NLP word representations and the Wittgenstein philosophy of language

I made an introductory talk on word embeddings in the past and this write-up is an extended version of the part about philosophical ideas behind word vectors. The aim of this article is to provide an introduction to Ludwig Wittgenstein’s main ideas on linguistics that are closely related to techniques that are distributional (I’ll talk what this means later) by design, such as word2vec [Mikolov et al., 2013], GloVe [Pennington et al., 2014], Skip-Thought Vectors [Kiros et al., 2015], among others.

One of the most interesting aspects of Wittgenstein is perhaps that fact that he had developed two very different philosophies during his life, and each of which had great influence. Something quite rare for someone who spent so much time working on these ideas and retreating even after the major influence they exerted, especially in the Vienna Circle. A true lesson of intellectual honesty, and in my opinion, one important legacy.

Wittgenstein was an avid reader of the Schopenhauer’s philosophy, and in the same way that Schopenhauer inherited his philosophy from Kant, especially regarding the division of what can be experimented (phenomena) or not (noumena), contrasting things as they appear for us from things as they are in themselves, Wittgenstein concluded that Schopenhauer philosophy was fundamentally right. He believed that in the noumena realm, we have no conceptual understanding and therefore we will never be able to say anything (without becoming nonsense), in contrast to the phenomena realm of our experience, where we can indeed talk about and try to understand. By adding secure foundations, such as logic, to the phenomenal world, he was able to reason about how the world is describable by language and thus mapping what are the limits of how and what can be expressed in language or in conceptual thought.

The first main theory of language from Wittgenstein, described in his Tractatus Logico-Philosophicus, is known as the “Picture theory of language” (aka Picture theory of meaning). This theory is based on an analogy with painting, where Wittgenstein realized that a painting is something very different than a natural landscape, however, a skilled painter can still represent the real landscape by placing patches or strokes corresponding to the natural landscape reality. Wittgenstein gave the name “logical form” to this set of relationships between the painting and the natural landscape. This logical form, the set of internal relationships common to both representations, is why the painter was able to represent reality because the logical form was the same in both representations (here I call both as “representations” to be coherent with Schopenhauer and Kant terms because the reality is also a representation for us, to distinguish between it and the thing-in-itself).

This theory was important, especially in our context (NLP), because Wittgenstein realized that the same thing happens with language. We are able to assemble words in sentences to match the same logical form of what we want to describe. The logical form was the core idea that made us able to talk about the world. However, later Wittgenstein realized that he had just picked a single task, out of the vast amount of tasks that language can perform and created a whole theory of meaning around it.

The fact is, language can do many other tasks besides representing (picturing) the reality. With language, as Wittgenstein noticed, we can give orders, and we can’t say that this is a picture of something. Soon as he realized these counter-examples, Wittgenstein abandoned the picture theory of language and adopted a much more powerful metaphor of a tool. And here we’re approaching the modern view of the meaning in language as well as the main foundational idea behind many modern Machine Learning techniques for word/sentence representations that works quite well. Once you realize that language works as a tool, if you want to understand the meaning of it, you just need to understand all the possible things you can do with it. And if you take for instance a word or concept in isolation, the meaning of it is the sum of all its uses, and this meaning is fluid and can have many different faces. This important thought can be summarized in the well-known quote below:

The meaning of a word is its use in the language.


One cannot guess how a word functions. One has to look at its use, and learn from that.

– Ludwig Wittgenstein, Philosophical Investigations

And indeed it makes complete sense because once you exhaust all the uses of a word, there is nothing left on it. Reality is also by far more fluid than usually thought, because:

Our language can be seen as an ancient city: a maze of little streets and squares, of old and new houses, and of houses with additions from various periods (…)

– Ludwig Wittgenstein, Philosophical Investigations

John R. Firth was a linguist also known for the popularization of this context-dependent nature of the meaning who also used Wittgenstein’s Philosophical Investigations as a recourse to emphasize the importance of the context in meaning, in which I quote below:

The placing of a text as a constituent in a context of situation contributes to the statement of meaning since situations are set up to recognize use. As Wittgenstein says, ‘the meaning of words lies in their use.’ (Phil. Investigations, 80, 109). The day-to-day practice of playing language games recognizes customs and rules. It follows that a text in such established usage may contain sentences such as ‘Don’t be such an ass !’, ‘You silly ass !’, ‘What an ass he is !’ In these examples, the word ass is in familiar and habitual company, commonly collocated with you silly-, he is a silly-, don’t be such an-. You shall know a word by the company it keeps ! One of the meanings of ass is its habitual collocation with such other words as those above quoted. Though Wittgenstein was dealing with another problem, he also recognizes the plain face-value, the physiognomy of words. They look at us ! ‘The sentence is composed of words and that is enough’.

– John R. Firth

This idea of learning the meaning of a word by the company it keeps is exactly what word2vec (and other count-based methods based on co-occurrence as well) is doing by means of data and learning on an unsupervised fashion with a supervised task that was by design built to predict context (or vice-versa, depending if you use skip-gram or cbow), which was also a source of inspiration for the Skip-Thought Vectors. Nowadays, this idea is also known as the “Distributional Hypothesis“, which is also being used on fields other than linguistics.

Now, it is quite amazing that if we look at the work by Neelakantan, et al., 2015, called “Efficient Non-parametric Estimation of Multiple Embeddings per Word in Vector Space“, where they mention about an important deficiency in word2vec in which each word type has only one vector representation, you’ll see that this has deep philosophical motivations if we relate it to the Wittgenstein and Firth ideas, because, as Wittgenstein noticed, the meaning of a word is unlikely to wear a single face and word2vec seems to be converging to an approximation of the average meaning of a word instead of capturing the polysemy inherent in language.

A concrete example of the multi-faceted nature of words can be seen in the example of the word “evidence”, where the meaning can be quite different to a historian, a lawyer and a physicist. The hearsay cannot count as evidence in a court while it is many times the only evidence that a historian has, whereas the hearsay doesn’t even arise in physics. Recent works such as ELMo [Peters, Matthew E. et al. 2018], which used different levels of features from a LSTM trained with a language model objective are also a very interesting direction with excellent results towards incorporating a context-dependent semantics into the word representations and breaking the tradition of shallow representations as seen in word2vec.

We’re in an exciting time where it is really amazing to see how many deep philosophical foundations are actually hidden in Machine Learning techniques. It is also very interesting that we’re learning a lot of linguistic lessons from Machine Learning experimentation, that we can see as important means for discovery that is forming an amazing virtuous circle. I think that we have never been self-conscious and concerned with language as in the past years.

I really hope you enjoyed reading this !

– Christian S. Perone

Cite this article as: Christian S. Perone, "NLP word representations and the Wittgenstein philosophy of language," in Terra Incognita, 23/05/2018,


Magee, Bryan. The history of philosophy. 1998.

Mikolov, Thomas et al. Efficient Estimation of Word Representations in Vector Space. 2013.

Pennington, Jeffrey et al. GloVe: Global Vectors for Word Representation. 2014.

Kiros, Ryan et al. Skip-Thought Vectors. 2015.

Neelakantan, Arvind et al. Efficient Non-parametric Estimation of Multiple Embeddings per Word in Vector Space. 2015.

Léon, Jacqueline. Meaning by collocation. The Firthian filiation of Corpus Linguistics. 2007.

PyTorch – Internal Architecture Tour

Update 28 Feb 2019: I added a new blog post with a slide deck containing the presentation I did for PyData Montreal.


This post is a tour around the PyTorch codebase, it is meant to be a guide for the architectural design of PyTorch and its internals. My main goal is to provide something useful for those who are interested in understanding what happens beyond the user-facing API and show something new beyond what was already covered in other tutorials.

Note: PyTorch build system uses code generation extensively so I won’t repeat here what was already described by others. If you’re interested in understanding how this works, please read the following tutorials:

Short intro to Python extension objects in C/C++

As you probably know, you can extend Python using C and C++ and develop what is called as “extension”. All the PyTorch heavy work is implemented in C/C++ instead of pure-Python. To define a new Python object type in C/C++, you define a structure like this one example below (which is the base for the autograd Variable class):

// Python object that backs torch.autograd.Variable
struct THPVariable {
    torch::autograd::Variable cdata;
    PyObject* backward_hooks;

As you can see, there is a macro at the beginning of the definition, called PyObject_HEAD, this macro’s goal is the standardization of Python objects and will expand to another structure that contains a pointer to a type object (which defines initialization methods, allocators, etc) and also a field with a reference counter.

There are two extra macros in the Python API called Py_INCREF() and Py_DECREF(), which are used to increment and decrement the reference counter of Python objects. Multiple entities can borrow or own a reference to other objects (the reference counter is increased), and only when this reference counter reaches zero (when all references get destroyed), Python will automatically delete the memory from that object using its garbage collector.

You can read more about Python C/++ extensions here.

Funny fact: it is very common in many applications to use small integer numbers as indexing, counters, etc. For efficiency, the official CPython interpreter caches the integers from -5 up to 256. For that reason, the statement a = 200; b = 200; a is b will be True, while the statement a = 300; b = 300; a is b will be False.

Zero-copy PyTorch Tensor to Numpy and vice-versa

PyTorch has its own Tensor representation, which decouples PyTorch internal representation from external representations. However, as it is very common, especially when data is loaded from a variety of sources, to have Numpy arrays everywhere, therefore we really need to make conversions between Numpy and PyTorch tensors. For that reason, PyTorch provides two methods called from_numpy() and numpy(), that converts a Numpy array to a PyTorch array and vice-versa, respectively. If we look the code that is being called to convert a Numpy array into a PyTorch tensor, we can get more insights on the PyTorch’s internal representation:

at::Tensor tensor_from_numpy(PyObject* obj) {
  if (!PyArray_Check(obj)) {
    throw TypeError("expected np.ndarray (got %s)", Py_TYPE(obj)->tp_name);

  auto array = (PyArrayObject*)obj;
  int ndim = PyArray_NDIM(array);
  auto sizes = to_aten_shape(ndim, PyArray_DIMS(array));
  auto strides = to_aten_shape(ndim, PyArray_STRIDES(array));
  // NumPy strides use bytes. Torch strides use element counts.
  auto element_size_in_bytes = PyArray_ITEMSIZE(array);
  for (auto& stride : strides) {
    stride /= element_size_in_bytes;

  // (...) - omitted for brevity

  void* data_ptr = PyArray_DATA(array);
  auto& type = CPU(dtype_to_aten(PyArray_TYPE(array)));
  return type.tensorFromBlob(data_ptr, sizes, strides, [obj](void* data) {
    AutoGIL gil;

(code from tensor_numpy.cpp)

As you can see from this code, PyTorch is obtaining all information (array metadata) from Numpy representation and then creating its own. However, as you can note from the marked line 18, PyTorch is getting a pointer to the internal Numpy array raw data instead of copying it. This means that PyTorch will create a reference for this data, sharing the same memory region with the Numpy array object for the raw Tensor data.

There is also an important point here: when Numpy array object goes out of scope and get a zero reference count, it will be garbage collected and destroyed, that’s why there is an increment in the reference counting of the Numpy array object at line 20.

After this, PyTorch will create a new Tensor object from this Numpy data blob, and in the creation of this new Tensor it passes the borrowed memory data pointer, together with the memory size and strides as well as a function that will be used later by the Tensor Storage (we’ll discuss this in the next section) to release the data by decrementing the reference counting to the Numpy array object and let Python take care of this object life cycle.

The tensorFromBlob() method will create a new Tensor, but only after creating a new “Storage” for this Tensor. The storage is where the actual data pointer will be stored (and not in the Tensor structure itself). This takes us to the next section about Tensor Storages.

Tensor Storage

The actual raw data of the Tensor is not directly kept in the Tensor structure, but on another structure called Storage, which in turn is part of the Tensor structure.

As we saw in the previous code from tensor_from_numpy(), there is a call for tensorFromBlob() that will create a Tensor from the raw data blob. This last function will call another function called storageFromBlob() that will, in turn, create a storage for this data according to its type. In the case of a CPU float type, it will return a new CPUFloatStorage instance.

The CPUFloatStorage is basically a wrapper with utility functions around the actual storage structure called THFloatStorage that we show below:

typedef struct THStorage
    real *data;
    ptrdiff_t size;
    int refcount;
    char flag;
    THAllocator *allocator;
    void *allocatorContext;
    struct THStorage *view;
} THStorage;

(code from THStorage.h)

As you can see, the THStorage holds a pointer to the raw data, its size, flags and also an interesting field called allocator that we’ll soon discuss. It is also important to note that there is no metadata regarding on how to interpret the data inside the THStorage, this is due to the fact that the storage is “dumb” regarding of its contents and it is the Tensor responsibility to know how to “view” or interpret this data.

From this, you already probably realized that we can have multiple tensors pointing to the same storage but with different views of this data, and that’s why viewing a tensor with a different shape (but keeping the same number of elements) is so efficient. This Python code below shows that the data pointer in the storage is being shared after changing the way Tensor views its data:

>>> tensor_a = torch.ones((3, 3))
>>> tensor_b = tensor_a.view(9)
>>> ==

As we can see in the example above, the data pointer on the storage of both Tensors are the same, but the Tensors represent a different interpretation of the storage data.

Now, as we saw in line 7 of the THFloatStorage structure, there is a pointer to a THAllocator structure there. And this is very important because it brings flexibility regarding the allocator that can be used to allocate the storage data. This structure is represented by the following code:

typedef struct THAllocator
  void* (*malloc)(void*, ptrdiff_t);
  void* (*realloc)(void*, void*, ptrdiff_t);
  void (*free)(void*, void*);
} THAllocator;

(code from THAllocator.h)

As you can see, there are three function pointer fields in this structure to define what an allocator means: a malloc, realloc and free. For CPU-allocated memory, these functions will, of course, relate to the traditional malloc/realloc/free POSIX functions, however, when we want a storage allocated on GPUs we’ll end up using the CUDA allocators such as the cudaMallocHost(), like we can see in the THCudaHostAllocator malloc function below:

static void *THCudaHostAllocator_malloc(void* ctx, ptrdiff_t size) {
  void* ptr;
  if (size < 0) THError("Invalid memory size: %ld", size);
  if (size == 0) return NULL;
  THCudaCheck(cudaMallocHost(&ptr, size));
  return ptr;

(code from THCAllocator.c)

You probably noticed a pattern in the repository organization, but it is important to keep in mind these conventions when navigating the repository, as summarized here (taken from the PyTorch lib readme):

  • TH = TorcH
  • THC = TorcH Cuda
  • THCS = TorcH Cuda Sparse
  • THCUNN = TorcH CUda Neural Network
  • THD = TorcH Distributed
  • THNN = TorcH Neural Network
  • THS = TorcH Sparse

This convention is also present in the function/class names and other objects, so it is important to always keep these patterns in mind. While you can find CPU allocators in the TH code, you’ll find CUDA allocators in the THC code.

Finally, we can see the composition of the main Tensor THTensor structure:

typedef struct THTensor
    int64_t *size;
    int64_t *stride;
    int nDimension;
    THStorage *storage;
    ptrdiff_t storageOffset;
    int refcount;
    char flag;
} THTensor;

(Code from THTensor.h)

And as you can see, the main THTensor structure holds the size/strides/dimensions/offsets/etc as well as the storage (THStorage) for the Tensor data.

We can summarize all this structure that we saw in the diagram below:

Now, once we have requirements such as multi-processing where we want to share tensor data among multiple different processes, we need a shared memory approach to solve it, otherwise, every time another process needs a tensor or even when you want to implement Hogwild training procedure where all different processes will write to the same memory region (where the parameters are), you’ll need to make copies between processes, and this is very inefficient. Therefore we’ll discuss in the next section a special kind of storage for Shared Memory.

Shared Memory

Shared memory can be implemented in many different ways depending on the platform support. PyTorch supports some of them, but for the sake of simplicity, I’ll talk here about what happens on MacOS using the CPU (instead of GPU). Since PyTorch supports multiple shared memory approaches, this part is a little tricky to grasp into since it involves more levels of indirection in the code.

PyTorch provides a wrapper around the Python multiprocessing module and can be imported from torch.multiprocessing. The changes they implemented in this wrapper around the official Python multiprocessing were done to make sure that everytime a tensor is put on a queue or shared with another process, PyTorch will make sure that only a handle for the shared memory will be shared instead of a new entire copy of the Tensor.

Now, many people aren’t aware of a Tensor method from PyTorch called share_memory_(), however, this function is what triggers an entire rebuild of the storage memory for that particular Tensor. What this method does is to create a region of shared memory that can be used among different processes. This function will, in the end, call this following function below:

static THStorage* THPStorage_(newFilenameStorage)(ptrdiff_t size)
  std::string handle = THPStorage_(__newHandle)();
  auto ctx = libshm_context_new(NULL, handle.c_str(), flags);
  return THStorage_(newWithAllocator)(size, &THManagedSharedAllocator, (void*)ctx);

(Code from StorageSharing.cpp)

And as you can see, this function will create another storage using a special allocator called THManagedSharedAllocator. This function first defines some flags and then it creates a handle which is a string in the format /torch_[process id]_[random number], and after that, it will then create a new storage using the special THManagedSharedAllocator. This allocator has function pointers to an internal PyTorch library called libshm, that will implement a Unix Domain Socket communication to share the shared memory region handles. This allocator is actual an especial case and it is a kind of “smart allocator” because it contains the communication control logic as well as it uses another allocator called THRefcountedMapAllocator that will be responsible for creating the actual shared memory region and call mmap() to map this region to the process virtual address space.

Note: when a method ends with a underscore in PyTorch, such as the method called share_memory_(), it means that this method has an in-place effect, and it will change the current object instead of creating a new one with the modifications.

I’ll now show a Python example of one processing using the data from a Tensor that was allocated on another process by manually exchanging the shared memory handle:

This is executed in the process A:

>>> import torch
>>> tensor_a = torch.ones((5, 5))
>>> tensor_a

 1  1  1  1  1
 1  1  1  1  1
 1  1  1  1  1
 1  1  1  1  1
 1  1  1  1  1
[torch.FloatTensor of size 5x5]

>>> tensor_a.is_shared()
>>> tensor_a = tensor_a.share_memory_()
>>> tensor_a.is_shared()
>>> tensor_a_storage =
>>> tensor_a_storage._share_filename_()
(b'/var/tmp/tmp.0.yowqlr', b'/torch_31258_1218748506', 25)

In this code, executed in the process A, we create a new Tensor of 5×5 filled with ones. After that we make it shared and print the tuple with the Unix Domain Socket address as well as the handle. Now we can access this memory region from another process B as shown below:

Code executed in the process B:

>>> import torch
>>> tensor_a = torch.Tensor()
>>> tuple_info = (b'/var/tmp/tmp.0.yowqlr', b'/torch_31258_1218748506', 25)
>>> storage = torch.Storage._new_shared_filename(*tuple_info)
>>> tensor_a = torch.Tensor(storage).view((5, 5))

 1  1  1  1  1
 1  1  1  1  1
 1  1  1  1  1
 1  1  1  1  1
 1  1  1  1  1
[torch.FloatTensor of size 5x5]

As you can see, using the tuple information about the Unix Domain Socket address and the handle we were able to access the Tensor storage from another process. If you change the tensor in this process B, you’ll also see that it will reflect in the process A because these Tensors are sharing the same memory region.

DLPack: a hope for the Deep Learning frameworks Babel

Now I would like to talk about something recent in the PyTorch code base, that is called DLPack. DLPack is an open standardization of an in-memory tensor structure that will allow exchange tensor data between frameworks, and what is quite interesting is that since this memory representation is standardized and very similar to the memory representation already in use by many frameworks, it will allow a zero-copy data sharing between frameworks, which is a quite amazing initiative given the variety of frameworks we have today without inter-communication among them.

This will certainly help to overcome the “island model” that we have today between tensor representations in MXNet, PyTorch, etc, and will allow developers to mix framework operations between frameworks and all the benefits that a standardization can bring to the frameworks.

The core of DLPack os a very simple structure called DLTensor, as shown below:

 * \brief Plain C Tensor object, does not manage memory.
typedef struct {
   * \brief The opaque data pointer points to the allocated data.
   *  This will be CUDA device pointer or cl_mem handle in OpenCL.
   *  This pointer is always aligns to 256 bytes as in CUDA.
  void* data;
  /*! \brief The device context of the tensor */
  DLContext ctx;
  /*! \brief Number of dimensions */
  int ndim;
  /*! \brief The data type of the pointer*/
  DLDataType dtype;
  /*! \brief The shape of the tensor */
  int64_t* shape;
   * \brief strides of the tensor,
   *  can be NULL, indicating tensor is compact.
  int64_t* strides;
  /*! \brief The offset in bytes to the beginning pointer to data */
  uint64_t byte_offset;
} DLTensor;

(code from dlpack.h)

As you can see, there is a data pointer for the raw data as well as shape/stride/offset/GPU vs CPU, and other metadata information about the data that the DLTensor pointing to.

There is also a managed version of the tensor that is called DLManagedTensor, where the frameworks can provide a context and also a “deleter” function that can be called by the framework who borrowed the Tensor to inform the other framework that the resources are no longer required.

In PyTorch, if you want to convert to or from a DLTensor format, you can find both C/C++ methods for doing that or even in Python you can do that as shown below:

import torch
from torch.utils import dlpack

t = torch.ones((5, 5))
dl = dlpack.to_dlpack(t)

This Python function will call the toDLPack function from ATen, shown below:

DLManagedTensor* toDLPack(const Tensor& src) {
  ATenDLMTensor * atDLMTensor(new ATenDLMTensor);
  atDLMTensor->handle = src;
  atDLMTensor->tensor.manager_ctx = atDLMTensor;
  atDLMTensor->tensor.deleter = &deleter;
  atDLMTensor-> = src.data_ptr();
  int64_t device_id = 0;
  if (src.type().is_cuda()) {
    device_id = src.get_device();
  atDLMTensor->tensor.dl_tensor.ctx = getDLContext(src.type(), device_id);
  atDLMTensor->tensor.dl_tensor.ndim = src.dim();
  atDLMTensor->tensor.dl_tensor.dtype = getDLDataType(src.type());
  atDLMTensor->tensor.dl_tensor.shape = const_cast<int64_t*>(src.sizes().data());
  atDLMTensor->tensor.dl_tensor.strides = const_cast<int64_t*>(src.strides().data());
  atDLMTensor->tensor.dl_tensor.byte_offset = 0;
  return &(atDLMTensor->tensor);

As you can see, it’s a pretty simple conversion, casting the metadata from the PyTorch format to the DLPack format and assigning a pointer to the internal Tensor data representation.

I really hope that more frameworks adopt this standard that will certainly give benefits to the ecosystem. It is also interesting to note that a potential integration with Apache Arrow would be amazing.

That’s it, I hope you liked this long post !

– Christian S. Perone

Cite this article as: Christian S. Perone, "PyTorch – Internal Architecture Tour," in Terra Incognita, 12/03/2018,

Privacy-preserving sentence semantic similarity using InferSent embeddings and secure two-party computation

Privacy-preserving Computation

Privacy-preserving computation or secure computation is a sub-field of cryptography where two (two-party, or 2PC) or multiple (multi-party, or MPC) parties can evaluate a function together without revealing information about the parties private input data to each other. The problem and the first solution to it were introduced in 1982 by an amazing breakthrough done by Andrew Yao on what later became known as the “Yao’s Millionaires’ problem“.

The Yao’s Millionaires Problem is where two millionaires, Alice and Bob, who are interested in knowing which of them is richer but without revealing to each other their actual wealth. In other words, what they want can be generalized as that: Alice and Bob want jointly compute a function securely, without knowing anything other than the result of the computation on the input data (that remains private to them).

To make the problem concrete, Alice has an amount A such as $10, and Bob has an amount B such as $ 50, and what they want to know is which one is larger, without Bob revealing the amount B to Alice or Alice revealing the amount A to Bob. It is also important to note that we also don’t want to trust on a third-party, otherwise the problem would just be a simple protocol of information exchange with the trusted party.

Formally what we want is to jointly evaluate the following function:

r = f(A, B)

Such as the private values A and B are held private to the sole owner of it and where the result r will be known to just one or both of the parties.

It seems very counterintuitive that a problem like that could ever be solved, but for the surprise of many people, it is possible to solve it on some security requirements. Thanks to the recent developments in techniques such as FHE (Fully Homomorphic Encryption), Oblivious Transfer, Garbled Circuits, problems like that started to get practical for real-life usage and they are being nowadays being used by many companies in applications such as information exchange, secure location, advertisement, satellite orbit collision avoidance, etc.

I’m not going to enter into details of these techniques, but if you’re interested in the intuition behind the OT (Oblivious Transfer), you should definitely read the amazing explanation done by Craig Gidney here. There are also, of course, many different protocols for doing 2PC or MPC, where each one of them assumes some security requirements (semi-honest, malicious, etc), I’m not going to enter into the details to keep the post focused on the goal, but you should be aware of that.

The problem: sentence similarity

What we want to achieve is to use privacy-preserving computation to calculate the similarity between sentences without disclosing the content of the sentences. Just to give a concrete example: Bob owns a company and has the description of many different projects in sentences such as: “This project is about building a deep learning sentiment analysis framework that will be used for tweets“, and Alice who owns another competitor company, has also different projects described in similar sentences. What they want to do is to jointly compute the similarity between projects in order to find if they should be doing partnership on a project or not, however, and this is the important point: Bob doesn’t want Alice to know the project descriptions and neither Alice wants Bob to be aware of their projects, they want to know the closest match between the different projects they run, but without disclosing the project ideas (project descriptions).

Sentence Similarity Comparison

Now, how can we exchange information about the Bob and Alice’s project sentences without disclosing information about the project descriptions ?

One naive way to do that would be to just compute the hashes of the sentences and then compare only the hashes to check if they match. However, this would assume that the descriptions are exactly the same, and besides that, if the entropy of the sentences is small (like small sentences), someone with reasonable computation power can try to recover the sentence.

Another approach for this problem (this is the approach that we’ll be using), is to compare the sentences in the sentence embeddings space. We just need to create sentence embeddings using a Machine Learning model (we’ll use InferSent later) and then compare the embeddings of the sentences. However, this approach also raises another concern: what if Bob or Alice trains a Seq2Seq model that would go from the embeddings of the other party back to an approximate description of the project ?

It isn’t unreasonable to think that one can recover an approximate description of the sentence given their embeddings. That’s why we’ll use the two-party secure computation for computing the embeddings similarity, in a way that Bob and Alice will compute the similarity of the embeddings without revealing their embeddings, keeping their project ideas safe.

The entire flow is described in the image below, where Bob and Alice shares the same Machine Learning model, after that they use this model to go from sentences to embeddings, followed by a secure computation of the similarity in the embedding space.

Diagram overview of the entire process.

Generating sentence embeddings with InferSent

Bi-LSTM max-pooling network. Source: Supervised Learning of Universal Sentence Representations from Natural Language Inference Data. Alexis Conneau et al.

InferSent is an NLP technique for universal sentence representation developed by Facebook that uses supervised training to produce high transferable representations.

They used a Bi-directional LSTM with attention that consistently surpassed many unsupervised training methods such as the SkipThought vectors. They also provide a Pytorch implementation that we’ll use to generate sentence embeddings.

Note: even if you don’t have GPU, you can have reasonable performance doing embeddings for a few sentences.

The first step to generate the sentence embeddings is to download and load a pre-trained InferSent model:

import numpy as np
import torch

# Trained model from:
GLOVE_EMBS = '../dataset/GloVe/glove.840B.300d.txt'
INFERSENT_MODEL = 'infersent.allnli.pickle'

# Load trained InferSent model
model = torch.load(INFERSENT_MODEL,
                   map_location=lambda storage, loc: storage)


Now we need to define a similarity measure to compare two vectors, and for that goal, I’ll the cosine similarity (I wrote a tutorial about this similarity measure here) since it’s pretty straightforward:

cos(\pmb x, \pmb y) = \frac {\pmb x \cdot \pmb y}{||\pmb x|| \cdot ||\pmb y||}

As you can see, if we have two unit vectors (vectors with norm 1), the two terms in the equation denominator will be 1 and we will be able to remove the entire denominator of the equation, leaving only:

cos(\hat{x}, \hat{y}) =\hat{x} \cdot\hat{y}

So, if we normalize our vectors to have a unit norm (that’s why the vectors are wearing hats in the equation above), we can make the computation of the cosine similarity become just a simple dot product. That will help us a lot in computing the similarity distance later when we’ll use a framework to do the secure computation of this dot product.

So, the next step is to define a function that will take some sentence text and forward it to the model to generate the embeddings and then normalize them to unit vectors:

# This function will forward the text into the model and
# get the embeddings. After that, it will normalize it
# to a unit vector.

def encode(model, text):
    embedding = model.encode([text])[0]
    embedding /= np.linalg.norm(embedding)
    return embedding

As you can see, this function is pretty simple, it feeds the text into the model, and then it will divide the embedding vector by the embedding norm.

Now, for practical reasons, I’ll be using integer computation later for computing the similarity, however, the embeddings generated by InferSent are of course real values. For that reason, you’ll see in the code below that we create another function to scale the float values and remove the radix point and converting them to integers. There is also another important issue, the framework that we’ll be using later for secure computation doesn’t allow signed integers, so we also need to clip the embeddings values between 0.0 and 1.0. This will of course cause some approximation errors, however, we can still get very good approximations after clipping and scaling with limited precision (I’m using 14 bits for scaling to avoid overflow issues later during dot product computations):

# This function will scale the embedding in order to
# remove the radix point.
def scale(embedding):
    SCALE = 1 << 14
    scale_embedding = np.clip(embedding, 0.0, 1.0) * SCALE
    return scale_embedding.astype(np.int32)

You can use floating-point in your secure computations and there are a lot of frameworks that support them, however, it is more tricky to do that, and for that reason, I used integer arithmetic to simplify the tutorial. The function above is just a hack to make it simple. It’s easy to see that we can recover this embedding later without too much loss of precision.

Now we just need to create some sentence samples that we’ll be using:

# The list of Alice sentences
alice_sentences = [
    'my cat loves to walk over my keyboard',
    'I like to pet my cat',

# The list of Bob sentences
bob_sentences = [
    'the cat is always walking over my keyboard',

And convert them to embeddings:

# Alice sentences
alice_sentence1 = encode(model, alice_sentences[0])
alice_sentence2 = encode(model, alice_sentences[1])

# Bob sentences
bob_sentence1 = encode(model, bob_sentences[0])

Since we have now the sentences and every sentence is also normalized, we can compute cosine similarity just by doing a dot product between the vectors:

>>>, alice_sentence1)

>>>, alice_sentence2)

As we can see, the first sentence of Bob is most similar (~0.87) with Alice first sentence than to the Alice second sentence (~0.62).

Since we have now the embeddings, we just need to convert them to scaled integers:

# Scale the Alice sentence embeddings
alice_sentence1_scaled = scale(alice_sentence1)
alice_sentence2_scaled = scale(alice_sentence2)

# Scale the Bob sentence embeddings
bob_sentence1_scaled = scale(bob_sentence1)

# This is the unit vector embedding for the sentence
>>> alice_sentence1
array([ 0.01698913, -0.0014404 ,  0.0010993 , ...,  0.00252409,
        0.00828147,  0.00466533], dtype=float32)

# This is the scaled vector as integers
>>> alice_sentence1_scaled
array([278,   0,  18, ...,  41, 135,  76], dtype=int32)

Now with these embeddings as scaled integers, we can proceed to the second part, where we’ll be doing the secure computation between two parties.

Two-party secure computation

In order to perform secure computation between the two parties (Alice and Bob), we’ll use the ABY framework. ABY implements many difference secure computation schemes and allows you to describe your computation as a circuit like pictured in the image below, where the Yao’s Millionaire’s problem is described:

Yao’s Millionaires problem. Taken from ABY documentation (

As you can see, we have two inputs entering in one GT GATE (greater than gate) and then a output. This circuit has a bit length of 3 for each input and will compute if the Alice input is greater than (GT GATE) the Bob input. The computing parties then secret share their private data and then can use arithmetic sharing, boolean sharing, or Yao sharing to securely evaluate these gates.

ABY is really easy to use because you can just describe your inputs, shares, gates and it will do the rest for you such as creating the socket communication channel, exchanging data when needed, etc. However, the implementation is entirely written in C++ and I’m not aware of any Python bindings for it (a great contribution opportunity).

Fortunately, there is an implemented example for ABY that can do dot product calculation for us, the example is here. I won’t replicate the example here, but the only part that we have to change is to read the embedding vectors that we created before instead of generating random vectors and increasing the bit length to 32-bits.

After that, we just need to execute the application on two different machines (or by emulating locally like below):

# This will execute the server part, the -r 0 specifies the role (server)
# and the -n 4096 defines the dimension of the vector (InferSent generates
# 4096-dimensional embeddings).
~# ./innerproduct -r 0 -n 4096

# And the same on another process (or another machine, however for another
# machine execution you'll have to obviously specify the IP).
~# ./innerproduct -r 1 -n 4096

And we get the following results:

Inner Product of alice_sentence1 and bob_sentence1  = 226691917
Inner Product of alice_sentence2 and bob_sentence1  = 171746521

Even in the integer representation, you can see that the inner product of the Alice’s first sentence and the Bob sentence is higher, meaning that the similarity is also higher. But let’s now convert this value back to float:

>>> SCALE = 1 << 14

# This is the dot product we should get
>>>, bob_sentence1)

# This is the inner product we got on secure computation
>>> 226691917 / SCALE**2.0

# This is the dot product we should get
>>>, bob_sentence1)

# This is the inner product we got on secure computation
>>> 171746521 / SCALE**2.0

As you can see, we got very good approximations, even in presence of low-precision math and unsigned integer requirements. Of course that in real-life you won’t have the two values and vectors, because they’re supposed to be hidden, but the changes to accommodate that are trivial, you just need to adjust ABY code to load only the vector of the party that it is executing it and using the correct IP addresses/port of the both parties.

I hope you liked it !

– Christian S. Perone

Cite this article as: Christian S. Perone, "Privacy-preserving sentence semantic similarity using InferSent embeddings and secure two-party computation," in Terra Incognita, 22/01/2018,

New prime on the block

The GIMPS (Great Internet Mersenne Prime Search) has confirmed yesterday the new largest known prime number: 277,232,917-1. This new largest known prime has 23,249,425 digits and is, of course, a Mersenne prime, prime numbers expressed in the form of 2n – 1, where the primality can be efficiently calculated using Lucas-Lehmer primality test.

One of the most asked questions about these largest primes is how the number of digits is calculated, given the size of these numbers (23,249,425 digits for the new largest known prime). And indeed there is a trick that avoids you to evaluate the number to calculate the number of digits, using Python you can just do:

>>> import numpy as np 
>>> a = 2
>>> b = 77232917
>>> num_digits = int(1 + b * np.log10(a))
>>> print(num_digits)

The reason why this works is that the log base 10 of a number is how many times this number should be divided by 10 to get to 1, so you get the number of digits after 1 and just need to add 1 back.

Another interesting fact is that we can also get the last digit of this very large number again without evaluating the entire number by using congruence. Since we’re interested in the number mod 10 and we know that the Mersenne prime has the form of 277,232,917-1, we can check that the powers 2n have an easy cycling pattern:

2^1 \equiv 2 \pmod{10}
2^2 \equiv 4 \pmod{10}
2^3 \equiv 8 \pmod{10}
2^4 \equiv 6 \pmod{10}
2^5 \equiv 2 \pmod{10}
2^6 \equiv 4 \pmod{10}
(… repeat)

Which means that powers of 2 mod 10 repeats at every 4 numbers, thus we just need to compute 77,232,917 mod 4, which is 1. Given that 2^{77,232,917} \equiv 2^1 \pmod{10} the part 277,232,917 ends in 2 and when you subtract 1 you end up with 1 as the last digit, as you can confirm by looking at the entire number (~10Mb zipfile).

– Christian S. Perone

The effective receptive field on CNNs

Given the interesting recent article on “The Emergence of a Fovea while Learning to Attend“, I decide to make a review of the paper written by Luo, Wenjie et al. called “Understanding the Effective Receptive Field in Deep Convolutional Neural Networks” where they introduced the idea of the “Effective Receptive Field” (ERF) and the surprising relationship with the foveal vision that arises naturally on Convolutional Neural Networks.

The receptive field in Convolutional Neural Networks (CNN) is the region of the input space that affects a particular unit of the network. Note that this input region can be not only the input of the network but also output from other units in the network, therefore this receptive field can be calculated relative to the input that we consider and also relative the unit that we are taking into consideration as the “receiver” of this input region. Usually, when the receptive field term is mentioned, it is taking into consideration the final output unit of the network (i.e. a single unit on a binary classification task) in relation to the network input (i.e. input image of the network).

It is easy to see that on a CNN, the receptive field can be increased using different methods such as: stacking more layers (depth), subsampling (pooling, striding), filter dilation (dilated convolutions), etc. In theory, when you stack more layers you can increase your receptive field linearly, however, in practice, things aren’t simple as we thought, as shown by Luo, Wenjie et al. article. In the article, they introduce the concept of the “Effective Receptive Field”, or ERF; the intuition behind the concept is that not all pixels in the receptive field contribute equally to the output unit’s response. When doing the forward pass, we can see that the central receptive field pixels can propagate their information to the output using many different paths, as they are part of multiple output unit’s calculations.

In the figure below, we can see in left the input pixels, after that we have a feature map calculated from the input pixels using a 3×3 convolution filter and then finally the output after another 3×3 filtering. The numbers inside the pixels on the left image represent how many times this pixel was part of a convolution step (each sliding step of the filter). As we can see, some pixels like the central ones will have their information propagated through many different paths in the network, while the pixels on the borders are propagated along a single path.

Receptive Field across 3 different layers using 3×3 filters.

By looking at the image above, it isn’t that surprising that the effective receptive field impact on the final output computation will look more like a Gaussian distribution instead of a uniform distribution. What is actually more even interesting is that this receptive field is dynamic and changes during the training. The impact of this on the backpropagation is that the central pixels will have a larger gradient magnitude when compared to the border pixels.

In the article written by Luo, Wenjie et al., they devised a way to quantify the effect on each input pixel of the network by calculating the quantity \frac{\partial y}{\partial x_{i, j}} that represents how much each pixel x_{i, j} contributes to the output y.

In the paper, they did experimentations to visualize the effective receptive field using multiple different architectures, activations, etc. I replicate here the ones that I found most interesting:

Figure 1 from the paper “Understanding the Effective Receptive Field in Deep Convolutional Neural Networks”, by Luo, Wenjie et al.

As we can see from the Figure 1 of the paper, where they compare the effect of the number of layers, initialization schemes, and different activations, the results are amazing. We can clearly see the Gaussian and also the sparsity added by the ReLU activations.

There are also some comparisons on Figure 3 of the paper, where CIFAR-10 and CamVid datasets were used to train the network.

Figure 3 of the paper “Understanding the Effective Receptive Field in Deep Convolutional Neural Networks”, by Luo, Wenjie et al.

As we can see, the size of the effective receptive field is very dynamic and it is increased by a large margin after the training, which implies, as stated by authors of the paper, that better initialization schemes can be employed to increase the receptive field in the beginning of the training. They actually developed a different initialization scheme and were able to get 30% training speed-up, however, these results weren’t consistent.

Foveal vision on reading activity. Image from

What is also very interesting, is that the effective receptive field has a very close relationship with the foveal vision of the human eye, which produces the sharp central vision, effect of the high-density region of cone cells (as shown in the image below) present in the eye fundus.

Fovea region on the human eye. Image from

Our central sharp vision also decays rapidly like the effective receptive field that is very similar to a Gaussian. It is amazing that this effect is also naturally present on the CNN networks.

PS: Just for the sake of curiosity, some birds that do complex aerial movements such as the hummingbird, have two foveas instead of a single one, which means that they have a sharp accurate vision not only on the central region but also on the sides.

I hope you enjoyed the post !

– Christian S. Perone

Cite this article as: Christian S. Perone, "The effective receptive field on CNNs," in Terra Incognita, 12/11/2017,

Benford’s law – Index

Since Benford’s law got some attention in the past years, I decided to make a list of the previous posts I made on the subject in the context of elections, fraud, corruption, universality and prime numbers:

Despesas de Custeio e Lei de Benford (June 2014 – in Portuguese)

Universality, primes and space communication (January 2014)

An analysis of Benford’s law applied to Twitter (August 2009)

Benford’s Law and the Iran’s election (June 2009)

Prime Numbers and the Benford’s Law (May 2009), checking user numbers against Benford’s Law (April 2009)

Benford’s Law meets Python and Apple Stock Prices (April 2009)

I hope you like it !

– Christian S. Perone

The same old historicism, now on AI

* This is a critical article regarding the presence of historicism in modern AI predictions for the future.

Ray Kurzweil

Perhaps you already read about the Technological Singularity, since it is one of the hottest predictions for the future (there is even a university with that name), especially after the past years’ development of AI, more precisely, after recent Deep Learning advancements that attracted a lot of attention (and bad journalism too). In his The Singularity is near (2005) book, Ray Kurzweil predicts that humans will transcend the “limitations of our biological bodies and brain”, stating also that “future machines will be human, even if they are not biological”. In other books, like The Age of Intelligent Machines (1990), he also predicts a new world government, computers passing Turing tests, exponential laws everywhere, and so on (not that hard to have a good recall rate with that amount of predictions right ?).

As science fiction, these predictions are pretty amazing, and many of them were very close to what happened in our “modern days” (and I also really love the works made by Arthur C. Clarke), however, there are a lot of people that are putting science clothes on what is called “futurism”, sometimes also called “future studies” or “futurology”, although as you can imagine, the last term is usually avoided due to some obvious reasons (sounds like astrology, and you don’t want to be linked to pseudo-science right ?).

In this post, I would like to talk not about the predictions. Personally, I think that these points of view are really relevant to our future, just like the serious research on ethics and moral in AI, but I would like to criticize a very particular aspect of the status of how these ideas are being diffused, and I like to make the point here very clear: I’m NOT criticizing the predictions themselves, NEITHER the importance of these predictions and different views of the future, but the status of these ideas, because it seems that there is a major comeback of a kind of historicism in this particular field that I would like to discuss.

There is a very subtle line on where it is very easy to transit from a personal prediction of historical events to a view where you pretend that these predictions have a scientific status. Some harsh critics were made in the past regarding the Technological Singularity, such as this one from Steven Pinker (2008):

(…) There is not the slightest reason to believe in a coming singularity. The fact that you can visualize a future in your imagination is not evidence that it is likely or even possible. Look at domed cities, jet-pack commuting, underwater cities, mile-high buildings, and nuclear-powered automobiles—all staples of futuristic fantasies when I was a child that have never arrived. Sheer processing power is not a pixie dust that magically solves all your problems. (…) –

– Steven Pinker, 2008

Steven Pinker is criticizing here an important aspect, that it is obvious but many people usually do not understand the implication of this: the fact that you can imagine something, isn’t a reason or evidence that this is possible. Just like the ontological argument was criticized in the past by Immanuel Kant, where we have the same kind of transition.

Karl Popper

However, what I would like to criticize here is the fact that a lot of futurists are postulating these predictions as if they have a scientific status, which is a gross misunderstanding of the scientific method that led to the development of the social historicism in the past, and that was hardly criticized by the philosopher Karl Popper in many different important works such as The Open Society and Its Enemies (1945) and on The Poverty of Historicism (1936) in the political context.

The historicism, as Popper describes, is characterized by the belief that once you have discovered the developmental laws (like the futurist exponential laws) of the history (or AI development), that would enable us to prophesy the destiny of man with a scientific status. Karl Popper found that the dangerous habit of historical prophecy, so widespread among our intellectual leaders, has various functions:

“It is always flattering to belong to the inner circle of the initiated, and to possess the unusual power of predicting the course of history. Besides, there is a tradition that intellectual leaders are gifted with such powers, and not to possess them may lead to the loss of caste. The danger, on the other hand, of their being unmasked as charlatans is very small, since they can always point out that it is certainly permissible to make less sweeping predictions; and the boundaries between these and augury are fluid.”

– Karl Popper, 1945

Recently, we were also able to witness the debate between Elon Musk and Mark Zuckerberg, where you’ll find all sort of criticism between each other, but little or no humility regarding the limits of these claims. Karl Popper mentions an important fact to consider in his The Open Society and Its Enemies book on the social context, that can also be certainly applied here as you’ll note:

(…) Such arguments may sound plausible enough. But plausibility is not a reliable guide in such matters. In fact, one should not enter into a discussion of these specious arguments before having considered the following question of method: Is it within the power of any social science to make such sweeping historical prophecies ? Can we expect to get more than the irresponsible reply of the soothsayer if we ask a man what the future has in store for mankind ?

– Karl Popper, 1945

With that said, we should always remember the importance of our future views and predictions, but we should also never forget the status of these predictions, and be always responsible for our diffusion of these claims. They aren’t scientific by any means, and we shouldn’t take them as that, especially when dangerous ideas such as urging for government regulation are being made based on these personal future prophecies.

I would like to close this post quoting Karl Popper:

The systematic analysis of historicism aims at something like scientific status. This book does not. Many of the opinions expressed are personal. What it owes to scientific method is largely the awareness of its limitations : it does not offer proofs where nothing can be proved, nor does it pretend to be scientific where it cannot give more than a personal point of view. It does not try to replace the old systems of philosophy by a new system. It does not try to add to all these volumes filled withwisdom, to the metaphysics of history and destiny, such as are fashionable nowadays. It rather tries to show that this prophetic wisdom is harmful, that the metaphysics of history impede the application of the piecemeal methods of science to the problems of social reform. And it further tries to show how we may become the makers of our fate when we have ceased to pose as its prophets.

Cite this article as: Christian S. Perone, "The same old historicism, now on AI," in Terra Incognita, 30/07/2017,

Introduction to Word Embeddings

Just publised a deck of slides for an Introduction to Word Embeddings, presented at our Machine Learning Meetup in Porto Alegre / RS / Brazil.

Here is the video (in Portugese):

Cite this article as: Christian S. Perone, "Introduction to Word Embeddings," in Terra Incognita, 08/02/2017,