Convergence theory for minimizing movements schemes

Flavien Léger

joint works with Pierre-Cyril Aubin-Frankowski,

Gabriele Todeschi, François-Xavier Vialard

What I will present

Theory for minimizing movement schemes in infinite dimensions and in nonsmooth (nondifferentiable) settings, with a movement limiter given by a general cost function.

Main motivation: optimization on a space of measures \(\mathcal{P}(M)\):

minimize   \(E\colon \mathcal{P}(M)\to\mathbb{R}\cup\{+\infty\}\)

Typical scheme:

\mu_{n+1}\in\operatorname*{argmin}_{\mu\in\mathcal{P}(M)} E(\mu)+\frac{1}{2\tau}D(\mu,\mu_n)

where \(D(\mu,\nu)=\)

transport cost:   \(W_2^2(\mu,\nu)\),  \(\mathcal{T}_c(\mu,\nu)\),...

Bregman divergence:   \(\operatorname{KL}(\mu,\nu)\),...

Csiszár divergence:   \(\int_M (\sqrt{\mu}-\sqrt{\nu})^2\),...

...

What I will present

Theory for minimizing movement schemes in infinite dimensions and in nonsmooth (nondifferentiable) settings, with a movement limiter given by a general cost function.

1. Formulations for implicit and explicit schemes in a general setting

More (not covered): forward–backward schemes, alternating minimization

2. Theory for rates of convergence based on convexity along specific paths, and generalized “\(L\)-smoothness” (“\(L\)-Lipschitz gradients”) for explicit scheme

Setting

\inf_{y\in Y}D(x,y)=0

Minimize  \(E\colon X\to\mathbb{R}\cup\{+\infty\}\),   where \(X\) is a set (set of measures, metric space...).

Use   \(D\colon X\times Y\to\mathbb{R}\cup\{+\infty\}\),   where \(Y\) is another set (often \(X=Y\)).

\begin{aligned} x_n & \in\operatorname*{argmin}_{x\in X}E(x)+D(x,y_n)\\ y_{n+1}&\in\operatorname*{argmin}_{y\in Y} D(x_n,y) \end{aligned}

Algorithm

(Implicit scheme)

Explicit minimizing movements

E(x)
D\colon X\times Y\to\mathbb{R}\cup\{+\infty\}
E\colon X\to\mathbb{R}\cup\{+\infty\}

\(\exists h\colon Y\to\mathbb{R}\cup\{+\infty\}\)

D(x,y)+h(y)
=\inf_{y\in Y}

Definition.

\(E\) is c-concave if

E
E

c-concave

not c-concave

D(x,y)+h(y)

generalizes “\(L\)-smoothness”

Explicit minimizing movements

\begin{aligned} y_{n+1}&\in\operatorname*{argmin}_{y\in Y}D(x_n,y)+h(y)\\ x_{n+1} & \in\operatorname*{argmin}_{x\in X}D(x,y_{n+1}) \end{aligned}
E(x)

(majorize)

(minimize)

E
D\colon X\times Y\to\mathbb{R}\cup\{+\infty\}
E\colon X\to\mathbb{R}\cup\{+\infty\}

\(\exists h\colon Y\to\mathbb{R}\cup\{+\infty\}\)

D(x,y)+h(y)
=\inf_{y\in Y}

Definition.

\(E\) is c-concave if

Algorithm.

(Explicit scheme)

Assume \(E\) c-concave.

(L–Aubin-Frankowski '23)

D(x,y)+h(y)

Explicit minimizing movements

\(X,Y\) smooth manifolds,   \(D\in C^1(X\times Y)\),   \(E\in C^1(X)\) c-concave

\begin{aligned} -\nabla_{\!x} D(x_n,y_{n+1})&=-\nabla E(x_n)\\ \nabla_{\!x} D(x_{n+1},y_{n+1})&=0 \end{aligned}

Under certain assumptions, the explicit scheme can be written as

x_{n+1}-x_n=-\tau\nabla f(x_n)
D(x,y)=\frac{1}{2\tau}\lVert x-y\rVert^2
x_{n+1}=\exp_{x_n}(-\tau\nabla f(x_n))
D(x,y)=\frac{1}{2\tau}d_M^2(x,y)
\nabla u(x_{n+1})-\nabla u(x_n)=-\nabla f(x_n)
D(x,y)=u(x)-u(y)-\nabla u(y)(x-y)
x_{n+1}-x_n=-\nabla^2u(x_n)^{-1}\nabla f(x_n)
D(x,y)=u(y)-u(x)-\nabla u(x)(y-x)

More: nonsmooth mirror descent, convergence rates for Newton

2. Convergence rates

EVI and convergence rates

Definition.

\forall x\in X,\quad E(x_n)+D(x_n,y_n)+(1+\mu)D(x,y_{n+1})\leq E(x)+D(x,y_n)

(Csiszár–Tusnády ’84)

(L–Aubin-Frankowski ’23)

Evolution Variational Inequality (or five-point property):

(\mu\geq 0)
x_n \in\operatorname*{argmin}_{x\in X}E(x)+D(x,y_n), \quad y_{n+1} \in\operatorname*{argmin}_{y\in Y} D(x_n,y)

If \((x_n,y_n)\) satisfy the EVI then

E(x_n)\leq E(x)+\frac{C(x,x_0,y_0)}{n}

sublinear rates when \(\mu=0\)

exponential rates when \(\mu>0\)

E(x_n)\leq E(x)+\frac{C(x,x_0,y_0,\mu)}{(1+\mu)^n-1}

Theorem.

(L–Aubin-Frankowski '23)

(Ambrosio–Gigli–Savaré ’05)

Variational c-segments and NNCC spaces

⏵  \(s\mapsto (x(s),\bar y)\) is a variational c-segment if \(D(x(s),\bar y)\) is finite and 

⏵  \((X\times Y,D)\) is a space with nonnegative cross-curvature (NNCC space) if variational c-segments always exist.

\(X, Y\) two arbitrary sets,   \(D\colon X\times Y\to\mathbb{R}\cup\{\pm\infty\}\).

(1-s)[D(x(0),\bar y)-D(x(0),y)]+s[D(x(1),\bar y)-D(x(1),y)].
\forall y\in Y\!,\quad D(x(s),\bar y)-D(x(s),y)\leq
x(0)
x(1)
\bar y

Definition.

(L–Todeschi–Vialard '24)

More: origins in regularity of optimal transport

(Ma–Trudinger–Wang ’05)

(Trudinger–Wang ’09)

(Kim–McCann ’10)

convexity of the set of c-concave functions

(Figalli–Kim–McCann '11)

Examples

Gromov–Wasserstein

Kullback–Leibler divergence, Hellinger, Fisher–Rao costs are NNCC

\mathcal{T}_c(\mu,\nu)=\inf_{\pi\in\Pi(\mu,\nu)}\int c(x,y)\,d\pi

Transport costs

\((\mathbb{G}\times\mathbb{G},\operatorname{GW}^2)\) is NCCC

\((\mathcal{P}(X)\times\mathcal{P}(Y),\mathcal{T}_c)\)  NNCC \(\iff\) \((X\times Y,c)\)   NNCC

(Polish spaces, lsc cost)

Ex: \(W_2^2\) on \(\mathbb{R}^n\), on \(\mathbb{S}^n\)...

\(\mathbf{X}=[X,f,\mu]\)   and   \(\mathbf{Y}=[Y,g,\nu]\in\mathbb{G}\)

\operatorname{GW}^2(\mathbf{X},\mathbf{Y})=\inf_{\pi\in\Pi(\mu,\nu)}\int\lvert f(x,x')-g(y,y')\rvert^2\,d\pi(x,y)\,d\pi(x',y')\,.

(L–Todeschi–Vialard '24)

Variational c-segments \(\approx\) generalized geodesics

Any Hilbert or Bregman cost is NNCC

Properties of NNCC spaces

Stable by products

Stable by quotients with “equidistant fibers”

Stable under Gromov–Hausdorff convergence

Metric cost \(c(x,y)=d^2(x,y)\)     NNCC\(\implies\)PC

d^2(x(s),\bar y )\leq (1-s)d^2(x(0),\bar y)+s\,d^2(x(1),\bar y)-s(1-s)d^2(x(0),x(1))

(connect to: Ambrosio–Gigli–Savaré ’05)

(connect to: Kim–McCann '12)

(connect to: Loeper ’09)

(L–Todeschi–Vialard '24)

Convergence rates

Suppose that for each \(x\in X\) and \(n\geq 0\),

Then sublinear (\(\mu=0\)) or linear (\(\mu>0\)) convergence rates. 

⏵ there exists a variational c-segment \(s\mapsto (x(s),y_n)\) on \((X\times Y,D)\) with \(x(0)=x_n\) and \(x(1)=x\)

⏵ \(s\mapsto E(x(s))-\mu \,D(x(s),y_{n+1})\) is convex

⏵ \(\displaystyle\lim_{s\to 0^+}\frac{D(x(s),y_{n+1})}{s}=0\)

Theorem.

(L–Aubin-Frankowski '23)

x_n \in\operatorname*{argmin}_{x\in X}E(x)+D(x,y_n), \quad y_{n+1} \in\operatorname*{argmin}_{y\in Y} D(x_n,y)
E(x_n)\leq E(x)+\frac{C(x,x_0,y_0)}{n}
E(x_n)\leq E(x)+\frac{C(x,x_0,y_0,\mu)}{(1+\mu)^n-1}

Thank you!