484 lines
17 KiB
Python
484 lines
17 KiB
Python
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import torch
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from .optimizer import Optimizer
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__all__ = ['LBFGS']
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def _cubic_interpolate(x1, f1, g1, x2, f2, g2, bounds=None):
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# ported from https://github.com/torch/optim/blob/master/polyinterp.lua
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# Compute bounds of interpolation area
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if bounds is not None:
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xmin_bound, xmax_bound = bounds
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else:
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xmin_bound, xmax_bound = (x1, x2) if x1 <= x2 else (x2, x1)
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# Code for most common case: cubic interpolation of 2 points
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# w/ function and derivative values for both
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# Solution in this case (where x2 is the farthest point):
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# d1 = g1 + g2 - 3*(f1-f2)/(x1-x2);
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# d2 = sqrt(d1^2 - g1*g2);
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# min_pos = x2 - (x2 - x1)*((g2 + d2 - d1)/(g2 - g1 + 2*d2));
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# t_new = min(max(min_pos,xmin_bound),xmax_bound);
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d1 = g1 + g2 - 3 * (f1 - f2) / (x1 - x2)
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d2_square = d1**2 - g1 * g2
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if d2_square >= 0:
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d2 = d2_square.sqrt()
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if x1 <= x2:
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min_pos = x2 - (x2 - x1) * ((g2 + d2 - d1) / (g2 - g1 + 2 * d2))
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else:
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min_pos = x1 - (x1 - x2) * ((g1 + d2 - d1) / (g1 - g2 + 2 * d2))
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return min(max(min_pos, xmin_bound), xmax_bound)
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else:
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return (xmin_bound + xmax_bound) / 2.
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def _strong_wolfe(obj_func,
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x,
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t,
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d,
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f,
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g,
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gtd,
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c1=1e-4,
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c2=0.9,
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tolerance_change=1e-9,
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max_ls=25):
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# ported from https://github.com/torch/optim/blob/master/lswolfe.lua
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d_norm = d.abs().max()
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g = g.clone(memory_format=torch.contiguous_format)
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# evaluate objective and gradient using initial step
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f_new, g_new = obj_func(x, t, d)
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ls_func_evals = 1
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gtd_new = g_new.dot(d)
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# bracket an interval containing a point satisfying the Wolfe criteria
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t_prev, f_prev, g_prev, gtd_prev = 0, f, g, gtd
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done = False
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ls_iter = 0
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while ls_iter < max_ls:
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# check conditions
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if f_new > (f + c1 * t * gtd) or (ls_iter > 1 and f_new >= f_prev):
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bracket = [t_prev, t]
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bracket_f = [f_prev, f_new]
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bracket_g = [g_prev, g_new.clone(memory_format=torch.contiguous_format)]
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bracket_gtd = [gtd_prev, gtd_new]
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break
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if abs(gtd_new) <= -c2 * gtd:
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bracket = [t]
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bracket_f = [f_new]
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bracket_g = [g_new]
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done = True
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break
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if gtd_new >= 0:
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bracket = [t_prev, t]
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bracket_f = [f_prev, f_new]
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bracket_g = [g_prev, g_new.clone(memory_format=torch.contiguous_format)]
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bracket_gtd = [gtd_prev, gtd_new]
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break
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# interpolate
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min_step = t + 0.01 * (t - t_prev)
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max_step = t * 10
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tmp = t
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t = _cubic_interpolate(
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t_prev,
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f_prev,
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gtd_prev,
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t,
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f_new,
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gtd_new,
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bounds=(min_step, max_step))
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# next step
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t_prev = tmp
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f_prev = f_new
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g_prev = g_new.clone(memory_format=torch.contiguous_format)
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gtd_prev = gtd_new
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f_new, g_new = obj_func(x, t, d)
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ls_func_evals += 1
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gtd_new = g_new.dot(d)
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ls_iter += 1
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# reached max number of iterations?
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if ls_iter == max_ls:
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bracket = [0, t]
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bracket_f = [f, f_new]
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bracket_g = [g, g_new]
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# zoom phase: we now have a point satisfying the criteria, or
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# a bracket around it. We refine the bracket until we find the
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# exact point satisfying the criteria
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insuf_progress = False
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# find high and low points in bracket
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low_pos, high_pos = (0, 1) if bracket_f[0] <= bracket_f[-1] else (1, 0)
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while not done and ls_iter < max_ls:
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# line-search bracket is so small
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if abs(bracket[1] - bracket[0]) * d_norm < tolerance_change:
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break
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# compute new trial value
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t = _cubic_interpolate(bracket[0], bracket_f[0], bracket_gtd[0],
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bracket[1], bracket_f[1], bracket_gtd[1])
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# test that we are making sufficient progress:
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# in case `t` is so close to boundary, we mark that we are making
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# insufficient progress, and if
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# + we have made insufficient progress in the last step, or
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# + `t` is at one of the boundary,
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# we will move `t` to a position which is `0.1 * len(bracket)`
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# away from the nearest boundary point.
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eps = 0.1 * (max(bracket) - min(bracket))
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if min(max(bracket) - t, t - min(bracket)) < eps:
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# interpolation close to boundary
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if insuf_progress or t >= max(bracket) or t <= min(bracket):
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# evaluate at 0.1 away from boundary
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if abs(t - max(bracket)) < abs(t - min(bracket)):
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t = max(bracket) - eps
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else:
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t = min(bracket) + eps
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insuf_progress = False
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else:
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insuf_progress = True
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else:
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insuf_progress = False
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# Evaluate new point
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f_new, g_new = obj_func(x, t, d)
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ls_func_evals += 1
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gtd_new = g_new.dot(d)
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ls_iter += 1
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if f_new > (f + c1 * t * gtd) or f_new >= bracket_f[low_pos]:
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# Armijo condition not satisfied or not lower than lowest point
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bracket[high_pos] = t
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bracket_f[high_pos] = f_new
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bracket_g[high_pos] = g_new.clone(memory_format=torch.contiguous_format)
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bracket_gtd[high_pos] = gtd_new
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low_pos, high_pos = (0, 1) if bracket_f[0] <= bracket_f[1] else (1, 0)
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else:
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if abs(gtd_new) <= -c2 * gtd:
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# Wolfe conditions satisfied
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done = True
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elif gtd_new * (bracket[high_pos] - bracket[low_pos]) >= 0:
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# old high becomes new low
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bracket[high_pos] = bracket[low_pos]
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bracket_f[high_pos] = bracket_f[low_pos]
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bracket_g[high_pos] = bracket_g[low_pos]
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bracket_gtd[high_pos] = bracket_gtd[low_pos]
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# new point becomes new low
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bracket[low_pos] = t
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bracket_f[low_pos] = f_new
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bracket_g[low_pos] = g_new.clone(memory_format=torch.contiguous_format)
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bracket_gtd[low_pos] = gtd_new
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# return stuff
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t = bracket[low_pos]
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f_new = bracket_f[low_pos]
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g_new = bracket_g[low_pos]
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return f_new, g_new, t, ls_func_evals
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class LBFGS(Optimizer):
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"""Implements L-BFGS algorithm.
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Heavily inspired by `minFunc
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<https://www.cs.ubc.ca/~schmidtm/Software/minFunc.html>`_.
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.. warning::
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This optimizer doesn't support per-parameter options and parameter
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groups (there can be only one).
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.. warning::
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Right now all parameters have to be on a single device. This will be
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improved in the future.
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.. note::
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This is a very memory intensive optimizer (it requires additional
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``param_bytes * (history_size + 1)`` bytes). If it doesn't fit in memory
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try reducing the history size, or use a different algorithm.
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Args:
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params (iterable): iterable of parameters to optimize. Parameters must be real.
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lr (float): learning rate (default: 1)
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max_iter (int): maximal number of iterations per optimization step
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(default: 20)
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max_eval (int): maximal number of function evaluations per optimization
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step (default: max_iter * 1.25).
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tolerance_grad (float): termination tolerance on first order optimality
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(default: 1e-7).
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tolerance_change (float): termination tolerance on function
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value/parameter changes (default: 1e-9).
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history_size (int): update history size (default: 100).
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line_search_fn (str): either 'strong_wolfe' or None (default: None).
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"""
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def __init__(self,
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params,
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lr=1,
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max_iter=20,
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max_eval=None,
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tolerance_grad=1e-7,
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tolerance_change=1e-9,
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history_size=100,
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line_search_fn=None):
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if max_eval is None:
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max_eval = max_iter * 5 // 4
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defaults = dict(
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lr=lr,
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max_iter=max_iter,
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max_eval=max_eval,
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tolerance_grad=tolerance_grad,
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tolerance_change=tolerance_change,
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history_size=history_size,
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line_search_fn=line_search_fn)
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super().__init__(params, defaults)
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if len(self.param_groups) != 1:
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raise ValueError("LBFGS doesn't support per-parameter options "
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"(parameter groups)")
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self._params = self.param_groups[0]['params']
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self._numel_cache = None
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def _numel(self):
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if self._numel_cache is None:
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self._numel_cache = sum(2 * p.numel() if torch.is_complex(p) else p.numel() for p in self._params)
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return self._numel_cache
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def _gather_flat_grad(self):
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views = []
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for p in self._params:
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if p.grad is None:
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view = p.new(p.numel()).zero_()
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elif p.grad.is_sparse:
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view = p.grad.to_dense().view(-1)
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else:
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view = p.grad.view(-1)
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if torch.is_complex(view):
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view = torch.view_as_real(view).view(-1)
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views.append(view)
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return torch.cat(views, 0)
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def _add_grad(self, step_size, update):
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offset = 0
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for p in self._params:
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if torch.is_complex(p):
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p = torch.view_as_real(p)
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numel = p.numel()
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# view as to avoid deprecated pointwise semantics
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p.add_(update[offset:offset + numel].view_as(p), alpha=step_size)
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offset += numel
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assert offset == self._numel()
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def _clone_param(self):
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return [p.clone(memory_format=torch.contiguous_format) for p in self._params]
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def _set_param(self, params_data):
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for p, pdata in zip(self._params, params_data):
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p.copy_(pdata)
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def _directional_evaluate(self, closure, x, t, d):
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self._add_grad(t, d)
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loss = float(closure())
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flat_grad = self._gather_flat_grad()
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self._set_param(x)
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return loss, flat_grad
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@torch.no_grad()
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def step(self, closure):
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"""Perform a single optimization step.
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Args:
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closure (Callable): A closure that reevaluates the model
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and returns the loss.
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"""
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assert len(self.param_groups) == 1
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# Make sure the closure is always called with grad enabled
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closure = torch.enable_grad()(closure)
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group = self.param_groups[0]
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lr = group['lr']
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max_iter = group['max_iter']
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max_eval = group['max_eval']
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tolerance_grad = group['tolerance_grad']
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tolerance_change = group['tolerance_change']
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line_search_fn = group['line_search_fn']
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history_size = group['history_size']
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# NOTE: LBFGS has only global state, but we register it as state for
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# the first param, because this helps with casting in load_state_dict
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state = self.state[self._params[0]]
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state.setdefault('func_evals', 0)
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state.setdefault('n_iter', 0)
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# evaluate initial f(x) and df/dx
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orig_loss = closure()
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loss = float(orig_loss)
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current_evals = 1
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state['func_evals'] += 1
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flat_grad = self._gather_flat_grad()
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opt_cond = flat_grad.abs().max() <= tolerance_grad
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# optimal condition
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if opt_cond:
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return orig_loss
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# tensors cached in state (for tracing)
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d = state.get('d')
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t = state.get('t')
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old_dirs = state.get('old_dirs')
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old_stps = state.get('old_stps')
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ro = state.get('ro')
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H_diag = state.get('H_diag')
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prev_flat_grad = state.get('prev_flat_grad')
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prev_loss = state.get('prev_loss')
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n_iter = 0
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# optimize for a max of max_iter iterations
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while n_iter < max_iter:
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# keep track of nb of iterations
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n_iter += 1
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state['n_iter'] += 1
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############################################################
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# compute gradient descent direction
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############################################################
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if state['n_iter'] == 1:
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d = flat_grad.neg()
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old_dirs = []
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old_stps = []
|
||
|
|
ro = []
|
||
|
|
H_diag = 1
|
||
|
|
else:
|
||
|
|
# do lbfgs update (update memory)
|
||
|
|
y = flat_grad.sub(prev_flat_grad)
|
||
|
|
s = d.mul(t)
|
||
|
|
ys = y.dot(s) # y*s
|
||
|
|
if ys > 1e-10:
|
||
|
|
# updating memory
|
||
|
|
if len(old_dirs) == history_size:
|
||
|
|
# shift history by one (limited-memory)
|
||
|
|
old_dirs.pop(0)
|
||
|
|
old_stps.pop(0)
|
||
|
|
ro.pop(0)
|
||
|
|
|
||
|
|
# store new direction/step
|
||
|
|
old_dirs.append(y)
|
||
|
|
old_stps.append(s)
|
||
|
|
ro.append(1. / ys)
|
||
|
|
|
||
|
|
# update scale of initial Hessian approximation
|
||
|
|
H_diag = ys / y.dot(y) # (y*y)
|
||
|
|
|
||
|
|
# compute the approximate (L-BFGS) inverse Hessian
|
||
|
|
# multiplied by the gradient
|
||
|
|
num_old = len(old_dirs)
|
||
|
|
|
||
|
|
if 'al' not in state:
|
||
|
|
state['al'] = [None] * history_size
|
||
|
|
al = state['al']
|
||
|
|
|
||
|
|
# iteration in L-BFGS loop collapsed to use just one buffer
|
||
|
|
q = flat_grad.neg()
|
||
|
|
for i in range(num_old - 1, -1, -1):
|
||
|
|
al[i] = old_stps[i].dot(q) * ro[i]
|
||
|
|
q.add_(old_dirs[i], alpha=-al[i])
|
||
|
|
|
||
|
|
# multiply by initial Hessian
|
||
|
|
# r/d is the final direction
|
||
|
|
d = r = torch.mul(q, H_diag)
|
||
|
|
for i in range(num_old):
|
||
|
|
be_i = old_dirs[i].dot(r) * ro[i]
|
||
|
|
r.add_(old_stps[i], alpha=al[i] - be_i)
|
||
|
|
|
||
|
|
if prev_flat_grad is None:
|
||
|
|
prev_flat_grad = flat_grad.clone(memory_format=torch.contiguous_format)
|
||
|
|
else:
|
||
|
|
prev_flat_grad.copy_(flat_grad)
|
||
|
|
prev_loss = loss
|
||
|
|
|
||
|
|
############################################################
|
||
|
|
# compute step length
|
||
|
|
############################################################
|
||
|
|
# reset initial guess for step size
|
||
|
|
if state['n_iter'] == 1:
|
||
|
|
t = min(1., 1. / flat_grad.abs().sum()) * lr
|
||
|
|
else:
|
||
|
|
t = lr
|
||
|
|
|
||
|
|
# directional derivative
|
||
|
|
gtd = flat_grad.dot(d) # g * d
|
||
|
|
|
||
|
|
# directional derivative is below tolerance
|
||
|
|
if gtd > -tolerance_change:
|
||
|
|
break
|
||
|
|
|
||
|
|
# optional line search: user function
|
||
|
|
ls_func_evals = 0
|
||
|
|
if line_search_fn is not None:
|
||
|
|
# perform line search, using user function
|
||
|
|
if line_search_fn != "strong_wolfe":
|
||
|
|
raise RuntimeError("only 'strong_wolfe' is supported")
|
||
|
|
else:
|
||
|
|
x_init = self._clone_param()
|
||
|
|
|
||
|
|
def obj_func(x, t, d):
|
||
|
|
return self._directional_evaluate(closure, x, t, d)
|
||
|
|
|
||
|
|
loss, flat_grad, t, ls_func_evals = _strong_wolfe(
|
||
|
|
obj_func, x_init, t, d, loss, flat_grad, gtd)
|
||
|
|
self._add_grad(t, d)
|
||
|
|
opt_cond = flat_grad.abs().max() <= tolerance_grad
|
||
|
|
else:
|
||
|
|
# no line search, simply move with fixed-step
|
||
|
|
self._add_grad(t, d)
|
||
|
|
if n_iter != max_iter:
|
||
|
|
# re-evaluate function only if not in last iteration
|
||
|
|
# the reason we do this: in a stochastic setting,
|
||
|
|
# no use to re-evaluate that function here
|
||
|
|
with torch.enable_grad():
|
||
|
|
loss = float(closure())
|
||
|
|
flat_grad = self._gather_flat_grad()
|
||
|
|
opt_cond = flat_grad.abs().max() <= tolerance_grad
|
||
|
|
ls_func_evals = 1
|
||
|
|
|
||
|
|
# update func eval
|
||
|
|
current_evals += ls_func_evals
|
||
|
|
state['func_evals'] += ls_func_evals
|
||
|
|
|
||
|
|
############################################################
|
||
|
|
# check conditions
|
||
|
|
############################################################
|
||
|
|
if n_iter == max_iter:
|
||
|
|
break
|
||
|
|
|
||
|
|
if current_evals >= max_eval:
|
||
|
|
break
|
||
|
|
|
||
|
|
# optimal condition
|
||
|
|
if opt_cond:
|
||
|
|
break
|
||
|
|
|
||
|
|
# lack of progress
|
||
|
|
if d.mul(t).abs().max() <= tolerance_change:
|
||
|
|
break
|
||
|
|
|
||
|
|
if abs(loss - prev_loss) < tolerance_change:
|
||
|
|
break
|
||
|
|
|
||
|
|
state['d'] = d
|
||
|
|
state['t'] = t
|
||
|
|
state['old_dirs'] = old_dirs
|
||
|
|
state['old_stps'] = old_stps
|
||
|
|
state['ro'] = ro
|
||
|
|
state['H_diag'] = H_diag
|
||
|
|
state['prev_flat_grad'] = prev_flat_grad
|
||
|
|
state['prev_loss'] = prev_loss
|
||
|
|
|
||
|
|
return orig_loss
|