Add the SDE variant of DPM-Solver and DPM-Solver++ (#3344)

* add SDE variant of DPM-Solver and DPM-Solver++

* add test

* fix typo

* fix typo

src/diffusers/schedulers/scheduling_dpmsolver_multistep.py

@@ -21,6 +21,7 @@ import numpy as np
 import torch
 
 from ..configuration_utils import ConfigMixin, register_to_config
+from ..utils import randn_tensor
 from .scheduling_utils import KarrasDiffusionSchedulers, SchedulerMixin, SchedulerOutput
 
 
@@ -70,6 +71,10 @@ class DPMSolverMultistepScheduler(SchedulerMixin, ConfigMixin):
     thresholding. Note that the thresholding method is unsuitable for latent-space diffusion models (such as
     stable-diffusion).
 
+    We also support the SDE variant of DPM-Solver and DPM-Solver++, which is a fast SDE solver for the reverse
+    diffusion SDE. Currently we only support the first-order and second-order solvers. We recommend using the
+    second-order `sde-dpmsolver++`.
+
     [`~ConfigMixin`] takes care of storing all config attributes that are passed in the scheduler's `__init__`
     function, such as `num_train_timesteps`. They can be accessed via `scheduler.config.num_train_timesteps`.
     [`SchedulerMixin`] provides general loading and saving functionality via the [`SchedulerMixin.save_pretrained`] and
@@ -103,10 +108,10 @@ class DPMSolverMultistepScheduler(SchedulerMixin, ConfigMixin):
             the threshold value for dynamic thresholding. Valid only when `thresholding=True` and
             `algorithm_type="dpmsolver++`.
         algorithm_type (`str`, default `dpmsolver++`):
-            the algorithm type for the solver. Either `dpmsolver` or `dpmsolver++`. The `dpmsolver` type implements the
-            algorithms in https://arxiv.org/abs/2206.00927, and the `dpmsolver++` type implements the algorithms in
-            https://arxiv.org/abs/2211.01095. We recommend to use `dpmsolver++` with `solver_order=2` for guided
-            sampling (e.g. stable-diffusion).
+            the algorithm type for the solver. Either `dpmsolver` or `dpmsolver++` or `sde-dpmsolver` or
+            `sde-dpmsolver++`. The `dpmsolver` type implements the algorithms in https://arxiv.org/abs/2206.00927, and
+            the `dpmsolver++` type implements the algorithms in https://arxiv.org/abs/2211.01095. We recommend to use
+            `dpmsolver++` or `sde-dpmsolver++` with `solver_order=2` for guided sampling (e.g. stable-diffusion).
         solver_type (`str`, default `midpoint`):
             the solver type for the second-order solver. Either `midpoint` or `heun`. The solver type slightly affects
             the sample quality, especially for small number of steps. We empirically find that `midpoint` solvers are
@@ -180,7 +185,7 @@ class DPMSolverMultistepScheduler(SchedulerMixin, ConfigMixin):
         self.init_noise_sigma = 1.0
 
         # settings for DPM-Solver
-        if algorithm_type not in ["dpmsolver", "dpmsolver++"]:
+        if algorithm_type not in ["dpmsolver", "dpmsolver++", "sde-dpmsolver", "sde-dpmsolver++"]:
             if algorithm_type == "deis":
                 self.register_to_config(algorithm_type="dpmsolver++")
             else:
@@ -212,7 +217,7 @@ class DPMSolverMultistepScheduler(SchedulerMixin, ConfigMixin):
         """
         # Clipping the minimum of all lambda(t) for numerical stability.
         # This is critical for cosine (squaredcos_cap_v2) noise schedule.
-        clipped_idx = torch.searchsorted(torch.flip(self.lambda_t, [0]), self.lambda_min_clipped)
+        clipped_idx = torch.searchsorted(torch.flip(self.lambda_t, [0]), self.config.lambda_min_clipped)
         timesteps = (
             np.linspace(0, self.config.num_train_timesteps - 1 - clipped_idx, num_inference_steps + 1)
             .round()[::-1][:-1]
@@ -338,10 +343,10 @@ class DPMSolverMultistepScheduler(SchedulerMixin, ConfigMixin):
         """
 
         # DPM-Solver++ needs to solve an integral of the data prediction model.
-        if self.config.algorithm_type == "dpmsolver++":
+        if self.config.algorithm_type in ["dpmsolver++", "sde-dpmsolver++"]:
             if self.config.prediction_type == "epsilon":
                 # DPM-Solver and DPM-Solver++ only need the "mean" output.
-                if self.config.variance_type in ["learned_range"]:
+                if self.config.variance_type in ["learned", "learned_range"]:
                     model_output = model_output[:, :3]
                 alpha_t, sigma_t = self.alpha_t[timestep], self.sigma_t[timestep]
                 x0_pred = (sample - sigma_t * model_output) / alpha_t
@@ -360,33 +365,42 @@ class DPMSolverMultistepScheduler(SchedulerMixin, ConfigMixin):
                 x0_pred = self._threshold_sample(x0_pred)
 
             return x0_pred
+
         # DPM-Solver needs to solve an integral of the noise prediction model.
-        elif self.config.algorithm_type == "dpmsolver":
+        elif self.config.algorithm_type in ["dpmsolver", "sde-dpmsolver"]:
             if self.config.prediction_type == "epsilon":
                 # DPM-Solver and DPM-Solver++ only need the "mean" output.
-                if self.config.variance_type in ["learned_range"]:
-                    model_output = model_output[:, :3]
-                return model_output
+                if self.config.variance_type in ["learned", "learned_range"]:
+                    epsilon = model_output[:, :3]
+                else:
+                    epsilon = model_output
             elif self.config.prediction_type == "sample":
                 alpha_t, sigma_t = self.alpha_t[timestep], self.sigma_t[timestep]
                 epsilon = (sample - alpha_t * model_output) / sigma_t
-                return epsilon
             elif self.config.prediction_type == "v_prediction":
                 alpha_t, sigma_t = self.alpha_t[timestep], self.sigma_t[timestep]
                 epsilon = alpha_t * model_output + sigma_t * sample
-                return epsilon
             else:
                 raise ValueError(
                     f"prediction_type given as {self.config.prediction_type} must be one of `epsilon`, `sample`, or"
                     " `v_prediction` for the DPMSolverMultistepScheduler."
                 )
 
+            if self.config.thresholding:
+                alpha_t, sigma_t = self.alpha_t[timestep], self.sigma_t[timestep]
+                x0_pred = (sample - sigma_t * epsilon) / alpha_t
+                x0_pred = self._threshold_sample(x0_pred)
+                epsilon = (sample - alpha_t * x0_pred) / sigma_t
+
+            return epsilon
+
     def dpm_solver_first_order_update(
         self,
         model_output: torch.FloatTensor,
         timestep: int,
         prev_timestep: int,
         sample: torch.FloatTensor,
+        noise: Optional[torch.FloatTensor] = None,
     ) -> torch.FloatTensor:
         """
         One step for the first-order DPM-Solver (equivalent to DDIM).
@@ -411,6 +425,20 @@ class DPMSolverMultistepScheduler(SchedulerMixin, ConfigMixin):
             x_t = (sigma_t / sigma_s) * sample - (alpha_t * (torch.exp(-h) - 1.0)) * model_output
         elif self.config.algorithm_type == "dpmsolver":
             x_t = (alpha_t / alpha_s) * sample - (sigma_t * (torch.exp(h) - 1.0)) * model_output
+        elif self.config.algorithm_type == "sde-dpmsolver++":
+            assert noise is not None
+            x_t = (
+                (sigma_t / sigma_s * torch.exp(-h)) * sample
+                + (alpha_t * (1 - torch.exp(-2.0 * h))) * model_output
+                + sigma_t * torch.sqrt(1.0 - torch.exp(-2 * h)) * noise
+            )
+        elif self.config.algorithm_type == "sde-dpmsolver":
+            assert noise is not None
+            x_t = (
+                (alpha_t / alpha_s) * sample
+                - 2.0 * (sigma_t * (torch.exp(h) - 1.0)) * model_output
+                + sigma_t * torch.sqrt(torch.exp(2 * h) - 1.0) * noise
+            )
         return x_t
 
     def multistep_dpm_solver_second_order_update(
@@ -419,6 +447,7 @@ class DPMSolverMultistepScheduler(SchedulerMixin, ConfigMixin):
         timestep_list: List[int],
         prev_timestep: int,
         sample: torch.FloatTensor,
+        noise: Optional[torch.FloatTensor] = None,
     ) -> torch.FloatTensor:
         """
         One step for the second-order multistep DPM-Solver.
@@ -470,6 +499,38 @@ class DPMSolverMultistepScheduler(SchedulerMixin, ConfigMixin):
                     - (sigma_t * (torch.exp(h) - 1.0)) * D0
                     - (sigma_t * ((torch.exp(h) - 1.0) / h - 1.0)) * D1
                 )
+        elif self.config.algorithm_type == "sde-dpmsolver++":
+            assert noise is not None
+            if self.config.solver_type == "midpoint":
+                x_t = (
+                    (sigma_t / sigma_s0 * torch.exp(-h)) * sample
+                    + (alpha_t * (1 - torch.exp(-2.0 * h))) * D0
+                    + 0.5 * (alpha_t * (1 - torch.exp(-2.0 * h))) * D1
+                    + sigma_t * torch.sqrt(1.0 - torch.exp(-2 * h)) * noise
+                )
+            elif self.config.solver_type == "heun":
+                x_t = (
+                    (sigma_t / sigma_s0 * torch.exp(-h)) * sample
+                    + (alpha_t * (1 - torch.exp(-2.0 * h))) * D0
+                    + (alpha_t * ((1.0 - torch.exp(-2.0 * h)) / (-2.0 * h) + 1.0)) * D1
+                    + sigma_t * torch.sqrt(1.0 - torch.exp(-2 * h)) * noise
+                )
+        elif self.config.algorithm_type == "sde-dpmsolver":
+            assert noise is not None
+            if self.config.solver_type == "midpoint":
+                x_t = (
+                    (alpha_t / alpha_s0) * sample
+                    - 2.0 * (sigma_t * (torch.exp(h) - 1.0)) * D0
+                    - (sigma_t * (torch.exp(h) - 1.0)) * D1
+                    + sigma_t * torch.sqrt(torch.exp(2 * h) - 1.0) * noise
+                )
+            elif self.config.solver_type == "heun":
+                x_t = (
+                    (alpha_t / alpha_s0) * sample
+                    - 2.0 * (sigma_t * (torch.exp(h) - 1.0)) * D0
+                    - 2.0 * (sigma_t * ((torch.exp(h) - 1.0) / h - 1.0)) * D1
+                    + sigma_t * torch.sqrt(torch.exp(2 * h) - 1.0) * noise
+                )
         return x_t
 
     def multistep_dpm_solver_third_order_update(
@@ -532,6 +593,7 @@ class DPMSolverMultistepScheduler(SchedulerMixin, ConfigMixin):
         model_output: torch.FloatTensor,
         timestep: int,
         sample: torch.FloatTensor,
+        generator=None,
         return_dict: bool = True,
     ) -> Union[SchedulerOutput, Tuple]:
         """
@@ -574,12 +636,21 @@ class DPMSolverMultistepScheduler(SchedulerMixin, ConfigMixin):
             self.model_outputs[i] = self.model_outputs[i + 1]
         self.model_outputs[-1] = model_output
 
+        if self.config.algorithm_type in ["sde-dpmsolver", "sde-dpmsolver++"]:
+            noise = randn_tensor(
+                model_output.shape, generator=generator, device=model_output.device, dtype=model_output.dtype
+            )
+        else:
+            noise = None
+
         if self.config.solver_order == 1 or self.lower_order_nums < 1 or lower_order_final:
-            prev_sample = self.dpm_solver_first_order_update(model_output, timestep, prev_timestep, sample)
+            prev_sample = self.dpm_solver_first_order_update(
+                model_output, timestep, prev_timestep, sample, noise=noise
+            )
         elif self.config.solver_order == 2 or self.lower_order_nums < 2 or lower_order_second:
             timestep_list = [self.timesteps[step_index - 1], timestep]
             prev_sample = self.multistep_dpm_solver_second_order_update(
-                self.model_outputs, timestep_list, prev_timestep, sample
+                self.model_outputs, timestep_list, prev_timestep, sample, noise=noise
             )
         else:
             timestep_list = [self.timesteps[step_index - 2], self.timesteps[step_index - 1], timestep]

tests/schedulers/test_scheduler_dpm_multi.py

@@ -167,16 +167,20 @@ class DPMSolverMultistepSchedulerTest(SchedulerCommonTest):
             self.check_over_configs(prediction_type=prediction_type)
 
     def test_solver_order_and_type(self):
-        for algorithm_type in ["dpmsolver", "dpmsolver++"]:
+        for algorithm_type in ["dpmsolver", "dpmsolver++", "sde-dpmsolver", "sde-dpmsolver++"]:
             for solver_type in ["midpoint", "heun"]:
                 for order in [1, 2, 3]:
                     for prediction_type in ["epsilon", "sample"]:
-                        self.check_over_configs(
-                            solver_order=order,
-                            solver_type=solver_type,
-                            prediction_type=prediction_type,
-                            algorithm_type=algorithm_type,
-                        )
+                        if algorithm_type in ["sde-dpmsolver", "sde-dpmsolver++"]:
+                            if order == 3:
+                                continue
+                        else:
+                            self.check_over_configs(
+                                solver_order=order,
+                                solver_type=solver_type,
+                                prediction_type=prediction_type,
+                                algorithm_type=algorithm_type,
+                            )
                         sample = self.full_loop(
                             solver_order=order,
                             solver_type=solver_type,