VI Diag

posteriors.vi.diag.build(log_posterior, optimizer, temperature=1.0, n_samples=1, stl=True, init_log_sds=0.0)

Builds a transform for variational inference with a diagonal Normal distribution over parameters.

Find \(\mu\) and diagonal \(\Sigma\) that mimimize \(\text{KL}(N(θ| \mu, \Sigma) || p_T(θ))\) where \(p_T(θ) \propto \exp( \log p(θ) / T)\) with temperature \(T\).

The log posterior and temperature are recommended to be constructed in tandem to ensure robust scaling for a large amount of data.

For more information on variational inference see Blei et al, 2017.

Parameters:

Name	Type	Description	Default
log_posterior	Callable[[TensorTree, Any], float]	Function that takes parameters and input batch and returns the log posterior (which can be unnormalised).	required
optimizer	GradientTransformation	TorchOpt functional optimizer for updating the variational parameters. Make sure to use lower case like torchopt.adam()	required
temperature	float	Temperature to rescale (divide) log_posterior.	1.0
n_samples	int	Number of samples to use for Monte Carlo estimate.	1
stl	bool	Whether to use the stick-the-landing estimator from (Roeder et al](https://arxiv.org/abs/1703.09194).	True
init_log_sds	TensorTree | float	Initial log of the square-root diagonal of the covariance matrix of the variational distribution. Can be a tree matching params or scalar.	0.0

Returns:

Type	Description
Transform	Diagonal VI transform instance.

Source code in posteriors/vi/diag.py

def build(
    log_posterior: Callable[[TensorTree, Any], float],
    optimizer: torchopt.base.GradientTransformation,
    temperature: float = 1.0,
    n_samples: int = 1,
    stl: bool = True,
    init_log_sds: TensorTree | float = 0.0,
) -> Transform:
    """Builds a transform for variational inference with a diagonal Normal
    distribution over parameters.

    Find $\\mu$ and diagonal $\\Sigma$ that mimimize $\\text{KL}(N(θ| \\mu, \\Sigma) || p_T(θ))$
    where $p_T(θ) \\propto \\exp( \\log p(θ) / T)$ with temperature $T$.

    The log posterior and temperature are recommended to be [constructed in tandem](../../log_posteriors.md)
    to ensure robust scaling for a large amount of data.

    For more information on variational inference see [Blei et al, 2017](https://arxiv.org/abs/1601.00670).

    Args:
        log_posterior: Function that takes parameters and input batch and
            returns the log posterior (which can be unnormalised).
        optimizer: TorchOpt functional optimizer for updating the variational
            parameters. Make sure to use lower case like torchopt.adam()
        temperature: Temperature to rescale (divide) log_posterior.
        n_samples: Number of samples to use for Monte Carlo estimate.
        stl: Whether to use the stick-the-landing estimator
            from (Roeder et al](https://arxiv.org/abs/1703.09194).
        init_log_sds: Initial log of the square-root diagonal of the covariance matrix
            of the variational distribution. Can be a tree matching params or scalar.

    Returns:
        Diagonal VI transform instance.
    """
    init_fn = partial(init, optimizer=optimizer, init_log_sds=init_log_sds)
    update_fn = partial(
        update,
        log_posterior=log_posterior,
        optimizer=optimizer,
        temperature=temperature,
        n_samples=n_samples,
        stl=stl,
    )
    return Transform(init_fn, update_fn)

posteriors.vi.diag.VIDiagState

Bases: NamedTuple

State encoding a diagonal Normal variational distribution over parameters.

Attributes:

Name	Type	Description
params	TensorTree	Mean of the variational distribution.
log_sd_diag	TensorTree	Log of the square-root diagonal of the covariance matrix of the variational distribution.
opt_state	OptState	TorchOpt state storing optimizer data for updating the variational parameters.
nelbo	tensor	Negative evidence lower bound (lower is better).
aux	Any	Auxiliary information from the log_posterior call.

Source code in posteriors/vi/diag.py

class VIDiagState(NamedTuple):
    """State encoding a diagonal Normal variational distribution over parameters.

    Attributes:
        params: Mean of the variational distribution.
        log_sd_diag: Log of the square-root diagonal of the covariance matrix of the
            variational distribution.
        opt_state: TorchOpt state storing optimizer data for updating the
            variational parameters.
        nelbo: Negative evidence lower bound (lower is better).
        aux: Auxiliary information from the log_posterior call.
    """

    params: TensorTree
    log_sd_diag: TensorTree
    opt_state: torchopt.typing.OptState
    nelbo: torch.tensor = torch.tensor([])
    aux: Any = None

posteriors.vi.diag.init(params, optimizer, init_log_sds=0.0)

Initialise diagonal Normal variational distribution over parameters.

optimizer.init will be called on flattened variational parameters so hyperparameters such as learning rate need to pre-specified through TorchOpt's functional API:

import torchopt

optimizer = torchopt.adam(lr=1e-2)
vi_state = init(init_mean, optimizer)

It's assumed maximize=False for the optimizer, so that we minimize the NELBO.

Parameters:

Name	Type	Description	Default
params	TensorTree	Initial mean of the variational distribution.	required
optimizer	GradientTransformation	TorchOpt functional optimizer for updating the variational parameters. Make sure to use lower case like torchopt.adam()	required
init_log_sds	TensorTree | float	Initial log of the square-root diagonal of the covariance matrix of the variational distribution. Can be a tree matching params or scalar.	0.0

Returns:

Type	Description
VIDiagState	Initial DiagVIState.

Source code in posteriors/vi/diag.py

def init(
    params: TensorTree,
    optimizer: torchopt.base.GradientTransformation,
    init_log_sds: TensorTree | float = 0.0,
) -> VIDiagState:
    """Initialise diagonal Normal variational distribution over parameters.

    optimizer.init will be called on flattened variational parameters so hyperparameters
    such as learning rate need to pre-specified through TorchOpt's functional API:

    ```
    import torchopt

    optimizer = torchopt.adam(lr=1e-2)
    vi_state = init(init_mean, optimizer)
    ```

    It's assumed maximize=False for the optimizer, so that we minimize the NELBO.

    Args:
        params: Initial mean of the variational distribution.
        optimizer: TorchOpt functional optimizer for updating the variational
            parameters. Make sure to use lower case like torchopt.adam()
        init_log_sds: Initial log of the square-root diagonal of the covariance matrix
            of the variational distribution. Can be a tree matching params or scalar.

    Returns:
        Initial DiagVIState.
    """
    if is_scalar(init_log_sds):
        init_log_sds = tree_map(
            lambda x: torch.full_like(x, init_log_sds, requires_grad=x.requires_grad),
            params,
        )

    opt_state = optimizer.init([params, init_log_sds])
    return VIDiagState(params, init_log_sds, opt_state)

posteriors.vi.diag.update(state, batch, log_posterior, optimizer, temperature=1.0, n_samples=1, stl=True, inplace=False)

Updates the variational parameters to minimize the NELBO.

Parameters:

Name	Type	Description	Default
state	VIDiagState	Current state.	required
batch	Any	Input data to log_posterior.	required
log_posterior	LogProbFn	Function that takes parameters and input batch and returns the log posterior (which can be unnormalised).	required
optimizer	GradientTransformation	TorchOpt functional optimizer for updating the variational parameters. Make sure to use lower case like torchopt.adam()	required
temperature	float	Temperature to rescale (divide) log_posterior.	1.0
n_samples	int	Number of samples to use for Monte Carlo estimate.	1
stl	bool	Whether to use the stick-the-landing estimator from (Roeder et al](https://arxiv.org/abs/1703.09194).	True
inplace	bool	Whether to modify state in place.	False

Returns:

Type	Description
VIDiagState	Updated DiagVIState.

Source code in posteriors/vi/diag.py

def update(
    state: VIDiagState,
    batch: Any,
    log_posterior: LogProbFn,
    optimizer: torchopt.base.GradientTransformation,
    temperature: float = 1.0,
    n_samples: int = 1,
    stl: bool = True,
    inplace: bool = False,
) -> VIDiagState:
    """Updates the variational parameters to minimize the NELBO.

    Args:
        state: Current state.
        batch: Input data to log_posterior.
        log_posterior: Function that takes parameters and input batch and
            returns the log posterior (which can be unnormalised).
        optimizer: TorchOpt functional optimizer for updating the variational
            parameters. Make sure to use lower case like torchopt.adam()
        temperature: Temperature to rescale (divide) log_posterior.
        n_samples: Number of samples to use for Monte Carlo estimate.
        stl: Whether to use the stick-the-landing estimator
            from (Roeder et al](https://arxiv.org/abs/1703.09194).
        inplace: Whether to modify state in place.

    Returns:
        Updated DiagVIState.
    """

    def nelbo_log_sd(m, lsd):
        sd_diag = tree_map(torch.exp, lsd)
        return nelbo(m, sd_diag, batch, log_posterior, temperature, n_samples, stl)

    with torch.no_grad(), CatchAuxError():
        nelbo_grads, (nelbo_val, aux) = grad_and_value(
            nelbo_log_sd, argnums=(0, 1), has_aux=True
        )(state.params, state.log_sd_diag)

    updates, opt_state = optimizer.update(
        nelbo_grads,
        state.opt_state,
        params=[state.params, state.log_sd_diag],
        inplace=inplace,
    )
    mean, log_sd_diag = torchopt.apply_updates(
        (state.params, state.log_sd_diag), updates, inplace=inplace
    )

    if inplace:
        tree_insert_(state.nelbo, nelbo_val.detach())
        return state._replace(aux=aux)

    return VIDiagState(mean, log_sd_diag, opt_state, nelbo_val.detach(), aux)

posteriors.vi.diag.nelbo(mean, sd_diag, batch, log_posterior, temperature=1.0, n_samples=1, stl=True)

Returns the negative evidence lower bound (NELBO) for a diagonal Normal variational distribution over the parameters of a model.

Monte Carlo estimate with n_samples from q. $$ \text{NELBO} = - 𝔼_{q(θ)}[\log p(y|x, θ) + \log p(θ) - \log q(θ) * T]) $$ for temperature \(T\).

log_posterior expects to take parameters and input batch and return a scalar as well as a TensorTree of any auxiliary information:

log_posterior_eval, aux = log_posterior(params, batch)

The log posterior and temperature are recommended to be constructed in tandem to ensure robust scaling for a large amount of data and variable batch size.

Parameters:

Name	Type	Description	Default
mean	dict	Mean of the variational distribution.	required
sd_diag	dict	Square-root diagonal of the covariance matrix of the variational distribution.	required
batch	Any	Input data to log_posterior.	required
log_posterior	LogProbFn	Function that takes parameters and input batch and returns the log posterior (which can be unnormalised).	required
temperature	float	Temperature to rescale (divide) log_posterior.	1.0
n_samples	int	Number of samples to use for Monte Carlo estimate.	1
stl	bool	Whether to use the stick-the-landing estimator from (Roeder et al](https://arxiv.org/abs/1703.09194).	True

Returns:

Type	Description
Tuple[float, Any]	The sampled approximate NELBO averaged over the batch.

Source code in posteriors/vi/diag.py

def nelbo(
    mean: dict,
    sd_diag: dict,
    batch: Any,
    log_posterior: LogProbFn,
    temperature: float = 1.0,
    n_samples: int = 1,
    stl: bool = True,
) -> Tuple[float, Any]:
    """Returns the negative evidence lower bound (NELBO) for a diagonal Normal
    variational distribution over the parameters of a model.

    Monte Carlo estimate with `n_samples` from q.
    $$
    \\text{NELBO} = - 𝔼_{q(θ)}[\\log p(y|x, θ) + \\log p(θ) - \\log q(θ) * T])
    $$
    for temperature $T$.

    `log_posterior` expects to take parameters and input batch and return a scalar
    as well as a TensorTree of any auxiliary information:

    ```
    log_posterior_eval, aux = log_posterior(params, batch)
    ```

    The log posterior and temperature are recommended to be [constructed in tandem](../../log_posteriors.md)
    to ensure robust scaling for a large amount of data and variable batch size.

    Args:
        mean: Mean of the variational distribution.
        sd_diag: Square-root diagonal of the covariance matrix of the
            variational distribution.
        batch: Input data to log_posterior.
        log_posterior: Function that takes parameters and input batch and
            returns the log posterior (which can be unnormalised).
        temperature: Temperature to rescale (divide) log_posterior.
        n_samples: Number of samples to use for Monte Carlo estimate.
        stl: Whether to use the stick-the-landing estimator
            from (Roeder et al](https://arxiv.org/abs/1703.09194).

    Returns:
        The sampled approximate NELBO averaged over the batch.
    """
    sampled_params = diag_normal_sample(mean, sd_diag, sample_shape=(n_samples,))
    if stl:
        mean = tree_map(lambda x: x.detach(), mean)
        sd_diag = tree_map(lambda x: x.detach(), sd_diag)

    # Don't use vmap for single sample, since vmap doesn't work with lots of models
    if n_samples == 1:
        single_param = tree_map(lambda x: x[0], sampled_params)
        log_p, aux = log_posterior(single_param, batch)
        log_q = diag_normal_log_prob(single_param, mean, sd_diag)
    else:
        log_p, aux = vmap(log_posterior, (0, None), (0, 0))(sampled_params, batch)
        log_q = vmap(diag_normal_log_prob, (0, None, None))(
            sampled_params, mean, sd_diag
        )
    return -(log_p - log_q * temperature).mean(), aux

posteriors.vi.diag.sample(state, sample_shape=torch.Size([]))

Single sample from diagonal Normal distribution over parameters.

Parameters:

Name	Type	Description	Default
state	VIDiagState	State encoding mean and log standard deviations.	required
sample_shape	Size	Shape of the desired samples.	Size([])

Returns:

Type	Description
TensorTree	Sample(s) from Normal distribution.

Source code in posteriors/vi/diag.py

def sample(state: VIDiagState, sample_shape: torch.Size = torch.Size([])) -> TensorTree:
    """Single sample from diagonal Normal distribution over parameters.

    Args:
        state: State encoding mean and log standard deviations.
        sample_shape: Shape of the desired samples.

    Returns:
        Sample(s) from Normal distribution.
    """
    sd_diag = tree_map(torch.exp, state.log_sd_diag)
    return diag_normal_sample(state.params, sd_diag, sample_shape=sample_shape)