## FrankWolfe.jl

Julia implementation for various Frank-Wolfe and Conditional Gradient variants
Author ZIB-IOL
Popularity
24 Stars
Updated Last
2 Years Ago
Started In
January 2021

# Frank-Wolfe and Conditional Gradient algorithms

This package defines a generic interface and several implementations for Frank-Wolfe algorithms. The main entry point is the frank_wolfe function running the algorithm.

FrankWolfe.frank_wolfe(f, grad!, lmo, x0, max_iteration=1000, line_search=FrankWolfe.Agnostic(), verbose=true)

where f(x) is the objective function, grad!(storage, x) is the inplace gradient. lmo is a structure implementing the Linear Minimization Oracle interface presented below.

## LMO

Several oracles are implemented, all are subtypes of LinearMinimizationOracle and implement the method:

compute_extreme_point(lmo::LMO, direction; kwargs...) -> v

which takes a minimization direction and returns the point v minimizing in the direction over the set represented by the LMO.

## Implemented Methods

### LMOs

Several common LMOs are available out-of-the-box

• Probability simplex
• Unit simplex
• K-sparse polytope
• K-norm ball
• L_p-norm ball
• Birkhoff polytope

See https://arxiv.org/pdf/2010.07243.pdf and https://arxiv.org/abs/2101.10040 for details

Moreover:

• you can simply define your own LMOs directly
• you can use an LP oracle defined via an LP solver (e.g., glop, scip, soplex) with MathOptInferface

### General Features

#### Multi-precision

All algorithms can run in various precisions modes: Float16, Float32, Float64, BigFloat and also for rationals based on various integer types Int32, Int64, BigInt (see e.g., the approximate CarathΓ©odory example)

#### Line Search Strategies

Most common strategies and some more particular ones:

• Agnostic: 2/(2+t) rule for FW
• Nonconvex: 1/sqrt rule for nonconvex function and vanilla FW
• Fixed: fixed stepsize of a given value. Useful for nonconvex and stochastic or more generally when we know the total number of iterations
• Short-step rule: basically minimizing the smoothness inequality -> requires knowledge of (an estimate of) L
• Golden ratio linesearch
• Backtracking line search
• Rational Short-step rule: some as short-step rule but all computations are kept rational if inputs are rational. useful for the rational variants
• Adaptive FW: starts with an estimate for L and then refine it dynamically (see https://arxiv.org/pdf/1806.05123.pdf and also the survey (forthcoming))

#### Callbacks

All top-level algorithms can take an optional callback argument, which must be a function taking a named tuple as argument of the form:

state = (
t = t,
primal = primal,
dual = primal - dual_gap,
dual_gap = phi_value,
time = (time_ns() - time_start) / 1e9,
x = x,
v = vertex,
active_set_length = active_set_length,
)

Some fields of the named tuple can vary, but the 6 first are always present in this order. The callback can be used to log additional information or store some values of interest in an external array. If a callback is passed, the trajectory keyword is ignored since it is a special case of callback pushing the 5 first elements of the state to an array returned from the algorithm.

#### Other

• Emphasis: All solvers support emphasis (parameter Emphasis) to either exploit vectorized linear algebra or be memory efficient, e.g., for extreme large-scale instances
• Various caching strategies for the lazy implementations. Unbounded cache sizes (can get slow), bounded cache sizes as well as early returns once any sufficient vertex is found in the cache.
• (to come:) when the LMO can compute dual prices then the Frank-Wolfe algorithms return dual prices for the (approximately) optimal solutions (see https://arxiv.org/abs/2101.02087)
• optionally all algorithms can be endowed with gradient momentum. This might help convergence especially in the stochastic context.

## Cool Examples

### Approximate CarathΓ©odory with rational arithmetic

Example: examples/approximateCaratheodory.jl

We can solve the approximate CarathΓ©odory problem with rational arithmetic to obtain rational approximations; see https://arxiv.org/abs/1911.04415 for some background about approximate CarathΓ©odory and Conditioanl Gradients. We consider the simple instance of approximating the 0 over the probability simplex here:

with n = 100.

Vanilla Frank-Wolfe Algorithm.
EMPHASIS: blas STEPSIZE: rationalshortstep EPSILON: 1.0e-7 max_iteration: 100 TYPE: Rational{BigInt}

βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
Type     Iteration         Primal           Dual       Dual Gap           Time
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
I             0   1.000000e+00  -1.000000e+00   2.000000e+00   1.540385e-01
FW            10   9.090909e-02  -9.090909e-02   1.818182e-01   2.821186e-01
FW            20   4.761905e-02  -4.761905e-02   9.523810e-02   3.027964e-01
FW            30   3.225806e-02  -3.225806e-02   6.451613e-02   3.100331e-01
FW            40   2.439024e-02  -2.439024e-02   4.878049e-02   3.171654e-01
FW            50   1.960784e-02  -1.960784e-02   3.921569e-02   3.244207e-01
FW            60   1.639344e-02  -1.639344e-02   3.278689e-02   3.326185e-01
FW            70   1.408451e-02  -1.408451e-02   2.816901e-02   3.418239e-01
FW            80   1.234568e-02  -1.234568e-02   2.469136e-02   3.518750e-01
FW            90   1.098901e-02  -1.098901e-02   2.197802e-02   3.620287e-01
Last                 1.000000e-02   1.000000e-02   0.000000e+00   4.392171e-01
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ

0.600608 seconds (3.83 M allocations: 111.274 MiB, 12.97% gc time)

Output type of solution: Rational{BigInt}


The solution returned is rational as we can see and in fact the exactly optimal solution:

x = Rational{BigInt}[1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100, 1//100]


### Large-Scale problems

Example: examples/large_scale.jl

The package is built to scale well, for those conditional gradients variants that can scale well. For exampple, Away-Step Frank-Wolfe and Pairwise Conditional Gradients do in most cases not scale well because they need to maintain active sets and maintaining them can be very expensive. Similarly, line search methods might become prohibitive at large sizes. However if we consider scale-friendly variants, e.g., the vanilla Frank-Wolfe algorithm with the agnostic step size rule or short step rule, then these algorithms can scale well to extreme sizes esentially only limited by the amount of memory available. However even for these methods that tend to scale well, allocation of memory itself can be very slow when you need to allocate gigabytes of memory for a single gradient computation.

The package is build to support extreme sizes with a special memory efficient emphasis emphasis=FrankWolfe.memory, which minimizes very expensive allocation memory and performs as many operations as possible in-place.

Here is an example of a run with 1e9 variables. Each gradient is around 7.5 GB in size. Here is the output of the run broken down into pieces:

Size of single vector (Float64): 7629.39453125 MB
Testing f... 100%|βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ| Time: 0:00:23
Testing grad... 100%|ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ| Time: 0:00:23
Testing lmo... 100%|βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ| Time: 0:00:29
Testing dual gap... 100%|ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ| Time: 0:00:46
Testing update... (Emphasis: blas) 100%|βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ| Time: 0:01:35
Testing update... (Emphasis: memory) 100%|βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ| Time: 0:00:58
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
Time                   Allocations
ββββββββββββββββββββββ   βββββββββββββββββββββββ
Tot / % measured:           278s / 31.4%            969GiB / 30.8%

Section           ncalls     time   %tot     avg     alloc   %tot      avg
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
update (blas)         10    36.1s  41.3%   3.61s    149GiB  50.0%  14.9GiB
lmo                   10    18.4s  21.1%   1.84s     0.00B  0.00%    0.00B
grad                  10    12.8s  14.6%   1.28s   74.5GiB  25.0%  7.45GiB
f                     10    12.7s  14.5%   1.27s   74.5GiB  25.0%  7.45GiB
update (memory)       10    5.00s  5.72%   500ms     0.00B  0.00%    0.00B
dual gap              10    2.40s  2.75%   240ms     0.00B  0.00%    0.00B
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ


The above is the optional benchmarking of the oracles that we provide to understand how fast crucial parts of the algorithms are, mostly notably oracle evaluations, the update of the iterate and the computation of the dual gap. As you can see if you compare update (blas) vs. update (memory), the normal update when we use BLAS requires an additional 14.9GB of memory on top of the gradient etc whereas the update (memory) (the memory emphasis mode) does not consume any extra memory. This is also reflected in the computational times: the BLAS version requires 3.61 seconds on average to update the iterate, while the memory emphasis version requires only 500ms. In fact none of the crucial components in the algorithm consume any memory when run in memory efficient mode. Now let us look at the actual footprint of the whole algorithm:

Vanilla Frank-Wolfe Algorithm.
EMPHASIS: memory STEPSIZE: agnostic EPSILON: 1.0e-7 MAXITERATION: 1000 TYPE: Float64
MOMENTUM: nothing GRADIENTTYPE: Nothing
WARNING: In memory emphasis mode iterates are written back into x0!

βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
Type     Iteration         Primal           Dual       Dual Gap           Time         It/sec
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
I             0   1.000000e+00  -1.000000e+00   2.000000e+00   8.783523e+00   0.000000e+00
FW           100   1.326732e-02  -1.326733e-02   2.653465e-02   4.635923e+02   2.157068e-01
FW           200   6.650080e-03  -6.650086e-03   1.330017e-02   9.181294e+02   2.178342e-01
FW           300   4.437059e-03  -4.437064e-03   8.874123e-03   1.372615e+03   2.185609e-01
FW           400   3.329174e-03  -3.329180e-03   6.658354e-03   1.827260e+03   2.189070e-01
FW           500   2.664003e-03  -2.664008e-03   5.328011e-03   2.281865e+03   2.191190e-01
FW           600   2.220371e-03  -2.220376e-03   4.440747e-03   2.736387e+03   2.192672e-01
FW           700   1.903401e-03  -1.903406e-03   3.806807e-03   3.190951e+03   2.193703e-01
FW           800   1.665624e-03  -1.665629e-03   3.331253e-03   3.645425e+03   2.194532e-01
FW           900   1.480657e-03  -1.480662e-03   2.961319e-03   4.099931e+03   2.195159e-01
FW          1000   1.332665e-03  -1.332670e-03   2.665335e-03   4.554703e+03   2.195533e-01
Last          1000   1.331334e-03  -1.331339e-03   2.662673e-03   4.559822e+03   2.195261e-01
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ

4560.661203 seconds (7.41 M allocations: 112.121 GiB, 0.01% gc time)


As you can see the algorithm ran for about 4600 secs (single-thread run) allocating 112.121 GiB of memory throughout. So how does this average out to the per-iteration cost in terms of memory: 112.121 / 7.45 / 1000 = 0.0151 so about 15.1MiB per iteration which is much less than the size of the gradient and in fact only stems from the reporting here.

NB. This example highlights also one of the great features of first-order methods and conditional gradients in particular: We have dimension-independent convergence rates. In fact, we contract the primal gap as 2LD^2 / (t+2) (for the simple agnostic rule) and, e.g., if the feasible region is the probability simplex with D = sqrt(2) and the function has bounded Lipschitzness, e.g., the function || x - xp ||^2 has L = 2, then the convergence rate is completely independent of the input size. The only thing that limits scaling is how much memory you have available and whether you can stomach the (linear) per-iteration cost.

## Citing

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