Abstract. In 2020 we are celebrating the 1/3 century anniversary of my first publication on
metalearning or learning to learn:
diploma thesis of 1987 [META1]
For its cover I drew a robot that bootstraps itself (image above).
[META1] was the first in a long series of publications on this topic, which became hot
in the 2010s
Here I also summarize our work on
meta-reinforcement learning with
self-modifying policies since 1994 [METARL2-9]
gradient descent-based metalearning in artificial neural networks
since 1992 [FASTMETA1-5]
asymptotically optimal metalearning for curriculum learning since 2002 [OOPS1-3]
mathematically optimal metalearning through the
self-referential Gödel Machine since 2003 [GM3-9]
Meta-RL combined with artificial curiosity and intrinsic motivation
since 1990/1997 (Sec. 6),
and recent work of 2020 (Sec. 7).
The most widely used machine learning algorithms were invented and hardwired by humans.
Can we also construct metalearning (or meta-learning) algorithms that can learn better learning
algorithms, to build truly self-improving AIs without any limits other than the limits of computability and physics?
This question has been a main drive of my research since
my 1987 diploma thesis on this topic [META1].
First note that metalearning is sometimes confused with simple transfer learning
from one training set to another (see N(eur)IPS 2016 slides).
a standard deep feedforward neural network (NN) can
transfer-learn to learn new images faster through pre-training on other image sets, e.g., [TRA12].
True metalearning is much more than that, and also
much more than just learning to adjust hyper-parameters
such as mutation rates in evolution strategies.
True metalearning is about encoding the initial learning algorithm
in a universal programming
language (e.g., on a recurrent neural network or RNN), with primitive
instructions that allow for modifying the code itself
in arbitrary computable fashion. We surround this self-referential, self-modifying code
by a recursive framework that ensures that only "useful" self-modifications survive,
to allow for
Recursive Self-Improvement (RSI),
e.g., Sec. 2, Sec. 5.
Metalearning may be
the most ambitious but also the most
rewarding goal of machine learning.
There are few limits to what
a good metalearner will learn.
Where appropriate, it will learn to
learn by analogy, by chunking, by planning,
by subgoal generation, by combinations
thereof—you name it.
1. Meta-Evolution and PSALMs (1987)
In 1987, we published
[GP87] [GP] what I think was the first paper on
Genetic Programming or GP
for evolving programs of unlimited size written in
a universal programming language [GOD] [GOD34] [CHU] [TUR] [POS].
In the same year, Sec. 2 of my diploma thesis [META1]
applied such GP to itself, to recursively evolve
better GP methods. There was not only a meta-level but also a meta-meta-level and a meta-meta-meta-level etc.
I called this Meta-Evolution.
Sec. 4 of [META1] also introduced metalearning
Prototypical Self-Referential Associating Learning Mechanisms (PSALMs) for payoff maximisation
or Reinforcement Learning (RL).
This was a first kind of meta-RL.
2. Meta-Reinforcement Learning with Self-Modifying Policies (1994-)
In 1994, I proposed another
type of meta-RL called incremental self-improvement [METARL2] for
general purpose RL machines
with a single life consisting of
a single lifelong trial. That is, unlike in
traditional RL, there is no assumption of repeatable independent trials, and the RL
agent is never reset. It is driven by a
(SMP) which is a modifiable probability distribution over programs
written in a universal programming language [GOD] [GOD34] [CHU] [TUR] [POS], to allow for arbitrary computations.
The learning algorithm of an SMP
is part of the SMP itself—SMPs can modify the way
they modify themselves. The credit assignment process has to take into account that early self-modifications are setting the stage
for later ones.
A method called
Environment-Independent Reinforcement Acceleration (EIRA) [METARL4] or
Success-Story Algorithm [METARL7-9]
forces SMPs to come
up with better and better self-modification algorithms that continually improve reward intake per time [METARL2-9]. This worked well in challenging experiments, although compute back then was 100,000 times more expensive than today. See
talk slides (2003)
N(eur)IPS WS 2018 overview slides.
As I have frequently pointed out since 1990 [AC90],
the connection strengths or weights of
an artificial neural network (NN) should be viewed as its program.
Inspired by Gödel's universal self-referential formal systems [GOD] [GOD34],
I built NNs whose outputs are programs or weight matrices of other NNs [FAST0-2],
and even self-referential recurrent NNs (RNNs)
that can run and inspect their own weight change algorithms or learning algorithms [FASTMETA1-5].
A difference to Gödel's work was that my universal programming language was not based on the integers,
but on real-valued weights, such that
each NN's output is differentiable with respect to its program.
That is, a simple program generator (the efficient
gradient descent procedure [BP1]—compare [BP2] [BPA] [BP4] [R7])
can compute a direction in program space where one may find a better program [AC90],
in particular, a
better program-generating program [FAST0-2].
Much of my work since 1989 has exploited this fact.
3. Gradient-Based NNs Learn to Program Other NNs (1991) and Themselves (1992)
Successful learning in deep architectures
started in 1965
Ivakhnenko & Lapa published the first general, working learning algorithms for deep multilayer perceptrons with arbitrarily many hidden layers. Their nets already contained the now popular multiplicative gates
[DEEP1-2] [DL1] [DL2],
an essential ingredient of what was later called NNs with
dynamic links or
v. d. Malsburg was the first to explicitly emphasize the importance of NNs with such rapidly changing connections
[FAST]; others followed [T20].
However, these authors did not yet have an end-to-end differentiable system that learns by gradient descent to quickly manipulate the fast weight storage. Such a system I published in 1991 [FAST0] [FAST1].
There a slow NN learns to control the weights of a separate fast NN.
That is, I separated storage and control like in traditional computers,
but in a fully neural way (rather than in a hybrid fashion [PDA1] [PDA2] [DNC]).
(Compare my related work on
what's now sometimes called
Synthetic Gradients [NAN1-5].)
Then I showed how fast weights can be used for meta-learning or
"learning to learn."
In references [FASTMETA1-5] since 1992, the slow RNN and the fast RNN are identical.
The RNN can see its own errors or reward signals called eval(t+1) in the image (from [FASTMETA5]).
The initial weight of each connection is trained by gradient descent, but during a training episode, each connection can be addressed
and read and modified by the RNN itself through O(log n) special output units, where n is the number of connections—see time-dependent vectors mod(t), anal(t), Δ(t), val(t+1) in the image. That is, each connection's weight may rapidly change, and the network becomes self-referential in the sense that it can in principle run arbitrary computable weight change algorithms or learning algorithms (for all of its weights) on itself.
In 1993, I simplified this through gradient descent-based, active control of fast weights through 2D tensors or outer product updates [FAST2] (compare our more recent work on this [FAST3] [FAST3a]).
was to get many more temporal variables under massively parallel end-to-end differentiable control than what's possible in standard RNNs of the same size: O(H2) instead of O(H), where H is the number of hidden units (compare Sec. 8 of [MIR] and item (7) of Sec. XVII of
also explicitly addressed the learning of
internal spotlights of attention
in end-to-end differentiable networks [FAST2] [ATT].
In 2001, my former student
used gradient descent in LSTM networks [LSTM1] instead of traditional
RNNs to metalearn
fast online learning
algorithms for nontrivial classes of functions, such as all quadratic
functions of two variables [HO1].
Today, the most famous end-to-end differentiable fast weight-based NN [FAST0] is actually our vanilla LSTM network of 2000 [LSTM2] (compare Sec. 4 & Sec. 8 of [MIR]), whose forget gates learn to control the fast weights on self-recurrent connections of internal LSTM cells. All the
major IT companies are now massively using
vanilla LSTM [DL4].
4. Asymptotically Optimal Metalearning for Curriculum Learning (2002-)
In 2002, I introduced a general and asymptotically time-optimal type of curriculum learning, that is, solving one problem after another, efficiently searching the space of programs that compute solution candidates, including those programs that organize and manage and adapt and reuse earlier acquired knowledge [OOPS1-3]. The
Optimal Ordered Problem Solver
(OOPS) draws inspiration from Levin's
Universal Search [OPT]
designed for single problems. It spends part of the total search time for a new problem on testing programs that exploit previous solution-computing programs in computable ways. If the new problem can be solved faster by copy-editing/invoking previous code than by solving the new problem from scratch, then OOPS will find this out. If not, then at least the previous solutions will not cause much harm. I introduced an efficient, recursive,
backtracking-based way of implementing OOPS
on realistic computers with limited storage. Experiments illustrated how OOPS can greatly profit from metalearning or metasearching, that is, searching for faster search procedures [OOPS1-2]. The image shows my poster on OOPS at N(eur)IPS 2003.
5. Self-Improving Gödel Machine (2003-)
The self-referential meta-RL system of Sec. 2 above (1994-) justified its
self-modifications through growing statistical evidence of subsequent reward accelerations.
But it was not guaranteed to execute theoretically optimal self-improvements.
This motivated my
Gödel Machine [GM3-9], which
was the first fully self-referential universal [UNI]
metalearner that was indeed optimal in a certain mathematical sense.
Typically it uses the somewhat less general
Optimal Ordered Problem Solver
[OOPS1-2] (Sec. 4)
for finding provably optimal self-improvements.
The metalearning Gödel Machine is inspired by Kurt Gödel, the founder of theoretical computer science
in the early 1930s [GOD] [GOD34].
He introduced the first universal coding language. It was
based on the integers,
and allows for formalizing the operations of any digital computer in axiomatic form.
Gödel used it to represent both data (such as axioms and theorems) and programs (such as proof-generating sequences of operations on the data).
He famously constructed formal statements that talk about the computation of other formal statements, especially self-referential statements which imply that they are not provable by any computational theorem prover. Thus he identified fundamental limits of mathematics and theorem proving and computing and Artificial Intelligence (AI) [GOD].
This had enormous impact on science and philosophy of the 20th century.
Furthermore, much of early AI in the 1940s-70s was actually about theorem proving and deduction in Gödel style through expert systems and logic programming.
Compare Sec. 18 of [MIR] and Sec. IV of [T20].
A Gödel Machine [GM6] is a general RL machine that will rewrite any part of its own code as soon as it has found a proof that the rewrite is useful, where the problem-dependent utility function and the hardware and the entire initial code are described by axioms encoded in an initial proof searcher which is also part of the initial code. While the machine is
interacting with its environment (initially in a suboptimal way),
the searcher systematically and efficiently tests computable proof techniques (programs whose outputs are proofs) until it finds a provably useful, computable self-rewrite. I showed that such a self-rewrite is globally optimal—no local maxima!—since the code first had to prove that it is not useful to continue the proof search for alternative self-rewrites. Unlike previous non-self-referential methods based on hardwired proof searchers, the Gödel Machine not only boasts an optimal order of complexity but can optimally reduce any slowdowns hidden by the O()-notation, provided the utility of such speed-ups is provable at all [GM3-9].
Before I continue the discussion of metalearning,
let me first explain RL with intrinsic motivation.
My popular principle of adversarial
from 1990 [AC90, AC90b] [AC20] (see also surveys [AC09] [AC10])
is now widely used not only for exploration in RL but also
for image synthesis [AC20] [T20]. It
works as follows. One NN (the controller) probabilistically generates outputs, another NN (the world model) sees those outputs and predicts environmental reactions to them. Using gradient descent, the world model NN minimizes its error, while the generator NN tries to make outputs that maximize this error. One net's loss is the other net's gain.
So the controller is intrinsically motivated to generate output actions or experiments that
yield data from which the world model can still learn something.
(GANs are a special case of this where the environment simply returns 1 or 0 depending on whether the generator's output is in a given set [AC20]; compare [R2] and Sec. 5 of [MIR] and Sec. XVII of
6. Meta-RL plus Artificial Curiosity and Intrinsic Motivation (1990, 1997-)
The Section "A Connection to Meta-Learning" in [AC90] (1990) already pointed out:
"A model network can be used not only for predicting the controller's inputs but also for predicting its future outputs. A perfect model of this kind would model the internal changes of the control network. It would predict the evolution of the controller, and thereby the effects of the gradient descent procedure itself. In this case, the flow of activation in the model network would model the weight changes of the control network. This in turn comes close to the notion of learning how to learn."
The paper [AC90] also introduced
planning with recurrent NNs (RNNs) as world models,
and high-dimensional reward signals.
Unlike in traditional RL,
those reward signals were also used as informative inputs to the controller NN
learning to execute actions that maximise cumulative reward
(see also Sec. 13 of [MIR]
and Sec. 5 of [DEC]).
This is important for metalearning: an NN that cannot see its own errors or rewards cannot learn
a better way of using such signals as inputs for self-invented learning algorithms.
A few years later, I combined the Meta-RL of Sec. 2 and
Adversarial Artificial Curiosity in a single system [AC97, AC99, AC02].
It generates computational experiments in form of programs whose execution may change both an external environment and the RL agent's internal state. An experiment has a binary outcome: either a particular effect happens, or it doesn't. Experiments are collectively proposed by two reward-maximizing adversarial policies. Both can predict and bet on experimental outcomes before they happen. Once such an outcome is actually observed, the winner will get a positive reward proportional to the bet, and the loser a negative reward of equal magnitude. So each policy is motivated to create experiments whose yes/no outcomes surprise the other policy. The latter in turn is motivated to learn something about the world that it did not yet know, such that it is not outwitted again.
Using Meta-RL with
self-modifying policies [METARL2-9]
the system learns when to learn and what to learn [AC97, AC99, AC02]. It
will also minimize the computational cost of learning new skills,
provided both brains receive a small
negative reward for each computational step, which
introduces a bias towards simple still surprising experiments (reflecting simple still unsolved problems). This may facilitate hierarchical construction of more and more complex experiments, including those yielding external reward (if there is any). In fact, this type of artificial creativity may not only drive artificial scientists and artists [AC06-09], but can also accelerate the intake of external reward [AC97] [AC02],
intuitively because a better understanding of the world can help to solve certain problems faster.
The more recent, intrinsically motivated
PowerPlay RL system (2011) [PP] [PP1]
can use the meatalearning
(Sec. 4) to
continually invent on its own new goals and tasks,
incrementally learning to become a more and more general problem solver in an active, partially unsupervised or self-supervised fashion.
RL robots with high-dimensional video inputs and intrinsic motivation (like in PowerPlay) learned to explore in 2015 [PP2].
7. Most Recent Work on Metalearning (2020)
My PhD student Imanol Schlag et al. [FASTMETA7] augmented an LSTM with an associative Fast Weight Memory (FWM). Through differentiable operations at every step of a given input sequence, the LSTM updates and maintains compositional associations of former observations stored in the rapidly changing FWM weights. The model is trained end-to-end by gradient descent and yields excellent performance on compositional language reasoning problems, small-scale word-level language modelling, and meta-RL for
partially observable environments [FASTMETA7].
Our recent MetaGenRL (2020) [METARL10] meta-learns
novel RL algorithms applicable to environments that significantly differ from those used for training.
MetaGenRL searches the space of low-complexity loss functions that describe such learning algorithms.
See the blog post of my PhD student Louis Kirsch.
This principle of searching for simple learning algorithms is also applicable to fast weight architectures.
Our recent Variable Shared Meta Learning (VS-ML) merges weight sharing and sparsity in meta-learning RNNs [FASTMETA6].
This allows for encoding the learning algorithm by few parameters although it has many time-varying
variables—compare [FAST2] (Sec. 3).
VS-ML combines end-to-end differentiable fast weights [FAST1-3a] (Sec. 3) and learning algorithms encoded in the activations of LSTMs [HO1].
Some of these activations can be interpreted as NN weights updated by the LSTM dynamics.
LSTMs with shared sparse entries in their weight matrix discover learning algorithms that generalize to new datasets.
The meta-learned learning algorithms do not require explicit gradient calculation.
VS-ML in RNNs can also learn to implement the famous backpropagation learning algorithm
purely in the end-to-end differentiable forward dynamics of RNNs [FASTMETA6].
Thanks to several expert reviewers for useful comments. Since science is about self-correction, let me know under firstname.lastname@example.org if you can spot any remaining error. The contents of this article may be used for educational and non-commercial purposes, including articles for Wikipedia and similar sites.
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In S. Thrun and L. Pratt, eds.,
Learning to learn, Kluwer, pages 293-309, 1997.
A general method for incremental self-improvement
and multiagent learning.
In X. Yao, editor, Evolutionary Computation: Theory and Applications.
Chapter 3, pp.81-123, Scientific Publ. Co., Singapore,
L. Kirsch, S. van Steenkiste, J. Schmidhuber. Improving Generalization in Meta Reinforcement Learning using Neural Objectives. International Conference on Learning Representations, 2020.
J. Schmidhuber. Bias-Optimal Incremental Problem Solving.
In S. Becker, S. Thrun, K. Obermayer, eds.,
Advances in Neural Information Processing Systems 15, N(eur)IPS'15, MIT Press, Cambridge MA, p. 1571-1578, 2003.
Optimal Ordered Problem Solver.
Machine Learning, 54, 211-254, 2004.
OOPS source code in crystalline format.
Schmidhuber, J., Zhumatiy, V. and Gagliolo, M. Bias-Optimal
Incremental Learning of Control Sequences for Virtual Robots. In Groen,
F., Amato, N., Bonarini, A., Yoshida, E., and Kroese, B., editors:
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on Intelligent Autonomous Systems, IAS-8, Amsterdam,
The Netherlands, pp. 658-665, 2004.
J. Schmidhuber (2003).
Goedel Machines: Self-Referential Universal Problem Solvers Making Provably Optimal Self-Improvements.
J. Schmidhuber (2006).
Fully Self-Referential Optimal Universal Self-Improvers.
In B. Goertzel and C. Pennachin, eds.: Artificial
General Intelligence, p. 199-226, 2006.
J. Schmidhuber (2009).
Ultimate Cognition à la Gödel.
Cognitive Computation 1(2):177-193, 2009. PDF.
T. Schaul and J. Schmidhuber. Metalearning. Scholarpedia, 5(6):4650, 2010.
[LSTM1] S. Hochreiter, J. Schmidhuber. Long Short-Term Memory. Neural Computation, 9(8):1735-1780, 1997. PDF.
Based on [LSTM0]. More.
[LSTM2] F. A. Gers, J. Schmidhuber, F. Cummins. Learning to Forget: Continual Prediction with LSTM. Neural Computation, 12(10):2451-2471, 2000.
[The "vanilla LSTM architecture" that everybody is using today, e.g., in Google's Tensorflow.]
D. Ciresan, U. Meier, J. Schmidhuber.
Transfer Learning for Latin and Chinese Characters with Deep Neural Networks.
Proc. IJCNN 2012, p 1301-1306, 2012.
[BP1] S. Linnainmaa. The representation of the cumulative rounding error of an algorithm as a Taylor expansion of the local rounding errors. Master's Thesis (in Finnish), Univ. Helsinki, 1970.
See chapters 6-7 and FORTRAN code on pages 58-60.
See also BIT 16, 146-160, 1976.
H. J. Kelley. Gradient Theory of Optimal Flight Paths. ARS Journal, Vol. 30, No. 10, pp. 947-954, 1960.
[BP2] P. J. Werbos. Applications of advances in nonlinear sensitivity analysis. In R. Drenick, F. Kozin, (eds): System Modeling and Optimization: Proc. IFIP,
[Extending thoughts in his 1974 thesis.]
[BP4] J. Schmidhuber.
Who invented backpropagation?
More in [DL2].
Ivakhnenko, A. G. and Lapa, V. G. (1965). Cybernetic Predicting Devices. CCM Information Corporation. [First working Deep Learners with many layers, learning internal representations.]
Ivakhnenko, Alexey Grigorevich. The group method of data of handling; a rival of the method of stochastic approximation. Soviet Automatic Control 13 (1968): 43-55.
Ivakhnenko, A. G. (1971). Polynomial theory of complex systems. IEEE Transactions on Systems, Man and Cybernetics, (4):364-378.
[T20] J. Schmidhuber (2020). Critique of 2018 Turing Award.
[DL1] J. Schmidhuber, 2015.
Deep Learning in neural networks: An overview. Neural Networks, 61, 85-117.
[DL2] J. Schmidhuber, 2015.
[DL4] J. Schmidhuber, 2017. Our impact on the world's most valuable public companies: 1. Apple, 2. Alphabet (Google), 3. Microsoft, 4. Facebook, 5. Amazon ...
[OPT] J. Schmidhuber (2004).
Optimal Universal Search.
[UNI] J. Schmidhuber (2004).
Theory of universal learning machines and universal AI.
[MIR] J. Schmidhuber (10/4/2019). Deep Learning: Our Miraculous Year 1990-1991. See also arxiv:2005.05744 (May 2020).
[DEC] J. Schmidhuber (02/20/2020). The 2010s: Our Decade of Deep Learning / Outlook on the 2020s.
[ATT] J. Schmidhuber (2020). End-to-End Differentiable Sequential Neural Attention 1990-93.