The Thirty-Third AAAI Conference on Artificial Intelligence (AAAI-19)
State Abstraction as Compression in Apprenticeship Learning
David Abel,
1Dilip Arumugam,
2Kavosh Asadi,
1Yuu Jinnai,
1Michael L. Littman,
1Lawson L.S. Wong
31Department of Computer Science, Brown University
2Department of Computer Science, Stanford University
3College of Computer and Information Science, Northeastern University
Abstract
State abstraction can give rise to models of environments that are both compressed and useful, thereby enabling efficient sequential decision making. In this work, we offer the first formalism and analysis of the trade-off between compression and performance made in the context of state abstraction for Apprenticeship Learning. We build on Rate-Distortion the-ory, the classic Blahut-Arimoto algorithm, and the Informa-tion Bottleneck method to develop an algorithm for com-puting state abstractions that approximate the optimal trade-off between compression and performance. We illustrate the power of this algorithmic structure to offer insights into ef-fective abstraction, compression, and reinforcement learning through a mixture of analysis, visuals, and experimentation.
1
Introduction
Reinforcement Learning (RL) poses a challenging problem. Agents must learn about their environment through high-dimensional and often noisy observations while receiving sparse and delayed evaluative feedback. The ability to un-derstand one’s surroundings well enough to support effec-tive decision making under these conditions is a remarkable feat, and arguably a hallmark of intelligent behavior. To this end, a long-standing goal of RL is to endow decision-making agents with the ability to acquire and exploit abstract models for use in decision making, drawing inspiration from human cognition (Tenenbaum et al. 2011).
One path toward realizing this goal is to make use ofstate
abstraction, which describes methods for compressing the environment’s state space to distill complex problems into simpler forms (Dietterich 2000b; Andre and Russell 2002; Li, Walsh, and Littman 2006). Critically, the degree of com-pression induced by an abstraction trades off directly with its capacity to represent good behavior. If the abstraction throws away too much information, the resulting abstract model will fail to preserve essential characteristics of the original task (Abel, Hershkowitz, and Littman 2016). Thus, care must be taken to identify state abstractions that balance between an appropriate degree of compression and adequate representational power.
Information Theory offers foundational results about the limits of compression (Shannon 1948). The core of the
the-Copyright c2019, Association for the Advancement of Artificial
Intelligence (www.aaai.org). All rights reserved.
ory clarifies how to communicate in the presence of noise, culminating in seminal results about the nature of commu-nication and compression that helped establish the science and engineering practices of computation. Of particular rel-evance to our agenda is Rate-Distortion theory, which stud-ies the trade-off between a code’s ability to compress (rate) and represent the original signal (distortion) (Shannon 1948; Berger 1971). Cognitive neuroscience has suggested that perception and generalization are tied to efficient compres-sion (Attneave 1954; Sims 2016; 2018), termed the “efficient coding hypothesis” by Barlow (1961).
The goal of this work is to understand the role of information-theoretic compression in state abstraction for sequential decision making. We draw a parallel between
state abstractionas used in reinforcement learning and
com-pressionas understood in information theory. We build on
the seminal work of Shannon (1948), Blahut (1972), Ari-moto (1972) and Tishby, Pereira, and Bialek (1999), and draw inspiration from related work on understanding the re-lationship between abstraction and compression (Botvinick et al. 2015; Solway et al. 2014). While the perspective we introduce is intended to be general, we focus our study in
two ways. First, we investigate onlystate abstraction,
de-ferring discussion of temporal (Sutton, Precup, and Singh 1999), action (Hauskrecht et al. 1998), and hierarchical ab-straction (Dayan and Hinton 1993; Dietterich 2000a) to fu-ture work. Second, we address the learning problem when
a demonstrator is available, as in Apprenticeship
Learn-ing (Atkeson and Schaal 1997; Abbeel and Ng 2004; Argall et al. 2009), which simplifies aspects of our model.
Concretely, we introduce a new objective function that explicitly balances state-compression and performance. Our main result proves this objective is upper bounded by a vari-ant of the Information Bottleneck objective adapted to se-quential decision making. We introduce Deterministic In-formation Bottleneck for State abstraction (DIBS), an
algo-rithm that outputs a lossy state abstraction optimizing the
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⇡ (a|s )
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d(⇡s
d, ⇡
s )
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sha1_base64="dZjrwtvQaCwzznTAk4G9E8cRZxI=">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</latexit><latexit sha1_base64="dZjrwtvQaCwzznTAk4G9E8cRZxI=">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</latexit>
Decoder Encoder
Source
p(x)
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p(z|x)
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d(x,x)˜
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:Z !X˜
(a) RD Setting
5 Rate
Distortion Achievable
Unachievable
Rate
Distortion Achievable
Unachievable
(b) RD Lower Bound
Figure 1: The usual Rate-Distortion setting (left) and the lower bound on the Rate-Distortion trade-off (right)
2
Background
Reinforcement Learning (RL) is the problem of an agent
learning to make decisions in an environment through inter-action alone. We assume the standard formalism: an agent interacts with a Markov Decision Process (MDP) to max-imize long-term expected discounted reward. Additionally, we focus on Apprenticeship Learning, wherein an agent
ob-serves the policyπEof an expert who is trying to maximize
an unknown reward function. For more on MDPs, see (Put-erman 2014), for RL, see (Sutton and Barto 2018), and for Apprenticeship Learning, see (Atkeson and Schaal 1997; Abbeel and Ng 2004; Argall et al. 2009).
Abstraction serves as a powerful tool for lowering the
complexity of decision making. Abstraction appears in sev-eral forms: state, action/temporal, and hierarchical. In all cases, abstraction is used to distill the agent’s representa-tion of the environment to informarepresenta-tion that is most useful for making effective decisions. Naturally, state abstraction is tightly connected to compression. State abstraction en-compasses methods that aggregate states together to form abstract states (Whitt 1978; Bertsekas and Castanon 1989; Singh, Jaakkola, and Jordan 1995; McCallum 1996; Dean, Givan, and Leach 1997; Dietterich 2000b; Andre and Rus-sell 2002; Givan, Dean, and Greig 2003; Ferns, Panangaden, and Precup 2004; Li, Walsh, and Littman 2006; Jiang, Singh, and Lewis 2014; Hostetler, Fern, and Dietterich 2014; Abel, Hershkowitz, and Littman 2016; Abel et al. 2018; Ta¨ıga, Courville, and Bellemare 2018). Given an MDP with
state spaceS, a state abstraction is a functionφthat projects
ground statess ∈ S to abstract statessφ ∈ Sφ, where
typ-ically|Sφ| |S|. In this work, we follow the formalisms
of Li, Walsh, and Littman (2006) and Abel et al. (2018).
Information Theory studies communication in the
pres-ence of noise (Shannon 1948). In Shannon’s words: “The fundamental problem of communication is that of reproduc-ing at one point either exactly or approximately a message selected at another point.” Information Theory typically in-vestigates coder-decoder pairs and their capacity to faith-fully communicate messages with zero or low error, even in the presence of noise.
Rate-Distortion (RD) Theory studies the trade-off
be-tween a coder-decoder pair’s ability to compress a signal
and the pair’s ability to faithfully reproduce the original signal. The typical RD setting is pictured in Figure 1a: an
information source generates x ∈ X, which is coded via
p(z|x)forz∈ Z, and decoded via a deterministic function
f :Z →X˜.Distortionis defined with respect to some
cho-sen distortion metric,d : X ×X →˜ R≥0, where typically
X = ˜X. The informationrate,R, denotes the number of bits
in each code word. So, with a coding alphabetZ˜={0,1}n,
the rate isn. Shannon and Kolmogorov (see Berger (1971)
for more background) offer a lower bound on the trade-off
between Rate and Distortion: given a level of distortion,D,
the following function defines the smallest rate that achieves
expected distortion of at mostD:
R(D) = min
p(˜x|x):E[d(x,x)]≤D˜ I(X; ˜X), (1)
whereI(X; ˜X)is the mutual information between random
variablesXandX˜:
I(X; ˜X) := X
x∈X X
˜
x∈X˜
p(x,˜x) log p(x,x)˜
p(x)p(˜x). (2)
Intuitively, Equation 1 tells us that as we add bits to our code, we can more faithfully reconstruct our source messages. The curve displayed in Figure 1b shows an example lower bound of the trade-off between Rate and Distortion expressed by Equation 1.
For a given information source, it is natural to consider how one might compute the coder-decoder pair that achieves one of the minimal points defined by the Rate-Distortion function. Finding this point presents the following optimiza-tion problem:
min
p(˜x|x)I(X| {z; ˜X})
Rate
+βEp(x,˜x)[d(x,x)]˜
| {z }
Distortion
, (3)
with a Lagrange multiplier β ∈ R≥0 expressing the
rela-tive preference between preserving rate and distortion. Asβ
gets closer to 0, rate becomes more important, while as β
approaches∞, minimizing distortion is prioritized.
