Contents

# Limited-Memory Broyden-Fletcher-Goldfarb-Shanno Algorithm

The limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm [Byrd2015] follows the algorithmic framework of an iterative solver with the algorithm-specific transformation
T
and set of intrinsic parameters
S
t
defined for the memory parameter
m
, frequency of curvature estimates calculation
L
, and step-length sequence
α
t
> 0, algorithm-specific vector U and power d of Lebesgue space defined as follows:

## Transformation

:
where
H
is an approximation of the inverse Hessian matrix computed from
m
correction pairs by the Hessian Update Algorithm.
Convergence check:

## Intrinsic Parameters

For the LBFGS algorithm, the set of intrinsic parameters
S
t
includes the following:
• Correction pairs (
s
j
,
y
j
)
• Correction index
k
in the buffer that stores correction pairs
• Index of last iteration
t
of the main loop from the previous run
• Average value of arguments for the previous
L
iterations
• Average value of arguments for the last
L
iterations
Below is the definition and update flow of the intrinsic parameters (
s
i
,
y
i
). The index is set and remains zero for the first 2
L
-1 iterations of the main loop. Starting with iteration 2
L
, the algorithm executes the following steps for each of
L
iterations of the main loop:
1. k
:=
k
+1
2. Choose a set of indices without replacement:
I
H
= {
i
1
,
i
2
, ... ,
i
bH
}, 1 ≤
i
l
<
n
,
l
∈ {1, ...,
b
H
}, |
I
H
| =
b
H
=
correctionPairBatchSize
.
3. Compute the sub-sampled Hessian
at the point
for the objective function using Hessians of its terms
4. Compute the correction pairs (
s
k
,
y
k
):
• The set
S
k
of intrinsic parameters is updated once per
L
iterations of the major loop and remains unchanged between iterations with the numbers that are multiples of
L
• A cyclic buffer stores correction pairs. The algorithm fills the buffer with pairs one-by-one. Once the buffer is full, it returns to the beginning and overwrites the previous correction pairs.

## Hessian Update Algorithm

This algorithm computes the approximation of the inverse Hessian matrix from the set of correction pairs [Byrd2015]).
For a given set of correction pairs (
s
j
,
y
j
),
j
=
k
- min (
k
,
m
) +1, ...,
k
:
1. Set
2. Iterate
j
from
k
- min (
k
,
m
)+1 until
k
:
3. Return
H

#### Product and Performance Information

1

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Notice revision #20110804