6.1 Episodic Memory

6.1 Episodic Memory#

In the previous chapters, learning was based on gradually adjusting weights in a neural network. However, humans have the ability to learn from a single experience without the need for repetition. One could argue that this “one-shot” learning can be achieved by a high learning rate. However, a high learning rate can lead to catastrophic forgetting, where the network forgets previously learned associations.

Consider for example, the Rumelhart Semantic Network from the previous chapter. If we first train the network to associate various birds with flying, and then train the network with a single example of a penguin with a very high learning rate, the network will forget the previous association with birds and flying. In general, we can mitigate this problem by using interleaved training, where we mix examples from the bird-category with the penguin example. However, this doesn’t reflect human learning, where we can learn from a single example without forgetting previous associations.

McClelland et al, 1995 proposed two complemantary learning systems: A slow learning system that learns gradually from repetition (in form of weight adjustments in the Neocortex) and a fast learning system that learns from single experiences (in form of episodic memory in the hippocampus). Here, we will explore how such a episodic memory system can be modeled in PsyNeuLink.

Installation and Setup

If the following code fails, you might have to restart the kernel/session and run it again. This is a known issue when installing PsyNeulink in google colab.

%%capture
%pip install psyneulink


import psyneulink as pnl
import numpy as np
# from torch import nn
# import torch
import matplotlib.pyplot as plt

Episodic Memory - Python Implementation#

We start with implementing a epispdic memory as python dictionary. The memory stores key-value pairs, that can be retrieved by querying the memory with a key:

em_dict = {
    'morning': 'breakfast',
    'afternoon': 'lunch',
    'evening': 'dinner'
}

We can retrieve memory by accessing the dictionary with a key:

food = em_dict['morning']
print(food)

breakfast

However, this implementation is limited in that the key has to be an exact match. For example, we can not just access “late afternoon” and get “brunch”:

food = em_dict['late afternoon']

---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
Cell In[4], line 1
----> 1 food = em_dict['late afternoon']

KeyError: 'late afternoon'

Also, to make this more general, instead of strings, we can use (one-hot encoded) vectors as keys and values. Let’s build our own episodic memory that stores key-value pairs as list:

morning_key = np.array([1, 0, 0])  # morning
morning_value = np.array([0, 0, 1])  # breakfast

afternoon_key = np.array([0, 1, 0])  # afternoon
afternoon_value = np.array([1, 0, 0])  # lunch

evening_key = np.array([0, 0, 1])  # evening
evening_value = np.array([0, 0, 1])  # dinner

em = np.array([(morning_key, morning_value), (afternoon_key, afternoon_value), (evening_key, evening_value)])
print(em)

[[[1 0 0]
  [0 0 1]]

 [[0 1 0]
  [1 0 0]]

 [[0 0 1]
  [0 0 1]]]

def plot(memory):
    """
    Plot the episodic memory as a matrix
    """
    def flatten(el):
        x = el[0]
        for i in el[1]:
            x = np.append(x, i)
        return x

    # Create a 6-row matrix (3 keys and 8 values) by padding the rows appropriately
    episodic_memory_matrix = [flatten(el) for el in memory]

    plt.imshow(episodic_memory_matrix, cmap='Reds', aspect='auto')

    labels = [f"Key {i}" for i in range(len(episodic_memory_matrix[0])//2)] + [f"Value {i}" for i in range(len(episodic_memory_matrix[0])//2)]
    plt.xticks(ticks=range(len(episodic_memory_matrix[0])),
               labels=labels, rotation=45)
    plt.yticks(ticks=range(len(episodic_memory_matrix)),
               labels=[f"Memory Entry {i}" for i in range(len(episodic_memory_matrix))])
    # Add vertical dashed line to separate groups

    plt.axvline(x=(len(labels)-1)/2, color='black', linestyle='--', linewidth=1)
    plt.grid(visible=True)
    plt.title('Episodic Memory')
    plt.xlabel('Columns')
    plt.ylabel('Rows')
    plt.show()


#
plot(em)
../../../../../_images/6135a29e486c6f3b05d70dfd225f10b9be2d37847cd355a2924690fbca25a1c4.png

To retrieve a value, we can first search for the key that is most similar to the query key and return the corresponding value. A good measure for similarity between two vectors is the dot product.

# We use a query key that the model has never seen before (for exampl a interpolation between the morning and afternoon key)
query_key = np.array([.9, .1, 0])  # between morning and afternoon

matches = [np.dot(query_key, key) for key, _ in em]

# our matches are:
print('Matches: ',matches)

# and the index with the highest match is:
max_match_index = matches.index(max(matches))
print('Index where key matches the most:', max_match_index)

# matched key:
key = em[max_match_index][0]
print('Matched key:', key)

# so the value is:
value = em[max_match_index][1]
print('Matched value:', value)

Matches:  [0.9, 0.1, 0.0]
Index where key matches the most: 0
Matched key: [1 0 0]
Matched value: [0 0 1]

As expected, the most close match to “between morning and afternoon” is “morning” and we get the value “breakfast”

We can go a step further, instead of using the argmax of the matches, we can also weigh the values by their respective match. But to do that, we first need to normalize the matches (in this case we use the softmax function):

import math
import numpy as np

def softmax(x):
    # calculate the sum
    _sum = 0
    for el in x:
        _sum += math.exp(el)
    res = []
    for el in x:
        res.append(math.exp(el) / _sum)
    return res

query_key = [1., 1., 0.]  # half way between morning and afternoon

matches = [np.dot(query_key, key) for key, _ in em]
print('Matches:', matches)

soft_maxed_matches = softmax(matches)
print('Soft maxed Matches:', soft_maxed_matches)

weighted_values = np.sum(np.array([soft_maxed_matches[i] * np.array(value) for i, (_, value) in enumerate(em)]), axis=0)
print('Weighted Values:', weighted_values)

Matches: [1.0, 1.0, 0.0]
Soft maxed Matches: [0.4223187982515182, 0.4223187982515182, 0.15536240349696362]
Weighted Values: [0.4223188 0.        0.5776812]

Here, instead of choosing the best key (that is in memory), we calculate a weighted between the values stored in memory. The weights, in this case are similarity scores between the query key and the stored key in memory.

🎯 Exercise 1

Instead of the above implementation of the softmax, in use cases, we use a softmax that masks values below a certain threshold. Why is this necessary? What would happen if we didn’t mask values?

💡 Hint

In our toy example we have only a few memory slots. However, in a most scenarios, we want to model a large number of episodic memory slots. In this case, the match_weight vector will have a large number of entries with a lot of zeros (or near zeros). Think about why that is problematic.

✅ Solution 1

The “flattening” effect of the softmax function is dependent on the length of the vector. For example, try running the following code:

res = softmax(torch.tensor([1] + [0.01] * 10))
res_2 = softmax(torch.tensor([1] + [0.01] * 100))
print(res[0])
print(res_2[0])

🎯 Exercise 2

Try implementing a larger memory with more key value pairs. How does the memory retrieval hold up without masking?

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