Gaussian concentration of smooth functions #
Let μ be a centred Gaussian measure on a separable Banach space E, and let f : E → ℝ be a
C¹ function whose derivative has variance at most v under μ at almost every point, that is
Var[fderiv ℝ f x; μ] ≤ v. Then f concentrates around its mean with Gaussian tails: f - 𝔼 f
is sub-Gaussian with variance proxy π² v / 4, and
μ {x | ε ≤ |f x - 𝔼 f|} ≤ 2 exp (-2 ε² / (π² v)).
For the standard Gaussian measure on a finite-dimensional inner product space the variance of a
linear functional is its squared norm, so the hypothesis holds with v = K² when ‖f' x‖ ≤ K,
that is, when the C¹ function f is K-Lipschitz. Estimates of this kind, transported to the
Euclidean sphere, are the concentration of measure behind Dvoretzky's theorem on almost Euclidean
sections of normed spaces.
The proof is the interpolation argument of Maurey and Pisier. For (x, y) ∈ E × E and the
rotation R θ (x, y) = (cos θ • x + sin θ • y, -sin θ • x + cos θ • y), the first coordinate of
R θ (x, y) runs along a quarter circle from x to y with velocity the second coordinate, so
(ContDiff.sub_eq_integral_fderiv_rotation)
f y - f x = ∫ θ in 0..π/2, f' (R θ (x, y)).1 (R θ (x, y)).2.
Averaging over θ and applying Jensen's inequality to the exponential bounds
exp (t (f y - f x)) by an average of exp (t π / 2 · f' (R θ (x, y)).1 (R θ (x, y)).2).
The product μ ⊗ μ is invariant under each rotation R θ
(ProbabilityTheory.IsGaussian.map_rotation_eq_self), so after integration every θ
contributes the same amount, which is an exponential moment of the Gaussian variable
f' x y in y. Jensen's inequality in y then gives the exponential-moment estimate
𝔼 exp (t (f - 𝔼 f)) ≤ exp (π² t² v / 8), and Chernoff's bound
(ProbabilityTheory.HasSubgaussianMGF.measure_ge_le) gives the tails.
Main results #
ProbabilityTheory.IsGaussian.lintegral_exp_mul_sub_le: the Maurey–Pisier inequality∫∫ exp (t (f y - f x)) dμ(x) dμ(y) ≤ ∫ exp (π² t² / 8 · Var[f' x; μ]) dμ(x).ProbabilityTheory.IsGaussian.hasSubgaussianMGF_sub_integral_of_variance_fderiv_le:f - 𝔼 fis sub-Gaussian with variance proxyπ² v / 4.ProbabilityTheory.IsGaussian.measureReal_setOf_le_abs_sub_integral_le_of_variance_fderiv_le: the two-sided tail bound.TauCeti.Probability.measureReal_stdGaussian_setOf_le_abs_sub_integral_le: the tail bound2 exp (-2 ε² / (π² K²))for the standard Gaussian measure and‖f'‖ ≤ K.
References #
- G. Pisier, Probabilistic methods in the geometry of Banach spaces, in Probability and Analysis (Varenna, 1985), Lecture Notes in Math. 1206, Springer, 1986, Theorem 2.2.
- F. Albiac and N. J. Kalton, Topics in Banach Space Theory, §13.2.
The Maurey–Pisier inequality. For a centred Gaussian measure μ and a C¹ function f,
∫∫ exp (t (f y - f x)) dμ(x) dμ(y) ≤ ∫ exp (π² t² / 8 · Var[f' x; μ]) dμ(x).
The Maurey–Pisier inequality when the variances of the derivative are at most v almost
everywhere.
A C¹ function whose derivative has variance at most v almost everywhere under a centred
Gaussian measure has finite exponential moments of every order.
A C¹ function whose derivative has variance at most v almost everywhere under a centred
Gaussian measure is integrable.
Gaussian concentration, exponential-moment form. If f is C¹ and its derivative has
variance at most v under a centred Gaussian measure μ at almost every point, then f - 𝔼 f is
sub-Gaussian with variance proxy π² v / 4: 𝔼 exp (t (f - 𝔼 f)) ≤ exp (π² t² v / 8).
Gaussian concentration. If f is C¹ and its derivative has variance at most v under
a centred Gaussian measure μ at almost every point, then for ε ≥ 0,
μ {x | ε ≤ |f x - 𝔼 f|} ≤ 2 exp (-2 ε² / (π² v)).
Under the standard Gaussian measure, a C¹ function f with ‖f' x‖ ≤ K everywhere (that
is, a C¹ function which is K-Lipschitz) is sub-Gaussian about its mean with variance proxy
π² K² / 4.
Gaussian concentration for the standard Gaussian measure. If f is C¹ with
‖f' x‖ ≤ K everywhere, then for ε ≥ 0,
γ {x | ε ≤ |f x - 𝔼 f|} ≤ 2 exp (-2 ε² / (π² K²)) for the standard Gaussian measure γ.