Learnability Beyond Uniform Convergence Shai Shalev-Shwartz School of CS and Engineering, The Hebrew University of Jerusalem ”Algorithmic Learning Theory”, Lyon 2012 Joint work with: N. Srebro, O. Shamir, K. Sridharan (COLT’09,JMLR’11) A. Daniely, S. Sabato, S. Ben-David (COLT’11) A. Daniely, S. Sabato (NIPS’12) Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 1 / 34
The Fundamental Theorem of Learning Theory For Binary Classification Uniform trivial trivial Learnable Learnable Convergence with ERM VC’71 NFL (W’96) Finite VC VC = Vapnik and Chervonenkis, W = Wolpert Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 2 / 34
The Fundamental Theorem of Learning Theory For Regression Uniform trivial trivial Learnable Learnable Convergence with ERM BLW’96,ABCH’97 Finite fat- KS’94,BLW’96,ABCH’97 shattering BLW = Bartlett, Long, Williamson. ABCH = Alon, Ben-David, Cesa-Bianchi, Hausler. KS = Kearns and Schapire Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 3 / 34
For general learning problems? Uniform trivial trivial Learnable Learnable Convergence with ERM ? Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 4 / 34
For general learning problems? Uniform trivial trivial Learnable Learnable Convergence with ERM X Not true Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 4 / 34
For general learning problems? Uniform trivial trivial Learnable Learnable Convergence with ERM X Not true Not true in “Convex learning problems” ! Not true even in “multiclass categorization” ! What is learnable ? How to learn ? Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 4 / 34
Outline Definitions 1 Learnability without uniform convergence 2 Characterizing Learnability using Stability 3 Characterizing Multiclass Learnability 4 Analyzing specific, practically relevant, classes 5 Open Questions 6 Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 5 / 34
The General Learning Setting (Vapnik) Hypothesis class H Examples domain Z with unknown distribution D Loss function ℓ : H × Z → R Given: Training set S ∼ D m Goal: Solve: min h ∈H L ( h ) where L ( h ) = E z ∼D [ ℓ ( h, z )] in the P robably (w.p. ≥ 1 − δ ) A pproximately C orrect (up to ǫ ) sense Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 6 / 34
The General Learning Setting (Vapnik) Hypothesis class H Examples domain Z with unknown distribution D Loss function ℓ : H × Z → R Given: Training set S ∼ D m Goal: Solve: min h ∈H L ( h ) where L ( h ) = E z ∼D [ ℓ ( h, z )] in the P robably (w.p. ≥ 1 − δ ) A pproximately C orrect (up to ǫ ) sense m Training loss: L S ( h ) = 1 � ℓ ( h, z i ) m i =1 Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 6 / 34
Examples Binary classification: Z = X × { 0 , 1 } h ∈ H is a predictor h : X → { 0 , 1 } ℓ ( h, ( x, y )) = 1 [ h ( x ) � = y ] Multiclass categorization: Z = X × Y h ∈ H is a predictor h : X → Y ℓ ( h, ( x, y )) = 1 [ h ( x ) � = y ] k -means clustering: Z = R d H ⊂ ( R d ) k specifies k cluster centers ℓ (( µ 1 , . . . , µ k ) , z ) = min j � µ j − z � Density Estimation: h is a parameter of a density p h ( z ) ℓ ( h, z ) = − log p h ( z ) Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 7 / 34
Learnability, ERM, Uniform convergence Uniform Convergence : For m ≥ m UC ( ǫ, δ ) S ∼D m [ ∀ h ∈ H , | L S ( h ) − L ( h ) | ≤ ǫ ] ≥ 1 − δ P Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 8 / 34
Learnability, ERM, Uniform convergence Uniform Convergence : For m ≥ m UC ( ǫ, δ ) S ∼D m [ ∀ h ∈ H , | L S ( h ) − L ( h ) | ≤ ǫ ] ≥ 1 − δ P Learnable : ∃A s.t. for m ≥ m PAC ( ǫ, δ ) , � � L ( A ( S )) ≤ min h ∈H L ( h ) + ǫ ≥ 1 − δ P S ∼D m Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 8 / 34
Learnability, ERM, Uniform convergence Uniform Convergence : For m ≥ m UC ( ǫ, δ ) S ∼D m [ ∀ h ∈ H , | L S ( h ) − L ( h ) | ≤ ǫ ] ≥ 1 − δ P Learnable : ∃A s.t. for m ≥ m PAC ( ǫ, δ ) , � � L ( A ( S )) ≤ min h ∈H L ( h ) + ǫ ≥ 1 − δ P S ∼D m ERM : An algorithm that returns A ( S ) ∈ argmin h ∈H L S ( h ) Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 8 / 34
Learnability, ERM, Uniform convergence Uniform Convergence : For m ≥ m UC ( ǫ, δ ) S ∼D m [ ∀ h ∈ H , | L S ( h ) − L ( h ) | ≤ ǫ ] ≥ 1 − δ P Learnable : ∃A s.t. for m ≥ m PAC ( ǫ, δ ) , � � L ( A ( S )) ≤ min h ∈H L ( h ) + ǫ ≥ 1 − δ P S ∼D m ERM : An algorithm that returns A ( S ) ∈ argmin h ∈H L S ( h ) Learnable by arbitrary ERM (with rate m ERM ( ǫ, δ ) ) Like “Learnable” but A should be an ERM. Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 8 / 34
For Binary Classification Uniform trivial trivial Learnable Learnable Convergence with ERM VC’71 NFL (W’96) Finite VC VC( H ) log(1 /δ ) m UC ( ǫ, δ ) ≈ m ERM ( ǫ, δ ) ≈ m PAC ( ǫ, δ ) ≈ ǫ 2 Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 9 / 34
Outline Definitions 1 Learnability without uniform convergence 2 Characterizing Learnability using Stability 3 Characterizing Multiclass Learnability 4 Analyzing specific, practically relevant, classes 5 Open Questions 6 Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 10 / 34
Counter Example — Stochastic Convex Optimization Consider the family of problems: H is a convex set with max h ∈H � h � ≤ 1 For all z , ℓ ( h, z ) is convex and Lipschitz w.r.t. h Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 11 / 34
Counter Example — Stochastic Convex Optimization Consider the family of problems: H is a convex set with max h ∈H � h � ≤ 1 For all z , ℓ ( h, z ) is convex and Lipschitz w.r.t. h Claim: Problem is learnable by the rule: m 2 � h � 2 + 1 λ m � argmin ℓ ( h, z i ) m h ∈H i =1 No uniform convergence Not learnable by ERM Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 11 / 34
Counter Example — Stochastic Convex Optimization Proof (of “not learnable by arbitrary ERM”) 1 -Mean + missing features Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 12 / 34
Counter Example — Stochastic Convex Optimization Proof (of “not learnable by arbitrary ERM”) 1 -Mean + missing features z = ( α, x ) , α ∈ { 0 , 1 } d , x ∈ R d , � x � ≤ 1 �� i α i ( h i − x i ) 2 ℓ ( h, ( α, x )) = Take P [ α i = 1] = 1 / 2 , P [ x = µ ] = 1 Let h ( i ) be s.t. � 1 − µ j if j = i h ( i ) = j µ j o.w. If d is large enough, exists i such that h ( i ) is an ERM √ But L ( h ( i ) ) ≥ 1 / 2 Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 12 / 34
Counter Example — Stochastic Convex Optimization Proof (of “not even learnable by a unique ERM”) Perturb the loss a little bit: �� α i ( h i − x i ) 2 + ǫ � 2 − i ( h i − 1) 2 ℓ ( h, ( α, x )) = i i Now loss is strictly convex — unique ERM But the unique ERM does not generalize (as before) Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 13 / 34
For general learning problems? Uniform trivial trivial Learnable Learnable Convergence with ERM X Not true Not true in “Convex learning problems” ! ✓ Not true even in “multiclass categorization” ! Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 14 / 34
Counter Example — Multiclass X – a set, Y = { 0 , 1 , 2 , . . . , 2 |X| − 1 } Let n : 2 X → Y be defined by binary encoding H = { h T : T ⊂ X} where � 0 x / ∈ T h T ( x ) = n ( T ) x ∈ T Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 15 / 34
Counter Example — Multiclass X – a set, Y = { 0 , 1 , 2 , . . . , 2 |X| − 1 } Let n : 2 X → Y be defined by binary encoding H = { h T : T ⊂ X} where � 0 x / ∈ T h T ( x ) = n ( T ) x ∈ T Claim: No uniform convergence: m UC ≥ |X| /ǫ Target function is h ∅ For any training set S , take T = X \ S L S ( h T ) = 0 but L ( h T ) = P [ T ] Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 15 / 34
Counter Example — Multiclass X – a set, Y = { 0 , 1 , 2 , . . . , 2 |X| − 1 } Let n : 2 X → Y be defined by binary encoding H = { h T : T ⊂ X} where � 0 x / ∈ T h T ( x ) = n ( T ) x ∈ T Claim: H is Learnable: m PAC ≤ 1 ǫ Let T be the target A ( S ) = h T if ( x, n ( T )) ∈ S A ( S ) = h ∅ if S = { ( x 1 , 0) , . . . , ( x m , 0) } In the 1st case, L ( A ( S )) = 0 . In the 2nd case, L ( A ( S )) = P [ T ] With high probability, if P [ T ] > ǫ then we’ll be in the 1st case Shai Shalev-Shwartz (Hebrew U) Learnability Beyond Uniform Convergence Oct’12 15 / 34
Recommend
More recommend
