The Interplay of Sampling and Optimization
in High Dimension
Santosh Vempala, Georgia Tech
Deep Learning
Neural nets can represent any real-valued function
approximately.
What can they learn efficiently?
I donβt know.
High-dimensional problems
What is the complexity of computational problems as the
dimension grows?
Dimension = number of variables
Typically, size of input is a function of the dimension.
Two fundamental problems: Optimization, Sampling
Problem 1: Optimization
Input: function f: π π β π specified by an oracle,
point x, error parameter Ξ΅ .
Output: point y such that
π π¦ β₯ maxπ β π
Examples: max π β π₯ π . π‘. π΄π₯ β₯ π , min π₯ π . π‘. π₯ β πΎ.
Problem 2: Sampling
Input: function f: π π β π +, β« π < β, specified by an
oracle, a point x, error parameter Ξ΅.
Output: A point y from a distribution within distance Ξ΅of distribution with density proportional to f.
Examples: π π₯ = 1πΎ π₯ , π π₯ = πβπ π₯ 1πΎ(π₯)
High-dimensional problems
Optimization
Sampling
Also:
Integration (volume)
Learning
Rounding
β¦
All intractable in general, even to approximate.
Q. What structure makes high-dimensional problems computationally tractable? (polytime solvable)
High-dimensional breatkthroughs
P1. Optimization. Find minimum of f over the set S.
Ellipsoid algorithm works when
S is a convex set and f is a convex function.
P2. Integration. Find the integral of f.
Dyer-Frieze-Kannan algorithm works when
f is the indicator function of a convex set.
Convexity(Indicator functions of) Convex sets:
βπ₯, π¦ β π π, π β 0,1 , π₯, π¦ β πΎ ππ₯ + 1 β π π¦ β πΎ
Concave functions:
π ππ₯ + 1 β π π¦ β₯ ππ π₯ + 1 β π π π¦
Logconcave functions:
π ππ₯ + 1 β π π¦ β₯ π π₯ π π π¦ 1βπ
Quasiconcave functions:
π ππ₯ + 1 β π π¦ β₯ min π π₯ , π π¦
Star-shaped sets:
βπ₯ β π π . π‘. βπ¦ β π, ππ₯ + 1 β π π¦ β π
Tractable cases: cONVEx++
Problem
Input type
Optimizationπππ/πππ π π : π β π²
Sampling
π βΌ π(π)
Convex Linear: Kha79
Convex: GLS83
DFK89
Logconcave GLS83(implicit) Lipshitz:AK91
General: LV03
Stochastic/approx.
convex
BV03,KV03,LV06,
BLNR15, FGV15
KV03, LV06
Star-shaped X CDV2010
Efficient sampling gives efficient volume computation, integration
for logconcave functions.
Computational model
Well-guaranteed Membership oracle:
Compact set K is given by
a membership oracle: answers YES/NO to βπ₯ β πΎ? "
a point π₯0 β πΎ
Numbers r, R s.t. π₯0 + ππ΅π β πΎ β π π΅π
Well-guaranteed Function oracle An oracle that returns π(π₯) for any π₯ β π π
A point π₯0 with π π₯0 β₯ π½
Numbers r, R s.t.
π₯0 + ππ΅π β πΏπ1
8and π 2 = πΈπ π β π
2
A closer look: polytime convex optimization
Linear programs Khachiyan79 Ellipsoid
Karmarkar83 Interior-point method (IPM)
Renegar88, Vaidya90 IPM
Dunagan-V04 Perceptron+rescaling
Kelner-Spielman05 Simplex+rescaling
β¦.
Lee-Sidford13,15 IPM
Convex programs GLS83* Ellipsoid++
Vaidya89* Volumetric center
NN94** IPM (universal barrier)
Bertsimas-V.02* Random walk+cutting plane
Kalai-V06* Simulated annealing
Lee-Sidford-Wong15# hybrid:IPM+cutting plane
*: need only a membership/function oracle!
**: yes, even this.
#: needs separation oracle: NO answer comes with violating hyperplane
A closer look: polytime sampling
Convex bodies DFK89 grid walk n20
LS91,93 ball walk π7
KLS97 ball walk π3
LV03 hit-and-run
Logconcave functions AK91 grid walk π8
LS93 ball walk π7
LV03,06 hit-and-run π3
Polytopes KLS,LV Ball/hit-and-run π3 ππ
KN09 Dikin walk mn πππβ1
Lee-V16+ Geodesic walk πππ πππβ1
This talk: interplay of sampling and optimization
Part I: Optimization via Sampling
Cutting Plane Method
Simulated Annealing
Interior Point Method
Part II: Sampling via Optimization (ideas)
Ball walk, hit-and-run
Dikin walk
Geodesic walk
Faster polytope sampling
Part III: Open questions
You want interplay?
SAMPLING OPTIMIZATION
ANNEALING BARRIERS
Convex feasibility from separation
Input: Separation oracle for a convex body K, r, R.
Output: A point x in K.
Complexity: #oracle calls and #arithmetic operations.
To be efficient, complexity of an algorithm should be bounded by poly(n, log(R/r)).
Q. Which sequence of points to query?
Each query either solves the problem or restricts the remaining set via the violating linear inequality.
(Convex optimization Convex Feasibility via binary search)
Centroid cutting-plane algorithm
[Levin β65]. Use centroid of surviving set as query point in each iteration.
Thm [Grunbaum β60]. For any halfspace H containing the centroid of a convex body K,
π£ππ πΎ β© π» β₯1
ππ£ππ πΎ .
#queries = O(nlog(R/r)).
Best possible.
Problem: how to find centroid?
#P-hard! [Rademacher 2007]
Randomized cutting plane
[Bertsimas-V. 02]
1. Let z=0, P = βπ , π π.
2. Does π§ β πΎ? If yes, output K.
3. If no, let H = π₯ βΆ πππ₯ β€ πππ§ be a halfspace containing K.
4. Let π β π β© π».
5. Sample π₯1, π₯2, β¦ , π₯π uniformly from P.
6. Let π§ β1
π π₯π . Go to Step 2.
Optimization via Sampling
Thm [BV02]. For any convex body K and halfspace H containing
the average of m random points from K,
πΈ(π£ππ πΎ β© π» ) β₯1
πβ
π
ππ£ππ πΎ .
Thm. [BV02] Convex feasibility can be solved using O(n log R/r)
oracle calls.
Ellipsoid takes π2, Vaidyaβs algorithm also takes O(n log R/r).
Optimization from membership
Sampling suggests a conceptually very simple algorithm.
Simulated Annealing [Kalai-V.04]
To optimize π consider a sequence π0, π1, π2, β¦ ,with ππ more and more concentrated near the optimum.
ππ π₯ = πβπ‘π π,π₯
Corresponding distributions:
ππ‘π π₯ =πβπ‘π π,π₯
β«πΎ πβπ‘πβ¨π,π₯β©ππ₯
Lemma. πΈππ‘π β π₯ β€ min π β π₯ +
π
π‘.
So going up to π‘ =π
πsuffices to obtain an π approximation.
Simulated Annealing [Kalai-V.04]
ππ‘π π₯ =πβπ‘π π,π₯
β«πΎ πβπ‘πβ¨π,π₯β©ππ₯
Lemma. πΈππ‘π β π₯ β€ min π β π₯ +
π
π‘.
Proof: First reduce to cone, only makes it worse.
For cone, πΈ π β π₯ =β« π¦πβπ‘π¦π¦πβ1ππ¦
β« πβπ‘π¦π¦πβ1ππ¦=
π!
π‘π+1
πβ1 !
π‘π
=π
π‘.
Optimization via Simulated Annealing
β’ min π, π₯ = max π π₯ = πββ¨π,π₯β©
β’ Can replace π, π₯ with any convex function
For π = 1,β¦ ,π:
β’ ππ π = π(π)ππ
β’ π0 =π
2π , ππ =
π
π
β’ ππ+1= ππ 1 +1
π
β’ Sample with density prop. to ππ π .
Output argmax π(π).
Complexity of Sampling
Thm. [KLS97] For a convex body, the ball walk with an M-
warm start reaches an (independent) nearly random point
in poly(n, R, M) steps.
π = supπ0 π
π πππ π = πΈπ0
π0 π₯
π π₯
Thm. [LV03]. Same holds for arbitrary logconcave density
functions. Complexity is πβ(π2π2π 2).
Isotropic transformation: π = π π ; Warm start: π = π(1).
Simulated Annealing
To optimize π = π, π₯ , take a sequence π‘0, π‘1, π‘2, β¦ , with corresponding distributions:
ππ π₯ =πβπ‘π π,π₯
β«πΎ πβπ‘πβ¨π,π₯β©ππ₯
For sampling to be efficient, consecutive distributions must
have significant overlap in πΏ2 distance:
ππ/ππ+1 2 = πΈππ
ππ π₯
ππ+1 π₯= π 1
Maintain near-isotropy, which is implied by ππ+1/ππ 2 = π 1 .
Lemma. For π‘π+1 = 1 +1
ππ‘π , ππ/ππ+1 2, ππ+1/ππ 2 < 5.
Hence, βΌ π phases. Best possible even if we assume only (a) logconcavedistributions and (b) overlap in TV distance [KV03].
Interplay of sampling and optimization
Part I: Optimization via Sampling
Cutting Plane Method
Simulated Annealing
Interior Point Method
Part II: Sampling via Optimization (ideas)
Ball walk, hit-and-run
Dikin walk
Geodesic walk
Faster polytope sampling
Part III: Open questions
Interior point method
ππ π₯ = minπΎ
π‘π β π π₯ + π(π₯)
π(π₯): objective function, π π₯ = β¨π, π₯β© for LP.
π(π₯): is smooth convex function βbarrierβ, blows up on the boundary of K.
π‘0 = 1, π‘π+1 = 1 +1
ππ‘π
Step: π₯π β argminππ(π₯)
Q1. How to implement a step?
A1. Newton method
Q2. Number of steps?
A2. π π for π-self-concordant π.
Barriers mollify boundary effects.
Interior point method
ππ π₯ = minπΎ
π‘π β π π₯ + π(π₯)
Log barrier for polytopes has π β€ π
π π₯ = β
π=1
π
log(π΄π β π₯ β ππ)
Universal barrier (Nemirovski-Nesterov94) has π = π π :
π π₯ = log π£ππ πΎ β π₯ π
Entropic barrier (Bubeck-Eldan) has π = 1 + π 1 π
π π₯ = supπn
βπ β π₯ β log πΎ
πβπβ π₯ππ₯
Canonical barrier (Hildebrand) has π β€ π + 1
IPM complexity
π π = π π for general convex bodies via Universal,
Entropic or Canonical barriers.
The Universal and Entropic barriers can be computed via
sampling and integration over convex bodies and
logconcave densities.
But there is an even closer connection!
Annealing | IPM
Entropic barrier IPM is equivalent to Simulated Annealing
πΈππ‘π₯ = argmin π‘β¨π, π₯β© + π(π₯)
Desired step of IPM is computed exactly by Simulated Annealing in one phase.
Thm[AH16]. Simulated annealing path = IPM Central path
Two very different analyses of the same algorithm: one with calculus (self-concordance), the other with probability.
This talk: interplay of sampling and optimization
Part I: Optimization via Sampling
Cutting Plane Method
Simulated Annealing
Interior Point Method
Part II: Sampling via Optimization (ideas)
Ball walk, hit-and-run
Dikin walk
Geodesic walk
Faster polytope sampling
Part III: Open questions
How to Sample?
Ball walk
At x, pick random y from π₯ + πΏπ΅π
if y is in K, go to y
Boundary Effect #1:
Step size cannot be too large,
else rejection probability becomes too high!
Hit-and-run
[Boneh, Smith]
At x, pick a random chord L through x
go to a uniform random point y on L
Boundary Effect #1:
Average chord length is small!
Larger steps?
Step size for both walks, even in isotropic position, is
π1
πleading to mixing in π π2 steps.
This constraint is due to points near the boundary
Idea: take larger steps for points in the interior
Boundary Effect #2: Then stationary distribution is not
uniform.
Dikin walk with barrier π At π₯, pick next step from Ellipsoid defined by π(π₯):
πΈππ π₯ = π¦: π¦ β π₯ π»2π(π₯) = π¦ β π₯ ππ»2π π₯ π¦ β π₯ β€ 1 .
Alternatively, π¦ βΌ π π₯,1
ππ»2π π₯
β1
Apply Metropolis filter to make steady state uniform
Log barrier: π π₯ = β π=1π log π΄π₯ β π π and π»2π π₯ = π΄πππ₯
β2π΄
Thm. [Kannan-Narayanan09] For any polytope in Rπ with m facets, the Dikin walk with the log barrier mixes in πβ(ππ) steps from a warm start. Each step takes π πππ€β1
time.
Faster than hit-and-run when not too many constraints.
Dikin walk
Dikin ellipsoid is fully contained in K
Idea: Pick next step y from a blown-up Dikin ellipsoid.
Can afford to blow up by ~ π/ logπ . WHP π¦ β πΎ.
But, rejection probability of filter becomes too high!
Markov chains State space K , next step distribution ππ’ . associated with each point u in K.
Stationary distribution Q, ergodic βflowβ defined as
Ξ¦ π΄ = π΄
ππ’ πΎ\A ππ(π’)
For a stationary distribution, we have Ξ¦ π΄ = Ξ¦(πΎ\A)
Conductance:
π π΄ =β«π΄ ππ’ πΎ\A ππ(π’)
minπ π΄ , π πΎ\Aπ = inf π(π΄)
Thm. [LS93] ππ‘: distribution after t steps
π = supπ΄βπΎ
π0 π΄
π π΄: πππ ππ‘, π β€ π 1 β
π2
2
π‘
π = πΈπ0
π0 π₯
π π₯: πππ ππ‘, π β€ π +
π
π1 β
π2
2
π‘
βπ > 0
Conductance
Consider an arbitrary measurable subset S.
Need to show that the escape probability from S is large.
(Smoothness of 1-step distribution) Points that do not cross over are far from each other i.e., nearby points have large overlap in 1-step distributions
(Isoperimetry) Large subsets have large boundaries
Isoperimetry
π π3 β₯π
π·π(π1, π2)minπ π1 , π π2
π 2= πΈπ π₯ β π₯ 2
A = πΈ((π₯ β π₯)(π₯ β π₯)π) : covariance matrix of π
π 2 = πΈπ π₯ β π₯ 2 = ππ π΄ =
π
ππ(π΄)
Thm. [KLS95]. π π3 β₯π
π π(π1, π2)min π π1 , π(π2)
Convergence of ball walk
Thm. [LS93, KLS97] If S is convex, then the ball walk with
an M-warm start reaches an (independent) nearly random
point in poly(n, D, M) steps.
Strictly speaking, this is not rapid mixing!
How to get the first random point?
Better dependence on diameter D?
Convergence of hit-and-run
Cross-ratio distance:
ππΎ π’, π£ =π’ β π£ π β π
π β π’ π£ β π
Thm. [L98;LV04] ππ π3 β₯ ππΎ π1, π2 ππ π1 ππ(π2)
Conductance = Ξ©1
ππ·
Thm [LV04]. Hit-and-run mixes in polynomial time from any
starting point inside a convex body.
Leads to πβ π3 mixing.
KLS hyperplane conjecture
π΄ = πΈ(π₯π₯π)
Conj. [KLS95]. ππ π3 β₯π
π1 π΄π(π1, π2)min ππ π1 , ππ(π2)
β’ Could improve sampling complexity by a factor of n
β’ Implies well-known conjectures in convex geometry: slicing conjecture and
thin-shell conjecture
β’ [CV13] True for the product of a logconcave function and a Gaussian.
β’ Best case scenario: π(π2).
This talk: interplay of sampling and optimization
Part I: Optimization via Sampling
Cutting Plane Method
Simulated Annealing
Interior Point Method
Part II: Sampling via Optimization (ideas)
Ball walk, hit-and-run
Dikin walk
Geodesic walk
Faster polytope sampling
Part III: Open questions
Can we sample faster?
Brownian motion SDE:
ππ₯π‘ = π π₯π‘, π‘ ππ‘ + π΄ π₯π‘, π‘ πππ‘
Roughly speaking, next step is from infinitesimal Gaussian π(π, π΄ π₯π‘, π‘ )
Each point π₯ β πΎ has its own local scaling (metric)
Thm. [Fokker-Planck] Diffusion equation of SDE is π
ππ‘π π₯, π‘ =
1
2π» β π΄ π₯, π‘ π»π π₯, π‘
i.e.,
π
ππ‘π π₯, π‘ = β
π
ππ
ππ₯ππ π₯, π‘ π π₯, π‘ +
1
2
π
π
π
ππ2
ππ₯πππ₯π[π΄ππ π₯, π‘ π π₯, π‘ ]
For any metric, SDE gives diffusion equation.
When π΄ = πΌ, this is the heat equation: π
ππ‘π π₯, π‘ =
1
2π»2π(π₯, π‘).
Which metric to use? Natural choice: metric defined by Hessian of a barrier function.
Why? It was useful in optimization to move quickly in the polytope, by avoiding boundaries.
For barrier π, diffusion equation is π
ππ‘π π₯, π‘ =
1
2π» β π»2π π₯
β1π»π(π₯, π‘)
and the SDE is
ππ₯π‘ = π π₯π‘ ππ‘ + π»2π π₯π‘β1/2
πππ‘
with π π₯π‘ = β1
2π»2π π₯
β1π» log det π»2π(π₯), a Newton step!
But how to simulate Brownian motion? Studied in practice, but not much from a complexity perspective.
Discretization: which coordinate system?
ππ₯π‘ = π π₯π‘ ππ‘ + π π₯π‘ πππ‘
Euler-Maruyama: For small h,
π₯π‘+β = π₯π‘ + π π₯π‘ β + π π₯π‘ π€π‘ β
Not so great:
Euclidean coordinates are arbitrary
Approximation is weaker if local metric π(π₯π‘) varies a lot
Is there a more natural coordinate system? That
effectively keeps local metric constant?
Enter Riemannian manifolds
n-dimensional manifold M is an n-dimensional surface in π π for some π > π.
We are going to map the polytope to a manifold.
Idea: distances will be shortest paths on manifold
Examples: flight paths on earth, light in relativity
Enter Riemannian manifolds
n-dimensional manifold M is an n-dimensional surface.
Each point π has a linear tangent space πππ of dimension n, the local linear approximation of M at p. Tangents of curves in M lie in πππ.
The inner product in πππ depends on p: π’, π£ π
Enter Riemannian manifolds
Each point π has a linear tangent space πππ.
The inner product in πππ depends on p: π’, π£ π
Length of a curve π: 0,1 β π is
πΏ π = 0
1 π
ππ‘π π‘
π(π‘)
ππ‘
Distance between x,y in M is the infimum over all paths in M between x and y. This is the Riemannian metric.
Riemannian manifold/metric
The inner product in πππ depends on p: π’, π£ π
Length of a curve π: 0,1 β π is
πΏ π = β«01 π
ππ‘π π‘
π(π‘)ππ‘
Metric: ππ π₯, π¦ = inf πΏ(πππ‘β π₯, π¦ )
Geodesic: curve πΎ: π, π β π that has
constant speed: π
ππ‘πΎ π‘
πΎ π‘is a constant
is a locally shortest path
Riemannian manifold/metric
Geodesic: curve πΎ: π, π β π that has
constant speed: π
ππ‘πΎ π‘
πΎ π‘is a constant
is a locally shortest path
Exponential map expπ: πππ β π is defined as expπ(π£) = πΎπ£(1), πΎπ£: unique geodesic from p with initial velocity π£.
Locally, expπβ1(π₯) gives x in βnormalβ coordinates at p.
Hessian manifold
Local inner product is defined by Hessian:
π’, π£ π = π’ππ»2π π π£
π£π
= π£π»2π(π)
Lemma. Let πΉ = expπ₯0β1, where expπ₯: ππ₯π β π.Then the SDE
ππ₯π‘ = π π₯π‘ ππ‘ + π π₯π‘ πππ‘
with π π₯ = π»2π π₯β1/2
implies
ππΉ π₯0 =1
2π π₯0 ππ‘ + π π₯0 ππ0.
Ergo: Geodesic walk
In tangent plane at x,
1. pick π€ βΌ ππ₯(0, πΌ), i.e. mean standar Gassian in β. βπ₯
2. Compute π¦ = expπ₯β
2π π₯ + βπ€
3. Compute π€β s.t. π₯ = expπ¦β
2π π¦ + βπ€β²
4. Accept with probability Min 1,π π¦ β π₯
π π₯ β π¦
How to compute geodesic and rejection probability?
π€β
w
Implementing one step Geodesic equation for π π₯ = β π=1
π log π΄π₯ β π π
Let ππ₯ = π·πππ π΄π₯ β π , π΄π₯ = ππ₯β1π΄
πΎβ²β² = π΄πΎππ΄πΎ
β1π΄πΎ
π π΄πΎπΎβ² 2
And the probability π π₯ β π¦
are both second-order ODEs.
We show this can be solved efficiently to inverse polynomial accuracy by the Collocation Method, for any Hessian manifold under smoothness assumptions.
π(πππβ1) per step for log barrier
w
Mixing of Geodesic walk
Thm 1. [Lee-V16] For log barrier, Geodesic walk mixes in π ππ0.75 steps.
This is a special case of more general theorem:
Thm 2. [Lee-V16] For any Hessian manifold, for small
enough step size h, Geodesic walk mixes in ππΊ2
βsteps,
where G is the expansion of metric wrt to Hilbert metric.
Q. How large can the step size be?
7-parameter mixing theoremConvergence for general Hessian manifolds is based on:
π·0 = sup π π₯π₯
: maximum norm of drift
π·1 = supπ
ππ‘π πΎ π‘
πΎ(π‘)
2: smoothness of drift norm
π·2 = sup π»π π π₯π₯
: smoothness of drift
πΊ1 = sup (logdet π πΎ π‘β²β²β²
: smoothness of volume element of local metric g.
πΊ2 = supπ π₯,π¦
ππ» π₯,π¦: smoothness of metric (ππ»: Hilbert dist)
π 1 : stability of Jacobi field, π 2 : smoothness of Ricci curvature
Thm. Suppose β β€ π min1
ππ·0π 1
23
,1
π·2,
1
ππ 1,
1
π13π·1
23
,1
ππΊ1
23
,1
ππ 2
23}
.
Then, the geodesic walk has conductance Ξ©β
πΊ2and mixes in π
πΊ22
βsteps.
7-parameter mixing for log barrierConvergence for general Hessian manifolds is based on:
π·0 = sup π π₯π₯
: maximum norm of drift π π
π·1 = supπ
ππ‘π πΎ π‘
πΎ(π‘)
2: smoothness of drift norm π π β
π·2 = sup π»π π π₯π₯
: smoothness of drift π π
πΊ1 = sup (logdet π πΎ π‘β²β²β²
: smoothness of volume element of local metric g π β
πΊ2 = supπ π₯,π¦
ππ» π₯,π¦: smoothness of metric (ππ»: Hilbert dist) π π = π π
π 1 : stability of Jacobi field π1
π, π 2 : smoothness of Ricci curvature π πβ
Thm (log barrier). Suppose β β€ π min1
ππ·0π 1
23
,1
π·2,
1
ππ 1,
1
π13π·1
23
,1
ππΊ1
23
,1
ππ 2
23}
. π πβ0.75
Then, the geodesic walk has conductance Ξ©β
πΊ2and mixes in π
πΊ22
βsteps. π ππ0.75
Proof outline
Need to show:
Rejection probability is small
1-step distributions are smooth
Isoperimetry is good: 1
πΊ2=
1
πfor log barrier
Follows from isoperimetry for hit-and-run
1-step distribution
Lemma. For any π₯ β π, β > 0, the probability density of the
one-step distribution (before rejection sampling) is:
ππ₯ π¦ =
expπ₯ π£π₯ =π¦
det π expπ₯ π£π₯β1
det π π¦
2πβ ππβ
12β π£π₯β
β2π π₯
π₯
2
.
Pick π£π₯ βΌ ππ₯(0, πΌ),
then apply expπ₯(π£π₯),
then account for new metric at π¦
1-step distribution
ππ₯ π¦ =
expπ₯ π£π₯ =π¦
det π expπ₯ π£π₯β1
det π π¦
2πβ ππβ
12β
π£π₯ββ2π π₯
π₯
2
.
Under conditions on h,
Lemma 1. logππ₯ π¦
ππ¦ π₯<
1
4.
(rejection probability is small)
Lemma 2. πππ ππ₯, ππ¦ < 0.31415926
(nearby points have large overlap in one-step distributions)
You can implement this!
Next steps
Analyze Geodesic walk for Lee-Sidford barrier function
Improve analysis further to allow larger step size
Higher order simulation, anyone?
Open questions: Algorithms
Faster LP/convex optimization?
Faster optimization with a membership oracle?
Faster sampling: geodesic walk, reflection walk, coordinate
hit-and-run, β¦
Open questions: Geometry
How true is the KLS conjecture?
infπ β€
πΎ
2
ππ
π
A weaker conjecture:1
π ππ(π΄)>
1
π ππ π΄ 2 1/4=
1
π΄πΉ
1/2>
1
π1 π΄
Open questions: Geometry
ππ π = infπ>0
π£ππ π₯: π π₯, π¦ β€ π
π β min π£ππ π , π£ππ(πΎ β π)
Generalized KLS.
Question: For any Hessian metric, is ππ achieved to within
a constant factor by a hyperplane cut?
Open questions: Riemannian manifolds
Learning: learn metric approximately?
Testing:
Modeling:
Voronoi diagrams?
Note: General relativity is Einstein-Kahler metric, this is
induced by the canonical barrier!
Open questions: Probability
Q1. Does ball walk mix rapidly starting at a single nice point, e.g., the centroid?
Q2. When to stop? How to check convergence to stationarity on the fly? Does it suffice to check that the measures of all halfspaces have converged?
(Note: poly(n) sample can estimate all halfspace measures approximately)
Open questions: Algorithms
How efficiently can we learn a polytope P given only random points?
With O(mn) points, cannot βseeβ structure, but enough information to estimate the polytope! Algorithms?
For convex bodies:
[KOS][GR] need 2Ξ© π points to learn P
[Eldan] need 2ππeven to estimate the volume of P
Open questions: Algorithms
Can we estimate the volume of an explicit polytope in
deterministic polynomial time?
π΄π₯ β₯ π
Thank you!
And to:
Ravi Kannan
Laci Lovasz
Adam Kalai
Ben Cousins
Yin Tat Lee