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MTL

minnormsolver

This script includes code adapted from the 'impartial-vaes' repository with minor modifications. The original code can be found at: https://github.com/adrianjav/impartial-vaes

Credit to the original authors: Adrian Javaloy, Maryam Meghdadi, and Isabel Valera for their valuable work.

MinNormLinearSolver

Bases: Module

Solves the min norm problem in case of 2 vectors (lies on a line).

Source code in vambn/modelling/mtl/minnormsolver.py 
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class MinNormLinearSolver(nn.Module):
    """Solves the min norm problem in case of 2 vectors (lies on a line)."""

    def __init__(self):
        super().__init__()

    @torch.no_grad()
    def forward(self, v1v1, v1v2, v2v2):
        """
        Solver execution on scalar products of 2 vectors.

        Args:
            v1v1 (float): Scalar product <V1, V1>.
            v1v2 (float): Scalar product <V1, V2>.
            v2v2 (float): Scalar product <V2, V2>.

        Returns:
            tuple: A tuple containing:
                - gamma (float): Min-norm solution c = (gamma, 1. - gamma).
                - cost (float): The norm of min-norm point.
        """
        if v1v2 >= v1v1:
            return 1.0, v1v1
        if v1v2 >= v2v2:
            return 0.0, v2v2
        gamma = (v2v2 - v1v2) / (v1v1 + v2v2 - 2 * v1v2 + 1e-8)
        cost = v2v2 + gamma * (v1v2 - v2v2)
        return gamma, cost

forward(v1v1, v1v2, v2v2)

Solver execution on scalar products of 2 vectors.

Parameters:

Name Type Description Default
v1v1 float

Scalar product .

 required
v1v2 float

Scalar product .

 required
v2v2 float

Scalar product .

 required

Returns:

Name Type Description
tuple

A tuple containing: - gamma (float): Min-norm solution c = (gamma, 1. - gamma). - cost (float): The norm of min-norm point.
Source code in vambn/modelling/mtl/minnormsolver.py 
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@torch.no_grad()
def forward(self, v1v1, v1v2, v2v2):
    """
    Solver execution on scalar products of 2 vectors.

    Args:
        v1v1 (float): Scalar product <V1, V1>.
        v1v2 (float): Scalar product <V1, V2>.
        v2v2 (float): Scalar product <V2, V2>.

    Returns:
        tuple: A tuple containing:
            - gamma (float): Min-norm solution c = (gamma, 1. - gamma).
            - cost (float): The norm of min-norm point.
    """
    if v1v2 >= v1v1:
        return 1.0, v1v1
    if v1v2 >= v2v2:
        return 0.0, v2v2
    gamma = (v2v2 - v1v2) / (v1v1 + v2v2 - 2 * v1v2 + 1e-8)
    cost = v2v2 + gamma * (v1v2 - v2v2)
    return gamma, cost

MinNormPlanarSolver

Bases: Module

Solves the min norm problem in case the vectors lie on the same plane.

Source code in vambn/modelling/mtl/minnormsolver.py 
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class MinNormPlanarSolver(nn.Module):
    """Solves the min norm problem in case the vectors lie on the same plane."""

    def __init__(self, n_tasks):
        """
        Initializes the MinNormPlanarSolver.

        Args:
            n_tasks (int): Number of tasks/vectors.
        """
        super().__init__()
        i_grid = torch.arange(n_tasks)
        j_grid = torch.arange(n_tasks)
        ii_grid, jj_grid = torch.meshgrid(i_grid, j_grid)
        i_triu, j_triu = np.triu_indices(n_tasks, 1)

        self.register_buffer("n", torch.tensor(n_tasks))
        self.register_buffer("i_triu", torch.from_numpy(i_triu))
        self.register_buffer("j_triu", torch.from_numpy(j_triu))
        self.register_buffer("ii_triu", ii_grid[i_triu, j_triu])
        self.register_buffer("jj_triu", jj_grid[i_triu, j_triu])
        self.register_buffer("one", torch.ones(self.ii_triu.shape))
        self.register_buffer("zero", torch.zeros(self.ii_triu.shape))

    @torch.no_grad()
    def line_solver_vectorized(self, v1v1, v1v2, v2v2):
        """
        Linear case solver, but for collection of vector pairs (Vi, Vj).

        Args:
            v1v1 (Tensor): Vector of scalar products <Vi, Vi>.
            v1v2 (Tensor): Vector of scalar products <Vi, Vj>.
            v2v2 (Tensor): Vector of scalar products <Vj, Vj>.

        Returns:
            tuple: A tuple containing:
                - gamma (Tensor): Vector of min-norm solution c = (gamma, 1. - gamma).
                - cost (Tensor): Vector of the norm of min-norm point.
        """
        gamma = (v2v2 - v1v2) / (v1v1 + v2v2 - 2 * v1v2 + 1e-8)
        gamma = gamma.where(v1v2 < v2v2, self.zero)
        gamma = gamma.where(v1v2 < v1v1, self.one)

        cost = v2v2 + gamma * (v1v2 - v2v2)
        cost = cost.where(v1v2 < v2v2, v2v2)
        cost = cost.where(v1v2 < v1v1, v1v1)
        return gamma, cost

    @torch.no_grad()
    def forward(self, grammian):
        """
        Planar case solver, when Vi lies on the same plane.

        Args:
            grammian (Tensor): Grammian matrix G[i, j] = [<Vi, Vj>], G is a nxn tensor.

        Returns:
            Tensor: Coefficients c = [c1, ... cn] that solves the min-norm problem.
        """
        vivj = grammian[self.ii_triu, self.jj_triu]
        vivi = grammian[self.ii_triu, self.ii_triu]
        vjvj = grammian[self.jj_triu, self.jj_triu]

        gamma, cost = self.line_solver_vectorized(vivi, vivj, vjvj)
        offset = torch.argmin(cost)
        i_min, j_min = self.i_triu[offset], self.j_triu[offset]
        sol = torch.zeros(self.n, device=grammian.device)
        sol[i_min], sol[j_min] = gamma[offset], 1.0 - gamma[offset]
        return sol

__init__(n_tasks)

Initializes the MinNormPlanarSolver.

Parameters:

Name Type Description Default
n_tasks int

Number of tasks/vectors.

 required
Source code in vambn/modelling/mtl/minnormsolver.py 
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def __init__(self, n_tasks):
    """
    Initializes the MinNormPlanarSolver.

    Args:
        n_tasks (int): Number of tasks/vectors.
    """
    super().__init__()
    i_grid = torch.arange(n_tasks)
    j_grid = torch.arange(n_tasks)
    ii_grid, jj_grid = torch.meshgrid(i_grid, j_grid)
    i_triu, j_triu = np.triu_indices(n_tasks, 1)

    self.register_buffer("n", torch.tensor(n_tasks))
    self.register_buffer("i_triu", torch.from_numpy(i_triu))
    self.register_buffer("j_triu", torch.from_numpy(j_triu))
    self.register_buffer("ii_triu", ii_grid[i_triu, j_triu])
    self.register_buffer("jj_triu", jj_grid[i_triu, j_triu])
    self.register_buffer("one", torch.ones(self.ii_triu.shape))
    self.register_buffer("zero", torch.zeros(self.ii_triu.shape))

forward(grammian)

Planar case solver, when Vi lies on the same plane.

Parameters:

Name Type Description Default
grammian Tensor

Grammian matrix G[i, j] = [], G is a nxn tensor.

 required

Returns:

Name Type Description
Tensor

Coefficients c = [c1, ... cn] that solves the min-norm problem.
Source code in vambn/modelling/mtl/minnormsolver.py 
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@torch.no_grad()
def forward(self, grammian):
    """
    Planar case solver, when Vi lies on the same plane.

    Args:
        grammian (Tensor): Grammian matrix G[i, j] = [<Vi, Vj>], G is a nxn tensor.

    Returns:
        Tensor: Coefficients c = [c1, ... cn] that solves the min-norm problem.
    """
    vivj = grammian[self.ii_triu, self.jj_triu]
    vivi = grammian[self.ii_triu, self.ii_triu]
    vjvj = grammian[self.jj_triu, self.jj_triu]

    gamma, cost = self.line_solver_vectorized(vivi, vivj, vjvj)
    offset = torch.argmin(cost)
    i_min, j_min = self.i_triu[offset], self.j_triu[offset]
    sol = torch.zeros(self.n, device=grammian.device)
    sol[i_min], sol[j_min] = gamma[offset], 1.0 - gamma[offset]
    return sol

line_solver_vectorized(v1v1, v1v2, v2v2)

Linear case solver, but for collection of vector pairs (Vi, Vj).

Parameters:

Name Type Description Default
v1v1 Tensor

Vector of scalar products .

 required
v1v2 Tensor

Vector of scalar products .

 required
v2v2 Tensor

Vector of scalar products .

 required

Returns:

Name Type Description
tuple

A tuple containing: - gamma (Tensor): Vector of min-norm solution c = (gamma, 1. - gamma). - cost (Tensor): Vector of the norm of min-norm point.
Source code in vambn/modelling/mtl/minnormsolver.py 
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@torch.no_grad()
def line_solver_vectorized(self, v1v1, v1v2, v2v2):
    """
    Linear case solver, but for collection of vector pairs (Vi, Vj).

    Args:
        v1v1 (Tensor): Vector of scalar products <Vi, Vi>.
        v1v2 (Tensor): Vector of scalar products <Vi, Vj>.
        v2v2 (Tensor): Vector of scalar products <Vj, Vj>.

    Returns:
        tuple: A tuple containing:
            - gamma (Tensor): Vector of min-norm solution c = (gamma, 1. - gamma).
            - cost (Tensor): Vector of the norm of min-norm point.
    """
    gamma = (v2v2 - v1v2) / (v1v1 + v2v2 - 2 * v1v2 + 1e-8)
    gamma = gamma.where(v1v2 < v2v2, self.zero)
    gamma = gamma.where(v1v2 < v1v1, self.one)

    cost = v2v2 + gamma * (v1v2 - v2v2)
    cost = cost.where(v1v2 < v2v2, v2v2)
    cost = cost.where(v1v2 < v1v1, v1v1)
    return gamma, cost

MinNormSolver

Bases: Module

Solves the min norm problem in the general case.

Source code in vambn/modelling/mtl/minnormsolver.py 
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class MinNormSolver(nn.Module):
    """Solves the min norm problem in the general case."""

    def __init__(self, n_tasks, max_iter=250, stop_crit=1e-6):
        """
        Initializes the MinNormSolver.

        Args:
            n_tasks (int): Number of tasks/vectors.
            max_iter (int, optional): Maximum number of iterations. Defaults to 250.
            stop_crit (float, optional): Stopping criterion. Defaults to 1e-6.
        """
        super().__init__()
        self.n = n_tasks
        self.linear_solver = MinNormLinearSolver()
        self.planar_solver = MinNormPlanarSolver(n_tasks)

        n_grid = torch.arange(n_tasks)
        i_grid = torch.arange(n_tasks, dtype=torch.float32) + 1
        ii_grid, jj_grid = torch.meshgrid(n_grid, n_grid)

        self.register_buffer("n_ts", torch.tensor(n_tasks))
        self.register_buffer("i_grid", i_grid)
        self.register_buffer("ii_grid", ii_grid)
        self.register_buffer("jj_grid", jj_grid)
        self.register_buffer("zero", torch.zeros(n_tasks))
        self.register_buffer("stop_crit", torch.tensor(stop_crit))

        self.max_iter = max_iter
        self.two_sol = nn.Parameter(torch.zeros(2))
        self.two_sol.require_grad = False

    @torch.no_grad()
    def projection_to_simplex(self, gamma):
        """
        Projects gamma to the simplex.

        Args:
            gamma (Tensor): The input tensor to project.

        Returns:
            Tensor: The projected tensor.
        """
        sorted_gamma, indices = torch.sort(gamma, descending=True)
        tmp_sum = torch.cumsum(sorted_gamma, 0)
        tmp_max = (tmp_sum - 1.0) / self.i_grid

        non_zeros = torch.nonzero(tmp_max[:-1] > sorted_gamma[1:])
        if non_zeros.shape[0] > 0:
            tmax_f = tmp_max[:-1][non_zeros[0][0]]
        else:
            tmax_f = tmp_max[-1]
        return torch.max(gamma - tmax_f, self.zero)

    @torch.no_grad()
    def next_point(self, cur_val, grad):
        """
        Computes the next point in the optimization.

        Args:
            cur_val (Tensor): Current value.
            grad (Tensor): Gradient.

        Returns:
            Tensor: The next point.
        """
        proj_grad = grad - (torch.sum(grad) / self.n_ts)
        lt_zero = torch.nonzero(proj_grad < 0)
        lt_zero = lt_zero.view(lt_zero.numel())
        gt_zero = torch.nonzero(proj_grad > 0)
        gt_zero = gt_zero.view(gt_zero.numel())
        tm1 = -cur_val[lt_zero] / proj_grad[lt_zero]
        tm2 = (1.0 - cur_val[gt_zero]) / proj_grad[gt_zero]

        t = torch.tensor(1.0, device=grad.device)
        tm1_gt_zero = torch.nonzero(tm1 > 1e-7)
        tm1_gt_zero = tm1_gt_zero.view(tm1_gt_zero.numel())
        if tm1_gt_zero.shape[0] > 0:
            t = torch.min(tm1[tm1_gt_zero])

        tm2_gt_zero = torch.nonzero(tm2 > 1e-7)
        tm2_gt_zero = tm2_gt_zero.view(tm2_gt_zero.numel())
        if tm2_gt_zero.shape[0] > 0:
            t = torch.min(t, torch.min(tm2[tm2_gt_zero]))

        next_point = proj_grad * t + cur_val
        next_point = self.projection_to_simplex(next_point)
        return next_point

    @torch.no_grad()
    def forward(self, vecs):
        """
        General case solver using simplex projection algorithm.

        Args:
            vecs (Tensor): 2D tensor V, where each row is a vector Vi.

        Returns:
            Tensor: Coefficients c = [c1, ... cn] that solves the min-norm problem.
        """
        if self.n == 1:
            return vecs[0]
        if self.n == 2:
            v1v1 = torch.dot(vecs[0], vecs[0])
            v1v2 = torch.dot(vecs[0], vecs[1])
            v2v2 = torch.dot(vecs[1], vecs[1])
            self.two_sol[0], cost = self.linear_solver(v1v1, v1v2, v2v2)
            self.two_sol[1] = 1.0 - self.two_sol[0]
            return self.two_sol.clone()

        grammian = torch.mm(vecs, vecs.t())
        sol_vec = self.planar_solver(grammian)

        ii, jj = self.ii_grid, self.jj_grid
        for iter_count in range(self.max_iter):
            grad_dir = -torch.mv(grammian, sol_vec)
            new_point = self.next_point(sol_vec, grad_dir)

            v1v1 = (sol_vec[ii] * sol_vec[jj] * grammian[ii, jj]).sum()
            v1v2 = (sol_vec[ii] * new_point[jj] * grammian[ii, jj]).sum()
            v2v2 = (new_point[ii] * new_point[jj] * grammian[ii, jj]).sum()

            gamma, cost = self.linear_solver(v1v1, v1v2, v2v2)
            new_sol_vec = gamma * sol_vec + (1 - gamma) * new_point
            change = new_sol_vec - sol_vec
            if torch.sum(torch.abs(change)) < self.stop_crit:
                return sol_vec
            sol_vec = new_sol_vec
        return sol_vec

__init__(n_tasks, max_iter=250, stop_crit=1e-06)

Initializes the MinNormSolver.

Parameters:

Name Type Description Default
n_tasks int

Number of tasks/vectors.

 required
max_iter int

Maximum number of iterations. Defaults to 250.

 250
stop_crit float

Stopping criterion. Defaults to 1e-6.

 1e-06
Source code in vambn/modelling/mtl/minnormsolver.py 
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def __init__(self, n_tasks, max_iter=250, stop_crit=1e-6):
    """
    Initializes the MinNormSolver.

    Args:
        n_tasks (int): Number of tasks/vectors.
        max_iter (int, optional): Maximum number of iterations. Defaults to 250.
        stop_crit (float, optional): Stopping criterion. Defaults to 1e-6.
    """
    super().__init__()
    self.n = n_tasks
    self.linear_solver = MinNormLinearSolver()
    self.planar_solver = MinNormPlanarSolver(n_tasks)

    n_grid = torch.arange(n_tasks)
    i_grid = torch.arange(n_tasks, dtype=torch.float32) + 1
    ii_grid, jj_grid = torch.meshgrid(n_grid, n_grid)

    self.register_buffer("n_ts", torch.tensor(n_tasks))
    self.register_buffer("i_grid", i_grid)
    self.register_buffer("ii_grid", ii_grid)
    self.register_buffer("jj_grid", jj_grid)
    self.register_buffer("zero", torch.zeros(n_tasks))
    self.register_buffer("stop_crit", torch.tensor(stop_crit))

    self.max_iter = max_iter
    self.two_sol = nn.Parameter(torch.zeros(2))
    self.two_sol.require_grad = False

forward(vecs)

General case solver using simplex projection algorithm.

Parameters:

Name Type Description Default
vecs Tensor

2D tensor V, where each row is a vector Vi.

 required

Returns:

Name Type Description
Tensor

Coefficients c = [c1, ... cn] that solves the min-norm problem.
Source code in vambn/modelling/mtl/minnormsolver.py 
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@torch.no_grad()
def forward(self, vecs):
    """
    General case solver using simplex projection algorithm.

    Args:
        vecs (Tensor): 2D tensor V, where each row is a vector Vi.

    Returns:
        Tensor: Coefficients c = [c1, ... cn] that solves the min-norm problem.
    """
    if self.n == 1:
        return vecs[0]
    if self.n == 2:
        v1v1 = torch.dot(vecs[0], vecs[0])
        v1v2 = torch.dot(vecs[0], vecs[1])
        v2v2 = torch.dot(vecs[1], vecs[1])
        self.two_sol[0], cost = self.linear_solver(v1v1, v1v2, v2v2)
        self.two_sol[1] = 1.0 - self.two_sol[0]
        return self.two_sol.clone()

    grammian = torch.mm(vecs, vecs.t())
    sol_vec = self.planar_solver(grammian)

    ii, jj = self.ii_grid, self.jj_grid
    for iter_count in range(self.max_iter):
        grad_dir = -torch.mv(grammian, sol_vec)
        new_point = self.next_point(sol_vec, grad_dir)

        v1v1 = (sol_vec[ii] * sol_vec[jj] * grammian[ii, jj]).sum()
        v1v2 = (sol_vec[ii] * new_point[jj] * grammian[ii, jj]).sum()
        v2v2 = (new_point[ii] * new_point[jj] * grammian[ii, jj]).sum()

        gamma, cost = self.linear_solver(v1v1, v1v2, v2v2)
        new_sol_vec = gamma * sol_vec + (1 - gamma) * new_point
        change = new_sol_vec - sol_vec
        if torch.sum(torch.abs(change)) < self.stop_crit:
            return sol_vec
        sol_vec = new_sol_vec
    return sol_vec

next_point(cur_val, grad)

Computes the next point in the optimization.

Parameters:

Name Type Description Default
cur_val Tensor

Current value.

 required
grad Tensor

Gradient.

 required

Returns:

Name Type Description
Tensor

The next point.
Source code in vambn/modelling/mtl/minnormsolver.py 
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@torch.no_grad()
def next_point(self, cur_val, grad):
    """
    Computes the next point in the optimization.

    Args:
        cur_val (Tensor): Current value.
        grad (Tensor): Gradient.

    Returns:
        Tensor: The next point.
    """
    proj_grad = grad - (torch.sum(grad) / self.n_ts)
    lt_zero = torch.nonzero(proj_grad < 0)
    lt_zero = lt_zero.view(lt_zero.numel())
    gt_zero = torch.nonzero(proj_grad > 0)
    gt_zero = gt_zero.view(gt_zero.numel())
    tm1 = -cur_val[lt_zero] / proj_grad[lt_zero]
    tm2 = (1.0 - cur_val[gt_zero]) / proj_grad[gt_zero]

    t = torch.tensor(1.0, device=grad.device)
    tm1_gt_zero = torch.nonzero(tm1 > 1e-7)
    tm1_gt_zero = tm1_gt_zero.view(tm1_gt_zero.numel())
    if tm1_gt_zero.shape[0] > 0:
        t = torch.min(tm1[tm1_gt_zero])

    tm2_gt_zero = torch.nonzero(tm2 > 1e-7)
    tm2_gt_zero = tm2_gt_zero.view(tm2_gt_zero.numel())
    if tm2_gt_zero.shape[0] > 0:
        t = torch.min(t, torch.min(tm2[tm2_gt_zero]))

    next_point = proj_grad * t + cur_val
    next_point = self.projection_to_simplex(next_point)
    return next_point

projection_to_simplex(gamma)

Projects gamma to the simplex.

Parameters:

Name Type Description Default
gamma Tensor

The input tensor to project.

 required

Returns:

Name Type Description
Tensor

The projected tensor.
Source code in vambn/modelling/mtl/minnormsolver.py 
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@torch.no_grad()
def projection_to_simplex(self, gamma):
    """
    Projects gamma to the simplex.

    Args:
        gamma (Tensor): The input tensor to project.

    Returns:
        Tensor: The projected tensor.
    """
    sorted_gamma, indices = torch.sort(gamma, descending=True)
    tmp_sum = torch.cumsum(sorted_gamma, 0)
    tmp_max = (tmp_sum - 1.0) / self.i_grid

    non_zeros = torch.nonzero(tmp_max[:-1] > sorted_gamma[1:])
    if non_zeros.shape[0] > 0:
        tmax_f = tmp_max[:-1][non_zeros[0][0]]
    else:
        tmax_f = tmp_max[-1]
    return torch.max(gamma - tmax_f, self.zero)

moo

This script includes code adapted from the 'impartial-vaes' repository with minor modifications. The original code can be found at: https://github.com/adrianjav/impartial-vaes

Credit to the original authors: Adrian Javaloy, Maryam Meghdadi, and Isabel Valera for their valuable work.

MOOForLoop

Bases: Module

A PyTorch Module for Multiple Objective Optimization (MOO) within a loop.

Source code in vambn/modelling/mtl/moo.py 
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class MOOForLoop(nn.Module):
    """A PyTorch Module for Multiple Objective Optimization (MOO) within a loop."""

    inputs: Optional[torch.Tensor]

    def __init__(self, num_heads: int, moo_method: Optional[nn.Module] = None):
        """
        Initialize the MOOForLoop module.

        Args:
            num_heads (int): Number of heads for extending the input.
            moo_method (nn.Module, optional): The MOO method to be used. Default is None.
        """
        super().__init__()

        self._moo_method = [moo_method]
        self.num_heads = num_heads
        self.inputs = None
        self.outputs = None

        if self.moo_method is not None:
            self.register_full_backward_hook(MOOForLoop._hook)

    @property
    def moo_method(self):
        """Get the MOO method."""
        return self._moo_method[0]

    def _hook(
        self, grads_input: Tuple[torch.Tensor], grads_output: Any
    ) -> Tuple[torch.Tensor]:
        """
        Hook function to replace gradients with MOO directions.

        Args:
            grads_input (Tuple[torch.Tensor]): Gradients of the module's inputs.
            grads_output (Any): Gradients of the module's outputs.

        Returns:
            Tuple[torch.Tensor]: Modified gradients.
        """
        moo_directions = self.moo_method(
            grads_output[0], self.inputs, self.outputs
        )
        self.outputs = None

        original_norm = grads_output[0].sum(dim=0).norm(p=2)
        moo_norm = moo_directions.sum(dim=0).norm(p=2).clamp_min(1e-10)
        moo_directions.mul_(original_norm / moo_norm)

        return (moo_directions.sum(dim=0),)

    def forward(self, z: torch.Tensor) -> torch.Tensor:
        """
        Forward pass. Extend the input to the number of heads and store it.

        Args:
            z (torch.Tensor): Input tensor.

        Returns:
            torch.Tensor: Extended input tensor.
        """
        extended_shape = [self.num_heads] + [-1 for _ in range(z.ndim)]
        if self.moo_method.requires_input and z.requires_grad:
            self.inputs = z.detach()
        extended_z = z.unsqueeze(0).expand(extended_shape)
        return extended_z

    def __str__(self) -> str:
        return f"MOOForLoop({self.moo_method})"

moo_method property

Get the MOO method.

__init__(num_heads, moo_method=None)

Initialize the MOOForLoop module.

Parameters:

Name Type Description Default
num_heads int

Number of heads for extending the input.

 required
moo_method Module

The MOO method to be used. Default is None.

 None
Source code in vambn/modelling/mtl/moo.py 
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def __init__(self, num_heads: int, moo_method: Optional[nn.Module] = None):
    """
    Initialize the MOOForLoop module.

    Args:
        num_heads (int): Number of heads for extending the input.
        moo_method (nn.Module, optional): The MOO method to be used. Default is None.
    """
    super().__init__()

    self._moo_method = [moo_method]
    self.num_heads = num_heads
    self.inputs = None
    self.outputs = None

    if self.moo_method is not None:
        self.register_full_backward_hook(MOOForLoop._hook)

forward(z)

Forward pass. Extend the input to the number of heads and store it.

Parameters:

Name Type Description Default
z Tensor

Input tensor.

 required

Returns:

Type Description
Tensor

torch.Tensor: Extended input tensor.
Source code in vambn/modelling/mtl/moo.py 
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def forward(self, z: torch.Tensor) -> torch.Tensor:
    """
    Forward pass. Extend the input to the number of heads and store it.

    Args:
        z (torch.Tensor): Input tensor.

    Returns:
        torch.Tensor: Extended input tensor.
    """
    extended_shape = [self.num_heads] + [-1 for _ in range(z.ndim)]
    if self.moo_method.requires_input and z.requires_grad:
        self.inputs = z.detach()
    extended_z = z.unsqueeze(0).expand(extended_shape)
    return extended_z

MooMulti

Bases: Module

A PyTorch Module for Multiple Objective Optimization (MOO) within a loop.

Source code in vambn/modelling/mtl/moo.py 
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class MooMulti(nn.Module):
    """A PyTorch Module for Multiple Objective Optimization (MOO) within a loop."""

    inputs: Optional[torch.Tensor]

    def __init__(
        self, num_modules: int, moo_method: Optional[nn.Module] = None
    ):
        """
        Initialize the MooMulti module.

        Args:
            num_modules (int): Number of heads for extending the input.
            moo_method (nn.Module, optional): The MOO method to be used. Default is None.
        """
        super().__init__()

        self._moo_method = [moo_method]
        self.num_heads = num_modules
        self.inputs = None
        self.outputs = None

        if self.moo_method is not None:
            self.register_full_backward_hook(MooMulti._hook)

    @property
    def moo_method(self):
        """Get the MOO method."""
        return self._moo_method[0]

    def _hook(
        self, grads_input: Tuple[torch.Tensor], grads_output: Any
    ) -> Tuple[torch.Tensor]:
        """
        Hook function to replace gradients with MOO directions.

        Args:
            grads_input (Tuple[torch.Tensor]): Gradients of the module's inputs.
            grads_output (Any): Gradients of the module's outputs.

        Returns:
            Tuple[torch.Tensor]: Modified gradients.
        """
        moo_directions = self.moo_method(
            grads_output[0], self.inputs, self.outputs
        )
        self.outputs = None

        if grads_output[0].shape != moo_directions.shape:
            raise ValueError(
                f"MOO directions shape {moo_directions.shape} does not match grads_output shape {grads_output[0].shape}"
            )

        original_norm = grads_output[0].norm(p=2)
        moo_norm = moo_directions.norm(p=2).clamp_min(1e-10)
        scaling_factor = original_norm / moo_norm
        scaled_moo_directions = moo_directions * scaling_factor

        if grads_input[0].shape != scaled_moo_directions.shape:
            raise ValueError(
                f"Scaled MOO directions shape {scaled_moo_directions.shape} does not match grads_input shape {grads_input[0].shape}"
            )
        return (scaled_moo_directions,)

    def forward(self, z: torch.Tensor) -> torch.Tensor:
        """
        Forward pass. Extend the input to the number of heads and store it.

        Args:
            z (torch.Tensor): Input tensor.

        Returns:
            torch.Tensor: Extended input tensor.
        """
        return z

    def __str__(self) -> str:
        return f"MooMulti({self.moo_method})"

moo_method property

Get the MOO method.

__init__(num_modules, moo_method=None)

Initialize the MooMulti module.

Parameters:

Name Type Description Default
num_modules int

Number of heads for extending the input.

 required
moo_method Module

The MOO method to be used. Default is None.

 None
Source code in vambn/modelling/mtl/moo.py 
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def __init__(
    self, num_modules: int, moo_method: Optional[nn.Module] = None
):
    """
    Initialize the MooMulti module.

    Args:
        num_modules (int): Number of heads for extending the input.
        moo_method (nn.Module, optional): The MOO method to be used. Default is None.
    """
    super().__init__()

    self._moo_method = [moo_method]
    self.num_heads = num_modules
    self.inputs = None
    self.outputs = None

    if self.moo_method is not None:
        self.register_full_backward_hook(MooMulti._hook)

forward(z)

Forward pass. Extend the input to the number of heads and store it.

Parameters:

Name Type Description Default
z Tensor

Input tensor.

 required

Returns:

Type Description
Tensor

torch.Tensor: Extended input tensor.
Source code in vambn/modelling/mtl/moo.py 
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def forward(self, z: torch.Tensor) -> torch.Tensor:
    """
    Forward pass. Extend the input to the number of heads and store it.

    Args:
        z (torch.Tensor): Input tensor.

    Returns:
        torch.Tensor: Extended input tensor.
    """
    return z

MultiMOOForLoop

Bases: Module

A PyTorch Module for applying multiple MOOForLoop modules in parallel.

Source code in vambn/modelling/mtl/moo.py 
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class MultiMOOForLoop(nn.Module):
    """A PyTorch Module for applying multiple MOOForLoop modules in parallel."""

    def __init__(self, num_heads: int, moo_methods: Sequence[nn.Module]):
        """
        Initialize the MultiMOOForLoop module.

        Args:
            num_heads (int): Number of heads for each MOOForLoop.
            moo_methods (Sequence[nn.Module]): List of MOO methods to be used.
        """
        super().__init__()

        self.num_inputs = len(moo_methods)
        self.loops = [MOOForLoop(num_heads, method) for method in moo_methods]

    def forward(self, *args) -> Generator[torch.Tensor, None, None]:
        """
        Forward pass. Applies each MOOForLoop to its corresponding input.

        Args:
            *args (torch.Tensor): Variable number of input tensors.

        Returns:
            Generator: A generator of extended input tensors after applying MOOForLoop.
        """
        if len(args) != self.num_inputs:
            raise ValueError(
                f"Expected {self.num_inputs} inputs, got {len(args)} instead."
            )
        return (loop(z) for z, loop in zip(args, self.loops))

__init__(num_heads, moo_methods)

Initialize the MultiMOOForLoop module.

Parameters:

Name Type Description Default
num_heads int

Number of heads for each MOOForLoop.

 required
moo_methods Sequence[Module]

List of MOO methods to be used.

 required
Source code in vambn/modelling/mtl/moo.py 
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def __init__(self, num_heads: int, moo_methods: Sequence[nn.Module]):
    """
    Initialize the MultiMOOForLoop module.

    Args:
        num_heads (int): Number of heads for each MOOForLoop.
        moo_methods (Sequence[nn.Module]): List of MOO methods to be used.
    """
    super().__init__()

    self.num_inputs = len(moo_methods)
    self.loops = [MOOForLoop(num_heads, method) for method in moo_methods]

forward(*args)

Forward pass. Applies each MOOForLoop to its corresponding input.

Parameters:

Name Type Description Default
*args Tensor

Variable number of input tensors.

 ()

Returns:

Name Type Description
Generator Generator[Tensor, None, None]

A generator of extended input tensors after applying MOOForLoop.
Source code in vambn/modelling/mtl/moo.py 
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def forward(self, *args) -> Generator[torch.Tensor, None, None]:
    """
    Forward pass. Applies each MOOForLoop to its corresponding input.

    Args:
        *args (torch.Tensor): Variable number of input tensors.

    Returns:
        Generator: A generator of extended input tensors after applying MOOForLoop.
    """
    if len(args) != self.num_inputs:
        raise ValueError(
            f"Expected {self.num_inputs} inputs, got {len(args)} instead."
        )
    return (loop(z) for z, loop in zip(args, self.loops))

setup_moo(hparams, num_tasks)

Setup the multi-task learning module.

Parameters:

Name Type Description Default
hparams List[MtlMethodParams]

MTL method parameters.

 required
num_tasks int

Number of tasks to perform.

 required

Raises:

Type Description
ValueError

If invalid method name is provided.

Returns:

Type Description
Module

nn.Module: Module for MTL objective.
Source code in vambn/modelling/mtl/moo.py 
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def setup_moo(hparams: List[MtlMethodParams], num_tasks: int) -> nn.Module:
    """
    Setup the multi-task learning module.

    Args:
        hparams (List[MtlMethodParams]): MTL method parameters.
        num_tasks (int): Number of tasks to perform.

    Raises:
        ValueError: If invalid method name is provided.

    Returns:
        nn.Module: Module for MTL objective.
    """
    if len(hparams) == 0:
        return mtl.Identity()

    modules = []
    for obj in hparams:
        try:
            method = mtl.MtlMethods[obj.name].value
        except KeyError:
            raise ValueError(f"Invalid method name: {obj.name}")

        if obj.name in ["nsgd"]:
            modules.append(method(num_tasks=num_tasks, update_at=obj.update_at))
        elif obj.name in ["gradnorm"]:
            modules.append(
                method(
                    num_tasks=num_tasks,
                    alpha=obj.alpha,
                    update_at=obj.update_at,
                )
            )
        elif obj.name in ["cagrad"]:
            modules.append(method(alpha=obj.alpha))
        elif obj.name in ["graddrop"]:
            modules.append(method(leakage=[0.2] * num_tasks))
        else:
            modules.append(method())

    return mtl.Compose(*modules) if len(modules) != 0 else None

mtl

This script includes code adapted from the 'impartial-vaes' repository with minor modifications. The original code can be found at: https://github.com/adrianjav/impartial-vaes

Credit to the original authors: Adrian Javaloy, Maryam Meghdadi, and Isabel Valera for their valuable work.

CAGrad

Bases: MOOMethod

CAGrad method for multiple objective optimization.

Source code in vambn/modelling/mtl/mtl.py 
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class CAGrad(MOOMethod):
    """CAGrad method for multiple objective optimization."""

    requires_input: bool = False

    def __init__(self, alpha: float):
        """
        Initialize CAGrad method.

        Args:
            alpha: Alpha parameter for CAGrad.
        """
        super(CAGrad, self).__init__()
        self.alpha = alpha

    def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
        """
        Compute new gradients using CAGrad method.

        Args:
            grads: Gradients tensor.
            inputs: Input tensor.
            outputs: Output tensor.

        Returns:
            New gradients tensor.
        """
        shape = grads.size()
        num_tasks = len(grads)
        grads = grads.flatten(start_dim=1).t()

        GG = grads.t().mm(grads).cpu()
        g0_norm = (GG.mean() + 1e-8).sqrt()

        x_start = np.ones(num_tasks) / num_tasks
        bnds = tuple((0, 1) for _ in x_start)
        cons = {"type": "eq", "fun": lambda x: 1 - sum(x)}

        A = GG.numpy()
        b = x_start.copy()
        c = (self.alpha * g0_norm + 1e-8).item()

        def objfn(x):
            return (
                x.reshape(1, num_tasks).dot(A).dot(b.reshape(num_tasks, 1))
                + c
                * np.sqrt(
                    x.reshape(1, num_tasks).dot(A).dot(x.reshape(num_tasks, 1))
                    + 1e-8
                )
            ).sum()

        res = minimize(objfn, x_start, bounds=bnds, constraints=cons)
        w_cpu = res.x

        ww = torch.Tensor(w_cpu).to(grads.device)
        gw = (grads * ww.view(1, -1)).sum(1)
        gw_norm = gw.norm()
        lmbda = c / (gw_norm + 1e-8)
        g = (grads + lmbda * gw.unsqueeze(1)) / num_tasks

        g = g.t().reshape(shape)
        grads = g

        return grads

__init__(alpha)

Initialize CAGrad method.

Parameters:

Name Type Description Default
alpha float

Alpha parameter for CAGrad.

 required
Source code in vambn/modelling/mtl/mtl.py 
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def __init__(self, alpha: float):
    """
    Initialize CAGrad method.

    Args:
        alpha: Alpha parameter for CAGrad.
    """
    super(CAGrad, self).__init__()
    self.alpha = alpha

forward(grads, inputs, outputs)

Compute new gradients using CAGrad method.

Parameters:

Name Type Description Default
grads Tensor

Gradients tensor.

 required
inputs Tensor

Input tensor.

 required
outputs Tensor

Output tensor.

 required

Returns:

Type Description
Tensor

New gradients tensor.
Source code in vambn/modelling/mtl/mtl.py 
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def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
    """
    Compute new gradients using CAGrad method.

    Args:
        grads: Gradients tensor.
        inputs: Input tensor.
        outputs: Output tensor.

    Returns:
        New gradients tensor.
    """
    shape = grads.size()
    num_tasks = len(grads)
    grads = grads.flatten(start_dim=1).t()

    GG = grads.t().mm(grads).cpu()
    g0_norm = (GG.mean() + 1e-8).sqrt()

    x_start = np.ones(num_tasks) / num_tasks
    bnds = tuple((0, 1) for _ in x_start)
    cons = {"type": "eq", "fun": lambda x: 1 - sum(x)}

    A = GG.numpy()
    b = x_start.copy()
    c = (self.alpha * g0_norm + 1e-8).item()

    def objfn(x):
        return (
            x.reshape(1, num_tasks).dot(A).dot(b.reshape(num_tasks, 1))
            + c
            * np.sqrt(
                x.reshape(1, num_tasks).dot(A).dot(x.reshape(num_tasks, 1))
                + 1e-8
            )
        ).sum()

    res = minimize(objfn, x_start, bounds=bnds, constraints=cons)
    w_cpu = res.x

    ww = torch.Tensor(w_cpu).to(grads.device)
    gw = (grads * ww.view(1, -1)).sum(1)
    gw_norm = gw.norm()
    lmbda = c / (gw_norm + 1e-8)
    g = (grads + lmbda * gw.unsqueeze(1)) / num_tasks

    g = g.t().reshape(shape)
    grads = g

    return grads

Compose

Bases: MOOMethod

Compose multiple MOO methods.

Parameters:

Name Type Description Default
modules MOOMethod

List of MOO methods to compose.

 ()

Attributes:

Name Type Description
methods ModuleList

List of MOO methods.
requires_input bool

Flag indicating if input is required.
Source code in vambn/modelling/mtl/mtl.py 
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class Compose(MOOMethod):
    """
    Compose multiple MOO methods.

    Args:
        modules (MOOMethod): List of MOO methods to compose.

    Attributes:
        methods (nn.ModuleList): List of MOO methods.
        requires_input (bool): Flag indicating if input is required.

    """

    def __init__(self, *modules: MOOMethod):
        super().__init__()
        self.methods = nn.ModuleList(modules)
        self.requires_input = any([m.requires_input for m in modules])

    def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
        """
        Apply composed MOO methods sequentially.

        Args:
            grads (torch.Tensor): Gradients tensor.
            inputs (torch.Tensor): Input tensor.
            outputs (torch.Tensor): Output tensor.

        Returns:
            torch.Tensor: Modified gradients.
        """
        for module in self.methods:
            grads = module(grads, inputs, outputs)
        return grads

forward(grads, inputs, outputs)

Apply composed MOO methods sequentially.

Parameters:

Name Type Description Default
grads Tensor

Gradients tensor.

 required
inputs Tensor

Input tensor.

 required
outputs Tensor

Output tensor.

 required

Returns:

Type Description
Tensor

torch.Tensor: Modified gradients.
Source code in vambn/modelling/mtl/mtl.py 
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def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
    """
    Apply composed MOO methods sequentially.

    Args:
        grads (torch.Tensor): Gradients tensor.
        inputs (torch.Tensor): Input tensor.
        outputs (torch.Tensor): Output tensor.

    Returns:
        torch.Tensor: Modified gradients.
    """
    for module in self.methods:
        grads = module(grads, inputs, outputs)
    return grads

GradDrop

Bases: MOOMethod

Gradient Dropout (GradDrop) method for MOO.

Parameters:

Name Type Description Default
leakage List[float]

List of leakage rates for each task.

 required

Attributes:

Name Type Description
leakage List[float]

List of leakage rates for each task.
Source code in vambn/modelling/mtl/mtl.py 
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class GradDrop(MOOMethod):
    """Gradient Dropout (GradDrop) method for MOO.

    Args:
        leakage (List[float]): List of leakage rates for each task.

    Attributes:
        leakage (List[float]): List of leakage rates for each task.

    """

    requires_input: bool = True

    def __init__(self, leakage: List[float]):
        """
        Initialize GradDrop method.

        Args:
            leakage (List[float]): List of leakage rates for each task.

        Raises:
            AssertionError: If any leakage rate is not in the range [0, 1].

        """
        super(GradDrop, self).__init__()
        assert all(
            [0 <= x <= 1 for x in leakage]
        ), "All leakages should be in the range [0, 1]"
        self.leakage = leakage

    def forward(
        self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor
    ) -> torch.Tensor:
        """
        Compute new gradients using GradDrop method.

        Args:
            grads (torch.Tensor): Gradients tensor.
            inputs (torch.Tensor): Input tensor.
            outputs (torch.Tensor): Output tensor.

        Returns:
            torch.Tensor: New gradients tensor.

        Raises:
            AssertionError: If the number of leakage parameters does not match the number of task gradients.

        """
        assert len(self.leakage) == len(
            grads
        ), "Leakage parameters should match the number of task gradients"
        sign_grads = [None for _ in range(len(grads))]
        for i in range(len(grads)):
            sign_grads[i] = inputs.sign() * grads[i]
            if len(grads[0].size()) > 1:  # It is batch-separated
                sign_grads[i] = grads[i].sum(dim=0, keepdim=True)

        odds = 0.5 * (
            1 + sum(sign_grads) / (sum(map(torch.abs, sign_grads)) + 1e-15)
        ).clamp(0, 1)
        assert odds.size() == sign_grads[0].size()  # pytype: disable=attribute-error

        new_grads = []
        samples = torch.rand(odds.size(), device=grads[0].device)
        for i in range(len(grads)):
            mask_i = torch.where(
                (odds > samples) & (sign_grads[i] > 0)  # pytype: disable=unsupported-operands
                | (odds < samples) & (sign_grads[i] < 0),  # pytype: disable=unsupported-operands
                torch.ones_like(odds),
                torch.zeros_like(odds),
            )
            mask_i = torch.lerp(
                mask_i, torch.ones_like(mask_i), self.leakage[i]
            )
            assert mask_i.size() == odds.size()
            new_grads.append(mask_i * grads[i])

        return torch.stack(new_grads, dim=0)

__init__(leakage)

Initialize GradDrop method.

Parameters:

Name Type Description Default
leakage List[float]

List of leakage rates for each task.

 required

Raises:

Type Description
AssertionError

If any leakage rate is not in the range [0, 1].
Source code in vambn/modelling/mtl/mtl.py 
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def __init__(self, leakage: List[float]):
    """
    Initialize GradDrop method.

    Args:
        leakage (List[float]): List of leakage rates for each task.

    Raises:
        AssertionError: If any leakage rate is not in the range [0, 1].

    """
    super(GradDrop, self).__init__()
    assert all(
        [0 <= x <= 1 for x in leakage]
    ), "All leakages should be in the range [0, 1]"
    self.leakage = leakage

forward(grads, inputs, outputs)

Compute new gradients using GradDrop method.

Parameters:

Name Type Description Default
grads Tensor

Gradients tensor.

 required
inputs Tensor

Input tensor.

 required
outputs Tensor

Output tensor.

 required

Returns:

Type Description
Tensor

torch.Tensor: New gradients tensor.

Raises:

Type Description
AssertionError

If the number of leakage parameters does not match the number of task gradients.
Source code in vambn/modelling/mtl/mtl.py 
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def forward(
    self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor
) -> torch.Tensor:
    """
    Compute new gradients using GradDrop method.

    Args:
        grads (torch.Tensor): Gradients tensor.
        inputs (torch.Tensor): Input tensor.
        outputs (torch.Tensor): Output tensor.

    Returns:
        torch.Tensor: New gradients tensor.

    Raises:
        AssertionError: If the number of leakage parameters does not match the number of task gradients.

    """
    assert len(self.leakage) == len(
        grads
    ), "Leakage parameters should match the number of task gradients"
    sign_grads = [None for _ in range(len(grads))]
    for i in range(len(grads)):
        sign_grads[i] = inputs.sign() * grads[i]
        if len(grads[0].size()) > 1:  # It is batch-separated
            sign_grads[i] = grads[i].sum(dim=0, keepdim=True)

    odds = 0.5 * (
        1 + sum(sign_grads) / (sum(map(torch.abs, sign_grads)) + 1e-15)
    ).clamp(0, 1)
    assert odds.size() == sign_grads[0].size()  # pytype: disable=attribute-error

    new_grads = []
    samples = torch.rand(odds.size(), device=grads[0].device)
    for i in range(len(grads)):
        mask_i = torch.where(
            (odds > samples) & (sign_grads[i] > 0)  # pytype: disable=unsupported-operands
            | (odds < samples) & (sign_grads[i] < 0),  # pytype: disable=unsupported-operands
            torch.ones_like(odds),
            torch.zeros_like(odds),
        )
        mask_i = torch.lerp(
            mask_i, torch.ones_like(mask_i), self.leakage[i]
        )
        assert mask_i.size() == odds.size()
        new_grads.append(mask_i * grads[i])

    return torch.stack(new_grads, dim=0)

GradNorm

Bases: GradNormBase

Gradient Normalization (GradNorm) method for MOO.

Parameters:

Name Type Description Default
GradNormBase class

Base class for GradNorm.

 required

Attributes:

Name Type Description
requires_input bool

Flag indicating whether input is required.

Methods:

Name Description
forward

Compute new gradients using GradNorm method.
Source code in vambn/modelling/mtl/mtl.py 
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class GradNorm(GradNormBase):
    """Gradient Normalization (GradNorm) method for MOO.

    Args:
        GradNormBase (class): Base class for GradNorm.

    Attributes:
        requires_input (bool): Flag indicating whether input is required.

    Methods:
        forward: Compute new gradients using GradNorm method.

    """

    requires_input: bool = False

    def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
        """
        Compute new gradients using GradNorm method.

        Args:
            grads (torch.Tensor): Gradients tensor.
            inputs (torch.Tensor): Input tensor.
            outputs (torch.Tensor): Output tensor.

        Returns:
            torch.Tensor: New gradients tensor.
        """
        return self._forward(grads, outputs)

forward(grads, inputs, outputs)

Compute new gradients using GradNorm method.

Parameters:

Name Type Description Default
grads Tensor

Gradients tensor.

 required
inputs Tensor

Input tensor.

 required
outputs Tensor

Output tensor.

 required

Returns:

Type Description
Tensor

torch.Tensor: New gradients tensor.
Source code in vambn/modelling/mtl/mtl.py 
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def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
    """
    Compute new gradients using GradNorm method.

    Args:
        grads (torch.Tensor): Gradients tensor.
        inputs (torch.Tensor): Input tensor.
        outputs (torch.Tensor): Output tensor.

    Returns:
        torch.Tensor: New gradients tensor.
    """
    return self._forward(grads, outputs)

GradNormBase

Bases: MOOMethod

Base class for Gradient Normalization (GradNorm) method.

Source code in vambn/modelling/mtl/mtl.py 
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class GradNormBase(MOOMethod):
    """Base class for Gradient Normalization (GradNorm) method."""

    initial_values: torch.Tensor
    counter: torch.Tensor

    def __init__(self, num_tasks: int, alpha: float, update_at: int = 20):
        """
        Initialize GradNormBase method.

        Args:
            num_tasks (int): Number of tasks.
            alpha (float): Alpha parameter for GradNorm.
            update_at (int): Update interval.
        """
        super(GradNormBase, self).__init__()
        self.epsilon = 1e-5
        self.num_tasks = num_tasks
        self.weight_ = nn.Parameter(torch.ones([num_tasks]), requires_grad=True)
        self.alpha = alpha
        self.update_at = update_at
        self.register_buffer("initial_values", torch.ones(self.num_tasks))
        self.register_buffer("counter", torch.zeros([]))

    @property
    def weight(self) -> torch.Tensor:
        """
        Compute normalized weights.

        Returns:
            torch.Tensor: Normalized weights.
        """
        ws = self.weight_.exp().clamp(self.epsilon, float("inf"))
        norm_coef = self.num_tasks / ws.sum()
        return ws * norm_coef

    def _forward(self, grads: torch.Tensor, values: List[float]) -> torch.Tensor:
        """
        Compute new gradients using GradNorm method.

        Args:
            grads (torch.Tensor): Gradients tensor.
            values (List[float]): Values for each task.

        Returns:
            torch.Tensor: New gradients tensor.
        """
        if self.initial_values is None or self.counter == self.update_at:
            self.initial_values = torch.tensor(values)
        self.counter += 1

        with torch.enable_grad():
            grads_norm = grads.flatten(start_dim=1).norm(p=2, dim=1)
            mean_grad_norm = (
                torch.mean(batch_product(grads_norm, self.weight), dim=0)
                .detach()
                .clone()
            )

            values = [
                x / y.clamp_min(self.epsilon)
                for x, y in zip(values, self.initial_values)
            ]
            average_value = torch.mean(torch.stack(values))

            loss = grads.new_zeros([])
            for i, [grad, value] in enumerate(zip(grads_norm, values)):
                r_i = value / average_value.clamp_min(self.epsilon)
                loss += torch.abs(
                    grad * self.weight[i]
                    - mean_grad_norm * torch.pow(r_i, self.alpha)
                )
            loss.backward()

        with torch.no_grad():
            new_grads = batch_product(grads, self.weight.detach())
        return new_grads

weight: torch.Tensor property

Compute normalized weights.

Returns:

Type Description
Tensor

torch.Tensor: Normalized weights.

__init__(num_tasks, alpha, update_at=20)

Initialize GradNormBase method.

Parameters:

Name Type Description Default
num_tasks int

Number of tasks.

 required
alpha float

Alpha parameter for GradNorm.

 required
update_at int

Update interval.

 20
Source code in vambn/modelling/mtl/mtl.py 
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def __init__(self, num_tasks: int, alpha: float, update_at: int = 20):
    """
    Initialize GradNormBase method.

    Args:
        num_tasks (int): Number of tasks.
        alpha (float): Alpha parameter for GradNorm.
        update_at (int): Update interval.
    """
    super(GradNormBase, self).__init__()
    self.epsilon = 1e-5
    self.num_tasks = num_tasks
    self.weight_ = nn.Parameter(torch.ones([num_tasks]), requires_grad=True)
    self.alpha = alpha
    self.update_at = update_at
    self.register_buffer("initial_values", torch.ones(self.num_tasks))
    self.register_buffer("counter", torch.zeros([]))

GradNormModified

Bases: GradNormBase

Modified Gradient Normalization (GradNorm) method for MOO.

Uses task-gradient convergence instead of task loss convergence.

Attributes:

Name Type Description
requires_input bool

Indicates whether the method requires input tensor.

Methods:

Name Description
forward

Compute new gradients using modified GradNorm method.
Source code in vambn/modelling/mtl/mtl.py 
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class GradNormModified(GradNormBase):
    """
    Modified Gradient Normalization (GradNorm) method for MOO.

    Uses task-gradient convergence instead of task loss convergence.

    Attributes:
        requires_input (bool): Indicates whether the method requires input tensor.

    Methods:
        forward(grads, inputs, outputs): Compute new gradients using modified GradNorm method.

    """

    requires_input: bool = False

    def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
        """
        Compute new gradients using modified GradNorm method.

        Args:
            grads (torch.Tensor): Gradients tensor.
            inputs (torch.Tensor): Input tensor.
            outputs (torch.Tensor): Output tensor.

        Returns:
            torch.Tensor: New gradients tensor.
        """
        return self._forward(grads, grads.flatten(start_dim=1).norm(p=2, dim=1))

forward(grads, inputs, outputs)

Compute new gradients using modified GradNorm method.

Parameters:

Name Type Description Default
grads Tensor

Gradients tensor.

 required
inputs Tensor

Input tensor.

 required
outputs Tensor

Output tensor.

 required

Returns:

Type Description
Tensor

torch.Tensor: New gradients tensor.
Source code in vambn/modelling/mtl/mtl.py 
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def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
    """
    Compute new gradients using modified GradNorm method.

    Args:
        grads (torch.Tensor): Gradients tensor.
        inputs (torch.Tensor): Input tensor.
        outputs (torch.Tensor): Output tensor.

    Returns:
        torch.Tensor: New gradients tensor.
    """
    return self._forward(grads, grads.flatten(start_dim=1).norm(p=2, dim=1))

GradVac

Bases: MOOMethod

Gradient Vaccination (GradVac) method for MOO.

Source code in vambn/modelling/mtl/mtl.py 
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class GradVac(MOOMethod):
    """Gradient Vaccination (GradVac) method for MOO."""

    requires_input: bool = False

    def __init__(self, decay: float):
        """
        Initialize GradVac method.

        Args:
            decay: Decay rate for EMA.
        """
        super(GradVac, self).__init__()
        self.decay = decay

    def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
        """
        Compute new gradients using GradVac method.

        Args:
            grads: Gradients tensor.
            inputs: Input tensor.
            outputs: Output tensor.

        Returns:
            New gradients tensor.
        """

        def vac_projection(u: torch.Tensor, v: torch.Tensor, pre_ema: float, post_ema: float) -> torch.Tensor:
            norm_u = torch.dot(u, u).sqrt()
            norm_v = torch.dot(v, v).sqrt()

            numer = norm_u * (
                pre_ema * math.sqrt(1 - post_ema**2)
                - post_ema * math.sqrt(1 - pre_ema**2)
            )
            denom = norm_v * math.sqrt(1 - pre_ema**2)

            return numer / denom.clamp_min(1e-15) * v

        size = grads.size()[1:]
        num_tasks = grads.size(0)

        grads_list = [g.flatten() for g in grads]
        ema = [[0 for _ in range(num_tasks)] for _ in range(num_tasks)]

        new_grads = []
        for i in range(num_tasks):
            grad_i = grads_list[i]
            for j in np.random.permutation(num_tasks):
                if i == j:
                    continue
                grad_j = grads_list[j]
                cos_sim = torch.cosine_similarity(grad_i, grad_j, dim=0)
                if cos_sim < ema[i][j]:
                    grad_i = grad_i + vac_projection(
                        grad_i, grad_j, ema[i][j], cos_sim
                    )
                    assert id(grads_list[i]) != id(grad_i), "Aliasing!"
                ema[i][j] = (1 - self.decay) * ema[i][j] + self.decay * cos_sim
            new_grads.append(grad_i.reshape(size))

        return torch.stack(new_grads, dim=0)

__init__(decay)

Initialize GradVac method.

Parameters:

Name Type Description Default
decay float

Decay rate for EMA.

 required
Source code in vambn/modelling/mtl/mtl.py 
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def __init__(self, decay: float):
    """
    Initialize GradVac method.

    Args:
        decay: Decay rate for EMA.
    """
    super(GradVac, self).__init__()
    self.decay = decay

forward(grads, inputs, outputs)

Compute new gradients using GradVac method.

Parameters:

Name Type Description Default
grads Tensor

Gradients tensor.

 required
inputs Tensor

Input tensor.

 required
outputs Tensor

Output tensor.

 required

Returns:

Type Description
Tensor

New gradients tensor.
Source code in vambn/modelling/mtl/mtl.py 
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def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
    """
    Compute new gradients using GradVac method.

    Args:
        grads: Gradients tensor.
        inputs: Input tensor.
        outputs: Output tensor.

    Returns:
        New gradients tensor.
    """

    def vac_projection(u: torch.Tensor, v: torch.Tensor, pre_ema: float, post_ema: float) -> torch.Tensor:
        norm_u = torch.dot(u, u).sqrt()
        norm_v = torch.dot(v, v).sqrt()

        numer = norm_u * (
            pre_ema * math.sqrt(1 - post_ema**2)
            - post_ema * math.sqrt(1 - pre_ema**2)
        )
        denom = norm_v * math.sqrt(1 - pre_ema**2)

        return numer / denom.clamp_min(1e-15) * v

    size = grads.size()[1:]
    num_tasks = grads.size(0)

    grads_list = [g.flatten() for g in grads]
    ema = [[0 for _ in range(num_tasks)] for _ in range(num_tasks)]

    new_grads = []
    for i in range(num_tasks):
        grad_i = grads_list[i]
        for j in np.random.permutation(num_tasks):
            if i == j:
                continue
            grad_j = grads_list[j]
            cos_sim = torch.cosine_similarity(grad_i, grad_j, dim=0)
            if cos_sim < ema[i][j]:
                grad_i = grad_i + vac_projection(
                    grad_i, grad_j, ema[i][j], cos_sim
                )
                assert id(grads_list[i]) != id(grad_i), "Aliasing!"
            ema[i][j] = (1 - self.decay) * ema[i][j] + self.decay * cos_sim
        new_grads.append(grad_i.reshape(size))

    return torch.stack(new_grads, dim=0)

IMTLG

Bases: MOOMethod

IMTLG method for multiple objective optimization.

Source code in vambn/modelling/mtl/mtl.py 
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class IMTLG(MOOMethod):
    """IMTLG method for multiple objective optimization."""

    requires_input: bool = False

    def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
        """
        Compute new gradients using IMTLG method.

        Args:
            grads (torch.Tensor): Gradients tensor.
            inputs (torch.Tensor): Input tensor.
            outputs (torch.Tensor): Output tensor.

        Returns:
            torch.Tensor: New gradients tensor.
        """
        flatten_grads = grads.flatten(start_dim=1)
        num_tasks = len(grads)
        if num_tasks == 1:
            return grads

        grad_diffs, unit_diffs = [], []
        for i in range(1, num_tasks):
            grad_diffs.append(flatten_grads[0] - flatten_grads[i])
            unit_diffs.append(
                unitary(flatten_grads[0]) - unitary(flatten_grads[i])
            )
        grad_diffs = torch.stack(grad_diffs, dim=0)
        unit_diffs = torch.stack(unit_diffs, dim=0)

        DU_T = torch.einsum("ik,jk->ij", grad_diffs, unit_diffs)
        DU_T_inv = torch.pinverse(DU_T)

        alphas = torch.einsum(
            "i,ki,kj->j", grads[0].flatten(), unit_diffs, DU_T_inv
        )
        alphas = torch.cat(
            (1 - alphas.sum(dim=0).unsqueeze(dim=0), alphas), dim=0
        )

        return batch_product(grads, alphas)

forward(grads, inputs, outputs)

Compute new gradients using IMTLG method.

Parameters:

Name Type Description Default
grads Tensor

Gradients tensor.

 required
inputs Tensor

Input tensor.

 required
outputs Tensor

Output tensor.

 required

Returns:

Type Description
Tensor

torch.Tensor: New gradients tensor.
Source code in vambn/modelling/mtl/mtl.py 
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def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
    """
    Compute new gradients using IMTLG method.

    Args:
        grads (torch.Tensor): Gradients tensor.
        inputs (torch.Tensor): Input tensor.
        outputs (torch.Tensor): Output tensor.

    Returns:
        torch.Tensor: New gradients tensor.
    """
    flatten_grads = grads.flatten(start_dim=1)
    num_tasks = len(grads)
    if num_tasks == 1:
        return grads

    grad_diffs, unit_diffs = [], []
    for i in range(1, num_tasks):
        grad_diffs.append(flatten_grads[0] - flatten_grads[i])
        unit_diffs.append(
            unitary(flatten_grads[0]) - unitary(flatten_grads[i])
        )
    grad_diffs = torch.stack(grad_diffs, dim=0)
    unit_diffs = torch.stack(unit_diffs, dim=0)

    DU_T = torch.einsum("ik,jk->ij", grad_diffs, unit_diffs)
    DU_T_inv = torch.pinverse(DU_T)

    alphas = torch.einsum(
        "i,ki,kj->j", grads[0].flatten(), unit_diffs, DU_T_inv
    )
    alphas = torch.cat(
        (1 - alphas.sum(dim=0).unsqueeze(dim=0), alphas), dim=0
    )

    return batch_product(grads, alphas)

Identity

Bases: MOOMethod

Identity MOO method that returns the input gradients unchanged.

Source code in vambn/modelling/mtl/mtl.py 
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class Identity(MOOMethod):
    """Identity MOO method that returns the input gradients unchanged."""

    def forward(
        self,
        grads: torch.Tensor,
        inputs: Optional[torch.Tensor],
        outputs: Optional[torch.Tensor],
    ) -> torch.Tensor:
        """
        Return the input gradients unchanged.

        Args:
            grads (torch.Tensor): Input gradients.
            inputs (torch.Tensor, optional): Input tensor.
            outputs (torch.Tensor, optional): Output tensor.

        Returns:
            torch.Tensor: Unchanged input gradients.
        """
        return grads

forward(grads, inputs, outputs)

Return the input gradients unchanged.

Parameters:

Name Type Description Default
grads Tensor

Input gradients.

 required
inputs Tensor

Input tensor.

 required
outputs Tensor

Output tensor.

 required

Returns:

Type Description
Tensor

torch.Tensor: Unchanged input gradients.
Source code in vambn/modelling/mtl/mtl.py 
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def forward(
    self,
    grads: torch.Tensor,
    inputs: Optional[torch.Tensor],
    outputs: Optional[torch.Tensor],
) -> torch.Tensor:
    """
    Return the input gradients unchanged.

    Args:
        grads (torch.Tensor): Input gradients.
        inputs (torch.Tensor, optional): Input tensor.
        outputs (torch.Tensor, optional): Output tensor.

    Returns:
        torch.Tensor: Unchanged input gradients.
    """
    return grads

MGDAUB

Bases: MOOMethod

MGDA-UB method for multiple objective optimization.

Source code in vambn/modelling/mtl/mtl.py 
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class MGDAUB(MOOMethod):
    """MGDA-UB method for multiple objective optimization."""

    requires_input: bool = False

    def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
        """
        Compute new gradients using MGDA-UB method.

        Args:
            grads (torch.Tensor): Gradients tensor.
            inputs (torch.Tensor): Input tensor.
            outputs (torch.Tensor): Output tensor.

        Returns:
            torch.Tensor: New gradients tensor.
        """
        epsilon: float = 1e-3
        shape: Tuple[int] = grads.size()[1:]
        grads = grads.flatten(start_dim=1).unsqueeze(dim=1)

        weights, min_norm = MinNormSolver.find_min_norm_element(
            grads.unbind(dim=0)
        )
        weights = [min(w, epsilon) for w in weights]

        grads = torch.stack(
            [g.reshape(shape) * w for g, w in zip(grads, weights)], dim=0
        )
        return grads

forward(grads, inputs, outputs)

Compute new gradients using MGDA-UB method.

Parameters:

Name Type Description Default
grads Tensor

Gradients tensor.

 required
inputs Tensor

Input tensor.

 required
outputs Tensor

Output tensor.

 required

Returns:

Type Description
Tensor

torch.Tensor: New gradients tensor.
Source code in vambn/modelling/mtl/mtl.py 
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def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
    """
    Compute new gradients using MGDA-UB method.

    Args:
        grads (torch.Tensor): Gradients tensor.
        inputs (torch.Tensor): Input tensor.
        outputs (torch.Tensor): Output tensor.

    Returns:
        torch.Tensor: New gradients tensor.
    """
    epsilon: float = 1e-3
    shape: Tuple[int] = grads.size()[1:]
    grads = grads.flatten(start_dim=1).unsqueeze(dim=1)

    weights, min_norm = MinNormSolver.find_min_norm_element(
        grads.unbind(dim=0)
    )
    weights = [min(w, epsilon) for w in weights]

    grads = torch.stack(
        [g.reshape(shape) * w for g, w in zip(grads, weights)], dim=0
    )
    return grads

MOOMethod

Bases: Module

Base class for multiple objective optimization (MOO) methods.

Source code in vambn/modelling/mtl/mtl.py 
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class MOOMethod(nn.Module, metaclass=ABCMeta):
    """Base class for multiple objective optimization (MOO) methods."""

    requires_input: bool = False

    def __init__(self):
        super().__init__()

    @abstractmethod
    def forward(
        self,
        grads: torch.Tensor,
        inputs: Optional[torch.Tensor],
        outputs: Optional[torch.Tensor],
    ) -> torch.Tensor:
        """
        Computes the new task gradients based on the original ones.

        Given K gradients of size D, returns a new set of K gradients of size D based on some criterion.

        Args:
            grads (torch.Tensor): Tensor of size K x D with the different gradients.
            inputs (torch.Tensor, optional): Tensor with the input of the forward pass (if requires_input is set to True).
            outputs (torch.Tensor, optional): Tensor with the K outputs of the module (not used currently).

        Returns:
            torch.Tensor: A tensor of the same size as `grads` with the new gradients to use during backpropagation.
        """
        raise NotImplementedError("You need to implement the forward pass.")

forward(grads, inputs, outputs) abstractmethod

Computes the new task gradients based on the original ones.

Given K gradients of size D, returns a new set of K gradients of size D based on some criterion.

Parameters:

Name Type Description Default
grads Tensor

Tensor of size K x D with the different gradients.

 required
inputs Tensor

Tensor with the input of the forward pass (if requires_input is set to True).

 required
outputs Tensor

Tensor with the K outputs of the module (not used currently).

 required

Returns:

Type Description
Tensor

torch.Tensor: A tensor of the same size as grads with the new gradients to use during backpropagation.
Source code in vambn/modelling/mtl/mtl.py 
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@abstractmethod
def forward(
    self,
    grads: torch.Tensor,
    inputs: Optional[torch.Tensor],
    outputs: Optional[torch.Tensor],
) -> torch.Tensor:
    """
    Computes the new task gradients based on the original ones.

    Given K gradients of size D, returns a new set of K gradients of size D based on some criterion.

    Args:
        grads (torch.Tensor): Tensor of size K x D with the different gradients.
        inputs (torch.Tensor, optional): Tensor with the input of the forward pass (if requires_input is set to True).
        outputs (torch.Tensor, optional): Tensor with the K outputs of the module (not used currently).

    Returns:
        torch.Tensor: A tensor of the same size as `grads` with the new gradients to use during backpropagation.
    """
    raise NotImplementedError("You need to implement the forward pass.")

MinNormSolver

Solver for finding the minimum norm solution in the convex hull of vectors.

Source code in vambn/modelling/mtl/mtl.py 
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class MinNormSolver:
    """Solver for finding the minimum norm solution in the convex hull of vectors."""

    MAX_ITER = 250
    STOP_CRIT = 1e-5

    @staticmethod
    def _min_norm_element_from2(v1v1: float, v1v2: float, v2v2: float) -> tuple:
        """
        Analytical solution for min_{c} |cx_1 + (1-c)x_2|_2^2.

        Args:
            v1v1: <x1, x1>.
            v1v2: <x1, x2>.
            v2v2: <x2, x2>.

        Returns:
            tuple: Coefficients and cost for the minimum norm element.
        """
        if v1v2 >= v1v1:
            gamma = 0.999
            cost = v1v1
            return gamma, cost
        if v1v2 >= v2v2:
            gamma = 0.001
            cost = v2v2
            return gamma, cost
        gamma = -1.0 * ((v1v2 - v2v2) / (v1v1 + v2v2 - 2 * v1v2))
        cost = v2v2 + gamma * (v1v2 - v2v2)
        return gamma, cost

    @staticmethod
    def _min_norm_2d(vecs: list, dps: dict) -> tuple:
        """
        Find the minimum norm solution as a combination of two points in 2D.

        Args:
            vecs: List of vectors.
            dps: Dictionary to store dot products.

        Returns:
            tuple: Solution and updated dot products.
        """
        dmin = float("inf")
        for i in range(len(vecs)):
            for j in range(i + 1, len(vecs)):
                if (i, j) not in dps:
                    dps[(i, j)] = sum(
                        torch.dot(vecs[i][k], vecs[j][k]).item()
                        for k in range(len(vecs[i]))
                    )
                    dps[(j, i)] = dps[(i, j)]
                if (i, i) not in dps:
                    dps[(i, i)] = sum(
                        torch.dot(vecs[i][k], vecs[i][k]).item()
                        for k in range(len(vecs[i]))
                    )
                if (j, j) not in dps:
                    dps[(j, j)] = sum(
                        torch.dot(vecs[j][k], vecs[j][k]).item()
                        for k in range(len(vecs[i]))
                    )
                c, d = MinNormSolver._min_norm_element_from2(
                    dps[(i, i)], dps[(i, j)], dps[(j, j)]
                )
                if d < dmin:
                    dmin = d
                    sol = [(i, j), c, d]
        return sol, dps

    @staticmethod
    def _projection2simplex(y: np.ndarray) -> np.ndarray:
        """
        Project y onto the simplex.

        Args:
            y: Input array.

        Returns:
            Projected array.
        """
        m = len(y)
        sorted_y = np.flip(np.sort(y), axis=0)
        tmpsum = 0.0
        tmax_f = (np.sum(y) - 1.0) / m
        for i in range(m - 1):
            tmpsum += sorted_y[i]
            tmax = (tmpsum - 1) / (i + 1.0)
            if tmax > sorted_y[i + 1]:
                tmax_f = tmax
                break
        return np.maximum(y - tmax_f, np.zeros(y.shape))

    @staticmethod
    def _next_point(cur_val: np.ndarray, grad: np.ndarray, n: int) -> np.ndarray:
        """
        Compute the next point for the projected gradient descent.

        Args:
            cur_val: Current value.
            grad: Gradient.
            n: Dimension of the problem.

        Returns:
            Next point.
        """
        proj_grad = grad - (np.sum(grad) / n)
        tm1 = -1.0 * cur_val[proj_grad < 0] / proj_grad[proj_grad < 0]
        tm2 = (1.0 - cur_val[proj_grad > 0]) / proj_grad[proj_grad > 0]

        t = 1
        if len(tm1[tm1 > 1e-7]) > 0:
            t = np.min(tm1[tm1 > 1e-7])
        if len(tm2[tm2 > 1e-7]) > 0:
            t = min(t, np.min(tm2[tm2 > 1e-7]))

        next_point = proj_grad * t + cur_val
        next_point = MinNormSolver._projection2simplex(next_point)
        return next_point

    @staticmethod
    def find_min_norm_element(vecs: List) -> Tuple | None:
        """
        Find the minimum norm element in the convex hull of vectors.

        Args:
            vecs: List of vectors.

        Returns:
            Minimum norm element and its cost.
        """
        dps = {}
        init_sol, dps = MinNormSolver._min_norm_2d(vecs, dps)

        n = len(vecs)
        sol_vec = np.zeros(n)
        sol_vec[init_sol[0][0]] = init_sol[1]
        sol_vec[init_sol[0][1]] = 1 - init_sol[1]

        if n < 3:
            return sol_vec, init_sol[2]

        iter_count = 0

        grad_mat = np.zeros((n, n))
        for i in range(n):
            for j in range(n):
                grad_mat[i, j] = dps[(i, j)]

        while iter_count < MinNormSolver.MAX_ITER:
            grad_dir = -1.0 * np.dot(grad_mat, sol_vec)
            new_point = MinNormSolver._next_point(sol_vec, grad_dir, n)
            v1v1 = sum(
                sol_vec[i] * sol_vec[j] * dps[(i, j)]
                for i in range(n)
                for j in range(n)
            )
            v1v2 = sum(
                sol_vec[i] * new_point[j] * dps[(i, j)]
                for i in range(n)
                for j in range(n)
            )
            v2v2 = sum(
                new_point[i] * new_point[j] * dps[(i, j)]
                for i in range(n)
                for j in range(n)
            )
            nc, nd = MinNormSolver._min_norm_element_from2(v1v1, v1v2, v2v2)
            new_sol_vec = nc * sol_vec + (1 - nc) * new_point
            change = new_sol_vec - sol_vec
            if np.sum(np.abs(change)) < MinNormSolver.STOP_CRIT:
                return sol_vec, nd
            sol_vec = new_sol_vec

find_min_norm_element(vecs) staticmethod

Find the minimum norm element in the convex hull of vectors.

Parameters:

Name Type Description Default
vecs List

List of vectors.

 required

Returns:

Type Description
Tuple | None

Minimum norm element and its cost.
Source code in vambn/modelling/mtl/mtl.py 
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@staticmethod
def find_min_norm_element(vecs: List) -> Tuple | None:
    """
    Find the minimum norm element in the convex hull of vectors.

    Args:
        vecs: List of vectors.

    Returns:
        Minimum norm element and its cost.
    """
    dps = {}
    init_sol, dps = MinNormSolver._min_norm_2d(vecs, dps)

    n = len(vecs)
    sol_vec = np.zeros(n)
    sol_vec[init_sol[0][0]] = init_sol[1]
    sol_vec[init_sol[0][1]] = 1 - init_sol[1]

    if n < 3:
        return sol_vec, init_sol[2]

    iter_count = 0

    grad_mat = np.zeros((n, n))
    for i in range(n):
        for j in range(n):
            grad_mat[i, j] = dps[(i, j)]

    while iter_count < MinNormSolver.MAX_ITER:
        grad_dir = -1.0 * np.dot(grad_mat, sol_vec)
        new_point = MinNormSolver._next_point(sol_vec, grad_dir, n)
        v1v1 = sum(
            sol_vec[i] * sol_vec[j] * dps[(i, j)]
            for i in range(n)
            for j in range(n)
        )
        v1v2 = sum(
            sol_vec[i] * new_point[j] * dps[(i, j)]
            for i in range(n)
            for j in range(n)
        )
        v2v2 = sum(
            new_point[i] * new_point[j] * dps[(i, j)]
            for i in range(n)
            for j in range(n)
        )
        nc, nd = MinNormSolver._min_norm_element_from2(v1v1, v1v2, v2v2)
        new_sol_vec = nc * sol_vec + (1 - nc) * new_point
        change = new_sol_vec - sol_vec
        if np.sum(np.abs(change)) < MinNormSolver.STOP_CRIT:
            return sol_vec, nd
        sol_vec = new_sol_vec

MtlMethods

Bases: Enum

Enumeration of available multi-task learning methods.

Source code in vambn/modelling/mtl/mtl.py 
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class MtlMethods(Enum):
    """Enumeration of available multi-task learning methods."""

    imtlg = IMTLG
    nsgd = NSGD
    gradnorm = GradNormModified
    pcgrad = PCGrad
    mgda_ub = MGDAUB
    identity = Identity
    cagrad = CAGrad
    graddrop = GradDrop

NSGD

Bases: MOOMethod

Normalized Stochastic Gradient Descent (NSGD) method for MOO.

Source code in vambn/modelling/mtl/mtl.py 
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class NSGD(MOOMethod):
    """Normalized Stochastic Gradient Descent (NSGD) method for MOO."""

    initial_grads: torch.Tensor
    requires_input: bool = False

    def __init__(self, num_tasks: int, update_at: int = 20):
        """
        Initialize NSGD method.

        Args:
            num_tasks (int): Number of tasks.
            update_at (int): Update interval.
        """
        super().__init__()
        self.num_tasks = num_tasks
        self.update_at = update_at
        self.register_buffer("initial_grads", torch.ones(num_tasks))
        self.counter = 0

    def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
        """
        Compute new gradients using NSGD method.

        Args:
            grads (torch.Tensor): Gradients tensor.
            inputs (torch.Tensor): Input tensor.
            outputs (torch.Tensor): Output tensor.

        Returns:
            torch.Tensor: New gradients tensor.
        """
        grad_norms = grads.flatten(start_dim=1).norm(dim=1)

        if self.initial_grads is None or self.counter == self.update_at:
            self.initial_grads = grad_norms

        self.counter += 1

        conv_ratios = grad_norms / self.initial_grads.clamp_min(1e-15)
        alphas = conv_ratios / conv_ratios.sum().clamp_min(1e-15)
        alphas = alphas / alphas.sum()

        weighted_sum_norms = (alphas * grad_norms).sum()
        grads = batch_product(
            grads, weighted_sum_norms / grad_norms.clamp_min(1e-15)
        )
        return grads

__init__(num_tasks, update_at=20)

Initialize NSGD method.

Parameters:

Name Type Description Default
num_tasks int

Number of tasks.

 required
update_at int

Update interval.

 20
Source code in vambn/modelling/mtl/mtl.py 
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def __init__(self, num_tasks: int, update_at: int = 20):
    """
    Initialize NSGD method.

    Args:
        num_tasks (int): Number of tasks.
        update_at (int): Update interval.
    """
    super().__init__()
    self.num_tasks = num_tasks
    self.update_at = update_at
    self.register_buffer("initial_grads", torch.ones(num_tasks))
    self.counter = 0

forward(grads, inputs, outputs)

Compute new gradients using NSGD method.

Parameters:

Name Type Description Default
grads Tensor

Gradients tensor.

 required
inputs Tensor

Input tensor.

 required
outputs Tensor

Output tensor.

 required

Returns:

Type Description
Tensor

torch.Tensor: New gradients tensor.
Source code in vambn/modelling/mtl/mtl.py 
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def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
    """
    Compute new gradients using NSGD method.

    Args:
        grads (torch.Tensor): Gradients tensor.
        inputs (torch.Tensor): Input tensor.
        outputs (torch.Tensor): Output tensor.

    Returns:
        torch.Tensor: New gradients tensor.
    """
    grad_norms = grads.flatten(start_dim=1).norm(dim=1)

    if self.initial_grads is None or self.counter == self.update_at:
        self.initial_grads = grad_norms

    self.counter += 1

    conv_ratios = grad_norms / self.initial_grads.clamp_min(1e-15)
    alphas = conv_ratios / conv_ratios.sum().clamp_min(1e-15)
    alphas = alphas / alphas.sum()

    weighted_sum_norms = (alphas * grad_norms).sum()
    grads = batch_product(
        grads, weighted_sum_norms / grad_norms.clamp_min(1e-15)
    )
    return grads

PCGrad

Bases: MOOMethod

Projected Conflicting Gradient (PCGrad) method for MOO.

Attributes:

Name Type Description
requires_input bool

Indicates whether the method requires input tensor.
Source code in vambn/modelling/mtl/mtl.py 
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class PCGrad(MOOMethod):
    """Projected Conflicting Gradient (PCGrad) method for MOO.

    Attributes:
        requires_input (bool): Indicates whether the method requires input tensor.
    """

    requires_input: bool = False

    def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
        """
        Compute new gradients using PCGrad method.

        Args:
            grads (torch.Tensor): Gradients tensor.
            inputs (torch.Tensor): Input tensor.
            outputs (torch.Tensor): Output tensor.

        Returns:
            torch.Tensor: New gradients tensor.
        """
        size = grads.size()[1:]
        num_tasks = grads.size(0)
        grads_list = [g.flatten() for g in grads]

        new_grads = [None for _ in range(num_tasks)]
        for i in np.random.permutation(num_tasks):
            grad_i = grads_list[i]
            for j in np.random.permutation(num_tasks):
                if i == j:
                    continue
                grad_j = grads_list[j]
                if torch.cosine_similarity(grad_i, grad_j, dim=0) < 0:
                    grad_i = grad_i - projection(grad_i, grad_j)
                    assert id(grads_list[i]) != id(grad_i), "Aliasing!"
            new_grads[i] = grad_i.reshape(size)

        return torch.stack(new_grads, dim=0)

forward(grads, inputs, outputs)

Compute new gradients using PCGrad method.

Parameters:

Name Type Description Default
grads Tensor

Gradients tensor.

 required
inputs Tensor

Input tensor.

 required
outputs Tensor

Output tensor.

 required

Returns:

Type Description
Tensor

torch.Tensor: New gradients tensor.
Source code in vambn/modelling/mtl/mtl.py 
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def forward(self, grads: torch.Tensor, inputs: torch.Tensor, outputs: torch.Tensor) -> torch.Tensor:
    """
    Compute new gradients using PCGrad method.

    Args:
        grads (torch.Tensor): Gradients tensor.
        inputs (torch.Tensor): Input tensor.
        outputs (torch.Tensor): Output tensor.

    Returns:
        torch.Tensor: New gradients tensor.
    """
    size = grads.size()[1:]
    num_tasks = grads.size(0)
    grads_list = [g.flatten() for g in grads]

    new_grads = [None for _ in range(num_tasks)]
    for i in np.random.permutation(num_tasks):
        grad_i = grads_list[i]
        for j in np.random.permutation(num_tasks):
            if i == j:
                continue
            grad_j = grads_list[j]
            if torch.cosine_similarity(grad_i, grad_j, dim=0) < 0:
                grad_i = grad_i - projection(grad_i, grad_j)
                assert id(grads_list[i]) != id(grad_i), "Aliasing!"
        new_grads[i] = grad_i.reshape(size)

    return torch.stack(new_grads, dim=0)

divide(numer, denom)

Numerically stable division.

Parameters:

Name Type Description Default
numer Tensor

Numerator tensor.

 required
denom Tensor

Denominator tensor.

 required

Returns:

Type Description

torch.Tensor: Result of numerically stable division.
Source code in vambn/modelling/mtl/mtl.py 
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def divide(numer, denom):
    """
    Numerically stable division.

    Args:
        numer (torch.Tensor): Numerator tensor.
        denom (torch.Tensor): Denominator tensor.

    Returns:
        torch.Tensor: Result of numerically stable division.
    """
    epsilon = 1e-15
    return (
        torch.sign(numer)
        * torch.sign(denom)
        * torch.exp(
            torch.log(numer.abs() + epsilon) - torch.log(denom.abs() + epsilon)
        )
    )

gradient_normalizers(grads, losses, normalization_type)

Compute gradient normalizers based on the specified normalization type.

Parameters:

Name Type Description Default
grads dict

A dictionary of gradients.

 required
losses dict

A dictionary of losses.

 required
normalization_type str

The type of normalization ('l2', 'loss', 'loss+', 'none').

 required

Returns:

Type Description
dict

A dictionary of gradient normalizers.
Source code in vambn/modelling/mtl/mtl.py 
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def gradient_normalizers(grads: dict, losses: dict, normalization_type: str) -> dict:
    """
    Compute gradient normalizers based on the specified normalization type.

    Args:
        grads: A dictionary of gradients.
        losses: A dictionary of losses.
        normalization_type: The type of normalization ('l2', 'loss', 'loss+', 'none').

    Returns:
        A dictionary of gradient normalizers.
    """
    gn = {}
    if normalization_type == "l2":
        for t in grads:
            gn[t] = np.sqrt(np.sum([gr.pow(2).sum().item() for gr in grads[t]]))
    elif normalization_type == "loss":
        for t in grads:
            gn[t] = losses[t]
    elif normalization_type == "loss+":
        for t in grads:
            gn[t] = losses[t] * np.sqrt(
                np.sum([gr.pow(2).sum().item() for gr in grads[t]])
            )
    elif normalization_type == "none":
        for t in grads:
            gn[t] = 1.0
    else:
        print("ERROR: Invalid Normalization Type")
    return gn

norm(tensor)

Compute the L2 norm of a tensor along the last dimension.

Parameters:

Name Type Description Default
tensor Tensor

Input tensor.

 required

Returns:

Type Description

torch.Tensor: L2 norm of the input tensor.
Source code in vambn/modelling/mtl/mtl.py 
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def norm(tensor):
    """
    Compute the L2 norm of a tensor along the last dimension.

    Args:
        tensor (torch.Tensor): Input tensor.

    Returns:
        torch.Tensor: L2 norm of the input tensor.
    """
    return tensor.norm(p=2, dim=-1, keepdim=True)

projection(u, v)

Project vector u onto vector v.

Parameters:

Name Type Description Default
u Tensor

Vector to be projected.

 required
v Tensor

Vector onto which u is projected.

 required

Returns:

Type Description

torch.Tensor: Projection of u onto v.
Source code in vambn/modelling/mtl/mtl.py 
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def projection(u, v):
    """
    Project vector u onto vector v.

    Args:
        u (torch.Tensor): Vector to be projected.
        v (torch.Tensor): Vector onto which u is projected.

    Returns:
        torch.Tensor: Projection of u onto v.
    """
    numer = torch.dot(u, v)
    denom = torch.dot(v, v)

    return numer / denom.clamp_min(1e-15) * v

unitary(tensor)

Normalize the tensor to unit norm.

Parameters:

Name Type Description Default
tensor Tensor

Input tensor.

 required

Returns:

Type Description

torch.Tensor: Unitary (normalized) tensor.
Source code in vambn/modelling/mtl/mtl.py 
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def unitary(tensor):
    """
    Normalize the tensor to unit norm.

    Args:
        tensor (torch.Tensor): Input tensor.

    Returns:
        torch.Tensor: Unitary (normalized) tensor.
    """
    return divide(tensor, norm(tensor) + 1e-15)

parameters

MtlMethodParams dataclass

Params and method description for multi-task learning.

Attributes:

Name Type Description
name str

Name of the MTL method.
update_at Optional[int]

Update interval, specific to certain methods.
alpha Optional[float]

Alpha parameter, specific to certain methods.
Source code in vambn/modelling/mtl/parameters.py 
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@dataclass
class MtlMethodParams:
    """
    Params and method description for multi-task learning.

    Attributes:
        name (str): Name of the MTL method.
        update_at (Optional[int]): Update interval, specific to certain methods.
        alpha (Optional[float]): Alpha parameter, specific to certain methods.
    """

    name: str
    update_at: Optional[int] = None
    alpha: Optional[float] = None

    def __post_init__(self):
        """
        Post-initialization to set default values for specific methods.
        """
        if self.name == "nsgd":
            if self.update_at is None:
                self.update_at = 1
        elif self.name == "gradnorm":
            if self.update_at is None:
                self.update_at = 1
            if self.alpha is None:
                self.alpha = 1.0
        elif self.name == "pcgrad":
            if self.update_at is None:
                self.update_at = 1
        elif self.name == "cagrad":
            if self.alpha is None:
                self.alpha = 10

__post_init__()

Post-initialization to set default values for specific methods.

Source code in vambn/modelling/mtl/parameters.py 
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def __post_init__(self):
    """
    Post-initialization to set default values for specific methods.
    """
    if self.name == "nsgd":
        if self.update_at is None:
            self.update_at = 1
    elif self.name == "gradnorm":
        if self.update_at is None:
            self.update_at = 1
        if self.alpha is None:
            self.alpha = 1.0
    elif self.name == "pcgrad":
        if self.update_at is None:
            self.update_at = 1
    elif self.name == "cagrad":
        if self.alpha is None:
            self.alpha = 10

utils

This script includes code adapted from the 'impartial-vaes' repository with minor modifications. The original code can be found at: https://github.com/adrianjav/impartial-vaes

Credit to the original authors: Adrian Javaloy, Maryam Meghdadi, and Isabel Valera for their valuable work.

batch_product(batch, weight)

Multiplies each slice of the first dimension of batch by the corresponding scalar in the weight vector.

Parameters:

Name Type Description Default
batch Tensor

Tensor of size [B, ...].

 required
weight Tensor

Tensor of size [B].

 required

Returns:

Type Description

torch.Tensor: A tensor such that result[i] = batch[i] * weight[i].
Source code in vambn/modelling/mtl/utils.py 
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def batch_product(batch: torch.Tensor, weight: torch.Tensor):
    r"""
    Multiplies each slice of the first dimension of batch by the corresponding scalar in the weight vector.

    Args:
        batch (torch.Tensor): Tensor of size [B, ...].
        weight (torch.Tensor): Tensor of size [B].

    Returns:
        torch.Tensor: A tensor such that `result[i] = batch[i] * weight[i]`.
    """
    assert batch.size(0) == weight.size(0)
    return (batch.T * weight.T).T