Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Efficient Memoization for Approximate Function
Evaluation over Sequence Arguments
AAIM 2014
Tamal Biswas and Kenneth W. Regan
University at Buffalo (SUNY)
9 July 2014
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Problem
1
Space of points x
= (x1 ; : : : ; xN ),
where N is not small (N
50).
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Problem
= (x1 ; : : : ; xN ),
1
Space of points x
2
Need to evaluate y
=
f (x ) for M
where N is not small (N
=
millions of x .
50).
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Problem
= (x1 ; : : : ; xN ),
1
Space of points x
2
Need to evaluate y
3
Each eval of f is expensive. Many repetitive evals.
=
f (x ) for M
where N is not small (N
=
millions of x .
50).
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Problem
= (x1 ; : : : ; xN ),
1
Space of points x
2
Need to evaluate y
3
Each eval of f is expensive. Many repetitive evals.
4
=
f (x ) for M
where N is not small (N
=
millions of x .
However, the target function (y1 ; : : : ; yM ) tolerates
approximation:
50).
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Problem
= (x1 ; : : : ; xN ),
1
Space of points x
2
Need to evaluate y
3
Each eval of f is expensive. Many repetitive evals.
4
=
f (x ) for M
where N is not small (N
=
millions of x .
However, the target function (y1 ; : : : ; yM ) tolerates
approximation:
Could be linear:
50).
= sumj aj yj . For mean or quantile statistics.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Problem
= (x1 ; : : : ; xN ),
1
Space of points x
2
Need to evaluate y
3
Each eval of f is expensive. Many repetitive evals.
4
=
f (x ) for M
where N is not small (N
=
50).
millions of x .
However, the target function (y1 ; : : : ; yM ) tolerates
approximation:
Could be linear: = sumj aj yj . For mean or quantile statistics.
Could be non-linear but well-behaved, e.g., logistic regression.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Problem
= (x1 ; : : : ; xN ),
1
Space of points x
2
Need to evaluate y
3
Each eval of f is expensive. Many repetitive evals.
4
=
f (x ) for M
where N is not small (N
=
50).
millions of x .
However, the target function (y1 ; : : : ; yM ) tolerates
approximation:
Could be linear: = sumj aj yj . For mean or quantile statistics.
Could be non-linear but well-behaved, e.g., logistic regression.
5
Two other helps: f is smooth and bounded.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Problem
= (x1 ; : : : ; xN ),
1
Space of points x
2
Need to evaluate y
3
Each eval of f is expensive. Many repetitive evals.
4
=
f (x ) for M
where N is not small (N
=
50).
millions of x .
However, the target function (y1 ; : : : ; yM ) tolerates
approximation:
Could be linear: = sumj aj yj . For mean or quantile statistics.
Could be non-linear but well-behaved, e.g., logistic regression.
5
6
Two other helps: f is smooth and bounded.
And we need good approximation to ( ) (only) under
distributions D (x ) controlled by a few model-specific parameters,
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Euclidean Grid Case
Pre-compute—or memoize—f (u ) for gridpoints u. Given x not in the
grid, several ideas:
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Euclidean Grid Case
Pre-compute—or memoize—f (u ) for gridpoints u. Given x not in the
grid, several ideas:
1
Find nearest neighbor u, use f (u ) as y. Not good enough
approximation.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Euclidean Grid Case
Pre-compute—or memoize—f (u ) for gridpoints u. Given x not in the
grid, several ideas:
1
2
Find nearest neighbor u, use f (u ) as y. Not good enough
approximation.
Write x = k bk uk , use y = k bk f (uk ).
P
P
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Euclidean Grid Case
Pre-compute—or memoize—f (u ) for gridpoints u. Given x not in the
grid, several ideas:
1
2
Find nearest neighbor u, use f (u ) as y. Not good enough
approximation.
Write x = k bk uk , use y = k bk f (uk ). Seems better. But N is
not small.
P
P
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Euclidean Grid Case
Pre-compute—or memoize—f (u ) for gridpoints u. Given x not in the
grid, several ideas:
1
2
3
Find nearest neighbor u, use f (u ) as y. Not good enough
approximation.
Write x = k bk uk , use y = k bk f (uk ). Seems better. But N is
not small.
Use any neighbor u and do Taylor expansion.
P
P
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Euclidean Grid Case
Pre-compute—or memoize—f (u ) for gridpoints u. Given x not in the
grid, several ideas:
1
2
3
Find nearest neighbor u, use f (u ) as y. Not good enough
approximation.
Write x = k bk uk , use y = k bk f (uk ). Seems better. But N is
not small.
Use any neighbor u and do Taylor expansion.
P
P
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Taylor Expansion
f (x ) = f (u ) +
X̀(x
i =1
i
ui )
@f
@ xi
(u ) +
1
2
X(x
i ;j
i
ui )(xj
uj )
@2f
@ ui @ uj
+
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Taylor Expansion
f (x ) = f (u ) +
X̀(x
i =1
1
i
ui )
@f
@ xi
(u ) +
1
2
X(x
i ;j
i
ui )(xj
uj )
@2f
@ ui @ uj
+
Given that f is fairly smooth as well as bounded, and the grid is
fine enough, we can ignore quadratic and higher terms.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Taylor Expansion
f (x ) = f (u ) +
X̀(x
i =1
i
ui )
@f
@ xi
(u ) +
1
2
X(x
i
ui )(xj
uj )
i ;j
@2f
@ ui @ uj
+
1
Given that f is fairly smooth as well as bounded, and the grid is
fine enough, we can ignore quadratic and higher terms.
2
Problem is that now we need to memoize all fi (u ) =
data!
@f
(u ).
@ xi
50x
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Taylor Expansion
f (x ) = f (u ) +
X̀(x
i =1
i
ui )
@f
@ xi
(u ) +
1
2
X(x
i
ui )(xj
uj )
i ;j
@2f
@ ui @ uj
+
1
Given that f is fairly smooth as well as bounded, and the grid is
fine enough, we can ignore quadratic and higher terms.
2
Problem is that now we need to memoize all fi (u ) =
data!
3
Main Question: Can we “cheat” by shortcutting the partials?
@f
(u ).
@ xi
50x
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Taylor Expansion
f (x ) = f (u ) +
X̀(x
i =1
i
ui )
@f
@ xi
(u ) +
1
2
X(x
i
ui )(xj
uj )
i ;j
@2f
@ ui @ uj
+
1
Given that f is fairly smooth as well as bounded, and the grid is
fine enough, we can ignore quadratic and higher terms.
2
Problem is that now we need to memoize all fi (u ) =
data!
3
Main Question: Can we “cheat” by shortcutting the partials?
4
If f were linear, obviously
@f
=
@ xi
constant.
@f
(u ).
@ xi
50x
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Taylor Expansion
f (x ) = f (u ) +
X̀(x
i =1
i
ui )
@f
@ xi
(u ) +
1
2
X(x
i
ui )(xj
uj )
i ;j
@2f
@ ui @ uj
+
1
Given that f is fairly smooth as well as bounded, and the grid is
fine enough, we can ignore quadratic and higher terms.
2
Problem is that now we need to memoize all fi (u ) =
data!
3
Main Question: Can we “cheat” by shortcutting the partials?
4
If f were linear, obviously
5
What if the grid is warped “similarly” to f ?
@f
=
@ xi
constant.
@f
(u ).
@ xi
50x
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Context: Decision-Making Model at Chess
1
Domain: A set of decision-making situations t .
Chess game turns
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Context: Decision-Making Model at Chess
1
Domain: A set of decision-making situations t .
Chess game turns
2
Inputs: Values vi for every option at turn t .
Computer values of moves mi
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Context: Decision-Making Model at Chess
1
Domain: A set of decision-making situations t .
Chess game turns
2
Inputs: Values vi for every option at turn t .
Computer values of moves mi
3
Parameters: s ; c ; : : : denoting skills and levels.
Trained correspondence to chess Elo rating E
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Context: Decision-Making Model at Chess
1
Domain: A set of decision-making situations t .
Chess game turns
2
Inputs: Values vi for every option at turn t .
Computer values of moves mi
3
4
Parameters: s ; c ; : : : denoting skills and levels.
Trained correspondence to chess Elo rating E
Defines fallible agent P (s ; c ; : : : ).
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Context: Decision-Making Model at Chess
1
Domain: A set of decision-making situations t .
Chess game turns
2
Inputs: Values vi for every option at turn t .
Computer values of moves mi
3
4
5
Parameters: s ; c ; : : : denoting skills and levels.
Trained correspondence to chess Elo rating E
Defines fallible agent P (s ; c ; : : : ).
Main Output: Probabilities pt ;i for P (s ; c ; : : : ) to select option i at
time t .
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Context: Decision-Making Model at Chess
1
Domain: A set of decision-making situations t .
Chess game turns
2
Inputs: Values vi for every option at turn t .
Computer values of moves mi
3
4
5
6
Parameters: s ; c ; : : : denoting skills and levels.
Trained correspondence to chess Elo rating E
Defines fallible agent P (s ; c ; : : : ).
Main Output: Probabilities pt ;i for P (s ; c ; : : : ) to select option i at
time t .
Derived Outputs:
Aggregate statistics: move-match MM, average error AE, . . .
Projected confidence intervals for those statistics.
“Intrinsic Performance Ratings” (IPR’s).
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
How the Model Operates
1
Use analysis data and parameters s ; c ; : : : to compute “perceived
inferiorities” xi 2 [0:0; 1:0] of each of N possible moves. Let
ai = 1 xi .
( x1 =
0:0 x2
x3 xN ) (a1 = 1:0 a2 aN 0)
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
How the Model Operates
1
Use analysis data and parameters s ; c ; : : : to compute “perceived
inferiorities” xi 2 [0:0; 1:0] of each of N possible moves. Let
ai = 1 xi .
( x1 =
2
0:0 x2
x3 xN ) (a1 = 1:0 a2 aN 0)
For a fixed function h, solve
h (pi )
h (p1 ) =
ai subject to
P
N
i =1 pi =
1:
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
How the Model Operates
1
Use analysis data and parameters s ; c ; : : : to compute “perceived
inferiorities” xi 2 [0:0; 1:0] of each of N possible moves. Let
ai = 1 xi .
( x1 =
2
3
0:0 x2
x3 xN ) (a1 = 1:0 a2 aN 0)
For a fixed function h, solve
h (pi )
h (p1 ) =
ai subject to
=
1 (a h (p ))
i
1
It suffices to compute p1 ; then pi
h
P
N
i =1 pi =
1:
is relatively easy.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
How the Model Operates
1
Use analysis data and parameters s ; c ; : : : to compute “perceived
inferiorities” xi 2 [0:0; 1:0] of each of N possible moves. Let
ai = 1 xi .
( x1 =
2
3
4
0:0 x2
x3 xN ) (a1 = 1:0 a2 aN 0)
For a fixed function h, solve
h (pi )
h (p1 ) =
ai subject to
=
1 (a h (p ))
i
1
It suffices to compute p1 ; then pi
( si )c
h
P
N
i =1 pi =
1:
is relatively easy.
Model uses ai = e
, where i is the scaled difference in value
between the best move and the i -th best move. Also fairly cheap.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
How the Model Operates
1
Use analysis data and parameters s ; c ; : : : to compute “perceived
inferiorities” xi 2 [0:0; 1:0] of each of N possible moves. Let
ai = 1 xi .
( x1 =
2
3
4
5
0:0 x2
x3 xN ) (a1 = 1:0 a2 aN 0)
For a fixed function h, solve
h (pi )
h (p1 ) =
ai subject to
=
1 (a h (p ))
i
1
It suffices to compute p1 ; then pi
h
P
N
i =1 pi =
1:
is relatively easy.
( si )c
Model uses ai = e
, where i is the scaled difference in value
between the best move and the i -th best move. Also fairly cheap.
But y
=
p1
=
f (x ) may require expensive iterative approximation.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
How the Model Operates
1
Use analysis data and parameters s ; c ; : : : to compute “perceived
inferiorities” xi 2 [0:0; 1:0] of each of N possible moves. Let
ai = 1 xi .
( x1 =
2
3
4
0:0 x2
x3 xN ) (a1 = 1:0 a2 aN 0)
For a fixed function h, solve
h (pi )
h (p1 ) =
ai subject to
=
1 (a h (p ))
i
1
It suffices to compute p1 ; then pi
h
P
N
i =1 pi =
1:
is relatively easy.
( si )c
Model uses ai = e
, where i is the scaled difference in value
between the best move and the i -th best move. Also fairly cheap.
=
p1
=
f (x ) may require expensive iterative approximation.
5
But y
6
Note f is symmetric, so x can be an ordered sequence.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Side Note and Pure-Math Problem
1
The simple function h (p ) = p, so pi
=
P
ai
i
ai
, works poorly.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Side Note and Pure-Math Problem
1
2
The simple function h (p ) = p, so pi
Much better is h (p ) =
1
log(1=p ) .
=
P
ai
i
ai
, works poorly.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Side Note and Pure-Math Problem
1
The simple function h (p ) = p, so pi
2
Much better is h (p ) =
3
Gives pi
=
=
P
1
log(1=p ) .
p1bi , where bi
1
= 1=ai = 1 x .
i
ai
i
ai
, works poorly.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Side Note and Pure-Math Problem
1
The simple function h (p ) = p, so pi
=
P
ai
i
ai
, works poorly.
1
log(1=p ) .
2
Much better is h (p ) =
3
Gives pi
4
Problem: Given y and b1 ; : : : ; bN , find p such that
=
p1bi , where bi
p b1
1
= 1=ai = 1 x .
i
+p
b2
+
+ pb
N
=
y:
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Side Note and Pure-Math Problem
1
The simple function h (p ) = p, so pi
=
P
ai
i
ai
, works poorly.
1
log(1=p ) .
2
Much better is h (p ) =
3
Gives pi
4
Problem: Given y and b1 ; : : : ; bN , find p such that
=
p1bi , where bi
p b1
5
For N
=
1, simply p
=
1
= 1=ai = 1 x .
i
+p
b2
+
+ pb
N
=
y:
py. So this generalizes taking roots.
b
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Side Note and Pure-Math Problem
1
The simple function h (p ) = p, so pi
=
P
ai
i
ai
, works poorly.
1
log(1=p ) .
2
Much better is h (p ) =
3
Gives pi
4
Problem: Given y and b1 ; : : : ; bN , find p such that
=
p1bi , where bi
1
= 1=ai = 1 x .
i
p b1
=
5
For N
1, simply p
6
We have y
=
=
1 and b1
+p
b2
+
+ pb
N
=
y:
py. So this generalizes taking roots.
b
=
1. Can this be solved without iteration?
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Side Note and Pure-Math Problem
1
The simple function h (p ) = p, so pi
=
P
ai
i
ai
, works poorly.
1
log(1=p ) .
2
Much better is h (p ) =
3
Gives pi
4
Problem: Given y and b1 ; : : : ; bN , find p such that
=
p1bi , where bi
1
= 1=ai = 1 x .
i
p b1
=
1, simply p
=
+p
b2
+
+ pb
N
=
y:
py. So this generalizes taking roots.
5
For N
6
We have y
7
Also: Current Expansion uses data for each depth d.
=
1 and b1
b
=
1. Can this be solved without iteration?
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Axioms and Properties
1
2
Suppose x = (0:0; 0:0; 0:0; 0:0; 1:0; 1:0; : : : ; 1:0).
This means four equal-optimal moves, all others lose instantly.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Axioms and Properties
1
2
3
Suppose x = (0:0; 0:0; 0:0; 0:0; 1:0; 1:0; : : : ; 1:0).
This means four equal-optimal moves, all others lose instantly.
The model will give p1 = p2 = p3 = p4 = 0:25, all other pi = 0.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Axioms and Properties
1
2
3
4
Suppose x = (0:0; 0:0; 0:0; 0:0; 1:0; 1:0; : : : ; 1:0).
This means four equal-optimal moves, all others lose instantly.
The model will give p1 = p2 = p3 = p4 = 0:25, all other pi = 0.
Axiom: Influence of poor moves tapers off:
@f
As xi ! 1:0;
! 0:
@ xi
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Axioms and Properties
1
2
3
4
5
Suppose x = (0:0; 0:0; 0:0; 0:0; 1:0; 1:0; : : : ; 1:0).
This means four equal-optimal moves, all others lose instantly.
The model will give p1 = p2 = p3 = p4 = 0:25, all other pi = 0.
Axiom: Influence of poor moves tapers off:
@f
As xi ! 1:0;
! 0:
@ xi
Axiom: Influence of lower-ranked moves becomes less:
@f
@f
For i < j ;
<
:
@ xj @ xi
(Not quite what was meant. . . )
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Axioms and Properties
1
2
3
4
5
6
Suppose x = (0:0; 0:0; 0:0; 0:0; 1:0; 1:0; : : : ; 1:0).
This means four equal-optimal moves, all others lose instantly.
The model will give p1 = p2 = p3 = p4 = 0:25, all other pi = 0.
Axiom: Influence of poor moves tapers off:
@f
As xi ! 1:0;
! 0:
@ xi
Axiom: Influence of lower-ranked moves becomes less:
@f
@f
For i < j ;
<
:
@ xj @ xi
(Not quite what was meant. . . )
“Universal Guess”: In the first Taylor term, use
@f
1
1
ai = (1 xi ):
@ xi
i
i
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Tapered Grid
1
Grid needs higher precision near 0.0 in any coordinate, less near 1.0.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Tapered Grid
1
Grid needs higher precision near 0.0 in any coordinate, less near 1.0.
2
Grid needs less precision also for higher coordinates i .
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Tapered Grid
1
Grid needs higher precision near 0.0 in any coordinate, less near 1.0.
2
Grid needs less precision also for higher coordinates i .
3
Given x = (x1 ; : : : ; xN ), how to define “nearest” gridpoint
u = (u1 ; : : : ; uN )?
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Tapered Grid
1
Grid needs higher precision near 0.0 in any coordinate, less near 1.0.
2
Grid needs less precision also for higher coordinates i .
3
4
Given x = (x1 ; : : : ; xN ), how to define “nearest” gridpoint
u = (u1 ; : : : ; uN )?
How to define a good bounding set u ; v ; : : : ?
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
The Tapered Grid
1
Grid needs higher precision near 0.0 in any coordinate, less near 1.0.
2
Grid needs less precision also for higher coordinates i .
3
Given x = (x1 ; : : : ; xN ), how to define “nearest” gridpoint
u = (u1 ; : : : ; uN )?
4
How to define a good bounding set u ; v ; : : : ?
5
How to make the computation of nearby gridpoints efficient?
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Strategies
Given x
1
= (x1 ; x2 ; : : : ; xN ),
Bounds x + and x are well-defined by rounding each coordinate
up/down to a gridpoint.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Strategies
Given x
1
2
= (x1 ; x2 ; : : : ; xN ),
Bounds x + and x are well-defined by rounding each coordinate
up/down to a gridpoint.
“Nearest Neighbor” is defined by a nondecreasing sequence
using only values from x + and x . Always u1 = 0:0 = x1+ = x1 .
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Strategies
Given x
1
= (x1 ; x2 ; : : : ; xN ),
Bounds x + and x are well-defined by rounding each coordinate
up/down to a gridpoint.
2
“Nearest Neighbor” is defined by a nondecreasing sequence
using only values from x + and x . Always u1 = 0:0 = x1+ = x1 .
3
Start with x , but “round up” when the rounding-down deficiency
exceeds some weighted threshold.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Strategies
Given x
1
= (x1 ; x2 ; : : : ; xN ),
Bounds x + and x are well-defined by rounding each coordinate
up/down to a gridpoint.
2
“Nearest Neighbor” is defined by a nondecreasing sequence
using only values from x + and x . Always u1 = 0:0 = x1+ = x1 .
3
Start with x , but “round up” when the rounding-down deficiency
exceeds some weighted threshold.
4
Once you have “rounded up,” you can use same gridpoint value, but
cannot “round down” again until x values come above it.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Strategies
Given x
1
= (x1 ; x2 ; : : : ; xN ),
Bounds x + and x are well-defined by rounding each coordinate
up/down to a gridpoint.
2
“Nearest Neighbor” is defined by a nondecreasing sequence
using only values from x + and x . Always u1 = 0:0 = x1+ = x1 .
3
Start with x , but “round up” when the rounding-down deficiency
exceeds some weighted threshold.
4
Once you have “rounded up,” you can use same gridpoint value, but
cannot “round down” again until x values come above it.
5
Like a heuristic for solving Knapsack problems.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Strategies
Given x
1
= (x1 ; x2 ; : : : ; xN ),
Bounds x + and x are well-defined by rounding each coordinate
up/down to a gridpoint.
2
“Nearest Neighbor” is defined by a nondecreasing sequence
using only values from x + and x . Always u1 = 0:0 = x1+ = x1 .
3
Start with x , but “round up” when the rounding-down deficiency
exceeds some weighted threshold.
4
Once you have “rounded up,” you can use same gridpoint value, but
cannot “round down” again until x values come above it.
5
Like a heuristic for solving Knapsack problems.
6
Refinements which we have not yet fully explored include working
backward from i = N (too).
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Strategies
Given x
1
= (x1 ; x2 ; : : : ; xN ),
Bounds x + and x are well-defined by rounding each coordinate
up/down to a gridpoint.
2
“Nearest Neighbor” is defined by a nondecreasing sequence
using only values from x + and x . Always u1 = 0:0 = x1+ = x1 .
3
Start with x , but “round up” when the rounding-down deficiency
exceeds some weighted threshold.
4
Once you have “rounded up,” you can use same gridpoint value, but
cannot “round down” again until x values come above it.
5
Like a heuristic for solving Knapsack problems.
6
Refinements which we have not yet fully explored include working
backward from i = N (too).
7
Combine with “universal gradient” idea, or even ignore said idea.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Results So Far. . .
1
We ran experiments under a randomized distribution D in which
r 2 [xi 1 ; 1] is sampled uniformly and
xi
= xi 1 + (r
xi
1)
(capped at xi
= 1:0):
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Results So Far. . .
1
We ran experiments under a randomized distribution D in which
r 2 [xi 1 ; 1] is sampled uniformly and
xi
2
= xi 1 + (r
xi
1)
(capped at xi
= 1:0):
That is, we make each move randomly slightly inferior to the
previous one. We choose according to N , to make expectation
xi 1:0 as i nears N .
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Results So Far. . .
1
We ran experiments under a randomized distribution D in which
r 2 [xi 1 ; 1] is sampled uniformly and
xi
2
3
= xi 1 + (r
xi
1)
(capped at xi
= 1:0):
That is, we make each move randomly slightly inferior to the
previous one. We choose according to N , to make expectation
xi 1:0 as i nears N .
Results under D are good: 3-place precision on (: : : ) given
2-place to 1-place precision on grid.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Results So Far. . .
1
We ran experiments under a randomized distribution D in which
r 2 [xi 1 ; 1] is sampled uniformly and
xi
2
3
4
= xi 1 + (r
xi
1)
(capped at xi
= 1:0):
That is, we make each move randomly slightly inferior to the
previous one. We choose according to N , to make expectation
xi 1:0 as i nears N .
Results under D are good: 3-place precision on (: : : ) given
2-place to 1-place precision on grid.
Results on real chess data. . .
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Results So Far. . .
1
We ran experiments under a randomized distribution D in which
r 2 [xi 1 ; 1] is sampled uniformly and
xi
2
3
4
= xi 1 + (r
xi
1)
(capped at xi
= 1:0):
That is, we make each move randomly slightly inferior to the
previous one. We choose according to N , to make expectation
xi 1:0 as i nears N .
Results under D are good: 3-place precision on (: : : ) given
2-place to 1-place precision on grid.
Results on real chess data. . . still a work in progress.
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Results for NN+UG
Efficient Memoization for Approximate Function Evaluation over Sequence Arguments
Results for Just NN