add direct
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direct/README.md
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direct/README.md
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325
direct/birect.py
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direct/birect.py
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#!/usr/bin/env python3
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# based on hamming_exact3.py, but with all debug stuff removed
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def birect(
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obj,
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lo,
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hi,
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*,
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min_diag,
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min_error=None,
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max_evals=None,
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max_iters=None,
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by_longest=False,
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pruning=0,
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F=float,
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):
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assert len(lo) == len(hi), "dimensional mismatch"
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assert not (
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min_error is None and max_evals is None and max_iters is None
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), "at least one stopping condition must be specified"
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# variables prefixed with v_ are to be one-dimensional vectors. [a, b, c]
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# variables prefixed with vw_ are to be two-dimensional vectors. [[a, b], [c, d]]
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# variables prefixed with w_ are to be one-dimensional vectors of pairs.
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# aside: xmin should actually be called v_xmin, but it's not!
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def fun(w_t):
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# xs = [l + (h - l) * t for t in w_t]
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v_x = [F(num) / F(den) for num, den in w_t]
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v_x = [l * (1 - t) + h * t for l, h, t in zip(lo, hi, v_x)]
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res = obj(v_x)
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return res
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def ab_to_lu(w_a, w_b):
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w_l = [(a[0] + a[0] + b[0], (1 << a[1]) * 3) for a, b in zip(w_a, w_b)]
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w_u = [(a[0] + b[0] + b[0], (1 << b[1]) * 3) for a, b in zip(w_a, w_b)]
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return w_l, w_u
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dims = len(lo)
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w_a, w_b = [(0, 0)] * dims, [(1, 0)] * dims
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w_l, w_u = ab_to_lu(w_a, w_b)
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fl = fun(w_l)
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fu = fun(w_u)
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if fl <= fu:
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xmin, fmin = w_l, fl
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else:
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xmin, fmin = w_u, fu
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imin = 0 # index of the minimum -- only one point so far, so it's that one.
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# sample coordinates and their values:
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vw_a, vw_b = [w_a], [w_b]
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v_fl = [fl] # remember that "l" and "u" are arbitrary shorthand used by the paper,
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v_fu = [fu] # and one isn't necessarily above or below the other.
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v_active = {0: True} # indices of hyper-rectangles that have yet to be subdivided
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v_depth = [0] # increments every split
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n = 1 # how many indices are in use
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del w_a, w_b, w_l, w_u, fl, fu # prevent accidental re-use
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def precision_met():
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return min_error is not None and fmin <= min_error
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def no_more_evals():
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return max_evals is not None and n + 1 >= max_evals
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avg = lambda a, b: (a + b) / 2 # TODO: delete me!
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diff = lambda a, b: -abs(a - b) / 2 # TODO: delete me!
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cycle_funs = [min, max, avg, diff] # TODO: delete me!
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cycle_funs = [min, avg, max, diff, max, avg] # TODO: delete me!
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cycle_funs = [min] # TODO: delete me!
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# interesting. using cycle_funs = [max] seems optimal for objective2210.
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def gather_potential(v_i):
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# crappy algorithm for finding the convex hull of the plot where
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# x = diameter of hyper-rectangle
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# y = minimum loss of the two points (v_fl, v_fu) within it
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cycle_fun = cycle_funs[outer % len(cycle_funs)] # TODO: delete me!
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# TODO: make this faster. use a sorted queue and peek at the best for each depth.
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# start by finding the arg-minimum for each set of equal-diameter rects.
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bests = {} # mapping of depth to arg-minimum value (i.e. its index)
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for i in v_i:
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fl, fu = v_fl[i], v_fu[i]
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f = cycle_fun(fl, fu) # TODO: use min(fl, fu)!
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depth = v_depth[i]
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if cycle_fun == diff: # TODO: delete me!
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# f = f * (1 << depth) # TODO: delete me!
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f = f / diagonal_cache[depth] # TODO: delete me!
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if by_longest:
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depth = depth // dims * dims
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# assert depth < depth_limit
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best = bests.get(depth)
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# TODO: handle f == best case.
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if best is None or f < best[1]:
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bests[depth] = (i, f)
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if len(bests) == 1: # nothing to compare
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return [i for i, f in bests.values()]
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asc = sorted(bests.items(), key=lambda t: -t[0]) # sort by length, ascending
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# first, remove points that slope downwards.
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# this yields a pareto front, which isn't necessarily convex.
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old = asc
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new = [old[-1]]
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smallest = old[-1][1][1]
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for i in reversed(range(len(old) - 1)):
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f = old[i][1][1]
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if f <= smallest:
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smallest = f
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new.append(old[i])
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new = new[::-1]
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# second, convert depths to lengths.
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if by_longest: # TODO: does this branch make any difference?
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new = [(longest_cache[depth],) + t for depth, t in new]
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else:
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new = [(diagonal_cache[depth],) + t for depth, t in new]
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# third, remove points that fall under a previous slope.
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old = new
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skip = [False] * len(old)
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for i in range(len(old)):
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if skip[i]:
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continue
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len0, i0, f0 = old[i]
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smallest_slope = None
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for j in range(i + 1, len(old)):
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if skip[j]:
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continue
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len1, i1, f1 = old[j]
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num = f1 - f0
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den = len1 - len0
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# this factor of 3/2 comes from the denominator;
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# each length should be multiplied by 2/3:
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# the furthest relative distance from a corner to a center point.
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slope = num / den # * F(3 / 2)
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if smallest_slope is None:
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smallest_slope = slope
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elif slope < smallest_slope:
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for k in range(i + 1, j):
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skip[k] = True
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smallest_slope = slope
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new = [entry for entry, skipping in zip(old, skip) if not skipping]
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if pruning:
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v_f = sorted(min(fl, fu) for fl, fu in zip(v_fl, v_fu))
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fmedian = v_f[len(v_f) // 2]
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offset = fmin - pruning * (fmedian - fmin)
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start = 0
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K_slope = None
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for i in range(len(new)):
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len0, i0, f0 = new[i]
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new_slope = (f0 - offset) / len0
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if K_slope is None or new_slope < K_slope:
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# if new_slope >= 0:
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K_slope = new_slope
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start = i
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# if start: print(end=f"[starting at {i} with slope {K_slope:.3f}]")
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new = new[start:]
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return [i for len, i, f in new]
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def determine_longest(w_a, w_b):
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# the index of the dimension is used as a tie-breaker (considered longer).
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# e.g. a box has lengths (2, 3, 3). the index returned is then 1.
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# TODO: alternate way of stating that comment: biased towards smaller indices.
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longest = 0
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invlen = None
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for i, (a, b) in enumerate(zip(w_a, w_b)):
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den_a = 1 << a[1]
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den_b = 1 << b[1]
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den = max(den_a, den_b) # TODO: always the same, right?
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if invlen is None or den < invlen:
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invlen = den
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longest = i
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return longest
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def split_it(i, which, *, w_a, w_b, d):
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nonlocal n, xmin, fmin
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new, n = n, n + 1 # designate an index for the new hyper-rectangle
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w_new_a = w_a.copy()
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w_new_b = w_b.copy()
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den = w_a[d][1] # should be equal to w_b[d][1] as well
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num_a = w_a[d][0]
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num_b = w_b[d][0]
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if which:
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w_new_a[d] = (num_b + num_b, den + 1) # swap
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w_new_b[d] = (num_a + num_b, den + 1) # slide
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else:
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w_new_a[d] = (num_a + num_b, den + 1) # slide
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w_new_b[d] = (num_a + num_a, den + 1) # swap
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w_l, w_u = ab_to_lu(w_new_a, w_new_b)
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fl = fun(w_l) if which else v_fl[i]
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fu = v_fu[i] if which else fun(w_u)
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vw_a.append(w_new_a)
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vw_b.append(w_new_b)
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v_fl.append(fl)
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v_fu.append(fu)
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v_depth.append(v_depth[i] + 1)
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if which:
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if fl < fmin:
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xmin, fmin = w_l, fl
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else:
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if fu < fmin:
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xmin, fmin = w_u, fu
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return new
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def split_rectangles(v_i): # returns new indices
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v_new = []
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for i in v_i:
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w_a = vw_a[i]
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w_b = vw_b[i]
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d = determine_longest(w_a, w_b)
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v_new.append(split_it(i, 0, w_a=w_a, w_b=w_b, d=d))
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if precision_met() or no_more_evals():
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break
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v_new.append(split_it(i, 1, w_a=w_a, w_b=w_b, d=d))
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if precision_met() or no_more_evals():
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break
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assert len(vw_a) == n, "internal error: vw_a has invalid length"
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assert len(vw_b) == n, "internal error: vw_b has invalid length"
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assert len(v_fl) == n, "internal error: v_fl has invalid length"
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assert len(v_fu) == n, "internal error: v_fu has invalid length"
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assert len(v_depth) == n, "internal error: v_depth has invalid length"
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return v_new
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def precompute_diagonals_by_limit(limit):
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diags = []
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w_a = vw_a[0].copy()
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w_b = vw_b[0].copy()
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for depth in range(limit):
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sq_dist = 0
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for a, b in zip(w_a, w_b):
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delta = b[0] - a[0]
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sq_dist += (delta * delta) << (2 * (limit - a[1]))
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diags.append(sq_dist)
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d = depth % dims
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a_d = w_a[d]
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b_d = w_b[d]
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large = max(a_d[0], b_d[0])
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small = min(a_d[0], b_d[0])
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w_a[d] = (large * 2 - 0, a_d[1] + 1)
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w_b[d] = (small * 2 + 1, b_d[1] + 1)
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return [F(diag) ** F(0.5) / F(1 << limit) for diag in diags]
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def precompute_diagonals_by_length(limit):
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diags, longests = [], []
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w_a = vw_a[0].copy()
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w_b = vw_b[0].copy()
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for depth in range(1_000_000):
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bits = max(max(a[1], b[1]) for a, b in zip(w_a, w_b))
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sq_dist, longest = 0, 0
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for a, b in zip(w_a, w_b):
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delta = b[0] - a[0]
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sq_dist += (delta * delta) << (2 * (bits - a[1]))
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longest = max(longest, abs(delta) << (bits - a[1]))
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diag = F(sq_dist) ** F(0.5) / F(1 << bits)
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if diag < limit:
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break
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longest = F(longest) / F(1 << bits)
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diags.append(diag)
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longests.append(longest)
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d = depth % dims
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a_d = w_a[d]
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b_d = w_b[d]
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large = max(a_d[0], b_d[0])
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small = min(a_d[0], b_d[0])
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w_a[d] = (large * 2 - 0, a_d[1] + 1)
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w_b[d] = (small * 2 + 1, b_d[1] + 1)
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return diags, longests
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diagonal_cache, longest_cache = precompute_diagonals_by_length(min_diag)
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depth_limit = len(diagonal_cache)
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diagonal_cache = precompute_diagonals_by_limit(depth_limit)
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for outer in range(1_000_000 if max_iters is None else max_iters):
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if precision_met() or no_more_evals():
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break
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v_potential = gather_potential(v_active)
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v_new = split_rectangles(v_potential)
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for j in v_potential:
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del v_active[j]
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for j in v_new:
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if v_depth[j] < depth_limit:
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v_active[j] = True
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tmin = [F(x[0]) / F(x[1]) for x in xmin]
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argmin = [l * (1 - t) + h * t for l, h, t in zip(lo, hi, tmin)]
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return argmin, fmin
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if __name__ == "__main__":
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from arsd_objectives import objective2210
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import numpy as np
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F = np.float64
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res = birect(
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lambda a: objective2210(np.array(a, F)),
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[0, 0],
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[5, 5],
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min_diag=F(1e-8 / 5),
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# max_evals=50_000,
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max_evals=2_000,
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F=F,
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by_longest=True,
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)
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print("", "birect result:", *res, sep="\n")
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print("", "double checked:", objective2210(np.array(res[0], F)), sep="\n")
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