilevel optimization problem, we denote our method as DMLCBO.\n\nDefine $\\alpha\\in(0,1)$ and $\\beta\\in(0,1)$ . For the lower-level problem, we can utilize the following projected gradient method with the momentum-based gradient estimator and adaptive step size to update y,\n\n$$\ny_{k+1}=(1-\\eta_{k})y_{k}+\\eta_{k}\\mathcal{P}y(y_{k}-\\frac{\\tau}{\\sqrt{\\|v_{k}\\|}+G_{0}}v_{k}),v_{k+1}=(1-\\beta)v_{k}+\\beta\\nabla_{y}g(x_{k},y_{k}),\n$$\n\nwhere $\\eta_{k}>0,\\tau>0 . G_{0}>0$ is used to avoid to prevent the molecule from being equal to 0. Here, we initialize $v_{1}=g(x_{1},y_{1})$ . Similarly, for the upper-level problem, we can utilize the following gradient method with the momentum-based gradient estimator and adaptive step size to update x,\n\n$$\nx_{k+1}=(1-\\eta_{k})x_{k}+\\eta_{k}\\mathcal{P}_{\\lambda}(x_{k}-\\frac{\\gamma}{\\sqrt{\\|w_{k}\\|}+G_{0}}w_{k}),\\ w_{k+1}=(1-\\alpha)w_{k}+\\alpha\\bar{\\nabla}f_{\\delta}(x_{k},y_{k};\\bar{\\xi}_{k}),\n$$\n\nand we initialize $w_{1}=\\bar{\\nabla}f_{\\delta}(x_{1},y_{1};\\bar{\\xi}_{1})$\n\nIn this section, we discuss the convergence performance of our DMLCBO (All the detailed proofs are presented in our Appendix). We follow the theoretical analysis framework in (Huang & Huang, 2021; Huang et al., 2020; Shi et al., 2022; Khanduri et al., 2021) (For easy understanding, we also provide a route map of the analysis in Appendix E). Before proceeding to the main results, we present an additional assumption and a useful lemma.\n\n$$\n\\mathbf{A s s u m p t i o n 5 .~}W e\\ln a v e\\ c_{l}\\leq 1/(\\sqrt{\\|v_{k}\\|}+G_{0})\\leq c_{u}\\ a n d\\ c_{l}\\leq 1/(\\sqrt{\\|w_{k}\\|}+G_{0})\\leq c_{u}.\n$$\n\nThe above assumption is used to bind the adaptive terms in our method which has been widely used in convergence analysis (Shi et al., 2022; Guo et al., 2021b). Since the gradient estimation is bounded, we can easily bound v and u for fixed momentum parameters. Even if the condition in Assumption 5 is not satisfied, we can also use the clipping method to make v and u bounded and finally make Assumption 5 hold.\n\nThen, we discuss the metric used to evaluate the convergence performance. When dealing with constrained optimization problems, one proper method is to utilize $\\mathcal{G}\\left(x_{k},\\nabla F(x_{k}),\\hat{\\gamma}\\right)=$ $\\frac{1}{\\hat{\\gamma}}\\left(x_{k}-\\mathcal{P}_{\\mathcal{X}}\\left(x_{k}-\\hat{\\gamma}\\nabla F(x_{k})\\right)\\right)$ as a metric to assess the convergence performance, where $\\hat{\\gamma}=$ $\\gamma/(\\sqrt{\\|w_{k}\\|}+G_{0})$ . We can bound $\\boldsymbol{\\mathcal{G}}\\left(\\boldsymbol{x}_{k},\\boldsymbol{\\nabla}F(\\boldsymbol{x}_{k}),\\boldsymbol{\\hat{\\gamma}}\\right)$ in following lemma,\n\nLemma 1. Define metric $\\mathcal{G}\\left(\\boldsymbol{x}_{k},\\nabla F(\\boldsymbol{x}_{k}),\\hat{\\gamma}\\right)~=~\\frac{1}{\\hat{\\gamma}}\\left(\\boldsymbol{x}_{k}-\\mathcal{P}_{\\mathcal{X}}\\left(\\boldsymbol{x}_{k}-\\hat{\\gamma}\\nabla F(\\boldsymbol{x}_{k})\\right)\\right)$ , and let $\\hat{\\gamma}\\ =$ $\\frac{\\gamma}{\\sqrt{\\|w_{k}\\|}+G_{0}}$ and $L_{0}=\\operatorname*{max}(L_{1}(\\frac{\\sqrt{d_{2}}}{\\delta}),L_{2}(\\frac{\\sqrt{d_{2}}}{\\delta}))$ and $R_{k}=\\nabla f_{\\delta}(x,y)-\\mathbb{E}[\\bar{\\nabla}f_{\\delta}(x,y;\\bar{\\xi})]$ , we\n\nhave\n\n$$\n\\begin{array}{ r }{\\|\\mathcal{G}\\left(x_{k},\\nabla F(x_{k}),\\widehat{\\gamma}\\right)\\|\\leq\\mathcal{M}_{k},}\\end{array}\\tag{2}\n$$\n\n$$\n\\begin{array}{ l }{{w h e r e~\\displaystyle{\\mathcal{M}_{k}}=\\left\\|w_{k}-\\nabla f(x_{k},y_{k})-R_{k}\\right\\|+\\left\\|R_{k}\\right\\|+L_{0}\\left\\|y^{*}(x_{k})-y_{k}\\right\\|+\\displaystyle{\\frac{1}{\\gamma c_{l}}}\\left\\|x_{k}-\\tilde{x}_{k+1}\\right\\|+}}\\\\ {{\\displaystyle{\\frac{\\delta L}{2\\mu_{g}}}C_{g x y}(d_{2}+3)^{3/2}C_{f y}(1+\\displaystyle{\\frac{1}{\\mu_{g}}}(1-\\eta\\mu_{g}))a n d\\tilde{x}_{k+1}}=\\mathcal{P}_{\\mathcal{X}}(x_{k}-{\\displaystyle{\\frac{\\gamma}{\\sqrt{\\left\\|w_{k}\\right\\|+G_{0}}}}w_{k})}.}\\end{array}\n$$\n\nHence, using the above lemma, we can define a new metric $\\mathcal{M}_{k}$ to discuss the convergence of our method. In the scenario where $\\mathcal{M}_{k}$ tends towards zero, we can easily observe that $\\|\\mathcal{G}(x_{k},\\nabla F(x_{k}),\\hat{\\gamma})\\|$ also tends towards zero.\n\nThen, turning back to the convergence analysis, using the above assumptions and lemmas, we can obtain the following theorem (For ease of reading, some parameter Settings are omitted here. The specific parameters can be found in Appendix G.2):\n\nTheorem 2. Under Assumptions 3, 4 5 and Lemma 2, setting $Q~=~{\\frac{1}{\\mu_{g}\\eta}}\\ln{\\frac{C_{g x y}C_{f y}K}{\\mu_{g}}},~\\eta_{k}~=$\n\n$$\n\\frac{t}{(m+k)^{1/2}},\\alpha=c_{1}\\eta_{k},\\beta=c_{2}\\eta_{k},t>0,w e h a\\nu e\n$$\n\n$$\n\\frac{1}{K}\\sum_{k=1}^{K}\\mathbb{E}[\\frac{1}{2}\\mathcal{M}_{k}]\\leq\\frac{2 m^{1/4}\\sqrt{G}}{\\sqrt{K t}}+\\frac{2\\sqrt{G}}{(K t)^{1/4}}+\\frac{\\delta L}{4\\mu_{g}}C_{g x y}(d_{2}+3)^{3/2}C_{f y}(1+\\frac{1}{\\mu_{g}}(1-\\eta\\mu_{g})),\\tag{3}\n$$\n\nwhere $c_{1},c_{2}$ and G are defined in the appendix.\n\nRemark 1. Let $t=\\mathcal{O}(1),a=0,m=\\mathcal{O}(1)$ and $\\ln(m+K)=\\tilde{\\mathcal{O}}(1)$ , we have $\\sqrt{G}=\\tilde{\\mathcal{O}}(\\sqrt{d_{2}})$ In addition, let $\\delta=\\mathcal{O}(\\epsilon d_{2}^{-3/2})$ Thus, our proposed DMLCBO can converge to a stationary point at the rate of $\\tilde{\\mathcal{O}}(\\frac{\\sqrt{d_{2}}}{K^{1/4}})$ Then, let $a={\\frac{1}{8}},$ r randomly sampled from $\\{0,1,\\cdots,K\\}$ , and $\\mathbb{E}[\\frac{1}{2}\\mathcal{M}_{r}]=\\frac{1}{K}\\sum_{k=1}^{K}\\mathbb{E}[\\frac{1}{2}\\mathcal{M}_{k}]=\\tilde{\\mathcal{O}}(\\frac{\\sqrt{d_{2}}}{K^{1/4}})\\le\\epsilon,$ we have $K=\\tilde{\\mathcal{O}}(\\frac{d_{2}^{2}}{\\epsilon^{4}})$ .", "theorems": [{"local_id": "T1", "statement": "Lemma 3. Under Assumptions $3,4$ and Lemma $^{2,}$ we have\n\n$$\n\\|\\nabla F_{\\delta}(x_{k})-w_{k}\\|^{2}\\leq 2 L_{1}^{2}(\\frac{\\sqrt{d_{2}}}{\\delta})\\|y^{*}(x_{k})-y_{k}\\|^{2}+2\\|\\nabla f_{\\delta}(x_{k},y_{k})-w_{k}\\|^{2}.\\tag{4}\n$$", "proof": {"steps": [{"n": 1, "claim": "\\quad\\|\\nabla F_{\\delta}(x_{k})-w_{k}\\|^{2}"}, {"n": 2, "claim": "=\\|\\nabla f_{\\delta}(x_{k},y^{*}(x_{k}))-\\nabla f_{\\delta}(x_{k},y_{k})+\\nabla f_{\\delta}(x_{k},y_{k})-w_{k}\\|^{2}"}, {"n": 3, "claim": "\\le 2\\|\\nabla f_{\\delta}(x_{k},y^{*}(x_{k}))-\\nabla f_{\\delta}(x_{k},y_{k})\\|^{2}+2\\|\\nabla f_{\\delta}(x_{k},y_{k})-w_{k}\\|^{2}"}, {"n": 4, "claim": "\\le 2 L_{1}^{2}(\\frac{\\sqrt{d_{2}}}{\\delta})\\|y^{*}(x_{k})-y_{k}\\|^{2}+2\\|\\nabla f_{\\delta}(x_{k},y_{k})-w_{k}\\|^{2}"}]}}, {"local_id": "T2", "statement": "Under Assumptions 3, 4, 5 and Lemma 2, set $Q=\\frac{1}{\\mu_g\\eta}\\ln\\frac{C_{gxy}C_{fy}K}{\\mu_g}$, $\\eta_k=\\frac{t}{(m+k)^{1/2}}$, $\\alpha=c_1\\eta_k$, $\\beta=c_2\\eta_k$, $t>0$. With $\\mathcal{G}(x_k,\\nabla F(x_k),\\hat\\gamma)=\\frac{1}{\\hat\\gamma}(x_k-\\mathcal{P}_{\\mathcal{X}}(x_k-\\hat\\gamma\\nabla F(x_k)))$, $\\hat\\gamma=\\gamma/(\\sqrt{\\|w_k\\|}+G_0)$, $L_0=\\max(L_1(\\frac{\\sqrt{d_2}}{\\delta}),L_2(\\frac{\\sqrt{d_2}}{\\delta}))$, Theorem 2 gives $\\frac{1}{K}\\sum_{k=1}^K\\mathbb{E}[\\frac12\\mathcal{M}_k]\\le\\frac{2m^{1/4}\\sqrt{G}}{\\sqrt{Kt}}+\\frac{2\\sqrt{G}}{(Kt)^{1/4}}+\\frac{\\delta L}{4\\mu_g}C_{gxy}(d_2+3)^{3/2}C_{fy}(1+\\frac{1}{\\mu_g}(1-\\eta\\mu_g))$ (Eqn 3). Then DMLCBO converges to $\\|\\nabla F(x)\\|\\le\\epsilon$ at rate $\\tilde{\\mathcal{O}}(\\sqrt{d_2}/K^{1/4})$, hence $K=\\tilde{\\mathcal{O}}(d_2^2/\\epsilon^4)$. $G=\\frac{\\Phi_1-\\Phi^*}{\\gamma c_l}+\\frac{17t}{4K^2}(m+K)^{1/2}+\\frac{4}{3tK^2}(m+K)^{3/2}+(m\\sigma_f^2(d_2))t^2\\ln(m+K)$, $\\Phi_1=F_\\delta(x_1)+\\frac{10L_0^ac_l}{\\tau\\mu_gc_u}\\|y_1-y_2\\|^2+\\dots$, $\\gamma\\le\\min\\{\\frac{1}{L_0^{2-a}},\\dots\\}$, $\\frac{125L_0^2}{3\\mu_g^2}\\le c_2\\le\\frac{m^{1/2}}{t}$, $\\frac{9}{2}\\le c_1\\le\\frac{m^{1/2}}{t}$, $L_1(\\frac{\\sqrt{d_2}}{\\delta})=L_f+\\frac{L_{gxy}C_{fy}}{\\mu_g}+\\frac{C_{gxy}C_{fy}}{\\mu_g}(\\frac{2\\sqrt{d_2}}{\\delta}+1)+\\dots$", "proof": {"steps": [{"n": 1, "claim": "$\\frac{1}{K}\\sum_{k=1}^K\\mathbb{E}[\\frac12\\mathcal{M}_k]\\le\\frac{2m^{1/4}\\sqrt{G}}{\\sqrt{Kt}}+\\frac{2\\sqrt{G}}{(Kt)^{1/4}}+\\frac{\\delta L}{4\\mu_g}C_{gxy}(d_2+3)^{3/2}C_{fy}(1+\\frac{1}{\\mu_g}(1-\\eta\\mu_g))$ (Eqn 3)."}, {"n": 2, "claim": "Let $t=\\mathcal{O}(1)$, $m=\\mathcal{O}(1)$, $\\ln(m+K)=\\tilde{\\mathcal{O}}(1)$. Then $\\sqrt{G}=\\tilde{\\mathcal{O}}(\\sqrt{d_2})$, giving rate $\\tilde{\\mathcal{O}}(\\sqrt{d_2}/K^{1/4})$ for the first two terms."}, {"n": 3, "claim": "Let $\\delta=\\mathcal{O}(\\epsilon d_2^{-3/2})$; then $\\frac{\\delta L}{4\\mu_g}C_{gxy}(d_2+3)^{3/2}C_{fy}(1+\\frac{1}{\\mu_g}(1-\\eta\\mu_g))\\le\\epsilon$."}, {"n": 4, "claim": "Setting $\\tilde{\\mathcal{O}}(\\sqrt{d_2}/K^{1/4})\\le\\epsilon$ gives $K=\\tilde{\\mathcal{O}}(d_2^2/\\epsilon^4)$."}]}}, {"local_id": "T3", "statement": "Lemma 4. (Lipschitz continous of $\\nabla F_{\\delta}(x).$ Under Assumptions 3, 4 and Lemma 2, we have $\\nabla F_{\\delta}(x)$ is Lipschitz continuous w.r.t $x,$ such that\n\n$$\n\\|\\nabla F_{\\delta}(x_{1})-\\nabla F_{\\delta}(x_{2})\\|\\le L_{F_{\\delta}}(\\frac{\\sqrt{d_{2}}}{\\delta})\\|x_{1}-x_{2}\\|\\tag{5}\n$$\n\n$$\nL_{F_{\\delta}}(\\frac{\\sqrt{d_{2}}}{\\delta})=\\frac{L_{g}}{\\mu_{g}}L_{1}(\\frac{\\sqrt{d_{2}}}{\\delta})+L_{2}(\\frac{\\sqrt{d_{2}}}{\\delta}),L_{1}(\\frac{\\sqrt{d_{2}}}{\\delta})=L_{f}+\\frac{L_{g x y}C_{f y}}{\\mu_{g}}+\\frac{C_{g x y}C_{f y}}{\\mu_{g}}(\\frac{2\\sqrt{d_{2}}}{\\delta}+\\eta\\frac{2\\sqrt{d_{2}}}{\\delta}L_{g})+C_{g x y}C_{f y}(\\frac{1}{\\eta\\mu_{g}^{2}}(\\frac{2\\sqrt{d_{2}}}{\\delta}+\\eta\\frac{2\\sqrt{d_{2}}}{\\delta}L_{g})(1-\\eta\\mu_{g})+\\frac{L_{g y y}}{\\mu_{g}^{2}})+\\frac{C_{g x y}}{\\mu_{g}}L_{f}\n$$\n\nand\n\n$$\nL_{2}(\\frac{\\sqrt{d_{2}}}{\\delta})=L_{f}+\\frac{L_{g x y}C_{f y}}{\\mu_{g}}+\\frac{C_{g x y}C_{f y}}{\\mu_{g}}\\eta\\frac{2\\sqrt{d_{2}}}{\\delta}L_{g}+C_{g x y}C_{f y}(\\eta\\frac{1}{\\mu_{g}^{2}}\\frac{2\\sqrt{d_{2}}}{\\delta}L_{g}(1-\\eta\\mu_{g})+\\frac{L_{g y y}}{\\mu_{g}^{2}})+\\frac{C_{g x y}}{\\mu_{g}}L_{f}.\n$$", "proof": {"steps": [{"n": 1, "claim": "\\quad\\|\\nabla F_{\\delta}(x_{1})-\\nabla F_{\\delta}(x_{2})\\|"}, {"n": 2, "claim": "=\\|\\nabla f_{\\delta}(x_{1},y^{*}(x_{1}))-\\nabla f_{\\delta}(x_{2},y^{*}(x_{2}))\\|"}, {"n": 3, "claim": "\\le\\|\\nabla f_{\\delta}(x_{1},y^{*}(x_{1}))-\\nabla f_{\\delta}(x_{1},y^{*}(x_{2}))\\|+\\|\\nabla f_{\\delta}(x_{1},y^{*}(x_{2}))-\\nabla f_{\\delta}(x_{2},y^{*}(x_{2}))\\|"}, {"n": 4, "claim": "For the first term, we have"}, {"n": 5, "claim": "\\|\\nabla f_{\\delta}(x_{1},y^{*}(x_{1}))-\\nabla f_{\\delta}(x_{1},y^{*}(x_{2}))\\|\\le L_{1}(\\frac{\\sqrt{d_{2}}}{\\delta})\\|y^{*}(x_{1})-y^{*}(x_{2})\\|\\le\\frac{L_{g}}{\\mu_{g}}L_{1}(\\frac{\\sqrt{d_{2}}}{\\delta})\\|x_{1}-x_{2}\\|"}, {"n": 6, "claim": "Thus, we have"}, {"n": 7, "claim": "\\|\\nabla F_{\\delta}(x_{1})-\\nabla F_{\\delta}(x_{2})\\|\\le\\left(L_{2}(\\frac{\\sqrt{d_{2}}}{\\delta})+\\frac{L_{g}}{\\mu_{g}}L_{1}(\\frac{\\sqrt{d_{2}}}{\\delta})\\right)\\|x_{1}-x_{2}\\|"}]}}], "target_local_id": "T2", "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "label": "flawed", "error_step": 2, "error_class6": ["computation_or_constant", "invalid_bound_or_inequality", "missing_assumption"], "error_explanation": "The claim $\\sqrt{G}=\\tilde{\\mathcal{O}}(\\sqrt{d_2})$ is false for the original potential function. In the original proof the constant/potential $G$ inherits $L_0=\\max(L_1(\\sqrt{d_2}/\\delta),L_2(\\sqrt{d_2}/\\delta))$, so $G$ scales like $\\sqrt{d_2}/\\delta$ (and higher powers), not $\\tilde{\\mathcal{O}}(d_2)$. Hence $\\sqrt{G}$ is NOT $\\tilde{\\mathcal{O}}(\\sqrt{d_2})$; it actually carries a $1/\\delta$ factor. Propagating the true dependence, Eqn 11 implies $K=\\tilde{\\mathcal{O}}(d_2^2/(\\delta^4\\epsilon^4))$, and with the required $\\delta=\\mathcal{O}(\\epsilon d_2^{-3/2})$ (so $\\delta^4=\\epsilon^4 d_2^{-6}$) this becomes $K=\\tilde{\\mathcal{O}}(d_2^8/\\epsilon^8)$ — an 8th-power dependence on both $d_2$ and $\\epsilon$, far worse than the advertised $\\tilde{\\mathcal{O}}(d_2^2\\epsilon^{-4})$. Recovering $\\tilde{\\mathcal{O}}(d_2^2/\\epsilon^4)$ required replacing the potential with a new $\\Phi_{k+1}$ and setting $a=0$ to genuinely decouple $G$ from $1/\\delta$."}
{"id": "KSvRZFCy7s", "context": "Definition 1 (Differential Privacy (Dwork & Roth, 2014)).\nDefinition 1 (Differential Privacy (Dwork & Roth, 2014)). A randomized algorithm M $i s \\ \\varepsilon -$ differentially private if for any neighboring datasets D and D' and any measurable subset $S \\subseteq$ range(M), we have\n\n$$\n\\mathbb{P}\\left\\{\\mathcal{M}(D)\\in S\\right\\}\\leq e^{\\varepsilon}\\mathbb{P}\\left\\{\\mathcal{M}(D^{\\prime})\\in S\\right\\},\n$$\n\nwhere the probability is with respect to the randomness of M.\n\n---\n\nNeighboring datasets (Section 1.1):\nIn this paper, we use Definition 1 on data matrix $\\mathbf{X}\\in\\mathbb{R}^{d\\times n}$ . We say two data matrices $\\mathbf{X},\\mathbf{X}^{\\prime}$ are neighboring datasets if X and $\\mathbf{X}^{\\prime}$ differ on only one column. We follow the setting and notation in (He et al., 2023) as follows. let $(\\Omega,\\rho)$ be a metric space. Consider a dataset $\\mathbf{X}=[X_{1}^{-},\\ldots,X_{n}]\\in\\Omega^{n}$ We aim to construct a computationally efficient differentially private randomized algorithm that outputs synthetic data $\\mathbf{Y}=\\hat{[}Y_{1},\\ldots,Y_{n}^{\\cdot}]\\in\\Omega^{m}$ such that the two empirical measures\n\n$$\n\\mu_{\\mathbf{X}}=\\frac{1}{n}\\sum_{i=1}^{n}\\delta_{X_{i}}\\quad\\mathrm{and}\\quad\\mu_{\\mathbf{Y}}=\\frac{1}{m}\\sum_{i=1}^{m}\\delta_{Y_{i}}\n$$\n\nare close to each other. Here $\\delta_{X_{i}}$ denotes the Dirac measure centered on $X_{i}$ . We measure the utility of the output by $\\mathbb{E}W_{1}(\\mu_{\\mathbf{X}},\\mu_{\\mathbf{Y}})$ , where the expectation is taken over the randomness of the algorithm.\n\n---\n\nCentered covariance matrix M, eq (1):\n1Construct a private covariance matrix $\\widehat{\\bf M}$ The private covariance is constructed by adding a Laplacian random matrix to a centered covariance matrix M defined as\n\n$$\n\\mathbf{M}={\\frac{1}{n-1}}\\sum_{i=1}^{n}(X_{i}-{\\overline{{X}}})(X_{i}-{\\overline{{X}}})^{\\mathsf{T}},\\quad{\\mathrm{where}}\\quad{\\overline{{X}}}={\\frac{1}{n}}\\sum_{i=1}^{n}X_{i}.\\tag{1}\n$$\n\n---\n\nAlgorithm 1 (Low-dimensional Synthetic Data) summary:\nAlgorithm 1 Low-dimensional Synthetic Data   \nInput: True data matrix $\\mathbf{X}=[X_{1},\\ldots,X_{n}],X_{i}\\in [0,1]^{d}$ , privacy parameter ε.   \nPrivate covariance matrix Apply Algorithm 2 to X with privacy parameter $\\varepsilon / 3$ to obtain a   \nprivate covariance matrix $\\widehat{\\bf M}$   \nPrivate linear projection Choose a target dimension $d^{\\prime}.$ Apply Algorithm 3 with privacy param  \neter $\\varepsilon / 3$ to project $\\mathbf{X}$ onto a private $d^{\\prime}.$ -dimensional linear subspace. Save the private mean   \n$\\overline{{X}}_{\\mathrm{{p r i v}}}$ .   \nLow-dimensional synthetic data Use subroutine in Section 3 to generate $\\varepsilon/3–\\mathrm{DP}$ synthetic data   \nX' of size m depending on $d^{\\prime}=2\\operatorname{or}d^{\\prime}\\geq 3.$   \nAdding the private mean vector Shift the data back by $X_{i}^{\\prime\\prime}=X_{i}+{\\overline{{X}}}_{\\mathrm{priv}}$   \nMetric projection Define $f:\\mathbb{R}\\to [0,1]$ such that   \n$f(x)={\\left\\{\\begin{array}{ l l }{0}&{{\\mathrm{~i f~}}x<0;}\\\\ {x}&{{\\mathrm{~i f~}}x\\in [0,1];}\\\\ {1}&{{\\mathrm{~i f~}}x>1 .}\\end{array}\\right.}$   \nThen, for $v\\in\\mathbb{R}^{d}.$ , we define $f(v)$ to be the result of applying $f$ to each coordinate of $v .$   \nOutput: Synthetic data $\\mathbf{Y}=[f(X_{1}^{\\prime\\prime}),\\dots,f(X_{m}^{\\prime\\prime})]$\n\n---\n\nAlgorithm 2 (Private Covariance Matrix):\n```latex\nAlgorithm 2 Private Covariance Matrix\nInput: Matrix $\\mathbf{X}=[X_{1},\\ldots,X_{n}],$ , privacy parameter $\\varepsilon,$ and variance parameter $\\begin{array}{ r }{\\sigma=\\frac{3 d^{2}}{{\\varepsilon}n}}\\end{array}$ .\nComputing the covariance matrix Compute the mean $\\begin{array}{ r }{\\overline{{X}}=\\frac{1}{n}\\sum_{i=1}^{n}X_{i}}\\end{array}$ and the centered covari\nance matrix M.\nGenerating a Laplacian random matrix Generate i.i.d. independent random variables $\\lambda_{i j}$ ∼\n$\\mathrm{Lap}(\\bar{\\sigma}),i\\leq\\bar{j}$ Define a symmetric matrix A such that\n$\\mathbf{A}_{i j}=\\mathbf{A}_{j i}={\\left\\{\\begin{array}{ l l }{\\lambda_{i j}\\qquad}&{{\\mathrm{i f~}}i<j;}\\\\ {2\\lambda_{i i}\\qquad}&{{\\mathrm{i f~}}i=j,}\\end{array}\\right.}$\nOutput: The noisy covariance matrix $\\widehat{\\mathbf{M}}=\\mathbf{M}+\\mathbf{A}.$\n```\n\n---\n\nAlgorithm 4 (PMM Subroutine):\n```latex\nAlgorithm 4 PMM Subroutine\nInput: dataset ${\\widehat{\\mathbf{X}}}=({\\widehat{X}}_{1},\\ldots,{\\widehat{X}}_{n})$ in the region\n$\\Omega^{\\prime}=\\{a_{1}\\widehat{v}_{1}+\\cdot\\cdot\\cdot+a_{d^{\\prime}}\\widehat{v}_{d^{\\prime}}\\mid a_{1}^{2}+\\cdot\\cdot\\cdot+a_{d^{\\prime}}^{2}\\leq R\\}.$\nBinary partition Let $r=\\lceil\\log_{2}(\\varepsilon n)\\rceil$ and $\\sigma_{j}=\\varepsilon^{-1}\\cdot 2^{{\\frac{1}{2}}\\left(1-{\\frac{1}{d^{\\prime}}}\\right)\\left(r-j\\right)}$ . Enlarge the region $\\Omega^{\\prime}$ into\n$\\Omega_{\\mathrm{PMM}}=\\{a_{1}\\widehat{v}_{1}+\\cdot\\cdot\\cdot+a_{d^{\\prime}}\\widehat{v}_{d^{\\prime}}\\ |\\ a_{i}\\in [-R,R],\\forall i\\in [d^{\\prime}]\\}.$\nBuild a binary partition $\\{\\Omega_{\\theta}\\}_{\\theta\\in\\{0,1\\}}\\ L_{\\Sigma^{r}}\\mathrm{on}\\Omega_{\\mathrm{PMM}}.$\nNoisy count For any θ, count the number of data in the region $\\Omega_{\\theta}$ denoted by $n_{\\theta}=\\left|\\widehat{\\mathbf{X}}\\cap\\Omega_{\\theta}\\right|$\nand let $n_{\\theta}^{\\prime}=(n_{\\theta}+\\lambda_{\\theta})_{+}$ , where $\\lambda_{\\theta}$ are independent integer Laplacian random variables with\n$\\lambda\\sim\\operatorname{Lap}_{\\mathbb{Z}}(\\sigma_{|\\theta|})$ , and |θ| is the length of the vector θ.\nConsistency Enforce consistency of $\\{n_{\\theta}^{\\prime}\\}_{\\theta\\in\\{0,1\\}\\leq r}$\nOutput: Synthetic data $\\mathbf{X}^{\\prime}$ randomly sampled from $\\{\\Omega_{\\theta}\\}_{\\theta\\in\\{0,1\\}^{r}}.$\n```\n\n---\n\nSection 3.2 setup (private mean vector and metric projection):\n3.2 Adding a Private Mean Vector and Metric Projection\n\nSince we shift the data by its private mean before projection, we need to add another private mean vector back, which shifts the dataset $\\widehat{\\mathbf X}$ to a new private affine subspace close to the original dataset X. The output data vectors in $\\mathbf{X}^{\\prime\\prime}$ (defined in Algorithm 1) are not necessarily inside $[0,1]^{d}$ . The subsequent metric projection enforces all synthetic data inside $[0,1]^{d}$ . Importantly, this post-processing step does not have privacy costs.\n\nAfter metric projection, dataset Y from the output of Algorithm 1 is close to an affine subspace, as shown in the next proposition. Notably, (2) shows that the metric projection step does not cause the largest accuracy loss among all subroutines.\n\n---\n\nDefinition 2 (p-Wasserstein distance):\nDefinition 2 (p-Wasserstein distance). Consider a metric space $(\\Omega,\\rho)$ . The p-Wasserstein distance (see e.g., (Villani, 2009) for more details) between two probability measures $\\mu,$ ν is defined as\n\n$$\nW_{p}(\\mu,\\nu): =\\left(\\operatorname*{inf}_{\\gamma\\in\\Gamma(\\mu,\\nu)}\\int_{\\Omega\\times\\Omega}\\rho(x,y)^{p}\\mathrm{d}\\gamma(x,y)\\right)^{1/p},\n$$\n\nwhere $\\Gamma(\\mu,\\nu)$ is the set of all couplings of µ and $\\nu .$\n\n---\n\nLemma 1 (Theorem 3.16 in Dwork & Roth, 2014):\nLemma 1 (Theorem 3.16 in (Dwork & Roth, 2014)). Suppose $\\mathcal{M}_{i}$ is $\\varepsilon_{i}{-}d i f f{\\dot{\\epsilon}}$ erentially private for $i=1,\\ldots,m$ , then the sequential composition $x\\mapsto(\\tilde{\\mathcal{M}}_{1}(x),\\dot{\\mathrm{,~.~.~,~}}\\tilde{\\mathcal{M}}_{m}(x))\\mathrm{~}i s\\stackrel{\\sim}{\\sum_{i=1}^{m}}\\varepsilon_{i}\\dot{\\mathrm{-}}d\\dot{i}f f e r e n t i\\dot{a}l l y$ private.\n\n---\n\nLemma 2 (Theorem 1 in Dwork et al., 2006):\nLemma 2 (Theorem 1 in (Dwork et al., 2006)). Suppose a randomized algorithm $\\mathcal{M}_{1}(x):\\Omega^{n}\\to$ $\\mathcal{R}_{1}$ is ε1-differentially private, and $\\mathcal{M}_{2}(x,y):\\Omega^{n^{\\prime}}\\times\\mathcal{R}_{1}\\mathcal{R}_{2}$ is ${\\varepsilon}_{2}{-}d i f f$ erentially private with respect to the first component for any fixed y. Then the sequential composition\n\n$$\nx\\mapsto(\\mathcal{M}_{1}(x),\\mathcal{M}_{2}(x,\\mathcal{M}_{1}(x)))\n$$\n\nis $(\\varepsilon_{1}+\\varepsilon_{2})\\ – d i f f$ erentially private.\n\n---\n\nDecomposition of Algorithm 1 used in the proof of Theorem 1 (Sec. C.5):\nProof. We can decompose Algorithm 1 into the following steps:\n\n1. $\\mathcal{M}_{1}(\\mathbf{X})=\\widehat{\\mathbf{M}}$ is to compute the private covariance matrix with Algorithm $2 .$\n\n2. $\\mathcal{M}_{2}(\\mathbf{X})=\\overline{{{X}}}+\\lambda$ is to compute the private sample mean.\n\n3. $\\mathcal{M}_{3}(\\mathbf{X},y,\\Sigma)$ for fixed $y$ and $\\Sigma,$ is to project the shifted data $\\{X_{i}-y\\}_{i=1}^{n}$ to the first $d^{\\prime}$ principal components of $\\Sigma$ and apply a certain differentially private subroutine (we choose y and $\\Sigma$ as the output of $\\mathcal{M}_{2}$ and $\\mathcal{M}_{1}$ , respectively). This step outputs synthetic data $\\mathbf{{\\bar{X}}^{\\prime}}=(X_{1}^{\\prime},\\ldots,X_{m}^{\\prime}) $ on a linear subspace.\n\n4. $\\mathcal{M}_{4}(\\mathbf{X},\\mathbf{X}^{\\prime})$ is to shift the dataset to $\\{X_{i}^{\\prime}+\\overline{{X}}_{\\mathrm{priv}}\\}_{i=1}^{m}$\n\n5.Metric projection.\n\nIt suffices to show that the data before metric projection has already been differentially private. We will need to apply Lemma 2 several times.\n\n---\n\nEquation (3) context, from the proof of Theorem 2 (Sec. C.3):\nSince $\\lambda=(\\lambda_{1},\\ldots,\\lambda_{d})$ is a Laplacian random vector with i.i.d. $\\mathrm{Lap}(1/(\\varepsilon n))$ entries,\n\n$$\n\\mathbb{E}\\Vert\\lambda\\Vert_{2}^{2}=\\sum_{j=1}^{d}\\mathbb{E}\\big|\\lambda_{j}\\big|^{2}=\\frac{2 d}{\\varepsilon^{2}n^{2}}.\\tag{3}\n$$\n\n---\n\nError decomposition in the accuracy analysis of Algorithm 1 (Sec. C.6):\nSimilar to privacy analysis, we will decompose the algorithm into several steps. Suppose that\n\n1. $\\mathbf{X}-(\\overline{X}+\\lambda)\\mathbf{1}^{\\mathsf T}$ denotes the shifted data $\\{X_i-\\overline{X}-\\lambda\\}_{i=1}^n$;\n\n2. $\\widehat{\\mathbf{X}}$ is the data after projection to the private linear subspace;\n\n3. $\\mathbf{X}'$ is the output of the synthetic data subroutine in Section 3;\n\n4. $\\mathbf{X}''=\\mathbf{X}'+(\\overline{X}+\\lambda')\\mathbf{1}^{\\mathsf T}$ denotes the data shifted back;\n\n5. $\\mathcal{M}(\\mathbf{X})$ is the data after metric projection, which is the output of the whole algorithm.\n\nApplying the triangle inequality of Wasserstein distance to the other steps of the algorithm, we have\n\n$$\n\\begin{aligned}\nW_1(\\mu_{\\mathbf{X}},\\mu_{\\mathbf{X}''}) &= W_1(\\mu_{\\mathbf{X}-\\overline{X}\\mathbf{1}^{\\mathsf T}},\\mu_{\\mathbf{X}'+\\lambda'\\mathbf{1}^{\\mathsf T}})\\\\\n&\\le W_1(\\mu_{\\mathbf{X}-\\overline{X}\\mathbf{1}^{\\mathsf T}},\\mu_{\\widehat{\\mathbf{X}}})+W_1(\\mu_{\\widehat{\\mathbf{X}}},\\mu_{\\mathbf{X}'})+W_1(\\mu_{\\mathbf{X}'},\\mu_{\\mathbf{X}'+\\lambda'})\\\\\n&\\le W_1(\\mu_{\\mathbf{X}-\\overline{X}\\mathbf{1}^{\\mathsf T}},\\mu_{\\widehat{\\mathbf{X}}})+W_1(\\mu_{\\widehat{\\mathbf{X}}},\\mu_{\\mathbf{X}'})+\\|\\lambda'\\|_\\infty .\n\\end{aligned}\\tag{9}\n$$\n\nHence\n\n$$\n\\begin{aligned}\n&\\mathbb{E}W_1(\\mu_{\\mathbf{X}},\\mu_{\\mathcal{M}(\\mathbf{X})})\\\\\n&\\le 2\\mathbb{E}W_1(\\mu_{\\mathbf{X}},\\mu_{\\mathbf{X}'+(\\overline{X}+\\lambda')\\mathbf{1}^{\\mathsf T}})\\\\\n&\\le 2\\mathbb{E}W_1(\\mu_{\\mathbf{X}-\\overline{X}\\mathbf{1}^{\\mathsf T}},\\mu_{\\widehat{\\mathbf{X}}})+2\\mathbb{E}W_1(\\mu_{\\widehat{\\mathbf{X}}},\\mu_{\\mathbf{X}'})+2\\mathbb{E}\\|\\lambda'\\|_\\infty\\\\\n&\\le 2\\sqrt{2\\sum_{i>d'}\\sigma_i(\\mathbf{M})}+2\\sqrt{\\frac{Cd'd^{2.5}}{\\varepsilon n}}+2C\\sqrt{\\frac{d}{d'}}(\\varepsilon n)^{-1/d'}+\\frac{2C\\log d}{\\varepsilon n}\\\\\n&\\lesssim \\sqrt{\\sum_{i>d'}\\sigma_i(\\mathbf{M})}+\\sqrt{\\frac{d}{d'}}(\\varepsilon n)^{-1/d'}+\\sqrt{\\frac{d'd^{2.5}}{\\varepsilon n}},\n\\end{aligned}\\tag{10}\n$$", "theorems": [{"local_id": "T1", "statement": "Lemma 3 (Stability of noisy projection). Let X be a d × n matrix and A be a d × d Hermitian matrix. Let $\\begin{array}{ r }{\\mathbf{M}=\\frac{1}{n}\\mathbf{X}\\mathbf{X}^{\\mathsf{T}}}\\end{array}$ with eigenvalues $\\sigma_{1}\\geq\\sigma_{2}\\geq\\cdot\\cdot\\cdot\\geq\\sigma_{d}.$ Let $\\begin{array}{ r }{\\widehat{\\mathbf{M}}=\\frac{1}{n}\\mathbf{X}\\mathbf{X}^{\\mathsf{T}}+\\mathbf{A},}\\end{array}$ $\\widehat{\\mathbf{V}}_{d^{\\prime}}$ be a $d\\times d^{\\prime}$ matrix whose columns are the first d' orthonormal eigenvectors of ${\\widehat{\\mathbf{M}}},$ and $\\mathbf{Y}=\\widehat{\\mathbf{V}}_{d^{\\prime}}\\widehat{\\mathbf{V}}_{d^{\\prime}}^{\\mathsf{T}}\\mathbf{X}$ Let $\\mu_{\\mathbf{X}}$ and $\\mu_{\\mathbf{Y}}$ be the empirical measures of column vectors of X and Y, respectively. Then\n\n$$\nW_{2}^{2}(\\mu_{\\mathbf{X}},\\mu_{\\mathbf{Y}})\\leq\\frac{1}{n}\\|\\mathbf{X}-\\mathbf{Y}\\|_{F}^{2}\\leq\\sum_{i>d^{\\prime}}\\sigma_{i}+2 d^{\\prime}\\|\\mathbf{A}\\|.\\tag{4}\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof. Let $\\widehat{v}_{1},\\ldots,\\widehat{v}_{d}$ be a set of orthonormal eigenvectors for $\\widehat{\\bf M}$ with the corresponding eigenvalues $\\widehat{\\sigma}_{1},\\ldots,\\widehat{\\sigma}_{d}$ . Define four matrices whose column vectors are eigenvectors:\n\n$$\n\\mathbf{V}=[v_{1},\\ldots,v_{d}],\\qquad\\widehat{\\mathbf{V}}=[\\widehat{v}_{1},\\ldots,\\widehat{v}_{d}],\n$$"}, {"n": 2, "claim": "$$\n\\mathbf{V}_{d^{\\prime}}=[v_{1},\\ldots,v_{d^{\\prime}}],\\qquad\\widehat{\\mathbf{V}}_{d^{\\prime}}=[\\widehat{v}_{1},\\ldots,\\widehat{v}_{d^{\\prime}}].\n$$"}, {"n": 3, "claim": "By orthogonality, the following identities hold:\n\n$$\n\\sum_{i=1}^{d}\\|v_{i}^{\\mathsf{T}}\\mathbf{X}\\|_{2}^{2}=\\sum_{i=1}^{d}\\|\\widehat{v}_{i}^{\\mathsf{T}}\\mathbf{X}\\|_{2}^{2}=\\|\\mathbf{X}\\|_{F}^{2}.\n$$\n\n$$\n\\sum_{i>d^{\\prime}}\\|v_{i}^{\\mathsf{T}}\\mathbf{X}\\|_{2}^{2}=\\|\\mathbf{X}-\\mathbf{V}_{d^{\\prime}}\\mathbf{V}_{d^{\\prime}}^{\\mathsf{T}}\\mathbf{X}\\|_{F}^{2}.\n$$\n\n$$\n\\sum_{i>d^{\\prime}}\\|\\widehat{v}_{i}^{\\mathsf{T}}\\mathbf{X}\\|_{2}^{2}=\\|\\mathbf{X}-\\widehat{\\mathbf{V}}_{d^{\\prime}}\\widehat{\\mathbf{V}}_{d^{\\prime}}^{\\mathsf{T}}\\mathbf{X}\\|_{F}^{2}.\n$$"}, {"n": 4, "claim": "Separating the top d' eigenvectors from the rest, we obtain\n\n$$\n\\sum_{i\\leq d^{\\prime}}\\|v_{i}^{\\mathsf{T}}\\mathbf{X}\\|_{2}^{2}+\\|\\mathbf{X}-\\mathbf{V}_{d^{\\prime}}\\mathbf{V}_{d^{\\prime}}^{\\mathsf{T}}\\mathbf{X}\\|_{F}^{2}=\\sum_{i\\leq d^{\\prime}}\\|\\widehat{v}_{i}^{\\mathsf{T}}\\mathbf{X}\\|_{2}^{2}+\\|\\mathbf{X}-\\widehat{\\mathbf{V}}_{d^{\\prime}}\\widehat{\\mathbf{V}}_{d^{\\prime}}^{\\mathsf{T}}\\mathbf{X}\\|_{F}^{2}.\n$$"}, {"n": 5, "claim": "Therefore\n\n$$\n\\begin{array}{ r l }{\\|{\\mathbf{X}}-\\widehat{{\\mathbf{V}}}_{d^{\\prime}}\\widehat{{\\mathbf{V}}}_{d^{\\prime}}^{\\mathsf{T}}{\\mathbf{X}}\\|_{F}^{2}=\\displaystyle\\sum_{i\\leq d^{\\prime}}\\|v_{i}^{\\mathsf{T}}{\\mathbf{X}}\\|_{2}^{2}-\\displaystyle\\sum_{i\\leq d^{\\prime}}\\|\\widehat{v}_{i}^{\\mathsf{T}}{\\mathbf{X}}\\|_{2}^{2}+\\|{\\mathbf{X}}-{\\mathbf{V}}_{d^{\\prime}}{\\mathbf{V}}_{d^{\\prime}}^{\\mathsf{T}}{\\mathbf{X}}\\|_{F}^{2}}&{}\\\\ {=n\\displaystyle\\sum_{i\\leq d^{\\prime}}\\sigma_{i}-n\\displaystyle\\sum_{i\\leq d^{\\prime}}\\widehat{v}_{i}^{\\mathsf{T}}{\\mathbf{M}}\\widehat{v}_{i}+n\\displaystyle\\sum_{i>d^{\\prime}}\\sigma_{i}}&{}\\\\ {=n\\displaystyle\\sum_{i\\leq d^{\\prime}}\\sigma_{i}-n\\displaystyle\\sum_{i\\leq d^{\\prime}}\\widehat{v}_{i}^{\\mathsf{T}}(\\widehat{\\mathbf{M}}-{\\mathbf{A}})\\widehat{v}_{i}+n\\displaystyle\\sum_{i>d^{\\prime}}\\sigma_{i}}&{}\\\\ {=n\\displaystyle\\sum_{i\\leq d^{\\prime}}(\\sigma_{i}-\\widehat{\\sigma}_{i})+n\\displaystyle\\operatorname{tr}({\\mathbf{A}}\\widehat{\\mathbf{V}}_{d^{\\prime}}\\widehat{\\mathbf{V}}_{d^{\\prime}}^{\\mathsf{T}})+n\\displaystyle\\sum_{i>d^{\\prime}}\\sigma_{i}.}&{}\\end{array}\\tag{5}\n$$"}, {"n": 6, "claim": "By Weyl's inequality, for $i\\leq d^{\\prime}$\n\n$$\n|\\sigma_{i}-{\\widehat{\\sigma}}_{i}|\\leq\\|\\mathbf{A}\\|.\\tag{6}\n$$"}, {"n": 7, "claim": "By von Neumann's trace inequality,\n\n$$\n\\mathrm{tr}(A\\widehat{\\mathbf{V}}_{d^{\\prime}}\\widehat{\\mathbf{V}}_{d^{\\prime}}^{\\mathsf{T}})\\leq\\sum_{i=1}^{d^{\\prime}}\\sigma_{i}({\\mathbf{A}}).\\tag{7}\n$$"}, {"n": 8, "claim": "From (5), (6), and (7),\n\n$$\n\\frac 1 n\\|{\\bf X}-\\widehat{{\\bf V}}_{d^{\\prime}}\\widehat{{\\bf V}}_{d^{\\prime}}^{\\top}{\\bf X}\\|_{F}^{2}\\leq\\sum_{i>d^{\\prime}}\\sigma_{i}+d^{\\prime}\\|{\\bf A}\\|+\\sum_{i=1}^{d^{\\prime}}\\sigma_{i}({\\bf A})\\leq\\sum_{i>d^{\\prime}}\\sigma_{i}+2 d^{\\prime}\\|{\\bf A}\\|.\n$$"}, {"n": 9, "claim": "Let $Y_{i}$ be the i-th column of Y. We have\n\n$$\nW_{2}^{2}(\\mu_{\\mathbf{X}},\\mu_{\\mathbf{Y}})\\leq{\\frac{1}{n}}\\sum_{i=1}^{n}\\|X_{i}-Y_{i}\\|_{2}^{2}={\\frac{1}{n}}\\|\\mathbf{X}-\\mathbf{Y}\\|_{F}^{2}.\n$$"}]}}, {"local_id": "T2", "statement": "Proposition 1. The subroutine Algorithm 4 is ε-differentially private. For any $d^{\\prime}\\geq 2,$ with the input as the projected data $\\widehat{\\mathbf{X}}$ and the range $\\Omega^{\\prime}$ with radius R, Algorithm 4 has an accuracy bound\n\n$$\n\\mathbb{E}W_{1}(\\mu_{\\widehat{\\mathbf{X}}},\\mu_{\\mathbf{X}^{\\prime}})\\leq C R(\\varepsilon n)^{-1/d^{\\prime}},\n$$\n\nwhere the expectation is taken with respect to the randomness of the synthetic data subroutine, conditioned on R.", "proof": {"steps": [{"n": 1, "claim": "The privacy guarantee follows from (He et al., 2023, Theorem 1.1)."}, {"n": 2, "claim": "For accuracy, note that the region Ω' is a subregion of a d'-dimensional ball. Algorithm 4 enlarges the region to a $\\bar{d}^{\\prime}.$ -dimensional hypercube with side length 2R."}, {"n": 3, "claim": "By re-scaling the size of the hypercube and applying (He et al., 2023, Corollary 4.4), we obtain the accuracy bound."}]}}, {"local_id": "T3", "statement": "Consider a data matrix $\\mathbf{X}=[X_1,\\ldots,X_n]\\in\\mathbb{R}^{d\\times n}$ with each $X_i\\in[0,1]^d$, where two data matrices are neighboring if they differ on only one column, and a randomized algorithm $\\mathcal{M}$ is $\\varepsilon$-differentially private if $\\mathbb{P}\\{\\mathcal{M}(D)\\in S\\}\\leq e^{\\varepsilon}\\,\\mathbb{P}\\{\\mathcal{M}(D')\\in S\\}$ for all neighboring $D,D'$ and measurable $S$. Algorithm 1 releases synthetic data via steps $\\mathcal{M}_1$ (private covariance matrix $\\widehat{\\mathbf{M}}$, Alg. 2), $\\mathcal{M}_2(\\mathbf{X})=\\overline{X}+\\lambda$ (private sample mean, with $\\overline{X}=\\frac{1}{n}\\sum_{i=1}^n X_i$ and $\\lambda\\in\\mathbb{R}^d$ having i.i.d. $\\mathrm{Lap}(1/(\\varepsilon n))$ components), $\\mathcal{M}_3$ (private projection onto the top $d'$ principal components plus a DP synthetic-data subroutine), $\\mathcal{M}_4$ (shifting the data back by $\\overline{X}_{\\mathrm{priv}}$), and a final metric projection. Theorem 1 (Privacy): Algorithm 1 is $\\varepsilon$-differentially private.", "proof": {"steps": [{"n": 1, "claim": "Algorithm 1 can be decomposed into steps $\\mathcal{M}_1(\\mathbf{X})=\\widehat{\\mathbf{M}}$ (private covariance via Algorithm 2), $\\mathcal{M}_2(\\mathbf{X})=\\overline{X}+\\lambda$ (private sample mean), $\\mathcal{M}_3(\\mathbf{X},y,\\Sigma)$ (project the shifted data $\\{X_i-y\\}$ onto the first $d'$ principal components of $\\Sigma$ plus a DP subroutine), $\\mathcal{M}_4(\\mathbf{X},\\mathbf{X}')$ (shift the dataset to $\\{X_i'+\\overline{X}_{\\mathrm{priv}}\\}$), and a final metric projection."}, {"n": 2, "claim": "It suffices to show that the data before the metric projection is already differentially private."}, {"n": 3, "claim": "The private-mean release $\\mathcal{M}_2(\\mathbf{X})=\\overline{X}+\\lambda$ uses noise $\\lambda=(\\lambda_1,\\ldots,\\lambda_d)$ with i.i.d. $\\mathrm{Lap}(1/(\\varepsilon n))$ entries, and $\\mathbb{E}\\|\\lambda\\|_2^2=\\sum_{j=1}^d\\mathbb{E}|\\lambda_j|^2=\\frac{2d}{\\varepsilon^2 n^2}$ (eq (3))."}, {"n": 4, "claim": "Each of $\\mathcal{M}_1,\\mathcal{M}_2,\\mathcal{M}_3,\\mathcal{M}_4$ is $\\varepsilon/4$-differentially private with respect to the input $\\mathbf{X}$ while fixing the other input variables."}, {"n": 5, "claim": "The private sample mean $\\mathcal{M}_2(\\mathbf{X})=\\overline{X}+\\lambda$ with entry-wise noise $\\mathrm{Lap}(1/(\\varepsilon n))$ attains its stated $\\varepsilon/4$-differential privacy."}, {"n": 6, "claim": "The composition $\\mathcal{M}_4(\\mathbf{X},\\mathcal{M}_3(\\mathbf{X},\\mathcal{M}_2(\\mathbf{X}),\\mathcal{M}_1(\\mathbf{X})))$ satisfies $\\varepsilon$-differential privacy, and hence Algorithm 1 is $\\varepsilon$-differentially private."}]}}, {"local_id": "T4", "statement": "Theorem 3. Algorithm 2 is ε-differentially private.", "proof": {"steps": [{"n": 1, "claim": "Proof. Before applying the definition of differential privacy, we compute the entries of M explicitly. One can easily check that\n\n$$\n\\mathbf{M}=\\frac{1}{n}\\sum_{k=1}^{n}X_{k}X_{k}^{\\mathsf{T}}-\\frac{1}{n(n-1)}\\sum_{k\\neq\\ell}X_{k}X_{\\ell}^{\\mathsf{T}}.\\tag{8}\n$$"}, {"n": 2, "claim": "Now, if there are neighboring datasets X and $\\mathbf{X}^{\\prime}.$ ,suppose $X_{k}=(X_{k}^{(1)},\\ldots,X_{k}^{(d)})^{\\top}$ is a column vector in X and $X_{k}^{\\prime}=(X_{k}^{\\prime}{}^{(1)},\\ldots,X_{k}^{\\prime}{}^{(d)})^{\\mathsf{T}}$ is a column vector in $\\mathbf{X}^{\\prime}.$ , and all other column vectors are the same. Let M and $\\ddot{\\mathbf{M}}^{\\prime}$ be the covariance matrix of X and $\\mathbf{X}^{\\prime}$ , respectively. Then we consider the density function ratio for the output of Algorithm 2 with input X and $\\mathbf{X}^{\\prime}\\colon$\n\n$$\n\\begin{array}{ l }{\\displaystyle\\frac{\\mathrm{den}_{\\boldsymbol{A}}(\\widetilde{\\mathbf{M}}-\\mathbf{M})}{\\mathrm{den}_{\\boldsymbol{A}}(\\widetilde{\\mathbf{M}}-\\mathbf{M^{\\prime}})}=\\displaystyle\\prod_{i<j}\\frac{\\mathrm{den}_{\\boldsymbol{A}j}\\left((\\widetilde{\\mathbf{M}}-\\mathbf{M})_{i j}\\right)}{\\mathrm{den}_{\\boldsymbol{A}j}\\left((\\widetilde{\\mathbf{M}}-\\mathbf{M^{\\prime}})_{i j}\\right)}\\displaystyle\\prod_{i=j}\\frac{\\mathrm{den}_{2\\boldsymbol{A}j}\\left((\\widetilde{\\mathbf{M}}-\\mathbf{M})_{i j}\\right)}{\\mathrm{den}_{2\\boldsymbol{A}j}\\left((\\widetilde{\\mathbf{M}}-\\mathbf{M^{\\prime}})_{i j}\\right)}}\\\\ {=\\displaystyle\\prod_{i<j}\\frac{\\exp\\left(-\\frac{|(\\widetilde{\\mathbf{M}}-\\mathbf{M})_{i j}|}{\\sigma}\\right)}{\\exp\\left(-\\frac{|(\\widetilde{\\mathbf{M}}-\\mathbf{M^{\\prime}})_{i j}|}{\\sigma}\\right)}\\displaystyle\\prod_{i}\\frac{\\exp\\left(-\\frac{|(\\widetilde{\\mathbf{M}}-\\mathbf{M})_{i i}|}{2\\sigma}\\right)}{\\exp\\left(-\\frac{|(\\widetilde{\\mathbf{M}}-\\mathbf{M^{\\prime}})_{i i}|}{2\\sigma}\\right)}}\\\\ {\\leq\\exp\\left(\\displaystyle\\sum_{i<j}\\left|\\mathbf{M}_{i j}-\\mathbf{M}_{i j}^{\\prime}\\right| /\\sigma+\\displaystyle\\sum_{i}\\left|\\mathbf{M}_{i i}-\\mathbf{M}_{i i}^{\\prime}\\right| /(2\\sigma)\\right)}\\\\ {=\\exp\\left(\\displaystyle\\frac{1}{2\\sigma}\\displaystyle\\sum_{i,j}\\left|\\mathbf{M}_{i j}-\\mathbf{M}_{i j}^{\\prime}\\right|\\right).}\\end{array}\n$$"}, {"n": 3, "claim": "As the datasets differs on only one data $X_{k},$ ,consider all entry containing $X_{k}$ in (8), we have\n\n$$\n\\begin{array}{ r }{\\left|{\\bf M}_{i j}-{\\bf M}_{i j}^{\\prime}\\right|\\leq\\displaystyle\\frac{1}{n}\\Big|X_{k}^{(i)}X_{k}^{(j)}-{X_{k}^{\\prime}}^{(i)}X_{k}^{\\prime}{}^{(j)}\\Big| +\\frac{1}{n(n-1)}\\displaystyle\\sum_{\\ell\\neq k}\\left|X_{k}^{(i)}-{X_{k}^{\\prime}}^{(i)}\\right|X_{\\ell}^{(j)}}\\\\ {+\\frac{1}{n(n-1)}\\displaystyle\\sum_{\\ell\\neq k}X_{\\ell}^{(i)}\\Big|X_{k}^{(j)}-X_{k}^{\\prime}{}^{(j)}\\Big|}\\end{array}\n$$\n\n$$\n\\leq{\\frac{2}{n}}+{\\frac{2}{n(n-1)}}\\cdot 2(n-1)={\\frac{6}{n}}.\n$$"}, {"n": 4, "claim": "Therefore, substituting the result in the probability ratio implies\n\n$$\n\\frac{\\mathrm{den}_{A}(\\widehat{\\mathbf{M}}-\\mathbf{M})}{\\mathrm{den}_{A}(\\widehat{\\mathbf{M}}-\\mathbf{M}^{\\prime})}\\leq\\exp\\left(\\frac{1}{2\\sigma}\\cdot d^{2}\\cdot\\frac{6}{n}\\right)=\\exp\\left(\\frac{3 d^{2}}{\\sigma n}\\right),\n$$"}, {"n": 5, "claim": "and when $\\begin{array}{ r }{\\sigma=\\frac{3 d^{2}}{{\\varepsilon}n}}\\end{array}$ , Algorithm 2 is ε-differentially private."}]}}, {"local_id": "T5", "statement": "Proposition 2 (Y is close to an affne subspace). The function $f:\\mathbb{R}^{d}[0,1]^{d}$ is the metric projection to $[0,1]^{d}$ with respect $t o\\parallel\\cdot\\parallel_{\\infty}$ and the accuracy error for the metric projection step in Algorithm 1 is dominated by the error of the previous steps:\n\n$$\nW_{1}(\\mu_{\\mathbf{Y}},\\mu_{\\mathbf{X}^{\\prime\\prime}})\\leq W_{1}(\\mu_{\\mathbf{X}},\\mu_{\\mathbf{X}^{\\prime\\prime}}),\\tag{2}\n$$\n\nwhere the dataset $\\mathbf{X}^{\\prime\\prime}$ defined in Algorithm 1 is in a d'-dimensional affine subspace. And we have\n\n$$\n\\mathbb{E}W_{1}(\\mu_{\\mathbf{Y}},\\mu_{\\mathbf{X}^{\\prime\\prime}})\\lesssim_{d}\\sqrt{\\sum_{i>d^{\\prime}}\\sigma_{i}(\\mathbf{M})}+(\\varepsilon n)^{-1/d^{\\prime}}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof. For the function f defined in Algorithm 1, we know $f(x)$ is the closest real number to x in the region $[0,1]$ for any $x\\in\\mathbb{R}$ Furthermore, if $v\\in\\mathbb{R}^{d}$ is a vector, then $f(v)$ is the closest vector to v in $[\\breve{0},1]^{d}$ with respect to $\\|\\cdot\\|_{\\infty}$ . Thus $f:{\\mathbb{R}^{d}}\\to [0,1]^{d}$ is indeed a metric projection to $[0,1]^{d}$"}, {"n": 2, "claim": "We first assume that the synthetic data $\\mathbf{X}^{\\prime\\prime}$ also has size n. Then for any column vector $X_{i}^{\\prime\\prime}$ , we know that $Y_{i}=f(X_{i}^{\\prime\\prime})$ is its closest vector in $[0,1]^{d}$ under the $\\ell^{\\infty}$ metric. For the data $X_{1},X_{2},\\ldots,X_{n},$ suppose that the solution to the optimal transportation problem for $W_{1}(\\mu_{\\mathbf{X}},\\mu_{\\mathbf{X}^{\\prime\\prime}})$ is to match $X_{\\tau(i)}$ with $X_{i}^{\\prime\\prime}$ , where τ is a permutation on [n]. Then\n\n$$\nW_{1}(\\mu_{{\\bf Y}},\\mu_{{\\bf X}^{\\prime\\prime}})\\leq\\frac{1}{n}\\sum_{i=1}^{n}\\bigl\\|Y_{i}-X_{i}^{\\prime\\prime}\\bigr\\|_{\\infty}\\leq\\frac{1}{n}\\sum_{i=1}^{n}\\Bigl\\|X_{\\tau(i)}-X_{i}^{\\prime\\prime}\\Bigr\\|_{\\infty}=W_{1}(\\mu_{{\\bf X}},\\mu_{{\\bf X}^{\\prime\\prime}}).\n$$"}, {"n": 3, "claim": "In general, if the synthetic dataset has m data points and $m \\neq n$ , we can split the points and regard both the true dataset and synthetic dataset as of size mn, then it's easy to check that the inequality still holds."}, {"n": 4, "claim": "The expectation bound follows from (9) and (10)."}]}}], "target_local_id": "T3", "topic_domain": "Others", "topic_subtopic": "Robustness, Privacy & Security", "label": "flawed", "error_step": 4, "error_class6": ["invalid_bound_or_inequality", "computation_or_constant", "missing_assumption"], "error_explanation": "Step 4 already asserts the false claim before Step 5 restates it: it says \"Each of M1,M2,M3,M4 is epsilon/4-differentially private with respect to the input X while fixing the other input variables\" -- an unqualified blanket claim that includes M2 (the private-mean release). This is false for the same reason given for Step 5: the L1-sensitivity of the d-dimensional mean X-bar is d/n, not 1/n, so releasing X-bar with epsilon-DP requires entry-wise noise Lap(d/(epsilon*n)), not the Lap(1/(epsilon*n)) used in the proof (per eq. C.7). Using the too-small noise scale means M2 only achieves (d*epsilon)-DP, a factor d too weak, so Step 4's blanket epsilon/4-DP claim already fails for M2 -- this is the same false statement that the record previously located at Step 5, but Step 5 is merely a restatement /elaboration of the identical error already asserted at Step 4, making Step 4 the first genuine error. (Reviewer quote and the authors' own admission of the underlying d/n-vs-1/n sensitivity error are unchanged and still apply.)"}
{"id": "KTS3QeWxRQq", "context": "2.1 Variational Autoencoder and Theoretical Analysis In VAE, ELBO is maximized instead of maximizing the log-likelihood directly. Let ${\\pmb x} \\in \\mathbb{R}^m$ be a point in a dataset. The original VAE model consists of a latent variable with fixed prior ${\\pmb z} \\sim p({\\pmb z}) = \\mathcal{N}({\\pmb z}; {\\pmb 0}, {\\pmb I}_n) \\in \\mathbb{R}^n$, a parametric encoder $\\mathrm{Enc}_\\phi:{\\pmb x}\\Rightarrow{\\pmb z}$, and a parametric decoder $\\mathrm{Dec}_\\theta : {\\pmb z} \\Rightarrow \\hat{\\pmb x}$. In the encoder, $q_\\phi({\\pmb z}|{\\pmb x}) = \\mathcal{N}({\\pmb z}; {\\pmb \\mu}_{({\\pmb x})}, {\\pmb \\sigma}_{({\\pmb x})})$ is provided by estimating parameters ${\\pmb \\mu}_{({\\pmb x})}$ and ${\\pmb \\sigma}_{({\\pmb x})}$. Let $L_{\\pmb x}$ be a local cost at data ${\\pmb x}$. Then, ELBO is described by $$ \\mathrm{ELBO} = E_{{\\pmb x} \\sim p({\\pmb x})} \\left[ E_{{\\pmb z} \\sim q_\\phi({\\pmb z}|{\\pmb x})} [\\log p_\\theta({\\pmb x}|{\\pmb z})] - D_{\\mathrm{KL}}(q_\\phi({\\pmb z}|{\\pmb x}) \\| p({\\pmb z})) \\right] .\\tag{14} $$ In $E_{{\\pmb x} \\sim p({\\pmb x})}[\\,\\cdot\\,]$, the first term $E_{{\\pmb z} \\sim q_\\phi({\\pmb z}|{\\pmb x})}[\\,\\cdot\\,]$ is called the reconstruction loss. The second term $D_{\\mathrm{KL}}(\\cdot)$ is a Kullback–Leibler (KL) divergence. Let $\\mu_{j({\\pmb x})}$, $\\sigma_{j({\\pmb x})}$, and $D_{\\mathrm{KL}j({\\pmb x})}$ be $j$-th dimensional values of ${\\pmb \\mu}_{({\\pmb x})}$, ${\\pmb \\sigma}_{({\\pmb x})}$, and KL divergence. Then $D_{\\mathrm{KL}}(\\cdot)$ is derived as: $$ D_{\\mathrm{KL}}(\\cdot)=\\sum_{j=1}^{n}D_{\\mathrm{KL}j({\\pmb x})},\\quad \\text{where } D_{\\mathrm{KL}j({\\pmb x})}=\\frac{1}{2}\\left({\\mu_{j({\\pmb x})}}^2+{\\sigma_{j({\\pmb x})}}^2-\\log{\\sigma_{j({\\pmb x})}}^2-1\\right). $$ $D({\\pmb x}, \\hat{\\pmb x})$ denotes a metric such as sum square error (SSE) and binary cross-entropy (BCE) as log-likelihoods of Gaussian and Bernoulli distributions, respectively. In training VAE, the next objective is used instead of Eq. 14, where $\\beta$ is a parameter to control the trade-off (Higgins et al., 2017). $$ L_{\\pmb x} = E_{{\\pmb z} \\sim q_\\phi({\\pmb z}|{\\pmb x})} [D({\\pmb x}, \\hat{\\pmb x})] + \\beta D_{\\mathrm{KL}}(\\cdot) . $$ 3.1 Hypothesis of Mapping VAE to the Implicit Orthonormal Transform Figure 1 shows the mapping of VAE to the implicit isometric embedding. Assume the data manifold is smooth and differentiable. Let $S_{\\mathrm{input}}\\mathopen{}\\mathclose\\bgroup\\left(\\mathsf{C}\\bar{\\mathbb{R^{m}}}\\aftergroup\\egroup\\right)$ be an input space of the dataset. $D({\\pmb x},{\\acute{\\pmb x}})$ denotes a metric for points x, $\\acute{\\pmb{x}}\\in S_{\\mathrm{input}}.$ Using the second order Taylor expansion, $D({\\pmb x},{\\pmb x}+\\delta{\\pmb x})$ can be approximated by tδx $G_{x}\\delta x$ , where $G_{x}$ and δx are an x dependent positive definite Hermitian metric tensor and an arbitrary infinitesimal displacement in $S_{\\mathrm{{i n p u t}}}$ , respectively. The derivations of $G_{x}$ for SSE, BCE, and SSIM are shown in Appendix H.2. Next, an implicit isometric embedding space $S_{\\mathrm{Iso}}(\\subset\\mathbb{R}^{m})$ is introduced like the isometric latent space in RaDOGAGA (Kato et al., 2020), such that the entropy of data representation is minimum in the inner product space of $G_{x}$ Let y and $y_{j}$ be a point in $S_{\\mathrm{{I s o}}}$ and its $j$ -th component, respectively. Because of the isometricity, $p({\\pmb x})\\simeq p({\\pmb y})$ will hold. We will also show the posterior variance of each dimensional component $y_{j}$ is a constant $\\beta / 2$ In addition, the variance of $y_{j}$ will show the importance like PCA when the data manifold has a disentangled feature by nature in the metric space of $G_{x}$ and the prior covariance is diagonal. Then, $S_{\\mathrm{{I s o}}}$ is nonlinearly scaled to the VAE's anisometric orthogonal space $S_{\\mathrm{VAE}}(\\subset\\ \\mathbb{R}^{n})$ on a variable-by-variable basis. Let z be a point in $S_{\\mathrm{VAE}}$ , and $z_{j}$ denotes the j-th component of z. Let $p(y_{j})$ and $p(z_{j})$ be the probability distribution of the j-th variable in $S_{\\mathrm{{I s o}}}$ and $S_{\\mathrm{VAE}}$ . Each variable $y_{j}$ is nonlinearly scaled $\\mathrm{to}~z_{j}$ ,such that $\\mathrm{d}z_{j}/\\mathrm{d}y_{j}=\\bar{p}(y_{j})/p(z_{j})$ to fit the cumulative distribution. $\\mathrm{d}z_{j}/\\mathrm{d}y_{j}$ is $\\sigma_{j(\\pmb{x})}/\\sqrt{\\beta/2}$ , the ratio of posterior's standard deviations for $z_{j}$ and $y_{j}$ , such that KL divergences in both spaces are equivalent. In addition, dimensional components whose KL divergences are zero can be discarded because such dimensions have no information. 3.2 Reformulation of Objective to the Form Using $\\partial\\pmb{x}/\\partial z_{j}$ and $\\partial\\pmb{x}/\\partial z$ We reformulate the objective $L_{x}$ to the form using $\\partial\\pmb{x}/\\partial z_{j}$ and ∂x $/ \\partial z$ . Here, the dimensions of x and z, i.e., m and n, are set as the same. The condition to reduce n is shown in section 3.3. Reformulation of $D(\\pmb{x},\\hat{\\pmb{x}})$ loss: In accordance with Kato et al. (2020), the loss $D(\\pmb{x},\\hat{\\pmb{x}})$ can be decomposed into $D(\\breve{\\pmb{x}},\\hat{\\pmb{x}})+D(\\pmb{x},\\breve{\\pmb{x}})$ where denotes $\\mathrm{Dec}_{\\theta}(\\pmb{\\mu}_{(\\pmb{x})})$ The first term $\\boldsymbol{D}(\\breve{\\pmb{x}},\\hat{\\pmb{x}})$ is a distortion between the decoded values of ${\\pmb\\mu}_{({\\pmb x})}$ with and without noise ${\\pmb\\sigma}_{({\\pmb x})}$ . We call this term as a coding loss. This term is expanded as follows. δx denotes $\\hat{\\mathbf{x}}-\\breve{\\mathbf{x}}$ .Then, $D(\\breve{\\pmb x},\\hat{\\pmb x})$ term can be approximated by tδ $\\mathbf{\\Pi}_{G_{x}\\delta\\breve{\\mathbf{x}}}$ . Let $\\mathbf{\\boldsymbol{x}}_{z_{j}}$ be ∂x/∂zj at $z_{j}=\\mu_{j(\\pmb{x})}$ , and $\\delta z_{j}\\sim\\mathcal{N}(0,\\overset{\\cdot}{\\sigma}_{j(\\pmb{x})})$ be an added noise in $z_{j}$ . Then, δx is approximated by $\\begin{array}{ r }{\\delta\\breve{\\pmb{x}}\\simeq\\sum_{j=1}^{m}\\delta z_{j}\\ \\pmb{x}_{z_{j}}}\\end{array}$ . Because $\\delta z_{j}$ and $\\delta z_{k}$ for $j \\neq k$ are uncorrelated, the average of $D(\\breve{\\pmb x},\\hat{\\pmb x})$ over $z\\sim q_{\\phi}(z|\\mathbf{x})$ can be finally reformulated by $$ E_{z\\sim q_{\\phi}(z|\\mathbf{x})}\\left[D(\\check{\\mathbf{x}},\\hat{\\mathbf{x}})\\right]\\simeq E_{z\\sim q_{\\phi}(z|\\mathbf{x})}\\left[{^t\\delta\\check{\\mathbf{x}}\\ G_{\\mathbf{x}}\\delta\\check{\\mathbf{x}}}\\right]\\simeq\\sum_{j=1}^{n}\\sigma_{j(\\mathbf{x})}{^2\\ ^{t}}x_{z_{j}}G_{x}x_{z_{j}}.\\tag{3} $$ The second term $D({\\pmb x},{\\breve{\\pmb x}})$ is a loss between the input data and $\\mathrm{Dec}_{\\theta}(\\pmb{\\mu}_{(\\pmb{x})})$ . We call this term a transform loss. We presume VAE is analogous to the Wiener filter (Wiener, 1964; Jin et al., 2003) where the coding loss is regarded as an added noise. From the Wiener filter theory, the ratio between the transform loss and coding loss is close to the ratio between the coding loss and the variance of the input data. The coding loss, approximately $n \\beta / 2$ as in Eq. 4, should be smaller than the variance of the input data to capture meaningful information. Thus the transform loss, usually small, is not considered in the following discussion. Appendix B explains the detail in a simple 1-dimensional VAE. We show the exhaustive and quantitative evaluation of coding loss and transform loss in the toy dataset in appendix E.2 to validate this approximation. Reformulation of KL divergence: When $\\sigma_{j(x)}\\ll 1,{\\sigma_{j(x)}}^{2}\\ll-\\log{\\sigma_{j(x)}}^{2}$ is observed. For example, when ${\\sigma_{j(\\pmb x)}}^{2}<0.1$ , we have $-({\\sigma_{j(x)}}^{2}/\\log{\\sigma_{j(x)}}^{2})<0.05$ . In such dimensions, $D_{\\mathrm{KL}j(x)}$ can be approximated as Eq. 5 by ignoring the ${\\dot{\\sigma}_{j(\\pmb{x})}}^{2}$ term and setting $p(\\mu_{j({\\pmb x})})$ to $\\mathcal{N}(z_{j};0,1)$ •• $$ D_{\\mathrm{KL}j(x)}\\simeq\\frac{1}{2}\\left(\\mu_{j(x)}{}^{2}-\\log\\sigma_{j(x)}{}^{2}-1\\right)= -\\log\\left(\\sigma_{j(x)}p\\big(\\mu_{j(x)}\\big)\\right)-\\frac{\\log 2\\pi e}{2}.\\tag{5} $$ Eq. 5 can be considered as a rate of entropy coding for a symbol with mean $\\mu_{j({\\pmb x})}$ allowing quantization noise ${\\sigma_{j(\\pmb x)}}^{2}$ , as shown in Appendix H.3. Thus, in the dimension with meaningful information, ${\\sigma_{j(\\pmb x)}}^{2}$ is much smaller than the prior variance 1, and the approximation in Eq.5 is reasonable.", "theorems": [{"local_id": "T1", "statement": "Minimum entropy of implicit isometric representation: Let $L_{\\operatorname*{min}{x}}^{\\prime}$ be a minimum of $L_{x}^{\\prime}$ at x. $D_{\\operatorname*{min}{\\pmb x}}$ and $R_{\\operatorname*{min}{x}}$ denote a coding loss and KL divergence in $L_{\\mathrm{min}\\alpha}^{\\prime},$ respectively. By applying Eqs. 6-7 and $p(z_{j})=(\\mathrm{d}y_{j}/\\mathrm{d}z_{j})p(y_{j})$ to Eqs. 3 and 8, the following equations are derived: $$ L_{\\operatorname*{min}\\ x}^{\\prime}=D_{\\operatorname*{min}\\ x}+\\beta R_{\\operatorname*{min}\\ x},{\\mathrm{~w h e r e~}}D_{\\operatorname*{min}\\ x}={\\frac{n\\beta}{2}},R_{\\operatorname*{min}\\ x}= -\\log p(y)-{\\frac{n\\log\\left(\\beta\\pi e\\right)}{2}}.\\tag{9} $$ Here, $D_{\\operatorname*{min}{\\boldsymbol x}}$ is derived as $\\begin{array}{ r }{\\sum_{j=1}^{n}(\\beta/2)\\ ^{t}x_{y_{j}}G_{x}x_{y_{j}}=n\\beta/2}\\end{array}$ ,implying each dimensional posterior variance of the implicit isometric variable is a constant $\\beta / 2$ In addition, $\\exp(-L_{\\operatorname*{min}{x}}^{\\prime}/\\beta)=$ $p(\\pmb{y})\\exp(\\mathrm{C o n s t .})\\propto p(\\pmb{y})\\simeq p(\\pmb{x})$ will hold in the inner product space of $G_{x}$ from the isometricity. By averaging $L_{\\mathrm{min}\\alpha}^{\\prime}$ over $\\begin{array}{ r }{\\pmb{x}\\sim p(\\pmb{x})}\\end{array}$ and approximating this average by the integration over $\\mathbf{\\mu}_{\\mathbf{\\mu}_{y}\\sim}$ $p(\\pmb{y})$ , the global minimum $L_{\\mathrm{G}}^{\\prime}$ is derived as: $$ L_{\\mathrm{G}}^{\\prime}=D_{\\mathrm{G}}+\\beta R_{\\mathrm{G}}\\mathrm{,~w h e r e~}D_{\\mathrm{G}}=\\frac{n\\beta}{2}\\mathrm{,~}R_{\\mathrm{G}}=\\operatorname*{min}_{p(y)}\\left(-\\int p(y)\\log p(y)\\mathrm{d}y\\right)-\\frac{n\\log\\left(\\beta\\pi e\\right)}{2}.\\tag{4} $$", "proof": {"steps": [{"n": 1, "claim": "$$ L_{\\operatorname*{min}\\ x}^{\\prime}=D_{\\operatorname*{min}\\ x}+\\beta R_{\\operatorname*{min}\\ x},{\\mathrm{~w h e r e~}}D_{\\operatorname*{min}\\ x}={\\frac{n\\beta}{2}},R_{\\operatorname*{min}\\ x}= -\\log p(y)-{\\frac{n\\log\\left(\\beta\\pi e\\right)}{2}}.\\tag{9} $$"}, {"n": 2, "claim": "Here, $D_{\\operatorname*{min}{\\boldsymbol x}}$ is derived as $\\begin{array}{ r }{\\sum_{j=1}^{n}(\\beta/2)\\ ^{t}x_{y_{j}}G_{x}x_{y_{j}}=n\\beta/2}\\end{array}$ ,implying each dimensional posterior variance of the implicit isometric variable is a constant $\\beta / 2$ In addition, $\\exp(-L_{\\operatorname*{min}{x}}^{\\prime}/\\beta)=$ $p(\\pmb{y})\\exp(\\mathrm{C o n s t .})\\propto p(\\pmb{y})\\simeq p(\\pmb{x})$ will hold in the inner product space of $G_{x}$ from the isometricity. By averaging $L_{\\mathrm{min}\\alpha}^{\\prime}$ over $\\begin{array}{ r }{\\pmb{x}\\sim p(\\pmb{x})}\\end{array}$ and approximating this average by the integration over $\\mathbf{\\mu}_{\\mathbf{\\mu}_{y}\\sim}$ $p(\\pmb{y})$ , the global minimum $L_{\\mathrm{G}}^{\\prime}$ is derived as:"}, {"n": 3, "claim": "$$ L_{\\mathrm{G}}^{\\prime}=D_{\\mathrm{G}}+\\beta R_{\\mathrm{G}}\\mathrm{,~w h e r e~}D_{\\mathrm{G}}=\\frac{n\\beta}{2}\\mathrm{,~}R_{\\mathrm{G}}=\\operatorname*{min}_{p(y)}\\left(-\\int p(y)\\log p(y)\\mathrm{d}y\\right)-\\frac{n\\log\\left(\\beta\\pi e\\right)}{2}.\\tag{4} $$"}, {"n": 4, "claim": "The term $-\\textstyle\\int p({\\pmb y})\\log p({\\pmb y})\\mathrm{d}{\\pmb y}$ in $R_{\\mathrm{G}}$ is the entropy of $\\mathbf{\\pmb{y}}.$ Thus, the optimal implicit isometric space is derived such that the entropy of data representation is minimum in the inner product space of $G_{x}$"}]}}, {"local_id": "T2", "statement": "$\\mathbf{\\boldsymbol{x}}_{y_{j}}$ denotes ${\\partial\\pmb x}/{\\partial y_{j}}$ . By applying $\\pmb{x}_{z_{j}}=\\mathrm{d}y_{j}/\\mathrm{d}z_{j}\\ \\pmb{x}_{y_{j}}$ to Eq. 6, $\\mathbf{\\boldsymbol{x}}_{y_{j}}$ shows the isometric property (Han & Hong, 2006; Kato et al., 2020) in the inner product space with a metric tensor $G_{x}$ as follows: $$ {}^{t}x_{y_{j}}G_{x}x_{y_{k}}=\\delta_{j k}.\\tag{10} $$", "proof": {"steps": [{"n": 1, "claim": "$$ \\frac{\\mathrm{d}L_{\\pmb{x}}^{\\prime}}{\\mathrm{d}{\\pmb x}_{z_{j}}}=2{\\sigma_{j(\\pmb{x})}}^{2}G_{x}\\pmb{x}_{z_{j}}-\\frac{\\beta}{\\mathrm{det}\\left(\\partial\\pmb{x}/\\partial z\\right)}\\tilde{\\pmb{x}}_{z_{j}}.\\tag{11} $$"}, {"n": 2, "claim": "Note that ${}^{t}{\\pmb x}_{z_{k}}\\cdot{\\tilde{\\pmb x}}_{z_{j}}=\\operatorname*{det}(\\partial{\\pmb x}/\\partial z)\\delta_{j k}$ holds by the cofactor's property. Here, · denotes the dot product, and $\\delta_{j k}$ denotes the Kronecker delta. By setting Eq. 11 to zero and multiplying $^t{\\pmb{x}}_{z_{k}}$ from the left, the condition to minimize $L_{x}^{\\prime}$ is derived by the next orthogonal form of $\\pmb{x}_{z_{j}}$ $$ ({2\\sigma_{j(\\pmb x)}}^{2}/\\beta)\\ ^{t}{\\pmb x}_{z_{k}}{\\pmb G}_{x}{\\pmb x}_{z_{j}}=\\delta_{j k}.\\tag{6} $$"}, {"n": 3, "claim": "Here, the diagonal posterior covariance is the key for orthogonality. Next, implicit latent variable y and its j-th dimensional component $y_{j}$ are introduced. Set $y_{j}$ to zero at $z_{j}=0$ .The derivative between $y_{j}$ and $z_{j}\\ a\\ t\\ \\mu_{j(\\pmb{x})}$ is defined by $$ \\frac{\\mathrm{d}y_{j}}{\\mathrm{d}z_{j}}\\Big|_{z_{j}=\\mu_{j(\\pmb{x})}}=\\sqrt{\\frac{\\beta}{2}}{\\sigma_{j(\\pmb{x})}}^{-1}.\\tag{7} $$"}, {"n": 4, "claim": "$\\mathbf{\\boldsymbol{x}}_{y_{j}}$ denotes ${\\partial\\pmb x}/{\\partial y_{j}}$ . By applying $\\pmb{x}_{z_{j}}=\\mathrm{d}y_{j}/\\mathrm{d}z_{j}\\ \\pmb{x}_{y_{j}}$ to Eq. 6, $\\mathbf{\\boldsymbol{x}}_{y_{j}}$ shows the isometric property (Han & Hong, 2006; Kato et al., 2020) in the inner product space with a metric tensor $G_{x}$ as follows: $$ {}^{t}x_{y_{j}}G_{x}x_{y_{k}}=\\delta_{j k}.\\tag{10} $$"}]}}, {"local_id": "T3", "statement": "Final objective form: From Eqs. 3 and 8, the objective $L_x'$ to minimise is derived as: $$ L_x' = \\sum_{j=1}^n \\sigma_{j(x)}^2\\, {}^t x_{z_j} G_x x_{z_j} - \\beta \\log\\big(p(x)\\, \\det(\\partial x/\\partial z) \\prod_{j=1}^n \\sigma_{j(x)}\\big) - C_{D_{\\mathrm{KL}}} .\\tag{12} $$ $C_{D_{\\mathrm{KL}}} = \\tfrac{n}{2}\\log 2\\pi e$, $\\det(\\partial x/\\partial z)$ the Jacobian determinant at $z=\\mu_{(x)}$.", "proof": {"steps": [{"n": 1, "claim": "$D_{\\mathrm{KL}j(x)} \\simeq -\\log(\\sigma_{j(x)} p(\\mu_{j(x)})) - \\tfrac{\\log 2\\pi e}{2}$. Setting $p(\\mu_{(x)}) = \\prod_{j=1}^n p(\\mu_{j(x)})$: $D_{\\mathrm{KL}}(\\cdot) \\simeq -\\log\\big(p(\\mu_{(x)}) \\prod_{j=1}^n \\sigma_{j(x)}\\big) - C_{D_{\\mathrm{KL}}}$, $C_{D_{\\mathrm{KL}}} = \\tfrac{n}{2}\\log 2\\pi e$."}, {"n": 2, "claim": "$p(\\mu_{(x)}) = p(x)\\,\\det(\\partial x/\\partial z)$, $\\det(\\partial x/\\partial z)$ the Jacobian determinant at $z = \\mu_{(x)}$."}, {"n": 3, "claim": "$$\nD_{\\mathrm{KL}}(\\cdot) \\simeq -\\log\\big(p(x)\\, \\det(\\partial x/\\partial z) \\prod_{j=1}^n \\sigma_{j(x)}\\big) - C_{D_{\\mathrm{KL}}}.\\tag{8}\n$$"}, {"n": 4, "claim": "$$\nL_x' = \\sum_{j=1}^n \\sigma_{j(x)}^2\\, {}^t x_{z_j} G_x x_{z_j} - \\beta \\log\\big(p(x)\\, \\det(\\partial x/\\partial z) \\prod_{j=1}^n \\sigma_{j(x)}\\big) - C_{D_{\\mathrm{KL}}}.\\tag{12}\n$$"}]}}, {"local_id": "T4", "statement": "By summing both terms, ELBO at x can be estimated as $$ \\begin{array}{ l l l }{ELBO}&{=}&{E_{x\\sim p(x)}[E_{z\\sim q_{\\phi}(z\\mid x)}[\\log p_{\\theta}({x}\\mid z)]-D_{\\mathrm{KL}}(\\cdot)]}\\\\ &{\\simeq}&{E_{x\\sim p(x)}[\\log p(\\boldsymbol{y})]}\\\\ &{\\simeq}&{E_{x\\sim p(x)}[\\log p(\\boldsymbol{x})].}\\end{array}\\tag{13} $$", "proof": {"steps": [{"n": 1, "claim": "$$ E_{z\\sim q_{\\phi}(z|x)}[\\log p_{\\theta}(x|z)]=\\int q_{\\phi}(z|x)\\log p_{\\theta}(x|z)\\mathrm{d}z=\\int q_{\\phi}(y|x)\\log p_{\\theta}(x|y)\\mathrm{d}y .\\tag{15} $$"}, {"n": 2, "claim": "Let $\\pmb{\\mu}_{y(\\pmb{x})}$ be a implicit isometric variable corresponding to $\\mu_{(\\pmb{x})}$ . Because the posterior variance in each isometric latent variable is a constant $\\beta/2,q_{\\phi}({\\pmb y}|{\\pmb x})\\simeq\\mathcal{N}({\\pmb y};{\\pmb\\mu}_{{\\pmb y}({\\pmb x})},(\\beta/2){\\pmb I}_{n})$ will hold. If $\\beta / 2$ is small, $p({\\hat{\\pmb x}})\\simeq p(\\pmb x)$ will hold. Then, the next equation will hold also using isometricity;"}, {"n": 3, "claim": "$$ p_{\\theta}({\\pmb x}|{\\pmb z})=p_{\\theta}({\\pmb x}|{\\pmb y})=p_{\\theta}({\\pmb x}|{\\hat{\\pmb x}})=p({\\hat{\\pmb x}}|{\\pmb x})p({\\pmb x})/p({\\hat{\\pmb x}})\\simeq p({\\hat{\\pmb x}}|{\\pmb x})\\simeq q_{\\phi}({\\pmb y}|{\\pmb x}).\\tag{16} $$"}, {"n": 4, "claim": "Thus the reconstruction loss is estimated as: $$ \\begin{array}{ r c l }{E_{z\\sim q_{\\phi}(z|x)}[\\log p_{\\theta}(x|z)]}&{\\sim}&{\\displaystyle\\int\\mathcal{N}(y;\\mu_{y(x)},(\\beta/2)I_{n})\\log\\mathcal{N}(y;\\mu_{y(x)},(\\beta/2)I_{n})\\mathrm{d}y}\\\\ &{=}&{\\displaystyle-(n/2)\\log(\\beta\\pi e).}\\end{array}\\tag{17} $$"}, {"n": 5, "claim": "From Eq. 9, KL divergence is derived as: $$ D_{\\mathrm{KL}}(\\cdot)=R_{\\mathrm{min}{\\pmb x}}= -\\log p({\\pmb y})-(n/2)\\log(\\beta\\pi e).\\tag{18} $$"}, {"n": 6, "claim": "By summing both terms, ELBO at x can be estimated as $$ \\begin{array}{ l l l }{ELBO}&{=}&{E_{x\\sim p(x)}[E_{z\\sim q_{\\phi}(z\\mid x)}[\\log p_{\\theta}({x}\\mid z)]-D_{\\mathrm{KL}}(\\cdot)]}\\\\ &{\\simeq}&{E_{x\\sim p(x)}[\\log p(\\boldsymbol{y})]}\\\\ &{\\simeq}&{E_{x\\sim p(x)}[\\log p(\\boldsymbol{x})].}\\end{array}\\tag{13} $$"}]}}], "target_local_id": "T3", "topic_domain": "Deep Learning Theory", "topic_subtopic": "Representation & Expressivity", "label": "flawed", "error_step": 2, "error_class6": ["invalid_bound_or_inequality", "others", "missing_assumption"], "error_explanation": "The change-of-variables relation for probability densities is $p(\\mu_{(x)}) = p(x)\\,|\\det(\\partial x/\\partial z)|$ — the density ratio is the ABSOLUTE VALUE of the Jacobian determinant, not the bare determinant. Writing $p(\\mu_{(x)}) = p(x)\\,\\det(\\partial x/\\partial z)$ (Step 2) is only valid when $\\det(\\partial x/\\partial z) > 0$; a Jacobian determinant need not be positive. If $\\det(\\partial x/\\partial z) < 0$ this equates a nonnegative density with a negative quantity, and it propagates to Eq. 8 where $p(x)\\,\\det(\\partial x/\\partial z)\\prod_j \\sigma_{j(x)}$ becomes negative, making its logarithm undefined and invalidating the objective. The correct relation and objective use $|\\det(\\partial x/\\partial z)|$; the author admission confirms 'Eq.8 is incorrect. We revise it to the absolute value of the Jacobian determinant.'"}
{"id": "NIzeVwedJzB", "context": "Problem formulation. Let $D=(x_{1},\\ldots,x_{n})$ be a database of size $n,$ where each of $i,i\\in\\lceil n\\rceil$ is a data instance. $x_{i}$ is a d-dimensional vector, $x_{i}\\in\\mathbb{R}^{d}$ and w.l.o.g., normalized to be $| |x_{i}| |_{2}\\in [\\mathrm{0 .\\bar{1}}]$\n\nGaussian mechanism G with variance $\\sigma^{2}$ is applied to each $x_{i}.$ resulting in $\\tilde{x_{i}}$ .Then, all $\\tilde{x_{i}}\\mathrm{\\tilde{s}}$ undergo shuffling; let $S_{n}:\\mathcal{V}^{n}\\mathcal{V}^{n}$ be the shuffling operation which takes $\\tilde{x_{i}}\\mathrm{:s}$ as input and outputs a uniformly and randomly permuted $\\tilde{x_{i}}\\mathrm{:s}$ . The randomization mechanism M of interest is therefore\n\n$$\n\\mathcal{M}(D)=S_{n}(\\mathcal{G}(x_{1}),\\ldots,\\mathcal{G}(x_{n})).\n$$\n\nOur purpose is to characterize and quantify the DP of M through RDP.\n\nDefinition 1 (Rényi Differential Privacy (RDP) (Mironov, 2017)). A randomization mechanism $\\mathcal{M}:\\mathcal{D}\\mathcal{S}$ is said to satisfy $(\\lambda,\\epsilon){\\tt RDP}$ for $\\lambda\\in(1,\\infty)$ , if for any adjacent databases D, $D^{\\prime}\\in\\mathcal{D}$ the Rényi divergence of order λ between $\\mathcal{M}(D)$ and $\\mathcal{M}(D^{\\prime})$ , as defined below, is upper-bounded by e:\n\n$$\nD_{\\lambda}(\\mathcal{M}(D)| |\\mathcal{M}(D^{\\prime})): =\\frac{1}{\\lambda-1}\\log\\left(\\mathbb{E}_{y\\sim\\mathcal{M}(D^{\\prime})}\\left[\\left(\\frac{\\mathcal{M}(D)(y)}{\\mathcal{M}(D^{\\prime})(y)}\\right)^{\\lambda}\\right]\\right)\\leq\\epsilon,\\tag{1}\n$$\n\nwhere $\\mathcal{M}(\\mathcal{D})(y)$ denotes M taking $D$ as input to output y with certain probability.\n\nDefinition 2 (Gaussian mechanism). Given $x\\in\\mathbb{R}^{d}$ , the Gaussian mechanism applied to x is a mechanism with parameter σ that adds zero mean isotropic Gaussian perturbation of variance $\\sigma^{2}$ to x, outputting $\\mathcal{M}(x)=x+\\mathcal{N}(0,\\sigma^{2}I_{d})$ .\n\n3 The RDP of Shuffle Gaussian\n\nWe first derive the Rényi divergence (of order λ) of the shuffle Gaussian mechanism in this section. Subsequently, an upper bound on the notion is given, followed by a description of the numerical technique used for evaluating the RDP.\n\nThe main result of this paper is given by the following theorem.\n\nUpper bound of shuffle Gaussian RDP. Using the above exact expression, we next give an upper bound on the shuffle Gaussian RDP.\n\nWe first consider the mechanism where a fixed number of data instances, $m,$ is sampled randomly from n data instances to participate in training. Each data instance is randomized with Gaussian noises before being shuffled. We call this overall procedure the subsampled (without replacement) shuffle (Girgis et al., 2021b) Gaussian mechanism. Let subsamp $\\mathbf{\\Delta}_{m}^{n}:\\mathcal{V}^{n}\\mathcal{V}^{m}$ be the subsampling operation mentioned above. Then, the mechanism can be written as\n\n$$\n{\\mathcal{M}}({\\mathcal{D}})=S_{n}\\circ\\operatorname{subsamp}_{m}^{n}({\\mathcal{G}}(x_{1}),\\dots,{\\mathcal{G}}(x_{n})).\n$$\n\nThe RDP of the subsampled shuffle Gaussian mechanism is given by the following theorem.", "theorems": [{"local_id": "T1", "statement": "Corollary 2 (Upper bound of shuffle Gaussian RDP). The shuffle Gaussian RDP (λ) is upperbounded by $\\lambda/2\\dot{\\sigma}^{2}$", "proof": {"steps": [{"n": 1, "claim": "Applying the Cauchy-Schwartz inequality, $\\begin{array}{ r }{\\sum_{i=1}^{n}k_{i}^{2}\\le(\\sum_{i=1}^{n}k_{i})^{2}=\\lambda^{2}}\\end{array}$ , to Equation 2,"}, {"n": 2, "claim": "$$\n\\begin{array}{ l }{{\\displaystyle\\frac{e^{-\\lambda/2\\sigma^{2}}}{n^{\\lambda}}\\sum_{{k_{1}+\\ldots+k_{n}=\\lambda;\\atop k_{1},\\ldots,k_{n}\\geq 0}\\atop k_{1},\\ldots,k_{n}\\geq 0}\\binom{\\lambda}{k_{1},\\ldots,k_{n}}e^{\\sum_{i=1}^{n}k_{i}^{2}/2\\sigma^{2}}\\leq\\frac{e^{-\\lambda/2\\sigma^{2}}}{n^{\\lambda}}\\sum_{{k_{1}+\\ldots+k_{n}=\\lambda;\\atop k_{1},\\ldots,k_{n}\\geq 0}\\atop k_{1},\\ldots,k_{n}\\geq 0}\\binom{\\lambda}{k_{1},\\ldots,k_{n}}e^{\\lambda^{2}/2\\sigma^{2}}}}\\\\ {{=e^{(\\lambda^{2}-\\lambda)/2\\sigma^{2}},}}\\end{array}\n$$"}, {"n": 3, "claim": "noting that $\\begin{array}{ l }{{\\sum_{k_{1}+\\ldots+k_{n}=\\lambda;}{\\binom{\\lambda}{k_{1},\\ldots,k_{n}}}=n^{\\lambda}}}\\end{array}$ . Substituting this to Equation 1, we get the desired expression."}]}}, {"local_id": "T2", "statement": "Let $D=(x_1,\\ldots,x_n)$ be a database of $n$ data instances $x_i\\in\\mathbb{R}^d$ with $\\|x_i\\|_2\\in[0,1]$, and let $\\mathcal{M}(D)=S_n(\\mathcal{G}(x_1),\\ldots,\\mathcal{G}(x_n))$ be the shuffle Gaussian mechanism, where the Gaussian mechanism $\\mathcal{G}$ adds zero-mean isotropic Gaussian noise of variance $\\sigma^2$ to each instance and $S_n$ uniformly randomly permutes the noisy instances. For any two adjacent databases $D,D'$ (differing by replacement of one instance) with $n$ instances, the Rényi divergence of order $\\lambda$ between $\\mathcal{M}(D)$ and $\\mathcal{M}(D')$ is $$D_{\\lambda}(\\mathcal{M}(D)\\|\\mathcal{M}(D'))=\\frac{1}{\\lambda-1}\\log\\left(\\frac{e^{-\\lambda/2\\sigma^2}}{n^{\\lambda}}\\sum_{\\substack{k_1+\\ldots+k_n=\\lambda;\\\\k_1,\\ldots,k_n\\geq 0}}\\binom{\\lambda}{k_1,\\ldots,k_n}e^{\\sum_{i=1}^{n}k_i^2/2\\sigma^2}\\right).$$", "proof": {"steps": [{"n": 1, "claim": "It suffices, without loss of generality, to consider the specific canonical adjacent databases $D,D'\\in\\mathbb{R}^n$ with one-dimensional instances $D=(0,\\ldots,0)$ and $D'=(1,0,\\ldots,0)$; equivalently in vector notation $D=0$ (the all-zeros $n$-vector) and $D'=e_1$ (a unit vector)."}, {"n": 2, "claim": "The shuffled output of $D$ distributes as $\\mathcal{M}(D)\\sim(\\mathcal{N}(0,\\sigma^2),\\ldots,\\mathcal{N}(0,\\sigma^2))$, an $n$-tuple of $\\mathcal{N}(0,\\sigma^2)$, and by the homogeneity of $D$ this is written $\\mathcal{M}(D)\\sim\\mathcal{N}(0,\\sigma^2 I_n)$."}, {"n": 3, "claim": "The shuffled output of $D'$ distributes as the uniform mixture $\\mathcal{M}(D')\\sim\\frac{1}{n}(\\mathcal{N}(e_1,\\sigma^2 I_n)+\\ldots+\\mathcal{N}(e_n,\\sigma^2 I_n))$."}, {"n": 4, "claim": "The neighbouring distributions of interest are $\\mathcal{M}(D)\\sim\\mathcal{N}(0,\\sigma^2 I_n)$ and $\\mathcal{M}(D')\\sim\\frac{1}{n}(\\mathcal{N}(e_1,\\sigma^2 I_n)+\\ldots+\\mathcal{N}(e_n,\\sigma^2 I_n))$ (Equation 3)."}, {"n": 5, "claim": "Using the Gaussian density $\\mathbb{P}(\\mathcal{N}(0,\\sigma^2))=\\exp[-x^2/2\\sigma^2]/\\sqrt{2\\pi\\sigma^2}$ and the mixture density for $\\mathcal{M}(D')$, the Rényi divergence integrand for the pair reduces to a power of a sum of exponentials, namely $\\left(\\sum_{i=1}^{n}\\exp[(2x_i-1)/2\\sigma^2]\\right)^{\\lambda}$ (Equation 4)."}, {"n": 6, "claim": "By the multinomial theorem, $\\left(\\sum_{i=1}^{n}\\exp[(2x_i-1)/2\\sigma^2]\\right)^{\\lambda}=\\sum_{\\substack{k_1+\\ldots+k_n=\\lambda;\\\\k_1,\\ldots,k_n\\geq 0}}\\binom{\\lambda}{k_1,\\ldots,k_n}\\prod_{i=1}^{n}\\exp[k_i(2x_i-1)/2\\sigma^2]$, where $\\binom{\\lambda}{k_1,\\ldots,k_n}=\\frac{\\lambda!}{k_1!\\cdots k_n!}$ and $k_i\\in\\mathbb{Z}^{+}$ (Equation 5)."}, {"n": 7, "claim": "Taking the expectation and simplifying yields $\\frac{1}{n^{\\lambda}}\\sum_{\\substack{k_1+\\ldots+k_n=\\lambda;\\\\k_1,\\ldots,k_n\\geq 0}}\\binom{\\lambda}{k_1,\\ldots,k_n}\\prod_{i=1}^{n}\\exp[(k_i^2-k_i)/2\\sigma^2]=\\frac{e^{-\\lambda/2\\sigma^2}}{n^{\\lambda}}\\sum_{\\substack{k_1+\\ldots+k_n=\\lambda;\\\\k_1,\\ldots,k_n\\geq 0}}\\binom{\\lambda}{k_1,\\ldots,k_n}e^{\\sum_{i=1}^{n}k_i^2/2\\sigma^2}$ (Equation 2)."}, {"n": 8, "claim": "Combining Equation 2 with Equation 1 gives $D_{\\lambda}(\\mathcal{M}(D)\\|\\mathcal{M}(D'))=\\frac{1}{\\lambda-1}\\log\\left(\\frac{e^{-\\lambda/2\\sigma^2}}{n^{\\lambda}}\\sum_{\\substack{k_1+\\ldots+k_n=\\lambda;\\\\k_1,\\ldots,k_n\\geq 0}}\\binom{\\lambda}{k_1,\\ldots,k_n}e^{\\sum_{i=1}^{n}k_i^2/2\\sigma^2}\\right)$, i.e. Equation 6, as desired."}]}}, {"local_id": "T3", "statement": "Theorem 3 (Subsampled Shuffle Gaussian RDP). Let n be the total number of users, m be the number of subsampled users in each round of training, and γ be the subsampling rate, $\\gamma = m / n$ Also let $\\epsilon_{m}^{\\mathrm{{\\tiny\\breve{S}G}}}(\\lambda)$ be the shuffle Gaussian RDP given in Theorem 2. The subsampled shuffle Gaussian mechanism satisfies $(\\lambda,\\epsilon_{\\gamma,m}^{\\mathrm{{S S G}}}(\\lambda))– R D P,$ where\n\n$$\n\\epsilon_{\\gamma,m}^{\\mathrm{SG}}(\\lambda)\\leq\\frac{1}{\\lambda-1}\\log\\left(1+\\gamma^{2}{\\binom{\\lambda}{2}}\\operatorname*{min}\\left\\{4(e^{\\epsilon_{m}^{\\mathrm{SG}}(2)}-1),2e^{\\epsilon_{m}^{\\mathrm{SG}}(2)}\\right\\}+\\sum_{j=3}^{\\lambda}2\\gamma^{j}{\\binom{\\lambda}{j}}e^{(j-1)\\epsilon_{m}^{\\mathrm{SG}}(j)}\\right)\\tag{7}\n$$", "proof": {"steps": [{"n": 1, "claim": "The proof is a direct application of Theorem 2 and the subsampled RDP results of Wang et al. (2019)."}, {"n": 2, "claim": "The full proof can be found in Appendix B."}]}}], "target_local_id": "T2", "topic_domain": "Others", "topic_subtopic": "Robustness, Privacy & Security", "label": "flawed", "error_step": 1, "error_class6": ["logical_gap_or_missing_case", "others", "missing_assumption"], "error_explanation": "Step 1 claims that computing the Rényi divergence for the single canonical adjacent pair $D=(0,\\ldots,0), D'=(1,0,\\ldots,0)$ suffices 'without loss of generality' for arbitrary adjacent databases, justifying this by normalizing/applying a unitary (affine) transform to the data instances. This is invalid because that affine normalization does NOT commute with the shuffling operation $S_n$: once the per-instance data are permuted by $S_n$, a fixed affine transform applied before shuffling no longer yields the canonical configuration. Hence the RDP computed for the canonical pair neither equals nor upper-bounds the RDP for general adjacent databases, so the result holds only for the specific pair $D=(0,\\ldots,0), D'=(1,0,\\ldots,0)$ and not in the generality Theorem 1 (Eq. 4) claims. The authors conceded this and withdrew the submission; no general fix exists."}
{"id": "Q4B6g_ubd39", "context": "Sequential approximation as a functional approximation problem. The goal of sequential modeling is to learn a relationship between an input sequence $\\textbf{ x }=\\ \\{\\mathbf{x}_{t}\\}$ and a corresponding output sequence $\\{y_{t}\\}$ . The time index t can be discrete or continuous. For ease of analysis, we adopt the continuous-time setting in (Li et al., 2021), where $t \\in \\mathbb R$ This is also a natural setting for irregularly sampled time series (Lechner & Hasani, 2020). We consider the d-dimensional input space $\\breve{\\mathscr{X}}=\\breve{C}_{0 . T . K}^{1}(\\mathbb{R},\\mathbb{R}^{d})$ , which is continuously differentiable with respect to time and vanishes at infinity in the sense that $\\mathbf{x}_{t}~=~\\mathbf{x}_{\\infty}$ for some $0 < T \\leq t \\leq \\infty$ , with the supremum norm $\\begin{array}{ r }{\\|\\mathbf{x}\\|_{\\mathcal{X}}: =\\operatorname*{sup}_{t\\in\\mathbb{R}}\\|\\mathbf{x}_{t}\\|_{\\infty}}\\end{array}$ Without loss of generality, we also assume $\\mathbf{x}_{t}~=~0$ for $t \\leq 0$ and the derivatives of input is bounded: $\\|\\frac{d{\\bf x}_{t}}{d t}\\|_{\\infty}\\leq K,\\forall t$\nWe assume the input and output sequences are related by a family of underlying functionals\n\n$$\ny_{t}=H_{t}(\\{\\mathbf{x}_{t}\\}),\\quad t\\in\\mathbb{R},\\tag{2}\n$$\n\nwhere $H_{t}:\\mathcal{X}\\mapsto\\mathbb{R}$ is a functional. Now, the sequential approximation problem can be formulated as the approximation of target functionals $\\{H_{t}:t\\in\\mathbb{R}\\}$ by model functionals $\\{\\widetilde{H}_{t}:t\\in\\mathbb{R}\\}$ , where $\\widetilde{H}_{t}(\\cdot)$ is contained in certain hypothesis space $\\widetilde{\\mathcal{H}},$ such as various classes of neural networks. We say a sequence of target functionals $\\{H_{t}:t\\stackrel{\\cdot}{\\in}\\mathbb{R}\\}$ can be uniformly approximated by model functionals, if for any tolerance $\\epsilon > 0$ , there exists $\\{\\widetilde{H}_{t}:t\\in\\mathbb{R}\\}\\in\\widetilde{\\mathcal{H}}$ such that\n\n$$\n\\operatorname*{sup}_{t\\in\\mathbb{R}}\\operatorname*{sup}_{\\|\\mathbf{x}\\|\\mathbf{x}\\leq 1}|H_{t}(\\mathbf{x})-\\widetilde{H}_{t}(\\mathbf{x})|\\leq\\epsilon .\\tag{3}\n$$\nThe continuous-time RNN architecture. First, we introduce the same continuous-time formulation of RNNs as (Li et al., 2021):\n\n$$\n\\frac{d h_{t}}{d t}=\\sigma(W h_{t}+U{\\bf x}_{t}),\\tag{4}\n$$\n\n$$\n\\hat{y}_{t}=c^{\\top}h_{t}.\\tag{5}\n$$\n\nHere, $\\hat{y}_{t}\\in\\mathbb R$ is the prediction, and $\\boldsymbol{h}_{t}\\in\\mathbb{R}^{m}$ denotes the hidden state. As a common practice, we set the initial hidden state $h_{0}=0 .{}^{1}$ The hyper-parameter m is also known as the hidden dimension of recurrent neural networks. For different hidden dimensions $m$ , the RNN is parameterized by trainable weights $(c,W,U)$ , where $c\\in\\mathbb{R}^{m}$ is the readout, $W\\in\\mathbb{R}^{m\\times m}$ is the recurrent kernel and $U\\in\\mathbb{R}^{m\\times d}$ is the input kernel. Obviously, the complexity of the RNN hypothesis space is characterized by the hidden dimension m. The nonlinearity arises from the activation $\\sigma(\\cdot)$ , which is a scalar nonlinear function performed element-wisely, such as tanh, hardtanh, sigmoid, ReLU and so on. The hypothesis space of RNNs with different activations are denoted by $\\mathcal{H}^{\\sigma(\\cdot)}$\nDefinition 1 Let $\\{H_{t}:t\\in\\mathbb{R}\\}$ be a sequence of functionals.\n\n1. (Linear) $H_{t}$ is linear if for any $\\lambda,\\lambda^{\\prime}\\in\\mathbb{R}a n d\\mathbf{x},\\mathbf{x}^{\\prime}\\in\\mathcal{X},H_{t}(\\lambda\\mathbf{x}+\\lambda^{\\prime}\\mathbf{x}^{\\prime})=\\lambda H_{t}(\\mathbf{x})+$ $\\lambda^{\\prime}H_{t}(\\mathbf{x}^{\\prime})$\n\n2. (Continuous) $H_{t}$ is continuous $\\begin{array}{ r }{i f\\operatorname*{sup}_{\\{\\mathbf{x}\\in\\mathcal{X},\\|\\mathbf{x}\\|_{\\mathcal{X}}\\leq 1\\}}|H_{t}(\\mathbf{x})| <\\infty .}\\end{array}$\n\n3. (Time-homogeneous) $\\{H_{t}:t\\in\\mathbb{R}\\}$ is time-homogeneous if the input-output relationship is invariant about time shift: $H_{t+\\tau}(\\mathbf{\\dot{x}}(\\tau))=H_{t}(\\mathbf{x})$ for all $t,\\tau\\in\\mathbb{R}$ ,where $\\mathbf{x}(\\tau)$ is defined by $\\mathbf{x}(\\tau)_{s}=\\mathbf{x}_{s+\\tau}$\n\n4.(Causal) $H_{t}$ is causal if it does not depend on future values of the input. That is, $i f{\\bf x},{\\bf x}^{\\prime}$ satisfy $\\mathbf{x}_{t}=\\mathbf{x}_{t}^{\\prime}f o r a n y t\\le t_{0},$ then $H_{t}(\\mathbf{x})=H_{t}(\\mathbf{x}^{\\prime})f o r a n y t\\leq t_{0}$\n\n5. (Regular) $H_{t}$ is regular if for any sequence $\\{\\mathbf{x}^{(n)}:n\\in\\mathbf{N}\\}$ such that $\\mathbf{x}_{s}^{(n)}\\to 0$ for almost every $s\\in\\mathbb{R}$ , then lim $\\mathsf{i}_{n\\to\\infty}H_{t}(\\mathbf{x}^{(n)})=0$\n4.2.1 Homogeneous Functionals Cannot Be Approximated by Tanh RNNs\n\nConsider the homogeneous functionals with degree $p > 0 \\colon$\n\n$$\nH_{t}(\\kappa\\mathbf{x})=\\kappa^{p}H_{t}(\\mathbf{x}),\\qquad\\forall\\kappa .\\tag{6}\n$$\n\n![](images/4582a475af29888dfb43612de4e67c2baec9e776310c29024de26a1299c56215.jpg)  \n(a) Exponential-sum target\n\n![](images/de13cb5bd6cfe9830e927873dcfd917097ad93a0f91d2a67eb2bde69e65ac9d0.jpg)  \n() Airy function target  \nFigure 2: Numerical illustrations for the difficulty of approximating homogeneous functionals by RNNs. The memory $\\rho_{1}$ and $\\rho_{2}$ are defined in Equation (7) and (8).\n\nAn example is the monomial functional $\\begin{array}{ r }{H_{t}(\\mathbf{x})=\\int_{0}^{\\infty}(\\mathbf{x}_{t-s}^{p})^{\\top}\\pmb{\\rho}_{s}d s}\\end{array}$ . This monomial functional is linear if and only $\\mathrm{if}\\ p=1$\n\nWe show that the sequence of homogeneous functionals with degree $p > 1$ cannot be universally approximated by nonlinear RNNs with the tanh activation.\n\nProposition 1 There exists a sequence of homogeneous functionals $\\{H_{t}:t\\in\\mathbb{R}\\}$ with an homogeneity degree $p > 1$ and an $\\epsilon > 0$ , such that there is no tanh RNN $\\{\\widetilde{H}_{t}:t\\in\\mathbb{R}\\}$ approximating $\\bar{\\{H_{t}:t\\in\\mathbb{R}\\}}$ uniformly with an approximation error smaller than e.\nTheorem 1 (Universal approximation for linear functionals by linear RNNs (Li et al., 2021)) Let $\\{H_{t}:t\\in\\mathbb{R}\\}$ be a family of linear, continuous, causal, regular and time-homogeneous functionals on X. Then, for any $\\epsilon > 0$ , there exists $\\{\\bar{H}_{t}:t\\in\\mathbb{R}\\}\\in\\tilde{\\mathcal{H}}^{\\operatorname*{linear}}$ such that\n\n$$\n\\operatorname*{sup}_{t\\in\\mathbb{R}}\\|H_{t}-\\bar{H}_{t}\\|\\equiv\\operatorname*{sup}_{t\\in\\mathbb{R}}\\operatorname*{sup}_{\\|\\mathbf{x}\\|_{\\boldsymbol{x}}\\leq 1}|H_{t}(\\mathbf{x})-\\bar{H}_{t}(\\mathbf{x})|\\leq\\epsilon .\\tag{9}\n$$\nDefine the maps $t\\mapsto H_{t}(\\mathbf{e}_{i})$ for $i=1,\\ldots,d$ , where $\\mathbf{e}_{i}$ is the constant input signal defined by $\\mathbf{e}_{i}=e_{i}\\mathbf{1}_{\\{t\\geq 0\\}}$ with $e_{i}$ as the standard basis vector in $\\mathbb{R}^{d}$ .\nTheorem 2 (Approximation rates for linear functionals by linear RNNs (Li et al., 2021)) Assume the same conditions as Theorem 1. Consider the output of constant signals\n\n$$\ny_{i}(t)=H_{t}(\\mathbf{e}_{i}),\\quad i=1,\\ldots,d .\n$$\n\nSuppose there exists constants $\\alpha\\in\\mathbb{N}_{+},\\beta,\\gamma>0$ such that $fori=1,\\ldots,d,k=1,\\ldots,\\alpha+1$ $y_{i}(t)\\in C^{(\\alpha+1)}(\\mathbb{R})$ and\n\n$$\n\\begin{array}{ l }{{e^{\\beta t}y_{i}^{(k)}(t)=o(1)\\quad a s t\\to+\\infty,}}\\\\ {{\\displaystyle\\operatorname*{sup}_{t\\geq 0}\\frac{|e^{\\beta t}y_{i}^{(k)}(t)|}{\\beta^{k}}\\leq\\gamma .}}\\end{array}\n$$\n\nThen, there exists a universal constant $C(\\alpha)$ only_depending on α, such that for any $m\\in\\mathbb{N}_{+}$ , there exists a sequence of width-m RNN functionals $\\{\\bar{H}_{t}:t\\in\\mathbb{R}\\}\\in\\bar{\\mathcal{H}}_{m}^{\\operatorname*{linear}}$ such that\n\n$$\nR(m)=\\operatorname*{sup}_{t\\in\\mathbb{R}}\\|{H}_{t}-\\bar{H}_{t}\\|\\equiv\\operatorname*{sup}_{t\\in\\mathbb{R}}\\operatorname*{sup}_{\\|{\\bf x}\\||{x}\\leq 1}|{H}_{t}({\\bf x})-\\bar{H}_{t}({\\bf x})|\\leq\\frac{C(\\alpha)\\gamma d}{\\beta m^{\\alpha}}.\\tag{10}\n$$\nProposition 2 (Universal approximation for linear RNNs by nonlinear RNNs) For any linear $R N N\\left\\{\\hat{H}_{t}:t\\in\\mathbb{R}\\right\\}$ there exists a sequence of nonlinear RNN $\\{\\widetilde{H}_{t}:t\\in\\mathbb{R}\\}$ with activations satisfying Equation (11) that can approximate the linear RNN with the following error bound:\n\n$$\n\\operatorname*{sup}_{t\\in\\mathbb{R}}\\operatorname*{sup}_{\\|\\mathbf{x}\\|_{\\mathcal{X}}\\leq 1}|\\bar{H}_{t}(\\mathbf{x})-\\widetilde{H}_{t}(\\mathbf{x})|\\leq\\frac{\\epsilon}{2}.\\tag{12}\n$$", "theorems": [{"local_id": "T1", "statement": "Theorem 3 (Universal approximation of linear functionals) Let $\\{H_{t}:t\\in\\mathbb{R}\\}$ be a family of linear, continuous, causal, regular and time-homogeneous functionals on X. Assume the activation $\\sigma(\\cdot)$ is continuous and monotonically-increasing, and satisfies the following assumptions\n\n$$\n\\sigma(0)=0,\\sigma^{\\prime}(0)=1,|\\sigma^{\\prime}(z)|\\leq M,|\\sigma^{\\prime}(z)-\\sigma^{\\prime}(0)|\\leq M|\\sigma(z)|,|\\sigma^{\\prime\\prime}(z)|\\leq M|\\sigma^{\\prime}(z)|\\tag{11}\n$$\n\nfor some constant $M > 0$ Then, for any $\\epsilon>0,$ there exists a sequence of nonlinear RNNs $\\{\\widetilde{H}_{t}$ $t\\in\\mathbb{R}\\}\\in\\widetilde{\\mathcal{H}}^{\\sigma}$ such that\n\n$$\n\\operatorname*{sup}_{t\\in\\mathbb{R}}\\|H_{t}-\\widetilde{H}_{t}\\|\\equiv\\operatorname*{sup}_{t\\in\\mathbb{R}}\\operatorname*{sup}_{\\|\\mathbf{x}\\|_{\\mathcal{X}}\\leq 1}|H_{t}(\\mathbf{x})-\\widetilde{H}_{t}(\\mathbf{x})|\\leq\\epsilon .\\tag{13}\n$$\n\nOne can verify that tanh, which is commonly-used in practice for RNNs, is an activation satisfying the conditions (11). The proof of Theorem 3 is given in Appendix A.3.", "proof": {"steps": [{"n": 1, "claim": "By the universal approximation theorem for linear functionals by linear RNNs (Theorem 1), and the universal approximation theorem for linear RNNs by tanh RNNs (Proposition 2), we have"}, {"n": 2, "claim": "$$\n\\operatorname*{sup}_{t\\in\\mathbb{R}}\\|H_{t}-\\hat{H}_{t}\\|\\equiv\\operatorname*{sup}_{t\\in\\mathbb{R}}\\operatorname*{sup}_{\\|\\mathbf{x}\\|_{\\boldsymbol{x}}\\leq 1}|H_{t}(\\mathbf{x})-\\hat{H}_{t}(\\mathbf{x})|\\leq\\frac{\\epsilon}{2},\n$$"}, {"n": 3, "claim": "$$\n\\operatorname*{sup}_{t\\in\\mathbb{R}}\\|\\hat{H}_{t}-\\widetilde{H}_{t}\\|\\equiv\\operatorname*{sup}_{t\\in\\mathbb{R}}\\operatorname*{sup}_{\\|\\mathbf{x}\\|_{x}\\leq 1}|\\hat{H}_{t}(\\mathbf{x})-\\widetilde{H}_{t}(\\mathbf{x})|\\leq\\frac{\\epsilon}{2},\n$$"}, {"n": 4, "claim": "which yields the desired estimate"}, {"n": 5, "claim": "$$\n\\operatorname*{sup}_{t\\in\\mathbb{R}}\\|H_{t}-\\widetilde{H}_{t}\\|\\equiv\\operatorname*{sup}_{t\\in\\mathbb{R}}\\operatorname*{sup}_{\\|\\mathbf{x}\\|_{\\mathcal{X}}\\leq 1}|H_{t}(\\mathbf{x})-\\widetilde{H}_{t}(\\mathbf{x})|\\leq\\epsilon .\n$$"}]}}, {"local_id": "T2", "statement": "Theorem 4 (Approximation rate of linear functionals) Assume the same conditions as in Theorem 3. Consider the output of constant signals\n\n$$\ny_{i}(t)=H_{t}(\\mathbf{e}_{i}),\\quad i=1,\\ldots,d .\\tag{14}\n$$\n\nSuppose there exist constants $\\alpha\\in\\mathbb{N}_{+},\\beta,\\gamma>0,$ such that $fori=1,\\ldots,d,k=1,\\ldots,\\alpha+1$ $y_{i}(t)\\in C^{(\\alpha+1)}(\\mathbb{R})$ , and\n\n$$\ne^{\\beta t}y_{i}^{(k)}(t)=o(1)\\quad a s t\\to+\\infty,\\tag{15}\n$$\n\n$$\n\\operatorname*{sup}_{t\\geq 0}\\frac{|e^{\\beta t}y_{i}^{(k)}(t)|}{\\beta^{k}}\\leq\\gamma .\\tag{16}\n$$\n\nThen, there exists a universal constant C(α) only depending on α, such that for any m $\\in\\mathbb{N}_{+}$ , there exists a sequence of width-m nonlinear RNN functionals $\\{\\widetilde{H}_{t}:t\\in\\mathbb{R}\\}\\in\\widetilde{\\mathcal{H}}_{m}^{\\sigma}$ such that\n\n$$\nR(m)=\\operatorname*{sup}_{t\\in\\mathbb{R}}\\|{H}_{t}-\\widetilde{H}_{t}\\|\\equiv\\operatorname*{sup}_{t\\in\\mathbb{R}}\\operatorname*{sup}_{\\|{\\mathbf x}\\|{x}\\leq 1}|{H}_{t}({\\mathbf x})-\\widetilde{H}_{t}({\\mathbf x})|\\leq\\frac{C(\\alpha)\\gamma d}{\\beta m^{\\alpha}}.\\tag{17}\n$$\n\nThe proof is included in Appendix A.4. Theorem 4 shows that, as long as the activation function $\\sigma(\\cdot)$ satisfies the conditions in Equation (11), such as the tanh activation, nonlinear RNNs can obtain the same approximation rate as linear RNNs.", "proof": {"steps": [{"n": 1, "claim": "By the approximation rate theorem for linear functionals by linear RNNs (Theorem 2), and proper rescaling from the universal approximation theorem for linear RNNs by tanh RNNs (Proposition 2), we have"}, {"n": 2, "claim": "$$\n\\operatorname*{sup}_{t\\in\\mathbb{R}}\\|H_{t}-\\bar{H}_{t}\\|\\equiv\\operatorname*{sup}_{t\\in\\mathbb{R}}\\operatorname*{sup}_{\\|\\mathbf{x}\\|_{x}\\leq 1}|H_{t}(\\mathbf{x})-\\bar{H}_{t}(\\mathbf{x})|\\leq\\frac{C(\\alpha)\\gamma d}{\\beta m^{\\alpha}},\n$$"}, {"n": 3, "claim": "$$\n\\operatorname*{sup}_{t\\in\\mathbb{R}}\\|\\bar{H}_{t}-\\widetilde{H}_{t}\\|\\equiv\\operatorname*{sup}_{t\\in\\mathbb{R}}\\operatorname*{sup}_{\\|\\mathbf{x}\\|_{x}\\leq 1}|\\bar{H}_{t}(\\mathbf{x})-\\widetilde{H}_{t}(\\mathbf{x})|\\leq\\epsilon\\frac{C(\\alpha)\\gamma d}{\\beta m^{\\alpha}},\n$$"}, {"n": 4, "claim": "which gives"}, {"n": 5, "claim": "$$\n\\operatorname*{sup}_{t\\in\\mathbb{R}}\\|{\\cal H}_{t}-\\widetilde{{\\cal H}}_{t}\\|\\equiv\\operatorname*{sup}_{t\\in\\mathbb{R}}\\operatorname*{sup}_{\\|{\\bf x}\\|_{\\cal x}\\leq 1}|{\\cal H}_{t}({\\bf x}-\\widetilde{{\\cal H}}_{t}({\\bf x}))|\\leq(1+\\epsilon)\\frac{C(\\alpha)\\gamma d}{\\beta m^{\\alpha}}.\n$$"}, {"n": 6, "claim": "Let the quantity M in Proposition 2 go to infinity, we get $\\epsilon  0 .$ This completes the proof."}]}}, {"local_id": "T3", "statement": "Consider the $d$-dimensional input space $\\mathcal{X}$ of continuously differentiable signals vanishing at infinity, with supremum norm $\\|\\mathbf{x}\\|_{\\mathcal{X}}:=\\sup_{t\\in\\mathbb{R}}\\|\\mathbf{x}_t\\|_\\infty$, and the continuous-time tanh RNN hypothesis space defined by $\\frac{dh_t}{dt}=\\sigma(Wh_t+U\\mathbf{x}_t)$, $\\hat y_t=c^\\top h_t$ (with $\\sigma=\\tanh$, $h_0=0$). Uniform approximation of target functionals $\\{H_t\\}$ by model functionals $\\{\\widetilde H_t\\}$ means: for any $\\epsilon>0$ there exists $\\{\\widetilde H_t\\}$ with $\\sup_{t\\in\\mathbb{R}}\\sup_{\\|\\mathbf{x}\\|_{\\mathcal{X}}\\le 1}|H_t(\\mathbf{x})-\\widetilde H_t(\\mathbf{x})|\\le\\epsilon$. A family $\\{H_t\\}$ is homogeneous of degree $p>0$ if $H_t(\\kappa\\mathbf{x})=\\kappa^p H_t(\\mathbf{x})$ for all $\\kappa$. Claim (Proposition 1): There exists a sequence of homogeneous functionals $\\{H_t : t\\in\\mathbb{R}\\}$ with a homogeneity degree $p>1$ and an $\\epsilon>0$, such that there is no tanh RNN $\\{\\widetilde H_t : t\\in\\mathbb{R}\\}$ approximating $\\{H_t : t\\in\\mathbb{R}\\}$ uniformly with an approximation error smaller than $\\epsilon$.", "proof": {"steps": [{"n": 1, "claim": "Assume, for contradiction, the universal approximation theorem holds for homogeneous functionals by tanh RNNs. Then for any $\\epsilon>0$ there exists a tanh RNN $(c,W,U)$ with $\\sup_t\\sup_{\\|\\mathbf{x}\\|_{\\mathcal{X}}\\le 1}|H_t(\\mathbf{x})-\\widetilde H_t(\\mathbf{x})|\\le\\epsilon$."}, {"n": 2, "claim": "Take an input $\\mathbf{x}$ with $H_t(\\mathbf{x})\\neq 0$ for $t\\ge 0$; WLOG $H_t(\\mathbf{x})>0$ for $t\\ge 0$."}, {"n": 3, "claim": "Set $\\epsilon=\\frac12|H_t(\\kappa\\mathbf{x})|=\\frac12|\\kappa^2 H_t(\\mathbf{x})|$, with the constant $\\kappa$ to be fixed later."}, {"n": 4, "claim": "By the (assumed) universal approximation theorem, there exists $\\widetilde H_t$ such that for any $t$: $|H_t(\\kappa\\mathbf{x})-\\widetilde H_t(\\kappa\\mathbf{x})|\\le\\epsilon$ and $|H_t(\\mathbf{x})-\\widetilde H_t(\\mathbf{x})|\\le\\epsilon$."}, {"n": 5, "claim": "Construct a linear RNN $\\bar H_t$ with the same parameters $(c,W,U)$ (i.e. the RNN with the tanh nonlinearity removed). Then $\\bar H_t(\\mathbf{x})=\\widetilde H_t(\\mathbf{x})\\equiv 0$ for $t\\le 0$."}, {"n": 6, "claim": "There always exists a $t>0$ such that, for $\\delta=\\frac12$: $|\\bar H_t(\\kappa\\mathbf{x})-\\widetilde H_t(\\kappa\\mathbf{x})|\\le\\delta|\\widetilde H_t(\\kappa\\mathbf{x})|$ and $|\\bar H_t(\\mathbf{x})-\\widetilde H_t(\\mathbf{x})|\\le\\delta|\\widetilde H_t(\\mathbf{x})|$; hence $\\bar H_t(\\mathbf{x})\\ge(1-\\delta)\\widetilde H_t(\\mathbf{x})$."}, {"n": 7, "claim": "Take $\\kappa=\\frac{1}{10},\\ \\delta=\\frac12$; then by continuity $(1-\\delta)(1-\\frac12\\kappa^2)>(1+\\delta)\\frac1\\kappa(\\kappa^2+\\frac12\\kappa^2)$."}, {"n": 8, "claim": "$\\bar H_t(\\mathbf{x})\\ge(1-\\delta)\\widetilde H_t(\\mathbf{x})\\ge(1-\\delta)\\big(H_t(\\mathbf{x})-\\frac12\\kappa^2 H_t(\\mathbf{x})\\big)>(1+\\delta)\\frac1\\kappa\\big(\\kappa^2 H_t(\\mathbf{x})+\\frac12\\kappa^2 H_t(\\mathbf{x})\\big)=(1+\\delta)\\frac1\\kappa\\big(H_t(\\kappa\\mathbf{x})+\\frac12\\kappa^2 H_t(\\mathbf{x})\\big)\\ge(1+\\delta)\\frac1\\kappa\\widetilde H_t(\\kappa\\mathbf{x})\\ge\\frac1\\kappa\\bar H_t(\\kappa\\mathbf{x})$."}, {"n": 9, "claim": "However, the linear RNN satisfies exactly $\\bar H_t(\\mathbf{x})=\\frac1\\kappa\\bar H_t(\\kappa\\mathbf{x})$, and $\\bar H_t(\\mathbf{x})>(1-\\delta)(1-\\frac12\\kappa^2)H_t(\\mathbf{x})>0$, contradicting the strict inequality $\\bar H_t(\\mathbf{x})>\\frac1\\kappa\\bar H_t(\\kappa\\mathbf{x})$ derived in Step 8."}]}}, {"local_id": "T4", "statement": "Theorem 5 (Inverse approximation theorem) Assume that the activation function $\\sigma(\\cdot)$ is tanh, and $\\{H_{t}:t\\in\\mathbb{R}\\}$ is a family of linear, continuous, causal, regular and time-homogeneous functionals on X. Consider the output of constant signals\n\n$$\ny_{i}(t)=H_{t}(\\mathbf{e}_{i})\\in C^{(\\alpha+1)}(\\mathbb{R}),\\quad i=1,\\dots,d,\\alpha\\in\\mathbb{N}_{+}.\\tag{18}\n$$\n\nSuppose there exists a sequence of nonlinear RNNs $\\{\\widetilde{H}_{t}:t\\in\\mathbb{R}\\}\\in\\widetilde{\\mathcal{H}}^{\\sigma}$ with an increasing hidden dimension m approximating $\\{H_{t}:t\\in\\mathbb{R}\\}$ in the following sense:\n\n$$\n\\operatorname*{lim}_{m\\to\\infty}\\operatorname*{sup}_{t\\geq 0}|\\tilde{y}_{i,m}^{(k)}(t)-y_{i}^{(k)}(t)| =0,\\quad i=1,\\ldots,d,k=1,\\ldots,\\alpha+1,\\tag{19}\n$$\n\nwhere\n\n$$\n\\widetilde{y}_{i,m}(t)=\\widetilde{H}_{t}(\\mathbf{e}_{i}),\\quad i=1,\\ldots,d .\\tag{20}\n$$\n\nMoreover, assume that the recurrent kernel matrix $W_{m}$ in the nonlinear RNN is a special Hurwitz matrix, that is, there exists a diagonal positive-definite matrix $D_{m}$ and a negative-definite matrix $N_{m}$ such that $W_{m}=D_{m}N_{m}$ . Define $n_{m}=\\mathrm{max}_{j\\in [m]}$ Re(λj), where $\\{\\lambda_{j}\\}_{j=1}^{m}$ collects all the eigenvalues of $N_{m}$ . Define $\\begin{array}{ r }{d_{m}=\\operatorname*{min}_{j\\in [m]}(D_{m})_{j j}.}\\end{array}$ Assume that the parameters in nonlinear RNNs are uniformly bounded and there exits constants $\\bar{\\beta},\\theta>0$ such that lim $\\begin{array}{ r }{\\operatorname*{sup}_{m\\infty}\\frac{n_{m}}{\\theta d_{m}}< -\\beta}\\end{array}$ , then we have\n\n$$\ne^{\\beta t}y_{i}^{(k)}(t)=o(1)a s t\\to+\\infty,\\quad i=1,\\ldots,d,k=1,\\ldots,\\alpha+1 .\\tag{21}\n$$\n\nHere, we only give the proof sketch to show the meaning and implications of Theorem 5. For the detailed proof, see Appendix A.7. Similar results can also be obtained for the sigmoid activation.", "proof": {"steps": [{"n": 1, "claim": "Without loss of generality, we give the proof for $\\alpha \\ : = \\ : 0$ and $k \\ = \\ 1$ The dynamics of hidden states of tanh RNNs are described by the following differential equations:\n\n$$\n\\begin{array}{ r }{\\frac{d h_{t}}{d t}=\\sigma(W h_{t}+U{\\bf x}_{t}),}\\\\ {h_{0}=0 .\\qquad}\\end{array}\n$$"}, {"n": 2, "claim": "Define $\\begin{array}{ r }{v_{t}=\\frac{d h_{t}}{d t}}\\end{array}$ , then we have $v_{t}=\\sigma(W h_{t}+U\\mathbf{x}_{t})$ . Notice that $\\sigma^{\\prime}(z)=1-\\sigma(z)^{2}$ holds for the tanh activation, the dynamics of $v_{t}$ satisfies\n\n$$\n\\begin{array}{ l }{\\displaystyle\\frac{d v_{t}}{d t}=\\sigma^{\\prime}({\\boldsymbol{W}}h_{t}+{\\boldsymbol{U}}\\mathbf{x}_{t})({\\boldsymbol{W}}v_{t}+{\\boldsymbol{U}}\\frac{d\\mathbf{x}_{t}}{d t})}\\\\ {\\displaystyle\\ }\\\\ {\\displaystyle=(I-\\mathrm{diag}(\\sigma({\\boldsymbol{W}}h_{t}+{\\boldsymbol{U}}\\mathbf{x}_{t}))^{2})({\\boldsymbol{W}}v_{t}+{\\boldsymbol{U}}\\frac{d\\mathbf{x}_{t}}{d t}),}\\\\ {\\displaystyle\\ }\\\\ {\\displaystyle=(I-\\mathrm{diag}(v_{t})^{2})({\\boldsymbol{W}}v_{t}+{\\boldsymbol{U}}\\frac{d\\mathbf{x}_{t}}{d t}),}\\\\ {\\displaystyle v_{0}=0 .}\\end{array}\n$$"}, {"n": 3, "claim": "We also consider a special class of constant inputs $\\pmb{x}_{t}=x_{0}1_{\\{t\\geq 0\\}},x_{0}\\in\\mathbb{R}^{d}.$ Then for any $t > 0$\n\n$$\n\\begin{array}{ c }{\\frac{d v_{t}}{d t}=(I-\\mathrm{diag}(v_{t})^{2})W v_{t},}\\\\ {v_{0}=\\sigma(W x_{0}).}\\end{array}\n$$"}, {"n": 4, "claim": "The proof is based on the construction of the Lyapunov function. Define\n\n$$\nV(v)=\\sum_{i=1}^{m}\\sum_{j=1}^{\\infty}\\frac{1}{2 j D_{i i}}v_{i}^{2 j},\n$$\n\nwhere $v_{i}$ is the i-th coordinate of v. It is straightforward to verify that $V(v)\\geq 0$ , and $V(v)=0$ if and only if $v = 0$"}, {"n": 5, "claim": "Since the special Hurwitz matrix W can be decomposed into $W = D N .$ ,where D is a positive diagonal matrix and N is a negative definite matrix. We can derive the global stability of $v_{t}$ as\n\nfollows:\n\n$$\n\\begin{array}{ l }{\\displaystyle\\frac{d V(v_{t})}{d t}=\\sum_{i=1}^{m}\\sum_{j=1}^{\\infty}\\frac{1}{D_{i i}}v_{i,t}^{2 j-1}\\frac{d v_{i,t}}{d t}}\\\\ {\\displaystyle\\quad\\quad=v_{t}^{\\top}D^{-1}\\bigg(\\sum_{j=1}^{\\infty}\\mathrm{diag}(v_{t})^{2 j-2}\\bigg)\\frac{d v_{t}}{d t}}\\\\ {\\displaystyle\\quad=v_{t}^{\\top}D^{-1}(I-\\mathrm{diag}(v_{t})^{2})^{-1}(I-\\mathrm{diag}(v_{t})^{2})W v_{t}}\\\\ {\\displaystyle\\quad=v_{t}^{\\top}D^{-1}W v_{t}}\\\\ {\\displaystyle\\quad=v_{t}^{\\top}N v_{t}\\leq 0 .}\\end{array}\n$$"}, {"n": 6, "claim": "For any bounded input, we get $\\left\\|v_{0}\\right\\|\\le 1-\\iota$ for some $\\iota > 0$ Therefore, we further have\n\n$$\n\\frac{d V(v_{t,m})}{d t}\\leq-\\alpha_{m}V(v_{t,m}),\n$$\n\nwhere m is the hidden dimension, $\\begin{array}{ r }{\\alpha_{m}=\\frac{n_{m}}{\\theta d_{m}}}\\end{array}$ with $\\textstyle\\theta=\\sum_{j=1}^{\\infty}{\\frac{1}{2 j}}(1-\\iota)^{2 j-2}$ as a constant related to t. Define $\\beta=\\mathrm{inf}_{m\\in\\mathbb{N}_{+}}\\alpha_{m}$ ."}, {"n": 7, "claim": "Based on the Lyapunov exponential stability theorem, we establish the exponential stability of $v_{t,m},$ i.e. $e^{\\alpha_{m}t}v_{t,m}=o(1)$ . Since $\\tilde{y}_{i,m}^{(1)}(t)=c^{\\top}v_{t,m}$ , we get $e^{\\beta t}\\tilde{y}_{i,m}^{(1)}(t)=$ o(1) for all $m\\in\\mathbb{N}_{+}$ , which gives $e^{\\beta t}y_{i}^{(1)}(t)=o(1)$ for $i=1,\\cdots,d.$"}]}}], "target_local_id": "T3", "topic_domain": "Deep Learning Theory", "topic_subtopic": "Representation & Expressivity", "label": "flawed", "error_step": 3, "error_class6": ["circular_or_unproven", "logical_gap_or_missing_case", "missing_assumption"], "error_explanation": "Step 3 sets $\\epsilon = \\tfrac12|H_t(\\kappa\\mathbf{x})|$ with $t$ still unbound -- no $t$ has been fixed anywhere in Steps 1-2. Step 4 then invokes the universal-approximation assumption as though $\\epsilon$ were already a well-defined positive real, but $H_t(\\mathbf{x})$ is generically not constant in $t$, so $\\epsilon$ cannot actually be pinned down until $t$ is known -- yet $t$ is only fixed in step 6, *after* the RNN built from this ill-defined $\\epsilon$ already exists. This is a genuine quantifier/scope circularity ($\\epsilon$ must depend on the $t$ that hasn't been chosen yet), not mere terseness. The authors' own withdrawal note confirms this exact issue: 'the selection of $\\epsilon$ should not depend on $\\kappa$ and $t_0$' -- precisely describing step 3's construction. Step 6 (the previously-labeled error step) only inherits this problem; it is not itself the first failure."}
{"id": "SJTB5GZCb", "context": "We consider a physical system specified by a state variable s and a parameter variable θ. The system is also influenced by an external input $\\mathrm{v},\\mathrm{e}.\\mathrm{g}.$ in the supervised setting $\\mathrm{~v~}=\\mathrm{~(x,y)~}$ where y is the target that the system wants to predict given x. Let $s\\mapsto\\mu(\\theta,\\mathrm{v},s)$ be a vector field in the state space and $C(\\theta,\\mathrm{v},s)$ a cost function. We assume that the state dynamics induced by µ converges to a stable fixed point $s_{\\theta,\\mathrm{v}}^{0},$ corresponding to the \"prediction\" from the model and characterized by\n\n$$\n\\begin{array}{ r }{\\mu\\left(\\boldsymbol{\\theta},\\mathrm{v},\\boldsymbol{s}_{\\boldsymbol{\\theta},\\mathrm{v}}^{0}\\right)=0 .}\\end{array}\\tag{1}\n$$\n\nThe objective function that we want to optimize is the cost at the fixed point\n\n$$\n\\begin{array}{ r }{J(\\theta,\\mathrm{v}): =C\\left(\\theta,\\mathrm{v},s_{\\theta,\\mathrm{v}}^{0}\\right).}\\end{array}\\tag{2}\n$$\n\nNote the distinction between J and $C { : }$ the cost function is defined for any state s whereas the objective function is the cost at the fixed point. Before defining ν we first introduce the \"extended vector field\"\n\n$$\n\\mu^{\\beta}(\\boldsymbol{\\theta},{\\mathrm v},s): =\\mu(\\boldsymbol{\\theta},{\\mathrm v},s)-\\beta\\ \\frac{\\partial C}{\\partial s}(\\boldsymbol{\\theta},{\\mathrm v},s),\\tag{3}\n$$\n\nwhere $\\beta$ is a real-valued scalar called \"influence parameter\". Then we extend the notion of fixed point for any value of $\\beta$ For any $\\beta$ we define the β-fixed point $s_{\\theta,\\mathrm{v}}^{\\beta}$ such that\n\n$$\n\\mu^{\\beta}\\left(\\theta,\\mathrm{v},s_{\\theta,\\mathrm{v}}^{\\beta}\\right)=0 .\\tag{4}\n$$\n\nUnder mild regularity conditions on µ and $C .$ , the implicit function theorem ensures that, for a fixed data sample v, the funtion $(\\theta,\\beta)\\mapsto s_{\\theta,\\mathrm{v}}^{\\beta}$ is differentiable.\n\nNow for every θ and v we define the vector $\\nu(\\theta,\\mathrm{v})$ in the parameter space as\n\n$$\n\\nu(\\theta,{\\bf v}): = -\\frac{\\partial C}{\\partial\\theta}\\left(\\theta,{\\bf v},s_{\\theta,\\mathrm{v}}^{0}\\right)+\\frac{\\partial\\mu}{\\partial\\theta}\\left(\\theta,{\\bf v},s_{\\theta,\\mathrm{v}}^{0}\\right)\\cdot\\left.\\frac{\\partial s_{\\theta,\\mathrm{v}}^{\\beta}}{\\beta}\\right|_{\\beta=0}.\\tag{5}\n$$\n\nIn this section, we attempt to understand why the proposed algorithm is experimentally found to optimize the objective function J (Figure 3). We say that $W$ is a \"good parameter\" if:\n\n1. for any initial state for the neurons, the state dynamics $\\begin{array}{ r }{\\frac{d h}{d t}=\\mu\\left(W,\\mathbf{x},h\\right)}\\end{array}$ converges to a fixed point - a condition required for the algorithm to be correctly defined,\n\n2. the scalar product $\\frac{\\partial J}{\\partial W}\\cdot\\nu(W)$ at the point $W$ is negative - a desirable condition for the algorithm to optimize the objective function J.\n\nExperiments show that the dynamics of h (almost) always converges to a fixed point and that J consistently decreases (Figure 3). This means that, during training, as the parameter W follows the update rule $\\Delta W\\propto\\nu(W)$ , all values of W that the network takes are \"good parameters\". In this section we attempt to explain why.\n\nTheorem 3. The gradient of J can be expressed in terms of µ and C as\n\n$$\n\\frac{\\partial J}{\\partial W}= -\\frac{\\partial C}{\\partial h}\\cdot\\left(\\frac{\\partial\\mu}{\\partial h}\\right)^{-1}\\cdot\\frac{\\partial\\mu}{\\partial W}.\\tag{6}\n$$\n\nSimilarly, the vector field ν (Eq. 7) is equal to\n\n$$\n\\nu(W)=\\frac{\\partial C}{\\partial h}\\cdot\\left(\\left(\\frac{\\partial\\mu}{\\partial h}\\right)^{T}\\right)^{-1}\\cdot\\frac{\\partial\\mu}{\\partial W}.\\tag{8}\n$$\n\nIn these expressions, all terms are evaluated at the fixed point $h^{0}$\n\nTheorem 3 is proved in Appendix A. Note that the formulas show that $\\nu(W)$ is related to $\\frac{\\partial J}{\\partial W}$ and that the angle between these two vectors is directly linked to the \"degree of symmetry\" of the Jacobian of $\\mu .$ .\n\nAn important particular case is the setting of Equilibrium Propagation (Scellier & Bengio, 2017), in which the vector field $\\mu$ is a gradient field $\\begin{array}{ r }{\\mu= -\\frac{\\partial E}{\\partial h}}\\end{array}$ , meaning that it derives from an energy function $E .$ In this case the Jacobian of $\\mu$ is symmetric since it is the Hessian of E.\n\nWe argue that the set of \"good parameters\" covers a large proportion of the weight space and that they contain the matrices W that present a form of symmetry or \"alignment\". In the next subsection, we discuss how this form of symmetry may arise from the learning procedure itself.\n\nExperiments show that a form of symmetry between feedforward and feedback weights arises from the learning procedure itself (Figure 3). Although the causes for this phenomenon aren't understood very well yet, it is worth pointing out that similar observations have been made in previous work and different settings.", "theorems": [{"local_id": "T1", "statement": "Proposition 1. Let $s^{0}$ be a stable fixed point of the vector field $s\\mapsto\\mu(s)$ , in the sense that $\\left(s-s^{0}\\right)$ $\\mu\\left(s\\right)<0$ for s in the neighborhood of $s^{0}$ (i.e. the vector field at s points towards $s^{0})$ Then the Jacobian of µ at the fixed point $\\frac{\\partial\\mu}{\\partial s}\\left(s^{0}\\right)$ is negative, in the sense that\n\n$$\n\\forall v,\\qquad v\\cdot\\frac{\\partial\\mu}{\\partial s}\\left(s^{0}\\right)\\cdot v\\leq 0 .\\tag{9}\n$$", "proof": {"steps": [{"n": 1, "claim": "Let v be a vector in the state space, $\\alpha > 0$ a positive scalar and let $s: =s^{0}+$ αv."}, {"n": 2, "claim": "For α small enough, the vector s is in the region of stability of $\\bar{s}^{0}$ ."}, {"n": 3, "claim": "Using a first order Taylor expansion and the fixed point condition $\\mu\\left(s^{0}\\right)=\\check{0}$ we get\n\n$$\n0>{\\bigl(}s-s^{0}{\\bigr)}\\cdot\\mu(s)\\tag{10}\n$$"}, {"n": 4, "claim": "$$\n=\\alpha v\\cdot\\mu\\left(s^{0}+\\alpha v\\right)\\tag{11}\n$$"}, {"n": 5, "claim": "$$\n=\\alpha v\\cdot{\\frac{\\partial\\mu}{\\partial s}}\\left(s^{0}\\right)\\cdot\\alpha v+o\\left(\\alpha^{2}\\right)\\tag{12}\n$$"}, {"n": 6, "claim": "as $\\alpha  0$ Hence the result."}]}}, {"local_id": "T2", "statement": "Lemma 1. Let $s\\mapsto\\mu^{\\beta}(\\theta,s)$ be a differentiable vector field, and $s_{\\theta}^{\\beta}$ a fixed point characterized by\n\n$$\n\\mu^{\\beta}\\left(\\theta,s_{\\theta}^{\\beta}\\right)=0 .\\tag{13}\n$$\n\nThen the partial derivatives of the fixed point are given by\n\n$$\n\\frac{\\partial s_{\\theta}^{\\beta}}{\\partial\\theta}= -\\left(\\frac{\\partial\\mu^{\\beta}}{\\partial s}\\left(\\theta,s_{\\theta}^{\\beta}\\right)\\right)^{-1}\\cdot\\frac{\\partial\\mu^{\\beta}}{\\partial\\theta}\\left(\\theta,s_{\\theta}^{\\beta}\\right)\\tag{14}\n$$\n\nand\n\n$$\n\\frac{\\partial s_{\\theta}^{\\beta}}{\\partial\\beta}= -\\left(\\frac{\\partial\\mu^{\\beta}}{\\partial s}\\left(\\theta,s_{\\theta}^{\\beta}\\right)\\right)^{-1}\\cdot\\frac{\\partial\\mu^{\\beta}}{\\partial\\beta}\\left(\\theta,s_{\\theta}^{\\beta}\\right).\\tag{15}\n$$", "proof": {"steps": [{"n": 1, "claim": "First we differentiate the fixed point equation Eq. 13 with respect to θ:\n\n$$\n{\\frac{d}{d\\theta}}(13)\\Rightarrow{\\frac{\\partial\\mu^{\\beta}}{\\partial\\theta}}\\left(\\theta,s_{\\theta}^{\\beta}\\right)+{\\frac{\\partial\\mu^{\\beta}}{\\partial s}}\\left(\\theta,s_{\\theta}^{\\beta}\\right)\\cdot{\\frac{\\partial s_{\\theta}^{\\beta}}{\\partial\\theta}}=0.\\tag{16}\n$$"}, {"n": 2, "claim": "Rearranging the terms we get Eq. 14."}, {"n": 3, "claim": "Similarly we differentiate the fixed point equation Eq. 13 with respect to $\\beta \\colon$\n\n$$\n{\\frac{d}{d\\beta}}(13)\\Rightarrow{\\frac{\\partial\\mu^{\\beta}}{\\partial\\beta}}\\left(\\theta,s_{\\theta}^{\\beta}\\right)+{\\frac{\\partial\\mu^{\\beta}}{\\partial s}}\\left(\\theta,s_{\\theta}^{\\beta}\\right)\\cdot{\\frac{\\partial s_{\\theta}^{\\beta}}{\\partial\\beta}}=0.\\tag{17}\n$$"}, {"n": 4, "claim": "Rearranging the terms we get Eq. 15."}]}}, {"local_id": "T3", "statement": "Consider a network of continuous-valued neurons with state $s=(s_1,s_2,\\dots)$, firing rate $\\rho(s_i)$, and untied (non-symmetric) synaptic weights $W_{ij}$, whose vector field is $\\mu_i(W,s):=\\sum_j W_{ij}\\rho(s_j)-s_i$ with dynamics $\\frac{ds}{dt}=\\mu(W,s)$. In the supervised setting the units split into clamped inputs $\\mathbf{x}$ and dynamic units $h$; assuming the free dynamics $\\frac{dh}{dt}=\\mu(W,\\mathbf{x},h)$ converges to a free fixed point $h^0$, the objective is the cost at that fixed point, $J:=C(h^0,\\mathrm{y})$ with $C(h,\\mathrm{y}):=\\tfrac12\\|\\mathrm{y}-h_0\\|^2$. With the extended vector field $\\mu^\\beta:=\\mu-\\beta\\frac{\\partial C}{\\partial h}$, weakly-clamped fixed point $h^\\beta$, and $\\nu(W):=\\frac{\\partial\\mu}{\\partial W}(W,\\mathbf{x},h^0)\\cdot\\left.\\frac{\\partial h^\\beta}{\\partial\\beta}\\right|_{\\beta=0}$ (Eq. 7), the paper claims (Theorem 3): the gradient of $J$ can be expressed as $\\frac{\\partial J}{\\partial W}=-\\frac{\\partial C}{\\partial h}\\cdot\\left(\\frac{\\partial\\mu}{\\partial h}\\right)^{-1}\\cdot\\frac{\\partial\\mu}{\\partial W}$ (Eq. 6) and $\\nu(W)=\\frac{\\partial C}{\\partial h}\\cdot\\left(\\left(\\frac{\\partial\\mu}{\\partial h}\\right)^{T}\\right)^{-1}\\cdot\\frac{\\partial\\mu}{\\partial W}$ (Eq. 8), all terms evaluated at $h^0$; and, on this basis, that the STDP-compatible update $\\Delta W\\propto\\nu(W)$ optimizes $J$, because during training every $W$ visited is a 'good parameter' — meaning the dynamics converges AND $\\frac{\\partial J}{\\partial W}\\cdot\\nu(W)<0$ — and the set of good parameters covers a large proportion of weight space.", "proof": {"steps": [{"n": 1, "claim": "$\\frac{\\partial J}{\\partial\\theta}=\\frac{\\partial C}{\\partial\\theta}+\\frac{\\partial C}{\\partial s}\\cdot\\frac{\\partial s_\\theta^0}{\\partial\\theta}$ (Eq. 18), for $J(\\theta,\\mathrm{v})=C(\\theta,\\mathrm{v},s_\\theta^0)$."}, {"n": 2, "claim": "$\\frac{\\partial s_\\theta^\\beta}{\\partial\\theta}=-\\left(\\frac{\\partial\\mu^\\beta}{\\partial s}(\\theta,s_\\theta^\\beta)\\right)^{-1}\\cdot\\frac{\\partial\\mu^\\beta}{\\partial\\theta}(\\theta,s_\\theta^\\beta)$ and $\\frac{\\partial s_\\theta^\\beta}{\\partial\\beta}=-\\left(\\frac{\\partial\\mu^\\beta}{\\partial s}(\\theta,s_\\theta^\\beta)\\right)^{-1}\\cdot\\frac{\\partial\\mu^\\beta}{\\partial\\beta}(\\theta,s_\\theta^\\beta)$ (Lemma 1, Eqs. 14–15)."}, {"n": 3, "claim": "$\\frac{\\partial J}{\\partial\\theta}=\\frac{\\partial C}{\\partial\\theta}-\\frac{\\partial C}{\\partial s}\\cdot\\left(\\frac{\\partial\\mu}{\\partial s}\\right)^{-1}\\cdot\\frac{\\partial\\mu}{\\partial\\theta}$ (Eq. 19), specializing to Eq. 6 (with $\\frac{\\partial C}{\\partial W}=0$) $\\frac{\\partial J}{\\partial W}=-\\frac{\\partial C}{\\partial h}\\cdot\\left(\\frac{\\partial\\mu}{\\partial h}\\right)^{-1}\\cdot\\frac{\\partial\\mu}{\\partial W}$, evaluated at $h^0$."}, {"n": 4, "claim": "$\\nu=-\\frac{\\partial C}{\\partial\\theta}+\\frac{\\partial C}{\\partial s}\\cdot\\left(\\left(\\frac{\\partial\\mu}{\\partial s}\\right)^{T}\\right)^{-1}\\cdot\\frac{\\partial\\mu}{\\partial\\theta}$ (Eq. 20), specializing to Eq. 8 $\\nu(W)=\\frac{\\partial C}{\\partial h}\\cdot\\left(\\left(\\frac{\\partial\\mu}{\\partial h}\\right)^{T}\\right)^{-1}\\cdot\\frac{\\partial\\mu}{\\partial W}$, evaluated at $h^0$."}, {"n": 5, "claim": "Comparing Eq. 6 and Eq. 8, $\\nu(W)=-\\frac{\\partial J}{\\partial W}$ holds when the Jacobian $\\frac{\\partial\\mu}{\\partial h}$ is symmetric (e.g. gradient/energy-based dynamics with tied weights); more generally the angle between $\\nu(W)$ and $\\frac{\\partial J}{\\partial W}$ is governed by the 'degree of symmetry' of $\\frac{\\partial\\mu}{\\partial h}$."}, {"n": 6, "claim": "Experiments (Figure 3) show that the dynamics of $h$ almost always converges to a fixed point and that $J$ consistently decreases during training."}, {"n": 7, "claim": "Therefore, during training, as $W$ follows the update rule $\\Delta W\\propto\\nu(W)$, all values of $W$ that the network takes are 'good parameters' (the dynamics converges AND $\\frac{\\partial J}{\\partial W}\\cdot\\nu(W)<0$), and the set of good parameters covers a large proportion of the weight space — those matrices $W$ that present a form of symmetry or 'alignment' — with this symmetry arising from the learning procedure itself."}, {"n": 8, "claim": "Hence the STDP-compatible update $\\Delta W\\propto\\nu(W)$ optimizes the objective function $J$."}]}}, {"local_id": "T4", "statement": "Theorem 4. The gradient of the objective function is equal to\n\n$$\n\\frac{\\partial J}{\\partial\\theta}=\\frac{\\partial C}{\\partial\\theta}-\\frac{\\partial C}{\\partial s}\\cdot\\left(\\frac{\\partial\\mu}{\\partial s}\\right)^{-1}\\cdot\\frac{\\partial\\mu}{\\partial\\theta}\\tag{19}\n$$\n\nand the vector field v is equal to\n\n$$\n\\nu= -\\frac{\\partial C}{\\partial\\theta}+\\frac{\\partial C}{\\partial s}\\cdot\\left(\\left(\\frac{\\partial\\boldsymbol{\\mu}}{\\partial s}\\right)^{T}\\right)^{-1}\\cdot\\frac{\\partial\\boldsymbol{\\mu}}{\\partial\\theta}.\\tag{20}\n$$\n\nAll the factors on the right-hand sides of Eq. 19 and Eq. 20 are evaluated at the fixed point $s_{\\theta}^{0}.$", "proof": {"steps": [{"n": 1, "claim": "Let us compute the gradient of the objective function with respect to θ."}, {"n": 2, "claim": "Using the chain rule of differentiation we get\n\n$$\n\\frac{\\partial J}{\\partial\\theta}=\\frac{\\partial C}{\\partial\\theta}+\\frac{\\partial C}{\\partial s}\\cdot\\frac{\\partial s_{\\theta}^{0}}{\\partial\\theta}.\\tag{18}\n$$"}, {"n": 3, "claim": "Hence Eq. 19 follows from Eq. 14 evaluated at $\\beta = 0$ ."}, {"n": 4, "claim": "Similarly, the expression for the vector field ν (Eq. 20) follows from its definition (Eq. 5), the identity Eq. 15 evaluated at $\\beta = 0$ and the fact that $\\begin{array}{ r }{\\frac{\\partial\\mu^{\\beta}}{\\partial\\beta}= -\\frac{\\partial C}{\\partial s}}\\end{array}$"}]}}, {"local_id": "T5", "statement": "Proposition 2. Let $s_{\\theta}^{\\beta}$ be a stable fixed point of the extended vector field $\\begin{array}{ r }{\\mu^{\\beta}=\\mu-\\beta\\frac{\\partial C}{\\partial s}}\\end{array}$ . Then the derivative of the function\n\n$$\n\\beta\\mapsto C\\left(\\theta,s_{\\theta}^{\\beta}\\right)\\tag{21}\n$$\n\nat $\\beta = 0$ is non-positive.", "proof": {"steps": [{"n": 1, "claim": "Multiplying both sides of Eq. 17 on the left by $\\mathrm{~-~}{\\left(\\frac{\\partial s_{\\theta}^{\\beta}}{\\partial\\beta}\\right)}^{T}$ and rearranging the terms, we get\n\n$$\n-\\left(\\frac{\\partial s_{\\theta}^{\\beta}}{\\partial\\beta}\\right)^{T}\\cdot\\frac{\\partial\\mu^{\\beta}}{\\partial\\beta}=\\left(\\frac{\\partial s_{\\theta}^{\\beta}}{\\partial\\beta}\\right)^{T}\\cdot\\frac{\\partial\\mu^{\\beta}}{\\partial s}\\cdot\\frac{\\partial s_{\\theta}^{\\beta}}{\\partial\\beta}\\leq 0 .\\tag{22}\n$$"}, {"n": 2, "claim": "The inequality holds because $\\frac{\\partial\\mu^{\\beta}}{\\partial s}\\left(\\theta,s_{\\theta}^{\\beta}\\right)$ is negative as $s_{\\theta}^{\\beta}$ is a stable fixed point of $\\mu^{\\beta}$ (Eq. 9)."}, {"n": 3, "claim": "Since $\\begin{array}{ r }{\\frac{\\partial\\mu^{\\beta}}{\\partial\\beta}= -\\frac{\\partial C}{\\partial s}}\\end{array}$ , the left-hand side, for $\\beta = 0$ , represents the derivative of\n\n$$\n\\beta\\mapsto C\\left(\\theta,s_{\\theta}^{\\beta}\\right).\\tag{23}\n$$"}]}}], "target_local_id": "T3", "topic_domain": "Deep Learning Theory", "topic_subtopic": "Training Dynamics & Implicit Bias", "label": "flawed", "error_step": 7, "error_class6": ["circular_or_unproven", "logical_gap_or_missing_case", "missing_assumption"], "error_explanation": "This step asserts, without proof, that the untied-weight learning procedure automatically drives $W$ into (and keeps it in) the symmetric/aligned regime in which $\\frac{\\partial J}{\\partial W}\\cdot\\nu(W)<0$, and hence that every $W$ visited during training is a 'good parameter'. But Theorem 1 only establishes that $\\nu(W)=-\\frac{\\partial J}{\\partial W}$ when $\\frac{\\partial\\mu}{\\partial h}$ is symmetric (Eq. 17 vs. Eq. 18 differ by the transpose in the inverted Jacobian). For general untied weights the Jacobian is not symmetric, so alignment does not follow from the identity. The pivotal facts a learning guarantee needs — (a) that the untied-weight dynamics $\\frac{dh}{dt}=\\mu(W,\\mathbf{x},h)$ converges to a fixed point for the parameters visited during training, and (b) that the update $\\Delta W\\propto\\nu(W)$ provably converges $W$ to the symmetric-weight condition — are never proved; they are only asserted from Figure 3 and analogy. The argument is therefore incomplete/circular: it presupposes the very alignment/convergence it must establish, so the central claim that the algorithm 'optimizes J' is not proven. (The authors concede this in the review response.)"}
{"id": "SkxpDT4YvS", "context": "We consider the Markov decision processes $M=(S,\\mathcal{A},\\mathcal{P},\\mathcal{R},\\rho_{0},\\gamma)$ , where S is state space, $\\mathcal{A}$ is action space; At time t, the agent is in a state $S_{t}\\in S$ and takes an action $A_{t}\\in{\\mathcal{A}}.$ then it receives a feedback $R_{t+1};P_{\\textrm{ e s ' }}^{a}=P(s^{'}|s,a)\\in\\mathcal{P}$ is the probability of the state transition from s to s' under taking $a\\in{\\mathcal{A}};$ The bounded reward function $\\mathcal{R}:\\mathcal{S}\\times\\mathcal{A}\\to [-R,R],\\mathcal{R}_{s}^{a}\\mapsto\\mathbb{E}[R_{t+1}|S_{t}=s,A_{t}=$ $a\\};\\rho_{0}:\\mathcal{S}\\to [0,1]$ is the initial state distribution and $\\gamma\\in(0,1)$ is discounted factor. Policy $\\pi_{\\boldsymbol{\\theta}}(a|\\boldsymbol{s})$ is a probability distribution on $\\boldsymbol{s}\\times\\boldsymbol{A}$ with the parameter $\\theta\\in\\mathbb{R}^{p}$ . Let $\\tau=\\{s_{t},a_{t},r_{t+1}\\}_{t=0}^{H_{\\tau}}$ be a trajectory, where $s_{0}\\sim\\rho_{0}(s_{0}),a_{t}\\sim\\pi_{\\theta}(\\cdot|s_{t}),r_{t+1}=\\mathcal{R}(s_{t},a_{t}),s_{t+1}\\sim P(\\cdot|s_{t},a_{t})$ , and $H_{\\tau}$ is the finite horizon of 7. The expected return $J(\\pi_{\\theta})$ is defined as:\n\n$$\nJ(\\theta)\\stackrel{\\mathrm{def}}{=}J(\\pi_{\\theta})=\\int_{\\tau}P(\\tau|\\theta)R(\\tau)\\mathrm{d}\\tau=\\mathbb{E}_{\\tau\\sim\\pi_{\\theta}}[R(\\tau)],\\tag{3}\n$$\n\nwhere $\\begin{array}{ r }{P(\\tau|\\theta)=\\rho_{0}(s_{0})\\prod_{t=0}^{H_{\\tau}}P(s_{t+1}|s_{t},a_{t})\\pi_{\\theta}(a_{t}|s_{t})}\\end{array}$ is the probability of generating $\\tau,R(\\tau)=$ $\\scriptstyle\\sum_{t=0}^{H_{\\tau}}\\gamma^{t}r_{t+1}$ is the accumulated discounted return.\n\nLet ${\\mathcal{I}}(\\theta)= -J(\\theta)$ , the central problem of policy-based RL is to solve the problem:\n\n$$\n\\theta^{*}=\\arg\\operatorname*{max}_{\\theta}J(\\theta)\\Longleftrightarrow\\theta^{*}=\\arg\\operatorname*{min}_{\\theta}\\mathcal{I}(\\theta).\\tag{4}\n$$\n\nComputing the $\\nabla J(\\theta)$ analytically, we have\n\n$$\n\\nabla J(\\theta)=\\mathbb{E}_{\\tau\\sim\\pi_{\\theta}}[\\sum_{t=0}^{H_{\\tau}}\\nabla_{\\theta}\\log\\pi_{\\theta}(a_{t}|s_{t})R(\\tau)].\\tag{5}\n$$\n\nFor any trajectory $\\tau,$ let $\\begin{array}{ r }{g(\\tau|\\theta)=\\sum_{t=0}^{H_{\\tau}}\\nabla_{\\theta}\\log\\pi_{\\theta}(a_{t}|s_{t})R(\\tau)}\\end{array}$ , which is an unbiased estimator of $\\nabla J(\\theta)$ . Vanilla policy gradient (vPG) is a straightforward way to solve problem (4):\n\n$$\n\\theta\\theta+\\alpha g(\\tau|\\theta),\n$$\n\nwhere α is step size.\n\nAssumption 2. (Sutton et al., 2000; Papini et al., 2018) For each pair $(s,a)$ , any $\\theta\\in\\mathbb{R}^{p}$ , and all components i, j, there exists positive constants G, F s.t.,\n\n$$\n|\\nabla_{\\theta_{i}}\\log\\pi_{\\theta}(a|s)|\\leq G,\\quad|\\frac{\\partial^{2}}{\\partial\\theta_{i}\\partial\\theta_{j}}\\log\\pi_{\\theta}(a|s)|\\leq F .\\tag{6}\n$$\n\nAccording to the Lemma B.2 of (Papini et al., 2018), Assumption 2 implies $\\nabla J(\\theta)$ is L-Lipschiz, i.e. $\\lvert\\lvert\\nabla J(\\theta_{1})-\\nabla J(\\theta_{2})\\rvert\\rvert\\leq L\\lvert\\lvert\\theta_{1}-\\bar{\\theta}_{2}\\rvert\\rvert$ , where\n\n$$\nL=R H(H G^{2}+F)/(1-\\gamma),\\tag{7}\n$$\n\nBesides, Assumption 2 implies the following property of the policy gradient estimator.\n\nLemma 3 (Properties of stochastic differential estimators (Shen et al., 2019)). Under Assumption 2, for any policy πθ and $\\tau\\sim\\pi_{\\theta;}$ ,we have\n\n$$\n\\|g(\\tau|\\theta)-\\nabla J(\\theta)\\|^{2}\\leq\\frac{G^{2}R^{2}}{(1-\\gamma)^{4}}\\stackrel{\\mathrm{def}}{=}\\sigma^{2}.\\tag{8}\n$$\n\nDefinition 2 (Proximal Operator (Moreau, 1965)). T is a function defined on a closed convex $x,$ and $\\alpha>0 .{\\ M}_{\\alpha,T}^{\\psi}(z)$ is the proximal operator of T, which is defined as:\n\n$$\n\\mathcal{M}_{\\alpha,T}^{\\psi}(z)=\\mathop{\\mathrm{arg}}_{x\\in\\mathcal{X}}\\operatorname*{min}\\{T(x)+\\frac{1}{\\alpha}D_{\\psi}(x,z)\\},\\tag{9}\n$$\n\nwhere $\\psi(x)$ is a continuously-differentiable, ζ-strictly convex function: $\\langle x-y,\\nabla\\psi(x)-\\nabla\\psi(y)\\rangle\\geq$ $\\zeta\\|x-y\\|^{2},\\zeta>0,D_{\\psi}$ is Bregman distance: $D_{\\psi}(x,y)=\\psi(x)-\\psi(y)-\\langle\\nabla\\psi(y),x-y\\rangle,\\forall x,y\\in\\mathcal{X}.$\n\nStochastic Mirror Descent The SMD solves (10) by generating an iterative solution as follows,\n\n$$\n\\boldsymbol{\\theta}_{t+1}=\\mathcal{M}_{\\alpha_{t},\\ell(\\theta)}^{\\psi}(\\boldsymbol{\\theta}_{t})=\\arg\\operatorname*{min}_{\\boldsymbol{\\theta}\\in D_{\\boldsymbol{\\theta}}}\\{\\langle\\boldsymbol{g}_{t},\\boldsymbol{\\theta}\\rangle+\\frac{1}{\\alpha_{t}}D_{\\psi}(\\boldsymbol{\\theta},\\boldsymbol{\\theta}_{t})\\},\\tag{11}\n$$\n\nwhere $\\alpha_{t}>0$ is step-size, $\\ell(\\theta)=\\left. g_{t},\\theta\\right.$ is the first-order approximation of f (θ) at $\\theta_{t},g_{t}=g(\\theta_{t},\\xi_{t})$ is stochastic subgradient such that $g(\\theta_{t}^{'})=\\mathbb{E}[g(\\theta_{t},\\xi_{t})]\\stackrel{\\cdot}{\\in}\\partial f(\\theta)|_{\\theta=\\theta_{t}},\\{\\bar{\\xi_{t}}\\}_{t\\geq 0}$ represents a draw from distribution $\\mu,$ and $\\partial f(\\theta)=\\{g|f(\\theta)-f(\\omega)\\leq g^{T}(\\theta-\\omega),\\forall\\omega\\in\\mathbf{dom}(\\bar{f})\\}$\n\nIf we choose $\\psi({x})=\\frac{1}{2}\\|{x}\\|_{2}^{2}$ ,which implies $D_{\\psi}(x,y)=\\frac{1}{2}\\|x-y\\|_{2}^{2}$ , since then iteration (11) is the proximal gradient (Rockafellar, 1976) view of SGD. Thus, SMD is a generalization of SGD.\n\nConvergence Criteria: Bregman Gradient Bregman gradient is a generation of projected gradient (Ghadimi et al., 2016). Recently, Zhang & He (2018); Davis & Grimmer (2019) develop it to measure the convergence of an algorithm for the non-convex optimization problem. Evaluating the difference between each candidate solution x and its proximity is the critical idea of Bregman gradient to measure the stationarity of x. Specifically, let X be a closed convex set on $\\mathbb{R}^{n},\\alpha\\stackrel{\\smile}{>}0,\\breve{T}(x)$ is defined on X. The Bregman gradient of T at $x\\in\\mathcal{X}$ is:\n\n$$\n\\mathcal{G}_{\\alpha,T}^{\\psi}(x)=\\alpha^{-1}(x-\\mathcal{M}_{\\alpha,T}^{\\psi}(x)),\\tag{12}\n$$\n\nwhere $\\mathcal{M}_{\\alpha,T}^{\\psi}(\\cdot)$ is defined in Eq.(9). If $\\psi(x)={\\frac{1}{2}}\\|x\\|_{2}^{2}$ , then $x^{*}$ is a critical point of T if and only if $\\mathcal{G}_{\\alpha,T}^{\\psi}(x^{*})=\\nabla T(x^{*})=0$ (Bauschke et al. (2011);Theorem 27.1). Thus, Bregman gradient (12) is a generalization of gradient. The following Remark 2 is helpful for us to understand the significance of Bregman gradient, and it gives us some insights to understand this convergence criterion.\n\nAlgorithm 1 Mirror Policy Optimization Algorithm (MPO)   \n1: Initialize: parameter $\\theta_{1}$ , step-sizea ${\\bf\\chi}_{k}>0,g_{0}=0,$ parametric policy $\\pi_{\\boldsymbol{\\theta}}(a|\\boldsymbol{s})$ , and map ψ.   \n2: for k = 1 to N do   \n3: Generate a trajectory $\\tau_{k}=\\{s_{t},a_{t},r_{t+1}\\}_{t=0}^{H_{\\tau_{k}}}\\sim\\pi_{\\theta_{k}}$ , temporary variable $g_{0}=0$   \n$$\ng_{k}\\leftarrow\\sum_{t=0}^{H_{\\tau_{k}}}\\nabla_{\\theta}\\log\\pi_{\\theta}(a_{t}|s_{t})R(\\tau_{k})|_{\\theta=\\theta_{k}}\\tag{14}\n$$\n\n$$\n\\hat{g}_{k}\\leftarrow\\frac{1}{k}g_{k}+(1-\\frac{1}{k})g_{k-1}\\tag{15}\n$$\n\n$$\n\\theta_{k+1}\\leftarrow\\arg\\operatorname*{min}_{\\omega}\\{\\langle-\\hat{g}_{k},\\omega\\rangle+\\frac{1}{\\alpha_{k}}D_{\\psi}(\\omega,\\theta_{k})\\}\\tag{16}\n$$\n\n4: end for   \n5: Output: ${\\tilde{\\theta}}_{N}$ according to (13).\n\n3.2 An Implementation with Convergent Guarantee\n\nIn this section, we propose a convergent implementation defined as follows, for each step k:\n\n$$\n\\theta_{k+1}=\\mathcal{M}_{\\alpha_{k},\\langle-\\hat{g}_{k},\\theta\\rangle}^{\\psi}(\\theta_{k})=\\arg\\operatorname*{min}_{\\theta\\in\\Theta}\\{\\langle-\\hat{g}_{k},\\theta\\rangle+\\frac{1}{\\alpha_{k}}D_{\\psi}(\\theta,\\theta_{k})\\},\\tag{17}\n$$\n\nwhere $\\hat{g}_{\\boldsymbol k}$ is an arithmetic mean of previous episodes' gradient estimate $\\{g(\\tau_{i}|\\theta_{i})\\}_{i=1}^{k}\\colon$\n\n$$\n\\hat{g}_{k}=\\frac{1}{k}\\sum_{i=1}^{k}g(\\tau_{i}|\\theta_{i}).\\tag{18}\n$$\n\nA Proof of Theorem 1\n\nLet $f(x)$ be a L-smooth function defined on $\\mathbb{R}^{n}$ , i.e $\\|\\nabla f(x)-\\nabla f(y)\\|\\leq L\\|x-y\\|$ . Then, for $\\forall x,y\\in\\mathbb{R}^{n}$ , the following holds\n\n$$\n\\|f(x)-f(y)-\\langle\\nabla f(y),x-y\\rangle\\|\\leq{\\frac{L}{2}}\\|x-y\\|^{2}.\\tag{19}\n$$\n\nThe following Lemma 4 is useful for our proof.\n\nLemma 4 (Ghadimi et al. (2016), Lemma 1 and Proposition 1). Let X be a closed convex set in $\\mathbb{R}^{d},h:\\mathcal{X} $ R be a convex function, but possibly nonsmooth, and $D_{\\psi}:\\mathcal{X}\\times\\mathcal{X}\\mathbb{R}i s$ Bregman divergence. Moreover, define\n\n$$\n\\begin{array}{ l }{{x^{+}=\\displaystyle\\arg\\operatorname*{min}_{u\\in\\mathcal X}\\left\\{\\langle g,u\\rangle+\\frac{1}{\\eta}D_{\\psi}(u,x)+h(u)\\right\\}}}\\\\ {{P_{\\mathcal X}(x,g,\\eta)=\\displaystyle\\frac{1}{\\eta}(x-x^{+}),}}\\end{array}\\tag{20}\n$$\n\nwhere $g\\in\\mathbb{R}^{d},x\\in\\mathcal{X},$ and $\\eta > 0 .$ Then, the following statement holds\n\n$$\n\\langle g,P_{\\mathcal{X}}(x,g,\\eta)\\rangle\\geq\\zeta\\|P_{\\mathcal{X}}(x,g,\\eta)\\|^{2}+\\frac{1}{\\eta}[h(x^{+})-h(x)],\\tag{21}\n$$\n\nwhere $\\zeta$ is a positive constant determined by ψ (i.e. ψ is a a continuously-differentiable and $\\zeta -$ strictly conyex function) that satisfying $\\langle x-y,\\nabla\\psi(x)-\\nabla\\psi(y)\\rangle\\geq\\zeta\\|x-y\\|^{2}$ . Moreover, for any $g_{1},g_{2}\\in\\mathbb{R}^{d}$ , the following statement holds\n\n$$\n\\lVert P\\chi(x,g_{1},\\eta)-P\\chi(x,g_{2},\\eta)\\rVert\\leq\\frac{1}{\\zeta}\\lVert g_{1}-g_{2}\\rVert .\\tag{22}\n$$\n\nAlgorithm 2 Variance-Reduced Mirror Policy Optimization (VRMPO).   \n1: Initialize: Policy $\\pi_{\\boldsymbol{\\theta}}(a|s)$ with parameter ${\\widetilde{\\theta}}_{0}.$ ,mirror map ψ,step-size $\\alpha_{k}>0$ , epoch size $K,m$   \n2: for $k = 1$ to K do   \n3: $\\begin{array}{ r }{\\theta_{k,0}=\\tilde{\\theta}_{k-1},}\\end{array}$ generate $\\mathcal{T}_{k}=\\{\\tau_{i}\\}_{i=1}^{N_{1}}\\sim\\pi\\theta_{k,\\mathrm{c}}$   \n4: $\\theta_{k,1}=\\theta_{k,0}-\\alpha_{k}G_{k,0},$ where $\\begin{array}{ r }{G_{k,0}= -\\hat{\\nabla}_{N_{1}}J(\\theta_{k,0})= -\\frac{1}{N_{1}}\\sum_{i=1}^{N_{1}}g(\\tau_{i}|\\theta_{k,0})}\\end{array}$   \n5: for $t = 1$ to $m - 1$ do   \n6: Generate $\\{\\tau_{j}\\}_{j=1}^{N_{2}}\\sim\\pi_{\\theta_{k,t}}$   \n$G_{k,t}=G_{k,t-1}+\\frac{1}{N_{2}}\\sum_{j=1}^{N_{2}}(-g(\\tau_{j}|\\theta_{k,t})+g(\\tau_{j}|\\theta_{k,t-1})),$ (23)   \n$\\theta_{k,t+1}=\\arg\\operatorname*{min}_{\\omega}\\{\\langle G_{k,t},\\omega\\rangle+\\frac{1}{\\alpha_{k}}D_{\\psi}(\\omega,\\theta_{k,t})\\}$ (24)   \n7: end for   \n8: $\\tilde{\\theta}_{k}=\\theta_{k,t}$ with t chosen uniformly randomly from $\\{0,1,...,m\\}$   \n9end for   \n10: Output: $\\tilde{\\theta}_{K}$\n", "theorems": [{"local_id": "T1", "statement": "Lemma 5. Let $\\zeta>\\frac{5}{32}$, the batch size of the trajectories of the outer loop $N_1=\\Big(\\frac{1}{8L\\zeta^2}+\\frac{1}{2(\\zeta-\\frac{5}{32})}\\big(1+\\frac{1}{32\\zeta^2}\\big)\\Big)\\frac{\\sigma^2}{\\epsilon^2}$, the iteration times of the inner loop $m-1=N_2=\\sqrt{\\Big(\\frac{1}{8L\\zeta^2}+\\frac{1}{2(\\zeta-\\frac{5}{32})}\\big(1+\\frac{1}{32\\zeta^2}\\big)\\Big)}\\frac{\\sigma}{\\epsilon}$, and step size $\\alpha_k=\\frac{1}{4L}$. For each $k$ and $t$, $G_{k,0}$ and $\\theta_{k,0}$ are generated by Algorithm 2, then the following holds: $\\mathbb{E}\\|\\nabla\\mathcal{I}(\\theta_{k,0})-G_{k,0}\\|^2\\le\\Big(\\frac{1}{8L\\zeta^2}+\\frac{1}{2(\\zeta-\\frac{5}{32})}\\big(1+\\frac{1}{32\\zeta^2}\\big)\\Big)^{-1}\\epsilon^2.$ (25)", "proof": {"steps": [{"n": 1, "claim": "$\\mathbb{E}\\|\\nabla\\mathcal{I}(\\theta_{k,0})-G_{k,0}\\|^2=\\mathbb{E}\\|\\nabla J(\\theta_{k,0})-\\frac{1}{N_1}\\sum_{i=1}^{N_1}g(\\tau_i|\\theta_{k,0})\\|^2$"}, {"n": 2, "claim": "$=\\frac{1}{N_1^2}\\sum_{i=1}^{N_1}\\mathbb{E}\\|\\nabla J(\\theta_{k,0})-g(\\tau_i|\\theta_{k,0})\\|^2$"}, {"n": 3, "claim": "$\\le\\frac{\\sigma^2}{N_1}$"}, {"n": 4, "claim": "$=\\Big(\\frac{1}{8L\\zeta^2}+\\frac{1}{2(\\zeta-\\frac{5}{32})}\\big(1+\\frac{1}{32\\zeta^2}\\big)\\Big)^{-1}\\epsilon^2\\;(=\\epsilon_1^2)$"}]}}, {"local_id": "T2", "statement": "Under Assumption 2 (for each pair $(s,a)$, any $\\theta\\in\\mathbb{R}^p$, and all components $i,j$, there exist positive constants $G,F$ with $|\\nabla_{\\theta_i}\\log\\pi_\\theta(a|s)|\\le G$ and $|\\frac{\\partial^2}{\\partial\\theta_i\\partial\\theta_j}\\log\\pi_\\theta(a|s)|\\le F$), with total trajectories $\\{\\tau_k\\}_{k=1}^N$, consider the sequence $\\{\\theta_k\\}_{k=1}^N$ generated by Algorithm 1 (MPO), whose update is $\\theta_{k+1}=\\arg\\min_{\\theta\\in\\Theta}\\{\\langle-\\hat g_k,\\theta\\rangle+\\frac{1}{\\alpha_k}D_\\psi(\\theta,\\theta_k)\\}$ with the averaged estimator $\\hat g_k=\\frac{1}{k}\\sum_{i=1}^k g_i$ and $g_i=\\sum_{t=0}^{H_{\\tau_i}}\\nabla_\\theta\\log\\pi_\\theta(a_t|s_t)R(\\tau_i)|_{\\theta=\\theta_i}$. Let the output $\\tilde\\theta_N=\\theta_n$ follow the distribution $P(\\tilde\\theta_N=\\theta_n)=\\frac{\\zeta\\alpha_n-L\\alpha_n^2}{\\sum_{k=1}^N(\\zeta\\alpha_k-L\\alpha_k^2)}$ of Eq.(13), let $0<\\alpha_k<\\frac{\\zeta}{L}$, $\\ell(g,u)=\\langle g,u\\rangle$, where $L=RH(HG^2+F)/(1-\\gamma)$ (Eq.7) and $\\sigma^2=\\frac{G^2R^2}{(1-\\gamma)^4}$ (Eq.8), and $\\zeta>0$ is the strong-convexity constant of $\\psi$, and $\\mathcal{G}^\\psi_{\\alpha,T}(x)=\\alpha^{-1}(x-\\mathcal{M}^\\psi_{\\alpha,T}(x))$ is the Bregman gradient. Then $\\mathbb{E}[\\|\\mathcal{G}^\\psi_{\\alpha_n,\\ell(-g_n,\\theta_n)}(\\theta_n)\\|^2]\\le\\frac{\\big(J(\\theta^*)-J(\\theta_1)\\big)+\\frac{\\sigma^2}{\\zeta}\\sum_{k=1}^N\\frac{\\alpha_k}{k}}{\\sum_{k=1}^N(\\zeta\\alpha_k-L\\alpha_k^2)}$ (Eq.26), and with $\\alpha_k=\\zeta/2L$ this yields $\\mathbb{E}[\\|\\mathcal{G}^\\psi_{\\alpha_n,\\ell(-\\hat g_n,\\theta_n)}(\\theta_n)\\|^2]\\le\\frac{4L(J(\\theta^*)-J(\\theta_1))+2\\sigma^2\\sum_{k=1}^N\\frac{1}{k}}{N\\zeta^2}=\\mathcal{O}(\\ln N/N)$, i.e. MPO converges.", "proof": {"steps": [{"n": 1, "claim": "The descent bound (Eq.27-28) reduces the analysis to controlling the error term $\\epsilon_k=-\\hat g_k-\\nabla\\mathcal{I}(\\theta_k)$."}, {"n": 2, "claim": "Recall the estimators $g_k=\\sum_{t=0}^{H_{\\tau_k}}\\nabla_\\theta\\log\\pi_\\theta(a_t|s_t)R(\\tau_k)$ and $\\hat g_k=\\frac{1}{k}\\sum_{i=1}^k g_i$."}, {"n": 3, "claim": "$\\mathbb{E}[-g_k]=\\mathbb{E}[-\\hat g_k]=-\\nabla J(\\theta_k)=\\nabla\\mathcal{I}(\\theta_k)$ (Eq.29)."}, {"n": 4, "claim": "Letting $\\mathcal{F}_k$ be the $\\sigma$-field generated by all random variables before round $k$ and $\\tilde\\epsilon_k=g_k-\\nabla\\mathcal{I}(\\theta_k)$, Eq.(29) implies $\\mathbb{E}\\big[\\langle\\epsilon_k,\\mathcal{G}^\\psi_{\\alpha_k,\\ell(-\\nabla J(\\theta_k),\\theta_k)}(\\theta_k)\\rangle\\,\\big|\\,\\mathcal{F}_{k-1}\\big]=\\mathbb{E}\\big[\\langle\\tilde\\epsilon_k,\\mathcal{G}^\\psi_{\\alpha_k,\\ell(-\\nabla J(\\theta_k),\\theta_k)}(\\theta_k)\\rangle\\,\\big|\\,\\mathcal{F}_{k-1}\\big]=0$."}, {"n": 5, "claim": "Setting $\\delta_s=\\sum_{t=1}^s\\tilde\\epsilon_t$, we have $\\mathbb{E}[\\langle\\delta_s,\\tilde\\epsilon_{s+1}\\rangle\\,|\\,\\delta_s]=0$ (Eq.30), hence $\\mathbb{E}[\\|\\delta_k\\|^2]=\\sum_{t=1}^k\\mathbb{E}\\|\\tilde\\epsilon_t\\|^2$ (Eq.31)."}, {"n": 6, "claim": "By Lemma 3 and Eq.(31), $\\mathbb{E}[\\|\\epsilon_k\\|^2]=\\frac{1}{k^2}\\sum_{t=1}^k\\mathbb{E}\\|\\tilde\\epsilon_t\\|^2\\le\\frac{\\sigma^2}{k}$ (Eq.32)."}, {"n": 7, "claim": "Substituting Eq.(32) into Eq.(33) and summing over $k$ gives $\\sum_{k=1}^N(\\zeta\\alpha_k-L\\alpha_k^2)\\mathbb{E}[\\|\\mathcal{G}^\\psi_{\\alpha_k,\\ell(-\\hat g_k,\\theta_k)}(\\theta_k)\\|^2]\\le\\mathcal{I}(\\theta_1)-\\mathcal{I}^*+\\frac{\\sigma^2}{\\zeta}\\sum_{k=1}^N\\frac{\\alpha_k}{k}$, which yields Eq.(26)."}]}}, {"local_id": "T3", "statement": "Lemma 6 (Lemma1 (Fang et al., 2018)). Under Assumption $I,G_{k,t}$ is generated according to (23), $\\theta_{k,t}$ is generated according to (24), then for any $0 \\leq t \\leq m$ , the following holds\n\n$$\n\\mathbb{E}[\\|G_{k,t}-\\nabla{\\mathcal{I}}(\\theta_{k,t})\\|^{2}]\\leq{\\frac{L^{2}}{N_{2}}}\\mathbb{E}[\\|\\theta_{k,t}-\\theta_{k,t-1}\\|^{2}]+\\mathbb{E}[\\|G_{k,t-1}-{\\mathcal{I}}(\\theta_{k,t-1})\\|^{2}].\\tag{34}\n$$", "proof": {"steps": [{"n": 1, "claim": "Telescoping $Eq.(5)$ over t from 1 to the time t, then the following holds"}, {"n": 2, "claim": "$$\n\\mathbb{E}[\\|G_{k,t}-\\nabla{\\mathcal{I}}(\\theta_{k,t})\\|^{2}]\\leq\\sum_{i=1}^{t}{\\frac{L^{2}}{N_{2}}\\mathbb{E}[\\|\\theta_{k,i+1}-\\theta_{k,i}\\|^{2}]}+\\mathbb{E}[\\|G_{k-1,0}-\\nabla{\\mathcal{I}}({\\widetilde{\\theta}}_{k-1})\\|^{2}].\\tag{35}\n$$"}]}}], "target_local_id": "T2", "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "label": "flawed", "error_step": 3, "error_class6": ["logical_gap_or_missing_case", "missing_assumption", "invalid_bound_or_inequality"], "error_explanation": "Step 3 claims the AVERAGED estimator $\\hat g_k=\\frac{1}{k}\\sum_{i=1}^k g(\\tau_i|\\theta_i)$ is an unbiased estimator of $\\nabla J(\\theta_k)$, i.e. $\\mathbb{E}[-\\hat g_k]=-\\nabla J(\\theta_k)$. This is false: $\\hat g_k$ averages single-trajectory gradient estimates evaluated at DIFFERENT iterates $\\theta_1,\\dots,\\theta_k$, so $\\mathbb{E}[\\hat g_k]=\\frac{1}{k}\\sum_{i=1}^k\\nabla J(\\theta_i)$, a running average of past policy gradients, not $\\nabla J(\\theta_k)$. Only the single-trajectory estimator is unbiased at the current point: $\\mathbb{E}[g_k]=\\mathbb{E}[g(\\tau_k|\\theta_k)]=\\nabla J(\\theta_k)$. Consequently the error term $\\epsilon_k=-\\hat g_k-\\nabla\\mathcal{I}(\\theta_k)$ is NOT zero-mean, so every later step that treats $\\epsilon_k$ as zero-mean (the conditional-expectation vanishing in step 4 and the martingale/telescoping variance bound $\\mathbb{E}\\|\\epsilon_k\\|^2\\le\\sigma^2/k$ in steps 5-6) fails, and the convergence conclusion Eq.(24) does not go through. A valid proof would have to separately control the bias $\\frac{1}{k}\\sum_{i=1}^k\\nabla J(\\theta_i)-\\nabla J(\\theta_k)$, which the paper never does."}
{"id": "VSKV3GykuE", "context": "Assumption 1 (Lipschitz Gradient). Function f is differentiable, and there exists $L > 0$ such that\n\n$$\n\\|\\nabla f(W)-\\nabla f(V)\\|\\leq L\\|W-V\\|,\\qquad\\forall W,V\\in\\mathbb{R}^{m\\times n},\n$$\n\nwhere |·| denotes the Frobenius matrix norm, the gradient is computed w.r.t. the trace inner product.\n\nDefinition 1 (Left Sketch). By a \"left sketch\" (of rank r) we refer to the update rule\n\n$$\n\\Delta W=\\frac{\\alpha}{r}B_{S}\\hat{A},\n$$\n\nwhere $B_{S}\\sim{\\mathcal{D}}_{B}$ is sampled from some fixed distribution over matrices of dimensions $n\\times r,$ and only the matrix Á is adjustable.\n\nDefinition 2 (Right Sketch). By a \"right sketch\" (of rank r) we refer to the update rule\n\n$$\n\\Delta W={\\frac{\\alpha}{r}}{\\hat{B}}A_{S},\n$$\n\nwhere $A_{S}\\sim{\\mathcal{D}}_{A}$ is sampled from some fixed distribution over matrices of dimensions $r\\times m,$ and only the matrix B is adjustable.\n\n5 Theory\n\n5.1 Derivation of the Update Step\n\nWithout loss of generality, let us focus on the Left Sketch scheme (Definition 1). Specifically, for each model in the chain, the update rule is given as follows:\n\n$$\n\\boldsymbol{W}^{t+1}=\\boldsymbol{W}^{t}+\\frac{\\alpha}{r}B_{S}^{t}\\boldsymbol{\\hat{A}^{t}}.\n$$\n\nNext, we apply the Lipschitz gradient condition (Assumption 1) to the loss function $f \\colon$\n\n$$\nf(U)\\leq f(V)+\\langle\\nabla f(V),U-V\\rangle+\\frac{L}{2}\\|U-V\\|_{F}^{2},\\quad\\forall U,V\\in\\mathbb{R}^{m\\times n}\n$$\n\n$$\n\\boldsymbol{W}^{t+1}=\\boldsymbol{W}^{t}-\\gamma H_{B}^{t}\\nabla f(\\boldsymbol{W}^{t}),\\tag{5}\n$$\n\nwhere $H_{B}^{t}=B_{S}^{t}\\left((B_{S}^{t})^{\\top}(B_{S}^{t})\\right)^{\\dagger}(B_{S}^{t})^{\\top}$ is projection matrix and $\\begin{array}{ r }{\\frac{\\alpha}{r}\\eta=\\gamma}\\end{array}$ .Similarly, we can obtain the update for Right Sketch scheme (Definition 2):\n\n$$\n\\boldsymbol{W}^{t+1}=\\boldsymbol{W}^{t}-\\gamma\\nabla f(\\boldsymbol{W}^{t})(\\boldsymbol{A}_{S}^{t})^{\\top}\\left(\\boldsymbol{A}_{S}^{t}(\\boldsymbol{A}_{S}^{t})^{\\top}\\right)^{\\dagger}\\boldsymbol{A}_{S}^{t}=\\boldsymbol{W}^{t}-\\gamma\\nabla f(\\boldsymbol{W}^{t})\\boldsymbol{H}_{A}^{t},\\tag{3}\n$$\n\n5.2 Convergence Results\n\nAssumption 2. Consider a projection matrix H generated by Left Sketch 1 or Right Sketch 2. Assume that the sampling distributions $\\mathcal{D}_{S}^{B}$ and $\\hat{\\mathcal{D}}_{S}^{A}$ are such that the smallest eigenvalue of the expected projection matrix H generated by sampled matrix is positive:\n\n$$\n\\begin{array}{ r }{\\lambda_{\\operatorname*{min}}^{H}=\\lambda_{\\operatorname*{min}}\\left[\\mathbb{E}\\left[H\\right]\\right]>0 .}\\end{array}\n$$\n\nRemark. Assumption 2 is easily satisfied. Let H be the projection matrix as defined below Equation (3) and assume that the A matrices are drawn from an isotropic distribution (the rows of A are isotropic). Then H is the projection onto the rank of A, which is a subspace of dimension r distributed isotropically in $\\mathbb{R}^{\\bar{n}}$ . The matrix $\\mathbb{E}[H]$ is then invariant under rotations, so must be a scalar multiple of the identity. By taking traces, one finds that $\\textstyle\\mathbb{E}[H]={\\frac{r}{n}}I$ so $\\begin{array}{ r }{\\lambda_{\\operatorname*{min}}^{H}=\\frac{r}{n}}\\end{array}$\n\nAssumption 3. Function f is bounded from below by an infimum $f^{\\star}\\in\\mathbb{R}.$\n\nAssumption 4 (PL-condition). Function f satisfies the Polyak-Lojasiewicz $(PL)$ condition with parameter $\\mu > 0 i f$\n\n$$\n\\frac{1}{2}\\|\\nabla f(W)\\|^{2}\\geq\\mu\\left(f(W)-f^{\\star}\\right)\n$$\n\nfor all $W\\in\\mathbb{R}^{m\\times n}$ , where $f^{\\star}=\\operatorname{inf}f,$ assumed to be finite.\n\nD Analysis of RAC-LoRA with Random Reshuffling\n\nAssumption 5. There exist nonnegative constants $\\sigma \\geq 0$ such that for any $W^{t}\\in\\mathbb{R}^{m\\times n}$ we have,\n\n$$\n\\frac{1}{N}\\sum_{i=1}^{n}\\left\\|\\nabla f_{i}\\left(W^{t}\\right)-\\nabla f\\left(W^{t}\\right)\\right\\|^{2}\\leq\\sigma^{2}.\n$$\n\nWe consider a method belonging to the class of data permutation methods which is the RR algorithm. In each epoch t of RR, we sample indices $\\pi_{0},\\pi_{1},...,\\pi_{N-1}$ without replacement from $\\left\\{1,2,\\ldots,N\\right\\},\\mathrm{i . e .,}\\left\\{\\pi_{0},{\\pi_{1}},\\ldots,{\\pi_{N-1}}\\right\\}$ is a random permutation of the set $\\{1,2\\dots\\hat{N}\\}$ and proceed with N iterates of the form:\n\n$$\n\\boldsymbol{W}_{i+1}^{t}=\\boldsymbol{W}_{i}^{t}-\\gamma\\boldsymbol{H}_{B}^{t}\\nabla\\boldsymbol{f}(\\boldsymbol{W}_{i}^{t}).\n$$\n\nWe then set $W^{t+1}=W_{N}^{t}$ , and repeat the process for a total of T LoRA blocks. We can derive the effective step:\n\n$$\n\\boldsymbol{W}^{t+1}=\\boldsymbol{W}^{t}-\\gamma H_{B}^{t}\\sum_{i=0}^{N-1}\\nabla f(\\boldsymbol{W}_{i}^{t})=\\boldsymbol{W}^{t}-\\gamma H_{B}^{t}N\\boldsymbol{\\hat{g}}^{t},\\tag{4}\n$$\n\nwhere $\\begin{array}{ r }{\\hat{g}^{t}=\\frac{1}{N}\\sum_{i=0}^{N-1}\\nabla f(W_{i}^{t})}\\end{array}$", "theorems": [{"local_id": "T1", "statement": "Corollary 1. Suppose that Assumption 1 and Assumption 2 hold. Suppose further that the sketches are isotropic, so that $\\mathbb{E}[H^t]=\\tfrac{r}{n}I$ and $\\lambda_{\\max}^H:=\\lambda_{\\max}(\\mathbb{E}[H^t])=\\tfrac{r}{n}$; consequently the denominator $1-\\lambda_{\\max}[\\mathbb{E}[I-H^t]]-\\tfrac14\\lambda_{\\max}^H=\\tfrac34\\,\\tfrac{r}{n}>0$. Suppose that a stepsize $\\gamma > 0$ is chosen such that $\\begin{array}{ r }{\\gamma\\le\\frac{1}{2 L N}}\\end{array}$ . Let the updates have a form of several gradient steps (variance $\\sigma^{2}=0)$ We choose the output of the method $\\widetilde{W}^{T}$ uniformly at random from $W^{0},W^{1},\\dots,W^{T-1}$ Then, the iterate $\\widetilde{W}^{T}$ of RAC-LoRA method (Algorithm 1) with several GD updates (Equation 5) satisfy\n\n$$\n\\mathbb{E}\\left[\\left\\|\\nabla f(\\widetilde{W}^{T})\\right\\|^{2}\\right]\\leq\\frac{2}{\\gamma N T}\\frac{f(W^{0})-f^{\\star}}{\\left(1-\\lambda_{\\operatorname*{max}}\\left[\\mathbb{E}\\left[I-H^{t}\\right]\\right]-\\frac{1}{4}\\lambda_{\\operatorname*{max}}^{H}\\right)}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Suppose that a stepsize $\\gamma > 0$ is chosen such that $\\begin{array}{ r }{\\gamma\\le\\frac{1}{2 L N}}\\end{array}$ ."}, {"n": 2, "claim": "Let the updates have a form of several gradient steps (variance $\\sigma^{2}=0)$"}, {"n": 3, "claim": "We choose the output of the method $\\widetilde{W}^{T}$ uniformly at random from $W^{0},W^{1},\\dots,W^{T-1}$"}, {"n": 4, "claim": "Then, the iterate $\\widetilde{W}^{T}$ of RAC-LoRA method (Algorithm 1) with several GD updates (Equation 5) satisfy"}, {"n": 5, "claim": "\\mathbb{E}\\left[\\left.\\nabla f(\\widetilde{W}^{T})\\right.^{2}\\right]\\leq\\frac{2}{\\gamma N T}\\frac{f(W^{0})-f^{\\star}}{\\left(1-\\lambda_{\\operatorname*{max}}\\left[\\mathbb{E}\\left[I-H^{t}\\right]\\right]-\\frac{1}{4}\\lambda_{\\operatorname*{max}}^{H}\\right)}."}, {"n": 6, "claim": "Given that the step size is divided by the number of gradient steps allocated for each LoRA block, employing multiple gradient steps for a single LoRA block does not provide any significant benefits."}, {"n": 7, "claim": "This observation suggests that a single gradient step is adequate for each LoRA block."}]}}, {"local_id": "T2", "statement": "Consider the optimization problem $\\min_{\\Delta W \\in \\mathbb{R}^{m\\times n}} f(W^0 + \\Delta W)$ with the LoRA parametrization $W = W^0 + \\frac{\\alpha}{r}BA$. Suppose $f$ is differentiable with $L$-Lipschitz gradient (Assumption 1: $\\|\\nabla f(W) - \\nabla f(V)\\| \\le L\\|W-V\\|$ for all $W,V$, Frobenius norm), that the expected projection matrix $H$ generated by the Left/Right Sketch has positive smallest eigenvalue $\\lambda_{\\min}^H = \\lambda_{\\min}[\\mathbb{E}[H]] > 0$ (Assumption 2), and that $f$ is bounded below by an infimum $f^\\star$ (Assumption 3). Then, for the RAC-LoRA algorithm (Algorithm 1) with GD updates (Equation 5 or 3) and stepsize $0 < \\gamma \\le \\frac{1}{L}$, the iterates satisfy $\\mathbb{E}\\left[\\|\\nabla f(\\widetilde{W}^T)\\|^2\\right] \\le \\frac{2(f(W^0)-f^\\star)}{\\lambda_{\\min}^H \\gamma T}$, where the output $\\widetilde{W}^T$ is chosen uniformly at random from $W^0, W^1, \\dots, W^{T-1}$.", "proof": {"steps": [{"n": 1, "claim": "For the Left Sketch scheme (Definition 1), the update rule for each model in the chain is $W^{t+1} = W^t + \\frac{\\alpha}{r} B_S^t \\hat{A}^t$."}, {"n": 2, "claim": "$f(U) \\le f(V) + \\langle \\nabla f(V), U-V\\rangle + \\frac{L}{2}\\|U-V\\|_F^2$ for all $U, V \\in \\mathbb{R}^{m\\times n}$."}, {"n": 3, "claim": "We instantiate the descent lemma with $U = W^t,\\ V = B_S^t \\hat{A}^t$, taking $\\eta \\le \\frac{1}{L}$."}, {"n": 4, "claim": "$f(W^{t+1}) \\le f(W^t) + \\langle \\nabla f(W^t), B_S^t \\hat{A}^t\\rangle + \\frac{L}{2}\\|B_S^t \\hat{A}^t\\|_F^2$."}, {"n": 5, "claim": "$f(W^{t+1}) \\le f(W^t) + \\langle (B_S^t)^\\top \\nabla f(W^t), \\hat{A}^t\\rangle + \\frac{1}{2\\eta}\\langle (B_S^t)^\\top B_S^t \\hat{A}^t, \\hat{A}^t\\rangle$."}, {"n": 6, "claim": "Minimizing the right-hand side in $\\hat{A}^t$ (gradient zero: $(B_S^t)^\\top \\nabla f(W^t) + \\frac{1}{\\eta}(B_S^t)^\\top(B_S^t)\\hat{A}^t = 0$) gives $\\hat{A}^t = -\\eta\\left((B_S^t)^\\top(B_S^t)\\right)^{\\dagger}(B_S^t)^\\top \\nabla f(W^t)$."}, {"n": 7, "claim": "$W^{t+1} = W^t + \\frac{\\alpha}{r} B_S^t \\hat{A}^t = W^t - \\gamma H_B^t \\nabla f(W^t)$, with $H_B^t = B_S^t\\left((B_S^t)^\\top(B_S^t)\\right)^{\\dagger}(B_S^t)^\\top$ and $\\frac{\\alpha}{r}\\eta = \\gamma$."}]}}, {"local_id": "T3", "statement": "Theorem 2. Let Assumptions 1, 2 and 4 hold, and let the stepsize satisfy $\\begin{array}{ r }{0<\\gamma\\le\\frac{1}{L}}\\end{array}$ . Then, for each $T \\geq 0$ , the iterates of RAC-LoRA (Algorithm 1) with GD updates (Equation 5 or 3) satisfy\n\n$$\n\\mathbb{E}\\left[f(W^{T})\\right]-f^{\\star}\\le\\left(1-\\gamma\\mu\\lambda_{\\operatorname*{min}}^{H}\\right)^{T}\\left(f(W^{0})-f^{\\star}\\right).\n$$", "proof": {"steps": [{"n": 1, "claim": "The proof is provided for Left Sketch (Definition 1). The result for Right Sketch (Definition 2) can be derived by following the same steps."}, {"n": 2, "claim": "Proof. We start from the inequality 6:"}, {"n": 3, "claim": "\\mathbb{E}\\left[f(W^{t+1})\\mid W^{t}\\right]-f^{\\star}\\le f(W^{t})-f^{\\star}-\\frac{\\gamma}{2}\\lambda_{\\operatorname*{min}}^{H_{B}}\\left\\|\\nabla f(W^{t})\\right\\|^{2}.\\tag{6}"}, {"n": 4, "claim": "Using PL condition $\\left\\|\\nabla f(W^{t})\\right\\|^{2}\\geq 2\\mu\\left(f(W^{t})-f^{\\star}\\right)$ we have"}, {"n": 5, "claim": "\\mathbb{E}\\left[f(W^{t+1})\\vert W^{t}\\right]-f^{\\star}\\le f(W^{t})-f^{\\star}-\\gamma\\mu\\lambda_{\\operatorname*{min}}^{H_{B}}\\left(f(W^{t})-f^{\\star}\\right)"}, {"n": 6, "claim": "\\le\\left(1-\\gamma\\mu\\lambda_{\\operatorname*{min}}^{H_{B}}\\right)\\left(f(W^{t})-f^{\\star}\\right)."}, {"n": 7, "claim": "Once we unroll the recursion we get"}, {"n": 8, "claim": "\\mathbb{E}\\left[f(W^{T})\\right]-f^{\\star}\\leq\\left(1-\\gamma\\mu\\lambda_{\\operatorname*{min}}^{H_{B}}\\right)^{T}\\left(f(W^{0})-f^{\\star}\\right)."}]}}], "target_local_id": "T2", "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "label": "flawed", "error_step": 3, "error_class6": ["logical_gap_or_missing_case", "others", "computation_or_constant"], "error_explanation": "Step 3 instantiates the descent lemma with U=W^t, V=B_S^t A_hat^t (dropping the alpha/r scaling). This substitution's literal output is f(W^t) <= f(B_S^t A_hat^t)+..., which cannot bound f(W^{t+1}); the correct choice is U=W^{t+1}, V=W^t with U-V=(alpha/r)B_S^t A_hat^t. The wrong substitution is the root defect; Step 4's invalid inequality is only its downstream manifestation."}
{"id": "XYK1eGjahp", "context": "This section presents and explains our main results on Transformers' capability in deductive reasoning and backtracking with CoT. To rigorously state our results, we first formally define decision problems, decision procedures, and what it means for a model to \"solve\" a decision problem using CoT:\n\nAlgorithm 1: Greedy Decoding\nInput: Model $M:\\mathcal{V}^{*}\\to\\mathcal{V}$ , prompt $s_{1:n}=(s_{1},s_{2},\\ldots,s_{n})$ , stop tokens $\\mathcal{E}\\subseteq\\mathcal{V}$ , $t \\leftarrow n$\n1: while $t \\leftarrow t + 1$ do\n2: $s_{t}\\leftarrow M(s_{1:t-1})$ ; // Obtain model output and append to string\n3: if $s_{t}\\in\\mathcal{E}$ return $s_{1:t}$\n4: end\n\nIn this paper, we refer to the full generated sequence of tokens as the Chain-of-Thought, and the number of chain-of-thought tokens in Algorithm 1 is $t - n$ .\n\nDefinition 1 (Decision Problem). Let V be a vocabulary, $\\Sigma \\subseteq \\nu$ be an alphabet, $L\\subseteq\\Sigma^{*}$ be a set of valid input strings. We say that a mapping $f:L\\{0,1\\}$ is a decision problem defined on L.\n\nDefinition 2 (Decision Procedure). We say that an algorithm A is a decision procedure for the decision problem f, if given any input string x from $L,A$ halts and outputs 1 if $f(x)=1$ , and halts and outputs 0 if $f(x)=0$\n\nDefinition 3 (Autoregressive Decision Procedure). For any map $M:\\mathcal{V}^{*}\\mathcal{V}.$ ,which we refer to as an auto-regressive next-token prediction model, and $\\mathcal{E}=\\{\\bar{\\mathcal{E}}_{0},\\bar{\\mathcal{E}}_{1}\\}\\subset\\mathcal{V}$ , define procedure $\\mathcal{A}_{M,\\mathcal{E}}$ as follows: For any input $s_{1:n},$ run Algorithm 1 with stop tokens $\\mathcal{E}.\\ A_{M,\\mathcal{E}}$ outputs 0 if $s_{1:t}$ ends with ${\\mathcal{E}}_{0}$ and $\\mathcal{A}_{M,\\mathcal{E}}$ output 1 otherwise. We say M autoregressively decides decision problem f if there is some $\\mathcal{E}\\subset\\mathcal{V}$ for which $\\mathcal{A}_{M,\\mathcal{E}}$ decides $f .$\n\nDefinition 4 ($3\\text{-}\\mathrm{SAT}_{p,c}$). Let $\\mathrm{DIMACS}(p,c)$ denote the set of valid DIMACS encodings of 3-SAT instances with at most p variables and c clauses with a prepended [BOS] token and an appended [SEP] token. Define $3\\text{-}\\mathrm{SAT}_{p,c} : \\mathrm{DIMACS}(p,c) \\to \\{0,1\\}$ as the problem of deciding whether the 3-SAT formula encoded in the input in $\\mathrm{DIMACS}(p,c)$ encoding is satisfiable.\n\nWith the above definition, we're ready to present a formal statement of our theoretical construction of a Transformer model that performs SAT Solving:\n\nRemarks on Theorem 1\n\nThe upper bound on the CoT length $p\\cdot 2^{p+1}$ is a worst-case upper bound which assumes that the model is unable to make any logical deductions have to try all $2^{p}$ assignments. However, this upper bound is never reached in practice, and in Figure 4 we show that the number of CoT tokens is no greater than $8p\\cdot 2^{0.58p}$ for most formulas. If the number of backtracking steps is bounded by T then the CoT is no longer than $(2p+1)(T+1)$\n\nPositional encodings are not included in the number of parameters. The positional encoding at position i is the numerical value i at a particular dimension.\n\nEach parameter can be represented with $O(p+\\log c)$ bits\n\nIn this section, we present adapted versions of several lemmas from Feng et al. (2023). Specifically, an MLP with ReGLU can exactly simulate ReLU, linear operations, and multiplication without error. For Self-attention lemmas, we directly adapt from Feng et al. (2023).\n\nThis section shows several lemmas showing the capabilities of the self-attention operation and MLP layers to approximate high-level vector operations. These high-level operations are later used as building blocks for the Transformer SAT-solver. Specifically, with appropriate weight configurations, a 2-layer MLP with ReGLU activation $f({\\pmb x})=\\bar{\\pmb W_{2}}[({\\pmb W_{1}\\pmb x}+{\\pmb b})\\otimes\\bar{\\mathrm{\\ r e l u}(\\pmb V x+\\pmb c)}]$ can approximate the following vector operations for arbitrary input x:\n\nSimulate a 2-layer MLP with ReLU activation: $W_{\\mathrm{2}}\\mathrm{ReLU}(W_{1}^{\\prime}x+b_{1}^{\\prime})+b_{2}^{\\prime}$\n\n•Simulate any linear operation Wx\n\nSimulate element-wise multiplication: $\\pmb{x}_{1}\\otimes\\pmb{x}_{2}$", "theorems": [{"local_id": "T1", "statement": "Lemma 1 (Simulating Linear Operations with ReGLU MLP). A 2-layer MLP with ReGLU activation can compute any linear operation $f({\\pmb x})={\\pmb W}{\\pmb x}+{\\pmb b}.$", "proof": {"steps": [{"n": 1, "claim": "Set the weights and biases as: $$ \\begin{array}{ c }{{W_{1}=W,\\quad b_{1}=b,}}\\\\ {{\\quad V=0,\\quad b_{2}={\\bf 1},}}\\\\ {{W_{2}=I,\\quad b={\\bf 0}.}}\\end{array} $$"}, {"n": 2, "claim": "Since $V x+b_{2}=b_{2}=\\mathbf{1}$ , we have: $$ \\mathrm{ReLU}(V x+b_{2})=\\mathrm{ReLU}({\\bf 1})={\\bf 1}. $$"}, {"n": 3, "claim": "Then, the ReGLU MLP computes: $$ f(\\pmb{x})=I\\left[\\left(\\pmb{W}\\pmb{x}+\\pmb{b}\\right)\\otimes\\pmb{1}\\right]=\\pmb{W}\\pmb{x}+\\pmb{b}. $$"}]}}, {"local_id": "T2", "statement": "Theorem 1 (Decoder-only Transformers can solve SAT). For any $p,c\\in\\mathbb{N}^{+}$ , there exists a Transformer model M : $\\nu^{*}\\to\\nu$ that autoregressively decides ${\\mathrm{3-S A T}}_{p,c}$ in no more than $p\\cdot 2^{p+1}$ CoT iterations. M requires L = 7 layers, $H = 5$ heads, $d_{e m b}={\\cal O}(p)$ , and $O(p^{2})$ parameters.", "proof": {"steps": [{"n": 1, "claim": "The construction uses adapted versions of lemmas from Feng et al. (2023) as basic building blocks. Here we provide a proof sketch of the core operations in our theoretical construction."}, {"n": 2, "claim": "Proof Sketch In Figure 1 we illustrate the CoT process used by our theoretical construction, which uses CoT tokens to simulate various operations including unit propagation (i.e., logical deduction), variable decision, and backtracking."}, {"n": 3, "claim": "To process clauses and partial assignments with attention operations, the initial layers of the theoretical construction compute the binary vector encodings of clauses and partial assignments and store them in the hidden states. We formally define the encoding scheme for clauses and partial assignments below:\n\nDefinition 5 (Encoding of clause). Let $C$ be a clause. Define encoding $E(C)\\in\\{0,1\\}^{2p}$ of clause $C$ as the following: For $v\\in[p]$, $E(C)_v=1$ iff $x_v$ is a literal in $C$, and $E(C)_{p+v}=1$ iff $\\neg x_v$ is a literal in $C$. All positions in $E(C)$ are 0 otherwise.\n\nDefinition 6 (Encoding of partial assignment). Let $A:\\{x_1,\\ldots,x_p\\}\\to\\{\\mathrm{True},\\mathrm{False},\\mathrm{None}\\}$ be a partial assignment. Define encoding $E(A)\\in\\{0,1\\}^{2p}$ of clause $C$ as the following: For $v\\in[p]$, $E(A)_v=1$ iff $A(x_v)=\\mathrm{True}$ , and $E(A)_{p+v}=1$ iff $A(x_v)=\\mathrm{False}$. All positions in $E(C)$ are 0 otherwise.\n\nWe also define a variant of the partial assignment encoding as an affine Transformers of $E(A)$, which sets both positions corresponding to a variable to 1 if the variable is unassigned:\n\nProposition 1. For partial assignment $A$, define $E_{\\textit{not-false}}(A)=M_{\\textit{not-false}}\\cdot E(A)+\\mathbf{1}^{2p}$ where $M_{\\textit{not-false}}=\\begin{bmatrix}\\mathbf{0}&-\\mathbf{I}_p\\\\-\\mathbf{I}_p&\\mathbf{0}\\end{bmatrix}\\in\\mathbb{R}^{2p\\times 2p}$ and $\\mathbf{1}^{2p}$ is the all ones vector. Then, for $v\\in[p]$:\n$$\nE_{\\textit{not-false}}(A)_v=1 \\text{ iff } A(x_v)\\in\\{\\mathrm{True},\\mathrm{None}\\},\\quad E_{\\textit{not-false}}(A)_{p+v}=1 \\text{ iff } A(x_v)\\in\\{\\mathrm{False},\\mathrm{None}\\}.\n$$\n\nWe now show that the relationship between a 3-SAT formula and a partial assignment can be established using their binary encoding:\n\nLemma 4. Let $F$ be a 3-SAT formula over variables $\\{x_1,\\ldots,x_p\\}$ with $c$ clauses $\\{C_1,\\ldots,C_c\\}$ and $A$ a partial assignment defined on variables $\\{x_1,\\ldots,x_p\\}$, then the following properties hold:\n\n1. Satisfiability Checking: The partial assignment $A$ satisfies the formula $F$ if and only if:\n$$\n\\forall i\\in[c],\\quad E(C_i)\\cdot E(A)\\geq 1.\n$$\n\n2. Conflict Detection: The partial assignment $A$ contradicts the formula $F$ if and only if:\n$$\n\\exists i\\in[c],\\quad E(C_i)\\cdot E_{\\textit{not-false}}(A)=0.\n$$\n\n3. Deduction: If partial assignment $A$ does not contradict formula $F$, then\n\n(a) A variable $x_v$ is implied to be true under $A$ and $F$ if:\n$$\n\\exists i\\in[c],\\quad E(C_i)\\cdot E_{\\textit{not-false}}(A)\\leq 1,\\quad E(C_i)_v=1,\\quad \\text{and}\\quad E(A)_v=E(A)_{p+v}=0.\n$$\n\n(b) A variable $x_v$ is implied to be false under $A$ and $F$ if :\n$$\n\\exists i\\in[c],\\quad E(C_i)\\cdot E_{\\textit{not-false}}(A)\\leq 1,\\quad E(C_i)_{p+v}=1,\\quad \\text{and}\\quad E(A)_v=E(A)_{p+v}=0.\n$$\n\nRecall that an attention head computes a query and key vector from the hidden states and the attention weight between two positions is based on the dot product between the query vector of the source position and the key vector of the target position. If the Transformer weights are configured such that the query vectors are Figure 1 is $E(A)$ or $E_{\\mathrm{not\\text{-}false}}(A)$ for partial assignments $A$ in the Chain-of-Thought illustrated, and the key vectors are $E(C_i)$ for positions of clauses $C_i$ in the formula, then the attention weight (before softmax) would be proportional to $E(C_i)\\cdot E(A)$ or $E(C_i)\\cdot E_{\\mathrm{not\\text{-}false}}(A)$ respectively, which are values crucial for the operations in 4. We can then scale the attention weights so that the the attention weights focus on only the extremal values of $E(C_i)\\cdot E(A)$ or $E(C_i)\\cdot E_{\\mathrm{not\\text{-}false}}(A)$. We illustrate the consequence of this correlation with the following informal lemma, which considers an idealized input that contains only the positions with encoding vectors and auxiliary values:\n\nLemma 5 (Parallel Processing of Clauses, Informal). Let $F$ be a 3-SAT formula over variables $\\{x_1,\\ldots,x_p\\}$ with $c$ clauses $\\{C_1,\\ldots,C_c\\}$ and $A$ a partial assignment defined on variables $\\{x_1,\\ldots,x_p\\}$. Let\n$$\nX_{encoding}=\\begin{bmatrix}0&1&1\\\\E(C_1)&0&1\\\\\\vdots&\\vdots&\\vdots\\\\E(C_c)&0&1\\\\E(A)&0&1\\end{bmatrix}\\in\\mathbb{R}^{(c+2)\\times(2p+2)}\n$$\nwhich includes encoding of clauses in $F$ and partial assignment $A$ as well as added auxiliary values. Let $\\mathbf{1}_{A\\models F}$ denote the indicator variable of whether $A$ satisfy formula $F$, $\\mathbf{1}_{A\\not\\models F}$ denote the indicator variable of whether $A$ constradict $F$, and $e_{UP}\\in 0,1^{2p}$ denote the encoding of all variable assignments that can be deduced from $A$ and $F$, then with $X_{encoding}$ as input and any $1>\\epsilon>0$ there exists:\n\n- An attention head that outputs $\\mathbf{1}_{A\\models F}$ with approximation error bounded by $\\epsilon$\n- An attention head that outputs $\\mathbf{1}_{A\\not\\models F}$ with approximation error bounded by $\\epsilon$\n- An attention head followed by a MLP layer that outputs $e_{UP}$ with $\\|\\cdot\\|_\\infty$ error bounded by $\\epsilon$\n\nand all weight values are bounded by $O(poly(p,c,\\log(1/\\epsilon)))$\n\nLemma 5 essentially shows that, when given the binary encoding of clauses and a partial assignment, a single Transformer layer can perform satisfiability checking, conflict detection, and deduction over all clauses in the formula in parallel, which is the core reasoning our theoretical construction uses drastically less CoT tokens than step-wise simulation of Turing Machines.\n\nThe remaining parts of the construction performs indexing operations that translates DIMACS encodings into our encoding of clauses and partial assignments and selects the correct output token from the results of the operations described in Lemma 5."}]}}, {"local_id": "T3", "statement": "There exists a decoder-only Transformer $M:\\mathcal{V}^*\\to\\mathcal{V}$ that solves 3-SAT via Chain-of-Thought reasoning, i.e. that reliably decides whether an arbitrary 3-SAT formula in DIMACS encoding is satisfiable.", "proof": {"steps": [{"n": 1, "claim": "By Theorem 1, for any $p,c \\in \\mathbb{N}^+$ there exists $M:\\mathcal{V}^*\\to\\mathcal{V}$ autoregressively deciding $3\\text{-}\\mathrm{SAT}_{p,c}$ in at most $p\\cdot 2^{p+1}$ CoT iterations, with $L=7$, $H=5$, $d_{emb}=\\mathcal{O}(p)$, $\\mathcal{O}(p^2)$ parameters."}, {"n": 2, "claim": "By Definition 4, for each fixed $(p,c)$ this $M$ decides $3\\text{-}\\mathrm{SAT}_{p,c}$: it determines satisfiability of every formula with at most $p$ variables and $c$ clauses in $\\mathrm{DIMACS}(p,c)$."}, {"n": 3, "claim": "The constructions above share one architecture template ($L=7$, $H=5$, $d_{emb}=\\mathcal{O}(p)$, $\\mathcal{O}(p^2)$ parameters), doing logical deduction on all clauses in parallel and backtracking via CoT."}, {"n": 4, "claim": "Hence there exists a single decoder-only Transformer $M$ that solves 3-SAT and reliably decides arbitrary SAT instances via CoT."}]}}, {"local_id": "T4", "statement": "Lemma 2 (Simulating a 2-Layer ReLU MLP with ReGLU Activation). A 2-layer MLP with ReGLU activation function can simulate any 2-layer MLP with ReLU activation function.", "proof": {"steps": [{"n": 1, "claim": "Set the weights and biases of the ReGLU MLP as follows: $$ \\begin{array}{ c }{{W_{1}={\\bf 0},\\quad b_{1}={\\bf 1},}}\\\\ {{\\nonumber}}\\\\ {{V=W_{1}^{\\prime},\\quad b_{2}=b_{1}^{\\prime},}}\\\\ {{W_{2}=W_{2}^{\\prime},\\quad b=b_{2}^{\\prime}.}}\\end{array} $$"}, {"n": 2, "claim": "Then, the ReGLU MLP computes: $$ f(\\pmb{x})=\\pmb{W}_{2}^{\\prime}\\left[(\\mathbf{0}\\cdot\\pmb{x}+\\mathbf{1})\\otimes\\mathrm{ReLU}(\\pmb{W}_{1}^{\\prime}\\pmb{x}+\\pmb{b}_{1}^{\\prime})\\right]+\\pmb{b}_{2}^{\\prime}. $$"}, {"n": 3, "claim": "Simplifying: $$ \\begin{array}{ r }{f(\\pmb{x})=W_{2}^{\\prime}\\left[\\mathbf{1}\\otimes\\mathrm{ReLU}(W_{1}^{\\prime}\\pmb{x}+b_{1}^{\\prime})\\right]+b_{2}^{\\prime}=W_{2}^{\\prime}\\mathrm{ReLU}(W_{1}^{\\prime}\\pmb{x}+b_{1}^{\\prime})+b_{2}^{\\prime}=g(\\pmb{x}).}\\end{array} $$"}]}}, {"local_id": "T5", "statement": "Lemma 3 (Element-wise Multiplication via ReGLU MLP). A 2-layer MLP with ReGLU activation can compute the element-wise multiplication of two input vectors x1 and x2, that is, $$ f(\\pmb{x})=\\pmb{x}_{1}\\otimes\\pmb{x}_{2}, $$ where $\\pmb{x}=[\\pmb{x}_{1};\\pmb{x}_{2}]$ denotes the concatenation of $\\scriptstyle{\\mathbf{\\dot{x}}}_{1}$ and $\\mathbf{\\boldsymbol{x}}_{2}$", "proof": {"steps": [{"n": 1, "claim": "Set the weights and biases: $$ W_{1}=\\begin{bmatrix}I_{n}\\\\ I_{n}\\end{bmatrix},\\qquad b_{1}=\\mathbf{0}_{2 n}, $$"}, {"n": 2, "claim": "$$ {\\cal V}=\\begin{bmatrix}I_{n}\\\\ -I_{n}\\end{bmatrix},\\qquad b_{2}=\\mathbf{0}_{2 n}, $$"}, {"n": 3, "claim": "$$ W_{2}=\\left[I_{n}\\quad-I_{n}\\right],\\quad b=\\mathbf{0}_{n}. $$"}, {"n": 4, "claim": "Compute: $$ \\pmb{W}_{1}\\pmb{x}+\\pmb{b}_{1}=\\begin{bmatrix}\\pmb{x}_{1}\\\\ \\pmb{x}_{1}\\end{bmatrix}, $$ $$ V x+b_{2}=\\begin{bmatrix}x_{2}\\\\ -x_{2}\\end{bmatrix}, $$ $$ \\operatorname{ReLU}(V x+b_{2})=\\begin{bmatrix}\\operatorname{ReLU}({\\pmb x}_{2})\\\\ \\operatorname{ReLU}(-{\\pmb x}_{2})\\end{bmatrix}. $$"}, {"n": 5, "claim": "The element-wise product: $$ (\\pmb{W}_{1}\\pmb{x}+\\pmb{b}_{1})\\otimes\\mathrm{ReLU}(\\pmb{V}\\pmb{x}+\\pmb{b}_{2})=\\begin{bmatrix}\\pmb{x}_{1}\\otimes\\mathrm{ReLU}(\\pmb{x}_{2})\\\\ \\pmb{x}_{1}\\otimes\\mathrm{ReLU}(-\\pmb{x}_{2})\\end{bmatrix}. $$"}, {"n": 6, "claim": "Compute the output: $$ \\begin{array}{ r l }&{f(\\pmb{x})=W_{2}\\left[\\left(W_{1}\\pmb{x}+\\pmb{b}_{1}\\right)\\otimes\\mathrm{ReLU}(V\\pmb{x}+\\pmb{b}_{2})\\right]+b}\\\\ &{\\qquad=\\pmb{x}_{1}\\otimes\\mathrm{ReLU}(\\pmb{x}_{2})-\\pmb{x}_{1}\\otimes\\mathrm{ReLU}(-\\pmb{x}_{2})}\\\\ &{\\qquad=\\pmb{x}_{1}\\otimes\\left(\\mathrm{ReLU}(\\pmb{x}_{2})-\\mathrm{ReLU}(-\\pmb{x}_{2})\\right)}\\\\ &{\\qquad=\\pmb{x}_{1}\\otimes\\pmb{x}_{2}.}\\end{array} $$"}]}}], "target_local_id": "T3", "topic_domain": "Deep Learning Theory", "topic_subtopic": "Representation & Expressivity", "label": "flawed", "error_step": 4, "error_class6": ["logical_gap_or_missing_case", "others", "missing_assumption"], "error_explanation": "Step 4 performs an illegitimate quantifier swap: it converts the '$\\forall (p,c)\\,\\exists M$' guarantee of Theorem 4.5 into '$\\exists M\\,\\forall\\text{ instances}$', asserting a single fixed Transformer that decides arbitrary 3-SAT. Theorem 4.5 only builds, for each bounded size $(p,c)$, an input-size-dependent model $M$ (with $d_{emb}=\\mathcal{O}(p)$ and $\\mathcal{O}(p^2)$ parameters that grow with $p$) that decides only $3\\text{-}\\mathrm{SAT}_{p,c}$, i.e. instances of at most $p$ variables and $c$ clauses. This is a non-uniform / bounded-input result; it does not imply the existence of one uniform Transformer solving general SAT. The step silently drops the boundedness (non-uniformity) assumption. The correct statement is that one constructs a non-uniform class of decoder-only Transformers, one per input size, deciding all 3-SAT instances of at most $p$ variables and $c$ clauses."}
{"id": "XcAJ0qsMgh", "context": "2 Preliminaries\n\nNeural ODE and Continuous Normalizing Flow: A Neural ODE is a continuous model where the trajectory of data is modeled as the solution of an ordinary differential equation (ODE). Formally, in $\\mathbf{\\mathbb{R}}^{d}$ , given an input $x(t_{0})=x_{0}$ at time $t_{0}$ , the transformation to the output $x(T)$ is governed by:\n\n$$\n\\frac{d x(t)}{d t}=\\mathbf v(x(t),t),\\tag{2}\n$$\n\nwhere $\\mathbf{v}(x(t),t)$ represents the velocity field, which is of the same dimension as $x(t)$ and is parameterized by a neural network with input $x(t)$ and t.\n\nA Continuous Normalizing Flow (CNF) is a class of normalizing flows where the transformation of a probability density from a base distribution $p(x)$ (at $t=0)$ to a target distribution $q(x)$ (at $t=T)$ is governed by a Neural ODE. The marginal density of $x(t)$ , denoted as $\\rho(x,t)$ , evolves according to the continuity equation derived from the ODE in Eq. (2). This continuity equation is written as:\n\n$$\n\\partial_{t}\\rho(x,t)+\\nabla\\cdot(\\rho(x,t)\\mathbf{v}(x,t))=0,\\quad\\rho(x,0)=p(x),\\tag{3}\n$$\n\nwhere the divergence $\\nabla\\cdot(\\rho\\boldsymbol{v})$ accounts for the change in density as the flow evolves over time.\n\nDynamic Optimal Transport $(O T){\\mathrm{{z}}}$ The Benamou-Brenier equation (Benamou & Brenier, 2000) below provides the dynamic formulation of Optimal Transport $\\tau .$\n\n$$\n\\begin{array}{ r l }&{\\underset{\\rho,v}{\\operatorname*{inf}}\\int_{0}^{1}\\mathbb{E}_{\\boldsymbol{x}(t)\\sim\\rho(\\cdot,t)}\\|\\mathbf{v}(\\boldsymbol{x}(t),t)\\|^{2}d t}\\\\ &{\\mathrm{s . t .}\\quad\\partial_{t}\\rho+\\nabla\\cdot(\\rho v)=0,\\quad\\rho(\\cdot,0)=p,\\quad\\rho(\\cdot,1)=q,}\\end{array}\\tag{4}\n$$\n\nThe optimization problem seeks to find the optimal transport map that moves mass from the base density p to the target density $q,$ subject to the continuity equation (3) to ensure that $\\rho(\\cdot,t)$ evolves as a valid probability density over time. Additionally, the constraint $\\rho(\\cdot,1)=q$ ensures that the target density is reached by the end of the time horizon. The time horizon is scaled to $[0,1]$\n\n3 Annealing Flow Model\n\nThe annealing philosophy (Gelfand et al., 1990; Neal, 2001; Sorkin, 1991; Van Groenigen & Stein, 1998) refers to gradually transitioning an initial fattened distribution to the target distribution as the temperature decreases. Building on this idea, we introduce Annealing Flow (AF), a sampling algorithm that learns a continuous normalizing flow to gradually map an initial easy-to-sample density $\\pi_{0}(x)$ to the target density $q(x)$ through a set of intermediate distributions.\n\nWe define $q(x)=Z\\tilde{q}(x)$ where $\\tilde{q}(x)$ represents the unnormalized target distribution given in explicit form. Next, we define a sequence of intermediate distributions $f_{k}(x)$ that interpolate between an easy-to-sample initial distribution $\\pi_{0}(x)$ (e.g., a Gaussian) and the target $q(x)$ . These intermediate distributions are formulated as:\n\n$$\nf_{k}(x)=\\pi_{0}(x)^{1-\\beta_{k}}q(x)^{\\beta_{k}}=Z_{k}{\\tilde{f}}_{k}(x),\\tag{5}\n$$\n\nHere $\\tilde{f}_{k}(x)=\\pi_{0}(x)^{1-\\beta_{k}}\\tilde{q}(x)^{\\beta_{k}}$ , and $\\beta_{k}$ is an increasing sequence with $\\beta_{0}=0$ and $\\beta_{K}=1$ . This formulation ensures that $\\tilde{f}_{0}(x)=\\pi_{0}(x)$ and $\\tilde{f}_{K}(x)=\\tilde{q}(x)$ . The sequence $0=\\beta_{0}<\\beta_{1}<\\cdots<$ $\\beta_{K}=1$ controls the gradual transition between the two distributions.\n\nThe above construction aligns with the annealing philosophy. As $\\beta_{k}$ increases, $\\tilde{f}_{k}(x)$ gradually sharpens toward the target $\\bar{\\tilde{q}(\\boldsymbol{x})}$ , starting from the initially flattened distribution around $\\pi_{0}(x)$ . These annealed densities serve as a bridge, providing a gradual flow path from the easy-to-sample distribution $\\pi_{0}(x)$ to the target density $q(x)$ . Figure 1 provides an intuitive illustration of this process, where $\\pi_{0}(x)$ is a standard Gaussian, and $q(x)$ is a Gaussian mixture model with six modes.\n\n![](images/82cdf27af71021dd9b4b3d8ce786559c738e404ba29c89657bc779fff8abf6ab.jpg)  \n(a) β0 = 0\n\n(b) β1 = 1/3  \n![](images/f229f472135185f0ec6bf61d6b59350bde1b12c0e342265fc74b77e84292e578.jpg)  \n(c) β2 = 2/3\n\n![](images/252ee1aa0708106fcf553c2275a7b9f4740d803fe077a1606a684439e02ecf96.jpg)  \n(d) β3 = 1  \nFigure 1: Illustration of the Annealing Flow Map, with a set of intermediate distributions from $\\pi_{0}(x)=N(0,I_{2})$ to $q(x)$ , a GMM with 6 modes.\n\n3.1 Optimal Transport Map\n\nWe aim to learn a continuous optimal transport map between an easy-to-sample distribution $\\pi_{0}(x)$ and the target distribution $q(x)$ . Once trained, users simply sample $\\{x^{(i)}(0)\\}_{i=1}^{n}\\sim\\pi_{0}(x)$ , and the transport map pushes them to $\\{x^{(i)}(1)\\}_{i=1}^{n}\\sim q(x)$ . The transport map $\\tau$ evolves the density according to (3), which in turn drives the evolution of the sample $x\\bar{(t)}$ following the ODE in (2):\n\n$$\n\\mathcal{T}(x(t))=x(0)+\\int_{0}^{t}\\mathbf{v}(x(s),s)d s,\\quad t\\in [0,1].\\tag{6}\n$$\n\nWe divide the time horizon $[0,1]$ of $\\tau$ into K intervals $[t_{k-1},t_{k}]$ for $k=1,2,\\dots,K$ , where $t_{0}=$ 0 and $t_{K}=1$ Guided by the annealing flow path defined in (5), the continuous flow map $\\tau$ gradually transforms the density from $f_{0}(\\bar{\\boldsymbol{x}})$ to $\\dot{f_{1}}(x)$ over $[0,t_{1}]$ , and continues this process until $f_{K-1}(x)$ is transformed into $f_{K}(x)~=~q(x)$ over $[t_{K-1},t_{K}]$ Figure 1 shows this progression with two intermediate distributions. For clarity, we denote $\\mathcal{T}_{k}(x)$ as the segment of the continuous normalizing flow during $[t_{k-1},t_{k}]$ , which pushes the density from $f_{k-1}(x)$ to $f_{k}(x)$\n\n3.2 Objective of Annealing Flow Net\n\nAnnealing Flow aims to learn each transport map $\\mathcal{T}_{k}$ based on dynamic OT objective (4) over the time horizon $[t_{k-1},t_{k}]$ , where the velocity field ${\\mathbf{v}}_{\\mathbf{k}}(x(t),t)$ is learned using a neural network. The terminal condition $\\rho(\\cdot,1)=q$ in (4) can be relaxed by introducing a KullbackLeibler (KL) divergence term (see, for instance, Ruthotto et al. (2020)). Consequently, minimizing the objective (4) for dynamic optimal transport $\\mathcal T_{k}:f_{k-1}(x)\\to f_{k}(x)$ can be reduced to solving the following problem:\n\n$$\n\\mathcal{T}_{k}=\\arg\\operatorname*{min}_{\\mathcal{T}}\\left\\{\\mathrm{KL}(\\mathcal{T}_{\\#}f_{k-1}\\|f_{k})+\\gamma\\int_{t_{k-1}}^{t_{k}}\\mathbb{E}_{x(t)\\sim\\rho_{k}(\\cdot,t)}\\|\\mathbf{v_{k}}(x(t),t)\\|^{2}dt\\right\\},\\tag{7}\n$$\n\nsubject to $\\rho_{k}(x(t),t)$ and ${\\mathbf{v}}_{\\mathbf{k}}(x(t),t)$ evolving according to (3). Here, $\\gamma > 0$ is a regularization parameter, ${\\mathbf{v}}_{\\mathbf{k}}(x(t),t)$ denotes the velocity field during the k-th time interval $[t_{k-1},t_{k}]$ , and $\\mathrm{KL}(\\dot{T}_{\\#}f_{k-1}\\|f_{k})$ represents the KL divergence between the push-forward density $\\bar{\\mathcal{T}_{\\#}}f_{k-1}$ and the target density $f_{k}$ . Additionally, the constraint (3) ensures that $x(t)$ follows the ODE trajectory defined by (2) during $t\\in [t_{k-1},\\dot{t}_{k}]$ , which is given by:\n\n$$\nx(t)=x(t_{k-1})+\\int_{t_{k-1}}^{t}\\mathbf{v_{k}}(x(s),s)d s,\\quad t\\in [t_{k-1},t_{k}].\\tag{8}\n$$\n\nWe can rewrite $\\tilde{f}_{k}(x)=Z e^{E_{k}(x)}$ , where $E_{k}(x)$ is the energy function, with the associated unnormalized energy given by $\\tilde{E}_{k}(x)= -\\log\\tilde{f}_{k}$ The following proposition shows that once we have obtained samples from $f_{k-1}(x)$ , the KL divergence in (7) can be computed exactly based on ${\\mathbf{v}}_{\\mathbf{k}}(x(t),t)$ and $\\tilde{E}_{k}(x)$ Therefore, learning an optimal transport map $\\mathcal{T}_{k}$ reduces to learning the optimal ${\\mathbf{v}}_{\\mathbf{k}}(x(t),t)$ . The proof is provided in Appendix A.1.\n\nProposition 2 (KL-Divergence Decomposition) Given the unnormalized density $f_{k-1}$ ,the $K L -$ Divergence between T# $f_{k-1}$ and $f_{k}$ is equivalent to:\n\n$$\nK L({\\mathcal{T}}_{\\#}f_{k-1}\\|f_{k})=c+\\mathbb{E}_{x(t_{k-1})\\sim f_{k-1}}\\left[{\\tilde{E}}_{k}(x(t_{k}))-\\int_{t_{k-1}}^{t_{k}}{\\boldsymbol{\\nabla}}\\cdot\\mathbf{v_{k}}(x(s),s)d s\\right],\\tag{9}\n$$\n\nup to a constant c that is independent of ${\\bf v_{k}}(x(s),s)$\n\nGiven $x(t_{k-1})$ from $f_{k-1}(x)$ , the value of $x(t_{k})$ inside the energy function $\\tilde{E}_{k}$ can be calculated as shown in equation (8). Additionally, according to the proposition below, the second term in the objective (7) can be relaxed as a discretized sum. The proof is provided in Appendix A.1.\n\nProposition 3 (Wasserstein Distance Discretization) Let x(t) be particle trajectories driven by a smooth velocity field ${\\mathbf{v}}_{\\mathbf{k}}(x(t),t)$ over the time interval $[t_{k-1},t_{k}]$ , where $h_{k}=t_{k}-t_{k-1}.$ Assume that $\\mathbf{v}_{\\mathbf{k}}(x,t)$ is Lipschitz continuous in both x and t. By dividing $[t_{k-1},t_{k}]$ into S equal miniintervals with grid points $t_{k-1,s}$ (where $s=0,1,\\ldots,S$ and $t_{k-1,0}=t_{k-1},t_{k-1,S}=t_{k})$ , we have:\n\n$$\n\\int_{t_{k-1}}^{t_{k}}\\mathbb{E}_{x(t)}\\left[\\|\\mathbf{v_{k}}(x(t),t)\\|^{2}\\right]d t=\\frac{S}{h_{k}}\\sum_{s=0}^{S-1}\\mathbb{E}\\left[\\|x(t_{k-1,s+1})-x(t_{k-1,s})\\|^{2}\\right]+O(h_{k}^{2}/S).\\tag{10}\n$$\n\nAs $h_{k}0 o r S\\infty,$ ,the error term $O(h_{k}^{2}/S)$ becomes negligible.\n\nOne can observe that the RHS of (10) can be interpreted as the discretized sum of the squared Wasserstein-2 distance. The dynamic $W_{2}$ regularization encourages smooth transitions from $f_{k-1}$ to $f_{k}$ with minimal transport cost, promoting efficient mode exploration.\n\nNext, by incorporating Propositions 2 and 3 into objective (7), the final objective becomes:\n\n$$\n\\operatorname*{min}_{\\mathbf{v}_{k}(\\cdot,t)}\\mathbb{E}_{\\boldsymbol{x}(t_{k-1})\\sim f_{k-1}}\\left[\\tilde{E}_{k}(\\boldsymbol{x}(t_{k}))-\\int_{t_{k-1}}^{t_{k}}\\nabla\\cdot\\mathbf{v_{k}}(\\boldsymbol{x}(s),s)d s+\\alpha\\sum_{s=0}^{S-1}\\|\\boldsymbol{x}(t_{k-1,s+1})-\\boldsymbol{x}(t_{k-1,s})\\|^{2}\\right].\\tag{11}\n$$\n\nHere, $\\alpha~=~\\gamma S/h_{k}$ and ${\\mathbf{v}}_{\\mathbf{k}}(x(s),s)$ is learned by a neural network. We break the time interval $[t_{k-1},t_{k}]$ into S mini-intervals, and $x(t_{k-1,s+1})$ is computed as in equation (8).\n\nAfter learning, connecting the Annealing Flow nets together yields a smooth flow map $\\mathcal{T}:\\mathcal{T}_{1} $ $\\mathcal{T}_{2}\\to\\cdots\\to\\overline{{\\mathcal{T}_{K}}}$ ,which transforms samples from $\\pi_{0}(x)$ to the target $q(x)$ . Please see Section 4.2 for efficient sampling of Annealing Flow and its comparisons with other sampling methods.\n\n3.3 Properties of Learned Velocity Field\n\nThe objective in (11) can be reformulated as shown below when $h_{k}=t_{k}-t_{k-1}\\to 0$ . The proof is provided in Appendix A.2.\n\nProposition 4 (Objective Reformulation) Denote $h_{k}=t_{k}-t_{k-1} $ ,and let $\\mathbf{s_{k}}=\\nabla$ log $f_{k}(x)$ denote the score function of $f_{k}$ As $h_{k}\\ \\to\\ 0$ and with $\\begin{array}{ r }{\\gamma=\\frac{1}{2}}\\end{array}$ (so that $\\begin{array}{ r }{\\alpha=\\frac{S}{2 h_{k}}),}\\end{array}$ the objective in (11) becomes equivalent to the following:\n\n$$\n\\operatorname*{min}_{\\mathbf{v}_{\\mathbf{k}}=\\mathbf{v}_{\\mathbf{k}}(\\cdot,0)}\\mathbb{E}_{x\\sim f_{k-1}}\\left[-T_{f_{k}}\\mathbf{v}_{\\mathbf{k}}+\\frac{1}{2}\\|\\mathbf{v}_{\\mathbf{k}}\\|^{2}\\right],\\quad T_{f_{k}}\\mathbf{v}_{\\mathbf{k}}: =\\mathbf{s_{k}}\\cdot\\mathbf{v}_{\\mathbf{k}}+\\nabla\\cdot\\mathbf{v_{k}}.\\tag{12}\n$$\n\nDefine $\\begin{array}{ r l r }{L^{2}(f_{k-1})}&{=}&{\\left\\{v:\\mathbb{R}^{d}\\to\\mathbb{R}^{d}\\ \\big\\vert\\int_{\\mathbb{R}^{d}}\\|\\mathbf{v}(x)\\|^{2}f_{k-1}(x)d x<\\infty\\right\\}}\\end{array}$ as the $L^{2}$ space over $(\\mathbb{R}^{d},f_{k-1}(x)d x)$ . We can then establish the following property, with proofs provided in Appendix $\\mathrm{{A . 2}}\\mathrm{{:}}$\n\nProposition 5 (Optimal Velocity Field as Score Difference) Suppose $h_{k}\\to 0$ and $\\begin{array}{ r }{\\gamma=\\frac{1}{2}}\\end{array}$ Let $f_{k-1}$ and $f_{k}$ be continuously differentiable on $\\mathbb{R}^{d}.$ Assume that $\\nabla\\cdot\\mathbf{v_{k}}(x)$ exists for all $x\\in\\mathbb{R}^{d}$ , and $\\nabla\\cdot{\\mathbf{v_{k}}}(x),{\\mathbf{s_{k-1}}}$ and sk belong to $L^{2}(f_{k-1})$ . Assume that the components of $\\mathbf{v_{k}}$ are independent and $\\operatorname*{lim}_{\\|x\\|\\to\\infty}f_{k-1}(x)\\|\\mathbf{v_{k}}(x)\\|_{2}=0 .$ Under these conditions, the minimizer of (11) is:\n\n$$\n{\\mathbf{v_{k}}}^{*}={\\mathbf{s_{k}}}-{\\mathbf{s_{k-1}}}.\\tag{13}\n$$\n\nTherefore, the infinitesimal optimal $\\mathbf{v_{k}}^{*}$ is equal to the difference between score function of the next density, $f_{k}$ , and the current density, $f_{k-1}$ . This suggests that when the two intermediate densities are sufficiently close, i.e., when the number of $\\beta_{k}$ is large enough, the optimal velocity field equals the difference between the score functions. By adding more intermediate densities, one can construct a sufficiently smooth transport map $\\tau$ that exactly learns the mapping between each pair of densities.\n\nAdditionally, one can observe that when each $\\tilde{f}_{k}(x)$ is set to the target $q(x)$ , i.e., when all $\\beta_{k}$ are set to 1, and the second term in the objective (7) is relaxed to static $\\bar{W_{2}}$ regularization, the objective of Annealing Flow becomes equivalent to Wasserstein gradient flow. This is detailed in Appendix B.", "theorems": [{"local_id": "T1", "statement": "Proposition 3. (Wasserstein Distance Discretization) Let $x(t)$ be particle trajectories driven by a smooth velocity field $\\mathbf{v}_k(x(t),t)$ over the time interval $[t_{k-1},t_k]$, where $h_k = t_k - t_{k-1}$. Assume that $\\mathbf{v}_k(x,t)$ is Lipschitz continuous in both $x$ and $t$. By dividing $[t_{k-1},t_k]$ into $S$ equal mini-intervals with grid points $t_{k-1,s}$ (where $s=0,1,\\ldots,S$ and $t_{k-1,0}=t_{k-1}$, $t_{k-1,S}=t_k$), the following approximation holds: $$\\int_{t_{k-1}}^{t_k} \\mathbb{E}_{x(t)}\\left[\\|\\mathbf{v}_k(x(t),t)\\|^2\\right] dt = \\frac{S}{h_k} \\sum_{s=0}^{S-1} \\mathbb{E}\\left[\\|x(t_{k-1,s+1})-x(t_{k-1,s})\\|^2\\right] + O(h_k^2/S).$$ As $h_k\\to 0$ or $S\\to\\infty$, the error term $O(h_k^2/S)$ becomes negligible.", "proof": {"steps": [{"n": 1, "claim": "Divide $[t_{k-1},t_k]$ into $S$ equal mini-intervals of length $\\delta t=h_k/S$, with grid points $t_{k-1,s}=t_{k-1}+s\\cdot\\delta t$, $s=0,\\ldots,S$."}, {"n": 2, "claim": "Within each mini-interval, Taylor-expand $x$ around $t_{k-1,s}$: $$x(t_{k-1,s+1}) = x(t_{k-1,s}) + \\mathbf{v}_k(x(t_{k-1,s}),t_{k-1,s})\\,\\delta t + \\frac{1}{2}\\frac{d\\mathbf{v}_k}{dt}\\,\\delta t^2 + O(\\delta t^3).$$"}, {"n": 3, "claim": "Here $d\\mathbf{v}_k/dt$ denotes the total (material) derivative of $\\mathbf{v}_k$ along the trajectory, $d\\mathbf{v}_k/dt=\\partial_t \\mathbf{v}_k+(\\nabla_x \\mathbf{v}_k)\\mathbf{v}_k$."}, {"n": 4, "claim": "$$\\|x(t_{k-1,s+1})-x(t_{k-1,s})\\|^2 = \\|\\mathbf{v}_k(x(t_{k-1,s}),t_{k-1,s})\\|^2\\, \\delta t^2 + O(\\delta t^3).$$"}, {"n": 5, "claim": "The residual after the $\\|\\mathbf{v}_k\\|^2\\delta t^2$ term is genuinely $O(\\delta t^3)$ and becomes negligible as $\\delta t\\to 0$."}, {"n": 6, "claim": "$$\\sum_{s=0}^{S-1} \\mathbb{E}\\left[\\|x(t_{k-1,s+1})-x(t_{k-1,s})\\|^2\\right] = \\delta t^2 \\sum_{s=0}^{S-1} \\mathbb{E}\\left[\\|\\mathbf{v}_k(x(t_{k-1,s}),t_{k-1,s})\\|^2\\right] + O(S\\cdot\\delta t^3).$$"}, {"n": 7, "claim": "$$\\int_{t_{k-1}}^{t_k} \\mathbb{E}\\left[\\|\\mathbf{v}_k(x(t),t)\\|^2\\right] dt = \\delta t \\sum_{s=0}^{S-1} \\mathbb{E}\\left[\\|\\mathbf{v}_k(x(t_{k-1,s}),t_{k-1,s})\\|^2\\right] + O(S\\cdot\\delta t^2) = \\frac{1}{\\delta t} \\sum_{s=0}^{S-1} \\mathbb{E}\\left[\\|x(t_{k-1,s+1})-x(t_{k-1,s})\\|^2\\right] + O(S\\cdot\\delta t^2),$$ which is the claim since $1/\\delta t=S/h_k$ and $O(S\\cdot\\delta t^2)=O(h_k^2/S)$."}]}}, {"local_id": "T2", "statement": "Proposition 4. (Objective Reformulation) Denote $h_{k}=t_{k}-t_{k-1}$ , and let $\\mathbf{s_{k}}=\\nabla$ log $f_{k}(x)$ denote the score function of $f_{k}$ As $h_{k}\\ \\to\\ 0$ and with $\\begin{array}{ r }{\\gamma=\\frac{1}{2}}\\end{array}$ (so that $\\begin{array}{ r }{\\alpha=\\frac{S}{2 h_{k}}),}\\end{array}$ the objective in (11) becomes equivalent to the following:\n\n$$\n\\operatorname*{min}_{\\mathbf{v}_{\\mathbf{k}}=\\mathbf{v}_{\\mathbf{k}}(\\cdot,0)}\\mathbb{E}_{x\\sim f_{k-1}}\\left[-T_{f_{k}}\\mathbf{v}_{\\mathbf{k}}+\\frac{1}{2}\\|\\mathbf{v}_{\\mathbf{k}}\\|^{2}\\right],\\quad T_{f_{k}}\\mathbf{v}_{\\mathbf{k}}: =\\mathbf{s_{k}}\\cdot\\mathbf{v}_{\\mathbf{k}}+\\nabla\\cdot\\mathbf{v_{k}}.\n$$", "proof": {"steps": [{"n": 1, "claim": "From the Neural ODE (2) and using Taylor's expansion, we obtain:\n\n$$\nx(t_{k})-x(t_{k-1})=\\int_{t_{k-1}}^{t_{k}}\\mathbf{v_{k}}(x(s),s)d s=h_{k}\\mathbf{v_{k}}(x(t_{k-1}),t_{k-1})+O(h_{k}^{2})\n$$"}, {"n": 2, "claim": "Next, by performing Taylor expansion of $\\tilde{E}_{k}(x(t_{k}))$ around $t_{k-1} $\n\n$$\n\\begin{array}{ r l }&{\\tilde{E}_{k}(x(t_{k}))=\\tilde{E}_{k}(x(t_{k-1}))+(x(t_{k})-x(t_{k-1}))\\nabla\\tilde{E}_{k}(x(t_{k-1}))+O(h_{k}^{2})}\\\\ &{\\qquad=\\tilde{E}_{k}(x(t_{k-1}))+h_{k}\\nabla\\tilde{E}_{k}(x(t_{k-1}))\\cdot{\\mathbf{v_{k}}}(x(t_{k-1}),t_{k-1})+O(h_{k}^{2})}\\end{array}\n$$"}, {"n": 3, "claim": "Besides, we also have that:\n\n$$\n\\int_{t_{k-1}}^{t_{k}}\\nabla\\cdot{\\mathbf{v_{k}}}(x(s),s)d s=h_{k}\\nabla\\cdot{\\mathbf{v_{k}}}(x(t_{k-1}),t_{k-1})+O(h_{k}^{2}).\n$$"}, {"n": 4, "claim": "As $h_{k}\\to 0 .$ , we no longer need to divide the time interval, $\\mathrm{i . e .,}S=1$ . By defining the score function as $\\mathbf{s_{k}}=\\nabla\\log f_{k}= -\\nabla\\tilde{E}_{k}$ , the objective function (11) can be then approximated as:"}, {"n": 5, "claim": "$$\n\\begin{array}{ r l }&{\\mathbb{E}_{x\\sim f_{k-1}}\\left[\\tilde{E}_{k}(x(t_{k}))-\\int_{t_{k-1}}^{t_{k}}\\nabla\\cdot\\mathbf{v_{k}}(x(s),s)d s+\\frac{1}{2 h_{k}}\\|x(t_{k})-x(t_{k-1})\\|^{2}\\right]}\\\\ &{=\\mathbb{E}_{x\\sim f_{k-1}}\\Bigg [\\Big(\\tilde{E}_{k}(x(t_{k-1}))-h_{k}\\mathbf{s_{k}}(x(t_{k-1}))\\cdot\\mathbf{v_{k}}(x(t_{k-1}),t_{k-1})+O(h_{k}^{2})\\Big)}\\\\ &{\\qquad-\\left(h_{k}\\nabla\\cdot\\mathbf{v_{k}}(x(t_{k-1}),t_{k-1})+O(h_{k}^{2})\\right)+\\frac{1}{2 h_{k}}\\|h_{k}\\mathbf{v_{k}}(x(t_{k-1}))+O(h_{k}^{2})\\|^{2}\\Bigg]}\\\\ &{=\\mathbb{E}_{x\\sim f_{k-1}}\\Bigg [\\tilde{E}_{k}(x)+h_{k}\\left(-\\mathbf{s_{k}}(x)\\cdot\\mathbf{v_{k}}(x,t_{k-1})-\\nabla\\cdot\\mathbf{v_{k}}(x,t_{k-1})+\\frac{1}{2}\\|\\mathbf{v_{k}}(x,t_{k-1})\\|^{2}\\right)+O(h_{k}^{2})\\Bigg]}\\end{array}\n$$"}, {"n": 6, "claim": "Since $\\mathbb{E}_{x(t_{k-1})\\sim f_{k-1}}[\\tilde{E}_{k}(x(t_{k-1}))]$ is independent of $\\mathbf{v}_{\\mathbf{k}}(x,t)$ , as $h_{k}\\to 0$ , the minimization of the leading term is equivalent to:"}, {"n": 7, "claim": "$$\n\\operatorname*{min}_{\\mathbf{v}_{\\mathbf{k}}=\\mathbf{v}_{\\mathbf{k}}(\\cdot,0)}\\mathbb{E}_{x\\sim f_{k-1}}\\left[-T_{f_{k}}\\mathbf{v}_{\\mathbf{k}}+\\frac{1}{2}\\|\\mathbf{v}_{\\mathbf{k}}\\|^{2}\\right],\\quad T_{f_{k}}\\mathbf{v}_{\\mathbf{k}}: =\\mathbf{s_{k}}\\cdot\\mathbf{v}_{\\mathbf{k}}+\\nabla\\cdot\\mathbf{v_{k}}.\n$$"}]}}, {"local_id": "T3", "statement": "Let $\\pi_0(x)$ be an easy-to-sample base distribution and $q(x)=Z\\tilde{q}(x)$ a target, with intermediate annealed distributions $f_k(x)=\\pi_0(x)^{1-\\beta_k}q(x)^{\\beta_k}=Z_k\\tilde{f}_k(x)$ ($0=\\beta_0<\\beta_1<\\cdots<\\beta_K=1$) and unnormalized energy $\\tilde{E}_k(x)=-\\log\\tilde{f}_k(x)$. Let $\\mathcal{T}_k$ be the continuous transport map on $[t_{k-1},t_k]$ whose marginal density $\\rho(x,t)$ evolves by the continuity equation $\\partial_t\\rho+\\nabla\\cdot(\\rho\\mathbf{v_k})=0$ with particles following $\\frac{dx(t)}{dt}=\\mathbf{v_k}(x(t),t)$, so that $\\mathcal{T}_{\\#}f_{k-1}(x)=\\rho(x,t_k)$. Proposition 2 (KL-Divergence Decomposition): given the unnormalized density $f_{k-1}$, the KL-Divergence between $\\mathcal{T}_{\\#}f_{k-1}$ and $f_k$ is equivalent to $\\mathrm{KL}(\\mathcal{T}_{\\#}f_{k-1}\\|f_k)=c+\\mathbb{E}_{x(t_{k-1})\\sim f_{k-1}}\\left[\\tilde{E}_k(x(t_k))-\\int_{t_{k-1}}^{t_k}\\nabla\\cdot\\mathbf{v_k}(x(s),s)ds\\right]$, up to a constant $c$ independent of $\\mathbf{v_k}$.", "proof": {"steps": [{"n": 1, "claim": "$\\rho(x,t)$ denotes the density evolution under $\\mathcal{T}$ as in (3)."}, {"n": 2, "claim": "$\\mathcal{T}_{\\#}f_{k-1}(x)=\\rho(x,t_k)$."}, {"n": 3, "claim": "$\\mathrm{KL}(\\mathcal{T}_{\\#}f_{k-1}\\|f_k)=\\mathbb{E}_{x\\sim f_{k-1}}\\left[\\log\\frac{\\mathcal{T}_{\\#}f_{k-1}(x)}{f_k(x)}\\right]=\\mathbb{E}_{x\\sim f_{k-1}}\\left[\\log\\mathcal{T}_{\\#}f_{k-1}(x)-\\log f_k(x)\\right]$."}, {"n": 4, "claim": "$\\mathrm{KL}(\\mathcal{T}_{\\#}f_{k-1}\\|f_k)=\\mathbb{E}_{x\\sim\\rho(x,t_{k-1})}\\left[\\log\\mathcal{T}_{\\#}f_{k-1}(x(t_k))+\\tilde{E}_k(x(t_k))\\right]-\\log Z_k$."}, {"n": 5, "claim": "$\\frac{d}{ds}\\log\\rho(x(s),s)=\\frac{\\nabla\\rho\\cdot\\mathbf{v_k}-\\nabla\\cdot(\\rho\\mathbf{v_k})}{\\rho}\\big|_{(x(s),s)}=\\frac{\\nabla\\rho\\cdot\\mathbf{v_k}-(\\nabla\\rho\\cdot\\mathbf{v_k}+\\rho\\nabla\\cdot\\mathbf{v_k})}{\\rho}=-\\nabla\\cdot\\mathbf{v_k}(x(s),s)$."}, {"n": 6, "claim": "$\\log\\mathcal{T}_{\\#}f_{k-1}(x(t_k))=\\log\\rho(x(t_k),t_k)=\\log\\rho(x(t_{k-1}),t_{k-1})-\\int_{t_{k-1}}^{t_k}\\nabla\\cdot\\mathbf{v_k}(x(s),s)ds$."}, {"n": 7, "claim": "$\\mathrm{KL}(\\mathcal{T}_{\\#}f_{k-1}\\|f_k)=\\mathbb{E}_{x\\sim\\rho(x,t_{k-1})}\\left[\\log\\rho(x(t_{k-1}),t_{k-1})-\\int_{t_{k-1}}^{t_k}\\nabla\\cdot\\mathbf{v_k}(x(s),s)ds+\\tilde{E}_k(x(t_k))\\right]-\\log Z_k$."}, {"n": 8, "claim": "$\\mathrm{KL}(\\mathcal{T}_{\\#}f_{k-1}\\|f_k)=c+\\mathbb{E}_{x\\sim f_{k-1}}\\left[\\tilde{E}_k(x(t_k))-\\int_{t_{k-1}}^{t_k}\\nabla\\cdot\\mathbf{v_k}(x(s),s)ds\\right]$."}]}}, {"local_id": "T4", "statement": "Proposition 5: (Optimal Velocity Field as Score Difference) Suppose $h_{k}\\to 0$ and $\\begin{array}{ r }{\\gamma=\\frac{1}{2}}\\end{array}$ Let $f_{k-1}$ and $f_{k}$ be continuously differentiable on $\\mathbb{R}^{d}$ Assume that $\\nabla\\cdot\\mathbf{v_{k}}(x)$ exists for all $x\\in\\mathbb{R}^{d}$ , and $\\nabla\\cdot{\\mathbf{v_{k}}}(x),{\\mathbf{s_{k-1}}}$ and $\\mathbf{s_{k}}$ belong to $L^{2}(f_{k-1})$ . Assume that the components of $\\mathbf{v_{k}}$ are independent and $\\begin{array}{ r }{\\operatorname*{lim}_{\\|\\boldsymbol{x}\\|\\infty}f_{k-1}(\\boldsymbol{x})\\|\\mathbf{v_{k}}(\\boldsymbol{x})\\|_{2}=0 .}\\end{array}$ Under these conditions, the minimizer of (11) is:\n\n$$\n\\mathbf{v_{k}}^{*}=\\mathbf{s_{k}}-\\mathbf{s_{k-1}}.\n$$", "proof": {"steps": [{"n": 1, "claim": "Under the assumptions that $h_{k}\\to 0$ and $\\begin{array}{ r }{\\gamma=\\frac{1}{2}}\\end{array}$ , we begin by considering the equivalent minimization objective derived in Proposition 4:\n\n$$\n\\operatorname*{min}_{\\mathbf{v_{k}}}J(\\mathbf{v_{k}}): =\\operatorname*{min}_{\\mathbf{v_{k}}}\\mathbb{E}_{x\\sim f_{k-1}}\\left[-T_{f_{k}}\\mathbf{v_{k}}+\\frac{1}{2}\\|\\mathbf{v_{k}}\\|^{2}\\right],\\quad T_{f_{k}}\\mathbf{v_{k}}: =\\mathbf{s_{k}}\\cdot\\mathbf{v_{k}}+\\nabla\\cdot\\mathbf{v_{k}}.\n$$"}, {"n": 2, "claim": "Expanding the objective functional, we have:\n\n$$\n\\mathbb{E}_{x\\sim f_{k-1}}\\left[-\\mathbf{s_{k}}\\cdot\\mathbf{v_{k}}-\\nabla\\cdot\\mathbf{v_{k}}+{\\frac{1}{2}}\\|\\mathbf{v_{k}}\\|^{2}\\right]=\\int_{\\mathbb{R}^{d}}f_{k-1}(x)\\left(-\\mathbf{s_{k}}(x)\\cdot\\mathbf{v_{k}}(x)-\\nabla\\cdot\\mathbf{v_{k}}(x)+{\\frac{1}{2}}\\|\\mathbf{v_{k}}(x)\\|^{2}\\right)d x{\\mathrm{~i s~}}\n$$"}, {"n": 3, "claim": "Define $B_{r}=\\{x\\in\\mathbb{R}^{d}:\\|x\\|\\leq r\\}$ , and let $\\partial B_{r}$ denote the boundary of $B_{r}$ ,which is the sphere of radius r. Under the assumption that lim $\\|x\\|{}\\infty f_{k-1}(x)\\|\\mathbf{v_{k}}(x)\\|_{2}=0$ ,we have the following:\n\n$$\n\\begin{array}{ r l }{\\Big|\\displaystyle\\int_{\\mathbb{R}^{d}}\\nabla\\cdot(f_{k-1}\\mathbf{v_{k}})d x\\Big| =\\displaystyle\\operatorname*{lim}_{r\\to\\infty}\\Big|\\displaystyle\\int_{B_{r}}\\nabla\\cdot(f_{k-1}\\mathbf{v_{k}})d x\\Big|}&{}\\\\ &{=\\displaystyle\\operatorname*{lim}_{r\\to\\infty}\\Big|\\displaystyle\\int_{\\partial\\{x\\in\\mathbb{R}^{d}:\\|x\\|<r\\}}f_{k-1}(x)\\mathbf{v_{k}}(x)\\cdot\\mathbf{n}(x)d S(x)\\Big|}\\\\ &{\\leq\\displaystyle\\operatorname*{lim}_{r\\to\\infty}\\displaystyle\\int_{\\partial\\{x\\in\\mathbb{R}^{d}:\\|x\\|<r\\}}f_{k-1}\\|\\mathbf{v_{k}}\\|_{2}\\|\\mathbf{n_{k}}\\|_{2}d S(x)}\\\\ &{=\\displaystyle\\operatorname*{lim}_{r\\to\\infty}\\displaystyle\\int_{\\partial\\{x\\in\\mathbb{R}^{d}:\\|x\\|<r\\}}f_{k-1}\\|\\mathbf{v_{k}}\\|_{2}d S(x)}\\\\ &{=0}\\end{array}\n$$"}, {"n": 4, "claim": "Therefore, $\\begin{array}{ r }{\\int_{\\mathbb{R}^{d}}\\nabla\\cdot\\left(f_{k-1}\\mathbf{v_{k}}\\right)d x=0}\\end{array}$ . Next, we further expand the divergence theorem:\n\n$$\n\\begin{array}{ r l }&{0=\\displaystyle\\int_{\\mathbb{R}^{d}}\\nabla\\cdot(f_{k-1}(x)\\mathbf{v_{k}}(x))d x}\\\\ &{\\quad=\\displaystyle\\int_{\\mathbb{R}^{d}}f_{k-1}(x)\\nabla\\cdot\\mathbf{v_{k}}(x)d x+\\displaystyle\\int_{\\mathbb{R}^{d}}\\mathbf{v_{k}}(x)\\cdot\\nabla f_{k-1}(x)d x}\\\\ &{\\quad=\\displaystyle\\int_{\\mathbb{R}^{d}}f_{k-1}(x)\\nabla\\cdot\\mathbf{v_{k}}(x)d x+\\displaystyle\\int_{\\mathbb{R}^{d}}\\mathbf{v_{k}}(x)\\cdot\\mathbf{s_{k-1}}(x)f_{k-1}(x)d x}\\end{array}\n$$"}, {"n": 5, "claim": "Substitute the result back into the objective functional, we have:\n\n$$\n\\begin{array}{ r l r }&{}&{\\mathbb{E}_{x\\sim f_{k-1}}\\left[-{\\bf s_{k}}\\cdot{\\bf v_{k}}-\\nabla\\cdot{\\bf v_{k}}+\\frac{1}{2}\\|{\\bf v_{k}}\\|^{2}\\right]=\\int_{\\mathbb{R}^{d}}f_{k-1}(x)\\left(-{\\bf s_{k}}(x)\\cdot{\\bf v_{k}}(x)-\\nabla\\cdot{\\bf v_{k}}(x)+\\frac{1}{2}\\|{\\bf v_{k}}(x)\\|^{2}\\right)d x}\\\\ &{}&{=\\int_{\\mathbb{R}^{d}}f_{k-1}(x)\\left(({\\bf s_{k-1}}(x)-{\\bf s_{k}}(x))\\cdot{\\bf v_{k}}(x)+\\frac{1}{2}\\|{\\bf v_{k}}(x)\\|^{2}\\right)d x .}\\end{array}\n$$"}, {"n": 6, "claim": "The integrand does not involve $\\nabla{v_{k,j}}(x),j\\ =\\ 1,\\cdot\\cdot\\cdot d$ and higher-order derivatives. Assuming the components $v_{k,j},j=1,\\cdots,d$ of $\\mathbf{v_{k}}$ are independent, we can take the functional derivative component-wise and set them to zero:"}, {"n": 7, "claim": "$$\n{\\frac{\\delta J}{\\delta\\mathbf{v_{k}}}}=f_{k-1}\\left(\\mathbf{v_{k}}+\\left(\\mathbf{s_{k-1}}-\\mathbf{s_{k}}\\right)\\right)=0,\n$$"}, {"n": 8, "claim": "Since $f_{k-1}>0$ for all x, this implies:\n\n$$\n{\\mathbf{v_{k}}}^{*}={\\mathbf{s_{k}}}-{\\mathbf{s_{k-1}}}.\n$$"}]}}], "target_local_id": "T3", "topic_domain": "Others", "topic_subtopic": "Optimal Transport & Sampling", "label": "flawed", "error_step": 3, "error_class6": ["logical_gap_or_missing_case", "others"], "error_explanation": "The KL divergence is defined between (normalized) distributions as $D(p\\|q)=\\mathbb{E}_{X\\sim p}[\\log p(X)/q(X)]$, so $\\mathrm{KL}(\\mathcal{T}_{\\#}f_{k-1}\\|f_k)$ must take the expectation with respect to its FIRST argument, the pushforward density $\\mathcal{T}_{\\#}f_{k-1}=\\rho(\\cdot,t_k)$. The step instead takes the expectation over the source density $f_{k-1}$ (i.e. $\\mathbb{E}_{x\\sim f_{k-1}}[\\cdot]$, in the non-working unnormalized/tilde notation), which is a different, incorrect quantity and not the KL divergence between $\\mathcal{T}_{\\#}f_{k-1}$ and $f_k$. The correct opening is $\\mathrm{KL}(\\mathcal{T}_{\\#}f_{k-1}\\|f_k)=\\mathbb{E}_{x\\sim\\rho(x,t_k)}[\\log\\frac{\\mathcal{T}_{\\#}f_{k-1}(x)}{f_k(x)}]$ with $\\rho(t_k)=\\mathcal{T}_{\\#}f_{k-1}$; the subsequent change of variables back to $x(t_{k-1})\\sim f_{k-1}$ is only valid because the correct starting measure is the pushforward $\\rho(\\cdot,t_k)$ pushed back along the flow."}
{"id": "Xo2E217_M4n", "context": "In this section, we develop a theoretical analysis to study the effectiveness of our proposed defense in a simple but representative FL setting. It consists of the following: (i) developing upper and lower bounds quantifying the cross-entropy loss changes on backdoored and clean data in the settings of with and without the defense (Theorem); (i) showing a sufficient condition on the quality of trigger recovery such that the proposed defense is provably effective in mitigating backdoor attacks (Theorem 1); (ii) following (ii), we show that inference with confidence thresholding on models trained with our proposed defense can provably reduce the backdoor attack success rate while maintaining similar accuracy on clean data. During analysis, we leverage inequality (Mitrinovic & Vasi, 1970) and use variable substitution to get the upper-bound and lower-bound. To the best of our knowledge, our analysis is new to this field due to the novel modeling of trigger recovery and model hardening schemes in our proposed defense method. These findings are also consistent with our empirical results in more complex settings.\n\nLearning Objective and Setting. Suppose the k-th device holds the $n_{k}$ training samples: $\\left\\{\\mathbf{x}_{k,1},\\mathbf{x}_{k,2},\\cdot\\cdot\\cdot,\\mathbf{x}_{k,n_{k}}\\right\\}$ , where $\\mathbf{x}~\\in~\\mathbb{R}^{1\\times\\bar{d}_{x}}$ . The model has one layer of weights $\\boldsymbol{W}^{\\setminus}\\in\\mathbb{R}^{\\hat{d}_{x}\\times I}$ Label $\\mathbf{q}\\in\\mathbb{R}^{1\\times I}$ is a one-hot vector for I classes. In this work, we consider the following distributed optimization problem: minw $\\begin{array}{ r }{\\left\\{F(W)=\\sum_{k=1}^{N}g_{k}F_{k}(W)\\right\\}}\\end{array}$ where N is the number of devices, $g_{k}$ is the weight of the k-th device such that $g_{k}\\geq 0$ and $\\textstyle\\sum_{k=1}^{N}g_{k}\\ =\\ 1$ , and $F(\\cdot)$ the objective function. The local objective $F_{k}(\\cdot)$ is defined by $\\begin{array}{ r }{F_{k}(W)=\\frac{1}{n_{k}}\\sum_{j=1}^{n_{k}}\\mathcal{L}(W;\\mathbf x_{k,j})}\\end{array}$ where $\\mathcal{L}(\\cdot;\\cdot)$ is loss function. In the global server, we can write the global softmax cross-entropy loss function as $\\begin{array}{ r }{\\mathcal{L}_{g l o b a l}= -\\sum_{i=1}^{I}q_{i}}\\end{array}$ . logsoftmax $\\begin{array}{ r }{({\\bf x}W)_{i}= -\\sum_{i=1}^{I}q_{i}\\cdot l o g\\big(\\frac{e^{({\\bf x}W)_{i}}}{\\sum_{t=1}^{I}e^{({\\bf x}W)_{t}}}\\big)=}\\end{array}$ $\\begin{array}{ r }{-\\sum_{i=1}^{I}q_{i}(\\mathbf{x}W)_{i}+l o g(\\sum_{t=1}^{I}e^{(\\mathbf{x}W)_{t}})}\\end{array}$ , i and t as label index for the I classes. Appendix A.16 describes all used symbols in our paper.\n\nIn the theoretical analysis, we assume that we are under the FedAvg protocol (McMahan et al., 2017). In order to simplify analysis without losing generality, we assume there is one global server and two clients: one benign and the other malicious. We conduct the analysis on multi-class classification using logistic regression. The model is composed of one linear layer with the softmax function and the cross-entropy loss.\n\nTheorem 2 develops the upper and lower bounds quantifying the loss changes on backdoored and clean data in the settings with and without the defense.\n\nTheorem 2 (Bounds on Loss Changes). Let $\\mathcal{L}_{g}^{\\prime}$ denote the global model loss with defense, $\\mathcal{L}_{g}$ without defense, let $\\Delta W=W^{\\prime}-W$ denote the weight differences with and without defense. The loss difference with and without defense can be upper and lower bounded by\n\n$$\n\\operatorname*{min}_{t}(\\mathbf{x}\\Delta W)_{t}-\\sum_{i=1}^{I}q_{i}(\\mathbf{x}\\Delta W)_{i}\\leq\\mathcal{L}_{g}^{\\prime}-\\mathcal{L}_{g}\\leq\\operatorname*{max}_{t}(\\mathbf{x}\\Delta W)_{t}-\\sum_{i=1}^{I}q_{i}(\\mathbf{x}\\Delta W)_{i}\\tag{1}\n$$\n\nThe detailed proof is provided in Appendix A.14. The above theorem bounds the loss changes with and without the defense. From CauchySchwarz inequality (Mitrinovic & Vasi, 1970), we derive the lower bound and upper bounds of loss changes. Intuitively, given the parameter W of the linear layer, the k-th device holds $n_{k}$ training samples ${\\bf x}_{k,n_{k}}$ , the loss differences of with and without defense must be larger than the lower bound $\\Delta$ min _loss as mint $\\begin{array}{ r }{(\\mathbf{x}_{k,n_{k}}\\Delta W)_{t}{-}\\sum_{i=1}^{I}q_{i}(\\mathbf{x}_{k,n_{k}}\\Delta W)_{}}\\end{array}$ , and smaller than the upper bound $\\Delta$ max loss as maxt $\\begin{array}{ r }{(\\mathbf{x}_{k,n_{k}}\\Delta W)_{t}-\\sum_{i=1}^{I}q_{i}(\\mathbf{x}_{k,n_{k}}\\Delta W)_{i}}\\end{array}$\n\nTo facilitate the analysis of attack success rate (ASR) and clean accuracy (ACC) changes, intuitively, we aim to analyze how much the ASR at least will be reduced and how much the ACC will at most be maintained. Thus, we studied this lower bound (∆ min _loss) on backdoor data, which indicates the minimal improvements on the backdoor defense that reduce the ASR. Similarly we studied the upper bound (∆ max loss) for clean data, as they indicates the worst-case accuracy degradation. Denote backdoor samples as $n_{b}$ and clean samples as $n_{c}$ Note that backdoor samples can be written as $\\mathbf{x}_{s}+\\delta$ On benign data, we have $\\Delta$ max loss $\\begin{array}{ r }{\\mathbf{x}_{s},\\Delta\\operatorname*{max}\\_l o s s=\\sum_{s=1}^{n_{c}}\\operatorname*{max}_{t}(\\mathbf{x}_{s}\\Delta W)_{t}-\\sum_{s=1}^{n_{c}}\\sum_{i=1}^{I}q_{s,i}(\\mathbf{x}_{s}\\Delta W)_{i}}\\end{array}$\n\nNext, we aim to develop a sufficient condition on the quality of trigger recovery such that the proposed defense is provably effective in mitigating backdoor attack and in the meantime maintaining similar accuracy on clean data, based on Theorem 2.\n\nTheorem 1 (General Robustness Condition). Let α =\n\n$$\n\\frac{\\eta_{r}\\sum_{s=1}^{n_{b}}\\sum_{i=1}^{I}(q_{s,i}^{*}-q_{s,i})\\{\\mathbf{z}_{\\mathbf{s}}\\sum_{j=1}^{n_{1}}[\\mathbf{z_{j}}^{T}(\\mathbf{q_{j}}-p(\\mathbf{z_{j}}))]\\}_{i}}{\\mathbf{b}\\left\\{\\eta_{r}\\sum_{s=1}^{n_{b}}\\sum_{i=1}^{I}(q_{s,i}^{*}-q_{s,i})\\{\\sum_{j=1}^{n_{1}}[\\mathbf{z_{j}}^{T}(\\mathbf{q_{j}}-p(\\mathbf{z_{j}}))]\\}_{i}\\right\\}}\n$$\n\nwhere $\\mathbf{b}=[b_{1},...,b_{d}],$ d is the sample dimension,\n\nlet $\\begin{array}{ r }{b_{v}=\\mathrm{sign}\\left\\{\\eta_{r}\\sum_{s=1}^{n_{b}}\\sum_{i=1}^{I}(q_{s,i}^{*}-q_{s,i})\\sum_{j=1}^{n_{1}}[{\\bf z_{j}}^{T}{\\bf q_{j}}-p({\\bf z_{j}})]]_{i,v}\\right\\}}\\end{array}$ , on all dimensions v of the vector. For all $\\|\\epsilon\\|_{\\infty}\\leq\\alpha,$ we have $\\Delta$ min loss $\\geq 0 .$\n\nAnd we have $\\Delta$ max $\\begin{array}{ r }{\\mathit{\\_l o s s}\\leq\\eta_{r}\\sum_{s=1}^{n_{c}}\\sum_{i=1}^{I}(q_{s,i}^{*}-q_{s,i})\\mathbf{x_{s}}\\sum_{j=1}^{n_{1}}[\\mathbf{z_{j}}^{T}(\\mathbf{q_{j}}-p(\\mathbf{z_{j}}))]_{i}.}\\end{array}$\n\nThe detailed proof is provided in Appendix A.15. We denote $q_{s}^{*}$ as a one-hot vector for sample s with I dimensions. Its i-th dimension is defined as $q_{s,i}^{*}$ , and $q_{s,i}^{*}=1\\mathrm{if}i=$ arg mint $[({\\bf x}_{s}+\\delta)(\\dot{W^{\\prime}}-W)]_{t}.$ Denote δ as ground truth trigger, e as difference of reversed trigger and ground truth trigger, $\\eta_{r}$ as the learning rate (a.k.a. step size) in round r. Denote z as $\\mathbf{x}+\\delta+\\epsilon$ for simplicity, which is the benign sample stamped with the recovered trigger. Note that $\\Delta$ min loss $\\geq 0$ indicates that the defense is provably effective than without defense. Since benign local clients' training can increase global model backdoor loss, and they have positive effects on mitigating malicious poisoning effect, the second condition $\\Delta$ max loss $\\begin{array}{ r }{\\leq\\eta_{r}\\sum_{s=1}^{n_{c}}\\sum_{i=1}^{I}(q_{s,i}^{*}-q_{s,i})\\mathbf{x_{s}}\\sum_{j=1}^{n_{1}}[\\mathbf{z_{j}}^{T}(\\mathbf{q_{j}}-p(\\mathbf{z_{j}}))]}\\end{array}$ i indicates that the defense is provably guarantee maintaining similar accuracy on clean data, similarly, here $i=a r g m a x_{t}[\\mathbf{x}_{s}(\\hat{W}^{\\prime}-W)]_{t}$ (with a slight abuse of the notation $q_{s}^{*})$ .\n\nCorollary 2. Assume $\\epsilon$ satisfies Theorem 1, let $n_{b}$ as backdoored samples, $n_{c}$ as benign samples, $\\tau$ as confidence threshold. Then the number of backdoored samples that are rejected is $\\bar{R_{b d}}=\\bar{R_{b}^{\\prime}}{-}R_{b}$ the number of benign samples that are rejected is $R_{b n}=R_{c}^{\\prime}-R_{c}$ :\n\n$R_{b}^{\\prime}$ and $R_{b}$ denote the rejected backdoor samples with and without defense. With defense, $R_{b}^{\\prime}$ is $\\begin{array}{ r }{\\sum_{j=1}^{n_{b}}\\mathbf{1}(\\dot{\\mathcal{L}}_{g}+\\Delta}\\end{array}$ min loss $>\\mathcal{L}_{\\tau})$ ; and without defense, $R_{b}$ is $\\begin{array}{ r }{\\sum_{i=1}^{n_{b}}\\mathbf{1}(\\mathcal{L}_{g}>\\mathcal{L}_{\\tau})}\\end{array}$ . Thus, the exact value of rejected backdoored samples can be calculated through $\\dot{R_{b d}}=R_{b}^{\\prime}-R_{b}$\n\nSimilarly, $R_{c}^{\\prime}$ and $R_{c}$ denote the rejected clean(benign) samples with and without defense. With defense, $\\begin{array}{ r }{R_{c}^{\\prime}\\operatorname{is}\\sum_{j=1}^{n_{c}}\\mathbf{1}(\\mathcal{L}_{g}+\\Delta\\operatorname*{max}_{-}loss>\\mathcal{L}_{\\tau})}\\end{array}$ and without defense, $R_{c}$ is $\\begin{array}{ r }{\\sum_{j=1}^{n_{c}}\\mathbf{1}(\\mathcal{L}_{g}>\\mathcal{L}_{\\tau})}\\end{array}$ Thus, the exact value of rejected benign samples can be calculated through $R_{b n}=\\dot{R}_{c}^{\\prime}-R_{c}.$", "theorems": [{"local_id": "T1", "statement": "Theorem 1 (Clean-Data Loss Change Upper Bound). Under the same FedAvg setting as Theorem 2 (one server, one benign and one malicious client; multi-class classification via a single linear layer $W$, softmax, cross-entropy), the benign client hardens on $n_1$ samples $\\{\\mathbf{x}_j\\}$ with labels $q_j$ using the recovered trigger $\\delta+\\epsilon$; write $z_j=\\mathbf{x}_j+\\delta+\\epsilon$. Let $W,W'$ be the without/with-defense weights and $\\Delta W=W'-W$. For the $n_c$ clean test samples $\\mathbf{x}_s$ with labels $q_s$, define the aggregate clean-data upper bound $\\Delta\\max\\_loss=\\sum_{s=1}^{n_c}\\max_t(\\mathbf{x}_s\\Delta W)_t-\\sum_{s=1}^{n_c}\\sum_{i=1}^{I}q_{s,i}(\\mathbf{x}_s\\Delta W)_i$, and let $q^*_s$ be the one-hot vector with $q^*_{s,i}=1$ iff $i=\\arg\\max_t(\\mathbf{x}_s\\Delta W)_t$. Then $\\mathcal{L}_g'-\\mathcal{L}_g\\le\\Delta\\max\\_loss\\le\\eta_r\\sum_{s=1}^{n_c}\\sum_{i=1}^{I}(q^*_{s,i}-q_{s,i})\\mathbf{x}_s\\sum_{j=1}^{n_1}[z_j^T(q_j-p(z_j))]_i$.", "proof": {"steps": [{"n": 1, "claim": "For each clean sample $\\mathbf{x}_s$ with label $q_s$, Theorem 2 gives $\\mathcal{L}_g'-\\mathcal{L}_g\\le\\max_t(\\mathbf{x}_s\\Delta W)_t-\\sum_{i=1}^{I}q_{s,i}(\\mathbf{x}_s\\Delta W)_i$."}, {"n": 2, "claim": "Summing over the $n_c$ clean samples, $\\Delta\\max\\_loss=\\sum_s\\max_t(\\mathbf{x}_s\\Delta W)_t-\\sum_s\\sum_i q_{s,i}(\\mathbf{x}_s\\Delta W)_i=\\sum_s\\sum_i(q^*_{s,i}-q_{s,i})(\\mathbf{x}_s\\Delta W)_i$."}, {"n": 3, "claim": "One FedAvg SGD hardening step gives the exact weight update $\\Delta W=W'-W=\\eta_r\\sum_{j=1}^{n_1}z_j^T(q_j-p(z_j))$ with $z_j=\\mathbf{x}_j+\\delta+\\epsilon$."}, {"n": 4, "claim": "Substituting Step 3 into Step 2: $\\Delta\\max\\_loss=\\eta_r\\sum_s\\sum_i(q^*_{s,i}-q_{s,i})[\\mathbf{x}_s\\sum_j z_j^T(q_j-p(z_j))]_i$, hence $\\mathcal{L}_g'-\\mathcal{L}_g\\le\\eta_r\\sum_s\\sum_i(q^*_{s,i}-q_{s,i})\\mathbf{x}_s\\sum_j[z_j^T(q_j-p(z_j))]_i$."}]}}, {"local_id": "T2", "statement": "Under FedAvg with one server and two clients (one benign, one malicious), multi-class classification via a single linear layer $W$, softmax, cross-entropy loss: let $\\mathcal{L}_g'$ be the global loss with defense, $\\mathcal{L}_g$ without, $\\Delta W = W'-W$. For sample $\\mathbf{x}$ with label $\\mathbf{q}$, Theorem 2 gives $\\operatorname*{min}_t(\\mathbf{x}\\Delta W)_t - \\sum_{i=1}^{I}q_i(\\mathbf{x}\\Delta W)_i \\le \\mathcal{L}_g'-\\mathcal{L}_g \\le \\operatorname*{max}_t(\\mathbf{x}\\Delta W)_t - \\sum_{i=1}^{I}q_i(\\mathbf{x}\\Delta W)_i$. Generalizing to the $n_b$ backdoor samples $\\mathbf{x}_s+\\delta$ with labels $q_{s,i}$: the aggregate lower bound is $\\Delta\\min\\_loss = \\operatorname*{min}_t\\big[\\sum_{s=1}^{n_b}(\\mathbf{x}_s+\\delta)\\Delta W\\big]_t - \\sum_{s=1}^{n_b}\\sum_{i=1}^{I}q_{s,i}[(\\mathbf{x}_s+\\delta)\\Delta W]_i$, the base case for the Theorem 1 condition $\\Delta\\min\\_loss \\ge 0$.", "proof": {"steps": [{"n": 1, "claim": "By Theorem 2, for sample $\\mathbf{x}$ with label $\\mathbf{q}$: $\\operatorname*{min}_t(\\mathbf{x}\\Delta W)_t - \\sum_{i=1}^{I}q_i(\\mathbf{x}\\Delta W)_i \\le \\mathcal{L}_g'-\\mathcal{L}_g \\le \\operatorname*{max}_t(\\mathbf{x}\\Delta W)_t - \\sum_{i=1}^{I}q_i(\\mathbf{x}\\Delta W)_i$."}, {"n": 2, "claim": "Apply this to the $n_b$ backdoor samples $\\mathbf{x}_s+\\delta$ with labels $q_{s,i}$ to get an aggregate lower bound $\\Delta\\min\\_loss$."}, {"n": 3, "claim": "$\\Delta\\min\\_loss = \\operatorname*{min}_t\\big[\\sum_{s=1}^{n_b}(\\mathbf{x}_s+\\delta)\\Delta W\\big]_t - \\sum_{s=1}^{n_b}\\sum_{i=1}^{I}q_{s,i}[(\\mathbf{x}_s+\\delta)\\Delta W]_i$."}, {"n": 4, "claim": "This $\\Delta\\min\\_loss$ is the base case for Theorem 1's $\\Delta\\min\\_loss \\ge 0$ and for Corollary 2."}]}}, {"local_id": "T3", "statement": "Theorem 2 (Bounds on Loss Changes). Let $\\mathcal{L}_{g}^{\\prime}$ denote the global model loss with defense, $\\mathcal{L}_{g}$ without defense, let $\\Delta W=W^{\\prime}-W$ denote the weight differences with and without defense. The loss difference with and without defense can be upper and lower bounded by\n\n$$\n\\operatorname*{min}_{t}(\\mathbf{x}\\Delta W)_{t}-\\sum_{i=1}^{I}q_{i}(\\mathbf{x}\\Delta W)_{i}\\leq\\mathcal{L}_{g}^{\\prime}-\\mathcal{L}_{g}\\leq\\operatorname*{max}_{t}(\\mathbf{x}\\Delta W)_{t}-\\sum_{i=1}^{I}q_{i}(\\mathbf{x}\\Delta W)_{i}\\tag{1}\n$$", "proof": {"steps": [{"n": 1, "claim": "\\begin{array}{ l }{{\\displaystyle{\\mathcal L}_{g}^{\\prime}-{\\mathcal L}_{g}= -\\sum_{i=1}^{I}q_{i}({\\bf x}W^{\\prime})_{i}+l o g(\\sum_{t=1}^{I}e^{({\\bf x}W^{\\prime})_{t}})+\\sum_{i=1}^{I}q_{i}({\\bf x}W)_{i}-l o g(\\sum_{t=1}^{I}e^{({\\bf x}W)_{t}})}\\ ~}\\\\ {{\\displaystyle~= -\\sum_{i=1}^{I}q_{i}[(W^{\\prime}-W){\\bf x}]_{i}+l o g(\\sum_{t=1}^{I}e^{({\\bf x}W^{\\prime})_{t}})-l o g(\\sum_{t=1}^{I}e^{({\\bf x}W)_{t}})}\\ ~}\\\\ {{\\displaystyle~= -\\sum_{i=1}^{I}q_{i}[(W^{\\prime}-W){\\bf x}]_{i}+l o g(\\frac{\\sum_{t=1}^{I}e^{({\\bf x}W^{\\prime})_{t}}}{\\sum_{t=1}^{I}e^{({\\bf x}W)_{t}}})}}\\end{array}\\tag{3}"}, {"n": 2, "claim": "Since both $\\textstyle\\sum_{t=1}^{I}e^{(\\mathbf{x}W^{\\prime})_{i}}$ and $\\textstyle\\sum_{t=1}^{I}e^{(\\mathbf{x}W)_{\\tau}}$ are a sequence of positive numbers. Then from Mitrinovic & Vasi (1970) we can have an inequality of"}, {"n": 3, "claim": "\\operatorname*{min}_{1\\leq t\\leq I}\\frac{e^{(\\mathbf{x}W^{\\prime})_{t}}}{e^{(\\mathbf{x}W)_{t}}}\\leq\\frac{\\sum_{t=1}^{I}e^{(\\mathbf{x}W^{\\prime})_{t}}}{\\sum_{t=1}^{I}e^{(\\mathbf{x}W)_{t}}}\\leq\\operatorname*{max}_{1\\leq t\\leq I}\\frac{e^{(\\mathbf{x}W^{\\prime})_{t}}}{e^{(\\mathbf{x}W)_{t}}}\\tag{4}"}, {"n": 4, "claim": "If we denote $m=\\operatorname*{min}_{t}\\frac{e^{(\\mathbf{x}W^{\\prime})_{t}}}{e^{(\\mathbf{x}W)_{t}}}$ and $M=\\operatorname*{max}_{t}\\frac{e^{(\\mathbf{x}W^{\\prime})_{t}}}{e^{(\\mathbf{x}W)_{t}}}$ , then we have successively\n\n$$\n\\begin{array}{ r l }{m\\leq\\frac{e^{(\\mathbf{x}W^{\\prime})_{t}}}{e^{(\\mathbf{x}W)_{t}}}}&{\\leq M}\\\\ {m\\cdot e^{(\\mathbf{x}W)_{t}}\\leq e^{(\\mathbf{x}W^{\\prime})_{t}}}&{\\leq M\\cdot e^{(\\mathbf{x}W)_{t}}}\\\\ {m\\cdot\\sum_{t=1}^{I}e^{(\\mathbf{x}W)_{t}}\\leq\\sum_{t=1}^{I}e^{(\\mathbf{x}W^{\\prime})_{t}}}&{\\leq M\\cdot\\sum_{t=1}^{I}e^{(\\mathbf{x}W)_{t}}}\\\\ {m\\leq\\frac{\\sum_{t=1}^{I}e^{(\\mathbf{x}W^{\\prime})_{t}}}{\\sum_{t=1}^{I}e^{(\\mathbf{x}W)_{t}}}\\leq M}&{}\\end{array}\n$$\n\nBy using log function monotonicity property, we can get an inequality of"}, {"n": 5, "claim": "l o g m\\le l o g(\\frac{\\sum_{t=1}^{I}e^{(\\mathbf{x}W^{\\prime})_{t}}}{\\sum_{t=1}^{I}e^{(\\mathbf{x}W)_{t}}})\\le l o g M\\tag{5}"}, {"n": 6, "claim": "So the deduction of $\\mathcal{L}_{g}^{\\prime}-\\mathcal{L}_{g}$ can be written as:"}, {"n": 7, "claim": "l o g\\operatorname*{min}_{t}\\frac{e^{(\\mathbf{x}W^{\\prime})_{t}}}{e^{(\\mathbf{x}W)_{t}}}-\\sum_{i=1}^{I}q_{i}[\\mathbf{x}(W^{\\prime}-W)]_{i}\\le\\mathcal{L}_{g}^{\\prime}-\\mathcal{L}_{g}\\le l o g\\operatorname*{max}_{t}\\frac{e^{(\\mathbf{x}W^{\\prime})_{t}}}{e^{(\\mathbf{x}W)_{t}}}-\\sum_{i=1}^{I}q_{i}[\\mathbf{x}(W^{\\prime}-W)]_{i}\\tag{6}"}]}}, {"local_id": "T4", "statement": "Corollary 2. Assume $\\epsilon$ satisfies Theorem 1, let $n_{b}$ as backdoored samples, $n_{c}$ as benign samples, $\\tau$ as confidence threshold. Then the number of backdoored samples that are rejected is $\\bar{R_{b d}}=\\bar{R_{b}^{\\prime}}{-}R_{b}$ the number of benign samples that are rejected is $R_{b n}=R_{c}^{\\prime}-R_{c}$ :", "proof": {"steps": [{"n": 1, "claim": "With defense, $R_{b}^{\\prime}$ is $\\begin{array}{ r }{\\sum_{j=1}^{n_{b}}\\mathbf{1}(\\dot{\\mathcal{L}}_{g}+\\Delta}\\end{array}$ min loss $>\\mathcal{L}_{\\tau})$ ; and without defense, $R_{b}$ is $\\begin{array}{ r }{\\sum_{i=1}^{n_{b}}\\mathbf{1}(\\mathcal{L}_{g}>\\mathcal{L}_{\\tau})}\\end{array}$ ."}, {"n": 2, "claim": "Thus, the exact value of rejected backdoored samples can be calculated through $\\dot{R_{b d}}=R_{b}^{\\prime}-R_{b}$"}, {"n": 3, "claim": "Similarly, $R_{c}^{\\prime}$ and $R_{c}$ denote the rejected clean(benign) samples with and without defense."}, {"n": 4, "claim": "With defense, $\\begin{array}{ r }{R_{c}^{\\prime}\\operatorname{is}\\sum_{j=1}^{n_{c}}\\mathbf{1}(\\mathcal{L}_{g}+\\Delta\\operatorname*{max}_{-}loss>\\mathcal{L}_{\\tau})}\\end{array}$ and without defense, $R_{c}$ is $\\begin{array}{ r }{\\sum_{j=1}^{n_{c}}\\mathbf{1}(\\mathcal{L}_{g}>\\mathcal{L}_{\\tau})}\\end{array}$"}]}}], "target_local_id": "T2", "topic_domain": "Others", "topic_subtopic": "Robustness, Privacy & Security", "label": "flawed", "error_step": 3, "error_class6": ["invalid_bound_or_inequality", "others", "logical_gap_or_missing_case"], "error_explanation": "The min operator and the summation over samples do not commute: in general $\\operatorname*{min}_t\\sum_{s=1}^{n_b} f_s(t) \\ge \\sum_{s=1}^{n_b}\\operatorname*{min}_t f_s(t)$, because a single common minimizing class $t$ cannot minimize every per-sample term simultaneously. Theorem 1 only guarantees the per-sample lower bound $\\operatorname*{min}_t[(\\mathbf{x}_s+\\delta)\\Delta W]_t - \\sum_{i}q_{s,i}[(\\mathbf{x}_s+\\delta)\\Delta W]_i \\le \\mathcal{L}_g'-\\mathcal{L}_g$ for each sample, so summing the valid per-sample bounds yields $\\sum_{s=1}^{n_b}\\operatorname*{min}_t[(\\mathbf{x}_s+\\delta)\\Delta W]_t - \\ldots$ (the min INSIDE the sum). Replacing that with $\\operatorname*{min}_t\\big[\\sum_{s=1}^{n_b}(\\mathbf{x}_s+\\delta)\\Delta W\\big]_t$ (min OUTSIDE the sum) is NOT a valid lower bound on the aggregate loss change, so the robustness condition $\\Delta\\min\\_loss \\ge 0$ that Theorem 2 derives from it is not justified. The correct expression is $\\Delta\\min\\_loss = \\sum_{s=1}^{n_b}\\operatorname*{min}_t[(\\mathbf{x}_s+\\delta)\\Delta W]_t - \\sum_{s=1}^{n_b}\\sum_{i=1}^{I}q_{s,i}[(\\mathbf{x}_s+\\delta)\\Delta W]_i$."}
{"id": "ceOdspvoaEA", "context": "3.1 Notation\n\nWe assume an MDP with states s, actions a, initial state distribution $p_{0}(s_{0})$ , dynamics $p(s^{\\prime}\\mid s,a)$ , and reward function $r(s,a).^{1}$ We assume episodes always have infinite length (i.e., there are no terminal states). Without loss of generality, we assume rewards are positive, adding a positive constant to all\n\nrewards can make them all positive without changing the optimal policy. We will learn a Markovian policy $\\pi(a\\mid s)$ to maximize the expected discounted sum of rewards:\n\n$$\n\\operatorname*{max}_{\\pi}\\mathbb{E}_{\\pi(\\tau)}\\left[\\sum_{t=0}^{\\infty}\\gamma^{t}r(s_{t},a_{t})\\mid s_{0}\\sim p_{0}(s_{0})\\right],\n$$\n\nwhere $\\begin{array}{ r }{\\pi(\\tau)=p(s_{0})\\prod_{t=0}^{\\infty}\\pi(a_{t}\\ \\mid\\ s_{t})p(s_{t+1}\\ \\mid\\ s_{t},a_{t})}\\end{array}$ is the probability of policy π sampling an infinite-length trajectory ${\\boldsymbol\\tau}=(s_{0},a_{0},\\cdots)$ . We define Q-values for policy $\\pi(a\\mid s)$ as\n\n$$\nQ^{\\pi}(s,a)=\\mathbb{E}_{\\pi(\\tau)}\\left[\\sum_{t=0}^{\\infty}\\gamma^{t}r(s_{t},a_{t})\\mid s_{0}=s,a_{0}=a\\right].\n$$\n\nNote that the reward being positive implies that the Q-values are also positive, $Q^{\\pi}(s,a)>0$ Since we focus on the offline setting, we will consider two policies: $\\beta(\\boldsymbol{a}\\ \\bar{|}\\ \\boldsymbol{s})$ is the behavioral policy that collected the dataset, and $\\pi(a\\mid s)$ is the online policy output by the algorithm that attempts to maximize the rewards. We will use $\\dot{p}(s,a,s^{\\prime})$ to denote the empirical distribution of transitions in an offline dataset, and $p(s,a)$ and $p(s)$ denote the corresponding marginal distributions. The behavioral policy is defined as $\\beta(a\\mid s)=p(a\\mid s)$\n\n3.4 Classifier Actor Critic\n\nTo support our analysis, we will introduce a new actor-critic algorithm. This algorithm is similar to prior work, but trains the critic using a cross entropy loss instead of an MSE loss. We introduce this algorithm not because we expect it to perform better than existing actor-critic methods, but rather because it allows us to make precise a connection between actor and critic regularization. This method treats the value function like a classifier, so we will call it classifier actor critic. We will then introduce actor-regularized and critic-regularized versions of this method. The subsequent section (Sec. 4) will show that these two regularized methods learn the same policy.\n\nThe key to our analysis will be to treat Q-values like probabilities, so we define the critic loss in terms of a cross-entropy loss, similar to prior work (Kalashnikov et al., 2018; Eysenbach et al., 2021). Recalling that Q-values are positive (Sec. 3.1), we transform the Q-values to have the correct range by using $\\breve{\\frac{Q}{Q+1}}\\in [0,1)$ . We will minimize the cross-entropy loss applied to the transformed Q-values:\n\n$$\n\\begin{array}{ r l }&{\\mathbb{E}_{p(s,a)}\\left[\\mathcal{C E}\\left(\\frac{Q(s,a)}{Q(s,a)+1};\\frac{y^{\\pi,Q_{t}}(s,a)}{y^{\\pi,Q_{t}}(s,a)+1}\\right)\\right]}\\\\ &{= -\\mathbb{E}_{p(s,a)}\\left[\\frac{y^{\\pi,Q_{t}}(s,a)}{y^{\\pi,Q_{t}}(s,a)+1}\\log\\frac{Q(s,a)}{Q(s,a)+1}+\\frac{1}{y^{\\pi,Q_{t}}(s,a)+1}\\log\\frac{1}{Q(s,a)+1}\\right]}\\\\ &{\\overset{\\mathrm{c o n s t .}}{=}-\\mathbb{E}_{p(s,a)}\\left[y^{\\pi,Q_{t}}(s,a)\\log\\frac{Q(s,a)}{Q(s,a)+1}+\\log\\frac{1}{Q(s,a)+1}\\right]\\triangleq\\mathcal{L}_{\\mathrm{ciits}}(Q,y^{\\pi,Q_{t}}),}\\end{array}\\tag{1}\n$$\n\nIn the last line we scale both the positive and negative term by $y^{\\pi,Q_{t}}(s,a)+1$ , a choice that does not change the optimal classifier but reduces notational clutter. When the TD target can be computed exactly, solving this optimization problem results in performing one SARSA update: $Q(s,a)\\gets r(s,a)+\\gamma Q(\\bar{s^{\\prime}},a^{\\prime})$ (see Lemma 1). Thus, by solving this optimization problem many times, each time using the previous Q-value to compute the TD targets, we will converge to the correct Q-values (see Lemma 1). The actor objective is to maximize the expected log of the Q-values:\n\n$$\n\\operatorname*{max}_{\\pi}\\mathcal{L}_{\\mathrm{actor}}(\\pi)\\triangleq\\mathbb{E}_{p(s)\\pi(a|s)}\\left[\\log(Q^{\\pi}(s,a))\\right],\\mathrm{where}Q^{\\pi}(s,a)=\\operatorname*{lim}_{t\\to\\infty}Q_{t}(s,a)\\tag{2}\n$$\n\n$$\n\\mathrm{and}Q_{t+1}=\\operatorname*{argmin}_{Q}\\mathcal L_{\\mathrm{critic}}(Q,y^{\\pi,Q_{t}}).\n$$", "theorems": [{"local_id": "T1", "statement": "Lemma 2. Assume that states and actions are tabular (discrete and finite), that rewards are positive, and that TD targets can be computed exactly (without sampling). Incrementally update the critic by minimizing a sequence of regularized critic losses using policy  and hyperparameter $\\lambda = 1 .$\n\n$$\nQ_{t+1}\\underset{Q}{\\arg\\operatorname*{min}}\\mathcal{L}_{critic}^{r}(Q,y^{\\pi,Q_{t}}).\n$$\n\nIn the limit, this sequence of Q-functions will converge to the Q-values for the behavioral policy $(\\beta(a\\mid s))$ ,weighted by the ratio of the behavioral and online policies:\n\n$$\n\\operatorname*{lim}_{t\\to\\infty}Q_{t}(s,a)=\\frac{Q^{\\beta}(s,a)\\beta(a\\mid s)}{\\pi(a\\mid s)}for all states s and actions a.\n$$", "proof": {"steps": [{"n": 1, "claim": "Step 0. We start by rearranging the regularized critic objective:\n\n$$\n\\begin{aligned}\n\\mathcal{L}_{\\mathrm{critic}}^r(Q, y^{\\pi, Q_t}) & \\triangleq \\mathcal{L}_{\\mathrm{critic}}(Q, y^{\\pi, Q_t}) + \\bigg(\\mathbb{E}_{p(s)\\pi(a \\mid s)}\\left[ \\log (Q(s, a) + 1)\\right] - \\mathbb{E}_{p(s)\\beta(a \\mid s)}\\left[ \\log (Q(s, a) + 1) \\right] \\bigg) \\\\\n&= -\\mathbb{E}_{p(s, a)}\\left[y^{\\pi, Q_t}(s, a) \\log \\frac{Q(s, a)}{Q(s, a) + 1} + \\log \\frac{1}{Q(s, a) + 1} \\right] \\\\\n& \\qquad + \\bigg(\\mathbb{E}_{p(s)\\pi(a \\mid s)}\\left[ \\log (Q(s, a) + 1)\\right] - \\mathbb{E}_{p(s)\\beta(a \\mid s)}\\left[ \\log (Q(s, a) + 1) \\right] \\bigg) \\\\\n&= -\\mathbb{E}_{p(s, a)}\\left[y^{\\pi, Q_t}(s, a) \\log \\frac{Q(s, a)}{Q(s, a) + 1} + \\cancel{\\log \\frac{1}{Q(s, a) + 1}} \\right] \\\\\n& \\qquad - \\bigg(\\mathbb{E}_{p(s)\\pi(a \\mid s)}\\left[ \\log \\frac{1}{Q(s, a) + 1}\\right] - \\cancel{\\mathbb{E}_{p(s)\\beta(a \\mid s)}\\left[ \\log \\frac{1}{Q(s, a) + 1} \\right]} \\bigg) \\\\\n&= -\\mathbb{E}_{p(s, a)}\\left[y^{\\pi, Q_t}(s, a) \\log \\frac{Q(s, a)}{Q(s, a) + 1}\\right] - \\mathbb{E}_{p(s)\\pi(a \\mid s)}\\left[ \\log \\frac{1}{Q(s, a) + 1} \\right].\n\\end{aligned}\n$$"}, {"n": 2, "claim": "For the cancelation on the third line, we used the fact that $p(s,a)=p(s)\\beta(a\\mid s)$"}, {"n": 3, "claim": "Step 1. To start, note that the regularized critic update is equivalent to a weighted classification loss: positive examples are sampled $(s,a)\\sim p(s)\\beta(a\\mid s)$ and receive weight $\\frac{y^{\\overline{{\\pi}},Q_{t}}\\left(s,a\\right)}{y^{\\pi,Q_{t}}\\left(s,a\\right)+1}$ , and negative examples are sampled $(s,a)\\sim p(s)\\pi(a\\mid s)$ and receive weight $\\frac{1}{y^{\\pi,Q_{t}}(s,a)+1}$ The Bayes' optimal classifier is given by\n\n$$\n\\frac{Q(s, a)}{Q(s, a) + 1} = \\frac{\\frac{y^{\\pi, Q_t}(s, a)}{y^{\\pi, Q_t}(s, a) + 1}p(s)\\beta(a \\mid s)}{\\frac{y^{\\pi, Q_t}(s, a)}{y^{\\pi, Q_t}(s, a) + 1}p(s)\\beta(a \\mid s) + \\frac{1}{y^{\\pi, Q_t}(s, a) + 1} p(s) \\pi(a \\mid s)} = \\frac{y^{\\pi, Q_t}(s, a) \\beta(a \\mid s)}{y^{\\pi, Q_t}(s, a) \\beta(a \\mid s) + \\pi(a \\mid s)}.\n$$"}, {"n": 4, "claim": "Solving for $Q(s,a)$ on the left hand side, the optimal value for $Q(s,a)$ is given by\n\n$$\nQ(s, a) = y^{\\pi, Q_t}(s, a)\\frac{\\beta(a \\mid s)}{\\pi(a \\mid s)} = (r(s, a) + \\mathbb{E}_{p(s' \\mid s, a) \\pi(a' \\mid s')}[Q_t(s', a')]) \\frac{\\beta(a \\mid s)}{\\pi(a \\mid s)}.\n$$"}, {"n": 5, "claim": "This equation tells us what each update for the regularized critic loss does.\n\nStep 2. To analyze these updates, we define $\\begin{array}{ r }{\\tilde{Q}(s,a)\\triangleq Q_{t}\\big(s,a\\big)\\frac{\\pi(a|s)}{\\beta(a|s)}}\\end{array}$ . Then these updates can be written using $\\tilde{Q}(s,a)$ as\n\n$$\n\\tilde{Q}(s, a)\\frac{\\beta(a \\mid s)}{\\pi(a \\mid s)} = \\left(r(s, a) + \\mathbb{E}_{p(s' \\mid s, a) \\pi(a' \\mid s')}\\left[\\tilde{Q}(s', a')\\frac{\\beta(a' \\mid s')}{\\pi(a' \\mid s')} \\right] \\right)\\frac{\\beta(a \\mid s)}{\\pi(a \\mid s)},\n$$"}, {"n": 6, "claim": "which can be simplified to\n\n$$\n\\tilde{Q}(s, a) = r(s, a) + \\mathbb{E}_{p(s' \\mid s, a) \\beta(a' \\mid s')}\\left[\\tilde{Q}(s', a')\\right].\n$$"}, {"n": 7, "claim": "Note that the ratio $\\frac{\\beta(a^{\\prime}|s^{\\prime})}{\\pi(a^{\\prime}|s^{\\prime})}$ inside the expectation acts like an importance weight, so that the expectation over $\\pi(a^{\\prime}|s^{\\prime})$ becomes an expectation over $\\beta(a^{\\prime}\\mid s^{\\prime})$ Thus, the regularized critic updates are equivalent to perform policy evaluation on $\\bar{Q}(s,a)$ . An immediately consequence is that the regularized critic updates converge, and they converge to $\\tilde{Q}^{*}(s,a)=Q^{\\beta}(s,a)$"}, {"n": 8, "claim": "Step 3. Finally, we translate these convergence results for $\\tilde{Q}(s,a)$ into convergence results for $Q(s,a)$ . Written in terms of the original Q-values, we see that the optimal critic for the regularized critic update is\n\n$$\nQ^*(s, a) = \\tilde{Q}^*(s, a) \\frac{\\beta(a \\mid s)}{\\pi(a \\mid s)} = Q^\\beta(s, a) \\frac{\\beta(a \\mid s)}{\\pi(a \\mid s)}.\n$$"}]}}, {"local_id": "T2", "statement": "Assume an MDP with positive rewards, so that Q-values are positive, $Q^\\pi(s,a)>0$. Define the classifier critic loss $\\mathcal{L}_{\\mathrm{critic}}(Q,y^{\\pi,Q_t})=-\\mathbb{E}_{p(s,a)}\\left[y^{\\pi,Q_t}(s,a)\\log\\frac{Q(s,a)}{Q(s,a)+1}+\\log\\frac{1}{Q(s,a)+1}\\right]$ with TD target $y^{\\pi,Q_t}(s,a)\\triangleq r(s,a)+\\gamma\\,\\mathbb{E}_{p(s'\\mid s,a)\\pi(a'\\mid s')}[Q_t(s',a')]$. Minimizing the classifier critic loss $\\mathcal{L}_{\\mathrm{critic}}(Q,y^{\\pi,Q})$ over $Q$ yields the true Q-values of the policy: the optimal $Q$ of this single objective satisfies $Q(s,a)=Q^\\pi(s,a)$ for all states $s$ and actions $a$.", "proof": {"steps": [{"n": 1, "claim": "$\\frac{Q(s,a)}{Q(s,a)+1}\\leftarrow \\frac{y^{\\pi,Q}(s,a)}{y^{\\pi,Q}(s,a)+1}$ gives the minimizer of $\\mathcal{L}_{\\mathrm{critic}}(Q,y^{\\pi,Q})$ over the single fixed objective in $Q$."}, {"n": 2, "claim": "Hence $Q(s,a)=y^{\\pi,Q}(s,a)$."}, {"n": 3, "claim": "$Q(s,a)=r(s,a)+\\gamma\\,\\mathbb{E}_{p(s'\\mid s,a)\\pi(a'\\mid s')}[Q(s',a')]$."}, {"n": 4, "claim": "$Q(s,a)=Q^\\pi(s,a)$ for all $s,a$."}]}}, {"local_id": "T3", "statement": "Theorem 1. Let a behavioral policy $\\beta(a\\mid s)$ be given and let $Q^{\\beta}(s,a)$ be the corresponding value function. Let π(a | s) be an arbitrary policy (typically learned) with support constrained to $\\breve{\\beta}(a\\mid s)$ $(i . e .,\\pi(a\\mid s)>0\\stackrel{\\cdot}{\\Longrightarrow}\\beta(a\\mid s)>\\stackrel{\\cdot}{0})\\stackrel{\\cdot}{\\Longrightarrow}0$ Let $Q_{r}^{\\bar{\\pi}}(s,a)$ be the critic obtained by the regularized critic update (Eq. 3) to this policy with $\\lambda = 1$ . Then critic regularization results in the same policy as one-step RL:\n\n$$\n\\begin{array}{ r }{\\mathbb{E}_{\\pi(a|s)}\\left[\\log Q_{r}^{\\pi}(s,a)\\right]=\\mathbb{E}_{\\pi(a|s)}\\left[\\log Q^{\\beta}(s,a)+\\log\\beta(a\\mid s)-\\log\\pi(a\\mid s)\\right]\\quad for all states~s .}\\end{array}\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof."}, {"n": 2, "claim": "\\log Q^{*}(s,a)=\\log\\left(Q^{\\beta}(s,a){\\frac{\\beta(a\\mid s)}{\\pi(a\\mid s)}}\\right)=\\log Q^{\\beta}(s,a)+\\log\\beta(a\\mid s)-\\log\\pi(a\\mid s)."}, {"n": 3, "claim": "We note that our proof does not account for stochastic and function approximation errors. However, if we assume that the TD updates are deterministic (e.g., as they are in deterministic MDPs), then the updates for classifier actor-critic are identical to those of Q-learning (Lemma 1)."}, {"n": 4, "claim": "Thus, it immediately inherits any theoretical results regarding the propagation of errors for Q-learning."}]}}, {"local_id": "T4", "statement": "Theorem 2. Let policy $\\pi(a\\mid s)$ be given, let $Q^{\\beta}(s,a)$ be the Q-function of the behavioral policy, and let $Q_{r}^{\\lambda_{T D}}\\left(s,a,\\lambda_{critic}\\right)$ be the critic obtained by the λcritic-weighted regularized critic update $(Eq.\\ 4)$ using TD targets $y^{\\lambda_{T D}}(s,a).\\ I f\\lambda_{critic}=\\lambda_{T D}=\\lambda_{K L}$ , then the $\\lambda_{K L^{-}W e i g}/ $ ated actor loss (Eq. 5) is equivalent to the un-regularized policy objective using the regularized critic:\n\n$$\n\\begin{array}{ r l }&{{\\mathbb E}_{p(s)\\pi(a\\mid s)}\\left[\\log Q(s,a)+\\log\\beta(a\\mid s)-\\log\\left((1-\\lambda_{K L})\\pi(a\\mid s)+\\lambda_{K L}\\beta(a\\mid s)\\right)\\right]}\\\\ &{\\ ={\\mathbb E}_{\\pi(a\\mid s)}\\left[\\log Q_{r}^{\\lambda_{T D}}(s,a,\\lambda_{critic})\\right]\\qquad for all states~s .}\\end{array}\n$$", "proof": {"steps": [{"n": 1, "claim": "$$\n\\begin{array}{ l }{\\displaystyle\\frac{Q(s,a)}{Q(s,a)+1}=\\frac{\\frac{y^{\\lambda_{\\mathrm{Im}}}(s,a)}{y^{\\lambda_{\\mathrm{Im}}}(s,a)+1}p(s)\\beta(a\\mid s)}{\\frac{y^{\\lambda_{\\mathrm{Im}}}(s,a)}{y^{\\lambda_{\\mathrm{Im}}}(s,a)+1}p(s)\\beta(a\\mid s)+\\frac{1}{y^{\\lambda_{\\mathrm{Im}}}(s,a)+1}p(s)\\left((1-\\lambda_{\\mathrm{critic}})\\pi(a\\mid s)+\\lambda_{\\mathrm{critic}}\\beta(a\\mid s)\\right)}}\\\\ {=\\frac{y^{\\lambda_{\\mathrm{Im}}}(s,a)\\beta(a\\mid s)}{y^{\\lambda_{\\mathrm{Im}}}(s,a)\\beta(a\\mid s)+(1-\\lambda_{\\mathrm{critic}})\\pi(a\\mid s)+\\lambda_{\\mathrm{critic}}\\beta(a\\mid s)}.}\\end{array}\n$$"}, {"n": 2, "claim": "Solving for $Q(s,a)$ on the left hand side, the optimal value for $Q(s,a)$ is given by\n\n$$\n\\begin{aligned}\nQ(s, a) &= y^{\\lambda_{\\mathrm{TD}}}(s, a)\\frac{\\beta(a \\mid s)}{(1 - \\lambda_{\\mathrm{critic}})\\pi(a \\mid s) + \\lambda_{\\mathrm{critic}} \\beta(a \\mid s)} \\\\\n&= (r(s, a) + \\mathbb{E}_{p(s' \\mid s, a), a' \\sim (1 - \\lambda_{\\mathrm{TD}}), \\pi(\\cdot \\mid s') + \\lambda_{\\mathrm{TD}} \\beta(\\cdot \\mid s)}[Q(s', a')]) \\frac{\\beta(a \\mid s)}{(1 - \\lambda_{\\mathrm{critic}})\\pi(a \\mid s) + \\lambda_{\\mathrm{critic}} \\beta(a \\mid s)}.\n\\end{aligned} \\tag{6}\n$$"}, {"n": 3, "claim": "Note that the next action $a^{\\prime}$ is sampled from a mixture policy defined by $\\lambda_{\\mathrm{TD}}$ . This equation tells us what each update for the λ-weighted regularized critic loss does.\n\nTo analyze these updates, we define\n\n$$\n\\tilde{Q}(s, a) \\triangleq Q(s, a) \\frac{(1 - \\lambda_{\\mathrm{critic}})\\pi(a \\mid s) + \\lambda_{\\mathrm{critic}} \\beta(a \\mid s)}{\\beta(a \\mid s)}.\n$$"}, {"n": 4, "claim": "Like before, the ratio $\\frac{\\beta(a^{\\prime}|s^{\\prime})}{(1-\\lambda_{\\mathrm{TD}})\\pi(a^{\\prime}|s^{\\prime})+\\lambda_{\\mathrm{TD}}\\beta(a^{\\prime}|s^{\\prime})}$ can act like an importance weight. When $\\lambda_{\\mathrm{TD}}=$ $\\lambda_{\\mathrm{critic}}$ , then this importance weight cancels with the sampling distribution, providing the following identity:\n\n$$\n\\begin{aligned}\n& \\mathbb{E}_{p(s' \\mid s, a), a' \\sim (1 - \\lambda_{\\mathrm{TD}}), \\pi(\\cdot \\mid s') + \\lambda_{\\mathrm{TD}} \\beta(\\cdot \\mid s)}[Q(s', a')] \\\\\n&= \\mathbb{E}_{p(s' \\mid s, a), a' \\sim (1 - \\lambda_{\\mathrm{TD}}), \\pi(\\cdot \\mid s') + \\lambda_{\\mathrm{TD}} \\beta(\\cdot \\mid s)} \\left[\\tilde{Q}(s, a)\\frac{\\beta(a \\mid s)}{(1 - \\lambda_{\\mathrm{critic}})\\pi(a \\mid s) + \\lambda_{\\mathrm{critic}} \\beta(a \\mid s)} \\right] \\\\\n&= \\mathbb{E}_{p(s' \\mid s, a), a' \\sim \\beta(\\cdot \\mid s')}[\\tilde{Q}(s, a)].\n\\end{aligned}\n$$"}, {"n": 5, "claim": "Substituting this identity in Eq. 6, we can write the updates using $\\tilde{Q}(s,a)$\n\n$$\n\\begin{aligned}\n& \\tilde{Q}(s, a)\\frac{\\beta(a \\mid s)}{(1 - \\lambda_{\\mathrm{critic}})\\pi(a \\mid s) + \\lambda_{\\mathrm{critic}} \\beta(a \\mid s)} \\\\\n& = \\left(r(s, a) + \\mathbb{E}_{p(s' \\mid s, a), a' \\sim \\beta(\\cdot \\mid s')}[\\tilde{Q}(s, a)] \\right)\\frac{\\beta(a \\mid s)}{(1 - \\lambda_{\\mathrm{critic}})\\pi(a \\mid s) + \\lambda_{\\mathrm{critic}} \\beta(a \\mid s)},\n\\end{aligned}\n$$"}, {"n": 6, "claim": "which can be simplified to\n\n$$\n\\tilde{Q}(s, a) = r(s, a) + \\mathbb{E}_{p(s' \\mid s, a), a' \\sim \\beta(\\cdot \\mid s')}[\\tilde{Q}(s, a)].\n$$"}, {"n": 7, "claim": "We then translate these convergence results for $\\tilde{Q}(s,a)$ into convergence results for $Q(s,a)$ . Written in terms of the original Q-values, we see that the optimal critic for the regularized critic update is\n\n$$\nQ^*(s, a) = Q^\\beta(s, a) \\frac{\\beta(a \\mid s)}{(1 - \\lambda_{\\mathrm{critic}})\\pi(a \\mid s) + \\lambda_{\\mathrm{critic}} \\beta(a \\mid s)}.\n$$"}, {"n": 8, "claim": "Finally, substitute this Q-function into the un-regularized actor loss, we see that the result is equivalent to the λ-weighted actor loss:\n\n$$\n\\mathbb{E}_{p(s) \\pi(a \\mid s)}\\left[\\log Q^*(s, a) \\right] = \\mathbb{E}_{p(s) \\pi(a \\mid s)}\\bigg[\\log Q^\\beta(s, a) + \\underbrace{\\log \\beta(a \\mid s) - \\log \\left((1 - \\lambda_{\\mathrm{KL}}) \\pi(a \\mid s) + \\lambda_{\\mathrm{KL}} \\beta(a \\mid s) \\right)}_{\\text{$\\lambda$-weighted actor regularizer}}\\bigg]\n$$"}]}}], "target_local_id": "T2", "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "label": "flawed", "error_step": 1, "error_class6": ["others", "logical_gap_or_missing_case", "circular_or_unproven"], "error_explanation": "Treating $\\frac{Q}{Q+1}\\leftarrow\\frac{y^{\\pi,Q}}{y^{\\pi,Q}+1}$ as a valid minimization of the single fixed objective $\\mathcal{L}_{\\mathrm{critic}}(Q,y^{\\pi,Q})$ is wrong: the TD target $y^{\\pi,Q}$ is itself a (nonlinear) function of the very $Q$ being optimized. The equation $\\frac{Q}{Q+1}=\\frac{y^{\\pi,Q}}{y^{\\pi,Q}+1}$ is only the per-step loss/update (labels held fixed), not the stationarity condition of $\\min_Q \\mathcal{L}_{\\mathrm{critic}}(Q,y(Q))$; minimizing over the outer $Q$ while holding the $Q$ inside $y(Q)$ fixed is not a correct minimization and can get stuck at a non-optimal fixed point rather than at $Q^\\pi$ (a reviewer's counterexample $\\min (y(Q)-1)^2+(Q-2)^2$ gets stuck at $Q=2$). The claim must instead be about the LIMIT of the sequence of incremental updates $Q_{t+1}\\gets\\arg\\min_Q\\mathcal{L}_{\\mathrm{critic}}(Q,y^{\\pi,Q_t})$ (target from the previous iterate $Q_t$), which equals the standard policy-evaluation iteration and hence converges to $Q^\\pi$. The authors conceded the lemma as stated was wrong."}
{"id": "gV3wdEOGy_V", "context": "We introduce the contrastive learning methods based on the instance discrimination task (Wu et al., 2018; Ye et al., 2019; He et al., 2020; Chen et al., 2020), with a particular focus on the recent state-of-the-art method, MoCo (He et al., 2020). Let $\\mathbf{X}=\\{\\mathbf{x}_{n}\\}_{n=1}^{N}$ be a set of images without the ground-truth labels, and each of the datapoint ${\\bf x}_{n}$ is assigned with a unique surrogate label $y_{n}\\in\\{1,2,...,N\\}$ such that $y_{n}\\neq y_{j},\\forall j\\neq n^{2}$ To learn representations in an unsupervised manner, instance discrimination considers a discriminative classifier that maps the given image to its surrogate label. Suppose that we have two encoder networks $f_{\\theta}$ and $f_{\\theta^{\\prime}}$ that generate $\\ell_{2}\\cdot$ normalized embeddings $\\mathbf{v}_{y_{n}}\\in\\mathbb{R}^{d}$ and $\\mathbf{f}_{n}\\in\\mathbb{R}^{d}$ , respectively, given the image ${\\bf x}_{n}$ with the surrogate label $y_{n}$ . We show the parameters of the networks in the subscript, and images are transformed by a stochastic data augmentation module before passing to the networks (please see Appendix D). We can model the probability classifier with:\n\n$$\np(\\mathbf{Y}|\\mathbf{X})=\\prod_{n=1}^{N}p(y_{n}|\\mathbf{x}_{n})=\\prod_{n=1}^{N}\\frac{\\exp(\\mathbf{v}_{y_{n}}^{\\top}\\mathbf{f}_{n}/\\tau)}{\\sum_{i=1}^{N}\\exp(\\mathbf{v}_{i}^{\\top}\\mathbf{f}_{n}/\\tau)},\\tag{4}\n$$\n\nUnsupervised clustering aims to partition a dataset X with N observations into K clusters. We introduce the latent variable $z_{n}\\in\\{1,2,...,K\\}$ to be the cluster label of the image ${\\bf x}_{n}$ and naturally extend Eq. (4) to Mixture of Contrastive Experts (MiCE). The formulation explicitly introduces a mixture model to capture the latent semantic structures, which is inspired by the mixture of experts (MoE) framework (Jacobs et al., 1991). In Eq. $(5),p\\big(y_{n}\\big|\\mathbf{x}_{n},z_{n}\\big)$ is one of the experts that learn to discriminate a subset of instances and $p(z_{n}|\\mathbf{x}_{n})$ is a gating function that partitions the dataset into subsets according to the latent semantics by routing the given input to one or a few experts. With a divide-and-conquer principle, the experts are often highly specialized in particular images that share similar semantics, which improves the learning efficiency. Notably, MiCE is generic to the choice of the underlying contrastive methods (Wu et al., 2018; He et al., 2020; Chen et al., 2020), while in this paper, we focus on an instance based on MoCo. Also, please see Fig. 1 for an illustration of MiCE with three experts.\n\nGating function. The gating function organizes the instance discrimination task into K simpler subtasks by weighting the experts based on the semantics of the input image. We define $g_{\\psi}$ as an encoder network that outputs an embedding for each input image. We denote the output vector for image ${\\bf x}_{n}$ as ${\\bf g}_{n}\\in\\mathbb{R}^{d}$ . The gating function is then parameterized as:\n\n$$\np(z_{n}|\\mathbf{x}_{n})=\\frac{\\exp(\\omega_{z_{n}}^{\\top}\\mathbf{g}_{n}/\\kappa)}{\\sum_{k=1}^{K}\\exp(\\omega_{k}^{\\top}\\mathbf{g}_{n}/\\kappa)},\\tag{6}\n$$\n\nwhere κ is the temperature, and $\\omega=\\{\\omega_{k}\\}_{k=1}^{K}$ represent the gating prototypes. All prototypes and image embeddings are 2-normalized in the $\\mathbb{R}^{d}$ space. Hence, the gating function performs a soft partitioning of the dataset based on the cosine similarity between the image embeddings and the gating prototypes. We can view it as a prototype-based discriminative clustering module, whereas we obtain the cluster labels using posterior inference to consider additional information in the experts.\n\nExperts. In MiCE, every expert learns to solve the instance discrimination subtask arranged by the gating function. We define the expert in terms of the unnormalized model $\\Phi(\\cdot)$ following Wu et al. (2018); He et al. (2020). Therefore, the probability of the image ${\\bf x}_{n}$ being recognized as the $y_{n}$ -th one by the $z_{n}$ -th expert is formulated as follows:\n\n$$\np(y_{n}|\\mathbf{x}_{n},z_{n})=\\frac{\\Phi(\\mathbf{x}_{n},y_{n},z_{n})}{Z(\\mathbf{x}_{n},z_{n})},\\tag{7}\n$$\n\nwhere $\\begin{array}{ r }{Z(\\mathbf{x}_{n},z_{n})=\\sum_{i=1}^{N}\\Phi(\\mathbf{x}_{n},y_{i},z_{n})}\\end{array}$ is a normalization constant that is often computationally intractable.\n\n5 Inference and Learning\n\nWe first discuss the evidence lower bound (ELBO), the single objective used in MiCE, in Sec. 5.1. Then, we present a scalable variant of the Expectation-Maximization (EM) algorithm (Dempster et al., 1977) to deal with the non-trivial inference and learning of MiCE in Sec. 5.2. Lastly, in Sec. 5.3, we show that a naïve two-stage baseline, in which we run a spherical k-means algorithm on the embeddings learned by MoCo, is a special case of MiCE.\n\n5.1 Evidence Lower Bound (ELBO)\n\nThe parameters to update include the parameters θ, ψ of the student and gating network respectively, and the expert prototypes $\\pmb{\\mu}=\\{\\pmb{\\mu}\\}_{k=1}^{K}$ The learning objective of MiCE is to maximize the evidence lower bound (ELBO) of the log conditional likelihood of the entire dataset. The ELBO of the datapoint n is given by:\n\n$$\n\\begin{array}{ r l }&{\\log p(y_{n}|\\mathbf{x}_{n})\\geq\\mathcal{L}(\\pmb{\\theta},\\psi,\\pmb{\\mu};\\mathbf{x}_{n},y_{n})}\\\\ &{\\qquad: =\\mathbb{E}_{q(z_{n}|\\mathbf{x}_{n},y_{n})}[\\log p(y_{n}|\\mathbf{x}_{n},z_{n};\\pmb{\\theta},\\pmb{\\mu})]-D_{\\mathrm{KL}}(q(z_{n}|\\mathbf{x}_{n},y_{n})\\|p(z_{n}|\\mathbf{x}_{n};\\psi)),}\\end{array}\\tag{8}\n$$\n\nwhere $q(z_{n}|\\mathbf{x}_{n},y_{n})$ is a variational distribution to infer the latent cluster label given the observed data. The first term in Eq. (8) encourages $q(z_{n}|\\mathbf{x}_{n},y_{n})$ to be high for the experts that are good at discriminating the input images. Intuitively, it can relief the potential degeneracy issue (Caron et al, 2018; Ji et al., 2019), where all images are assigned to the same cluster. This is because a degenerated posterior puts the pressure of discriminating all images on a single expert, which may result in a looser ELBO. The second term in Eq. (8) is the Kullback-Leibler divergence between the variational distribution and the distribution defined by the gating function. With this term, the gating function is refined during training and considers the capability of the experts when partitioning data. Notably, MiCE does not rely on auxiliary loss or regularization terms as many prior methods (Haeusser et al., 2018; Shiran & Weinshall, 2019; Wu et al., 2019; Niu et al., 2020) do.\n\n\nAssumption 1. The gating temperature $\\kappa  \\infty$ such that for all k, the prior\n\n$$\np(z_{n}|\\mathbf{x}_{n})=\\frac{1}{K},\\quad\\forall n .\n$$\n\nAssumption 2. There is only a single output layer for both student network $f_{\\theta}$ and teacher network $f_{\\pmb{\\theta}^{\\prime}}$ respectively. The resulting embeddings are used across K expert, to be specific, for all possible cases,\n\n$$\n\\Phi(\\mathbf{x}_{n},y_{n},k)=\\exp\\left(\\mathbf{v}_{y_{n}}^{\\top}(\\mathbf{f}_{n}+\\pmb{\\mu}_{k})/\\tau\\right),\n$$\n\nwhere $\\mathbf{f}_{n}=f_{\\pmb{\\theta}}(\\mathbf{x}_{n})\\in\\mathbb{R}^{d},\\mathbf{v}_{y_{n}}^{\\top}=f_{\\pmb{\\theta}^{\\prime}}(\\mathbf{x}_{n})\\in\\mathbb{R}^{d}.$\n\nAssumption 3. The unnormalized model $\\Phi(\\cdot)$ considers only the instance-level information such that for all possible cases,\n\n$$\n\\Phi(\\mathbf{x}_{n},y_{n},z_{n})=\\exp\\left({\\bf v}_{y_{n},z_{n}}^{\\top}\\mathbf{f}_{n,z_{n}}/\\tau\\right).\n$$\n", "theorems": [{"local_id": "T1", "statement": "Theorem 1. If Assumptions 1-3 hold, the overall output of MiCE becomes $$p(y_n|\\mathbf{x}_n) = \\frac{\\exp(\\mathbf{v}_{y_n}^\\top \\mathbf{f}_n/\\tau)}{\\sum_{i=1}^N \\exp(\\mathbf{v}_i^\\top \\mathbf{f}_n/\\tau)}.$$", "proof": {"steps": [{"n": 1, "claim": "The expert is $p(y_n|\\mathbf{x}_n,z_n) = \\Phi(\\mathbf{x}_n,y_n,z_n)/Z(\\mathbf{x}_n,z_n)$. By Assumption 2 (single shared output layer), $\\Phi(\\mathbf{x}_n,y_n,k) = \\exp(\\mathbf{v}_{y_n}^\\top(\\mathbf{f}_n+\\pmb{\\mu}_k)/\\tau)$, so $$p(y_n|\\mathbf{x}_n,z_n) = \\frac{\\exp(\\mathbf{v}_{y_n}^\\top(\\mathbf{f}_n+\\pmb{\\mu}_{z_n})/\\tau)}{\\sum_{i=1}^N \\exp(\\mathbf{v}_i^\\top(\\mathbf{f}_n+\\pmb{\\mu}_{z_n})/\\tau)}.$$"}, {"n": 2, "claim": "By Assumption 3 the unnormalized model considers only instance-level information, $\\Phi(\\mathbf{x}_n,y_n,z_n)=\\exp(\\mathbf{v}_{y_n}^\\top \\mathbf{f}_n/\\tau)$, i.e. it does not depend on the cluster offset $\\pmb{\\mu}_{z_n}$. Substituting this $\\pmb{\\mu}$-free $\\Phi$ gives $$p(y_n|\\mathbf{x}_n,z_n) = \\frac{\\exp(\\mathbf{v}_{y_n}^\\top \\mathbf{f}_n/\\tau)}{\\sum_{i=1}^N \\exp(\\mathbf{v}_i^\\top \\mathbf{f}_n/\\tau)},$$ which is independent of $z_n$."}, {"n": 3, "claim": "Marginalizing over the latent cluster, $p(y_n|\\mathbf{x}_n) = \\sum_{k=1}^K p(z_n=k|\\mathbf{x}_n)\\, p(y_n|\\mathbf{x}_n,z_n=k)$. By Assumption 1 ($\\kappa\\to\\infty$) $p(z_n=k|\\mathbf{x}_n)=1/K$, and since the expert is independent of $z_n$, $$p(y_n|\\mathbf{x}_n) = \\sum_{k=1}^K \\frac{1}{K} \\cdot \\frac{\\exp(\\mathbf{v}_{y_n}^\\top \\mathbf{f}_n/\\tau)}{\\sum_{i=1}^N \\exp(\\mathbf{v}_i^\\top \\mathbf{f}_n/\\tau)} = \\frac{\\exp(\\mathbf{v}_{y_n}^\\top \\mathbf{f}_n/\\tau)}{\\sum_{i=1}^N \\exp(\\mathbf{v}_i^\\top \\mathbf{f}_n/\\tau)}.$$"}]}}, {"local_id": "T2", "statement": "Given the instance-discrimination classifier of Eq. (4), the gating function $p(z_n|\\mathbf{x}_n) = \\frac{\\exp(\\omega_{z_n}^\\top \\mathbf{g}_n/\\kappa)}{\\sum_{k=1}^K \\exp(\\omega_k^\\top \\mathbf{g}_n/\\kappa)}$ (Eq. 6), and the expert $p(y_n|\\mathbf{x}_n,z_n) = \\frac{\\Phi(\\mathbf{x}_n,y_n,z_n)}{Z(\\mathbf{x}_n,z_n)}$ (Eq. 7), the MiCE joint model over surrogate labels $\\mathbf{Y}$ and latent cluster labels $\\mathbf{Z}$ factorizes as $p(\\mathbf{Y},\\mathbf{Z}|\\mathbf{X}) = \\prod_{n=1}^N \\prod_{k=1}^K p(y_n, z_n = k|\\mathbf{x}_n)$, decomposing into gating-times-expert Mixture-of-Experts form, on which the ELBO (Eq. 8), E-step, and M-step are derived.", "proof": {"steps": [{"n": 1, "claim": "Introduce latent cluster label $z_n \\in \\{1,\\dots,K\\}$ for each $\\mathbf{x}_n$, extending Eq. (4) to a joint $p(\\mathbf{Y},\\mathbf{Z}|\\mathbf{X})$."}, {"n": 2, "claim": "$p(\\mathbf{Y},\\mathbf{Z}|\\mathbf{X}) = \\prod_{n=1}^N \\prod_{k=1}^K p(y_n, z_n = k|\\mathbf{x}_n)$."}, {"n": 3, "claim": "$p(y_n, z_n=k|\\mathbf{x}_n) = p(z_n=k|\\mathbf{x}_n)\\, p(y_n|\\mathbf{x}_n, z_n=k)$."}, {"n": 4, "claim": "$p(z_n|\\mathbf{x}_n)$ is the gating distribution of Eq. (6); $p(y_n|\\mathbf{x}_n,z_n)$ is the expert of Eq. (7)."}, {"n": 5, "claim": "Gating $p(z_n|\\mathbf{x}_n)$ routes to experts $p(y_n|\\mathbf{x}_n,z_n)$: a Mixture-of-Experts."}, {"n": 6, "claim": "ELBO (Eq. 8), E-step (Eqs. 9-11), M-step (Eqs. 12-14) follow from this factorization."}]}}, {"local_id": "T3", "statement": "Theorem 2 (Convergence of MiCE). Assume that the log conditional likelihood of the observed data log $p(\\mathbf{Y}|\\mathbf{X};\\pmb{\\theta},\\psi,\\pmb{\\mu})$ has an upper bound, the approximated ELBO $\\widetilde{\\mathcal{L}}(\\pmb{\\theta},\\psi,\\pmb{\\mu};\\mathbf{X},\\mathbf{Y})$ of MiCE is then upper bounded and it will converge to a certain value $\\mathcal{L}^{\\ast}$ with the proposed EM algorithm.", "proof": {"steps": [{"n": 1, "claim": "\\begin{array}{ r l }&{\\widetilde{\\mathcal{L}}(\\theta,\\psi,\\mu;\\mathbf{x}_{n},y_{n})}\\\\ &{=\\mathbb{E}_{q(z_{n}|\\mathbf{x}_{n},y_{n})}\\left[\\log\\frac{\\Phi\\left(\\mathbf{x}_{n},y_{n},z_{n};\\theta,\\mu\\right)}{\\hat{Z}\\left(\\mathbf{x}_{n},z_{n};\\theta,\\mu\\right)}\\right]-D_{\\mathrm{K L}}(q(z_{n}|\\mathbf{x}_{n},y_{n})| |p(z_{n}|\\mathbf{x}_{n};\\psi))}\\\\ &{=\\mathbb{E}_{q(z_{n}|\\mathbf{x}_{n},y_{n})}\\left[\\log\\frac{\\Phi\\left(\\mathbf{x}_{n},y_{n},z_{n};\\theta,\\mu\\right)}{Z\\left(\\mathbf{x}_{n},z_{n};\\theta,\\mu\\right)}+\\log\\frac{Z\\left(\\mathbf{x}_{n},z_{n};\\theta,\\mu\\right)}{\\hat{Z}\\left(\\mathbf{x}_{n},z_{n};\\theta,\\mu\\right)}\\right]-D_{\\mathrm{K L}}(q(z_{n}|\\mathbf{x}_{n},y_{n})| |p(z_{n}|\\mathbf{x}_{n};\\psi))}\\end{array}"}, {"n": 2, "claim": "=\\mathcal{L}(\\pmb\\theta,\\psi,\\mu;\\mathbf x_{n},y_{n})+\\mathbb{E}_{q(z_{n}|\\mathbf x_{n},y_{n})}\\left[\\log\\frac{Z(\\mathbf{x}_{n},z_{n};\\pmb\\theta,\\pmb\\mu)}{\\hat{Z}(\\mathbf{x}_{n},z_{n};\\pmb\\theta,\\pmb\\mu)}\\right]."}, {"n": 3, "claim": "Recall that the normalization constant is\n\n$$\nZ(\\mathbf{x}_{n},z_{n};\\pmb\\theta,\\pmb\\mu)=\\sum_{i=1}^{N}\\mathrm{exp}\\left(\\mathbf{v}_{y_{i},z_{n}}^{\\top}\\left(\\mathbf{f}_{n,z_{n}}+\\pmb\\mu_{z_{n}}\\right)/\\tau\\right),\n$$"}, {"n": 4, "claim": "and the approximated normalization constant is\n\n$$\n\\hat{Z}(\\mathbf{x}_{n},z_{n};\\pmb\\theta,\\pmb\\mu)=\\sum_{j=1}^{\\nu}\\mathrm{exp}\\left(\\mathbf{q}_{j,z_{n}}^{\\top}\\left(\\mathbf{f}_{n,z_{n}}+\\pmb\\mu_{z_{n}}\\right)/\\tau\\right),\\tag{As in Eq. (10}\n$$\n\nwhere $\\mathbf{q}$ is a queue that stores the outputs of the teacher network as described in the main text."}, {"n": 5, "claim": "If we use the teacher embeddings of the entire dataset as the queue, then the two normalization constants cancel each other and we obtain the original ELBO. In such cases, the convergence can be proved following the standard EM algorithm (Dempster et al., 1977; Wu, 1983)."}, {"n": 6, "claim": "Otherwise, we can also bound the approximated ELBO. Since that all student and teacher embeddings and expert prototypes are $\\ell_{2}\\cdot$ -normalized, we can bound the log ratio term independent of the choice of the queue:\n\n$$\n\\frac{-2}{\\tau}\\leq\\mathbf{v}_{y_{i},z_{n}}^{\\top}\\left(\\mathbf{f}_{n,z_{n}}+\\mu_{z_{n}}\\right)/\\tau\\leq\\frac{2}{\\tau},\n$$"}, {"n": 7, "claim": "which implies\n\n$$\nZ(\\mathbf{x}_{n},z_{n};\\pmb\\theta,\\pmb\\mu)\\leq N\\exp(\\frac{2}{\\tau}),\n$$"}, {"n": 8, "claim": "and\n\n$$\n\\hat{Z}(\\mathbf{x}_{n},z_{n};\\pmb\\theta,\\pmb\\mu)\\geq\\nu\\exp(\\frac{-2}{\\tau}).\n$$"}, {"n": 9, "claim": "Given that they are both positive, we can get\n\n$$\n\\log\\frac{Z(\\mathbf{x}_{n},z_{n};\\pmb\\theta,\\pmb\\mu)}{\\hat{Z}(\\mathbf{x}_{n},z_{n};\\pmb\\theta,\\pmb\\mu)}\\leq\\log N-\\log\\nu+\\frac{4}{\\tau}\n$$"}, {"n": 10, "claim": "Therefore, the approximated ELBO is bounded by:\n\n$$\n\\begin{array}{ l }{\\displaystyle\\widetilde{\\mathcal{L}}(\\pmb{\\theta},\\psi,\\mu;\\mathbf{x}_{n},y_{n})=\\mathcal{L}(\\pmb{\\theta},\\psi,\\mu;\\mathbf{x}_{n},y_{n})+\\mathbb{E}_{q(z_{n}|\\mathbf{x}_{n},y_{n})}\\left[\\log\\frac{Z(\\mathbf{x}_{n},z_{n};\\pmb{\\theta},\\pmb{\\mu})}{\\hat{Z}(\\mathbf{x}_{n},z_{n};\\pmb{\\theta},\\mu)}\\right]}\\\\ {\\leq\\log p(y_{n}|\\mathbf{x}_{n};\\pmb{\\theta},\\psi,\\pmb{\\mu})+\\log N-\\log\\nu+\\frac{4}{\\tau}.}\\end{array}\n$$"}, {"n": 11, "claim": "Similar to the proof of original EM (Wu, 1983), since the approximated ELBO will not decrease in expectation during training, the approximated ELBO of MiCE will converge to a certain value $\\mathcal{L}^{*}$"}]}}], "target_local_id": "T2", "topic_domain": "Deep Learning Theory", "topic_subtopic": "Representation & Expressivity", "label": "flawed", "error_step": 2, "error_class6": ["logical_gap_or_missing_case", "missing_assumption", "others"], "error_explanation": "Taking the bare product $\\prod_{k=1}^K p(y_n, z_n=k|\\mathbf{x}_n)$ multiplies together the probabilities of $z_n$ taking every cluster value $k$ simultaneously, so it double-counts across clusters and does not equal $p(\\mathbf{Y},\\mathbf{Z}|\\mathbf{X})$. The bare product is a valid joint only under a HARD assignment, where exactly one $k$ has $p(z_n=k|\\mathbf{x}_n)=1$ and the rest vanish. But the gating function (Eq. 4) is an explicitly SOFT categorical distribution placing nonzero mass on multiple experts, so this hard-assignment assumption is missing. The correct factor requires the selector exponent $\\mathbb{1}(z_n=k)$: $p(\\mathbf{Y},\\mathbf{Z}|\\mathbf{X}) = \\prod_{n=1}^N \\prod_{k=1}^K p(y_n, z_n=k|\\mathbf{x}_n)^{\\mathbb{1}(z_n=k)}$, so that for the realized cluster only the $k=z_n$ term contributes (exponent 1) and every other term becomes $(\\cdot)^0=1$, recovering the intended single-expert factor $p(z_n|\\mathbf{x}_n)\\,p(y_n|\\mathbf{x}_n,z_n)$."}
{"id": "hac6DzbMa7", "context": "3 Theoretical Analysis of the CF and KT Problems\n\nTask Incremental Learning (TIL). Let T tasks be $\\mathbb{T}=\\{t\\}_{t=1}^{T}$ , which are learned sequentially. Each task has a training dataset with its task descriptor $t,\\mathbb{D}_{t}=\\{(\\boldsymbol{x}_{t,i},t),\\boldsymbol{y}_{t,i})\\}_{i=1}^{N_{t}}$ , where $\\pmb{x}_{t,i}\\in X$ is the input data and $\\pmb{y}_{t,i}\\in Y_{t}\\in Y$ is its class label. The goal of TIL is to construct a predictor $f \\colon$ $X\\times\\mathbb{T}\\to Y$ to predict the class label $y\\in Y_{t}$ for (x, t) (a given test instance x from task t).\n\nKnowledge Transfer (KT). Let $\\mathbb{T}_{s i m}/\\mathbb{T}_{d i s}$ be a set of similar/dissimilar tasks of the current task t $(\\mathbb{T}_{s i m},\\mathbb{T}_{d i s}\\subseteq\\mathbb{T},\\mathbb{T}_{d i s}=\\mathbb{T}-\\mathbb{T}_{s i m})$ .The learner should transfer the knowledge learned in the past forward and leverages it to learn t better, and additionally, the learning of t should also improve the tasks in $\\mathbb{T}_{s i m}$ (backward KT). (Ke et al., 2020) suggested that a TIL model/method should satisfy the basic requirements: (1) overcoming CF and (2) performing FKT and/or BKT to improve the performance of the TIL model across similar tasks.\n\n3.1 Theoretical Analysis of the CF Problem\n\nGiven a fixed capacity DNN model with L layers, its weight matrix is denoted by $\\mathbb{W}=\\{\\mathbb{W}^{l}\\}_{l=1}^{L}$ where $\\mathbb{W}^{l}$ is the layer-wise weight matrix for layer l. Let $\\mathcal{L}(\\mathbb{W}_{t})$ be the loss function, $\\mathrm{e . g .}$ , crossentropy loss, where $\\mathbb{W}_{t}$ is the weight matrix of the model after learning task $t\\in [1,T]$\n\nInput, Gradient, and Weight Spaces. Given a DNN model, we define the space of the input data as the Input Space denoted by $S_{i n},$ and the space of the gradient span as the Gradient Space $S_{g}$ We call the space/domain of the weight matrix of a model as its Weight Space denoted by $S_{u}$ •\n\nWhen learning a new task t, the weight matrix can be updated by SGD (Amari, 1993) as follows:\n\n$$\n\\mathbb{W}_{t}=\\mathbb{W}_{t-1}-\\lambda\\nabla\\mathbb{W}_{t},\\mathbb{W}_{t-1},\\mathbb{W}_{t}\\in S_{w},\\nabla\\mathbb{W}_{t}\\in S_{g}\\tag{3}\n$$\n\nwhere λ is the learning rate and $\\nabla\\mathbb{W}_{t}$ is the gradient with respect to $\\mathbb{W}_{t}$ in task t learning. Eq.(3) implies the followings: (1) the weight matrix $\\mathbb{W}_{t}\\left(\\mathbb{W}_{t-1}\\right)$ records the knowledge learned from task t $(t-1),(2)$ with the quantity of $\\lambda\\nabla\\mathbb{W}_{t}.$ , the new task t learning may interfere with the knowledge (stored in $\\mathbb{W}_{t-1})$ learned from previous tasks $\\leq t - 1$ , which can cause CF. And with further theoretical analysis, we present and prove Theorem 2 below, which has not been reported in the existing literature. Please see the proof of Theorem 2 in Appendix A.\n\nTheorem 2. The learning of the new task $t \\leq T$ will not interfere with the knowledge (weights $W_{t-1})$ of the previous learned $t - 1$ tasks (i.e., there is no CF) if and only if 1) task t occupies its own independent weight subspace $W_{t}$ that is orthogonal to all other weight subspaces of the previous tasks in the whole weight space $S_{w}$ of the model, and 2) the modified weights spanned by the orthogonal gradient to the previous learned tasks are imposed in its weight subspace $\\mathbf{}W_{t}$ of task t.\n\nTheorem 2 shows that when learning task t, the weight modification should happen in its own weight subspace $W_{t}$ to ensure the weight-level orthogonality between tasks rather than only the gradient-level orthogonality. The weight space and gradient space, although dependent, are two different spaces. Thus existing OG-based methods cannot guarantee zero CF as they don't satisfy the second condition in Theorem 2. Moreover, an orthogonal weight subspace based (OW-based for short) TIL method following Theorem 2 also yields the benefit stated in Lemma 2 below.\n\nLemma 2. The total number of continually learnable tasks of the OG-based methods is determined by the rank value of its weight matrix in a fixed-size network (Zeng et al., 2019). With the rank value as the metric, the upper bound of the minimum number of learnable tasks of the OW-based TIL method is far greater than that of the OG-based methods (see Theorems 3-4 and its proof in Appendix A).\n\n3.2 Theoretical Analysis of the KT Problem\n\nWe first introduce some definitions and then explore what factors cause positive or negative KT.\n\nForward Negative Knowledge Transfer Margin (FNM). Given two similar tasks i and $t\\left(i<t\\right)$ in the TIL setting, let $\\epsilon_{t}$ be the test error of task t. $f^{\\prime}(i,t)$ denotes that task t performs its learning with the help of the knowledge of the previous task i, and $f^{\\prime}(\\varnothing,t)$ otherwise. Then, negative forward knowledge transfer (FKT) happens when $\\epsilon_{t}(f^{\\prime}(i,t))>\\epsilon_{t}(f^{\\prime}(\\emptyset,t))$ .\n\nBackward Negative Knowledge Transfer Margin (BNM). Given two similar tasks i and $t\\left(i<t\\right)$ in the TIL setting, let $\\epsilon_{i}(f^{\\prime}(i))$ be the test error of task i before task t learning, and $\\epsilon_{i}^{\\prime}(f^{\\prime}(i,t))$ be the test error of task i after task t has been learned, then negative backward knowledge transfer (BKT) happens when $\\epsilon_{i}^{\\prime}(f^{\\prime}(i,t))>\\epsilon_{i}(f^{\\prime}(i))$ . Thus, the negative FKT and BKT margins can be defined as:\n\n$$\nF N M=\\epsilon_{t}(f^{\\prime}(i,t))-\\epsilon_{t}(f^{\\prime}(\\emptyset,t));B N M=\\epsilon_{i}^{\\prime}(f^{\\prime}(i,t))-\\epsilon_{i}(f^{\\prime}(i)),i<t,t\\in [2,T]\\tag{4}\n$$\n\nProposed FNM/BNM-based KT Metrics. From Eq. (4) , it is clear that the degree of forward negative KT/backward negative KT can be evaluated by the FNM/BNM, and that the negative KT occurs if the FNM/BNM is positive. As $\\epsilon_{t}$ is inversely proportional to the test accuracy of task t (denoted by $A_{t})$ and FNM/BNM may not always be computable, the degree of FWT and BWT across similar tasks i and t in CL can be evaluated by the following Eq.(5)\n\n$$\nF W T=A_{t}(f^{\\prime}(i,t))-A_{t}(f^{\\prime}(\\emptyset,t));B W T=A_{i}^{\\prime}(f^{\\prime}(i,t))-A_{i}(f^{\\prime}(i)),i<t,t\\in [2,T]\\tag{5}\n$$\n\nwhere $A_{i}^{\\prime}$ is the test accuracy of task i after task t learning. The greater the positive/negative value of FWT/BWT, the greater the quantity of positive/negative FWT/BWT.\n\nTheoretical Bound for KT. Given two similar tasks i and $t(i,t\\in [1,T])$ in CL, we now analyze the theoretical bound for KT between similar tasks i and t in their TIL so as to investigate the factors that lead to forward or backward positive/negative KT between them.\n\nLet a hypothesis be a mapping function of a neural network for classification of TIL $h:X\\times\\mathbb{T}\\to Y$ According to the test data distribution $\\mathcal{D^{\\prime}}_{t}$ of task t, the test error showing that the hypothesis h disagrees with its labeling function $l_{t}$ (which can also be a hypothesis) is defined as:\n\n$$\n\\epsilon_{t}(h,l_{t})=\\mathbb{E}_{\\mathbf{x}\\sim\\mathcal{D^{\\prime}}_{t}}\\left[|h(\\mathbf{x},t)-l_{t}(\\mathbf{x})|\\right]\\tag{6}\n$$\n\nFor simplicity, we also denote the risk or error of hypothesis h on task t by $\\epsilon_{t}(h)\\left(=\\epsilon_{t}(h,l_{t})\\right)$ .With the assumption that the training and the test data are i.i.d (independently identically distributed), we can get its empirical error $\\hat{\\epsilon}_{t}(h)$ from the training data. Let the divergence of the test data distributions of $\\mathcal{\\hat{D^{\\prime}}}_{i}$ and $\\mathcal{D}_{~i}^{{\\prime}}$ be $d(\\mathcal{D}_{\\textit{ i }}^{\\prime},\\mathcal{D}_{\\textit{ t }}^{\\prime})$ of tasks i and t, where $d(.)$ can be calculated by a similarity/distance metric. Then we can derive and prove the following Theorem 5.\n\nTheorem 5. The theoretical bounds for FWT and BWT of tasks i and t are respectively as follows:\n\n$$\n\\epsilon_{t}(h)\\leq\\epsilon_{i}(h)+d(\\mathcal{D}_{i}^{\\prime},\\mathcal{D}_{t}^{\\prime})+\\operatorname*{min}\\{\\mathbb{E}_{\\mathcal{D}_{i}^{\\prime}}(\\mathbb{S}),\\mathbb{E}_{\\mathcal{D}_{t}^{\\prime}}(\\mathbb{S})\\};\\epsilon_{i}(h)\\leq\\epsilon_{t}(h)+d(\\mathcal{D}_{i}^{\\prime},\\mathcal{D}_{t}^{\\prime})+\\operatorname*{min}\\{\\mathbb{E}_{\\mathcal{D}_{i}^{\\prime}}(\\mathbb{S}),\\mathbb{E}_{\\mathcal{D}_{t}^{\\prime}}(\\mathbb{S})\\}\\tag{7}\n$$\n\nwhere $\\mathbb{S}= |l_{i}(\\mathbf{x})-l_{t}(\\mathbf{x})|$ represents the absolute difference of the test results on text data x of tasks i and t. The proof is given in Appendix A.\n\nFrom Theorem 2, we observe the followings: (1) In the forward/backward KT process, two additional losses are introduced, (i) the loss due to the divergence of the test data distributions of tasks i and t (the second term on the right side of Eq.(7), and (i) the loss due to the difference of data classification results of the tasks (the third term). (2) It is clear that the necessary and sufficient conditions for elimination of negative forward/backward KT is that the errors introduced by the above two terms are zero. (3) The less the two additional losses above are, the greater the gain of FWT or BWT will be. In practice, however, Eq.(7) may not always be computable as it is impossible to get the test data during model training. Thus, with the assumption that the training and test data are i.i.d, we can employ the empirical errors $\\hat{\\epsilon}_{t}(h)$ and $\\hat{\\epsilon}_{i}(h)$ to approximate $\\epsilon_{t}(h)$ and $\\epsilon_{i}(h)$ using training data.\n\nMoreover, related KT researches (Zhang et al., 2022; Prado & Riddle, 2022; Wang et al., 2021; Riemer et al., 2019; Lin et al., 2022a;b) have proven the following Theorem 6.\n\nTheorem 6. Low similarity or negatively correlated tasks will result in negative KT. Only high similarity tasks or positively correlated tasks can achieve positive KT.", "theorems": [{"local_id": "T1", "statement": "Lemma 2. The total number of continually learnable tasks of the OG-based methods is determined by the rank value of its weight matrix in a fixed-size network (Zeng et al., 2019). With the rank value as the metric, the upper bound of the minimum number of learnable tasks of the OW-based TIL method is far greater than that of the OG-based methods (see Theorems 3-4 and its proof in Appendix A).", "proof": {"steps": [{"n": 1, "claim": "The total number of continually learnable tasks of the OG-based methods is determined by the rank value of its weight matrix in a fixed-size network (Zeng et al., 2019)."}, {"n": 2, "claim": "With the rank value as the metric, the upper bound of the minimum number of learnable tasks of the OW-based TIL method is far greater than that of the OG-based methods (see Theorems 3-4 and its proof in Appendix A)."}]}}, {"local_id": "T2", "statement": "Consider Task Incremental Learning over $T$ tasks $\\mathbb{T}=\\{t\\}_{t=1}^{T}$ learned sequentially, with a fixed-capacity $L$-layer DNN whose layer-wise weight matrices are $\\mathbb{W}=\\{\\mathbb{W}^l\\}_{l=1}^{L}$, where $\\mathbb{W}_t$ is the weight matrix after learning task $t$, updated by SGD $\\mathbb{W}_t=\\mathbb{W}_{t-1}-\\lambda\\nabla\\mathbb{W}_t$ with $\\mathbb{W}_{t-1},\\mathbb{W}_t\\in S_w$ (weight space) and $\\nabla\\mathbb{W}_t\\in S_g$ (gradient space). Theorem 2: The learning of the new task $t\\leq T$ will not interfere with the knowledge (weights $W_{t-1}$) of the previous learned $t-1$ tasks (i.e., there is no catastrophic forgetting) if and only if 1) task $t$ occupies its own independent weight subspace $W_t$ that is orthogonal to all other weight subspaces of the previous tasks in the whole weight space $S_w$ of the model, and 2) the modified weights spanned by the orthogonal gradient to the previous learned tasks are imposed in its weight subspace $W_t$ of task $t$.", "proof": {"steps": [{"n": 1, "claim": "The weight matrix $\\mathbb{W}_{t-1}$ records the knowledge of the previous $t-1$ tasks, and the SGD update $\\mathbb{W}_t=\\mathbb{W}_{t-1}-\\lambda\\nabla\\mathbb{W}_t$ perturbs it by $\\lambda\\nabla\\mathbb{W}_t$, so learning task $t$ may interfere with that stored knowledge and cause CF."}, {"n": 2, "claim": "To avoid this interference, when learning task $t$ the weight modification should happen inside task $t$'s own weight subspace $W_t$, ensuring weight-level (not merely gradient-level) orthogonality between tasks; because the weight space and gradient space, though dependent, are different spaces, gradient-level orthogonality alone does not entail weight-level orthogonality."}, {"n": 3, "claim": "Therefore there is no CF if and only if conditions 1) and 2) hold: task $t$ occupies an independent weight subspace $W_t$ orthogonal to all previous tasks' weight subspaces in $S_w$, and its modified weights (spanned by the orthogonal gradient) are imposed inside $W_t$; i.e., weight-level orthogonality between tasks guarantees complete CF elimination."}, {"n": 4, "claim": "Condition 2 is realized by the restricted orthogonal-gradient update Eq.(8): $W_t^l=W_t^l-\\lambda\\nabla_{W_t^l}\\mathcal{L}$ with $\\nabla_{W_t^l}\\mathcal{L}=\\nabla_{W_t^l}\\mathcal{L}-(\\nabla_{W_t^l}\\mathcal{L})B_{t-1}^l(B_{t-1}^l)^{\\mathsf{T}}$, which projects out the previous tasks' common bases $B_{t-1}^l$ and confines each layer's update to the OIWS $W_t^l$ orthogonal to previous subspaces."}, {"n": 5, "claim": "Consequently STIL, and any OW-based method following Theorem 2, maximally eliminates CF, and only weight-level orthogonality between tasks can guarantee CF elimination — including throughout the full procedure with backward knowledge transfer via $\\mathcal{L}_{sim}$ (Eq.(9)-(10))."}]}}, {"local_id": "T3", "statement": "Theorem 5. The theoretical bounds for FWT and BWT of tasks i and t are respectively as follows:\n\n$$\n\\epsilon_{t}(h)\\leq\\epsilon_{i}(h)+d(\\mathcal{D}_{i}^{\\prime},\\mathcal{D}_{t}^{\\prime})+\\operatorname*{min}\\{\\mathbb{E}_{\\mathcal{D}_{i}^{\\prime}}(\\mathbb{S}),\\mathbb{E}_{\\mathcal{D}_{t}^{\\prime}}(\\mathbb{S})\\};\\epsilon_{i}(h)\\leq\\epsilon_{t}(h)+d(\\mathcal{D}_{i}^{\\prime},\\mathcal{D}_{t}^{\\prime})+\\operatorname*{min}\\{\\mathbb{E}_{\\mathcal{D}_{i}^{\\prime}}(\\mathbb{S}),\\mathbb{E}_{\\mathcal{D}_{t}^{\\prime}}(\\mathbb{S})\\}\\tag{7}\n$$\n\nwhere $\\mathbb{S}= |l_{i}(\\mathbf{x})-l_{t}(\\mathbf{x})|$ represents the absolute difference of the test results on text data x of tasks i and t.", "proof": {"steps": [{"n": 1, "claim": "$$\n\\epsilon_{t}(h)\\leq\\epsilon_{i}(h)+d(\\mathcal{D}_{i}^{\\prime},\\mathcal{D}_{t}^{\\prime})+\\operatorname*{min}\\{\\mathbb{E}_{\\mathcal{D}_{i}^{\\prime}}(\\mathbb{S}),\\mathbb{E}_{\\mathcal{D}_{t}^{\\prime}}(\\mathbb{S})\\};\\epsilon_{i}(h)\\leq\\epsilon_{t}(h)+d(\\mathcal{D}_{i}^{\\prime},\\mathcal{D}_{t}^{\\prime})+\\operatorname*{min}\\{\\mathbb{E}_{\\mathcal{D}_{i}^{\\prime}}(\\mathbb{S}),\\mathbb{E}_{\\mathcal{D}_{t}^{\\prime}}(\\mathbb{S})\\}\\tag{7}\n$$"}, {"n": 2, "claim": "where $\\mathbb{S}= |l_{i}(\\mathbf{x})-l_{t}(\\mathbf{x})|$ represents the absolute difference of the test results on text data x of tasks i and t."}]}}, {"local_id": "T4", "statement": "Theorem 6. Low similarity or negatively correlated tasks will result in negative KT. Only high similarity tasks or positively correlated tasks can achieve positive KT.", "proof": {"steps": [{"n": 1, "claim": "Low similarity or negatively correlated tasks will result in negative KT."}, {"n": 2, "claim": "Only high similarity tasks or positively correlated tasks can achieve positive KT."}]}}], "target_local_id": "T2", "topic_domain": "Deep Learning Theory", "topic_subtopic": "Training Dynamics & Implicit Bias", "label": "flawed", "error_step": 3, "error_class6": ["missing_assumption", "logical_gap_or_missing_case", "others"], "error_explanation": "The step asserts an unconditional necessary-and-sufficient guarantee: satisfying conditions 1) and 2) (task $t$ occupying an orthogonal weight subspace $W_t$ and imposing its modified weights inside $W_t$) yields 'no CF'. This silently omits two required assumptions. (i) Nonlinearity: weight-level orthogonality between per-task subspaces only preserves output separation for a single linear layer; in a multi-layer DNN with nonlinear (e.g. ReLU) activations, the propagated hidden-layer values lose the orthogonality property through the activations up to the final logit layer that produces the prediction, so orthogonal weight subspaces do not guarantee no interference at the output. (ii) Backward KT: the method built on this theorem (STIL) performs backward knowledge transfer via $\\mathcal{L}_{sim}$ (Eq.(8)-(9)), which modifies the weights of previously learned similar tasks and thereby destroys the assumed weight orthogonality — so zero forgetting cannot be guaranteed, as the authors themselves concede. Properly scoped, the claim holds only for a linear/final-layer classifier AND in the pure CF-prevention regime with no modification of previous tasks' weights; under backward KT it must be weakened from 'guarantees zero CF' to 'maximally reduces (but does not eliminate) CF'."}
{"id": "lkIRFglmTp", "context": "Modeling the decision process as a POMDP allows us to compute the TD(λ) fixed point in closed form for any parameter λ and reactive policy π:\n\n$$\nQ_{\\pi}^{\\lambda}=W{\\Bigl(}I-\\gamma T{\\bigl(}\\lambda\\Pi^{S}+(1-\\lambda)\\Phi W^{\\Pi}{\\bigr)}{\\Bigr)}^{-1}R^{S A},\\tag{1}\n$$\n\nwhere $Q_{\\pi}^{\\lambda}$ is an $\\Omega \\times A$ matrix, and the right hand side is comprised of the following tensors: state weights W $(\\Omega\\times S)$ containing probabilities $\\mathrm{Pr}(s|\\omega)$ (see Appendix A for the formal definition); identity I (an $S \\times A \\times S \\times A$ tensor with $I_{s a s^{\\prime}a^{\\prime}}=\\delta_{s s^{\\prime}}\\delta_{a a^{\\prime}});$ latent transition dynamics $T\\left(S\\times A\\times S\\right);$ effective policy over latent states $\\Pi^{S}(S\\times S\\times A$ ; see Appendix A); observation function $\\Phi\\left(S\\times\\Omega\\right)$ containing probabilities $\\operatorname*{Pr}(\\omega|s)$ state-action weights $W^{\\Pi}\\left(\\Omega\\times S\\times A\\right)$ containing probabilities $\\mathrm{Pr}(s,a|\\omega);$ ; and latent rewards $\\mathring{R}^{S A}\\ (S\\times A)$ Notation: Each product between tensors contracts one adjacent index, e.g. for tensors A and $\\begin{array}{ r }{B,(A B)_{i j l m}=\\sum_{k}\\bar{A_{i j k}}B_{k l m}}\\end{array}$ , with the exception of the products involving $R^{S A}$ and T on the right hand side, which contract two indices. This derivation is given in Appendix A, and follows the Markov version by Sutton (1988). Setting λ = 0 or λ = 1 recovers either the TD or MC value function above, respectively.\n\nWe have shown that, under partially observability, there may be a discrepancy between $Q_{\\pi}^{\\lambda}$ value functions for two different λ parameters due to the implicit Markov assumption in TD(λ). We call this difference the λ-discrepancy, and we will use it as a measure of non-Markovianity.\n\nDefinition 2 The λ-discrepancy $\\Delta Q_{\\pi}^{\\lambda_{1},\\lambda_{2}}$ is the weighted $L^{2}$ norm of the difference between two action-value functions estimated by TD(λ) using different values of λ:\n\n$$\n\\Delta Q_{M,\\pi}^{\\lambda_{1},\\lambda_{2}}: =\\left\\|Q_{\\pi}^{\\lambda_{1}}-Q_{\\pi}^{\\lambda_{2}}\\right\\|_{2,\\pi}\n$$\n\nwhere M is the POMDP model, and the norm weighting is as described in Appendix E.2.\n\nBecause norms are positive definite, to analyze the case in Theorem 2 where the λ-discrepancy is 0 for all policies, it suffices to consider the expression inside the norm of Definition 2:\n\n$$\n{\\cal W}\\Big(A_{\\pi}^{\\lambda_{1}}-A_{\\pi}^{\\lambda_{2}}\\Big)R^{S A},\\quad\\mathrm{w h e r e~}A_{\\pi}^{\\lambda}=\\Big(I-\\gamma T\\left(\\lambda\\Pi^{S}+(1-\\lambda)\\Phi W^{\\Pi}\\right)\\Big)^{-1}.\\tag{2}\n$$\n\nThe only ways for this equation to be zero are when the difference term $A_{\\pi}^{\\lambda_{1}}-A_{\\pi}^{\\lambda_{2}}$ is zero (which we will show to be an MDP), or in the special case where this difference term is projected away by the outer terms W and/or $R^{\\dot{S}A}$ . We first consider when the two inner terms—which are the only terms that depend on λ—are equal, i.e.:\n\n$$\nA_{\\pi}^{\\lambda_{1}}=A_{\\pi}^{\\lambda_{2}}.\\tag{3}\n$$\n\nIn this case the system is a block MDP, a POMDP / MDP hybrid model featuring Markov observations, each of which can only be produced by one unique latent state (Du et al., 2019). Block MDPs are thus Markov over both states and observations.\n\nA TD(λ) Fixed Point\n\nHere we derive the fixed point of the $\\mathrm{TD}(\\lambda)$ action-value update rule in a POMDP. First, define the expected return given initial observation $\\omega_{0}$ and initial action $a_{0}$ as\n\n$$\n\\begin{aligned}\n\\mathbb{E}_\\Pi(G^n|\\omega_0,a_0) = {}& \\sum_{s_0}\\mathbb{P}(s_0|\\omega_0)\\sum_{s_1}\\mathbb{P}(s_1|s_0,a_0)\\sum_{r}\\mathbb{P}(r_0|s_0,a_0,s_1)r_0 \\\\\n&+\\gamma\\sum_{s_0}\\mathbb{P}(s_0|\\omega_0)\\sum_{s_1}\\mathbb{P}(s_1|s_0,a_0)\\sum_{\\omega_1}\\sum_{a_1}\\sum_{s_2}\\mathbb{P}(\\omega_1|s_1)\\mathbb{P}(a_1|\\omega_1)\\mathbb{P}(s_2|s_1,a_1)\\sum_{r_1}\\mathbb{P}(r_1|s_1,a_1,s_2)r_1 \\\\\n&+\\gamma^2\\sum_{s_0}\\mathbb{P}(s_0|\\omega_0)\\sum_{s_1}\\mathbb{P}(s_1|s_0,a_0)\\sum_{\\omega_1}\\sum_{a_1}\\sum_{s_2}\\mathbb{P}(\\omega_1|s_1)\\mathbb{P}(a_1|\\omega_1)\\mathbb{P}(s_2|s_1,a_1) \\\\\n&\\qquad *\\sum_{\\omega_2}\\sum_{a_2}\\sum_{s_3}\\mathbb{P}(\\omega_2|s_2)\\mathbb{P}(a_2|\\omega_2)\\mathbb{P}(s_3|s_2,a_2)\\sum_{r_2}\\mathbb{P}(r_2|s_2,a_2,s_3)r_2 \\\\\n&+\\dots\n\\end{aligned}\n$$\n\nNotation: We use non-standard notation here for tensor contractions to avoid being overly verbose. Unless otherwise noted, all contractions contract 1 index except those involving $R^{S A}$ and $\\mathcal{T}$ on the right hand side and those with $W^{A}$ on the left hand side, which contract 2 indices.", "theorems": [{"local_id": "T1", "statement": "Theorem 2 For any POMDP and for any fixed λ and λ', either $\\Delta Q_{\\pi}^{\\lambda,\\lambda^{\\prime}}\\neq 0$ for almost all policies π or $\\Delta Q_{\\pi}^{\\lambda,\\lambda^{\\prime}}=0 f o r$ all policies π.", "proof": {"steps": [{"n": 1, "claim": "\n$$\n\\Delta Q_{\\pi}: =\\left\\|Q_{\\pi}^{\\lambda}-Q_{\\pi}^{\\lambda^{\\prime}}\\right\\|_{2,\\pi}=\\left\\|\\left(W^{\\Pi}\\Big(A_{\\pi}^{\\lambda}-A_{\\pi}^{\\lambda^{\\prime}}\\Big)R^{S A}\\right)\\cdot w_{\\mathbb{-}}\\right\\|_{2}\n$$"}, {"n": 2, "claim": "where $A_{\\pi}^{\\lambda}=\\left(I-\\gamma T\\left(\\lambda\\Pi^{S}+(1-\\lambda)\\Phi W^{\\Pi}\\right)\\right)^{-1}$ and $w_{\\pi}$ is a weight vector of probabilities dependent on π defined as $w_{\\pi}(\\omega,a)=(1,\\pi(a|\\dot{\\omega}))$ . Let U be the largest open set in the space of stochastic $\\Omega \\times A$ matrices, considered as a subset of $\\mathbb{R}^{\\Omega(A-1)}$ . Now consider the lambda discrepancy as a function of the policy π. In other words, we define\n\n$$\n\\begin{array}{ c }{{\\Delta Q:U\\to\\mathbb{R}}}\\\\ {{\\pi\\mapsto Q_{\\pi}^{\\lambda}-Q_{\\pi}^{\\lambda^{\\prime}}}}\\end{array}\n$$"}, {"n": 3, "claim": "Let U be an open subset of $\\mathbb{R}^{n}$ . We say that a function $f:U\\to\\mathbb{R}$ is real analytic on U if for all $x \\in U$ $f$ can be written as a convergent power series in some neighborhood of x. For this proof, we willutilize the following facts: 1) the composition of analytic functions is analytic (Krantz and Parks, 2002), 2) the quotient of two analytic functions is analytic where the denominator is nonzero, 3) a real analytic function on a domain $\\dot{U}$ is either identically 0 or only zero on a set of measure 0 (Mityagin, 2020)."}, {"n": 4, "claim": "We will also use the fact that for A an invertible matrix, each entry $A_{i j}^{-1}$ is analytic in the entries of A where the entries of A yield a nonzero determinant. We can prove this by first writing $A^{-1}=\\operatorname*{det}(A)^{-1}\\operatorname{adj}(A)=\\operatorname*{det}(A)^{-1}C^{T}$ where adj A is the adjugate of $A$ and $C$ is the cofactor matrix of A. Each entry of the cofactor matrix is a cofactor that is polynomial in the entries of A, and is therefore analytic in them. Therefore, each entry of $A^{-1}$ is the quotient of two analytic functions and is therefore analytic except where det $A = 0$"}, {"n": 5, "claim": "Next, we will show that $\\Delta Q$ is an analytic function. Note that the variable terms in the equation are $W,\\dot{W}^{\\Pi},R^{S A},T,\\Pi^{S}$ , and Φ. $R^{S A},T$ , and Φ are constant with respect to π. $\\begin{array}{ r }{\\Pi_{i l j}^{S}=\\sum_{k}\\dot{\\delta}_{i l}\\Phi_{i k}\\pi_{k j},}\\end{array}$ so each entry of $\\Pi^{S}$ is analytic on U in the entries of π. Likewise, $\\begin{array}{ r }{P_{i j}=\\sum_{k,a}\\Phi_{i k}\\mathrm{{\\bar{\\pi}}}_{k a}T_{i a j}}\\end{array}$ is analytic on U . Therefore, the state-occupancy counts $c=\\mu_{0}+\\gamma P^{T}\\mu_{0}+\\gamma^{2}(P^{T})^{2}\\mu_{0}+\\cdot\\cdot\\cdot=(I-\\gamma P^{T})^{-1}\\mu_{0}$ are the composition of analytic functions and thus analytic on $\\begin{array}{ r }{U_{\\cdot}W_{i j}=\\frac{\\Phi_{j i}c_{j}}{\\sum_{k}\\Phi_{k i}c_{k}}}\\end{array}$ is analytic on U for the same reason, and the denominator of $\\begin{array}{ r }{W_{i j},\\sum_{k}\\Phi_{k i}c_{k}}\\end{array}$ , is nonzero for all observations able to be observed with nonzero probability. Finally, $\\Delta Q$ is then a composition of analytic functions on U and thus analytic itself."}]}}, {"local_id": "T2", "statement": "For any POMDP with latent state space $S$, observations $\\Omega$, actions $A$, discount $\\gamma$, latent transition dynamics $T\\,(S\\times A\\times S)$, and observation function $\\Phi\\,(S\\times\\Omega)$ with $\\Phi_{s,\\omega}=\\mathbb{P}(\\omega|s)$, and any reactive policy $\\pi$: define state weights $W\\,(\\Omega\\times S)$ with $W_{\\omega,s}=\\mathbb{P}(s|\\omega)$, the effective policy over latent states $\\Pi^S\\,(S\\times S\\times A)$ (the tensor form of $\\Phi\\cdot\\pi$), and latent rewards $R^{SA}=\\sum_{s'}T_{s,a,s'}R^{SAS}_{s,a,s'}$. The TD($\\lambda$) fixed point $Q^\\lambda$ satisfies $Q^\\lambda = W\\big((I-\\lambda\\gamma T\\Pi^S)^{-1}(R^{SA}+(1-\\lambda)\\gamma T\\Phi\\Pi\\, Q^\\lambda)\\big)$, equivalently (Eq. 1) $Q^\\lambda_\\pi = W\\big(I-\\gamma T(\\lambda\\Pi^S+(1-\\lambda)\\Phi W^\\Pi)\\big)^{-1} R^{SA}$.", "proof": {"steps": [{"n": 1, "claim": "$\\mathbb{E}_\\Pi(G^n|\\omega_0,a_0) = \\sum_{s_0} W_{\\omega_0,s_0}\\sum_{s_1} T_{s_0,a_0,s_1} R_{s_0,a_0} + \\gamma\\sum_{s_0} W_{\\omega_0,s_0}\\sum_{s_1} T_{s_0,a_0,s_1}\\sum_{\\omega_1}\\sum_{a_1}\\sum_{s_2}\\Phi_{s_1,\\omega_1}\\pi_{\\omega_1,a_1} T_{s_1,a_1,s_2} R_{s_1,a_1} + \\dots$"}, {"n": 2, "claim": "$n$-step bootstrap replaces the $\\gamma^n$ term by a $Q$-value; for $n=2$: $Q(\\omega_0,a_0) \\gets \\sum_{s_0} W_{\\omega_0,s_0}\\sum_{s_1} T_{s_0,a_0,s_1} R_{s_0,a_0} + \\gamma(\\dots) + \\gamma^2\\sum_{s_0}W_{\\omega_0,s_0}\\sum_{s_1}T_{s_0,a_0,s_1}\\sum_{\\omega_1}\\sum_{a_1}\\sum_{s_2}\\Phi_{s_1,\\omega_1}\\pi_{\\omega_1,a_1}T_{s_1,a_1,s_2}*\\sum_{\\omega_2}\\Phi_{s_2,\\omega_2}\\sum_{a_2}\\Pi_{\\omega_2,a_2}Q(\\omega_2,a_2)$."}, {"n": 3, "claim": "In general, $Q_n := W\\big(\\sum_{k=0}^{n-1}(\\gamma T\\Pi^S)^k R^{SA} + \\gamma(\\gamma T\\Pi^S)^{n-1} T\\Phi\\Pi\\, Q_n\\big)$, where $\\Pi^S=\\Phi\\cdot\\pi$ and $R^{SA}=\\sum T_{s,a,s'}R^{SAS}$."}, {"n": 4, "claim": "$Q^\\lambda = (1-\\lambda)\\sum_{n=1}^\\infty \\lambda^{n-1} Q_n = (1-\\lambda)\\sum_{n=1}^\\infty \\lambda^{n-1} W\\big(\\sum_{k=0}^{n-1}(\\gamma T\\Pi^S)^k R^{SA} + \\gamma(\\gamma T\\Pi^S)^{n-1} T\\Phi\\Pi\\, Q^\\lambda\\big)$."}, {"n": 5, "claim": "Value part: $(1-\\lambda)W\\big(\\sum_{n=1}^\\infty (\\lambda\\gamma T\\Pi^S)^{n-1}\\big)\\gamma T\\Phi\\Pi\\, Q^\\lambda = (1-\\lambda)W(I-\\lambda\\gamma T\\Pi^S)^{-1}\\gamma T\\Phi\\Pi\\, Q^\\lambda$."}, {"n": 6, "claim": "Reward part: $(1-\\lambda)W\\sum_{k=0}^\\infty \\frac{1}{1-\\lambda}(\\lambda\\gamma T\\Pi^S)^k R^{SA} = W(I-\\lambda\\gamma T\\Pi^S)^{-1} R^{SA}$."}, {"n": 7, "claim": "$Q^\\lambda = W\\big((I-\\lambda\\gamma T\\Pi^S)^{-1}(R^{SA}+(1-\\lambda)\\gamma T\\Phi\\Pi\\, Q^\\lambda)\\big)$."}, {"n": 8, "claim": "$Q^\\lambda_\\pi = W\\big(I-\\gamma T(\\lambda\\Pi^S+(1-\\lambda)\\Phi W^\\Pi)\\big)^{-1} R^{SA}$ (Eq. 1); recovers TD at $\\lambda=0$, MC at $\\lambda=1$."}]}}, {"local_id": "T3", "statement": "Lemma 1 For any POMDP, $\\Delta Q_{\\pi}^{0,1}=0$ if for all initial observations $o^{0}$ and actions $a^{0},$ and all horizons T,\n\n$$\n\\mathbb{E}(r^{T}|\\omega^{0},a^{0})=\\sum_{\\omega^{1},\\ldots,\\omega^{T}}\\mathbb{E}(r^{T}|\\omega^{T})\\mathbb{P}(\\omega^{1}|\\omega^{0},a^{0})\\prod_{t=1}^{T-1}\\mathbb{P}(\\omega^{t+1}|\\omega^{t}).\n$$\n\nFurthermore, the converse holds for almost all choices of the discount factor γ.", "proof": {"steps": [{"n": 1, "claim": "Let $A^{\\lambda}=\\left(I-\\gamma T\\left(\\lambda\\Pi^{S}+(1-\\lambda)\\Phi W^{\\Pi}\\right)\\right)^{-1}. Q^{\\lambda}=Q^{\\lambda^{\\prime}}\\operatorname{iff}W(A^{\\lambda}-A^{\\lambda^{\\prime}})R^{S A}=0$ Expanding $A^{\\lambda}$ and $A^{\\lambda^{\\prime}}$ into power series, we have"}, {"n": 2, "claim": "$$\n\\begin{array}{ r l }&{0=\\left(W R^{S A}+\\gamma W T\\left(\\lambda\\Pi^{S}+(1-\\lambda)\\Phi W^{\\Pi}\\right)R^{S A}+\\right.}\\\\ &{\\left.\\gamma^{2}W T\\left(\\lambda\\Pi^{S}+(1-\\lambda)\\Phi W^{\\Pi}\\right)T\\left(\\lambda\\Pi^{S}+(1-\\lambda)\\Phi W^{\\Pi}\\right)R^{S A}+\\ldots\\right)}\\\\ &{-\\left(W R^{S A}+\\gamma W T\\left(\\lambda^{\\prime}\\Pi^{S}+(1-\\lambda^{\\prime})\\Phi W^{\\Pi}\\right)R^{S A}+\\right.}\\\\ &{\\left.\\gamma^{2}W T\\left(\\lambda^{\\prime}\\Pi^{S}+(1-\\lambda^{\\prime})\\Phi W^{\\Pi}\\right)T\\left(\\lambda^{\\prime}\\Pi^{S}+(1-\\lambda^{\\prime})\\Phi W^{\\Pi}\\right)R^{S A}+\\ldots\\right)}\\end{array}\n$$"}, {"n": 3, "claim": "We observe that it is potentially possible to get cases of zero λ-discrepancy from pairs of terms in this equation cancelling out. Concretely, this means\n\n$$\n\\begin{array}{ r }{W T\\left(\\lambda\\Pi^{S}+(1-\\lambda)\\Phi W^{\\Pi}\\right)R^{S A}W T\\Pi^{S}R^{S A}-W T\\Phi W^{\\Pi}R^{S A}=0}\\\\ {W T\\Pi^{S}T\\Pi^{S}R^{S A}-W T\\Phi W^{\\Pi}T\\Phi W^{\\Pi}R^{S A}=0}\\end{array}\n$$"}, {"n": 4, "claim": "This can occur in one particularly nice way if $\\lambda = 0$ and $\\lambda^{\\prime}=1$ . In this case, there is no λ-discrepancy precisely when\n\n$$\n\\begin{array}{ r l }&{0=\\left(W R^{S A}+\\gamma W T\\Pi^{S}R^{S A}+\\gamma^{2}W T\\Pi^{S}T\\Pi^{S}R^{S A}+\\ldots\\right)}\\\\ &{\\ -\\left(W R^{S A}+\\gamma W T\\Phi W^{\\Pi}R^{S A}+\\gamma^{2}W T\\Phi W^{\\Pi}T\\Phi W^{\\Pi}R^{S A}+\\ldots\\right)}\\\\ &{\\ =\\gamma\\left(W T\\Pi^{S}R^{S A}-W T\\Phi W^{\\Pi}R^{S A}\\right)}\\\\ &{+\\gamma^{2}\\left(W T\\Pi^{S}T\\Pi^{S}R^{S A}-W T\\Phi W^{\\Pi}T\\Phi W^{\\Pi}R^{S A}\\right)}\\\\ &{+\\ldots .}\\end{array}\\tag{5}\n$$"}, {"n": 5, "claim": "This occurs when each pair of n-step return terms cancel, or\n\n$$\n\\begin{array}{ c }{{W T\\Pi^{S}R^{S A}-W T\\Phi W^{\\Pi}R^{S A}=0}}\\\\ {{W T\\Pi^{S}T\\Pi^{S}R^{S A}-W T\\Phi W^{\\Pi}T\\Phi W^{\\Pi}R^{S A}=0}}\\\\ {{\\ldots\\qquad}}\\end{array}\n$$"}, {"n": 6, "claim": "Interpreting this in terms of probabilities, this says that for all initial observations $\\omega^{0}$ , initial actions $a^{0}.$ , and horizons $T,$\n\n$$\n\\mathbb{E}(r^{T}|\\omega^{0},a^{0})=\\sum_{\\omega^{1},\\dots,\\omega^{T}}\\mathbb{E}(r^{T}|\\omega^{T})\\mathbb{P}(\\omega^{1}|\\omega^{0},a^{0})\\prod_{t=1}^{T-1}\\mathbb{P}(\\omega^{t+1}|\\omega^{t})\\tag{4}\n$$"}, {"n": 7, "claim": "In particular, this applies when $R^{S A}$ is a constant matrix, as in this case, the expected rewards are all equal. Therefore, when $R^{S A}$ is constant, $Q^{1}=Q^{0}$"}, {"n": 8, "claim": "Next, we prove that the converse holds for almost all $\\gamma_{\\cdot}$ Consider what happens if $\\gamma$ is not fixed with the POMDP, but is instead allowed to vary. In that case, we have that the function $f$ from γ to the λ-discrepancy is an analytic function of $\\gamma \\colon$\n\n$$\n\\begin{array}{ r l }&{f:\\gamma\\mapsto\\gamma\\left(W T\\Pi^{S}R^{S A}-W T\\Phi W^{\\Pi}R^{S A}\\right)}\\\\ &{\\quad\\quad\\quad+\\gamma^{2}\\left(W T\\Pi^{S}T\\Pi^{S}R^{S A}-W T\\Phi W^{\\Pi}T\\Phi W^{\\Pi}R^{S A}\\right)}\\\\ &{\\quad\\quad\\quad+\\ldots}\\end{array}\\tag{6}\n$$"}]}}], "target_local_id": "T2", "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "label": "flawed", "error_step": 3, "error_class6": ["others", "circular_or_unproven", "logical_gap_or_missing_case"], "error_explanation": "The $n$-step bootstrapped quantity $Q_n$ is an UPDATE operator that must take, as an explicit argument, the value matrix $Q$ off which it bootstraps. Writing $Q_n := W\\big(\\sum_{k=0}^{n-1}(\\gamma T\\Pi^S)^k R^{SA} + \\gamma(\\gamma T\\Pi^S)^{n-1} T\\Phi\\Pi\\, Q_n\\big)$ with no argument (and with its own name $Q_n$ appearing inside the bootstrap term) conflates the update rule with its fixed point: it makes the equation self-referential rather than a definition of an operator. This renders the ensuing step ill-defined, because when $Q^\\lambda = (1-\\lambda)\\sum_{n=1}^\\infty \\lambda^{n-1} Q_n$ is expanded, the inner $Q_n$ is 'directly replaced by $Q^\\lambda$ without any argument' — but there is no argument slot to substitute into. Correctly, $Q_n$ must be $Q_n(Q) := W\\big(\\sum_{k=0}^{n-1}(\\gamma T\\Pi^S)^k R^{SA} + \\gamma(\\gamma T\\Pi^S)^{n-1} T\\Phi\\Pi\\, Q\\big)$, the TD($\\lambda$) update is $Q \\gets (1-\\lambda)\\sum_{n=1}^\\infty \\lambda^{n-1} Q_n(Q)$, and $Q^\\lambda$ is the fixed point of that update. This is grounded in the reviewer's objection and the authors' explicit concession that the definition was wrong (though the final result is unchanged)."}
{"id": "lydPkW4lfz", "context": "C.1 Setup\n\nWe consider the following optimization problem:\n\n$$\n\\operatorname*{min}_{w\\in\\mathbb{R}^{d}}\\left\\{F(w): =\\frac{1}{M}\\sum_{m=1}^{M}F_{m}(w): =\\frac{1}{M}\\sum_{m=1}^{M}\\mathbb{E}_{\\xi\\sim\\mathcal{D}_{m}}f(w;\\xi)\\right\\}\\tag{1}\n$$\n\nAssumption 2.\n\n$f(\\cdot,\\xi)$ is convex and H-smooth for every ξ.\n\nThere exists $F_{*}\\in\\mathbb{R}$ such that $F({\\boldsymbol{w}})\\geq F_{*}$ for every w $\\in\\mathbb{R}^{d}.$\n\nFor all $\\pmb{w}\\in\\mathbb{R}^{d}\\colon\\mathbb{E}_{\\xi\\sim\\mathcal{D}_{i}}\\left[\\nabla f(\\pmb{w},\\xi)\\right]=\\nabla F_{i}(\\pmb{w}).$\n\nNote that we do not assume that f achieves its infimum at some point in the domain.\n\nAssumption 3 (Stochastic Gradient Variance).\n\n(a) (Global) There exists $\\sigma \\geq 0$ such that for all w $\\in\\mathbb{R}^{d}$ and $m \\in [ M ] .$\n\n$$\n\\begin{array}{ r }{\\mathbb{E}_{\\xi\\sim\\mathcal{D}_{m}}\\left[\\|\\nabla f(\\pmb{w},\\xi)-\\nabla F_{m}(\\pmb{w})\\|^{2}\\right]\\leq\\sigma^{2}.}\\end{array}\\tag{2}\n$$\n\n(b) (Local) For every $\\pmb{u}\\in\\mathbb{R}^{d}$ , there exists $\\boldsymbol{\\sigma}(\\boldsymbol{u})\\geq 0$ such that for all $m \\in [ M ]$\n\n$$\n\\mathbb{E}_{\\xi\\sim\\mathcal{D}_{m}}\\left[\\|\\nabla f(\\pmb{u},\\xi)-\\nabla F_{m}(\\pmb{u})\\|^{2}\\right]\\leq\\sigma^{2}(\\pmb{u}).\\tag{3}\n$$\n\nAssumption 4 (Objective Heterogeneity).\n\n(a) (Global): There exists $\\zeta \\geq 0$ such that for all $\\pmb{w}\\in\\mathbb{R}^{d}$ and $m \\in [ M ] .$\n\n$$\n\\begin{array}{ r }{\\|\\nabla F_{m}({\\pmb w})-\\nabla F({\\pmb w})\\|\\le\\zeta .}\\end{array}\\tag{4}\n$$\n\n(b) (Local): For every $\\pmb{u}\\in\\mathbb{R}^{d},$ there exists $\\zeta({\\pmb u})\\geq 0$ such that for all $m \\in [ M ] .$\n\n$$\n\\|\\nabla F_{m}(\\boldsymbol{u})-\\nabla F(\\boldsymbol{u})\\|\\leq\\zeta(\\boldsymbol{u}).\\tag{5}\n$$\n\nAlgorithm 4 Local SGD\n\nInput: Initialization $\\bar{\\pmb w}_{0}\\in\\mathbb{R}^{d}$ ,rounds $R\\in\\mathbb{N},$ local steps $K\\in\\mathbb{N}.$ learning rate $\\eta > 0 .$ averaging   \nweights $\\{\\alpha_{r,k}\\}_{r,k}$   \n1: for $r=0,1,\\ldots,R-1$ do   \n2: for $m \\in [ M ]$ do   \n3: ${\\pmb w}_{r,0}^{m}\\leftarrow\\bar{{\\pmb w}}_{r}$   \n4: for $\\mathbf{\\bar{\\boldsymbol{k}}}=0,\\dots,K-1$ do   \n5: Sample $\\xi_{r,k}^{m}\\sim\\mathcal{D}_{m}$   \n6: $\\pmb{w}_{r,k+1}^{m}\\leftarrow\\pmb{w}_{r,k}^{m}-\\eta\\nabla f(\\pmb{w}_{r,k}^{m};\\xi_{r,k}^{m})$   \n7: end for   \n8: end for   \n9: $\\begin{array}{ r }{\\bar{\\pmb{w}}_{r+1}\\leftarrow\\frac{1}{M}\\sum_{m=1}^{M}\\pmb{w}_{r,K}^{m}}\\end{array}$   \n10: end for   \n11: return $\\begin{array}{ r }{\\hat{\\pmb{w}}=\\sum_{r=0}^{R-1}\\sum_{k=0}^{K-1}\\alpha_{r,k}\\left(\\frac{1}{M}\\sum_{m=1}^{M}\\pmb{w}_{r,k}^{m}\\right)}\\end{array}$\n\nLocal SGD for the above optimization problem is defined in Algorithm 4.\n\nC.2 Statement of General Convergence Results\n\nTheorems 1 and 2 below are proven by modifying two existing analyses of Local SGD Woodworth et al. (2020b); Koloskova et al. (2020) which use global and local assumptions (respectively) on stochastic gradient variance and objective heterogeneity, by removing the assumption that the global objective F has a minimizer ${\\pmb w}_{\\ast}.$ The resulting rates match the corresponding rates from the original analyses, up to an additional additive term proportional to $F({\\pmb u})-\\overset{*}{F_{*}}$ . The convex combination weights $\\{\\alpha_{r,k}\\}_{r,k}$ are specified separately in each proof.\n\nTheorem 1. Let $B=\\|\\pmb{\\bar{w}}_{0}-\\pmb{u}\\|$ .Under Assumptions 2, 3(a), and 4(a), for any $\\pmb{u}\\in\\mathbb{R}^{d}$ , there exists a choice of η such that Local SGD satisfies\n\n$$\n\\mathbb{E}\\left[F(\\pmb{\\hat{w}})-F_{*}\\right]\\leq\\mathcal{O}\\left(\\frac{H B^{2}}{K R}+\\frac{\\sigma B}{\\sqrt{M K R}}+\\frac{(H\\zeta^{2}B^{4})^{1/3}}{R^{2/3}}+\\frac{(H\\sigma^{2}B^{4})^{1/3}}{K^{1/3}R^{2/3}}+\\left(F(u)-F_{*}\\right)\\right).\\tag{6}\n$$\n\nTheorem 2. Let $B=\\|\\pmb{\\bar{w}}_{0}-\\pmb{u}\\|.$ Under Assumptions 2, 3(b), and 4(b), for any $\\pmb{u}\\in\\mathbb{R}^{d}$ , there exists a choice of η such that Local SGD satisfies\n\n$$\n\\begin{array}{ r l }&{\\mathbb{E}[F(\\hat{w})-F_{*}]\\overset{\\le}\\delta}\\\\ &{\\quad{\\mathcal{O}}\\left(\\frac{H B^{2}}{R}+\\frac{\\sigma(u)B}{\\sqrt{M K R}}+\\frac{(H\\sigma^{2}(u)B^{4})^{1/3}}{K^{1/3}R^{2/3}}+\\frac{(H\\zeta^{2}(u)B^{4})^{1/3}}{R^{2/3}}+R(F(u)-F_{*})\\right).}\\end{array}\\tag{7}\n$$\n\nProofs are given in Appendices C.3 and C.4, respectively.", "theorems": [{"local_id": "T1", "statement": "Lemma 1. For every $\\pmb{u}\\in\\mathbb{R}^{d},t\\geq 0$ and $m \\in [ M ] .$\n\n$$\n\\begin{array}{ r l }&{\\mathbb{E}_{\\xi_{t}^{m}}\\left[\\|\\nabla f(w_{t}^{m};\\xi_{t}^{m})-\\nabla F_{m}(w_{t}^{m})\\|^{2}\\right]}\\\\ &{\\quad\\le 3\\sigma^{2}(u)+3 H^{2}\\left\\|w_{t}^{m}-\\bar{w}_{t}\\right\\|^{2}+6 H(F_{m}(\\bar{w}_{t})-F_{m}(u))-6 H\\langle\\bar{w}_{t}-u,\\nabla F_{m}(u)\\rangle .}\\end{array}\\tag{8}\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof. We can decompose:"}, {"n": 2, "claim": "$$\n\\nabla f({\\boldsymbol w}_{t}^{m};{\\boldsymbol\\xi}_{t}^{m})-\\nabla F_{m}({\\boldsymbol w}_{t}^{m})=(\\nabla f({\\boldsymbol w}_{t}^{m};{\\boldsymbol\\xi}_{t}^{m})-\\nabla f(\\bar{{\\boldsymbol w}}_{t};{\\boldsymbol\\xi}_{t}^{m})-\\nabla F_{m}({\\boldsymbol w}_{t}^{m})+\\nabla F_{m}(\\bar{{\\boldsymbol w}}_{t}))\\tag{9}\n$$"}, {"n": 3, "claim": "$$\n+\\left(\\nabla f(\\bar{w}_{t};\\xi_{t}^{m})-\\nabla f(\\pmb{u};\\xi_{t}^{m})-\\nabla F_{m}(\\bar{w}_{t})+\\nabla F_{m}(\\pmb{u})\\right)\\tag{10}\n$$"}, {"n": 4, "claim": "$$\n+\\left(\\nabla f(\\pmb{u};\\xi_{t}^{m})-\\nabla F_{m}(\\pmb{u})\\right),\\tag{11}\n$$"}, {"n": 5, "claim": "So\n\n$$\n\\mathbb{E}_{\\xi_{t}^{m}}\\left[\\|\\nabla f(\\pmb{w}_{t}^{m};\\xi_{t}^{m})-\\nabla F_{m}(\\pmb{w}_{t}^{m})\\|^{2}\\right]\\tag{12}\n$$"}, {"n": 6, "claim": "$$\n\\leq 3\\mathbb{E}_{\\xi_{t}^{m}}\\left[\\|\\nabla f({\\pmb w}_{t}^{m};\\xi_{t}^{m})-\\nabla f(\\bar{{\\pmb w}}_{t};\\xi_{t}^{m})-\\nabla F_{m}({\\pmb w}_{t}^{m})+\\nabla F_{m}(\\bar{{\\pmb w}}_{t})\\|^{2}\\right]\\tag{13}\n$$"}, {"n": 7, "claim": "$$\n+3\\mathbb{E}_{\\xi_{t}^{m}}\\left[\\left\\|\\nabla f(\\bar{w}_{t};\\xi_{t}^{m})-\\nabla f(u;\\xi_{t}^{m})-\\nabla F_{m}(\\bar{w}_{t})+\\nabla F_{m}(u)\\right\\|^{2}\\right]+3\\mathbb{E}_{\\xi_{t}^{m}}\\left[\\left\\|\\nabla f(u;\\xi_{t}^{m})-\\nabla F_{m}(u)\\right\\|^{2}\\right]\\tag{14}\n$$"}, {"n": 8, "claim": "$$\n\\begin{array}{ r l }&{\\overset{(i)}{\\leq}3\\mathbb{E}_{\\xi_{t}^{m}}\\left[\\|\\nabla f(w_{t}^{m};\\xi_{t}^{m})-\\nabla f(\\bar{w}_{t};\\xi_{t}^{m})\\|^{2}\\right]+3\\mathbb{E}_{\\xi_{t}^{m}}\\left[\\|\\nabla f(\\bar{w}_{t};\\xi_{t}^{m})-\\nabla f(u;\\xi_{t}^{m})\\|^{2}\\right]}\\end{array}\\tag{15}\n$$"}, {"n": 9, "claim": "$$\n+3\\mathbb{E}_{\\xi_{t}^{m}}\\left[\\left\\|\\nabla f(\\pmb{u};\\xi_{t}^{m})-\\nabla F_{m}(\\pmb{u})\\right\\|^{2}\\right]\\tag{16}\n$$"}, {"n": 10, "claim": "$$\n\\overset{(i i)}{\\leq}3 H^{2}\\mathbb{E}_{\\xi_{t}^{m}}\\left[\\left\\|w_{t}^{m}-\\bar{w}_{t}\\right\\|^{2}\\right]+6 H\\mathbb{E}_{\\xi_{t}^{m}}\\left[f(\\bar{w}_{t};\\xi_{t}^{m})-f(u;\\xi_{t}^{m})-\\langle\\bar{w}_{t}-u,\\nabla f(u;\\xi_{t}^{m})\\right]+3\\sigma^{2}(u)\\tag{17}\n$$"}, {"n": 11, "claim": "$$\n=3 H^{2}\\left.\\boldsymbol{w}_{t}^{m}-\\bar{\\boldsymbol{w}}_{t}\\right.^{2}+6 H(F_{m}(\\bar{\\boldsymbol{w}}_{t})-F_{m}(\\boldsymbol{u}))-6 H\\langle\\bar{\\boldsymbol{w}}_{t}-\\boldsymbol{u},\\nabla F_{m}(\\boldsymbol{u})\\rangle+3\\sigma^{2}(\\boldsymbol{u}),\\tag{18}\n$$\n\nwhere (i) uses the fact that E $\\left[\\left\\|X-\\mathbb{E}[X]\\right\\|^{2}\\right]\\leq\\mathbb{E}\\left[\\left\\|X\\right\\|^{2}\\right]$ , and (ii) uses the fact that $f(\\cdot,\\xi_{t}^{m})$ is smooth and convex together with Lemma 2."}]}}, {"local_id": "T2", "statement": "Lemma 3. Suppose $w\\in\\mathbb R^d$ such that $y_{mi}\\langle x_{mi},w\\rangle\\ge 0$ for all $m\\in[M],i\\in[n]$. Then $\\lVert\\nabla F(w)\\rVert\\ge\\frac{\\gamma}{2}F(w)$, where $\\gamma$ denotes the maximum margin of the combined dataset.", "proof": {"steps": [{"n": 1, "claim": "$w_*$ is the max-margin classifier with $\\|w_*\\|=1$ and $y_{mi}\\langle x_{mi},w_*\\rangle\\ge\\gamma$; moreover $\\ell'(z)=-1/(1+e^{z})<0$ so $-\\ell'(z)=|\\ell'(z)|$."}, {"n": 2, "claim": "$\\|\\nabla F(w)\\|=\\langle\\nabla F(w),\\nabla F(w)/\\|\\nabla F(w)\\|\\rangle\\ge\\langle\\nabla F(w),z/\\|z\\|\\rangle$ for any $z\\ne0$."}, {"n": 3, "claim": "Choosing $z=-w_*$ (a unit vector) yields $\\|\\nabla F(w)\\|\\ge\\langle\\nabla F(w),-w_*\\rangle=-\\langle\\nabla F(w),w_*\\rangle$."}, {"n": 4, "claim": "$-\\langle\\nabla F(w),w_*\\rangle=\\frac{1}{Mn}\\sum_{m=1}^M\\sum_{i=1}^n|\\ell'(y_{mi}\\langle w,x_{mi}\\rangle)|\\,y_{mi}\\langle x_{mi},w_*\\rangle$."}, {"n": 5, "claim": "$\\overset{(i)}{\\ge}\\frac{\\gamma}{Mn}\\sum_{m=1}^M\\sum_{i=1}^n|\\ell'(y_{mi}\\langle w,x_{mi}\\rangle)|$."}, {"n": 6, "claim": "$\\overset{(ii)}{\\ge}\\frac{\\gamma}{2Mn}\\sum_{m=1}^M\\sum_{i=1}^n\\ell(y_{mi}\\langle w,x_{mi}\\rangle)$."}, {"n": 7, "claim": "$=\\frac{\\gamma}{2}F(w)$, where (i) uses the definition of $w_*$ and (ii) uses $\\ell(z)\\le 2|\\ell'(z)|$ (valid for all $z\\ge 0$) together with the condition $y_{mi}\\langle w,x_{mi}\\rangle\\ge 0$."}]}}, {"local_id": "T3", "statement": "Consider the distributed optimization problem $\\min_{w\\in\\mathbb R^d}\\{F(w):=\\frac1M\\sum_{m=1}^M F_m(w):=\\frac1M\\sum_{m=1}^M \\mathbb E_{\\xi\\sim\\mathcal D_m} f(w;\\xi)\\}$ solved by Local SGD (Algorithm 4), with averaged iterate $\\bar w_t=\\frac1M\\sum_{m=1}^M w_t^m$ at absolute timestep $t=Kr+k$. Suppose Assumption 2 holds: $f(\\cdot,\\xi)$ is convex and $H$-smooth for every $\\xi$; there exists $F_*\\in\\mathbb R$ with $F(w)\\ge F_*$ for all $w$; and $\\mathbb E_{\\xi\\sim\\mathcal D_m}[\\nabla f(w,\\xi)]=\\nabla F_m(w)$. Suppose Assumption 3(a) holds: there exists $\\sigma\\ge 0$ such that $\\mathbb E_{\\xi\\sim\\mathcal D_m}[\\|\\nabla f(w,\\xi)-\\nabla F_m(w)\\|^2]\\le\\sigma^2$ for all $w$ and $m\\in[M]$. Then (Lemma 4): if $\\eta\\le 1/(4H)$, for any $\\pmb u\\in\\mathbb R^d$, $\\mathbb E[F(\\bar w_t)-F(u)] \\le \\frac{1}{\\eta}\\big(\\mathbb E[\\|\\bar w_t-u\\|^2]-\\mathbb E[\\|\\bar w_{t+1}-u\\|^2]\\big) + \\frac{\\eta\\sigma^2}{M} + \\frac{3H}{2M}\\sum_{m=1}^M \\mathbb E[\\|w_t^m-\\bar w_t\\|^2] + (F(u)-F_*).$", "proof": {"steps": [{"n": 1, "claim": "Starting from $\\|\\bar w_{t+1}-u\\|^2$ (Eq. 19) and taking conditional expectation, the quantity $A=\\|\\bar w_t-u-\\tfrac{\\eta}{M}\\sum_m\\nabla F_m(w_t^m)\\|^2$ decomposes (Eq. 20) as $A = \\|\\bar w_t-u\\|^2 + \\tfrac{\\eta^2}{M^2}B_1 + \\tfrac{2\\eta}{M}B_2$."}, {"n": 2, "claim": "$B_1 \\le 4HM^2(F(\\bar w_t)-F_*) + 2H^2 M\\sum_{m=1}^M \\|w_t^m-\\bar w_t\\|^2$ (Eq. 21)."}, {"n": 3, "claim": "$B_2 \\le -M(F(\\bar w_t)-F(u)) + \\tfrac{H}{2}\\sum_{m=1}^M \\|\\bar w_t-w_t^m\\|^2$ (Eq. 22)."}, {"n": 4, "claim": "Plugging the bounds on $B_1$ and $B_2$ back into Eq. 23 (Eqs. 24-25) gives $A+\\tfrac{\\eta^2\\sigma^2}{M} \\le \\|\\bar w_t-u\\|^2 + 4\\eta^2 H(F(\\bar w_t)-F_*) - 2\\eta(F(\\bar w_t)-F(u)) + \\left(\\tfrac{\\eta H}{M}+\\tfrac{2\\eta^2 H^2}{M}\\right)\\sum_{m=1}^M \\|\\bar w_t-w_t^m\\|^2 + \\tfrac{\\eta^2\\sigma^2}{M}.$"}, {"n": 5, "claim": "Using $\\eta\\le 1/(4H)$, combine the two $(F(\\bar w_t)-F(u))$-containing terms: bound $-(2\\eta-4\\eta^2 H)(F(\\bar w_t)-F(u))$ by $-\\eta(F(\\bar w_t)-F(u))$, i.e. decompose $4\\eta^2 H(F(\\bar w_t)-F_*)=4\\eta^2 H(F(\\bar w_t)-F(u))+4\\eta^2 H(F(u)-F_*)$ and apply $4\\eta H\\le 1$, yielding (Eq. 26) $A+\\tfrac{\\eta^2\\sigma^2}{M} \\le \\|\\bar w_t-u\\|^2 + \\eta(F(\\bar w_t)-F_*) + \\tfrac{3\\eta H}{2M}\\sum_{m=1}^M \\|w_t^m-\\bar w_t\\|^2$."}, {"n": 6, "claim": "$\\mathbb E[F(\\bar w_t)-F(u)] \\le \\frac{1}{\\eta}\\big(\\mathbb E[\\|\\bar w_t-u\\|^2]-\\mathbb E[\\|\\bar w_{t+1}-u\\|^2]\\big) + \\frac{\\eta\\sigma^2}{M} + \\frac{3H}{2M}\\sum_{m=1}^M \\mathbb E[\\|w_t^m-\\bar w_t\\|^2] + (F(u)-F_*)$ (Eq. 27 / 28)."}]}}, {"local_id": "T4", "statement": "Lemma 5. For any $a > 0$ and $n>0,$ if\n\n$$\nx\\geq\\operatorname*{max}\\left\\{2,\\frac{(2 n)^{n}a}{n}\\log^{n}(n^{n}a)\\right\\},\\tag{29}\n$$\n\nthen\n\n$$\n{\\frac{x}{\\log^{n}x}}\\geq a .\\tag{30}\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof. The desired inequality is equivalent to\n\n$$\n{\\frac{x^{1/n}}{\\log x}}\\geq a^{1/n}\\tag{31}\n$$"}, {"n": 2, "claim": "$$\n{\\frac{x^{1/n}}{n\\log x^{1/n}}}\\geq a^{1/n}\\tag{32}\n$$"}, {"n": 3, "claim": "$$\n{\\frac{x^{1/n}}{\\log x^{1/n}}}\\geq n a^{1/n}.\\tag{33}\n$$"}, {"n": 4, "claim": "So denoting y $y=x^{1/n}$ and $b=n a^{1/n}$ , we want to show that\n\n$$\n{\\frac{y}{\\log y}}\\geq b .\\tag{34}\n$$"}, {"n": 5, "claim": "From the definition of y and b,\n\n$$\ny=x^{1/n}=\\operatorname*{max}\\left\\{2^{1/n},2 n a^{1/n}\\log(n a^{1/n})\\right\\}=\\operatorname*{max}\\left\\{2^{1/n},2 b\\log(b)\\right\\}.\\tag{35}\n$$"}, {"n": 6, "claim": "We consider two cases depending on the magnitude of b. If $\\textit{ b }\\leq \\textit{ e }$ , then we are done, since $z/\\log(z)\\ \\geq\\ e$ for every $z \\ > \\ 1$ , so that $y/\\log(y)\\geq b.$ Otherwise, 2b $\\log(b)\\geq 2 e\\geq 2^{1/n}$ , so that the second term of the max in the definition is larger, i.e. $y=2b\\log(b)$ . Therefore"}, {"n": 7, "claim": "$$\n\\frac{y}{\\log y}=\\frac{2 b\\log(b)}{\\log(2 b\\log(b))}=\\frac{2 b\\log(b)}{\\log(b)+\\log(2\\log(b))}\\stackrel{(i)}{\\ge}\\frac{2 b\\log(b)}{2\\log(b)}=b,\\tag{36}\n$$"}, {"n": 8, "claim": "where (i) used log $(b)>0$ together with $\\forall z:\\log(z)\\leq z/2$ to show $\\log(2\\log(b))\\leq\\log(b)$ .This proves Equation 34 in the second case, so that it always holds."}]}}], "target_local_id": "T3", "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "label": "flawed", "error_step": 5, "error_class6": ["invalid_bound_or_inequality", "missing_assumption", "logical_gap_or_missing_case"], "error_explanation": "The step uses $\\eta\\le 1/(4H)$ to bound $-(2\\eta-4\\eta^2 H)(F(\\bar w_t)-F(u))$ by $-\\eta(F(\\bar w_t)-F(u))$ (equivalently, to split $4\\eta^2 H(F(\\bar w_t)-F_*)$ into an $\\eta(F(\\bar w_t)-F(u))$ term). Shrinking the coefficient of $F(\\bar w_t)-F(u)$ this way preserves the inequality direction only when $F(\\bar w_t)-F(u)\\ge 0$. But Lemma 17 is asserted for an ARBITRARY comparator $\\pmb u\\in\\mathbb R^d$ (separable logistic regression has no global minimizer), so $F(\\bar w_t)-F(u)$ can be negative; then the inequality flips and the bound is invalid. The authors acknowledged this is a genuine proof bug. The correct order is to first apply $4\\eta H\\le 1$ to the always-nonnegative term $4\\eta^2 H(F(\\bar w_t)-F_*)\\le \\eta(F(\\bar w_t)-F_*)$, and only afterward decompose $\\eta(F(\\bar w_t)-F_*)=\\eta(F(\\bar w_t)-F(u))+\\eta(F(u)-F_*)$."}
{"id": "nJnky5K944", "context": "We use bold lowercase letters to represent vectors and bold uppercase letters to represent matrices. For any vector $v\\in\\mathbb{R}^{a}$ , we denote by $v_{i}$ the i-th element of v. For any matrix $\\pmb{A}\\in\\overset{\\cdot}{\\mathbb{R}}^{a\\times b}$ , we denote its i-th row by $A_{i,:},$ its k-th column by $A_{:,k}$ and the element at its i-th row and k-th column by $A_{i,k}$ For any positive integer m $\\in\\mathbb{N}_{+},[m]$ represents the set $\\{1,\\ldots,m\\}$ . For any real numbers $a<b,$ $[a,b]$ represents the interval $\\{x\\in\\mathbb{R}^{\\cdot}|\\stackrel{\\cdot}{a}\\leq x\\leq b\\},(-\\infty,a)$ represents $\\{x\\in\\mathbb{R}\\mid x<a\\}$ , and $(b,\\infty)$ represents $\\{x\\in\\mathbb{R}\\mid x>b\\}$ . Let $\\sigma_{S}[\\pmb{v}]$ and $\\sigma_{H}[\\pmb{v}]$ for any input vector v be the softmax function and hardmax function, respectively. Note that when there are multiple indices with maximum values, the hardmax function is defined such that the sum of the values at these indices equals one. By abuse of notation, for any input matrix $A,\\sigma_{S}\\left[A\\right]$ and $\\sigma_{H}\\left[\\boldsymbol{A}\\right]$ are defined as column-wise softmax and column-wise hardmax, respectively. We denote the ReLU activation function by $\\sigma_{R}$ . Unlike $\\sigma_{S}$ and $\\sigma_{H},\\sigma_{R}$ is always an element-wise operator, regardless of whether the input is a vector or a matrix. Let | · |lbe the $\\bar{\\ell}^{2}$ norm and $\\|\\cdot\\|_{p}\\left(1\\overset{\\_}{\\leq}p<\\infty\\right)$ be the $\\ell^{p\\ }$ norm.\n\n2.2 Transformer Block\n\nTransformer was first introduced in Vaswani et al. (2017). Here we follow the definitions adopted in Kim et al. (2023): the Transformer block is composed of the self-attention mechanism and the feed-forward neural network, each accompanied by a skip connection. Given an input sequence $\\boldsymbol{Z}\\in\\mathbb{R}^{d\\times n}$ − composed of n tokens each with an embedding dimension of size $d,$ a dot-product self-attention mechanism with h heads outputs the following values:\n\n$$\n\\mathcal{F}_{S}^{(S A)}(Z)=Z+\\sum_{i=1}^{h}W_{i}^{(O)}\\left(W_{i}^{(V)}Z\\right)\\sigma_{S}\\left[\\left(W_{i}^{(K)}Z\\right)^{\\top}\\left(W_{i}^{(Q)}Z\\right)\\right]\\in\\mathbb{R}^{d\\times n},\\tag{1}\n$$\n\nwhere ${\\bf W}_{i}^{(V)},{\\bf W}_{i}^{(K)},{\\bf W}_{i}^{(Q)}\\in\\mathbb{R}^{s\\times d}$ and $W_{i}^{(O)}\\in\\mathbb{R}^{d\\times s}$ are the weight matrices, and s is the head size. In contrast, given an input $H\\in\\mathbb{R}^{d\\times n}$ , the output of feed-forward neural network with a skip connection at index $k \\in [ n ]$ is\n\n$$\n\\mathcal{F}^{(F F)}(H)_{:,k}=H_{:,k}+W^{(2)}\\sigma_{R}\\left[W^{(1)}H_{:,k}+b^{(1)}\\right]+b^{(2)}\\in\\mathbb{R}^{d},\n$$\n\nwhere q is the hidden dimension, $W^{(1)}\\in\\mathbb{R}^{q\\times d}$ and $W^{(2)}\\in\\mathbb{R}^{d\\times q}$ are weight matrices, and $b^{(1)}\\in\\mathbb{R}^{q}$ and $b^{(2)}$ are bias terms. On the basis of the above definition, the Transformer block is represented as a composition of a self-attention mechanism and a feed-forward neural network: for any input sequence $Z\\in\\mathbb{R}^{d\\times n}$ , composed of n tokens each with an embedding dimension of size d, the Transformer block $\\mathcal{F}:\\mathbb{R}^{d\\times n}\\to\\mathbb{R}^{d\\times n}$ outputs\n\n$$\n\\mathcal{F}(Z)=\\mathcal{F}^{(F F)}\\left(\\mathcal{F}_{S}^{(S A)}(Z)\\right).\n$$\n\n3.1 Problem Setting\n\nLet $(\\pmb{X}^{(1)},\\pmb{Y}^{(1)}),\\dots,(\\pmb{X}^{(1)},\\pmb{Y}^{(1)})\\subset\\mathbb{R}^{d\\times n}\\times [\\pmb{C}]^{d\\times n}$ be an N input-output pairs of sequences, each of which consists of a sequence $\\pmb{X}^{(i)}$ of n tokens with embedding dimension d, and an output $\\mathbf{\\nabla}\\mathbf{\\mathbf{{Y}}}^{(i)}$ , where $Y_{:,k}^{(i)}$ corresponds to the label of the token $X_{:,k}^{(i)}$ at index k. In addition, we define the i-th vocabulary set for $\\begin{array}{ r }{i\\in [N]\\mathrm{~b y~}\\mathcal{V}^{(i)}=\\bigcup_{k\\in [n]}X_{:,k}^{(i)}\\subset\\mathbb{R}^{d}}\\end{array}$ , and the whole vocabulary set $\\nu$ is defined by $\\begin{array}{ r }{\\mathcal{V}=\\bigcup_{i\\in [N]}\\mathcal{V}^{(i)}\\subset\\mathbb{R}^{d}}\\end{array}$\n\n3.2 Background\n\nDefinition 2 (Tokenwise Separatedness). Let $m\\in\\mathbb{N}$ and $Z^{(1)},\\ldots,Z^{(N)}\\in\\mathbb{R}^{m\\times n}$ be input sequences. Then, ${Z}^{(1)},\\ldots,{Z}^{\\bar{(N)}}$ are called tokenwise $(r_{\\operatorname*{min}},r_{\\operatorname*{max}},\\delta)$ -separated if the following three conditions hold.\n\n1. For any $i \\in [ N ]$ and $k\\in [n],\\left\\|Z_{:,k}^{(i)}\\right\\|>r_{\\operatorname*{min}}$ holds.\n\n2. For any $i \\in [ N ]$ and $k\\in [n],\\left\\|Z_{:,k}^{(i)}\\right\\|<r_{\\operatorname*{max}}$ holds.\n\n3. For any $i,j\\in [N]$ and $k,l\\in [n]$ with $Z_{:,k}^{(i)}\\neq Z_{:,l}^{(j)},\\left. Z_{:,k}^{(i)}-Z_{:,l}^{(j)}\\right.>\\delta$ holds.\n\nNote that we refer to ${Z}^{(1)},\\ldots,{Z}^{(N)}$ as tokenwise $(r_{\\operatorname*{max}},\\epsilon)$ -separated instead if the sequences satisfy conditions 2 and 3.\n\nDefinition 3 (Contextual Mapping). Let $\\pmb{X}^{(1)},\\ldots,\\pmb{X}^{(N)}\\in\\mathbb{R}^{d\\times n}$ be input sequences. Then, a map $q:\\mathbb{R}^{d\\times n}\\bar{\\mathbb{R}}^{d\\times n}$ is called an $(r,\\delta)$ -contextual mapping if the following two conditions hold:\n\n1. For any $i \\in [ N ]$ and $k\\in [n],\\left\\|q\\left(\\mathbf{X}^{\\left(i\\right)}\\right)_{:,k}\\right\\|<r$ holds.\n\n2. For any $i,j\\in\\mathsf{\\Gamma}[N]$ and $k,l~\\in~[n]$ such that $\\mathcal{V}^{(i)}~\\neq~\\mathcal{V}^{(j)}$ or $X_{:,k}^{(i)}~\\neq~X_{:,l}^{(j)}$ $\\left\\|\\boldsymbol{q}\\left(\\boldsymbol{X}^{(i)}\\right)_{:,k}-\\boldsymbol{q}\\left(\\boldsymbol{X}^{(j)}\\right)_{:,l}\\right\\|>\\delta$ holds.\n\nIn particular, $\\boldsymbol{q}(\\boldsymbol{X}^{(i)})$ for $i \\in [ N ]$ is called a context id of $\\pmb{X}^{(i)}$\n\n3.3 Self-Attention with Hardmax\n\nIn previous studies analyzing the memorization capacity of Transformers (Yun et al., 2019; Kim et al., 2023), the softmax function is taken to be an approximation of the hardmax function. However, we show here that the attention block with the hardmax function is not a contextual mapping.\n\nFirst we define the attention block with the hardmax function: for an input sequence $\\boldsymbol{Z}\\in\\mathbb{R}^{d\\times n}$ , the attention with the hardmax function is calculated as\n\n$$\n\\mathcal{F}_{H}^{(S A)}(Z)=Z+\\sum_{i=1}^{h}W_{i}^{(O)}\\left(W_{i}^{(V)}Z\\right)\\sigma_{H}\\left[\\left(W_{i}^{(K)}Z\\right)^{\\top}\\left(W_{i}^{(Q)}Z\\right)\\right],\\tag{2}\n$$\n\nwhere $\\pmb{W}_{i}^{(V)},\\pmb{W}_{i}^{(K)},\\pmb{W}_{i}^{(Q)}\\in\\mathbb{R}^{s\\times d}$ and $W_{i}^{(O)}\\in\\mathbb{R}^{d\\times s}$ are the weight matrices\n\nThe following theorem holds for such a model. The proof is in Appendix A.1.\n\n3.4 Self-Attention with Softmax\n\nIn this subsection, we show that a softmax-based 1-layer attention block with low-rank weight matrices is a contextual mapping for almost all input sequences.\n\n4.1 Memorization Capacity of One-Layer Transformer\n\nAs a first application of Theorem 1, we prove that a 1-layer Transformer can completely memorize finite samples, each of which has no duplicate token. This result emphasizes that in contrast to the proof of Kim et al. (2023), which requires 2n self-attention layers for Transformer memorization, one layer of self-attention is actually sufficient. In addition, it is worth noting that the hardmaxbased Transformers do not have a memorization capacity, which is implied straightforwardly from Theorem 2.\n\nIn addition, it is straightforward to show that a 1-layer Transformer with trainable positional encodings has a memorization capacity for arbitrary input sequences possibly with duplicate tokens.\n\nB.2 Properties of Boltzmann Operator\n\nFor any vector $\\pmb{a}=(a_{1},\\ldots,a_{n})\\in\\mathbb{R}^{n}$ , let denote $\\pmb{p}=(p_{1},\\ldots,p_{n})=\\sigma_{S}[\\pmb{a}]$ . In addition, we define the Boltzmann operator, partition function, and entropy as follows.\n\n$$\n\\mathbf{boltz}(\\pmb{a})=\\sum_{i=1}^{n}a_{i}p_{i},\n$$\n\n$$\n\\mathcal{L}(\\pmb{a})=\\log\\left(\\sum_{i=1}^{n}e^{a_{i}}\\right),\n$$\n\n$$\n\\mathcal{S}(\\pmb{p})= -\\sum_{i=1}^{n}p_{i}\\log p_{i}.\n$$", "theorems": [{"local_id": "T1", "statement": "Let $\\pmb{a}^{(1)},\\ldots,\\pmb{a}^{(m)} \\in \\mathbb{R}^n$ be tokenwise $(r,\\delta)$-separated vectors with no duplicate element in each vector and $\\delta > 2 \\log n + 3$. Then the outputs of the Boltzmann operator are $(r,\\delta')$-separated, that is, $|\\mathbf{boltz}(\\pmb{a}^{(i)})| \\le r$ and $|\\mathbf{boltz}(\\pmb{a}^{(i)}) - \\mathbf{boltz}(\\pmb{a}^{(j)})| > \\delta' = (\\log n)^2 e^{-2r}$ hold for each $i,j \\in [m]$ such that $\\pmb{a}^{(i)}$ and $\\pmb{a}^{(j)}$ are not permutations of one another (equivalently, their entry-sets $\\{a^{(i)}_1,\\ldots,a^{(i)}_n\\} \\neq \\{a^{(j)}_1,\\ldots,a^{(j)}_n\\}$ differ).", "proof": {"steps": [{"n": 1, "claim": "Since the Boltzmann operator is permutation invariant, assume WLOG $a_1 > \\cdots > a_n$ and $b_1 > \\cdots > b_n$ (no duplicate element $\\Rightarrow$ strict)."}, {"n": 2, "claim": "Since the Boltzmann operator can be regarded as weighting averaging, we have $|\\mathbf{boltz}(\\pmb{a})| \\le \\max(|a_1|,|a_n|) < r$."}, {"n": 3, "claim": "For $\\delta'$-separatedness, since $\\pmb{a}$ and $\\pmb{b}$ have different entry-sets, their descending sorts differ, so there exists $k \\in \\{0,\\ldots,n-1\\}$ with $(a_1,\\ldots,a_k)=(b_1,\\ldots,b_k)$ and $a_{k+1} > b_{k+1}$."}, {"n": 4, "claim": "Then, the following auxiliary lemma of the source implies that we have $|\\mathbf{boltz}(\\pmb{a}) - \\mathbf{boltz}(\\pmb{b})| > (\\log n)^2 e^{-(a_1 - b_{k+1})}$. [Auxiliary lemma: Let $\\pmb{a},\\pmb{b}\\in\\mathbb R^n$ be tokenwise $\\delta$-separated vectors in a decreasing order with no duplicate element in each vector and $\\delta > 2\\log n + 3$, that is, $i > j \\Rightarrow a_i - a_j,\\ b_i - b_j > \\delta$ and $a_i \\neq b_j \\Rightarrow |a_i - b_j| > \\delta$ for any $i,j\\in[n]$. In addition, suppose there exists $k\\in\\{0,\\ldots,n-1\\}$ such that $(a_1,\\ldots,a_k)=(b_1,\\ldots,b_k)$ and $a_{k+1} > b_{k+1}$. Then, the outputs of the Boltzmann operator are $(\\log n)^2 e^{-(a_1-b_{k+1})}$-separated, that is, $|\\mathbf{boltz}(\\pmb{a}) - \\mathbf{boltz}(\\pmb{b})| > (\\log n)^2 e^{-(a_1-b_{k+1})}$ holds.]"}, {"n": 5, "claim": "Since $\\pmb{a},\\pmb{b}$ are tokenwise $(r,\\delta)$-separated, $a_1 - b_{k+1} < 2r$, hence $|\\mathbf{boltz}(\\pmb{a}) - \\mathbf{boltz}(\\pmb{b})| > (\\log n)^2 e^{-(a_1-b_{k+1})} > (\\log n)^2 e^{-2r} = \\delta'$."}]}}, {"local_id": "T2", "statement": "Let $\\pmb{X}^{(1)},\\ldots,\\pmb{X}^{(N)}\\in\\mathbb{R}^{d\\times n}$ be input sequences with no duplicate word token in each sequence, i.e. $\\pmb{X}_{:,k}^{(i)}\\neq\\pmb{X}_{:,l}^{(i)}$ for any $i\\in[N]$ and $k,l\\in[n]$ with $k\\neq l$. Assume that $\\pmb{X}^{(1)},\\ldots,\\pmb{X}^{(N)}$ are tokenwise $(r_{\\min},r_{\\max},\\epsilon)$-separated (Definition 2). Here the $i$-th vocabulary set is $\\mathcal{V}^{(i)}=\\bigcup_{k\\in[n]}\\pmb{X}_{:,k}^{(i)}\\subset\\mathbb{R}^d$ and the whole vocabulary is $\\mathcal{V}=\\bigcup_{i\\in[N]}\\mathcal{V}^{(i)}$. Then there exist weight matrices $\\pmb{W}^{(O)}\\in\\mathbb{R}^{d\\times s}$ and $\\pmb{W}^{(V)},\\pmb{W}^{(K)},\\pmb{W}^{(Q)}\\in\\mathbb{R}^{s\\times d}$ whose ranks are all $1$ such that the one-layer single-head self-attention with softmax $\\mathcal{F}_S^{(SA)}$ (with $h=1$), given by $\\mathcal{F}_S^{(SA)}(Z)=Z+\\pmb{W}^{(O)}(\\pmb{W}^{(V)}Z)\\sigma_S[(\\pmb{W}^{(K)}Z)^\\top(\\pmb{W}^{(Q)}Z)]$, is an $(r,\\delta)$-contextual mapping (Definition 3) for $\\pmb{X}^{(1)},\\ldots,\\pmb{X}^{(N)}$, with $r=r_{\\max}+\\tfrac{\\epsilon}{4}$ and $\\delta=\\frac{2(\\log n)^2\\epsilon^2 r_{\\min}}{r_{\\max}^2(|\\mathcal{V}|+1)^4(2\\log n+3)\\pi d}\\exp\\!\\left(-(|\\mathcal{V}|+1)^4\\frac{(2\\log n+3)\\pi d r_{\\max}^2}{4\\epsilon r_{\\min}}\\right)$.", "proof": {"steps": [{"n": 1, "claim": "For each token $\\pmb{v}\\in\\mathcal{V}$, it suffices to identify, by some $\\pmb{v}$-dependent sequence id, which input sequences contain $\\pmb{v}$; a context id is then built from such ids via $\\pmb{W}^{(O)}$ and the skip connection."}, {"n": 2, "claim": "Construct $\\pmb{W}^{(V)},\\pmb{W}^{(K)},\\pmb{W}^{(Q)}\\in\\mathbb{R}^{1\\times d}$ and $\\epsilon>0$ with $\\left(\\pmb{W}^{(V)}\\pmb{X}^{(i)}\\right)\\sigma_S\\!\\left[(\\pmb{W}^{(K)}\\pmb{X}^{(i)})^\\top(\\pmb{W}^{(Q)}\\pmb{X}_{:,k}^{(i)})\\right]-\\left(\\pmb{W}^{(V)}\\pmb{X}^{(j)}\\right)\\sigma_S\\!\\left[(\\pmb{W}^{(K)}\\pmb{X}^{(j)})^\\top(\\pmb{W}^{(Q)}\\pmb{X}_{:,l}^{(j)})\\right]>\\epsilon$ for all distinct $i,j\\in[N]$, $k,l\\in[n]$ with $\\pmb{X}_{:,k}^{(i)}=\\pmb{X}_{:,l}^{(j)}$."}, {"n": 3, "claim": "Take $\\pmb{W}^{(V)}=\\pmb{W}^{(K)}=\\pmb{W}^{(Q)}=\\pmb{w}^\\top$ with $\\pmb{w}\\in\\mathbb{R}^d$ s.t. $c\\|\\pmb{v}_a-\\pmb{v}_b\\|\\leq|\\pmb{w}^\\top\\pmb{v}_a-\\pmb{w}^\\top\\pmb{v}_b|\\leq\\|\\pmb{v}_a-\\pmb{v}_b\\|$, $0<c<1$, for all $\\pmb{v}_a,\\pmb{v}_b\\in\\mathcal{V}$."}, {"n": 4, "claim": "With $\\pmb{t}=\\pmb{X}_{:,k}^{(i)}=\\pmb{X}_{:,l}^{(j)}$: $\\left|\\pmb{w}^\\top\\pmb{t}\\right|\\cdot\\left|(\\pmb{w}^\\top\\pmb{X}^{(i)})\\sigma_S[(\\pmb{w}^\\top\\pmb{X}^{(i)})^\\top(\\pmb{w}^\\top\\pmb{t})]-(\\pmb{w}^\\top\\pmb{X}^{(j)})\\sigma_S[(\\pmb{w}^\\top\\pmb{X}^{(j)})^\\top(\\pmb{w}^\\top\\pmb{t})]\\right|\\geq\\left|(\\pmb{a}^{(i)})^\\top\\sigma_S[\\pmb{a}^{(i)}]-(\\pmb{a}^{(j)})^\\top\\sigma_S[\\pmb{a}^{(j)}]\\right|$, $\\pmb{a}^{(i)}=(\\pmb{w}^\\top\\pmb{X}^{(i)})^\\top(\\pmb{w}^\\top\\pmb{t})$, $\\pmb{a}^{(j)}=(\\pmb{w}^\\top\\pmb{X}^{(j)})^\\top(\\pmb{w}^\\top\\pmb{t})\\in\\mathbb{R}^n$."}, {"n": 5, "claim": "Suffices to show separability of $\\mathbf{boltz}:\\mathbb{R}^n\\to\\mathbb{R}$, $\\pmb{a}\\mapsto\\pmb{a}^\\top\\sigma_S[\\pmb{a}]$."}]}}, {"local_id": "T3", "statement": "Let $\\epsilon>0,r_{\\mathrm{max}}>r_{\\mathrm{min}}>0$ and $(\\pmb{X}^{(1)},\\pmb{\\dot{Y}}^{(1)}),\\dots,(\\pmb{X}^{(N)},\\pmb{Y}^{(\\hat{N})})\\ \\dot{\\subset}\\ \\mathbb{R}^{d\\times n}\\times\\ \\dot{[\\pmb{C}]}^{d\\times n}$ be sequences of input-output-pairs such that $\\bar{\\pmb X}^{(1)},\\dots,\\bar{\\pmb X}^{(N)}$ are tokenwise $(r_{\\operatorname*{min}},r_{\\operatorname*{max}},\\epsilon)$ -separated input sequences with no duplicate token in each sentence and consistently labeled, that is, $\\pmb{Y}_{:,k}^{(i)}=\\pmb{Y}_{:,l}^{(j)}$ holds for any $i,j\\in [N]$ and $k,l\\in [n]$ such that $\\mathcal{V}^{(i)}=\\mathcal{V}^{(j)}$ and $\\pmb{X}_{:,k}^{(i)}=\\pmb{X}_{:,l}^{(j)}$\n\nThen, there exist $4(s+d)+d(2nN+d)$ weight parameters such that for any $i \\in [ N ]$\n\n$$\n\\mathcal{F}\\left(\\boldsymbol{X}^{(i)}\\right)=\\mathcal{F}^{(F F)}\\left(\\mathcal{F}_{S}^{(S A)}\\left(\\boldsymbol{X}^{(i)}\\right)\\right)=\\boldsymbol{Y}^{(i)}\\tag{4}\n$$\n\nholds.", "proof": {"steps": [{"n": 1, "claim": "$$\n\\begin{array}{ r l }&{r=r_{\\operatorname*{max}}+\\frac{\\epsilon}{4},}\\\\ &{\\delta=\\frac{\\epsilon r_{\\operatorname*{min}}\\log n}{r_{\\operatorname*{max}}^{2}(|\\mathcal{V}| +1)^{4}(2\\log n+3)\\pi d}}\\\\ &{\\qquad\\cdot\\exp\\left(-(|\\mathcal{V}| +1)^{4}\\frac{(2\\log n+3)\\pi d r_{\\operatorname*{max}}^{2}}{4\\epsilon r_{\\operatorname*{min}}}\\right).}\\end{array}\\tag{5}\n$$"}, {"n": 2, "claim": "So what remains to do is to associate each context id with the corresponding output label using a feed-forward neural network $\\mathcal{F}^{(F F)}$ . Construction of such a network is a typical memorization task of a one-hidden-layer feed-forward neural network. Here we adopt the implementation from Zhang et al. (2016)."}, {"n": 3, "claim": "In this case, since the possible number of context ids is upper-bounded by $n N$ , the required parameters for the FF layer with output dimension d is at most ${\\dot{\\boldsymbol{d}}}\\times(2 n{\\boldsymbol{N}}+{\\dot{\\boldsymbol{d}}})$ (Zhang et al., 2016)."}, {"n": 4, "claim": "As for the self-attention layer, rank-1 weight matrices $W^{(O)}\\in\\mathbb{R}^{d\\times s}$ and $\\mathbf{\\Delta}\\mathbf{W}^{(V)},\\mathbf{W}^{(K)},\\mathbf{W}^{(Q)}\\in\\mathbb{R}^{s\\times d}$ all require $s + d$ parameters each. Thus, the number of parameters of the self-attention layer is $4(\\mathrm{s}+\\mathrm{d})$ ."}]}}, {"local_id": "T4", "statement": "Let $n\\in\\mathbb{N}_{+},i\\in [n]$ and $\\pmb{a}=(a_{1},\\ldots,a_{n})\\in\\mathbb{R}^{n}$ . Then, the derivative of the Boltzmann operator boltz $:({\\pmb a})={\\pmb a}^{\\top}\\sigma_{S}[{\\pmb a}]$ with respect $a_{i}$ is\n\n$$\n{\\frac{\\partial}{\\partial a_{i}}}\\mathbf{boltz}(\\pmb{a})=p_{i}(1+\\log p_{i}+\\pmb{S}(\\pmb{p})).\\tag{6}\n$$\n\nIn particular, the Boltzmann operator is monotonically decreasing with respect to $a_{i}$ whenever\n\n$$\na_{i}<\\operatorname*{max}_{j\\in [n]}a_{j}-\\log n-1\\tag{7}\n$$\n\nholds.", "proof": {"steps": [{"n": 1, "claim": "Since\n\n$$\n\\begin{array}{ l }{{\\displaystyle{\\mathcal E}(a)-\\mathcal S(p)=\\log\\left(\\sum_{j=1}^{n}e^{a_{j}}\\right)+\\sum_{k=1}^{n}p_{k}\\log p_{k}}}\\\\ {{\\displaystyle\\qquad=\\sum_{k=1}^{n}p_{k}\\log\\left(p_{k}\\sum_{j=1}^{n}e^{a_{j}}\\right)}}\\\\ {{\\displaystyle\\qquad=\\sum_{k=1}^{n}p_{k}\\log e^{a_{k}}=\\sum_{k=1}^{n}p_{k}a_{k}=\\mathbf b\\mathrm{oltz}(a)},}\\end{array}\\tag{8}\n$$"}, {"n": 2, "claim": "we have\n\n$$\n{\\frac{\\partial}{\\partial a_{i}}}\\mathbf{boltz}(\\mathbf{a})={\\frac{\\partial}{\\partial a_{i}}}{\\mathcal{L}}(\\mathbf{a})-{\\frac{\\partial}{\\partial a_{i}}}S(\\mathbf{p}).\n$$"}, {"n": 3, "claim": "Notice that\n\n(9)\n\n$$\n\\begin{array}{ l }{\\displaystyle\\frac{\\partial p_{i}}{\\partial a_{j}}=\\frac{\\partial}{\\partial a_{j}}\\frac{e^{a_{i}}}{\\sum_{k=1}^{n}e^{a_{k}}}}\\\\ {=\\frac{\\frac{\\partial}{\\partial a_{j}}e^{a_{i}}\\cdot\\sum_{k=1}^{n}e^{a_{k}}-e^{a_{i}}\\cdot\\frac{\\partial}{\\partial a_{j}}\\sum_{k=1}^{n}e^{a_{k}}}{\\left(\\sum_{k=1}^{n}e^{a_{k}}\\right)^{2}}}\\\\ {=\\frac{\\delta_{i j}e^{a_{j}}\\cdot\\sum_{k=1}^{n}e^{a_{k}}-e^{a_{i}}\\cdot e^{a_{j}}}{\\left(\\sum_{k=1}^{n}e^{a_{k}}\\right)^{2}}=p_{j}(\\delta_{i j}-p_{i})}\\end{array}\\tag{10}\n$$\n\nholds for any $i,j\\in [n]$ .Here $\\delta_{i j}$ is the Kronecker delta, that is,\n\n$$\n\\delta_{i j}={\\left\\{\\begin{array}{ l l }{0}&{{\\mathrm{i f~}}i\\neq j,}\\\\ {1}&{{\\mathrm{i f~}}i=j .}\\end{array}\\right.}\\tag{11}\n$$"}, {"n": 4, "claim": "Thus,\n\n$$\n\\frac{\\partial\\mathcal{L}(\\pmb{a})}{\\partial a_{i}}=\\frac{e^{a_{i}}}{\\sum_{j=1}^{n}e^{a_{j}}}=p_{i},\\tag{12}\n$$\n\n$$\n\\begin{array}{ r l }{\\displaystyle\\frac{\\partial\\mathcal{S}(p)}{\\partial\\alpha_{i}}=\\sum_{j=1}^{n}\\frac{\\partial\\mathcal{S}(p)}{\\partial\\nu_{j}}\\cdot\\frac{\\partial p_{j}}{\\partial\\alpha_{i}}}\\\\ {\\displaystyle}&{=\\sum_{j=1}^{n}\\frac{\\partial}{\\partial p_{j}}\\left(-\\sum_{k=1}^{n}p_{k}\\log p_{k}\\right)\\cdot p_{i}(\\delta_{j i}-p_{j})}\\\\ {\\displaystyle}&{=\\sum_{j=1}^{n}(-\\log p_{j}-1)\\cdot p_{i}(\\delta_{j i}-p_{j})}\\\\ {\\displaystyle}&{= -p_{i}\\sum_{j=1}^{n}\\left[\\delta_{j i}(\\log p_{i}+1)-p_{j i}\\log p_{j}-p_{j i}\\right]}\\\\ {\\displaystyle}&{= -p_{i}(\\log p_{i}+1+\\mathcal{S}(p)-1)}\\\\ {\\displaystyle}&{= -p_{i}(\\log p_{i}+\\mathcal{S}(p)).}\\end{array}\\tag{13}\n$$"}, {"n": 5, "claim": "Plugging equation 12 and equation 13 into equation 9, we have\n\n$$\n\\begin{array}{ r }{\\displaystyle\\frac{\\partial}{\\partial a_{i}}\\mathbf{boltz}(\\pmb{a})=\\frac{\\partial}{\\partial a_{i}}\\mathcal{L}(\\pmb{a})-\\frac{\\partial}{\\partial a_{i}}\\mathcal{S}(\\pmb{p})}\\\\ {\\displaystyle=p_{i}+p_{i}(\\log p_{i}+\\mathcal{S}(\\pmb{p}))}\\\\ {\\displaystyle=p_{i}(1+\\log p_{i}+\\mathcal{S}(\\pmb{p})).}\\end{array}\\tag{14}\n$$"}, {"n": 6, "claim": "In particular, the derivative of the Boltzmann operator at index i is negative when\n\n$$\n\\begin{array}{ c }{{1+S(p)+\\log{p_{i}}<0\\Leftrightarrow 1+S(p)+\\left(a_{i}-{\\mathcal{L}}(a)\\right)<0}}\\\\ {{\\Leftrightarrow a_{i}<{\\mathcal{L}}(a)-S(p)-1 .}}\\end{array}\\tag{15}\n$$\n\nSince $\\begin{array}{ r }{\\operatorname*{max}_{j\\in [n]}a_{j}\\ \\leq\\ \\mathcal{L}(\\pmb{a})(\\mathtt{p.72}}\\end{array}$ , Boyd & Vandenberghe (2004)) and $S({\\pmb{p}})\\ \\leq$ log n, we have $\\frac{\\partial}{\\partial a_{i}}$ bolt $\\mathbf{z}(a)<0$ whenever\n\n$$\na_{i}<\\operatorname*{max}_{j\\in [n]}a_{j}-\\log n-1\\tag{16}\n$$\n\nholds."}]}}], "target_local_id": "T2", "topic_domain": "Deep Learning Theory", "topic_subtopic": "Representation & Expressivity", "label": "flawed", "error_step": 2, "error_class6": ["logical_gap_or_missing_case", "invalid_bound_or_inequality", "others"], "error_explanation": "The separability condition demanded in this step is stated with a bare (non-absolute-valued) difference and without the hypothesis $\\mathcal{V}^{(i)}\\neq\\mathcal{V}^{(j)}$, and as written it is impossible to satisfy. (1) The left-hand side $\\text{LHS}(i,k;j,l)=(W^{(V)}X^{(i)})\\sigma_S[\\cdots X_{:,k}^{(i)}\\cdots]-(W^{(V)}X^{(j)})\\sigma_S[\\cdots X_{:,l}^{(j)}\\cdots]$ is antisymmetric under swapping $(i,k)\\leftrightarrow(j,l)$. If $\\text{LHS}(i,k;j,l)>\\epsilon$, then the same required inequality applied to the swapped indices gives $\\text{LHS}(j,l;i,k)=-\\text{LHS}(i,k;j,l)>\\epsilon$, i.e. $\\text{LHS}(i,k;j,l)<-\\epsilon$, a direct contradiction; hence no such $\\epsilon>0$ can exist. The fix is to wrap the LHS in absolute-value bars $|\\cdots|>\\epsilon$, which is symmetric and consistent. (2) Contextual mapping (Definition 2, condition 2) only requires separating context ids when the contexts genuinely differ ($\\mathcal{V}^{(i)}\\neq\\mathcal{V}^{(j)}$ or $X_{:,k}^{(i)}\\neq X_{:,l}^{(j)}$); without the $\\mathcal{V}^{(i)}\\neq\\mathcal{V}^{(j)}$ hypothesis the step demands a strict gap even between identical tokens sitting in identical context sets, where no separation can or should hold. The correct condition adds both the absolute value and the hypothesis $\\mathcal{V}^{(i)}\\neq\\mathcal{V}^{(j)}$."}
{"id": "rJlhYa4FPB", "context": "Our key insight is that the relation between factors in a representation with data can be described from the perspective of containing information. Since each factor in a disentangled representation corresponds to a single factor of variation in data, each factor only contains the information of the corresponding factor of variation in data. In other words, different factors in a disentangled representation contain absolutely different information about data, and the containing information about data in a single factor is irrelevant to other factors. Motivated by this insight, we aim at proposing an information-based assumption to describe the relation between factors in a disentangled representation with factors of variation in data. For mathematical analysis of the information terms, we adopt mutual information to measure the information of a variable contained by another:\n\n$$\nI(\\mathbf{x};\\mathbf{z})=H(\\mathbf{z})-H(\\mathbf{z}|\\mathbf{x})\\tag{5}\n$$\n\nwhere entropy $H(\\mathbf{z})= -\\mathbb{E}_{q(\\mathbf{z})}[\\log q(\\mathbf{z})]$ and conditional entropy $H(\\mathbf{z}|\\mathbf{x})= -\\mathbb{E}_{q(\\mathbf{z},\\mathbf{x})}[\\log q(\\mathbf{z}|\\mathbf{x})]$ are measures of uncertainty. Hence mutual information measures the uncertainty reduction of one variable when another is given. The mutual information satisfies the symmetry $I(\\mathbf{x;z})=I(\\mathbf{z;x})$\n\nTherefore, for a single factor $\\mathbf{Z}_{j}$ in a disentangled representation $\\mathbf{z},$ its containing information about the corresponding underlying factor of variation in data x can be expressed as $I({\\bf z};{\\bf x})$ . To represent the irrelevance of $\\mathbf{Z}_{j}$ to other information in data, we involve conditional mutual information:\n\n$$\nI(\\mathbf{z}_{j};\\mathbf{x}|\\mathbf{z}_{i\\in S_{-j}})=H(\\mathbf{z}_{j}|\\mathbf{z}_{i\\in S_{-j}})-H(\\mathbf{z}_{j}|\\mathbf{x},\\mathbf{z}_{i\\in S_{-j}})\\tag{6}\n$$\n\nwhere $j\\in\\{1,2,...,J\\},S_{-j}$ is any subset of $\\{1,\\dotsc,j-1,j+1,\\dotsc,J\\}$ and $\\mathbf{z}_{i\\in S_{-j}}$ denotes all factors with subscript index in $S_{-j}$ .The conditional mutual information is the difference of two conditional entropy, representing the containing information of $\\mathbf{Z}_{j}$ on the corresponding underlying factor of variation in data x when other factors $\\mathbf{z}_{i\\in S_{-j}}$ are given. Considering the corresponding factors of variation in data of $\\mathbf{z}_{i\\in S_{-j}}$ , they are fixed when $\\mathbf{z}_{i\\in S_{-}}$ are given. Hence the conditional mutual information term above can describe the their relation with $\\mathbf{Z}_{j}$\n\nAssumption. Suppose $\\mathbf{z}\\in\\mathbb{R}^{J}$ is a disentangled representation of data x and $I({\\bf z};{\\bf x})>0$ , then for any single factor $\\mathbf{z}_{j},$ its containing information in data is irrelevant to other factors $z_{i\\in S_{-j}},i . e .$\n\n$$\nI(\\mathbf{z}_{j};\\mathbf{x}|\\mathbf{z}_{i\\in S_{-j}})=I(\\mathbf{z}_{j};\\mathbf{x})\\tag{7}\n$$\n\nwhere $j\\in\\{1,2,\\dots,J\\}$ and $S_{-j}$ is any subset of $\\{1,\\dotsc,j-1,j+1,\\dotsc,J\\}$\n\nOn condition of independence, the proposed assumption is equivalent to conditional independence. In this section we analyze the inductive biases about conditional independence, and then propose a simple architecture to improve disentanglement based on our assumption and the inductive biases.\n\nFigure 1: Structure of vanilla encoder and split encoder. The blue and short-dash box denotes sharing-parameter block, and the red and long-dash box represents no-sharing-parameter block.\n\nMotivated by our proposed assumption, we find that there are some inductive biases on encoders of VAEs to ensure conditional independence. Considering $q(\\mathbf{z})$ is pushed to match a factorized prior $p(\\mathbf{z})$ in VAE-based models and thus encourages independence and the common assumption is satisfied, this discovery shows that our assumption is also satisfied in these models.\n\nIn VAEs, the encoders encode data into a Gaussian distribution of representation $q(\\mathbf{z}|\\mathbf{x})$ via reparameterization trick. Specifically, the data is encoded into a mean $\\mu(\\mathbf{x})$ and a variance $\\sigma(\\mathbf{x})$ , then the representation is sampled from $\\mathcal{N}(\\mathbf{z};\\pmb{\\mu}(\\mathbf{x}),\\sigma^{2}(\\mathbf{x})\\pmb{I})$ with reparameterization trick:\n\n$$\nz\\sim q(\\mathbf{z}|\\mathbf{x})\\Leftrightarrow z=\\mu(\\mathbf{x})+\\sigma(\\mathbf{x})\\odot\\epsilon,\\epsilon\\sim\\mathcal{N}(0,I)\\tag{8}\n$$\n\nFor analyzing the outputs of those encoders, we divide them into two blocks: if the parameters of several layers are shared for different factors in a representation, the block consists of these layers is called \"sharing-parameter block\". Conversely, those layers without shared parameters for different factors form \"no-sharing-parameter block\". In vanilla VAE, the last layers in encoders are fullyconnected nets, which are no-sharing-parameter block (see Fig. 1).", "theorems": [{"local_id": "T1", "statement": "Lemma 1. For any subset S $of\\left\\{1,2,\\dots,J\\right\\}$ , we have: $$\nI(\\mathbf{z}_{i\\in S};\\mathbf{x})-\\sum_{i\\in S}I(\\mathbf{z}_{i};\\mathbf{x})=\\mathbb{E}_{q(\\mathbf{x})}[D_{\\mathrm{KL}}(q(\\mathbf{z}_{i\\in S}|\\mathbf{x})|\\big\\lVert\\prod_{i\\in S}q(\\mathbf{z}_{i}|\\mathbf{x}))\\big]-D_{\\mathrm{KL}}(q(\\mathbf{z}_{i\\in S})|\\big\\lVert\\prod_{i\\in S}q(\\mathbf{z}_{i})\\big\\rVert\\tag{9}\n$$", "proof": {"steps": [{"n": 1, "claim": "Proof: $$ I(\\mathbf{z}_{i\\in S};\\mathbf{x})-\\sum_{i\\in S}I(\\mathbf{z}_{i};\\mathbf{x})=\\mathbb{E}_{q(\\mathbf{z}_{i\\in S},\\mathbf{x})}[\\log\\frac{q(\\mathbf{z}_{i\\in S},\\mathbf{x})}{q(\\mathbf{z}_{i\\in S})q(\\mathbf{x})}]-\\sum_{i\\in S}\\mathbb{E}_{q(\\mathbf{z}_{i},\\mathbf{x})}[\\log\\frac{q(\\mathbf{z}_{i},\\mathbf{x})}{q(\\mathbf{z}_{i})q(\\mathbf{x})}] $$"}, {"n": 2, "claim": "$$ =\\mathbb{E}_{q(\\mathbf{z}_{i\\in S},\\mathbf{x})}[\\log\\frac{q(\\mathbf{z}_{i\\in S},\\mathbf{x})}{q(\\mathbf{z}_{i\\in S})q(\\mathbf{x})}-\\sum_{i\\in S}\\log\\frac{q(\\mathbf{z}_{i},\\mathbf{x})}{q(\\mathbf{z}_{i})q(\\mathbf{x})}] $$"}, {"n": 3, "claim": "$$ =\\mathbb{E}_{q(\\mathbf{z}_{i\\in S},\\mathbf{x})}[\\log\\frac{q(\\mathbf{z}_{i\\in S}|\\mathbf{x})}{q(\\mathbf{z}_{i\\in S})}-\\sum_{i\\in S}\\log\\frac{q(\\mathbf{z}_{i}|\\mathbf{x})}{q(\\mathbf{z}_{i})}] $$"}, {"n": 4, "claim": "$$ =\\mathbb{E}_{q(\\mathbf{z}_{i\\in S},\\mathbf{x})}[\\log\\frac{q(\\mathbf{z}_{i\\in S}|\\mathbf{x})\\prod_{i\\in S}q(\\mathbf{z}_{i})}{q(\\mathbf{z}_{i\\in S})\\prod_{i\\in S}q(\\mathbf{z}_{i}|\\mathbf{x})}] $$"}, {"n": 5, "claim": "$$ =\\mathbb{E}_{q(\\mathbf{z}_{i\\in S},\\mathbf{x})}[\\log\\frac{q(\\mathbf{z}_{i\\in S}|\\mathbf{x})}{\\prod_{i\\in S}q(\\mathbf{z}_{i}|\\mathbf{x})}-\\log\\frac{q(\\mathbf{z}_{i\\in S})}{\\prod_{i\\in S}q(\\mathbf{z}_{i})}] $$\n\n$$ =\\mathbb{E}_{q(\\mathbf{x})}[D_{\\mathrm{KL}}(q(\\mathbf{z}_{i\\in S}|\\mathbf{x})\\|\\prod_{i\\in S}q(\\mathbf{z}_{i}|\\mathbf{x}))]-D_{\\mathrm{KL}}(q(\\mathbf{z}_{i\\in S})\\|\\prod_{i\\in S}q(\\mathbf{z}_{i})) $$"}]}}, {"local_id": "T2", "statement": "In VAEs the encoder outputs $z=\\mu(\\mathbf{x})+\\sigma(\\mathbf{x})\\odot\\epsilon$, $\\epsilon\\sim\\mathcal{N}(0,I)$ (Eq. 8), where $j\\in\\{1,2,\\dots,J\\}$ indexes the factors of $\\mathbf{z}$. The encoder's last layers form a 'no-sharing-parameter block' (parameters not shared across factors). Claim: this block, together with the independence of the reparameterization noise, is a source of the conditional independence $q(\\mathbf{z}|\\mathbf{x})=\\prod_{j=1}^{J} q(\\mathbf{z}_j|\\mathbf{x})$ of the representation factors; removing the no-sharing-parameter block violates conditional independence.", "proof": {"steps": [{"n": 1, "claim": "Encoder samples $z=\\mu(\\mathbf{x})+\\sigma(\\mathbf{x})\\odot\\epsilon$, $\\epsilon\\sim\\mathcal{N}(0,I)$ (Eq. 8)."}, {"n": 2, "claim": "The encoder's last (fully-connected) layers constitute a no-sharing-parameter block (parameters not shared across factors); vanilla VAE, Fig. 1."}, {"n": 3, "claim": "Factors output by the no-sharing-parameter block are independent conditioned on $\\mathbf{x}$, hence the factors of $\\mu(\\mathbf{x})$ and of $\\sigma(\\mathbf{x})$ are conditionally independent."}, {"n": 4, "claim": "$q(\\mu|\\mathbf{x})=\\prod_{j=1}^{J} q(\\mu_j|\\mathbf{x})$, $q(\\sigma|\\mathbf{x})=\\prod_{j=1}^{J} q(\\sigma_j|\\mathbf{x})$, $q(\\epsilon)=\\prod_{j=1}^{J} q(\\epsilon_j)$ (Eq. 10)."}, {"n": 5, "claim": "Hence $q(\\mathbf{z}|\\mathbf{x})=\\prod_{j=1}^{J} q(\\mathbf{z}_j|\\mathbf{x})$ (Eq. 11), originating from the no-sharing-parameter block and the independent reparameterization noise."}, {"n": 6, "claim": "Removing the no-sharing-parameter block (sharing-parameter encoder) violates conditional independence and degrades disentanglement."}]}}, {"local_id": "T3", "statement": "Lemma 2. $I(\\mathbf{z}_{j};\\mathbf{x}|\\mathbf{z}_{i\\in S_{-j}})=I(\\mathbf{z}_{j};\\mathbf{x})$ for any $j$ is equivalent to the following equation:\n\n$$\nI(\\mathbf{z}_{i\\in S};\\mathbf{x})=I(\\mathbf{z}_{j};\\mathbf{x})+I(\\mathbf{z}_{i\\in S_{-j}};\\mathbf{x})\n$$\n\nwhere $S=\\{j\\}\\cup S_{-j}$ is any subset of $\\{1,2,\\ldots,J\\},j$ is any single element in $S .$", "proof": {"steps": [{"n": 1, "claim": "For further understanding, here we derive it from scratch using Bayes rule:"}, {"n": 2, "claim": "\\begin{array}{ r l }&{I(\\mathbf{z}_{j};\\mathbf{x}|\\mathbf{z}_{i\\in S_{-j}})=\\mathbb{E}_{q(\\mathbf{z}_{i\\in S},\\mathbf{x})}[\\log\\frac{q(\\mathbf{z}_{j})\\mathbf{x}|\\mathbf{z}_{i\\in S_{-j}}}{q(\\mathbf{z}_{j}|\\mathbf{z}_{i\\in S_{-j}})q(\\mathbf{x}|\\mathbf{z}_{i\\in S_{-j}})}]}\\\\ &{\\qquad=\\mathbb{E}_{q(\\mathbf{z}_{i\\in S},\\mathbf{x})}[\\log\\frac{q(\\mathbf{x}|\\mathbf{z}_{i\\in S})}{q(\\mathbf{x}|\\mathbf{z}_{i\\in S_{-j}})}]}\\\\ &{\\qquad=\\mathbb{E}_{q(\\mathbf{z}_{i\\in S},\\mathbf{x})}[\\log\\frac{q(\\mathbf{x},\\mathbf{z}_{i\\in S})q(\\mathbf{z}_{i\\in S_{-j}})}{q(z_{i\\in S})q(\\mathbf{x},\\mathbf{z}_{i\\in S_{-j}})}]}\\\\ &{\\qquad=\\mathbb{E}_{q(\\mathbf{z}_{i\\in S},\\mathbf{x})}[\\log\\frac{q(\\mathbf{x},\\mathbf{z}_{i\\in S})}{q(z_{i\\in S})q(\\mathbf{x})}-\\log\\frac{q(\\mathbf{x},\\mathbf{z}_{i\\in S_{-j}})}{q(\\mathbf{z}_{i\\in S_{-j}})q(\\mathbf{x})}]}\\\\ &{\\qquad=\\mathbb{E}_{q(\\mathbf{z}_{i\\in S},\\mathbf{x})}[\\log\\frac{q(\\mathbf{x},\\mathbf{z}_{i\\in S})}{q(z_{i\\in S})q(\\mathbf{x})}-\\log\\frac{q(\\mathbf{x},\\mathbf{z}_{i\\in S_{-j}})}{q(\\mathbf{z}_{i\\in S_{-j}})q(\\mathbf{x})}]}\\\\ &{\\qquad=I(\\mathbf{z}_{i\\in S},\\mathbf{x})-I(\\mathbf{z}_{i\\in S_{-j},\\mathbf{x}})}\\end{array}"}, {"n": 3, "claim": "Hence we have:"}, {"n": 4, "claim": "I(\\mathbf{z}_{j};\\mathbf{x}|\\mathbf{z}_{i\\in S_{-j}})=I(\\mathbf{z}_{j};\\mathbf{x})\\Leftrightarrow I(\\mathbf{z}_{i\\in S};\\mathbf{x})=I(\\mathbf{z}_{j};\\mathbf{x})+I(\\mathbf{z}_{i\\in S_{-j}};\\mathbf{x})"}]}}], "target_local_id": "T2", "topic_domain": "Deep Learning Theory", "topic_subtopic": "Representation & Expressivity", "label": "flawed", "error_step": 3, "error_class6": ["logical_gap_or_missing_case", "others", "missing_assumption"], "error_explanation": "Step 3 asserts that the factors output by the 'no-sharing-parameter block' are independent conditioned on the input, and hence that the factors of $\\mu(\\mathbf{x})$ and of $\\sigma(\\mathbf{x})$ are conditionally independent. This attribution is false. Eq. 9, $q(\\mathbf{z}|\\mathbf{x})=\\prod_j q(\\mathbf{z}_j|\\mathbf{x})$, is the defining property of the factorized (diagonal) Gaussian variational family and follows ENTIRELY from the factorized reparameterization noise $q(\\epsilon)=\\prod_j q(\\epsilon_j)$ (Eq. 7); it does not follow from, and does not require, parameters being un-shared across factors. Parameter sharing and (conditional) independence are different concepts: if one sets every mean equal, $\\mu_j=f(\\mathbf{x})$ for all $j$ (all parameters fully shared, so no no-sharing-parameter block at all), Eq. 9 still holds because the per-coordinate noise is independent. Thus the no-sharing-parameter block is neither necessary nor a source of conditional independence, and the premise that a shared-parameter encoder violates conditional independence is false. Correctly, conditional independence comes solely from the factorized noise of the chosen variational family; a valid experiment must genuinely remove the no-sharing-parameter block (e.g. sharing the last fully-connected layer's bias across factors), not merely replace FC layers with conv layers whose last-layer parameters are still un-shared across factors. Steps 5-6 only restate and apply this same invalid attribution."}
{"id": "rJma2bZCW", "context": "We follow a similar (though not identical) theoretical setup to (Mandt et al., 2017), approximating SGD with a continuous-time stochastic process, which we now outline.\n\nLet us consider a model parameterized by $\\pmb{\\theta}=\\{\\theta_{1},...,\\theta_{q}\\}$ . For $N$ training examples $\\mathbf{\\Psi}_{{\\pmb{x}}_{i},i}\\in\\mathbf{\\Psi}$ $\\{1,...,N\\}$ , the loss function, $L(\\theta)$ , and the corresponding gradient ${\\bf g}(\\pmb{\\theta})$ , are defined based on the sum over the loss values for all training examples. Stochastic gradients ${\\bf g}^{(S)}(\\pmb{\\theta})$ arise when we consider a batch B of size $S < N$ of random indices drawn uniformly from $\\{1,...,N\\}$ and form an (unbiased) estimate of loss and gradient based on the corresponding subset of training examples\n\n$$\nL^{(S)}(\\pmb{\\theta})=\\frac{1}{S}\\sum_{n\\in\\mathcal{B}}l(\\pmb{\\theta},\\pmb{x}_{n})\\ ,\\qquad\\mathbf{g}^{(S)}(\\pmb{\\theta})=\\frac{\\partial}{\\partial\\pmb{\\theta}}L^{(S)}(\\pmb{\\theta})\\ .\n$$\n\nWe consider stochastic gradient descent (SGD) with learning rate η, as defined by the update rule\n\n$$\n\\pmb{\\theta}(t+1)=\\pmb{\\theta}(t)-\\eta\\pmb{g}^{(S)}(\\pmb{\\theta})\\ .\n$$\n\nWe now make the following assumptions:\n\n(1 By the central limit theorem (CLT), we assume the noise in the stochastic gradient is Gaussian with covariance matrix $\\scriptstyle{\\frac{1}{S}}\\mathbf{C}({\\boldsymbol{\\theta}})$\n\n$$\n{\\bf g}^{(S)}(\\pmb{\\theta})={\\bf g}(\\pmb{\\theta})+\\frac{1}{\\sqrt{S}}\\Delta{\\bf g}(\\pmb{\\theta}),\\mathrm{where}\\Delta{\\bf g}(\\pmb{\\theta})\\sim N(0,{\\bf C}(\\pmb{\\theta})).\n$$\n\nWe note that the covariance is symmetric positive-semidefinite, and so can be decomposed into the product of two matrices $\\mathbf{C}(\\pmb\\theta)=\\mathbf{\\dot{B}}(\\pmb\\theta)\\mathbf{B}^{\\top}(\\pmb\\theta)$\n\n(2) We assume the discrete process of SGD can be approximated by the continuous time limit of the following stochastic differential equation (known as a Langevin equation)\n\n$$\n\\frac{d\\pmb{\\theta}}{d t}= -\\eta\\pmb{g}(\\pmb{\\theta})+\\frac{\\eta}{\\sqrt{S}}\\mathbf{B}(\\pmb{\\theta})\\mathbf{f}(t)\\tag{3}\n$$\n\nwhere $\\mathbf f(t)$ is a normalized Gaussian time-dependent stochastic term.\n\nNote that the continuous time approximation of SGD as a stochastic differential equation has been shown to hold in a weak approximation on the condition that the learning rate is small (Li et al., 2017).\n\nNote that we have not made Assumption 4 of (Mandt et al., 2017), where they assume the loss can be globally approximated by a quadratic. Instead, we allow for a general loss function, which can have many local minima.\n\nThe Langevin equation is a stochastic differential equation, and we are interested in its equilibrium distribution which gives insights into the behavior of SGD and the properties of the points it converges to. Assuming isotropic noise, the Langevin equation is well known to have a Gibbs-Boltzmann distribution as its equilibrium distribution. This equilibrium distribution can be derived by finding the stationary solution of the Fokker-Planck equation, with detailed balance, which governs the evolution of the probability density of the value of the parameters with time. The Fokker-Planck equation and its derivation is standard in the statistical physics literature. In Appendix A we give the equation in the machine learning context in Eq. (4) and for completeness of presentation we also give its derivation. In Appendix C we restate the standard proofs of the stationary distribution of a Langevin system, and provide the resulting Gibbs-Boltzmann equilbirium distribution here, using the notation of this paper:\n\nDiscussion: Here $P(\\pmb\\theta)$ defines the density over the parameter space. The above result says that if we run SGD for long enough (under the assumptions made regarding the SGD sufficiently matching the infinitesimal limit), then the probability of the parameters being in a particular state asymptotically follows the above density. Note, that $\\textstyle n\\equiv{\\frac{\\eta}{S}}$ is a measure of the noise in the system set by the choice of learning rate η and batch size S. The fact that the loss is divided by n emphasizes that the higher the noise $n,$ the less granular the loss surface appears to SGD. The gradient variance $\\mathbf{C}(\\pmb\\theta)$ on the other hand is determined by the dataset and model priors (e.g. architecture, model parameterization, batch normalization etc.). This reveals an important area of investigation, i.e., how different architectures and model parameterizations affect the gradient's covariance structure. We note that in the analysis above, the assumption of the gradient covariance $\\mathbf{C}(\\pmb\\theta)$ being fixed and isotropic in the parameter space is unrealistic. However it is a simplification that enables straightforward insights regarding the relationship of the noise, batch size and learning rate in the Gibbs-Boltzmann equilibrium. We empirically show that various predictions based on this relationship hold in practice.\n\nReturning to SGD as an optimization method, we can ask, given the probability density $P(\\pmb\\theta)$ can we derive the probability of ending at a given minimum, $\\pmb{\\theta}_{A}$ , which we will denote by lowercase $p_{A}={\\tilde{p}}_{A}C$ , where C is a normalization constant which is the same for all minima (the unnormalized probability ${\\tilde{p}}_{A}$ A is all we are interested in when estimating the relative probability of finishing in a given minimum compared to another one). This probability is derived in Appendix D, and given in the following theorem, which is the core insight from our theory.\n\nDiscussion: For this analysis, we qualitatively categorize a minima $\\theta_{A}$ by its loss $L_{A}$ (depth) and the determinant of the Hessian det $\\mathbf{H}_{A}$ (a larger determinant implies a sharper minima). The above result shows that the probability of landing in a specific minimum depends on three factors - learning rate, batch-size and covariance of the gradients. The two factors that we directly control only appear in the ratio given by the noise $n = \\eta / S$ Note that the proof of this result utilizes a Laplace Approximation in which the loss near a given minimum can be approximated using a second order Taylor series in order to evaluate an integral. We emphasize this is not the same as globally treating the loss as a quadratic.\n\nIn this appendix we derive the Fokker-Planck equation. For any stochastic differential equation, since the evolution is noisy we can't say exactly where in parameter space the parameter values will be at any given time. But we can talk about the probability $P(\\boldsymbol{\\theta},t|\\boldsymbol{\\theta}_{0},t_{0})$ that a parameter takes a certain value θ at a certain time t, given that it started at $\\theta_{0},t_{0}$ That is captured by the Fokker-Planck equation, which reads\n\n$$\n\\frac{\\partial P(\\pmb{\\theta},t)}{\\partial t}=\\nabla_{\\pmb{\\theta}}\\cdot\\left[\\eta\\pmb{g}(\\pmb{\\theta})P(\\pmb{\\theta},t)+\\frac{\\eta^{2}}{2 S}\\nabla_{\\pmb{\\theta}}\\cdot [\\mathbf{C}(\\pmb{\\theta})P(\\pmb{\\theta},t)]\\right].\\tag{4}\n$$\n\nIn this appendix we will not be interested in pure mathematical rigour, but intend the proof to add intuition for the machine learning audience.\n\nFor brevity we will sometimes write the probability just as $P(\\theta,t)$ . We will sometimes make use of tensor index notation, where a tensor is denoted by its components, (for example $\\theta_{i}$ are the components of the vector $\\pmb\\theta)_{\\backslash}$ , and we will use the index summation convention, where a repeated index is to be summed over.\n\nIn order to prove this is the equilibrium distribution, we need to solve the Fokker-Planck equation (4) with the left hand side set equal to zero (which would give a stationary distribution) and further we require for equilibrium that detailed balance holds. To do this, we begin by writing the Fokker-Planck equation in a slightly different form, making use of a probability current J, defined\n\n$$\n{\\pmb J}\\equiv\\eta{\\pmb g}({\\pmb\\theta})P({\\pmb\\theta},t)+\\frac{\\eta^{2}}{2 S}\\nabla_{{\\pmb\\theta}}\\cdot({\\bf C}({\\pmb\\theta})P({\\pmb\\theta},t))\\tag{5}\n$$\n\nin which the Fokker-Planck equation becomes\n\n$$\n\\frac{\\partial P(\\pmb{\\theta},t)}{\\partial t}=\\nabla_{\\pmb{\\theta}}\\cdot\\pmb{J}.\\tag{6}\n$$\n\nWe emphasize this is not the same as Assumption 4 of (Mandt et al., 2017), where they assume the loss can be globally approximated by a quadratic. In contrast, we allow for a general loss, with multiple minima, and locally to each minima approximate the loss by its second-order Taylor expansion in order to evaluate the integral in the Laplace method.\n\nThe distribution $P_{A}(\\pmb\\theta)$ is a probability density, while we are interested in the discrete set of probabilities of ending at a given minimum, which we will denote by lowercase $p_{A}$ . To calculate this discrete set of probabilities, for each minimum we need to integrate this stationary distribution over an interval containing the minimum.", "theorems": [{"local_id": "T1", "statement": "For any stochastic differential equation, since the evolution is noisy we can't say exactly where in parameter space the parameter values will be at any given time. But we can talk about the probability $P(\\boldsymbol{\\theta},t|\\boldsymbol{\\theta}_{0},t_{0})$ that a parameter takes a certain value θ at a certain time t, given that it started at $\\theta_{0},t_{0}$ That is captured by the Fokker-Planck equation, which reads\n\n$$\n\\frac{\\partial P(\\pmb{\\theta},t)}{\\partial t}=\\nabla_{\\pmb{\\theta}}\\cdot\\left[\\eta\\pmb{g}(\\pmb{\\theta})P(\\pmb{\\theta},t)+\\frac{\\eta^{2}}{2 S}\\nabla_{\\pmb{\\theta}}\\cdot [\\mathbf{C}(\\pmb{\\theta})P(\\pmb{\\theta},t)]\\right].\\tag{4}\n$$", "proof": {"steps": [{"n": 1, "claim": "We start with the stochastic differential equation\n\n$$\n\\frac{d\\pmb{\\theta}}{d t}= -\\eta\\pmb{g}(\\pmb{\\theta})+\\frac{\\eta}{\\sqrt{S}}\\mathbf{B}(\\pmb{\\theta})\\mathbf{f}(t).\\tag{7}\n$$"}, {"n": 2, "claim": "A formal expression for the probability for a given noise function f is given by $P(\\pmb\\theta,t)=\\delta(\\pmb\\theta-\\pmb\\theta_{\\mathbf f})$ but since we don't know the noise, we instead average over all possible noise functions to get\n\n$$\nP(\\pmb\\theta,t)=\\mathbb{E}[\\delta(\\pmb\\theta-\\pmb\\theta_{\\mathbf f})].\\tag{8}\n$$"}, {"n": 3, "claim": "We now consider a small step in time $\\delta t,$ working at first order in $\\delta t .$ and ask how far the parameters move, denoted δθ, which is given by integrating (7)\n\n$$\n\\delta\\pmb{\\theta}= -\\eta\\pmb{g}\\delta t+\\frac{\\eta}{\\sqrt{S}}\\mathbf{B}\\int_{t}^{t+\\delta t}\\mathbf{f}(t^{\\prime})d t^{\\prime}\\tag{9}\n$$"}, {"n": 4, "claim": "We now look at expectations of δθ. Using the fact that the noise is normalized Gaussian, $\\mathbb{E}({\\bf f}(t))=0$ ,we get (switching to index notation for clarity)\n\n$$\n\\mathbb{E}(\\delta\\theta_{i})= -\\eta g_{i}\\delta t\\tag{10}\n$$\n\nand using that the noise is normalized Gaussian again, we have that $\\mathbb{E}({\\bf f}(t){\\bf f}(t^{\\prime}))={\\bf I}\\delta(t-t^{\\prime})$ leading to\n\n$$\n\\mathbb{E}(\\delta\\theta_{i}\\delta\\theta_{j})=\\frac{\\eta^{2}}{S}C_{i j}\\delta t+\\mathcal{O}(\\delta t^{2}).\\tag{11}\n$$"}, {"n": 5, "claim": "If at time t we are at position $\\pmb{\\theta}^{\\prime}$ and end up at position $\\pmb{\\theta}=\\pmb{\\theta}^{\\prime}+\\delta\\pmb{\\theta}$ at time $t + \\delta t$ , then we can take ) with $\\theta_{\\mathbf{f}}=\\theta^{\\prime}$ and Taylor expand it in $\\delta\\theta_{i}$\n\n$$\nP(\\pmb\\theta,t+\\delta t|\\pmb\\theta^{\\prime},t)=\\left(1+\\mathbb{E}(\\delta\\pmb\\theta_{i})\\frac{\\partial}{\\partial\\theta_{i}^{\\prime}}+\\frac{1}{2}\\mathbb{E}(\\delta\\pmb\\theta_{i}\\delta\\pmb\\theta_{j})\\frac{\\partial^{2}}{\\partial\\theta_{i}^{\\prime}\\partial\\theta_{j}^{\\prime}}+\\mathcal{O}(\\delta t^{2})\\right)\\delta(\\pmb\\theta-\\pmb\\theta^{\\prime}).\\tag{12}\n$$"}, {"n": 6, "claim": "We will use the following identity, called the Chapman-Kolmogorov equation, which reads\n\n$$\nP(\\theta,t+\\delta t|\\theta_{0},t_{0})=\\int_{-\\infty}^{+\\infty}d\\theta^{\\prime}P(\\theta,t+\\delta t|\\theta^{\\prime},t^{\\prime})P(\\theta^{\\prime},t^{\\prime}|\\theta_{0},t_{0})\\tag{13}\n$$\n\nfor any $t^{\\prime},$ such that $t_{0}\\leq t^{\\prime}\\leq t+\\delta t$ This identity is an integral version of the chain rule of probability, stating there are multiple paths of getting from an initial position $\\pmb{\\theta}_{0}$ to $\\pmb \\theta$ at time $t + \\delta t$ and one should sum all of these paths."}, {"n": 7, "claim": "We will now substitute (12) into the first term on the right hand side of (13) with $t^{\\prime}$ set to be $t,$ and apply integration by parts (assuming vanishing boundary conditions at infinity) to move the derivatives off of the delta functions and onto the other terms. We end up with\n\n$$\n\\begin{array}{ r l }&{P(\\pmb{\\theta},t+\\delta t|\\pmb{\\theta}^{\\prime},t)=P(\\pmb{\\theta},t|\\pmb{\\theta}_{0},t_{0})-\\displaystyle\\frac{\\partial}{\\partial\\theta_{i}}\\left(\\mathbb{E}(\\delta\\theta_{i})P(\\pmb{\\theta},t|\\pmb{\\theta}_{0},t_{0})\\right)}\\\\ &{\\qquad+\\displaystyle\\frac{1}{2}\\frac{\\partial^{2}}{\\partial\\theta_{i}^{\\prime}\\partial\\theta_{j}^{\\prime}}\\left(\\mathbb{E}(\\delta\\theta_{i}\\delta\\theta_{j})P(\\pmb{\\theta},t|\\pmb{\\theta}_{0},t_{0})\\right).}\\end{array}\\tag{14}\n$$"}, {"n": 8, "claim": "We can then take the first term on the right hand side of (14) to the other side, insert (10) and (11), divide by δt and take the limit $\\delta t  0$ ,getting a partial derivative with respect to time on the left hand side, leading directly to the Fokker-Planck equation quoted in the text (4) (where we have reverted back to vector notation from index notation for conciseness)\n\n$$\n\\frac{\\partial P(\\pmb{\\theta},t)}{\\partial t}=\\nabla_{\\pmb{\\theta}}\\cdot\\left[\\eta\\pmb{g}(\\pmb{\\theta})P(\\pmb{\\theta},t)+\\frac{\\eta^{2}}{2 S}\\nabla_{\\pmb{\\theta}}\\cdot [\\mathbf{C}(\\pmb{\\theta})P(\\pmb{\\theta},t)]\\right].\\tag{15}\n$$"}]}}, {"local_id": "T2", "statement": "Consider SGD with learning rate $\\eta$ and batch size $S<N$, whose update $\\pmb\\theta(t+1)=\\pmb\\theta(t)-\\eta\\mathbf{g}^{(S)}(\\pmb\\theta)$ is approximated by the Langevin SDE $\\frac{d\\pmb\\theta}{dt}=-\\eta\\mathbf{g}(\\pmb\\theta)+\\frac{\\eta}{\\sqrt{S}}\\mathbf{B}(\\pmb\\theta)\\mathbf{f}(t)$ (Eq. 3), where the stochastic gradient noise is Gaussian with covariance $\\frac{1}{S}\\mathbf{C}(\\pmb\\theta)$, $\\mathbf{C}(\\pmb\\theta)=\\mathbf{B}(\\pmb\\theta)\\mathbf{B}^\\top(\\pmb\\theta)$, and $\\mathbf{f}(t)$ is a normalized Gaussian time-dependent term. Assume that the gradient covariance is isotropic, i.e. $\\mathbf{C}(\\pmb\\theta)=\\sigma^2(\\pmb\\theta)\\mathbf{I}$. Then the equilibrium distribution of the SDE (Eq. 3) is $P(\\pmb\\theta)=P_0\\exp\\left(-\\frac{2L(\\pmb\\theta)}{n\\sigma^2}\\right)$, where $n\\equiv\\eta/S$ and $P_0$ is a normalization constant (well defined for loss functions with $L_2$ regularization).", "proof": {"steps": [{"n": 1, "claim": "The current is $\\pmb J\\equiv\\eta\\mathbf{g}(\\pmb\\theta)P(\\pmb\\theta,t)+\\frac{\\eta^2}{2S}\\nabla_{\\pmb\\theta}\\cdot(\\mathbf{C}(\\pmb\\theta)P(\\pmb\\theta,t))$ (Eq. 5), so $\\partial_tP=\\nabla_{\\pmb\\theta}\\cdot\\pmb J$ (Eq. 6)."}, {"n": 2, "claim": "With $\\mathbf{C}(\\pmb\\theta)=\\sigma^2(\\pmb\\theta)\\mathbf{I}$, $\\mathbf{C}(\\pmb\\theta)P=\\sigma^2(\\pmb\\theta)P$."}, {"n": 3, "claim": "So $\\nabla_{\\pmb\\theta}\\cdot(\\sigma^2(\\pmb\\theta)P)=\\sigma^2\\nabla_{\\pmb\\theta}P$, giving $\\pmb J=\\eta\\mathbf{g}(\\pmb\\theta)P(\\pmb\\theta,t)+\\frac{\\eta^2\\sigma^2}{2S}\\nabla_{\\pmb\\theta}P(\\pmb\\theta,t)$ (Eq. 16)."}, {"n": 4, "claim": "Equilibrium requires detailed balance, $\\pmb J=0$: $\\eta\\mathbf{g}(\\pmb\\theta)P=-\\frac{\\eta^2\\sigma^2}{2S}\\nabla_{\\pmb\\theta}P$."}, {"n": 5, "claim": "$\\nabla_{\\pmb\\theta}\\log P=-\\frac{2S}{\\eta\\sigma^2}\\mathbf{g}(\\pmb\\theta)=-\\frac{2S}{\\eta\\sigma^2}\\nabla_{\\pmb\\theta}L(\\pmb\\theta)$."}, {"n": 6, "claim": "Integrating, $P(\\pmb\\theta)=P_0\\exp\\left(-\\frac{2S}{\\eta\\sigma^2}L(\\pmb\\theta)\\right)=P_0\\exp\\left(-\\frac{2L(\\pmb\\theta)}{n\\sigma^2}\\right)$ (Eq. 17)."}]}}, {"local_id": "T3", "statement": "Theorem 3 (Probability of ending in region near minima $\\pmb{\\theta}_{A})$ . Assume the loss has a series of separated local minima. Consider one such minima, with Hessian $\\mathbf{H}_{A}$ and loss $L_{A}$ at a minimum $\\theta_{A}$ . Then the unnormalized probability of ending in a region near minima $\\pmb{\\theta}_{A}$ is\n\n$$\n\\tilde{p}_{A}=\\frac{1}{\\sqrt{\\operatorname*{det}\\mathbf{H}_{A}}}\\exp\\left(-\\frac{2 L_{A}}{n\\sigma^{2}}\\right)\\tag{18}\n$$\n\nwhere $\\begin{array}{ r }{n=\\frac{\\eta}{S}}\\end{array}$ is the noise used in the SGD process to reach $\\theta_{A}$", "proof": {"steps": [{"n": 1, "claim": "We work locally near $\\theta_{A}$ and take the following approximation of the loss function, since it is near a minimum,\n\n$$\nL(\\pmb\\theta)\\Big|_{R_{A}}\\approx L_{A}+\\frac{1}{2}(\\pmb\\theta-\\pmb\\theta_{A})^{\\top}\\mathbf{H}_{A}(\\pmb\\theta-\\pmb\\theta_{A}).\\tag{19}\n$$"}, {"n": 2, "claim": "Integrating (20) in some region $R_{A}$ around $\\theta_{A}$ and using (19)\n\n$$\np_{A}\\approx P_{0}\\int_{R_{A}}\\exp{\\left(-\\frac{2 S}{\\eta\\sigma^{2}}L(\\pmb\\theta)\\Big|_{R_{A}}\\right)}\\tag{21}\n$$\n\n$$\n\\approx P_{0}\\int_{R_{A}}\\exp{\\left(-\\frac{2 S}{\\eta\\sigma^{2}}\\left[L_{A}+\\frac{1}{2}(\\pmb{\\theta}-\\pmb{\\theta}_{A})^{\\top}\\mathbf{H}_{A}(\\pmb{\\theta}-\\pmb{\\theta}_{A})\\right]\\right)}\\tag{22}\n$$"}, {"n": 3, "claim": "$$\n\\approx\\tilde{P}_{0}\\exp\\left(-\\frac{2 S L_{A}}{\\eta\\sigma^{2}}\\right)\\sqrt{\\frac{1}{\\operatorname*{det}\\mathbf{H}_{A}}}\\tag{23}\n$$\n\nwhere in the last line we assume the region is large enough that an approximation to the full Gaussian integral can be used (i.e. that the tails don’t contribute, which is fine as long as the region is sufficiently larger than $\\operatorname*{det}\\mathbf{H}_{A}$ ) – note that the region can’t be too large, otherwise we would invalidate our local assumption. The picture is that the minima are sufficiently far apart that the region can be taken sufficiently large for this approximation to be valid. Note that $\\tilde{P}_{0}$ is different to $P_{0}$ and includes the normalization factors from performing the Gaussian integral, which are the same for all minima."}, {"n": 4, "claim": "Since we are interested in relative probabilities between different minima, so we can consider the unnormalized probability, dropping factors that are common amongst all minima we get\n\n$$\n\\tilde{p}_{A}=\\exp\\left(-\\frac{2 S L_{A}}{\\eta\\sigma^{2}}\\right)\\sqrt{\\frac{1}{\\operatorname*{det}\\mathbf{H}_{A}}}\\tag{24}\n$$"}]}}], "target_local_id": "T2", "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "label": "flawed", "error_step": 3, "error_class6": ["missing_assumption", "logical_gap_or_missing_case", "computation_or_constant"], "error_explanation": "The step pulls $\\sigma^2$ out of the divergence, writing $\\nabla_{\\pmb\\theta}\\cdot(\\sigma^2(\\pmb\\theta)P)=\\sigma^2\\nabla_{\\pmb\\theta}P$. But when $\\sigma=\\sigma(\\pmb\\theta)$ depends on $\\pmb\\theta$, the product rule gives $\\nabla_{\\pmb\\theta}\\cdot(\\sigma^2(\\pmb\\theta)P)=\\sigma^2(\\pmb\\theta)\\nabla_{\\pmb\\theta}P+P\\,\\nabla_{\\pmb\\theta}\\sigma^2(\\pmb\\theta)$. Eq. 20 silently drops the second term $P\\,\\nabla_{\\pmb\\theta}\\sigma^2(\\pmb\\theta)$, so the equality is unjustified and can hold only approximately (even if $\\nabla\\sigma$ is bounded). The clean Eq. 20 — and hence the exact $\\pmb J=0$ Gibbs solution — is valid only under the missing assumption that $\\sigma^2$ is constant (independent of $\\pmb\\theta$), which the theorem fails to state."}
{"id": "uHv20yi8saL", "context": "Dec-MDPs. We consider a fully cooperative multi-agent task in which a team of cooperative agents choose sequential actions in a stochastic environment. It can be modeled as a decentralized Markov decision process (Dec-MDP), defined by a tuple $\\{\\mathcal{N},\\mathcal{S},\\mathcal{A},P,r,\\rho_{0},\\gamma\\}$ , where $\\mathcal{N}\\triangleq\\{1,\\dots,N\\}$ denotes the set of N agents and s $\\in\\mathcal{S}\\triangleq\\mathcal{S}^{1}\\times\\mathcal{S}^{2}\\times\\ldots\\times\\mathcal{S}^{N}$ describes the joint state of the environment. The initial state $s_{0}\\sim\\rho_{0}$ is drawn from distribution $\\rho_{0}$ , and at each time step t, all agents $k \\in \\mathcal N$ choose simultaneously one action $a_{t}^{k}\\in\\mathcal{A}^{k}$ , yielding a joint action $\\begin{array}{ r }{\\pmb{a}_{t}\\triangleq a_{t}^{1}\\times a_{t}^{2}\\times ...\\times a_{t}^{N}\\in\\mathcal{A}\\triangleq}\\end{array}$ $\\mathcal{A}^{1}\\times\\mathcal{A}^{2}\\times\\ldots\\times\\mathcal{A}^{N}$ . After executing the joint action $\\mathbf{}\\mathbf{a}_{t}$ in state $\\mathbf{}_{s_{t},}$ the next state $\\pmb{s}_{t+1}\\sim P(\\pmb{s}_{t},\\pmb{a}_{t})$ is drawn from transition kernel P and a collaborative reward $r_{t}=r(s_{t},{\\boldsymbol{a}}_{t})$ is returned. In a Dec-MDP, each agent $k \\in \\mathcal N$ has a local state $s_{t}^{k}\\in S^{k}$ , and chooses its actions with a decentralized policy $a_{t}^{k}\\sim\\pi_{k}(\\cdot|s_{t}^{k})$ based only on its local state. The collaborating team of agents aims to learn a joint policy, $\\begin{array}{ r }{\\mathbf{\\sigma}:\\left(\\mathbf{a}_{t}|\\mathbf{s}_{t}\\right)\\triangleq\\prod_{k=1}^{N}\\pi_{k}\\big(a_{t}^{k}|s_{t}^{k}\\big)}\\end{array}$ , that maximizes their expected discounted return, $\\begin{array}{ r }{\\eta(\\pmb{\\pi})\\triangleq\\mathbb{E}_{(\\pmb{s}_{t},\\pmb{a}_{t})}[\\sum_{t=0}^{\\infty}\\gamma^{t}r_{t}]}\\end{array}$ , where $\\gamma\\in [0,1)$ is a discount factor.\n\nLet the advantage function be $A_{\\pi}(s,a)=Q_{\\pi}(s,a)-V_{\\pi}(s)$ ; the following useful identity expresses the expected return of another policy  in terms of the advantage over π (Kakade & Langford, 2002): $\\begin{array}{ r }{\\eta(\\tilde{\\pi})=\\eta(\\pi)+\\sum_{s}\\rho_{\\tilde{\\pi}}(s)\\sum_{a}\\tilde{\\pi}(a|s)A_{\\pi}(s,a)}\\end{array}$ , where $\\rho_{\\tilde{\\pi}}(s)$ is the state distribution induced by π.\n\nDefine $\\begin{array}{ r }{D_{\\mathrm{TV}}^{\\operatorname*{max}}(\\pi,\\tilde{\\pi})\\triangleq\\operatorname*{max}_{s}D_{\\mathrm{TV}}\\big(\\pi(\\cdot|s),\\tilde{\\pi}(\\cdot|s)\\big)}\\end{array}$ , where $D_{\\mathrm{TV}}$ is the total variation (TV) divergence.\n\nTo analyze the non-stationarity in decentralized policy optimization, we first consider the Markov chain for the local state $s^{k}$ induced by the underlying MDP for agent k. When all agents’ policies are updated from $\\pi_{1},\\dots,\\pi_{N}$ to $\\tilde{\\pi}_{1},\\dots,\\tilde{\\pi}_{N}$ , the state transition distribution of this Markov chain also shifts. We denote such transition shift from $s^{k}$ to $\\tilde{s}^{k}$ for agent k as\n\n$$\n\\Delta_{\\pi_{1},\\dots,\\pi_{N}}^{\\tilde{\\pi}_{1},\\dots,\\tilde{\\pi}_{N}}(\\tilde{s}^{k}|s^{k})\\triangleq\\sum_{a^{k}}\\big [p_{\\tilde{\\pi}_{1},\\dots,\\tilde{\\pi}_{N}}(\\tilde{s}^{k}|s^{k},a^{k})\\tilde{\\pi}_{k}(a^{k}|s^{k})-p_{\\pi_{1},\\dots,\\pi_{N}}(\\tilde{s}^{k}|s^{k},a^{k})\\pi_{k}(a^{k}|s^{k})\\big],\n$$\n\nwhere $p_{\\pi_{1},\\dots,\\pi_{N}}$ and $p_{\\tilde{\\pi}_{1},\\dots,\\tilde{\\pi}_{N}}$ refer to the transition dynamics before and update π is updated.\n\nDefine the surrogate objective for decentralized policy $\\pi_{k}$ as\n\n$$\nL_{\\pi_{1},\\pi_{2},...,\\pi_{N}}^{(k)}(\\tilde{\\pi}_{k^{\\prime}})=\\sum_{s^{k}}\\rho_{\\pi_{1},\\pi_{2},...,\\pi_{N}}(s^{k})\\sum_{a^{k}}\\tilde{\\pi}_{k^{\\prime}}(a^{k}|s^{k})A_{\\pi_{k^{\\prime}}}(s^{k},a^{k}),\n$$\n\nand the expected return of decentralized policy $\\pi_{k}$ as\n\n$$\n\\eta_{\\pi_{1},\\ldots,\\pi_{N}}(\\pi_{k})=\\mathbb{E}_{(s^{k},a^{k})\\sim\\rho_{\\pi_{1},\\ldots,\\pi_{N}}(s^{k},a^{k})}\\left[r_{k}(s^{k},a^{k})\\right].\n$$\n\nIn practice, $r_{k}(s^{k},a^{k})$ is usually unknown. As the state transition shift decomposes into the sum of transition shifts caused by each decentralized policy, we can bound this state transition shift with a centralized trust region constraint as in equation 1.\n\nProposition 1. In a Dec-MDP, $\\begin{array}{ r }{D_{\\mathrm{TV}}^{\\operatorname*{max}}(\\pi,\\tilde{\\pi})\\leq\\sum_{k=1}^{N}D_{\\mathrm{TV}}^{\\operatorname*{max}}(\\pi_{k},\\tilde{\\pi}_{k})}\\end{array}$\n\nThis proposition is a direct result of the fact that the joint policy in a Dec-MDP factors as a product of decentralized policies, i.e., $\\begin{array}{ r }{\\pi=\\prod_{k=1}^{N}\\pi_{k}}\\end{array}$\n\nProposition 2. $\\begin{array}{ r }{D_{\\mathrm{TV}}^{\\operatorname*{max}}(\\pi_{k},\\tilde{\\pi}_{k})=\\operatorname*{max}_{s\\in\\mathcal{S}}\\sum_{\\tilde{\\pi}_{k}(a^{k}\\mid s^{k})\\geq\\pi_{k}(a^{k}\\mid s^{k})}\\big [\\tilde{\\pi}_{k}(a^{k}\\vert s^{k})-\\pi_{k}(a^{k}\\vert s^{k})\\big].}\\end{array}$\n\nThis useful identity follows from a property of $\\begin{array}{ r }{D_{\\mathrm{TV}}\\colon D_{\\mathrm{TV}}(\\mu(x),\\nu(x))=\\sum_{\\mu(x)>\\nu(x)}[\\mu(x)-\\nu(x)]}\\end{array}$ where $\\mu$ and v are two distributions. This proposition indicates that a decentralized trust region is also defined by the sum of probability differences over a special subset. We use this to upper bound the trust region in the following analysis.\n\nAssumption 2. Assume the advantage function $A_{\\pi_{k}}(s_{k},a_{k})$ converges to a fixed point for ∀k.", "theorems": [{"local_id": "T1", "statement": "Theorem 2. If independent ratios $\\lambda_k \\triangleq \\frac{\\tilde\\pi_k(a^k|s^k)}{\\pi_k(a^k|s^k)}$ are within the range $[1-\\epsilon_k, 1+\\epsilon_k]$ for $\\forall k \\in \\mathcal{N}$, then the following bound holds: $D_{\\mathrm{TV}}^{\\max}(\\pi_k(\\cdot|s), \\tilde\\pi_k(\\cdot|s)) < \\epsilon_k;\\ D_{\\mathrm{TV}}^{\\max}(\\pi(\\cdot|s), \\tilde\\pi(\\cdot|s)) < \\sum_{k=1}^N \\epsilon_k.$", "proof": {"steps": [{"n": 1, "claim": "$D_{\\mathrm{TV}}^{\\max}(\\pi_k,\\tilde\\pi_k)=\\max_{s^k}\\sum_{a^k:\\tilde\\pi_k(a^k|s^k)>\\pi_k(a^k|s^k)}[\\tilde\\pi_k(a^k|s^k)-\\pi_k(a^k|s^k)]$."}, {"n": 2, "claim": "$D_{\\mathrm{TV}}^{\\max}(\\pi_k,\\tilde\\pi_k)\\le \\epsilon_k\\,\\max_{s^k}\\sum_{a^k:\\tilde\\pi_k>\\pi_k}\\pi_k(a^k|s^k)$."}, {"n": 3, "claim": "$\\sum_{a^k:\\tilde\\pi_k>\\pi_k}\\pi_k(a^k|s^k)<1$ whenever $\\tilde\\pi_k\\ne\\pi_k$, hence $D_{\\mathrm{TV}}^{\\max}(\\pi_k,\\tilde\\pi_k)<\\epsilon_k$."}, {"n": 4, "claim": "The complementary representation $D_{\\mathrm{TV}}^{\\max}(\\pi_k,\\tilde\\pi_k)=\\max_{s^k}\\sum_{a^k:\\tilde\\pi_k<\\pi_k}[\\pi_k-\\tilde\\pi_k]$ with the floor $\\lambda_k\\ge 1-\\epsilon_k$ yields the identical bound $<\\epsilon_k$."}, {"n": 5, "claim": "$D_{\\mathrm{TV}}^{\\max}(\\pi,\\tilde\\pi)\\le\\sum_{k=1}^N D_{\\mathrm{TV}}^{\\max}(\\pi_k,\\tilde\\pi_k)<\\sum_{k=1}^N\\epsilon_k$."}]}}, {"local_id": "T2", "statement": "Consider a Dec-MDP $\\{\\mathcal{N},\\mathcal{S},\\mathcal{A},P,r,\\rho_0,\\gamma\\}$ with $N$ agents, joint policy $\\pi(a|s)=\\prod_{k=1}^N\\pi_k(a^k|s^k)$, centralized advantage $A_\\pi(s,a)=Q_\\pi(s,a)-V_\\pi(s)$, and $\\gamma\\in[0,1)$. Let the surrogate objective for decentralized policy $\\pi_k$ be $L_{\\pi_1,\\dots,\\pi_N}^{(k)}(\\tilde\\pi_{k'})=\\sum_{s^k}\\rho_{\\pi_1,\\dots,\\pi_N}(s^k)\\sum_{a^k}\\tilde\\pi_{k'}(a^k|s^k)A_{\\pi_{k'}}(s^k,a^k)$ and let $\\eta_{\\pi_1,\\dots,\\pi_N}(\\pi_k)=\\mathbb{E}_{(s^k,a^k)\\sim\\rho_{\\pi_1,\\dots,\\pi_N}}[r_k(s^k,a^k)]$ be the expected return of $\\pi_k$. Let $\\alpha=D_{\\mathrm{TV}}^{\\max}(\\pi,\\tilde\\pi)\\triangleq\\max_s D_{\\mathrm{TV}}(\\pi(\\cdot|s),\\tilde\\pi(\\cdot|s))$ and $\\epsilon=\\max_{k\\in\\mathcal{N}}\\max_{s^k,a^k}|A_{\\pi_k}(s^k,a^k)|$. Then the following bound holds for all $k$: $\\eta_{\\tilde\\pi_1,\\dots,\\tilde\\pi_N}(\\tilde\\pi_k)\\geq \\eta_{\\pi_1,\\dots,\\pi_N}(\\pi_k)+\\sum_{k'=1}^N L_{\\pi_1,\\dots,\\pi_N}^{(k)}(\\tilde\\pi_{k'})-\\frac{4\\epsilon\\gamma}{(1-\\gamma)^2}\\alpha^2$.", "proof": {"steps": [{"n": 1, "claim": "Define $G^{s_i}=(1+\\gamma P^{s_i}_{\\pi_1,\\dots,\\pi_N}+(\\gamma P^{s_i}_{\\pi_1,\\dots,\\pi_N})^2+\\dots)=(1-\\gamma P^{s_i}_{\\pi_1,\\dots,\\pi_N})^{-1}$ and $\\tilde G^{s_i}=(1-\\gamma P^{s_i}_{\\tilde\\pi_1,\\dots,\\tilde\\pi_N})^{-1}$, with $\\rho$ a density (vector) and $r$ a reward (dual vector), so that $\\eta(\\pi)=rG\\rho_0$ and $\\eta(\\tilde\\pi)=r\\tilde G\\rho_0$; let $\\Delta(s_i)=P^{s_i}_{\\tilde\\pi_1,\\dots,\\tilde\\pi_N}-P^{s_i}_{\\pi_1,\\dots,\\pi_N}$."}, {"n": 2, "claim": "$\\tilde G^{s_i}-G^{s_i}=\\gamma G^{s_i}\\Delta(s_i)G^{s_i}+\\gamma^2 G^{s_i}\\Delta(s_i)G^{s_i}\\Delta(s_i)\\tilde G^{s_i}$."}, {"n": 3, "claim": "$\\eta_{\\tilde\\pi_1,\\dots,\\tilde\\pi_N}(\\tilde\\pi_i)-\\eta_{\\pi_1,\\dots,\\pi_N}(\\pi_i)=r(\\tilde G^{s_i}-G^{s_i})\\rho_0=\\gamma rG^{s_i}\\Delta(s_i)G^{s_i}\\rho_0+\\gamma^2 rG^{s_i}\\Delta(s_i)G^{s_i}\\Delta(s_i)\\tilde G^{s_i}\\rho_0$."}, {"n": 4, "claim": "The leading term is $\\gamma rG^{s_i}\\Delta(s_i)G^{s_i}\\rho_0=\\sum_{s_i}\\rho(s_i)\\sum_{s_i'}(p_{\\tilde\\pi}(s_i'|s_i)-p_\\pi(s_i'|s_i))\\gamma v(s_i')$, and by Proposition 3 the transition shift $\\Delta$ decomposes into $N$ per-policy terms $\\Delta_{\\tilde\\pi_1,\\dots,\\tilde\\pi_{j-1},\\pi_j,\\dots,\\pi_N}^{\\tilde\\pi_1,\\dots,\\tilde\\pi_{j-1},\\pi_j,\\dots,\\pi_N}(s_i'|s_i)$ (eq 2)."}, {"n": 5, "claim": "(eq 3) One summation term equals $\\sum_{s_i}\\rho(s_i)\\sum_{s_i'}\\sum_{a_i}(p_{\\tilde\\pi_1,\\dots,\\tilde\\pi_{j-1},\\pi_j,\\dots,\\pi_N}(s_i'|s_i,a_i)\\tilde\\pi_j(a_i|s_i)-p_{\\tilde\\pi_1,\\dots,\\tilde\\pi_{j-1},\\pi_j,\\dots,\\pi_N}(s_i'|s_i,a_i)\\pi_j(a_i|s_i))\\gamma v(s_i')$."}, {"n": 6, "claim": "(eq 4) $=\\sum_{s_i}\\rho(s_i)\\sum_{a_i}(\\tilde\\pi_j(a_i|s_i)-\\pi_j(a_i|s_i))\\sum_{s_i'}p_{\\tilde\\pi_1,\\dots,\\tilde\\pi_{j-1},\\pi_j,\\dots,\\pi_N}(s_i'|s_i,a_i)\\gamma v(s_i')$."}, {"n": 7, "claim": "(eq 5) $=\\sum_{s_i}\\rho(s_i)\\sum_{a_i}(\\tilde\\pi_j(a_i|s_i)-\\pi_j(a_i|s_i))[r(s_i)+\\sum_{s_i'}p_{\\tilde\\pi_1,\\dots,\\tilde\\pi_{j-1},\\pi_j,\\dots,\\pi_N}(s_i'|s_i,a_i)\\gamma v(s_i')-v_{\\tilde\\pi_1,\\dots,\\tilde\\pi_{j-1},\\ \\pi_j,\\dots,\\pi_N}(s_i)]$."}, {"n": 8, "claim": "(eq 6) $=\\sum_{s_i}\\rho(s_i)\\sum_{a_i}(\\tilde\\pi_j(a_i|s_i)-\\pi_j(a_i|s_i))A_{\\pi_j}(s_i,a_i)$, using the definition $A_{\\pi_j}(s_i,a_i)\\triangleq r(s_i)+\\sum_{s_i'}p_{\\tilde\\pi_1,\\dots,\\tilde\\pi_{j-1},\\pi_j,\\dots,\\pi_N}(s_i'|s_i,a_i)\\gamma v(s_i')-v_{\\tilde\\pi_1,\\dots,\\tilde\\pi_{j-1},\\ \\pi_j,\\dots,\\pi_N}(s_i)$ with baseline $v_{\\tilde\\pi_1,\\dots,\\tilde\\pi_{j-1},\\ \\pi_j,\\dots,\\pi_N}(s_i)\\triangleq r(s_i)+\\gamma\\sum_{s_i'}p(s_i'|s_i,a_i)\\sum_{a_i}\\pi_j(a_i|s_i)v(s_i')$."}, {"n": 9, "claim": "(eq 7) $=L_{\\pi_1,\\pi_2,\\dots,\\pi_N}^{(i)}(\\tilde\\pi_j)-L_{\\pi_1,\\pi_2,\\dots,\\pi_N}^{(i)}(\\pi_j)$."}, {"n": 10, "claim": "Summing over the $N$ per-policy terms and using $L_{\\pi_1,\\dots,\\pi_N}^{(i)}(\\pi_j)=0$ for all $j$ gives $\\gamma rG^{s_i}\\Delta(s_i)G^{s_i}\\rho_0=L^{(i)}(\\tilde\\pi_1)+L^{(i)}(\\tilde\\pi_2)+\\dots+L^{(i)}(\\tilde\\pi_N)=\\sum_{j}L_{\\pi_1,\\dots,\\pi_N}^{(i)}(\\tilde\\pi_j)$."}, {"n": 11, "claim": "The second-order term is bounded: $\\gamma^2|rG^{s_i}\\Delta(s_i)G^{s_i}\\Delta(s_i)\\tilde G^{s_i}\\rho_0|\\leq\\gamma\\|rG^{s_i}\\Delta(s_i)\\|_\\infty\\|G^{s_i}\\Delta(s_i)\\tilde G^{s_i}\\rho_0\\|_1\\leq\\frac{4\\epsilon\\gamma}{(1-\\gamma)^2}\\alpha^2$."}, {"n": 12, "claim": "$\\eta_{\\tilde\\pi_1,\\dots,\\tilde\\pi_N}(\\tilde\\pi_k)\\geq \\eta_{\\pi_1,\\dots,\\pi_N}(\\pi_k)+\\sum_{k'=1}^N L_{\\pi_1,\\dots,\\pi_N}^{(k)}(\\tilde\\pi_{k'})-\\frac{4\\epsilon\\gamma}{(1-\\gamma)^2}\\alpha^2$ for all $k$."}]}}, {"local_id": "T3", "statement": "Proposition 3. In a Dec-MDP, the transition shift $\\Delta_{\\pi_{1},\\dots,\\pi_{N}}^{\\tilde{\\pi}_{1},\\dots,\\tilde{\\pi}_{N}}(\\tilde{s}^{k}|s^{k})$ decomposes as follows:\n\n$$\n\\Delta_{\\pi_{1},\\dots,\\pi_{N}}^{\\tilde{\\pi}_{1},\\dots,\\tilde{\\pi}_{N}}(\\tilde{s}^{k}|s^{k})=\\Delta_{\\pi_{1},\\pi_{2},\\dots,\\pi_{N}}^{\\tilde{\\pi}_{1},\\pi_{2},\\dots,\\pi_{N}}(\\tilde{s}^{k}|s^{k})+\\Delta_{\\tilde{\\pi}_{1},\\pi_{2},\\pi_{3},\\dots,\\pi_{N}}^{\\tilde{\\pi}_{1},\\tilde{\\pi}_{2},\\pi_{3},\\dots,\\pi_{N}}(\\tilde{s}^{k}|s^{k})+\\ldots+\\Delta_{\\tilde{\\pi}_{1},\\dots,\\tilde{\\pi}_{N-1},\\pi_{N}}^{\\tilde{\\pi}_{1},\\dots,\\tilde{\\pi}_{N-1},\\tilde{\\pi}_{N}}(\\tilde{s}^{k}|s^{k}).\n$$", "proof": {"steps": [{"n": 1, "claim": "$$\n\\begin{aligned}&\\Delta_{\\pi^{1},\\dots,\\pi^{N}}^{\\tilde{\\pi}^{1},\\dots,\\tilde{\\pi}^{N}}(\\tilde{s}^{k}|s^{k})\\\\ &=\\sum_{\\substack{\\tilde{s}^{-k},s^{-k}\\\\ a^{k},a^{-k}}}p(\\tilde{s}^{k},\\tilde{s}^{-k}|s^{k},s^{-k},a^{k},a^{-k})\\big [\\tilde{\\pi}^{k}(a^{k}|s^{k})\\tilde{\\pi}^{-k}(a^{-k}|s^{-k})-\\pi^{k}(a^{k}|s^{k})\\pi^{-k}(a^{-k}|s^{-k})\\big]\\\\ &=\\sum_{\\substack{\\tilde{s}^{-k},s^{-k}\\\\ a^{k},a^{-k}}}p(\\tilde{s}^{k},\\tilde{s}^{-k}|s^{k},s^{-k},a^{k},a^{-k})\\cdot\\big [\\underbrace{\\tilde{\\pi}^{k}(a^{k}|s^{k})\\tilde{\\pi}^{-k}(a^{-k}|s^{-k})-\\tilde{\\pi}^{k}(a^{k}|s^{k})\\pi^{-k}(a^{-k}|s^{-k})}_{\\mathrm{exogenous}}\\\\ &\\quad+\\underbrace{\\tilde{\\pi}^{k}(a^{k}|z^{k})\\pi^{-k}(a^{-k}|s^{-k})-\\pi^{k}(a^{k}|s^{k})\\pi^{-k}(a^{-k}|s^{-k})}_{\\mathrm{endogenous}}\\big].\\end{aligned}\n$$"}, {"n": 2, "claim": "The above decomposition can be repeated such that the exogenous part can be translated into endogenous parts that are specific to each agent."}, {"n": 3, "claim": "Specifically, repeat the decomposition for the exogenous part by considering agent $\\displaystyle{k^{\\prime}(k^{\\prime}\\neq k)}.$"}, {"n": 4, "claim": "$$\n\\begin{array}{ r l }&{\\tilde{\\pi}^{k}(a^{k}|s^{k})\\tilde{\\pi}^{-k}(a^{-k}|s^{-k})-\\tilde{\\pi}^{k}(a^{k}|s^{k})\\pi^{-k}(a^{-k}|s^{-k})}\\\\ &{=\\tilde{\\pi}^{k}(a^{k}|s^{k})\\big [\\tilde{\\pi}^{k^{\\prime}}(a^{k^{\\prime}}|s^{k^{\\prime}})\\tilde{\\pi}^{-\\{k,k^{\\prime}\\}}(a^{-\\{k,k^{\\prime}\\}}|s^{-\\{k,k^{\\prime}\\}})-\\pi^{k^{\\prime}}(a^{k^{\\prime}}|s^{k^{\\prime}})\\pi^{-\\{k,k^{\\prime}\\}}(a^{-\\{k,k^{\\prime}\\}}|s^{-\\{k,k^{\\prime}\\}})}\\\\ &{\\tilde{\\pi}^{k}(a^{k}|s^{k})\\big [\\underbrace{\\tilde{\\pi}^{k^{\\prime}}(a^{k^{\\prime}}|s^{k^{\\prime}})\\tilde{\\pi}^{-\\{k,k^{\\prime}\\}}(a^{-\\{k,k^{\\prime}\\}}|s^{-\\{k,k^{\\prime}\\}})-\\tilde{\\pi}^{k^{\\prime}}(a^{k^{\\prime}}|s^{k^{\\prime}})\\pi^{-\\{k,k^{\\prime}\\}}(a^{-\\{k,k^{\\prime}\\}}|s^{-\\{k,k^{\\prime}\\}})}_{\\pi^{k}-\\mathrm{exogenous}}}\\\\ &{\\quad+\\underbrace{\\tilde{\\pi}^{k^{\\prime}}(a^{k^{\\prime}}|s^{k^{\\prime}})\\pi^{-\\{k,k^{\\prime}\\}}(a^{-\\{k,k^{\\prime}\\}}|s^{-\\{k,k^{\\prime}\\}})-\\pi^{k^{\\prime}}(a^{k^{\\prime}}|s^{k^{\\prime}})\\pi^{-\\{k,k^{\\prime}\\}}(a^{-\\{k,k^{\\prime}\\}}|s^{-\\{k,k^{\\prime}\\}})}_{\\pi^{k}\\mathrm{endogenous}}.}\\end{array}\n$$"}, {"n": 5, "claim": "So on and so forth, one can decompose $\\Delta_{\\pi^{1},\\dots,\\pi^{N}}^{\\tilde{\\pi}^{1},\\dots,\\tilde{\\pi}^{N}}(\\tilde{s}^{k}|s^{k})$ as follows:"}, {"n": 6, "claim": "$$\n\\begin{array}{ r }{\\Delta_{\\pi^{1},\\ldots,\\pi^{N}}^{\\bar{\\pi}^{1},\\ldots,\\bar{\\pi}^{N}}(\\tilde{s}^{k}|s^{k})=\\Delta_{\\pi^{1},\\pi^{2},\\ldots,\\pi^{N}}^{\\bar{\\pi}^{1},\\pi^{2},\\ldots,\\pi^{N}}(\\tilde{s}^{k}|s^{k})+\\Delta_{\\pi^{1},\\pi^{2},\\pi^{3},\\ldots,\\pi^{N}}^{\\bar{\\pi}^{1},\\bar{\\pi}^{2},\\pi^{3},\\ldots,\\pi^{N}}(\\tilde{s}^{k}|s^{k})+\\ldots+\\Delta_{\\bar{\\pi}^{1},\\ldots,\\bar{\\pi}^{N-1},\\pi^{N}}^{\\bar{\\pi}^{1},\\ldots,\\bar{\\pi}^{N-1},\\bar{\\pi}^{N}}(\\tilde{s}^{k}|s^{k}),}\\end{array}\n$$"}]}}], "target_local_id": "T2", "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "label": "flawed", "error_step": 8, "error_class6": ["circular_or_unproven", "logical_gap_or_missing_case"], "error_explanation": "In going from eq (13) to eq (14) the proof asserts that the bracketed quantity $r(s_i)+\\sum_{s_i'}p_{\\tilde\\pi_1,\\dots,\\tilde\\pi_{j-1},\\pi_j,\\dots,\\pi_N}(s_i'|s_i,a_i)\\gamma v(s_i')-v_{\\tilde\\pi_1,\\dots,\\tilde\\pi_{j-1}}(s_i)$ equals the advantage $A_{\\pi_j}(s_i,a_i)$. This is not true: the equality is forced only by an ad-hoc redefinition of $A_{\\pi_j}$ with a baseline $v_{\\tilde\\pi_1,\\dots,\\tilde\\pi_{j-1}}(s_i)$ engineered so that $L^{(i)}(\\pi_j)=0$. The resulting object is not an advantage function (it does not match the IPPO/MAPPO advantage) — 'it does not have any actual meaning, just a function by definition.' Crucially, the true value $v(s_i')$ in eq (11) is action-independent and, as the matrix notation $G^{s_i}\\rho_0$ shows, is NOT changed by the policy update, so the value $v(s_i')$ inside the bracket differs from the value used to form the baseline; the $v$ for state $s'$ and the $v$ for state $s$ are different functions. Hence the second term in eq (15) does not vanish and the claim $L^{(i)}(\\pi_j)=0$ fails. This breaks the telescoping $\\gamma rG^{s_i}\\Delta(s_i)G^{s_i}\\rho_0=\\sum_j L^{(i)}(\\tilde\\pi_j)$ and therefore invalidates the entire lower bound of Theorem 2. The step also silently discards the exogenous (non-stationary) transition shift that is the paper's central difficulty. The authors ultimately conceded this confusion during rebuttal and provided no correct replacement."}
{"id": "wqi85OBVLE", "context": "In RL, the task environment is modeled as an MDP $M=(S,A,T,R,\\gamma)$ , where S is the state space, A is the action space, $T:S\\times A\\times S[0,1]$ is the transition function, $R : S \\times A $ R is the reward function, and γ is the discount factor. At every step t, the RL agent observes state $s_{t}$ and takes an action $a_{t}\\in A$ As a result, the agent progresses to state $s_{t+1}$ with probability $T(s_{t},a_{t},s_{t+1})$ , and receives a reward $r_{t}$ The goal is to search for a policy that maximizes the expected cumulative reward. We use π to denote a policy as a mapping from S to A. The Q function of the optimal policy $\\pi^{*}$ is defined by:\n\n$$\nQ_{R}^{*}(s,a)=\\operatorname*{max}_{\\pi}\\left[\\mathbb{E}\\left[\\sum_{t=0}^{\\infty}\\gamma^{t}r_{t}|s_{0}=s,a_{0}=a,\\pi\\right]\\right]\\tag{2}\n$$\n\nOur solution to RA is related to a variant of the Q function, which we refer to as Q-min (note the min operator below), denoted by $Q_{R}^{\\mu}$ •.\n\n$$\nQ_{R}^{\\mu}(s,a)=\\operatorname*{min}_{\\pi}\\left[\\mathbb{E}\\left[\\sum_{t=0}^{\\infty}\\gamma^{t}r_{t}|s_{0}=s,a_{0}=a,\\pi\\right]\\right]\\tag{3}\n$$\n\nIntuitively, $Q_{R}^{\\mu}$ above represents the Q function of the policy that leads to the minimum expected return, in contrast to ${Q}_{R}^{\\bar{*}}$ that corresponds to the maximum expected return. Furthermore, the following lemma establishes a connection between $Q_{R}^{\\mu}$ and a max form of Q. Throughout the paper, proofs, if omitted, are included in the appendix.\n\nLemma 2.\n\n$$\nQ_{R}^{\\mu}(s,a)= -Q_{-R}^{*}(s,a)\\tag{4}\n$$\n\nwhere $Q_{-R}^{*}(s,a)$ denotes the Q function of the optimal policy in Eq. 2 under —R.\n\nDefinition (Reward Adaptation (RA)). Given a set of source behaviors trained under an MDP $M\\setminus R=(S,A,T,\\cdot,\\gamma)$ with different source reward functions, compute the optimal policy for a target reward function under $M \\setminus R$ .\n\nDefinition 1 (Reward Adaptation with Q Variants). Given a reward adaptation problem where $Q_{R_{i}}^{*}$ and $Q_{-R_{i}}^{*}$ are accessible for each source domain indexed by i, compute the optimal policy under a target reward function that is a polynomial function of the source reward functions:\n\n$$\n\\mathcal{R}=\\sum_{i_{1}+i_{2}+i_{3}+\\ldots i_{n}\\leq m}a_{i_{1}i_{2}\\ldots i_{n}}R_{1}^{i_{1}}R_{2}^{i_{2}}\\cdots R_{n}^{i_{n}}\\tag{5}\n$$\n\nwhere $R_{i}$ is the source reward function for the $i^{t h}$ domain.\n\nTo derive bounds under R, we start with simpler forms of Eq. 5 and then combine the results. In the following, we denote the upper bound as $\\mathbb{Q}_{\\mathcal{R}}^{\\ast}$ and the lower bound as $\\mathbb{Q}_{\\mathcal{R}}^{\\mu}$ . We refer to a reward function as positive when all rewards are non-negative. We also assume in the following that the influence of discounting can be safely ignored, e.g., when MDPs with absorbing states are considered.\n\nLemma 6. When $\\mathcal{R}=R^m$ and $R$ is positive, an upper and lower bounds of the expected return under $\\mathcal{R}$ are given, respectively, by:\n\n$$\n\\mathbb{Q}_{\\mathcal{R}}^{*}=(Q_R^{*})^{m}\\ge Q_{R^m}^{*}=Q_{\\mathcal{R}}^{*}\n$$\n\n$$\n\\mathbb{Q}_{\\mathcal{R}}^{\\mu}=-|Q_{-R}^{*}|^{m}\\le -Q_{-R^m}^{*}=Q_{\\mathcal{R}}^{\\mu}\n$$\n\nLemma 5. When $\\mathcal{R}=R_i\\times R_j$, and both $R_i$ and $R_j$ are positive, an upper and lower bounds of the expected return under $\\mathcal{R}$ is given, respectively, by:\n\n$$\n\\mathbb{Q}_{\\mathcal{R}}^{*}=Q_{R_i}^{*}\\times Q_{R_j}^{*}\\ge Q_{R_iR_j}^{*}=Q_{\\mathcal{R}}^{*}\n$$\n\n$$\n\\mathbb{Q}_{\\mathcal{R}}^{\\mu}=-|Q_{-R_i}^{*}|\\times|Q_{-R_j}^{*}|\\le -Q_{-R_iR_j}^{*}=Q_{\\mathcal{R}}^{\\mu}\n$$\n\nIntuitively, the smaller the distance between the bounds is, the more pruning opportunities there may be. Even though we cannot tighten the bounds without additional information, in the following, we show that we can \"effectively\" achieve the same effect by reward shaping Ng et al. (1999). Reward shaping has been widely used to support more informative reward functions to guide learning. In our case, however, we apply it with the objective to reduce the distance between the computed bounds.\n\nA shaping function of a reward function R has the form of $\\mathcal{F}=\\gamma *\\Phi(s^{\\prime})-\\Phi(s)$ following Ng et al. (1999), where Φ is referred to as a potential function. The shaped reward function, denoted by $R_{F},$ satisfies $R_{F}=R+\\mathcal{F}$ Applying reward shaping this way does not affect the optimal policy, i.e., the optimal policy remains invariant. Next, we first establish the effects of applying reward shaping to the target reward function R on $Q_{\\mathcal{R}}^{\\ast}$ and $Q_{\\mathcal{R}}^{\\mu}$ below.", "theorems": [{"local_id": "T1", "statement": "Lemma 3: When $\\mathcal{R}=a R_{i}+b R_{j}\\ (a>0,b>0)$ , an upper and lower bounds of the expected return under R are given, respectively, by:\n\n$$\n\\mathbb{Q}_{\\mathcal{R}}^{*}=a Q_{R_{i}}^{*}+b Q_{R_{j}}^{*}\\ge Q_{a R_{i}+b R_{j}}^{*}=Q_{\\mathcal{R}}^{*}\\tag{6}\n$$\n\n$$\n\\mathbb{Q}_{\\mathcal{R}}^{\\mu}= -(a Q_{-R_{i}}^{*}+b Q_{-R_{j}}^{*})\\le-Q_{-(a R_{i}+b R_{j})}^{*}=Q_{\\mathcal{R}}^{\\mu}\\tag{7}\n$$", "proof": {"steps": [{"n": 1, "claim": "From this, we can derive that\n\n$$\n\\begin{array}{ l }{{Q_{a R}^{*}=\\underset{\\pi}{\\operatorname*{max}}\\left[\\mathbb{E}\\left[a r_{0}+\\gamma a r_{1}+...+\\gamma^{n}a r_{n}\\middle|s_{0}=s,a_{0}=a,\\pi\\right]\\right]}}\\\\ {{\\ =a Q_{R}^{*}}}\\end{array}\\tag{9}\n$$"}, {"n": 2, "claim": "Denote the optimal policy for $Q_{a R_{i}+b R_{j}}^{*}$ as $\\pi_{i j}^{*}$ , given $a > 0$ and $b > 0$ , we can derive that\n\n$$\na Q_{R_{i}}^{*}+b Q_{R_{j}}^{*}\\geq a Q_{R_{i}}(\\pi_{i j}^{*})+b Q_{R_{j}}(\\pi_{i j}^{*})=Q_{a R_{i}+b R_{j}}^{*}\\tag{10}\n$$"}, {"n": 3, "claim": "Since we do not require positive reward above, to derive a lower bound we simply replace R with R above, and we have\n\n$$\n(a Q_{-R_{i}}^{*}+b Q_{-R_{j}}^{*})\\geq Q_{-(a R_{i}+b R_{j})}^{*}\\tag{11}\n$$"}, {"n": 4, "claim": "which can then be used to derive the lower bound by multiplying both sides by —1."}]}}, {"local_id": "T2", "statement": "In an MDP $M = (S,A,T,R,\\gamma)$, let $Q_{\\mathcal{R}}^{*}(s,a) = \\max_{\\pi} \\mathbb{E}[\\sum_{t=0}^{\\infty} \\gamma^{t} r_{t} \\mid s_0=s, a_0=a, \\pi]$ be the optimal Q function and $Q_{\\mathcal{R}}^{\\mu}(s,a) = \\min_{\\pi} \\mathbb{E}[\\sum_{t=0}^{\\infty} \\gamma^{t} r_{t} \\mid s_0=s, a_0=a, \\pi]$ be the Q-min function. Let $\\mathcal{F} = \\gamma \\cdot \\Phi(s') - \\Phi(s)$ be a potential-based shaping function (Ng et al. 1999) with potential $\\Phi$, and let $\\mathcal{R}_{\\mathcal{F}} = \\mathcal{R} + \\mathcal{F}$ be the shaped reward. Assume the influence of discounting can be safely ignored (e.g., MDPs with absorbing states). Then after applying reward shaping to $\\mathcal{R}$ with $\\mathcal{F}$, the shaped optimal and Q-min functions are $Q_{\\mathcal{R}_{\\mathcal{F}}}^{*}(s,a) = Q_{\\mathcal{R}}^{*}(s,a) - \\Phi(s)$ (Eq. 12) and $Q_{\\mathcal{R}_{\\mathcal{F}}}^{\\mu}(s,a) = Q_{\\mathcal{R}}^{\\mu}(s,a) + \\Phi(s)$ (Eq. 13).", "proof": {"steps": [{"n": 1, "claim": "$Q_{\\mathcal{R}_{\\mathcal{F}}}^{*}(s,a) = Q_{\\mathcal{R}}^{*}(s,a) - \\Phi(s)$"}, {"n": 2, "claim": "$Q_{\\mathcal{R}_{\\mathcal{F}}}^{\\mu}(s,a) = -\\left[Q_{-\\mathcal{R}_{\\mathcal{F}}}^{*}(s,a)\\right]$"}, {"n": 3, "claim": "$Q_{\\mathcal{R}_{\\mathcal{F}}}^{\\mu}(s,a) = -\\left[Q_{-\\mathcal{R}}^{*}(s,a) - \\Phi(s)\\right]$"}, {"n": 4, "claim": "$Q_{\\mathcal{R}_{\\mathcal{F}}}^{\\mu}(s,a) = -Q_{-\\mathcal{R}}^{*}(s,a) + \\Phi(s)$"}, {"n": 5, "claim": "$Q_{\\mathcal{R}_{\\mathcal{F}}}^{\\mu}(s,a) = Q_{\\mathcal{R}}^{\\mu}(s,a) + \\Phi(s)$"}]}}, {"local_id": "T3", "statement": "Theorem 2: Given R in the form of Eq. 5 where every $a_{i_{1}i_{2}... i_{n}}>0$ and every source reward function is positive, an upper and lower bounds of the expected return under $\\mathcal{R}$ are given, respectively, by:\n\n$$\n\\mathbb{Q}_{\\mathcal{R}}^{*}=\\sum_{i_{1}+i_{2}+i_{3}+\\ldots i_{n}\\leq m}a_{i_{1}i_{2}\\ldots i_{n}}|Q_{R_{1}}^{*}|^{i_{1}}\\times|Q_{R_{2}}^{*}|^{i_{2}}\\times\\ldots\\times|Q_{R_{n}}^{*}|^{i_{n}}\\geq Q_{\\mathcal{R}}^{*}\n$$\n\n$$\n\\mathbb{Q}_{\\mathcal{R}}^{\\mu}= -\\left[\\sum_{i_{1}+i_{2}+i_{3}+\\ldots i_{n}\\leq m}a_{i_{1}i_{2}\\ldots i_{n}}|Q_{-R_{1}}^{*}|^{i_{1}}\\times|Q_{-R_{2}}^{*}|^{i_{2}}\\times\\ldots\\times|Q_{-R_{n}}^{*}|^{i_{n}}\\right]\\leq Q_{\\mathcal{R}}^{\\mu}\n$$", "proof": {"steps": [{"n": 1, "claim": "Substituting $R_{k}^{i_{k}}=\\bar{R_{k}}$ in Eq. 14, we obtain\n\n$$\nQ_{a_{i_{1}i_{2}\\ldots i_{n}}R_{1}^{i_{1}}\\times R_{2}^{i_{2}}\\ldots R_{n}^{i_{n}}}^{*}\\leq a_{i_{1}i_{2}\\ldots i_{n}}|Q_{R_{1}}^{*}|^{i_{1}}\\times|Q_{R_{2}}^{*}|^{i_{2}}\\times\\ldots\\times|Q_{R_{n}}^{*}|^{i_{n}}\\mathrm{[L e m m a~6]}\\tag{15}\n$$"}, {"n": 2, "claim": "Now we can derive the following for $\\begin{array}{ r }{{\\mathcal{R}}=\\sum_{i_{1}+i_{2}+i_{3}+... i_{n}\\leq m}a_{i_{1}i_{2}... i_{n}}R_{1}^{i_{1}}R_{2}^{i_{2}}\\cdot\\cdot\\cdot R_{n}^{i_{n}}{\\mathrm{.}}}\\end{array}$\n\n$$\nQ_{{\\mathcal R}}^{*}\\leq\\sum_{\\substack{i_{1}+i_{2}+i_{3}+\\ldots i_{n}\\leq m}}a_{i_{1}i_{2}\\ldots i_{n}}|Q_{{\\cal R}_{1}}^{*}|^{i_{1}}\\times|Q_{{\\cal R}_{2}}^{*}|^{i_{2}}\\times\\ldots\\times|Q_{{\\cal R}_{n}}^{*}|^{i_{n}}\\quad\\quad [\\mathrm{L e m m a~}3]\\tag{16}\n$$"}, {"n": 3, "claim": "Similarly we can derive the lower bound:\n\n$$\n\\mathbb{Q}_{\\mathcal{R}}^{\\mu}= -\\left[\\sum_{i_{1}+i_{2}+i_{3}+... i_{n}\\leq m}a_{i_{1}i_{2}... i_{n}}|Q_{-R_{1}}^{*}|^{i_{1}}\\times|Q_{-R_{2}}^{*}|^{i_{2}}\\times ...\\times|Q_{-R_{n}}^{*}|^{i_{n}}\\right]\\leq Q_{\\mathcal{R}}^{\\mu}\\tag{17}\n$$"}]}}, {"local_id": "T4", "statement": "Theorem 3 (optimality). Given a problem of reward adaptation with Q variants (Def. 1), the optimal policies under R remain invariant under Q-Manipulation.", "proof": {"steps": [{"n": 1, "claim": "To retain all optimal policies, it must be satisfied that none of the optimal actions under each state are pruned."}, {"n": 2, "claim": "Assuming that a pruned action $\\widehat{a}$ under s is an optimal action, we must have\n\n$$\n\\forall a~Q_{\\mathcal{R}_{\\mathcal{F}}}^{*}(s,a)\\leq Q_{\\mathcal{R}_{\\mathcal{F}}}^{*}(s,\\widehat{a})\n$$"}, {"n": 3, "claim": "Given that Q-Manipulation only prunes an action $\\widehat{a}$ under s when $\\exists a$ $\\mathbb{Q}_{\\mathcal{R}_{\\mathcal{F}}}^{\\mu}(s,a)>\\mathbb{Q}_{\\mathcal{R}_{\\mathcal{F}}}^{*}(s,\\widehat{a})$ , we can derive that\n\n$$\n\\begin{array}{ r }{Q_{\\mathcal{R}_{\\mathcal{F}}}^{\\mu}(s,a)\\geq\\mathbb{Q}_{\\mathcal{R}_{\\mathcal{F}}}^{\\mu}(s,a)>\\mathbb{Q}_{\\mathcal{R}_{\\mathcal{F}}}^{*}(s,\\widehat{a})\\geq Q_{\\mathcal{R}_{\\mathcal{F}}}^{*}(s,\\widehat{a})\\geq Q_{\\mathcal{R}_{\\mathcal{F}}}^{*}(s,a),}\n\\end{array}\n$$\n\nresulting in a contradiction that\n\n$$\nQ_{\\mathcal{R}_{\\mathcal{F}}}^{\\mu}(s,a)>Q_{\\mathcal{R}_{\\mathcal{F}}}^{*}(s,a)\n$$"}, {"n": 4, "claim": "As a result, we know that all optimal actions and hence policies are retained."}]}}], "target_local_id": "T2", "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "label": "flawed", "error_step": 3, "error_class6": ["computation_or_constant", "logical_gap_or_missing_case", "others"], "error_explanation": "The step asserts $Q_{-\\mathcal{R}_{\\mathcal{F}}}^{*}(s,a) = Q_{-\\mathcal{R}}^{*}(s,a) - \\Phi(s)$ by reusing Eq. (13). This is wrong because the shaping function applied to the negated reward $-\\mathcal{R}$ is $-\\mathcal{F}$ (its potential is $-\\Phi$, since $-\\mathcal{R}_{\\mathcal{F}} = -\\mathcal{R} + (-\\mathcal{F})$). By Ng et al. (1999) the correct value is $Q_{-\\mathcal{R}_{\\mathcal{F}}}^{*}(s,a) = Q_{-\\mathcal{R}}^{*}(s,a) + \\Phi(s)$, giving a $+\\Phi(s)$ term rather than $-\\Phi(s)$. Propagating the correct sign through the outer negation yields $Q_{\\mathcal{R}_{\\mathcal{F}}}^{\\mu}(s,a) = -[Q_{-\\mathcal{R}}^{*}(s,a) + \\Phi(s)] = Q_{\\mathcal{R}}^{\\mu}(s,a) - \\Phi(s)$, i.e. the lower bound shifts DOWN by $\\Phi(s)$, the SAME direction as the upper bound in Eq. (13) — not the opposite direction. The proof effectively uses two different shaping functions ($\\mathcal{F}$ for $\\mathcal{R}$ but implicitly $+\\mathcal{F}$ instead of $-\\mathcal{F}$ for $-\\mathcal{R}$). A counter-example: in a 1-state, 1-action MDP all policies coincide, so min and max Q-values are equal and stay equal after adding a constant potential, contradicting Eq. (14)."}
{"id": "zWy1uxjDdZJ", "context": "We consider the problem of verifying the local robustness of piecewise-linear neural networks for a given $\\ell_p$ bound. Precisely, given a point, $x$, network, $F$, and norm bound, $\\epsilon$, this entails determining whether Equation 2 holds. $$\\forall x' . \\|x - x'\\|_p \\le \\epsilon \\Longrightarrow F(x) = F(x') \\tag{2}$$ In this section we give a high-level overview of our proposed algorithm, and present some implementation heuristics that improve its performance (Section 2.2). We also propose a variant (Section 2.3) to compute certified lower bounds of the robustness radius. Correctness proofs for all of the algorithms discussed in this section are provided in Appendix A. Because our algorithm applies to arbitrary $\\ell_{p}$ norms, we use the un-subscripted notation $\\| \\cdot \\|$ to refer to a general $\\ell_{p}$ norm for the remainder of this section. We propose the Fast Geometric Projections (FGP) algorithm, which takes a model, $F,$ an input, x, and a bound, $\\epsilon,$ and either proves that Equation 2 is satisfied (i.e., e-local robustness), finds an adversarial input at distance less than  from x, or returns unknown. Our algorithm assumes $F(x)=\\operatorname{argmax}\\{f(x)\\}$ , where $f:\\mathbb{R}^{m}\\mathbb{R}^{n}$ is the function computed by a neural network composed of linear transformations with ReLU activations $(\\mathrm{i . e .,}f$ is a feed-forward ReLU network). The algorithm relies on an analysis of all possible activation regions around x. An activation region is a maximal set of inputs having the same activation pattern of ReLU nonlinearities. Formally, let $f_{u}(x)$ denote the pre-activation value of neuron u in network $F$ when evaluating x. We say that neuron u is activated if $f_{u}(x)\\geq 0$ . An activation pattern, A, is a Boolean function over neurons that characterizes whether each neuron is activated. Then the activation region, A, associated with pattern A is the set of inputs that realize A: $\\bar{A}: =\\{x\\mid\\forall u .(f_{u}(x)\\geq 0)\\iff A(u)\\}$ Because we assume that F is a piecewise-linear composition of linear transformations and ReLU activations, we can associate the activation status $A(u)$ of any neuron with a closed half-space of $\\mathbb{R}^{m}$ (Jordan et al., 2019). The activation constraint, $C_{u}$ , for neuron u and pattern A is the linear inequality $w_{u}^{T}x+b_{u}\\leq 0$ , where $w_{u}$ and $b_{u}$ satisfy $\\forall x\\in\\mathbb{R}^{m}. w_{u}^{T}x+b\\leq 0\\iff A(u)$ . The coefficients, $w_{u}$ are equal to the gradient of $f_{u}$ with respect to its inputs, evaluated at a point in Å. Crucially, because the gradient is the same at every point in Å, the constraints can be computed from the activation pattern alone via backpropagation. More details on this computation are given in Appendix B. The intersection of these constraints yields the activation region ${\\bar{A}}.$ and the facets of Å correspond to the non-redundant constraints. The convexity of activation regions follows from this observation, as does the fact that the decision boundaries are also linear constraints of the form $f(x)_{i}\\geq f(x)_{j}$ for classes i and $j$ . The FGP algorithm performs a search of all activation regions that might be at distance less than from x. We begin by analyzing the region, ${\\bar{A}}_{0}$ , associated with the activation pattern of the input, x, as follows. First, we check to see if ${\\bar{A}}_{0}$ contains a decision boundary, C, that is at distance less than  from x; if so, we take the projection, $p,$ from x onto C. Intuitively, the projection is an input satisfying C that is at minimal distance from $x,$ i.e., $\\forall x^{\\prime}. C(x^{\\prime})\\Longrightarrow\\|x-p\\|\\dot{\\leq}\\|\\dot{x}-x^{\\prime}\\|$ . Similarly we define the distance, d, from x to a constraint or decision boundary, C, as $d(x,C): =\\mathrm{min}_{x^{\\prime}:C(x^{\\prime})}\\left\\|x-x^{\\prime}\\right\\|$ , i.e., d is the distance from x to its projection onto C. If p does not have the same class as x according to $F$ (or lies directly on the decision boundary between classes), then p is an adversarial example. $\\operatorname{If}p$ has the same class as x, it means that the projection to the decision boundary is outside of the activation region we are currently analyzing; however, this is not sufficient to conclude that no point on the decision boundary is inside both the current activation region and the e-ball (see Figure 1b). Therefore, we return unknown. Otherwise, we collect all activation constraints in ${\\bar{A}}_{0};$ there is one such activation constraint per neuron u in the network; thus each constraint corresponds to the neighboring region, ${\\bar{A}}_{0}^{u}$ , which has the same activation pattern as ${\\bar{A}}_{0},$ except with neuron u flipped, i.e., $\\ \\bar{A}_{0}(u^{\\prime})\\ \\bar{\\neq}\\ \\bar{A}_{0}^{u}(u^{\\prime})\\iff u=u^{\\prime}$ For each constraint $\\bar{C}_{u},$ we check whether it might be at distance less than e from x using a geometric projection. Although $C_{u}$ is only valid in the polytope corresponding to the activation region ${\\bar{A}}_{0}$ , we compute the distance to the hyperplane corresponding to the infinite prolongation of $C_{u}$ . As such, we only obtain a lower bound on the distance from x to $C_{u}.$ If the computed distance is smaller than e, we enqueue the corresponding activation region if it has not been searched yet. Thus the queue contains unexplored regions that might be at distance less than  from x. The algorithm continues to analyze each enqueued region in the same way, each explored region enqueuing neighbouring regions that might be at distance less than  from x, until the queue is empty. Exhausting the queue means that we did not find any adversarial examples in any activation region that might intersect with the -ball centered at x. We therefore conclude that x is locally e-robust. We show that when the FGP algorithm returns not_robust, there exists an adversarial example, and when it returns robust, the model is locally robust at x, with radius e. However, the algorithm may also return unknown, in which case we do not claim anything about the robustness of the model. A complete proof of Theorem 2 is provided in Appendix A.1. As mentioned previously, our focus is on feed-forward ReLU networks. Such networks contain an ordered hierarchy of layers. Therefore, for a neuron $u$, the activation constraint corresponding to the condition $f_u(x) > 0$ only depends on the activations of neurons in previous layers. In particular, if we fix the activations of all neurons at previous layers, then this constraint becomes linear. Our crucial observation is that, if two activations patterns $A$ and $A'$ differ only in the activation of one neuron, $u$, in layer $N$, the activation constraints for all neurons up to layer $N$ remain identical in $A$ and $A'$, as they do not depend on neurons deeper in the network such as $u$. We leverage this hierarchy to enhance FGP with a tree-based exploration of the network to prove that an input $x$ is $\\epsilon$-robust. Each node in our exploration tree corresponds to the activation pattern of neurons at a layer. Nodes are ordered similarly to the layers in the network: if layer $N$ has $k$ neurons, the vertices at depth $N-1$ have degree $2^k$, with one child for each possible activation pattern of these $k$ neurons. Leaf nodes give a complete characterization of an activation pattern for the entire network. Leaves that share an ancestor at depth $N$ in the tree correspond to activation patterns that share an *activation prefix*, i.e., the activations of all neurons in the layers up to layer $N$ are the same in both activation patterns. An illustration of such a tree is shown in panel (a) of the figure below. The version of the FGP algorithm given in Section 2.1 systematically explores all regions that might be at distance less than $\\epsilon$ from $x$; a region $\\bar{A}_i$ is explored if it is reachable from a previously explored region (starting with the one containing $x$) by flipping the activation of exactly one neuron. Exploration of $\\bar{A}_i$ consists of collecting all activation constraints and decision boundaries at distance less than $\\epsilon$ from $x$ w.r.t the activation pattern $A_i$, and enqueuing the corresponding neighboring regions to analyze them. In our optimized variant, when enqueuing a region, $\\bar{A}_{i+1}$, we additionally keep track of its predecessor in the exploration, $\\bar{A}_i$, i.e., the region from which $\\bar{A}_{i+1}$ was discovered. Crucially, when exploring $\\bar{A}_{i+1}$, we only enqueue neighboring regions belonging to the subtree of the lowest common ancestor (LCA) of $\\bar{A}_i$ and $\\bar{A}_{i+1}$ in the tree. Supposing neuron $u$ was flipped to obtain $A_{i+1}$ from $A_i$, this corresponds to considering only activation constraints in $\\bar{A}_{i+1}$ that are associated with the layer $u$ belongs to or deeper. To understand how this approach helps prune unnecessary regions that the basic FGP algorithm would otherwise have searched, an example is provided in panel (b) of the figure below. In practice, for efficiency, we do not construct the tree in memory. Instead, it is sufficient to use a regular queue of regions to explore, enriched with information about which layer the last flipped neuron belongs to. When exploring a region, we filter out all collected constraints that belong to earlier layers. This preserves the inherent parallelism from the basic version of FGP, leading to a more efficient implementation. In the correctness argument for this tree-based variant, the paper fixes the following notation: we consider the case where we are exploring a region $\\bar{A}$, reached from a previously explored region $\\bar{A}_{prev}$ by flipping a neuron $u$ at layer $N$. Figure: (a) Illustration of the \"tree\" of activation pattern prefixes for the network shown on the left. Each leaf corresponds to an activation pattern, each level corresponds to a layer, and internal nodes correspond to activation pattern prefixes at the respective layer. (b) Example of the tree-based exploration on a simple network with two internal neurons. The constraint for neuron 1, $C_1$, is linear and the constraint for neuron 2, $C_2$, is piecewise linear, as it depends on the activation of neuron 1. The initial region, $\\bar{A}_{00}$, has been explored, its neighbor obtained by flipping neuron 1, $\\bar{A}_{10}$, is enqueued, and its neighbor obtained by flipping neuron 2, $\\bar{A}_{01}$, is being explored. The LCA in the tree of $\\bar{A}_{01}$ and its predecessor in the exploration ($\\bar{A}_{00}$) is shown in red. $\\bar{A}_{11}$ (which is not in the $\\epsilon$-ball) is not searched from $\\bar{A}_{01}$, despite the fact that $C_1$ is in the $\\epsilon$-ball, because it is not a child of the LCA in the tree. We now consider the related problem of finding a lower bound on the local robustness of the network. A variant of the FGP algorithm can provide certified lower bounds by using a priority queue; constraints are enqueued with priority corresponding to their distance from x, such that the closest constraint is always at the front of the queue. We keep track of the current certified lower bound in a variable, β. At each iteration, we set $\\boldsymbol{\\beta}$ to the maximum of the old β and the distance from x to the constraint at the front of the queue. The algorithm terminates either when all constraints at distance less than the initially specified  were handled, or when a decision boundary for which the initial algorithm returns unknown or not_robust is found. In the first case, we return e; in the second, we return the value stored in $\\beta$ at this iteration. The proof of this variant is similar to the proof of the FGP algorithm. It relies on the following loop invariant, which we prove in Appendix A.2. We define a path of activation regions as a sequence, $\\bar{A}_{0},\\ldots,\\bar{A}_{k}.$ , of activation regions such that the underlying activation patterns $\\bar{A_{i}}$ and $A_{i+1}$ differ in exactly one neuron for all i, and there exists at least one input, $x_{i},$ that has activation pattern $A_{i}$ for all i.", "theorems": [{"local_id": "T1", "statement": "Theorem 2 (general position). Assume the ReLU network is in general position: distinct neurons have distinct pre-activation boundary hyperplanes $\\{z : f_u(z)=0\\}$, and their arrangement is simple (any $k$ distinct such boundaries meet in codimension $k$). Then (1) when the FGP algorithm returns not_robust, there exists an adversarial example $p$ with $\\|x - p\\| \\le \\epsilon$ and $F(x) \\neq F(p)$; (2) when it returns robust, $x$ is locally robust with radius $\\epsilon$ (Equation 2 holds).", "proof": {"steps": [{"n": 1, "claim": "If FGP returns not_robust, the claim is immediate; the substance is the robust case, where we show x is locally robust with radius epsilon by contradiction."}, {"n": 2, "claim": "Assume for contradiction FGP returns robust yet some $p$ has $\\|x-p\\| \\le \\epsilon$ and $F(x) \\neq F(p)$; let $A_x$, $A_p$ be the activation regions of $x$ and $p$."}, {"n": 3, "claim": "A path of activation regions is a sequence $A_0,\\ldots,A_k$ in which consecutive activation patterns differ in exactly one neuron and each $A_i$ is realized by some input $x_i$."}, {"n": 4, "claim": "The proof rests on three facts: (1) a path $P$ from $A_x$ to $A_p$ exists with every region containing an input within $\\epsilon$ of $x$ and (if $A_x \\neq A_p$) all regions distinct; (2) FGP visits every region of $P$; (3) any visited region containing an adversarial input makes FGP return not_robust or unknown."}, {"n": 5, "claim": "Existence of a path $P$ from $A_x$ to $A_p$, in general position, all traversed regions within $\\epsilon$ of $x$ and pairwise distinct. Consider the segment going from $x$ to $p$ in a straight line. As $\\|x - p\\| \\le \\epsilon$, all points on this segment are also at distance less than $\\epsilon$. As $f$ is a neural network with ReLU activations, $f$ is a continuous function, as are the activation functions, $f_u$ for each of its internal neurons, $u$. Furthermore, input points at the boundary between two activation regions (i.e., $z$, such that $f_u(z) = 0$) belong to both activation regions. Therefore, listing all activation regions on the segment between $x$ and $p$ yields a path, $P$, from $A_x$ to $A_p$, with each region on the path containing an input point, $x_i$, on the segment, such that $\\|x - x_i\\| \\le \\epsilon$. That each $A_i, A_j$ in $P$ is unique follows from the convexity of activation regions: If there exist $A_i = A_j$ in $P$ such that there exists an $A_k$ in between, then $A_i = A_j$ cannot be convex, as there exists a line segment with its end points in $A_i$ and $A_j$ that is not entirely contained within the region $A_i = A_j$. This ensures that paths are of finite length."}, {"n": 6, "claim": "FGP visits every region of P when it returns robust, by induction on path length. We proceed by induction on the length of paths induced (similarly to as above) by a line segment included in the $\\epsilon$-ball centered on $x$. In the base case, if the path is of length one, then it contains only $A_x$, and the claim holds trivially since FGP starts by visiting $A_x$. In the inductive case, let us assume that for any path with length at most $k$ induced by a segment, $s$, beginning at $x$ with $\\|s\\| \\le \\epsilon$, FGP visits all regions in the path. Now consider a path, $P' = A_0, \\ldots, A_k$, of length $k+1$, induced by a segment, $s'$, beginning at $x$ with $\\|s'\\| \\le \\epsilon$. Since $A_{k-1}$ is on the path, there exists a point, $x_{k-1}$, on $s'$ such that the sub-segment from $x$ to $x_{k-1}$ induces a path of length $k$; thus we can apply our induction hypothesis to conclude that FGP visits $A_0, \\ldots, A_{k-1}$. Now, since $A_{k-1}$ and $A_k$ are neighbors in $P'$, they must share some boundary, $C$, that is intersected by $s'$. Thus, since $\\|s'\\| \\le \\epsilon$, $d(x, C) \\le \\epsilon$; thus when FGP visits $A_{k-1}$, it will add $A_k$ to the queue via $C$. Therefore, since FGP returns robust only when all regions in the queue have been visited, FGP will visit $A_k$."}, {"n": 7, "claim": "If a visited region holds an adversarial example, FGP returns not_robust or unknown. If $p$ in $A_p$ is an adversarial example, then $F(x) \\neq F(p)$. By continuity of $f$, this means that there exists an input, $p'$ on the segment from $x$ to $p$ that is exactly on a decision boundary, $C$. As $\\|x - p'\\| \\le \\|x - p\\| \\le \\epsilon$, $C$ must have been analyzed when exploring $A_p$. However, when analyzing a decision boundary, FGP always returns either not_robust or unknown."}]}}, {"local_id": "T2", "statement": "The tree-based exploration heuristic, which prunes which neighboring activation regions are enqueued, is sound: every region filtered out when exploring from a region is either unreachable within the $\\epsilon$-ball or is still explored through a different path in the tree. Hence pruning cannot cause FGP to skip a region that could contain an adversarial example, and Theorem 2 continues to hold for the tree-based variant.", "proof": {"steps": [{"n": 1, "claim": "It suffices to consider adjacent activation regions differing by exactly one flipped neuron, and trace how a filtered-out region is recovered through another branch of the tree."}, {"n": 2, "claim": "$\\bar{A}_{10}$ is visited by the tree search."}, {"n": 3, "claim": "From $\\bar{A}_{10}$, the neighboring region $\\bar{A}_{11}$ is in turn explored (enqueued) by the search."}, {"n": 4, "claim": "So a region $\\bar{A}_{11}$ pruned when exploring from some other branch (e.g. from $\\bar{A}_{prev}/\\bar{A}$) is still reachable, since it is explored through the $\\bar{A}_{10}$ branch."}, {"n": 5, "claim": "Hence no reachable region within the $\\epsilon$-ball is dropped by the heuristic, and the tree-based variant preserves the soundness guarantee of Theorem 2."}]}}, {"local_id": "T3", "statement": "Loop Invariant 2. (1) All activation regions at distance less than β from x were previously visited, (2) β is always smaller than e, and (3) there is no adversarial point at distance less than $\\beta .$", "proof": {"steps": [{"n": 1, "claim": "Loop Invariant 2 trivially holds on entry of the loop since bound is initialized to 0. Proving that the invariant is maintained is more interesting."}, {"n": 2, "claim": "Suppose that bound increases to $\\mathrm{\\it~b o u n d^{\\prime}}$ . It must then be shown that there is no unvisited activation region at distance less than bound'."}, {"n": 3, "claim": "We proceed again by contradiction: assume there was such a region, ${\\bar{A}},$ containing a point, $p,$ such that $\\lvert | x - p \\rvert | \\leq$ bound'. Again, let us consider the segment from x to $p,$ and the path, $P_{\\mathrm{:}}$ it induces."}, {"n": 4, "claim": "Let us consider ${\\bar{A}}_{i}.$ , the first region of $P$ at distance greater than bound that was not previously visited. If no such region exists, then Å is at distance less than bound from $x,$ and so by our induction hypothesis, it was already visited."}, {"n": 5, "claim": "otherwise, $\\bar{A}_{i-1}$ was visited, and the activation constraint, $C,$ between $\\bar{A}_{i-1}$ and $\\bar{A}_{i}$ is such that $d(x,C)\\leq b o u n d^{\\prime}\\leq\\epsilon$ . Therefore, $C$ (which leads to ${\\bar{A}}_{i})$ was already added to the queue with priority less than bound', and by virtue of the priority queue, it was explored before the current iteration, which yields a contradiction."}, {"n": 6, "claim": "If bound does not increase, the invariant still trivially holds. This case can happen because our computation of the distance to constraints is an underapproximation of the true distance to the feasible portion of the constraint."}]}}], "target_local_id": "T2", "topic_domain": "Others", "topic_subtopic": "Robustness, Privacy & Security", "label": "flawed", "error_step": 3, "error_class6": ["circular_or_unproven", "logical_gap_or_missing_case", "missing_assumption"], "error_explanation": "The step asserts that once $\\bar{A}_{10}$ is visited, the neighboring region $\\bar{A}_{11}$ is enqueued and explored from $\\bar{A}_{10}$ — but this is only assumed, never proved. The tree-based heuristic prunes which neighbors are enqueued, so visiting $\\bar{A}_{10}$ does not entail that $\\bar{A}_{11}$ is reached from it. The counterexample shows $\\bar{A}_{11}$ may fail to be explored from $\\bar{A}_{10}$: exactly as boundary $GC$ between $\\bar{A}_{prev}$ and $\\bar{A}'$ does not pass inside the $\\epsilon$-ball (so $\\bar{A}'$ is never explored from $\\bar{A}_{prev}$), $\\bar{A}_{11}$ can be unreachable via the $\\bar{A}_{10}$ branch. If such a region is also pruned on every other path, a reachable region that could contain an adversarial example is dropped, and the pruned search can return robust when an adversarial example actually exists. The proof needed, but never established, the assumption that every pruned region is recovered along some retained path; hence the soundness claim (Theorem 1 for the tree-based variant) is unsound. There is no valid fix — the authors removed the tree-based exploration heuristic entirely."}