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1.Computing an exact quantilal policy
discussion post by Vadim Kosoy 45 days ago | discuss
2.Quantilal control for finite MDPs
post by Vadim Kosoy 57 days ago | Ryan Carey, Alex Appel and Abram Demski like this | discuss

We introduce a variant of the concept of a “quantilizer” for the setting of choosing a policy for a finite Markov decision process (MDP), where the generic unknown cost is replaced by an unknown penalty term in the reward function. This is essentially a generalization of quantilization in repeated games with a cost independence assumption. We show that the “quantilal” policy shares some properties with the ordinary optimal policy, namely that (i) it can always be chosen to be Markov (ii) it can be chosen to be stationary when time discount is geometric (iii) the “quantilum” value of an MDP with geometric time discount is a continuous piecewise rational function of the parameters, and it converges when the discount parameter \(\lambda\) approaches 1. Finally, we demonstrate a polynomial-time algorithm for computing the quantilal policy, showing that quantilization is not qualitatively harder than ordinary optimization.

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3.More precise regret bound for DRL
post by Vadim Kosoy 156 days ago | Alex Appel likes this | discuss

We derive a regret bound for DRL reflecting dependence on:

  • Number of hypotheses

  • Mixing time of MDP hypotheses

  • The probability with which the advisor takes optimal actions

That is, the regret bound we get is fully explicit up to a multiplicative constant (which can also be made explicit). Currently we focus on plain (as opposed to catastrophe) and uniform (finite number of hypotheses, uniform prior) DRL, although this result can and should be extended to the catastrophe and/or non-uniform settings.

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4.Why DRL doesn't work for arbitrary environments
discussion post by Vadim Kosoy 178 days ago | discuss
5.Catastrophe Mitigation Using DRL (Appendices)
discussion post by Vadim Kosoy 187 days ago | discuss
6.Catastrophe Mitigation Using DRL
post by Vadim Kosoy 191 days ago | 3 comments

Previously we derived a regret bound for DRL which assumed the advisor is “locally sane.” Such an advisor can only take actions that don’t lose any value in the long term. In particular, if the environment contains a latent catastrophe that manifests with a certain rate (such as the possibility of an UFAI), a locally sane advisor has to take the optimal course of action to mitigate it, since every delay yields a positive probability of the catastrophe manifesting and leading to permanent loss of value. This state of affairs is unsatisfactory, since we would like to have performance guarantees for an AI that can mitigate catastrophes that the human operator cannot mitigate on their own. To address this problem, we introduce a new form of DRL where in every hypothetical environment the set of uncorrupted states is divided into “dangerous” (impending catastrophe) and “safe” (catastrophe was mitigated). The advisor is then only required to be locally sane in safe states, whereas in dangerous states certain “leaking” of long-term value is allowed. We derive a regret bound in this setting as a function of the time discount factor, the expected value of catastrophe mitigation time for the optimal policy, and the “value leak” rate (i.e. essentially the rate of catastrophe occurrence). The form of this regret bound implies that in certain asymptotic regimes, the agent attains near-optimal expected utility (and in particular mitigates the catastrophe with probability close to 1), whereas the advisor on its own fails to mitigate the catastrophe with probability close to 1.

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7.Delegative Reinforcement Learning with a Merely Sane Advisor
post by Vadim Kosoy 270 days ago | discuss

Previously, we defined a setting called “Delegative Inverse Reinforcement Learning” (DIRL) in which the agent can delegate actions to an “advisor” and the reward is only visible to the advisor as well. We proved a sublinear regret bound (converted to traditional normalization in online learning, the bound is \(O(n^{2/3})\)) for one-shot DIRL (as opposed to standard regret bounds in RL which are only applicable in the episodic setting). However, this required a rather strong assumption about the advisor: in particular, the advisor had to choose the optimal action with maximal likelihood. Here, we consider “Delegative Reinforcement Learning” (DRL), i.e. a similar setting in which the reward is directly observable by the agent. We also restrict our attention to finite MDP environments (we believe these results can be generalized to a much larger class of environments, but not to arbitrary environments). On the other hand, the assumption about the advisor is much weaker: the advisor is only required to avoid catastrophic actions (i.e. actions that lose value to zeroth order in the interest rate) and assign some positive probability to a nearly optimal action. As before, we prove a one-shot regret bound (in traditional normalization, \(O(n^{3/4})\)). Analogously to before, we allow for “corrupt” states in which both the advisor and the reward signal stop being reliable.

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8.On the computational feasibility of forecasting using gamblers
discussion post by Vadim Kosoy 313 days ago | discuss
9.Improved formalism for corruption in DIRL
discussion post by Vadim Kosoy 319 days ago | discuss
10.Delegative Inverse Reinforcement Learning
post by Vadim Kosoy 330 days ago | Alex Appel likes this | 11 comments

We introduce a reinforcement-like learning setting we call Delegative Inverse Reinforcement Learning (DIRL). In DIRL, the agent can, at any point of time, delegate the choice of action to an “advisor”. The agent knows neither the environment nor the reward function, whereas the advisor knows both. Thus, DIRL can be regarded as a special case of CIRL. A similar setting was studied in Clouse 1997, but as far as we can tell, the relevant literature offers few theoretical results and virtually all researchers focus on the MDP case (please correct me if I’m wrong). On the other hand, we consider general environments (not necessarily MDP or even POMDP) and prove a natural performance guarantee.

The use of an advisor allows us to kill two birds with one stone: learning the reward function and safe exploration (i.e. avoiding both the Scylla of “Bayesian paranoia” and the Charybdis of falling into traps). We prove that, given certain assumption about the advisor, a Bayesian DIRL agent (whose prior is supported on some countable set of hypotheses) is guaranteed to attain most of the value in the slow falling time discount (long-term planning) limit (assuming one of the hypotheses in the prior is true). The assumption about the advisor is quite strong, but the advisor is not required to be fully optimal: a “soft maximizer” satisfies the conditions. Moreover, we allow for the existence of “corrupt states” in which the advisor stops being a relevant signal, thus demonstrating that this approach can deal with wireheading and avoid manipulating the advisor, at least in principle (the assumption about the advisor is still unrealistically strong). Finally we consider advisors that don’t know the environment but have some beliefs about the environment, and show that in this case the agent converges to Bayes-optimality w.r.t. the advisor’s beliefs, which is arguably the best we can expect.

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11.Learning incomplete models using dominant markets
post by Vadim Kosoy 436 days ago | Jessica Taylor likes this | discuss

This post is formal treatment of the idea outlined here.

Given a countable set of incomplete models, we define a forecasting function that converges in the Kantorovich-Rubinstein metric with probability 1 to every one of the models which is satisfied by the true environment. This is analogous to Blackwell-Dubins merging of opinions for complete models, except that Kantorovich-Rubinstein convergence is weaker than convergence in total variation. The forecasting function is a dominant stochastic market for a suitably constructed set of traders.

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12.Dominant stochastic markets
post by Vadim Kosoy 443 days ago | discuss

We generalize the formalism of dominant markets to account for stochastic “deductive processes,” and prove a theorem regarding the asymptotic behavior of such markets. In a following post, we will show how to use these tools to formalize the ideas outlined here.

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13.A measure-theoretic generalization of logical induction
discussion post by Vadim Kosoy 497 days ago | Jessica Taylor and Scott Garrabrant like this | discuss
14.Towards learning incomplete models using inner prediction markets
discussion post by Vadim Kosoy 504 days ago | Jessica Taylor and Paul Christiano like this | 4 comments
15.Subagent perfect minimax
discussion post by Vadim Kosoy 506 days ago | discuss
16.Minimax and dynamic (in)consistency
discussion post by Vadim Kosoy 532 days ago | discuss
17.Minimax forecasting
post by Vadim Kosoy 532 days ago | 2 comments

This post continues the research programme of attacking the grain of truth problem by departure from the Bayesian paradigm. In the previous post, I suggested using Savage’s minimax regret decision rule, but here I fall back to the simple minimax decision rule. This is because the mathematics is considerably simpler, and minimax should be sufficient to get IUD play in general games and Nash equilibrium in zero-sum two-player games. I hope to build on these results to get analogous results for minimax regret in the future.

We consider “semi-Bayesian” agents following the minimax expected utility decision rule, in oblivious environments with full monitoring (a setting that we will refer to as “forecasting”). This setting is considered in order to avoid the need to enforce exploration, as a preparation for analysis of general environments. We show that such agents satisfy a certain asymptotic optimality theorem. Intuitively, this theorem means that whenever the environment satisfies an incomplete model that is included in the prior, the agent will eventually learn this model i.e. extract at least as much utility as can be guaranteed for this model.

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18.Attacking the grain of truth problem using Bayes-Savage agents
discussion post by Vadim Kosoy 584 days ago | Paul Christiano likes this | discuss
19.IRL is hard
post by Vadim Kosoy 621 days ago | 6 comments

We show that assuming the existence of public-key cryptography, there is an environment in which Inverse Reinforcement Learning is computationally intractable, even though the “teacher” agent, the environment and the utility functions are computable in polynomial-time and there is only 1 bit of information to learn.

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20.Stabilizing logical counterfactuals by pseudorandomization
post by Vadim Kosoy 831 days ago | Abram Demski likes this | 2 comments

Previously, we discussed the construction of logical counterfactuals in the language of optimal predictors. These counterfactuals were found to be well-behaved when a certain non-degeneracy condition is met which can be understood as a bound on the agent’s ability to predict itself. We also demonstrated that desired game-theoretic behavior seems to require randomization (thermalizing instead of maximizing) which has to be logical randomization to implement metathreat game theory by logical counterfactuals. Both of these considerations suggest that the agent has to pseudorandomize (randomize in the logical uncertainty sense) its own behavior. Here, we show how to implement this pseudorandomization and prove it indeed guarantees the non-degeneracy condition.


The proofs of the results are given in Appendix A.

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21.Logical counterfactuals for random algorithms
post by Vadim Kosoy 872 days ago | Abram Demski, Nate Soares and Patrick LaVictoire like this | discuss

Updateless decision theory was informally defined by Wei Dai in terms of logical conditional expected utility, where the condition corresponds to an algorithm (the agent) producing a given output (action or policy). This kind of conditional expected values can be formalised by optimal predictors. However, since optimal predictor systems which are required to apply optimal predictors to decision theory generally have random advice, we need counterfactuals well-defined for random algorithms i.e. algorithms that produce different outputs with different probabilities depending on internal coin tosses. We propose to define these counterfactuals by a generalization of the notion of conditional expected utility which amounts to linear regression of utility with respect to the probabilities of different outputs in the space of “impossible possible worlds.” We formalise this idea by introducing “relative optimal predictors,” prove the analogue of the conditional probability formula (which takes matrix form) and uniqueness theorems.


We start by explaining the analogous construction in classical probability theory and proceed to defining the logical counterpart in the Results section.

Consider \(\zeta\) a probability measure on some space, a random variable \(u\) representing utility, a finite set \(\mathcal{A}\) representing possible actions and another random variable \(p\) taking values in \([0,1]^{\mathcal{A}}\) and satisfying \(\sum_{a \in \mathcal{A}} p_a = 1\) representing the probabilities of taking different actions. For a deterministic algorithm, \(p\) takes values \(\{0,1\}^{\mathcal{A}}\) allowing defining conditional expected utility as

\[u_a := \operatorname{E}_\zeta[u \mid p_a = 1] = \frac{\operatorname{E}_\zeta[u p_a]}{\operatorname{E}_\zeta[p_a]}\]

In the general case, it is tempting to consider

\[\operatorname{E}_{\zeta \ltimes p}[u \mid a] = \frac{\operatorname{E}_\zeta[u p_a]}{\operatorname{E}_\zeta[p_a]}\]

where \(\zeta \ltimes p\) stands for the semidirect product of \(\zeta\) with \(p\), the latter regarded as a Markov kernel with target \(\mathcal{A}\). However, this would lead to behavior similar to EDT since conditioning by \(a\) is meaningful even for a single “world” (i.e. completely deterministic \(u\) and \(p\)). Instead, we select \(u^* \in {\mathbb{R}}^{\mathcal{A}}\) that minimizes \(\operatorname{E}_\zeta[(u - p^t u^*)^2]\) (we regard elements of \({\mathbb{R}}^{\mathcal{A}}\) as column vectors so \(p^t\) is a row vector). This means \(u^*\) has to satisfy the matrix equation

\[\operatorname{E}_\zeta[p p^t] u^* = \operatorname{E}_\zeta[u p]\]

The solution to this equation is only unique when \(\operatorname{E}_\zeta[p p^t]\) is non-degenerate. This corresponds to requiring positive probability of the condition for usual conditional expected values. In case \(p\) takes values in \(\{0,1\}^{\mathcal{A}}\), \(u^*\) is the usual conditional expected value.


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22.Implementing CDT with optimal predictor systems
post by Vadim Kosoy 889 days ago | Patrick LaVictoire likes this | 2 comments

We consider transparent games between bounded CDT agents (“transparent” meaning each player has a model of the other players). The agents compute the expected utility of each possible action by executing an optimal predictor of a causal counterfactual, i.e. an optimal predictor for a function that evaluates the other players and computes the utility for the selected action. Since the agents simultaneously attempt to predict each other, the optimal predictors form an optimal predictor system for the reflective system comprised by the causal counterfactuals of all agents. We show that for strict maximizers, the resulting outcome is a bounded analogue of an approximate Nash equilibrium, i.e. a strategy which is an optimal response within certain resource constraints up to an asymptotically small error. For “thermalizers” (agents that choose an action with probability proportional to \(2^{\frac{u}{T}}\)), we get a similar result with expected utility \(\operatorname{E}_s[u]\) replaced by “free utility” \(\operatorname{E}_s[u]+T \operatorname{H}(s)\). Thus, such optimal predictor systems behave like bounded counterparts of reflective oracles.


The proofs for this section are given in Appendix A.

We redefine \(\mathcal{E}_{2(ll,\phi)}\) and \(\mathcal{E}_{2(ll)}\) to be somewhat smaller proto-error spaces which nevertheless yield the same existence theorems as before. This is thanks to Lemma A.1.

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23.Reflection with optimal predictors
post by Vadim Kosoy 913 days ago | Patrick LaVictoire likes this | discuss

A change in terminology: It is convenient when important concepts have short names. The concept of an “optimal predictor scheme” seems much more important than its historical predecessor, the “optimal predictor”. Therefore “optimal predictor schemes” will be henceforth called just “optimal predictors” while the previous concept of “optimal predictor” might be called “flat optimal predictor”.

We study systems of computations which have access to optimal predictors for each other. We expect such systems to play an important role in decision theory (where self-prediction is required to define logical counterfactuals and mutual prediction is required for a collection of agents in a game) and Vingean reflection (where the different computations correspond to different successor agents). The previously known existence theorems for optimal predictors are not directly applicable to this case. To overcome this we prove new, specifically tailored existence theorems.

The Results section states the main novelties, Appendix A contains adaptations of old theorems, Appendix B proves selected claims from Appendix A and Appendix C proves the novel results.


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24.Superrationality in arbitrary games
discussion post by Vadim Kosoy 935 days ago | Jessica Taylor, Nate Soares, Patrick LaVictoire, Scott Garrabrant and Stuart Armstrong like this | 5 comments
25.Bounded Solomonoff induction using optimal predictor schemes
post by Vadim Kosoy 929 days ago | Patrick LaVictoire likes this | discuss

Most of the content of this post was covered by the talk I gave in Los Angeles MIRIx in October, minus the proofs and a minor amendment of Theorem 1 (the role of \(\Delta_{sqp,\phi}^2\)).

We define variants of the concept of generatable distributional estimation problem and show that these variants also admits a uniformly universal optimal predictor scheme. We show how to use this to implement a form of bounded Solomonoff induction.


We have previously defined a “word ensemble” to be a collection \(\{\mu^k\}_{k \in {\mathbb{N}}}\) of probability measures on \({{\{ 0, 1 \}^*}}\) s.t. for some polynomial \(p\), \(\operatorname{supp}\mu^k \subseteq {{\{ 0, 1 \}^{\leq{p(k)}}}}\). This was convenient when the formalism was based on Boolean circuits but unnecessary for Turing machines. It is enough to assume that the Turing machine is allowed to read only the beginning of the input and thus halt in time arbitrarily smaller than the length of the input. In the following we will use “word ensemble” to mean an arbitrary sequence of probability measures on \({{\{ 0, 1 \}^*}}\), allow such word ensembles in distributional estimation problems etc.

All proofs are in the Appendix.

We start by defining “\(\Delta(log)\)-sampler” and “\(\Delta(log)\)-generator” for \(\Delta\) an error space of rank 2 (they were previously defined for an error space of rank 1). Fix such an error space.

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