I'm trying to train a neural network for classification, but the labels I have are rather noisy (around 30% of the labels are wrong).

The cross-entropy loss indeed works, but I was wondering are there any alternatives more effective in this case? or is cross-entropy loss the optimal?

I'm not sure but I'm thinking of somewhat "clipping" the cross-entropy loss, such that the loss for one data point will be no greater than some upper bound, will that work?

Thanks!

**Update**  
According to Lucas' answer, I got the following for the derivatives for the prediction output $y$ and input of the softmax function $z$. So I guess essentially it is adding a smoothing term $\frac{3}{7N}$ to the derivatives.  
$$p_i=0.3/N+0.7y_i$$
$$l=-\sum t_i\log(p_i)$$
$$\frac{\partial l}{\partial y_i}=-t_i\frac{\partial\log(p_i)}{\partial p_i}\frac{\partial p_i}{\partial y_i}=-0.7\frac{t_i}{p_i}=-\frac{t_i}{\frac{3}{7N}+y_i}$$
$$\frac{\partial l}{\partial z_i}=0.7\sum_j\frac{t_j}{p_j}\frac{\partial y_j}{\partial z_i}=y_i\sum_jt_j\frac{y_j}{\frac{3}{7N}+y_j}-t_i\frac{y_i}{\frac{3}{7N}+y_i}$$
Derivatives for the original cross-entropy loss:
$$\frac{\partial l}{\partial y_i}=-\frac{t_i}{y_i}$$
$$\frac{\partial l}{\partial z_i}=y_i-t_i$$
Please let me know if I'm wrong. Thanks!

**Update**  
I just happened to read [a paper by Google][1] that applies the same formula as in Lucas' answer but with different interpretations. 

In Section 7 Model Regularization via Label Smoothing

> This (the cross entropy loss), however, can cause two problems. First, it may result in
> over-fitting: if the model learns to assign full probability to the
> groundtruth label for each training example, it is not guaranteed to
> generalize. Second, it encourages the differences between the largest
> logit and all others to become large, and this, combined with the
> bounded gradient $∂l/∂z_k$, reduces the ability of the model to adapt.
> Intuitively, this happens because the model becomes too confident
> about its predictions.

But **instead of adding the smoothing term to the predictions, they added it to the ground truth**, which turned out to be helpful.

> [![enter image description here][2]][2]
>
> In our ImageNet experiments with K = 1000 classes, we used u(k) =
> 1/1000 and $\epsilon$ = 0.1. For ILSVRC 2012, we have found a consistent
> improvement of about 0.2% absolute both for top-1 error and the top-5
> error.


  [1]: https://arxiv.org/pdf/1512.00567.pdf
  [2]: https://i.sstatic.net/jCRF5.png