To date, all torch use circumstances we’ve mentioned right here have been in deep studying. Nevertheless, its automated differentiation function is helpful in different areas. One outstanding instance is numerical optimization: We will use torch to seek out the minimal of a operate.

In truth, operate minimization is precisely what occurs in coaching a neural community. However there, the operate in query usually is much too advanced to even think about discovering its minima analytically. Numerical optimization goals at increase the instruments to deal with simply this complexity. To that finish, nonetheless, it begins from features which can be far much less deeply composed. As an alternative, they’re hand-crafted to pose particular challenges.

This put up is a primary introduction to numerical optimization with torch. Central takeaways are the existence and usefulness of its L-BFGS optimizer, in addition to the influence of working L-BFGS with line search. As a enjoyable add-on, we present an instance of constrained optimization, the place a constraint is enforced by way of a quadratic penalty operate.

To heat up, we take a detour, minimizing a operate “ourselves” utilizing nothing however tensors. This may turn into related later, although, as the general course of will nonetheless be the identical. All adjustments might be associated to integration of optimizers and their capabilities.

Operate minimization, DYI strategy

To see how we will reduce a operate “by hand”, let’s attempt the enduring Rosenbrock operate. It is a operate with two variables:

[
f(x_1, x_2) = (a – x_1)^2 + b * (x_2 – x_1^2)^2
]

, with (a) and (b) configurable parameters typically set to 1 and 5, respectively.

In R:

library(torch)

a <- 1
b <- 5

rosenbrock <- operate(x) {
  x1 <- x[1]
  x2 <- x[2]
  (a - x1)^2 + b * (x2 - x1^2)^2
}

Its minimal is situated at (1,1), inside a slim valley surrounded by breakneck-steep cliffs:

Determine 1: Rosenbrock operate.

Our purpose and technique are as follows.

We wish to discover the values (x_1) and (x_2) for which the operate attains its minimal. Now we have to start out someplace; and from wherever that will get us on the graph we comply with the unfavorable of the gradient “downwards”, descending into areas of consecutively smaller operate worth.

Concretely, in each iteration, we take the present ((x1,x2)) level, compute the operate worth in addition to the gradient, and subtract some fraction of the latter to reach at a brand new ((x1,x2)) candidate. This course of goes on till we both attain the minimal – the gradient is zero – or enchancment is under a selected threshold.

Right here is the corresponding code. For no particular causes, we begin at (-1,1) . The training charge (the fraction of the gradient to subtract) wants some experimentation. (Attempt 0.1 and 0.001 to see its influence.)

num_iterations <- 1000

# fraction of the gradient to subtract 
lr <- 0.01

# operate enter (x1,x2)
# that is the tensor w.r.t. which we'll have torch compute the gradient
x_star <- torch_tensor(c(-1, 1), requires_grad = TRUE)

for (i in 1:num_iterations) {

  if (i %% 100 == 0) cat("Iteration: ", i, "n")

  # name operate
  worth <- rosenbrock(x_star)
  if (i %% 100 == 0) cat("Worth is: ", as.numeric(worth), "n")

  # compute gradient of worth w.r.t. params
  worth$backward()
  if (i %% 100 == 0) cat("Gradient is: ", as.matrix(x_star$grad), "nn")

  # handbook replace
  with_no_grad({
    x_star$sub_(lr * x_star$grad)
    x_star$grad$zero_()
  })
}

Iteration:  100 
Worth is:  0.3502924 
Gradient is:  -0.667685 -0.5771312 

Iteration:  200 
Worth is:  0.07398106 
Gradient is:  -0.1603189 -0.2532476 

...
...

Iteration:  900 
Worth is:  0.0001532408 
Gradient is:  -0.004811743 -0.009894371 

Iteration:  1000 
Worth is:  6.962555e-05 
Gradient is:  -0.003222887 -0.006653666 

Whereas this works, it actually serves as an example the precept. With torch offering a bunch of confirmed optimization algorithms, there is no such thing as a want for us to manually compute the candidate (mathbf{x}) values.

Operate minimization with torch optimizers

As an alternative, we let a torch optimizer replace the candidate (mathbf{x}) for us. Habitually, our first attempt is Adam.

Adam

With Adam, optimization proceeds so much sooner. Reality be instructed, although, selecting an excellent studying charge nonetheless takes non-negligeable experimentation. (Attempt the default studying charge, 0.001, for comparability.)

num_iterations <- 100

x_star <- torch_tensor(c(-1, 1), requires_grad = TRUE)

lr <- 1
optimizer <- optim_adam(x_star, lr)

for (i in 1:num_iterations) {
  
  if (i %% 10 == 0) cat("Iteration: ", i, "n")
  
  optimizer$zero_grad()
  worth <- rosenbrock(x_star)
  if (i %% 10 == 0) cat("Worth is: ", as.numeric(worth), "n")
  
  worth$backward()
  optimizer$step()
  
  if (i %% 10 == 0) cat("Gradient is: ", as.matrix(x_star$grad), "nn")
  
}

Iteration:  10 
Worth is:  0.8559565 
Gradient is:  -1.732036 -0.5898831 

Iteration:  20 
Worth is:  0.1282992 
Gradient is:  -3.22681 1.577383 

...
...

Iteration:  90 
Worth is:  4.003079e-05 
Gradient is:  -0.05383469 0.02346456 

Iteration:  100 
Worth is:  6.937736e-05 
Gradient is:  -0.003240437 -0.006630421 

It took us a couple of hundred iterations to reach at an honest worth. It is a lot sooner than the handbook strategy above, however nonetheless quite a bit. Fortunately, additional enhancements are attainable.

L-BFGS

Among the many many torch optimizers generally utilized in deep studying (Adam, AdamW, RMSprop …), there’s one “outsider”, a lot better identified in basic numerical optimization than in neural-networks house: L-BFGS, a.okay.a. Restricted-memory BFGS, a memory-optimized implementation of the Broyden–Fletcher–Goldfarb–Shanno optimization algorithm (BFGS).

BFGS is probably essentially the most broadly used among the many so-called Quasi-Newton, second-order optimization algorithms. Versus the household of first-order algorithms that, in deciding on a descent path, make use of gradient data solely, second-order algorithms moreover take curvature data into consideration. To that finish, actual Newton strategies truly compute the Hessian (a pricey operation), whereas Quasi-Newton strategies keep away from that value and, as a substitute, resort to iterative approximation.

Trying on the contours of the Rosenbrock operate, with its extended, slim valley, it’s not tough to think about that curvature data would possibly make a distinction. And, as you’ll see in a second, it actually does. Earlier than although, one notice on the code. When utilizing L-BFGS, it’s essential to wrap each operate name and gradient analysis in a closure (calc_loss(), within the under snippet), for them to be callable a number of instances per iteration. You’ll be able to persuade your self that the closure is, in truth, entered repeatedly, by inspecting this code snippet’s chatty output:

num_iterations <- 3

x_star <- torch_tensor(c(-1, 1), requires_grad = TRUE)

optimizer <- optim_lbfgs(x_star)

calc_loss <- operate() {

  optimizer$zero_grad()

  worth <- rosenbrock(x_star)
  cat("Worth is: ", as.numeric(worth), "n")

  worth$backward()
  cat("Gradient is: ", as.matrix(x_star$grad), "nn")
  worth

}

for (i in 1:num_iterations) {
  cat("Iteration: ", i, "n")
  optimizer$step(calc_loss)
}

Iteration:  1 
Worth is:  4 
Gradient is:  -4 0 

Worth is:  6 
Gradient is:  -2 10 

...
...

Worth is:  0.04880721 
Gradient is:  -0.262119 -0.1132655 

Worth is:  0.0302862 
Gradient is:  1.293824 -0.7403332 

Iteration:  2 
Worth is:  0.01697086 
Gradient is:  0.3468466 -0.3173429 

Worth is:  0.01124081 
Gradient is:  0.2420997 -0.2347881 

...
...

Worth is:  1.111701e-09 
Gradient is:  0.0002865837 -0.0001251698 

Worth is:  4.547474e-12 
Gradient is:  -1.907349e-05 9.536743e-06 

Iteration:  3 
Worth is:  4.547474e-12 
Gradient is:  -1.907349e-05 9.536743e-06 

Although we ran the algorithm for 3 iterations, the optimum worth actually is reached after two. Seeing how nicely this labored, we attempt L-BFGS on a harder operate, named flower, for fairly self-evident causes.

(But) extra enjoyable with L-BFGS

Right here is the flower operate. Mathematically, its minimal is close to (0,0), however technically the operate itself is undefined at (0,0), because the atan2 used within the operate isn’t outlined there.

a <- 1
b <- 1
c <- 4

flower <- operate(x) {
  a * torch_norm(x) + b * torch_sin(c * torch_atan2(x[2], x[1]))
}

Determine 2: Flower operate.

We run the identical code as above, ranging from (20,20) this time.

num_iterations <- 3

x_star <- torch_tensor(c(20, 0), requires_grad = TRUE)

optimizer <- optim_lbfgs(x_star)

calc_loss <- operate() {

  optimizer$zero_grad()

  worth <- flower(x_star)
  cat("Worth is: ", as.numeric(worth), "n")

  worth$backward()
  cat("Gradient is: ", as.matrix(x_star$grad), "n")
  
  cat("X is: ", as.matrix(x_star), "nn")
  
  worth

}

for (i in 1:num_iterations) {
  cat("Iteration: ", i, "n")
  optimizer$step(calc_loss)
}

Iteration:  1 
Worth is:  28.28427 
Gradient is:  0.8071069 0.6071068 
X is:  20 20 

...
...

Worth is:  19.33546 
Gradient is:  0.8100872 0.6188223 
X is:  12.957 14.68274 

...
...

Worth is:  18.29546 
Gradient is:  0.8096464 0.622064 
X is:  12.14691 14.06392 

...
...

Worth is:  9.853705 
Gradient is:  0.7546976 0.7025688 
X is:  5.763702 8.895616 

Worth is:  2635.866 
Gradient is:  -0.7407354 -0.6717985 
X is:  -1949.697 -1773.551 

Iteration:  2 
Worth is:  1333.113 
Gradient is:  -0.7413024 -0.6711776 
X is:  -985.4553 -897.5367 

Worth is:  30.16862 
Gradient is:  -0.7903821 -0.6266789 
X is:  -21.02814 -21.72296 

Worth is:  1281.39 
Gradient is:  0.7544561 0.6563575 
X is:  964.0121 843.7817 

Worth is:  628.1306 
Gradient is:  0.7616636 0.6480014 
X is:  475.7051 409.7372 

Worth is:  4965690 
Gradient is:  -0.7493951 -0.662123 
X is:  -3721262 -3287901 

Worth is:  2482306 
Gradient is:  -0.7503822 -0.6610042 
X is:  -1862675 -1640817 

Worth is:  8.61863e+11 
Gradient is:  0.7486113 0.6630091 
X is:  645200412672 571423064064 

Worth is:  430929412096 
Gradient is:  0.7487153 0.6628917 
X is:  322643460096 285659529216 

Worth is:  Inf 
Gradient is:  0 0 
X is:  -2.826342e+19 -2.503904e+19 

Iteration:  3 
Worth is:  Inf 
Gradient is:  0 0 
X is:  -2.826342e+19 -2.503904e+19 

This has been much less of a hit. At first, loss decreases properly, however all of the sudden, the estimate dramatically overshoots, and retains bouncing between unfavorable and optimistic outer house ever after.

Fortunately, there’s something we will do.

L-BFGS with line search

Taken in isolation, what a Quasi-Newton technique like L-BFGS does is decide the most effective descent path. Nevertheless, as we simply noticed, an excellent path isn’t sufficient. With the flower operate, wherever we’re, the optimum path results in catastrophe if we keep on it lengthy sufficient. Thus, we want an algorithm that fastidiously evaluates not solely the place to go, but additionally, how far.

For that reason, L-BFGS implementations generally incorporate line search, that’s, a algorithm indicating whether or not a proposed step size is an effective one, or must be improved upon.

Particularly, torch’s L-BFGS optimizer implements the Robust Wolfe situations. We re-run the above code, altering simply two traces. Most significantly, the one the place the optimizer is instantiated:

optimizer <- optim_lbfgs(x_star, line_search_fn = "strong_wolfe")

And secondly, this time I discovered that after the third iteration, loss continued to lower for some time, so I let it run for 5 iterations. Right here is the output:

Iteration:  1 
...
...

Worth is:  -0.8838741 
Gradient is:  3.742207 7.521572 
X is:  0.09035123 -0.03220009 

Worth is:  -0.928809 
Gradient is:  1.464702 0.9466625 
X is:  0.06564617 -0.026706 

Iteration:  2 
...
...

Worth is:  -0.9991404 
Gradient is:  39.28394 93.40318 
X is:  0.0006493925 -0.0002656128 

Worth is:  -0.9992246 
Gradient is:  6.372203 12.79636 
X is:  0.0007130796 -0.0002947929 

Iteration:  3 
...
...

Worth is:  -0.9997789 
Gradient is:  3.565234 5.995832 
X is:  0.0002042478 -8.457939e-05 

Worth is:  -0.9998025 
Gradient is:  -4.614189 -13.74602 
X is:  0.0001822711 -7.553725e-05 

Iteration:  4 
...
...

Worth is:  -0.9999917 
Gradient is:  -382.3041 -921.4625 
X is:  -6.320081e-06 2.614706e-06 

Worth is:  -0.9999923 
Gradient is:  -134.0946 -321.2681 
X is:  -6.921942e-06 2.865841e-06 

Iteration:  5 
...
...

Worth is:  -0.9999999 
Gradient is:  -3446.911 -8320.007 
X is:  -7.267168e-08 3.009783e-08 

Worth is:  -0.9999999 
Gradient is:  -3419.361 -8253.501 
X is:  -7.404627e-08 3.066708e-08 

It’s nonetheless not excellent, however so much higher.

Lastly, let’s go one step additional. Can we use torch for constrained optimization?

Quadratic penalty for constrained optimization

In constrained optimization, we nonetheless seek for a minimal, however that minimal can’t reside simply anyplace: Its location has to meet some variety of extra situations. In optimization lingo, it needs to be possible.

For instance, we stick with the flower operate, however add on a constraint: (mathbf{x}) has to lie exterior a circle of radius (sqrt(2)), centered on the origin. Formally, this yields the inequality constraint

[
2 – {x_1}^2 – {x_2}^2 <= 0
]

A option to reduce flower and but, on the similar time, honor the constraint is to make use of a penalty operate. With penalty strategies, the worth to be minimized is a sum of two issues: the goal operate’s output and a penalty reflecting potential constraint violation. Use of a quadratic penalty, for instance, ends in including a a number of of the sq. of the constraint operate’s output:

# x^2 + y^2 >= 2
# 2 - x^2 - y^2 <= 0
constraint <- operate(x) 2 - torch_square(torch_norm(x))

# quadratic penalty
penalty <- operate(x) torch_square(torch_max(constraint(x), different = 0))

A priori, we will’t understand how massive that a number of needs to be to implement the constraint. Due to this fact, optimization proceeds iteratively. We begin with a small multiplier, (1), say, and improve it for so long as the constraint continues to be violated:

penalty_method <- operate(f, p, x, k_max, rho = 1, gamma = 2, num_iterations = 1) {

  for (okay in 1:k_max) {
    cat("Beginning step: ", okay, ", rho = ", rho, "n")

    reduce(f, p, x, rho, num_iterations)

    cat("Worth: ",  as.numeric(f(x)), "n")
    cat("X: ",  as.matrix(x), "n")
    
    current_penalty <- as.numeric(p(x))
    cat("Penalty: ", current_penalty, "n")
    if (current_penalty == 0) break
    
    rho <- rho * gamma
  }

}

reduce(), known as from penalty_method(), follows the same old proceedings, however now it minimizes the sum of the goal and up-weighted penalty operate outputs:

reduce <- operate(f, p, x, rho, num_iterations) {

  calc_loss <- operate() {
    optimizer$zero_grad()
    worth <- f(x) + rho * p(x)
    worth$backward()
    worth
  }

  for (i in 1:num_iterations) {
    cat("Iteration: ", i, "n")
    optimizer$step(calc_loss)
  }

}

This time, we begin from a low-target-loss, however unfeasible worth. With yet one more change to default L-BFGS (specifically, a lower in tolerance), we see the algorithm exiting efficiently after twenty-two iterations, on the level (0.5411692,1.306563).

x_star <- torch_tensor(c(0.5, 0.5), requires_grad = TRUE)

optimizer <- optim_lbfgs(x_star, line_search_fn = "strong_wolfe", tolerance_change = 1e-20)

penalty_method(flower, penalty, x_star, k_max = 30)

Beginning step:  1 , rho =  1 
Iteration:  1 
Worth:  0.3469974 
X:  0.5154735 1.244463 
Penalty:  0.03444662 

Beginning step:  2 , rho =  2 
Iteration:  1 
Worth:  0.3818618 
X:  0.5288152 1.276674 
Penalty:  0.008182613 

Beginning step:  3 , rho =  4 
Iteration:  1 
Worth:  0.3983252 
X:  0.5351116 1.291886 
Penalty:  0.001996888 

...
...

Beginning step:  20 , rho =  524288 
Iteration:  1 
Worth:  0.4142133 
X:  0.5411959 1.306563 
Penalty:  3.552714e-13 

Beginning step:  21 , rho =  1048576 
Iteration:  1 
Worth:  0.4142134 
X:  0.5411956 1.306563 
Penalty:  1.278977e-13 

Beginning step:  22 , rho =  2097152 
Iteration:  1 
Worth:  0.4142135 
X:  0.5411962 1.306563 
Penalty:  0 

Conclusion

Summing up, we’ve gotten a primary impression of the effectiveness of torch’s L-BFGS optimizer, particularly when used with Robust-Wolfe line search. In truth, in numerical optimization – versus deep studying, the place computational pace is way more of a difficulty – there’s infrequently a cause to not use L-BFGS with line search.

We’ve then caught a glimpse of find out how to do constrained optimization, a process that arises in lots of real-world functions. In that regard, this put up feels much more like a starting than a stock-taking. There’s a lot to discover, from common technique match – when is L-BFGS nicely suited to an issue? – by way of computational efficacy to applicability to totally different species of neural networks. Evidently, if this conjures up you to run your personal experiments, and/or should you use L-BFGS in your personal tasks, we’d love to listen to your suggestions!

Thanks for studying!

Appendix

Rosenbrock operate plotting code

library(tidyverse)

a <- 1
b <- 5

rosenbrock <- operate(x) {
  x1 <- x[1]
  x2 <- x[2]
  (a - x1)^2 + b * (x2 - x1^2)^2
}

df <- expand_grid(x1 = seq(-2, 2, by = 0.01), x2 = seq(-2, 2, by = 0.01)) %>%
  rowwise() %>%
  mutate(x3 = rosenbrock(c(x1, x2))) %>%
  ungroup()

ggplot(information = df,
       aes(x = x1,
           y = x2,
           z = x3)) +
  geom_contour_filled(breaks = as.numeric(torch_logspace(-3, 3, steps = 50)),
                      present.legend = FALSE) +
  theme_minimal() +
  scale_fill_viridis_d(path = -1) +
  theme(side.ratio = 1)

Flower operate plotting code

a <- 1
b <- 1
c <- 4

flower <- operate(x) {
  a * torch_norm(x) + b * torch_sin(c * torch_atan2(x[2], x[1]))
}

df <- expand_grid(x = seq(-3, 3, by = 0.05), y = seq(-3, 3, by = 0.05)) %>%
  rowwise() %>%
  mutate(z = flower(torch_tensor(c(x, y))) %>% as.numeric()) %>%
  ungroup()

ggplot(information = df,
       aes(x = x,
           y = y,
           z = z)) +
  geom_contour_filled(present.legend = FALSE) +
  theme_minimal() +
  scale_fill_viridis_d(path = -1) +
  theme(side.ratio = 1)

Photograph by Michael Trimble on Unsplash