# 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

*and set of intrinsic parameters defined for the memory parameter*T

*, frequency of curvature estimates calculation*m

*, and step-length sequence , algorithm-specific vector*L

*and power*U

*of Lebesgue space defined as follows:*d

## Transformation

where

*is an approximation of the inverse Hessian matrix computed from m correction pairs by the Hessian Update Algorithm.*H

Convergence check:

## Intrinsic Parameters

For the LBFGS algorithm, the set of intrinsic parameters
includes the following:

- Correction pairs
- 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
. The index is set and
remains zero for the first 2L-1 iterations of the main loop.
Starting with iteration

*, the algorithm executes the following steps for each of L iterations of the main loop:*2L

- Choose a set of indices without replacement: , , , .
- Compute the sub-sampled Hessianat the point for the objective function using Hessians of its terms
- Compute the correction pairs :

- The set of intrinsic parameters is updated once per
iterations of the major loop and remains unchanged between iterations with the numbers that are multiples ofLL - 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
,
:

- Set
- Iterate
from untilj:k - ReturnH

## Computation

The limited-memory BFGS algorithm is a special case of an iterative
solver. For parameters, input, and output of iterative solvers, see Computation.

Algorithm Input

In addition to the input of the iterative solver,
the limited-memory BFGS algorithm accepts the following optional input:

OptionalDataID | Input |
---|---|

correctionPairs | A numeric table of size
where the rows represent correction pairs
and s . The row correctionPairs[j],
, is a correction vector
, and the row correctionPairs[j],
, is a correction
vector
.y |

correctionIndices | A numeric table of size
with 32-bit integer indexes. The first value
is the index of correction pair , the second value is the index of last
iteration t from the previous run.k |

averageArgumentLIterations | A numeric table of size
, where row 0 represents average arguments
for previous iterations, and row 1 represents average arguments for
last L iterations. These values are required to compute L correction
vectors in the next step.s |

Algorithm Parameters

In addition to parameters of the iterative solver,
the limited-memory BFGS algorithm has the following parameters:

Parameter | Default Value | Description |
---|---|---|

algorithmFPType | float | The floating-point type that the algorithm uses for intermediate computations. Can be float or double . |

method | defaultDense | Performance-oriented computation method |

batchIndices | NULL | The numeric table of size
with 32-bit integer
indices of terms in the objective function to be used in step
2 of the limited-memory BFGS algorithm. If no indices are provided, the
implementation generates random indices. This parameter can be an object of any class derived from NumericTable ,
except for PackedTriangularMatrix , PackedSymmetricMatrix , and CSRNumericTable . |

batchSize | 10 | The number of observations to compute the stochastic gradient. The
implementation of the algorithm ignores this parameter if the
batchIndices numeric table is provided. If BatchSize equals the number of terms in the objective function, no
random sampling is performed and all terms are used to calculate the
gradient. |

correctionPairBatchSize | 100 | The number of observations to compute the sub-sampled Hessian for
correction pairs computation. The implementation of the
algorithm ignores this parameter if the correctionPairIndices numeric
table is provided. If correctionPairBatchSize equals the number of terms in the objective
function, no random sampling is performed and all terms are used to
calculate the Hessian matrix. |

correctionPairIndices | NULL | The numeric table of size
with
32-bit integer indices to be used instead of random values. If no indices are provided, the
implementation generates random indices. This parameter can be an object of any class derived from NumericTable ,
except for PackedTriangularMatrix , PackedSymmetricMatrix , and CSRNumericTable .If the algorithm runs with no optional input data,
rows
of the table are used. Otherwise, it can use one more row,
in total. |

m | 10 | The memory parameter. The maximum number of correction pairs that define
the approximation of the Hessian matrix. |

L | 10 | The number of iterations between calculations of the curvature estimates. |

stepLengthSequence | A numeric table of size
that contains the default step length equal to .1 | The numeric table of size
or
. The contents of the table depend on its size: - : values of the step-length sequence for .
- : the value of step length at each iteration
..note:
The recommended data type for storing the step-length sequence is the
floating-point type, either float or double, that the algorithm uses in
intermediate computations. |

engine | SharePtr< engines:: mt19937:: Batch>() | Pointer to the random number generator engine that is used internally
for random choosing terms from the objective function. |

Algorithm Output

In addition to the output of the iterative solver, the limited-memory
BFGS algorithm calculates the following optional results:

OptionalDataID | Output |
---|---|

correctionPairs | A numeric table of size
where the rows represent correction pairs
and s . The row correctionPairs[j],
, is a correction vector
, and the row correctionPairs[j],
, is a correction
vector
.y |

correctionIndices | A numeric table of size
with 32-bit integer indexes. The first value
is the index of correction pair , the second value is the index of last
iteration t from the previous run.k |

averageArgumentLIterations | A numeric table of size
, where row 0 represents average arguments
for previous iterations, and row 1 represents average arguments for
last L iterations. These values are required to compute L correction
vectors in the next step.s |

Examples

C++ (CPU)

Batch Processing:

Java*

There is no support for Java on GPU.

Batch Processing:

Python*

Batch Processing: