From hyper-parameter optimisation to meta-learning
Cuong Nguyen 
Meta-learning, also known as learn-how-to-learning, has been being studied from 1980s (Schmidhuber 1987; Naik and Mammone 1992), and recently attracted much attention from the research community. Meta-learning is a technique in transfer learning — a learning paradigm that utilises knowledge gained from past experience to facilitate the learning in the future. Due to being defined implicitly
, meta -learning is often confused with other transfer learning techniques, e.g. fine-tuning, multi-task learning, domain adaptation and continual learning. The purpose of this post is to formulate meta-learning explicitly via empirical Bayes, and in particular hyper-parameter optimisation, to differentiate meta-learning from those common transfer learning approaches.
This post is structured as follows: First, we define some terminologies used in general transfer learning and review hyper-parameter optimisation in single-task setting. We then formulate meta-learning as an extension of hyper-parameter optimisation in multi-task setting. Finally, we show the differences between meta-learning and other transfer-learning approaches.
1 Background
1.1 Data generation model of a task
A data point of a task indexed by i \in \mathbb{N} consists of an input \mathbf{x}_{ij} \in \mathcal{X} \subseteq \mathbb{R}^{d} and a corresponding label \mathbf{y}_{ij} \in \mathcal{Y} with j \in \mathbb{N}. For simplicity, only two families of tasks – regression and classification – are considered in this thesis. As a result, the label is defined as \mathcal{Y} \subseteq \mathbb{R} for regression and as \mathcal{Y} = \{0, 1, \ldots, C - 1\} for classification, where C is the number of classes.
Each data point in a task can be generated in 2 steps:
- generate the input \mathbf{x}_{ij} by sampling from some probability distribution \mathcal{D}_{i},
- determine the label \mathbf{y}_{ij} = f(\mathbf{x}_{ij}), where f_{i}: \mathcal{X} \to \mathcal{Y} is the
correct
labelling function.
Both the probability distribution \mathcal{D}_{i} and the labelling function f_{i} are unknown to the learning agent during training, and the aim of the supervised learning is to use the generated data to infer such labelling function f.
For simplicity, we denote (\mathbf{x}_{ij}, \mathbf{y}_{ij}) \sim (\mathcal{D}_{i}, f_{i}) as the data generation model of task i-th.
1.2 Task instance
Definition 1 (Hospedales et al. 2021)
A task or a task instance \mathcal{T}_{i} consists of an unknown associated data generation model (\mathcal{D}_{i}, f_{i}), and a loss function \ell_{i}, denoted as: \mathcal{T}_{i} = \{(\mathcal{D}_{i}, f_{i}), \ell_{i}\}.
Remark. The loss function \ell_{i} is defined abstractly, and can be either:
negative log-likelihood (NLL): - \ln p(y_{ij} | \mathbf{x}_{ij}, \mathbf{w}_{i}), corresponding to maximum likelihood estimation. This type of loss is quite common in practice, for example:
- mean squared error (MSE) in regression
- cross-entropy in classification
variational-free energy (negative evidence lower-bound) — corresponding to the objective function in variational inference.
To solve a task \mathcal{T}_{i}, one needs to obtain an optimal task-specific model {h(.; \mathbf{w}_{i}^{*}): \mathcal{X} \to \mathcal{Y}}, parameterised by \mathbf{w}^{*}_{i} \in \mathcal{W} \subseteq \mathbb{R}^{n}, which minimises a loss function \ell_{i} on the data of that task: \mathbf{w}_{i}^{*} = \arg\min_{\mathbf{w}_{i}} \mathbb{E}_{(\mathbf{x}_{ij}, \mathbf{y}_{ij}) \sim (\mathcal{D}_{i}, f_{i})} \left[ \ell_{i} (\mathbf{x}_{ij}, \mathbf{y}_{ij}; \mathbf{w}_{i}) \right].
In practice, since both \mathcal{D}_{i} and f_{i} are unknown, the data generation model is replaced by a dataset consisting of a finite number of data-points generated according to the data generation model (\mathcal{D}_{i}, f_{i}), denoted as S_{i} = \{\mathbf{x}_{ij}, \mathbf{y}_{ij}\}_{j=1}^{m_{i}}. The objective to solve that task is often known as empirical risk minimisation: \mathbf{w}^{\mathrm{ERM}}_{i} = \arg\min_{\mathbf{w}_{i}} \frac{1}{m_{i}} \sum_{j = 1}^{m_{i}} \left[ \ell_{i} (\mathbf{x}_{ij}, \mathbf{y}_{ij}; \mathbf{w}_{i}) \right]. \tag{1}
Since the loss function used is the same for each task family, e.g. \ell is NLL or variational-free energy, the subscript on the loss function is, therefore, dropped, and the loss is denoted as \ell throughout this chapter. Furthermore, given the commonality of the loss function across all tasks, a task can, therefore, be simply represented by either its data generation model (\mathcal{D}_{i}, f_{i}) or the associated dataset S_{i}.
1.3 Hyper-parameter optimisation
In single-task setting, the common way to tune
or optimise a hyper-parameter is to split a given dataset S_{i} into two disjoint subsets:
\begin{aligned}
S_{i}^{(t)} \cup S_{i}^{(v)} & = S_{i}\\
S_{i}^{(t)} \cap S_{i}^{(v)} & = \varnothing,
\end{aligned}
where:
- S_{i}^{(t)} = \left\{ \left( \mathbf{x}_{ij}^{(t)}, y_{ij}^{(t)} \right) \right\}_{j=1}^{m_{i}^{(t)}} is the training (or support) subset,
- S_{i}^{(v)} = \left\{ \left( \mathbf{x}_{ij}^{(v)}, y_{ij}^{(v)} \right) \right\}_{j=1}^{m_{i}^{(v)}} is the validation (or query) subset.
Note that with this definition, m_{i}^{(t)} + m_{i}^{(v)} = m_{i}, and m_{i}^{(t)} and m_{i}^{(v)} are not necessarily identical.
The subset S_{i}^{(t)} is used to train the model parameter of interest \mathbf{w}_{i}, while the subset S_{i}^{(v)} is used to validate the hyper-parameter, denoted by \theta (we provide examples of the hyper-parameter in Section Formulation of meta-learning). Mathematically, hyper-parameter optimisation in the single-task setting can be written as the following bi-level optimisation: \begin{aligned} & \min_{\theta} \frac{1}{m_{i}^{(v)}} \sum_{k = 1}^{m_{i}^{(v)}} \ell \left( \mathbf{x}_{ik}^{(v)}, y_{ik}^{(v)}; \mathbf{w}_{i}^{*} (\theta) \right)\\ & \text{s.t.: } \mathbf{w}_{i}^{*} (\theta) = \arg\min_{\mathbf{w}_{i}} \frac{1}{m_{i}^{(t)}} \sum_{j = 1}^{m_{i}^{(t)}} \ell \left( \mathbf{x}_{ij}^{(t)}, y_{ij}^{(t)}; \mathbf{w}_{i} (\theta) \right). \end{aligned}
We can extend the hyper-parameter optimisation from the two data subsets S_{i}^{(t)} and S_{i}^{(v)} to the general data generation model as the following: \begin{aligned} & \min_{\theta} \mathbb{E}_{\left( \mathbf{x}_{ik}^{(v)}, y_{ik}^{(v)} \right) \sim \left( \mathcal{D}_{i}^{(v)}, f_{i} \right)} \left[ \ell \left( \mathbf{x}_{ik}^{(v)}, y_{ik}^{(v)}; \mathbf{w}_{i}^{*} (\theta) \right) \right]\\ & \text{s.t.: } \mathbf{w}_{i}^{*} (\theta) = \arg\min_{\mathbf{w}_{i}} \mathbb{E}_{\left( \mathbf{x}_{ik}^{(t)}, y_{ik}^{(t)} \right) \sim \left( \mathcal{D}_{i}^{(t)}, f_{i} \right)} \left[ \ell \left( \mathbf{x}_{ij}^{(t)}, y_{ij}^{(t)}; \mathbf{w}_{i} (\theta) \right) \right], \end{aligned} where \mathcal{D}_{i}^{(t)} and \mathcal{D}_{i}^{(v)} are the probability distributions of training and validation input data, respectively, and they are not necessarily identical.
Formulation of meta-learning
The setting of the meta-learning problem considered in this paper follows the task environment (Baxter 2000) that describes the unknown distribution p(\mathcal{D}, f) over a family of tasks. Each task \mathcal{T}_{i} is sampled from this task environment and can be represented as \left( \mathcal{D}_{i}^{(t)}, \mathcal{D}_{i}^{(v)}, f_{i} \right), where \mathcal{D}_{i}^{(t)} and \mathcal{D}_{i}^{(v)} are the probability of training and validation input data, respectively, and are not necessarily identical. The aim of meta-learning is to use T training tasks to train a meta-learning model that can be fine-tuned to perform well on an unseen task sampled from the same task environment.
Such meta-learning methods use meta-parameters to model the common latent structure of the task distribution p(\mathcal{D}, f). In this thesis, we consider meta-learning as an extension of hyper-parameter optimisation in single-task learning, where the hyper-parameter of interest — often called meta-parameter — is shared across many tasks. Similar to hyper-parameter optimisation presented in subsection hyper-parameter-optimisation, the objective of meta-learning is also a bi-level optimisation: \begin{aligned} & \min_{\theta} \textcolor{red}{\mathbb{E}_{\mathcal{T}_{i} \sim p \left( \mathcal{D}, f_{i} \right)}} \mathbb{E}_{ \left( \mathbf{x}_{ik}^{(v)}, y_{ik}^{(v)} \right) \sim \left( \mathcal{D}_{i}^{(v)}, f_{i} \right)} \left[ \ell \left( \mathbf{x}_{ik}^{(v)}, y_{ik}^{(v)}; \mathbf{w}_{i}^{*}(\theta) \right) \right]\\ & \text{s.t.: } \mathbf{w}^{*}_{i}(\theta) = \arg\min_{\mathbf{w}_{i}} \mathbb{E}_{\left( \mathbf{x}_{ij}^{(t)}, y_{ij}^{(t)} \right) \sim \left( \mathcal{D}_{i}^{(t)}, f_{i} \right)} \left[ \ell \left( \mathbf{x}_{ij}^{(t)}, y_{ij}^{(t)}; \mathbf{w}_{i}(\theta) \right) \right]. \end{aligned} \tag{2}
The difference between meta-learning and hyper-parameter optimisation is that the meta-parameter (also known as hyper-parameter) \theta is shared across all tasks sampled from the task environment p(\mathcal{D}, f) as highlighted in red colour in Equation 2.
In practice, the meta-parameter (or shared hyper-parameter) \theta can be chosen as one of the followings:
- learning rate of gradient-based optimisation used to minimise the lower level objective function in Equation 2 to learn \mathbf{w}_{i}^{*} \left(\theta\right) (Z. Li et al. 2017),
- initialisation of model parameter (Finn, Abbeel, and Levine 2017),
- data representation or feature extractor (Vinyals et al. 2016; Snell, Swersky, and Zemel 2017),
- optimiser used to optimise the lower-level in Equation 2.
In this post, the meta-parameter \theta is assumed to be the initialisation of model parameters. Formulation, derivation and analysis in the subsequent sections and chapters will, therefore, revolve around this assumption. Note that the analysis can be straight-forwardly extended to other types of meta-parameters with slight modifications.
In general, the objective function of meta-learning in Equation 2 can be solved by gradient-based optimisation, such as gradient descent. Due to the nature of the bi-level optimisation, the optimisation are often carried out in two steps. The first step is to adapt (or fine-tuned) the meta-parameter \theta to the task-specific parameter \mathbf{w}_{i}(\theta). This corresponds to the optimisation in the lower-level, and can be written as: \mathbf{w}_{i}^{*}(\theta) = \theta - \alpha \mathbb{E}_{\left( \mathbf{x}_{ij}^{(t)}, y_{ij}^{(t)} \right) \sim \left( \mathcal{D}_{i}^{(t)}, f_{i} \right)} \left[ \nabla_{\theta} \ell \left( \mathbf{x}_{ij}^{(t)}, \mathbf{y}_{ij}^{(t)}; \mathbf{w}_{i}(\theta) \right) \right], \tag{3} where \alpha is a hyper-parameter denoting the learning rate for task \mathcal{T}_{i}. For simplicity, the adaptation step in Equation 3} is carried out with only one gradient descent update.
The second step is to minimise the validation loss induced by the locally-optimal task-specific parameter \mathbf{w}_{i}^{*}(\theta) evaluated on the validation subset w.r.t. the meta-parameter \theta. This corresponds to the upper-level optimisation, and can be expressed as: \theta \gets \theta - \gamma \mathbb{E}_{\mathcal{T}_{i} \sim p(\mathcal{D}, f)} \mathbb{E}_{ \left( \mathbf{x}_{ik}^{(v)}, \mathbf{y}_{ik}^{(v)} \right) \sim \left( \mathcal{D}_{i}^{(v)}, f_{i} \right)} \left[ \nabla_{\theta} \ell \left( \mathbf{x}_{ij}^{(v)}, \mathbf{y}_{ij}^{(v)}; \mathbf{w}_{i}^{*}(\theta) \right) \right], \tag{4} where \gamma is another hyper-parameter representing the learning rate to learn \theta.
The general algorithm of meta-learning using gradient-based optimisation is shown in Algorithm 1.
\begin{algorithm}
\caption{Training procedure of meta-learning in general}
\begin{algorithmic}
\PROCEDURE{Training}{task environment $p(\mathcal{D}, f)$, learning rates $\gamma$ and $\alpha$}
\STATE initialise meta-parameter $\theta$
\WHILE{$\theta$ not converged}
\STATE sample a mini-batch of $T$ tasks from task environment $p\left( \mathcal{D}, f \right)$
\FOR{each task $\mathcal{T}_{i}, i \in \{1, \ldots, T\}$}
\STATE sample two data subsets $S_{i}^{(t)}$ and $S_{i}^{(v)}$ from task $\mathcal{T}_{i} = (\mathcal{D}_{i}^{(t)}, \mathcal{D}_{i}^{(v)}, f_{i})$
\STATE adapt meta-parameter to task $\mathcal{T}_{i}$: $\mathbf{w}_{i}^{*} \left( \theta \right) = \theta - \frac{\alpha}{m_{i}^{(t)}} \sum_{j = 1}^{m_{i}^{(t)}} \nabla_{\theta} \left[ \ell \left( \mathbf{x}_{ij}^{(t)}, y_{ij}^{(t)}; \theta \right)\right]$ \COMMENT{Eq. @eq-task_adaptation_sgd}}
\ENDFOR
\STATE update meta-parameter: $\theta \gets \theta - \frac{\gamma}{T} \sum_{i=1}^{T} \frac{1}{m_{i}^{(v)}} \sum_{k=1}^{m_{i}^{(v)}} \nabla_{\theta} \left[\ell \left( \mathbf{x}_{ik}^{(v)}, y_{ik}^{(v)}; \mathbf{w}_{i}^{*} \left( \theta \right) \right) \right]$ \COMMENT{Eq. @eq-meta_parameter_update_sgd}}
\ENDWHILE
\STATE \textbf{return} the trained meta-parameter $\theta$
\ENDPROCEDURE
\end{algorithmic}
\end{algorithm}
1.4 Second-order meta-learning
As shown in Equation 4, the optimisation for the meta-parameter \theta requires the gradient of the validation loss averaged across T tasks. Given that each task-specific parameter \mathbf{w}_{i}^{*} is a function of \theta due to the lower-level optimisation in Equation 3, the gradient of interest can be expanded as: \begin{aligned} & \mathbb{E}_{\mathcal{T}_{i} \sim p \left( \mathcal{D}, f \right)} \mathbb{E}_{\left( \mathbf{x}_{ik}^{(v)}, y_{ik}^{(v)} \right) \sim \left( \mathcal{D}_{i}^{(v)}, f_{i} \right)} \left[ \nabla_{\theta} \ell \left( \mathbf{x}_{ik}^{(v)}, y_{ik}^{(v)}; \mathbf{w}_{i}^{*}(\theta) \right) \right]\\ & = \mathbb{E}_{\mathcal{T}_{i} \sim p \left( \mathcal{D}, f \right)} \mathbb{E}_{\left( \mathbf{x}_{ik}^{(v)}, y_{ik}^{(v)} \right) \sim \left( \mathcal{D}_{i}^{(v)}, f_{i} \right)} \left[ \nabla_{\theta}^{\top} \mathbf{w}_{i}^{*} \left( \theta \right) \times \nabla_{\mathbf{w}_{i}^{*}(\theta)} \ell \left( \mathbf{x}_{ik}^{(v)}, y_{ik}^{(v)}; \mathbf{w}_{i}^{*}(\theta) \right) \right]\\ & = \mathbb{E}_{\mathcal{T}_{i} \sim p \left( \mathcal{D}, f \right)} \left\{ \left[ \mathbf{I} - \alpha \mathbb{E}_{ \left( \mathbf{x}_{ij}^{(t)}, y_{ij}^{(t)} \right) \sim \left( \mathcal{D}_{i}^{(t)}, f_{i} \right)} \left[ \textcolor{red}{\nabla_{\theta}^{2} \ell \left( \mathbf{x}_{ij}^{(t)}, y_{ij}^{(t)}; \theta \right)} \right] \right] \right.\\ & \quad \times \left. \mathbb{E}_{\left( \mathbf{x}_{ik}^{(v)}, y_{ik}^{(v)} \right) \sim \left( \mathcal{D}_{i}^{(v)}, f_{i} \right)} \left[ \textcolor{green}{\nabla_{\mathbf{w}_{i}^{*}(\theta)} \ell \left( \mathbf{x}_{ik}^{(v)}, y_{ik}^{(v)}; \mathbf{w}_{i}^{*}(\theta) \right)} \right] \right\}, \end{aligned} where the first equality is due to chain rule, and the second equality is the result that differentiates the gradient update in Equation 3. Note that in the second equality, we remove the transpose notation since the corresponding matrix is symmetric.
Thus, naively implementing such gradient would require to calculate the Hessian matrix \textcolor{Red}{\nabla_{\theta}^{2} \ell \left( \mathbf{x}_{ij}^{(t)}, y_{ij}^{(t)}; \theta \right)}, resulting in an intractable procedure for large models, such as deep neural networks. To obtain a more efficient implementation, one can utilise the Hessian-vector product (Pearlmutter 1994) between the gradient vector \textcolor{green}{\nabla_{\mathbf{w}_{i}^{*}(\theta)} \ell \left( \mathbf{x}_{ik}^{(v)}, \mathbf{y}_{ik}^{(v)}; \mathbf{w}_{i}^{*}(\theta) \right)} and the Hessian matrix \textcolor{red}{\nabla_{\theta}^{2} \ell \left( \mathbf{x}_{ij}^{(t)}, y_{ij}^{(t)}; \theta \right)} to efficiently calculate the gradient of the validation loss w.r.t. \theta.
Another way to calculate the gradient of the validation loss w.r.t. the meta-parameter \theta is to use implicit differentiation (Domke 2012; Rajeswaran et al. 2019; Lorraine, Vicol, and Duvenaud 2020). This approach is more advantaged since it does not need to stores the computational graph and takes gradient via chain rule. Such implicit differentiation technique reduces the memory usage and therefore, allows to work with large-scale models. However, the trade-off is the increasing computational time to apply the chain rule to calculate the gradient of interest.
Nevertheless, the implementations that compute the exact gradient of the validation loss w.r.t. \theta without approximation are often referred to as second-order
meta-learning.
1.5 First-order meta-learning
In practice, the Hessian matrix \textcolor{Red}{\nabla_{\theta}^{2} \ell \left( \mathbf{x}_{ij}^{(t)}, y_{ij}^{(t)}; \theta \right)} is often omitted from the calculation to simplify the update for the meta-parameter \theta (Finn, Abbeel, and Levine 2017). The resulting gradient consists of only the gradient of validation loss \textcolor{Green}{\nabla_{\mathbf{w}_{i}^{*}(\theta)} \ell \left( \mathbf{x}_{ik}^{(v)}, y_{ij}^{(v)}; \mathbf{w}_{i}^{*}(\theta) \right)}, which is more efficient to calculate with a single forward-pass if auto differentiation is used. This approximation is often referred as first-order
meta-learning, and the gradient of interest can be presented as:
\begin{aligned}
& \mathbb{E}_{\mathcal{T}_{i} \sim p \left( \mathcal{D}, f \right)} \mathbb{E}_{\left( \mathbf{x}_{ik}^{(v)}, y_{ik}^{(v)} \right) \sim \left(\mathcal{D}_{i}^{(v)}, f_{i} \right)} \left[ \nabla_{\theta} \ell \left( \mathbf{x}_{ij}^{(v)}, y_{ik}^{(v)}; \mathbf{w}_{i}^{*}(\theta) \right) \right] \\
& \approx \mathbb{E}_{\mathcal{T}_{i} \sim p \left( \mathcal{D}, f \right)} \mathbb{E}_{\left( \mathbf{x}_{ik}^{(v)}, y_{ik}^{(v)} \right) \sim \left( \mathcal{D}_{i}^{(v)}, f_{i} \right)} \left[ \textcolor{Green}{\nabla_{\mathbf{w}_{i}^{*}(\theta)} \ell \left( \mathbf{x}_{ij}^{(v)}, y_{ik}^{(v)}; \mathbf{w}_{i}^{*}(\theta) \right)} \right].
\end{aligned}
REPTILE [Nichol, Achiam, and Schulman (2018)} — a variant first-order meta-learning — approximates further the gradient of validation loss \textcolor{Green}{\nabla_{\mathbf{w}_{i}^{*}(\theta)} \ell \left( \mathbf{x}_{ij}^{(v)}, y_{ik}^{(v)}; \mathbf{w}_{i}^{*}(\theta) \right)} by the difference \theta - \mathbf{w}_{i}^{*}, resulting in a much simpler approximation: \mathbb{E}_{\mathcal{T}_{i} \sim p \left( \mathcal{D}, f \right)} \mathbb{E}_{\left( \mathbf{x}_{ik}^{(v)}, \mathbf{y}_{ik}^{(v)} \right) \sim \left( \mathcal{D}_{i}^{(v)}, f_{i} \right)} \left[ \nabla_{\theta} \ell \left( \mathbf{x}_{ik}^{(v)}, y_{ik}^{(v)}; \mathbf{w}_{i}^{*}(\theta) \right) \right] = \theta - \mathbb{E}_{\mathcal{T}_{i} \sim p \left( \mathcal{D}, f \right)} \left[ \mathbf{w}_{i}^{*}(\theta) \right].
2 Differentiation from other transfer learning approaches
In this section, some popular transfer learning methods are described with their objective functions to purposely distinguish from meta-learning.
2.1 Fine-tuning
Fine-tuning is the most common technique in neural network based transfer learning (Pratt et al. 1991; Yosinski et al. 2014) where the last or a couple of last layers in a neural network pre-trained on a source task are replaced and fine-tuned on a target task. Formally, if g(.; \mathbf{w}_{0}) is denoted as the forward function of the shared layers with shared parameters \mathbf{w}_{0}, where \mathbf{w}_{s} and \mathbf{w}_{t} are the parameters of the remaining layers h specifically trained on source and target tasks, respectively, then the objective of fine-tuning can be expressed as: \begin{aligned} & \min_{\mathbf{w}_{t}} \mathbb{E}_{(\mathbf{x}_{t}, \mathbf{y}_{t}) \sim \mathcal{T}_{t}} \left[ \ell \left( h\left( g\left( \mathbf{x}_{t}; \mathbf{w}_{0}^{*} \right); \mathbf{w}_{t} \right), \mathbf{y}_{t} \right) \right] \\ & \text{s.t.: } \mathbf{w}_{0}^{*}, \mathbf{w}_{s}^{*} = \arg\min_{\mathbf{w}_{0}, \mathbf{w}_{s}} \mathbb{E}_{(\mathbf{x}_{s}, \mathbf{y}_{s}) \sim \mathcal{T}_{s}} \left[ \ell \left( h \left( g\left( \mathbf{x}_{s}; \mathbf{w}_{0} \right); \mathbf{w}_{s} \right), \mathbf{y}_{s} \right) \right], \end{aligned} \tag{5}
where \mathbf{x}_{s}, \mathbf{y}_{s} and \mathbf{x}_{t}, \mathbf{y}_{t} are the data sampled from the source task \mathcal{T}_{s} and target task \mathcal{T}_{t}, respectively.
Although the objective of fine-tuning shown in Equation 5 is still a bi-level optimisation, it is easier to solve than the one in meta-learning due to the following reasons:
- The objective in fine-tuning has only one constrain corresponding to one source task, while meta-learning has several constrains corresponding to multiple training tasks.
- In fine-tuning, \mathbf{w}_{t} and \mathbf{w}_{0} are inferred separately, while in meta-learning, the task-specific parameter is a function of the meta-parameter, resulting in a more complicated correlation.
The downside of fine-tuning is the requirement of a reasonable number of training examples on the target task to fine-tune \mathbf{w}_{t}. In contrast, meta-learning leverages the knowledge extracted from several training tasks to quickly adapt to a new task with only a few training examples.
2.2 Domain adaptation and generalisation
Domain adaptation or domain-shift refers to the case when the joint data-label distribution on source and target are different, denoted as p_{s} \left( \mathcal{D}, f \right) \neq p_{t} \left( \mathcal{D}, f \right), or simply p_{s}(\mathbf{x}, \mathbf{y}) \neq p_{t}(\mathbf{x}, \mathbf{y})~(Heckman 1979; Shimodaira 2000; Japkowicz and Stephen 2002; Daume III and Marcu 2006; Ben-David et al. 2007). The aim of domain adaptation is to leverage the model trained on source domain to available data in the target domain, so that the model adapted to the target domain can perform reasonably well. In other words, domain adaptation relies on a data transformation g(., .; \mathbf{w}_{0}): \mathcal{X} \times \mathcal{Y} \to \mathcal{X}^{\prime} \times \mathcal{Y}^{\prime} that produces a domain-invariant latent space. Mathematically, the transformation g is obtained by minimising a divergence between the two transformed data distribution: \begin{aligned} & \min_{\mathbf{w}_{0}} \mathrm{Divergence} \left[ p\left( \mathbf{x}_{s}^{\prime}, \mathbf{y}_{s}^{\prime} \right) || p\left( \mathbf{x}_{t}^{\prime}, \mathbf{y}_{t}^{\prime} \right) \right]\\ & \text{s.t.: } \left( \mathbf{x}_{i}^{\prime}, \mathbf{y}_{i}^{\prime} \right) = g \left( \mathbf{x}_{i}, \mathbf{y}_{i}; \mathbf{w}_{0} \right), i \in \{s, t\}. \end{aligned} \tag{6}
After obtaining the transformation g, one can simply train a model using the transformed data of the source domain, and then use that model to make predictions on the target domain.
Given the optimisation in~Equation 6, domain adaptation is different from meta-learning due to the following reasons:
- Domain adaptation assumes a shift in the task environments that generate source and target tasks, while meta-learning is based on the assumption of same task generation.
- Domain adaptation utilises information of data from target domain, while meta-learning does not have such access.
In general, meta-learning learns a shared prior or hyper-parameters to generalise for unseen tasks, while domain adaptation produces a model to solve a particular task in a specified target domain. Recently, there is a variance of domain adaptation, named domain generalisation, where the aim is to learn a domain-invariant model without any information of target domain. In this view, domain generalisation is very similar to meta-learning, and there are some works that employ meta-learning algorithms for domain generalisation (D. Li et al. 2018; Y. Li et al. 2019).
2.3 Multi-task learning
Multi-task learning learns several related auxiliary tasks and a target task simultaneously to exploit the diversity of task representation to regularise and improve the performance on the target task~(Caruana 1997). If the input \mathbf{x} is assumed to be the same across T extra tasks and the target task \mathcal{T}_{T + 1}, then the objective of multi-task learning can be expressed as: \min_{\mathbf{w}_{0}, \{\mathbf{w}_{i}\}_{i = 1}^{T + 1}} \frac{1}{T + 1} \sum_{i = 1}^{T + 1} \ell_{i} \left( h_{i} \left( g\left( \mathbf{x}; \mathbf{w}_{0} \right); \mathbf{w}_{i} \right), \mathbf{y}_{i} \right), \tag{7} where \mathbf{y}_{i}, \ell_{i} and h_{i} are the label, loss function and the classifier for task \mathcal{T}_{i}, respectively, and g(., \mathbf{w}_{0}) is the shared feature extractor for T + 1 tasks.
Multi-task learning is often confused with meta-learning due to their similar nature extracting information from many tasks. However, the objective function of multi-task learning in Equation 7 is a single-level optimisation for the shared parameter \mathbf{w}_{0} and multiple task-specific classifier \{\mathbf{w}_{i}\}_{i = 1}^{T + 1}. It is, therefore, not as complicated as a bi-level optimisation seen in meta-learning as shown in Equation 2. Furthermore, multi-task learning aims to solve a number of specific tasks known during training (referred to as target tasks), while meta-learning targets the generalisation for unseen tasks in the future.
2.4 Continual learning
Continual or life-long learning refers to a situation where a learning agent has access to a continuous stream of tasks available over time, and the number of tasks to be learnt is not pre-defined~(Chen and Liu 2018; Parisi et al. 2019). The aim is to accommodate the knowledge extracted from one-time observed tasks to accelerate the learning of new tasks without catastrophically forgetting old tasks~(French 1999). In this sense, continual learning is very similar to meta-learning. However, continual learning most likely focuses on systematic design to acquire new knowledge in such a way that prevents interfering to the existing one, while meta-learning is more about algorithmic design to learn the new knowledge more efficiently. Thus, we cannot mathematically distinguish their differences as done in sub-sections Fine-tuning, Domain adaptation and generalisation and Multi-task learning . Nevertheless, continual learning criteria, especially catastrophic forgetting, can be encoded into meta-learning objective to advance further continual learning performance~(Al-Shedivat et al. 2018; Nagabandi et al. 2019).
3 Summary
In general, meta-learning is an extension of hyper-parameter optimisation in multi-task setting. The objective function of meta-learning is, therefore, a bi-level optimisation, where the lower-level is to adapt the meta-parameter to a task, while the upper-level is to evaluate how well the meta-parameter performs across T tasks. Given such mathematical formulation, we can easily distinguish meta-learning from some common transfer learning approaches, such as fine-tuning, multi-task learning, domain adaptation and continual learning.
Hope that this post would give another perspective of meta-learning. I’ll see you in the next post about probabilistic methods in meta-learning.
4 References
Reuse
Citation
@online{nguyen2021,
author = {Nguyen, Cuong},
title = {From Hyper-Parameter Optimisation to Meta-Learning},
date = {2021-11-22},
url = {https://cnguyen10.github.io/posts/meta-learning},
langid = {en}
}