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Differentiable Programs with Neural Libraries
Figure 1: Components of an illustrative NTPT program for learning loopy programs that measure path length (path len) through a maze of street sign images The learned program (parameterized by instr and goto) must control the position (X, Y) of an agent on a grid of (W H) street sign images each of size (w h) The agent has a single register of
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Differentiable Programs with Neural Libraries
Alexander L. Gaunt
1Marc Brockschmidt1Nate Kushman1Daniel Tarlow2
AbstractWe develop a framework for combining differen- tiable programming languages with neural net- works. Using this framework we create end-to- end trainable systems that learn to write inter- pretable algorithms with perceptual components.We explore the benefits of inductive biases for
strong generalization and modularity that come from the program-like structure of our models. In particular, modularity allows us to learn a library of (neural) functions which grows and improves as more tasks are solved. Empirically, we show that this leads to lifelong learning systems that transfer knowledge to new tasks more effectively than baselines.1. Introduction
Recently, there has been much work on learning algorithms using neural networks. Following the idea of the Neural Tur- ing Machine (Graves et al., 2014), this work has focused on extending neural networks with interpretable components that are differentiable versions of traditional computer com- ponents, such as external memories, stacks, and discrete functional units. However, trained models are not easily interpreted as the learned algorithms are embedded in the weights of a monolithic neural network. In this work we flip the roles of the neural network and differentiable computer architecture. We considerinterpretablecontroller architec- tures which express algorithms using differentiable pro- gramming languages (Gaunt et al.,2016; Riedel et al.,2016; Bunel et al., 2016). In our framework, these controllers can execute discrete functional units (such as those considered by past work), but also have access to a library of trainable, uninterpretable neural network functional units. The sys- tem is end-to-end differentiable such that the source code representation of the algorithm is jointly induced with the parameters of the neural function library. In this paper we1 Canada (work done while at Microsoft). Correspondence to:Alexander L. Gaunt.
Proceedings of the34thInternational Conference on Machine Learning, Sydney, Australia, PMLR 70, 2017. Copyright 2017 by the author(s). explore potential advantages of this class of hybrid model over purely neural systems, with a particular emphasis on lifelong learning systems that learn from weak supervision. We concentrate onperceptual programming by example (PPBE) tasks that have both algorithmic and perceptual ele- ments to exercise the traditional strengths of program-like and neural components. Examples of this class of task in- clude navigation tasks guidedbyimages ornaturallanguage Using an illustrative set of PPBE tasks we aim to emphasize two specific benefits of our hybrid models: First, the source code representation in the controller allows modularity: the neural components are small functions that specialize to different tasks within the larger program struc- ture. It is easy to separate and share these functional units to transfer knowledge between tasks. In contrast, the absence of well-defined functions in purely neural solutions makes effective knowledge transfer more difficult, leading to prob- lems such as catastrophic forgetting in multitask and life- long learning (McCloskey & Cohen, 1989; Ratcliff, 1990). In our experiments, we consider a lifelong learning setting in which we train the system on asequenceof PPBE tasks that share perceptual subtasks. Second, the source code representation enforces an induc- tive bias that favors learning solutions that exhibit strong generalization. For example, once a suitable control flow structures (e.g., aforloop) for a list manipulation prob- lem was learned on short examples, it trivially generalizes to lists of arbitrary length. In contrast, although some neu- ral architectures demonstrate a surprising ability to general- ize, the reasons for this generalization are not fully under- stood (Zhang et al., 2017) and generalization performance invariably degrades as inputs become increasingly distinct from the training data. This paper is structured as follows. We first present a lan- guage, calledNEURALTERPRET(NTPT), for specifying hybrid source code/neural network models (Sec. 2), and then introduce a sequence of PPBE tasks (Sec. 3). Our NTPT models and purely neural baselines are described in Sec. 4 and 5 respectively. The experimental results are presented in Sec. 6. Differentiable Programs with Neural Libraries# Discrete operations @Runtime([max_int], max_int) defINC(a): return(a + 1) % max_int @Runtime([max_int], max_int) defDEC(a): return(a -1) % max_int @Runtime([W, 5], W) defMOVE_X(x, dir): ifdir== 1: return(x + 1) % W de elifdir== 3: return(x -1) % W dd else: returnx @Runtime([H, 5], H) defMOVE_Y(y, dir): ifdir== 2: return(y -1) % H d` elifdir== 4: return(y + 1) % H da else: returny # Helper functions @Runtime([5],2) defeq_zero(dir): return1 ifdir== 0 else0 # Learned operations defLOOK(img): pass # constants max_int= 15; n_instr= 3; T = 45 W = 5; H = 3; w = 28; h = 28
# variables img_grid= InputTensor(w, h)[W, H] init_X= Input(W) init_Y= Input(H) final_X= Output(W) final_Y= Output(H) path_len= Output(max_int) is_halted_at_end= Output(2) instr= Param(4)[n_instr] goto= Param(n_instr)[n_instr]X = Var(W)[T]
Y = Var(H)[T]
dir= Var(5)[T] reg= Var(max_int)[T] instr_ptr= Var(n_instr)[T] is_halted= Var(2)[T] # initializationX[0].set_to(init_X)
Y[0].set_to(init_Y)
dir[0].set_to(1) reg[0].set_to(0) instr_ptr[0].set_to(0) fort inrange(T -1): is_halted[t].set_to(eq_zero(dir[t])) ifis_halted[t] == 1: # halted dir[t + 1].set_to(dir[t])X[t + 1].set_to(X[t])
Y[t + 1].set_to(Y[t])
reg[t + 1].set_to(reg[t]) instr_ptr[t + 1].set_to(instr_ptr[t]) elifis_halted[t] == 0: # not halted withinstr_ptr[t] asi: ifinstr[i] == 0: # INC reg[t + 1].set_to(INC(reg[t])) elifinstr[i] == 1: # DEC reg[t + 1].set_to(DEC(reg[t])) else: reg[t + 1].set_to(reg[t]) ifinstr[i] == 2: # MOVEX[t + 1].set_to(MOVE_X(X[t], dir[t]))
Y[t + 1].set_to(MOVE_Y(Y[t], dir[t]))
else:X[t + 1].set_to(X[t])
Y[t + 1].set_to(Y[t])
ifinstr[i] == 3: # LOOK with X[t] as x: with Y[t] as y: dir[t + 1].set_to(LOOK(img_grid[y,x])) else: dir[t + 1].set_to(dir[t]) instr_ptr[t + 1].set_to(goto[i]) final_X.set_to(X[T -1]) final_Y.set_to(X[T -1]) path_len.set_to(reg[T w1]) is_halted_at_end.set_to(ishalted[T -2]) Instruction SetDeclaration & initializationExecution modelInput-output
data setimg_grid= init_X= 0 init_Y= 1 final_X= 4 final_Y= 2 path_len= 7 instr= [3,2,0] goto= [1,2,0] L0 if not halted: dir= LOOK halt if dir==0 gotoL1 L1 if not halted:MOVE(dir)
gotoL2 L2 if not halted: reg= INC(reg) gotoL0Solution
LOOK:=
Figure 1: Components of an illustrative NTPT program for learning loopy programs that measure path length (pathlen)
through a maze of street sign images. The learned program (parameterized byinstrandgoto) must control the position
(X,Y) of an agent on a grid of (WH) street sign images each of size (wh). The agent has a single register of memory
(reg) and learns to interpret street signs using theLOOKneural function. Our system produces a solution consisting of a
correctly inferred program and a trained neural network (see Supplementary Material). Learnable components are shown in
blue and the NTPT extensions to theTERPRETlanguage are highlighted. The red path on theimggridshows the desired
behavior and is not provided at training time.