## 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 ) 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

- The function approximator is learned from a dataset of input data and successful solutions
- 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 (or ) 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 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

- 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

^{^}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.