def cyclical_sgld_integrator(
    rng_key: jax.Array,
    init_step_size: float,
    cycle_length: int,
    exploration_ratio: float,
    preconditioner: Preconditioner,
) -> GradientTransformation:
    """Optax implementation of the Cyclical SGLD integrator.

    Parameters
    ----------
        rng_key: jax.Array
            An initial random number generator.
        init_step_size: float
            The initial step size.
        cycle_length: int
            The length of each exploration/sampling cycle, in steps.
        exploration_ratio: float
            The fraction of steps to allocate to the mode exploration phase.
        preconditioner: Preconditioner
            See :class:`Preconditioner` for reference.
    """
    step_schedule = cyclical_cosine_schedule_with_const_burnin(
        init_step_size=init_step_size,
        burnin_steps=0,
        cycle_length=cycle_length,
    )
    # SGHMC with no momentum is equivalent to SGLD
    sgld = sghmc_integrator(
        momentum_decay=0.0,
        momentum_resample_steps=None,
        rng_key=rng_key,
        step_schedule=step_schedule,
        preconditioner=preconditioner,
    )
    sgd = optax.sgd(learning_rate=1.0)

    def init_fn(params):
        return OptaxCyclicalSGLDState(
            sgd_state=sgd.init(params),
            sgld_state=sgld.init(params),
        )

    def update_fn(gradient, state, *_):
        def sgd_step():
            step_size = step_schedule(state.sgld_state.count)
            preconditioner_state = preconditioner.update_preconditioner(
                gradient, state.sgld_state.preconditioner_state
            )
            new_sgld_state = state.sgld_state._replace(
                count=state.sgld_state.count + 1,
                preconditioner_state=preconditioner_state,
            )
            rescaled_gradient = jax.tree_map(
                lambda g: -1.0 * step_size * g,
                gradient,
            )
            updates, new_sgd_state = sgd.update(rescaled_gradient, state.sgd_state)
            updates = preconditioner.multiply_by_m_inv(updates, preconditioner_state)
            new_state = OptaxCyclicalSGLDState(
                sgd_state=new_sgd_state,
                sgld_state=new_sgld_state,
            )
            return updates, new_state

        def sgld_step():
            updates, new_sgld_state = sgld.update(gradient, state.sgld_state)
            new_state = OptaxCyclicalSGLDState(
                sgd_state=state.sgd_state,
                sgld_state=new_sgld_state,
            )
            return updates, new_state

        updates, state = jax.lax.cond(
            ((state.sgld_state.count % cycle_length) / cycle_length)
            >= exploration_ratio,
            sgld_step,
            sgd_step,
        )
        return updates, state

    return GradientTransformation(init_fn, update_fn)
Optax implementation of the Cyclical SGLD integrator.

    Parameters
    ----------
        rng_key: jax.Array
            An initial random number generator.
        init_step_size: float
            The initial step size.
        cycle_length: int
            The length of each exploration/sampling cycle, in steps.
        exploration_ratio: float
            The fraction of steps to allocate to the mode exploration phase.
        preconditioner: Preconditioner
            See :class:`Preconditioner` for reference.
    
fortuna/prob_model/posterior/sgmcmc/cyclical_sgld/cyclical_sgld_integrator.py