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arXiv:1606.01885 June 2016 arXiv:1703.00441 March 2017 This domain is prone to overfiKng and underfiKng. If we want to do well on a single objec/ve func/on: Consider an algorithm that memorizes the op/mum. This is the best op/mizer since it gets to the op/mum in one step. If we want to do well on all objec/ve func/ons: Given any op/mizer, we can always construct an objec/ve func/on that it performs poorly on. Goal: Do well on a class of objec/ve func/ons with similar geometry, e.g.: Logis/c regression loss func/ons Neural net classifica/on loss func/ons Op/miza/on problems are ubiquitous in science and engineering. Devising a new op/miza/on algorithm manually is challenging. Is there a beWer way? If the mantra of machine learning is to learn what is tradi/onally manually designed… Why not learn the op?miza?on algorithm itself? The predic/on of the neural net at any point in /me affects the inputs that it sees in the future. This violates the i.i.d. assump/on in supervised learning. Compounding errors: A policy trained using supervised learning does not know how to recover from previous mistakes. A supervised learner that makes a mistake with probability incurs a cumula/ve error of , rather than . (Ross and Bagnell, 2010) Proper?es of the Learning Problem Given: a set of training objec/ve func/ons , a distribu/on for ini/alizing the iterate and a metaloss that measures the quality of the iterates . An op/miza/on algorithm takes an objec/ve func/on and an ini/al iterate as input and produces a sequence of iterates . Goal: learn such that is minimized. We choose Formula?on Ke Li Jitendra Malik {ke.li,malik}@eecs.berkeley.edu Learning to Op?mize Introduc?on SeLng Future Work Challenges f 1 ,...,f n F D L(f,x (1) ,...,x (T ) ) x (1) ,...,x (T ) A f x (0) x (1) ,...,x (T ) A E f F ,x (0) D h L(f, A (f,x (0) )) i L(f,x (1) ,...,x (T ) )= T X i=1 f (x (i) ) Input: Recent history of iterates, gradients and objec/ve values Output: Step vector Searching over the space of op/miza/on algorithms reduces to learning the parameters of the neural net. Parameterizing Op?miza?on Algorithms Gradient Descent Momentum Learned Algorithm Neural Net φ(·)= -γ 0 @ i-1 X j =0 i-1-j rf (x (j ) ) 1 A φ(·)= φ(·)= -γ rf (x (i-1) ) O(T 2 ) O(T ) The goal of RL is to find: where the expecta/on is taken w.r.t. The method we use is Guided Policy Search (Levine and Abbeel, 2014), which alternates between compu/ng target trajectories and training the policy to replicate them. More precisely, it solves: Reinforcement Learning = arg min E s 0 ,a 0 ,s 1 ,...,s T " T X t=0 c(s t ) # q (s 0 ,a 0 ,s 1 ,...,s T )= p 0 (s 0 ) T -1 Y t=0 ( a t | s t ,t) p ( s t+1 | s t ,a t ) Cost State Ac/on Ini/al State Distribu/on Policy Dynamics Time Horizon Φ(·) State f (x (i) ) Ac/on Policy Cost Experiments We trained op/mizers for the following classes of lowdimensional op/miza/on problems: Logis/c Regression (Convex) Robust Linear Regression (Nonconvex) Small Neural Net Classifier (Nonconvex) Trained on a set of random problems. Tested on a different set of random problems. Logis+c Regression: Robust Linear Regression: Small Neural Net: Wider Architecture: Noisier Gradients: Wider Architecture and Noisier Gradients: Wider Architecture and Longer Time Horizon: Credit: John Schulman min ,E " T X t=0 c(s t ) # s.t. ( a t | s t ,t; )= ( a t | s t ; ) 8a t ,s t ,t Learning to Op?mize Neural Nets (hWps:// arxiv.org /abs/1703.00441 ) Trained op/mizer on the experience of training neural net on MNIST (a single objec/ve func/on). Tested it on the problems of training a neural net on Toronto Faces, CIFAR10 and CIFAR100. Δx
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Page 1: THIS SIDEBAR DOES NOT PRINT—) LearningtoOpmize( QUICK ...ke.li/papers/lto_iclr17_poster.pdfThis PowerPoint 2007 template produces a 42”x90” presentation poster. You can use it

(—THIS SIDEBAR DOES NOT PRINT—) D E S I G N G U I D E

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ORIGINAL   DISTORTED  

Corner  handles  

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 prin

/ng  qu

ality

 

Bad  prin/n

g  qu

ality

 

Q U I C K S TA R T ( c o n t . )

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©  2015  PosterPresenta/ons.com  2117  Fourth  Street  ,  Unit  C                  Berkeley  CA  94710  [email protected]  

arXiv:1606.01885  

June  2016  arXiv:1703.00441  

March  2017  

•  This  domain  is  prone  to  overfiKng  and  underfiKng.    •  If  we  want  to  do  well  on  a  single  objec/ve  func/on:  –  Consider  an  algorithm  that    memorizes  the  op/mum.    

–  This  is  the  best  op/mizer    since  it  gets  to  the  op/mum    in  one  step.  

•  If  we  want  to  do  well  on  all  objec/ve  func/ons:  –  Given  any  op/mizer,  we  can  always  construct  an  objec/ve  func/on  that  it  performs  poorly  on.    

 

•  Goal:  Do  well  on  a  class  of  objec/ve  func/ons  with  similar  geometry,  e.g.:  –  Logis/c  regression  loss  func/ons  –  Neural  net  classifica/on  loss  func/ons  

•  Op/miza/on  problems  are  ubiquitous  in  science  and  engineering.    

•  Devising  a  new  op/miza/on  algorithm  manually  is  challenging.  Is  there  a  beWer  way?  

•  If  the  mantra  of  machine  learning  is  to  learn  what  is  tradi/onally  manually  designed…  

Why  not  learn  the  op?miza?on  algorithm  itself?  

•  The  predic/on  of  the  neural  net  at  any  point  in  /me  affects  the  inputs  that  it  sees  in  the  future.    

•  This  violates  the  i.i.d.  assump/on  in  supervised  learning.  

•  Compounding  errors:    A  policy  trained  using    supervised  learning    does  not  know  how  to    recover  from  previous    mistakes.    

•  A  supervised  learner  that  makes  a  mistake  with  probability            incurs  a  cumula/ve  error  of                            ,  rather  than                          .  (Ross  and  Bagnell,  2010)  

Proper?es  of  the  Learning  Problem  •  Given:  a  set  of  training  objec/ve  func/ons                                                  ,  a  distribu/on              for  ini/alizing  the  iterate  and  a  meta-­‐loss                                                                  that  measures  the  quality  of  the  iterates                                                    .          

•  An  op/miza/on  algorithm              takes  an  objec/ve  func/on          and  an  ini/al  iterate                    as  input  and  produces  a  sequence  of  iterates                                                    .  

•  Goal:  learn                such  that                                                                                                  is  minimized.      

•  We  choose    

Formula?on  

Ke  Li                        Jitendra  Malik  {ke.li,malik}@eecs.berkeley.edu

Learning  to  Op?mize  

Introduc?on   SeLng   Future  Work  

Challenges  

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•  Input:  Recent  history  of  iterates,  gradients  and  objec/ve  values  

•  Output:  Step  vector  •  Searching  over  the  space  of  op/miza/on  algorithms  reduces  to  learning  the  parameters  of  the  neural  net.    

Parameterizing  Op?miza?on  Algorithms  

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•  The  goal  of  RL  is  to  find:      where  the  expecta/on  is  taken  w.r.t.  

•  The  method  we  use  is  Guided  Policy  Search  (Levine  and  Abbeel,  2014),  which  alternates  between  compu/ng  target  trajectories  and  training  the  policy  to  replicate  them.  More  precisely,  it  solves:  

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Experiments  •  We  trained  op/mizers  for  the  following  classes  of  low-­‐dimensional  op/miza/on  problems:  –  Logis/c  Regression  (Convex)    –  Robust  Linear  Regression  (Non-­‐convex)  –  Small  Neural  Net  Classifier  (Non-­‐convex)    

•  Trained  on  a  set  of  random  problems.    •  Tested  on  a  different  set  of  random  problems.              

Logis+c  Regression:

Robust  Linear  Regression:                                    Small  Neural  Net:  

Wider  Architecture:

Noisier  Gradients:

Wider  Architecture  and  Noisier  Gradients:

Wider  Architecture  and  Longer  Time  Horizon:

Credit:  John  Schulman

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Learning  to  Op?mize  Neural  Nets  (hWps://arxiv.org/abs/1703.00441)  

 

•  Trained  op/mizer  on  the  experience  of  training  neural  net  on  MNIST  (a  single  objec/ve  func/on).    

•  Tested  it  on  the  problems  of  training  a  neural  net  on      Toronto  Faces,                      CIFAR-­‐10          and        CIFAR-­‐100.    

             

             

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