BUG: `ADVI` - arguments `start` and `start_sigma` have inconsistent keys for transformed random variables

[trendelkampschroer]

Describe the issue:

There are a couple of issues with the current design:
i) The keys of start and start_sigma are inconsistent as can be seen in example below.
ii) I can specify arbitrary (str-valued) keys for start_sigma, if they are not in the model, then the default zero initialisation is used for the rho variable in the variational approximation.
iii) The np.log(np.expm1(np.abs(sigma))) transformation from sigma to rho in method create_shared_parameters from class MeanFieldGroup is also somewhat surprising.
iv) It is also unclear how the values generated from advi.approx.mean.eval and advi.approx.std.eval relate to the variational parameters mu and rho returned by method create_shared_params and/or with the free random variables beta and sigma of the model. For example I think that tracker["mean"][-1][-1] (cf. example below) is the variational parameter corresponding to the variable sigma_log__.
v) In pm.fit arguments start and start_sigma are ignored if an instance of ADVI is passed as method.

Reproduceable code example:

Minimal example below 


import numpy as np
import pandas as pd
import pymc as pm


def generate_data(num_samples: int) -> pd.DataFrame:
    rng = np.random.default_rng(seed=42)
    beta = 1.0
    sigma = 10.0
    x = rng.normal(loc=0.0, scale=1.0, size=num_samples)
    y = beta * x + sigma * rng.normal(size=num_samples)
    return pd.DataFrame({"x": x, "y": y})


def make_model(frame: pd.DataFrame) -> pm.Model:
    with pm.Model() as model:
        # Data
        x = pm.Data("x", frame["x"])
        y = pm.Data("y", frame["y"])

        # Prior
        beta = pm.Normal("beta", sigma=10.0)
        sigma = pm.HalfNormal("sigma", sigma=20.0)

        # Linear model
        mu = beta * x

        # Likelihood
        pm.Normal("y_obs", mu=mu, sigma=sigma, observed=y)
    return model

if __name__ == "__main__":
    frame = generate_data(10000)
    model = make_model(frame)
    with model:
        advi = pm.ADVI(
            start={"beta": 1.0, "sigma": 10.0},
            start_sigma={"beta": 0.5, "sigma_log__": 0.5}
        )
       tracker = Tracker(
            mean=advi.approx.mean.eval,
            std=advi.approx.std.eval
        )
        approx = pm.fit(
            n=1_000_000,
            method=advi,
            callbacks=[
                CheckParametersConvergence(diff="relative", tolerance=1e-3),
                CheckParametersConvergence(diff="absolute", tolerance=1e-3),
                tracker
            ],
        )

This is from class MeanFieldGroup

    def create_shared_params(self, start=None, start_sigma=None):
        # NOTE: `Group._prepare_start` uses `self.model.free_RVs` to identify free variables and
        # `DictToArrayBijection` to turn them into a flat array, while `Approximation.rslice` assumes that the free
        # variables are given by `self.group` and that the mapping between original variables and flat array is given
        # by `self.ordering`. In the cases I looked into these turn out to be the same, but there may be edge cases or
        # future code changes that break this assumption.
        start = self._prepare_start(start)
        rho1 = np.zeros((self.ddim,))

        if start_sigma is not None:
            for name, slice_, *_ in self.ordering.values():
                sigma = start_sigma.get(name)
                if sigma is not None:
                    rho1[slice_] = np.log(np.expm1(np.abs(sigma)))
        rho = rho1

        return {
            "mu": pytensor.shared(pm.floatX(start), "mu"),
            "rho": pytensor.shared(pm.floatX(rho), "rho"),
        }

I think this simple example is already hitting the problematic edge case mentioned in the NOTE. Also the line sigma = start_sigma.get(name) is problematic as passing start_sigma with wrong keys will never raise an error.

This is in contrast to parameter start where a wrong key will raise, e.g. using "beta_foo" instead of "beta"

Traceback (most recent call last):
  File ".../univariate.py", line 167, in <module>
    advi = pm.ADVI(
           ^^^^^^^^
  File ".../pymc/variational/inference.py", line 471, in __init__
    super().__init__(MeanField(*args, **kwargs))
                     ^^^^^^^^^^^^^^^^^^^^^^^^^^
  File ".../pymc/variational/approximations.py", line 339, in __init__
    super().__init__(groups, model=kwargs.get("model"))
  File ".../pymc/variational/opvi.py", line 1229, in __init__
    rest.__init_group__(unseen_free_RVs)
  File ".../pytensor/configparser.py", line 44, in res
    return f(*args, **kwargs)
           ^^^^^^^^^^^^^^^^^^
  File ".../pymc/variational/approximations.py", line 74, in __init_group__
    self.shared_params = self.create_shared_params(
                         ^^^^^^^^^^^^^^^^^^^^^^^^^^
  File ".../pymc/variational/approximations.py", line 85, in create_shared_params
    start = self._prepare_start(start)
            ^^^^^^^^^^^^^^^^^^^^^^^^^^
  File ".../pymc/variational/opvi.py", line 760, in _prepare_start
    ipfn = make_initial_point_fn(
           ^^^^^^^^^^^^^^^^^^^^^^
  File ".../pymc/initial_point.py", line 134, in make_initial_point_fn
    sdict_overrides = convert_str_to_rv_dict(model, overrides or {})
                      ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File ".../pymc/initial_point.py", line 49, in convert_str_to_rv_dict
    initvals[model[key]] = initval
             ~~~~~^^^^^
  File ".../pymc/model/core.py", line 1575, in __getitem__
    raise e
  File ".../pymc/model/core.py", line 1570, in __getitem__
    return self.named_vars[key]
           ~~~~~~~~~~~~~~~^^^^^
KeyError: 'beta_foo'

Also this block in pm.fit is problematic since it will ignore start and start_sigma if an instance of e.g. ADVI is passed as method argument to instead of the string 'advi'

 _select = dict(advi=ADVI, fullrank_advi=FullRankADVI, svgd=SVGD, asvgd=ASVGD)
    if isinstance(method, str):
        method = method.lower()
        if method in _select:
            inference = _select[method](model=model, **inf_kwargs)
        else:
            raise KeyError(f"method should be one of {set(_select.keys())} or Inference instance")
    elif isinstance(method, Inference):
        inference = method
...

Initially this made debugging of the actual issue even more contrived. Also it is not clear if I can use any instance of ADVI to get tracking for mean and std or if it has to be the same instance that is passed to pm.fit.

I'd be happy to make a PR with a fix, but I am lacking familiarity with the APIs/concepts mentioned in the NOTE, i.e.

`Group._prepare_start` uses `self.model.free_RVs` to identify free variables and
        # `DictToArrayBijection` to turn them into a flat array, while `Approximation.rslice` assumes that the free
        # variables are given by `self.group` and that the mapping between original variables and flat array is given
        # by `self.ordering`.

I'd be super glad to receive guidance so that I can work on this. I think that a fix could significantly improve the API and usability of the initialisation for ADVI.

Context for the issue:

cf. https://discourse.pymc.io/t/variational-fit-advi-initialisation/15940/4 for more context. Also thanks a lot @jessegrabowski for already very helpful suggestions.

A working example, that demonstrates the quirks of the current state is below. The example demonstrates how to use the "final state" (variational parameters mu, rho???) to initialise a fit for a model with slightly enlarged data set, but I wouldn't know how to implement this for a complicated (hierarchical) model with many (transformed) random variables. As the transformation need to be applied in order to use the "final state" from the first fit as initialisation of the second fit.

import numpy as np
import pandas as pd
import pymc as pm


def generate_data(num_samples: int) -> pd.DataFrame:
    rng = np.random.default_rng(seed=42)
    beta = 1.0
    sigma = 10.0
    x = rng.normal(loc=0.0, scale=1.0, size=num_samples)
    y = beta * x + sigma * rng.normal(size=num_samples)
    return pd.DataFrame({"x": x, "y": y})


def make_model(frame: pd.DataFrame) -> pm.Model:
    with pm.Model() as model:
        # Data
        x = pm.Data("x", frame["x"])
        y = pm.Data("y", frame["y"])

        # Prior
        beta = pm.Normal("beta", sigma=10.0)
        sigma = pm.HalfNormal("sigma", sigma=20.0)

        # Linear model
        mu = beta * x

        # Likelihood
        pm.Normal("y_obs", mu=mu, sigma=sigma, observed=y)
    return model


if __name__ == "__main__":
    num_samples = 10_000
    frame = generate_data(num_samples=num_samples)
    model = make_model(frame)
    with model:
        advi = pm.ADVI(
        )
        tracker = Tracker(
            mean=advi.approx.mean.eval,
            std=advi.approx.std.eval
        )
        t0 = time.time()
        approx = pm.fit(
            n=1_000_000,
            method=advi,
            callbacks=[
                CheckParametersConvergence(diff="relative", tolerance=1e-3),
                CheckParametersConvergence(diff="absolute", tolerance=1e-3),
                tracker
            ],
        )
        t = time.time() - t0
        print(f"Time for fit is {t:.3f}s.")

    frame_new = pd.concat([frame, generate_data(100)], axis=0)
    model = make_model(frame_new)
    with model:
        advi = pm.ADVI(
            start={"beta_foo": tracker["mean"][-1][0], "sigma": np.exp(tracker["mean"][-1][1])},
            start_sigma={"beta": tracker["std"][-1][0], "sigma_log__": tracker["std"][-1][1]}
        )
        tracker = Tracker(
            mean=advi.approx.mean.eval,
            std=advi.approx.std.eval
        )
        t0 = time.time()
        approx = pm.fit(
            n=1_000_000,
            method=advi,
            callbacks=[
                CheckParametersConvergence(diff="relative", tolerance=1e-3),
                CheckParametersConvergence(diff="absolute", tolerance=1e-3),
                tracker
            ],
        )
        t = time.time() - t0
        print(f"Time for fit is {t:.3f}s.")

This is the trace for the initial ADVI fit

and here for the model with data updated and parameters initialised

As we can see we can save quite a lot of computation by using the final parameters from the old fit as the initial guess for the new fit.

Error message:

No response

PyMC version information:

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[fonnesbeck]

Thanks for this. Yeah, the start values should really be associated with fit rather than the ADVI object.

[jessegrabowski]

cc @ferrine

[ferrine]

It's also problematic because some approximations might not have a very clear correspondence to variables: e.g. Normalizing Flows or other low rank approximations, the initialization parameters of ADVI just happen to coincide with model parameters. For every approximation I think there is a special way to initialize parameters. If we consider to rethink the API, not error messages, this should be taken into account

[trendelkampschroer]

@ferrine thanks for chiming in, would it make sense to ask the user to directly specify the parameters of the variational model (mu, rho) and provide some means to convert a dictionary with initial guess in terms of the (free) model parameters to the variational parameters. This would obviously be a large breaking change to the API and the types that are used internally.

I'd like to move forward with a less invasive solution that at least ensures that start and start_sigma are consistent in terms of the keys that are used. Also if possible I'd want an API that enables me to get the final values for the variational parameters from a fitted approximation and a clear recipe of turning them back into an initial guess for a new fit.