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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. , NO. 4
gradually consider less smoothing versions, until the target
objective of interest. This strategy also shares the same spirit
with simulated annealing. As illustrated in Fig 3, continuation
methods provide a sequence of optimization objectives,
starting with a heavily smoothed objective for which it
is easy to find a global minimum, and tracking the local
minima throughout the training. In this way, continuation
methods guide the training towards better regions in param-
eter space, i.e., as shown in Fig 3, the local minima learned
from easier objectives have better generalization ability and
are more likely to approximate global minima. Moreover,
from the view of transfer learning, this continuation strategy
can also be regarded as a sequence of unsupervised pre-
training [6]: training on the preceding objectives could act
as a pre-training process which both helps optimization and
provides regularization on succeeding objectives.
Fig. 3. Illustration of the continuation method from [5], which is the
essence of the CL [6]. It starts from optimizing a heavily smoothed
version of the objective, and gradually moves to the target objective.
Tracking the local minima throughout the training guides the model
towards better parameter space and makes it more generalizable.
Additionally, recent studies provide more theoretical
evidence for the convergence speedup in CL from the opti-
mization perspective. Weinshall et al. [123] prove a theorem
On the other hand, researchers also analyze the CL
mechanism from the perspective of data distribution. In
the era of deep learning, large-scale data sources are re-
quired for training, which are collected and annotated by
company users, the web, and crowd-sourcing systems. This
big data collection brings noisy data that is less cogniz-
able or wrongly annotated. In the CL setting, the noisy
data corresponds to harder examples in the datasets while
the cleaner data form the easier part. Since CL strategy
encourages training more on the easier data, an intuitive
hypothesis is that CL learner wastes less time with the
harder and noisy examples to achieve faster training [6].
This hypothesis reveals the denoising efficacy of CL on noisy
data.
To have a closer look at this denoising mechanism,
Gong et al. [27] provide a theory based on the assumption
that there exists deviation between training and testing
distributions caused by noisy/wrongly-annotated training
data. Intuitively, training and target/testing distributions
share a common high-confidence annotated region with
large density, which corresponds to the easier examples
in CL. Therefore, to start training from easier examples
by CL strategy actually simulates learning from this high-
confidence common region (as an approximation to the
target distribution), which guides the learning towards the
expected target while reduces the negative impacts from
low-confidence noisy examples. This data distribution per-
spective of CL is illustrated in Fig 4. The common density
peak (at the center of the x-axis) of training and target
distributions P
train
(x) and P
target
(x) in the left part refers
to the common high-confidence area, while the heavy tail
of P
train
(x) demonstrates the relatively more noisy data in
training distribution. The right part illustrates the sequence
of weight functions in CL, which initially assigns small
values to the noisy tails and much larger values in the
common easy area, and gradually moves to equal weights
for all examples. Based on the above analysis, the authors
formulate P
target
(x) as the weighted expression of P
train
(x).
A follow-up theory clarifies that CL essentially minimizes
an upper bound of the expected risk under target distribu-
tion, and this bound shows that we could approach the task
of minimizing the expected risk on P
target
(x) by taking the
core idea of CL: gradually taking relatively easy examples
according to the curriculum and minimizing the empirical
risk on these examples.
Fig. 4. Illustration of the CL from the data distribution perspective [27].
The left part demonstrates the data distribution shifts from the easy
subset (the solid curve, which is assumed to approximate the testing
distribution P
target
(x) well) to the full training set P
train
(x) (the red dashed
curve). The right part shows the corresponding weighting scheme to
enable this distribution shift. The center peak of curves refers to the
high-confidence clean data, while the tails refer to the noisy data in the
distributions. As shown in the left part, P
target
(x) is cleaner than P
train
(x).
3.2 Suitable Application Scenes of CL
Based on the above analysis on why CL is effective, we
can categorize the motivations for applying CL into two
groups: to guide, regularizing the training towards better
regions in parameter space (with steeper gradients) as from
the perspective of the optimization problem, and to de-
noise, focusing on high-confidence easier area to alleviate
the interference of noisy data as from the perspective of
data distribution. Not surprisingly, most of the existing
application scenes of CL can be classified into these two
groups, as demonstrated in Table 1.
The application scenes based on the “to guide” motiva-
tion often involve difficult target tasks where direct training
on these tasks results in poor performance or slow conver-
gence. CL strategies are adopted to guide the training from
easier tasks or smoother versions of objectives to the target
tasks. For instance, in sparse-reward RL, direct training
on the final tasks rarely gets any positive rewards, which
hinders agent learning. Therefore, researchers propose to
take the CL strategy and manually [72] or automatically [20]
design a sequence of auxiliary (sub)tasks/goals from easy to
hard to guide the training. In multi-task learning, learning
all the tasks simultaneously or in random order often leads
to unsatisfactory performance. To yield performance gains,
CL strategies are adopted to automatically choose the easier
tasks which are more related to the previous one [83] or
can bring more learning progress to the model training [29],