RuntimeError: einsum(): the number of subscripts in the equation (3) does not match the number of dimensions (4) for operand 0 and no ellipsis was given

[QiyaoWei]

Dear authors,

I am writing to raise an issue about the broyden method. My environment is python=3.8 + torch=1.9.0 (which I fear might be too high). I attach an example code and the error below. Any help would be appreciated!

from scipy.optimize import root
import time
from termcolor import colored
import torch
import torch.nn as nn
import torch.nn.functional as F
from torch.nn import Parameter


device = torch.device("cuda:0" if torch.cuda.is_available() else "cpu")

# toy network
class ResNetLayer(nn.Module):
    def __init__(self):
        super(ResNetLayer, self).__init__()
        self.conv1 = nn.Conv2d(3, 9, 3, padding=3//2)
        self.conv2 = nn.Conv2d(9, 3, 5, padding=5//2)
        
    def forward(self, z, x):
        y = F.relu(self.conv1(z))
        return F.relu(z + F.relu(x + self.conv2(y)))



def _safe_norm(v):
    if not torch.isfinite(v).all():
        return np.inf
    return torch.norm(v)


def scalar_search_armijo(phi, phi0, derphi0, c1=1e-4, alpha0=1, amin=0):
    ite = 0
    phi_a0 = phi(alpha0)    # First do an update with step size 1
    if phi_a0 <= phi0 + c1*alpha0*derphi0:
        return alpha0, phi_a0, ite

    # Otherwise, compute the minimizer of a quadratic interpolant
    alpha1 = -(derphi0) * alpha0**2 / 2.0 / (phi_a0 - phi0 - derphi0 * alpha0)
    phi_a1 = phi(alpha1)

    # Otherwise loop with cubic interpolation until we find an alpha which
    # satisfies the first Wolfe condition (since we are backtracking, we will
    # assume that the value of alpha is not too small and satisfies the second
    # condition.
    while alpha1 > amin:       # we are assuming alpha>0 is a descent direction
        factor = alpha0**2 * alpha1**2 * (alpha1-alpha0)
        a = alpha0**2 * (phi_a1 - phi0 - derphi0*alpha1) - \
            alpha1**2 * (phi_a0 - phi0 - derphi0*alpha0)
        a = a / factor
        b = -alpha0**3 * (phi_a1 - phi0 - derphi0*alpha1) + \
            alpha1**3 * (phi_a0 - phi0 - derphi0*alpha0)
        b = b / factor

        alpha2 = (-b + torch.sqrt(torch.abs(b**2 - 3 * a * derphi0))) / (3.0*a)
        phi_a2 = phi(alpha2)
        ite += 1

        if (phi_a2 <= phi0 + c1*alpha2*derphi0):
            return alpha2, phi_a2, ite

        if (alpha1 - alpha2) > alpha1 / 2.0 or (1 - alpha2/alpha1) < 0.96:
            alpha2 = alpha1 / 2.0

        alpha0 = alpha1
        alpha1 = alpha2
        phi_a0 = phi_a1
        phi_a1 = phi_a2

    # Failed to find a suitable step length
    return None, phi_a1, ite


def line_search(update, x0, g0, g, nstep=0, on=True):
    """
    `update` is the propsoed direction of update.
    Code adapted from scipy.
    """
    tmp_s = [0]
    tmp_g0 = [g0]
    tmp_phi = [torch.norm(g0)**2]
    s_norm = torch.norm(x0) / torch.norm(update)

    def phi(s, store=True):
        if s == tmp_s[0]:
            return tmp_phi[0]    # If the step size is so small... just return something
        x_est = x0 + s * update
        g0_new = g(x_est)
        phi_new = _safe_norm(g0_new)**2
        if store:
            tmp_s[0] = s
            tmp_g0[0] = g0_new
            tmp_phi[0] = phi_new
        return phi_new
    
    if on:
        s, phi1, ite = scalar_search_armijo(phi, tmp_phi[0], -tmp_phi[0], amin=1e-2)
    if (not on) or s is None:
        s = 1.0
        ite = 0

    x_est = x0 + s * update
    if s == tmp_s[0]:
        g0_new = tmp_g0[0]
    else:
        g0_new = g(x_est)
    return x_est, g0_new, x_est - x0, g0_new - g0, ite

def rmatvec(part_Us, part_VTs, x):
    # Compute x^T(-I + UV^T)
    # x: (N, 2d, L')
    # part_Us: (N, 2d, L', threshold)
    # part_VTs: (N, threshold, 2d, L')
    if part_Us.nelement() == 0:
        return -x
    xTU = torch.einsum('bij, bijd -> bd', x, part_Us)   # (N, threshold)
    return -x + torch.einsum('bd, bdij -> bij', xTU, part_VTs)    # (N, 2d, L'), but should really be (N, 1, (2d*L'))


def matvec(part_Us, part_VTs, x):
    # Compute (-I + UV^T)x
    # x: (N, 2d, L')
    # part_Us: (N, 2d, L', threshold)
    # part_VTs: (N, threshold, 2d, L')
    if part_Us.nelement() == 0:
        return -x
    VTx = torch.einsum('bdij, bij -> bd', part_VTs, x)  # (N, threshold)
    return -x + torch.einsum('bijd, bd -> bij', part_Us, VTx)     # (N, 2d, L'), but should really be (N, (2d*L'), 1)


def broyden(f, x0, threshold, eps=1e-3, stop_mode="rel", ls=False, name="unknown"):
    bsz, total_hsize, H, W = x0.size()
    seq_len = H * W
    g = lambda y: f(y) - y
    dev = x0.device
    alternative_mode = 'rel' if stop_mode == 'abs' else 'abs'
    
    x_est = x0           # (bsz, 2d, L')
    gx = g(x_est)        # (bsz, 2d, L')
    nstep = 0
    tnstep = 0
    
    # For fast calculation of inv_jacobian (approximately)
    Us = torch.zeros(bsz, total_hsize, seq_len, threshold).to(dev)     # One can also use an L-BFGS scheme to further reduce memory
    VTs = torch.zeros(bsz, threshold, total_hsize, seq_len).to(dev)
    update = -matvec(Us[:,:,:,:nstep], VTs[:,:nstep], gx)      # Formally should be -torch.matmul(inv_jacobian (-I), gx)
    prot_break = False
    
    # To be used in protective breaks
    protect_thres = (1e6 if stop_mode == "abs" else 1e3) * seq_len
    new_objective = 1e8

    trace_dict = {'abs': [],
                  'rel': []}
    lowest_dict = {'abs': 1e8,
                   'rel': 1e8}
    lowest_step_dict = {'abs': 0,
                        'rel': 0}
    nstep, lowest_xest, lowest_gx = 0, x_est, gx

    while nstep < threshold:
        x_est, gx, delta_x, delta_gx, ite = line_search(update, x_est, gx, g, nstep=nstep, on=ls)
        nstep += 1
        tnstep += (ite+1)
        abs_diff = torch.norm(gx).item()
        rel_diff = abs_diff / (torch.norm(gx + x_est).item() + 1e-9)
        diff_dict = {'abs': abs_diff,
                     'rel': rel_diff}
        trace_dict['abs'].append(abs_diff)
        trace_dict['rel'].append(rel_diff)
        for mode in ['rel', 'abs']:
            if diff_dict[mode] < lowest_dict[mode]:
                if mode == stop_mode: 
                    lowest_xest, lowest_gx = x_est.clone().detach(), gx.clone().detach()
                lowest_dict[mode] = diff_dict[mode]
                lowest_step_dict[mode] = nstep

        new_objective = diff_dict[stop_mode]
        if new_objective < eps: break
        if new_objective < 3*eps and nstep > 30 and np.max(trace_dict[stop_mode][-30:]) / np.min(trace_dict[stop_mode][-30:]) < 1.3:
            # if there's hardly been any progress in the last 30 steps
            break
        if new_objective > trace_dict[stop_mode][0] * protect_thres:
            prot_break = True
            break

        part_Us, part_VTs = Us[:,:,:,:nstep-1], VTs[:,:nstep-1]
        vT = rmatvec(part_Us, part_VTs, delta_x)
        u = (delta_x - matvec(part_Us, part_VTs, delta_gx)) / torch.einsum('bij, bij -> b', vT, delta_gx)[:,None,None]
        vT[vT != vT] = 0
        u[u != u] = 0
        VTs[:,nstep-1] = vT
        Us[:,:,:,nstep-1] = u
        update = -matvec(Us[:,:,:,:nstep], VTs[:,:nstep], gx)

    # Fill everything up to the threshold length
    for _ in range(threshold+1-len(trace_dict[stop_mode])):
        trace_dict[stop_mode].append(lowest_dict[stop_mode])
        trace_dict[alternative_mode].append(lowest_dict[alternative_mode])

    return {"result": lowest_xest,
            "lowest": lowest_dict[stop_mode],
            "nstep": lowest_step_dict[stop_mode],
            "prot_break": prot_break,
            "abs_trace": trace_dict['abs'],
            "rel_trace": trace_dict['rel'],
            "eps": eps,
            "threshold": threshold}

test = torch.rand(128, 3, 32, 32).to(device)
f = ResNetLayer().to(device)
zz = broyden(lambda Z : f(Z,test), torch.zeros_like(test), threshold=100, eps=1e-1)['result']

Traceback (most recent call last):
  File "mwe.py", line 251, in <module>
    zz = broyden(lambda Z : f(Z,test), torch.zeros_like(test), threshold=100, eps=1e-1)['result']
  File "mwe.py", line 188, in broyden
    u = (delta_x - matvec(part_Us, part_VTs, delta_gx)) / torch.einsum('bij, bij -> b', vT, delta_gx)[:,None,None]
  File "/opt/conda/lib/python3.8/site-packages/torch/functional.py", line 299, in einsum
    return _VF.einsum(equation, operands)  # type: ignore[attr-defined]
RuntimeError: einsum(): the number of subscripts in the equation (3) does not match the number of dimensions (4) for operand 0 and no ellipsis was given

[jerrybai1995]

Hi @QiyaoWei ,

Thanks for your interest in our work. It seems like you are using a 4D tensor rather than a 3D tensor. You should do the following instead:

bsz, C = 128, 3
H, W = 32, 32
test = torch.rand(128, 3, 32, 32).to(device)
layer = ResNetLayer().to(device)
f = lambda x: layer(x.view(bsz, C, H, W)).view(bsz, C, -1)
zz = broyden(f, torch.zeros_like(test).view(bsz, C, -1), threshold=100, eps=1e-1)['result']

As a side note, I suggest adding some normalization layers to your ResNetLayer design. Without normalizations it is likely that this layer will not converge with random initialization.