Learning Interaction Laws in Particle- and Agent-based Systems · 2020-04-24 · Learning Interaction Laws. Estimation/Learning for ODE systems No randomness: this is an approximation

Learning Interaction Laws in Particle- and Agent-based Systems

Mauro Maggioni

Johns Hopkins University

Joint work with Fei Lu, Sui Tang, Ming Zhong, Jason Miller

Felix Munoz, https://www.youtube.com/watch?v=OxYn3e_imhA

10/2/13 10:30 AM

Page 1 of 1http://upload.wikimedia.org/wikipedia/commons/1/1c/Air_Force_Research_Laboratory.svg

AFOSR NSF

BBC Blue Planet

https://www.youtube.com/watch?v=15B8qN9dre4

From https://www.youtube.com/watch?v=gJhn7WmXWVY Felix Munoz, https://www.youtube.com/watch?v=OxYn3e_imhA

Problem: “Given trajectories of dynamical system of interacting agents, learnthe interaction rules.”Applications: biological systems, particle systems.Further goals: hypothesis testing for agent-based systems; transfer learning;agents on networks; collaborative and competitive games; learning dictionariesfor complex dynamical systems.

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Renewed interest in learning ODE’s and PDE’s in Applied Math and Engineer-ing e.g. S. Osher, H. Shae↵er, N. Kutz, Y. Kevrekidis, D. Giannakis, C. Shutte, R. Ward...; also in the ML community, e.g. Grzeszczuk, Chang, Battaglia,Tenenbaum, especially in view of applications to control/reinforcement learning.

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Learning Interaction Laws

Estimation/Learning for ODE systems

No randomness: this is an approximation problem.

Randomness: this is a statistical problem. Sources of randomness:· initial conditions x(0) are random, e.g ⇠i.i.d µ0, a prob. meas. on RD;· the observations are corrupted by noise.

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Suppose we have a system driven by of ODEs in the form

x(t) = f(x(t)) ,x 2 RD, f : RD ! RD

and we are given observations of positions and velocities

(x(m)(tl), x

(m)(tl))l=1,...,L;m=1,...,M ,

where 0 = t1 < · · · < tL = T , and m indexes trajectories corresponding to

di↵erent initial conditions at, say, t = 0.

Objective: construct an estimator f that is close to f .

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Given (x(m)(tl), x(m)(tl))l=1,...,L;m=1,...,M , with x(m)(0) ⇠i.i.d. µ0, we want toapproximate the unknown f in x(t) = f(x(t)).View this as a regression problem: in regression one is given pairs {(zi, f(zi) +⌘i)}ni=1, with zi ⇠i.i.d. ⇢, ⇢ a probability measure on RD, and outputs an esti-mator fn such that fn ! f as n ! +1.

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Curse of dimensionality: D in the denominator of the rate makes the rate tooslow to be useful in applications.

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Nonparametric regression

Well-understood problem; there exist nonparametric estimators that, only as-

suming f is (say) s-Holder and that ⇢ has compact support, satisfy

E[||fn � f ||L2(⇢)] Cn� s2s+D .

They may be constructed as piecewise polynomials.

Without further assumption, no other estimator exists that achieves a better

rate, in the min-max sense.

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Given (x(m)(tl), x(m)(tl))l=1,...,L;m=1,...,M , with x(m)(0) ⇠i.i.d. µ0, we want toapproximate the unknown f in x(t) = f(x(t)).View this as a regression problem: in regression one is given pairs {(zi, f(zi) +⌘i)}ni=1, with zi ⇠i.i.d. ⇢, ⇢ a probability measure on RD, and outputs an esti-mator fn such that fn ! f as n ! +1.

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Nonparametric regressionSuppose we have a system driven by of ODEs in the form

x(t) = f(x(t)) ,x 2 RD, f : RD ! RD


(x(m)(tl), x

(m)(tl))l=1,...,L;m=1,...,M ,




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x(t) = f(x(t)) ,x 2 RD, f : RD ! RD


(x(m)(tl), x

(m)(tl))l=1,...,L;m=1,...,M ,





Map our problem to regression? zi = x(m)(tl) and f(zi) = x(m)

(tl) (so i runsover the index sets for l and m).

Unfortunately: no independence over l (observations over the same deterministic

trajectory). There is independence, for each fixed l, over m.

Even if we pretended to have independence, without further assumptions on f ,besides s-Holder regularity, the best attainable rate is n� s

2s+D , where n = LM .

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x(t) = f(x(t)) ,x 2 RD, f : RD ! RD


(x(m)(tl), x

(m)(tl))l=1,...,L;m=1,...,M ,





Map our problem to regression? zi = x(m)(tl) and f(zi) = x(m)

(tl) (so i runsover the index sets for l and m).

Unfortunately: no independence over l (observations over the same deterministic

trajectory). There is independence, for each fixed l, over m.

Even if we pretended to have independence, without further assumptions on f ,besides s-Holder regularity, the best attainable rate is n� s

2s+D , where n = LM .

<latexit sha1_base64="Lbs69WneNr6KNlKfzJO6JksdQKc=">AAAD8XicbZJbT9RAFMdLVwXXC6CPvpxATXYjNLvERGIkkqgJD5JgwgIJu2ym01M60s40M9OFpem38MUHjfHVb+Ob38bTUiO3eTo51//5nQmyRBjb6/2ZcVt37t6bnbvffvDw0eP5hcUne0blmuOAq0Tpg4AZTITEgRU2wYNMI0uDBPeDk3dVfH+C2ggld+00w1HKjqWIBGeWXONFd3YolZAhSgvbLAPqC5lWVJ6CVaDxWKOpqt 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</latexit>

For a system of N agents in Rd, D = Nd is typically very large, making theabove unfeasible. Further assumptions are needed.

<latexit sha1_base64="tofjQsqRFOKFFIpvSg18ZNmuoAU=">AAACfXicbVFNb9NAEF2brxK+Ahy5jIgrAYosu6pUDq1UCVRxqgoibaUkROv1OFllP6zddSUryr/gl3Hjr3CBcZoDtMzp6b3ZmTdvi1pJH7LsZxTfuXvv/oOdh71Hj588fdZ//uLc28YJHAmrrLssuEclDY6CDAova4dcFwoviuWHTr+4QuelNV9DW+NU87mRlRQ8EDXrf58YK02JJsCJdcDBtz6gBltBcpoAn5PiQRpIJpqHRV Gsvqy/lckQko9Hp2UC0gONpXFKtUCLWlDczXEImi+lmUNYIPDCXiE0pkLuJRlL4aRxJNA67xtdd048cIdgEEss01l/kKXZpuA2yLdgwLZ1Nuv/mJRWNJrMCkVDx3lWh+mKuyCFwnVv0nisuVjSOWOChmv009UmvTXsElNCRddXlmLYsH+/WHHtfasL6uwi8De1jvyfNm5C9X66kqZuAhpxvahqFATbZYZQSociUG6l5MJJ8gpiwR0XgT6sRyHkN0++Dc730nw/3f+8Nzg+3Maxw16x1+wNy9kBO2af2BkbMcF+RRC9jd5Fv+PdeBin161xtH3zkv1T8cEfTMq/wA==</latexit>

Interesting extensions to:- higher-order systems,- stochastic systems,- agents of di↵erent types,- varying environment....

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<latexit sha1_base64="JkDiJj6eCZ/GraJ5tbJ/RRSShL0=">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</latexit>

Agent-based systemsParticle- and agent-based systems are driven by ODEs with special structure.

A widely used model:

x(m)i =

1

N

NX

i0=1

�(||x(m)i � x(m)

i0 ||)(x(m)i0 � x(m)

i )

Given observations {(xi, xi)}Ni=1 at di↵erent times {tl}Ll=1 and/or for di↵erent

initial conditions {x(m)(0)}Mm=1, we want to learn the interaction kernel �.

Di↵erent limits: N ! +1 (mean-field limit, joint work with M. Fornasier and

M. Bongini), M ! +1 (joint current work with F. Lu, M. Zhong and S. Tang).

<latexit sha1_base64="RTF4DIdwJ3Y138b8ptsJB3rYeuE=">AAAEsnicbVNLb9NAEHabAMW8WjhyGVEjEpG4SVUJVFGpQCkc2lJE01atU2u9XidL1mtrd500cvz/OHPj37DrhL5XsjSexzcz38wEKaNStVp/5+Yr1Xv3Hyw8tB89fvL02eLS80OZZAKTDk5YIo4DJAmjnHQUVYwcp4KgOGDkKBh8NvajIRGSJvxAjVPSjVGP04hipLTKX5r/7fGE8pBwBftIKIoZaQLiIaCe1jUNdghyLBWJJS 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</latexit>

The Mean-field limit

Considering the measure µN(t) = 1

N

PNi=1 �xi(t), we may let N ! +1 to obtain

(under suitable regularity assumptions on �) the mean field equations

@tµ(t) = �r ·✓✓

��0(|| · ||)|| · || ⇤ µ(t)

◆µ(t)

◆, µ(0) = µ0 .

This is also a gradient flow for the energy J (µ) =RRd⇥d �(||x�y||)dµ(x)dµ(y)

on the space of probability measures with Wasserstein distance.

This was studied in Inferring Interaction Rules from Observations of Evolutive

Systems I: The Variational Approach, M. Bongini, M. Fornasier, M. Hansen,

and MM, M3S, 2017<latexit sha1_base64="a209SLQMQUjieqn9EnzrBfG1JoI=">AAAE23icbVNLj9xEEHZ2BgjDI5tw5FLKGjEDsyN7OSRCRMpDoCzSLkuSfUjbs1bbLs+00u423e2dWF7nwgGEuPLHuPEz+AdUe0aQB63RuLq6uur7vqpOKymsi6K/rm0Nhu+8+97190cffPjRxze2b946sbo2GR5nWmpzlnKLUig8dsJJPKsM8jKVeJo+f+TPTy/RWKHVM9dUOC/5QolCZNyRK7m59TdTWqgclYNHWlmRoxFqAW 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</latexit><latexit sha1_base64="a209SLQMQUjieqn9EnzrBfG1JoI=">AAAE23icbVNLj9xEEHZ2BgjDI5tw5FLKGjEDsyN7OSRCRMpDoCzSLkuSfUjbs1bbLs+00u423e2dWF7nwgGEuPLHuPEz+AdUe0aQB63RuLq6uur7vqpOKymsi6K/rm0Nhu+8+97190cffPjRxze2b946sbo2GR5nWmpzlnKLUig8dsJJPKsM8jKVeJo+f+TPTy/RWKHVM9dUOC/5QolCZNyRK7m59TdTWqgclYNHWlmRoxFqAW 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</latexit><latexit sha1_base64="a209SLQMQUjieqn9EnzrBfG1JoI=">AAAE23icbVNLj9xEEHZ2BgjDI5tw5FLKGjEDsyN7OSRCRMpDoCzSLkuSfUjbs1bbLs+00u423e2dWF7nwgGEuPLHuPEz+AdUe0aQB63RuLq6uur7vqpOKymsi6K/rm0Nhu+8+97190cffPjRxze2b946sbo2GR5nWmpzlnKLUig8dsJJPKsM8jKVeJo+f+TPTy/RWKHVM9dUOC/5QolCZNyRK7m59TdTWqgclYNHWlmRoxFqAW 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</latexit><latexit sha1_base64="a209SLQMQUjieqn9EnzrBfG1JoI=">AAAE23icbVNLj9xEEHZ2BgjDI5tw5FLKGjEDsyN7OSRCRMpDoCzSLkuSfUjbs1bbLs+00u423e2dWF7nwgGEuPLHuPEz+AdUe0aQB63RuLq6uur7vqpOKymsi6K/rm0Nhu+8+97190cffPjRxze2b946sbo2GR5nWmpzlnKLUig8dsJJPKsM8jKVeJo+f+TPTy/RWKHVM9dUOC/5QolCZNyRK7m59TdTWqgclYNHWlmRoxFqAW 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</latexit>

Rewriting

xi =1

N

X

i0

�(||xi � xi0 ||)(xi0 � xi) =1

N

X

i0

�0(||xi � xi0 ||)||xi � xi0 ||

(xi � xi0)

we see this is the gradient flow of the energy JN (X) = 12N

PNi,i0=1 �(||xi�xi0 ||).

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Measures on pairwise distances

Nonparametric inference of interaction laws in systems of agents from trajectory data, Fei Lu, Ming Zhong, Sui Tang, and MM PNAS July 16, 2019

Observations: {(xi, xi)(m)

(tl)}N,L,Mi=1,l=1,m=1, where x(m)

(0) ⇠ µ0 for some µ0 on

Rd. Note that each state of the system is in RdN

.

All we want however is the one-dimensional interaction kernel � in the equations

x(m)i0 (t) =

1

N

NX

i0=1

�(||x(m)i0 (t)� x(m)

i (t)||| {z }

r(m)

ii0 (t)

)(x(m)i0 (t)� x(m)

i (t)) .

<latexit sha1_base64="wxAV2UIAxkMcrbVQkCB0NZyUvzc=">AAAEEXicjVPLbtQwFE1neJTwaAtLNhYN6kSajpKqEghRqYgNCxgKog+pnkZOctNYdeJgO52OMvkFNvwKGxYgxJYdO/4GOxOgrwWWIh2fe8+177lxWDAqlef9mut0r1y9dn3+hn3z1u07C4tLd3ckL0UE2xFnXOyFRAKjOWwrqhjsFQJIFjLYDY+em/juMQhJef5OTQoYZeQwpwmNiNJUsNRZwTmneQy5Qq9DCeK4CcgnyMFVr8 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</latexit>

For a fixed t = tl and m, we cannot just solve for �(rii0), as we have N(N�1)/2unknowns and only dN knowns. We have to leverage observations in time.

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At time scale [0, T ], we define the probability measure on R+:

⇢LT (r) := Ex(0)⇠µ0

1�N2

�L

LX

l=1

NX

i,i0=1,i<i0

�r(m)

ii0 (tl)(r) .

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Many short time trajectories learning

1 1.5 2 2.5 3 3.5r, pairwise distances

-1200

-1000

-800

-600

-400

-200

0

True Learned

Example. The Lennard Jones force is thederivative of the potential

VLJ(r) = 4✏⇣�

�r

�12 ��r

�6⌘.

Right figure: In blue the LJ �,superimposed to an empirical estimate of ⇢LT ,for a system of N = 7 agents, and L, T small.

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Measures on pairwise distancesObservations: {(xi, xi)

(m)(tl)}N,L,M

i=1,l=1,m=1, where x(m)(0) ⇠ µ0 for some µ0 on

Rd. Note that each state of the system is in RdN

.

All we want however is the one-dimensional interaction kernel � in the equations

x(m)i0 (t) =

1

N

NX

i0=1

�(||x(m)i0 (t)� x(m)

i (t)||| {z }

r(m)

ii0 (t)

)(x(m)i0 (t)� x(m)

i (t)) .

<latexit sha1_base64="wxAV2UIAxkMcrbVQkCB0NZyUvzc=">AAAEEXicjVPLbtQwFE1neJTwaAtLNhYN6kSajpKqEghRqYgNCxgKog+pnkZOctNYdeJgO52OMvkFNvwKGxYgxJYdO/4GOxOgrwWWIh2fe8+177lxWDAqlef9mut0r1y9dn3+hn3z1u07C4tLd3ckL0UE2xFnXOyFRAKjOWwrqhjsFQJIFjLYDY+em/juMQhJef5OTQoYZeQwpwmNiNJUsNRZwTmneQy5Qq9DCeK4CcgnyMFVr8 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</latexit>

At time scale [0, T ], we define the probability measure on R+:

⇢LT (r) := Ex(0)⇠µ0

1�N2

�L

LX

l=1

NX

i,i0=1,i<i0

�r(m)

ii0 (tl)(r) .

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The estimator

For H linear subspace, this is a least squares problem (Gauss, Legendre); thesubspace serves as a regularizer.

<latexit sha1_base64="ZnDRCgIOSXKHoHtTBEbne7Jdnvc=">AAACbnicbVFNixNBEO0Zv9b4lVXw4CIWZoUIS5jZi4IeFgTdg4cVzO5CEkJNpyZptqd77OpeiEOO/kFv/gYv/gR7skF016IPj1dVr6peF7VW7LPsR5Jeu37j5q2t2507d+/df9DdfnjMNjhJQ2m1dacFMmllaOiV13RaO8Kq0HRSnL1r8yfn5FhZ89kva5pUODeqVBJ9pKbdb2NjlZmR8fDeOtgdV+gXEnVzuNqFVhQdcCi4Rk l74BeKIT4ETcge+EtARwy1s3FgBf0PGJj34CPNycwcvXwTW+iPADC581iOrYKjedDo1Fdyg2m3lw2ydcBVkG9AT2ziaNr9Pp5ZGaq4t9TIPMqz2k8adF5JTavOODDFiWc4p1GEBiviSbO2awUvIjODMp5b2nj3mv27o8GKeVkVsbJ1gy/nWvJ/uVHw5etJo0wdPBl5MagMGryF1nuYKUfS62UEKJ2Ku4JcoEPp4w91ogn55ZOvguP9QZ4N8k/7vYO3Gzu2xI54LvoiF6/EgTgUR2IopPiZbCdPkp3kV/o4fZo+uyhNk03PI/FPpP3fNua62w==</latexit><latexit sha1_base64="ZnDRCgIOSXKHoHtTBEbne7Jdnvc=">AAACbnicbVFNixNBEO0Zv9b4lVXw4CIWZoUIS5jZi4IeFgTdg4cVzO5CEkJNpyZptqd77OpeiEOO/kFv/gYv/gR7skF016IPj1dVr6peF7VW7LPsR5Jeu37j5q2t2507d+/df9DdfnjMNjhJQ2m1dacFMmllaOiV13RaO8Kq0HRSnL1r8yfn5FhZ89kva5pUODeqVBJ9pKbdb2NjlZmR8fDeOtgdV+gXEnVzuNqFVhQdcCi4Rk l74BeKIT4ETcge+EtARwy1s3FgBf0PGJj34CPNycwcvXwTW+iPADC581iOrYKjedDo1Fdyg2m3lw2ydcBVkG9AT2ziaNr9Pp5ZGaq4t9TIPMqz2k8adF5JTavOODDFiWc4p1GEBiviSbO2awUvIjODMp5b2nj3mv27o8GKeVkVsbJ1gy/nWvJ/uVHw5etJo0wdPBl5MagMGryF1nuYKUfS62UEKJ2Ku4JcoEPp4w91ogn55ZOvguP9QZ4N8k/7vYO3Gzu2xI54LvoiF6/EgTgUR2IopPiZbCdPkp3kV/o4fZo+uyhNk03PI/FPpP3fNua62w==</latexit><latexit sha1_base64="ZnDRCgIOSXKHoHtTBEbne7Jdnvc=">AAACbnicbVFNixNBEO0Zv9b4lVXw4CIWZoUIS5jZi4IeFgTdg4cVzO5CEkJNpyZptqd77OpeiEOO/kFv/gYv/gR7skF016IPj1dVr6peF7VW7LPsR5Jeu37j5q2t2507d+/df9DdfnjMNjhJQ2m1dacFMmllaOiV13RaO8Kq0HRSnL1r8yfn5FhZ89kva5pUODeqVBJ9pKbdb2NjlZmR8fDeOtgdV+gXEnVzuNqFVhQdcCi4Rk l74BeKIT4ETcge+EtARwy1s3FgBf0PGJj34CPNycwcvXwTW+iPADC581iOrYKjedDo1Fdyg2m3lw2ydcBVkG9AT2ziaNr9Pp5ZGaq4t9TIPMqz2k8adF5JTavOODDFiWc4p1GEBiviSbO2awUvIjODMp5b2nj3mv27o8GKeVkVsbJ1gy/nWvJ/uVHw5etJo0wdPBl5MagMGryF1nuYKUfS62UEKJ2Ku4JcoEPp4w91ogn55ZOvguP9QZ4N8k/7vYO3Gzu2xI54LvoiF6/EgTgUR2IopPiZbCdPkp3kV/o4fZo+uyhNk03PI/FPpP3fNua62w==</latexit><latexit sha1_base64="ZnDRCgIOSXKHoHtTBEbne7Jdnvc=">AAACbnicbVFNixNBEO0Zv9b4lVXw4CIWZoUIS5jZi4IeFgTdg4cVzO5CEkJNpyZptqd77OpeiEOO/kFv/gYv/gR7skF016IPj1dVr6peF7VW7LPsR5Jeu37j5q2t2507d+/df9DdfnjMNjhJQ2m1dacFMmllaOiV13RaO8Kq0HRSnL1r8yfn5FhZ89kva5pUODeqVBJ9pKbdb2NjlZmR8fDeOtgdV+gXEnVzuNqFVhQdcCi4Rk l74BeKIT4ETcge+EtARwy1s3FgBf0PGJj34CPNycwcvXwTW+iPADC581iOrYKjedDo1Fdyg2m3lw2ydcBVkG9AT2ziaNr9Pp5ZGaq4t9TIPMqz2k8adF5JTavOODDFiWc4p1GEBiviSbO2awUvIjODMp5b2nj3mv27o8GKeVkVsbJ1gy/nWvJ/uVHw5etJo0wdPBl5MagMGryF1nuYKUfS62UEKJ2Ku4JcoEPp4w91ogn55ZOvguP9QZ4N8k/7vYO3Gzu2xI54LvoiF6/EgTgUR2IopPiZbCdPkp3kV/o4fZo+uyhNk03PI/FPpP3fNua62w==</latexit>

Observations: {(x(m)i , x(m)

i )(tl)}N,L,MI=1,l=1,m=1, for M di↵erent IC’s, from

x(m)i (t) =

1

N

X

i0

�(||x(m)i0 (t)� x(m)

i (t)||)(x(m)i0 (t)� x(m)

i (t)) =: f�(x(m)i (t)) .

<latexit sha1_base64="phOYvYEbousRhjBrvUglkiDJORs=">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</latexit>

Consider the empirical error functional

EL,M (') :=1

LMN

L,M,NX

l,m,i=1

��x(m)i (tl)� f'(x

(m)i (tl))

��2 .

Our estimator is defined as a minimizer of EL,M over ' 2 H, a suitable hypoth-

esis space of functions on R+, with dim(H) = n (with n = n(M)):

�L,M,H := arg min'2H

EL,M (') .<latexit sha1_base64="x1etkuIn8sSjZ5PGdHlFjeJJqgY=">AAAEQXiclVNLb9NAEHYTHiW8WjhyWVEjOSJEcTiAKiJVqpB6aKAg+pDq1Fqv1/Gq3rW1uy4Ny/41LvwDbty5cAAhrlwYOymkLT2wp/HMfPPNfDOOiowp3et9Xmg0L12+cnXxWuv6jZu3bi8t39lReSkJ3SZ5lsu9CCuaMUG3NdMZ3SskxTzK6G50uF7Fd4+oVCwXb/SkoCOOx4IljGANrnC5sRuInImYCo3Wc6FYTCXSKUWUF0 xCWoaolLlESSlIBcFZy3VbAcc6haB5bkOz2RlazwSaHuu6ISNpbIMjLIuU2TZaHaAgkZgY35rN4QsbqJKHJuvwDhsg3x5U+A64IzYO3gdxrk1dPUrMsbUhOzAeb1tPh1n70UkgAdYL+Dz0Fz2FhqwGt6cEB/2g04UJXpYSUaUZJMN0TKGYJqBhjLBCGHEmGGfvQIs8Qe65aV2Ug6gQmJIGTPxJ2bBuBwqokmkMS0DppMhBTwUMqsCEVgVPtFQoF7PqUWRe2/AhYN8ynaKYcdebq9keCBd5dcgVA+EN2257FRYxr0KUldSaIMU6qISYttqZK2JXB/WX5AbLsQ1gyNBcpOPpmex/LHyqb7i00uv26ofOG/7MWHFmbytc+gS7JyWHSyQZVmrf7xV6BJ1qRjJqW0GpKAh4iMd0H0yBOVUjU/Nb9AA8MUqqO83hkmvvPMJgrtSER5BZDaLOxirnv2L7pU6ejgwTRampIFOipMyQzlH1O8GiJCU6m4CBiWTQKyIphmvX8NNVIvhnRz5v7PS7/uNu/1V/Ze3ZTI5F555z3/Ec33nirDkbzpaz7ZDGh8aXxrfG9+bH5tfmj+bPaWpjYYa565x6zV+/Adgtblo=</latexit>

Coercivity condition

Lemma. Coercivity =) unique minimizer of limM!+1 EL,M (') over ' 2 H<latexit sha1_base64="1ouX9bf3ii2TrnoyeT1jz3t1cGo=">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</latexit><latexit sha1_base64="1ouX9bf3ii2TrnoyeT1jz3t1cGo=">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</latexit><latexit sha1_base64="1ouX9bf3ii2TrnoyeT1jz3t1cGo=">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</latexit><latexit sha1_base64="1ouX9bf3ii2TrnoyeT1jz3t1cGo=">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</latexit>

EL,M (') :=1

LMN

L,M,NX

l,m,i=1

��x(m)i (tl)� f'(x

(m)i (tl))

��2,


EL,M (') .<latexit sha1_base64="4JaLFjl/emTPRdZ2c+tgG1hq0vs=">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</latexit><latexit sha1_base64="4JaLFjl/emTPRdZ2c+tgG1hq0vs=">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</latexit><latexit sha1_base64="4JaLFjl/emTPRdZ2c+tgG1hq0vs=">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</latexit><latexit sha1_base64="4JaLFjl/emTPRdZ2c+tgG1hq0vs=">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</latexit>

We shall assume that the unknown interaction kernel � is in the admissible classKR,S := {' 2 C1(R+) : supp' ⇢ [0, R], supr2[0,R] |'(r)|+ |'0(r)| S}.Coercivity condition: 8' : '(·)· 2 H, for cL > 0

cL,N,Hk'(·) · k2L2(⇢LT )

1

NL

L,NX

l,i=1

E�� 1

N

NX

i0=1

'(rii0(tl))rii0(tl)��2 .

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The coercivity constant cL,N,H also controls the condition number of the least

squares problem yielding �L,M,H.<latexit sha1_base64="9t2lcdmg2nvf/j27x+woAUgq4L0=">AAACh3icbVFNb9QwEHVCoe3ytcCRi8UuEkLVkpSq9MChqJceABWp21barFYTZ7Kx6tipPakURftX+FHc+Dd10j3QlpEsPb2Z8Zt5k1ZKOoqiv0H4aOPxk82t7cHTZ89fvBy+en3mTG0FToVRxl6k4FBJjVOSpPCisghlqvA8vTzq8ufXaJ00+pSaCuclLLXMpQDy1GL4O9FG6gw18dMCuTBohbyW1HioHYHnx2LRft/5uZOUQI UA1R6vVmMOypmuhqxRjlPfqzPZ/cp1XaZoucl7XiE44u6qBouOV9b42UreSFSZ1Es+TgqgpCpkp/LjrspkMRxFk6gP/hDEazBi6zhZDP8kmRF16RcSCpybxVFF8xYsSaFwNUhqhxWIS1jizEMNJbp52/u44u89k/HcWP/84j37b0cLpXNNmfrKbkp3P9eR/8vNasoP5q3UVU2oxa1QXitOhndH4Zm0KEg1HoCw3kTBRQEWBPnTDbwJ8f2VH4Kz3UkcTeJfu6PDr2s7tthb9o59YDH7wg7ZMTthUyaCjeBj8DnYC7fDT+F+eHBbGgbrnjfsToTfbgBBl8Rg</latexit><latexit sha1_base64="9t2lcdmg2nvf/j27x+woAUgq4L0=">AAACh3icbVFNb9QwEHVCoe3ytcCRi8UuEkLVkpSq9MChqJceABWp21barFYTZ7Kx6tipPakURftX+FHc+Dd10j3QlpEsPb2Z8Zt5k1ZKOoqiv0H4aOPxk82t7cHTZ89fvBy+en3mTG0FToVRxl6k4FBJjVOSpPCisghlqvA8vTzq8ufXaJ00+pSaCuclLLXMpQDy1GL4O9FG6gw18dMCuTBohbyW1HioHYHnx2LRft/5uZOUQI UA1R6vVmMOypmuhqxRjlPfqzPZ/cp1XaZoucl7XiE44u6qBouOV9b42UreSFSZ1Es+TgqgpCpkp/LjrspkMRxFk6gP/hDEazBi6zhZDP8kmRF16RcSCpybxVFF8xYsSaFwNUhqhxWIS1jizEMNJbp52/u44u89k/HcWP/84j37b0cLpXNNmfrKbkp3P9eR/8vNasoP5q3UVU2oxa1QXitOhndH4Zm0KEg1HoCw3kTBRQEWBPnTDbwJ8f2VH4Kz3UkcTeJfu6PDr2s7tthb9o59YDH7wg7ZMTthUyaCjeBj8DnYC7fDT+F+eHBbGgbrnjfsToTfbgBBl8Rg</latexit><latexit sha1_base64="9t2lcdmg2nvf/j27x+woAUgq4L0=">AAACh3icbVFNb9QwEHVCoe3ytcCRi8UuEkLVkpSq9MChqJceABWp21barFYTZ7Kx6tipPakURftX+FHc+Dd10j3QlpEsPb2Z8Zt5k1ZKOoqiv0H4aOPxk82t7cHTZ89fvBy+en3mTG0FToVRxl6k4FBJjVOSpPCisghlqvA8vTzq8ufXaJ00+pSaCuclLLXMpQDy1GL4O9FG6gw18dMCuTBohbyW1HioHYHnx2LRft/5uZOUQI UA1R6vVmMOypmuhqxRjlPfqzPZ/cp1XaZoucl7XiE44u6qBouOV9b42UreSFSZ1Es+TgqgpCpkp/LjrspkMRxFk6gP/hDEazBi6zhZDP8kmRF16RcSCpybxVFF8xYsSaFwNUhqhxWIS1jizEMNJbp52/u44u89k/HcWP/84j37b0cLpXNNmfrKbkp3P9eR/8vNasoP5q3UVU2oxa1QXitOhndH4Zm0KEg1HoCw3kTBRQEWBPnTDbwJ8f2VH4Kz3UkcTeJfu6PDr2s7tthb9o59YDH7wg7ZMTthUyaCjeBj8DnYC7fDT+F+eHBbGgbrnjfsToTfbgBBl8Rg</latexit><latexit sha1_base64="9t2lcdmg2nvf/j27x+woAUgq4L0=">AAACh3icbVFNb9QwEHVCoe3ytcCRi8UuEkLVkpSq9MChqJceABWp21barFYTZ7Kx6tipPakURftX+FHc+Dd10j3QlpEsPb2Z8Zt5k1ZKOoqiv0H4aOPxk82t7cHTZ89fvBy+en3mTG0FToVRxl6k4FBJjVOSpPCisghlqvA8vTzq8ufXaJ00+pSaCuclLLXMpQDy1GL4O9FG6gw18dMCuTBohbyW1HioHYHnx2LRft/5uZOUQI UA1R6vVmMOypmuhqxRjlPfqzPZ/cp1XaZoucl7XiE44u6qBouOV9b42UreSFSZ1Es+TgqgpCpkp/LjrspkMRxFk6gP/hDEazBi6zhZDP8kmRF16RcSCpybxVFF8xYsSaFwNUhqhxWIS1jizEMNJbp52/u44u89k/HcWP/84j37b0cLpXNNmfrKbkp3P9eR/8vNasoP5q3UVU2oxa1QXitOhndH4Zm0KEg1HoCw3kTBRQEWBPnTDbwJ8f2VH4Kz3UkcTeJfu6PDr2s7tthb9o59YDH7wg7ZMTthUyaCjeBj8DnYC7fDT+F+eHBbGgbrnjfsToTfbgBBl8Rg</latexit>

Bias/variance trade-offEL,M (') :=

1

LMN

L,M,NX

l,m,i=1

��x(m)i (tl)� f'(x

(m)i (tl))

��2,


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H<latexit sha1_base64="4RtK6QM+0V0Vok7kz4/ND513bWM=">AAAB9HicbVBNTwIxFHyLX4hfqEcvjWDiieziQQ8eSLxwxETABDakW7rQ0HbXtktCNvwOLx40xqs/xpv/xi7sQcFJmkxm3subThBzpo3rfjuFjc2t7Z3ibmlv/+DwqHx80tFRoghtk4hH6jHAmnImadsww+ljrCgWAafdYHKX+d0pVZpF8sHMYuoLPJIsZAQbK/nVvsBmTDBPm/PqoFxxa+4CaJ14OalAjtag/NUfRiQRVBrCsdY9z42Nn2JlGOF0XuonmsaYTPCI9iyVWFDtp4vQc3RhlSEKI2WfNGih/t5IsdB6JgI7mWXUq14m/uf1EhPe+CmTcWKoJMtDYcKRiVDWABoyRYnhM0swUcxmRWSMFSbG9lSyJXirX14nnXrNu6rV7+uVxm1eRxHO4BwuwYNraEATWtAGAk/wDK/w5kydF+fd+ViOFpx85xT+wPn8ATcYkbY=</latexit>

L2(⇢LT )<latexit sha1_base64="f9tdKc5vOeacQn2AYYqkeuWs1kg=">AAAB+HicbVC7TsMwFHV4lvJogJHFokUqS5WEAQaGSiwMHYrUl9SmkeO6rVXHjmwHqUT9EhYGEGLlU9j4G9w2A7Qc6UpH59yre+8JY0aVdpxva2Nza3tnN7eX3z84PCrYxyctJRKJSRMLJmQnRIowyklTU81IJ5YERSEj7XByN/fbj0QqKnhDT2PiR2jE6ZBipI0U2IVSre+Ve3Isgka/dlkK7KJTcRaA68TNSBFkqAf2V28gcBIRrjFDSnVdJ9Z+iqSmmJFZvpcoEiM8QSPSNZSjiCg/XRw+gxdGGcChkKa4hgv190SKIqWmUWg6I6THatWbi/953UQPb/yU8jjRhOPlomHCoBZwngIcUEmwZlNDEJbU3ArxGEmEtckqb0JwV19eJy2v4l5VvAevWL3N4siBM3AOysAF16AK7kEdNAEGCXgGr+DNerJerHfrY9m6YWUzp+APrM8fkDWRtA==</latexit>

�<latexit sha1_base64="YH8YoR1OUrOE79cnNfk06KX7kK0=">AAAB7XicbVA9SwNBEJ3zM8avqKXNYiJYhbtYaGERsLGMYD4gOcLeZi9Zs7e77O4J4ch/sLFQxNb/Y+e/cZNcoYkPBh7vzTAzL1KcGev7397a+sbm1nZhp7i7t39wWDo6bhmZakKbRHKpOxE2lDNBm5ZZTjtKU5xEnLaj8e3Mbz9RbZgUD3aiaJjgoWAxI9g6qVXpqRGr9Etlv+rPgVZJkJMy5Gj0S1+9gSRpQoUlHBvTDXxlwwxrywin02IvNVRhMsZD2nVU4ISaMJtfO0XnThmgWGpXwqK5+nsiw4kxkyRynQm2I7PszcT/vG5q4+swY0KllgqyWBSnHFmJZq+jAdOUWD5xBBPN3K2IjLDGxLqAii6EYPnlVdKqVYPLau2+Vq7f5HEU4BTO4AICuII63EEDmkDgEZ7hFd486b14797HonXNy2dO4A+8zx/Lr46X</latexit>

�<latexit sha1_base64="PzcUNFOPDX6T1JidKB7SftwmObQ=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69BFvBU0nqQQ8eCl48VrCt2ISy2W6apZtN2J0IJfRfePGgiFf/jTf/jds2B219MPB4b4aZeUEquEbH+bZKa+sbm1vl7crO7t7+QfXwqKuTTFHWoYlI1ENANBNcsg5yFOwhVYzEgWC9YHwz83tPTGmeyHucpMyPyUjykFOCRnqsexFBL414fVCtOQ1nDnuVuAWpQYH2oPrlDROaxUwiFUTrvuuk6OdEIaeCTStepllK6JiMWN9QSWKm/Xx+8dQ+M8rQDhNlSqI9V39P5CTWehIHpjMmGOllbyb+5/UzDK/8nMs0QybpYlGYCRsTe/a+PeSKURQTQwhV3Nxq04goQtGEVDEhuMsvr5Jus+FeNJp3zVrruoijDCdwCufgwiW04Bba0AEKEp7hFd4sbb1Y79bHorVkFTPH8AfW5w/WVpBY</latexit>

bias

variancePH�

<latexit sha1_base64="Zx88y8ijpLOLyNSqbRDdHW5pPOA=">AAAB/nicbVDLSsNAFL3xWesrKq7cBFvBVUnqQhcuCm66rGAf0IQwmU7aoZNJmJkIJQT8FTcuFHHrd7jzb5y0WWjrgYHDOfdyz5wgYVQq2/421tY3Nre2KzvV3b39g0Pz6Lgn41Rg0sUxi8UgQJIwyklXUcXIIBEERQEj/WB6V/j9RyIkjfmDmiXEi9CY05BipLTkm6f1jp+5EVITjFjWznM3mdC6b9bshj2HtUqcktSgRMc3v9xRjNOIcIUZknLo2InyMiQUxYzkVTeVJEF4isZkqClHEZFeNo+fWxdaGVlhLPTjypqrvzcyFEk5iwI9WQSVy14h/ucNUxXeeBnlSaoIx4tDYcosFVtFF9aICoIVm2mCsKA6q4UnSCCsdGNVXYKz/OVV0ms2nKtG875Za92WdVTgDM7hEhy4hha0oQNdwJDBM7zCm/FkvBjvxsdidM0od07gD4zPH/2qlXs=</latexit>

bias

variance

Pick dimH an increasing function of M ,to attain the minimum of the sum ofbias (squared) and variance.

<latexit sha1_base64="w0wh09Hgj9PNrelbp/KwTKEEISg=">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</latexit>

+ coercivity<latexit sha1_base64="9hKYDy/1zGXwaFmAIsRz73NCQI8=">AAAB83icbVA9SwNBEJ2LXzF+RS1tFoMgCOEuFlpYBGwsI5gPSI6wt5lLluztHbt7gSPkb9hYKGLrn7Hz37hJrtDEBwOP92aYmRckgmvjut9OYWNza3unuFva2z84PCofn7R0nCqGTRaLWHUCqlFwiU3DjcBOopBGgcB2ML6f++0JKs1j+WSyBP2IDiUPOaPGSr0rwmJUjE+4yfrlilt1FyDrxMtJBXI0+uWv3iBmaYTSMEG17npuYvwpVYYzgbNSL9WYUDamQ+xaKmmE2p8ubp6RC6sMSBgrW9KQhfp7YkojrbMosJ0RNSO96s3F/7xuasJbf8plkhqUbLkoTAUxMZkHQAZcITMis4Qyxe2thI2ooszYmEo2BG/15XXSqlW962rtsVap3+VxFOEMzuESPLiBOjxAA5rAIIFneIU3J3VenHfnY9lacPKZU/gD5/MHv9GReg==</latexit>

Unlike regression, we do not have access to values of �, but only observationsthat are linear functions of �; coercivity allows us to invert in a stable fashion.

<latexit sha1_base64="mTBTTiCZctjAcDyUZ+Q/A/rD3Ng=">AAACn3icbVFNb9NAEF2brxI+msIRDgMpEocqssMBpHKo4AASCAWJtEVJFI3X43iV9a61H66iqH+LH8KNf8PaiRC0zGVH783bmXmT1VJYlyS/ovjGzVu37+zd7d27/+Dhfv/g0anV3nCacC21Oc/QkhSKJk44See1IawySWfZ6n3LnzVkrNDqm1vXNK9wqUQhOLoALfo/ZkoLlZNyMFFSrAgMLQ3ZVnAEFwS5BqUdlNgQIOeBAa ehQenJgi7gcFaX4vAIMu9AK7kGnVkyTfd9KC3RARqCdj40UHjFt8wf6TFwTYaLRrg1oJT6woLvmggVJnfhAQTrMKwEBdoyyIeL/iAZJl3A9STdJQO2i/Gi/3OWa+6rsCiXaO00TWo336Bxgku67M28pRr5Cpc0DanCiux80/l7CS8CkkOhw/w6GNWhfys2WFm7rrJQWaEr7VWuBf/HTb0r3sw3QtXekeLbRoWX7fLtsSAXhrgLpuYCuRFhVuAlGuQunLQXTEivrnw9OR0N01fD0dfR4OTtzo499oQ9Zy9Zyl6zE/aRjdmE8ehp9C76FH2On8Uf4i/xeFsaRzvNY/ZPxN9/A31PzMg=</latexit>

decreases as dimH increases; depends only onapproximation properties of H

<latexit sha1_base64="B7nOwL5uGOtRZCc032SSun24waQ=">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</latexit>

increases as dimH increases, for fixed M ;measures randomnesso of � 2 H

<latexit sha1_base64="Kad7QFTW0WpDPmGDdbAiXo/5sD4=">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</latexit>

KR,S<latexit sha1_base64="tMBdQYp4cZvwmCxE4Yurxt89nyo=">AAAB/HicbVC7TsMwFHXKq5RXoCOLRYvEgKqkDDAwVGJBYimPPqQ2ihzXaa06TmQ7SFEUfoWFAYRY+RA2/ganzQAtR7J0dM69usfHixiVyrK+jdLK6tr6RnmzsrW9s7tn7h90ZRgLTDo4ZKHoe0gSRjnpKKoY6UeCoMBjpOdNr3K/90iEpCF/UElEnACNOfUpRkpLrlmtDwOkJhix9CZz07vT+6zumjWrYc0Al4ldkBoo0HbNr+EoxHFAuMIMSTmwrUg5KRKKYkayyjCWJEJ4isZkoClHAZFOOgufwWOtjKAfCv24gjP190aKAimTwNOTeVK56OXif94gVv6Fk1IexYpwPD/kxwyqEOZNwBEVBCuWaIKwoDorxBMkEFa6r4ouwV788jLpNhv2WaN526y1Los6yuAQHIETYINz0ALXoA06AIMEPINX8GY8GS/Gu/ExHy0ZxU4V/IHx+QP5+5RO</latexit>

Main TheoremTheorem. Let {Hn}n ✓ H be a sequence of subspaces of L1

[0, R], with

dim(Hn) c0n and inf'2Hn k'(·)� �(·)kL1([0,R]) c1n�s, for some constants

c0, c1, s > 0. It exists, for example, if � is s-Holder regular.

Choose n⇤ = (M/logM)1

2s+1 : then for some C = C(c0, c1, R, S)

E[kb�L,M,Hn⇤ (·) ·��(·) · kL2(⇢TL)]

C

cL,N,H

✓logM

M

◆ s2s+1

.<latexit sha1_base64="mSD9IRDIH6WlwO8G8A1ZVPGe2jA=">AAAEjXicbVNtb9s2EFZjb+u0l6bbx305NBpgd4onp1i7Fm1RwBiWAcnQdUlbwHQEijpZRCXSIam2AcNfs3+0b/s3Iy0nTdYdIOJ4r8891BWrhmuTZf/c2BoMP/n0s5ufx198+dXXt7Zvf/NSy04xPGaykep1QTU2XOCx4abB1yuFtC0afFW8mQX/q7eoNJfiyJytcNHSpeAVZ9R4U3576y8iJBclCmNJUcFRjVJh6ybxARpIiC UtNTWjjd13uSDh012h0eDpFU8CBQIFjacdCoYgKwhRK8pQh0tycGIJF5U5c/MsfbFIUnjHTQ2+QSiiWlvy1o2u9RoDafAUWJ6BSICKEuIk1MgteUvVqubgb3AtxZHzjW9EWCnNeJd80Ml5bg9OehSjNYqxg4seUxAndlc7D6ySCrRsEZgU2lBhNCQeRBqiUtDwNEsmEP9mAN/759F9Ar6n7arBFLgfNjRNgPs8nezukzuyKVGBwmXXUDWJ40vGYVZLqdHTIPK7T0b20P1oSSOXcOjGnrBKUWanzu7pH6bOJY/A1Cg+4EtmT2ajgGwN7EX65ziJSYFLLixt+FLcdXFPT1HYX9ycnAN5x0usqQkAPRnpYXqVPutBOHfBVjiv0RfOnsO9EVG1zA9OjjyDi57CHuzMWRYK/361sHMhpDKjPmYzoB+WKL6szeWk+mJSkk6AoCgvx8i3d7JJthb4WJlulJ1oI8/z7b9JKVnXeo5ZQ7WeT7OVWViqDGcNuph0Gv3P+YYuce5VQVvUC7veJgffe0u5prmS/o3W1qsZlrZan7WFjwxD6v/6gvH/fPPOVD8vLBerzvgt6RtVXQNGQlhNKLlCZpozr1CmuMcKrKaeGuMXuCfhYZD7lyN/rLzcm0zvTe79sbfz7PGGjpvRd9GdaBRNowfRs2g/eh4dR2wQD7LBw8Gj4a3hT8PHw6d96NaNTc630TUZ/vovbaF2lg==</latexit>

In the end solving the minimization problem is just a least-squares problem in n

dimensions. Algorithms for computing the estimator run in time O(N2Ld ·M),

and we expect to be able to also run them in online fashion.

<latexit sha1_base64="OEIIYdLd7sy3Cj+EEww8fFyGYJo=">AAAC1HicbVLJbhNBEO0ZtmA2A0cuJWykIIFlW5HgwCGIC0gsiYSdSLaJ2j01dpNehq6eBGMigRBXPo4b38BPUDMxAhLq9PrV/qqnhdEUu90fSXrm7LnzF9YuNi5dvnL1WvP6jSH5MigcKG982J1KQqMdDqKOBneLgNJODe5M959U/p0DDKS9ex0XBU6snDmdayUjU3vNn2PntcvQRXjmIM4R0GVA3hxoN6vfVjtt9Yc6Horgub IFTfC2pAgSDEqK9+ldKQPSH7+DtmtDpi26qjd14LGZ+aDj3BLkPoDytijj7yZIUVsZmQ+lq7L5idB+tf7yTf95NlaZj/DibvseSJ7ukOPfF6giRA9TBMktKygN+TqfK9YjeFfJArmkOc/Q2Wu2up1ubXAa9FagJVa2tdf8Ps68KnmHqIwkGvW6RZwsZYhaGTxqjEvCQqp9OcMRQyct0mRZH+UI7jCT1avmntWt2b8zltISLeyUI3nzOZ30VeT/fKMy5g8nS+1YPnTquFFemkqB6sIsemBtzIKBVKy4VqDmMkgV+R80WITeyZVPg2G/09vobGz3W5uPVnKsiVvitlgXPfFAbIqnYksMhEq2k8PkU/I5HaYf0y/p1+PQNFnl3BT/WPrtF7104QY=</latexit>

· The good: Rate in M is optimal, in fact even optimal in the case ofregression, where we would be given (rm,�(rm))Mm=1.

· The bad: no dependency on L. Numerical examples: suggest in somecases that e↵ective sample size is LM = #obs.

<latexit sha1_base64="3Qz5bTB4U5kqjkSmyGR7NzllBds=">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</latexit>

We chooseHn to be the space of piecewise linear functions on a uniform partitionof cardinality n of [0, Rmax] (estimated supp.⇢TL), for n = n⇤.

<latexit sha1_base64="zr06k3iws8XABJFrOU0WapJTtOE=">AAACoHicbVFNb9NAEF2brxK+AhwR0qgxUkGVZVdI9ABSpV5yKKJFTVMRB2u9Hjerrnet3TUQWfld/A9u/BvGaQ7QMqenNx9v5k3RKOl8kvwOwlu379y9t3V/8ODho8dPhk+fnTnTWoETYZSx5wV3qKTGiZde4XljkdeFwmlxedjnp9/QOmn0qV82OK/5hZaVFNwTlQ9/ZtpIXaL2MEUQC2McQpTV3C8EV914lesIvIECwS8QXM MFgqmgkSjwu6TiXplbqFot+pEOjAYOLYkYW0PDLW1FfN8kuC2l5kr6JUQ0l6holux+zjsS/LGaR7CDzksSxxJc2zRxlNmFyY++nkavd4EGUtsHnb+J4nw4SuJkHXATpBswYps4zoe/stKItqZLheLOzdKk8fOuX08oXA2y1iEdd8kvcEZQ8xrdvFsbvIJXxJRr/cqQU2v2746O184t64Iqe+fc9VxP/i83a321P++kblqPWlwJVa3qHe+/BaW0KLxaEuDCkpGCfsQtF55+OiAT0usn3wRne3H6Nt4/2RsdvN/YscVesG22w1L2jh2wMTtmEyaCl8FhcBR8DLfDcfgpPLkqDYNNz3P2T4Rf/gCbK8rN</latexit>

Main Theorem

12 13 14 15 16 17 18 19 20 21log2(M)

-9

-8.5

-8

-7.5

-7

-6.5

-6

-5.5

log 2(e

rror

)

Learning rate errors slope -0.36

Example. The Lennard Jones kernelis not admissible, yet since particles rarelyget very close to each other, we obtain aconvergence rate close to optimal.

<latexit sha1_base64="L7wefGRngeRSo2U1nws1rYphWuk=">AAACvnicbVFLi9RAEO7E1zq+Rj16KRwED0tI9qKghwERRDyssLO7MDMMlU5lpp1+hO7O7oaQPyl48N/YmQ26DwsaPr6qr+qr6rySwvk0/R3Fd+7eu/9g7+Ho0eMnT5+Nn784dqa2nGbcSGNPc3QkhaaZF17SaWUJVS7pJN9+6vMnZ2SdMPrINxUtFa61KAVHH6jV+NdCG6EL0r5d5CV8vkBVSeoSONoQfCOt0Rbw1WhysCWrSY 5GfxUgHLQLUq02vusACyWcE2HyPjTkwQnNCSq0XnAZ9BYtyQauNliHsuCuAS6NI/AGCPkGjN+Q3YdzApN7FBrwmoobHURr6ttb9PRPbSovFMpkNZ6kSboLuA2yAUzYEIer8c9FYXitQnsu0bl5llZ+2Q7eu9GidlQh3+Ka5gFqVOSW7e78HbwJTAGlseHt7AX2qqJF5Vyj8lCp0G/czVxP/i83r335ftkKXdU+LHs5qKxlv2n/l1AIS9yHmxYCuRXBK/ANWuQ+/PgoHCG7ufJtcHyQZGmSfT+YTD8O59hjr9hr9pZl7B2bsi/skM0Yjz5EGP2ItvE0LmMVm8vSOBo0L9m1iC/+AFLK2V8=</latexit><latexit sha1_base64="L7wefGRngeRSo2U1nws1rYphWuk=">AAACvnicbVFLi9RAEO7E1zq+Rj16KRwED0tI9qKghwERRDyssLO7MDMMlU5lpp1+hO7O7oaQPyl48N/YmQ26DwsaPr6qr+qr6rySwvk0/R3Fd+7eu/9g7+Ho0eMnT5+Nn784dqa2nGbcSGNPc3QkhaaZF17SaWUJVS7pJN9+6vMnZ2SdMPrINxUtFa61KAVHH6jV+NdCG6EL0r5d5CV8vkBVSeoSONoQfCOt0Rbw1WhysCWrSY 5GfxUgHLQLUq02vusACyWcE2HyPjTkwQnNCSq0XnAZ9BYtyQauNliHsuCuAS6NI/AGCPkGjN+Q3YdzApN7FBrwmoobHURr6ttb9PRPbSovFMpkNZ6kSboLuA2yAUzYEIer8c9FYXitQnsu0bl5llZ+2Q7eu9GidlQh3+Ka5gFqVOSW7e78HbwJTAGlseHt7AX2qqJF5Vyj8lCp0G/czVxP/i83r335ftkKXdU+LHs5qKxlv2n/l1AIS9yHmxYCuRXBK/ANWuQ+/PgoHCG7ufJtcHyQZGmSfT+YTD8O59hjr9hr9pZl7B2bsi/skM0Yjz5EGP2ItvE0LmMVm8vSOBo0L9m1iC/+AFLK2V8=</latexit><latexit sha1_base64="L7wefGRngeRSo2U1nws1rYphWuk=">AAACvnicbVFLi9RAEO7E1zq+Rj16KRwED0tI9qKghwERRDyssLO7MDMMlU5lpp1+hO7O7oaQPyl48N/YmQ26DwsaPr6qr+qr6rySwvk0/R3Fd+7eu/9g7+Ho0eMnT5+Nn784dqa2nGbcSGNPc3QkhaaZF17SaWUJVS7pJN9+6vMnZ2SdMPrINxUtFa61KAVHH6jV+NdCG6EL0r5d5CV8vkBVSeoSONoQfCOt0Rbw1WhysCWrSY 5GfxUgHLQLUq02vusACyWcE2HyPjTkwQnNCSq0XnAZ9BYtyQauNliHsuCuAS6NI/AGCPkGjN+Q3YdzApN7FBrwmoobHURr6ttb9PRPbSovFMpkNZ6kSboLuA2yAUzYEIer8c9FYXitQnsu0bl5llZ+2Q7eu9GidlQh3+Ka5gFqVOSW7e78HbwJTAGlseHt7AX2qqJF5Vyj8lCp0G/czVxP/i83r335ftkKXdU+LHs5qKxlv2n/l1AIS9yHmxYCuRXBK/ANWuQ+/PgoHCG7ufJtcHyQZGmSfT+YTD8O59hjr9hr9pZl7B2bsi/skM0Yjz5EGP2ItvE0LmMVm8vSOBo0L9m1iC/+AFLK2V8=</latexit><latexit sha1_base64="L7wefGRngeRSo2U1nws1rYphWuk=">AAACvnicbVFLi9RAEO7E1zq+Rj16KRwED0tI9qKghwERRDyssLO7MDMMlU5lpp1+hO7O7oaQPyl48N/YmQ26DwsaPr6qr+qr6rySwvk0/R3Fd+7eu/9g7+Ho0eMnT5+Nn784dqa2nGbcSGNPc3QkhaaZF17SaWUJVS7pJN9+6vMnZ2SdMPrINxUtFa61KAVHH6jV+NdCG6EL0r5d5CV8vkBVSeoSONoQfCOt0Rbw1WhysCWrSY 5GfxUgHLQLUq02vusACyWcE2HyPjTkwQnNCSq0XnAZ9BYtyQauNliHsuCuAS6NI/AGCPkGjN+Q3YdzApN7FBrwmoobHURr6ttb9PRPbSovFMpkNZ6kSboLuA2yAUzYEIer8c9FYXitQnsu0bl5llZ+2Q7eu9GidlQh3+Ka5gFqVOSW7e78HbwJTAGlseHt7AX2qqJF5Vyj8lCp0G/czVxP/i83r335ftkKXdU+LHs5qKxlv2n/l1AIS9yHmxYCuRXBK/ANWuQ+/PgoHCG7ufJtcHyQZGmSfT+YTD8O59hjr9hr9pZl7B2bsi/skM0Yjz5EGP2ItvE0LmMVm8vSOBo0L9m1iC/+AFLK2V8=</latexit>


· The bad: no dependency on L.<latexit sha1_base64="jU/zB7ammE84ZCg95eBasoSM71Y=">AAAC03icbVJdaxNBFJ1dv2r8ivroy8Ws0EIIu31RSoWCLz5YiNK0hWwMs7N3k6HzsczMpsQlKOKrf843/4K/wpk0ira97MDh3Htnzz13ilpw69L0ZxTfuHnr9p2tu5179x88fNR9/OTY6sYwHDEttDktqEXBFY4cdwJPa4NUFgJPirM3IX+yQGO5VkduWeNE0pniFWfUeWra/ZUrzVWJykEnL3DGVcsdSv4JV508oHGSs1K7ZA JHc4SZ1uUefKAOgStIDhPgFnTtuKSiH6iKMge4QPWHDaTzncyLBF2BwZlBG/T04XyOBuHcf7oRJRT+fh5ak20zlf28nvMAdnamrXydrT4eJoPrNBXUS1IaSqwxTMKWoP0d70K1J/7OM+320kG6DrgKsg3okU0Mp90fealZI705TFBrx1lau0lLjeNMBH8aizVlZ3SGYw8VlWgn7XonK3jhmRIqbfzx5q7ZfztaKq1dysJXSurm9nIukNflxo2rXk1arurG+VkvflQ1ApyGsGAouUHmxNIDygz3WoHNqfF78c+g403ILo98FRzvDrJ0kL3f7R3sb+zYIs/Ic7JNMvKSHJC3ZEhGhEXDaBF9jr7Eo7iNv8bfLkrjaNPzlPwX8fffo8LdqQ==</latexit><latexit sha1_base64="jU/zB7ammE84ZCg95eBasoSM71Y=">AAAC03icbVJdaxNBFJ1dv2r8ivroy8Ws0EIIu31RSoWCLz5YiNK0hWwMs7N3k6HzsczMpsQlKOKrf843/4K/wpk0ira97MDh3Htnzz13ilpw69L0ZxTfuHnr9p2tu5179x88fNR9/OTY6sYwHDEttDktqEXBFY4cdwJPa4NUFgJPirM3IX+yQGO5VkduWeNE0pniFWfUeWra/ZUrzVWJykEnL3DGVcsdSv4JV508oHGSs1K7ZA JHc4SZ1uUefKAOgStIDhPgFnTtuKSiH6iKMge4QPWHDaTzncyLBF2BwZlBG/T04XyOBuHcf7oRJRT+fh5ak20zlf28nvMAdnamrXydrT4eJoPrNBXUS1IaSqwxTMKWoP0d70K1J/7OM+320kG6DrgKsg3okU0Mp90fealZI705TFBrx1lau0lLjeNMBH8aizVlZ3SGYw8VlWgn7XonK3jhmRIqbfzx5q7ZfztaKq1dysJXSurm9nIukNflxo2rXk1arurG+VkvflQ1ApyGsGAouUHmxNIDygz3WoHNqfF78c+g403ILo98FRzvDrJ0kL3f7R3sb+zYIs/Ic7JNMvKSHJC3ZEhGhEXDaBF9jr7Eo7iNv8bfLkrjaNPzlPwX8fffo8LdqQ==</latexit><latexit sha1_base64="jU/zB7ammE84ZCg95eBasoSM71Y=">AAAC03icbVJdaxNBFJ1dv2r8ivroy8Ws0EIIu31RSoWCLz5YiNK0hWwMs7N3k6HzsczMpsQlKOKrf843/4K/wpk0ira97MDh3Htnzz13ilpw69L0ZxTfuHnr9p2tu5179x88fNR9/OTY6sYwHDEttDktqEXBFY4cdwJPa4NUFgJPirM3IX+yQGO5VkduWeNE0pniFWfUeWra/ZUrzVWJykEnL3DGVcsdSv4JV508oHGSs1K7ZA JHc4SZ1uUefKAOgStIDhPgFnTtuKSiH6iKMge4QPWHDaTzncyLBF2BwZlBG/T04XyOBuHcf7oRJRT+fh5ak20zlf28nvMAdnamrXydrT4eJoPrNBXUS1IaSqwxTMKWoP0d70K1J/7OM+320kG6DrgKsg3okU0Mp90fealZI705TFBrx1lau0lLjeNMBH8aizVlZ3SGYw8VlWgn7XonK3jhmRIqbfzx5q7ZfztaKq1dysJXSurm9nIukNflxo2rXk1arurG+VkvflQ1ApyGsGAouUHmxNIDygz3WoHNqfF78c+g403ILo98FRzvDrJ0kL3f7R3sb+zYIs/Ic7JNMvKSHJC3ZEhGhEXDaBF9jr7Eo7iNv8bfLkrjaNPzlPwX8fffo8LdqQ==</latexit><latexit sha1_base64="jU/zB7ammE84ZCg95eBasoSM71Y=">AAAC03icbVJdaxNBFJ1dv2r8ivroy8Ws0EIIu31RSoWCLz5YiNK0hWwMs7N3k6HzsczMpsQlKOKrf843/4K/wpk0ira97MDh3Htnzz13ilpw69L0ZxTfuHnr9p2tu5179x88fNR9/OTY6sYwHDEttDktqEXBFY4cdwJPa4NUFgJPirM3IX+yQGO5VkduWeNE0pniFWfUeWra/ZUrzVWJykEnL3DGVcsdSv4JV508oHGSs1K7ZA JHc4SZ1uUefKAOgStIDhPgFnTtuKSiH6iKMge4QPWHDaTzncyLBF2BwZlBG/T04XyOBuHcf7oRJRT+fh5ak20zlf28nvMAdnamrXydrT4eJoPrNBXUS1IaSqwxTMKWoP0d70K1J/7OM+320kG6DrgKsg3okU0Mp90fealZI705TFBrx1lau0lLjeNMBH8aizVlZ3SGYw8VlWgn7XonK3jhmRIqbfzx5q7ZfztaKq1dysJXSurm9nIukNflxo2rXk1arurG+VkvflQ1ApyGsGAouUHmxNIDygz3WoHNqfF78c+g403ILo98FRzvDrJ0kL3f7R3sb+zYIs/Ic7JNMvKSHJC3ZEhGhEXDaBF9jr7Eo7iNv8bfLkrjaNPzlPwX8fffo8LdqQ==</latexit>

Theorem. Let {Hn}n ✓ H be a sequence of subspaces of L1[0, R], with





E[kb�L,M,Hn⇤ (·) ·��(·) · kL2(⇢TL)]

C

cL,N,H

✓logM

M

◆ s2s+1


125 250 500 1000 2000 4000 8000L

8000

4000

2000

1000

500

250

125

63

32

1

M

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

Main Theorem


· The bad: no dependency on L.<latexit sha1_base64="jU/zB7ammE84ZCg95eBasoSM71Y=">AAAC03icbVJdaxNBFJ1dv2r8ivroy8Ws0EIIu31RSoWCLz5YiNK0hWwMs7N3k6HzsczMpsQlKOKrf843/4K/wpk0ira97MDh3Htnzz13ilpw69L0ZxTfuHnr9p2tu5179x88fNR9/OTY6sYwHDEttDktqEXBFY4cdwJPa4NUFgJPirM3IX+yQGO5VkduWeNE0pniFWfUeWra/ZUrzVWJykEnL3DGVcsdSv4JV508oHGSs1K7ZA JHc4SZ1uUefKAOgStIDhPgFnTtuKSiH6iKMge4QPWHDaTzncyLBF2BwZlBG/T04XyOBuHcf7oRJRT+fh5ak20zlf28nvMAdnamrXydrT4eJoPrNBXUS1IaSqwxTMKWoP0d70K1J/7OM+320kG6DrgKsg3okU0Mp90fealZI705TFBrx1lau0lLjeNMBH8aizVlZ3SGYw8VlWgn7XonK3jhmRIqbfzx5q7ZfztaKq1dysJXSurm9nIukNflxo2rXk1arurG+VkvflQ1ApyGsGAouUHmxNIDygz3WoHNqfF78c+g403ILo98FRzvDrJ0kL3f7R3sb+zYIs/Ic7JNMvKSHJC3ZEhGhEXDaBF9jr7Eo7iNv8bfLkrjaNPzlPwX8fffo8LdqQ==</latexit><latexit sha1_base64="jU/zB7ammE84ZCg95eBasoSM71Y=">AAAC03icbVJdaxNBFJ1dv2r8ivroy8Ws0EIIu31RSoWCLz5YiNK0hWwMs7N3k6HzsczMpsQlKOKrf843/4K/wpk0ira97MDh3Htnzz13ilpw69L0ZxTfuHnr9p2tu5179x88fNR9/OTY6sYwHDEttDktqEXBFY4cdwJPa4NUFgJPirM3IX+yQGO5VkduWeNE0pniFWfUeWra/ZUrzVWJykEnL3DGVcsdSv4JV508oHGSs1K7ZA JHc4SZ1uUefKAOgStIDhPgFnTtuKSiH6iKMge4QPWHDaTzncyLBF2BwZlBG/T04XyOBuHcf7oRJRT+fh5ak20zlf28nvMAdnamrXydrT4eJoPrNBXUS1IaSqwxTMKWoP0d70K1J/7OM+320kG6DrgKsg3okU0Mp90fealZI705TFBrx1lau0lLjeNMBH8aizVlZ3SGYw8VlWgn7XonK3jhmRIqbfzx5q7ZfztaKq1dysJXSurm9nIukNflxo2rXk1arurG+VkvflQ1ApyGsGAouUHmxNIDygz3WoHNqfF78c+g403ILo98FRzvDrJ0kL3f7R3sb+zYIs/Ic7JNMvKSHJC3ZEhGhEXDaBF9jr7Eo7iNv8bfLkrjaNPzlPwX8fffo8LdqQ==</latexit><latexit sha1_base64="jU/zB7ammE84ZCg95eBasoSM71Y=">AAAC03icbVJdaxNBFJ1dv2r8ivroy8Ws0EIIu31RSoWCLz5YiNK0hWwMs7N3k6HzsczMpsQlKOKrf843/4K/wpk0ira97MDh3Htnzz13ilpw69L0ZxTfuHnr9p2tu5179x88fNR9/OTY6sYwHDEttDktqEXBFY4cdwJPa4NUFgJPirM3IX+yQGO5VkduWeNE0pniFWfUeWra/ZUrzVWJykEnL3DGVcsdSv4JV508oHGSs1K7ZA JHc4SZ1uUefKAOgStIDhPgFnTtuKSiH6iKMge4QPWHDaTzncyLBF2BwZlBG/T04XyOBuHcf7oRJRT+fh5ak20zlf28nvMAdnamrXydrT4eJoPrNBXUS1IaSqwxTMKWoP0d70K1J/7OM+320kG6DrgKsg3okU0Mp90fealZI705TFBrx1lau0lLjeNMBH8aizVlZ3SGYw8VlWgn7XonK3jhmRIqbfzx5q7ZfztaKq1dysJXSurm9nIukNflxo2rXk1arurG+VkvflQ1ApyGsGAouUHmxNIDygz3WoHNqfF78c+g403ILo98FRzvDrJ0kL3f7R3sb+zYIs/Ic7JNMvKSHJC3ZEhGhEXDaBF9jr7Eo7iNv8bfLkrjaNPzlPwX8fffo8LdqQ==</latexit><latexit sha1_base64="jU/zB7ammE84ZCg95eBasoSM71Y=">AAAC03icbVJdaxNBFJ1dv2r8ivroy8Ws0EIIu31RSoWCLz5YiNK0hWwMs7N3k6HzsczMpsQlKOKrf843/4K/wpk0ira97MDh3Htnzz13ilpw69L0ZxTfuHnr9p2tu5179x88fNR9/OTY6sYwHDEttDktqEXBFY4cdwJPa4NUFgJPirM3IX+yQGO5VkduWeNE0pniFWfUeWra/ZUrzVWJykEnL3DGVcsdSv4JV508oHGSs1K7ZA JHc4SZ1uUefKAOgStIDhPgFnTtuKSiH6iKMge4QPWHDaTzncyLBF2BwZlBG/T04XyOBuHcf7oRJRT+fh5ak20zlf28nvMAdnamrXydrT4eJoPrNBXUS1IaSqwxTMKWoP0d70K1J/7OM+320kG6DrgKsg3okU0Mp90fealZI705TFBrx1lau0lLjeNMBH8aizVlZ3SGYw8VlWgn7XonK3jhmRIqbfzx5q7ZfztaKq1dysJXSurm9nIukNflxo2rXk1arurG+VkvflQ1ApyGsGAouUHmxNIDygz3WoHNqfF78c+g403ILo98FRzvDrJ0kL3f7R3sb+zYIs/Ic7JNMvKSHJC3ZEhGhEXDaBF9jr7Eo7iNv8bfLkrjaNPzlPwX8fffo8LdqQ==</latexit>

Numerical results suggest that thee↵ective sample size shouldscale linearly in L, at least up to a point.

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E[kb�L,M,Hn⇤ (·) ·��(·) · kL2(⇢TL)]

C

cL,N,H

✓logM

M

◆ s2s+1


Main Theorem

5 10 15 20 25 30N

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Coe

civi

ty c

onst

ant

N/(N-1) 2Gaussian

5 10 15 20 25 30N

N/(N-1) 2Uniform

5 10 15 20 25 30N

N/(N-1) 2Spherical Uniform

cL,N,H can be as small as N�1N2 , but in fact we conjecture that under some

general conditions it is independent of N when evaluated on compact subspacesH ⇢ L2(⇢LT ). We can prove this in special cases, for L = 1 and µ0 exchangeableGaussian.

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E[kb�L,M,Hn⇤ (·) ·��(·) · kL2(⇢TL)]

C

cL,N,H

✓logM

M

◆ s2s+1


Main TheoremTheorem. Let {Hn}n ✓ H be a sequence of subspaces of L1

[0, R], with





E[kb�L,M,Hn⇤ (·) ·��(·) · kL2(⇢TL)]

C

cL,N,H

✓logM

M

◆ s2s+1


Learning interaction kernels in heterogeneous systems of agents from multiple trajectories, F. Lu, MM, S. Tang, arxiv 1910.04832

This result may be extended to heterogeneous agent systems (e.g. prey-predator)[Lu, Tang, MM, on arXiV].

It may also be extended to multi-variable interaction kernels, depending on moreobservables than pairwise distances, as well as second-order systems (work inprogress).

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Errors on trajectories

Proposition. Assume b�(k · k)· 2 Lip(Rd), with Lipschitz constant CLip. LetbX(t) and X(t) be the solutions of systems with kernels b� and � respectively,started from the same initial condition. Then for each trajectory

supt2[0,T ]

kbX(t)�X(t)k2 2Te8T2C2

Lip

Z T

0

��X(t)� f�(X(t))��2dt ,

and on average w.r.t. the distribution µ0 of initial conditions:

Eµ0 [ supt2[0,T ]

kbX(t)�X(t)k] C(T,CLip)pNk�(·) ·��(·) · kL2(⇢T ) ,

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Standard arguments yield bounds on trajectories between trajectories of thetrue system and those of the system driven by the estimated interaction kernel.

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Example 2nd order Prey-Predator system. Left: the interaction kernels and⇢TL’s. Right: trajectories of the true system (left col.) and learned system (rightcol.) with an initial condition from training data (top) and a new one (bottom).

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Examples: 2nd order systems







Testing hypotheses for agent systems

ExampleWe want to test if a 2nd order system is driven by energy or alignmentinteractions. Left: we learn a general model (with both types of interaction)on a system with only energy interaction terms: we obtain �A is u 0. Right:learning on a system with only alignment term yields �E u 0.

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Example We want to test if a system is governed by 1st or 2nd order interac-tions. We are able to tell thedi↵erence reliably, by testingthe predictions of the learnedmodels on trajectories.

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Emerging behaviorsOrganized collecting stable patterns at large spatial/temporal scale.

Simple, local interaction kernels can learn to complex, organized behavior.

Most of the above is ill-defined, and quotes needed a.e.

Examples include flocking of bird, milling of fish, synchronization in systems of

oscillators (neurons, frogs, ...), etc...<latexit sha1_base64="rvZNSTGr0gh2/B8wi1PrIE5JQFA=">AAADW3icbVLfaxNBEL4kamOsmio++TJYhArxTOKDPvhQEMEXsaJpC0kIc3tzyZK93evOXmgM/Sd90gf/FXEuDWpaB5b7+Gbmmx83SWE0h273R63euHHz1k7zduvO7t1799t7D47ZlV7RQDnj/GmCTEZbGgQdDJ0WnjBPDJ0k87eV/2RBnrWzX8KyoHGOU6szrTAINdmrnY2s0zYlG+Cjn6LVXykF0TWkgrZT4ICiBQWGQN4yYA CDfkrAQmk0LwLlhfNogBUailutv4KfdV4Y6oBx4gJtRQFVVRfmokWGQaEFQ+gtBCdVq/DzDrg/jSQ0w4V2fkv2g+MALoMwI8DELQg0gzbmeUqZrCHtANoUzkoXiMESpSKE8XZr786xKiZ5VpkyJcikyXk1sAgn2otILpIbItM86wAvrZp5J52tlyepQrHMz1WMYyUJGJxnOLBUSiB3IPNuKp84jp91gIISMGnvd+Pu2uA66G3AfrSxo0n72yh1qsylb2WQedjrFmG8Qh+0MnTRGpVMBao5Tmko0GJOPF6tb+MCngqTQua8PJl7zf6bscKceZknEpljmPFVX0X+zzcsQ/Z6vNK2KANZdVkoK031I6tDg1R7OSGzFIDKa+kV1AyrA5BzbMkSeldHvg6O+3HvZdz/1N8/fLNZRzN6HD2JDqJe9Co6jN5HR9EgUrXvtV/1nXqz/rPRaLQau5eh9dom52G0ZY1HvwFIOxBR</latexit>

In general di�cult to characterize and predict; however if robust, we may hopeto recover them with systems driven by estimated interaction kernels.

<latexit sha1_base64="QVo14/EGvNGK7/ItHC5slFu9Gjg=">AAACinicbVFNixNBEO0ZP3aNq0Y9eikMggcNMxFcZT0s6EFvK5jdhSSEnp6apEl/DN01G8awP8a/5M1/Y002oO5a0PB49apf9euiNjpSlv1K0lu379zd27/Xu3/w4OGj/uMnp9E3QeFYeePDeSEjGu1wTJoMntcBpS0MnhWrj13/7AJD1N59o7bGmZULpyutJDE17/+YOq9diY7gi4MFOgzSQKkrljSGgDyopQxSEQb9HUG6Et ig1IqOYOnXyJeDrgCCL5pIr2CNYGXLrRq74YDKdxJaooW1piXENhLaCGXQF+igaAEjaSsJS9CObdiLV4MVBocmDuf9QTbMtgU3Qb4DA7Grk3n/57T0qrH8JmVkjJM8q2m2kYG0MnjZmzYRa6lWcoEThk5ajLPNNspLeMFMCZUPfDiTLfv3xEbaGFtbsJJXXsbrvY78X2/SUPVuttGubgidujKqGtNl1P0LR85RkWkZSBU07/on+NjjEPLrT74JTkfD/M1w9HU0OP6wi2NfPBPPxUuRi0NxLD6LEzEWKtlLXidvk8P0IB2l79OjK2ma7Gaein8q/fQbVBLF3g==</latexit>

BBC Blue PlanetFelix Munoz, https://www.youtube.com/watch?v=OxYn3e_imhA

Ming Zhong, Jason Miller

https://www.youtube.com/watch?v=15B8qN9dre4

Emerging behaviors: flocking

(*) F. Cucker, J. G. Dong,Avoiding collisions in flocks,IEEE Transactions onAutomatic Control, 2010.

<latexit sha1_base64="PoOe9utUW21xkDXZcHGsz/dOMmM=">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</latexit>

The governing equations of Cucker-Dong (*) dynamics are,

xi = �bi(t)xi +

NX

i0=1

⇥ai,i0(x)(xi0 � xi) + f(||xi � xi0 ||2)(xi0 � xi)

⇤.

Here ai,i0(x) = H(1 + ||xi0 � xi||2)��; bi : [0,1) ! [0,1) is a bounded and

uniformly continuous damping function, and f : (�,1) ! [0,1) is a non-

increasing C1repulsion function integrable at +1.

<latexit sha1_base64="UQiCSNS+I3ghuAi2h+Tv7TFdPpk=">AAAEGXicbVNLb9NAEHYTHiU82sKRy4ga1aFJiMMBVKhUqRx6QkXqS4qTaL1eJ6vau2YfiMjx3+DCX+HCAYQ4wol/wzgNkDRdydLsPL7xzPdtmCVcm3b790qleu36jZurt2q379y9t7a+cf9ES6soO6YykeosJJolXLBjw03CzjLFSBom7DQ83y/jp++Z0lyKIzPOWC8lQ8FjTolB12Cj8jQQkouICQNHIwZDidmCiyGwd3aao0 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</latexit>

Emerging behaviors: Fish mill patternsThe governing equations of fish milling dynamics in R2 of (*) are

mixi = ↵xi � �||xi||2xi � ~rUi ,

with Ui a potential for the interaction of the ith agent with the other agents:Ui =

PNi0=1(�Cae�||xi�xi0 ||/`a + Cre�||xi�xi0 ||/`r ).

<latexit sha1_base64="nbhEUzUi2hLgdTu1s9dZu8XxXTY=">AAADpXicnVJNbxMxEN0mfJTloy0cuYzaRW2hDUk4gBCVKvXSC6itmrZSNll5HW/WqtdebG8h8vqX8S+48W+wNxEiLSfmYI3fvBnPm3FaMqp0t/trpdW+d//Bw9VH4eMnT5+trW88v1CikpgMsGBCXqVIEUY5GWiqGbkqJUFFyshlen3k45c3RCoq+LmelWRUoCmnGcVIOyjZWPkRc0H5hHAN5zmBqXBsTvkUyNeq4SgQGWRU5V 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</latexit>

(*) Y. Li Chuang, M. R. D’Orsogna,

D. Marthaler, A. L. Bertozzi, L. S.

Chayes, Physica D: Nonlinear Phe-

nomena 232 (2007)<latexit sha1_base64="MWbgJViP+MoRi1wOdzvoErF1cn8=">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</latexit>

Parametrized Interaction KernelsWe consider the 5 innermost planets and the Sun in our solar system, movingaccording to Newton’s law:

xi(t) =NX

i0=1

Gmi0

||xi0 � xi||3(xi0 � xi) ,

where mi are the unknown masses, and G is the known gravitational constant.<latexit sha1_base64="Ta92GLr1D++PxY5P2xEcM33mTcI=">AAADOXicdVJLb9NAELbNq4RXC0cuIxLUVgpR0gqBEJUqcSinUgRtKtVptFlPklX2Ye2OGyLHf4sL/4IbEhcOIMSVP8DaDa8WRrL17fft7M58s4NUCkft9ocwunDx0uUrS1dr167fuHlreeX2gTOZ5bjPjTT2cMAcSqFxnwRJPEwtMjWQ2B1MnpV69wStE0a/plmKPcVGWgwFZ+Sp/kr4ItZG6AQ1QReBG+1EghZojNB42AChNV plHEEqmUZywHRSqa8y7VXwhYAzkvn/zBGqJihzIvQIGOfGJiUiA7s4JaNXHUg2fVJrNGqxU8yN8zhJDOXxYAhvir5Yo3XYgthlqp+L1a1OcbwL8dAynu/EJGSCuSpKpSjy+fxnVrmGB/DrkPn8eLNY+5+6XsTNZlnAdIzWt/j7XNEA5pmytUxPtJlq8CU6dM2q5caON8NV8qk4suxEUGUjk5VxxDS1+sv1dqtdBZwHnQWoB4vY6y+/jxPDM+UnwKW/8KjTTqmXM0uCSyxqceYwZXzCRnjkoWYKXS+vJl/Afc8kMDTWf36CFftnRs6UczM18DsVo7E7q5Xkv7SjjIaPe7nQaUao+elFw0yWsyyfESTCIic584BxK3ytwMfMj4r8Y6t5EzpnWz4PDjZanc3WxsuN+vbThR1Lwd3gXrAWdIJHwXbwPNgL9gMevg0/hp/DL9G76FP0Nfp2ujUKFzl3gr8i+v4DU4AHCQ==</latexit>

We do not know that the agents follow a common law, modulo parameters(mass), so we treat all of them as being of di↵erent type, with interactions �ii0 .

<latexit sha1_base64="3EL0Ygx2m7pf8cxxF3h7Ns7IFG0=">AAAClHicbVDLattAFB2pr9R9xGmhm24udUpTMEZyF8kiC5dQ6KqkUMcB25ir0cgaPA8xM4oxwl/Uv+muf9Mrx4s26YUZDue+zj1ZpaQPSfI7ih88fPT4ycHTzrPnL14edo9eXXlbOy7G3CrrrjP0QkkjxkEGJa4rJ1BnSkyy1UWbn9wI56U1P8KmEnONSyMLyTEQtej+nBkrTS5MgImA3IKxAVbGriGUGOgTgEvKeiisUkQjcK u1NaBw3Qdt81pZqNChFoH2wIlG7z/2wVtYCwikJQAqBbZoh2lAD5mQZtkSuSwK4drdrbQ+rGUoQRqag7yV5+F4VpVy0Uj5YXs8WHR7ySDZBdwH6R702D4uF91fs9zyWtMGrkjWNE2qMG/QBcmV2HZmtRcV8hVdOCVo6AY/b3ambuE9MTld7eiRwh37d0eD2vuNzqhSYyj93VxL/i83rUNxNm+kqeogDL9dVNQKgt3ZQK44wYPaEEDuJGkFXmJrCfnbIRPSuyffB1fDQfppMPw+7I3O93YcsLfsHTthKTtlI/aVXbIx49FRdBqNos/xm/g8voi/3JbG0b7nNfsn4m9/AMxRx60=</latexit>

We will then discover the relationships between these interaction kernels, namelythat they are all multiple of 1/r3, and the multiple is mass.

<latexit sha1_base64="AYV+RDqG6dnESzL8LPknqdTrdsg=">AAAChXicbVFLb9NAEF6bR0t4hXLkMiJF6qEEO0XQAxKVuHAsEmkqJSEar8fNKvuwdtdUUdR/wq/ixr9h7EYIWkZaaXa+b17fFLVWIWbZryS9c/fe/Z3dB72Hjx4/edp/tncWXOMljaXTzp8XGEgrS+Oooqbz2hOaQtOkWH1q8cl38kE5+zWua5obvLCqUhIjhxb9HzPrlC3JRpgQXCqtIS7JQqmCdJzY/sCT7uhhqeoABcVLYg ojgUDZSB5lC8OKvCUdDsGiIb1mBsaWtgb0BMi1TaOjqjWBq2A/f+O/He0fAtqya/MHVAEMhjBc9AfZMOsMbjv51hmIrZ0u+j9npZON4XWk5grTPKvjfIM+KqnpqjdrAtUoV3hBU3bbMcN806l4Ba84UkLlPD+Wo4v+nbFBE8LaFMw0GJfhJtYG/4dNm1gdzzfK1k0kK68bVQ0L7aA9CWvtSUbWq1QoveJZQS6xFZUP12MR8psr33bORsP8aDj6MhqcfNjKsSteiJfiQOTivTgRn8WpGAuZpMlBkiejdCd9nb5N311T02Sb81z8Y+nH39LvwoY=</latexit>

Discovering the parameters

We are able to estimateto identify, in a fullynonparametric fashion,1/r3.

<latexit sha1_base64="ghEVuPrmfX4zOxXAM2X2rHS3wks=">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</latexit>

50 100 150 200

10-7

10-6

10-5

50 100 150 200

10-7

10-6

10-5

We also estimate masses quite accurately.

<latexit sha1_base64="+F944b+prYHzPJZs/dDdO6tfALI=">AAACHHicbVDLSgMxFM34rPU16tJNsAiuykwV7MJFwY3LCvYB7VAymds2NJOMSUYYSj/Ejb/ixoUiblwI/o1pOwttPRA4Ofd9woQzbTzv21lZXVvf2CxsFbd3dvf23YPDppapotCgkkvVDokGzgQ0DDMc2okCEoccWuHoehpvPYDSTIo7kyUQxGQgWJ9RYqzUc8+7QjIRgTC4BZhwLTFow2JiAMdEa9D4PmX2QyhNlVV5Vu65Ja/szYCXiZ+TEspR77mf3UjSNLZTKLdNO76XmGBMlGGUw6TYTTUkhI7IADqWChKDDsaz4yb41CoR7ktln91ypv6uGJNY6ywObabdeqgXY1Pxv1gnNf1qMGYiSQ0IOh/UTzk2Ek+dwhFTQA3PLCFUMbsrpkOiCDXWz6I1wV88eZk0K2X/oly9rZRqV7kdBXSMTtAZ8tElqqEbVEcNRNEjekav6M15cl6cd+djnrri5DVH6A+crx8V3KH9</latexit>

Gravity KernelsEstimated interaction kernels between Sun, Mercury, Venus, Earth and Mars,�ii0 , as a function of pairwise distance.

<latexit sha1_base64="z9mJuRMJ3bMJTon+ARP+oa3H+z8=">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</latexit>

Sun

Mercury

Venus

Earth

Mars

Sun Mercury Venus Earth Mars

Gravity Kernels - cleanedEstimated interaction kernels between Sun, Mercury, Venus, Earth and Mars,�ii0 , as a function of pairwise distance.

<latexit sha1_base64="z9mJuRMJ3bMJTon+ARP+oa3H+z8=">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</latexit>

Sun Mercury Venus Earth Mars

Sun

Mercury

Venus

Earth

Mars

Trajectories from gravity kernelsTrajectory predictions are accurate (relative errors O(10�4) with M = 500and daily observations for about 6 months), exhibit stability, and approximateenergy conservation.

<latexit sha1_base64="85zyjkya4kNrXjK7olq7MGaZkN4=">AAACqHicbVHbbtNAEF2bWwmXBnjkZURcKUUlsqvS9gGkSrwgIUSQkqYoCdF4PU6Wrnet3XWoFeXb+Afe+Bs2FyRomaejMzNnZs6kpRTWxfGvILx1+87dezv3Gw8ePnq823zy9NzqynDqcy21uUjRkhSK+k44SRelISxSSYP08t0qP5iTsUKrnqtLGhc4VSIXHJ2nJs0fI6WFykg56Bn8RtxpU4OXyARfVVhAQwDIeWXQEbQNSd 86JyBjtLEQfWon8dfFq6PlfgTfhZtB9PHt6ziOAFUGGQpZg04tmTlu9HJtAFNdOYiOIyi0cjO7fwB0NROpcGAdpkIKVx+sBbAsjb4SxWo2KTLTGrhX+SPXmTRbcSdeB9wEyRa02Da6k+bPUaZ5VfiLuURrh0lcuvECjRNc0rIxqiyVyC9xSkMPFRZkx4u10UvY80y2viD3e8Oa/btjgYW1dZH6Sr/xzF7Prcj/5YaVy0/HC6HKypHim0F5JcFpWH0NMmH8a7yVmUBuhN8V+AwNcud/2/AmJNdPvgnODzvJUef082Hr7M3Wjh32nL1gbZawE3bG3rMu6zMe7AUfgl7QD1+G3XAQftmUhsG25xn7J8L0N5MAzyg=</latexit>

Trajectories from gravity kernelsTrajectory predictions are accurate (relative errors O(10�4) with M = 500and daily observations for about 6 months), exhibit stability, and approximateenergy conservation.

<latexit sha1_base64="85zyjkya4kNrXjK7olq7MGaZkN4=">AAACqHicbVHbbtNAEF2bWwmXBnjkZURcKUUlsqvS9gGkSrwgIUSQkqYoCdF4PU6Wrnet3XWoFeXb+Afe+Bs2FyRomaejMzNnZs6kpRTWxfGvILx1+87dezv3Gw8ePnq823zy9NzqynDqcy21uUjRkhSK+k44SRelISxSSYP08t0qP5iTsUKrnqtLGhc4VSIXHJ2nJs0fI6WFykg56Bn8RtxpU4OXyARfVVhAQwDIeWXQEbQNSd 86JyBjtLEQfWon8dfFq6PlfgTfhZtB9PHt6ziOAFUGGQpZg04tmTlu9HJtAFNdOYiOIyi0cjO7fwB0NROpcGAdpkIKVx+sBbAsjb4SxWo2KTLTGrhX+SPXmTRbcSdeB9wEyRa02Da6k+bPUaZ5VfiLuURrh0lcuvECjRNc0rIxqiyVyC9xSkMPFRZkx4u10UvY80y2viD3e8Oa/btjgYW1dZH6Sr/xzF7Prcj/5YaVy0/HC6HKypHim0F5JcFpWH0NMmH8a7yVmUBuhN8V+AwNcud/2/AmJNdPvgnODzvJUef082Hr7M3Wjh32nL1gbZawE3bG3rMu6zMe7AUfgl7QD1+G3XAQftmUhsG25xn7J8L0N5MAzyg=</latexit>

Trajectory of Earth and its Moon

Trajectories for inner Solar System

Conclusions

10/2/13 10:30 AM

Page 1 of 1http://upload.wikimedia.org/wikipedia/commons/1/1c/Air_Force_Research_Laboratory.svg

AFOSR NSF

Nonparametric inference of interaction laws in systems of agents from trajectory data, P.N.A.S., F. Lu, S. Tang, M. Zhong, MM, July 2019.

Data-driven Discovery of Emergent Behaviors in Collective Dynamics, MM, J.Miller, M. Zhong, arXiV 1912.11123, submitted.

Learning interaction kernels in heterogeneous systems of agents from multiple trajectories, F. Lu, MM, S. Tang, ArXiv 1910.04832, submitted.

· Learning agent-based type system may be performed e�ciently, nonparametri-

cally, at least in special cases, notwithstanding the high-dimensional state space.

· Important generalizations: 1st- and 2nd-order, multi-type; more general inter-

action kernels.

· Hypothesis testing; transfer learning; dictionary learning for dynamical sys-

tems.

· Many open problems. E.g.: quantifying information needed for learning;

stochasticity; hidden variables; general interaction kernels; ...

· Many applications: biological systems, particle systems, learning forces in

molecular systems, ...

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https://pages.jh.edu/~mzhong5/LearningDynamics.htmlCode on github; examples and movies:

Documents

Learning Interaction Laws in Particle- and Agent-based Systems · 2020-04-24 · Learning Interaction Laws. Estimation/Learning for ODE systems No randomness: this is an approximation