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  "body": "\n## Definition\n\nFor each dimension $d \\ge 1$, let $F_d$ be a class of real functions on $[0, 1]^d$ and let $S_d \\colon F_d \\to \\R$ be a functional, such as the integral $S_d(f) = \\int_{[0,1]^d} f$.\nAn algorithm with $n$ function values computes $f(x_1), \\dots, f(x_n)$, where each point may depend on the earlier values, and outputs a number computed from these values.\nIts worst-case error is the supremum of $\\abs{S_d(f) - A(f)}$ over $f \\in F_d$.\nLet $n(\\varepsilon, d)$ be the smallest $n$ such that some algorithm with $n$ function values has worst-case error at most $\\varepsilon$.\nThe family of problems $(S_d, F_d)$ *suffers from the curse of dimensionality* if there are constants $C, \\gamma, \\varepsilon_0 \u003e 0$ as follows.\nFor all $\\varepsilon \\le \\varepsilon_0$ and infinitely many $d$,\n$$n(\\varepsilon, d) \\ge C (1 + \\gamma)^d.$$\n\n## Remarks\n\nExample: let $F_d$ be the functions $f$ on $[0, 1]^d$ with $\\abs{f} \\le 1$ and $\\abs{f(x) - f(y)} \\le \\norm{x - y}_\\infty$, and let $S_d$ be the integral.\nTake an algorithm with $n \\le 2^{d-1}$ values, and let $P$ be the points it uses for the input $0$.\nThe function $g(x) = \\min\\big(1, \\operatorname{dist}_\\infty(x, P)\\big)$ lies in $F_d$ and vanishes on $P$, so the algorithm uses the same points and outputs the same number for $g$, $-g$ and $0$.\nThe set where $g \u003c \\frac14$ is covered by $n$ cubes of side $\\frac12$, of total volume at most $n 2^{-d} \\le \\frac12$, so $\\int g \\ge \\frac14 \\cdot \\frac12 = \\frac18$.\nHence the error is at least $\\frac18$ for $g$ or for $-g$, and $n(\\varepsilon, d) \u003e 2^{d-1}$ for $\\varepsilon \u003c \\frac18$: the problem suffers from the curse.\n\nNon-example: let $F_d$ be the functions $f(x) = h(x_1)$ that depend only on the first coordinate, where $h \\colon [0, 1] \\to \\R$ satisfies $\\abs{h} \\le 1$ and $\\abs{h(s) - h(t)} \\le \\abs{s - t}$.\nA [[quadrature rule|ADSVMCY77A]] in the first variable alone reaches error $\\varepsilon$ with a number of points independent of $d$, so no exponential lower bound holds.\n\nIn the information-based complexity literature the error is often normalized by the error of the zero algorithm; this changes constants but not the idea.\n",
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