Differientiable Sampling and Argmax

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WIP, last updated: 2019.12.6
Introduction
Softmax is a commonly used function for turning an unnormalized log probability into a normalized probability (or categorical distribution).
π=softmax(o)=eo∑jeoj,oj∈(−∞,+∞)\mathbf{\pi} = \text{softmax}(\mathbf{o}) = \frac{e^{\mathbf{o}}}{\sum_{j} e^{o_j}},\\o_j \in (-\infty, +\infty)π=softmax(o)=∑j​eoj​eo​,oj​∈(−∞,+∞)
Say o\mathbf{o}o
 is the output of a neural network before softmax, we call o\mathbf{o}o
 the unnormalized log probability.
After softmax, we usually sample from this categorical distribution, or taking an argmax function to select the index. However, one can notice that neither the sampling nor the argmax is differientiable.
Researchers have proposed several works to make this possible. I am going to discuss them here.
Sampling
I will introduce Gumbel Softmax [1611.01144], which have made the sampling procedure differentiable.
Gumbel Max
First, we need to introduce Gumbel Max. In short, Gumbel Max is a trick to use gumbel distribution to sample a categorical distribution.
Say we want to sample from a categorical distribution π\mathbf{\pi}π
.
The usual way of doing this is using π\piπ
 to separate [0,1][0, 1][0,1]
 into intervals, sampling from a uniform distribution U∼[0,1]\text{U} \sim[0, 1]U∼[0,1]
, and see where it locates.
The Gumbel Max trick provides an alternative way of doing this. It use Reparameterization Trick to avoid the stochastic node during backpropagation.
y=arg⁡max⁡i(oi+gi)y = \arg \max_{i} (o_i +g_i)y=argimax​(oi​+gi​)
where gi∼Gumbel(0,1)g_i \sim \text{Gumbel}(0, 1)gi​∼Gumbel(0,1)
, which can be sampled by −log⁡(−log⁡(Uniform[0,1]))-\log(-\log(\text{Uniform}[0, 1]))−log(−log(Uniform[0,1]))
.
We can prove that yyy
 is distributed according to π\mathbf{\pi}π
.
Prove that
​y=arg⁡max⁡i(oi+gi)y = \arg \max_{i} (o_i +g_i)y=argmaxi​(oi​+gi​)
, where gi∼Gumbel(0,1)g_i \sim \text{Gumbel}(0, 1)gi​∼Gumbel(0,1)

which can be sampled by −log⁡(−log⁡(Uniform[0,1]))-\log(-\log(\text{Uniform}[0, 1]))−log(−log(Uniform[0,1]))

is distributed with π=softmax(oi)=eoi∑jeoj\pi = \text{softmax}(o_i) = \frac{e^{o_i}}{\sum{j} e^{o_j}}π=softmax(oi​)=∑jeoj​eoi​​
​
Prerequisites
Gumbel Distribution (param by location ****μ\muμ
, and scale β>0\beta>0β>0
) (wikipedia)
CDF: F(x;μ,β)=e−e(x−μ)/βF(x; \mu, \beta) = e^{-e^{(x-\mu)/\beta}}F(x;μ,β)=e−e(x−μ)/β

PDF: f(x;μ,β)=1βe−(z+e−z),z=x−μβf(x; \mu, \beta) = \frac{1}{\beta} e^{-(z+e^{-z})}, z = \frac{x-\mu}{\beta}f(x;μ,β)=β1​e−(z+e−z),z=βx−μ​

Mean: E(X)=μ+γβ,γ≈0.5772\text{E}(X) = \mu+\gamma\beta, \gamma \approx 0.5772E(X)=μ+γβ,γ≈0.5772
is the Euler–Mascheroni constant.
Quantile Function: Q(p)=μ−βlog⁡(−log⁡(p))Q(p) = \mu-\beta \log(-\log(p))Q(p)=μ−βlog(−log(p))
(Quantile Function is used to sample random variables from a distribution given CDF, it is also called inverse CDF)
Proof
We actually want to prove that Gumbel(μ=oi,β=1)\text{Gumbel}(\mu=o_i, \beta=1)Gumbel(μ=oi​,β=1)

is distributed with πi=eoi∑jeoj\pi_i = \frac{e^{o_i}}{\sum_{j} e^{o_j}}πi​=∑j​eoj​eoi​​
.
We can find that Gumbel(μ=oi,β=1)\text{Gumbel}(\mu=o_i, \beta=1)Gumbel(μ=oi​,β=1)
 has the following PDF and CDF
f(x;μ,1)=e−(x−μ)–e−(x−μ)F(x;μ,1)=e−e−(x−μ)\begin{align} f(x; \mu, 1) &= e^{-(x-\mu) – e^{-(x-\mu)}}\\ F(x; \mu, 1) &= e^{-e^{-(x-\mu)}} \end{align}f(x;μ,1)F(x;μ,1)​=e−(x−μ)–e−(x−μ)=e−e−(x−μ)​​
.Then, the probability that all other πj≠i\pi_{j \neq i}πj=i​
 are less than πi\pi_iπi​
 is:
Pr⁡(πi is the largest∣πi,{oj})=∏j≠ie−e−(πi−oj)\Pr(\pi_i ~\text{is the largest} | \pi_i, \{o_{j}\}) = \prod_{j \neq i} e^{-e^{-(\pi_i - o_j)}}Pr(πi​ is the largest∣πi​,{oj​})=j=i∏​e−e−(πi​−oj​)
We know the marginal distribution over πi\pi_iπi​
 and we are able to integrate it out to find the overall probability: (p(x)=∫yp(x,y)dy=∫yp(x∣y)p(y)dyp(x) = \int_y p(x,y) dy = \int_y p(x|y) p(y) dyp(x)=∫y​p(x,y)dy=∫y​p(x∣y)p(y)dy
)
Pr⁡(i is largest∣{oj})=∫e−(πi−oi)−e−(πi−oi)×∏j≠ie−e−(πi−oj)dπi=∫e−πi+oi−e−πi∑jeojdπi=eoi∑jeoj\begin{align} \Pr(\text{$i$ is largest}|\{o_{j}\}) &= \int e^{-(\pi_i-o_i)-e^{-(\pi_i-o_i)}} \times \prod_{j\neq i}e^{-e^{-(\pi_i-o_j)}} \mathrm{d}\pi_i \\ &=\int e^{-\pi_i + o_i -e^{-\pi_i} \sum_{j} e^{o_j}}\mathrm{d}\pi_i \\ &=\frac{e^{o_i}}{\sum_{j}e^{o_j}} \end{align}Pr(i is largest∣{oj​})​=∫e−(πi​−oi​)−e−(πi​−oi​)×j=i∏​e−e−(πi​−oj​)dπi​=∫e−πi​+oi​−e−πi​∑j​eoj​dπi​=∑j​eoj​eoi​​​​
which is exactly a softmax probablity. QED.
Reference: https://lips.cs.princeton.edu/the-gumbel-max-trick-for-discrete-distributions/****
Gumbel Softmax
Notice that there is still an argmax in Gumbel Max, which still makes it indifferentiable. Therefore, we use a softmax function to approximate this argmax procedure.
y=e(oi+gi)/τ∑je(oj+gj)/τ\mathbf{y} = \frac{e^{(o_i+g_i) / \tau}}{\sum_{j}e^{(o_j+g_j) / \tau}}y=∑j​e(oj​+gj​)/τe(oi​+gi​)/τ​
where τ∈(0,∞)\tau \in (0, \infty)τ∈(0,∞)
 is a temparature hyperparameter.
We note that the output of Gumbel Softmax function here is a vector which sum to 1, which somewhat looks like a one-hot vector (but it's not).
So by far, this does not actually replace the argmax function.
To actually get a pure one-hot vector, we need to use a Straight-Through (ST) Gumbel Trick.
Let's directly see an implementation of Gumbel Softmax in PyTorch
(We use the hard mode, soft mode does not get a pure one-hot vector).
def gumbel_softmax(logits, tau=1, hard=False, eps=1e-10, dim=-1):
    # type: (Tensor, float, bool, float, int) -> Tensor
    r"""
    Samples from the Gumbel-Softmax distribution (`Link 1`_  `Link 2`_) and optionally discretizes.
​
    Args:
      logits: `[..., num_features]` unnormalized log probabilities
      tau: non-negative scalar temperature
      hard: if ``True``, the returned samples will be discretized as one-hot vectors,
            but will be differentiated as if it is the soft sample in autograd
      dim (int): A dimension along which softmax will be computed. Default: -1.
​
    Returns:
      Sampled tensor of same shape as `logits` from the Gumbel-Softmax distribution.
      If ``hard=True``, the returned samples will be one-hot, otherwise they will
      be probability distributions that sum to 1 across `dim`.
​
    .. note::
      This function is here for legacy reasons, may be removed from nn.Functional in the future.
​
    .. note::
      The main trick for `hard` is to do  `y_hard - y_soft.detach() + y_soft`
​
      It achieves two things:
      - makes the output value exactly one-hot
      (since we add then subtract y_soft value)
      - makes the gradient equal to y_soft gradient
      (since we strip all other gradients)
​
    Examples::
        >>> logits = torch.randn(20, 32)
        >>> # Sample soft categorical using reparametrization trick:
        >>> F.gumbel_softmax(logits, tau=1, hard=False)
        >>> # Sample hard categorical using "Straight-through" trick:
        >>> F.gumbel_softmax(logits, tau=1, hard=True)
​
    .. _Link 1:
        https://arxiv.org/abs/1611.00712
    .. _Link 2:
        https://arxiv.org/abs/1611.01144
    """
    if eps != 1e-10:
        warnings.warn("`eps` parameter is deprecated and has no effect.")
​
    gumbels = -torch.empty_like(logits).exponential_().log()  # ~Gumbel(0,1)
    gumbels = (logits + gumbels) / tau  # ~Gumbel(logits,tau)
    y_soft = gumbels.softmax(dim)
​
    if hard:
        # Straight through.
        index = y_soft.max(dim, keepdim=True)[1]
        y_hard = torch.zeros_like(logits).scatter_(dim, index, 1.0)
        ret = y_hard - y_soft.detach() + y_soft
    else:
        # Reparametrization trick.
        ret = y_soft
    return ret
When fowarding, the code use an argmax to get an actual one-hot vector.
And it uses ret = y_hard - y_soft.detach() + y_soft, y_hard has no grad, and by minusing y_soft.detach() and adding y_soft, it achieves a grad from y_soft without modifying the forwarding value.
So eventually, we are able to get a pure one-hot vector in forward pass, and a grad when back propagating, which makes the sampling procedure differientiable.
Finally, let's look at how τ\tauτ
affects the sampling procedure. The below image shows the sampling distribution (which is also called the Concrete Distribution [1611.00712]) and one random sample instance when using different hyperparameter τ\tauτ
.
image from https://arxiv.org/abs/1611.01144
when τ→0\tau \rightarrow 0τ→0
, the softmax becomes an argmax and the Gumbel-Softmax distribution becomes the categorical distribution. During training, we let τ>0\tau > 0τ>0
 to allow gradients past the sample, then gradually anneal the temperature τ\tauτ
 (but not completely to 0, as the gradients would blow up).
from Eric Jang. https://blog.evjang.com/2016/11/tutorial-categorical-variational.html​
Argmax
How to make argmax differentiable?
Intuitively, the Straight-Through Trick is also applicable for softmax+argmax (or softargmax + argmax).
I am still not sure, needs more digging in the literature.
Some have introduced the soft-argmax function. It doesn't actually makes it differentiable, but use a continuous function to approximate the softmax+argmax procedure.
π=soft-argmax(o)=eβo∑jeβoj\mathbf{\pi} = \text{soft-argmax}(\mathbf{o}) = \frac{e^{\mathbf{\beta o}}}{\sum_{j} e^{\beta o_j}}π=soft-argmax(o)=∑j​eβoj​eβo​
where β\betaβ
 can be a large value to make π\mathbf{\pi}π
 very much "look like" a one-hot vector.
Discussion
Goal
softmax + argmax is used for classification, we only want the index with the highest probability.
gumbel softmax + argmax is used for sampling, we may want to sample an index not with the highest probability.
Deterministic
softmax + argmax is deterministic. Get the index with the highest probablity.
gumbel softmax + argmax is stochastic. We need to sample from a gumbel distribution in the beginning.
Output vector
softmax and gumbel softmax aboth output a vector sum to 1.
softmax outputs a normalized probability distribution.
gumbel softmax outputs a sample somewhat more similar to a one-hot vector.(can be controlled by τ\tauτ
)
Straight-Through Trick can actually be applied to both softmax + argmax and gumbel softmax + argmax, which can make both of them differentiable. (?)
Reference
Gumbel Softmax [1611.01144]​
Concrete Distribution (Gumbel Softmax Distribution) [1611.00712]​
Eric Jang official blog: https://blog.evjang.com/2016/11/tutorial-categorical-variational.html​
PyTorch Implementation of Gumbel Softmax: https://pytorch.org/docs/stable/nn.functional.html#torch.nn.functional.gumbel_softmax​
​https://timvieira.github.io/blog/post/2014/07/31/gumbel-max-trick/​
​https://lips.cs.princeton.edu/the-gumbel-max-trick-for-discrete-distributions/​
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Introduction
Sampling
Gumbel Max
Gumbel Softmax
Argmax
Discussion
Reference