Efficient Optimization for Rank-Based Loss Functions

Pritish Mohapatra, Michal RolĂnek, C.V. Jawahar, Vladimir Kolmogorov, M. Pawan Kumar.

**Concept Graph & Resume using Claude 3 Opus | Chat GPT4o | Llama 3:**

graph LR
classDef ranking fill:#f9d4d4, font-weight:bold, font-size:14px
classDef pipeline fill:#d4f9d4, font-weight:bold, font-size:14px
classDef evaluation fill:#d4d4f9, font-weight:bold, font-size:14px
classDef learning fill:#f9f9d4, font-weight:bold, font-size:14px
classDef optimization fill:#f9d4f9, font-weight:bold, font-size:14px
classDef quicksort fill:#d4f9f9, font-weight:bold, font-size:14px
classDef results fill:#f9d4d4, font-weight:bold, font-size:14px
A[Efficient Optimization for

Rank-Based Loss Functions] --> B[Rank images by

query relevance 1] A --> C[Standard ranking process 2] A --> D[Evaluate with AP,

NDCG measures 3] A --> E[Learn with differentiable

losses like zero-one 4] A --> F[Rank-based loss better,

computationally expensive 5] F --> G[Assign scores, determine

rankings, solve optimization 6] G --> H[Most violating ranking

for efficient gradient 7] A --> I[QS suitable losses

for efficient optimization 8] I --> J[Loss depends on

interleaving rank 9] J --> K[AP, NDCG losses

are QS suitable 10] I --> L[Loss decomposable onto

negative samples 11] I --> M[Loss depends only

on interleaving pattern 12] I --> N[Multiple most violating

rankings exist 13] I --> O[Constraints on ranks

based on scores 14] I --> P[Induce ordering, find

optimal interleaving 15] A --> Q[Baseline: sort positives,

negatives, find interleaving 16] A --> R[Quicksort: sort positives,

assign negatives recursively 17] R --> S[Same-rank negatives get

same rank free 18] R --> T[Complexity: p log p

+ n log p 19] Q --> U[Worse complexity than

quicksort: n log n + pn 20] A --> V[AP/NDCG loss improves

Pascal action classification 21] A --> W[Quicksort scales well

vs baseline 22] A --> X[AP loss improves

Pascal VOC detection >7% 23] A --> Y[AP/NDCG loss improves

CIFAR-10, quicksort faster 24] A --> Z[Rank-based loss good

for ranking performance 25] Z --> AA[But expensive to

optimize generally 26] Z --> AB[QS suitable losses

enable efficient optimization 27] AB --> AC[Performance boost without

computation time increase 28] A --> AD[Other scores like

F-score unexplored 29] A --> AE[Assumes binary labels,

pairwise prefs unconsidered 30] class A,B,C ranking class D,E,F,G,H,Z,AA optimization class I,J,K,L,M,N,O,P evaluation class Q,R,S,T,U quicksort class V,W,X,Y,AC results class AD,AE learning

Rank-Based Loss Functions] --> B[Rank images by

query relevance 1] A --> C[Standard ranking process 2] A --> D[Evaluate with AP,

NDCG measures 3] A --> E[Learn with differentiable

losses like zero-one 4] A --> F[Rank-based loss better,

computationally expensive 5] F --> G[Assign scores, determine

rankings, solve optimization 6] G --> H[Most violating ranking

for efficient gradient 7] A --> I[QS suitable losses

for efficient optimization 8] I --> J[Loss depends on

interleaving rank 9] J --> K[AP, NDCG losses

are QS suitable 10] I --> L[Loss decomposable onto

negative samples 11] I --> M[Loss depends only

on interleaving pattern 12] I --> N[Multiple most violating

rankings exist 13] I --> O[Constraints on ranks

based on scores 14] I --> P[Induce ordering, find

optimal interleaving 15] A --> Q[Baseline: sort positives,

negatives, find interleaving 16] A --> R[Quicksort: sort positives,

assign negatives recursively 17] R --> S[Same-rank negatives get

same rank free 18] R --> T[Complexity: p log p

+ n log p 19] Q --> U[Worse complexity than

quicksort: n log n + pn 20] A --> V[AP/NDCG loss improves

Pascal action classification 21] A --> W[Quicksort scales well

vs baseline 22] A --> X[AP loss improves

Pascal VOC detection >7% 23] A --> Y[AP/NDCG loss improves

CIFAR-10, quicksort faster 24] A --> Z[Rank-based loss good

for ranking performance 25] Z --> AA[But expensive to

optimize generally 26] Z --> AB[QS suitable losses

enable efficient optimization 27] AB --> AC[Performance boost without

computation time increase 28] A --> AD[Other scores like

F-score unexplored 29] A --> AE[Assumes binary labels,

pairwise prefs unconsidered 30] class A,B,C ranking class D,E,F,G,H,Z,AA optimization class I,J,K,L,M,N,O,P evaluation class Q,R,S,T,U quicksort class V,W,X,Y,AC results class AD,AE learning

**Resume: **

**1.-** Ranking problem: Given images and a query, rank images by relevance to the query

**2.-** Standard ranking pipeline: Collect data, learn ranking model, assign scores, sort samples

**3.-** Evaluating ranking models: Use ranking measures like Average Precision (AP) or Normalized Discounted Cumulative Gain (NDCG)

**4.-** Learning model parameters: Popular to use differentiable loss functions like zero-one loss

**5.-** Optimizing rank-based loss functions directly can give better performance but is computationally expensive

**6.-** Expensive gradient computation procedure: Assign scores, determine candidate rankings, solve optimization problem

**7.-** Most violating ranking: Optimal solution to the optimization problem, needed for efficient gradient computation

**8.-** QS suitable loss functions: Rank-based loss functions amenable to efficient optimization

**9.-** Interleaving rank: Number of positive samples preceding a negative sample

**10.-** AP loss and NDCG loss are QS suitable

**11.-** Negative decomposability property: Loss additively decomposable onto negative samples

**12.-** Interleaving dependence property: Loss depends only on interleaving pattern of negatives and positives

**13.-** Multiple possible most violating rankings exist

**14.-** Partial ordering structure: Constraints on interleaving ranks based on scores

**15.-** Gradient computation steps: Induce partial ordering, find optimal interleaving pattern

**16.-** Baseline algorithm: Completely sort positives (p log p) and negatives (n log n), find interleaving (np)

**17.-** Quicksort-flavored algorithm: Sort positives (p log p), assign negatives optimal rank recursively (n log p)

**18.-** Negative samples between those with same rank get same rank for free

**19.-** Quicksort-flavored complexity: p log p + n log p + p log n ~ n log p

**20.-** Baseline complexity: n log n + pn, worse than quicksort-flavored

**21.-** Empirical performance on Pascal action classification: AP/NDCG loss improves performance, quicksort time comparable to zero-one loss

**22.-** Good scaling of quicksort algorithm compared to baseline as number of samples increases

**23.-** Weakly supervised object detection on Pascal VOC: AP loss improves mean performance >7%

**24.-** Training deep model on CIFAR-10: AP/NDCG loss improves performance, quicksort faster than baseline

**25.-** Optimizing rank-based loss good for ranking model performance

**26.-** But expensive to optimize rank-based loss in general

**27.-** QS suitable rank-based losses enable efficient optimization

**28.-** Improvement in performance without additional computational time

**29.-** Applicability to other ranking scores like F-score or mean reciprocity rank not yet explored

**30.-** Approach assumes zero-one labeling ground truth, extension to pairwise preferences not considered

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