Beren's "Deconfusing Direct vs Amortised Optimisation"

𝕮𝖎𝖓𝖊𝖗𝖆

Preamble

I heavily recommend @beren's "Deconfusing Direct vs Amortised Optimisation". It's a very important conceptual clarification that has changed how I think about many issues bearing on technical AI safety.

Currently, it's the most important blog post I've read this year.

This sequence (if I get around to completing it) is an attempt to draw more attention to Beren's conceptual frame and its implications for how to think about issues of alignment and agency.

This first post presents a distillation of the concept, and subsequent posts explore its implications.

Two Approaches to Optimisation

Beren introduces a taxonomy categorising intelligent systems according to the kind of optimisation they are performing. I think it's more helpful to think of these as two ends of a spectrum as opposed to distinct discrete categories; sophisticated real world intelligent systems (e.g. humans) appear to be a hybrid of the two approaches.

Direct Optimisers

Systems that perform inference by directly choosing actions^[1] to optimise some objective function
- Responses are computed on the fly and individually for each input
Direct optimisers perform inference by answering the question: "what action maximises or minimises this objective function ([discounted] cumulative reward and loss respectively)?"
Examples: AIXI, MCTS, model-based reinforcement learning, other "planning" systems

Naively, direct optimisers can be understood as computing (an approximation of) (or $arg min$ ) for a suitable objective function during inference.

Amortised Optimisers

Systems that learn to approximate a function^[2] during training and perform inference by evaluating the output of the learned function on their inputs.
- The function approximator is learned from a dataset of input data and successful solutions
  - Amortised optimisation converts an inference problem to a supervised learning problem
- It's called "amortised optimisation" because while learning the policy is expensive, the cost of inference is amortised over all evaluations of the learned policy
Amortised optimisers can be seen as performing inference by answering the question "what output (e.g. action, probability distribution over tokens) does this learned function (policy, predictive model) return for this input (agent state, prompt)?"
Examples: model free reinforcement learning, LLMs, most supervised & self supervised(?) learning systems

Naively, amortised optimisers can be understood as evaluating a (fixed) learned function; they're not directly computing $arg max$ (or $arg min$ ) for any particular objective function during inference.

Differences

Aspect	Direct Optimization	Amortized Optimization
Problem Solving	Computes optimal responses "on the fly"	Evaluates the learned function approximator on the given input
Computational Approach	Searches through a solution space	Learns a function approximator
Runtime Cost	Higher, as it requires in-depth search for a suitable solution	Lower, as it only needs a forward pass through the function approximator
Scalability with Compute	Scales by expanding search depth	Scales by better approximating the posterior distribution
Convergence	In the limit of arbitrary compute, the system's policy converges to $arg max \| \| arg min$ of the appropriate objective function	In the limit of arbitrary compute, the system's policy converges to the best description of the training dataset
Performance	More favourable in "simple" domains	More favourable in "rich" domains
Data Efficiency	Little data needed for high performance (e.g. an MCTS agent can attain strongly superhuman performance in Chess/Go given only the rules and sufficient compute)	Requires (much) more data for high performance (e.g. an amortised agent necessarily needs to observe millions of chess games to learn skilled play)
Generalization	Dependent on search depth and compute	Dependent on the learned function approximator/training dataset
Alignment Focus	Emphasis on safe reward function design	Emphasis on reward function and dataset design
Out-of-Distribution Behavior	Can diverge arbitrarily from previous behavior	Constrained by the learned function approximator
Examples	AIXI, MCTS, model-based RL	Supervised learning, model-free RL, GPT models

Some Commentary

Direct optimisation is feasible in "simple" (narrow problem domains, deterministic, discrete, fully observable/perfect information, etc.) environments (e.g. tic-tac-toe, chess, go) but unwieldy in "rich" (complex/high dimensional problem domains, continuous, stochastic, large state/action spaces, partially observable/imperfect information, etc.) environments (e.g. the real world).
- The limitations of direct optimisation in rich environments seem complexity theoretic, so better algorithms won't fix them
In practice some systems use a hybrid of the two approaches with most cognition performed in an amortised manner but planning deployed when necessary (e.g. system 2 vs system 1 in humans)
Hybrid systems can be "bootstrapped" in both directions
- A planner can be initialised with amortised policies, or an amortised value model could be used to prune subtrees of a planner's search that are unlikely to be fruitful
  - This approach is used in Alpha Go and similar systems
- Likewise, direct optimisation can be used to improve the data we are training the function approximator on

^{^}
Or strategies, plans, probabilities, categories, etc.; any "output" of the system.
^{^}
Beren:
I would add that this function is usually the solution to the objective solved by some form of direct optimiser. I.e. your classifier learns the map from input -> label.

Effective Altruism Forum
EA Forum