pytorch pyro 贝叶斯神经网络 bnn beyesean neure network svi 定制SVI目标和培训循环，变更推理

定制SVI目标和培训循环¶

Pyro支持各种基于优化的贝叶斯推理方法，包括Trace_ELBO作为SVI(随机变分推理)的基本实现。参见文件（documents的简写）有关各种SVI实现和SVI教程的更多信息I, 二，以及罗马数字3了解SVI的背景。

在本教程中，我们将展示高级用户如何修改和/或增加变分目标(或者:损失函数)以及由Pyro提供的训练步骤实现，以支持特殊的用例。

1. 基本SVI用法

   1. 较低层次的模式

2. 示例:自定义正则化

3. 示例:调整损失

4. 例如:贝塔VAE

5. 示例:混合优化器

6. 示例:自定义ELBO

7. 示例:KL退火

基本SVI用法¶

我们首先回顾一下SVI烟火中的物体。我们假设用户已经定义了一个model和一个guide。然后，用户创建一个优化器和一个SVI对象:

optimizer = pyro.optim.Adam({"lr": 0.001, "betas": (0.90, 0.999)})
svi = pyro.infer.SVI(model, guide, optimizer, loss=pyro.infer.Trace_ELBO())

然后可以通过调用svi.step(...)。对...的争论step()然后被传递给model和guide.

较低层次的模式¶

上述模式的好处在于，它允许Pyro为我们处理各种细节，例如:

• pyro.optim.Adam动态创建一个新的torch.optim.Adam每当遇到新参数时优化器

• SVI.step()渐变步骤之间的零渐变

如果我们想要更多的控制，我们可以直接操纵各种可微损失方法ELBO班级。例如，这个优化循环:

svi = pyro.infer.SVI(model, guide, optimizer, loss=pyro.infer.Trace_ELBO())
for i in range(n_iter):
    loss = svi.step(X_train, y_train)

相当于这个低级模式:

loss_fn = lambda model, guide: pyro.infer.Trace_ELBO().differentiable_loss(model, guide, X_train, y_train)
with pyro.poutine.trace(param_only=True) as param_capture:
    loss = loss_fn(model, guide)
params = set(site["value"].unconstrained()
                for site in param_capture.trace.nodes.values())
optimizer = torch.optim.Adam(params, lr=0.001, betas=(0.90, 0.999))
for i in range(n_iter):
    # compute loss
    loss = loss_fn(model, guide)
    loss.backward()
    # take a step and zero the parameter gradients
    optimizer.step()
    optimizer.zero_grad()

示例:自定义正则化¶

假设我们想在SVI损失中加入一个定制的正则项。使用上面的使用模式，这很容易做到。首先我们定义正则项:

def my_custom_L2_regularizer(my_parameters):
    reg_loss = 0.0
    for param in my_parameters:
        reg_loss = reg_loss + param.pow(2.0).sum()
    return reg_loss

那么我们唯一需要做的改变就是:

- loss = loss_fn(model, guide)
+ loss = loss_fn(model, guide) + my_custom_L2_regularizer(my_parameters)

示例:剪裁渐变¶

对于一些模型，损耗梯度可能在训练期间爆炸，导致溢出和NaN价值观。防止这种情况的一种方法是使用渐变剪辑。中的优化器pyro.optim拿一本可选的字典clip_args这允许将梯度范数或梯度值剪切到给定的界限内。

要更改上面的基本示例:

- optimizer = pyro.optim.Adam({"lr": 0.001, "betas": (0.90, 0.999)})
+ optimizer = pyro.optim.Adam({"lr": 0.001, "betas": (0.90, 0.999)}, {"clip_norm": 10.0})

还可以通过修改上述低级模式来手动实现梯度裁剪的其他变体。

示例:调整损失¶

根据优化算法，损失的规模可能重要，也可能无关紧要。假设我们想在对损失函数进行微分之前，根据数据点的数量对其进行缩放。这很容易做到:

- loss = loss_fn(model, guide)
+ loss = loss_fn(model, guide) / N_data

请注意，在SVI的情况下，损失函数中的每一项都是来自模型或指南的对数概率，同样的效果可以通过使用poutine.scale。例如，我们可以使用poutine.scale装饰器缩放模型和指南:

@poutine.scale(scale=1.0/N_data)
def model(...):
    pass

@poutine.scale(scale=1.0/N_data)
def guide(...):
    pass

例如:贝塔VAE¶

我们也可以使用poutine.scale以构建非标准的ELBO变分目标，其中，例如，KL散度相对于期望的对数似然性被不同地缩放。特别是对于βVAE，KL散度用一个因子来缩放beta:

def model(data, beta=0.5):
    z_loc, z_scale = ...
    with pyro.poutine.scale(scale=beta)
        z = pyro.sample("z", dist.Normal(z_loc, z_scale))
    pyro.sample("obs", dist.Bernoulli(...), obs=data)

def guide(data, beta=0.5):
    with pyro.poutine.scale(scale=beta)
        z_loc, z_scale = ...
        z = pyro.sample("z", dist.Normal(z_loc, z_scale))

有了这种模型的选择，并引导对应于潜在变量的测井密度z来构建变分目标

svi = pyro.infer.SVI(model, guide, optimizer, loss=pyro.infer.Trace_ELBO())

将被缩放一倍beta，导致KL发散，其同样由beta.

示例:混合优化器¶

中的各种优化器pyro.optim允许用户在每个参数的基础上指定优化设置(如学习率)。但是如果我们要对不同的参数使用不同的优化算法呢？我们可以使用Pyro的MultiOptimizer(见下文)，但我们也可以实现同样的事情，如果我们直接操纵differentiable_loss:

adam = torch.optim.Adam(adam_parameters, {"lr": 0.001, "betas": (0.90, 0.999)})
sgd = torch.optim.SGD(sgd_parameters, {"lr": 0.0001})
loss_fn = pyro.infer.Trace_ELBO().differentiable_loss
# compute loss
loss = loss_fn(model, guide)
loss.backward()
# take a step and zero the parameter gradients
adam.step()
sgd.step()
adam.zero_grad()
sgd.zero_grad()

为了完整起见，我们还展示了如何使用多重优化器，这使我们能够结合多个烟火优化。请注意，由于MultiOptimizer使用torch.autograd.grad引擎盖下(而不是torch.Tensor.backward())，它的界面略有不同；特别是step()方法也将参数作为输入。

def model():
    pyro.param('a', ...)
    pyro.param('b', ...)
    ...

adam = pyro.optim.Adam({'lr': 0.1})
sgd = pyro.optim.SGD({'lr': 0.01})
optim = MixedMultiOptimizer([(['a'], adam), (['b'], sgd)])
with pyro.poutine.trace(param_only=True) as param_capture:
    loss = elbo.differentiable_loss(model, guide)
params = {'a': pyro.param('a'), 'b': pyro.param('b')}
optim.step(loss, params)

示例:自定义ELBO¶

在前三个例子中，我们绕过了创建SVI对象提供的可微分损失函数ELBO实施。我们可以做的另一件事是创造定制ELBO实现，并将它们传递到SVI机械。例如，简化版本的Trace_ELBO损失函数可能如下所示:

# note that simple_elbo takes a model, a guide, and their respective arguments as inputs
def simple_elbo(model, guide, *args, **kwargs):
    # run the guide and trace its execution
    guide_trace = poutine.trace(guide).get_trace(*args, **kwargs)
    # run the model and replay it against the samples from the guide
    model_trace = poutine.trace(
        poutine.replay(model, trace=guide_trace)).get_trace(*args, **kwargs)
    # construct the elbo loss function
    return -1*(model_trace.log_prob_sum() - guide_trace.log_prob_sum())

svi = SVI(model, guide, optim, loss=simple_elbo)

请注意，这基本上就是elbo实施于"迷你烟火"看起来像。

示例:KL退火¶

在深度学习中，KL退火是一种在训练过程中逐渐减少KL散度损失权重的技术，这有助于模型更好地学习数据分布。这种方法在处理深度马尔可夫模型时尤其有用，因为它允许模型在训练初期不过分关注潜在变量的细节，而是更多地关注数据的整体结构。随着训练的进行，逐渐增加KL散度的权重，使得模型能够细化其对潜在变量的推断。

在实现KL退火时，可以通过定义一个定制的损失函数来实现，或者使用像poutine.scale这样的工具来动态调整损失函数中的KL散度项的权重。例如，在Pyro这个概率编程库中，可以使用poutine.scale装饰器来按比例缩小KL散度项，从而实现退火效果。这种方法提供了一种灵活的方式来控制模型在训练过程中对潜在变量分布的推断强度。

在实际应用中，KL退火可以帮助模型避免过早地陷入局部最优解，并且有助于提高模型对数据的泛化能力。通过逐渐调整损失函数中各项的权重，模型可以在训练的不同阶段关注不同的学习目标，最终达到更好的学习效果。

此外，KL退火也可以视为一种正则化技术，它通过控制模型复杂度来防止过拟合。在训练初期，较小的KL散度权重允许模型在不完全匹配潜在变量分布的情况下进行学习，这有助于模型探索更广泛的参数空间。随着训练的进行，逐渐增大的KL散度权重则迫使模型更加精确地学习数据分布，从而提高模型的预测准确性。

总的来说，KL退火是一种有效的技术，可以在深度学习模型的训练中发挥作用，特别是在处理复杂的数据分布和结构时。通过合理地设计退火策略，可以显著提高模型的性能和泛化能力。

爬山法是完完全全的贪心法，每次都鼠目寸光的选择一个当前最优解，因此只能搜索到局部的最优值。模拟退火其实也是一种贪心算法，但是它的搜索过程引入了随机因素。模拟退火算法以一定的概率 来接受一个比当前解要差的解，因此有可能 会跳出这个局部的最优解，达到全局的最优解。以图1为例，模拟退火算法在搜索到局部最优解A后，会以一定的概率接受到E的移动。也许经过几次这样的不是局部最优的移动后会到达D点，于是就跳出了局部最大值A。

模拟退火算法描述：

若J( Y(i+1) )>= J( Y(i) ) (即移动后得到更优解)，则总是接受该移动

若J( Y(i+1) )< J( Y(i) ) (即移动后的解比当前解要差)，则以一定的概率接受移动，而且这个概率随着时间推移逐渐降低（逐渐降低才能趋向稳定）

这里的"一定的概率"的计算参考了金属冶炼的退火过程，这也是模拟退火算法名称的由来。

根据热力学的原理，在温度为T时，出现能量差为dE的降温的概率为P(dE)，表示为：

**　　　　P(dE) = exp( dE/(kT) )**

其中k是一个常数，exp表示自然指数，且dE<0。这条公式说白了就是：温度越高，出现一次能量差为dE的降温的概率就越大；温度越低，则出现降温的概率就越小。又由于dE总是小于0（否则就不叫退火了），因此dE/kT < 0 ，所以P(dE)的函数取值范围是(0,1) 。

随着温度T的降低，P(dE)会逐渐降低。

我们将一次向较差解的移动看做一次温度跳变过程，我们以概率P(dE)来接受这样的移动。

关于爬山算法与模拟退火，有一个有趣的比喻：

爬山算法：兔子朝着比现在高的地方跳去。它找到了不远处的最高山峰。但是这座山不一定是珠穆朗玛峰。这就是爬山算法，它不能保证局部最优值就是全局最优值。

模拟退火：兔子喝醉了。它随机地跳了很长时间。这期间，它可能走向高处，也可能踏入平地。但是，它渐渐清醒了并朝最高方向跳去。这就是模拟退火

(http://www.cnblogs.com/heaad/ 转载请注明)

在......里深度马尔可夫模型教程ELBO变分目标在训练期间被修改。特别地，潜在随机变量之间的各种KL-散度项相对于观察数据的对数概率按比例缩小(即退火)。在本教程中，这是通过使用poutine.scale。我们可以通过定义一个定制的损失函数来完成同样的事情。后一种选择并不是一种非常优雅的模式，但是我们还是包含了它，以显示我们所拥有的灵活性。

def simple_elbo_kl_annealing(model, guide, *args, **kwargs):
    # get the annealing factor and latents to anneal from the keyword
    # arguments passed to the model and guide
    annealing_factor = kwargs.pop('annealing_factor', 1.0)
    latents_to_anneal = kwargs.pop('latents_to_anneal', [])
    # run the guide and replay the model against the guide
    guide_trace = poutine.trace(guide).get_trace(*args, **kwargs)
    model_trace = poutine.trace(
        poutine.replay(model, trace=guide_trace)).get_trace(*args, **kwargs)

    elbo = 0.0
    # loop through all the sample sites in the model and guide trace and
    # construct the loss; note that we scale all the log probabilities of
    # samples sites in `latents_to_anneal` by the factor `annealing_factor`
    for site in model_trace.values():
        if site["type"] == "sample":
            factor = annealing_factor if site["name"] in latents_to_anneal else 1.0
            elbo = elbo + factor * site["fn"].log_prob(site["value"]).sum()
    for site in guide_trace.values():
        if site["type"] == "sample":
            factor = annealing_factor if site["name"] in latents_to_anneal else 1.0
            elbo = elbo - factor * site["fn"].log_prob(site["value"]).sum()
    return -elbo

svi = SVI(model, guide, optim, loss=simple_elbo_kl_annealing)
svi.step(other_args, annealing_factor=0.2, latents_to_anneal=["my_latent"])

以前的然后

Customizing SVI objectives and training loops¶

Pyro provides support for various optimization-based approaches to Bayesian inference, with Trace_ELBO serving as the basic implementation of SVI (stochastic variational inference). See the docs for more information on the various SVI implementations and SVI tutorials I, II, and III for background on SVI.

In this tutorial we show how advanced users can modify and/or augment the variational objectives (alternatively: loss functions) and the training step implementation provided by Pyro to support special use cases.

1. Basic SVI Usage

   1. A Lower Level Pattern

2. Example: Custom Regularizer

3. Example: Scaling the Loss

4. Example: Beta VAE

5. Example: Mixing Optimizers

6. Example: Custom ELBO

7. Example: KL Annealing

Basic SVI Usage¶

We first review the basic usage pattern of SVI objects in Pyro. We assume that the user has defined a model and a guide. The user then creates an optimizer and an SVI object:

optimizer = pyro.optim.Adam({"lr": 0.001, "betas": (0.90, 0.999)})
svi = pyro.infer.SVI(model, guide, optimizer, loss=pyro.infer.Trace_ELBO())

Gradient steps can then be taken with a call to svi.step(...). The arguments to step() are then passed to model and guide.

A Lower-Level Pattern¶

The nice thing about the above pattern is that it allows Pyro to take care of various details for us, for example:

• pyro.optim.Adam dynamically creates a new torch.optim.Adam optimizer whenever a new parameter is encountered

• SVI.step() zeros gradients between gradient steps

If we want more control, we can directly manipulate the differentiable loss method of the various ELBO classes. For example, this optimization loop:

svi = pyro.infer.SVI(model, guide, optimizer, loss=pyro.infer.Trace_ELBO())
for i in range(n_iter):
    loss = svi.step(X_train, y_train)

is equivalent to this low-level pattern:

loss_fn = lambda model, guide: pyro.infer.Trace_ELBO().differentiable_loss(model, guide, X_train, y_train)
with pyro.poutine.trace(param_only=True) as param_capture:
    loss = loss_fn(model, guide)
params = set(site["value"].unconstrained()
                for site in param_capture.trace.nodes.values())
optimizer = torch.optim.Adam(params, lr=0.001, betas=(0.90, 0.999))
for i in range(n_iter):
    # compute loss
    loss = loss_fn(model, guide)
    loss.backward()
    # take a step and zero the parameter gradients
    optimizer.step()
    optimizer.zero_grad()

Example: Custom Regularizer¶

Suppose we want to add a custom regularization term to the SVI loss. Using the above usage pattern, this is easy to do. First we define our regularizer:

def my_custom_L2_regularizer(my_parameters):
    reg_loss = 0.0
    for param in my_parameters:
        reg_loss = reg_loss + param.pow(2.0).sum()
    return reg_loss

Then the only change we need to make is:

- loss = loss_fn(model, guide)
+ loss = loss_fn(model, guide) + my_custom_L2_regularizer(my_parameters)

Example: Clipping Gradients¶

For some models the loss gradient can explode during training, leading to overflow and NaN values. One way to protect against this is with gradient clipping. The optimizers in pyro.optim take an optional dictionary of clip_args which allows clipping either the gradient norm or the gradient value to fall within the given limit.

To change the basic example above:

- optimizer = pyro.optim.Adam({"lr": 0.001, "betas": (0.90, 0.999)})
+ optimizer = pyro.optim.Adam({"lr": 0.001, "betas": (0.90, 0.999)}, {"clip_norm": 10.0})

Further variants of gradient clipping can also be implemented manually by modifying the low-level pattern described above.

Example: Scaling the Loss¶

Depending on the optimization algorithm, the scale of the loss may or not matter. Suppose we want to scale our loss function by the number of datapoints before we differentiate it. This is easily done:

- loss = loss_fn(model, guide)
+ loss = loss_fn(model, guide) / N_data

Note that in the case of SVI, where each term in the loss function is a log probability from the model or guide, this same effect can be achieved using poutine.scale. For example we can use the poutine.scale decorator to scale both the model and guide:

@poutine.scale(scale=1.0/N_data)
def model(...):
    pass

@poutine.scale(scale=1.0/N_data)
def guide(...):
    pass

Example: Beta VAE¶

We can also use poutine.scale to construct non-standard ELBO variational objectives in which, for example, the KL divergence is scaled differently relative to the expected log likelihood. In particular for the Beta VAE the KL divergence is scaled by a factor beta:

def model(data, beta=0.5):
    z_loc, z_scale = ...
    with pyro.poutine.scale(scale=beta)
        z = pyro.sample("z", dist.Normal(z_loc, z_scale))
    pyro.sample("obs", dist.Bernoulli(...), obs=data)

def guide(data, beta=0.5):
    with pyro.poutine.scale(scale=beta)
        z_loc, z_scale = ...
        z = pyro.sample("z", dist.Normal(z_loc, z_scale))

With this choice of model and guide the log densities corresponding to the latent variable z that enter into constructing the variational objective via

svi = pyro.infer.SVI(model, guide, optimizer, loss=pyro.infer.Trace_ELBO())

will be scaled by a factor of beta, resulting in a KL divergence that is likewise scaled by beta.

Example: Mixing Optimizers¶

The various optimizers in pyro.optim allow the user to specify optimization settings (e.g. learning rates) on a per-parameter basis. But what if we want to use different optimization algorithms for different parameters? We can do this using Pyro's MultiOptimizer (see below), but we can also achieve the same thing if we directly manipulate differentiable_loss:

adam = torch.optim.Adam(adam_parameters, {"lr": 0.001, "betas": (0.90, 0.999)})
sgd = torch.optim.SGD(sgd_parameters, {"lr": 0.0001})
loss_fn = pyro.infer.Trace_ELBO().differentiable_loss
# compute loss
loss = loss_fn(model, guide)
loss.backward()
# take a step and zero the parameter gradients
adam.step()
sgd.step()
adam.zero_grad()
sgd.zero_grad()

For completeness, we also show how we can do the same thing using MultiOptimizer, which allows us to combine multiple Pyro optimizers. Note that since MultiOptimizer uses torch.autograd.grad under the hood (instead of torch.Tensor.backward()), it has a slightly different interface; in particular the step() method also takes parameters as inputs.

def model():
    pyro.param('a', ...)
    pyro.param('b', ...)
    ...

adam = pyro.optim.Adam({'lr': 0.1})
sgd = pyro.optim.SGD({'lr': 0.01})
optim = MixedMultiOptimizer([(['a'], adam), (['b'], sgd)])
with pyro.poutine.trace(param_only=True) as param_capture:
    loss = elbo.differentiable_loss(model, guide)
params = {'a': pyro.param('a'), 'b': pyro.param('b')}
optim.step(loss, params)

Example: Custom ELBO¶

In the previous three examples we bypassed creating a SVI object and directly manipulated the differentiable loss function provided by an ELBO implementation. Another thing we can do is create custom ELBO implementations and pass those into the SVI machinery. For example, a simplified version of a Trace_ELBO loss function might look as follows:

# note that simple_elbo takes a model, a guide, and their respective arguments as inputs
def simple_elbo(model, guide, *args, **kwargs):
    # run the guide and trace its execution
    guide_trace = poutine.trace(guide).get_trace(*args, **kwargs)
    # run the model and replay it against the samples from the guide
    model_trace = poutine.trace(
        poutine.replay(model, trace=guide_trace)).get_trace(*args, **kwargs)
    # construct the elbo loss function
    return -1*(model_trace.log_prob_sum() - guide_trace.log_prob_sum())

svi = SVI(model, guide, optim, loss=simple_elbo)

Note that this is basically what the elbo implementation in "mini-pyro" looks like.

Example: KL Annealing¶

In the Deep Markov Model Tutorial the ELBO variational objective is modified during training. In particular the various KL-divergence terms between latent random variables are scaled downward (i.e. annealed) relative to the log probabilities of the observed data. In the tutorial this is accomplished using poutine.scale. We can accomplish the same thing by defining a custom loss function. This latter option is not a very elegant pattern but we include it anyway to show the flexibility we have at our disposal.

def simple_elbo_kl_annealing(model, guide, *args, **kwargs):
    # get the annealing factor and latents to anneal from the keyword
    # arguments passed to the model and guide
    annealing_factor = kwargs.pop('annealing_factor', 1.0)
    latents_to_anneal = kwargs.pop('latents_to_anneal', [])
    # run the guide and replay the model against the guide
    guide_trace = poutine.trace(guide).get_trace(*args, **kwargs)
    model_trace = poutine.trace(
        poutine.replay(model, trace=guide_trace)).get_trace(*args, **kwargs)

    elbo = 0.0
    # loop through all the sample sites in the model and guide trace and
    # construct the loss; note that we scale all the log probabilities of
    # samples sites in `latents_to_anneal` by the factor `annealing_factor`
    for site in model_trace.values():
        if site["type"] == "sample":
            factor = annealing_factor if site["name"] in latents_to_anneal else 1.0
            elbo = elbo + factor * site["fn"].log_prob(site["value"]).sum()
    for site in guide_trace.values():
        if site["type"] == "sample":
            factor = annealing_factor if site["name"] in latents_to_anneal else 1.0
            elbo = elbo - factor * site["fn"].log_prob(site["value"]).sum()
    return -elbo

svi = SVI(model, guide, optim, loss=simple_elbo_kl_annealing)
svi.step(other_args, annealing_factor=0.2, latents_to_anneal=["my_latent"])