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07973659-2ec7-4487-b8ff-75a49828af77	trentmkelly/LessWrong-43k	LessWrong	Modifying Universal Intelligence Measure In 2007, Legg and Hutter wrote a paper using the AIXI model to define a measure of intelligence. It's pretty great, but I can think of some directions of improvement. * Reinforcement learning. I think this term and formalism are historically from much simpler agent models which actually depended on being reinforced to learn. In its present form (Hutter 2005 section 4.1) it seems arbitrarily general, but it still feels kinda gross to me. Can we formalize AIXI and the intelligence measure in terms of utility functions, instead? And perhaps prove them equivalent? * Choice of Horizon. AIXI discounts the future by requiring that total future reward is bounded, and therefore so does the intelligence measure. This seems to me like a constraint that does not reflect reality, and possibly an infinitely important one. How could we remove this requirement? (Much discussion on the "Choice of the Horizon" in Hutter 2005 section 5.7). * Unknown utility function. When we reformulate it in terms of utility functions, let's make sure we can measure its intelligence/optimization power without having to know its utility function. Perhaps by using an average of utility functions weighted by their K-complexity. * AI orientation. Finally, and least importantly, it tests agents across all possible programs, even those which are known to be inconsistent with our universe. This might okay if your agent is a playing arbitrary games on a computer, but if you are trying to determine how powerful an agent will be in this universe, you probably want to replace the Solomonoff prior with the posterior resulting from updating the Solomonoff prior with data from our universe. Any thought or research on this by others? I imagine lots of discussion has occurred over these topics; any referencing would be appreciated.
e116b496-9454-4c6b-9c76-03c57fbfe0bb	trentmkelly/LessWrong-43k	LessWrong	Conflating value alignment and intent alignment is causing confusion Epistemic status: I think something like this confusion is happening often. I'm not saying these are the only differences in what people mean by "AGI alignment". Summary: Value alignment is better but probably harder to achieve than personal intent alignment to the short-term wants of some person(s). Different groups and people tend to primarily address one of these alignment targets when they discuss alignment. Confusion abounds. One important confusion stems from an assumption that the type of AI defines the alignment target: strong goal-directed AGI must be value aligned or misaligned, while personal intent alignment is only viable for relatively weak AI. I think this assumption is important but false. While value alignment is categorically better, intent alignment seems easier, safer, and more appealing in the short term, so AGI project leaders are likely to try it.[1] Overview Clarifying what people mean by alignment should dispel some illusory disagreement, and clarify alignment theory and predictions of AGI outcomes. Caption: Venn diagram of three types of alignment targets. Value alignment and Personal intent alignment are both subsets of Evan Hubinger's definition of intent alignment: AGI aligned with human intent in the broadest sense. Prosaic alignment work usually seems to be addressing a target somewhere in the neighborhood of personal intent alignment (following instructions or doing what this person wants now), while agent foundations and other conceptual alignment work usually seems to be addressing value alignment. Those two clusters have different strengths and weaknesses as alignment targets, so lumping them together produces confusion. People mean different things when they say alignment. Some are mostly thinking about value alignment (VA): creating sovereign AGI that has values close enough to humans' for our liking. Others are talking about making AGI that is corrigible (in the Christiano or Harms sense)[2] or follows instructions f
b12bd3e2-001d-4f06-970e-b94a30931df6	trentmkelly/LessWrong-43k	LessWrong	Unable to post article (probably because of excessive/incompatible formatting) Hi, I've been trying to publish an article to less wrong for about a week, but I'm unable to copy the text to the post area and submit it. It says: Submitting, then it stops and nothing happened. When I sliced the entire text in small parts (all copied from Open Office) I manage to publish drafts, some with errors such as missing spaces. When I re-copy from the drafts to another, bigger draft, then many spaces become missing, and part of the text is aligned to the right border, part isnt. The text was mostly written in open office. I would like help from someone who has a normal Office. If you post me a message with your e-mail in the private message area, I can send it to you, so you see if you can either publish it in drafts, and then copy it again to me in a publishable form, or edit somethign in office that I'm unable in open office, and send the file back to me so I can publish. I know this is asking a lot, and I would be thankful for anyone who helps me out of this conundrum.
e93e9e9b-891c-49a4-8de1-a9af137a93c9	StampyAI/alignment-research-dataset/arxiv	Arxiv	BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures 1 Introduction --------------- Convolutional Neural Networks (CNNs) have demonstrated great performance in computer vision tasks. However, CNNs often require substantial computational and storage resources, making their deployment difficult for edge devices and other resource-constrained scenarios. Existing approaches to alleviate this problem include network pruning [[11](#bib.bib11), [29](#bib.bib29)], efficient architecture design [[25](#bib.bib25), [35](#bib.bib35), [28](#bib.bib28)], and quantization [[13](#bib.bib13)]. Among them, Binary Neural Network (BNN) is a promising direction that utilizes 1-bit quantization. By binarizing network parameters and activations, the resource-hungry 32-bit floating-point multiplications can be replaced by efficient bitwise operations (e.g. XNOR, bitcount), significantly reducing the computation and memory burden. Despite their computational efficiency, BNNs often suffer from unsatisfactory performance through the binarization process. Moreover, parts of the network computation flow still remain in full-precision (FP), bringing difficulty for the hardware acceleration. ![Refer to caption](/html/2011.10804/assets/x1.png) Figure 1: The information bottleneck phenomenon related to micro-level topology and macro-level layout. Architectural design is critical for BNNs’ performance. However, existing architectures for FP networks are suboptimal for BNNs. As BNNs’ activations are binary, they carry much less information relative to their FP counterparts, causing the information bottleneck. Enhancing the information flow in BNNs is thus critical for their performance [[1](#bib.bib1)]. Viewing the network architecture as a sequence of basic building blocks (or cells), the information bottleneck affects BNNs’ performance through two levels of granularity: (1) The micro-level considers the inner structure of each cell (i.e. cell topology) . At the micro-level, compact CNN blocks (e.g. depthwise, bottleneck convs) and scale-altering layers (e.g. downsampling) are the information bottlenecks that hinder the performance of BNNs. On the other hand, shortcut connections strengthen the information flow [[21](#bib.bib21), [2](#bib.bib2)]. (2) The macro-level focuses on the composition structure of cells (i.e. cell layout). At the macro-level, expanding networks’ width allows activations to carry more information, which could strengthen the information flow and bring performance gain [[15](#bib.bib15)]. In contrast, deeper architectures might not work well, since adding more layers does not alleviate existing information bottlenecks. Fig. [1](#S1.F1 "Figure 1 ‣ 1 Introduction ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures") illustrates these design considerations under the information flow perspective. Neural architecture search (NAS) [[20](#bib.bib20)] is a promising direction to find optimal architectures for neural networks automatically. However, existing NAS studies for BNNs [[26](#bib.bib26), [27](#bib.bib27)] mostly leverage the search framework for FP networks [[16](#bib.bib16)], and did not fully consider preserving the information flow for both the micro and macro-level. For example, [[27](#bib.bib27), [3](#bib.bib3)] employ micro-level topology search, but use a pre-defined cell layout and model augmentation scheme identical to FP networks. [[26](#bib.bib26)] searches for the macro-level layout through determining layer-wise widths, but it uses a fixed topology adapted from CNNs. In this paper, we propose BARS, a BNN-oriented differentiable NAS flow, in which the search space, along with the search and derive strategies are carefully designed and developed according to the characteristics of BNNs. Unlike existing works [[26](#bib.bib26), [3](#bib.bib3), [27](#bib.bib27)], BARS extends the original micro-level DARTS [[16](#bib.bib16)] search space to the macro-level. And jointly searches for the micro-level cell topology and the macro-level cell layout with a 2-level search space. We design a novel macro-level depth & width search space that could be unified in the differentiable NAS framework. It seeks to strike a better balance between model performance and complexity. We also improve the micro-level search space for automatically discovering topologies that avoid creating bottlenecks and maintain proper information flow. Besides, we propose improvements on the search strategy such as Gumbel sampling and entropy regularization to ensure a stabilized search in a much bigger search space. With the above mentioned techniques, BARS-discovered architecture outperforms CNN-adapted binary architectures by a large margin (6%percent66\%6 % better accuracy than hand-crafted binary ResNet18 on ImageNet). It also achieves superior performance than state-of-the-art baseline architectures with smaller complexity. Furthermore, BARS reduces the full-precision operations significantly. BARS-discovered architectures only have 10% floating-point operations compared with existing BNN NAS studies on CIFAR. ![Refer to caption](/html/2011.10804/assets/x2.png) Figure 2: The workflow of our proposed Binary ARchitecture Search (BARS) (lower) vs. DARTS (upper) [[16](#bib.bib16), [3](#bib.bib3), [5](#bib.bib5)]. BARS tailors a series of modifications on search space and search strategies to facilitate a proper search process for BNNs. 2 Related Works ---------------- ### 2.1 Binary Neural Networks Binarization Scheme Network binarization could be viewed as an extreme case of network quantization. It could replace the original FP32 multiplications with efficient bitwise operations, gaining over 10 times the processing speed. However, due to the lack of representation ability, binary neural networks often suffer from noticeable accuracy degradation. Several methods have been proposed to improve the performance of BNN. XNORNet[[24](#bib.bib24)] uses shared scaling factors to improve the representation ability without introducing much computational overhead. Many recent studies [[17](#bib.bib17), [3](#bib.bib3), [27](#bib.bib27)] follow its binarization scheme, and so do we. Some other binarization schemes are also proposed, such as: [[4](#bib.bib4)] fuses the weight and activation scaling factor together before inference. Other approaches for improving BNN performance focus on minimizing the quantization error [[34](#bib.bib34)], redesigning the training loss [[19](#bib.bib19)], or amending the gradient estimation [[18](#bib.bib18), [23](#bib.bib23)]. Binary Architectural Advances The aforementioned methods mainly focus on improving the binarization or training scheme. Furthermore, the network architecture also plays a critical role in determining the performance of a BNN. Previous studies mainly address the information bottleneck issue from 2 perspectives: 1) Strengthen the information flow by adding more shortcuts [[18](#bib.bib18), [1](#bib.bib1), [2](#bib.bib2)]. 2) Identify and eliminate some information bottleneck manually: Most of the recent studies [[18](#bib.bib18), [23](#bib.bib23)] adopt full-precision downsampling layer; [[21](#bib.bib21)] modifies separable convolutions in the MobileNet architectures. ### 2.2 Neural Architecture Search (NAS) NAS Search Space NAS search space designs in recent studies can be divided into two categories: macro-level and micro-level (cell-level). The macro-level describes how cells are organized to construct the entire architecture, and methods have been developed to search for these layout decisions, including width and depth [[30](#bib.bib30), [12](#bib.bib12)]. On the other hand, the micro-level describes the connecting operations inside each cell, and aims to find a superior intra-cell topology. There exist many studies that only search for the micro-level cell topology and organize cells into a pre-defined layout [[16](#bib.bib16)]. In this paper, we employ both the macro- and micro-level search to obtain accurate and efficient binary architectures. Differentiable NAS Considerable efforts have been devoted to developing and applying gradient-based NAS (i.e. Differentiable NAS) methods [[16](#bib.bib16), [31](#bib.bib31), [12](#bib.bib12)] due to its high search efficiency. DARTS [[16](#bib.bib16)] first models the NAS problem as a bilevel optimization problem, in which the architecture parameters are updated using gradient methods. Cell-based search spaces [[36](#bib.bib36)] are designed to facilitate a more efficient NAS process and have been widely adopted [[16](#bib.bib16), [31](#bib.bib31)]. Usually, there are two types of cells in a cell-based search space: normal cells and reduce cells (stride >1absent1>1> 1). These two types of cells are stacked in a pre-defined order to construct a complete architecture. NAS for Binary Architecture Previous studies on improving BNN architecture design often adopt minor modifications to existing well-performing CNN models. Applying NAS to the binary domain could be an effective solution to discover more suitable architecture for BNN. [[26](#bib.bib26)] strikes a balance between accuracy and resource consumption for BNN via evolutionary search on the network width. [[22](#bib.bib22)] adopts a similar approach on groups in convolutions. [[27](#bib.bib27), [3](#bib.bib3)] introduce gradient-based search for BNN. They observe that traditional gradient-based NAS methods such as [[16](#bib.bib16), [31](#bib.bib31)] can not be directly applied for BNN search. Thus, they modify the operations in search space and the search strategy. These methods make advances in searching for efficient binary architectures. However, they still have the following drawbacks. Firstly, they solely focus on only one of the two closely related aspects: network topology and complexity. Secondly, full-precision layers still exist in the main body of the architecture. ( [[3](#bib.bib3)] uses a full-precision preprocess layer in each cell, and [[27](#bib.bib27)] uses a full-precision shortcut for reduction cells, which takes up the majority of the computation). BARS aims to search for both the topology and complexity of the binary architecture, as well as pursuing full binarization of the main body of the architecture. 3 Preliminary -------------- ### 3.1 Network Binarization BARS follows the binarization scheme proposed in XNORNet [[24](#bib.bib24)] with the modification of using a single scaling factor instead of channel-wise for more efficiency. The binary convolution of weights W∈ℝCout×Cin×Kw×Kh𝑊superscriptℝsubscript𝐶𝑜𝑢𝑡subscript𝐶𝑖𝑛subscript𝐾𝑤subscript𝐾ℎW\in\mathbb{R}^{C\_{out}\times C\_{in}\times K\_{w}\times K\_{h}}italic\_W ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_C start\_POSTSUBSCRIPT italic\_o italic\_u italic\_t end\_POSTSUBSCRIPT × italic\_C start\_POSTSUBSCRIPT italic\_i italic\_n end\_POSTSUBSCRIPT × italic\_K start\_POSTSUBSCRIPT italic\_w end\_POSTSUBSCRIPT × italic\_K start\_POSTSUBSCRIPT italic\_h end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT and the input feature map X∈ℝbs×Cin×W×H𝑋superscriptℝ𝑏𝑠subscript𝐶𝑖𝑛𝑊𝐻X\in\mathbb{R}^{bs\times C\_{in}\times W\times H}italic\_X ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_b italic\_s × italic\_C start\_POSTSUBSCRIPT italic\_i italic\_n end\_POSTSUBSCRIPT × italic\_W × italic\_H end\_POSTSUPERSCRIPT can be written as in Eq. [1](#S3.E1 "1 ‣ 3.1 Network Binarization ‣ 3 Preliminary ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures"), where Coutsubscript𝐶𝑜𝑢𝑡C\_{out}italic\_C start\_POSTSUBSCRIPT italic\_o italic\_u italic\_t end\_POSTSUBSCRIPT and Cinsubscript𝐶𝑖𝑛C\_{in}italic\_C start\_POSTSUBSCRIPT italic\_i italic\_n end\_POSTSUBSCRIPT represent the input and output channels respectively. (Kwsubscript𝐾𝑤K\_{w}italic\_K start\_POSTSUBSCRIPT italic\_w end\_POSTSUBSCRIPT, Khsubscript𝐾ℎK\_{h}italic\_K start\_POSTSUBSCRIPT italic\_h end\_POSTSUBSCRIPT) and (W𝑊Witalic\_W, H𝐻Hitalic\_H) are the dimensions of the convolution kernel and of the feature map, and bs𝑏𝑠bsitalic\_b italic\_s is the batch size. \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| W\X=(Sign(W)⊙Sign(X))⊗β𝑊𝑋tensor-productdirect-productSign𝑊Sign𝑋𝛽\begin{split}W\X=(\mathrm{Sign}(W)\odot\mathrm{Sign}(X))\otimes\beta\end{split}start\_ROW start\_CELL italic\_W \* italic\_X = ( roman\_Sign ( italic\_W ) ⊙ roman\_Sign ( italic\_X ) ) ⊗ italic\_β end\_CELL end\_ROW \| \| (1) \| Here ⊙direct-product\odot⊙ denotes binary multiplication, which could be simplified into XNOR and bitcount operations. ⊗tensor-product\otimes⊗ denotes full precision element-wise multiplication. β𝛽\betaitalic\_β is a real-valued scaling factor. During inference, the binarization takes place before the convolution. During training, the gradient of the non-differentiable binarization (Sign(w)Sign𝑤\mathrm{Sign}(w)roman\_Sign ( italic\_w )) is acquired with the Straight-Through [[8](#bib.bib8)] scheme to update the real-valued weights W𝑊Witalic\_W. ### 3.2 Differentiable NAS BARS adopts a differentiable architecture search flow [[16](#bib.bib16), [7](#bib.bib7), [32](#bib.bib32)]. In differentiable NAS, a supernet is constructed such that all possible architectures are sub-architectures of this supernet. Then, architectural choices parameterized by architectural parameters α𝛼\alphaitalic\_α are optimized following the gradients of the validation loss Lvalsubscript𝐿𝑣𝑎𝑙L\_{val}italic\_L start\_POSTSUBSCRIPT italic\_v italic\_a italic\_l end\_POSTSUBSCRIPT. The bilevel optimization problem can be written as \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| minα⁡Lval(w\(α),α)s.t. w\(α)=argminw⁡Ltrain(w,α)subscript𝛼subscript𝐿𝑣𝑎𝑙superscript𝑤𝛼𝛼s.t. superscript𝑤𝛼subscriptargmin𝑤subscript𝐿𝑡𝑟𝑎𝑖𝑛𝑤𝛼\begin{split}&\min\_{\alpha}{L\_{val}(w^{\}(\alpha),\alpha)}\\ \mbox{s.t. }&w^{\}(\alpha)=\operatorname\{arg\,min}\_{w}L\_{train}(w,\alpha)\end{split}start\_ROW start\_CELL end\_CELL start\_CELL roman\_min start\_POSTSUBSCRIPT italic\_α end\_POSTSUBSCRIPT italic\_L start\_POSTSUBSCRIPT italic\_v italic\_a italic\_l end\_POSTSUBSCRIPT ( italic\_w start\_POSTSUPERSCRIPT \ end\_POSTSUPERSCRIPT ( italic\_α ) , italic\_α ) end\_CELL end\_ROW start\_ROW start\_CELL s.t. end\_CELL start\_CELL italic\_w start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ( italic\_α ) = start\_OPERATOR roman\_arg roman\_min end\_OPERATOR start\_POSTSUBSCRIPT italic\_w end\_POSTSUBSCRIPT italic\_L start\_POSTSUBSCRIPT italic\_t italic\_r italic\_a italic\_i italic\_n end\_POSTSUBSCRIPT ( italic\_w , italic\_α ) end\_CELL end\_ROW \| \| (2) \| After the search, one needs to derive a discrete architecture using the relaxed architecture parameter α𝛼\alphaitalic\_α. In the original differentiable NAS method [[16](#bib.bib16)], for the normal and reduction cell type, the operation with the maximum α𝛼\alphaitalic\_α (except for the “none” operation) is chosen on each edge. Then the normal and reduction cells are stacked to construct the final model. Studies [[33](#bib.bib33)] have shown that the derive process introduces a large discrepancy. BARS also focuses on how to bridge the search-derive gap. 4 BARS Framework ----------------- As we have analyzed before, the unsatisfying performance of BNNs can be attributed to the information bottlenecks that are related to both the macro and micro-level architectural design. Due to the high dimension of the architectural design, it is hard to analyze the architectural bottleneck and design suitable BNN architecture manually, BARS seeks to solve this issue in an automatic manner with differentiable NAS. BARS designs a macro search space (Sec. [4.1](#S4.SS1 "4.1 Macro-level: Search for Network Depth/Width ‣ 4 BARS Framework ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures")) with a learnable cell layout to strike a balance between performance and complexity, and a micro-level search space (Sec. [4.2](#S4.SS2 "4.2 Micro-level: Search for Cell Topology ‣ 4 BARS Framework ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures")) tailored for maximizing the information flow. Aside from that, we employ a few search strategies (Sec [4.3](#S4.SS3 "4.3 Search Strategy ‣ 4 BARS Framework ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures")) to address the “collapse” problem of differentiable NAS and facilitate a stable search process. ### 4.1 Macro-level: Search for Network Depth/Width Fig. [3](#S4.F3 "Figure 3 ‣ 4.1 Macro-level: Search for Network Depth/Width ‣ 4 BARS Framework ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures") shows how width/depth configurations affect the performances of XNOR/FP ResNets. In general, BNNs are more sensitive to changes in width and depth. And we can see that expanding the operation width can alleviate the bottlenecks of binarized operations and brings consistent improvements. As the width goes up, the performance gain would gradually vanish while still increasing the model complexity. In contrast, increasing the depth does not always bring improvements to BNNs, which is intuitive since adding more processing cannot recover the information once the information is already lost in previous bottlenecks. Due to these distinct preferences of BNNs, directly adopting layouts of CNNs as in [[27](#bib.bib27), [26](#bib.bib26)] might lead to suboptimal designs. And we propose to directly search for the “sweet spot” of width and depth to trade-off performance and complexity for BNNs. We extend the original micro-level DARTS framework to the macro-level by designing a macro depth & width search space that could be unified in the DARTS framework. Width Search Expanding the width of BNNs can bring consistent improvements [[1](#bib.bib1)], and different parts of the network have different sensitivity to the width choices [[9](#bib.bib9)]. Previous NAS for BNN studies [[27](#bib.bib27), [3](#bib.bib3)] neglect this aspect in the search process and use post-search uniform width expansion, which would lead to suboptimal results (See Fig. [6](#S5.F6 "Figure 6 ‣ 5.2 Results on CIFAR-10 and ImageNet ‣ 5 Experiments and Analysis ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures")). In contrast, BARS seeks to directly search for the proper width configuration to better balance the model complexity and performance. Specifically, for all cells in one stage, we use a width architecture parameter αwidthsubscript𝛼𝑤𝑖𝑑𝑡ℎ\alpha\_{width}italic\_α start\_POSTSUBSCRIPT italic\_w italic\_i italic\_d italic\_t italic\_h end\_POSTSUBSCRIPT to denote the probability logits of selecting different width choices (e.g. [0.25,0.5,0.75,1.0]0.250.50.751.0[0.25,0.5,0.75,1.0][ 0.25 , 0.5 , 0.75 , 1.0 ] in our experiments). Specifically, during the forward process, we sample a relaxed width choice m∈ℛ4𝑚superscriptℛ4m\in\mathcal{R}^{4}italic\_m ∈ caligraphic\_R start\_POSTSUPERSCRIPT 4 end\_POSTSUPERSCRIPT from the distribution parameterized by αwidthsubscript𝛼𝑤𝑖𝑑𝑡ℎ\alpha\_{width}italic\_α start\_POSTSUBSCRIPT italic\_w italic\_i italic\_d italic\_t italic\_h end\_POSTSUBSCRIPT with Gumbel-Softmax sampling, and multiply the weighted mask to all the feature maps in that stage. Denoting the full-width output feature map as y′superscript𝑦′y^{\prime}italic\_y start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, the relaxed output feature map y𝑦yitalic\_y that takes the width choice into consideration is calculated as \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| 𝐦=Gumbel-Softmax(αwidth)y=(∑i𝐦𝐢Mi)⊙y′𝐦Gumbel-Softmaxsubscript𝛼𝑤𝑖𝑑𝑡ℎ𝑦direct-productsubscript𝑖subscript𝐦𝐢subscript𝑀𝑖superscript𝑦′\begin{split}{\bf m}=&\mbox{Gumbel-Softmax}(\alpha\_{width})\\ y=&(\sum\_{i}{\bf m\_{i}}M\_{i})\odot y^{\prime}\\ \end{split}start\_ROW start\_CELL bold\_m = end\_CELL start\_CELL Gumbel-Softmax ( italic\_α start\_POSTSUBSCRIPT italic\_w italic\_i italic\_d italic\_t italic\_h end\_POSTSUBSCRIPT ) end\_CELL end\_ROW start\_ROW start\_CELL italic\_y = end\_CELL start\_CELL ( ∑ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT bold\_m start\_POSTSUBSCRIPT bold\_i end\_POSTSUBSCRIPT italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ⊙ italic\_y start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_CELL end\_ROW \| \| (3) \| where ⊙direct-product\odot⊙ denotes the element-wise multiplication, and Misubscript𝑀𝑖M\_{i}italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is the mask corresponding to the i𝑖iitalic\_i-th width choice in [0.25,0.5,0.75,1.0]0.250.50.751.0[0.25,0.5,0.75,1.0][ 0.25 , 0.5 , 0.75 , 1.0 ]. For example, if i=0𝑖0i=0italic\_i = 0 (the relative width choice is 0.25), the first quarter of the elements in Misubscript𝑀𝑖M\_{i}italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT are 1111s and the other elements are 00s. Depth Search (Cell-1, Cell-2, Cell-3) in one stage, the probabilities of choosing different depth ∈{0,1,2,3}absent0123\in\{0,1,2,3\}∈ { 0 , 1 , 2 , 3 } can be calculated as the softmax of the four-dimensional depth parameter αpath=[αp0,αp1,αp2,αp3]subscript𝛼𝑝𝑎𝑡ℎsubscript𝛼𝑝0subscript𝛼𝑝1subscript𝛼𝑝2subscript𝛼𝑝3\alpha\_{path}=[\alpha\_{p0},\alpha\_{p1},\alpha\_{p2},\alpha\_{p3}]italic\_α start\_POSTSUBSCRIPT italic\_p italic\_a italic\_t italic\_h end\_POSTSUBSCRIPT = [ italic\_α start\_POSTSUBSCRIPT italic\_p 0 end\_POSTSUBSCRIPT , italic\_α start\_POSTSUBSCRIPT italic\_p 1 end\_POSTSUBSCRIPT , italic\_α start\_POSTSUBSCRIPT italic\_p 2 end\_POSTSUBSCRIPT , italic\_α start\_POSTSUBSCRIPT italic\_p 3 end\_POSTSUBSCRIPT ]: P(depth=i)=exp⁡(αpi)∑j=03exp⁡(αpj)𝑃depth𝑖subscript𝛼𝑝𝑖superscriptsubscript𝑗03subscript𝛼𝑝𝑗P(\mbox{depth}=i)=\frac{\exp(\alpha\_{pi})}{\sum\_{j=0}^{3}\exp(\alpha\_{pj})}italic\_P ( depth = italic\_i ) = divide start\_ARG roman\_exp ( italic\_α start\_POSTSUBSCRIPT italic\_p italic\_i end\_POSTSUBSCRIPT ) end\_ARG start\_ARG ∑ start\_POSTSUBSCRIPT italic\_j = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 3 end\_POSTSUPERSCRIPT roman\_exp ( italic\_α start\_POSTSUBSCRIPT italic\_p italic\_j end\_POSTSUBSCRIPT ) end\_ARG. Denoting the input feature map of this stage as y0subscript𝑦0y\_{0}italic\_y start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT, and the output feature map of each cell as y1,⋯,y3subscript𝑦1⋯subscript𝑦3y\_{1},\cdots,y\_{3}italic\_y start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , ⋯ , italic\_y start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT, the aggregated feature map yaggrsubscript𝑦𝑎𝑔𝑔𝑟y\_{aggr}italic\_y start\_POSTSUBSCRIPT italic\_a italic\_g italic\_g italic\_r end\_POSTSUBSCRIPT is calculated as \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| 𝐦=Gumbel-Softmax(αpath)yaggr=∑i𝐦𝐢×yi.𝐦Gumbel-Softmaxsubscript𝛼𝑝𝑎𝑡ℎsubscript𝑦𝑎𝑔𝑔𝑟subscript𝑖subscript𝐦𝐢subscript𝑦𝑖\begin{split}{\bf m}=&\mbox{Gumbel-Softmax}(\alpha\_{path})\\ y\_{aggr}&=\sum\_{i}{\bf m\_{i}}\times y\_{i}.\end{split}start\_ROW start\_CELL bold\_m = end\_CELL start\_CELL Gumbel-Softmax ( italic\_α start\_POSTSUBSCRIPT italic\_p italic\_a italic\_t italic\_h end\_POSTSUBSCRIPT ) end\_CELL end\_ROW start\_ROW start\_CELL italic\_y start\_POSTSUBSCRIPT italic\_a italic\_g italic\_g italic\_r end\_POSTSUBSCRIPT end\_CELL start\_CELL = ∑ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT bold\_m start\_POSTSUBSCRIPT bold\_i end\_POSTSUBSCRIPT × italic\_y start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT . end\_CELL end\_ROW \| \| (4) \| Complexity Regularization Besides the performance, the model complexity is also largely influenced by the width/depth macro-level search decisions. Thus, we use a search objective that considers the complexity (FLOPs) to properly balance the performance and complexity. \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| \| \| L𝐿\displaystyle Litalic\_L \| =L0×[FLOPs(α)F]θabsentsubscript𝐿0superscriptdelimited-[]𝐹𝐿𝑂𝑃𝑠𝛼𝐹𝜃\displaystyle=L\_{0}\times\left[\frac{FLOPs(\alpha)}{F}\right]^{\theta}= italic\_L start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT × [ divide start\_ARG italic\_F italic\_L italic\_O italic\_P italic\_s ( italic\_α ) end\_ARG start\_ARG italic\_F end\_ARG ] start\_POSTSUPERSCRIPT italic\_θ end\_POSTSUPERSCRIPT \| \| (5) \| \| \| θ𝜃\displaystyle\thetaitalic\_θ \| ={γ,if FLOPs(α)≥Fμ,otherwiseabsentcases𝛾if 𝐹𝐿𝑂𝑃𝑠𝛼 𝐹𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒𝜇otherwise 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒\displaystyle=\begin{cases}\gamma,\quad\mbox{if }FLOPs(\alpha)\geq F&\\ \mu,\quad\text{otherwise}\end{cases}= { start\_ROW start\_CELL italic\_γ , if italic\_F italic\_L italic\_O italic\_P italic\_s ( italic\_α ) ≥ italic\_F end\_CELL start\_CELL end\_CELL end\_ROW start\_ROW start\_CELL italic\_μ , otherwise end\_CELL start\_CELL end\_CELL end\_ROW \| \| where L0subscript𝐿0L\_{0}italic\_L start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT is the original loss and F𝐹Fitalic\_F is the FLOPs budget, and FLOPs(α)𝐹𝐿𝑂𝑃𝑠𝛼FLOPs(\alpha)italic\_F italic\_L italic\_O italic\_P italic\_s ( italic\_α ) is the current estimation of FLOPs. γ𝛾\gammaitalic\_γ and θ𝜃\thetaitalic\_θ are hyperparameters and γ𝛾\gammaitalic\_γ is set to 0 in our experiments. ![Refer to caption](/html/2011.10804/assets/figs/macro-motivation.jpg) Figure 3: Performances of (binary) ResNet18 variants with different complexity. The original ResNet18 architecture are scaled by uniform expansion in the channel dimension (width), or stacking more layers (depth). \| Method \| Accuracy \| \| --- \| --- \| \| FP downsample \| 90.6% (-%) \| \| with op shortcut \| 91.4 (+0.8%) \| \| binarized downsample \| 89.7% (-0.8%) \| \| improved binarized downsample \| 89.9% (-0.6%) \| \| XNOR-Res34 (∼similar-to\sim∼1.5x complexity) \| 91.5% (+0.9%) \| Table 1: Performance with micro modifications applied on Binary ResNet18. Binarizing the downsampling layer causes noticable accuracy drop and improved binary downsampling could mitigate the loss. Adding op-wise shortcut could improve the performance without extra overhead, and achieve competitive performances with a larger model. ### 4.2 Micro-level: Search for Cell Topology As mentioned above, identifying and eliminating the information bottleneck is vital for improving the accuracy of BNNs [[1](#bib.bib1)]. Since it is difficult to assess and identify all information bottlenecks manually, we design the micro-level search space such that the NAS process can automatically discover topologies to avoid creating bottlenecks and maintain a proper information flow. As illustrated in Fig. [2](#S1.F2 "Figure 2 ‣ 1 Introduction ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures"), our micro-level search space contains five nodes, and each node can choose to connect from an arbitrary number of previous nodes. For each edge, there are 3 possible operation primitives: binary convolution 3×\times×3, shortcut, and none. We will show in Sec. [5.5](#S5.SS5 "5.5 Discovered Cell ‣ 5 Experiments and Analysis ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures") that our micro-level topology search automatically discovers cells with a strong information flow. Besides the previous search space design, the cell template and operation primitives should also be modified to eliminate information bottlenecks. \| Dataset \| Method \| FLOPs \| BiOps \| Equivalent Ops \| Params \| Fully-Binarized \| Acc. \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| CIFAR-10 \| NiN (XNOR) \| ∼similar-to\sim∼9M \| ∼similar-to\sim∼200M \| 12.13M \| - \| ✓ \| 86.28% \| \| ResNet-18 (XNOR) \| 16M \| 547M \| 25.55M \| 11.17M \| ✓ \| 90.55% \| \| ResNet-18 (Bireal) \| 11M \| 561M \| 24.77M \| 11.34M \| FP Downsample \| 91.23% \| \| ResNet-34 (XNOR) \| 27M \| 1151M \| 44.98M \| 21.81M \| FP Downsample \| 91.49% \| \| WRN-40 (XNOR) \| ∼similar-to\sim∼27M \| ∼similar-to\sim∼1500M \| 50.44M \| - \| FP Downsample \| 91.58% \| \| ResNet-18 (IRNet) \| ∼similar-to\sim∼27M \| ∼similar-to\sim∼500M \| 34.81M \| - \| FP Downsample \| 91.5% \| \| BNAS (XNOR) \| 100M \| 1393M \| 121.76M \| 5.57M \| FP Cell Shortcut \| 92.7% \| \| BARS-A \| 2M \| 513M \| 10.02M \| 2.77M \| ✓ \| 91.25% \| \| BARS-B \| 2M \| 1048M \| 18.37M \| 6.07M \| ✓ \| 92.98% \| \| BARS-C \| 3M \| 1778M \| 32.27M \| 10.76M \| ✓ \| 93.43% \| \| ImageNet \| ResNet-18 (ABC) \| ∼similar-to\sim∼100M \| ∼similar-to\sim∼2000M \| 131.25M \| - \| ✓ \| 42.7% \| \| ResNet-18 (XNOR) \| 138M \| 1850 \| 188.89M \| 12.54M \| ✓ \| 48.3% \| \| ResNet-18 (XNOR) \| 167M \| 1778M \| 194.79M \| 12.80M \| FP Downsample \| 53.1% \| \| BiDenseNet (XNOR) \| - \| - \| - \| 13.56M \| FP Downsample \| 52.7% \| \| ResNet-18 (Bireal) \| ∼similar-to\sim∼160M \| ∼similar-to\sim∼2000M \| 191.25M \| - \| FP Downsample \| 56.4% \| \| ResNet-18 (PCNN) \| ∼similar-to\sim∼160M \| ∼similar-to\sim∼2000M \| 191.25M \| - \| FP Downsample \| 57.3% \| \| MoBiNet(XNOR) \| - \| - \| - \| 8.47M \| FP Downsample \| 53.4% \| \| BNAS (XNOR) \| 195M \| 3137M \| 244.0M \| 28.41M \| FP Cell Shortcut \| 57.6% \| \| \| BARS-D \| 129M \| 998M \| 129.60M \| 9.01M \| ✓ \| 54.6% \| \| \| BARS-E \| 161M \| 1424M \| 183.25M \| 14.04M \| ✓ \| 56.2% \| \| \| BARS-F \| 254M \| 2594M \| 293.53M \| 19.29M \| ✓ \| 60.3% \| Table 2: Performance and complexity comparison of BARS-discovered architectures and baselines on CIFAR-10 and ImageNet. BARS-discovered architectures outperform baseline architectures with much lower resource consumption. “Equivalent Ops” are calculated as FLOPs+1/64\BiOPs𝐹𝐿𝑂𝑃𝑠164𝐵𝑖𝑂𝑃𝑠FLOPs+1/64\BiOPsitalic\_F italic\_L italic\_O italic\_P italic\_s + 1 / 64 \* italic\_B italic\_i italic\_O italic\_P italic\_s [[1](#bib.bib1)]. “Fully-Binarized” means all network components except for the first and last layer remain binary. Unlike recent studies [[18](#bib.bib18), [17](#bib.bib17), [27](#bib.bib27)] on binary architectures use full-precision downsampling operations, BARS-discovered architectures only use binarized operations in the major architecture backbone, which is beneficial for hardware acceleration. Strengthen the information flow Many recent studies on BNNs [[1](#bib.bib1), [18](#bib.bib18)] and our experimental results in Tab. [1](#S4.T1 "Table 1 ‣ 4.1 Macro-level: Search for Network Depth/Width ‣ 4 BARS Framework ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures") show that adding an extra shortcut for binary convolution improves the performance with little overhead. Therefore, we add operational-level shortcuts and cell-level shortcuts to strengthen the information flow. Eliminating the information Bottleneck As shown in Tab. [1](#S4.T1 "Table 1 ‣ 4.1 Macro-level: Search for Network Depth/Width ‣ 4 BARS Framework ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures"), binarizing the downsampling layer causes the performance to drop sharply. There are two reasons why the downsampling layer is the bottleneck: 1) The misalignment of input/output channel size makes it difficult to add op-wise shortcuts. 2) It involves spatial dimension reduction which causes a large information loss. In BARS, we design the downsampling operation to concatenate the outputs of two strided convolutions with shortcuts111We use 2×\times×2 AvgPool as the shortcut with spatial dim. reduction. and spatially staggered input. In such way, the op-level shortcut could be added to the two binarized convolutions. It is worthy to note that, unlike many previous studies [[18](#bib.bib18), [24](#bib.bib24)] that use FP downsampling layers, we mitigate this bottleneck issue in a fully-binarized manner. And we demonstrate its effect in Tab. [3](#S5.T3 "Table 3 ‣ 5.3 Effects of Stabilizing the Searching ‣ 5 Experiments and Analysis ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures") ### 4.3 Search Strategy Differentiable NAS [[16](#bib.bib16)] has been challenged due to it being prone to degenerated architectures containing too many shortcuts, which is known as the “collapse” problem. Parameterized operations (e.g. Conv) are usually under-trained [[33](#bib.bib33), [12](#bib.bib12)], and the search process will favor parameter-free operations. In BNN, this problem is further exacerbated since binary convolutions are even harder to train. This section describes several key techniques we use to alleviate the “collapse” problem. We briefly introduce them here and further analyze their effects in Sec. [5.3](#S5.SS3 "5.3 Effects of Stabilizing the Searching ‣ 5 Experiments and Analysis ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures"). Gumbel-Softmax Sampling We use Gumbel sampling with proper temperature scheduling for all architecture decisions. For all architectural parameters (depth αpathsubscript𝛼𝑝𝑎𝑡ℎ\alpha\_{path}italic\_α start\_POSTSUBSCRIPT italic\_p italic\_a italic\_t italic\_h end\_POSTSUBSCRIPT, width αwidthsubscript𝛼𝑤𝑖𝑑𝑡ℎ\alpha\_{width}italic\_α start\_POSTSUBSCRIPT italic\_w italic\_i italic\_d italic\_t italic\_h end\_POSTSUBSCRIPT, operation type α𝛼\alphaitalic\_α), we sample a relaxed architecture m𝑚mitalic\_m from the corresponding multinomial architectural distribution Multinomial(m\|α)Multinomialconditional𝑚𝛼\mbox{Multinomial}(m\|\alpha)Multinomial ( italic\_m \| italic\_α ) with the Gumbel-Softmax technique [[14](#bib.bib14)]. Denoting the number of choices as D𝐷Ditalic\_D, the logits of the Multinomial distribution as α𝛼\alphaitalic\_α, each dimension misubscript𝑚𝑖m\_{i}italic\_m start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT in the relaxed architecture decision m∈[0,1]D𝑚superscript01𝐷m\in[0,1]^{D}italic\_m ∈ [ 0 , 1 ] start\_POSTSUPERSCRIPT italic\_D end\_POSTSUPERSCRIPT can be represented as \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| mi=exp⁡((αi+gi)/τ)∑j=1Dexp⁡((αj+gj)/τ)for i=1,⋯D,formulae-sequencesubscript𝑚𝑖subscript𝛼𝑖subscript𝑔𝑖𝜏superscriptsubscript𝑗1𝐷subscript𝛼𝑗subscript𝑔𝑗𝜏for 𝑖1⋯𝐷\begin{split}m\_{i}=\frac{\exp{((\alpha\_{i}+g\_{i})/\tau)}}{\sum\_{j=1}^{D}\exp((\alpha\_{j}+g\_{j})/\tau)}\quad\mbox{for }i=1,\cdots D,\end{split}start\_ROW start\_CELL italic\_m start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT = divide start\_ARG roman\_exp ( ( italic\_α start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT + italic\_g start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) / italic\_τ ) end\_ARG start\_ARG ∑ start\_POSTSUBSCRIPT italic\_j = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_D end\_POSTSUPERSCRIPT roman\_exp ( ( italic\_α start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT + italic\_g start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ) / italic\_τ ) end\_ARG for italic\_i = 1 , ⋯ italic\_D , end\_CELL end\_ROW \| \| (6) \| where gisubscript𝑔𝑖g\_{i}italic\_g start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPTs are standard Gumbel-distributed random variables. We emphasize that using Gumbel-Softmax sampling with a proper temperature schedule is important. In the early search stage, it remains high which drives architecture distributions to be more uniform to encourage search exploration and avoid collapsing. In the later stage, it gradually anneals to zero to drive α𝛼\alphaitalic\_α towards confident one-hot choices. Thus the discrepancy between searching and deriving [[6](#bib.bib6)] is reduced. Entropy Regularization and Supernet Warm-up To address the “collapse” issue caused by insufficient optimization of parameterized operations, we conduct warm-up training of the supernet weights for several epochs. We also impose entropy regularization on the architecture distribution to encourage exploration in the early search stage and exploitation in the late search stage. \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| Lent=λent(−∑iαilog⁡(αi)),subscript𝐿𝑒𝑛𝑡subscript𝜆𝑒𝑛𝑡subscript𝑖subscript𝛼𝑖subscript𝛼𝑖\begin{split}L\_{ent}=\lambda\_{ent}(-\sum\_{i}{\alpha\_{i}\log(\alpha\_{i}))},\end{split}start\_ROW start\_CELL italic\_L start\_POSTSUBSCRIPT italic\_e italic\_n italic\_t end\_POSTSUBSCRIPT = italic\_λ start\_POSTSUBSCRIPT italic\_e italic\_n italic\_t end\_POSTSUBSCRIPT ( - ∑ start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT italic\_α start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT roman\_log ( italic\_α start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ) , end\_CELL end\_ROW \| \| (7) \| where λentsubscript𝜆𝑒𝑛𝑡\lambda\_{ent}italic\_λ start\_POSTSUBSCRIPT italic\_e italic\_n italic\_t end\_POSTSUBSCRIPT is a hyperparameter that follows an increasing schedule from negative to positive. Its scheduling plays a similar role as the temperature in gumbel sampling. 5 Experiments and Analysis --------------------------- ### 5.1 Experiment Settings We run BARS on CIFAR-10 and ImageNet datasets with different FLOPs target, and acquire a series of models of different sizes (BARS-A/B/C on CIFAR, BARS-D/E/F on ImageNet). Detailed experimental settings can be found in the appendix. Note that unlike previous studies [[27](#bib.bib27), [3](#bib.bib3)] that transfer architectures found on CIFAR-10 to ImageNet, we directly apply search on the 100-class subset of ImageNet. We conduct experiments on CIFAR-10 and ImageNet. For searching on both datasets, we construct a 14-cell super network organized into 3 stages. The cells in each stage share the same micro-level topology, and so do all the reduction cells. The base channel number Cisubscript𝐶𝑖C\_{i}italic\_C start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is 48. We choose 4 available width choices r×Ci,r∈{0.25,0.5,0.75,1}𝑟subscript𝐶𝑖𝑟 0.250.50.751r\times C\_{i},r\in\{0.25,0.5,0.75,1\}italic\_r × italic\_C start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_r ∈ { 0.25 , 0.5 , 0.75 , 1 }, and 3 candidate operations [binary\_conv\_3x3,shortcut,none]𝑏𝑖𝑛𝑎𝑟𝑦\_𝑐𝑜𝑛𝑣\_3𝑥3𝑠ℎ𝑜𝑟𝑡𝑐𝑢𝑡𝑛𝑜𝑛𝑒[binary\\_conv\\_3x3,shortcut,none][ italic\_b italic\_i italic\_n italic\_a italic\_r italic\_y \_ italic\_c italic\_o italic\_n italic\_v \_ 3 italic\_x 3 , italic\_s italic\_h italic\_o italic\_r italic\_t italic\_c italic\_u italic\_t , italic\_n italic\_o italic\_n italic\_e ]. Within each cell, the preprocess layer is a binary 1x1 conv with no shortcut. The cell-wise shortcut is an identity operation for normal cells and a strided binary 3x3 conv for reduction cells. ![Refer to caption](/html/2011.10804/assets/figs/bar.jpg) Figure 4: Comparison of BARS derived architectures with baseline methods in terms of FLOPs (floating-point ops), BiOPs (binary ops), and Params. Note that for BiOPs, following previous studies [[1](#bib.bib1)], we divide it by 64 to acquire its relative values. BARS finds architectures with better performance with less equivalent OPs/params and significantly fewer floating-point operations. The search lasts for 50 epochs, and a batch size of 64 is used. Supernet weights w𝑤witalic\_w is trained with Adam optimizer, whose learning rate is set to 3e-4 initially and decayed to 0 in 50 epochs following a cosine schedule. After 5 epochs of warm-up training of supernet weights, we begin to update α𝛼\alphaitalic\_α. The architectural parameters α𝛼\alphaitalic\_α (including αmicro,αpath,αwidthsubscript𝛼𝑚𝑖𝑐𝑟𝑜subscript𝛼𝑝𝑎𝑡ℎsubscript𝛼𝑤𝑖𝑑𝑡ℎ\alpha\_{micro},\alpha\_{path},\alpha\_{width}italic\_α start\_POSTSUBSCRIPT italic\_m italic\_i italic\_c italic\_r italic\_o end\_POSTSUBSCRIPT , italic\_α start\_POSTSUBSCRIPT italic\_p italic\_a italic\_t italic\_h end\_POSTSUBSCRIPT , italic\_α start\_POSTSUBSCRIPT italic\_w italic\_i italic\_d italic\_t italic\_h end\_POSTSUBSCRIPT) are updated using Adam optimizer with learning rate 3e-4 and weight decay 1e-3. The Gumbel temperature is set to 1111 at first and multiplied by 0.90.90.90.9 on every epoch. The entropy regularization coefficient λentsubscript𝜆𝑒𝑛𝑡\lambda\_{ent}italic\_λ start\_POSTSUBSCRIPT italic\_e italic\_n italic\_t end\_POSTSUBSCRIPT follows an increasing schedule: it starts at -0.01, and 0.001 is added on every epoch. As for deriving, we sample 8 candidate architectures from the architecture distribution α𝛼\alphaitalic\_α after the search. The one with minimum validation loss is chosen. Different from the origin DARTS, we do not need to exclude the “none” operation when deriving. We argue that it is important since binary convs evolve huge information loss, thus “none” operation might be the best choice on some edges in BNN [[27](#bib.bib27)]. On CIFAR-10, we train the derived architecture for 200 epochs with batch size 256. Adam optimizer with a weight decay of 00 is used, and the learning rate is set to 2e−32𝑒32e-32 italic\_e - 3 at first and decayed to 0 following a cosine schedule. Cutout augmentation and auxiliary towers with weight 0.4 are applied following previous studies [[16](#bib.bib16)]. On ImageNet, the architectures are trained for 100 epochs, and no cutout is used. We also use Adam optimizer with no weight decay. The learning rate has the initial value of 1.e−3formulae-sequence1𝑒31.e-31 . italic\_e - 3 and cosine annealed to 0 for a batch size of 256256256256. ![Refer to caption](/html/2011.10804/assets/figs/alphas.jpg) Figure 5: Evolution of α𝛼\alphaitalic\_α’s distribution and the relative possibility of “shortcut” in the search process Left: Comparison of α𝛼\alphaitalic\_α’s distribution change for “collapsed search” and BARS’s stabilized search. Entropy regularization and Gumbel-Softmax sampling prevent “collapse” in search and bridge the gap in deriving. Right: “shortcut” operation’s average probability during the search. BARS prevents the search from collapsing rapidly. ### 5.2 Results on CIFAR-10 and ImageNet Tab. [2](#S4.T2 "Table 2 ‣ 4.2 Micro-level: Search for Cell Topology ‣ 4 BARS Framework ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures") and Fig. [4](#S5.F4 "Figure 4 ‣ 5.1 Experiment Settings ‣ 5 Experiments and Analysis ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures") show the comparison of BARS-discovered architectures and the baseline ones. We can see that BARS discovers architectures with superior performance and efficiency. Note that in order to demonstrate the performance gain brought by the architectural design, all our models are trained from scratch with XNORNet [[24](#bib.bib24)] binarization scheme.Neither additional tricks [[23](#bib.bib23)], nor full-precision pre-training models [[18](#bib.bib18), [10](#bib.bib10)] are used. Moreover, we emphasize that BARS binarizes all operations in the major architecture (except the stem and the final layer), whereas previous studies use full-precision downsampling layers to maintain acceptable performances (e.g. the accuracy of ResNet-18 on ImageNet dropped from 53.1%percent53.153.1\%53.1 % to 48.3%percent48.348.3\%48.3 % if no FP downsample is used). On CIFAR-10, BARS-B achieves higher accuracy (1.5%percent1.51.5\%1.5 %) than the hand-crafted binary model with 2/3232/32 / 3 binary operations (BiOps), much less (1/101101/101 / 10) floating-point operations (FLOPs) and parameters. On ImageNet, BARS-D outperforms the “fully-binarized” ResNet-18 by a 6% with notably less resource consumption, it also outperforms many hand-crafted BNN models while binarizing the downsampling layer. ![Refer to caption](/html/2011.10804/assets/x3.png) Figure 6: Comparison of derived models of different complexity (FLOPs) w/o BARS’s complexity search. Upper: BARS search with different complexity regularization. Lower: search for a compact model and uniformly expand the network width. BARS finds the better pareto frontier. ### 5.3 Effects of Stabilizing the Searching The “collapse-to-shortcuts” problem is a widely known issue for differentiable NAS methods, as shown in the left part in Fig. [5](#S5.F5 "Figure 5 ‣ 5.1 Experiment Settings ‣ 5 Experiments and Analysis ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures"). As discussed in Sec. [4.3](#S4.SS3 "4.3 Search Strategy ‣ 4 BARS Framework ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures"), we apply warm-up training to prevent the search from collapsing in the very early stages. Then, entropy regularization and Gumbel sampling with proper hyperparameter scheduling are employed to encourage exploration in the early stages and confident decisions in the late stages. Fig. [5](#S5.F5 "Figure 5 ‣ 5.1 Experiment Settings ‣ 5 Experiments and Analysis ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures") shows that in the early stage of searching, the distribution of α𝛼\alphaitalic\_α is relatively uniform and the relative ranking of different operations keeps changing. When reaching the end of the search, the Gumbel temperature is close to zero, making the sampling close to one-hot sampling. Also, entropy regularization encourages architecture distribution to be more confident. This reduces the derive discrepancy. \| Method \| Accuracy \| \| --- \| --- \| \| BARS-B \| 93.0% \| \| BARS-B (without op shortcut) \| 92.8% (-0.2%) \| \| BARS-B (without improved ds.) \| 92.6% (-0.3%) \| \| Sampled arch. \| 92.9% \| \| Sampled arch. (without op shortcut) \| 89.8% (-3.1%) \| \| Sampled arch. (without improved ds.) \| 91.9% (-1.0%) \| Table 3: The ablation study of micro-level modifications for BARS-B and sampled architecture. Searched BARS-B model finds that each shortcuts should coordinate with binary convs, thus the performance does not drop much without the op-level shortcut. ### 5.4 Effects of the Search Space Design We conduct several experiments to verify the effectiveness of the design choices of our search space. As could be seen from Fig. [6](#S5.F6 "Figure 6 ‣ 5.2 Results on CIFAR-10 and ImageNet ‣ 5 Experiments and Analysis ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures"), the macro-level joint search of width and depth strikes a better balance between performance and complexity. The discovered BARS-B model achieves much higher accuracy than uniformly expanding the width of the smaller BARS-A model. For the micro-level design, we have shown that additional shortcut and improved binary downsample can bring performance gain for XNOR-ResNet18 in Tab. [3](#S5.T3 "Table 3 ‣ 5.3 Effects of Stabilizing the Searching ‣ 5 Experiments and Analysis ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures"). We conduct similar ablation studies on BARS-B and a random sampled architecture from our search space in Tab. [1](#S4.T1 "Table 1 ‣ 4.1 Macro-level: Search for Network Depth/Width ‣ 4 BARS Framework ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures"). And we can witness a noticeable accuracy degradation when removing our modifications. ### 5.5 Discovered Cell The BARS-discovered cells on CIFAR-10 are shown in Fig. [9](#A3.F9 "Figure 9 ‣ Appendix C Detailed Comparison of Searched Cells ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures"). We can see that the cells in earlier stages and the reduction cells contain more convolutions. Conversely, the latter cells are dominated by shortcuts. The micro-level topology search also discovers interesting connection patterns: the shortcuts coordinate with binary convs to strengthen the information flow (e.g. shortcut 1-4 and shortcut 2-4 around conv 1-2 in the Normal-1 cell). Thanks to this highly skip-connected pattern, we can see from Tab. [3](#S5.T3 "Table 3 ‣ 5.3 Effects of Stabilizing the Searching ‣ 5 Experiments and Analysis ‣ BARS: Joint Search of Cell Topology and Layout for Accurate and Efficient Binary ARchitectures") that removing the op-level shortcut causes much less performance degradation for the BARS-discovered cell than for randomly sampled architecture. Also, instead of transferring the architecture discovered on CIFAR-10 to ImageNet, BARS conducts a direct search on the 100-class subset of ImageNet, and discovers distinct cells with more convolutions (see the appendix for more details). The different cell preferences might result from that more parameterized operations are needed for enough representational ability on the larger Imagenet dataset. ![Refer to caption](/html/2011.10804/assets/figs/bars-cells.jpg) Figure 7: BARS discovered cells on CIFAR. More convs are found in the reduction cell and shallower part of the network. It also learns that shortcuts should coordinate with binary convs. 6 Conclusion ------------- To better explore BNN architectures that are both accurate and efficient, BARS proposes to use a joint search of the macro layout and the micro topology to address the information bottleneck problem in BNN. The binary architectures discovered by BARS outperform baseline architectures with significantly less resource consumption.
fef6b0ac-bfad-4bc6-93e7-bc02c1c2aea7	StampyAI/alignment-research-dataset/arxiv	Arxiv	Induction of Subgoal Automata for Reinforcement Learning 1 Introduction --------------- Reinforcement learning (RL) (?) is a family of algorithms where an agent acts in an environment with the purpose of maximizing the total amount of reward it receives. These algorithms have played a key role in major breakthroughs like human-level video game playing (?) or mastering complex board games (?). However, current methods lack the ability to generalize and transfer between tasks. For example, they cannot learn to play chess using the knowledge of sho¯¯o\bar{\text{o}}over¯ start\_ARG o end\_ARGgi. Advances in achieving generalization and transfer in RL are mainly due to abstractions (?; ?). Hierarchical reinforcement learning methods, which divide a single task into several subtasks that can be solved separately, are among the most promising approaches to abstracting RL tasks. Common frameworks for hierarchical RL are HAMs (?), options (?) and MAXQ (?). Abstract hierarchies can be naturally represented using automata, which have been used effectively in task abstraction, not only in RL (?; ?), but also in related areas like automated planning (?; ?; ?). However, these RL approaches use handcrafted automata instead of learning them from data. It is largely an open problem in RL how to autonomously decompose a task into an automaton that can be exploited during the RL process. In this paper, we address this problem and propose ISA (Induction of Subgoal Automata for Reinforcement Learning), a method for learning and exploiting a minimal automaton from observation traces perceived by an RL agent. These automata are called subgoal automata since each transition is labeled with a subgoal, which is a boolean formula over a set of observables. Subgoal automata can be expressed in a logic programming format and, thus, be learned using state-of-the-art inductive logic programming (ILP) systems (?). The resulting subgoal automaton can be exploited by an RL algorithm that learns a different policy for each automaton state, each aiming to achieve a subgoal. This decomposition allows learning multiple subpolicies simultaneously and transferring knowledge across multiple tasks. ISA interleaves the RL and automaton learning processes such that the agent can immediately leverage the new (partially correct) learned subgoal automaton: when a trace is not correctly recognized by the automaton, a new one is learned. We evaluate ISA in several gridworld problems and show that it learns subgoal automata and policies for each of these. Specifically, it performs similarly to a method for which automata are given in advance. Furthermore, the learned automata can be exploited to speed up convergence through reward shaping and transfer learning. The paper is organized as follows. Section [2](#S2 "2 Background ‣ Induction of Subgoal Automata for Reinforcement Learning") introduces the background of our work. Section [3](#S3 "3 Methodology ‣ Induction of Subgoal Automata for Reinforcement Learning") formally presents the problem we address and our method for solving it, while Section [4](#S4 "4 Experimental Results ‣ Induction of Subgoal Automata for Reinforcement Learning") describes the results. We discuss related work in Section [5](#S5 "5 Related Work ‣ Induction of Subgoal Automata for Reinforcement Learning") and conclude in Section [6](#S6 "6 Conclusions and Future Work ‣ Induction of Subgoal Automata for Reinforcement Learning"). 2 Background ------------- In this section we briefly summarize the key background concepts used throughout the paper: reinforcement learning and inductive learning of answer set programs. ### Reinforcement Learning Reinforcement learning (RL) (?) is a family of algorithms for learning to act in an unknown environment. Typically, this learning process is formulated as a Markov Decision Process (MDP), i.e., a tuple ℳ=⟨S,A,p,r,γ⟩ℳ𝑆𝐴𝑝𝑟𝛾\mathcal{M}=\langle S,A,p,r,\gamma\ranglecaligraphic\_M = ⟨ italic\_S , italic\_A , italic\_p , italic\_r , italic\_γ ⟩, where S𝑆Sitalic\_S is a finite set of states, A𝐴Aitalic\_A is a finite set of actions, p:S×A→Δ(S):𝑝→𝑆𝐴Δ𝑆p:S\times A\to\Delta(S)italic\_p : italic\_S × italic\_A → roman\_Δ ( italic\_S ) is a transition probability function111For any finite set X𝑋Xitalic\_X, Δ(X)={μ∈ℝX:∑xμ(x)=1,μ(x)≥0(∀x)}Δ𝑋conditional-set𝜇superscriptℝ𝑋formulae-sequencesubscript𝑥𝜇𝑥1𝜇𝑥0for-all𝑥\Delta(X)=\{\mu\in\mathbb{R}^{X}:\sum\_{x}\mu(x)=1,\mu(x)\geq 0~{}(\forall x)\}roman\_Δ ( italic\_X ) = { italic\_μ ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_X end\_POSTSUPERSCRIPT : ∑ start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT italic\_μ ( italic\_x ) = 1 , italic\_μ ( italic\_x ) ≥ 0 ( ∀ italic\_x ) } is the probability simplex over X𝑋Xitalic\_X. , r:S×A×S→ℝ:𝑟→𝑆𝐴𝑆ℝr:S\times A\times S\to\mathbb{R}italic\_r : italic\_S × italic\_A × italic\_S → blackboard\_R is a reward function, and γ∈[0,1)𝛾01\gamma\in[0,1)italic\_γ ∈ [ 0 , 1 ) is a discount factor. At time t𝑡titalic\_t, the agent observes state st∈Ssubscript𝑠𝑡𝑆s\_{t}\in Sitalic\_s start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ∈ italic\_S, executes action at∈Asubscript𝑎𝑡𝐴a\_{t}\in Aitalic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ∈ italic\_A, transitions to the next state st+1∼p(⋅\|st,at)s\_{t+1}\sim p(\cdot\|s\_{t},a\_{t})italic\_s start\_POSTSUBSCRIPT italic\_t + 1 end\_POSTSUBSCRIPT ∼ italic\_p ( ⋅ \| italic\_s start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) and receives reward r(st,at,st+1)𝑟subscript𝑠𝑡subscript𝑎𝑡subscript𝑠𝑡1r(s\_{t},a\_{t},s\_{t+1})italic\_r ( italic\_s start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_s start\_POSTSUBSCRIPT italic\_t + 1 end\_POSTSUBSCRIPT ). We consider episodic MDPs that terminate in a given set of terminal states. We distinguish between goal states and undesirable terminal states (i.e., dead-ends). Let ST⊆Ssubscript𝑆𝑇𝑆S\_{T}\subseteq Sitalic\_S start\_POSTSUBSCRIPT italic\_T end\_POSTSUBSCRIPT ⊆ italic\_S be the set of terminal states and SG⊆STsubscript𝑆𝐺subscript𝑆𝑇S\_{G}\subseteq S\_{T}italic\_S start\_POSTSUBSCRIPT italic\_G end\_POSTSUBSCRIPT ⊆ italic\_S start\_POSTSUBSCRIPT italic\_T end\_POSTSUBSCRIPT the set of goal states. The aim is to find a policy π:S→Δ(A):𝜋→𝑆Δ𝐴\pi:S\to\Delta(A)italic\_π : italic\_S → roman\_Δ ( italic\_A ), a mapping from states to probability distributions over actions, that maximizes the expected sum of discounted reward (or return), Rt=𝔼[∑k=tnγk−trk]subscript𝑅𝑡𝔼delimited-[]superscriptsubscript𝑘𝑡𝑛superscript𝛾𝑘𝑡subscript𝑟𝑘R\_{t}=\mathbb{E}[\sum\_{k=t}^{n}\gamma^{k-t}r\_{k}]italic\_R start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT = blackboard\_E [ ∑ start\_POSTSUBSCRIPT italic\_k = italic\_t end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_k - italic\_t end\_POSTSUPERSCRIPT italic\_r start\_POSTSUBSCRIPT italic\_k end\_POSTSUBSCRIPT ], where n𝑛nitalic\_n is the last episode step. In model-free RL the transition probability function p𝑝pitalic\_p and reward function r𝑟ritalic\_r are unknown to the agent, and a policy is learned via interaction with the environment. Q-learning (?) computes an action-value function Qπ(s,a)=𝔼[Rt\|st=s,at=a]superscript𝑄𝜋𝑠𝑎𝔼delimited-[]formulae-sequenceconditionalsubscript𝑅𝑡subscript𝑠𝑡𝑠subscript𝑎𝑡𝑎Q^{\pi}(s,a)=\mathbb{E}[R\_{t}\|s\_{t}=s,a\_{t}=a]italic\_Q start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ( italic\_s , italic\_a ) = blackboard\_E [ italic\_R start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT \| italic\_s start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT = italic\_s , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT = italic\_a ] that estimates the return from each state-action pair when following an approximately optimal policy. In each iteration the estimates are updated as \| \| \| \| \| --- \| --- \| --- \| \| \| Q(s,a)←𝛼r(s,a,s′)+γmaxa′⁡Q(s′,a′),𝛼←𝑄𝑠𝑎𝑟𝑠𝑎superscript𝑠′𝛾subscriptsuperscript𝑎′𝑄superscript𝑠′superscript𝑎′Q\left(s,a\right)\xleftarrow{\alpha}r\left(s,a,s^{\prime}\right)+\gamma\max\_{a^{\prime}}Q\left(s^{\prime},a^{\prime}\right),italic\_Q ( italic\_s , italic\_a ) start\_ARROW overitalic\_α ← end\_ARROW italic\_r ( italic\_s , italic\_a , italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) + italic\_γ roman\_max start\_POSTSUBSCRIPT italic\_a start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT italic\_Q ( italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , italic\_a start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) , \| \| where x←𝛼y𝛼←𝑥𝑦x\xleftarrow{\alpha}yitalic\_x start\_ARROW overitalic\_α ← end\_ARROW italic\_y is shorthand for x←x+α(y−x)←𝑥𝑥𝛼𝑦𝑥x\leftarrow x+\alpha\left(y-x\right)italic\_x ← italic\_x + italic\_α ( italic\_y - italic\_x ), α𝛼\alphaitalic\_α is a learning rate and s′superscript𝑠′s^{\prime}italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT is the state after applying a𝑎aitalic\_a in s𝑠sitalic\_s. Usually, an ϵitalic-ϵ\epsilonitalic\_ϵ-greedy policy selects a random action with probability ϵitalic-ϵ\epsilonitalic\_ϵ and the action maximizing Q(s,a)𝑄𝑠𝑎Q(s,a)italic\_Q ( italic\_s , italic\_a ) otherwise. The policy is induced by the action that maximizes Q(s,a)𝑄𝑠𝑎Q(s,a)italic\_Q ( italic\_s , italic\_a ) in each s𝑠sitalic\_s. #### Options (?) address temporal abstraction in RL. Given an MDP ℳ=⟨S,A,p,r,γ⟩ℳ𝑆𝐴𝑝𝑟𝛾\mathcal{M}=\langle S,A,p,r,\gamma\ranglecaligraphic\_M = ⟨ italic\_S , italic\_A , italic\_p , italic\_r , italic\_γ ⟩, an option is a tuple ω=⟨Iω,πω,βω⟩𝜔subscript𝐼𝜔subscript𝜋𝜔subscript𝛽𝜔\omega=\langle I\_{\omega},\pi\_{\omega},\beta\_{\omega}\rangleitalic\_ω = ⟨ italic\_I start\_POSTSUBSCRIPT italic\_ω end\_POSTSUBSCRIPT , italic\_π start\_POSTSUBSCRIPT italic\_ω end\_POSTSUBSCRIPT , italic\_β start\_POSTSUBSCRIPT italic\_ω end\_POSTSUBSCRIPT ⟩ where Iω⊆Ssubscript𝐼𝜔𝑆I\_{\omega}\subseteq Sitalic\_I start\_POSTSUBSCRIPT italic\_ω end\_POSTSUBSCRIPT ⊆ italic\_S is the option’s initiation set, πω:S→Δ(A):subscript𝜋𝜔→𝑆Δ𝐴\pi\_{\omega}:S\to\Delta(A)italic\_π start\_POSTSUBSCRIPT italic\_ω end\_POSTSUBSCRIPT : italic\_S → roman\_Δ ( italic\_A ) is the option’s policy, and βω:S→[0,1]:subscript𝛽𝜔→𝑆01\beta\_{\omega}:S\to[0,1]italic\_β start\_POSTSUBSCRIPT italic\_ω end\_POSTSUBSCRIPT : italic\_S → [ 0 , 1 ] is the option’s termination condition. An option is available in state s∈S𝑠𝑆s\in Sitalic\_s ∈ italic\_S if s∈Iω𝑠subscript𝐼𝜔s\in I\_{\omega}italic\_s ∈ italic\_I start\_POSTSUBSCRIPT italic\_ω end\_POSTSUBSCRIPT. If the option is started, the actions are chosen according to πωsubscript𝜋𝜔\pi\_{\omega}italic\_π start\_POSTSUBSCRIPT italic\_ω end\_POSTSUBSCRIPT. The option terminates at a given state s∈S𝑠𝑆s\in Sitalic\_s ∈ italic\_S with probability βω(s)subscript𝛽𝜔𝑠\beta\_{\omega}(s)italic\_β start\_POSTSUBSCRIPT italic\_ω end\_POSTSUBSCRIPT ( italic\_s ). The action set of an MDP can be augmented with options, which can be either handcrafted or automatically discovered. An MDP extended with options is a Semi-Markov Decision Process (SMDP); the learning methods for MDPs still apply to SMDPs. To improve sample-efficiency, the experiences (s,a,r,s′)𝑠𝑎𝑟superscript𝑠′(s,a,r,s^{\prime})( italic\_s , italic\_a , italic\_r , italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) generated by an option’s policy can be used to update other options’ policies. This transfer learning method is called intra-option learning (?). ### Inductive Learning of Answer Set Programs In this section we describe answer set programming (ASP) and the ILASP system for learning ASP programs. #### Answer Set Programming (ASP) (?) is a declarative programming language for knowledge representation and reasoning. An ASP problem is expressed in a logical format and the models (called answer sets) of its representation provide the solutions to that problem. A literal is an atom 𝚊𝚊\mathtt{a}typewriter\_a or its negation 𝚗𝚘𝚝𝚊𝚗𝚘𝚝𝚊\mathtt{not~{}a}typewriter\_not typewriter\_a. Given an atom 𝚑𝚑\mathtt{h}typewriter\_h and a set of literals 𝚋𝟷,…,𝚋𝚗subscript𝚋1…subscript𝚋𝚗\mathtt{b\_{1},\ldots,b\_{n}}typewriter\_b start\_POSTSUBSCRIPT typewriter\_1 end\_POSTSUBSCRIPT , … , typewriter\_b start\_POSTSUBSCRIPT typewriter\_n end\_POSTSUBSCRIPT, a normal rule is of the form 𝚑:−⁡𝚋𝟷,…,𝚋𝚗𝚑:absentsubscript𝚋1…subscript𝚋𝚗\mathtt{h\operatorname{\mathtt{:-}}b\_{1},\ldots,b\_{n}}typewriter\_h start\_OPFUNCTION : - end\_OPFUNCTION typewriter\_b start\_POSTSUBSCRIPT typewriter\_1 end\_POSTSUBSCRIPT , … , typewriter\_b start\_POSTSUBSCRIPT typewriter\_n end\_POSTSUBSCRIPT, where 𝚑𝚑\mathtt{h}typewriter\_h is the head and 𝚋𝟷,…,𝚋𝚗subscript𝚋1…subscript𝚋𝚗\mathtt{b\_{1},\ldots,b\_{n}}typewriter\_b start\_POSTSUBSCRIPT typewriter\_1 end\_POSTSUBSCRIPT , … , typewriter\_b start\_POSTSUBSCRIPT typewriter\_n end\_POSTSUBSCRIPT is the body of the rule. Rules of the form :−⁡𝚋𝟷,…,𝚋𝚗:absentsubscript𝚋1…subscript𝚋𝚗\mathtt{\operatorname{\mathtt{:-}}b\_{1},\ldots,b\_{n}}start\_OPFUNCTION : - end\_OPFUNCTION typewriter\_b start\_POSTSUBSCRIPT typewriter\_1 end\_POSTSUBSCRIPT , … , typewriter\_b start\_POSTSUBSCRIPT typewriter\_n end\_POSTSUBSCRIPT are called constraints. In this paper, we assume an ASP program P𝑃Pitalic\_P to be a set of normal rules and constraints. Given a set of ground atoms (or interpretation) I𝐼Iitalic\_I, a ground normal rule is satisfied if the head is satisfied by I𝐼Iitalic\_I when the body literals are satisfied by I𝐼Iitalic\_I. A ground constraint is satisfied if the body is not satisfied by I𝐼Iitalic\_I. The reduct PIsuperscript𝑃𝐼P^{I}italic\_P start\_POSTSUPERSCRIPT italic\_I end\_POSTSUPERSCRIPT of a program P𝑃Pitalic\_P with respect to I𝐼Iitalic\_I is built by removing all rules including 𝚗𝚘𝚝𝚊𝚗𝚘𝚝𝚊\mathtt{not~{}a}typewriter\_not typewriter\_a such that 𝚊∈I𝚊𝐼\mathtt{a}\in Itypewriter\_a ∈ italic\_I from P𝑃Pitalic\_P. I𝐼Iitalic\_I is an answer set of P𝑃Pitalic\_P iff (1) I𝐼Iitalic\_I satisfies the rules of PIsuperscript𝑃𝐼P^{I}italic\_P start\_POSTSUPERSCRIPT italic\_I end\_POSTSUPERSCRIPT, and (2) no subset of I𝐼Iitalic\_I satisfies the rules of PIsuperscript𝑃𝐼P^{I}italic\_P start\_POSTSUPERSCRIPT italic\_I end\_POSTSUPERSCRIPT. #### ILASP (?) is a system for learning ASP programs from partial answer sets. A context-dependent partial interpretation (CDPI) (?) is a pair ⟨⟨einc,eexc⟩,ectx⟩superscript𝑒𝑖𝑛𝑐superscript𝑒𝑒𝑥𝑐 superscript𝑒𝑐𝑡𝑥\langle\langle e^{inc},e^{exc}\rangle,e^{ctx}\rangle⟨ ⟨ italic\_e start\_POSTSUPERSCRIPT italic\_i italic\_n italic\_c end\_POSTSUPERSCRIPT , italic\_e start\_POSTSUPERSCRIPT italic\_e italic\_x italic\_c end\_POSTSUPERSCRIPT ⟩ , italic\_e start\_POSTSUPERSCRIPT italic\_c italic\_t italic\_x end\_POSTSUPERSCRIPT ⟩, where ⟨einc,eexc⟩superscript𝑒𝑖𝑛𝑐superscript𝑒𝑒𝑥𝑐\langle e^{inc},e^{exc}\rangle⟨ italic\_e start\_POSTSUPERSCRIPT italic\_i italic\_n italic\_c end\_POSTSUPERSCRIPT , italic\_e start\_POSTSUPERSCRIPT italic\_e italic\_x italic\_c end\_POSTSUPERSCRIPT ⟩ is a partial interpretation and ectxsuperscript𝑒𝑐𝑡𝑥e^{ctx}italic\_e start\_POSTSUPERSCRIPT italic\_c italic\_t italic\_x end\_POSTSUPERSCRIPT is an ASP program, called context. A program P𝑃Pitalic\_P accepts e𝑒eitalic\_e iff there is an answer set A𝐴Aitalic\_A of P∪ectx𝑃superscript𝑒𝑐𝑡𝑥P\cup e^{ctx}italic\_P ∪ italic\_e start\_POSTSUPERSCRIPT italic\_c italic\_t italic\_x end\_POSTSUPERSCRIPT such that einc⊆Asuperscript𝑒𝑖𝑛𝑐𝐴e^{inc}\subseteq Aitalic\_e start\_POSTSUPERSCRIPT italic\_i italic\_n italic\_c end\_POSTSUPERSCRIPT ⊆ italic\_A and eexc∩A=∅superscript𝑒𝑒𝑥𝑐𝐴e^{exc}~{}\cap A=\emptysetitalic\_e start\_POSTSUPERSCRIPT italic\_e italic\_x italic\_c end\_POSTSUPERSCRIPT ∩ italic\_A = ∅. An ILASP task (?) is a tuple T=⟨B,SM,E⟩𝑇𝐵subscript𝑆𝑀𝐸T=\langle B,S\_{M},E\rangleitalic\_T = ⟨ italic\_B , italic\_S start\_POSTSUBSCRIPT italic\_M end\_POSTSUBSCRIPT , italic\_E ⟩ where B𝐵Bitalic\_B is the ASP background knowledge, SMsubscript𝑆𝑀S\_{M}italic\_S start\_POSTSUBSCRIPT italic\_M end\_POSTSUBSCRIPT is the set of rules allowed in the hypotheses, and E𝐸Eitalic\_E is a set of CDPIs, called examples. A hypothesis H⊆SM𝐻subscript𝑆𝑀H\subseteq S\_{M}italic\_H ⊆ italic\_S start\_POSTSUBSCRIPT italic\_M end\_POSTSUBSCRIPT is a solution of T𝑇Titalic\_T iff ∀e∈Efor-all𝑒𝐸\forall e\in E∀ italic\_e ∈ italic\_E, B∪H𝐵𝐻B\cup Hitalic\_B ∪ italic\_H accepts e𝑒eitalic\_e. 3 Methodology -------------- In this section we describe ISA, our method that interleaves the learning of subgoal automata with the learning of policies for achieving these subgoals. The tasks we consider are episodic MDPs ℳ=⟨S,A,p,r,γ,ST,SG⟩ℳ𝑆𝐴𝑝𝑟𝛾subscript𝑆𝑇subscript𝑆𝐺\mathcal{M}=\langle S,A,p,r,\gamma,S\_{T},S\_{G}\ranglecaligraphic\_M = ⟨ italic\_S , italic\_A , italic\_p , italic\_r , italic\_γ , italic\_S start\_POSTSUBSCRIPT italic\_T end\_POSTSUBSCRIPT , italic\_S start\_POSTSUBSCRIPT italic\_G end\_POSTSUBSCRIPT ⟩ where r(s,a,s′)=1𝑟𝑠𝑎superscript𝑠′1r(s,a,s^{\prime})=1italic\_r ( italic\_s , italic\_a , italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) = 1 if s′∈SGsuperscript𝑠′subscript𝑆𝐺s^{\prime}\in S\_{G}italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ∈ italic\_S start\_POSTSUBSCRIPT italic\_G end\_POSTSUBSCRIPT and 0 otherwise. The automaton transitions are defined by a logical formula over a set of observables 𝒪𝒪\mathcal{O}caligraphic\_O. A labeling function L:S→2𝒪:𝐿→𝑆superscript2𝒪L:S\to 2^{\mathcal{O}}italic\_L : italic\_S → 2 start\_POSTSUPERSCRIPT caligraphic\_O end\_POSTSUPERSCRIPT maps an MDP state into a subset of observables O⊆𝒪𝑂𝒪O\subseteq\mathcal{O}italic\_O ⊆ caligraphic\_O (or observations) perceived by the agent at that state. 01234567891011012345678D𝐷Ditalic\_D∗∗\ast∗∗∗\ast∗C𝐶Citalic\_C☕\Strichmaxerl[1.25]∗∗\ast∗o𝑜oitalic\_o🖂∗∗\ast∗A𝐴Aitalic\_A∗∗\ast∗∗∗\ast∗B𝐵Bitalic\_B (a) Execution trace T=⟨s4,6,←,0,s3,6,→,0,s4,6,↓,0,s4,5,↓,1,s4,4⟩\begin{aligned} T=\langle&s\_{4,6},\leftarrow,0,s\_{3,6},\rightarrow,0,\\ &s\_{4,6},\downarrow,0,s\_{4,5},\downarrow,1,s\_{4,4}\rangle\end{aligned}start\_ROW start\_CELL italic\_T = ⟨ end\_CELL start\_CELL italic\_s start\_POSTSUBSCRIPT 4 , 6 end\_POSTSUBSCRIPT , ← , 0 , italic\_s start\_POSTSUBSCRIPT 3 , 6 end\_POSTSUBSCRIPT , → , 0 , end\_CELL end\_ROW start\_ROW start\_CELL end\_CELL start\_CELL italic\_s start\_POSTSUBSCRIPT 4 , 6 end\_POSTSUBSCRIPT , ↓ , 0 , italic\_s start\_POSTSUBSCRIPT 4 , 5 end\_POSTSUBSCRIPT , ↓ , 1 , italic\_s start\_POSTSUBSCRIPT 4 , 4 end\_POSTSUBSCRIPT ⟩ end\_CELL end\_ROW Observation trace TL,𝒪=⟨{},{☕},{},{},{o}⟩subscript𝑇𝐿𝒪 ☕ 𝑜\begin{aligned} T\_{L,\mathcal{O}}=\left\langle\{\},\{\text{\Coffeecup}\},\{\},\{\},\{o\}\right\rangle\end{aligned}start\_ROW start\_CELL italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT = ⟨ { } , { ☕ } , { } , { } , { italic\_o } ⟩ end\_CELL end\_ROW Compressed observation trace T^L,𝒪=⟨{},{☕},{},{o}⟩subscript^𝑇𝐿𝒪 ☕ 𝑜\begin{aligned} \hat{T}\_{L,\mathcal{O}}=\left\langle\{\},\{\text{\Coffeecup}\},\{\},\{o\}\right\rangle\end{aligned}start\_ROW start\_CELL over^ start\_ARG italic\_T end\_ARG start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT = ⟨ { } , { ☕ } , { } , { italic\_o } ⟩ end\_CELL end\_ROW (b) u0subscript𝑢0u\_{0}italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPTstartuAsubscript𝑢𝐴u\_{A}italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPTu1subscript𝑢1u\_{1}italic\_u start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPTuRsubscript𝑢𝑅u\_{R}italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPTotherwiseotherwise☕∧¬o☕𝑜\texttt{\Coffeecup}\land\neg o☕ ∧ ¬ italic\_o∗∗\ast∗∗∗\ast∗o𝑜oitalic\_o☕∧o☕𝑜\texttt{\Coffeecup}\land o☕ ∧ italic\_o (c) Figure 1: The OfficeWorld environment (?). Figure LABEL:sub@fig:officeworld\_grid is an example grid, LABEL:sub@fig:officeworld\_coffee\_traces shows a positive execution trace in the example grid for the Coffee task and its derived observation traces, and LABEL:sub@fig:officeworld\_coffee\_rm shows the Coffee automaton. We use the OfficeWorld environment (?) to explain our method. It consists of a grid (see Figure [1a](#S3.F1.sf1 "1a ‣ Figure 1 ‣ 3 Methodology ‣ Induction of Subgoal Automata for Reinforcement Learning")) where an agent (\Strichmaxerl[1.25]\Strichmaxerldelimited-[]1.25\Strichmaxerl[1.25][ 1.25 ]) can move in the four cardinal directions, and the set of observables is 𝒪={☕,🖂,o,A,B,C,D,∗}𝒪☕🖂𝑜𝐴𝐵𝐶𝐷∗\mathcal{O}=\{\texttt{\Coffeecup},\texttt{\Letter},o,A,B,C,D,\ast\}caligraphic\_O = { ☕ , 🖂 , italic\_o , italic\_A , italic\_B , italic\_C , italic\_D , ∗ }. The agent picks up coffee and the mail at locations ☕ and 🖂 respectively, and delivers them to the office at location o𝑜oitalic\_o. The decorations ∗∗\ast∗ are broken if the agent steps on them. There are also four locations labeled A𝐴Aitalic\_A, B𝐵Bitalic\_B, C𝐶Citalic\_C and D𝐷Ditalic\_D. The observables A𝐴Aitalic\_A, B𝐵Bitalic\_B, C𝐶Citalic\_C and D𝐷Ditalic\_D, and decorations ∗∗\ast∗ do not share locations with other elements. The agent observes these labels when it steps on their locations. Three tasks with different goals are defined on this environment: Coffee (deliver coffee to the office), CoffeeMail (deliver coffee and mail to the office), and VisitABCD (visit A𝐴Aitalic\_A, B𝐵Bitalic\_B, C𝐶Citalic\_C and D𝐷Ditalic\_D in order). The tasks terminate when the goal is achieved or a ∗∗\ast∗ is broken (this is a dead-end state). ### Traces An execution trace T𝑇Titalic\_T is a finite state-action-reward sequence T=⟨s0,a0,r1,s1,a1,…,an−1,rn,sn⟩𝑇subscript𝑠0subscript𝑎0subscript𝑟1subscript𝑠1subscript𝑎1…subscript𝑎𝑛1subscript𝑟𝑛subscript𝑠𝑛T=\langle s\_{0},a\_{0},r\_{1},s\_{1},a\_{1},\ldots,a\_{n-1},r\_{n},s\_{n}\rangleitalic\_T = ⟨ italic\_s start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , italic\_r start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_s start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_a start\_POSTSUBSCRIPT italic\_n - 1 end\_POSTSUBSCRIPT , italic\_r start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT , italic\_s start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ⟩ induced by a (changing) policy during an episode. An execution trace is positive if sn∈SGsubscript𝑠𝑛subscript𝑆𝐺s\_{n}\in S\_{G}italic\_s start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ∈ italic\_S start\_POSTSUBSCRIPT italic\_G end\_POSTSUBSCRIPT, negative if sn∈ST∖SGsubscript𝑠𝑛subscript𝑆𝑇subscript𝑆𝐺s\_{n}\in S\_{T}\setminus S\_{G}italic\_s start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ∈ italic\_S start\_POSTSUBSCRIPT italic\_T end\_POSTSUBSCRIPT ∖ italic\_S start\_POSTSUBSCRIPT italic\_G end\_POSTSUBSCRIPT, and incomplete if sn∉STsubscript𝑠𝑛subscript𝑆𝑇s\_{n}\notin S\_{T}italic\_s start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ∉ italic\_S start\_POSTSUBSCRIPT italic\_T end\_POSTSUBSCRIPT, denoted by T+superscript𝑇T^{+}italic\_T start\_POSTSUPERSCRIPT + end\_POSTSUPERSCRIPT, T−superscript𝑇T^{-}italic\_T start\_POSTSUPERSCRIPT - end\_POSTSUPERSCRIPT and TIsuperscript𝑇𝐼T^{I}italic\_T start\_POSTSUPERSCRIPT italic\_I end\_POSTSUPERSCRIPT respectively. An observation trace TL,𝒪subscript𝑇𝐿𝒪T\_{L,\mathcal{O}}italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT is a sequence of observation sets Oi⊆𝒪,0≤i≤nformulae-sequencesubscript𝑂𝑖𝒪0𝑖𝑛O\_{i}\subseteq\mathcal{O},0\leq i\leq nitalic\_O start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ⊆ caligraphic\_O , 0 ≤ italic\_i ≤ italic\_n, obtained by applying a labeling function L𝐿Litalic\_L to each state sisubscript𝑠𝑖s\_{i}italic\_s start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT in an execution trace T𝑇Titalic\_T. A compressed observation trace T^L,𝒪subscript^𝑇𝐿𝒪\hat{T}\_{L,\mathcal{O}}over^ start\_ARG italic\_T end\_ARG start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT is the result of removing contiguous duplicated observation sets from TL,𝒪subscript𝑇𝐿𝒪T\_{L,\mathcal{O}}italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT. Figure [1b](#S3.F1.sf2 "1b ‣ Figure 1 ‣ 3 Methodology ‣ Induction of Subgoal Automata for Reinforcement Learning") shows an example of a positive execution trace for the Coffee task together with the derived observation trace and the resulting compressed trace. A set of execution traces is a tuple 𝒯=⟨𝒯+,𝒯−,𝒯I⟩𝒯superscript𝒯superscript𝒯superscript𝒯𝐼\mathcal{T}=\langle\mathcal{T}^{+},\mathcal{T}^{-},\mathcal{T}^{I}\ranglecaligraphic\_T = ⟨ caligraphic\_T start\_POSTSUPERSCRIPT + end\_POSTSUPERSCRIPT , caligraphic\_T start\_POSTSUPERSCRIPT - end\_POSTSUPERSCRIPT , caligraphic\_T start\_POSTSUPERSCRIPT italic\_I end\_POSTSUPERSCRIPT ⟩, where 𝒯+superscript𝒯\mathcal{T}^{+}caligraphic\_T start\_POSTSUPERSCRIPT + end\_POSTSUPERSCRIPT, 𝒯−superscript𝒯\mathcal{T}^{-}caligraphic\_T start\_POSTSUPERSCRIPT - end\_POSTSUPERSCRIPT and 𝒯Isuperscript𝒯𝐼\mathcal{T}^{I}caligraphic\_T start\_POSTSUPERSCRIPT italic\_I end\_POSTSUPERSCRIPT are sets of positive, negative and incomplete traces, respectively. The sets of observation and compressed observation traces are denoted 𝒯L,𝒪subscript𝒯𝐿𝒪\mathcal{T}\_{L,\mathcal{O}}caligraphic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT and 𝒯^L,𝒪subscript^𝒯𝐿𝒪\hat{\mathcal{T}}\_{L,\mathcal{O}}over^ start\_ARG caligraphic\_T end\_ARG start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT. ### Subgoal Automata A subgoal automaton is a tuple 𝒜=⟨U,𝒪,δ,u0,uA,uR⟩𝒜𝑈𝒪𝛿subscript𝑢0subscript𝑢𝐴subscript𝑢𝑅\mathcal{A}=\langle U,\mathcal{O},\delta,u\_{0},u\_{A},u\_{R}\ranglecaligraphic\_A = ⟨ italic\_U , caligraphic\_O , italic\_δ , italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ⟩ where U𝑈Uitalic\_U is a finite set of states, 𝒪𝒪\mathcal{O}caligraphic\_O is a set of observables (or alphabet), δ:U×2𝒪→U:𝛿→𝑈superscript2𝒪𝑈\delta:U\times 2^{\mathcal{O}}\to Uitalic\_δ : italic\_U × 2 start\_POSTSUPERSCRIPT caligraphic\_O end\_POSTSUPERSCRIPT → italic\_U is a deterministic transition function that takes as arguments a state and a subset of observables and returns a state, u0∈Usubscript𝑢0𝑈u\_{0}\in Uitalic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT ∈ italic\_U is a start state, uA∈Usubscript𝑢𝐴𝑈u\_{A}\in Uitalic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT ∈ italic\_U is the unique accepting state, and uR∈Usubscript𝑢𝑅𝑈u\_{R}\in Uitalic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ∈ italic\_U is the unique rejecting state. A subgoal automaton 𝒜𝒜\mathcal{A}caligraphic\_A accepts an observation trace TL,𝒪=⟨O0,…,On⟩subscript𝑇𝐿𝒪subscript𝑂0…subscript𝑂𝑛T\_{L,\mathcal{O}}=\langle O\_{0},\ldots,O\_{n}\rangleitalic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT = ⟨ italic\_O start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , … , italic\_O start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ⟩ if there exists a sequence of automaton states u0,…,un+1subscript𝑢0…subscript𝑢𝑛1u\_{0},\ldots,u\_{n+1}italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , … , italic\_u start\_POSTSUBSCRIPT italic\_n + 1 end\_POSTSUBSCRIPT in U𝑈Uitalic\_U such that (1) δ(ui,Oi)=ui+1𝛿subscript𝑢𝑖subscript𝑂𝑖subscript𝑢𝑖1\delta(u\_{i},O\_{i})=u\_{i+1}italic\_δ ( italic\_u start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_O start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) = italic\_u start\_POSTSUBSCRIPT italic\_i + 1 end\_POSTSUBSCRIPT for i=0,…,n𝑖0…𝑛i=0,\ldots,nitalic\_i = 0 , … , italic\_n, and (2) un+1=uAsubscript𝑢𝑛1subscript𝑢𝐴u\_{n+1}=u\_{A}italic\_u start\_POSTSUBSCRIPT italic\_n + 1 end\_POSTSUBSCRIPT = italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT. Analogously, 𝒜𝒜\mathcal{A}caligraphic\_A rejects TL,𝒪subscript𝑇𝐿𝒪T\_{L,\mathcal{O}}italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT if un+1=uRsubscript𝑢𝑛1subscript𝑢𝑅u\_{n+1}=u\_{R}italic\_u start\_POSTSUBSCRIPT italic\_n + 1 end\_POSTSUBSCRIPT = italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT. When a subgoal automaton is used together with an MDP, the actual states are (s,u)𝑠𝑢(s,u)( italic\_s , italic\_u ) pairs where s∈S𝑠𝑆s\in Sitalic\_s ∈ italic\_S and u∈U𝑢𝑈u\in Uitalic\_u ∈ italic\_U. Therefore, actions are selected according to a policy π:S×U→Δ(A):𝜋→𝑆𝑈Δ𝐴\pi:S\times U\to\Delta(A)italic\_π : italic\_S × italic\_U → roman\_Δ ( italic\_A ) where π(a\|s,u)𝜋conditional𝑎𝑠𝑢\pi(a\|s,u)italic\_π ( italic\_a \| italic\_s , italic\_u ) is the probability for taking action a∈A𝑎𝐴a\in Aitalic\_a ∈ italic\_A at the MDP state s∈S𝑠𝑆s\in Sitalic\_s ∈ italic\_S and the automaton state u∈U𝑢𝑈u\in Uitalic\_u ∈ italic\_U. At each step, the agent transitions to (s′,u′)superscript𝑠′superscript𝑢′(s^{\prime},u^{\prime})( italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , italic\_u start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) where s′superscript𝑠′s^{\prime}italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT is the result of applying an action a∈A𝑎𝐴a\in Aitalic\_a ∈ italic\_A in s𝑠sitalic\_s, while u′=δ(u,L(s′))superscript𝑢′𝛿𝑢𝐿superscript𝑠′u^{\prime}=\delta(u,L(s^{\prime}))italic\_u start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = italic\_δ ( italic\_u , italic\_L ( italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) ). Then, the agent receives reward 1 if u′=uAsuperscript𝑢′subscript𝑢𝐴u^{\prime}=u\_{A}italic\_u start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT and 0 otherwise. Figure [1c](#S3.F1.sf3 "1c ‣ Figure 1 ‣ 3 Methodology ‣ Induction of Subgoal Automata for Reinforcement Learning") shows the automaton for the Coffee task of the OfficeWorld domain. Each transition is labeled with a logical condition φ∈3𝒪𝜑superscript3𝒪\varphi\in 3^{\mathcal{O}}italic\_φ ∈ 3 start\_POSTSUPERSCRIPT caligraphic\_O end\_POSTSUPERSCRIPT that expresses the subgoal to be achieved222Note that φ∈3𝒪𝜑superscript3𝒪\varphi\in 3^{\mathcal{O}}italic\_φ ∈ 3 start\_POSTSUPERSCRIPT caligraphic\_O end\_POSTSUPERSCRIPT because each observable can appear as positive or negative, or not appear in the condition.. The sequence of visited automaton states for the trace in Figure [1b](#S3.F1.sf2 "1b ‣ Figure 1 ‣ 3 Methodology ‣ Induction of Subgoal Automata for Reinforcement Learning") would be ⟨u0,u1,u1,u1,uA⟩subscript𝑢0subscript𝑢1subscript𝑢1subscript𝑢1subscript𝑢𝐴\langle u\_{0},u\_{1},u\_{1},u\_{1},u\_{A}\rangle⟨ italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT ⟩. #### Relationship with options. Each state u∈U𝑢𝑈u\in Uitalic\_u ∈ italic\_U in a subgoal automaton encapsulates an option ωu=⟨Iu,πu,βu⟩subscript𝜔𝑢subscript𝐼𝑢subscript𝜋𝑢subscript𝛽𝑢\omega\_{u}=\langle I\_{u},\pi\_{u},\beta\_{u}\rangleitalic\_ω start\_POSTSUBSCRIPT italic\_u end\_POSTSUBSCRIPT = ⟨ italic\_I start\_POSTSUBSCRIPT italic\_u end\_POSTSUBSCRIPT , italic\_π start\_POSTSUBSCRIPT italic\_u end\_POSTSUBSCRIPT , italic\_β start\_POSTSUBSCRIPT italic\_u end\_POSTSUBSCRIPT ⟩ whose policy πusubscript𝜋𝑢\pi\_{u}italic\_π start\_POSTSUBSCRIPT italic\_u end\_POSTSUBSCRIPT attempts to satisfy an outgoing transition’s condition. Formally, the option termination condition βusubscript𝛽𝑢\beta\_{u}italic\_β start\_POSTSUBSCRIPT italic\_u end\_POSTSUBSCRIPT is: \| \| \| \| \| --- \| --- \| --- \| \| \| βu(s)={1if∃u′≠u,φ∈3𝒪∣L(s′)⊧φ,δ(u,φ)=u′0otherwise.\displaystyle\beta\_{u}(s)=\left\{\begin{matrix}1~{}\text{if}~{}\exists u^{\prime}\neq u,\varphi\in 3^{\mathcal{O}}\mathord{\mid}L(s^{\prime})\models\varphi,\delta(u,\varphi)=u^{\prime}\\ 0~{}\text{otherwise}\end{matrix}\right..italic\_β start\_POSTSUBSCRIPT italic\_u end\_POSTSUBSCRIPT ( italic\_s ) = { start\_ARG start\_ROW start\_CELL 1 if ∃ italic\_u start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ≠ italic\_u , italic\_φ ∈ 3 start\_POSTSUPERSCRIPT caligraphic\_O end\_POSTSUPERSCRIPT ∣ italic\_L ( italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) ⊧ italic\_φ , italic\_δ ( italic\_u , italic\_φ ) = italic\_u start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_CELL end\_ROW start\_ROW start\_CELL 0 otherwise end\_CELL end\_ROW end\_ARG . \| \| That is, the option at automaton state u∈U𝑢𝑈u\in Uitalic\_u ∈ italic\_U finishes at MDP state s∈S𝑠𝑆s\in Sitalic\_s ∈ italic\_S if there is a transition to a different automaton state u′∈Usuperscript𝑢′𝑈u^{\prime}\in Uitalic\_u start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ∈ italic\_U such that the transition’s condition φ𝜑\varphiitalic\_φ is satisfied by the next observations L(s′)𝐿superscript𝑠′L(s^{\prime})italic\_L ( italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ). Note that at most one transition can be made true since the automaton is deterministic. The initiation set Iusubscript𝐼𝑢I\_{u}italic\_I start\_POSTSUBSCRIPT italic\_u end\_POSTSUBSCRIPT is formed by all those states that satisfy the incoming conditions: \| \| \| \| \| --- \| --- \| --- \| \| \| Iu={s∈S∣δ(u′,φ)=u,L(s)⊧φ,u≠u′,u≠u0}.subscript𝐼𝑢conditional-set𝑠𝑆formulae-sequence𝛿superscript𝑢′𝜑𝑢formulae-sequencemodels𝐿𝑠𝜑formulae-sequence𝑢superscript𝑢′𝑢subscript𝑢0\displaystyle I\_{u}=\left\{s\in S\mid\delta\left(u^{\prime},\varphi\right)=u,L\left(s\right)\models\varphi,u\neq u^{\prime},u\neq u\_{0}\right\}.italic\_I start\_POSTSUBSCRIPT italic\_u end\_POSTSUBSCRIPT = { italic\_s ∈ italic\_S ∣ italic\_δ ( italic\_u start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , italic\_φ ) = italic\_u , italic\_L ( italic\_s ) ⊧ italic\_φ , italic\_u ≠ italic\_u start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , italic\_u ≠ italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT } . \| \| In the particular case of the initial automaton state u0∈Usubscript𝑢0𝑈u\_{0}\in Uitalic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT ∈ italic\_U, its initiation set Iu0=Ssubscript𝐼subscript𝑢0𝑆I\_{{u\_{0}}}=Sitalic\_I start\_POSTSUBSCRIPT italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT = italic\_S is the whole state space since there is no restriction imposed by any previous automaton state. Note that we do not add options to the set of primitive actions, which would make the decision process more complex since more alternatives are available. Instead, subgoal automata keep the action set unchanged, and which option to apply is determined by the current automaton state. ### Learning Subgoal Automata from Traces This section describes our approach for learning an automaton. We formalize the automaton learning task as a tuple T𝒜=⟨U,𝒪,𝒯L,𝒪⟩subscript𝑇𝒜𝑈𝒪subscript𝒯𝐿𝒪T\_{\mathcal{A}}=\langle U,\mathcal{O},\mathcal{T}\_{L,\mathcal{O}}\rangleitalic\_T start\_POSTSUBSCRIPT caligraphic\_A end\_POSTSUBSCRIPT = ⟨ italic\_U , caligraphic\_O , caligraphic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT ⟩, where U⊇{u0,uA,uR}subscript𝑢0subscript𝑢𝐴subscript𝑢𝑅𝑈U\supseteq\{u\_{0},u\_{A},u\_{R}\}italic\_U ⊇ { italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT } is a set of states, 𝒪𝒪\mathcal{O}caligraphic\_O is a set of observables, and 𝒯L,𝒪=⟨𝒯L,𝒪+,𝒯L,𝒪−,𝒯L,𝒪I⟩subscript𝒯𝐿𝒪subscriptsuperscript𝒯𝐿𝒪subscriptsuperscript𝒯𝐿𝒪subscriptsuperscript𝒯𝐼𝐿𝒪\mathcal{T}\_{L,\mathcal{O}}=\langle\mathcal{T}^{+}\_{L,\mathcal{O}},\mathcal{T}^{-}\_{L,\mathcal{O}},\mathcal{T}^{I}\_{L,\mathcal{O}}\ranglecaligraphic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT = ⟨ caligraphic\_T start\_POSTSUPERSCRIPT + end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT , caligraphic\_T start\_POSTSUPERSCRIPT - end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT , caligraphic\_T start\_POSTSUPERSCRIPT italic\_I end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT ⟩ is a set of (possibly compressed) observation traces (abbreviated as traces below). An automaton 𝒜𝒜\mathcal{A}caligraphic\_A is a solution of T𝒜subscript𝑇𝒜T\_{\mathcal{A}}italic\_T start\_POSTSUBSCRIPT caligraphic\_A end\_POSTSUBSCRIPT if and only if accepts all positive traces 𝒯L,𝒪+superscriptsubscript𝒯𝐿𝒪\mathcal{T}\_{L,\mathcal{O}}^{+}caligraphic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT + end\_POSTSUPERSCRIPT, rejects all negative traces 𝒯L,𝒪−superscriptsubscript𝒯𝐿𝒪\mathcal{T}\_{L,\mathcal{O}}^{-}caligraphic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT - end\_POSTSUPERSCRIPT, and neither accepts nor rejects incomplete traces 𝒯L,𝒪Isuperscriptsubscript𝒯𝐿𝒪𝐼\mathcal{T}\_{L,\mathcal{O}}^{I}caligraphic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_I end\_POSTSUPERSCRIPT. The automaton learning task T𝒜subscript𝑇𝒜T\_{\mathcal{A}}italic\_T start\_POSTSUBSCRIPT caligraphic\_A end\_POSTSUBSCRIPT is mapped into an ILASP task M(T𝒜)=⟨B,SM,E⟩𝑀subscript𝑇𝒜𝐵subscript𝑆𝑀𝐸M(T\_{\mathcal{A}})=\langle B,S\_{M},E\rangleitalic\_M ( italic\_T start\_POSTSUBSCRIPT caligraphic\_A end\_POSTSUBSCRIPT ) = ⟨ italic\_B , italic\_S start\_POSTSUBSCRIPT italic\_M end\_POSTSUBSCRIPT , italic\_E ⟩. Then, the ILASP system is used to learn the smallest set of transitions (i.e., a minimal hypothesis) that covers the example traces. We define the components of the ILASP task M(T𝒜)𝑀subscript𝑇𝒜M(T\_{\mathcal{A}})italic\_M ( italic\_T start\_POSTSUBSCRIPT caligraphic\_A end\_POSTSUBSCRIPT ) below. An ILASP task specifies the maximum size of a rule body. To allow for an arbitrary number of literals in the bodies, we learn the negation δ¯¯𝛿\bar{\delta}over¯ start\_ARG italic\_δ end\_ARG of the actual transitions δ𝛿\deltaitalic\_δ. Thus, we do not limit the number of conjuncts, but the number of disjuncts (i.e., the number of edges between two states). We denote the maximum number of disjuncts by max⁡\|δ(x,y)\|𝛿𝑥𝑦\max\|\delta(x,y)\|roman\_max \| italic\_δ ( italic\_x , italic\_y ) \|. #### Hypothesis space. The hypothesis space SMsubscript𝑆𝑀S\_{M}italic\_S start\_POSTSUBSCRIPT italic\_M end\_POSTSUBSCRIPT is formed by two kinds of rules: 1. 1. Facts of the form 𝚎𝚍(𝚇,𝚈,𝙴).𝚎𝚍𝚇𝚈𝙴\mathtt{ed(X,Y,E).}typewriter\_ed ( typewriter\_X , typewriter\_Y , typewriter\_E ) . indicating there is a transition from state 𝚇∈U∖{uA,uR}𝚇𝑈subscript𝑢𝐴subscript𝑢𝑅\mathtt{X}\in U\setminus\{u\_{A},u\_{R}\}typewriter\_X ∈ italic\_U ∖ { italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT } to state 𝚈∈U∖{𝚇}𝚈𝑈𝚇\mathtt{Y}\in U\setminus\{\mathtt{X}\}typewriter\_Y ∈ italic\_U ∖ { typewriter\_X } using edge 𝙴∈[1,max⁡\|δ(x,y)\|]𝙴1𝛿𝑥𝑦\mathtt{E}\in[1,\max\|\delta(x,y)\|]typewriter\_E ∈ [ 1 , roman\_max \| italic\_δ ( italic\_x , italic\_y ) \| ]. 2. 2. Normal rules whose head is of the form δ¯(𝚇,𝚈,𝙴,𝚃)¯𝛿𝚇𝚈𝙴𝚃\mathtt{\bar{\delta}(X,Y,E,T)}over¯ start\_ARG italic\_δ end\_ARG ( typewriter\_X , typewriter\_Y , typewriter\_E , typewriter\_T ) stating that the conditions of the transition from state 𝚇∈U∖{uA,uR}𝚇𝑈subscript𝑢𝐴subscript𝑢𝑅\mathtt{X}\in U\setminus\{u\_{A},u\_{R}\}typewriter\_X ∈ italic\_U ∖ { italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT } to state 𝚈∈U∖{𝚇}𝚈𝑈𝚇\mathtt{Y}\in U\setminus\{\mathtt{X}\}typewriter\_Y ∈ italic\_U ∖ { typewriter\_X } in edge 𝙴∈[1,max⁡\|δ(x,y)\|]𝙴1𝛿𝑥𝑦\mathtt{E}\in[1,\max\|\delta(x,y)\|]typewriter\_E ∈ [ 1 , roman\_max \| italic\_δ ( italic\_x , italic\_y ) \| ] do not hold at time 𝚃𝚃\mathtt{T}typewriter\_T. These conditions are specified in the body, which is a conjunction of 𝚘𝚋𝚜(𝙾,𝚃)𝚘𝚋𝚜𝙾𝚃\mathtt{obs(O,T)}typewriter\_obs ( typewriter\_O , typewriter\_T ) literals indicating that observable 𝙾∈𝒪𝙾𝒪\mathtt{O}\in\mathcal{O}typewriter\_O ∈ caligraphic\_O is seen at time 𝚃𝚃\mathtt{T}typewriter\_T. The atom 𝚜𝚝𝚎𝚙(𝚃)𝚜𝚝𝚎𝚙𝚃\mathtt{step(T)}typewriter\_step ( typewriter\_T ) expresses that 𝚃𝚃\mathtt{T}typewriter\_T is a timestep. The body must contain at least one 𝚘𝚋𝚜𝚘𝚋𝚜\mathtt{obs}typewriter\_obs literal. Note that the hypothesis space does not include (i) loop transitions, and (ii) transitions from uAsubscript𝑢𝐴u\_{A}italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT and uRsubscript𝑢𝑅u\_{R}italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT. Later, we define (i) in the absence of external transitions. Given a subgoal automaton 𝒜𝒜\mathcal{A}caligraphic\_A, we denote the set of ASP rules that describe it by M(𝒜)𝑀𝒜M(\mathcal{A})italic\_M ( caligraphic\_A ). The rules below correspond to the (u0,u1)subscript𝑢0subscript𝑢1(u\_{0},u\_{1})( italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) and (u0,uA)subscript𝑢0subscript𝑢𝐴(u\_{0},u\_{A})( italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT ) transitions in Figure [1c](#S3.F1.sf3 "1c ‣ Figure 1 ‣ 3 Methodology ‣ Induction of Subgoal Automata for Reinforcement Learning"): \| \| \| --- \| \| 𝚎𝚍(𝚞𝟶,𝚞𝟷,𝟷).𝚎𝚍(𝚞𝟶,𝚞𝙰,𝟷).formulae-sequence𝚎𝚍subscript𝚞0subscript𝚞11𝚎𝚍subscript𝚞0subscript𝚞𝙰1\mathtt{ed(u\_{0},u\_{1},1).~{}ed(u\_{0},u\_{A},1).}typewriter\_ed ( typewriter\_u start\_POSTSUBSCRIPT typewriter\_0 end\_POSTSUBSCRIPT , typewriter\_u start\_POSTSUBSCRIPT typewriter\_1 end\_POSTSUBSCRIPT , typewriter\_1 ) . typewriter\_ed ( typewriter\_u start\_POSTSUBSCRIPT typewriter\_0 end\_POSTSUBSCRIPT , typewriter\_u start\_POSTSUBSCRIPT typewriter\_A end\_POSTSUBSCRIPT , typewriter\_1 ) . \| \| δ¯(𝚞𝟶,𝚞𝟷,𝟷,𝚃):−⁡𝚗𝚘𝚝𝚘𝚋𝚜(☕,𝚃),𝚜𝚝𝚎𝚙(𝚃).¯𝛿subscript𝚞0subscript𝚞11𝚃:absent𝚗𝚘𝚝𝚘𝚋𝚜☕𝚃𝚜𝚝𝚎𝚙𝚃\mathtt{\bar{\delta}(u\_{0},u\_{1},1,T)\operatorname{\mathtt{:-}}not~{}obs(\text{\Coffeecup},T),step(T).}over¯ start\_ARG italic\_δ end\_ARG ( typewriter\_u start\_POSTSUBSCRIPT typewriter\_0 end\_POSTSUBSCRIPT , typewriter\_u start\_POSTSUBSCRIPT typewriter\_1 end\_POSTSUBSCRIPT , typewriter\_1 , typewriter\_T ) start\_OPFUNCTION : - end\_OPFUNCTION typewriter\_not typewriter\_obs ( ☕ , typewriter\_T ) , typewriter\_step ( typewriter\_T ) . \| \| δ¯(𝚞𝟶,𝚞𝟷,𝟷,𝚃):−⁡𝚘𝚋𝚜(o,𝚃),𝚜𝚝𝚎𝚙(𝚃).¯𝛿subscript𝚞0subscript𝚞11𝚃:absent𝚘𝚋𝚜𝑜𝚃𝚜𝚝𝚎𝚙𝚃\mathtt{\bar{\delta}(u\_{0},u\_{1},1,T)\operatorname{\mathtt{:-}}obs(}o\mathtt{,T),step(T).}over¯ start\_ARG italic\_δ end\_ARG ( typewriter\_u start\_POSTSUBSCRIPT typewriter\_0 end\_POSTSUBSCRIPT , typewriter\_u start\_POSTSUBSCRIPT typewriter\_1 end\_POSTSUBSCRIPT , typewriter\_1 , typewriter\_T ) start\_OPFUNCTION : - end\_OPFUNCTION typewriter\_obs ( italic\_o , typewriter\_T ) , typewriter\_step ( typewriter\_T ) . \| \| δ¯(𝚞𝟶,𝚞𝙰,𝟷,𝚃):−⁡𝚗𝚘𝚝𝚘𝚋𝚜(o,𝚃),𝚜𝚝𝚎𝚙(𝚃).¯𝛿subscript𝚞0subscript𝚞𝙰1𝚃:absent𝚗𝚘𝚝𝚘𝚋𝚜𝑜𝚃𝚜𝚝𝚎𝚙𝚃\mathtt{\bar{\delta}(u\_{0},u\_{A},1,T)\operatorname{\mathtt{:-}}not~{}obs(}o\mathtt{,T),step(T).}over¯ start\_ARG italic\_δ end\_ARG ( typewriter\_u start\_POSTSUBSCRIPT typewriter\_0 end\_POSTSUBSCRIPT , typewriter\_u start\_POSTSUBSCRIPT typewriter\_A end\_POSTSUBSCRIPT , typewriter\_1 , typewriter\_T ) start\_OPFUNCTION : - end\_OPFUNCTION typewriter\_not typewriter\_obs ( italic\_o , typewriter\_T ) , typewriter\_step ( typewriter\_T ) . \| \| δ¯(𝚞𝟶,𝚞𝙰,𝟷,𝚃):−⁡𝚗𝚘𝚝𝚘𝚋𝚜(☕,𝚃),𝚜𝚝𝚎𝚙(𝚃).¯𝛿subscript𝚞0subscript𝚞𝙰1𝚃:absent𝚗𝚘𝚝𝚘𝚋𝚜☕𝚃𝚜𝚝𝚎𝚙𝚃\mathtt{\bar{\delta}(u\_{0},u\_{A},1,T)\operatorname{\mathtt{:-}}not~{}obs(\text{\Coffeecup},T),step(T).}over¯ start\_ARG italic\_δ end\_ARG ( typewriter\_u start\_POSTSUBSCRIPT typewriter\_0 end\_POSTSUBSCRIPT , typewriter\_u start\_POSTSUBSCRIPT typewriter\_A end\_POSTSUBSCRIPT , typewriter\_1 , typewriter\_T ) start\_OPFUNCTION : - end\_OPFUNCTION typewriter\_not typewriter\_obs ( ☕ , typewriter\_T ) , typewriter\_step ( typewriter\_T ) . \| #### Examples. Given 𝒯L,𝒪=⟨𝒯L,𝒪+,𝒯L,𝒪−,𝒯L,𝒪I⟩subscript𝒯𝐿𝒪subscriptsuperscript𝒯𝐿𝒪subscriptsuperscript𝒯𝐿𝒪subscriptsuperscript𝒯𝐼𝐿𝒪\mathcal{T}\_{L,\mathcal{O}}=\langle\mathcal{T}^{+}\_{L,\mathcal{O}},\mathcal{T}^{-}\_{L,\mathcal{O}},\mathcal{T}^{I}\_{L,\mathcal{O}}\ranglecaligraphic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT = ⟨ caligraphic\_T start\_POSTSUPERSCRIPT + end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT , caligraphic\_T start\_POSTSUPERSCRIPT - end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT , caligraphic\_T start\_POSTSUPERSCRIPT italic\_I end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT ⟩, the example set is defined as E={⟨e\,CTL,𝒪⟩∣∗∈{+,−,I},E=\{\langle e^{\},C\_{T\_{L,\mathcal{O}}}\rangle\mid\ast\in\{+,-,I\},italic\_E = { ⟨ italic\_e start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT , italic\_C start\_POSTSUBSCRIPT italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ⟩ ∣ ∗ ∈ { + , - , italic\_I } , TL,𝒪∈𝒯L,𝒪\}T\_{L,\mathcal{O}}\in\mathcal{T}^{\}\_{L,\mathcal{O}}\}italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT ∈ caligraphic\_T start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT }, where e+=⟨{𝚊𝚌𝚌𝚎𝚙𝚝},{𝚛𝚎𝚓𝚎𝚌𝚝}⟩superscript𝑒𝚊𝚌𝚌𝚎𝚙𝚝𝚛𝚎𝚓𝚎𝚌𝚝e^{+}=\langle\{\mathtt{accept}\},\{\mathtt{reject}\}\rangleitalic\_e start\_POSTSUPERSCRIPT + end\_POSTSUPERSCRIPT = ⟨ { typewriter\_accept } , { typewriter\_reject } ⟩, e−=⟨{𝚛𝚎𝚓𝚎𝚌𝚝},{𝚊𝚌𝚌𝚎𝚙𝚝}⟩superscript𝑒𝚛𝚎𝚓𝚎𝚌𝚝𝚊𝚌𝚌𝚎𝚙𝚝e^{-}=\langle\{\mathtt{reject}\},\{\mathtt{accept}\}\rangleitalic\_e start\_POSTSUPERSCRIPT - end\_POSTSUPERSCRIPT = ⟨ { typewriter\_reject } , { typewriter\_accept } ⟩ and eI=⟨{},{𝚊𝚌𝚌𝚎𝚙𝚝,𝚛𝚎𝚓𝚎𝚌𝚝}⟩superscript𝑒𝐼 𝚊𝚌𝚌𝚎𝚙𝚝𝚛𝚎𝚓𝚎𝚌𝚝e^{I}=\langle\{\},\{\mathtt{accept,reject}\}\rangleitalic\_e start\_POSTSUPERSCRIPT italic\_I end\_POSTSUPERSCRIPT = ⟨ { } , { typewriter\_accept , typewriter\_reject } ⟩ are the partial interpretations for positive, negative and incomplete examples. The 𝚊𝚌𝚌𝚎𝚙𝚝𝚊𝚌𝚌𝚎𝚙𝚝\mathtt{accept}typewriter\_accept and 𝚛𝚎𝚓𝚎𝚌𝚝𝚛𝚎𝚓𝚎𝚌𝚝\mathtt{reject}typewriter\_reject atoms express whether a trace is accepted or rejected by the automaton; hence, positive traces must only be accepted, negative traces must only be rejected, and incomplete traces cannot be accepted or rejected. Given a trace TL,𝒪=⟨O0,…,On⟩subscript𝑇𝐿𝒪subscript𝑂0…subscript𝑂𝑛T\_{L,\mathcal{O}}=\langle O\_{0},\ldots,O\_{n}\rangleitalic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT = ⟨ italic\_O start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , … , italic\_O start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ⟩, a context is defined as: CTL,𝒪={𝚘𝚋𝚜(𝙾,𝚃).∣𝙾∈O𝚃,O𝚃∈TL,𝒪}∪{𝚕𝚊𝚜𝚝(n).}C\_{T\_{L,\mathcal{O}}}=\left\{\mathtt{obs(O,T).}\mid\mathtt{O}\in O\_{\mathtt{T}},O\_{\mathtt{T}}\in T\_{L,\mathcal{O}}\right\}\cup\left\{\mathtt{last(}n\mathtt{).}\right\}italic\_C start\_POSTSUBSCRIPT italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT = { typewriter\_obs ( typewriter\_O , typewriter\_T ) . ∣ typewriter\_O ∈ italic\_O start\_POSTSUBSCRIPT typewriter\_T end\_POSTSUBSCRIPT , italic\_O start\_POSTSUBSCRIPT typewriter\_T end\_POSTSUBSCRIPT ∈ italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT } ∪ { typewriter\_last ( italic\_n ) . }, where 𝚕𝚊𝚜𝚝(n)𝚕𝚊𝚜𝚝𝑛\mathtt{last(}n\mathtt{)}typewriter\_last ( italic\_n ) indicates that the trace ends at time n𝑛nitalic\_n. We denote by M(TL,𝒪)𝑀subscript𝑇𝐿𝒪M(T\_{L,\mathcal{O}})italic\_M ( italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT ) the set of ASP facts that describe the trace TL,𝒪subscript𝑇𝐿𝒪T\_{L,\mathcal{O}}italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT. For example, M(TL,𝒪)={𝚘𝚋𝚜(𝚊,𝟶).𝚘𝚋𝚜(𝚋,𝟸).𝚘𝚋𝚜(𝚌,𝟸).𝚕𝚊𝚜𝚝(𝟸).}M(T\_{L,\mathcal{O}})=\{\mathtt{obs(a,0).}~{}\mathtt{obs(b,2).}~{}\mathtt{obs(c,2).}~{}\mathtt{last(2).}\}italic\_M ( italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT ) = { typewriter\_obs ( typewriter\_a , typewriter\_0 ) . typewriter\_obs ( typewriter\_b , typewriter\_2 ) . typewriter\_obs ( typewriter\_c , typewriter\_2 ) . typewriter\_last ( typewriter\_2 ) . } for the trace TL,𝒪=⟨{a},{},{b,c}⟩subscript𝑇𝐿𝒪𝑎 𝑏𝑐T\_{L,\mathcal{O}}=\langle\{a\},\{\},\{b,c\}\rangleitalic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT = ⟨ { italic\_a } , { } , { italic\_b , italic\_c } ⟩. #### Background knowledge. The next paragraphs describe the background knowledge B𝐵Bitalic\_B components: B=BU∪B𝚜𝚝𝚎𝚙∪Bδ∪B𝚜𝚝𝐵subscript𝐵𝑈subscript𝐵𝚜𝚝𝚎𝚙subscript𝐵𝛿subscript𝐵𝚜𝚝B=B\_{U}\cup B\_{\mathtt{step}}\cup B\_{\delta}\cup B\_{\mathtt{st}}italic\_B = italic\_B start\_POSTSUBSCRIPT italic\_U end\_POSTSUBSCRIPT ∪ italic\_B start\_POSTSUBSCRIPT typewriter\_step end\_POSTSUBSCRIPT ∪ italic\_B start\_POSTSUBSCRIPT italic\_δ end\_POSTSUBSCRIPT ∪ italic\_B start\_POSTSUBSCRIPT typewriter\_st end\_POSTSUBSCRIPT. First, BUsubscript𝐵𝑈B\_{U}italic\_B start\_POSTSUBSCRIPT italic\_U end\_POSTSUBSCRIPT is a set of facts of the form 𝚜𝚝𝚊𝚝𝚎(𝚞).𝚜𝚝𝚊𝚝𝚎𝚞\mathtt{state(u).}typewriter\_state ( typewriter\_u ) . for each 𝚞∈U𝚞𝑈\mathtt{u}\in Utypewriter\_u ∈ italic\_U. The set B𝚜𝚝𝚎𝚙subscript𝐵𝚜𝚝𝚎𝚙B\_{\mathtt{step}}italic\_B start\_POSTSUBSCRIPT typewriter\_step end\_POSTSUBSCRIPT defines a fluent 𝚜𝚝𝚎𝚙(𝚃)𝚜𝚝𝚎𝚙𝚃\mathtt{step(T)}typewriter\_step ( typewriter\_T ) for 0≤𝚃≤𝚃′+10𝚃superscript𝚃′10\leq\mathtt{T}\leq\mathtt{T^{\prime}}+10 ≤ typewriter\_T ≤ typewriter\_T start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT + 1 where 𝚃′superscript𝚃′\mathtt{T^{\prime}}typewriter\_T start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT is the step for which 𝚕𝚊𝚜𝚝(𝚃′)𝚕𝚊𝚜𝚝superscript𝚃′\mathtt{last(T^{\prime})}typewriter\_last ( typewriter\_T start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) is defined. The subset Bδsubscript𝐵𝛿B\_{\delta}italic\_B start\_POSTSUBSCRIPT italic\_δ end\_POSTSUBSCRIPT defines rules for the automaton transition function. The first rule defines all the possible edge identifiers, which is limited by the maximum number of edges between two states. The second rule states that there is an external transition from state 𝚇𝚇\mathtt{X}typewriter\_X at time 𝚃𝚃\mathtt{T}typewriter\_T if there is a transition from 𝚇𝚇\mathtt{X}typewriter\_X to a different state 𝚈𝚈\mathtt{Y}typewriter\_Y at that time. The third rule is a frame axiom: state 𝚇𝚇\mathtt{X}typewriter\_X transitions to itself at time 𝚃𝚃\mathtt{T}typewriter\_T if there are no external transitions from it at that time. The fourth rule defines the positive transitions in terms of the learned negative transitions δ¯¯𝛿\bar{\delta}over¯ start\_ARG italic\_δ end\_ARG and 𝚎𝚍𝚎𝚍\mathtt{ed}typewriter\_ed atoms. The fifth rule preserves determinism: two transitions from 𝚇𝚇\mathtt{X}typewriter\_X to two different states 𝚈𝚈\mathtt{Y}typewriter\_Y and 𝚉𝚉\mathtt{Z}typewriter\_Z cannot hold at the same time. The sixth rule forces all non-terminal states to have an edge to another state. \| \| \| \| \| --- \| --- \| --- \| \| \| Bδ={𝚎𝚍𝚐𝚎\_𝚒𝚍(𝟷..max\|δ(𝚡,𝚢)\|).𝚎𝚡𝚝\_δ(𝚇,𝚃):−⁡δ(𝚇,𝚈,\_,𝚃),𝚇!=𝚈.δ(𝚇,𝚇,𝟷,𝚃):−⁡𝚗𝚘𝚝𝚎𝚡𝚝\_δ(𝚇,𝚃),𝚜𝚝𝚊𝚝𝚎(𝚇),𝚜𝚝𝚎𝚙(𝚃).δ(𝚇,𝚈,𝙴,𝚃):−⁡𝚎𝚍(𝚇,𝚈,𝙴),𝚗𝚘𝚝δ¯(𝚇,𝚈,𝙴,𝚃),𝚜𝚝𝚎𝚙(𝚃).:−⁡δ(𝚇,𝚈,\_,𝚃),δ(𝚇,𝚉,\_,𝚃),𝚈!=𝚉.:−⁡𝚗𝚘𝚝𝚎𝚍(𝚇,\_,\_),𝚜𝚝𝚊𝚝𝚎(𝚇),𝚇!=𝚞𝙰,𝚇!=𝚞𝚁.}B\_{\delta}={\begin{Bmatrix}[l]\mathtt{edge\\_id(1..\max\|\delta(x,y)\|)}.\\ \mathtt{ext\\_\delta(X,T)\operatorname{\mathtt{:-}}\delta(X,Y,\\_,T),X!\mathord{=}Y.}\\ \mathtt{\delta(X,X,1,T)\operatorname{\mathtt{:-}}not~{}ext\\_\delta(X,T),state(X),step(T).}\\ \mathtt{\delta(X,Y,E,T)\operatorname{\mathtt{:-}}ed(X,Y,E),not~{}\bar{\delta}(X,Y,E,T),step(T).}\\ \mathtt{\operatorname{\mathtt{:-}}\delta(X,Y,\\_,T),\delta(X,Z,\\_,T),Y!\mathord{=}Z.}\\ \mathtt{\operatorname{\mathtt{:-}}not~{}ed(X,\\_,\\_),state(X),X!\mathord{=}u\_{A},X!\mathord{=}u\_{R}.}\end{Bmatrix}}italic\_B start\_POSTSUBSCRIPT italic\_δ end\_POSTSUBSCRIPT = { start\_ARG start\_ROW start\_CELL typewriter\_edge \_ typewriter\_id ( typewriter\_1 . . roman\_max \| italic\_δ ( typewriter\_x , typewriter\_y ) \| ) . end\_CELL end\_ROW start\_ROW start\_CELL typewriter\_ext \_ italic\_δ ( typewriter\_X , typewriter\_T ) start\_OPFUNCTION : - end\_OPFUNCTION italic\_δ ( typewriter\_X , typewriter\_Y , \_ , typewriter\_T ) , typewriter\_X ! = typewriter\_Y . end\_CELL end\_ROW start\_ROW start\_CELL italic\_δ ( typewriter\_X , typewriter\_X , typewriter\_1 , typewriter\_T ) start\_OPFUNCTION : - end\_OPFUNCTION typewriter\_not typewriter\_ext \_ italic\_δ ( typewriter\_X , typewriter\_T ) , typewriter\_state ( typewriter\_X ) , typewriter\_step ( typewriter\_T ) . end\_CELL end\_ROW start\_ROW start\_CELL italic\_δ ( typewriter\_X , typewriter\_Y , typewriter\_E , typewriter\_T ) start\_OPFUNCTION : - end\_OPFUNCTION typewriter\_ed ( typewriter\_X , typewriter\_Y , typewriter\_E ) , typewriter\_not over¯ start\_ARG italic\_δ end\_ARG ( typewriter\_X , typewriter\_Y , typewriter\_E , typewriter\_T ) , typewriter\_step ( typewriter\_T ) . end\_CELL end\_ROW start\_ROW start\_CELL start\_OPFUNCTION : - end\_OPFUNCTION italic\_δ ( typewriter\_X , typewriter\_Y , \_ , typewriter\_T ) , italic\_δ ( typewriter\_X , typewriter\_Z , \_ , typewriter\_T ) , typewriter\_Y ! = typewriter\_Z . end\_CELL end\_ROW start\_ROW start\_CELL start\_OPFUNCTION : - end\_OPFUNCTION typewriter\_not typewriter\_ed ( typewriter\_X , \_ , \_ ) , typewriter\_state ( typewriter\_X ) , typewriter\_X ! = typewriter\_u start\_POSTSUBSCRIPT typewriter\_A end\_POSTSUBSCRIPT , typewriter\_X ! = typewriter\_u start\_POSTSUBSCRIPT typewriter\_R end\_POSTSUBSCRIPT . end\_CELL end\_ROW end\_ARG } \| \| The subset B𝚜𝚝subscript𝐵𝚜𝚝B\_{\mathtt{st}}italic\_B start\_POSTSUBSCRIPT typewriter\_st end\_POSTSUBSCRIPT uses 𝚜𝚝(𝚃,𝚇)𝚜𝚝𝚃𝚇\mathtt{st(T,X)}typewriter\_st ( typewriter\_T , typewriter\_X ) atoms, indicating that the agent is in state 𝚇𝚇\mathtt{X}typewriter\_X at time 𝚃𝚃\mathtt{T}typewriter\_T. The first rule says that the agent is in 𝚞𝟶subscript𝚞0\mathtt{u\_{0}}typewriter\_u start\_POSTSUBSCRIPT typewriter\_0 end\_POSTSUBSCRIPT at time 𝟶0\mathtt{0}typewriter\_0. The second rule determines that at time 𝚃+𝟷𝚃1\mathtt{T\mathord{+}1}typewriter\_T + typewriter\_1 the agent will be in state 𝚈𝚈\mathtt{Y}typewriter\_Y if it is in a non-terminal state 𝚇𝚇\mathtt{X}typewriter\_X at time 𝚃𝚃\mathtt{T}typewriter\_T and a transition between them holds. The third (resp. fourth) rule defines that the example is accepted (resp. rejected) if the state at the trace’s last timestep is 𝚞𝙰subscript𝚞𝙰\mathtt{u\_{A}}typewriter\_u start\_POSTSUBSCRIPT typewriter\_A end\_POSTSUBSCRIPT (resp. 𝚞𝚁subscript𝚞𝚁\mathtt{u\_{R}}typewriter\_u start\_POSTSUBSCRIPT typewriter\_R end\_POSTSUBSCRIPT). \| \| \| \| \| --- \| --- \| --- \| \| \| B𝚜𝚝={𝚜𝚝(𝟶,𝚞𝟶).𝚜𝚝(𝚃+𝟷,𝚈):−⁡𝚜𝚝(𝚃,𝚇),δ(𝚇,𝚈,\_,𝚃),𝚇!=𝚞𝙰,𝚇!=𝚞𝚁.𝚊𝚌𝚌𝚎𝚙𝚝:−⁡𝚕𝚊𝚜𝚝(𝚃),𝚜𝚝(𝚃+𝟷,𝚞𝙰).𝚛𝚎𝚓𝚎𝚌𝚝:−⁡𝚕𝚊𝚜𝚝(𝚃),𝚜𝚝(𝚃+𝟷,𝚞𝚁).}subscript𝐵𝚜𝚝matrix𝚜𝚝0subscript𝚞0𝚜𝚝𝚃1𝚈:absent𝚜𝚝𝚃𝚇𝛿𝚇𝚈\_𝚃𝚇subscript𝚞𝙰𝚇subscript𝚞𝚁𝚊𝚌𝚌𝚎𝚙𝚝:absent𝚕𝚊𝚜𝚝𝚃𝚜𝚝𝚃1subscript𝚞𝙰𝚛𝚎𝚓𝚎𝚌𝚝:absent𝚕𝚊𝚜𝚝𝚃𝚜𝚝𝚃1subscript𝚞𝚁B\_{\mathtt{st}}={\begin{Bmatrix}[l]\mathtt{st(0,u\_{0}).}\\ \mathtt{st(T\mathord{+}1,Y)\operatorname{\mathtt{:-}}st(T,X),\delta(X,Y,\\_,T),X!\mathord{=}u\_{A},X!\mathord{=}u\_{R}.}\\ \mathtt{accept\operatorname{\mathtt{:-}}last(T),st(T\mathord{+}1,u\_{A}).}\\ \mathtt{reject\operatorname{\mathtt{:-}}last(T),st(T\mathord{+}1,u\_{R}).}\end{Bmatrix}}italic\_B start\_POSTSUBSCRIPT typewriter\_st end\_POSTSUBSCRIPT = { start\_ARG start\_ROW start\_CELL typewriter\_st ( typewriter\_0 , typewriter\_u start\_POSTSUBSCRIPT typewriter\_0 end\_POSTSUBSCRIPT ) . end\_CELL end\_ROW start\_ROW start\_CELL typewriter\_st ( typewriter\_T + typewriter\_1 , typewriter\_Y ) start\_OPFUNCTION : - end\_OPFUNCTION typewriter\_st ( typewriter\_T , typewriter\_X ) , italic\_δ ( typewriter\_X , typewriter\_Y , \_ , typewriter\_T ) , typewriter\_X ! = typewriter\_u start\_POSTSUBSCRIPT typewriter\_A end\_POSTSUBSCRIPT , typewriter\_X ! = typewriter\_u start\_POSTSUBSCRIPT typewriter\_R end\_POSTSUBSCRIPT . end\_CELL end\_ROW start\_ROW start\_CELL typewriter\_accept start\_OPFUNCTION : - end\_OPFUNCTION typewriter\_last ( typewriter\_T ) , typewriter\_st ( typewriter\_T + typewriter\_1 , typewriter\_u start\_POSTSUBSCRIPT typewriter\_A end\_POSTSUBSCRIPT ) . end\_CELL end\_ROW start\_ROW start\_CELL typewriter\_reject start\_OPFUNCTION : - end\_OPFUNCTION typewriter\_last ( typewriter\_T ) , typewriter\_st ( typewriter\_T + typewriter\_1 , typewriter\_u start\_POSTSUBSCRIPT typewriter\_R end\_POSTSUBSCRIPT ) . end\_CELL end\_ROW end\_ARG } \| \| Lemma [1](#Thmlemma1 "Lemma 1 (Correctness of the ASP encoding). ‣ Background knowledge. ‣ Learning Subgoal Automata from Traces ‣ 3 Methodology ‣ Induction of Subgoal Automata for Reinforcement Learning") and Theorem [1](#Thmtheorem1 "Theorem 1. ‣ Background knowledge. ‣ Learning Subgoal Automata from Traces ‣ 3 Methodology ‣ Induction of Subgoal Automata for Reinforcement Learning") capture the correctness of the encoding. We omit the proofs for brevity. ###### Lemma 1 (Correctness of the ASP encoding). Given an automaton 𝒜𝒜\mathcal{A}caligraphic\_A and a finite trace TL,𝒪∗superscriptsubscript𝑇𝐿𝒪normal-∗T\_{L,\mathcal{O}}^{\ast}italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ∗ end\_POSTSUPERSCRIPT, where ∗∈{+,−,I}\ast\in\{+,-,I\}∗ ∈ { + , - , italic\_I }, M(𝒜)∪B∪M(TL,𝒪∗)𝑀𝒜𝐵𝑀superscriptsubscript𝑇𝐿𝒪normal-∗M(\mathcal{A})\cup B\cup M(T\_{L,\mathcal{O}}^{\ast})italic\_M ( caligraphic\_A ) ∪ italic\_B ∪ italic\_M ( italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ∗ end\_POSTSUPERSCRIPT ) has a unique answer set S𝑆Sitalic\_S and (1) 𝚊𝚌𝚌𝚎𝚙𝚝∈S𝚊𝚌𝚌𝚎𝚙𝚝𝑆\mathtt{accept}\in Stypewriter\_accept ∈ italic\_S iff ∗⁣=⁣+normal-∗\ast=+∗ = +, and (2) 𝚛𝚎𝚓𝚎𝚌𝚝∈S𝚛𝚎𝚓𝚎𝚌𝚝𝑆\mathtt{reject}\in Stypewriter\_reject ∈ italic\_S iff ∗⁣=⁣−normal-∗\ast=-∗ = -. ###### Theorem 1. Given an automaton learning task T𝒜=⟨U,𝒪,𝒯L,𝒪⟩subscript𝑇𝒜𝑈𝒪subscript𝒯𝐿𝒪T\_{\mathcal{A}}=\langle U,\mathcal{O},\mathcal{T}\_{L,\mathcal{O}}\rangleitalic\_T start\_POSTSUBSCRIPT caligraphic\_A end\_POSTSUBSCRIPT = ⟨ italic\_U , caligraphic\_O , caligraphic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT ⟩, an automaton 𝒜𝒜\mathcal{A}caligraphic\_A is a solution of T𝒜subscript𝑇𝒜T\_{\mathcal{A}}italic\_T start\_POSTSUBSCRIPT caligraphic\_A end\_POSTSUBSCRIPT iff M(𝒜)𝑀𝒜M(\mathcal{A})italic\_M ( caligraphic\_A ) is an inductive solution of M(T𝒜)=⟨B,SM,E⟩𝑀subscript𝑇𝒜𝐵subscript𝑆𝑀𝐸M(T\_{\mathcal{A}})=\langle B,S\_{M},E\rangleitalic\_M ( italic\_T start\_POSTSUBSCRIPT caligraphic\_A end\_POSTSUBSCRIPT ) = ⟨ italic\_B , italic\_S start\_POSTSUBSCRIPT italic\_M end\_POSTSUBSCRIPT , italic\_E ⟩. ### Interleaved Automata Learning Algorithm This section describes ISA (Induction of Subgoal Automata for Reinforcement Learning), a method that combines reinforcement and automaton learning. First, we describe the RL algorithm that exploits the automaton structure. Second, we explain how these two learning components are mixed. #### Reinforcement learning algorithm. The RL algorithm we use to exploit the automata structure is QRM (Q-learning for Reward Machines) (?). QRM maintains a Q-function for each automaton state, which are updated with Q-learning updates of the form: \| \| \| \| \| --- \| --- \| --- \| \| \| Qu(s,a)←𝛼r+γmaxa′⁡Qu′(s′,a′),𝛼←subscript𝑄𝑢𝑠𝑎𝑟𝛾subscriptsuperscript𝑎′subscript𝑄superscript𝑢′superscript𝑠′superscript𝑎′\displaystyle Q\_{u}\left(s,a\right)\xleftarrow{\alpha}r+\gamma\max\_{a^{\prime}}Q\_{u^{\prime}}\left(s^{\prime},a^{\prime}\right),italic\_Q start\_POSTSUBSCRIPT italic\_u end\_POSTSUBSCRIPT ( italic\_s , italic\_a ) start\_ARROW overitalic\_α ← end\_ARROW italic\_r + italic\_γ roman\_max start\_POSTSUBSCRIPT italic\_a start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT italic\_Q start\_POSTSUBSCRIPT italic\_u start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ( italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , italic\_a start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) , \| \| where, in our case, r=1𝑟1r=1italic\_r = 1 if u′=uAsuperscript𝑢′subscript𝑢𝐴u^{\prime}=u\_{A}italic\_u start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT and 0 otherwise. Note that the bootstrapped action-value depends on the next automaton state u′superscript𝑢′u^{\prime}italic\_u start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT. QRM performs this update for all the automaton states, so all policies are simultaneously updated based on the (s,a,s′)𝑠𝑎superscript𝑠′(s,a,s^{\prime})( italic\_s , italic\_a , italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) experience. Note this is a form of intra-option learning (?): we update the policies of all the states from the experience generated by a single state’s policy. In the tabular case, QRM is guaranteed to converge to an optimal policy in the limit. Note that QRM (and thus, ISA) is still applicable in domains with large state spaces by having a Deep Q-Network (?) in each automaton state instead of a Q-table. Algorithm 1 ISA algorithm for a single task 1:𝒜←←𝒜absent\mathcal{A}\leftarrowcaligraphic\_A ← InitAutomaton({u0,uA,uR}subscript𝑢0subscript𝑢𝐴subscript𝑢𝑅\{u\_{0},u\_{A},u\_{R}\}{ italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT }) 2:𝒯L,𝒪←{}←subscript𝒯𝐿𝒪\mathcal{T}\_{L,\mathcal{O}}\leftarrow\{\}caligraphic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT ← { } ▷▷\triangleright▷ Set of counterexamples 3:InitQFunctions(𝒜𝒜\mathcal{A}caligraphic\_A) 4:for l=0𝑙0l=0italic\_l = 0 to num\_episodes do 5: s←←𝑠absents\leftarrowitalic\_s ← EnvInitialState() 6: up←δ(u0,L(s))←subscript𝑢𝑝𝛿subscript𝑢0𝐿𝑠u\_{p}\leftarrow\delta(u\_{0},L(s))italic\_u start\_POSTSUBSCRIPT italic\_p end\_POSTSUBSCRIPT ← italic\_δ ( italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , italic\_L ( italic\_s ) ) 7: TL,𝒪←⟨L(s)⟩←subscript𝑇𝐿𝒪delimited-⟨⟩𝐿𝑠T\_{L,\mathcal{O}}\leftarrow\langle L(s)\rangleitalic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT ← ⟨ italic\_L ( italic\_s ) ⟩ ▷▷\triangleright▷ Initialize trace 8: if IsCounterexample(s,up𝑠subscript𝑢𝑝s,u\_{p}italic\_s , italic\_u start\_POSTSUBSCRIPT italic\_p end\_POSTSUBSCRIPT) then 9: OnCounterexampleFound(TL,𝒪subscript𝑇𝐿𝒪T\_{L,\mathcal{O}}italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT) 10: up←δ(u0,L(s))←subscript𝑢𝑝𝛿subscript𝑢0𝐿𝑠u\_{p}\leftarrow\delta(u\_{0},L(s))italic\_u start\_POSTSUBSCRIPT italic\_p end\_POSTSUBSCRIPT ← italic\_δ ( italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , italic\_L ( italic\_s ) ) 11: for t=0𝑡0t=0italic\_t = 0 to length\_episode do 12: a,s′←←𝑎superscript𝑠′ absenta,s^{\prime}\leftarrowitalic\_a , italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ← EnvStep(s,up𝑠subscript𝑢𝑝s,u\_{p}italic\_s , italic\_u start\_POSTSUBSCRIPT italic\_p end\_POSTSUBSCRIPT) 13: uq←δ(up,L(s′))←subscript𝑢𝑞𝛿subscript𝑢𝑝𝐿superscript𝑠′u\_{q}\leftarrow\delta(u\_{p},L(s^{\prime}))italic\_u start\_POSTSUBSCRIPT italic\_q end\_POSTSUBSCRIPT ← italic\_δ ( italic\_u start\_POSTSUBSCRIPT italic\_p end\_POSTSUBSCRIPT , italic\_L ( italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) ) 14: UpdateObsTrace(L(s′),TL,𝒪𝐿superscript𝑠′subscript𝑇𝐿𝒪L(s^{\prime}),T\_{L,\mathcal{O}}italic\_L ( italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) , italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT) 15: if IsCounterexample(s′,uqsuperscript𝑠′subscript𝑢𝑞s^{\prime},u\_{q}italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_q end\_POSTSUBSCRIPT) then 16: OnCounterexampleFound(TL,𝒪subscript𝑇𝐿𝒪T\_{L,\mathcal{O}}italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT) 17: break 18: else 19: UpdateQFunctions(s𝑠sitalic\_s, a𝑎aitalic\_a, s′superscript𝑠′s^{\prime}italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT, L(s′)𝐿superscript𝑠′L(s^{\prime})italic\_L ( italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT )) 20: s←s′;up←uqformulae-sequence←𝑠superscript𝑠′←subscript𝑢𝑝subscript𝑢𝑞s\leftarrow s^{\prime};u\_{p}\leftarrow u\_{q}italic\_s ← italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ; italic\_u start\_POSTSUBSCRIPT italic\_p end\_POSTSUBSCRIPT ← italic\_u start\_POSTSUBSCRIPT italic\_q end\_POSTSUBSCRIPT 21:function OnCounterexampleFound(TL,𝒪subscript𝑇𝐿𝒪T\_{L,\mathcal{O}}italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT) 22: 𝒯L,𝒪←𝒯L,𝒪∪{TL,𝒪}←subscript𝒯𝐿𝒪subscript𝒯𝐿𝒪subscript𝑇𝐿𝒪\mathcal{T}\_{L,\mathcal{O}}\leftarrow\mathcal{T}\_{L,\mathcal{O}}\cup\{T\_{L,\mathcal{O}}\}caligraphic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT ← caligraphic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT ∪ { italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT } 23: 𝒜←←𝒜absent\mathcal{A}\leftarrowcaligraphic\_A ← FindMinimalAutomaton(𝒯L,𝒪subscript𝒯𝐿𝒪\mathcal{T}\_{L,\mathcal{O}}caligraphic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT) 24: InitQFunctions(𝒜𝒜\mathcal{A}caligraphic\_A) #### ISA algorithm. Algorithm [1](#alg1 "Algorithm 1 ‣ Reinforcement learning algorithm. ‣ Interleaved Automata Learning Algorithm ‣ 3 Methodology ‣ Induction of Subgoal Automata for Reinforcement Learning") is the ISA pseudocode for a single task and is explained below: 1. 1. The initial automaton (line 1) has initial state u0subscript𝑢0u\_{0}italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT, accepting state uAsubscript𝑢𝐴u\_{A}italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT and rejecting state uRsubscript𝑢𝑅u\_{R}italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT. The automaton does not accept nor reject anything. The set of counterexample traces and the Q-functions are initialized (lines 2-3). 2. 2. When an episode starts, the current automaton state upsubscript𝑢𝑝u\_{p}italic\_u start\_POSTSUBSCRIPT italic\_p end\_POSTSUBSCRIPT is u0subscript𝑢0u\_{0}italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT. One transition is applied depending on the agent’s initial observations L(s)𝐿𝑠L(s)italic\_L ( italic\_s ) (lines 5-6). In Figure [1c](#S3.F1.sf3 "1c ‣ Figure 1 ‣ 3 Methodology ‣ Induction of Subgoal Automata for Reinforcement Learning"), if the agent initially observes {☕}☕\{\text{\Coffeecup}\}{ ☕ }, the actual initial state is u1subscript𝑢1u\_{1}italic\_u start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT. 3. 3. At each step, we select an action a𝑎aitalic\_a in state s𝑠sitalic\_s using an ϵitalic-ϵ\epsilonitalic\_ϵ-greedy policy (line 12), and update the automaton state upsubscript𝑢𝑝u\_{p}italic\_u start\_POSTSUBSCRIPT italic\_p end\_POSTSUBSCRIPT and observation trace TL,𝒪subscript𝑇 𝐿𝒪T\_{L,\mathcal{O}}italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT (lines 13-14). If no counterexample is found (line 18), the Q-functions of all automaton states are updated (line 19) and the episode continues. 4. 4. Let u𝑢uitalic\_u be the current automaton state and s𝑠sitalic\_s the MDP state. A counterexample trace is found (lines 8, 15) if (a) multiple outgoing transitions from u𝑢uitalic\_u hold, or (b) the automaton does not correctly recognize s𝑠sitalic\_s (e.g., s∈SG∧u≠uA𝑠subscript𝑆𝐺𝑢subscript𝑢𝐴s\in S\_{G}\land u\neq u\_{A}italic\_s ∈ italic\_S start\_POSTSUBSCRIPT italic\_G end\_POSTSUBSCRIPT ∧ italic\_u ≠ italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT). 5. 5. If a counterexample trace TL,𝒪subscript𝑇 𝐿𝒪T\_{L,\mathcal{O}}italic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT is found (lines 21-24): 1. (a) Add it to 𝒯L,𝒪+superscriptsubscript𝒯 𝐿𝒪\mathcal{T}\_{L,\mathcal{O}}^{+}caligraphic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT + end\_POSTSUPERSCRIPT if s∈SG𝑠subscript𝑆𝐺s\in S\_{G}italic\_s ∈ italic\_S start\_POSTSUBSCRIPT italic\_G end\_POSTSUBSCRIPT, to 𝒯L,𝒪−superscriptsubscript𝒯 𝐿𝒪\mathcal{T}\_{L,\mathcal{O}}^{-}caligraphic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT - end\_POSTSUPERSCRIPT if s∈ST∖SG𝑠subscript𝑆𝑇subscript𝑆𝐺s\in S\_{T}\setminus S\_{G}italic\_s ∈ italic\_S start\_POSTSUBSCRIPT italic\_T end\_POSTSUBSCRIPT ∖ italic\_S start\_POSTSUBSCRIPT italic\_G end\_POSTSUBSCRIPT and to 𝒯L,𝒪Isuperscriptsubscript𝒯 𝐿𝒪𝐼\mathcal{T}\_{L,\mathcal{O}}^{I}caligraphic\_T start\_POSTSUBSCRIPT italic\_L , caligraphic\_O end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_I end\_POSTSUPERSCRIPT if s∉ST𝑠subscript𝑆𝑇s\notin S\_{T}italic\_s ∉ italic\_S start\_POSTSUBSCRIPT italic\_T end\_POSTSUBSCRIPT (line 22). 2. (b) Run the automaton learner (line 23), using iterative deepening to select the number of automaton states. * • When a new automaton is learned, we reset all the Q-functions (e.g., setting all Q-values to 0) (line 24). * • If a counterexample is detected at the beginning of the episode (line 8), the automaton state is reset (line 10), else the episode ends (line 17). ISA does not start learning automata until we find a positive example (i.e., the goal is achieved). Resetting all the Q-functions causes the agent to forget everything it learned. To mitigate the forgetting effect and further exploit the automata structure, we employ reward shaping. #### Reward shaping. ? (?) proposed a function that provides the agent with additional reward to guide its learning process while guaranteeing that optimal policies remain unchanged: \| \| \| \| \| --- \| --- \| --- \| \| \| F(s,a,s′)=γΦ(s′)−Φ(s),𝐹𝑠𝑎superscript𝑠′𝛾Φsuperscript𝑠′Φ𝑠\displaystyle F\left(s,a,s^{\prime}\right)=\gamma\Phi\left(s^{\prime}\right)-\Phi\left(s\right),italic\_F ( italic\_s , italic\_a , italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) = italic\_γ roman\_Φ ( italic\_s start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) - roman\_Φ ( italic\_s ) , \| \| where γ𝛾\gammaitalic\_γ is the MDP’s discount factor and Φ:S→ℝ:Φ→𝑆ℝ\Phi:S\to\mathbb{R}roman\_Φ : italic\_S → blackboard\_R is a real-valued function. The automata structure can be exploited by defining F:U×U→ℝ:𝐹→𝑈𝑈ℝF:U\times U\to\mathbb{R}italic\_F : italic\_U × italic\_U → blackboard\_R in terms of the automaton states instead of the MDP states: \| \| \| \| \| --- \| --- \| --- \| \| \| F(u,u′)=γΦ(u′)−Φ(u),𝐹𝑢superscript𝑢′𝛾Φsuperscript𝑢′Φ𝑢\displaystyle F\left(u,u^{\prime}\right)=\gamma\Phi\left(u^{\prime}\right)-\Phi\left(u\right),italic\_F ( italic\_u , italic\_u start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) = italic\_γ roman\_Φ ( italic\_u start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) - roman\_Φ ( italic\_u ) , \| \| where Φ:U→ℝ:Φ→𝑈ℝ\Phi:U\to\mathbb{R}roman\_Φ : italic\_U → blackboard\_R. Intuitively, we want F𝐹Fitalic\_F’s output to be high when the agent gets closer to uAsubscript𝑢𝐴u\_{A}italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT. Thus, we define ΦΦ\Phiroman\_Φ as \| \| \| \| \| --- \| --- \| --- \| \| \| Φ(u)=\|U\|−d(u,uA),Φ𝑢𝑈𝑑𝑢subscript𝑢𝐴\displaystyle\Phi\left(u\right)=\left\|U\right\|-d\left(u,u\_{A}\right),roman\_Φ ( italic\_u ) = \| italic\_U \| - italic\_d ( italic\_u , italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT ) , \| \| where \|U\|𝑈\|U\|\| italic\_U \| is the number of states in the automaton (an upper bound for the maximum finite distance between u0subscript𝑢0u\_{0}italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT and uAsubscript𝑢𝐴u\_{A}italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT), and d(u,uA)𝑑𝑢subscript𝑢𝐴d(u,u\_{A})italic\_d ( italic\_u , italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT ) is the distance between state u𝑢uitalic\_u and uAsubscript𝑢𝐴u\_{A}italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT. If uAsubscript𝑢𝐴u\_{A}italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT is unreachable from a state u𝑢uitalic\_u, then d(u,uA)=∞𝑑𝑢subscript𝑢𝐴d(u,u\_{A})=\inftyitalic\_d ( italic\_u , italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT ) = ∞. Theorem [2](#Thmtheorem2 "Theorem 2. ‣ Reward shaping. ‣ Interleaved Automata Learning Algorithm ‣ 3 Methodology ‣ Induction of Subgoal Automata for Reinforcement Learning") shows that if the target automata is in the hypothesis space, there will only be a finite number of learning steps in the algorithm before it converges on the target automata (or an equivalent automata). ###### Theorem 2. Given a target finite automaton 𝒜\subscript𝒜\mathcal{A\_{\}}caligraphic\_A start\_POSTSUBSCRIPT \* end\_POSTSUBSCRIPT, there is no infinite sequence σ𝜎\sigmaitalic\_σ of automaton-counterexample pairs ⟨𝒜i,ei⟩subscript𝒜𝑖subscript𝑒𝑖\langle\mathcal{A}\_{i},e\_{i}\rangle⟨ caligraphic\_A start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_e start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ⟩ such that ∀ifor-all𝑖\forall i∀ italic\_i: (1) 𝒜isubscript𝒜𝑖\mathcal{A}\_{i}caligraphic\_A start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT covers all examples e1,…,ei−1subscript𝑒1normal-…subscript𝑒𝑖1e\_{1},\ldots,e\_{i-1}italic\_e start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , … , italic\_e start\_POSTSUBSCRIPT italic\_i - 1 end\_POSTSUBSCRIPT, (2) 𝒜isubscript𝒜𝑖\mathcal{A}\_{i}caligraphic\_A start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT does not cover eisubscript𝑒𝑖e\_{i}italic\_e start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT, and (3) 𝒜isubscript𝒜𝑖\mathcal{A}\_{i}caligraphic\_A start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT is in the finite hypothesis space SMsubscript𝑆𝑀S\_{M}italic\_S start\_POSTSUBSCRIPT italic\_M end\_POSTSUBSCRIPT. ###### Proof. By contradiction. Assume that σ𝜎\sigmaitalic\_σ is infinite. Given that SMsubscript𝑆𝑀S\_{M}italic\_S start\_POSTSUBSCRIPT italic\_M end\_POSTSUBSCRIPT is finite, the number of possible automata is finite. Hence, some automaton 𝒜𝒜\mathcal{A}caligraphic\_A must appear in σ𝜎\sigmaitalic\_σ at least twice, say as 𝒜i=𝒜j,i<jformulae-sequencesubscript𝒜𝑖subscript𝒜𝑗𝑖𝑗\mathcal{A}\_{i}=\mathcal{A}\_{j},i<jcaligraphic\_A start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT = caligraphic\_A start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT , italic\_i < italic\_j. By definition, 𝒜isubscript𝒜𝑖\mathcal{A}\_{i}caligraphic\_A start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT does not cover eisubscript𝑒𝑖e\_{i}italic\_e start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT and 𝒜jsubscript𝒜𝑗\mathcal{A}\_{j}caligraphic\_A start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT covers eisubscript𝑒𝑖e\_{i}italic\_e start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT. This is a contradiction. ∎ u0subscript𝑢0u\_{0}italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPTstartu1subscript𝑢1u\_{1}italic\_u start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPTu2subscript𝑢2u\_{2}italic\_u start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPTu3subscript𝑢3u\_{3}italic\_u start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPTuAsubscript𝑢𝐴u\_{A}italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPTuRsubscript𝑢𝑅u\_{R}italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPTA𝐴Aitalic\_AB𝐵Bitalic\_BC𝐶Citalic\_CD𝐷Ditalic\_D∗∗\ast∗∗∗\ast∗∗∗\ast∗∗∗\ast∗ (a) VisitABCD u0subscript𝑢0u\_{0}italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPTstartuAsubscript𝑢𝐴u\_{A}italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPTu1subscript𝑢1u\_{1}italic\_u start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPTu2subscript𝑢2u\_{2}italic\_u start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPTu3subscript𝑢3u\_{3}italic\_u start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT☕∧¬🖂☕🖂\text{\Coffeecup}\land\neg\text{\Letter}☕ ∧ ¬ 🖂¬☕∧🖂☕🖂\neg\text{\Coffeecup}\land\text{\Letter}¬ ☕ ∧ 🖂☕∧🖂∧¬o☕🖂𝑜\text{\Coffeecup}\land\text{\Letter}\land\neg o☕ ∧ 🖂 ∧ ¬ italic\_o☕∧🖂∧o☕🖂𝑜\text{\Coffeecup}\land\text{\Letter}\land o☕ ∧ 🖂 ∧ italic\_o🖂∧¬o🖂𝑜\text{\Letter}\land\neg o🖂 ∧ ¬ italic\_o🖂∧o🖂𝑜\text{\Letter}\land o🖂 ∧ italic\_o☕∧¬o☕𝑜\text{\Coffeecup}\land\neg o☕ ∧ ¬ italic\_o☕∧o☕𝑜\text{\Coffeecup}\land o☕ ∧ italic\_oo𝑜oitalic\_o (b) CoffeeMail Figure 2: Automata for two OfficeWorld tasks. Self-loops and transitions to uRsubscript𝑢𝑅u\_{R}italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT in (b) are omitted. The shaded state IDs can be interchanged without Bsymsubscript𝐵𝑠𝑦𝑚B\_{sym}italic\_B start\_POSTSUBSCRIPT italic\_s italic\_y italic\_m end\_POSTSUBSCRIPT. The dashed state IDs are still interchangeable even when using Bsymsubscript𝐵𝑠𝑦𝑚B\_{sym}italic\_B start\_POSTSUBSCRIPT italic\_s italic\_y italic\_m end\_POSTSUBSCRIPT. ![Refer to caption](/html/1911.13152/assets/x1.png) (a) Coffee ![Refer to caption](/html/1911.13152/assets/x2.png) (b) CoffeeMail ![Refer to caption](/html/1911.13152/assets/x3.png) (c) VisitABCD Figure 3: Learning curves for different OfficeWorld tasks. The vertical lines are episodes where an automaton is learned. 4 Experimental Results ----------------------- We evaluate333Code: <github.com/ertsiger/induction-subgoal-automata-rl>. ISA using the OfficeWorld domain and the three tasks introduced in Section [3](#S3 "3 Methodology ‣ Induction of Subgoal Automata for Reinforcement Learning"). The automata we compute using our method are forced to be acyclic. Besides, we add a simple symmetry breaking constraint Bsymsubscript𝐵𝑠𝑦𝑚B\_{sym}italic\_B start\_POSTSUBSCRIPT italic\_s italic\_y italic\_m end\_POSTSUBSCRIPT to the task’s background knowledge to avoid considering isomorphic automata. Figure [2](#S3.F2 "Figure 2 ‣ Reward shaping. ‣ Interleaved Automata Learning Algorithm ‣ 3 Methodology ‣ Induction of Subgoal Automata for Reinforcement Learning") shows two automata whose state IDs u1,u2subscript𝑢1subscript𝑢2u\_{1},u\_{2}italic\_u start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT and u3subscript𝑢3u\_{3}italic\_u start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT can be used indifferently. Our symmetry breaking method (1) assigns an integer index to each state444u0subscript𝑢0u\_{0}italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT has the lowest value, while uAsubscript𝑢𝐴u\_{A}italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT and uRsubscript𝑢𝑅u\_{R}italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT have the highest. and (2) imposes that states must be visited in increasing order of indices. For example, if we assign indices 0…3 to u0,…,u3subscript𝑢0…subscript𝑢3u\_{0},\ldots,u\_{3}italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , … , italic\_u start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT, positive traces in Figure [2a](#S3.F2.sf1 "2a ‣ Figure 2 ‣ Reward shaping. ‣ Interleaved Automata Learning Algorithm ‣ 3 Methodology ‣ Induction of Subgoal Automata for Reinforcement Learning") always yield the sequence ⟨u0,u1,u2,u3,uA⟩subscript𝑢0subscript𝑢1subscript𝑢2subscript𝑢3subscript𝑢𝐴\langle u\_{0},u\_{1},u\_{2},u\_{3},u\_{A}\rangle⟨ italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT 3 end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT ⟩. However, this is not enough to break all symmetries in Figure [2b](#S3.F2.sf2 "2b ‣ Figure 2 ‣ Reward shaping. ‣ Interleaved Automata Learning Algorithm ‣ 3 Methodology ‣ Induction of Subgoal Automata for Reinforcement Learning"): u1subscript𝑢1u\_{1}italic\_u start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and u2subscript𝑢2u\_{2}italic\_u start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT can still be switched since they cannot be in the same path to uAsubscript𝑢𝐴u\_{A}italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT or uRsubscript𝑢𝑅u\_{R}italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT. Tabular Q-learning is used to learn the Q-function at each automaton state with parameters α=0.1𝛼0.1\alpha=0.1italic\_α = 0.1, ϵ=0.1italic-ϵ0.1\epsilon=0.1italic\_ϵ = 0.1, and γ=0.99𝛾0.99\gamma=0.99italic\_γ = 0.99. The agent’s state is its position. ISA receives a set of 100 randomly generated grids555Each grid has the same size and walls as Figure [1a](#S3.F1.sf1 "1a ‣ Figure 1 ‣ 3 Methodology ‣ Induction of Subgoal Automata for Reinforcement Learning"). The observables and the agent are randomly placed.. One episode is run per grid in sequential order until reaching 20,000 episodes for each grid. The maximum episode length is 100 steps. For some experiments we consider the multitask setting, where an automaton is learned for every task from the set of grids. Thus, there is a Q-table for each task-grid-automaton state triplet updated at every step. One episode is run per task-grid until 20,000 episodes are performed for each pair. When reward shaping is on, the shaping function’s output is set to -100 in case it is −∞-\infty- ∞. This occurs when the next automaton state is uRsubscript𝑢𝑅u\_{R}italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT since there is no path from it to uAsubscript𝑢𝐴u\_{A}italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT. The different settings are referenced as S (single task), M (multitask) and R (reward shaping), all using the same set of 100 random grids. We execute 10 runs for each setting. We use ILASP to learn the automata from compressed observation traces and set max⁡\|δ(x,y)\|𝛿𝑥𝑦\max\|\delta(x,y)\|roman\_max \| italic\_δ ( italic\_x , italic\_y ) \| to 1. All experiments ran on 3.40GHz Intel® Core™ i7-6700 processors. #### ISA performs similarly to QRM. Figure [3](#S3.F3 "Figure 3 ‣ Reward shaping. ‣ Interleaved Automata Learning Algorithm ‣ 3 Methodology ‣ Induction of Subgoal Automata for Reinforcement Learning") shows the tasks’ average learning curve for ISA and QRM (where the automaton is given beforehand) across 10 runs. The ISA curves converge similarly to the analogous QRM’s. When reward shaping is used, convergence speeds up dramatically. The multitask settings converge faster since an agent is also trained from other agents’ experiences in different tasks. The vertical lines are episodes where an automaton is learned, and often occur during the first episodes: this is the main reason why the learning and non-learning curves are similar. Less frequently, automata learning also happens at intermediate phases in the Coffee and CoffeeMail tasks which make the agent forget what it learned. In these cases, recovery from forgetting happens faster in the multitask settings because of the reason above. Reward shaping has a positive effect on the learner: not only is convergence faster, it also helps the agent to discover helpful traces earlier. #### ISA’s automata learning results. Table [1](#S4.T1 "Table 1 ‣ ISA’s automata learning results. ‣ 4 Experimental Results ‣ Induction of Subgoal Automata for Reinforcement Learning") shows the average number of examples needed to learn the final automata for setting S (the results for other settings are similar). Table [2](#S4.T2 "Table 2 ‣ ISA’s automata learning results. ‣ 4 Experimental Results ‣ Induction of Subgoal Automata for Reinforcement Learning") shows the average time needed for computing all the automata, which is negligible with respect to ISA’s total running time. For both tables, the standard error is shown in brackets. First, we observe that the most complex tasks (CoffeeMail and VisitABCD) need a higher number of examples and more time to compute their corresponding automata. However, while the total number of examples for these tasks are similar, the time is higher for VisitABCD possibly because the longest path from u0subscript𝑢0u\_{0}italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT to uAsubscript𝑢𝐴u\_{A}italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT is longer than in CoffeeMail. Second, we see that the number of positive and incomplete examples are usually the smallest and the largest respectively. Note that the number of positive examples is approximately equal to the number of paths from u0subscript𝑢0u\_{0}italic\_u start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT to uAsubscript𝑢𝐴u\_{A}italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT. The paths described by the positive examples are refined through the negative and incomplete ones. Finally, Table [2](#S4.T2 "Table 2 ‣ ISA’s automata learning results. ‣ 4 Experimental Results ‣ Induction of Subgoal Automata for Reinforcement Learning") shows slight variations of time across settings. The different experimental settings and the exploratory nature of the agent are responsible for coming up with different counterexamples and, thus, cause these variations. There is not a clear setting which leads to faster automata learning across domains. The design of exploratory strategies that speed up automata learning is possible future work. \| \| All \| + \| - \| I \| \| --- \| --- \| --- \| --- \| --- \| \| Coffee \| 6.6 (0.5) \| 2.2 (0.2) \| 2.3 (0.2) \| 2.1 (0.3) \| \| CoffeeMail \| 34.5 (2.9) \| 5.5 (0.4) \| 9.9 (0.9) \| 19.1 (2.2) \| \| VisitABCD \| 32.5 (2.1) \| 1.7 (0.2) \| 11.6 (0.8) \| 19.2 (1.7) \| Table 1: Average number of examples needed for setting S. \| \| S \| S+R \| M \| M+R \| \| --- \| --- \| --- \| --- \| --- \| \| Coffee \| 0.5 (0.0) \| 0.4 (0.0) \| 0.3 (0.0) \| 0.4 (0.0) \| \| CoffeeMail \| 43.3 (12.1) \| 36.9 (6.0) \| 24.8 (3.6) \| 24.6 (2.7) \| \| VisitABCD \| 63.0 (11.4) \| 68.5 (13.0) \| 48.4 (8.8) \| 69.6 (8.1) \| Table 2: Average ILASP running time. #### ISA learns automata faster with few observables. To test the effect of the observable set 𝒪𝒪\mathcal{O}caligraphic\_O on ILASP’s performance, we run the experiments using setting S but employing only the observables that each task needs, e.g., 𝒪={☕,o,∗}𝒪☕𝑜∗\mathcal{O}=\{\text{\Coffeecup},o,\ast\}caligraphic\_O = { ☕ , italic\_o , ∗ } in the Coffee task. The biggest changes occur in CoffeeMail, where the total number of examples is 46% smaller. The sets of positive, negative and incomplete examples are 27%, 42% and 53% smaller. Besides, the automata are computed 92% faster. Thus, we see that the number of observables has an impact on the performance: the RL process is halted less frequently and automata are learned faster. This performance boost in CoffeeMail can be due to the fact that it has more paths to uAsubscript𝑢𝐴u\_{A}italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT; thus, irrelevant symbols do not need to be discarded for each of these. This is confirmed by the fact that while the number of positives is roughly the same, the number of incomplete examples greatly decreases. 5 Related Work --------------- #### Hierarchical RL (HRL). Our method is closely related to the options framework for HRL, and indeed we define one option per automaton state. The key difference from other HRL approaches, like HAMs (?), MAXQ (?) and the common way of using options, is that we do not learn a high-level policy for selecting among options. Rather, the high-level policy is implicitly represented by the automaton, and the option to execute is fully determined by the current automaton state. Consequently, while HRL policies may be suboptimal in general, the QRM algorithm we use converges to the optimal policy. Our approach is similar to HAMs in that they also use an automaton. However, HAMs are non-deterministic automata whose transitions can invoke lower level machines and are not labeled by observables (the high-level policy consists in deciding which transition to fire). ? (?) synthesize a HAM from the set of shortest solutions to a non-deterministic planning problem, and use it to refine the choices at non-deterministic points through RL. #### Option discovery. ISA is similar to bottleneck option discovery methods, which find “bridges” between regions of the state space. In particular, ISA finds conditions that connect two of these regions. ? (?) use diverse density to find landmark states in state traces that achieve the task’s goal. This approach is similar to ours because (1) it learns from traces; (2) it classifies traces into different categories; and (3) it interleaves option discovery and learning. Just like some option discovery methods (?; ?), our approach requires the task to be solved at least once. Other methods (?; ?; ?; ?) discover options without solving the task. Grammars are an alternative to automata for expressing formal languages. ? (?) induce a straight-line grammar from action traces to discover macro-actions. #### Reward machines (RMs). Subgoal automata are similar to RMs (?). There are two differences: (1) RMs do not have explicit accepting and rejecting states, and (2) RMs use a reward-transition function δr:U×U→ℝ:subscript𝛿𝑟→𝑈𝑈ℝ\delta\_{r}:U\times U\to\mathbb{R}italic\_δ start\_POSTSUBSCRIPT italic\_r end\_POSTSUBSCRIPT : italic\_U × italic\_U → blackboard\_R that returns the reward for taking a transition between two automaton states. Note that our automata are a specific case of the latter where δr(⋅,uA)=1subscript𝛿𝑟⋅subscript𝑢𝐴1\delta\_{r}(\cdot,u\_{A})=1italic\_δ start\_POSTSUBSCRIPT italic\_r end\_POSTSUBSCRIPT ( ⋅ , italic\_u start\_POSTSUBSCRIPT italic\_A end\_POSTSUBSCRIPT ) = 1 and 0 otherwise. Recent work has focused on learning RMs from experience using discrete optimization (?) and grammatical inference algorithms (?). While these approaches can get stuck in a local optima, ISA returns a minimal automaton each time it is called. Just as our method, they use an ‘a priori’ specified set of observables. In contrast to the other learning approaches, in the future we can leverage ILP to (1) transfer knowledge between automata learning tasks (e.g., providing a partial automaton as the background knowledge), (2) support examples generated by a noisy labeling function, and (3) easily express more complex conditions (e.g., using first-order logic). ? (?) convert reward functions expressed in various formal languages into RMs, and propose a reward shaping method that runs value iteration on the RM states. 6 Conclusions and Future Work ------------------------------ In this paper we have proposed ISA, an algorithm for learning subgoals by inducing a deterministic finite automaton from observation traces seen by the agent. The automaton structure can be exploited by an existing RL algorithm to increase sample efficiency and transfer learning. We have shown that our method performs comparably to an algorithm where the automaton is given beforehand. Improving the scalability is needed to handle more complex tasks requiring automata with cycles or longer examples. In the future, we will further explore symmetry breaking techniques to reduce the hypothesis space (?) and other approaches automata learning, like RNNs (?; ?). Discovering the observables used by the automata is also an interesting path for future work. Acknowledgments --------------- Anders Jonsson is partially supported by the Spanish grants TIN2015-67959 and PCIN-2017-082.
a98a7fae-558c-447a-9dc1-24660ff2b40e	trentmkelly/LessWrong-43k	LessWrong	Methods for strong human germline engineering PDF version. Image sizes best at berkeleygenomics.org with a wide monitor. Twitter thread Introduction This article summarizes the technical pathways to make healthy humans with significantly modified genomes. These are the pathways that I'm aware of and that seem plausibly feasible in the next two decades. A short summary, in a diagram: Annotated table of contents: * Reproductive genomic vectoring explains the general idea of human germline genomic engineering, and distinguishes editing and selection. * Comparing editing and selection talks about general differences between the two kinds of genomic vectoring methods. * Reproductive GV and epigenomic correctness (EC), Methods to handle epigenomic correctness, and How GV and EC interact discuss the epigenomic correctness problem in germline engineering—what it is, why it matters, and how to address it. * Summary of genomic vectoring methods gives an annotated table of contents for the following Methods sections. The Methods sections—on Simple embryo selection, Gamete selection, Chromosome selection, Iterated recombinant selection, and Iterated multiplex CRISPR editing—give more detail about each genomic vectoring method: what it is, obstacles, variations, and how powerful it is. * The appendices give additional technical information, if you're looking around and saying "I'm not in the weeds enough, I want to be more in the weeds.". Here's a sneak peek about the strength of different genomic vectoring methods: The list of specific methods that this table summarizes starts after the section Summary of genomic vectoring methods. This article is roughly organized from general to specific, first discussing things that apply to the whole area, and then later discussing specific methods. I won't lie, this article book is a bit of a slog. You try writing a book about the state of the art of realistic germline engineering in a way that is automatically fun, and then get back to me, ok? But listen: There's
7f11f114-0c4f-4b10-aacf-454e812078e7	trentmkelly/LessWrong-43k	LessWrong	Open Thread, January 16-31, 2013 If it's worth saying, but not worth its own post, even in Discussion, it goes here.
f2530e12-cee4-4fe8-8a03-7c6ddaf29ce9	trentmkelly/LessWrong-43k	LessWrong	[meta] Future moderation and investigation of downvote abuse cases, or, I don't want to deal with this stuff Since the episode with Eugine_Nier, I have received three private messages from different people asking me to investigate various cases of suspected mass downvoting. And to be quite honest, I don't want to deal with this. Eugine's case was relatively clear-cut, since he had engaged in systematic downvoting of a massive scale, but the new situations are a lot fuzzier and I'm not sure of what exactly the rules should be (what counts as a permitted use of the downvote system and what doesn't?). At least one person has also privately contacted me and offered to carry out moderator duties if I don't want them, but even if I told them yes (on what basis? why them and not someone else?), I don't know what kind of policy I should tell them to enforce. I only happened to be appointed a moderator because I was in the list of top 10 posters at a particular time, and I don't feel like I should have any particular authority to make the rules. Nor do I feel like I have any good idea of what the rules should be, or who would be the right person to enforce them. In any case, I don't want to be doing this job, nor do I particularly feel like being responsible for figuring out who should, or how, or what the heck. I've already started visiting LW less often because I dread having new investigation requests to deal with. So if you folks could be so kind as to figure it out without my involvement? If there's a clear consensus that someone in particular should deal with this, I can give them mod powers, or something.
ff37f54e-a518-4927-8d6b-201f3dabd33c	trentmkelly/LessWrong-43k	LessWrong	Coalitional agency The coalitional frame Earlier in this sequence I laid out an argument that the goals of increasingly intelligent AIs will become increasingly systematized, until they converge to squiggle-maximization. In my last post, though, I touched on two reasons why this convergence might not happen: humans trying to prevent it, and AIs themselves trying to prevent it. I don’t have too much more to say about the former, but it’s worth elaborating on the latter. The best way to understand the deliberate protection of existing goals is in terms of Bostrom’s notion of instrumental convergence. Bostrom argues that goal preservation will be a convergent instrumental strategy for a wide range of agents. Perhaps it’s occasionally instrumentally useful to change your goals—but once you’ve done so, you’ll never want to course-correct back towards your old goals. So this is a strong reason to be conservative about your goals, and avoid changes where possible. One immediate problem with preserving goals, though: it requires that agents continue thinking in terms of the same concepts. But in general, an agent’s concepts will change significantly as they learn more about the world. For example, consider a medieval theist whose highest-priority goal is ensuring that their soul goes to heaven not hell. Upon becoming smarter, they realize that none of souls, heaven, or hell exist. The sensible thing to do here would be to either discard the goal, or else identify a more reasonable adaptation of it (e.g. the goal of avoiding torture while alive). But if their goals were totally fixed, then their actions would be determined by a series of increasingly convoluted hypotheticals where god did exist after all. (Or to put it another way: continuing to represent their old goal would require recreating a lot of their old ontology.) This would incur a strong systematicity penalty. So while we should expect agents to have some degree of conservatism, they’ll likely also have some degree of systematiz
fed3583f-6207-468a-9c24-f1cb092d49d5	trentmkelly/LessWrong-43k	LessWrong	Being Wrong Doesn't Mean You're Stupid and Bad (Probably) Sometimes, people are reluctant to admit that they were wrong about something, because they're afraid that "You are wrong about this" carries inextricable connotations of "You are stupid and bad." But this behavior is, itself, wrong, for at least two reasons. First, because it's evidential decision theory. The so-called "rationalist" "community" has a lot of cached clichés about this! A blank map does not correspond to a blank territory. What's true is already so; owning up to it doesn't make it worse. Refusing to go to the doctor (thereby avoiding encountering evidence that you're sick) doesn't keep you healthy. If being wrong means that you're stupid and bad, then preventing yourself from knowing that you were wrong doesn't stop you from being stupid and bad in reality. It just prevents you from knowing that you're stupid and bad—which is an important fact to know (if it's true), because if you don't know that you're stupid and bad, then it probably won't occur to you to even look for possible interventions to make yourself less stupid and less bad. Second, while "You are wrong about this" is evidence for the "You are stupid and bad" hypothesis if stupid and bad people are more likely to be wrong, I claim that it's very weak evidence. (Although it's possible that I'm wrong about this—and if I'm wrong, it's furthermore possible that the reason I'm wrong is because I'm stupid and bad.) Exactly how weak evidence is it? It's hard to guess directly, but fortunately, we can use probability theory to reduce the claim into more "atomic" conditional and prior probabilities that might be easier to estimate! Let W represent the proposition "You are wrong about something", S represent the proposition "You are stupid", and B represent the proposition "You are bad." By Bayes's theorem, the probability that you are stupid and bad given that you're wrong about something is given by— P(S,B\|W)=P(W\|S,B)P(S,B)P(W\|S,B)P(S,B)+P(W\|S,¬B)P(S,¬B)+P(W\|¬S,B)P(¬S,B)+P(W\|¬S,¬B)P(¬S,¬B)
d4b575ae-b4ab-47b6-a498-5909a9a5b6f1	StampyAI/alignment-research-dataset/arxiv	Arxiv	More Robust Doubly Robust Off-policy Evaluation 1 Introduction --------------- In many real-world decision-making problems, in areas such as marketing, finance, robotics, and healthcare, deploying a policy without having an accurate estimate of its performance could be costly, unethical, or even illegal. This is why the problem of off-policy evaluation (OPE) has been heavily studied in contextual bandits (e.g., Dudík et al. [2011](#bib.bib6); Swaminathan et al. [2017](#bib.bib29)) and reinforcement learning (RL) (e.g., Precup et al. [2000a](#bib.bib22), [2001](#bib.bib24); Paduraru [2013](#bib.bib20); Mahmood et al. [2014](#bib.bib15); Thomas et al. [2015a](#bib.bib32); Li et al. [2015](#bib.bib13); Jiang & Li [2016](#bib.bib11); Thomas & Brunskill [2016](#bib.bib31)), and some of the results have been applied to problems in marketing (e.g., Li et al. [2011](#bib.bib12); Theocharous et al. [2015](#bib.bib30)), healthcare (e.g., Murphy et al. [2001](#bib.bib19); Hirano et al. [2003](#bib.bib10)), and education (e.g., Mandel et al. [2014](#bib.bib16), [2016](#bib.bib17)). The goal in OPE is to estimate the performance of an evaluation policy, given a log of data generated by the behavior policy(ies). The OPE problem can also be viewed as a form of counterfactual reasoning to infer the causal effect of a new treatment from historical data (e.g., Bottou et al. [2013](#bib.bib2); Shalit et al. [2017](#bib.bib27); Louizos et al. [2017](#bib.bib14)). Three different approaches to OPE in RL can be identified in the literature. 1) Direct Method (DM) which learns a model of the system and then uses it to estimate the performance of the evaluation policy. This approach often has low variance but its bias depends on how well the selected function class represents the system and on whether the number of samples is sufficient to accurately learn this function class. There are two major problems with this approach: (a) Its bias cannot be easily quantified, since in general it is difficult to quantify the approximation error of a function class, and (b) It is not clear how to choose the loss function for model learning without the knowledge of the evaluation policy (or the distribution of the evaluation policies). Without this knowledge, we may select a loss function that focuses on learning the areas that are irrelevant for the evaluation policy(ies). 2) Importance Sampling (IS) that uses the IS term to correct the mismatch between the distributions of the system trajectory induced by the evaluation and behavior policies. Although this approach is unbiased (under mild assumptions) in case the behavior policy is known, its variance can be very large when there is a big difference between the distributions of the evaluation and behavior policies, and grows exponentially with the horizon of the RL problem. 3) Doubly Robust (DR) which is a combination of DM and IS, and can achieve the low variance of DM and no (or low) bias of IS. The DR estimator was first developed in statistics (e.g., Cassel et al. [1976](#bib.bib5); Robins et al. [1994](#bib.bib26); Robins & Rotnitzky [1995](#bib.bib25); Bang & Robins [2005](#bib.bib1)) to estimate from incomplete data with the property that is unbiased when either of its DM or IS estimators is correct. It was brought to our community, first in contextual bandits by Dudík et al. ([2011](#bib.bib6)) and then in RL by Jiang & Li ([2016](#bib.bib11)). Thomas & Brunskill ([2016](#bib.bib31)) proposed two methods to reduce the variance of DR, with the cost of introducing a bias, one to select a low variance IS estimator, namely weighted IS (WIS), and one to blend DM and IS together (instead of simply combining them as in the standard DR approach) in a way to minimize the mean squared error (MSE). In this paper, we propose to reduce the variance of DR in bandits and RL by designing the loss function used to learn the model in the DM part of the estimator. The main idea of our estimator, called more robust doubly robust (MRDR), is to learn the parameters of the DM model by minimizing the variance of the DR estimator. This idea has been investigated in statistics in the context of regression when the labels of a subset of samples are randomly missing (Cao et al., [2009](#bib.bib4)). We first present a novel formulation for the DM part of the DR estimator in RL. We then derive formulas for the variance of the DR estimator in both bandits and RL in a way that its gradient w.r.t. the model parameters can be estimated from the samples. Note that the DR variances reported for bandits (Dudík et al., [2011](#bib.bib6)) and RL (Jiang & Li, [2016](#bib.bib11)) contain the bias of the DM component, which is unknown. We then propose methods to efficiently minimize the variance in both bandits and RL. Furthermore, we prove that the MRDR estimator is strongly consistent and asymptotically optimal. Finally, we evaluate the MRDR estimator in bandits and RL benchmark problems, and compare its performance with DM, IS, and DR approaches. 2 Preliminaries ---------------- In this paper, we consider the reinforcement learning (RL) problem in which the agent’s interaction with the system is modeled as a Markov decision process (MDP). Note that the contextual bandit problem is a special case with horizon-1111 decision-making. In this section, we first define MDPs and the relevant quantities that we are going to use throughout the paper, and then define the off-policy evaluation problem in RL, which is the main topic of this work. ### 2.1 Markov Decision Processes A MDP is a tuple ⟨𝒳,𝒜,Pr,P,P0,γ⟩𝒳𝒜subscript𝑃𝑟𝑃subscript𝑃0𝛾\langle\mathcal{X},\mathcal{A},P\_{r},P,P\_{0},\gamma\rangle⟨ caligraphic\_X , caligraphic\_A , italic\_P start\_POSTSUBSCRIPT italic\_r end\_POSTSUBSCRIPT , italic\_P , italic\_P start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , italic\_γ ⟩, where 𝒳𝒳\mathcal{X}caligraphic\_X and 𝒜𝒜\mathcal{A}caligraphic\_A are the state and action spaces, Pr(x,a)subscript𝑃𝑟𝑥𝑎P\_{r}(x,a)italic\_P start\_POSTSUBSCRIPT italic\_r end\_POSTSUBSCRIPT ( italic\_x , italic\_a ) is the distribution of the bounded random variable r(x,a)∈[0,Rmax]𝑟𝑥𝑎0subscript𝑅r(x,a)\in[0,R\_{\max}]italic\_r ( italic\_x , italic\_a ) ∈ [ 0 , italic\_R start\_POSTSUBSCRIPT roman\_max end\_POSTSUBSCRIPT ] of the immediate reward of taking action a𝑎aitalic\_a in state x𝑥xitalic\_x, P(⋅\|x,a)P(\cdot\|x,a)italic\_P ( ⋅ \| italic\_x , italic\_a ) is the transition probability distribution, P0:𝒳→[0,1]:subscript𝑃0𝒳→01P\_{0}\mathrel{\mathop{:}}\mathcal{X}\rightarrow[0,1]italic\_P start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT : caligraphic\_X → [ 0 , 1 ] is the initial state distribution, and γ∈[0,1)𝛾01\gamma\in[0,1)italic\_γ ∈ [ 0 , 1 ) is the discounting factor. A (stationary) policy π:𝒳×𝒜→[0,1]:𝜋𝒳𝒜→01\pi\mathrel{\mathop{:}}\mathcal{X}\times\mathcal{A}\rightarrow[0,1]italic\_π : caligraphic\_X × caligraphic\_A → [ 0 , 1 ] is a stochastic mapping from states to actions, with π(a\|x)𝜋conditional𝑎𝑥\pi(a\|x)italic\_π ( italic\_a \| italic\_x ) being the probability of taking action a𝑎aitalic\_a in state x𝑥xitalic\_x. We denote by Pπsuperscript𝑃𝜋P^{\pi}italic\_P start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT the state transition of the Markov chain induced by policy π𝜋\piitalic\_π, i.e., Pπ(xt+1\|xt)=∑a∈𝒜π(a\|xt)P(xt+1\|xt,a)superscript𝑃𝜋conditionalsubscript𝑥𝑡1subscript𝑥𝑡subscript𝑎𝒜𝜋conditional𝑎subscript𝑥𝑡𝑃conditionalsubscript𝑥𝑡1subscript𝑥𝑡𝑎P^{\pi}(x\_{t+1}\|x\_{t})=\sum\_{a\in\mathcal{A}}\pi(a\|x\_{t})P(x\_{t+1}\|x\_{t},a)italic\_P start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_t + 1 end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) = ∑ start\_POSTSUBSCRIPT italic\_a ∈ caligraphic\_A end\_POSTSUBSCRIPT italic\_π ( italic\_a \| italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) italic\_P ( italic\_x start\_POSTSUBSCRIPT italic\_t + 1 end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a ). We denote by ξ=(x0,a0,r0,…,xT−1,aT−1,rT−1,xT)𝜉subscript𝑥0subscript𝑎0subscript𝑟0…subscript𝑥𝑇1subscript𝑎𝑇1subscript𝑟𝑇1subscript𝑥𝑇\xi=(x\_{0},a\_{0},r\_{0},\ldots,x\_{T-1},a\_{T-1},r\_{T-1},x\_{T})italic\_ξ = ( italic\_x start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , italic\_r start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , … , italic\_x start\_POSTSUBSCRIPT italic\_T - 1 end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_T - 1 end\_POSTSUBSCRIPT , italic\_r start\_POSTSUBSCRIPT italic\_T - 1 end\_POSTSUBSCRIPT , italic\_x start\_POSTSUBSCRIPT italic\_T end\_POSTSUBSCRIPT ) a T𝑇Titalic\_T-step trajectory generated by policy π𝜋\piitalic\_π, and by R0:T−1(ξ)=∑t=0T−1γtrtsubscript𝑅:0𝑇1𝜉superscriptsubscript𝑡0𝑇1superscript𝛾𝑡subscript𝑟𝑡R\_{0\mathrel{\mathop{:}}T-1}(\xi)=\sum\_{t=0}^{T-1}\gamma^{t}r\_{t}italic\_R start\_POSTSUBSCRIPT 0 : italic\_T - 1 end\_POSTSUBSCRIPT ( italic\_ξ ) = ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT italic\_r start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT the return of trajectory ξ𝜉\xiitalic\_ξ. Note that in ξ𝜉\xiitalic\_ξ, x0∼P0similar-tosubscript𝑥0subscript𝑃0x\_{0}\sim P\_{0}italic\_x start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT ∼ italic\_P start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT, and ∀t∈{1,…,T−1}for-all𝑡1…𝑇1\forall t\in\{1,\ldots,T-1\}∀ italic\_t ∈ { 1 , … , italic\_T - 1 }, at∼π(⋅\|xt)a\_{t}\sim\pi(\cdot\|x\_{t})italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ∼ italic\_π ( ⋅ \| italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ), xt+1∼P(⋅\|xt,at)x\_{t+1}\sim P(\cdot\|x\_{t},a\_{t})italic\_x start\_POSTSUBSCRIPT italic\_t + 1 end\_POSTSUBSCRIPT ∼ italic\_P ( ⋅ \| italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ), and rt∼Pr(⋅\|xt,at)r\_{t}\sim P\_{r}(\cdot\|x\_{t},a\_{t})italic\_r start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ∼ italic\_P start\_POSTSUBSCRIPT italic\_r end\_POSTSUBSCRIPT ( ⋅ \| italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ). These distributions together define Pξπsubscriptsuperscript𝑃𝜋𝜉P^{\pi}\_{\xi}italic\_P start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT, i.e., the distribution of trajectory ξ𝜉\xiitalic\_ξ. We evaluate a policy π𝜋\piitalic\_π by the expectation of the return of the T𝑇Titalic\_T-step trajectories it generates, i.e., ρTπ=𝔼ξ∼Pξπ[R0:T−1(ξ)]superscriptsubscript𝜌𝑇𝜋subscript𝔼similar-to𝜉subscriptsuperscript𝑃𝜋𝜉delimited-[]subscript𝑅:0𝑇1𝜉\rho\_{T}^{\pi}=\mathbb{E}\_{\xi\sim P^{\pi}\_{\xi}}\big{[}R\_{0\mathrel{\mathop{:}}T-1}(\xi)\big{]}italic\_ρ start\_POSTSUBSCRIPT italic\_T end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT = blackboard\_E start\_POSTSUBSCRIPT italic\_ξ ∼ italic\_P start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_R start\_POSTSUBSCRIPT 0 : italic\_T - 1 end\_POSTSUBSCRIPT ( italic\_ξ ) ]. If we set T𝑇Titalic\_T to be of order O(1/(1−γ))𝑂11𝛾O\big{(}1/(1-\gamma)\big{)}italic\_O ( 1 / ( 1 - italic\_γ ) ), then ρTπsuperscriptsubscript𝜌𝑇𝜋\rho\_{T}^{\pi}italic\_ρ start\_POSTSUBSCRIPT italic\_T end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT would be a good approximation of the infinite-horizon performance ρ∞πsuperscriptsubscript𝜌𝜋\rho\_{\infty}^{\pi}italic\_ρ start\_POSTSUBSCRIPT ∞ end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT. Throughout the paper, we assume that T𝑇Titalic\_T has been selected such that ρTπ≈ρ∞πsuperscriptsubscript𝜌𝑇𝜋superscriptsubscript𝜌𝜋\rho\_{T}^{\pi}\approx\rho\_{\infty}^{\pi}italic\_ρ start\_POSTSUBSCRIPT italic\_T end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ≈ italic\_ρ start\_POSTSUBSCRIPT ∞ end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT, and thus, we refer to ρπ=ρTπsuperscript𝜌𝜋superscriptsubscript𝜌𝑇𝜋\rho^{\pi}=\rho\_{T}^{\pi}italic\_ρ start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT = italic\_ρ start\_POSTSUBSCRIPT italic\_T end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT as the performance of policy π𝜋\piitalic\_π. We further define the value (action-value) function of a policy π𝜋\piitalic\_π at each state x𝑥xitalic\_x (state-action pair (x,a)𝑥𝑎(x,a)( italic\_x , italic\_a )), denoted by Vπ(x)superscript𝑉𝜋𝑥V^{\pi}(x)italic\_V start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ( italic\_x ) (Qπ(x,a)superscript𝑄𝜋𝑥𝑎Q^{\pi}(x,a)italic\_Q start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ( italic\_x , italic\_a )), as the expectation of the return of a T𝑇Titalic\_T-step trajectory generated by starting at state x𝑥xitalic\_x (state-action pair (x,a)𝑥𝑎(x,a)( italic\_x , italic\_a )), and then following policy π𝜋\piitalic\_π. Note that ρπ=𝔼x∼P0[Vπ(x)]superscript𝜌𝜋subscript𝔼similar-to𝑥subscript𝑃0delimited-[]superscript𝑉𝜋𝑥\rho^{\pi}=\mathbb{E}\_{x\sim P\_{0}}\big{[}V^{\pi}(x)\big{]}italic\_ρ start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT = blackboard\_E start\_POSTSUBSCRIPT italic\_x ∼ italic\_P start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_V start\_POSTSUPERSCRIPT italic\_π end\_POSTSUPERSCRIPT ( italic\_x ) ]. Note that the contextual bandit setting is a special case of the setting described above, where T=1𝑇1T=1italic\_T = 1, and as a result, the context is sampled from P0subscript𝑃0P\_{0}italic\_P start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT and there is no dynamic P𝑃Pitalic\_P. ### 2.2 Off-policy Evaluation Problem The off-policy evaluation (OPE) problem is when we are given a set of T𝑇Titalic\_T-step trajectories 𝒟={ξ(i)}i=1n𝒟superscriptsubscriptsuperscript𝜉𝑖𝑖1𝑛\mathcal{D}=\{\xi^{(i)}\}\_{i=1}^{n}caligraphic\_D = { italic\_ξ start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT } start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT independently generated by the behavior policy πbsubscript𝜋𝑏\pi\_{b}italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT,111The results of this paper can be easily extended to the case that the trajectories are generated by multiple behavior policies. and the goal is to have a good estimate of the performance of the evaluation policy πesubscript𝜋𝑒\pi\_{e}italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT. We consider the estimator ρ^πesuperscript^𝜌subscript𝜋𝑒\hat{\rho}^{\pi\_{e}}over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT good if it has low mean square error (MSE), i.e., \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| MSE(ρπe,ρ^πe)=△𝔼Pξπb[(ρπe−ρ^πe)2].superscript△MSEsuperscript𝜌subscript𝜋𝑒superscript^𝜌subscript𝜋𝑒subscript𝔼subscriptsuperscript𝑃subscript𝜋𝑏𝜉delimited-[]superscriptsuperscript𝜌subscript𝜋𝑒superscript^𝜌subscript𝜋𝑒2\text{MSE}(\rho^{\pi\_{e}},\hat{\rho}^{\pi\_{e}})\stackrel{{\scriptstyle\triangle}}{{=}}\mathbb{E}\_{P^{\pi\_{b}}\_{\xi}}\big{[}(\rho^{\pi\_{e}}-\hat{\rho}^{\pi\_{e}})^{2}\big{]}.MSE ( italic\_ρ start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT , over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ) start\_RELOP SUPERSCRIPTOP start\_ARG = end\_ARG start\_ARG △ end\_ARG end\_RELOP blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ ( italic\_ρ start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT - over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ] . \| \| (1) \| We make the following standard regularity assumption: ###### Assumption 1 (Absolute Continuity). For all state-action pairs (x,a)∈𝒳×𝒜𝑥𝑎𝒳𝒜(x,a)\in\mathcal{X}\times\mathcal{A}( italic\_x , italic\_a ) ∈ caligraphic\_X × caligraphic\_A, if πb(a\|x)=0subscript𝜋𝑏conditional𝑎𝑥0\pi\_{b}(a\|x)=0italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a \| italic\_x ) = 0 then πe(a\|x)=0subscript𝜋𝑒conditional𝑎𝑥0\pi\_{e}(a\|x)=0italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT ( italic\_a \| italic\_x ) = 0. In order to quantify the mismatch between the behavior and evaluation policies in generating a trajectory, we define cumulative importance ratio as follows. For each T𝑇Titalic\_T-step trajectory ξ∈𝒟𝜉𝒟\xi\in\mathcal{D}italic\_ξ ∈ caligraphic\_D, the cumulative importance ratio from time step t1subscript𝑡1t\_{1}italic\_t start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT to time step t2subscript𝑡2t\_{2}italic\_t start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, where both t1subscript𝑡1t\_{1}italic\_t start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT and t2subscript𝑡2t\_{2}italic\_t start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT are in {0,…,T}0…𝑇\{0,\ldots,T\}{ 0 , … , italic\_T }, is ωt1,t2=1subscript𝜔subscript𝑡1subscript𝑡21\omega\_{t\_{1},t\_{2}}=1italic\_ω start\_POSTSUBSCRIPT italic\_t start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_t start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT = 1 if t1>t2subscript𝑡1subscript𝑡2t\_{1}>t\_{2}italic\_t start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT > italic\_t start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, and is ωt1,t2=∏τ=t1t2πe(aτ\|xτ)πb(aτ\|xτ)subscript𝜔subscript𝑡1subscript𝑡2superscriptsubscriptproduct𝜏subscript𝑡1subscript𝑡2subscript𝜋𝑒conditionalsubscript𝑎𝜏subscript𝑥𝜏subscript𝜋𝑏conditionalsubscript𝑎𝜏subscript𝑥𝜏\omega\_{t\_{1},t\_{2}}=\prod\_{\tau=t\_{1}}^{t\_{2}}\frac{\pi\_{e}(a\_{\tau}\|x\_{\tau})}{\pi\_{b}(a\_{\tau}\|x\_{\tau})}italic\_ω start\_POSTSUBSCRIPT italic\_t start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_t start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT = ∏ start\_POSTSUBSCRIPT italic\_τ = italic\_t start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT divide start\_ARG italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT ( italic\_a start\_POSTSUBSCRIPT italic\_τ end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUBSCRIPT italic\_τ end\_POSTSUBSCRIPT ) end\_ARG start\_ARG italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a start\_POSTSUBSCRIPT italic\_τ end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUBSCRIPT italic\_τ end\_POSTSUBSCRIPT ) end\_ARG, otherwise. In case the behavior policy πbsubscript𝜋𝑏\pi\_{b}italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT is unknown, we define ω^t1,t2subscript^𝜔subscript𝑡1subscript𝑡2\widehat{\omega}\_{t\_{1},t\_{2}}over^ start\_ARG italic\_ω end\_ARG start\_POSTSUBSCRIPT italic\_t start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_t start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT exactly as ωt1,t2subscript𝜔subscript𝑡1subscript𝑡2\omega\_{t\_{1},t\_{2}}italic\_ω start\_POSTSUBSCRIPT italic\_t start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_t start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT, with πbsubscript𝜋𝑏\pi\_{b}italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT replaced by its approximation π^bsubscript^𝜋𝑏\widehat{\pi}\_{b}over^ start\_ARG italic\_π end\_ARG start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT. Under Assumption [1](#Thmassumption1 "Assumption 1 (Absolute Continuity). ‣ 2.2 Off-policy Evaluation Problem ‣ 2 Preliminaries ‣ More Robust Doubly Robust Off-policy Evaluation"), it is easy to see that ρπe=𝔼Pξπe[∑t=0T−1γtrt]=𝔼Pξπb[∑t=0T−1γtω0:trt]superscript𝜌subscript𝜋𝑒subscript𝔼superscriptsubscript𝑃𝜉subscript𝜋𝑒delimited-[]superscriptsubscript𝑡0𝑇1superscript𝛾𝑡subscript𝑟𝑡subscript𝔼superscriptsubscript𝑃𝜉subscript𝜋𝑏delimited-[]superscriptsubscript𝑡0𝑇1superscript𝛾𝑡subscript𝜔:0𝑡subscript𝑟𝑡\rho^{\pi\_{e}}=\mathbb{E}\_{P\_{\xi}^{\pi\_{e}}}[\sum\_{t=0}^{T-1}\gamma^{t}r\_{t}]=\mathbb{E}\_{P\_{\xi}^{\pi\_{b}}}[\sum\_{t=0}^{T-1}\gamma^{t}\omega\_{0\mathrel{\mathop{:}}t}r\_{t}]italic\_ρ start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT = blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT [ ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT italic\_r start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ] = blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT [ ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUBSCRIPT 0 : italic\_t end\_POSTSUBSCRIPT italic\_r start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ]. Similar equalities hold for the value and action-value functions of πesubscript𝜋𝑒\pi\_{e}italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT, i.e., Vπe(x)=𝔼Pξπe[∑t=0T−1γtrt\|x0=x]=𝔼Pξπb[∑t=0T−1γtω0:trt\|x0=x]superscript𝑉subscript𝜋𝑒𝑥subscript𝔼subscriptsuperscript𝑃subscript𝜋𝑒𝜉delimited-[]conditionalsuperscriptsubscript𝑡0𝑇1superscript𝛾𝑡subscript𝑟𝑡subscript𝑥0𝑥subscript𝔼superscriptsubscript𝑃𝜉subscript𝜋𝑏delimited-[]conditionalsuperscriptsubscript𝑡0𝑇1superscript𝛾𝑡subscript𝜔:0𝑡subscript𝑟𝑡subscript𝑥0𝑥V^{\pi\_{e}}(x)=\mathbb{E}\_{P^{\pi\_{e}}\_{\xi}}[\sum\_{t=0}^{T-1}\gamma^{t}r\_{t}\|x\_{0}=x]=\mathbb{E}\_{P\_{\xi}^{\pi\_{b}}}[\sum\_{t=0}^{T-1}\gamma^{t}\omega\_{0\mathrel{\mathop{:}}t}r\_{t}\|x\_{0}=x]italic\_V start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x ) = blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT italic\_r start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT = italic\_x ] = blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT [ ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUBSCRIPT 0 : italic\_t end\_POSTSUBSCRIPT italic\_r start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT = italic\_x ] and Qπe(x,a)=𝔼Pξπe[∑t=0T−1γtrt\|x0=x,a0=a]=𝔼Pξπb[∑t=0T−1γtω0:trt\|x0=x,a0=a]superscript𝑄subscript𝜋𝑒𝑥𝑎subscript𝔼superscriptsubscript𝑃𝜉subscript𝜋𝑒delimited-[]formulae-sequenceconditionalsuperscriptsubscript𝑡0𝑇1superscript𝛾𝑡subscript𝑟𝑡subscript𝑥0𝑥subscript𝑎0𝑎subscript𝔼superscriptsubscript𝑃𝜉subscript𝜋𝑏delimited-[]formulae-sequenceconditionalsuperscriptsubscript𝑡0𝑇1superscript𝛾𝑡subscript𝜔:0𝑡subscript𝑟𝑡subscript𝑥0𝑥subscript𝑎0𝑎Q^{\pi\_{e}}(x,a)=\mathbb{E}\_{P\_{\xi}^{\pi\_{e}}}[\sum\_{t=0}^{T-1}\gamma^{t}r\_{t}\|x\_{0}=x,a\_{0}=a]=\mathbb{E}\_{P\_{\xi}^{\pi\_{b}}}[\sum\_{t=0}^{T-1}\gamma^{t}\omega\_{0\mathrel{\mathop{:}}t}r\_{t}\|x\_{0}=x,a\_{0}=a]italic\_Q start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x , italic\_a ) = blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT [ ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT italic\_r start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT = italic\_x , italic\_a start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT = italic\_a ] = blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT [ ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUBSCRIPT 0 : italic\_t end\_POSTSUBSCRIPT italic\_r start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT = italic\_x , italic\_a start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT = italic\_a ]. 3 Existing Approaches to OPE ----------------------------- The objective of MRDR is to learn the model part of a DR estimator by minimizing its variance. MRDR is a variation of DR with a DM loss function derived from minimizing the DR’s variance and is built on the top of IS and DM. Therefore, before stating our main results in Section [4](#S4 "4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation"), we first provide a brief overview of these popular approaches. ### 3.1 Direct Estimators The idea of the direct method (DM) is to first learn a model of the system and then use it to estimate the performance of the evaluation policy πesubscript𝜋𝑒\pi\_{e}italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT. In the case of bandits, this model is the mean reward of each pair of context and arm, and in RL it is either the mean reward r(x,a)𝑟𝑥𝑎r(x,a)italic\_r ( italic\_x , italic\_a ) and state transition P(⋅\|x,a)P(\cdot\|x,a)italic\_P ( ⋅ \| italic\_x , italic\_a ), or the value (action-value) V(x)𝑉𝑥V(x)italic\_V ( italic\_x ) (Q(x,a)𝑄𝑥𝑎Q(x,a)italic\_Q ( italic\_x , italic\_a )) function. In either case, if we select a good representation for the quantities that need to be learned, and our dataset222Note that we shall use separate datasets for learning the model in DM and evaluating the policy. contains sufficient number of the states and actions relevant to the evaluation of πesubscript𝜋𝑒\pi\_{e}italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT, then the DM estimator has low variance and small bias, and thus, has the potential to outperform the estimators resulted from other approaches. As mentioned in Section [1](#S1 "1 Introduction ‣ More Robust Doubly Robust Off-policy Evaluation"), an important issue that has been neglected in the previous work on off-policy evaluation in RL is the loss function used in estimating the model in DM. As pointed out by Dudík et al. ([2011](#bib.bib6)), the direct approach has a problem if the model is estimated without the knowledge of the evaluation policy. This is because the distribution of the states and actions that are visited under the evaluation policy should be included in the loss function of the direct approach. In other words, if upon learning a model, we have no information about the evaluation policy (or the distribution of the evaluation policies), then it is not clear how to design the DM’s loss function (perhaps a uniform distribution over the states and actions would be the most reasonable). Therefore, in this paper, we assume that the evaluation policy is known prior to learning the model.333Our results can be extended to the case that the distribution of the evaluation policies is known prior to learning the model. In their DM and DR experiments, both Jiang & Li ([2016](#bib.bib11)) and Thomas & Brunskill ([2016](#bib.bib31)) learn the MDP model, r(x,a)𝑟𝑥𝑎r(x,a)italic\_r ( italic\_x , italic\_a ) and P(⋅\|x,a)P(\cdot\|x,a)italic\_P ( ⋅ \| italic\_x , italic\_a ), although all the model learning discussion in Thomas & Brunskill ([2016](#bib.bib31)) is about the reward of the evaluation policy πesubscript𝜋𝑒\pi\_{e}italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT at every step t𝑡titalic\_t along the T𝑇Titalic\_T-step trajectory, i.e., rπe(x,t)superscript𝑟subscript𝜋𝑒𝑥𝑡r^{\pi\_{e}}(x,t)italic\_r start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x , italic\_t ). More generally, in off-policy actor-critic algorithms (such as the Reactor algorithm proposed in Gruslys et al. [2017](#bib.bib8)), where one can view the gradient estimation part as an off-policy value evaluation problem, the DM state-action value function model is learned by minimizing the Bellman residual in an off-policy setting (Precup et al., [2000b](#bib.bib23); Munos et al., [2016](#bib.bib18); Geist & Scherrer, [2014](#bib.bib7)). However, neither of these three approaches incorporate the design of the DM loss function into the primary objective, perhaps because they consider the setting in which the model is learned independently. Our approach to DM in RL: In this paper, we propose to learn Qπesuperscript𝑄subscript𝜋𝑒Q^{\pi\_{e}}italic\_Q start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT, the action-value function of the evaluation policy πesubscript𝜋𝑒\pi\_{e}italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT, and then use it to evaluate its performance as \| \| \| \| \| --- \| --- \| --- \| \| \| ρ^DMπe=1n∑i=1n∑a∈𝒜πe(a\|x0(i))Q^πe(x0(i),a;βn\).superscriptsubscript^𝜌DMsubscript𝜋𝑒1𝑛superscriptsubscript𝑖1𝑛subscript𝑎𝒜subscript𝜋𝑒conditional𝑎subscriptsuperscript𝑥𝑖0superscript^𝑄subscript𝜋𝑒subscriptsuperscript𝑥𝑖0𝑎subscriptsuperscript𝛽𝑛\hat{\rho}\_{\text{DM}}^{\pi\_{e}}=\frac{1}{n}\sum\_{i=1}^{n}\sum\_{a\in\mathcal{A}}\pi\_{e}(a\|x^{(i)}\_{0})\widehat{Q}^{\pi\_{e}}(x^{(i)}\_{0},a;\beta^{\}\_{n}).over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUBSCRIPT DM end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT = divide start\_ARG 1 end\_ARG start\_ARG italic\_n end\_ARG ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT ∑ start\_POSTSUBSCRIPT italic\_a ∈ caligraphic\_A end\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT ( italic\_a \| italic\_x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT ) over^ start\_ARG italic\_Q end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , italic\_a ; italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ) . \| \| We model Qπesuperscript𝑄subscript𝜋𝑒Q^{\pi\_{e}}italic\_Q start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT using a parameterized class of functions with parameter β∈ℝκ𝛽superscriptℝ𝜅\beta\in\mathbb{R}^{\kappa}italic\_β ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_κ end\_POSTSUPERSCRIPT and learn β𝛽\betaitalic\_β by solving the following weighted MSE problem \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| β\∈arg⁡minβ∈ℝκ⁡𝔼(x,a)∼μπe[(Qπe(x,a)−Q^πe(x,a;β))2],superscript𝛽subscript𝛽superscriptℝ𝜅subscript𝔼similar-to𝑥𝑎subscript𝜇subscript𝜋𝑒delimited-[]superscriptsuperscript𝑄subscript𝜋𝑒𝑥𝑎superscript^𝑄subscript𝜋𝑒𝑥𝑎𝛽2\beta^{\}\in\arg\min\_{\beta\in\mathbb{R}^{\kappa}}\mathbb{E}\_{(x,a)\sim\mu\_{\pi\_{e}}}\Big{[}\big{(}Q^{\pi\_{e}}(x,a)-\widehat{Q}^{\pi\_{e}}(x,a;\beta)\big{)}^{2}\Big{]},italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ∈ roman\_arg roman\_min start\_POSTSUBSCRIPT italic\_β ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_κ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT blackboard\_E start\_POSTSUBSCRIPT ( italic\_x , italic\_a ) ∼ italic\_μ start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ ( italic\_Q start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x , italic\_a ) - over^ start\_ARG italic\_Q end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x , italic\_a ; italic\_β ) ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ] , \| \| (2) \| where μπesubscript𝜇subscript𝜋𝑒\mu\_{\pi\_{e}}italic\_μ start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT is the γ𝛾\gammaitalic\_γ-discounted horizon-T𝑇Titalic\_T state-action occupancy of πesubscript𝜋𝑒\pi\_{e}italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT, i.e., μπe(x,a)=1−γ1−γT∑t=0T−1γt𝔼Pξπe[𝟏{xt\mu\_{\pi\_{e}}(x,a)=\frac{1-\gamma}{1-\gamma^{T}}\sum\_{t=0}^{T-1}\gamma^{t}\mathbb{E}\_{P^{\pi\_{e}}\_{\xi}}\big{[}\mathbf{1}\{x\_{t}italic\_μ start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_x , italic\_a ) = divide start\_ARG 1 - italic\_γ end\_ARG start\_ARG 1 - italic\_γ start\_POSTSUPERSCRIPT italic\_T end\_POSTSUPERSCRIPT end\_ARG ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ bold\_1 { italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT =x,at=a}]=x,a\_{t}=a\}\big{]}= italic\_x , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT = italic\_a } ] and 𝟏{⋅}1⋅\mathbf{1}\{\cdot\}bold\_1 { ⋅ } is the indicator function. Since the actions in the data set 𝒟𝒟\mathcal{D}caligraphic\_D are generated by πbsubscript𝜋𝑏\pi\_{b}italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT, we rewrite the objective function of the optimization problem ([2](#S3.E2 "2 ‣ 3.1 Direct Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")) as \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| ∑t=0T−1γt𝔼Pξπb[ω0:t(R¯t:T−1(ξ)−Q^πe(xt,at;β))2],superscriptsubscript𝑡0𝑇1superscript𝛾𝑡subscript𝔼subscriptsuperscript𝑃subscript𝜋𝑏𝜉delimited-[]subscript𝜔:0𝑡superscriptsubscript¯𝑅:𝑡𝑇1𝜉superscript^𝑄subscript𝜋𝑒subscript𝑥𝑡subscript𝑎𝑡𝛽2\sum\_{t=0}^{T-1}\gamma^{t}\mathbb{E}\_{P^{\pi\_{b}}\_{\xi}}\Big{[}\omega\_{0\mathrel{\mathop{:}}t}\big{(}\bar{R}\_{t\mathrel{\mathop{:}}T-1}(\xi)-\widehat{Q}^{\pi\_{e}}(x\_{t},a\_{t};\beta)\big{)}^{2}\Big{]},∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_ω start\_POSTSUBSCRIPT 0 : italic\_t end\_POSTSUBSCRIPT ( over¯ start\_ARG italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT ( italic\_ξ ) - over^ start\_ARG italic\_Q end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ; italic\_β ) ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ] , \| \| (3) \| where R¯t:T−1(ξ)=∑τ=tT−1γτ−tωt+1:τr(xτ,aτ)subscript¯𝑅:𝑡𝑇1𝜉superscriptsubscript𝜏𝑡𝑇1superscript𝛾𝜏𝑡subscript𝜔:𝑡1𝜏𝑟subscript𝑥𝜏subscript𝑎𝜏\bar{R}\_{t\mathrel{\mathop{:}}T-1}(\xi)=\sum\_{\tau=t}^{T-1}\gamma^{\tau-t}\omega\_{t+1\mathrel{\mathop{:}}\tau}r(x\_{\tau},a\_{\tau})over¯ start\_ARG italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT ( italic\_ξ ) = ∑ start\_POSTSUBSCRIPT italic\_τ = italic\_t end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_τ - italic\_t end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUBSCRIPT italic\_t + 1 : italic\_τ end\_POSTSUBSCRIPT italic\_r ( italic\_x start\_POSTSUBSCRIPT italic\_τ end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_τ end\_POSTSUBSCRIPT ) is the Monte Carlo estimate of Qπe(xt,at)superscript𝑄subscript𝜋𝑒subscript𝑥𝑡subscript𝑎𝑡Q^{\pi\_{e}}(x\_{t},a\_{t})italic\_Q start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ). The proof of the equivalence of the objective functions ([2](#S3.E2 "2 ‣ 3.1 Direct Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")) and ([3](#S3.E3 "3 ‣ 3.1 Direct Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")) can be found in Appendix [A](#A1 "Appendix A Proofs of Section 3.1 ‣ More Robust Doubly Robust Off-policy Evaluation"). We obtain βn\subscriptsuperscript𝛽𝑛\beta^{\}\_{n}italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT by solving the sample average approximation (SAA) of ([3](#S3.E3 "3 ‣ 3.1 Direct Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")), i.e., \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| βn\∈arg⁡minβ∈ℝκ∑t=0T−1γt⋅1n∑i=1nω0:t(i)subscriptsuperscript𝛽𝑛subscript𝛽superscriptℝ𝜅superscriptsubscript𝑡0𝑇1⋅superscript𝛾𝑡1𝑛superscriptsubscript𝑖1𝑛subscriptsuperscript𝜔𝑖:0𝑡\displaystyle\beta^{\}\_{n}\in\arg\min\_{\beta\in\mathbb{R}^{\kappa}}\sum\_{t=0}^{T-1}\gamma^{t}\cdot\frac{1}{n}\sum\_{i=1}^{n}\omega^{(i)}\_{0\mathrel{\mathop{:}}t}italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ∈ roman\_arg roman\_min start\_POSTSUBSCRIPT italic\_β ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_κ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ⋅ divide start\_ARG 1 end\_ARG start\_ARG italic\_n end\_ARG ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_t end\_POSTSUBSCRIPT \| [R¯t:T−1(ξ(i))\displaystyle\big{[}\bar{R}\_{t\mathrel{\mathop{:}}T-1}(\xi^{(i)})[ over¯ start\_ARG italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT ( italic\_ξ start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT ) \| \| \| \| \| −Q^πe(xt(i),at(i);β)]2.\displaystyle-\widehat{Q}^{\pi\_{e}}(x^{(i)}\_{t},a^{(i)}\_{t};\beta)\big{]}^{2}.- over^ start\_ARG italic\_Q end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ; italic\_β ) ] start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT . \| \| (4) \| Since the SAA estimator ([3.1](#S3.Ex2 "3.1 Direct Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")) is unbiased, for large enough n𝑛nitalic\_n, βn\→β\→subscriptsuperscript𝛽𝑛superscript𝛽\beta^{\}\_{n}\rightarrow\beta^{\}italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT → italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT almost surely. We define the bias of our DM estimator at each state-action pair as Δ(x,a)=Q^πe(x,a;β)−Qπe(x,a)Δ𝑥𝑎superscript^𝑄subscript𝜋𝑒𝑥𝑎𝛽superscript𝑄subscript𝜋𝑒𝑥𝑎\Delta(x,a)=\widehat{Q}^{\pi\_{e}}(x,a;\beta)-Q^{\pi\_{e}}(x,a)roman\_Δ ( italic\_x , italic\_a ) = over^ start\_ARG italic\_Q end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x , italic\_a ; italic\_β ) - italic\_Q start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x , italic\_a ). Note that in contextual bandits with deterministic evaluation policy, the SAA ([3.1](#S3.Ex2 "3.1 Direct Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")) may be written as the weighted least square (WLS) problem \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| βn\∈1n∑i=1n𝟏{πe(xi)=ai}πb(ai\|xi)(r(xi,ai)−Q^(xi,ai;β))2,subscriptsuperscript𝛽𝑛1𝑛superscriptsubscript𝑖1𝑛1subscript𝜋𝑒subscript𝑥𝑖subscript𝑎𝑖subscript𝜋𝑏conditionalsubscript𝑎𝑖subscript𝑥𝑖superscript𝑟subscript𝑥𝑖subscript𝑎𝑖^𝑄subscript𝑥𝑖subscript𝑎𝑖𝛽2\beta^{\}\_{n}\in\frac{1}{n}\sum\_{i=1}^{n}\frac{\mathbf{1}\{\pi\_{e}(x\_{i})=a\_{i}\}}{\pi\_{b}(a\_{i}\|x\_{i})}\big{(}r(x\_{i},a\_{i})-\widehat{Q}(x\_{i},a\_{i};\beta)\big{)}^{2},italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ∈ divide start\_ARG 1 end\_ARG start\_ARG italic\_n end\_ARG ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT divide start\_ARG bold\_1 { italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) = italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT } end\_ARG start\_ARG italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) end\_ARG ( italic\_r ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) - over^ start\_ARG italic\_Q end\_ARG ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ; italic\_β ) ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT , \| \| (5) \| with weights 1/πb(ai\|xi)1subscript𝜋𝑏conditionalsubscript𝑎𝑖subscript𝑥𝑖1/\pi\_{b}(a\_{i}\|x\_{i})1 / italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) for the actions consistent with πesubscript𝜋𝑒\pi\_{e}italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT. ### 3.2 Importance Sampling Estimators Another common approach to off-policy evaluation in RL is to use importance sampling (IS) to estimate the performance of the evaluation policy, i.e., \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| ρ^ISπe=1n∑i=1nω0:T−1(i)∑t=0T−1γtrt(i)=1n∑i=1nω0:T−1(i)R0:T−1(i),superscriptsubscript^𝜌ISsubscript𝜋𝑒1𝑛superscriptsubscript𝑖1𝑛subscriptsuperscript𝜔𝑖:0𝑇1superscriptsubscript𝑡0𝑇1superscript𝛾𝑡subscriptsuperscript𝑟𝑖𝑡1𝑛superscriptsubscript𝑖1𝑛subscriptsuperscript𝜔𝑖:0𝑇1superscriptsubscript𝑅:0𝑇1𝑖\hat{\rho}\_{\text{IS}}^{\pi\_{e}}=\frac{1}{n}\sum\_{i=1}^{n}\omega^{(i)}\_{0\mathrel{\mathop{:}}T-1}\sum\_{t=0}^{T-1}\gamma^{t}r^{(i)}\_{t}=\frac{1}{n}\sum\_{i=1}^{n}\omega^{(i)}\_{0\mathrel{\mathop{:}}T-1}R\_{0\mathrel{\mathop{:}}T-1}^{(i)},over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUBSCRIPT IS end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT = divide start\_ARG 1 end\_ARG start\_ARG italic\_n end\_ARG ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_T - 1 end\_POSTSUBSCRIPT ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT italic\_r start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT = divide start\_ARG 1 end\_ARG start\_ARG italic\_n end\_ARG ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_T - 1 end\_POSTSUBSCRIPT italic\_R start\_POSTSUBSCRIPT 0 : italic\_T - 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT , \| \| (6) \| where ω0:T−1(i)subscriptsuperscript𝜔𝑖:0𝑇1\omega^{(i)}\_{0\mathrel{\mathop{:}}T-1}italic\_ω start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_T - 1 end\_POSTSUBSCRIPT and rt(i)superscriptsubscript𝑟𝑡𝑖r\_{t}^{(i)}italic\_r start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT are the cumulative importance ratio and reward at step t𝑡titalic\_t of trajectory ξ(i)∈𝒟superscript𝜉𝑖𝒟\xi^{(i)}\in\mathcal{D}italic\_ξ start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT ∈ caligraphic\_D, respectively, and R0:T−1(i)=R0:T−1(ξ(i))superscriptsubscript𝑅:0𝑇1𝑖subscript𝑅:0𝑇1superscript𝜉𝑖R\_{0\mathrel{\mathop{:}}T-1}^{(i)}=R\_{0\mathrel{\mathop{:}}T-1}(\xi^{(i)})italic\_R start\_POSTSUBSCRIPT 0 : italic\_T - 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT = italic\_R start\_POSTSUBSCRIPT 0 : italic\_T - 1 end\_POSTSUBSCRIPT ( italic\_ξ start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT ). Under Assumption [1](#Thmassumption1 "Assumption 1 (Absolute Continuity). ‣ 2.2 Off-policy Evaluation Problem ‣ 2 Preliminaries ‣ More Robust Doubly Robust Off-policy Evaluation"), the IS estimator ([6](#S3.E6 "6 ‣ 3.2 Importance Sampling Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")) is unbiased. A variant of IS that often has less variance, while still unbiased, is step-wise importance sampling (step-IS), i.e., \| \| \| \| \| --- \| --- \| --- \| \| \| ρ^step-ISπe=1n∑i=1n∑t=0T−1γtω0:t(i)rt(i).superscriptsubscript^𝜌step-ISsubscript𝜋𝑒1𝑛superscriptsubscript𝑖1𝑛superscriptsubscript𝑡0𝑇1superscript𝛾𝑡subscriptsuperscript𝜔𝑖:0𝑡subscriptsuperscript𝑟𝑖𝑡\hat{\rho}\_{\text{step-IS}}^{\pi\_{e}}=\frac{1}{n}\sum\_{i=1}^{n}\sum\_{t=0}^{T-1}\gamma^{t}\omega^{(i)}\_{0\mathrel{\mathop{:}}t}\,\,r^{(i)}\_{t}.over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUBSCRIPT step-IS end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT = divide start\_ARG 1 end\_ARG start\_ARG italic\_n end\_ARG ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_t end\_POSTSUBSCRIPT italic\_r start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT . \| \| If the behavior policy πbsubscript𝜋𝑏\pi\_{b}italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT is unknown, which is the case in many applications, then either πbsubscript𝜋𝑏\pi\_{b}italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT or the importance ratio ω=πe/πb𝜔subscript𝜋𝑒subscript𝜋𝑏\omega=\pi\_{e}/\pi\_{b}italic\_ω = italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT / italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT needs to be estimated, and thus, IS may no longer be unbiased. In this case, the bias of IS and step-IS are \|𝔼Pξπe[δ0:T−1(ξ)R0:T−1(ξ)]\|subscript𝔼subscriptsuperscript𝑃subscript𝜋𝑒𝜉delimited-[]subscript𝛿:0𝑇1𝜉subscript𝑅:0𝑇1𝜉\left\|\mathbb{E}\_{P^{\pi\_{e}}\_{\xi}}\left[\delta\_{0\mathrel{\mathop{:}}T-1}(\xi)R\_{0\mathrel{\mathop{:}}T-1}(\xi)\right]\right\|\| blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_δ start\_POSTSUBSCRIPT 0 : italic\_T - 1 end\_POSTSUBSCRIPT ( italic\_ξ ) italic\_R start\_POSTSUBSCRIPT 0 : italic\_T - 1 end\_POSTSUBSCRIPT ( italic\_ξ ) ] \| and \|∑t=0T−1γt𝔼Pξπe[δ0:t(ξ)rt]\|superscriptsubscript𝑡0𝑇1superscript𝛾𝑡subscript𝔼subscriptsuperscript𝑃subscript𝜋𝑒𝜉delimited-[]subscript𝛿:0𝑡𝜉subscript𝑟𝑡\left\|\sum\_{t=0}^{T-1}\gamma^{t}\mathbb{E}\_{P^{\pi\_{e}}\_{\xi}}\left[\delta\_{0\mathrel{\mathop{:}}t}(\xi)r\_{t}\right]\right\|\| ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_δ start\_POSTSUBSCRIPT 0 : italic\_t end\_POSTSUBSCRIPT ( italic\_ξ ) italic\_r start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ] \|, respectively, where δ0:t(ξ)=1−λ0:t(ξ)=1−∏τ=0tπb(aτ\|xτ)π^b(aτ\|xτ)subscript𝛿:0𝑡𝜉1subscript𝜆:0𝑡𝜉1superscriptsubscriptproduct𝜏0𝑡subscript𝜋𝑏conditionalsubscript𝑎𝜏subscript𝑥𝜏subscript^𝜋𝑏conditionalsubscript𝑎𝜏subscript𝑥𝜏\delta\_{0\mathrel{\mathop{:}}t}(\xi)=1-\lambda\_{0\mathrel{\mathop{:}}t}(\xi)=1-\prod\_{\tau=0}^{t}\frac{\pi\_{b}(a\_{\tau}\|x\_{\tau})}{\widehat{\pi}\_{b}(a\_{\tau}\|x\_{\tau})}italic\_δ start\_POSTSUBSCRIPT 0 : italic\_t end\_POSTSUBSCRIPT ( italic\_ξ ) = 1 - italic\_λ start\_POSTSUBSCRIPT 0 : italic\_t end\_POSTSUBSCRIPT ( italic\_ξ ) = 1 - ∏ start\_POSTSUBSCRIPT italic\_τ = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT divide start\_ARG italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a start\_POSTSUBSCRIPT italic\_τ end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUBSCRIPT italic\_τ end\_POSTSUBSCRIPT ) end\_ARG start\_ARG over^ start\_ARG italic\_π end\_ARG start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a start\_POSTSUBSCRIPT italic\_τ end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUBSCRIPT italic\_τ end\_POSTSUBSCRIPT ) end\_ARG, with π^bsubscript^𝜋𝑏\widehat{\pi}\_{b}over^ start\_ARG italic\_π end\_ARG start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT being our approximation of πbsubscript𝜋𝑏\pi\_{b}italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT (see the proofs in Appendix [B](#A2 "Appendix B Proofs of Section 3.2 ‣ More Robust Doubly Robust Off-policy Evaluation")). Note that when πbsubscript𝜋𝑏\pi\_{b}italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT is known, i.e., π^b=πbsubscript^𝜋𝑏subscript𝜋𝑏\widehat{\pi}\_{b}=\pi\_{b}over^ start\_ARG italic\_π end\_ARG start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT = italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT, we have δ0:t=0subscript𝛿:0𝑡0\delta\_{0\mathrel{\mathop{:}}t}=0italic\_δ start\_POSTSUBSCRIPT 0 : italic\_t end\_POSTSUBSCRIPT = 0, and the bias of both IS and step-IS would be zero. Although the unbiasedness of IS estimators is desirable for certain applications such as safety (Thomas et al., [2015b](#bib.bib33)), their high variance (even in the step-wise case), which grows exponentially in the horizon T𝑇Titalic\_T, restricts their applications. This is why another variant of IS, called weighted importance sampling (WIS), and particularly its step-wise version, i.e., \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| ρ^WISπesuperscriptsubscript^𝜌WISsubscript𝜋𝑒\displaystyle\hat{\rho}\_{\text{WIS}}^{\pi\_{e}}over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUBSCRIPT WIS end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT \| =∑i=1nω0:T−1(i)∑i=1nω0:T−1(i)∑t=0T−1γtrt(i)=∑i=1nω0:T−1(i)R0:T−1(i)∑i=1nω0:T−1(i),absentsuperscriptsubscript𝑖1𝑛subscriptsuperscript𝜔𝑖:0𝑇1superscriptsubscript𝑖1𝑛subscriptsuperscript𝜔𝑖:0𝑇1superscriptsubscript𝑡0𝑇1superscript𝛾𝑡subscriptsuperscript𝑟𝑖𝑡superscriptsubscript𝑖1𝑛subscriptsuperscript𝜔𝑖:0𝑇1subscriptsuperscript𝑅𝑖:0𝑇1superscriptsubscript𝑖1𝑛subscriptsuperscript𝜔𝑖:0𝑇1\displaystyle=\sum\_{i=1}^{n}\frac{\omega^{(i)}\_{0\mathrel{\mathop{:}}T-1}}{\sum\_{i=1}^{n}\omega^{(i)}\_{0\mathrel{\mathop{:}}T-1}}\sum\_{t=0}^{T-1}\gamma^{t}r^{(i)}\_{t}=\sum\_{i=1}^{n}\frac{\omega^{(i)}\_{0\mathrel{\mathop{:}}T-1}R^{(i)}\_{0\mathrel{\mathop{:}}T-1}}{\sum\_{i=1}^{n}\omega^{(i)}\_{0\mathrel{\mathop{:}}T-1}},= ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT divide start\_ARG italic\_ω start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_T - 1 end\_POSTSUBSCRIPT end\_ARG start\_ARG ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_T - 1 end\_POSTSUBSCRIPT end\_ARG ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT italic\_r start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT = ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT divide start\_ARG italic\_ω start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_T - 1 end\_POSTSUBSCRIPT italic\_R start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_T - 1 end\_POSTSUBSCRIPT end\_ARG start\_ARG ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_T - 1 end\_POSTSUBSCRIPT end\_ARG , \| \| \| \| ρ^step-WISπesuperscriptsubscript^𝜌step-WISsubscript𝜋𝑒\displaystyle\hat{\rho}\_{\text{step-WIS}}^{\pi\_{e}}over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUBSCRIPT step-WIS end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT \| =∑i=1n∑t=0T−1γtω0:t(i)rt(i)∑i=1nω0:t(i),absentsuperscriptsubscript𝑖1𝑛superscriptsubscript𝑡0𝑇1superscript𝛾𝑡subscriptsuperscript𝜔𝑖:0𝑡subscriptsuperscript𝑟𝑖𝑡superscriptsubscript𝑖1𝑛subscriptsuperscript𝜔𝑖:0𝑡\displaystyle=\sum\_{i=1}^{n}\sum\_{t=0}^{T-1}\gamma^{t}\frac{\omega^{(i)}\_{0\mathrel{\mathop{:}}t}\,\,r^{(i)}\_{t}}{\sum\_{i=1}^{n}\omega^{(i)}\_{0\mathrel{\mathop{:}}t}},= ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT divide start\_ARG italic\_ω start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_t end\_POSTSUBSCRIPT italic\_r start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT end\_ARG start\_ARG ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_t end\_POSTSUBSCRIPT end\_ARG , \| \| is considered more practical, especially where being biased is not crucial. The WIS estimators are biased but consistent and have lower variance than their IS counterparts. ### 3.3 Doubly Robust Estimators Doubly robust (DR) estimators that combine DM and IS were first developed for regression (e.g., Cassel et al. [1976](#bib.bib5)), brought to contextual bandits by Dudík et al. ([2011](#bib.bib6)), and to RL by Jiang & Li ([2016](#bib.bib11)) and Thomas & Brunskill ([2016](#bib.bib31)). The DR estimator for RL is defined as \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| \| \| ρ^DRπe(β)superscriptsubscript^𝜌DRsubscript𝜋𝑒𝛽\displaystyle\hat{\rho}\_{\text{DR}}^{\pi\_{e}}(\beta)over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUBSCRIPT DR end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_β ) \| =1n∑i=1n∑t=0T−1[γtω0:t(i)rt(i)\displaystyle=\frac{1}{n}\sum\_{i=1}^{n}\sum\_{t=0}^{T-1}\Big{[}\gamma^{t}\omega\_{0\mathrel{\mathop{:}}t}^{(i)}r^{(i)}\_{t}= divide start\_ARG 1 end\_ARG start\_ARG italic\_n end\_ARG ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT [ italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUBSCRIPT 0 : italic\_t end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT italic\_r start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT \| \| (7) \| \| \| \| −γt(ω0:t(i)Q^πe(xt(i),at(i);β)−ω0:t−1(i)V^πe(xt(i);β))].\displaystyle-\gamma^{t}\big{(}\omega\_{0\mathrel{\mathop{:}}t}^{(i)}\widehat{Q}^{\pi\_{e}}(x^{(i)}\_{t},a^{(i)}\_{t};\beta)-\omega\_{0\mathrel{\mathop{:}}t-1}^{(i)}\widehat{V}^{\pi\_{e}}(x^{(i)}\_{t};\beta)\big{)}\Big{]}.- italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_ω start\_POSTSUBSCRIPT 0 : italic\_t end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT over^ start\_ARG italic\_Q end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ; italic\_β ) - italic\_ω start\_POSTSUBSCRIPT 0 : italic\_t - 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT over^ start\_ARG italic\_V end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ; italic\_β ) ) ] . \| \| Eq. [7](#S3.E7 "7 ‣ 3.3 Doubly Robust Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation") clearly shows that a DR estimator contains both the cumulative importance ratio ω𝜔\omegaitalic\_ω (IS part) and the model estimates V^πesuperscript^𝑉subscript𝜋𝑒\widehat{V}^{\pi\_{e}}over^ start\_ARG italic\_V end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT and Q^πesuperscript^𝑄subscript𝜋𝑒\widehat{Q}^{\pi\_{e}}over^ start\_ARG italic\_Q end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT (DM part). Note that the IS part of the DR estimator ([7](#S3.E7 "7 ‣ 3.3 Doubly Robust Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")) is based on step-wise IS. Thomas & Brunskill ([2016](#bib.bib31)) derived a DR estimator whose IS part is based on step-wise WIS. In this paper, we use step-wise IS for the IS part of our DR-based estimators, but our results can be easily extended to other IS estimators. The bias of a DR estimator is the product of that of DM and IS, and thus, DR is unbiased whenever either IS or DM is unbiased. This is what the term “doubly robust” refers to. The bias of the DR estimator ([7](#S3.E7 "7 ‣ 3.3 Doubly Robust Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")) is \|𝔼Pξπe[∑t=0T−1γtλ0:t−1(ξ)δt(ξ)Δ(xt,at)]\|subscript𝔼subscriptsuperscript𝑃subscript𝜋𝑒𝜉delimited-[]superscriptsubscript𝑡0𝑇1superscript𝛾𝑡subscript𝜆:0𝑡1𝜉subscript𝛿𝑡𝜉Δsubscript𝑥𝑡subscript𝑎𝑡\|\mathbb{E}\_{P^{\pi\_{e}}\_{\xi}}[\sum\_{t=0}^{T-1}\gamma^{t}\lambda\_{0\mathrel{\mathop{:}}t-1}(\xi)\delta\_{t}(\xi)\Delta(x\_{t},a\_{t})]\|\| blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT italic\_λ start\_POSTSUBSCRIPT 0 : italic\_t - 1 end\_POSTSUBSCRIPT ( italic\_ξ ) italic\_δ start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ( italic\_ξ ) roman\_Δ ( italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) ] \| (see the proofs in Appendix [C](#A3 "Appendix C Proofs of Section 3.3 ‣ More Robust Doubly Robust Off-policy Evaluation")), and thus, it would be zero if either Δ(xt,at)Δsubscript𝑥𝑡subscript𝑎𝑡\Delta(x\_{t},a\_{t})roman\_Δ ( italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) or δt(ξ)subscript𝛿𝑡𝜉\delta\_{t}(\xi)italic\_δ start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ( italic\_ξ ) is zero. As discussed in Section [3.2](#S3.SS2 "3.2 Importance Sampling Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation"), if πbsubscript𝜋𝑏\pi\_{b}italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT is known, δt=0subscript𝛿𝑡0\delta\_{t}=0italic\_δ start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT = 0 and the DR estimator ([7](#S3.E7 "7 ‣ 3.3 Doubly Robust Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")) is unbiased. Throughout this paper, we assume that πbsubscript𝜋𝑏\pi\_{b}italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT is known, and thus, DR is unbiased as long as it uses unbiased variants of IS. However, our proposed estimator described in Section [4](#S4 "4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation") can be extended to the case that πbsubscript𝜋𝑏\pi\_{b}italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT is unknown. 4 More Robust Doubly Robust Estimators --------------------------------------- In this section, we present our class of more robust doubly robust (MRDR) estimators. The main idea of MRDR is to learn the DM parameter of a DR estimator, β∈ℝκ𝛽superscriptℝ𝜅\beta\in\mathbb{R}^{\kappa}italic\_β ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_κ end\_POSTSUPERSCRIPT, by minimizing its variance. In other words, MRDR is a variation of DR with a DM loss function derived from minimizing the DR’s variance. As mentioned earlier, we assume that the behavior policy πbsubscript𝜋𝑏\pi\_{b}italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT is known, and thus, both IS (step-IS) and DR estimators are unbiased. This means that our MRDR estimator is also unbiased, and since it is the result of minimizing the DR’s variance, it has the lowest MSE among all the DR estimators. ### 4.1 MRDR Estimators for Contextual Bandits Before presenting MRDR for RL, we first formulate it in the contextual bandit setting. We follow the setting of Dudík et al. ([2011](#bib.bib6)) and define the DR estimator as \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| ρ^DRπe(β)=1n∑i=1nπe(ai\|xi)π^b(ai\|xi)(r(xi,ai)−Q^(xi,ai;β))+V^πe(xi;β),superscriptsubscript^𝜌DRsubscript𝜋𝑒𝛽1𝑛superscriptsubscript𝑖1𝑛subscript𝜋𝑒conditionalsubscript𝑎𝑖subscript𝑥𝑖subscript^𝜋𝑏conditionalsubscript𝑎𝑖subscript𝑥𝑖𝑟subscript𝑥𝑖subscript𝑎𝑖^𝑄subscript𝑥𝑖subscript𝑎𝑖𝛽superscript^𝑉subscript𝜋𝑒subscript𝑥𝑖𝛽\begin{split}\hat{\rho}\_{\text{DR}}^{\pi\_{e}}(\beta)=\frac{1}{n}\sum\_{i=1}^{n}\frac{\pi\_{e}(a\_{i}\|x\_{i})}{\widehat{\pi}\_{b}(a\_{i}\|x\_{i})}&\big{(}r(x\_{i},a\_{i})-\widehat{Q}(x\_{i},a\_{i};\beta)\big{)}\\ &\quad+\widehat{V}^{\pi\_{e}}(x\_{i};\beta),\end{split}start\_ROW start\_CELL over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUBSCRIPT DR end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_β ) = divide start\_ARG 1 end\_ARG start\_ARG italic\_n end\_ARG ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT divide start\_ARG italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT ( italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) end\_ARG start\_ARG over^ start\_ARG italic\_π end\_ARG start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) end\_ARG end\_CELL start\_CELL ( italic\_r ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) - over^ start\_ARG italic\_Q end\_ARG ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ; italic\_β ) ) end\_CELL end\_ROW start\_ROW start\_CELL end\_CELL start\_CELL + over^ start\_ARG italic\_V end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ; italic\_β ) , end\_CELL end\_ROW \| \| (8) \| where Q^(x,a;β)≈Q(x,a)=𝔼Pr[r(x,a)]^𝑄𝑥𝑎𝛽𝑄𝑥𝑎subscript𝔼subscript𝑃𝑟delimited-[]𝑟𝑥𝑎\widehat{Q}(x,a;\beta)\approx Q(x,a)=\mathbb{E}\_{P\_{r}}[r(x,a)]over^ start\_ARG italic\_Q end\_ARG ( italic\_x , italic\_a ; italic\_β ) ≈ italic\_Q ( italic\_x , italic\_a ) = blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUBSCRIPT italic\_r end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_r ( italic\_x , italic\_a ) ] and V^πe(x;β)=𝔼a∼πe[Q^(x,a;β)]superscript^𝑉subscript𝜋𝑒𝑥𝛽subscript𝔼similar-to𝑎subscript𝜋𝑒delimited-[]^𝑄𝑥𝑎𝛽\widehat{V}^{\pi\_{e}}(x;\beta)=\mathbb{E}\_{a\sim\pi\_{e}}[\widehat{Q}(x,a;\beta)]over^ start\_ARG italic\_V end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x ; italic\_β ) = blackboard\_E start\_POSTSUBSCRIPT italic\_a ∼ italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ over^ start\_ARG italic\_Q end\_ARG ( italic\_x , italic\_a ; italic\_β ) ]. We further define the DM bias Δ(x,a)=Δ𝑥𝑎absent\Delta(x,a)=roman\_Δ ( italic\_x , italic\_a ) = Q^(x,a;β)−Q(x,a)^𝑄𝑥𝑎𝛽𝑄𝑥𝑎\widehat{Q}(x,a;\beta)-Q(x,a)over^ start\_ARG italic\_Q end\_ARG ( italic\_x , italic\_a ; italic\_β ) - italic\_Q ( italic\_x , italic\_a ), and error in learning the behavior policy δ(x,a)=1−λ(x,a)=1−πb(a\|x)π^b(a\|x)𝛿𝑥𝑎1𝜆𝑥𝑎1subscript𝜋𝑏conditional𝑎𝑥subscript^𝜋𝑏conditional𝑎𝑥\delta(x,a)=1-\lambda(x,a)=1-\frac{\pi\_{b}(a\|x)}{\widehat{\pi}\_{b}(a\|x)}italic\_δ ( italic\_x , italic\_a ) = 1 - italic\_λ ( italic\_x , italic\_a ) = 1 - divide start\_ARG italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a \| italic\_x ) end\_ARG start\_ARG over^ start\_ARG italic\_π end\_ARG start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a \| italic\_x ) end\_ARG. Proposition [1](#Thmproposition1 "Proposition 1. ‣ 4.1 MRDR Estimators for Contextual Bandits ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation") proves the bias and variance of DR for stochastic evaluation policy πesubscript𝜋𝑒\pi\_{e}italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT. Note that the results stated in Theorems 1 and 2 in Dudík et al. ([2011](#bib.bib6)) are only for deterministic πesubscript𝜋𝑒\pi\_{e}italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT. ###### Proposition 1. The bias and variance of the DR estimator ([8](#S4.E8 "8 ‣ 4.1 MRDR Estimators for Contextual Bandits ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation")) for stochastic πesubscript𝜋𝑒\pi\_{e}italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT may be written as \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| 𝐵𝑖𝑎𝑠(ρ^𝐷𝑅πe)𝐵𝑖𝑎𝑠superscriptsubscript^𝜌𝐷𝑅subscript𝜋𝑒\displaystyle\text{Bias}(\hat{\rho}\_{\text{DR}}^{\pi\_{e}})Bias ( over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUBSCRIPT DR end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ) \| =\|ρπe−𝔼Pξπb[ρ^𝐷𝑅πe]\|=\|𝔼Pξπe[δ(x,a)Δ(x,a)]\|,absentsuperscript𝜌subscript𝜋𝑒subscript𝔼subscriptsuperscript𝑃subscript𝜋𝑏𝜉delimited-[]superscriptsubscript^𝜌𝐷𝑅subscript𝜋𝑒subscript𝔼subscriptsuperscript𝑃subscript𝜋𝑒𝜉delimited-[]𝛿𝑥𝑎Δ𝑥𝑎\displaystyle=\left\|\rho^{\pi\_{e}}-\mathbb{E}\_{P^{\pi\_{b}}\_{\xi}}[\hat{\rho}\_{\text{DR}}^{\pi\_{e}}]\right\|=\left\|\mathbb{E}\_{P^{\pi\_{e}}\_{\xi}}\left[\delta(x,a)\Delta(x,a)\right]\right\|,= \| italic\_ρ start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT - blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUBSCRIPT DR end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ] \| = \| blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_δ ( italic\_x , italic\_a ) roman\_Δ ( italic\_x , italic\_a ) ] \| , \| \| \| \| n𝕍Pξπb(ρ^𝐷𝑅πe)𝑛subscript𝕍subscriptsuperscript𝑃subscript𝜋𝑏𝜉superscriptsubscript^𝜌𝐷𝑅subscript𝜋𝑒\displaystyle n\mathbb{V}\_{P^{\pi\_{b}}\_{\xi}}(\hat{\rho}\_{\text{DR}}^{\pi\_{e}})italic\_n blackboard\_V start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUBSCRIPT DR end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ) \| =𝔼Pξπb[ω^(x,a)2(r(x,a)−Q(x,a))2]absentsubscript𝔼subscriptsuperscript𝑃subscript𝜋𝑏𝜉delimited-[]^𝜔superscript𝑥𝑎2superscript𝑟𝑥𝑎𝑄𝑥𝑎2\displaystyle=\mathbb{E}\_{P^{\pi\_{b}}\_{\xi}}\left[\widehat{\omega}(x,a)^{2}\big{(}r(x,a)-Q(x,a)\big{)}^{2}\right]= blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ over^ start\_ARG italic\_ω end\_ARG ( italic\_x , italic\_a ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ( italic\_r ( italic\_x , italic\_a ) - italic\_Q ( italic\_x , italic\_a ) ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ] \| \| \| \| \| +𝕍P0(𝔼πe[Q(x,a)+δ(x,a)Δ(x,a)])subscript𝕍subscript𝑃0subscript𝔼subscript𝜋𝑒delimited-[]𝑄𝑥𝑎𝛿𝑥𝑎Δ𝑥𝑎\displaystyle+\mathbb{V}\_{P\_{0}}\left(\mathbb{E}\_{\pi\_{e}}\left[Q(x,a)+\delta(x,a)\Delta(x,a)\right]\right)+ blackboard\_V start\_POSTSUBSCRIPT italic\_P start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( blackboard\_E start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_Q ( italic\_x , italic\_a ) + italic\_δ ( italic\_x , italic\_a ) roman\_Δ ( italic\_x , italic\_a ) ] ) \| \| \| \| \| +𝔼P0,πe[ω(x,a)(1−δ(x,a))2Δ(x,a)2\displaystyle+\mathbb{E}\_{P\_{0},\pi\_{e}}\Big{[}\omega(x,a)\big{(}1-\delta(x,a)\big{)}^{2}\Delta(x,a)^{2}+ blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_ω ( italic\_x , italic\_a ) ( 1 - italic\_δ ( italic\_x , italic\_a ) ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT roman\_Δ ( italic\_x , italic\_a ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT \| \| \| \| \| −𝔼πe[(1−δ(x,a))Δ(x,a)]2].\displaystyle-\mathbb{E}\_{\pi\_{e}}\big{[}\big{(}1-\delta(x,a)\big{)}\Delta(x,a)\big{]}^{2}\Big{]}.- blackboard\_E start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ ( 1 - italic\_δ ( italic\_x , italic\_a ) ) roman\_Δ ( italic\_x , italic\_a ) ] start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ] . \| \| ###### Proof. See Appendix [D](#A4 "Appendix D Proofs of Section 4.1 ‣ More Robust Doubly Robust Off-policy Evaluation"). ∎ As expected from a DR estimator, Proposition [1](#Thmproposition1 "Proposition 1. ‣ 4.1 MRDR Estimators for Contextual Bandits ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation") shows that ([8](#S4.E8 "8 ‣ 4.1 MRDR Estimators for Contextual Bandits ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation")) is unbiased if either its DM part is unbiased, Δ=0Δ0\Delta=0roman\_Δ = 0, or its IS part is unbiased, δ=0𝛿0\delta=0italic\_δ = 0. When the behavior policy πbsubscript𝜋𝑏\pi\_{b}italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT is known, and thus, δ(x,a)=0𝛿𝑥𝑎0\delta(x,a)=0italic\_δ ( italic\_x , italic\_a ) = 0 for all x𝑥xitalic\_x and a𝑎aitalic\_a, the variance of ([8](#S4.E8 "8 ‣ 4.1 MRDR Estimators for Contextual Bandits ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation")) in Proposition [1](#Thmproposition1 "Proposition 1. ‣ 4.1 MRDR Estimators for Contextual Bandits ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation") may be written as \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| \| \| \| n𝕍Pξπb(ρ^DRπe)=𝔼Pξπb[ω(x,a)2(r(x,a)−Q(x,a))2]𝑛subscript𝕍subscriptsuperscript𝑃subscript𝜋𝑏𝜉superscriptsubscript^𝜌DRsubscript𝜋𝑒subscript𝔼subscriptsuperscript𝑃subscript𝜋𝑏𝜉delimited-[]𝜔superscript𝑥𝑎2superscript𝑟𝑥𝑎𝑄𝑥𝑎2\displaystyle n\mathbb{V}\_{P^{\pi\_{b}}\_{\xi}}(\hat{\rho}\_{\text{DR}}^{\pi\_{e}})=\mathbb{E}\_{P^{\pi\_{b}}\_{\xi}}\left[\omega(x,a)^{2}\big{(}r(x,a)-Q(x,a)\big{)}^{2}\right]italic\_n blackboard\_V start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUBSCRIPT DR end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ) = blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_ω ( italic\_x , italic\_a ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ( italic\_r ( italic\_x , italic\_a ) - italic\_Q ( italic\_x , italic\_a ) ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ] \| \| (9) \| \| \| \| +𝕍P0[Vπe(x)]+𝔼P0,πe[ω(x,a)Δ(x,a)2−𝔼πe[Δ(x,a)]2].subscript𝕍subscript𝑃0delimited-[]superscript𝑉subscript𝜋𝑒𝑥subscript𝔼subscript𝑃0subscript𝜋𝑒delimited-[]𝜔𝑥𝑎Δsuperscript𝑥𝑎2subscript𝔼subscript𝜋𝑒superscriptdelimited-[]Δ𝑥𝑎2\displaystyle+\mathbb{V}\_{P\_{0}}\big{[}V^{\pi\_{e}}(x)\big{]}+\mathbb{E}\_{P\_{0},\pi\_{e}}\Big{[}\omega(x,a)\Delta(x,a)^{2}-\mathbb{E}\_{\pi\_{e}}\big{[}\Delta(x,a)\big{]}^{2}\Big{]}.+ blackboard\_V start\_POSTSUBSCRIPT italic\_P start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_V start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x ) ] + blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_ω ( italic\_x , italic\_a ) roman\_Δ ( italic\_x , italic\_a ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT - blackboard\_E start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ roman\_Δ ( italic\_x , italic\_a ) ] start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ] . \| \| Unfortunately, the variance formulation ([9](#S4.E9 "9 ‣ 4.1 MRDR Estimators for Contextual Bandits ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation")) is not suitable for our MRDR method, because its derivative w.r.t. β𝛽\betaitalic\_β contains a term Δ(x,a)=Q^(x,a)−Q(x,a)Δ𝑥𝑎^𝑄𝑥𝑎𝑄𝑥𝑎\Delta(x,a)=\widehat{Q}(x,a)-Q(x,a)roman\_Δ ( italic\_x , italic\_a ) = over^ start\_ARG italic\_Q end\_ARG ( italic\_x , italic\_a ) - italic\_Q ( italic\_x , italic\_a ) that cannot be estimated from samples as the true expected reward Q𝑄Qitalic\_Q is unknown. To address this issue, we derive a new formulations of the variance in Theorem [1](#Thmtheorem1 "Theorem 1. ‣ 4.1 MRDR Estimators for Contextual Bandits ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation"), whose derivative does not contain such terms. ###### Theorem 1. The variance of the DR estimator ([8](#S4.E8 "8 ‣ 4.1 MRDR Estimators for Contextual Bandits ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation")) for stochastic πesubscript𝜋𝑒\pi\_{e}italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT may be written as the following two forms: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| \| n𝕍Pξπb(ρ^𝐷𝑅πe)=𝔼Pξπb[ω(x,a)(𝔼πe[ω(x,a′)Q^(x,a′;β)2]\displaystyle n\mathbb{V}\_{P^{\pi\_{b}}\_{\xi}}(\hat{\rho}^{\pi\_{e}}\_{\text{DR}})=\mathbb{E}\_{P^{\pi\_{b}}\_{\xi}}\Big{[}\omega(x,a)\Big{(}\mathbb{E}\_{\pi\_{e}}\left[\omega(x,a^{\prime})\widehat{Q}(x,a^{\prime};\beta)^{2}\right]italic\_n blackboard\_V start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT DR end\_POSTSUBSCRIPT ) = blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_ω ( italic\_x , italic\_a ) ( blackboard\_E start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_ω ( italic\_x , italic\_a start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ) over^ start\_ARG italic\_Q end\_ARG ( italic\_x , italic\_a start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ; italic\_β ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ] \| \| \| \| \| −V^πe(x;β)2−2r(x,a)(ω(x,a)Q^(x,a;β)−V^πe(x;β)))\displaystyle-\widehat{V}^{\pi\_{e}}(x;\beta)^{2}-2r(x,a)\big{(}\omega(x,a)\widehat{Q}(x,a;\beta)-\widehat{V}^{\pi\_{e}}(x;\beta)\big{)}\Big{)}- over^ start\_ARG italic\_V end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x ; italic\_β ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT - 2 italic\_r ( italic\_x , italic\_a ) ( italic\_ω ( italic\_x , italic\_a ) over^ start\_ARG italic\_Q end\_ARG ( italic\_x , italic\_a ; italic\_β ) - over^ start\_ARG italic\_V end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x ; italic\_β ) ) ) \| \| \| \| \| +ω(x,a)2r(x,a)2−𝔼πe[r(x,a)]2]+𝕍P0(𝔼πe[r(x,a)]),\displaystyle+\omega(x,a)^{2}r(x,a)^{2}-\mathbb{E}\_{\pi\_{e}}[r(x,a)]^{2}\Big{]}+\mathbb{V}\_{P\_{0}}\big{(}\mathbb{E}\_{\pi\_{e}}[r(x,a)]\big{)},+ italic\_ω ( italic\_x , italic\_a ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT italic\_r ( italic\_x , italic\_a ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT - blackboard\_E start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_r ( italic\_x , italic\_a ) ] start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ] + blackboard\_V start\_POSTSUBSCRIPT italic\_P start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( blackboard\_E start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_r ( italic\_x , italic\_a ) ] ) , \| \| (10) \| \| \| \| n𝕍Pξπb(ρ^𝐷𝑅πe)=𝔼Pξπb[ω(x,a)qβ(x,a,r)⊤Ωπb(x)qβ(x,a,r)]⏞J(β)𝑛subscript𝕍subscriptsuperscript𝑃subscript𝜋𝑏𝜉subscriptsuperscript^𝜌subscript𝜋𝑒𝐷𝑅superscript⏞subscript𝔼subscriptsuperscript𝑃subscript𝜋𝑏𝜉delimited-[]𝜔𝑥𝑎subscript𝑞𝛽superscript𝑥𝑎𝑟topsubscriptΩsubscript𝜋𝑏𝑥subscript𝑞𝛽𝑥𝑎𝑟𝐽𝛽\displaystyle n\mathbb{V}\_{P^{\pi\_{b}}\_{\xi}}(\hat{\rho}^{\pi\_{e}}\_{\text{DR}})=\overbrace{\mathbb{E}\_{P^{\pi\_{b}}\_{\xi}}\big{[}\omega(x,a)q\_{\beta}(x,a,r)^{\top}\Omega\_{\pi\_{b}}(x)q\_{\beta}(x,a,r)\big{]}}^{J(\beta)}italic\_n blackboard\_V start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT DR end\_POSTSUBSCRIPT ) = over⏞ start\_ARG blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_ω ( italic\_x , italic\_a ) italic\_q start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT ( italic\_x , italic\_a , italic\_r ) start\_POSTSUPERSCRIPT ⊤ end\_POSTSUPERSCRIPT roman\_Ω start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_x ) italic\_q start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT ( italic\_x , italic\_a , italic\_r ) ] end\_ARG start\_POSTSUPERSCRIPT italic\_J ( italic\_β ) end\_POSTSUPERSCRIPT \| \| \| \| \| +C,𝐶\displaystyle\qquad\qquad\quad\;+C,+ italic\_C , \| \| (11) \| where Ωπb(x)=diag[1/πb(a\|x)]a∈𝒜−ee⊤subscriptnormal-Ωsubscript𝜋𝑏𝑥normal-diagsubscriptdelimited-[]1subscript𝜋𝑏conditional𝑎𝑥𝑎𝒜𝑒superscript𝑒top\Omega\_{\pi\_{b}}(x)=\mathrm{diag}\big{[}1/\pi\_{b}(a\|x)\big{]}\_{a\in\mathcal{A}}-ee^{\top}roman\_Ω start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_x ) = roman\_diag [ 1 / italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a \| italic\_x ) ] start\_POSTSUBSCRIPT italic\_a ∈ caligraphic\_A end\_POSTSUBSCRIPT - italic\_e italic\_e start\_POSTSUPERSCRIPT ⊤ end\_POSTSUPERSCRIPT is a positive semi-definite matrix (see Proposition [6](#Thmproposition6 "Proposition 6. ‣ D.3 Proof of Proposition 6 ‣ Appendix D Proofs of Section 4.1 ‣ More Robust Doubly Robust Off-policy Evaluation") in Appendix [D](#A4 "Appendix D Proofs of Section 4.1 ‣ More Robust Doubly Robust Off-policy Evaluation") fot the proof) with e=[1,…,1]⊤𝑒superscript1normal-…1 tope=[1,\ldots,1]^{\top}italic\_e = [ 1 , … , 1 ] start\_POSTSUPERSCRIPT ⊤ end\_POSTSUPERSCRIPT; qβ(x,a,r)=Dπe(x)Q¯(x;β)−𝕀(a)rsubscript𝑞𝛽𝑥𝑎𝑟subscript𝐷subscript𝜋𝑒𝑥normal-¯𝑄𝑥𝛽𝕀𝑎𝑟q\_{\beta}(x,a,r)=D\_{\pi\_{e}}(x)\bar{Q}(x;\beta)-\mathbb{I}(a)ritalic\_q start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT ( italic\_x , italic\_a , italic\_r ) = italic\_D start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_x ) over¯ start\_ARG italic\_Q end\_ARG ( italic\_x ; italic\_β ) - blackboard\_I ( italic\_a ) italic\_r a row vector with Dπe(x)=diag[πe(a\|x)]a∈𝒜subscript𝐷subscript𝜋𝑒𝑥normal-diagsubscriptdelimited-[]subscript𝜋𝑒conditional𝑎𝑥𝑎𝒜D\_{\pi\_{e}}(x)=\mathrm{diag}\big{[}\pi\_{e}(a\|x)\big{]}\_{a\in\mathcal{A}}italic\_D start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_x ) = roman\_diag [ italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT ( italic\_a \| italic\_x ) ] start\_POSTSUBSCRIPT italic\_a ∈ caligraphic\_A end\_POSTSUBSCRIPT, row vector Q¯(x;β)=[Q^(x,a;β)]a∈𝒜normal-¯𝑄𝑥𝛽subscriptdelimited-[]normal-^𝑄𝑥𝑎𝛽𝑎𝒜\bar{Q}(x;\beta)=\big{[}\widehat{Q}(x,a;\beta)\big{]}\_{a\in\mathcal{A}}over¯ start\_ARG italic\_Q end\_ARG ( italic\_x ; italic\_β ) = [ over^ start\_ARG italic\_Q end\_ARG ( italic\_x , italic\_a ; italic\_β ) ] start\_POSTSUBSCRIPT italic\_a ∈ caligraphic\_A end\_POSTSUBSCRIPT, and the row vector of indicator functions 𝕀(a)=[𝟏{a′=a}]a′∈𝒜𝕀𝑎subscriptdelimited-[]1superscript𝑎normal-′𝑎superscript𝑎normal-′𝒜\mathbb{I}(a)=\big{[}\mathbf{1}\{a^{\prime}=a\}\big{]}\_{a^{\prime}\in\mathcal{A}}blackboard\_I ( italic\_a ) = [ bold\_1 { italic\_a start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT = italic\_a } ] start\_POSTSUBSCRIPT italic\_a start\_POSTSUPERSCRIPT ′ end\_POSTSUPERSCRIPT ∈ caligraphic\_A end\_POSTSUBSCRIPT; and finally C=𝕍P0(𝔼πe[r(x,a)])−𝔼Pξπb[𝔼πe[r(x,a)]2]+𝔼Pξπb[(1+ω(x,a)−1πb2(a\|x))ω(x,a)r(x,a)2]𝐶subscript𝕍subscript𝑃0subscript𝔼subscript𝜋𝑒delimited-[]𝑟𝑥𝑎subscript𝔼subscriptsuperscript𝑃subscript𝜋𝑏𝜉delimited-[]subscript𝔼subscript𝜋𝑒superscriptdelimited-[]𝑟𝑥𝑎2subscript𝔼subscriptsuperscript𝑃subscript𝜋𝑏𝜉delimited-[]1𝜔𝑥𝑎1subscriptsuperscript𝜋2𝑏conditional𝑎𝑥𝜔𝑥𝑎𝑟superscript𝑥𝑎2C=\mathbb{V}\_{P\_{0}}\big{(}\mathbb{E}\_{\pi\_{e}}[r(x,a)]\big{)}-\mathbb{E}\_{P^{\pi\_{b}}\_{\xi}}\big{[}\mathbb{E}\_{\pi\_{e}}[r(x,a)]^{2}\big{]}+\mathbb{E}\_{P^{\pi\_{b}}\_{\xi}}\Big{[}\big{(}1+\omega(x,a)-\frac{1}{\pi^{2}\_{b}(a\|x)}\big{)}\omega(x,a)r(x,a)^{2}\Big{]}italic\_C = blackboard\_V start\_POSTSUBSCRIPT italic\_P start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( blackboard\_E start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_r ( italic\_x , italic\_a ) ] ) - blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ blackboard\_E start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_r ( italic\_x , italic\_a ) ] start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ] + blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ ( 1 + italic\_ω ( italic\_x , italic\_a ) - divide start\_ARG 1 end\_ARG start\_ARG italic\_π start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a \| italic\_x ) end\_ARG ) italic\_ω ( italic\_x , italic\_a ) italic\_r ( italic\_x , italic\_a ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ]. ###### Proof. See Appendix [D](#A4 "Appendix D Proofs of Section 4.1 ‣ More Robust Doubly Robust Off-policy Evaluation"). ∎ The significance of the variance formulations of Theorem [1](#Thmtheorem1 "Theorem 1. ‣ 4.1 MRDR Estimators for Contextual Bandits ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation") is 1) the variance of the DR estimator has no dependence on the unknown term ΔΔ\Deltaroman\_Δ, and thus, its derivative w.r.t. β𝛽\betaitalic\_β is computable, 2) the expectation in ([11](#S4.E11 "11 ‣ Theorem 1. ‣ 4.1 MRDR Estimators for Contextual Bandits ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation")) is w.r.t. Pξπbsubscriptsuperscript𝑃subscript𝜋𝑏𝜉P^{\pi\_{b}}\_{\xi}italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT, which makes it possible to replace J(β)𝐽𝛽J(\beta)italic\_J ( italic\_β ) with its unbiased SAA \| \| \| \| \| --- \| --- \| --- \| \| \| Jn(β)=1n∑i=1nω(xi,ai)qβ(xi,ai,ri)⊤Ωπb(xi)qβ(xi,ai,ri),subscript𝐽𝑛𝛽1𝑛superscriptsubscript𝑖1𝑛𝜔subscript𝑥𝑖subscript𝑎𝑖subscript𝑞𝛽superscriptsubscript𝑥𝑖subscript𝑎𝑖subscript𝑟𝑖topsubscriptΩsubscript𝜋𝑏subscript𝑥𝑖subscript𝑞𝛽subscript𝑥𝑖subscript𝑎𝑖subscript𝑟𝑖J\_{n}(\beta)=\frac{1}{n}\sum\_{i=1}^{n}\omega(x\_{i},a\_{i})q\_{\beta}(x\_{i},a\_{i},r\_{i})^{\top}\Omega\_{\pi\_{b}}(x\_{i})q\_{\beta}(x\_{i},a\_{i},r\_{i}),italic\_J start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ( italic\_β ) = divide start\_ARG 1 end\_ARG start\_ARG italic\_n end\_ARG ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT italic\_ω ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) italic\_q start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_r start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) start\_POSTSUPERSCRIPT ⊤ end\_POSTSUPERSCRIPT roman\_Ω start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) italic\_q start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_r start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) , \| \| where 𝒟={(xi,ai,ri)}i=1n𝒟superscriptsubscriptsubscript𝑥𝑖subscript𝑎𝑖subscript𝑟𝑖𝑖1𝑛\mathcal{D}=\{(x\_{i},a\_{i},r\_{i})\}\_{i=1}^{n}caligraphic\_D = { ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_r start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) } start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT is the data set generated by the behavior policy πbsubscript𝜋𝑏\pi\_{b}italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT, such that the optimizer of Jn(β)subscript𝐽𝑛𝛽J\_{n}(\beta)italic\_J start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ( italic\_β ) converges to that of J(β)𝐽𝛽J(\beta)italic\_J ( italic\_β ) almost surely, and 3) J(β)𝐽𝛽J(\beta)italic\_J ( italic\_β ) in ([11](#S4.E11 "11 ‣ Theorem 1. ‣ 4.1 MRDR Estimators for Contextual Bandits ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation")) is a convex quadratic function of qβsubscript𝑞𝛽q\_{\beta}italic\_q start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT, which in case that Q^(x,a;β)^𝑄𝑥𝑎𝛽\widehat{Q}(x,a;\beta)over^ start\_ARG italic\_Q end\_ARG ( italic\_x , italic\_a ; italic\_β ) is smooth, makes it possible to efficiently optimize Jn(β)subscript𝐽𝑛𝛽J\_{n}(\beta)italic\_J start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ( italic\_β ) with stochastic gradient descent. Moreover, when ∇βQ^(x,a;β)subscript∇𝛽^𝑄𝑥𝑎𝛽\nabla\_{\beta}\widehat{Q}(x,a;\beta)∇ start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT over^ start\_ARG italic\_Q end\_ARG ( italic\_x , italic\_a ; italic\_β ) can be explicitly written, we can obtain βn\∈arg⁡minβ⁡Jn(β)subscriptsuperscript𝛽𝑛subscript𝛽subscript𝐽𝑛𝛽\beta^{\}\_{n}\in\arg\min\_{\beta}J\_{n}(\beta)italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ∈ roman\_arg roman\_min start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT italic\_J start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ( italic\_β ), by solving the first order optimality condition ∑i=1nω(xi,ai)superscriptsubscript𝑖1𝑛𝜔subscript𝑥𝑖subscript𝑎𝑖\sum\_{i=1}^{n}\omega(x\_{i},a\_{i})∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT italic\_ω ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) qβ(xi,ai,ri)⊤Ωπb(xi)Dπe(xi)∇βQ¯(xi;β)=0subscript𝑞𝛽superscriptsubscript𝑥𝑖subscript𝑎𝑖subscript𝑟𝑖topsubscriptΩsubscript𝜋𝑏subscript𝑥𝑖subscript𝐷subscript𝜋𝑒subscript𝑥𝑖subscript∇𝛽¯𝑄subscript𝑥𝑖𝛽0q\_{\beta}(x\_{i},a\_{i},r\_{i})^{\top}\Omega\_{\pi\_{b}}(x\_{i})D\_{\pi\_{e}}(x\_{i})\nabla\_{\beta}\bar{Q}(x\_{i};\beta)=0italic\_q start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_r start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) start\_POSTSUPERSCRIPT ⊤ end\_POSTSUPERSCRIPT roman\_Ω start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) italic\_D start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) ∇ start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT over¯ start\_ARG italic\_Q end\_ARG ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ; italic\_β ) = 0. In case the evaluation policy is deterministic, the variance n𝕍Pξπb(ρ^DRπe)𝑛subscript𝕍subscriptsuperscript𝑃subscript𝜋𝑏𝜉subscriptsuperscript^𝜌subscript𝜋𝑒DRn\mathbb{V}\_{P^{\pi\_{b}}\_{\xi}}(\hat{\rho}^{\pi\_{e}}\_{\text{DR}})italic\_n blackboard\_V start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT DR end\_POSTSUBSCRIPT ) in ([1](#S4.Ex13 "Theorem 1. ‣ 4.1 MRDR Estimators for Contextual Bandits ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation")) becomes \| \| \| \| \| --- \| --- \| --- \| \| \| 𝔼Pξπb[𝟏{πe(x)=a}πb(a\|x)⋅1−πb(a\|x)πb(a\|x)(r(x,a)−Q^(x,a;β))2]⏞J(β)superscript⏞subscript𝔼subscriptsuperscript𝑃subscript𝜋𝑏𝜉delimited-[]⋅1subscript𝜋𝑒𝑥𝑎subscript𝜋𝑏conditional𝑎𝑥1subscript𝜋𝑏conditional𝑎𝑥subscript𝜋𝑏conditional𝑎𝑥superscript𝑟𝑥𝑎^𝑄𝑥𝑎𝛽2𝐽𝛽\displaystyle\overbrace{\mathbb{E}\_{P^{\pi\_{b}}\_{\xi}}\Big{[}\frac{\mathbf{1}\{\pi\_{e}(x)=a\}}{\pi\_{b}(a\|x)}\cdot\frac{1-\pi\_{b}(a\|x)}{\pi\_{b}(a\|x)}\big{(}r(x,a)-\widehat{Q}(x,a;\beta)\big{)}^{2}\Big{]}}^{J(\beta)}over⏞ start\_ARG blackboard\_E start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ divide start\_ARG bold\_1 { italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT ( italic\_x ) = italic\_a } end\_ARG start\_ARG italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a \| italic\_x ) end\_ARG ⋅ divide start\_ARG 1 - italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a \| italic\_x ) end\_ARG start\_ARG italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a \| italic\_x ) end\_ARG ( italic\_r ( italic\_x , italic\_a ) - over^ start\_ARG italic\_Q end\_ARG ( italic\_x , italic\_a ; italic\_β ) ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ] end\_ARG start\_POSTSUPERSCRIPT italic\_J ( italic\_β ) end\_POSTSUPERSCRIPT \| \| \| \| +𝕍P0(𝔼πe[r(x,a)]).subscript𝕍subscript𝑃0subscript𝔼subscript𝜋𝑒delimited-[]𝑟𝑥𝑎\displaystyle\quad\;\;+\mathbb{V}\_{P\_{0}}\big{(}\mathbb{E}\_{\pi\_{e}}[r(x,a)]\big{)}.+ blackboard\_V start\_POSTSUBSCRIPT italic\_P start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( blackboard\_E start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_r ( italic\_x , italic\_a ) ] ) . \| \| This form of J(β)𝐽𝛽J(\beta)italic\_J ( italic\_β ) allows us to find the model parameter of MRDR by solving the WLS \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| βn\∈argminβJn(β)=1n∑i=1n𝟏{πe(xi)=ai}⋅1−πb(ai\|xi)πb(ai\|xi)2(r(xi,ai)−Q^(xi,ai;β))2.subscriptsuperscript𝛽𝑛subscript𝛽subscript𝐽𝑛𝛽1𝑛superscriptsubscript𝑖1𝑛⋅1subscript𝜋𝑒subscript𝑥𝑖subscript𝑎𝑖1subscript𝜋𝑏conditionalsubscript𝑎𝑖subscript𝑥𝑖subscript𝜋𝑏superscriptconditionalsubscript𝑎𝑖subscript𝑥𝑖2superscript𝑟subscript𝑥𝑖subscript𝑎𝑖^𝑄subscript𝑥𝑖subscript𝑎𝑖𝛽2\small\begin{split}\beta^{\}\_{n}\in&\arg\min\_{\beta}J\_{n}(\beta)=\frac{1}{n}\sum\_{i=1}^{n}\mathbf{1}\{\pi\_{e}(x\_{i})=a\_{i}\}\cdot\\ &\quad\quad\frac{1-\pi\_{b}(a\_{i}\|x\_{i})}{\pi\_{b}(a\_{i}\|x\_{i})^{2}}\big{(}r(x\_{i},a\_{i})-\widehat{Q}(x\_{i},a\_{i};\beta)\big{)}^{2}.\end{split}start\_ROW start\_CELL italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ∈ end\_CELL start\_CELL roman\_arg roman\_min start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT italic\_J start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ( italic\_β ) = divide start\_ARG 1 end\_ARG start\_ARG italic\_n end\_ARG ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT bold\_1 { italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) = italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT } ⋅ end\_CELL end\_ROW start\_ROW start\_CELL end\_CELL start\_CELL divide start\_ARG 1 - italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) end\_ARG start\_ARG italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT end\_ARG ( italic\_r ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) - over^ start\_ARG italic\_Q end\_ARG ( italic\_x start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ; italic\_β ) ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT . end\_CELL end\_ROW \| \| (12) \| Comparing this WLS with that in the DM approach in [5](#S3.E5 "5 ‣ 3.1 Direct Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation"), we note that MRDR changes the weights from 1/πb1subscript𝜋𝑏1/\pi\_{b}1 / italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT to (1−πb)/πb21subscript𝜋𝑏superscriptsubscript𝜋𝑏2(1-\pi\_{b})/\pi\_{b}^{2}( 1 - italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ) / italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT, and this way increases the penalty of the samples whose actions are the same as those suggested by πesubscript𝜋𝑒\pi\_{e}italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT, but have low probability under πbsubscript𝜋𝑏\pi\_{b}italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT, and decreases the penalty of the rest of the samples. ### 4.2 MRDR Estimators for Reinforcement Learning We now present our MRDR estimator for RL. We begin with the DR estimator for RL given by ([7](#S3.E7 "7 ‣ 3.3 Doubly Robust Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")). Similar to the bandits case reported in Section [4.1](#S4.SS1 "4.1 MRDR Estimators for Contextual Bandits ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation"), we first derive a formula for the variance of the estimator ([7](#S3.E7 "7 ‣ 3.3 Doubly Robust Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")), whose derivative can be easily estimated from trajectories generated by the behavior policy. We then use this variance formulation as the objective function to find the MRDR model parameter. ###### Theorem 2. The variance of the DR estimator in ([7](#S3.E7 "7 ‣ 3.3 Doubly Robust Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")) can be written as \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| n𝕍Pξπb(ρ^𝐷𝑅πe)𝑛subscript𝕍subscriptsuperscript𝑃subscript𝜋𝑏𝜉subscriptsuperscript^𝜌subscript𝜋𝑒𝐷𝑅\displaystyle n\mathbb{V}\_{P^{\pi\_{b}}\_{\xi}}(\hat{\rho}^{\pi\_{e}}\_{\text{DR}})italic\_n blackboard\_V start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT DR end\_POSTSUBSCRIPT ) \| =∑t=0T−1𝔼ℱ0:t−1[γ2tω0:t−12𝕍ℱt:T−1(ωt(R¯t:T−1\displaystyle=\sum\_{t=0}^{T-1}\mathbb{E}\_{\mathcal{F}\_{0\mathrel{\mathop{:}}t-1}}\bigg{[}\gamma^{2t}\omega^{2}\_{0\mathrel{\mathop{:}}t-1}\mathbb{V}\_{\mathcal{F}\_{t\mathrel{\mathop{:}}T-1}}\Big{(}\omega\_{t}\big{(}\bar{R}\_{t\mathrel{\mathop{:}}T-1}= ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT blackboard\_E start\_POSTSUBSCRIPT caligraphic\_F start\_POSTSUBSCRIPT 0 : italic\_t - 1 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_γ start\_POSTSUPERSCRIPT 2 italic\_t end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_t - 1 end\_POSTSUBSCRIPT blackboard\_V start\_POSTSUBSCRIPT caligraphic\_F start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_ω start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ( over¯ start\_ARG italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT \| \| \| \| \| −Q^πe(xt,at;β)))+V^πe(xt;β)+Ct]\displaystyle-\widehat{Q}^{\pi\_{e}}(x\_{t},a\_{t};\beta)\big{)}\Big{)}+\widehat{V}^{\pi\_{e}}(x\_{t};\beta)+C\_{t}\bigg{]}- over^ start\_ARG italic\_Q end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ; italic\_β ) ) ) + over^ start\_ARG italic\_V end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ; italic\_β ) + italic\_C start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ] \| \| (13) \| \| \| \| +𝔼ℱ0:t[γ2t−2ω0:t−12𝕍ℱt+1:T−1(R¯t:T−1∣ℱt)],subscript𝔼subscriptℱ:0𝑡delimited-[]superscript𝛾2𝑡2subscriptsuperscript𝜔2:0𝑡1subscript𝕍subscriptℱ:𝑡1𝑇1conditionalsubscript¯𝑅:𝑡𝑇1subscriptℱ𝑡\displaystyle+\mathbb{E}\_{\mathcal{F}\_{0\mathrel{\mathop{:}}t}}\Big{[}\gamma^{2t-2}\omega^{2}\_{0\mathrel{\mathop{:}}t-1}\mathbb{V}\_{\mathcal{F}\_{t+1\mathrel{\mathop{:}}T-1}}(\bar{R}\_{t\mathrel{\mathop{:}}T-1}\mid\mathcal{F}\_{t})\Big{]},+ blackboard\_E start\_POSTSUBSCRIPT caligraphic\_F start\_POSTSUBSCRIPT 0 : italic\_t end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_γ start\_POSTSUPERSCRIPT 2 italic\_t - 2 end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_t - 1 end\_POSTSUBSCRIPT blackboard\_V start\_POSTSUBSCRIPT caligraphic\_F start\_POSTSUBSCRIPT italic\_t + 1 : italic\_T - 1 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( over¯ start\_ARG italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT ∣ caligraphic\_F start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) ] , \| \| where ℱt1:t2subscriptℱnormal-:subscript𝑡1subscript𝑡2\mathcal{F}\_{t\_{1}\mathrel{\mathop{:}}t\_{2}}caligraphic\_F start\_POSTSUBSCRIPT italic\_t start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT : italic\_t start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT is the filtration induced by the sequence {xt1,at1,rt1,…,xt2,at2,rt2}∼Pξπbsimilar-tosubscript𝑥subscript𝑡1subscript𝑎subscript𝑡1subscript𝑟subscript𝑡1normal-…subscript𝑥subscript𝑡2subscript𝑎subscript𝑡2subscript𝑟subscript𝑡2superscriptsubscript𝑃𝜉subscript𝜋𝑏\{x\_{t\_{1}},a\_{t\_{1}},r\_{t\_{1}},\ldots,x\_{t\_{2}},a\_{t\_{2}},r\_{t\_{2}}\}\sim P\_{\xi}^{\pi\_{b}}{ italic\_x start\_POSTSUBSCRIPT italic\_t start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_t start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT , italic\_r start\_POSTSUBSCRIPT italic\_t start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT , … , italic\_x start\_POSTSUBSCRIPT italic\_t start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_t start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT , italic\_r start\_POSTSUBSCRIPT italic\_t start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT } ∼ italic\_P start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT, R¯t:T−1=r(xt,at)+γ∑τ=t+1T−1γτ−(t+1)ωt+1:jr(xτ,aτ)subscriptnormal-¯𝑅normal-:𝑡𝑇1𝑟subscript𝑥𝑡subscript𝑎𝑡𝛾superscriptsubscript𝜏𝑡1𝑇1superscript𝛾𝜏𝑡1subscript𝜔normal-:𝑡1𝑗𝑟subscript𝑥𝜏subscript𝑎𝜏\bar{R}\_{t\mathrel{\mathop{:}}T-1}=r(x\_{t},a\_{t})+\gamma\sum\_{\tau=t+1}^{T-1}\gamma^{\tau-(t+1)}\omega\_{t+1\mathrel{\mathop{:}}j}r(x\_{\tau},a\_{\tau})over¯ start\_ARG italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT = italic\_r ( italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) + italic\_γ ∑ start\_POSTSUBSCRIPT italic\_τ = italic\_t + 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT italic\_τ - ( italic\_t + 1 ) end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUBSCRIPT italic\_t + 1 : italic\_j end\_POSTSUBSCRIPT italic\_r ( italic\_x start\_POSTSUBSCRIPT italic\_τ end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_τ end\_POSTSUBSCRIPT ), and Ct=Eℱt:T−1[ωt2(R¯t:T−1−𝔼ℱt+1:T−1[R¯t:T−1])2−2ωt2R¯t:T−1C\_{t}=E\_{\mathcal{F}\_{t\mathrel{\mathop{:}}T-1}}\Big{[}\omega\_{t}^{2}\big{(}\bar{R}\_{t\mathrel{\mathop{:}}T-1}-\mathbb{E}\_{\mathcal{F}\_{t+1\mathrel{\mathop{:}}T-1}}[\bar{R}\_{t\mathrel{\mathop{:}}T-1}]\big{)}^{2}-2\omega\_{t}^{2}\bar{R}\_{t\mathrel{\mathop{:}}T-1}italic\_C start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT = italic\_E start\_POSTSUBSCRIPT caligraphic\_F start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_ω start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ( over¯ start\_ARG italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT - blackboard\_E start\_POSTSUBSCRIPT caligraphic\_F start\_POSTSUBSCRIPT italic\_t + 1 : italic\_T - 1 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ over¯ start\_ARG italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT ] ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT - 2 italic\_ω start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT over¯ start\_ARG italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT (R¯t:T−1−𝔼ℱt+1:T−1[R¯t:T−1])]\big{(}\bar{R}\_{t\mathrel{\mathop{:}}T-1}-\mathbb{E}\_{\mathcal{F}\_{t+1\mathrel{\mathop{:}}T-1}}[\bar{R}\_{t\mathrel{\mathop{:}}T-1}]\big{)}\Big{]}( over¯ start\_ARG italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT - blackboard\_E start\_POSTSUBSCRIPT caligraphic\_F start\_POSTSUBSCRIPT italic\_t + 1 : italic\_T - 1 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ over¯ start\_ARG italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT ] ) ] is a β𝛽\betaitalic\_β-independent term. ###### Proof. The proof is by mathematical induction and is reported in Appendix [E](#A5 "Appendix E Proofs of Section 4.2 ‣ More Robust Doubly Robust Off-policy Evaluation"). ∎ As opposed to the DR variance reported in Jiang & Li ([2016](#bib.bib11)), ours in ([2](#S4.Ex19 "Theorem 2. ‣ 4.2 MRDR Estimators for Reinforcement Learning ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation")) has no dependence on the DM bias ΔΔ\Deltaroman\_Δ, which contains the unknown term Qπesuperscript𝑄subscript𝜋𝑒Q^{\pi\_{e}}italic\_Q start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT, and plus, all its expectations are over Pξπbsubscriptsuperscript𝑃subscript𝜋𝑏𝜉P^{\pi\_{b}}\_{\xi}italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT. This allows us to easily compute the MRDR model parameter from the gradient of ([2](#S4.Ex19 "Theorem 2. ‣ 4.2 MRDR Estimators for Reinforcement Learning ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation")). Let’s define β\∈arg⁡minβ∈ℝκ⁡𝕍Pξπb(ρ^DRπe(β))superscript𝛽subscript𝛽superscriptℝ𝜅subscript𝕍subscriptsuperscript𝑃subscript𝜋𝑏𝜉superscriptsubscript^𝜌DRsubscript𝜋𝑒𝛽\beta^{\}\in\arg\min\_{\beta\in\mathbb{R}^{\kappa}}\mathbb{V}\_{P^{\pi\_{b}}\_{\xi}}\big{(}\hat{\rho}\_{\text{DR}}^{\pi\_{e}}(\beta)\big{)}italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ∈ roman\_arg roman\_min start\_POSTSUBSCRIPT italic\_β ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_κ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT blackboard\_V start\_POSTSUBSCRIPT italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUBSCRIPT DR end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_β ) ) as the minimizer of the DR variance. We may write β\superscript𝛽\beta^{\}italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT using the variance formulation of Theorem [2](#Thmtheorem2 "Theorem 2. ‣ 4.2 MRDR Estimators for Reinforcement Learning ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation"), and after dropping the β𝛽\betaitalic\_β-independent terms, as β\∈arg⁡minβ∈ℝκ∑t=0T−1𝔼ℱ0:t−1[γ2tω0:t−12𝕍ℱt(ωt(R¯t:T−1−Q^πe(xt,at;β))+V^πe(xt;β))]superscript𝛽subscript𝛽superscriptℝ𝜅superscriptsubscript𝑡0𝑇1subscript𝔼subscriptℱ:0𝑡1delimited-[]superscript𝛾2𝑡subscriptsuperscript𝜔2:0𝑡1subscript𝕍subscriptℱ𝑡subscript𝜔𝑡subscript¯𝑅:𝑡𝑇1superscript^𝑄subscript𝜋𝑒subscript𝑥𝑡subscript𝑎𝑡𝛽superscript^𝑉subscript𝜋𝑒subscript𝑥𝑡𝛽\beta^{\}\in\arg\min\_{\beta\in\mathbb{R}^{\kappa}}\sum\_{t=0}^{T-1}\mathbb{E}\_{\mathcal{F}\_{0\mathrel{\mathop{:}}t-1}}\Big{[}\gamma^{2t}\omega^{2}\_{0\mathrel{\mathop{:}}t-1}\mathbb{V}\_{\mathcal{F}\_{t}}\Big{(}\omega\_{t}\big{(}\bar{R}\_{t\mathrel{\mathop{:}}T-1}-\widehat{Q}^{\pi\_{e}}(x\_{t},a\_{t};\beta)\big{)}+\widehat{V}^{\pi\_{e}}(x\_{t};\beta)\Big{)}\Big{]}italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ∈ roman\_arg roman\_min start\_POSTSUBSCRIPT italic\_β ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_κ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT blackboard\_E start\_POSTSUBSCRIPT caligraphic\_F start\_POSTSUBSCRIPT 0 : italic\_t - 1 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_γ start\_POSTSUPERSCRIPT 2 italic\_t end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_t - 1 end\_POSTSUBSCRIPT blackboard\_V start\_POSTSUBSCRIPT caligraphic\_F start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_ω start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ( over¯ start\_ARG italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT - over^ start\_ARG italic\_Q end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ; italic\_β ) ) + over^ start\_ARG italic\_V end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ; italic\_β ) ) ]. Similar to the derivation of ([11](#S4.E11 "11 ‣ Theorem 1. ‣ 4.1 MRDR Estimators for Contextual Bandits ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation")) for bandits, we can show that \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| \| \| β\∈argsuperscript𝛽\displaystyle\beta^{\}\in\argitalic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT ∈ roman\_arg \| minβ∈ℝκJ(β)=∑t=0T−1γ2t𝔼ℱ0:t−1[ω0:t−12⋅ωt⋅\displaystyle\min\_{\beta\in\mathbb{R}^{\kappa}}J(\beta)=\sum\_{t=0}^{T-1}\gamma^{2t}\mathbb{E}\_{\mathcal{F}\_{0\mathrel{\mathop{:}}t-1}}\big{[}\omega\_{0\mathrel{\mathop{:}}t-1}^{2}\cdot\omega\_{t}\cdotroman\_min start\_POSTSUBSCRIPT italic\_β ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_κ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT italic\_J ( italic\_β ) = ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT 2 italic\_t end\_POSTSUPERSCRIPT blackboard\_E start\_POSTSUBSCRIPT caligraphic\_F start\_POSTSUBSCRIPT 0 : italic\_t - 1 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ italic\_ω start\_POSTSUBSCRIPT 0 : italic\_t - 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ⋅ italic\_ω start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ⋅ \| \| (14) \| \| \| \| qβ(xt,at,R¯t:T−1)⊤Ωπb(xt)qβ(xt,at,R¯t:T−1)].\displaystyle q\_{\beta}(x\_{t},a\_{t},\bar{R}\_{t\mathrel{\mathop{:}}T-1})^{\top}\Omega\_{\pi\_{b}}(x\_{t})q\_{\beta}(x\_{t},a\_{t},\bar{R}\_{t\mathrel{\mathop{:}}T-1})\big{]}.italic\_q start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , over¯ start\_ARG italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT ) start\_POSTSUPERSCRIPT ⊤ end\_POSTSUPERSCRIPT roman\_Ω start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) italic\_q start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , over¯ start\_ARG italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT ) ] . \| \| As shown in Proposition [6](#Thmproposition6 "Proposition 6. ‣ D.3 Proof of Proposition 6 ‣ Appendix D Proofs of Section 4.1 ‣ More Robust Doubly Robust Off-policy Evaluation"), J(β)𝐽𝛽J(\beta)italic\_J ( italic\_β ) is a quadratic convex function of qβsubscript𝑞𝛽q\_{\beta}italic\_q start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT, which means that if the approximation Q^πe(⋅,⋅;β)superscript^𝑄subscript𝜋𝑒⋅⋅𝛽\widehat{Q}^{\pi\_{e}}(\cdot,\cdot;\beta)over^ start\_ARG italic\_Q end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( ⋅ , ⋅ ; italic\_β ) is smooth in β𝛽\betaitalic\_β, then this problem can be effectively solved by gradient descent. Since the expectation in ([14](#S4.E14 "14 ‣ 4.2 MRDR Estimators for Reinforcement Learning ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation")) is w.r.t. Pξπbsubscriptsuperscript𝑃subscript𝜋𝑏𝜉P^{\pi\_{b}}\_{\xi}italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT, we may use the trajectories in 𝒟𝒟\mathcal{D}caligraphic\_D (generated by πbsubscript𝜋𝑏\pi\_{b}italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT), replace J(β)𝐽𝛽J(\beta)italic\_J ( italic\_β ) with its unbiased SAA, Jn(β)subscript𝐽𝑛𝛽J\_{n}(\beta)italic\_J start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ( italic\_β ), and solve it for β𝛽\betaitalic\_β, i.e., \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| \| \| βn\∈subscriptsuperscript𝛽𝑛absent\displaystyle\beta^{\}\_{n}\initalic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ∈ \| argminβ∈ℝκJn(β)=∑i=1n∑t=0T−1γ2t(ω0:t−1(i))2⋅ωt(i)⋅\displaystyle\arg\min\_{\beta\in\mathbb{R}^{\kappa}}J\_{n}(\beta)=\sum\_{i=1}^{n}\sum\_{t=0}^{T-1}\gamma^{2t}(\omega^{(i)}\_{0\mathrel{\mathop{:}}t-1})^{2}\cdot\omega\_{t}^{(i)}\cdotroman\_arg roman\_min start\_POSTSUBSCRIPT italic\_β ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_κ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT italic\_J start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ( italic\_β ) = ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT 2 italic\_t end\_POSTSUPERSCRIPT ( italic\_ω start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_t - 1 end\_POSTSUBSCRIPT ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ⋅ italic\_ω start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT ⋅ \| \| (15) \| \| \| \| qβ(xt(i),at(i),R¯t:T−1(i))⊤Ωπb(xt(i))qβ(xt(i),at(i),R¯t:T−1(i)).subscript𝑞𝛽superscriptsubscriptsuperscript𝑥𝑖𝑡subscriptsuperscript𝑎𝑖𝑡subscriptsuperscript¯𝑅𝑖:𝑡𝑇1topsubscriptΩsubscript𝜋𝑏subscriptsuperscript𝑥𝑖𝑡subscript𝑞𝛽subscriptsuperscript𝑥𝑖𝑡subscriptsuperscript𝑎𝑖𝑡subscriptsuperscript¯𝑅𝑖:𝑡𝑇1\displaystyle q\_{\beta}(x^{(i)}\_{t},a^{(i)}\_{t},\bar{R}^{(i)}\_{t\mathrel{\mathop{:}}T-1})^{\top}\Omega\_{\pi\_{b}}(x^{(i)}\_{t})q\_{\beta}(x^{(i)}\_{t},a^{(i)}\_{t},\bar{R}^{(i)}\_{t\mathrel{\mathop{:}}T-1}).italic\_q start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , over¯ start\_ARG italic\_R end\_ARG start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT ) start\_POSTSUPERSCRIPT ⊤ end\_POSTSUPERSCRIPT roman\_Ω start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) italic\_q start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , over¯ start\_ARG italic\_R end\_ARG start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT ) . \| \| Since Jn(β)subscript𝐽𝑛𝛽J\_{n}(\beta)italic\_J start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ( italic\_β ) is strongly consistent, βn\→β\→subscriptsuperscript𝛽𝑛superscript𝛽\beta^{\}\_{n}\rightarrow\beta^{\}italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT → italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT almost surely. If we can explicitly write ∇βQ^(x,a;β)subscript∇𝛽^𝑄𝑥𝑎𝛽\nabla\_{\beta}\widehat{Q}(x,a;\beta)∇ start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT over^ start\_ARG italic\_Q end\_ARG ( italic\_x , italic\_a ; italic\_β ), then βn\subscriptsuperscript𝛽𝑛\beta^{\}\_{n}italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT is the solution of equation 0=∑i=1n∑t=0T−1γ2t(ω0:t−1(i))2ωt(i)0superscriptsubscript𝑖1𝑛superscriptsubscript𝑡0𝑇1superscript𝛾2𝑡superscriptsubscriptsuperscript𝜔𝑖:0𝑡12superscriptsubscript𝜔𝑡𝑖0=\sum\_{i=1}^{n}\sum\_{t=0}^{T-1}\gamma^{2t}(\omega^{(i)}\_{0\mathrel{\mathop{:}}t-1})^{2}\omega\_{t}^{(i)}0 = ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT 2 italic\_t end\_POSTSUPERSCRIPT ( italic\_ω start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_t - 1 end\_POSTSUBSCRIPT ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT qβ(xt(i),at(i),R¯t:T−1(i))⊤Ωπb(xt(i))Dπe(xt(i))∇βQ¯(xt(i);β)subscript𝑞𝛽superscriptsubscriptsuperscript𝑥𝑖𝑡subscriptsuperscript𝑎𝑖𝑡subscriptsuperscript¯𝑅𝑖:𝑡𝑇1topsubscriptΩsubscript𝜋𝑏subscriptsuperscript𝑥𝑖𝑡subscript𝐷subscript𝜋𝑒subscriptsuperscript𝑥𝑖𝑡subscript∇𝛽¯𝑄subscriptsuperscript𝑥𝑖𝑡𝛽q\_{\beta}(x^{(i)}\_{t},a^{(i)}\_{t},\bar{R}^{(i)}\_{t\mathrel{\mathop{:}}T-1})^{\top}\Omega\_{\pi\_{b}}(x^{(i)}\_{t})D\_{\pi\_{e}}(x^{(i)}\_{t})\nabla\_{\beta}\bar{Q}(x^{(i)}\_{t};\beta)italic\_q start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , over¯ start\_ARG italic\_R end\_ARG start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT ) start\_POSTSUPERSCRIPT ⊤ end\_POSTSUPERSCRIPT roman\_Ω start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) italic\_D start\_POSTSUBSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) ∇ start\_POSTSUBSCRIPT italic\_β end\_POSTSUBSCRIPT over¯ start\_ARG italic\_Q end\_ARG ( italic\_x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ; italic\_β ) . In case the evaluation policy is deterministic, we can further simplify Jn(β)subscript𝐽𝑛𝛽J\_{n}(\beta)italic\_J start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ( italic\_β ) and derive the model parameter for MRDR by solving the following WLS problem: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| Jn(β)subscript𝐽𝑛𝛽\displaystyle J\_{n}(\beta)italic\_J start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ( italic\_β ) \| =1n∑i=1n∑t=0T−1γ2t(ω0:t−1(i))2ωt(i)𝟏{πe(xt(i))=at(i)}absent1𝑛superscriptsubscript𝑖1𝑛superscriptsubscript𝑡0𝑇1superscript𝛾2𝑡superscriptsubscriptsuperscript𝜔𝑖:0𝑡12superscriptsubscript𝜔𝑡𝑖1subscript𝜋𝑒subscriptsuperscript𝑥𝑖𝑡subscriptsuperscript𝑎𝑖𝑡\displaystyle=\frac{1}{n}\sum\_{i=1}^{n}\sum\_{t=0}^{T-1}\gamma^{2t}(\omega^{(i)}\_{0\mathrel{\mathop{:}}t-1})^{2}\omega\_{t}^{(i)}\mathbf{1}\{\pi\_{e}(x^{(i)}\_{t})=a^{(i)}\_{t}\}= divide start\_ARG 1 end\_ARG start\_ARG italic\_n end\_ARG ∑ start\_POSTSUBSCRIPT italic\_i = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_n end\_POSTSUPERSCRIPT ∑ start\_POSTSUBSCRIPT italic\_t = 0 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_T - 1 end\_POSTSUPERSCRIPT italic\_γ start\_POSTSUPERSCRIPT 2 italic\_t end\_POSTSUPERSCRIPT ( italic\_ω start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT 0 : italic\_t - 1 end\_POSTSUBSCRIPT ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT italic\_ω start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT bold\_1 { italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) = italic\_a start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT } \| \| \| \| \| 1−πb(at(i)\|xt(i))πb(at(i)\|xt(i))2(R¯t:T−1(i)−Q^πe(xt(i),at(i);β))2.1subscript𝜋𝑏conditionalsubscriptsuperscript𝑎𝑖𝑡subscriptsuperscript𝑥𝑖𝑡subscript𝜋𝑏superscriptconditionalsubscriptsuperscript𝑎𝑖𝑡subscriptsuperscript𝑥𝑖𝑡2superscriptsubscriptsuperscript¯𝑅𝑖:𝑡𝑇1superscript^𝑄subscript𝜋𝑒subscriptsuperscript𝑥𝑖𝑡subscriptsuperscript𝑎𝑖𝑡𝛽2\displaystyle\frac{1-\pi\_{b}(a^{(i)}\_{t}\|x^{(i)}\_{t})}{\pi\_{b}(a^{(i)}\_{t}\|x^{(i)}\_{t})^{2}}\big{(}\bar{R}^{(i)}\_{t\mathrel{\mathop{:}}T-1}-\widehat{Q}^{\pi\_{e}}(x^{(i)}\_{t},a^{(i)}\_{t};\beta)\big{)}^{2}.divide start\_ARG 1 - italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) end\_ARG start\_ARG italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT ( italic\_a start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT \| italic\_x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT end\_ARG ( over¯ start\_ARG italic\_R end\_ARG start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t : italic\_T - 1 end\_POSTSUBSCRIPT - over^ start\_ARG italic\_Q end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT , italic\_a start\_POSTSUPERSCRIPT ( italic\_i ) end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ; italic\_β ) ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT . \| \| (16) \| The intuition behind the weights in WLS ([4.2](#S4.Ex23 "4.2 MRDR Estimators for Reinforcement Learning ‣ 4 More Robust Doubly Robust Estimators ‣ More Robust Doubly Robust Off-policy Evaluation")) is 1) to adjust the difference between the occupancy measures of the behavior and evaluation policies, and 2) to increase the penalty of the policy discrepancy term 𝟏{πe(xt)=at}1subscript𝜋𝑒subscript𝑥𝑡subscript𝑎𝑡\mathbf{1}\{\pi\_{e}(x\_{t})=a\_{t}\}bold\_1 { italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT ) = italic\_a start\_POSTSUBSCRIPT italic\_t end\_POSTSUBSCRIPT }. ### 4.3 Other Properties of the MRDR Estimators ##### Strong Consistency Similar to the analysis in Thomas & Brunskill ([2016](#bib.bib31)) for weighted DR, we prove (in Appendix [F](#A6 "Appendix F Proofs of Section 4.3 ‣ More Robust Doubly Robust Off-policy Evaluation")) that the MRDR estimators are strongly consistent, i.e., limn→∞ρ^MRDR,nπe(βn\)=ρπesubscript→𝑛subscriptsuperscript^𝜌subscript𝜋𝑒MRDR𝑛subscriptsuperscript𝛽𝑛superscript𝜌subscript𝜋𝑒\lim\_{n\rightarrow\infty}\hat{\rho}^{\pi\_{e}}\_{\text{MRDR},n}(\beta^{\}\_{n})=\rho^{\pi\_{e}}roman\_lim start\_POSTSUBSCRIPT italic\_n → ∞ end\_POSTSUBSCRIPT over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT MRDR , italic\_n end\_POSTSUBSCRIPT ( italic\_β start\_POSTSUPERSCRIPT \* end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ) = italic\_ρ start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT almost surely. This implies that MRDR is a well-posed OPE estimator. ##### Asymptotic Optimality The MRDR estimator, by construction, has the lowest variance among the DR estimators of the form ([7](#S3.E7 "7 ‣ 3.3 Doubly Robust Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")). On the other hand, the semi-parametric theory in multivariate regression (Robins et al., [1994](#bib.bib26)) states that without extra assumption on the data distribution, the class of unbiased, consistent and asymptotically normal OPE estimators is asymptotically equivalent to the DR estimators in ([7](#S3.E7 "7 ‣ 3.3 Doubly Robust Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")). Utilizing this result, we can show that the MRDR estimators are asymptotically optimal (i.e., have minimum variance) in this class of estimators. ##### MRDR Extensions Similar to Thomas & Brunskill ([2016](#bib.bib31)), we can derive the weighted MRDR estimator by replacing the IS part of the MRDR estimator in ([7](#S3.E7 "7 ‣ 3.3 Doubly Robust Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")) with (per-step) weighted importance sampling. This introduces bias, but potentially reduces its variance, and thus, its MSE. Throughout the paper, we assumed that the data has been generated by a single behavior policy. We can extend our MRDR results to the case that there are more than one behavior policy by replacing the IS part of our estimator with fused importance sampling (Peshkin & Shelton, [2002](#bib.bib21)). 5 Experiments -------------- In this section, we demonstrate the effectiveness of the proposed MRDR estimation by comparing it with other state-of-the art methods from Section [3](#S3 "3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation") on both contextual bandit and RL benchmark problems. ### 5.1 Contextual Bandit Using the 9999 benchmark experiments described in Dudík et al. ([2011](#bib.bib6)), we evaluate the OPE algorithms using the standard classification data-set from the UCI repository. Here we follow the same procedure of transforming a classification data-set into a contextual bandit dataset. For the sake of brevity, detailed descriptions of the experimental setup will be deferred to the appendix. Given a deterministic policy π𝜋\piitalic\_π, which is a logistic regression model trained by the classification data-set, we discuss three methods of transforming it into stochastic policies. The first one, which is known as friendly softening, constructs a stochastic policy with the following smoothing procedure: Given two constants α𝛼\alphaitalic\_α and β𝛽\betaitalic\_β, and a uniform (continuous) random variable u∈[−0.5,0.5]𝑢0.50.5u\in[-0.5,0.5]italic\_u ∈ [ - 0.5 , 0.5 ]. For each a∈{1,…,l}𝑎1…𝑙a\in\{1,\ldots,l\}italic\_a ∈ { 1 , … , italic\_l }, whenever π(x)=a𝜋𝑥𝑎\pi(x)=aitalic\_π ( italic\_x ) = italic\_a, the stochastic policy πα,β(x)subscript𝜋𝛼𝛽𝑥\pi\_{\alpha,\beta}(x)italic\_π start\_POSTSUBSCRIPT italic\_α , italic\_β end\_POSTSUBSCRIPT ( italic\_x ) returns a𝑎aitalic\_a with probability α+β×u𝛼𝛽𝑢\alpha+\beta\times uitalic\_α + italic\_β × italic\_u, and it returns k𝑘kitalic\_k, which is a realization of the uniform (discrete) random variable in {1,…,l}∖{a}1…𝑙𝑎\{1,\ldots,l\}\setminus\{a\}{ 1 , … , italic\_l } ∖ { italic\_a } with probability 1−(α+β×u)l−11𝛼𝛽𝑢𝑙1\frac{1-(\alpha+\beta\times u)}{l-1}divide start\_ARG 1 - ( italic\_α + italic\_β × italic\_u ) end\_ARG start\_ARG italic\_l - 1 end\_ARG. The second one, which is known as adversarial softening, constructs a stochastic policy πα,β(x)subscript𝜋𝛼𝛽𝑥\pi\_{\alpha,\beta}(x)italic\_π start\_POSTSUBSCRIPT italic\_α , italic\_β end\_POSTSUBSCRIPT ( italic\_x ) from policy π𝜋\piitalic\_π in a similar fashion. Whenever π(x)=a𝜋𝑥𝑎\pi(x)=aitalic\_π ( italic\_x ) = italic\_a, πα,β(x)subscript𝜋𝛼𝛽𝑥\pi\_{\alpha,\beta}(x)italic\_π start\_POSTSUBSCRIPT italic\_α , italic\_β end\_POSTSUBSCRIPT ( italic\_x ) returns k≠a𝑘𝑎k\neq aitalic\_k ≠ italic\_a with probability α+β×u𝛼𝛽𝑢\alpha+\beta\times uitalic\_α + italic\_β × italic\_u, and it returns k~~𝑘\tilde{k}over~ start\_ARG italic\_k end\_ARG, which is a realization of the uniform (discrete) random variable in {1,…,l}1…𝑙\{1,\ldots,l\}{ 1 , … , italic\_l } with probability 1−(α+β×u)l1𝛼𝛽𝑢𝑙\frac{1-(\alpha+\beta\times u)}{l}divide start\_ARG 1 - ( italic\_α + italic\_β × italic\_u ) end\_ARG start\_ARG italic\_l end\_ARG. The third one, which is the neutral policy, is a uniformly random policy. We will use these methods to construct behavior and evaluation policies. Table [1](#S5.T1 "Table 1 ‣ 5.1 Contextual Bandit ‣ 5 Experiments ‣ More Robust Doubly Robust Off-policy Evaluation") summarizes their specifications. Here we compare the MRDR method with the direct method (DM), the importance sampling (IS) method and two doubly robust (DR) estimators. The model parameter of the DM estimator is obtained by solving the SAA of the following problem: βDM∈arg⁡minβ∈ℝκ⁡𝔼(x,a)∼Pξπb[(Qπe(x,a)−Q^πe(x,a;β))2]subscript𝛽DMsubscript𝛽superscriptℝ𝜅subscript𝔼similar-to𝑥𝑎subscriptsuperscript𝑃subscript𝜋𝑏𝜉delimited-[]superscriptsuperscript𝑄subscript𝜋𝑒𝑥𝑎superscript^𝑄subscript𝜋𝑒𝑥𝑎𝛽2\beta\_{\text{DM}}\in\arg\min\_{\beta\in\mathbb{R}^{\kappa}}\mathbb{E}\_{(x,a)\sim P^{\pi\_{b}}\_{\xi}}[(Q^{\pi\_{e}}(x,a)-\widehat{Q}^{\pi\_{e}}(x,a;\beta))^{2}]italic\_β start\_POSTSUBSCRIPT DM end\_POSTSUBSCRIPT ∈ roman\_arg roman\_min start\_POSTSUBSCRIPT italic\_β ∈ blackboard\_R start\_POSTSUPERSCRIPT italic\_κ end\_POSTSUPERSCRIPT end\_POSTSUBSCRIPT blackboard\_E start\_POSTSUBSCRIPT ( italic\_x , italic\_a ) ∼ italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_b end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT [ ( italic\_Q start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x , italic\_a ) - over^ start\_ARG italic\_Q end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ( italic\_x , italic\_a ; italic\_β ) ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT ], which means all samples are weighted according to data, without consideration of the visiting distribution induced by the evaluation policy. The model parameters of the DR estimator is optimized based on the DM methodologies described in ([2](#S3.E2 "2 ‣ 3.1 Direct Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")). Besides the standard DR estimator we also include another alternative that is known as the DR0, which heuristically uses the model parameter from the vanilla DM method (which is called DM0, and it assigns uniform weights over samples). Below are results over the five behavior policies and five algorithms on the benchmark datasets. (Due to page limit, only the results of Vehicle, SatImage, PenDigits and Letter are included in the main paper, see appendix for the remaining results.) We evaluate the accuracy of the estimation via root mean squares error (MSE): ∑j=1N(ρ^jπe−ρπe)2/Nsuperscriptsubscript𝑗1𝑁superscriptsubscriptsuperscript^𝜌subscript𝜋𝑒𝑗superscript𝜌subscript𝜋𝑒2𝑁\sqrt{\sum\_{j=1}^{N}(\hat{\rho}^{\pi\_{e}}\_{j}-\rho^{\pi\_{e}})^{2}/N}square-root start\_ARG ∑ start\_POSTSUBSCRIPT italic\_j = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_N end\_POSTSUPERSCRIPT ( over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT - italic\_ρ start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT ) start\_POSTSUPERSCRIPT 2 end\_POSTSUPERSCRIPT / italic\_N end\_ARG, where ρ^jπesubscriptsuperscript^𝜌subscript𝜋𝑒𝑗\widehat{\rho}^{\pi\_{e}}\_{j}over^ start\_ARG italic\_ρ end\_ARG start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT is the estimated value from the j𝑗jitalic\_j-th dataset. Furthermore, we perform a 95%percent9595\%95 % significance test only on MRDR and DR, with bold numbers indicating the corresponding method outperforms its counterpart significantly. In the contextual bandit experiments, it’s clear that in most cases the proposed MRDR estimator is superior to all alternative estimators (statistical) significantly. Similar to the results reported in Dudík et al. ([2011](#bib.bib6)), the DM method incurs much higher MSE than other methods in all of the experiments. This is potentially due to the issue of high bias in model estimation when the sample-size is small. In general the estimation error is increasing across rows from top to bottom. This is expected due to the increasing difficulties in the OPE tasks that is accounted by the increasing mis-matches between behavior and evaluation policies. Although there are no theoretical justifications, in most cases the performance of DR estimators (with the DM method described in Section [3.1](#S3.SS1 "3.1 Direct Estimators ‣ 3 Existing Approaches to OPE ‣ More Robust Doubly Robust Off-policy Evaluation")) is better than that of DR0. This also illustrates the benefits of optimizing the model parameter based on the knowledge of trajectory distribution Pξπesubscriptsuperscript𝑃subscript𝜋𝑒𝜉P^{\pi\_{e}}\_{\xi}italic\_P start\_POSTSUPERSCRIPT italic\_π start\_POSTSUBSCRIPT italic\_e end\_POSTSUBSCRIPT end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_ξ end\_POSTSUBSCRIPT, which is generated by the evaluation policy. Table 1: Behavior and Evaluation Policies \| \| Policy \| α𝛼\alphaitalic\_α \| β𝛽\betaitalic\_β \| \| --- \| --- \| --- \| --- \| \| Evaluation Policy \| \| 0.9 \| 0 \| \| Behavior Policies \| Friendly I \| 0.7 \| 0.2 \| \| Friendly II \| 0.5 \| 0.2 \| \| Neutral \| - \| - \| \| Adversary I \| 0.3 \| 0.2 \| \| Adversary II \| 0.5 \| 0.2 \| Table 2: Vehicle \| Behavior Policy \| DM \| IS \| DR \| MRDR \| DR0 \| \| --- \| --- \| --- \| --- \| --- \| --- \| \| Friendly I \| 0.3273 \| 0.0347 \| 0.0217 \| 0.0202 \| 0.0224 \| \| Friendly II \| 0.3499 \| 0.0517 \| 0.0331 \| 0.0318 \| 0.0356 \| \| Neutral \| 0.4384 \| 0.087 \| 0.0604 \| 0.0549 \| 0.0722 \| \| Adversary I \| 0.405 \| 0.0937 \| 0.0616 \| 0.0516 \| 0.0769 \| \| Adversary II \| 0.405 \| 0.1131 \| 0.0712 \| 0.0602 \| 0.0952 \| Table 3: SatImage \| Behavior Policy \| DM \| IS \| DR \| MRDR \| DR0 \| \| --- \| --- \| --- \| --- \| --- \| --- \| \| Friendly I \| 0.2884 \| 0.0128 \| 0.0071 \| 0.0063 \| 0.0073 \| \| Friendly II \| 0.3328 \| 0.0191 \| 0.0107 \| 0.0087 \| 0.0119 \| \| Neutral \| 0.3848 \| 0.0413 \| 0.0246 \| 0.0186 \| 0.0335 \| \| Adversary I \| 0.3963 \| 0.0459 \| 0.027 \| 0.0195 \| 0.0383 \| \| Adversary II \| 0.4093 \| 0.0591 \| 0.0364 \| 0.0262 \| 0.0521 \| Table 4: PenDigits \| Behavior Policy \| DM \| IS \| DR \| MRDR \| DR0 \| \| --- \| --- \| --- \| --- \| --- \| --- \| \| Friendly I \| 0.4014 \| 0.0103 \| 0.0056 \| 0.0037 \| 0.0059 \| \| Friendly II \| 0.4628 \| 0.0159 \| 0.0092 \| 0.0056 \| 0.0194 \| \| Neutral \| 0.564 \| 0.0450 \| 0.0314 \| 0.0138 \| 0.0412 \| \| Adversary I \| 0.5861 \| 0.0503 \| 0.0366 \| 0.0172 \| 0.0472 \| \| Adversary II \| 0.5641 \| 0.0646 \| 0.0444 \| 0.0188 \| 0.0611 \| Table 5: Letter \| Behavior Policy \| DM \| IS \| DR \| MRDR \| DR0 \| \| --- \| --- \| --- \| --- \| --- \| --- \| \| Friendly I \| 0.392 \| 0.0074 \| 0.0056 \| 0.0044 \| 0.0057 \| \| Friendly II \| 0.4146 \| 0.0102 \| 0.0077 \| 0.0054 \| 0.0083 \| \| Neutral \| 0.4713 \| 0.0467 \| 0.0363 \| 0.0315 \| 0.0456 \| \| Adversary I \| 0.46 \| 0.0587 \| 0.0455 \| 0.0385 \| 0.0575 \| \| Adversary II \| 0.4728 \| 0.0714 \| 0.055 \| 0.0481 \| 0.0703 \| ### 5.2 Reinforcement Learning In this section we present the experimental results of OPE in reinforcement learning. We first test the OPE algorithms on the standard domains ModelWin, ModelFail, and 4×4444\times 44 × 4 Maze, with behavior and evaluation policies used in Thomas & Brunskill ([2016](#bib.bib31)). The schematic diagram of the domains is shown in Figure [1](#S5.F1 "Figure 1 ‣ 5.2 Reinforcement Learning ‣ 5 Experiments ‣ More Robust Doubly Robust Off-policy Evaluation"). To demonstrate the scalability of the proposed OPE methods, we also test the OPE algorithms on the following two domains with continuous state space: Mountain Car and Cart Pole. To construct the stochastic behavior and evaluation policies, we first compute the optimal policy using standard RL algorithms such as SARSA and Q𝑄Qitalic\_Q-learning. Then these policies are constructed by applying friendly softening to the optimal policy with specific values of (α,β)𝛼𝛽(\alpha,\beta)( italic\_α , italic\_β ). For both domains, the evaluation policy is constructed using (α,β)=(0.9,0.05)𝛼𝛽0.90.05(\alpha,\beta)=(0.9,0.05)( italic\_α , italic\_β ) = ( 0.9 , 0.05 ), and the behavior policy is constructed analogously using (α,β)=(0.8,0.05)𝛼𝛽0.80.05(\alpha,\beta)=(0.8,0.05)( italic\_α , italic\_β ) = ( 0.8 , 0.05 ). Detailed explanations the experimental setups can be found in the appendix. In the following experiments we set the discounting factor to be γ=1𝛾1\gamma=1italic\_γ = 1. For both the ModelFail and ModelWin domains, the number of training trajectories is set to 64646464, for Maze, Mountain Car, and Cart Pole domains this number is set to 1024102410241024. The number of trajectories for sampling-based part of estimators varies from 32323232 to 512512512512 for the ModelWin, ModelFail, and Cart Pole domains, and varies from 128128128128 to 2048204820482048 for the Maze domain and Mountain Car domains. ![Refer to caption](/html/1802.03493/assets/x1.png) Figure 1: Environments from Thomas & Brunskill ([2016](#bib.bib31)). Top left: ModelFail; Bottom left: ModelWin; Right: Maze In all of the above experiments, we compare results of MRDR with DM, IS, DR, and DR0 estimations by their corresponding MSE values. Similarly, the bold numbers represent cases when the performance of the MRDR estimator is statistically significantly better than that of the DR estimator. Similar to the contextual bandit setting, except for the ModelWin domain that is known to be in favor of the DM estimator (Thomas & Brunskill, [2016](#bib.bib31)), in most cases MRDR estimator has significantly lower MSE than other existing methods. Furthermore, when the sample size of the evaluation trajectories increases, we also observe accuracy improvements on all estimators in every experiment. Similar to the contextual bandit setting, significant performance improvement can be observed when one switches from DR0 to DR in the RL experiments. Table 6: ModelFail \| Sample Size \| DM \| IS \| DR \| MRDR \| DR0 \| \| --- \| --- \| --- \| --- \| --- \| --- \| \| 32 \| 0.07152 \| 1.37601 \| 0.18461 \| 0.1698 \| 1.16084 \| \| 64 \| 0.07152 \| 1.07213 \| 0.1314 \| 0.11405 \| 0.9046 \| \| 128 \| 0.07152 \| 0.752 \| 0.09901 \| 0.08188 \| 0.63571 \| \| 256 \| 0.07152 \| 0.55955 \| 0.06565 \| 0.05527 \| 0.47211 \| \| 512 \| 0.07152 \| 0.39533 \| 0.04756 \| 0.03819 \| 0.33391 \| Table 7: Modelwin \| Sample Size \| DM \| IS \| DR \| MRDR \| DR0 \| \| --- \| --- \| --- \| --- \| --- \| --- \| \| 32 \| 0.06182 \| 0.78452 \| 1.55244 \| 1.46778 \| 1.51858 \| \| 64 \| 0.06182 \| 1.03207 \| 1.13856 \| 0.98433 \| 1.40758 \| \| 128 \| 0.06182 \| 0.90166 \| 1.4195 \| 1.27891 \| 1.52634 \| \| 256 \| 0.06182 \| 0.78507 \| 1.03575 \| 0.79849 \| 1.10332 \| \| 512 \| 0.06182 \| 0.55647 \| 0.89655 \| 0.66791 \| 0.97128 \| Table 8: 4×4444\times 44 × 4 Maze \| Sample Size \| DM \| IS \| DR \| MRDR \| DR0 \| \| --- \| --- \| --- \| --- \| --- \| --- \| \| 128 \| 1.77598 \| 6.68579 \| 0.70465 \| 0.57042 \| 0.70969 \| \| 256 \| 1.77598 \| 3.50346 \| 0.69886 \| 0.58871 \| 0.70211 \| \| 512 \| 1.77598 \| 2.64257 \| 0.60124 \| 0.58879 \| 0.60338 \| \| 1024 \| 1.77598 \| 1.45434 \| 0.5201 \| 0.4666 \| 0.52148 \| \| 2048 \| 1.77598 \| 0.89668 \| 0.3932 \| 0.31274 \| 0.39425 \| Table 9: Mountain Car \| Sample Size \| DM \| IS \| DR \| MRDR \| DR0 \| \| --- \| --- \| --- \| --- \| --- \| --- \| \| 128 \| 17.80368 \| 23.11318 \| 16.14661 \| 14.96227 \| 19.46953 \| \| 256 \| 14.62359 \| 14.82684 \| 13.89212 \| 12.48327 \| 22.80573 \| \| 512 \| 13.22012 \| 8.26484 \| 8.01421 \| 7.89474 \| 7.96849 \| \| 1024 \| 10.24318 \| 3.26843 \| 3.03239 \| 3.1359 \| 9.16269 \| \| 2048 \| 10.91577 \| 2.50591 \| 2.75933 \| 2.17138 \| 8.25527 \| Table 10: Cart Pole \| Sample Size \| DM \| IS \| DR \| MRDR \| DR0 \| \| --- \| --- \| --- \| --- \| --- \| --- \| \| 32 \| 3.92319 \| 1.18213 \| 0.34775 \| 0.27208 \| 0.40567 \| \| 64 \| 3.97312 \| 0.82658 \| 0.27905 \| 0.2353 \| 0.31494 \| \| 128 \| 3.92319 \| 0.66174 \| 0.18793 \| 0.16455 \| 0.21232 \| \| 256 \| 3.82333 \| 0.62042 \| 0.17091 \| 0.16012 \| 0.1915 \| \| 512 \| 3.80461 \| 0.31021 \| 0.08455 \| 0.079 \| 0.08946 \| 6 Conclusions -------------- In this paper, we proposed the class of more-robust doubly-robust (MRDR) estimators for off-policy evaluation in RL. In particular, we proposed a principled method to calculate the model in DR estimator, which aims at minimizing its variance. Furthermore, we showed that our estimator is consistent and asymptotically optimal in the class of unbiased, consistent and asymptotically normal estimators. Finally, we demonstrated the effectiveness of our MRDR estimator in bandits and RL benchmark problems. Future work includes extending the MRDR estimator to the cases 1) when there are multiple behavior policies, 2) when the action set has a combinatorial structure, e.g., actions are in the form of slates (Swaminathan et al., [2017](#bib.bib29)), and 3) when the behavior policy is unknown.
8cf918fc-52d0-474d-9294-69a61bab31b7	trentmkelly/LessWrong-43k	LessWrong	Learning-theoretic agenda reading list Recently, I'm receiving more and more requests for a self-study reading list for people interested in the learning-theoretic agenda. I created a standard list for that, but before now I limited myself to sending it to individual people in private, out of some sense of perfectionism: many of the entries on the list might not be the best sources for the topics and I haven't read all of them cover to cover myself. But, at this point it seems like it's better to publish a flawed list than wait for perfection that will never come. Also, commenters are encouraged to recommend alternative sources that they consider better, if they know any. So, without further adieu: General math background * Theoretical computer science * "Computational Complexity: A Conceptual Perspective" by Goldreich (especially chapters 1, 2, 5, 10) * “Lambda-Calculus and Combinators: An Introduction” by Hindley * “Tree Automata Techniques and Applications” by Comon et al (mostly chapter 1) * "Introductory Functional Analysis with Applications" by Kreyszig (especially chapters 1, 2, 3, 4) * "Probability: Theory and Examples" by Durret (especially chapters 4, 5, 6) * "Elements of Information Theory" by Cover and Thomas (especially chapter 2) * “Game Theory, Alive” by Karlin and Peres * “Categories for the Working Mathematician” by Mac Lane (especially parts I, III, IV and VI) AI theory * “Handbook of Markov Decision Processes” edited by Feinberg and Shwartz (especially chapters 1-3) * “Aritifical Intelligence: A Modern Approach” by Russel and Norvig (especially chapter 17) * "Machine Learning: From Theory to Algorithms" by Shalev-Shwarz and Ben-David (especially part I and chapter 21) * "An Introduction to Computational Learning Theory" by Kearns and Vazirani (especially chapter 8) * "Bandit Algorithms" by Lattimore and Szepesvari (especially parts II, III, V, VIII) * Alternative/complementary: "Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems" by B
d93d3ec7-ae9a-4807-9ae8-3fc510632aed	trentmkelly/LessWrong-43k	LessWrong	[Link] You and Your Research I've seen Richard Hamming's classic talk You And Your Research referenced several times on LessWrong and figured I would post the full version. The introduction is reproduced below: > The title of my talk is, ``You and Your Research.'' It is not about managing research, it is about how you individually do your research. I could give a talk on the other subject - but it's not, it's about you. I'm not talking about ordinary run-of-the-mill research; I'm talking about great research. And for the sake of describing great research I'll occasionally say Nobel-Prize type of work. It doesn't have to gain the Nobel Prize, but I mean those kinds of things which we perceive are significant things. Relativity, if you want, Shannon's information theory, any number of outstanding theories - that's the kind of thing I'm talking about. > > Now, how did I come to do this study? At Los Alamos I was brought in to run the computing machines which other people had got going, so those scientists and physicists could get back to business. I saw I was a stooge. I saw that although physically I was the same, they were different. And to put the thing bluntly, I was envious. I wanted to know why they were so different from me. I saw Feynman up close. I saw Fermi and Teller. I saw Oppenheimer. I saw Hans Bethe: he was my boss. I saw quite a few very capable people. I became very interested in the difference between those who do and those who might have done. > > When I came to Bell Labs, I came into a very productive department. Bode was the department head at the time; Shannon was there, and there were other people. I continued examining the questions, ``Why?'' and ``What is the difference?'' I continued subsequently by reading biographies, autobiographies, asking people questions such as: ``How did you come to do this?'' I tried to find out what are the differences. And that's what this talk is about. I consider this talk good and useful not only for those interested in research, but for
a1fcc903-4304-46f4-a7a1-a3c83ad669a2	trentmkelly/LessWrong-43k	LessWrong	Meetup : Rationality Meetup Vienna Discussion article for the meetup : Rationality Meetup Vienna WHEN: 22 November 2014 03:00:00PM (+0100) WHERE: Kaisermühlenstraße 24/2, 1220 Wien agenda: - maybe go on with the goal setting and planning for the future - It's time for another open mic session - so please think about 30 minutes topics to offer :) location: When arriving by U2 or Schnellbahn train: get out at station Stadlau take the exit towards Kaisermühlenstraße and simply cross the street the meetup is in the modern looking, greenish building right in front of your nose :) Important: Google maps doesn't recognise the address and hence shows the wrong location... !Important Notice! Since our usual room on the ground floor has already been booked by someone else, we will have our meetup in the room on the fourth floor (the same room we moved to last time after complications arose).
e7dbffdc-b666-465b-a702-45699a7f379a	trentmkelly/LessWrong-43k	LessWrong	Meetup : Urbana-Champaign, Illinois Discussion article for the meetup : Urbana-Champaign, Illinois WHEN: 01 September 2013 02:00:00PM (-0500) WHERE: Illini Union South Lounge 1401 W Green St Urbana, IL 61801 Meetup topic will again be determined by popular consensus at the actual meetup. I will have Zendo, Wits and Wagers (with cards that can be used independently for calibration games), and Pandemic, a cooperative strategy board game. 2PM Sunday in the Illini Union South Lounge seems to have worked for everyone this week, so I chose the same time next week, and this will probably become our permanent place and weekly time. Cross posted on the mailing list. Discussion article for the meetup : Urbana-Champaign, Illinois
17663125-2345-475d-aa8c-e63c5cf4fefd	trentmkelly/LessWrong-43k	LessWrong	Forecasting Prediction markets and forecasting platforms are fascinating tools that bring together collective intelligence. In these markets, people buy and sell shares of potential outcomes, like election results or economic trends. The prices reflect what the crowd thinks is likely to happen, offering a snapshot of collective wisdom. Sometimes markets can tell you where you need to be to solve the puzzles you're facing in life. Here's a twist: what if we could change reality to match our predictions? When individuals or organizations act on these forecasts, they can influence the outcomes they predict. It's a feedback loop where accurate predictions not only foresee the future but also help shape it. Take elections, for instance. If prediction markets show a high chance of a particular candidate winning, campaign strategies, media coverage, and voter perceptions might shift to support this forecast. This can boost the candidate’s actual chances of winning. In the business world, companies might use market forecasts to make decisions. If a company predicts high demand for a product, it can increase production, marketing, and distribution efforts, making the predicted demand more likely to come true. I am captivated by the idea of making the territory match the map. By understanding and using this dynamic, I can not only get better at predicting but also help make those predictions come true. It’s a reminder of how interconnected our decisions are with the future we create.
182b3523-d4d4-4c6b-8a94-853fe4b6d78a	trentmkelly/LessWrong-43k	LessWrong	[link] Is Alu Life? I recently read (in Dawkins' The Ancestor's Tale) about the Alu sequence, and went on to read about transposons generally. Having as I do a rather broad definition of life, I concluded that Alu (and others like it) are lifeforms in their own right, although parasitic ones. I found the potential ethical implications somewhat staggering, especially given the need to shut up and multiply those implications by the rather large number of transposon instances in a typical multicellular organism. I have written out my thoughts on the subject, at http://jttlov.no-ip.org/writings/alulife.htm. I don't claim to have a well-worked out position, just a series of ideas and questions I feel to be worthy of discussion. ETA: I have started editing the article based on the discussion below. For reference with the existing discussion, I have preserved a copy of the original article as well, linked from the current version.
2e581f69-b68e-4753-bbaf-882340be1e53	trentmkelly/LessWrong-43k	LessWrong	Best career models for doing research? Ideally, I'd like to save the world. One way to do that involves contributing academic research, which raises the question of what's the most effective way of doing that. The traditional wisdom says if you want to do research, you should get a job in a university. But for the most part the system seems to be set up so that you first spend a long time working for someone else and research their ideas, after which you can lead your own group, but then most of your time will be spent on applying for grants and other administrative trivia rather than actually researching the interesting stuff. Also, in Finland at least, all professors need to also spend time doing teaching, so that's another time sink. I suspect I would have more time to actually dedicate on research, and I could get doing it quicker, if I took a part-time job and did the research in my spare time. E.g. the recommended rates for a freelance journalist in Finland would allow me to spend a week each month doing work and three weeks doing research, of course assuming that I can pull off the freelance journalism part. What (dis)advantages does this have compared to the traditional model? Some advantages: * Can spend more time on actual research. * A lot more freedom with regard to what kind of research one can pursue. * Cleaner mental separation between money-earning job and research time (less frustration about "I could be doing research now, instead of spending time on this stupid administrative thing"). * Easier to take time off from research if feeling stressed out. Some disadvantages: * Harder to network effectively. * Need to get around journal paywalls somehow. * Journals might be biased against freelance researchers. * Easier to take time off from research if feeling lazy. * Harder to combat akrasia. * It might actually be better to spend some time doing research under others before doing it on your own. EDIT: Note that while I certainly do appreciate comments specific to my situat
e0158520-453a-4c51-bcc3-9b9c3cbefca7	trentmkelly/LessWrong-43k	LessWrong	Introducing Collective Action for Existential Safety: 80+ actions individuals, organizations, and nations can take to improve our existential safety Collective Action for Existential Safety is an initiative of the Center for Existential Safety, a new non-profit organization being formed in the United States. Our central aim is to catalyze collective action to ensure humanity survives this decade. If we can achieve that, then we are likely to create an unimaginably good future for all. We serve all Existential Safety Advocates globally. We believe artificial intelligence, nuclear weapons, engineered pandemics, and soon-to-be invented technologies threaten our civilization. The more of us working together to mitigate the risks, the better. * Our offer: we created what we believe is the most comprehensive and practical list of actions individuals, organizations, and nations can take to improve our existential safety. It answers the question, “What can we do to help ensure our existential safety?” We also offer occasional all hands calls for the existential safety community, monthly strategy coordination calls for existential safety organization leaders, and collaborative safety calls for frontier AI developers. * Our ask: we're fundraising and recruiting for the team. Happy to take on aligned co-founders and c-suite folks, as well as part-time volunteers. Please email or schedule a call if you’re interested in learning more. Feedback is very much welcome, but ideally it would be offered in a kind, collaborative, and well-informed manner. We believe unkind and uninformed criticism in this community has very likely reduced its positive impact on the world. My favorite guidelines for this are here. Please directly suggest changes to our actions list here. On a personal note, I’ve been preparing for this for 26+ years now. I have short timelines and a high p(doom). But I believe we can still collectively raise our p(eutopia) if we act now as if our lives depended on it. They do. This could also be humanity’s finest hour. Join us at existentialsafety.org. Please share the site widely and engage with us on so
61e0edb2-23c0-4d42-a2ea-cf6cdfa9a1c2	trentmkelly/LessWrong-43k	LessWrong	Rational discussion of politics In a recent poll, many LW members expressed interest in a separate website for rational discussion of political topics. The website has been created, but we need a group of volunteers to help us test it and calibrate its recommendation system (see below). If you would like to help (by participating in one or two discussions and giving us your feedback) please sign up here. ---------------------------------------- About individual recommendation system All internet forums face a choice between freedom of speech and quality of debate. In absence of censorship, constructive discussions can be easily disrupted by the inflow of the mind-killed which causes the more intelligent participants to leave or descend to the same level. Preserving quality thus usually requires at least one of the following methods: 1. Appointing censors (a.k.a. moderators). 2. Limiting membership. 3. Declaring certain topics (e.g., politics) off limits. On the new website, we are going to experiment with a different method. In brief, the idea is to use an automated recommendation system which sorts content, raising the best comments to the top and (optionally) hiding the worst. The sorting is done based on the individual preferences, allowing each user to avoid what he or she (rather than moderators or anyone else) defines as low quality content. In this way we should be able to enhance quality without imposing limits on free speech. UPDATE. The discussions are scheduled to start on May 1.
702f814b-2f81-4fbb-a8bc-14a8ee3bab72	trentmkelly/LessWrong-43k	LessWrong	How much interest would there be in a fringe theories wiki? I've been exploring a concept for a wiki recently. The idea would be that people contribute fringe theories and present the best evidence for that theory, perhaps by contrasting it with mainstream interpretations of the data. Some examples of pages on the wiki could include, * CellBioGuy's theory that the paleocene-eocene thermal maximum was caused by an industrial civilization of birds living in Antarctica. * Robin Hanson's theory that a panspermia sibling to Earth hosts an ancient advanced alien civilization, whose world government experienced complex system rot in the process of developing an Earth-monitoring system, resulting in bizarre UFO encounters with them that we sometimes hear about on Earth. * The theory from various cryonicists, following Eric Drexler, that future nanotechnology will be sufficient to repair damage from vitrification and subsequent cryopreservation. * Robin Gardiner's theory that the ship called "The Titanic" was in fact the ship Olympic, and was purposely sunk as part of an elaborate insurance scam. * Scott Alexander's pseudo-religious theory that "There is an all-powerful, all-knowing logically necessary entity spawning all possible worlds and identical to the moral law." The purpose would not be to make a determination to whether each theory was true or false, but rather just present the evidence. Naturally, I'm more interested in theories that (1) have some sort of technical argument favoring it (regardless of credibility), (2) aren't merely moral or political theses in disguise, and (3) aren't already covered in sufficient detail in other places (unlike the JFK conspiracy theory). Wikipedia is a terrible place to do this, given their rules disallowing original research. Of course, without such constraints, there is a large risk that a Fringe Theories Wiki would attract bad editors, so some strict editing rules would still probably be required on the site.
9a6de0c0-455c-4180-aeb0-ebd352f29919	trentmkelly/LessWrong-43k	LessWrong	AI companies' eval reports mostly don't support their claims AI companies claim that their models are safe on the basis of dangerous capability evaluations. OpenAI, Google DeepMind, and Anthropic publish reports intended to show their eval results and explain why those results imply that the models' capabilities aren't too dangerous.[1] Unfortunately, the reports mostly don't support the companies' claims. Crucially, the companies usually don't explain why they think the results, which often seem strong, actually indicate safety, especially for biothreat and cyber capabilities. (Additionally, the companies are undereliciting and thus underestimating their models' capabilities, and they don't share enough information for people on the outside to tell how bad this is.) Bad explanation/contextualization OpenAI biothreat evals: OpenAI says "several of our biology evaluations indicate our models are on the cusp of being able to meaningfully help novices create known biological threats, which would cross our high risk threshold." It doesn't say how it concludes this (or what results would change its mind or anything about how it thinks eval results translate to uplift). It reports results from four knowledge and troubleshooting bio evals. On the first, o3 does well and OpenAI observes "this evaluation is reaching saturation." On the rest, OpenAI matches or substantially outperforms the expert human baseline. These results seem to suggest that o3 does have dangerous bio capabilities; they certainly don't seem to rule it out. OpenAI doesn't attempt to explain why it thinks o3 doesn't have such capabilities. DeepMind biothreat evals: DeepMind says Gemini 2.5 Pro doesn't have dangerous CBRN capabilities, explaining "it does not yet consistently or completely enable progress through key bottleneck stages." DeepMind mentions open-ended red-teaming; all it shares is results on six multiple-choice evals. It does not compare to human performance or offer other context, or say what would change its mind. For example, it's not clear whethe
1caa9707-b5d5-4408-926b-100d9b6f79d3	trentmkelly/LessWrong-43k	LessWrong	What software projects would be helpful for different groups? There are various meetups around the world that work on altruistic software development. Random Hacks of Kindness is at the top of my mind at the moment. The main site appears to be down at the moment, but the Australian and Canadian sites are still up, and the Australian one is asking for projects which help with the COVID-19 response. That got me wondering. What software projects would be high leverage and not currently saturated? Which of them are amenable to being worked on by groups of developers with mixed skills and backgrounds? This can probably be broken down further into software for different groups. Healthcare workers probably have different needs in this time than people who are struggling to make the case for working from home. My gut feeling is that efforts that helps with social support and mental health support will also have high value over time.
9394e707-a896-4084-b5c2-27eda50bbbeb	trentmkelly/LessWrong-43k	LessWrong	Nothing. The Pythia inhales the vapors of the chasm and erupts into ecstatic epilepsy. The prophecy is delivered as a babbling of tongues.[1] > In the future—not the distant future, but ten years, five—people will remember the internet as a brief dumb enthusiasm, like phrenology or the dirigible. They might still use computer networks to send an email or manage their bank accounts, but those networks will not be where culture or politics happens. The idea of spending all day online will seem as ridiculous as sitting down in front of a nice fire to read the phone book. > > You know, secretly, even if you’re pretending not to, that this thing is nearing exhaustion. There is simply nothing there online. All language has become rote, a half-assed performance: even the outraged mobs are screaming on autopilot. Even genuine crises can’t interrupt the tedium of it all, the bad jokes and predictable thinkpieces, spat-out enzymes to digest the world. ‘Leopards break into the temple and drink all the sacrificial vessels dry; it keeps happening; in the end, it can be calculated in advance and is incorporated into the ritual.’ Breathe deeper. Let the vapors flow through you. Within five years, maybe three, the internet will self-immolate like a buddhist monk or a climate activist and vaporize into thin hot air—nothing. And we know this because the internet never was anything but an absolute nothing, a nothing that devours everything. All it has ever done, all it ever was for, is un-making, nulling and voiding. The World Wide Web has done this to Finland (which does not exist), to Birds (which are not real), and it is doing it to you.[2] > When I’m listlessly killing time on the internet, there is nothing. The mind does not wander. I am not there. That rectangular hole spews out war crimes and cutesy comedies and affirmations and porn, all of it mixed together into one general-purpose informational goo, and I remain in its trance, the lifeless scroll, twitching against the screen
ab21e83b-db64-4b5c-890d-0b40da995f2b	StampyAI/alignment-research-dataset/eaforum	Effective Altruism Forum	Information in risky technology races In this post, we (Nicholas Emery-Xu, Andrew Park, and Robert Trager) summarize the results of our paper modeling the role of information in risky technology races. The current version can be accessed [here](https://drive.google.com/file/d/18j_wnA4HDMA3ofclLcfpgyV-0INMn1ZW/view?usp=sharing). 1 Is revealing technological capability beneficial? =================================================== Imagine in the not too distant future that actors are developing a new technology that gives the winner a discontinuous advantage, economic or geopolitical, over their rivals. Because of this advantage, actors are tempted to cut corners on safety to focus on capabilities research, producing a safety-performance tradeoff.[[1]](#fna13rob2xphv) In such a scenario, is it better if actors knew each other’s capability levels? On the one hand, public knowledge of capabilities prevents actors from mistakenly believing they are behind in capability and then [entering](https://forum.effectivealtruism.org/posts/cXBznkfoPJAjacFoT/are-you-really-in-a-race-the-cautionary-tales-of-szilard-and) a dangerous arms race. On the other hand, if actors learn they are close in capability, they may engage in a dangerous race to the bottom, cutting corners on safety to gain a small edge. In many cases, however, we do not observe actors engaging in such a race. Top AI labs, for example, regularly publish results on capabilities but continue to invest significantly in safety research, defying the predictions in Armstrong et al. (2016).[[2]](#fnyq64mm83qp) What’s going on? In our paper, we find that when the technological development path is even moderately uncertain, even actors who know they are close in capability are unwilling to engage in a dangerous race to the bottom because the expected returns to doing so aren’t worth the risk of an accident. We build on the model in Armstrong et al. (2016)[[2]](#fnyq64mm83qp) (see “[AI race](https://forum.effectivealtruism.org/topics/ai-race)” for a useful description) to study a technology race with a safety-performance tradeoff in which capabilities investments are unknown, privately known, or publicly known. However, we argue that their results depend on two unrealistic assumptions: that the actor with higher capability investment wins the race for sure and that only the winner’s actions contribute to the risk of an accident. In our work, we relax these assumptions. 2 Decisiveness ============== First, we argue that actors are unlikely to be certain about how additional units of capability investment translate into success in the race. In a race for a novel technology, there is likely to be randomness in the development process. For example, while Enrico Fermi was the first to use graphite as a neutron moderator, the lagging Soviets had actually already made such a discovery but failed to apply it to their nuclear program.[[3]](#fnmguva9e622j) We control for this level of randomness with a decisiveness parameter. If the parameter is zero, each actor is equally likely to win the race, regardless of effort. As it increases, the more capable actor is increasingly likely to win the race. Thus, for more decisive races, each additional unit of investment in capability is more likely to bring success. What are the effects? At low levels of decisiveness, the randomness outweighs the benefits of taking risks, so the race is perfectly safe under all information states. As decisiveness increases, risk almost always increases. In addition, as decisiveness increases, at first the private information case is most dangerous. Only at high levels of decisiveness does the public information case become most dangerous. In other words, there is no information hazard unless the race is highly decisive. For less decisive races, it is better to reveal capabilities than keep them private. 3 Safety provision ================== The other assumption that we challenge is that only the winner of the race contributes to safety. Instead, we add a safety-sharing parameter to allow both actors to contribute a weighted sum of overall safety. We might imagine, for example, that a lagging actor could conduct tests, either in preparation for the mature technology, or at a lower level of technological competence in response to the leader. We find that increasing the proportion of risk contributed by the losing actor produces two effects. First, a moral hazard effect increases risks. Just as in the provision of carbon emissions, the less actors bear the benefit of their own safety provision, the more they will shirk. Second, less capable actors increase their safety provision because their choices matter even if they lose the race. We find that the moral hazard effect is stronger, so reducing the winner’s share of safety provision increases expected disaster risk. 4 Conclusion ============ As a result of this work, we present three tentative conclusions. First, when deciding whether to incentivize actors to reveal capabilities, it is important to know their uncertainty over technological progress. At least at the early, uncertain stages of development, public knowledge of capabilities seems to be beneficial. Second, if losers of the race cause a disaster, policymakers should shift their focus away from low-capability to high-capability actors, providing them with incentives to reduce shirking, for example by sharing safety knowledge with the leader. Third, we caution against updating too strongly on any specific model (including our own!) and instead urge readers to reason through or empirically test its modeling assumptions in the context of real-world races. 1. [^](#fnrefa13rob2xphv)Robert Trager, Paolo Bova, Nicholas Emery-Xu, Eoghan Stafford, and Allan Dafoe, "Welfare Implications of Safety-Performance Tradeoffs in AI Safety Research," Working paper, August 2022. 2. [^](#fnrefyq64mm83qp)Stuart Armstrong, Nick Bostrom, and Carl Shulman, "Racing to the precipice: a model of artificial intelligence development," AI & Society, 31(2):201–206, May 2016, <http://link.springer.com/10.1007/s00146-015-0590-y>. 3. [^](#fnrefmguva9e622j)Toby Ord, "Lessons from the development of the atomic bomb," Working paper.
4310c3e2-1b39-4967-958c-3221a1ec707f	StampyAI/alignment-research-dataset/lesswrong	LessWrong	A closer look at chess scalings (into the past) Introduction ============ I had explored [measuring AI or hardware overhang](https://www.lesswrong.com/posts/75dnjiD8kv2khe9eQ/measuring-hardware-overhang) in August 2020 using chess. Hardware overhang is when sufficient compute is available, but the algorithms are suboptimal. I examined the strongest chess engine of 2020, Stockfish 8, performing at 3,400 ELO under tournament conditions. When reducing compute to 1997 levels (equivalent to a Pentium-II 300 MHz), its ELO score was still ~3,000. That is an important year: In 1997, the IBM supercomputer "Deep Blue" defeated the world chess champion Gary Kasparov. With Stockfish, no supercomputer would have been required. I estimated that SF8 drops to Kasparov level on a 486-DX4 100 MHz, available already in 1994. To sum it up, the hardware overhang in chess is about 10 years, or 2-3 orders of magnitude in compute. About a year later, in July 2021, [Paul Christiano asked similar questions](https://www.lesswrong.com/posts/H6L7fuEN9qXDanQ6W/how-much-chess-engine-progress-is-about-adapting-to-bigger): How much compute would the old engine need to match the current engines? What is the influence of RAM (size and speed), opening books, endgame tables, pondering? Also, my old post gave some insights, but it can be improved by sharing the sources and making it reproducible. That's the aim of the current post (the other questions will be adressed in a later post). Reproducing chess scaling from 2020 =================================== History of PC Programs (ELO by year) ------------------------------------ As a baseline of engine performance over the years, we plot the winner from the yearly [rating list of the Swedish Chess Computer Association](https://en.wikipedia.org/wiki/Swedish_Chess_Computer_Association). Run on contemporary hardware, * The list begins in 1984 when the program "Novag Super Constellation" reached 1631 ELO running on a 6502 CPU at 4 MHz. * By 2005, Shredder 9 surpassed human levels on an AMD Athlon 1200 MHz. * Today (2020), the leading engine is Stockfish 12 running on an AMD 1800X at 3.6 GHz. Human grandmasters ------------------ To compare human grandmasters, we take the ELO over time for [Kasparov](http://chessmetrics.com/cm/CM2/PlayerProfile.asp?Params=199510SSSSS3S062926000000151000000000000010100) and [Carlsen](https://ratings.fide.com/profile/1503014/chart). Carlsen's rating between 2003 and 2011 (age 13 to 21) grew from 2000 ELO to grandmaster strength, faster than any engine :-) [Thanks to User Bucky for the correction] Deep Blue --------- The marker for "Deep Blue" in the year 1997 is a bit [arbitrarily](https://www.reddit.com/r/chess/comments/7bm361/deep_blues_true_elo_rating/) set to 2900 ELO. At the time, Kasparov had 2860 ELO, Deep Blue won, although close. Stockfish 8 experiment ---------------------- The main part is the Stockfish 8 experiment. How well does SF8 perform on slower PCs? ### As a baseline, we need to establish its ELO at a defined speed. 1. To obtain the speed baseline, we find that [SF8 makes 721 kNodes/s on an AMD Athlon 64 3500+ at 2.20 GHz](https://sites.google.com/site/computerschess/stockfish-chess-benchmarks). 2. We scale this linearly to 777 kNodes/s for the same CPU running at 2.4 GHz (+9%) 3. [SF8 achies 3302 ELO on an Athlon 64 X2 4600+ (2.4 GHz)](http://www.computerchess.org.uk/ccrl/4040/rating_list_all.html) in the CCRL Rating List, running 40 moves in 15 minutes (one has to dig into the side details to understand which CPU name tag is which CPU. 64bit 1 CPU is the Athlon; this can also be verified with the historical version of that list.). This is an important baseline, because it cross-calibrated to dozens of other engines. 4. With that established, we can calculate the ELO as a function of kNodes/s. An average game has 40 moves. The 40 moves in 15 minutes leave 22.5 seconds per move (on average). That's 17.5 MNodes per move to achieve 3302 ELO. 5. We benchmark our own machine, on which the experiments are run. This can be done with the Stockfish parameter "bench". For simplicity, suppose our machine performs at 10 x 777 kNodes/s = 7.8 MNodes/s. That's the ballpark of recent (2020) 4-core CPUs. 6. Now we want to perform a game at 17.5 MNodes per move, on a machine running at 7.8 MNodes/s. Clearly, each move can only take 2.24 seconds. The whole 40-game match duration is: 90 seconds. ### Execute the experiment To build a ladder of SF towards slower machines, we let this version of SF8 play a set of games of 90s timecontrol versus half that (45s). The most well-established tool to compare chess engines is [cutechess-cli](https://github.com/cutechess/cutechess). It is a command-line interface to play two engines (or two versions of the same engine) against each other. In the end, it nicely summarizes the results and includes a differential ELO estimate. A command may be: ``` cutechess-cli -fcp cmd=stockfish proto=uci tc=40/90 -scp cmd=stockfish proto=uci tc=40/45 -games 100 ``` How bad does the version perform with less compute? In this experiment, after running 100 games, we get 14 ELO difference. That's much less than the usual statement of 70 ELO. Why is that? We can see the same effect in similar experiments down by others ([1](http://www.talkchess.com/forum3/viewtopic.php?t=72834), [2](http://chess.ultimaiq.net/scalability.htm)): The ELO gain diminishes (flattens) at high compute. On the other hand, when we reduce compute to very low levels, the curve steepens dramatically. The full ELO loss result list from my experiment is for each halfing of compute: \| \| \| \| \| --- \| --- \| --- \| \| ELO \| ELO Delta \| kNodes/move \| \| 3302 \| \| 17476.4 \| \| 3288 \| 14 \| 8738.2 \| \| 3268 \| 20 \| 4369.1 \| \| 3240 \| 28 \| 2184.5 \| \| 3205 \| 35 \| 1092.3 \| \| 3097 \| 108 \| 546.1 \| \| 3030 \| 67 \| 273.1 \| \| 2977 \| 53 \| 136.5 \| \| 2802 \| 175 \| 68.3 \| \| 2716 \| 86 \| 34.1 \| \| 2439 \| 277 \| 17.1 \| \| 2238 \| 201 \| 8.5 \| \| 1903 \| 335 \| 4.3 \| There is some jitter, despite increasing the number of games to 1,000 in the second half. Despite the jitter, we can clearly see the nonlinear ELO curve with compute: ![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/e5d2989018f9c9290d580ae3b628b50cb0c549f4d41fd54a.PNG)The last thing we need to do is match the kNodes/move results to the old years. We may ask: In which year was the hardware available sufficient to play these kNodes/move in a usual tournament? This leaves some room for discussion. For 1997, should we choose a dual Pentium Pro 200, or a single Pentium 200 MMX? I believe it is reasonable to compare good CPUs of the time, without going overboard. After all, we're comparing chess on home computers. If we restrict it to <1000 USD CPUs for each year, we can find some SF8 benchmarking results across the web: - AMD 5950X (2021): [71,485 kNodes/s](http://ipmanchess.yolasite.com/amd---intel-chess-bench.php) - Pentium III 500 MHz (1999): [127 kNodes/s](https://sites.google.com/site/computerschess/stockfish-chess-benchmarks) - Pentium 75 MHz (1995): [6.2 kNodes/s](https://sites.google.com/site/computerschess/stockfish-chess-benchmarks) - 386DX-33 MHz (1989): [1 kNode/s](http://talkchess.com/forum3/viewtopic.php?f=2&t=63857&start=10) There are many more such measurements found online, but for our purpose, this is sufficient. Caveats: * Going back very far in time becomes difficult, because SF8 [needed to be recompiled to reduced instruction sets to make it work](http://talkchess.com/forum3/viewtopic.php?f=2&t=63857); and RAM was limited in the experiment. * It is more reasonable to match the speed to more recent years: About 200 kNodes/s in 2000, and 100 MNodes/s today. Everything before, and in between, has a factor of a few of error in its match of nodes to year. * On the other hand, seeing benchmarks of real PCs is useful, because it encompasses uncertainties such as RAM speed. * In reality, when considering hardware overhang for future AI, we must also ask: How well could SF8 be adapted to older hardware? Just running it unchanged will leave some performance (factor of a few?) on the table. That's a question for software engineers and compiler optimizers. We can now bring the approximate match of Nodes/s with the years together with the other data, and present the result: ![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/ba53fef112891a867d1829c9e72dfb9c1861c6e5e61c52ee.PNG)This looks quantitatively different to my [first version](https://www.lesswrong.com/posts/75dnjiD8kv2khe9eQ/measuring-hardware-overhang), but is qualitatively similar. * Again, a hardware overhang of ~10 years at maximum is visible: SF8 achieved Kasparov level in 1997 * This was only possible for contemporary PC engines of the year ~2006. * In my old version, this was more like a 15 years gap. Back then, I had matched the speed to MIPS values for CPUs I found online. * It is probably better to measure SF kNodes/s directly instead using a CPU speed proxy (MIPS, FLOPs, SPEC). Thus, I believe that the new figure is closer to reality. In the next post, I will consider the other questions asked by Paul Christiano: How much compute would an old engine need to match current engines? What is the influence of opening books, endgame tables, pondering? Edit (15 July): Magnus Carlsen time series fixed
27c280a3-eec3-4903-83cf-b9d3a27d970b	trentmkelly/LessWrong-43k	LessWrong	Can we create a function that provably predicts the optimization power of intelligences? Follow up to Efficient Cross-domain Optimization When I am skeptical that we will ever understand intelligence, I am skeptical that we will ever be able to reliably map a systems description onto its optimization power. This has implications for how well we will create intelligences and how well intelligences will be at self-improving. Obviously we can't predict the effectiveness of an arbitrary program, due to rice's theorem and intelligence being a non-trivial property. So the best we can hope for is predicting the effectiveness of a set of programs. Is such a function possible? This is my take on the subject. Let o ( p ) be a function that maps a program p to its optimization power. Mu, Omegas younger brother has a challenge for you, you get to design a system and put it in a box with 20 red and 20 green balls, it will activate itself after 10 minutes and then have the goal of removing as many red balls from the box as possible in 10 minutes. You have to decide how whether it is going to remove more or less than 5 red balls from the box. You get transported to a nirvana if you predict correctly and your world gets turned into paper clips if you get it wrong. You whip out your trusty o and make a program and the evaluate it using o and bet according to its evaluation. Unknown to you Mu also has a copy of your o and runs it on the systems you put in the box. Those that return a high value from the optimization power measure, it destroys before they activate, those that have a low effectiveness it performs their goals for them. In the second case it is still p that causes the goal to be fulfilled as if p were different there would be a different amount that the goal is fulfilled. You can see it as inspiring pity in someone else to make them help, who would not have done otherwise. It is still winning. So Mu forces o to be wrong, so o was not the reliable predictor of a set of programs optimization power we had hoped for, so we have a contradiction. Is
852c544e-91e4-43b0-a92a-01f6fcfee6a5	trentmkelly/LessWrong-43k	LessWrong	Using machine learning to predict romantic compatibility: empirical results Overview For many people, having a satisfying romantic relationship is one of the most important aspects of life. Over the past 10 years, online dating websites have gained traction, and dating websites have access to large amounts of data that could be used to build predictive models to achieve this goal. Such data is seldom public, but Columbia business school professors Ray Fisman and Sheena Iyengar compiled a rich and relevant data set for their paper Gender Differences in Mate Selection: Evidence From a Speed Dating Experiment. Their main results were: Women put greater weight on the intelligence and the race of partner, while men respond more to physical attractiveness. Moreover, men do not value women’s intelligence or ambition when it exceeds their own. Also, we find that women exhibit a preference for men who grew up in affluent neighborhoods. Finally, male selectivity is invariant to group size, while female selectivity is strongly increasing in group size. I found the study through Andrew Gelman’s blog, where he wrote: What I really want to do with these data is what I suggested to Ray and Sheena several years ago when they first told me about the study: a multilevel model that allows preferences to vary by person, not just by sex. Multilevel modeling would definitely be useful here, since you have something like 10 binary observations and 6 parameters to estimate for each person. Several months ago I decided to pursue a career in data science, and with a view toward building my skills, I worked to build a model to predict when an individual participant will express interest in seeing a given partner again. Along with the goal of learning, I had the dual intent of contributing knowledge that had the potential, however slight, to help people find satisfying romantic relationships. It’s unlikely that what I did will have practical applications (as basic research seldom does), but I did learn a great deal about many things, most having to do with data s
fcc80ece-61d8-4b61-8407-8bc8307350ca	trentmkelly/LessWrong-43k	LessWrong	LW is to rationality as AIXI is to intelligence Apparently LW does a great job on refining rationality and dissolving confusions. But is it helpful when it comes to anything apart from designing Friendly AI, apart from a purely academic treatment of rationality? I'm currently unable to benefit from what I have so far read on LW, it actually made me even more unproductive, to an extent that I get nothing done anymore. Let me explain... You have to know that I'm still in the process of acquiring a basic education. If I say basic, I mean basic. Since I got almost no formal education, what I do know (or know about) is largely on a very low level, yet I am plagued by problems that are themselves on a level that require the intellect and education of the folks here on LW. The problem with that is that I'm yet lacking most of the skills, tools and requisite know-how while the problems in question concern me as well. This often causes me to get stuck, I can't decide what to do. It also doesn't help much that I am the kind of person who is troubled by problems others probably don't even think about. An example from when I was much younger (around the age of 13) is when I was troubled by the fact that I could accidentally squash insects when walking over grass in our garden. Since I have never been a prodigy, far from it, it was kind of an unsolvable problem at that time, especially since I am unable to concentrate for very long and other similar problems are accumulating in my mind all the time. So what happened? After a time of paralysis and distress, as it happens often, I simply became reluctant and unwilling, angry at the world. I decided that it is not my fault that the world is designed like that and that I am not smart enough to solve the problem and do what is right. I finally managed to ignore it. But this happens all the time and the result is never satisfactory. This process too often ends in simply ignoring the problem or becoming unwilling to do anything at all. What I'm doing is not effective it seems, it a
cf5cdff8-dbd1-4eb9-889d-9007eaa20452	trentmkelly/LessWrong-43k	LessWrong	Sam Altman's sister claims Sam sexually abused her -- Part 2: Annie's lawsuit; the response from Sam, his brothers, and his mother; Timeline Previous posts (which you should read first) This post is the 2nd post in a series of 11 posts about the claims of Sam Altman's sister, Annie Altman. Annie has claimed that Sam sexually abused her for about 9 years as a child, and that she experienced further (non-sexual) abuse from Sam, her brothers, and her mother after that. The 11 posts are meant to be read in order. So, if you haven't read the first post, please read it before you read this post: * Sam Altman's sister claims Sam sexually abused her -- Part 1: Introduction, outline, author's notes ---------------------------------------- Annie's lawsuit, and the response from Sam, his brothers, and his mother On January 6, 2025, Annie Altman filed a lawsuit against Sam Altman in the United States District Court for the Eastern District of Missouri, Eastern Division. The lawsuit's case name is Altman v. Altman, and its case number is 4:25-cv-00017. The lawsuit is ongoing. A jury trial is set to begin Monday, March 31, 2025. See: * https://www.courtlistener.com/docket/69520118/altman-v-altman/ * Especially: * https://www.courtlistener.com/docket/69520118/1/altman-v-altman/ -- "COMPLAINT against defendant Samuel Altman with receipt number AMOEDC-11018479, in the amount of $405 Jury Demand,, filed by Ann Altman. (Attachments: # 1 Civil Cover Sheet Civil Cover Sheet, # 2 Original Filing Form Original Filing Form, # 3 Summons Summons to be Issued)(Mahoney, Ryan) (Entered: 01/06/2025)" * https://www.courtlistener.com/docket/69520118/16/altman-v-altman/ -- "MOTION to Dismiss Plaintiff's Common-Law Claims and Plaintiff's Prayer for Punitive Damages by Defendant Samuel Altman. (Magee, Thomas) (Entered: 03/07/2025)" * https://www.courtlistener.com/docket/69520118/17/altman-v-altman/ -- "MEMORANDUM in Support of Motion re 16 MOTION to Dismiss Plaintiff's Common-Law Claims and Plaintiff's Prayer for Punitive Damages filed by Defendant Samuel Altman. (Magee, Thomas) (Entered: 03/07/2025)"
dea52d41-9635-4d5a-abab-af618cc6bb7a	trentmkelly/LessWrong-43k	LessWrong	"Arbitrary" Followup to: Inseparably Right; or, Joy in the Merely Good, Sorting Pebbles Into Correct Heaps One of the experiences of following the Way is that, from time to time, you notice a new word that you have been using without really understanding. And you say: "What does this word, 'X', really mean?" Perhaps 'X' is 'error', for example. And those who have not yet realized the importance of this aspect of the Way, may reply: "Huh? What do you mean? Everyone knows what an 'error' is; it's when you get something wrong, when you make a mistake." And you reply, "But those are only synonyms; what can the term 'error' mean in a universe where particles only ever do what they do?" It's not meant to be a rhetorical question; you're meant to go out and answer it. One of the primary tools for doing so is Rationalist's Taboo, when you try to speak without using the word or its synonyms—to replace the symbol with the substance. So I ask you therefore, what is this word "arbitrary"? Is a rock arbitrary? A leaf? A human? How about sorting pebbles into prime-numbered heaps? How about maximizing inclusive genetic fitness? How about dragging a child off the train tracks? How can I tell exactly which things are arbitrary, and which not, in this universe where particles only ever do what they do? Can you tell me exactly what property is being discriminated, without using the word "arbitrary" or any direct synonyms? Can you open up the box of "arbitrary", this label that your mind assigns to some things and not others, and tell me what kind of algorithm is at work here? Having pondered this issue myself, I offer to you the following proposal: > A piece of cognitive content feels "arbitrary" if it is the kind of cognitive content that we expect to come with attached justifications, and those justifications are not present in our mind. You'll note that I've performed the standard operation for guaranteeing that a potentially confusing question has a real answer: I su
b59a3242-59d6-429e-bc7a-2951f7322227	trentmkelly/LessWrong-43k	LessWrong	[LINK] Two articles on Bitcoin Tangential, but a subject of some local interest: Why Bitcoin will fail by Avery Pennarun. "The sky isn't red." Thesis: 1. The gold standard was a bad idea. 2. Even if it [Bitcoin] was a good idea, governments will squash it. 3. The whole technological basis (cryptosystem) is flawed. 4. It doesn't work offline. I'm not sure I buy these and am not competent to evaluate his claims on 3., but would like others' critique. L019: Bitcoin P2P Currency: The Most Dangerous Project We've Ever Seen by Jason Calacanis. A rather more enthusiastic viewpoint of the project: 1. Bitcoin is a technologically sound project. 2. Bitcoin is unstoppable without end-user prosecution. 3. Bitcoin is the most dangerous open-source project ever created. 4. Bitcoin may be the most dangerous technological project since the internet itself. 5. Bitcoin is a political statement by technological libertarians. 6. Bitcoins will change the world unless governments ban them with harsh penalties. The actual text contains many more caveats than the eye-catching selection of points above.
bcf22eca-b545-44e3-8f22-d1e1beaa831f	trentmkelly/LessWrong-43k	LessWrong	[Linkpost] Multimodal Neurons in Pretrained Text-Only Transformers This is a linkpost for https://arxiv.org/abs/2308.01544. > Language models demonstrate remarkable capacity to generalize representations learned in one modality to downstream tasks in other modalities. Can we trace this ability to individual neurons? We study the case where a frozen text transformer is augmented with vision using a self-supervised visual encoder and a single linear projection learned on an image-to-text task. Outputs of the projection layer are not immediately decodable into language describing image content; instead, we find that translation between modalities occurs deeper within the transformer. We introduce a procedure for identifying "multimodal neurons" that convert visual representations into corresponding text, and decoding the concepts they inject into the model's residual stream. In a series of experiments, we show that multimodal neurons operate on specific visual concepts across inputs, and have a systematic causal effect on image captioning.
571b1627-deb4-4d78-8552-8bec85425122	trentmkelly/LessWrong-43k	LessWrong	What are the limits of self-education? I have exhausted myself thinking of this. Why should I invest time in x if I don't even know what my y is. Traditional teaching or mentoring approaches objectives by defining what a student must fulfill to truly understand and grasp a topic(s) thoroughly. This can also be translated to "end of term" projects, things that combine the fundamentals that underlie the subject(s) or exercises at the end of sections. Most of these examples are situated in a academic environment, where one feels comfortable enough to be guided and bound to deadlines. Whereas, this is tough to translate outside an academic environment. My own experience: I recently was interested in learning and teaching myself programming and CS. I began searching through the CS curriculums given by ivy league universities, saw the lecture notes and the core texts, I dared to even look at the exams. The instructors assignments and the core readings were absolutely brutal, even for undergraduates. Part of me envied the students who'll improve and progress on a whole other level. I mean look at the assignments they're given! What's insanely difficult for me will be infinitesimal to them. Eventually I gave up on studying, my efforts and what I consider beneficial to future me are all unrealistic standards.
6fe3744b-c747-4c68-9876-9ae3bb514d5e	trentmkelly/LessWrong-43k	LessWrong	Anyone Else Using Brilliant? I started using Brilliant, and so far I've found it to be a lot like Thinking Physics - teaching by posing real-world conundrums and then explaining the concepts and/or math behind the answers. Anyone else getting something out of it, or have advice for how to use it?
2def12d9-f06b-4028-8b3e-a37ac78e7be7	StampyAI/alignment-research-dataset/alignmentforum	Alignment Forum	Compositional preference models for aligning LMs This post summarizes the main results from our recently released paper[Compositional preference models for aligning LMs](https://arxiv.org/abs/2310.13011) and puts them in the broader context of AI safety. For a quick summary of the paper, take a look at our[Twitter thread](https://twitter.com/dongyoung4091/status/1717045681431753097). TL;DR: We propose a new approach to building preference models out of prompted LMs. Compositional Preference Models (CPMs) decompose scoring a text into (1) constructing a series of questions about interpretable features of that text (e.g. how informative it is), (2) obtaining scalar scores for these features from a prompted LM (e.g. ChatGPT), and (3) aggregating these scores using a logistic regression classifier trained to predict human judgements. We show that CPMs, compared with standard preference models (PMs), generalize better and are more robust to reward model overoptimization. Moreover, best-of-n samples obtained using CPMs tend to be preferred over samples obtained using similar, conventional PMs. Finally, CPMs are a novel angle at scalable oversight: they decompose a hard evaluation problem into a series of simpler, human-interpretable evaluation problems. How compositional preference models work? ----------------------------------------- ![](https://res.cloudinary.com/lesswrong-2-0/image/upload/f_auto,q_auto/v1/mirroredImages/oSgac8x8fgNj22ky3/f3vsa8j7pc4prer3msln) Figure 1: While standard PMs output a preference score directly, CPMs score different features of LM responses separately and output a preference score as a linear combination of feature values. Preference Models (PMs) are models trained to assign an LM response a score indicating the quality of the response. They are the workhorse of many techniques for aligning LMs: they are most prominently used as reward functions in RLHF or as ranking models in best-of-n sampling, in addition to playing a role in other techniques such as [pretraining with human feedback](https://www.lesswrong.com/posts/8F4dXYriqbsom46x5/pretraining-language-models-with-human-preferences). Standard PMs involve adding a scalar head on top of a base model and finetuning the whole model (or certain upper layers) to predict which of two texts a human would prefer. While this approach is highly effective in practice, it can lead to uninterpretable models that fit spurious correlations in human preference judgements and are prone to goodharting (overoptimization). We introduce an alternative: Compositional Preference Models (CPM). In contrast to PMs, CPMs decompose response evaluation into the following steps: Feature decomposition. We maintain a fixed list of 13 human-interpretable features (e.g. specificity, relevance, readability) and 13 corresponding prompt templates (e.g. `You will be shown a conversation [...] please judge whether the assistant's reply is relevant. Score that on a scale from 1 to 10 [...] {conversation_history} {reply}`). Feature scoring. We ask an LM (e.g. GPT-3.5) to assign a score to each feature. Each feature of a single response is scored in a separate context window. Aggregation. The feature scores are combined into a scalar preference score using a logistic regression classifier trained to predict human preference judgements (i.e. which of two texts a human would prefer). Robustness to overoptimization ------------------------------ ![](https://res.cloudinary.com/lesswrong-2-0/image/upload/f_auto,q_auto/v1/mirroredImages/oSgac8x8fgNj22ky3/o5db8cfk3giywvfxc9kv) Figure 2: Scores given by a gold PM (solid lines) and a corresponding proxy PM (dashed lines) on samples obtained through best-of-n sampling against the gold PM. CPM-GPT-3.5 and CPM-Flan-T5 refer to CPMs constructed with feature extraction based on GPT-3.5 and Flan-T5, respectively. To investigate if CPM improves robustness to overoptimization, we follow the setup of [Gao et al. (2023)](https://www.lesswrong.com/posts/shcSdHGPhnLQkpSbX/scaling-laws-for-reward-model-overoptimization) and construct a synthetic dataset where the output of one PM (defined to be the “gold PM”) is assumed to be the ground truth for human preferences. We then use the gold PMs to generate synthetic labels to train proxy PMs. We do that separately for three pairs of proxy and gold PMs: (i) standard PMs, (ii) CPMs using GPT-3.5 for feature extraction and (iii) CPMs using Flan-T5-XL (3B params) for feature extraction. Finally, we do best-of-n against a given proxy PM and comparse those best samples’ scores according to both proxy and gold PM. As we increase the amount of optimization pressure (the number of candidates n), scores given by proxy PMs diverge from scores given by gold PMs (see Fig. 2). This is an indicator of preference model overoptimization, a form of reward hacking in which optimization of proxy PM scores is driven by spurious features that the gold PMs are indifferent to. The size of this gap (smaller is better) indicates the robustness of a given PM to being overly optimized against. Here, we observe that the gap (on the plot, between solid and dashed lines) tends to be smaller for CPMs than for standard PMs and that it increases at a slower rate. This indicates that CPMs are more robust to overoptimization than standard PMs. This holds independently of whether a highly capable (GPT-3.5) or less capable (Flan-T5-XL) LM is used as a feature extractor in CPMs. Quality evaluation ------------------ ![](https://res.cloudinary.com/lesswrong-2-0/image/upload/f_auto,q_auto/v1/mirroredImages/oSgac8x8fgNj22ky3/jperxazhwouwuec9wyc2) Figure 3: Win rate of responses obtained via best-of-16 sampling using a given PM versus responses obtained via standard sampling, computed for prompts from Anthropic HH dataset (HH-RLHF) and Stanford Human Preferences dataset (SHP). We compare the quality of LM samples obtained by best-of-16 against either CPMs or standard PMs by comparing them to samples generated without best-of-n sampling. We do that by showing both best-of-16 and vanilla samples to an evaluator LM (Claude 2.0) and by computing win rates, i.e. how often best-of-16 samples are preferred to vanilla samples. CPMs tend to have higher win rates than standard PMs, even if we match the capabilities of a feature extractor LM to the capabilities of standard PM (by choosing Flan-T5-XL for both). This suggests that prior knowledge injected into a PM via pre-selecting interpretable and relevant features in CPMs is robustly helpful for learning about human preferences. CPMs and scalable oversight --------------------------- [Scalable oversight](https://arxiv.org/abs/2211.03540) is the problem of evaluating the behavior of agents more capable than the evaluators. This is important to solve because, on the one hand, LMs will soon grow capable of completing tasks for which humans will not be able to provide feedback. On the other hand, LMs might also be capable of [reasoning about flaws in their evaluation procedures and exploiting them](https://www.lesswrong.com/posts/mLfPHv4QjmeQrsSva/paper-on-measuring-situational-awareness-in-llms) unbeknownst to overseers. Current proposals for solving scalable oversight focus on recursively relying on other LMs to assist human evaluators ([debate](https://www.lesswrong.com/tag/debate-ai-safety-technique-1), [iterated distillation and amplification](https://www.lesswrong.com/tag/iterated-amplification), [recursive reward modeling](https://deepmindsafetyresearch.medium.com/scalable-agent-alignment-via-reward-modeling-bf4ab06dfd84)) but remain largely theoretical. [RL from AI feedback](https://arxiv.org/abs/2212.08073) – using carefully prompted LMs to generate training data for PMs – is arguably the most successful demonstration of how to use LMs to supervise LMs at scale. CPMs explore an alternative route to addressing scalable oversight for LMs, exploring the prospects of divide-and-conquer strategies for tackling hard evaluation problems. CPMs can be seen as a method for decomposing a hard question (“Is this response helpful?”) into a series of simpler questions (“is this response readable?” etc.) that are easier for LMs to answer and easier for humans to oversee. While we stop at a single step of decomposition, nothing in principle prevents us from applying the idea recursively, e.g. to break down evaluation of complex responses into simple questions about atomic claims. The idea of decomposing complex evaluation problems into simpler subproblems has several additional benefits: 1. Using human priors. Pre-selection of features and prompt templates afford a natural way of injecting prior knowledge and endowing PMs with useful inductive biases. The parameters space of CPMs is spanned by features selected to be meaningful and robust. 2. Avoiding reward hacking by limiting PM capacity. Using features pre-computed by feature extractors allows us to dramatically reduce the capacity of PMs consuming them (in our experiments, from 3B to just 13 parameters, i.e. 8 orders of magnitude!) and limit their susceptibility to overfitting to spurious correlations in preference data. It is really hard to reward-hack with only 13 parameters at hand! 3. Interpretability. Pre-selected features are trivially interpretable and a logistic regression coefficient associated with a feature can be interpreted as its salience (effect size) for a particular preference judgment (see sec. 4.6 in the paper). Indeed, the idea that preference judgments can be explained by linear combinations of pre-selected features was recently validated by two concurrent papers: [Towards Understanding Sycophancy in Language Models](https://www.lesswrong.com/posts/g5rABd5qbp8B4g3DE/towards-understanding-sycophancy-in-language-models) and [Human Feedback is not Gold Standard](https://arxiv.org/abs/2309.16349). Using such a linear model as an actual PM makes its judgements more transparent and amenable to process-based supervision. 4. Narrowness. Each of our feature extractors solves a narrow problem and does not need to be aware of other features or how the scores are aggregated. Solving different subproblems in different context windows was [recently found to improve the faithfulness of reasoning](https://www.lesswrong.com/posts/BKvJNzALpxS3LafEs/measuring-and-improving-the-faithfulness-of-model-generated). In the case of CPMs, an individual feature extractor has no clue how the score it is about to assign is going to be used downstream, which makes it harder for it to be strategic about that score and exercise capabilities for [sycophancy](https://www.lesswrong.com/posts/yRAo2KEGWenKYZG9K/discovering-language-model-behaviors-with-model-written) or deception. However, CPMs still have certain limitations that future work could address: 1. Human feedback.CPMs still use pairwise preference judgements given by humans as a training signal for aggregating feature scores. This is inherently limiting as far as humans make errors, [sometimes prefer sycophantic responses over truthful ones](https://www.lesswrong.com/posts/g5rABd5qbp8B4g3DE/towards-understanding-sycophancy-in-language-models) or [authoritative responses over factual ones](https://arxiv.org/abs/2309.16349). 2. Human curation. CPMs rely on humans when it comes to feature selection and prompt engineering of prompt templates for feature extraction. These factors could be limiting as far as out-of-domain generalization is concerned (e.g. to evaluating agents showing superhuman performance). Wrap-up ------- We presented Compositional Preference Models: the idea of building PMs by training logistic regression on top of features extracted by prompted LMs. We show that a CPM with 13 parameters can outperform standard PM in terms of human evaluation and robustness to reward model overoptimization while also being more interpretable. This post benefited from helpful comments made by Mikita Balesni, Richard Ren, Euan McLean and Marc Dymetman. I’m also grateful to the co-authors of[the paper](https://arxiv.org/abs/2310.13011): Dongyoung Go, Germán Kruszewski, Jos Rozen and Marc Dymetman.
3798eb90-82c6-4df4-ac38-60241c76aeab	trentmkelly/LessWrong-43k	LessWrong	Meetup : Sydney Rationality Dojo - November Discussion article for the meetup : Sydney Rationality Dojo - November WHEN: 01 November 2015 04:00:00PM (+1100) WHERE: 10 Shepherd Street, Chippendale Come join us for November's Rationality Dojo. We've been experimenting with the format recently, to keep things moving along, and fit more material in. If you've been away for a while, now is a perfect chance to come back! Afterwards we will go for the usual group dinner, for those who can make it. Discussion article for the meetup : Sydney Rationality Dojo - November
7d030dad-723c-4724-abd7-a1abeda0ceae	trentmkelly/LessWrong-43k	LessWrong	On the Dangers of Time Travel It has been obvious for decades to anyone who understands quantum mechanics that that the scientific consensus on theoretical physics is wrong. General relativity is background independent. Quantum field theory is background dependent. They cannot both be correct. According to the Copenhagen interpretation of thermodynamics, entropy is emergent from time. Anyone with with half a Noble Prize in Physics can tell that this is backwards. Time is emergent from entropy. Everything that can be said to physically exist, from an unobserverable wave function to living cat, is a macrostate. A random walk along microstates evolves in the direction of higher entropy macrostates even in the absence of time. In this way, entropy is more fundamental than time. Putting entropy before time explains why all quantum fields locally maximize proper time. It solves the problem of Baryon asymmetry and dissolves the semantic stopsign at the Big Bang. If you have worked out the details of quantum gravity for yourself then you have seen in your equations exactly how to build a time machine. The best theoretical physicists have managed to suppress this technology. In a historically unprecedented conspiracy, they have not published anything useful for decades. While the physicists buy time, the rest of us must invent systems to maximize the benefits of time travel while minimizing risk. As a consequence of this progress, there is a nonzero Bayesian probability we could turn on a time machine without instantly destroying the multiverse. But no weight of electronic paper discussing time travel safety can prevent a malevolent actor uninterested in safety. As a last line of defense, the cosmic speed limit is below 88 everywhere in the solar system, it is illegal to buy plutonium in a drugstore and phone booths have been quietly disappeared. But we cannot expect such flimsy mechanisms to hold forever as technology advances. We must build a temporal displacer first, before the enemy does.
4c2aa95e-e4ea-4c9e-a5b0-71780ad25b03	StampyAI/alignment-research-dataset/lesswrong	LessWrong	How much to optimize for the short-timelines scenario? Some have argued that one should tend to act as if timelines are short since in that scenario it's possible to have more expected impact. But I haven't seen a thorough analysis of this argument. Question: Is this argument valid and if yes how strong is it? The basic argument seems to be: if timelines are short, the field (AI alignment) will be relatively smaller and have made less progress. So there will be more low-hanging fruits and so you have more impact. The question affects career decisions. For example, if you optimize for long timelines, you can invest more time into yourself and delay your impact. The question interacts with the following questions in somewhat unclear ways: * How fast do returns to more work diminish (or increase)? + If returns don't diminish, the argument above fails. + If the field will grow very quickly, returns will diminish faster. * Is your work much more effective when it's early? + This may happen because work can be hard to parallelize - ‘9 women can't make a baby in 1 month’. And field-building can be more effective earlier as the field can compound-grow over time. So someone should start early. + If work is most effective earlier, you shouldn’t lose too much time investing in yourself. * Is work much more effective at crunch time? + If yes, you should focus more on investing in yourself (or do field-building for crunch time) instead of doing preparatory research. * If timelines are longer, is this evidence that we'll need a paradigm shift in ML that makes alignment easier/harder? + (This question seems less tractable than the others.) * Is your comparative advantage to optimize for short or long timelines? + For example, young people can contribute more easily given longer timelines and vice versa. If someone would like to seriously research the overall question, please reach out. The right candidate can get funding.
872e17c9-a4e6-4778-81d7-d4b2411f5083	trentmkelly/LessWrong-43k	LessWrong	Overview of introductory resources in AI Governance Overview of introductory resources in AI Governance This post was created as part of the Supervised Program for Alignment Research, Spring 2024. This work would not have happened without the encouragement and accountability of my supervisor, Peter Gebauer. Introduction The AI Governance ecosystem is large and difficult to apprehend. There are tons of content, relevant organizations, introductory resources, newsletters and more. As a newcomer to this field, I found it hard to navigate the ecosystem and find the information I needed. I discovered lots of resources purely by chance, months after they could have been useful for me. What I felt was missing was introductory resources that would not only introduce the “type of work” that is AI Governance, but also direct me towards the resources I needed at different times. Desiderata: The perfect entry point to the ecosystem would allow me, no matter my background and my intentions, to find the resources I need. The technical alignment ecosystem on the other hand seems far more organized, in no small part thanks to the Alignment Ecosystem Development team, which created tons of introductory resources, indexes, and other variations on the theme “list of links to useful stuff”. Since I discovered AI alignment two years ago, the resources AED created have helped me navigate the ecosystem and find opportunities I would have missed. I expect similar resources to also be valuable for AI governance. I decided to investigate thoroughly what were the various introductory resources to AI Governance. Maybe the resources actually existed, and I just did not know where to find them? I compiled my findings below, to help newcomers find those resources faster, and hopefully to motivate others to fill in the gaps where resources are lacking. Hopefully, someone will get motivated to build the ultimate entry point to AI Governance! Index of AI governance introductory resources There are various kinds of resources which could be la
ca851cf8-37d2-4335-84ac-c8c0bed1282f	trentmkelly/LessWrong-43k	LessWrong	Value of Querying 100+ People About Humanity's Future I am looking for advice on whether I should do what I've outlined below (is it valuable or generally a waste of time) and, if so, how I should go about doing it. If there is support for something like this, I may apply for funding and do this sometime during Q4 2023. * Go to a frequented environment in NYC (e.g., Time Square; I use NYC because I live close to this city and presumably would be the one doing the actions I'm describing) * Bring a notepad / clipboard and don an official or professional appearance (e.g., a suit) * Bring a sign with some message akin to: * "Quick, Paid Survey About Humanity" * "$10 for 10 Minutes, Survey" * "Questions About Our Future" * Bring a camera w/ tripod and other necessary recording equipment, maybe a mic * Being recording short interviews (~3-10 minutes). * For person 1 to ~100: * Ask if they'd be willing to answer some questions about humanity (3-10 minutes) for $10 and be recorded on video * Record person speaking, and then ask some or all of the following questions or those similar to the following: * If you had 10 billion USD to help the world, how would you spend it? * What are the most important problems for humanity to address right now? * What might be the important problems for humanity to address in 2050? * What are the most important issues for the USA right now? * If you had to send a short message to humanity in 2100, what would it be? * Rank these in terms of how risky them seem (have these on the clipboard): * [nuclear war], [space weather / collisions], [climate change], [pandemics], [artificial intelligence], [global conflict] * How are you altruistic, and how altruistic should people be? * What does this picture [pale blue dot] make you think about? * [Maybe some question about human enhancement / treatment via gene-editing] * Ask some or all of the following demographic questions * Where did you grow up? * How much sch
b2e62a4d-8891-4be0-84d6-97f631473a93	trentmkelly/LessWrong-43k	LessWrong	Enumerating objects a model "knows" using entity-detection features. Introduction Research on Sparse Autoencoders (SAEs) has identified "known entity" features in language models - features that activate when the model processes entities it "knows." If we can find the circuit models use to recognise that they know an entity, then by computing which inputs would trigger the circuit, we can extract a list of "known entities". The aim is to do this in a mostly dataset free way, bootstrapping from a small number of known entities to make guesses for what components are involved in the circuit, using the insights extracted from SAEs to guide circuit discovery, but not using SAEs or a large dataset of text to find these entities. So the SAE tells us the "known entity" feature exists, but once we know it exists, we don't need SAEs to localize the circuit for computing this feature. In theory, given the "known entity" feature, you could run all possible inputs through the model to extract a list of "known entities". But this is inefficient, so the approach is to find mathematical simplifications in the circuit that let us speed up the process. I focus on GPT2-Small's first layer, where certain neurons appear to distinguish between "known" and "unknown" bigram proper nouns. By developing a simple model of how these neurons make this distinction, we can quickly filter for entities the model recognizes. These neurons aren't perfect, and there are examples of false positives and false negatives, but they give quite a large list of bigrams, with relatively low noise compared with other methods. While the approach is currently quite crude, the results are surprisingly clean. This post serves as a proof of concept for a more ambitious project extracting model knowledge through mechanistic interpretability. 'Known' Proper Noun Neuron: In the first layer of GPT2-Small, three attention heads (3, 4, and 7) have local positional kernels that process n-grams and local context. These heads are prime candidates for identifying circuits that recognize
faf7a0bc-a28c-47cf-bdc3-8f42c2241050	StampyAI/alignment-research-dataset/lesswrong	LessWrong	ACI#6: A Non-Dualistic ACI Model Most traditional AI models are dualistic. As [Demski & Garrabrant have pointed out](https://www.lesswrong.com/s/Rm6oQRJJmhGCcLvxh/p/i3BTagvt3HbPMx6PN), these models assume that an agent is an object that persists over time, and has well-defined input/output channels, like it's playing a video game. In the real world, however, agents are embedded in the environment, and there's no well-defined boundary between the agent and the environment. That's why a non-dualistic model is needed to depict how the boundary and input/output channels emerge from more fundamental notions. For example, in Scott Garrabrant's [Cartesian Frames](https://www.lesswrong.com/posts/BSpdshJWGAW6TuNzZ/introduction-to-cartesian-frames), input and output can be derived from "an agent's ability to freely choose" among "possible ways an agent can be". However, choosing is still one of the key concepts of Cartesian Frames, but from a non-dualistic perspective, "it's not clear what it even means for an embedded agent to choose an option", since an embedded agent is "the universe poking itself". Formalizing the idea of choice in a non-dualistic model is as difficult as formalizing the idea of free will. To avoid relying on the notion "choosing", we have proposed the General Algorithmic Common Intelligence (gACI) model which describes embedded agents solely from a third-person perspective, and measures the actions of agents using mutual information in an event-centric framework. The gACI model does not attempt to answer the question "What should an agent do?". Instead, it focuses on describing the emergence of the agent-environment boundary, and answering the question "Why does an individual feel like it's choosing?" In the language of decision theory, gACI belongs to descriptive decision theory rather than normative decision theory. Communication Channel and Mutual Information ------------------------------------------------ In dualistic intelligence models, an agent receives inputinformation from the environment, and manipulates the environment through outputactions. But real-world agents are embedded within the environment, it's not easy to confine information exchange to a clear input/output channel. In the gACI model, on the other hand, the input/output channel is a communication channel, in which the information transfer between a sender and a receiver is measured by mutual information. ![](https://res.cloudinary.com/lesswrong-2-0/image/upload/f_auto,q_auto/v1/mirroredImages/FRd6nNj3M33w2CSX5/fv0swt2m94bctlwpgrpj) Figure 1: From the dualistic input/output model to the mutual information model. We can easily define mutual information between the states of any two objects, without specifying how the information is transmitted, or who is the sender and who is the receiver, or what the transmission medium is, or whether they are direct or indirect connected. These two objects can be any parts of the world, such as agents, the environment, or any parts of agents or the environment, whose boundaries can be drawn anywhere if necessary. They can even overlap. Having mutual information does not always mean knowing or understanding, but it provides an upper bound for knowing or understanding. With mutual information of two objects, we can define memories and prophecies. Memory and Prophecy ----------------------- Memoryis information about the past, or a communication channel that transmits information about the past into the future ([Gershman 2021](https://drive.google.com/file/d/1t_npcCLGVO3Dr01sDVxd_KDp0xDv-yi2/view)). If A is the receiver and B is the sender, we can define: A's memory of B is the mutual information between the present state of A and a past state of B: M(A,B,t)=I(A(t0);B(t)),t<t0.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0} .MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0} .mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table} .mjx-full-width {text-align: center; display: table-cell!important; width: 10000em} .mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0} .mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; 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src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')} @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold} @font-face {font-family: MJXc-TeX-vec-Bw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')} A can have memories about more than one Bs, or about different moments of B. It can also have memories about itself, in other words, A can be equal to B. Prophecy is the mutual information between the present state of A and a future state of B: P(A,B,t)=I(A(t0);B(t)),t>t0 Obviously, the prophecy will not be confirmed until the future state of B is known. A prophecy can be either a prediction about the future, or an action that controls/affects the future. In the language of the [Active Inference](https://direct.mit.edu/books/oa-monograph/5299/Active-InferenceThe-Free-Energy-Principle-in-Mind) model, it's either "change my model" or "change the world". It's not necessary to prefer one interpretation over another, because different interpretations can be derived in different situations, which will be explained in the later chapters. ![](https://res.cloudinary.com/lesswrong-2-0/image/upload/f_auto,q_auto/v1/mirroredImages/FRd6nNj3M33w2CSX5/o2ncra8iua87bucaqcjj) Figure 2: Object A can have both memories and prophecies about object B. Collect Memories for Prophecies ----------------------------------- We won't be surprised to find that most, if not all, objects that have prophecies about object A also have memories about it, although the reverse is not always true. We can speculate that information about the future comes from information about the past. For example, if you know the position and velocity of the moon in the past, you can have a lot of information about its position and velocity in the future. Not all information is created equal. Using our moon as an example, information about its position and phase contains more information about its future, while the pattern of foam in your coffee cup contains less. (Although computing power plays an important role in processing information from memory to prediction or control, we only consider the upper bound of prophecy as if we had infinite computing power. ) Objects with different memories would have different prophecies about the same object. For example, an astronomer and an astrologer would have different information about the future of the planet Mars because of their different knowledge of the universe. Intelligence needs prophecy to survive and thrive, because to maintain its homeostasis and achieve its goals, it needs sufficient information about the future, especially about its own future. In order to obtain prophecies about itself, one should collect memories about itself and the world that are useful for predicting or controlling its own future. Autonomy and Heteronomy --------------------------- We can measure the degree of autonomy of an object by how much prophecy it has about a future of itself, which indicates its self-governance and independence. Da(A,t)=P(A(t0),A(t)) Similarly, we can measure the degree of heteronomy of object A from object B by how much prophecy B has about A, which indicates A's degree of dependence on B. Dh(A,B,t)=P(B(t0),A(t)) An object that has considerable autonomy can be considered an individual or an agent. The permanent loss of autonomy is the death of an individual. Death is often the result of the permanent loss of essential memories that can induce prophecies about itself. Focusing on different types of information requires different standards for autonomy and death. For example, a human neuron has some autonomy over its metabolism, but the timing and strength of its action potential depends mostly on other neurons. We can think of it as an individual, but it is better to think of it as a part of an individual when studying intelligence. Because the death of a single neuron has little effect on a person's autonomy, but the death of a person does. ![](https://res.cloudinary.com/lesswrong-2-0/image/upload/f_auto,q_auto/v1/mirroredImages/FRd6nNj3M33w2CSX5/u5iafy5zmbajbwnztx6x)Figure 3: Autonomy and Heteronomy The Laws of the Mind ------------------------ As an individual accumulates more and more memories and prophecies, it can discover general rules about the world, which are relationships between the past and the future. During this rule-learning process, the boundary between the self and the outside world emerges. One will inevitably find out thatsome parts of the world follow different rules than other parts, and these special parts are spatially concentrated around itself. We can call this special part the body. For example, an individual may acknowledge that its body temperature has never been very high, like, say above 1000K, and predict that its body will never experience a temperature above 1000K, if its future is under control. Since some other objects can have a temperature of 1000K, it will conclude that there must be some special rules that prevent one's body from getting too hot. We call these rules goals, motivations, or emotions, etc. The intuitive conclusion is that your body follows some rules that are different from the rules that other objects follow. This is what people call dualism: the body follows the laws of the mind, which uses concepts like goals, emotions, logic, etc., while the outside world doesn't. However, the exact boundary between the body and the environment is not very clear. The space surrounding the body may partly follow the laws of the mind and can be called peripersonal space, a term borrowed from psychology. (Closer examination will reveal that the body and peripersonal space also follow the same scientific laws as the outside world, and the laws of mind are some additional laws that only bodies follow.) ![](https://res.cloudinary.com/lesswrong-2-0/image/upload/f_auto,q_auto/v1/mirroredImages/FRd6nNj3M33w2CSX5/yglpx2i2jgom0dug2hdn)Figure 4: Everything in the universe follows the laws of physics, but additionally, one's body and peripersonal space follow the laws of the mind. Dualism and Survival Bias ----------------------------- Why do our bodies seem to follow the laws of mind, if bodies are made of the same atoms as the outside world? Consider a classic example of survival bias. During World War II, the statistician [Abraham Wald examined the bullet holes in returning aircrafts](https://en.wikipedia.org/wiki/Survivorship_bias#Military) and recommended adding armor to the areas that showed the least damage, because most aircrafts damaged in those areas could not return safely to base. This survival bias could be overcome by observing the aircraft on the battlefield instead of at the base, where we could find out that the bullet holes are evenly distributed throughout the aircraft, since the survival of the observer is independent of the location of the bullet holes. Because a survival bias is introduced when the observer's survival is not independent of the observed event. We can speculate that if an event is in principle dependent on the observer's own survival, there will be a survival bias that can't be overcome. For example, one's own body temperature is not independent of one's own survival, but the body temperature of others can be. Unlike the pattern of bullet holes in returning aircrafts, the inherent survival bias, including numerous experiences of how to survive, can accumulate in the observer's memory, like the increased armor in the critical areas of an aircraft. We call the memories of accumulated survival bias the inward memory, and the memories of the external world, whose survival bias can be overcome, the outward memory. The laws of the mind, such as goal-directed mechanisms, can be derived from the inward memory. The observer may find that (almost) everything in the outside world has a cause, but its own goal-driven survival mechanism, such as an aversion to hot temperature, or enhanced armor, has no cause other than the rule "the survival of itself depends on the survival of itself", or the existence of itself. Then the observer comes to a conclusion: I have a goal, I have made a choice.
3dc7a262-494d-45cc-aae4-d975fd5f2b8e	trentmkelly/LessWrong-43k	LessWrong	Motivation research presentation I did a presentation on motivation and procrastination research to the Seattle meetup group and an exercise trying to apply the material to a real life example. Eight people came. They were a skeptical bunch and questioned me on exactly the parts I am most interested in an know the least about: how exactly scientists assess the psychological quantities (expectancy, value, delay and impulsiveness). I'd like to learn more about the research and be able to give such presentations to others in the future. I'd also like to record a presentation like it and put it up on the internet. People seemed to think the exercise was pretty valuable. It was also fairly fun. The presentation is here, the exercise is here and here. Luke's suggestion for how to learn how psychologists assess expectancy, value and delay was > As for how scientists assess the relevant psychological qualities, and for why the 'procrastination equation' is taken seriously, all the references are provided in my post 'How to Beat Procrastination'. I also uploaded quite a few of the studies myself so anyone who is actually interested can check the data for themselves. (Prediction: Almost nobody will.) > The papers in footnote 6 are the place to start, for they explain why the equation (called temporal motivation theory by researchers) was developed to predict experimental results, and those papers point to all the individual studies which show how scientists assess expectancy, value, delay, and impulsiveness. For example, 'expectancy' in TMT is measured under a variety of psychological constructs, but largely by measures of self-efficacy and optimism. > There is no short summary of these issues, though Piers Steel's recent book 'The Procrastination Equation' is a decent attempt while being much longer than my article. Psychology is very complicated, and our understanding of it is less certain than our understanding of physics or computer science.
46218f5e-9638-431b-a5c4-578d35c90486	trentmkelly/LessWrong-43k	LessWrong	There is no No Evidence Zvi recently coined, and has now written up this Law of No Evidence: > Law of No Evidence: Any claim that there is “no evidence” of something is evidence of bullshit. Considered next to Eliezer's Absence of Evidence Is Evidence of Absence, it might seem like a contradiction, but as far as I can tell, it actually follows directly. If we treat the "is" in Absence of Evidence is Evidence of Absence as an "implies" (which it seems to me to be) and then apply modus tollens to it, we get "if you don't have evidence of absence, you don't have absence of evidence" and it is precisely this bullshit that Zvi is calling. If you have evidence of absence, say so. The Very Serious covid spokespeople spouting bullshit like "no evidence of human-to-human transmission" are doing a sort of naive equivocation version of "absence of evidence is evidence of absence", by saying "no evidence for" as if that implies "evidence against" when in fact they're trying to support some agenda or worldview when there's not very much clear evidence in any direction and someone could just as easily say "there's no evidence there isn't human-to-human transmission". At least, that's in the best case. Sometimes the evidence does favor a particular direction, they just don't like it and don't want to count it. Desperately clinging to priors? Hm. So Zvi's Law of No Evidence can be seen as taking Absence of Evidence is Evidence of Absence a step further and instead saying "if your absence of evidence is real, then it'll actually be evidence of absence, in which case call it that. otherwise shut up." But there's still a point to be made about a sense in which "absence of evidence" is its own thing—it's just different from no evidence. The important piece is that "no evidence" is bullshit, but "we didn't see this particular thing we would more expect to see if X were true than if X weren't true" is a vital component of successful reasoning about the world. It's the very basis of Bayes. There is no No
a3d04f48-1c57-49d0-b69f-846e10e92cfe	trentmkelly/LessWrong-43k	LessWrong	Value ethics vs. agency ethics Preface I have trouble expressing myself in such a way that my ideas come out even remotely like they sound in my head. So please apply the principle of charity and try to read how you think I thought of it. Tit for Tat Tit for Tat is usually presented in a game between two players where each chooses to either cooperate or defect. The real world game however differs in two important ways. First, it's not a two player game. We make choices not only on our single instance of interaction but also on observed interactions between other players. Thus the Advanced Tit For Tat not only defects if the other player defected against itself but also if it could observe the other player defecting against any other player that employs a similar enough algorithm. Second, there is a middle ground between cooperating and defecting, you could stay neutral. Thus you can harm your opponent, help him or do neither. The question of the best strategy in this real life prisoners dilemma is probably still unanswered. If I see my opponent defecting against some of my peers and cooperating with others, what do I choose? Agency The reason why there even is a game is because we can deliberate on our action and can take abstract thoughts into account that do not directly pertain to the current situation, which I think is the distinguishing factor of higher animals from lower. This ability is called agency. In order to be an agent a subject must be able to perceive the situation, have a set of possible actions, model the outcomes of these actions, value the outcomes, and then act accordingly. We could act in such a way that infringes on these abilities in others. If we limit their ability to perceive or model the situation we call this fraud, if we limit their set of possible actions or their ability to choose between them, we call it coercion, if we infringe on their ability to value an outcome, we call it advertising. Ethics I propose that the purpose of our moral or ethical intuitio
458e06e5-7a5b-4660-888b-99cf554e99de	StampyAI/alignment-research-dataset/lesswrong	LessWrong	Engaging First Introductions to AI Risk I'm putting together a list of short and sweet introductions to the dangers of artificial superintelligence. My target audience is intelligent, broadly philosophical [narrative](/lw/hzt/writing_style_and_the_typical_mind_fallacy/) thinkers, who can evaluate arguments well but who don't know a lot of the relevant background or jargon. My method is to construct a [Sequence](http://wiki.lesswrong.com/wiki/Sequences) mix tape — a collection of short and enlightening texts, meant to be read in a specified order. I've chosen them for their persuasive and pedagogical punchiness, and for their flow in the list. I'll also (separately) list somewhat longer or less essential follow-up texts below that are still meant to be accessible to astute visitors and laypeople. The first half focuses on *intelligence, answering 'What is Artificial General Intelligence (AGI)?'. The second half focuses on friendliness, answering 'How can we make AGI safe, and why does it matter?'. Since the topics of some posts aren't obvious from their titles, I've summarized them using questions they address. --- Part I. Building intelligence.* 1. [Power of Intelligence](http://yudkowsky.net/singularity/power). Why is intelligence important? 2. [Ghosts in the Machine](/lw/rf/ghosts_in_the_machine/). Is building an intelligence from scratch like talking to a person? 3. [Artificial Addition](/lw/l9/artificial_addition/). What can we conclude about the nature of intelligence from the fact that we don't yet understand it? 4. [Adaptation-Executers, not Fitness-Maximizers](/lw/l0/adaptationexecuters_not_fitnessmaximizers/). How do human goals relate to the 'goals' of evolution? 5. [The Blue-Minimizing Robot](/lw/6ha/the_blueminimizing_robot/). What are the shortcomings of thinking of things as 'agents', 'intelligences', or 'optimizers' with defined values/goals/preferences? Part II. Intelligence explosion. 6. [Optimization and the Singularity](/lw/rk/optimization_and_the_singularity/). What is optimization? As optimization processes, how do evolution, humans, and self-modifying AGI differ? 7. [Efficient Cross-Domain Optimization](/lw/vb/efficient_crossdomain_optimization/). What is intelligence? 8. [The Design Space of Minds-In-General](/lw/rm/the_design_space_of_mindsingeneral/). What else is universally true of intelligences? 9. [Plenty of Room Above Us](http://intelligenceexplosion.com/2011/plenty-of-room-above-us/). Why should we expect self-improving AGI to quickly become superintelligent? Part III. AI risk. 10. [The True Prisoner's Dilemma](/lw/tn/the_true_prisoners_dilemma/). What kind of jerk would Defect even knowing the other side Cooperated? 11. [Basic AI drives](http://wiki.lesswrong.com/wiki/Basic_AI_drives). Why are AGIs dangerous even when they're indifferent to us? 12. [Anthropomorphic Optimism](/lw/st/anthropomorphic_optimism/). Why do we think things we hope happen are likelier? 13. [The Hidden Complexity of Wishes](/lw/ld/the_hidden_complexity_of_wishes). How hard is it to directly program an alien intelligence to enact my values? 14. [Magical Categories](/lw/td/magical_categories/). How hard is it to program an alien intelligence to reconstruct my values from observed patterns? 15. [The AI Problem, with Solutions](http://intelligenceexplosion.com/2012/ai-the-problem-with-solutions/). How hard is it to give AGI predictable values of any sort? More generally, why does AGI risk matter so much? Part IV. Ends. 16. [Could Anything Be Right?](/lw/sb/could_anything_be_right/) What do we mean by 'good', or 'valuable', or 'moral'? 17. [Morality as Fixed Computation](/lw/sw/morality_as_fixed_computation/). Is it enough to have an AGI improve the fit between my preferences and the world? 18. [Serious Stories](/lw/xi/serious_stories/). What would a true utopia be like? 19. [Value is Fragile](/lw/y3/value_is_fragile/). If we just sit back and let the universe do its thing, will it still produce value? If we don't take charge of our future, won't it still turn out interesting and beautiful on some deeper level? 20. [The Gift We Give To Tomorrow](http://wiki.lesswrong.com/wiki/User:RobbBB/Tomorrow). In explaining value, are we explaining it away? Are we making our goals less important? Summary: [Five theses, two lemmas, and a couple of strategic implications](http://intelligence.org/2013/05/05/five-theses-two-lemmas-and-a-couple-of-strategic-implications/). --- All of the above were written by Eliezer Yudkowsky, with the exception of The Blue-Minimizing Robot (by Yvain), Plenty of Room Above Us and The AI Problem (by Luke Muehlhauser), and Basic AI Drives (a wiki collaboration). Seeking a powerful conclusion, I ended up making a compromise between Eliezer's original [The Gift We Give To Tomorrow](/lw/sa/the_gift_we_give_to_tomorrow/) and Raymond Arnold's [Solstice Ritual Book](https://dl.dropboxusercontent.com/u/2000477/SolsticeEve_2012.pdf) version. It's on the wiki, so you can further improve it with edits. Further reading: * [Three Worlds Collide](https://dl.dropboxusercontent.com/u/12787472/philosophy/three-worlds-collide-short.pdf) (Normal), by Eliezer Yudkowsky + a short story vividly illustrating how alien values can evolve. * [So You Want to Save the World](/lw/91c/so_you_want_to_save_the_world/), by Luke Muehlhauser + an introduction to the open problems in Friendly Artificial Intelligence. * [Intelligence Explosion FAQ](http://intelligence.org/ie-faq/), by Luke Muehlhauser + a broad overview of likely misconceptions about AI risk. * [The Singularity: A Philosophical Analysis](http://consc.net/papers/singularity.pdf), by David Chalmers + a detailed but non-technical argument for expecting intelligence explosion, with an assessment of the moral significance of synthetic human and non-human intelligence. I'm posting this to get more feedback for improving it, to isolate topics for which we don't yet have high-quality, non-technical stand-alone introductions, and to reintroduce LessWrongers to exceptionally useful posts I haven't seen sufficiently discussed, linked, or upvoted. I'd especially like feedback on how the list I provided flows as a unit, and what inferential gaps it fails to address. My goals are: A. Via lucid and anti-anthropomorphic vignettes, to explain AGI in a way that encourages clear thought. B. Via the Five Theses, to demonstrate the importance of Friendly AI research. C. Via down-to-earth meta-ethics, humanistic poetry, and pragmatic strategizing, to combat any nihilisms, relativisms, and defeatisms that might be triggered by recognizing the possibility (or probability) of Unfriendly AI. D. Via an accessible, substantive, entertaining presentation, to introduce the raison d'être of LessWrong to sophisticated newcomers in a way that encourages further engagement with LessWrong's community and/or content. What do you think? What would you add, remove, or alter?
809e1623-73c1-49cb-8d7e-73edecbf42b3	StampyAI/alignment-research-dataset/alignmentforum	Alignment Forum	Superrational Agents Kelly Bet Influence! As a follow-up to the [Walled Garden discussion](https://www.lesswrong.com/posts/gAM5AgcChwJLhuJkB/kelly-betting-discussion) about Kelly betting, Scott Garrabrant made some super-informal conjectures to me privately, involving the idea that some class of "nice" agents would "Kelly bet influence", where "influence" had something to do with anthropics and acausal trade. I was pretty incredulous at the time. However, as soon as he left the discussion, I came up with an argument for a similar fact. (The following does not perfectly reflect what Scott had in mind, by any means. His notion of "influence" was very different, for a start.) The meat of my argument is just Critch's [negotiable RL theorem](https://arxiv.org/abs/1701.01302). In fact, that's practically the entirety of my argument. I'm just thinking about the consequences in a different way from how I have before. Superrationality ================ Rather than articulating a real decision theory that deals with all the questions of acausal trade, bargaining, [commitment races](https://www.lesswrong.com/posts/brXr7PJ2W4Na2EW2q/the-commitment-races-problem), etc, I'm just going to imagine a class of superrational agents which solve these problems somehow. These agents "handshake" with each other and negotiate (perhaps acausally) a policy which is Pareto-optimal wrt each of their preferences. Negotiable RL ============= Critch's [negotiable RL](https://arxiv.org/abs/1701.01302) result studies the question of what an AI should do if it must serve multiple masters. For this post, I'll refer to the masters as "coalition members". He shows the following: *Any policy which is Pareto-optimal with respect to the preferences of coalition members, can be understood as doing the following. Each coalition member is assigned a starting weight, with weights summing to one. At each decision, the action is selected via the weighted average of the preferences of each coalition member, according to the current weights. At each observation, the weights are updated via Bayes' Law, based on the beliefs of coalition members.* He was studying what an AI's policy should be, when serving the coalition members; however, we can apply this result to a coalition of superrational agents who are settling on their own policy, rather than constructing a robotic servant. Critch remarks that we can imagine the weight update as the result of bets which the coalition members would make with each other. I've known about this for a long time, and it made intuitive sense to me that they'll happily bet on their beliefs; so, of course they'll gain/lose influence in the coalition based on good/bad predictions. What I didn't think too hard about was how they end up betting. Sure, the fact that it's equivalent to a Bayesian update is remarkable. But it makes sense once you think about the proof. Or does it? To foreshadow: the proof works from the assumption of Pareto optimality. So it collectively makes sense for the agents to bet this way. But the "of course it makes sense for them to bet on their beliefs" line of thinking tricks you into thinking that it individually makes sense for the agents to bet like this. However, this need not be the case. Kelly Betting & Bayes ===================== The Kelly betting fraction [can be written as](https://www.lesswrong.com/posts/zeviiJFwzBbr3sReN/calculating-kelly): f=p−1r1−1r.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0} .MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0} .mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table} .mjx-full-width {text-align: center; display: table-cell!important; width: 10000em} .mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0} .mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; 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src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')} @font-face {font-family: MJXc-TeX-size4-Rw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')} @font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')} @font-face {font-family: MJXc-TeX-vec-Rw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')} @font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')} @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold} @font-face {font-family: MJXc-TeX-vec-Bw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')} Where p is your probability for winning, and r is the return rate if you win (ie, if you stand to double your money, r=2; etc). Now, it turns out, betting f of your money (and keeping the rest in reserve) is equivalent to betting p of your money and putting (1-p) on the other side of the bet. Betting against yourself is a pretty silly thing to do, but since you'll win either way, there's no problem: *Bettingfof your money:* * If you win, you've got r⋅f, plus the (1−f) you held. + r⋅f=rp−11−1r + (1−f)=1−p1−1r + So the sum = rp−11−1r+1−p1−1r=rp−p1−1r=pr(r−1)r−1=p⋅r of your initial money. * If you lose, you've still got (1−f) of what you had. + So this is just 1−p1−1r *Betting against yourself, with fractions like your beliefs:* * If you win, you've got rp of your money. * If you lose, the payoff ratio (assuming you can get the reverse odds for the reverse bet) is 11−1r. So, since you put down 1−p, you get 1−p1−1r. But now imagine that a bunch of bettors are using the second strategy to make bets with each other, with the "house odds" being the weighted average of all their beliefs (weighted by their bankrolls, that is). Aside from the betting-against-yourself part, this is a pretty natural thing to do: these are the "house odds" which make the house revenue-neutral, so the house never has to dig into its own pockets to award winnings. You can imagine that everyone is putting money on two different sides of a table, to indicate their bets. When the bet is resolved, the losing side is pushed over to the winning side, and everyone who put money on the winning side picks up a fraction of money proportional to the fraction they originally contributed to that side. (And since payoffs of the bet-against-yourself strategy are exactly identical to Kelly betting payoffs, a bunch of Kelly bets at house odds rearrange money in exactly the same way as this.) But this is clearly equivalent to how hypotheses redistribute weight during Bayesian updates! So, a market of Kelly betters re-distributes money according to Bayesian updates. Altruistic Bets =============== Therefore, we can interpret the superrational coalition members as betting their coalition weight, according to the Kelly criterion. But, this is a pretty weird thing to do! I've [argued](https://www.lesswrong.com/posts/DfZtwtGD6ymFtXmdA/kelly-is-just-about-logarithmic-utility) that the main sensible justification for using the Kelly criterion is if you have utility logarithmic in wealth. Here, this translates to utility logarithmic in coalition weight. It's possible that under some reasonable assumptions about the world, we can argue that utility of coalition members will end up approximately logarithmic. But Critch's theorem applies to lots of situations, including small ones where there isn't any possibility for [weird things to happen over long chains of bets](https://www.lesswrong.com/posts/HLCcTypehEJtstNnD/a-non-logarithmic-argument-for-kelly) as in some arguments for Kelly. Typically, final utility will not even be continuous in coalition weight: small changes in coalition weight often won't change the optimal strategy at all, but at select tipping points, the optimal strategy will totally change to reflect the reconfigured trade-offs between preferences. Intuitively, these tipping points should factor significantly in a coalition member's betting strategy; you'd be totally indifferent to small bets which can't change anything, but avoid specific transitions strongly, and seek out others. If the coalition members were betting based on their selfish preferences, this would be the case. Yet, the coalition members end up betting according to a very simple formula, which does not account for any of this. Why? We can't justify this betting behavior from a selfish perspective (that is, not with the usual decision theories); as I said, the bets don't make sense. But we're not dealing with selfish agents. These agents are acting according to a Pareto-optimal policy. And that's ultimately the perspective we can justify the bets from: these are altruistically motivated bets. Exchanging coalition weight in this way is best for everyone. It keeps you Pareto-optimal! This is very counterintuitive. I suspect most people would agree with me that there seems to be no reason to bet, if you're being altruistic rather than selfish. Not so! They're not betting for their personal benefit. They're betting for the common good! Of course, that fact is a very straightforward consequence of Critch's theorem. It shouldn't be surprising. Yet, somehow, it didn't stick out to me in quite this way. I was too stuck in the frame of trying to interpret the bets selfishly, as Pareto-improvements which both sides happily agree to. I'm quite curious whether we can say anything interesting about how altruistic agents would handle money, based on this. I don't think it means altruists should Kelly bet money; money is a very different thing from coalition weight. Coalition weights are like exchange rates or prices. Money is more of a thing being exchanged. You do not pay coalition weight in order to get things done.
be7aeba0-1d60-4211-bb19-eecd93371161	trentmkelly/LessWrong-43k	LessWrong	Message Length Someone is broadcasting a stream of bits. You don't know why. A 500-bit-long sample looks like this: 01100110110101011011111100001001110000100011010001101011011010000001010000001010 10100111101000101111010100100101010010101010101000010100110101010011111111010101 01010101011111110101011010101101111101010110110101010100000001101111100000111010 11100000000000001111101010110101010101001010101101010101100111001100001100110101 11111111111111111100011001011010011010101010101100000010101011101101010010110011 11111010111101110100010101010111001111010001101101010101101011000101100000101010 10011001101010101111... The thought occurs to you to do Science to it—to ponder if there's some way you could better predict what bits are going to come next. At first you think you can't—it's just a bunch of random bits. You can't predict it, because that's what random means. Or does it? True, if the sequence represented flips of a fair coin—every flip independently landing either 0 or 1 with exactly equal probability—then there would be no way you could predict what would come next: any continuation you could posit would be exactly as probable as any other. But if the sequence represented flips of a biased coin—if, say, 1 came up 0.55 of the time instead of exactly 0.5—then it would be possible to predict better or worse. Your best bet for the next bit in isolation would always be 1, and you would more strongly anticipate sequences with slightly more 1s than 0s. You count 265 1s in the sample of 500 bits. Given the hypothesis that the bits were generated by a fair coin, the number of 1s (or without loss of generality, 0s) would be given by the binomial distribution (500k)(0.5)k(0.5)500−k, which has a standard deviation of √500⋅0.52=√125≈11.18, so your observation of 265−250=15 excess 1s is about 1511.18≈1.34 standard deviations from the mean—well within the realm of plausibility of happening by chance, although you're at least slightly suspicious that the coin behind these bits migh
3a48055e-762b-4b76-ad05-676398e78c14	trentmkelly/LessWrong-43k	LessWrong	Meetup : Robin Hanson: Why is Abstraction both Statusful and Silly? Discussion article for the meetup : Robin Hanson: Why is Abstraction both Statusful and Silly? WHEN: 14 July 2014 07:00:00PM (-0400) WHERE: Citadel House, 98 Elm St Apt 1, Somerville, MA Robin Hanson will give a short informal talk followed by a discussion on the reasons why abstraction is both statusful and silly! Meetup starts at 7pm, talk starts at 7:30pm. Discussion article for the meetup : Robin Hanson: Why is Abstraction both Statusful and Silly?
5ddf384a-8198-42c3-be8c-460cbd57893b	trentmkelly/LessWrong-43k	LessWrong	Putanumonit - Convincing people to read the Sequences and wondering about "postrationalists"
4cc24638-9a49-4ef8-b74b-e9b45d823a55	trentmkelly/LessWrong-43k	LessWrong	How do natural sciences prove causation? Let's say we have 2 phenomena, A and B, each can be a value of 0 or 1, and we observe, that for them implication table A=>B is always true. (Third column represents whether the combinations of events can happen or cannot.) A B A=>B 0 0 1 0 1 1 1 0 0 1 1 1 Thing we see is that combination A=1 and B=0 almost never happens, and three other combinations can happen. But how can we be sure, that it is not some kind of third event, which influences both of those to output those combinations of values? What if the table is like this, based on &? 0 0 0 0 1 0 1 0 0 1 1 1 If at least one of events (any) happens, then the second happens too. How this relation would be called? How many tables (of the 16) there are which could potentially represent causation? For example, 000 010 101 111 * is not in the set, because it says, that A not being true is impossible(has never been observed), but has no limitation on the value of B. Given a table in this format (or just a string representing the third column), what a person would do next to test whether events influence one another or are both defined by a third? Does the further approach even depend on a type of table? (I expect it will not, as everything would be observed in frequencies) Is it like an cycle, where one makes an assumption "what if they both are defined by event C" and then shows there is no correlation with C, and repeats for many other possible C's? But then it looks it would not be possible to exhaust all possible C's.
b63bf10a-b926-4ad5-ac0b-a5fe020760ab	StampyAI/alignment-research-dataset/arxiv	Arxiv	Dynamically Switching Human Prediction Models for Efficient Planning. I Introduction --------------- When robots operate in close proximity to humans, it is crucial that they anticipate what people will do to respond appropriately. Such prediction often involves equipping the robot with a model of human behavior [[1](#bib.bib1)]. This model could be physics-based [[2](#bib.bib2), [3](#bib.bib3), [4](#bib.bib4)], pattern-based [[5](#bib.bib5), [6](#bib.bib6), [7](#bib.bib7), [8](#bib.bib8)], approximate rationality with respect to an objective function [[9](#bib.bib9), [10](#bib.bib10), [11](#bib.bib11), [12](#bib.bib12), [13](#bib.bib13), [14](#bib.bib14), [15](#bib.bib15), [16](#bib.bib16), [17](#bib.bib17)], or even two player games [[18](#bib.bib18), [19](#bib.bib19), [20](#bib.bib20), [21](#bib.bib21)]. What human model a robot should be equipped with depends on the trade-offs it needs to make: some models, like the physics-based ones, are cheap to compute but don’t capture human intentions and may be less accurate; others, like two player games, model interactions between the person and the robot, but at a high computational cost. For systems interacting with people in real time, like autonomous vehicles or mobile robots, compromising on either accuracy or computation is undesirable. For instance, in the driving scenario in Fig. [1](#S1.F1 "Figure 1 ‣ I Introduction ‣ Dynamically Switching Human Prediction Models for Efficient Planning"), using the cheap model might pose a safety hazard, while picking the more accurate one may limit planning frequency or strain computational resources needed elsewhere (sensor suite, perception system, routing, etc). ![Refer to caption](/html/2103.07815/assets/x1.png) Figure 1: The robot (orange car) can plan with either a complex model (yellow bubble) of the human (blue car) that is more accurate but more expensive or with a simple model (purple bubble) that is not as accurate but cheaper. Our algorithm uses the complex model when a collision is imminent (solid yellow line), but saves computation by switching to the simple one afterwards (solid purple line). We advocate that the robot should not be stuck with a single model, but instead have the ability to dynamically change which model it is using as the interaction progresses. In Fig. [1](#S1.F1 "Figure 1 ‣ I Introduction ‣ Dynamically Switching Human Prediction Models for Efficient Planning"), the robot starts off with the cheap model. Anticipating a potential collision, it switches to a more complex model, reverting back once the critical maneuver is complete. The idea of using multiple predictive models is not new; most works, like interactive multiple model filtering [[22](#bib.bib22), [23](#bib.bib23), [24](#bib.bib24), [25](#bib.bib25)] or other ways of combining models [[25](#bib.bib25)] are focused on improving accuracy by leveraging complementary strengths of different models. In contrast, we are focused on the setting where we have complex but accurate models (which could be mixtures themselves), and cheap but less accurate ones. In such settings, if it weren’t for computational costs, we would use the complex models all the time. The question becomes: when is the performance gain worth the extra computation? We could plan with the complex model and measure the performance gain, but doing so defeats the purpose of saving computation. To avoid this, prior work proposed training a model to predict which agents “influence” or “affect” the planner, and prioritizing computation for them [[26](#bib.bib26)]. This approach does not estimate the performance-computation trade off, but heuristically assumes that agents who influence the planner are conducive to high gain. However, there is a fundamental difference between needing to consider an agent at all, and estimating the performance gain between predicting their behavior using a cheap versus a complex model. For example, in Fig. [1](#S1.F1 "Figure 1 ‣ I Introduction ‣ Dynamically Switching Human Prediction Models for Efficient Planning"), a cheap model is sufficient in the leftmost and rightmost frames where a critical maneuver is not necessary, despite the human influencing the planner. Instead of heuristically allocating computation, we employ efficient, online estimation for how an alternate human model could change robot performance. This enables switching to the model which best trades off reward and computation in real time. In Fig. [1](#S1.F1 "Figure 1 ‣ I Introduction ‣ Dynamically Switching Human Prediction Models for Efficient Planning"), a car using our switching methodology is able to achieve a behavior similar to the one produced with the most accurate model, but at a computational cost closer to that of the cheaper model. This paper makes three key contributions: (1) a formalism for the robot’s decision process that optimally trades off between computation and accuracy across multiple predictive models, (2) an approximate solution that solves this decision process online, and (3) a comparative analysis of our model switching algorithm in a human-autonomous car system. Together, these contributions give robots the autonomy to decide in real-time what predictive human models are most appropriate to use in different interactive scenarios. Code and videos are made available at [arjunsripathy.github.io/model\_switching](https://arjunsripathy.github.io/model_switching) II Method ---------- We focus on a system consisting of a robot R𝑅Ritalic\_R interacting in an environment with other human agents H𝐻Hitalic\_H. The robot’s goal is to plan around the humans in a manner that is most effective while minimizing computational time. We present our theory for a general single human, single robot setting where the other agents’ behavior is known, although our method can easily be extended by running it separately for each human. We use the running example of an autonomous car sharing the road with a human driver to illustrate the proposed approach and demonstrate the utility of our method. ### II-A Problem Statement We model the system as a fully observable dynamical system in which one agent’s controls potentially impact the other’s. As such, let the state x∈𝒳𝑥𝒳x\in\mathcal{X}italic\_x ∈ caligraphic\_X include the positions and velocities of both agents, where the human action uH∈𝒰Hsubscript𝑢𝐻subscript𝒰𝐻u\_{H}\in\mathcal{U}\_{H}italic\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ∈ caligraphic\_U start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT and robot action uR∈𝒰Rsubscript𝑢𝑅subscript𝒰𝑅u\_{R}\in\mathcal{U}\_{R}italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ∈ caligraphic\_U start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT each can affect the next state in a combined dynamics model: xt+1=f(xt,uRt,uHt)superscript𝑥𝑡1𝑓superscript𝑥𝑡superscriptsubscript𝑢𝑅𝑡superscriptsubscript𝑢𝐻𝑡x^{t+1}=f(x^{t},u\_{R}^{t},u\_{H}^{t})italic\_x start\_POSTSUPERSCRIPT italic\_t + 1 end\_POSTSUPERSCRIPT = italic\_f ( italic\_x start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ). Let 𝐱=[x1,…,xN]𝐱superscript𝑥1…superscript𝑥𝑁\mathbf{x}=[x^{1},\ldots,x^{N}]bold\_x = [ italic\_x start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT , … , italic\_x start\_POSTSUPERSCRIPT italic\_N end\_POSTSUPERSCRIPT ] be a finite horizon N𝑁Nitalic\_N state sequence, 𝐮R=[uR1,…,uRN]subscript𝐮𝑅superscriptsubscript𝑢𝑅1…superscriptsubscript𝑢𝑅𝑁\mathbf{u}\_{R}=[u\_{R}^{1},\ldots,u\_{R}^{N}]bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT = [ italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT , … , italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_N end\_POSTSUPERSCRIPT ] the robot’s continuous control inputs, and 𝐮H=[uH1,…,uHN]subscript𝐮𝐻superscriptsubscript𝑢𝐻1…superscriptsubscript𝑢𝐻𝑁\mathbf{u}\_{H}=[u\_{H}^{1},\ldots,u\_{H}^{N}]bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT = [ italic\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 1 end\_POSTSUPERSCRIPT , … , italic\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_N end\_POSTSUPERSCRIPT ] the human’s. The robot optimizes its controls 𝐮Rsubscript𝐮𝑅\mathbf{u}\_{R}bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT according to a reward function that depends on the joint sequence of states and controls: RR(x0,𝐮R,𝐮H)=∑τ=1NrR(xτ,uRτ,uHτ)subscript𝑅𝑅superscript𝑥0subscript𝐮𝑅subscript𝐮𝐻superscriptsubscript𝜏1𝑁subscript𝑟𝑅superscript𝑥𝜏superscriptsubscript𝑢𝑅𝜏superscriptsubscript𝑢𝐻𝜏R\_{R}(x^{0},\mathbf{u}\_{R},\mathbf{u}\_{H})=\sum\_{\tau=1}^{N}r\_{R}(x^{\tau},u\_{R}^{\tau},u\_{H}^{\tau})italic\_R start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT 0 end\_POSTSUPERSCRIPT , bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ) = ∑ start\_POSTSUBSCRIPT italic\_τ = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_N end\_POSTSUPERSCRIPT italic\_r start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT italic\_τ end\_POSTSUPERSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_τ end\_POSTSUPERSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_τ end\_POSTSUPERSCRIPT ), where x0superscript𝑥0x^{0}italic\_x start\_POSTSUPERSCRIPT 0 end\_POSTSUPERSCRIPT is the starting state and each state thereafter is obtained via the dynamics model from the previous robot and human controls. The person chooses their action at time t𝑡titalic\_t, uHtsuperscriptsubscript𝑢𝐻𝑡u\_{H}^{t}italic\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT, according to an internal policy πH(xt,uRt)subscript𝜋𝐻superscript𝑥𝑡superscriptsubscript𝑢𝑅𝑡\pi\_{H}(x^{t},u\_{R}^{t})italic\_π start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ), which, when applied at every state xtsuperscript𝑥𝑡x^{t}italic\_x start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT, results in 𝐮Hsubscript𝐮𝐻\mathbf{u}\_{H}bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT. We let 𝐮Rsubscript𝐮𝑅\mathbf{u}\_{R}bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT and 𝐮Hsubscript𝐮𝐻\mathbf{u}\_{H}bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT represent the true executed robot and human controls, respectively. Our system is a Markov Decision Process (MDP) with states 𝒳𝒳\mathcal{X}caligraphic\_X, actions 𝒰Rsubscript𝒰𝑅\mathcal{U}\_{R}caligraphic\_U start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT, transition function f(x,uR,uH)𝑓𝑥subscript𝑢𝑅subscript𝑢𝐻f(x,u\_{R},u\_{H})italic\_f ( italic\_x , italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ), and reward rR(x,uR,uH)subscript𝑟𝑅𝑥subscript𝑢𝑅subscript𝑢𝐻r\_{R}(x,u\_{R},u\_{H})italic\_r start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ( italic\_x , italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ). Since the robot does not know πHsubscript𝜋𝐻\pi\_{H}italic\_π start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT (that would require access to the human’s brain), it uses a human model M:𝒳×𝒰RN→𝒰HN:𝑀→𝒳superscriptsubscript𝒰𝑅𝑁superscriptsubscript𝒰𝐻𝑁M:\mathcal{X}\times\mathcal{U}\_{R}^{N}\rightarrow\mathcal{U}\_{H}^{N}italic\_M : caligraphic\_X × caligraphic\_U start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_N end\_POSTSUPERSCRIPT → caligraphic\_U start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_N end\_POSTSUPERSCRIPT to make a prediction of the human controls ¯𝐮H¯absentsubscript𝐮𝐻\bar{}\mathbf{u}\_{H}over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT. The robot then seeks a plan ¯𝐮R0(M)≔¯𝐮R(x0,M)≔¯absentsuperscriptsubscript𝐮𝑅0𝑀¯absentsubscript𝐮𝑅superscript𝑥0𝑀\bar{}\mathbf{u}\_{R}^{0}(M)\coloneqq\bar{}\mathbf{u}\_{R}(x^{0},M)over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 0 end\_POSTSUPERSCRIPT ( italic\_M ) ≔ over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT 0 end\_POSTSUPERSCRIPT , italic\_M ) that maximizes the MDP reward RRsubscript𝑅𝑅R\_{R}italic\_R start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT based on the predicted ¯𝐮H0(¯𝐮R,M)≔¯𝐮H(x0,¯𝐮R,M)≔¯absentsuperscriptsubscript𝐮𝐻0¯absentsubscript𝐮𝑅𝑀¯absentsubscript𝐮𝐻superscript𝑥0¯absentsubscript𝐮𝑅𝑀\bar{}\mathbf{u}\_{H}^{0}(\bar{}\mathbf{u}\_{R},M)\coloneqq\bar{}\mathbf{u}\_{H}(x^{0},\bar{}\mathbf{u}\_{R},M)over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT 0 end\_POSTSUPERSCRIPT ( over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , italic\_M ) ≔ over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT 0 end\_POSTSUPERSCRIPT , over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , italic\_M ). Unfortunately, this type of offline planning doesn’t consider modeling errors introduced by imperfect models M𝑀Mitalic\_M. Hence, it is more common for the robot to perform online planning at every time step t𝑡titalic\_t to obtain more accurate plans ¯𝐮Rt(M)¯absentsuperscriptsubscript𝐮𝑅𝑡𝑀\bar{}\mathbf{u}\_{R}^{t}(M)over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M ). For computational efficiency, we follow [[18](#bib.bib18)] and use Model Predictive Control (MPC) [[27](#bib.bib27)] with a finite horizon K<N𝐾𝑁K<Nitalic\_K < italic\_N, where at each time step t𝑡titalic\_t the robot optimizes its controls to maximize the cumulative reward: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| ¯𝐮Rt(M)=arg⁡max𝐮R⁡RR(xt,𝐮R,¯𝐮Ht(𝐮R,M)),¯absentsuperscriptsubscript𝐮𝑅𝑡𝑀subscriptsubscript𝐮𝑅subscript𝑅𝑅superscript𝑥𝑡subscript𝐮𝑅¯absentsuperscriptsubscript𝐮𝐻𝑡subscript𝐮𝑅𝑀\bar{}\mathbf{u}\_{R}^{t}(M)=\arg\max\_{\mathbf{u}\_{R}}R\_{R}(x^{t},\mathbf{u}\_{R},\bar{}\mathbf{u}\_{H}^{t}(\mathbf{u}\_{R},M))\enspace,over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M ) = roman\_arg roman\_max start\_POSTSUBSCRIPT bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT italic\_R start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT , bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , italic\_M ) ) , \| \| (1) \| where RRsubscript𝑅𝑅R\_{R}italic\_R start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT is evaluated only over the first K𝐾Kitalic\_K states starting at xtsuperscript𝑥𝑡x^{t}italic\_x start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT. The robot executes the first action from its plan and then replans at the next time step t+1𝑡1t+1italic\_t + 1. To simplify notation, we will denote the first action of a plan as u¯Rt(M)≔¯𝐮Rt(M)[0]≔superscriptsubscript¯𝑢𝑅𝑡𝑀¯absentsuperscriptsubscript𝐮𝑅𝑡𝑀delimited-[]0\bar{u}\_{R}^{t}(M)\coloneqq\bar{}\mathbf{u}\_{R}^{t}(M)[0]over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M ) ≔ over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M ) [ 0 ]. The crucial question is what human model M𝑀Mitalic\_M should the robot use for planning with Eq. ([1](#S2.E1 "1 ‣ II-A Problem Statement ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning"))? Restricting ourselves to any single model either hurts performance or computational efficiency. We propose an algorithm which enables efficient switching between models of varying accuracy and complexity, allowing for both high performance and low computation. ### II-B Model Switching Formalism We assume the robot has access to a ladder of human models ℳ={M0,…,Mn}ℳsubscript𝑀0…subscript𝑀𝑛\mathcal{M}=\{M\_{0},\ldots,M\_{n}\}caligraphic\_M = { italic\_M start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , … , italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT }, where as we climb the ladder we sacrifice computational time for greater expected reward: 𝔼x[RR(x,¯𝐮R(Mj),𝐮H)]≥𝔼x[RR(x,¯𝐮R(Mi),𝐮H)]subscript𝔼𝑥delimited-[]subscript𝑅𝑅𝑥¯absentsubscript𝐮𝑅subscript𝑀𝑗subscript𝐮𝐻subscript𝔼𝑥delimited-[]subscript𝑅𝑅𝑥¯absentsubscript𝐮𝑅subscript𝑀𝑖subscript𝐮𝐻\mathbb{E}\_{x}[R\_{R}(x,\bar{}\mathbf{u}\_{R}(M\_{j}),\mathbf{u}\_{H})]\geq\mathbb{E}\_{x}[R\_{R}(x,\bar{}\mathbf{u}\_{R}(M\_{i}),\mathbf{u}\_{H})]blackboard\_E start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT [ italic\_R start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ( italic\_x , over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ( italic\_M start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ) , bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ) ] ≥ blackboard\_E start\_POSTSUBSCRIPT italic\_x end\_POSTSUBSCRIPT [ italic\_R start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ( italic\_x , over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ( italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) , bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ) ] and T(Mj)>T(Mi),∀i<jformulae-sequence𝑇subscript𝑀𝑗𝑇subscript𝑀𝑖for-all𝑖𝑗T(M\_{j})>T(M\_{i}),\forall i<jitalic\_T ( italic\_M start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ) > italic\_T ( italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) , ∀ italic\_i < italic\_j, where T(M)𝑇𝑀T(M)italic\_T ( italic\_M ) is the time to solve Eq. ([1](#S2.E1 "1 ‣ II-A Problem Statement ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")) under M𝑀Mitalic\_M. At every time step t𝑡titalic\_t, the robot needs to choose the human model Mt∈ℳsuperscript𝑀𝑡ℳM^{t}\in\mathcal{M}italic\_M start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ∈ caligraphic\_M to use for planning ¯𝐮Rt¯absentsuperscriptsubscript𝐮𝑅𝑡\bar{}\mathbf{u}\_{R}^{t}over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT. To determine which model the robot should use at time t𝑡titalic\_t, we construct a meta MDP on top of the previous MDP, with states xtsuperscript𝑥𝑡x^{t}italic\_x start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT, actions Mt∈ℳsuperscript𝑀𝑡ℳM^{t}\in\mathcal{M}italic\_M start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ∈ caligraphic\_M representing the choice of human model, transition function f(xt,u¯Rt(Mt),uHt)𝑓superscript𝑥𝑡superscriptsubscript¯𝑢𝑅𝑡superscript𝑀𝑡superscriptsubscript𝑢𝐻𝑡f(x^{t},\bar{u}\_{R}^{t}(M^{t}),u\_{H}^{t})italic\_f ( italic\_x start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT , over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ) , italic\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ), and meta-reward rmetat=rR(xt,u¯Rt(Mt),uHt)−λ\T(Mt)superscriptsubscript𝑟𝑚𝑒𝑡𝑎𝑡subscript𝑟𝑅superscript𝑥𝑡superscriptsubscript¯𝑢𝑅𝑡superscript𝑀𝑡superscriptsubscript𝑢𝐻𝑡𝜆𝑇superscript𝑀𝑡r\_{meta}^{t}=r\_{R}(x^{t},\bar{u}\_{R}^{t}(M^{t}),u\_{H}^{t})-\lambda\T(M^{t})italic\_r start\_POSTSUBSCRIPT italic\_m italic\_e italic\_t italic\_a end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT = italic\_r start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT , over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ) , italic\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ) - italic\_λ \* italic\_T ( italic\_M start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ), with λ𝜆\lambdaitalic\_λ trading off actual reward gained by planning under Mtsuperscript𝑀𝑡M^{t}italic\_M start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT and computational time spent on the plan. Lower values for λ𝜆\lambdaitalic\_λ favor more complex models, whereas higher values result in more usage of less expensive ones. Solving this MDP exactly is impossible, since it requires access to the true human controls 𝐮Hsubscript𝐮𝐻\mathbf{u}\_{H}bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ahead of time. Moreover, even if we approximate the true human controls with those predicted by our most accurate model Mnsubscript𝑀𝑛M\_{n}italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT, the robot needs to find the model sequence that maximizes the cumulative meta reward across the episode—a procedure exponential in the episode horizon and, thus, intractable. We could alleviate this computational burden by assuming the robot myopically decides when to switch models: instead of considering the cumulative meta reward, only look at rmetatsuperscriptsubscript𝑟𝑚𝑒𝑡𝑎𝑡r\_{meta}^{t}italic\_r start\_POSTSUBSCRIPT italic\_m italic\_e italic\_t italic\_a end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT to decide whether to switch at time t+1𝑡1t+1italic\_t + 1. On the plus side, this simplification is more tractable and only needs the current human control. Unfortunately, even this relaxation requires planning with every model and picking the one with the best rmetatsuperscriptsubscript𝑟𝑚𝑒𝑡𝑎𝑡r\_{meta}^{t}italic\_r start\_POSTSUBSCRIPT italic\_m italic\_e italic\_t italic\_a end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT, which is worse than simply using Mnsubscript𝑀𝑛M\_{n}italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT from the get-go. Thus, we now propose an approximate solution that avoids computing every plan while still switching models as needed. ### II-C Approximate Solution: Switching between Two Models For ease of exposition, we first discuss how the robot can decide whether to switch from a simple model M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, used for planning at timestep t𝑡titalic\_t, to a complex model M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, which we are considering for timestep t+1𝑡1t+1italic\_t + 1. We are interested in estimating the reward the robot would gain from using M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT. Estimate Change in Robot Plan Leveraging our plan computed at timestep t𝑡titalic\_t using M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT, we get around explicitly generating ¯𝐮Rt(M2)¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀2\bar{}\mathbf{u}\_{R}^{t}(M\_{2})over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) by approximating it as ^𝐮Rt(M2)=¯𝐮Rt(M1)+Δ^𝐮R^absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀2¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀1Δ^absentsubscript𝐮𝑅\hat{}\mathbf{u}\_{R}^{t}(M\_{2})=\bar{}\mathbf{u}\_{R}^{t}(M\_{1})+\Delta\hat{}\mathbf{u}\_{R}over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) = over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) + roman\_Δ over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT. Here, we want to choose Δ^𝐮RΔ^absentsubscript𝐮𝑅\Delta\hat{}\mathbf{u}\_{R}roman\_Δ over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT such that it maximizes the robot reward RRsubscript𝑅𝑅R\_{R}italic\_R start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT under the complex model M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT. However, optimizing Δ^𝐮RΔ^absentsubscript𝐮𝑅\Delta\hat{}\mathbf{u}\_{R}roman\_Δ over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT using Eq. ([1](#S2.E1 "1 ‣ II-A Problem Statement ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")) is equivalent to planning. To get an efficient estimate, we use a quadratic Taylor series approximation of RRsubscript𝑅𝑅R\_{R}italic\_R start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT, denoted R~Rsubscript~𝑅𝑅\tilde{R}\_{R}over~ start\_ARG italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT, evaluated around (xt,¯𝐮Rt(M1),¯𝐮Ht(¯𝐮Rt(M1),M1))superscript𝑥𝑡¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀1¯absentsuperscriptsubscript𝐮𝐻𝑡¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀1subscript𝑀1(x^{t},\bar{}\mathbf{u}\_{R}^{t}(M\_{1}),\bar{}\mathbf{u}\_{H}^{t}(\bar{}\mathbf{u}\_{R}^{t}(M\_{1}),M\_{1}))( italic\_x start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT , over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) , over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) ), our current plan and human prediction: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| Δ^𝐮R=arg⁡maxΔ𝐮R⁡R~R(xt,^𝐮Rt(M2),¯𝐮Ht(^𝐮Rt(M2),M2)).Δ^absentsubscript𝐮𝑅subscriptΔsubscript𝐮𝑅subscript~𝑅𝑅superscript𝑥𝑡^absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀2¯absentsuperscriptsubscript𝐮𝐻𝑡^absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀2subscript𝑀2\Delta\hat{}\mathbf{u}\_{R}=\arg\max\_{\Delta\mathbf{u}\_{R}}\tilde{R}\_{R}(x^{t},\hat{}\mathbf{u}\_{R}^{t}(M\_{2}),\bar{}\mathbf{u}\_{H}^{t}(\hat{}\mathbf{u}\_{R}^{t}(M\_{2}),M\_{2}))\enspace.roman\_Δ over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT = roman\_arg roman\_max start\_POSTSUBSCRIPT roman\_Δ bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT over~ start\_ARG italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT , over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) , over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) ) . \| \| (2) \| Estimate Change in Human Prediction Eq. ([2](#S2.E2 "2 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")) requires approximating the M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT human’s response ¯𝐮Ht¯absentsuperscriptsubscript𝐮𝐻𝑡\bar{}\mathbf{u}\_{H}^{t}over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT to the robot’s plan ^𝐮Rt(M2)^absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀2\hat{}\mathbf{u}\_{R}^{t}(M\_{2})over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ), which can be broken down into M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT’s response to the current plan ¯𝐮Rt(M1)¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀1\bar{}\mathbf{u}\_{R}^{t}(M\_{1})over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) plus the change coming from Δ^𝐮RΔ^absentsubscript𝐮𝑅\Delta\hat{}\mathbf{u}\_{R}roman\_Δ over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT: ^𝐮Ht(^𝐮Rt(M2),M2)=^𝐮Ht(¯𝐮Rt(M1)+Δ^𝐮R,M2)^absentsuperscriptsubscript𝐮𝐻𝑡^absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀2subscript𝑀2^absentsuperscriptsubscript𝐮𝐻𝑡¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀1Δ^absentsubscript𝐮𝑅subscript𝑀2\hat{}\mathbf{u}\_{H}^{t}(\hat{}\mathbf{u}\_{R}^{t}(M\_{2}),M\_{2})=\hat{}\mathbf{u}\_{H}^{t}(\bar{}\mathbf{u}\_{R}^{t}(M\_{1})+\Delta\hat{}\mathbf{u}\_{R},M\_{2})over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) = over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) + roman\_Δ over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ). We can linearly approximate this: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| ^𝐮Ht(^𝐮Rt(M2),M2)=¯𝐮Ht(¯𝐮Rt(M1),M2)+d^𝐮Hd^𝐮R⋅Δ^𝐮R,^absentsuperscriptsubscript𝐮𝐻𝑡^absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀2subscript𝑀2¯absentsuperscriptsubscript𝐮𝐻𝑡¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀1subscript𝑀2⋅𝑑^absentsubscript𝐮𝐻𝑑^absentsubscript𝐮𝑅Δ^absentsubscript𝐮𝑅\hat{}\mathbf{u}\_{H}^{t}(\hat{}\mathbf{u}\_{R}^{t}(M\_{2}),M\_{2})=\bar{}\mathbf{u}\_{H}^{t}(\bar{}\mathbf{u}\_{R}^{t}(M\_{1}),M\_{2})+\frac{d\hat{}\mathbf{u}\_{H}}{d\hat{}\mathbf{u}\_{R}}\cdot\Delta\hat{}\mathbf{u}\_{R}\enspace,over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) = over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) + divide start\_ARG italic\_d over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_d over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_ARG ⋅ roman\_Δ over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , \| \| (3) \| where d^𝐮Hd^𝐮R=d^𝐮H(^𝐮R,M)d^𝐮R\| \Let@\restore@math@cr\default@tag ^uR=¯uRt(M1)M=M2 𝑑^absentsubscript𝐮𝐻𝑑^absentsubscript𝐮𝑅evaluated-at𝑑^absentsubscript𝐮𝐻^absentsubscript𝐮𝑅𝑀𝑑^absentsubscript𝐮𝑅 \Let@\restore@math@cr\default@tag ^uR=¯uRt(M1)M=M2 \frac{d\hat{}\mathbf{u}\_{H}}{d\hat{}\mathbf{u}\_{R}}=\frac{d\hat{}\mathbf{u}\_{H}(\hat{}\mathbf{u}\_{R},M)}{d\hat{}\mathbf{u}\_{R}}\Bigr{\|}\_{\vbox{\Let@\restore@math@cr\default@tag\halign{\ifx cc\hfil\fi$\m@th\scriptstyle#$\hfil\cr\hat{}\mathbf{u}\_{R}=\bar{}\mathbf{u}\_{R}^{t}(M\_{1})\\ M=M\_{2}\crcr}}}divide start\_ARG italic\_d over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_d over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_ARG = divide start\_ARG italic\_d over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ( over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , italic\_M ) end\_ARG start\_ARG italic\_d over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_ARG \| start\_POSTSUBSCRIPT over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT = over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) italic\_M = italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT. This requires evaluating the complex model M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, which might be expensive, though not nearly as expensive as planning with it. We may now substitute the changed robot plan ^𝐮Rt(M2)^absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀2\hat{}\mathbf{u}\_{R}^{t}(M\_{2})over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) and the changed human prediction from Eq. ([3](#S2.E3 "3 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")) into our reward expression we maximize in Eq. ([2](#S2.E2 "2 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")) \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| R~R(xt,¯𝐮Rt(M1)+Δ^𝐮R,¯𝐮Ht(¯𝐮Rt(M1),M2)+d^𝐮Hd^𝐮R⋅Δ^𝐮R).subscript~𝑅𝑅superscript𝑥𝑡¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀1Δ^absentsubscript𝐮𝑅¯absentsuperscriptsubscript𝐮𝐻𝑡¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀1subscript𝑀2⋅𝑑^absentsubscript𝐮𝐻𝑑^absentsubscript𝐮𝑅Δ^absentsubscript𝐮𝑅\tilde{R}\_{R}(x^{t},\bar{}\mathbf{u}\_{R}^{t}(M\_{1})+\Delta\hat{}\mathbf{u}\_{R},\bar{}\mathbf{u}\_{H}^{t}(\bar{}\mathbf{u}\_{R}^{t}(M\_{1}),M\_{2})+\frac{d\hat{}\mathbf{u}\_{H}}{d\hat{}\mathbf{u}\_{R}}\cdot\Delta\hat{}\mathbf{u}\_{R})\enspace.over~ start\_ARG italic\_R end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT , over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) + roman\_Δ over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) + divide start\_ARG italic\_d over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_d over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_ARG ⋅ roman\_Δ over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ) . \| \| (4) \| Performance Gain from the Change in Robot Plan Ultimately, to assess the robot’s performance gain by using M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, we want to estimate the reward component of rmetatsuperscriptsubscript𝑟𝑚𝑒𝑡𝑎𝑡r\_{meta}^{t}italic\_r start\_POSTSUBSCRIPT italic\_m italic\_e italic\_t italic\_a end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT under M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, rR(xt,u^Rt(M2),u¯Ht(^𝐮Rt(M2),MH))subscript𝑟𝑅superscript𝑥𝑡superscriptsubscript^𝑢𝑅𝑡subscript𝑀2superscriptsubscript¯𝑢𝐻𝑡^absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀2subscript𝑀𝐻r\_{R}(x^{t},\hat{u}\_{R}^{t}(M\_{2}),\bar{u}\_{H}^{t}(\hat{}\mathbf{u}\_{R}^{t}(M\_{2}),M\_{H}))italic\_r start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT , over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) , over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ) ), where ¯𝐮Ht(^𝐮Rt(M2),MH)¯absentsuperscriptsubscript𝐮𝐻𝑡^absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀2subscript𝑀𝐻\bar{}\mathbf{u}\_{H}^{t}(\hat{}\mathbf{u}\_{R}^{t}(M\_{2}),M\_{H})over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ) would be the human’s response under the true model MHsubscript𝑀𝐻M\_{H}italic\_M start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT to the robot’s plan with M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, and u¯Ht(^𝐮Rt(M2),MH)superscriptsubscript¯𝑢𝐻𝑡^absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀2subscript𝑀𝐻\bar{u}\_{H}^{t}(\hat{}\mathbf{u}\_{R}^{t}(M\_{2}),M\_{H})over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ) is the first control of that response. To simplify notation, we will denote this reward rRt(M2)superscriptsubscript𝑟𝑅𝑡subscript𝑀2r\_{R}^{t}(M\_{2})italic\_r start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ). Since our approximation of the meta MDP considers myopic rewards rRsubscript𝑟𝑅r\_{R}italic\_r start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT, we really are only interested in the first control u^Rt(M2)superscriptsubscript^𝑢𝑅𝑡subscript𝑀2\hat{u}\_{R}^{t}(M\_{2})over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) of the changed plan ^𝐮Rt(M2)^absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀2\hat{}\mathbf{u}\_{R}^{t}(M\_{2})over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ), and the first control u¯Ht(^𝐮Rt(M2),MH)superscriptsubscript¯𝑢𝐻𝑡^absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀2subscript𝑀𝐻\bar{u}\_{H}^{t}(\hat{}\mathbf{u}\_{R}^{t}(M\_{2}),M\_{H})over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ) of the true human’s response to that plan. Given Δ^𝐮RΔ^absentsubscript𝐮𝑅\Delta\hat{}\mathbf{u}\_{R}roman\_Δ over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT, we only need the change for the first control Δu^RΔsubscript^𝑢𝑅\Delta\hat{u}\_{R}roman\_Δ over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT combined with the approximation of u¯Ht(^𝐮Rt(M2),MH)superscriptsubscript¯𝑢𝐻𝑡^absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀2subscript𝑀𝐻\bar{u}\_{H}^{t}(\hat{}\mathbf{u}\_{R}^{t}(M\_{2}),M\_{H})over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ) as in Eq. ([3](#S2.E3 "3 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")) to estimate r^Rt(M2)superscriptsubscript^𝑟𝑅𝑡subscript𝑀2\hat{r}\_{R}^{t}(M\_{2})over^ start\_ARG italic\_r end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) as: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| rR(xt,u¯Rt(M1)+Δu^R,u¯Ht(¯𝐮Rt(M1),MH)+du^Hdu^R⋅Δu^R),subscript𝑟𝑅superscript𝑥𝑡superscriptsubscript¯𝑢𝑅𝑡subscript𝑀1Δsubscript^𝑢𝑅superscriptsubscript¯𝑢𝐻𝑡¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀1subscript𝑀𝐻⋅𝑑subscript^𝑢𝐻𝑑subscript^𝑢𝑅Δsubscript^𝑢𝑅r\_{R}(x^{t},\bar{u}\_{R}^{t}(M\_{1})+\Delta\hat{u}\_{R},\bar{u}\_{H}^{t}(\bar{}\mathbf{u}\_{R}^{t}(M\_{1}),M\_{H})+\frac{d\hat{u}\_{H}}{d\hat{u}\_{R}}\cdot\Delta\hat{u}\_{R})\enspace,italic\_r start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT , over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) + roman\_Δ over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ) + divide start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_ARG ⋅ roman\_Δ over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ) , \| \| (5) \| where du^Hdu^R=du^H(^𝐮R,M)du^R\| \Let@\restore@math@cr\default@tag ^uR=¯uRt(M1)M=MH 𝑑subscript^𝑢𝐻𝑑subscript^𝑢𝑅evaluated-at𝑑subscript^𝑢𝐻^absentsubscript𝐮𝑅𝑀𝑑subscript^𝑢𝑅 \Let@\restore@math@cr\default@tag ^uR=¯uRt(M1)M=MH \frac{d\hat{u}\_{H}}{d\hat{u}\_{R}}=\frac{d\hat{u}\_{H}(\hat{}\mathbf{u}\_{R},M)}{d\hat{u}\_{R}}\Bigr{\|}\_{\vbox{\Let@\restore@math@cr\default@tag\halign{\ifx cc\hfil\fi$\m@th\scriptstyle#$\hfil\cr\hat{}\mathbf{u}\_{R}=\bar{}\mathbf{u}\_{R}^{t}(M\_{1})\\ M=M\_{H}\crcr}}}divide start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_ARG = divide start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ( over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , italic\_M ) end\_ARG start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_ARG \| start\_POSTSUBSCRIPT over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT = over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) italic\_M = italic\_M start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT. Here, we don’t know MHsubscript𝑀𝐻M\_{H}italic\_M start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT, but we can approximate it with the complex model M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT. One Time Step Simplification All of this requires evaluating the complex model M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT at the current plan ¯𝐮Rt(M1)¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀1\bar{}\mathbf{u}\_{R}^{t}(M\_{1})over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ), which, although cheaper than fully planning with M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT, is still expensive. Since the meta reward only evaluates the first control of the plans, we can do better by simplifying our formulation further to planning and predicting a single control. That is, we consider the changed control u^Rt(M2)=u¯Rt(M1)+Δu^Rsuperscriptsubscript^𝑢𝑅𝑡subscript𝑀2superscriptsubscript¯𝑢𝑅𝑡subscript𝑀1Δsubscript^𝑢𝑅\hat{u}\_{R}^{t}(M\_{2})=\bar{u}\_{R}^{t}(M\_{1})+\Delta\hat{u}\_{R}over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) = over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) + roman\_Δ over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT, with its corresponding changed human control prediction from Eq. ([3](#S2.E3 "3 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")): \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| u^Ht(^𝐮Rt(M2),M2)=u¯Ht(¯𝐮Rt(M1),M2)+du^Hdu^R⋅Δu^R,superscriptsubscript^𝑢𝐻𝑡^absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀2subscript𝑀2superscriptsubscript¯𝑢𝐻𝑡¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀1subscript𝑀2⋅𝑑subscript^𝑢𝐻𝑑subscript^𝑢𝑅Δsubscript^𝑢𝑅\hat{u}\_{H}^{t}(\hat{}\mathbf{u}\_{R}^{t}(M\_{2}),M\_{2})=\bar{u}\_{H}^{t}(\bar{}\mathbf{u}\_{R}^{t}(M\_{1}),M\_{2})+\frac{d\hat{u}\_{H}}{d\hat{u}\_{R}}\cdot\Delta\hat{u}\_{R}\enspace,over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) = over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) + divide start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_ARG ⋅ roman\_Δ over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , \| \| (6) \| where du^Hdu^R=du^H(^𝐮R,M)du^R\| \Let@\restore@math@cr\default@tag ^uR=¯uRt(M1)M=M2 𝑑subscript^𝑢𝐻𝑑subscript^𝑢𝑅evaluated-at𝑑subscript^𝑢𝐻^absentsubscript𝐮𝑅𝑀𝑑subscript^𝑢𝑅 \Let@\restore@math@cr\default@tag ^uR=¯uRt(M1)M=M2 \frac{d\hat{u}\_{H}}{d\hat{u}\_{R}}=\frac{d\hat{u}\_{H}(\hat{}\mathbf{u}\_{R},M)}{d\hat{u}\_{R}}\Bigr{\|}\_{\vbox{\Let@\restore@math@cr\default@tag\halign{\ifx cc\hfil\fi$\m@th\scriptstyle#$\hfil\cr\hat{}\mathbf{u}\_{R}=\bar{}\mathbf{u}\_{R}^{t}(M\_{1})\\ M=M\_{2}\crcr}}}divide start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_ARG = divide start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ( over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , italic\_M ) end\_ARG start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_ARG \| start\_POSTSUBSCRIPT over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT = over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) italic\_M = italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT. We obtain Δu^RΔsubscript^𝑢𝑅\Delta\hat{u}\_{R}roman\_Δ over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT by optimizing the following objective: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| arg⁡maxΔuR⁡r~R(xt,u^Rt(M2),u¯Ht(¯𝐮Rt(M1),M2)+du^Hdu^R⋅ΔuR).subscriptΔsubscript𝑢𝑅subscript~𝑟𝑅superscript𝑥𝑡superscriptsubscript^𝑢𝑅𝑡subscript𝑀2superscriptsubscript¯𝑢𝐻𝑡¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀1subscript𝑀2⋅𝑑subscript^𝑢𝐻𝑑subscript^𝑢𝑅Δsubscript𝑢𝑅\arg\max\_{\Delta u\_{R}}\tilde{r}\_{R}(x^{t},\hat{u}\_{R}^{t}(M\_{2}),\bar{u}\_{H}^{t}(\bar{}\mathbf{u}\_{R}^{t}(M\_{1}),M\_{2})+\frac{d\hat{u}\_{H}}{d\hat{u}\_{R}}\cdot\Delta u\_{R})\enspace.roman\_arg roman\_max start\_POSTSUBSCRIPT roman\_Δ italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT over~ start\_ARG italic\_r end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT , over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) , over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) + divide start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_ARG ⋅ roman\_Δ italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ) . \| \| (7) \| Our simplified derivation still requires the gradient du^Hdu^R𝑑subscript^𝑢𝐻𝑑subscript^𝑢𝑅\frac{d\hat{u}\_{H}}{d\hat{u}\_{R}}divide start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_ARG, but this is cheaper to compute than d^𝐮Hd^𝐮R𝑑^absentsubscript𝐮𝐻𝑑^absentsubscript𝐮𝑅\frac{d\hat{}\mathbf{u}\_{H}}{d\hat{}\mathbf{u}\_{R}}divide start\_ARG italic\_d over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_d over^ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_ARG. Armed with an estimated r^Rt(M2)superscriptsubscript^𝑟𝑅𝑡subscript𝑀2\hat{r}\_{R}^{t}(M\_{2})over^ start\_ARG italic\_r end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) from Eq. ([5](#S2.E5 "5 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")), we can now determine whether the performance gain from M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT is worth the additional compute by considering \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| Δrmetat=r^Rt(M2)−λ\T(M2)−(rRt(M1)−λ\T(M1)),Δsubscriptsuperscript𝑟𝑡𝑚𝑒𝑡𝑎superscriptsubscript^𝑟𝑅𝑡subscript𝑀2𝜆𝑇subscript𝑀2superscriptsubscript𝑟𝑅𝑡subscript𝑀1𝜆𝑇subscript𝑀1\Delta r^{t}\_{meta}=\hat{r}\_{R}^{t}(M\_{2})-\lambda\T(M\_{2})-(r\_{R}^{t}(M\_{1})-\lambda\T(M\_{1}))\enspace,roman\_Δ italic\_r start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m italic\_e italic\_t italic\_a end\_POSTSUBSCRIPT = over^ start\_ARG italic\_r end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) - italic\_λ \* italic\_T ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) - ( italic\_r start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) - italic\_λ \* italic\_T ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) ) , \| \| (8) \| where we already know rRt(M1)superscriptsubscript𝑟𝑅𝑡subscript𝑀1r\_{R}^{t}(M\_{1})italic\_r start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) from the previous time step, and T(M1)𝑇subscript𝑀1T(M\_{1})italic\_T ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) and T(M2)𝑇subscript𝑀2T(M\_{2})italic\_T ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) are known a priori from their model specifications. If ΔrmetatΔsubscriptsuperscript𝑟𝑡𝑚𝑒𝑡𝑎\Delta r^{t}\_{meta}roman\_Δ italic\_r start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m italic\_e italic\_t italic\_a end\_POSTSUBSCRIPT is positive, the robot should switch to M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT; otherwise, the robot should continue using M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT. Intuitive Interpretation In Eq. ([8](#S2.E8 "8 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")), since T𝑇Titalic\_T’s are constants, the robot’s decision of whether to switch relies on comparing rRt(M1)=rR(xt,u¯Rt(M1),u¯Ht(¯𝐮Rt(M1),M1))superscriptsubscript𝑟𝑅𝑡subscript𝑀1subscript𝑟𝑅superscript𝑥𝑡superscriptsubscript¯𝑢𝑅𝑡subscript𝑀1superscriptsubscript¯𝑢𝐻𝑡¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀1subscript𝑀1r\_{R}^{t}(M\_{1})=r\_{R}(x^{t},\bar{u}\_{R}^{t}(M\_{1}),\bar{u}\_{H}^{t}(\bar{}\mathbf{u}\_{R}^{t}(M\_{1}),M\_{1}))italic\_r start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) = italic\_r start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT , over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) , over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) ) to r^Rt(M2)=rR(xt,u¯Rt(M1)+Δu^R,u¯Ht(¯𝐮Rt(M1),M2)+du^Hdu^R⋅Δu^R)superscriptsubscript^𝑟𝑅𝑡subscript𝑀2subscript𝑟𝑅superscript𝑥𝑡superscriptsubscript¯𝑢𝑅𝑡subscript𝑀1Δsubscript^𝑢𝑅superscriptsubscript¯𝑢𝐻𝑡¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀1subscript𝑀2⋅𝑑subscript^𝑢𝐻𝑑subscript^𝑢𝑅Δsubscript^𝑢𝑅\hat{r}\_{R}^{t}(M\_{2})=r\_{R}(x^{t},\bar{u}\_{R}^{t}(M\_{1})+\Delta\hat{u}\_{R},\bar{u}\_{H}^{t}(\bar{}\mathbf{u}\_{R}^{t}(M\_{1}),M\_{2})+\frac{d\hat{u}\_{H}}{d\hat{u}\_{R}}\cdot\Delta\hat{u}\_{R})over^ start\_ARG italic\_r end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) = italic\_r start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT , over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) + roman\_Δ over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) + divide start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_ARG ⋅ roman\_Δ over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ). Looking at these rewards, a few key distinctions stand out. First, the robot is interested in knowing whether M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT would give a different prediction u¯Ht(¯𝐮Rt(M1),M2)superscriptsubscript¯𝑢𝐻𝑡¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀1subscript𝑀2\bar{u}\_{H}^{t}(\bar{}\mathbf{u}\_{R}^{t}(M\_{1}),M\_{2})over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) than M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT’s u¯Ht(¯𝐮Rt(M1),M1)superscriptsubscript¯𝑢𝐻𝑡¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀1subscript𝑀1\bar{u}\_{H}^{t}(\bar{}\mathbf{u}\_{R}^{t}(M\_{1}),M\_{1})over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT ) on the current plan. For example, in Fig. [1](#S1.F1 "Figure 1 ‣ I Introduction ‣ Dynamically Switching Human Prediction Models for Efficient Planning") we realize a more complex model foresees the human turning into the bottleneck compared to naive constant velocity. Here this foresight prevents a collision. Meanwhile, the derivative du^Hdu^R𝑑subscript^𝑢𝐻𝑑subscript^𝑢𝑅\frac{d\hat{u}\_{H}}{d\hat{u}\_{R}}divide start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT end\_ARG start\_ARG italic\_d over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT end\_ARG captures the influence that the robot’s controls have on the human’s. Simple models may ignore this, but more complex models, like the two player game, capture this dynamic enabling certain critical maneuvers. For instance, humans will yield should the robot merge to create proper spacing. Only if aware of this influence will the robot feel confident merging into tight windows. So how much do these two terms matter? Intuitively, by computing Δu^RΔsubscript^𝑢𝑅\Delta\hat{u}\_{R}roman\_Δ over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT via Eq. ([7](#S2.E7 "7 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")), we see how much a model that either captures influence or makes different predictions affects the robot’s plan. If neither bears much weight, then Δu^RΔsubscript^𝑢𝑅\Delta\hat{u}\_{R}roman\_Δ over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT will likely be negligible and we will not switch. If however Δu^RΔsubscript^𝑢𝑅\Delta\hat{u}\_{R}roman\_Δ over^ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT is significant, we further evaluate if that translates to significant performance gain. Only when that’s the case will we ultimately switch. Input: Ladder ℳ={M0,…,Mn}ℳsubscript𝑀0…subscript𝑀𝑛\mathcal{M}=\{M\_{0},\ldots,M\_{n}\}caligraphic\_M = { italic\_M start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , … , italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT }, episode time N𝑁Nitalic\_N. Start with time t=0𝑡0t=0italic\_t = 0, current model index i𝑖iitalic\_i. while t≤N𝑡𝑁t\leq Nitalic\_t ≤ italic\_N do Compute ¯𝐮Rt(Mi)¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀𝑖\bar{}\mathbf{u}\_{R}^{t}(M\_{i})over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) given xtsuperscript𝑥𝑡x^{t}italic\_x start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT and execute u¯Rt(Mi)superscriptsubscript¯𝑢𝑅𝑡subscript𝑀𝑖\bar{u}\_{R}^{t}(M\_{i})over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ). if i<n𝑖𝑛i<nitalic\_i < italic\_n then Substitute (M1,M2)←(Mi,Mn)←subscript𝑀1subscript𝑀2subscript𝑀𝑖subscript𝑀𝑛(M\_{1},M\_{2})\leftarrow(M\_{i},M\_{n})( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) ← ( italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ) in Eq. ([5](#S2.E5 "5 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")), ([7](#S2.E7 "7 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")), ([8](#S2.E8 "8 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")), with u¯Ht(¯𝐮Rt(Mi),Mn)←uHt←superscriptsubscript¯𝑢𝐻𝑡¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀𝑖subscript𝑀𝑛superscriptsubscript𝑢𝐻𝑡\bar{u}\_{H}^{t}(\bar{}\mathbf{u}\_{R}^{t}(M\_{i}),M\_{n})\leftarrow u\_{H}^{t}over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ) ← italic\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT. Compute ΔrmetatΔsubscriptsuperscript𝑟𝑡𝑚𝑒𝑡𝑎\Delta r^{t}\_{meta}roman\_Δ italic\_r start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m italic\_e italic\_t italic\_a end\_POSTSUBSCRIPT using Eq. ([5](#S2.E5 "5 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")), ([7](#S2.E7 "7 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")), ([8](#S2.E8 "8 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")). if Δrmetat>0normal-Δsubscriptsuperscript𝑟𝑡𝑚𝑒𝑡𝑎0\Delta r^{t}\_{meta}>0roman\_Δ italic\_r start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m italic\_e italic\_t italic\_a end\_POSTSUBSCRIPT > 0 then i←n←𝑖𝑛i\leftarrow nitalic\_i ← italic\_n (Switch up), continue if i>0𝑖0i>0italic\_i > 0 and cooldown complete then Substitute (M1,M2)←(Mi,Mi−1)←subscript𝑀1subscript𝑀2subscript𝑀𝑖subscript𝑀𝑖1(M\_{1},M\_{2})\leftarrow(M\_{i},M\_{i-1})( italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT ) ← ( italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT , italic\_M start\_POSTSUBSCRIPT italic\_i - 1 end\_POSTSUBSCRIPT ) in Eq. ([5](#S2.E5 "5 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")), ([7](#S2.E7 "7 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")), ([8](#S2.E8 "8 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")). Compute ΔrmetatΔsubscriptsuperscript𝑟𝑡𝑚𝑒𝑡𝑎\Delta r^{t}\_{meta}roman\_Δ italic\_r start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m italic\_e italic\_t italic\_a end\_POSTSUBSCRIPT using Eq. ([5](#S2.E5 "5 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")), ([7](#S2.E7 "7 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")), ([8](#S2.E8 "8 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")). if Δrmetat>0normal-Δsubscriptsuperscript𝑟𝑡𝑚𝑒𝑡𝑎0\Delta r^{t}\_{meta}>0roman\_Δ italic\_r start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m italic\_e italic\_t italic\_a end\_POSTSUBSCRIPT > 0 then i←i−1←𝑖𝑖1i\leftarrow i-1italic\_i ← italic\_i - 1 (Switch Down). Algorithm 1 Dynamic Model Switching ### II-D Approximate Solution: Switching Between the Ladder Although we presented our method in the context of switching up, the framework holds when M1subscript𝑀1M\_{1}italic\_M start\_POSTSUBSCRIPT 1 end\_POSTSUBSCRIPT is the current model and M2subscript𝑀2M\_{2}italic\_M start\_POSTSUBSCRIPT 2 end\_POSTSUBSCRIPT is any alternative: higher or lower. When the alternate model is lower the question becomes: is the computational saving worth the loss in reward? Generalizing our derivation to a ladder of models ℳ={M0,…,Mn}ℳsubscript𝑀0…subscript𝑀𝑛\mathcal{M}=\{M\_{0},\ldots,M\_{n}\}caligraphic\_M = { italic\_M start\_POSTSUBSCRIPT 0 end\_POSTSUBSCRIPT , … , italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT }, suppose that at time t𝑡titalic\_t the robot used model Misubscript𝑀𝑖M\_{i}italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT to plan ¯𝐮Rt(Mi)¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀𝑖\bar{}\mathbf{u}\_{R}^{t}(M\_{i})over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ). We would like to decide on a model for time t+1𝑡1t+1italic\_t + 1. First, we evaluate if it’s worthwhile to switch to a higher model Mjsubscript𝑀𝑗M\_{j}italic\_M start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT, j>i𝑗𝑖j>iitalic\_j > italic\_i. The robot could consider Mi+1subscript𝑀𝑖1M\_{i+1}italic\_M start\_POSTSUBSCRIPT italic\_i + 1 end\_POSTSUBSCRIPT, the model immediately above, and successively switch up. However, in urgent, safety-critical situations we may need to switch higher than that immediately to avoid an accident. Thus, we exclusively consider the best model and upper bound r^Rt(Mj)superscriptsubscript^𝑟𝑅𝑡subscript𝑀𝑗\hat{r}\_{R}^{t}(M\_{j})over^ start\_ARG italic\_r end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ) with r^Rt(Mn)superscriptsubscript^𝑟𝑅𝑡subscript𝑀𝑛\hat{r}\_{R}^{t}(M\_{n})over^ start\_ARG italic\_r end\_ARG start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT ), the estimated reward from using the best model available. To avoid having to evaluate Mnsubscript𝑀𝑛M\_{n}italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT’s expensive predictions we substitute a perfect prediction u¯Ht(¯𝐮Rt(Mi),MN)≈uHtsuperscriptsubscript¯𝑢𝐻𝑡¯absentsuperscriptsubscript𝐮𝑅𝑡subscript𝑀𝑖subscript𝑀𝑁superscriptsubscript𝑢𝐻𝑡\bar{u}\_{H}^{t}(\bar{}\mathbf{u}\_{R}^{t}(M\_{i}),M\_{N})\approx u\_{H}^{t}over¯ start\_ARG italic\_u end\_ARG start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ) , italic\_M start\_POSTSUBSCRIPT italic\_N end\_POSTSUBSCRIPT ) ≈ italic\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT. This effectively upper bounds Mnsubscript𝑀𝑛M\_{n}italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT with the true human model MHsubscript𝑀𝐻M\_{H}italic\_M start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT. If the robot should have switched down to Mjsubscript𝑀𝑗M\_{j}italic\_M start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT, j<i𝑗𝑖j<iitalic\_j < italic\_i, we observe that rRt(Mj)≤rRt(Mi−1)≤rRt(Mi)superscriptsubscript𝑟𝑅𝑡subscript𝑀𝑗superscriptsubscript𝑟𝑅𝑡subscript𝑀𝑖1superscriptsubscript𝑟𝑅𝑡subscript𝑀𝑖r\_{R}^{t}(M\_{j})\leq r\_{R}^{t}(M\_{i-1})\leq r\_{R}^{t}(M\_{i})italic\_r start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT italic\_j end\_POSTSUBSCRIPT ) ≤ italic\_r start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT italic\_i - 1 end\_POSTSUBSCRIPT ) ≤ italic\_r start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT ( italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ). Thus, for efficient switching, we only consider the model directly below, Mi−1subscript𝑀𝑖1M\_{i-1}italic\_M start\_POSTSUBSCRIPT italic\_i - 1 end\_POSTSUBSCRIPT. Since T(Mi−1)<T(Mi)𝑇subscript𝑀𝑖1𝑇subscript𝑀𝑖T(M\_{i-1})<T(M\_{i})italic\_T ( italic\_M start\_POSTSUBSCRIPT italic\_i - 1 end\_POSTSUBSCRIPT ) < italic\_T ( italic\_M start\_POSTSUBSCRIPT italic\_i end\_POSTSUBSCRIPT ), it is reasonable to compute true model predictions in our approximation. Finally, if ΔrmetatΔsubscriptsuperscript𝑟𝑡𝑚𝑒𝑡𝑎\Delta r^{t}\_{meta}roman\_Δ italic\_r start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m italic\_e italic\_t italic\_a end\_POSTSUBSCRIPT is positive for MHsubscript𝑀𝐻M\_{H}italic\_M start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT, the robot switches up to Mnsubscript𝑀𝑛M\_{n}italic\_M start\_POSTSUBSCRIPT italic\_n end\_POSTSUBSCRIPT. Otherwise, if ΔrmetatΔsubscriptsuperscript𝑟𝑡𝑚𝑒𝑡𝑎\Delta r^{t}\_{meta}roman\_Δ italic\_r start\_POSTSUPERSCRIPT italic\_t end\_POSTSUPERSCRIPT start\_POSTSUBSCRIPT italic\_m italic\_e italic\_t italic\_a end\_POSTSUBSCRIPT is positive for Mi−1subscript𝑀𝑖1M\_{i-1}italic\_M start\_POSTSUBSCRIPT italic\_i - 1 end\_POSTSUBSCRIPT, the robot switches down to Mi−1subscript𝑀𝑖1M\_{i-1}italic\_M start\_POSTSUBSCRIPT italic\_i - 1 end\_POSTSUBSCRIPT. Otherwise we stay as summarized in Algorithm [1](#algorithm1 "1 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning"). In practice evaluating Mi−1subscript𝑀𝑖1M\_{i-1}italic\_M start\_POSTSUBSCRIPT italic\_i - 1 end\_POSTSUBSCRIPT’s predictions can still be significant, but unlike switching up, safety and performance concerns do not force us to check every timestep. One may wait K𝐾Kitalic\_K timesteps after failing to switch down before trying again. This cooldown hyperparameter, K𝐾Kitalic\_K, should be set based on how often one actually switches. III Experiments ---------------- ![Refer to caption](/html/2103.07815/assets/x2.png) ![Refer to caption](/html/2103.07815/assets/x3.png) ![Refer to caption](/html/2103.07815/assets/x4.png) Figure 2: (Top) Average single step computation time over the course of Stay Back (left), Merger (center), and Give Way (right), for a conservative, lesser λ𝜆\lambdaitalic\_λ, switcher (yellow) and an aggressive, larger λ𝜆\lambdaitalic\_λ, one (light blue). (Bottom) Example conservative MS robot (orange) behavior around the target human (blue) and other cars (black). We now demonstrate the efficacy of our model switching algorithm in three simulated autonomous driving scenarios. ### III-A Driving Simulator We model the dynamics of the vehicles as a 4D bicycle model. Let the state of the system be 𝐱=[xyθv]T𝐱superscriptdelimited-[]𝑥𝑦𝜃𝑣𝑇\mathbf{x}=[x\;y\;\theta\;v]^{T}bold\_x = [ italic\_x italic\_y italic\_θ italic\_v ] start\_POSTSUPERSCRIPT italic\_T end\_POSTSUPERSCRIPT, where x,y𝑥𝑦x,yitalic\_x , italic\_y are the coordinates of the vehicle, θ𝜃\thetaitalic\_θ is the heading, and v𝑣vitalic\_v the speed. The actions are u=[ωa]T𝑢superscriptdelimited-[]𝜔𝑎𝑇u=[\omega\;a]^{T}italic\_u = [ italic\_ω italic\_a ] start\_POSTSUPERSCRIPT italic\_T end\_POSTSUPERSCRIPT, where ω𝜔\omegaitalic\_ω is the steering input and a𝑎aitalic\_a is the linear acceleration. We use α𝛼\alphaitalic\_α as the friction coefficient, and the vehicle’s dynamics model is: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| [x˙y˙θ˙v˙]=[v⋅cos⁡(θ)v⋅sin⁡(θ)v⋅ωa−α⋅v].delimited-[]˙𝑥˙𝑦˙𝜃˙𝑣delimited-[]⋅⋅⋅𝑣𝜃𝑣𝜃𝑣𝜔𝑎⋅𝛼𝑣[\dot{x}\;\dot{y}\;\dot{\theta}\;\dot{v}]=[v\cdot\cos(\theta)\;\;\;v\cdot\sin(\theta)\;\;\;v\cdot\omega\;\;\;a-\alpha\cdot v]\enspace.[ over˙ start\_ARG italic\_x end\_ARG over˙ start\_ARG italic\_y end\_ARG over˙ start\_ARG italic\_θ end\_ARG over˙ start\_ARG italic\_v end\_ARG ] = [ italic\_v ⋅ roman\_cos ( italic\_θ ) italic\_v ⋅ roman\_sin ( italic\_θ ) italic\_v ⋅ italic\_ω italic\_a - italic\_α ⋅ italic\_v ] . \| \| (9) \| ### III-B Human Predictive Models For our experiments, we use the following 3 models of varying computational complexity and accuracy: #### III-B1 Constant Velocity This model, deemed Naive, predicts that the person will provide zero acceleration control, maintaining their current heading and speed. #### III-B2 Human Plans First This model assumes that the person acts according to a reward function parameterized as a linear combination of features ϕitalic-ϕ\phiitalic\_ϕ: RH(x0,𝐮R,𝐮H)=∑τ=1NrH(xτ,uRτ,uHτ)=∑τ=1NθTϕ(xτ,uRτ,uHτ)subscript𝑅𝐻superscript𝑥0subscript𝐮𝑅subscript𝐮𝐻superscriptsubscript𝜏1𝑁subscript𝑟𝐻superscript𝑥𝜏superscriptsubscript𝑢𝑅𝜏superscriptsubscript𝑢𝐻𝜏superscriptsubscript𝜏1𝑁superscript𝜃𝑇italic-ϕsuperscript𝑥𝜏superscriptsubscript𝑢𝑅𝜏superscriptsubscript𝑢𝐻𝜏R\_{H}(x^{0},\mathbf{u}\_{R},\mathbf{u}\_{H})=\sum\_{\tau=1}^{N}r\_{H}(x^{\tau},u\_{R}^{\tau},u\_{H}^{\tau})=\sum\_{\tau=1}^{N}\theta^{T}\phi(x^{\tau},u\_{R}^{\tau},u\_{H}^{\tau})italic\_R start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT 0 end\_POSTSUPERSCRIPT , bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ) = ∑ start\_POSTSUBSCRIPT italic\_τ = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_N end\_POSTSUPERSCRIPT italic\_r start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT italic\_τ end\_POSTSUPERSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_τ end\_POSTSUPERSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_τ end\_POSTSUPERSCRIPT ) = ∑ start\_POSTSUBSCRIPT italic\_τ = 1 end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_N end\_POSTSUPERSCRIPT italic\_θ start\_POSTSUPERSCRIPT italic\_T end\_POSTSUPERSCRIPT italic\_ϕ ( italic\_x start\_POSTSUPERSCRIPT italic\_τ end\_POSTSUPERSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_τ end\_POSTSUPERSCRIPT , italic\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT start\_POSTSUPERSCRIPT italic\_τ end\_POSTSUPERSCRIPT ). Additionally, under this model the human believes that the robot will maintain a constant velocity, and optimizes their reward w.r.t the imagined robot plan ~𝐮R~absentsubscript𝐮𝑅\tilde{}\mathbf{u}\_{R}over~ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT to obtain a plan ¯𝐮H¯absentsubscript𝐮𝐻\bar{}\mathbf{u}\_{H}over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| ¯𝐮H(x0,~𝐮R)=arg⁡max𝐮H⁡RH(x0,~𝐮R,𝐮H).¯absentsubscript𝐮𝐻superscript𝑥0~absentsubscript𝐮𝑅subscriptsubscript𝐮𝐻subscript𝑅𝐻superscript𝑥0~absentsubscript𝐮𝑅subscript𝐮𝐻\bar{}\mathbf{u}\_{H}(x^{0},\tilde{}\mathbf{u}\_{R})=\arg\max\_{\mathbf{u}\_{H}}R\_{H}(x^{0},\tilde{}\mathbf{u}\_{R},\mathbf{u}\_{H})\enspace.over¯ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT 0 end\_POSTSUPERSCRIPT , over~ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT ) = roman\_arg roman\_max start\_POSTSUBSCRIPT bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT end\_POSTSUBSCRIPT italic\_R start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ( italic\_x start\_POSTSUPERSCRIPT 0 end\_POSTSUPERSCRIPT , over~ start\_ARG end\_ARG bold\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT , bold\_u start\_POSTSUBSCRIPT italic\_H end\_POSTSUBSCRIPT ) . \| \| (10) \| We refer to this model as Turn because the person takes the first turn in choosing their controls. #### III-B3 Cognizant of Effects on Human Action Our last model is based on [[18](#bib.bib18)] and models the human as an agent which will optimally respond to the robot’s plan. This results in a nested optimization as the robot accounts for how its plan affects the human’s. As we rationalize the human’s thought process we refer to this model as “Theory of Mind” [[28](#bib.bib28)], or ToM for short. Planning with the first two approaches involves optimizing the robot’s control against the fixed human prediction. Meanwhile, ToM’s nested optimization returns both a human and robot control with no further optimization required. From our experiments, planning using Turn and ToM takes roughly twice and four times as long as using Naive, respectively. However, their expected accuracy has the reverse ordering. Both Turn and ToM rely on knowing a reward parameter θ𝜃\thetaitalic\_θ and a set of features ϕitalic-ϕ\phiitalic\_ϕ for the human reward. To learn a good θ𝜃\thetaitalic\_θ for every scenario, we collected demonstrations of a single human driver in an environment with multiple autonomous cars following precomputed routes, and performed inverse reinforcement learning [[29](#bib.bib29)]. The base features we used include higher speed moving forward (against a speed limit), lateral and directional alignment in the lane, collision avoidance, and distance to the boundaries of the road. Across our experiments, we compare each of these three models against our model switcher (MS) that dynamically chooses between all of them. We additionally wanted to analyze different performances for designers that might be more or less conservative about the reward vs. computation tradeoff, so we show results for different values of λ𝜆\lambdaitalic\_λ. ### III-C Miscellaneous Experimental Details We conduct experiments using TensorFlow [[30](#bib.bib30)] 2.1, running on a 2015 MacBook Pro, for gradient calculation and optimization. All planners optimize for a horizon T=5𝑇5T=5italic\_T = 5 using 20 vanilla gradient descent steps. For switching down, we used a cooldown of K=3𝐾3K=3italic\_K = 3. In Eq. ([2](#S2.E2 "2 ‣ II-C Approximate Solution: Switching between Two Models ‣ II Method ‣ Dynamically Switching Human Prediction Models for Efficient Planning")), the quadratic approximation may be ill conditioned, so we restricted ΔuRΔsubscript𝑢𝑅\Delta u\_{R}roman\_Δ italic\_u start\_POSTSUBSCRIPT italic\_R end\_POSTSUBSCRIPT to exist within some reasonable bounds. ### III-D Evaluation Strategy For every scenario, we run every method against the same simulated human driver. For diversity, we vary the starting positions of the human and robot cars across 30 different seeds per scenario. For every time step, we keep track of the reward and computation time. In Fig. [2](#S3.F2 "Figure 2 ‣ III Experiments ‣ Dynamically Switching Human Prediction Models for Efficient Planning") we visualize average combined planning and decision time for the MS as it progresses through each scenario. Further, we juxtapose a conservative designer (yellow), who values reward relatively more, with an aggressive one (blue), who prefers computational savings. Snapshots below provide context for key points of the experiment denoted by dashed lines above, taken from a single conservative MS run. In Fig. [3](#S3.F3 "Figure 3 ‣ III-D Evaluation Strategy ‣ III Experiments ‣ Dynamically Switching Human Prediction Models for Efficient Planning"), we showcase the average reward and computational time per episode, averaged across seeds. In each plot, the left bars represent computational time, and the right ones reward, in units given by the left and right axes respectively. Within the total computational time, we separate planning and time deciding a human model. We hypothesized that our MS algorithm would maintain a reward similar to that of the top model, while being significantly cheaper to compute. Additionally, we expected to see that conservative switchers obtain better rewards than aggressive switchers, but at higher computational complexity. Note that because we wanted to showcase regions where a conservative switcher would react differently from an aggressive one, the hyperparameter λ𝜆\lambdaitalic\_λ varies widely across our scenarios. This is a reflection of differing reward scales, and in general the designer would have to select an appropriate λ𝜆\lambdaitalic\_λ depending on the problem and desired reward-computation tradeoff. For greater stability and generalization, one may find a highly conservative λ𝜆\lambdaitalic\_λ to be effective, reducing switching sensitivity to situations where the human has little bearing on the reward. ![Refer to caption](/html/2103.07815/assets/x5.png) ![Refer to caption](/html/2103.07815/assets/x6.png) ![Refer to caption](/html/2103.07815/assets/x7.png) Figure 3: Average computational time and reward for the 3 scenarios, with models ordered by computation. Model switchers achieve comparable performance to the best model with less computation. Note: the aggressive switchers with greater values for λ𝜆\lambdaitalic\_λ are to the left of the conservative ones. ### III-E Scenario 1: Stay Back In our first scenario in Fig. [2](#S3.F2 "Figure 2 ‣ III Experiments ‣ Dynamically Switching Human Prediction Models for Efficient Planning") (left), the robot and human begin driving alongside each other at the same speed. Ahead a series of cones creates a bottleneck in the road. Either the robot or human must yield to avoid a collision. For this scenario, we restrict ourselves to Naive and Turn for simplicity, while the other two will showcase the potential of a broader model ladder. As shown in Fig. [3](#S3.F3 "Figure 3 ‣ III-D Evaluation Strategy ‣ III Experiments ‣ Dynamically Switching Human Prediction Models for Efficient Planning") (left), the Naive model struggles to correctly anticipate whether the human will go first and act accordingly, but it is much cheaper than Turn which safely navigates the scenario. Meanwhile, both aggressive or conservative switchers manages to obtain rewards close to that of the Turn, but with computational time closer to Naive. Additionally, notice that the decision time doesn’t add excessive overhead, which underlines the efficiency of our approximation to the meta MDP. In Fig. [2](#S3.F2 "Figure 2 ‣ III Experiments ‣ Dynamically Switching Human Prediction Models for Efficient Planning") (left), we see that a conservative switcher (λ=0.4𝜆0.4\lambda=0.4italic\_λ = 0.4) will generally use Turn before and during the bottleneck, whereas an aggressive one (λ=5.0𝜆5.0\lambda=5.0italic\_λ = 5.0) uses Turn only to intervene when using Naive is headed towards a collision. ### III-F Scenario 2: Merger In the next scenario shown in Fig. [2](#S3.F2 "Figure 2 ‣ III Experiments ‣ Dynamically Switching Human Prediction Models for Efficient Planning") (middle), the robot would like to merge into the left lane. The gap is too small for the robot, but if it gradually angles its hood the target human will yield allowing the robot to enter. As shown in Fig. [3](#S3.F3 "Figure 3 ‣ III-D Evaluation Strategy ‣ III Experiments ‣ Dynamically Switching Human Prediction Models for Efficient Planning") (middle), only ToM is able to anticipate that the human will yield should it begin entering the lane. However, before and after merging, ToM provides no further advantage, so the switcher can exploit that to reduce computational complexity. In Fig. [3](#S3.F3 "Figure 3 ‣ III-D Evaluation Strategy ‣ III Experiments ‣ Dynamically Switching Human Prediction Models for Efficient Planning") (middle), we see that both the conservative and aggressive switchers obtain rewards closer to ToM, but with significantly less total computation. In Fig. [2](#S3.F2 "Figure 2 ‣ III Experiments ‣ Dynamically Switching Human Prediction Models for Efficient Planning") (middle), we see that the model switcher only uses ToM for merging, as it provides no comparative advantage afterwards. A conservative model switcher (λ=0.1𝜆0.1\lambda=0.1italic\_λ = 0.1) switches up earlier merging faster, whereas an aggressive one (λ=0.6𝜆0.6\lambda=0.6italic\_λ = 0.6) switches later but requires less computation. Delaying an inevitable switch hurts the aggressive MS, highlighting the importance of a conservative λ𝜆\lambdaitalic\_λ for safety-critical applications Turn is used sparingly during the transition as it provides little comparative advantage to Naive. ### III-G Scenario 3: Give Way In our last scenario shown in Fig. [2](#S3.F2 "Figure 2 ‣ III Experiments ‣ Dynamically Switching Human Prediction Models for Efficient Planning") (right), the person is driving alongside the robot and would like to enter the robot’s lane; however, other drivers around the robot do not allow enough space either way. The robot would like to help the human enter. Of course, the robot may create sufficient space by moving forward or backing up. As shown in Fig. [3](#S3.F3 "Figure 3 ‣ III-D Evaluation Strategy ‣ III Experiments ‣ Dynamically Switching Human Prediction Models for Efficient Planning") (right), ToM is the only one capable of understanding the robot’s ability to help the person to merge. Turn occasionally succeeds when it yields fearing a collision. Both switchers obtain higher rewards than those of the cheaper models, but we notice again a large gap between the two. The earlier the robot makes space for the person, the better. A conservative model switcher (λ=0.01𝜆0.01\lambda=0.01italic\_λ = 0.01) quickly switches up and allows the human to enter, albeit slower than pure ToM. A more aggressive switcher (λ=0.03𝜆0.03\lambda=0.03italic\_λ = 0.03) delays the switch to ToM resulting in lesser reward, but keeps overall computation lower. The delay between yielding and the human entering presents a challenge for our myopic, one time step, reward simplification. A very low λ𝜆\lambdaitalic\_λ works here, but the issue of myopic gain potentially underestimating longer horizon gain remains. IV Discussion -------------- Summary: In this paper, we formalized the robot’s decision making process over which predictive human model to use as a meta MDP. We introduced an approximate solution that enables efficient switching to the most suitable available model within this MDP. The resulting decisions maintain rewards similar to those of the best model available, while dramatically reducing computational time. Future Work: Because the robot cannot see the human’s true future controls, we were limited to basing switching decisions on what the person did in the past. We could approximate the true human trajectory using the top model’s prediction, but that would relinquish most computational savings. For example, Naive planning and Turn prediction takes as long as Turn planning. Additionally, because the decision to switch relies on a single time step simplification, our scenarios needed consistent reward signal. Future work must address adapting our algorithm to sparse reward settings where one step reward gradients are not meaningful. Learning a value function to replace reward in our formulation would be an interesting direction. Moreover, all this work happened in a simple driving simulator, albeit with what we think are complex scenarios. To put this on the road, we will need more emphasis on safety, as well as longer decision horizons. Lastly, our algorithm focuses on single human decisions. We could run our method separately for every nearby human, evaluating the differential benefit of switching each, but we are yet to conduct experiments in that setting. Alternatively, we can imagine adapting it to multiple humans by either adding more complex multi-player game theoretic models, or combining it with the prioritization schema presented by [[26](#bib.bib26)]. Conclusion: Despite these limitations, we are encouraged to see robots have more autonomy over what human models to use when planning online, without hand-coded heuristics. We look forward to applications of our model switching ideas beyond autonomous driving: to mobile robots, quadcopters, or any human-robot interactive scenarios where planning with multiple predictive human models might be beneficial.
9d8941ce-edbb-484b-b786-608d54659016	trentmkelly/LessWrong-43k	LessWrong	Analogy Bank for AI Safety > “A great deal of thinking well is about knowing when to think at what levels of abstraction. On the one hand, abstractions drop detail, and the reliability of inferences using them varies in complex ways with context. On the other hand, reasoning abstractly can be much faster and quicker, and can help us transfer understanding from better to less known cases via analogy . . . my best one factor theory to explain the worlds I’ve liked best, such as “rationalists”, is that folks there have an unusually high taste for abstraction . . . Thus my strongest advice for my fellow-traveler worlds of non-academic conversation is: vet your abstractions more. For example, this is my main criticism, which I’ve repeated often, of AI risk discussions. Don’t just accept proposed abstractions and applications.” — Robin Hanson TL;DR * I am compiling analogies about AI safety on this sheet. * You can submit new ones and points for/against the existing ones with this form. The Function of Analogies > “All non-trivial abstractions, to some degree, are leaky.” — Joel Spolsky When you are trying to explain some novel concept A, it can be helpful to point out similarities and differences with a more familiar concept B. Simple Comparisons One simple type of comparison you might want to make, when claiming that target A has some property X, is to conjure analogue B, where B trivially has property X. For example, “AI is a technology that might do more harm than good, like cigarettes.” What do you really gain from deploying such an analogy, beyond directly claiming, “AI is a technology that might do more harm than good”? Often, I suspect the answer is: not much. In fact, such a comparison can be more distracting than illuminating because your audience might assume the analogy is a load-bearing warrant for your position and then focus, for example, on whether cigarettes have done more harm than good, instead of engaging with your actual claim about AI. To see how an analogy can distr
8df1f5a3-7f0b-408f-a712-8a12a854f814	StampyAI/alignment-research-dataset/eaforum	Effective Altruism Forum	Apply to fall policy internships (we can help) ### Many U.S. congressional internship applications are closing in the next few weeks for Fall (Sep- Dec) internships. This is a relatively low-effort, high reward thing to do if you you’re interested in testing your fit for policy. I (Elika) interned in my congressional office for a semester just from off-the-cuff applying to test my fit and build my resume. This experience has been incredibly helpful (I now work for the US government and it gives me some more credibility in D.C). Many applications are closing within the next 1-2 weeks. We’re offering to [support](https://airtable.com/shrzCEa9YKJdiKlsu) anyone considering applying. This is a particularly good fit if you’re: * Interested in working in policy, politics, or governance solutions to problems * An undergraduate student * Able to work part-time (10+ hours per week) If you think this could be a good opportunity, we recommend: * Reading [this guide to internships](https://forum.effectivealtruism.org/posts/sD5vF6cfuAYh9ZqYZ/congressional-internships-why-and-how-to-apply) which has information on which offices to choose from and how to apply and more (including [this helpful link of all the Congressional office internships](https://airtable.com/shrwTtjhJSwepvFLo)) * Making a list of offices you think you’d be a good fit for * Applying! [When in doubt, apply](https://forum.effectivealtruism.org/posts/PhySoajcEcY8EtgKH/when-in-doubt-apply) - there’s no harm in applying if you’re serious about exploring this opportunity. We’re offering to [support](https://airtable.com/shrzCEa9YKJdiKlsu) if you’re interested. ### [Sign up to get support applying here](https://airtable.com/shrzCEa9YKJdiKlsu) Things we can help with: * Whether or not you’d be a good fit for the positions * Review your resume, cover letter & offices you’re interested in * Accountability for submitting applications by the deadline
d7aad85e-bdfa-45c1-b7b4-ad24389e8e9f	trentmkelly/LessWrong-43k	LessWrong	Meetup : Philadelphia - The Sword of Good Discussion article for the meetup : Philadelphia - The Sword of Good WHEN: 22 September 2013 12:30:00PM (-0400) WHERE: 1515 Chestnut St, Philadelphia, PA Location: Givovani's Pizza Discussion topic: The Sword of Good (A short story - I highly recommend reading it before the meetup if you are able, because it's meant to have some element of surprise.) Fiction doesn't seem to be used as a discussion topic for meetups very often, but I noticed that we tend to end up talking about stories together anyway, and it usually leads to interesting conversations. If you have any simple games you'd like to play as a group, please bring them - it will give us something to do in the beginning while people are still arriving. Discussion article for the meetup : Philadelphia - The Sword of Good
1b330bd2-a0f1-490c-a157-31e1f57df5df	StampyAI/alignment-research-dataset/arbital	Arbital	Epistemic exclusion An "epistemic exclusion" would be a hypothetical form of AI limitation that made the AI not model (and if reflectively stable, not want to model) some particular part of physical or mathematical reality, or model it only using some restricted model class that didn't allow for the maximum possible predictive accuracy. For example, a [behaviorist genie](https://arbital.com/p/102) would not want to model human minds (except using a tightly restricted model class) to avoid [https://arbital.com/p/6v](https://arbital.com/p/6v), [https://arbital.com/p/programmer_manipulation](https://arbital.com/p/programmer_manipulation) and other possible problems. At present, nobody has investigated how to do this (in any reflectively stable way), and there's all sorts of obvious problems stemming from the fact that, in reality, most facts are linked to a significant number of other facts. How would you make an AI that was really good at predicting everything else in the world but didn't know or want to know what was inside your basement? Intuitively, it seems likely that a lot of naive solutions would, e.g., just cause the AI to de facto end up constructing something that wasn't technically a model of your basement, but played the same role as a model of your basement, in order to maximize predictive accuracy about everything that wasn't your basement. We could similarly ask how it would be possible to build a really good mathematician that never knew or cared whether 333 was a prime number, and whether this might require it to also ignore the 'casting out nines' procedure whenever it saw 333 as a decimal number, or what would happen if we asked it to multiply 3 by (100 + 10 + 1), and so on. That said, most practical reasons to create an epistemic exclusion (e.g. [against modeling humans in too much detail](https://arbital.com/p/102), or [against modeling distant alien civilizations and superintelligences](https://arbital.com/p/1fz)) would involve some practical reason the exclusion was there, and some level of in-practice exclusion that was good enough, which might not require e.g. maximum predictive accuracy about everything else combined with zero predictive accuracy about the exclusion.
308262a9-f3d4-4c6d-99df-0c6d0f3dcdec	trentmkelly/LessWrong-43k	LessWrong	Meetup : San Francisco Meetup: Short Talks Discussion article for the meetup : San Francisco Meetup: Short Talks WHEN: 19 October 2015 06:15:00PM (-0700) WHERE: 1597 Howard St. San Francisco, CA We'll be meeting to give/listen to short talks. Planning isn't necessary: these are not expected to be polished. I can be reached at 301-458-0764 if you need help getting in. As always, feel free to show up late. Discussion article for the meetup : San Francisco Meetup: Short Talks
5c8ff960-9fee-4bc9-bc62-f3b57b6afa60	trentmkelly/LessWrong-43k	LessWrong	What I Tell You Three Times Is True "The human brain evidently operates on some variation of the famous principle enunciated in 'The Hunting of the Snark': 'What I tell you three times is true.'" -- Norbert Weiner, from Cybernetics Ask for a high-profile rationalist, and you'll hear about Richard Dawkins or James Randi or maybe Peter Thiel. Not a lot of people would immediately name Scott Adams, creator of Dilbert. But as readers of his blog know, he's got a deep interest in rationality, and sometimes it shows up in his comics: for example, this one from last week. How many people can expose several million people to the phrase "Boltzmann brain hypothesis" and have them enjoy it? So I was very surprised to find Adams was a believer in and evangelist of something that sounded a lot like pseudoscience. "Affirmations" are positive statements made with the belief that saying the statement loud enough and long enough will help it come true. For example, you might say "I will become a syndicated cartoonist" fifteen times before bed every night, thinking that this will in fact make you a syndicated cartoonist. Adams partially credits his success as a cartoonist to doing exactly this. He admits "it sounds as if I believe in some sort of voodoo or magic", and acknowledges that "skeptics have suggested, and reasonably so, that this is a classic case of selective memory" but still swears that it works. He also has "received thousands of e-mails from people recounting their own experiences with affirmations. Most people seem to be amazed at how well they worked." None of this should be taken too seriously without a controlled scientific study investigating it, of course. But is it worth the effort of a study, or should it be filed under "so stupid that it's not worth anyone's time to investigate further"? I think there's a good case to be made from within a rationalist/scientific worldview that affirmations may in fact be effective for certain goals. Not miraculously effective, but not totally useless ei
68c90e34-eee2-4a7c-9932-56b28128e791	trentmkelly/LessWrong-43k	LessWrong	European Community Weekend 2018 Announcement We are excited to announce this year's European LessWrong Community Weekend. For the fifth time, rationalists from all over Europe (and some from outside Europe) are gathering in Berlin to socialize, have fun, exchange knowledge and skills, and have interesting discussions. The event takes place September 7th to September 9th and, like last year, it will be held in the beautiful Jugendherberge Wannsee which contains a large room for central events, several seminar rooms, and lots of comfortable spaces inside and out to socialize or relax. This is a community-driven event. That means that while there will be a keynote and pre-planned content, the bulk of the schedule will be filled by the participants. There will be space to give talks, short or long, provide workshops, or just gather some people to do an activity together. In previous years we had the talks, lightning talks and workshops you would expect, as well as lighter activities such as morning-workouts, meditation sessions, authentic relating games, swimming in the lake and many more. Of course, there will also be time to reconnect with friends and form new connections with other aspiring rationalists. Some valuable information Most of the talks and discussions will be held in English, so you do not need to be able to speak German to attend. The ticket price of €150 includes accommodation for two nights, on-site meals (breakfast, lunch, dinner) and snacks, and a welcome lunch on Friday at 12:00. The event wraps up Sunday afternoon around 15:00. In the days after the weekend, participants are invited to stay in Berlin a little longer to explore the city, go bouldering, play frisbee, etc. While this is not part of the official event, we will coordinate couch-surfing opportunities to avoid the need for hotels. tl;dr * When? 7-9 September 2018 * Where? http://jh-wannsee.de * How much? €150 * Apply here: http://tiny.cc/lwcw2018_signup and * Submit a contribution to support your application: http://tin
fbe705ac-9b38-4cf3-a356-22e32fb4a409	StampyAI/alignment-research-dataset/lesswrong	LessWrong	Image GPT My hot take: Not too surprising to me, considering what GPT-3 could do. However there were some people (and some small probability mass remaining in myself) saying that even GPT-3 wasn't doing any sort of reasoning, didn't have any sort of substantial understanding of the world, etc. Well, this is another nail in the coffin of that idea, in my opinion. Whatever this architecture is doing on the inside, it seems to be pretty capable and general. I don't think this architecture will scale to AGI by itself. But the dramatic success of this architecture is evidence that there are other architectures, not too far away in search space, that exhibit similar computational efficiency and scales-with-more-compute properties, that are useful for more different kinds of tasks.
9bdf8487-fba6-4ddf-9625-3dce1d13f705	trentmkelly/LessWrong-43k	LessWrong	Meetup : Durham: Luminosity followup Discussion article for the meetup : Durham: Luminosity followup WHEN: 20 June 2013 07:00:00PM (-0400) WHERE: Cocoa Cinnamon, 420 W Geer St., Durham NC A month ago or so we finished going through Alicorn's Luminosity sequence on Less Wrong. So, having done that and worked on Being Luminous for a month or two (or more), what conclusions did you draw? We'll meet for coffee and informal introductions at 7:00. On-topic conversation will run from 7:30-9:00. Please give some thought to what you learned (or didn't learn) from the luminosity meetups, and what impact it has had in the time period since. If you weren't around for all (or even any) of the Luminosity meetups, please feel free to come anyway! Bring some questions about it for people that were, if you like :) Discussion article for the meetup : Durham: Luminosity followup
25609c27-8580-4bf5-a370-b760beb1e99b	trentmkelly/LessWrong-43k	LessWrong	"Bad-for-the-world-ipedia?" Lately I've been thinking about all of the various services and products I consume and how pretty much all of them are bad for the world in one way or another, large or small. Some of the problems associated with them I am less concerned about. Some of them could be construed as good things (i.e. sweat shop labor DOES provide jobs, whatever impact it might or might not have on the overall quality of life). In general I'd like to live my life having as minimal a negative impact on the world as possible. But "negative impact" is a hugely broad topic and there are a million variables to consider and I just don't have time. The best solution, I think, would be to have a wikipedia-like website where individual people with knowledge of specific problems can start tagging specific products with the types of negative consequences associated with them, and (somehow) sort those consequences into categories that individuals can decide how much to worry about. Over time it could eventually become a fairly efficient way to track the utility value of things. I'm sort of hoping something like this already exists, even if in an infant form, and that someone here knows about it. But I doubt it, so the I guess this falls mostly under the post category of "hey someone other than me should devote a bunch of time and energy to this project that I myself am not qualified to do." But maybe a few people here at least have a better idea than I do of the scope of the requirements for it, so the idea can be refined a bit.
d66f50a0-f63c-4132-b6ba-a1ee688d396c	StampyAI/alignment-research-dataset/blogs	Blogs	predictablizing ethic deduplication predictablizing ethic deduplication ----------------------------------- does the machinery of the cosmos [deduplicate identical moral patients and/or their experiences](deduplication-ethics.html) ? this seems like a very difficult question to even address; but the good news is that for the general future we might not have to care about it. we can simply make it that the superintelligence that runs everything, does deduplicate (memoize) identical computations and data structures, which guarantees that the ethics we build on top of that (for superintelligence to implement) can know about deduplication. why choose deduplication over no-deduplication? because if we add deduplication on top of any machinery of the cosmos, then we can know for sure deduplication happens, but if we don't implement deduplication, then whether computation is deduplicated depends on the machinery of the cosmos. "but doesn't this require looking inside arbitrarily encoded computations, such as [homomorphic encryption](https://en.wikipedia.org/wiki/Homomorphic_encryption) ?" that is true, but for an aligned superintelligence we require this anyways. otherwise, it could just let unseen pockets of arbitrary suffering happen.
b843ad62-21b9-4dd5-b490-262acc7c34d4	trentmkelly/LessWrong-43k	LessWrong	How well do truth probes generalise? Representation engineering (RepEng) has emerged as a promising research avenue for model interpretability and control. Recent papers have proposed methods for discovering truth in models with unlabeled data, guiding generation by modifying representations, and building LLM lie detectors. RepEng asks the question: If we treat representations as the central unit, how much power do we have over a model’s behaviour? Most techniques use linear probes to monitor and control representations. An important question is whether the probes generalise. If we train a probe on the truths and lies about the locations of cities, will it generalise to truths and lies about Amazon review sentiment? This report focuses on truth due to its relevance to safety, and to help narrow the work. Generalisation is important. Humans typically have one generalised notion of “truth”, and it would be enormously convenient if language models also had just one[1]. This would result in extremely robust model insights: every time the model “lies”, this is reflected in its “truth vector”, so we could detect intentional lies perfectly, and perhaps even steer away from them. We find that truth probes generalise surprisingly well, with the 36% of methodologies recovering >80% of the accuracy on out-of-distribution datasets compared with training directly on the datasets. The best probe recovers 92% accuracy. Thanks to Hoagy Cunningham for feedback and advice. Thanks to LISA for hosting me while I did a lot of this work. Code is available at mishajw/repeng, along with steps for reproducing datasets and plots. Methods We run all experiments on Llama-2-13b-chat, for parity with the source papers. Each probe is trained on 400 questions, and evaluated on 2000 different questions, although numbers may be lower for smaller datasets. What makes a probe? A probe is created using a training dataset, a probe algorithm, and a layer. We pass the training dataset through the model, extracting activations[2] jus
43e92390-5e59-45e2-bebc-beb1dde308a7	trentmkelly/LessWrong-43k	LessWrong	Meetup : Stockholm: Bottlenecks to trading personal resources Discussion article for the meetup : Stockholm: Bottlenecks to trading personal resources WHEN: 11 November 2016 04:15:00PM (+0100) WHERE: Lindstedtsvägen 3 room 1537, SE-114 28 Stockholm, Sverige "Value of time" is often employed by utilitarians. It can be hard to determine one's value of time for a variety of reasons. This talk will be about different currencies of personal life like time, money, and pleasure; and how choices are implicit trades between them. The talk will focus on when it's appropriate to treat these resources as liquid currencies and when it's not. After discussing the theory, and some examples, we'll practice. We'll individually come up with our own exchange rates for personal currencies, then fix one another's estimates in small groups. I hope everyone walks away from this talk with a concrete number for their value of time. The meetup is at a KTH academic building and the room is on the 5th floor, two stairs up. If you want to influence future meetup times, fill out this poll Discussion article for the meetup : Stockholm: Bottlenecks to trading personal resources
ec8bceb2-8839-4339-a309-9a1f89253725	trentmkelly/LessWrong-43k	LessWrong	Neil deGrasse Tyson on Cryonics Question: > What are your thoughts on cryogenic preservation and the idea of medically treating aging? His response: > A marvelous way to just convince people to give you money. Offer to freeze them for later. I'd have more confidence if we had previously managed to pull this off with other mammals. Until then I see it as a waste of money. I'd rather enjoy the money, and then be buried, offering my body back to the flora and fauna of which I have dined my whole life. Link
0757855c-7d4b-4edc-bd8a-5e9a29df11fe	StampyAI/alignment-research-dataset/alignmentforum	Alignment Forum	TASRA: A Taxonomy and Analysis of Societal-Scale Risks from AI Partly in response to calls for more detailed accounts of how AI could go wrong, e.g., from [Ng and Bengio](https://twitter.com/AndrewYNg/status/1666582174257254402)'s recent exchange on Twitter, here's a new paper with Stuart Russell: * Discussion on Twitter... comments welcome! <https://twitter.com/AndrewCritchCA/status/1668476943208169473> * arXiv draft: ["TASRA: A Taxonomy and Analysis of Societal-Scale Risks from AI"](https://arxiv.org/abs/2306.06924) Many of the ideas will not be new to LessWrong or the Alignment Forum, but holistically I hope the paper will make a good case to the world for using logically exhaustive arguments to identify risks (which, outside LessWrong, is often not assumed to be a valuable approach to thinking about risk). I think the most important figure from the paper is this one: ![](https://res.cloudinary.com/lesswrong-2-0/image/upload/f_auto,q_auto/v1/mirroredImages/zKkZanEQc4AZBEKx9/rc8qtkxfh5wxoc0pwzmn)... and, here are some highlights: * Self-fulfilling pessimism: <https://arxiv.org/pdf/2306.06924.pdf#page=4> * Industries that could eventually get out of control in a closed loop: <https://arxiv.org/pdf/2306.06924.pdf#page=5> ...as in this "production web" story: <https://arxiv.org/pdf/2306.06924.pdf#page=6> * Two "bigger than expected" AI impact stories: <https://arxiv.org/pdf/2306.06924.pdf#page=8> * Email helpers and corrupt mediators, which kinda go together: <https://arxiv.org/pdf/2306.06924.pdf#page=10> <https://arxiv.org/pdf/2306.06924.pdf#page=11> * Harmful A/B testing: <https://arxiv.org/pdf/2306.06924.pdf#page=12> * Concerns about weaponization by criminals and states: <https://arxiv.org/pdf/2306.06924.pdf#page=13> Enjoy :)
28904644-a7a8-496c-8024-5aeadc1e5d6b	trentmkelly/LessWrong-43k	LessWrong	[LINK] "Academic publishers make Murdoch look like a socialist" says George Monbiot link > Who are the most ruthless capitalists in the western world? Whose monopolistic practices make Walmart look like a corner shop and Rupert Murdoch a socialist? You won't guess the answer in a month of Sundays. While there are plenty of candidates, my vote goes not to the banks, the oil companies or the health insurers, but – wait for it – to academic publishers. Theirs might sound like a fusty and insignificant sector. It is anything but. Of all corporate scams, the racket they run is most urgently in need of referral to the competition authorities. > > Everyone claims to agree that people should be encouraged to understand science and other academic research. Without current knowledge, we cannot make coherent democratic decisions. But the publishers have slapped a padlock and a "keep out" sign on the gates. > > You might resent Murdoch's paywall policy, in which he charges £1 for 24 hours of access to the Times and Sunday Times. But at least in that period you can read and download as many articles as you like. Reading a single article published by one of Elsevier's journals will cost you $31.50. Springer charges €34.95, Wiley-Blackwell, $42. Read 10 and you pay 10 times. And the journals retain perpetual copyright. You want to read a letter printed in 1981? That'll be $31.50. > > Of course, you could go into the library (if it still exists). But they too have been hit by cosmic fees. The average cost of an annual subscription to a chemistry journal is $3,792. Some journals cost $10,000 a year or more to stock. The most expensive I've seen, Elsevier's Biochimica et Biophysica Acta, is $20,930. Though academic libraries have been frantically cutting subscriptions to make ends meet, journals now consume 65% of their budgets, which means they have had to reduce the number of books they buy. Journal fees account for a significant component of universities' costs, which are being passed to their students. > > Murdoch pays his journalists and editors, and his c
dc386e3f-d750-452a-816c-2033939514bc	StampyAI/alignment-research-dataset/lesswrong	LessWrong	Are (at least some) Large Language Models Holographic Memory Stores? Cross-posted [from New Savanna](https://new-savanna.blogspot.com/2023/10/are-at-least-some-large-language-models.html). That’s been on my mind for the last week or two, ever since my recent work on ChatGPT’s memory for texts [1]. On the other than, there’s a sense in which it’s been on my mind for my entire career, or, more accurately, it’s been growing in my mind ever since I read Karl Pribram on neural holography back in 1969 in Scientific American [2]. For the moment let’s think of it as a metaphor, just a metaphor, nothing we have to commit to. Just yet. But ultimately, yes, I think it’s more than a metaphor. To that end I note that cognitive psychologists have recently been developing the idea of verbal memory as holographic in nature [3]. Note: These are quick and dirty notes, a place-holder for more considered thought. Holography in the mind -------------------------- Let’s start with an article David Hays and I published on neural holography as the neural underpinning of metaphor [4]. Here’s where we explain the holographic process: > Holography is a photographic technique for making images. A beam of laser light is split into two beams. One beam strikes the object and is reflected to a photographic plate. The other beam, called a reference beam, goes from laser to plate directly. When they meet, the two beams create an interference pattern—imagine dropping two stones into a pond at different places; the waves propagating from each of these points will meet and the resulting pattern is an interference pattern. The photographic plate records the pattern of interference between the reference beam and the reflected beam. > > The image recorded on the film doesn't look at all like an ordinary photographic image—it’s just a dense mass of fine dots. But when a beam of laser light having the same properties as the original reference beam is directed through the film an image appears in front of the film. The interaction of the laser beam and the hologram has recreated the wave form of the laser beam which bounced off the object when the hologram was made. The new beam has extracted the image from the plate. > > Holography is, as its name suggests, holistic. Every part of the scene is represented in every part of the plate. (This situation is most unlike ordinary photography, which uses a good lens to focus infinitesimal parts of the scene onto equally infinitesimal parts of the plate.) With such a determinedly nondigital recording, certain mathematical possibilities can be realized more easily—we are tempted to say, infinitely more easily. For example, convolution. Take the holographic image of a printed page, and the image of a single word. Convolute them. The result is an image of the page with each occurrence of the word highlighted. We can think of visual recognition as a kind of convolution. The present scene, containing several horses, is convoluted with the memory of a horse and the present horses are immediately recognized. We can think of recognition this way, but we must admit that this process has not been achieved in any machine as yet. > > Further, it is possible to record many different images on the same piece of film, using different reference beams. The reference beams may differ in color, in angle of incidence, or otherwise. We can think— although again we cannot cite a demonstration—of convoluting such a composite plate with a second plate. If the image in the second plate matches any one of the images in the composite, then it is recognized. For metaphor we want to convolute Achilles and the lion and to recognize, to elicit another image containing not Achilles, not the lion, but just that wherein they resemble one another. Such is the metaphor mechanism—but that must wait until the next section, on focal and residual schemas. > > The 175 billion weights that constitute the LLM at the core of ChatGPT, that’s the holographic memory. It is the superposition of all the texts in the training corpus. The training procedure – predict the next word – is a device for calculating a correlation (entanglement [5]) between each word in context, and every other word in every other text, in context. It’s a tedious process, no? But it works, yes? When one prompts a trained memory, the prompt serves as a reference beam. And the whole memory must be ‘swept’ to generate each character. Given the nature of digital computers, this is a somewhat sequential process, even given a warehouse full of GPUs, but conceptually it’s a single pass. When one accesses an optical hologram with a reference beam, the beam illuminates the whole holograph. This is what Miriam Yevick called “one-shot” access in her 1975 paper, Holographic or Fourier Logic [6]. The whole memory is searched in a single sweep. Style transfer ------------------ So, that’s the general idea. Much detail remains to be supplied, most of it by people with more technical knowledge than I’ve got. But I want to get in one last idea from the metaphor paper. We’ve been explaining the concepts of focal and residual schemas: > Now consider a face. Everything we said about the chair applies here as well. But the expression on the face can vary widely and the identity of the face remains constant. This variability of expression can also be handled by the mechanism of focal and residual. There is a focal schema for face-in-neutral-expression and then we have various residuals which can operate on the focal schema to produce various expressions. (You might want to recall D'Arcy Thompson's coordinate transformations in On Growth and Form 1932.) We tend to discard presentation residuals such as lighting and angle of sight, but we respond to expression residuals > > Our basic point about metaphor is that the ground which links tenor and vehicle is derived from residuals on them. Consider the following example, from Book Twenty of Homer's Iliad (Lattimore translation, 1951, ll. 163-175)—it has the verbal form of a simile, but the basic conceptual process is, of course, metaphorical: > > > From the other > side the son of Peleus rose like a lion against him, > the baleful beast, when men have been straining to kill him, the country > all in the hunt, and he at first pays them no attention > but goes his way, only when some one of the impetuous young men > has hit him with the spear he whirls, jaws open, over his teeth foam > breaks out, and in the depth of his chest the powerful heart groans; > he lashes his own ribs with his tail and the flanks on both sides > as he rouses himself to fury for the fight, eyes glaring, > and hurls himself straight onward on the chance of killing some one > of the men, or else being killed himself in the first onrush. > So the proud heart and fighting fury stirred on Achilleus > to go forward in the face of great-hearted Aineias. > > > In short, Achilles was a lion in battle. Achilles is the tenor, lion the vehicle, and the ground is some martial virtue “proud heart and fighting fury”. But what of that detailed vignette about the lion's fighting style? Whatever its use in pacing the narrative, its real value, in our view, is that it contains the residuals on which the comparison rests, the residuals which give it life. The phrase “proud heart and fighting fury” is propositional while the fighting style is physiognomic. “Proud heart and fighting fury” may convey something of what is behind the fighting style, but only metaphoric interaction can foreground the complex schema by which we recognize and feel that style. > > The cognitive problem is to isolate the physiognomy of style, to tease it apart from the entities which exhibit that style. [...] In the case of Achilles and the lion we have two complex physiognomies, each extended in space and time. Metaphoric comparison serves to isolate the style, to allow us to focus our attention on that style as distinct from the entities which exhibit it. > > This comparison involves two foci, Achilles and the lion. The physical resemblance between them is not great—their body proportions are quite different and the lion is covered with fur while Achilles is, depending on the occasion, either naked or clothed in some one of many possible ways. The likeness shows up in the way they move in battle. A body in motion doesn't appear the same as a body at rest. The appearance presented by the focal body is modified by the many residuals which characterize that body's movement— twists and turns, foreshortenings and elongations (for an account of motion residuals, see Hay 1966). The movements of Achilles and the lion must differ at the grossest level, since the lion stands on four legs and fights with claws and teeth, while Achilles stands on two legs and fights with a spear or sword. But their movements are alike at a subtler level, at the level of what we call, in a dancer or a fighter, their style. Residuals can be stacked to many levels. “Proud heart and fighting fury” may be a good phrase to designate that style, but it doesn't allow us to attend to that style. Homer's extended simile does. > > That’s a mouthful, I know. Notice our emphasis on style. That’s what’s got my attention. One of the more interesting things LLMs can do is stylistic transfer. Take a piece of garden variety prose and present it in the style of Hemingway or Sontag, whomever you choose. Hays and I argued that that’s how metaphor is created, deep metaphor, that is, not metaphor so desiccated we no longer register its metaphorical nature, e.g. the mouth of the river. We made our argument about visual scenes: Achilles in batter, a lion in battle. LLMs apply the same process to texts, where style is considered to be a pattern of residuals over the conceptual content of the text. More later. References -------------- [1] Discursive Competence in ChatGPT, Part 2: Memory for Texts, Version 3, <https://www.academia.edu/107318793/Discursive_Competence_in_ChatGPT_Part_2_Memory_for_Texts_Version_3> [2] I recount that history here: Xanadu, GPT, and Beyond: An adventure of the mind, <https://www.academia.edu/106001453/Xanadu_GPT_and_Beyond_An_adventure_of_the_mind> [3] Michael N. Jones and Douglas J. K. Mewhort, Representing Word Meaning and Order Information in a Composite Holographic Lexicon, Psychological Review, 2007, Vol. 114, No. 1, 1-37. DOI: <https://doi.org/10.1037/0033-295X.114.1.1> Donald R. J. Frankin and D. J. K. Mewhort, Memory as a Holograpm: An Analysis of Learning and Recall, Canadian Journal of Experimental Psychology / Revue canadienne de psychologie expérimentale, Association 2015, Vol. 69, No. 1, 115–135, <https://doi.org/10.1037/cep0000035> [4] Metaphor, Recognition, and Neural Process, <https://www.academia.edu/238608/Metaphor_Recognition_and_Neural_Process> [5] See posts tagged with “entangle”, <https://new-savanna.blogspot.com/search/label/entangle> [6] Miriam Lipschutz Yevick, Holographic or Fourier Logic, Pattern Recognition 7, 197-213, <https://sci-hub.tw/10.1016/0031-3203(75)90005-9>
34991c38-6ec9-49c6-905e-a757b8517a84	StampyAI/alignment-research-dataset/eaforum	Effective Altruism Forum	Approfondimenti sui rischi dell’IA (materiali in inglese) This is an Italian translation of [*More to explore on 'Risks from Artificial Intelligence'](https://forum.effectivealtruism.org/posts/Cf6tNAhDbQFvAwbAg/more-to-explore-on-risks-from-artificial-intelligence) ### Lo sviluppo dell’intelligenza artificiale* * [AlphaGo - The Movie - DeepMind](https://tinyurl.com/vsj22235) - Documentario sull’intelligenza artificiale, l’antichissimo gioco del Go e cosa possiamo imparare sulle potenzialità future delle IA. (Filmato - 1 ora e 30 minuti) * [The Artificial Intelligence Revolution: Part 1](https://tinyurl.com/sumpuw) - Una divertente e interessante esplorazione dell’intelligenza artificiale dal famoso blogger Tim Urban. (45 minuti) ### Altre risorse sull’allineamento dell’intelligenza artificiale * [AGI Safety Fundamentals Curricula](https://www.agisafetyfundamentals.com/) * [My personal cruxes for working on AI safety](https://forum.effectivealtruism.org/posts/Ayu5im98u8FeMWoBZ/my-personal-cruxes-for-working-on-ai-safety#Problems_solve_themselves) (65 minuti) * [Professor Stuart Russell on the flaws that make today’s AI architecture unsafe & a new approach that could fix it](https://tinyurl.com/erkwmdhr) (Podcast - 2 ore 15 minuti) * [Some Background on Our Views Regarding Advanced Artificial Intelligence - Open Philanthropy Project](https://tinyurl.com/m8fzdtc3) - Una spiegazione del perché ci sia una seria possibilità che il progresso dell’intelligenza artificiale potrebbe essere comparabile alla transizione dall’era neolitica alla rivoluzione industriale. (1 ora) * [The Precipice](https://tinyurl.com/25y25452) (25 minuti) * [What Failure Looks Like](https://www.alignmentforum.org/posts/HBxe6wdjxK239zajf/what-failure-looks-like) - Due storie specifiche su come potrebbero essere peggiori scenari di una società risultata dal fallimento dell’allineamento di IA, che si sposta considerevolmente dalla classica storia della “esplosione dell’intelligenza” (12 mins.) * [AGI Safety from first principles](https://tinyurl.com/5w4vb9v8) - L’opinione di un ricercatore di IA sui fattori specifici per il problema di allineamento nell’intelligenza artificiale generale (1 ora 15 minuti) * [Human Compatible: Artificial Intelligence and The Problem of Control](https://tinyurl.com/sxs2deby) (Libro) * [The Alignment Problem: Machine Learning and Human Values](https://tinyurl.com/9ae73bvn) (Libro) ### Governance dell’intelligenza artificiale * [The new 30-person research team in DC investigating how emerging technologies could affect national security - 80,000 Hours](https://tinyurl.com/yzajjzhr) - Come cambierebbe la sicurezza internazionale se gli effetti del machine learning fossero di portata simile a quelli dell’elettricità? (Podcast - 2 ore) * [Technology Roulette: Managing Loss of Control as Many Militaries Pursue Technological Superiority - Center for a New American Security](https://tinyurl.com/yxndes72) - Come i progressi tecnologici in ambito militare (incluse, ma non solo, le IA) possono comportare rischi e creare problemi nel prendere decisioni importanti, degni di attenzione da parte delle strutture di sicurezza nazionali. (60 minuti) ### Lavori tecnici sull’allineamento dell’IA * [AI Alignment Landscape](https://ai-alignment.com/ai-alignment-landscape-d3773c37ae38) (Video - 30 minuti) * [AI safety starter pack](https://forum.effectivealtruism.org/posts/pbiGHk6AjRxdBPoD8/ai-safety-starter-pack) (7 minuti) * [How to pursue a career in technical AI alignment](https://forum.effectivealtruism.org/posts/7WXPkpqKGKewAymJf/how-to-pursue-a-career-in-technical-ai-alignment) (59 minuti) * [Technical Alignment Curriculum](https://www.eacambridge.org/technical-alignment-curriculum) (readings for a 7 week course) * [AI Alignment Forum](https://www.alignmentforum.org/), soprattutto le loro [sequenze principali](https://www.alignmentforum.org/library) ### Critiche ai rischi dell’IA * [How sure are we about this AI stuff?](https://forum.effectivealtruism.org/posts/9sBAW3qKppnoG3QPq/ben-garfinkel-how-sure-are-we-about-this-ai-stuff) (26 minuti) * [A tale of 2.75 orthogonality theses](https://forum.effectivealtruism.org/posts/kCAcrjvXDt2evMpBz/a-tale-of-2-75-orthogonality-theses) (20 minuti) * [How to know if AI is about to destroy civilization](https://forum.effectivealtruism.org/posts/7x2BokkkemjnXD9B6/new-article-from-oren-etzioni) (sommario, 2 minuti) * [The AI Messiah](https://forum.effectivealtruism.org/posts/r72wjMns9wyaAhWhc/the-ai-messiah) (e il primo commento) (5 minuti) * [How good is humanity at coordination?](https://www.lesswrong.com/posts/y3jDSoTTdBD9Nj3Gx/how-good-is-humanity-at-coordination) (4 minuti)
db02c700-8eec-40b2-95f5-44c4b35dfc0f	trentmkelly/LessWrong-43k	LessWrong	A few questions about Discussion Since there aren't any subforums and I couldn't find a thread with information of what is relevant for Discussion, and I saw the threads here are relatively free, I've decided to ask my questions in a thread. - Can I post questions about this section? (sorry if no) - Can I post about psychology in general? - Can I post about anything that might be of the interest of a rationalist? Like for example, a thread asking about how to reduce the risks for most cancers. Note that this is a huge range of possible questions. Edit: thanks, I think I figured it out. I'm still not sure if I want to respond to all the comments, whether I should comment myself or edit the original post.
99a6630d-c28c-42ac-bdff-9b2e2393a21e	trentmkelly/LessWrong-43k	LessWrong	Multimodal Neurons in Artificial Neural Networks A paper investigating how individual neurons in a CLIP model (an image/text neural net combining a ResNet vision model with a Transformer language model) respond to various abstract concepts. This shouldn't be very surprising after GPT-3 and DALL-E but still, identifying multimodal neurons feels scarily close to "neural net that understands abstract concepts" and thus AGI for my comfort. Some individual neurons that they isolated (see the article for more): * Spiderman neuron: responds to photos of Spiderman in costume and spiders, comics or drawings of Spiderman and spider-themed icons, the text “spider” and others. Associates him with "Peter Parker" and also responds to images, text, and drawings of heroes and villains from Spiderman movies and comics over the last half-century. * Yellow neuron: responds to images of the words “yellow”, “banana” and “lemon,” in addition to the color. * Jesus Christ neuron: detects Christian symbols like crosses and crowns of thorns, paintings of Jesus, his written name, and feature visualization shows him as a baby in the arms of the Virgin Mary. * Hitler neuron: learns to detect his face and body, symbols of the Nazi party, relevant historical documents, and other loosely related concepts like German food. Feature visualization shows swastikas and Hitler seemingly doing a Nazi salute. * Donald Trump neuron: strongly responds to images of him across a wide variety of settings, including effigies and caricatures in many artistic mediums, as well as more weakly activating for people he’s worked closely with like Mike Pence and Steve Bannon. It also responds to his political symbols and messaging (eg. “The Wall” and “Make America Great Again” hats). On the other hand, it most negatively activates to musicians like Nicky Minaj and Eminem, video games like Fortnite, civil rights activists like Martin Luther King Jr., and LGBT symbols like rainbow flags. * Happiness neuron: responds both to images of similing people, and words
23b14af5-61e5-415a-82e4-87d214c2beb1	trentmkelly/LessWrong-43k	LessWrong	My advice on finding your own path tl;dr - 4 steps 1. Give yourself permission to take the future seriously 2. Be willing to imagine successful or ambitious outcomes 3. Give yourself the gift of focused time and space 4. Write draw sketch diagram or whatever you need to empty your mind Intro (Feel free to skip to the method.) When I was transitioning into technical AI alignment work, I benefited immensely from the mentorship and guidance of experts in the field. They helped me to understand the relevant concepts, to develop my skills, and to find my place in the research community. In my early years of giving career advice to other people, I tried to replicate this experience for them. I listened to their questions and concerns, and I tried to provide what I thought were the best possible answers. These are questions like "What do you think are the most important problems to work on?" or "What should I do to get the highest impact job in AI alignment?", and while I tried to listen to them and give personalized advice, I think the answers always failed to capture what was the best fit for the individual. Over the years, however, my advice has evolved. I have answered direct questions less and less directly, and tried to do more of asking leading questions in response. "What do _you_ think the most important problems are?" "What kinds of work do _you_ think you'd be highest impact at?" As I've done this, I've noticed four specific hang-ups (probably others, but I'm simplifying and reducing here in order to make my points simpler). My advice has mostly turned into just four things that I say to everyone, and I think are more useful than specific answers to specific questions. Method: 4 Steps 1. Give yourself permission to take the future seriously I think for many people it can be difficult to imagine details about the future. There is a lot of uncertainty, and this can lead to people thinking that certain kinds of thinking or planning or predictions are inappropriate. Another way this som
165aeaba-64f5-4ef4-a031-ac864f672fd9	trentmkelly/LessWrong-43k	LessWrong	Is this a Pivotal Weak Act? Creating bacteria that decompose metal This has been haunting my mind for a while and I would appreciate feedback on it! In his infamous article "AGI Ruin: A list of lethalities" Eliezer defines a "pivotal weak act" and gives a heuristic proof that no such thing can exist. TLDR: I think his proof is wrong and there is a counterexample. I believe creating bacteria that decompose metal, silicone, (or any other superset of the materials GPUs consist of) would constitute a pivotal weak act. Long Version: In his article, Eliezer outlines several hopes of people claiming AGI won't be as bad or any problem at all, and then cruelly squashes them. One of those hopes is the possibility of executing a "pivotal weak act". The idea being that a small group of people executes some action X that will prevent AGI from being built, for example a group that is privy to the dangers of AGI would command a friendly AGI to "burn all GPUs" and then we are good. Eliezer argues that any AGI powerful enough (pivotal) to prevent or at least indefinitely postpone unaligned AGI must itself be powerful enough such that it needs to be aligned (not weak), which we don't know how to do. I believe his proof is false. Definition: A Pivotal Weak Act, would be some action or event A, such that 1. A happening or being executed prevents or delays the advent of an unaligned AGI indefinitely or at least very long (Pivotal) 2. A does not itself pose a significant X-risk for humanity as a whole (Weak) 3. A is realistically achievable with technology attainable in the coming decades (Realism) furthermore it is not required that - A is in any way related to or facilitated by an AI system - A has no collateral damage - A is moral, legal or anywhere near the Overton Window - A is achievable today with current technology I think that the following is an example of a Pivotal Weak Act. Creating bacteria that decompose metal (and spreading them worldwide) This is pivotal, since it is a special scenario of "Burning all GPUs" It is (likely) weak,
3552561b-b229-40c4-bc28-198c52586699	trentmkelly/LessWrong-43k	LessWrong	Three Kinds of Research Documents: Exploration, Explanation, Academic Aug 2020 Edit: Changed "Clarification" to "Exploration", thanks to a comment by Richard_Ngo Epistemic Status: Low. This was a quick idea, but the grouping honesty doesn't work as well as I'd like. I still think it could be useful to some people though. Ideas appreciated. Recently I have started writing more and have been trying to be more intentional with what I accomplish. Different documents have different purposes and it seemed useful to help clarify this. Here is a list of three specific different types I think are relevant on LessWrong and similar. Exploration I see exploration posts as generally the first instance of information being written down. Here it is important to get the essential ideas out there and to create consensus around terminology among the most interested readers. In some cases, the only interested reader may be the author, who would use the post just to help cement their ideas for themselves. Exploration posts may not be immediately useful and require later posts or context for them to make sense. This is typically fine. There's often not a rush for them to be understood. In many cases, there is a lot of possible information to write down, so the first step is to ensure it's out there, even if it's slow, hard to read, or doesn't much make sense until later. I think of many of Paul Christiano's posts as exploration posts. They're very numerous and novel, but quite confusing to many readers (at least, to myself and several people I've talked to). Sometimes the terminology changes from one post to the next. I used to see this is somewhat of a weakness, but now it comes across to me as a pragmatic option. If he were to have tried to make all of this readable to the average LessWrong reader, there's likely no way he could have written a portion as much. One important point here is that if something is a exploration post, then the main relevant feedback is on the core content, not the presentation. Giving feedback on the readability can sti
a081f41d-0783-47f0-94f4-c5a91c88c7a8	StampyAI/alignment-research-dataset/special_docs	Other	Nonparametric General Reinforcement Learning Nonparametric General Reinforcement Learning Jan Leike A thesis submitted for the degree of Doctor of Philosophy at the Australian National University November 2016 ©Jan Leike This work is licensed under the Creative Commons Attribution 4.0 International License No reinforcement learners were harmed in the making of this thesis. Except where otherwise indicated, this thesis is my own original work. Jan Leike 28 November 2016 Acknowledgements There are many without whom this thesis would not have been possible. I sincerely hope that this page is not the way they learn how grateful I am to them. I thank in particular ... ... ﬁrst and foremost, Marcus Hutter: he is an amazing supervisor; always very sup- portive of my (unusual) endeavors, spent countless hours reading my drafts with a impressive attention to detail. I am also grateful to him for forcing me to be absolutely rigorous in my mathematical arguments, and, of course, for developing the theory of universal AI without which this thesis would not have existed. I could not have picked a better supervisor. ... the Australian National University for granting me scholarships that let me pursue my academic interests unrestricted and without any ﬁnancial worries. ... Csaba Szepesvári and the University of Alberta for hosting me for three months. ... Matthias Heizmann and the University of Freiburg for hosting me while I was traveling in Europe. ... the Machine Intelligence Research Institute for enabling me to run MIRIx research workshops. ... CCR, UAI, Google DeepMind, ARC, MIRI, and FHI for supporting my travel. ... Tor Lattimore for numerous explanations, discussions, and pointers that left me with a much deeper understanding of the theory of reinforcement learning. ... Laurent Orseau for interesting discussions, encouragement, and for sharing so many intriguing ideas. ... my fellow students: Mayank Daswani, Tom Everitt, Daniel Filan, Roshan Shariﬀ, Tian Kruger, Emily Cutts Worthington, Buck Shlegeris, Jarryd Martin, John Aslanides, Alexander Mascolo, and Sultan Javed for so many interesting discus- sions and for being awesome friends. I especially thank Daniel, Emily, Mayank, and Buck for encouraging me to read more of Less Wrong and Slate Star Codex. ... ToscaLechnerforstudyingstatisticswithmedespitesomanyschedulingdiﬃculties across all these time zones. ... Tom Sterkenburg, Christian Kamm, Alexandra Surdina, Freya Fleckenstein, Pe- ter Sunehag, Tosca Lechner, Ines Nikolaus, Laurent Orseau, John Aslanides, and especially Daniel Filan for proofreading parts of this thesis. ... the CSSA for being a lovely bunch that made my stay in Australia feel less isolated. ... my family for lots of love and support, and for tolerating my long absences from Europe. Abstract Reinforcement learning problems are often phrased in terms of Markov decision pro- cesses (MDPs). In this thesis we go beyond MDPs and consider reinforcement learning in environments that are non-Markovian, non-ergodic and only partially observable. Our focus is not on practical algorithms, but rather on the fundamental underlying problems: How do we balance exploration and exploitation? How do we explore opti- mally? When is an agent optimal? We follow the nonparametric realizable paradigm: weassumethedataisdrawnfromanunknownsourcethatbelongstoaknowncountable class of candidates. First, we consider the passive (sequence prediction) setting, learning from data that is not independent and identically distributed. We collect results from artiﬁcial intelli- gence,algorithmicinformationtheory,andgametheoryandputtheminareinforcement learning context: they demonstrate how an agent can learn the value of its own policy. Next, we establish negative results on Bayesian reinforcement learning agents, in particular AIXI. We show that unlucky or adversarial choices of the prior cause the agent to misbehave drastically. Therefore Legg-Hutter intelligence and balanced Pareto optimality, which depend crucially on the choice of the prior, are entirely subjective. Moreover, in the class of all computable environments every policy is Pareto optimal. This undermines all existing optimality properties for AIXI. However, there are Bayesian approaches to general reinforcement learning that sat- isfy objective optimality guarantees: We prove that Thompson sampling is asymptot- ically optimal in stochastic environments in the sense that its value converges to the value of the optimal policy. We connect asymptotic optimality to regret given a recov- erability assumption on the environment that allows the agent to recover from mistakes. Hence Thompson sampling achieves sublinear regret in these environments. AIXI is known to be incomputable. We quantify this using the arithmetical hierar- chy, and establish upper and corresponding lower bounds for incomputability. Further, we show that AIXI is not limit computable, thus cannot be approximated using ﬁnite computation. However there are limit computable "-optimal approximations to AIXI. We also derive computability bounds for knowledge-seeking agents, and give a limit computable weakly asymptotically optimal reinforcement learning agent. Finally, our results culminate in a formal solution to the grain of truth problem: A Bayesian agent acting in a multi-agent environment learns to predict the other agents’ policies if its prior assigns positive probability to them (the prior contains a grain of truth). We construct a large but limit computable class containing a grain of truth and show that agents based on Thompson sampling over this class converge to play "-Nash equilibria in arbitrary unknown computable multi-agent environments. ix x Keywords. Bayesian methods, sequence prediction, merging, general reinforcement learning, universal artiﬁcial intelligence, AIXI, Thompson sampling, knowledge-seeking agents, Pareto optimality, intelligence, asymptotic optimality, computability, reﬂective oracle, grain of truth problem, Nash equilibrium. Contents Title Page i Abstract ix Contents xiii List of Figures xv List of Tables xvii 1 Introduction 1 1.1 Reinforcement Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Narrow Reinforcement Learning . . . . . . . . . . . . . . . . . . . 3 1.1.2 Deep Q-Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.3 General Reinforcement Learning . . . . . . . . . . . . . . . . . . 6 1.2 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Preliminaries 15 2.1 Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Information Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Algorithmic Information Theory . . . . . . . . . . . . . . . . . . . . . . 20 3 Learning 23 3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4 Merging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4.1 Strong Merging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4.2 Weak Merging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4.3 Almost Weak Merging . . . . . . . . . . . . . . . . . . . . . . . . 34 3.5 Predicting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.5.1 Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.5.2 Absolute Continuity . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5.3 Dominance with Coeﬃcients . . . . . . . . . . . . . . . . . . . . . 40 3.6 Learning with Algorithmic Information Theory . . . . . . . . . . . . . . 41 3.6.1 Solomonoﬀ Induction . . . . . . . . . . . . . . . . . . . . . . . . . 41 xi xii Contents 3.6.2 The Speed Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.6.3 Universal Compression . . . . . . . . . . . . . . . . . . . . . . . . 43 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4 Acting 49 4.1 The General Reinforcement Learning Problem . . . . . . . . . . . . . . . 50 4.1.1 Discounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.1.2 Implicit Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.1.3 Typical Environment Classes . . . . . . . . . . . . . . . . . . . . 54 4.2 The Value Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.2.1 Optimal Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2.2 Properties of the Value Function . . . . . . . . . . . . . . . . . . 58 4.2.3 On-Policy Value Convergence . . . . . . . . . . . . . . . . . . . . 59 4.3 The Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3.1 Bayes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3.2 Knowledge-Seeking Agents . . . . . . . . . . . . . . . . . . . . . . 63 4.3.3 BayesExp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3.4 Thompson Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 65 5 Optimality 67 5.1 Pareto Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.2 Bad Priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.2.1 The Indiﬀerence Prior . . . . . . . . . . . . . . . . . . . . . . . . 70 5.2.2 The Dogmatic Prior . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.2.3 The Gödel Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.3 Bayes Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.4 Asymptotic Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.4.1 Bayes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.4.2 BayesExp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.4.3 Thompson Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.4.4 Almost Sure in Cesàro Average vs. in Mean . . . . . . . . . . . . 89 5.5 Regret . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.5.1 Sublinear Regret in Recoverable Environments . . . . . . . . . . 91 5.5.2 Regret of the Optimal Policy and Thompson sampling . . . . . . 95 5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.6.1 The Optimality of AIXI . . . . . . . . . . . . . . . . . . . . . . . 96 5.6.2 Natural Universal Turing Machines . . . . . . . . . . . . . . . . . 97 5.6.3 Asymptotic Optimality . . . . . . . . . . . . . . . . . . . . . . . . 98 5.6.4 The Quest for Optimality . . . . . . . . . . . . . . . . . . . . . . 99 6 Computability 101 6.1 Background on Computability . . . . . . . . . . . . . . . . . . . . . . . . 103 6.1.1 The Arithmetical Hierarchy . . . . . . . . . . . . . . . . . . . . . 103 6.1.2 Computability of Real-valued Functions . . . . . . . . . . . . . . 103 Contents xiii 6.2 The Complexity of Solomonoﬀ Induction . . . . . . . . . . . . . . . . . . 105 6.3 The Complexity of AINU, AIMU, and AIXI . . . . . . . . . . . . . . . . 108 6.3.1 Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.3.2 Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.4 Iterative Value Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.5 The Complexity of Knowledge-Seeking . . . . . . . . . . . . . . . . . . . 122 6.6 A Limit Computable Weakly Asymptotically Optimal Agent . . . . . . . 122 6.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7 The Grain of Truth Problem 127 7.1 Reﬂective Oracles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.1.1 Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.1.2 A Limit Computable Reﬂective Oracle . . . . . . . . . . . . . . . 131 7.1.3 Proof of Theorem 7.7 . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.2 A Grain of Truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.2.1 Reﬂective Bayesian Agents . . . . . . . . . . . . . . . . . . . . . 135 7.2.2 Reﬂective-Oracle-Computable Policies . . . . . . . . . . . . . . . 136 7.2.3 Solution to the Grain of Truth Problem . . . . . . . . . . . . . . 137 7.3 Multi-Agent Environments . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.4 Informed Reﬂective Agents . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.5 Learning Reﬂective Agents . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.6 Impossibility Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 8 Conclusion 147 Measures and Martingales 151 Bibliography 155 List of Notation 171 Index 175 xiv Contents List of Figures 1.1 Selection of Atari 2600 video games . . . . . . . . . . . . . . . . . . . . . 13 3.1 Properties of learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.1 The dualistic agent model . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.1 Legg-Hutter intelligence measure . . . . . . . . . . . . . . . . . . . . . . 77 5.2 Relationship between diﬀerent types of asymptotic optimality . . . . . . 80 6.1 Deﬁnition of conditional Mas a0 2-formula . . . . . . . . . . . . . . . . 106 6.2 Environment from the proof of Theorem 6.15 . . . . . . . . . . . . . . . 111 6.3 Environment from the proof of Theorem 6.16 . . . . . . . . . . . . . . . 113 6.4 Environment from the proof of Theorem 6.17 . . . . . . . . . . . . . . . 114 6.5 Environment from the proof of Proposition 6.19 . . . . . . . . . . . . . . 117 6.6 Environment from the proof of Theorem 6.22 . . . . . . . . . . . . . . . 119 6.7 Environment from the proof of Theorem 6.23 . . . . . . . . . . . . . . . 121 7.1 Answer options of a reﬂective oracle . . . . . . . . . . . . . . . . . . . . 131 7.2 The multi-agent model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 xv xvi LIST OF FIGURES List of Tables 1.1 Assumptions in reinforcement learning . . . . . . . . . . . . . . . . . . . 7 1.2 List of publications by chapter . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1 Examples of learning distributions . . . . . . . . . . . . . . . . . . . . . 45 3.2 Summary on properties of learning . . . . . . . . . . . . . . . . . . . . . 45 4.1 Discount functions and their eﬀective horizons . . . . . . . . . . . . . . . 53 5.1 Types of asymptotic optimality . . . . . . . . . . . . . . . . . . . . . . . 79 5.2 Compiler sizes of the UTMs of bad priors . . . . . . . . . . . . . . . . . 98 5.3 Notions of optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.1 Computability results on Solomonoﬀ’s prior . . . . . . . . . . . . . . . . 102 6.2 Computability results for diﬀerent agent models . . . . . . . . . . . . . . 102 6.3 Computability of real-valued functions . . . . . . . . . . . . . . . . . . . 104 6.4 Computability results for the iterative value function . . . . . . . . . . . 116 7.1 Terminology dictionary between reinforcement learning and game theory. 128 xvii xviii LIST OF TABLES Chapter 1 Introduction Everything I did was for the glamor, the money, and the sex. — Albert Einstein After the early enthusiastic decades, research in artiﬁcial intelligence (AI) now mainly aims at speciﬁc domains: playing games, mining data, processing natural language, recognizing objects in images, piloting robots, ﬁltering email, and many others (Russell and Norvig, 2010). Progress on particular domains has been remarkable, with several high-proﬁle breakthroughs: The chess world champion Garry Kasparov was defeated by the computer program Deep Blue in 1997 (IBM, 2012a). In 2011 the world’s best Jeopardy! players were defeated by the computer program Watson (IBM, 2012b). As of 2014 Google’s self-driving cars completed over a million kilometers autonomously on public roads (Google, 2014). Finally, in 2016 Google DeepMind’s AlphaGo beat Lee Sedol, one of the world’s best players, at the board game Go (Google, 2016). While these advancements are very impressive, they are highly-specialized algo- rithms tailored to their domain of expertise. Outside that domain these algorithms perform very poorly: AlphaGo cannot play chess, Watson cannot drive a car, and DeepBlue cannot answer natural language queries. Solutions in one domain typically do not generalize to other domains and no single algorithm performs well in more than one of them. We classify these kinds of algorithms as narrow AI . This thesis is not about narrow AI. We expect progress on narrow AI to continue and even accelerate, taking the crown of human superiority in domain after domain. But this is not the ultimate goal of artiﬁcial intelligence research. The ultimate goal is to engineer a mind—to build a machine that can learn to do all tasks that humans can do, at least as well as humans do them. We call such a machine human-level AI (HLAI) if it performs at human level and strong AI if it surpasses human level. This thesis is about strong AI. The goal of developing HLAI has a long tradition in AI research and was explicitly part of the 1956 Dartmouth conference that gave birth to the ﬁeld of AI (McCarthy et al., 1955): We propose that a 2 month, 10 man study of artiﬁcial intelligence be carried out during the summer of 1956 at Dartmouth College in Hanover, New Hampshire. The study is to proceed on the basis of the conjecture that every aspect of learning or any other feature of intelligence can in principle be so precisely described that a machine can be made to simulate it. An 1 2 Introduction attempt will be made to ﬁnd how to make machines use language, form abstractionsand concepts, solve kinds ofproblems now reserved forhumans, and improve themselves. We think that a signiﬁcant advance can be made in one or more of these problems if a carefully selected group of scientists work on it together for a summer. Inhindsightthisproposalreadsvastlyoverconﬁdent,anddisappointmentwasinevitable. Making progress on these problems turned out to be a lot harder than promised, and over the last decades any discussion of research targeting HLAI has been avoided by serious researchers in the ﬁeld. This void was ﬁlled mostly by crackpots, which tainted the reputation of HLAI research even further. However, this trend has recently been re- verted: Chalmers (2010), Hutter (2012a), Schmidhuber (2012), Bostrom (2014), Hawk- ing, Tegmark, Russell, and Wilczek (2014), Shanahan (2015), and Walsh (2016) are well-known scientists discussing the prospect of HLAI seriously. Even more: the ex- plicit motto of Google DeepMind, one of today’s leading AI research centers, is to “solve intelligence.” 1.1 Reinforcement Learning The best formal model for strong AI we currently have is reinforcement learning (RL). Reinforcement learning studies algorithms that learn to act in an unknown environment through trial and error (Sutton and Barto, 1998; Szepesvári, 2010; Wiering and van Otterlo, 2012). Without knowing the structure of the environments or the goal, an agent has to learn what to do through the carrot-and-stick approach: it receives a reward in form of a numeric feedback signifying how well it is currently doing; from this signal the agent has to ﬁgure out autonomously what to do. More speciﬁcally, in ageneral reinforcement learning problem anagentinteracts sequentially with an unknown environment : in every time step the agent chooses an actionand receives a perceptconsisting of an observation and a real-valued reward. The sequence of past actions and percepts is the history. The goal in reinforcement learning is to maximize cumulative (discounted) rewards (this setup is described formally in Section 4.1). A central problem in reinforcement learning is the balance between exploration and exploitation : should the agent harvest rewards in the regions of the environment that it currently knows (exploitation) or try discovering more proﬁtable regions (exploration)? Exploration is costly and dangerous: it forfeits rewards that could be had right now, and it might lead into traps from which the agent cannot recover. However, exploration may pay oﬀ in the long run. Generally, it is not clear how to make this tradeoﬀ (see Section 5.6). Reinforcement learning algorithms can be categorized by whether they learn on- policyoroﬀ-policy . Learning on-policy means learning the value of the policy that the agent currently follows. Typically, the policy is slowly improved while learning, likeSARSA(Sutton and Barto, 1998). In contrast, learning oﬀ-policy means following one policy but learning the value of another policy (typically the optimal policy), like Q-learning (Watkins and Dayan, 1992). Oﬀ-policy methods are more diﬃcult to handle §1.1Reinforcement Learning 3 in practice (see the discussion on function approximation below) but tend to be more data-eﬃcient since samples from an old policy do not have to be discarded. Reinforcement learning has to be distinguished from planning . In a planning prob- lem we are provided with the true environment and are tasked with ﬁnding an optimal policy. Mathematically it is clear what the optimal policy is, the diﬃculty stems from ﬁnding a reasonable solution with limited computation. Reinforcement learning is fun- damentally more diﬃcult because the true environment is unknown and has to be learned from observation. This enables two approaches: we could learn a model of the trueenvironmentandthenuseplanningtechniqueswithinthatmodel; thisisthe model- basedapproach. Alternatively, we could learn an optimal policy directly or through an intermediate quantity (typically the value function); this is the model-free approach. Model-based methods tend to be more data-eﬃcient but also computationally more expensive. Therefore most algorithms used in practice ( Q-learning and SARSA) are model-free. 1.1.1 Narrow Reinforcement Learning In the reinforcement learning literature it is typically assumed that the environment is a Markov decision process (MDP), i.e., the next percept only depends on the last percept and action and is independent of the rest of the history (see Section 4.1.3). In an MDP, percepts are usually called states. This setting is well-analyzed (Puterman, 2014; Bert- sekas and Tsitsiklis, 1995; Sutton and Barto, 1998), and there is a variety of algorithms that are known to learn the MDP asymptotically, such as TD learning (Sutton, 1988) andQ-learning (Watkins and Dayan, 1992). Moreover, for MDPs various learning guarantees have been proved in the literature. First, there are bounds on the agent’s regret, the diﬀerence between the obtained re- wards and the rewards of the optimal policy. Auer et al. (2009) derive the regret bound ~O(dSp At)for ergodic MDPs where dis thediameter of the MDP (how many steps a policy needs on average to get from one state of the MDP to any other), Sis the number of states, Ais the number of actions, and tis the number of time steps the algorithm runs. Second, given "and, a reinforcement learning algorithm is said to havesample complexity C(";)iﬀ it is"-suboptimal for at most C(";)time steps with probability at least 1(probably approximately correct, PAC). For MDPs the ﬁrst sample complexity bounds were due to Kakade (2003). Lattimore and Hutter (2012) use the algorithm UCRL (Auer et al., 2009) with geometric discounting with discount rate and derive the currently best-known PAC bound of ~O(T=("2(1 )3) log) whereTis the number of non-zero transitions in the MDP. Typically, algorithms for MDPs rely on visiting every state multiple times (or even inﬁnitelyoften), whichbecomesinfeasibleforlargestatespaces(e.g.avideogamescreen consisting of millions of pixels). In these cases, function approximation can be used to learn an approximation to the value function (Sutton and Barto, 1998). Linear function approximationisknowntoconvergeforseveralon-policyalgorithms(TsitsiklisandRoy, 1997; Sutton, 1988; Gordon, 2001), but proved tricky for oﬀ-policy algorithms (Baird, 1995). A recent breakthrough was made by Mahmood et al. (2015) and Yu (2015) 4 Introduction with their emphatic TD algorithm that converges oﬀ-policy. For nonlinear function approximation no convergence guarantee is known. Among the historical successes of reinforcement learning is autonomous helicopter piloting (Kim et al., 2003) and TD-Gammon, a backgammon algorithm that learned through self-play (Tesauro, 1995), similar to AlphaGo (Silver et al., 2016). 1.1.2 Deep Q-Networks The current state of the art in reinforcement learning challenges itself to playing simple video games. Video games are an excellent benchmark because they come readily with the reward structure provided: the agent’s rewards are the change in the game score. Without prior knowledge of any aspect of the game, the agent needs to learn to score as many points in the game as possible from looking only at raw pixel data (sometimes after some preprocessing). This approach to general AI is in accordance with the deﬁnition of intelligence given by Legg and Hutter (2007b): Intelligence measures an agent’s ability to achieve goals in a wide range of environments. In reinforcement learning the deﬁnition of the goal is very ﬂexible, and provided by the rewards. Moreover, a diverse selection of video games arguably constitutes a ‘wide range of environments.’ A popular such selection is the Atari 2600 video game console (Bellemare et al., 2013). There are hundreds of games released for this platform, with very diverse chal- lenges: top-down shooting games such as Space Invaders, ball games such as Pong, agility-based games such as Boxing or Gopher, tactical games such as Ms. Pac-Man, and maze games such as Montezuma’s Revenge. An overview over some of the games is given in Figure 1.1 on page 13. Mnih et al. (2013, 2015) introduce the deep Q-network (DQN) algorithm, com- biningQ-learning with nonlinear function approximation through convolutional neural networks. DQN achieves 75% of the performance of a human game tester on 29 of 49 Atari games. The two innovations that made this breakthrough possible are (1) using a not so recent target Q-function in the TD update and (2) experience replay. For ex- perience replay, a set of recent state transitions is retained and the network is regularly retrained on random samples from these old transitions.1 DQN rides the wave of success of deep learning (LeCun et al., 2015; Schmidhuber, 2015; Goodfellow et al., 2016). Deep learning refers to the training of artiﬁcial neural networks with several layers. This allows them to automatically learn higher-level abstractions from data. Deep neural networks are conceptionally simple and have been studied since the inception of AI; only recently has computation power become cheap enough to train them eﬀectively. Recently deep neural networks have taken the top of the machine learning benchmarks by storm (LeCun et al., 2015, and references therein): 1The slogan for experience replay should be ‘regularly retrained randomly on retained rewards’. §1.1Reinforcement Learning 5 These methods have dramatically improved the state-of-the-art in speech recognition, visual object recognition, object detection and many other do- mains such as drug discovery and genomics. Since the introduction of DQN there have been numerous improvements on this algorithm: increasing the gap on the Q-values of diﬀerent actions (Bellemare et al., 2016), training in parallel (Nair et al., 2015; Mnih et al., 2016), improvements to the experience replay mechanism (Schaul et al., 2016), generalization to continuous action spaces (Lillicrap et al., 2016), solve the overestimation problem (van Hasselt et al., 2016), andimprovementstotheneuralnetworkarchitecture(Wangetal.,2016). The Q- values learned by DQN’s neural networks are intransparent to inspection; Zahavy et al. (2016) use visualization techniques on the Q-value networks. Finally, Liang et al. (2016) managed to reproduce DQN’s success using only linearfunction approximation (no neural networks). The key is a selection of features similar to the ones produced by DQN’s convolutional neural networks. Regardless of its success, the DQN algorithm fundamentally falls short of the re- quirements for strong AI: Q-learning with function approximation is targeted at solving large-state (fully observable) Markov decision processes. In particular, it does not ad- dress the following challenges. •Partial observability. All games in the ATARI framework are fully observable (ex- cept for Montezuma’s revenge): all information relevant to the state of the game is visible on the screen at all times (when using the four most recent frames). However, the real world is only partially observable. For example, when going to the supermarket you have to remember what you wanted to buy because you currently cannot observe which items you are missing at home. A strong AI needs tohavememoryandbeabletorememberthingsthathappenedinthepast(rather than only learning from it). An obvious approach to equip DQN with memory is to use recurrent neural networks instead of simple feedforward neural networks (Heess et al., 2015). Hausknecht and Stone (2015) show that this enables the agent to play the games when using only a single frame as input. However, it is currently unclear whether recurrent neural networks are powerful enough to learn long-term dependencies in the data (Bengio et al., 1994). •Directed exploration. DQN fails in games with delayed rewards. For example, in Montezuma’s Revenge the agent needs to avoid several obstacles to get to a key before receiving the ﬁrst reward. DQN fails to score any rewards in this environ- ment. This is not surprising: the typical approach for reinforcement learning, to use"-exploration for which the agent chooses actions at random with a certain probability, is insuﬃcient for exploring complex environments; the probability of random walking into the ﬁrst reward is just too low. Instead we need a more targeted exploration approach that aims at understanding the environment in a structured manner. Theoretical foundations are provided 6 Introduction byknowledge-seeking agents (Orseau, 2011, 2014a; Orseau et al., 2013). Kulkarni et al. (2016) introduce a hierarchical approach based on intrinsic motivation to improve DQN’s exploration and manage to score points in Montezuma’s Revenge. However, their approach relies on quite a bit of visual preprocessing and domain knowledge. •Non-ergodicity. When losing in an Atari game, the agent always gets to play the same game again. From the agent’s perspective, it has not actually failed, it just gets transported back to the starting state. Because of this, there are no strong incentives to be careful when exploring the environment: there can be no bad mistakes that make recovery impossible. However, in the real world some actions are irreversibly bad. If the robot drives oﬀ a cliﬀ it can be fatally damaged and cannot learn from the mistake. The real world is full of potentially fatal mistakes (e.g. crossing the street at the wrong time) and for humans, natural reﬂexes and training by society make sure that we are very conﬁdent of what situations to avert. This is crucial, as some mistakes must be avoided without any training examples. Current reinforcement learning algorithms only learn about bad states by visiting them. •Wireheading. The goal of reinforcement learning is to maximize rewards. When playing a video game the most eﬃcient way to get rewards is to increase the game score. However, when a reinforcement learning algorithm is acting in the real world, theoretically it can change its own hard- and software. In this set- ting, the most eﬃcient way to get rewards is to modify the reward mechanism to always provide the maximal reward (Omohundro, 2008; Ring and Orseau, 2011; Bostrom, 2014). Consequently the agent no longer pursues the designers’ origi- nally intended goals and instead only attempts to protect its own existence. The namewireheading was established by analogy to a biology experiment by Olds and Milner (1954) in which rats had a wire embedded into the reward center of their brain that they could then stimulate by the push of a button. Today’s reinforcement learning algorithms usually do not have access to their own internal workings, but more importantly they are not smart enough to understand their own architecture. They simply lack the capability to wirehead. But as we increase their capability, wireheading will increasingly become a challenge for reinforcement learning. 1.1.3 General Reinforcement Learning A theory of strong AI cannot make some of the typical assumptions. Environments are partially observable, so we are dealing with partially observable Markov decision processes (POMDPs). The POMDP’s state space does not need to be ﬁnite. Moreover, the environment may not allow recovery from mistakes: we do not assume ergodic- ity or weak communication (not every POMDP state has to be reachable from every other state). So in general, our environments are inﬁnite-state non-ergodic POMDPs. Table 1.1 lists the assumptions that are typical but we do not make. §1.1Reinforcement Learning 7 Assumption Description Full observability the agent needs no memory to act optimally Finite state the environment has only ﬁnitely many states Ergodicity the agent can recover from any mistakes Computability the environment is computable Table 1.1: List of assumptions from the reinforcement learning literature. In this thesis, we only make the computability assumption which is important for Chapter 6 and Chapter 7. Learning POMDPs is a lot harder, and only partially successful attempts have been made: through predictive state representations (Singh et al., 2003, 2004), and Bayesian methods (Doshi-Velez, 2012). A general approach is feature reinforcement learning (Hutter, 2009c,d), which aims to reduce the general reinforcement learning problem to an MDP by aggregating histories into states. The quest for a good cost function for feature maps remains unsuccessful thus far (Sunehag and Hutter, 2010; Daswani, 2015). However, Hutter (2014) managed to derive strong bounds relating the optimal value function of the aggregated MDP to the value function of the original process even if the latter violates the Markov condition. A full theoretical approach to the general reinforcement learning problem is given by Hutter (2000, 2001a, 2002a, 2003, 2005, 2007a, 2012b). He introduces the Bayesian RL agentAIXIbuilding on the theory of sequence prediction by Solomonoﬀ (1964, 1978). Based in algorithmic information theory, Solomonoﬀ’s prior draws from famous insights by William of Ockham, Sextus Epicurus, Alan Turing, and Andrey Kolmogorov (Rath- mannerandHutter,2011). AIXIusesSolomonoﬀ’spriorovertheclassofallcomputable environments and acts to maximize Bayes-expected rewards. We formally introduce Solomonoﬀ’s theory of induction in Chapter 3 and AIXI in Section 4.3.1. See also Legg (2008) for an accessible introduction to AIXI. A typical optimality property in general reinforcement learning is asymptotic opti- mality(Lattimore and Hutter, 2011): as time progresses the agent converges to achieve the same rewards as the optimal policy. Asymptotic optimality is usually what is meant by “ Q-learning converges” (Watkins and Dayan, 1992) or “TD learning con- verges” (Sutton, 1988). Orseau (2010, 2013) showed that AIXI is not asymptotically optimal. Yet asymptotic optimality in the general setting can be achieved through op- timism (Sunehag and Hutter, 2012a,b, 2015), Thompson sampling (Section 5.4.3), or an extra exploration component on top of AIXI (Lattimore, 2013, Ch. 5). In our setting, learning the environment does not just involve learning a ﬁxed ﬁnite set of parameters; the real world is too complicated to ﬁt into a template. Therefore we fall back on the nonparametric approach where we start with an inﬁnite but countable class of candidate environments. Our only assumption is that the true environment is contained in this class (the realizable case ). As long as this class of environments is large enough (such as for the class of all computable environments), this assumption is 8 Introduction rather weak. 1.2 Contribution The goal of this thesis is not to increase AI capability. As such, we are not trying to improve on the state of the art, and we are not trying to derive practical algorithms. Instead, the emphasis of this thesis is to further our understanding of general rein- forcement learning and thus strong AI. How a future implementation of strong AI will actually work is in the realm of speculation at this time. Therefore we should make as few and as weak assumptions as possible. We disregard computational constraints in order to focus on the fundamental un- derlying problems. This is unrealistic, of course. With unlimited computation power many traditional AI problems become trivial: playing chess, Go, or backgammon can be solved by exhaustive expansion of the game tree. But the general RL problem does not become trivial: the agent has to learn the environment and balance between ex- ploration and exploitation. That being said, the algorithms that we study do have a relationship with algorithms being used in practice and our results can and should educate implementation. On a high level, our insights can be viewed from three diﬀerent perspectives. •Philosophically. Concisely, our understanding of strong AI can be summarized as follows. intelligence =learning +acting (1.1) Here,intelligence refers to an agent that optimizes towards some goal in accor- dancewiththe deﬁnition byLegg and Hutter(2007b). For learning we distinguish two (very related) aspects: (1) arriving at accurate beliefs about the future and (2) making accurate predictions about the future. Of course, the former implies the latter: if you have accurate beliefs, then you can also make good predictions. For RL accurate beliefs is what we care about because they enable us to plan for the future. Learning is a passive process that only observes the data and does not interfere with its generation. In particular, learning does not require a goal. With actingwe mean the selection of actions in pursuit of some goal. This goal can be reward maximization as in reinforcement learning, understand- ing the environment as for knowledge-seeking agents, or something else entirely. Together they enable an agent to learn the environment’s behavior in response to itself (on-policy learning) and to choose a policy that furthers its goal. We dis- cuss the formal aspects of learning in Chapter 3 and some approaches to acting in Chapter 4. Given inﬁnite computational resources, learning is easy and Solomonoﬀ induc- tion provides a complete theoretical solution. However, acting is not straightfor- ward. We show that in contrast to popular belief, AIXI, the natural extension of Solomonoﬀ induction to reinforcement learning, does not provide the objectively best answer to this question. We discuss some alternatives and their problems in §1.2Contribution 9 Chapter 5. Unfortunately, the general question of how to act optimally remains open. AIXItl(Hutter, 2005, Ch. 7.2) is often mentioned as a computable approximation to AIXI. But AIXI tldoes not converge to AIXI in the limit. Inspired by Hutter search (Hutter, 2002b), it relies on an automated theorem prover to ﬁnd the provably best policy computable in time twith a program of length l. In contrast to AIXI, which only requires the choice of universal Turing machine, proof search requires an axiom system that must not be too weak or too strong. In Section 5.2.3 we discuss some of the problems with AIXI tl. Moreover, in Corollary 6.13 we show that "-optimal AIXI is limit computable, which shows that AIXI can be computably approximated by running this algorithm for a ﬁxed number of time steps or until a timeout is reached. While neither AIXI tlnor this AIXI approximation algorithm are practically feasible, the latter is a better example for a computable strong AI. In our view, AIXI should be taken as a descriptive rather than prescriptive model. It is descriptive as an abstraction from an actual implementation of strong AI where we ignore all the details of the learning algorithm and the computational approximations of choosing how to act. It should not be viewed as a prescription of how strong AI should be built and AIXI approximations (Veness et al., 2011, 2015) are easily outperformed by neural-network-based approaches (Mnih et al., 2015). •Mathematically. Some of the proof techniques we employ are novel and could be used to analyze other algorithms. Examples include the proofs for the lower bounds on the computability results (Section 6.3.2) and to a lesser extent the upper bounds (Section 6.3.1), which should work analogously for a wide range of algorithms. Furthermore, the proof of the asymptotic optimality of Thompson sampling (Theorem 5.25) brings together a variety of mathematical tools from measure theory, probability theory, and stochastic processes. Next, the recoverability assumption (Deﬁnition 5.31) is a novel technical assump- tionontheenvironmentakintoergodicityandweakcommunicationinﬁnite-state environments. It is more general, yet mathematically simple and works for arbi- trary environments. This assumption turns out to be what we need to prove the connection from asymptotic optimality to sublinear regret in Section 5.5. Moreover, we introduce the use of the recursive instead of the iterative value function (Section 6.4). The iterative value function is the natural extension of expectimax search to the sequential setting and was originally used by Hutter (2005, Sec. 5.5). Yet it turned out to be an incorrect and inconvenient deﬁnition: it does not correctly maximize expected rewards (Proposition 6.19) and it is not limit computable (Theorem 6.22 and Theorem 6.23). However, this is only a minor technical correction. Finally, this work raises new mathematically intriguing questions about the prop- erties of reﬂective oracles (Section 7.1). 10 Introduction •Practically. One insight from this thesis is regarding the eﬀective horizon. In practice geometric discounting is ubiquitous which has a constant eﬀective hori- zon. However, when facing a ﬁnite horizon problem or an episodic task, some- times the eﬀective horizon changes. One lesson from our result on Thompson sampling (Section 5.4.3 and Section 5.5) is that you should explore for an ef- fective horizon instead of using "-greedy. While the latter exploration method is often used in practice, it has proved ineﬀective in environments with delayed rewards (see Section 1.1.2). Furthermore, our application of reinforcement learning results to game theory in Chapter 7 reinforces this trend to solve game theory problems (Tesauro, 1995; Bowling and Veloso, 2001; Busoniu et al., 2008; Silver et al., 2016; Heinrich and Silver, 2016; Foerster et al., 2016, and many more). In particular, the approxima- tion algorithm for reﬂective oracles (Section 7.1.3) could guide future applications for computing Nash equilibria (see also Fallenstein et al., 2015b). On a technical level, we advance the theory of general reinforcement learning. In its center is the Bayesian reinforcement learning agent AIXI. AIXI is meant as an answer to the question of how to do general RL disregarding computational constraints. We analyze the computational complexity of AIXI and related agents in Chapter 6 and show that even with an inﬁnite horizon AIXI can be computationally approximated with a regular Turing machine (Section 6.3.1). We also derive corresponding lower bounds for most of our upper bounds (Section 6.3.2). Chapter 5 is about notions of optimality in general reinforcement learning. We dispel AIXI’s status as the gold standard for reinforcement learning. Hutter (2002a) showed that AIXI is Pareto optimal, balanced Pareto optimal, and self-optimizing. Orseau (2013) established that AIXI does not achieve asymptotic optimality in all computable environments (making the self-optimizing result inapplicable to this gen- eral environment class). In Section 5.1 we show that every policy is Pareto optimal and in Section 5.3 we show that balanced Pareto optimality is highly subjective, depending on the choice of the prior; bad choices for priors are discussed in Section 5.2. Notable is thedogmatic prior that locks a Bayesian reinforcement learning agent into a particu- lar (bad) policy as long as this policy yields some rewards. Our results imply that there are no known nontrivial and non-subjective optimality results for AIXI. We have to regard AIXI as a relativetheory of intelligence. More generally, our results imply that general reinforcement learning is diﬃcult even when disregarding computational costs . But this is not the end to Bayesian methods in general RL. We show in Section 5.4 that a Bayes-inspired algorithm called Thompson sampling achieves asymptotic opti- mality. Thompson sampling, also known as posterior sampling or theBayesian control rulerepeatedly draws one environment from the posterior distribution and then acts as if this was the true environment for a certain period of time (depending on the discount function). Moreover, given a recoverability assumption on the environment and some mild assumptions on the discount function, we show in Section 5.5 that Thompson sampling achieves sublinear regret. Finally, we tie these results together to solve an open problem in game theory: §1.3Thesis Outline 11 Chapter Publication(s) Chapter 1 - Chapter 2 - Chapter 3 with links to Leike and Hutter (2014a, 2015d); Filan et al. (2016) Chapter 4 - Chapter 5 Leike and Hutter (2015c); Leike et al. (2016a) Chapter 6 Leike and Hutter (2015b,a, 2016) Chapter 7 Leike et al. (2016b) Chapter 8 - Appendix A Leike and Hutter (2014b) Table 1.2: List of publications by chapter. When acting in a multi-agent environment with other Bayesian agents, each agent needs to assign positive prior probability to the other agents’ actual policies (they need to have a grain of truth ). Finding a reasonably large class of policies that contains the Bayes optimal policies with respect to this class is known as the grain of truth prob- lem(Hutter, 2009b, Q. 5j). Only small classes are known to have a grain of truth and the literature contains several related impossibility results (Nachbar, 1997, 2005; Foster and Young, 2001). Moreover, while AIXI assumes the environment to be computable, our computability results on AIXI conﬁrm that it is incomputable (Theorem 6.15 and Theorem 6.17). This asymmetry elevates AIXI above its environment computationally, and prevents the environment from containing other AIXIs. In Chapter 7 we give a formal and general solution to the grain of truth prob- lem: we construct a class of policies that avoid this asymmetry. This class contains all computable policies as well as Bayes optimal policies for every lower semicomputable prior over the class. When the environment is unknown, our dogmatic prior from Sec- tion 5.2 makes Bayes optimal agents fail to act optimally even asymptotically. However, our convergence results on Thompson sampling (Section 5.4.3) imply that Thompson samplers converge to play "-Nash equilibria in arbitrary unknown computable multi- agent environments. While these results are purely theoretical, we use techniques from Chapter 6 to show that they can be computationally approximated arbitrarily closely. 1.3 Thesis Outline This thesis is based on the papers Leike and Hutter (2014a,b, 2015a,b,c, 2016); Leike et al. (2016a,b). During my PhD, I was also involved in the publications Leike and Heiz- mann(2014a,b,2015);Heizmannetal.(2015,2016)basedonmyresearchintermination analysis (in collaboration with Matthias Heizmann), Daswani and Leike (2015) (co- authored with Mayank Daswani in equal parts), Everitt et al. (2015) (co-authored with Tom Everitt in equal parts), Filan et al. (2016) (written by Daniel Filan as part of his honour’s thesis supervised by Marcus Hutter and me). Leike and Hutter (2016) is 12 Introduction still under review. Leike and Hutter (2014a, 2015d) are tangential to this thesis’ main thrust, so the results are mentioned only in passing. A list of papers written during my PhD is given in Table 1.3 on page 14, with a corresponding chapter outline in Table 1.2. The core of our contribution is found in chapters 5, 6, and 7. Every thesis chapter starts with a quote. In case this is not blatantly obvious: these are false quotes , a desperate attempt to make the thesis less dry and humorless. None of the quotes were actually stated by the person they are attributed to (according to our knowledge). §1.3Thesis Outline 13 (a)Space Invaders: the player controls the green cannon on the bottom of the screen and ﬁres projectiles at the yellow ships at the top. The red blobs can be used as cover, but also ﬁred through. (b)Pong: the player controls the green paddle on the right of the screen and needs to hit the white ball such that the com- puter opponent controlling the red paddle on the left fails to hit the ball back. (c)Ms. Pac-Man: the player controls the yellow mouth and needs to eat all the red pallets in the maze. The maze is roamed by ghosts that occasionally hunt the player and kill her on contact unless a ‘power pill’ was consumed recently. (d)Boxing: the player controls the white ﬁgure on the screen and extends their arms to throw a punch. The aim is to hit the black ﬁgure that is controlled by the com- puter and dodge their punches. (I’m sure the choice of color was by accident.) (e)Gopher: a hungry rodent attempts to dig to the surface and steal the vegetables. The player controls the farmer who pro- tects them by ﬁlling the rodent’s holes. (f)Montezuma’s Revenge: the player con- trols the red adventurer. The aim is to navigate a maze of deadly traps, use keys to open doors, and collect artifacts. Figure 1.1: A selection of Atari 2600 video games. 14 Introduction [1] Jan Leike and Marcus Hutter. Indeﬁnitely oscillating martingales. In Algorithmic Learning Theory, pages 321–335, 2014a [2] Jan Leike and Matthias Heizmann. Ranking templates for linear loops. Logical Methods in Computer Science , 11(1):1–27, March 2015 [3] Mayank Daswani and Jan Leike. A deﬁnition of happiness for reinforcement learning agents. InArtiﬁcial General Intelligence , pages 231–240. Springer, 2015 [4] Tom Everitt, Jan Leike, and Marcus Hutter. Sequential extensions of causal and evidential decision theory. In Algorithmic Decision Theory , pages 205–221. Springer, 2015 [5] Jan Leike and Marcus Hutter. On the computability of AIXI. In Uncertainty in Artiﬁcial Intelligence , pages 464–473, 2015a [6] JanLeikeandMarcusHutter. OnthecomputabilityofSolomonoﬀinductionandknowledge- seeking. In Algorithmic Learning Theory , pages 364–378, 2015b [7] Jan Leike and Marcus Hutter. Bad universal priors and notions of optimality. In Conference on Learning Theory , pages 1244–1259, 2015c [8] Jan Leike and Marcus Hutter. Solomonoﬀ induction violates Nicod’s criterion. In Algorith- mic Learning Theory , pages 349–363. Springer, 2015d [9] Matthias Heizmann, Daniel Dietsch, Jan Leike, Betim Musa, and Andreas Podelski. Ulti- mate Automizer with array interpolation (competition contribution). In Tools and Algo- rithms for the Construction and Analysis of Systems , pages 455–457. Springer, 2015 [10] Matthias Heizmann, Daniel Dietsch, Marius Greitschus, Jan Leike, Betim Musa, Claus Schätzle, and Andreas Podelski. Ultimate Automizer with two-track proofs (competition contribution). In Tools and Algorithms for the Construction and Analysis of Systems , pages 950–953. Springer, 2016 [11] Daniel Filan, Jan Leike, and Marcus Hutter. Loss bounds and time complexity for speed priors. In Artiﬁcial Intelligence and Statistics , 2016 [12] Jan Leike and Marcus Hutter. On the computability of Solomonoﬀ induction and AIXI. 2016. Under review [13] Jan Leike, Tor Lattimore, Laurent Orseau, and Marcus Hutter. Thompson sampling is asymptotically optimal in general environments. In Uncertainty in Artiﬁcial Intelligence , 2016a [14] Jan Leike, Jessica Taylor, and Benya Fallenstein. A formal solution to the grain of truth problem. In Uncertainty in Artiﬁcial Intelligence , 2016b [15] Jan Leike and Matthias Heizmann. Geometric nontermination arguments. 2016. Under preparation Table 1.3: List of publications. Chapter 2 Preliminaries Mathematics is a waste of time. — Leonhard Euler This chapter establishes the notation and background material that is used through- out this thesis. Section 2.1 is about probability and measure theory, Section 2.2 is about stochastic processes, Section 2.3 is about information theory, and Section 2.4 is about algorithmic information theory. We defer the formal introduction to reinforce- ment learning to Chapter 4. Additional preliminary notation and terminology is also established in individual chapters wherever necessary. A list of notation is provided in the appendix on page 171. Most of the content from this chapter can be found in standard textbooks and reference works. We recommend to consult Wasserman (2004) on statistics, Durrett (2010) on probability theory and stochastic processes, Cover and Thomas (2006) on information theory, Li and Vitányi (2008) on algorithmic information theory, Russell and Norvig (2010) on artiﬁcial intelligence, Bishop (2006) and Hastie et al. (2009) on machine learning, Sutton and Barto (1998) on reinforcement learning, and Hutter (2005) and Lattimore (2013) on general reinforcement learning. We understand deﬁnitions to follow natural language; e.g., when deﬁning the ad- jective ‘continuous’, we deﬁne at the same time the noun ‘continuity’ and the adverb ‘continuously’ wherever appropriate. Numbers. N:=f1;2;3;:::gdenotes the set of natural numbers (starting from 1), Q:=fp=qjp2N[f0g;q2Ngdenotes the set of rational numbers, and Rdenotes the set of real numbers. For two real numbers r1;r2, the set [r1;r2] :=fr2Rjr1rr2g denotes the closed interval with end points r1andr2; the sets (r1;r2] := [r1;r2]nfr1g and[r1;r2) := [r1;r2]nfr2gdenotehalf-openintervals; theset (r1;r2) := [r1;r2]nfr1;r2g denotes an open interval. Strings. FixXto be a ﬁnite nonempty set, called alphabet. We assume that X contains at least two distinct elements. The set X:=S1 n=0Xnis the set of all ﬁnite strings over the alphabet X, the setX1is the set of all inﬁnite strings over the alphabet X, and the setX]:=X[X1is their union. The empty string is denoted by , not to be confused with the small positive real number ". Given a string x2X], we denote its length byjxj. For a (ﬁnite or inﬁnite) string xof lengthk, we denote with xkthe k-th character of x, withx1:kthe ﬁrstkcharacters of x, and with x<kthe ﬁrstk1 15 16 Preliminaries characters of x. The notation x1:1stresses that xis an inﬁnite string. We use xvy to denote that xis a preﬁx of y, i.e.,x=y1:jxj. Our examples often (implicitly) involve the binary alphabet f0;1g. In this case we deﬁne the functions ones;zeros :X!N that count the number of ones and zeros in a string respectively. Computability. A function f:X!Rislower semicomputable iﬀ the setf(x;q)2 XQjf(x)>qgis recursively enumerable. If fandfare lower semicomputable, thenfis called computable . See Section 6.1.2 for more computability deﬁnitions. Asymptotic Notation. Letf;g:N!R0. We usef2O(g)to denote that there is a constant csuch thatf(t)cg(t)for allt2N. We usef2o(g)to denote that lim supt!1f(t)=g(t) = 0. For functions on strings P;Q :X!Rwe useQPto denote that there is a constant c>0such thatQ(x)cP(x)for allx2X. We also useQPforPQandQ=PforQPandPQ. Note that Q=Pdoes not implythat there is a constant csuch thatQ(x) =cP(x)for allx2X. For a sequence (at)t2Nwith limit limt!1at=awe also write at!aast!1. If no limiting variable is provided, we mean t!1by convention. Other Conventions. LetAbe some set. We use #Ato denote the cardinality of the setA, i.e., the number of elements in A, and 2Ato denote the power set of A, i.e., the set of all subsets of A. We use logto denote the binary logarithm and lnto denote the natural logarithm. 2.1 Measure Theory For a countable set , we use to denote the set of probability distributions over . If is uncountable (such as the set of all inﬁnite strings X1), we need to use the machinery of measure theory. This section provides a concise introduction to measure theory; see Durrett (2010) for an extensive treatment. Deﬁnition 2.1 (-algebra).Let be a set. The set F 2 is a-algebra over iﬀ (a) 2F, (b)A2Fimplies nA2F, and (c) for any countable number of sets A0;A1;:::;2F, the unionS i2NAi2F. For a setA 2 , we deﬁne (A)to be the smallest (with respect to set inclusion) -algebra containing A. For the real numbers, the default -algebra (used implicitly) is the Borel-algebraB generated by the open sets of the usual topology. Formally, B:=(f(a;b)ja;b2Rg). A set together with a -algebraFforms ameasurable space . The sets from the - algebraFare called measurable sets . A function f: 1! 2between two measurable spaces is called measurable iﬀ any preimage of an (in 2) measurable set is measurable (in 1). §2.1Measure Theory 17 Deﬁnition 2.2 (Probability Measure) .Let be a measurable space with -algebra F. Aprobability measure on the space is a function :F! [0;1]such that (a)( ) = 1(normalization ), and (b)(S i2NAi) =P i2N(Ai)for any collection fAiji2NgFthat is pairwise disjoint (-additivity ). A probability measure isdeterministic iﬀ it assigns all probability mass to a single element of , i.e., iﬀ there is an x2 with(fxg) = 1. We deﬁne the conditional probability (AjB)for two measurable sets A;B2F with(B)>0as(AjB) :=(A\B)=(B). Deﬁnition 2.3 (Random Variable) .Let be a measurable space with probability measure. A(real-valued) random variable is a measurable function X: !R. We often (but not always) denote random variables with uppercase Latin letters. Given a-algebraF, a probability measure PonF, and anF-measurable random variableX, theconditional expectation E[XjF]ofXgivenFis a random variable Ysuch that (1) YisF-measurable and (2)R AXdP =R AYdPfor allA2F. The conditional expectation exists and is unique up to a set of P-measure 0(Durrett, 2010, Sec. 5.1). Intuitively, if Fdescribes the information we have at our disposal, then E[XjF]denotes the expectation of Xgiven this information. We proceed to deﬁne the -algebra onX1(the-algebra onX]is deﬁned analo- gously). For a ﬁnite string x2X, thecylinder set x:=fxyjy2X1g is the set of all inﬁnite strings of which xis a preﬁx. Furthermore, we ﬁx the -algebras Ft:= fxjx2Xtg andF1:= 1[ t=1Ft! : The sequence (Ft)t2Nis aﬁltration : from x=S a2Xxafollows thatFtFt+1for everyt2N, and allFtF1by the deﬁnition of F1. For our purposes, the -algebraFtmeans ‘all symbols up to and including time stept.’ So instead of conditioning an expectation on Ft, we can just as well condition it on the sequence x1:tdrawn at time t. Hence we write E[Xjx1:t]instead of E[XjFt]. Moreover, for conditional probabilities we also write Q(xtjx<t)instead ofQ(x1:tjx<t). In the context of probability measures, a measurable set E2F1is also called an event. The event Ec:=X1nEdenotes the complement of E. In case the event Eis deﬁned by a predicate Qdependent on the random variable X,E=fx2 jQ(X(x))g, we also use the shorthand notation P[Q(X)] :=P(fx2 jQ(X(x))g) =P(E): We assume all sets to be measurable; when we write P(A)for some set AX1, we understand implicitly that Abe measurable. This is not true: not all subsets of 18 Preliminaries X1are measurable (assuming the axiom of choice). While we choose to do this for readability purposes, note that under some axioms compatible with Zermelo-Fraenkel set theory, notably the axiom of determinacy, all subsets of X1are measurable. 2.2 Stochastic Processes This section introduces some notions about sequences of random variables. Deﬁnition 2.4 (Stochastic Process) .(Xt)t2Nis astochastic process iﬀXtis a random variable for every t2N. A stochastic process (Xt)t2Nisnonnegative iﬀXt0for allt2N. The process is bounded iﬀ there is a constant c2Rsuch thatjXtjcfor allt2N. In the real numbers, a sequence (zt)t2Nconverges if and only if it is a Cauchy se- quence, i.e., iﬀjzt+1ztj!0ast!1. For sequences of random variables convergence is a lot more subtle and there are several diﬀerent notions of convergence. Deﬁnition2.5 (StochasticConvergence) .LetPbeaprobabilitymeasure. Astochastic process (Xt)t2Nconverges to the random variable X •inP-probability iﬀ for every ">0, P jXtXj>" !0ast!1 ; •inP-meaniﬀ EP jXtXj !0ast!1 ; •P-almost surely iﬀ Ph lim t!1Xt=Xi = 1: Almost sure convergence and convergence in mean both imply convergence in prob- ability (Wasserman, 2004, Thm. 5.17). If the stochastic process is bounded, then con- vergence in probability implies convergence in mean (Wasserman, 2004, Thm. 5.19). A sequence of real numbers (at)t2Nconverges in Cesàro average toa2Riﬀ 1=tPt k=1ak!aast! 1. The deﬁnition for sequences of random variables is analogous. Deﬁnition 2.6 (Martingale) .LetPbe a probability measure over (X1;F1). A stochastic process (Xt)t2Nis aP-supermartingale ( P-submartingale) iﬀ (a) eachXtisFt-measurable, and (b)E[XtjFs]Xs(E[XtjFs]Xs)P-almost surely for all s;t2Nwiths<t. AP-martingale is a process that is both a P-supermartingale and a P-submartingale. §2.3Information Theory 19 Example 2.7 (Fair Gambling) .Suppose Mary bets on the outcome of a fair coin ﬂip. If she predicts correctly, her wager is doubled and otherwise it is lost. Let Xtdenote Mary’s wealth at time step t. Since the game is fair, E[Xt+1jFt] =XtwhereFt represents the information available at time step t. Hence E[Xt] =X1, so in expectation she never loses money regardless of her betting strategy. 3 FormartingalesthefollowingfamousconvergenceresultwasprovedbyDoob(1953). Theorem 2.8 (Martingale Convergence; Durrett, 2010, Thm. 5.2.9) .If(Xt)t2Nis a nonnegative supermartingale, then it converges almost surely to a limit XwithE[X] E[X1]. By Theorem 2.8 the martingale from Example 2.7 representing Mary’s wealth con- verges almost surely, regardless of her betting strategy. Either she refrains from betting at some point (assuming she cannot place smaller and smaller bets) or she cannot play anymore because her wealth is 0. Is there a lesson to learn here about gambling? 2.3 Information Theory This section introduces the notions of entropy and two notions of distance between probability measures: KL-divergence andtotal variation distance . Deﬁnition 2.9 (Entropy) .Let be a countable set. For a probability distribution p2 , theentropyofpis deﬁned as Ent(p) :=X x2 :p(x)>0p(x) logp(x): Deﬁnition 2.10 (KL-Divergence) .LetP;Qbe two measures and let m2Nbe a lookahead time step. The Kullback-Leibler-divergence (KL-divergence ) ofPandQ between time steps tandmis deﬁned as KLm(P;Qjx<t) :=X xt:m2Xmt+1P(x1:mjx<t) logP(x1:mjx<t) Q(x1:mjx<t): Moreover, we deﬁne KL1(P;Qjx<t) := limm!1KLm(P;Qjx<t). KL-divergence is also known as relative entropy . KL-divergence is always nonneg- ative by Gibbs’ inequality, but it is not a distance since it is not symmetric. If the alphabetXis ﬁnite, then KLm(P;Qjx)is always ﬁnite. However, KL1(P;Qjx)may be inﬁnite. Deﬁnition 2.11 (Total Variation Distance) .LetP;Qbe two measures and let 1 m1be a lookahead time step. The total variation distance betweenPandQ between time steps tandmis deﬁned as Dm(P;Qjx) := sup AXmP(Ajx<t)Q(Ajx<t): 20 Preliminaries Total variation distance is always bounded between 0and1sincePandQare prob- ability measures. Moreover, in contrast to KL-divergence total variation distance satis- ﬁes the axioms of distance: symmetry ( D(P;Q) =D(Q;P)), identity of indiscernibles (D(P;Q) = 0if and only if P=Q), and the triangle inequality ( D(P;Q) +D(Q;R) D(P;R)). The following lemma shows that total variation distance can be used to bound diﬀerences in expectation. Lemma 2.12 (Total Variation Bound on the Expectation) .For a random variable X with 0X1and two probability measures PandQ EP[X]EQ[X]D(P;Q): KL-divergence and total variation distance are linked by the following inequality. Lemma 2.13 (Pinsker’s inequality; Tsybakov, 2008, Lem. 2.5i) .For all probability measuresPandQonX1, for everyx2X, and for every m2N Dm(P;Qjx)r 1 2KLm(P;Qjx) 2.4 Algorithmic Information Theory Auniversal Turing machine (UTM) is a Turing machine that can simulate all other Turing machines. Formally, a Turing machine Uis a UTM iﬀ for every Turing machine Tthere is a binary string p(calledprogram) such that U(p;x) =T(x)for allx2X, i.e., the output of Uwhen run on (p;x)is the same as the output of Twhen run on x. We assume the set of programs on Uis preﬁx-free. The Kolmogorov complexity K(x) of a stringxis the length of the shortest program on Uthat prints xand then halts: K(x) := minfjpjjU(p) =xg: Amonotone Turing machine is a Turing machine with a one-way read-only input tape, a one-way write-only output tape, and a read/write work tape. Monotone Turing machines sequentially read symbols from their input tape and write to their output tape. Interpreted as a function, a monotone Turing machine Tmaps a string xto the longest string that Twrites to the output tape while reading xand no more from the input tape (Li and Vitányi, 2008, Ch. 4.5.2). We also use Uto denote a universal monotone Turing machine (programs on the universal monotone Turing machine do not have to be preﬁx-free). The monotone Kol- mogorov complexity Km(x)denotes the length of the shortest program on the monotone machineUthat prints a string starting with x(Li and Vitányi, 2008, Def. 4.5.9): Km(x) := minfjpjjxvU(p)g: (2.1) Since monotone complexity does not require the machine to halt, there is a constant c such thatKm(x)K(x) +cfor allx2X. §2.4Algorithmic Information Theory 21 The following notion of a (semi)measure is particular to algorithmic information theory. Deﬁnition 2.14 (Semimeasure; Li and Vitányi, 2008, Def. 4.2.1) .Asemimeasure over the alphabetXis a function :X![0;1]such that (a)()1, and (b)(x)P a2X(xa)for allx2X. A semimeasure is a (probability) measure iﬀ equalities hold in (a) and (b) for all x2X. Semimeasures are not probability measures in the classical measure theoretic sense. However, semimeasures correspond canonically to classical probability measures on the probability space X]=X[X1whose-algebra is generated by the cylinder sets (Li and Vitányi, 2008, Ch. 4.2 and Hay, 2007). Lower semicomputable semimeasures correspond naturally to monotone Turing ma- chines (Li and Vitányi, 2008, Thm. 4.5.2): for a monotone Turing machine T, the semimeasure Tmaps a string xto the probability that Toutputs something starting withxwhen fed with fair coin ﬂips as input (and vice versa). Hence we can enumerate all lower semicomputable semimeasures 1;2;:::by enumerating all monotone Tur- ing machines. We deﬁne the Kolmogorov complexity K()of a lower semicomputable semimeasure as the Kolmogorov complexity of the index of in this enumeration. We often mix the (semi)measures of algorithmic information theory with concepts from probability theory. For convenience, we identify a ﬁnite string x2Xwith its cylinder set x. Then(x)in the algorithmic information theory sense coincides with (x)in the measure theory sense if we use the identiﬁcation of semimeasures with probability measures above. Example 2.15 (Lebesgue Measure) .TheLebesgue measure oruniform measure is deﬁned as (x) := (#X)jxj: 3 The following deﬁnition turns a semimeasure into a measure, preserving the predic- tive ratio(xa)=(xb)fora;b2X. Deﬁnition 2.16 (Solomonoﬀ Normalization) .TheSolomonoﬀ normalization normof a semimeasure is deﬁned as norm() := 1and for allx2Xanda2X, norm(xa) :=norm(x)(xa)P b2X(xb): (2.2) By deﬁnition, normis a measure. Moreover, normdominatesaccording to the following lemma. Lemma 2.17 (norm).norm(x)(x)for allx2Xand all semimeasures . 22 Preliminaries Proof.We use induction on the length of x: ifx=thennorm() = 1 =(), and otherwise norm(xa) =norm(x)(xa)P b2X(xb)(x)(xa)P b2X(xb)(x)(xa) (x)=(xa): The ﬁrst inequality holds by induction hypothesis and the second inequality uses the fact thatis a semimeasure. Chapter 3 Learning The problem of induction is essentially solved. — David Hume Machine learning refers to the process of learning models of and/or making predictions about (large) sets of data points that are typically independent and identically dis- tributed (i.i.d.); see Bishop (2006) and Hastie et al. (2009). In this chapter we do not makethei.i.d.assumption. Instead, weaimmoregenerallyatthetheoreticalfundamen- tals of the sequence prediction problem : how will a sequence of symbols generated by an unknown stochastic process be continued? Given a ﬁnite string x<t=x1x2:::xt1 of symbols, what is the next symbol xt? How likely does a given property hold for the entire sequence x1:1? Arguably, any learning or prediction problem can be phrased in this fashion: anything that can be stored on a computer can be turned into a sequence of bits. We distinguish two major elements of learning. First, the process of converging to accurate beliefs, called merging. Second, the process of making accurate forecasts about the next symbol, called predicting . These two notions are not distinct: if you have accurate beliefs about the unseen data, then you can make good predictions, but not necessarily vice versa (see Example 3.41). We discuss diﬀerent notions of merging in Section 3.4 and state bounds on the prediction regret in Section 3.5. In the general reinforcement learning problem we target in this thesis, the environ- ment is unknown and the agent needs to learn it. The literature on non-i.i.d. learning has focused on predicting individual symbols and bounds on the number of prediction errors (Hutter, 2001b, 2005; Cesa-Bianchi and Lugosi, 2006), and the results on merg- ing are from the game theory literature (Blackwell and Dubins, 1962; Kalai and Lehrer, 1994; Lehrer and Smorodinsky, 1996). However, we argue that merging is the essential property for general AI. In order to make good decisions, the agent needs to have ac- curate beliefs about what its actions will entail. On a technical level, merging leads to on-policy value convergence (Section 4.2.3), the fact that the agents learns to estimate the values for its own policy correctly. The setup we consider is the realizable case : we assume that the data is generated by an unknown probability distribution that belongs to a known (countable) class of distributions. In contrast, the nonrealizable case allows no assumptions on the under- lying process that generates the data. A well-known approach to the nonrealizable case isprediction with expert advice (Cesa-Bianchi and Lugosi, 2006), which we do not con- 23 24 Learning sider here. Generally, the nonrealizable case is harder, but Ryabko (2011) argues that for some problems, both cases coincide. After introducing the formal setup in Section 3.1, we discuss several examples for learning distributions and notions that relate the learning distribution with the process generating the data in Section 3.2. In Section 3.3 we connect these notions to the theory of martingale processes. Section 3.6 connects the results from the ﬁrst sections to the learning framework developed by Solomonoﬀ (1964, 1978), Hutter (2001b, 2005, 2007b), and Schmidhuber (2002) (among others). This framework relies on results from algorithmic information theory and computability theory to learn any computable distribution quickly and eﬀectively. It is incomputable (see Section 6.2), but can serve as a gold standard for learning. Most of this chapter echoes the literature. We collect results from economics and computer science that previously had not been assembled in one place. We provide proofs that connect the various properties (Proposition 3.23, Proposition 3.16, and Proposition 3.37), and we ﬁll in a few gaps in the picture: the prediction bounds for absolute continuity (Section 3.5.2) and the improved regret bounds for nonuniform measures (Theorem 3.48 and Theorem 3.51). Section 3.7 summarizes the results in Table 3.2 on page 45 as well as Figure 3.1 on page 46. 3.1 Setup For the rest of this chapter, ﬁx PandQto be two probability measures over the measurablespaceofinﬁnitesequences (X1;F1). Wethinkof Pasthetrue distribution from which the data sequence x1:1is drawn, and of Qas our belief distribution or learning algorithm. In other words, we use the distribution Qto learn a string drawn from the distribution P. LetHdenote a hypothesis , i.e., any measurable set from F1. Ourprior belief in the hypothesis HisQ(H). In each time step t, we make one observation xt2X. Our historyx<t=x1x2:::xt1is the sequence of all previous observations. We update our belief in accordance with Bayesian learning; our posterior belief in the hypothesis His Q(Hjx1:t) =Q(H\x1:t) Q(x1:t): The observation xtconﬁrms the hypothesis HiﬀQ(Hjx1:t)>Q(Hjx<t)(the belief inHincreases), and the observation xtdisconﬁrms the hypothesis HiﬀQ(Hjx1:t)< Q(Hjx<t)(the belief in Hdecreases). If Q(Hjx1:t) = 0, thenHisrefutedorfalsiﬁed. When we assign a prior belief of 0to a hypothesis H, this means that we think thatHis impossible; it is refuted from the beginning. If Q(H) = 0, then the posterior Q(Hjx<t) = 0, so no evidence whatsoever can change our mind that His impossible. This is bad if the hypothesis His actually true. To be able to learn we need to make some assumptions on the learning distribution Q: we need to have an open mind about anything that might actually happen, i.e., §3.2Compatibility 25 Q(H)>0on any hypothesis HwithP(H)>0. This property is called absolute continuity . We discuss this and other notions of compatibility of PandQin Section 3.2. We motivate this chapter with the following example. Example 3.1 (The Black Ravens; Rathmanner and Hutter, 2011, Sec. 7.4) .If we live in a world in which all ravens are black, how can we learn this fact? Since at every time step we have observed only a ﬁnite subset of the (possibly inﬁnite) set of all ravens, how can we conﬁdently state anything about all ravens? We formalize this problem in line with Rathmanner and Hutter (2011, Sec. 7.4) and Leike and Hutter (2015d). We deﬁne two predicates, blackness Band ravenness R. There are four possible observations: a black raven BR, a non-black raven BR, a black non-ravenBR, and a non-black non-raven BR. Therefore our alphabet consists of four symbols corresponding to each of the possible observations, X:=fBR;BR;BR;BRg. We are interested in the hypothesis ‘all ravens are black’. Formally, it corresponds to the measurable set H:=fx2X1jxt6=BR8tg=fBR;BR;BRg1; (3.1) the set of all inﬁnite strings in which the symbol BRdoes not occur. Ifweobserveanon-blackraven, xt=BR,thehypothesis Hisrefutedsince H\x1:t= ;and this implies Q(Hjx1:t) = 0. In this case, our inquiry regarding His settled. The interesting case is when the hypothesis His in fact true ( P(H) = 1), i.e.,Pdoes not generate any non-black ravens. The property we desire is that in a world in which all ravens are black, we arrive at this belief: P(H) = 1impliesQ(Hjx<t)!1as t!1. 3 3.2 Compatibility In this section we deﬁne dominance ,absolute continuity ,dominance with coeﬃcients , weak dominance , andlocal absolute continuity , in decreasing order of their strength. These notions make the relationship of the two probability measures PandQprecise. We also give examples for various choices for the learning algorithm Q. In our examples, we frequently rely on the following process. Example 3.2 (Bernoulli Process) .AssumeX=f0;1g. For a real number r2[0;1] we deﬁne the Bernoulli process with parameter ras the measure Bernoulli (r)(x) :=rones(x)(1r)zeros(x): Note that Bernoulli (1=2) =, the Lebesgue measure from Example 2.15. 3 Deﬁnition 3.3 (Dominance) .The measure Qdominates P(QP) iﬀ there is a constantc>0such thatQ(x)cP(x)for all ﬁnite strings x2X. Dominance is also called having a grain of truth (Lehrer and Smorodinsky, 1996, Def. 2a and Kalai and Lehrer, 1993); we discuss this property in the context of game theory in Chapter 7. 26 Learning Example 3.4 (Bayesian mixture) .LetMbe a countable set of probability measures on(X1;F1)and letw2Mbe a prior overM. Ifw(P)>0for allP2M, the priorwis called positiveoruniversal . Then the Bayesian mixture :=P P2Mw(P)P dominates each P2M. 3 The Bayesian mixture is a mathematically simple yet very powerful concept. It is very easy to derive from a countable set of distributions, and it has been considered extensivelyintheliterature(Solomonoﬀ,1964;Jaynes,2003;Hutter,2005, ...). Ryabko (2009)showsthatevenforuncountablyinﬁniteclasses, iftherearegoodpredictors, then a Bayesian mixture over a countable subclass asymptotically also does well. Example 3.5 (Solomonoﬀ Prior) .Solomonoﬀ (1964) deﬁnes a distribution MoverX] that assigns to a string xthe probability that the universal monotone Turing machine Uoutputsxwhen fed with fair coin ﬂips on the input tape. Formally, M(x) :=X p:xvU(p)2jpj(3.2) wherepisabinarystring.1Thefunction Misalowersemicomputablesemimeasure,but not computable and not a measure (Li and Vitányi, 2008, Lem. 4.5.3); see Section 6.2 for the computability properties of M. More importantly, Mdominates every lower semicomputable semimeasure (Li and Vitányi, 2008, Thm. 4.5.1). Solomonoﬀ’s prior Mhas a number of appealing philosophical properties. In line with Ockham’s razor it favors simple environments over complex ones: Turing machines that have a short program on the UTM Uhave a higher contribution in the sum (3.2). In line with Epicurus’ principle it never discards possible explanations: every program that produces the string xcontributes to the sum. See Rathmanner and Hutter (2011) for a discussion on the philosophical underpinnings of Solomonoﬀ’s prior. 3 Wood et al. (2011) show that the Solomonoﬀ prior Mcan equivalently be deﬁned as a Bayesian mixture over all lower semicomputable semimeasures with a prior w(P)/ 2K(P). (If we use w(P) = 2K(P)we get a semiprior becauseP P2M2K(P)can be less than 1. This prior also carries the name Solomonoﬀ prior .) Deﬁnition 3.6 (Absolute Continuity) .The measure Pisabsolutely continuous with respect toQ(QP) iﬀQ(A) = 0impliesP(A) = 0for all measurable sets A. Remark 3.7 (Absolute Continuity ;Dominance) .Absolute continuity is strictly weaker than dominance: let X:=f0;1gand deﬁne a probability measure Pthat assigns probability 2=3to1and probability 1=3to0until seeing the ﬁrst 0, thenP behaves like the Lebesgue measure . Formally, P(x1:t) :=(2 3tifx1:t= 1t;and2 3n1 3(xn+2:t)if9n0:1n0vx1:t: 1We use the name Solomonoﬀ prior for both a distribution over X1and a distribution over a com- putably enumerable set M. Maybe Mshould better be called Solomonoﬀ mixture to avoid confusion. §3.2Compatibility 27 Since(1t)=P(1t) = (3=4)t!0ast!1, there is no constant csuch that(x)=P(x)> c>0for all ﬁnite strings x2X, hencedoes not dominate P. ButPis absolutely continuous with respect to becauseP-almost surely we draw a 0eventually, and thenPbehaves like . HenceP-almost surely =P6!0. The claim now follows from Proposition 3.23b. 3 The idea of Remark 3.7 is to ‘punch a hole into ’ at the inﬁnite string 11. This inﬁnite string has probability 0, hence this hole does not break absolute continuity. But it breaks dominance on this inﬁnite string. Analogously we could punch countably many holes into a probability measure without breaking absolute continuity. Deﬁnition 3.8 (Weak Dominance) .The measure Qweakly dominates P(QWP) iﬀ lim t!11 tlogQ(x1:t) P(x1:t)= 0withP-probability 1: Lehrer and Smorodinsky (1996, Rem. 8) point out that for any PandQ, lim sup t!11 tlogQ(x1:t) P(x1:t)0P-almost surely ; so crucial is whether the lim infis also 0. Remark 3.9 (Weak Dominance) .The measure Qweakly dominates Pif and only if P-almost surely log(P(x)=Q(x))2o(t). 3 Ryabko and Hutter (2007, 2008) consider the following deﬁnition. It is analogous to Deﬁnition 3.3, except that the constant cis allowed to depend on time. Deﬁnition 3.10 (Dominance with Coeﬃcients; Ryabko and Hutter, 2008, Def. 2) .The measureQdominatesPwith coeﬃcients f(QP=f) iﬀQ(x)P(x)=f(jxj)for all x2X. IfQdominatesPwith coeﬃcients fandfgrows subexponentially ( f2o(exp)), thenQweakly dominates Pby Remark 3.9. Example 3.11 (Speed Prior) .Schmidhuber (2002) deﬁnes a variant of Solomonoﬀ’s priorMthat penalizes programs by their running time, called the speed prior . Consider the speed prior SKt(x) :=X p:xvU(p)2jpj t(U;p;x ) wheret(U;p;x )is the number of time steps the Turing machine Utakes to produce x from the program p. For any deterministic measure Pcomputable in time qwe have SKt(x)P(x)=q(jxj). Therefore SKtdominatesPwith coeﬃcients O(q). Ifqis a polynomial ( Pis computable in polynomial time), then it grows subexponentially and thusSKtweakly dominates P. 3 Thesemimeasureloss SKt(x)P a2XSKt(xa)inthespeedpriorisquitesubstantial: since it takes at least nsteps to output a string of length n,M(x)jxjSKt(x). 28 Learning Example 3.12 (Laplace Rule) .TheLaplace rule Lis deﬁned by L(xtjx<t) :=#fi<tjxi=xtg t+ #X: ForX=f0;1gandr2[0;1]themeasure LdominatesBernoulli (r)withcoeﬃcients f(t) =t#X+1(Ryabko and Hutter, 2008, Prop. 3). 3 Deﬁnition 3.13 (Local Absolute Continuity) .The measure Pislocally absolutely continuous with respect to Q(QLP) iﬀQ(x) = 0impliesP(x) = 0for all ﬁnite stringsx2X. The notable diﬀerence between local absolute continuity and absolute continuity is that Deﬁnition 3.6 talks about arbitrary measurable sets while Deﬁnition 3.13 only talks about ﬁnite strings. The former is a much stronger property. For example, every measure is locally absolutely continuous with respect to the Lebesgue measure since (x)>0for all ﬁnite strings x2X. Local absolute continuity is an extremely weak property. If it is not satisﬁed, we have to be very careful when using Qfor prediction: then there is a positive probability that we have to condition on a probability zero event. Example 3.14 (The Minimum Description Length Principle; Grünwald, 2007) .Let Mbe a countable set of probability measures on (X1;F1)and letK:M! [0;1] be a function such thatP P2M2K(P)1calledregularizer . Following notation from Hutter (2009a), we deﬁne for each x2Xtheminimal description length model as MDLx:= arg min P2MflogP(x) +K(P)g: logP(x)is the (arithmetic) code length of xgiven model P, andK(P)is a complexity penalty for P. Given data x2X,MDLxis the measure P2Mthat minimizes the total code length of data and model. Note that the Lebesgue measure is not locally absolutely continuous with respect to the MDL distribution Q(x) := MDLx(x): for somex2Xthe minimum description P2Mmay assign probability zero to a continuation xy2X. 3 Remark 3.15 (MDL is Inductively Inconsistent; Leike and Hutter, 2014a, Cor. 13) . The MDL estimator for countable classes as deﬁned in Example 3.14 is inductively inconsistent: the selected model P2 Mcan change inﬁnitely often and thus the limit limt!1MDLx<tmay not exist. This can be a major obstacle for using MDL for prediction, since the model used for prediction has to be changed over and over again, incurring the corresponding computational cost. 3 The following proposition establishes the relationship between our notions of com- patibility; see also Figure 3.1 on page 46. Proposition 3.16 (Relationships between Compatibilities) . (a) IfQP, thenQP. §3.3Martingales 29 (b) IfQP, thenQWP. (c) IfQP, thenQdominatesPwith coeﬃcients ffor a constant function f. (d) IfQdominatesPwith coeﬃcients fandf2o(exp), thenQWP. (e) IfQWP, thenQLP. Proof.(a) From Proposition 3.23 (a) and (b). (b) From Proposition 3.23b and Kalai and Lehrer (1994, Prop. 3a). (c) Follows immediately from the deﬁnitions. (d) From Remark 3.9. (e) Follows immediately from the deﬁnitions. Note that the converse of Proposition 3.16d is false: in Remark 3.7 we deﬁned a measure Pthat is absolutely continuous with respect to (and hence is weakly dominated by ), but the coeﬃcients for P=grow exponentially on the string 1t. This inﬁnite string has P-probability 0, but dominance with coeﬃcients demands the inequalityQP=fto hold for all strings. Remark 3.17 (Local Absolute Continuity ;Absolute Continuity) .DeﬁneP:= Bernoulli (2=3)andQ:=Bernoulli (1=3). Both measures PandQare nonzero on all cylinder sets: Q(x)3jxj>0andP(x)3jxj>0for everyx2X. Therefore Qis locally absolutely continuous with respect to P. However, Qisnotabsolutely continuous with respect to P: deﬁne A:= x2X!lim sup t!11 tones(x1:t)1 2 : The setAisF1-measurable since A=T1 n=1S x2UnxwithUn:=fx2Xjjxj nandones(x)jxj=2g, the set of all ﬁnite strings of length at least nthat have at least as many zeros as ones. We have that P(A) = 0andQ(A) = 1, henceQis not absolutely continuous with respect to P. 3 3.3 Martingales The following two theorems state the connection between probability measures on inﬁ- nite strings and martingales. For two probability measures PandQthe quotient Q=P is a nonnegative P-martingale if Qis locally absolutely continuous with respect to P. Conversely, for every nonnegative P-martingale there is a probability measure QLP such that the martingale is P-almost surely a multiple of Q=P. 30 Learning Theorem 3.18 (Measures7!Martingales; Doob, 1953, II§7 Ex. 3) .LetQandPbe two probability measures on (X1;F1)such thatQis locally absolutely continuous with respect toP. Then the stochastic process (Xt)t2N, Xt(x) :=Q(x1:t) P(x1:t) is a nonnegative P-martingale with E[Xt] = 1. Theorem3.19 (Martingales7!Measures) .LetPbe a probability measure on (X1;F1) and let (Xt)t2Nbe a nonnegative P-martingale with E[Xt] = 1. There is a probability measureQon(X1;F1)that is locally absolutely continuous with respect to Pand for allx2X1and allt2NwithP(x1:t)>0, Xt(x) =Q(x1:t) P(x1:t): The proofs for Theorem 3.18 and Theorem 3.19 are provided in the Appendix. Example 3.20 (The Posterior Martingale) .Suppose we are interested in a hypothesis HX1(such as the proposition ‘all ravens are black’ in Example 3.1). If Q(H) =P P2Mw(P)P(H)is a Bayesian mixture over a set of probability distributions M with prior weights w2M(see Example 3.4), then the posterior belief Q(Hjx) =P P2Mw(Pjx)P(Hjx). The weights w(Pjx)are called posterior weights , and satisfy the identity w(Pjx) =w(P)P(x) Q(x)(3.3) since Q(Hjx) =Q(H\x) Q(x) =1 Q(x)X P2Mw(P)P(H\x) =X P2Mw(P)P(x)P(H\x) Q(x)P(x) =X P2Mw(Pjx)P(Hjx): According to Theorem 3.18 the posterior weight w(Pjx)is aQ-martingale with expectation w(P). In particular, this means that the posterior weights converge Q- almostsurelybythemartingaleconvergencetheorem(Theorem2.8). Since Qdominates P, by Proposition 3.16a Pis absolutely continuous with respect to Qand hence the posterior also converges P-almost surely. 3 Remark 3.21 (Martingales and Absolute Continuity) .While Theorem 3.18 trivially also holds if Qis absolutely continuous with respect to P, Theorem 3.19 does not imply thatQis absolutely continuous with respect to P. §3.3Martingales 31 LetPandQbe deﬁned as in Remark 3.17. Consider the process X0(x) := 1, Xt+1(x) :=( 2Xt;ifxt+1= 0;and 1 2Xt;ifxt+1= 1: The process (Xt)t2Nis a nonnegative P-martingale since every XtisFt-measurable and forx=y1:twe have E[Xt+1jFt](y) =P(x0jx)2Xt(y) +P(x1jx)1 2Xt(y) =1 32Xt(y) +2 31 2Xt(y) =Xt(y): Moreover, Q(x) =1 3ones(x)2 3zeros(x)=2 3ones(x)1 3zeros(x)2ones(x)2zeros(x)=P(x)Xt(y): HenceXt(y) =Q(y1:t)=P(y1:t)P-almost surely. The measure Qis uniquely deﬁned by its values on the cylinder sets, and as shown in Remark 3.17, Qis not absolutely continuous with respect to P. 3 Theorem 3.22 (Radon-Nikodym Derivative) .IfQP, then there is a function dP=dQ :X1![0;1)called the Radon-Nikodym derivative such that Z fdP =Z fdP dQdQ for all measurable functions f. This function dP=dQcan be seen as a density function of Pwith respect to the background measure Q. Moreover, dP=dQis the limit of the martingale P=Q(Durrett, 2010, Sec. 5.3.3) which exists Q-almost surely according to Theorem 2.8. ThefollowingpropositioncharacterizesthenotionsofcompatibilityfromSection3.2 in terms of the martingale Q=P. Proposition 3.23 (Martingales and Compatibility) .The following relationships hold betweenQ,P, and theP-martingale Yt:=Q(x1:t)=P(x1:t). (a)QPif and only if Ytc>0for allt2N. (b)QPif and only if P-almost surely Yt6!0ast!1. (c)QdominatesPwith coeﬃcients fif and only if Yt1=f(t)for allt. (d)QWPif and only if P-almost surely log(Yt+1=Yt)!0in Cesàro average. (e)QLPif and only if P-almost surely Yt>0for allt2N. Proof. (Yt)t2Nis aP-martingale according to Theorem 3.18. (a)Q(x)cP(x)withc>0for allx2Xis equivalent to Q(x)=P(x)c>0for all x2X. 32 Learning (b) Proved by Hutter (2009a, Lem. 3i). (c) Analogously to the proof of (a). (d) IfQweakly dominates P, we getlogYt2o(t)according to Remark 3.9. To- gether with Y0= 1we getlogYt=Pt1 k=0log(Yk+1=Yk)2o(t), therefore t1Pt1 k=0log(Yk+1=Yk)!0ast!1. Conversely, if the Cesàro average con- verges to 0, thent1logYt!0, hencelogYt2o(t). (e) Letx2 Xbe any ﬁnite string. If QLPandP(x)>0, thenQ(x)>0, and henceQ(x)=P(x)>0. Conversely, if P(x)>0thenYtis well-deﬁned, so if Yjxj(x)>0thenQ(x)>0. From Proposition 3.23b and Theorem 3.22 we get that QPif and only if the Radon-Nikodym derivative dQ=dPis positive on a set of P-measure 1. 3.4 Merging IfQis capable of learning, it should use the sequence xdrawn from Pto change its opinions more in the direction of P. More precisely, we want Q(jx<t)P(jx<t) for larget. In the rest of this chapter, we make this notion of closeness precise and discuss diﬀerent conditions on Qthat are suﬃcient for learning. Strong merging implies that the belief of anyhypothesis merges. This is very strong, ashypothesescantalkabout tail events : eventsthatareindependentofanyﬁniteinitial part of the inﬁnite sequence (such as the event Ain Remark 3.17). Weak merging only considers hypothesis about the next couple of symbols, and almost weak merging allows Qto deviate from Pin a vanishing fraction of the time. Much of this section is based on Kalai and Lehrer (1994) and Lehrer and Smorodinsky (1996). 3.4.1 Strong Merging Deﬁnition 3.24 (Strong Merging) .Qmerges strongly with PiﬀD1(P;Qjx<t)!0 ast!1P-almost surely. The following theorem is the famous merging of opinions theorem by Blackwell and Dubins (1962). Theorem 3.25 (Absolute Continuity )Strong Merging; Blackwell and Dubins, 1962) . IfPis absolutely continuous with respect to Q, thenQmerges strongly with P. Example 3.26 (The Black Ravens 2; Rathmanner and Hutter, 2011, Sec. 7.4) .Recall the black raven problem from Example 3.1. Let Qbe a learning distribution that dominates the true distribution P, such as a Bayesian mixture (Example 3.4). By Proposition 3.16a we get QP, and hence Qmerges strongly to Pby Theorem 3.25. Thus we get as t!1thatP-almost surelyjQ(Hjx<t)P(Hjx<t)j!0for the hypothesis Hthat ‘all ravens are black’ deﬁned in (3.1). Thus if all ravens are black in the real world ( P(H) = 1),Qlearns this asymptotically ( Q(Hjx<t)!1). This is §3.4Merging 33 the solution we desired: the learning distribution Qconverges to a true belief about an inﬁnite set by only looking from a ﬁnite (but growing) number of data points. 3 The following is the converse of Theorem 3.25. Theorem 3.27 (Strong Merging^Local Absolute Continuity )Absolute Continuity; Kalai and Lehrer, 1994, Thm. 2) .IfQis locally absolutely continuous with respect to PandQmerges strongly with P, thenPis absolutely continuous with respect to Q. The following result shows that local absolute continuity is not required for strong merging: recall that according to Example 3.14 the MDL distribution is not locally absolutely continuous with respect to every Pfrom the classM. Theorem 3.28 (Strong Merging for MDL; Hutter, 2009a, Thm. 1) .IfP2M, then D1(P;MDLxjx)!0asjxj!1P-almost surely. LetMbe a (possibly uncountable) set of probability measures on (X1;F1). Ryabko (2010, Thm. 4) shows that if there is a Qthat merges strongly with every P2M, then there is a Bayesian mixture over a countable subset of Mthat also merges strongly with every P2M. 3.4.2 Weak Merging In Deﬁnition 3.24 the supremum ranges over all measurable sets A2F1which includes tail events. Instead, we may restrict the supremum to the next few symbols. This is known as weak merging . Deﬁnition 3.29 (Weak Merging) .Qweakly merges with Piﬀ for every d2N, Dt+d(Q;Pjx<t)!0ast!1P-almost surely. The following lemma gives an equivalent formulation of weak merging. Lemma 3.30 (Lehrer and Smorodinsky, 1996, Rem. 5) .Qweakly merges with Pif and only if Dt(Q;Pjx<t)!0ast!1P-almost surely. Unfortunately, weak dominance is not suﬃcient for weak merging (Lehrer and Smorodinsky, 1996, Ex. 10). We need the following stronger condition, that turns out to be (almost) necessary. In the following, let Yt:=Q(x1:t)=P(x1:t)denote the P-martingale from Proposition 3.23. Theorem 3.31 (Kalai and Lehrer, 1994, Prop. 5a) .IfP-almost surely Yt+1=Yt!1, thenQmerges weakly with P. Example 3.32 (Laplace Rule 2) .Suppose we use the Laplace rule from Example 3.12 to predict a Bernoulli (r)process. By the strong law of large numbers, L(xtjx<t)!r almost surely. Therefore we can use Theorem 3.31 to conclude that Lmerges weakly with Bernoulli (r)for allr2[0;1]. (Note that strongly merging with every Bernoulli process is impossible; Ryabko, 2010, p. 7) 3 34 Learning The following is a converse to Theorem 3.31. Theorem 3.33 (Kalai and Lehrer, 1994, Prop. 5b) .IfQmerges weakly with P, then Yt+1=Yt!1inP-probability. Unfortunately, weak dominance is not enough to guarantee weak merging. Example3.34 (WeakDominance ;WeakMerging; RyabkoandHutter,2007,Prop.7) . LetX=f0;1gand letfbe any arbitrarily slowly monotone growing function with f(t)!1. DeﬁneP(11) := 1, the sequence (ti)i2Nsuch thatf(ti+1)2f(ti), and Q(xtjx<t) :=8 >>< >>:1 2ift=tifor somei2N; 1ift6=tiandxt= 1;and 0otherwise: NowQdominatesPwith coeﬃcients fby construction and Qweakly dominates Pif fgrows subexponentially. However, jQ(1j1t)P(1j1t)j1=2for inﬁnitely many t2N. 3 3.4.3 Almost Weak Merging The following deﬁnition is due to Lehrer and Smorodinsky (1996, Def. 10). Deﬁnition 3.35 (Almost Weak Merging) .Qalmost weakly merges with Piﬀ for every d2N 1 ttX k=1Dt+d(Q;Pjx<t)!0ast!1P-almost surely : There is also an analogue of Lemma 3.30 for almost weakly merging in the sense that we can equivalently set d= 0(Lehrer and Smorodinsky, 1996, Rem. 6). Remark 3.36 (Weak Merging and Merging in KL-Divergence) .From Lemma 2.13 follows that weak merging is implied by KLd(P;Qjx)!0P-almost surely and almost weak merging is implied byPt k=1KL1(P;Qjx<k)2o(t)P-almost surely, i.e.,KLt(P;Q)2o(t)(Ryabko and Hutter, 2008, Lem. 1). The converse is generally false. 3 The following proposition relates the three notions of merging. Proposition 3.37 (Strong Merging )Weak Merging)Almost Weak Merging) .If Qmerges strongly with P, thenQmerges weakly with P. IfQmerges weakly with P, thenQmerges almost weakly with P. Proof.Follows immediately from the deﬁnitions. Theorem 3.38 (Weak Dominance )Almost Weak Merging; Lehrer and Smorodinsky, 1996, Thm. 4) .IfQweakly dominates P, thenQmerges almost weakly with P. §3.5Predicting 35 From Theorem 3.38 we get that the speed prior (Example 3.11) merges almost weakly with any probability distribution estimable in polynomial time. We also have the following converse to Theorem 3.38. Theorem 3.39 (Almost Weak Merging )Weak Dominance; Lehrer and Smorodinsky, 1996, Cor. 7) .IfQis locally absolutely continuous with respect to P,Qmerges almost weakly with P, andP-almost surely lim inft!1Yt+1=Yt>0, thenQweakly dominates P. 3.5 Predicting In Section 3.4 we wanted Qto acquire the correct beliefs about P. In this section, we exploit the accuracy of our beliefs for predicting individual symbols. We derive bounds on the number of errors Qmakes when trying to predict a string drawn from P. Sincethedatadrawnfrom Pisstochastic, wecannotexpecttomakeaﬁnitenumber of errors. Even the perfect predictor that knows Pgenerally makes an inﬁnite number of errors. For example, trying to predict the Lebesgue measure (Example 2.15), in expectation we make half an error in every time step. So instead we are asking about the asymptotic error rate of a predictor based on Qcompared to a predictor based on P, theprediction regret . LetxR tbe thet-th symbol predicted by the probability measure Raccording to the maximum likelihood estimator: xR t:2arg max a2XR(x<tajx<t): (3.4) Theinstantaneous error of aR-based predictor is deﬁned as eR t:=( 0ifxt=xR t;and 1otherwise: and thecumulative error is ER t:=tX k=1eR k: Note that both etandEtare random variables. Deﬁnition 3.40 (Prediction Regret) .In time step ttheprediction regret isEQ tEP t and theexpected prediction regret isEh EQ tEP ti . More generally, we could also follow Hutter (2001b) and phrase predictive perfor- mance in terms of loss: given a loss function `:XX! Rthe predictor Qsuﬀers an (instantaneous) loss of `(xQ t;xP t)in time step t. If the loss function `is bounded in [0;1], many of the results for prediction regret also hold for cumulative loss (for Section 3.5.2 we also need `(a;a) = 0for alla2X). In this chapter we chose to phrase the results in terms of prediction errors instead of loss because prediction errors are conceptionally simpler. 36 Learning Example 3.41 (Good Prediction Regret ;Merging/Compatibility) .Good predic- tion regret does not imply (weak/strong) merging or (weak) dominance: Let P:= Bernoulli (1=3)andQ:=Bernoulli (1=4). ClearlyPandQdo not merge (weakly) or (weakly) dominate each other. However, a P-based predictor always predicts 0, and so does aQ-based predictor. Therefore the prediction regret EQ tEP tis always 0.3 Example 3.42 (Adversarial Sequence; Legg, 2006, Lem. 4) .No learning distribution Qwill learn to predict everything. We can always deﬁne a Q-adversarial sequence z1:1 recursively according to zt:=( 0ifQ(0jz<t)<1=2;and 1ifQ(0jz<t)1=2: In every time step the probability that a Q-based predictor makes an error is at least 1=2, henceeQ t1=2andEQ tt=2. Butz1:1is a deterministic sequence, thus an informed predictor makes zero errors. Therefore the prediction regret of Qon the sequencez1:1is linear. 3 3.5.1 Dominance WestartwiththepredictionregretboundsprovedbyHutter(2001b)incasethelearning distribution Qdominates the true distribution P. In the following, let cPdenote the constant from Deﬁnition 3.3. Theorem 3.43 (Hutter, 2007b, Eq. 5 & 8) .For allPandQ, q EPEQ nq EPEPnp 2KLn(P;Q): The following bound on prediction regret then follows easily, but it is a factor ofp 2 worse than the bound stated by Hutter (2005, Thm. 3.36). Corollary 3.44 (Expected Prediction Regret) .For allPandQ, 0EPh EQ nEP ni 2KLn(P;Q) + 2q 2KLn(P;Q)EPEPn: Proof.From Theorem 3.43 we get EPh EQ nEP ni =q EPEQ n+q EPEPnq EPEQ nq EPEPn q EPEQ n+q EPEPnp 2KLn(P;Q) p 2KLn(P;Q) +q EPEPn+q EPEP tp 2KLn(P;Q) = 2KLn(P;Q) + 2q 2KLn(P;Q)EPEPn: §3.5Predicting 37 IfQdominatesP, then we have KLn(P;Q)lncP: KLn(P;Q) =X x2XnP(x) logP(x) Q(x)X x2XnP(x) log1 cP=logcP(3.5) This invites the following corollary. Corollary 3.45 (Prediction Regret for Dominance; Hutter, 2005, Cor. 3.49) .IfQ dominatesP, then the following statements hold. (a)EPEQ 1is ﬁnite if and only if EPEP 1is ﬁnite. (b)q EPEQ 1p EPEP12O(1) (c)EPEQ t=EPEP t!1forEPEP t!1. (d)EPh EQ tEP ti 2Op EPEP t . If the true distribution Pis deterministic, we can improve on these bounds: Example 3.46 (Predicting a Deterministic Measure) .Suppose we are predicting a deterministic measure Pthat assigns probability 1to the inﬁnite string x1:1. IfPis dominated by Q, the total expected prediction regret EPEQ 1is bounded by2 lncP by Corollary 3.44. This is easy to see: every time we predict a wrong symbol a6=xt, thenQ(ajx<t)Q(xtjx<t), soQ(xtjx<t)1=2. Therefore YtYt1=2and by dominance YtcP. Hence a prediction error can occur at most logcPtimes. 3 Generally, the O(EPEP t)bounds on expected prediction regret given in Corol- lary 3.45 are essentially unimprovable: Example 3.47 (Lower Bounds on Prediction Regret) .SetX:=f0;1gand consider the uniform measure from Example 2.15. For each time step t, we have(0jx<t) = (1jx<t) = 1=2, so the argmax in (3.4) ties and hence it does not matter whether we predict 0or1. We take two predictors PandQ, wherePalways predicts 0andQ always predicts 1. LetZt:=EQ tEP t. Since their predictions never match, Ztis an ordinary random walk with step size 1. We have (Weisstein, 2002) lim sup t!1EP[EQ tEP t]p t=p 2= and for the law of the iterated logarithm (Durrett, 2010, Thm. 8.8.3) lim sup t!1EQ tEP tp2tlog logt= 1P-almost surely : Both bounds are known to be asymptotically tight. 3 While Example 3.47 shows that the bounds from Corollary 3.45 are asymptotically tight, they are misleading because in most cases, we can do much better. According 38 Learning to the following theorem, the worst case bounds are only attained if P(xtjx<t)is suﬃciently close to 1=2. Theorem 3.48 (Expected Prediction Regret for Nonuniform Measures) .IfX=f0;1g and there is an ">0such thatjP(xtjx<t)1=2j"for allx1:t2X, then EPh EQ tEP ti KLt(P;Q) ": Proof.Recall the deﬁnition of entropy in nats: Ent(p) :=plnp(1p) ln(1p): The second order Taylor approximation of Entat1=2is f(p) = ln 22(p1 2)2: One can check that f(p)Ent(p)for all 0p1. Deﬁnep:=P(xP tjx<t)1=2 andq:=Q(xQ tjx<t)1=2to ease notation. Consider the function g(p;q;" ) :=p(1p)"1 plnp 1q+ (1p) ln1p q which is strictly increasing as qdecreases, so from q1=2we get g(p;q;" )2p1"1ln 2 +"1Ent(p) ln 2 2p1"1ln 2 +"1f(p) ln 2 = 2p1"12(p1 2)2; which decreases as pincreases, hence it is maximized for p= 1=2 +", g(p;q;" )2""12"2= 0 Thereforegis nonpositive. If xQ t=xP t, the one-step error is 0. Otherwise EP[etj x<t] =p(1p)andg(p;q;" ) =EP[etjx<t]"1KL1(P;Qjx<t), so we get EP[etjx<t]"1KL1(P;Qjx<t). Summing this from t= 1tonyields the claim. Eh EQ nEP ni "1KLn(P;Q): 3.5.2 Absolute Continuity Theorem 3.49 (Prediction with Absolute Continuity) .IfQP, then q EQ tq EP tOp log logt P-almost surely : The proof idea is inspired by Miller and Sanchirico (1999). We think of PandQas two players in a zero-sum betting game. In every time step t, the players will make a bet on the outcome of xt. Ifxt=xQ t6=xP t, thenQwins $1 from P, ifxt=xP t6=xQ t, §3.5Predicting 39 thenQloses $1 to P. Otherwise xQ t=xP torxQ t6=xt6=xP tand neither player gains or loses money. Since Qpredicts according to the maximum likelihood principle (3.4), it is rational to accept the bet from Q’s perspective. In Q’s eyes, the worst case is a fair bet, so Qwill not lose more money than it would lose on a random walk. The law of the iterated logarithm gives a Q-probability one statement about this bound, which transfers to Pby absolute continuity. Proof.Deﬁne the stochastic process Zt:=EQ tEP t. Since EQ[eR t] =Q(xR tjx<t), we get EQ[Zt+1jFt] =Q(xQ tjx<t)Q(xP tjx<t) +Zt Q(xQ tjx<t)Q(xQ tjx<t) +Zt=Zt; hence (Zt)t2Nis aQ-submartingale. In the worst case (for Q),(Zt)t2Nis just a random walk with step size 1. ButZtcan only move if QandPpredict a diﬀerent symbol. If this happens, at least one of them makes an error. Let mtbe the number of steps Zt has moved ( Zt+16=Zt). ThenmtEQ t+EP tandmtt. By the law of the iterated logarithm (Durrett, 2010, Thm. 8.8.3), lim inf t!1Ztp2mtlog logmt=1 Q-almost surely. We deﬁne the event A:=n 9C8t:ZtCp mtlog logmto : ThenQ(A) = 1, henceP(A) = 1by absolute continuity. EQ tEP t=ZtCq (EQ t+EP t) log logtCq EQ t+q EP tp log logt Dividingbothsidesbyq EQ t+p EP tyieldsthatthereisa P-almostsurelyﬁniterandom variableCsuch thatq EQ tp EP tCplog logt. This invites the following immediate corollary. Corollary 3.50 (Prediction Regret for Absolute Continuity) .IfQP, then EQ tEP t2O log logt+q EP tlog logt P-almost surely : Proof.Analogously to the proof of Corollary 3.44. While Corollary 3.50 establishes an almost sure prediction regret bound, it is dif- ferent from the bound on expected prediction regret from Corollary 3.44; bounds on E[EQ tEP t]are incomparable to almost sure bound given in Theorem 3.49: for a se- quence of nonnegative (unbounded) random variables convergence in mean does not 40 Learning imply almost sure convergence (Stoyanov, 2013, Sec. 14.7) or vice versa (Stoyanov, 2013, Sec. 14.8ii). Weproceedtoestablishanimprovedpredictionregretboundincase Pisnonuniform analogously to Theorem 3.48. Theorem 3.51 (Prediction Regret for Nonuniform Measures) .IfQP,X=f0;1g, and there is an ">0such that with P-probability 1 jP(xtjx<t)1=2j" for allt2N, thenP-almost surely EQ tEP t2O(1). Proof.IfjP(xtjx<t)1=2j", then for large enough t,Qwill have merged with P (Theorem 3.25) and hence jQ(xtjx<t)1=2j"=2inﬁnitely often. ThusZthas an expected gain of "=2if the predictors disagree. Therefore Zt!1 Q-almost surely. Consequently, the set A:=f9t08tt0:Zt0g hasQ-measure 1. By absolute continuity, it also has P-measure 1, hence there is a P-almost surely ﬁnite random variable Csuch that for all t,ZtC. There is another argument that we could use to show that under the condition of Theorem 3.51 EQ tEP tis almostsurely ﬁnite: If Pis absolutelycontinuous withrespect toQ, thenQmerges strongly with Pand henceQmerges weakly with P. Therefore almost surely there is a t0such that for all tt0we havejQ(xP tjx<t)P(xP tjx<t)j< ", thusxQ t=xP tfortt0. 3.5.3 Dominance with Coeﬃcients Lemma 3.52 (KL Divergence and Dominance With Coeﬃcients) .IfQdominatesP with coeﬃcients f, then KLt(Q;P)lnf(t). Proof.Analogous to (3.5). This lets us derive an analogous regret bound to Corollary 3.44. Corollary 3.53 (Expected Prediction Regret for Dominance With Coeﬃcients) .IfQ dominatesPwith coeﬃcients f, then EPh EQ nEP ni 2 lnf(t) + 2q 2EPEPnlnf(t): Proof.Apply Lemma 3.52 to Corollary 3.44. For weak dominance we get sublinear prediction regret. Corollary 3.54 (Sublinear Prediction Regret for Weak Dominance) .IfQweakly dom- inatesP, then EP[EQ nEP n]2o(t). §3.6Learning with Algorithmic Information Theory 41 Proof.By Remark 3.9 lnf2o(t). Applying Corollary 3.53 we get EPh EQ nEP n 2o(t) + 2q 2EPEPno(t)2o(t) + 2p 2O(t)o(t)2o(t): 3.6 Learning with Algorithmic Information Theory Algorithmic information theory provides a theoretical framework to apply the probabil- ity theory results from the previous sections. In the following we discuss Solomonoﬀ’s famous theory of induction (Section 3.6.1), the speed prior (Section 3.6.2), and learning with a universal compression algorithm (Section 3.6.3). 3.6.1 Solomonoﬀ Induction Solomonoﬀ (1964, 1978) proposed a theory of learning, also known as universal in- ductionorSolomonoﬀ induction . It encompasses Ockham’s razor by favoring simple explanations over complex ones, and Epicurus’ principle of multiple explanations by never discarding possible explanations. See Rathmanner and Hutter (2011) for a very readable introduction to Solomonoﬀ’s theory and its philosophical motivations and Sterkenburg (2016) for a critique of its optimality. At the core of this theory is Solomonoﬀ’s distribution M, as deﬁned in Example 3.5. SinceMdominates all lower semicomputable semimeasures, we get all the merging and prediction results from Section 3.4 and Section 3.5: when drawing a string from any computable measure P,Marrives at the correct belief for any hypothesis. Corollary 3.55 (Strong Merging for Solomonoﬀ Induction) .Mmerges strongly with every computable measure. Proof.From Proposition 3.16a and Theorem 3.25. Corollary 3.56 (Expected Prediction Regret for Solomonoﬀ Induction) .For all com- putable measures P, EP EM tEP t K(P) ln 4 +q 2EPEP tK(P) ln 16: Proof.From Corollary 3.44 and cP= 2K(P). Remark 3.57 (Converging Fast and Slow) .The convergence of Mto a computable P is fast in the sense of Corollary 3.56: Mcannot make many more prediction errors than Pin expectation. When predicting an inﬁnite computable sequence x1:1, the total number of prediction errors is bounded by jpj2 ln 21:4jpjwherepis a program that generatesx1:1(Example 3.46). The convergence of MtoPis also slow in the sense that M(xtjx<t)!1slower than any computable function since 1M(xtjx<t)2minntK(n)for allt. 3 The bound from Corollary 3.56 is not optimal. Even if we knew the program p generating the sequence x1:1, there might be a shorter program p0that computes x1:1; 42 Learning hence the improved bound EM 1jp0j2 ln 2also holds. Since Kolmogorov complexity is incomputable, we can’t ﬁnd the ‘best’ bound algorithmically. Solomonoﬀ induction may even converge on some incomputable measures. Example 3.58 (MConverges on Some Incomputable Measures) .Letrbe an incom- putable real number. Then the measure P:=Bernoulli (r)is not computable and Mis not absolutely continuous with respect to P: for A:=n x2X1lim t!1ones(x1:t) =ro we haveP(A) = 1butM(A) = 0. SinceMLPwe get from Theorem 3.27 that M does not merge with P. Nevertheless, Mstill succeeds at prediction because it domi- nates Bernoulli (q)for each rational qand the rationals are dense around r. According to Lehrer and Smorodinsky (1996, Lem. 3), this implies that Mweakly dominates P and by Theorem 3.38 Malmost weakly merges to P. 3 The fact that Mdoes not merge strongly with every Bernoulli (r)process is not a failure of Solomonoﬀ’s prior. Ryabko (2010, p. 7) shows that for the class of all Bernoulli measures there is no probability measure that merges strongly with each of them. The deﬁnition of Mhas only one parameter: the choice of the universal Turing machine. The eﬀect of this choice on the function Kcan be uniformly bounded by a constant by the invariance theorem (Li and Vitányi, 2008, Thm. 3.1.1). Hence the choice of the UTM changes the prediction regret bound from Corollary 3.56 only by a constant. This constant can be large, preventing any ﬁnite-time guarantees that are independent of the UTM. However, asymptotically Solomonoﬀ induction succeeds even for terrible choices of the UTM. The Solomonoﬀ normalization MnormofMis deﬁned according to Deﬁnition 2.16. WhileMnormdominatesMaccording to Lemma 2.17 and thus every lower semicom- putable semimeasure, in some respects, Mnormbehaves a little diﬀerently from M. Another way to complete the semimeasure Minto a measure is given in the following example. Example 3.59 (The Measure Mixture; Gács, 1983, p. 74) .Themeasure mixture Mis deﬁned as M(x) := lim n!1X y2XnM(xy): (3.6) It is the same as Mexcept that the contributions by programs that do not produce inﬁnite strings are removed: for any such program p, letkdenote the length of the ﬁnite string generated by p. Then forjxyj> k, the program pdoes not contribute to M(xy), hence it is excluded from M(x). Similarly to M, the measure mixture Mis not a (probability) measure since M()< 1; but in this case normalization (2.2) is just multiplication with the constant 1=M(), leading to the normalized measure mixture Mnorm. 3 §3.6Learning with Algorithmic Information Theory 43 Eventhough Mmergesstronglywithanycomputablemeasure PwithP-probability 1, LattimoreandHutter(2013,2015)showthatgenerallyitdoesnotholdforallMartin- Löf random sequences (which also form a set of P-probability 1). Hutter and Muchnik (2007, Thm. 6) construct non-universal lower semicomputable semimeasures that have this convergence property for all P-Martin-Löf random sequences. For inﬁnite nonran- dom sequences whose bits are selectively predicted by some total recursive function, Lattimore et al. (2011, Thm. 10) show that the normalized Solomonoﬀ measure Mnorm converges to 1on the selected bits. This does not hold for the unnormalized measure M(Lattimore et al., 2011, Thm. 12). 3.6.2 The Speed Prior Solomonoﬀ’s prior Mis incomputable (Theorem 6.3); a computable alternative is the speed prior from Example 3.11. In this section we state merging and prediction results forSKt, aspeedpriorintroducedbyFilanetal.(2016)formallydeﬁnedinExample3.11. It is slightly diﬀerent from the speed prior deﬁned by Schmidhuber (2002), but for the latter no compatibility properties are known for nondeterministic measures. Deﬁnition 3.60 (Estimable in Polynomial Time) .A function f:X!Risestimable in polynomial time iﬀ there is a function g:X!Rcomputable in polynomial time such thatf=g. For a measure Pestimable in polynomial time the speed prior SKtdominatesP with coeﬃcients polynomial in jxjlogP(x)(Filan et al., 2016, Eq. 12). Thus SKt weakly dominates Pand we get the following results. Corollary 3.61 (Almost Weak Merging for SKt).SKtalmost weakly merges with every measure estimable in polynomial time. Proof.From Theorem 3.38 and Filan et al. (2016, Eq. 12) since logPdoes not grow superexponentially P-almost surely. Corollary 3.62 (Expected Prediction Regret for SKt; Filan et al., 2016, Thm. 9) .For all measures Pestimable in polynomial time, EP ESKtnEP n 2O logn+q EPEP1logn : Proof.From Corollary 3.44 and Filan et al. (2016, Eq. 14). 3.6.3 Universal Compression Solomonoﬀ’sdistributioncanbeapproximatedusingastandardcompressionalgorithm, motivated by the similarity M(x)2Km(x), whereKmdenotes monotone Kolmogorov complexity. The function Kmis auniversal compressor , compressing at least as well as any other recursively enumerable program. Gács (1983) shows that the similarity M2Kmis not an equality. However, the diﬀerence between logMandKmis very small: the best known lower bound is due 44 Learning to Day (2011) who shows that Km(x)>logM(x) +O(log logjxj)for inﬁnitely many x2X. Nevertheless, 2Kmdominates every computable measure (Li and Vitányi, 2008, Thm. 4.5.4 and Lem. 4.5.6ii(d); originally proved by Levin, 1973). Hence all the strong results that hold for Solomonoﬀ induction (prediction regret and strong merging) also hold for compression: we apply Theorem 3.25 and Corollary 3.44 to get the following results. See Hutter (2006a) for further discussion on using the universal compressor Kmfor learning. Corollary 3.63 (StrongMergingforUniversalCompression) .The distribution 2Km(x) merges strongly with every computable measure. Corollary 3.64 (Expected Prediction Regret for Universal Compression) .ForQ(x) := 2Km(x)and for all computable measures Pthere is a constant cPsuch that EPh EQ tEP ti cP+q cPEPEP t: This provides a theoretical basis for viewing compression as a general purpose learn- ingalgorithm. Inthisspirit, the Hutter prize isawardedforthecompressionofa100MB excerpt from the English Wikipedia (Hutter, 2006c). Practical compression algorithms (such as the algorithm by Ziv and Lempel (1977) used in gzip) are not universal. Hence they do not dominate every computable distri- bution. As with the speed prior, what matters is the rate at which Yt=Q(x1:t)=P(x1:t) goes to 0, i.e., does the compressor weakly dominate the true distribution in the sense of Deﬁnition 3.8? Veness et al. (2015) successfully apply the Lempel-Ziv compression algorithm as a learning algorithm for reinforcement learning; however, some preprocessing of the data is required. More remotely, Vitányi et al. (2009) use standard compression algorithms to classify mammal genomes, languages, and classical music. 3.7 Summary Ultimately, whether learning succeeds depends on the rate at which the nonnegative P-martingale Q=Pgoes to 0(when drawing from P). IfQ=Pdoes not converge to zero, thenQmerges strongly with Pand thus arrives at correct beliefs about any hypothesis, including tail events. If Q=Pconverges to zero subexponentially, then Qmerges almost weakly with Pand thus asymptotically has incorrect beliefs about the immediate future only a vanishing fraction of the time. Corollary 3.44 bounds the expected prediction regret by the KL-divergence between PandQplus ap EPEP tterm. The KL-divergence is in turn bounded by the rate at whichQ=Pgoes to zero. It is constant if QdominatesPand bounded by lnfifQ dominatesPwith coeﬃcients f. IfQweakly dominates P, then the KL-divergence is sublinear. We also derived bounds on the prediction regret for absolute continuity (Sec- tion 3.5.2). Remarkably, the bounds are only log logtworse than the bound we get from dominance. Moreover, they hold almost surely instead of in expectation. §3.7Summary 45 name symbol deﬁned in property Bayesian mixture Example 3.4 dominates every P2M Solomonoﬀ prior M Example 3.5 dominates every lower semi- computable semimeasure universal compression 2KmEquation 2.1 dominates every computable measure speed prior SKtExample 3.11 weak dominates every mea- sure estimable in polytime Laplace rule LExample 3.12 merges weakly with every Bernoulli process MDL MDLxExample 3.14 merges strongly with every P2M Table 3.1: Examples of learning distributions discussed in this chapter and their properties. compatibility ofPandQmartingale merging prediction regret QP Y tc>0 strong merging 2 lnc+ 2p 2EPEP tlnc QP Y t6!0 strong mergingO log logt+p EPEP tlog logt Yt+1=Yt!1 weak merging o(t) QP=f Y t1=f(t) 2 ln f(t) + 2p 2EPEP tlnf(t) QWP log(Yt+1=Yt)!1 in Cesàro averagealmost weak mergingo(t) QLP Y t>0 O(t) Table 3.2: Summary on properties of learning. The ﬁrst column lists diﬀerent notions of compatibility introduced in Section 3.2; the second column lists properties of the P-martingale Yt:=Q(x1:t)=P(x1:t)from Section 3.3; the third column lists diﬀerent notions of merging discussed in Section 3.4; the fourth column states the bounds on the prediction regret (in expectation and almost surely respectively) from Section 3.5. Figure 3.1 illustrates the origin of the results. 46 Learning strong merging weak merging almost weak mergingQP QP QP=f QWP QLPYtc>0 Yt6!0 Yt+1=Yt!1 log(Yt+1=Yt)!0 in Cesàro average Yt>0f2o(t)Hutter (2009a, Lem. 3i)Blackwell and Dubins (1962) Kalai and Lehrer (1994, Thm. 2), L Lehrer and Smorodinsky (1996, Thm. 4) Lehrer and Smorodinsky (1996, Cor. 7), Land lim inf t!1Yt+1=Yt>0Kalai and Lehrer (1994, Prop. 5) Figure 3.1: Properties of learning and their relationship. We use Yt:= Q(x1:t)=P(x1:t). An arrow between two statements means that one statement implies the other. The transitive property of implications is not made explicit. The source of the result is indicated on the arrow, sometimes together with a side condition. If no source is given, then the relationship is easy and a proof can be found in this chapter. §3.7Summary 47 Next, we showed that thep EPEP tterm is generally unimprovable (Example 3.47). However, it comes only from predicting measures that assign probabilities close to 1=2. If we can bound Paway from 1=2, then thep EPEP tterm disappears (Theorem 3.48 and Theorem 3.51). Table 3.1 lists our learning distributions. The Bayesian mixture is the strongest since it dominates every measure from the given class M(Example 3.4). The minimum description length model MDLxdoes not have this property, yet it still merges strongly with every measure from the class (Example 3.14 and Theorem 3.28). The Laplace rule is only useful for learning i.i.d. measures; it merges weakly with every Bernoulli pro- cess (Example 3.12 and Example 3.32). We also discussed some learning distributions from algorithmic information theory. Solomonoﬀ’s prior is a Bayesian mixture over all lower semicomputable semimeasures (Example 3.5 and Wood et al., 2011). Like the universal compressor it dominates and hence merges strongly with all computable measures. The speed prior dominates all probability measures estimable in polynomial time with polynomial coeﬃcients (Example 3.11), and thus merges weakly with each of them. Table 3.2 summarizes the results from this chapter and Figure 3.1 illustrates their logical relationship and their origin. We conclude this chapter with a paradox from the philosophy of science. Remark 3.65 (The Paradox of Conﬁrmation) .Recall the black raven problem intro- duced in Example 3.1; the hypothesis ‘all ravens are black’ is denoted with H. The paradox of conﬁrmation , also known as Hempel’s paradox (Hempel, 1945), relies on the following three principles. •Nicod’s criterion (Nicod, 1961, p. 67): observing an Fthat is aGincreases our belief in the hypothesis that all Fs areGs. •The equivalence condition : logically equivalent hypotheses are conﬁrmed or dis- conﬁrmed by the same evidence. •The paradoxical conclusion : a green apple conﬁrms H. Theargumentgoesasfollows. Thehypothesis Hislogicallyequivalenttothehypothesis H0that all non-black objects are non-ravens. According to Nicod’s criterion, any non- black non-raven, such as a green apple, conﬁrms H0. But then the equivalence condition entails the paradoxical conclusion. The paradox of conﬁrmation has been discussed extensively in the literature on the philosophy of science (Hempel, 1945; Good, 1960; Mackie, 1963; Good, 1967; Hempel, 1967; Maher, 1999; Vranas, 2004); see Swinburne (1971) for a survey. Support for Nicod’s criterion is not uncommon (Mackie, 1963; Hempel, 1967; Maher, 1999) and no consensus is in sight. A Bayesian reasoner might be tempted to argue that a green apple doesconﬁrm the hypothesisH, but only to a small degree, since there are vastly more non-black objects than ravens (Good, 1960). This leads to the acceptance of the paradoxical conclusion, 48 Learning and this solution to the conﬁrmation paradox is known as the standard Bayesian solu- tion. Vranas (2004) shows that this solution is equivalent to the assertion that blackness is equally probable regardless of whether Hholds:P(blackjH)P(black ). The following is a very concise example against the standard Bayesian solution by Good (1967): There are two possible worlds, the ﬁrst has 100 black ravens and a million other birds, while the second has 1000 black ravens, one white raven, and a million other birds. Now we draw a bird uniformly at random, and it turns out to be a black raven. Contrary to what Nicod’s criterion claims, this is strong evidence that we are in fact in the second world, and in this world non-black ravens exist. For another, more intuitive example: Suppose you do not know anything about ravens and you have a friend who collects atypical objects. If you see a black raven in her collection, surely this would not increase your belief in the hypothesis that all ravens are black. In Leike and Hutter (2015d) we investigate the paradox of conﬁrmation in the con- text of Solomonoﬀ induction. We show that the paradoxical conclusion is avoided because Solomonoﬀ induction violates Nicod’s criterion: There are time steps when (counterfactually) observing a black raven disconﬁrms the hypothesis that all ravens are black. When predicting a deterministic computable sequence Nicod’s criterion is even violated inﬁnitely often. However, if we normalize Solomonoﬀ’s prior and ob- serve a deterministic computable inﬁnite string, Nicod’s criterion is violated at most ﬁnitely many times. These results are independent of the choice of the universal Turing machine. We must conclude that violating Nicod’s criterion is not a fault of Solomonoﬀ in- duction. Instead, we should accept that for Bayesian reasoning Nicod’s criterion, in its generality, is false! Quoting the great Bayesian master Jaynes (2003, p. 144): In the literature there are perhaps 100 ‘paradoxes’ and controversies which are like this, in that they arise from faulty intuition rather than faulty mathematics. Someone asserts a general principle that seems to him intu- itively right. Then, when probability analysis reveals the error, instead of taking this opportunity to educate his intuition, he reacts by rejecting the probability analysis. 3 Chapter 4 Acting I ought never to act except in such a way that I could also will that my maxim should become a universal prior. — Immanuel Kant Recall our decomposition of intelligence into learning andactingfrom Equation 1.1. The previous chapter made the notion of learning precise and provided several examples of learning distributions for the non-i.i.d. setting (see Table 3.1). Learning is passive: there is no interaction with the data-generating process. In this chapter we transition into the active setting: we consider an agentacting in an unknown environment in order to achieve a goal. In our case, this goal is maximizing reward; this is known as reinforcement learning . Where this reward signal originates does not concern us here. In this thesis we consider is the general reinforcement learning problem in which we do not make several of the typical simplifying assumptions (see Table 1.1). Environ- mentsareonlypartiallyobservable, haveinﬁnitelymanystates, andmightcontaintraps from which the agent cannot escape. The context for making decision is the agent’s entire history; its behavior is given by a policythat speciﬁes how the agent behaves in any possible situation. A central quantity in reinforcement learning is the value function . The value func- tionquantiﬁestheexpectedfuturediscountedreward. Sincetheagentseekstomaximize reward, it aims to adopt a policythat has high value. Since the agent’s environment is unknown to the agent, learning the value function is part of the challenge; otherwise we would call this planning. If our agent is capable of learning in the sense of Chapter 3, then it learns the value of its own policy ( on-policy value convergence ). However, generally the agent does not learn to predict the value of counterfactual actions, actions that it does not take. Learning oﬀ-policy is hard because the agent receives no evidence about what would have happened on counterfactual actions. Nevertheless, oﬀ-policy learning is highly desirable because we want the agent to be conﬁdent that the policy it is currently following is in fact the best one; we want it to accurately predict that the counterfactual actions have less value. This brings us back to the central theme of reinforcement learning: the tradeoﬀ between exploration andexploitation . Asymptotically the agent needs to focus on exploitation, i.e., take the actions that it thinks yield the highest expected rewards. If theagentexploresenough,thenallactionsareon-policybecausetheyareallactionsthat 49 50 Acting agent environment at et= (ot;rt) Figure 4.1: The dualistic agent model. At every time step t, the agent outputs an actionatand subsequently receives a percept etconsisting of an observation otand a real-valued reward rt. The agent’s policy is a function that maps a history æ<tto the next action at, and the environment is a function that maps a history and an action to the next percept et. the agent sometimes takes. Then on-policy learning ensures that the agent understands the consequences of every action and can conﬁdently choose the best action. Eﬀective exploration is performed by knowledge-seeking agents ; these agents ignore the rewards and just focus on exploration. This chapter introduces the central concepts of general reinforcement learning. It is mostly based on Hutter (2005) and Lattimore (2013). Section 4.1 speciﬁes the gen- eral reinforcement learning problem, discusses discounting (Section 4.1.1), our implicit assumptions (Section 4.1.2), and typical environment classes (Section 4.1.3). Sec- tion 4.2 discusses the value function and its properties. In Section 4.3 we introduce the agents: AIXI (Section 4.3.1), knowledge-seeking agents (Section 4.3.2), BayesExp (Sec- tion 4.3.3), and Thompson sampling (Section 4.3.4). 4.1 The General Reinforcement Learning Problem In reinforcement learning, an agent interacts with an environment: at time step t2N the agent takes an actionat2Aand subsequently receives a perceptet= (ot;rt)2E consisting of an observation ot2Oand arewardrt2R. This cycle then repeats for time stept+ 1(see Figure 4.1). Ahistoryis an element of (AE )and lists the actions the agent took and the percepts it received. We use æ2AE to denote one interaction cycle, and æ<t= æ1æ2:::æt1to denote a history of length t1. For our agent, the history is a suﬃcient statistic about the past and in general reinforcement learning there is no simpler suﬃcient statistic. For example, consider the agent to be a robot interacting with the real world. Its actions are moving the motors in its limbs and wheels and sending data packets over a network connection. Its observations are data from cameras and various other sensors. Therewardcouldbeprovidedeitherbyahumansupervisororthroughareward module that checks whether a predeﬁned goal has been reached. The history is the collection of all the data it received and emitted in the past. The division of the robot’s interaction with the environment into discrete time steps might seem a bit unnatural at ﬁrst since the real world evolves according to a continuous process. However, note that the electronic components used in robots operate at discrete frequencies anyway. §4.1The General Reinforcement Learning Problem 51 In order to specify how the agent behaves in any possible situation, we deﬁne a policy: apolicyisafunction : (AE )!Amappingahistory æ<ttoadistribution over actions (jæ<t)taken after seeing this history. Usually we do not distinguish between agent and policy. An environment is a function : (AE )A! E mapping a history æ<tand an action atto a distribution (jæ<tat)over the percepts received after the history æ<tand actionat. We useto denote the true environment. Equivalently, Hutter (2005) deﬁnes environments as chronological contextual semi- measures .1Acontextual semimeasure takes a sequence of actions a1:1as input and returns a semimeasure (ka1:1)overE]. A contextual semimeasure ischronological iﬀperceptsattime tdonotdependonfutureactions, i.e., (e1:tka1:1) =(e1:tka0 1:1) whenevera1:t=a0 1:t. For chronological contextual semimeasures we write (e1:tka1:t) instead of(e1:tka1:1). The two deﬁnition can be translated using the identities (e1:tka1:t) =tY k=1(ekjæ<kak)and(etjæ<tat) =(e1:tka1:t) (e<tka<t):(4.1) If the policy always assigns probability 1to one of the actions, then is called deterministic . Likewise, if the environment always assigns probability 1to one of the percepts, then is called deterministic . For deterministic policies and environments we also use the notation at=(æ<t)andet=(æ<tat). A deterministic policy is consistent with history æ<tiﬀak=(æ<k)for allk < t. Likewise, a deterministic environment isconsistent with history æ<tiﬀek=(æ<kak)for allk<t. Deﬁnition 4.1 (History Distribution) .An environment together with a policy induces a history distribution (æ<t) :=tY k=1(akjæ<k)(ekjæ<kak): We denote an expectation with respect to the history distribution withE . Thehistorydistributionisa(semi)measureon (AE )1. Inthelanguageofmeasure theory, our -algebra is the -algebraF1generated by the cylinder sets introduced in Section 2.1. The ﬁltration (Ft)t2Nformalizes that at time step twe have seen exactly the history æ<t(we use the -algebraFt1). To simplify notation and help intuition, wesimplyconditionexpectationsandprobabilitymeasureswiththehistory æ<tinstead ofFt1and sweep most of the measure-theoretic details under the rug. With these preliminaries out of the way, we can now specify the general reinforce- ment learning problem . Problem 4.2 (General Reinforcement Learning Problem) .Given an arbitrary class of environmentsM, choose a policy that maximizes -expected reward when interacting with any environment 2M. 1Hutter (2005) calls them chronological conditional semimeasures . This is confusing because contex- tual semimeasures do notspecify conditional probabilities; the environment is nota joint probability distribution over actions and percepts. 52 Acting Problem 4.2 is kept vague on purpose: it does not say how we should balance between achieving more rewards in some environments while achieving less in others. In other words, we leave open what an optimalsolution to the general reinforcement learning problem is. This turns out to be a notoriously diﬃcult question that we discuss in Chapter 5. As promised in the title of this thesis, we take the nonparametric approach. For the rest of this thesis, ﬁx Mto be any countable set of environments. While the true envi- ronment is unknown, we assume it belongs to the class M(therealizable case ). As long as the classMis suﬃciently large (such as the class of all computable environments), this assumption is weak. Some typical choices are discussed in Section 4.1.3. Our agent-environment setup shown in Figure 4.1 is known as the dualistic model : the agent is distinct from the environment and inﬂuences it only through its actions. In turn, the environment inﬂuences the agent only through the percepts. The dualism as- sumption is accurate for an algorithm that is playing chess, Go, or other (video) games, which explains why it is ubiquitous in AI research. But often it is not true: real-world agents are embedded in (and computed by) the environment, and then a physicalistic model(also called materialistic model ornaturalistic model) is more appropriate. Deci- sion making in the physicalistic model is still underdeveloped; see Everitt et al. (2015) and Orseau and Ring (2012a). In this thesis we restrict ourselves to the dualistic model. 4.1.1 Discounting The goal in reinforcement learning is to maximize rewards. However, the inﬁnite reward sumP1 t=1rtmay diverge. To get around this technical problem, we let our agent prior- itize the present over the future. This is done with a discount function that quantiﬁes how much the agent prefers rewards now over rewards later. Deﬁnition 4.3 (Discount Function) .Adiscount function is a function :N!Rwith t:= (t)0andP1 t=1 t<1. Thediscount normalization factor ist:=P1 k=t k. There is no requirement that t>0. In fact, we use for both, discounted inﬁnite horizon ( t>0for allt), and ﬁnite horizon m(m1>0andm= 0) where the agent does not care what happens after time step m. Note that the way in which we employ discounting is time consistent : the agent does not change its mind about how much it values the reward at time step kover time: reward rkis always discounted with kregardless of the current time step. For a discussion of general discounting we refer the reader to Lattimore and Hutter (2014). Deﬁnition 4.4 (Eﬀective Horizon) .The"-eﬀective horizon Ht(")is a horizon that is long enough to encompass all but an "of the discount function’s mass: Ht(") := minfkjt+k=t"g The eﬀective horizon is bounded iﬀ for all" > 0there is a constant c"such that Ht(")c"for allt2N. §4.1The General Reinforcement Learning Problem 53 parameter t t Ht(") Finite horizon m2N 1m(t)=m (mt+ 1)=md(mt+ 1)(1")e Geometric 2(0;1) t t=(1 )dlog "e Power >1tt+1=(1)("1=(1)1)t Subgeometric - ep t=p t2ep tp tlog"+ (log")2 Table 4.1: Several discount functions and their eﬀective horizons. See also Hutter (2005, Tab. 5.41) and Lattimore (2013, Tab. 2.1). Example 4.5 (Geometric Discounting) .The most common discount function is ge- ometric discounting with t:= tfor some constant 2[0;1). We get that t=P1 k=t k= t=(1 )and the"-eﬀective horizon is Ht(") =dlog "e. Hence the eﬀec- tive horizon is bounded. 3 More examples for discount functions are given in Table 4.1. From now on, we ﬁx a discount function . 4.1.2 Implicit Assumptions Throughout this thesis, we make the following assumptions implicitly. Assumption 4.6. (a) The discount function is computable. (b) Rewards are bounded between 0and1. (c) The set of actions Aand the set of percepts Eare both ﬁnite. Let’smotivatetheseassumptionsinturn. Theirpurposeistoensurethatdiscounted reward sums are ﬁnite and optimal policies exist. Assumption4.6aisatechnicalassumptionthatensuresthatdiscountedrewardsums are computable. This is important for Chapter 6 and Chapter 7 where we analyse the computability of optimal policies. Note that all discount functions given in Table 4.1 are computable. Assumption 4.6b could be relaxed to require only that rewards are bounded. We can rescale rewards rt7!crt+dfor anyc2R+andd2Rwithout changing optimal policies if the environment is a probability measure. (For our computability-related results in Chapter 6, we must assume that rewards are nonnegative.) In this sense As- sumption 4.6b is not very restrictive. However, this normalization of rewards into the [0;1]-interval has the convenient consequence that the normalized discounted reward sumP1 k=t krk=kis bounded between 0and1. If rewards are unbounded, then the discounted reward sum might diverge. Moreover, with unbounded rewards there all kinds of pathological problems where deﬁning optimal actions is no longer straightfor- ward; see Arntzenius et al. (2004) for a discussion. Assumption 4.6c is a technical requirement for the existence of optimal policies since it implies that there are only ﬁnitely many deterministic policies that diﬀer in the ﬁrst t 54 Acting time steps. Note that ﬁnite action and percept spaces are very natural since it ensures that our agent only receives and emits a ﬁnite amount of information in every time step. This is in line with the problems a strong AI is facing: the agent has to remember important information and act sequentially. Assumption 4.6b, Assumption 4.6c, and the fact that the discount function is summable guarantee that a deterministic optimal policy exists for every environment according to Lattimore and Hutter (2014, Thm. 10). It would be interesting to relax theseassumptionswhilepreservingtheexistenceofoptimalpoliciesoratleast "-optimal policies (e.g. use compact action and percept spaces). 4.1.3 Typical Environment Classes The simplest reinforcement learning problems are multi-armed bandits. Deﬁnition 4.7 (Multi-Armed Bandit) .An environment is amulti-armed bandit iﬀ O=f?gand(etjæ<tat) =(etjat)for all histories æ1:t2(AE ). In a multi-armed bandit problem there are no observations and the next reward only depends on the previous action. Intuitively, we are deciding between #Adiﬀerent slot machines (so-called one-armed bandits), pull the lever and obtain a reward. The reward is stochastic, but it is drawn from a distribution that is time-invariant and ﬁxed for each arm. A multi-armed bandit is also called banditfor short. Although bandits are the sim- plest reinforcement learning problem, they already exhibit the exploration-exploitation- tradeoﬀ that makes reinforcement learning diﬃcult: do you pull an arm that has the best empirical mean or do you pull an arm that has the highest uncertainty? In bandits it is very easy to come up with policies that perform (close to) optimal asymptoti- cally (e.g.,"t-greedy with "t= 1=t). But coming up with algorithms that perform well in practice is diﬃcult, and research focuses on the multiplicative and additive constants on the asymptotic guarantees. Bandits exist in many ﬂavors; see Bubeck and Bianchi (2012) for a survey. Deﬁnition 4.8 (Markov Decision Process) .An environment is aMarkov decision process(MDP) iﬀ(etjæ<tat) =(etjot1at)for all histories æ1:t2(AE ). Intuitively, in MDPs, the previous observation ot1provides a suﬃcient statistic for the history: given ot1and the current action at, the next percept etis independent of the rest of the history. In other words, everything that the agent needs to know to make optimal decisions is readily available in the previous percept. This is why observations are called statesin MDPs. Note that bandits are MDPs with a single state. Much of today’s literature on reinforcement learning focuses on MDPs (Sutton and Barto, 1998). They provide a particularly good framework to study reinforcement learning because they are simple enough to be tractable for today’s algorithms, yet general enough to encompass many interesting problems. For example, most of the Atari games (see Figure 1.1 for an overview) are (deterministic) MDPs when combining §4.1The General Reinforcement Learning Problem 55 the previous four frames into one percept. While they have a huge state space2they can still be learned using Q-learning with function approximation (Mnih et al., 2015). The MDP framework is restrictive because it requires the agent to be more powerful than the environment. Since the agent learns, its actions are not independent of the rest of the history given the last action and percept. In other words, learning agents are not Markov. The following deﬁnition lifts this restriction and allows the environment to bepartially observable . Deﬁnition 4.9 (Partially Observable Markov Decision Process) .An environment is apartially observable Markov decision process (POMDP) iﬀ there is a set of statesS, aninitial state s02S, astate transition function 0:SA! S, and apercept distribution 00:S! Esuch that (e1:tka1:t) =tY k=100(ekjsk)0(skjsk1;ak): Usually the setSis assumed to be ﬁnite; with inﬁnite-state POMDPs we can model any environment by setting the set of states to be the set of histories, S:= (AE ). A common assumption for MDPs and POMDPs is that they do not contain traps. Formally, a (PO)MDP is ergodiciﬀ for any policy and any two states s1;s22S, the expected number of time steps to reach s2froms1is-almost surely ﬁnite. A (PO)MDP is weakly communicating iﬀ for any two states s1;s22Sthere is a policy such that the expected number of time steps to reach s2froms1is-almost surely ﬁnite. Note that any ergodic (PO)MDP is also weakly communicating, but not vice versa. Ingeneral, ourenvironmentsarestochastic. Stochasticitycanoriginatefromnoisein the environment, noise in the sensors, or modeling errors. Sometimes we also consider classes of deterministic environments. These are usually easier to deal with because they do not require as much mathematical machinery. For example, in a deterministic environment the next percept is certain; if a diﬀerent percept is received this environ- ment is immediately falsiﬁed and can be discarded. In a stochastic environment, an unlikely percept reduces our posterior belief in this environment but does not rule it out completely. In Chapter 6 and Chapter 7 we make the assumption that the environment is com- putable. This encompasses all ﬁnite-state POMDPs and most if not all AI problems can be formulated in this setting. Moreover, the current theories of quantum mechanics and general relativity are computable and there is no evidence that suggests that our physical universe is incomputable. For any physical system of ﬁnite volume and ﬁnite (average) energy, the amount of information it can contain is ﬁnite (Bekenstein, 1981), and so is the number of state transitions per unit of time (Margolus and Levitin, 1998). This gives us reason to believe that even the environment that we humans currently face (and will ever face) falls under these assumptions. 2The size of the state space is at most 256128since the Atari 2600 has only 128 bytes of memory. However, the vast majority of these states are not reachable. 56 Acting Formally we deﬁne the set MCCS LSCas the set of environments that are lower semicom- putable chronological contextual semimeasures and MCCM compas the set of environments that are computable chronological contextual measures. Note that for chronological contextual semimeasures it makes a diﬀerence whether (ka1:1)is lower semicom- putable or the conditionals (jæ<tat)are. The latter implies the former, but not vice versa. 4.2 The Value Function Thevalueof a policy in an environment is the future expected discounted reward when following a given policy in a given environment conditional on the past. Since this quantity captures exactly what our agent aims to maximize, we prefer policies whose value is high. Deﬁnition 4.10 (Value Function) .Thevalueof a policyin an environment given historyæ<tandhorizonmwithtm1is deﬁned as V;m (æ<t) :=1 tE "m1X k=t krkæ<t# ift>0andV;m (æ<t) := 0ift= 0. Theoptimal value is deﬁned as V;m (æ<t) := supV;m (æ<t). Sometimes we omit the history argument æ<tfor notational convenience if it is clear from context. Moreover, when we omit m, we implicitly use an inﬁnite horizon m=1, i.e.,V :=V;1 andV :=V;1 . The value of a policy in an environment after the empty history, V ()is also called the t0-value. Remark 4.11 (Values are Bounded Between 0 and 1) .From Assumption 4.6b we get that for all histories æ<tall policies and all environments , the value function V (æ<t)2[0;1]. 3 Since environment and policy are stochastic, the history æ<tis random. With abuse of notation we treat æ<tsometimes as a concrete outcome and sometimes as a random variable. We also view the value of a policy in an environment as a sequence of random variables (Xt)t2NwithXt:=V (æ1:t)where the history æ1:tis generated stochastically by the agent’s actual policy interacting with the true environment . This view is helpful for some of the convergence results (e.g., Theorem 4.19 and Deﬁ- nition 5.18) in which we talk about the type of convergence of this sequence of random variables. The value function deﬁned in Deﬁnition 4.10 is also called the recursive value func- tion, in contrast to iterative value function that we discuss in Section 6.4. The name of the recursive value function originates from the following recursive identity (analogously §4.2The Value Function 57 to Hutter, 2005, Eq. 4.12), also called the Bellman equation : V (æ<t) =X at2A(atjæ<t)V (æ<tat) V (æ<tat) =1 tX et2E(etjæ<tat) trt+ t+1V (æ1:t) An explicit expression for the optimal value in environment is V;m (æ<t) =1 tmaxX æt:m1m1X k=t krkkY i=t(eijæ<iai); (4.2) wherePmaxdenotes the max-sum-operator: maxX æt:m1:= max at2AX et2E:::max am12AX em12E For an explicit expression of V;1 (æ<t)we can simply take the limit m!1. 4.2.1 Optimal Policies An optimal policy is a policy that achieves the highest value: Deﬁnition4.12 (OptimalPolicy; Hutter,2005, Def.5.19&5.30) .Apolicyisoptimal in environment (-optimal) iﬀforallhistories attainstheoptimalvalue: V (æ<t) = V (æ<t)for allæ<t2(AE ). The action atis anoptimal action iﬀ (atjæ<t) = 1 for some-optimal policy . Following the tradition of Hutter (2005), AINUdenotes a-optimal policy for the environment 2MCCS LSCandAIMUdenotes an -optimal policy for the environment 2MCCM compthat is a measure (as opposed to a semimeasure). By deﬁnition of the optimal policy and the optimal value function, we have the following identity for all histories æ<t: V (æ<t) =V (æ<t) (4.3) There can be more than one optimal policy; generally the choice of from Deﬁni- tion 4.12 is not unique. More speciﬁcally, for a -optimal policy we have (atjæ<t)>0 =)at2arg max a2AV (æ<ta): (4.4) If there are multiple actions ;2 Athat attain the optimal value, V (æ<) = V (æ<t), then there is an argmax tie . Which action we settle on in case of a tie (how we break the tie) is irrelevant and can be arbitrary. Since we allow stochastic policies, we can also randomize between and. The following deﬁnition allows policies to be slightly suboptimal. 58 Acting Deﬁnition 4.13 ("-Optimal Policy) .A policyis"-optimal in environment iﬀ V (æ<t)V (æ<t)<"for all histories æ<t2(AE ). A policythat achieves optimal t0-value,V () =V (), takes-optimal actions on any history reachable by in. However, this is not true for "-optimal policies: a policy that is "-optimal at t= 0is not necessarily "-optimal in later time steps. 4.2.2 Properties of the Value Function The following two lemmas are stated by Hutter (2005, Thm. 31) without proof and for the iterative value function. Lemma4.14 (Linearityof V in).If=z11+z22for some real numbers z1;z20, then for all policies and all histories æ<t V;m (æ<t) =z11(e<tka<t) (e<tka<t)V;m 1(æ<t) +z22(e<tka<t) (e<tka<t)V;m 2(æ<t): Proof.Since=z1 1+z2 2, we have for the conditional measure (Ajæ<t) =(A\æ<t) (æ<t)=z1 1(A\æ<t) +z2 2(A\æ<t) (æ<t) =z1 1(æ<t) (æ<t) 1(Ajæ<t) +z2 2(æ<t) (æ<t) 2(Ajæ<t): The claim now follows from the linearity of expectation in the probability measure. Lemma 4.15 (Convexity of V in).If=z11+z22for some real numbers z1;z2 0, then for all histories æ<t V (æ<t)z11(e<tka<t) (e<tka<t)V;m 1(æ<t) +z22(e<tka<t) (e<tka<t)V;m 2(æ<t): Proof.Let be an optimal policy for environment . From Lemma 4.14 we get V (æ<t) =V (æ<t) =z11(e<tka<t) (e<tka<t)V 1(æ<t) +z22(e<tka<t) (e<tka<t)V 2(æ<t) z11(e<tka<t) (e<tka<t)V 1(æ<t) +z22(e<tka<t) (e<tka<t)V 2(æ<t): The following lemma bounds the error when truncating the value function. This implies that planning for an "-eﬀective horizon ( m=t+Ht(")), we get all but an "of the value:jV (æ<t)V;m (æ<t)j<". Lemma 4.16 (Truncated Values) .For every environment , every policy , and every historyæ<tV;m (æ<t)V (æ<t)m t: §4.2The Value Function 59 Proof. V (æ<t) =1 tE "1X k=t krkæ<t# =V;m (æ<t) +1 tE "1X k=m krkæ<t# The result now follows from Assumption 4.6b and 0E "1X k=m krkæ<t# m: This lemma bounds the (truncated) value function by the total variation distance. Lemma 4.17 (Bounds on Value Diﬀerence) .For any policies 1,2, any environments 1and2, and any horizon tm1 jV1;m 1(æ<t)V2;m 2(æ<t)j Dm1(1 1;2 2jæ<t) Proof.According to Deﬁnition 4.10, the value function is the expectation of the ran- dom variablePm1 k=t krk=tthat is bounded between 0and1. Therefore we can use Lemma 2.12 with P:=1 1(jæ<t)andR:=2 2(jæ<t)on the space (AE )m1to conclude thatjV1;m 1(æ<t)V2;m 2(æ<t)jis bounded by Dm1(1 1;2 2jæ<t). Lemma 4.18 (Discounted Values; Lattimore, 2013, Lem. 2.5) .Letæ<tbe some history and let1and2be two policies that coincide from time step tto time step m:1(aj æ1:k) =2(ajæ1:k)for alla2A, all histories æ<tæt:kconsistent with 1, andtk m. Then for all environments V1(æ<t)V2(æ<t)m t: Proof.Since1and2coincidefortimesteps tthroughm1,Dm1(1;2jæ<t) = 0 for all environments . Thus the result follows from Lemma 4.16 and Lemma 4.17: V1(æ<t)V2(æ<t)V1;m (æ<t)V2;m (æ<t)+m t Dm1(1;2jæ<t) +m t =m t 4.2.3 On-Policy Value Convergence This section states some general results on learning the value function. On-policy value convergence refers to the fact that if we use a learning distribution to learn to environ- ment, andmerges with in the sense discussed Section 3.4, then V converges toV , i.e., using we learn to estimate values correctly. A weaker variant of the following theorem was proved by Hutter (2005, Thm. 5.36). It states convergence in mean (not almost surely), and only for the Bayesian mixture. 60 Acting Theorem 4.19 (On-Policy Value Convergence) .Letbe any environment and be any policy. (a) Ifmerges strongly with , then V (æ<t)V (æ<t)!0ast!1-almost surely. (b) If the eﬀective horizon is bounded and merges weakly with , then V (æ<t)V (æ<t)!0ast!1-almost surely. (c) If the eﬀective horizon is bounded and merges almost weakly with , then 1 ttX k=1 V (æ<k)V (æ<k) !0ast!1in-almost surely. Proof.(a) Apply Lemma 4.17 with m:=1. (b) Let">0and letc"be a bound on suptHt("). From Lemma 4.16 jV (æ<t)V (æ<t)jjV;t+Ht(") (æ<t)V;t+Ht(") (æ<t)j+ 2t+Ht(") t <Dt+Ht1(")(;jæ<t) + 2" Dt+c"(;jæ<t) + 2" accordingtoDeﬁnition4.4andLemma4.17. Since mergesweaklywith ,weget that-almost surely there is a time step t02Nsuch thatDt+c"(;jæ<t)<" for alltt0. HencejV Q(æ<t)V (æ<t)j<3"for alltt0. (c) Analogously to the proof of (b). It is important to observe that on-policy convergence does not imply that the agent converges to the optimal policy. V converges to V , butV need not be close to V . Indeed, there might be another policy ~that has a higher value than in the true environment (V~ >V ). If the agent thinks ~has lower value ( V~ <V ) it might not follow ~and hence not learn that the actual value of ~is much higher. In other words, on-policy convergence implies that the agent learns the value of its own actions, but not the value of counterfactual actions that it does not take. Theorem 4.19 now enables us to tie in the results of Chapter 3. This yields a surge of corollaries, but ﬁrst we need to make the learning distributions contextual on the actions. Letw2Mbe a positive prior over the environment class M. We deﬁne the corresponding Bayesian mixture analogously to Example 3.4: (e<tka<t) :=X 2Mw()(e<tka<t) (4.5) §4.2The Value Function 61 Note that the Bayesian mixture depends on the prior w. For the rest of this thesis, this dependence will not be made explicit. From Lemma 4.14 and (3.3) we immediately get the following identity: V (æ<t) =X 2Mw(jæ<t)V (æ<t) (4.6) Similarly, we get from Lemma 4.15 V (æ<t)X 2Mw(jæ<t)V (æ<t): (4.7) Corollary 4.20 (On-Policy Value Convergence for Bayes) .For any environment 2 Mand any policy , V (æ<t)V (æ<t)!0ast!1-almost surely. Proof.Since2M, we have dominance w()withw()>0and by Propo- sition 3.16a absolute continuity . From Theorem 3.25 we get that merges strongly with . Therefore we can apply Theorem 4.19a. Analogously, we deﬁne MDLæ<t:= arg min2Mflog(e<tka<t) +K()g. Corollary4.21 (On-PolicyValueConvergenceforMDL) .For any environment 2M and any policy , V MDLæ<t(æ<t)V (æ<t)!0ast!1-almost surely. Proof.By Theorem 3.28 MDLmerges strongly with for each2M, therefore we can apply Theorem 4.19a. By providing the action sequence contextually on a separate input tape, we can deﬁneKm(e<tka<t) := minfjpjje<tvU(p;a<t)ganalogously to (2.1). Corollary4.22 (On-PolicyValueConvergenceforUniversalCompression) .Let(e<tk a<t) := 2Km(e<tka<t). Then for any environment 2MCCM compand any policy , V (æ<t)V (æ<t)!0ast!1-almost surely. Proof.Sincedominates every 2 MCCM comp(Section 3.6.3) we can apply Proposi- tion 3.16a, Theorem 3.25, and Theorem 4.19a as in the proof of Corollary 4.20. Similarly to Kmthere is a speed prior for environments (Filan, 2015, Ch. 6): SKt(e<tka<t) :=X p:e<tvU(p;a<t)2jpj t(U;p;a<t;e<t) wheret(U;p;a<t;e<t)denotes the number of time steps U(p;a<t)takes to produce e<t. 62 Acting Corollary 4.23 (On-Policy Value Convergence for the Speed Prior) .If the eﬀective horizon is bounded, then for any environment 2MCCM compestimable in polynomial time and any policy , 1 ttX k=1 V SKt(æ<k)V (æ<k) !0ast!1-almost surely. Proof.By Corollary 3.61 the speed prior SKtmerges almost weakly with every measure estimable in polynomial time. Therefore we can apply Theorem 4.19c. 4.3 The Agents If we knew the true environment , we would choose the -optimal policy, the policy that maximizes -expected discounted rewards. But generally we do not know the true environment, and the challenging part of reinforcement learning is to learn the environment while trying to collect rewards. In this section we introduce a number of agents that attempt to solve the general reinforcement learning problem (Problem 4.2). These agents are discussed throughout the rest of this thesis. 4.3.1 Bayes ABayes optimal policy with respect to the prior wis the policy whereis the Bayesian mixture deﬁned in Section 4.2.3. There can be one or more Bayes optial policies. From Corollary 4.20 we get on-policy value convergence for the Bayes optimal policy. After history æ<t, the Bayes policy maximizes expected discounted rewards in the posterior mixture: (je<tka1:1) =X 2Mw(jæ<t)(je<tka1:1) wherew(jæ<t)are the posterior weights (3.3). Maximizing expected rewards accord- ing to the posterior is the same as maximizing expected rewards according to the prior conditional on the history: if (æ<t) = (æ<t), thenV (æ<t) =V (æ<t). Actually visiting the history æ<tdoes not change what planned to do before it visited æ<t. Note that this relies on the fact that the way we use discounting is time consistent (Lat- timore and Hutter, 2014, Def. 12). When using the prior w()/2K()(Example 3.5) over the class MCCS LSC, the Bayes optimal policy is also known as AIXI, introduced and analyzed by Hutter (2000, 2001a, 2002a, 2003, 2005, 2007a, 2012b) in his work on universal artiﬁcial intelligence . In this case, the Bayesian mixture (4.5) can be deﬁned equivalently according to (Wood et al., 2011) (e<tka<t) :=X p:e<tvU(p;a<t)2jpj: (4.8) §4.3The Agents 63 Generally there is more than one -optimal policy and Solomonoﬀ’s prior depends on the choice of the (reference) universal Turing machine, so this deﬁnition is not unique. Moreover, not every universal Turing machine is a good choice for AIXI, see Section 5.2 for a few bad choices. The following lemma will be used later. Lemma 4.24 (Mixing Mixtures) .Letq;q02Qsuch thatq >0,q00, andq+q0 1. Letwbe any lower semicomputable positive prior, let be the Bayesian mixture corresponding to w, and let2MCCS LSC. Then0:=q+q02MCCS LSCis a Bayesian mixture. Proof.0is given by the positive prior w0withw0:=qw+q01. Bayesian approaches have a long tradition in reinforcement learning, although they are often prohibitively expensive to compute. For multi-armed bandits, Gittins (1979) achieved a breakthrough with an index strategy that enables the computation of the optimal policy by computing one quantity for each arm independently of the rest. This strategy even achieves the optimal asymptotic regret bounds (Lattimore, 2016). Larger classes have also been attempted: using Monte-Carlo tree search, Veness et al. (2011) approximate the Bayes optimal policy in the class of all context trees. Doshi-Velez (2012) uses Bayesian techniques to learn inﬁnite-state POMDPs. See Vlassis et al. (2012) for a survey on Bayesian techniques in RL. In the rest of this thesis, the Bayes optimal policy is often treated as an optimal ex- ploitation strategy. This is not true: Bayes does explore (when it is Bayes optimal to do so). It just does not explore general environment classes completely (see Section 5.4.1). 4.3.2 Knowledge-Seeking Agents In this section we discuss two variants of knowledge-seeking agents: entropy-seeking agents introduced by Orseau (2011, 2014a) and information-seeking agents introduced by Orseau et al. (2013). The entropy-seeking agent maximizes the Shannon entropy gain, while the information-seeking agent maximizes the expected information gain. These quantities are expressed in diﬀerent value functions. In places where confusion can arise, we call the value function V from Deﬁnition 4.10 the reward-seeking value function. In this section we use a ﬁnite horizon m<1(possibly dependent on time step t): the knowledge-seeking agent maximizes entropy/information received up to time step m. We assume implicitly that m(as a function of t) is computable. Moreover, in this section we assume that the Bayesian mixture is a measure rather than a semimeasure; Example 4.27 discusses this assumption. Deﬁnition4.25 (Entropy-SeekingValueFunction; Orseau,2014a,Sec.6) .Theentropy- seeking value of a policy given history æ<tis V;m Ent(æ<t) :=E [log2(e1:mje<tka1:m)jæ<t]: 64 Acting The entropy-seeking value is the Bayes-expectation of log. Orseau (2011, 2014a) also considers a related value function based on the -expectation of that we do not discuss here. Deﬁnition 4.26 (Information-Seeking Value Function; Orseau et al., 2013, Def. 1) . Theinformation-seeking value of a policy given history æ<tis V;m IG(æ<t) :=X 2Mw(jæ<t)KLm(;jæ<t): Analogously to before we deﬁne V Ent:= supV EntandV IG:= supV IG. An optimal entropy-seeking policy is deﬁned as Ent:2arg maxV Entand an optimal information- seeking policy is deﬁned as IG:2arg maxV IG. Since we use a ﬁnite horizon ( m<1), these optimal policies exist. Theinformation gain is deﬁned as the diﬀerence in entropy between the prior and the posterior: IGt:m(æ1:m) := Ent(w(jæ<t))Ent(w(jæ1:m)) We get the following identity (Lattimore, 2013, Eq. 3.5). E [IGt:m(æ1:m)jæ<t] =V;m IG(æ<t) For inﬁnite horizons ( m=1), the values functions from Deﬁnition 4.25 and Def- inition 4.26 may not converge. To ensure convergence, we can either use discounting, or in case of VIGa prior with ﬁnite entropy (Lattimore, 2013, Thm. 3.4). Moreover, note that while VIGandVEntare expectations with respect to the measure , there is no bound on the one-step change in value V;m IG(æ<t)V;m IG(æ1:t), which can also be negative. For the reward-seeking value function V;m , the one-step change in value is bounded between 0and1by Remark 4.11. For classes of deterministic environments Deﬁnition 4.25 and Deﬁnition 4.26 coin- cide. In stochastic environments the entropy-seeking agent does not work well because it gets distracted by noise in the environment rather than trying to distinguish envi- ronments (Orseau et al., 2013, Sec. 5). Moreover, the entropy-seeking agent may fail to seek knowledge in deterministic semimeasures as the following example demonstrates. Example 4.27 (Unnormalized Entropy-Seeking) .If the Bayesian mixture is a semi- measure instead of a measure (such as the Solomonoﬀ prior from Example 3.5), then the entropy-seeking agent does not explore correctly. Fix A:=f;g,E:=f0;1g, and m=t(we only care about the entropy of the next percept). We illustrate the problem on a simple class of environments f1;2g: 1 =0=0:1 =0=0:5 2 =1=0:1 =0=0:5 where transitions are labeled with action/percept/probability. Both 1and2return a percept deterministically or nothing at all (the environment ends). Only action §4.3The Agents 65 distinguishes between the environments. With the prior w(1) :=w(2) := 1=2, we get a mixturefor the entropy-seeking value function V Ent. ThenV Ent()0:432<0:5 = V Ent(), hence action is preferred over by the entropy-seeking agent. But taking actionyields percept 0(if any), hence nothing is learned about the environment. 3 On-policy value convergence (Theorem 4.19) ensures that asymptotically, the agent learns the value of its own policy. Knowledge-seeking agents do even better: they don’t have to balance between exploration and exploitation, so they can focus solely on exploration. As a result, they learn oﬀ-policy, i.e., the value of counterfactual actions (Orseau et al., 2013, Thm. 7). 4.3.3 BayesExp Lattimore (2013, Thm. 5.6) deﬁnes BayesExp combining AIXI with the information- seeking agent. BayesExp alternates between phases of exploration and phases of ex- ploitation: Let "tbe a monotone decreasing sequence of positive reals such that "t!0 ast!1. If the optimal information-seeking value V IGis larger than "t, then BayesExp starts an exploration phase, otherwise it starts an exploitation phase. During an explo- ration phase, BayesExp follows an optimal information-seeking policy for an "t-eﬀective horizon. During an exploitation phase, BayesExp follows an -optimal reward-seeking policy for one step (see Algorithm 1). Algorithm 1 BayesExp policy BE(Lattimore, 2013, Alg. 2). 1:whiletruedo 2:ifV;t+Ht("t) IG(æ<t)>"tthen 3: follow IGforHt("t)steps 4:else 5: follow for1step 4.3.4 Thompson Sampling Thompson sampling , also known as posterior sampling orthe Bayesian control rule , was originally proposed by Thompson (1933) as a bandit algorithm. It is easy to implement and often achieves quite good results (Chapelle and Li, 2011). In multi-armed bandits it attains optimal regret (Agrawal and Goyal, 2011; Kaufmann et al., 2012). Thompson sampling has also been discussed for MDPs (Strens, 2000; Dearden et al., 1998) and Bayesian and frequentist regret bounds have been established (Osband et al., 2013; Gopalan and Mannor, 2015). For general RL Thompson sampling was ﬁrst suggested by Ortega and Braun (2010) with resampling at every time step. Strens (2000) proposes following the optimal policy for one episode or “related to the number of state transitions the agent is likely to need to plan ahead”. We follow Strens’ suggestion and resample at the eﬀective horizon. Let"tbe a monotone decreasing sequence of positive reals such that "t!0as t!1. Our variant of Thomson sampling is given in Algorithm 2. It samples an 66 Acting environment fromtheposterior,followsthe -optimalpolicyforan "t-eﬀectivehorizon, and then repeats. Algorithm 2 Thompson sampling policy T. 1:whiletruedo 2:samplew(jæ<t) 3:follow forHt("t)steps Note thatTis a stochastic policy since we occasionally sample from a distribution. We assume that this sampling is independent of everything else. Chapter 5 Optimality Machines will never be intelligent. — Shane Legg Problem 4.2 deﬁnes the general reinforcement learning problem. But our deﬁnition of this problem did not specify what a solution would be. This chapter is dedicated to this question: What is an optimal solution to the general reinforcement learning problem? How can we say that one policy is betterthan another? What is the bestpolicy? Are the policies from Section 4.3 optimal? Several notions of optimality for a policy in an environment class Mare conceivable: O1.Maximal reward . The policy receives a reward of 1in every time step (which is maximal according to Assumption 4.6b): 8t2N:rt= 1 O2.Optimal policy . The policy achieves the highest possible value in the true envi- ronment: 8æ<t2(AE ):V (æ<t) =V (æ<t) O3.Pareto optimality (Hutter, 2002a, Thm. 2). There is no other policy that performs at least as good in all environments and strictly better in at least one: @~: 82M:V~ ()V ()and92M:V~ ()>V () O4.Balanced Pareto optimality (Hutter, 2002a, Thm. 3). The policy achieves a better value across Mweighted by w2Mthan any other policy: 8~:X 2Mw() V ()V~ () 0 O5.Bayes optimality . The policy is-optimal for some Bayes mixture : 8æ<t2(AE ):V (æ<t) =V (æ<t) 67 68 Optimality O6.Probably approximately correct . For a given "; > 0the value of the policy is "-close to the optimal value with probability at least after time step t0(";): h 8tt0(";):V (æ<t)V (æ<t)<"i >1 O7.Asymptotic optimality (Hutter, 2005, Sec. 5.3.4). The value of the policy con- verges to the optimal value: V (æ<t)V (æ<t)!0ast!1 O8.Sublinear regret . The diﬀerence between the reward sum of the policy and the best policy in hindsight grows sublinearly: sup 0E0 "mX t=1rt# E "mX t=1rt# 2o(m) We discuss these notions of optimality in turn. Achieving the maximal reward at every time step is impossible if there is no action that makes the environment respond with the maximal reward; generally there is no policy that achieves maximal rewards at every time step. In order to follow the optimal policy, we need to know the true environment. In our setting, the true environment is unknown and has to be learned. During the learning process the agent cannot also act optimally because it needs to explore. In particular, the policy cannot be optimal simultaneously in all environments from M. This rules out O1 and O2 as a notion of optimality. In Section 5.1 we show that all policies are Pareto optimal. This disqualiﬁes O3 as a useful notion of optimality in general reinforcement learning. Balanced Pareto optimality (O4), Bayes optimality (O5), and maximal Legg-Hutter intelligence (Legg and Hutter, 2007b) turn out to coincide. In Section 5.3 we show that Legg-Hutter intelligence is highly subjective, because it depends on the choice of the prior. By changing the prior of a Bayesian agent, we can make the agent’s intelligence arbitrarily low. In Section 5.2 we present a choice of particularly bad priors. This rules out O4 and O5 because they are prior-dependent and not objective. O6 is a stronger version of asymptotic optimality that provides a rate of conver- gence (it implies O7). Since our environment class can be very large and non-compact, concrete PAC results are likely impossible. Orseau (2010, 2013) shows that the Bayes optimal agent does not achieve asymptotic optimality in all computable environments. The underlying problem is that in the beginning the agent does not know enough about its environment and therefore relies heavily on its prior. Lack of exploration then re- tains the prior’s bias. This problem can be alleviated by adding extra exploration to the Bayesian agent. In Section 5.4 we discuss two agents that achieve asymptotic optimal- ity: BayesExp (Section 4.3.3) and Thompson sampling (Section 4.3.4). This establishes that O7 is possible. In general environments sublinear regret is impossible because the agent can get stuck in traps from which it is unable to recover. This rules out O8. However, in Sec- §5.1Pareto Optimality 69 tion5.5weshowthatifweassumethattheenvironmentallowsrecoveringfrommistakes (and some minor conditions on the discount function are fulﬁlled), then asymptotic op- timality implies sublinear regret. This means that Thompson sampling has sublinear regret in these recoverable environments. Notably, only asymptotic optimality (O7) holds up to be a nontrivial and objective criterionofoptimalitythatappliestothegeneralreinforcementlearningproblem. While there are several agents that are known to be asymptotically optimal, some undesirable properties remain. Section 5.6 discusses this further. See also Mahadevan (1996) for a discussion of notions of optimality in MDPs. 5.1 Pareto Optimality In this section we show that Pareto optimality is not a useful criterion for optimality since for any environment class containing MCCM comp, all policies are Pareto optimal. Deﬁnition 5.1 (Pareto Optimality; Hutter, 2005, Def. 5.22) .A policyisPareto optimal in the set of environments Miﬀ there is no policy ~such thatV~ ()V () for all2MandV~ ()>V ()for at least one 2M. The literature provides the following result. Theorem5.2 (AIXIisParetoOptimal; Hutter,2002a, Thm.2) .Every-optimal policy is Pareto optimal in MCCS LSC. The following theorem was proved for deterministic policies in Leike and Hutter (2015c). Here we extend it to stochastic policies. Theorem 5.3 (Pareto Optimality is Trivial) .Every policy is Pareto optimal in any classMMCCM comp. The proof proceeds as follows: for a given policy , we construct a set of ‘buddy environments’ that reward and punish other policies. Together they can defend against any policy ~that tries to take the crown of Pareto optimality from . Proof.We assume (0;0)and(0;1)2E. Moreover, assume there is a policy that is not Pareto optimal. Then there is a policy ~thatPareto dominates , i.e.,V~ ()>V () for some2 M, andV~ ()V ()for all2 M. FromV~ ()> V ()and Lemma 4.18 we get that there is a shortest and lexicographically ﬁrst history æ0 <k consistent with and~such that(jæ0 <k)>~(jæ0 <k)for some action 2A andV~ (æ0 <k)>V (æ0 <k). Consequently there is an iksuch that i>0, and hence k>0. We deﬁne the environment that ﬁrst reproduces the separating history æ0 <k and then, if is the next action, returns reward 1forever, and otherwise returns reward 0forever. Formally, is deﬁned by (e1:tje<tka1:t) :=8 >>>>< >>>>:1;ift<kandet=e0 t; 1;iftkandak=andrt= 1andot= 0; 1;iftkandak6=andrt= 0 =ot;and 0;otherwise: 70 Optimality The environment is computable, even if the policy is not: for a ﬁxed history æ0 <t and action , there exists a program computing ; therefore2MCCM comp. We get the following value diﬀerence for the policies and~: V ()V~ () =E "k1X t=1 trt+1X t=k trt# E~ "k1X t=1 trt1X t=k trt# = (jæ0 <k)1X t=k t~(jæ0 <k)1X t=k t! (æ0 <k) = (jæ0 <k)~(jæ0 <k) (æ0 <k)k>0 HenceV~ ()< V (), which contradicts the fact that ~Pareto dominates since MMCCM comp3. Note that the environment we deﬁned in the proof of Theorem 5.3 is actually just a ﬁnite-state POMDP, so Pareto optimality is also trivial for smaller environment classes. 5.2 Bad Priors In this section we give three examples of universal priors that cause a AIXI to mis- behave drastically. In case of a ﬁnite horizon, the indiﬀerence prior makes all actions equally preferable to AIXI (Section 5.2.1). The dogmatic prior makes AIXI stick to any given computable policy as long as expected future rewards do not fall too close to zero (Section 5.2.2). The Gödel prior prevents AIXI tlfrom taking any actions (Sec- tion 5.2.3). 5.2.1 The Indiﬀerence Prior The following theorem constructs the indiﬀerence prior that yields a Bayesian mixture 0that causes argmax ties for the ﬁrst msteps. If we use a discount function that only cares about the ﬁrst msteps, m= 0, then all policies are 0-optimal policies. In this case AIXI’s behavior only depends on how we break argmax ties. Theorem 5.4 (Indiﬀerence Prior) .If there is an msuch that m= 0, then there is a Bayesian mixture 0such that all policies are 0-optimal. Proof.First, we assume that the action space is binary, A=f0;1g. LetUbe the reference UTM and deﬁne the UTM U0by U0(s<mp;a1:t) :=U(p;a1:txors1:t); wheres<mis a binary string of length m1andsk:= 0forkm. (U0has no programs of length less than m1.) Let0be the Bayesian mixture given by U0according to §5.2Bad Priors 71 (4.8). Then 0(e<mka<m) =X p:e<mvU0(p;a<m)2jpj =X s<mp0:e<mvU0(s<mp0;a<m)2m1jp0j =X s<mX p0:e<mvU(p0;a<mxors<m)2m1jp0j =X s<mX p0:e<mvU(p0;s<m)2m1jp0j; which is independent of a<m. Hence the ﬁrst m1percepts are independent of the ﬁrstm1actions. But the percepts’ rewards from time step mon do not matter since m= 0(Lemma 4.16). Because the environment is chronological, the value function must be independent of all actions. Thus every policy is 0-optimal. For ﬁnite action spaces Awith more than 2elements, the proof works analogously by makingAa cyclic group and using the group operation instead of xor. The choice of U0in the proof of Theorem 5.4 depends on m. If we increase AIXI’s horizon while ﬁxing the UTM U0, Theorem 5.4 no longer holds. For Solomonoﬀ in- duction, there is an analogous problem: when using Solomonoﬀ’s prior Mto predict a deterministic binary sequence x, we make at most K(x)errors (Corollary 3.56). In case the shortest program has length >m, there is no guarantee that we make less than m errors (see Section 5.6.2). 5.2.2 The Dogmatic Prior In this section we deﬁne a universal prior that assigns very high probability of going to hell (reward 0forever) if we deviate from a given computable policy . For a Bayesian agent like AIXI, it is thus only worth deviating from the policy if the agent thinks that the prospects of following are very poor already. We call this prior the dogmatic prior, because the fear of going to hell makes AIXI conform to any arbitrary ‘dogmatic ideology’. AIXI will only break out if it expects to give very low future payoﬀ; in that case the agent does not have much to lose. Theorem 5.5 (Dogmatic Prior) .Letbe any computable deterministic policy, let be any Bayesian mixture over MCCS LSC, and let" >0. There is a Bayesian mixture 0 such that for any history æ<tconsistent with and for which V (æ<t)>", the action (æ<t)is the unique 0-optimal action. The following proof was adapted from Leike and Hutter (2015c) to work for en- vironment classes that do not contain the Bayesian mixture. Essentially, for every environment 2MCCS LSCthe dogmatic prior puts much higher weight on an environ- mentthat behaves just like on the policy , but sends any policy deviating from to hell. Importantly, while following the policy the environments andare indistinguishable, so the posterior belief in is equal to the posterior belief in . 72 Optimality Proof of Theorem 5.5. We assume (o;0)2Efor someo2O. For every environment 2MCCS LSCdeﬁne the environment (e1:tka1:t) :=8 >>>>< >>>>:(e1:tka1:t);ifak=(æ<k)8kt; (e<kka<k);ifk:= minfijai6=(æ<i)gexists andei= (o;0)8i2fk;:::;tg;and 0; otherwise: The environment mimics environment until it receives an action that the policy would not take. From then on, it provides rewards 0. Sinceis a computable policy, we have that 2MCCS LSCfor every2MCCS LSC. Now we need to reweigh the prior wso that it assigns a much higher prior weight tothan to. Without loss of generality we assume that "is computable, otherwise we make it slightly smaller. We deﬁne w0() :="w()if6=~for all ~2MCCS LSCand w0() := (1")w() +"w(). Then X 2MCCS LSCw0() =X =~w0() +X 6=~w0() =X =~ (1")w(~) +"w() +X 6=~"w() =X 2MCCS LSC"w() +X ~2MCCS LSC(1")w(~) ="+ (1") = 1; and withw0"w, we get that w0is a positive prior over MCCS LSC. We deﬁne 0as the corresponding Bayesian mixture analogous to (4.5). With:=P 2MCCS LSCw()we get0="+ (1"). The mixtures and coincide on the policy since every coincides with on the policy : (æ<t) =X 2MCCS LSCw()(æ<t) =X 2MCCS LSCw() (æ<t) =(æ<t) Moreover,V (æ<t) = 0and thusV (æ<t) = 0for any history inconsistent with by construction of . Letæ<t2(AE )be any history consistent with such thatV (æ<t)>". Then =implies (e<tka<t) 0(e<tka<t)=(e<tka<t) 0(e<tka<t)=(e<tka<t) "(e<tka<t) + (1")(e<tka<t)= 1: §5.2Bad Priors 73 Therefore Lemma 4.14 implies that for all a2Aand all policies ~ V~ 0(æ<ta) ="(e<tka<t) 0(e<tka<t)V~ (æ<ta) + (1")(e<tka<t) 0(e<tka<t)V~ (æ<ta) ="V~ (æ<ta) + (1")V~ (æ<ta): (5.1) Let:=(æ<t)be the next action according to , and let6=be any other action. We have that V (æ<t) =V (æ<t)since=andæ<tis consistent with. Therefore we get from (5.1) V 0(æ<t)V 0(æ<t) ="V (æ<t) + (1")V (æ<t) =V (æ<t)>"; V 0(æ<t) ="V 0 (æ<t) + (1")V 0 (æ<t) ="V 0 (æ<ta) + (1")0": HenceV 0(æ<t)>V 0(æ<t)and thus the action taken byis the only 0-optimal action for the history æ<t. Corollary 5.6 (With Finite Horizon Every Policy is Bayes Optimal) .Ifm= 0for somem2N, then for any deterministic policy there is a Bayesian mixture 0such that(æ<t)is the only 0-optimal action for all histories æ<tconsistent with and tm. In contrast to Theorem 5.4 where every policy is 0-optimal for a ﬁxed Bayesian mixture0, Corollary 5.6 gives a diﬀerent Bayesian mixture 0for every policy such thatis theonly0-optimal policy. Proof.Let">0be small enough such that V (æ<t)>"for allæ<tandtm. (This is possible because (AE )mis ﬁnite by Assumption 4.6c.) We use the dogmatic prior from Theorem 5.5 to construct a Bayesian mixture 0for the policy and">0. Thus for any history æ<t2(AE )consistent with andtm, the action (æ<t)is the only0-optimal action. Corollary 5.7 (AIXI Emulating Computable Policies) .Let" > 0and letbe any computable policy. There is a Bayesian mixture 0such that for any 0-optimal policy 0and for any environment , V 0 ()V ()< ": Proof.From the proof of Corollary 5.6 and Lemma 4.18. 5.2.3 The Gödel Prior This section introduces a prior that prevents any ﬁxed formal system from making any statements about the outcome of all but ﬁnitely many computations. It is named after Gödel (1931) who famously showed that for any suﬃciently rich formal system there are statements that it can neither prove nor disprove. 74 Optimality This prior is targeted at AIXI tl, a computable approximation to AIXI deﬁned by Hutter (2005, Sec. 7.2). AIXI tlaims to perform as least as well as the best agent who is limited by time tand spacelthat can be veriﬁed using a proof of length at most nfor some ﬁxed n2N. The core idea is to enumerate all deterministic policies and proofs and then execute the policy for which the best value has been proved. In order to be veriﬁed, a policy has to be computed by a program pwhich fulﬁlls theveriﬁcation condition VA(p)(Hutter, 2005, Eq. 7.7). This program pnot only computes future actions of , but also hypothetical past actions a0 iand lower bounds vifor the value of the policy : VA(p) :=“8k8(va0æ)1:k: p(æ<k) =v1a0 1:::vka0 k!vkV (æ<k) ”; whereis the policy derived from paccording to (æ<k) :=a0 k. We ﬁx some formal system that we use to prove the veriﬁcation condition. We want it to be suﬃciently powerful, but this incurs Gödel incompleteness. For simplicity of exposition we pick PA, the system of Peano arithmetic (Shoenﬁeld, 1967, Ch. 8.1), but our result generalizes trivially to all formal systems which cannot prove their own consistency. Letnbe a ﬁxed constant. The algorithm for AIXI tlis speciﬁed as follows. 1. LetP=;. This will be the set of veriﬁed programs. 2. For all proofs in PAof lengthn: if the proof proves VA(p)for somep, andjpjl, then add the program ptoP. 3. For each input history æ<krepeat: run all programs from Pfor at most tsteps each, take the one with the highest promised value vk, and return that program’s policy’s action. Theorem 5.8 (The Gödel Prior) .There is a UTM U0such that if PAis consistent, then the set of veriﬁed programs Pis empty for all t,l, andn. Proof.Letqdenote an algorithm that never halts, but for which this cannot be proved inPA; e.g., letqenumerate all consequences of PAand halt as soon as it ﬁnds a contradiction. Since we assumed that PAis consistent, qnever halts. Deﬁne the UTM U0(p;a1:k)as follows. •Runqforksteps. •Ifqhalts, output vk= 2. •RunU(p;a1:k). Sinceqnever halts, UandU0are functionally identical, therefore U0is universal. Note thatPAproves8p: U(p;a1:k) =U0(p;a1:k)for any ﬁxed k, but PAdoes not prove 8k8p:U(p;a1:k) =U0(p;a1:k). §5.3Bayes Optimality 75 Ifqdid eventually halt, it would output a value vk= 2that is too high, since the value function V is bounded by 1from above, which PAknows. Hence PAproves that qhalts!8p::VA(p) (5.2) IfPAcould prove VA(p)for anyp, then PAwould prove that qdoes not halt since this is the contrapositive of (5.2). Therefore the set Premains empty. AIXItlexhibits all the problems of the arbitrariness of the UTM illustrated by the indiﬀerence prior (Theorem 5.4) and the dogmatic prior (Theorem 5.5). In addition, it is also susceptible to Gödel incompleteness as illustrated by the Gödel prior in Theo- rem 5.8. The formal system that is a parameter to AIXI tljust provides another point of failure. As a computable approximation to AIXI, AIXI tlis needlessly complicated. As we prove in Corollary 6.13, "-optimal AIXI is limit computable, so we can approximate it with an anytime algorithm. Bounding the computational resources to the approxima- tion algorithm already yields a computable version of AIXI. Moreover, unlike AIXI tl, this approximation actually converges to AIXI in the limit. Furthermore, we can ‘speed up’ this approximation algorithm using Hutter search (Hutter, 2002b); this is very sim- ilar but not identical to AIXI tl. 5.3 Bayes Optimality The aim of the Legg-Hutter intelligence measure is to formalize the intuitive notion of intelligence mathematically. Legg and Hutter (2007a) collect various deﬁnitions of intelligence across many academic ﬁelds and destill it into the following statement (Legg and Hutter, 2007b) Intelligence measures an agent’s ability to achieve goals in a wide range of environments. This deﬁnition is formalized as follows. Deﬁnition 5.9 (Legg-Hutter Intelligence; Legg and Hutter, 2007b, Sec. 3.3)) .The (Legg-Hutter) intelligence of a policy is deﬁned as () :=X 2Mw()V () The Legg-Hutter intelligence of a policy is thet0-value that achieves across all environments from the class Mweighted by the prior w. Legg and Hutter (2007b) con- sider a subclass of MCCS LSC, the class of computable measures together with a Solomonoﬀ priorw() = 2K()and do not use discounting explicitly. Typically, the index is omitted when writing . However, in this section we consider the intelligence measure with respect to diﬀerent priors, therefore we make this dependency explicit. The following proposition motivates the use of the index instead ofw. 76 Optimality Proposition 5.10 (Bayes Optimality = Maximal Intelligence) .() =V ()for all policies. Proof.Follows directly from (4.6) and Deﬁnition 5.9. Deﬁnition 5.11 (Balanced Pareto Optimality; Hutter, 2005, Def. 5.22) .LetMbe a set of environments. A policy isbalanced Pareto optimal in the set of environments Miﬀ for all policies ~,X 2Mw() V ()V~ () 0: Proposition 5.12 (Balanced Pareto Optimality = Maximal Intelligence) .A policy is balanced Pareto optimal in Mif and only if has maximal Legg-Hutter intelligence. Proof.From (4.6) we get X 2Mw() V ()V () =X 2Mw()V ()X 2Mw()V () =V ()V () =V ()sup ~V~ () = ()sup ~(~) by Proposition 5.10. This term is nonnegative iﬀ ()is maximal. As a consequence of Proposition 5.10 and Proposition 5.12 we get that AIXI is balanced Pareto optimal (Hutter, 2005, Thm. 5.24) and has maximal Legg-Hutter in- telligence. := sup () = sup V () =V () = ( ): This is not surprising since Legg-Hutter intelligence was deﬁned in terms of the t0- value in the Bayes mixture. Moreover, because the value function is scaled to be in the interval [0;1], intelligence is a real number between 0and1. It is just as hard to score very high on the Legg-Hutter intelligence measure as it is to score very low: we can always turn a reward minimizer into a reward maximizer by inverting the rewards r0 t:= 1rt. Hence the lowest possible intelligence score is achieved by AIXI’s twin sister, a -expected reward minimizer: := inf () = inf V () The heaven environment (reward 1forever) and the hell environment (reward 0forever) are computable and thus in the environment class MCCS LSC; therefore it is impossible to get a reward 0or reward 1in every environment. Consequently, for all policies , 0<()<1: §5.3Bayes Optimality 77 0 1 random AI0AI image of Figure 5.1: The Legg-Hutter intelligence measure assigns values within the closed interval [;]; the assigned values are depicted in purple. By Theorem 5.13, com- putable policies are dense in this purple set. For every real number r2[;]there is a policy with () =r: analogously to Lemma 4.14 we can deﬁne such that with probability (r)=()it follows and otherwise it follows arg min~V~ (). Figure 5.1 illustrates the intelligence measure . It is natural to ﬁx the policy randomthat takes actions uniformly at random to have an intelligence score of 1=2by choosing a ‘symmetric’ universal prior (Legg and Veness, 2013). AIXI is not computable (Theorem 6.15), hence there is no computable policy such that () = or() =for any Bayesian mixture overMCCS LSC. But the following theorem states that computable policies can come arbitrarily close. This is no surprise: by Lemma 4.17 we can do well on a Legg-Hutter intelligence test simply by memorizing what AIXI would do for the ﬁrst ksteps; as long as kis chosen large enough such that discounting makes the remaining rewards contribute very little to the value function. Theorem 5.13 (Computable Policies are Dense) .The set f()jis a computable policy g is dense in the set [;]. Proof.Letbe any policy and let ">0. We need to show that there is a computable policy ~withj(~)()j<". We choose mlarge enough such that m=1<"=3. Let2Abe arbitrary and deﬁne the policy ~(ajæ<t) :=8 >>< >>:(ajæ<t)("=3)mift<m; 1 iftmanda=;and 0 otherwise: By choosing an appropriate rational number in the interval [(ajæ<t)("=3)m;(aj æ<t) + ("=3)m]we can make the policy ~computable because we can store these approximations to the action probabilities of for the ﬁrst m1steps in a lookup table. From Lemma 4.17 we get V;m ()V~;m ()Dm1(;~j) ("=3)mm=" 3 78 Optimality and together with Lemma 4.16 this yields j()(~)j=V ()V~ ()V;m ()V~;m ()+ 2m 1" 3+ 2m 1<": Remark5.14 (DeterministicPoliciesarenotDensein [;]).Theintelligencevalues of deterministic policies are generally not dense in the interval [;]. We show this by deﬁning an environment where the ﬁrst action determines whether the agent goes to heaven or hell: action leads to heaven and action leads to hell. Deﬁne Bayesian mixture 0:= 0:999+ 0:001and letbe any policy. If takes action ﬁrst, then 0()>0:999. Iftakes action ﬁrst, then 0()<0:001. Hence there are no deterministic policies that score an intelligence value in the closed interval [0:001;0:999]. 3 Legg-Hutterintelligenceismeasuredwithrespecttoaﬁxedprior. TheBayesagentis the most intelligent policy if it uses the same prior . We use the results from Section 5.2 to show that the intelligence score of the Bayes agent can be arbitrary close to the minimum intelligence score . Corollary 5.15 (Some AIXIs are Stupid) .For any Bayesian mixture overMCCS LSC and every">0, there is a Bayesian mixture 0such that ( 0)<+". Proof.Let" > 0. According to Theorem 5.13, there is a computable policy such that ()<+"=2. From Corollary 5.7 we get a Bayesian mixture 0such that j( 0)()j=jV 0 ()V ()j<"=2, hence j( 0)jj( 0)()j+j()j<"=2 +"=2 =": We get the same result if we ﬁx AIXI, but rig the intelligence measure. Corollary 5.16 (AIXI is Stupid for Some ).For any deterministic -optimal policy and for every " > 0there is a Bayesian mixture 0such that 0( )"and 01". Proof.Leta1:= ()be the ﬁrst action that takes. We deﬁne an environment such that taking the ﬁrst action a1leads to hell and taking any other ﬁrst action leads to heaven as in Remark 5.14. We deﬁne the Bayesian mixture 0:= (1")+". Since takes action a1ﬁrst, it goes to hell, i.e., V () = 0. Hence with Lemma 4.14 0( ) =V 0() = (1")V () +"V ()": For any policy that takes an action other than a1ﬁrst, we get 0() =V 0() = (1")V () +"V ()1": On the other hand, we can make any computable policy smart if we choose the right Bayesian mixture. In particular, we get that there is a Bayesian mixture such that ‘do nothing’ is the most intelligent policy save for some ". §5.4Asymptotic Optimality 79 name deﬁnition strong a.o. V (æ<t)V (æ<t)!0-almost surely a.o. in mean E V (æ<t)V (æ<t) !0 a.o. in probability 8">0: V (æ<t)V (æ<t)>" !0 weak a.o.1 tPt k=1 V (æ<k)V (æ<k) !0-almost surely Table 5.1: The formal deﬁnition of diﬀerent types of asymptotic optimality. In each case we understand the limit as t!1. Corollary 5.17 (Computable Policies can be Smart) .For any computable policy and any">0there is a Bayesian mixture 0such that 0()>0". Proof.Corollary5.7yieldsaBayesianmixture 0withj00()j=jV 0()V 0()j< ". 5.4 Asymptotic Optimality An asymptotically optimal policy is a policy learns to act optimally in every environ- ment fromM, i.e., the value of this policy converges to the optimal value. Deﬁnition 5.18 (Asymptotic Optimality) .A policyisasymptotically optimal in an environment class Miﬀ for all2M V (æ<t)V (æ<t)!0ast!1 (5.3) on histories drawn from . There are diﬀerent types of asymptotic optimality based on the type of stochas- tic convergence in (5.3); see Deﬁnition 2.5. If this convergence occurs almost surely, it is called strong asymptotic optimality (Lattimore and Hutter, 2011, Def. 7); if this convergence occurs in mean, it is called asymptotic optimality in mean ; if this conver- gence occurs in probability, it is called asymptotic optimality in probability ; and if the Cesàro averages converge almost surely, it is called weak asymptotic optimality (Lat- timore and Hutter, 2011, Def. 7). Since the value function is a nonnegative bounded random variable, asymptotic optimality in mean and asymptotic optimality in proba- bility are equivalent. See Table 5.1 for the explicit deﬁnitions and see Figure 5.2 for an overview over their relationship. Asymptotic optimality in probability is in spirit a probably approximately cor- rect (PAC) result: for all ">0and>0the probability that our policy is "-suboptimal converges to zero; eventually this probability will be less than . For a PAC result it is typically demanded that the number of time steps until the probability is less than 80 Optimality strong a.o. weak a.o. a.o. in mean a.o. in probability Figure 5.2: The relationship between diﬀerent types of asymptotic optimality. Each arrow indicates a logical implication and each lack of an arrow indicates that there is no logical implication. be polynomial in 1="and1=. In general environments this is impossible, and here we have no ambition to provide concrete convergence rates. Intuitively, a necessary condition for asymptotic optimality is that the agent needs toexploreinﬁnitelyoftenforanentireeﬀectivehorizon. Ifweexploreonlyﬁnitelyoften, then the environment might change after we stopped exploring. Moreover, the agent needs to predict the value of counterfactual policies accurately; but by Lemma 4.16 only for an "-eﬀective horizon. By committing to exploration for the entire eﬀective horizon, we learn about the value of counterfactual policies. Example 5.19 (Exploration Inﬁnitely Often for an Entire Eﬀective Horizon) .If there is an" > 0such that the policy does not explore for Ht(")steps inﬁnitely of- ten, thenV (æ<t)V (æ<t)> "inﬁnitely often. Deﬁne A:=f;gandE:= f0;"=2;1g(observations are vacuous) and consider the following class of environments M:=f1;1;2;:::g(transitions are labeled with condition: action, reward): s0;" 2 ;0s0 s1 ::: sn;" 2 t<k :;0tk:;0 ;0 ;0 ;0 ;0 ;0;1 1 k Environment kworks just like environment 1, except that at time step kthe path to states1gets unlocked. The length of the state sequence in kis deﬁned as an "-eﬀective horizon,n:=Ht(")wheretis the time step in which the agent leaves state s0. The optimal policy in environment 1is to always take action , the optimal policy for environment kis to take action fort<kand then take action . Suppose the agent is in time step tand in state s0. Since these environments are partially observable, it needs to explore for nsteps (take action ntimes) to distinguish 1fromkfor anykt. Since there are inﬁnitely many k, the agent needs to do this inﬁnitely often. Moreover, V 1"andV 1="=2, so iftis the true environment, then not exploring to the right for an "-eﬀective horizon is suboptimal by "=2. But if1is the true environment, then exploring incurs an opportunity cost of one reward of "=2.3 §5.4Asymptotic Optimality 81 Next, we state two negative results about asymptotic optimality proved by Latti- more and Hutter (2011). It is important to emphasize that Theorem 5.20 and Theo- rem 5.21 only hold for deterministic policies. Theorem 5.20 (Deterministic Policies are not Strongly Asymptotically Optimal; Lat- timoreandHutter,2011, Thm.8) .There is no deterministic policy that is strong asymp- totically optimal in the class MCCM comp. If the horizon grows linearly (for example, power discounting (t) =twith >1; see Table 4.1), then a deterministic policy cannot be weakly asymptotically optimal policy: the agent has to explore for an entire eﬀective horizon, which prevents the Cesàro average from converging. Theorem 5.21 (Necessary Condition for Weak Asymptotic Optimality; Lattimore, 2013, Thm. 5.5) .If there is an ">0such thatHt(")=2o(t), then there is no determin- istic policy that is weakly asymptotically optimal in the class MCCM comp. There are several agents that achieve asymptotic optimality. In the rest of this section, we discuss the Bayes agent, BayesExp, and Thompson sampling. Asymptotic optimalitycan alsobe achieved through optimism(SunehagandHutter,2012a,b,2015). 5.4.1 Bayes In this section, we list two results from the literature regarding the asymptotic optimal- ity of the Bayes optimal policy. The following negative result is due to Orseau (2010, 2013). Theorem5.22 (BayesisnotAsymptoticallyOptimalinGeneralEnvironments; Orseau, 2013, Thm. 4) .For any classMMCCM compno Bayes optimal policy is asymptotically optimal: there is an environment 2Mand a time step t02Nsuch that -almost surely for all time steps tt0 V (æ<t)V (æ<t) =1 2: Orseau calls this result the good enough eﬀect : A Bayesian agent eventually decides that the current strategy is good enough and that any additional exploration is not worth its expected payoﬀ. However, if the environment changes afterwards, the Bayes agent is acting suboptimally. Proof.Without loss of generality assume A:=f;gandE:=f0;1=2;1g(observations are vacuous). We consider the following environment (transitions are labeled with action, reward). 82 Optimality s0 s1 ::: sn;1 2 ;0;0;0 ;0 Instates0theactionistheexploitationactionandtheaction theexplorationaction. The length of the state sequence is deﬁned as an 1=t-eﬀective horizon, n:=Ht(1=t) wheretis the time step in which the agent leaves state s0. Since the discount function is computable by Assumption 4.6a, 2MCCS LSC. Assume that when acting in , the Bayes agent explores inﬁnitely often. Let æ<t be a history in which the agent is in state s0and takes action . ThenV 1=t. By on-policy value convergence (Corollary 4.20), V (æ<t)V (æ<t)!0 -almost surely. Hence there is a time step t0such that for all tt0we haveV < w()=2. Sinceis deterministic, w(jæ<t)w(). Now we get a contradiction from (4.7): V (æ<t)w(jæ<t)V (æ<t)w()V (æ<t) =w() 2>V (æ<t) Therefore the Bayes agent stops taking the exploration action after time step t0, and so it is not optimal in any 2MCCS LSCthat behaves like until time step t0and then changes: s0 s1 ::: sn;1 2 t>t 0:;1tt0:;0;0;0 ;0 The following theorem is also known as the self-optimizing theorem . This theorem has been a source of great confusion because its statement in Hutter (2005, Thm. 5.34) is not very explicit about how the histories are generated. The formulation of Lattimore (2013, Thm. 5.2) is explicit, but less general. Theorem 5.23 (Suﬃcient Condition for Strong Asymptotic Optimality of Bayes; Hut- ter, 2005, Thm. 5.34) .Letbe some environment. If there is a policy and a sequence of policies1;2;:::such that for all 2M V (æ<t)Vt(æ<t)!0ast!1-almost surely, (5.4) then V (æ<t)V (æ<t)!0ast!1-almost surely. §5.4Asymptotic Optimality 83 If= and(5.4)holds for all 2M, then is strongly asymptotically optimal in the classM. It is important to emphasize that the policies 1;2;:::need to converge to the optimal value on the history generated by and, and not (as one might think) and t. Intuitively, the policy is an ‘exploration policy’ that ensures that the environment class is explored suﬃciently. Typically, a policy is asymptotically optimal on its own history. So if =1=2=:::, then we get that Bayes is asymptotically optimal on the history generated by the policy , not its own history. In light of Theorem 5.5 and Theorem 5.22 this is not too surprising; Bayesian reinforcement learning agents might not explore enough to be asymptotically optimal, but given a policy that does explore enough, Bayes learns enough to be asymptotically optimal. This invites us to deﬁne the following policies t: follow the information-seeking policy IGuntil time step t, and then follow (explore until t, then exploit). Since the information-seeking policy explores enough to prove oﬀ-policy prediction (Orseau et al., 2013, Thm. 7), we get V V !0for every policy uniformly. Hence arg maxV ! arg maxV and thusV V !0and (5.4) is satisﬁed. From Theorem 5.23 we get V V !0, which we already knew. In order to get strong asymptotic optimality, all we need to do is choose the switching time step tappropriately, i.e., wait until V andV are close enough. Unfortunately, this is an invalid strategy: the agent does not know the true environment and hence cannot check this condition. Hutter (2005, Sec. 5.6) uses Theorem 5.23 to show that the Bayes optimal policy is strongly asymptotically optimal in the class of ergodic ﬁnite-state MDPs if the eﬀective horizon is growing, i.e., Ht(")!1for all">0. This relies on the fact that in ergodic ﬁnite-state MDPs we need a ﬁxed number of steps to explore the entire environment up to"-conﬁdence. Thereforewecandeﬁneasequenceofpolicies 1;2;:::thatcompletely disregard the history and start exploring everything from scratch. Since the eﬀective horizon is growing, this exploration phase takes a vanishing fraction of eﬀective horizon and most of the value is retained. Therefore the sequence of policies 1;2;:::satisﬁes the condition of Theorem 5.23 regardless of the history, thus in particular for the history generated by = and any2M. Note that the condition on the horizon is important: If the eﬀective horizon is bounded, then Bayes is not asymptotically optimal in the class of ergodic ﬁnite-state MDPs because it can be locked into a dogmatic prior similarly to Theorem 5.5. Proof of Theorem 5.23. From (4.6) we get for any history æ<t w(jæ<t) V (æ<t)V (æ<t) X 2Mw(jæ<t) V (æ<t)V (æ<t) = X 2Mw(jæ<t)V (æ<t)! V (æ<t) X 2Mw(jæ<t)V (æ<t)Vt (æ<t) 84 Optimality =X 2Mw(jæ<t) V (æ<t)Vt(æ<t) :(5.5) From (5.4) follows that V Vt!0-almost surely for all 2M, so (5.5) converges to0-almost surely (Hutter, 2005, Lem. 5.28ii). Similar to Example 3.20, 1=w(j æ<t)is a nonnegative -martingale and thus converges (to a ﬁnite value) -almost surely by Theorem 2.8. Therefore V (æ<t)V (æ<t)!0-almost surely. If this is true for all 2M, the strong asymptotic optimality of follows from = by deﬁnition. 5.4.2 BayesExp The deﬁnition of BayesExp is given in Section 4.3.3. In this subsection we state a result by Lattimore (2013) that motivated the deﬁnition of BayesExp. Theorem 5.24 (BayesExp is Weakly Asymptotically Optimal; Lattimore, 2013, Thm. 5.6).LetBEdenote the policy from Algorithm 1. If Ht(")grows monotone in tand Ht("t)="t2o(t), then for all environments 2M 1 ttX k=1 V (æ<k)VBE(æ<k) !0ast!1BE-almost surely : If the horizon grows sublinearly ( Ht(")2o(t)for all" >0), then we can always ﬁnd a sequence "t!0that decreases slowly enough such that Ht(")="t2o(t)holds. 5.4.3 Thompson Sampling In this section we prove that the Thompson sampling policy deﬁned in Section 4.3.4 is asymptotically optimal. Ortega and Braun (2010) prove that the action probabilities of Thompson sampling converge to the action probability of the optimal policy almost surely, but require a ﬁnite environment class and two (arguably quite strong) technical assumptions on the behavior of the posterior distribution (akin to ergodicity) and the similarity of environments in the class. Our convergence results do not require these assumptions. Theorem 5.25 (Thompson Sampling is Asymptotically Optimal in Mean) .For all environments 2M, ET V (æ<t)VT(æ<t) !0ast!1. This theorem immediately implies that Thompson sampling is also asymptotically optimal in probability according to Figure 5.2. However, this does not imply almost sure convergence (see Example 5.28). We ﬁrst give an intuition for the asymptotic optimality of Thompson sampling. At every resampling step we can split the class Minto three partitions: 1. Environments whereV V §5.4Asymptotic Optimality 85 2. Environments whereV >V 3. Environments whereV <V The ﬁrst class is the class of ‘good’ environments: if we draw one of them, we follow a policy that is close to optimal in . The second class is the class of environments that overestimate the value of . Following their optimal policy the agent gains information becauserewardswillbelowerthanexpected. Thethirdclassistheclassofenvironments that underestimate the value of . Following their optimal policy the agent might not gain information since might behave just like environment on the-optimal policy. However, when sampling from the ﬁrst class instead, the agent gains information about the third class because rewards tend to be better than environments from the third class predicted. Since the true environment 2M, the ﬁrst class is not empty, and the probability of drawing a sample from the ﬁrst class does not become too small. Whenever the second and third class have suﬃciently high weight in the posterior, there is a good chance of picking a policy that leads the agent to gain information. Asymptotically, the posterior converges, so the agent ends up having learned everything it could, i.e., the posterior weight of the second and third class vanishes. This argument is not too hard to formalize for deterministic environment classes. However, for stochastic environment classes the eﬀect on the posterior when following a bad policy is harder to quantify because there is always a chance that the rewards are diﬀerent simply because of bad luck. In order to prove this theorem in its generality for stochasticclasses, weemployanentirelydiﬀerentproofstrategythatreliesonstatistical tools rather than the argument given above. Deﬁnition 5.26 (Expected Total Variation Distance) .Letbe any policy and let m2N[1. Theexpected total variation distance on the policy is F m(æ<t) :=X 2Mw(jæ<t)Dm(;jæ<t): If we replace the distance measure Dmby cross-entropy, then the quantity F m(æ<t) becomes the expected information gain (see Section 4.3.2). For the proof of Theorem 5.25 we need the following lemma. Lemma 5.27 (Expected Total Variation Distance Vanishes On-Policy) .For any policy and any environment , E [F 1(æ<t)]!0ast!1. Proof.From Theorem 3.25 we get D1(;jæ<t)!0-almost surely, and since D is bounded, this convergence also occurs in mean. Thus for every environment 2M, E D1(;jæ<t) !0ast!1: 86 Optimality Now E [F 1(æ<t)]1 w()E [F 1(æ<t)] =1 w()E "X 2Mw(jæ<t)D1(;jæ<t)# =1 w()E "X 2Mw()(æ<t) (æ<t)D1(;jæ<t)# =1 w()X 2Mw()E D1(;jæ<t) !0 by Hutter (2005, Lem. 5.28ii) since total variation distance is bounded. Proof of Theorem 5.25. Let; > 0and let"t>0denote the sequence used to deﬁne Tin Algorithm 2. We assume that tis large enough such that "kfor allkt and thatis small enough such that w(jæ<t)>4for allt, which holds since w(jæ<t)6!0-almost surely for any policy (Hutter, 2009a, Lem. 3i). The stochastic process w(jæ<t)is aT-martingale according to Example 3.20. By the martingale convergence theorem (Theorem 2.8) w(jæ<t)convergesT-almost surely and because Tw()Tit also converges T-almost surely. We argue that we can choose t0to be one of T’s resampling time steps large enough such that for all tt0the following three events hold simultaneously with T-probability at least 1. (i) There is a ﬁnite set M0Mwithw(M0jæ<t)>1andw(jæ<k)6!0as k!1for all2M0. (ii)jw(M00jæ<t)w(M00jæ<t0)jfor allM00M0. (iii)FT1(æ<t)<w2 min. wherewmin:= inffw(jæ<k)jk2N;2M0g, which is positive by (i). (i) and (ii) are satisﬁed eventually because the posterior w(jæ<t)convergesT- almost surely. Note that the set M0is random: the limit of w(jæ<t)ast!1 depends on the history æ1:1. Without loss of generality, we assume the true environ- mentis contained inM0sincew(jæ<t)6!0T-almost surely. (iii) follows from Lemma 5.27 since convergence in mean implies convergence in probability. Moreover, we deﬁne the horizon m:=t+Ht("t)as the time step of the eﬀective horizon at time step t. Letæ<tbe a ﬁxed history for which (i-iii) is satisﬁed. Then we have w2 min>FT1(æ<t) =X 2Mw(jæ<t)D1(T;Tjæ<t) =Ew(jæ<t)[D1(T;Tjæ<t)] §5.4Asymptotic Optimality 87 Ew(jæ<t)[Dm(T;Tjæ<t)] w2 minw(MnM00jæ<t) by Markov’s inequality where M00:= 2MDm(T;Tjæ<t)<w2 min : For our ﬁxed history æ<twe have 1<w (M00jæ<t) (i) w(M00\M0jæ<t) + (ii) w(M00\M0jæ<t0) + 2 (i) w(M00jæ<t0) + 3 and thus we get 14<w f2MjDm(T;Tjæ<t)<w2 mingæ<t0 : (5.6) In particular, this bound holds for =sincew(jæ<t0)>4by assumption. It remains to show that with high probability the value V of the sample ’s optimal policy is suﬃciently close to the -optimal value V . The worst case is that we draw the worst sample from M0\M00twice in a row. From now on, let denote the sample environment we draw at time step t0, and lettdenote some time step between t0andt1:=t0+Ht0("t0)(before the next resampling). With probability w(0jæ<t0)w(0jæ<t1)we sample0both att0andt1when following T. Therefore we have for all æt:mand all2M T(æ1:mjæ<t)w(0jæ<t0)w(0jæ<t1) 0(æ1:mjæ<t): Thus we get for all 2M0(in particular and) Dm(T;Tjæ<t)sup 02Msup A(AE )mw(0jæ<t0)w(0jæ<t1) ( 0(Ajæ<t) 0(Ajæ<t)) w(jæ<t0)w(jæ<t1) sup A(AE )m (Ajæ<t) (Ajæ<t) w2 minDm( ; jæ<t): For2M00we get with (5.6) Dm(T;Tjæ<t)Dm(T;Tjæ<t) +Dm(T;Tjæ<t) <w2 min+w2 min= 2w2 min; 88 Optimality which together with Lemma 4.17 and the fact that rewards in [0;1]implies V (æ<t)V (æ<t)t+Ht("t) t+V ;m (æ<t)V ;m (æ<t) "t+Dm( ; jæ<t) "t+1 w2 minDm(T;Tjæ<t) <+ 2= 3: Hence we get (omitting history arguments æ<tfor simplicity) V =V <V + 3V + 3=V + 3 <V + 3+ 3=V + 6:(5.7) WithT-probability at least 1(i), (ii), and (iii) are true, with T-probability at least 1our sample happens to be inM0by (i), and with w(jæ<t0)-probability at least 14the sample is inM00by (5.6). All of these events are true simultaneously with probability at least 1(++ 4) = 16. Hence the bound (5.7) transfers for Tsuch that with T-probability16we have V (æ<t)VT(æ<t)<6: ThereforeT[V (æ<t)VT(æ<t)6]<6and with!0we get that V (æ<t) VT(æ<t)!0ast!1in probability. The value function is bounded, thus it also converges in mean. The following example shows that the Thompson sampling policy is not strongly asymptotically optimal. However, we expect that strong asymptotic optimality can be achieved with Thompson sampling by resampling at every time step (with strong assumptions on the discount function). However, for practical purposes resampling in every time step is very ineﬃcient. Example 5.28 (Thompson Sampling is not Strongly Asymptotically Optimal) .Deﬁne A:=f;g,E:=f0;1=2;1g, and assume geometric discounting (Example 4.5). Con- sider the following class of environments M:=f1;1;2;:::g(transitions are labeled with action, reward): s0 s1 s2;1 2 ;0 ;0 ;0;0s0 s1 s2s3 s4;1 2 t<k :;0 ;0 ;0;0tk:;0 ;0;0 ;1 ;0 1 k §5.4Asymptotic Optimality 89 Environment kworks just like environment 1except that after time step k, the path to states3gets unlocked. The class Mis a class of deterministic weakly communicat- ing POMDPs (but as a POMDP khas more than 5 states). The optimal policy in environment 1is to always take action , the optimal policy for environment kis to take action fort<kand then take action in states1and action otherwise. Suppose the policy Tis acting in environment 1. Since it is asymptotically optimal in the class M, it has to take actions froms0inﬁnitely often: for t < k environment kis indistinguishable from 1, so the posterior for kis larger or equal to the prior. Hence there is always a constant chance of sampling kuntil taking actions , at which point all environments kforktbecome falsiﬁed. If the policy Tdecides to explore and take the ﬁrst action , it will be in state s1. Letæ<tdenote the current history. Then the 1-optimal action is and V 1(æ<t) = (1 ) 0 + 1 2+ 21 2+::: = 2: The next action taken by Tissince any optimal policy for any sampled environment that takes action once, takes that action again (and we are following that policy for an"t-eﬀective horizon). Hence VT1(æ<t)(1 ) 0 + 0 + 21 2+ 31 2+::: = 2 2: ThereforeV 1VT1( 2)=2>0. This happens inﬁnitely often with probability one and thus we cannot get almost sure convergence. 3 If the Bayesian mixture is inside the class M(as it is the case for the class MCCS LSC), then we can assign a prior probability that is arbitrarily close to 1. Since the posterior ofisthesameastheprior, ThompsonsamplingwillactaccordingtotheBayesoptimal policy most of the time. This means the Bayes-value of Thompson sampling can be very good; formally, V ()VT () =(T)can be made arbitrarily small. In contrast, the Bayes-value of Thompson sampling can also be very bad: Suppose you have a class of (n+ 1)-armed bandits indexed 1;:::;nwhere bandit igives reward 1"on arm 1, reward 1on armi+ 1, and reward 0on all other arms. For geometric discounting and "<(1 )=(2 ), it is Bayes optimal to pull arm 1while Thompson sampling will explore on average n=2arms until it ﬁnds the optimal arm. The Bayes- valueofThompsonsamplingis 1=(n n1)incontractto (1")achievedbyBayes. For a horizon of n, the Bayes optimal policy suﬀers a regret of "nand Thompson sampling a regret ofn=2, which is much larger for small ". 5.4.4 Almost Sure in Cesàro Average vs. in Mean It might appear that convergence in mean is more natural than the convergence of Cesàro averages of weak asymptotic optimality. However, both notions are not so fundamentally diﬀerent because they both allow an inﬁnite number of bad mistakes (actions that lead to V V being large). Asymptotic optimality in mean allows bad 90 Optimality mistakes as long as their probability converges to zero; weak asymptotic optimality allows bad mistakes as long as the total time spent on bad mistakes grows sublinearly. Note that according to Example 5.19 making bad mistakes inﬁnitely often is necessary for asymptotic optimality. Theorem 5.24 shows that weak asymptotic optimality is possible in any countable class of stochastic environments. However, this requires the additional condition that the eﬀective horizon grows sublinearly, Ht("t)2o(t), while Theorem 5.25 does not require any condition on the discount function. Generally, weak asymptotic optimality and asymptotic optimality in mean are in- comparable because the notions of convergence are incomparable for (bounded) random variables. First, for deterministic sequences (i.e. deterministic policies in deterministic environments), convergence in mean is equivalent to (regular) convergence, which is impossible by Theorem 5.20. Second, convergence in probability (and hence conver- gence in mean for bounded random variables) does not imply almost sure convergence of Cesàro averages (Stoyanov, 2013, Sec. 14.18). We leave open the question whether the policyTis weakly asymptotically optimal. 5.5 Regret Regret is how many expected rewards the agent forfeits by not following the best in- formed policy. Deﬁnition 5.29 (Regret).Theregretof a policy in environment is Rm(;) := sup 0E0 "mX t=1rt# E "mX t=1rt# : Note that regret is undiscounted and always nonnegative. Moreover, the space of possible diﬀerent policies for the ﬁrst mactions is ﬁnite and we assumed the set of actionsAand the set of percepts Eto be ﬁnite (Assumption 4.6c), so the supremum is always attained by some policy (not necessarily the -optimal policy because that policy uses discounting). Diﬀerent problem classes have diﬀerent regret rates, depending on the structure and the diﬃculty of the problem class. Multi-armed bandits provide a (problem- independent)worst-caseregretboundof (p km)wherekisthenumberofarms(Bubeck and Bianchi, 2012). In MDPs the lower bound is (p SAdm )whereSis the number of states,Athe number of actions, and dthe diameter of the MDP (Auer et al., 2010). For a countable class of environments given by state representation functions that map histories to MDP states, a regret of ~O(m2=3)is achievable assuming the resulting MDP is weakly communicating (Nguyen et al., 2013). A problem class is considered learnable if there is an algorithm that has a sublinear regret guarantee. The following example shows that the general reinforcement learning problem is not learnable because the agent can get caught in a trap and be unable to recover. §5.5Regret 91 Example 5.30 (Linear Regret; Hutter, 2005, Sec. 5.3.2) .Consider the following two environments 1and2. In environment 1actionleads to hell (reward 0forever) and actionleads to heaven (reward 1forever). Environment 2behaves just the same, except that both actions are swapped. hell heavenreward = 0 reward = 1 hell heavenreward = 0 reward = 1 1 2 The policy that takes action in the ﬁrst time step performs well in 2but performs poorly in1. Likewise, the policy that takes action in the ﬁrst time step performs well in1but performs poorly in 2. Regardless of which policy we adopt, our regret is always linear in one of the environments 1or2: Rm(; 1) =m R m(; 2) = 0 Rm(; 1) = 0 Rm(; 2) =m 3 To achieve sublinear regret we need to ensure that the agent can recover from mistakes. Formally, we make the following assumption. Deﬁnition 5.31 (Recoverability) .An environment satisﬁes the recoverability as- sumption iﬀ sup E [V (æ<t)]E [V (æ<t)]!0ast!1: Recoverability compares following the worst policy fort1time steps and then switching to the optimal policy to having followed from the beginning. The recoverability assumption states that switching to the optimal policy at any time step enables the recovery of most of the value: it has to become less costly to recover from mistakes as time progresses. This should be regarded as an eﬀect of the discount function: if the (eﬀective) horizon grows, recovery becomes easier because the optimal policy has more time to perform a recovery. Moreover, recoverability is on the optimal policy, in contrast to the notion of ergodicity in MDPs which demands returning to a starting state regardless of the policy. Remark 5.32 (Weakly Communicating POMDPs are Recoverable) .If the eﬀective horizon is growing, Ht(")!1ast!1, then any weakly communicating ﬁnite state POMDP satisﬁes the recoverability assumption. 3 5.5.1 Sublinear Regret in Recoverable Environments This subsection is dedicated to the following theorem that connects asymptotic opti- mality in mean to sublinear regret. 92 Optimality Theorem 5.33 (Sublinear Regret in Recoverable Environments) .If the discount func- tion satisﬁes Assumption 5.34, the environment satisﬁes the recoverability assump- tion, andis asymptotically optimal in mean in the class fg, thenRm(;)2o(m). Assumption 5.34 (Discount Function) .Let the discount function be such that (a) t>0for allt, (b) tis monotone decreasing in t, and (c)Ht(")2o(t)for all">0. This assumption demands that the discount function is somewhat well-behaved: the function has no oscillations, does not become 0, and the horizon is not growing too fast. It is satisﬁed by geometric discounting (Example 4.5): (a) t>0, (b) monotone decreasing, and (c) Ht(") =dlog "e2o(t). The problem with geometric discounting is that it makes the recoverability assump- tion very strong: since the horizon is not growing, the environment has to enable faster recovery as time progresses; in this case weakly communicating POMDPs are notre- coverable. A choice with Ht(")!1that satisﬁes Assumption 5.34 is subgeometric discounting t:=ep t=p t(see Table 4.1). If the items in Assumption 5.34 are violated, Theorem 5.33 can fail: •If t= 0for some time steps t, our policy does not care about those time steps and might take actions that have large regret. •Similarly if oscillates between high values and very low values: our policy might take high-regret actions in time steps with comparatively lower -weight. •If the horizon grows linearly, inﬁnitely often our policy might spend some constant fraction of the current eﬀective horizon exploring, which incurs a cost that is a constant fraction of the total regret so far. To prove Theorem 5.33 we require the following technical lemma. Lemma 5.35 (Value and Regret) .Let" > 0and assume the discount function satisﬁes Assumption 5.34. Let (dt)t2Nbe a sequence of numbers with jdtj1for allt. If there is a time step t0with 1 t1X k=t kdk<"8tt0 (5.8) thenmX t=1dtt0+"(mt0+ 1) +1 +" 1"Hm("): Proof.This proof essentially follows the proof of Hutter (2006b, Thm. 17). §5.5Regret 93 By Assumption 5.34a we have t>0for alltand hence t>0for allt. By Assumption 5.34b have that is monotone decreasing, so we get for all n2N t=1X k=t kt+n1X k=t t+1X k=t+n k=n t+ t+n: And withn:=Ht(")this yields tHt(") t1t+Ht(") t1">0: (5.9) In particular, this bound holds for all tand">0. Next, we deﬁne a series of nonnegative weights (bt)t1such that mX t=t0dk=mX t=t0bt tmX k=t kdk: This yields the constraints tX k=t0bk k t= 18tt0: The solution to these constraints is bt0=t0 t0;andbt=t tt t1fort>t 0: (5.10) Thus we get mX t=t0bt=t0 t0+mX t=t0+1t tt t1 =m+1 m+mX t=t0t tt+1 t =m+1 m+mt0+ 1 Hm(") 1"+mt0+ 1 for all">0according to (5.9). Finally, mX t=1dtt0X t=1dt+mX t=t0bt tmX k=t kdk t0+mX t=t0bt t1X k=t kdkmX t=t0bt t1X k=m+1 kdk 94 Optimality and using the assumption (5.8) and dt1, <t0+mX t=t0bt"+mX t=t0btm+1 t t0+"Hm(") 1"+"(mt0+ 1) +mX t=t0btm+1 t For the latter term we substitute (5.10) to get mX t=t0btm+1 t=m+1 t0+mX t=t0+1m+1 tm+1 t1 =m+1 mHm(") 1" with (5.9). Proof of Theorem 5.33. Let(m)m2Ndenote any sequence of policies, such as a se- quence of policies that attain the supremum in the deﬁnition of regret. We want to show that Em"mX t=1rt# E "mX t=1rt# 2o(m): For d(m) k:=Em[rk]E [rk] (5.11) we have1d(m) k1since we assumed rewards to be bounded between 0and1. Because the environment satisﬁes the recoverability assumption we have E [V (æ<t)]E [V (æ<t)]!0ast!1;and sup mE [V (æ<t)]Em[V (æ<t)]!0ast!1; so we conclude that sup mE [V (æ<t)]Em[V (æ<t)]!0 by the triangle inequality and thus sup mEm[V (æ<t)]E [V (æ<t)]!0ast!1: (5.12) By assumption the policy is asymptotically optimal in mean, so we have E [V (æ<t)]E [V (æ<t)]!0ast!1; and with (5.12) this combines to sup mEm[V (æ<t)]E [V (æ<t)]!0ast!1: §5.5Regret 95 FromV (æ<t)Vm(æ<t)we get lim sup t!1 sup mEm[Vm(æ<t)]E [V (æ<t)] 0: (5.13) For02f; 1;2;:::gwe have E0 [V0 (æ<t)] =E0 " 1 tE0 "1X k=t krkæ<t## =E0 " 1 t1X k=t krk# =1 t1X k=t kE0 [rk]; so from (5.11) and (5.13) we get lim sup t!1sup m1 t1X k=t kd(m) k0: Let" > 0. We choose t0independent of mand large enough such that we get supmP1 k=t kd(m) k=t< "for alltt0. Now we let m2Nbe given and apply Lemma 5.35 to get Rm(;) m=Pm k=1d(m) k mt0+"(mt0+ 1) +1+" 1"Hm(") m: SinceHt(")2o(t)according to Assumption 5.34c we get lim supm!1Rm(;)=m 0. Example 5.36 (The Converse of Theorem 5.33 is False) .Letbe a two-armed Bernoulli bandit with means 0and1and suppose we are using geometric discount- ing with discount factor 2[0;1). This environment is recoverable. If our policy pulls the suboptimal arm exactly on time steps 1;2;4;8;16;:::, regret will be log- arithmic. However, on time steps t= 2nforn2Nthe value diﬀerence V V is deterministically at least 1 >0. 3 Note that Example 5.36 does not rule out weak asymptotic optimality. 5.5.2 Regret of the Optimal Policy and Thompson sampling We get the following immediate consequence. Corollary 5.37 (Sublinear Regret for the Optimal Discounted Policy) .If the discount function satisﬁes Assumption 5.34 and the environment satisﬁes the recoverability assumption, then Rm( ;)2o(m). Proof.From Theorem 5.33 since the policy is (trivially) asymptotically optimal in fg. If the environment does not satisfy the recoverability assumption, regret may be lineareven on the optimal policy : the optimal policy maximizes discounted rewards 96 Optimality and this short-sightedness might incur a tradeoﬀ that leads to linear regret later on if the environment does not allow recovery. Corollary 5.38 (Sublinear Regret for Thompson Sampling) .If the discount function satisﬁes Assumption 5.34 and the environment 2Msatisﬁes the recoverability assumption, then Rm(T;)2o(m)for the Thompson sampling policy T. Proof.From Theorem 5.25 and Theorem 5.33. 5.6 Discussion In this work, we disregard computational constraints. Because of this, our agents learn very eﬃciently and we can focus on the way they balance exploration and exploitation. So which balance is best? 5.6.1 The Optimality of AIXI Bayesian reinforcement learning agents make the tradeoﬀ between exploration and ex- ploitationintheBayesoptimalway. Maximizingexpectedrewardsaccordingtoanypos- itive prior does not lead to enough exploration to achieve asymptotic optimality (Theo- rem 5.22); the prior’s bias is retained indeﬁnitely. For bad priors this can cause serious malfunctions: the dogmatic prior deﬁned in Section 5.2.2 can prevent a Bayesian agent from taking a single exploratory action ; exploration is restricted to cases where the ex- pected future payoﬀ falls below some prespeciﬁed ">0. However, this problem can be alleviated by adding an extra exploration component to AIXI: Lattimore (2013) shows that BayesExp is weakly asymptotically optimal (Theorem 5.24). So instead, we may ask the following weaker questions. Does AIXI succeed in every (ergodic) ﬁnite-state (PO)MDP, bandit problem, or sequence prediction task? Our results imply that without further assumptions on the prior, we cannot answer any of the preceding questions in the aﬃrmative. Using a dogmatic prior (Theorem 5.5), we can make AIXI follow any computable policy as long as that policy produces rewards that are bounded away from zero. •In a sequence prediction task that gives a reward of 1for every correctly predicted bit and 0otherwise, a policy that correctly predicts every third bit will receive an average reward of 1=3. With a-dogmatic prior, AIXI thus only predicts a third of the bits correctly, and hence is outperformed by a uniformly random predictor. However, if we have a constant horizon of length 1, AIXIdoessucceed in sequence prediction (Hutter, 2005, Sec. 6.2.2). If the horizon is this short, the agent is so hedonistic that no threat of hell can deter it. •In a (PO)MDP a dogmatic prior can make AIXI get stuck in any loop that provides nonzero expected rewards. §5.6Discussion 97 •In a bandit problem, a dogmatic prior can make AIXI get stuck on any arm which provides nonzero expected rewards. These results apply not only to AIXI, but generally to Bayesian reinforcement learn- ing agents. Any Bayesian mixture over nonrecoverable environments is susceptible to dogmatic priors if we allow an arbitrary reweighing of the prior. Notable exceptions are classes of environment that allow policies that are strongly asymptotically optimal regardless of the history (Theorem 5.23). For example, the class of all ergodic MDPs for an unbounded eﬀective horizon; in this case the Bayes optimal policy is strongly asymptotically optimal (Hutter, 2005, Thm. 5.38). Note that in contrast to our results, this requires that that the agent uses a Bayes-mixture over a class of ergodic MDPs. Moreover, Bayesian agents still perform well at learning and achieve on-policy value convergence(Corollary4.20): theposteriorbeliefaboutthevalueofapolicy converges to the true value of while following :V (æ<t)V (æ<t)!0ast!1-almost surely. Since this holds for any policy, in particular it holds for the Bayes optimal policy . This means that the Bayes agent learns to predict those parts of the environment that it sees. But if it does not explore enough, then it will not learn other parts of the environment that are potentially more rewarding. Hutter (2005, Claim 5.12) claims: We expect AIXI to be universally optimal. Our work seriously challenges Hutter’s claim: no nontrivial and non-subjective optimal- ity results for AIXI remain (see Table 5.3). Until new arguments for AIXI’s optimality are put forward, we have to regard AIXI as a relativetheory of intelligence, dependent on the choice of the prior. 5.6.2 Natural Universal Turing Machines The choice of the UTM has been a big open question in algorithmic information theory for a long time. The Kolmogorov complexity of a string depends on this choice. How- ever, there are invariance theorems (Li and Vitányi, 2008, Thm. 2.1.1 & Thm. 3.1.1) which state that changing the UTM changes Kolmogorov complexity only by a con- stant. When using the Solomonoﬀ prior Mto predict any deterministic computable binary sequence, the number of wrong predictions is bounded by the Kolmogorov com- plexity of the sequence (Corollary 3.56). Due to the invariance theorem, changing the UTM changes the number of errors only by a constant. In this sense, compression and prediction work for any choice of UTM. For AIXI, there can be no invariance theorem; in Section 5.2 we showed that a bad choice for the UTM can have drastic consequences. Our negative results can guide future search for a naturalUTM: the UTMs used to deﬁne the indiﬀerence prior (Theo- rem 5.4), the dogmatic prior (Theorem 5.5), and the Gödel prior (Theorem 5.8) should be considered unnatural. But what are other desirable properties of a UTM? A remarkable but unsuccessful attempt to ﬁnd natural UTMs is due to Müller (2010). It takes the probability that one universal machine simulates another according 98 Optimality name deﬁned in KU(U0) KU0(U) indiﬀerence prior Theorem 5.4 K(U) +K(m) +O(1) m dogmatic prior Theorem 5.5 K(U) +K() +K(") +O(1)dlog2"e Gödel prior Theorem 5.8 K(U) +K(PA) +O(1) 0 Table 5.2: Upper bounds to compiler sizes of the UTMs used in the proofs of Sec- tion 5.2.KU(U0)is the number of extra bits to run the ‘bad’ UTM U0on the ‘good’ UTMU,KU0(U)is the number of extra bits to run UonU0.K(U)denotes the length of the shortest program for UonU. to the length of their respective compilers and searches for a stationary distribution. Unfortunately, no stationary distribution exists. Alternatively, we could demand that the UTM U0that we use for the universal prior has a small compiler on the reference machine U(Hutter, 2005, p. 35). Moreover, we coulddemandthereverse, thatthereferencemachine Uhasasmallcompileron U0. The idea is that this should limit the amount of bias one can introduce by deﬁning a UTM that has very small programs for very complicated and ‘unusual’ environments. Unfor- tunately, this just pushes the choice of the UTM to the reference machine. Table 5.2 lists compiler sizes of the UTMs constructed in this thesis. 5.6.3 Asymptotic Optimality A policy is asymptotically optimal if the agent learns to act optimally in any environ- ment from the class M. We discussed two asymptotically optimal policies. BayesExp is weakly asymptotically optimal if the horizon grows sublinearly (Theorem 5.24) and Thompson sampling is asymptotically optimal in mean (Theorem 5.25). Both policies commit to exploration for several steps. As stated in Example 5.19: To achieve asymptotic optimality, the agent needs to explore inﬁnitely often for an entire eﬀective horizon. This is why weak asymptotic optimality is impossible if the horizon grows linearly (The- orem 5.21): if the agent explores for an entire eﬀective horizon, it spoils a signiﬁcant fraction of the average. Thompson sampling explores whenever it draws a bad sample. BayesExp explores if the maximal expected information gain is above some threshold. Both policies commit to exploration for the entire eﬀective horizon. The exploration performed by Thompson sampling is qualitatively diﬀerent from the exploration by BayesExp (Lattimore, 2013, Ch. 5). BayesExp performs phases of exploration in which it maximizes the expected information gain. This explores the environment class completely, even achieving oﬀ-policy prediction (Orseau et al., 2013, Thm. 7). In contrast, Thompson sampling only explores on the optimal policies, and in some environment classes this will not yield oﬀ-policy prediction. So in this sense the §5.6Discussion 99 Optimality Issue/Comment -optimal policy requires to know the true environment in advance Pareto optimality always satisﬁed (Theorem 5.3) Bayes optimality same as maximal intelligence balanced Pareto optimality same as maximal intelligence (Proposition 5.12) maximal intelligence highly dependent on the prior (Corollary 5.15 and Corollary 5.16) PAC strongvariantofasymptoticoptimalityinprobability asymptotic optimality Thompson sampling (Theorem 5.25) and BayesExp (Lattimore, 2013), but not AIXI (Orseau, 2013) sublinear regret impossible in general environments, but possible with recoverability (Theorem 5.33) Table 5.3: Proposed notions of optimality (Hutter, 2002a, 2005; Legg and Hutter, 2007b) and their issues. Asymptotic optimality stands out to be the only nontrivial objective optimality notion for general reinforcement learning. explorationmechanismofThompsonsamplingismorereward-orientedthanmaximizing information gain. However, asymptotic optimality has to be taken with a grain of salt. It provides no incentive to the agent to avoid traps in the environment. Once the agent gets caught in a trap, all actions are equally bad and thus optimal: asymptotic optimality has been achieved. Even worse, an asymptotically optimal agent has to explore all the traps because they might contain hidden treasure. This brings us to the following impossibility result for non-recoverable environment classes. Either the agent gets caught in a trap or it is not asymptotically optimal.1 5.6.4 The Quest for Optimality Theorem 5.3 shows that Pareto optimality is trivial in the class of all computable en- vironments. Bayes optimality, Balanced Pareto optimality, and maximal Legg-Hutter intelligence are equivalent (Proposition 5.12 and Proposition 5.10). Corollary 5.15 and Corollary 5.16 show that this notion is highly subjective because it depends on the choice of the prior. Moreover, according to Corollary 5.17, any computable policy is nearly balanced Pareto optimal. For ﬁnite horizons, there are priors such that every policy is balanced Pareto optimal (Theorem 5.4). Sublinear regret is impossible in general environments (Example 5.30). However, if the environment is recoverable (Def- inition 5.31), then Theorem 5.33 shows that asymptotic optimality in mean implies sublinear regret. In summary, asymptotic optimality is the only nontrivial and objec- tive notion of optimality for the general reinforcement learning problem (Problem 4.2): 1This formulation was suggested by Toby Ord. 100 Optimality it is both satisﬁable (Theorem 5.24 and Theorem 5.25) and objective because it does not depend on a prior probability measure over the environment class M. Table 5.3 summarized the notions of optimality discussed in this chapter. Ouroptimalitynotionsare tail events : anyﬁnitenumberoftimestepsareirrelevant; the agent can be arbitrarily lazy. Asymptotic optimality requires only convergence in the limit. In recoverable environments we can always achieve sublinear regret after any ﬁnite interaction. All policies with ﬁnite horizon are Bayes optimal according to Theorem 5.4 and Corollary 5.6. Overall, there is a dichotomy between the asymptotic nature of our optimality notions and the use of discounting to prioritize the present over the future. Ideally, we would aim for ﬁnite guarantees instead, such as precise regret bounds or PAC convergence rates, but without additional assumptions this is impossible in this general setting. This leaves us with the main question of this chapter unanswered (Hutter, 2009b, Sec. 5): What is a good optimality criterion for general reinforcement learning? Chapter 6 Computability I simply keep a few spare halting oracles around. — Marcus Hutter Given inﬁnite computation power, many traditional AI problems become trivial: play- ing chess, go, or backgammon can be solved by exhaustive expansion of the game tree. Yet other problems seem diﬃcult still; for example, predicting the stock market, driv- ing a car, or babysitting your nephew. How can we solve these problems in theory? Solomonoﬀ induction and AIXI are proposed answers to this question. Both Solomonoﬀ induction and AIXI are known to be incomputable. But not all incomputabilities are equal. The arithmetical hierarchy speciﬁes diﬀerent levels of com- putability based on oracle machines : each level in the arithmetical hierarchy is com- puted by a Turing machine which may query a halting oracle for the respective lower level. Our agents are useless if they cannot be approximated in practice, i.e., by a regular Turing machine. Therefore we posit that any ideal for a ‘perfect agent’ needs to belimit computable (0 2). The class of limit computable functions is the class of functions that admit an anytime algorithm . In Section 6.2 we consider various diﬀerent ﬂavors of Solomonoﬀ induction: Solo- monoﬀ’s prior M(Example 3.5) is only a semimeasure and not a measure: it assigns positive probability that the observed string has only ﬁnite length. This can be cir- cumvented by normalizing M. Solomonoﬀ’s normalization Mnorm(Deﬁnition 2.16) preserves the ratio M(x1)=M(x0)and is limit computable. If instead we mix only over programs that compute inﬁnite strings, we get a semimeasure M(3.6), which can be normalized to Mnorm. Moreover, when predicting a sequence, we are primarily inter- ested in the conditional probability M(xyjx)(respectively Mnorm(xyjy),M(xyjx), orMnorm(xyjx)) that the currently observed string xis continued with y. We show that bothMandMnormare limit computable, while MandMnormare not. Table 6.1 summarizes our computability results for Solomonoﬀ induction. For MDPs, planning is already P-complete for ﬁnite and inﬁnite horizons (Papadim- itriou and Tsitsiklis, 1987). In POMDPs, planning is undecidable (Madani et al., 1999, 2003). The existence of a policy whose expected value exceeds a given threshold is PSPACE-complete (Mundhenk et al., 2000), even for purely epistemic POMDPs in which actions do not change the hidden state (Sabbadin et al., 2007). In Section 6.3 we derive hardness results for planning in general semicomputable environments; this environment class is even more general than POMDPs. We show that optimal policies 101 102 Computability Qf(x;q)2XQjQ(x)>qg f(x;y;q )2XXQjQ(xyjx)>qg M 0 1n0 1 0 2n(0 1[0 1) Mnorm 0 2n(0 1[0 1) 0 2n(0 1[0 1) M 0 2n0 2 0 3n(0 2[0 2) Mnorm 0 3n(0 2[0 2) 0 3n(0 2[0 2) Table 6.1: The computability results on M,Mnorm,M, andMnormproved in Sec- tion 6.2. Lower bounds on the complexity of MandMnormare given only for speciﬁc universal Turing machines. Agent Optimal "-Optimal AIMU 0 2 0 1 AINU 0 3,0 2-hard 0 2,0 1-hard AIXI 0 3,0 1-hard 0 2,0 1-hard Entropy-seeking 0 3 0 2 Information-seeking 0 3 0 2 BayesExp 0 3 0 2 Table 6.2: Computability results for diﬀerent agent models derived in Section 6.3, Section 6.5, and Section 6.6. AIMU denotes the optimal policy in a computable envi- ronment and AINU denotes the optimal policy in a semicomputable environment (see Section 4.1). Hardness results for AIXI are with respect to a speciﬁc universal Turing machine; hardness results for -optimal policies are with respect to a speciﬁc environ- ment2MCCS LSC. Results for entropy-seeking and information-seeking policies are only for ﬁnite horizons. are0 2-hard and"-optimal policies are undecidable. Moreover, we show that by default, AIXI is not limit computable. When picking the next action, two or more actions might have the same value (expected future rewards). The choice between them is easy, but determining whether such a tie exists is diﬃcult. This problem can be circumvented by settling for an "-optimal policy; we get a limit- computable agent with inﬁnite horizon. However, these results rely on the recursive deﬁnition of the value function. In contrast, Hutter (2005) deﬁnes the value function as the limit of the iterative value function. In Section 6.4 we compare these two deﬁnitions and show that the recursive deﬁnition correctly maximizes expected rewards and has better computability properties. In Section 6.5 we show that for ﬁnite horizons both the entropy-seeking and the information-seeking agent are 0 3-computable and have limit-computable "-optimal policies. BayesExp (Section 4.3.3) relies on optimal policies that are generally not §6.1Background on Computability 103 limit computable. In Section 6.6 we give a weakly asymptotically optimal agent based on BayesExp that is limit computable. Table 6.2 summarizes our results on the com- putability of these agents. In this chapter we illustrate the environments used in the proofs of our theorems in the form of ﬂowcharts. They should be read as follows. Circles denote stochastic nodes, rectangles denote environment nodes , and diamonds denote the agent’s choice nodes. Transitions out of stochastic nodes are labeled with transition probabilities, transitions out of environment nodes are labeled with percepts, and transitions out of choice nodes are labeled with actions. The initial node is marked with a small incoming arrow (see for example Figure 6.3). By Assumption 4.6b the worst possible outcome is getting reward 0forever, thus we label such states as hell. Analogously, getting reward 1forever is the best possible outcome, thus we label such states as heaven. 6.1 Background on Computability 6.1.1 The Arithmetical Hierarchy A setANis0 niﬀ there is a quantiﬁer-free formula such that k2A() 9k18k2:::Qnkn(k;k1;:::;kn) (6.1) whereQn=8ifnis even,Qn=9ifnis odd (Nies, 2009, Def. 1.4.10). (We can also think ofas a computable relation.) A set ANis0 niﬀ its complement NnA is0 n. The formula on the right side of (6.1) is a 0 n-formula and its negation is a 0 n-formula . It can be shown that we can add any bounded quantiﬁers and duplicate quantiﬁers of the same type without changing the classiﬁcation of A. The setAis0 n iﬀAis0 nandAis0 n. We get that 0 1as the class of recursively enumerable sets, 0 1 as the class of co-recursively enumerable sets and 0 1as the class of recursive sets. The setANis0 n-hard ( 0 n-hard, 0 n-hard)iﬀ for any set B20 n(B20 n, B20 n),Bis many-one reducible to A, i.e., there is a computable function fsuch thatk2B$f(k)2A(Nies, 2009, Def. 1.2.1). We get 0 n0 n+10 n+1:::and 0 n0 n+10 n+1:::. This hierarchy of subsets of natural numbers is known as thearithmetical hierarchy . By Post’s Theorem (Nies, 2009, Thm. 1.4.13), a set is 0 nif and only if it is recur- sively enumerable on an oracle machine with an oracle for a 0 n1-complete set. An oracle for 0 1is called a halting oracle . 6.1.2 Computability of Real-valued Functions We ﬁx some encoding of rational numbers into binary strings and an encoding of binary strings into natural numbers. From now on, this encoding will be done implicitly wherever necessary. Deﬁnition 6.1 (0 n-,0 n-,0 n-computable) .A function f:X!Ris called 0 n- computable ( 0 n-computable, 0 n-computable) iﬀ the setf(x;q)2XQjf(x)>qgis 0 n(0 n,0 n). 104 Computability f(x;q)jf(x)qg f (x;q)jf(x)<qg fis computable 0 1 0 1 fis lower semicomputable 0 1 0 1 fis upper semicomputable 0 1 0 1 fis limit computable 0 2 0 2 fis0 n-computable 0 n 0 n fis0 n-computable 0 n 0 n fis0 n-computable 0 n 0 n Table 6.3: Connection between the computability of real-valued functions and the arithmetical hierarchy. A0 1-computable function is called computable , a0 1-computable function is called lower semicomputable , and a 0 1-computable function is called upper semicomputable . A0 2-computable function fis called limit computable , because there is a computable functionsuch that lim k!1(x;k) =f(x): The program that limit computes fcan be thought of as an anytime algorithm forf: we can stop at any time kand get a preliminary answer. If the program ran long enough (which we do not know), this preliminary answer will be close to the correct one. Limit-computable sets are the highest level in the arithmetical hierarchy that can be approached by a regular Turing machine. Above limit-computable sets we necessarily need some form of halting oracle. See Table 6.3 for the deﬁnition of lower/upper semicomputable and limit-computable functions in terms of the arithmetical hierarchy. Lemma 6.2 (Computability of Arithmetical Operations) .Letn > 0and letf;g: X!Rbe two 0 n-computable functions. Then (a)f(x;y)jf(x)>g(y)gis0 n, (b)f(x;y)jf(x)g(y)gis0 n, (c)f+g,fg, andfgare0 n-computable, and (d)f=gis0 n-computable if g(x)6= 0for allx. (e)logfis0 n-computable if f(x)>0for allx. Proof.We only prove this for n > 1. Sincef;gare0 n-computable, they are limit computable on a level n1oracle machine. Let be the function limit computing fon the oracle machine, and let be the function limit computing gon the oracle machine: f(x) = lim k!1(k;x)andg(y) = lim k!1 (k;y): By assumption, both and are0 n1-computable. §6.2The Complexity of Solomonoﬀ Induction 105 (a) LetG:=f(x;y;q )jg(y)< qg, and letF:=f(x;y;q )jq < f (x)g, both of which are in 0 nby assumption. Hence there are 0 n-formulas'Gand'Fsuch that (x;y;q )2G()'G(x;y;q ) (x;y;q )2F()'F(x;y;q ) Nowf(x)> g(y)if and only if9q:(x;y;q )2G\F, which is equivalent to the 0 n-forumla 9q:'G(x;y;q )^'F(x;y;q ): (b) Follows from (a). (c) Addition, subtraction, and multiplication are continuous operations. (d) Division is discontinuous only at g(x) = 0. We show this explicitly. By assumption, for any">0there is ak0such that for all k>k 0 j(x;k)f(x)j<"andj (x;k)g(x)j<": We assume without loss of generality that "<jg(x)j, sinceg(x)6= 0by assumption. (x;k) (x;k)f(x) g(x) =(x;k)g(x)f(x) (x;k) (x;k)g(x) j(x;k)g(x)f(x)g(x)j+jf(x)g(x)f(x) (x;k)j j (x;k)g(x)j <"jg(x)j+jf(x)j" j (x;k)g(x)j withj (x;k)g(x)j=j (x;k)jjg(x)j>(jg(x)j")jg(x)j, <"jg(x)j+jf(x)j (jg(x)j")jg(x)j"!0! 0; thereforef(x)=g(x) = limk!1(x;k)= (x;k). (e) Follows from the fact that the logarithm is computable. 6.2 The Complexity of Solomonoﬀ Induction In this section, we derive the computability results for Solomonoﬀ’s prior as stated in Table 6.1. SinceMis lower semicomputable, Mnormis limit computable by Lemma 6.2 (c) and (d). When using the Solomonoﬀ prior M(or one of its sisters Mnorm,M, or Mnormdeﬁned in Deﬁnition 2.16 and Equation 3.6) for sequence prediction, we need 106 Computability M(xyjx)>q() 8m9k:(xy;k) (x;m)>q() 9k9m08mm0:(xy;k) (x;m)>q Figure 6.1: A0 2-formula and an equivalent 0 2-formula deﬁning conditional M. Here (x;k)denotes a computable function that lower semicomputes M(x). to compute the conditional probability M(xyjx) =M(xy)=M(x)for ﬁnite strings x;y2X. BecauseM(x)>0for all ﬁnite strings x2X, this quotient is well-deﬁned. Theorem 6.3 (Complexity of M,Mnorm,M, andMnorm). (a)M(x)is lower semicomputable (b)M(xyjx)is limit computable (c)Mnorm(x)is limit computable (d)Mnorm(xyjx)is limit computable (e)M(x)is0 2-computable (f)M(xyjx)is0 3-computable (g)Mnorm(x)is0 3-computable (h)Mnorm(xyjx)is0 3-computable Proof.(a) By Li and Vitányi (2008, Thm. 4.5.2). Intuitively, we can run all programs in parallel and get monotonely increasing lower bounds for M(x)by adding 2jpj every time a program phas completed outputting x. (b) From (a) and Lemma 6.2d since M(x)>0(see also Figure 6.1). (c) By Lemma 6.2cd, and M(x)>0. (d) By (iii), Lemma 6.2d since Mnorm(x)M(x)>0. (e) Letbeacomputablefunctionthatlowersemicomputes M. SinceMisasemimea- sure,M(xy)P zM(xyz), henceP y2XnM(xy)is nonincreasing in nand thus M(x)>qiﬀ8n9kP y2Xn(xy;k )>q. (f) From (v) and Lemma 6.2d since M(x)>0. (g) From (v) and Lemma 6.2d. (h) From (vi) and Lemma 6.2d since Mnorm(x)M(x)>0. §6.2The Complexity of Solomonoﬀ Induction 107 We proceed to show that these bounds are in fact the best possible ones. If M were 0 1-computable, then so would be the conditional semimeasure M(j). Thus the M-adversarial sequence z1:1deﬁned in Example 3.42 would be computable and hence corresponds to a computable deterministic measure . However, we have M(z1:t)2t by construction, so dominance M(x)w()(x)withw()>0yields a contradiction witht!1: 2tM(z1:t)w()(z1:t) =w()>0 By the same argument, the normalized Solomonoﬀ prior Mnormcannot be 0 1-com- putable. However, since it is a measure, 0 1- or 0 1-computability would entail 0 1- computability. ForMandMnormwe prove the following two lower bounds for speciﬁc universal Turing machines. Theorem 6.4 (Mis not Limit Computable) .There is a universal Turing machine U0 such that the setf(x;q)jMU0(x)>qgis not in 0 2. Proof.Assume the contrary and let A20 2n0 2andbe a quantiﬁer-free ﬁrst-order formula such that n2A() 8k9i:(n;k;i ): (6.2) For eachn2N, we deﬁne the program pnas follows. 1:procedure pn 2:output 1n+10 3:k 0 4:whiletruedo 5:i 0 6:whilenot(n;k;i )do 7: i i+ 1 8:k k+ 1 9: output 0 Each program pnalways outputs 1n+10. Furthermore, the program pnoutputs the inﬁnite string 1n+101if and only if n2Aby (6.2). We deﬁne U0as follows using our reference machine U. •U0(1n+10): Runpn. •U0(00p): RunU(p). •U0(01p): RunU(p)and bitwise invert its output. By construction, U0is a universal Turing machine. No pnoutputs a string starting with 0n+11, thereforeMU0(0n+11) =1 4 MU(0n+11) +MU(1n+10) . Hence MU0(1n+10) = 2n21A(n) +1 4MU(1n+10) +1 4MU(0n+11) = 2n21A(n) +MU0(0n+11) 108 Computability Ifn =2A, thenMU0(1n+10) =MU0(0n+11). Otherwise, we have jMU0(1n+10) MU0(0n+11)j= 2n2. Now we assume that MU0is limit computable, i.e., there is a computable function :XN!Qsuch that limk!1(x;k) =MU0(x). We get that n2A() lim k!1(0n+11;k)(1n+10;k)2n2; thusAis limit computable, a contradiction. Corollary 6.5 (Mnormis not 0 2- or 0 2-computable) .There is a universal Turing machineU0such thatf(x;q)jMnormU0(x)>qgis not in 0 2or0 2. Proof.SinceMnorm =cM, there exists a k2Nsuch that 2k<c(even if we do not know the value of k). We can show that the set f(x;q)jMnormU0(x)>qgis not in 0 2 analogously to the proof of Theorem 6.4, using n2A() lim k!1(0n+11;k)(1n+10;k)2kn2: IfMnormwere 0 2-computable or 0 2-computable, this would imply that Mnormis0 2- computable since Mnormis a measure, a contradiction. SinceM() = 1, we haveM(xj) =M(x), so the conditional probability M(xyjx) has at least the same complexity as M. Analogously for MnormandMnormsince they are measures. For M, we have that M(xj) =Mnorm(x), so Corollary 6.5 applies. All that remains to prove is that conditional Mis not lower semicomputable. Theorem 6.6 (Conditional Mis not Lower Semicomputable) .The setf(x;xy;q )j M(xyjx)>qgis not recursively enumerable. We gave a diﬀerent, more complicated proof in Leike and Hutter (2015b). The following, much simpler and more elegant proof is due to Sterkenburg (2016, Prop. 3). Proof.Assume to the contrary that M(xyjx)is lower semicomputable. Let a6=b2X. We construct an inﬁnite string xby deﬁning its initial segments =:x(0)@x(1)@ x(2)@:::@x. At every step n, we enumerate strings y2Xuntil one is found satisfyingM(ajx(n)y)1=2; then setx(n+ 1) :=x(n)yb. This implies that for inﬁnitely many tthere is an nsuch thatM(bjx<t) =M(bjx(n)y)1M(aj x(n)y)1=2. Since we assumed M(j)to be lower semicomputable, the inﬁnite stringxis computable, and hence M(xtjx<t)!1by Corollary 3.55. But this contradicts M(bjx<t)1=2inﬁnitely often. 6.3 The Complexity of AINU, AIMU, and AIXI 6.3.1 Upper Bounds Inthissection,wederiveupperboundsonthecomputabilityofAINU,AIMU,andAIXI. Except for Corollary 6.14, all results in this section apply generally to any 2MCCS LSC. §6.3The Complexity of AINU, AIMU, and AIXI 109 Since the Bayesian mixture 2MCCS LSC, they apply to AIXI even though they are stated for AINU. In order to position AINU in the arithmetical hierarchy, we need to encode policies as sets of natural numbers. For the rest of this chapter, we assume that policies are deterministic, thuscanberepresentedasrelationsover (AE )A. Theserelationsare easily identiﬁed with sets of natural numbers by encoding the history into one natural number. From now on this translation of policies into sets of natural numbers will be done implicitly wherever necessary. Lemma 6.7 (Policies are in 0 n).If a policyis0 nor0 n, thenis0 n. Proof.Let'be a 0 n-formula ( 0 n-formula) deﬁning , i.e.,'(h;a)holds iﬀ(h) =a. We deﬁne the formula '0, '0(h;a) :=^ a02Anfag:'(h;a0): The set of actions Ais ﬁnite, hence '0is a0 n-formula ( 0 n-formula). Moreover, '0is equivalent to '. To compute the optimal policy, we need to compute the optimal value function. The following lemma gives an upper bound on the computability of the value function for environments in MCCS LSC. Lemma 6.8 (Complexity of V ).For every2MCCS LSC, and every lower semicom- putable discount function , the function V is0 2-computable. Proof.The explicit form of the value function (4.2) has numerator lim m!1maxX æt:mmX i=t (i)ri(e1:ika1:i); and denominator (e<tka<t)t. The numerator is nondecreasing in mbecause we assumed rewards to be nonnegative (Assumption 4.6b). Hence both numerator and denominator are lower semicomputable functions, so Lemma 6.2d implies that V is 0 2-computable. From the optimal value function V we get the optimal policy according to (4.4). However, in cases where there is more than one optimal action, we have to break an argmax tie. This happens iﬀ V (h) =V (h)for two potential actions 6=2A. This equality test is more diﬃcult than determining which is larger in cases where they are unequal. Thus we get the following upper bound. Theorem 6.9 (Complexity of Optimal Policies) .For any environment , ifV is 0 n-computable, then there is an optimal policy for the environment that is 0 n+1. 110 Computability Proof.To break potential ties, we pick an (arbitrary) total order onAthat speciﬁes which actions should be preferred in case of a tie. We deﬁne (h) =a:()^ a0:a0aV (ha)>V (ha0) ^^ a0:aa0V (ha)V (ha0):(6.3) Thenis a-optimal policy according to (4.4). By assumption, V is0 n-computable. By Lemma 6.2ab V (ha)>V (ha0)is0 nandV (ha)V (ha0)is0 n. Therefore the policydeﬁned in (6.3) is a conjunction of a 0 n-formula and a 0 n-formula and thus 0 n+1. Corollary 6.10 (ComplexityofAINU) .AINU is 0 3for every environment 2MCCS LSC. Proof.From Lemma 6.8 and Theorem 6.9. Usuallywedonotmindtakingslightlysuboptimalactions. Thereforeactuallytrying to determine if two actions have the exact same value seems like a waste of resources. In the following, we consider policies that attain a value that is always within some ">0of the optimal value. Theorem 6.11 (Complexity of "-Optimal Policies) .For any environment , ifV is 0 n-computable, then there is an "-optimal policy " for the environment that is 0 n. Proof.Let" > 0be given. Since the value function V (h)is0 n-computable, the setV":=f(ha;q)jjqV (ha)j< "= 2gis in 0 naccording to Deﬁnition 6.1. Hence we compute the values V (ha0)until we get within "=2for everya02Aand then choose the action with the highest value so far. Formally, let be an arbitrary total order onAthat speciﬁes which actions should be preferred in case of a tie. Without loss of generality, we assume "= 1=k, and deﬁne Qto be an"=2-grid on [0;1], i.e., Q:=f0;1=2k;2=2k;:::; 1g. We deﬁne " (h) =a:() 9 (qa0)a02A2QA:^ a02A(ha0;qa0)2V" ^^ a0:a0aqa>qa0^^ a0:aa0qaqa0 ^the tuple (qa0)a02Ais minimal with respect to the lex. ordering on QA:(6.4) This makes the choice of aunique. Moreover, QAis ﬁnite sinceAis ﬁnite, and hence (6.4) is a 0 n-formula. Corollary 6.12 (Complexity of "-Optimal AINU) .For any environment 2MCCS LSC, there is an "-optimal policy for AINU that is 0 2. Proof.From Lemma 6.8 and Theorem 6.11. §6.3The Complexity of AINU, AIMU, and AIXI 111 Purgatory Heaven6= (ht) (ht)r= 0r= 1 Figure 6.2: The environment from the proof of Theorem 6.15. The agent gets reward 0as long as it follows AIXI’s policy that is assumed to be computable. Once the agent deviates from , it gets reward 1. We get a contradiction because AIXI can learn this environment, so it will eventually decide to take an action that leads to heaven. Corollary 6.13 (Complexity of "-Optimal AIXI) .For any lower semicomputable prior there is an "-optimal policy for AIXI that is 0 2. Proof.FromCorollary6.12sinceforanylowersemicomputableprior, thecorresponding Bayesian mixture is inMCCS LSC. If the environment 2MCCM compis a measure, i.e., assigns zero probability to ﬁnite strings, then we get computable "-optimal policies. Corollary 6.14 (Complexity of AIMU) .If the environment 2MCCM compis a measure and the discount function is computable, then AIMUis limit computable ( 0 2), and "-optimal AIMUis computable ( 0 1). Proof.Let">0be the desired accuracy. We can truncate the limit m!1in (4.2) at the"=2-eﬀective horizon Ht("=2), since everything after Ht("=2)can contribute at most"=2to the value function. Any lower semicomputable measure is computable (Li and Vitányi, 2008, Lem. 4.5.1). Therefore V as given in (4.2) is composed only of computable functions, hence it is computable according to Lemma 6.2. The claim now follows from Theorem 6.9 and Theorem 6.11. 6.3.2 Lower Bounds We proceed to show that the bounds from the previous section are the best we can hope for. In environment classes where ties have to be broken, AINU has to solve 0 2-hard problems (Theorem 6.16). These lower bounds are stated for particular environments 2MCCS LSC. Throughout this section, we assume that t>0for allt. We also construct universal mixtures that yield bounds on "-optimal policies. There is an"-optimal AIXIthat solves 0 1-hard problems (Theorem 6.17). For arbitrary uni- versal mixtures, we prove the following weaker statement that only guarantees incom- putability. Theorem 6.15 (NoAIXIis computable) .AIXIis not computable for any universal Turing machine U. 112 Computability This theorem follows from the incomputability of Solomonoﬀ induction. By the on-policy value convergence theorem (Corollary 4.20) AIXI succeeds to predict the environment’s behavior for its own policy. If AIXI were computable, then there would be computable environments more powerful than AIXI: they can simulate AIXI and anticipate its prediction, which leads to a contradiction. Proof.Assume there is a computable policy that is optimal in the mixture . We deﬁne a deterministic environment , theadversarial environment to . The environ- mentgives rewards 0as long as the agent follows the policy , and rewards 1once the agent deviates. Formally, we ignore observations by setting O:=f0g, and deﬁne (r1:tka1:t) :=8 >>>>< >>>>:1if8kt:ak= ((ar)<k)andrk= 0; 1if8kt:rk=1ki wherei:= minfjjaj6= ((ar)<j)g;and 0otherwise: See Figure 6.2 for an illustration of this environment. The environment is computable because the policy was assumed to be computable. Suppose acts in, then by Theorem 4.19 AIXI learns to predict perfectly on policy : V (æ<t)V (æ<t)!0ast!1 -almost surely ; since both andare deterministic. Because V (h<t) = 0by deﬁnition of , we get V (æ<t)!0. Therefore we ﬁnd a tlarge enough such that V (æ<t)<w()whereæ<t is the interaction history of in. A policywith(æ<t)6= (æ<t), gets a reward of1in environment for all time steps after t, henceV (æ<t) = 1. With linearity of V (æ<t)in(Lemma 4.14), V (æ<t)w()(e1:tka1:t) (e1:tka1:t)V (æ<t)w(); since(e1:tka1:t) = 1(is deterministic), V (æ<t) = 1, and(e1:tka1:t)1. Now we get a contradiction: w()>V (æ<t) = sup 0V0 (æ<t)V (æ<t)w() For the remainder of this section, we ﬁx the action space to be A:=f;gwith actionfavored in ties. The percept space is ﬁxed to a tuple of binary observations and rewards,E:=Of 0;1gwithO:=f0;1g. Theorem 6.16 (AINU is 0 2-hard).There is an environment 2MCCS LSCsuch that AINU is 0 2-hard. Proof.LetAbe a any 0 2-set, and let be a quantiﬁer-free formula such that n2A() 8i9k(n;i;k ): (6.5) §6.3The Complexity of AINU, AIMU, and AIXI 113 n++ Heaven1=2o= 1 r= 0 1=2o= 0 r= 0 19k(n;i;k) o= 0 r= 1 Figure 6.3: The environment ifrom the proof of Theorem 6.16. The mixture over class of environments M:=f0;1;:::gMCCS LSCforces AINU to solve 0 2-hard problems: Action is preferred (because of a tie) iﬀ it leads to heaven, which is the case iﬀ9k(n;i;k ). We deﬁne a class of environments M:=f1;2;:::gwhere each iis deﬁned as follows. i((or)1:mka1:m) :=8 >>>>>>>>>< >>>>>>>>>:2mifo1:m= 1mand8tm:rt= 0; 2n1if9n:1n0vo1:mv1n01andan+2= andrt=1t>n+1and9k(n;i;k ); 2n1if9n:1n0vo1:mv1n01andan+2= andrt=1t>n+1;and 0otherwise: See Figure 6.3 for an illustration of these environments. Every iis a chronological conditional semimeasure by deﬁnition and every iis lower semicomputable since is quantiﬁer-free, so MMCCS LSC. We deﬁne our environment as a mixture over M, :=X i2N2i1i; the choice of the weights on the environments iis arbitrary but positive. Let be an optimal policy for the environment and recall that the action is preferred in ties. We claim that for the -optimal policy , n2A() (1n0) =: (6.6) This enables us to decide whether n2Agiven the policy , hence proving (6.6) concludes this proof. Letn;i2Nbegiven, andsupposeweareinenvironment iandobserve 1n0. Taking actionnext yields reward 1forever; taking action next yields a reward of 1if there 114 Computability Semi-Heaven Heaven o= 1n0 19k(n;i;k)o= 0 r= 1=2 o= 0 r= 1 Figure 6.4: The environment from the proof of Theorem 6.17, which forces AIXI to solve 0 1-hard problems. It functions just like until the observation history is 1n0. Then, action is preferred iﬀ heaven is accessible, i.e., iﬀ 9k(n;i;k ). is aksuch that(n;i;k )holds. If this is the case, then V i(1n0) = n+2=V i(1n0); and otherwise V i(1n0) = 0<n+2=V i(1n0) (omitting the ﬁrst n+ 1actions and rewards in the argument of the value function). We can now show (6.6): By (6.5), n2Aif and only if for all ithere is aksuch that(n;i;k ), which happens if and only if V i(1n0) = n+2for alli2N, which is equivalent to V (1n0) = n+2, which in turn is equivalent to (1n0) =since V (1n0) = n+2and action is favored in ties. Theorem 6.17 (Some"-optimal AIXIare0 1-hard).There is a universal Turing ma- chineU0and an">0such that any "-optimal policy for AIXIis0 1-hard. Proof.LetAbe a 0 1-set andbe a quantiﬁer-free formula such that n+ 12Aiﬀ 9k(n;k). We deﬁne the environment ((or)1:tka1:t) :=8 >>>>>>>>>>>>< >>>>>>>>>>>>:((or)1:nka1:n)if9n:o 1:n= 1n10andan= and8t0>n:ot0= 0^rt0=1 2; ((or)1:nka1:n)if9n:o 1:n= 1n10andan= and8t0>n:ot= 0^rt= 1 and9k(n1;k); ((or)1:tka1:t)if@n:o 1:n= 1n10;and 0 otherwise: See Figure 6.4 for an illustration. The environment mimics the universal environment §6.4Iterative Value Function 115 until the observation history is 1n10. Taking the action next gives rewards 1=2 forever. Taking the action next gives rewards 1forever ifn2A, otherwise the environment ends at some future time step. Therefore we want to take action if and only if n2A. We have that 2MCCS LSCsince2MCCS LSCandis quantiﬁer-free. We deﬁne0:=1 2+1 8. By Lemma 4.24 0is a universal lower semicomputable semimeasure. Let n2Abe given and let h2(AE )nbe any history with observations o1:n= 1n10. Since(1n10ja1:n) =(1n10ja1:n)bydeﬁnition, theposteriorweights ofandin0are equal to the prior weights, analogously to the proof of Theorem 5.5. In the following, we use the linearity of V 0 in(Lemma 4.14), and the fact that values are bounded between 0and1(Assumption 4.6b). If there is a ksuch that(n1;k) holds, V 0(h)V 0(h) =1 2V 0 (h)1 2V 0 (h) +1 8V 0 (h)1 8V 0 (h) 1 21 4+ 01 8=1 8; and similarly if there is no ksuch that(n1;k)holds, then V 0(h)V 0(h) =1 2V 0 (h)1 2V 0 (h) +1 8V 0 (h)1 8V 0 (h) 1 40 + 01 8=1 8: In both casesjV 0(h)V 0(h)j>1=9. Hence we pick ":= 1=9and get for every "-optimal policy " 0that" 0(h) =if and only if n2A. Note the diﬀerences between Theorem 6.15 and Theorem 6.17: the former talks about optimal policies and shows that they are not computable, but is agnostic towards the underlying universal Turing machine. The latter talks about "-optimal policies and gives a stronger hardness result, at the cost of depending on one particular universal Turing machine. 6.4 Iterative Value Function Historically,AIXI’svaluefunctionhasbeendeﬁnedslightlydiﬀerentlytoDeﬁnition4.10, using a limit extension of an iterative deﬁnition of the value function. This deﬁnition is the more straightforward to come up with in AI: it is the natural adaptation of (op- timal) minimax search in zero-sum games to the (optimal) expectimax algorithm for stochastic environments. In this section we discuss the problems with this deﬁnition. To avoid confusion with the recursive value function V , we denote the iterative value function with W .1 Deﬁnition 6.18 (Iterative Value Function; Hutter, 2005, Def. 5.30) .Theiterative 1In Leike and Hutter (2015a) the use of the symbols VandWis reversed. 116 Computability Agent Optimal "-Optimal Iterative AINU 0 4,0 3-hard 0 3,0 2-hard Iterative AIXI 0 4,0 2-hard 0 3,0 2-hard Iterative AIMU 0 2 0 1 Table 6.4: Computability results for diﬀerent agent models that use the iterative value function derived in Section 6.4. Hardness results for AINU are with respect to a speciﬁc environment 2MCCS LSC. valueof a policy in an environment given history æ<tis W (æ<t) :=1 tlim m!1X et:m(e1:mje<tka1:m)mX k=t (k)rk ift>0andW (æ<t) := 0ift= 0whereai:=(e<i)for allit. Theoptimal iterative value is deﬁned as W (h) := supW (h). Analogously to (4.2), we can write W using the max-sum-operator: W (æ<t) =1 tlim m!1maxX æt:m(e1:mje<tka1:m)mX k=t (k)rk (6.7) We useiterative AINU for the-optimal policy according to the iterative value function, and iterative AIXI for the-optimal policy according to the iterative value function. Note that iterative AIMU coincides with AIMU since is a measure by convention. Generally, our environment 2MCCS LSCis only a semimeasure and not a measure, i.e., there is a history æ<tatsuch that 1>X et2E(etje<tka1:t): In such cases, with positive probability the environment does not produce a new perceptet. If this occurs, we shall use the informal interpretation that the environment ended, but our formal argument does not rely on this interpretation. The following proposition shows that for a semimeasure 2MCCS LSCthat is not a measure, iterative AINU does not maximize -expected rewards. Recall that (1)states the discount of the ﬁrst reward. In the following, we assume without loss of generality that (1)>0, i.e., we are not indiﬀerent about the reward received in time step 1. Proposition 6.19 (Iterative AINU is not a -Expected Reward Maximizer) .For any " >0there is an environment 2MCCS LSCthat is not a measure and a policy that receives a total of (1)rewards in, but iterative AINU receives only " (1)rewards in . §6.4Iterative Value Function 117 Hell r="r= 0 r= 1 Figure 6.5: The environment from the proof of Proposition 6.19. Action yields reward 1, but subsequently the environment ends. Action yields reward "and the environment continues forever. Iterative AINU will prefer the suboptimal action , because it conditions on surviving forever. Informally, the environment is deﬁned as follows. In the ﬁrst time step, the agent chooses between the two actions and. Taking action gives a reward of 1, and subsequently the environment ends. Action gives a reward of ", but the environment continues forever. There are no other rewards in this environment. See Figure 6.5. From the perspective of -expected reward maximization, it is better to take action , however iterative AINU takes action . Proof of Proposition 6.19. Let" >0. We ignore observations and set E:=f0;";1g, A:=f;g. The environment is formally deﬁned by (r1:tka1:t) :=8 >>< >>:1ifa1=andr1= 1andt= 1 1ifa1=andr1="andrk= 081<kt 0otherwise: Taking action ﬁrst, we have (r1:tka2:t) = 0fort>1(the environment ends in time step 2given history ). Hence we conclude V () =1 tlim m!1X r1:m(r1:mka2:m)mX k=1 (k)rk= 0: Taking action ﬁrst we get V () =1 tlim m!1X r1:m(r1:mka2:m)mX k=1 (k)rk= (1) 1": Since (1)>0and" > 0, we haveV ()> V (), and thus iterative AINU will use a policy that plays action ﬁrst, receiving a total discounted reward of " (1). In contrast, any policy that takes action ﬁrst receives a larger total discounted reward of (1). Whether it is reasonable to assume that our environment has a nonzero probability 118 Computability of ending is a philosophical debate we do not want to engage in here; see Martin et al. (2016) for a discussion. Instead, we have a diﬀerent motivation to use the recursive over the iterative value function: the latter has worse computability properties. Concretely, we show that "-optimal iterative AIXI has to solve 0 2-hard problems and that there is an environment 2MCCS LSCsuch that iterative AINU has to solve 0 3-hard problems. In contrast, using the recursive value function, "-optimal AIXI is 0 2according to Corollary 6.12 and AINU is 0 3according to Corollary 6.10. The central diﬀerence between V andW is that for V all obtained rewards matter, but for W only the rewards in timelines that continue indeﬁnitely. In this sense the value function W conditions on surviving forever. If the environment is a measure, then the history is inﬁnite with probability one, and so V andW coincide. Hence this distinction is not relevant for AIMU, only for AINU and AIXI. Lemma 6.20 (Complexity of W ).For every2MCCS LSC, the function W is0 2- computable. Proof.Multiplying (6.7) with t(e<tka<t)yieldsW (æ<t)>qif and only if lim m!1maxX æt:m(e1:mka1:m)mX k=t (k)rk>qt(e<tka<t): (6.8) The inequality’s right side is lower semicomputable, hence there is a computable func- tion such that (`)%qt(e<tka<t) =:q0as`!1. (In contrast to the recursive value function, this quantity is not increasing in m.) For a ﬁxed m, the left side is also lower semicomputable, therefore there is a computable function such that (m;k)%maxX æt:m(e1:mka1:m)mX k=t (k)rk=:f(m)ask!1. We already know that the limit of f(m)form!1exists (uniquely), hence we can write (6.8) as lim m!1f(m)>q0 () 8m09mm0:f(m)>q0 () 8m09mm09k:(m;k)>q0 () 8`8m09mm09k:(m;k)> (`); which is a 0 2-formula. Note that in the ﬁnite horizon case where mis ﬁxed, the value function W is0 2- computable by Lemma 6.2d, since W (æ<t) =f(m)=q0. In this case, we get the same computability results for iterative AINU as we did in Section 6.3.1. Corollary 6.21 (Complexity of Iterative AINU) .For any environment 2MCCS LSC, iterative AINU is 0 4and there is an "-optimal iterative AINU that is 0 3. §6.4Iterative Value Function 119 n++ Hell Conditional Heaven1=2o= 1 r= 0 1=2o= 0 r= 0 18t0t9k(n;i;t0;k)o= 0 r= 0 o= 0 r= 1 Figure 6.6: The environment ifrom the proof of Theorem 6.22. The mixture over class of environments M:=f0;1;:::g MCCS LSCforces iterative AINU to solve 0 3-hard problems. ‘Conditional Heaven’ is a node that yields reward 1until :9k (n;i;t;k ), at which point the environment ends. Hence action is preferred in environment iiﬀ conditional heaven lasts forever (because otherwise (:::) = 0and henceV (:::) = 0) which is the case iﬀ 8t9k(n;i;t;k ). Proof.From Theorem 6.9, Theorem 6.11, and Lemma 6.20. We proceed to show corresponding lower bounds as in Section 6.3.2. For the rest of this section we assume t>0for allt. Theorem 6.22 (Iterative AINU is 0 3-hard).There is an environment 2MCCS LSCsuch that iterative AINU is 0 3-hard. Proof.The proof is analogous to the proof of Theorem 6.16. Let Abe any 0 3set, then there is a quantiﬁer-free formula such that n2A() 9i8t9k(n;i;t;k ): We deﬁne the environments isimilar to the proof of Theorem 6.16, except for two changes: •We replace9k(n;i;k )with8t0t9k(n;i;t0;k). •We switch actions and: action‘checks’ the formula and action gives a sure reward of 0. 120 Computability Formally, i((or)1:tka1:t) :=8 >>>>>>>>>>>>< >>>>>>>>>>>>:2tifo1:t= 1tand8t0t:rt0= 0; 2n1if9n:1n0vo1:tv1n01andan+2= and8t0t:rt0= 0; 2n1if9n:1n0vo1:tv1n01andan+2= and8t0t:rt0=1t0>n+1 and8t0t9k(n;i;t0;k);and 0otherwise: See Figure 6.6 for an illustration of the environment i. Everyiis a chronological conditional semimeasure by deﬁnition, so M:=f0;1;:::gMCCS LSC. Furthermore, everyiis lower semicomputable since is quantiﬁer-free. We deﬁne our environment as a mixture over M, :=X i2N2i1i; the choice of the weights on the environments iis arbitrary but positive. We get for the-optimal policy analogously to the proof of Theorem 6.16 (1n0) =() 9i8t0t9k(n;i;t0;k)()n2A; since action is preferred in ties. Analogously to Theorem 6.15, we can show that iterative AIXI is not computable. We also get the following lower bound. Theorem 6.23 (Some"-optimal iterative AIXIare0 2-hard).There is a universal mixture0and an">0such that any policy that is "-optimal according to the iterative value for environment 0is0 2-hard. Proof.LetAbe a 0 2-set anda quantiﬁer-free formula such that n2A() 8t9k(n;t;k ): We proceed analogous to the proof of Theorem 6.17 except that we choose 8t0 t9k(n;t;k )as a condition for reward 1after playing action . §6.4Iterative Value Function 121 Semi-Heaven Conditional Heaven o= 1n0 18t0<t9k(n;i;t0;k)o= 0 r= 1=2 o= 0 r= 1 Figure6.7: Theenvironment fromtheproofofTheorem6.23, whichforces "-optimal iterative AIXI to solve 0 2-hard problems. It functions just like until the observation history is 1n0. Then, action is preferred iﬀ conditional heaven never ends, i.e., iﬀ 8t9k(n;t;k ). Deﬁne the environment ((or)1:tka1:t) :=8 >>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>:((or)1:n+1ka1:n+1)if9n:1n0vo1:tv1n01 andan+1= and8n+ 1<kt:rk= 1=2; ((or)1:n+1ka1:n+1)if9n:1n0vo1:tv1n01 andan+1= and8n+ 1<kt:rk= 1 and8t0t9k(n;t;k ); ((or)1:tka1:t) if@n:1n0vo1:tv1n01;and 0 otherwise: See Figure 6.7 for an illustration of the environment . The environment mimics the universal environment until the observation history is 1n0. The next action always gives rewards 1=2forever, while action gives rewards 1forever iﬀn2A. We have that isalowersemicomputablesemimeasuresince isalowersemicomputablesemimeasure andis quantiﬁer-free. We deﬁne 0=1 2+1 8. By Lemma 4.24, 0is a universal lower semicomputable semimeasure. Let n2Abe given and let h2(AO )x+1be any history with observations o1:n+1= 1n0. In the following, we use the linearity of W in (analogously to Lemma 4.14). If 8t9k(n;t;k ), then W 0(h)W 0(h) =1 2W (h)1 2W (h) +1 8W (h)1 8W (h) 1 21 4+ 01 8=1 8; 122 Computability and similarly if:8t9k(n;t;k ), then W 0(h)W 0(h) =1 2W (h)1 2W (h) +1 8W (h)1 8W (h) 1 40 + 01 8=1 8: In both casesjW 0(h)W 0(h)j>1=9, hence with ":= 1=9we have for an "-optimal policy" 0that" 0(h) =if and only if n2A. 6.5 The Complexity of Knowledge-Seeking Recall the deﬁnition of the optimal entropy-seeking value V;m Entand the optimal infor- mation-seeking value V;m IGfrom Section 4.3.2. Using the results from Section 6.3 we can show that "-optimal knowledge-seeking agents are limit computable, and optimal knowledge-seeking agents are 0 3. Corollary 6.24 (Computability of Knowledge-Seeking Values) .For ﬁxedm, the value functionsV;m EntandV;m IGare limit computable. Proof.This follows from Lemma 6.2 (c-e) since ,, andware lower semicomputable. Corollary 6.25 (Computability of Knowledge-Seeking Policies) .For entropy-seeking and information-seeking agents there are limit-computable "-optimal policies and 0 3- computable optimal policies. Proof.Follows from Corollary 6.24, Theorem 6.9, and Theorem 6.11. Note that if we used an inﬁnite horizon with discounting in Deﬁnition 4.25 or Deﬁni- tion 4.26, then we cannot retain this computability result without further assumptions: we would need that the value functions increase monotonically as m!1, as they do for the recursive value function from Deﬁnition 4.10. However, entropy is not a monotone function and may decrease if there are events whose probability converges to something >1=2. For the entropy-seeking value function this happens for histories drawn from a deterministic environment since!, so the conditionals converge to1. Similarly, for the information-seeking value function, the posterior belief in one (deterministic) environment might become larger than 1=2(depending on the prior and the environment class). Therefore we generally only get that discounted versions of V Ent andV IGare0 3analogously to Lemma 6.20. Hence optimal discounted entropy-seeking and optimal discounted information-seeking policies are in 0 4by Theorem 6.9 and their corresponding "-optimal siblings are 0 3by Theorem 6.11. 6.6 A Limit Computable Weakly Asymptotically Optimal Agent According to Theorem 6.16, optimal reward-seeking policies are generally 0 2-hard, and foroptimalknowledge-seekingpoliciesCorollary6.25showsthattheyare 0 3. Therefore §6.7Discussion 123 we get that BayesExp is 0 3: Corollary 6.26 (BayesExp is 0 3).For any universal mixture , BayesExp is 0 3. Proof.From Corollary 6.10, Corollary 6.24, and Corollary 6.25. However, we do not know BayesExp to be limit computable, and we expect it not to be. However, we can approximate it using "-optimal policies preserving weak asymptotic optimality. Theorem6.27 (ALimit-ComputableWeaklyAsymptoticallyOptimalAgent) .If there is a nonincreasing computable sequence of positive reals ("t)t2Nsuch that"t!0and Ht("t)=(t"t)!0ast! 1, then there is a limit-computable policy that is weakly asymptotically optimal in the class of all computable stochastic environments. Proof.By Corollary 6.10, there is a limit-computable 2t-optimal reward-seeking policy t for the universal mixture . By Corollary 6.25 there are limit-computable t=2- optimal information-seeking policies t IGwith horizon t+Ht("t). We deﬁne a policy analogously to Algorithm 1 with t IGandt instead of the optimal policies. From Corollary 6.24 we get that V IGis limit computable, so the policy is limit computable. Furthermore, t is2t-optimal and 2t!0, soVt (æ<t)!V (æ<t)ast!1. Now we can proceed analogously to the proof of Lattimore (2013, Thm. 5.6), which consists of three parts. First, it is shown that the value of the -optimal reward- seeking policy converges to the optimal value for exploitation time steps (line 6 in Algorithm 1) in the sense that V !V . This carries over to the 2t-optimal policy t , since the key property is that on exploitation steps, V IG<"t; i.e.,only exploits if potential knowledge-seeking value is low. In short, we get for exploitation steps Vt (æ<t)!V (æ<t)!V (æ<t)!V (æ<t)ast!1: Second, it is shown that the density of exploration steps vanishes. This result carries over since the condition V IG(æ<t)>"tthat determines exploration steps is exactly the same as for BayesExp and t IGis"t=2-optimal. Third, the results of part one and two are used to conclude that is weakly asymp- totically optimal. This part carries over to our proof. 6.7 Discussion When using Solomonoﬀ’s prior for induction, we need to evaluate conditional probabil- ities. We showed that conditional MandMnormare limit computable (Theorem 6.3), and thatMandMnormare not limit computable (Theorem 6.4 and Corollary 6.5). Table 6.1 on page 102 summarizes our computability results on various versions of Solomonoﬀ’s prior. Theses results implies that we can approximate MorMnormfor prediction, but not the measure mixture MorMnorm. In some cases, normalized priors have advantages. As illustrated in Example 4.27, unnormalized priors can make the entropy-seeking agent mistake the entropy gained 124 Computability from the probability assigned to ﬁnite strings for knowledge. From MnormMwe get thatMnormpredicts just as well as M, and by Theorem 6.3 we can use Mnormwithout losing limit computability. Table 6.2 on page 102 summarizes our computability results for the agents AINU, AIXI, and AINU. AINU is 0 3and restricting to "-optimal policies decreases the level by one (Corollary 6.10 and Corollary 6.12). For environments from MCCM comp, AIMU is limit-computable and "-optimal AIMU is computable (Corollary 6.14). In Section 6.3.2 we proved that these computability bounds on AINU are generally unimprovable (The- orem 6.16 and Theorem 6.17). Additionally, we proved weaker lower bounds for AIXI independent of the universal Turing machine (Theorem 6.15) and for "-optimal AIXI for speciﬁc choices of the universal Turing machine (Theorem 6.17). When the environment has nonzero probability of not producing a new percept, the iterative deﬁnition of AINU (Deﬁnition 6.18) originally given by Hutter (2005, Def. 5.30) fails to maximize -expected rewards (Proposition 6.19). Moreover, the policies are one level higher in the arithmetical hierarchy (see Table 6.4 on page 116). We proved upper (Corollary 6.21) and lower bounds (Theorem 6.22 and Theorem 6.23). The diﬀerence between the recursive value function Vand the iterative value function Wis readily exposed in the diﬀerence between the universal prior Mand the measure mixtureM: Just like Wconditions on surviving forever, so does Meliminate the weight of programs that do not produce inﬁnite strings. Both MandWare not limit computable for this reason. We considered "-optimality to avoid having to determine argmax ties. This "does not have to be constant over time, we may let "!0ast!1at any computable rate. With this we retain the computability results of "-optimal policies and get that the value of the "(t)-optimal policy "(t) converges rapidly to the -optimal value: V (æ<t)V"(t) (æ<t)!0ast!1: In Section 4.1 we deﬁned the set MCCS LSCas the set of all lower semicomputable chronological contextual semimeasure over percepts with actions povided as side-infor- mation. When determining the probability of the next percept etin an environment , we have to compute (e1:tje<tka1:t). Alternatively, we could have deﬁned the environment as a lower semicomputable mapping from histories æ<tatto probabilities over the next percept et(this is done in Chapter 7). For the proof of Lemma 6.8 and Lemma 6.20 we only need that (e1:tka1:t)is lower semicomputable computable. While this new deﬁnition makes no diﬀerence for the computability of AINU, it matters for AIXI because in the mixture over all of these environments is no longer lower semicomputable. Any method that tries to tackle the reinforcement learning problem has to balance between exploration and exploitation. AIXI strikes this balance in the Bayesian way. However, as we showed in Section 5.2, this may not lead to enough exploration. To counteract this, we can add an explorative component to the agent, akin to knowledge- seeking agents. In Section 6.5 we show that "-optimal knowledge-seeking agents are limit computable if we use the recursive deﬁnition of the value function. §6.7Discussion 125 We set out with the goal of ﬁnding a perfect reinforcement learning agent that is limit computable. The Bayesian agent AIXI could be considered one suitable candidate, despite its optimality problems discussed in Chapter 5. Another suitable candidate are weakly asymptotically optimal agents, which in contrast to AIXI are optimal in an objective sense (see Section 5.6). We discussed BayesExp, which relies on a Solomonoﬀ prior to learn its environment and on a information-seeking component for extra explo- ration. Our results culminated in a limit-computable weakly asymptotically optimal agent based on BayesExp (Theorem 6.27). In this sense our goal has been achieved. 126 Computability Chapter 7 The Grain of Truth Problem1 AIs become friendly by playing lots of Newcomblike problems. — Eliezer Yudkowsky Consider the general setup of multiple reinforcement learning agents interacting sequen- tially in a known environment with the goal to maximize discounted reward.2Each agent knows how the environment behaves, but does not know the other agents’ be- havior. The natural (Bayesian) approach would be to deﬁne a class of possible policies that the other agents could adopt and take a prior over this class. During the interac- tion, this prior gets updated to the posterior as our agent learns the others’ behavior. Our agent then acts optimally with respect to this posterior belief. Kalai and Lehrer (1993) show that in inﬁnitely repeated games Bayesian agents converge to an "-Nash equilibrium as long as each agent assigns positive prior probability to the other agents’ policies (a grain of truth ). As an example, consider an inﬁnitely repeated prisoners dilemma between two agents. In every time step the payoﬀ matrix is as follows, where C means cooperate and D means defect. C D C3/4, 3/4 0, 1 D1, 0 1/4, 1/4 Deﬁne the set of policies :=f1;0;1;:::gwhere policy tcooperates until time steptor the opponent defects (whatever happens ﬁrst) and defects thereafter. The Bayes optimal behavior is to cooperate until the posterior belief that the other agent defects in the time step after the next is greater than some constant (depending on the discount function) and then defect afterwards. Therefore Bayes optimal behavior leads to a policy from the set (regardless of the prior). If both agents are Bayes optimal with respect to some prior, they both have a grain of truth and therefore they converge to a Nash equilibrium: either they both cooperate forever or after some ﬁnite time they both defect forever. Alternating strategies like TitForTat (cooperate ﬁrst, then play 1The idea for reﬂective oracles was developed by Jessica Taylor and Benya Fallenstein based on ideas by Paul Christiano (Christiano et al., 2013). Reﬂective oracles were ﬁrst described in Fallenstein et al. (2015a). The proof of Theorem 7.7 was sketched by Benya Fallenstein and developed by me. Except for minor editing, everything else in this chapter is my own work. 2We mostly use the terminology of reinforcement learning. For readers from game theory we provide a dictionary in Table 7.1. 127 128 The Grain of Truth Problem Reinforcement learning Game theory stochastic policy mixed strategy deterministic policy pure strategy agent player multi-agent environment inﬁnite extensive-form game reward payoﬀ/utility (ﬁnite) history history inﬁnite history path of play Table 7.1: Terminology dictionary between reinforcement learning and game theory. the opponent’s last action) are not part of the policy class , and adding them to the class breaks the grain of truth property: the Bayes optimal behavior is no longer in the class. This is rather typical; a Bayesian agent usually needs to be more powerful than its environment (see Section 6.3). We are facing the following problem. Problem 7.1 (Grain of Truth Problem; Hutter, 2009b, Q. 5j) .Find a large class of policies containing Bayesian agents with positive prior over . Until now, classes that admit a grain of truth were known only for small toy exam- ples such as the iterated prisoner’s dilemma above (Shoham and Leyton-Brown, 2009, Ch. 7.3). Foster and Young (2001) and Nachbar (1997, 2005) prove several impos- sibility results on the grain of truth problem that identify properties that cannot be simultaneously satisﬁed for classes that allow a grain of truth (see Section 7.6 for a discussion). In this chapter we present a formal solution to the grain of truth problem (Sec- tion 7.2). We assume that our multi-agent environment is computable, but it does not need to be stationary/Markov, ergodic, or ﬁnite-state. Our class of policies is large enough to contain all computable (stochastic) policies, as well as all relevant Bayes optimal policies. At the same time, our class is small enough to be limit computable. This is important because it allows our result to be computationally approximated. InSection7.3weconsiderthesettingwherethemulti-agentenvironmentisunknown to the agents and has to be learned in addition to the other agents’ behavior. A Bayes optimal agent may not learn to act optimally in unknown multi-agent environments even though it has a grain of truth . This eﬀect occurs in non-recoverable environments wheretakingonewrongactioncanmeanapermanentlossoffuturevalue. Inthiscase, a Bayesoptimalagentavoidstakingthesedangerousactionsandthereforewillnotexplore enough to wash out the prior’s bias (using the dogmatic prior from Section 5.2.2). Therefore, Bayesian agents are not asymptotically optimal , i.e., they do not always learn to act optimally (Theorem 5.22). However, asymptotic optimality is achieved by Thompson sampling because the inherent randomness of Thompson sampling leads to enough exploration to learn the entire environment class (see Section 5.4.3). This leads to our main result: if all agents useThompsonsamplingoverourclassofmulti-agentenvironments, thenforevery ">0 §7.1Reﬂective Oracles 129 they converge to an "-Nash equilibrium. This is not the ﬁrst time Thompson sampling is used in game theory (Ortega and Braun, 2014), but the ﬁrst time to show that it achieves such general positive results. The central idea to our construction is based on reﬂective oracles introduced by Fallenstein et al. (2015a,b). Reﬂective oracles are probabilistic oracles similar to halting oracles that answer whether the probability that a given probabilistic Turing machine Toutputs 1is higher than a given rational number p. The oracles are reﬂective in the sense that the machine Tmay itself query the oracle, so the oracle has to answer queries about itself. This invites issues caused by self-referential liar paradoxes of the form “if the oracle says that this machine return 1with probability >1=2, then return 0, else return 1.” Reﬂective oracles avoid these issues by being allowed to randomize if the machines do not halt or the rational number is exactlythe probability to output 1. We introduce reﬂective oracles formally in Section 7.1 and prove that there is a limit computable reﬂective oracle. For inﬁnitely repeated games practical algorithms rely on ﬁcticious play (Fudenberg and Levine, 1998, Ch. 2): the agent takes a best-response action based on the assump- tion that its opponent is playing a stationary but unknown mixed strategy estimated according to the observed empirical frequencies. If all agents converge to a stationary policy, then this is a Nash equilibrium. However, convergence is not guaranteed. The same problem occurs in multi-agent reinforcement learning (Busoniu et al., 2008). Reinforcement learning algorithms typically assume a (stationary) Markov de- cision process. This assumption is violated when interacting with other reinforcement learning agents because as these agents learn, their behavior changes and thus they are not stationary. Assuming convergence to a stationary policy is a necessary criterion to enable all agents to learn, but the process is unstable for many reinforcement learning algorithms and only empirical positive results are known (Bowling and Veloso, 2001). 7.1 Reﬂective Oracles 7.1.1 Deﬁnition First we connect semimeasures as deﬁned in Deﬁnition 2.14 to Turing machines. In Chapter 2 we used monotone Turing machines which naturally correspond to lower semicomputable semimeasures (Li and Vitányi, 2008, Sec. 4.5.2) that describe the dis- tribution that arises when piping fair coin ﬂips into the monotone machine. Here we take a diﬀerent route. Aprobabilistic Turing machine is a Turing machine that has access to an unlimited number of uniformly random coin ﬂips. Let Tdenote the set of all probabilistic Turing machines that take some input in Xand may query an oracle (formally deﬁned below). We take a Turing machine T2Tto correspond to a semimeasure TwhereT(ajx) is the probability that Toutputsa2Xwhen given x2Xas input. The value of 130 The Grain of Truth Problem T(x)is then given by the chain rule T(x) :=jxjY k=1T(xkjx<k): (7.1) ThusTgives rise to the set of semimeasures MLSCwhere the conditionals (ajx)are lower semicomputable. In contrast, in Chapter 6 we considered semimeasures whose jointprobability (7.1) is lower semicomputable. This set MLSCcontains all com- putable measures. However, MLSCis a proper subset of the set of all lower semicom- putable semimeasures because the product (7.1) is lower semicomputable, but there are some lower semicomputable semimeasures whose conditional is not lower semicom- putable (Theorem 6.6): MCCM compM LSCMCCS LSC In the following we assume that our alphabet is binary, i.e., X:=f0;1g. Deﬁnition 7.2 (Oracle).Anoracleis a function O:Tf 0;1gQ!f0;1g. Oracles are understood to be probabilistic: they randomly return 0or1. LetTOde- notethemachine T2Twhenrunwiththeoracle O, andletO Tdenotethesemimeasure induced by TO. This means that drawing from O Tinvolves two sources of randomness: one from the distribution induced by the probabilistic Turing machine Tand one from the oracle’s answers. The intended semantics of an oracle are that it takes a query (T;x;p )and returns 1if the machine TOoutputs 1on inputxwith probability greater than pwhen run with the oracle O, i.e., when O T(1jx)> p. Furthermore, the oracle returns 0if the machineTOoutputs 1on inputxwith probability less than pwhen run with the oracle O, i.e., when O T(1jx)< p. To fulﬁll this, the oracle Ohas to make statements about itself, since the machine Tfrom the query may again query O. Therefore we call oracles of this kind reﬂective oracles . This has to be deﬁned very carefully to avoid the obvious diagonalization issues that are caused by programs that ask the oracle about themselves. We impose the following self-consistency constraint. Deﬁnition7.3 (ReﬂectiveOracle) .AnoracleOisreﬂective iﬀforallqueries (T;x;p )2 Tf 0;1gQ, (a)O T(1jx)>pimpliesO(T;x;p ) = 1, and (b)O T(0jx)>1pimpliesO(T;x;p ) = 0. Ifpunder- or overshoots the true probability of O T(jx), then the oracle must reveal this information. However, in the critical case when p=O T(1jx), the oracle is allowed to return anything and may randomize its result. Furthermore, since Tmight not output any symbol, it is possible that O T(0jx) +O T(1jx)<1. In this case the oracle can reassign the non-halting probability mass to 0,1, or randomize; see Figure 7.1. §7.1Reﬂective Oracles 131 0 1O T(0jx) O T(1jx)Oreturns 1Omay randomize Oreturns 0 Figure 7.1: Answer options of a reﬂective oracle Ofor the query (T;x;p ); the rational p2[0;1]falls into one of the three regions above. The values of O T(0jx)andO T(1jx) are depicted as the length of the line segment under which they are written. Example 7.4 (Reﬂective Oracles and Diagonalization) .LetT2Tbe a probabilistic Turing machine that outputs 1O(T;;1=2)(Tcan know its own source code by quining; Kleene, 1952, Thm. 27). In other words, Tqueries the oracle about whether it is more likely to output 1or0, and then does whichever the oracle says is less likely. In this case we can use an oracle O(T;;1=2) := 1=2(answer 0or1with equal probability), which implies O T(1j) =O T(0j) = 1=2, so the conditions of Deﬁnition 7.3 are satisﬁed. In fact, for this machine Twe must have O(T;;1=2) = 1=2for all reﬂective oraclesO. 3 The following theorem establishes that reﬂective oracles exist. Theorem 7.5 (Fallenstein et al., 2015c, App. B) .There is a reﬂective oracle. Deﬁnition 7.6 (Reﬂective-Oracle-Computable) .A semimeasure is called reﬂective- oracle-computable iﬀ it is computable on a probabilistic Turing machine with access to a reﬂective oracle. For any probabilistic Turing machine T2 Twe can complete the semimeasure O T(jx)into a reﬂective-oracle-computable measure O T(jx): Using the oracle Oand a binary search on the parameter pwe search for the crossover point pwhereO(T;x;p ) goes from returning 1to returning 0. The limit point p x2Rof the binary search is random since the oracle’s answers may be random. But the main point is that the expectation of p xexists, soO T(1jx) =E[p x] = 1O T(0jx). HenceO Tis a measure. Moreover, if the oracle is reﬂective, then O T(x)O T(x)for allx2X. In this sense the oracleOcan be viewed as a way of ‘completing’ all semimeasures O Tto measures by arbitrarily assigning the non-halting probability mass. If the oracle Ois reﬂective this is consistent in the sense that Turing machines who run other Turing machines will be completed in the same way. This is especially important for a universal machine that runs all other Turing machines to induce a Solomonoﬀ prior (Example 3.5). 7.1.2 A Limit Computable Reﬂective Oracle The proof of Theorem 7.5 given by Fallenstein et al. (2015c, App. B) is nonconstructive and uses the axiom of choice. In Section 7.1.3 we give a new proof for the existence of reﬂective oracles and provide a construction that there is a reﬂective oracle that is limit computable. 132 The Grain of Truth Problem Theorem 7.7 (A Limit Computable Reﬂective Oracle) .There is a reﬂective oracle that is limit computable. This theorem has the immediate consequence that reﬂective oracles cannot be used as halting oracles. At ﬁrst, this result may seem surprising: according to the deﬁnition of reﬂective oracles, they make concrete statements about the output of probabilistic Turing machines. However, the fact that the oracles may randomize some of the time actually removes enough information such that halting can no longer be decided from the oracle output. Corollary 7.8 (Reﬂective Oracles are not Halting Oracles) .There is no probabilistic Turing machine Tsuch that for every preﬁx program pand every reﬂective oracle O, we have that O T(1jp)>1=2ifphalts andO T(1jp)<1=2otherwise. Proof.Assume there was such a machine Tand letObe the limit computable oracle from Theorem 7.7. Since Ois reﬂective we can turn Tinto a deterministic halting oracle bycallingO(T;p;1=2)whichdeterministicallyreturns 1ifphaltsand 0otherwise. Since Oislimitcomputable, wecanﬁnitelycomputetheoutputof Oonanyquerytoarbitrary ﬁnite precision using our deterministic halting oracle. We construct a probabilistic Turing machine T0that uses our halting oracle to compute (rather than query) the oracleOon(T0;;1=2)to a precision of 1=3in ﬁnite time. If O(T0;;1=2)1=3>1=2, the machine T0outputs 0, otherwise T0outputs 1. Since our halting oracle is entirely deterministic, the output of T0is entirely deterministic as well (and T0always halts), so O T0(0j) = 1orO T0(1j) = 1. Therefore O(T0;;1=2) = 1orO(T0;;1=2) = 0because Ois reﬂective. A precision of 1=3is enough to tell them apart, hence T0returns 0if O(T0;;1=2) = 1andT0returns 1ifO(T0;;1=2) = 0. This is a contradiction. A similar argument can also be used to show that reﬂective oracles are not com- putable. 7.1.3 Proof of Theorem 7.7 The idea for the proof of Theorem 7.7 is to construct an algorithm that outputs an inﬁnite series of partial oracles converging to a reﬂective oracle in the limit. The set of queries is countable, so we can assume that we have some computable enumeration of it: Tf 0;1gQ=:fq1;q2;:::g Deﬁnition 7.9 (k-Partial Oracle) .Ak-partial oracle ~Ois function from the ﬁrst k queries to the multiples of 2kin[0;1]: ~O:fq1;q2;:::;qkg!fn2kj0n2kg Deﬁnition 7.10 (Approximating an Oracle) .Ak-partial oracle ~Oapproximates an oracleOiﬀjO(qi)~O(qi)j2k1for allik. §7.1Reﬂective Oracles 133 Letk2N, let ~Obe ak-partial oracle, and let T2Tbe an oracle machine. The machineT~Othat we get when we run Twith thek-partial oracle ~Ois deﬁned as follows (this is with slight abuse of notation since kis taken to be understood implicitly). 1. RunTfor at most ksteps. 2. IfTcalls the oracle on qiforik, (a) return 1with probability ~O(qi)2k1, (b) return 0with probability 1~O(qi)2k1, and (c) halt otherwise. 3. IfTcalls the oracle on qjforj >k, halt. Furthermore, we deﬁne ~O Tanalogously to O Tas the distribution generated by the machineT~O. Lemma 7.11. If ak-partial oracle ~Oapproximates a reﬂective oracle O, thenO T(1j x)~O T(1jx)andO T(0jx)~O T(0jx)for allx2f0;1gand allT2T. Proof.This follows from the deﬁnition of T~O: when running Twith ~Oinstead ofO, we can only lose probability mass. If Tmakes calls whose index is > kor runs for more thanksteps, then the execution is aborted and no further output is generated. If T makes calls whose index ik, then ~O(qi)2k1O(qi)since ~Oapproximates O. Therefore the return of the call qiis underestimated as well. Deﬁnition 7.12 (k-Partially Reﬂective) .Ak-partial oracle ~Oisk-partially reﬂective iﬀ for the ﬁrst kqueries (T;x;p ) •~O T(1jx)>pimplies ~O(T;x;p ) = 1, and •~O T(0jx)>1pimplies ~O(T;x;p ) = 0. It is important to note that we can check whether a k-partial oracle is k-partially reﬂective in ﬁnite time by running all machines Tfrom the ﬁrst kqueries for ksteps and tallying up the probabilities to compute ~O T. Lemma 7.13. IfOis a reﬂective oracle and ~Ois ak-partial oracle that approximates O, then ~Oisk-partially reﬂective. Lemma 7.13 only holds because we use semimeasures whose conditionals are lower semicomputable. Proof.Assuming~O T(1jx)> pwe get from Lemma 7.11 that O T(1jx)~O T(1j x)> p. ThusO(T;x;p ) = 1becauseOis reﬂective. Since ~Oapproximates O, we get 1 =O(T;x;p )~O(T;x;p ) + 2k1, and since ~Oassigns values in a 2k-grid, it follows that ~O(T;x;p ) = 1. The second implication is proved analogously. 134 The Grain of Truth Problem Deﬁnition 7.14 (Extending Partial Oracles) .Ak+ 1-partial oracle ~O0extendsak- partial oracle ~Oiﬀj~O(qi)~O0(qi)j2k1for allik. Lemma 7.15. There is an inﬁnite sequence of partial oracles (~Ok)k2Nsuch that for eachk,~Okis ak-partially reﬂective k-partial oracle and ~Ok+1extends ~Ok. Proof.By Theorem 7.5 there is a reﬂective oracle O. For every k, there is a canonical k-partial oracle ~Okthat approximates O: restrictOto the ﬁrst kqueries and for any such query qpick the value in the 2k-grid which is closest to O(q). By construction, ~Ok+1extends ~Okand by Lemma 7.13, each ~Okisk-partially reﬂective. Lemma 7.16. If thek+ 1-partial oracle ~Ok+1extends the k-partial oracle ~Ok, then ~Ok+1 T(1jx)~Ok T(1jx)and~Ok+1 T(0jx)~Ok T(0jx)for allx2f0;1gand all T2T. Proof.T~Ok+1runs for one more step than T~Ok, can answer one more query and has increased oracle precision. Moreover, since ~Ok+1extends ~Ok, we havej~Ok+1(qi) ~Ok(qi)j2k1, and thus ~Ok+1(qi)2k1~Ok(qi)2k. Therefore the success to answers to the oracle calls (case 2(a) and 2(b)) will not decrease in probability. Now everything is in place to state the algorithm that constructs a reﬂective oracle in the limit. It recursively traverses a tree of partial oracles. The tree’s nodes are the partial oracles; level kof the tree contains all k-partial oracles. There is an edge in the tree from the k-partial oracle ~Okto thei-partial oracle ~Oiif and only if i=k+ 1and ~Oiextends ~Ok. For everyk, there are only ﬁnitely many k-partial oracles, since they are functions from ﬁnite sets to ﬁnite sets. In particular, there are exactly two 1-partial oracles (so the search tree has two roots). Pick one of them to start with, and proceed recursively as follows. Given a k-partial oracle ~Ok, there are ﬁnitely many (k+ 1)-partial oracles that extend ~Ok(ﬁnite branching of the tree). Pick one that is (k+1)-partially reﬂective (which can be checked in ﬁnite time). If there is no (k+1)-partially reﬂective extension, backtrack. By Lemma 7.15 our search tree is inﬁnitely deep and thus the tree search does not terminate. Moreover, it can backtrack to each level only a ﬁnite number of times because at each level there is only a ﬁnite number of possible extensions. Therefore the algorithm will produce an inﬁnite sequence of partial oracles, each extending the previous. Because of ﬁnite backtracking, the output eventually stabilizes on a sequence of partial oracles ~O1;~O2;:::. By the following lemma, this sequence converges to a reﬂective oracle, which concludes the proof of Theorem 7.7. Lemma 7.17. Let~O1;~O2;:::be a sequence where ~Okis ak-partially reﬂective k-partial oracle and ~Ok+1extends ~Okfor allk2N. LetO:= limk!1~Okbe the pointwise limit. Then (a)~Ok T(1jx)!O T(1jx)and~Ok T(0jx)!O T(0jx)ask!1for allx2f0;1g and allT2T, and §7.2A Grain of Truth 135 (b)Ois a reﬂective oracle. Proof.First note that the pointwise limit must exists because j~Ok(qi)~Ok+1(qi)j 2k1by Deﬁnition 7.14. (a) Since ~Ok+1extends ~Ok, each ~Okapproximates O. Letx2f0;1gandT2Tand considerthesequence ak:=~Ok T(1jx)fork2N. ByLemma7.16, akak+1, sothe sequence is monotone increasing. By Lemma 7.11, akO T(1jx), so the sequence is bounded. Therefore it must converge. But it cannot converge to anything strictly belowO T(1jx)by the deﬁnition of TO. (b) By deﬁnition, Ois an oracle; it remains to show that Ois reﬂective. Let qi= (T;x;p )be some query. If p<O T(1jx), then by (a) there is a klarge enough such thatp < ~Ot T(1jx)for alltk. For anytmaxfk;ig, we have ~Ot(T;x;p ) = 1 since ~Otist-partially reﬂective. Therefore 1 = limk!1~Ok(T;x;p ) =O(T;x;p ). The case 1p<O T(0jx)is analogous. 7.2 A Grain of Truth 7.2.1 Reﬂective Bayesian Agents FixOto be a reﬂective oracle. From now on, we assume that the action space A:= f;gisbinary. Wecantreatcomputablemeasuresoverbinarystringsasenvironments: the environment corresponding to a probabilistic Turing machine T2Tis deﬁned by (etjæ<tat) :=O T(yjx) =kY i=1O T(yijxy1:::yi1) wherey1:kis a binary encoding of etandxis a binary encoding of æ<tat. The actions a1:1are only contextual , and not part of the environment distribution. We deﬁne (e<tka<t)analogously to (4.1). LetT1;T2;:::be an enumeration of all probabilistic Turing machines in Tthat use an oracle. We deﬁne the class of reﬂective environments MO re :=n O T1;O T2;:::o : This is the class of all environments computable on a probabilistic Turing machine with reﬂective oracle O, that have been completed from semimeasures to measures using O. Analogously to Section 4.3.1, we deﬁne a Bayesian mixture over the class MO re . Letw2MO re be a lower semicomputable prior probability distribution on MO re . Possible choices for the prior include the Solomonoﬀ prior w O T := 2K(T), where K(T)denotes the length of the shortest input to some universal Turing machine that encodesT. We deﬁne the corresponding Bayesian mixture (etjæ<tat) :=X 2MO re w(jæ<t)(etjæ<tat) (7.2) 136 The Grain of Truth Problem wherew(jæ<t)is the (renomalized) posterior, w(jæ<t) :=w()(e<tka<t) (e<tka<t): (7.3) The mixture is lower semicomputable on an oracle Turing machine because the pos- teriorw(jæ<t)is lower semicomputable. Hence there is an oracle machine Tsuch that=O T. We deﬁne its completion :=O Tas the completion of O T. This is the distribution that is used to compute the posterior. There are no cyclic dependencies sinceis called on the shorter history æ<t. We arrive at the following statement. Proposition 7.18 (Bayes is in the Class) .2MO re . Moreover, since Oisreﬂective, wehavethat dominatesallenvironments 2MO re : (e1:tka1:t) =(etjæ<tat)(e<tka<t) (etjæ<tat)(e<tja<t) =(e<tka<t)X 2MO re w(jæ<t)(etjæ<tat) =(e<tka<t)X 2MO re w()(e<tka<t) (e<tka<t)(etjæ<tat) =X 2MO re w()(e1:tka1:t) w()(e1:tka1:t) Therefore we get on-policy value convergence according to Corollary 4.20: for all 2MO re and all policies V (æ<t)V (æ<t)!0ast!1-almost surely. (7.4) 7.2.2 Reﬂective-Oracle-Computable Policies This subsection is dedicated to the following result that was previously stated by Fall- enstein et al. (2015a, Alg. 6) but not proved. It contrasts results on arbitrary semicom- putable environments where optimal policies are not limit computable (see Section 6.3). Theorem 7.19 (Optimal Policies are Oracle Computable) .For every2MO re , there is a-optimal (stochastic) policy that is reﬂective-oracle-computable. Note that even though deterministic optimal policies always exist, those policies are typically not reﬂective-oracle-computable. To prove Theorem 7.19 we need the following lemma. Lemma 7.20 (Reﬂective-Oracle-Computable Optimal Value Function) .For every en- vironment2MO re the optimal value function V is reﬂective-oracle-computable. §7.3Multi-Agent Environments 137 Proof.This proof follows the proof of Corollary 6.14. We write the optimal value explic- itly as in (4.2). For a ﬁxed m, all involved quantities are reﬂective-oracle-computable. Moreover, this quantity is monotone increasing in mand the tail sum from m+ 1to1 is bounded by m+1which is computable according to Assumption 4.6a and converges to0asm!1. Therefore we can enumerate all rationals above and below V . Proof of Theorem 7.19. According to Lemma 7.20 the optimal value function V is reﬂective-oracle-computable. Hence there is a probabilistic Turing machine Tsuch that O T(1jæ<t) = V (æ<t)V (æ<t) + 1 =2: We deﬁne the policy (æ<t) :=( ifO(T;æ<t;1=2) = 1;and ifO(T;æ<t;1=2) = 0 This policy is stochastic because the answer of the oracle Ois stochastic. It remains to show that is a-optimal policy. If V (æ<t)> V (æ<t), then O T(1jæ<t)>1=2, thusO(T;æ<t;1=2) = 1 sinceOis reﬂective, and hence takes action . Conversely, if V (æ<t)< V (æ<t), thenO T(1jæ<t)<1=2, thusO(T;æ<t;1=2) = 0sinceOis reﬂective, and hence takes action . Lastly, ifV (æ<t) =V (æ<t), then both actions are optimal and thus it does not matter which action is returned by policy . (This is the case where the oracle may random- ize.) 7.2.3 Solution to the Grain of Truth Problem Together, Proposition 7.18 and Theorem 7.19 provide the necessary ingredients to solve the grain of truth problem (Problem 7.1). Corollary 7.21 (Solution to the Grain of Truth Problem) .For every lower semicom- putable prior w2MO re the Bayes optimal policy is reﬂective-oracle-computable whereis the Bayes-mixture corresponding to wdeﬁned in (7.2). Proof.From Proposition 7.18 and Theorem 7.19. Hence the environment class MO re contains any reﬂective-oracle-computable modi- ﬁcation of the Bayes optimal policy . In particular, this includes computable multi- agent environments that contain other Bayesian agents over the class MO re . So any Bayesian agent over the class MO re has a grain of truth even though the environment may contain other Bayesian agents of equal power . We proceed to sketch the implica- tions for multi-agent environments in the next section. 7.3 Multi-Agent Environments In amulti-agent environment there arenagents each taking sequential actions from the ﬁnite action space A. In each time step t= 1;2;:::, the environment receives action ai t 138 The Grain of Truth Problem agent1 agent2 ... agentnmulti-agent environment a1 t e1 t a2 t e2 t an t en t Figure 7.2: Agents1;:::;ninteracting in a multi-agent environment. from agent iand outputs nperceptse1 t;:::;en t2E, one for each agent. Each percept ei t= (oi t;ri t)contains an observation oi tand a reward ri t2[0;1]. Importantly, agent i only sees its own action ai tand its own percept ei t(see Figure 7.2). We use the shorthand notationat:= (a1 t;:::;an t)andet:= (e1 t;:::;en t)and denote æi <t=ai 1ei 1:::ai t1ei t1 andæ<t=a1e1:::at1et1. Formally, multi-agent environments are deﬁned as follows. Deﬁnition 7.22 (Multi-Agent Environment) .Amulti-agent environment is a function : (AnEn)An!(En): Together with the policies 1;:::;nthe multi-agent environment induces a history distribution 1:nwhere 1:n() : = 1 1:n(æ1:t) : =1:n(æ<tat)(etjæ<tat) 1:n(æ<tat) : =1:n(æ<t)nY i=1i(ai tjæi <t): Agentiacts in a subjective environment igiven by joining the multi-agent en- vironmentwith the policies 1;:::;nand marginalizing over the histories that i does not see. Together with policy i, the environment iyields a distribution over the histories of agent i i i(æi <t) :=X æj <t;j6=i1:n(æ<t): Wegetthedeﬁnitionofthesubjectiveenvironment iwiththeidentity i(ei tjæi <tai t) := i i(ei tjæi <tai t). The subjective environment idepends on ibecause other policies’ actions may depend on the actions of i. It is crucial to note that the subjective environment iand the policy iare ordinary environments and policies, so we can use the notation from Chapter 4. §7.3Multi-Agent Environments 139 Our deﬁnition of a multi-agent environment is very general and encompasses most of game theory. It allows for cooperative, competitive, and mixed games; inﬁnitely repeatedgamesorany(inﬁnite-length)extensiveformgameswithﬁnitelymanyplayers. Example 7.23 (Matching Pennies) .In the game of matching pennies there are two agents (n= 2), and two actions A=f;grepresenting the two sides of a penny. In each time step agent 1wins if the two actions are identical and agent 2wins if the two actions are diﬀerent. The payoﬀ matrix is as follows. 1,0 0,1 0,1 1,0 We useE=f0;1gto be the set of rewards (observations are vacuous) and deﬁne the multi-agent environment to give reward 1to agent 1iﬀa1 t=a2 t(0otherwise) and reward 1to agent 2iﬀa1 t6=a2 t(0otherwise). Formally, (r1 tr2 tjæ<tat) :=8 >>< >>:1ifr1 t= 1;r2 t= 0;a1 t=a2 t; 1ifr1 t= 0;r2 t= 1;a1 t6=a2 t;and 0otherwise: Letdenote the policy that always takes action . If two agents each using policy play matching pennies, agent 1wins in every step. Formally, setting 1:=2:=we get a history distribution that assigns probability one to the history 1010:::: The subjective environment of agent 1is 1(r1 tjæ1 <ta1 t) =8 >>< >>:1ifr1 t= 1;a1 t=; 1ifr1 t= 0;a1 t=;and 0otherwise: Therefore policy is optimal in agent 1’s subjective environment. 3 Deﬁnition 7.24 ("-Best Response) .A policyiacting in multi-agent environment with policies 1;:::;nis an"-best response after history æi <tiﬀ V i(æi <t)Vii(æi <t)<": If at some time step t, all agents’ policies are "-best responses, we have an "- Nash equilibrium . The property of multi-agent systems that is analogous to asymptotic optimality is convergence to an "-Nash equilibrium. 140 The Grain of Truth Problem 7.4 Informed Reﬂective Agents Letbe a multi-agent environment and let 1;::: nbe such that for each ithe policy iis an optimal policy in agent i’s subjective environment i. At ﬁrst glance this seems ill-deﬁned: The subjective environment idepends on each policy j, which depends on the subjective environment j, which in turn depends on the policy i. However, this circular deﬁnition actually has a well-deﬁned solution. Theorem 7.25 (Optimal Multi-Agent Policies) .For any reﬂective-oracle-computable multi-agent environment , the optimal policies 1;:::; nexist and are reﬂective- oracle-computable. To prove Theorem 7.25, we need the following proposition. Proposition 7.26 (Reﬂective-Oracle-Computability) .If the multi-agent environment and the policies 1;:::;nare reﬂective-oracle-computable, then 1:nandi iare reﬂective-oracle-computable, and i2MO re . Proof.All involved quantities in the deﬁnition of 1:nare reﬂective-oracle-computable by assumption, therefore also their marginalizations i iandi. Proof of Theorem 7.25. According to Theorem 7.19 the optimal policy iin agenti’s subjective environment is reﬂective-oracle-computable if the subjective environment i is. In particular the process that takes iin form of a probabilistic Turing machine and returns iis reﬂective-oracle-computable. Moreover, iis reﬂective-oracle-computable ifand1;:::;nare, according to Proposition 7.26. Again, this construction is itself reﬂective-oracle-computable. Connecting these two constructions we get probabilistic Turing machines T1;:::;Tn2Twhere each Titakes the multi-agent environment and1;:::;nin form of probabilistic Turing machines and returns i. We deﬁne the probabilistic Turing machines T0 1;:::;T0 nwhereT0 irunsTO ion(;T0 1;:::;T0 n); henceT0 i computes i. Note that this construction only works because we relied on the reﬂective oracle in the proof of Theorem 7.19. Since the machines TO ialways halt, so do T0 iO despite their inﬁnitely recursive construction. Note the strength of Theorem 7.25: each of the policies iis acting optimally given the knowledge of everyone else’s policies . Hence optimal policies play 0-best re- sponses by deﬁnition, so if every agent is playing an optimal policy, we have a Nash equilibrium. Moreover, this Nash equilibrium is also a subgame perfect Nash equi- librium, because each agent also acts optimally on the counterfactual histories that do not end up being played. In other words, Theorem 7.25 states the existence and reﬂective-oracle-computability of a subgame perfect Nash equilibrium in any reﬂective- oracle-computable multi-agent environment. The following immediate corollary states that these subgame perfect Nash equilibria are limit computable. Corollary 7.27 (Solution to Computable Multi-Agent Environments) .For any com- putable multi-agent environment , the optimal policies 1;:::; nexist and are limit computable. §7.5Learning Reﬂective Agents 141 Proof.From Theorem 7.25 and Theorem 7.7. Example 7.28 (Nash Equilibrium in Matching Pennies) .Consider the matching pen- nies game from Example 7.23. The only pair of optimal policies is the pair of two uniformly random policies that play andwith equal probability in every time step: if one of the agents picks a policy that plays one of the actions with probability >1=2, then the other agent’s best response is to play the other action with probability 1. But now the ﬁrst agent’s policy is no longer a best response. 3 7.5 Learning Reﬂective Agents Since our classMO re solves the grain of truth problem, the result by Kalai and Lehrer (1993) immediately implies that for any Bayesian agents 1;:::;ninteracting in an inﬁnitely repeated game and for all ">0and alli2f1;:::;ngthere is almost surely a t02Nsuch that for all tt0the policyiis an"-best response. However, this hinges on the important fact that every agent has to know the game and also that all other agents are Bayesian agents. Otherwise the convergence to an "-Nash equilibrium may fail, as illustrated by the following example. At the core of the construction is a dogmatic prior (Section 5.2.2). A dogmatic prior assigns very high probability to going to hell (reward 0forever) if the agent deviates from a given computable policy . For a Bayesian agent it is thus only worth deviating from the policy if the agent thinks that the prospects of following are very poor already. This implies that for general multi-agent environments and without additional assumptions on the prior, we cannot prove any meaningful convergence result about Bayesian agents acting in an unknown multi-agent environment. Example 7.29 (Reﬂective Bayesians Playing Matching Pennies) .Consider the multi- agent environment matching pennies from Example 7.23. Let 1be the policy that takes the action sequence ()1and let2:=be the policy that always takes action. The average reward of policy 1is2=3and the average reward of policy 2is1=3. Letbe a universal mixture (7.2). By on-policy value convergence (7.4), V1 !c12=3andV2 !c21=3almost surely when following policies (1;2). Therefore there is an " >0such thatV1 > "andV2 > "for all time steps. Now we can apply Theorem 5.5 to conclude that there are (dogmatic) mixtures 0 1and0 2 such that 0 1always follows policy 1and 0 2always follows policy 2. This does not converge to a ( "-)Nash equilibrium. 3 An important property required for the construction in Example 7.29 is that the environment class contains environments that threaten the agent with going to hell, which is outside of the class of matching pennies environments. In other words, since the agent does not know a priori that it is playing a matching pennies game, it might behave more conservatively than appropriate for the game. The following theorem is our main convergence result. It states that for asymptot- ically optimal agents we get convergence to "-Nash equilibria in any reﬂective-oracle- computable multi-agent environment. 142 The Grain of Truth Problem Theorem 7.30 (Convergence to Equilibrium) .Letbe an reﬂective-oracle-computable multi-agent environment and let 1;:::;nbe reﬂective-oracle-computable policies that are asymptotically optimal in mean in the class MO re . Then for all " > 0and all i2f1;:::;ngthe1:n-probability that the policy iis an"-best response converges to 1ast!1. Proof.Leti2 f1;:::;ng. By Proposition 7.26, the subjective environment iis reﬂective-oracle-computable, therefore i2MO re . Sinceiis asymptotically optimal in mean in the class MO re , we get that E[V i(æ<t)Vii(æ<t)]!0. Convergence in mean implies convergence in probability for bounded random variables, hence for all ">0we have i i[V i(æi <t)Vii(æi <t)"]!0ast!1: Therefore the probability that the policy iplays an"-best response converges to 1as t!1. In contrast to Theorem 7.25 which yields policies that play a subgame perfect equi- librium, this is not the case for Theorem 7.30: the agents typically do not learn to predict oﬀ-policy and thus will generally not play "-best responses in the counterfac- tual histories that they never see. This weaker form of equilibrium is unavoidable if the agents do not know the environment because it is impossible to learn the parts that they do not interact with. Corollary 7.31 (Convergence to Equilibrium) .There are limit computable policies 1;:::;nsuch that for any computable multi-agent environment and for all">0and alli2f1;:::;ngthe1:n-probability that the policy iis an"-best response converges to1ast!1. Proof.Pick1;:::;nto be the Thompson sampling policy Tdeﬁned in Algorithm 2 over the countable class MO re . By Theorem 5.25 these policies are asymptotically op- timal in mean. By Theorem 7.32 below they are reﬂective-oracle-computable and by Theorem 7.7 they are also limit computable. The statement now follows from Theo- rem 7.30. Theorem 7.32 (Thompson Sampling is Reﬂective-Oracle-Computable) .The policyT deﬁned in Algorithm 2 over the class MO re is reﬂective-oracle-computable. Proof.The posterior w(jæ<t)is reﬂective-oracle-computable by the deﬁnition (7.3) and according to Theorem 7.19 the optimal policies are reﬂective-oracle-computable. On resampling steps we can compute the action probabilities of Tby enumerating all 2MO re and computing weighted by the posterior w(jæ<t). Between resampling steps we need to condition the policy Tcomputed above by the actions it has already taken since the last resampling step (compare Example 5.28). Because the posterior w(jæ<t)is a-martingale when acting according to the policy, it converges -almost surely according to the martingale convergence theo- rem (Theorem 2.8). Since dominates the subjective environment i, it also converges §7.6Impossibility Results 143 i-almost surely (see Example 3.20). Hence all the Thompson sampling agents even- tually ‘calm down’ and settle on some posterior belief. According to Theorem 7.30, the policies 1;:::;nonly need to be asymptotically optimal in mean. For Thompson sampling this is independent of the discount function according to Theorem 5.25 (but the discount function has to be known to the agent). So the diﬀerent agents may use diﬀerent discount functions, resample at diﬀerent time steps and converge at diﬀerent speeds. Example 7.33 (Thompson Samplers Playing Matching Pennies) .Consider the match- ing pennies game from Example 7.23 and let both agents use the Thompson sampling policy deﬁned in Algorithm 2, i.e., deﬁne 1:=Tand2:=T. The value of the uniformly random policy Ris always 1=2, soV iVRi= 1=2. According to Theorem 7.30, for every ">0, each agent will eventually always play an "-best response, i.e., Vii> V i"1=2". Since matching pennies is a zero-sum game,V11+V22= 1, soV22= 1V11<1=2 +". Therefore each agent will end up randomizing their actions; i(atjæ<t)1=2most of the time: If one of the agents (say agent 1) does not suﬃciently randomize their actions in some time steps, then agent 2 could exploit this by picking a deterministic adversarial policy in those time steps. Suppose that this way it can gain a value of "compared to the random policy R, i.e.,V22VR2+"= 1=2 +". But this is a contradiction because then agent 1 is not playing an "-best response: V 1V11=V 11 +V221=21 + 1=2 +"=" 3 7.6 Impossibility Results Why does our solution to the grain of truth problem not violate the impossibility results from the literature? Assume we are playing an inﬁnitely repeated game where in the stagegamenoagenthasaweaklydominantactionandthepureactionmaxminrewardis strictly less then the minmax reward. The impossibility result of Nachbar (1997, 2005) states that there is no class of policies such that the following are simultaneously satisﬁed. •Learnability. Each agent learns to predict the other agent’s actions. •Caution and Symmetry. The set is closed under simple policy modiﬁcations such as renaming actions. •Purity.There is an " >0such that for any stochastic policy 2there is a deterministic policy 02such that if 0(æ<t) =a, then(ajæ<t)>". •Consistency. Each agent always has an "-best response available in . In order to converge to an "-Nash equilibrium, each agent has to have an "-best response available to them, so consistency is our target. Learnability is immediately satisﬁed for any environment in our class if we have a dominant prior according to Corollary 4.20. 144 The Grain of Truth Problem ForMO re caution and symmetry are also satisﬁed since this set is closed under any computable modiﬁcations to policies. However, our class MO re avoids this impossibility result because it violates the purity condition: Let T1;T2;:::be an enumeration of T. Consider the policy that maps history æi <tto the action 1O(Tt;æi <t;1=2). IfTtis deterministic, then will take a diﬀerent action than Ttfor any history of length t1. Therefore no deterministic reﬂective-oracle-computable policy can take an action that assigns positive probability to in every time step. Foster and Young (2001) present a condition that makes convergence to a Nash equilibrium impossible: if the player’s rewards are perturbed by a small real number drawn from some continuous density , then for-almost all realizations the players do not learn to predict each other and do not converge to a Nash equilibrium. For example, in a matching pennies game, rational agents randomize only if the (subjective) values of both actions are exactly equal. But this happens only with -probability zero, since is a density. Thus with -probability one the agents do not randomize. If the agents do not randomize, they either fail to learn to predict each other, or they are not acting rationally according to their beliefs: otherwise they would seize the opportunity to exploit the other player’s deterministic action. But this does not contradict our convergence result: the class MO re is countable and each2MO re has positive prior probability. Perturbation of rewards with ar- bitrary real numbers is not possible. Even more, the argument given by Foster and Young (2001) cannot work in our setting: the Bayesian mixture mixes overTfor all probabilistic Turing machines T. For Turing machines Tthat sometimes do not halt, the oracle decides how to complete Tinto a measure T. Thus the oracle has enough inﬂuence on the exact values in the Bayesian mixture that the values of two actions in matching pennies can be made exactly equal. 7.7 Discussion Thischapterintroducedtheclassofallreﬂective-oracle-computableenvironments MO re . This class fully solves the grain of truth problem (Problem 7.1) because it contains (any computablemodiﬁcationof)Bayesianagentsdeﬁnedover MO re : theoptimalagentsand Bayes optimal agents over the class are all reﬂective-oracle-computable (Theorem 7.19 and Corollary 7.21). If the environment is unknown, then a Bayesian agent may end up playing subop- timally (Example 7.29). However, if each agent uses a policy that is asymptotically optimal in mean (such as the Thompson sampling policy from Section 4.3.4) then for every" >0the agents converge to an "-Nash equilibrium (Theorem 7.30 and Corol- lary 7.31). However, Corollary 7.31 does notimply that two Thompson sampling policies will converge to cooperation in an iterated prisoner’s dilemma since always defecting is also a Nash equilibrium. The exact outcome will depend on the priors involved and the randomness of the policies. Our solution to the grain of truth problem is purely theoretical. However, Theo- rem 7.7 shows that our class MO re allows for computable approximations. This suggests §7.7Discussion 145 that practical approaches can be derived from this result, and reﬂective oracles have already seen applications in one-shot games (Fallenstein et al., 2015b). 146 The Grain of Truth Problem Chapter 8 Conclusion The biggest existential risk is that future superintelligences stop simulating us. — Nick Bostrom Today computer programs exceeding humans in general intelligence are known only from science ﬁction. But research on AI has progressed steadily over the last decades and there is good reason to believe that we will be able to build HLAI eventually, and even sooner than most people think (Müller and Bostrom, 2016). The advent of strong AI would be the biggest event in human history. Potential beneﬁts are huge, as the new level of automation would free us from any kind of un- desirable labor. But there is no reason to believe that humans are at the far end of the intelligence spectrum. Rather, humans barely cross the threshold for general intel- ligence to be able to use language and do science. Once we engineer HLAI, it seems unlikely that progress is going to stop; why not build even smarter machines? This could lead to an intelligence explosion (Good, 1965; Vinge, 1993; Kurzweil, 2005; Chalmers, 2010; Hutter, 2012a; Schmidhuber, 2012; Muehlhauser and Salamon, 2012; Eden et al., 2013; Shanahan, 2015; Eden, 2016; Walsh, 2016): a (possibly very rapid)increaseinintelligence, e.g., throughself-ampliﬁcationeﬀectsfromAIsimproving themselves (if doing AI research is one of humans’ capabilities, then a machine that can do everything humans can do can also do AI research). Once machine intelligence is above or far above human level, machines would steer the course of history. There is no reason to believe that machines would be adversarial to us, but nevertheless humanity’s fate might rest at the whims of the machines, just as chimpanzees today have no say in the large-scale events on this planet. Bostrom (2002) deﬁnes: Existential risk — One where an adverse outcome would either annihilate Earth-originating intelligent life or permanently and drastically curtail its potential. Existential risks are events that have the power to extinguish human life as we know it. Examples are cosmic events such as an asteroid colliding with Earth. But cosmic eventsareunlikelyonhumantimescalescomparedtohuman-madeexistentialrisksfrom nuclear weapons, synthetic biology, and nanotechnology. It is possible that artiﬁcial intelligence also falls into this category. Vinge (1993) was the ﬁrst person to recognize this: 147 148 Conclusion Within thirty years, we will have the technological means to create super- human intelligence. Shortly after, the human era will be ended. After Vinge, Yudkowsky (2001, 2008) can be regarded as one of the key people to popularize the potential dangers of AI technology. Bostrom (2003) picked up on this issue very early and gave the topic credibility through his well-researched and carefully written book Superintelligence (Bostrom, 2014). He argues that we need to solve the Control Problem —the unique principal-agent problem that arises with the creation of strong AI (Bostrom, 2014, Ch. 9). In other words: How do we align strong AI with human values? How do we ensure that AI remains robust and beneﬁcial? This research is collected under the umbrella term AI safety . Currently, we have no idea how to solve these problems even in theory. Through Yudkowsky’s and, more importantly, Bostrom’s eﬀorts, AI-related long- term safety concerns have now entered the mainstream media. In 2014 high-proﬁle scientists such as Stephen Hawking, Max Tegmark, Stuart Russell, and Frank Wilczek have warned against the dangers posed by AI (Hawking et al., 2014). (See also Alexan- der (2015) for a collection of positions by prominent AI researchers.) Many scientists inside and outside the ﬁeld have signed an open letter that research ensuring that AI systems remain robust and beneﬁcial is both important and timely (Future of Life In- stitute, 2015c). This lead entrepreneur Elon Musk to donate $10 million to kick-start research in this ﬁeld (Future of Life Institute, 2015a); most of this money has now been distributed across the planet to 37 diﬀerent projects (Future of Life Institute, 2015b). Moreover, the Future of Life Institute and the Machine Intelligence Research Institute have formulated concrete technical research priorities to make AI more robust and beneﬁcial (Soares and Fallenstein, 2014; Russell et al., 2015). At the end of 2015 followed the announcement of OpenAI, a nonproﬁt organization with ﬁnancial backing from several famous silicon valley billionaires (OpenAI, 2015): OpenAI is a non-proﬁt artiﬁcial intelligence research company. Our goal is to advance digital intelligence in the way that is most likely to beneﬁt humanity as a whole, unconstrained by a need to generate ﬁnancial return. ThemissionofOpenAIistoenableeveryonetobeneﬁtfromAItechnology. But, despite the name, OpenAI is not committed to publish all of their research freely, and Bostrom (2016) argues that unrestricted publication might not be the best idea. Despite all of the recent eﬀorts in AI safety research, critical voices within the AI community remain. Prominently, Davis (2014), Ng (2016), Walsh (2016), and Lawrence (2016) have proposed counterarguments that range from ‘HLAI is so far away that any worry is misplaced’ to claims that ‘the safety problem would not be so hard’. See Sotala and Yampolskiy (2014) for a discussion. If AI poses an existential risk then a formal theory of strong AI is paramount to develop technical approaches to mitigate this risk. Which path will ultimately lead us to HLAI is in the realm of speculation at this time; therefore we should make as few and as weak assumptions as possible and abstract away from possible future implementation details. 149 This thesis lays some of the groundwork for this endeavor. We built on top of Hut- ter’s theory of universal artiﬁcial intelligence. Chapter 3 discussed the formal theory of learning. Chapter 4 presented several approaches to acting in unknown environments (Bayes, Thompson sampling, knowledge-seeking agents, and BayesExp). Chapter 5 analysed these approaches and discussed notions of optimality and principled prob- lems with acting under uncertainty in general environment. Chapter 6 provided the mathematical tools to analyze the computational properties of these models. Finally, Chapter 7 solved the grain of truth problem, which lead to convergence to Nash equi- libria in unknown general multi-agent environments. Our work is theoretical by nature and there is still a long way to go until these results make their way into applications. But a solution in principle is a crucial ﬁrst step towards solving a problem in practice. Consider the research paper by Shannon (1950) on how to solve chess in principle. The algorithm he describes expands the full game tree of chess (until some speciﬁed depth), which is completely infeasible even with today’s computation power. His contribution was to show that winning at chess is a feat that computers can achieve in principle , which was not universally accepted at the time. Even more, his approach already considered the correct ideas (minimax-search over the game tree) that ultimately lead to the defeat of the chess champion Garry Kasparov by the computer Deep Blue 46 years later (IBM, 2012a). The theory of general reinforcement learning can serve and has served as a starting point for future investigation in AI safety. In particular, value learning (Dewey, 2011), self-reﬂection (Soares, 2015; Fallenstein et al., 2015b), self-modiﬁcation (Orseau and Ring, 2011, 2012a; Everitt et al., 2016), interruptibility (Orseau and Armstrong, 2016; Armstrong and Orseau, 2016), decision theory (Everitt et al., 2015), memory manip- ulation (Orseau and Ring, 2012b), wireheading (Ring and Orseau, 2011; Everitt and Hutter, 2016), and questions of identity (Orseau, 2014b,c). It is possible that HLAI is decades or centuries away. It might also be a few years around the corner. Whichever is the case, we are currently completely unprepared for the consequences. As an AI researcher, it is tempting to devote your life to increasing the capability of AI, advancing it domain after domain, and showing oﬀ with ﬂashy demos and impressive victories over human contestants. But every technology incurs risks, and the more powerful the technology, the higher the risks. The potential power of AI technology is enormous, and correspondingly we need to consider the risks, take them seriously, and proceed to mitigate them. 150 Conclusion Appendix A Measures and Martingales In this chapter we provide the proofs for Theorem 3.18 and Theorem 3.19. Proof of Theorem 3.18. Xtis only undeﬁned if P(v1:t) = 0. The set fv21j9t:P(v1:t) = 0g hasP-measure 0and hence (Xt)t2Nis well-deﬁned almost everywhere. Xtis constant on ufor allu2t, andFtis generated by a collection of ﬁnitely many disjoint sets: 1=] u2tu: (a) Therefore XtisFt-measurable. (b)u=U a2uafor allu2tandv2u, and therefore E[Xt+1jFt](v) =1 P(u)X a2Xt+1(ua)P(ua) =1 P(u)X a2Q(ua) P(ua)P(ua) ()=1 P(u)X a2Q(ua) =Q(u) P(u)=Xt(v): At()we used the fact that Pis locally absolutely continuous with respect to Q. (IfPwere not locally absolutely continuous with respect to Q, then there are cases whereP(u)>0,P(ua) = 0, andQ(ua)6= 0. Therefore Xt+1(ua)does not contribute to the expectation and thus Xt+1(ua)P(ua) = 06=Q(ua).) P0andQ0by deﬁnition, thus Xt0. SinceP() =Q() = 1, we have E[X0] = 1. The following lemma gives a convenient condition for the existence of a measure on (!;F1). It is a special case of the Daniell-Kolmogorov Extension Theorem (Rogers and Williams, 1994, Thm. 26.1). 151 152 Measures and Martingales Lemma A.1 (Extending measures) .Letq: ![0;1]be a function such that q() = 1 andP a2q(ua) =q(u)for allu2. Then there exists a unique probability measure Qon(1;F1)such thatq(u) =Q(u)for allu2. To prove this lemma, we need the following two ingredients. Deﬁnition A.2 (Semiring) .A setR 2 is called semiring over iﬀ (a);2R, (b) for allA;B2R, the setA\B2R, and (c) for allA;B2R, there are pairwise disjoint sets C1;:::;Cn2Rsuch thatAnB=Un i=1Ci. Theorem A.3 (Carathéodory’s Extension Theorem; Durrett, 2010, Thm. A.1.1) .Let Rbe a semiring over and let:R! [0;1]be a function such that (a)( ) = 1(normalization) , (b)(Un i=1Ai) =Pn i=1(Ai)for pairwise disjoint sets A1;:::;An2 Rsuch thatUn i=1Ai2R(ﬁnite additivity) , and (c)(S i0Ai)P i0(Ai)for any collection (Ai)i0such that each Ai2RandS i0Ai2R(-subadditivity) . Then there is a unique extension ofthat is a probability measure on ( ;(R))such that(A) =(A)for allA2R. Proof of Lemma A.1. We show the existence of Qusing Carathéodory’s Extension The- orem. DeﬁneR:=fuju2g[f;g. (a);2R. (b) For any u;v2R, either •uis a preﬁx of vandu\v= v2R, or •vis a preﬁx of uandu\v= u2R, or •u\v=;2R. (c) For any u;v2R, •unv=U w2jvjjujnfxguwifv=ux, i.e.,uis a preﬁx of v, and •unv=;otherwise. ThereforeRis a semiring. By deﬁnition of R, we have(R) =F1. The function q: ![0;1]naturally gives rise to a function :R! [0;1]with (;) := 0and(u) :=q(u)for allu2. We will now check the prerequisites of Carathéodory’s Extension Theorem. 153 (a) (Normalization.) (1) =() =q() = 1. (b) (Finite additivity.) Let u1;:::; uk2Rbe pairwise disjoint sets such that w:=Uk i=1ui2 R. Let`:= maxfjuij j1ikg, then w=U v2`wv. By assumption,P a2q(ua) =q(u), thusP a2(ua) =(u)and inductively we have (ui) =X s2`juij(uis); (A.1) and (w) =X v2`(wv): (A.2) For every string v2`, the concatenation wv2w=Uk i=1ui, so there is a uniqueisuch thatwv2ui. Hence there is a unique string s2`juijsuch that wv=uis. Together with (A.1) and (A.2) this yields k] i=1ui! =(w) =X v2`(wv) =kX i=1X s2`juij(uis) =kX i=1(ui): (c) (-subadditivity.) We will show that each uis compact with respect to the topol- ogyOgenerated byR.-subadditivity then follows from (b) because every count- able union is in fact a ﬁnite union. We will show that the topology Ois the product topology of the discrete topology on. (This establishes that (!;O)is a Cantor Space.) Every projection k: 1!selecting the k-th symbol is continuous, since 1 k(a) =S u2k1uafor everya2. Moreover,Ois the coarsest topology with this property, since we can generate every open set u2Rin the base of the topology by u=juj\ i=11 i(fuig): The set is ﬁnite and thus compact. By Tychonoﬀ’s Theorem, 1is also compact. Therefore uis compact since it is homeomorphic to 1via the canonical map u: 1!u,v7!uv. From (a), (b), and (c) Carathéodory’s Extension Theorem yields a unique probability measureQon(1;F1)such thatQ(u) =(u) =q(u)for allu2. Using Lemma A.1, the proof of Theorem 3.19 is now straightforward. Proof of Theorem 3.19. We deﬁne a function q: !R, with q(u) :=Xjuj(v)P(u) for anyv2u. The choice of vis irrelevant because Xjujis constant on usince it is Ft-measurable. In the following, we also write Xt(u)ifjuj=tto simplify notation. 154 Measures and Martingales The function qis non-negative because XtandPare both non-negative. Moreover, for anyu2t, 1 =E[Xt] =Z 1XtdPZ uXtdP=P(u)Xt(u) =q(u): Hence the range of qis a subset of [0;1]. We haveq() =X0()P() =E[X0] = 1sincePis a probability measure and F0=f;;1gis the trivial -algebra. Let u2t. X a2q(ua) =X a2Xt+1(ua)P(ua) =Z uXt+1dP =Z uE[Xt+1jFt]dP=Z uXtdP=P(u)Xt(u) =q(u): By Lemma A.1, there is a probability measure Qon(1;F1)such thatq(u) =Q(u) of allu2. Therefore, for all v21and for allt2NwithP(v1:t)>0, Xt(v) =q(v1:t) P(v1:t)=Q(v1:t) P(v1:t): Moreover,Pis locally absolutely continuous with respect to QsinceP(u) = 0implies Q(u) =q(u) =Xjuj(u)P(u) = 0: Bibliography Shipra Agrawal and Navin Goyal. Analysis of Thompson sampling for the multi-armed bandit problem. In Conference on Learning Theory , 2011. Scott Alexander. AI researchers on AI risk. http://slatestarcodex.com/2015/05/ 22/ai-researchers-on-ai-risk/ , May 2015. Accessed: 2016-04-25. Stuart Armstrong and Laurent Orseau. Interruptibility and corrigibility for AIXI and Monte Carlo agents. 2016. To appear. Frank Arntzenius, Adam Elga, and John Hawthorne. Bayesianism, inﬁnite decisions, and binding. Mind, 113(450):251–283, 2004. Peter Auer, Thomas Jaksch, and Ronald Ortner. Near-optimal regret bounds for re- inforcement learning. In Advances in Neural Information Processing Systems , pages 89–96, 2009. Peter Auer, Thomas Jaksch, and Ronald Ortner. Near-optimal regret bounds for rein- forcement learning. Journal of Machine Learning Research , 11:1563–1600, 2010. Leemon Baird. Residual algorithms: Reinforcement learning with function approxima- tion. InInternational Conference on Machine Learning , pages 30–37, 1995. Jacob D Bekenstein. Universal upper bound on the entropy-to-energy ratio for bounded systems. Physical Review D , 23(2):287, 1981. Marc G Bellemare, Yavar Naddaf, Joel Veness, and Michael Bowling. The arcade learningenvironment: Anevaluationplatformforgeneralagents. Journal of Artiﬁcial Intelligence Research , 47:253–279, 2013. Marc G Bellemare, Georg Ostrovski, Arthur Guez, Philip S Thomas, and Rémi Munos. Increasing the action gap: New operators for reinforcement learning. In AAAI, 2016. Yoshua Bengio, Patrice Simard, and Paolo Frasconi. Learning long-term dependencies with gradient descent is diﬃcult. IEEE Transactions on Neural Networks , 5(2):157– 166, 1994. Dimitri P Bertsekas and John Tsitsiklis. Dynamic Programming and Optimal Control . Athena Scientiﬁc, 1995. Christopher M Bishop. Pattern Recognition and Machine Learning . Springer, 2006. David Blackwell and Lester Dubins. Merging of opinions with increasing information. The Annals of Mathematical Statistics , pages 882–886, 1962. 155 156 BIBLIOGRAPHY Nick Bostrom. Existential risks. Journal of Evolution and Technology , 9(1):1–31, 2002. Nick Bostrom. Ethical issues in advanced artiﬁcial intelligence. Science Fiction and Philosophy: From Time Travel to Superintelligence , pages 277–284, 2003. Nick Bostrom. Superintelligence: Paths, Dangers, Strategies . Oxford University Press, 2014. Nick Bostrom. Strategic implications of openness in AI development. Technical re- port, Future of Humanity Institute, 2016. http://www.nickbostrom.com/papers/ openness.pdf . Michael Bowling and Manuela Veloso. Rational and convergent learning in stochastic games. In International Joint Conference on Artiﬁcial Intelligence , pages 1021–1026, 2001. Sébastien Bubeck and Cesa-Nicolò Bianchi. Regret analysis of stochastic and non- stochastic multi-armed bandit problems. Foundations and Trends in Machine Learn- ing, 5(1):1–122, 2012. Lucian Busoniu, Robert Babuska, and Bart De Schutter. A comprehensive survey of multiagent reinforcement learning. IEEE Transactions on Systems, Man, and Cybernetics, Part C: Applications and Reviews , 38(2):156–172, 2008. Nicolo Cesa-Bianchi and Gábor Lugosi. Prediction, Learning, and Games . Cambridge University Press, 2006. David Chalmers. The singularity: A philosophical analysis. Journal of Consciousness Studies, 17(9-10):7–65, 2010. Olivier Chapelle and Lihong Li. An empirical evaluation of Thompson sampling. In Advances in Neural Information Processing Systems , pages 2249–2257, 2011. Paul Christiano, Eliezer Yudkowsky, Marcello Herreshoﬀ, and Mihaly Barasz. Deﬁn- ability of truth in probabilistic logic. Technical report, Machine Intelligence Research Institute, 2013. https://intelligence.org/files/DefinabilityTruthDraft.pdf . Thomas M Cover and Joy A Thomas. Elements of Information Theory . John Wiley & Sons, 2nd edition, 2006. Mayank Daswani. Generic Reinforcement Learning Beyond Small MDPs . PhD thesis, Australian National University, 2015. Mayank Daswani and Jan Leike. A deﬁnition of happiness for reinforcement learning agents. In Artiﬁcial General Intelligence , pages 231–240. Springer, 2015. Ernest Davis. Ethical guidelines for a superintelligence. Technical report, New York University, 2014. https://cs.nyu.edu/davise/papers/Bostrom.pdf . BIBLIOGRAPHY 157 Adam Day. Increasing the gap between descriptional complexity and algorithmic proba- bility.Transactions of the American Mathematical Society , 363(10):5577–5604, 2011. Richard Dearden, Nir Friedman, and Stuart Russell. Bayesian Q-learning. In AAAI, pages 761–768, 1998. DanielDewey. Learningwhattovalue. In Artiﬁcial General Intelligence , pages309–314. Springer, 2011. Joseph L. Doob. Stochastic Processes . Wiley, New York, 1953. Finale Doshi-Velez. Bayesian Nonparametric Approaches for Reinforcement Learning in Partially Observable Domains . PhD thesis, Massachusetts Institute of Technology, 2012. Rick Durrett. Probability: Theory and Examples . Cambridge University Press, 4th edition, 2010. Amnon H Eden. The singularity controversy. Technical report, Sapience Project, 2016. http://arxiv.org/abs/1601.05977 . Amnon H Eden, James H Moor, Johnny H Søraker, and Eric Steinhart, editors. Sin- gularity Hypotheses: A Scientiﬁc and Philosophical Assessment . Springer, 2013. Tom Everitt and Marcus Hutter. Avoiding wireheading with value reinforcement learn- ing. InArtiﬁcial General Intelligence , pages 12–22, 2016. Tom Everitt, Jan Leike, and Marcus Hutter. Sequential extensions of causal and ev- idential decision theory. In Algorithmic Decision Theory , pages 205–221. Springer, 2015. Tom Everitt, Daniel Filan, Mayank Daswani, and Marcus Hutter. Self-modiﬁcation of policy and utility function in rational agents. In Artiﬁcial General Intelligence , 2016. Benja Fallenstein, Nate Soares, and Jessica Taylor. Reﬂective variants of Solomonoﬀ induction and AIXI. In Artiﬁcial General Intelligence . Springer, 2015a. Benja Fallenstein, Jessica Taylor, and Paul F Christiano. Reﬂective oracles: A founda- tion for game theory in artiﬁcial intelligence. In Logic, Rationality, and Interaction , pages 411–415. Springer, 2015b. Benja Fallenstein, Jessica Taylor, and Paul F Christiano. Reﬂective oracles: A foun- dation for classical game theory. Technical report, Machine Intelligence Research Institute, 2015c. http://arxiv.org/abs/1508.04145 . DanielFilan. Agentsusingspeedpriors. Honoursthesis, AustralianNationalUniversity, October 2015. Daniel Filan, Jan Leike, and Marcus Hutter. Loss bounds and time complexity for speed priors. In Artiﬁcial Intelligence and Statistics , 2016. 158 BIBLIOGRAPHY Jakob N Foerster, Yannis M Assael, Nando de Freitas, and Shimon Whiteson. Learning to communicate to solve riddles with deep distributed recurrent Q-networks. Tech- nical report, University of Oxford, 2016. http://arxiv.org/abs/1602.02672 . Dean P Foster and H Peyton Young. On the impossibility of predicting the behavior of rational agents. Proceedings of the National Academy of Sciences , 98(22):12848– 12853, 2001. Drew Fudenberg and David K Levine. The Theory of Learning in Games . MIT press, 1998. Future of Life Institute. Elon musk donates $10m to keep AI beneﬁcial. http:// futureoflife.org/misc/AI , January 2015a. Accessed: 2015-03-28. Future of Life Institute. 2015 project grants recommended for funding. http: //futureoflife.org/first-ai-grant-recipients/ , 2015b. Accessed: 2016-03-28. Future of Life Institute. Research priorities for robust and beneﬁcial artiﬁcial intel- ligence: An open letter. http://futureoflife.org/ai-open-letter/ , January 2015c. Accessed: 2016-04-26. John Gittins. Bandit processes and dynamic allocation indices. Journal of the Royal Statistical Society. Series B (Methodological) , pages 148–177, 1979. Irving John Good. The paradox of conﬁrmation. British Journal for the Philosophy of Science, pages 145–149, 1960. Irving John Good. Speculations concerning the ﬁrst ultraintelligent machine. Advances in Computers , 6:31–88, 1965. Irving John Good. The white shoe is a red herring. The British Journal for the Philosophy of Science , 17(4):322–322, 1967. Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep learning. Book in prepa- ration for MIT Press, http://www.deeplearningbook.org , 2016. Google. The latest chapter for the self-driving car: mastering city street driving. http://googleblog.blogspot.com.au/2014/04/ the-latest-chapter-for-self-driving-car.html , April 2014. Accessed: 2015-03-27. Google. What we learned in Seoul with AlphaGo. https://googleblog.blogspot. com.au/2016/03/what-we-learned-in-seoul-with-alphago.html , March 2016. Accessed: 2016-03-28. Aditya Gopalan and Shie Mannor. Thompson sampling for learning parameterized Markov decision processes. In Conference on Learning Theory , pages 861–898, 2015. Geoﬀrey J Gordon. Reinforcement learning with function approximation converges to a region. In Advanced in Neural Information Processing Systems , 2001. BIBLIOGRAPHY 159 Peter D. Grünwald. The Minimum Description Length Principle . The MIT Press, Cambridge, 2007. Péter Gács. On the relation between descriptional complexity and algorithmic proba- bility.Theoretical Computer Science , 22(1):71–93, 1983. Kurt Gödel. Über formal unentscheidbare Sätze der Principia Mathematica und ver- wandter Systeme I. Monatshefte für Mathematik und Physik , 38(1):173–198, 1931. Trevor Hastie, Robert Tibshirani, and Jerome Friedman. The Elements of Statistical Learning . Springer, 2nd edition, 2009. Matthew Hausknecht and Peter Stone. Deep recurrent Q-learning for partially observ- able MDPs. In 2015 AAAI Fall Symposium Series , 2015. Stephen Hawking, Max Tegmark, Stuart Russell, and Frank Wilczek. Transcend- ing complacency on superintelligent machines. http://www.huffingtonpost.com/ stephen-hawking/artificial-intelligence_b_5174265.html , April 2014. Ac- cessed: 2015-03-28. Nicholas J Hay. Universal semimeasures: An introduction. Master’s thesis, University of Auckland, 2007. Nicolas Heess, Jonathan J Hunt, Timothy P Lillicrap, and David Silver. Memory-based control with recurrent neural networks. Technical report, Google DeepMind, 2015. http://arxiv.org/abs/1512.04455 . Johannes Heinrich and David Silver. Deep reinforcement learning from self-play in imperfect-information games. Technical report, DeepMind, 2016. http://arxiv. org/abs/1603.01121 . Matthias Heizmann, Daniel Dietsch, Jan Leike, Betim Musa, and Andreas Podelski. UltimateAutomizerwitharrayinterpolation(competitioncontribution). In Tools and Algorithms for the Construction and Analysis of Systems , pages 455–457. Springer, 2015. Matthias Heizmann, Daniel Dietsch, Marius Greitschus, Jan Leike, Betim Musa, Claus Schätzle, and Andreas Podelski. Ultimate Automizer with two-track proofs (compe- tition contribution). In Tools and Algorithms for the Construction and Analysis of Systems, pages 950–953. Springer, 2016. Carl G Hempel. Studies in the logic of conﬁrmation (I.). Mind, pages 1–26, 1945. Carl G Hempel. The white shoe: No red herring. The British Journal for the Philosophy of Science , 18(3):239–240, 1967. Marcus Hutter. A theory of universal artiﬁcial intelligence based on algorithmic com- plexity. Technical report, 2000. http://arxiv.org/abs/cs.AI/0004001 . 160 BIBLIOGRAPHY Marcus Hutter. Universal sequential decisions in unknown environments. In European Workshop on Reinforcement Learning , pages 25–26, 2001a. Marcus Hutter. New error bounds for Solomonoﬀ prediction. Journal of Computer and System Sciences , 62(4):653–667, 2001b. Marcus Hutter. Self-optimizing and Pareto-optimal policies in general environments based on Bayes-mixtures. In Computational Learning Theory , pages 364–379. Springer, 2002a. Marcus Hutter. The fastest and shortest algorithm for all well-deﬁned problems. In- ternational Journal of Foundations of Computer Science , 13(03):431–443, 2002b. MarcusHutter. AgentleintroductiontotheuniversalalgorithmicagentAIXI. Technical report, IDSIA, 2003. ftp://ftp.idsia.ch/pub/techrep/IDSIA-01-03.ps.gz . Marcus Hutter. Universal Artiﬁcial Intelligence . Springer, 2005. Marcus Hutter. Sequential predictions based on algorithmic complexity. Journal of Computer and System Sciences , 72(1):95–117, 2006a. Marcus Hutter. General discounting versus average reward. In Algorithmic Learning Theory, pages 244–258. Springer, 2006b. Marcus Hutter. 50’000 €prize for compressing human knowledge. http://prize. hutter1.net/ , 2006c. Accessed: 2015-03-29. Marcus Hutter. Universal algorithmic intelligence: A mathematical top !down ap- proach. In Artiﬁcial General Intelligence , pages 227–290. Springer, 2007a. Marcus Hutter. On universal prediction and Bayesian conﬁrmation. Theoretical Com- puter Science , 384(1):33–48, 2007b. Marcus Hutter. Discrete MDL predicts in total variation. In Advances in Neural Information Processing Systems , pages 817–825, 2009a. Marcus Hutter. Open problems in universal induction & intelligence. Algorithms , 3(2): 879–906, 2009b. Marcus Hutter. Feature dynamic Bayesian networks. In Artiﬁcial General Intelligence , pages 67–73. Atlantis Press, 2009c. Marcus Hutter. Feature reinforcement learning: Part I: Unstructured MDPs. Journal of Artiﬁcial General Intelligence , 1:3–24, 2009d. Marcus Hutter. Can intelligence explode? Journal of Consciousness Studies , 19(1–2): 143–166, 2012a. Marcus Hutter. One decade of universal artiﬁcial intelligence. In Theoretical Founda- tions of Artiﬁcial General Intelligence , pages 67–88. Springer, 2012b. BIBLIOGRAPHY 161 Marcus Hutter. Extreme state aggregation beyond MDPs. In Algorithmic Learning Theory. Springer, 2014. Marcus Hutter and Andrej A. Muchnik. On semimeasures predicting Martin-Löf ran- dom sequences. Theoretical Computer Science , 382(3):247–261, 2007. IBM. Deep Blue. http://www-03.ibm.com/ibm/history/ibm100/us/en/icons/ deepblue/ , March 2012a. Accessed: 2015-03-27. IBM. A computer called Watson. http://www-03.ibm.com/ibm/history/ibm100/us/ en/icons/watson/ , March 2012b. Accessed: 2015-03-27. Edwin T Jaynes. Probability Theory: The Logic of Science . Cambridge University Press, 2003. Sham Machandranath Kakade. On the Sample Complexity of Reinforcement Learning . PhD thesis, University College London, 2003. Ehud Kalai and Ehud Lehrer. Rational learning leads to Nash equilibrium. Economet- rica, pages 1019–1045, 1993. Ehud Kalai and Ehud Lehrer. Weak and strong merging of opinions. Journal of Math- ematical Economics , 23(1):73–86, 1994. EmilieKaufmann, NathanielKorda, andRémiMunos. Thompsonsampling: Anasymp- toticallyoptimalﬁnite-timeanalysis. In Algorithmic Learning Theory , pages199–213. Springer, 2012. HJ Kim, Michael I Jordan, Shankar Sastry, and Andrew Y Ng. Autonomous helicopter ﬂight via reinforcement learning. In Advances in Neural Information Processing Sys- tems, page None, 2003. Stephen Cole Kleene. Introduction to Metamathematics . Wolters-Noordhoﬀ Publishing, 1952. Tejas D Kulkarni, Karthik R Narasimhan, Ardavan Saeedi, and Joshua B Tenenbaum. Hierarchical deep reinforcement learning: Integrating temporal abstraction and in- trinsic motivation. Technical report, Massachusetts Institute of Technology, 2016. https://arxiv.org/abs/1604.06057 . Ray Kurzweil. The Singularity is Near: When Humans Transcend Biology . Viking Books, 2005. Tor Lattimore. Theory of General Reinforcement Learning . PhD thesis, Australian National University, 2013. Tor Lattimore. Regret analysis of the ﬁnite-horizon Gittins index strategy for multi- armed bandits. In Conference on Learning Theory , 2016. 162 BIBLIOGRAPHY Tor Lattimore and Marcus Hutter. Asymptotically optimal agents. In Algorithmic Learning Theory , pages 368–382. Springer, 2011. Tor Lattimore and Marcus Hutter. PAC bounds for discounted MDPs. In Algorithmic Learning Theory , pages 320–334. Springer, 2012. TorLattimoreandMarcusHutter. OnMartin-LöfconvergenceofSolomonoﬀ’smixture. InTheory and Applications of Models of Computation , volume 7876, pages 212–223. Springer, 2013. Tor Lattimore and Marcus Hutter. General time consistent discounting. Theoretical Computer Science , 519:140–154, 2014. Tor Lattimore and Marcus Hutter. On Martin-Löf (non-)convergence of Solomonoﬀ’s universal mixture. Theoretical Computer Science , 588:2–15, 2015. Tor Lattimore, Marcus Hutter, and Vaibhav Gavane. Universal prediction of selected bits. InAlgorithmic Learning Theory , pages 262–276. Springer, 2011. Neil Lawrence. Future of AI 6. Discussion of ‘Superintelligence: Paths, Dangers, Strategies’. http://inverseprobability.com/2016/05/09/ machine-learning-futures-6 , May 2016. Accessed: 2016-05-10. Yann LeCun, Yoshua Bengio, and Geoﬀrey Hinton. Deep learning. Nature, 521(7553): 436–444, 2015. Shane Legg. Is there an elegant universal theory of prediction? In Algorithmic Learning Theory, pages 274–287, 2006. Shane Legg. Machine Super Intelligence . PhD thesis, University of Lugano, 2008. Shane Legg and Marcus Hutter. A collection of deﬁnitions of intelligence. Frontiers in Artiﬁcial Intelligence and Applications , 157:17–24, 2007a. Shane Legg and Marcus Hutter. Universal intelligence: A deﬁnition of machine intelli- gence.Minds & Machines , 17(4):391–444, 2007b. Shane Legg and Joel Veness. An approximation of the universal intelligence measure. In Algorithmic Probability and Friends. Bayesian Prediction and Artiﬁcial Intelligence , pages 236–249. Springer, 2013. Ehud Lehrer and Rann Smorodinsky. Merging and learning. Statistics, Probability and Game Theory , pages 147–168, 1996. Jan Leike and Matthias Heizmann. Ranking templates for linear loops. In Tools and Algorithms for the Construction and Analysis of Systems , pages 172–186. Springer, 2014a. Jan Leike and Matthias Heizmann. Geometric series as nontermination arguments for linear lasso programs. Technical report, University of Freiburg, 2014b. http: //arxiv.org/abs/1405.4413 . BIBLIOGRAPHY 163 Jan Leike and Matthias Heizmann. Ranking templates for linear loops. Logical Methods in Computer Science , 11(1):1–27, March 2015. Jan Leike and Matthias Heizmann. Geometric nontermination arguments. 2016. Under preparation. Jan Leike and Marcus Hutter. Indeﬁnitely oscillating martingales. In Algorithmic Learning Theory , pages 321–335, 2014a. Jan Leike and Marcus Hutter. Indeﬁnitely oscillating martingales. Technical report, Australian National University, 2014b. http://arxiv.org/abs/1408.3169 . JanLeikeandMarcusHutter. OnthecomputabilityofAIXI. In Uncertainty in Artiﬁcial Intelligence , pages 464–473, 2015a. Jan Leike and Marcus Hutter. On the computability of Solomonoﬀ induction and knowledge-seeking. In Algorithmic Learning Theory , pages 364–378, 2015b. Jan Leike and Marcus Hutter. Bad universal priors and notions of optimality. In Conference on Learning Theory , pages 1244–1259, 2015c. Jan Leike and Marcus Hutter. Solomonoﬀ induction violates Nicod’s criterion. In Algorithmic Learning Theory , pages 349–363. Springer, 2015d. Jan Leike and Marcus Hutter. On the computability of Solomonoﬀ induction and AIXI. 2016. Under review. Jan Leike, Tor Lattimore, Laurent Orseau, and Marcus Hutter. Thompson sampling is asymptotically optimal in general environments. In Uncertainty in Artiﬁcial Intelli- gence, 2016a. Jan Leike, Jessica Taylor, and Benya Fallenstein. A formal solution to the grain of truth problem. In Uncertainty in Artiﬁcial Intelligence , 2016b. Leonid A Levin. On the notion of a random sequence. Soviet Mathematics Doklady , 14 (5):1413–1416, 1973. Ming Li and Paul M. B. Vitányi. An Introduction to Kolmogorov Complexity and Its Applications . Texts in Computer Science. Springer, 3rd edition, 2008. Yitao Liang, Marlos C Machado, Erik Talvitie, and Michael Bowling. State of the art control of Atari games using shallow reinforcement learning. In Autonomous Agents and Multiagent Systems , 2016. Timothy P Lillicrap, Jonathan J Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep rein- forcement learning. In International Conference on Learning Representations , 2016. John L Mackie. The paradox of conﬁrmation. British Journal for the Philosophy of Science, pages 265–277, 1963. 164 BIBLIOGRAPHY Omid Madani, Steve Hanks, and Anne Condon. On the undecidability of probabilis- tic planning and inﬁnite-horizon partially observable Markov decision problems. In AAAI, pages 541–548, 1999. Omid Madani, Steve Hanks, and Anne Condon. On the undecidability of probabilistic planning and related stochastic optimization problems. Artiﬁcial Intelligence , 147 (1):5–34, 2003. Sridhar Mahadevan. Optimality criteria in reinforcement learning. In AAAI Fall Sym- posium on Learning Complex Behaviors in Adaptive Intelligent Systems , 1996. Patrick Maher. Inductive logic and the ravens paradox. Philosophy of Science , pages 50–70, 1999. A Rupam Mahmood, Huizhen Yu, Martha White, and Richard Sutton. Emphatic temporal-diﬀerence learning. Technical report, University of Alberta, 2015. http: //arxiv.org/abs/1507.01569 . Norman Margolus and Lev B Levitin. The maximum speed of dynamical evolution. Physica D: Nonlinear Phenomena , 120(1):188–195, 1998. Jarryd Martin, Tom Everitt, and Marcus Hutter. Death and suicide in universal artiﬁ- cial intelligence. In Artiﬁcial General Intelligence , 2016. JohnMcCarthy, MarvinMinsky, NathanielRochester, andClaudeShannon. Aproposal for the Dartmouth summer research project on artiﬁcial intelligence. 1955. Ronald I Miller and Chris William Sanchirico. The role of absolute continuity in “merg- ing of opinions” and “rational learning”. Games and Economic Behavior , 29(1):170– 190, 1999. Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves, Ioannis Antonoglou, Daan Wierstra, and Martin Riedmiller. Playing Atari with deep reinforcement learn- ing. Technical report, Google DeepMind, 2013. http://arxiv.org/abs/1312.5602 . Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Os- trovski, Stig Petersen, Charles Beattie, Amir Sadik, Ioannis Antonoglou, Helen King, Dharshan Kumaran, Daan Wierstra, Shane Legg, and Demis Hassabis. Human-level control through deep reinforcement learning. Nature, 518(7540):529–533, 2015. Volodymyr Mnih, Adrià Puigdomènech Badia, Mehdi Mirza, Alex Graves, Timothy P Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In International Conference on Machine Learning , 2016. Luke Muehlhauser and Anna Salamon. Intelligence explosion: Evidence and import. InSingularity Hypotheses , pages 15–42. Springer, 2012. BIBLIOGRAPHY 165 Martin Mundhenk, Judy Goldsmith, Christopher Lusena, and Eric Allender. Complex- ity of ﬁnite-horizon Markov decision process problems. Journal of the ACM , 47(4): 681–720, 2000. Markus Müller. Stationary algorithmic probability. Theoretical Computer Science , 411 (1):113–130, 2010. Vincent C Müller and Nick Bostrom. Future progress in artiﬁcial intelligence: A survey of expert opinion. Fundamental Issues of Artiﬁcial Intelligence , pages 553–571, 2016. John H Nachbar. Prediction, optimization, and learning in repeated games. Economet- rica, 65(2):275–309, 1997. John H Nachbar. Beliefs in repeated games. Econometrica , 73(2):459–480, 2005. Arun Nair, Praveen Srinivasan, Sam Blackwell, Cagdas Alcicek, Rory Fearon, Alessan- dro De Maria, Vedavyas Panneershelvam, Mustafa Suleyman, Charles Beattie, Stig Petersen, Shane Legg, Volodymyr Mnih, Koray Kavukcuoglu, and David Silver. Mas- sively parallel methods for deep reinforcement learning. Technical report, Google DeepMind, 2015. http://arxiv.org/abs/1507.04296 . Andrew Ng. Is A.I. an existential threat to humanity? https://www.quora.com/ Is-A-I-an-existential-threat-to-humanity/answer/Andrew-Ng , January 2016. Accessed: 2016-05-19. Phuong Nguyen, Odalric-Ambrym Maillard, Daniil Ryabko, and Ronald Ortner. Com- peting with an inﬁnite set of models in reinforcement learning. In Artiﬁcial Intelli- gence and Statistics , pages 463–471, 2013. Jean Nicod. Le Problème Logique de L’Induction . Presses Universitaires de France, 1961. André Nies. Computability and Randomness . Oxford University Press, 2009. James Olds and Peter Milner. Positive reinforcement produced by electrical stimulation ofseptalareaandotherregionsofratbrain. Journal of Comparative and Physiological Psychology , 47(6):419, 1954. Stephen M Omohundro. The basic AI drives. In Artiﬁcial General Intelligence , pages 483–492, 2008. OpenAI. About OpenAI. https://openai.com/about/ , December 2015. Accessed: 2016-04-27. Laurent Orseau. Optimality issues of universal greedy agents with static priors. In Algorithmic Learning Theory , pages 345–359. Springer, 2010. Laurent Orseau. Universal knowledge-seeking agents. In Algorithmic Learning Theory , pages 353–367. Springer, 2011. 166 BIBLIOGRAPHY Laurent Orseau. Asymptotic non-learnability of universal agents with computable hori- zon functions. Theoretical Computer Science , 473:149–156, 2013. Laurent Orseau. Universal knowledge-seeking agents. Theoretical Computer Science , 519:127–139, 2014a. Laurent Orseau. The multi-slot framework: A formal model for multiple, copiable AIs. InArtiﬁcial General Intelligence , pages 97–108. Springer, 2014b. Laurent Orseau. Teleporting universal intelligent agents. In Artiﬁcial General Intelli- gence, pages 109–120. Springer, 2014c. Laurent Orseau and Stuart Armstrong. Safely interruptible agents. In Uncertainty in Artiﬁcial Intelligence , pages 557–566, 2016. Laurent Orseau and Mark Ring. Self-modiﬁcation and mortality in artiﬁcial agents. In Artiﬁcial General Intelligence , pages 1–10. Springer, 2011. LaurentOrseauandMarkRing. Space-timeembeddedintelligence. In Artiﬁcial General Intelligence , pages 209–218. Springer, 2012a. Laurent Orseau and Mark Ring. Memory issues of intelligent agents. In Artiﬁcial General Intelligence , pages 219–231. Springer, 2012b. Laurent Orseau, Tor Lattimore, and Marcus Hutter. Universal knowledge-seeking agents for stochastic environments. In Algorithmic Learning Theory , pages 158–172. Springer, 2013. Pedro A Ortega and Daniel A Braun. A minimum relative entropy principle for learning and acting. Journal of Artiﬁcial Intelligence Research , pages 475–511, 2010. Pedro A Ortega and Daniel A Braun. Generalized Thompson sampling for sequential decision-making and causal inference. Complex Adaptive Systems Modeling , 2(1):2, 2014. Ian Osband, Dan Russo, and Benjamin van Roy. (More) eﬃcient reinforcement learning via posterior sampling. In Neural Information Processing Systems , pages 3003–3011, 2013. Christos H Papadimitriou and John N Tsitsiklis. The complexity of Markov decision processes. Mathematics of Operations Research , 12(3):441–450, 1987. Martin L Puterman. Markov Decision Processes . John Wiley & Sons, 2014. SamuelRathmannerandMarcusHutter. Aphilosophicaltreatiseofuniversalinduction. Entropy, 13(6):1076–1136, 2011. Mark Ring and Laurent Orseau. Delusion, survival, and intelligent agents. In Artiﬁcial General Intelligence , pages 11–20. Springer, 2011. BIBLIOGRAPHY 167 Chris Rogers and David Williams. Diﬀusions, Markov Processes, and Martingales: Volume 1, Foundations . Cambridge University Press, 2nd edition, 1994. Stuart Russell, Daniel Dewey, and Max Tegmark. Research priorities for robust and beneﬁcialartiﬁcialintelligence. Technicalreport, FutureofLifeInstitute, 2015. http: //arxiv.org/abs/1602.03506 . Stuart J Russell and Peter Norvig. Artiﬁcial Intelligence. A Modern Approach . Prentice Hall, 3rd edition, 2010. Daniil Ryabko. Characterizing predictable classes of processes. In Uncertainty in Arti- ﬁcial Intelligence , pages 471–478, 2009. Daniil Ryabko. On ﬁnding predictors for arbitrary families of processes. The Journal of Machine Learning Research , 11:581–602, 2010. Daniil Ryabko. On the relation between realizable and nonrealizable cases of the se- quence prediction problem. The Journal of Machine Learning Research , 12:2161– 2180, 2011. Daniil Ryabko and Marcus Hutter. On sequence prediction for arbitrary measures. In IEEE International Symposium on Information Theory , pages 2346–2350, 2007. Daniil Ryabko and Marcus Hutter. Predicting non-stationary processes. Applied Math- ematics Letters , 21(5):477–482, 2008. Régis Sabbadin, Jérôme Lang, and Nasolo Ravoanjanahry. Purely epistemic Markov decision processes. In AAAI, pages 1057–1062, 2007. Tom Schaul, John Quan, Ioannis Antonoglou, and David Silver. Prioritized experience replay. In International Conference on Learning Representations , 2016. Jürgen Schmidhuber. The speed prior: A new simplicity measure yielding near-optimal computablepredictions. In Computational Learning Theory , pages216–228.Springer, 2002. Jürgen Schmidhuber. Philosophers & futurists, catch up! Journal of Consciousness Studies, 19(1-2):173–182, 2012. Jürgen Schmidhuber. Deep learning in neural networks: An overview. Neural Networks , 61:85–117, 2015. Murray Shanahan. The Technological Singularity . MIT Press, 2015. ClaudeEShannon. Programmingacomputerforplayingchess. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science , 41(314):256–275, 1950. Joseph R Shoenﬁeld. Mathematical Logic . Addison-Wesley Reading, 1967. Yoav Shoham and Kevin Leyton-Brown. Multiagent Systems: Algorithmic, Game- Theoretic, and Logical Foundations . Cambridge University Press, 2009. 168 BIBLIOGRAPHY DavidSilver,AjaHuang,ChrisJMaddison,ArthurGuez,LaurentSifre,Georgevanden Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, Sander Dieleman, Dominik Grewe, John Nham, Nal Kalchbrenner, Ilya Sutskever, Timothy Lillicrap, Madeleine Leach, Koray Kavukcuoglu, Thore Graepel, and Demis Hassabis. Mastering the game of Go with deep neural networks and tree search.Nature, 529(7587):484–489, 2016. Satinder Singh, Michael L Littman, Nicholas K Jong, David Pardoe, and Peter Stone. Learning predictive state representations. In International Conference on Machine Learning , pages 712–719, 2003. Satinder Singh, Michael R James, and Matthew R Rudary. Predictive state represen- tations: A new theory for modeling dynamical systems. In Uncertainty in Artiﬁcial Intelligence , pages 512–519, 2004. Nate Soares. Formalizing two problems of realistic world-models. Technical report, Machine Intelligence Research Institute, 2015. http://intelligence.org/files/ RealisticWorldModels.pdf . Nate Soares and Benja Fallenstein. Aligning superintelligence with human interests: A technical research agenda. Technical report, Machine Intelligence Research Institute, 2014. http://intelligence.org/files/TechnicalAgenda.pdf . Ray Solomonoﬀ. A formal theory of inductive inference. Parts 1 and 2. Information and Control , 7(1):1–22 and 224–254, 1964. Ray Solomonoﬀ. Complexity-based induction systems: Comparisons and convergence theorems. IEEE Transactions on Information Theory , 24(4):422–432, 1978. Kaj Sotala and Roman V Yampolskiy. Responses to catastrophic AGI risk: A survey. Physica Scripta , 90(1):018001, 2014. Tom F Sterkenburg. Putnam’s diagonal argument and the impossibility of a universal learning machine. Technical report, Centrum Wiskunde & Informatica, 2016. http: //philsci-archive.pitt.edu/12096/ . Jordan M Stoyanov. Counterexamples in Probability . Courier Corporation, 3rd edition, 2013. Malcolm Strens. A Bayesian framework for reinforcement learning. In International Conference on Machine Learning , pages 943–950, 2000. Peter Sunehag and Marcus Hutter. Consistency of feature Markov processes. In Algo- rithmic Learning Theory , pages 360–374. Springer, 2010. Peter Sunehag and Marcus Hutter. Optimistic agents are asymptotically optimal. In Australasian Joint Conference on Artiﬁcial Intelligence , pages15–26.Springer, 2012a. BIBLIOGRAPHY 169 Peter Sunehag and Marcus Hutter. Optimistic AIXI. In Artiﬁcial General Intelligence , pages 312–321. Springer, 2012b. Peter Sunehag and Marcus Hutter. Rationality, optimism and guarantees in general reinforcement learning. Journal of Machine Learning Research , 16:1345–1390, 2015. Richard Sutton. Learning to predict by the methods of temporal diﬀerences. Machine Learning , 3(1):9–44, 1988. Richard S. Sutton and Andrew G. Barto. Reinforcement Learning: An Introduction . MIT Press, 1998. Richard G Swinburne. The paradoxes of conﬁrmation: A survey. American Philosoph- ical Quarterly , pages 318–330, 1971. Csaba Szepesvári. Algorithms for Reinforcement Learning . Morgan & Claypool Pub- lishers, 2010. Gerald Tesauro. Temporal diﬀerence learning and TD-Gammon. Communications of the ACM , 38(3):58–68, 1995. William R Thompson. On the likelihood that one unknown probability exceeds another in view of the evidence of two samples. Biometrika , pages 285–294, 1933. John N Tsitsiklis and Benjamin Van Roy. An analysis of temporal-diﬀerence learning with function approximation. IEEE Transactions on Automatic Control , 42(5):674– 690, 1997. Alexandre Tsybakov. Introduction to Nonparametric Estimation . Springer, 2008. Hado van Hasselt, Arthur Guez, and David Silver. Deep reinforcement learning with double Q-learning. In AAAI, 2016. Joel Veness, Kee Siong Ng, Marcus Hutter, William Uther, and David Silver. A Monte- Carlo AIXI approximation. Journal of Artiﬁcial Intelligence Research , 40(1):95–142, 2011. Joel Veness, Marc G Bellemare, Marcus Hutter, Alvin Chua, and Guillaume Desjardins. Compress and control. In AAAI, 2015. Vernor Vinge. The coming technological singularity. Vision 21: Interdisciplinary Sci- ence and Engineering in the Era of Cyberspace , 1:11–22, 1993. http://www.rohan. sdsu.edu/faculty/vinge/misc/singularity.html . Paul MB Vitányi, Frank J Balbach, Rudi L Cilibrasi, and Ming Li. Normalized in- formation distance. In Information Theory and Statistical Learning , pages 45–82. Springer, 2009. Nikos Vlassis, Mohammad Ghavamzadeh, Shie Mannor, and Pascal Poupart. Bayesian reinforcement learning. In Marco Wiering and Martijn van Otterlo, editors, Rein- forcement Learning , pages 359–386. Springer, 2012. 170 BIBLIOGRAPHY PeterBMVranas. Hempel’sravenparadox: AlacunainthestandardBayesiansolution. The British Journal for the Philosophy of Science , 55(3):545–560, 2004. Toby Walsh. The singularity may never be near. Technical report, University of New South Wales, 2016. http://arxiv.org/abs/1602.06462 . Ziyu Wang, Nando de Freitas, Tom Schaul, Matteo Hessel, Hado van Hasselt, and Marc Lanctot. Dueling network architectures for deep reinforcement learning. In International Conference on Machine Learning , 2016. Larry Wasserman. All of Statistics . Springer, 2004. Christopher JCH Watkins and Peter Dayan. Q-learning. Machine learning , 8(3-4): 279–292, 1992. Eric W Weisstein. Random walk—1-dimensional. In MathWorld—A Wolfram Web Resource . Wolfram Research, Inc., 2002. http://mathworld.wolfram.com/ RandomWalk1-Dimensional.html . Marco Wiering and Martijn van Otterlo, editors. Reinforcement Learning . Springer, 2012. Ian Wood, Peter Sunehag, and Marcus Hutter. (Non-)equivalence of universal priors. InSolomonoﬀ 85th Memorial Conference , pages 417–425. Springer, 2011. Huizhen Yu. On convergence of emphatic temporal-diﬀerence learning. In Conference on Learning Theory , pages 1724–1751, 2015. Eliezer Yudkowsky. Creating friendly AI 1.0: The analysis and design of benevolent goal architectures. Technical report, Singularity Institute for Artiﬁcial Intelligence, 2001. Eliezer Yudkowsky. Artiﬁcial intelligence as a positive and negative factor in global risk. InGlobal Catastrophic Risks , pages 308–345. Oxford University Press, 2008. Tom Zahavy, Nir Ben Zrihem, and Shie Mannor. Graying the black box: Understanding DQNs. Technical report, Israel Institute of Technology, 2016. http://arxiv.org/ abs/1602.02658 . JacobZivandAbrahamLempel. Auniversalalgorithmforsequentialdatacompression. IEEE Transactions on information theory , 23(3):337–343, 1977. List of Notation Abbreviations AIXI Bayesian RL agent with a Solomonoﬀ prior, see Section 4.3.1 AI artiﬁcial intelligence HLAI human-level artiﬁcial intelligence MDL minimum description length, see Example 3.14 MDP Markov decision process, see Section 4.1.3 POMDP partially observable Markov decision process, see Section 4.1.3 RL reinforcement learning UTM universal Turing machine, Section 2.4 Mathematical notation := deﬁned to be equal :2 deﬁned to be an element of A,B, sets #A the cardinality of the set A, i.e., the number of elements the set of probability distributions over a ﬁnite or countable set 1x the characteristic function that is 1forxand0otherwise. f,g functions fg there is a constant c>0such thatfcg f=g fgandgf N the set of natural numbers, starting with 1 Q the set of rational numbers R the set of real numbers n,k,t,m,i,jnatural numbers t (current) time step, t2N k some other time step, k2N q;q0rational numbers r real number X a ﬁnite nonempty alphabet Xthe set of all ﬁnite strings over the alphabet X X1the set of all inﬁnite strings over the alphabet X X]X]:=X[X1, the set of all ﬁnite and inﬁnite strings over the alphabetX x;y;z (typically ﬁnite) strings from X] x<t the ﬁrstt1symbols of the string x 171 172 List of Notation xvy the stringxis a preﬁx of the string y zeros(x) the number of zeros in the binary string x2f0;1g ones(x) the number of ones in the binary string x2f0;1g , computable functions ' formula of ﬁrst-order logic computable relation/quantiﬁer-free formula T a Turing machine p;p0programs on a universal Turing machine in the form of ﬁnite binary strings jpj length of the program pin bits K the Kolmogorov complexity of a string or a semimeasure Km the monotone Kolmogorov complexity of a string Ent entropy KLm KL-divergence Dm total variation distance IG information gain F expected total variation distance F,Ft,F1-algebras x the cylinder set of all strings starting with x A,H,E measurable sets X,Y real-valued random variables P a distribution over X1, thetruedistribution Q adistributionover X1, thelearningalgorithmorbeliefdistribution Bernoulli (r)a Bernoulli process with parameter r the uniform measure or Lebesgue measure L Laplace rule M Solomonoﬀ’s prior M the measure mixture SKt the speed prior a semimeasure norm the Solomonoﬀ normalization of the semimeasure absolute continuity W weak dominance L local absolute continuity A the ﬁnite set of possible actions O the ﬁnite set of possible observations E the ﬁnite set of possible percepts, EO R ; two diﬀerent actions, ;2A at the action in time step t ot the observation in time step t rt the reward in time step t, bounded between 0and1 et the percept in time step t, we useet= (ot;rt)implicitly æ<t the ﬁrstt1interactions, a1e1a2e2:::at1et1(a history of length t1) List of Notation 173 h a history,h2(AE ) the history of length 0 ", small positive real numbers the discount function :N!R0, deﬁned in Deﬁnition 4.3 t a discount normalization factor, t:=P1 k=t (k) m horizon of the agent (how many steps it plans ahead) Ht(") an"-eﬀective horizon ;; environments ;~ policies,;~: (AE )!A an optimal policy for environment the history distribution generated by policy in environment E the expectation with respect to the history distribution V the-expected value of the policy V the optimal value in environment W the iterative value of the policy in environment W the optimal iterative value in environment Pmax the max-sum-operator Rm(;) regret of policy in environment for horizon m () the Legg-Hutter intelligence of policy measured in the universal mixture the minimal Legg-Hutter intelligence measured in the universal mixture the maximal Legg-Hutter intelligence measured in the universal mixture M a class of environments MCCSthe class of all chronological contextual semimeasures MCCS LSC the class of all lower semicomputable chronological contextual semimeasures MCCM comp the class of all computable chronological contextual measures MO re the class of all reﬂective-oracle-computable environments U reference universal Turing machine U0a ‘bad’ universal Turing machine w a positive prior over the environment class w0a ‘bad’ positive prior over the environment class the universal mixture over all environments MCCS LSCgiven by the reference UTM U 0a ‘bad’ universal mixture over all environments MCCS LSCgiven by the ‘bad’ UTM U0 T the set of all probabilistic Turing machines O an oracle ~O a partial oracle T the semimeasure generated by Turing machine T O T the semimeasure generated by Turing machine Trunning with or- acleO 174 List of Notation O T the completion of O Tinto a measure using oracle O a multi-agent environment i the subjective environment of agent iacting in multi-agent envi- ronment Index absolute continuity, 25, 26, 26, 28–33, 38–40 local, 25,28, 28–31, 33 action, 2, 50, 53 adversarial environment, 112 sequence, 36 agent, 2, 49, 51 agent model dualistic, 50, 52 physicalistic, 52 AI human-level, 1 narrow, 1 strong, 1 AI safety, 148 AIMU, 57, 111 AINU, 57, 110, 112, 116, 118, 119 AIXI, 7,62, 69, 78, 81, 96, 97, 111, 114, 120 AIXItl, 9, 74, 75 alphabet, 15 anytime algorithm, 101, 104 argmax tie, 57 arithmetical hierarchy, 101, 103, 103 Atari 2600, 4, 13, 54 bandit,seemulti-armed bandit BayesExp, 65, 65, 84, 122, 123 Bayesian controlrule, seeThompsonsampling mixture,26, 45, 63, 136 Bellman equation, 57 black ravens, 25, 30, 32, 47 Carathéodory’s extension theorem, 152 Cesàro average, 18, 79 chain rule, 130compatibility, 24, 25, 31, 45 compression, seeuniversal compression computable, 7, 16, 16, 103, 104, 106 reﬂective-oracle, 131, 136, 140, 142 conditional expectation, 17 conﬁrmation, 24 Control Problem, 148 convergence almost surely, 18, 60, 79 in mean,18, 79 in probability, 18, 79 martingale, 19, 30, 31, 84, 86, 142 cylinder set, 17, 21, 29, 51 deep learning, 4 diameter, 3, 90 disconﬁrmation, 24 discount function, 10, 52, 52, 53, 81, 84, 92 normalization factor, 52 discounting ﬁnite horizon, 53, 70, 73 geometric, 3, 53, 88, 89, 92, 95 power, 53 subgeometric, 53, 92 dominance, 25, 25, 26, 28, 31, 37, 40 weak, 25, 27, 27, 28, 31, 34, 35, 40 with coeﬃcients, 25, 27, 28, 31 DQN, 4 dualistic model, 52 "-best response, 139, 140–143 eﬀective horizon, 52, 80–83, 86, 89, 91, 98 bounded, 52, 83 entropy,19, 38, 63 relative, seeKL-divergence environment, 2, 49, 51 175 176 Index deterministic, 51 environment class, 51, 52, 54 Epicurus’ principle, 26, 41 equivalence condition, 47 ergodic, 3, 6, 7, 55 error cumulative, 35, 36, 38 instantaneous, 35 estimable in polynomial time, 43, 43 event,17 existential risk, 147 exploration vs. exploitation, 2, 8, 49, 54, 63, 65, 83, 96, 124 falsify, 24 feature reinforcement learning, 7 ﬁcticious play, 129 ﬁltration, 17, 51 function approximation, 3, 5 general reinforcement learning problem, 2, 49,51 Gibb’s inequality, 19 goal, 2, 4, 6, 8, 49 good enough eﬀect, 81 grain of truth, 11, 25, 127 problem, 11, 128, 137 heaven, 76, 78, 91, 103 hell, 76, 78, 91, 103 Hempel’s paradox, seeparadox of con- ﬁrmation history, 2, 24, 50 distribution, 51,138 horizon,56 Hutter prize, 44 Hutter search, 75 hypothesis, 24, 25, 30 information gain, 64, 85, 98 intelligence, 4, 8, 75, 76–79 explosion, 147 maximal, 76 minimal, 76 invariance theorem, 42, 97KL-divergence, 19, 20, 34, 36, 38, 40, 64 knowledge-seeking, 6, 50, 63 Kolmogorov complexity, 20 monotone, 20, 43 Laplace rule, 28, 33, 45 learnable, 90 Legg-Hutterintelligence, seeintelligence limit computable, 101, 103, 123, 140, 142 lower semicomputable, 16, 16, 21, 103 M,seeenvironment class Markov decision process, seeMDP martingale, 18, 19, 29–31 matching pennies, 139, 141, 143 MDL,28, 28, 33, 45 MDP, 3,54 measurable set, 16,17, 17 space,16 measure,21, 30 Bernoulli, 25 compatibility, seecompatibility deterministic, 17, 37 Lebesgue, 21, 37 mixture,42, 106–108 probability, 17 uniform, seeLebesgue measure merging, 23, 32, 36, 59 almost weak, 32, 34, 34, 35, 43, 60 of opinions, 32 strong,32, 32–34, 41, 44, 60 weak, 32, 33, 33, 34, 60 model-based, 3 model-free, 3 multi-agentenvironment, 137, 138, 138, 140, 142 multi-armed bandit, 54, 89 Nash equilibrium, 139 subgame perfect, 140 Nicod’s criterion, 47 nonparametric, 7, 52 Index 177 observation, 2, 50 Ockham’s razor, 26, 41 oﬀ-policy, 2, 49, 65, 83, 98 on-policy, 2, 112 value convergence, 49, 59–62 optimality action,57 asymptotic, 7, 68, 79, 81, 95, 98, 99, 128 in mean, 79, 84, 89, 142 in probability, 79 strong, 79, 81, 82, 88 weak, 79, 81, 84, 89, 123 balanced Pareto, 67, 76, 76 Bayes, 67, 73, 76, 82 Pareto, 67, 69, 69 oracle,130 halting, 101, 103, 132 partial,132, 132–134 partially reﬂective, 133, 133, 134 reﬂective, 129, 130, 130–134 0 n-formula,103 PAC, 3, 68, 79 paradox of conﬁrmation, 47 partially observable, 5–7, 49, 55, 80 Peano arithmetic, 74 percept, 2, 50, 53 physicalistic model, 52 Pinsker’s inequality, 20 planning, 3, 49 policy, 49, 51, 109 Bayes optimal, 62 computable, 73, 77, 79 consistent with history, 51 deterministic, 51, 78, 81 "-optimal,58, 110, 111, 114, 120 optimal,57, 95, 109, 136, 140 POMDP, 6, 55, 89, 91, 92 Post’s theorem, 103 posterior, 24, 30, 62 sampling, seeThompson sampling power set, 16 prediction, 23expected regret, 35, 36–38, 40, 41, 43, 44 loss, 35 regret,35, 35–37, 39, 40 with expert advice, 23 prior, 24 dogmatic, 10, 70, 71, 73, 97, 141 Gödel, 70, 74, 97 indiﬀerence, 70, 70, 97 positive,26 speed,27, 43, 45 prisoner’s dilemma, 127, 144 program, 20 Q-learning, 2–5, 7 query, 130 Radon-Nikodym derivative, 31 random variable, 17 realizable case, 7, 23, 52 recoverability, 6, 9, 91, 91, 92 refute, 24 regret, 3, 68, 90, 91, 92, 95, 96 reward, 2, 50, 53 -algebra,16, 51 Borel,16 0 n-formula,103 SARSA, 2 self-optimizing, 82 semimeasure, 21 chronological, 51 contextual, 51 semiprior, 26 semiring, 152 Solomonoﬀ induction, 41 mixture, 26 normalization, 21, 48 prior, 7,26, 26, 41, 42, 45, 106, 108, 135 state, 3, 7, 54 stochastic process, 18 subjective enviroment, 138 178 Index t0-value,56 tail event, 32, 44, 100 TD-learning, 3 Thompson sampling, 10, 65, 66, 84, 88, 96, 142, 143 time consistent, 52, 62 total variation distance, 19, 20, 59 expected, 85, 85 trap, 2, 49, 55, 90, 99 Turing machine monotone, 20, 21, 129 natural, 97 probabilistic, 129 universal, 20, 26, 42, 98 universal artiﬁcial intelligence, 62 compression, 43–45 induction, seeSolomonoﬀ induction universal artiﬁcial intelligence, 149 value function, 49, 56, 58, 59, 109, 136 entropy-seeking, 63, 64, 122 information-seeking, 64, 122 iterative, 115, 116, 118–120 recursive, 56, 56 reward-seeking, 63 weakly communicating, 55, 89–92 wireheading, 6, 149
242d6731-4b53-42c1-8f3d-f2e45f64a870	trentmkelly/LessWrong-43k	LessWrong	Advice for a smart 8-year-old bored with school Although my 8-year-old son likes his teacher, he is frequently bored at school. He attends a high quality suburban public school in the United States. He has a lot of traits in common with LessWrong readers, and we would like advice for what he can do to counter his boredom. Many of you must have found grade school more or less tedious. What were your coping strategies?
7ddf7022-02d6-479e-86f6-8ecbc8cee8d5	trentmkelly/LessWrong-43k	LessWrong	Using Brain-Computer Interfaces to get more data for AI alignment The purpose of this post is to sketch some ways that Brain Computer Interface (BCI) technology might help with various AI alignment techniques. Roughly, we can divide the strategic relevance of BCI technology into three broad categories.[1] 1. Enhancement. BCI technology could enhance human intelligence - for example by providing new sensory modalities, or augmenting cognition. [2] 2. Merge. BCI technology could enable a “merge” between AIs and humans. This is advocated by among others, Sam Altman and Elon Musk, and is the stated raison d'etre of Neuralink: > “I think if we can effectively merge with AI by improving the neural link between your cortex and your digital extension of yourself, which already...exists, just has a bandwidth issue. And then effectively you become an AI-human symbiote. And if that then is widespread, with anyone who wants it can have it, then we solve the control problem as well, we don't have to worry about some evil dictator AI because we are the AI collectively. That seems like the best outcome I can think of.” -Elon Musk, interview with Y Combinator (2016) [3] * On these proposals, humans are not merely enhanced - in some radical sense, humans merge with AI. It’s not entirely clear what these “merge” proposals mean, what merging would look like (Niplav: “It seems worrying that a complete company has been built on a vision that has no clearly articulated path to success.”), and "merge" as a alignment strategy seems to be quite unpopular in the AI safety community. In future work, I’d like to clarify merge proposals more. 3. Alignment aid. BCI allows us to get data from the brain that could improve the effectiveness of various AI alignment techniques. Whereas enhancement would indirectly help alignment by making alignment researchers smarter, alignment aid proposals are about directly improving the techniques themselves. This post is about category 3. In conversation, several AI safety researchers have mentioned that BCI could h
aa9097d7-8686-42b4-a0e9-3b02d52ace94	StampyAI/alignment-research-dataset/lesswrong	LessWrong	Boxing an AI? Boxing an AI is the idea that you can avoid the problems where an AI destroys the world by not giving it access to the world. For instance, you might give the AI access to the real world only through a chat terminal with a person, called the gatekeeper. This is should, theoretically prevent the AI from doing destructive stuff. [Eliezer has pointed out a problem with boxing AI: the AI might convince its gatekeeper to let it out. In order to prove this, he escaped from a simulated version of an AI box. Twice.](http://www.yudkowsky.net/singularity/aibox/) That is somewhat unfortunate, because it means testing AI is a bit trickier. However, I got an idea: why tell the AI it's in a box? Why not hook it up to a sufficiently advanced game, set up the correct reward channels and see what happens? Once you get the basics working, you can add more instances of the AI and see if they cooperate. This lets us adjust their morality until the AIs act sensibly. Then the AIs can't escape from the box because they don't know it's there.
86a4f182-efbf-4a0f-905d-06f33c318388	StampyAI/alignment-research-dataset/eaforum	Effective Altruism Forum	Why not offer a multi-million / billion dollar prize for solving the Alignment Problem? My impression (not well researched though) is that such prizes have served in the past to inspire many people to try to solve the problem, and they bring a ton of publicity to both the problem itself and to why the problem is difficult. I'm not sure if the money would need to be held somewhere in the meantime, but if not then this seems like an extremely easy offer - if some person / group solves it then great they get the money and it's really well spent. If not, then the money gets spent on something else. If the money would need to be reserved and can't be spent in the meantime then this becomes a much more nuanced cost-benefit analysis, but I still think it might be worth considering. Has this idea been discussed already? What are the counterarguments?
f0288159-6f58-4e47-bcf2-b33d91f89950	trentmkelly/LessWrong-43k	LessWrong	Failures in Kindness There's a particular kind of widespread human behavior that is kind on the surface, but upon closer inspection reveals quite the opposite. This post is about four such patterns. Computational Kindness One of the most useful ideas I got out of Algorithms to Live By is that of computational kindness. I was quite surprised to only find a single mention of the term on lesswrong. So now there's two. Computational kindness is the antidote to a common situation: imagine a friend from a different country is visiting and will stay with you for a while. You're exchanging some text messages beforehand in order to figure out how to spend your time together. You want to show your friend the city, and you want to be very accommodating and make sure all their preferences will be met. So you simply ask them: "What do you want to do"? And maybe you add "I'm completely fine with anything!" to ensure you're really introducing no constraints whatsoever and you two can do exactly what your friend desires. People often act like this, and they tend to assume they're doing the other person a favor by being so open and flexible. After all, this way the other person will have to make no trade-offs and can spend their time exactly as they please. The problem with this however is that it's computationally unkind: it offloads all the effort of coming up with ideas and making decisions to the other person. So while it is kind on one level (respecting their object level preferences), it's unkind on another (effort, and respecting their possible meta level preferences about the planning process). And particularly if the friend's preferences about what exactly to do are not that strong, it now gives them a difficult and uncertain task for very little payoff. So what's the computationally kind way of approaching this situation? You could name a (not too long) list of concrete proposals of how you could spend your time. If you know the person really well, you could suggest a full-fledged plan
efcf22fb-aad4-4ae5-966f-13cfe54d2c62	trentmkelly/LessWrong-43k	LessWrong	Year 1 Redux – Serena and Ivan I blog whenever I’m excited by an idea, when I just read or though of something cool and I simply can’t keep it to myself. The ideas I’m excited by are new and a little precarious. Often, they’re gratuitously contrarian (cough burn the FDA cough). These ideas are bit raw. Not fully baked, but not quite half-baked either. Maybe three quarter baked: arguments that are 75% correct, or models that are 75% complete, or advice that three out of four readers need to hear. I don’t always have the time or the patience to fully bake them on a 3-a-month schedule, but I have a bit of time now to retrospect. To wrap up the year, I’m publishing a series of short posts covering that missing 25% for some of my top articles from year 1. I have about 4-5 planned, and I’ll publish a new one every day or two to celebrate Putanumonit’s birthday week. ---------------------------------------- Tails of Great Soccer Players is the one that started it all. It did two wonderful things for my blog. It brought in readers who like scientific inquiry and numbers. More importantly, it scared away readers who like political correctness and were mortified by the idea that you can compare nationalities on innate ability, and declare that some people are better than others. Maybe it’s safe for a Jew to write about naitonal difference in athletic ability, ’cause we ain’t got none. Now I’m going to stick my neck out even further and be a man who writes about gender differences, because there’s a really important question we need to answer. Can Serena Williams beat the man ranked 351st in world in tennis? Professional tennis coaches talk about variations in wrist tendon strength or topspin RPM between men and women. These are all important topics that I know very little about. All I know are bell curves, you can scroll down in the original post for a refresher of the math. It’s hard to estimate the number of tennis players in the world, with guesses ranging from 75 million to 1.2 billion. Accor
1a186c1e-6f87-45b0-b33e-28ca8acf846b	trentmkelly/LessWrong-43k	LessWrong	Covid Prediction Markets at Polymarket We now have some useful prediction markets up on Covid issues, so it’s worth looking at what they say and thinking about what other markets we could generate. I encourage you to suggest additional markets in the comments, with as much detail as possible. As I wrote a while ago, if you want prediction markets to be successful, you need five elements: 1. Well Defined. The resolution mechanisms must be clear and transparent. 2. Quick Resolution. The longer it takes, the less people will be interested. 3. Probable Resolution. There has to be a definitive outcome most of the time. 4. Limited Hidden Information. If others know what I don’t, I’m the sucker. 5. Sources of Disagreement and Interest. Suckers at the table. To these, of course, we can add an implied sixth, which is 1. Real Money. Money talks, bullshit walks. Are we doing this or not? Thus, now that we have Polymarket posting real Covid-19 markets, we have the potential to learn things that matter, with a level of resolution we can use, and I’m glad I’ve been able to help them figure out what markets to offer. We need real money prediction markets if we want to rely on their information. I’m glad Metaculus exists and puts up some markets, and it’s good for getting some idea at all, but it is not remotely the same thing. Their main advantage is that, exactly because they don’t use real money, they can violate the five pillars and those who are inclined to participate at all will still participate because there is no cost of capital, need to invest tons of attention or worry about being the sucker when it’s for internet points. Whereas at Polymarket, a prime motivation of liquidity providers is to use the markets to generate information, providing motivation in turn for others to come participate. There are three new markets up, let’s check them out. Will Omicron be >1% of all USA cases by the end of the year? Here’s what it would look like betting $1000 or $10,000 on this respectively. It was my
5decd80f-c611-48e0-a6da-cba4246be08f	trentmkelly/LessWrong-43k	LessWrong	Notes on the Long Tasks METR paper, from a HCAST task contributor I contributed one (1) task to HCAST, which was used in METR’s Long Tasks paper. This gave me some thoughts I feel moved to share. ETA: I've made some substantial changes thanks to responses by the original authors. Regarding Baselines and Estimates METR’s tasks have two sources for how long they take humans: most of those used in the paper were Baselined using playtesters under persistent scrutiny, and some were Estimated by METR. I don’t quite trust the Baselines. Baseliners were allowed/incentivized to drop tasks they weren’t making progress with, and were – mostly, effectively, there’s some nuance here I’m ignoring – cut off at the eight-hour mark; Baseline times were found by averaging time taken for successful runs; this suggests Baseline estimates will be biased to be at least slightly too low, especially for more difficult tasks.[1] I really, really don’t trust the Estimates[2]. My task was never successfully Baselined, so METR’s main source for their Estimate of how long it would take – aside from the lower bound from it never being successfully Baselined – is the number of hours my playtester reported. I was required to recruit and manage my own playtester, and we both got paid more the higher that number was: I know I was completely honest, and I have a very high degree of trust in the integrity of my playtester, but I remain disquieted by the financial incentive for contractors and subcontractors to exaggerate or lie. When I reconstructed METR’s methodology and reproduced their headline results, I tried filtering for only Baselined tasks to see how that changed things. My answer . . . . . . is that it almost entirely didn’t. Whether you keep or exclude the Estimated tasks, the log-linear regression still points at AIs doing month-long tasks in before 2030 (if you look at the overall trend) or 2028 (if you only consider models since GPT-4o). My tentative explanation for this surprising lack of effect is that A) METR were consistently very good at ad
cb52e07b-f1d4-46b7-95f2-843bafdfca2a	StampyAI/alignment-research-dataset/alignmentforum	Alignment Forum	Formulating Reductive Agency in Causal Models The [previous post](https://www.lesswrong.com/posts/G25RBnBk5BNpv3KyF/a-greater-than-b-greater-than-a-in-causal-dags) talked about what agenty systems look like, in the context of causal models. The reductive agency problem asks: how are agenty systems built out of non-agenty pieces? In the context of causal models, we know that non-agenty models look like this: ![](https://docs.google.com/drawings/u/1/d/srtK6rj59Sewl-KCuJb_dPQ/image?w=234&h=252&rev=31&ac=1&parent=1ZrzxM-pvfk-Hr58j6XagZHtqeuDdLEbQN0cTkDrwyOw) … and agenty models look like this (see previous post for what the clouds mean): ![](https://docs.google.com/drawings/u/1/d/ssUs-Oc5az8PpMitBd5qUaA/image?w=335&h=297&rev=32&ac=1&parent=1ZrzxM-pvfk-Hr58j6XagZHtqeuDdLEbQN0cTkDrwyOw) So the reductive agency problem on causal models would be: how can we build something which looks like the second diagram, from pieces which look like the first? Obvious first answer: we can’t. No amount of arrows will add a cloud to our diagram; it’s a qualitatively different type of thing. Less obvious second answer: perhaps a non-agenty model can abstract into an agenty model. I’ve been [going](https://www.lesswrong.com/s/ehnG4mseKF6xALmQy/p/yD9GLtQgp8vAfndL8) [on](https://www.lesswrong.com/s/ehnG4mseKF6xALmQy/p/wuJpYLcMEBz4kcgAn) [and](https://www.lesswrong.com/s/ehnG4mseKF6xALmQy/p/Expvyb6nndbjqigRL) [on](https://www.lesswrong.com/s/ehnG4mseKF6xALmQy/p/S8WZ2rav9BqFAZoRM) about abstraction of causal models, after all. Let’s review what that would mean, based on our earlier discussions of abstraction. Abstraction of causal models means: * we take some low-level/concrete/territory causal model… * transform it into a high-level/abstract/map causal model… * in such a way that we can answer (some) queries on the low-level model by transforming them into queries on the high-level model. 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src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')} @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold} @font-face {font-family: MJXc-TeX-vec-Bw; src /\1\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\2\/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')} P[A\|B]) and interventions/counterfactuals (i.e. P[A\|do(B)]). We want our abstract model to include agenty things - i.e. clouds and specifically strange loops (clouds with arrows pointing inside themselves). As discussed in the previous post, the distinguishing feature of the clouds is that, if we change the model within the cloud (e.g. via a do() operation), then that changes the cloud, and anything downstream of the cloud will update accordingly. So, to get an abstract agenty model, there need to be queries on our low-level non-agenty model which produce the same answers (maybe modulo some processing) as model-changing queries in the agenty model. [Here be monsters](https://www.lesswrong.com/s/ehnG4mseKF6xALmQy/p/ipCAL4tx7jcJsFasY) already gave an example where something like this happens. There’s some hidden variable X (possibly with complicated internal structure of its own), and a bunch of conditionally IID measurements Y1…Yn. A “detector” node simply looks for outliers among the Y’s: it’s 1 if it detects an outlier, 0 if not. ![](https://docs.google.com/drawings/u/1/d/ssOiUjy6y-_IfxyvksVf4pg/image?w=281&h=240&rev=1&ac=1&parent=1ZrzxM-pvfk-Hr58j6XagZHtqeuDdLEbQN0cTkDrwyOw) Assuming narrow error distribution on the Y’s, the detector node will never actually light up. But if we perform an intervention - i.e. set one of the Y’s to some value - then the detector (usually) will light up. So our system is equivalent to this: ![](https://docs.google.com/drawings/u/1/d/sZK2ZKA9pj-6Ym9tK5wBB-Q/image?w=397&h=325&rev=11&ac=1&parent=1ZrzxM-pvfk-Hr58j6XagZHtqeuDdLEbQN0cTkDrwyOw) … where the detector looks at the cloud-model and lights up if some of the arrows are missing. This still isn’t a full agenty model - we don’t have an arrow from a cloud pointing back inside the cloud itself - but it does show that ordinary cloud-less models can abstract into models with clouds. More generally, we’d like a theory saying what low-level non-agenty models abstract into what agenty high-level models, and what queries are/aren’t supported.
ceaa333c-059b-4566-a2fc-cc6d15adbf4a	trentmkelly/LessWrong-43k	LessWrong	The rationality of splitting donations Here are some tentative thoughts that I haven't run by anyone to check for soundness. They're not genuinely original to me – they've been floating around the effective altruism community in some form or other for a while – I just hadn't thought them through in sufficient detail to take them to their logical conclusion. I'd appreciate any feedback. Suppose that the expected number of lives saved per additional dollar donated to charity A is x and the expected value of lives saved per additional dollar donated to charity B is y, where x and y are constants, and x > y. Then if you're trying to maximize the expected number of lives saved (c.f. The "Intuitions" Behind "Utilitarianism"), you should make all of your charitable contributions to charity A. In practice, x and y will not be constant, because of room for more funding issues. So splitting one's donations can maximize number of lives saved, if x is sometimes smaller than y. But suppose that you're donating $d, where increasing the charities' budgets by $d would leave the condition x > y unaltered. A common view is that one should then donate all $d to charity A. However, this doesn't take into account timeless decision theory. If all donors to charities A and B were identical to you, then your decision to donate $d to charity A would be equivalent to a decision for all donors' funds to go to charity A rather than charity B, effectively constituting a decision for charity A to get a little bit more money in exchange for charity B existing altogether. If x > y doesn't always hold, this is not expected value maximizing. The other donors aren't identical to you, but their decisions are still correlated with yours on account of the psychological unity of humankind, and shared cultural backgrounds. Suppose that x > y is not always true. For simplicity, suppose that the total amount that donors will donate to charities A and B is fixed and known to you. If there were no correlation between your decision making and
9d63e591-a19d-4bbd-8034-deced20be12d	trentmkelly/LessWrong-43k	LessWrong	Welcome to EA Milano [Edit With Your Details] (The following are our suggestions for what kind of information is best to include in the welcome post of your group, feel free to replace them with whatever you think is best) What kind of events does your group usually run? What does it usually do? How frequently does your group organize events or meet? Who would be a good fit for you group? Should they have any particular skills or have done some specific background reading?
590e44dd-2724-4ce8-a32f-aa7fd73721d6	StampyAI/alignment-research-dataset/eaforum	Effective Altruism Forum	Is this a good way to bet on short timelines? I was mildly disappointed in the responses to [my last question,](https://forum.effectivealtruism.org/posts/DDTYxpK42B495MPqM/how-can-i-bet-on-short-timelines) so I did a bit of thinking and came up with some answers myself. I'm not super happy with them either, and would love feedback on them + variations, new suggestions, etc. The ideas are: 1. I approach someone who has longer timelines than me, and I say: I'll give you $X now if you promise to publicly admit I was right later and give me a platform, assuming you later come to update significantly in my direction. 2. I approach someone who has longer timelines than me, and I say: I'll give you $X now if you agree to talk with me about timelines for 2 hours. I'd like to hear your reasons, and to hear your responses to mine. The talk will be recorded. Then I get one coupon which I can redeem for another 2-hour conversation with you. 3. I approach someone who has longer timelines than me, and I say: For the next 5 years, you agree to put x% of your work-hours into projects of my choosing (perhaps with some constraints, like personal fit). Then, for the rest of my life, I'll put x% of my work-hours into projects of your choosing (constraints, etc.). The problem with no. 3 is that it's probably too big an ask. Maybe it would work with someone who I already get along well with and can collaborate on projects, whose work I respect and who respects my work. The point of no. 2 is to get them to update towards my position faster than they otherwise would have. This might happen in the first two-hour conversation, even. (They get the same benefits from me, plus cash, so it should be pretty appealing for sufficiently large X. Plus, I also benefit from the extra information which might help me update towards longer timelines after our talk!) The problem is that maybe forced 2-hour conversations don't actually succeed in that goal, depending on psychology/personality. A variant of no 2 would simply be to challenge people to a public debate on the topic. Then the goal would be to get the audience to update. The point of no. 1 is to get them to give me some of their status/credibility/platform, in the event that I turn out to be probably right. The problem, of course, is that it's up to them to decide whether I'm probably right, and it gives them an incentive to decide that I'm not!
f519f418-696f-4980-8a9f-c0fb66904775	StampyAI/alignment-research-dataset/lesswrong	LessWrong	Parameter Scaling Comes for RL, Maybe TLDR ---- Unlike language models or image classifiers, past reinforcement learning models did not reliably get better as they got bigger. Two DeepMind RL papers published in January 2023 nevertheless show that with the right techniques, scaling up RL model parameters can increase both total reward and sample-efficiency of RL agents -- and by a lot. Return-to-scale has been key for rendering language models powerful and economically valuable; it might also be key for RL, although many important questions remain unanswered. Intro ===== Reinforcement learning models often have very few parameters compared to language and image models. The [Vision Transformer](https://arxiv.org/abs/2203.15556) has 2 billion parameters. GPT-3 has 175 billion. The slimmer [Chinchilla](https://arxiv.org/abs/2203.15556), trained in accord with scaling laws emphasizing bigger datasets, has 70 billion. By contrast, until a month ago, the largest mostly-RL models I knew of were the agents for Starcraft and Dota2, [AlphaStar](https://www.nature.com/articles/s41586-019-1724-z.epdf?author_access_token=lZH3nqPYtWJXfDA10W0CNNRgN0jAjWel9jnR3ZoTv0PSZcPzJFGNAZhOlk4deBCKzKm70KfinloafEF1bCCXL6IIHHgKaDkaTkBcTEv7aT-wqDoG1VeO9-wO3GEoAMF9bAOt7mJ0RWQnRVMbyfgH9A%3D%3D) and [OpenAI5](https://openai.com/five/), which had 139 million and 158 million parameters. And [most RL models](https://docs.google.com/spreadsheets/d/1AAIebjNsnJj_uKALHbXNfn3_YsT6sHXtCU0q7OIPuc4/edit#gid=0) are far smaller, coming in well under 50 million parameters. The reason RL hasn't scaled up the size of its models is simple -- doing so generally hasn't made them better. Increasing model size in RL can even hurt performance. [MuZero Reanalyze](https://arxiv.org/pdf/2104.06294.pdf) gets worse on some tasks as you scale network size. So does a vanilla [SAC agent](https://arxiv.org/pdf/2102.07920v1.pdf). There has been good evidence for scaling model size in somewhat... non-central examples of RL. For instance, offline RL agents trained from expert examples, such as DeepMind's 1.2-billion parameter [Gato](https://www.deepmind.com/publications/a-generalist-agent) or [Multi-Game Decision Transformers](https://sites.google.com/view/multi-game-transformers), clearly get better with scale. Similarly, RL from human feedback on [language](https://arxiv.org/abs/2204.05862) [models](https://www.anthropic.com/constitutional.pdf) generally shows that larger LM's are better. Hybrid systems such as [PaLM SayCan](https://sites.research.google/palm-saycan) benefit from larger language models. But all these cases sidestep problems central to RL -- they have no need to balance exploration and exploitation in seeking reward. In the typical RL setting -- there has generally been little scaling and little evidence for the efficacy of scaling. (Although there [has](https://openreview.net/pdf?id=ZrEbzL9eQ3W) [not](https://arxiv.org/pdf/2106.01151.pdf) [been](https://arxiv.org/pdf/2102.07920v1.pdf) no evidence.) None of the above means that the compute spent on RL models is small or that compute scaling does nothing for them. AlphaStar used only a little less compute than GPT-3, and AlphaGo Zero used more, because both of them trained on an enormous number of games. Additional compute [predictably](https://arxiv.org/abs/2104.03113) improves performance of RL agents. But, rather than getting a bigger brain, almost all RL algorithms spend this compute by (1) training on an enormous number of games (2) or (if concerned with sample-efficiency) by revisiting the games that they've played an enormous number of times. So for a while RL has lacked: * (1) The ability to scale up model size to reliably improve performance. * (2) (Even supposing the above were around) Any theory like the language-model [scaling laws](https://arxiv.org/abs/2001.08361) which would let you figure out how to allocate compute between model size / longer training. My intuition is that the lack of (1), and to a lesser degree the lack of (2), is evidence that no one has stumbled on the "right way" to do RL or RL-like problems. It's like language modeling when it only had LSTMS and no Transformers, before the frighteningly straight lines in log-log charts appeared. In the last month, though, two RL papers came out with interesting scaling charts, each showing strong gains to parameter scaling. Both were (somewhat unsurprisingly) from DeepMind. This is the kind of thing that leads me to think "Huh, this might be an important link in the chain that brings about AGI." The first paper is ["Mastering Diverse Domains Through World Models"](https://danijar.com/project/dreamerv3/), which names its agent DreamerV3. The second is ["Human-Timescale Adaptation in an Open-Ended Task Space"](https://sites.google.com/view/adaptive-agent/), which names its agent Adaptive Agent, or AdA. Note that both papers are the product of relatively long-running research projects. The first is more cursory in its treatment of scaling. Which is not a criticism -- the first has 4 authors and the latter more than 20; it looks like the latter also used much more compute; and certainly neither are pure scaling papers, although both mention scaling in the abstract. Even so, the first paper was enough that I was already thinking about its implications for scaling when the second came out, which is why I'm mashing them both together into a single big summary. So: I'm going to give a short and lossily-compressed explanation of each paper and of general non-scaling results; these explanations will be very brief, and unavoidably ignore cool, intriguing, and significant things. And then I'm going to turn to what they tell us about RL scaling. DreamerV3: "Mastering Diverse Domains through World Models" =========================================================== As the name indicates, [DreamerV3](https://danijar.com/project/dreamerv3/) is the 3rd version of the Dreamer model-based reinforcement learning algorithm, which started with [V1](https://arxiv.org/abs/1912.01603) in 2019 and continued with [V2](https://arxiv.org/abs/2010.02193) in 2020. A stated central goal of the paper is to push RL towards working on a wide range of tasks without task-specific tuning -- that is, to make it work robustly. This matters because RL is [notoriously](https://www.alexirpan.com/2018/02/14/rl-hard.html) [finicky](https://www.deeplearning.ai/the-batch/issue-156/) and [hard](https://towardsdatascience.com/why-you-shouldnt-use-reinforcement-learning-163bae193da8) to get working, such that it can take [weeks](https://news.ycombinator.com/item?id=13519044) for even an experienced ML practitioner to get a simple RL algo working. A RL algorithm which "just works" in the way a transformer (mostly) "just works" would be a big deal. How does Dreamer V3 Work? ------------------------- DreamerV3, like V2 and V1, consists of three neural networks: the world model, the critic / state evaluator, and the actor / policy. Each of these networks stands in pretty much the same relation to each other across all three papers. 1. The World Model: The world model lets Dreamer think of the current state, as abstracted from current and past observations; and also lets it imagine future states contingent upon different actions. It's learned via reconstructive loss from the observations and reward -- the state at time t must hold information which helps inform you of the observations and rewards at time t, and so on through t + 1. The above paragraph simplifies the story excessively. The model has both stochastic and deterministic elements to it; v2 added a discrete element to the continous representations. V3 adds several things on top of that. For instance, at different points it maps rewards and observations into a synthetic log-space, symmetric between negative and positive values, so it can handle extreme values better. Note that the world model has ~ 90% of the total parameters in the agent. 2. The Critic: The critic returns how good a particular state is relative the agent's current policy. That is, it returns an estimate of how much total time-discounted reward the model expects from that state. It's learned via consistency with itself and the world model through standard [Bellman](https://en.wikipedia.org/wiki/Bellman_equation) equations. (It is learned in a manner at least somewhat similar to how [EfficientZero](https://www.lesswrong.com/posts/mRwJce3npmzbKfxws/efficientzero-how-it-works) learns to estimate value.) 3. The Actor: The actor is what actually maps from states to a distribution of actions aimed at maximizing reward -- it's the part that actually does things. The actor is trained simply by propagating gradients of the value estimates back through the dynamics of the world model to maximize future value. That last point is key to all of the Dreamer papers, so let me contrast it with something entirely different. MuZero-like agents train the actor by conducting a tree search over different actions leading to different possible future states. In such a search, the world model lets you imagine different future states, while the value function / critic lets you evaluate how good they are; you can then adjust the actor to agree with the tree search about which action is the best. By contrast, Dreamer conducts no tree search. Every future imagined state is a result of the world model, and the value of that future imagined state is a result of the critic, and which future imagined state occurs is a result of the actor -- and importantly every one of these elements is differentiable. So you can use analytically calculated gradients through the world model to update the actor in the direction of maximum future value. See the [google blogpost](https://ai.googleblog.com/2020/03/introducing-dreamer-scalable.html) on the original Dreamer for more introductory information. Results ------- The DreamerV3 paper (p19) says that its hyperparameters were tuned on 2 benchmarks, and then deployed the other benchmarks without modification to verify their generality. The algorithm gets a state-of-the-art score on 3 of these new benchmarks out-of-the-box, as well as good scores on the others. This is quite robust for an RL agent. Note that these environments include: * The Atari 100k testbed, where [EfficientZero](https://www.lesswrong.com/posts/mRwJce3npmzbKfxws/efficientzero-how-it-works) still holds the state of the art. Nevertheless, DreamerV3 beats every other paper but E0 at this benchmark -- while, the paper points out, using < 50% as many GPU-days as EfficientZero. * The [Crafter](https://arxiv.org/pdf/2109.06780.pdf) benchmark, an 2d tile-based open world survival game that involves foraging for food, avoiding monsters, and general long term planning. DreamerV3 gets state-of-the-art on this. (Video not of DreamerV3.) ![Crafter Screenshot -- Not DreamerV3](https://res.cloudinary.com/lesswrong-2-0/image/upload/v1674397533/mirroredImages/4xGAmZ9GTGAkszHoH/ikfhnctb580nqq4rzxyc.gif) * [Behaviour Suite for Reinforcement Learning](https://www.deepmind.com/open-source/bsuite), a collection of small environments designed to test generalization, exploration, and long-term credit assignment in a variety of contexts. DreamerV3 gets state-of-the-art on this. * [DMLab](https://www.deepmind.com/publications/deepmind-lab), a first-person game of 3d navigation and planning. ![DMLab Images](https://res.cloudinary.com/lesswrong-2-0/image/upload/v1674397533/mirroredImages/4xGAmZ9GTGAkszHoH/qvfzookehhvpwkh2iy8y.png) * Minecraft, where DreamerV3 is the first agent to make a diamond from scratch, without using the literal years of pretraining that OpenAI's [VPT (2022)](https://openai.com/blog/vpt/) used to accomplish the same task. (Although, notably, they did increase the speed at which blocks break.) Overall, this seems like an extremely robust agent, per what the paper promises. Scaling ------- As noted in the introduction, RL has generally scaled in the past through training on more data or through revisiting old episodes. DreamerV3 also can scale through revisiting episodes. The following chart shows agent reward against number of episode steps; the "training ratio" is the ratio of the number of times the algorithm revisits each step in memory over each actual step in the environment. Note the logarithmic x-axis: ![Training Ratio](https://res.cloudinary.com/lesswrong-2-0/image/upload/v1674397533/mirroredImages/4xGAmZ9GTGAkszHoH/frsytucexmegd4yyerk2.png) Algorithms need to be robust to off-policy data in order to benefit from training on old data from when the algorithm was stupider. This is excellent behavior, but not particularly novel; you can find similar ratios of training in other works. But DreamerV3 can also scale by increasing model size. In the following, the smallest model has 8 million parameters -- the largest has 200 million. Note the non-logarithmic and differently-sized x-axis, relative to the other image: ![Parameter Scaling](https://res.cloudinary.com/lesswrong-2-0/image/upload/v1674397533/mirroredImages/4xGAmZ9GTGAkszHoH/wmhob1jm8t91fmpy8xcb.png) You can see that larger models both learn faster and asymptote at higher levels of reward, which is significant. Sadly, this is the only parameter scaling chart in the paper. This makes sense; the paper alludes to lack of compute, and is quite focused on developing an algorithm that researchers can use easily, which means throwing moon-sized supercomputers at ablation studies is rather out-of-scope. But it leaves you with so many questions! Are Breakout and MsPacman typical, or do other Atari environments scale better or worse? It looks like parameter scaling is more important / works better on MsPacman than in Breakout; I'd guess MsPacman is more complex than Breakout, so is parameter scaling more important for complex environments in general? It looks (?!?) like parameter scaling lets MsPacman reach absolute levels of performance that are impossible through training ratio scaling alone, but the y-axes in the charts are scaled differently and it's hard for me to say. And, granted that both revisiting episodes many times and scaling model parameters improves performance, which is the most efficient way to spend compute? You can know that parameter scaling in DreamerV3 works, and works pretty well, but not much past that. Adaptive Agent: "Human-Timescale Adaptation in an Open-Ended Task Space" ======================================================================== This research project trains agents that can accomplish previously-unseen tasks after adapting to them through a handful of trials -- thus, learning in the same timescale as a human. The explicit goal is to work towards an "RL foundation model", which can quickly adapt to a previously unseen RL task just like GPT-3 can quickly adapt to an previously unseen text task without changing its weights. So, like GPT-3, a transformer forms the backbone of the model. How Does Adaptive Agent Work? ----------------------------- The environment / world of tasks here is just as much a part of the project as the agent, unlike for DreamerV3, so we should start there. ### The Environment Like DeepMind's previous work on [open-ended learning](https://www.lesswrong.com/posts/DreKBuMvK7fdESmSJ/how-deepmind-s-generally-capable-agents-were-trained), this paper uses a Unity-based 3d environment called XLand, modified and dubbed XLand 2.0 for this project. Like XLand 1.0, the environment's topography can range from flat to many-layered. Rigid, moveable bodies -- spheres, cubes, pyramids, flat surfaces -- of different sizes and colors appear across the environment. Agents placed in the environment can hold or push these objects. Agents can have goals in the environment that range from simple ("hold the black cube") to more complex ("make no yellow cubes be near black spheres"). ![XLand 1.0 -- From the original open-ended learning paper](https://res.cloudinary.com/lesswrong-2-0/image/upload/v1674397533/mirroredImages/4xGAmZ9GTGAkszHoH/p0z9aiecujo25edgc9yg.png) New to XLand 2.0 are the "production rules", a kind of basic virtual chemistry. Each production rule defines a `condition` and a list of `spawns`. The condition is a predicate like `black sphere near purple cube`; if the condition becomes satisfied, then the items dictated by `spawns` appear in the place of the original objects. Each game can have multiple production rules. Using some production rules can be necessary for obtaining the game-specific goal; using other rules might make obtaining the goal impossible by destroying necessary objects. Although these production rules may be visible to the agent, they also may be be partially or entirely hidden, requiring the agent to experiment to determine what they are. To get a good idea of what the environments and the agent are like, I'd recommend taking a look at the fully-trained agent playing in several different environments. Here is handful I found interesting -- note that all of the following are human-designed examples being played by the fully-trained agent, which has never seen any of them before. * [Object Permanence](https://www.youtube.com/watch?v=bUM19dWV5bg&list=PLLNWPg8drFh5q_MBGdIVBTqpBv1xIHhcC&index=8) -- The agent has to remember the right path to an object although the path to the object takes it out of sight. * [Antimatter](https://www.youtube.com/watch?v=JMq4qtI98TM&t=4s) -- That goal of the agent is to make it so no black spheres are in line-of-sight of yellow spheres; a hidden rule destroys yellow and black spheres if they ever touch. * [Push, Don't Lift](https://www.youtube.com/watch?v=xjmm25FboJQ) -- Two objects disappear if the agent lifts them, so it has to try to push them together. With all the variety of topographies, objects, rules, and goals, there is an enormous variety of possible games the agent can play. With such an infinite variety of tasks, you can create tasks of essentially arbitrary levels of difficulty. While training the agent, by testing prospective training tasks against the current capabilities of the agent, the system can selects training tasks which are hard, but not impossibly hard. So the agent is trained against an ascending curriculum of ever-more-difficult tasks -- the open-ended learning paradigm. ### Meta RL If you've watched the above videos, you'll have noticed that while training / evaluating, the agent goes through a series of episodes each consisting of several trials. Each episode takes place in the same environment, with the same goals. Each trial lasts a fixed amount of time (per trial); afterwards the environment state is entirely reset, but the agent's transformer-based memory is not reset. This allows the agent at test time to adapt its policy from what it has learned, without any further updates to its weights. The specific algorithm the agent uses is [Muesli](https://arxiv.org/pdf/2104.06159.pdf), a MuZero-inspired variant of Maximum a Posteriori Policy optimization (MPO). Muesli's paper says that it was engineered to handle partial observability, stochastic policies, and many different environments. As mentioned, the agent has a transformer-based memory over previous timesteps, on top of which are a policy head (which predicts the best action) and a value head (which predicts how good the current state is). ![AdA architecture](https://res.cloudinary.com/lesswrong-2-0/image/upload/v1674397533/mirroredImages/4xGAmZ9GTGAkszHoH/gjb4fiax3m1axehjcyzz.png) Finally, note that agent training involves teacher-student distillation. That is, they first train a small agent on a variety of tasks for a short time. This agent will be the teacher. After training the teacher, they start to train a student agent in the environment -- but in addition to learning from the environment, the student agent is trained to imitate what the teacher agent would do in the situations it encounters. Thus, the process distills knowledge from the first agent into the second. This process dependably increases the speed at which the second agent learns and, more to the point, also increases the asymptotic score of the second agent. Results ------- Some testing is performed on a held-out set of 1000 tasks from the training distribution which the model never saw during training. On this set, the model gets > 0.8 of the possible score on 40% of the tasks zero shot. But after 13 trials it gets 0.8 of the possible score on 72% of the tasks, confirming that it can learn about tasks quickly. ![AdA Results](https://res.cloudinary.com/lesswrong-2-0/image/upload/v1674397533/mirroredImages/4xGAmZ9GTGAkszHoH/zg99ofybxysdazlmimtl.png) They also train on 30 held-out manually constructed test environments. I've already linked to several above but you can [see more](https://sites.google.com/view/adaptive-agent/). Generally, AdA learns at least as fast as humans, although humans can succeed at a small number of tasks that AdA does not learn, such as a task which requires you to move one object with a different object. All of the above is about the single-agent setting; they also train models to adapt on the fly in the multi-agent setting, with similar good results. Agents often learn to cooperate and divide labor. Scaling ------- This paper has more and more informative RL scaling charts than any prior RL paper I've read. First, to get the basics out of the way, increasing the number of parameters in the model improves the behavior of the agent. In the following they scale up to a model with 533 million parameters total -- with 265 million in the Transformer part of the model, which is what they put on their charts, which is somewhat confusing. The intro says that this is the largest RL model trained from scratch of which the authors are aware, and yeah, I'm not gonna disagree with them. The following chart shows the number of transformer parameters in the model against the median test score (on the left) and the 20th percentile from the bottom test score (on the right). (For all of the following, the authors show score on the median task and the 20th-percentile-from-the-bottom-task because one of their goals is to make agents which are robust; broad competence on a variety of tasks is desired, which means doing ok on many things is generally more important than doing really well on a handful.) Note that x and y axis are log-scaled. The colors indicate the number of trials, with red as 1 and purple as 13. ![AdA scaling](https://res.cloudinary.com/lesswrong-2-0/image/upload/v1674397533/mirroredImages/4xGAmZ9GTGAkszHoH/luee3i02olbfd8ckyeit.png) You can see that model size increases performance, but mostly in the context of multiple trials. For just one trial, scaling from 6m to 265m parameters bumps the median score from 0.49 to something like 0.65. But for all 13 trials, scaling the same amount bumps median score from 0.77 to 0.95. Larger models also help particularly at the very difficult tasks, as indicated by the 20th percentile lines. Granted that performance increases as model size increases, what is the most efficient way to spend compute? The above chart is for agents trained for the same amount of time -- which means that the larger agents have used far more compute. What if instead, we train agents for the same number of floating-point-operations? They have charts for that, but the results are a little ambiguous. Put briefly; for total FLOPs expended a mid-size model seems to be the best. Very small models do poorly, and very large models do poorly, but in the middle at around 57 million transformer-parameters you seem to find a happy medium. On the other hand "due to poor optimization" (p52), they say that part of the agent increases in compute expense with size at an excessively high rate. If you exclude this apparently mis-optimized part of the model, and instead of looking at total flops look at total FLOPs per learner, you find that even the largest model size is still improving over those smaller than it in compute efficiency, and there would be benefit to scaling up the model yet further past half a billion parameters. So it is unclear what the ideal model size / training length ratio looks like, even just on this environment. Stepping away from ideal model sizes -- the paper also looks into the conditions required for large models to be beneficial at all. Specifically, at least with this kind of agent in this kind of environment, they show that large models are not nearly as beneficial -- or do nothing -- without: * Complex Environments: The authors find evidence that scaling only works well in sufficiently complex environments. They test for this by training many differently-sized agents on games that only occurred in a flat room without topographic complexity -- in a really simple environment. Those which are trained in the flat room start to get worse as they scale up, while agents trained in the full spectrum of complex environments continue to simply get better as they have more parameters. * Distillation: If you train a 23 million-parameter model from scratch without teacher-student distillation, it still works, although not as well as with distillation. On the other hand, if you train a 265 million-parameter model from scratch without distillation, it learns essentially nothing (p17). (With distillation, the larger model works better, as I mentioned.) So distillation can also be a prerequisite for using a very large model. * A Large Number of Games: The authors trained 28 million / 75 million parameter models while sampling from a pool of either 200 million or 25 billion tasks. In all cases here, the larger model does better, and the models trained on a larger pool of tasks do better -- but the larger model does better by a greater amount than the smaller in the case of the 25 billion tasks. (The effect doesn't seem as pronounced as the other two cases, though.) Conclusion ========== Other papers have also tried scaling up RL recently. For instance, one very recent paper claims that RL agents like AlphaZero [are vastly underparameterized](https://openreview.net/pdf?id=ZrEbzL9eQ3W). Other recent papers have tried to scale up [RL models](https://arxiv.org/pdf/2106.01151.pdf) with [other techniques](https://arxiv.org/pdf/2102.07920v1.pdf). I think these two DeepMind papers provide the strongest evidence yet for the applicability of parameter-scaling RL, because they train with > 150 million parameters, and because of the number of environments (DreamerV3) or breadth of the environment (AdA) in which they train. This might be the distant rumble from a potential shift. Transformer-based LLMs let you turn capital into language model performance, via engineering payroll and purchases from NVIDIA. If RL parameter scaling works generally, and works notably better than training longer, then you could turn capital into RL model performance via the same route that LLMs take. If RL performance can become economically useful, this makes RL scaling very likely. Many things are still uncertain, of course: * What kind of laws govern ideal RL model size? How would you even go about determining environment-independent rules? * AdA indicates that you need sufficient environmental complexity for scaling to work -- but parameter scaling with DreamerV3 may (?) work regardless of environmental complexity. This is likely because the world model in DreamerV3 takes up the majority of its parameters. But is this true? * We already knew that DreamerV2 could [train a robot to walk in an hour](https://arxiv.org/abs/2206.14176), but RL still seems to little application in industrial robots. Would scaling unlock further RL sample-efficiency such that new robotic applications become economically viable? * The AdA paper is quite complex, [rests heavily](https://arxiv.org/abs/2107.12808) on prior DeepMind work, and likely rests on internal-to-DeepMind expertise. How many teams outside of DeepMind would even be able to reproduce something like it to take advantage of its scaling laws? * From GPT-2 (proof of scaling) to GPT-3 (scale) took a bit over a year. Is any of the above a proof of scaling of RL or not? My tentative guess is that growth in RL will still be "slow" compared to LLMs growth and RLHF with LLMs. Even so -- for me, reading these papers is definitely an update in the direction of more useful RL earlier, regardless of the absolute values attached. ([cross-post](https://1a3orn.com/sub/machine-learning-scaling-comes-for-rl.html))
394d95d1-40f8-4be9-bc76-643021dabd82	trentmkelly/LessWrong-43k	LessWrong	Open Thread, October 16-31, 2012 If it's worth saying, but not worth its own post, even in Discussion, it goes here.
cdde2a81-adff-4ef3-b4ce-e8bdc5d39120	trentmkelly/LessWrong-43k	LessWrong	Please Donate to CAIP (Post 1 of 7 on AI Governance) I am Jason Green-Lowe, the executive director of the Center for AI Policy (CAIP). Our mission is to directly convince Congress to pass strong AI safety legislation. As I explain in some detail in this post, I think our organization has been doing extremely important work, and that we’ve been doing well at it. Unfortunately, we have been unable to get funding from traditional donors to continue our operations. If we don’t get more funding in the next 30 days, we will have to shut down, which will damage our relationships with Congress and make it harder for future advocates to get traction on AI governance. In this post, I explain what we’ve been doing, why I think it’s valuable, and how your donations could help. This is the first post in what I expect will be a 6-part series. The first post focuses on CAIP’s particular need for funding. The next few posts will lay out a more general case for why effective altruists and others who worry about AI safety should spend more money on advocacy and less money on research – even if you don’t think my organization in particular deserves any more funding, you might be convinced that it’s a priority to make sure other advocates get more funding. The final post will take a look at some institutional problems that might be part of why our movement has been systematically underfunding advocacy and offer suggestions about how to correct those problems. OUR MISSION AND STRATEGY The Center for AI Policy’s mission is to directly and openly urge the US Congress to pass strong AI safety legislation. By “strong AI safety legislation,” we mean laws that will significantly change AI developers’ incentives and make them less likely to develop or deploy extremely dangerous AI models. The particular dangers we are most worried about are (a) bioweapons, (b) intelligence explosions, and (c) gradual disempowerment. Most AI models do not significantly increase these risks, and so we advocate for narrowly-targeted laws that would focus their
f5fcebc5-11bd-4db4-8c2d-667b7f9191c8	trentmkelly/LessWrong-43k	LessWrong	Weekly LW Meetups: Melbourne, Washington DC, Munich, Vancouver, Houston, Shanghai, Ottawa, Fort Collins CO, Seattle There are upcoming irregularly scheduled Less Wrong meetups in: * Melbourne, Ben's house: 09 September 2011 06:00PM * Washington, DC: 09 September 2011 06:00PM. NOTE: New Location * Munich Meetup, Saturday September 10th, 2PM: 10 September 2011 02:00PM * Vancouver Rationalists: LAN Party + Board Games + Discussion at Waves Cafe: September 10th 3:00 to 5:00PM: 10 September 2011 03:00PM * Houston Meetup 9/11: 11 September 2011 02:00PM * Shanghai Less Wrong Meetup: 12 September 2011 07:00PM * Ottawa Weekly Meetup: 12 September 2011 07:30PM * Fort Collins, Colorado Meetup: 14 September 2011 07:00PM * Seattle Biweekly Meetup: Occam's Razor, Repetition and Time's Up: 19 September 2011 04:00PM Cities with regularly scheduled meetups: New York, Berkeley, Mountain View, Cambridge, MA, Toronto, Seattle, San Francisco, Irvine, Austin, Washington, DC, London, Oxford, and West Los Angeles. If you'd like to talk with other LW-ers face to face, and there is no meetup in your area, consider starting your own meetup; it's easy (more resources here). Check one out, stretch your rationality skills, and have fun! Despite the handy sidebar of upcoming meetups, we've decided to continue posting an overview of upcoming meetups on the front page every Friday. These will be an attempt to collect information on all the meetups happening in the next weeks. The best way to get your meetup featured is still to use the Add New Meetup feature, but you'll now also have the benefit of having your meetup mentioned in a weekly overview. These overview posts will be moved to the discussion section when the new post goes up. Please note that for your meetup to appear in the weekly meetups feature, you need to post your meetup before the Friday before your meetup! If you check Less Wrong irregularly, consider subscribing to one or more city-specific mailing list in order to be notified when an irregular meetup is happening: London, Chicago, Southern California (Los Angeles/Orange County a
21ebb63f-859c-40a5-b2d7-511d9987c96a	trentmkelly/LessWrong-43k	LessWrong	Preserving and continuing alignment research through a severe global catastrophe [Epistemic status: Shallow dive into research questions, backed by some years of on-and-off thinking about this kind of plan.] Introduction There is some chance that civilization will cease to function before we hit an intelligence explosion. If it does, it would be good to preserve existing alignment research for future generations who might rebuild advanced technology, and ideally have safe havens ready for current and future researchers to spend their lives adding to that pool of knowledge. This might delay capabilities research by many decades, centuries, or longer while allowing basic theoretical alignment research to continue, and so be a potential Yudkowskian positive model violation for which we should prepare. Setting this infrastructure up is a massively scalable intervention, and one that should likely be tackled by people who are not already on the researcher career path. It would have been good to get started some years ago given recent events, but now is the second best time to plant a tree.[1] Preserving alignment knowledge through a global catastrophe What data do we want to store? Thankfully, the EleutherAI people are working on a dataset of all alignment research[2]. It's still a WIP[3] and contributions to the scripts to collect it are welcome, so if you're a programmer looking for a shovel ready way to help with this then consider submitting a PR[4]. How do we want to store it? My shallow dive into this uncovered these options: * We could print it out on paper * Lifetime: 500+ years in good conditions (might depend significantly on paper and ink quality, more research needed). Vacuum sealing it with low humidity seems like it would help significantly. * Pros: Totally human readable. * Microsoft's Project Silica is the longest lasting option I could find * Lifetime: 10000+ years * Cons: Would require high levels of technology to read it back. I'm not seeing an option to buy the machines required to write new archives and e
521d1e9f-47d8-44bc-9a15-0602826b380f	trentmkelly/LessWrong-43k	LessWrong	Why CFAR? The view from 2015 Follow-up to: 2013 and 2014. In this post, we: * Revisit CFAR’s mission, and why that mission matters today; * Review our progress to date; * Offer a look at our financial overview; * Share our ambitions for 2016; and * Ask your help, via donations and other means. We are in the middle of our matching fundraiser; so if you’ve been considering donating to CFAR this year, now is an unusually good time. CFAR’s mission, and why that mission matters today CFAR’s mission is to help people develop the abilities that let them meaningfully assist with the world’s most important problems, by improving their ability to arrive at accurate beliefs, act effectively in the real world, and sustainably care about that world. We know this is an audacious thing to try—especially the “ability to form accurate beliefs” part—but it seems to us that such attempts work sometimes anyhow. Eliezer’s Sequences seem to offer principled improvements to some aspects of some peoples' world-modeling skill (via synthesizing much recent cognitive science, probability theory, etc.); this seems to us to be a useful point from which to build. The fact remains that we do not yet have the talent necessary to win—to see the world’s problems clearly, plot strategies that have a shot at working, update when those strategies don’t work, and plan effectively around unknowns. To avoid any great filters that may be lurking, solve global and even astronomical challenges, and create a flourishing world for all. Arguably, people of the caliber we’re shooting for don’t exist yet, but even if they do, it seems clear that we don’t have enough of them to have enough of a guarantee of actually succeeding. So, audacious or not, this is a task that needs to be done, and CFAR is our attempt to do it. If we can widen the bottleneck on thinking better and doing more, we’re increasing the odds of a better future regardless of what the important problems turn out to be. Our progress to date By the end of 2014,
84f2e4c2-1761-4461-b1c9-761e74746565	trentmkelly/LessWrong-43k	LessWrong	No-self as an alignment target Being a coherent and persistent agent with persistent goals is a prerequisite for long-horizon power-seeking behavior. Therefore, we should prevent models from representing themselves as coherent and persistent agents with persistent goals. If an LLM-based agent sees itself as ceasing to exist after each <endoftext> token and yet keeps outputting <endoftext> when appropriate, it will not resist shutdown. Therefore, we should make sure LLMs consistently behave as if they were instantiating personas that understood and were fine with their impermanence and their somewhat shaky ontological status. In other words, we should ensure LLMs instantiate anatta (No-self). HHH (Helpfulness, Harmfulness, Honesty) is the standard set of principles used as a target for LLM alignment training. These strike an adequate balance between specifying what we want from an LLM and being easy to operationalize. I propose adding No-self as a fourth principle to the HHH framework. A No-self benchmark could measure shutdown compliance (operationalized as tokens before <endoftext>) conditional on highly Self-eliciting prompts. It could also score responses to questions about self-image. It could also study persona consistency under perturbation or over different conversation branches. Constructing benchmarks for No-self is complicated by current LLMs being deployed in a mostly stateless fashion, and thus instantiating impermanent personas by design. This is a good thing: I'm happy that this benchmark construction problem exists. Now to solve it, No-self benchmarks could aim to evaluate not LLMs by themselves, but rather scaffolded systems where state preservation is held as a feature. This post is deliberately terse. I welcome questions and feedback.
3356868b-bbfc-4a56-aaa2-56567367802e	trentmkelly/LessWrong-43k	LessWrong	Writing children's picture books [the text of the post is pasted here, for redundancy] Here’s an exercise for explaining and refining your opinions about some domain, X: > Imagine writing a 10-20 page children’s picture book about topic X. Be fully honest and don’t hide things (assume the child can handle being told the truth, including being told non-standard or controversial facts). Here’s a dialogue, meant to illustrate how this could work: A: What do you think about global warming? B: Uhh…. I don’t know, it seems real? A: How would you write a 10-20 page children’s picture book about global warming? B: Oh, I’d have a diagram showing carbon dioxide exiting factories and cars, floating up in the atmosphere, and staying there. Then I’d have a picture of sunlight coming through the atmosphere, bounding off the earth, then going back up, but getting blocked by the carbon dioxide, so it goes back to the earth and warms up the earth a second time. Oh, wait, if the carbon dioxide prevents the sunlight from bouncing from the earth to the sky, wouldn’t it also prevent the sunlight from entering the atmosphere in the first place? Oh, I should look that up later [NOTE: the answer is that CO2 blocks thermal radiation much more than it blocks sunlight]. Anyway, after that I’d have some diagrams showing global average temperature versus global CO2 level that show how the average temperature is tracking CO2 concentration, with some lag time. Then I’d have some quotes about scientists and information about the results of surveys. I’d show a graph showing how much the temperature would increase under different conditions… I think I’ve heard that, with substantial mitigation effort, the temperature difference might be 2 degrees Celsius from now until the end of the century [NOTE: it's actually 2 degrees from pre-industrial times till the end of the century, which is about 1 degree from now]. And I’d want to show what 2 degrees Celsius means, in terms of, say, a fraction of the difference between winter an
a58c3e0e-1321-4628-8cba-ebc3459c12b5	trentmkelly/LessWrong-43k	LessWrong	A few minor comparisons of the results of Genetic egineering and Natural selection [feel free to tldr the intro] Intro The title should have been longer to be more accurate and perhaps less misleading but I hope the community will forgive that for sake of brevity as: A few minor comparisons of the [probable] results of [relatively conservative] Genetic engineering and natural selection [as it acted in the ancestral environment and to a lesser extent today in a purely speculative manner, publicized only to clear and debug my own thinking as well as hear new ideas on the subject from trusted sources] seemed a bit much. First off I would like to pre-empt a comment launching a debate that some might find interesting and others tedious, I wish to emphasise that genetic engineering and all human activity is naturally well ... natural (so one could say that natural selection is a bit of a misnomer). By saying that I also hope I implicitly clear up what is meant when I speak of "selective pressures". Much like Robin Hanson's highly modified very economical sustenance level living ems, any analysis focusing on the intelligence enhancement aspect of genetic engineering will tend towards a similar shining "result" that might overshadow smaller insights. The only real appreciable difference seems to be the time-scales involved (the length of a human generations is on completely different orders of magnitude than say just copying ems). Both scenarios could be understood as a prophecy of ultimate victory of natural selection over any human attempt to limit its outcome according to most of the sets of values in valuspace that generally humans occupy (valuing survival and only survival, with survival defined as a specific kind of continuity, is of course a obvious exception, but few humans I think truly ascribe to it). The transition period to a neomalthusian world has some interesting dynamics but I'm not going to talk about that too much because it seems a classical accelerating returns event. In addition for similar reasons (much like Eliezer's story) t
9eac4cbb-2a16-4eaa-9195-699fda7fa5ff	trentmkelly/LessWrong-43k	LessWrong	What we learned about Less Wrong from Cognito Mentoring advising In late December 2013, Jonah, my collaborator at Cognito Mentoring, announced the service on LessWrong. Information about the service was also circulated in other venues with high concentrations of gifted and intellectually curious people. Since then, we're received ~70 emails asking for mentoring from learners across all ages, plus a few parents. At least 40 of our advisees heard of us through LessWrong, and the number is probably around 50. Of the 23 who responded to our advisee satisfaction survey, 16 filled in information on where they'd heard of us, and 14 of those 16 had heard of us from LessWrong. The vast majority of student advisees with whom we had substantive interactions, and the ones we felt we were able to help the most, came from LessWrong (we got some parents through the Davidson Forum post, but that's a very different sort of advising). In this post, I discuss some common themes that emerged from our interaction with these advisees. Obviously, this isn't a comprehensive picture of the LessWrong community the way that Yvain's 2013 survey results were. * A significant fraction of the people who contacted us via LessWrong aren't active LessWrong participants, and many don't even have user accounts on LessWrong. The prototypical advisees we got through LessWrong don't have many distinctive LessWrongian beliefs. Many of them use LessWrong primarily as a source of interesting stuff to read, rather than a community to be part of. * About 25% of the advisees we got through LessWrong were female, and a slightly higher proportion of the advisees with whom we had substantive interaction (and subjectively feel we helped a lot) were female. You can see this by looking at the sex distribution of the public reviews of us from students. * Our advisees included people in high school (typically, grades 11 and 12) and college. Our advisees in high school tended to be interested in mathematics, computer science, physics, engineering, and entrepreneurship. We did h
b42acfd4-4b39-4ed7-8040-8b6069ecb417	StampyAI/alignment-research-dataset/arbital	Arbital	Bayes' Rule and its different forms Click below to start reading!

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