Gradient Episodic Memory for Continual Learning

Presented by

• Yu Xuan Lee
• Tsen Yee Heng

Background and Introduction

Supervised learning consist of a training set [math]\displaystyle{ D_{tx}=(x_i,y_i)^n_{i=1} }[/math], where [math]\displaystyle{ x_i \in \mathcal{X} }[/math] and [math]\displaystyle{ y_i \in \mathcal{Y} }[/math]. Empirical Risk Minimization (ERM) is one of the common supervised learning method used to minimize a loss function by having multiple passes over the training set.

[math]\displaystyle{ \frac{1}{|D_{tr}|}\textstyle \sum_{(x_i,y_i) \in D_{tr}} \ell (f(x_i),y_i) }[/math]

where [math]\displaystyle{ \ell :\mathcal {Y} \times \mathcal {Y} \to [0, \infty) }[/math]

Different to machine learning, datas are being observed sequentially, occurred recurrently, and stored limitedly for learning humans. Thus, the iid assumption is not applicable to ERM. One of the characteristics of ERM is "catastrophic forgetting", which is the problem of recalling past knowledge upon acquiring new ones. To overcome this problem, Gradient Episodic Memory (GEM) is introduced to alleviates forgetting on previous acquired knowledge, while solving new problems more efficiently.

Framework for Continual Learning

The feature vector [math]\displaystyle{ x_i \in \mathcal{X}_t }[/math], task descriptor [math]\displaystyle{ t_i \in \mathcal{T} }[/math], and target vector [math]\displaystyle{ y_i \in \mathcal{Y}_t }[/math] are the three main components of a continuum of data. Note that the continuum is locally iid where for every [math]\displaystyle{ (x_i, t_i, y_i) }[/math]

[math]\displaystyle{ (x_i,y_i) \overset{iid}{\sim} P_{t_i}(X,Y) }[/math]

The main mathematical purpose of continual learning is to obtain [math]\displaystyle{ f: \mathcal{X} \times \mathcal{Y} }[/math] where a target vector [math]\displaystyle{ y }[/math] must be inquired using a test pair [math]\displaystyle{ (x,t) }[/math].

Task Descriptor

Task descriptors are structured objects, describing how to solve each [math]\displaystyle{ i }[/math]-th task. They are integers [math]\displaystyle{ t_i=i \in \mathbb{Z} }[/math] which occurs in a collection where [math]\displaystyle{ t_1,...,t_n \in \mathcal{T} }[/math]. Most importantly, they could distinguish every same input [math]\displaystyle{ x_i }[/math] that have different target. To conclude, task descriptors carries crucial information of the example and able to distinguish different learning environment for similar examples.