Balancing Vectors in Any Norm Aleksandar (Sasho) Nikolov University - PowerPoint PPT Presentation

Balancing Vectors in Any Norm Aleksandar (Sasho) Nikolov University of Toronto Based on joint work with Daniel Dadush, Kunal Talwar, and Nicole Tomczak-Jaegermann Sasho Nikolov (U of T) Balancing Vectors 1 / 25

Introduction Outline Introduction 1 Volume Lower Bound 2 Factorization Upper Bounds 3 Conclusion 4 Sasho Nikolov (U of T) Balancing Vectors 2 / 25

Introduction Discrepancy − 1   1     1    1 1 1 0 0 0 0 0 0   1    − 1   1 1 0 1 1 0 0 0 0 0       1 =       0 0 0 1 0 0 1 0 0 0       − 1   0 0 0 1 1 1 1 1 1 − 1   1     1   − 1 disc( U , � · � ∞ ) = ε ∈{± 1 } N � U ε � ∞ min Sasho Nikolov (U of T) Balancing Vectors 3 / 25

Introduction Discrepancy − 1   1     1    1 1 1 0 0 0 0 0 0   1    − 1   1 1 0 1 1 0 0 0 0 0       1 =       0 0 0 1 0 0 1 0 0 0       − 1   0 0 0 1 1 1 1 1 1 − 1   1     1   − 1 disc( U , � · � ∞ ) = ε ∈{± 1 } N � U ε � ∞ min Natural to consider arbitrary norms: any norm can be written as � U · � ∞ . Sasho Nikolov (U of T) Balancing Vectors 3 / 25

Introduction Basic Bounds [Spencer, 1985; Gluskin, 1989]: For any matrix U ∈ { 0 , 1 } n × N , disc( U ) � √ n Sasho Nikolov (U of T) Balancing Vectors 4 / 25

Introduction Basic Bounds [Spencer, 1985; Gluskin, 1989]: For any matrix U ∈ { 0 , 1 } n × N , disc( U ) � √ n Implied by : For any u 1 , . . . , u N ∈ B n ∞ = [ − 1 , 1] n , there exist ε 1 , . . . , ε N ∈ {− 1 , +1 } s.t. � ε 1 u 1 + . . . + ε N u N � ∞ � √ n . Sasho Nikolov (U of T) Balancing Vectors 4 / 25

Introduction Basic Bounds [Spencer, 1985; Gluskin, 1989]: For any matrix U ∈ { 0 , 1 } n × N , disc( U ) � √ n Implied by : For any u 1 , . . . , u N ∈ B n ∞ = [ − 1 , 1] n , there exist ε 1 , . . . , ε N ∈ {− 1 , +1 } s.t. � ε 1 u 1 + . . . + ε N u N � ∞ � √ n . [Beck and Fiala, 1981]: For any matrix U ∈ { 0 , 1 } n × N with at most t ones per column, disc( U ) ≤ 2 t − 1 Sasho Nikolov (U of T) Balancing Vectors 4 / 25

Introduction Basic Bounds [Spencer, 1985; Gluskin, 1989]: For any matrix U ∈ { 0 , 1 } n × N , disc( U ) � √ n Implied by : For any u 1 , . . . , u N ∈ B n ∞ = [ − 1 , 1] n , there exist ε 1 , . . . , ε N ∈ {− 1 , +1 } s.t. � ε 1 u 1 + . . . + ε N u N � ∞ � √ n . [Beck and Fiala, 1981]: For any matrix U ∈ { 0 , 1 } n × N with at most t ones per column, disc( U ) ≤ 2 t − 1 Implied by : For any u 1 , . . . , u N ∈ B n 1 , there exist ε 1 , . . . , ε N ∈ {− 1 , +1 } s.t. � ε 1 u 1 + . . . + ε N u N � ∞ < 2. Sasho Nikolov (U of T) Balancing Vectors 4 / 25

Introduction Basic Bounds [Spencer, 1985; Gluskin, 1989]: For any matrix U ∈ { 0 , 1 } n × N , disc( U ) � √ n Implied by : For any u 1 , . . . , u N ∈ B n ∞ = [ − 1 , 1] n , there exist ε 1 , . . . , ε N ∈ {− 1 , +1 } s.t. � ε 1 u 1 + . . . + ε N u N � ∞ � √ n . [Beck and Fiala, 1981]: For any matrix U ∈ { 0 , 1 } n × N with at most t ones per column, disc( U ) ≤ 2 t − 1 Implied by : For any u 1 , . . . , u N ∈ B n 1 , there exist ε 1 , . . . , ε N ∈ {− 1 , +1 } s.t. � ε 1 u 1 + . . . + ε N u N � ∞ < 2. Most combinatorial discrepancy bounds are implied by geometric vector balancing arguments. Sasho Nikolov (U of T) Balancing Vectors 4 / 25

Introduction The Vector Balancing Problem Given u 1 , . . . , u N ∈ R n , and symmetric convex body K ⊂ R n ( K = − K ), find the smallest t such that ∃ ε 1 , . . . , ε N ∈ {− 1 , +1 } : ε 1 u 1 + . . . + ε N u N ∈ tK − u 1 + u 2 u 1 + u 2 u 1 − u 2 − u 1 − u 2 Sasho Nikolov (U of T) Balancing Vectors 5 / 25

Introduction The Vector Balancing Problem Given u 1 , . . . , u N ∈ R n , and symmetric convex body K ⊂ R n ( K = − K ), find the smallest t such that ∃ ε 1 , . . . , ε N ∈ {− 1 , +1 } : ε 1 u 1 + . . . + ε N u N ∈ tK − u 1 + u 2 u 1 + u 2 u 1 − u 2 − u 1 − u 2 Minkowski Norm : � x � K = inf { t : x ∈ tK } ; t = disc(( u i ) N i =1 , � · � K ). Sasho Nikolov (U of T) Balancing Vectors 5 / 25

Introduction The Vector Balancing Problem Given u 1 , . . . , u N ∈ R n , and symmetric convex body K ⊂ R n ( K = − K ), find the smallest t such that ∃ ε 1 , . . . , ε N ∈ {− 1 , +1 } : ε 1 u 1 + . . . + ε N u N ∈ tK − u 1 + u 2 u 1 + u 2 u 1 − u 2 − u 1 − u 2 Minkowski Norm : � x � K = inf { t : x ∈ tK } ; t = disc(( u i ) N i =1 , � · � K ). Vector Balancing Constant : worst case over sequences in C � � disc( U , � · � K ) : N ∈ N , u 1 , . . . , u N ∈ C , U = ( u i ) N vb( C , K ) = sup i =1 Sasho Nikolov (U of T) Balancing Vectors 5 / 25

Introduction Questions and Prior Results [Dvoretzky, 1963] “What can be said” about vb( K , K )? any and Grinberg, 1981] vb( K , K ) ≤ n for all K . [B´ ar´ Sasho Nikolov (U of T) Balancing Vectors 6 / 25

Introduction Questions and Prior Results [Dvoretzky, 1963] “What can be said” about vb( K , K )? any and Grinberg, 1981] vb( K , K ) ≤ n for all K . [B´ ar´ ∞ ) � √ n [Spencer, 1985; Gluskin, 1989] vb( B n ∞ , B n [Beck and Fiala, 1981] vb( B n 1 , B n ∞ ) < 2 Sasho Nikolov (U of T) Balancing Vectors 6 / 25

Introduction Questions and Prior Results [Dvoretzky, 1963] “What can be said” about vb( K , K )? any and Grinberg, 1981] vb( K , K ) ≤ n for all K . [B´ ar´ ∞ ) � √ n [Spencer, 1985; Gluskin, 1989] vb( B n ∞ , B n [Beck and Fiala, 1981] vb( B n 1 , B n ∞ ) < 2 [Banaszczyk, 1998] vb ( B n 2 , K ) ≤ 5 if K has Gaussian measure γ n ( K ) ≥ 1 2 os Problem : Prove or disprove vb( B n 2 , B n Koml´ ∞ ) � 1. ∞ ) � √ log 2 n . Banaszczyk’s theorem implies vb( B n 2 , B n Sasho Nikolov (U of T) Balancing Vectors 6 / 25

Introduction Vector Balancing and Rounding For any w ∈ [0 , 1] N , any U = ( u i ) N i =1 , u i ∈ C , and any symmetric convex K , there exists a x ∈ { 0 , 1 } N such that � Ux − Uw � K ≤ vb( C , K ) . Sasho Nikolov (U of T) Balancing Vectors 7 / 25

Introduction Our Results We initiate a systematic study of upper and lower bounds on vb( C , K ) and its computational complexity: Sasho Nikolov (U of T) Balancing Vectors 8 / 25

Introduction Our Results We initiate a systematic study of upper and lower bounds on vb( C , K ) and its computational complexity: A natural volumetric lower bound on vb( C , K ) is tight up to a O (log n ) factor. The proof implies an efficient algorithm to compute ε ∈ {− 1 , 1 } N given u 1 , . . . , u N ∈ C , so that � ε 1 u 1 + . . . + ε N u N � K � (1 + log n ) vb( C , K ). Also rounding version. Sasho Nikolov (U of T) Balancing Vectors 8 / 25

Introduction Our Results We initiate a systematic study of upper and lower bounds on vb( C , K ) and its computational complexity: A natural volumetric lower bound on vb( C , K ) is tight up to a O (log n ) factor. The proof implies an efficient algorithm to compute ε ∈ {− 1 , 1 } N given u 1 , . . . , u N ∈ C , so that � ε 1 u 1 + . . . + ε N u N � K � (1 + log n ) vb( C , K ). Also rounding version. An efficiently computable upper bound on vb( C , K ) is tight up to factors polynomial in log n . Based on an optimal application of Banaszczyks’ theorem. Implies an efficient approximation algorithm for vb( C , K ). Sasho Nikolov (U of T) Balancing Vectors 8 / 25

Introduction Our Results We initiate a systematic study of upper and lower bounds on vb( C , K ) and its computational complexity: A natural volumetric lower bound on vb( C , K ) is tight up to a O (log n ) factor. The proof implies an efficient algorithm to compute ε ∈ {− 1 , 1 } N given u 1 , . . . , u N ∈ C , so that � ε 1 u 1 + . . . + ε N u N � K � (1 + log n ) vb( C , K ). Also rounding version. An efficiently computable upper bound on vb( C , K ) is tight up to factors polynomial in log n . Based on an optimal application of Banaszczyks’ theorem. Implies an efficient approximation algorithm for vb( C , K ). The results extend to hereditary discrepancy with respect to arbitrary norms. Sasho Nikolov (U of T) Balancing Vectors 8 / 25

Introduction Our Results We initiate a systematic study of upper and lower bounds on vb( C , K ) and its computational complexity: A natural volumetric lower bound on vb( C , K ) is tight up to a O (log n ) factor. The proof implies an efficient algorithm to compute ε ∈ {− 1 , 1 } N given u 1 , . . . , u N ∈ C , so that � ε 1 u 1 + . . . + ε N u N � K � (1 + log n ) vb( C , K ). Also rounding version. An efficiently computable upper bound on vb( C , K ) is tight up to factors polynomial in log n . Based on an optimal application of Banaszczyks’ theorem. Implies an efficient approximation algorithm for vb( C , K ). The results extend to hereditary discrepancy with respect to arbitrary norms. Prior work [Bansal, 2010; Nikolov and Talwar, 2015] implies bounds which deteriorate with the number of facets of K . Sasho Nikolov (U of T) Balancing Vectors 8 / 25