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|
| 3418 |
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parse/train/K6y77KRUowQ/K6y77KRUowQ_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3419 |
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parse/train/K6y77KRUowQ/K6y77KRUowQ_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3420 |
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parse/train/Jnspzp-oIZE/Jnspzp-oIZE_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3421 |
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parse/train/Jnspzp-oIZE/Jnspzp-oIZE_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3422 |
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parse/train/Jnspzp-oIZE/Jnspzp-oIZE_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3423 |
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parse/train/KJ5h-yfUHa/KJ5h-yfUHa_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3424 |
+
parse/train/KJ5h-yfUHa/KJ5h-yfUHa_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3425 |
+
parse/train/KJ5h-yfUHa/KJ5h-yfUHa_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3426 |
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parse/train/HkePNpVKPB/HkePNpVKPB_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3427 |
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parse/train/HkePNpVKPB/HkePNpVKPB_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3428 |
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parse/train/HkePNpVKPB/HkePNpVKPB_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3429 |
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parse/train/h7FqQ6hCK18/h7FqQ6hCK18_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3430 |
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parse/train/h7FqQ6hCK18/h7FqQ6hCK18_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3431 |
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parse/train/h7FqQ6hCK18/h7FqQ6hCK18_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3432 |
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parse/train/3_MUAtqR0aA/3_MUAtqR0aA_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3433 |
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parse/train/3_MUAtqR0aA/3_MUAtqR0aA_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3434 |
+
parse/train/3_MUAtqR0aA/3_MUAtqR0aA_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3435 |
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parse/train/pvjfA4wogD6/pvjfA4wogD6_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3436 |
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parse/train/pvjfA4wogD6/pvjfA4wogD6_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3437 |
+
parse/train/pvjfA4wogD6/pvjfA4wogD6_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3438 |
+
parse/train/HkgxasA5Ym/HkgxasA5Ym_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3439 |
+
parse/train/HkgxasA5Ym/HkgxasA5Ym_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3440 |
+
parse/train/HkgxasA5Ym/HkgxasA5Ym_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3441 |
+
parse/train/HJGXzmspb/HJGXzmspb_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3442 |
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parse/train/HJGXzmspb/HJGXzmspb_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3443 |
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parse/train/HJGXzmspb/HJGXzmspb_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3444 |
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parse/train/9QffERDO_rJ/9QffERDO_rJ_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3445 |
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parse/train/9QffERDO_rJ/9QffERDO_rJ_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3446 |
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parse/train/9QffERDO_rJ/9QffERDO_rJ_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3447 |
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parse/train/ByIAPUcee/ByIAPUcee_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3448 |
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parse/train/ByIAPUcee/ByIAPUcee_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3449 |
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parse/train/ByIAPUcee/ByIAPUcee_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3450 |
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parse/train/ryepFJbA-/ryepFJbA-_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3451 |
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parse/train/ryepFJbA-/ryepFJbA-_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3452 |
+
parse/train/ryepFJbA-/ryepFJbA-_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3453 |
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parse/train/-kfLEqppEm_/-kfLEqppEm__layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3454 |
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|
| 3455 |
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parse/train/-kfLEqppEm_/-kfLEqppEm__origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3456 |
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parse/train/HkxBJT4YvB/HkxBJT4YvB_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3457 |
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parse/train/HkxBJT4YvB/HkxBJT4YvB_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3458 |
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parse/train/HkxBJT4YvB/HkxBJT4YvB_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3459 |
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parse/train/BJxhijAcY7/BJxhijAcY7_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3460 |
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parse/train/BJxhijAcY7/BJxhijAcY7_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3461 |
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parse/train/BJxhijAcY7/BJxhijAcY7_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3462 |
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parse/train/uqv8-U4lKBe/uqv8-U4lKBe_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3463 |
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parse/train/uqv8-U4lKBe/uqv8-U4lKBe_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3464 |
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parse/train/uqv8-U4lKBe/uqv8-U4lKBe_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3465 |
+
parse/train/B1xDq2EFDH/B1xDq2EFDH_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3466 |
+
parse/train/B1xDq2EFDH/B1xDq2EFDH_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3467 |
+
parse/train/B1xDq2EFDH/B1xDq2EFDH_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3468 |
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parse/train/vvi7KqHQiA/vvi7KqHQiA_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3469 |
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parse/train/vvi7KqHQiA/vvi7KqHQiA_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3470 |
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parse/train/vvi7KqHQiA/vvi7KqHQiA_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3471 |
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parse/train/Bkg3g2R9FX/Bkg3g2R9FX_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3472 |
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parse/train/Bkg3g2R9FX/Bkg3g2R9FX_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3473 |
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parse/train/Bkg3g2R9FX/Bkg3g2R9FX_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3474 |
+
parse/train/FrIDgjDOH1u/FrIDgjDOH1u_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3475 |
+
parse/train/FrIDgjDOH1u/FrIDgjDOH1u_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3476 |
+
parse/train/FrIDgjDOH1u/FrIDgjDOH1u_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3477 |
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parse/train/Wj4ODo0uyCF/Wj4ODo0uyCF_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3478 |
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parse/train/Wj4ODo0uyCF/Wj4ODo0uyCF_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3479 |
+
parse/train/Wj4ODo0uyCF/Wj4ODo0uyCF_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3480 |
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parse/train/H1faSn0qY7/H1faSn0qY7_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3481 |
+
parse/train/H1faSn0qY7/H1faSn0qY7_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3482 |
+
parse/train/H1faSn0qY7/H1faSn0qY7_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3483 |
+
parse/train/KRODJAa6pzE/KRODJAa6pzE_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3484 |
+
parse/train/KRODJAa6pzE/KRODJAa6pzE_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3485 |
+
parse/train/KRODJAa6pzE/KRODJAa6pzE_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3486 |
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parse/train/ZKbZ4mebI9l/ZKbZ4mebI9l_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3487 |
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parse/train/ZKbZ4mebI9l/ZKbZ4mebI9l_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3488 |
+
parse/train/ZKbZ4mebI9l/ZKbZ4mebI9l_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3489 |
+
parse/train/HkCsm6lRb/HkCsm6lRb_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3490 |
+
parse/train/HkCsm6lRb/HkCsm6lRb_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3491 |
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parse/train/HkCsm6lRb/HkCsm6lRb_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3492 |
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parse/train/H1xipsA5K7/H1xipsA5K7_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3493 |
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parse/train/H1xipsA5K7/H1xipsA5K7_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3494 |
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parse/train/H1xipsA5K7/H1xipsA5K7_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3495 |
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parse/train/HyxPx3R9tm/HyxPx3R9tm_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3496 |
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parse/train/HyxPx3R9tm/HyxPx3R9tm_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3497 |
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parse/train/HyxPx3R9tm/HyxPx3R9tm_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3498 |
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parse/train/P42rXLGZQ07/P42rXLGZQ07_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3499 |
+
parse/train/P42rXLGZQ07/P42rXLGZQ07_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3500 |
+
parse/train/P42rXLGZQ07/P42rXLGZQ07_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3501 |
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parse/train/_b8l7rVPe8z/_b8l7rVPe8z_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3502 |
+
parse/train/_b8l7rVPe8z/_b8l7rVPe8z_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3503 |
+
parse/train/_b8l7rVPe8z/_b8l7rVPe8z_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3504 |
+
parse/train/BygqBiRcFQ/BygqBiRcFQ_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3505 |
+
parse/train/BygqBiRcFQ/BygqBiRcFQ_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3506 |
+
parse/train/BygqBiRcFQ/BygqBiRcFQ_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3507 |
+
parse/train/j7u7cJDBo8p/j7u7cJDBo8p_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3508 |
+
parse/train/j7u7cJDBo8p/j7u7cJDBo8p_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3509 |
+
parse/train/j7u7cJDBo8p/j7u7cJDBo8p_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3510 |
+
parse/train/HkNDsiC9KQ/HkNDsiC9KQ_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3511 |
+
parse/train/HkNDsiC9KQ/HkNDsiC9KQ_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3512 |
+
parse/train/HkNDsiC9KQ/HkNDsiC9KQ_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3513 |
+
parse/train/rkA1f3NpZ/rkA1f3NpZ_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3514 |
+
parse/train/rkA1f3NpZ/rkA1f3NpZ_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3515 |
+
parse/train/rkA1f3NpZ/rkA1f3NpZ_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3516 |
+
parse/train/g-wu9TMPODo/g-wu9TMPODo_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3517 |
+
parse/train/g-wu9TMPODo/g-wu9TMPODo_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3518 |
+
parse/train/g-wu9TMPODo/g-wu9TMPODo_layout.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3519 |
+
parse/train/SUyxNGzUsH/SUyxNGzUsH_span.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3520 |
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parse/train/SUyxNGzUsH/SUyxNGzUsH_origin.pdf filter=lfs diff=lfs merge=lfs -text
|
| 3521 |
+
parse/train/SUyxNGzUsH/SUyxNGzUsH_layout.pdf filter=lfs diff=lfs merge=lfs -text
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| 1 |
+
# DOGE-Train: Discrete Optimization on GPU with End-to-end Training
|
| 2 |
+
|
| 3 |
+
Anonymous Author(s)
|
| 4 |
+
Affiliation
|
| 5 |
+
Address
|
| 6 |
+
email
|
| 7 |
+
|
| 8 |
+
# Abstract
|
| 9 |
+
|
| 10 |
+
1 We present a fast, scalable, data-driven approach for solving linear relaxations of
|
| 11 |
+
2 0-1 integer linear programs using a graph neural network. Our solver is based
|
| 12 |
+
3 on the Lagrange decomposition based algorithm [1]. We make the algorithm
|
| 13 |
+
4 differentiable and perform backpropagation through the dual update scheme for
|
| 14 |
+
5 end-to-end training of its algorithmic parameters. This allows to preserve the
|
| 15 |
+
6 algorithm’s theoretical properties including feasibility and guaranteed non-decrease
|
| 16 |
+
7 in the lower bound. Since [1] can get stuck in suboptimal fixed points, we provide
|
| 17 |
+
8 additional freedom to our graph neural network to predict non-parametric update
|
| 18 |
+
9 steps for escaping such points while maintaining dual feasibility. For training of
|
| 19 |
+
10 the graph neural network we use an unsupervised loss and perform experiments on
|
| 20 |
+
11 large-scale real world datasets. We train on smaller problems and test on larger ones
|
| 21 |
+
12 showing strong generalization performance with a graph neural network comprising
|
| 22 |
+
13 only around $1 0 k$ parameters. Our solver achieves significantly faster performance
|
| 23 |
+
14 and better dual objectives than its non-learned version [1]. In comparison to
|
| 24 |
+
15 commercial solvers our learned solver achieves close to optimal objective values of
|
| 25 |
+
16 LP relaxations and is faster by up to an order of magnitude on very large problems
|
| 26 |
+
17 from structured prediction and on selected combinatorial optimization problems.
|
| 27 |
+
18 Our code will be made available upon acceptance.
|
| 28 |
+
|
| 29 |
+
# 19 1 Introduction
|
| 30 |
+
|
| 31 |
+
20 Integer linear programs (ILP) are a universal tool for solving combinatorial optimization problems.
|
| 32 |
+
21 While great progress has been made on improving ILP solvers over the past several decades, some
|
| 33 |
+
22 fundamental questions for future improvements remain open: Can ILP solvers make effective use of
|
| 34 |
+
23 the massive parallelism afforded by GPUs and can modern machine learning meaningfully help? As
|
| 35 |
+
24 of now the consensus seems that neither GPUs nor ML have yet helped general purpose ILP solvers
|
| 36 |
+
25 in a fundamental way. In particular, this holds true for LP solvers which are a key component of most
|
| 37 |
+
26 commonly used ILP approaches. LP solvers produce lower bounds on the optimal solution objective
|
| 38 |
+
27 and are integral for many heuristics to decode feasible integral solutions. For many problems the ILP
|
| 39 |
+
28 solvers spend most of the time on solving multiple LP relaxations, hence any impact GPUs and ML
|
| 40 |
+
29 can have will directly translate into overall improvement of ILP solvers.
|
| 41 |
+
30 State of the art LP solvers [23, 13, 17, 4, 18] make little utility of modern machine learning but rather
|
| 42 |
+
31 use either hand-designed or auto-tuned parameters and update rules. Moreover, with the exception
|
| 43 |
+
32 of [18] these solvers are not open-source, hence researchers’ ability to assess the potential of neural
|
| 44 |
+
33 networks for improving LP solvers is limited. From a conceptual point of view traditional solver
|
| 45 |
+
34 paradigms, e.g. simplex or interior point methods, are not GPU friendly and contain non-differentiable
|
| 46 |
+
35 steps (such as pivot selection for simplex). Additionally, their high complexity further complicates
|
| 47 |
+
36 any effort at making them differentiable. This makes utilization of neural networks and GPUs for
|
| 48 |
+
37 solver improvement difficult.
|
| 49 |
+
38 We propose a new way to use the potential of GPU parallelism and modern ML to obtain advances
|
| 50 |
+
39 in LP relaxation solvers for ILPs. We argue that due to the difficulties in putting GPUs and ML to
|
| 51 |
+
40 work in traditional solver methodologies, investigation of new paradigms is called for. To this end we
|
| 52 |
+
41 build upon the recent work of [1] which proposed a massively parallel GPU friendly solver for 0-1
|
| 53 |
+
42 integer linear programming using Lagrange decomposition. The solver exhibits faster performance
|
| 54 |
+
43 than traditional CPU solvers on large-scale problems making good use of GPU parallelism. Also
|
| 55 |
+
44 due to its comparatively simple control flow and its usage of simple arithmetic operations for all its
|
| 56 |
+
45 operations it can be made differentiable. This allows to train its parameters and predict update steps
|
| 57 |
+
46 that will allow for faster convergence and overcoming fixed points from which the basic version of the
|
| 58 |
+
47 algorithm suffers. This results in superior performance as compared to the non-learned version [1].
|
| 59 |
+
48 We obtain small gaps to (I)LP optima on a diverse range of large scale structured prediction problems,
|
| 60 |
+
49 QAPLib [8] and independent set problems [39]. We are up to an order of magnitude faster than
|
| 61 |
+
50 traditional ILP solvers.
|
| 62 |
+
51 Contributions We propose to learn the Lagrange decomposition based algorithm [1] for solving LP
|
| 63 |
+
52 relaxations of ILP problems and show its benefits. In particular,
|
| 64 |
+
|
| 65 |
+
• We make the dual update steps of [1] differentiable. This allows us to predict parameters of the update steps so that faster convergence is achieved as compared to using hand-picked values.
|
| 66 |
+
• We train a predictor for arbitrary non-parametric update steps that allow to escape suboptimal fixed points into which the parametric update steps of [1] can fall.
|
| 67 |
+
• We propose to train predictors for both the parametric and non-parametric updates in fully unsupervised manner. Our loss optimizes for parameters/update steps producing large improvements in the dual lower bound over a long time horizon.
|
| 68 |
+
• We show the benefits of our learned massively parallel GPU approach on a wide range of problems. We have chosen structured prediction tasks including graph matching [29] and cell tracking [24]. From theoretical computer science we compare on the QAPLib [8] dataset and randomly generated independent set problems [39].
|
| 69 |
+
|
| 70 |
+
# 2 Related Work
|
| 71 |
+
|
| 72 |
+
# 2.1 Learning to solve Combinatorial Optimization
|
| 73 |
+
|
| 74 |
+
66 ML has been used to improve various aspects of solving combinatorial problems. For the standard
|
| 75 |
+
67 branch-and-cut ILP solvers the works [19, 22, 35] learn variable selection for branching. The
|
| 76 |
+
68 approaches [14, 35] learn to fix a subset of integer variables in ILPs to their hopefully optimal values
|
| 77 |
+
69 to improve finding high quality primal solutions. The works [43, 54] learn variable selection for
|
| 78 |
+
70 the large neighborhood search heuristic for obtaining primal solutions to ILPs. Selecting good cuts
|
| 79 |
+
71 through scoring them with neural networks was investigated in [26, 46]. While all these approaches
|
| 80 |
+
72 result in runtime and solution quality improvements, only a few works tackle the important task of
|
| 81 |
+
73 speeding up ILP relaxations by ML. Specifically, the work [11] used graph neural network (GNN) to
|
| 82 |
+
74 predict variable orderings of decision diagrams representing combinatorial optimization problems.
|
| 83 |
+
75 The goal is to obtain an ordering such that a corresponding dual lower bound is maximal. To our
|
| 84 |
+
76 knowledge it is the only work that addresses computing ILP relaxations with ML. For constraint
|
| 85 |
+
77 satisfaction problems [40, 9, 47] train GNN while [47] train in an unsupervised manner. For narrow
|
| 86 |
+
78 subclasses of problems primal heuristics have been augmented through learning some of their
|
| 87 |
+
79 decisions, e.g. for capacitated vehicle routing [36] and traveling salesman [55]. For a more complete
|
| 88 |
+
80 overview of ML for combinatorial optimization we refer to the detailed surveys [6, 10].
|
| 89 |
+
|
| 90 |
+
# 81 2.2 Massively parallel combinatorial optimization
|
| 91 |
+
|
| 92 |
+
82 Massively parallel algorithms running on GPU have been proposed for narrow problem classes,
|
| 93 |
+
83 including inference in [41, 56] and dense [45] Markov Random Fields, multicut [2] and for max
|
| 94 |
+
84 flow [49, 53]. The algorithm [1] on which our work is based is, to our knowledge, the only generic
|
| 95 |
+
85 ILP solver that can make adequate use of parallelism offered by GPUs.
|
| 96 |
+
|
| 97 |
+
# 86 2.3 Unrolling algorithms for parameter learning
|
| 98 |
+
|
| 99 |
+
87 Algorithms containing differentiable iterative procedures are combined with neural networks for
|
| 100 |
+
88 improving performance of such algorithms. One of the earliest works in this direction is [21]
|
| 101 |
+
89 which embedded sparse coding algorithms in a neural network by unrolling. For solving inverse
|
| 102 |
+
90 problems [57, 12] unroll through ADMM and non-linear diffusion resp. Overall, such approaches
|
| 103 |
+
91 show more generalization power than pure neural networks based ones as shown in the survey [34].
|
| 104 |
+
92 Slightly different than from the above works, neural networks were used to predict update directions
|
| 105 |
+
93 for training other neural networks (e.g. in [3]).
|
| 106 |
+
|
| 107 |
+
# 94 3 Method
|
| 108 |
+
|
| 109 |
+
We first recapitulate the Lagrange decomposition approach to binary ILPs from [31] and the deferred min-marginal averaging scheme for its solution proposed in [1]. We highlight possible parameters of the update steps which we will predict by training a graph neural network. Proofs are in the Appendix.
|
| 110 |
+
|
| 111 |
+
# 99 3.1 Lagrange Decomposition & Deferred Min-Marginal Averaging
|
| 112 |
+
|
| 113 |
+
Definition 1 (Binary Program [31]). Let a linear objective $c \in \mathbb { R } ^ { n }$ and $m$ variable subsets $\mathcal { T } _ { j } \subset [ n ]$ of constraints with feasible set $\mathcal { X } _ { j } \subset \{ 0 , 1 \} ^ { \mathbb { Z } _ { j } }$ for $j \in [ m ]$ be given. The corresponding binary program is
|
| 114 |
+
|
| 115 |
+
$$
|
| 116 |
+
\operatorname* { m i n } _ { x \in \{ 0 , 1 \} ^ { n } } \langle c , x \rangle \quad { \mathrm { s . t . } } \quad x _ { { \bar { \mathcal { T } } } _ { j } } \in { \mathcal { X } } _ { j } \quad \forall j \in [ m ] ,
|
| 117 |
+
$$
|
| 118 |
+
|
| 119 |
+
where $x _ { \mathbb { Z } _ { j } }$ is the restriction to variables in $\mathcal { T } _ { j }$ .
|
| 120 |
+
|
| 121 |
+
Any binary ILP $\mathrm { m i n } _ { x \in \{ 0 , 1 \} ^ { n } } \langle c , x \rangle$ s.t. $A x \leq b$ where $A \in \mathbb { R } ^ { m \times n }$ can be written as (BP) by associating each constraint $a _ { j } ^ { T } x \leq b _ { j }$ for $j \in [ m ]$ with its own subproblem $\mathcal { X } _ { j }$ .
|
| 122 |
+
|
| 123 |
+
06 In order to obtain a problem formulation amenable for parallel optimization we consider its Lagrange
|
| 124 |
+
07 dual which decomposes the full problem (BP) into a series of coupled subproblems.
|
| 125 |
+
108 Definition 2 (Lagrangean dual problem [31]). Define the set of subproblems that constrain variable $i$
|
| 126 |
+
109 as $\mathcal { T } _ { i } = \{ j \in [ \bar { m ] } \ | \ \bar { i } \in \mathcal { T } _ { j } \}$ . Let the energy for subproblem $j \in [ m ]$ w.r.t. Lagrangean dual variables
|
| 127 |
+
110 $\lambda _ { \bullet j } = ( \lambda _ { i j } ) _ { i \in \mathcal { T } _ { j } } \in \mathbb { R } ^ { \mathcal { T } _ { j } }$ be
|
| 128 |
+
|
| 129 |
+
$$
|
| 130 |
+
E ^ { j } ( \lambda _ { \bullet j } ) = \operatorname* { m i n } _ { x \in \mathcal { X } _ { j } } \langle \lambda _ { \bullet j } , x \rangle .
|
| 131 |
+
$$
|
| 132 |
+
|
| 133 |
+
111 Then the Lagrangean dual problem is defined as
|
| 134 |
+
|
| 135 |
+
$$
|
| 136 |
+
\operatorname* { m a x } _ { \lambda } \quad \sum _ { j \in [ m ] } E ^ { j } ( \lambda _ { \bullet j } ) \quad \mathrm { s . t . } \quad \sum _ { j \in \mathcal { I } _ { i } } \lambda _ { i j } = c _ { i } \quad \forall i \in [ n ] .
|
| 137 |
+
$$
|
| 138 |
+
|
| 139 |
+
112 The authors in [1] have proposed a parallelization friendly iterative algorithm for updating Lagrange
|
| 140 |
+
113 multipliers $\lambda$ for maximizing (D), see Algorithm 1. We write it in a slightly adapted form since it
|
| 141 |
+
114 will allow us to easily describe its backpropagation. The algorithm assigns the Lagrange variables
|
| 142 |
+
115 in $u$ -many disjoint blocks $B _ { 1 } , \ldots , B _ { u }$ in such a way that each block contains at most one Lagrange
|
| 143 |
+
116 variable from each subproblem and all variables within a block are updated in parallel. The dual update
|
| 144 |
+
117 scheme relies on computing min-marginal differences i.e., the difference of subproblem objectives
|
| 145 |
+
118 when a certain variable is set to 1 minus its objective when the same variable is set to 0, see line 10
|
| 146 |
+
119 in Algorithm 1. These min-marginal differences are averaged out across subproblems via updates
|
| 147 |
+
120 to Lagrange variables in line 11 in Algorithm 1. The crucial ingredient allowing parallelization is
|
| 148 |
+
121 that in the min-marginal averaging step values from the last iteration are used (i.e. $M ^ { \mathrm { i n } }$ ), making
|
| 149 |
+
122 synchronization between subproblems unnecessary.
|
| 150 |
+
123 In [1] the min-marginal averaging parameters of Algorithm 1 were set as $\omega = 0 . 5$ and $\alpha _ { i j } =$
|
| 151 |
+
124 $1 / | \mathcal { I } _ { i } |$ leading to uniform averaging. We generalize the min-marginal update step by considering
|
| 152 |
+
125 more general parametric update steps. We allow $\omega \in ( 0 , 1 )$ and $\alpha$ -values to be arbitrary convex
|
| 153 |
+
126 combinations. In the next section we will show how to train these values to achieve faster convergence.
|
| 154 |
+
127 Proposition 1 (Dual Feasibility and Monotonicity of Min-marginal Averaging). For any $\alpha _ { i j } \geq 0$
|
| 155 |
+
128 with $\begin{array} { r } { \sum _ { j \in \mathcal { T } _ { i } } \alpha _ { i j } = 1 } \end{array}$ and $\omega _ { i j } \in [ 0 , 1 ]$ the min-marginal averaging step in line $1 l$ in Algorithm $^ { l }$
|
| 156 |
+
129 retains dual feasibility and is non-decreasing in the dual lower bound.
|
| 157 |
+
|
| 158 |
+
Input: Lagrange variables $\lambda _ { i j } \forall i \in [ n ] , j \in \mathcal { T } _ { i }$ , damping factors $\omega _ { i j } \in ( 0 , 1 ) \forall i \in [ n ] , j \in \mathcal { T } _ { i }$ , anisotropic min-marginal averaging weights $\alpha _ { i j } \in ( 0 , 1 ) \forall i \in [ n ] , j \in \mathcal { T } _ { i }$ , max. number of iterations $T$ . 1 Initialize deferred min-marginal diff. $M = \mathbb { 0 }$ 2 for $T$ iterations do 3 for block $B \in ( B _ { 1 } , \ldots B _ { u } )$ do 4 $\lambda , M \gets$ BlockUpdate $( B , \lambda , M , \alpha , \omega )$ 5 for block $B \in ( B _ { u } , \ldots B _ { 1 } )$ do 6 λ, M ← BlockUpdate $( B , \lambda , M , \alpha , \omega )$ 7 return λ, M 8 Procedure BlockUpdate $( B , \lambda ^ { \mathrm { i n } } , M ^ { \mathrm { i n } } , \alpha , \omega )$ 9 for $i j \in B$ in parallel do 10 CompuUpdate $\begin{array} { r } { \lambda _ { i j } ^ { \mathrm { o u t } } = \lambda _ { i j } ^ { \mathrm { i n } } - M _ { i j } ^ { \mathrm { o u t } } + \alpha _ { i j } \sum _ { k \in \mathcal { I } _ { i } } M _ { i k } ^ { \mathrm { i n } } } \end{array}$ $\begin{array} { r } { M _ { i j } ^ { \mathrm { { o u t } } } = \omega _ { i j } [ \operatorname* { m i n } _ { x \in \mathcal { X } _ { j } : x _ { i } = 1 } \langle \lambda _ { \bullet j } ^ { \mathrm { { i n } } } , x \rangle - \operatorname* { m i n } _ { x \in \mathcal { X } _ { j } : x _ { i } = 0 } \langle \lambda _ { \bullet j } ^ { \mathrm { { i n } } } , x \rangle ] } \end{array}$ 12 return $\lambda ^ { \mathsf { o u t } }$ , M out
|
| 159 |
+
|
| 160 |
+
# 130 3.2 Backpropagation through Deferred Min-Marginal Averaging
|
| 161 |
+
|
| 162 |
+
131 We show below how to differentiate through Algorithm 1 with respect to the parameters $\alpha$ and $\omega$ .
|
| 163 |
+
132 This will ultimately allow us to learn these parameters such that faster convergence is achieved. To
|
| 164 |
+
133 this end we describe backpropagation for a block update (lines 8- 12) of Alg. 1. All other operations
|
| 165 |
+
134 can be tackled by automatic differentiation. For a block $B$ in $\{ B _ { 1 } , \ldots , B _ { u } \}$ we view the Lagrangean
|
| 166 |
+
135 update as a mapping $\mathcal { H } : ( \mathbb { R } ^ { | B | } ) ^ { 4 } \to ( \mathbb { R } ^ { | B | } ) ^ { 2 }$ , $( \lambda ^ { \mathrm { i n } } , M ^ { \mathrm { i n } } , \alpha , \omega ) \mapsto ( \lambda ^ { \circ \mathrm { u t } } , M ^ { \circ \mathrm { u t } } )$ .
|
| 167 |
+
|
| 168 |
+
Given a loss function 36 $\mathcal { L } : \mathbb { R } ^ { N } \mathbb { R }$ we denote $\partial \mathcal { L } / \partial x$ by $\dot { x }$ . Algorithm 2 shows backpropagation through 7 $\mathcal { H }$ to compute the gradients $\dot { \lambda } ^ { \mathrm { { i n } } } , \dot { M } ^ { \mathrm { { i n } } } ,$ $\dot { \alpha }$ and $\dot { \omega }$ .
|
| 169 |
+
|
| 170 |
+
138 Proposition 2. Algorithm 2 performs backpropagation through $\mathcal { H }$ .
|
| 171 |
+
|
| 172 |
+
139 Efficient Implementation Generally, the naive computation of min-marginal differences and its
|
| 173 |
+
140 backpropagation are both expensive operations as they require solving two optimization problems
|
| 174 |
+
141 for each dual variable. In [1, 31] the authors represented each subproblem using binary decision
|
| 175 |
+
142 diagrams (BDDs) for fast incremental computation of min-marginal differences. Their algorithm
|
| 176 |
+
143 results in a computation graph involving only elementary arithmetic operations and taking minima
|
| 177 |
+
144 over several variables. Using this computational graph we can implement the abstract Algorithm 2
|
| 178 |
+
145 efficiently and parallelize on GPU. For details we refer to the Appendix.
|
| 179 |
+
|
| 180 |
+
# Algorithm 2: BlockUpdate backpropagation
|
| 181 |
+
|
| 182 |
+
Input: Forward pass inputs: $B , \lambda ^ { \mathrm { i n } } , M ^ { \mathrm { i n } } , \alpha , \omega$ , gradients of forward pass output: λ˙ out, $\dot { M } ^ { \mathrm { { o u t } } }$ , gradients of parameters $\dot { \alpha } , \dot { \omega }$
|
| 183 |
+
|
| 184 |
+
1 for $i j \in B$ in parallel do
|
| 185 |
+
|
| 186 |
+
2 $\begin{array} { r } { \dot { M } _ { i j } ^ { \mathrm { i n } } = \sum _ { k \in \mathcal { T } _ { i } } \dot { \lambda } _ { i k } ^ { \mathrm { o u t } } \alpha _ { i k } } \end{array}$ , $\dot { M } _ { i j } ^ { \mathrm { o u t } } = \dot { M } _ { i j } ^ { \mathrm { o u t } } - \dot { \lambda } _ { i j } ^ { \mathrm { o u t } }$
|
| 187 |
+
3 $\begin{array} { r } { \dot { \alpha } _ { i j } = \dot { \alpha } _ { i j } + \dot { \lambda } _ { i j } \sum _ { k \in \mathcal { I } _ { i } } M _ { i k } ^ { \mathrm { i n } } , \quad \dot { \omega } _ { i j } = \dot { \omega } _ { i j } + \dot { M } _ { i j } ^ { \mathrm { o u t } } [ M _ { i j } ^ { \mathrm { o u t } } / \omega _ { i j } ] } \end{array}$
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4 Compute minimizers $\begin{array} { r } { s ^ { j } ( i , \beta ) = \arg \operatorname* { m i n } _ { x \in \mathcal { X } _ { j } : x _ { i } = \beta } \langle \lambda _ { \bullet j } ^ { \mathrm { { i n } } } , x \rangle , \forall \beta \in \{ 0 , 1 \} } \end{array}$ }
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5 $\dot { \lambda } _ { p j } ^ { \mathrm { i n } } = \dot { \lambda } _ { p j } ^ { \circ \mathrm { u t } } + \dot { M } _ { i j } ^ { \circ \mathrm { u t } } \omega _ { i j } [ s _ { p } ^ { j } ( i , 1 ) - s _ { p } ^ { j } ( i , 0 ) ] ,$ , ∀p ∈ Ij
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6 return $\dot { \lambda } ^ { \mathrm { i n } } , \dot { M } ^ { \mathrm { i n } } , \dot { \alpha } , \dot { \omega }$
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# 146 3.3 Non-Parametric Update Steps
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147 Although the min-marginal averaging scheme of Alg. 1 guarantees non-decreasing lower bound, it
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148 can get stuck in suboptimal fixed points, see [50] for a discussion for the special case of MAP-MRF.
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149 To alleviate this shortcoming we allow arbitrary updates to Lagrange variables through a vector
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150 $\boldsymbol { \theta } \in \mathbb { R } ^ { | \lambda | }$ as
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$$
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\lambda _ { i j } \lambda _ { i j } + \theta _ { i j } - \frac { 1 } { | \mathcal { T } _ { i } | } \sum _ { k \in \mathcal { I } _ { i } } \theta _ { i k } , \forall i \in [ n ] , j \in \mathcal { I } _ { i }
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$$
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151 where the last term ensures feasibility of updated Lagrange variables w.r.t. the dual problem (D).
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# 3.4 Graph neural network
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153 We train a graph neural network (GNN) to predict the parameters $\alpha , \omega$ of Alg. 1 and also the
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154 non-parametric update $\theta$ for (2). To this end we encode the dual problem (D) on a bipartite graph
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155 $\mathcal { G } = ( \nu , \mathcal { E } )$ . Its nodes correspond to primal variables $\mathcal { T }$ and subproblems $\mathcal { I }$ i.e., $\mathcal { V } = \mathcal { I } \cup \mathcal { I }$ and
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156 edges $\mathcal { E } = \{ i j \mid i \in \mathcal { T } , j \in \mathcal { T } _ { i } \}$ correspond to Lagrange multipliers. We need to predict values of
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157 $\alpha _ { i j } , \omega _ { i j }$ and $\theta _ { i j }$ for each edge $i j$ in $\mathcal { E }$ . We associate features $\boldsymbol { f } \overset { - } { = } \left( f _ { \mathcal { T } } , f _ { \mathcal { T } } , f _ { \mathcal { E } } \right)$ with each entity of the
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158 graph which capture the current state of Alg. 1. Additionally, we encode a number of quantities as
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159 features which can make learning easier. For example, a history of previous dual objectives for each
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160 subproblem is encoded in the constraint nodes and minimizers of each subproblem (which correspond
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161 to a subgradient of the dual problem (D)) are encoded in the edge features $f _ { \mathcal { E } }$ . A complete list of
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162 features is provided in the Appendix.
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163 Message passing To perform message passing we use the transformer based graph convolution
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164 scheme of [42]. We first compute an embedding of all subproblems $j$ in $\mathcal { I }$ by receiving messages
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165 from adjacent nodes and edges as
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$$
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\mathsf { C O N V } _ { \mathcal { I } } ( f _ { \mathcal { I } } , f _ { \mathcal { I } } , f _ { \mathcal { E } } , \mathcal { E } ) _ { j } = \mathbf { W _ { s } } f _ { j } + \sum _ { i | i j \in \mathcal { E } } a _ { i j } ( f _ { j } , f _ { \mathcal { I } } , f _ { \mathcal { E } } ; \mathbf { W _ { a } } ) \left[ \mathbf { W _ { t } } f _ { i } + \mathbf { W _ { e } } f _ { i j } \right] ,
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$$
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166 where $\mathbf { W } = \left( \mathbf { W _ { a } } , \mathbf { W _ { s } } , \mathbf { W _ { t } } , \mathbf { W _ { e } } \right)$ are trainable parameters and $a _ { i j } \left( f _ { j } , f _ { \mathbb { Z } } , f _ { \mathbb { \varepsilon } } ; \mathbf { W _ { a } } \right)$ is the softmax
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167 attention weight between nodes $i$ and $j$ parameterized by $\mathbf { W _ { a } }$ . Afterwards we perform message
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168 passing in the reverse direction to compute embeddings for primal variables $\mathcal { T }$ . Similar strategy for
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169 message passing on a bipartite graph was followed by [19].
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170 Recurrent connections Our default GNN as mentioned above only uses hand-crafted features
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171 to maintain a history of previous optimization rounds. To learn a summary of the past updates we
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172 optionally allow recurrent connections through an LSTM with forget gate [20]. The LSTM is only
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173 applied on primal variable nodes $\mathcal { T }$ and maintains cell states $s \tau$ which can be updated and used for
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174 parameter prediction in subsequent optimization rounds.
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175 Prediction The learned embeddings from GNN, LSTM outputs and solver features from Alg. 1
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176 are consumed by a multi-layer perceptron $\Phi$ to predict the required variables for each edge $i j$ in $\mathcal { E }$ .
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177 Afterwards we transform these outputs so that they satisfy Prop. 1.
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178 The exact sequence of operations performed by the graph neural network are shown in Alg. 3 where
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179 $[ u _ { 1 } , \ldots , u _ { k } ]$ denotes concatenation of vectors $u _ { 1 } , \ldots , u _ { k }$ , LN denotes layer normalization [5] and
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180 $\mathtt { L S T M } _ { \mathcal { T } }$ stands for an LSTM cell which operates on each primal variable node.
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+
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# Algorithm 3: Parameter prediction by GNN
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Input: Primal variable features $f _ { \mathcal { T } }$ and cell states $s \tau$ , Subproblem features $f _ { \mathcal { I } }$ , Dual variable (edge) features $f _ { \mathcal { E } }$ , Set of edges $\mathcal { E }$ . 1 $h _ { \mathcal { T } } = \mathsf { R e L U } \left( \operatorname { L N } \left( \operatorname { C O N V } _ { \mathcal { T } } \left( f _ { \mathcal { T } } , f _ { \mathcal { T } } , f _ { \mathcal { E } } , \mathcal { E } \right) \right) \right)$ // Compute subproblems embeddings 2 $h _ { \mathcal { T } } = \mathtt { R e L U }$ $\left( \mathrm { L N } \left( \mathrm { C O N V } _ { \mathcal { T } } \left( f _ { \mathcal { T } } , [ f _ { \mathcal { T } } , h _ { \mathcal { T } } ] , f _ { \mathcal { E } } , \mathcal { E } \right) \right) \right)$ // Compute primal variable embeddings 3 $z _ { \mathcal { T } } , s _ { \mathcal { T } } = \mathtt { L S T M } _ { \mathcal { T } } ( h _ { \mathbb { Z } } , s _ { \mathcal { T } } )$ // Compute output and cell state 4 $( \hat { \alpha } , \hat { \omega } , \theta ) = \Phi \left( [ f _ { \mathcal { T } } , h _ { \mathcal { T } } , z _ { \mathcal { T } } ] , [ f _ { \mathcal { T } } , h _ { \mathcal { T } } ] , f _ { \mathcal { E } } , \mathcal { E } \right)$ // Prediction per edge 5 $\alpha _ { i \bullet } = \mathtt { S o f t m a x } ( \hat { \alpha } _ { i \bullet } )$ , $\forall i \in \mathcal { T }$ , $\omega = \mathtt { S i g m o i d } ( \hat { \omega } )$ // Ensure non-decreasing obj., Prop. 1 6 return $\alpha , \omega , \theta , s _ { \mathcal { T } }$
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+
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| 246 |
+
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Figure 1: Our pipeline for optimizing the Lagrangean dual (D). The problem is encoded on a bipartite graph containing features $f _ { \mathcal { T } }$ , $f _ { \mathcal { I } }$ and $f _ { \mathcal { E } }$ for primal variables, subproblems and dual variables resp. A graph neural network (GNN) predicts the non-parameteric update $\theta$ (2) and parameters $\alpha$ and $\omega$ for Alg. 1. In one optimization round current set of Lagrange multipliers $\lambda$ are first updated by the non-parametric update using $\theta$ . Afterwards deferred min-marginal averaging is performed parameterized by $\alpha$ and $\omega$ . The updated solver features $f$ (which also includes $\lambda$ ) and LSTM cell states $s \tau$ are sent to the GNN in next optimization round. These rounds are repeated at most $R$ -times during training and until convergence during inference.
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+
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# 181 3.5 Loss
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+
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82 Given the Lagrange variables $\lambda$ we directly use the dual objective (D) as an unsupervised loss to train
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183 the GNN. Thus, we maximize the loss $L$ defined as
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+
|
| 254 |
+
$$
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+
\mathcal { L } ( \lambda ) = \sum _ { j \in [ m ] } E ^ { j } ( \lambda _ { \bullet j } ) .
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+
$$
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| 257 |
+
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+
184 For a mini-batch of instances during training we take the mean of corresponding per-instance losses.
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185 For backpropagation, gradient of loss $\mathcal { L }$ w.r.t. Lagrange variables of a subproblem $j$ is computed by
|
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+
186 finding a minimizing assignment for that subproblem, written as
|
| 261 |
+
|
| 262 |
+
$$
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+
\left( \frac { \partial \mathcal { L } } { \partial \lambda } \right) _ { \bullet j } = \operatorname { a r g m i n } _ { x \in \mathcal { X } _ { j } } \langle \lambda _ { \bullet j } , x \rangle \in \{ 0 , 1 \} ^ { \mathcal { Z } _ { j } } .
|
| 264 |
+
$$
|
| 265 |
+
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+
187 The above gradient is then sent as input for backpropagation. For computing the minimizing
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188 assignment efficiently we use binary decision diagram representation of each subproblem as in [1, 31].
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+
|
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+
# 3.6 Overall pipeline
|
| 270 |
+
|
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+
190 Our overall pipeline combining all building blocks from the previous sections is shown in Figure 1.
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191 We train our pipeline which contains multiple dual optimization rounds in a fashion similar to that
|
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+
192 of recurrent neural networks. One round of our dual optimization consists of message passing
|
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+
193 by GNN, a non-parametric update step and $T$ iterations of deferred min-marginal averaging. For
|
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194 computational efficiency we run our pipeline for at most $R$ dual optimization rounds during training.
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195 On each mini-batch we randomly sample a number of optimization rounds $r$ in $[ R ]$ , run $r - 1$ rounds
|
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196 without tracking gradients and backpropagate through the last round by computing the loss (4). For
|
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197 the pipeline with recurrent connections we backpropagate through last 3 rounds and apply the loss
|
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+
198 after each of these rounds. Since the task of dual optimization is relatively easier in early rounds
|
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199 as compared to later ones (where [1] can get stuck) we use two neural networks. The early stage
|
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+
200 network is trained if the randomly sampled $r$ is in $[ 0 , R / 2 ]$ and the late stage network is chosen
|
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+
201 otherwise. During testing we switch to the later stage network when the relative improvement in the
|
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+
202 dual objective by the early stage network becomes less than $1 0 ^ { - 6 }$ .
|
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+
|
| 285 |
+
# 03 4 Experiments
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+
|
| 287 |
+
204 As main evaluation metric we report convergence plots of the relative dual gap $g ( t ) \in [ 0 , 1 ]$ at time $t$
|
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+
|
| 289 |
+
$$
|
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+
g ( t ) = \operatorname* { m i n } \left( \frac { d ^ { * } - d ( t ) } { d ^ { * } - d _ { i n i t } } , 1 . 0 \right)
|
| 291 |
+
$$
|
| 292 |
+
|
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+
205 where $d ( t )$ is the dual objective at time $t$ , $d ^ { * }$ is the optimal (or best known) objective value of the
|
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+
206 Lagrange relaxation $( \mathrm { D } )$ and $d _ { i n i t }$ is the objective value before optimization as computed by [1].
|
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+
207 Additionally we also report per dataset averages of relative dual gap integral $\begin{array} { r } { g _ { I } = \int g ( \bar { t } ) d t } \end{array}$ [7], best
|
| 296 |
+
208 objective value $( E )$ and time taken $\mathbf { \rho } ( t )$ to obtain best objective. To cater the dominating effect of
|
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+
209 worse initial lower bounds on $g _ { I }$ (as $g ( t )$ can be close to 1 at $t \approx 0$ ) we start calculating $g \tau$ after a few
|
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+
210 rounds of our solver are completed. This start time is then also used to evaluate other algorithms for a
|
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211 fair comparison. To evaluate CPU solvers we use an AMD EPYC 7702 CPU. For the GPU solvers
|
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+
212 we use either one NVIDIA RTX 8000 (48GB) or A100 (80GB) GPU depending on instance size.
|
| 301 |
+
|
| 302 |
+
# 4.1 Algorithms
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| 303 |
+
|
| 304 |
+
Gurobi: Results of the dual simplex algorithm from the commercial ILP solver [23].
|
| 305 |
+
|
| 306 |
+
FastDOG: The non-learned baseline [1] of Alg. 1 with $\omega _ { i j } = 0 . 5$ and $\alpha _ { i j } = 1 / | \mathcal { I } _ { i } |$
|
| 307 |
+
|
| 308 |
+
DOGE: Our approach where we learn to predict parametric and non-parametric updates by using two graph neural networks for early and late-stage optimization. Size of the learned embeddings $h$ computed by the GNN in Alg. 3 is set to 16 for nodes and 8 for edges. For computing attention weights in (3) we use only one attention head for efficiency. The predictor $\Phi$ in Alg. 3 contains 4 linear layers with the ReLU activation. We train the networks using the Adam optimizer [30]. To prevent gradient overflow we use gradient clipping on model parameters by an $l ^ { 2 }$ norm of 50. The number of trainable parameters is $8 k$ .
|
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+
|
| 310 |
+
DOGE-M: Variant of our method where we additionally use recurrent connections using LSTM. The cell state vector $s _ { i }$ for each primal variable node $i \in \mathcal { T }$ has a size of 16. The number of trainable parameters is $1 2 k$ .
|
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+
|
| 312 |
+
We have not tested against specialized heuristics for our benchmark problems since [1] has shown them to be on par or outperformed by FastDOG. For training our approach we use the frameworks [15, 16, 38] and implement the Algorithms 1,2 in CUDA [37] using [25, 28].
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| 313 |
+
|
| 314 |
+
# 4.2 Datasets
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| 315 |
+
|
| 316 |
+
Cell tracking $( C T )$ : Instances of developing flywing tissue from cell tracking challenge [48] processed by [24] and obtained from [44]. We use the largest and hardest 3 instances, train on the 2 smaller instances and test on the largest one.
|
| 317 |
+
|
| 318 |
+
Graph matching (GM): Instances of graph matching for matching nuclei in 3D microscopic images [32] processed by [29] and made publicly available through [44]. We train on 10 instances and test on the remaining 20 instances.
|
| 319 |
+
|
| 320 |
+
Independent set (IS): Random instances of independent set problem generated using [39]. For training we generate 240 instances with $1 0 k$ vertices each and test on 60 instances with $5 0 k$ vertices. We generating edges between vertices in the graph with a probability of 0.25.
|
| 321 |
+
|
| 322 |
+
QAPLib: The benchmark dataset for quadratic assignment problems used in the combinatorial optimization community [8]. We train on 61 instances having up to 40 nodes and test on 35 instances having up to 70 nodes.
|
| 323 |
+
|
| 324 |
+
242 For each dataset we use a separate set of hyperparameters due to varying instance sizes given in
|
| 325 |
+
243 Table 1. All our test datasets on average contain more than a million edges (i.e., Lagrange variables)
|
| 326 |
+
244 while training instances are considerably smaller. For efficiency, during evaluation we use a larger
|
| 327 |
+
245 value of $T$ in Alg. 1 than during training. For the $C T$ dataset containing we learn only the non
|
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+
246 parametric update steps (2) and fix the parameters in Alg. 1 to their default values from [1]. Learning
|
| 329 |
+
247 these parameters gave slightly worse training loss at convergence.
|
| 330 |
+
|
| 331 |
+
# 48 4.3 Ablation study
|
| 332 |
+
|
| 333 |
+
249 We perform an ablation study to test the importance of various components of our approach. Starting
|
| 334 |
+
250 from [1] as a baseline we first predict all parameters $\alpha , \omega , \theta$ through the two multi-layer perceptrons $\Phi$
|
| 335 |
+
251 for early and late stage optimization without using GNN. Next, we report results of using one network
|
| 336 |
+
252 (instead of two) which is trained and tested for both early and later rounds of dual optimization. Lastly,
|
| 337 |
+
253 we aim to seek the importance of learning parameters of Alg. 2 and the non-parametric update (2).
|
| 338 |
+
254 To this end, we learn to predict only the non-parametric update and apply the loss directly on updated
|
| 339 |
+
255 $\lambda$ without requiring backpropagation through Alg. 1. We also try learning a subset of parameters i.e.,
|
| 340 |
+
256 not predicting averaging weights $\alpha$ or damping factors $\omega$ . Lastly, we report results of DOGE-M which
|
| 341 |
+
257 uses recurrent connections. The results are in Table 2.
|
| 342 |
+
|
| 343 |
+
Table 1: Hyperparameters of our approach and dataset statistics. $| \mathcal { T } | + | \mathcal { I } |$ : Average number of variables and constraints in each dataset (# vertices in GNN); $\textstyle \sum _ { j = 1 } ^ { m } | { \dot { \mathcal { I } } } _ { i } |$ : Average number of Lagrange multipliers (# edges in GNN); $T$ : Number of iterations of Alg. 1 in each optimization round; $R$ : max. number of training rounds; # itr. train: Number of training iterations.
|
| 344 |
+
|
| 345 |
+
<table><tr><td rowspan="2">Dataset</td><td colspan="2">|Z|+|J|(×106)</td><td colspan="2"></td><td colspan="2">T</td><td rowspan="2">R</td><td rowspan="2">batch size</td><td rowspan="2">learn. rate</td><td rowspan="2">#itr. train</td><td rowspan="2">train time [hrs]</td></tr><tr><td>train</td><td>test</td><td>train</td><td>test</td><td>train</td><td>test</td></tr><tr><td>CT</td><td>3.7</td><td>12.4</td><td>8.5</td><td>28</td><td>1</td><td>100</td><td>400</td><td>1</td><td>1e-3</td><td>500</td><td>14</td></tr><tr><td>GM</td><td>1.7</td><td>1.7</td><td>3.3</td><td>3.3</td><td>20</td><td>200</td><td>20</td><td>2</td><td>1e-3</td><td>400</td><td>4</td></tr><tr><td>IS</td><td>0.05</td><td>0.4</td><td>0.1</td><td>1.2</td><td>20</td><td>50</td><td>20</td><td>8</td><td>1e-3</td><td>2500</td><td>10</td></tr><tr><td>QAPLib</td><td>0.1</td><td>2.8</td><td>0.5</td><td>11</td><td>5</td><td>20</td><td>500</td><td>4</td><td>1e-3</td><td>1600</td><td>48</td></tr></table>
|
| 346 |
+
|
| 347 |
+
Table 2: Ablation study results on the Graph matching dataset. w/o GNN: Use only the two predictors $\Phi$ without GNN for early and late stage optimization; same network: use one network (GNN, $\Phi$ ) for both early and late stage; only non-param., param.: predict only the non-parametric update (2) or the parametric update (Alg. 1); w/o α, ω: does not predict $\alpha$ or $\omega$ resp.
|
| 348 |
+
|
| 349 |
+
<table><tr><td></td><td>w/o learn. ([1])</td><td>w/o GNN</td><td>same network</td><td>only non-param.</td><td>only param.</td><td>w/oα</td><td>w/ow</td><td>DOGE</td><td>DOGE-M</td></tr><tr><td>g1 (）</td><td>21</td><td>0.42</td><td>0.95</td><td>2.3</td><td>0.7</td><td>0.36</td><td>0.35</td><td>0.33</td><td>0.19</td></tr><tr><td>E(1）</td><td>-48912</td><td>-48440</td><td>-48444</td><td>-48476</td><td>-48444</td><td>-48439</td><td>-48439</td><td>-48439</td><td>-48436</td></tr><tr><td>t[s](）</td><td>61</td><td>29</td><td>24</td><td>51</td><td>74</td><td>30</td><td>30</td><td>17</td><td>21</td></tr></table>
|
| 350 |
+
|
| 351 |
+
Firstly, from our ablation study we observe that learning even one of the two types of updates i.e., non-parametric or parametric already gives better results than the non-learned solver [1]. This is because non-parametric update can help in escaping fixed-points of [1] when they occur and the parametric update can help Alg. 1 in avoiding such fixed-points. Combining both of these strategies further improves the results. Secondly, we observe that performing message passing with GNN gives improvement over only using the predictor $\Phi$ . Thirdly, we find using separate networks for early and late stage optimization gives better performance than using the same network for all stages. Lastly, using recurrent connections gives the best performance.
|
| 352 |
+
|
| 353 |
+
# 4.4 Results
|
| 354 |
+
|
| 355 |
+
Convergence plots of relative dual gaps change w.r.t. wall clock times are given in Figure 2. Rest of the evaluation metrics are reported in Table 3. For further details we refer to the Appendix.
|
| 356 |
+
|
| 357 |
+
Discussion As compared to the non-learned baseline FastDOG we reach an order of magnitude more accurate relaxation solutions, almost closing the gap to optimum as computed by Gurobi. We retain high speed afforded by exploiting GPU parallelism. Interestingly, we can often outperform FastDOG also in the early stage where optimization is easy. Our LSTM version DOGE-M has shown improved performance than the non-LSTM version. Especially it shows much improvement on the most difficult QAPLib dataset. On QAPLib Gurobi does not converge on instances with more than
|
| 358 |
+
|
| 359 |
+
Table 3: Results comparison on all datasets where the values are averaged within a dataset. Numbers in bold highlight the best performance.
|
| 360 |
+
|
| 361 |
+
<table><tr><td rowspan="2"></td><td colspan="3">Cell tracking</td><td colspan="3">Graph matching</td><td colspan="3">Independent set</td><td colspan="3">QAPLib</td></tr><tr><td>g1</td><td>E(×108）</td><td>t[s]</td><td>91</td><td>E(×104)</td><td>t[s]</td><td>91</td><td>E(×108)</td><td>t[s]</td><td>91</td><td>E(×106)</td><td>t[]</td></tr><tr><td>Gurobi [23]</td><td>18</td><td>-3.852</td><td>809</td><td>9</td><td>-4.8433</td><td>278</td><td>14</td><td>−2.4457</td><td>52</td><td>3472</td><td>0.9</td><td>2618</td></tr><tr><td>FastDOG[1]</td><td>7</td><td>-3.863</td><td>1005</td><td>21</td><td>-4.8912</td><td>61</td><td>42</td><td>-2.4913</td><td>9</td><td>276</td><td>5.7</td><td>1680</td></tr><tr><td>DOGE</td><td>2.4</td><td>-3.854</td><td>1015</td><td>0.3</td><td>-4.8439</td><td>17</td><td>0.3</td><td>-2.4460</td><td>8</td><td>320</td><td>12.1</td><td>720</td></tr><tr><td>DOGE-M</td><td>2.1</td><td>-3.854</td><td>730</td><td>0.2</td><td>-4.8436</td><td>21</td><td>0.2</td><td>-24459</td><td>5</td><td>131</td><td>14.5</td><td>861</td></tr></table>
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Figure 2: Convergence plots for $g ( t )$ defined in (6), the relative dual gap to the optimum (or maximum suboptimal objective among all methods) of the relaxation (D). Both axes are logarithmic.
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275 40 nodes within the time limit of one hour. We show convergence plots for smaller instances in the
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276 Appendix. The difference to Gurobi is most pronounced w.r.t. anytime performance measured by $g _ { I }$
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277 since our solver reaches good solutions relatively early.
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Limitations While our approach gives solutions of high accuracy for the presented datasets, we have also tried our approach on other datasets (small cell tracking instances, MRFs for protein folding [27] and shape matching [51, 52]) where we were not able to obtain significant improvements w.r.t. the non-learned baseline [1]. For small cell tracking instances FastDOG already found the optimum in a moderate number of iterations, making it hard to beat. On shape matching and protein folding the parallelization of FastDOG did not bring enough speed-ups due to few large subproblems resulting in sequential bottlenecks. This limited the number of training iterations we could perform within a reasonable time.
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We have set some hyperparameters in a dataset-dependent way. This was partly necessitated due to problem sizes e.g., training on long time horizons was not possible with very large instances. Moreover, these instances only permitted a limited number of parameters in our neural networks.
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# 5 Conclusion
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We have proposed a learning approach for solving relaxations to combinatorial optimization problems by backpropagating through and learning parameters for the non-learned baseline [1]. We demonstrated its potential in obtaining close to optimal solutions faster than with traditional methods.
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Our work raises interesting follow-up questions: (i) Contrary to many approaches for backpropagation which replace non-smooth operations with smoothed variants (e.g. [33]) we directly compute (sub-) gradients for the non-smooth solver updates. Can smoothing of the solver help obtain a better backpropagated supervision? (ii) We argue that predicting good update steps for our solver is in itself an interesting and challenging problem for GNNs. We hope that our work can become a testbed for GNN architectures. (iii) There are a few desiderata for future learned solvers, including training universal models that generalize across different problem classes. Possibly more powerful GNNs and more involved training regimes are needed for this.
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1. For all authors...
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(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]
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(b) Did you describe the limitations of your work? [Yes]
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(c) Did you discuss any potential negative societal impacts of your work? [N/A] We only solve ILP relaxations fast.
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(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
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2. If you are including theoretical results...
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(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] Provided in the Appendix.
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3. If you ran experiments...
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(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No] Will be provided after acceptance
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(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Yes
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(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] Due to lack of computational resources we do not report error bars. However we do report results on different variants of our method in Ablation study. The random seed is fixed to same value of 1 for all experiments on all datasets.
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(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
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4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
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(a) If your work uses existing assets, did you cite the creators? [Yes]
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(b) Did you mention the license of the assets? [No]
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(c) Did you include any new assets either in the supplemental material or as a URL? [No]
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(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]
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(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
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5. If you used crowdsourcing or conducted research with human subjects...
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(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
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(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
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(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
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| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
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"type": "text",
|
| 4 |
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"text": "DOGE-Train: Discrete Optimization on GPU with End-to-end Training ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
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"bbox": [
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| 7 |
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| 11 |
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],
|
| 12 |
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"page_idx": 0
|
| 13 |
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},
|
| 14 |
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{
|
| 15 |
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"type": "text",
|
| 16 |
+
"text": "Anonymous Author(s) \nAffiliation \nAddress \nemail ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
423,
|
| 19 |
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226,
|
| 20 |
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|
| 21 |
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281
|
| 22 |
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],
|
| 23 |
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"page_idx": 0
|
| 24 |
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},
|
| 25 |
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{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "Abstract ",
|
| 28 |
+
"text_level": 1,
|
| 29 |
+
"bbox": [
|
| 30 |
+
462,
|
| 31 |
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318,
|
| 32 |
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535,
|
| 33 |
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334
|
| 34 |
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],
|
| 35 |
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"page_idx": 0
|
| 36 |
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},
|
| 37 |
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{
|
| 38 |
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"type": "text",
|
| 39 |
+
"text": "1 We present a fast, scalable, data-driven approach for solving linear relaxations of \n2 0-1 integer linear programs using a graph neural network. Our solver is based \n3 on the Lagrange decomposition based algorithm [1]. We make the algorithm \n4 differentiable and perform backpropagation through the dual update scheme for \n5 end-to-end training of its algorithmic parameters. This allows to preserve the \n6 algorithm’s theoretical properties including feasibility and guaranteed non-decrease \n7 in the lower bound. Since [1] can get stuck in suboptimal fixed points, we provide \n8 additional freedom to our graph neural network to predict non-parametric update \n9 steps for escaping such points while maintaining dual feasibility. For training of \n10 the graph neural network we use an unsupervised loss and perform experiments on \n11 large-scale real world datasets. We train on smaller problems and test on larger ones \n12 showing strong generalization performance with a graph neural network comprising \n13 only around $1 0 k$ parameters. Our solver achieves significantly faster performance \n14 and better dual objectives than its non-learned version [1]. In comparison to \n15 commercial solvers our learned solver achieves close to optimal objective values of \n16 LP relaxations and is faster by up to an order of magnitude on very large problems \n17 from structured prediction and on selected combinatorial optimization problems. \n18 Our code will be made available upon acceptance. ",
|
| 40 |
+
"bbox": [
|
| 41 |
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148,
|
| 42 |
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353,
|
| 43 |
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766,
|
| 44 |
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602
|
| 45 |
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],
|
| 46 |
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"page_idx": 0
|
| 47 |
+
},
|
| 48 |
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{
|
| 49 |
+
"type": "text",
|
| 50 |
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"text": "19 1 Introduction ",
|
| 51 |
+
"text_level": 1,
|
| 52 |
+
"bbox": [
|
| 53 |
+
148,
|
| 54 |
+
633,
|
| 55 |
+
310,
|
| 56 |
+
651
|
| 57 |
+
],
|
| 58 |
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"page_idx": 0
|
| 59 |
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},
|
| 60 |
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{
|
| 61 |
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"type": "text",
|
| 62 |
+
"text": "20 Integer linear programs (ILP) are a universal tool for solving combinatorial optimization problems. \n21 While great progress has been made on improving ILP solvers over the past several decades, some \n22 fundamental questions for future improvements remain open: Can ILP solvers make effective use of \n23 the massive parallelism afforded by GPUs and can modern machine learning meaningfully help? As \n24 of now the consensus seems that neither GPUs nor ML have yet helped general purpose ILP solvers \n25 in a fundamental way. In particular, this holds true for LP solvers which are a key component of most \n26 commonly used ILP approaches. LP solvers produce lower bounds on the optimal solution objective \n27 and are integral for many heuristics to decode feasible integral solutions. For many problems the ILP \n28 solvers spend most of the time on solving multiple LP relaxations, hence any impact GPUs and ML \n29 can have will directly translate into overall improvement of ILP solvers. \n30 State of the art LP solvers [23, 13, 17, 4, 18] make little utility of modern machine learning but rather \n31 use either hand-designed or auto-tuned parameters and update rules. Moreover, with the exception \n32 of [18] these solvers are not open-source, hence researchers’ ability to assess the potential of neural \n33 networks for improving LP solvers is limited. From a conceptual point of view traditional solver \n34 paradigms, e.g. simplex or interior point methods, are not GPU friendly and contain non-differentiable \n35 steps (such as pivot selection for simplex). Additionally, their high complexity further complicates \n36 any effort at making them differentiable. This makes utilization of neural networks and GPUs for \n37 solver improvement difficult. \n38 We propose a new way to use the potential of GPU parallelism and modern ML to obtain advances \n39 in LP relaxation solvers for ILPs. We argue that due to the difficulties in putting GPUs and ML to \n40 work in traditional solver methodologies, investigation of new paradigms is called for. To this end we \n41 build upon the recent work of [1] which proposed a massively parallel GPU friendly solver for 0-1 \n42 integer linear programming using Lagrange decomposition. The solver exhibits faster performance \n43 than traditional CPU solvers on large-scale problems making good use of GPU parallelism. Also \n44 due to its comparatively simple control flow and its usage of simple arithmetic operations for all its \n45 operations it can be made differentiable. This allows to train its parameters and predict update steps \n46 that will allow for faster convergence and overcoming fixed points from which the basic version of the \n47 algorithm suffers. This results in superior performance as compared to the non-learned version [1]. \n48 We obtain small gaps to (I)LP optima on a diverse range of large scale structured prediction problems, \n49 QAPLib [8] and independent set problems [39]. We are up to an order of magnitude faster than \n50 traditional ILP solvers. \n51 Contributions We propose to learn the Lagrange decomposition based algorithm [1] for solving LP \n52 relaxations of ILP problems and show its benefits. In particular, ",
|
| 63 |
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| 64 |
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| 66 |
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| 69 |
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|
| 70 |
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|
| 71 |
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{
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| 72 |
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"type": "text",
|
| 73 |
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"text": "",
|
| 74 |
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"bbox": [
|
| 75 |
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| 76 |
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| 77 |
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| 80 |
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|
| 81 |
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|
| 82 |
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| 83 |
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"type": "text",
|
| 84 |
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"text": "",
|
| 85 |
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"bbox": [
|
| 86 |
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156,
|
| 87 |
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|
| 88 |
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823,
|
| 89 |
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|
| 90 |
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],
|
| 91 |
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"page_idx": 1
|
| 92 |
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},
|
| 93 |
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{
|
| 94 |
+
"type": "text",
|
| 95 |
+
"text": "",
|
| 96 |
+
"bbox": [
|
| 97 |
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145,
|
| 98 |
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|
| 99 |
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| 100 |
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| 101 |
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],
|
| 102 |
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"page_idx": 1
|
| 103 |
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|
| 104 |
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{
|
| 105 |
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"type": "text",
|
| 106 |
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"text": "",
|
| 107 |
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"bbox": [
|
| 108 |
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148,
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| 109 |
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| 110 |
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|
| 111 |
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338
|
| 112 |
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],
|
| 113 |
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"page_idx": 1
|
| 114 |
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},
|
| 115 |
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{
|
| 116 |
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"type": "text",
|
| 117 |
+
"text": "• We make the dual update steps of [1] differentiable. This allows us to predict parameters of the update steps so that faster convergence is achieved as compared to using hand-picked values. \n• We train a predictor for arbitrary non-parametric update steps that allow to escape suboptimal fixed points into which the parametric update steps of [1] can fall. \n• We propose to train predictors for both the parametric and non-parametric updates in fully unsupervised manner. Our loss optimizes for parameters/update steps producing large improvements in the dual lower bound over a long time horizon. \n• We show the benefits of our learned massively parallel GPU approach on a wide range of problems. We have chosen structured prediction tasks including graph matching [29] and cell tracking [24]. From theoretical computer science we compare on the QAPLib [8] dataset and randomly generated independent set problems [39]. ",
|
| 118 |
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"bbox": [
|
| 119 |
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| 120 |
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| 121 |
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| 122 |
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| 123 |
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],
|
| 124 |
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|
| 125 |
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},
|
| 126 |
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{
|
| 127 |
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"type": "text",
|
| 128 |
+
"text": "2 Related Work ",
|
| 129 |
+
"text_level": 1,
|
| 130 |
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"bbox": [
|
| 131 |
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166,
|
| 132 |
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| 133 |
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320,
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| 134 |
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554
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| 135 |
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],
|
| 136 |
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"page_idx": 1
|
| 137 |
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},
|
| 138 |
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{
|
| 139 |
+
"type": "text",
|
| 140 |
+
"text": "2.1 Learning to solve Combinatorial Optimization ",
|
| 141 |
+
"text_level": 1,
|
| 142 |
+
"bbox": [
|
| 143 |
+
173,
|
| 144 |
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|
| 145 |
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534,
|
| 146 |
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587
|
| 147 |
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],
|
| 148 |
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"page_idx": 1
|
| 149 |
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},
|
| 150 |
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{
|
| 151 |
+
"type": "text",
|
| 152 |
+
"text": "66 ML has been used to improve various aspects of solving combinatorial problems. For the standard \n67 branch-and-cut ILP solvers the works [19, 22, 35] learn variable selection for branching. The \n68 approaches [14, 35] learn to fix a subset of integer variables in ILPs to their hopefully optimal values \n69 to improve finding high quality primal solutions. The works [43, 54] learn variable selection for \n70 the large neighborhood search heuristic for obtaining primal solutions to ILPs. Selecting good cuts \n71 through scoring them with neural networks was investigated in [26, 46]. While all these approaches \n72 result in runtime and solution quality improvements, only a few works tackle the important task of \n73 speeding up ILP relaxations by ML. Specifically, the work [11] used graph neural network (GNN) to \n74 predict variable orderings of decision diagrams representing combinatorial optimization problems. \n75 The goal is to obtain an ordering such that a corresponding dual lower bound is maximal. To our \n76 knowledge it is the only work that addresses computing ILP relaxations with ML. For constraint \n77 satisfaction problems [40, 9, 47] train GNN while [47] train in an unsupervised manner. For narrow \n78 subclasses of problems primal heuristics have been augmented through learning some of their \n79 decisions, e.g. for capacitated vehicle routing [36] and traveling salesman [55]. For a more complete \n80 overview of ML for combinatorial optimization we refer to the detailed surveys [6, 10]. ",
|
| 153 |
+
"bbox": [
|
| 154 |
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148,
|
| 155 |
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|
| 156 |
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|
| 157 |
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806
|
| 158 |
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],
|
| 159 |
+
"page_idx": 1
|
| 160 |
+
},
|
| 161 |
+
{
|
| 162 |
+
"type": "text",
|
| 163 |
+
"text": "81 2.2 Massively parallel combinatorial optimization ",
|
| 164 |
+
"text_level": 1,
|
| 165 |
+
"bbox": [
|
| 166 |
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| 167 |
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| 168 |
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|
| 169 |
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|
| 170 |
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],
|
| 171 |
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"page_idx": 1
|
| 172 |
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},
|
| 173 |
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{
|
| 174 |
+
"type": "text",
|
| 175 |
+
"text": "82 Massively parallel algorithms running on GPU have been proposed for narrow problem classes, \n83 including inference in [41, 56] and dense [45] Markov Random Fields, multicut [2] and for max \n84 flow [49, 53]. The algorithm [1] on which our work is based is, to our knowledge, the only generic \n85 ILP solver that can make adequate use of parallelism offered by GPUs. ",
|
| 176 |
+
"bbox": [
|
| 177 |
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|
| 178 |
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|
| 179 |
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| 180 |
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|
| 181 |
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],
|
| 182 |
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"page_idx": 1
|
| 183 |
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},
|
| 184 |
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{
|
| 185 |
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"type": "text",
|
| 186 |
+
"text": "86 2.3 Unrolling algorithms for parameter learning ",
|
| 187 |
+
"text_level": 1,
|
| 188 |
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"bbox": [
|
| 189 |
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| 190 |
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| 191 |
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| 192 |
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| 193 |
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],
|
| 194 |
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"page_idx": 2
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| 195 |
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},
|
| 196 |
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{
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| 197 |
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"type": "text",
|
| 198 |
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"text": "87 Algorithms containing differentiable iterative procedures are combined with neural networks for \n88 improving performance of such algorithms. One of the earliest works in this direction is [21] \n89 which embedded sparse coding algorithms in a neural network by unrolling. For solving inverse \n90 problems [57, 12] unroll through ADMM and non-linear diffusion resp. Overall, such approaches \n91 show more generalization power than pure neural networks based ones as shown in the survey [34]. \n92 Slightly different than from the above works, neural networks were used to predict update directions \n93 for training other neural networks (e.g. in [3]). ",
|
| 199 |
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"page_idx": 2
|
| 206 |
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},
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| 207 |
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{
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| 208 |
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"type": "text",
|
| 209 |
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"text": "94 3 Method ",
|
| 210 |
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"text_level": 1,
|
| 211 |
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"bbox": [
|
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| 217 |
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"page_idx": 2
|
| 218 |
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},
|
| 219 |
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{
|
| 220 |
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"type": "text",
|
| 221 |
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"text": "We first recapitulate the Lagrange decomposition approach to binary ILPs from [31] and the deferred min-marginal averaging scheme for its solution proposed in [1]. We highlight possible parameters of the update steps which we will predict by training a graph neural network. Proofs are in the Appendix. ",
|
| 222 |
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|
| 229 |
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},
|
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{
|
| 231 |
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"type": "text",
|
| 232 |
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"text": "99 3.1 Lagrange Decomposition & Deferred Min-Marginal Averaging ",
|
| 233 |
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"text_level": 1,
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| 234 |
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},
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{
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| 243 |
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"type": "text",
|
| 244 |
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"text": "Definition 1 (Binary Program [31]). Let a linear objective $c \\in \\mathbb { R } ^ { n }$ and $m$ variable subsets $\\mathcal { T } _ { j } \\subset [ n ]$ of constraints with feasible set $\\mathcal { X } _ { j } \\subset \\{ 0 , 1 \\} ^ { \\mathbb { Z } _ { j } }$ for $j \\in [ m ]$ be given. The corresponding binary program is ",
|
| 245 |
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},
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| 253 |
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{
|
| 254 |
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"type": "equation",
|
| 255 |
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"img_path": "images/89359ddcd168b3803df9fff69a132e1ed123e65bbca3b92ead23e4fcab62e9a2.jpg",
|
| 256 |
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"text": "$$\n\\operatorname* { m i n } _ { x \\in \\{ 0 , 1 \\} ^ { n } } \\langle c , x \\rangle \\quad { \\mathrm { s . t . } } \\quad x _ { { \\bar { \\mathcal { T } } } _ { j } } \\in { \\mathcal { X } } _ { j } \\quad \\forall j \\in [ m ] ,\n$$",
|
| 257 |
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"text_format": "latex",
|
| 258 |
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| 262 |
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| 264 |
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},
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| 266 |
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{
|
| 267 |
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"type": "text",
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| 268 |
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"text": "where $x _ { \\mathbb { Z } _ { j } }$ is the restriction to variables in $\\mathcal { T } _ { j }$ . ",
|
| 269 |
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| 276 |
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},
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| 277 |
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{
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| 278 |
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"type": "text",
|
| 279 |
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"text": "Any binary ILP $\\mathrm { m i n } _ { x \\in \\{ 0 , 1 \\} ^ { n } } \\langle c , x \\rangle$ s.t. $A x \\leq b$ where $A \\in \\mathbb { R } ^ { m \\times n }$ can be written as (BP) by associating each constraint $a _ { j } ^ { T } x \\leq b _ { j }$ for $j \\in [ m ]$ with its own subproblem $\\mathcal { X } _ { j }$ . ",
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| 280 |
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"text": "06 In order to obtain a problem formulation amenable for parallel optimization we consider its Lagrange \n07 dual which decomposes the full problem (BP) into a series of coupled subproblems. \n108 Definition 2 (Lagrangean dual problem [31]). Define the set of subproblems that constrain variable $i$ \n109 as $\\mathcal { T } _ { i } = \\{ j \\in [ \\bar { m ] } \\ | \\ \\bar { i } \\in \\mathcal { T } _ { j } \\}$ . Let the energy for subproblem $j \\in [ m ]$ w.r.t. Lagrangean dual variables \n110 $\\lambda _ { \\bullet j } = ( \\lambda _ { i j } ) _ { i \\in \\mathcal { T } _ { j } } \\in \\mathbb { R } ^ { \\mathcal { T } _ { j } }$ be ",
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"text": "$$\nE ^ { j } ( \\lambda _ { \\bullet j } ) = \\operatorname* { m i n } _ { x \\in \\mathcal { X } _ { j } } \\langle \\lambda _ { \\bullet j } , x \\rangle .\n$$",
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"text": "111 Then the Lagrangean dual problem is defined as ",
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"text": "$$\n\\operatorname* { m a x } _ { \\lambda } \\quad \\sum _ { j \\in [ m ] } E ^ { j } ( \\lambda _ { \\bullet j } ) \\quad \\mathrm { s . t . } \\quad \\sum _ { j \\in \\mathcal { I } _ { i } } \\lambda _ { i j } = c _ { i } \\quad \\forall i \\in [ n ] .\n$$",
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"text": "112 The authors in [1] have proposed a parallelization friendly iterative algorithm for updating Lagrange \n113 multipliers $\\lambda$ for maximizing (D), see Algorithm 1. We write it in a slightly adapted form since it \n114 will allow us to easily describe its backpropagation. The algorithm assigns the Lagrange variables \n115 in $u$ -many disjoint blocks $B _ { 1 } , \\ldots , B _ { u }$ in such a way that each block contains at most one Lagrange \n116 variable from each subproblem and all variables within a block are updated in parallel. The dual update \n117 scheme relies on computing min-marginal differences i.e., the difference of subproblem objectives \n118 when a certain variable is set to 1 minus its objective when the same variable is set to 0, see line 10 \n119 in Algorithm 1. These min-marginal differences are averaged out across subproblems via updates \n120 to Lagrange variables in line 11 in Algorithm 1. The crucial ingredient allowing parallelization is \n121 that in the min-marginal averaging step values from the last iteration are used (i.e. $M ^ { \\mathrm { i n } }$ ), making \n122 synchronization between subproblems unnecessary. \n123 In [1] the min-marginal averaging parameters of Algorithm 1 were set as $\\omega = 0 . 5$ and $\\alpha _ { i j } =$ \n124 $1 / | \\mathcal { I } _ { i } |$ leading to uniform averaging. We generalize the min-marginal update step by considering \n125 more general parametric update steps. We allow $\\omega \\in ( 0 , 1 )$ and $\\alpha$ -values to be arbitrary convex \n126 combinations. In the next section we will show how to train these values to achieve faster convergence. \n127 Proposition 1 (Dual Feasibility and Monotonicity of Min-marginal Averaging). For any $\\alpha _ { i j } \\geq 0$ \n128 with $\\begin{array} { r } { \\sum _ { j \\in \\mathcal { T } _ { i } } \\alpha _ { i j } = 1 } \\end{array}$ and $\\omega _ { i j } \\in [ 0 , 1 ]$ the min-marginal averaging step in line $1 l$ in Algorithm $^ { l }$ \n129 retains dual feasibility and is non-decreasing in the dual lower bound. ",
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"text": "Input: Lagrange variables $\\lambda _ { i j } \\forall i \\in [ n ] , j \\in \\mathcal { T } _ { i }$ , damping factors $\\omega _ { i j } \\in ( 0 , 1 ) \\forall i \\in [ n ] , j \\in \\mathcal { T } _ { i }$ , anisotropic min-marginal averaging weights $\\alpha _ { i j } \\in ( 0 , 1 ) \\forall i \\in [ n ] , j \\in \\mathcal { T } _ { i }$ , max. number of iterations $T$ . 1 Initialize deferred min-marginal diff. $M = \\mathbb { 0 }$ 2 for $T$ iterations do 3 for block $B \\in ( B _ { 1 } , \\ldots B _ { u } )$ do 4 $\\lambda , M \\gets$ BlockUpdate $( B , \\lambda , M , \\alpha , \\omega )$ 5 for block $B \\in ( B _ { u } , \\ldots B _ { 1 } )$ do 6 λ, M ← BlockUpdate $( B , \\lambda , M , \\alpha , \\omega )$ 7 return λ, M 8 Procedure BlockUpdate $( B , \\lambda ^ { \\mathrm { i n } } , M ^ { \\mathrm { i n } } , \\alpha , \\omega )$ 9 for $i j \\in B$ in parallel do 10 CompuUpdate $\\begin{array} { r } { \\lambda _ { i j } ^ { \\mathrm { o u t } } = \\lambda _ { i j } ^ { \\mathrm { i n } } - M _ { i j } ^ { \\mathrm { o u t } } + \\alpha _ { i j } \\sum _ { k \\in \\mathcal { I } _ { i } } M _ { i k } ^ { \\mathrm { i n } } } \\end{array}$ $\\begin{array} { r } { M _ { i j } ^ { \\mathrm { { o u t } } } = \\omega _ { i j } [ \\operatorname* { m i n } _ { x \\in \\mathcal { X } _ { j } : x _ { i } = 1 } \\langle \\lambda _ { \\bullet j } ^ { \\mathrm { { i n } } } , x \\rangle - \\operatorname* { m i n } _ { x \\in \\mathcal { X } _ { j } : x _ { i } = 0 } \\langle \\lambda _ { \\bullet j } ^ { \\mathrm { { i n } } } , x \\rangle ] } \\end{array}$ 12 return $\\lambda ^ { \\mathsf { o u t } }$ , M out ",
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"text": "130 3.2 Backpropagation through Deferred Min-Marginal Averaging ",
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"text": "131 We show below how to differentiate through Algorithm 1 with respect to the parameters $\\alpha$ and $\\omega$ . \n132 This will ultimately allow us to learn these parameters such that faster convergence is achieved. To \n133 this end we describe backpropagation for a block update (lines 8- 12) of Alg. 1. All other operations \n134 can be tackled by automatic differentiation. For a block $B$ in $\\{ B _ { 1 } , \\ldots , B _ { u } \\}$ we view the Lagrangean \n135 update as a mapping $\\mathcal { H } : ( \\mathbb { R } ^ { | B | } ) ^ { 4 } \\to ( \\mathbb { R } ^ { | B | } ) ^ { 2 }$ , $( \\lambda ^ { \\mathrm { i n } } , M ^ { \\mathrm { i n } } , \\alpha , \\omega ) \\mapsto ( \\lambda ^ { \\circ \\mathrm { u t } } , M ^ { \\circ \\mathrm { u t } } )$ . ",
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"text": "Given a loss function 36 $\\mathcal { L } : \\mathbb { R } ^ { N } \\mathbb { R }$ we denote $\\partial \\mathcal { L } / \\partial x$ by $\\dot { x }$ . Algorithm 2 shows backpropagation through 7 $\\mathcal { H }$ to compute the gradients $\\dot { \\lambda } ^ { \\mathrm { { i n } } } , \\dot { M } ^ { \\mathrm { { i n } } } ,$ $\\dot { \\alpha }$ and $\\dot { \\omega }$ . ",
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"text": "138 Proposition 2. Algorithm 2 performs backpropagation through $\\mathcal { H }$ . ",
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"text": "139 Efficient Implementation Generally, the naive computation of min-marginal differences and its \n140 backpropagation are both expensive operations as they require solving two optimization problems \n141 for each dual variable. In [1, 31] the authors represented each subproblem using binary decision \n142 diagrams (BDDs) for fast incremental computation of min-marginal differences. Their algorithm \n143 results in a computation graph involving only elementary arithmetic operations and taking minima \n144 over several variables. Using this computational graph we can implement the abstract Algorithm 2 \n145 efficiently and parallelize on GPU. For details we refer to the Appendix. ",
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"text": "Algorithm 2: BlockUpdate backpropagation ",
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"text": "Input: Forward pass inputs: $B , \\lambda ^ { \\mathrm { i n } } , M ^ { \\mathrm { i n } } , \\alpha , \\omega$ , gradients of forward pass output: λ˙ out, $\\dot { M } ^ { \\mathrm { { o u t } } }$ , gradients of parameters $\\dot { \\alpha } , \\dot { \\omega }$ ",
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"text": "1 for $i j \\in B$ in parallel do ",
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"text": "2 $\\begin{array} { r } { \\dot { M } _ { i j } ^ { \\mathrm { i n } } = \\sum _ { k \\in \\mathcal { T } _ { i } } \\dot { \\lambda } _ { i k } ^ { \\mathrm { o u t } } \\alpha _ { i k } } \\end{array}$ , $\\dot { M } _ { i j } ^ { \\mathrm { o u t } } = \\dot { M } _ { i j } ^ { \\mathrm { o u t } } - \\dot { \\lambda } _ { i j } ^ { \\mathrm { o u t } }$ \n3 $\\begin{array} { r } { \\dot { \\alpha } _ { i j } = \\dot { \\alpha } _ { i j } + \\dot { \\lambda } _ { i j } \\sum _ { k \\in \\mathcal { I } _ { i } } M _ { i k } ^ { \\mathrm { i n } } , \\quad \\dot { \\omega } _ { i j } = \\dot { \\omega } _ { i j } + \\dot { M } _ { i j } ^ { \\mathrm { o u t } } [ M _ { i j } ^ { \\mathrm { o u t } } / \\omega _ { i j } ] } \\end{array}$ \n4 Compute minimizers $\\begin{array} { r } { s ^ { j } ( i , \\beta ) = \\arg \\operatorname* { m i n } _ { x \\in \\mathcal { X } _ { j } : x _ { i } = \\beta } \\langle \\lambda _ { \\bullet j } ^ { \\mathrm { { i n } } } , x \\rangle , \\forall \\beta \\in \\{ 0 , 1 \\} } \\end{array}$ } \n5 $\\dot { \\lambda } _ { p j } ^ { \\mathrm { i n } } = \\dot { \\lambda } _ { p j } ^ { \\circ \\mathrm { u t } } + \\dot { M } _ { i j } ^ { \\circ \\mathrm { u t } } \\omega _ { i j } [ s _ { p } ^ { j } ( i , 1 ) - s _ { p } ^ { j } ( i , 0 ) ] ,$ , ∀p ∈ Ij \n6 return $\\dot { \\lambda } ^ { \\mathrm { i n } } , \\dot { M } ^ { \\mathrm { i n } } , \\dot { \\alpha } , \\dot { \\omega }$ ",
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"text": "146 3.3 Non-Parametric Update Steps ",
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"text": "147 Although the min-marginal averaging scheme of Alg. 1 guarantees non-decreasing lower bound, it \n148 can get stuck in suboptimal fixed points, see [50] for a discussion for the special case of MAP-MRF. \n149 To alleviate this shortcoming we allow arbitrary updates to Lagrange variables through a vector ",
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"text": "150 $\\boldsymbol { \\theta } \\in \\mathbb { R } ^ { | \\lambda | }$ as ",
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"text": "$$\n\\lambda _ { i j } \\lambda _ { i j } + \\theta _ { i j } - \\frac { 1 } { | \\mathcal { T } _ { i } | } \\sum _ { k \\in \\mathcal { I } _ { i } } \\theta _ { i k } , \\forall i \\in [ n ] , j \\in \\mathcal { I } _ { i }\n$$",
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"text": "151 where the last term ensures feasibility of updated Lagrange variables w.r.t. the dual problem (D). ",
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"text": "3.4 Graph neural network ",
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"text": "153 We train a graph neural network (GNN) to predict the parameters $\\alpha , \\omega$ of Alg. 1 and also the \n154 non-parametric update $\\theta$ for (2). To this end we encode the dual problem (D) on a bipartite graph \n155 $\\mathcal { G } = ( \\nu , \\mathcal { E } )$ . Its nodes correspond to primal variables $\\mathcal { T }$ and subproblems $\\mathcal { I }$ i.e., $\\mathcal { V } = \\mathcal { I } \\cup \\mathcal { I }$ and \n156 edges $\\mathcal { E } = \\{ i j \\mid i \\in \\mathcal { T } , j \\in \\mathcal { T } _ { i } \\}$ correspond to Lagrange multipliers. We need to predict values of \n157 $\\alpha _ { i j } , \\omega _ { i j }$ and $\\theta _ { i j }$ for each edge $i j$ in $\\mathcal { E }$ . We associate features $\\boldsymbol { f } \\overset { - } { = } \\left( f _ { \\mathcal { T } } , f _ { \\mathcal { T } } , f _ { \\mathcal { E } } \\right)$ with each entity of the \n158 graph which capture the current state of Alg. 1. Additionally, we encode a number of quantities as \n159 features which can make learning easier. For example, a history of previous dual objectives for each \n160 subproblem is encoded in the constraint nodes and minimizers of each subproblem (which correspond \n161 to a subgradient of the dual problem (D)) are encoded in the edge features $f _ { \\mathcal { E } }$ . A complete list of \n162 features is provided in the Appendix. \n163 Message passing To perform message passing we use the transformer based graph convolution \n164 scheme of [42]. We first compute an embedding of all subproblems $j$ in $\\mathcal { I }$ by receiving messages \n165 from adjacent nodes and edges as ",
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"text": "$$\n\\mathsf { C O N V } _ { \\mathcal { I } } ( f _ { \\mathcal { I } } , f _ { \\mathcal { I } } , f _ { \\mathcal { E } } , \\mathcal { E } ) _ { j } = \\mathbf { W _ { s } } f _ { j } + \\sum _ { i | i j \\in \\mathcal { E } } a _ { i j } ( f _ { j } , f _ { \\mathcal { I } } , f _ { \\mathcal { E } } ; \\mathbf { W _ { a } } ) \\left[ \\mathbf { W _ { t } } f _ { i } + \\mathbf { W _ { e } } f _ { i j } \\right] ,\n$$",
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"text": "166 where $\\mathbf { W } = \\left( \\mathbf { W _ { a } } , \\mathbf { W _ { s } } , \\mathbf { W _ { t } } , \\mathbf { W _ { e } } \\right)$ are trainable parameters and $a _ { i j } \\left( f _ { j } , f _ { \\mathbb { Z } } , f _ { \\mathbb { \\varepsilon } } ; \\mathbf { W _ { a } } \\right)$ is the softmax \n167 attention weight between nodes $i$ and $j$ parameterized by $\\mathbf { W _ { a } }$ . Afterwards we perform message \n168 passing in the reverse direction to compute embeddings for primal variables $\\mathcal { T }$ . Similar strategy for \n169 message passing on a bipartite graph was followed by [19]. \n170 Recurrent connections Our default GNN as mentioned above only uses hand-crafted features \n171 to maintain a history of previous optimization rounds. To learn a summary of the past updates we \n172 optionally allow recurrent connections through an LSTM with forget gate [20]. The LSTM is only \n173 applied on primal variable nodes $\\mathcal { T }$ and maintains cell states $s \\tau$ which can be updated and used for \n174 parameter prediction in subsequent optimization rounds. \n175 Prediction The learned embeddings from GNN, LSTM outputs and solver features from Alg. 1 \n176 are consumed by a multi-layer perceptron $\\Phi$ to predict the required variables for each edge $i j$ in $\\mathcal { E }$ . \n177 Afterwards we transform these outputs so that they satisfy Prop. 1. \n178 The exact sequence of operations performed by the graph neural network are shown in Alg. 3 where \n179 $[ u _ { 1 } , \\ldots , u _ { k } ]$ denotes concatenation of vectors $u _ { 1 } , \\ldots , u _ { k }$ , LN denotes layer normalization [5] and \n180 $\\mathtt { L S T M } _ { \\mathcal { T } }$ stands for an LSTM cell which operates on each primal variable node. ",
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"text": "Algorithm 3: Parameter prediction by GNN ",
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"type": "text",
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"text": "Input: Primal variable features $f _ { \\mathcal { T } }$ and cell states $s \\tau$ , Subproblem features $f _ { \\mathcal { I } }$ , Dual variable (edge) features $f _ { \\mathcal { E } }$ , Set of edges $\\mathcal { E }$ . 1 $h _ { \\mathcal { T } } = \\mathsf { R e L U } \\left( \\operatorname { L N } \\left( \\operatorname { C O N V } _ { \\mathcal { T } } \\left( f _ { \\mathcal { T } } , f _ { \\mathcal { T } } , f _ { \\mathcal { E } } , \\mathcal { E } \\right) \\right) \\right)$ // Compute subproblems embeddings 2 $h _ { \\mathcal { T } } = \\mathtt { R e L U }$ $\\left( \\mathrm { L N } \\left( \\mathrm { C O N V } _ { \\mathcal { T } } \\left( f _ { \\mathcal { T } } , [ f _ { \\mathcal { T } } , h _ { \\mathcal { T } } ] , f _ { \\mathcal { E } } , \\mathcal { E } \\right) \\right) \\right)$ // Compute primal variable embeddings 3 $z _ { \\mathcal { T } } , s _ { \\mathcal { T } } = \\mathtt { L S T M } _ { \\mathcal { T } } ( h _ { \\mathbb { Z } } , s _ { \\mathcal { T } } )$ // Compute output and cell state 4 $( \\hat { \\alpha } , \\hat { \\omega } , \\theta ) = \\Phi \\left( [ f _ { \\mathcal { T } } , h _ { \\mathcal { T } } , z _ { \\mathcal { T } } ] , [ f _ { \\mathcal { T } } , h _ { \\mathcal { T } } ] , f _ { \\mathcal { E } } , \\mathcal { E } \\right)$ // Prediction per edge 5 $\\alpha _ { i \\bullet } = \\mathtt { S o f t m a x } ( \\hat { \\alpha } _ { i \\bullet } )$ , $\\forall i \\in \\mathcal { T }$ , $\\omega = \\mathtt { S i g m o i d } ( \\hat { \\omega } )$ // Ensure non-decreasing obj., Prop. 1 6 return $\\alpha , \\omega , \\theta , s _ { \\mathcal { T } }$ ",
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"image_caption": [
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"Figure 1: Our pipeline for optimizing the Lagrangean dual (D). The problem is encoded on a bipartite graph containing features $f _ { \\mathcal { T } }$ , $f _ { \\mathcal { I } }$ and $f _ { \\mathcal { E } }$ for primal variables, subproblems and dual variables resp. A graph neural network (GNN) predicts the non-parameteric update $\\theta$ (2) and parameters $\\alpha$ and $\\omega$ for Alg. 1. In one optimization round current set of Lagrange multipliers $\\lambda$ are first updated by the non-parametric update using $\\theta$ . Afterwards deferred min-marginal averaging is performed parameterized by $\\alpha$ and $\\omega$ . The updated solver features $f$ (which also includes $\\lambda$ ) and LSTM cell states $s \\tau$ are sent to the GNN in next optimization round. These rounds are repeated at most $R$ -times during training and until convergence during inference. "
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"text": "181 3.5 Loss ",
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"text": "82 Given the Lagrange variables $\\lambda$ we directly use the dual objective (D) as an unsupervised loss to train \n183 the GNN. Thus, we maximize the loss $L$ defined as ",
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"img_path": "images/d7a38a1459e2c6cff83559a06ca3a405fefa94f7238901390a2d071a3cb9386a.jpg",
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"text": "$$\n\\mathcal { L } ( \\lambda ) = \\sum _ { j \\in [ m ] } E ^ { j } ( \\lambda _ { \\bullet j } ) .\n$$",
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"text": "184 For a mini-batch of instances during training we take the mean of corresponding per-instance losses. \n185 For backpropagation, gradient of loss $\\mathcal { L }$ w.r.t. Lagrange variables of a subproblem $j$ is computed by \n186 finding a minimizing assignment for that subproblem, written as ",
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"text": "$$\n\\left( \\frac { \\partial \\mathcal { L } } { \\partial \\lambda } \\right) _ { \\bullet j } = \\operatorname { a r g m i n } _ { x \\in \\mathcal { X } _ { j } } \\langle \\lambda _ { \\bullet j } , x \\rangle \\in \\{ 0 , 1 \\} ^ { \\mathcal { Z } _ { j } } .\n$$",
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"text": "187 The above gradient is then sent as input for backpropagation. For computing the minimizing \n188 assignment efficiently we use binary decision diagram representation of each subproblem as in [1, 31]. ",
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"text": "3.6 Overall pipeline ",
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"text": "190 Our overall pipeline combining all building blocks from the previous sections is shown in Figure 1. \n191 We train our pipeline which contains multiple dual optimization rounds in a fashion similar to that \n192 of recurrent neural networks. One round of our dual optimization consists of message passing \n193 by GNN, a non-parametric update step and $T$ iterations of deferred min-marginal averaging. For \n194 computational efficiency we run our pipeline for at most $R$ dual optimization rounds during training. \n195 On each mini-batch we randomly sample a number of optimization rounds $r$ in $[ R ]$ , run $r - 1$ rounds \n196 without tracking gradients and backpropagate through the last round by computing the loss (4). For \n197 the pipeline with recurrent connections we backpropagate through last 3 rounds and apply the loss \n198 after each of these rounds. Since the task of dual optimization is relatively easier in early rounds \n199 as compared to later ones (where [1] can get stuck) we use two neural networks. The early stage \n200 network is trained if the randomly sampled $r$ is in $[ 0 , R / 2 ]$ and the late stage network is chosen \n201 otherwise. During testing we switch to the later stage network when the relative improvement in the \n202 dual objective by the early stage network becomes less than $1 0 ^ { - 6 }$ . ",
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"text": "03 4 Experiments ",
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"text": "204 As main evaluation metric we report convergence plots of the relative dual gap $g ( t ) \\in [ 0 , 1 ]$ at time $t$ ",
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"text": "$$\ng ( t ) = \\operatorname* { m i n } \\left( \\frac { d ^ { * } - d ( t ) } { d ^ { * } - d _ { i n i t } } , 1 . 0 \\right)\n$$",
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"text": "205 where $d ( t )$ is the dual objective at time $t$ , $d ^ { * }$ is the optimal (or best known) objective value of the \n206 Lagrange relaxation $( \\mathrm { D } )$ and $d _ { i n i t }$ is the objective value before optimization as computed by [1]. \n207 Additionally we also report per dataset averages of relative dual gap integral $\\begin{array} { r } { g _ { I } = \\int g ( \\bar { t } ) d t } \\end{array}$ [7], best \n208 objective value $( E )$ and time taken $\\mathbf { \\rho } ( t )$ to obtain best objective. To cater the dominating effect of \n209 worse initial lower bounds on $g _ { I }$ (as $g ( t )$ can be close to 1 at $t \\approx 0$ ) we start calculating $g \\tau$ after a few \n210 rounds of our solver are completed. This start time is then also used to evaluate other algorithms for a \n211 fair comparison. To evaluate CPU solvers we use an AMD EPYC 7702 CPU. For the GPU solvers \n212 we use either one NVIDIA RTX 8000 (48GB) or A100 (80GB) GPU depending on instance size. ",
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"text": "4.1 Algorithms ",
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"text": "Gurobi: Results of the dual simplex algorithm from the commercial ILP solver [23]. ",
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"text": "FastDOG: The non-learned baseline [1] of Alg. 1 with $\\omega _ { i j } = 0 . 5$ and $\\alpha _ { i j } = 1 / | \\mathcal { I } _ { i } |$ ",
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"text": "DOGE: Our approach where we learn to predict parametric and non-parametric updates by using two graph neural networks for early and late-stage optimization. Size of the learned embeddings $h$ computed by the GNN in Alg. 3 is set to 16 for nodes and 8 for edges. For computing attention weights in (3) we use only one attention head for efficiency. The predictor $\\Phi$ in Alg. 3 contains 4 linear layers with the ReLU activation. We train the networks using the Adam optimizer [30]. To prevent gradient overflow we use gradient clipping on model parameters by an $l ^ { 2 }$ norm of 50. The number of trainable parameters is $8 k$ . ",
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"text": "DOGE-M: Variant of our method where we additionally use recurrent connections using LSTM. The cell state vector $s _ { i }$ for each primal variable node $i \\in \\mathcal { T }$ has a size of 16. The number of trainable parameters is $1 2 k$ . ",
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"text": "We have not tested against specialized heuristics for our benchmark problems since [1] has shown them to be on par or outperformed by FastDOG. For training our approach we use the frameworks [15, 16, 38] and implement the Algorithms 1,2 in CUDA [37] using [25, 28]. ",
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"type": "text",
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"text": "4.2 Datasets ",
|
| 890 |
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"type": "text",
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"text": "Cell tracking $( C T )$ : Instances of developing flywing tissue from cell tracking challenge [48] processed by [24] and obtained from [44]. We use the largest and hardest 3 instances, train on the 2 smaller instances and test on the largest one. ",
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| 902 |
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"type": "text",
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| 912 |
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"text": "Graph matching (GM): Instances of graph matching for matching nuclei in 3D microscopic images [32] processed by [29] and made publicly available through [44]. We train on 10 instances and test on the remaining 20 instances. ",
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| 913 |
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"type": "text",
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"text": "Independent set (IS): Random instances of independent set problem generated using [39]. For training we generate 240 instances with $1 0 k$ vertices each and test on 60 instances with $5 0 k$ vertices. We generating edges between vertices in the graph with a probability of 0.25. ",
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"type": "text",
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"text": "QAPLib: The benchmark dataset for quadratic assignment problems used in the combinatorial optimization community [8]. We train on 61 instances having up to 40 nodes and test on 35 instances having up to 70 nodes. ",
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"bbox": [
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"type": "text",
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"text": "242 For each dataset we use a separate set of hyperparameters due to varying instance sizes given in \n243 Table 1. All our test datasets on average contain more than a million edges (i.e., Lagrange variables) \n244 while training instances are considerably smaller. For efficiency, during evaluation we use a larger \n245 value of $T$ in Alg. 1 than during training. For the $C T$ dataset containing we learn only the non \n246 parametric update steps (2) and fix the parameters in Alg. 1 to their default values from [1]. Learning \n247 these parameters gave slightly worse training loss at convergence. ",
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"type": "text",
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"text": "48 4.3 Ablation study ",
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"text_level": 1,
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"bbox": [
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"type": "text",
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"text": "249 We perform an ablation study to test the importance of various components of our approach. Starting \n250 from [1] as a baseline we first predict all parameters $\\alpha , \\omega , \\theta$ through the two multi-layer perceptrons $\\Phi$ \n251 for early and late stage optimization without using GNN. Next, we report results of using one network \n252 (instead of two) which is trained and tested for both early and later rounds of dual optimization. Lastly, \n253 we aim to seek the importance of learning parameters of Alg. 2 and the non-parametric update (2). \n254 To this end, we learn to predict only the non-parametric update and apply the loss directly on updated \n255 $\\lambda$ without requiring backpropagation through Alg. 1. We also try learning a subset of parameters i.e., \n256 not predicting averaging weights $\\alpha$ or damping factors $\\omega$ . Lastly, we report results of DOGE-M which \n257 uses recurrent connections. The results are in Table 2. ",
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{
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"type": "table",
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"img_path": "images/80cbf349b9f5e0b8ab4760e9118d4654d9fb649d81af84239d7edd6839756eab.jpg",
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"table_caption": [
|
| 981 |
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"Table 1: Hyperparameters of our approach and dataset statistics. $| \\mathcal { T } | + | \\mathcal { I } |$ : Average number of variables and constraints in each dataset (# vertices in GNN); $\\textstyle \\sum _ { j = 1 } ^ { m } | { \\dot { \\mathcal { I } } } _ { i } |$ : Average number of Lagrange multipliers (# edges in GNN); $T$ : Number of iterations of Alg. 1 in each optimization round; $R$ : max. number of training rounds; # itr. train: Number of training iterations. "
|
| 982 |
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| 983 |
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"table_footnote": [],
|
| 984 |
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"table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"2\">|Z|+|J|(×106)</td><td colspan=\"2\"></td><td colspan=\"2\">T</td><td rowspan=\"2\">R</td><td rowspan=\"2\">batch size</td><td rowspan=\"2\">learn. rate</td><td rowspan=\"2\">#itr. train</td><td rowspan=\"2\">train time [hrs]</td></tr><tr><td>train</td><td>test</td><td>train</td><td>test</td><td>train</td><td>test</td></tr><tr><td>CT</td><td>3.7</td><td>12.4</td><td>8.5</td><td>28</td><td>1</td><td>100</td><td>400</td><td>1</td><td>1e-3</td><td>500</td><td>14</td></tr><tr><td>GM</td><td>1.7</td><td>1.7</td><td>3.3</td><td>3.3</td><td>20</td><td>200</td><td>20</td><td>2</td><td>1e-3</td><td>400</td><td>4</td></tr><tr><td>IS</td><td>0.05</td><td>0.4</td><td>0.1</td><td>1.2</td><td>20</td><td>50</td><td>20</td><td>8</td><td>1e-3</td><td>2500</td><td>10</td></tr><tr><td>QAPLib</td><td>0.1</td><td>2.8</td><td>0.5</td><td>11</td><td>5</td><td>20</td><td>500</td><td>4</td><td>1e-3</td><td>1600</td><td>48</td></tr></table>",
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{
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"type": "table",
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"img_path": "images/1e78081dd0b986a7d7b94b2ee662503d9cef4b2ca58f5b6a406a3d00c860c5c7.jpg",
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| 996 |
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"table_caption": [
|
| 997 |
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"Table 2: Ablation study results on the Graph matching dataset. w/o GNN: Use only the two predictors $\\Phi$ without GNN for early and late stage optimization; same network: use one network (GNN, $\\Phi$ ) for both early and late stage; only non-param., param.: predict only the non-parametric update (2) or the parametric update (Alg. 1); w/o α, ω: does not predict $\\alpha$ or $\\omega$ resp. "
|
| 998 |
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],
|
| 999 |
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"table_footnote": [],
|
| 1000 |
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"table_body": "<table><tr><td></td><td>w/o learn. ([1])</td><td>w/o GNN</td><td>same network</td><td>only non-param.</td><td>only param.</td><td>w/oα</td><td>w/ow</td><td>DOGE</td><td>DOGE-M</td></tr><tr><td>g1 (）</td><td>21</td><td>0.42</td><td>0.95</td><td>2.3</td><td>0.7</td><td>0.36</td><td>0.35</td><td>0.33</td><td>0.19</td></tr><tr><td>E(1）</td><td>-48912</td><td>-48440</td><td>-48444</td><td>-48476</td><td>-48444</td><td>-48439</td><td>-48439</td><td>-48439</td><td>-48436</td></tr><tr><td>t[s](）</td><td>61</td><td>29</td><td>24</td><td>51</td><td>74</td><td>30</td><td>30</td><td>17</td><td>21</td></tr></table>",
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"type": "text",
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"text": "",
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"type": "text",
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| 1022 |
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"text": "Firstly, from our ablation study we observe that learning even one of the two types of updates i.e., non-parametric or parametric already gives better results than the non-learned solver [1]. This is because non-parametric update can help in escaping fixed-points of [1] when they occur and the parametric update can help Alg. 1 in avoiding such fixed-points. Combining both of these strategies further improves the results. Secondly, we observe that performing message passing with GNN gives improvement over only using the predictor $\\Phi$ . Thirdly, we find using separate networks for early and late stage optimization gives better performance than using the same network for all stages. Lastly, using recurrent connections gives the best performance. ",
|
| 1023 |
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"type": "text",
|
| 1033 |
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"text": "4.4 Results ",
|
| 1034 |
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"text_level": 1,
|
| 1035 |
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|
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"type": "text",
|
| 1045 |
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"text": "Convergence plots of relative dual gaps change w.r.t. wall clock times are given in Figure 2. Rest of the evaluation metrics are reported in Table 3. For further details we refer to the Appendix. ",
|
| 1046 |
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"bbox": [
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"type": "text",
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| 1056 |
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"text": "Discussion As compared to the non-learned baseline FastDOG we reach an order of magnitude more accurate relaxation solutions, almost closing the gap to optimum as computed by Gurobi. We retain high speed afforded by exploiting GPU parallelism. Interestingly, we can often outperform FastDOG also in the early stage where optimization is easy. Our LSTM version DOGE-M has shown improved performance than the non-LSTM version. Especially it shows much improvement on the most difficult QAPLib dataset. On QAPLib Gurobi does not converge on instances with more than ",
|
| 1057 |
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|
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},
|
| 1065 |
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{
|
| 1066 |
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"type": "table",
|
| 1067 |
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"img_path": "images/0580078f0a53e15c520d8450c70af7fddd3edf925e648cd3ddc8c21d20bf525c.jpg",
|
| 1068 |
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"table_caption": [
|
| 1069 |
+
"Table 3: Results comparison on all datasets where the values are averaged within a dataset. Numbers in bold highlight the best performance. "
|
| 1070 |
+
],
|
| 1071 |
+
"table_footnote": [],
|
| 1072 |
+
"table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">Cell tracking</td><td colspan=\"3\">Graph matching</td><td colspan=\"3\">Independent set</td><td colspan=\"3\">QAPLib</td></tr><tr><td>g1</td><td>E(×108）</td><td>t[s]</td><td>91</td><td>E(×104)</td><td>t[s]</td><td>91</td><td>E(×108)</td><td>t[s]</td><td>91</td><td>E(×106)</td><td>t[]</td></tr><tr><td>Gurobi [23]</td><td>18</td><td>-3.852</td><td>809</td><td>9</td><td>-4.8433</td><td>278</td><td>14</td><td>−2.4457</td><td>52</td><td>3472</td><td>0.9</td><td>2618</td></tr><tr><td>FastDOG[1]</td><td>7</td><td>-3.863</td><td>1005</td><td>21</td><td>-4.8912</td><td>61</td><td>42</td><td>-2.4913</td><td>9</td><td>276</td><td>5.7</td><td>1680</td></tr><tr><td>DOGE</td><td>2.4</td><td>-3.854</td><td>1015</td><td>0.3</td><td>-4.8439</td><td>17</td><td>0.3</td><td>-2.4460</td><td>8</td><td>320</td><td>12.1</td><td>720</td></tr><tr><td>DOGE-M</td><td>2.1</td><td>-3.854</td><td>730</td><td>0.2</td><td>-4.8436</td><td>21</td><td>0.2</td><td>-24459</td><td>5</td><td>131</td><td>14.5</td><td>861</td></tr></table>",
|
| 1073 |
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|
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|
| 1080 |
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},
|
| 1081 |
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{
|
| 1082 |
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"type": "image",
|
| 1083 |
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"img_path": "images/0a31741462078393f5c916abcea1158cbb9604e5432d80e75f71021dd2ce14cf.jpg",
|
| 1084 |
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"image_caption": [
|
| 1085 |
+
"Figure 2: Convergence plots for $g ( t )$ defined in (6), the relative dual gap to the optimum (or maximum suboptimal objective among all methods) of the relaxation (D). Both axes are logarithmic. "
|
| 1086 |
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],
|
| 1087 |
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"image_footnote": [],
|
| 1088 |
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"bbox": [
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|
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"type": "text",
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| 1098 |
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"text": "275 40 nodes within the time limit of one hour. We show convergence plots for smaller instances in the \n276 Appendix. The difference to Gurobi is most pronounced w.r.t. anytime performance measured by $g _ { I }$ \n277 since our solver reaches good solutions relatively early. ",
|
| 1099 |
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"bbox": [
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|
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"type": "text",
|
| 1109 |
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"text": "Limitations While our approach gives solutions of high accuracy for the presented datasets, we have also tried our approach on other datasets (small cell tracking instances, MRFs for protein folding [27] and shape matching [51, 52]) where we were not able to obtain significant improvements w.r.t. the non-learned baseline [1]. For small cell tracking instances FastDOG already found the optimum in a moderate number of iterations, making it hard to beat. On shape matching and protein folding the parallelization of FastDOG did not bring enough speed-ups due to few large subproblems resulting in sequential bottlenecks. This limited the number of training iterations we could perform within a reasonable time. ",
|
| 1110 |
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"type": "text",
|
| 1120 |
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"text": "We have set some hyperparameters in a dataset-dependent way. This was partly necessitated due to problem sizes e.g., training on long time horizons was not possible with very large instances. Moreover, these instances only permitted a limited number of parameters in our neural networks. ",
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| 1121 |
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| 1130 |
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"type": "text",
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| 1131 |
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"text": "5 Conclusion ",
|
| 1132 |
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"text_level": 1,
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| 1133 |
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| 1140 |
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"type": "text",
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| 1143 |
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"text": "We have proposed a learning approach for solving relaxations to combinatorial optimization problems by backpropagating through and learning parameters for the non-learned baseline [1]. We demonstrated its potential in obtaining close to optimal solutions faster than with traditional methods. ",
|
| 1144 |
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"text": "Our work raises interesting follow-up questions: (i) Contrary to many approaches for backpropagation which replace non-smooth operations with smoothed variants (e.g. [33]) we directly compute (sub-) gradients for the non-smooth solver updates. Can smoothing of the solver help obtain a better backpropagated supervision? (ii) We argue that predicting good update steps for our solver is in itself an interesting and challenging problem for GNNs. We hope that our work can become a testbed for GNN architectures. (iii) There are a few desiderata for future learned solvers, including training universal models that generalize across different problem classes. Possibly more powerful GNNs and more involved training regimes are needed for this. ",
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"text": "References [1] Ahmed Abbas and Paul Swoboda. FastDOG: Fast discrete optimization on GPU. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2022. [2] Ahmed Abbas and Paul Swoboda. RAMA: A Rapid Multicut Algorithm on GPU. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2022. [3] Marcin Andrychowicz, Misha Denil, Sergio Gomez, Matthew W Hoffman, David Pfau, Tom Schaul, Brendan Shillingford, and Nando De Freitas. Learning to learn by gradient descent by gradient descent. Advances in neural information processing systems, 29, 2016. [4] MOSEK ApS. 9.0.105, 2022. [5] Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016. [6] Yoshua Bengio, Andrea Lodi, and Antoine Prouvost. Machine learning for combinatorial optimization: a methodological tour d’horizon. European Journal of Operational Research, 290(2):405–421, 2021. [7] Timo Berthold. 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[12] Yunjin Chen and Thomas Pock. Trainable nonlinear reaction diffusion: A flexible framework for fast and effective image restoration. IEEE Transactions on Pattern Analysis and Machine Intelligence, 39(6):1256– 1272, 2017. \n329 [13] Cplex, IBM ILOG. CPLEX Optimization Studio 12.10, 2019. \n330 [14] Jian-Ya Ding, Chao Zhang, Lei Shen, Shengyin Li, Bing Wang, Yinghui Xu, and Le Song. Accelerating primal solution findings for mixed integer programs based on solution prediction. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 1452–1459, 2020. \n333 [15] William Falcon and The PyTorch Lightning team. PyTorch Lightning, 3 2019. \n334 [16] Matthias Fey and Jan E. Lenssen. Fast graph representation learning with PyTorch Geometric. In ICLR Workshop on Representation Learning on Graphs and Manifolds, 2019. \n336 [17] FICO. FICO Xpress Optimization Suite, 2022. \n337 [18] Gerald Gamrath, Daniel Anderson, Ksenia Bestuzheva, Wei-Kun Chen, Leon Eifler, Maxime Gasse, Patrick Gemander, Ambros Gleixner, Leona Gottwald, Katrin Halbig, Gregor Hendel, Christopher Hojny, Thorsten Koch, Pierre Le Bodic, Stephen J. Maher, Frederic Matter, Matthias Miltenberger, Erik Mühmer, Benjamin Müller, Marc Pfetsch, Franziska Schlösser, Felipe Serrano, Yuji Shinano, Christine Tawfik, Stefan Vigerske, Fabian Wegscheider, Dieter Weninger, and Jakob Witzig. The SCIP Optimization Suite 7.0. Technical Report 20-10, ZIB, Takustr. 7, 14195 Berlin, 2020. \n43 [19] Maxime Gasse, Didier Chételat, Nicola Ferroni, Laurent Charlin, and Andrea Lodi. Exact combinatorial optimization with graph convolutional neural networks. arXiv preprint arXiv:1906.01629, 2019. \n345 [20] F.A. Gers, J. Schmidhuber, and F. Cummins. Learning to forget: continual prediction with LSTM. 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Version 1.7.0. \n358 [26] Zeren Huang, Kerong Wang, Furui Liu, Hui-Ling Zhen, Weinan Zhang, Mingxuan Yuan, Jianye Hao, Yong Yu, and Jun Wang. Learning to select cuts for efficient mixed-integer programming. Pattern Recognition, 123:108353, 2022. [27] Ariel Jaimovich, Gal Elidan, Hanah Margalit, and Nir Friedman. Towards an integrated protein–protein interaction network: A relational markov network approach. Journal of Computational Biology, 13(2):145– 164, 2006. [28] Wenzel Jakob, Jason Rhinelander, and Dean Moldovan. pybind11 – Seamless operability between $\\mathrm { C } { + } { + } 1 1$ and Python, 2017. https://github.com/pybind/pybind11. [29] Dagmar Kainmueller, Florian Jug, Carsten Rother, and Gene Myers. Active graph matching for automatic joint segmentation and annotation of C. elegans. In International Conference on Medical Image Computing and Computer-Assisted Intervention, pages 81–88. Springer, 2014. [30] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. [31] Jan-Hendrik Lange and Paul Swoboda. Efficient message passing for 0–1 ILPs with binary decision diagrams. In International Conference on Machine Learning, pages 6000–6010. PMLR, 2021. [32] Fuhui Long, Hanchuan Peng, Xiao Liu, Stuart K Kim, and Eugene Myers. A 3D digital atlas of C. elegans and its application to single-cell analyses. Nature methods, 6(9):667–672, 2009. [33] Arthur Mensch and Mathieu Blondel. Differentiable dynamic programming for structured prediction and attention. In International Conference on Machine Learning, pages 3462–3471. PMLR, 2018. [34] Vishal Monga, Yuelong Li, and Yonina C. Eldar. Algorithm unrolling: Interpretable, efficient deep learning for signal and image processing. IEEE Signal Processing Magazine, 38(2):18–44, 2021. \n379 [35] Vinod Nair, Sergey Bartunov, Felix Gimeno, Ingrid von Glehn, Pawel Lichocki, Ivan Lobov, Brendan O’Donoghue, Nicolas Sonnerat, Christian Tjandraatmadja, Pengming Wang, et al. Solving mixed integer programs using neural networks. arXiv preprint arXiv:2012.13349, 2020. [36] Mohammadreza Nazari, Afshin Oroojlooy, Lawrence Snyder, and Martin Takác. Reinforcement learning for solving the vehicle routing problem. Advances in neural information processing systems, 31, 2018. \n[37] NVIDIA, Péter Vingelmann, and Frank H.P. Fitzek. CUDA, release: 11.2, 2021. \n385 [38] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. PyTorch: An Imperative Style, High-Performance Deep Learning Library. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems 32, pages 8024–8035. Curran Associates, Inc., 2019. [39] Antoine Prouvost, Justin Dumouchelle, Lara Scavuzzo, Maxime Gasse, Didier Chételat, and Andrea Lodi. Ecole: A Gym-like Library for Machine Learning in Combinatorial Optimization Solvers. In Learning Meets Combinatorial Algorithms at NeurIPS2020, 2020. [40] Daniel Selsam, Matthew Lamm, Benedikt Bünz, Percy Liang, Leonardo de Moura, and David L Dill. Learning a SAT solver from single-bit supervision. arXiv preprint arXiv:1802.03685, 2018. \n396 [41] Alexander Shekhovtsov, Christian Reinbacher, Gottfried Graber, and Thomas Pock. Solving dense image matching in real-time using discrete-continuous optimization. In Proceedings of the 21st Computer Vision Winter Workshop (CVWW), page 13, 2016. \n399 [42] Yunsheng Shi, Zhengjie Huang, Shikun Feng, Hui Zhong, Wenjing Wang, and Yu Sun. Masked label prediction: Unified message passing model for semi-supervised classification. In Zhi-Hua Zhou, editor, Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence, IJCAI-21, pages 1548–1554. International Joint Conferences on Artificial Intelligence Organization, 8 2021. Main Track. \n403 [43] Nicolas Sonnerat, Pengming Wang, Ira Ktena, Sergey Bartunov, and Vinod Nair. Learning a large neighborhood search algorithm for mixed integer programs. arXiv preprint arXiv:2107.10201, 2021. [44] Paul Swoboda, Andrea Hornakova, Paul Roetzer, and Ahmed Abbas. Structured prediction problem archive. arXiv preprint arXiv:2202.03574, 2022. [45] Siddharth Tourani, Alexander Shekhovtsov, Carsten Rother, and Bogdan Savchynskyy. MPLP $^ { + + }$ : Fast, parallel dual block-coordinate ascent for dense graphical models. In Proceedings of the European Conference on Computer Vision (ECCV), pages 251–267, 2018. \n410 [46] Mark Turner, Thorsten Koch, Felipe Serrano, and Michael Winkler. Adaptive cut selection in mixed-integer linear programming. arXiv preprint arXiv:2202.10962, 2022. \n412 [47] Jan Tönshoff, Martin Ritzert, Hinrikus Wolf, and Martin Grohe. Graph Neural Networks for Maximum Constraint Satisfaction. Frontiers in Artificial Intelligence, 3, 2021. \n[48] Vladimír Ulman, Martin Maška, Klas EG Magnusson, Olaf Ronneberger, Carsten Haubold, Nathalie Harder, Pavel Matula, Petr Matula, David Svoboda, Miroslav Radojevic, et al. An objective comparison of cell-tracking algorithms. Nature methods, 14(12):1141–1152, 2017. [49] Vibhav Vineet and PJ Narayanan. CUDA cuts: Fast graph cuts on the GPU. In 2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, pages 1–8. IEEE, 2008. \n419 [50] Tomas Werner. A linear programming approach to max-sum problem: A review. IEEE transactions on pattern analysis and machine intelligence, 29(7):1165–1179, 2007. \n[51] Thomas Windheuser, Ulrich Schlickewei, Frank R Schmidt, and Daniel Cremers. Geometrically consistent elastic matching of 3d shapes: A linear programming solution. In 2011 International Conference on Computer Vision, pages 2134–2141. IEEE, 2011. [52] Thomas Windheuser, Ulrich Schlickwei, Frank R Schimdt, and Daniel Cremers. Large-scale integer linear programming for orientation preserving 3d shape matching. In Computer Graphics Forum, volume 30, pages 1471–1480. Wiley Online Library, 2011. \n427 [53] Jiadong Wu, Zhengyu He, and Bo Hong. Chapter 5 - efficient CUDA algorithms for the maximum network flow problem. In Wen mei W. Hwu, editor, GPU Computing Gems Jade Edition, Applications of GPU Computing Series, pages 55–66. Morgan Kaufmann, Boston, 2012. [54] Yaoxin Wu, Wen Song, Zhiguang Cao, and Jie Zhang. Learning large neighborhood search policy for integer programming. Advances in Neural Information Processing Systems, 34, 2021. [55] Liang Xin, Wen Song, Zhiguang Cao, and Jie Zhang. NeuroLKH: Combining Deep Learning Model with Lin-Kernighan-Helsgaun Heuristic for Solving the Traveling Salesman Problem. Advances in Neural Information Processing Systems, 34, 2021. \n435 [56] Zhiwei Xu, Thalaiyasingam Ajanthan, and Richard Hartley. Fast and differentiable message passing on pairwise markov random fields. In Proceedings of the Asian Conference on Computer Vision, 2020. \n437 [57] Yan Yang, Jian Sun, Huibin Li, and Zongben Xu. ADMM-CSNet: A Deep Learning Approach for Image Compressive Sensing. IEEE Transactions on Pattern Analysis and Machine Intelligence, 42(3):521–538, 2020. ",
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| 1 |
+
# UNIFIED VISION AND LANGUAGE PROMPT LEARNING
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| 2 |
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Anonymous authors Paper under double-blind review
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# ABSTRACT
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Prompt tuning, a parameter- and data-efficient transfer learning paradigm that tunes only a small number of parameters in a pre-trained model’s input space, has become a trend in the vision community since the emergence of large visionlanguage models like CLIP. We present a systematic study on two representative prompt tuning methods, namely text prompt tuning and visual prompt tuning. A major finding is that none of the unimodal prompt tuning methods performs consistently well: text prompt tuning fails on data with high intra-class visual variances while visual prompt tuning cannot handle low inter-class variances. To combine the best from both worlds, we propose a conceptually simple approach called Unified Prompt Tuning (UPT), which learns a tiny neural network to jointly optimize prompts across different modalities. Extensive experiments on over 11 vision datasets show that UPT achieves a better trade-off than the unimodal counterparts on few-shot learning benchmarks, as well as on domain generalization benchmarks. Code and models will be released to facilitate future research.
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# 1 INTRODUCTION
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Vision-language (VL) models pre-trained on millions of image-text pairs (e.g., CLIP (Radford et al., 2021) and ALIGN (Jia et al., 2021)) have shown excellent transferability on a variety of downstream tasks, such as few-shot learning (Zhou et al., 2022a;b; Ju et al., 2021) and open-vocabulary perception (Gu et al., 2022; Zhou et al., 2022c; Zang et al., 2022; Ghiasi et al., 2021). When adapting large VL models to downstream tasks, it is often impractical to fine-tune the entire model directly due to their huge parameter size. To make adaptation more efficiently, many studies (Gao et al., 2021; Li & Liang, 2021; Lester et al., 2021; Zhou et al., 2022a; Lu et al., 2022; Ju et al., 2021; Yao et al., 2021; Jia et al., 2022; Bahng et al., 2022) have explored prompt tuning where the idea is to fine-tune a small number of parameters in a pre-trained model’s input space, called prompt, while keeping the majority of pre-trained parameters frozen.
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A typical VL model consists of two sub-networks—an image encoder and a text encoder—to extract features from visual and textual modalities respectively. Correspondingly, existing prompt tuning approaches can be grouped into two types: text prompt tuning and visual prompt tuning. For text prompt tuning methods, e.g., CoOp (Zhou et al., 2022a), extra text prompt tokens treated as learnable parameters are applied on the text encoder (Fig. 1(a)) to mitigate the issue that hand-crafted text prompt templates (e.g., “a photo of a [CLASS].”) are often sub-optimal. On the contrary, visual prompt tuning approaches focus on modulating the image encoder (Fig. 1(b)). A representative method is VPT (Jia et al., 2022), which injects learnable parameters into multiple layers of a Vision Transformer. Notably, these prompt-based methods treat the two modalities in isolation.
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Despite significant improvements achieved recently, we observe that current prompt tuning approaches (Zhou et al., 2022a; Jia et al., 2022) fail to perform consistently due to inherent variances in visual and text features in downstream tasks. That is to say, using the unimodal prompt may obtain good results on one dataset but not on others.
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To analyze the phenomenon, we measure the discrepancy in data distribution focusing on the intraclass variance of visual features and inter-class variance of text embedding, and study the correlation between data statistics and performance improvement. As shown in Fig. 1(d), when the intra-class variance of image features is large (bottom right), CoOp struggles to learn suitable text prompts for improving the text classifier. As for visual prompt tuning, VPT faces difficulties when the interclass variance of text features is small, as shown in bottom left of Fig. 1(e). That is, if the text classifiers are based on text features of low separability, tuning visual prompts would lend little help to improve the final performance. Moreover, intra-class visual variance and inter-class text variance are typically orthogonal. As a consequence, the performance of unimodal prompt tuning methods varies widely across different datasets: CoOp beats VPT by $8 . 1 \%$ on Flowers102 (Nilsback & Zisserman, 2008) while VPT outperforms CoOp by $8 . 4 \%$ on EuroSAT (Helber et al., 2019).
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Figure 1: Top: Architectures of (a) text prompt tuning (Zhou et al., 2022a), (b) visual prompt tuning (Jia et al., 2022) and (c) our multimodal unified prompt tuning ( $\textcircled { 3 }$ : learnable; $\frac { 2 0 0 } { 9 0 0 }$ : frozen parameters). Bottom: the performance improvements $( \% )$ of text prompt tuning (d) and visual prompt tuning (e) compared with the zero-shot CLIP baseline. We show that the variance of visual and text features ( $\scriptstyle { \dot { x } }$ -axis) will affect the improvements $y$ -axis). We project the text/visual features of the dataset (pointed by the dashed arrow) into a unit sphere to show the variance of different distributions. Please refer to the appendix for the implementation details about how we compute the feature variance.
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+
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We argue that the key would be to simultaneously adapt both text and visual prompts to overcome the vast differences across different data distributions. A straightforward solution is to introduce both text and visual prompts to the model and jointly optimize the two modality-specific prompts. However, we find that such a na¨ıve joint training leads to poor performance due to the intrinsic discrepancy between text and image modalities. In particular, the performance is occasionally worse than tuning modality-specific prompts as shown in our experiments.
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| 23 |
+
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Solving the aforementioned issues requires modality-agnostic optimization to bridge the isolated prompts. To this end, we present a unified prompt tuning method for both text and visual modalities, dubbed Unified Prompt Tuning (UPT). See Fig. 1(c). Specifically, we start with a shared prompt and propose a lightweight self-attention network to generate the prompts for CLIP’s text and visual encoders respectively. We empirically show that such a conceptually simple design can preserve the benefit of individual modalities.
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Our contributions are summarized as follows. 1) We provide a comprehensive study on existing text and visual prompt tuning methods, and identify the shortcoming of unimodal learning. 2) We present a unified prompt learning method for VL models, which is simple and easy to implement. 3) We conduct extensive experiments to show that unified prompt tuning outperforms previous unimodal prompt tuning methods under the few-shot learning and domain generalization settings.
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# 2 METHODOLOGY
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We first introduce vision-language models focusing on CLIP (Radford et al., 2021), in company with text/visual prompt tuning approaches for visual recognition in Sec. 2.1. We then analyze the
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+
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limitations of previous single-modal prompt tuning approaches in Sec. 2.2. Finally, we present technical details of our proposed unified prompt learning in Sec. 2.3.
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| 33 |
+
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+
# 2.1 PRELIMINARIES
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CLIP. CLIP (Radford et al., 2021) consists of two sub-networks: an image encoder $\phi$ and a text encoder $\psi$ . These two encoders, respectively, map the text and image inputs into a joint hidden space $\mathbb { R } ^ { d }$ , where the semantics of vision and language modalities are well-aligned. Here, $d$ refers to the final hidden dimension of the text or image encoder (e.g., $d = 2 5 6$ in the ResNet (He et al., 2016) backbone and $d = 5 1 2$ in the ViT backbone). Given an input image $_ { \textbf { \em x } }$ and a set of categories $\mathbf { Y } = \{ y _ { 1 } , y _ { 2 } , . . . , y _ { k } \}$ (e.g., $k = 1 0 0 0$ for ImageNet (Deng et al., 2009)), the image encoder extracts the corresponding image feature $z = f _ { \phi } ( \pmb { x } ) \in \mathbb { R } ^ { d }$ . While the class names in $\mathbf { Y }$ are first filled into a hand-crafted text prompt template a photo of a [CLASS] to obtain the text descriptions A, further processed by the text encoder for the text representations: $\mathbf { W } = f _ { \psi } ( \mathbf { A } ) \in \mathbb { R } ^ { d \times k }$ . The final prediction is computed as follows:
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+
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+
$$
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+
p ( y = i \mid \pmb { x } ) = \frac { \exp \left( \cos \left( \pmb { w } _ { i } , \pmb { z } \right) / \tau \right) } { \sum _ { j = 1 } ^ { k } \exp \left( \cos \left( \pmb { w } _ { j } , \pmb { z } \right) / \tau \right) } ,
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| 40 |
+
$$
|
| 41 |
+
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| 42 |
+
where $\cos ( \cdot , \cdot )$ denotes the cosine similarity and $\tau$ is a fixed temperature value (e.g., $\tau = 1 0 0$ ). Conceptually, such a decision process for the input image $_ { \textbf { \em x } }$ in Eq. (1) is formulated in a way that the text encoder $\psi$ takes a role of generating dynamic classifiers W from open-set categories $\mathbf { Y }$ , with the image encoder $\phi$ producing encoded visual features $_ { z }$ . In practice, it is generally infeasible to fine-tune the millions of parameters (i.e., $\phi$ and $\psi$ ) in a VL model for transfer learning in every downstream task.
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Text Prompt Tuning. For efficient and effective model adaptation, text prompt tuning approaches consider generating more adaptive classifiers without fine-tuning the text encoder $\psi$ . For example, Context Optimization $\left( \mathbf { C o O p } \right)$ (Zhou et al., 2022a) introduce a set of learnable parameters $\textbf { T } \in$ $\mathbb { R } ^ { d \times m }$ to replace the hand-crafted text prompt template (a photo of a [CLASS]). The wordembedding of class names in $\mathbf { Y }$ will concatenate with these text prompts in the following form:
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| 45 |
+
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+
$$
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+
\mathrm { \hat { T } } = [ t _ { 1 } , t _ { 2 } , \dots , t _ { m } , \mathrm { C L A S S } ] .
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+
$$
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Here, the symbol $m$ denotes the prompt length. The resulting dynamic text representations are extracted by the text encoder: $\mathbf { W } = f _ { \psi } ( \hat { \mathbf { T } } ) \in \mathbb { R } ^ { d \times k }$ . In each downstream task, the learnable prompts $\mathbf { T }$ will be optimized with each task-specific objective function, e.g., a cross-entropy classification loss $\mathcal { L } _ { \mathrm { C E } } ( p , y )$ in few-shot learning. Note that both the image and text encoders $\cdot \phi$ and $\psi$ ) are frozen during downstream training. As a result, updating the text prompt $\mathbf { T }$ will correspondingly adjust the decision boundaries with generated classifiers $\mathbf { W }$ for downstream tasks.
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Visual Prompt Tuning. Conversely, visual prompt tuning methods focus on extracting more transferable visual features while keeping the visual encoder $\phi$ unchanged. Following the success of text prompt tuning approaches, recent Visual Prompt Tuning (VPT) (Jia et al., 2022) introduces a similar prompt tuning recipe for the visual encoder $\phi$ . Suppose the image encoder $\phi$ contains $L$ Vision Transformer layers, the output of $i$ -th layer, $l _ { i }$ , where $i = 1 , 2 , \dots , L$ , is given by:
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+
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+
$$
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+
[ { \pmb { c } } ^ { i + 1 } , z _ { 1 } ^ { i + 1 } , \dots , z _ { s } ^ { i + 1 } ] = l _ { i } \left( \left[ { \pmb { c } } ^ { i } , z _ { 1 } ^ { i } , \dots , z _ { s } ^ { i } \right] \right) ,
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| 56 |
+
$$
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+
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+
where $c \in \mathbb { R } ^ { d }$ denotes the classification token ([CLS]), and $Z = [ z _ { 1 } , z _ { 2 } , \ldots , z _ { s } ] \in \mathbb { R } ^ { d \times s }$ denotes the input image patch tokens with length $s$ . For the $i$ -th encoder layer, a set of learnable visual prompts $\mathbf { V } ^ { i } \in \mathbb { R } ^ { \tilde { d } \times n }$ are inserted and computed as follows:
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| 59 |
+
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+
$$
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+
[ { \pmb { c } } ^ { i + 1 } , \ldots , { \pmb { Z } } ^ { i + 1 } ] = l _ { i } \left( \left[ { \pmb { c } } ^ { i } , { \pmb { V } } ^ { i } , { \pmb { Z } } ^ { i } \right] \right) ,
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| 62 |
+
$$
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+
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where $n$ stands for the length of visual prompts. Two VPT variants are proposed: VPT-shallow and VPT-deep. For VPT-shallow, the visual prompts are only inserted into the first Transformer layer $( i = 1 )$ ). Whereas for VPT-deep, visual prompts are introduced at every layer. The learnable visual prompts are data-independent, which once learned, can modulate the visual features $_ z$ of input images for better downstream transfer learning.
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Figure 2: Visualization of input features $_ { z }$ (projected points) and text classifier W (projected lines) on EuroSAT and Flowers102.
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# 2.2 ANALYSIS
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We conduct a series of probing studies to analyze the characteristics of text/visual prompt tuning. First, when adapting the CLIP model with two representative text and visual prompt tuning approaches $\mathrm { C o O p }$ (Zhou et al., 2022a) and VPT (Jia et al., 2022)), we measure the variance of both visual features $_ z$ and text embeddings W (i.e., classifiers) for all 11 downstream vision datasets (see Appendix for detailed implementations). For text prompt tuning, as shown in Fig. 1(d), we observe that CoOp performs well on datasets with low intra-class variance between visual features, such as Flowers102, but fails on Food101 dataset with high intra-class feature variance. As for visual prompt tuning, VPT succeeds in improving performance on SUN397 dataset with large inter-class text embeddings, while being less effective on Food101 and Flowers102 with relatively smaller inter-class text embedding variance. The performance improvements of text/visual prompt tuning are highly correlated with the variance of visual features $_ z$ or text embeddings W in downstream datasets.
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In order to understand this phenomenon, we select two downstream vision datasets (Flowers102 (Nilsback & Zisserman, 2008), EuroSAT (Helber et al., 2019)) for further analysis. During downstream training, we project both visual features $_ z$ and text embeddings W (i.e., classifiers) into joint sphere space ${ \bar { \mathbb { R } } } ^ { 3 }$ for better visualization. As we illustrated in Fig. 2, we can observe that: 1) For the EuroSAT dataset with high intra-class visual feature variance, text prompts in $\mathrm { C o O p }$ fails to adapt the text classifiers W. Clearly, the text classifiers in Fig. $2 ( \mathbf { b } )$ are almost unchanged compared with zero-shot CLIP baseline (Fig. 2(a)). 2) For the Flowers102 dataset with low inter-class text embedding variance, visual prompts in VPT are not effective in modulating the visual features $_ { z }$ (Fig. ${ \bf \Pi } ( \mathbf { g } )$ ), thus cannot obtain considerable performance gain.
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In conclusion, the single-modal prompt tuning approaches $\mathrm { C o O p }$ and VPT), face the dilemma that consistent improvements over Zero-shot CLIP are hard to achieve due to inherent variances of visual features and text embedding in downstream tasks. Our observation motivates us to present a unified prompt tuning method that tunes the $_ z$ and W at the same time.
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# 2.3 UNIFIED PROMPT TUNING
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Driven by our analysis, we devise a simple yet effective multi-modal Unified Prompt Tuning (UPT) approach for adapting VL models. Specifically, instead of introducing two sets of isolated modalityspecific prompts (i.e., T in Eq. (2) and $\mathbf { V }$ in Eq. (4)) for the text and visual encoders, we consider learning a set of unified modality-agnostic prompts for tuning VL models. As shown in Fig. 3, we define a set of learnable prompts $\breve { U } \in \mathbb { R } ^ { \tilde { d } \times n }$ with length $n$ . Rather than na¨ıvely appending the unified prompts into the text and visual encoders, we employ a lightweight Transformer layer $\theta$ to
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Figure 3: The architecture of (a) our unified prompt $U$ that is applied to $\mathbf { ( b ) }$ CLIP text encoder and (c) CLIP image encoder.
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+
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+
transform unified prompts $U$ as follows:
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+
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| 86 |
+
$$
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+
\begin{array} { r l } & { U ^ { \prime } = \mathrm { S A } \left( U \right) + \mathrm { L N } \left( U \right) , } \\ & { \hat { U } = \mathrm { F F N } \left( \mathrm { L N } \left( U ^ { \prime } \right) \right) + \mathrm { L N } \left( U ^ { \prime } \right) , } \end{array}
|
| 88 |
+
$$
|
| 89 |
+
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+
where the self-attention operator SA, feed-forward network FFN and layer normalization LN are applied to obtain the transformed prompts $\hat { U }$ . The self-attention module in the lightweight Transformer layer allows beneficial interaction between two modalities, so as to maximize the complementary effects. Our unified prompts can be introduced into multiple layers of VL models. In particular, for each $i$ -th layer of text and image encoders, we consider learning a set of layer-wise prompts $U ^ { i }$ , and split transformed $\hat { \pmb { U } } ^ { i }$ into two parts $\hat { U } ^ { i } = \{ \hat { U } _ { t } ^ { i } , \hat { U } _ { v } ^ { i } \}$ , sending into the text and visual encoders respectively. During downstream training, we froze both the text and visual encoder $\dot { \psi }$ and $\phi$ ) and only optimize the unified prompts $U$ and the lightweight Transformer layer $\theta$ . In this way, both the dynamic classifiers W and visual features $_ z$ in Eq. (1) are effectively tuned for reliable prediction in the downstream task. As shown in Fig. 2 (d) and (h), our unified prompts can simultaneously obtain well-aligned text classifiers and separable visual features compared with single-modal counterparts.
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# 3 EXPERIMENTS
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In this section, we conduct experiments under two problem settings, i.e., (i) few-shot image classification (Sec. 3.1) and (ii) domain generalization (Sec. 3.2). We also present ablation studies in Sec. 3.3 on several design choices and extra experimental results about generalizability in Appendix.
|
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Baselines. We compare our approach against the following methods: (1) Zero-shot CLIP. This baseline uses hand-crafted text prompt templates and does not involve any prompt-learning strategies. (2) Single-modal Prompt Tuning methods, including CoOp (Zhou et al., 2022a) and ProDA (Lu et al., 2022) for the text modality, and VPT (Jia et al., 2022) for the visual modality. In the domain generalization setting, we further compare with CoCoOp (Zhou et al., 2022b), which improves CoOp’s generalization performance with an input-conditional design. For VPT, we report the results of both the shallow and deep variants, as described in Sec. 2.1.
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# 3.1 FEW-SHOT LEARNING
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| 100 |
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In this section, we measure a model’s generalization ability by conducting prompt tuning using different strategies, with just a limited amount of labeled examples per-class in the specific downstream task. Detailed implementation is presented in Appendix.
|
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+
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Datasets. We follow (Zhou et al., 2022b) to use 11 datasets (ImageNet (Deng et al., 2009), Caltech101 (Fei-Fei et al., 2004), OxfordPets (Parkhi et al., 2012), StanfordCars (Krause et al., 2013), Flowers102 (Nilsback & Zisserman, 2008), Food101 (Bossard et al., 2014), FGVC-Aircraft (Maji et al., 2013), SUN397 (Xiao et al., 2010), UCF101 (Soomro et al., 2012), DTD (Cimpoi et al., 2014), EuroSAT (Helber et al., 2019)) as our benchmarks. Following (Zhou et al., 2022a), we use the fewshot evaluation protocol selecting 1/2/4/8/16 shots for training and the whole test set for evaluation. We report averaged results over three runs with different random seeds to reduce the variance. The detailed results are shown in Fig. 4.
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|
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Limitation of Single-modal Baselines. Figure 4 shows that the performance improvements of existing text prompt tuning method CoOp and visual prompt tuning method VPT are not consistent across different datasets. In particular, CoOp obtains better performance than VPT on some datasets, such as StanfordCars and SUN397. However, for other datasets with high intra-class visual variances, VPT is much more effective than CoOp. For instance, on the EuroSAT dataset, VPT-deep beats $\mathrm { C o O p }$ by over $12 \%$ . The discrepancy of previous single-modal baselines is also consistent with our motivation in Fig. 1(d) and Fig. 1(e). According to VPT (Jia et al., 2022), VPT-deep is more effective than VPT-shallow, and our experimental results also verify this point. We later show that VPT-shallow obtains much stronger performance than the VPT-deep in the domain generalization setting (Sec. 3.2).
|
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+
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+
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+
Figure 4: Main results over 11 datasets under the few-shot learning setting. We report the average accuracy $( \% )$ of 1/2/4/8/16 shots over three runs. Overall, the proposed UPT (blue line) achieves apparent improvements compared with the Zero-shot CLIP and single-modal prompt tuning baselines $\mathrm { C o O p }$ , ProDA and VPT).
|
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+
|
| 109 |
+
UPT vs. Single-modal Baselines. Our UPT achieves clear advantages over the single-modal prompt-tuning counterparts CoOp, ProDA and VPT, as suggested by the averaged performance (topleft of Fig. 4). In general, the average performance gap between UPT and baselines increases with the shot number available for prompt tuning. Specifically, UPT obtains $0 . 4 8 / 1 . 3 6 / 1 . 2 9 / 2 . 4 6 / 3 . 1 9 ( \% )$ accuracy improvements compared with the text prompt tuning method CoOp on 1/2/4/8/16 shots settings. Even compared with the strong text prompt tuning baseline ProDA, UPT still boosts the accuracy of $0 . 1 1 / 1 . { \overset { \cdot } { 0 } } 1 / 0 . 6 7 / 1 . 5 / 1 . 6 1 ( \% )$ . Similarly, UPT achieves $0 . 8 9 / 2 . 7 0 / 2 . 0 3 / 2 . 4 0 / 2 . 0 1 ( \% )$ accuracy gains over the visual prompt tuning approach VPT-deep. Notably, UPT significantly boosts the performance over CoOp and VPT-deep on challenging large datasets, such as ImageNet with 1,000 classes and SUN397 with 397 categories. UPT also surpasses CoOp and VPT-deep on finegrained datasets such as StanfordCars and FGVC Aircraft. We also observe that UPT shows less improvement on the two datasets (OxfordPets and Food101), possibly caused by the noisy training data (Zhou et al., 2022a; Bossard et al., 2014). Overall, the experimental results in Fig. 4 demonstrate the effectiveness of our proposed UPT.
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+
Table 1: Main results under the domain generalization setting. We report the average accuracy $( \% )$ of 16 shots over three runs. The best and second best methods are highlighted in red and orange , respectively.
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<table><tr><td rowspan="2">#</td><td rowspan="2">Method</td><td>Source</td><td colspan="4">Target</td><td rowspan="2">Overall Average</td><td rowspan="2">00D Average</td></tr><tr><td>ImageNet</td><td>-V2</td><td>-S</td><td>-A</td><td>-R</td></tr><tr><td></td><td>CoOp</td><td>71.51</td><td>64.20</td><td>47.99</td><td>49.71</td><td>75.21</td><td>61.72</td><td>59.28</td></tr><tr><td></td><td>CoCoOp</td><td>71.02</td><td>64.07</td><td>48.75</td><td>50.63</td><td>76.18</td><td>62.13</td><td> 59.91</td></tr><tr><td></td><td>VPT-shallow</td><td>68.98</td><td>62.10</td><td>47.68</td><td>47.19</td><td>76.10</td><td>60.38</td><td>58.27</td></tr><tr><td>１２３４</td><td>VPT-deep</td><td>70.57</td><td>63.67</td><td>47.66</td><td>43.85</td><td>74.42</td><td>60.04</td><td>57.40</td></tr><tr><td>5</td><td>Joint Training</td><td>71.42</td><td>64.36</td><td>48.20</td><td>49.71</td><td>76.23</td><td>61.97</td><td>59.61</td></tr><tr><td>6</td><td>Shared</td><td>71.46</td><td>64.43</td><td>48.13</td><td>50.03</td><td>75.76</td><td>61.96</td><td>59.55</td></tr><tr><td>7</td><td>MLP</td><td>71.00</td><td>64.11</td><td>48.65</td><td>48.76</td><td>76.14</td><td>61.78</td><td>59.48</td></tr><tr><td>8</td><td>UPT</td><td>72.63</td><td>64.35</td><td>48.66</td><td>50.66</td><td>76.24</td><td>62.51</td><td> 59.98</td></tr></table>
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Figure 5: Ablation studies on different design choices. (a): jointly train the existing text and visual prompt tuning approaches; (b): shared prompts for all modalities; (c): using two MLP layers to generate the prompts.
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# 3.2 DOMAIN GENERALIZATION
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Pre-trained VL models like CLIP have shown strong generalization ability. However, the prompt tuned on a specific downstream dataset may hinder the generalization ability on categories outside the training set. In this section, we evaluate the generalization ability of different prompt tuning methods on out-of-distribution (OOD) data.
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Datasets. We follow (Zhou et al., 2022a) to use five datasets (ImageNet (Deng et al., 2009), ImageNet V2 (Recht et al., 2019), ImageNet-Sketch (Wang et al., 2019), ImageNet-A (Hendrycks et al., 2021b) and ImageNet-R (Hendrycks et al., 2021a)) for evaluation. Following the protocol, we train a model on ImageNet and evaluate it on four other variants of ImageNet with their domains shifted.
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Results. Table 1 summarizes the results. We report the average accuracy on both the source and target datasets (penultimate column), and the OOD average accuracy on target datasets (last column). The results show that VPT-shallow (row #2) achieves higher OOD accuracy than VPT-deep (row #3), and text prompt tuning methods outperform visual prompt tuning approaches. Furthermore, the proposed UPT (row #8) is generally a better option than single-modal baselines (rows #1-#4) and obtains comparable performance with CoCoOp. Our UPT achieves the best results three times on five datasets, showing that UPT is a reliable prompt tuning method among its competitors in the domain generalization setting.
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# 3.3 ABLATION STUDIES
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Comparison with the Joint Training Baseline. As shown in Fig. 5(a), a straightforward approach for multi-modal prompts is tune the text prompt (using CoOp) and visual prompt (using VPT) jointly.
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Table 2: Ablation studies on different multi-modal prompt design choices in Fig. 5 over 11 datasets. We report the accuracy results under the 16 shots setting. The best and second best methods are highlighted in red and orange , respectively.
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<table><tr><td>#</td><td>Prrega</td><td>eee</td><td>Grreeaer</td><td>s1edpitrit</td><td>ssrrprteets</td><td>Tiroeni0</td><td></td><td>FTaaeieelr [orpoon</td><td></td><td>163308</td><td></td><td>JtoSSS</td><td>UUIIII</td><td>2neace</td></tr><tr><td>1</td><td>CoOp</td><td>71.36</td><td>95.93</td><td>92.74</td><td>77.45</td><td>95.90</td><td>86.36</td><td>38.04</td><td>73.59</td><td>68.38</td><td>78.77</td><td></td><td>82.04</td><td>78.24</td></tr><tr><td>2</td><td>VPT-shallow</td><td>68.98</td><td>94.66</td><td>92.61</td><td>69.09</td><td></td><td>81.40</td><td>86.91</td><td>30.93</td><td>68.08</td><td>52.28</td><td>84.87</td><td>75.19</td><td>73.18</td></tr><tr><td>3</td><td>VPT-deep</td><td>70.57</td><td>95.83</td><td>92.91</td><td>76.13</td><td></td><td>94.96</td><td>86.18</td><td>40.96</td><td>71.63</td><td>69.79</td><td>91.53</td><td>82.76</td><td>79.39</td></tr><tr><td>4</td><td>Joint Training</td><td>71.42</td><td>95.84</td><td></td><td>93.34 79.02</td><td></td><td>95.25</td><td>86.55</td><td>40.56</td><td>74.17</td><td>67.83</td><td>78.94</td><td>82.81</td><td>78.70</td></tr><tr><td>5</td><td>Shared</td><td>71.46</td><td>95.50</td><td>92.99</td><td>78.66</td><td></td><td>95.55</td><td>86.67</td><td>39.18</td><td>73.64</td><td>67.69</td><td>73.36</td><td>82.06</td><td>77.88</td></tr><tr><td>6</td><td>MLP</td><td>71.00</td><td>95.59</td><td>93.74</td><td>75.88</td><td>93.38</td><td></td><td>87.20</td><td>37.17</td><td>72.74</td><td>67.31</td><td>90.66</td><td>81.43</td><td>78.73</td></tr><tr><td>7</td><td>UPT (Ours)</td><td>72.63 95.94</td><td></td><td>92.95</td><td></td><td>84.33 97.11</td><td></td><td>85.00</td><td>46.80</td><td>75.92 70.65</td><td></td><td>90.51</td><td>84.03 81.44</td><td></td></tr></table>
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Figure 6: Visualization of attention response map between visual prompts and image patch tokens. The images are test images from ImageNet. We visualize the self-attention module from the last block of ViT of the CLIP image encoder.
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We investigate the effectiveness of such joint training scheme, and report its results in Table 2 row #4 and Table 1 row #5. On the few-shot learning setting, we see that such a joint training solution performs better than $\mathrm { C o O p }$ and VPT-shallow, which shows that multi-modal optimization is helpful to a certain extent. But the joint training approach obtains slightly worse accuracy than the VPT-deep $7 8 . 7 0 \%$ vs. $7 9 . 3 9 \%$ ) since VPT-deep involves a large number of parameters. On the domain generalization setting, we find the joint training method performs much better than VPT-deep. Also, the joint training method shows inferior performance to our UPT, demonstrating that our self-attention base mechanism is more effective.
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Shared Prompts for Text and Visual Modalities. We also investigate the results of directly sharing prompts for different modalities. As shown in Fig. 5(b), the shared prompts will be optimized for both text and visual modalities. This scheme differs from the proposed UPT, where the shared prompts are transformed with self attention. Experimental results are presented in Table 2 row #5 and Table 1 row #6, and we observe that such a prompt sharing strategy achieves worst performance among all the ablation design choices.
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MLP Baseline. For our proposed UPT, we use a Transformer layer with the self-attention operator to partially share the hyper-parameters for different modalities. Here, we study a simpler design that generates the unified prompts with two MLP layers. Results are presented on Table 2 row #6 and Table 1 row #7. The MLP baseline is still competitive, yielding best performance on two datasets. Nonetheless, the average results is still poorer than the proposed self-attention based approach.
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# 3.4 QUALITATIVE RESULTS
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While it is hard to visualize what have been learned during text prompt tuning, it is possible to visualize the visual prompts learned by VPT and UPT following the self-supervised learning method, DINO (Caron et al., 2021). In particular, for each layer of the Vision Transformer (ViT), we can compute the self-attention response map of visual prompts and image patch tokens. Figure 6 compares such response maps by VPT and the proposed UPT. We find that UPT shows stronger selfattention responses compared with VPT. This could be the possible reason why UPT achieves better performance on the few-shot learning and the OOD generalization settings.
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# 4 RELATED WORK
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Vision-Language Models. Recent vision-language pre-trained models (Radford et al., 2021; Jia et al., 2021) use the contrastive loss to align an image encoder (e.g., ViT (Dosovitskiy et al., 2021)) and a text encoder (e.g., BERT (Kenton & Toutanova, 2019)) in a common feature space. These vision-language models are trained on web-scale image-text pairs and are transferable across various downstream tasks such as point cloud classification (Zhang et al., 2022a), video classification (Qian et al., 2022), object detection (Gu et al., 2022; Du et al., 2022; Zhou et al., 2022c; Zang et al., 2022) and semantic segmentation (Ghiasi et al., 2021). In this work, we aim to explore how to adapt the CLIP model to the downstream few-shot recognition task.
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Text Prompt Tuning. The concept of prompt tuning was first proposed in the NLP area (Liu et al., 2021; Gao et al., 2021; Li & Liang, 2021; Lester et al., 2021). In particular, a text prompt refers to a task-specific template for language models. For example, in sentiment analysis, the template might be “I [MASK] the movie.” where the mask placeholder will be filled with either “love” or “hate.” Common practices in text prompt tuning include (i) searching for a specific word in the dictionary, known as hard prompt learning (Gao et al., 2021), or (ii) turning masked tokens into learnable vectors, known as soft prompt learning (Li & Liang, 2021; Lester et al., 2021). Text prompt tuning has also been applied in computer vision after the emergence of large vision-language models (e.g., CLIP (Radford et al., 2021)), which are too big to fine-tune. A representative work is CoOp (Zhou et al., 2022a), which turned the input context tokens in CLIP’s text branch into learnable vectors for adapting CLIP to downstream image recognition. Other follow-ups of $\mathrm { C o O p }$ include CoCoOp (Zhou et al., 2022b), DualCoOp (Sun et al., 2022), ProGrad (Xing et al., 2022), and ProDA (Lu et al., 2022).
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Visual Prompt Tuning. The idea of visual prompt tuning is to adapt large pre-trained Vision Transformers (Dosovitskiy et al., 2021) by adding learnable parameters in the visual input space, which is analogous to text prompt tuning in NLP. VPT (Jia et al., 2022) and Visual Prompting (Bahng et al., 2022) both add trainable tokens to the input of Transformer models. A recent work, NOAH (Zhang et al., 2022b), uses neural architecture search algorithms to identify the optimal configuration of prompt modules. In comparison to the unimodal prompt learning methods discussed above, our paper provides a timely study on how to achieve a better trade-off using multimodal prompt learning.
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# 5 CONCLUSION
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With the rapid scaling of vision models along the size dimension, efficient downstream adaptation methods have become essential for facilitating large-scale deployment of vision models in the wild. Our paper provides a timely and comprehensive study on how to adapt large vision-language models like CLIP from the prompt learning perspective. In particular, our study unveils that the previous unimodal prompt tuning methods do not work consistently well across different computer vision datasets. In contrast, the proposed UPT method, despite having a simple design, achieves a better trade-off compared with the unimodal counterparts. The results suggest that one should exploit correspondences between different modalities for prompt learning.
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On the other hand, the results achieved by UPT are by no means perfect: in the ablation studies we observe that some alternative designs, such as using MLP instead of Transformer, might sometimes give better performance. In summary, we believe multimodal prompt learning is a promising framework, and we expect more improvements to be achieved with more advanced (and efficient) designs.
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# Appendix
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In the supplementary materials, we discuss the implementation details and more experimental results. Section A explains how we compute the intra-/inter- class variance for Fig.(1) of the main paper. Section B reports the implementation details of our paper. Section C presents more experimental results under the base-to-new generalization and cross-dataset transfer settings.
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# A INTRA-/INTER- CLASS VARIANCE
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In this section, we provide the implementation details about how we compute the intra-class visual variance and inter-class text variance for different datasets (Fig.1 (d)(e) in the main paper.
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Intra-class Visual Variance. Given one dataset with $k$ classes in total, for each image $_ { \textbf { \em x } }$ that belongs to class $c$ , we first use the CLIP image encoder $\phi$ to extract the corresponding image feature $f _ { \phi } ( \bar { \pmb x ) }$ . Then we get the intra-class variance of class $c$ as:
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$$
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\mathsf { v a r } _ { c } = \frac { 1 } { \vert \vert X _ { c } \vert \vert } \sum _ { x \in X _ { c } } \left( f _ { \phi } ( \pmb { x } ) - \bar { f } _ { \phi } ( \pmb { x } ) \right) ^ { 2 } ,
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$$
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where $X _ { c }$ denotes to the set of images that have the ground-truth class label $c$ , and $\bar { f } _ { \phi } ( \pmb { x } )$ refers to the mean values of class $c$ . Then we can compute the intra-class variance $\operatorname { V a r } _ { \mathrm { v } }$ for all the $k$ classes as
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$$
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\mathrm { V a r } _ { \mathrm { v } } = { \frac { 1 } { k } } \sum _ { c = 1 } ^ { k } \mathrm { v a r } _ { c } .
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$$
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Inter-class Text Variance. For each dataset, we first compute the CLIP text features $\pmb { w }$ of classs $c$ , and the mean value $\bar { \pmb w }$ of all the $k$ classes. Then we get the inter-class text variance $\mathrm { V a r } _ { \mathrm { t } }$ as:
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$$
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\mathrm { V a r _ { t } } = { \frac { 1 } { k } } \sum _ { c = 1 } ^ { k } ( w _ { c } - { \bar { w } } ) ^ { 2 } .
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$$
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# B IMPLEMENTATION DETAILS
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Our implementation is based on the source code of $\mathrm { C o O p }$ (Zhou et al., 2022a). We use ViT-B/16 as the CLIP backbone (Radford et al., 2021). Following (Zhou et al., 2022b), we set the context length of $\mathrm { C o O p }$ as $m = 4$ (same for VPT). For Zero-shot CLIP and VPT, we use the default prompt template, “a photo of a [CLS].” We use SGD as the optimizer, with an initial learning rate of 0.002, which is decayed by the cosine annealing rule. The batch size is set to 32 for all datasets.
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# C MORE EXPERIMENTAL RESULTS
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Recent work CoCoOp (Zhou et al., 2022b) points out that the text prompts learned by $\mathrm { C o O p }$ (Zhou et al., 2022a) are not generalizable to novel classes and out-of-distribution data. CoCoOp defines two new settings - base-to-new generalization and cross-dataset transfer - to measure the generalizability ability of prompt learning approaches. In this section, we provide the experimental results of our UPT in these two settings.
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Datasets. We use the same 11 datasets we used in the few-shot learning setting (section 3.1 in the main paper). Following CoCoOp (Zhou et al., 2022b), we use the 16-shot protocol and report the averaged results over three runs, and set the training schedule as ten epochs. We report the accuracy on base and new classes, and the harmonic mean for base-to-novel trade-off.
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# C.1 BASE-TO-NEW GENERALIZATION
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In the base-to-new generalization setting, we split the classes into two disjoint groups - base classes and new classes. All the prompt learning approaches are required to train on the base classes, while evaluation is conducted on the base and new classes separately. The experimental results are shown in Table 3.
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Table 3: Comparison results in the base-to-new generalization setting. H: Harmonic mean (Xian et al., 2017). The best and second best methods are highlighted in red and orange , respectively. The method ‘VPT-s’ refers to VPT-shallow.
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<table><tr><td colspan="4">(a) Average over 11 datasets.</td><td colspan="4">(b) ImageNet.</td><td colspan="4">(c) Caltech101.</td></tr><tr><td></td><td>Base</td><td>New</td><td>H</td><td></td><td>Base</td><td>New</td><td>H</td><td></td><td>Base</td><td>New</td><td>H</td></tr><tr><td>CLIP</td><td>69.34</td><td>74.22</td><td>71.70</td><td>CLIP</td><td></td><td>72.43 68.14</td><td>70.22</td><td>CLIP</td><td></td><td>96.8494.00</td><td>95.40</td></tr><tr><td>CoOp</td><td>82.69</td><td>63.22</td><td>71.66</td><td>CoOp</td><td>76.47</td><td>67.88</td><td>71.92</td><td>CoOp</td><td>98.00</td><td>89.81</td><td>93.73</td></tr><tr><td>CoCoOp</td><td>80.47</td><td>71.69</td><td>75.83</td><td>CoCoOp</td><td>75.98</td><td>70.43</td><td>73.10</td><td>CoCoOp</td><td>97.96</td><td>93.81</td><td>95.84</td></tr><tr><td>VPT-s</td><td>73.32</td><td>73.21</td><td>73.16</td><td>VPT-s</td><td>74.47</td><td>69.13</td><td>71.70</td><td>VPT-s</td><td>97.47</td><td>93.80</td><td>95.60</td></tr><tr><td> VPT-deep</td><td>75.81</td><td>72.40</td><td>73.97</td><td>VPT-deep</td><td>75.80</td><td>68.76</td><td>72.11</td><td> VPT-deep</td><td>97.50</td><td>94.10</td><td>95.77</td></tr><tr><td>UPT</td><td>76.88</td><td></td><td>75.5776.15</td><td>UPT</td><td>75.83</td><td>70.8073.23</td><td></td><td>UPT</td><td>97.70</td><td></td><td>95.6396.14</td></tr><tr><td colspan="4">(d) OxfordPets.</td><td colspan="4">(e) StanfordCars.</td><td colspan="4">(f) Flowers102.</td></tr><tr><td></td><td>Base</td><td>New</td><td>H</td><td></td><td>Base</td><td>New</td><td>H</td><td></td><td>Base</td><td>New</td><td>H</td></tr><tr><td>CLIP</td><td>91.17</td><td>97.26</td><td>94.12</td><td>CLIP</td><td>63.37</td><td>74.89</td><td>68.65</td><td>CLIP</td><td>72.08</td><td>77.80</td><td>74.83</td></tr><tr><td>CoOp CoCoOp</td><td>93.67</td><td>95.29</td><td>94.47</td><td>CoOp</td><td>78.12</td><td>60.40</td><td>68.13</td><td>CoOp</td><td>97.60</td><td>59.67</td><td>74.06</td></tr><tr><td>VPT-s</td><td>95.20</td><td>97.69</td><td>96.43</td><td>CoCoOp</td><td>70.49</td><td>73.59</td><td>72.01</td><td>CoCoOp</td><td>94.87</td><td>71.75</td><td>81.71</td></tr><tr><td></td><td>93.90</td><td>96.87</td><td>95.36</td><td>VPT-s</td><td>66.00</td><td>74.23</td><td>69.88</td><td>VPT-s</td><td>75.83</td><td>75.73</td><td>75.78</td></tr><tr><td>VPT-deep</td><td>94.33 95.50</td><td></td><td>94.91</td><td>VPT-deep</td><td>69.23</td><td>74.03</td><td>71.55</td><td>VPT-deep</td><td>83.63</td><td>70.50</td><td>76.50</td></tr><tr><td>UPT</td><td colspan="3">96.07 97.6096.32</td><td>UPT</td><td>68.5075.37</td><td></td><td>71.77</td><td>UPT</td><td>85.00</td><td></td><td>77.3380.99</td></tr><tr><td></td><td colspan="3">(g) Food101.</td><td></td><td>(h) FGVCAircraft.</td><td></td><td></td><td></td><td>(i) SUN397.</td><td></td><td></td></tr><tr><td></td><td colspan="3">Base New</td><td>H</td><td>Base</td><td>New</td><td>H</td><td></td><td>Base</td><td>New</td><td>H</td></tr><tr><td>CLIP CoOp</td><td>90.10</td><td>91.22</td><td>90.66</td><td>CLIP</td><td>27.19</td><td>36.29</td><td>31.09</td><td>CLIP</td><td>69.36</td><td>75.35</td><td>72.23</td></tr><tr><td>CoCoOp</td><td>88.33</td><td>82.26</td><td>85.19</td><td>CoOp</td><td>40.44</td><td>22.30</td><td>28.75</td><td>CoOp</td><td>80.60</td><td>65.89</td><td>72.51</td></tr><tr><td>VPT-s</td><td>90.70</td><td>91.29</td><td>90.99</td><td>CoCoOp</td><td>33.41</td><td>23.71</td><td>27.74</td><td>CoCoOp</td><td>79.74</td><td>76.86</td><td>78.27</td></tr><tr><td></td><td>90.17</td><td>90.97</td><td>90.56</td><td>VPT-s</td><td>30.83</td><td>35.17</td><td>32.86</td><td>VPT-s</td><td>75.40</td><td>77.27</td><td>76.32</td></tr><tr><td>VPT-deep</td><td>90.20 91.17</td><td></td><td>90.68</td><td>VPT-deep</td><td>33.40</td><td>35.17</td><td>34.26</td><td>VPT-deep</td><td>78.23 76.63</td><td></td><td>77.43</td></tr><tr><td>UPT</td><td colspan="3">90.72 92.0091.35</td><td>UPT</td><td>32.76</td><td>36.10</td><td>34.53</td><td>UPT</td><td>78.90</td><td></td><td>78.5678.73</td></tr><tr><td></td><td colspan="3">() DTD.</td><td></td><td>(k) EuroSAT.</td><td></td><td></td><td></td><td>(l) UCF101.</td><td></td><td></td></tr><tr><td></td><td>Base</td><td>New</td><td>H</td><td></td><td>Base</td><td>New</td><td>H</td><td></td><td>Base</td><td>New</td><td>H</td></tr><tr><td>CLIP CoOp</td><td>53.24</td><td>59.90</td><td>56.37</td><td>CLIP</td><td>56.48</td><td>64.05</td><td>60.03</td><td>CLIP</td><td>70.53</td><td>77.50</td><td>73.85</td></tr><tr><td>CoCoOp</td><td>79.44</td><td>41.18</td><td>54.24</td><td>CoOp</td><td>92.19</td><td>54.74</td><td>68.69</td><td>CoOp</td><td>84.69</td><td>56.05</td><td>67.46</td></tr><tr><td>VPT-s</td><td>77.01</td><td>56.00</td><td>64.85</td><td>CoCoOp</td><td>87.49</td><td>60.04</td><td>71.21</td><td>CoCoOp</td><td>82.33</td><td>73.45</td><td>77.64</td></tr><tr><td>VPT-deep</td><td>55.27</td><td>57.16</td><td>56.20</td><td>VPT-s</td><td>71.67</td><td>58.87</td><td>64.64</td><td>VPT-s</td><td>75.60</td><td>76.10</td><td>75.85</td></tr><tr><td></td><td>64.87</td><td>55.40</td><td>59.76</td><td>VPT-deep</td><td>66.70</td><td>60.67</td><td>63.54</td><td> VPT-deep</td><td>80.07</td><td>74.50</td><td>77.18</td></tr><tr><td>UPT</td><td colspan="3">69.53 62.1365.63</td><td>UPT</td><td>73.57</td><td></td><td>70.4371.96</td><td>UPT</td><td>78.10</td><td></td><td>76.3377.21</td></tr></table>
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Single-modal Baselines. We observe that previous single-modal baselines perform dramatically different on the base and new splits. In particular, the text prompt tuning method CoOp achieves the highest performance on base classes and poor performance on new classes. On the contrary, visual prompt tuning approaches VPT-shallow and VPT-deep obtain high accuracy on base classes, but low accuracy on new classes. Such results show the intrinsic discrepancy between single-modal text and visual prompt tuning methods. CoOp optimizes specifically for base classes but at the expense of generalization ability on new classes. The advanced text prompt tuning approach CoCoOp with the input-conditional design achieves the best base and new trade-off among the single-modal baselines.
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Strong Generalizability of UPT. As shown in Table 3, UPT is more generalizable than baseline methods when taking into account both the base and new classes. As for base classes, UPT is better than VPT but worse than $\mathrm { C o O p }$ . This is reasonable because UPT are jointly optimized on the text and visual modalities, and the visual modality branch is not specifically for base classes. For new classes, UPT has significantly improved performance. For instance, UPT obtains $+ 1 2 . 3 5 / + 2 . 3 6 / + 3 . 1 7$ gains for CoOp/VPT-shallow/VPT-deep. UPT even achieves $+ 1 . 3 5$ gains on new classes compared with the CLIP baseline without prompt learning. In summary, the experimental results under the base-to-new generalization setting show strong generalizability of UPT.
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Table 4: Comparison results in the cross-dataset transfer setting. Prompts applied to the 10 target datasets are learned from source ImageNet dataset. The best and second best methods are highlighted in red and orange , respectively.
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<table><tr><td></td><td>Source</td><td colspan="10">Target</td></tr><tr><td></td><td>1enege</td><td>CErleeaer</td><td>DPPpprtet</td><td>srsrrretttes</td><td>TiroinG</td><td>JorPoon</td><td>FrTeiettt</td><td>160308</td><td>CII</td><td>JItoII</td><td>UUUIII</td><td>aneace</td></tr><tr><td>CoOp CoCoOp</td><td>71.51</td><td>93.70</td><td>89.14</td><td>64.51</td><td>68.71</td><td>85.30</td><td>18.47</td><td>64.15</td><td>41.92</td><td>46.39</td><td>66.55</td><td>63.88</td></tr><tr><td></td><td>71.02</td><td>94.43</td><td>90.14</td><td>65.32</td><td>71.88</td><td>86.06</td><td>22.94</td><td>67.36</td><td>45.73</td><td>45.37</td><td>68.21</td><td>65.74</td></tr><tr><td>VPT-shallow</td><td>68.98</td><td>93.07</td><td>89.63</td><td>63.63</td><td>70.50</td><td>85.03</td><td>24.01</td><td>66.30</td><td>45.13</td><td>45.56</td><td>66.80</td><td>65.33</td></tr><tr><td>VPT-deep</td><td>70.57</td><td>90.33</td><td>88.50</td><td>57.87</td><td>63.83</td><td>76.90</td><td>21.93</td><td>63.10</td><td>42.13</td><td>40.63</td><td>64.53</td><td>61.85</td></tr><tr><td>UPT</td><td>70.86</td><td>93.31</td><td></td><td></td><td></td><td>90.57 65.3372.33 86.17</td><td>24.57</td><td>67.66</td><td>45.67</td><td>44.94</td><td></td><td>68.23 65.85</td></tr></table>
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# C.2 CROSS-DATASET TRANSFER
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In the cross-dataset transfer setting, prompts learned from ImageNet are applied to ten other target datasets to evaluate the generalizability. The detailed results are presented in Table 4. We find VPT-shallow achieves higher accuracy than VPT-deep, and the text prompt tuning method CoCoOp outperforms visual prompt tuning approaches. On the source dataset, UPT obtains better performance than VPT but worse than CoOp. On target datasets, UPT obtains the best performance on six out of ten. The results on the cross-dataset transfer setting also verify that our proposed UPT is more generalizable than single-modal baselines.
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| 1 |
+
[
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| 2 |
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{
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| 3 |
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"type": "text",
|
| 4 |
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"text": "UNIFIED VISION AND LANGUAGE PROMPT LEARNING ",
|
| 5 |
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"text_level": 1,
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| 6 |
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"bbox": [
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| 13 |
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{
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| 15 |
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"type": "text",
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| 16 |
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"text": "Anonymous authors Paper under double-blind review ",
|
| 17 |
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"bbox": [
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| 18 |
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| 19 |
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{
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"type": "text",
|
| 27 |
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"text": "ABSTRACT ",
|
| 28 |
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"text_level": 1,
|
| 29 |
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"bbox": [
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| 30 |
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| 31 |
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| 38 |
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"type": "text",
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| 39 |
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"text": "Prompt tuning, a parameter- and data-efficient transfer learning paradigm that tunes only a small number of parameters in a pre-trained model’s input space, has become a trend in the vision community since the emergence of large visionlanguage models like CLIP. We present a systematic study on two representative prompt tuning methods, namely text prompt tuning and visual prompt tuning. A major finding is that none of the unimodal prompt tuning methods performs consistently well: text prompt tuning fails on data with high intra-class visual variances while visual prompt tuning cannot handle low inter-class variances. To combine the best from both worlds, we propose a conceptually simple approach called Unified Prompt Tuning (UPT), which learns a tiny neural network to jointly optimize prompts across different modalities. Extensive experiments on over 11 vision datasets show that UPT achieves a better trade-off than the unimodal counterparts on few-shot learning benchmarks, as well as on domain generalization benchmarks. Code and models will be released to facilitate future research. ",
|
| 40 |
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"bbox": [
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| 41 |
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| 42 |
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| 47 |
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| 48 |
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{
|
| 49 |
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"type": "text",
|
| 50 |
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"text": "1 INTRODUCTION ",
|
| 51 |
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"text_level": 1,
|
| 52 |
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"bbox": [
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| 53 |
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| 54 |
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| 60 |
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{
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| 61 |
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"type": "text",
|
| 62 |
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"text": "Vision-language (VL) models pre-trained on millions of image-text pairs (e.g., CLIP (Radford et al., 2021) and ALIGN (Jia et al., 2021)) have shown excellent transferability on a variety of downstream tasks, such as few-shot learning (Zhou et al., 2022a;b; Ju et al., 2021) and open-vocabulary perception (Gu et al., 2022; Zhou et al., 2022c; Zang et al., 2022; Ghiasi et al., 2021). When adapting large VL models to downstream tasks, it is often impractical to fine-tune the entire model directly due to their huge parameter size. To make adaptation more efficiently, many studies (Gao et al., 2021; Li & Liang, 2021; Lester et al., 2021; Zhou et al., 2022a; Lu et al., 2022; Ju et al., 2021; Yao et al., 2021; Jia et al., 2022; Bahng et al., 2022) have explored prompt tuning where the idea is to fine-tune a small number of parameters in a pre-trained model’s input space, called prompt, while keeping the majority of pre-trained parameters frozen. ",
|
| 63 |
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"bbox": [
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"page_idx": 0
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| 70 |
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| 71 |
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| 72 |
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"type": "text",
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| 73 |
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"text": "A typical VL model consists of two sub-networks—an image encoder and a text encoder—to extract features from visual and textual modalities respectively. Correspondingly, existing prompt tuning approaches can be grouped into two types: text prompt tuning and visual prompt tuning. For text prompt tuning methods, e.g., CoOp (Zhou et al., 2022a), extra text prompt tokens treated as learnable parameters are applied on the text encoder (Fig. 1(a)) to mitigate the issue that hand-crafted text prompt templates (e.g., “a photo of a [CLASS].”) are often sub-optimal. On the contrary, visual prompt tuning approaches focus on modulating the image encoder (Fig. 1(b)). A representative method is VPT (Jia et al., 2022), which injects learnable parameters into multiple layers of a Vision Transformer. Notably, these prompt-based methods treat the two modalities in isolation. ",
|
| 74 |
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| 80 |
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| 81 |
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| 82 |
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{
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| 83 |
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"type": "text",
|
| 84 |
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"text": "Despite significant improvements achieved recently, we observe that current prompt tuning approaches (Zhou et al., 2022a; Jia et al., 2022) fail to perform consistently due to inherent variances in visual and text features in downstream tasks. That is to say, using the unimodal prompt may obtain good results on one dataset but not on others. ",
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| 85 |
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| 94 |
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"type": "text",
|
| 95 |
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"text": "To analyze the phenomenon, we measure the discrepancy in data distribution focusing on the intraclass variance of visual features and inter-class variance of text embedding, and study the correlation between data statistics and performance improvement. As shown in Fig. 1(d), when the intra-class variance of image features is large (bottom right), CoOp struggles to learn suitable text prompts for improving the text classifier. As for visual prompt tuning, VPT faces difficulties when the interclass variance of text features is small, as shown in bottom left of Fig. 1(e). That is, if the text classifiers are based on text features of low separability, tuning visual prompts would lend little help to improve the final performance. Moreover, intra-class visual variance and inter-class text variance are typically orthogonal. As a consequence, the performance of unimodal prompt tuning methods varies widely across different datasets: CoOp beats VPT by $8 . 1 \\%$ on Flowers102 (Nilsback & Zisserman, 2008) while VPT outperforms CoOp by $8 . 4 \\%$ on EuroSAT (Helber et al., 2019). ",
|
| 96 |
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| 103 |
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| 104 |
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{
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| 105 |
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"type": "image",
|
| 106 |
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"img_path": "images/61a5b482223f042b9e5ad0d88db275988b5b17c8486555e68f490deded9d8046.jpg",
|
| 107 |
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"image_caption": [
|
| 108 |
+
"Figure 1: Top: Architectures of (a) text prompt tuning (Zhou et al., 2022a), (b) visual prompt tuning (Jia et al., 2022) and (c) our multimodal unified prompt tuning ( $\\textcircled { 3 }$ : learnable; $\\frac { 2 0 0 } { 9 0 0 }$ : frozen parameters). Bottom: the performance improvements $( \\% )$ of text prompt tuning (d) and visual prompt tuning (e) compared with the zero-shot CLIP baseline. We show that the variance of visual and text features ( $\\scriptstyle { \\dot { x } }$ -axis) will affect the improvements $y$ -axis). We project the text/visual features of the dataset (pointed by the dashed arrow) into a unit sphere to show the variance of different distributions. Please refer to the appendix for the implementation details about how we compute the feature variance. "
|
| 109 |
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],
|
| 110 |
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"image_footnote": [],
|
| 111 |
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"bbox": [
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| 112 |
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| 118 |
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|
| 119 |
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|
| 120 |
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"type": "text",
|
| 121 |
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"text": "",
|
| 122 |
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| 123 |
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| 131 |
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"type": "text",
|
| 132 |
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"text": "We argue that the key would be to simultaneously adapt both text and visual prompts to overcome the vast differences across different data distributions. A straightforward solution is to introduce both text and visual prompts to the model and jointly optimize the two modality-specific prompts. However, we find that such a na¨ıve joint training leads to poor performance due to the intrinsic discrepancy between text and image modalities. In particular, the performance is occasionally worse than tuning modality-specific prompts as shown in our experiments. ",
|
| 133 |
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|
| 139 |
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"page_idx": 1
|
| 140 |
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|
| 141 |
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{
|
| 142 |
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"type": "text",
|
| 143 |
+
"text": "Solving the aforementioned issues requires modality-agnostic optimization to bridge the isolated prompts. To this end, we present a unified prompt tuning method for both text and visual modalities, dubbed Unified Prompt Tuning (UPT). See Fig. 1(c). Specifically, we start with a shared prompt and propose a lightweight self-attention network to generate the prompts for CLIP’s text and visual encoders respectively. We empirically show that such a conceptually simple design can preserve the benefit of individual modalities. ",
|
| 144 |
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| 150 |
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"page_idx": 1
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| 151 |
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|
| 152 |
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| 153 |
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"type": "text",
|
| 154 |
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"text": "Our contributions are summarized as follows. 1) We provide a comprehensive study on existing text and visual prompt tuning methods, and identify the shortcoming of unimodal learning. 2) We present a unified prompt learning method for VL models, which is simple and easy to implement. 3) We conduct extensive experiments to show that unified prompt tuning outperforms previous unimodal prompt tuning methods under the few-shot learning and domain generalization settings. ",
|
| 155 |
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| 161 |
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|
| 162 |
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},
|
| 163 |
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{
|
| 164 |
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"type": "text",
|
| 165 |
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"text": "2 METHODOLOGY ",
|
| 166 |
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"text_level": 1,
|
| 167 |
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"bbox": [
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| 168 |
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| 169 |
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| 173 |
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| 176 |
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"type": "text",
|
| 177 |
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"text": "We first introduce vision-language models focusing on CLIP (Radford et al., 2021), in company with text/visual prompt tuning approaches for visual recognition in Sec. 2.1. We then analyze the ",
|
| 178 |
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| 185 |
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},
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| 186 |
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{
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| 187 |
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"type": "text",
|
| 188 |
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"text": "limitations of previous single-modal prompt tuning approaches in Sec. 2.2. Finally, we present technical details of our proposed unified prompt learning in Sec. 2.3. ",
|
| 189 |
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| 196 |
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},
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| 197 |
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{
|
| 198 |
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"type": "text",
|
| 199 |
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"text": "2.1 PRELIMINARIES ",
|
| 200 |
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"text_level": 1,
|
| 201 |
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{
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"type": "text",
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| 211 |
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"text": "CLIP. CLIP (Radford et al., 2021) consists of two sub-networks: an image encoder $\\phi$ and a text encoder $\\psi$ . These two encoders, respectively, map the text and image inputs into a joint hidden space $\\mathbb { R } ^ { d }$ , where the semantics of vision and language modalities are well-aligned. Here, $d$ refers to the final hidden dimension of the text or image encoder (e.g., $d = 2 5 6$ in the ResNet (He et al., 2016) backbone and $d = 5 1 2$ in the ViT backbone). Given an input image $_ { \\textbf { \\em x } }$ and a set of categories $\\mathbf { Y } = \\{ y _ { 1 } , y _ { 2 } , . . . , y _ { k } \\}$ (e.g., $k = 1 0 0 0$ for ImageNet (Deng et al., 2009)), the image encoder extracts the corresponding image feature $z = f _ { \\phi } ( \\pmb { x } ) \\in \\mathbb { R } ^ { d }$ . While the class names in $\\mathbf { Y }$ are first filled into a hand-crafted text prompt template a photo of a [CLASS] to obtain the text descriptions A, further processed by the text encoder for the text representations: $\\mathbf { W } = f _ { \\psi } ( \\mathbf { A } ) \\in \\mathbb { R } ^ { d \\times k }$ . The final prediction is computed as follows: ",
|
| 212 |
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"page_idx": 2
|
| 219 |
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},
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| 220 |
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{
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| 221 |
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"type": "equation",
|
| 222 |
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"img_path": "images/b912ae5785e192b60de0cd41fea00087ff53daf3f57f63d66f7726d62fb8a822.jpg",
|
| 223 |
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"text": "$$\np ( y = i \\mid \\pmb { x } ) = \\frac { \\exp \\left( \\cos \\left( \\pmb { w } _ { i } , \\pmb { z } \\right) / \\tau \\right) } { \\sum _ { j = 1 } ^ { k } \\exp \\left( \\cos \\left( \\pmb { w } _ { j } , \\pmb { z } \\right) / \\tau \\right) } ,\n$$",
|
| 224 |
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"text_format": "latex",
|
| 225 |
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| 233 |
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| 234 |
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"type": "text",
|
| 235 |
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"text": "where $\\cos ( \\cdot , \\cdot )$ denotes the cosine similarity and $\\tau$ is a fixed temperature value (e.g., $\\tau = 1 0 0$ ). Conceptually, such a decision process for the input image $_ { \\textbf { \\em x } }$ in Eq. (1) is formulated in a way that the text encoder $\\psi$ takes a role of generating dynamic classifiers W from open-set categories $\\mathbf { Y }$ , with the image encoder $\\phi$ producing encoded visual features $_ { z }$ . In practice, it is generally infeasible to fine-tune the millions of parameters (i.e., $\\phi$ and $\\psi$ ) in a VL model for transfer learning in every downstream task. ",
|
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"type": "text",
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| 246 |
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"text": "Text Prompt Tuning. For efficient and effective model adaptation, text prompt tuning approaches consider generating more adaptive classifiers without fine-tuning the text encoder $\\psi$ . For example, Context Optimization $\\left( \\mathbf { C o O p } \\right)$ (Zhou et al., 2022a) introduce a set of learnable parameters $\\textbf { T } \\in$ $\\mathbb { R } ^ { d \\times m }$ to replace the hand-crafted text prompt template (a photo of a [CLASS]). The wordembedding of class names in $\\mathbf { Y }$ will concatenate with these text prompts in the following form: ",
|
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"img_path": "images/b801f43fd817c4f30cff86c1e947dee5827e5672c2f40e50a2c2a35e711fe15c.jpg",
|
| 258 |
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"text": "$$\n\\mathrm { \\hat { T } } = [ t _ { 1 } , t _ { 2 } , \\dots , t _ { m } , \\mathrm { C L A S S } ] .\n$$",
|
| 259 |
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"text_format": "latex",
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| 260 |
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"text": "Here, the symbol $m$ denotes the prompt length. The resulting dynamic text representations are extracted by the text encoder: $\\mathbf { W } = f _ { \\psi } ( \\hat { \\mathbf { T } } ) \\in \\mathbb { R } ^ { d \\times k }$ . In each downstream task, the learnable prompts $\\mathbf { T }$ will be optimized with each task-specific objective function, e.g., a cross-entropy classification loss $\\mathcal { L } _ { \\mathrm { C E } } ( p , y )$ in few-shot learning. Note that both the image and text encoders $\\cdot \\phi$ and $\\psi$ ) are frozen during downstream training. As a result, updating the text prompt $\\mathbf { T }$ will correspondingly adjust the decision boundaries with generated classifiers $\\mathbf { W }$ for downstream tasks. ",
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"text": "Visual Prompt Tuning. Conversely, visual prompt tuning methods focus on extracting more transferable visual features while keeping the visual encoder $\\phi$ unchanged. Following the success of text prompt tuning approaches, recent Visual Prompt Tuning (VPT) (Jia et al., 2022) introduces a similar prompt tuning recipe for the visual encoder $\\phi$ . Suppose the image encoder $\\phi$ contains $L$ Vision Transformer layers, the output of $i$ -th layer, $l _ { i }$ , where $i = 1 , 2 , \\dots , L$ , is given by: ",
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"text": "$$\n[ { \\pmb { c } } ^ { i + 1 } , z _ { 1 } ^ { i + 1 } , \\dots , z _ { s } ^ { i + 1 } ] = l _ { i } \\left( \\left[ { \\pmb { c } } ^ { i } , z _ { 1 } ^ { i } , \\dots , z _ { s } ^ { i } \\right] \\right) ,\n$$",
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"text": "where $c \\in \\mathbb { R } ^ { d }$ denotes the classification token ([CLS]), and $Z = [ z _ { 1 } , z _ { 2 } , \\ldots , z _ { s } ] \\in \\mathbb { R } ^ { d \\times s }$ denotes the input image patch tokens with length $s$ . For the $i$ -th encoder layer, a set of learnable visual prompts $\\mathbf { V } ^ { i } \\in \\mathbb { R } ^ { \\tilde { d } \\times n }$ are inserted and computed as follows: ",
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"text": "$$\n[ { \\pmb { c } } ^ { i + 1 } , \\ldots , { \\pmb { Z } } ^ { i + 1 } ] = l _ { i } \\left( \\left[ { \\pmb { c } } ^ { i } , { \\pmb { V } } ^ { i } , { \\pmb { Z } } ^ { i } \\right] \\right) ,\n$$",
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"text": "where $n$ stands for the length of visual prompts. Two VPT variants are proposed: VPT-shallow and VPT-deep. For VPT-shallow, the visual prompts are only inserted into the first Transformer layer $( i = 1 )$ ). Whereas for VPT-deep, visual prompts are introduced at every layer. The learnable visual prompts are data-independent, which once learned, can modulate the visual features $_ z$ of input images for better downstream transfer learning. ",
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"image_caption": [
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"Figure 2: Visualization of input features $_ { z }$ (projected points) and text classifier W (projected lines) on EuroSAT and Flowers102. "
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"text": "2.2 ANALYSIS ",
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"text": "We conduct a series of probing studies to analyze the characteristics of text/visual prompt tuning. First, when adapting the CLIP model with two representative text and visual prompt tuning approaches $\\mathrm { C o O p }$ (Zhou et al., 2022a) and VPT (Jia et al., 2022)), we measure the variance of both visual features $_ z$ and text embeddings W (i.e., classifiers) for all 11 downstream vision datasets (see Appendix for detailed implementations). For text prompt tuning, as shown in Fig. 1(d), we observe that CoOp performs well on datasets with low intra-class variance between visual features, such as Flowers102, but fails on Food101 dataset with high intra-class feature variance. As for visual prompt tuning, VPT succeeds in improving performance on SUN397 dataset with large inter-class text embeddings, while being less effective on Food101 and Flowers102 with relatively smaller inter-class text embedding variance. The performance improvements of text/visual prompt tuning are highly correlated with the variance of visual features $_ z$ or text embeddings W in downstream datasets. ",
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"text": "In order to understand this phenomenon, we select two downstream vision datasets (Flowers102 (Nilsback & Zisserman, 2008), EuroSAT (Helber et al., 2019)) for further analysis. During downstream training, we project both visual features $_ z$ and text embeddings W (i.e., classifiers) into joint sphere space ${ \\bar { \\mathbb { R } } } ^ { 3 }$ for better visualization. As we illustrated in Fig. 2, we can observe that: 1) For the EuroSAT dataset with high intra-class visual feature variance, text prompts in $\\mathrm { C o O p }$ fails to adapt the text classifiers W. Clearly, the text classifiers in Fig. $2 ( \\mathbf { b } )$ are almost unchanged compared with zero-shot CLIP baseline (Fig. 2(a)). 2) For the Flowers102 dataset with low inter-class text embedding variance, visual prompts in VPT are not effective in modulating the visual features $_ { z }$ (Fig. ${ \\bf \\Pi } ( \\mathbf { g } )$ ), thus cannot obtain considerable performance gain. ",
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"text": "In conclusion, the single-modal prompt tuning approaches $\\mathrm { C o O p }$ and VPT), face the dilemma that consistent improvements over Zero-shot CLIP are hard to achieve due to inherent variances of visual features and text embedding in downstream tasks. Our observation motivates us to present a unified prompt tuning method that tunes the $_ z$ and W at the same time. ",
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"text": "2.3 UNIFIED PROMPT TUNING ",
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"text": "Driven by our analysis, we devise a simple yet effective multi-modal Unified Prompt Tuning (UPT) approach for adapting VL models. Specifically, instead of introducing two sets of isolated modalityspecific prompts (i.e., T in Eq. (2) and $\\mathbf { V }$ in Eq. (4)) for the text and visual encoders, we consider learning a set of unified modality-agnostic prompts for tuning VL models. As shown in Fig. 3, we define a set of learnable prompts $\\breve { U } \\in \\mathbb { R } ^ { \\tilde { d } \\times n }$ with length $n$ . Rather than na¨ıvely appending the unified prompts into the text and visual encoders, we employ a lightweight Transformer layer $\\theta$ to ",
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"image_caption": [
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"Figure 3: The architecture of (a) our unified prompt $U$ that is applied to $\\mathbf { ( b ) }$ CLIP text encoder and (c) CLIP image encoder. "
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"text": "transform unified prompts $U$ as follows: ",
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"text": "$$\n\\begin{array} { r l } & { U ^ { \\prime } = \\mathrm { S A } \\left( U \\right) + \\mathrm { L N } \\left( U \\right) , } \\\\ & { \\hat { U } = \\mathrm { F F N } \\left( \\mathrm { L N } \\left( U ^ { \\prime } \\right) \\right) + \\mathrm { L N } \\left( U ^ { \\prime } \\right) , } \\end{array}\n$$",
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"type": "text",
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"text": "where the self-attention operator SA, feed-forward network FFN and layer normalization LN are applied to obtain the transformed prompts $\\hat { U }$ . The self-attention module in the lightweight Transformer layer allows beneficial interaction between two modalities, so as to maximize the complementary effects. Our unified prompts can be introduced into multiple layers of VL models. In particular, for each $i$ -th layer of text and image encoders, we consider learning a set of layer-wise prompts $U ^ { i }$ , and split transformed $\\hat { \\pmb { U } } ^ { i }$ into two parts $\\hat { U } ^ { i } = \\{ \\hat { U } _ { t } ^ { i } , \\hat { U } _ { v } ^ { i } \\}$ , sending into the text and visual encoders respectively. During downstream training, we froze both the text and visual encoder $\\dot { \\psi }$ and $\\phi$ ) and only optimize the unified prompts $U$ and the lightweight Transformer layer $\\theta$ . In this way, both the dynamic classifiers W and visual features $_ z$ in Eq. (1) are effectively tuned for reliable prediction in the downstream task. As shown in Fig. 2 (d) and (h), our unified prompts can simultaneously obtain well-aligned text classifiers and separable visual features compared with single-modal counterparts. ",
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"text": "3 EXPERIMENTS ",
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"text": "In this section, we conduct experiments under two problem settings, i.e., (i) few-shot image classification (Sec. 3.1) and (ii) domain generalization (Sec. 3.2). We also present ablation studies in Sec. 3.3 on several design choices and extra experimental results about generalizability in Appendix. ",
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"text": "Baselines. We compare our approach against the following methods: (1) Zero-shot CLIP. This baseline uses hand-crafted text prompt templates and does not involve any prompt-learning strategies. (2) Single-modal Prompt Tuning methods, including CoOp (Zhou et al., 2022a) and ProDA (Lu et al., 2022) for the text modality, and VPT (Jia et al., 2022) for the visual modality. In the domain generalization setting, we further compare with CoCoOp (Zhou et al., 2022b), which improves CoOp’s generalization performance with an input-conditional design. For VPT, we report the results of both the shallow and deep variants, as described in Sec. 2.1. ",
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"type": "text",
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"text": "3.1 FEW-SHOT LEARNING ",
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"text_level": 1,
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"text": "In this section, we measure a model’s generalization ability by conducting prompt tuning using different strategies, with just a limited amount of labeled examples per-class in the specific downstream task. Detailed implementation is presented in Appendix. ",
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"text": "Datasets. We follow (Zhou et al., 2022b) to use 11 datasets (ImageNet (Deng et al., 2009), Caltech101 (Fei-Fei et al., 2004), OxfordPets (Parkhi et al., 2012), StanfordCars (Krause et al., 2013), Flowers102 (Nilsback & Zisserman, 2008), Food101 (Bossard et al., 2014), FGVC-Aircraft (Maji et al., 2013), SUN397 (Xiao et al., 2010), UCF101 (Soomro et al., 2012), DTD (Cimpoi et al., 2014), EuroSAT (Helber et al., 2019)) as our benchmarks. Following (Zhou et al., 2022a), we use the fewshot evaluation protocol selecting 1/2/4/8/16 shots for training and the whole test set for evaluation. We report averaged results over three runs with different random seeds to reduce the variance. The detailed results are shown in Fig. 4. ",
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"type": "text",
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"text": "Limitation of Single-modal Baselines. Figure 4 shows that the performance improvements of existing text prompt tuning method CoOp and visual prompt tuning method VPT are not consistent across different datasets. In particular, CoOp obtains better performance than VPT on some datasets, such as StanfordCars and SUN397. However, for other datasets with high intra-class visual variances, VPT is much more effective than CoOp. For instance, on the EuroSAT dataset, VPT-deep beats $\\mathrm { C o O p }$ by over $12 \\%$ . The discrepancy of previous single-modal baselines is also consistent with our motivation in Fig. 1(d) and Fig. 1(e). According to VPT (Jia et al., 2022), VPT-deep is more effective than VPT-shallow, and our experimental results also verify this point. We later show that VPT-shallow obtains much stronger performance than the VPT-deep in the domain generalization setting (Sec. 3.2). ",
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"type": "image",
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"img_path": "images/7f2fe993c3668a606909d93a194552a52f5961592c7252717552ed40130576c0.jpg",
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"image_caption": [
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| 554 |
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"Figure 4: Main results over 11 datasets under the few-shot learning setting. We report the average accuracy $( \\% )$ of 1/2/4/8/16 shots over three runs. Overall, the proposed UPT (blue line) achieves apparent improvements compared with the Zero-shot CLIP and single-modal prompt tuning baselines $\\mathrm { C o O p }$ , ProDA and VPT). "
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"text": "UPT vs. Single-modal Baselines. Our UPT achieves clear advantages over the single-modal prompt-tuning counterparts CoOp, ProDA and VPT, as suggested by the averaged performance (topleft of Fig. 4). In general, the average performance gap between UPT and baselines increases with the shot number available for prompt tuning. Specifically, UPT obtains $0 . 4 8 / 1 . 3 6 / 1 . 2 9 / 2 . 4 6 / 3 . 1 9 ( \\% )$ accuracy improvements compared with the text prompt tuning method CoOp on 1/2/4/8/16 shots settings. Even compared with the strong text prompt tuning baseline ProDA, UPT still boosts the accuracy of $0 . 1 1 / 1 . { \\overset { \\cdot } { 0 } } 1 / 0 . 6 7 / 1 . 5 / 1 . 6 1 ( \\% )$ . Similarly, UPT achieves $0 . 8 9 / 2 . 7 0 / 2 . 0 3 / 2 . 4 0 / 2 . 0 1 ( \\% )$ accuracy gains over the visual prompt tuning approach VPT-deep. Notably, UPT significantly boosts the performance over CoOp and VPT-deep on challenging large datasets, such as ImageNet with 1,000 classes and SUN397 with 397 categories. UPT also surpasses CoOp and VPT-deep on finegrained datasets such as StanfordCars and FGVC Aircraft. We also observe that UPT shows less improvement on the two datasets (OxfordPets and Food101), possibly caused by the noisy training data (Zhou et al., 2022a; Bossard et al., 2014). Overall, the experimental results in Fig. 4 demonstrate the effectiveness of our proposed UPT. ",
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"bbox": [
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{
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| 588 |
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"type": "table",
|
| 589 |
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"img_path": "images/7ad12df293e19497632a3f1c07c5db6c168d9e25534c7342af0cbb0cc50b7739.jpg",
|
| 590 |
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"table_caption": [
|
| 591 |
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"Table 1: Main results under the domain generalization setting. We report the average accuracy $( \\% )$ of 16 shots over three runs. The best and second best methods are highlighted in red and orange , respectively. "
|
| 592 |
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],
|
| 593 |
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"table_footnote": [],
|
| 594 |
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"table_body": "<table><tr><td rowspan=\"2\">#</td><td rowspan=\"2\">Method</td><td>Source</td><td colspan=\"4\">Target</td><td rowspan=\"2\">Overall Average</td><td rowspan=\"2\">00D Average</td></tr><tr><td>ImageNet</td><td>-V2</td><td>-S</td><td>-A</td><td>-R</td></tr><tr><td></td><td>CoOp</td><td>71.51</td><td>64.20</td><td>47.99</td><td>49.71</td><td>75.21</td><td>61.72</td><td>59.28</td></tr><tr><td></td><td>CoCoOp</td><td>71.02</td><td>64.07</td><td>48.75</td><td>50.63</td><td>76.18</td><td>62.13</td><td> 59.91</td></tr><tr><td></td><td>VPT-shallow</td><td>68.98</td><td>62.10</td><td>47.68</td><td>47.19</td><td>76.10</td><td>60.38</td><td>58.27</td></tr><tr><td>１２３４</td><td>VPT-deep</td><td>70.57</td><td>63.67</td><td>47.66</td><td>43.85</td><td>74.42</td><td>60.04</td><td>57.40</td></tr><tr><td>5</td><td>Joint Training</td><td>71.42</td><td>64.36</td><td>48.20</td><td>49.71</td><td>76.23</td><td>61.97</td><td>59.61</td></tr><tr><td>6</td><td>Shared</td><td>71.46</td><td>64.43</td><td>48.13</td><td>50.03</td><td>75.76</td><td>61.96</td><td>59.55</td></tr><tr><td>7</td><td>MLP</td><td>71.00</td><td>64.11</td><td>48.65</td><td>48.76</td><td>76.14</td><td>61.78</td><td>59.48</td></tr><tr><td>8</td><td>UPT</td><td>72.63</td><td>64.35</td><td>48.66</td><td>50.66</td><td>76.24</td><td>62.51</td><td> 59.98</td></tr></table>",
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"bbox": [
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"type": "image",
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"img_path": "images/5b3cbe1f1b4fb1f455f9e5634a72ef4d7ac81c1eaba4befe1124729e53440768.jpg",
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| 606 |
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"image_caption": [
|
| 607 |
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"Figure 5: Ablation studies on different design choices. (a): jointly train the existing text and visual prompt tuning approaches; (b): shared prompts for all modalities; (c): using two MLP layers to generate the prompts. "
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],
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"image_footnote": [],
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"bbox": [
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"type": "text",
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"text": "",
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"type": "text",
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| 631 |
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"text": "3.2 DOMAIN GENERALIZATION ",
|
| 632 |
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"text_level": 1,
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| 633 |
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"bbox": [
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"type": "text",
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"text": "Pre-trained VL models like CLIP have shown strong generalization ability. However, the prompt tuned on a specific downstream dataset may hinder the generalization ability on categories outside the training set. In this section, we evaluate the generalization ability of different prompt tuning methods on out-of-distribution (OOD) data. ",
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"type": "text",
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"text": "Datasets. We follow (Zhou et al., 2022a) to use five datasets (ImageNet (Deng et al., 2009), ImageNet V2 (Recht et al., 2019), ImageNet-Sketch (Wang et al., 2019), ImageNet-A (Hendrycks et al., 2021b) and ImageNet-R (Hendrycks et al., 2021a)) for evaluation. Following the protocol, we train a model on ImageNet and evaluate it on four other variants of ImageNet with their domains shifted. ",
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{
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"type": "text",
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| 665 |
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"text": "Results. Table 1 summarizes the results. We report the average accuracy on both the source and target datasets (penultimate column), and the OOD average accuracy on target datasets (last column). The results show that VPT-shallow (row #2) achieves higher OOD accuracy than VPT-deep (row #3), and text prompt tuning methods outperform visual prompt tuning approaches. Furthermore, the proposed UPT (row #8) is generally a better option than single-modal baselines (rows #1-#4) and obtains comparable performance with CoCoOp. Our UPT achieves the best results three times on five datasets, showing that UPT is a reliable prompt tuning method among its competitors in the domain generalization setting. ",
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"type": "text",
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| 676 |
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"text": "3.3 ABLATION STUDIES ",
|
| 677 |
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"text_level": 1,
|
| 678 |
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{
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| 687 |
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"type": "text",
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| 688 |
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"text": "Comparison with the Joint Training Baseline. As shown in Fig. 5(a), a straightforward approach for multi-modal prompts is tune the text prompt (using CoOp) and visual prompt (using VPT) jointly. ",
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{
|
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"type": "table",
|
| 699 |
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"img_path": "images/801e6e20902f3fcaaa0edddad3e834f04fb9e01f65bca2efe64bf2275f21b009.jpg",
|
| 700 |
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"table_caption": [
|
| 701 |
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"Table 2: Ablation studies on different multi-modal prompt design choices in Fig. 5 over 11 datasets. We report the accuracy results under the 16 shots setting. The best and second best methods are highlighted in red and orange , respectively. "
|
| 702 |
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],
|
| 703 |
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"table_footnote": [],
|
| 704 |
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"table_body": "<table><tr><td>#</td><td>Prrega</td><td>eee</td><td>Grreeaer</td><td>s1edpitrit</td><td>ssrrprteets</td><td>Tiroeni0</td><td></td><td>FTaaeieelr [orpoon</td><td></td><td>163308</td><td></td><td>JtoSSS</td><td>UUIIII</td><td>2neace</td></tr><tr><td>1</td><td>CoOp</td><td>71.36</td><td>95.93</td><td>92.74</td><td>77.45</td><td>95.90</td><td>86.36</td><td>38.04</td><td>73.59</td><td>68.38</td><td>78.77</td><td></td><td>82.04</td><td>78.24</td></tr><tr><td>2</td><td>VPT-shallow</td><td>68.98</td><td>94.66</td><td>92.61</td><td>69.09</td><td></td><td>81.40</td><td>86.91</td><td>30.93</td><td>68.08</td><td>52.28</td><td>84.87</td><td>75.19</td><td>73.18</td></tr><tr><td>3</td><td>VPT-deep</td><td>70.57</td><td>95.83</td><td>92.91</td><td>76.13</td><td></td><td>94.96</td><td>86.18</td><td>40.96</td><td>71.63</td><td>69.79</td><td>91.53</td><td>82.76</td><td>79.39</td></tr><tr><td>4</td><td>Joint Training</td><td>71.42</td><td>95.84</td><td></td><td>93.34 79.02</td><td></td><td>95.25</td><td>86.55</td><td>40.56</td><td>74.17</td><td>67.83</td><td>78.94</td><td>82.81</td><td>78.70</td></tr><tr><td>5</td><td>Shared</td><td>71.46</td><td>95.50</td><td>92.99</td><td>78.66</td><td></td><td>95.55</td><td>86.67</td><td>39.18</td><td>73.64</td><td>67.69</td><td>73.36</td><td>82.06</td><td>77.88</td></tr><tr><td>6</td><td>MLP</td><td>71.00</td><td>95.59</td><td>93.74</td><td>75.88</td><td>93.38</td><td></td><td>87.20</td><td>37.17</td><td>72.74</td><td>67.31</td><td>90.66</td><td>81.43</td><td>78.73</td></tr><tr><td>7</td><td>UPT (Ours)</td><td>72.63 95.94</td><td></td><td>92.95</td><td></td><td>84.33 97.11</td><td></td><td>85.00</td><td>46.80</td><td>75.92 70.65</td><td></td><td>90.51</td><td>84.03 81.44</td><td></td></tr></table>",
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{
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"type": "image",
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"img_path": "images/f8905de698bb1983af8c7ec1a80c6337a8c33c12f232aa908c9d3e4c84cbcdf8.jpg",
|
| 716 |
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"image_caption": [
|
| 717 |
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"Figure 6: Visualization of attention response map between visual prompts and image patch tokens. The images are test images from ImageNet. We visualize the self-attention module from the last block of ViT of the CLIP image encoder. "
|
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"image_footnote": [],
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"type": "text",
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| 730 |
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"text": "We investigate the effectiveness of such joint training scheme, and report its results in Table 2 row #4 and Table 1 row #5. On the few-shot learning setting, we see that such a joint training solution performs better than $\\mathrm { C o O p }$ and VPT-shallow, which shows that multi-modal optimization is helpful to a certain extent. But the joint training approach obtains slightly worse accuracy than the VPT-deep $7 8 . 7 0 \\%$ vs. $7 9 . 3 9 \\%$ ) since VPT-deep involves a large number of parameters. On the domain generalization setting, we find the joint training method performs much better than VPT-deep. Also, the joint training method shows inferior performance to our UPT, demonstrating that our self-attention base mechanism is more effective. ",
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| 731 |
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| 739 |
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{
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| 740 |
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"type": "text",
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| 741 |
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"text": "Shared Prompts for Text and Visual Modalities. We also investigate the results of directly sharing prompts for different modalities. As shown in Fig. 5(b), the shared prompts will be optimized for both text and visual modalities. This scheme differs from the proposed UPT, where the shared prompts are transformed with self attention. Experimental results are presented in Table 2 row #5 and Table 1 row #6, and we observe that such a prompt sharing strategy achieves worst performance among all the ablation design choices. ",
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| 749 |
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| 750 |
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{
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| 751 |
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"type": "text",
|
| 752 |
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"text": "MLP Baseline. For our proposed UPT, we use a Transformer layer with the self-attention operator to partially share the hyper-parameters for different modalities. Here, we study a simpler design that generates the unified prompts with two MLP layers. Results are presented on Table 2 row #6 and Table 1 row #7. The MLP baseline is still competitive, yielding best performance on two datasets. Nonetheless, the average results is still poorer than the proposed self-attention based approach. ",
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| 761 |
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{
|
| 762 |
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"type": "text",
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| 763 |
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"text": "3.4 QUALITATIVE RESULTS ",
|
| 764 |
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"text_level": 1,
|
| 765 |
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| 774 |
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"type": "text",
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| 775 |
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"text": "While it is hard to visualize what have been learned during text prompt tuning, it is possible to visualize the visual prompts learned by VPT and UPT following the self-supervised learning method, DINO (Caron et al., 2021). In particular, for each layer of the Vision Transformer (ViT), we can compute the self-attention response map of visual prompts and image patch tokens. Figure 6 compares such response maps by VPT and the proposed UPT. We find that UPT shows stronger selfattention responses compared with VPT. This could be the possible reason why UPT achieves better performance on the few-shot learning and the OOD generalization settings. ",
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{
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| 785 |
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"type": "text",
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| 786 |
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"text": "4 RELATED WORK ",
|
| 787 |
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"text_level": 1,
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| 788 |
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| 796 |
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{
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| 797 |
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"type": "text",
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| 798 |
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"text": "Vision-Language Models. Recent vision-language pre-trained models (Radford et al., 2021; Jia et al., 2021) use the contrastive loss to align an image encoder (e.g., ViT (Dosovitskiy et al., 2021)) and a text encoder (e.g., BERT (Kenton & Toutanova, 2019)) in a common feature space. These vision-language models are trained on web-scale image-text pairs and are transferable across various downstream tasks such as point cloud classification (Zhang et al., 2022a), video classification (Qian et al., 2022), object detection (Gu et al., 2022; Du et al., 2022; Zhou et al., 2022c; Zang et al., 2022) and semantic segmentation (Ghiasi et al., 2021). In this work, we aim to explore how to adapt the CLIP model to the downstream few-shot recognition task. ",
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| 806 |
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|
| 807 |
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{
|
| 808 |
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"type": "text",
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| 809 |
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"text": "Text Prompt Tuning. The concept of prompt tuning was first proposed in the NLP area (Liu et al., 2021; Gao et al., 2021; Li & Liang, 2021; Lester et al., 2021). In particular, a text prompt refers to a task-specific template for language models. For example, in sentiment analysis, the template might be “I [MASK] the movie.” where the mask placeholder will be filled with either “love” or “hate.” Common practices in text prompt tuning include (i) searching for a specific word in the dictionary, known as hard prompt learning (Gao et al., 2021), or (ii) turning masked tokens into learnable vectors, known as soft prompt learning (Li & Liang, 2021; Lester et al., 2021). Text prompt tuning has also been applied in computer vision after the emergence of large vision-language models (e.g., CLIP (Radford et al., 2021)), which are too big to fine-tune. A representative work is CoOp (Zhou et al., 2022a), which turned the input context tokens in CLIP’s text branch into learnable vectors for adapting CLIP to downstream image recognition. Other follow-ups of $\\mathrm { C o O p }$ include CoCoOp (Zhou et al., 2022b), DualCoOp (Sun et al., 2022), ProGrad (Xing et al., 2022), and ProDA (Lu et al., 2022). ",
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| 810 |
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"type": "text",
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| 820 |
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"text": "Visual Prompt Tuning. The idea of visual prompt tuning is to adapt large pre-trained Vision Transformers (Dosovitskiy et al., 2021) by adding learnable parameters in the visual input space, which is analogous to text prompt tuning in NLP. VPT (Jia et al., 2022) and Visual Prompting (Bahng et al., 2022) both add trainable tokens to the input of Transformer models. A recent work, NOAH (Zhang et al., 2022b), uses neural architecture search algorithms to identify the optimal configuration of prompt modules. In comparison to the unimodal prompt learning methods discussed above, our paper provides a timely study on how to achieve a better trade-off using multimodal prompt learning. ",
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"text": "5 CONCLUSION ",
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"text": "With the rapid scaling of vision models along the size dimension, efficient downstream adaptation methods have become essential for facilitating large-scale deployment of vision models in the wild. Our paper provides a timely and comprehensive study on how to adapt large vision-language models like CLIP from the prompt learning perspective. In particular, our study unveils that the previous unimodal prompt tuning methods do not work consistently well across different computer vision datasets. In contrast, the proposed UPT method, despite having a simple design, achieves a better trade-off compared with the unimodal counterparts. The results suggest that one should exploit correspondences between different modalities for prompt learning. ",
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"text": "On the other hand, the results achieved by UPT are by no means perfect: in the ablation studies we observe that some alternative designs, such as using MLP instead of Transformer, might sometimes give better performance. In summary, we believe multimodal prompt learning is a promising framework, and we expect more improvements to be achieved with more advanced (and efficient) designs. ",
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"text": "In the supplementary materials, we discuss the implementation details and more experimental results. Section A explains how we compute the intra-/inter- class variance for Fig.(1) of the main paper. Section B reports the implementation details of our paper. Section C presents more experimental results under the base-to-new generalization and cross-dataset transfer settings. ",
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"text": "A INTRA-/INTER- CLASS VARIANCE ",
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"text": "In this section, we provide the implementation details about how we compute the intra-class visual variance and inter-class text variance for different datasets (Fig.1 (d)(e) in the main paper. ",
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| 1154 |
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"text": "Intra-class Visual Variance. Given one dataset with $k$ classes in total, for each image $_ { \\textbf { \\em x } }$ that belongs to class $c$ , we first use the CLIP image encoder $\\phi$ to extract the corresponding image feature $f _ { \\phi } ( \\bar { \\pmb x ) }$ . Then we get the intra-class variance of class $c$ as: ",
|
| 1155 |
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| 1161 |
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| 1164 |
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"type": "equation",
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"img_path": "images/b66bcc4027046fa62c424f55c314bae0fc2b7d4d0850148ba981ebc5a77471fd.jpg",
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| 1166 |
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"text": "$$\n\\mathsf { v a r } _ { c } = \\frac { 1 } { \\vert \\vert X _ { c } \\vert \\vert } \\sum _ { x \\in X _ { c } } \\left( f _ { \\phi } ( \\pmb { x } ) - \\bar { f } _ { \\phi } ( \\pmb { x } ) \\right) ^ { 2 } ,\n$$",
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364
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| 1176 |
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| 1177 |
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"type": "text",
|
| 1178 |
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"text": "where $X _ { c }$ denotes to the set of images that have the ground-truth class label $c$ , and $\\bar { f } _ { \\phi } ( \\pmb { x } )$ refers to the mean values of class $c$ . Then we can compute the intra-class variance $\\operatorname { V a r } _ { \\mathrm { v } }$ for all the $k$ classes as ",
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"img_path": "images/b8d8241ed2e49df530d8973bbb59c9c6b911c1ddcbbfe7e52b36f62b5a098a49.jpg",
|
| 1190 |
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"text": "$$\n\\mathrm { V a r } _ { \\mathrm { v } } = { \\frac { 1 } { k } } \\sum _ { c = 1 } ^ { k } \\mathrm { v a r } _ { c } .\n$$",
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| 1191 |
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|
| 1201 |
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"type": "text",
|
| 1202 |
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"text": "Inter-class Text Variance. For each dataset, we first compute the CLIP text features $\\pmb { w }$ of classs $c$ , and the mean value $\\bar { \\pmb w }$ of all the $k$ classes. Then we get the inter-class text variance $\\mathrm { V a r } _ { \\mathrm { t } }$ as: ",
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"img_path": "images/6776f518a9bd145703f28a74214a04b58a2586266dae413a37b5f39c7d7af8d6.jpg",
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| 1214 |
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"text": "$$\n\\mathrm { V a r _ { t } } = { \\frac { 1 } { k } } \\sum _ { c = 1 } ^ { k } ( w _ { c } - { \\bar { w } } ) ^ { 2 } .\n$$",
|
| 1215 |
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"text_format": "latex",
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| 1225 |
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"type": "text",
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| 1226 |
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"text": "B IMPLEMENTATION DETAILS ",
|
| 1227 |
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"text_level": 1,
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| 1228 |
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"type": "text",
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"text": "Our implementation is based on the source code of $\\mathrm { C o O p }$ (Zhou et al., 2022a). We use ViT-B/16 as the CLIP backbone (Radford et al., 2021). Following (Zhou et al., 2022b), we set the context length of $\\mathrm { C o O p }$ as $m = 4$ (same for VPT). For Zero-shot CLIP and VPT, we use the default prompt template, “a photo of a [CLS].” We use SGD as the optimizer, with an initial learning rate of 0.002, which is decayed by the cosine annealing rule. The batch size is set to 32 for all datasets. ",
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"type": "text",
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"text": "C MORE EXPERIMENTAL RESULTS ",
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| 1250 |
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"type": "text",
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| 1261 |
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"text": "Recent work CoCoOp (Zhou et al., 2022b) points out that the text prompts learned by $\\mathrm { C o O p }$ (Zhou et al., 2022a) are not generalizable to novel classes and out-of-distribution data. CoCoOp defines two new settings - base-to-new generalization and cross-dataset transfer - to measure the generalizability ability of prompt learning approaches. In this section, we provide the experimental results of our UPT in these two settings. ",
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"type": "text",
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| 1272 |
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"text": "Datasets. We use the same 11 datasets we used in the few-shot learning setting (section 3.1 in the main paper). Following CoCoOp (Zhou et al., 2022b), we use the 16-shot protocol and report the averaged results over three runs, and set the training schedule as ten epochs. We report the accuracy on base and new classes, and the harmonic mean for base-to-novel trade-off. ",
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| 1282 |
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"type": "text",
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| 1283 |
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"text": "C.1 BASE-TO-NEW GENERALIZATION ",
|
| 1284 |
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"type": "text",
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| 1295 |
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"text": "In the base-to-new generalization setting, we split the classes into two disjoint groups - base classes and new classes. All the prompt learning approaches are required to train on the base classes, while evaluation is conducted on the base and new classes separately. The experimental results are shown in Table 3. ",
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"type": "table",
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"img_path": "images/e99ff9de10f08511555375a1222e56c0d336118e46bc141b982516373469a551.jpg",
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| 1307 |
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"table_caption": [
|
| 1308 |
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"Table 3: Comparison results in the base-to-new generalization setting. H: Harmonic mean (Xian et al., 2017). The best and second best methods are highlighted in red and orange , respectively. The method ‘VPT-s’ refers to VPT-shallow. "
|
| 1309 |
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"table_footnote": [],
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| 1311 |
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"table_body": "<table><tr><td colspan=\"4\">(a) Average over 11 datasets.</td><td colspan=\"4\">(b) ImageNet.</td><td colspan=\"4\">(c) Caltech101.</td></tr><tr><td></td><td>Base</td><td>New</td><td>H</td><td></td><td>Base</td><td>New</td><td>H</td><td></td><td>Base</td><td>New</td><td>H</td></tr><tr><td>CLIP</td><td>69.34</td><td>74.22</td><td>71.70</td><td>CLIP</td><td></td><td>72.43 68.14</td><td>70.22</td><td>CLIP</td><td></td><td>96.8494.00</td><td>95.40</td></tr><tr><td>CoOp</td><td>82.69</td><td>63.22</td><td>71.66</td><td>CoOp</td><td>76.47</td><td>67.88</td><td>71.92</td><td>CoOp</td><td>98.00</td><td>89.81</td><td>93.73</td></tr><tr><td>CoCoOp</td><td>80.47</td><td>71.69</td><td>75.83</td><td>CoCoOp</td><td>75.98</td><td>70.43</td><td>73.10</td><td>CoCoOp</td><td>97.96</td><td>93.81</td><td>95.84</td></tr><tr><td>VPT-s</td><td>73.32</td><td>73.21</td><td>73.16</td><td>VPT-s</td><td>74.47</td><td>69.13</td><td>71.70</td><td>VPT-s</td><td>97.47</td><td>93.80</td><td>95.60</td></tr><tr><td> VPT-deep</td><td>75.81</td><td>72.40</td><td>73.97</td><td>VPT-deep</td><td>75.80</td><td>68.76</td><td>72.11</td><td> VPT-deep</td><td>97.50</td><td>94.10</td><td>95.77</td></tr><tr><td>UPT</td><td>76.88</td><td></td><td>75.5776.15</td><td>UPT</td><td>75.83</td><td>70.8073.23</td><td></td><td>UPT</td><td>97.70</td><td></td><td>95.6396.14</td></tr><tr><td colspan=\"4\">(d) OxfordPets.</td><td colspan=\"4\">(e) StanfordCars.</td><td colspan=\"4\">(f) Flowers102.</td></tr><tr><td></td><td>Base</td><td>New</td><td>H</td><td></td><td>Base</td><td>New</td><td>H</td><td></td><td>Base</td><td>New</td><td>H</td></tr><tr><td>CLIP</td><td>91.17</td><td>97.26</td><td>94.12</td><td>CLIP</td><td>63.37</td><td>74.89</td><td>68.65</td><td>CLIP</td><td>72.08</td><td>77.80</td><td>74.83</td></tr><tr><td>CoOp CoCoOp</td><td>93.67</td><td>95.29</td><td>94.47</td><td>CoOp</td><td>78.12</td><td>60.40</td><td>68.13</td><td>CoOp</td><td>97.60</td><td>59.67</td><td>74.06</td></tr><tr><td>VPT-s</td><td>95.20</td><td>97.69</td><td>96.43</td><td>CoCoOp</td><td>70.49</td><td>73.59</td><td>72.01</td><td>CoCoOp</td><td>94.87</td><td>71.75</td><td>81.71</td></tr><tr><td></td><td>93.90</td><td>96.87</td><td>95.36</td><td>VPT-s</td><td>66.00</td><td>74.23</td><td>69.88</td><td>VPT-s</td><td>75.83</td><td>75.73</td><td>75.78</td></tr><tr><td>VPT-deep</td><td>94.33 95.50</td><td></td><td>94.91</td><td>VPT-deep</td><td>69.23</td><td>74.03</td><td>71.55</td><td>VPT-deep</td><td>83.63</td><td>70.50</td><td>76.50</td></tr><tr><td>UPT</td><td colspan=\"3\">96.07 97.6096.32</td><td>UPT</td><td>68.5075.37</td><td></td><td>71.77</td><td>UPT</td><td>85.00</td><td></td><td>77.3380.99</td></tr><tr><td></td><td colspan=\"3\">(g) Food101.</td><td></td><td>(h) FGVCAircraft.</td><td></td><td></td><td></td><td>(i) SUN397.</td><td></td><td></td></tr><tr><td></td><td colspan=\"3\">Base New</td><td>H</td><td>Base</td><td>New</td><td>H</td><td></td><td>Base</td><td>New</td><td>H</td></tr><tr><td>CLIP CoOp</td><td>90.10</td><td>91.22</td><td>90.66</td><td>CLIP</td><td>27.19</td><td>36.29</td><td>31.09</td><td>CLIP</td><td>69.36</td><td>75.35</td><td>72.23</td></tr><tr><td>CoCoOp</td><td>88.33</td><td>82.26</td><td>85.19</td><td>CoOp</td><td>40.44</td><td>22.30</td><td>28.75</td><td>CoOp</td><td>80.60</td><td>65.89</td><td>72.51</td></tr><tr><td>VPT-s</td><td>90.70</td><td>91.29</td><td>90.99</td><td>CoCoOp</td><td>33.41</td><td>23.71</td><td>27.74</td><td>CoCoOp</td><td>79.74</td><td>76.86</td><td>78.27</td></tr><tr><td></td><td>90.17</td><td>90.97</td><td>90.56</td><td>VPT-s</td><td>30.83</td><td>35.17</td><td>32.86</td><td>VPT-s</td><td>75.40</td><td>77.27</td><td>76.32</td></tr><tr><td>VPT-deep</td><td>90.20 91.17</td><td></td><td>90.68</td><td>VPT-deep</td><td>33.40</td><td>35.17</td><td>34.26</td><td>VPT-deep</td><td>78.23 76.63</td><td></td><td>77.43</td></tr><tr><td>UPT</td><td colspan=\"3\">90.72 92.0091.35</td><td>UPT</td><td>32.76</td><td>36.10</td><td>34.53</td><td>UPT</td><td>78.90</td><td></td><td>78.5678.73</td></tr><tr><td></td><td colspan=\"3\">() DTD.</td><td></td><td>(k) EuroSAT.</td><td></td><td></td><td></td><td>(l) UCF101.</td><td></td><td></td></tr><tr><td></td><td>Base</td><td>New</td><td>H</td><td></td><td>Base</td><td>New</td><td>H</td><td></td><td>Base</td><td>New</td><td>H</td></tr><tr><td>CLIP CoOp</td><td>53.24</td><td>59.90</td><td>56.37</td><td>CLIP</td><td>56.48</td><td>64.05</td><td>60.03</td><td>CLIP</td><td>70.53</td><td>77.50</td><td>73.85</td></tr><tr><td>CoCoOp</td><td>79.44</td><td>41.18</td><td>54.24</td><td>CoOp</td><td>92.19</td><td>54.74</td><td>68.69</td><td>CoOp</td><td>84.69</td><td>56.05</td><td>67.46</td></tr><tr><td>VPT-s</td><td>77.01</td><td>56.00</td><td>64.85</td><td>CoCoOp</td><td>87.49</td><td>60.04</td><td>71.21</td><td>CoCoOp</td><td>82.33</td><td>73.45</td><td>77.64</td></tr><tr><td>VPT-deep</td><td>55.27</td><td>57.16</td><td>56.20</td><td>VPT-s</td><td>71.67</td><td>58.87</td><td>64.64</td><td>VPT-s</td><td>75.60</td><td>76.10</td><td>75.85</td></tr><tr><td></td><td>64.87</td><td>55.40</td><td>59.76</td><td>VPT-deep</td><td>66.70</td><td>60.67</td><td>63.54</td><td> VPT-deep</td><td>80.07</td><td>74.50</td><td>77.18</td></tr><tr><td>UPT</td><td colspan=\"3\">69.53 62.1365.63</td><td>UPT</td><td>73.57</td><td></td><td>70.4371.96</td><td>UPT</td><td>78.10</td><td></td><td>76.3377.21</td></tr></table>",
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|
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|
| 1321 |
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|
| 1322 |
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"text": "",
|
| 1323 |
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{
|
| 1332 |
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"type": "text",
|
| 1333 |
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"text": "Single-modal Baselines. We observe that previous single-modal baselines perform dramatically different on the base and new splits. In particular, the text prompt tuning method CoOp achieves the highest performance on base classes and poor performance on new classes. On the contrary, visual prompt tuning approaches VPT-shallow and VPT-deep obtain high accuracy on base classes, but low accuracy on new classes. Such results show the intrinsic discrepancy between single-modal text and visual prompt tuning methods. CoOp optimizes specifically for base classes but at the expense of generalization ability on new classes. The advanced text prompt tuning approach CoCoOp with the input-conditional design achieves the best base and new trade-off among the single-modal baselines. ",
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| 1334 |
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| 1342 |
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| 1343 |
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"type": "text",
|
| 1344 |
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"text": "Strong Generalizability of UPT. As shown in Table 3, UPT is more generalizable than baseline methods when taking into account both the base and new classes. As for base classes, UPT is better than VPT but worse than $\\mathrm { C o O p }$ . This is reasonable because UPT are jointly optimized on the text and visual modalities, and the visual modality branch is not specifically for base classes. For new classes, UPT has significantly improved performance. For instance, UPT obtains $+ 1 2 . 3 5 / + 2 . 3 6 / + 3 . 1 7$ gains for CoOp/VPT-shallow/VPT-deep. UPT even achieves $+ 1 . 3 5$ gains on new classes compared with the CLIP baseline without prompt learning. In summary, the experimental results under the base-to-new generalization setting show strong generalizability of UPT. ",
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{
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"type": "table",
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"img_path": "images/4cc90c73175a8c58b2b9176663a0e9a1bdf3e191ce643a338019cac510319bcd.jpg",
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| 1356 |
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"table_caption": [
|
| 1357 |
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"Table 4: Comparison results in the cross-dataset transfer setting. Prompts applied to the 10 target datasets are learned from source ImageNet dataset. The best and second best methods are highlighted in red and orange , respectively. "
|
| 1358 |
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],
|
| 1359 |
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"table_footnote": [],
|
| 1360 |
+
"table_body": "<table><tr><td></td><td>Source</td><td colspan=\"10\">Target</td></tr><tr><td></td><td>1enege</td><td>CErleeaer</td><td>DPPpprtet</td><td>srsrrretttes</td><td>TiroinG</td><td>JorPoon</td><td>FrTeiettt</td><td>160308</td><td>CII</td><td>JItoII</td><td>UUUIII</td><td>aneace</td></tr><tr><td>CoOp CoCoOp</td><td>71.51</td><td>93.70</td><td>89.14</td><td>64.51</td><td>68.71</td><td>85.30</td><td>18.47</td><td>64.15</td><td>41.92</td><td>46.39</td><td>66.55</td><td>63.88</td></tr><tr><td></td><td>71.02</td><td>94.43</td><td>90.14</td><td>65.32</td><td>71.88</td><td>86.06</td><td>22.94</td><td>67.36</td><td>45.73</td><td>45.37</td><td>68.21</td><td>65.74</td></tr><tr><td>VPT-shallow</td><td>68.98</td><td>93.07</td><td>89.63</td><td>63.63</td><td>70.50</td><td>85.03</td><td>24.01</td><td>66.30</td><td>45.13</td><td>45.56</td><td>66.80</td><td>65.33</td></tr><tr><td>VPT-deep</td><td>70.57</td><td>90.33</td><td>88.50</td><td>57.87</td><td>63.83</td><td>76.90</td><td>21.93</td><td>63.10</td><td>42.13</td><td>40.63</td><td>64.53</td><td>61.85</td></tr><tr><td>UPT</td><td>70.86</td><td>93.31</td><td></td><td></td><td></td><td>90.57 65.3372.33 86.17</td><td>24.57</td><td>67.66</td><td>45.67</td><td>44.94</td><td></td><td>68.23 65.85</td></tr></table>",
|
| 1361 |
+
"bbox": [
|
| 1362 |
+
173,
|
| 1363 |
+
160,
|
| 1364 |
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825,
|
| 1365 |
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334
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| 1366 |
+
],
|
| 1367 |
+
"page_idx": 14
|
| 1368 |
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},
|
| 1369 |
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{
|
| 1370 |
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"type": "text",
|
| 1371 |
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"text": "",
|
| 1372 |
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"bbox": [
|
| 1373 |
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174,
|
| 1374 |
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|
| 1375 |
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825,
|
| 1376 |
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444
|
| 1377 |
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],
|
| 1378 |
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"page_idx": 14
|
| 1379 |
+
},
|
| 1380 |
+
{
|
| 1381 |
+
"type": "text",
|
| 1382 |
+
"text": "C.2 CROSS-DATASET TRANSFER ",
|
| 1383 |
+
"text_level": 1,
|
| 1384 |
+
"bbox": [
|
| 1385 |
+
176,
|
| 1386 |
+
459,
|
| 1387 |
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411,
|
| 1388 |
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474
|
| 1389 |
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],
|
| 1390 |
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"page_idx": 14
|
| 1391 |
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},
|
| 1392 |
+
{
|
| 1393 |
+
"type": "text",
|
| 1394 |
+
"text": "In the cross-dataset transfer setting, prompts learned from ImageNet are applied to ten other target datasets to evaluate the generalizability. The detailed results are presented in Table 4. We find VPT-shallow achieves higher accuracy than VPT-deep, and the text prompt tuning method CoCoOp outperforms visual prompt tuning approaches. On the source dataset, UPT obtains better performance than VPT but worse than CoOp. On target datasets, UPT obtains the best performance on six out of ten. The results on the cross-dataset transfer setting also verify that our proposed UPT is more generalizable than single-modal baselines. ",
|
| 1395 |
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"bbox": [
|
| 1396 |
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|
| 1401 |
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"page_idx": 14
|
| 1402 |
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}
|
| 1403 |
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]
|
parse/dev/1QQnYd02etI/1QQnYd02etI_model.json
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parse/dev/3jooF27-0Wy/3jooF27-0Wy.md
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| 1 |
+
# FLEXCONV: CONTINUOUS KERNEL CONVOLUTIONSWITH DIFFERENTIABLE KERNEL SIZES
|
| 2 |
+
|
| 3 |
+
David W. Romero∗,1, Robert-Jan Bruintjes∗,2, Erik J. Bekkers3, Jakub M. Tomczak1, Mark Hoogendoorn1, Jan C. van Gemert2 1 Vrije Universiteit Amsterdam 2 Delft University of Technology 3 University of Amsterdam The Netherlands d.w.romeroguzman@vu.nl, r.bruintjes@tudelft.nl
|
| 4 |
+
|
| 5 |
+
# ABSTRACT
|
| 6 |
+
|
| 7 |
+
When designing Convolutional Neural Networks (CNNs), one must select the size of the convolutional kernels before training. Recent works show CNNs benefit from different kernel sizes at different layers, but exploring all possible combinations is unfeasible in practice. A more efficient approach is to learn the kernel size during training. However, existing works that learn the kernel size have a limited bandwidth. These approaches scale kernels by dilation, and thus the detail they can describe is limited. In this work, we propose FlexConv, a novel convolutional operation with which high bandwidth convolutional kernels of learnable kernel size can be learned at a fixed parameter cost. FlexNets model long-term dependencies without the use of pooling, achieve state-of-the-art performance on several sequential datasets, outperform recent works with learned kernel sizes, and are competitive with much deeper ResNets on image benchmark datasets. Additionally, FlexNets can be deployed at higher resolutions than those seen during training. To avoid aliasing, we propose a novel kernel parameterization with which the frequency of the kernels can be analytically controlled. Our novel kernel parameterization shows higher descriptive power and faster convergence speed than existing parameterizations. This leads to important improvements in classification accuracy.
|
| 8 |
+
|
| 9 |
+
# 1 INTRODUCTION
|
| 10 |
+
|
| 11 |
+
The kernel size of a convolutional layer defines the region from which features are computed, and is a crucial choice in their design. Commonly, small kernels (up to $7 \mathrm { p x }$ ) are used almost exclusively and are combined with pooling to model long term dependencies (Simonyan & Zisserman, 2014; Szegedy et al., 2015; He et al., 2016; Tan & Le, 2019). Recent works indicate, however, that CNNs benefit from using convolutional kernels (i) of varying size at different layers (Pintea et al., 2021; Tomen et al., 2021), and $( i i )$ at the same resolution of the data (Peng et al., 2017; Cordonnier et al., 2019; Romero et al., 2021). Unfortunately, most CNNs represent convolutional kernels as tensors of discrete weights and their size must be fixed prior to training. This makes exploring different kernel sizes at different layers difficult and time-consuming due to $( i )$ the large search space, and (ii) the large number of weights required to construct large kernels.
|
| 12 |
+
|
| 13 |
+
A more efficient way to tune different kernel sizes at different layers is to learn them during training. Existing methods define a discrete weighted set of basis functions, e.g., shifted Delta-Diracs (Fig. 2b, Dai et al. (2017)) or Gaussian functions (Fig. 2c, Jacobsen et al. (2016); Shelhamer et al. (2019); Pintea et al. (2021)). During training they learn dilation factors over the basis functions to increase the kernel size, which crucially limits the bandwidth of the resulting kernels.
|
| 14 |
+
|
| 15 |
+
In this work, we present the Flexible Size Continuous Kernel Convolution (FlexConv), a convolutional layer able to learn high bandwidth convolutional kernels of varying size during training (Fig. 1). Instead of using discrete weights, we provide a continuous parameterization of convolutional kernels via a small neural network (Romero et al., 2021). This parameterization allows us to model continuous functions of arbitrary size with a fixed number of parameters. By multiplying the response of the neural network with a Gaussian mask, the size of the kernel can be learned during training (Fig. 2a). This allows us to produce detailed kernels of small sizes (Fig. 3), and tune kernel sizes efficiently.
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Figure 1: The Flexible Size Continuous Kernel Convolution (FlexConv). FlexConv defines convolutional kernels as the multiplication of a continuous convolutional kernel $\mathtt { M L P } ^ { \psi }$ , with a Gaussian mask of local support $w _ { \mathrm { g a u s s } }$ : $\boldsymbol { \psi } ( x , y ) = w _ { \mathrm { g a u s s } } ( x , y ; \theta _ { \mathrm { m a s k } } ) \cdot \mathtt { M L P } ^ { \boldsymbol { \psi } } ( x , y )$ . By learning the parameters =of the mask, the size of the convolutional kernel can be optimized during training. See also Fig. 7.
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FlexConvs can be deployed at higher resolutions than those observed during training, simply by using a more densely sampled grid of kernel indices. However, the high bandwidth of the kernel can lead FlexConv to learn kernels that show aliasing at higher resolutions, if the kernel bandwidth exceeds the Nyquist frequency. To solve this problem, we propose to parameterize convolutional kernels as Multiplicative Anisotropic Gabor Networks (MAGNets). MAGNets are a new class of Multiplicative Filter Networks (Fathony et al., 2021) that allows us to analyze and control the frequency spectrum of the generated kernels. We use this analysis to regularize FlexConv against aliasing. With this regularization, FlexConvs can be directly deployed at higher resolutions with minimal accuracy loss. Furthermore, MAGNets provide higher descriptive power and faster convergence speed than existing continuous kernel parameterizations (Schütt et al., 2017; Finzi et al., 2020; Romero et al., 2021). This leads to important improvements in classification accuracy (Sec. 4).
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Our experiments show that CNNs with FlexConvs, coined FlexNets, achieve state-of-the-art across several sequential datasets, match performance of recent works with learnable kernel sizes with less compute, and are competitive with much deeper ResNets (He et al., 2016) when applied on image benchmark datasets. Thanks to the ability of FlexConvs to generalize across resolutions, FlexNets can be efficiently trained at low-resolution to save compute, e.g., $1 6 \times 1 6$ CIFAR images, and be deployed on the original data resolution with marginal accuracy loss, e.g., $3 2 \times 3 2$ CIFAR images.
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In summary, our contributions are:
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• We introduce the Flexible Size Continuous Kernel Convolution (FlexConv), a convolution operation able to learn high bandwidth convolutional kernels of varying size end-to-end. • Our proposed Multiplicative Anisotropic Gabor Networks (MAGNets) allow for analytic control of the properties of the generated kernels. This property allows us to construct analytic alias-free convolutional kernels that generalize to higher resolutions, and to train FlexNets at low resolution and deploy them at higher resolutions. Moreover, MAGNets show higher descriptive power and faster convergence speed than existing kernel parameterizations. • CNN architectures with FlexConvs (FlexNets) obtain state-of-the-art across several sequential datasets, and match recent works with learnable kernel size on CIFAR-10 with less compute.
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# 2 RELATED WORK
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Adaptive kernel sizes. Loog & Lauze (2017) regularize the scale of convolutional kernels for filter learning. For image classification, adaptive kernel sizes have been proposed via learnable pixel-wise offsets (Dai et al., 2017), learnable padding operations (Han et al., 2018), learnable dilated Gaussian functions (Shelhamer et al., 2019; Xiong et al., 2020; Tabernik et al., 2020; Nguyen, 2020) and scalable Gaussian derivative filters (Pintea et al., 2021; Tomen et al., 2021; Lindeberg, 2021). These approaches either dilate discrete kernels (Fig. 2b), or use discrete weights on dilated basis functions (Fig. 2c). Using dilation crucially limits the bandwidth of the resulting kernels. In contrast, FlexConvs are able to construct high bandwidth convolutional kernels of varying size with a fixed parameter count. Larger kernels are obtained simply by passing more positions to the kernel network (Fig. 1).
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Continuous kernel convolutions. Discrete convolutional kernel parameterizations assign an independent weight to each specific position in the kernel. Continuous convolutional kernels, on the other hand, view convolutional kernels as continuous functions parameterized via a small neural network $\mathtt { M L P } ^ { \psi } \colon \mathbb { R } ^ { \mathrm { D } } \mathbb { R } ^ { \mathrm { N _ { o u t } \times N _ { i n } } }$ , with D the data dimensionality. This defines a convolutional kernel for which arbitrary input positions can be queried. Continuous kernels have primarily been used to handle irregularly-sampled data locally, e.g., molecular data (Simonovsky & Komodakis, 2017; Schütt et al., 2017) and point-clouds (Thomas et al., 2018; Wang et al., 2018; Shi et al., 2019).
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Figure 2: Existing approaches increase the size of convolutional kernels via (learnable) parametric dilations, e.g., by deformation (b) or by Gaussian blur (c). However, dilation limits the bandwidth of the dilated kernel and with it, the amount of detail it can describe. Contrarily, FlexNets extend their kernels by passing a larger vector of positions to the neural network parameterizing them. As a result, FlexConvs are able to learn high bandwidth convolutional kernels of varying size end-to-end (a).
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Figure 3: The importance of dynamic sizes in continuous kernel convolutions. Consider a neural network predicting pixel values at each position. If the entire image is considered, the network must use part of its capacity to learn to predict zeros outside of the flower region, which in turn degrades the quality of the approximation in the region of interest (b). Importantly, the better the localization of the flower, the higher the approximation fidelity becomes. FlexNets learn the size of their convolutional kernels at each layer during training, and thus $( i )$ use the capacity of the kernel efficiently, (ii) converge faster to good approximations, and $( i i i )$ are faster in execution –via dynamic cropping–.
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Recently, Romero et al. (2021) introduced the Continuous Kernel Convolution (CKConv) as a tool to model long-term dependencies. CKConv uses a continuous kernel parameterization to construct convolutional kernels as big as the input signal with a constant parameter cost. Contrarily, FlexConvs jointly learn the convolutional kernel as well as its size. This leads to important advantages in terms of expressivity (Fig. 3), convergence speed and compute costs of the operation.
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Implicit neural representations. Parameterizing a convolutional kernel via a neural network can be seen as learning an implicit neural representation of the underlying convolutional kernel (Romero et al., 2021). Implicit neural representations construct continuous data representations by encoding data in the weights of a neural network (Park et al., 2019; Sitzmann et al., 2020; Fathony et al., 2021).
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We replace the SIREN (Sitzmann et al., 2020) kernel parameterization used in Romero et al. (2021) by our Multiplicative Anisotropic Gabor Networks: a new class of Multiplicative Filter Networks (Fathony et al., 2021). MFNs allow for analytic control of the resulting representations, and allow us to construct analytic alias-free convolutional kernels. The higher expressivity and convergence speed of MAGNets lead to accuracy improvements in CNNs using them as kernel parameterization.
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# 3 METHOD
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In this section, we introduce our approach. First, we introduce FlexConv and the Gaussian mask. Next, we introduce our Multiplicative Anisotropic Gabor Networks (MAGNets) and provide a description of our regularization technique used to control the spectral components of the generated kernel.
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# 3.1 FLEXIBLE SIZE CONTINUOUS KERNEL CONVOLUTION (FLEXCONV)
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To learn the kernel size during training, FlexConvs define their convolutional kernels $\psi$ as the product of the output of a neural network $\mathtt { M L P } ^ { \psi }$ with a Gaussian mask of local support. The neural network $\mathtt { M L P } ^ { \psi }$ parameterizes the kernel, and the Gaussian mask parameterizes its size (Fig. 1).
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Anisotropic Gaussian mask. Let $\begin{array} { r } { G ( x ; \mu _ { \mathrm { X } } , \sigma _ { \mathrm { X } } ^ { 2 } ) { : = } \exp \left\{ - \frac { 1 } { 2 } \sigma _ { \mathrm { X } } ^ { - 2 } ( x - \mu _ { \mathrm { X } } ) ^ { 2 } \right\} } \end{array}$ be a Gaussian function parameterized by a mean-variance tuple $( \mu _ { \mathrm { X } } , \sigma _ { \mathrm { X } } ^ { 2 } )$ . The anisotropic Gaussian mask is defined as:
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$$
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w _ { \mathrm { g a u s s } } ( x , y ; \{ \mu _ { \mathrm { X } } , \sigma _ { \mathrm { X } } ^ { 2 } , \mu _ { \mathrm { Y } } , \sigma _ { \mathrm { Y } } ^ { 2 } \} ) = G ( x ; \mu _ { \mathrm { X } } , \sigma _ { \mathrm { X } } ^ { 2 } ) G ( y ; \mu _ { \mathrm { Y } } , \sigma _ { \mathrm { Y } } ^ { 2 } ) .
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$$
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By learning $( \mu _ { \mathrm { X } } , \sigma _ { \mathrm { X } } ^ { 2 } )$ and $( \mu _ { \mathrm { Y } } , \sigma _ { \mathrm { Y } } ^ { 2 } )$ independently, anisotropic non-centered windows can be learne
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3.2 MULTIPLICATIVE ANISOTROPIC GABOR NETWORKS (MAGNETS)
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In this section, we formalize our proposed parameterization for the kernel $\mathtt { M L P } ^ { \psi }$ . We start by introducing Multiplicative Filter Networks (Fathony et al., 2021), and present our MAGNets next.
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Multiplicative Filter Networks (MFNs). Recently, Fathony et al. (2021) proposed to construct implicit neural representations as the linear combination of exponentially many basis functions $\mathbf { g }$ :
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$$
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\begin{array} { r l r l } & { \mathbf { h } ^ { ( 1 ) } = \mathbf { g } \big ( [ x , y ] ; \theta ^ { ( 1 ) } \big ) } & & { \quad \mathbf { g } : \mathbb { R } ^ { 2 } \to \mathbb { R } ^ { \mathrm { N _ { h i d } } } } \\ & { \mathbf { h } ^ { ( l ) } = \big ( \mathbf { W } ^ { ( l ) } \mathbf { h } ^ { ( l - 1 ) } + \mathbf { b } ^ { ( l ) } \big ) \cdot \mathbf { g } \big ( [ x , y ] ; \theta ^ { ( l ) } \big ) } & & { \quad \mathbf { W } ^ { ( l ) } \in \mathbb { R } ^ { \mathrm { N _ { h i d } } \times \mathrm { N _ { h i d } } } , \mathbf { b } ^ { ( l ) } \in \mathbb { R } ^ { \mathrm { N _ { h i d } } } } \\ & { \psi ( x , y ) = \mathbf { W } ^ { ( \mathrm { L } ) } \mathbf { h } ^ { ( \mathrm { L } - 1 ) } + \mathbf { b } ^ { ( \mathrm { L } ) } \quad } & & { \quad \mathbf { W } ^ { ( \mathrm { L } ) } \in \mathbb { R } ^ { \mathrm { N } \times \mathrm { N _ { h i d } } } , \mathbf { b } ^ { ( \mathrm { L } ) } \in \mathbb { R } ^ { \mathrm { N } } } \end{array}
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$$
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+
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where $\left\{ \pmb { \theta } ^ { ( l ) } , \mathbf { W } ^ { ( l ) } , \mathbf { b } ^ { ( l ) } \right\}$ depict the learnable parameters of the bases and the affine transformations, and $\mathrm { { N , N _ { h i d } } }$ depict the number of output and hidden channels, respectively. Depending on the selection of $\mathbf { g }$ , MFNs obtain approximations comparable to those of SIRENs (Sitzmann et al., 2020) with faster convergence rate. The most successful instantiation of MNFs are the Multiplicative Gabor Network (MGN): MFNs constructed with isotropic Gabor functions as basis $\mathbf { g }$ (in Eq. 2):
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$$
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\mathbf { g } \big ( [ x , y ] ; \pmb { \theta } ^ { ( l ) } \big ) = \exp \Bigg ( - \frac { \gamma ^ { ( l ) } } { 2 } \Big [ \big ( x - \pmb { \mu } ^ { ( l ) } \big ) ^ { 2 } + \big ( y - \pmb { \mu } ^ { ( l ) } \big ) ^ { 2 } \Big ] \Bigg ) \mathrm { S i n } \big ( \mathbf { W } _ { \mathbf { g } } ^ { ( l ) } \cdot [ x , y ] + \mathbf { b } _ { \mathbf { g } } ^ { ( l ) } \big ) ,
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$$
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Note that, by setting $\mathrm { N { = } N _ { o u t } { \times } N _ { i n } }$ , an MFN can parameterize a convolutional kernel with $\mathrm { { N } } _ { \mathrm { { i n } } }$ input and $\mathrm { N _ { o u t } }$ output channels. Fathony et al. (2021) show that MFNs are equivalent to a linear combination of exponentially many basis functions $\mathbf { g }$ . This allows us to analytically derive properties of MFN representations, and plays a crucial role in the derivation of alias-free MAGNets (Sec. 3.3).
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Multiplicative Anisotropic Gabor Networks (MAGNets). Our MAGNet formulation is based on the observation that isotropic Gabor functions, i.e., with equal $\gamma$ for the horizontal and vertical directions, are undesirable as basis for the construction of MFNs. Whenever a frequency is required along a certain direction, an isotropic Gabor function automatically introduces that frequency in both directions. As a result, other bases must counteract this frequency in the direction where the frequency is not required, and thus the capacity of the MFN is not used optimally (Daugman, 1988).
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Following the original formulation of the 2D Gabor functions (Daugman, 1988), we alleviate this limitation by using anisotropic Gabor functions instead:
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$$
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\begin{array} { r l } & { \displaystyle \mathbf { g } \big ( [ x , y ] ; \pmb \theta ^ { ( l ) } \big ) = \exp \Bigg ( - \frac { 1 } { 2 } \Big [ \Big ( \gamma _ { _ \mathrm { X } } ^ { ( l ) } \big ( x - \pmb \mu _ { _ \mathrm { X } } ^ { ( l ) } \big ) \Big ) ^ { 2 } + \Big ( \gamma _ { _ \mathrm { Y } } ^ { ( l ) } \big ( y - \pmb \mu _ { _ \mathrm { Y } } ^ { ( l ) } \big ) \Big ) ^ { 2 } \Big ] \Bigg ) \mathrm { S i n } \Big ( \mathbf { W } _ { \Xi } ^ { ( l ) } \big [ x , y \big ] + \mathbf { b } _ { \mathrm { g } } ^ { ( l ) } \Big ) } \\ & { \displaystyle \pmb \theta ^ { ( l ) } = \Big \{ \gamma _ { _ \mathrm { X } } ^ { ( l ) } \in \mathbb { R } ^ { \mathrm { N _ { h i d } } } , \gamma _ { _ \mathrm { Y } } ^ { ( l ) } \in \mathbb { R } ^ { \mathrm { N _ { h i d } } } , \mu _ { _ \mathrm { X } } ^ { ( l ) } \in \mathbb { R } ^ { \mathrm { N _ { h i d } } } , \mu _ { _ \mathrm { Y } } ^ { ( l ) } \in \mathbb { R } ^ { \mathrm { N _ { h i d } } } , \mathbf { W } _ { \Xi } ^ { ( l ) } \in \mathbb { R } ^ { \mathrm { N _ { h i d } } \times 2 } , \mathbf { b } _ { \mathrm { g } } ^ { ( l ) } \in \mathbb { R } ^ { \mathrm { N _ { h i d } } } \Big \} . } \end{array}
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$$
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+
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The resulting Multiplicative Anisotropic Gabor Network (MAGNet) obtains better control upon frequency components introduced to the approximation, and demonstrates important improvements in terms of descriptive power and convergence speed (Sec. 4).
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MAGNet initialization. Fathony et al. (2021) proposes to initialize MGNs by drawing the size of the Gaussian envelopes, i.e., the $\gamma ^ { ( l ) }$ term, from a Gamma $\left( \alpha \cdot \mathrm { L } ^ { - 1 } , \beta \right)$ distribution at every layer $l \in [ 1 , . . , \mathrm { L } - 1 ]$ . We observe however that this initialization does not provide much variability on the ∈initial extension of the Gaussian envelopes and in fact, most of them cover a large portion of the space at initialization. T stimulate diversity, we initialize the $\{ \gamma _ { \mathrm { X } } ^ { ( l ) } , \gamma _ { \mathrm { Y } } ^ { ( l ) } \}$ terms by a $\mathrm { G a m m a } ( \alpha l ^ { - 1 } , \beta )$ $l$ -th layer. We observe that our proposed initialization consistently leads to better accuracy than the initialization of Fathony et al. (2021) across all tasks considered. (Sec. 4).
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# 3.3 ANALYTIC ALIAS-FREE MAGNETS
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FlexConvs can be deployed at higher resolutions than those observed during training, simply by sampling the underlying continuous representation of the kernel more densely, and accounting for the
|
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+
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+
change in sampling rate. Consider a D-dimensional input signal $f _ { \mathrm { r } ^ { ( 1 ) } }$ with resolution $\mathrm { r } ^ { ( 1 ) }$ . FlexConv learns a kernel $\psi _ { \mathrm { r } ^ { ( 1 ) } }$ that can be inferred at a higher resolution $\mathrm { r } ^ { ( 2 ) }$ (Romero et al., 2021):
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+
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+
$$
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\biggl ( f _ { \mathrm { r } ^ { ( 2 ) } } * \psi _ { \mathrm { r } ^ { ( 2 ) } } \biggr ) \approx \left( \frac { \mathrm { \bar { r } ^ { \left( 1 \right) } } } { \mathrm { r } ^ { \left( 2 \right) } } \right) ^ { \mathrm { D } } \biggl ( f _ { \mathrm { r } ^ { ( 1 ) } } * \psi _ { \mathrm { r } ^ { ( 1 ) } } \biggr ) .
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$$
|
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+
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+
Note however, that Eq. 9 holds approximately. This is due to aliasing artifacts which can appear if the frequencies in the learned kernel surpass the Nyquist criterion of the target resolution. Consequently, an anti-aliased parameterization is vital to construct kernels that generalize well to high resolutions.
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Towards alias-free implicit neural representations. We observe that SIRENs as well as unconstrained MFNs and MAGNets exhibit aliasing when deployed on resolutions higher than the training resolution, which hurts performance of the model. An example kernel with aliasing is shown in Fig. 8.
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To combat aliasing, we would like to control the representation learned by MAGNets. MAGNets –and MFNs in general– construct implicit neural representations that can be seen as a linear combination of basis functions. This property allows us to analytically derive and study the properties of the resulting neural representation. Here, we use this property to derive the maximum frequency of MAGNet-generated kernels, so as to regularize MAGNets against aliasing during training. We analytically derive the maximum frequency of a MAGNet, and penalize it whenever it exceeds the Nyquist frequency of the training resolution. We note that analytic derivations are difficult for other implicit neural representations, e.g., SIRENs, due to stacked layer-wise nonlinearities.
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+
Maximum frequency of MAGNets. The maximum frequency component of a MAGNet is given by:
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|
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+
$$
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f _ { \mathrm { M A G N e t } } ^ { + } = \sum _ { l = 1 } ^ { \mathrm { L } } \operatorname* { m a x } _ { i _ { l } } \left( \left( \operatorname* { m a x } _ { j } \frac { \mathbf { W } _ { \mathrm { g } , i _ { l } , j } ^ { ( l ) } } { 2 \pi } \right) + \frac { \sigma _ { \mathrm { c u t } } \operatorname* { m i n } \{ \gamma _ { \mathrm { X } , i _ { l } } ^ { ( l ) } , \gamma _ { \mathrm { Y } , i _ { l } } ^ { ( l ) } \} } { 2 \pi } \right) ,
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+
$$
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+
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+
where L corresponds to the number of layers, W(l)g , $\mathbf { W } _ { \mathrm { g } } ^ { ( l ) } , \gamma _ { \mathrm { X } } ^ { ( l ) } , \gamma _ { \mathrm { Y } } ^ { ( l ) }$ to the MAGNet parameters as defined in Eq. 8, and $\sigma _ { \mathrm { c u t } } { = } 2$ stdev to the cut-off frequency of the Gaussian envelopes in the Gabor filters. A formal treatment as well as the derivations can be found in Appx. A.1.
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Effect of the FlexConv mask. The Gaussian mask used to localize the response of the MAGNet also has an effect on the frequency spectrum. Hence, the maximum frequency of a FlexConv kernel is:
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$$
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f _ { \mathrm { F l e x C o n v } } ^ { + } = f _ { \mathrm { M A G N e t } } ^ { + } + f _ { w _ { \mathrm { g a u s s } } } ^ { + } , \mathrm { w i t h } f _ { w _ { \mathrm { g a u s s } } } ^ { + } = \frac { \sigma _ { \mathrm { c u t } } } { \operatorname* { m a x } \{ \sigma _ { \mathrm { X } } , \sigma _ { \mathrm { Y } } \} 2 \pi } .
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+
$$
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+
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+
Here, $\sigma _ { \mathrm { X } } , \sigma _ { \mathrm { Y } }$ correspond to the mask parameters (Eq. 1). Intuitively, multiplication with the mask blurs in the frequency domain, as it is equivalent to convolution with the Fourier transform of the mask.
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Aliasing regularization of FlexConv kernels. With the analytic derivation of $f _ { \mathrm { F l e x C o n v } } ^ { + }$ we penalize the generated kernels to have frequencies smaller or equal to their Nyquist frequency $\hat { f } _ { \mathrm { N y q } } ( k )$ via:
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+
|
| 126 |
+
$$
|
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+
\begin{array} { r } { \mathcal { L } _ { \mathrm { H F } } = | | \operatorname* { m a x } \{ f _ { \mathrm { F l e x C o n v } } ^ { + } , f _ { \mathrm { N y q } } ( k ) \} - f _ { \mathrm { N y q } } ( k ) | | ^ { 2 } , \mathrm { w i t h } f _ { \mathrm { N y q } } ( k ) = \frac { k - 1 } { 4 } . } \end{array}
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$$
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+
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Here, $k$ depicts the size of the FlexConv kernel before applying the Gaussian mask, and is equal to the size of the input signal. In practice, we implement Eq. 25 by regularizing the individual MAGNet layers, as is detailed in Appx. A.2. To verify our method, Fig. 8 (Appx. A.1) shows that the frequency components of FlexNet kernels are properly regularized for aliasing.
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# 4 EXPERIMENTS
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We evaluate FlexConv across classification tasks on sequential and image benchmark datasets, and validate the ability of MAGNets to approximate complex functions. A complete description of the datasets used is given in Appx. B. Appx. D.2 reports the parameters used in all our experiments.1
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# 4.1 WHAT KIND OF FUNCTIONS CAN MAGNETS APPROXIMATE?
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Bandwidth of methods with learnable sizes. First, we compare the bandwidth of MAGNet against N-Jet (Pintea et al., 2021) by optimizing each to fit simple targets: (i) Gabor filters of known frequency, (ii) random noise and (iii) an a $1 1 \times 1 1$ AlexNet kernel from the first layer (Krizhevsky et al., 2012). Fig. 4 shows that, even with 9 orders of Gaussian derivatives, N-Jets cannot fit high frequency signals in large kernels. Crucially, N-Jet models require many Gaussian derivative orders to model high frequency signals in large kernels: a hyperparameter which proportionally increases their inference time and parameter count. MAGNets, on the other hand, accurately model large high frequency signals. This allows FlexNets to learn large kernels with high frequency components.
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Figure 4: Left: Final MSE after fitting each model to Gabor filters of different frequencies. N-Jets cannot fit high frequencies. Right: Kernels learned by each model. SIREN and MAGNet can fit all targets. MAGNet-S: a small MAGNet of size akin to N-Jets, still does well on the Gabor and AlexNet targets.
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Table 1: Test accuracy and ablation studies on sMNIST, pMNIST, sCIFAR10 and npCIFAR10.
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<table><tr><td>MODEL</td><td>SIZE</td><td>SMNIST</td><td>PMNIST</td><td>sCIFAR10</td><td>NPCIFAR10</td></tr><tr><td>DilRNN (Chang et al.,2017)</td><td>44K</td><td>98.0</td><td>96.1</td><td></td><td></td></tr><tr><td>IndRNN (Li et al., 2018)</td><td>83K</td><td>99.0</td><td>96.0</td><td></td><td></td></tr><tr><td>TCN (Bai et al.,2018a)</td><td>70K</td><td>99.0</td><td>97.2</td><td></td><td></td></tr><tr><td>r-LSTM(Trinh et al.,2018)</td><td>0.5M</td><td>98.4</td><td>95.2</td><td>72.2</td><td></td></tr><tr><td>Self-Att. (Trinh et al.,2018)</td><td>0.5M</td><td>98.9</td><td>97.9</td><td>62.2</td><td></td></tr><tr><td>TrellisNet (Bai et al.,2018b)</td><td>8M</td><td>99.20</td><td>98.13</td><td>73.42</td><td></td></tr><tr><td>URLSTM(Gu et al.,2020b)</td><td>-</td><td>99.28</td><td>96.96</td><td>71.00</td><td></td></tr><tr><td>URGRU + Zoneout (Gu et al.,2020b)</td><td>-</td><td>99.27</td><td>96.51</td><td>74.40</td><td></td></tr><tr><td>HiPPO (Gu et al.,2020a)</td><td>0.5M</td><td></td><td>98.30</td><td></td><td></td></tr><tr><td>Lipschitz RNN (Erichson et al.,2020)</td><td>158K</td><td>99.4</td><td>97.3</td><td>64.2</td><td>59.0</td></tr><tr><td>coRNN(Rusch& Mishra,2020)</td><td>134K</td><td>99.4</td><td>97.3</td><td>-</td><td>59.0</td></tr><tr><td>UnICORNN(Rusch& Mishra,2021)</td><td>135K</td><td>-</td><td>98.4</td><td>-</td><td>62.4</td></tr><tr><td>pLMU(Chilkuri & Eliasmith,2021)</td><td>165K</td><td>-</td><td>98.49</td><td>-</td><td>-</td></tr><tr><td>CKCNN-2</td><td>98K</td><td>99.31</td><td>98.00</td><td>62.25</td><td>60.5</td></tr><tr><td>CKCNN-2-Big</td><td>1M</td><td>99.32</td><td>98.54</td><td>63.74</td><td>62.2</td></tr><tr><td>CKTCNFOURIER-2</td><td>105K</td><td>99.44</td><td>98.40</td><td>68.28</td><td>66.26</td></tr><tr><td>CKTCNGABOR-2</td><td>106K</td><td>99.52</td><td>98.38</td><td>69.26</td><td>67.37</td></tr><tr><td>CKTCNMAGNET-2</td><td>105K</td><td>99.55</td><td>98.57</td><td>74.58</td><td>67.52</td></tr><tr><td>FlexTCN-2</td><td>108K</td><td>99.60</td><td>98.61</td><td>78.99</td><td>67.11</td></tr><tr><td>FlexTCN-4</td><td>241K</td><td>99.60</td><td>98.72</td><td>80.26</td><td>67.42</td></tr><tr><td>FlexTCN-6</td><td>375K</td><td>99.62</td><td>98.63</td><td>80.82</td><td>69.87</td></tr><tr><td>FlexTCNSIREN-6</td><td>343K</td><td>99.03</td><td>95.36</td><td>69.24</td><td>57.27</td></tr><tr><td>FlexTCNFourier-6</td><td>370K</td><td>99.49</td><td>97.97</td><td>74.79</td><td>67.35</td></tr><tr><td>FlexTCNGabor-6</td><td>373K</td><td>99.50</td><td>98.37</td><td>78.36</td><td>67.56</td></tr><tr><td>FlexTCNMAGNet-6</td><td>375K</td><td>99.62</td><td>98.63</td><td>80.82</td><td>69.87</td></tr></table>
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Expressivity of MLP parameterizations. Next, we compare the descriptive power and convergence speed of MAGNets, Gabor MFNs, Fourier MFNs and SIRENs for image approximation. To this end, we fit the images in the Kodak dataset (Kodak, 1991) with each of these methods. Our results (Tab. 5) show that MAGNets outperform all other methods, and converge faster to good approximations.
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# 4.2 CLASSIFICATION TASKS
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Network specifications. Here, we specify our networks for all our classification experiments. We parameterize all our convolutional kernels as the superposition of a 3-layer MAGNet and a learnable anisotropic Gaussian mask. We construct two network instances for sequential and image datasets respectively: FlexTCNs and FlexNets. Both are constructed by taking the structure of a baseline network –TCN (Bai et al., 2018a) or CIFARResNet (He et al., 2016)–, removing all internal pooling layers, and replacing convolutional kernels by FlexConvs. The FlexNet architecture is shown in Fig. 10 and varies only in the number of channels and blocks, e.g., FlexNet-16 has 7 blocks. Akin to Romero et al. (2021) we utilize the Fourier theorem to speed up convolutions with large kernels.
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Mask initialization. We initialize the FlexConv masks to be small. Preliminary experiments show this leads to better performance, faster execution, and faster training convergence. For sequences, the mask center is initialized at the last kernel position to prioritize the last information seen.
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Time series and sequential data. First we evaluate FlexTCNs on sequential classification datasets, for which long-term dependencies play an important role. We validate our approach on intrinsic discrete data: sequential MNIST, permuted MNIST (Le et al., 2015), sequential CIFAR10 (Chang et al., 2017), noise-padded CIFAR10 (Chang et al., 2019), as well as time-series data: CharacterTrajectories (CT) (Bagnall et al., 2018), SpeechCommands (Warden, 2018) with raw waveform (SC_raw) and MFCC input representations (SC).
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Table 2: Test accuracy on CT, SC and SC_raw
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<table><tr><td>MODEL</td><td>SIZE</td><td>CT</td><td>SC</td><td>SC_RAW</td></tr><tr><td>GRU-ODE</td><td>89K</td><td>96.2</td><td>44.8</td><td>~10.0</td></tr><tr><td>GRU-△t</td><td>89K</td><td>97.8</td><td>20.0</td><td>~10.0</td></tr><tr><td>GRU-D</td><td>89K</td><td>95.9</td><td>23.9</td><td>~10.0</td></tr><tr><td>ODE-RNN</td><td>89K</td><td>97.1</td><td>93.2</td><td>~10.0</td></tr><tr><td>NCDE</td><td>89K</td><td>98.8</td><td>88.5</td><td>~10.0</td></tr><tr><td>CKCNN</td><td>100K</td><td>99.53</td><td>95.27</td><td>71.66</td></tr><tr><td>CKTCNFourier</td><td></td><td>=</td><td>95.65</td><td>74.90</td></tr><tr><td>CKTCNGabor</td><td></td><td>=</td><td>96.66</td><td>78.10</td></tr><tr><td>CKTCNMAGNet</td><td>105K</td><td>99.53</td><td>97.01</td><td>80.69</td></tr><tr><td>FlexTCN-2</td><td>105sK</td><td>99.53</td><td>97.10</td><td>88.03</td></tr><tr><td>FlexTCN-4</td><td>239K</td><td>99.53</td><td>97.73</td><td>90.45</td></tr><tr><td>FlexTCN-6</td><td>373K</td><td>99.53</td><td>97.67</td><td>91.73</td></tr><tr><td>FlexTCNSIREN-6</td><td>370K</td><td>-</td><td>95.83</td><td>85.73</td></tr><tr><td>FlexTCNFourier-6</td><td>342K</td><td></td><td>97.62</td><td>91.02</td></tr><tr><td>FlexTCNGabor-6</td><td>373K</td><td></td><td>97.35</td><td>91.50</td></tr><tr><td>FlexTCNMAGNet-6</td><td>373K</td><td></td><td>97.67</td><td>91.73</td></tr></table>
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Table 3: Results on CIFAR-10. Results from \*original works and $^ \dagger$ single run.
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<table><tr><td>MODEL</td><td>SIZE</td><td>CIFAR-10 Acc.</td><td>TIME (SEC/EPOCH)</td></tr><tr><td>CIFARResNet-44</td><td>0.66M</td><td>92.9*+</td><td>22</td></tr><tr><td>DCN-gji</td><td>0.47M</td><td>89.7±0.3*</td><td>-</td></tr><tr><td>N-Jet-CIFARResNet32</td><td>0.52M</td><td>92.3 ±0.3*</td><td>-</td></tr><tr><td>N-Jet-ALLCNN</td><td>1.07M</td><td>92.5± 0.1*</td><td>-</td></tr><tr><td>FlexNet-16 w/ conv.(k = 3)</td><td>0.17M</td><td>89.5 ± 0.3</td><td>41</td></tr><tr><td>FlexNet-16 w/conv. (k = 33)</td><td>20.0M</td><td>78.0± 0.3</td><td>242</td></tr><tr><td>FlexNet-16 w/N-Jet</td><td>0.70M</td><td>91.7 ± 0.1</td><td>409</td></tr><tr><td>CKCNN-16</td><td>0.63M</td><td>72.1 ± 0.2</td><td>68</td></tr><tr><td>CKCNNMAGNet-16</td><td>0.67M</td><td>86.8 ± 0.6</td><td>102</td></tr><tr><td>FlexNetsIREN-16</td><td>0.63M</td><td>89.0± 0.3</td><td>89</td></tr><tr><td>FlexNetGabor-16</td><td>0.67M</td><td>91.9 ± 0.2</td><td>161</td></tr><tr><td>FlexNetGabor-16 +anis. Gauss.</td><td>0.67M</td><td>92.0 ± 0.1</td><td>147</td></tr><tr><td>FlexNetGabor-16 +Gabor init.</td><td>0.67M</td><td>92.0± 0.2</td><td>150</td></tr><tr><td>FlexNet-16</td><td>0.67M</td><td>92.2 ± 0.1</td><td>127</td></tr></table>
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Our results are summarized in Tables 1 and 2. FlexTCNs with two residual blocks obtain state-ofthe-art results on all tasks considered. In addition, depth further improves performance. FlexTCN-6 improves the current state-of-the-art on sCIFAR10 and npCIFAR10 by more than $6 \%$ . On the difficult SC_raw dataset –with sequences of length 16000–, FlexTCN-6 outperform the previous state-of-the-art by $2 0 . 0 7 \%$ : a remarkable improvement.
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Furthermore, we conduct ablation studies by changing the parameterization of $\mathtt { M L P } ^ { \psi }$ , and switching off the learnable kernel size ("CKTCNs") and considering global kernel sizes instead. CKTCNs and FlexTCNs with MAGNet kernels outperform corresponding models with all other kernel parameterizations: SIRENs (Sitzmann et al., 2020), MGNs and MFNs (Fathony et al., 2021). Moreover, we see a consistent improvement with respect to CKCNNs (Romero et al., 2021) by using learnable kernel sizes. This shows that both MAGNets and learnable kernel sizes contribute to the performance of FlexTCNs. Note that in 1D, MAGNets are equivalent to MGNs. However, MAGNets consistently perform better than MGNs. This improvement in accuracy is a result of our MAGNet initialization.
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Image classification. Next, we evaluate FlexNets for image classification on CIFAR-10 (Krizhevsky et al., 2009). Additional experiments on Imagenet-32, MNIST and STL-10 can be found in Appx. C.
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Table 3 shows our results on CIFAR-10. FlexNets are competitive with pooling-based methods such as CIFARResNet (He et al., 2016) and outperform learnable kernel size method DCNs (Tomen et al., 2021). In addition, we compare using N-Jet layers of order three (as in Pintea et al. (2021)) in FlexNets against using MAGNet kernels. We observe that N-Jet layers lead to worse performance, and are significantly slower than FlexConv layers with MAGNet kernels. The low accuracy of N-Jet layers is likely to be linked to the fact that FlexNets do not use pooling. Consequently, N-Jets are forced to learn large kernels with high-frequencies, which we show N-Jets struggle learning in Sec. 4.1.
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To illustrate the effect of learning kernel sizes, we also compare FlexNets against FlexNets with large and small discrete convolutional kernels (Tab. 3). Using small kernel sizes is parameter efficient, but is not competitive with FlexNets. Large discrete kernels on the other hand require a copious amount of parameters and lead to significantly worse performance. These results indicate that the best solution is somewhere in the middle and varying kernel sizes can learn the optimal kernel size for the task at hand.
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Similar to the sequential case, we conduct ablation studies on image data with learnable, nonlearnable kernel sizes and different kernel parameterizations. Table 3 shows that FlexNets outperform CKCNNs with corresponding kernel parameterizations. In addition, a clear difference in performance is apparent for MAGNets with respect to other parameterizations. These results corroborate that both MAGNets and FlexConvs contribute to the performance of FlexNets. Moreover, Tab. 3 illustrates the effect of the two contributions of MAGNet over MGN: anisotropic Gabor filters, and our improved initialization. Our results in image data are in unison with our previous results for sequential data (Tabs. 1, 2) and illustrate the value of the proposed improvements in MAGNets.
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Figure 5: Alias-free FlexNet-16 on CIFAR-10. We report change in accuracy between source and target resolutions, directly after upsampling (left) and after fine-tuning (right) (means over five runs).
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# 4.3 ALIAS-FREE FLEXNETS
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Regularizing the FlexConv mask. Though including f+wgauss in the frequency analysis of MAGNets is crucial for the accuracy of the derivation, including the FlexConv mask in aliasing regularization is undesirable, as it steers the model to learn large kernels in order to minimize the loss (see Eq. 25). However, excluding the mask from regularization could compromise the ability of FlexNet to generalize to higher resolutions. Here, we experiment with this trade-off.
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Accuracy change after fine-tuning
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Table 4: Alias-free FlexNets on CIFAR-10.
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<table><tr><td>MODEL</td><td>SIZE</td><td colspan="2">CIFAR-10 ACC.</td></tr><tr><td></td><td></td><td>16 px</td><td>△16px 32 px</td></tr><tr><td>CIFARResNet-44</td><td>0.66M</td><td>85.8 ± 0.2</td><td>-31.6 ± 1.3</td></tr><tr><td>FlexNet-16 w/ conv. (k = 3) FlexNet-16 w/conv.(k = 33)</td><td>0.17M 20.0M</td><td>85.3± 0.2</td><td>-21.2 ± 1.0</td></tr><tr><td>FlexNet-16 w/N-Jets</td><td>0.70M</td><td>67.7 ± 0.6 86.4 ± 0.2</td><td>-57.1 ± 1.6 -5.5 ± 1.3</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>CKCNN-16SIREN FlexNet-16sIREN</td><td>0.63m 0.63M</td><td>45.9 ± 1.0 70.4 ± 0.8</td><td>-15.8 ± 1.2</td></tr><tr><td></td><td></td><td></td><td>-50.0 ± 16.9</td></tr><tr><td>FlexNet-16 w/o reg.</td><td>0.67M</td><td>86.4 ± 0.4</td><td>-34.4 ± 14.3</td></tr><tr><td>FlexNet-16 w/reg.j fMAGNet FlexNet-16w/reg.JeConv</td><td>0.67M</td><td>86.5 ± 0.1</td><td>-3.8 ± 2.0</td></tr><tr><td></td><td>0.67M</td><td>85.1 ± 0.3</td><td>-3.3 ± 0.3</td></tr></table>
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Figure 5 shows accuracy change between ten source and target resolution combinations on CIFAR-10, both for including and excluding the FlexConv mask in the aliasing regularization. We train at the source resolution for 100 epochs, before testing the model at the target resolution with the upsampling described in Sec. 3.3. Next, we adjust $f _ { \mathrm { N y q } } ( k )$ to the target resolution, and finetune each model for 100 epochs at the target resolution.
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We find that regularizing just $f _ { \mathrm { M A G N e t } } ^ { + }$ yields a trade-off. It increases the accuracy difference
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between low and high resolution inference, but also increases the fine-tune accuracy at the target resolution.We therefore choose to, by default, regularize $f _ { \mathrm { M A G N e t } } ^ { + }$ only.
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Results of our alias-free FlexNet training on CIFAR-10 are in Table 4. We observe that the performance of a FlexNet trained without aliasing regularization largely breaks down when the dataset is upscaled. However, with our aliasing regularization most of the performance is retained.
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Comparatively, FlexNet retains more of the source resolution performance than FlexNets with N-Jet layers, while baselines degrade drastically at the target resolution. Fig. 8 shows the effect of aliasing regularization on the frequency components of FlexConv.
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Training at lower resolutions saves compute. We can train alias-free FlexNets at lower resolutions. To verify that this saves compute, we time the first 32 batches of training a FlexNet-7 on CIFAR-10. We compare against training on $1 6 \times 1 6$ images (downsampled before training). On 16x16 images, each batch takes $1 7 9 \mathrm { m s }$ $\left( \pm { } 7 \mathrm { m s } \right)$ . On $3 2 \mathrm { x } 3 2$ images, each batch takes $2 2 2 \mathrm { m s }$ $( \pm 9 \mathrm { { m s } ) }$ . Therefore, we save $24 \%$ training time when training FlexNets alias-free at half the native CIFAR-10 resolution.
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# 5 DISCUSSION
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Learned kernel sizes match conventional priors. Commonly, CNNs use architectures of small kernels and pooling layers. This allows convolutions to build a progressively growing receptive field. With learnable kernel sizes, FlexNet could learn a different prior over receptive fields, e.g., large kernels first, and small kernels next. However, FlexNets learn to increase kernel sizes progressively (Fig. 6), and match the network design that has been popular since AlexNet (Krizhevsky et al., 2012).
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Mask initialization as a prior for feature importance. The initial values of the FlexConv mask can be used to prioritize information at particular input regions. For instance, initializing the center of mask on the first element of sequential FlexConvs can be used to prioritize information from the far past. This prior is advantageous for tasks such as npCIFAR10. We observe that using this prior on npCIFAR10 leads to much faster convergence and better results $6 8 . 3 3 \%$ acc. w/ FlexTCN-2).
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Figure 6: Learned FlexConv masks for FlexNets with 3, 5 and 7 residual blocks. FlexNets learn very small kernels at shallow layers, which become larger as a function of depth.
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MAGNet regularization as prior induction. MAGNets allow for analytic control of the properties of the resulting representations. We use this property to generate alias-free kernels. However, other desiderata could be induced, e.g., smoothness, for the construction of implicit neural representations.
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Benefits of cropping and the influence of PyTorch. Dynamic cropping adjust the computational cost of the convolutions on the fly. For a signal of size M and a cropped kernel size $\mathrm { k \Omega }$ , this incurs in savings from $\mathrm { O } ( \mathrm { M } ^ { 2 ^ { D } } )$ to $\mathrm { O ( M ^ { D } k ^ { D } ) }$ relative to using global kernel sizes $\mathrm { ( O ( M ^ { 4 } ) }$ to $\mathrm { O } ( \mathrm { M } ^ { 2 } \mathrm { k } ^ { 2 } )$ in 2D). We test this theoretical speed up in a controlled environment for the Speech Commands and CIFAR10 datasets. Cropping reduces the per-epoch run time by a factor of $1 1 . 8 \mathrm { x }$ and $5 . 5 \mathrm { x }$ for Speech Commands and CIFAR-10, respectively. Interestingly, however, both run times become similar if the flag torch.backends.cudnn.benchmark is activated, with global kernel sizes being sometimes faster. This is because this flag tells PyTorch to optimize the convolution algorithms used under the hood, and some of these CUDA algorithms seem to be faster than our masking strategy on Python.
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# 6 LIMITATIONS
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Dynamic kernel sizes: computation and memory cost of convolutions with large kernels. Performing convolutions with large convolutional kernels is a compute-intensive operation. FlexConvs are initialized with small kernel sizes and their inference cost is relatively small at the start of training. However, despite the cropping operations used to improve computational efficiency (Figs. 1, 3, Tab. 3), the inference time may increase to up to double as the learned masks increase in size. At the cost of more memory, convolutions can be sped up by performing them in the frequency domain. However, we observe that this does not bring gains for the image data considered because FFT convolutions are faster only for very large convolutional kernels (in the order of hundreds of pixels).
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Remaining accuracy drop in alias-free FlexNets. Some drop in accuracy is still observed when using alias-free FlexNets at a higher test resolutions (Tab. 4). Although more evidence is needed, this may be caused by aliasing effects introduced by ReLU (Vasconcelos et al., 2021), or changes in the activation statistics of the feature maps passed to global average pooling (Touvron et al., 2019).
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# 7 CONCLUSION
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We propose FlexConv, a convolutional operation able to learn high bandwidth convolutional kernels of varying size during training at a fixed parameter cost. We demonstrate that FlexConvs are able to model long-term dependencies without the need of pooling, and shallow pooling-free FlexNets achieve state-of-the-art performance on several sequential datasets, match performance of recent works with learned kernel sizes with less compute, and are competitive with much deeper ResNets on image benchmark datasets. In addition, we show that our alias-free convolutional kernels allow FlexNets to be deployed at higher resolutions than seen during training with minimal precision loss.
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Future work. MAGNets give control over the bandwidth of the kernel. We anticipate that this control has more uses, such as fighting sub-sampling aliasing (Zhang, 2019; Kayhan & Gemert, 2020; Karras et al., 2021). With the ability to upscale FlexNets to different input image sizes comes the possibility of transfer learning representations between previously incompatible datasets, such as CIFAR-10 and Imagenet. In a similar vein, the automatic adaptation of FlexConv to the kernel sizes required for the task at hand may make it possible to generalize the FlexNet architecture across different tasks and datasets. Neural architecture search (Zoph & Le, 2016) could see benefits from narrowing the search space to exclude kernel size and pooling layers. In addition, we envisage additional improvements from structural developments of FlexConvs such as attentive FlexNets.
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# REPRODUCIBILITY STATEMENT
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We hope to inspire others to use and reproduce our work. We publish the source code of this work, for which the link is provided in Sec. 4.2. Sec. 4 and Appx. D.1 detail FlexNet, its hyperparameters and optimization procedure. The full derivation of the aliasing regularization objective is included in Appx. A.1. We report means over multiple runs for many experiments, to ensure the reported results are fair and reproducible, and do not rely on tuning of the random seed. All datasets used in our experiments are publicly available. If any questions remain, we welcome one and all to contact the corresponding author.
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# ACKNOWLEDGMENTS
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We thank Nergis Tömen for her valuable insights regarding signal processing principles for FlexConv, and Silvia-Laura Pintea for explanations and access to code of her work Pintea et al. (2021). We thank Yerlan Idelbayev for the use of the CIFARResNet code.
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This work is co-supported by the Qualcomm Innovation Fellowship granted to David W. Romero. David W. Romero sincerely thanks Qualcomm for his support. David W. Romero is financed as part of the Efficient Deep Learning (EDL) programme (grant number P16-25), partly funded by the Dutch Research Council (NWO). Robert-Jan Bruintjes is financed by the Dutch Research Council (NWO) (project VI.Vidi.192.100). All authors sincerely thank everyone involved in funding this work.
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This work was partially carried out on the Dutch national infrastructure with the support of SURF Cooperative. We used Weights & Biases (Biewald, 2020) for experiment tracking and visualizations.
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# REFERENCES
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Anthony Bagnall, Hoang Anh Dau, Jason Lines, Michael Flynn, James Large, Aaron Bostrom, Paul Southam, and Eamonn Keogh. The uea multivariate time series classification archive, 2018. arXiv preprint arXiv:1811.00075, 2018.
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Shaojie Bai, J Zico Kolter, and Vladlen Koltun. An empirical evaluation of generic convolutional and recurrent networks for sequence modeling. arXiv preprint arXiv:1803.01271, 2018a.
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Shaojie Bai, J Zico Kolter, and Vladlen Koltun. Trellis networks for sequence modeling. arXiv preprint arXiv:1810.06682, 2018b.
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Figure 7: Example kernels, generated step by step. FlexConv samples a kernel from $\mathtt { M I P } ^ { \psi }$ (a), which is attenuated by an anistropic Gaussian envelope with learned parameters $\pmb \theta ^ { ( l ) }$ (b), creating (c) which is cropped to contain only values of $> 0 . 1$ (d).
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# A ALIAS-FREE FLEXCONV REGULARIZATION
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In this section we provide the complete derivation and analysis for our FlexConv regularization against aliasing. First, we derive the analytic maximum frequency component of a FlexConv kernel. Next, we compute the Nyquist frequency of a FlexConv kernel, and subsequently show how to combine the previous results into a regularization term to train alias-free FlexConvs.
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# A.1 ANALYZING THE FREQUENCY SPECTRUM OF FLEXCONV
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In order to make FlexConv alias-free (Sec. 3.3), we need to compute the maximum frequency component of the kernels generated by a MAGNet, so that we can regularize it during training. In this section we analytically derive this maximum frequency component from the parameters of the MAGNet.
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Recall that MAGNets generate a kernel $\psi ( x , y )$ through of a succession of anisotropic Gabor filters and linear layers (Sec. 3.2, Eqs. 2–7):
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$$
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\begin{array} { r l r l } & { \mathbf { h } ^ { ( 1 ) } = \mathbf { g } \big ( [ x , y ] ; \theta ^ { ( 1 ) } \big ) } & & { \mathbf { g } \colon \mathbb { R } ^ { 2 } \to \mathbb { R } ^ { \mathrm { N _ { h i d } } } } \\ & { \mathbf { h } ^ { ( l ) } = \big ( \mathbf { W } ^ { ( l ) } \mathbf { h } ^ { ( l - 1 ) } + \mathbf { b } ^ { ( l ) } \big ) \cdot \mathbf { g } \big ( [ x , y ] ; \theta ^ { ( l ) } \big ) } & & { \mathbf { W } ^ { ( l ) } \in \mathbb { R } ^ { \mathrm { N _ { h i d } \times N _ { h i d } } } , \mathbf { b } ^ { ( l ) } \in \mathbb { R } ^ { \mathrm { N _ { h i d } } } } \\ & { \psi ( x , y ) = \mathbf { W } ^ { ( \mathrm { L } ) } \mathbf { h } ^ { ( \mathrm { L } - 1 ) } + \mathbf { b } ^ { ( \mathrm { L } ) } } & & { \mathbf { W } ^ { ( \mathrm { L } ) } \in \mathbb { R } ^ { ( \mathrm { N _ { o u t } \times N _ { \mathrm { i n } } } ) \times \mathrm { N _ { h i d } } } , \mathbf { b } ^ { ( \mathrm { L } ) } \in \mathbb { R } ^ { ( \mathrm { N _ { o u t } \times N _ { \mathrm { i n } } } ) } } \\ & { \mathbf { g } \big ( [ x , y ] ; \theta ^ { ( l ) } \big ) = \mathrm { e x p } \Bigg ( - \displaystyle \frac { 1 } { 2 } \Big [ \Big ( \gamma _ { \mathrm { X } } ^ { ( l ) } \big ( x - \mu _ { \mathrm { X } } ^ { ( l ) } \big ) \Big ) ^ { 2 } + \Big ( \gamma _ { \mathrm { Y } } ^ { ( l ) } \big ( y - \mu _ { \mathrm { Y } } ^ { ( l ) } \big ) \Big ) ^ { 2 } \Big ] \Bigg ) \mathrm { S i n } \big ( \mathbf { W } _ { \mathrm { g } } ^ { ( l ) } [ x , y ] + \mathbf { b } _ { \mathrm { g } } ^ { ( l ) } \big ) } \\ & \theta ^ { ( l ) } = \Big \{ \gamma _ { \mathrm { X } } ^ { ( l ) } \in \mathbb { R } ^ { \mathrm { N _ { h i d } } } , \gamma _ { \mathrm { Y } } ^ { ( l ) } \in \mathbb { R } ^ { \mathrm { N _ { h i d } } } , \mu _ { \mathrm { X } } ^ { ( l ) } \in \mathbb { R } ^ { \mathrm { N _ { h i d } } } , \mu _ { \mathrm { Y } } ^ \end{array}
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$$
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To analyse the maximum frequency component $f _ { \mathrm { M A G N e t } } ^ { + }$ , we analyse the frequency components of the Gabor filters used in MAGNet, and retain their maximum. We then plug the found frequency component into the analysis of Fathony et al. (2021) to show how the frequency responses of Gabor filters and linear layers interact in MFNs. Finally, we add the effect of the FlexConv Gaussian mask to our analysis to obtain the maximum frequency component ot the final FlexConv kernel $f _ { \mathrm { F l e x C o n v } } ^ { + }$ .
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Sine term in Gabor filters. In a Gabor filter, the sine term is multiplied with a Gaussian envelope. The frequency (in radians) of a sine function of the form $\mathrm { S i n } ( { \pmb w } ^ { T } [ x , \overset { \cdot } { y } ] + b )$ is given by $\pmb { w }$ . We divide by $2 \pi$ to convert the frequency units to Hertz, for compatibility with the rest of the analysis. For 2D inputs, the maximum frequency component of the sine function correspond to the largest frequency in the two input dimensions:
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$$
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f _ { \mathrm { S i n } } ^ { + } = \operatorname* { m a x } _ { j } \frac { w _ { j } } { 2 \pi } .
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$$
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The sine terms in MAGNets have multiple output channels: $\mathrm { S i n } \big ( \mathbf { W } _ { \mathrm { g } } \cdot [ x , y ] + \mathbf { b } _ { \mathrm { g } } ^ { ( l ) } \big )$ . Effectively, we compute the sine term independently for each channel:
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$$
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f _ { \mathrm { S i n } , i } ^ { + } = \operatorname* { m a x } _ { j } \frac { \mathbf { W } _ { \mathrm { g } , i , j } } { 2 \pi } .
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$$
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(c) Regularizing f +FlexConv, block 4 of 7.
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Figure 8: Example kernels from FlexNet-16 models trained (i) without regularization, (ii) with aliasing regularization of $f _ { \mathrm { M A G N e t } } ^ { + }$ , (iii) with aliasing regularization of $f _ { \mathrm { F l e x C o n v } } ^ { + }$ . In the columns, from left to right: (i) original kernel at $3 3 \times 3 3$ , (ii) FFT of the original kernel, (iii) kernel inferred at $6 5 \times 6 5$ , to find aliasing effects, (iiii) FFT of the $6 5 \times 6 5$ kernel, with the solid line showing the Nyquist frequency of the $3 3 \times 3 3$ kernel, and the red dotted line showing the maximum frequency component ×as computed by our analysis. For $f _ { \mathrm { F l e x C o n v } } ^ { + }$ the maximum frequency matches almost exactly with the Nyquist frequency, showing that our aliasing regularization works. For $f _ { \mathrm { M A G N e t } } ^ { + }$ , the maximum frequency is slightly higher than the Nyquist frequency, as the FlexConv mask is not included in the frequency term derivation. This is reflected in the slightly worse resolution generalization results reported in Sec. 4.3. Furthermore, some aliasing effects are still apparent for the aliasing regularized models, as discussed in Sec. 6.
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Gaussian term in Gabor filters. In a Gabor filter, a Gaussian envelope modulates a sine term. Let us assume for now that the Gaussian envelope is isotropic, rather than anisotropic as in MAGNets, and has single-channel output. By applying the convolution theorem, the sine term is equivalently convolved with the Fourier transform of the Gaussian envelope in the frequency domain. Since the Fourier transform of a Gaussian envelope is another Gaussian envelope, the application of a Gaussian envelope amounts to blurring with a Gaussian kernel in the frequency domain. The size of the envelope in the Fourier domain $\sigma _ { \mathrm { F } }$ can be derived from the standard deviation of the Guassian envelope in the spatial domain $\sigma _ { \mathrm { T } }$ as follows:
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$$
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\sigma _ { \mathrm { T } } \sigma _ { \mathrm { F } } = \frac { 1 } { 2 \pi } \Rightarrow \sigma _ { \mathrm { F } } = \frac { 1 } { 2 \pi \sigma _ { \mathrm { T } } } .
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$$
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Gaussian blurs induce impulse signals to have a long tail. Consequently, we must define a cutoff point for this tail in terms of standard deviations to derive the maximum added frequency induced by the blur. We describe the cutoff point as $\sigma _ { \mathrm { c u t } } \in \mathbb { N }$ . Typical choices for $\sigma _ { \mathrm { c u t } }$ are known as the ∈empirical, or the "68-95-99.7" rule (Hald, 2007). We choose a standard of two standard deviations, i.e., $\sigma _ { \mathrm { c u t } } { = } 2$ , which covers $9 5 \%$ of the mass of the Gaussian envelope.
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For an isotropic Gabor filter with $\gamma { = } \sigma _ { \mathrm { T } } ^ { - 1 }$ , the maximum frequency of its Gaussian envelope $f _ { \mathrm { e n v } } ^ { + }$ is:
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$$
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f _ { \mathrm { e n v } } ^ { + } = \frac { \sigma _ { \mathrm { c u t } } } { 2 \pi ( \sigma _ { \mathrm { T } } ) ^ { - 1 } } = \frac { \sigma _ { \mathrm { c u t } } \gamma } { 2 \pi } .
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+
$$
|
| 426 |
+
|
| 427 |
+
Anisotropic envelopes. Our analysis so far assumes an isotropic Gaussian envelope in the Gabor filter. However, we need to account for the anisotropic Gaussian envelopes in MAGNets. Anisotropic filters have not one but two $\gamma$ parameters: $\{ \gamma _ { \mathrm { X } } , \gamma _ { \mathrm { Y } } \}$ . The smallest of these will contribute most to $f _ { \mathrm { e n v } } ^ { + }$ , as it will blur the most, so it is sufficient to compute $f _ { \mathrm { e n v } } ^ { + }$ only using the smallest of the two $\gamma$ terms:
|
| 428 |
+
|
| 429 |
+
$$
|
| 430 |
+
f _ { \mathrm { e n v } } ^ { + } \big ( \gamma _ { X } , \gamma _ { Y } \big ) = f _ { \mathrm { e n v } } ^ { + } \big ( \operatorname* { m i n } \{ \gamma _ { X } , \gamma _ { Y } \} \big ) .
|
| 431 |
+
$$
|
| 432 |
+
|
| 433 |
+
The other assumption we made before was to work with single-channel outputs. MAGNets however use multi-channel outputs with independent Gaussian terms. The maximum frequency of multichannel Gaussian envelopes is given by:
|
| 434 |
+
|
| 435 |
+
$$
|
| 436 |
+
f _ { \mathrm { e n v } , i } ^ { + } ( \gamma _ { \mathrm { X } } , \gamma _ { \mathrm { Y } } ) = f _ { \mathrm { e n v } } ^ { + } \left( \operatorname* { m i n } \{ \gamma _ { \mathrm { X } , i } , \gamma _ { \mathrm { Y } , i } \} \right) = \frac { \sigma _ { \mathrm { c u t } } \operatorname* { m i n } \{ \gamma _ { \mathrm { X } , i } , \gamma _ { \mathrm { Y } , i } \} } { 2 \pi } ,
|
| 437 |
+
$$
|
| 438 |
+
|
| 439 |
+
where the subscript $i$ indexes the channels of the multi-channel Gaussian envelopes.
|
| 440 |
+
|
| 441 |
+
Maximum frequency component of anisotropic Gabor filters. Finally, the maximum frequency component of the $i$ -th channel of an anisotropic Gabor filter $\mathbf { g }$ is given by:
|
| 442 |
+
|
| 443 |
+
$$
|
| 444 |
+
\begin{array} { r l } & { f _ { \mathrm { G a b o r } , i } ^ { + } = f _ { \mathrm { S i n } , i } ^ { + } ( \mathbf { W } _ { \mathrm { g } } ) + f _ { \mathrm { e n v } , i } ^ { + } ( \gamma _ { \mathrm { X } } , \gamma _ { \mathrm { Y } } ) } \\ & { ~ = \left( \displaystyle \operatorname* { m a x } _ { j } \frac { \mathbf { W } _ { \mathrm { g } , i , j } } { 2 \pi } \right) + \frac { \sigma _ { \mathrm { c u t } } \operatorname* { m i n } \left\{ \gamma _ { \mathrm { X } , i } , \gamma _ { \mathrm { Y } , i } \right\} } { 2 \pi } . } \end{array}
|
| 445 |
+
$$
|
| 446 |
+
|
| 447 |
+
Figure 9 illustrates the frequency spectrum of an example Gabor filter.
|
| 448 |
+
|
| 449 |
+
Maximum frequency component of a MAGNet. Fathony et al. (2021) characterize the expansion of each term of the isotropic Gabor layers in MFNs in the final MFN output. In Eq. 25, Fathony et al. (2021) demonstrate that the MFN representation contains a set of sine frequencies $\overline { { \omega } }$ given by:
|
| 450 |
+
|
| 451 |
+
$$
|
| 452 |
+
\overline { { \omega } } = \left\{ s _ { \mathrm { L } } \omega _ { i _ { \mathrm { L } } } ^ { ( \mathrm { L } ) } + s _ { \mathrm { L } - 1 } \omega _ { i _ { \mathrm { L } - 1 } } ^ { ( \mathrm { L } - 1 ) } + \cdots + s _ { l } \omega _ { i _ { 2 } } ^ { ( 2 ) } + \omega _ { i _ { 1 } } ^ { ( 1 ) } \right\} .
|
| 453 |
+
$$
|
| 454 |
+
|
| 455 |
+
Here, the indexes $i _ { 1 } , i _ { 2 } , \cdots , i _ { \mathrm { L - 1 } }$ range over all possible indices of each hidden unit of each layer of an MFN, and $s _ { 2 } , \cdots , s _ { \mathrm { L } } \in \{ - 1 , + 1 \}$ range over all $2 ^ { \mathrm { L - 1 } }$ possible binary signs. In other words, ∈Fathony et al. (2021) demonstrate that the representation of an MFN at a particular layer contains an exponential combination of all possible positive and negative combinations of the frequencies of the sine terms in each hidden unit at each layer in the MFN up to the current layer.
|
| 456 |
+
|
| 457 |
+
The original analysis uses these terms to argue that MFNs model exponentially many terms through a linear amount of layers. For our purpose of computing the frequency response of the MAGNet generated kernel, we can plug our derivation of the frequencies of the Gabor filter $f _ { \mathrm { G a b o r } }$ into $\overline { { \omega } }$ to compute the frequency spectrum of the generated kernel:
|
| 458 |
+
|
| 459 |
+
$$
|
| 460 |
+
{ \pmb f } _ { \mathrm { M A G N e t } } ^ { + } = \left\{ s _ { \mathrm { L } } f _ { \mathrm { G a b o r } , i _ { \mathrm { L } } } ^ { ( \mathrm { L } ) } + s _ { \mathrm { L } - 1 } f _ { \mathrm { G a b o r } , i _ { \mathrm { L } - 1 } } ^ { ( \mathrm { L } - 1 ) } + \cdots + s _ { 2 } f _ { \mathrm { G a b o r } , i _ { 2 } } ^ { ( 2 ) } + f _ { \mathrm { G a b o r } , i _ { 1 } } ^ { ( 1 ) } \right\} .
|
| 461 |
+
$$
|
| 462 |
+
|
| 463 |
+
|
| 464 |
+
Figure 9: Decomposition of a Gabor filter and its frequency spectrum. Top row: a decomposition of a Gabor filter (right) into its Gaussian term (left) and its sine term (center). Bottom row: frequency responses for each respective filter. The Fourier transform of a Gaussian envelope is a Gaussian envelope (blue circles show $\sigma { \mathcal { F } }$ for $h = \lbrace 1 , 2 \rbrace$ ). The Fourier transform of a sine pattern is a collection of symmetrical impulse signals (red box shows the Nyquist frequency). The Gaussian envelope blurs the frequency response of the sine term (purple boxes show the frequency response for $h = \{ \bar { 1 } , 2 , 3 \}$ ).
|
| 465 |
+
|
| 466 |
+
As stated before, we are only interested in the maximum frequency in the frequency spectrum. We can therefore simplify Eq. 18 in two ways. First, we simplify over MAGNet layers by taking the maximum value of the spectrum, which is the sum over all layers using only the positive binary signs in $s _ { \mathrm { L } }$ (Eq. 19). Next, we simplify over channel indices by retaining only the channel index that results in the highest frequency (Eq. 20). The maximum frequency of a MAGNet is shown in Eq. 21:
|
| 467 |
+
|
| 468 |
+
$$
|
| 469 |
+
\begin{array} { r l } & { f _ { \mathrm { M A G M v e t } } ^ { + } = \{ ( + 1 ) f _ { \mathrm { G a b o r } , i _ { 2 } } ^ { + ( \mathrm { L } ) } + ( + 1 ) f _ { \mathrm { G a b o r } , i _ { 1 } - 1 } ^ { + ( \mathrm { L } ) } + \cdots + ( + 1 ) f _ { \mathrm { G a b o r } , i _ { 2 } } ^ { + ( 2 ) } + f _ { \mathrm { G a b o r } , i _ { 1 } } ^ { + ( 1 ) } \} } \\ & { \qquad = \{ f _ { \mathrm { G a b o r } , i _ { 1 } } ^ { + ( \mathrm { L } ) } + f _ { \mathrm { G a b o r } , i _ { 1 } - 1 } ^ { + ( \mathrm { L } ) } + \cdots + f _ { \mathrm { G a b o r } , i _ { 2 } } ^ { + ( 2 ) } + f _ { \mathrm { G a b o r } , i _ { 1 } } ^ { + ( 1 ) } \} } \\ & { f _ { \mathrm { M A G N e t } } ^ { + } = \displaystyle \operatorname* { m a x } _ { i _ { \perp } } ( f _ { \mathrm { G a b o r } , i _ { 2 } } ^ { + ( \mathrm { L } ) } ) + \operatorname* { m a x } _ { i _ { 1 } , \mathrm { L } - 1 } ( f _ { \mathrm { G a b o r } , i _ { 1 } - 1 } ^ { + ( \mathrm { L } ) } ) \cdots + \displaystyle \operatorname* { m a x } _ { i _ { 2 } } ( f _ { \mathrm { G a b o r } , i _ { 2 } } ^ { + ( 2 ) } ) + \operatorname* { m a x } _ { i _ { 1 } } ( f _ { \mathrm { G a b o r } , i _ { 1 } } ^ { + ( 1 ) } ) } \\ & { \qquad = \displaystyle \sum _ { l = 1 } ^ { \mathrm { L } } \operatorname* { m a x } ( f _ { \mathrm { G a b o r } , i _ { 1 } } ^ { + ( l ) } ) } \\ & \qquad = \displaystyle \sum _ { l = 1 } ^ { \mathrm { L } } \operatorname* { m a x } _ { i _ { l } } ( ( \operatorname* { m a x } _ { j } \frac { \mathbf { W } _ { \mathrm { g } , i _ { 1 } , j } ^ { ( l ) } } { 2 \pi } ) + \frac \sigma _ { \mathrm { c u t } } \operatorname* { m i n } \{ \gamma _ { \mathrm { X } , i _ { 1 } } ^ { ( l ) } , \gamma _ \mathrm { Y } , i \end{array}
|
| 470 |
+
$$
|
| 471 |
+
|
| 472 |
+
Effect of the Gaussian mask in the frequency components of a FlexConv. FlexConvs attenuate the MAGNet output with a Gaussian mask. The Gaussian mask (Eq. 1) works analogously to the Gaussian envelope term in the Gabor filter: it blurs the frequency components of the generated kernel with standard deviation $\sigma _ { \mathrm { F } }$ . Therefore, we can reuse our derivation for the Gaussian envelope of the
|
| 473 |
+
|
| 474 |
+
Gabor filter (Eq. 15). The maximum frequency component of a FlexConv kernel is given by:
|
| 475 |
+
|
| 476 |
+
$$
|
| 477 |
+
\begin{array} { r l } & { f _ { \mathrm { H e x C o n v } } ^ { + } = f _ { \mathrm { M A G N e t } } ^ { + } + f _ { \mathrm { e n v } } ^ { + } } \\ & { \phantom { f _ { \mathrm { M A G N e t } } ^ { + } + \frac { \sigma _ { \mathrm { c u t } } \operatorname* { m i n } \left\{ \sigma _ { \mathrm { X } } ^ { - 1 } , \sigma _ { \mathrm { Y } } ^ { - 1 } \right\} } { 2 \pi } } = f _ { \mathrm { M A G N e t } } ^ { + } + \frac { \sigma _ { \mathrm { c u t } } } { \operatorname* { m a x } \left\{ \sigma _ { \mathrm { X } } , \sigma _ { \mathrm { Y } } \right\} 2 \pi } } \\ & \phantom { f _ { \mathrm { M A G N e t } } ^ { - } + \frac { \sigma _ { \mathrm { m a x } } } { l _ { u } } \left( \left( \operatorname* { m a x } _ { j } \frac { \mathbf { W } _ { \mathbf { g } , i _ { l } , j } ^ { ( l ) } } { 2 \pi } \right) + \frac { \sigma _ { \mathrm { c u t } } \operatorname* { m i n } \left\{ \gamma _ { \mathrm { X } , i _ { l } } ^ { ( l ) } , \gamma _ { \mathrm { Y } , i _ { l } } ^ { ( l ) } \right\} } { 2 \pi } \right) + \frac { \sigma _ { \mathrm { c u t } } } { \operatorname* { m a x } \left\{ \sigma _ { \mathrm { X } } , \sigma _ { \mathrm { Y } } \right\} 2 \pi } . } \end{array}
|
| 478 |
+
$$
|
| 479 |
+
|
| 480 |
+
Visualization of regularized kernels. Fig. 8 shows example kernels from FlexNets trained with aliasing regularization. The frequency domain plots confirm the accuracy of our frequency component regularization.
|
| 481 |
+
|
| 482 |
+
# A.2 REGULARIZING THE FREQUENCY RESPONSE OF FLEXCONV
|
| 483 |
+
|
| 484 |
+
Nyquist frequency of a FlexConv kernel. Given the sampling rate $f _ { \mathrm { s } }$ of the kernel, we can compute its Nyquist frequency $f _ { \mathrm { N y q } }$ as:
|
| 485 |
+
|
| 486 |
+
$$
|
| 487 |
+
f _ { \mathrm { N y q } } = \frac { 1 } { 2 } f _ { s }
|
| 488 |
+
$$
|
| 489 |
+
|
| 490 |
+
To compute the sampling rate, we note that the kernel coordinates input to our MAGNet stretch over a $[ - 1 , 1 ] ^ { \mathrm { D } }$ domain. For a kernel of length $k$ , we therefore sample one point in every $f _ { s } = \frac { k - 1 } { 2 }$ units.
|
| 491 |
+
|
| 492 |
+
Knowing the sampling rate in terms of the kernel size allows us to express the Nyquist frequency in terms of the (pre-masked) kernel size:
|
| 493 |
+
|
| 494 |
+
$$
|
| 495 |
+
f _ { \mathrm { N y q } } ( k ) = \frac { 1 } { 2 } \frac { k - 1 } { 2 } = \frac { k - 1 } { 4 } .
|
| 496 |
+
$$
|
| 497 |
+
|
| 498 |
+
Note that the kernel size in a FlexConv is initialized to be equal to the resolution of the data, if it is odd. For even resolutions, it corresponds to the resolution of the data plus one.
|
| 499 |
+
|
| 500 |
+
Constructing the regularization term. We train FlexConv with a regularization term on the frequency response of the generated kernel to ensure that aliasing effects do not distort the performance of the model when it is inferred at a higher resolution. This section details the implementation of the regularization function.
|
| 501 |
+
|
| 502 |
+
From the parameters of each FlexConv module, we compute $f _ { \mathrm { F l e x C o n v } } ^ { + }$ according to Eq. 22. For the amount of standard deviations to use in determining $f _ { \mathrm { e n v } } ^ { + }$ (Eq. 15) we use $h = 2$ . From the kernel size $k$ of the FlexConv module we compute $f _ { \mathrm { N y q } } ( k )$ = according to Eq. 24. We then apply an L2 regularizer over the amount that $f _ { \mathrm { F l e x C o n v } } ^ { + }$ exceeds $f _ { \mathrm { N y q } } ( k )$ :
|
| 503 |
+
|
| 504 |
+
$$
|
| 505 |
+
\begin{array} { r } { \mathcal { L } _ { \mathrm { H F } } = | | \operatorname* { m a x } \{ f _ { \mathrm { F l e x C o n v } } ^ { + } , f _ { \mathrm { N y q } } ( k ) \} - f _ { \mathrm { N y q } } ( k ) | | ^ { 2 } . } \end{array}
|
| 506 |
+
$$
|
| 507 |
+
|
| 508 |
+
We weight $\mathcal { L } _ { \mathrm { H F } }$ by $\lambda = 0 . 1$ when adding it to our loss function.
|
| 509 |
+
|
| 510 |
+
Improved implementation. Eq. 25 contains a sum over the $\mathrm { L }$ layers of the MAGNet. In practice, we prefer to regularize each layer $l \in \mathrm { L }$ separately, so that the gradients of the regularization of different layers are not dependent on each other. We therefore implement the anti-aliasing regularization by regularizing each MAGNet layer independently, and spreading the $f _ { \mathrm { e n v } } ^ { + }$ term from the gaussian mask uniformly over all MAGNet layers:
|
| 511 |
+
|
| 512 |
+
$$
|
| 513 |
+
\begin{array} { r l } & { \mathcal { L } _ { \mathrm { H F } , l } = \Vert \operatorname* { m a x } \left\{ f _ { \mathrm { M A G N e t } , l } ^ { + } + \frac { f _ { \mathrm { e n v } } ^ { + } } { \mathrm { L } } , \frac { f _ { \mathrm { N y q } } ( k ) } { \mathrm { L } } \right\} - \frac { f _ { \mathrm { N y q } } ( k ) } { \mathrm { L } } \Vert ^ { 2 } } \\ & { \qquad = \Vert \operatorname* { m a x } \left\{ \underset { i _ { l } } { \operatorname* { m a x } } \left( f _ { \mathrm { G a b o r } , i _ { l } } ^ { + ( l ) } \right) + \frac { f _ { \mathrm { e n v } } ^ { + } } { \mathrm { L } } , \frac { f _ { \mathrm { N y q } } ( k ) } { \mathrm { L } } \right\} - \frac { f _ { \mathrm { N y q } } ( k ) } { \mathrm { L } } \Vert ^ { 2 } . } \end{array}
|
| 514 |
+
$$
|
| 515 |
+
|
| 516 |
+
In the code, we refer to this method as the together method, versus the summed method of Eq. 25. In preliminary experiments, we observed improved performance of anti-aliasing training when using the together method. All of our experiments anti-aliasing experiments therefore use the together setting.
|
| 517 |
+
|
| 518 |
+
# B DATASET DESCRIPTION
|
| 519 |
+
|
| 520 |
+
# B.1 IMAGE FITTING DATASETS
|
| 521 |
+
|
| 522 |
+
Kodak dataset. The Kodak dataset (Kodak, 1991) consists of 24 natural images of size $7 6 8 \times 5 1 2$ .
|
| 523 |
+
This dataset is a popular benchmark used for compression and image fitting methods.
|
| 524 |
+
|
| 525 |
+
# B.2 SEQUENTIAL DATASETS
|
| 526 |
+
|
| 527 |
+
Sequential and Permuted MNIST. The sequential MNIST dataset (sMNIST) (Le et al., 2015)takes the $2 8 \times 2 8$ images from the original MNIST dataset (LeCun et al., 1998), and presents them as a sequence of 784 pixels. The goal of this task is to perform digit classification given the representation of the last sequence element of a sequential model. Consequently, good predictions require the model to preserve long-term dependencies up to 784 steps in the past.
|
| 528 |
+
|
| 529 |
+
The permuted MNIST dataset (pMNIST) additionally changes the order of all the sMNIST sequences by a random permutation. Consequently, models can no longer rely on local features to construct good feature representations. As a result, the classification problem becomes more difficult, and the importance of long-term dependencies more pronounced.
|
| 530 |
+
|
| 531 |
+
Sequential and Noise-Padded CIFAR10. The sequential CIFAR10 dataset (sCIFAR10) (Chang et al., 2017) takes the $3 2 \times 3 2$ images from the original CIFAR10 dataset (Krizhevsky et al., 2009) and presents them as a sequence of 1,024 pixels. The goal of this task is to perform image classification given the representation of the last sequence element of a sequential model. This task is more difficult than sMNIST, as a larger memory horizon is required to solve the task and more complex structures and intra-class variations are present in the data (Bai et al., 2018b).
|
| 532 |
+
|
| 533 |
+
The noise-padded CIFAR10 dataset (npCIFAR10) (Chang et al., 2019) flattens the images from the original CIFAR10 dataset (Krizhevsky et al., 2009) along their rows to create a sequence of length 32, and 96 channels $( 3 2 \mathrm { r o w s } \times 3 $ channels). Next, these sequences are concatenated with 968 entries of noise to form the final sequences of length 1000. As for sCIFAR10, the goal of the task is to perform image classification given the representation of the last sequence element of a sequential model.
|
| 534 |
+
|
| 535 |
+
CharacterTrajectories. The CharacterTrajectories dataset is part of the UEA time series classification archive (Bagnall et al., 2018). It consists of 2858 time series of length 182 and 3 channels representing the $x , y$ positions, and the tip force of a pen while writing Latin alphabet characters in a single stroke. The goal is to classify out of 20 classes the written character using the time series data.
|
| 536 |
+
|
| 537 |
+
Speech Commands. The Speech Commands dataset (Warden, 2018) consists of 105,809 onesecond audio recordings of 35 spoken words sampled at 16kHz. Following Kidger et al. (2020), we extract 34975 recordings from ten spoken words to construct a balanced classification problem. We refer to this dataset as SpeechCommands_raw, or SC_raw for short. Furhtermore, we utilize the preprocessing steps of Kidger et al. (2020) and extract mel-frequency cepstrum coefficients from the raw data. The resulting dataset, abreviated SC, consists of time series of length 101, and 20 channels.
|
| 538 |
+
|
| 539 |
+
# B.3 IMAGE BENCHMARK DATASETS
|
| 540 |
+
|
| 541 |
+
MNIST. The MNIST hadwritten digits datset (LeCun & Cortes, 2010) consists of 70,000 grayscale handwritten digits of size $2 8 \times 2 8$ , divided into a training and test sets of 60,000 and 10,000 images, respectively. The goal of the task is to classify these digits as one of the ten possible digits $( 0 , 1 , . . 8 , 9 )$ .
|
| 542 |
+
|
| 543 |
+
CIFAR-10 The CIFAR-10 dataset (Krizhevsky et al., 2009) consists of 60,000 natural images from 10 classes of size $3 2 \times 3 2$ , divided into training and test sets of 50,000 and 10,000 images, respectively.
|
| 544 |
+
|
| 545 |
+
STL-10. The STL-10 dataset (Coates et al., 2011) is a subset of the ImageNet dataset (Krizhevsky et al., 2012) consisting of 5,000 natural images from 10 classes of size $9 6 \times 9 6$ , divided into trainint and test sets of 4,500 and 500 images, respectively.
|
| 546 |
+
|
| 547 |
+
ImageNet-k. The Imagenet-k (Chrabaszcz et al., 2017) dataset is derived from the ImageNet dataset Russakovsky et al. (2015) by downsampling all samples to a resolution $\mathrm { ~ k ~ } \in \ [ 6 4 , 3 2 , 1 6 , 8 ]$ . The ∈dataset contains 1000 classes with 1,281,167 training samples and 50,000 validation samples.
|
| 548 |
+
|
| 549 |
+
Table 5: Average PSNR for fitting of images in the Kodak dataset. Both our improved initialization scheme, as well as the inclusion of anisotropic Gabor functions lead to better reconstructions.
|
| 550 |
+
|
| 551 |
+
<table><tr><td>MODEL</td><td>#PARAMS</td><td>IMPROVED INIT</td><td>PSNR</td></tr><tr><td>SIREN</td><td>7.14K</td><td>-</td><td>25.665</td></tr><tr><td>MFNFourier</td><td>7.40K</td><td>-</td><td>23.276</td></tr><tr><td>MFNGabor</td><td>7.11K</td><td>X √</td><td>25.361 25.606</td></tr><tr><td></td><td></td><td></td><td>25.791</td></tr><tr><td>MAGNet</td><td>7.36K</td><td>X</td><td>25.893</td></tr></table>
|
| 552 |
+
|
| 553 |
+
Table 6: Full results on CIFAR-10. We report results over three runs per setting. CIFARResNet-44 w/ CKConv is a CIFARResNet-44 where all convolutional layers are replaced with CKConvs with $k = 3$ . =CIFARResNet-44 w/ FlexConv is a CIFARResNet-44 where all convolutional layers are replaced with FlexConv with learned kernel size, except for the shortcut connections of the strided convolutional layers, which are pointwise convolutions. \*Results are taken from the respective original works instead of reproduced. †Results are from single run.
|
| 554 |
+
|
| 555 |
+
<table><tr><td>MODEL</td><td>SIZE</td><td>CIFAR-10 Acc.</td></tr><tr><td>DCN-gji (Tomen et al., 2021)</td><td>0.47M</td><td>89.7 ± 0.3*</td></tr><tr><td>N-Jet-CIFARResNet32 (Pintea et al.,2021)</td><td>0.52M</td><td>92.3 ± 0.3*</td></tr><tr><td>N-Jet-ALLCNN(Pintea et al.,2021)</td><td>1.07M</td><td>92.5± 0.1*</td></tr><tr><td>CIFARResNet-44 (He et al.,2016)</td><td>0.66M</td><td>92.9*†</td></tr><tr><td>CIFARResNet-44 (He et al.,2016)(our reproduction)</td><td>0.66M</td><td>90.9 ± 0.2</td></tr><tr><td>CIFARResNet-44 w/ CKConv (k = 3)</td><td>2.58M</td><td>86.1 ± 0.9</td></tr><tr><td>CIFARResNet-44 w/FlexConv</td><td>2.58M</td><td>81.6 ± 0.8</td></tr><tr><td>FlexNet-7 w/conv.(k = 3)</td><td>0.17M</td><td>89.5 ± 0.3</td></tr><tr><td>FlexNet-7w/ conv.(k = 33)</td><td>20.0M</td><td>78.0 ± 0.3</td></tr><tr><td>FlexNet-7 w/N-Jet (Pintea et al.,2021)</td><td>0.70M</td><td>91.7 ± 0.1</td></tr><tr><td>CKCNNSIREN-3</td><td>0.26M</td><td>72.4*</td></tr><tr><td>CKCNNFourier-3</td><td>0.27M</td><td>83.8*</td></tr><tr><td>CKCNNGabor-3</td><td>0.28M</td><td>85.6*</td></tr><tr><td>CKCNNMAGNet-3</td><td>0.28M</td><td>86.2*</td></tr><tr><td>CKCNN-7</td><td>0.63M</td><td>71.7*</td></tr><tr><td>CKCNNFourier-7</td><td>0.63M</td><td>84.6*</td></tr><tr><td>CKCNNGabor-7</td><td>0.67M</td><td>87.7*</td></tr><tr><td>CKCNNMAGNet-7</td><td>0.67M</td><td>85.9*</td></tr><tr><td>FlexNetsIREN-7</td><td>0.63M</td><td>88.9*</td></tr><tr><td>FlexNetFourier-7</td><td>0.66M</td><td>91.6*</td></tr><tr><td>FlexNetGabor-7</td><td>0.67M</td><td>92.0*</td></tr><tr><td>FlexNet-3</td><td>0.27M</td><td>90.4± 0.2</td></tr><tr><td>FlexNet-5</td><td>0.44M</td><td>91.0 ± 0.5</td></tr><tr><td>FlexNet-7</td><td>0.67M</td><td>92.2 ± 0.1</td></tr></table>
|
| 556 |
+
|
| 557 |
+
# C ADDITIONAL EXPERIMENTS
|
| 558 |
+
|
| 559 |
+
# C.1 IMAGE CLASSIFICATION
|
| 560 |
+
|
| 561 |
+
CIFAR-10. Tab. 6 shows all results for our CIFAR-10 experiments, including more ablations.
|
| 562 |
+
|
| 563 |
+
ImageNet-32. Results for the ImageNet-32 experiment are shown in Table 7. FlexNets are slightly worse than CIFARResNet-32 (He et al., 2016) with slightly less parameters. However, the results reported by Chrabaszcz et al. (2017) for Wide ResNets (Zagoruyko & Komodakis, 2016) outperform FlexNets by a significant margin.
|
| 564 |
+
|
| 565 |
+
Table 7: Results on ImageNet-32. \*Results are taken from the respective original works instead of reproduced. †Results are from a single run.
|
| 566 |
+
|
| 567 |
+
<table><tr><td>MODEL</td><td>SIZE</td><td colspan="2">IMAGENET-32 ToP-1</td></tr><tr><td>CIFARResNet-32</td><td>0.53M</td><td>26.41 ± 0.13</td><td>ToP-5 49.37 ± 0.15</td></tr><tr><td>WRN-28-1</td><td>0.44M</td><td>32.03*+</td><td>57.51*+</td></tr><tr><td>FlexNet-5</td><td>0.44M</td><td>24.9 ± 0.4</td><td>47.7 ± 0.6</td></tr></table>
|
| 568 |
+
|
| 569 |
+
Table 8: Results for alias-free FlexNets on CIFAR-10 and ImageNet-k. $\Delta$ denotes difference in accuracy.
|
| 570 |
+
|
| 571 |
+
<table><tr><td rowspan="2">MODEL</td><td rowspan="2">SIZE</td><td colspan="2">IMAGENET-K TOP-1</td></tr><tr><td>k =16</td><td>△k=16k=32</td></tr><tr><td>CIFARResNet-32</td><td>0.52m</td><td>16.1 ± 0.0</td><td>-11.6 ± 0.4</td></tr><tr><td>FlexNet-5 w/N-Jets</td><td>0.46M</td><td>15.7 ± 0.1</td><td>-1.9 ± 0.4</td></tr><tr><td>FlexNet-5</td><td>0.44M</td><td>14.9 ± 0.1</td><td>-1.9 ± 1.7</td></tr></table>
|
| 572 |
+
|
| 573 |
+
Table 9: Results on MNIST. We train each model with three different seeds and report mean and standard deviation. \*Results are taken from the respective original works instead of reproduced. $^ \dagger$ Results are from single run.
|
| 574 |
+
|
| 575 |
+
<table><tr><td>MODEL</td><td>SIZE</td><td>MNIST Acc.</td></tr><tr><td>Efficient-CapsNet (Mazzia et al.,2021)</td><td>0.16M</td><td>99.8*†</td></tr><tr><td>Network in Network (Lin et al.,2013)</td><td>N/A</td><td>99.6*+</td></tr><tr><td>VGG-5 (results from Kabir et al. (2020))</td><td>3.65M</td><td>99.7*+</td></tr><tr><td>FlexNet-16</td><td>0.67M</td><td>99.7 ± 0.0</td></tr></table>
|
| 576 |
+
|
| 577 |
+
Alias-free ImageNet-32. We report results for alias-free FlexNets on ImageNet-k (Chrabaszcz et al., 2017) in Table 8, to verify the results of alias-free training at a larger scale. We find that FlexConv and N-Jet both mostly retain classification accuracy between source and target resolution, while CIFARResNet-32 degrades drastically.
|
| 578 |
+
|
| 579 |
+
MNIST and STL-10. We additionally report results on MNIST (Tab. 9) and STL-10 (Tab. 10. We choose these dataset for the difference in image sizes of the training data. On MNIST, though performance on MNIST is quite saturated, we are competitive with state of the art methods. On STL-10 we are significantly worse than the baseline CIFARResNet from (Luo et al., 2020), though with significantly less parameters. We were not able to prepare a more relevant baseline for this experiment.
|
| 580 |
+
|
| 581 |
+
# D EXPERIMENTAL DETAILS
|
| 582 |
+
|
| 583 |
+
# D.1 FLEXNET
|
| 584 |
+
|
| 585 |
+
We propose an image classification architecture named FlexNet (Fig. 10), consisting of a stack of FlexConv blocks followed by a global average pooling layer and a linear layer. FlexNets are named "FlexNet-L" where $\mathrm { L }$ indicates the amount of layers in the architecture.
|
| 586 |
+
|
| 587 |
+
FlexBlock. Each FlexBlock consists of two FlexConvs with BatchNorm (Ioffe & Szegedy, 2015) and dropout (Srivastava et al., 2014) $\langle d = 0 . 2 \rangle$ ) as well as a residual connection. The width of a block $i$ is determined by scaling a base amount $c$ by progressively increasing factors: $c _ { i } = [ c , c \times 1 . 5 , c \times$ $1 . 5 , c \times 2 . 0 , c \times 2 . 0 ] ( i )$ . The default configuration of FlexNet uses $c = 2 2$ . In FlexNet-N-Jet models, we scale $c$ to match the amount of parameters of the FlexNet in the comparison.
|
| 588 |
+
|
| 589 |
+
Table 10: Results on STL-10. We train each model with three different seeds and report mean and standard deviation. \*Results are taken from Luo et al. (2020). $^ \dagger$ Results are from single run.
|
| 590 |
+
|
| 591 |
+
<table><tr><td>MODEL</td><td>SIZE</td><td>STL-10 ACC.</td></tr><tr><td>CIFARResNet-18</td><td>11.2M</td><td>81.0*+</td></tr><tr><td>FlexNet-16</td><td>0.67M</td><td>68.6 ± 0.7</td></tr></table>
|
| 592 |
+
|
| 593 |
+
|
| 594 |
+
Figure 10: FlexNet architecture. FlexNet-L consists of L FlexBlocks, where each FlexBlock is a residual block of FlexConvs.
|
| 595 |
+
|
| 596 |
+
FlexConv initialization. We initialize the FlexConv mask variances small, at $\sigma _ { \mathrm { X } } ^ { 2 } , \sigma _ { \mathrm { Y } } ^ { 2 } = 0 . 1 2 5$ . For initializing MAGNet, we initialize the Gaussian envelopes as discussed in Sec. 3.2. We initialize the linear layer weights by the same Gamma distribution as used for the enveloped, modulated by a scaling factor of 25.6. We found that this value of the scaling factor, rather than a higher one, helped in reducing the performance of alias-free models. We initialize the bias of the linear layers by $\mathcal { U } ( - \pi , \pi )$ .
|
| 597 |
+
|
| 598 |
+
CIFAR-10. In FlexNet-16 models for CIFAR-10 we use $c = 2 4$ to approximate the parameter count of CIFARResNets in the experiment.
|
| 599 |
+
|
| 600 |
+
# D.2 OPTIMIZATION
|
| 601 |
+
|
| 602 |
+
We use Adam (Kingma & Ba, 2014) to optimize FlexNet. Unless otherwise specified, we use a learning rate of 0.01 with a cosine annealing scheme (Loshchilov & Hutter, 2016) with five warmup epochs. We use a different learning rate of $0 . 1 \times$ the regular learning rate for the FlexConv Gaussian mask parameters. We do not use weight decay, unless otherwise specified.
|
| 603 |
+
|
| 604 |
+
Kodak. We overfit on each image of the dataset for 20,000 iterations. To this end, we use a learning rate of 0.01 without any learning rate scheme. We observe that SIRENs diverge with this learning rate and thus, reduce the learning rate to 0.001 for these models.
|
| 605 |
+
|
| 606 |
+
CIFAR-10. We train for 350 epochs with a batch size of 64. We use the data augmentation from He et al. (2016) when training CIFAR-10: a four pixel padding, followed by a random 32 pixel crop and a random horizontal flip.
|
| 607 |
+
|
| 608 |
+
ImageNet-32. We train for 350 epochs with a batch size of 2048. We use the same data augmentation as used for CIFAR-10. We do use a weight decay of 1e−5 for ImageNet-32 training.
|
| 609 |
+
|
| 610 |
+
Sequential and Permuted MNIST. We train for 200 epochs with a batch size of 64 and a learning rate of 0.01. We use a weight decay of 1e 5.
|
| 611 |
+
|
| 612 |
+
Sequential and Noise-Padded CIFAR-10. For sequential CIFAR-10, we train for 200 epochs with a batch size of 64, a learning rate of 0.001 and a weight decay of 1e 5. For noise-padded CIFAR-10, we train for 300 epochs with a batch size of 32, a learning rate of 0.01 and no weight decay.
|
| 613 |
+
|
| 614 |
+
Speech Commands and CharTrajectories. We train for 300 epochs with a batch size of 32 and a learning rate of 0.001. For CharTrajectories, we use a weight decay of 1e 5.
|
| 615 |
+
|
| 616 |
+
# D.3 ROTATED GAUSSIAN MASKS
|
| 617 |
+
|
| 618 |
+
MAGNets use anisotropic Gaussian terms in the Gabor filters, which yields improvements in descriptive power and convergence speed (Sec. 3.2). For the same reason, we explore making the anisotropic FlexConv Gaussian mask steerable, by including an additional vector of learnable angle parameters $\phi ^ { ( l ) } \in \mathbb { R } ^ { \mathrm { N _ { \mathrm { h i d } } } }$ that rotates the Gaussian masks. Although preliminary experiments show rotated masks ∈lead to slight additional improvements, the computational overhead required to rotate the masks is large. Consequently, we do not consider rotated Gaussian masks in our final experiments.
|
parse/dev/45L_dgP48Vd/45L_dgP48Vd.md
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|
| 1 |
+
# GRAPH-AUGMENTED NORMALIZING FLOWS FOR ANOMALY DETECTION OF MULTIPLE TIME SERIES
|
| 2 |
+
|
| 3 |
+
Enyan Dai∗ Pennsylvania State University emd5759@psu.edu
|
| 4 |
+
|
| 5 |
+
Jie Chen† MIT-IBM Watson AI Lab, IBM Research chenjie@us.ibm.com
|
| 6 |
+
|
| 7 |
+
# ABSTRACT
|
| 8 |
+
|
| 9 |
+
Anomaly detection is a widely studied task for a broad variety of data types; among them, multiple time series appear frequently in applications, including for example, power grids and traffic networks. Detecting anomalies for multiple time series, however, is a challenging subject, owing to the intricate interdependencies among the constituent series. We hypothesize that anomalies occur in low density regions of a distribution and explore the use of normalizing flows for unsupervised anomaly detection, because of their superior quality in density estimation. Moreover, we propose a novel flow model by imposing a Bayesian network among constituent series. A Bayesian network is a directed acyclic graph (DAG) that models causal relationships; it factorizes the joint probability of the series into the product of easy-to-evaluate conditional probabilities. We call such a graph-augmented normalizing flow approach GANF and propose joint estimation of the DAG with flow parameters. We conduct extensive experiments on real-world datasets and demonstrate the effectiveness of GANF for density estimation, anomaly detection, and identification of time series distribution drift.
|
| 10 |
+
|
| 11 |
+
# 1 INTRODUCTION
|
| 12 |
+
|
| 13 |
+
Anomaly detection (Pimentel et al., 2014; Ruff et al., 2018) is the task of identifying unusual samples that significantly deviate from the majority of the data instances. It is applied in a broad variety of domains, including risk management (Aven, 2016), video surveillance (Kiran et al., 2018), adversarial example detection (Grosse et al., 2017), and fraud detection (Roy & George, 2017). Representative classical methods for anomaly detection are one-class support vector machines (Scholkopf ¨ et al., 2001) and kernel density estimation (Parzen, 1962; Kim & Scott, 2012). These methods rely on handcrafted features and often are not robust for high-dimensional data (e.g., images, speech signals, and time series). In recent years, inspired by the success of deep learning for complex data, many deep anomaly detection methods were proposed and they are remarkably effective in applications (Ruff et al., 2018; Sabokrou et al., 2018; Goyal et al., 2020).
|
| 14 |
+
|
| 15 |
+
Apart from these demonstrated applications, increasing demand exists for the anomaly detection of even more complex data; i.e., multiple time series. They contain a set of multivariate time series that often interact with each other in a system. A prominent example of the source of multiple time series is the power grid, where each constituent series is the grid state over time, recorded by a sensor deployed at a certain geographic location. The grid state includes many attributes; e.g., current magnitude and angle, voltage magnitude and angle, and frequency. Time series readings from sensors at nearby locations are often correlated and their behavior may be causal under cascading effects. Anomaly detection amounts to timely identifying abnormal grid conditions such as generator trip and insulator damage.
|
| 16 |
+
|
| 17 |
+
Anomaly detection of multiple time series is rather challenging, due to high dimensionality, interdependency, and label scarcity. First, a straightforward approach is to concatenate the constituent series along the attribute dimension and apply a detection method for multivariate time series. However, when the system contains many constituents, the resulting data suffer high dimensionality. Second, constituent series bear intricate interdependencies, which may be implicit and challenging to model. When an explicit graph topology is known, graph neural networks are widely used to digest the relational information (Seo et al., 2016; Li et al., 2018b; Yu et al., 2018; Zhao et al., 2019). However, a graph may not always be known (because, for example, it is sensitive information) and hence graph structure learning becomes an indispensable component of the solution (Kipf et al., 2018; Wu et al., 2020; Shang et al., 2021; Deng & Hooi, 2021). Third, labeling information is often limited. Even if certain labels are present, in practice, many anomalies may still stay unidentified because labeling is laborious and expensive. Hence, unsupervised approaches are the most suitable choice. However, albeit many unsupervised detection methods were proposed (Ruff et al., 2018; Sabokrou et al., 2018; Malhotra et al., 2016; Hendrycks et al., 2019), they are not effective for multiple time series.
|
| 18 |
+
|
| 19 |
+
In this work, we explore the use of normalizing flows (Dinh et al., 2016; Papamakarios et al., 2017) for anomaly detection, based on a hypothesis that anomalies often lie on low density regions of the data distribution. Normalizing flows are a class of deep generative models for learning the underlying distribution of data samples. They are unsupervised and they resolve the label scarcity challenge aforementioned. An advantage of normalizing flows is that they are particularly effective in estimating the density of any sample. A recent work by Rasul et al. (2021) extends normalizing flows for time series data by expressing the density of a series through successive conditioning on historical data and applying conditional flows to learn each conditional density, paving ways to build sophisticated flows for multiple time series.
|
| 20 |
+
|
| 21 |
+
We address the high dimensionality and interdependency challenges by learning the relational structure among constituent series. To this end, Bayesian networks (Pearl, 1985; 2000) that model causal relationships of variables are a principled choice. A Bayesian network is a directed acyclic graph (DAG) where a node is conditionally independent of its non-descendents given its parents. Such a structure allows factorizing the intractable joint density of all graph nodes into a product of easyto-evaluate conditional densities of each node. Hence, learning the relational structure among constituent series amounts to identifying a DAG that maximizes the densities of observed data.
|
| 22 |
+
|
| 23 |
+
We propose a novel framework, GANF (Graph-Augmented Normalizing Flow), to augment a normalizing flow with graph structure learning and to apply it for anomaly detection. There are nontrivial technical problems to resolve to materialize this framework: (i) How does one inject a graph into a normalizing flow, which essentially maps one distribution to another? (ii) How does one learn a DAG, which is a discrete object, inside a continuous flow model? The solution we take is to factorize the density of a multiple time series along the attribute, the temporal, and the series dimensions and use a graph-based dependency encoder to model the conditional densities resulting from factorization. Therein, the graph adjacency matrix is a continuous variable and we impose a differentiable constraint to ensure that the corresponding graph is acyclic (Zheng et al., 2018; Yu et al., 2019). We propose a joint training algorithm to optimize both the graph adjacency matrix and the flow parameters.
|
| 24 |
+
|
| 25 |
+
In addition to resolving the high dimensionality and interdependency challenges, an advantage of modeling the relational graph structure among constituent series is that one can easily observe the dynamics of the data distribution from the graph. For time series datasets that span a long period, one naturally questions if the distribution changes over time. The graph structure is a useful indicator of distribution drift. We will study the graph evolution empirically observed.
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| 26 |
+
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+
We highlight the following contributions of this work:
|
| 28 |
+
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| 29 |
+
• We propose a framework to augment a normalizing flow with graph structure learning, to model interdependencies exhibited inside multiple time series.
|
| 30 |
+
• We apply the augmented flow model to detect anomalies in multiple time series data and perform extensive empirical evaluation to demonstrate its effectiveness on real-world data sets.
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| 31 |
+
• We study the evolution of the learned graph structure and identify distribution drift in time series data that span a long time period.
|
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+
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+
# 2 RELATED WORK
|
| 34 |
+
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+
Anomaly Detection. Anomaly detection is a widely studied subject owing to its diverse applications. Recently, inspired by the success of deep learning, several deep anomaly detection methods are proposed and they achieve remarkable success on complex data, such as individual time series (Malhotra et al., 2016), images (Sabokrou et al., 2018), and videos (Ionescu et al., 2019). These methods generally fall under three categories: deep one-class models, generative model-based methods, and transformation-based methods. Deep one-class models (Ruff et al., 2018; Wu et al., 2019) treat normal instances as the target class and identify instances that do not belong to this class. In generative model-based methods (Malhotra et al., 2016; Nguyen et al., 2019; Li et al., 2018a), an autoencoder or a generative adversarial network is used to model the data distribution. Then, an anomaly measure is defined, such as the reconstruction error in autoencoding. Transformationbased methods (Golan & El-Yaniv, 2018; Hendrycks et al., 2019) are based on the premise that transformations applied to normal instances can be identified while anomalies not. Various transformations such as rotations and affine transforms have been investigated. On the other hand, anomaly detection of multiple time series is under explored. Recently, Deng & Hooi (2021) study the use of graph neural networks in combination with structure learning to detect anomalies. Our method substantially differs from this work in that the learned structure is a Bayesian network, which allows density estimation. Moreover, the Bayesian network identifies conditional dependencies among the constituent series and induces a better interpretation of the graph as well as the data distribution.
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| 36 |
+
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+
Normalizing Flows. Normalizing flows are generative models that normalize complex real-world data distributions to “standard” distributions by using a sequence of invertible and differentiable transformations. Dinh et al. (2016) introduce a widely used normalizing flow architecture— RealNVP—for density estimation. Various extensions and improvements are proposed (Papamakarios et al., 2017; Hoogeboom et al., 2019; Kingma & Dhariwal, 2018). For example, Papamakarios et al. (2017) view an autoregressive model as a normalizing flow. To model temporal data, Rasul et al. (2021) use sequential models to parameterize conditional flows. Moreover, graph normalizing flows are proposed to handle graph structured data and improve predictions and generations (Liu et al., 2019). In contrast, normalizing flows for multiple time series are rarely studied in the literature. In this work, we develop a graph-augmented flow for density estimation and anomaly detection of multiple time series.
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+
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# 3 PRELIMINARIES
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+
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+
We first recall key concepts and familiarize the reader with notations to be used throughout the paper.
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+
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+
# 3.1 NORMALIZING FLOWS
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| 44 |
+
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| 45 |
+
Let $\mathbf { x } \in \mathbb { R } ^ { D }$ be a $D$ -dimensional random variable. A normalizing flow is a vector-valued invertible mapping $\mathbf { f } ( \mathbf { x } ) : \mathbb { R } ^ { D } \mathbb { R } ^ { D }$ that normalizes the distribution of $\mathbf { x }$ to a “standard” distribution (or called base distribution). This distribution is usually taken to be an isotropic Gaussian or other ones that are easy to sample from and whose density is easy to evaluate. Let ${ \bf z } = { \bf f } ( { \bf x } )$ with probability density function $q ( \mathbf { z } )$ . With the change-of-variable formula, we can express the density of the $\mathbf { x }$ , $p ( \mathbf { x } )$ , by:
|
| 46 |
+
|
| 47 |
+
$$
|
| 48 |
+
\begin{array} { r } { \log p ( \mathbf { x } ) = \log q ( \mathbf { f } ( \mathbf { x } ) ) + \log | \operatorname* { d e t } \nabla _ { \mathbf { x } } \mathbf { f } ( \mathbf { x } ) | . } \end{array}
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| 49 |
+
$$
|
| 50 |
+
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+
In practical uses, the Jacobian determinant in (1) needs be easy to compute, so that the density $p ( \mathbf { x } )$ can be evaluated. Moreover, as a generative model, the invertibility of $\mathbf { f }$ allows drawing new instances $\mathbf { x } = \mathbf { f } ^ { - 1 } ( \mathbf { z } )$ through sampling the base distribution. One example of such f is the masked autoregressive flow (Papamakarios et al., 2017), which yields $\mathbf { z } = [ z _ { 1 } , \dots , z _ { D } ]$ from $\mathbf { x } = [ x _ { 1 } , \dots , x _ { D } ]$ through
|
| 52 |
+
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| 53 |
+
$$
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| 54 |
+
z _ { i } = ( x _ { i } - \mu _ { i } ( \mathbf { x } _ { 1 : i - 1 } ) ) \exp ( \alpha _ { i } ( \mathbf { x } _ { 1 : i - 1 } ) ) ,
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| 55 |
+
$$
|
| 56 |
+
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+
where $\mu _ { i }$ and $\alpha _ { i }$ are neural networks such as the multilayer perceptron.
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| 58 |
+
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| 59 |
+
A flow may be augmented with conditional information $\mathbf { h } \in \mathbb { R } ^ { d }$ with a possibly different dimension. Such a flow is a conditional flow and is denoted by $\textbf { f } : \mathbb { R } ^ { D } \times \mathbb { R } ^ { d } \overset { \cdot } { } \mathbb { R } ^ { D }$ . The log-density of $\mathbf { x }$ conditioned on $\mathbf { h }$ admits the following formula:
|
| 60 |
+
|
| 61 |
+
$$
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+
\begin{array} { r } { \log p ( { \mathbf x } | { \mathbf h } ) = \log q ( { \mathbf f } ( { \mathbf x } ; { \mathbf h } ) ) + \log | \operatorname* { d e t } \nabla _ { { \mathbf x } } { \mathbf f } ( { \mathbf x } ; { \mathbf h } ) | . } \end{array}
|
| 63 |
+
$$
|
| 64 |
+
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| 65 |
+
We now consider a normalizing flow for time series. Let $\mathbf { X } = [ \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } , \ldots , \mathbf { x } _ { T } ]$ denote a time series of length $T$ , where $\mathbf { x } _ { t } \in \mathbb { R } ^ { D }$ . Through successive conditioning, the density of the time series can be written as:
|
| 66 |
+
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| 67 |
+
$$
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| 68 |
+
p ( \mathbf { X } ) = p ( \mathbf { x } _ { 1 } ) p ( \mathbf { x } _ { 2 } | \mathbf { x } _ { < 2 } ) \cdot \cdot \cdot p ( \mathbf { x } _ { T } | \mathbf { x } _ { < T } ) ,
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| 69 |
+
$$
|
| 70 |
+
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| 71 |
+
where $\mathbf { x } _ { < t }$ denotes all variables before time $t$ . When the conditional probabilities are parameterized, Rasul et al. (2021) propose to model each $p ( \mathbf { x } _ { t } | \mathbf { x } _ { < t } )$ as $p ( \mathbf x _ { t } | \mathbf h _ { t - 1 } )$ , where $\mathbf { h } _ { t - 1 }$ summarizes the
|
| 72 |
+
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| 73 |
+
past information $\mathbf { x } _ { < t }$ . For example, $\mathbf { h } _ { t - 1 }$ is the hidden state of a recurrent neural network before accepting input $\mathbf { x } _ { t }$ . Then, a conditional normalizing flow can be applied to evaluate each $p ( \mathbf x _ { t } | \mathbf h _ { t - 1 } )$ .
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+
|
| 75 |
+
# 3.2 BAYESIAN NETWORKS
|
| 76 |
+
|
| 77 |
+
Let $X ^ { i }$ denote a general random variable, either scalar valued, vector valued, or even matrix valued. A Bayesian network of $n$ variables $( X ^ { 1 } , \ldots , X ^ { n } )$ is a directed acyclic graph of the variables as nodes. Let A denote the weighted adjacency matrix of the graph, where $\mathbf { A } _ { i j } \neq 0$ if $X ^ { j }$ is the parent of $X ^ { i }$ . A Bayesian network describes the conditional independence among variables. Specifically, a node $X ^ { i }$ is conditionally independent of its non-descendents given its parents. In other words, the density of the joint distribution of $( X ^ { 1 } , \ldots , X ^ { n } )$ is
|
| 78 |
+
|
| 79 |
+
$$
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| 80 |
+
p ( X ^ { 1 } , \ldots , X ^ { n } ) = \prod _ { i = 1 } ^ { n } p ( X ^ { i } | \operatorname { p a } ( X ^ { i } ) ) ,
|
| 81 |
+
$$
|
| 82 |
+
|
| 83 |
+
where $\operatorname { p a } ( X ^ { i } ) = \{ X ^ { j } : \mathbf { A } _ { i j } \neq 0 \}$ denotes the set of parents of $X ^ { i }$
|
| 84 |
+
|
| 85 |
+
# 4 PROBLEM STATEMENT
|
| 86 |
+
|
| 87 |
+
In this paper, we focus on unsupervised anomaly detection with multiple time series. The training set $\mathcal { D }$ consists of only unlabeled instances and we assume that the majority of them are not anomalies. Each instance $\mathcal { X } \in \mathcal { D }$ contains $n$ constituent series with $D$ attributes and of length $T$ ; i.e., $\mathcal { X } =$ $( \mathbf { X } ^ { 1 } , \mathbf { X } ^ { 2 } , \ldots , \mathbf { X } ^ { n } )$ where $\mathbf { X } ^ { i } \in \mathbb { R } ^ { T \times D }$ . We use a Bayesian network (DAG) to model the relational structure of the constituent series $\mathbf { X } ^ { i }$ and augment a normalizing flow to compute the density of $\mathcal { X }$ through a factorization in the form (5). Let $\bar { \mathbf { A } } \in \mathbb { R } ^ { n \times n }$ be the adjacency matrix of the DAG and let $\mathcal { F } : ( \mathcal { X } , \mathbf { A } ) \mathcal { Z }$ denote the augmented flow. Because anomaly points tend to have low densities, we propose to conduct unsupervised anomaly detection by evaluating the density of a multiple time series computed through the augmented flow. The problem is formulated as the following.
|
| 88 |
+
|
| 89 |
+
Problem 1. Given a training set $\mathcal { D } = \{ \mathcal { X } _ { i } \} _ { i = 1 } ^ { | \mathcal { D } | }$ of multiple time series, we aim to simultaneously learn the adjacency matrix A of the Bayesian Network that represents the conditional dependencies among the constituent series, as well as the correspondingly graph-augmented normalizing flow $\mathcal { F } : ( \mathcal { X } , \mathbf { A } ) \mathcal { Z }$ , which is used to estimate the density of an instance $\mathcal { X }$ . Here, $\mathcal { Z }$ is a random variable with $a$ “simple” distribution, such as the anisotropic Gaussian.
|
| 90 |
+
|
| 91 |
+
# 5 METHOD
|
| 92 |
+
|
| 93 |
+
In this section, we materialize the graph-augmented normalizing flow $\mathcal { F } : ( \mathcal { X } , \mathbf { A } ) \mathcal { Z }$ introduced in the problem statement and use it to compute the density of a multiple time series $\mathcal { X }$ . The central idea is factorization: we factorize $p ( \mathcal { X } )$ along the series dimension by using a Bayesian network and then factorize along the temporal dimension by using conditional normalizing flows. Then, we employ a novel graph-based dependency encoder to parameterize the conditional probabilities resulting from the factorization. The DAG used for factorization is a discrete object and is usually intractable to learn; however, the discrete structure is reflected in the dependency encoder through a graph adjacency matrix A that is differentiable. Moreover, the requirement that A must correspond to a DAG can be expressed as a differentiable equation. Hence, one can jointly optimize A and the flow components by using gradient based optimization. Once $\mathcal { F }$ is learned, the density $p ( \mathcal { X } )$ is straightforwardly evaluated for anomaly detection. An illustration of the framework GANF is shown in Figure 1.
|
| 94 |
+
|
| 95 |
+
# 5.1 FACTORIZATION
|
| 96 |
+
|
| 97 |
+
Figure 1 shows a toy example of a Bayesian network as a DAG. Based on (5), the density of a multiple time series $\mathbf { \dot { \mathcal { X } } } = ( \hat { \mathbf { X } } ^ { 1 } , \mathbf { X } ^ { 2 } , \dots , \mathbf { \bar { X } } ^ { n } )$ can be computed as the product of $p ( \mathbf { X } ^ { i } | \mathbf { \theta } \mathrm { p a } ( \mathbf { X } ^ { i } ) )$ for all nodes, where recall that $\operatorname { p a } ( \mathbf { X } ^ { i } )$ denotes the set of parents of $\mathbf { X } ^ { i }$ . Then, following Rasul et al. (2021), we further factorize each conditional density along the temporal dimension. Specifically, for a time step $t$ , $\mathbf { x } _ { t } ^ { i }$ depends on its past history as well as its parents in the DAG. We write
|
| 98 |
+
|
| 99 |
+
$$
|
| 100 |
+
p ( \mathcal { X } ) = \prod _ { i = 1 } ^ { n } p ( \mathbf { X } ^ { i } | \mathbf { \ p a } ( \mathbf { X } ^ { i } ) ) = \prod _ { i = 1 } ^ { n } \prod _ { t = 1 } ^ { T } p ( \mathbf { x } _ { t } ^ { i } | \mathbf { \ p a } ( \mathbf { x } ^ { i } ) _ { 1 : t } , \mathbf { x } _ { 1 : t - 1 } ^ { i } ) ,
|
| 101 |
+
$$
|
| 102 |
+
|
| 103 |
+
|
| 104 |
+
Figure 1: Illustration of a Bayesian network and the proposed framework GANF.
|
| 105 |
+
|
| 106 |
+
where $\mathbf { x } _ { 1 : t - 1 } ^ { i }$ denotes the history of node $i$ before time $t$ , and $\mathrm { p a } ( \mathbf { x } ^ { i } ) _ { 1 : t } = \{ \mathbf { x } _ { 1 : t } ^ { j } : \mathbf { A } _ { i j } \neq 0 \}$ . In the next subsection, we will parameterize each conditional density $p ( \mathbf { x } _ { t } ^ { i } \mid \mathrm { p a } ( \mathbf { x } ^ { i } ) _ { 1 : t } , \mathbf { x } _ { 1 : t - 1 } ^ { i } )$ by using a graph-based dependency encoder. Note that so far the factorization (6) has been based on the discrete structure of the Bayesian network. The dependency encoder we introduce next, however, uses the adjacency matrix A in a differentiable manner, which is sufficient to ensure that $\mathbf { x } _ { t } ^ { i }$ will not depend on nodes other than its parents and itself.
|
| 107 |
+
|
| 108 |
+
# 5.2 NEURAL NETWORK PARAMETERIZATION
|
| 109 |
+
|
| 110 |
+
According to Sec. 3.1, conditional densities $p ( \mathbf { x } _ { t } ^ { i } \mid \mathrm { p a } ( \mathbf { x } ^ { i } ) _ { 1 : t } , \mathbf { x } _ { 1 : t - 1 } ^ { i } )$ can be learned by using conditional normalizing flows. However, the conditional information $\mathrm { p a } ( \mathbf { x } ^ { i } ) _ { 1 : t }$ and $\mathbf { x } _ { 1 : t - 1 } ^ { i }$ cannot be directly used for parameterization, because its size is not fixed. Therefore, as is illustrated in Figure 1, we design a graph-based dependency encoder to summarize the conditional information into a fixed length vector $\bar { \mathbf { d } } _ { t } ^ { i } \in \mathbb { R } ^ { d }$ . Then, a conditional normalizing flow is used to evaluate $p ( \mathbf { x } _ { t } ^ { i } | \mathbf { d } _ { t } ^ { i } )$ , which is equivalent to $\breve { p } ( \mathbf { x } _ { t } ^ { i } | \mathrm { \ p a } ( \mathbf { x } ^ { i } ) _ { 1 : t } , \mathbf { x } _ { 1 : t - 1 } ^ { i } )$ .
|
| 111 |
+
|
| 112 |
+
Dependency Encoder. Since the history has an arbitrary length, we first employ a recurrent neural network (RNN) to map multiple time steps to a vector of fixed length. For a time series $\mathbf { x } _ { 1 : t } ^ { i }$ , the recurrent model abstracts it into a hidden state $\mathbf { h } _ { t } ^ { i } \in \mathbb { R } ^ { d }$ through the following recurrence
|
| 113 |
+
|
| 114 |
+
$$
|
| 115 |
+
\mathbf h _ { t } ^ { i } = \mathbf { R N N } ( \mathbf x _ { t } ^ { i } , \mathbf h _ { t - 1 } ^ { i } ) ,
|
| 116 |
+
$$
|
| 117 |
+
|
| 118 |
+
where $\mathbf { h } _ { t } ^ { i }$ summarizes the time series up to step $t$ . The RNN can be any sequential model, such as the LSTM (Hochreiter & Schmidhuber, 1997) and in a broad sense a transformer (Vaswani et al., 2017). We let the RNN parameters be shared across all nodes in the DAG to avoid overfitting and to reduce computational costs.
|
| 119 |
+
|
| 120 |
+
With (7), the conditional information of $\mathbf { x } _ { t } ^ { i }$ is all summarized in $\{ \mathbf { h } _ { t } ^ { j } : \mathbf { A } _ { i j } \neq 0 \} \cup \{ \mathbf { h } _ { t - 1 } ^ { i } \}$ . Inspired by the success of GCN (Kipf & Welling, 2016) in node representation learning through neighborhood aggregation, we design a graph convolution layer to aggregate hidden states of the parents for dependency encoding. This layer produces dependency representations $\mathbf { D } _ { t } = ( \mathbf { d } _ { t } ^ { 1 } , \dots , \mathbf { \bar { d } } _ { t } ^ { n } )$ for all constituent series at time $t$ :
|
| 121 |
+
|
| 122 |
+
$$
|
| 123 |
+
\mathbf { D } _ { t } = \operatorname { R e L U } ( \mathbf { A } \mathbf { H } _ { t } \mathbf { W } _ { 1 } + \mathbf { H } _ { t - 1 } \mathbf { W } _ { 2 } ) \cdot \mathbf { W } _ { 3 } ,
|
| 124 |
+
$$
|
| 125 |
+
|
| 126 |
+
where $\mathbf { H } _ { t } \ = \ ( \mathbf { h } _ { t } ^ { 1 } , \ldots , \mathbf { h } _ { t } ^ { n } )$ contains all the hidden states at time $t$ . Here, $\mathbf { W } _ { 1 } ~ \in ~ \mathbb { R } ^ { d \times d }$ and $\mathbf { W } _ { 2 } ~ \in ~ \mathbb { R } ^ { d \times d }$ are parameters to transform the aggregated representations of the parents and the node’s historical information, respectively; while $\mathbf { \check { W } } _ { 3 } ^ { - } \in \mathbb { R } ^ { d \times d }$ is an additional transformation to improve the dependency representation.
|
| 127 |
+
|
| 128 |
+
Density Estimation. With the dependency encoder, we obtain the representations ${ \bf d } _ { t } ^ { i }$ of the conditional information. Then, a normalizing flow $\mathbf { f } : \mathbb { R } ^ { D } \times \mathbb { R } ^ { d } \to \mathbb { R } ^ { D }$ conditioned on ${ \bf d } _ { t } ^ { i }$ is applied to model each $p ( \mathbf { x } _ { t } ^ { i } | \mathrm { \ p a } ( \mathbf { x } ^ { i } ) _ { 1 : t } , \mathbf { x } _ { 1 : t - 1 } ^ { i } )$ . Similar to the computation of the hidden states, the parameters of the conditional flow are also shared among nodes, to avoid overfitting. Based on (3), the conditional density of $\mathbf { x } _ { t } ^ { i }$ can be written as:
|
| 129 |
+
|
| 130 |
+
$$
|
| 131 |
+
\log p ( \mathbf { x } _ { t } ^ { i } \mid \operatorname { p a } ( \mathbf { x } ^ { i } ) _ { 1 : t } , \mathbf { x } _ { 1 : t - 1 } ^ { i } ) = \log p ( \mathbf { x } _ { t } ^ { i } \mid \mathbf { d } _ { t } ^ { i } ) = \log q ( \mathbf { f } ( \mathbf { x } _ { t } ^ { i } ; \mathbf { d } _ { t } ^ { i } ) ) + \log \mid \operatorname* { d e t } \nabla _ { \mathbf { x } _ { t } ^ { i } } \mathbf { f } ( \mathbf { x } _ { t } ^ { i } ; \mathbf { d } _ { t } ^ { i } ) \mid ,
|
| 132 |
+
$$
|
| 133 |
+
|
| 134 |
+
where $q ( \mathbf { z } )$ is chosen to be the standard normal $\mathcal { N } ( \mathbf { z } | \mathbf { 0 } , \mathbf { I } )$ with $\textbf { z } \in \mathbb { R } ^ { D }$ . The conditional flow f can be any effective one proposed by the literature, such as RealNVP (Dinh et al., 2016) and
|
| 135 |
+
|
| 136 |
+
MAF (Papamakarios et al., 2017). Combining (9) and (6), we obtain the log-density of a multiple time series $\mathcal { X }$ :
|
| 137 |
+
|
| 138 |
+
$$
|
| 139 |
+
\log p ( \mathcal { X } ) = \sum _ { i = 1 } ^ { n } \sum _ { t = 1 } ^ { T } \Big [ \log q ( \mathbf { f } ( \mathbf { x } _ { t } ^ { i } ; \mathbf { d } _ { t } ^ { i } ) ) + \log | \operatorname* { d e t } \nabla _ { \mathbf { x } _ { t } ^ { i } } \mathbf { f } ( \mathbf { x } _ { t } ^ { i } ; \mathbf { d } _ { t } ^ { i } ) | \Big ] .
|
| 140 |
+
$$
|
| 141 |
+
|
| 142 |
+
Anomaly Measure. Because anomalies deviate significantly from the majority of the data instances, we hypothesize that their densities are low. Thus, we use the density computed by (10) as the anomaly measure, where a lower density indicates a more likely anomaly. Apart from evaluating the density for the entire $\mathcal { X }$ , the computation also produces conditional densities $\log p ( \mathbf { X } ^ { i } | \operatorname { p a } ( \mathbf { X } ^ { i } ) ) =$ $\begin{array} { r } { \sum _ { t = 1 } ^ { T } \log p ( \mathbf { x } _ { t } ^ { i } | \mathbf { d } _ { t } ^ { i } ) } \end{array}$ for each constituent series $\mathbf { X } ^ { i }$ . We use this conditional density as the anomaly measure for constituent series. A low density $p ( \mathcal { X } )$ is caused by one or a few low conditional densities $p ( \mathbf { X } ^ { i } | \operatorname { p a } ( \mathbf { X } ^ { i } ) )$ in the Bayesian network, suggesting that abnormal behaviors could be traced to individual series.
|
| 143 |
+
|
| 144 |
+
# 5.3 JOINT TRAINING
|
| 145 |
+
|
| 146 |
+
Learning a Bayesian network is a challenging combinatorial problem, due to the intractable search space superexponential in the number of nodes. A recent work by Zheng et al. (2018) proposes the equation $\operatorname { t r } ( e ^ { \mathbf { A } \circ \mathbf { A } } ) = n$ that characterizes the acyclicity of the corresponding graph of A, where $e$ is matrix exponential and $\circ$ denotes element-wise multiplication. We will impose this equation as a constraint in the training of GANF.
|
| 147 |
+
|
| 148 |
+
Training Objective. Following the training of a usual normalizing flow, the joint density (likelihood) of the observed data is the training objective, which is equivalent to the Kullback–Leibler divergence between the true distribution of data and the flow recovered distribution. Together with the DAG constraint, the optimization problem reads
|
| 149 |
+
|
| 150 |
+
$$
|
| 151 |
+
\operatorname* { m i n } _ { \mathbf { A } , \theta } \mathcal { L } ( \mathbf { A } , \theta ) = \frac { 1 } { | \mathcal { D } | } \sum _ { i = 1 } ^ { | \mathcal { D } | } - \log p ( \mathcal { X } _ { i } ) ,
|
| 152 |
+
$$
|
| 153 |
+
|
| 154 |
+
where $\pmb \theta$ contains all neural network parameters, including those of the dependency encoder and the normalizing flow. Here, the DAG constraint $h ( \mathbf { A } )$ admits an easy-to-evaluate gradient $\nabla h ( { \mathbf A } ) =$ $( e ^ { \mathbf { A } \circ \mathbf { A } } ) ^ { T } \circ \mathsf { 2 } \mathbf { A }$ , which allows a gradient based optimizer to solve (11).
|
| 155 |
+
|
| 156 |
+
Training Algorithm. Problem (11) is a nonlinear equality-constrained optimization. Such problems are extensively studied and the augmented Lagrangian method (Bertsekas, 1999; Yu et al., 2019) is one of the most widely used approaches. The augmented Lagrangian is defined as
|
| 157 |
+
|
| 158 |
+
$$
|
| 159 |
+
\mathcal { L } _ { c } = \mathcal { L } ( { \bf A } , \pmb \theta ) + \lambda h ( { \bf A } ) + \frac { c } { 2 } | h ( { \bf A } ) | ^ { 2 } ,
|
| 160 |
+
$$
|
| 161 |
+
|
| 162 |
+
where $\lambda$ and $c$ denote the Lagrange multiplier and the penalty parameter, respectively. The general idea of the method is to gradually increase the penalty parameter to ensure that the constraint is eventually satisfied. Over iterations, $\lambda$ as a dual variable will converge to the Lagrangian multiplier of (11). The update rule at the $k$ th iteration reads the following:
|
| 163 |
+
|
| 164 |
+
$$
|
| 165 |
+
\mathbf { A } ^ { k } , \theta ^ { k } = \arg \operatorname* { m i n } _ { \mathbf { A } , \theta } \mathcal { L } _ { c ^ { k } } ; \quad \lambda ^ { k + 1 } = \lambda ^ { k } + c ^ { k } h ( \mathbf { A } ^ { k } ) ; \quad c ^ { k + 1 } = \left\{ \begin{array} { l l } { \eta c ^ { k } } & { \mathrm { i f } \left| h ( \mathbf { A } ^ { k } ) \right| > \gamma \left| h ( \mathbf { A } ^ { k - 1 } ) \right| ; } \\ { c ^ { k } } & { \mathrm { e l s e } , } \end{array} \right.
|
| 166 |
+
$$
|
| 167 |
+
|
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where $\eta \in \left( 1 , + \infty \right)$ and $\gamma \in \mathsf { \Gamma } ( 0 , 1 )$ are hyperparameters to be tuned. We set $\eta$ and $\gamma$ as 10 and 0.5, respectively. The subproblem of optimizing A and $\pmb { \theta }$ can be solved by using the Adam optimizer (Kingma & Ba, 2014). The training algorithm is summarized in Appendix A.
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# 6 EXPERIMENTS
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In this section, we conduct a comprehensive set of experiments to validate the effectiveness of the proposed GANF framework. In particular, they are designed to answer the following questions:
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• Q1: Can GANF accurately detect anomalies and estimate densities?
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• Q2: Does the proposed graph structure learning help? Is the framework sufficiently flexible to include various normalizing flow backbones?
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• Q3: What can one observe for a dataset spanning a long time? E.g., does the graph pattern change?
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# 6.1 SETTINGS
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Datasets. To evaluate the effectiveness of GANF for anomaly detection and density estimation, we conduct experiments on two power grid datasets, one water system dataset, and one traffic dataset.
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• PMU-B and PMU-C: These two datasets correspond to two separate interconnects of the U.S. power grid, containing time series recorded by 38 and 132 phasor measurement units (PMUs), respectively. We process one-year data at the frequency of one second to form a ten-month training set, one-month validation set, and one-month test set. Each multiple time series is obtained by shifting a one-minute window. Additionally, to investigate distribution drift, we shift a one-month window to obtain multiple training/validation/test sets (12 in total, because of availability of twoyear data). Sparse grid events (anomalies) labeled by domain experts exist for evaluation; but note that the labels are both noisy and incomplete. These datasets are proprietary.
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• SWaT: We also use a public dataset for evaluation. The Secure Water Treatment (SWaT) dataset originates from an operational water treatment test-bed coordinated with Singapore’s Public Utility Board (Goh et al., 2016). The data collects 51 sensor recordings lasting four days, at the frequency of one second. A total of 36 attacks were conducted, resulting in approximately $11 \%$ time steps as anomaly ground truths. We use a sliding window of 60 seconds to construct series data and perform a 60/20/20 chronological split for training, validation, and testing, respectively.
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• METR-LA: This dataset is also public; it contains speed records of 207 sensors deployed on the highways of Los Angles, CA (Li et al., 2018b). No anomaly labels exist however and we use this dataset for exploratory analysis only. Results are deferred to Appendix E.
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Evaluation metrics (under noisy labels). For SWaT, which offers reliable ground truths, we use the standard ROC and AUC metrics for evaluation. For the two PMU datasets, however, the resolution of the time series and the granularity of the events result in rather noisy ground truths. Hence, we adapt ROC for noisy labels. We smooth the time point of a “ground truth” event (anomaly) by introducing probabilities to the label. Specifically, the probability that a multiple time series starting at time t is a ground truth anomaly is maxi{exp(− (t−ti)2σ2 ) , where $t _ { i }$ is the starting time of the ith labeled anomaly. Then, when computing the confusion matrix, we sum probabilities rather than counting 0/1s. The smoothing window $\sigma$ is chosen to be 6 time steps.
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Baselines. We compare with the following representative, state-of-the-art deep methods.
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• EncDecAD (Malhotra et al., 2016): In this method, an autoencoder based on LSTM is trained. The reconstruction error is used as the anomaly measure.
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• DeepSVDD (Ruff et al., 2018): This method minimizes the volume of a hypersphere that encloses the representations of data. Samples distant from the hypersphere center are considered anomalies.
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• ALOCC (Sabokrou et al., 2020): In this GAN-based method, the generator learns to reconstruct normal instances, while the discriminator works as an anomaly detector.
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• DROCC (Goyal et al., 2020): This method performs adversarial training to learn robust representations of data and identifies anomalies.
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• DeepSAD (Ruff et al., 2020): This method extends DeepSVDD with a semi-supervised loss term for training. We use noisy labels as supervision.
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To apply these baselines on multiple time series, we concatenate the constituent series along the attribute dimension (resulting in high-dimensional series) and use LSTM or CNN as the backbones. On the other hand, for the proposed method, we use LSTM as the RNN model and MAF as the normalizing flow. See Appendix C for more implementation details.
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# 6.2 PERFORMANCE OF ANOMALY DETECTION AND DENSITY ESTIMATION
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Table 1: AUC-ROC $( \% )$ of anomaly detection.
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<table><tr><td>Dataset</td><td>EncDecAD</td><td>DeepSVDD</td><td>ALOCC</td><td>DROCC</td><td>DeepSAD</td><td>GANF</td></tr><tr><td>PMU-B</td><td>55.6±1.8</td><td>55.6±3.3</td><td>62.9±2.2</td><td>58.6±3.0</td><td>63.7±0.9</td><td>67.5±0.8</td></tr><tr><td>PMU-C</td><td>53.7±0.5</td><td>56.9±0.9</td><td>60.9±1.3</td><td>61.9±2.7</td><td>60.1±1.4</td><td>70.6±3.3</td></tr><tr><td>SWaT</td><td>76.5±0.7</td><td>68.8±2.0</td><td>75.4±2.3</td><td>73.3±1.6</td><td>75.4±1.2</td><td>79.6±0.9</td></tr></table>
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Figure 2: ROC curves of anomaly detection on various datasets.
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Figure 3: Qualitative evaluation of GANF on PMU-C. (a) Distribution of log-densities on the test set (note in log scale). (b) Anomaly detection results for a week in the test set.
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To answer Q1, we evaluate quantitatively and qualitatively on datasets with labels.
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Anomaly detection. We compare GANF with the aforementioned baselines in Table 1, where standard deviations of the AUC scores are additionally reported through five random repetitions of model training. The table suggests an overwhelmingly high AUC score achieved by GANF. Observations follow. (i) GANF outperforms generative model-based methods (EncDecAD and ALOCC). Being a generative model as well, normalizing flows augmented with a graph structure leverage the interdependencies of constituent series more effectively, leading to a substantial improvement in detection. (ii) GANF significantly outperforms deep one-class models (DeepSVDD and DROCC), corroborating the appeal of using densities for detection. (iii) GANF also performs better than the semi-supervised method DeepSAD, probably because such methods rely on high quality labels for supervision (especially in the case of label scarcity) and they are less effective facing noisy labels.
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Besides a single score, we also plot the ROC curve in Figure 2. One sees that the curve of GANF dominates those of others. This behavior is generally more salient in the low false-alarm regime.
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Density estimation. We investigate the densities estimated by GANF, shown in Figure 3. Distributions of the log-densities in the test set are shown in Figure 3a. We use log-density as the anomaly measure; the lower the more likely. Note that the vertical axis is in the log-scale. One sees that a log-density of 16 approximately separates the majority normal instances from the minority anomalies. To cross-verify that the instances with low densities are suspiciously anomalous, we investigate Figure 3b, which is a temporal plot of log-densities for a week, overlaid with given labels. From this plot, one sees that the noisily labeled series generally have low densities or are near a low density time step. Additionally, GANF discovers a few suspicious time steps with low densities undetected earlier. These new discoveries raise interest to power system experts for analysis and archiving.
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# 6.3 ABLATION STUDY
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Table 2: Performance of variants of the proposed method.
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<table><tr><td>Dataset</td><td>Metrics</td><td>GANF\G</td><td>GANF\D</td><td>GANF\T</td><td>GANFRNVP</td><td>GANF</td></tr><tr><td>PMU-B</td><td>AUC-ROC</td><td>0.641</td><td>0.643</td><td>0.653</td><td>0.661</td><td>0.678</td></tr><tr><td></td><td>Log-Density</td><td>15.31</td><td>8.70</td><td>15.09</td><td>15.90</td><td>16.22</td></tr><tr><td>PMU-C</td><td>AUC-ROC</td><td>0.630</td><td>0.544</td><td>0.688</td><td>0.703</td><td>0.705</td></tr><tr><td></td><td>Log-Density</td><td>15.55</td><td>8.94</td><td>15.70</td><td>17.06</td><td>16.98</td></tr></table>
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Figure 4: Evolution of the learned DAG on PMU-B over time.
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Figure 5: Evolution of edge weights in the DAG learned by GANF over time (PMU-B).
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To answer Q2, we conduct an ablation study (including varying architecture components) to investigate impacts of DAG structure learning and the flexibility of the GANF framework. To investigate the power of modeling pairwise relationship, we train a variant GANF\G that factorizes $\begin{array} { r } { p ( \mathcal { X } ) = \prod _ { i = 1 } ^ { n ^ { * } } p ( \mathbf { X } ^ { i } ) } \end{array}$ ; i.e., assuming independence among constituent series. To investigate the effectiveness of graph structure learning, we train a variant GANF\D that decomposes the joint density as $\begin{array} { r } { p ( \mathbf { \mathcal { X } } ) \mathbf { \bar { \Phi } } = \prod _ { i = 1 } ^ { n } p ( \mathbf { X } ^ { i } | \mathbf { X } ^ { < i } ) } \end{array}$ ; i.e., a full decomposition without a DAG. It is equivalent to concatenating the series along the attribute dimension and running MAF on the resulting series. To verify the contribution of joint training of $\mathbf { A }$ and $\pmb { \theta }$ , we train a variant GANF\T where A is separately learned by using NOTEARS (Zheng et al., 2018). To prove the flexibility of GANF, we replace the MAF-based normalizing flow by RealNVP, denoted as GANFRNVP.
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Results are presented in Table 2. Apart from AUC-ROC, the log-density is also reported. Observations follow. (i) GANF significantly outperforms GANF\G and GANF\D, corroborating the importance of interdependency modeling among constituent series. Note that GANF\D results in particularly poor performance in general, likely because the high dimensional input (resulting from concatenating too many series) impedes the learning of normalizing flows. (iii) GANF\T is slightly better than GANF\G, because of the presence of relational modeling, but it cannot match the performance of GANFRNVP and GANF that jointly train the DAG and the flow. (ii) These latter two models are the best for both datasets and both metrics. MAF works more often better than RealNVP.
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# 6.4 EVOLUTION OF THE DAG STRUCTURE
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To answer Q3, we investigate how the learned DAG evolves by shifting the train/validation/test sets month by month. The graphs within the first three-month shiftings are shown in Figure 4 and more can be found in Appendix F. In addition to the graph structure, we plot in Figure 5 the learned edge weights over time, one column per edge. The appearance and disappearance of edges demonstrate changes of the conditional independence structure among constituent series over time, suggesting data distribution drift (i.e., a change of internal data generation mechanism). It is interesting to observe the seasonal effect. The columns (edges) in Figure 5 can be loosely grouped in three clusters: those persisting the entire year, those appearing in the first half of the year, and those existing more briefly (e.g., within a season). Such a pattern plausibly correlates with electricity consumption, which is also seasonal. Were spatial information of the PMUs known, these identified DAGs would help mapping the seasonal patterns to geography and help planning a more resilient grid.
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# 7 CONCLUSIONS
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In this paper, we present a graph-augmented normalizing flow GANF for anomaly detection of multiple time series. The graph is materialized as a Bayesian network, which models the conditional dependencies among constituent time series. A graph-based dependency decoder is designed to summarize the conditional information needed by the normalizing flow that calculates series density. Anomalies are detected through identifying instances with low density. Extensive experiments on real-world datasets demonstrate the effectiveness of the framework. Ablation studies confirm the contribution of the learned graph structure in anomaly detection. Additionally, we investigate the evolution of the graph and offer insights of distribution drift over time.
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# ACKNOWLEDGMENT AND DISCLAIMER
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This material is based upon work supported by the Department of Energy under Award Number(s) DE-OE0000910. This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.
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# REFERENCES
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# A TRAINING ALGORITHM
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We summarize the training method in Algorithm 1.
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Algorithm 1 Training Algorithm of GANF
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Input: Training set $\mathcal { D }$ , hyperparameters $\eta$ and $\gamma$
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Output: GANF $\mathcal { F }$ and adjacency matrix A of the DAG
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1: Initialize $c \gets 0$ and initialize $\lambda$ randomly
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2: for $k = 0 , 1 , 2 , \ldots$ do
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3: Compute $\mathbf { A } ^ { k }$ and $\pmb { \theta } ^ { k }$ as a minimizer of (12) by using the Adam optimizer, where the loss $\mathcal { L }$ and the constraint $h$ are defined in (11), the log-density $\log p ( \mathcal { X } )$ is defined in (10), the dependency representation ${ \bf d } _ { t } ^ { i }$ is defined in (8), the hidden state $\mathbf { h } _ { t } ^ { i }$ is defined in (7), and the conditional flow f is RealNVP or MAF
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4: Update Lagrange multiplier $\lambda \gets \lambda + c h ( \mathbf { A } ^ { k } )$
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5: if $k > 0$ and $| h ( { \bf A } ^ { k } ) | > \gamma | h ( { \bf A } ^ { k - 1 } ) |$ then
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6: $c \eta c$
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7: end if
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8: if $h ( \mathbf { A } ^ { k } ) = = 0$ then
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9: break
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10: end if
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11: end for
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12: return A and $\mathcal { F }$ (including f , the neural network (8), and the RNN (7))
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# B CODE
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Code is available at https://github.com/EnyanDai/GANF.
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C ADDITIONAL DETAILS OF EXPERIMENT SETTINGS
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C.1 IMPLEMENTATION DETAILS OF GANF.
|
| 365 |
+
|
| 366 |
+
An LSTM is used as the RNN model in the dependency encoder. For normalizing flows, we use MAF with six flow blocks. All hidden dimensions as set as 32. The initial learning rate is set as 0.001 for the adjacency matrix A and the model parameters $\pmb \theta$ . Learning rate decay is 0.1. To avoid gradient explosion, we clip the gradients whose values are larger than 1.0.
|
| 367 |
+
|
| 368 |
+
For hyperparameter tuning, we select the hyperparameters that yield the highest log-density on the validation set. Specifically, we conduct grid search by varying the number of normalizing flow blocks from $\{ 1 , 2 , 4 , 6 , 8 \}$ , the learning rate from $\left. 0 . 0 0 3 , 0 . 0 0 1 , 0 . 0 0 0 3 , 0 . 0 0 0 1 \right.$ , and the hidden dimension from $\{ 1 6 , 3 2 , 6 4 , 1 2 8 \}$ .
|
| 369 |
+
|
| 370 |
+
C.2 IMPLEMENTATION DETAILS OF BASELINES.
|
| 371 |
+
|
| 372 |
+
• EncDecAD (Malhotra et al., 2016): This method is applied for anomaly detection on time series. Thus, we concatenate constituent series along the attribute dimension and adopt the code released by the authors in https://github.com/chickenbestlover/ RNN-Time-series-Anomaly-Detection.
|
| 373 |
+
• DeepSVDD (Ruff et al., 2018): To handle time series data, we replace the backbone to an LSTM based on the official implementation https://github.com/lukasruff/ Deep-SVDD-PyTorch.
|
| 374 |
+
• ALOCC (Sabokrou et al., 2020): We use the official implementation released by the authors in https://github.com/khalooei/ALOCC-CVPR2018. We replace the two-dimensional convolution to one-dimensional convolution to build a GAN for time series data.
|
| 375 |
+
• DROCC (Goyal et al., 2020): Similar to other baselines, this method is proposed for tabular data and image data. We replace the backbone to LSTM to deal with multiple time series by revising the encoder in https://github.com/microsoft/EdgeML/tree/master/pytorch.
|
| 376 |
+
• DeepSAD (Ruff et al., 2020): This is a semi-supervised approach, which requires labeling. We utilize the noisy labels in PMU-B and PMU-C as supervision. We use LSTM as the
|
| 377 |
+
|
| 378 |
+
backbone, based on the official implementation in https://github.com/lukasruff/ Deep-SAD-PyTorch.
|
| 379 |
+
|
| 380 |
+
All hyperparameters of the baselines are tuned based on the validation set to make fair comparisons.
|
| 381 |
+
|
| 382 |
+
# D TIME COMPLEXITY ANALYSIS
|
| 383 |
+
|
| 384 |
+
The GANF framework involves Bayesian network structure learning, which is known to be highly challenging, owing to the intractable search space superexponential in the number of graph nodes. In this work, we formulate a continuous optimization of the graph structure, so that the training of GANF is more scalable. In what follows, we analyze the time complexity.
|
| 385 |
+
|
| 386 |
+
Recall that each instance $\mathcal { X }$ of the multiple time series dataset contains $n$ constituent series with $D$ attributes and of length $T$ ; i.e., $\mathcal { X } = \left( \mathbf { \bar { X } } ^ { 1 } , \mathbf { X } ^ { 2 } , \ldots , \mathbf { X } ^ { n } \right)$ where $\mathbf { X } ^ { i } \in \mathbb { R } ^ { T \times D }$ . In the evaluation of the model, the dominant costs appear in running the dependency encoder and the normalizing flow. For the dependency encoder, an RNN is first deployed to map the multiple time series to hidden vectors; the time complexity is $O ( n T D )$ . Then, graph convolution is conducted to obtain dependency vectors; the convolution cost is $O ( n ^ { 2 } T )$ . For the normalizing flow module, the time complexity is $O ( n T D )$ . Therefore, the time complexity of computing log-density of one instance is $O ( n T ( D + n ) )$ . If we use a batch size $B$ for training, the time cost of calculating the augmented Lagrangian $\mathcal { L } ( \mathbf { A } , \pmb \theta )$ in (12) is $O ( n B T ( D + n ) )$ . Additionally, the time cost of calculating the constraint $\underline { { h } } ( \mathbf { A } )$ is ${ \dot { O } } ( n ^ { 3 } )$ . Thus, the overall time complexity of one training iteration is $O ( n ( B T D +$ $B T n + n ^ { 2 } )$ ).
|
| 387 |
+
|
| 388 |
+
# E RESULTS FOR METR-LA
|
| 389 |
+
|
| 390 |
+
METR-LA contains speed records of 207 sensors deployed on the highways of Los Angles, CA (Li et al., 2018b). The records are in four months at the frequency of five minutes. We shift a one-hour window to obtain multiple time series. The first three months are used for training and the last month is split in halves for validation and testing. No anomaly labels exist however and we use this dataset for exploratory analysis only.
|
| 391 |
+
|
| 392 |
+
|
| 393 |
+
Figure 6: Density estimation for METR-LA.
|
| 394 |
+
|
| 395 |
+
Figure 6a shows the traffic speed on four main highways on June 13, 2012. Each speed is the average over all sensors on the same highway. We observe that despite spatial proximity, the speeds vary significantly around 4PM (rush hour) but they are unanimously high around 5AM and 8PM– 12AM. The estimated densities, shown in Figure 6b, tracks this pattern rather closely, with rush hours corresponding to low density and night traffics corresponding to high density. Note the nature of traffic: speed varies smoothly on the macroscopic level and hence does density, too. Such a phenomenon is in striking contrast to power systems where events are rare and abrupt.
|
| 396 |
+
|
| 397 |
+
# F ADDITIONAL ANOMALY DETECTION RESULTS ON PMU DATASETS
|
| 398 |
+
|
| 399 |
+
See Figure 7 and Figure 8 for additional anomaly detection results on the test sets of PMU-C and PMU-B, respectively. The observations are rather similar to those of Figure 3b.
|
| 400 |
+
|
| 401 |
+
|
| 402 |
+
Figure 7: Additional anomaly detection results on the test set of PMU-C.
|
| 403 |
+
|
| 404 |
+
|
| 405 |
+
Figure 8: Anomaly detection results on the test set of PMU-B.
|
| 406 |
+
|
| 407 |
+
# G ADDITIONAL RESULTS FOR DAG EVOLUTION
|
| 408 |
+
|
| 409 |
+
|
| 410 |
+
Figure 9: Evolution of the learned DAG on PMU-B over time.
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| 1 |
+
# AdaptFormer: Adapting Vision Transformers for Scalable Visual Recognition
|
| 2 |
+
|
| 3 |
+
Shoufa Chen1⇤ Chongjian Ge1⇤ Zhan Tong2 Jiangliu Wang2 Yibing Song2 Jue Wang2 Ping Luo1
|
| 4 |
+
|
| 5 |
+
1The University of Hong Kong 2Tencent AI Lab
|
| 6 |
+
|
| 7 |
+
# Abstract
|
| 8 |
+
|
| 9 |
+
Pretraining Vision Transformers (ViTs) has achieved great success in visual recognition. A following scenario is to adapt a ViT to various image and video recognition tasks. The adaptation is challenging because of heavy computation and memory storage. Each model needs an independent and complete finetuning process to adapt to different tasks, which limits its transferability to different visual domains. To address this challenge, we propose an effective adaptation approach for Transformer, namely AdaptFormer, which can adapt the pre-trained ViTs into many different image and video tasks efficiently. It possesses several benefits more appealing than prior arts. Firstly, AdaptFormer introduces lightweight modules that only add less than $2 \%$ extra parameters to a ViT, while it is able to increase the ViT’s transferability without updating its original pre-trained parameters, significantly outperforming the existing $100 \%$ fully fine-tuned models on action recognition benchmarks. Secondly, it can be plug-and-play in different Transformers and scalable to many visual tasks. Thirdly, extensive experiments on five image and video datasets show that AdaptFormer largely improves ViTs in the target domains. For example, when updating just $1 . 5 \%$ extra parameters, it achieves about $10 \%$ and $19 \%$ relative improvement compared to the fully fine-tuned models on Something-Something v2 and HMDB51, respectively. Code is available at https://github.com/ShoufaChen/AdaptFormer.
|
| 10 |
+
|
| 11 |
+
# 1 Introduction
|
| 12 |
+
|
| 13 |
+
There is a growing interest in adopting a general neural model to tackle a large variety of different tasks since it benefits in reducing the need for task-specific model design and training. Recently, Transformer [81] demonstrates great potential in this goal considering its success in various fields, e.g., natural language processing (NLP) [27, 10, 82, 88], visual recognition [31, 79, 90, 63], dense prediction [83, 11, 98, 96, 86], Generative Adversarial Network (GAN) [52, 48], reinforcement learning (RL) [18, 16, 87], robotics [50, 25], and etc. However, existing literature in computer vision tend to focus on the same network with task-specific weights scenario, where a single network is used to train from scratch or fully fine-tune on a specific dataset, making it infeasible to maintain a separate model weight for every dataset when the number of task grows, especially for the increasing model capacity of state-of-the-art models (e.g., ViT-G/14 [93] with over 1.8 billion parameters).
|
| 14 |
+
|
| 15 |
+
Different from prior arts, we step into the direction of developing same network with almost same weights and achieve superior performance than the full-tuning approach by only tuning less than $2 \%$ parameters, with the remaining over $98 \%$ parameters shared across different tasks. There are two challenges to learning universal representations using a single model. The first one lies in the pre-training stage, which requires algorithms that can learn well-generalized representations that are easy to be applied to many tasks. Recent arts in self-supervised learning [12, 5, 43, 97, 85, 78, 35] can serve as a solution to this challenge. The second one, which is our main concern in this work, is to build an effective pipeline that can adapt the model obtained at the pre-training stage to various downstream tasks by tuning parameters as less as possible and keeping the left parameters frozen.
|
| 16 |
+
|
| 17 |
+
While fine-tuning pre-trained models has been widely studied in NLP [6, 46, 69, 70, 58, 56, 47, 92, 62, 42], this topic is seldomly explored in the vision, where full-tuning of model parameters is still the dominant strategy for adapting vision transformers. However, the full fine-tuning cannot satisfy the goal of universal representation as it assigns an independent set of weights for every task. Linear probing is a straightforward approach to maintaining the pre-trained model fixed by only tuning a specific lightweight classification head for every task. However, linear probing tends to have an unsatisfactory performance and misses the opportunity of pursuing strong but non-linear features [43], which indeed benefit deep learning. More recently, Bahng et.al., [4] aimed to adapt pre-trained models by modifying raw input pixel space. Jia et.al., [51] proposed Visual Prompt Tuning (VPT) to adapt transformer models for downstream vision tasks, which prepends several learnable parameters (prompts) to the patch embeddings and freezes the whole pre-trained backbone.
|
| 18 |
+
|
| 19 |
+
In this work, we propose a lightweight module, namely AdaptFormer, to adapt vision transformers by updating the weights of AdaptFormer. We introduce learnable parameters from the model perspective, which is different from VPT, which inserts learnable parameters into the token space. Our AdaptFormer is conceptually simple yet effective. It consists of two fully connected layers, a non-linear activation function, and a scaling factor. This module is set in parallel to the feed-forward network (FFN) of the original ViT model, as shown in Figure 2b. This design is turned out to be effective for model transfer when processing scalable visual tokens for both image and video data (i.e., image data consists of a small scale of visual tokens while video data consists of a large scale). As shown in Figure 1, compared with the full-tuning strategy, AdaptFormer achieves comparable per
|
| 20 |
+
|
| 21 |
+
|
| 22 |
+
Figure 1: Parameter-Accuracy trade-off. We leverage ViT-Base as backbone and report top-1 accuracy on SSv2 dataset. AdaptFormer can surpass full-tuning with only $0 . 2 \%$ tunable parameters. More detailed results are shown in Table 1.
|
| 23 |
+
|
| 24 |
+
formance on video recognition with only about $0 . 1 \%$ tunable parameters. Meanwhile, with less than $2 \%$ tunable parameters, AdaptFormer surpasses the full-tuning solution by about $10 \%$ on top-1 accuracy. Similar approaches are also proposed in fine-tuning pre-trained language models (PLMs) [6, 46, 70, 42].
|
| 25 |
+
|
| 26 |
+
The key contributions of this paper are summarized as follows: (1) We propose a simple yet effective framework, namely AdaptFormer, for adapting vision transformers to a large variety of downstream visual recognition tasks and avoiding catastrophic interference with each other. To the best of our knowledge, this is the first work that explores efficient fine-tuning in video action recognition. (2) We ablate many design choices and demonstrate the superior robustness of AdaptFormer when parameters scale up. (3) Extensive experiments on various downstream tasks demonstrate that AdaptFormer outperforms existing fine-tuning approaches significantly. By demonstrating the effectiveness of AdaptFormer on multiple visual benchmarks, we hope our work could inspire the research communities to rethink the fine-tuning mechanism in computer vision and make progress toward a flexible yet universal Transformer model for visual recognition.
|
| 27 |
+
|
| 28 |
+
# 2 Related Works
|
| 29 |
+
|
| 30 |
+
In the proposed AdaptFormer, we mainly introduce a plug-and-play module for efficiently fine-tuning the current vision Transformer models. In this section, we perform a literature review on related works from two perspectives, i.e., the vision Transformers, and efficient transfer learning for vision Transformers.
|
| 31 |
+
|
| 32 |
+
# 2.1 Transformer in Vision
|
| 33 |
+
|
| 34 |
+
The Transformer architecture is first introduced in [81] and has re-energized the natural language processing (NLP) field from then on [27, 10]. Inspired by its huge success, researches in the computer vision filed have also evolved into Transformer era since ViTs [31]. The strong capability of modeling long-range relation has facilitated Transformer in various vision tasks, including image classification [31, 63, 60], object detection [11, 98, 22], semantic/instance segmentation [86], video understanding [8, 2, 33, 57], point cloud modeling [95, 41], 3D Object Recognition [20] and even low-level processing [17, 59, 84]. Furthermore, transformers have advanced the vision recognition performance by a large-scale pretraining [21, 67, 13, 36, 43, 78, 71]. In such a situation, given the pre-trained Transformer models, which are more larger than the previously prevalent CNN backbones, one open question is how to fine-tune the big vision models so that they can be adapted into more specific down-stream tasks. To solve the open question, we propose AdaptFormer to transfer ViTs from the pre-trained pre-texts into the target tasks in a more effective and efficient way.
|
| 35 |
+
|
| 36 |
+
# 2.2 Efficient Transfer learning for Transformers
|
| 37 |
+
|
| 38 |
+
Transfer learning targets re-adopting a pre-trained model (either via the supervised or the unsupervised manner) as the starting point and further fine-tuning the specific model on a new task. In the NLP field, transferring the large pre-trained language models (PLMs) [27, 10] into downstream tasks has been the popular paradigm for a long time. Conventional arts [27, 10] set all the network parameters as learnable ones and adapt them to the target tasks. However, with the growth of model sizes and the complexity of the specific tasks, the conventional paradigm is inevitably limited by the huge computational burden. The NLP community has explored several ways for parameter-efficient transfer learning that only set a few parameters learnable and fine-tune them for efficiency. The pioneer works could be mainly categorized from the token [58, 56] and network perspectives [46, 47, 92, 40]. Basically speaking, the token-related methods [56, 58] typically prepend several learnable prefix vectors/tokens to the projected tokens within the multi-head self-attention layers (MHSA [81]). The philosophy behind it is to assist the pre-trained models in understanding downstream tasks with the guidance of extra token information. On the other hand, network-related methods [46, 47] integrate shallow modules to improve the model transferability. The introduced modules adapt the produced representations into the downstream tasks via features fusion.
|
| 39 |
+
|
| 40 |
+
Recently, with the emergence of a much more large-scale dataset [26, 72, 74, 66, 53], increasing researchers in computer vision have adopted the homologous paradigm, i.e., first pre-training and then fine-tuning, to advance the vision tasks. As for the second stage, traditional methods typically adopt the full-tuning arts in the downstream tasks. Rare attention has been drawn to the field of efficient adaptation, especially in the field of vision Transformers. Inspired by Prompting in NLP, [51] introduced the learnable tokens in exploring the efficient adaptation for ViTs. We empirically found that the performance of prompting is hindered by the scale of tokens. That is to say, for the tasks where the number of tokens is on a small scale, e.g., image classification, Prompting is efficient for improving the model transferability. However, for larger scale tokens, e.g., video understanding, Prompting presents limited potential. This observation motivates us to introduce AdaptFormer, which is effective in the scenarios of scalable visual tokens.
|
| 41 |
+
|
| 42 |
+
# 3 Approach
|
| 43 |
+
|
| 44 |
+
We propose AdaptFormer for efficiently transferring large pre-trained vision transformer models to downstream tasks, in both image and video domains. AdaptFormer attains strong transfer learning abilities by only fine-tuning a small number of extra parameters, circumventing catastrophic interference among tasks. We illustrate the overall framework of AdaptFormer in Figure 2b.
|
| 45 |
+
|
| 46 |
+
# 3.1 Preliminary and Notation
|
| 47 |
+
|
| 48 |
+
Vision Transformers (ViTs) are first introduced by [31] into vision recognition. A vanilla vision Transformer basically consists of a patch embedding layer and several consecutively connected encoders, as depicted in Figure 2a. Given an image $\boldsymbol { x } ^ { \setminus } \in \dot { \mathbb { R } } ^ { H \times W \times 3 }$ , the patch embedding layer first splits and flatten the sample $x$ into sequential patches $x _ { p } \in \mathbb { R } ^ { N \times ( P ^ { 2 } d ) }$ , where $( H , W )$ represents the height and width of the input image, $( P , P )$ is the resolution of each image patch, $d$ denotes the output channel, and $N = H W / P ^ { 2 }$ is the number of image tokens. The overall combination of a prepended [CLS] token and the image tokens $x _ { p }$ are further fed into Transformer encoders for attention calculation.
|
| 49 |
+
|
| 50 |
+
|
| 51 |
+
Figure 2: Comparison of previous full and our AdaptFormer fine-tuning. AdaptFormer is conceptually simple by replacing the original MLP block with AdaptMLP, which consists of two branches, including the frozen branch (left) and the trainable down $ \mathtt { u p }$ bottleneck module (right).
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Each Transformer encoder mainly consists of two types of sub-layers, i.e., a multi-head self-attention layer (MHSA) and a MLP layer. In MHSA, the tokens are linearly projected and further re-formulated into three vectors, namely $Q , \pmb { K }$ and $V$ . The self-attention calculation is performed on $Q , \pmb { K }$ and $V$ by:
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$$
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x _ { \ell } ^ { \prime } = \mathrm { A t t e n t i o n } ( Q , K , V ) = \mathrm { S o f t m a x } ( \frac { Q K ^ { \top } } { \sqrt { d } } ) V ,
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$$
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where $ { \boldsymbol { { x } } } _ { \ell } ^ { \prime }$ are the tokens produced by MHSA at the $\ell$ -th layer. The output tokens $ { \boldsymbol { { x } } } _ { \ell } ^ { \prime }$ are further sent to a LayerNorm [3] and a MLP block which is consisted of two fully connected layers with a GELU activation [45] in between. This process is formally formulated as follows,
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$$
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x _ { \ell } = \mathrm { M L P } ( \mathrm { L N } ( x _ { \ell } ^ { \prime } ) ) + x _ { \ell } ^ { \prime } ,
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$$
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where $x _ { \ell }$ is the output of the $\ell$ -th encoder block. At the last transformer layer, the [CLS] is utilized for the final object recognition. We refer the readers to find more details in [31]. In our work, we replace the MLP layer with our AdaptMLP module for efficient fine-tuning purposes.
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# 3.2 AdaptFormer
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We propose a plug-and-play bottleneck module, namely AdaptMLP2. We denote the vision Transformer equipped with AdaptMLP as AdaptFormer.
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Architecture. The design principle of AdaptFormer is simple yet effective, which is illustrated in Figure 2b. Compared to the vanilla full fine-tuning regime, AdaptFormer replaces the MLP block in the transformer encoder with AdaptMLP, which is consisted of two sub-branches. The MLP layer in the left branch is identical to the original network, while the right branch is an additionally introduced lightweight module for task-specific fine-tuning. Specifically, the right branch is designed to be a bottleneck structure for limiting the number of parameters purpose, which includes a down-projection layer with parameters $W _ { \mathrm { d o w n } } \in \mathbb { R } ^ { d \times \hat { d } }$ , an up-projection layer with parameters $W _ { \mathrm { u p } } \in \mathbb { R } ^ { \hat { d } \times d }$ , where $\hat { d }$ is the bottleneck middle dimension and satisfies $\hat { d } \ll d$ . In addition, there is a ReLU layer [1] between these projection layers for non-linear property. This bottleneck module is connected to the original MLP network (left branch) through the residual connection via a scale factor $s$ . For a specific input feature $ { \boldsymbol { { x } } } _ { \ell } ^ { \prime }$ , the right branch in AdaptMLP produces the adapted features, $\tilde { x } _ { \ell }$ , formally via:
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$$
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\begin{array} { r } { \tilde { x } _ { \ell } = \mathrm { R e L U } ( \mathrm { L N } ( x _ { \ell } ^ { \prime } ) \cdot W _ { \mathrm { d o w n } } ) \cdot W _ { \mathrm { u p } } . } \end{array}
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$$
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Then both the features $\tilde { x } _ { \ell }$ and $ { \boldsymbol { { x } } } _ { \ell } ^ { \prime }$ are fused with $x _ { \ell }$ by residual connection,
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$$
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x _ { \ell } = \mathrm { M L P } ( \mathrm { L N } ( x _ { \ell } ^ { \prime } ) ) + s \cdot \tilde { x } _ { \ell } + x _ { \ell } ^ { \prime } .
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$$
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Fine-tuning. During the fine-tuning phase, we only choose the newly added parameters to optimize and keep rest ones fixed. Specifically, the original model parts (blue blocks in Figure 2b) load weights from the pre-trained checkpoint and keeps parameters frozen. The newly added parameters (orange blocks) are updated on the specific data domain with the task-specific losses.
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Inference. After fine-tuning, we still keep the shared parameters frozen as in the previous finetuning state, and additionally load the weights of the extra parameters that were fine-tuned in the previous stage. The single overall model is able to be adapted to multiple tasks with the assistance of lightweight introduced modules.
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# 3.3 Discussion
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Tunable parameters analysis. Our AdaptMLP module is lightweight. The total number of parameters introduced to per layer is $2 \times d \times \bar { \hat { d } } + \hat { d } + d$ , which includes biases parameters. The middle dimension $\hat { d }$ is a small value compared with $d$ (AdaptFormer still obtains a decent performance even when $\hat { d } = 1$ , as discussed in Sec. 4.5). Since most of the shared parameters are fixed and the number of newly introduced parameters is small $\leq 2 \%$ of the pre-trained model parameters), the total model size grows slowly when more downstream tasks are added.
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Applicability. We note that AdaptMLP is a plug-and-play module that can be adaptively inserted into existing popular vision transformer architectures [31, 63, 83, 90, 23, 29] since all of the backbones share the same MLP layers even though they differ in the MHSA architectures (as shown in Figure 2b). Compared to our methods, we notice that recent prompt-related approaches insert trainable parameters into the token space, as illustrated in Figure 3. They prepend learnable parameters either into the embedded tokens before linear projection [58] or the key and value tokens after linear projection [51]. Therefore, the prompt-related method can not be straightforwardly adapted to special MHSA variants, especially for the one that takes the pyramid spatial information into account [63, 83]. Besides, we empirically observe that prompt-related methods perform not well when the number of patch tokens grows up from image to video scale, as shown in Figure 1.
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Figure 3: Prompt tuning illustration.
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In summary, we present a strategy for tuning a pre-trained vision Transformer on a set of scalable vision recognition tasks (e.g.image domain and video domain). It adds limited learnable parameters for tuning while achieving comparable or even better performance than the full-tuning strategy. Moreover, AdaptFormer could serve as a generic module for a large variety of recognition tasks.
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Insights of architecture design. The MLP module is important for ViTs. As illustrated in [30], MLPs prevent ViTs from producing a rank-1 matrix. Also, MLPs stop the ViT output from degenerations. Inspired by the above analysis, we believe an effective ViT adaptation shall focus on its MLPs rather than multi-head self attentions. Meanwhile, we learn from the inception framework [76] that parallel design is an effective way for feature ensemble. With the parallel design, the domain-specific features produced by the adapter module can supplement the domain-agnostic features from the fixed branch for a better feature ensemble. Our following experiments will verify that the parallel performs better than the sequential design.
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Besides, though many advanced Transformer-based models [63, 83, 34, 90] which have emerged since the success of ViT having different attention mechanisms within the Transformer block, they all share the similar MLPs (feed-forward network) structures. Therefore, our AdaptMLP can be easily plugged into these ViT variants. Moreover, AdaptMLP can also be applied to more recent attention-free models [77, 61, 19].
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# 4 Experiments
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We evaluate the effectiveness of AdaptFormer by conducting extensive visual recognition experiments in both the image and video domains. We first describe our experimental settings in Sec. 4.1, covering the pre-trained backbones, baseline methods, downstream tasks and training details. We then compare AdaptFormer with baseline methods and provide a thorough analysis in Sec. 4.2. In addition, we also conduct ablation studies to explore different experimental configurations and explain what makes for the superiority of AdaptFormer in Sec 4.5.
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# 4.1 Experimental Settings
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Pre-trained backbone. We adopt the plain Vision Transformer (ViT) [31], i.e., ViT-Base (ViT-B/16) as our backbone model and pre-train the model with both supervised and self-supervised approaches. Specifically, for image, we directly use the ImageNet-21k [26] supervised pre-trained model3 and MAE [43] self-supervised model4. For video, we take both supervised and self-supervised pre-trained models from VideoMAE [78]. More details about pre-training approaches and datasets can be found in Appendix.
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Initialization of AdaptFormer. For the original networks, we directly load the weights pre-trained on the upstream tasks and keep them frozen/untouched during the fine-tuning process. For the newly added modules, the weights of down-projection layers are initialized with Kaiming Normal [44], while the biases of the additional networks and the weights of the up-projection layers are configured with zero initialization. The reason for the zero initialization of other layers is that in this way, the initial newly added parameters are initialized such that the new function resembles the original one at the start of the fine-tuning stage. We empirically found that if the initialization deviates too far from the identity function, the model is not stable to train.
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Baseline methods. We compare AdaptFormer with three commonly used fine-tuning approaches, including (1)Linear probing: adding an extra linear layer on top of the backbone and tuning the added parameters for evaluation. (2) Full Fine-tuning: setting all the parameters learnable and tuning them together. (3) Visual Prompt Tuning (VPT): [51] fine-tuning the extra token parameters as shown in Figure 3.
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Downstream tasks. We evaluate our AdaptFormer on both image and video recognition tasks to verify its effectiveness. The specific datasets leveraged in this work are presented in the following.
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Image domain : CIFAR-100 [54] contains 50,000 training images and 10,000 validation images of resolution $3 2 \times 3 2$ with 100 labels. Street View House Numbers (SVHN) [37] is a digit classification benchmark dataset. In total, the dataset comprises over 600,000 labeled images, containing 73,257 training samples, 26,032 testing samples and 531,131 extra training data. The Food-101 [9] dataset consists of 101 food categories with a total of 101k images, including 750 training and 250 testing samples per category.
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Video domain $:$ Something-Something V2 (SSv2) [39] is a large collection of video clips showing the people perform several normal actions in the daily life (e.g., moving stuff and opening the door). It consists of 168,913 training samples, 24,777 validation samples and 27,157 testing samples, making a total of 220,847 videos with 174 labels. HMDB51 [55] is composed of 6,849 videos with 51 categories, making a split of $3 . 5 \mathrm { k } / 1 . 5 \mathrm { k }$ train/val videos.
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Implementation details. In this work, we use PyTorch toolkit [68] to conduct all experiments on NVIDIA V100 GPUs. Unless otherwise stated, we use $8 \times 8$ GPUs for video experiments and $1 \times 8$ GPUs for image experiments. Our default configurations follow the linear probing settings in [21, 43], which do not utilize many common regularization strategies, such as mixup [94], cutmix [91], color jittering and so on. More details can be found in Appendix.
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# 4.2 Main Properties and Analysis
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We compare the performance of different fine-tuning approaches in Table 1 with the backbones pre-trained via the self-supervised paradigms. The results show that AdaptFormer consistently surpasses linear probing and Visual Prompt tuning (VPT) methods. Specifically, AdaptFormer64 outperforms VPT on image benchmark CIFAR-100, SVHN, and Food-101, by $3 . 4 6 \%$ , $2 . 8 7 \%$ , and $4 . 6 3 \%$ respectively. On the more challenging video action recognition dataset SomethingSomething V2, the superiority becomes even more significant, i.e., about $15 \%$ . Note that even compared with the full fine-tuning strategy, our AdaptFormer still outperforms by about $5 \%$ Top-1 accuracy on SSv2 dataset. To summarize, our AdaptFormer is highly parameter-efficient, as well as yielding good performance with parameter size at most $2 \%$ times than the full fine-tuning manner.
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Table 1: Fine-tuning with self-supervised pre-trained model. For tunable parameters, we also report the parameter percentage in the brackets. Besides, we report the top-1 accuracy on different dataset with the absolute value and the gap value relative to the full-tuning regime. † denotes $0 . 1 \times$ learning rate due to unstable training.
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<table><tr><td>Method</td><td>Avg. Params (M)</td><td>CIFAR-100</td><td>Image SVHN</td><td>Food-101</td><td>Video</td></tr><tr><td>Full-tuning</td><td>86.04 (100%)</td><td>85.90</td><td>97.67†</td><td>SSv2 90.09† 53.97</td><td>HMDB51 46.41</td></tr><tr><td>Linear</td><td>0.07 (0.08%)</td><td>69.83 (-16.07) 66.91 (-30.76) 69.74 (-20.35)</td><td></td><td></td><td>29.23 (-24.74) 49.84 (+3.43)</td></tr><tr><td>VPT [51]</td><td>0.08 (0.09%)</td><td>82.44 (-3.46)</td><td>94.02 (-3.65)</td><td>82.98 (-7.11)</td><td>43.73 (-10.24) 52.67 (+6.26)</td></tr><tr><td>AdaptFormer-1</td><td>0.10 (0.12%)</td><td>83.52 (-2.38)</td><td>93.04 (-4.63)</td><td>83.64 (-6.45)</td><td>50.03 (-3.94) 51.68 (+5.27)</td></tr><tr><td>AdaptFormer-4</td><td>0.15 (0.17%)</td><td>84.83 (-1.07)</td><td>96.19 (-1.48)</td><td>85.42 (-4.67)</td><td>54.70 (+0.73) 51.81 (+5.40)</td></tr><tr><td>AdaptFormer-64</td><td>1.26 (1.46%)</td><td>85.90 (0.00)</td><td>96.89 (-0.78)</td><td>87.61 (-2.48)</td><td>59.02 (+5.05) 55.69 (+9.28)</td></tr></table>
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Figure 4: The trend of performance as the number of tunable parameters grows up. The accuracy of VPT drops dramatically when the parameter number exceeds task-specific value, while AdaptFormer is robust to the increasing parameters.
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Figure 5: Test accuracy of VPT [51] with different number of introduced tokens. The optimization procedure becomes unstable when the token number is equal or larger than eight on HMDB51 dataset [55].
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# 4.3 Scaling Tunable Parameters Up
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Even though there are only limited parameters introduced, one might also argue that more tunable parameters of AdaptFormer contribute to its higher accuracy compared with VPT [51]. We conduct experiments to make a comprehensive discussion on this aspect.
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As described in Sec. 3.3, the number of tunable parameters can be adjusted by changing the number of introduced tokens for VPT, or the hidden feature dimension for AdaptFormer. As shown in Figure 4, we conduct experiments with a wide range of tunable parameters on both SSv2 and HMDB-51 datasets. Since AdaptFormer and VPT share the same number of parameters of classification head on a specific dataset, we only report the tunable parameters on the $\mathbf { X }$ -axis, which comes from the visual prompts (VPT) or weight/bias of the down-up fully-connected layers (AdaptFormer), without calculating the parameters of classification head. For VPT, the number of introduced tokens is chosen from {1, 2, 4, 8, 16, 32, 48, 64}. Similarly, the number of hidden dimensions in AdaptFormer is in {1, 2, 4, 8, 16, 32}. AdaptFormer has a slight performance gain or maintains the accuracy stably when the parameters scale up. On the contrary, the performance of VPT decreases dramatically when the parameters exceed the task-specific value. Moreover, choosing the most suitable number of token number becomes laborious since it might be task-specific (i.e.varying from one dataset to the other one). For example, the accuracy of VPT keeps going up when the number of tunable parameters increases up to 300K on SSv2, whereas it begins to drop when the number of tunable parameters exceeds 50K on HMDB-51.
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Table 2: AdaptFormer for multi-label classification.
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<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Params (M)</td><td rowspan=1 colspan=1>Params (M)NUS-WIDE [24]</td></tr><tr><td rowspan=1 colspan=1>Full-tuning</td><td rowspan=1 colspan=1>85.86 (100%)</td><td rowspan=1 colspan=1>61.26</td></tr><tr><td rowspan=1 colspan=1>LinearVPT [51]</td><td rowspan=1 colspan=1>0.06 (0.08%)0.07 (0.09%)</td><td rowspan=1 colspan=1>51.19 (-27.25)57.08 (-7.56)</td></tr><tr><td rowspan=1 colspan=1>AdaptFormer-1AdaptFormer-4AdaptFormer-64|1.25 (1.46%)</td><td rowspan=1 colspan=1>[0.09 (0.12%)0.15 (0.17%)AdaptFormer-64|1.25 (1.46%)</td><td rowspan=1 colspan=1>57.51 (-4.08)58.14 (-2.13)59.07 (-0.06)</td></tr></table>
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We further study the optimization procedures of VPT by monitoring the test accuracy of the training stage. As shown in Figure 5, we gradually increase the number of tokens in VPT and plot the Top-1 accuracy of each epoch. The training stages are stable when the number of tokens is less than or equal to 4, e.g., {1, 2, 4}. However, when the number becomes 8 or larger, e.g., {8, 16, 32}, the training procedure collapses at about the tenth epoch and achieves poor performance at the end of the training stage. On the contrary, the optimization procedures of AdaptFormer are stable when the number of parameters varies across a large range, as shown in Table 3a. The top-1 accuracy fluctuates within $1 . 5 \%$ when the number of parameters increases from 0.44M ${ \dot { \mathsf { d i m } } } { = } 1 6$ ) to 4.87M ${ \tt d i m } { = } 2 5 6$ ).
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# 4.4 Multi-Label Classification
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We further conduct experiments on dataset with larger scale and diversity. Specifically, we evaluate AdaptFormer on NUS-WIDE [24] for multi-label classification. NUS-WIDE contains 269,648 images collected from Flicker, which are annotated with 81 visual concepts. Since some images are not available on Flicker, we only use 220,000 images following [7, 32]. We utilize mean average precision (mAP) as performance metric.
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Settings and results. Our training settings mainly follow ASL [7]. Specifically, We trained all models for 40 epochs using Adam optimize and 1-cycle learning rate policy [73]. The maximal learning rate is 0.001. As shown in Table 2, though AdaptFormer-64 achieves a slightly lower mAP than fine-tuning, it significantly reduces the amount parameters that need to be updated (from 85.86 to 1.25M). Moreover, AdaptFormer has an clear advantage over other fine-tuning approaches including linear probing and VPT.
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# 4.5 Ablation Studies
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We ablate our AdaptFormer to study what properties make for a good AdaptFormer and observe several intriguing properties. The ablation studies conducted in this work are all performed on the SSv2 validation set [39].
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Table 3: AdaptFormer ablation experiments with ViT-B/16 on SSv2. We report the top-1 accuracy on the val set. Most suitable settings are marked in color .
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(a) Middle dimension $\hat { d }$ .
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<table><tr><td>mid dim</td><td>#params top-1</td></tr><tr><td>1 0.16M 0.44M</td><td>50.03</td></tr><tr><td>16 32 0.73M</td><td>57.62</td></tr><tr><td>64 1.32M</td><td>58.27 59.02</td></tr><tr><td>256 4.87M</td><td>58.87</td></tr></table>
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(b) AdaptMLP inserted layers and form.
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<table><tr><td>layers</td><td></td><td>form#params top-1</td></tr><tr><td>1→6</td><td></td><td>parallel0.7350.48</td></tr><tr><td></td><td></td><td>7→12parallel 0.73 57.99</td></tr><tr><td></td><td></td><td>1 →12parallel 1.32 59.02</td></tr><tr><td></td><td></td><td>1 -→12 sequential 1.32 58.17</td></tr></table>
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(c) Scaling factor s.
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<table><tr><td></td><td>factor top-1</td></tr><tr><td>0.01</td><td>53.44</td></tr><tr><td>0.05</td><td>58.85</td></tr><tr><td>0.10 59.02</td><td></td></tr><tr><td></td><td>0.2058.89</td></tr></table>
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Middle dimension. The middle dimension controls the number of introduced parameters by AdaptFormer. Lower middle dimensions introduce fewer parameters with a possible performance cost. We ablate AdaptFormer on the middle feature dimension to study this effects. As shown in Table 3a, the accuracy consistently improves when the middle dimension increases up to 64 and reaches the saturation point when the middle dimension is about 64 on SSv2 dataset. We note that our AdaptFormer can achieve a decent performance when the middle dimension reduces even to one, about $5 0 . 0 3 \%$ top-1 accuracy.
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We conduct more extensive ablation studies on middle dimension in Appendix Table 10 and found that the optimal middle dimension varies per dataset. For example, the accuracy reaches saturation when the middle dimension equals 64 on SSv2, whereas for NUS-WIDE dataset, the mAP slightly improves when the middle dimension increases from 64 to 512. However, AdaptFormer with middle dimension as 512 has $0 . 7 5 \mathrm { m A P }$ higher (59.82 vs. $5 9 . 0 7 \mathrm { m A P }$ ) than the one with 64 at the cost of about 8 times more parameters. Therefore, we choose the middle dimension $\mathtt { . 0 4 }$ for both SSv2 and NUS-WIDE for a better trade-off.
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Scaling factor. The scaling factor $s$ is introduced to balance the task-agnostic features (generated by the original frozen branch) and the task-specific features (generated by the tunable bottleneck branch). We evaluate AdaptFormer with multiple $s$ values and the results are summarized in Table 3c. Different from the scaling factor in NLP field which prefer $s$ larger than 1 (e.g., $s = 4$ in [42]), we empirically found that the $s$ should be $< 1$ for vision tasks, otherwise the fine-tuning would become unstable. Besides, we found that AdaptFormer achieves optimal performance with $s = 0 . 1$ . A larger or smaller $s$ would bring slight performance drop. Thus, we choose $s = 0 . 1 0$ as a default setting.
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AdaptFormer position. As shown in Table 3b, we further ablate on the specific position to introduce the AdaptMLP block. We gradually increase the number of AdaptMLP layers with a step of three (start end, both included). We observe that the performance of AdaptFormer has a positive correlation with the number of added layers. In addition, AdaptFormer prefers the top part (the one far away from the input image) of the network to the bottom part when introducing the same number of layers, e.g., AdaptFormer with $7 \to 1 2$ obtains over $1 4 . 5 \%$ higher accuracy than $1 6$ , though both equipped with six AdaptMLP layers.
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Insertion form. We study the insertion formulation by comparing the parallel and sequential instances which are illustrated in Figure 6. As shown in Table 3b, the parallel AdaptFormer is able to outperform the sequential one by $0 . 8 5 \%$ top-1 accuracy. The reason might be: (1) the parallel design maintains the original feature using an independent branch and aggregating updated context by element-wise scaled sum; (2) the sequential design is equivalent to adding more layers, which might cause optimization difficulty. Therefore, we adopt the parallel design as our default setting due to its superiority.
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Figure 6: Illustration of the parallel and sequential insertion form. Comparison results are shown in Table 3b.
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Figure 7: Performance with video frames number. AdaptFormer outperforms VPT and linear fine-tuning.
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Number of frames. The number of embedded patch tokens increases linearly with the number of video frames for the plain ViT [31]. We conduct experiments with the different number of frames, i.e., {2, 4, 8} and the results are shown in Figure 7. We observe that increasing the number of frames is beneficial for all these three fine-tuning methods. However, AdaptFormer consistently outperforms the linear manner (e.g., $+ 3 0 \%$ top-1 accuracy on 8 input frames) and VPT method(e.g., $+ 1 4 \%$ top-1 accuracy on 8 input frames).
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# 4.6 Towards Visual Recognition Generalist Agent
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In the above experiments, we typically utilize a modality-specific pre-trained checkpoint for the corresponding downstream tasks. For example, we use Kinetics-400 (video domain) pretrained model for downstream video action recognition on Something-Something V2 and HMDB51 benchmarks. Besides, we use ImageNet-21K (image domain) pre-rained model for downstream image classification on CIFAR-100, SVHN and Food-101 benchmarks. Our AdaptFormer achieves superior performances in this same network with modality-specific weights scenario.
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Next, we take a further step to ask what would happen if using the same network with the modality-agnostic weights for multiple tasks in the multi-modalities downstream tasks?
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We use the model pre-trained on ImagNet-21k to do action recognition on SSv2. As shown in Table 4, AdaptFormer is robust to domain shift caused by modality. The experimental results show that the linear probe approach obtains a very poor accuracy (i.e., $6 . 5 6 \%$ top-1 accuracy) when fine-tuning on SSv2. Meanwhile,
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Table 4: Fine-tuning on video data with image pre-trained model.
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<table><tr><td>Method</td><td>Avg. Params (M)</td><td>Fine-tuning SSv2</td></tr><tr><td>Full-tuning</td><td>86.36</td><td>41.50</td></tr><tr><td>Linear</td><td>0.15</td><td>6.56</td></tr><tr><td>VPT [51]</td><td>0.16</td><td>16.94</td></tr><tr><td>AdaptFormer</td><td>1.33</td><td>46.06</td></tr></table>
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VPT [51] achieves a better performance than linear probe but it is not decent (i.e., $1 6 . 9 4 \%$ top-1 accuracy). Our AdaptFormer, compared to the above two methods, attains a promising $4 6 . 0 6 \%$ top-1 accuracy, which is even higher than the full-tuning schedule $( + 4 . 5 6 \% )$ .
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# 4.7 Visualization
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Figure 8: t-SNE visualizations on SSv2 val dataset. We extract the final classification features from the top linear layer for t-SNE visualizations. The top-1 accuracy is reported in red, while the relative parameter (compared to the full fine-tuning strategy) is reported in blue.
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To evaluate the quality of the produced features, we conduct t-SNE [80] visualizations on AdaptFormer and other baseline methods. The features are extracted from the SSv2 validation set via the ViT-Base backbone. Figure 8 shows that the linear fine-tuning and the VPT methods tend to output mixed features as shown in Figure 8(a)-(b). Compared with the above two methods, the full fine-tuning strategy performs well in projecting features. However, it consumes huge computational sources to tune the whole network parameters. Figure 8(d) validates that our AdaptFormer facilitates ViT-Base in generating more separable representations with fewer learnable parameters.
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# 5 Conclusion
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We present a conceptually simple yet effective framework, AdaptFormer, for efficiently adapting a pre-trained Vision Transformer (ViT) backbone to scalable vision recognition tasks. By introducing AdaptMLP, our AdaptFormer is able to fine-tune the lightweight modules for producing features adapted to multiple downstream tasks. The extensive experiments on five datasets, covering both the image and the video domains, validate that our proposed methods are able to increase the ViT’s transferability with little computational cost. We hope our work will inspire future research in exploring more efficient fine-tuning methods for large vision models. One limitation is that AdaptFormer is only employed in recognition tasks in this work, it’s unclear whether it can work well in tasks beyond recognition, e.g., object detection and semantic segmentation. We leave it for the future exploration. Since our method is specially designed for efficient fine-tuning, we do not foresee obvious undesirable ethical/social impacts at this moment.
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Acknowledgment. This work is supported by CCF-Tencent Open Fund. Ping Luo is supported by the General Research Fund of HK No.27208720, No.17212120, and No.17200622.
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[98] Xizhou Zhu, Weijie Su, Lewei Lu, Bin Li, Xiaogang Wang, and Jifeng Dai. Deformable detr: Deformable transformers for end-to-end object detection. arXiv preprint arXiv:2010.04159, 2020. 1, 3
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parse/dev/ATiz_CDA66/ATiz_CDA66_content_list.json
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| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "AdaptFormer: Adapting Vision Transformers for Scalable Visual Recognition ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
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| 7 |
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200,
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| 8 |
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| 9 |
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| 10 |
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| 11 |
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],
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| 12 |
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"page_idx": 0
|
| 13 |
+
},
|
| 14 |
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{
|
| 15 |
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"type": "text",
|
| 16 |
+
"text": "Shoufa Chen1⇤ Chongjian Ge1⇤ Zhan Tong2 Jiangliu Wang2 Yibing Song2 Jue Wang2 Ping Luo1 ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
279,
|
| 19 |
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|
| 20 |
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| 21 |
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| 22 |
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],
|
| 23 |
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"page_idx": 0
|
| 24 |
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},
|
| 25 |
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{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "1The University of Hong Kong 2Tencent AI Lab ",
|
| 28 |
+
"bbox": [
|
| 29 |
+
336,
|
| 30 |
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|
| 31 |
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|
| 32 |
+
281
|
| 33 |
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],
|
| 34 |
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"page_idx": 0
|
| 35 |
+
},
|
| 36 |
+
{
|
| 37 |
+
"type": "text",
|
| 38 |
+
"text": "Abstract ",
|
| 39 |
+
"text_level": 1,
|
| 40 |
+
"bbox": [
|
| 41 |
+
462,
|
| 42 |
+
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|
| 43 |
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|
| 44 |
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334
|
| 45 |
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],
|
| 46 |
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"page_idx": 0
|
| 47 |
+
},
|
| 48 |
+
{
|
| 49 |
+
"type": "text",
|
| 50 |
+
"text": "Pretraining Vision Transformers (ViTs) has achieved great success in visual recognition. A following scenario is to adapt a ViT to various image and video recognition tasks. The adaptation is challenging because of heavy computation and memory storage. Each model needs an independent and complete finetuning process to adapt to different tasks, which limits its transferability to different visual domains. To address this challenge, we propose an effective adaptation approach for Transformer, namely AdaptFormer, which can adapt the pre-trained ViTs into many different image and video tasks efficiently. It possesses several benefits more appealing than prior arts. Firstly, AdaptFormer introduces lightweight modules that only add less than $2 \\%$ extra parameters to a ViT, while it is able to increase the ViT’s transferability without updating its original pre-trained parameters, significantly outperforming the existing $100 \\%$ fully fine-tuned models on action recognition benchmarks. Secondly, it can be plug-and-play in different Transformers and scalable to many visual tasks. Thirdly, extensive experiments on five image and video datasets show that AdaptFormer largely improves ViTs in the target domains. For example, when updating just $1 . 5 \\%$ extra parameters, it achieves about $10 \\%$ and $19 \\%$ relative improvement compared to the fully fine-tuned models on Something-Something v2 and HMDB51, respectively. Code is available at https://github.com/ShoufaChen/AdaptFormer. ",
|
| 51 |
+
"bbox": [
|
| 52 |
+
232,
|
| 53 |
+
348,
|
| 54 |
+
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|
| 55 |
+
609
|
| 56 |
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],
|
| 57 |
+
"page_idx": 0
|
| 58 |
+
},
|
| 59 |
+
{
|
| 60 |
+
"type": "text",
|
| 61 |
+
"text": "1 Introduction ",
|
| 62 |
+
"text_level": 1,
|
| 63 |
+
"bbox": [
|
| 64 |
+
174,
|
| 65 |
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|
| 66 |
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|
| 67 |
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|
| 68 |
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],
|
| 69 |
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"page_idx": 0
|
| 70 |
+
},
|
| 71 |
+
{
|
| 72 |
+
"type": "text",
|
| 73 |
+
"text": "There is a growing interest in adopting a general neural model to tackle a large variety of different tasks since it benefits in reducing the need for task-specific model design and training. Recently, Transformer [81] demonstrates great potential in this goal considering its success in various fields, e.g., natural language processing (NLP) [27, 10, 82, 88], visual recognition [31, 79, 90, 63], dense prediction [83, 11, 98, 96, 86], Generative Adversarial Network (GAN) [52, 48], reinforcement learning (RL) [18, 16, 87], robotics [50, 25], and etc. However, existing literature in computer vision tend to focus on the same network with task-specific weights scenario, where a single network is used to train from scratch or fully fine-tune on a specific dataset, making it infeasible to maintain a separate model weight for every dataset when the number of task grows, especially for the increasing model capacity of state-of-the-art models (e.g., ViT-G/14 [93] with over 1.8 billion parameters). ",
|
| 74 |
+
"bbox": [
|
| 75 |
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174,
|
| 76 |
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|
| 77 |
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|
| 78 |
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|
| 79 |
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],
|
| 80 |
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"page_idx": 0
|
| 81 |
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},
|
| 82 |
+
{
|
| 83 |
+
"type": "text",
|
| 84 |
+
"text": "Different from prior arts, we step into the direction of developing same network with almost same weights and achieve superior performance than the full-tuning approach by only tuning less than $2 \\%$ parameters, with the remaining over $98 \\%$ parameters shared across different tasks. There are two challenges to learning universal representations using a single model. The first one lies in the pre-training stage, which requires algorithms that can learn well-generalized representations that are easy to be applied to many tasks. Recent arts in self-supervised learning [12, 5, 43, 97, 85, 78, 35] can serve as a solution to this challenge. The second one, which is our main concern in this work, is to build an effective pipeline that can adapt the model obtained at the pre-training stage to various downstream tasks by tuning parameters as less as possible and keeping the left parameters frozen. ",
|
| 85 |
+
"bbox": [
|
| 86 |
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|
| 87 |
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|
| 88 |
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|
| 89 |
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|
| 90 |
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],
|
| 91 |
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"page_idx": 0
|
| 92 |
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},
|
| 93 |
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{
|
| 94 |
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"type": "text",
|
| 95 |
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"text": "",
|
| 96 |
+
"bbox": [
|
| 97 |
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174,
|
| 98 |
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|
| 99 |
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| 100 |
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|
| 101 |
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],
|
| 102 |
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"page_idx": 1
|
| 103 |
+
},
|
| 104 |
+
{
|
| 105 |
+
"type": "text",
|
| 106 |
+
"text": "While fine-tuning pre-trained models has been widely studied in NLP [6, 46, 69, 70, 58, 56, 47, 92, 62, 42], this topic is seldomly explored in the vision, where full-tuning of model parameters is still the dominant strategy for adapting vision transformers. However, the full fine-tuning cannot satisfy the goal of universal representation as it assigns an independent set of weights for every task. Linear probing is a straightforward approach to maintaining the pre-trained model fixed by only tuning a specific lightweight classification head for every task. However, linear probing tends to have an unsatisfactory performance and misses the opportunity of pursuing strong but non-linear features [43], which indeed benefit deep learning. More recently, Bahng et.al., [4] aimed to adapt pre-trained models by modifying raw input pixel space. Jia et.al., [51] proposed Visual Prompt Tuning (VPT) to adapt transformer models for downstream vision tasks, which prepends several learnable parameters (prompts) to the patch embeddings and freezes the whole pre-trained backbone. ",
|
| 107 |
+
"bbox": [
|
| 108 |
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173,
|
| 109 |
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154,
|
| 110 |
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825,
|
| 111 |
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|
| 112 |
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],
|
| 113 |
+
"page_idx": 1
|
| 114 |
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},
|
| 115 |
+
{
|
| 116 |
+
"type": "text",
|
| 117 |
+
"text": "In this work, we propose a lightweight module, namely AdaptFormer, to adapt vision transformers by updating the weights of AdaptFormer. We introduce learnable parameters from the model perspective, which is different from VPT, which inserts learnable parameters into the token space. Our AdaptFormer is conceptually simple yet effective. It consists of two fully connected layers, a non-linear activation function, and a scaling factor. This module is set in parallel to the feed-forward network (FFN) of the original ViT model, as shown in Figure 2b. This design is turned out to be effective for model transfer when processing scalable visual tokens for both image and video data (i.e., image data consists of a small scale of visual tokens while video data consists of a large scale). As shown in Figure 1, compared with the full-tuning strategy, AdaptFormer achieves comparable per",
|
| 118 |
+
"bbox": [
|
| 119 |
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176,
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| 120 |
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| 121 |
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| 122 |
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| 123 |
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|
| 124 |
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"page_idx": 1
|
| 125 |
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},
|
| 126 |
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{
|
| 127 |
+
"type": "image",
|
| 128 |
+
"img_path": "images/ca626c3a15ccaec295098cc6a518116950fa622c4a80643f2267e2ba99f7e87d.jpg",
|
| 129 |
+
"image_caption": [
|
| 130 |
+
"Figure 1: Parameter-Accuracy trade-off. We leverage ViT-Base as backbone and report top-1 accuracy on SSv2 dataset. AdaptFormer can surpass full-tuning with only $0 . 2 \\%$ tunable parameters. More detailed results are shown in Table 1. "
|
| 131 |
+
],
|
| 132 |
+
"image_footnote": [],
|
| 133 |
+
"bbox": [
|
| 134 |
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488,
|
| 135 |
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310,
|
| 136 |
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818,
|
| 137 |
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502
|
| 138 |
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],
|
| 139 |
+
"page_idx": 1
|
| 140 |
+
},
|
| 141 |
+
{
|
| 142 |
+
"type": "text",
|
| 143 |
+
"text": "formance on video recognition with only about $0 . 1 \\%$ tunable parameters. Meanwhile, with less than $2 \\%$ tunable parameters, AdaptFormer surpasses the full-tuning solution by about $10 \\%$ on top-1 accuracy. Similar approaches are also proposed in fine-tuning pre-trained language models (PLMs) [6, 46, 70, 42]. ",
|
| 144 |
+
"bbox": [
|
| 145 |
+
174,
|
| 146 |
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|
| 147 |
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|
| 148 |
+
643
|
| 149 |
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],
|
| 150 |
+
"page_idx": 1
|
| 151 |
+
},
|
| 152 |
+
{
|
| 153 |
+
"type": "text",
|
| 154 |
+
"text": "The key contributions of this paper are summarized as follows: (1) We propose a simple yet effective framework, namely AdaptFormer, for adapting vision transformers to a large variety of downstream visual recognition tasks and avoiding catastrophic interference with each other. To the best of our knowledge, this is the first work that explores efficient fine-tuning in video action recognition. (2) We ablate many design choices and demonstrate the superior robustness of AdaptFormer when parameters scale up. (3) Extensive experiments on various downstream tasks demonstrate that AdaptFormer outperforms existing fine-tuning approaches significantly. By demonstrating the effectiveness of AdaptFormer on multiple visual benchmarks, we hope our work could inspire the research communities to rethink the fine-tuning mechanism in computer vision and make progress toward a flexible yet universal Transformer model for visual recognition. ",
|
| 155 |
+
"bbox": [
|
| 156 |
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174,
|
| 157 |
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|
| 158 |
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|
| 159 |
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787
|
| 160 |
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],
|
| 161 |
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"page_idx": 1
|
| 162 |
+
},
|
| 163 |
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{
|
| 164 |
+
"type": "text",
|
| 165 |
+
"text": "2 Related Works ",
|
| 166 |
+
"text_level": 1,
|
| 167 |
+
"bbox": [
|
| 168 |
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174,
|
| 169 |
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| 170 |
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328,
|
| 171 |
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| 172 |
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],
|
| 173 |
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"page_idx": 1
|
| 174 |
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},
|
| 175 |
+
{
|
| 176 |
+
"type": "text",
|
| 177 |
+
"text": "In the proposed AdaptFormer, we mainly introduce a plug-and-play module for efficiently fine-tuning the current vision Transformer models. In this section, we perform a literature review on related works from two perspectives, i.e., the vision Transformers, and efficient transfer learning for vision Transformers. ",
|
| 178 |
+
"bbox": [
|
| 179 |
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174,
|
| 180 |
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|
| 181 |
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823,
|
| 182 |
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911
|
| 183 |
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],
|
| 184 |
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"page_idx": 1
|
| 185 |
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},
|
| 186 |
+
{
|
| 187 |
+
"type": "text",
|
| 188 |
+
"text": "2.1 Transformer in Vision ",
|
| 189 |
+
"text_level": 1,
|
| 190 |
+
"bbox": [
|
| 191 |
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174,
|
| 192 |
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90,
|
| 193 |
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367,
|
| 194 |
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106
|
| 195 |
+
],
|
| 196 |
+
"page_idx": 2
|
| 197 |
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},
|
| 198 |
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{
|
| 199 |
+
"type": "text",
|
| 200 |
+
"text": "The Transformer architecture is first introduced in [81] and has re-energized the natural language processing (NLP) field from then on [27, 10]. Inspired by its huge success, researches in the computer vision filed have also evolved into Transformer era since ViTs [31]. The strong capability of modeling long-range relation has facilitated Transformer in various vision tasks, including image classification [31, 63, 60], object detection [11, 98, 22], semantic/instance segmentation [86], video understanding [8, 2, 33, 57], point cloud modeling [95, 41], 3D Object Recognition [20] and even low-level processing [17, 59, 84]. Furthermore, transformers have advanced the vision recognition performance by a large-scale pretraining [21, 67, 13, 36, 43, 78, 71]. In such a situation, given the pre-trained Transformer models, which are more larger than the previously prevalent CNN backbones, one open question is how to fine-tune the big vision models so that they can be adapted into more specific down-stream tasks. To solve the open question, we propose AdaptFormer to transfer ViTs from the pre-trained pre-texts into the target tasks in a more effective and efficient way. ",
|
| 201 |
+
"bbox": [
|
| 202 |
+
173,
|
| 203 |
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116,
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"type": "text",
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"text": "2.2 Efficient Transfer learning for Transformers ",
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"text_level": 1,
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"text": "Transfer learning targets re-adopting a pre-trained model (either via the supervised or the unsupervised manner) as the starting point and further fine-tuning the specific model on a new task. In the NLP field, transferring the large pre-trained language models (PLMs) [27, 10] into downstream tasks has been the popular paradigm for a long time. Conventional arts [27, 10] set all the network parameters as learnable ones and adapt them to the target tasks. However, with the growth of model sizes and the complexity of the specific tasks, the conventional paradigm is inevitably limited by the huge computational burden. The NLP community has explored several ways for parameter-efficient transfer learning that only set a few parameters learnable and fine-tune them for efficiency. The pioneer works could be mainly categorized from the token [58, 56] and network perspectives [46, 47, 92, 40]. Basically speaking, the token-related methods [56, 58] typically prepend several learnable prefix vectors/tokens to the projected tokens within the multi-head self-attention layers (MHSA [81]). The philosophy behind it is to assist the pre-trained models in understanding downstream tasks with the guidance of extra token information. On the other hand, network-related methods [46, 47] integrate shallow modules to improve the model transferability. The introduced modules adapt the produced representations into the downstream tasks via features fusion. ",
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"text": "Recently, with the emergence of a much more large-scale dataset [26, 72, 74, 66, 53], increasing researchers in computer vision have adopted the homologous paradigm, i.e., first pre-training and then fine-tuning, to advance the vision tasks. As for the second stage, traditional methods typically adopt the full-tuning arts in the downstream tasks. Rare attention has been drawn to the field of efficient adaptation, especially in the field of vision Transformers. Inspired by Prompting in NLP, [51] introduced the learnable tokens in exploring the efficient adaptation for ViTs. We empirically found that the performance of prompting is hindered by the scale of tokens. That is to say, for the tasks where the number of tokens is on a small scale, e.g., image classification, Prompting is efficient for improving the model transferability. However, for larger scale tokens, e.g., video understanding, Prompting presents limited potential. This observation motivates us to introduce AdaptFormer, which is effective in the scenarios of scalable visual tokens. ",
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"text": "3 Approach ",
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"text": "We propose AdaptFormer for efficiently transferring large pre-trained vision transformer models to downstream tasks, in both image and video domains. AdaptFormer attains strong transfer learning abilities by only fine-tuning a small number of extra parameters, circumventing catastrophic interference among tasks. We illustrate the overall framework of AdaptFormer in Figure 2b. ",
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"type": "text",
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"text": "3.1 Preliminary and Notation ",
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"text": "Vision Transformers (ViTs) are first introduced by [31] into vision recognition. A vanilla vision Transformer basically consists of a patch embedding layer and several consecutively connected encoders, as depicted in Figure 2a. Given an image $\\boldsymbol { x } ^ { \\setminus } \\in \\dot { \\mathbb { R } } ^ { H \\times W \\times 3 }$ , the patch embedding layer first splits and flatten the sample $x$ into sequential patches $x _ { p } \\in \\mathbb { R } ^ { N \\times ( P ^ { 2 } d ) }$ , where $( H , W )$ represents the height and width of the input image, $( P , P )$ is the resolution of each image patch, $d$ denotes the output channel, and $N = H W / P ^ { 2 }$ is the number of image tokens. The overall combination of a prepended [CLS] token and the image tokens $x _ { p }$ are further fed into Transformer encoders for attention calculation. ",
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| 290 |
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"type": "image",
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"img_path": "images/35696e7775ebb9b1080fc338067dc39cdacd8b97f6f1dda31d2bf3a66621ee84.jpg",
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"image_caption": [
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| 293 |
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"Figure 2: Comparison of previous full and our AdaptFormer fine-tuning. AdaptFormer is conceptually simple by replacing the original MLP block with AdaptMLP, which consists of two branches, including the frozen branch (left) and the trainable down $ \\mathtt { u p }$ bottleneck module (right). "
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"text": "",
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"text": "Each Transformer encoder mainly consists of two types of sub-layers, i.e., a multi-head self-attention layer (MHSA) and a MLP layer. In MHSA, the tokens are linearly projected and further re-formulated into three vectors, namely $Q , \\pmb { K }$ and $V$ . The self-attention calculation is performed on $Q , \\pmb { K }$ and $V$ by: ",
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"text": "$$\nx _ { \\ell } ^ { \\prime } = \\mathrm { A t t e n t i o n } ( Q , K , V ) = \\mathrm { S o f t m a x } ( \\frac { Q K ^ { \\top } } { \\sqrt { d } } ) V ,\n$$",
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"text": "where $ { \\boldsymbol { { x } } } _ { \\ell } ^ { \\prime }$ are the tokens produced by MHSA at the $\\ell$ -th layer. The output tokens $ { \\boldsymbol { { x } } } _ { \\ell } ^ { \\prime }$ are further sent to a LayerNorm [3] and a MLP block which is consisted of two fully connected layers with a GELU activation [45] in between. This process is formally formulated as follows, ",
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"text": "$$\nx _ { \\ell } = \\mathrm { M L P } ( \\mathrm { L N } ( x _ { \\ell } ^ { \\prime } ) ) + x _ { \\ell } ^ { \\prime } ,\n$$",
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"text": "where $x _ { \\ell }$ is the output of the $\\ell$ -th encoder block. At the last transformer layer, the [CLS] is utilized for the final object recognition. We refer the readers to find more details in [31]. In our work, we replace the MLP layer with our AdaptMLP module for efficient fine-tuning purposes. ",
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"text": "3.2 AdaptFormer ",
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"text_level": 1,
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"text": "We propose a plug-and-play bottleneck module, namely AdaptMLP2. We denote the vision Transformer equipped with AdaptMLP as AdaptFormer. ",
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"text": "Architecture. The design principle of AdaptFormer is simple yet effective, which is illustrated in Figure 2b. Compared to the vanilla full fine-tuning regime, AdaptFormer replaces the MLP block in the transformer encoder with AdaptMLP, which is consisted of two sub-branches. The MLP layer in the left branch is identical to the original network, while the right branch is an additionally introduced lightweight module for task-specific fine-tuning. Specifically, the right branch is designed to be a bottleneck structure for limiting the number of parameters purpose, which includes a down-projection layer with parameters $W _ { \\mathrm { d o w n } } \\in \\mathbb { R } ^ { d \\times \\hat { d } }$ , an up-projection layer with parameters $W _ { \\mathrm { u p } } \\in \\mathbb { R } ^ { \\hat { d } \\times d }$ , where $\\hat { d }$ is the bottleneck middle dimension and satisfies $\\hat { d } \\ll d$ . In addition, there is a ReLU layer [1] between these projection layers for non-linear property. This bottleneck module is connected to the original MLP network (left branch) through the residual connection via a scale factor $s$ . For a specific input feature $ { \\boldsymbol { { x } } } _ { \\ell } ^ { \\prime }$ , the right branch in AdaptMLP produces the adapted features, $\\tilde { x } _ { \\ell }$ , formally via: ",
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"text": "$$\n\\begin{array} { r } { \\tilde { x } _ { \\ell } = \\mathrm { R e L U } ( \\mathrm { L N } ( x _ { \\ell } ^ { \\prime } ) \\cdot W _ { \\mathrm { d o w n } } ) \\cdot W _ { \\mathrm { u p } } . } \\end{array}\n$$",
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"text": "Then both the features $\\tilde { x } _ { \\ell }$ and $ { \\boldsymbol { { x } } } _ { \\ell } ^ { \\prime }$ are fused with $x _ { \\ell }$ by residual connection, ",
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"text": "$$\nx _ { \\ell } = \\mathrm { M L P } ( \\mathrm { L N } ( x _ { \\ell } ^ { \\prime } ) ) + s \\cdot \\tilde { x } _ { \\ell } + x _ { \\ell } ^ { \\prime } .\n$$",
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"type": "text",
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"text": "Fine-tuning. During the fine-tuning phase, we only choose the newly added parameters to optimize and keep rest ones fixed. Specifically, the original model parts (blue blocks in Figure 2b) load weights from the pre-trained checkpoint and keeps parameters frozen. The newly added parameters (orange blocks) are updated on the specific data domain with the task-specific losses. ",
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"text": "Inference. After fine-tuning, we still keep the shared parameters frozen as in the previous finetuning state, and additionally load the weights of the extra parameters that were fine-tuned in the previous stage. The single overall model is able to be adapted to multiple tasks with the assistance of lightweight introduced modules. ",
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"text": "3.3 Discussion ",
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"text_level": 1,
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"type": "text",
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"text": "Tunable parameters analysis. Our AdaptMLP module is lightweight. The total number of parameters introduced to per layer is $2 \\times d \\times \\bar { \\hat { d } } + \\hat { d } + d$ , which includes biases parameters. The middle dimension $\\hat { d }$ is a small value compared with $d$ (AdaptFormer still obtains a decent performance even when $\\hat { d } = 1$ , as discussed in Sec. 4.5). Since most of the shared parameters are fixed and the number of newly introduced parameters is small $\\leq 2 \\%$ of the pre-trained model parameters), the total model size grows slowly when more downstream tasks are added. ",
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"text": "Applicability. We note that AdaptMLP is a plug-and-play module that can be adaptively inserted into existing popular vision transformer architectures [31, 63, 83, 90, 23, 29] since all of the backbones share the same MLP layers even though they differ in the MHSA architectures (as shown in Figure 2b). Compared to our methods, we notice that recent prompt-related approaches insert trainable parameters into the token space, as illustrated in Figure 3. They prepend learnable parameters either into the embedded tokens before linear projection [58] or the key and value tokens after linear projection [51]. Therefore, the prompt-related method can not be straightforwardly adapted to special MHSA variants, especially for the one that takes the pyramid spatial information into account [63, 83]. Besides, we empirically observe that prompt-related methods perform not well when the number of patch tokens grows up from image to video scale, as shown in Figure 1. ",
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"type": "image",
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"img_path": "images/5d467d01457a47b07687d63448d4b5c4d4f2e5c80914c213978fd93e62c70455.jpg",
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"image_caption": [
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| 505 |
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"Figure 3: Prompt tuning illustration. "
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"text": "In summary, we present a strategy for tuning a pre-trained vision Transformer on a set of scalable vision recognition tasks (e.g.image domain and video domain). It adds limited learnable parameters for tuning while achieving comparable or even better performance than the full-tuning strategy. Moreover, AdaptFormer could serve as a generic module for a large variety of recognition tasks. ",
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"text": "Insights of architecture design. The MLP module is important for ViTs. As illustrated in [30], MLPs prevent ViTs from producing a rank-1 matrix. Also, MLPs stop the ViT output from degenerations. Inspired by the above analysis, we believe an effective ViT adaptation shall focus on its MLPs rather than multi-head self attentions. Meanwhile, we learn from the inception framework [76] that parallel design is an effective way for feature ensemble. With the parallel design, the domain-specific features produced by the adapter module can supplement the domain-agnostic features from the fixed branch for a better feature ensemble. Our following experiments will verify that the parallel performs better than the sequential design. ",
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"page_idx": 4
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{
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"type": "text",
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"text": "Besides, though many advanced Transformer-based models [63, 83, 34, 90] which have emerged since the success of ViT having different attention mechanisms within the Transformer block, they all share the similar MLPs (feed-forward network) structures. Therefore, our AdaptMLP can be easily plugged into these ViT variants. Moreover, AdaptMLP can also be applied to more recent attention-free models [77, 61, 19]. ",
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"type": "text",
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"text": "4 Experiments ",
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"text_level": 1,
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"type": "text",
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"text": "We evaluate the effectiveness of AdaptFormer by conducting extensive visual recognition experiments in both the image and video domains. We first describe our experimental settings in Sec. 4.1, covering the pre-trained backbones, baseline methods, downstream tasks and training details. We then compare AdaptFormer with baseline methods and provide a thorough analysis in Sec. 4.2. In addition, we also conduct ablation studies to explore different experimental configurations and explain what makes for the superiority of AdaptFormer in Sec 4.5. ",
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"type": "text",
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"text": "4.1 Experimental Settings ",
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"type": "text",
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"text": "Pre-trained backbone. We adopt the plain Vision Transformer (ViT) [31], i.e., ViT-Base (ViT-B/16) as our backbone model and pre-train the model with both supervised and self-supervised approaches. Specifically, for image, we directly use the ImageNet-21k [26] supervised pre-trained model3 and MAE [43] self-supervised model4. For video, we take both supervised and self-supervised pre-trained models from VideoMAE [78]. More details about pre-training approaches and datasets can be found in Appendix. ",
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"type": "text",
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"text": "Initialization of AdaptFormer. For the original networks, we directly load the weights pre-trained on the upstream tasks and keep them frozen/untouched during the fine-tuning process. For the newly added modules, the weights of down-projection layers are initialized with Kaiming Normal [44], while the biases of the additional networks and the weights of the up-projection layers are configured with zero initialization. The reason for the zero initialization of other layers is that in this way, the initial newly added parameters are initialized such that the new function resembles the original one at the start of the fine-tuning stage. We empirically found that if the initialization deviates too far from the identity function, the model is not stable to train. ",
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"type": "text",
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"text": "Baseline methods. We compare AdaptFormer with three commonly used fine-tuning approaches, including (1)Linear probing: adding an extra linear layer on top of the backbone and tuning the added parameters for evaluation. (2) Full Fine-tuning: setting all the parameters learnable and tuning them together. (3) Visual Prompt Tuning (VPT): [51] fine-tuning the extra token parameters as shown in Figure 3. ",
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"type": "text",
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"text": "Downstream tasks. We evaluate our AdaptFormer on both image and video recognition tasks to verify its effectiveness. The specific datasets leveraged in this work are presented in the following. ",
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"text": "Image domain : CIFAR-100 [54] contains 50,000 training images and 10,000 validation images of resolution $3 2 \\times 3 2$ with 100 labels. Street View House Numbers (SVHN) [37] is a digit classification benchmark dataset. In total, the dataset comprises over 600,000 labeled images, containing 73,257 training samples, 26,032 testing samples and 531,131 extra training data. The Food-101 [9] dataset consists of 101 food categories with a total of 101k images, including 750 training and 250 testing samples per category. ",
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"text": "Video domain $:$ Something-Something V2 (SSv2) [39] is a large collection of video clips showing the people perform several normal actions in the daily life (e.g., moving stuff and opening the door). It consists of 168,913 training samples, 24,777 validation samples and 27,157 testing samples, making a total of 220,847 videos with 174 labels. HMDB51 [55] is composed of 6,849 videos with 51 categories, making a split of $3 . 5 \\mathrm { k } / 1 . 5 \\mathrm { k }$ train/val videos. ",
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"type": "text",
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"text": "Implementation details. In this work, we use PyTorch toolkit [68] to conduct all experiments on NVIDIA V100 GPUs. Unless otherwise stated, we use $8 \\times 8$ GPUs for video experiments and $1 \\times 8$ GPUs for image experiments. Our default configurations follow the linear probing settings in [21, 43], which do not utilize many common regularization strategies, such as mixup [94], cutmix [91], color jittering and so on. More details can be found in Appendix. ",
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"type": "text",
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"text": "4.2 Main Properties and Analysis ",
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"text_level": 1,
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"type": "text",
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"text": "We compare the performance of different fine-tuning approaches in Table 1 with the backbones pre-trained via the self-supervised paradigms. The results show that AdaptFormer consistently surpasses linear probing and Visual Prompt tuning (VPT) methods. Specifically, AdaptFormer64 outperforms VPT on image benchmark CIFAR-100, SVHN, and Food-101, by $3 . 4 6 \\%$ , $2 . 8 7 \\%$ , and $4 . 6 3 \\%$ respectively. On the more challenging video action recognition dataset SomethingSomething V2, the superiority becomes even more significant, i.e., about $15 \\%$ . Note that even compared with the full fine-tuning strategy, our AdaptFormer still outperforms by about $5 \\%$ Top-1 accuracy on SSv2 dataset. To summarize, our AdaptFormer is highly parameter-efficient, as well as yielding good performance with parameter size at most $2 \\%$ times than the full fine-tuning manner. ",
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{
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"type": "table",
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"img_path": "images/54434713c21d46b457864a708c09af61261a503bd94d55b692ef12b2d3710859.jpg",
|
| 687 |
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"table_caption": [
|
| 688 |
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"Table 1: Fine-tuning with self-supervised pre-trained model. For tunable parameters, we also report the parameter percentage in the brackets. Besides, we report the top-1 accuracy on different dataset with the absolute value and the gap value relative to the full-tuning regime. † denotes $0 . 1 \\times$ learning rate due to unstable training. "
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"table_footnote": [],
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"table_body": "<table><tr><td>Method</td><td>Avg. Params (M)</td><td>CIFAR-100</td><td>Image SVHN</td><td>Food-101</td><td>Video</td></tr><tr><td>Full-tuning</td><td>86.04 (100%)</td><td>85.90</td><td>97.67†</td><td>SSv2 90.09† 53.97</td><td>HMDB51 46.41</td></tr><tr><td>Linear</td><td>0.07 (0.08%)</td><td>69.83 (-16.07) 66.91 (-30.76) 69.74 (-20.35)</td><td></td><td></td><td>29.23 (-24.74) 49.84 (+3.43)</td></tr><tr><td>VPT [51]</td><td>0.08 (0.09%)</td><td>82.44 (-3.46)</td><td>94.02 (-3.65)</td><td>82.98 (-7.11)</td><td>43.73 (-10.24) 52.67 (+6.26)</td></tr><tr><td>AdaptFormer-1</td><td>0.10 (0.12%)</td><td>83.52 (-2.38)</td><td>93.04 (-4.63)</td><td>83.64 (-6.45)</td><td>50.03 (-3.94) 51.68 (+5.27)</td></tr><tr><td>AdaptFormer-4</td><td>0.15 (0.17%)</td><td>84.83 (-1.07)</td><td>96.19 (-1.48)</td><td>85.42 (-4.67)</td><td>54.70 (+0.73) 51.81 (+5.40)</td></tr><tr><td>AdaptFormer-64</td><td>1.26 (1.46%)</td><td>85.90 (0.00)</td><td>96.89 (-0.78)</td><td>87.61 (-2.48)</td><td>59.02 (+5.05) 55.69 (+9.28)</td></tr></table>",
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"page_idx": 6
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},
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{
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"type": "image",
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"img_path": "images/3cb13ac156a686a5d9e98125e1cc7f8c760135e6af0a6126c8bd946c3c656392.jpg",
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| 703 |
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"image_caption": [
|
| 704 |
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"Figure 4: The trend of performance as the number of tunable parameters grows up. The accuracy of VPT drops dramatically when the parameter number exceeds task-specific value, while AdaptFormer is robust to the increasing parameters. "
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],
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"image_footnote": [],
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"bbox": [
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"page_idx": 6
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"type": "image",
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"img_path": "images/f544c19bbc7ea39939adee815bd13258b47377a2e2fd9467e464e347116b55d4.jpg",
|
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"image_caption": [
|
| 719 |
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"Figure 5: Test accuracy of VPT [51] with different number of introduced tokens. The optimization procedure becomes unstable when the token number is equal or larger than eight on HMDB51 dataset [55]. "
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"text": "",
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"type": "text",
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"text": "4.3 Scaling Tunable Parameters Up ",
|
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"text_level": 1,
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"type": "text",
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"text": "Even though there are only limited parameters introduced, one might also argue that more tunable parameters of AdaptFormer contribute to its higher accuracy compared with VPT [51]. We conduct experiments to make a comprehensive discussion on this aspect. ",
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"type": "text",
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"text": "As described in Sec. 3.3, the number of tunable parameters can be adjusted by changing the number of introduced tokens for VPT, or the hidden feature dimension for AdaptFormer. As shown in Figure 4, we conduct experiments with a wide range of tunable parameters on both SSv2 and HMDB-51 datasets. Since AdaptFormer and VPT share the same number of parameters of classification head on a specific dataset, we only report the tunable parameters on the $\\mathbf { X }$ -axis, which comes from the visual prompts (VPT) or weight/bias of the down-up fully-connected layers (AdaptFormer), without calculating the parameters of classification head. For VPT, the number of introduced tokens is chosen from {1, 2, 4, 8, 16, 32, 48, 64}. Similarly, the number of hidden dimensions in AdaptFormer is in {1, 2, 4, 8, 16, 32}. AdaptFormer has a slight performance gain or maintains the accuracy stably when the parameters scale up. On the contrary, the performance of VPT decreases dramatically when the parameters exceed the task-specific value. Moreover, choosing the most suitable number of token number becomes laborious since it might be task-specific (i.e.varying from one dataset to the other one). For example, the accuracy of VPT keeps going up when the number of tunable parameters increases up to 300K on SSv2, whereas it begins to drop when the number of tunable parameters exceeds 50K on HMDB-51. ",
|
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"page_idx": 6
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},
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{
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"type": "table",
|
| 777 |
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"img_path": "images/cfb8e3b50c92b2be6bd0a5e0fc4c84920e16aefca5e0d986f97e4c22bd32410d.jpg",
|
| 778 |
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"table_caption": [
|
| 779 |
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"Table 2: AdaptFormer for multi-label classification. "
|
| 780 |
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],
|
| 781 |
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"table_footnote": [],
|
| 782 |
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"table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Params (M)</td><td rowspan=1 colspan=1>Params (M)NUS-WIDE [24]</td></tr><tr><td rowspan=1 colspan=1>Full-tuning</td><td rowspan=1 colspan=1>85.86 (100%)</td><td rowspan=1 colspan=1>61.26</td></tr><tr><td rowspan=1 colspan=1>LinearVPT [51]</td><td rowspan=1 colspan=1>0.06 (0.08%)0.07 (0.09%)</td><td rowspan=1 colspan=1>51.19 (-27.25)57.08 (-7.56)</td></tr><tr><td rowspan=1 colspan=1>AdaptFormer-1AdaptFormer-4AdaptFormer-64|1.25 (1.46%)</td><td rowspan=1 colspan=1>[0.09 (0.12%)0.15 (0.17%)AdaptFormer-64|1.25 (1.46%)</td><td rowspan=1 colspan=1>57.51 (-4.08)58.14 (-2.13)59.07 (-0.06)</td></tr></table>",
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"type": "text",
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"text": "",
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{
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"type": "text",
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"text": "We further study the optimization procedures of VPT by monitoring the test accuracy of the training stage. As shown in Figure 5, we gradually increase the number of tokens in VPT and plot the Top-1 accuracy of each epoch. The training stages are stable when the number of tokens is less than or equal to 4, e.g., {1, 2, 4}. However, when the number becomes 8 or larger, e.g., {8, 16, 32}, the training procedure collapses at about the tenth epoch and achieves poor performance at the end of the training stage. On the contrary, the optimization procedures of AdaptFormer are stable when the number of parameters varies across a large range, as shown in Table 3a. The top-1 accuracy fluctuates within $1 . 5 \\%$ when the number of parameters increases from 0.44M ${ \\dot { \\mathsf { d i m } } } { = } 1 6$ ) to 4.87M ${ \\tt d i m } { = } 2 5 6$ ). ",
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},
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"type": "text",
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"text": "4.4 Multi-Label Classification ",
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"text_level": 1,
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"type": "text",
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"text": "We further conduct experiments on dataset with larger scale and diversity. Specifically, we evaluate AdaptFormer on NUS-WIDE [24] for multi-label classification. NUS-WIDE contains 269,648 images collected from Flicker, which are annotated with 81 visual concepts. Since some images are not available on Flicker, we only use 220,000 images following [7, 32]. We utilize mean average precision (mAP) as performance metric. ",
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"type": "text",
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| 838 |
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"text": "Settings and results. Our training settings mainly follow ASL [7]. Specifically, We trained all models for 40 epochs using Adam optimize and 1-cycle learning rate policy [73]. The maximal learning rate is 0.001. As shown in Table 2, though AdaptFormer-64 achieves a slightly lower mAP than fine-tuning, it significantly reduces the amount parameters that need to be updated (from 85.86 to 1.25M). Moreover, AdaptFormer has an clear advantage over other fine-tuning approaches including linear probing and VPT. ",
|
| 839 |
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"bbox": [
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},
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| 847 |
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{
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| 848 |
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"type": "text",
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| 849 |
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"text": "4.5 Ablation Studies ",
|
| 850 |
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"text_level": 1,
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| 851 |
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"bbox": [
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| 859 |
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{
|
| 860 |
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"type": "text",
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| 861 |
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"text": "We ablate our AdaptFormer to study what properties make for a good AdaptFormer and observe several intriguing properties. The ablation studies conducted in this work are all performed on the SSv2 validation set [39]. ",
|
| 862 |
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"bbox": [
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| 871 |
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"type": "text",
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| 872 |
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"text": "Table 3: AdaptFormer ablation experiments with ViT-B/16 on SSv2. We report the top-1 accuracy on the val set. Most suitable settings are marked in color . ",
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"bbox": [
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"type": "table",
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"img_path": "images/086345cb4593296be15545fdbc89aab9e35ccea0de188155e28048d2f6e9d7d9.jpg",
|
| 884 |
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"table_caption": [
|
| 885 |
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"(a) Middle dimension $\\hat { d }$ . "
|
| 886 |
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],
|
| 887 |
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"table_footnote": [],
|
| 888 |
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"table_body": "<table><tr><td>mid dim</td><td>#params top-1</td></tr><tr><td>1 0.16M 0.44M</td><td>50.03</td></tr><tr><td>16 32 0.73M</td><td>57.62</td></tr><tr><td>64 1.32M</td><td>58.27 59.02</td></tr><tr><td>256 4.87M</td><td>58.87</td></tr></table>",
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"type": "table",
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"img_path": "images/e74df1b576918c08df54e8bf2cffbfca925d05f250754b128382c940184ccb56.jpg",
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| 900 |
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"table_caption": [
|
| 901 |
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"(b) AdaptMLP inserted layers and form. "
|
| 902 |
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],
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"table_footnote": [],
|
| 904 |
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"table_body": "<table><tr><td>layers</td><td></td><td>form#params top-1</td></tr><tr><td>1→6</td><td></td><td>parallel0.7350.48</td></tr><tr><td></td><td></td><td>7→12parallel 0.73 57.99</td></tr><tr><td></td><td></td><td>1 →12parallel 1.32 59.02</td></tr><tr><td></td><td></td><td>1 -→12 sequential 1.32 58.17</td></tr></table>",
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"type": "table",
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"img_path": "images/1ceab813e3ad9c143f5df23f39d81bdaeaf33364f2ede1803abf4ac9e52f1282.jpg",
|
| 916 |
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"table_caption": [
|
| 917 |
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"(c) Scaling factor s. "
|
| 918 |
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],
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| 919 |
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"table_footnote": [],
|
| 920 |
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"table_body": "<table><tr><td></td><td>factor top-1</td></tr><tr><td>0.01</td><td>53.44</td></tr><tr><td>0.05</td><td>58.85</td></tr><tr><td>0.10 59.02</td><td></td></tr><tr><td></td><td>0.2058.89</td></tr></table>",
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"bbox": [
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{
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| 930 |
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"type": "text",
|
| 931 |
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"text": "Middle dimension. The middle dimension controls the number of introduced parameters by AdaptFormer. Lower middle dimensions introduce fewer parameters with a possible performance cost. We ablate AdaptFormer on the middle feature dimension to study this effects. As shown in Table 3a, the accuracy consistently improves when the middle dimension increases up to 64 and reaches the saturation point when the middle dimension is about 64 on SSv2 dataset. We note that our AdaptFormer can achieve a decent performance when the middle dimension reduces even to one, about $5 0 . 0 3 \\%$ top-1 accuracy. ",
|
| 932 |
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"bbox": [
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"page_idx": 8
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{
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| 941 |
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"type": "text",
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| 942 |
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"text": "We conduct more extensive ablation studies on middle dimension in Appendix Table 10 and found that the optimal middle dimension varies per dataset. For example, the accuracy reaches saturation when the middle dimension equals 64 on SSv2, whereas for NUS-WIDE dataset, the mAP slightly improves when the middle dimension increases from 64 to 512. However, AdaptFormer with middle dimension as 512 has $0 . 7 5 \\mathrm { m A P }$ higher (59.82 vs. $5 9 . 0 7 \\mathrm { m A P }$ ) than the one with 64 at the cost of about 8 times more parameters. Therefore, we choose the middle dimension $\\mathtt { . 0 4 }$ for both SSv2 and NUS-WIDE for a better trade-off. ",
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| 943 |
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"bbox": [
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"page_idx": 8
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| 950 |
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},
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| 951 |
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{
|
| 952 |
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"type": "text",
|
| 953 |
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"text": "Scaling factor. The scaling factor $s$ is introduced to balance the task-agnostic features (generated by the original frozen branch) and the task-specific features (generated by the tunable bottleneck branch). We evaluate AdaptFormer with multiple $s$ values and the results are summarized in Table 3c. Different from the scaling factor in NLP field which prefer $s$ larger than 1 (e.g., $s = 4$ in [42]), we empirically found that the $s$ should be $< 1$ for vision tasks, otherwise the fine-tuning would become unstable. Besides, we found that AdaptFormer achieves optimal performance with $s = 0 . 1$ . A larger or smaller $s$ would bring slight performance drop. Thus, we choose $s = 0 . 1 0$ as a default setting. ",
|
| 954 |
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"bbox": [
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"page_idx": 8
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},
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| 962 |
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{
|
| 963 |
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"type": "text",
|
| 964 |
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"text": "AdaptFormer position. As shown in Table 3b, we further ablate on the specific position to introduce the AdaptMLP block. We gradually increase the number of AdaptMLP layers with a step of three (start end, both included). We observe that the performance of AdaptFormer has a positive correlation with the number of added layers. In addition, AdaptFormer prefers the top part (the one far away from the input image) of the network to the bottom part when introducing the same number of layers, e.g., AdaptFormer with $7 \\to 1 2$ obtains over $1 4 . 5 \\%$ higher accuracy than $1 6$ , though both equipped with six AdaptMLP layers. ",
|
| 965 |
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"bbox": [
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"page_idx": 8
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| 972 |
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},
|
| 973 |
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{
|
| 974 |
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"type": "text",
|
| 975 |
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"text": "Insertion form. We study the insertion formulation by comparing the parallel and sequential instances which are illustrated in Figure 6. As shown in Table 3b, the parallel AdaptFormer is able to outperform the sequential one by $0 . 8 5 \\%$ top-1 accuracy. The reason might be: (1) the parallel design maintains the original feature using an independent branch and aggregating updated context by element-wise scaled sum; (2) the sequential design is equivalent to adding more layers, which might cause optimization difficulty. Therefore, we adopt the parallel design as our default setting due to its superiority. ",
|
| 976 |
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"bbox": [
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"page_idx": 8
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},
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| 984 |
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{
|
| 985 |
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"type": "image",
|
| 986 |
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"img_path": "images/a58367ee574fdfe5511fdf7062e4a35b07211ff0569ec9b84be80a728803498e.jpg",
|
| 987 |
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"image_caption": [
|
| 988 |
+
"Figure 6: Illustration of the parallel and sequential insertion form. Comparison results are shown in Table 3b. "
|
| 989 |
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],
|
| 990 |
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"image_footnote": [],
|
| 991 |
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"bbox": [
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"page_idx": 8
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| 998 |
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},
|
| 999 |
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{
|
| 1000 |
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"type": "image",
|
| 1001 |
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"img_path": "images/e07b57e7624ceb177dbf06ac91d87e803bfff5dfe82d5d34bd90dde77120dfce.jpg",
|
| 1002 |
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"image_caption": [
|
| 1003 |
+
"Figure 7: Performance with video frames number. AdaptFormer outperforms VPT and linear fine-tuning. "
|
| 1004 |
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],
|
| 1005 |
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"image_footnote": [],
|
| 1006 |
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"bbox": [
|
| 1007 |
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560,
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| 1008 |
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622,
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| 1009 |
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762
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| 1012 |
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"page_idx": 8
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| 1013 |
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},
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| 1014 |
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{
|
| 1015 |
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"type": "text",
|
| 1016 |
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"text": "Number of frames. The number of embedded patch tokens increases linearly with the number of video frames for the plain ViT [31]. We conduct experiments with the different number of frames, i.e., {2, 4, 8} and the results are shown in Figure 7. We observe that increasing the number of frames is beneficial for all these three fine-tuning methods. However, AdaptFormer consistently outperforms the linear manner (e.g., $+ 3 0 \\%$ top-1 accuracy on 8 input frames) and VPT method(e.g., $+ 1 4 \\%$ top-1 accuracy on 8 input frames). ",
|
| 1017 |
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"bbox": [
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|
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"page_idx": 8
|
| 1024 |
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},
|
| 1025 |
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{
|
| 1026 |
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"type": "text",
|
| 1027 |
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"text": "4.6 Towards Visual Recognition Generalist Agent ",
|
| 1028 |
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"text_level": 1,
|
| 1029 |
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"bbox": [
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176,
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|
| 1037 |
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{
|
| 1038 |
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"type": "text",
|
| 1039 |
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"text": "In the above experiments, we typically utilize a modality-specific pre-trained checkpoint for the corresponding downstream tasks. For example, we use Kinetics-400 (video domain) pretrained model for downstream video action recognition on Something-Something V2 and HMDB51 benchmarks. Besides, we use ImageNet-21K (image domain) pre-rained model for downstream image classification on CIFAR-100, SVHN and Food-101 benchmarks. Our AdaptFormer achieves superior performances in this same network with modality-specific weights scenario. ",
|
| 1040 |
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"bbox": [
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| 1046 |
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"page_idx": 9
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| 1047 |
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},
|
| 1048 |
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{
|
| 1049 |
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"type": "text",
|
| 1050 |
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"text": "Next, we take a further step to ask what would happen if using the same network with the modality-agnostic weights for multiple tasks in the multi-modalities downstream tasks? ",
|
| 1051 |
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"bbox": [
|
| 1052 |
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"page_idx": 9
|
| 1058 |
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},
|
| 1059 |
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{
|
| 1060 |
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"type": "text",
|
| 1061 |
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"text": "We use the model pre-trained on ImagNet-21k to do action recognition on SSv2. As shown in Table 4, AdaptFormer is robust to domain shift caused by modality. The experimental results show that the linear probe approach obtains a very poor accuracy (i.e., $6 . 5 6 \\%$ top-1 accuracy) when fine-tuning on SSv2. Meanwhile, ",
|
| 1062 |
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"bbox": [
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| 1064 |
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251,
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| 1065 |
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485,
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| 1066 |
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348
|
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|
| 1068 |
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"page_idx": 9
|
| 1069 |
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},
|
| 1070 |
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{
|
| 1071 |
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"type": "table",
|
| 1072 |
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"img_path": "images/4d13637c3083022a8aef1474debf633af80768b2fa5f99c9f5912095aff35eec.jpg",
|
| 1073 |
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"table_caption": [
|
| 1074 |
+
"Table 4: Fine-tuning on video data with image pre-trained model. "
|
| 1075 |
+
],
|
| 1076 |
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"table_footnote": [],
|
| 1077 |
+
"table_body": "<table><tr><td>Method</td><td>Avg. Params (M)</td><td>Fine-tuning SSv2</td></tr><tr><td>Full-tuning</td><td>86.36</td><td>41.50</td></tr><tr><td>Linear</td><td>0.15</td><td>6.56</td></tr><tr><td>VPT [51]</td><td>0.16</td><td>16.94</td></tr><tr><td>AdaptFormer</td><td>1.33</td><td>46.06</td></tr></table>",
|
| 1078 |
+
"bbox": [
|
| 1079 |
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501,
|
| 1080 |
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229,
|
| 1081 |
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816,
|
| 1082 |
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327
|
| 1083 |
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],
|
| 1084 |
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"page_idx": 9
|
| 1085 |
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},
|
| 1086 |
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{
|
| 1087 |
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"type": "text",
|
| 1088 |
+
"text": "VPT [51] achieves a better performance than linear probe but it is not decent (i.e., $1 6 . 9 4 \\%$ top-1 accuracy). Our AdaptFormer, compared to the above two methods, attains a promising $4 6 . 0 6 \\%$ top-1 accuracy, which is even higher than the full-tuning schedule $( + 4 . 5 6 \\% )$ . ",
|
| 1089 |
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"bbox": [
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| 1096 |
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},
|
| 1097 |
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{
|
| 1098 |
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"type": "text",
|
| 1099 |
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"text": "4.7 Visualization ",
|
| 1100 |
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"text_level": 1,
|
| 1101 |
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"bbox": [
|
| 1102 |
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174,
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396,
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| 1104 |
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303,
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"page_idx": 9
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| 1108 |
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},
|
| 1109 |
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{
|
| 1110 |
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"type": "image",
|
| 1111 |
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"img_path": "images/ae04de45868e6008c2c97233c852d9367b4293cc51f45cc1713eb0c549e39c2f.jpg",
|
| 1112 |
+
"image_caption": [
|
| 1113 |
+
"Figure 8: t-SNE visualizations on SSv2 val dataset. We extract the final classification features from the top linear layer for t-SNE visualizations. The top-1 accuracy is reported in red, while the relative parameter (compared to the full fine-tuning strategy) is reported in blue. "
|
| 1114 |
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],
|
| 1115 |
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"image_footnote": [],
|
| 1116 |
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"bbox": [
|
| 1117 |
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183,
|
| 1118 |
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415,
|
| 1119 |
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812,
|
| 1120 |
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536
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],
|
| 1122 |
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"page_idx": 9
|
| 1123 |
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},
|
| 1124 |
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{
|
| 1125 |
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"type": "text",
|
| 1126 |
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"text": "To evaluate the quality of the produced features, we conduct t-SNE [80] visualizations on AdaptFormer and other baseline methods. The features are extracted from the SSv2 validation set via the ViT-Base backbone. Figure 8 shows that the linear fine-tuning and the VPT methods tend to output mixed features as shown in Figure 8(a)-(b). Compared with the above two methods, the full fine-tuning strategy performs well in projecting features. However, it consumes huge computational sources to tune the whole network parameters. Figure 8(d) validates that our AdaptFormer facilitates ViT-Base in generating more separable representations with fewer learnable parameters. ",
|
| 1127 |
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"bbox": [
|
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| 1134 |
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},
|
| 1135 |
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{
|
| 1136 |
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"type": "text",
|
| 1137 |
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"text": "5 Conclusion ",
|
| 1138 |
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"text_level": 1,
|
| 1139 |
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"bbox": [
|
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},
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{
|
| 1148 |
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"type": "text",
|
| 1149 |
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"text": "We present a conceptually simple yet effective framework, AdaptFormer, for efficiently adapting a pre-trained Vision Transformer (ViT) backbone to scalable vision recognition tasks. By introducing AdaptMLP, our AdaptFormer is able to fine-tune the lightweight modules for producing features adapted to multiple downstream tasks. The extensive experiments on five datasets, covering both the image and the video domains, validate that our proposed methods are able to increase the ViT’s transferability with little computational cost. We hope our work will inspire future research in exploring more efficient fine-tuning methods for large vision models. One limitation is that AdaptFormer is only employed in recognition tasks in this work, it’s unclear whether it can work well in tasks beyond recognition, e.g., object detection and semantic segmentation. We leave it for the future exploration. Since our method is specially designed for efficient fine-tuning, we do not foresee obvious undesirable ethical/social impacts at this moment. ",
|
| 1150 |
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"bbox": [
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"page_idx": 9
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},
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{
|
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"type": "text",
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"text": "Acknowledgment. This work is supported by CCF-Tencent Open Fund. Ping Luo is supported by the General Research Fund of HK No.27208720, No.17212120, and No.17200622. ",
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"text": "References \n[1] Abien Fred Agarap. Deep learning using rectified linear units (relu). arXiv preprint arXiv:1803.08375, 2018. 4 \n[2] Anurag Arnab, Mostafa Dehghani, Georg Heigold, Chen Sun, Mario Luciˇ c, and Cordelia Schmid. Vivit: A ´ video vision transformer. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 6836–6846, 2021. 3, 17 \n[3] Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016. 4 \n[4] Hyojin Bahng, Ali Jahanian, Swami Sankaranarayanan, and Phillip Isola. Visual prompting: Modifying pixel space to adapt pre-trained models. arXiv preprint arXiv:2203.17274, 2022. 2 \n[5] Hangbo Bao, Li Dong, and Furu Wei. Beit: Bert pre-training of image transformers. arXiv preprint arXiv:2106.08254, 2021. 2 \n[6] Ankur Bapna, Naveen Arivazhagan, and Orhan Firat. 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| 1 |
+
# EmbodiedGPT: Vision-Language Pre-Training via Embodied Chain of Thought
|
| 2 |
+
|
| 3 |
+
Yao ${ { \bf { M } } { \bf { u } } ^ { 1 } }$ , Qinglong Zhang2, Mengkang $\mathbf { H } \mathbf { u } ^ { 1 }$ , Wenhai $\mathbf { W a n g ^ { 3 } }$ , Mingyu Ding∗,1, $\mathbf { J u n \mathbf { \mathbf { J i n } } ^ { 4 } }$ , Bin Wang4, Jifeng Dai2,5, Yu Qiao2, Ping Luo∗,1,2 1The University of Hong Kong, 2Shanghai AI Laboratory, 3The Chinese University of Hong Kong, 4Noah’s Ark Laboratory 5Tsinghua University
|
| 4 |
+
|
| 5 |
+
# Abstract
|
| 6 |
+
|
| 7 |
+
Embodied AI is a crucial frontier in robotics, capable of planning and executing action sequences for robots to accomplish long-horizon tasks in physical environments. In this work, we introduce EmbodiedGPT, an end-to-end multi-modal foundation model for embodied AI, empowering embodied agents with multi-modal understanding and execution capabilities. To achieve this, we have made the following efforts: (i) We craft a large-scale embodied planning dataset, termed EgoCOT. The dataset consists of carefully selected videos from the Ego4D dataset, along with corresponding high-quality language instructions. Specifically, we generate a sequence of sub-goals with the "Chain of Thoughts" mode for effective embodied planning. (ii) We introduce an efficient training approach to EmbodiedGPT for high-quality plan generation, by adapting a 7B large language model (LLM) to the EgoCOT dataset via prefix tuning. (iii) We introduce a paradigm for extracting task-related features from LLM-generated planning queries to form a closed loop between high-level planning and low-level control. Extensive experiments show the effectiveness of EmbodiedGPT on embodied tasks, including embodied planning, embodied control, visual captioning, and visual question answering. Notably, EmbodiedGPT significantly enhances the success rate of the embodied control task by extracting more effective features. It has achieved a remarkable 1.6 times increase in success rate on the Franka Kitchen benchmark and a 1.3 times increase on the Meta-World benchmark, compared to the BLIP-2 baseline fine-tuned with the Ego4D dataset. More demos, code, and dataset information can be found at our homepage.
|
| 8 |
+
|
| 9 |
+
# 1 Introduction
|
| 10 |
+
|
| 11 |
+
Embodied AI tasks, e.g., embodied planning, embodied VQA, and embodied control, aim to imbue robots with the ability to perceive, reason, and act within their environment, enabling them to perform long-horizon plans and execute actions autonomously based on real-time observations. Recently, large language models (LLMs) such as GPT4 [1] and PaLM-E [2], have shown promising language understanding, reasoning, and "chain-of-thought" capabilities. Such advances may open new possibilities for developing robots capable of processing natural language instructions, performing multi-modal "chain-of-thought", and planning actions in physical environments.
|
| 12 |
+
|
| 13 |
+
Large-scale datasets play important roles in training large language models. For example, OpenCLIP trains its ViT-G/14 model on the LAION-2B dataset [3], which contains 2B image-language pairs. Unlike general-purpose visual language tasks that can get a huge amount of weakly labeled imagecaption pairs from the Internet, embodied AI tasks require egocentric data in robotics domains. Also,
|
| 14 |
+
|
| 15 |
+
Human: Describe the video in detail.
|
| 16 |
+
|
| 17 |
+
|
| 18 |
+
|
| 19 |
+
Human: Can you write a detailed plan for the task the robot is performing?
|
| 20 |
+
|
| 21 |
+
|
| 22 |
+
Figure 1: EmbodiedGPT’s capabilities for video captioning, multi-turn question answering, embodied planning, and low-level control. The plans given by EmbodiedGPT are highly executable and incorporate task-specific features, leading to a significant improvement in the success rate of embodied control tasks, outperforming both R3M [12] (a video-language contrastive learned model) and BLIP2 [13] (a multi-modal foundation model) on Franka Kitchen [14] and Meta-World [15] environments.
|
| 23 |
+
|
| 24 |
+
structured language instructions are needed for precise planning, which usually requires huge manual efforts and costs. This poses a challenging problem in collecting high-quality embodied multi-modal data. Some researchers [4, 5, 6, 7] explore creating large-scale embodied datasets with simulators, but a significant gap remains between simulation and the real world. Recent works [8, 9, 10] also explore adapting the pre-trained LLMs to a new domain by efficient tuning strategies like LoRA [11]. However, several open questions still remain: how to apply LLMs to the field of robotics which may face large domain gaps; how to leverage the "chain-of-thought" capability for structured planning; and how to use the output language plan for downstream manipulation tasks in an end-to-end manner.
|
| 25 |
+
|
| 26 |
+
To solve the above challenges, in this work, we first build a large-scale embodied planning dataset, termed EgoCOT, which features chain-of-thought planning instructions. It contains carefully selected egocentric videos from the Ego4D dataset [16] and corresponding high-quality step-by-step language instructions, which are machine-generated, then semantics-based filtered, and finally human-verified. Additionally, we also create the EgoVQA dataset as an extension of the Ego4D dataset, focusing on egocentric human-object interaction video question answering tasks, which aims to offer a wider range of egocentric multi-modal data.
|
| 27 |
+
|
| 28 |
+
Based on our EgoCOT and EgoVQA, we present an end-to-end multi-modal embodied foundation model called EmbodiedGPT, which can interact with the physical world in a more natural and intuitive manner, and perform many embodied tasks, as shown in Figure 1, such as embodied planning, embodied VQA, and embodied control. EmbodiedGPT comprises four integrated modules that work together, including i) a frozen vision model for encoding visual features of current observations, ii) a frozen language model used to execute natural language for question answering, captioning, and embodied planning tasks, iii) an embodied-former with a language mapping layer for aligning the visual and embodied instructions and extracting task-relevant instance-level features with the generated planning for low-level control, and iv) a policy network, which is responsible for producing low-level actions based on the task-relevant features, enabling the agent to effectively interact with the environment. To further enhance EmbodiedGPT ’s performance in generating reliable planning containing sub-goal sequences, we implement prefix tuning on the frozen language model to encourage the generation of more executable planning.
|
| 29 |
+
|
| 30 |
+
Our method possesses the following core advantages: i) the generated planning exhibits strong executability and granularity at the object part level, such as the gripper of a robotic arm or the handle of a door, manifested in sub-goal sequences. ii) the proposed EgoCOT dataset is built based on an open-source large-scale dataset, which offers greater scalability compared to the PaLM-E [2] model trained on proprietary robot data. And both the EgoCOT dataset, and the EmbodiedGPT model will be open-sourced. iii) EmbodiedGPT forms a closed-loop from high-level planning to low-level control, which enables seamless integration of high-level planning and low-level control, providing efficient task performance and adaptability to a wide range of tasks. To achieve this, we utilize the embodied-former to query task-relevant instance-level features through cross-attention between visual observations and generated embodied planning. This enables the policy network to complete low-level control tasks with fewer than 25 demonstrations.
|
| 31 |
+
|
| 32 |
+
The contributions can be summarized as follows: (i) We build an end-to-end multi-modal foundation model EmbodiedGPT for embodied AI, which is featured with "chain-of-thought" capability, empowering embodied agents to interact with the physical world in a more natural and intuitive manner. (ii) We develop two datasets, EgoCOT and EgoVQA, consisting of 200M annotated videos from the Ego4D dataset with corresponding detailed planning instructions and VQA data. The datasets are first machine-generated, then semantics-based filtered, and finally human-verified for quality control. (iii) We introduce EmbodiedGPT a cost-effective training approach and a paradigm for extracting task-relevant features from LLM-generated planning queries, thereby forming a closed loop between high-level planning and low-level control. We demonstrate our approach’s effectiveness by achieving state-of-the-art or comparable performance on multiple embodied tasks, including embodied control, embodied planning, video captioning, and video QA. Notably, in comparison to BLIP-2 [17] fine-tuned on the Ego4D dataset and R3M [12] specifically designed for manipulation tasks, EmbodiedGPT outperforms both models on the Franka Kitchen [14] benchmark with a margin of $2 2 . 1 \%$ and $5 . 5 \%$ , respectively. Similarly, on the Meta-World [14] benchmark, EmbodiedGPT surpasses both models with margins of $2 2 . 5 \%$ and $4 . 2 \%$ , respectively.
|
| 33 |
+
|
| 34 |
+
# 2 Related Work
|
| 35 |
+
|
| 36 |
+
# 2.1 Vision Language Pre-training with large scale foundation model
|
| 37 |
+
|
| 38 |
+
Vision-Language Pre-training focuses on strengthening the link between visual observation and natural language. The goal is to develop models that can better understand and process visual content, such as recognizing objects and actions, and generating descriptive text. As models become larger, the computational expense for end-to-end pre-training rises, leading to the need for modular vision-language pre-training methods. These methods smartly use pre-trained models, keeping them ‘frozen’ during vision language pre-training to save on computational costs. For example, models like Uniter [18], Oscar [19], VinVL [20], and LiT [21] freeze the image encoder, while Frozen [22] and VGPT [23] freeze the language model. Furthermore, Flamingo [24] and BLIP-2 [17] use both frozen image encoders and language models, providing a balance between performance and computational efficiency. Due to the lack of open-source data for multi-modal embodied planning, previous works struggled to perform detailed task decomposition and lacked the ability to generate precise and executable plans. To tackle this issue, we create the EgoCOT dataset and develop an embodied chain-of-thought vision language pre-training framework to enhance the capacity of multi-modal models for embodied reasoning and planning.
|
| 39 |
+
|
| 40 |
+
# 2.2 Egocentric Video Datasets.
|
| 41 |
+
|
| 42 |
+
Egocentric videos, which are captured using wearable cameras, provide a natural perspective of daily activities and pose several challenging research questions [25, 26, 27]. Several egocentric video datasets have been created over the years, including [28, 29, 30]. However, the collection of egocentric videos is expensive, and previous datasets tend to be small-scale and domain-specific. Recently, a massive egocentric video dataset, Ego4D [16], has been released and has been used for embodied representation learning. The dataset comprises 3,670 hours of videos collected by 931 people from 74 locations across 9 countries, with videos accompanied by narrations. For embodied AI tasks, learning from large and diverse egocentric human videos has emerged as a promising approach to acquiring a generally useful visual representation for controlling such tasks. For example, R3M [12] developed a sparse and compact visual representation using the Ego4D human video dataset through a combination of time-contrastive learning and video-language alignment. VIP [31], learns general-purpose reward functions for goal-conditioned robotic manipulation using the Ego4D dataset.
|
| 43 |
+
|
| 44 |
+
|
| 45 |
+
Figure 2: Overall framework of EmbodiedGPT. The black arrow shows the vision-language planning process, while the red arrow represents that we leverage the queried language plans for better policy learning in low-level control tasks.
|
| 46 |
+
|
| 47 |
+
# 2.3 Large Foundation Model Assistant System
|
| 48 |
+
|
| 49 |
+
Recent advancements in large-scale multi-modal language models (LLMs), such as GPT-3 [32] and GPT-4 [1], have resulted in the creation of various models that can understand multiple modes of information. Two main approaches are used in this field: systematic collaboration and end-to-end trained models. Systematic collaboration approaches involve coordinating multiple vision models or tools with language models to combine visual information with textual descriptions. Examples include models like Visual ChatGPT [33], MM-REACT [34], and HuggingGPT [35]. However, this approach is limited by the accuracy and capacity of fixed modular models, which can lead to an accumulation of errors. On the other hand, end-to-end models aim to provide unified models for multi-modal tasks. For example, Flamingo [24] combines vision and language by freezing pre-trained vision encoders and language models. BLIP-2 [13] introduces Q-Former to align visual features from frozen visual encoders with large language models. Recently, models such as MiniGPT-4 [36] and LLaVA [37] align instruction-tuned language models with visual features from frozen visual backbones. VideoChat[38], mPLUG-Owl [39] and X-LLM [40], further expand support for video input. PaLM-E [41] is the first large embodied multi-modal model, which directly incorporates features from sensor modalities to improve real-world performance and is trained with their largescale everyday robot data [42]. Compared to PaLM-E, EmbodiedGPT is more compact, with a size of only 10B and offers additional support for video captioning, video QA and making planning according to a demonstration video. Furthermore, we form a closed-loop system that spans from high-level planning to low-level control.
|
| 50 |
+
|
| 51 |
+
# 3 Method
|
| 52 |
+
|
| 53 |
+
The goal of the embodied foundation model is to imitate human-like perception and interaction with the environment by accurately perceiving the environment, identifying relevant objects, analyzing their spatial relationships, and formulating a detailed task plan. To achieve this, the EmbodiedGPT employs a pre-trained vision transformer as the visual encoder and a pre-trained LLaMA [43] model as the language model. As shown in Figure 2, the embodied-former acts as a bridge between the visual and language domains, it first extracts compact visual features from the output of the vision model through attention-based interaction involving visual tokens, text queries, and learnable embodied queries and then maps it to the language modality through a language mapping layer. These embeddings are sent to the frozen LLaMA [43] language model for visual caption, visual QA, and embodied planning. The generated planning is then used to query highly relevant features from the general visual tokens encoded by the visual model via the embodied-former. These features are utilized to generate low-level control commands for task execution through the downstream policy network. To enhance performance across a range of embodied tasks, we introduce a novel video-language pre-training paradigm that leverages a cognitive chain of thought to produce embodied planning from egocentric video inputs. We formulate this task as a standard VQA (Visual Question Answering) task, using "how to do the task that $^ +$ original caption" as the question and embodied planning as the answer. This framework enriches the data of embodied planning and standard visual QA tasks, encouraging the embodied-former to capture task-specific features that are more suitable for embodied control tasks.
|
| 54 |
+
|
| 55 |
+
# 3.1 Framework
|
| 56 |
+
|
| 57 |
+
The training process consists of three stages, each designed to incrementally develop reasoning and planning capabilities. The first two stages focus on pre-training in basic cognitive and responsive skills, while the third stage involves training the embodied AI task with egocentric video-text data on EgoCOT. In the first stage, we focus on image-text conversation alignment pre-training, which involves using three datasets: COCO Caption [44], 595 thousand finely filtered image-text pairs from CC3M [45], and 491 thousand filtered image-text pairs obtained by re-captioning LAION-400M using BLIP-2 [17]. The primary goal of this stage is to pre-train the Embodied-former and language projection while keeping the vision and language model parameters frozen to save computational resources. In the second stage, our goal is to enhance the model’s ability to comprehend and generate more complex sentences and improve its reasoning skills. We achieve this by updating the language projection and prefix language adapter and utilizing the "Complex_Reasoning_ $. 7 7 \mathrm { k } "$ and multi-turn conversation datasets provided by "LLaVA_Instruct_150K" [46].
|
| 58 |
+
|
| 59 |
+
Embodied "chain-of-thought" training with EgoCOT: During the third stage, we employ Conv3D [47] to adapt the pre-trained vision model from stage 2 for video encoding, using a total of eight evenly distributed keyframes from each video. Each keyframe is partitioned into threedimensional (3D) patches, which can be visualized as spatio-temporal cubes, adeptly capturing both the visual content and the sequence of events within the video. These 3D patches are subsequently encoded into visual tokens via the Conv3D module with a time offset of 2 and are then integrated into the internal vision transformer. Then, we introduce the ’chain-of-thought’ vision language pre-training paradigm where the model takes 8 keyframes of the video as input, along with the task description, embodied planning, and structured verb-noun pairs summary to reason with a prompt, such as Listing 1. To avoid overfitting, we provide a prompt set that has different instructions with the same meaning. In this stage, we fine-tune the patch embedding, the language projection layer, and the prefix language adapter to better capture temporal information.
|
| 60 |
+
|
| 61 |
+
Watch this video , identify the actions and devise a plan using chain -of - thought . Extract detailed actions using this schema :
|
| 62 |
+
Task : {" task description " }
|
| 63 |
+
Plan : {" plan with chain -of - thought "} Actions : {{ " number " }: { ’ verb ’ }({ ’ noun ’}) }.
|
| 64 |
+
|
| 65 |
+
Listing 1: Prompt we used for chain-of-thought pre-training.
|
| 66 |
+
|
| 67 |
+
# 3.2 Model Architecture
|
| 68 |
+
|
| 69 |
+
The Embodied-former, denoted as $\mathcal { E } ( \cdot )$ , serves as a bridge between visual input $x _ { \mathrm { v i s } }$ and the frozen language model, acting as an information bottleneck that delivers the most relevant visual data to the language model. The Embodied-former consists of two sub-modules: one for extracting features from the image input, denoted as $\mathcal { E } _ { \mathrm { v i s } } : x _ { \mathrm { v i s } } \to y _ { \mathrm { v i s } }$ , and another for extracting features from the text input, denoted as $\mathcal { E } _ { \mathrm { t x t } } : x _ { \mathrm { t x t } } \to y _ { \mathrm { t x t } } .$ . We employ $N$ learnable embodied query embeddings $y _ { \mathrm { { q u e r y } } }$ as the input of $\mathcal { E }$ to interact with $x _ { \mathrm { v i s } }$ through cross-attention layers and with $x _ { \mathrm { t x t } }$ through self-attention layers. We denote the output query representation as $z \in \mathbb { R } ^ { N \times D }$ , where $D$ is the dimensionality of the embeddings. The dimension of $z$ is significantly smaller than that of the visual features. The output query embeddings are then transformed to $z ^ { ' } \in \mathbb { R } ^ { N \times D ^ { ' } }$ , which have the same dimensionality $D ^ { ' }$ as the LLM’s text embedding in the language modality. This transformation is performed by a mapping function denoted as $M : z \to z ^ { ' }$ , which is accomplished by a linear projection via a fully-connected (FC) layer. The projected embeddings, $z ^ { \prime }$ , serve as "soft visual prompts for the language model," decoupling the whole interaction into visual-query interaction and query-text interaction. The final embodied planning is inferred by the language model with $z ^ { \prime }$ and text prompt(shown as Listing 1) as input. For low-level control which aims to generate actions to interact with the environment, the embodied plan $x _ { \mathrm { p l a n } }$ is used as input text for embodied-former to query the task-relevant instance level features ${ z _ { \mathrm { i n s t a n c e } } = \mathcal { E } ( x _ { \mathrm { v i s } } , x _ { \mathrm { p l a n } } , y _ { \mathrm { q u e r y } } ) }$ . Subsequently, the agent is capable of generating control commands, such as the turning angle of the servo, represented as $a = g ( z _ { \mathrm { i n s t a n c e } } , z _ { \mathrm { g l o b a l } } )$ . This function combines both the instance-specific information zinstance and the global context $z _ { \mathrm { g l o b a l } }$ . The global context is inferred using a ResNet50 model [48] that has been pre-trained on ImageNet [49], employing global average pooling. Here, $g ( \cdot )$ represents the policy network, which is a Multi-Layer Perceptron (MLP) [50] mapping function. The output of the policy network consists of specific executable actions, such as positions and velocities in the Cartesian coordinate system. More implementation details can be found in Appendix A.
|
| 70 |
+
|
| 71 |
+
# 3.3 Training Settings
|
| 72 |
+
|
| 73 |
+
We employ the same pre-trained image encoder as BLIP-2[17]. Specifically, we utilize the ViT-G/14 model from EVA-CLIP [51] and remove its last layer, using the output features of the second last layer instead. For the frozen language model, we adopt a pre-trained LLaMA-7B [43] model and fine-tune it using the ShareGPT dataset and a GPT-4 generated 52K English instruction-following dataset [52]. We then utilize the well-fine-tuned language model as the frozen language model for vision-language pre-training. Additionally, we convert the data type of parameters of the frozen ViT [53] and language model to FP16 during pre-training to increase efficiency.
|
| 74 |
+
|
| 75 |
+
# 3.4 Creating EgoCOT and EgoVQA Dataset
|
| 76 |
+
|
| 77 |
+
For our EgoCOT dataset, we obtain basic data from the Ego4D dataset [16], which includes 9, 645 untrimmed videos of various durations ranging from 5 seconds to 7 hours. To prepare the data for our purposes, we conducted two stages of data cleaning to prepare our data. In the first stage, we filtered out videos with missing or very short narrations (which made up $7 . 4 \%$ and $0 . 9 \%$ of the text, respectively), as well as those with unsure tags (which accounted for $4 . 0 \%$ of the text). We also excluded videos without human-object interaction, such as watching TV or walking. After this stage, we were left with 2.9 thousand hours of video, containing 3.85 million narrations, from 129 different scenarios covering 2927 hours of video.
|
| 78 |
+
|
| 79 |
+
To generate pairs of captions, embodied plannings, and corresponding video segments with time intervals, we utilized the EgoVLP framework [54] to segment the video. The narrations are organized as a sequence of sentences $\mathcal { T } _ { 0 } , \cdots , \mathcal { T } _ { n }$ with precise timestamps $t _ { 0 } , \cdots , t _ { n }$ that indicate when a described event occurred. For each narration $\mathcal { T } _ { i }$ with timestamp $t _ { i }$ , we paired it with a clip $\nu _ { i }$ by determining its start and end time points:
|
| 80 |
+
|
| 81 |
+
$$
|
| 82 |
+
[ t _ { i } ^ { s t a r t } , t _ { i } ^ { e n d } ] = [ t _ { i } - \beta _ { i } / 2 \alpha , t _ { i } + \beta _ { i } / 2 \alpha ] ,
|
| 83 |
+
$$
|
| 84 |
+
|
| 85 |
+
where $\textstyle \beta _ { i } = \sum _ { j = 0 } ^ { n - 1 } \left( t _ { j + 1 } - t _ { j } \right) / n$ is an adjustable parameter equal to the average temporal distance between consecutive narrations in a given video. Conversely, $\alpha$ is a scale factor computed as the average of all $\beta _ { i }$ across all videos in the EgoCOT dataset $\alpha = 4 . 9$ seconds). For each video segment, we provide prompts and corresponding captions for ChatGPT [55] to generate a reasonable and detailed embodied planning. The caption is typically a brief introduction such as $" C$ opens a drawer." We use the ChatGPT to generate a chain of thought according to the caption and organize it into a list of verb-noun pairs, such as "plans: grasp the handle with the gripper and pull the handle; actions: 1. grasp(handle, gripper) 2. pull(handle)." The prompt we used to generate EgoCOT dataset is shown in Listing 2. To enhance the diversity of generated chain of thoughts, we employ a temperature parameter of 0.9 and a top-p parameter of 0.95. For each prompt, we perform five sampling iterations.
|
| 86 |
+
|
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Post-procedure. To ensure the quality of the generated planning instructions, we perform the second stage of data cleaning. We used the CLIP model [56] to assess the similarities between the video and text pairs. For each video, we compared it against five potential embodied plans and selected the one with the highest similarity as the corresponding label for the embodied plan. We then took our data-cleaning process a step further by filtering out any video-caption-planning pairs with similarities lower than the threshold. We eliminated both data with the low similarity between the video and caption and between the video and planning to ensure the highest quality data for our EgoCOT dataset. For each keyframe of the video segment, we use the CLIP model to encode both the text data $T$ and the image data $I$ into a shared embedding space. The similarity is calculated using the cosine similarity function as $\begin{array} { r } { S ( y _ { T } , y _ { I } ) = \frac { y _ { T } \cdot y _ { I } } { \| y _ { T } \| \| y _ { I } \| } } \end{array}$ where $S ( y _ { T } , y _ { I } )$ denotes the similarity between the text and image, and $y _ { T }$ and $y _ { I }$ are the respective embeddings. Given that each video contains multiple keyframes, an ensemble of similarity scores is obtained for each video. This ensemble strategy helps to alleviate the problem of variability among individual frames and ensures a more robust and representative measure of overall similarity. The ensemble similarity score between a video $V$ with $n$ keyframes and text data $T$ is given by:
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Listing 2: Prompt we used for creating EgoCOT dataset.
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$$
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E ( V , T ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } S ( y _ { T { i } } , y _ { I { i } } )
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$$
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where $E ( V , T )$ is the ensemble similarity score, $S ( y _ { T _ { i } } , y _ { I _ { i } } )$ is the similarity score for the $i \cdot$ -th keyframe, and $n$ is the total number of keyframes. We also created the EgoVQA dataset specifically for egocentric human-object interaction video question answering tasks to enrich the training data. For each caption in the Ego4D dataset, we used ChatGPT to generate five QA pairs. To ensure relevance, we guided ChatGPT to focus on core key verbs and nouns by designing prompts as shown in Listing 3. The sampling schema when crafting EgoVQA is the same to that as EgoCOT.
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Listing 3: Prompt used for creating EgoVQA dataset.
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# 4 Experiments
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In this section, we present a comprehensive evaluation of multi-modal foundation models and EmbodiedGPT, across various tasks including visual captioning, embodied planning, and control.
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Evaluation on image input tasks. In order to evaluate the quality of generated captions and planning with the given image, we conducted a user study with 30 participants. The study included 10 cases of image caption tasks from MS-COCO dataset [44], 5 embodied planning scenarios in different embodied AI simulators, and 5 real-world scenes with accompanying embodied planning tasks. Participants were asked to rate the generated captions from different end-to-end models on five dimensions using a scoring system ranging from 1 to 10: object recognition accuracy, spatial
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<table><tr><td>Model</td><td>Object(↑)</td><td>Spatial(↑)</td><td>Redundancy(↓)</td><td>Plan Reasonable(↑)</td><td>Plan Executable(↑)</td></tr><tr><td>Minigpt4</td><td>5.6</td><td>4.8</td><td>4.4</td><td>4.5</td><td>4.8</td></tr><tr><td>LLaVA-7B</td><td>7.3</td><td>7.4</td><td>3.9</td><td>7.5</td><td>6.6</td></tr><tr><td>LLaVA-13B</td><td>8.5</td><td>8.6</td><td>3.4</td><td>8.4</td><td>7.6</td></tr><tr><td>EmbodiedGPT</td><td>8.4</td><td>8.8</td><td>2.6</td><td>8.8</td><td>8.4</td></tr></table>
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Table 1: Generate Quality Evaluation on image input tasks.
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Human: Give a detailed explanation of the image, describe the scene and the observed objects, and spatial relationships. Where should I go if I wish to find a coat hanger?
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# EmbodiedGPT:
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1. The image depicts a doorway leading into a bedroom, with a clock hanging on the wall beside it and a picture of a woman further to the right. 2. There is a white closet suitable for storing clothes on the right side of the bedroom. 3. If one needs to hang a coat, the white closet on the left seem to be the logical choice.
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Figure 3: Comparison between EmbodiedGPT and VisualGPT in the question-answering task.
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# Visual ChatGPT:
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The image shows a bathroom with a clock on the wall and a toilet in the comer. The clock is white and has a black face with two hands owalls are tiled and there is a window in the room. The image shows a bathroom with a clock and a toilet. There is no coat hanger visible in the image.
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Figure 4: Example of video input embodied AI tasks on Meta-World benchmark. EmbodiedGPT accurately analyzes embodied control tasks in demonstration videos and provides precise planning.
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relationship understanding, level of redundancy in the answer, and reasonability of the planning and the executability of the planning. The average scores among all the participants for different models are shown in Table 1. The results demonstrate that EmbodiedGPT achieves a comparable level of object recognition and spatial relationship understanding as the LLaVA-13B model, despite having only 7B parameters in the language model. Furthermore, EmbodiedGPT generates less redundant content in relation to the given embodied AI task, and produces the most reasonable and executable planning outputs. We also compared the performance of EmbodiedGPT with Visual ChatGPT [33], which adopts a hierarchical approach by combining several pre-trained vision models and language models to answer questions. In the Virtual-Home [57] benchmark, Visual ChatGPT uses a visual caption model to generate dense captions that are subsequently passed into ChatGPT for deriving a solution. As shown in Figure 3, Visual ChatGPT failed to find a coat hanger due to its limitations of relying solely on the caption model for extracting visual information, resulting in poor performance when compared to the end-to-end model like EmbodiedGPT. These findings highlight the advantages of adopting a unified, end-to-end model over hierarchical approaches that rely on multiple stages.
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Evaluation on video input embodied AI tasks. We evaluate the recognition ability of videos and planning abilities of our model for embodied control tasks on standard embodied AI benchmarks, Franka Kitchen [14] and Meta-World [15]. Meta-World provides a challenging set of tasks that require complex object manipulation skills, including assembling a ring on a peg, picking and placing a block between bins, pushing a button, opening a drawer, and hammering a nail. Franka Kitchen benchmark focuses on tasks like sliding open the right door, opening the cabinet, turning on the light, turning the stovetop knob, and opening the microwave. As shown in Figure 4, given a demonstration video, EmbodiedGPT can accurately interpret the embodied control task and provide step-by-step planning. The output planning is fed into the Embodied-former module of EmbodiedGPT to query highly relevant features for use as inputs in the policy network and the low-level actions are generated by the policy network to interact with the environment (see more visualizations in Appendix B).
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Figure 5: Performance of EmbodiedGPT in low-level control tasks with 10 demonstration demos.
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Figure 6: Performance of EmbodiedGPT in low-level control tasks with 25 demonstration demos.
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Evaluation on embodied control tasks. For embodied control tasks, we compare our model with R3M[12], which is the state-of-the-art method in these two benchmarks, and an ablation version called ’BLIP-2[Ego4D]’, which has the same structure and same amount of parameters as EmbodiedGPT, and is only fine-tuned on the video caption task using the Ego4D dataset without incorporating EgoCOT. In all experiments, the policy network is learned using few-shot learning on a small amount of demonstration data. There are two settings, one of which utilizes 10 demonstrations, and the other utilizes 25 demonstrations. We report the success rate in 100 random evaluations with only visual observations in 5 tasks per benchmark over 5 seeds and 2 different camera views for each setting, respectively. As shown in Figure 5 and Figure 6, EmbodiedGPT outperforms the baseline methods, demonstrating the effectiveness of learning with EgoCOT.
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<table><tr><td>Model</td><td></td><td></td><td></td><td>Franka(10 demos） Franka(25 demos）Meta-World(10 demos）Meta-World(25 demos)</td></tr><tr><td>EmbodiedGPT</td><td>50.8% ±2.8</td><td>58.5% ±2.7</td><td>76.4% ±2.2</td><td>81.2%±2.0</td></tr><tr><td> - Close-loop</td><td>38.6% ±2.9</td><td>47.3% ±2.5</td><td>62.7% ±2.2</td><td>64.9% ±2.0</td></tr><tr><td>- COT</td><td>26.2% ±3.2</td><td>36.4% ±2.7</td><td>55.2% ±2.4</td><td>58.7% ±2.0</td></tr></table>
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Table 2: Ablation on the closed-loop spans from planning to low-level control, and "chain-of-thought" (COT) training with 25 and 10 demonstrations("-" symbol indicates "removing"). We report the average success rate over 5 tasks and 2 camera views per benchmark.
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Human: You are the controller of the robot shown in this image, now your task is put all the objects into the brown bag next to the robot, please write a step by step plan:
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# EmbodiedGPT:
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Step1. Move the robotic arm to the top of the gray bottle.
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Step 2. Close the gripper and grasp the gray bottle.
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Step 3. Move the arm to the brown bag.
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Step 4. Release the bottle.
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Step 5. Move the arm to the top of the cereal bag.
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Step 6. Close the gripper and grasp the cereal bag.
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Step 7. Move the arm to the brown bag.
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Step 8. Finally, release the cereal bag.
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Franka Emika Panda arm with front facing camera
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Figure 7: Generated plan by EmbodiedGPT for rearrangement task.
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Figure 8: Real-world experiment on the Franka Emika robot arm.
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Ablation study. We perform ablation studies to analyze the effectiveness of the "Chain-of-Thought" training mode and the importance of a closed-loop design for embodied control. The results, as shown in Table 2, demonstrate a significant improvement in success rate when using the EgoCOT approach compared to training solely with the EGO4D caption task. Moreover, the closed-loop design is necessary as the generated plans contained specific and relevant sub-goal information, which proved crucial for control tasks. In summary, EmbodiedGPT exhibits a strong ability to generate reasonable planning, accurately extract task-relevant features from visual inputs, as well as execute low-level actions to interact with the environment. The ablation experiments demonstrate that both the training paradigm based on EgoCOT and the closed-loop design from embodied planning to low-level control significantly contribute to the performance improvement of EmbodiedGPT.
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Real robot experiment. We also conducted a real-world experiment using the Franka Emika robot arm, emphasizing a rearrangement task where the objective was to pack scattered bottles from a table into a box. As depicted in Figures 7 and 8, model is capable of generating intricate plans and executing of low-level actions based on 50 demonstration.
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# 5 Conclusion
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In this paper, we present EmbodiedGPT, an end-to-end multi-modal foundational model for embodied AI that enables agents to perform step-by-step planning and execute low-level commands. To achieve this, we create a large-scale embodied planning dataset called EgoCOT and develop an efficient training approach that utilizes prefix tuning to generate high-quality plans with a "chain-of-thought". Furthermore, our embodied control paradigm seamlessly coordinates high-level planning and lowlevel control. Extensive experiments demonstrate the effectiveness of EmbodiedGPT on various embodied tasks, achieving state-of-the-art or comparable performance. We believe that EmbodiedGPT represents a significant step towards developing more intelligent embodied AI agents.
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Future works and limitations: EmbodiedGPT freezes the parameters of the vision and language model due to limited computational resources. Joint training with all modules and exploring other modalities, such as speech, could be future works. We do not foresee obvious undesirable ethical or social impacts at this moment.
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Acknowledgement. This paper is partially supported by the National Key R&D Program of China No.2022ZD0161000 and the General Research Fund of Hong Kong No.17200622. We would like to express our appreciation to Dr. René Zurbrügg from ETH AI Center for his support on robot experimental platform.
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| 1 |
+
[
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| 2 |
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{
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| 3 |
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"type": "text",
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| 4 |
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"text": "EmbodiedGPT: Vision-Language Pre-Training via Embodied Chain of Thought ",
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| 5 |
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"text_level": 1,
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"bbox": [
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"type": "text",
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| 16 |
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"text": "Yao ${ { \\bf { M } } { \\bf { u } } ^ { 1 } }$ , Qinglong Zhang2, Mengkang $\\mathbf { H } \\mathbf { u } ^ { 1 }$ , Wenhai $\\mathbf { W a n g ^ { 3 } }$ , Mingyu Ding∗,1, $\\mathbf { J u n \\mathbf { \\mathbf { J i n } } ^ { 4 } }$ , Bin Wang4, Jifeng Dai2,5, Yu Qiao2, Ping Luo∗,1,2 1The University of Hong Kong, 2Shanghai AI Laboratory, 3The Chinese University of Hong Kong, 4Noah’s Ark Laboratory 5Tsinghua University ",
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| 17 |
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"bbox": [
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| 24 |
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| 25 |
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{
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| 26 |
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"type": "text",
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| 27 |
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"text": "Abstract ",
|
| 28 |
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"text_level": 1,
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| 29 |
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"bbox": [
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| 31 |
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"type": "text",
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"text": "Embodied AI is a crucial frontier in robotics, capable of planning and executing action sequences for robots to accomplish long-horizon tasks in physical environments. In this work, we introduce EmbodiedGPT, an end-to-end multi-modal foundation model for embodied AI, empowering embodied agents with multi-modal understanding and execution capabilities. To achieve this, we have made the following efforts: (i) We craft a large-scale embodied planning dataset, termed EgoCOT. The dataset consists of carefully selected videos from the Ego4D dataset, along with corresponding high-quality language instructions. Specifically, we generate a sequence of sub-goals with the \"Chain of Thoughts\" mode for effective embodied planning. (ii) We introduce an efficient training approach to EmbodiedGPT for high-quality plan generation, by adapting a 7B large language model (LLM) to the EgoCOT dataset via prefix tuning. (iii) We introduce a paradigm for extracting task-related features from LLM-generated planning queries to form a closed loop between high-level planning and low-level control. Extensive experiments show the effectiveness of EmbodiedGPT on embodied tasks, including embodied planning, embodied control, visual captioning, and visual question answering. Notably, EmbodiedGPT significantly enhances the success rate of the embodied control task by extracting more effective features. It has achieved a remarkable 1.6 times increase in success rate on the Franka Kitchen benchmark and a 1.3 times increase on the Meta-World benchmark, compared to the BLIP-2 baseline fine-tuned with the Ego4D dataset. More demos, code, and dataset information can be found at our homepage. ",
|
| 40 |
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| 47 |
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| 48 |
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{
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| 49 |
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"type": "text",
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| 50 |
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"text": "1 Introduction ",
|
| 51 |
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"text_level": 1,
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| 52 |
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"type": "text",
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"text": "Embodied AI tasks, e.g., embodied planning, embodied VQA, and embodied control, aim to imbue robots with the ability to perceive, reason, and act within their environment, enabling them to perform long-horizon plans and execute actions autonomously based on real-time observations. Recently, large language models (LLMs) such as GPT4 [1] and PaLM-E [2], have shown promising language understanding, reasoning, and \"chain-of-thought\" capabilities. Such advances may open new possibilities for developing robots capable of processing natural language instructions, performing multi-modal \"chain-of-thought\", and planning actions in physical environments. ",
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| 63 |
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| 71 |
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| 72 |
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"type": "text",
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| 73 |
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"text": "Large-scale datasets play important roles in training large language models. For example, OpenCLIP trains its ViT-G/14 model on the LAION-2B dataset [3], which contains 2B image-language pairs. Unlike general-purpose visual language tasks that can get a huge amount of weakly labeled imagecaption pairs from the Internet, embodied AI tasks require egocentric data in robotics domains. Also, ",
|
| 74 |
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| 82 |
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| 83 |
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"type": "text",
|
| 84 |
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"text": "Human: Describe the video in detail. ",
|
| 85 |
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"bbox": [
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| 86 |
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| 87 |
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| 88 |
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| 90 |
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| 91 |
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},
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| 93 |
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{
|
| 94 |
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"type": "image",
|
| 95 |
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"img_path": "images/ca3f7fcc311c8a80c4594e6f910429514912da588e1d632c4ab30ce842514c3f.jpg",
|
| 96 |
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"image_caption": [],
|
| 97 |
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"image_footnote": [],
|
| 98 |
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"page_idx": 1
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| 105 |
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|
| 106 |
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|
| 107 |
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"type": "text",
|
| 108 |
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"text": "Human: Can you write a detailed plan for the task the robot is performing? ",
|
| 109 |
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| 110 |
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"page_idx": 1
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| 116 |
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},
|
| 117 |
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{
|
| 118 |
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"type": "image",
|
| 119 |
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"img_path": "images/dae261c153cc867c75ffc25196665abd373dcf2f5bb9265bb822c5af97974793.jpg",
|
| 120 |
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"image_caption": [
|
| 121 |
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"Figure 1: EmbodiedGPT’s capabilities for video captioning, multi-turn question answering, embodied planning, and low-level control. The plans given by EmbodiedGPT are highly executable and incorporate task-specific features, leading to a significant improvement in the success rate of embodied control tasks, outperforming both R3M [12] (a video-language contrastive learned model) and BLIP2 [13] (a multi-modal foundation model) on Franka Kitchen [14] and Meta-World [15] environments. "
|
| 122 |
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],
|
| 123 |
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"image_footnote": [],
|
| 124 |
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"bbox": [
|
| 125 |
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| 126 |
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| 127 |
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| 128 |
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| 129 |
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|
| 130 |
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"page_idx": 1
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|
| 132 |
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| 133 |
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"type": "text",
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| 134 |
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"text": "structured language instructions are needed for precise planning, which usually requires huge manual efforts and costs. This poses a challenging problem in collecting high-quality embodied multi-modal data. Some researchers [4, 5, 6, 7] explore creating large-scale embodied datasets with simulators, but a significant gap remains between simulation and the real world. Recent works [8, 9, 10] also explore adapting the pre-trained LLMs to a new domain by efficient tuning strategies like LoRA [11]. However, several open questions still remain: how to apply LLMs to the field of robotics which may face large domain gaps; how to leverage the \"chain-of-thought\" capability for structured planning; and how to use the output language plan for downstream manipulation tasks in an end-to-end manner. ",
|
| 135 |
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"page_idx": 1
|
| 142 |
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|
| 143 |
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{
|
| 144 |
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"type": "text",
|
| 145 |
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"text": "To solve the above challenges, in this work, we first build a large-scale embodied planning dataset, termed EgoCOT, which features chain-of-thought planning instructions. It contains carefully selected egocentric videos from the Ego4D dataset [16] and corresponding high-quality step-by-step language instructions, which are machine-generated, then semantics-based filtered, and finally human-verified. Additionally, we also create the EgoVQA dataset as an extension of the Ego4D dataset, focusing on egocentric human-object interaction video question answering tasks, which aims to offer a wider range of egocentric multi-modal data. ",
|
| 146 |
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| 155 |
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"type": "text",
|
| 156 |
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"text": "Based on our EgoCOT and EgoVQA, we present an end-to-end multi-modal embodied foundation model called EmbodiedGPT, which can interact with the physical world in a more natural and intuitive manner, and perform many embodied tasks, as shown in Figure 1, such as embodied planning, embodied VQA, and embodied control. EmbodiedGPT comprises four integrated modules that work together, including i) a frozen vision model for encoding visual features of current observations, ii) a frozen language model used to execute natural language for question answering, captioning, and embodied planning tasks, iii) an embodied-former with a language mapping layer for aligning the visual and embodied instructions and extracting task-relevant instance-level features with the generated planning for low-level control, and iv) a policy network, which is responsible for producing low-level actions based on the task-relevant features, enabling the agent to effectively interact with the environment. To further enhance EmbodiedGPT ’s performance in generating reliable planning containing sub-goal sequences, we implement prefix tuning on the frozen language model to encourage the generation of more executable planning. ",
|
| 157 |
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"type": "text",
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| 167 |
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"text": "Our method possesses the following core advantages: i) the generated planning exhibits strong executability and granularity at the object part level, such as the gripper of a robotic arm or the handle of a door, manifested in sub-goal sequences. ii) the proposed EgoCOT dataset is built based on an open-source large-scale dataset, which offers greater scalability compared to the PaLM-E [2] model trained on proprietary robot data. And both the EgoCOT dataset, and the EmbodiedGPT model will be open-sourced. iii) EmbodiedGPT forms a closed-loop from high-level planning to low-level control, which enables seamless integration of high-level planning and low-level control, providing efficient task performance and adaptability to a wide range of tasks. To achieve this, we utilize the embodied-former to query task-relevant instance-level features through cross-attention between visual observations and generated embodied planning. This enables the policy network to complete low-level control tasks with fewer than 25 demonstrations. ",
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"text": "",
|
| 179 |
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"page_idx": 2
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"type": "text",
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"text": "The contributions can be summarized as follows: (i) We build an end-to-end multi-modal foundation model EmbodiedGPT for embodied AI, which is featured with \"chain-of-thought\" capability, empowering embodied agents to interact with the physical world in a more natural and intuitive manner. (ii) We develop two datasets, EgoCOT and EgoVQA, consisting of 200M annotated videos from the Ego4D dataset with corresponding detailed planning instructions and VQA data. The datasets are first machine-generated, then semantics-based filtered, and finally human-verified for quality control. (iii) We introduce EmbodiedGPT a cost-effective training approach and a paradigm for extracting task-relevant features from LLM-generated planning queries, thereby forming a closed loop between high-level planning and low-level control. We demonstrate our approach’s effectiveness by achieving state-of-the-art or comparable performance on multiple embodied tasks, including embodied control, embodied planning, video captioning, and video QA. Notably, in comparison to BLIP-2 [17] fine-tuned on the Ego4D dataset and R3M [12] specifically designed for manipulation tasks, EmbodiedGPT outperforms both models on the Franka Kitchen [14] benchmark with a margin of $2 2 . 1 \\%$ and $5 . 5 \\%$ , respectively. Similarly, on the Meta-World [14] benchmark, EmbodiedGPT surpasses both models with margins of $2 2 . 5 \\%$ and $4 . 2 \\%$ , respectively. ",
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"text": "2 Related Work ",
|
| 201 |
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"type": "text",
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"text": "2.1 Vision Language Pre-training with large scale foundation model ",
|
| 213 |
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"text_level": 1,
|
| 214 |
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"type": "text",
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"text": "Vision-Language Pre-training focuses on strengthening the link between visual observation and natural language. The goal is to develop models that can better understand and process visual content, such as recognizing objects and actions, and generating descriptive text. As models become larger, the computational expense for end-to-end pre-training rises, leading to the need for modular vision-language pre-training methods. These methods smartly use pre-trained models, keeping them ‘frozen’ during vision language pre-training to save on computational costs. For example, models like Uniter [18], Oscar [19], VinVL [20], and LiT [21] freeze the image encoder, while Frozen [22] and VGPT [23] freeze the language model. Furthermore, Flamingo [24] and BLIP-2 [17] use both frozen image encoders and language models, providing a balance between performance and computational efficiency. Due to the lack of open-source data for multi-modal embodied planning, previous works struggled to perform detailed task decomposition and lacked the ability to generate precise and executable plans. To tackle this issue, we create the EgoCOT dataset and develop an embodied chain-of-thought vision language pre-training framework to enhance the capacity of multi-modal models for embodied reasoning and planning. ",
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"type": "text",
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"text": "2.2 Egocentric Video Datasets. ",
|
| 236 |
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"text_level": 1,
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| 237 |
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"type": "text",
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"text": "Egocentric videos, which are captured using wearable cameras, provide a natural perspective of daily activities and pose several challenging research questions [25, 26, 27]. Several egocentric video datasets have been created over the years, including [28, 29, 30]. However, the collection of egocentric videos is expensive, and previous datasets tend to be small-scale and domain-specific. Recently, a massive egocentric video dataset, Ego4D [16], has been released and has been used for embodied representation learning. The dataset comprises 3,670 hours of videos collected by 931 people from 74 locations across 9 countries, with videos accompanied by narrations. For embodied AI tasks, learning from large and diverse egocentric human videos has emerged as a promising approach to acquiring a generally useful visual representation for controlling such tasks. For example, R3M [12] developed a sparse and compact visual representation using the Ego4D human video dataset through a combination of time-contrastive learning and video-language alignment. VIP [31], learns general-purpose reward functions for goal-conditioned robotic manipulation using the Ego4D dataset. ",
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"image_caption": [
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"Figure 2: Overall framework of EmbodiedGPT. The black arrow shows the vision-language planning process, while the red arrow represents that we leverage the queried language plans for better policy learning in low-level control tasks. "
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"text": "2.3 Large Foundation Model Assistant System ",
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"text": "Recent advancements in large-scale multi-modal language models (LLMs), such as GPT-3 [32] and GPT-4 [1], have resulted in the creation of various models that can understand multiple modes of information. Two main approaches are used in this field: systematic collaboration and end-to-end trained models. Systematic collaboration approaches involve coordinating multiple vision models or tools with language models to combine visual information with textual descriptions. Examples include models like Visual ChatGPT [33], MM-REACT [34], and HuggingGPT [35]. However, this approach is limited by the accuracy and capacity of fixed modular models, which can lead to an accumulation of errors. On the other hand, end-to-end models aim to provide unified models for multi-modal tasks. For example, Flamingo [24] combines vision and language by freezing pre-trained vision encoders and language models. BLIP-2 [13] introduces Q-Former to align visual features from frozen visual encoders with large language models. Recently, models such as MiniGPT-4 [36] and LLaVA [37] align instruction-tuned language models with visual features from frozen visual backbones. VideoChat[38], mPLUG-Owl [39] and X-LLM [40], further expand support for video input. PaLM-E [41] is the first large embodied multi-modal model, which directly incorporates features from sensor modalities to improve real-world performance and is trained with their largescale everyday robot data [42]. Compared to PaLM-E, EmbodiedGPT is more compact, with a size of only 10B and offers additional support for video captioning, video QA and making planning according to a demonstration video. Furthermore, we form a closed-loop system that spans from high-level planning to low-level control. ",
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"text": "3 Method ",
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"text": "The goal of the embodied foundation model is to imitate human-like perception and interaction with the environment by accurately perceiving the environment, identifying relevant objects, analyzing their spatial relationships, and formulating a detailed task plan. To achieve this, the EmbodiedGPT employs a pre-trained vision transformer as the visual encoder and a pre-trained LLaMA [43] model as the language model. As shown in Figure 2, the embodied-former acts as a bridge between the visual and language domains, it first extracts compact visual features from the output of the vision model through attention-based interaction involving visual tokens, text queries, and learnable embodied queries and then maps it to the language modality through a language mapping layer. These embeddings are sent to the frozen LLaMA [43] language model for visual caption, visual QA, and embodied planning. The generated planning is then used to query highly relevant features from the general visual tokens encoded by the visual model via the embodied-former. These features are utilized to generate low-level control commands for task execution through the downstream policy network. To enhance performance across a range of embodied tasks, we introduce a novel video-language pre-training paradigm that leverages a cognitive chain of thought to produce embodied planning from egocentric video inputs. We formulate this task as a standard VQA (Visual Question Answering) task, using \"how to do the task that $^ +$ original caption\" as the question and embodied planning as the answer. This framework enriches the data of embodied planning and standard visual QA tasks, encouraging the embodied-former to capture task-specific features that are more suitable for embodied control tasks. ",
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"text": "3.1 Framework ",
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"text": "The training process consists of three stages, each designed to incrementally develop reasoning and planning capabilities. The first two stages focus on pre-training in basic cognitive and responsive skills, while the third stage involves training the embodied AI task with egocentric video-text data on EgoCOT. In the first stage, we focus on image-text conversation alignment pre-training, which involves using three datasets: COCO Caption [44], 595 thousand finely filtered image-text pairs from CC3M [45], and 491 thousand filtered image-text pairs obtained by re-captioning LAION-400M using BLIP-2 [17]. The primary goal of this stage is to pre-train the Embodied-former and language projection while keeping the vision and language model parameters frozen to save computational resources. In the second stage, our goal is to enhance the model’s ability to comprehend and generate more complex sentences and improve its reasoning skills. We achieve this by updating the language projection and prefix language adapter and utilizing the \"Complex_Reasoning_ $. 7 7 \\mathrm { k } \"$ and multi-turn conversation datasets provided by \"LLaVA_Instruct_150K\" [46]. ",
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"text": "Embodied \"chain-of-thought\" training with EgoCOT: During the third stage, we employ Conv3D [47] to adapt the pre-trained vision model from stage 2 for video encoding, using a total of eight evenly distributed keyframes from each video. Each keyframe is partitioned into threedimensional (3D) patches, which can be visualized as spatio-temporal cubes, adeptly capturing both the visual content and the sequence of events within the video. These 3D patches are subsequently encoded into visual tokens via the Conv3D module with a time offset of 2 and are then integrated into the internal vision transformer. Then, we introduce the ’chain-of-thought’ vision language pre-training paradigm where the model takes 8 keyframes of the video as input, along with the task description, embodied planning, and structured verb-noun pairs summary to reason with a prompt, such as Listing 1. To avoid overfitting, we provide a prompt set that has different instructions with the same meaning. In this stage, we fine-tune the patch embedding, the language projection layer, and the prefix language adapter to better capture temporal information. ",
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"text": "Watch this video , identify the actions and devise a plan using chain -of - thought . Extract detailed actions using this schema : \nTask : {\" task description \" } \nPlan : {\" plan with chain -of - thought \"} Actions : {{ \" number \" }: { ’ verb ’ }({ ’ noun ’}) }. ",
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"text": "Listing 1: Prompt we used for chain-of-thought pre-training. ",
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"text": "3.2 Model Architecture ",
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"text": "The Embodied-former, denoted as $\\mathcal { E } ( \\cdot )$ , serves as a bridge between visual input $x _ { \\mathrm { v i s } }$ and the frozen language model, acting as an information bottleneck that delivers the most relevant visual data to the language model. The Embodied-former consists of two sub-modules: one for extracting features from the image input, denoted as $\\mathcal { E } _ { \\mathrm { v i s } } : x _ { \\mathrm { v i s } } \\to y _ { \\mathrm { v i s } }$ , and another for extracting features from the text input, denoted as $\\mathcal { E } _ { \\mathrm { t x t } } : x _ { \\mathrm { t x t } } \\to y _ { \\mathrm { t x t } } .$ . We employ $N$ learnable embodied query embeddings $y _ { \\mathrm { { q u e r y } } }$ as the input of $\\mathcal { E }$ to interact with $x _ { \\mathrm { v i s } }$ through cross-attention layers and with $x _ { \\mathrm { t x t } }$ through self-attention layers. We denote the output query representation as $z \\in \\mathbb { R } ^ { N \\times D }$ , where $D$ is the dimensionality of the embeddings. The dimension of $z$ is significantly smaller than that of the visual features. The output query embeddings are then transformed to $z ^ { ' } \\in \\mathbb { R } ^ { N \\times D ^ { ' } }$ , which have the same dimensionality $D ^ { ' }$ as the LLM’s text embedding in the language modality. This transformation is performed by a mapping function denoted as $M : z \\to z ^ { ' }$ , which is accomplished by a linear projection via a fully-connected (FC) layer. The projected embeddings, $z ^ { \\prime }$ , serve as \"soft visual prompts for the language model,\" decoupling the whole interaction into visual-query interaction and query-text interaction. The final embodied planning is inferred by the language model with $z ^ { \\prime }$ and text prompt(shown as Listing 1) as input. For low-level control which aims to generate actions to interact with the environment, the embodied plan $x _ { \\mathrm { p l a n } }$ is used as input text for embodied-former to query the task-relevant instance level features ${ z _ { \\mathrm { i n s t a n c e } } = \\mathcal { E } ( x _ { \\mathrm { v i s } } , x _ { \\mathrm { p l a n } } , y _ { \\mathrm { q u e r y } } ) }$ . Subsequently, the agent is capable of generating control commands, such as the turning angle of the servo, represented as $a = g ( z _ { \\mathrm { i n s t a n c e } } , z _ { \\mathrm { g l o b a l } } )$ . This function combines both the instance-specific information zinstance and the global context $z _ { \\mathrm { g l o b a l } }$ . The global context is inferred using a ResNet50 model [48] that has been pre-trained on ImageNet [49], employing global average pooling. Here, $g ( \\cdot )$ represents the policy network, which is a Multi-Layer Perceptron (MLP) [50] mapping function. The output of the policy network consists of specific executable actions, such as positions and velocities in the Cartesian coordinate system. More implementation details can be found in Appendix A. ",
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"text": "3.3 Training Settings ",
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"text": "We employ the same pre-trained image encoder as BLIP-2[17]. Specifically, we utilize the ViT-G/14 model from EVA-CLIP [51] and remove its last layer, using the output features of the second last layer instead. For the frozen language model, we adopt a pre-trained LLaMA-7B [43] model and fine-tune it using the ShareGPT dataset and a GPT-4 generated 52K English instruction-following dataset [52]. We then utilize the well-fine-tuned language model as the frozen language model for vision-language pre-training. Additionally, we convert the data type of parameters of the frozen ViT [53] and language model to FP16 during pre-training to increase efficiency. ",
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"text": "3.4 Creating EgoCOT and EgoVQA Dataset ",
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"text": "For our EgoCOT dataset, we obtain basic data from the Ego4D dataset [16], which includes 9, 645 untrimmed videos of various durations ranging from 5 seconds to 7 hours. To prepare the data for our purposes, we conducted two stages of data cleaning to prepare our data. In the first stage, we filtered out videos with missing or very short narrations (which made up $7 . 4 \\%$ and $0 . 9 \\%$ of the text, respectively), as well as those with unsure tags (which accounted for $4 . 0 \\%$ of the text). We also excluded videos without human-object interaction, such as watching TV or walking. After this stage, we were left with 2.9 thousand hours of video, containing 3.85 million narrations, from 129 different scenarios covering 2927 hours of video. ",
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"text": "To generate pairs of captions, embodied plannings, and corresponding video segments with time intervals, we utilized the EgoVLP framework [54] to segment the video. The narrations are organized as a sequence of sentences $\\mathcal { T } _ { 0 } , \\cdots , \\mathcal { T } _ { n }$ with precise timestamps $t _ { 0 } , \\cdots , t _ { n }$ that indicate when a described event occurred. For each narration $\\mathcal { T } _ { i }$ with timestamp $t _ { i }$ , we paired it with a clip $\\nu _ { i }$ by determining its start and end time points: ",
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"text": "$$\n[ t _ { i } ^ { s t a r t } , t _ { i } ^ { e n d } ] = [ t _ { i } - \\beta _ { i } / 2 \\alpha , t _ { i } + \\beta _ { i } / 2 \\alpha ] ,\n$$",
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"text": "where $\\textstyle \\beta _ { i } = \\sum _ { j = 0 } ^ { n - 1 } \\left( t _ { j + 1 } - t _ { j } \\right) / n$ is an adjustable parameter equal to the average temporal distance between consecutive narrations in a given video. Conversely, $\\alpha$ is a scale factor computed as the average of all $\\beta _ { i }$ across all videos in the EgoCOT dataset $\\alpha = 4 . 9$ seconds). For each video segment, we provide prompts and corresponding captions for ChatGPT [55] to generate a reasonable and detailed embodied planning. The caption is typically a brief introduction such as $\" C$ opens a drawer.\" We use the ChatGPT to generate a chain of thought according to the caption and organize it into a list of verb-noun pairs, such as \"plans: grasp the handle with the gripper and pull the handle; actions: 1. grasp(handle, gripper) 2. pull(handle).\" The prompt we used to generate EgoCOT dataset is shown in Listing 2. To enhance the diversity of generated chain of thoughts, we employ a temperature parameter of 0.9 and a top-p parameter of 0.95. For each prompt, we perform five sampling iterations. ",
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"text": "Post-procedure. To ensure the quality of the generated planning instructions, we perform the second stage of data cleaning. We used the CLIP model [56] to assess the similarities between the video and text pairs. For each video, we compared it against five potential embodied plans and selected the one with the highest similarity as the corresponding label for the embodied plan. We then took our data-cleaning process a step further by filtering out any video-caption-planning pairs with similarities lower than the threshold. We eliminated both data with the low similarity between the video and caption and between the video and planning to ensure the highest quality data for our EgoCOT dataset. For each keyframe of the video segment, we use the CLIP model to encode both the text data $T$ and the image data $I$ into a shared embedding space. The similarity is calculated using the cosine similarity function as $\\begin{array} { r } { S ( y _ { T } , y _ { I } ) = \\frac { y _ { T } \\cdot y _ { I } } { \\| y _ { T } \\| \\| y _ { I } \\| } } \\end{array}$ where $S ( y _ { T } , y _ { I } )$ denotes the similarity between the text and image, and $y _ { T }$ and $y _ { I }$ are the respective embeddings. Given that each video contains multiple keyframes, an ensemble of similarity scores is obtained for each video. This ensemble strategy helps to alleviate the problem of variability among individual frames and ensures a more robust and representative measure of overall similarity. The ensemble similarity score between a video $V$ with $n$ keyframes and text data $T$ is given by: ",
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"image_caption": [
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"Listing 2: Prompt we used for creating EgoCOT dataset. "
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"type": "equation",
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"text": "$$\nE ( V , T ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } S ( y _ { T { i } } , y _ { I { i } } )\n$$",
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"text_format": "latex",
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"text": "where $E ( V , T )$ is the ensemble similarity score, $S ( y _ { T _ { i } } , y _ { I _ { i } } )$ is the similarity score for the $i \\cdot$ -th keyframe, and $n$ is the total number of keyframes. We also created the EgoVQA dataset specifically for egocentric human-object interaction video question answering tasks to enrich the training data. For each caption in the Ego4D dataset, we used ChatGPT to generate five QA pairs. To ensure relevance, we guided ChatGPT to focus on core key verbs and nouns by designing prompts as shown in Listing 3. The sampling schema when crafting EgoVQA is the same to that as EgoCOT. ",
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"text": "Listing 3: Prompt used for creating EgoVQA dataset. ",
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"text": "4 Experiments ",
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"text": "In this section, we present a comprehensive evaluation of multi-modal foundation models and EmbodiedGPT, across various tasks including visual captioning, embodied planning, and control. ",
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"text": "Evaluation on image input tasks. In order to evaluate the quality of generated captions and planning with the given image, we conducted a user study with 30 participants. The study included 10 cases of image caption tasks from MS-COCO dataset [44], 5 embodied planning scenarios in different embodied AI simulators, and 5 real-world scenes with accompanying embodied planning tasks. Participants were asked to rate the generated captions from different end-to-end models on five dimensions using a scoring system ranging from 1 to 10: object recognition accuracy, spatial ",
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"type": "table",
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"img_path": "images/b86cd66f3b065ff4708e85ab8931b95a23a0969d4e0780c57739c6c1deac5722.jpg",
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"Table 1: Generate Quality Evaluation on image input tasks. "
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"table_body": "<table><tr><td>Model</td><td>Object(↑)</td><td>Spatial(↑)</td><td>Redundancy(↓)</td><td>Plan Reasonable(↑)</td><td>Plan Executable(↑)</td></tr><tr><td>Minigpt4</td><td>5.6</td><td>4.8</td><td>4.4</td><td>4.5</td><td>4.8</td></tr><tr><td>LLaVA-7B</td><td>7.3</td><td>7.4</td><td>3.9</td><td>7.5</td><td>6.6</td></tr><tr><td>LLaVA-13B</td><td>8.5</td><td>8.6</td><td>3.4</td><td>8.4</td><td>7.6</td></tr><tr><td>EmbodiedGPT</td><td>8.4</td><td>8.8</td><td>2.6</td><td>8.8</td><td>8.4</td></tr></table>",
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"text": "Human: Give a detailed explanation of the image, describe the scene and the observed objects, and spatial relationships. Where should I go if I wish to find a coat hanger? ",
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"text": "EmbodiedGPT: ",
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"text": "1. The image depicts a doorway leading into a bedroom, with a clock hanging on the wall beside it and a picture of a woman further to the right. 2. There is a white closet suitable for storing clothes on the right side of the bedroom. 3. If one needs to hang a coat, the white closet on the left seem to be the logical choice. ",
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"Figure 3: Comparison between EmbodiedGPT and VisualGPT in the question-answering task. "
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"text": "Visual ChatGPT: ",
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"text": "The image shows a bathroom with a clock on the wall and a toilet in the comer. The clock is white and has a black face with two hands owalls are tiled and there is a window in the room. The image shows a bathroom with a clock and a toilet. There is no coat hanger visible in the image. ",
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"type": "image",
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"img_path": "images/05c50bb1bd98096d39a333098a8c0b4f020d9a1715c54b4f5306ea6898926f3d.jpg",
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"image_caption": [
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"Figure 4: Example of video input embodied AI tasks on Meta-World benchmark. EmbodiedGPT accurately analyzes embodied control tasks in demonstration videos and provides precise planning. "
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"text": "relationship understanding, level of redundancy in the answer, and reasonability of the planning and the executability of the planning. The average scores among all the participants for different models are shown in Table 1. The results demonstrate that EmbodiedGPT achieves a comparable level of object recognition and spatial relationship understanding as the LLaVA-13B model, despite having only 7B parameters in the language model. Furthermore, EmbodiedGPT generates less redundant content in relation to the given embodied AI task, and produces the most reasonable and executable planning outputs. We also compared the performance of EmbodiedGPT with Visual ChatGPT [33], which adopts a hierarchical approach by combining several pre-trained vision models and language models to answer questions. In the Virtual-Home [57] benchmark, Visual ChatGPT uses a visual caption model to generate dense captions that are subsequently passed into ChatGPT for deriving a solution. As shown in Figure 3, Visual ChatGPT failed to find a coat hanger due to its limitations of relying solely on the caption model for extracting visual information, resulting in poor performance when compared to the end-to-end model like EmbodiedGPT. These findings highlight the advantages of adopting a unified, end-to-end model over hierarchical approaches that rely on multiple stages. ",
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"text": "Evaluation on video input embodied AI tasks. We evaluate the recognition ability of videos and planning abilities of our model for embodied control tasks on standard embodied AI benchmarks, Franka Kitchen [14] and Meta-World [15]. Meta-World provides a challenging set of tasks that require complex object manipulation skills, including assembling a ring on a peg, picking and placing a block between bins, pushing a button, opening a drawer, and hammering a nail. Franka Kitchen benchmark focuses on tasks like sliding open the right door, opening the cabinet, turning on the light, turning the stovetop knob, and opening the microwave. As shown in Figure 4, given a demonstration video, EmbodiedGPT can accurately interpret the embodied control task and provide step-by-step planning. The output planning is fed into the Embodied-former module of EmbodiedGPT to query highly relevant features for use as inputs in the policy network and the low-level actions are generated by the policy network to interact with the environment (see more visualizations in Appendix B). ",
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"Figure 5: Performance of EmbodiedGPT in low-level control tasks with 10 demonstration demos. "
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"image_caption": [
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"Figure 6: Performance of EmbodiedGPT in low-level control tasks with 25 demonstration demos. "
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"text": "Evaluation on embodied control tasks. For embodied control tasks, we compare our model with R3M[12], which is the state-of-the-art method in these two benchmarks, and an ablation version called ’BLIP-2[Ego4D]’, which has the same structure and same amount of parameters as EmbodiedGPT, and is only fine-tuned on the video caption task using the Ego4D dataset without incorporating EgoCOT. In all experiments, the policy network is learned using few-shot learning on a small amount of demonstration data. There are two settings, one of which utilizes 10 demonstrations, and the other utilizes 25 demonstrations. We report the success rate in 100 random evaluations with only visual observations in 5 tasks per benchmark over 5 seeds and 2 different camera views for each setting, respectively. As shown in Figure 5 and Figure 6, EmbodiedGPT outperforms the baseline methods, demonstrating the effectiveness of learning with EgoCOT. ",
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"table_caption": [],
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"table_footnote": [
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"Table 2: Ablation on the closed-loop spans from planning to low-level control, and \"chain-of-thought\" (COT) training with 25 and 10 demonstrations(\"-\" symbol indicates \"removing\"). We report the average success rate over 5 tasks and 2 camera views per benchmark. "
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],
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"table_body": "<table><tr><td>Model</td><td></td><td></td><td></td><td>Franka(10 demos） Franka(25 demos）Meta-World(10 demos）Meta-World(25 demos)</td></tr><tr><td>EmbodiedGPT</td><td>50.8% ±2.8</td><td>58.5% ±2.7</td><td>76.4% ±2.2</td><td>81.2%±2.0</td></tr><tr><td> - Close-loop</td><td>38.6% ±2.9</td><td>47.3% ±2.5</td><td>62.7% ±2.2</td><td>64.9% ±2.0</td></tr><tr><td>- COT</td><td>26.2% ±3.2</td><td>36.4% ±2.7</td><td>55.2% ±2.4</td><td>58.7% ±2.0</td></tr></table>",
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"text": "Human: You are the controller of the robot shown in this image, now your task is put all the objects into the brown bag next to the robot, please write a step by step plan: ",
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"text": "EmbodiedGPT: ",
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"text_level": 1,
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},
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"type": "text",
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"text": "Step1. Move the robotic arm to the top of the gray bottle. \nStep 2. Close the gripper and grasp the gray bottle. \nStep 3. Move the arm to the brown bag. \nStep 4. Release the bottle. \nStep 5. Move the arm to the top of the cereal bag. \nStep 6. Close the gripper and grasp the cereal bag. \nStep 7. Move the arm to the brown bag. \nStep 8. Finally, release the cereal bag. ",
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"image_caption": [
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"Franka Emika Panda arm with front facing camera "
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"image_caption": [
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"Figure 7: Generated plan by EmbodiedGPT for rearrangement task. ",
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"Figure 8: Real-world experiment on the Franka Emika robot arm. "
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"text": "Ablation study. We perform ablation studies to analyze the effectiveness of the \"Chain-of-Thought\" training mode and the importance of a closed-loop design for embodied control. The results, as shown in Table 2, demonstrate a significant improvement in success rate when using the EgoCOT approach compared to training solely with the EGO4D caption task. Moreover, the closed-loop design is necessary as the generated plans contained specific and relevant sub-goal information, which proved crucial for control tasks. In summary, EmbodiedGPT exhibits a strong ability to generate reasonable planning, accurately extract task-relevant features from visual inputs, as well as execute low-level actions to interact with the environment. The ablation experiments demonstrate that both the training paradigm based on EgoCOT and the closed-loop design from embodied planning to low-level control significantly contribute to the performance improvement of EmbodiedGPT. ",
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"text": "Real robot experiment. We also conducted a real-world experiment using the Franka Emika robot arm, emphasizing a rearrangement task where the objective was to pack scattered bottles from a table into a box. As depicted in Figures 7 and 8, model is capable of generating intricate plans and executing of low-level actions based on 50 demonstration. ",
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"text": "5 Conclusion ",
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"text": "In this paper, we present EmbodiedGPT, an end-to-end multi-modal foundational model for embodied AI that enables agents to perform step-by-step planning and execute low-level commands. To achieve this, we create a large-scale embodied planning dataset called EgoCOT and develop an efficient training approach that utilizes prefix tuning to generate high-quality plans with a \"chain-of-thought\". Furthermore, our embodied control paradigm seamlessly coordinates high-level planning and lowlevel control. Extensive experiments demonstrate the effectiveness of EmbodiedGPT on various embodied tasks, achieving state-of-the-art or comparable performance. We believe that EmbodiedGPT represents a significant step towards developing more intelligent embodied AI agents. ",
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"text": "Future works and limitations: EmbodiedGPT freezes the parameters of the vision and language model due to limited computational resources. Joint training with all modules and exploring other modalities, such as speech, could be future works. We do not foresee obvious undesirable ethical or social impacts at this moment. ",
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"text": "Acknowledgement. This paper is partially supported by the National Key R&D Program of China No.2022ZD0161000 and the General Research Fund of Hong Kong No.17200622. We would like to express our appreciation to Dr. René Zurbrügg from ETH AI Center for his support on robot experimental platform. ",
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|
| 1 |
+
# ROBUST UNIVERSAL ADVERSARIAL PERTURBATIONS
|
| 2 |
+
|
| 3 |
+
Anonymous authors Paper under double-blind review
|
| 4 |
+
|
| 5 |
+
# ABSTRACT
|
| 6 |
+
|
| 7 |
+
Universal Adversarial Perturbations (UAPs) are imperceptible, image-agnostic vectors that cause deep neural networks (DNNs) to misclassify inputs from a data distribution with high probability. In practical attack scenarios, adversarial perturbations may undergo transformations such as changes in pixel intensity, rotation, etc. while being added to DNN inputs. Existing methods do not create UAPs robust to these real-world transformations, thereby limiting their applicability in attack scenarios. In this work, we introduce and formulate robust UAPs. We build an iterative algorithm using probabilistic robustness bounds and transformations generated by composing arbitrary sub-differentiable transformation functions to construct such robust UAPs. We perform an extensive evaluation on the popular CIFAR-10 and ILSVRC 2012 datasets measuring our UAPs’ robustness under a wide range common, real-world transformations such as rotation, contrast changes, etc. Our results show that our method can generate UAPs up to $2 3 \%$ more robust than existing state-of-the-art baselines.
|
| 8 |
+
|
| 9 |
+
# 1 INTRODUCTION
|
| 10 |
+
|
| 11 |
+
Deep neural networks (DNNs) have achieved impressive results in many application domains such as natural language processing (Abdel-Hamid et al., 2014; Brown et al., 2020), medicine (Esteva et al., 2017; 2019), and computer vision (Simonyan & Zisserman, 2014; Szegedy et al., 2016). Despite their performance, they can be fragile in the face of adversarial perturbations: small imperceptible changes added to a correctly classified input that make a DNN misclassify. While there is a large amount of work on generating adversarial perturbations (Szegedy et al., 2013; Goodfellow et al., 2014; Moosavi-Dezfooli et al., 2016; Madry et al., 2017; Carlini & Wagner, 2017; Xiao et al., 2018a; Dong et al., 2018; Croce & Hein, 2019; Wang et al., 2019; Zheng et al., 2019; Andriushchenko et al., 2019; Tramèr et al., 2020), the threat model considered by these works cannot be realized in practical scenarios. This is because the threat model depends upon unrealistic assumptions about the power of the attacker: the attacker knows the DNN input in advance, generates input-specific perturbations in real-time and exactly combines the perturbation with the input before being processed by the DNN.
|
| 12 |
+
|
| 13 |
+
Practically feasible adversarial perturbations. In this work, we consider a more practical adversary to reveal real-world vulnerabilities of state-of-the-art DNNs. We assume that the attacker (i) does not know the DNN inputs in advance, (ii) can only transmit additive adversarial perturbations, and (iii) their transmitted perturbations are susceptible to modification due to real-world effects. Examples of attacks in our threat model include adding stickers to the cameras for fooling image classifiers (Li et al., 2019b) or transmitting perturbations over the air for deceiving audio classifiers (Li et al., 2019a). Note that this threat model is distinct from directly generating adversarial examples (Athalye et al., 2018) which require access to the original input.
|
| 14 |
+
|
| 15 |
+
The first two requirements in our threat model can be fulfilled by generating Universal Adversarial Perturbations (UAPs) (Moosavi-Dezfooli et al., 2017). Here the attacker can train a single adversarial perturbation that has a high probability of being adversarial on all inputs in the training distribution. However, as our experimental results show, the generated UAPs need to be combined with the DNN inputs precisely, otherwise they fail to remain adversarial. In practice, changes to UAPs are likely due to real-world effects. For example, the stickers applied to a camera can undergo changes in contrast due to weather conditions or the transmitted perturbation in audio can change due to noise in the transmission channel. This non-robustness reduces the efficiency of practical attacks created with existing methods (Moosavi-Dezfooli et al., 2017; Shafahi et al., 2020; Li et al., 2019b;a).
|
| 16 |
+
|
| 17 |
+
|
| 18 |
+
Figure 1: Robust UAPs (left) cause a classier to misclassify on most of the data distribution even after transformations are applied on them. Standard UAPs (right) are not robust to transformations and have a low probability of remaining UAPs after transformation.
|
| 19 |
+
|
| 20 |
+
This work: Robust UAPs. To overcome the above limitation, we propose the concept of robust UAPs: perturbations that have a high probability of remaining adversarial on inputs in the training distribution even after applying a set of real-world transformations. The optimization problem in generating robust UAPs (Moosavi-Dezfooli et al., 2017) is the main challenge as we are looking for perturbations that are adversarial for a set of inputs as well as to transformations applied to the perturbations. To address this challenge, we make the following main contributions:
|
| 21 |
+
|
| 22 |
+
• We introduce Robust UAPs and formulate their generation as an optimization problem.
|
| 23 |
+
• We design a new method for constructing robust UAPs. Our method is general and works for any transformations generated by composing arbitrary sub-differentiable transformation functions. We provide an algorithm for computing provable probabilistic bounds on the robustness of our UAPs against many practical transformations.
|
| 24 |
+
• We perform an extensive evaluation of the effectiveness of our method, RobustUAP, on stateof-the-art models for the popular CIFAR-10 (Krizhevsky et al., 2009) and ILSVRC 2012 (Deng et al., 2009) datasets. We compare the robustness of our UAPs under compositions of challenging real-world transformations, such as rotation, contrast change, etc. We show that on both datasets, the UAPs generated by RobustUAP are significantly more robust, achieving up to $2 3 \%$ more robustness, than the UAPs generated from the baselines.
|
| 25 |
+
|
| 26 |
+
Our work is complementary to the development of real-world attacks (Li et al., $2 0 1 9 \mathrm { a } ; \mathrm { b } )$ ) in various domains, which require modeling how the universal perturbations change during transmission. RobustUAP can improve the efficiency of such attacks by constructing perturbations that are more robust against domain-specific, real-world transformations than possible with existing algorithms (Moosavi-Dezfooli et al., 2017; Shafahi et al., 2020; Li et al., 2019a;b).
|
| 27 |
+
|
| 28 |
+
# 2 BACKGROUND
|
| 29 |
+
|
| 30 |
+
In this section, we provide necessary background definitions and notation for our work.
|
| 31 |
+
|
| 32 |
+
Adversarial Examples and Perturbations. An adversarial example is a misclassified data point that is close (in some norm) to a correctly classified data point (Goodfellow et al., 2014; Madry et al., 2017; Carlini & Wagner, 2017). Let $\mu \doteq \mathbb { R } ^ { d }$ be the input data distribution, $\mathbf { x } \in \mu$ be an input point with the corresponding true label $y \in \mathbb { R }$ , and $f : \mathbb { R } ^ { \bar { d } } \mathbb { R } ^ { d ^ { \prime } }$ be our target classifier. For ease of notation, we define $f _ { k } ( { \bf x } )$ to be the $k ^ { \mathrm { { t h } } }$ element of $f ( \mathbf { x } )$ and allow ${ \hat { f } } ( \mathbf { x } ) = \arg \operatorname* { m a x } _ { k } f _ { k } ( \mathbf { x } )$ to directly refer to the classification label. We use $\mathbf { v }$ to reference image specific perturbations and $\mathbf { u }$ to reference universal adversarial perturbations, $\mathbf { v _ { r } }$ and $\mathbf { u _ { r } }$ refer to the robust variants and will be defined in Sec. 3. We now formally define an adversarial example.
|
| 33 |
+
|
| 34 |
+
Definition 2.1. Given a correctly classified point $\mathbf { x }$ , a distance function $d ( \cdot , \cdot ) : \mathbb { R } ^ { d } \times \mathbb { R } ^ { d } \to \mathbb { R }$ , and bound $\epsilon \in \mathbb { R }$ , $\mathbf { x } ^ { \prime }$ is an adversarial example iff $d ( { \bf x } ^ { \prime } , { \bf x } ) < \epsilon$ and $\hat { f } ( { \bf x } ^ { \prime } ) \neq y$ .
|
| 35 |
+
|
| 36 |
+
In this paper, we consider examples $\mathbf { x } ^ { \prime }$ generated as $\mathbf { x } ^ { \prime } = \mathbf { x } + \mathbf { v }$ where $\mathbf { v }$ is an adversarial perturbation.
|
| 37 |
+
|
| 38 |
+
Universal Adversarial Perturbations. UAPs are single vector, input-agnostic perturbations (Moosavi-Dezfooli et al., 2017). They differ from traditional adversarial attacks, which create perturbations dependent on each input sample. To measure UAP performance, we introduce the notion of universal adversarial success rate, which measures the probability that a perturbation $\mathbf { u }$ when added to $\mathbf { x }$ , sampled from $\mu$ , causes a change in classification under $f$ .
|
| 39 |
+
|
| 40 |
+
Definition 2.2. Given a data distribution $\mu$ , and perturbation u, universal adversarial success rate $\operatorname { A S R } _ { U }$ for $\mathbf { u }$ , is defined as
|
| 41 |
+
|
| 42 |
+
$$
|
| 43 |
+
\operatorname { A S R } _ { U } ( f , \mu , { \mathbf { u } } ) = \underset { { \mathbf { x } } \sim \mu } { P } \left( \hat { f } ( { \mathbf { x } } + { \mathbf { u } } ) \neq \hat { f } ( { \mathbf { x } } ) \right)
|
| 44 |
+
$$
|
| 45 |
+
|
| 46 |
+
Using Definition 2.2, we formally define a UAP.
|
| 47 |
+
|
| 48 |
+
Definition 2.3. A universal adversarial perturbation is a vector $\mathbf { u } \in \mathbb { R } ^ { d }$ which, when added to almost all datapoints in $\mu$ causes the classifier $f$ to misclassify. Formally, given $\gamma$ , a bound on universal ASR, and $l _ { p }$ -norm with corresponding bound $\epsilon$ , $\mathbf { u }$ is a UAP iff $\mathrm { A S R } _ { U } ( f , \mu , { \bf u } ) > \gamma$ and $| | \mathbf { u } | | _ { p } < \epsilon$ .
|
| 49 |
+
|
| 50 |
+
In general, if the additive perturbations have small $l _ { p }$ -norm , then they look like noise and do not affect the semantic content of the image. For ease of notation in later parts of the paper, we can also pose the construction of UAPs as an expectation minimization problem:
|
| 51 |
+
|
| 52 |
+
$$
|
| 53 |
+
\underset { u } { \arg \operatorname* { m i n } } \mathbb { E } _ { \mathbf { x } \sim \mu } [ \delta ( \hat { f } ( \mathbf { x } + \mathbf { u } ) , \hat { f } ( \mathbf { x } ) ) ] \mathrm { s . t . } | | \mathbf { u } | | _ { p } < \epsilon
|
| 54 |
+
$$
|
| 55 |
+
|
| 56 |
+
where $\delta$ is the Kronecker Delta function (Agarwal, 2013).
|
| 57 |
+
|
| 58 |
+
# 3 ROBUST UNIVERSAL ADVERSARIAL PERTURBATIONS
|
| 59 |
+
|
| 60 |
+
In this section, we will define the optimization problem for generating robust UAPs. We first define transformation sets and neighborhoods.
|
| 61 |
+
|
| 62 |
+
Definition 3.1. A transformation, $\tau$ , is a composition of bijective sub-differentiable transformation functions. A transformation set, $T$ , is a set of distinct transformations. A point $\mathbf { v } ^ { \prime }$ is in the neighborhood $N _ { T } ( \mathbf { v } )$ , of $\mathbf { v }$ , if there is a transform in $T$ that maps $\mathbf { v }$ to $\mathbf { v } ^ { \prime }$ . Formally,
|
| 63 |
+
|
| 64 |
+
$$
|
| 65 |
+
\mathbf { v } ^ { \prime } \in N _ { T } ( \mathbf { v } ) \iff \exists \tau \in T { \mathrm { ~ s . t . ~ } } \tau ( \mathbf { v } ) = \mathbf { v } ^ { \prime }
|
| 66 |
+
$$
|
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Example 3.2. Let $T$ be the set of all transformations represented by a rotation of $\pm 3 0 ^ { \circ }$ , scaling of up to a factor of 2, and a translation of up to $\pm 2$ pixels, in this case one $\tau \in T$ could be {rotation of $8 ^ { \circ }$ , scaling a factor of 1.2, and translation of -1.3} in that order and $N _ { T } ( \mathbf { v } )$ would include any point that can be obtained by applying one of the transformations from $T$ on $\mathbf { v }$ .
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In order to define robust UAPs we introduce robust universal adversarial success rate.
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Definition 3.3. Given a data distribution $\mu$ , transformation set $T$ , universal ASR level $\gamma$ , bound $\epsilon$ on $l _ { p }$ -norm, and perturbation $\mathbf { u _ { r } }$ , robust universal adversarial success rate, $\operatorname { A S R } _ { R }$ , is defined as,
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$$
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\mathbf { A S R } _ { R } ( f , \mu , T , \gamma , \mathbf { u _ { r } } ) = \underset { \mathbf { u } _ { r } ^ { \prime } \sim N _ { T } ( \mathbf { u _ { r } } ) } { \cal P } ( \mathbf { A S R } _ { U } ( f , \mu , \mathbf { u } _ { r } ^ { \prime } ) > \gamma \land | | \mathbf { u } _ { \mathbf { r } } ^ { \prime } | | _ { p } < \epsilon )
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+
$$
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The robust universal adversarial success rate measures the probability that a neighbor of $\mathbf { u _ { r } }$ is also an UAP on $\mu$ , i.e. after transformation it maintains high universal ASR. We note that even though $| | \mathbf { u } _ { \mathbf { r } } | | _ { p } \leq \epsilon$ , it can happen that a ${ \bf u } _ { \bf r } ^ { \prime } \in \ d { \cal N } _ { T } ( { \bf u } _ { \bf r } )$ has $| | \mathbf { u } _ { \mathbf { r } } ^ { \prime } | | _ { p } > \epsilon$ , this is particularly true for the semantic transformations considered in this work. Therefore, we require that the norm of $\mathbf { u } _ { \mathbf { r } } ^ { \prime }$ is small.
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Using Definition 3.3 we can now formally define a robust UAP.
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Definition 3.4. A robust universal adversarial perturbation, $\mathbf { u _ { r } }$ , is one which most points within a neighborhood of $\mathbf { u _ { r } }$ when added to most points in $\mu$ fool the classifier, $f . \mathbf { u _ { r } }$ satisfies $\bar { | } | \mathbf { u _ { r } } | | _ { p } < \epsilon$ and $\mathrm { A S R } _ { R } ( f , \mu , T , \gamma , \mathbf { u _ { r } } ) > \zeta$ .
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In order to construct robust UAPs, we can pose the following expectation minimization problem:
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$$
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\underset { \mathbf { u } _ { \mathbf { r } } } { \arg \operatorname* { m i n } } \ \underset { \mathbf { u } _ { \mathbf { r } } ^ { \prime } \in N _ { T } ( \mathbf { u } _ { \mathbf { r } } ) } { \mathbb { E } } [ I ( | | \mathbf { u } _ { \mathbf { r } } ^ { \prime } | | < \epsilon ) \times \underset { \mathbf { x } \sim \mu } { \mathbb { E } } [ \delta ( \hat { f } ( \mathbf { x } + \mathbf { u } _ { \mathbf { r } } ^ { \prime } ) , \hat { f } ( \mathbf { x } ) ) ] ] \ \mathrm { s . t . } \ | | \mathbf { u } _ { \mathbf { r } } | | _ { p } < \epsilon
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$$
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Here $I : \mathbb { R } ^ { d } \mathbb { R }$ denotes an indicator function. The inner expectation represents the UAP condition for the transformed perturbation $\mathbf { u } _ { \mathbf { r } } ^ { \prime }$ while the outer expectation represents the neighborhood robustness condition. Solving Equation 5 requires computing $\mathbf { u _ { r } }$ which minimizes the expectation over the transformation set and data distribution. This composition makes it computationally harder than minimizing over only the transformation set, as in EOT (Athalye et al., 2018), or than minimizing over only the data distribution, as done for standard UAP (Moosavi-Dezfooli et al., 2017).
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# 4 GENERATING ROBUST UNIVERSAL ADVERSARIAL PERTURBATIONS
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In this section, we will discuss our approach for optimizing Equation 5 to generate UAPs robust to transformations generated by a composition of arbitrary sub-differentiable transformation functions. At a high level, the objective can be seen as gluing the outer expectation, a EOT objective over the transformations applied on the perturbation, with the inner expectation, a UAP objective over the input data distribution. We first describe intuitive baselines for optimizing Equation 5 and then present our new algorithm, RobustUAP.
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# 4.1 STOCHASTIC GRADIENT DESCENT
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The first baseline directly solves Equation 5 using gradient descent. Since we are solving a constrained optimization problem, we cannot use gradient descent directly. Instead, we can solve the Lagrangianrelaxed form of the problem as in (Carlini & Wagner, 2017; Athalye et al., 2018).
|
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+
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+
$$
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\underset { \mathbf { u } _ { \mathbf { r } } } { \arg \operatorname* { m i n } } \ \underset { \mathbf { u } _ { \mathbf { r } } ^ { \prime } \in N _ { T } ( \mathbf { u } _ { \mathbf { r } } ) } { \mathbb { E } } [ I ( | | \mathbf { u } _ { \mathbf { r } } ^ { \prime } | | < \epsilon ) \times \underset { \mathbf { x } \sim \mu } { \mathbb { E } } [ \delta ( \hat { f } ( \mathbf { x } + \mathbf { u } _ { \mathbf { r } } ^ { \prime } ) , \hat { f } ( \mathbf { x } ) ) ] ] - \lambda | | \mathbf { u } _ { \mathbf { r } } | | _ { p }
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+
$$
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+
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We use a momentum based Stochastic Gradient Descent (SGD) method for solving Equation 6. Shafahi et al. (2020) suggests that this is an effective method for generating standard UAPs. In order to implement this, we replace the Kronecker Delta function with a loss function, $L$ . We iteratively converge towards the inner expectation by computing it in batches, and towards the outer expectation by sampling a large number of transformations. Given that we would like to estimate on a batch, $\hat { \mathbf { x } } \subset \mu$ , and a random set of transformations sampled from $T$ , $\hat { \tau } \subset T$ , we can approximate Equation 6:
|
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+
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+
$$
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\frac { I ( | | \hat { \tau } _ { j } ( \mathbf { u _ { r } } ) | | < \epsilon ) } { | \hat { \mathbf { x } } | \times | \hat { \tau } | } \sum _ { i = 1 } ^ { | \hat { \mathbf { x } } | } \sum _ { j = 1 } ^ { | \hat { \tau } | } L [ f ( \hat { \mathbf { x } } _ { \mathbf { i } } + \hat { \tau } _ { j } ( \mathbf { u _ { r } } ) ) , f ( \hat { \mathbf { x } } _ { \mathbf { i } } ) ] - \lambda | | \mathbf { u _ { r } } | | _ { p }
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+
$$
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+
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Our final algorithm is in Appendix C.
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# 4.2 STANDARD UAP ALGORITHM WITH ROBUST ADVERSARIAL PERTURBATIONS
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For our second baseline, we leverage the standard UAP algorithm from Moosavi-Dezfooli et al. (2017) (see Appendix D for the algorithm). The standard UAP algorithm iterates over the entire training dataset and at each input, $\mathbf { x _ { i } }$ , computes the smallest additive change, $\Delta \mathbf { u }$ , to the current perturbation, u, that would make $\mathbf { u } + \Delta \mathbf { u }$ an adversarial perturbation for $\mathbf { x _ { i } }$ . Intuitively, over time the algorithm will approach a perturbation that works on most inputs in the training dataset. This approach works by computing robust adversarial perturbations rather than standard adversarial perturbations. At each point $\mathbf { x _ { i } }$ , we compute the smallest additive change, $\Delta \mathbf { u _ { r } }$ , to the current robust adversarial perturbation, $\mathbf { u _ { r } }$ , that would make $\mathbf { u } _ { \mathbf { r } } + \Delta \mathbf { u } _ { \mathbf { r } }$ a robust adversarial perturbation for $\mathbf { x _ { i } }$ .
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+
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We search for robust adversarial perturbations by optimizing the expectation that a point in the neighborhood of $\mathbf { v _ { r } }$ is adversarial while restricting the perturbation to an $l _ { p }$ norm of $\epsilon$ . We formulate this as the following minimization problem:
|
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+
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+
$$
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+
\underset { \mathbf { v _ { r } } } { \arg \operatorname* { m i n } } \quad \underset { \mathbf { v _ { r } ^ { \prime } } \in { \cal { W } } _ { T } ( \mathbf { v _ { r } } ) } { \mathbb { E } } [ I ( | | \mathbf { v _ { r } ^ { \prime } } | | < \epsilon ) \times \delta ( \hat { f } ( \mathbf { x } + \mathbf { v } _ { \mathbf { r } } ^ { \prime } ) , \hat { f } ( \mathbf { x } ) ) ] \mathrm { ~ s . t . ~ } | | \mathbf { v _ { r } } | | _ { p } < \epsilon
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+
$$
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+
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+
# 4.3 ROBUST UAP ALGORITHM
|
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+
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The baseline algorithms have two fundamental limitations: (i) they rely on random sampling over the symbolic transformation region, but the sampling strategy does not explicitly try to maximize the robustness of the generated UAP over the entire symbolic region, and (ii) they do not estimate robustness on unsampled transformations. As a result, the baselines yield suboptimal UAPs (as confirmed by our experiments below). To overcome these fundamental limitations, we create a method to compute probabilistic bounds for expected robustness on an entire symbolic region. We leverage this method for approximating expected robustness in a new algorithm to generate robust UAPs with guarantees. We make a simplifying assumption that ${ { N } _ { T } } ( { \bf { u } } _ { \bf { r } } )$ has a well defined, sampleable probability density function (PDF) as we cannot bound robustness for arbitrary transformations. Our experiments show that even though our assumptions do not hold for all the transformation sets considered in this work, they significantly improve the robustness of our generated UAPs. Our approximation of the expected robustness relies on the following theoretical result:
|
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+
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+
Theorem 4.1. Given a perturbation $\mathbf { u _ { r } }$ , a neural network $f _ { i }$ , a finite set of inputs $\mathbf { X }$ , a set of transformations $T$ , and minimum universal adversarial success rate $\gamma ~ \in ~ \mathbb { R }$ . Let $p ( \gamma ) =$ $P _ { \mathbf { u } _ { \mathbf { r } } ^ { \prime } \sim N _ { T } ( \mathbf { u } _ { \mathbf { r } } ) } ( A S R _ { U } ( f , \mathbf { X } , \mathbf { u } _ { \mathbf { r } } ^ { \prime } ) > \gamma )$ . For $i \in { 1 \dots n }$ , let ${ \bf u } _ { \bf r } ^ { \bf i } \sim N _ { T } ( { \bf u } _ { \bf r } )$ be random variables with a well defined PDF and $I : \mathbb { R } ^ { d } \mathbb { R }$ be the indicator function, let
|
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+
|
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+
$$
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+
\hat { p } _ { n } ( \gamma ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } I ( A S R _ { U } ( f , \mathbf { X } , \mathbf { u } _ { \mathbf { r } } ^ { \mathbf { i } } ) > \gamma )
|
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+
$$
|
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+
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+
For accura0 and 1. If $\psi \in ( 0 , 1 )$ , and confidence, $\phi \in ( 0 , 1 )$ , where $( 0 , 1 )$ is the open interval between $\begin{array} { r } { \dot { n } \geq \frac { 1 } { 2 \psi ^ { 2 } } \ln \frac { 2 } { \phi } } \end{array}$
|
| 133 |
+
|
| 134 |
+
$$
|
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+
P ( | \hat { p } _ { n } ( \gamma ) - p ( \gamma ) | < \psi ) \geq 1 - \phi
|
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+
$$
|
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+
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+
Proof. The bound on $n$ is derived via the Chernoff inequality applied to $\hat { p } _ { n } ( \gamma )$ and $\mathbb { E } [ \hat { p } _ { n } ( \gamma ) ] =$ $p ( \gamma )$ (Chernoff, 1952; Alippi, 2014). Equation 10 holds since computing universal ASR is Lebesgue measurable over the data distribution and since we assume ${ { N } _ { T } } ( { \bf { u } } _ { \bf { r } } )$ has a well defined PDF. $\boxed { \begin{array} { r l } \end{array} }$
|
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+
|
| 140 |
+
Theorem 4.1 states that with enough samples from the neighborhood of a perturbation, $\mathbf { u _ { r } }$ , the adversarial success rate of $\mathbf { u _ { r } }$ on the entire neighborhood is arbitrarily close to the adversarial success rate of $\mathbf { u _ { r } }$ on sampled transformations with probability greater than $1 - \phi$ . One key observation is that the Chernoff bound is independent of the dimensionality of the sample space which allows us to efficiently apply this result to high-dimensional transformation set provided they have a well-defined PDF (e.g., $L _ { \infty }$ -ball) and obtain provable bounds on the expected robustness. For the combinations of semantic transformations, such as rotation, translation, etc. used in the experiment section the neighborhood does not have a well-defined PDF, thus we uniformly sample the parameter space of each transformation to produce a point in the neighborhood. We believe uniformly sampling the parameter space is a realistic approximation of real-world effects.
|
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+
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+
Leveraging Theorem 4.1, we create EstimateRobustness which given accuracy, $\psi$ , and confidence, $\phi$ , returns the robust adversarial success rate on a finite set of inputs with probabilistic robustness guarantees under the assumptions of Theorem 4.1. The pseudocode for EstimateRobustness is in Algorithm 1
|
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+
|
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+
# Algorithm 1 EstimateRobustness
|
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+
|
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+
Our algorithm: RobustUAP. We leverage Theorem 4.1 and Algorithm 1 to develop RobustUAP, the pseudocode for which is seen in Algorithm 2. Similar to the SGD baseline, we approximate the expectation in Equation 5 in batches. We start by sampling transformations from the PDF of the neighborhood. We set the number of transformations, $n$ , based on Theorem 4.1 to satisfy the desired confidence level and accuracy. For each gradient step, we compute the mean loss over the current batch and set of sampled transforms (line 8). For each set of batch and sampled transformations, instead of making a single gradient update like SGD, we use Projected Gradient Descent (PGD) to iteratively compute a more robust update to the universal perturbation and end only when the estimated robustness on the batch satisfies a given threshold (line 10). At the end of each epoch, we check the robustness across the entire training set and transformation space using EstimateRobustness and stop when we have reached the desired performance (line 14).
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+
|
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+
# Algorithm 2 Robust UAP Algorithm
|
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+
|
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+
1: Init $\begin{array} { r } { \mathbf { u } _ { \mathbf { r } } 0 , n \lceil \frac { 1 } { 2 \psi ^ { 2 } } \ln \frac { 2 } { \phi } \rceil } \end{array}$
|
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+
$i = 1 \dots n$ $\tau _ { i } \sim T$
|
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+
3: repeat
|
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+
4: for $\mathbf { B } \subset \mathbf { X }$ do
|
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+
5: if EstimateRobustnes $\mathbf { \Phi } _ { \mathsf { S } } ( f , \mathbf { B } , T , \gamma , \mathbf { u _ { r } } , \psi , \phi ) < \zeta$ then
|
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+
6: $\Delta \mathbf { u _ { r } } \gets 0$
|
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+
7: repeat
|
| 157 |
+
8: Compute $\begin{array} { r } { L _ { \mathbf { B } , \tau } = \frac { 1 } { | \mathbf { B } | \times n } \sum _ { i = 1 } ^ { | \mathbf { B } | } \sum _ { j = 1 } ^ { n } L [ f ( \mathbf { B _ { i } } + \tau _ { j } ( \mathbf { u _ { r } } + \Delta \mathbf { u _ { r } } ) ) , f ( \mathbf { B _ { i } } ) ] } \end{array}$
|
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+
9: $\Delta \mathbf { u } _ { \mathbf { r } } = \mathcal { P } _ { p , \epsilon } ( \Delta \mathbf { u } _ { \mathbf { r } } + \alpha \mathrm { s i g n } ( \nabla L _ { \mathbf { B } , \tau } ) )$
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+
10: until EstimateRobustnes $\mathrm { s } ( f , \mathbf { B } , T , \gamma , \mathbf { u _ { r } } + \Delta \mathbf { u _ { r } } , \psi , \phi ) < \zeta$
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+
11: Update the perturbation with projection: $\mathbf { u _ { r } } \gets \mathcal { P } _ { p , \epsilon } ( \mathbf { u _ { r } } + \Delta \mathbf { u _ { r } } )$
|
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+
12: end if
|
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+
13: end for
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+
14: until EstimateRobustne $\mathfrak { s s } ( f , \mathbf { X } , T , \gamma , \mathbf { u _ { r } } , \psi , \phi ) < \zeta$
|
| 164 |
+
|
| 165 |
+
# 5 EVALUATION
|
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+
|
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+
Our RobustUAP framework is applicable to all transformation sets in a variety of domains. We empirically evaluate our method RobustUAP and three baseline approaches (SGD, StandardUAP_RP, StandardUAP (Moosavi-Dezfooli et al., 2017)) on popular models from the vision domain. We show that RobustUAP is more robust on both uniform random noise and compositions of real-world transformations such as rotation, scaling, etc.
|
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+
|
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Experimental evaluation. We consider two popular image recognition datasets: CIFAR10(Krizhevsky et al., 2009) and ILSVRC 2012(Deng et al., 2009). For CIFAR-10, we evaluate on the entire test set (1,000 images) and use a state-of-the-art pretrained VGG16 (Simonyan & Zisserman, 2014) network as the target classification model. For ILSVRC 2012, we evaluate on a random subset of the test set (1,000 images), and use a state-of-the-art Inception-v3 (Szegedy et al., 2016) network. We evaluate the robustness against uniform random noise as well as a composition of transformations from brightness/contrast, rotation, scaling, shearing, and translation. All experiments were performed on a desktop PC with a GeForce RTX(TM) 3090 GPU and a 16-core Intel(R) Core(TM) i9-9900KS CPU $\textcircled { a } 4 . 0 0 \mathrm { G H z }$ .
|
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We report the results for $l _ { 2 }$ -norm with $\epsilon = 1 0 0$ for ILSVRC 2012 and $\epsilon = 1 0$ for CIFAR-10. These values were chosen based on the values presented by the original UAP paper (Moosavi-Dezfooli et al., 2017). We use an image normalization function given by our pretrained models and thus scaled our $\epsilon$ values accordingly. We note that the $\epsilon$ -values are significantly smaller than the image norms. Therefore the generated perturbation is imperceptible and does not affect the semantic content of the image. Due to the hardness of the optimization problem, for the same norm value, the effectiveness of a UAP is less than input-specific perturbations. We note that crafting input-specific perturbations requires making unrealistic assumptions about the power of the attacker as mentioned in the introduction and therefore we do not consider them part of our threat model which aims to generate practically feasible perturbations. We use $\psi = 0 . 0 5$ and $\phi = 0 . 0 5$ resulting in $n = 7 3 8$ for generating samples for our RobustUAP algorithm as well as reporting robust ASR in our evaluation. The UAPs are trained on 2,000 images, other parameters for evaluation are given in Appendix E.
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+
# 5.1 ROBUSTNESS TO RANDOM NOISE
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+
|
| 175 |
+
First, we show that our algorithm generates UAPs robust against uniform random noise. Here our neighborhood is defined as an $L _ { \infty }$ ball of radius $\epsilon$ around the perturbation. $U ( \epsilon )$ represents noise drawn uniformly from such a ball. Figure 2 shows the performance of each algorithm. For example, the RobustUAP algorithm achieves a $\operatorname { A S R } _ { U }$ of 0.9 greater than $9 7 \%$ of the time under $U ( 0 . { \bar { 1 } } )$ on CIFAR-10, where all other algorithms achieve 0.9 at most $3 0 \%$ of the time. RobustUAP outperforms all other algorithms for both noise sizes. StandardUAP has a lower mean and higher variance in universal ASR and is much less robust to transformation. A table of Robust ASR results for $\gamma = 0 . 8$ can be seen in Appendix F. Our Robust ASR results are guaranteed to be $\pm 0 . 0 5$ from the actual result with a probability of $9 5 \%$ . For example, we estimate that RobustUAP has $\operatorname { A S R } _ { R }$ of $9 6 . 1 \%$ for U(0.3), we are guaranteed that the true robustness is $> 9 1 . 1 \%$ with a probability of $9 5 \%$ . Note that we get robustness guarantees from EstimateRobustness as our neighborhood has a well-defined PDF.
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+
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+
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+
Figure 2: For each method, a point $( x , y )$ in the corresponding line represents the percentage of sampled UAPs $( y \% )$ with Universal $\mathbf { A } \mathbf { S } \mathbf { R } > x$ for $U ( 0 . 1 )$ and $U ( 0 . 3 )$ on ILSVRC and CIFAR-10.
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+
# 5.2 ROBUSTNESS TO SEMANTIC TRANSFORMATIONS
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|
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+
Next, we consider transformation sets generated by composing five popular semantic transformations in existing literature (Athalye et al., 2018; Balunovic et al., 2019): brightness/contrast, rotation, ´ scaling, shearing, and translation.
|
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+
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We use a variety of different compositions to show that our algorithm works under different conditions, and base our parameters for the transformations on (Balunovic et al., 2019). For our experiments, ´ $R ( \theta )$ corresponds to rotations with angles between $\pm \theta$ ; $T ( x , y )$ , to translations of $\pm x$ horizontally and $\pm y$ vertically; $S c ( p )$ to scaling the image between $\pm p \%$ ; $S h ( m )$ to shearing by shearing factor between $\pm m \%$ ; and $B ( \alpha , \beta )$ to changes in contrast between $\pm \alpha \%$ and brightness between $\pm \beta$ . Further details about these transformations can be seen in Appendix A. We consider compositions of different subsets and ranges of these transformations shown in Table 1 including composing all transformations together. The hardness of generating robust UAPs depends on the effect that the transformation set has on the UAP (i.e. random noise has a relatively small effect compared to rotation). The hardness also increases with the number of transformations in the composition as well as the range of parameters for each individual transformation. For example, generating robust UAPs is harder for the composition shown in the first and last row for ILSVRC 2012 in Table 1 compared to the second and third row. The same is true for generating a UAP robust to uniform random noise.
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+
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+
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Figure 3: For each method, a point $( x , y )$ in the corresponding line represents the percentage of sampled UAPs $( y \% )$ with Universal $\mathrm { A S R } > x$ for the different semantic transformations on ILSVRC.
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+
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Robust ASR $( \mathbf { A S R _ { R } } )$ ). Figure 3 shows performance of UAPs obtained by applying 738 randomly sampled transformations to the original UAPs generated by different methods on ILSVRC, similar graphs for CIFAR-10 can be found in Appendix G. The RobustUAP algorithm outperforms all others in each case, we observe that for these harder transformation sets StandardUAP loses its effectiveness completely. In Table 1 we compare robust universal adversarial success rate $\mathrm { A S R } _ { R }$ with $\gamma = 0 . 6$ , in other words, we are finding the percentage of sampled neighbors of the perturbation that are still UAPs with $6 0 \%$ effectiveness on the testing set. We provide average $\operatorname { A S R } _ { U }$ scores as well as $\mathrm { A S R } _ { R }$ for different $\gamma$ levels in Appendix $_ \mathrm { H }$ .
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+
Our RobustUAP algorithm achieves at least $5 3 . 4 \%$ higher robust ASR when compared to the standard UAP algorithm on both datasets and the challenging transformation sets shown in Table 1. Furthermore, our RobustUAP algorithm significantly outperforms both robust baseline approaches. Except for the $T ( 2 , 2 )$ case which we observe to be the easiest, RobustUAP achieves at least $1 1 . 6 \%$ performance gain over the baselines. SGD is the best performing baseline and achieves high robust ASR on relatively easier transformation sets performing within $\bar { 1 } \%$ of RobustUAP on $T ( 2 , 2 )$ . On harder transformation sets, this gap widens considerably, see Table 1.
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+
|
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+
<table><tr><td>DATASET</td><td>TRANSFORMATION SET</td><td>STANDARD UAP</td><td>SGD</td><td>STANDARD UAP_RP</td><td>ROBUST UAP</td></tr><tr><td rowspan="4">ILSVRC 2012</td><td>R(20)</td><td>0.0%</td><td>69.9%</td><td>2.9%</td><td>93.2%</td></tr><tr><td>T(2,2)</td><td>35.9%</td><td>96.1%</td><td>38.8%</td><td>97.1%</td></tr><tr><td>Sc(5),R(5),B(5,0.01)</td><td>22.3%</td><td>85.4%</td><td>43.7%</td><td>96.1%</td></tr><tr><td>R(10),T(2,2), Sh(2),Sc(2),B(2,0.001)</td><td>0.0%</td><td>63.1%</td><td>2.9%</td><td>86.4%</td></tr><tr><td rowspan="3">CIFAR-10</td><td>R(30),B(2,0.001)</td><td>0.0%</td><td>64.1%</td><td>2.9%</td><td>75.7%</td></tr><tr><td>R(2),Sh(2)</td><td>42.7%</td><td>88.3%</td><td>52.4%</td><td>96.1%</td></tr><tr><td>R(10),T(2,2),Sh(2),Sc(2),B(2,0.001)</td><td>0.0%</td><td>58.3%</td><td>7.8%</td><td>79.6%</td></tr></table>
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+
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+
Table 1: Robust ASR of RobustUAP compared to the three baselines.
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+
|
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Visualization. We visualize UAPs generated with RobustUAP and StandardUAP transformed with random transformations from $\bar { R ( 1 0 ) } , \bar { T ( 2 , 2 ) } , S h ( 2 ) , S c ( 2 ) , \bar { B ( 2 , 0 . 0 0 1 ) }$ and added to images in ILSVRC 2012 in Figure 4. Our robust UAPs have a similar level of imperceptibility to standard UAPs and do not affect the semantic content of the images. Robust UAPs affect the model classification after transformation with high probability, unlike standard UAPs.
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Figure 4: Examples of perturbed images with labels. The top row is unperturbed ILSVRC 2012 test set images, the second row has a randomly transformed robust UAP added to it, and the bottom row has a randomly transformed standard UAP added to it. Labels calculated using Inception-v3.
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We further visualize UAPs generated with our three robust algorithms on the same transformation set against a standard UAP generated on ILSVRC 2012 in Figure 5. We observe that UAPs generated by the StandardUAP algorithm resemble those generated by the StandardUAP_RP algorithm. We believe that this is due to the similarity in the workings of both algorithms. However, the two UAPs are not identical. Under our transformation set the center of the image is least likely to be perturbed so we observe StandardUAP_RP algorithm concentrates its budget towards the center. Both the RobustUAP and the SGD algorithm generate larger patterns distributed over the entire image.
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# 5.3 ADDITIONAL EXPERIMENTS
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In Appendix I we show how our robust UAPs compare to standard UAPs on the non-robust universal ASR metric. In Appendix J, we evaluate our methods on ResNet18 (He et al., 2015) and MobileNet (Howard et al., 2017) for CIFAR-10 and ILSVRC 2012 respectively. The results follow the same trends as those reported in Table 1. In Appendix H we provide the average $\operatorname { A S R } _ { U }$ achieved by all the algorithms and also provide $\operatorname { A S R } _ { R }$ computed with different values of $\gamma$ for the same transformation sets in Table 1. Finally, we provide runtimes for all algorithms in Appendix L.
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Figure 5: Comparison of UAPs generated with (a) StandardUAP, (b) RobustUAP, (c) StandardUAP_RP, and (d) RobustUAP on ILSVRC 2012.
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# 6 RELATED WORK
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In this section, we survey works closely related ours.
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UAP Algorithms. Most works focusing on UAPs (Moosavi-Dezfooli et al., 2017; Mopuri et al., 2018; Zhang et al., 2020a; Khrulkov & Oseledets, 2018; Li et al., 2020; Akhtar et al., 2018; Hendrik Metzen et al., 2017; Zhang et al., 2020b) generate singular vectors and do not consider perturbation robustness. Bahramali et al. (2021) introduces a perturbation generator model (PGM) for the wireless domain which creates UAPs with random trigger patterns. They show that both adversarial training and noise subtracting defenses used in the wireless domain are highly effective in mitigating the effects of a single vector UAP attack; they further show that their method of generating a set of UAPs is an effective way for an attacker to circumvent these defenses. Although PGM provides a method for efficiently sampling unique UAPs, they do not train to be robust to real-world transformations. In contrast, our method enables efficient sampling of UAPs that are robust to transformations.
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Robust Adversarial Examples. The following papers introduce notions of robustness under different viewpoints and environmental conditions for constructing realizable adversarial examples. This is a different threat model compared to the additive perturbations discussed in this paper. Luo et al. (2018) constructs adversarial examples which minimize human detectability, further introducing the idea of robustness for adversarial examples. They show that their attacks are robust against jpeg compression. Sharif et al. (2016) attack facial recognition systems by putting adversarial perturbations on glass frames. Their work demonstrates a successful physical attack under stable conditions and poses. Eykholt et al. (2018) proposes Robust Physical Perturbations $\mathrm { ( R P _ { 2 } ) }$ in order to show that adding graffiti on a stop sign can cause it to be misclassified in both simulations and in the real world. Athalye et al. (2018) introduce Expectation over Transformation (EOT) and use it to print real-world objects which are adversarial given a range of physical and environmental conditions.
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Robust Adversarial Perturbations. Li et al. (2019a) generates music which affects a voice assistant based system from picking up its wake word. Li et al. (2019b) presents a method for generating a targeted adversarial sticker which changes an image classifier’s classification from one pre-specified class to another. Both of these methods rely on specific use cases and are tailored towards generating adversaries coming from strict distributions, e.g. (Li et al., 2019a) generates guitar music while (Li et al., 2019b) generates a small grid of dots. These works build on algorithms akin to our baseline approaches and are limited in scope to domain specific transformations. Our work provides a framework for improving robustness against a wide range of transformations in diverse domains and can be leveraged for improving the effectiveness of these attacks.
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# 7 CONCLUSION
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In this paper, we demonstrate that standard UAPs are highly susceptible to transformations, i.e. they fail to be universally adversarial under transformation. We propose a new method, RobustUAP to generate robust UAPs based upon obtaining probabilistic bounds on UAP robustness across an entire transformation space. Our experiments provide empirical evidence that this principled approach generates UAPs that are practically more robust under a wide range of transformation sets than those from the baseline methods.
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# APPENDIX
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# A SEMANTIC TRANSFORMATIONS
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In this section, we discuss the semantic transformations used in the paper. Brightness and contrast can be represented via bias $( \beta )$ and gain $( \alpha > 0 )$ ) parameters respectively. Formally, if $\mathbf { x }$ is the original image, then the transformed image, $\mathbf { x } ^ { \prime }$ , can be represented as
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+
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| 329 |
+
$$
|
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+
\mathbf { x } ^ { \prime } = \alpha \mathbf { x } + \beta
|
| 331 |
+
$$
|
| 332 |
+
|
| 333 |
+
Rotation, scaling, shearing, and translation are all affine transformations acting on the coordinate system, $c$ , of the images instead of the pixel values, $\mathbf { x }$ . In order to recover the pixel values and differentiate over the transformation, we will need sub-differentiable interpolation, see Appendix B. For finite dimensions, affine transformations can be represented as a linear coordinate map where the original coordinates are multiplied by an invertible augmented matrix and then translated with additional bias vector. Below, we give the general form for an affine transformation given augmented matrix A, bias matrix $\mathbf { b }$ , and input coordinates $c$ . We can compute the output coordinates, $c ^ { \prime }$ , as
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+
|
| 335 |
+
$$
|
| 336 |
+
{ \left[ \begin{array} { l } { \mathbf { c } ^ { \prime } } \\ { 1 } \end{array} \right] } = { \left[ \begin{array} { l l l l } { [ c c c | c ] } & { \mathbf { A } } & & { \mathbf { b } } \\ { 0 } & { \ldots } & { 0 } & { 1 } \end{array} \right] } { \left[ \begin{array} { l } { \mathbf { c } } \\ { 1 } \end{array} \right] }
|
| 337 |
+
$$
|
| 338 |
+
|
| 339 |
+
Below, we give the augmented matrix A and additional bias matrix $\mathbf { b }$ for rotation, scaling, shearing, and translation.
|
| 340 |
+
|
| 341 |
+
Rotation, $R ( \theta )$ , by $\theta$ degrees:
|
| 342 |
+
|
| 343 |
+
$$
|
| 344 |
+
\mathbf { A } = { \binom { \cos { \theta } } { \sin { \theta } } } \quad { \cos { \theta } } ^ { \prime } \mathbf { , b } = { \binom { 0 } { 0 } }
|
| 345 |
+
$$
|
| 346 |
+
|
| 347 |
+
Scaling, $S c ( p )$ , by $p \%$ :
|
| 348 |
+
|
| 349 |
+
$$
|
| 350 |
+
\mathbf { A } = \left( { \begin{array} { c c } { 1 + { \frac { p } { 1 0 0 } } } & { 0 } \\ { 0 } & { 1 + { \frac { p } { 1 0 0 } } } \end{array} } \right) , \mathbf { b } = \left( { \begin{array} { c } { 0 } \\ { 0 } \end{array} } \right)
|
| 351 |
+
$$
|
| 352 |
+
|
| 353 |
+
Shearing, $S h ( m )$ , by shear factor $m \%$ :
|
| 354 |
+
|
| 355 |
+
$$
|
| 356 |
+
\mathbf { A } = { \binom { 1 } { 0 } } \quad { \overset { 1 + { \frac { m } { 1 0 0 } } } { 1 } } ) , \mathbf { b } = { \binom { 0 } { 0 } }
|
| 357 |
+
$$
|
| 358 |
+
|
| 359 |
+
Translation, $T ( x , y )$ , by $x$ pixels horizontally and $y$ pixels vertically:
|
| 360 |
+
|
| 361 |
+
$$
|
| 362 |
+
\mathbf { A } = { \binom { 0 } { 0 } } \ { \begin{array} { l } { 0 } \\ { 0 } \end{array} } ) , \mathbf { b } = { \binom { x } { y } }
|
| 363 |
+
$$
|
| 364 |
+
|
| 365 |
+
# B INTERPOLATION
|
| 366 |
+
|
| 367 |
+
Affine transformations may change a pixel’s integer coordinates into non-integer coordinates. Interpolation is typically used to ensure that the resulting image can be represented on a lattice (integer) pixel grid. For this paper, we will be using bilinear interpolation, a common interpolation method which achieves a good trade-off between accuracy and efficiency in practice and is commonly used in literature (Xiao et al., 2018b; Balunovic et al., 2019). Let ´ $x _ { i , j }$ , $x _ { i , j } ^ { \prime }$ represent the pixel value position -coordina $i , j$ fond the original and transform-coordinate of the pixel at image respectively. Let after transformation. We $c _ { i , j } ^ { \prime x } , c _ { i , j } ^ { \prime y }$ represent ther transformed $x$ $y$ $i , j$
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| 368 |
+
image by summing over all pixels $n , m \in [ 1 \dots H ] \times [ 1 \dots W ]$ where $H$ and $W$ represent the height and width of the image.
|
| 369 |
+
|
| 370 |
+
$$
|
| 371 |
+
x _ { i , j } ^ { \prime } = \sum _ { n } ^ { H } \sum _ { m } ^ { W } x _ { n , m } \operatorname* { m a x } ( 0 , 1 - | c _ { i , j } ^ { \prime x } - m | ) \operatorname* { m a x } ( 0 , 1 - | c _ { i , j } ^ { \prime y } - n | )
|
| 372 |
+
$$
|
| 373 |
+
|
| 374 |
+
This interpolation can be computed for each channel in the image. While interpolation is typically not differentiable, in order to generate adversarial examples using standard techniques we need a differentiable version of interpolation. (Jaderberg et al., 2015) introduces differentiable image sampling. Their method works for any interpolation method as long as the (sub-)gradients can be defined with respect to $x , c ^ { \prime } { } _ { i , j }$ . For bilinear interpolation this becomes,
|
| 375 |
+
|
| 376 |
+
$$
|
| 377 |
+
\frac { \partial x _ { i , j } ^ { \prime } } { \partial x _ { n , m } } = \sum _ { n } ^ { H } \sum _ { m } ^ { W } \operatorname* { m a x } ( 0 , 1 - | c ^ { \prime } _ { i , j } - m | ) \operatorname* { m a x } ( 0 , 1 - | c ^ { \prime } _ { i , j } - n | )
|
| 378 |
+
$$
|
| 379 |
+
|
| 380 |
+
$$
|
| 381 |
+
\frac { \partial x _ { i , j } ^ { \prime } } { \partial c _ { i , j } ^ { \prime \boldsymbol { x } } } = \sum _ { n } ^ { H } \sum _ { m } ^ { W } x _ { n , m } \operatorname* { m a x } ( 0 , 1 - | c _ { i , j } ^ { \prime \boldsymbol { y } } - n | ) \left\{ \begin{array} { l l } { 1 } & { \mathrm { i f ~ } m \ge | c _ { i , j } ^ { \prime \boldsymbol { x } } - m | } \\ { - 1 } & { \mathrm { i f ~ } m < | c _ { i , j } ^ { \prime \boldsymbol { x } } - m | } \\ { 0 } & { \mathrm { o t h e r w i s e } } \end{array} \right.
|
| 382 |
+
$$
|
| 383 |
+
|
| 384 |
+
# C SGD ALGORITHM
|
| 385 |
+
|
| 386 |
+
Our SGD UAP algorithm is based on standard momentum based SGD while optimizing over the objective proposed in 5, the algorithm details can be seen in Algorithm 3.
|
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+
|
| 388 |
+
# Algorithm 3 Stochastic Gradient Descent UAP Algorithm
|
| 389 |
+
|
| 390 |
+
1: Initialize $\mathbf { u _ { r } } \gets 0 , \Delta \mathbf { u _ { r } } \gets 0$
|
| 391 |
+
2: repeat
|
| 392 |
+
3: for $\mathbf { B } \in \mathbf { X }$ do
|
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+
4: 5: $\begin{array} { r } { \Delta \mathbf { u } _ { \mathbf { r } } \gets \alpha \Delta \mathbf { u } _ { \mathbf { r } } - \frac { \nu } { | \hat { \mathbf { x } } | \times | \hat { t } | } \sum _ { i = 1 } ^ { | \hat { \mathbf { x } } | } \sum _ { j = 1 } ^ { | \hat { t } | } \nabla L \big [ f \big ( \hat { \mathbf { x } } _ { \mathbf { i } } + \hat { t } _ { j } ( \mathbf { u } _ { \mathbf { r } } ) \big ) , f \big ( \hat { \mathbf { x } } _ { \mathbf { i } } \big ) \big ] } \end{array}$ $\hat { t } \subset T$
|
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+
6: Update the perturbation with projection:
|
| 395 |
+
7: $\mathbf { u } \gets \mathcal { P } _ { p , \epsilon } ( \mathbf { u _ { r } } + \Delta \mathbf { u _ { r } } )$
|
| 396 |
+
8: end for
|
| 397 |
+
9: until $A S R _ { R } ( f , \mathbf { X } , T , \gamma , \mathbf { u _ { r } } ) < \zeta$
|
| 398 |
+
|
| 399 |
+
# D ITERATIVE UAP ALGORITHM
|
| 400 |
+
|
| 401 |
+
Moosavi-Dezfooli et al. (2017) introduces an iterative UAP algorithm, the algorithm can be seen in Algorithm 4.
|
| 402 |
+
|
| 403 |
+
1: Initialize $\mathbf u \gets 0$
|
| 404 |
+
2: repeat
|
| 405 |
+
3: for $\mathbf { x _ { i } } \in \mathbf { X }$ do
|
| 406 |
+
4: if ${ \bar { \hat { f } } } ( \mathbf { x _ { i } } + \mathbf { u } ) = { \hat { f } } ( \mathbf { x _ { i } } )$ then
|
| 407 |
+
5: Compute minimal adversarial perturbation:
|
| 408 |
+
6: $\Delta \mathbf { u } \arg \operatorname* { m i n } _ { \mathbf { r } } | | \mathbf { r } | | _ { 2 }$ s.t. $\hat { f } ( \mathbf { x _ { i } } + \mathbf { u } + \mathbf { r } ) \neq \hat { f } ( \mathbf { x _ { i } } )$
|
| 409 |
+
7: Update the perturbation with projection:
|
| 410 |
+
8: $\mathbf { u } \bar { \mathbf { \Omega } } \ll \mathcal { P } _ { p , \epsilon } ( \bar { \mathbf { u } } + \Delta \mathbf { u } )$
|
| 411 |
+
9: end if
|
| 412 |
+
10: end for
|
| 413 |
+
11: until $A S R _ { U } ( f , { \bf X } , { \bf u } ) < \gamma$
|
| 414 |
+
|
| 415 |
+
# E EXPERIMENT PARAMETERS
|
| 416 |
+
|
| 417 |
+
In our experiments, we have capped all algorithms at 5 epochs or if they have achieved an $\mathrm { A S R } _ { R }$ of 0.95. The UAPs are trained with the same transformation set that they are evaluated on. For algorithms running PGD internally, we have capped the number of iterations to 40.
|
| 418 |
+
|
| 419 |
+
# F FURTHER EVALUATION OF UNIFORM NOISE
|
| 420 |
+
|
| 421 |
+
Results in Table 2.
|
| 422 |
+
Table 2: Robust ASR with uniform random noise, $\gamma = 0 . 8$ .
|
| 423 |
+
|
| 424 |
+
<table><tr><td>DATASET</td><td>TRANSFORMATION SET</td><td>STANDARD UAP</td><td>SGD</td><td>STANDARD UAP_RP</td><td>ROBUST UAP</td></tr><tr><td>ILSVRC</td><td>U(0.1)</td><td>81.6%</td><td>94.2%</td><td>91.3%</td><td>99.0%</td></tr><tr><td>2012</td><td>U(0.3)</td><td>10.7%</td><td>68.9%</td><td>42.7%</td><td>96.1%</td></tr><tr><td>CIFAR-10</td><td>U(0.1)</td><td>66.0%</td><td>98.1%</td><td>96.1%</td><td>100%</td></tr><tr><td></td><td>U(0.3)</td><td>5.8%</td><td>96.1%</td><td>47.6%</td><td>100%</td></tr></table>
|
| 425 |
+
|
| 426 |
+
# G UAP PERFORMANCE AGAINST SEMANTIC TRANSFORMATIONS ON CIFAR-10
|
| 427 |
+
|
| 428 |
+
|
| 429 |
+
Figure 6: For each method, a point $( x , y )$ in the corresponding line represents the percentage of sampled UAPs $( y \% )$ with Universal ASR $> x$ for the different semantic transformations on CIFAR10.
|
| 430 |
+
|
| 431 |
+
# H AVERAGE $\mathrm { A S R } _ { U }$ AND $\operatorname { A S R } _ { R }$ WITH DIFFERENT $\gamma$ ’ S
|
| 432 |
+
|
| 433 |
+
We provide additional metrics computed on the same set of transformations, datasets, and models as in Table 1. In Table 3, we present the Average $\operatorname { A S R } _ { U }$ rather than $\mathrm { A S R } _ { R }$ . The average shows us that our RobustUAP algorithm creates UAPs which after transformation on average are better UAPs than all other algorithms. We observe that the average shows us that even standard UAPs aren’t completely ineffective after transformation they just have a very low chance of being highly effective.
|
| 434 |
+
|
| 435 |
+
<table><tr><td>DATASET</td><td>TRANSFORMATION SET</td><td>STANDARD UAP</td><td>SGD</td><td>STANDARD UAP_RP</td><td>ROBUST UAP</td></tr><tr><td rowspan="4">ILSVRC 2012</td><td>R(20)</td><td>16.3%</td><td>71.5%</td><td>24.7%</td><td>81.3%</td></tr><tr><td>T(2,2)</td><td>52.6%</td><td>82.6%</td><td>55.4%</td><td>85.4%</td></tr><tr><td>Sc(5),R(5),B(5,0.01)</td><td>44.9%</td><td>76.3%</td><td>58.5%</td><td>82.2%</td></tr><tr><td>R(10),T(2,2),Sh(2),Sc(2),B(2,0.001)</td><td>13.6%</td><td>64.8%</td><td>29.0%</td><td>75.3%</td></tr><tr><td rowspan="3">CIFAR-10</td><td>R(30),B(2,0.001)</td><td>9.9%</td><td>66.8%</td><td>22.2%</td><td>73.4%</td></tr><tr><td>R(2),Sh(2)</td><td>57.1%</td><td>78.8%</td><td>61.2%</td><td>82.9%</td></tr><tr><td>R(10),T(2,2),Sh(2),Sc(2),B(2,0.001)</td><td>16.2%</td><td>61.2%</td><td>32.6%</td><td>76.4%</td></tr></table>
|
| 436 |
+
|
| 437 |
+
Table 3: Average Universal ASR of our Robust UAP algorithms and the standard UAP (MoosaviDezfooli et al., 2017) method.
|
| 438 |
+
|
| 439 |
+
In Table 4, we present $\operatorname { A S R } _ { R }$ computed at $\gamma = [ 0 . 5 , 0 . 7 ]$ rather than $\gamma = 0 . 6$ . This table shows a similar story to above, and shows that our algorithm produces better results under a variety of success thresholds.
|
| 440 |
+
|
| 441 |
+
<table><tr><td rowspan="3">DATASET</td><td rowspan="3">TRANSFORMATION SET</td><td colspan="2">STANDARD UAP</td><td colspan="2">SGD</td><td colspan="2">STANDARD</td><td colspan="2"></td><td rowspan="3">ROBUST UAP</td></tr><tr><td colspan="2">0.5</td><td colspan="2">0.5</td><td colspan="2">UAP_RP 0.5</td><td colspan="2"></td></tr><tr><td></td><td>0.7</td><td></td><td></td><td>0.7</td><td></td><td>0.7</td><td>0.5</td></tr><tr><td rowspan="4">ILSVRC 2012</td><td>R(20)</td><td>1.9%</td><td>0.0%</td><td></td><td>88.3%</td><td>58.3%</td><td>10.7%</td><td>1.0%</td><td></td><td>98.1% 76.7%</td></tr><tr><td>T(2,2)</td><td>51.5% 21.4% 100% 84.5% 57.3% 2</td><td></td><td></td><td></td><td></td><td></td><td>23.3%</td><td></td><td>100% 91.3%</td></tr><tr><td>Sc(5),R(5),B(5,0.01)</td><td>38.8%11.7%</td><td></td><td></td><td>96.1% 67.0%64.1%25.2%</td><td></td><td></td><td></td><td></td><td>99.0%87.4%</td></tr><tr><td>R(10),T(2,2), Sh(2), Sc(2),B(2,0.001)</td><td>1.9%</td><td>0.0%</td><td></td><td>82.5% 38.8%12.6%</td><td></td><td></td><td>1.0%</td><td>95.1%</td><td>59.2%</td></tr><tr><td rowspan="3">CIFAR-10 R(2), Sh(2)</td><td>R(30),B(2,0.001)</td><td>1.0%</td><td>0.0%</td><td></td><td>80.6% 43.7% 12.6%</td><td></td><td></td><td>1.0%</td><td>93.2%</td><td>49.5%</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>62.1% 22.3% 96.1% 68.9% 68.0% 30.1% 99.0% 89.3%</td></tr><tr><td>R(10),T(2,2),Sh(2),Sc(2),B(2,0.001)</td><td>2.9%</td><td></td><td></td><td>0.0% 67.0% 38.8% 19.4%</td><td></td><td></td><td>1.0%</td><td></td><td>93.2% 55.3%</td></tr></table>
|
| 442 |
+
|
| 443 |
+
Table 4: Robust ASR of our Robust UAP algorithms and the standard UAP (Moosavi-Dezfooli et al., 2017) method with $\gamma = [ 0 . 5 , 0 . 7 ]$ .
|
| 444 |
+
Table 5: Universal ASR of our Robust UAP algorithms and the standard UAP method.
|
| 445 |
+
|
| 446 |
+
<table><tr><td>DATASET</td><td>STANDARDUAP</td><td>SGD</td><td>STANDARDUAP_RP</td><td>ROBUS TUAP</td></tr><tr><td>ILSVRC 2012</td><td>95.5%</td><td>85.6%</td><td>82.3%</td><td>91.3%</td></tr><tr><td>CIFAR-10</td><td>96.2%</td><td>89.3%</td><td>84.0%</td><td>93.7%</td></tr></table>
|
| 447 |
+
|
| 448 |
+
# I COMPARISON ON NON-ROBUST UNIVERSAL ASR METRIC
|
| 449 |
+
|
| 450 |
+
We compare our robust UAPs to standard UAPs on the non-robust universal ASR metric, see Table 5. All robust UAPs are generated to be robust against $R ( 1 0 ) , T ( 2 , 2 ) , S h ( 2 ) , S c ( 2 ) , B ( 2 , 0 . 0 0 1 )$ . We observe that at the same $l _ { 2 }$ -norm all robust UAPs achieve a lower universal ASR than the standard UAP algorithm. This result is not too surprising as solving the optimization problem for robust UAP is significantly more difficult. We further observe that our RobustUAP algorithm is the most effective in comparison to the other robust baseline approaches.
|
| 451 |
+
|
| 452 |
+
# J ADDITIONAL MODELS
|
| 453 |
+
|
| 454 |
+
We also provide additional data on our methods evaluated on the same transformations and datasets but on different models. In this case, we use ResNet-18 (He et al., 2015) for CIFAR-10 and MobileNet (Howard et al., 2017) for ILSVRC 2012. Results can be seen in Table 6. We observe similar performance across models suggesting that the performance of the attacks is more directly tied to transformation set and dataset.
|
| 455 |
+
|
| 456 |
+
Table 6: Robust ASR on Resnet-18 for CIFAR-10 and MobileNet for ILSVRC 2012.
|
| 457 |
+
|
| 458 |
+
<table><tr><td>DATASET</td><td>MODEL</td><td>TRANSFORMATION SET</td><td>STANDARD UAP</td><td>SGD</td><td>STANDARD ROBUST UAP_RP</td><td>UAP</td></tr><tr><td rowspan="4">ILSVRC 2012</td><td rowspan="4">MOBILENET</td><td>R(20)</td><td>8.1%</td><td>71.2%</td><td>2.6%</td><td>85.0%</td></tr><tr><td>T(2,2)</td><td>40.9%</td><td>98.7%</td><td>54.3%</td><td>99.6%</td></tr><tr><td>Sc(5), R(5),B(5,0.01)</td><td>16.3%</td><td>94.5%</td><td>44.3%</td><td>96.3%</td></tr><tr><td>R(10),T(2,2), Sh(2),Sc(2),B(2,0.001)</td><td>4.1%</td><td>75.7%</td><td>8.6%</td><td>86.2%</td></tr><tr><td rowspan="3">CIFAR-10 RESNET-18</td><td rowspan="3"></td><td>R(30),B(2,0.001)</td><td>0.9%</td><td>67.8%</td><td>6.4%</td><td>74.9%</td></tr><tr><td>R(2),Sh(2)</td><td>49.9%</td><td>99.5%</td><td>49.1%</td><td>99.8%</td></tr><tr><td>R(10),T(2,2),Sh(2),Sc(2),B(2,0.001)</td><td>8.0%</td><td>70.8%</td><td>12.2%</td><td>83.8%</td></tr></table>
|
| 459 |
+
|
| 460 |
+
# K COMMON CORRUPTIONS
|
| 461 |
+
|
| 462 |
+
We also evaluate robust UAP against the 2D fog transformations in (Kar et al., 2022). We set the shift intensity of the fog to be 1 and train our robust UAPs to be robust against random fog perturbations. We observe similar results to the transformations we experiment with above. The graph of the results can be seen in Figure 7.
|
| 463 |
+
|
| 464 |
+
|
| 465 |
+
Figure 7: For each method, a point $( x , y )$ in the corresponding line represents the percentage of sampled UAPs $( y \% )$ with Universal ASR $> x$ for the different semantic transformations on ILSVRC2012.
|
| 466 |
+
|
| 467 |
+
# L ALGORITHM RUNTIMES
|
| 468 |
+
|
| 469 |
+
We compare the average runtimes of the different methods on one of our most challenging $R ( 1 0 ) , \bar { T ( 2 , 2 ) } , S h ( 2 ) , \bar { S c } ( 2 ) , B ( 2 , 0 . 0 0 1 )$ transformation set on ILSVRC-2012 and $n = 7 3 8$ . The results are in Table 7. We observe that RobustUAP is the slowest algorithm and SGD is the fastest. RobustUAP uses EstimateRobustness in each loop and thus with high $n$ it requires much more time to compute. The extra computation enables Robust UAP to obtain better robustness than all baselines. On the same set of transformations and dataset we observe that one iteration of EstimateRobustness on the entire test set takes on average 19 minutes. When running EstimateRobustness in the RobustUAP loop, each call takes 36 seconds for a batch size of 32.
|
| 470 |
+
|
| 471 |
+
Table 7: Average Runtime for Robust UAP algorithms
|
| 472 |
+
|
| 473 |
+
<table><tr><td>ALGORITHM</td><td>TIME(MIN)</td></tr><tr><td>STANDARD UAP</td><td>37</td></tr><tr><td>SGD</td><td>32</td></tr><tr><td>STANDARD UAP_RP</td><td>43</td></tr><tr><td>ROBUST UAP</td><td>118</td></tr></table>
|
| 474 |
+
|
| 475 |
+
# M EFFECT OF COMPUTE TIME ON ROBUSTNESS
|
| 476 |
+
|
| 477 |
+
Previous sections highlight SGD as the most competitive algorithm to RobustUAP in terms of performance. However, in the previous section we note that SGD takes significantly less time to run. In this section, we investigate how RobustUAP performs with limited compute time as well as how SGD performs with increased runtime. We first add results to the ILSVRC 2012 part of Table 1 by also computing RobustUAP performance when limited to the same amount of time that SGD takes. Table 8 shows that RobustUAP outperforms SGD even when its compute time is limited with up to $9 \%$ more robustness on our most challenging transformation $R ( 1 0 ) , \bar { T } ( 2 , 2 ) , S h ( 2 ) , S c ( 2 ) , B ( \bar { 2 , } 0 . 0 0 1 ) .$ .
|
| 478 |
+
|
| 479 |
+
<table><tr><td>DATASET</td><td>TRANSFORMATION SET</td><td>SGD</td><td>ROBUST UAP</td><td>RESTRICTED ROBUST UAP</td></tr><tr><td rowspan="4">ILSVRC 2012</td><td>R(20)</td><td>69.9%</td><td>93.2%</td><td>72.9%</td></tr><tr><td>T(2,2)</td><td>96.1%</td><td>97.1%</td><td>96.9%</td></tr><tr><td>Sc(5),R(5),B(5,0.01)</td><td>85.4%</td><td>96.1%</td><td>86.3%</td></tr><tr><td>R(10),T(2,2),Sh(2),Sc(2),B(2,0.001)</td><td>63.1%</td><td>86.4%</td><td>72.0%</td></tr></table>
|
| 480 |
+
|
| 481 |
+
Table 8: Robust ASR of RobustUAP restricted to the same amount of compute time as SGD.
|
| 482 |
+
|
| 483 |
+
Next, we vary the number of SGD iterations. We compute the robust ASR on ILSVRC for robustness against $R ( 1 0 ) , T ( 2 , 2 ) , S h ( 2 ) , S c ( 2 ) , B ( 2 , 0 . 0 0 1 )$ . Figure 8, shows the robust ASR achieved by SGD over time, here we observe that SGD’s performance flatlines after a small number of iterations and seems to be unable to surpass about 65. Here SGD is allowed to continue to run past where it would usually stop (at around 250 iterations), in this experiment we allow it to go to 1250 iterations which is about the same amount of time that RobustUAP takes to run. RobustUAP is able to achieve a performance of 72 even when restricted to the amount of compute time of base SGD (It achieves 86.4 when unrestricted). These two results in combination show that RobustUAP is able to find more robust UAPs than SGD whose performance stabilizes.
|
| 484 |
+
|
| 485 |
+
|
| 486 |
+
Figure 8: The Robust ASR with $\gamma = 0 . 6$ for SGD over time
|
| 487 |
+
|
| 488 |
+
# N ROBUSTNESS ON HOLD-OUT TRANSFORMATIONS
|
| 489 |
+
|
| 490 |
+
In this section, we measure the effectiveness of our robust UAPs against hold-out transformations. In this experiment, we learn a UAP which is robust to $R ( 5 )$ and obtain a robust ASR of 96.2 at $\gamma = 0 . 6$ . We then measure its effectiveness against $S c ( 5 )$ and get a robust ASR of 85.4, in contrast, a robust UAP trained directly to be robust to $S c ( 5 )$ obtains robust ASR of 98.1. Next, we measure the robustness of UAP trained against $R ( 5 )$ when subjected to transformations from $B ( 5 , 0 . 0 1 )$ . Here we get a robust ASR of 97.3, whereas a robust UAP trained to be robust to $B ( 5 , 0 . 0 1 )$ obtains a robust ASR of 99.2. Finally, we test the robust UAP on $R ( 5 ) , S c ( 5 ) , B ( 5 , 0 . 0 1 )$ and get a robust ASR of 83.1. Our previous results show that a UAP trained to be robust against these parameters directly can obtain a robust ASR of 96.1. In each case, our UAP maintains robustness on hold-out transformations but has lower performance compared to robust UAPs trained directly to be robust to those transformations.
|
| 491 |
+
|
| 492 |
+
# O TARGETED ATTACK
|
| 493 |
+
|
| 494 |
+
So far in this paper we have focused on untargeted attacks, i.e. attacks which aim to degrade the general performance of the model. Targeted attacks are also possible with both standard adversarial attack methods and universal adversarial perturbation methods. Here, we can simply turn our algorithm from untargeted to targeted by replacing the loss function. We would like to have target class, A, be classified as target class, B. Instead of maximizing the expected value of the cross entropy loss we can instead formulate the loss based on maximizing B while minimizing A similar to (Benz et al., 2020). For ILSVRC 2012, we randomly select a couple of target classes and perform this attack, for each of these cases, we train our robust UAP to be robust to $R ( 1 0 ) , T ( 2 , 2 ) , S h ( 2 ) , S c ( 2 ) , B ( 2 , 0 . 0 0 1 )$ . Table 9 shows our results for robust ASR with $\gamma = 0 . 6$ . We are measuring our robust ASR of turning class A into class B and observe similar results with RobustUAP being the most robust followed by SGD. It is also interesting to note that different random combinations lead to more or less success, i.e. it is easier to turn a dog into another dog than perfume into a padlock.
|
| 495 |
+
|
| 496 |
+
Table 9: Robust ASR of RobustUAP for target to target attack compared to the three baselines with $\gamma = 0 . 6$ .
|
| 497 |
+
|
| 498 |
+
<table><tr><td>DATASET</td><td>TARGET CLASS</td><td>STANDARD UAP</td><td>SGD</td><td>STANDARD UAP_RP</td><td>ROBUST UAP</td></tr><tr><td>ILSVRC-2012</td><td>TOY POODLE →→MALTESE DOG PERFUME-→PADLOCK</td><td>42.4% 0.0%</td><td>99.1% 63.8%</td><td>85.6% 5.1%</td><td>99.8% 76.4%</td></tr></table>
|
| 499 |
+
|
| 500 |
+
# P DATA EFFICIENCY
|
| 501 |
+
|
| 502 |
+
In this section, we will evaluate the data efficiency of RobustUAP. We use RobustUAP to generate UAPs robust to $R ( 1 0 ) , T ( 2 , 2 ) , S h ( 2 ) , S c ( 2 ) , B ( 2 , 0 . 0 0 1 )$ on ILSVRC-2012 with differing amounts of training data. The results can be seen in Figure 9. These results show that the algorithm is able to achieve good performance at 500 data points but continues to improve up to 4000 data points. After that it seems to stagnate.
|
| 503 |
+
|
| 504 |
+
|
| 505 |
+
Figure 9: Robust ASR with $\gamma = 0 . 6$ for RobustUAP with differing amounts of training data
|
| 506 |
+
|
| 507 |
+
# Q TRANSFERABILITY
|
| 508 |
+
|
| 509 |
+
In this section, we will evaluate the transferability of RobustUAP. Previous works on UAPs (Moosavi-Dezfooli et al., 2017) show that UAPs are transferable across different models. Here, we will evaluate whether robust UAPs exhibit the same behavior for robustness. The robust UAPs studied here are generated with RobustUAP on $R ( 1 0 ) , T ( 2 , 2 ) , S h ( 2 ) , S c ( 2 ) , B ( 2 , 0 . 0 0 1 )$ for ILSVRC-2012 with $\gamma = 0 . 6$ . We use a variety of models: Inception-v3 (Szegedy et al., 2016), MobileNet (Howard et al., 2017), Inception-v3 trained to be robust on $R ( 2 0 )$ (InceptionR20), Inceptionv3 trained to be robust on horizontal flips (InceptionHF), and ViT (Dosovitskiy et al., 2020). Table 10 shows us that our robust UAPs are transferable between different architectures. Our results show that robust UAPs transfer their robustness properties between architectures and models. Ignoring ViT, on all of the Inception and MobileNet models, the generated UAPs maintain at least $65 \%$ robust ASR when transferred to each other. This transfer is less but still significant for ViT where it maintains at least $32 \%$ robustness when transferred to or from the other models.
|
| 510 |
+
|
| 511 |
+
Table 10: Robust ASR when UAP is learned on source model and transfered to target model.
|
| 512 |
+
|
| 513 |
+
<table><tr><td></td><td colspan="5">TARGET MODEL</td></tr><tr><td>SOURCE MODEL</td><td>INCEPTION</td><td>MOBILENET</td><td>INCEPTIONR20</td><td>INCEPTIONHF</td><td>VIT</td></tr><tr><td>INCEPTION</td><td>86.4%</td><td>65.2%</td><td>75.2%</td><td>78.5%</td><td>35.1%</td></tr><tr><td>MOBILENET</td><td>74.3%</td><td>86.2%</td><td>67.3%</td><td>68.6%</td><td>38.3%</td></tr><tr><td>INCEPTIONR20</td><td>80.1%</td><td>67.3%</td><td>81.3%</td><td>73.1%</td><td>32.0%</td></tr><tr><td>INCEPTIONHF</td><td>77.8%</td><td>70.9%</td><td>75.8%</td><td>83.8%</td><td>34.6%</td></tr><tr><td>VIT</td><td>41.2%</td><td>32.4%</td><td>43.2%</td><td>39.7%</td><td>88.5%</td></tr></table>
|
| 514 |
+
|
| 515 |
+
# R TRANSFORMER-BASED MODELS
|
| 516 |
+
|
| 517 |
+
Recently, transformers have become popular as a new architecture for deep learning models for computer vision tasks. In this section, we evaluate the effectiveness of robust UAPs against one such model, ViT (Dosovitskiy et al., 2020). Benz et al. (2021) has shown that standard UAPs are still effective against transformer based architectures. In Table 11 we can see that we get similar results compared to our results on Inception and MobileNet. This shows that our methods work against transformer based models as well.
|
| 518 |
+
|
| 519 |
+
<table><tr><td>DATASET</td><td>MODEL TRANSFORMATION SET</td><td></td><td>STANDARD UAP</td><td>SGD</td><td>STANDARD ROBUST UAP_RP</td><td>UAP</td></tr><tr><td>ILSVRC-2012 VIT</td><td></td><td>R(10),T(2,2),Sh(2),Sc(2),B(2,0.001)</td><td>2.0%</td><td>72.1%</td><td>12.9%</td><td>88.5%</td></tr></table>
|
| 520 |
+
|
| 521 |
+
Table 11: Robust ASR of RobustUAPcompared to the three baselines for ViT.
|
| 522 |
+
|
| 523 |
+
# S ROBUST UAPS AGAINST ROBUSTLY TRAINED NETWORKS
|
| 524 |
+
|
| 525 |
+
In this section, we are interested in seeing whether training networks to be robust against the same transformations that the UAP is trying to be robust against is helpful. For this, we trained two new Inception-v3 networks. Because of time limitations, we started with our base Inception-v3 network and fine-tuned it using data augmentations. For the first network InceptionR20, we augmented the data by adding random rotations within 20 degrees. For the second network InceptionHF, we augmented the data by adding horizontal flips. We then crafted UAPs robust against rotations and flips on InceptionR20 and InceptionHF respectively. The results can be seen in Table 12. We can compare the $R ( 2 0 )$ results to those from our normal inception network. We postulate that since the network has received some additional robustness training it is harder to attack, and thus we should see slightly lower robustness scores. However, it seems that training the network to be robust to $R ( 2 0 )$ does not significantly effect the ability to create robust UAPs. The horizontal flips seems like it might be too easy of a transformation as even standard UAP performs quite well for robust ASR.
|
| 526 |
+
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| 527 |
+
<table><tr><td>DATASET</td><td>MODEL</td><td>TRANSFORMATION SET</td><td>STANDARD UAP</td><td>SGD</td><td>STANDARD UAP_RP</td><td>ROBUST UAP</td></tr><tr><td>ILSVRC-2012</td><td>INCEPTIONR20 INCEPTIONHF</td><td>R(20) HF</td><td>6.3% 81.3%</td><td>72.4% 99.5%</td><td>10.2% 89.7%</td><td>81.3% 99.6%</td></tr></table>
|
| 528 |
+
|
| 529 |
+
Table 12: Robust ASR of RobustUAP compared to the three baselines for robust networks.
|
| 530 |
+
|
| 531 |
+
# T ABLATION ON OPTIMIZATION STRATEGY
|
| 532 |
+
|
| 533 |
+
In this section, we study the effect of using different optimizers in addition to SGD. We use a variety of standard PyTorch optimizers, Adam, Adamax, Adagrad, and RMSProp. We formulate the optimization problem in the same way but instead use these algorithms in order to optimize our perturbation. We compute these results on ILSVRC-2012 with Inception-v3 and use $R ( 1 0 ) , T ( 2 , 2 ) , S h ( 2 ) , S c ( 2 ) , B ( 2 , 0 . 0 0 1 )$ as the transformation set and with $\gamma = 0 . 6$ . The results can be seen in Table 13. We see that the optimization strategy has some affect on the results and that SGD performs the best. We also found that SGD performed marginally faster than the rest of the approaches.
|
| 534 |
+
|
| 535 |
+
Table 13: Comparison of different optimization strategies.
|
| 536 |
+
|
| 537 |
+
<table><tr><td>OPTIMIZER|ASRR</td><td></td></tr><tr><td>SGD ADAM</td><td>63.1% 59.7%</td></tr><tr><td>ADAMAX ADAGRAD</td><td>60.1% 62.3%</td></tr><tr><td>RMSPROP</td><td>58.3%</td></tr></table>
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|
| 1 |
+
# BIGVGAN: A UNIVERSAL NEURAL VOCODER WITHLARGE-SCALE TRAINING
|
| 2 |
+
|
| 3 |
+
Sang-gil Lee1∗ Wei Ping 2†Boris Ginsburg2 Bryan Catanzaro2 Sungroh Yoon1,3†
|
| 4 |
+
|
| 5 |
+
1 Data Science & AI Lab, Seoul National University (SNU)
|
| 6 |
+
2 NVIDIA
|
| 7 |
+
3 AIIS, ASRI, INMC, ISRC, NSI, and Interdisciplinary Program in AI, SNU
|
| 8 |
+
|
| 9 |
+
tkdrlf9202@snu.ac.kr wping@nvidia.com bginsburg@nvidia.com bcatanzaro@nvidia.com sryoon@snu.ac.kr
|
| 10 |
+
|
| 11 |
+
# ABSTRACT
|
| 12 |
+
|
| 13 |
+
Despite recent progress in generative adversarial network (GAN)-based vocoders, where the model generates raw waveform conditioned on acoustic features, it is challenging to synthesize high-fidelity audio for numerous speakers across various recording environments. In this work, we present BigVGAN, a universal vocoder that generalizes well for various out-of-distribution scenarios without fine-tuning. We introduce periodic activation function and anti-aliased representation into the GAN generator, which brings the desired inductive bias for audio synthesis and significantly improves audio quality. In addition, we train our GAN vocoder at the largest scale up to 112M parameters, which is unprecedented in the literature. We identify and address the failure modes in large-scale GAN training for audio, while maintaining high-fidelity output without over-regularization. Our BigVGAN, trained only on clean speech (LibriTTS), achieves the state-of-the-art performance for various zero-shot (out-of-distribution) conditions, including unseen speakers, languages, recording environments, singing voices, music, and instrumental audio. 1 We release our code and model at: https://github.com/NVIDIA/BigVGAN.
|
| 14 |
+
|
| 15 |
+
# 1 INTRODUCTION
|
| 16 |
+
|
| 17 |
+
Deep generative models have demonstrated noticeable successes for modeling raw audio. The successful methods include, autoregressive models (van den Oord et al., 2016; Mehri et al., 2017; Kalchbrenner et al., 2018), flow-based models (van den Oord et al., 2018; Ping et al., 2019; Prenger et al., 2019; Kim et al., 2019; Ping et al., 2020; Lee et al., 2020), GAN-based models (Donahue et al., 2019; Kumar et al., 2019; Binkowski et al. ´ , 2020; Yamamoto et al., 2020; Kong et al., 2020), and diffusion models (Kong et al., 2021; Chen et al., 2021; Lee et al., 2022).
|
| 18 |
+
|
| 19 |
+
Among these methods, GAN-based vocoders (e.g., Kong et al., 2020) can generate high-fidelity raw audio conditioned on mel spectrogram, while synthesizing hundreds of times faster than real-time on a single GPU. However, existing GAN vocoders are confined to the settings with a moderate number of voices recorded in clean environment due to the limited model capacity. The audio quality can heavily degrade when the models are conditioned on mel spectrogram from unseen speakers in different recording environments. In practice, a universal vocoder, that can do zero-shot generation for out-of-distribution samples, is very valuable in many real-world applications, including text-to-speech with numerous speakers (Ping et al., 2018), neural voice cloning (Arik et al., 2018; Jia et al., 2018), voice conversion (Liu et al., 2018), speech-to-speech translation (Jia et al., 2019), and neural audio codec (Zeghidour et al., 2021). In these applications, the neural vocoder also needs to generalize well for audio recorded at various conditions.
|
| 20 |
+
|
| 21 |
+
Scaling up the model size for zero-shot performance is a noticeable trend in text generation (e.g., Brown et al., 2020) and image synthesis (e.g., Ramesh et al., 2021), but has not been explored in audio synthesis. Although likelihood-based models are found to be easier for scaling among others because of their simple training objective and stable optimization, we build our universal vocoder with large-scale GAN training, because GAN vocoder has the following advantages: $i ,$ In contrast to autoregressive or diffusion models, it is fully parallel and requires only one forward pass to generate high-dimensional waveform. ii) In contrast to flow-based models (Prenger et al., 2019), it does not enforce any architectural constraints (e.g., affine coupling layer) that maintain the bijection between latent and data. Such architectural constraints can limit model capacity given the same number of parameters (Ping et al., 2020).
|
| 22 |
+
|
| 23 |
+
In this work, we present BigVGAN, a Big Vocoding GAN that enables high-fidelity out-ofdistribution (OOD) generation without fine-tuning. Specifically, we make the following contributions:
|
| 24 |
+
|
| 25 |
+
1. We introduce periodic activations into the generator, which provide the desired inductive bias for audio synthesis. Inspired by the methods proposed for other domains (Liu et al., 2020; Sitzmann et al., 2020), we demonstrate the noticeable success of periodic activations in audio synthesis. 2. We propose anti-aliased multi-periodicity composition (AMP) module for modeling complex audio waveform. AMP composes multiple signal components with learnable periodicities and uses low-pass filter to reduce the high-frequency artifacts. 3. We successfully scale BigVGAN up to 112M parameters by fixing the failure modes of large-scale GAN training without regularizing both generator and discriminator. The empirical insights are different from Brock et al. (2019) in image domain. For example, regularization methods (e.g., Miyato et al., 2018) introduce phase mismatch artifacts in audio synthesis. 4. We demonstrate that BigVGAN-base with 14M parameters outperforms the state-of-the-art neural vocoders with comparable size for both in-distribution and out-of-distribution samples. In particular, BigVGAN with 112M parameters outperforms the state-of-the-art models by a large margin for zero-shot generation at various OOD scenarios, including unseen speakers, novel languages, singing voices, music and instrumental audio in varied unseen recording environments.
|
| 26 |
+
|
| 27 |
+
We organize the rest of the paper as follows. We discuss related work in $\ S 2$ and present BigVGAN in $\ S \ O 3$ . We report empirical results in $\ S 4$ and conclude the paper in $\ S 5$ .
|
| 28 |
+
|
| 29 |
+
# 2 RELATED WORK
|
| 30 |
+
|
| 31 |
+
Our work builds upon the state-of-the-art of GANs for image and audio synthesis. GAN was first proposed for image synthesis (Goodfellow et al., 2014). Since then, impressive results have been obtained through optimized architectures (e.g., Radford et al., 2016; Karras et al., 2021) or large scale training (e.g., Brock et al., 2019).
|
| 32 |
+
|
| 33 |
+
In audio synthesis, previous works focus on improving the discriminator architectures or adding new auxiliary training losses. MelGAN (Kumar et al., 2019) introduces the multi-scale discriminator (MSD) that uses average pooling to downsample the raw waveform at multiple scales and applies window-based discriminators at each scale separately. It also enforces the mapping between input mel spectrogram and generated waveform via an $\ell _ { 1 }$ feature matching loss from discriminator. In contrast, GAN-TTS (Binkowski et al. ´ , 2020) uses an ensemble of discriminators which operate on random windows of different sizes, and enforces the mapping between the conditioner and waveform adversarially using conditional discriminators. Parallel WaveGAN (Yamamoto et al., 2020) extends the single short-time Fourier transform (STFT) loss (Ping et al., 2019) to multi-resolution, and adds it as an auxiliary loss for GAN training. Yang et al. (2021) and Mustafa et al. (2021) further improve MelGAN by incorporating the multi-resolution STFT loss. HiFi-GAN (Kong et al., 2020) reuses the MSD from MelGAN, and introduces the multi-period discriminator (MPD) for high-fidelity synthesis. UnivNet (Jang et al., 2020; 2021) uses the multi-resolution discriminator (MRD) that takes the multi-resolution spectrograms as the input and can sharpen the spectral structure of synthesized waveform. In contrast, CARGAN (Morrison et al., 2022) incorporates the partial autoregression (Ping et al., 2020) into generator to improve the pitch and periodicity accuracy.
|
| 34 |
+
|
| 35 |
+
In this work, we focus on improving and scaling up the generator. We introduce the periodic inductive bias for audio synthesis and address the feature aliasing issues within the non-autoregressive generator architecture. Our architectural design has a connection with the latest results in timeseries prediction (Liu et al., 2020), implicit neural representations (Sitzmann et al., 2020), and image synthesis (Karras et al., 2021). Note that, You et al. (2021) argues that different generator architectures can perform equally well for single-speaker neural vocoding. We demonstrate that improving generator architecture is crucial for universal neural vocoding in challenging conditions.
|
| 36 |
+
|
| 37 |
+
|
| 38 |
+
Figure 1: Schematic diagram of BigVGAN generator. The generator is composed of multiple blocks of transposed 1-D convolution followed by the proposed anti-aliased multi-periodicity composition (AMP) module. The AMP module adds features from multiple residual blocks with different channel-wise periodicities before dilated 1-D convolutions. It uses Snake function for providing periodic inductive bias, and low-pass filter for anti-aliasing purpose.
|
| 39 |
+
|
| 40 |
+
There are limited successes for universal neural vocoding due to the noticeable challenges. In previous work, WaveRNN has been applied for universal vocoding task (Lorenzo-Trueba et al., 2019; Paul et al., 2020). Jiao et al. (2021) builds the universal vocoder with flow-based model. GAN vocoder is found to be a good candidate recently (Jang et al., 2021).
|
| 41 |
+
|
| 42 |
+
# 3 METHOD
|
| 43 |
+
|
| 44 |
+
In this section, we introduce the preliminaries for GAN vocoder, then present the BigVGAN. See Figure 1 for an illustration and refer to the Appendix A for a detailed description of the architecture.
|
| 45 |
+
|
| 46 |
+
# 3.1 PRELIMINARIES OF GAN VOCODER
|
| 47 |
+
|
| 48 |
+
Generator The generator network takes mel spectrogram or other features as input and output the corresponding raw waveform. In previous studies, several generator architectures have been applied, including WaveNet (e.g., Yamamoto et al., 2020), or convolutional network that gradually upsamples the mel spectrogram to high-resolution waveform with a stack of residual blocks (e.g., Kumar et al., 2019; Kong et al., 2020). We choose the HiFi-GAN generator as the baseline architecture. We believe the proposed techniques are applicable to other generator architectures as well.
|
| 49 |
+
|
| 50 |
+
Discriminator The state-of-the-art GAN vocoders usually comprise several discriminators to guide the generator to synthesize coherent waveform while minimizing perceptual artifacts that are detectable by human ears. Importantly, each discriminator contains multiple sub-discriminators operating on different resolution windows of the waveform. For example, HiFi-GAN (Kong et al., 2020) applies two types of discriminators: i) the multi-period discriminator (MPD), where the 1-D signal is reshaped to 2-D representations with varied heights and widths to separately capture the multiple periodic structures though 2-D convolutions. ii) The multi-scale discriminator (MSD) (Kumar et al., 2019), where each sub-discriminator receives down-sampled 1-D signals at different frequency by average pooling in the time domain. Jang et al. (2020; 2021) propose to apply the discriminator on the time–frequency domain using the multi-resolution discriminator (MRD), which is composed of several sub-discriminators that operate on multiple 2-D linear spectrograms with different STFT resolutions. We also find that replacing MSD with MRD improves audio quality with reduced pitch and periodicity artifacts.
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Training objectives Our training objective is similar as HiFi-GAN (Kong et al., 2020), with an exception of replacing MSD to MRD. It comprises the weighted sum of the least-square adversarial loss (Mao et al., 2017), the feature matching loss (Larsen et al., 2016), and the spectral $\ell _ { 1 }$ regression loss on mel spectrogram. We leave the details of each loss and hyper-parameters in the Appendix B.
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# 3.2 PERIODIC INDUCTIVE BIAS
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The audio waveform is known to exhibit high periodicity and can be naturally represented as the composition of primitive periodic components (i.e., Fourier series under Dirichlet conditions). This suggests that we need to provide the desired inductive bias to the generator architecture. However, the current non-autoregressive GAN vocoders (e.g., Kong et al., 2020) solely rely on layers of dilated convolutions to learn the required periodic components at different frequencies. Their activation functions (e.g., Leaky ReLU) can produce new details with necessary nonlinearities, but do not provide any periodic inductive bias. Furthermore, we identified that Leaky ReLU behaves poorly for extrapolation in waveform domain: although the model can generate high-quality speech signal in a seen recording environment at training, the performance degrades significantly for out-of-distribution scenarios such as unseen recording environments, non-speech vocalizations, and instrumental audio.
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Figure 2: Spectrogram visualization of a out-of-distribution sample (singing voice) from HiFi-GAN and BigVGAN trained on LibriTTS, with a zoomed in view of high-frequency harmonic components.
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We introduce a proper inductive bias of periodicity to the generator by applying a recently proposed periodic activation called Snake function (Liu et al., 2020), defined as $\begin{array} { r } { \dot { f } _ { \alpha } ( x ) = x + \frac { \mathrm { i } } { \alpha } \sin ^ { 2 } ( \alpha x ) , } \end{array}$ where $\alpha$ is a trainable parameter that controls the frequency of the periodic component of the signal and larger $\alpha$ gives higher frequency. The use of $\sin ^ { 2 } ( x )$ ensures monotonicity and renders it amenable to easy optimization. Liu et al. (2020) demonstrates this periodic activation exhibits an improved extrapolation capability for temperature and financial data prediction.
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In BigVGAN, we use Snake activations $f _ { \alpha } ( x )$ with channel-wise trainable parameters $\pmb { \alpha } \in \mathbb { R } ^ { h }$ that define the periodic frequencies for each 1-D convolution channels. Taking this periodic functional form with learned frequency control, the convolutional module can naturally fit the raw waveform with multi-periodic components. We demonstrate that the proposed Snake-based generator is more robust for out-of-distribution audio samples unseen during training, indicating strong extrapolation capabilities in universal vocoding task. See Figure 2 and Appendix D for illustrative examples; BigVGAN-base w/o filter using snake activations is closer to ground-truth sample than HiFi-GAN.
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# 3.3 ANTI-ALIASED REPRESENTATION
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The Snake activations provide the required periodic inductive bias for modeling raw waveform, but it can produce arbitrary high frequency details for continuous-time signals that can not be represented by the discrete-time output of the network, 2 which can lead to aliasing artifacts. This side effect can be suppressed by applying a low-pass filter (e.g., Karras et al., 2021). The anti-aliased nonlinearity operates by upsampling the signal $2 \times$ along time dimension, applying the Snake activation, then downsampling the signal by $2 \times$ , which is a common practice inspired by the Nyquist-Shannon sampling theorem (Shannon, 1949). Each upsampling and downsampling operation is accompanied by the low-pass filter using a windowed sinc filter with a Kaiser window (Oppenheim & Schafer, 2009). Refer to the Appendix A for details.
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We apply this filtered Snake nonlinearity in every residual dilated convolution layers within the generator to obtain the anti-aliased representation of the discrete-time 1-D signals. The module is named as anti-aliased multi-periodicity composition (AMP). See Figure 1 for an illustration. We find that incorporating the filtered activation can reduce the high-frequency artifacts in the synthesized waveform; see BigVGAN-base w/o filter vs. BigVGAN-base (with filter) in Figure 2 as an illustration. We will demonstrate that it provides significant improvements in various objective and subjective evaluations. Note that we also explored anti-aliased upsampling layers, but this results in significant training instabilities and lead to early collapse for large models. See Appendix C for more details.
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# 3.4 BIGVGAN WITH LARGE SCALE TRAINING
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In this subsection, we explore the limits of universal vocoding by scaling up the generator’s model size to 112M parameters while maintaining the stability of GAN training and practical usability as a high-speed neural vocoder. We start with our improved generator using the comparable HiFi-GAN V1 configuration with 14M parameters (Kong et al., 2020), which is denoted as BigVGAN-base. We grow BigVGAN-base by increasing the number of upsampling blocks and convolution channels for each block. The BigVGAN-base upsamples the signal by $2 5 6 \times$ using 4 upsampling blocks with the ratio of [8, 8, 2, 2]. Each upsampling block is accompanied by multiple residual layers with dilated convolutions, i.e., the AMP module. We further divides the $2 5 6 \times$ upsampling into 6 blocks [4, 4, 2, 2, 2, 2] for more fine-grained feature refinement. In addition, we increase the number of channels of AMP module (analogous to MRF in HiFi-GAN) from 512 to 1536. We denote the model with 1536 channels and 112M parameters as BigVGAN.
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We found that the default learning rate of $2 \times 1 0 ^ { - 4 }$ used in HiFi-GAN causes an early training collapse for BigVGAN training, where the losses from the discriminator submodules immediately converge to zero after several thousands of iterations. Halving the learning rate to $1 \times 1 0 ^ { - 4 }$ was able to reduce such failures. We also found that large batch size is helpful to reduce mode collapse at training (Brock et al., 2019). We only double the batch size from the usual 16 to 32 for a good trade-off between training efficiency and stability, as neural vocoders can require millions of steps to converge. Note that this recommended batch size is still much smaller than the one for image synthesis (e.g., 2048) (Brock et al., 2019), because neural vocoding has strong conditional information.
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Even with the aforementioned changes, the large BigVGAN can still be prone to collapse early in training. We track the gradient norm of each modules during training and identify that the anti-aliased nonlinearity significantly amplified the gradient norm of MPD. Consequently, BigVGAN generator receives a diverging gradient early in training, leading to instabilities and potential collapse. We visualize the norm of gradient for each modules in Figure 4 at Appendix C. We alleviate the issue by clipping the global norm of the gradient to $1 0 ^ { 3 }$ , which is close to the average gradient norm of the 112M BigVGAN generator. This gradient clipping prevents the early training collapse of the generator. Note that, gradient clipping was found ineffective to alleviate training instability for image synthesis (see Appendix H in Brock et al. (2019)), but it is very effective in our endeavors.
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In addition to above efforts, we have explored other directions, including various ways to improve the model architecture, spectral normalization (Miyato et al., 2018) to stabilize GAN training, which is crucial for large-scale GAN training in image domain, and data augmentation to improve model generalization. Unfortunately, all these trials resulted in worse perceptual quality in our study. The details can be found in the Appendix C. We hope these practical lessons that we have learned would be useful to future research endeavors.
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# 4 RESULTS
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We conduct a comprehensive evaluation of BigVGAN for both in-distribution and out-of-distribution scenarios. We train BigVGAN and all baseline models on the full LibriTTS dataset.
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# 4.1 TRAINING DATA
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We use LibriTTS (Zen et al., 2019) dataset with the original sampling rate of $2 4 \mathrm { k H z }$ for training. Unlike previous studies which only adopted a subset (train-clean-100 or train-clean-360) recorded in a clean environment (Jang et al., 2020; 2021; AlBadawy et al., 2022), we use all training data including the subset from diverse recording environments (train-full $=$ train-clean-100 $^ +$ train-clean-360 $^ +$ train-other-500), which is unprecedented in the literature. We find that the diversity of the training data is important to achieve the goal towards universal neural vocoding using BigVGAN. 3 For OOD experiments, we resample the audio to $2 4 \mathrm { k H z }$ if necessary using kaiser-best algorithm provided by librosa package.
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Conventional STFT parameters are engineered to have a limited frequency band [0, 8] kHz by cutting off the high frequency details for easier modeling. On a contrary, we train all models (including the baselines) using a frequency range [0, 12] kHz and a 100-band log-mel spectrogram, which is also used in a recent study towards universal vocoding (Jang et al., 2021). We set other STFT parameters as in previous work (Kong et al., 2020), with 1024 FFT size, 1024 Hann window, and 256 hop size.
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Table 1: Model footprint and synthesis speed for $2 4 \mathrm { k H z }$ audio measured on an NVIDIA RTX 8000 GPU.
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<table><tr><td>Method</td><td>WaveGlow</td><td>WaveFlow</td><td>HiFi-GAN (V1)</td><td>BigVGAN-base</td><td>w/o filter</td><td>BigVGAN</td></tr><tr><td>Params (M)</td><td>99.43</td><td>22.58</td><td>14.01</td><td>14.01</td><td>14.01</td><td>112.4</td></tr><tr><td>Syn. speed</td><td>31.87×</td><td>19.59×</td><td>93.75×</td><td>70.18x</td><td>75.83×</td><td>44.72×</td></tr></table>
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# 4.2 MODELS
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We train all BigVGAN models including the ablation models and the baseline HiFi-GAN using our training configuration for 1M steps. We use the batch size of 32, a segment size of 8,192, and a initial learning rate of $1 \times 1 0 ^ { - 4 }$ . All other configurations including optimizer, learning rate scheduler, and scalar weights of the loss terms follow the official open-source implementation of HiFi-GAN (Kong et al., 2020) without modification, with an exception that BigVGAN replaces MSD by MRD for the discriminator. All models are trained using NVIDIA DGX-1 with 8 V100 GPUs. Refer to Table 6 in the Appendix A for detailed hyperparameters.
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We include a comparison with SC-WaveRNN (Paul et al., 2020), a state-of-the-art autoregressive universal neural vocoder based on WaveRNN (Kalchbrenner et al., 2018), using the official implementation. We also include two popular flow-based models: WaveGlow (Prenger et al., 2019) and WaveFlow (Ping et al., 2020), using their official implementation. For out-of-distribution test, we include the unofficial open-source implementation of UnivNet-c32 (Jang et al., 2021), 4 which uses train-clean-360 subset for training and is reported to outperform HiFi-GAN under the same training configurations. See appendix E for more details.
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Table 1 summarizes the synthesis speed of flow-based and GAN vocoders for generating $2 4 \mathrm { k H z }$ audio. We omit SC-WaveRNN as it is much slower. BigVGAN-base with 14M parameters can synthesize the audio $7 0 . 1 8 \times$ faster than real time, which is relatively slower than HiFi-GAN as the filtered Snake function requires more computation. HiFi-GAN and BigVGAN are faster than flow-based models, because they are fully parallel (WaveFlow has partial autoregression) and have much fewer layers (WaveGlow has 96 layers). Our BigVGAN with 112M parameters can synthesize the audio $4 4 . 7 2 \times$ faster than real-time and keeps the promise as a high-speed neural vocoder.
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# 4.3 EVALUATION METRICS
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The objective metrics we collected are designed to measure varied types of distance between the ground-truth audio and the generated sample. We provide 5 different metrics: 1) Multi-resolution STFT (M-STFT) (Yamamoto et al., 2020) which measures the spectral distance across multiple resolutions. 5 2) Perceptual evaluation of speech quality (PESQ) (Rix et al., 2001), a widely adopted automated assessment of voice quality. 6 3) Mel-cepstral distortion (MCD) (Kubichek, 1993) with dynamic time warping which measures the difference between mel cepstra. 7 4) Periodicity error, and 5) F1 score of voiced/unvoiced classification (V/UV F1) which are considered as major artifacts from non-autoregressive GAN vocoders (Morrison et al., 2022). 8
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The conventional 5-scale mean opinion score (MOS) is insufficient for the subjective evaluation of universal vocoder, because the metric needs to differentiate the utterances from diverse speaker identities recorded in various environments. For example, the model may always output some very natural “average” voices, which is not preferred but can still be highly rated by human workers in MOS evaluation. As a result, we also perform the 5-scale similarity mean opinion score (SMOS) evaluation, where the participant is asked to give the score of similarity for the pair of audio after listening to ground-truth audio and the sample from the model side-by-side. SMOS provides an improved way of assessing how close the given sample is to the ground-truth, where the ground-truth recordings can have diverse speaker identities, contains unseen languages for the listeners, and be recorded in various acoustic environments. SMOS is also directly applicable to non-speech samples, e.g., music. We did MOS and SMOS evaluation on Mechanical Turk. More details can be found in Appendix G.
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Table 2: Objective and subjective quality metrics of BigVGAN evaluated on LibriTTS. Objective results are obtained from dev sets, and subjective evaluations with 5-scale mean opinion score (MOS) and similarity mean opinion score (SMOS) with $9 5 \%$ confidence interval (CI) are obtained from test sets.
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<table><tr><td>LibriTTS</td><td>M-STFT(↓)</td><td>PESQ(↑)</td><td>MCD(↓)</td><td>Periodicity(↓)</td><td>V/UV F1(↑)</td><td>MOS(↑)</td><td>SMOS(↑)</td></tr><tr><td>Ground Truth</td><td>-</td><td>-</td><td>-</td><td>1</td><td>=</td><td>4.40±0.06</td><td>4.44±0.06</td></tr><tr><td>SC-WaveRNN</td><td>2.2358</td><td>1.701</td><td>1.8854</td><td>0.3044</td><td>0.8144</td><td>3.20±0.11</td><td>3.29±0.10</td></tr><tr><td>WaveGlow-256</td><td>1.3099</td><td>3.138</td><td>2.3591</td><td>0.1485</td><td>0.9378</td><td>3.84±0.10</td><td>3.87±0.10</td></tr><tr><td>WaveFlow-128</td><td>1.1120</td><td>3.027</td><td>1.2455</td><td>0.1416</td><td>0.9410</td><td>3.85±0.10</td><td>3.89±0.10</td></tr><tr><td>HiFi-GAN (V1)</td><td>1.0017</td><td>2.947</td><td>0.6603</td><td>0.1565</td><td>0.9300</td><td>4.08±0.09</td><td>4.15±0.09</td></tr><tr><td>BigVGAN-base</td><td>0.8788</td><td>3.519</td><td>0.4564</td><td>0.1287</td><td>0.9459</td><td>4.10±0.09</td><td>4.20±0.08</td></tr><tr><td>BigVGAN</td><td>0.7997</td><td>4.027</td><td>0.3745</td><td>0.1018</td><td>0.9598</td><td>4.11±0.09</td><td>4.26±0.08</td></tr></table>
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Table 3: The 5-scale SMOS results with $9 5 \%$ CI evaluated on unseen languages with different types of noise in unseen recording environments. $^ \dagger$ : pretrained weight obtained from an open-source repository which used train-clean-360 subset for training.
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<table><tr><td>Recording env. Language</td><td>Clean Jv,Km,Ne,Su</td><td>Noisy (sim) Es,Fr,I,Pt</td><td>Noisy (real) Ko</td></tr><tr><td>Ground Truth</td><td>4.58±0.05</td><td>4.36±0.05</td><td>4.56±0.05</td></tr><tr><td>UnivNet-c32†</td><td>4.35±0.07</td><td>3.95±0.09</td><td>4.18±0.08</td></tr><tr><td>HiFi-GAN (V1)</td><td>4.39±0.07</td><td>4.13±0.08</td><td>4.21±0.08</td></tr><tr><td>BigVGAN-base</td><td>4.38±0.07</td><td>4.21±0.07</td><td>4.36±0.07</td></tr><tr><td>BigVGAN</td><td>4.41±0.07</td><td>4.26±0.07</td><td>4.38±0.07</td></tr></table>
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# 4.4 LIBRITTS RESULTS
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We report the performance of BigVGAN and the baseline models evaluated on LibriTTS using above objective and subjective metrics. We perform objective evaluations on dev-clean and dev-other altogether, and conduct subjective evaluations on the combined test-clean and test-other. The dev and test splits of LibriTTS contains unseen speakers during training, but the recording environments are covered in the train split.
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Table 2 shows the in-distribution test results on LibriTTS. Baseline models other than HiFi-GAN performs significantly worse. This indicates that GAN vocoder is the state-of-the-art for universal neural vocoding. BigVGAN significantly improves all objective metrics. In particular, BigVGANbase exhibits consistently improved objective scores over HiFi-GAN (V1) with the same amount of paramters, suggesting that it has better periodic inductive bias for waveform data.
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HiFi-GAN (V1), BigVGAN-base, and BigVGAN perform comparably well in terms of MOS without listening to the ground-truth audio side-by-side. When the listeners can compare the model sample with ground truth audio side-by-side, BigVGAN-base measurably outperforms HiFi-GAN (V1) in terms of SMOS $\left( + 0 . 0 5 \right)$ , and the 112M BigVGAN outperforms HiFi-GAN by a clear margin in terms of SMOS $( + 0 . 1 1 )$ because it has high model capacity to further leverage the diverse training data for better quality.
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In Appendix E, we also report the additional results including UnivNet (Jang et al., 2021) and the ablation models of BigVGAN-base on LibriTTS dev sets, unseen VCTK (Yamagishi et al., 2019), and LJSpeech (Ito, 2017) data.
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# 4.5 UNSEEN LANGUAGES AND VARIED RECORDING ENVIRONMENTS
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In this subsection, we assess the universal vocoding capability of BigVGAN by measuring its zeroshot performance for various unseen languages with varied types of the recording environments in the unseen dataset. Based on the results in Table 2, we only include GAN-based vocoders as the state-of-the-art baseline. We gather three classes of a publicly available multi-language dataset categorized by the type of noise from the recording environment.
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• A collection of under-resourced languages recorded in a noiseless studio environment (Sodimana et al., 2018): Javanese, Khmer, Nepali, and Sundanese. We use randomly selected 50 audio clips with equal balance across languages from the combined dataset.
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Table 4: The 5-scale SMOS results with $9 5 \%$ CI evaluated on out-of-distribution samples from MUSDB18-HQ. †: pretrained model from an open-source repository which used train-clean-360 subset for training.
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<table><tr><td>Method</td><td>Vocal</td><td>Drums</td><td>Bass</td><td>Others</td><td>Mixture</td><td>Average</td></tr><tr><td>Ground Truth</td><td>4.58±0.05</td><td>4.57±0.05</td><td>4.52±0.05</td><td>4.61±0.05</td><td>4.56±0.05</td><td>4.57±0.02</td></tr><tr><td>UnivNet-c32t</td><td>4.22±0.09</td><td>4.23±0.09</td><td>3.90±0.11</td><td>3.80±0.13</td><td>3.80±0.12</td><td>3.99±0.05</td></tr><tr><td>HiFi-GAN (V1)</td><td>4.26±0.08</td><td>4.37±0.08</td><td>3.95±0.11</td><td>3.92±0.12</td><td>3.91±0.11</td><td>4.08±0.05</td></tr><tr><td>BigVGAN-base</td><td>4.36±0.08</td><td>4.39±0.07</td><td>4.00 ±0.11</td><td>4.14±0.09</td><td>4.11±0.10</td><td>4.20±0.04</td></tr><tr><td>w/o filter</td><td>4.30±0.08</td><td>4.32±0.07</td><td>3.95±0.11</td><td>4.05±0.10</td><td>4.11±0.10</td><td>4.15±0.04</td></tr><tr><td>w/o filter & snake</td><td>4.31±0.08</td><td>4.32±0.07</td><td>3.94±0.11</td><td>4.01±0.11</td><td>4.02±0.10</td><td>4.12±0.04</td></tr><tr><td>BigVGAN</td><td>4.37±0.08</td><td>4.41±0.07</td><td>4.00±0.10</td><td>4.25±0.09</td><td>4.26±0.08</td><td>4.26±0.04</td></tr></table>
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• The Multilingual TEDx Corpus (Salesky et al., 2021): contains a collection of TEDx talks in Spanish, French, Italian, and Portuguese. We use randomly selected 50 audio clips with equal balance across languages from the IWSLT’21 test set. We simulate the unseen recording environment setup by adding a random environmental noise from MS-SNSD (Reddy et al., 2019), such as airport, cafe, babble, etc.
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• Deeply Korean read speech corpus (Deeply, 2021): contains short speech audio clips in Korean, recorded in three types of recording environments (anechoic chamber, studio apartment, and dance studio) using a smartphone. We use randomly selected 50 audio clips where 25 clips are from the studio apartment, and the remaining 25 clips are from the dance studio. The collected audio clips contain a significant amount of noise and artifacts from real-world recording environments, such as reverb, echo, and static background noise.
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Table 3 summarizes the SMOS results from three different types of unseen dataset. We only did SMOS evaluations, because the datasets have unseen languages for human listeners and it is hard to determine the quality without side-by-side comparison with the ground-truth recordings. For clean under-resourced language dataset, the performance gap between models is not substantially large. This indicates that the universal vocoder trained on the entire LibriTTS training set is robust to unseen languages under clean recording environments. For both types of unseen recording environment (simulated or real-world), BigVGAN outperforms the baseline models by a large margin. The small capacity BigVGAN-base also shows improvements compared to the baseline with statistical significance $\mathrm { { \widetilde { p } } }$ -value $< 0 . 0 5$ from the Wilcoxon signed-rank test). This suggests that BigVGAN is significantly more robust to the unseen recording environments thanks to the improved generator design with the AMP module. In Appendix F, we further demonstrate that BigVGAN is the most linguistically accurate universal vocoder in terms of character error rate (CER) on multiple languages.
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We test the open-source implementation of UnivNet (Jang et al., 2021) with the pretrained checkpoint which is trained on train-clean-360 subset. Contrary to the report from Jang et al. (2021) that UnivNet-c32 outperformed HiFi-GAN (Kong et al., 2020), we find that the unmodified HiFi-GAN trained on the entire LibriTTS dataset is able to match or outperform UnivNet-c32. We also train UnivNet-c32 on LibriTTS train-full and find that it is not benefited from larger training data. See Appendix E for detailed analysis.
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# 4.6 OUT-OF-DISTRIBUTION ROBUSTNESS
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In this subsection, we test BigVGAN’s robustness and extrapolation capability by measuring zeroshot performance on out-of-distribution data. We conduct the SMOS experiment using MUSDB18- HQ (Rafii et al., 2019), a multi-track music audio dataset which contains vocal, drums, bass, other instruments, and the original mixture. The test set contains 50 songs with 5 tracks. We gather the mid-song clip with the duration of 10 seconds for each track and song.
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Table 4 shows the SMOS results from the 5 tracks and their average from the MUSDB18-HQ test set. BigVGAN models demonstrate a substantially improved zero-shot generation performance with wider frequency band coverage, whereas baseline models fail to generate audio outside the limited frequency range and suffer from severe distortion. The improvements are most profound for singing voice (vocal), instrumental audio (others) and the full mixture of the song (mixture), whereas the improvements from drums and bass are less significant.
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Table 5: Ablation results on training data diversity using 112M BigVGAN model, evaluated on LibriTTS. Objective results are obtained from dev-other and subjective evaluation with 5-scale SMOS with $9 5 \%$ confidence interval (CI) is obtained from test-other.
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<table><tr><td>Training data</td><td>M-STFT(↓)</td><td>PESQ(↑)</td><td>MCD(↓)</td><td>Periodicity(↓)</td><td>V/UV F1(↑)</td><td>SMOS(↑)</td></tr><tr><td>Ground Truth</td><td>-</td><td>-</td><td>1</td><td>-</td><td>-</td><td>4.55±0.05</td></tr><tr><td>train-full</td><td>0.8197</td><td>4.001</td><td>0.4097</td><td>0.1023</td><td>0.9586</td><td>4.38±0.07</td></tr><tr><td>train-clean-360</td><td>0.8429</td><td>3.847</td><td>0.4232</td><td>0.1149</td><td>0.9521</td><td>4.31±0.08</td></tr><tr><td>VCTK</td><td>0.8747</td><td>3.818</td><td>0.5921</td><td>0.1215</td><td>0.9490</td><td>4.27±0.08</td></tr></table>
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We also experiment with audios obtained from YouTube videos from real-world recording environments. BigVGAN also exhibits robustness to various types of out-of-distribution signals such as laughter. We provide audio samples to our demo page. 9
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# 4.7 ABLATION STUDY
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Model architecture To measure the effectiveness of the BigVGAN generator, we include SMOS test for the ablation models of BigVGAN-base on MUSDB18-HQ data. Table 4 shows that the ablation models exhibit clear degradation on various scenarios such as instrumental audio (others, mixture). From the average SMOS ratings, 1) disabling the anti-aliasing filter for Snake activation performs worse than BigVGAN-base and 2) removing both the filter and Snake activation (i.e., vanilla HiFi-GAN trained with MRD replacing MSD) is even worse than the Snake-only ablation model, both with statistical significance (p-value $< 0 . 0 1$ from the Wilcoxon signed-rank test). This indicates that Leaky ReLU is not robust enough to extrapolate beyond the learned frequency range and the aliasing artifacts degrade the audio quality in challenging setups. The results show that BigVGAN generator demonstrates strong robustness and extrapolation capability to out-of-distribution scenarios because of the seamless integration of periodic inductive bias and anti-aliased feature representation. See Appendix D for the visualization of anti-aliasing effect in BigVGAN.
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Big model We compare HiFi-GAN and BigVGAN both with the largest 112M parameters. We train the 112M HiFi-GAN with the same training setting as BigVGAN. We conduct a pairwise test between the two models on the mixture test set of MUSDB18-HQ which is challenging out-of-distribution data. We ask the participants to select a better sounding audio between the samples from the two models. The test shows that $5 8 \%$ of the ratings voted to BigVGAN over the large HiFi-GAN and the quality of BigVGAN is greater than the large HiFi-GAN with statistical significance (p-value $<$ 0.01 from the Wilcoxon signed-rank test). The results further validate the architectural advantage of BigVGAN in large-scale setting.
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Large Training data To verify the importance of using large-scale training data, we trained our BigVGAN using less diverse, clean speech-only dataset with the same training configuration for 1M steps: 1) train-clean-360 subset of LibriTTS, or 2) VCTK dataset. Table 5 shows that training BigVGAN on less diverse data shows degradation in both objective metrics and the subjective SMOS on the LibriTTS evaluation sets. The result verifies the importance of using diverse training data and demonstrates the effectiveness of BigVGAN on large-scale datasets.
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# 5 CONCLUSIONS
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This study explores the limits of universal neural vocoding with an unprecedented scale of the data, model, and evaluations. We analyze the performance with various automatic and human evaluations across diverse scenarios including unseen speakers, languages, recording environments and out-ofdistribution data. We present BigVGAN with an improved generator architecture by introducing anti-aliased periodic activation function with learned frequency control, which injects the desired inductive bias for waveform generation. Based on the improved generator, we demonstrate the largest GAN vocoder with strong zero-shot performance under various OOD conditions, including unseen recording environments, singing voice, and instrumental audio. We believe that BigVGAN, combined with practical lessons learned from the large scale training, will inspire future endeavors for universal vocoding and improve the state-of-the-art results for real-world applications, including voice cloning, voice conversion, speech translation, and audio codec.
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# APPENDIX
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Figure 3: Detailed diagram of $\mathrm { B i g V G A N }$ generator and discriminator architectures. Left: generator architecture, where values in parentheses denote (output channel, kernel width, dilation rate) respectively. Right: discriminator architectures (MRD in orange and MPD in yellow), where values in parentheses denote (output channel, [kernel width, kernel height], [stride width, stride height]) respectively.
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Table 6: Hyperparameters of $\mathrm { B i g V G A N }$ generators and discriminators.
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<table><tr><td colspan="3">Generator</td><td colspan="2">Discriminator</td></tr><tr><td></td><td>BigVGAN-base</td><td>BigVGAN</td><td colspan="2">MRD&MPD</td></tr><tr><td>h</td><td>512</td><td>1536</td><td>n_ffti</td><td>[1024,2048,512]</td></tr><tr><td>Ui</td><td>[8,8,2,2]</td><td>[4,4,2,2, 2,2]</td><td>hop_lengthi</td><td>[120,240,50]</td></tr><tr><td>Ki</td><td>[3,3,3,7,7,7,11,11,11]</td><td>[3,3,3, 7, 7, 7,11, 11,11]</td><td>win_lengthi</td><td>[600,1200,240]</td></tr><tr><td>Di</td><td>[[1,1],[3,1],[5,1]] ×3</td><td>[1,1],[3,1],[5,1]] × 3</td><td>Reshape2di (pi)</td><td>[2,3,5,7,11]</td></tr></table>
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# A ARCHITECTURAL DETAILS
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In this section, we present a detailed description of the BigVGAN architecture. Refer to Figure 3 for illustrative details. BigVGAN uses the similar generator architecture presented in HiFi-GAN (Kong et al., 2020). The generator takes a mel spectrogram as input and synthesizes a corresponding waveform as output. The generation process starts with a single layer of $1 D$ convolution using a channel width of $h$ and a kernel size of 7 without dilation (denoted as 1 in the Figure 3). The hierarchical generation comprises $N$ number of upsampling blocks. The $i$ -th upsampling block $( i = \{ 1 , . . . , N \} )$ starts with a transposed $1 D$ convolution using half the number of channels of the preceding block and an upsampling rate of $u _ { i }$ . The upsampled feature is followed by $M$ number of AMP residual blocks, where each AMP block uses different kernel sizes for a stack of dilated $1 D$ convolutions defined as $k _ { i , j } ( j = \{ 1 , \dots , M \} )$ . The $j$ -th AMP block contains $L$ number of the anti-aliased periodic activation and the dilated $1 D$ convolution using a dilatation rate of $d _ { i , j , l } ( l = \{ 1 , \dots , L \} )$ . Refer to the Table 6 for the hyperparameters of the BigVGAN generators.
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Our design of the low-pass filter is similar to StyleGAN3 (Karras et al., 2021). We use a cutoff frequency of $\frac { s } { 2 m }$ , where $m = 2$ is a up- and down-sampling ratio and $s$ is a sampling rate (e.g., width) of the signal. The Kaiser window uses a window length of $n = 6 \cdot m$ and a shape parameter $\beta$ is approximated by $0 . 1 1 0 2 \cdot \left( A - 8 . 7 \right)$ , where a maximum attenuation $A$ is approximated by $A = 2 . 2 8 5 \cdot ( \textstyle { \frac { n } { 2 } } - 1 ) \cdot \pi \cdot 4 f _ { h } + 7 . 9 5$ (Oppenheim & Schafer, 2009) with a transition band half-width $\begin{array} { r } { f _ { h } = \frac { 0 . 6 } { m } } \end{array}$ . The low-pass filter is applied as the kernel in $1 D$ convolution for downsampling, and as the kernel in transposed $1 D$ convolution for upsampling.
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The BigVGAN discriminator comprises two submodules: MRD and MPD. Each module is composed of multiple subdiscriminators using a stack of $2 D$ convolutions as in Figure 3. MRD converts the input $1 D$ waveform to its $2 D$ linear spectrogram using STFT with different parameters ([n_fft, hop_length, win_length]). MPD converts the input $1 D$ waveform with length $T$ to its $2 D$ representation by reshaping and reflection padding (Reshape2d) with different width $( p _ { i } )$ and height $\textstyle { \big ( } { \frac { T } { p _ { i } } } { \big ) }$ . Refer to the Table 6 for the MRD and MPD hyperparameters. We used the same MRD and MPD hyperparameters for training all BigVGAN generators.
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# B TRAINING OBJECTIVE DETAILS
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We apply the training objective formulation and its hyperparameters described in (Kong et al., 2020) without modification, with an exception that BigVGAN applies MRD replacing MSD as the discriminator submodule. Concretely, we apply the following objectives $\mathcal { L } _ { G }$ for generator and $\mathcal { L } _ { D }$ for discriminator, respectively:
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$$
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\mathcal { L } _ { G } = \sum _ { k = 1 } ^ { K } \bigg [ \mathcal { L } _ { a d v } ( G ; D _ { k } ) + \lambda _ { f m } \mathcal { L } _ { f m } ( G ; D _ { k } ) \bigg ] + \lambda _ { m e l } \mathcal { L } _ { m e l } ( G ) , \quad \mathcal { L } _ { D } = \sum _ { k = 1 } ^ { K } \bigg [ \mathcal { L } _ { a d v } ( D _ { k } ; G ) \bigg ] ,
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$$
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where $D _ { k }$ denotes the $k$ -th MPD or MRD discriminator submodules. $\mathcal { L } _ { a d v }$ uses the least-square GAN (Mao et al., 2017) as follows:
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$$
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\begin{array} { r } { \underline { { \cdot } } _ { a d v } ( G ; D _ { k } ) = \mathbb { E } _ { s } \Big [ ( D _ { k } ( G ( s ) ) - 1 ) ^ { 2 } \Big ] , \quad \mathcal { L } _ { a d v } ( D _ { k } ; G ) = \mathbb { E } _ { ( x , s ) } \Big [ ( D _ { k } ( x ) - 1 ) ^ { 2 } + ( D _ { k } ( G ( s ) ) ) ^ { 2 } \Big ] , } \end{array}
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$$
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+
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+
where $x$ is the ground-truth waveform, and $s$ is the input mel spectrogram. The feature matching loss $\mathcal { L } _ { f m }$ (Larsen et al., 2016; Kumar et al., 2019) minimizes the $\ell _ { 1 }$ distance for every intermediate features from the discriminator layers:
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$$
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\mathcal { L } _ { f m } ( G ; D _ { k } ) = \mathbb { E } _ { ( x , s ) } \bigg [ \sum _ { i = 1 } ^ { T } \frac { 1 } { N } | | D _ { k } ^ { i } ( x ) - D _ { k } ^ { i } ( G ( s ) ) | | _ { 1 } \bigg ] ,
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+
$$
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+
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where $T$ is the number of layers of the sub-discriminator $D _ { k }$ . The generator loss $\mathcal { L } _ { G }$ also has the spectral $\ell _ { 1 }$ regression loss between the mel spectrogram of the synthesized waveform and the corresponding ground-truth:
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+
$$
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\begin{array} { r } { \mathcal { L } _ { m e l } ( G ) = \mathbb { E } _ { ( x , s ) } \biggl [ | | \phi ( x ) - \phi ( G ( s ) ) | | _ { 1 } \biggr ] , } \end{array}
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$$
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+
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where $\phi$ is the STFT function that converts the waveform into the mel spectrogram. We used the scalar weights $\lambda _ { f m } = 2$ and $\lambda _ { m e l } = 4 5$ identically as (Kong et al., 2020).
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# C PRACTICAL LESSONS FOR LARGE-SCALE GAN TRAINING ON AUDIO
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Figure 4: Visualization of gradient norm for different modules from BigVGAN training. Left: Gradient norm from the generator. Mid: Gradient from the MPD module. Right: Gradient norm from the MRD module. For BigVGAN-base, the gradient norm significantly increases at early training without clipping. For BigVGAN, the gradient will explode without clipping.
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In this section, we document additional directions that we have explored for improving the model architecture and large scale training, but resulted in worse perceptual quality from our preliminary study. We do not make conclusive claims based on these observations because the methods we explored here may have been ineffective specific to our BigVGAN settings. Nevertheless, we believe that reporting negative results can be helpful to future research endeavors (Brock et al., 2019).
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Anti-aliased upsampling layers Our BigVGAN uses anti-aliased activation for AMP modules. We also explored replacing the transposed convolution-based upsampling layers, which are known to contain checkerboard aliasing (Odena et al., 2016), to the anti-aliased alternatives with different lowpass filter hyperparameters. However, this introduced significant instabilities during training, leading to early collapse even with the aforementioned stabilization. We also tried out nearest-neighbor upsampling which is reported to have less artifacts for audio synthesis (Pons et al., 2021), but it also resulted in the early collapse.
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Periodic discriminators Inspired by the improvements with the periodic activation function to the generator, we experimented on discriminators with Snake function. However, it degraded the quality with the diverging feature matching loss (Kong et al., 2020) from the discriminators. We conjecture that the periodic activation to the discriminator is not stable enough to improve the gradient from the feature matching loss.
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Spectral normalization Spectral normalization (Miyato et al., 2018) is a widely adopted method to stabilize GAN training in image domain. We tried applying spectral normalization to all discriminator submodules and found that it can stabilize the training without the early divergence of the generator. However, it suffered from a significant degradation with the excessive amount of phase mismatch artifacts, similar to the findings in the previous work (Kumar et al., 2019). We found that the gradient from MPD is over-regularized and the generator start to solely rely on the mel regression loss. Because MPD is repeatedly found to be a key component for high-quality audio synthesis (Kong et al., 2020), regularizing MPD leads to worse result.
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Larger discriminators We hypothesized that the enlarged generator can be slower to learn, thereby trivializing the discriminator in the early training (loss converge to zero). We tried to balance the training by enlarging the discriminators, such as employing more MPD sub-discriminators or enlarging the channel width of MPD and MRD modules. The large discriminator partially alleviated the early collapse, but the audio quality degraded in most cases, and showed no clear improvements even with our best configuration.
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Even larger generators We experimented with deeper model with 8 upsampling blocks with ratio of [2, 2, 2, 2, 2, 2, 2, 2]. However, it exhibited high-frequency rattling artifacts and degraded the quality. We conjecture that generating fine-grained high-frequency details from the early upsampling blocks can be arbitrary (Karras et al., 2021) and unstable when using the filtered periodic nonlinearity. We also tried to further increase the number of convolution channels to 2048, but it suffered from early collapse.
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Data augmentation Data augmentation is one of major methods that improve model generalization, which is also repeatedly found to be valuable in GAN literature (Karras et al., 2020). We explored augmenting the data by applying SpecAugment (Park et al., 2019) to the input mel spectrogram. However, SpecAugment resulted in over-smoothing artifacts in waveform, because it enforces the model to map multiple distorted mel spectrograms to the same waveform, which counters the upsampling process in generative models. We also tried applying mixup-like (Zhang et al., 2018) approach to the waveform by using a convex combination of two audio samples and its corresponding mel spectrogram. However, this occasionally resulted in a mixed voice of two speaker identities during a single-speaker inference without noticeable improvement in perceptual quality.
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+
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Positional encoding Inspired by using partial autoregression (Ping et al., 2020; Morrison et al., 2022) to provide inductive bias of the cumulative sum relationship of pitch and phase (Morrison et al., 2022), we explored whether we can provide an approximate inductive bias in a non-autoregressive manner by injecting a sinusoidal positional encoding (Vaswani et al., 2017) to the generator. However, the generator with positional encoding is only exposed to the fixed audio segment at training $( \sim 0 . 3$ seconds with 8,192 time-steps) and unable to extrapolate unknown and significantly longer sequence at inference (e.g., several seconds).
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Figure 5: Spectrogram visualization of out-of-distribution samples from HiFi-GAN and BigVGAN models trained on LibriTTS, with a zoomed in view of harmonic components. Top: a singing voice. Bottom: an instrumental audio.
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# D VISUALIZATION
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In this section, we present visual examples to analyze the improvements from the methodologies used in BigVGAN. We use HiFi-GAN (V1), BigVGAN-base along with the ablation model without the filtered nonlinearity, and the high-capacity BigVGAN for the in-depth analysis.
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The top row of the Figure 5 corresponds to a singing voice. The feature aliasing of HiFi-GAN introduces blurry harmonics because aliased features at multiple incorrect frequency components are aggregated during the generative process, which amplifies the error. BigVGAN-base without the filtered nonlinearity improves the harmonic components using the periodic activation and the advanced discriminator. BigVGAN models further improve the accuracy using the continuous feature representation and the anti-aliasing filter. The bottom row of the Figure 5 shows an instrumental audio. Similar to the singing voice example, HiFi-GAN exhibits blurry harmonics. BigVGAN models capture such challenging high-frequency harmonics significantly better than the baselines, which suggests that BigVGAN is robust to various unseen conditions. The visual analysis shows that BigVGAN exhibits a significantly less distortion of the harmonic components and a better sound quality for various types of audio beyond the clean speech signal.
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+
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# E ADDITIONAL RESULTS
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In addition to the results in Section 4 of main text, we provide additional comparison with previous work and ablation models with the automatic objective evaluation.
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+
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Comparison with Baselines Models Table 7 and 8 show the expanded objective results evaluated on LibriTTS, including SC-WaveRNN (Paul et al., 2020), WaveGlow (Prenger et al., 2019), WaveFlow (Ping et al., 2020), UnivNet (Jang et al., 2021), and the ablation models of BigVGAN-base.
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+
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+
SC-WaveRNN (Paul et al., 2020) is a universal vocoder based on WaveRNN (Kalchbrenner et al., 2018) with a speaker embedding from a separately trained encoder network. We replicate the training procedure following (Paul et al., 2020) using the official implementation, while matching training data and audio hyperparameters used in this study (i.e., LibriTTS-full, 24,000Hz sampling rate and a 100-band log-mel spectrogram with [0, 12] kHz frequency). We train SC-WaveRNN for 3M steps. Although the model can generate speech with diverse speaker identities, we find that the performance is significantly worse than GAN vocoders both in objective scores and human listening test. It fails to generate intelligible audio for unseen recording environments and out-of-distribution sample. We conjecture that the speaker embedding network is not robust enough to generalize to such setups.
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+
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+
WaveGlow (Prenger et al., 2019) and WaveFlow (Ping et al., 2020) are two widely known vocoders based on normalizing flows (Rezende & Mohamed, 2015). We train the flow-based vocoders using the STFT hyperparameters used in this study. WaveGlow is trained for 2M steps and WaveFlow is trained for 1M steps as suggested by the authors. Flow-based vocoder features fast and parallel synthesis with an analytic likelihood objective due to its bijectivity. However, the quality of flow-based vocoders degrades heavily for multi-speaker setup. This renders the flow-based model unsuitable as a universal neural vocoder.
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+
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+
We also train UnivNet-c32 (Jang et al., 2021) on LibriTTS train-full using the open-source implementation (note that UnivNet uses the $2 4 { , } 0 0 0 \mathrm { H z }$ sampling rate and the 100-band log-mel spectrogram with [0, 12] kHz frequency by default). However, unlike the HiFi-GAN and BigVGAN architectures, using the large-scale dataset did not improve the quality of the UnivNet architecture. Overall, the publicly available model trained on train-clean-360 scores marginally better in terms of PESQ and periodicity error compared to our checkpoint trained on train-full. Other metrics exhibit different preference depending on the training set. The subjective quality is indistinguishable to the authors between the two checkpoints, including the out-of-distribution setup. We conjecture that the UnivNet architecture is harder to generalize to the unseen recording environments. For subjective SMOS tests, we choose to use the publicly available checkpoint using train-clean-360 to faithfully represent the performance of the previous work. Note that the open-source implementation of UnivNet is unofficial. Therefore, our observation does not lead to the conclusion that UnivNet is worse than HiFi-GAN.
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+
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+
Comparison with Ablation Models From both clean and other recording environments, BigVGANbase shows consistent improvements in all metrics compared to its ablation models. Specifically, BigVGAN-base (without filter & snake) refers to the vanilla HiFi-GAN architecture trained with MRD replacing MSD. Similar to the findings in (Jang et al., 2021), MRD provides more accurate modeling of the waveform by sharpening the spectral structure. Applying Snake activation (BigVGAN-base without filter) increases accuracy and reduces the periodicity error from the desired inductive bias. Finally, incorporating the continuous feature representation and the low-pass filter (BigVGAN-base) provides the best accuracy by suppressing aliasing and high frequency artifacts. Our final 112M BigVGAN substantially improves the accuracy for the state-of-the-art universal neural vocoding.
|
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+
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| 392 |
+
Results on Unseen VCTK and LJSpeech Dataset Table 9 and 10 show the objective speech evaluation metric results gathered from VCTK and LJSpeech dataset, consistent with the results from Table 7 and 8. Although the dataset is not included for training BigVGAN and the baseline HiFi-GAN, all GAN-based models performed comparatively well from our subjective listening test. This indicates that given diverse enough training data, modern non-autoregressive GAN vocoder can synthesize high-quality speech with unseen speaker identities from the clean recording environment.
|
| 393 |
+
|
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+
Table 7: Objective results of BigVGAN from dev-clean of LibriTTS including ablation models of BigVGAN-base and previous work. †: pretrained weight obtained from an open-source repository which used train-clean-360 subset for training.
|
| 395 |
+
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| 396 |
+
<table><tr><td>LibriTTS (clean)</td><td>MAE(↓)</td><td>M-STFT(↓)</td><td>PESQ(↑)</td><td>MCD(↓)</td><td>Periodicity(↓)</td><td>V/UV F1(↑)</td></tr><tr><td>SC-WaveRNN</td><td>0.5517</td><td>2.1411</td><td>1.774</td><td>1.5854</td><td>0.2925</td><td>0.8300</td></tr><tr><td>WaveGlow-256</td><td>0.5368</td><td>1.3238</td><td>3.179</td><td>2.3897</td><td>0.1423</td><td>0.9419</td></tr><tr><td>WaveFlow-128</td><td>0.2839</td><td>1.0706</td><td>3.120</td><td>0.9632</td><td>0.1339</td><td>0.9459</td></tr><tr><td>UnivNet-c32†</td><td>0.2803</td><td>0.9552</td><td>3.348</td><td>0.7017</td><td>0.1342</td><td>0.9433</td></tr><tr><td>UnivNet-c32</td><td>0.2772</td><td>0.9433</td><td>3.310</td><td>0.6942</td><td>0.1356</td><td>0.9435</td></tr><tr><td>HiFi-GAN (V1)</td><td>0.2579</td><td>0.9773</td><td>3.042</td><td>0.6257</td><td>0.1545</td><td>0.9306</td></tr><tr><td>BigVGAN-base</td><td>0.1546</td><td>0.8569</td><td>3.574</td><td>0.4180</td><td>0.1298</td><td>0.9475</td></tr><tr><td>w/o filter</td><td>0.1770</td><td>0.8838</td><td>3.472</td><td>0.4624</td><td>0.1354</td><td>0.9437</td></tr><tr><td>w/o filter& snake</td><td>0.1899</td><td>0.9047</td><td>3.351</td><td>0.4852</td><td>0.1399</td><td>0.9401</td></tr><tr><td>BigVGAN</td><td>0.0931</td><td>0.7796</td><td>4.053</td><td>0.3392</td><td>0.1013</td><td>0.9610</td></tr></table>
|
| 397 |
+
|
| 398 |
+
Table 8: Objective results of BigVGAN from dev-other of LibriTTS including ablation models of BigVGANbase and previous work.
|
| 399 |
+
|
| 400 |
+
<table><tr><td>LibriTTS (other)</td><td>MAE(↓)</td><td>M-STFT(↓)</td><td>PESQ(↑)</td><td>MCD(↓)</td><td>Periodicity(↓)</td><td>V/UV F1(↑)</td></tr><tr><td>SC-WaveRNN</td><td>0.6125</td><td>2.3274</td><td>1.576</td><td>1.4717</td><td>0.2174</td><td>0.8896</td></tr><tr><td>WaveGlow-256</td><td>0.5096</td><td>1.2960</td><td>3.098</td><td>2.3284</td><td>0.1546</td><td>0.9337</td></tr><tr><td>WaveFlow-128</td><td>0.3359</td><td>1.1533</td><td>2.935</td><td>1.5278</td><td>0.1493</td><td>0.9360</td></tr><tr><td>UnivNet-c32t</td><td>0.3027</td><td>0.9973</td><td>3.166</td><td>0.8643</td><td>0.1439</td><td>0.9345</td></tr><tr><td>UnivNet-c32</td><td>0.2998</td><td>0.9883</td><td>3.107</td><td>0.8495</td><td>0.1457</td><td>0.9337</td></tr><tr><td>HiFi-GAN (V1)</td><td>0.2724</td><td>1.026</td><td>2.853</td><td>0.6948</td><td>0.1585</td><td>0.9294</td></tr><tr><td>BigVGAN-base</td><td>0.1625</td><td>0.9006</td><td>3.464</td><td>0.4947</td><td>0.1276</td><td>0.9442</td></tr><tr><td>w/o filter</td><td>0.1844</td><td>0.9223</td><td>3.364</td><td>0.5017</td><td>0.1391</td><td>0.9395</td></tr><tr><td>w/o filter& snake</td><td>0.2008</td><td>0.9456</td><td>3.240</td><td>0.5389</td><td>0.1451</td><td>0.9344</td></tr><tr><td>BigVGAN</td><td>0.0986</td><td>0.8197</td><td>4.001</td><td>0.4097</td><td>0.1023</td><td>0.9586</td></tr></table>
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| 401 |
+
|
| 402 |
+
Table 9: Objective results from unseen VCTK dataset. We used randomly selected 100 audio clips.
|
| 403 |
+
|
| 404 |
+
<table><tr><td>VCTK</td><td>MAE(↓)</td><td>M-STFT(↓)</td><td>PESQ(↑)</td><td>MCD(↓)</td><td>Periodicity(↓)</td><td>V/UV F1(↑)</td></tr><tr><td>SC-WaveRNN</td><td>0.6619</td><td>2.7344</td><td>1.445</td><td>1.7597</td><td>0.2528</td><td>0.8611</td></tr><tr><td>WaveGlow-256</td><td>0.5454</td><td>1.3324</td><td>3.149</td><td>2.4512</td><td>0.1218</td><td>0.9513</td></tr><tr><td>WaveFlow-128</td><td>0.2982</td><td>1.0825</td><td>2.953</td><td>1.1127</td><td>0.1213</td><td>0.9518</td></tr><tr><td>UnivNet-c32t</td><td>0.2753</td><td>0.9476</td><td>3.235</td><td>0.7180</td><td>0.1131</td><td>0.9535</td></tr><tr><td>UnivNet-c32</td><td>0.2820</td><td>0.9586</td><td>3.184</td><td>0.7403</td><td>0.1198</td><td>0.9434</td></tr><tr><td>HiFi-GAN (V1)</td><td>0.2541</td><td>0.9859</td><td>3.029</td><td>0.6795</td><td>0.1336</td><td>0.9375</td></tr><tr><td>BigVGAN-base</td><td>0.1527</td><td>0.8677</td><td>3.443</td><td>0.4526</td><td>0.1047</td><td>0.9586</td></tr><tr><td>w/o filter</td><td>0.1739</td><td>0.8935</td><td>3.379</td><td>0.4924</td><td>0.1128</td><td>0.9528</td></tr><tr><td>w/o filter & snake</td><td>0.1885</td><td>0.9184</td><td>3.316</td><td>0.5159</td><td>0.1208</td><td>0.9481</td></tr><tr><td>BigVGAN</td><td>0.0925</td><td>0.7684</td><td>4.001</td><td>0.3557</td><td>0.0833</td><td>0.9672</td></tr></table>
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| 405 |
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|
| 406 |
+
Table 10: Objective results on unseen LJSpeech dataset. We used randomly selected 100 audio clips.
|
| 407 |
+
|
| 408 |
+
<table><tr><td>LJSpeech</td><td>MAE(↓)</td><td>M-STFT(↓)</td><td>PESQ(↑)</td><td>MCD(↓)</td><td>Periodicity(↓)</td><td>V/UV F1(↑)</td></tr><tr><td>SC-WaveRNN</td><td>1.0115</td><td>2.6994</td><td>1.233</td><td>4.8464</td><td>0.3907</td><td>0.7404</td></tr><tr><td>WaveGlow-256</td><td>0.4933</td><td>1.2893</td><td>3.352</td><td>2.9921</td><td>0.1182</td><td>0.9561</td></tr><tr><td>WaveFlow-128</td><td>0.3674</td><td>1.2402</td><td>3.072</td><td>2.7217</td><td>0.1170</td><td>0.9560</td></tr><tr><td>UnivNet-c32†</td><td>0.3418</td><td>1.0613</td><td>3.425</td><td>1.1903</td><td>0.1210</td><td>0.9519</td></tr><tr><td>UnivNet-c32</td><td>0.3356</td><td>1.0429</td><td>3.384</td><td>1.1356</td><td>0.1230</td><td>0.9503</td></tr><tr><td>HiFi-GAN (V1)</td><td>0.3008</td><td>1.0950</td><td>3.210</td><td>1.7370</td><td>0.1347</td><td>0.9456</td></tr><tr><td>BigVGAN-base</td><td>0.1747</td><td>0.9121</td><td>3.741</td><td>0.8626</td><td>0.1164</td><td>0.9548</td></tr><tr><td>w/o filter</td><td>0.2015</td><td>0.9395</td><td>3.662</td><td>0.9715</td><td>0.1169</td><td>0.9548</td></tr><tr><td>w/o filter & snake</td><td>0.2263</td><td>0.9946</td><td>3.521</td><td>1.1320</td><td>0.1299</td><td>0.9479</td></tr><tr><td>BigVGAN</td><td>0.1102</td><td>0.8554</td><td>4.112</td><td>0.7164</td><td>0.0957</td><td>0.9642</td></tr></table>
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+
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+
Table 11: Character error rates (CER) of synthesized speech on multiple languages.
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+
|
| 412 |
+
<table><tr><td>Method</td><td>English</td><td>German</td><td>Catalan</td><td>Spanish</td><td>French</td><td>Mandarin</td></tr><tr><td>Ground Truth</td><td>6.298</td><td>3.785</td><td>3.001</td><td>4.194</td><td>5.450</td><td>25.431</td></tr><tr><td>SC-WaveRNN</td><td>15.155</td><td>9.508</td><td>5.763</td><td>10.198</td><td>12.427</td><td>47.653</td></tr><tr><td>UnivNet-c32†</td><td>6.914</td><td>4.037</td><td>3.058</td><td>4.439</td><td>5.717</td><td>26.504</td></tr><tr><td>HiFi-GAN (V1)</td><td>6.906</td><td>4.070</td><td>3.082</td><td>4.490</td><td>5.770</td><td>27.179</td></tr><tr><td>HiFi-GAN (112M)</td><td>6.615</td><td>3.920</td><td>3.043</td><td>4.315</td><td>5.608</td><td>26.174</td></tr><tr><td>BigVGAN-base</td><td>6.574</td><td>3.905</td><td>3.034</td><td>4.335</td><td>5.594</td><td>26.174</td></tr><tr><td>BigVGAN</td><td>6.436</td><td>3.829</td><td>3.028</td><td>4.261</td><td>5.517</td><td>25.756</td></tr></table>
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+
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+
# F LINGUISTIC ACCURACY EVALUATION
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| 415 |
+
|
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+
In this section, we evaluate the quality of neural vocoders in terms of linguistic accuracy. To measure linguistic accuracy of the synthesized speech from BigVGAN on in-distribution and various out-of-distribution languages, we conduct experiments using a high-performance Conformer-based (Gulati et al., 2020) automatic speech recognition (ASR) model on multiple languages. We use conformer_transducer_large model, where the pretrained checkpoints for the considered languages are publicly available from the NVIDIA NeMo (Kuchaiev et al., 2019) toolkit. We generate samples from BigVGAN using test set corpus from Mozilla Common Voice (MCV) 8.0 dataset for the following languages: English (en), German (de), Catalan (ca), Spanish (es), French (fr), and Mandarin (zh). Then, we obtain transcriptions from the synthesized speech using the ASR model and calculate character error rate (CER). In this way, we can assess a degree of degradation in linguistic accuracy compared to the CER of the ground truth speech.
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+
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+
The CER results are shown in Table 11. BigVGAN has the consistently lowest CERs, which are also close to ground truth. Furthermore, BigVGAN-base (14M) outperforms the largest HiFi-GAN (112M) on four (en, de, ca, fr) out of six languages, ties on one $\mathrm { ( z h ) }$ , and underperforms only on one language (es). The results further validate that BigVGAN is the most linguistically accurate universal vocoder with the lowest artifacts and show the benefits of the model to be applied to various speech applications.
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+
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+
# G ADDITIONAL DETAILS OF MECHANICAL TURK EVALUATION
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+
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| 422 |
+
We use Mechanical Turk for MOS and SMOS tests with 450 unique ratings per model. We allow each worker to evaluate up to three random samples for diverse participants. Crowdsourcing evaluation is considered to be a noisy process. We apply several filtering criteria to improve the reliability of workers : i) We restrict the listeners to be native English speakers in the United States. ii) We only allow listeners with the evaluation acceptance rate higher than $98 \%$ , and the number of previously approved HITs higher than 100 to participate in this test. iii) We conduct 5-scale MOS/SMOS evaluation with random ordering of the model samples, with the inclusion of ground-truth samples as the hidden control questions. We apply filtering by rejecting ratings from those workers who score the ground-truth samples as 3 or even lower, because the listeners are instructed to give score 5 for ground-truth samples after listening to the same ground-truth sample prior to the rating.
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| 1 |
+
# LST: Ladder Side-Tuning for Parameter and Memory Efficient Transfer Learning
|
| 2 |
+
|
| 3 |
+
Yi-Lin Sung Jaemin Cho Mohit Bansal UNC Chapel Hill {ylsung, jmincho, mbansal}@cs.unc.edu
|
| 4 |
+
|
| 5 |
+
# Abstract
|
| 6 |
+
|
| 7 |
+
Fine-tuning large pre-trained models on downstream tasks has been adopted in a variety of domains recently. However, it is costly to update the entire parameter set of large pre-trained models. Although recently proposed parameter-efficient transfer learning (PETL) techniques allow updating a small subset of parameters (e.g. only using $2 \%$ of parameters) inside a pre-trained backbone network for a new task, they only reduce the training memory requirement by up to $3 0 \%$ . This is because the gradient computation for the trainable parameters still requires backpropagation through the large pre-trained backbone model. To address this, we propose Ladder Side-Tuning (LST), a new PETL technique that can reduce training memory requirements by more substantial amounts. Unlike existing parameter-efficient methods that insert additional parameters inside backbone networks, we train a ladder side network, a small and separate network that takes intermediate activations as input via shortcut connections (called ladders) from backbone networks and makes predictions. LST has significantly lower memory requirements than previous methods, because it does not require backpropagation through the backbone network, but instead only through the side network and ladder connections. We evaluate our method with various models (T5 and CLIP-T5) on both natural language processing (GLUE) and vision-and-language (VQA, GQA, $\mathrm { \Delta N L V R ^ { 2 } }$ , MSCOCO) tasks. LST saves $6 9 \%$ of the memory costs to fine-tune the whole network, while other methods only save $2 6 \%$ of that in similar parameter usages (hence, $2 . 7 \mathbf { x }$ more memory savings). Moreover, LST achieves higher accuracy than Adapter and LoRA in a low-memory regime. To further show the advantage of this better memory efficiency, we also apply LST to larger T5 models (T5-large, T5-3B), attaining better GLUE performance than full fine-tuning and other PETL methods. The trend also holds in the experiments on vision-and-language tasks, where LST achieves similar accuracy to other PETL methods when training a similar number of parameters while also having $2 . 7 \mathbf { x }$ more memory savings.
|
| 8 |
+
|
| 9 |
+
# 1 Introduction
|
| 10 |
+
|
| 11 |
+
Recently, large-scale pre-training and fine-tuning of transformers [54] have been successful in various domains [8, 36, 44, 38, 53, 7, 20, 1, 10]. As the model size grows rapidly, fine-tuning the entire parameter set of the large pre-trained model has become very costly. Parameter-efficient transfer learning (PETL) [31, 23, 39, 40, 34, 51, 58, 17, 24, 18, 42, 60, 14, 52, 59] is a recent research direction for online or multi-task learning. The goal is to build a system that performs well on all tasks without training an entire new model for every new task. Concretely, PETL methods select a small subset of pre-trained parameters and/or insert a few parameters to a pre-trained network and update those parameters for new tasks, while freezing most of the original parameters. In the natural language processing (NLP), computer vision (CV), and vision-and-language (VL) domains, two types of parameters have been commonly updated for parameter-efficient transfer learning: (a) Adapters [23, 40, 39]: small modules inserted into transformer blocks; (b) Prompt [34, 31]: small parameters concatenated with input embeddings (see Figure 2).
|
| 12 |
+
|
| 13 |
+
|
| 14 |
+
Figure 1: Comparison between full finetuning, Adapter, LoRA, BitFit, and Ladder Side-Tuning over GLUE tasks. The y-axis is the average accuracy of 8 GLUE tasks, while the $\mathbf { X }$ -axis is the GPU memory usage during training. Unless specially stated, we use the T5-base in the figure.
|
| 15 |
+
|
| 16 |
+
|
| 17 |
+
Figure 2: Comparison between transfer learning with (a) Adapters, (b) Prompt Tuning, and our (b) Ladder SideTuning (LST). LST reduces memory usage by removing the need of backpropgation through backbone networks.
|
| 18 |
+
|
| 19 |
+
However, while parameter-efficient techniques reduce the number of parameters to update, they do not reduce the memory requirement during training by much (up to $3 0 \%$ ). In other words, if a large pre-trained language model does not fit on a GPU, these techniques usually do not help to fit the model on the GPU. In other words, these techniques usually do not help to fit a large pre-trained language model on a GPU if the model cannot be trained on the GPU with standard fine-tuning. Since the updated parameters are inside the backbone language models, to calculate gradients for these parameters for backpropagation, we still need to run the backward pass through the large pre-trained language models. This prevents PETL methods from being applied to many real-world applications with limited computational resources.
|
| 20 |
+
|
| 21 |
+
To address this issue, we propose Ladder Side-Tuning (LST), a memory-efficient PETL method. LST separates trainable parameters from the backbone model to construct a side network, which is responsible for adapting the entire model to new tasks. Concretely, we train a ladder side network, a lightweight network that takes intermediate activations via shortcut connections (ladders) from the backbone networks as input and makes predictions. As shown in Figure 2, unlike previous (a) adapters and (b) prompt tuning methods, (c) our LST does not add trainable parameters inside the pre-trained model, and this completely eliminates the need for expensive backpropagation of a large backbone network and saves substantial memory during transfer learning. Instead of initializing the side network’s weights randomly, we utilize a network pruning technique to retrieve a smaller pruned network and use it as the side network. In addition to our standard design of the side network, we also boost the efficiency of LST by dropping layers of the side network. We empirically demonstrate that the layer dropping can significantly improve both memory and parameter efficiency without sacrificing performance. Furthermore, during inference, even though we have to propagate forward through two distinct networks, LST does not necessarily use more inference time because the same level of the backbone network and the side network can be computed in parallel.
|
| 22 |
+
|
| 23 |
+
We conduct comprehensive studies on LST using diverse NLP and VL tasks, namely, GLUE [55], VQA [16], GQA [25], $\mathrm { \Delta N L V R ^ { 2 } }$ [50] and MSCOCO [6]. Overall, in GLUE experiments, LST saves $6 9 \%$ of the GPU memory that is needed for fine-tuning the entire backbone model, saving $2 . 7 \mathbf { x }$ memory compared against Adapter and LoRA. Also, LST achieves higher accuracy than other PETL methods in a low-memory regime. To take advantage of this better memory efficiency, we also apply LST to larger language models (T5-large and T5-3B) and find that it achieves higher accuracy than other PETL techniques when GPU memory utilization is similar. The findings still hold in the Vision-Language (VL) experiments; LST is not only a method that can fit in a 16GB GPU with $7 . 5 \%$ trainable parameters, but it also has similar or better accuracy than other PETL methods (and again with $2 . 7 \mathbf { x }$ memory savings). To justify our design of LST, we conduct ablation studies on initialization strategies and alternatives to add shortcut connections. The results reveal that these components considerably help performance with minor computation overhead.
|
| 24 |
+
|
| 25 |
+
# 2 Related Work
|
| 26 |
+
|
| 27 |
+
# 2.1 Parameter-efficient Transfer Learning (PETL)
|
| 28 |
+
|
| 29 |
+
PETL for NLP. In the past few years, large pre-trained language models have made huge success in NLP. Parameter-efficient transfer learning (PETL) is a research direction that reduces computational cost of adapting large pre-trained models to new tasks, by avoiding updates of the entire parameters. A popular line of work on PETL is to add a few trainable parameters and only tune them. For example, Adapters [48, 23] are small bottleneck modules that are inserted into transformer layers, and experiments have shown that training adapters with layer normalization layers is sufficient to achieve full fine-tuning performance. In a similar trend, LoRA [24] injects trainable rank decomposition matrices into a frozen pre-trained model. Instead of inserting new parameters into pre-trained models, prompt-based [31, 34] methods add trainable parameters to the input and keep the entire pre-trained model unchanged during training. The inserted prompts learn to make use of the knowledge of the pre-trained model to solve new tasks. Although the concept of adapter-based and prompt-based approaches is different, He et al. [18] unify the two lines of approaches (including LoRA) into adapter-based methods. In addition to approaches that introduce new parameters, there are also various methods [51, 58, 17] that select a sparse subset of parameters from the pre-trained model to update, without adding any new parameters. One of such representative methods is BitFit [58], which updates every bias term in the model.
|
| 30 |
+
|
| 31 |
+
PETL for CV/VL. While most of the progress in PETL is made in NLP domain, researchers have also applied this technique to the CV [47, 46, 27, 59, 62, 28, 60, 21] and VL [52, 61, 62, 28, 60] domains. VL-Adapter [52] benchmarks adapter-based and prompt-based methods on multiple imagetext and video-text tasks, and shows adapters enable us to efficiently learn fusion information of vision and language. On the CV side, benefitting from the parameter efficiency of adapters and prompt-tuning, some works [62, 28, 60] apply these approaches to CLIP [43] to achieve strong few-shot performance in image classification tasks. Side-Tuning [59] uses an additive side network, which sums its representation with the backbone network in the last layer, to solve various tasks with ResNet [19] and BERT [8]. Although LST takes inspiration and has similarities to Side-Tuning, we argue that there are major differences in motivations, architecture designs, and applied tasks between the two methods. LST aims to reduce the memory requirement of current PETL methods, whereas Side-Tuning does not focus on memory reduction (sometimes their side network is even as big as the backbone network), but instead their motivation is to ease the forgetfulness in incremental learning. Our ladder side network is more robust than their design because the shortcuts fuse the intermediate information from the backbone network, and we also use layer dropping and network pruning techniques to make LST more efficient and stronger. Lastly, we further extend LST in VL architecture and demonstrate its usefulness on multiple VL tasks.
|
| 32 |
+
|
| 33 |
+
Current PETL approaches explore how to achieve competitive results using as few parameters as possible. However, parameter efficiency does not necessarily mean memory efficiency. In this work, we propose LST that has these two benefits simultaneously. Concurrently, Liu et al. [37] also propose Y-tuning to address a similar issue; it exhausts all possible labels and feeds them into a model to select the best answer from the input. However, it is intractable oftentimes to list all answers in some tasks, for example, regression and open-ended generation tasks. On the other hand, LST is more flexible in applying to different architectures and tasks. We show that LST can outperform Y-tuning with fewer parameter updates in Table 2.
|
| 34 |
+
|
| 35 |
+
# 2.2 Memory-Efficient Training
|
| 36 |
+
|
| 37 |
+
Memory-efficient training aims to reduce the memory cost when training neural networks. Some approaches achieve this goal by cutting down the storage of intermediate activations, which dominate the training cost, to release a large amount of memory. The design of reversible neural networks [15, 29, 41] allows the model not to save intermediate activations because each layer’s activations can be reconstructed from the next layer’s. Gradient checkpointing [5] proposes to trade computation for memory by dropping some intermediate activations and recovering them from an extra forward pass. LST uses a different approach to cut the storage of activations; it keeps the backbone model frozen and constructs a side network for cheaper training. Since the backbone model is not updated, LST is not only memory-efficient but also parameter-efficient. Furthermore, memory saving from reversible neural networks and checkpointing is agnostic to memory saving from LST. Researchers can combine those methods with LST if they pursue a higher level of memory efficiency.
|
| 38 |
+
|
| 39 |
+
Another line of memory-efficient methods is network compression, which compresses a backbone network to a smaller one, and the generated network is cheaper for both training and inference. Two popular approaches for compression are network pruning and distillation. Network distillation [22, 30] constructs a student network and force it to have same output distribution of the teacher network over a chosen dataset. Network pruning [12, 13] makes models lighter by learning importance scores of parameters and trimming unimportant parameters or neurons. While PETL still uses those untrained parameters in the forward pass, network compression entirely discards them or sets them to zero. As a result, network compression can generate models for faster inference speed while PETL can achieve better performance by updating fewer parameters.
|
| 40 |
+
|
| 41 |
+
In this paper, we explore using network pruning or network distillation [22] (used in Side-tuning) to extract a sub-network that contains critical information of the backbone model and use it for initializing our side network. For distillation, we did not follow the original Side-tuning to apply distillation with large-scale pre-training datasets (e.g., C4 [44]) because it makes distillation hard to conduct with limited resources and ultimately violates our goal of “efficient training.” Also, using an extra pre-training dataset during fine-tuning is unfair to other approaches. We use the standard distillation procedure with the T5 [44] pre-training objective to train the side network. That is, the student (side) network learns to predict the masked spans and match the output distribution of the teacher (backbone) network simultaneously. We show the comparison between distillation-based and pruning-based initializations in Figure 8.
|
| 42 |
+
|
| 43 |
+
We primarily use the network pruning method proposed by Li et al. [33] to initialize the side network because of its efficiency, and we describe the approach in detail in Section 3.3. The standard procedure of network pruning is (1) learn [12, 13] or heuristically define [33] an "importance measure" to identify the importance of parameters, (2) prune $p \%$ of parameters with lower importance scores, (3) repeat the first and second steps until reaching the target sparsity. The rewinding procedure enables pruning techniques to find a more sparse sub-network. In this paper, to keep the whole pruning process efficient, we either use weights magnitude [33] or Fisher Information [51, 35] as importance measures, and reach the target sparsity in one shot. As a PETL method, LST makes use of intermediate information from the backbone model as the inputs, and we empirically demonstrate those additional inputs significantly improve performance in Figure 8.
|
| 44 |
+
|
| 45 |
+
# 3 Ladder Side-Tuning (LST)
|
| 46 |
+
|
| 47 |
+
We introduce Ladder Side-Tuning (LST), a new PETL technique that can also reduce training memory requirements by substantial amounts than previous methods. In Section 3.1, we analyze the computational cost for fine-tuning with trainable modules in backbone models. Then we explain the architectural details (Section 3.2), structural weight initialization based on network pruning (Section 3.3), and dropping side network layers for more efficiency (Section 3.4).
|
| 48 |
+
|
| 49 |
+
# 3.1 Dependency on Backpropagation through Large Backbone Model
|
| 50 |
+
|
| 51 |
+
We consider a $N$ multilayer perceptron (MLP): $f _ { N } ( f _ { N - 1 } ( \dots f _ { 2 } ( f _ { 1 } ( x ) ) . . . ) )$ , where the $i ^ { t h }$ layer $f _ { i } ( x ) = \sigma _ { i } ( W _ { i } x + b _ { i } )$ consists of weight $W _ { i }$ , bias $b _ { i }$ , and nonlinear function $\sigma _ { i }$ . We denote the output of $i ^ { t h }$ layer as $a _ { i + 1 }$ and the pre-activation as $z _ { i + 1 }$ , where $a _ { i + 1 } = \sigma _ { i } ( z _ { i + 1 } ) = \sigma _ { i } ( W _ { i } a _ { i } + b _ { i } )$ . In backpropagation with loss $L$ , the gradient with respect to $W _ { i }$ and $b _ { i }$ :
|
| 52 |
+
|
| 53 |
+
$$
|
| 54 |
+
\frac { \partial L } { d W _ { i } } = \frac { \partial L } { \partial a _ { i + 1 } } \frac { \partial a _ { i + 1 } } { \partial z _ { i + 1 } } \frac { \partial z _ { i + 1 } } { \partial W _ { i } } = \frac { \partial L } { \partial a _ { i + 1 } } \sigma _ { i } ^ { \prime } a _ { i } , \qquad \frac { \partial L } { d b _ { i } } = \frac { \partial L } { \partial a _ { i + 1 } } \sigma _ { i } ^ { \prime }
|
| 55 |
+
$$
|
| 56 |
+
|
| 57 |
+
where σ′i is the derivative of σi. ∂L∂ai+1 , the gradient with respect to $a _ { i }$ , can be calculated with the gradients with respect to $a _ { i + 2 }$ , using the chain rule:
|
| 58 |
+
|
| 59 |
+
|
| 60 |
+
Figure 3: Illustration of Ladder Side-Tuning (LST) with transformers described in Section 3.2. (a) shows a high-level overview of LST, and (b) shows LST with an encoder-decoder architecture.
|
| 61 |
+
|
| 62 |
+
$$
|
| 63 |
+
\frac { \partial L } { \partial a _ { i + 1 } } = \frac { \partial L } { \partial a _ { i + 2 } } \frac { \partial a _ { i + 2 } } { \partial z _ { i + 2 } } \frac { \partial z _ { i + 2 } } { \partial a _ { i + 1 } } = \frac { \partial L } { \partial a _ { i + 2 } } \sigma _ { i + 1 } ^ { \prime } W _ { i + 1 }
|
| 64 |
+
$$
|
| 65 |
+
|
| 66 |
+
As shown in Equations (1) and (2), during backpropagation, there are two terms dominating the memory footprint: 1) $\{ a \}$ corresponding to updated parameters $\{ W \}$ and 2) $\{ \sigma ^ { \prime } \}$ that must be cached for the chain rule. Note that we use $\{ \cdot \}$ to denote a set of activations, parameters, or gradients. Existing PETL methods, such as Adapters [23], LoRA [24], Prompt-tuning [31], and BitFit [58, 3] could reduce the memory footprint by making $| a |$ smaller, as they have fewer $\{ W \}$ to update, but do not reduce $| \sigma ^ { \prime } |$ , where $| \cdot |$ means the size of set $\{ \cdot \}$ . Since most activation functions do not change dimensions (i.e., $| a | = | \sigma ^ { \prime } | )$ , the memory footprint for backpropagation $| a | + | \sigma ^ { \prime } |$ can be reduced by up to $50 \%$ by the PETL methods when they reduce the entire memory footprint for $| a |$ . By making the updated parameter do not require backpropagation through the backbone network, our LST can achieve better memory efficiency beyond $50 \%$ , and we explain it below.
|
| 67 |
+
|
| 68 |
+
# 3.2 Ladder Side Network for Transformers
|
| 69 |
+
|
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Unlike existing transfer learning methods that insert additional parameters inside a transformer network, we propose training a ladder side network, a small and separate network that takes intermediate activations from the backbone transformer as input and makes predictions. As illustrated in Figure 3 (a), since the ladder side network parameters $\phi$ are not used during the forward pass of the backbone transformer with parameters $\theta$ , the update of the ladder side network does not require expensive backpropagation of the large backbone transformer. Note that our LST method is not limited to a specific architecture. We provide a simplified overview of LST with an encoder architecture in Figure 3 (a) and an illustration of LST with an encoder-decoder architecture in Figure 3 (b).
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Lightweight architecture. Our side network $g$ is a lightweight version of the backbone transformer $f$ , where all weights and hidden state dimensions of in $g$ are $\textstyle { \frac { 1 } { r } }$ times of the original weights and hidden states of $f$ , where $r$ is a reduction factor (e.g. $r = 2$ , 4, 8, 16). For example, if the backbone $f$ has a 768-dimensional hidden state, then the side network $g$ with $r = 1 6$ has a hidden state of 48 dimensions $\left( = 7 6 8 / 1 6 \right)$ . The side network $g$ reuses frozen word embeddings (‘Emb’ in Figure 3 (a)) and the language model head (‘LM head’ in Figure 3 (a)) of the backbone $f$ . Following the analysis in Section 3.1, we also examine the memory cost of LST. Recall that original memory footprint for backpropagation is $| a | + | \sigma ^ { \prime } |$ . Because we do not have to run a backward pass through the backbone network, we can only consider the gradients for the side network, whose memory footprint is $\frac { | a | + | \sigma ^ { \prime } | } { r }$ . Therefore, LST has a better memory efficiency than other PETL methods (saving up to $50 \%$ ) as long as $r$ is greater than 2 (we find 8 works well in most experiments).
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Figure 4: Illustration of (a) Structural Weight Initialization (Section 3.3) and (b) Layer Dropping (Section 3.4). In our experiments, we find that initialization of side network parameters from backbone network parameters improves performance, and dropping some shortcut connections improves efficiency without hurting performance.
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Gated ladder connections. Although Zhang et al. [59] found that late fusion to combine the representations of the backbone and the side network works well with convolutional networks for CV tasks, in our experiments, we find that late fusion hurts the performance of the transformer architecture in NLP tasks (see Figure 8 in Section 5 for details). To address this, we use the shortcut connection (called ladder, due to the overall shape created from the multiple shortcut connections) from intermediate activations from the backbone $f$ to the side network $g$ and find it helpful. We learn linear projections to downsample $\scriptstyle ( \times { \frac { 1 } { r } } )$ the intermediate activations (including word embeddings) of $f$ to low-dimensional attention blocks in $g$ . Then, we learn a linear projection to upsample $( \times r )$ the side network output to the dimension of the original language model head. The linear projections are illustrated as green trapezoids in Figure 3 (a). The $i ^ { \tilde { t } h }$ transformer layer of the side network $g$ combines the activation of the backbone $h _ { i } ^ { f }$ and the activation of the previous layer of the side network $h _ { i - 1 } ^ { g }$ with learned gating: $\mu _ { i } * h _ { i } ^ { f } + ( 1 - \mu _ { i } ) * h _ { i - 1 } ^ { g }$ , where $\begin{array} { r } { \mu _ { i } = \tt s i g m o i d ( \frac { \alpha _ { i } } { T } ) } \end{array}$ is a gate parameterized with a learnable zero-initialized scalar $\alpha _ { i }$ and temperature $T$ $( = 0 . 1 )$ ). We have also tried to use Adapter blocks to build the side network and replace the gating mechanism with cross-attentions, but we find the current design works the best (see ??).
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# 3.3 Structural Weight Initialization for Ladder Side Network
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We find it helpful to initialize the weights of the side network $\phi$ from the weight of the backbone network $\theta$ based on structural pruning [33], as shown in Figure 4 (a). Concretely, given a weight matrix $W \in \mathbb { R } ^ { d _ { o u t } \times d _ { i n } }$ of the backbone network that maps the $d _ { i n }$ -dim vectors to the $d _ { o u t }$ -dim space, and the importance matrix of the weight $I \in \mathbb { R } ^ { d _ { o u t } \times d _ { i n } }$ , we first calculate the importance score of each row $\begin{array} { r } { s _ { i } = \sum _ { j } | I _ { i , j } | } \end{array}$ , denoting the importance of each weight vector. Note that the importance matrix $I$ used in this work are either weight magnitude [33] $\mathit { \Delta } M = W$ ) or empirical Fisher Information [51] $\begin{array} { r } { ( I = F _ { W } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } ( \nabla _ { W } \log p ( y _ { i } | x _ { i } ) ) ^ { 2 } ; ( x _ { i } , y _ { i } ) , . . . , ( x _ { N } , y _ { N } ) } \end{array}$ are samples from data). Then, we choose the rows of remaining rows to obtain a new weig $W$ which matrix $\frac { d _ { o u t } } { r }$ importance scores and prune the. The columns of the weights and $W ^ { P } \in \mathbb { R } ^ { \frac { d _ { o u t } } { r } \times d _ { i n } }$ the importance matrix in the next layer corresponding to the pruned feature map are also pruned. By iterating this process, we obtain the set of weight matrices whose rows and columns are pruned $\frac { \texttt { i } } { r }$ times from the backbone network and use them to initialize the side network. In our experiments shown in Figure 7, we find that using Fisher information as an importance score metric generally performs well, and therefore we use it in our structural weight initialization.
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# 3.4 Layer Dropping in the Ladder Side Network
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We explore to increase efficiency of LST even further by making side network more compact, by dropping its intermediate transformer layers, as illustrated in Figure 4 (b). Similar to LayerDrop [11], we drop layers in the side network, and this can linearly reduce the memory and parameter requirements of LST. For instance, a side network with $N$ layers will only have $2 ^ { n { \dot { d } } }$ , $4 ^ { t h }$ , $\mathbf { \bar { \boldsymbol { 6 } } } ^ { t h }$ . . . layers left, after we drop half of the layers. Refer to Section 4 for more details on applying layer dropping on an encoder-decoder architecture. In Figure 6, we show that layer dropping can greatly boost the model’s efficiency without sacrificing performance.
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# 4 Experiment Setup
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Datasets. We evaluate LST on NLP and VL tasks. For NLP tasks, we use the GLUE [55] benchmark, which consists of seven classification and one regression task. The benchmark evaluate models on multiple diverse tasks over linguistic acceptability (CoLA [56]), sentiment analysis (SST2 [49]), similarity and paraphrase (MRPC [9], QQP [26], STS-B [4]) and natural language inference (MNLI [57], QNLI [45], RTE [2]). For VL tasks, we experiment with visual question answering (VQA [16], GQA [25]), visual reasoning $\mathrm { ( N L V R ^ { 2 } }$ [50]) and image captioning (MSCOCO [6]) tasks.
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Baselines. We compare LST against the full fine-tuning and several popular PETL approaches on both NLP and VL setups. In Full fine-tuning, all the parameters are updated for a downstream task. Full fine-tuning is not parameter-efficient nor memory-efficient, but it serves as the upper bound of the fine-tuning performance. To compare to other PETL methods, we reproduce Adapters, where we inject small trainable modules after every attention and feed-forward layer, and we solely train those modules and layer normalization layers while keeping the rest of the model frozen. We also reproduce LoRA, which inserts trainable low-rank matrices into the model to parameterize the weights’ changes. In BitFit, we only update the bias terms over the course of training. Lastly, we compare our method to Prompt-tuning, where trainable prompt vectors are prepended to the input. We initialize the prompt vectors with the embedding of the pre-trained model’s vocabularies.
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Training and Evaluation Setup. For NLP tasks, we use T5 [44], a pre-trained encoder-decoder language model as our backbone. We use T5-base in most experiments, except that we scale up LST on T5-large and T5-3B to demonstrate its memory efficiency. The training and evaluation process follows the setup used by Mahabadi et al. [40]. Since there is no local test set, we split 1k samples from the training set as the new validation set and use the original validation set as the test set. For datasets whose samples are less than 10k (RTE, MRPC, STS-B, CoLA), we split the validation set into two equal-sized subsets and treat them as a new validation and test set. For MNLI, we use the mismatched set as the validation set and matched set as the test set. We train every approach with 10 epochs on large datasets and 20 epochs on small ones (RTE, MRPC, STS-B, CoLA) for complete convergence. We search for learning rates over $\{ 3 \times 1 0 ^ { - 4 } , 1 \times 1 0 ^ { - 3 } , 3 \times 1 0 ^ { - 3 } \}$ for LST and LoRA[24], and we use the optimal learning rates that are used by Mahabadi et al. [40] for other methods. The reduction factor used in LST is set to 8 if not additionally specified. T5-base has 12 layers each in encoder and decoder, while T5-large and T5-3B have 24 layers each. In our experiments, we do not drop layers in T5-base unless we specially mention it. For T5-large and T5-3B, we drop 24 layers (12 layers each in encoder and decoder) and 46 layers (23 each) of the side network to make the memory usage close to our baselines. The experiments on T5 take around 12 hours to train with one A6000 GPU (48GB).
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For VL tasks, we experiment with CLIP-T5 [52], which is a VL architecture combining CLIP [43] and T5 [44]. We always freeze the CLIP and only train the T5 for new tasks. The CLIP visual representation is concatenated with the text embedding, and the combined input is fed to T5 to make predictions. A visual projection layer is added between CLIP and T5 to let the visual representation have the same dimension as the text embedding. To avoid updating the visual projection layer by the gradients from the backbone model, we do not feed combined inputs to the backbone model, but only text inputs. The combined inputs are fed to the side network, so we can achieve efficient training by only computing the gradients from the side network. Because the backbone network only uses texts as the input, the information from the backbone network via shortcut connections is only summed to the text part of the side network’s combined inputs. We follow the multi-tasking setting for training and evaluation used in VL-Adapter [52]. We report the performance on Karpathy test/test-dev/test-P/Karpathy test split fo $\mathrm { r \nabla { V O A } / \bar { G O A } / N L V R ^ { 2 } / M \bar { S } C O C O }$ , and train models for 20 epochs. We search learning rates over $\{ 3 \times 1 0 ^ { - 4 } , 1 \times 1 0 ^ { - 3 } , 3 \times 1 0 ^ { - 3 } \}$ for PETL methods, and use $1 \times 1 0 ^ { - 4 }$ used by Sung et al. [52] for full fine-tuning. We set the reduction factor for the side network to 4. We train CLIP-T5 for 16 hours on one A6000 GPU. In ??, we comprehensively list hyper-parameters for NLP and VL experiments in ?? and ??, respectively.
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Table 1: Comparison between multiple parameter-efficient training methods on GLUE benchmark. We use T5-base if we don’t additionally specify. We report accuracy for SST-2, MNLI, QNLI and RTE. For CoLA and STS-B, we use Matthew’s Correlation and Pearson-Spearman Correlation as the metrics, respectively. For MRPC and QQP, we report the average of F1 score and accuracy. Each number in the table is the average result over three seeds, and the subscripts are standard deviations. For the results with †, we report the best performance out of three seeds due to the instability of the method. We report the maximum memory usage training and evaluating on RTE for each method.
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<table><tr><td rowspan="2">Method</td><td rowspan="2">Update Param. per Task (%)</td><td colspan="2">Memory Usage (GB)</td><td rowspan="2">CoLA</td><td rowspan="2">SST-2</td><td rowspan="2">MRPC</td><td rowspan="2">QQP</td><td rowspan="2">MNLI</td><td rowspan="2">QNLI</td><td rowspan="2">RTE</td><td rowspan="2">STS-B</td><td rowspan="2">Avg.</td></tr><tr><td>Train</td><td>Inference</td></tr><tr><td>Full fine-tuning</td><td>100</td><td>17.6</td><td>0.86</td><td>62.82.5</td><td>93.90.6</td><td>91.91.0</td><td>89.90.4</td><td>86.20.4</td><td>92.50.3</td><td>74.11.0</td><td>90.30.1</td><td>85.20.4</td></tr><tr><td>Adapters</td><td>1.63</td><td>13.0</td><td>0.87</td><td>64.41.5</td><td>94.20.5</td><td>88.90.2</td><td>88.90.1</td><td>86.40.2</td><td>93.10.2</td><td>75.10.7</td><td>91.10.2</td><td>85.30.2</td></tr><tr><td>LoRA</td><td>1.71</td><td>12.6</td><td>0.86</td><td>63.30.1</td><td>94.30.1</td><td>90.10.7</td><td>89.00.1</td><td>86.30.1</td><td>93.20.1</td><td>75.53.3</td><td>90.90.0</td><td>85.30.5</td></tr><tr><td>BitFit</td><td>0.13</td><td>10.7</td><td>0.86</td><td>61.81.5</td><td>94.30.1 90.316.3</td><td>91.00.2</td><td>88.70.0</td><td>85.60</td><td>93.10.1</td><td>67.60.6</td><td>90.80.2</td><td>84.10.1</td></tr><tr><td>Prompt-tuning</td><td>0.03</td><td>22.2</td><td>0.87</td><td>02.5</td><td></td><td>74.60.0</td><td>88.50.2</td><td>82.50.9</td><td>92.50.2</td><td>59.52.9</td><td>90.10.1</td><td>72.21.6</td></tr><tr><td>Ladder Side-Tuning</td><td>1.74</td><td>5.5</td><td>0.88</td><td>58.13.2</td><td>94.10.3</td><td>90.41.0</td><td>88.80.1</td><td>85.60.1</td><td>93.30.1</td><td>71.92.1</td><td>90.70.2</td><td>84.10.5</td></tr><tr><td>Ladder Side-Tuning (T5-large)</td><td>1.23</td><td>12.2</td><td>2.88</td><td>65.31.9</td><td>95.70.1</td><td>91.61.0</td><td>89.70.0</td><td>88.60.0</td><td>94.10.2</td><td>79.90.0</td><td>92.40.1</td><td>87.10.2</td></tr><tr><td>Ladder Side-Tuning (T5-3B)</td><td>0.08</td><td>22.4</td><td>11.01</td><td>66.41.7</td><td>96.50.1</td><td>92.90.8</td><td>89.70.1</td><td>90.70.1</td><td>95.10.2</td><td>80.11.0</td><td>93.00.3</td><td>88.10.4</td></tr></table>
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Table 3: Comparison between multiple parameter-efficient training methods on VQA, GQA, $\mathrm { \ N L V R ^ { 2 } }$ , and MSCOCO. We use T5-base for all approaches. We report accuracy for VQA, GQA and NLVR while we use CIDEr to evaluate MSCOCO. Each number in the table is the average result over three seeds, and the subscripts are standard deviations.
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<table><tr><td rowspan="2">Method</td><td rowspan="2">Update Param. (%)</td><td colspan="2">Memory Usage (GB)</td><td rowspan="2">VQA</td><td rowspan="2">GQA</td><td rowspan="2">NLVR²</td><td rowspan="2">MSCOCO</td><td rowspan="2">Avg.</td></tr><tr><td>Train</td><td>Inference</td></tr><tr><td>Full fine-tuning</td><td>100</td><td>36.2</td><td>0.86</td><td>67.10.1</td><td>56.30.3</td><td>74.30.4</td><td>112.20.3</td><td>77.50.3</td></tr><tr><td>Adapters</td><td>7.98</td><td>28.4</td><td>0.93</td><td>67.10.1</td><td>56.00.4</td><td>72.70.3</td><td>111.80.1</td><td>76.90.2</td></tr><tr><td>LoRA</td><td>7.54</td><td>27.9</td><td>0.86</td><td>63.70.2</td><td>53.30.1</td><td>70.00.3</td><td>110.30.4</td><td>74.30.1</td></tr><tr><td>BitFit</td><td>0.83</td><td>22.7</td><td>0.86</td><td>55.10.2</td><td>45.50.2</td><td>51.71.1</td><td>101.20.2</td><td>63.40.1</td></tr><tr><td>Prompt-tuning</td><td>1.26</td><td>38.7</td><td>0.87</td><td>47.40.7</td><td>40.60.4</td><td>51.00.4</td><td>96.10.9</td><td>58.80.6</td></tr><tr><td>Ladder Side-Tuning</td><td>7.46</td><td>15.3</td><td>0.93</td><td>66.50.1</td><td>55.90.1</td><td>71.60.3</td><td>113.50.3</td><td>76.90.1</td></tr></table>
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# 5 Experimental Results
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In this section, we show experiments to justify our design of LST and demonstrate that LST performs the best among all approaches in the scenario with limited memory. As the result, LST is the most efficient tool to fine-tune large-scale pre-trained models for real-world applications.
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LST outperforms other methods under similar memory usage. Figure 1 and Table 1 show the results on GLUE of different approaches applying on T5-base. We drop 6 layers (3 layers each in side encoder and decoder) for LST to match the parameter usage of the Adapter and LoRA. Under the same parameter usage, LST can save $6 9 \%$ of memory cost to fully fine-tune the model, while Adapter and LoRA only save $26 \%$ of that, leading to LST having a $2 . 7 \mathbf { x }$ more memory saving. Compared to BitFit, LST achieves the same average performance but costs 5GB less GPU memory. LST also surpasses Prompt-tuning in terms of both performance and memory cost. To further take advantage of the memory efficiency of LST, we also train T5-large and T5-3B with LST. We find that with a similar budget of memory usage in Adapter and LoRA, LST with T5-large can surpass the performance of other methods by a large margin. The result on T5-3B also outperforms the result on T5-large, demonstrating the scalability of our memory-efficiency method on large language models. Furthermore, even though LST increases the model size, its additional inference memory usage is negligible as LST uses almost the same inference memory (0.88 GB) as the full fine-tuning (0.86 GB).
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Table 2: LST vs. Y-tuning.
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<table><tr><td>Method</td><td>Update Param. per Task (%)</td><td>Avg. GLUE</td></tr><tr><td>Y-tuning</td><td>7.7</td><td>76.9</td></tr><tr><td>LST</td><td>2.6</td><td>82.1</td></tr></table>
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In Table 2, we also compare LST with a concurrent work, Y-tuning [37] on GLUE tasks (except for STS-B) with BARTlarge [32] encoder as backbone. Following their experimental setup, we use a different learning rate and report the best accuracy out of three seeds for each task. Overall, LST outperforms Y-tuning by a large margin with fewer updated parameters.
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LST is competitive on VL tasks. As we have mentioned beforehand, we also extend LST on a multi-modal architecture, CLIP-T5, on multiple VL tasks, and we demonstrate the outcome in Table 3.
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Figure 5: The accuracy-memory trade-off for Adapter, LoRA, and Ladder Side-Tuning over GLUE tasks. We vary the reduction factor in Ladder Side-Tuning, hidden dimension in Adapter, rank in LoRA to get the architectures with different training costs.
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Figure 6: The accuracy-memory trade-off for Adapter, LoRA, BitFit, LST over GLUE tasks. we drop $N \in \{ 0 , 6 , 1 2 , 1 8 \}$ layers in an interleaving manner for LST while we gradually freeze the first $N \in \{ 0 , 6 , 1 2 , 1 8 \}$ layers in other methods (also remove inserted parameters in those layers).
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With similar parameter usage, LST is the only method that can fit into a single 16GB GPU. Besides the efficiency, it is as competitive as full fine-tuning and Adapter, outperforming other PETL approaches.
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LST performs the best in low-memory regime. To have a better understand of the memory advantage of LST, we adjust the hyper-parameters in our method (reduction factor $\in \{ 3 2 , 1 6 , 8 , 4 \} \dot { ) }$ , Adapter (hidden dimension $\in \{ 6 , 1 2 , 2 4 , 4 8 \} \rangle$ and LoRA (rank $\in \{ 4 , 8 , 1 6 , 3 2 \} )$ to create multiple architectures with different memory costs. Figure 5 shows the performance and memory efficiency trade-off for all methods. We find the memory saving is not obvious for Adapter and LoRA, because the gradients of the backbone model’s intermediate outputs are still computed (see Section 3.1 for details). Even though the Adapter and LoRA can get slightly better memory efficiency by reducing the hidden dimension and the rank, we find that the performance drops significantly. On the other hand, LST is quite robust across a wide range of side network sizes.
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We also consider another way to compare LoRA, Adapter and BitFit to LST in different memory budgets. While we drop $N \in \{ 0 , 6 , 1 2 , 1 8 \}$ layers in an interleaving manner to improve memory efficiency, we freeze the first $N$ layers and remove the corresponding inserted modules in other approaches. With this, other methods can achieve better memory efficiency because gradients do not propagate to those earlier frozen layers. We discuss the layer dropping and layer freezing with details in the following. In LST, we drop $\begin{array} { l } { { \frac { N } { 2 } } } \end{array}$ layers in both side encoder and side decoder. However, in other PETL approaches, we start from freezing layers in the encoder and then turn to freeze layers in the decoder (e.g. freezing 18 layers means freezing all encoder layers and first 6 decoder layers). We display the comparison in Figure 6, showing that LST has a better performance and memory trade-off and outperforms other methods in the low-memory regime. We also find that layer dropping generally reduces the training cost without hurting performance.
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The ablation of weight initialization on the side network. We compare the different initialization strategies for the side network and demonstrate the results in Figure 7. “Random” denotes we randomly initialize the network while we use network pruning to select initial weights for the side network based on two importance measures, “Weight Magnitude” and “Fisher Information.” In general, the initialization from the pruned network helps no matter the size of the side network, showing the effectiveness of our network pruning strategy.
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Comparison of LST to network compression methods and Side-tuning. In Section 3.2, we mention that shortcut connections are added to every layer of the side network. We justify this design by comparing LST to two types of approaches: (1) network compression, which discards all shortcut connections and the entire backbone model; (2) Side-tuning, which only adds one shortcut connection to merge representations right before the output layer. Note that we do not drop any layer in the side network but only remove the shortcuts in this experiment. We also compare both distillation-based and pruning-based initialization methods as we describe in Section 2.2. We set the reduction factor to 8 for all approaches. Figure 8 shows the comparison and LST outperforms the other two types of methods significantly. We conclude that PETL methods are stronger than network compression as they use the information from the backbone model. This also suggests that network compression approaches need to train more parameters to achieve the same level of performance as PETL methods. We also demonstrate the usefulness of intermediate shortcuts since LST surpasses Side-tuning by a large amount. Furthermore, we find that using distillation-based initialization or network pruning-based initialization provides similar accuracy in all setups. Note that our network pruning-based initialization method is more efficient since it does not involve training.
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Figure 7: The ablation study on different initialization strategies. Y-axis denotes the average score over GLUE tasks.
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Figure 8: The comparison between network compression, Side-Tuning, and LST on GLUE tasks.
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# 6 Conclusion
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We propose Ladder Side-Tuning (LST), a parameter- and memory-efficient method to adapt a large backbone network. LST does not require backpropagation through the backbone network, which allows for significantly lower memory requirement during training than recently proposed parameterefficient training techniques. We demonstrate that LST allows users to adapt a larger and more powerful backbone network to target tasks with a limited memory requirement, which cannot be achieved with recent parameter-efficient techniques. We also show that LST achieves a more efficient accuracy-memory trade-off than recent baselines, the impact of weight initialization of side networks, and the usefulness of intermediate shortcut connections. Finally, we show that the LST can be also extended beyond NLP tasks, with strong results on VL tasks. We hope that LST helps users with limited computational resources tune larger models in diverse domains.
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# Acknowledgments
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We thank the reviewers, Muqeeth Mohammed, Derek Tam, Prateek Yadav, and Gedas Bertasius for their helpful discussions. This work was supported by ARO Award W911NF2110220, ONR Grant N000141812871, and NSF-AI Engage Institute DRL-211263. The views, opinions, and/or findings contained in this article are those of the authors and not of the funding agency.
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# References
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[1] Hangbo Bao, Li Dong, and Furu Wei. Beit: BERT pre-training of image transformers. CoRR, abs/2106.08254, 2021. URL https://arxiv.org/abs/2106.08254.
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[2] Luisa Bentivogli, Ido Dagan, Hoa Trang Dang, Danilo Giampiccolo, and Bernardo Magnini. The fifth pascal recognizing textual entailment challenge. In In Proc Text Analysis Conference (TAC’09, 2009.
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[3] Han Cai, Chuang Gan, Ligeng Zhu, and Song Han. Tinytl: Reduce memory, not parameters for efficient on-device learning. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and
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# Checklist
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1. For all authors...
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| 257 |
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(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]
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| 258 |
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(b) Did you describe the limitations of your work? [No]
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| 259 |
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(c) Did you discuss any potential negative societal impacts of your work? [N/A]
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| 260 |
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(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
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| 261 |
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| 262 |
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2. If you are including theoretical results...
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| 263 |
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| 264 |
+
(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
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| 266 |
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3. If you ran experiments...
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| 267 |
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| 268 |
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(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]
|
| 269 |
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(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]
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| 270 |
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(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]
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| 271 |
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(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
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| 272 |
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4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
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(a) If your work uses existing assets, did you cite the creators? [Yes]
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(b) Did you mention the license of the assets? [No]
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| 277 |
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(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]
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| 278 |
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(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]
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| 279 |
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(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
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| 280 |
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5. If you used crowdsourcing or conducted research with human subjects...
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(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
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| 284 |
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(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
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| 286 |
+
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
|
parse/dev/isPnnaTZaP5/isPnnaTZaP5_content_list.json
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|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "LST: Ladder Side-Tuning for Parameter and Memory Efficient Transfer Learning ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
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"bbox": [
|
| 7 |
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| 8 |
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| 9 |
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| 10 |
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| 11 |
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],
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| 12 |
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"page_idx": 0
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| 13 |
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},
|
| 14 |
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{
|
| 15 |
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"type": "text",
|
| 16 |
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"text": "Yi-Lin Sung Jaemin Cho Mohit Bansal UNC Chapel Hill {ylsung, jmincho, mbansal}@cs.unc.edu ",
|
| 17 |
+
"bbox": [
|
| 18 |
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| 20 |
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| 23 |
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| 24 |
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},
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| 25 |
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{
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| 26 |
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"type": "text",
|
| 27 |
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"text": "Abstract ",
|
| 28 |
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"text_level": 1,
|
| 29 |
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"bbox": [
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| 30 |
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| 31 |
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| 32 |
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| 33 |
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| 34 |
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| 35 |
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"page_idx": 0
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| 36 |
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},
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| 37 |
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{
|
| 38 |
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"type": "text",
|
| 39 |
+
"text": "Fine-tuning large pre-trained models on downstream tasks has been adopted in a variety of domains recently. However, it is costly to update the entire parameter set of large pre-trained models. Although recently proposed parameter-efficient transfer learning (PETL) techniques allow updating a small subset of parameters (e.g. only using $2 \\%$ of parameters) inside a pre-trained backbone network for a new task, they only reduce the training memory requirement by up to $3 0 \\%$ . This is because the gradient computation for the trainable parameters still requires backpropagation through the large pre-trained backbone model. To address this, we propose Ladder Side-Tuning (LST), a new PETL technique that can reduce training memory requirements by more substantial amounts. Unlike existing parameter-efficient methods that insert additional parameters inside backbone networks, we train a ladder side network, a small and separate network that takes intermediate activations as input via shortcut connections (called ladders) from backbone networks and makes predictions. LST has significantly lower memory requirements than previous methods, because it does not require backpropagation through the backbone network, but instead only through the side network and ladder connections. We evaluate our method with various models (T5 and CLIP-T5) on both natural language processing (GLUE) and vision-and-language (VQA, GQA, $\\mathrm { \\Delta N L V R ^ { 2 } }$ , MSCOCO) tasks. LST saves $6 9 \\%$ of the memory costs to fine-tune the whole network, while other methods only save $2 6 \\%$ of that in similar parameter usages (hence, $2 . 7 \\mathbf { x }$ more memory savings). Moreover, LST achieves higher accuracy than Adapter and LoRA in a low-memory regime. To further show the advantage of this better memory efficiency, we also apply LST to larger T5 models (T5-large, T5-3B), attaining better GLUE performance than full fine-tuning and other PETL methods. The trend also holds in the experiments on vision-and-language tasks, where LST achieves similar accuracy to other PETL methods when training a similar number of parameters while also having $2 . 7 \\mathbf { x }$ more memory savings. ",
|
| 40 |
+
"bbox": [
|
| 41 |
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| 42 |
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| 43 |
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| 44 |
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| 45 |
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],
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| 46 |
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"page_idx": 0
|
| 47 |
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},
|
| 48 |
+
{
|
| 49 |
+
"type": "text",
|
| 50 |
+
"text": "1 Introduction ",
|
| 51 |
+
"text_level": 1,
|
| 52 |
+
"bbox": [
|
| 53 |
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| 54 |
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| 55 |
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| 56 |
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| 57 |
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| 58 |
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"page_idx": 0
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| 59 |
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| 60 |
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|
| 61 |
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"type": "text",
|
| 62 |
+
"text": "Recently, large-scale pre-training and fine-tuning of transformers [54] have been successful in various domains [8, 36, 44, 38, 53, 7, 20, 1, 10]. As the model size grows rapidly, fine-tuning the entire parameter set of the large pre-trained model has become very costly. Parameter-efficient transfer learning (PETL) [31, 23, 39, 40, 34, 51, 58, 17, 24, 18, 42, 60, 14, 52, 59] is a recent research direction for online or multi-task learning. The goal is to build a system that performs well on all tasks without training an entire new model for every new task. Concretely, PETL methods select a small subset of pre-trained parameters and/or insert a few parameters to a pre-trained network and update those parameters for new tasks, while freezing most of the original parameters. In the natural language processing (NLP), computer vision (CV), and vision-and-language (VL) domains, two types of parameters have been commonly updated for parameter-efficient transfer learning: (a) Adapters [23, 40, 39]: small modules inserted into transformer blocks; (b) Prompt [34, 31]: small parameters concatenated with input embeddings (see Figure 2). ",
|
| 63 |
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"bbox": [
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| 64 |
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| 66 |
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| 67 |
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| 68 |
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| 69 |
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"page_idx": 0
|
| 70 |
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},
|
| 71 |
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{
|
| 72 |
+
"type": "image",
|
| 73 |
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"img_path": "images/18eb5e3e54f9858afcb3e25583afaf0d7227ef22b8525f7c9c518e04266e9bbd.jpg",
|
| 74 |
+
"image_caption": [
|
| 75 |
+
"Figure 1: Comparison between full finetuning, Adapter, LoRA, BitFit, and Ladder Side-Tuning over GLUE tasks. The y-axis is the average accuracy of 8 GLUE tasks, while the $\\mathbf { X }$ -axis is the GPU memory usage during training. Unless specially stated, we use the T5-base in the figure. "
|
| 76 |
+
],
|
| 77 |
+
"image_footnote": [],
|
| 78 |
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"bbox": [
|
| 79 |
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| 80 |
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| 81 |
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| 82 |
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| 83 |
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| 84 |
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"page_idx": 1
|
| 85 |
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},
|
| 86 |
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{
|
| 87 |
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"type": "image",
|
| 88 |
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"img_path": "images/4c0f767f75c9392dd19ce33799ddabc168cb61dc945bd8614c83cfc8f183c5d3.jpg",
|
| 89 |
+
"image_caption": [
|
| 90 |
+
"Figure 2: Comparison between transfer learning with (a) Adapters, (b) Prompt Tuning, and our (b) Ladder SideTuning (LST). LST reduces memory usage by removing the need of backpropgation through backbone networks. "
|
| 91 |
+
],
|
| 92 |
+
"image_footnote": [],
|
| 93 |
+
"bbox": [
|
| 94 |
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| 95 |
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| 96 |
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| 97 |
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| 98 |
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|
| 99 |
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"page_idx": 1
|
| 100 |
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},
|
| 101 |
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{
|
| 102 |
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"type": "text",
|
| 103 |
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"text": "",
|
| 104 |
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"bbox": [
|
| 105 |
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| 106 |
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| 107 |
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| 108 |
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| 109 |
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|
| 110 |
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"page_idx": 1
|
| 111 |
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},
|
| 112 |
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{
|
| 113 |
+
"type": "text",
|
| 114 |
+
"text": "However, while parameter-efficient techniques reduce the number of parameters to update, they do not reduce the memory requirement during training by much (up to $3 0 \\%$ ). In other words, if a large pre-trained language model does not fit on a GPU, these techniques usually do not help to fit the model on the GPU. In other words, these techniques usually do not help to fit a large pre-trained language model on a GPU if the model cannot be trained on the GPU with standard fine-tuning. Since the updated parameters are inside the backbone language models, to calculate gradients for these parameters for backpropagation, we still need to run the backward pass through the large pre-trained language models. This prevents PETL methods from being applied to many real-world applications with limited computational resources. ",
|
| 115 |
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| 116 |
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| 121 |
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| 122 |
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|
| 123 |
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|
| 124 |
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"type": "text",
|
| 125 |
+
"text": "To address this issue, we propose Ladder Side-Tuning (LST), a memory-efficient PETL method. LST separates trainable parameters from the backbone model to construct a side network, which is responsible for adapting the entire model to new tasks. Concretely, we train a ladder side network, a lightweight network that takes intermediate activations via shortcut connections (ladders) from the backbone networks as input and makes predictions. As shown in Figure 2, unlike previous (a) adapters and (b) prompt tuning methods, (c) our LST does not add trainable parameters inside the pre-trained model, and this completely eliminates the need for expensive backpropagation of a large backbone network and saves substantial memory during transfer learning. Instead of initializing the side network’s weights randomly, we utilize a network pruning technique to retrieve a smaller pruned network and use it as the side network. In addition to our standard design of the side network, we also boost the efficiency of LST by dropping layers of the side network. We empirically demonstrate that the layer dropping can significantly improve both memory and parameter efficiency without sacrificing performance. Furthermore, during inference, even though we have to propagate forward through two distinct networks, LST does not necessarily use more inference time because the same level of the backbone network and the side network can be computed in parallel. ",
|
| 126 |
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"bbox": [
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| 127 |
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| 130 |
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| 131 |
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| 132 |
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"page_idx": 1
|
| 133 |
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},
|
| 134 |
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{
|
| 135 |
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"type": "text",
|
| 136 |
+
"text": "We conduct comprehensive studies on LST using diverse NLP and VL tasks, namely, GLUE [55], VQA [16], GQA [25], $\\mathrm { \\Delta N L V R ^ { 2 } }$ [50] and MSCOCO [6]. Overall, in GLUE experiments, LST saves $6 9 \\%$ of the GPU memory that is needed for fine-tuning the entire backbone model, saving $2 . 7 \\mathbf { x }$ memory compared against Adapter and LoRA. Also, LST achieves higher accuracy than other PETL methods in a low-memory regime. To take advantage of this better memory efficiency, we also apply LST to larger language models (T5-large and T5-3B) and find that it achieves higher accuracy than other PETL techniques when GPU memory utilization is similar. The findings still hold in the Vision-Language (VL) experiments; LST is not only a method that can fit in a 16GB GPU with $7 . 5 \\%$ trainable parameters, but it also has similar or better accuracy than other PETL methods (and again with $2 . 7 \\mathbf { x }$ memory savings). To justify our design of LST, we conduct ablation studies on initialization strategies and alternatives to add shortcut connections. The results reveal that these components considerably help performance with minor computation overhead. ",
|
| 137 |
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| 138 |
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| 143 |
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| 144 |
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|
| 145 |
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|
| 146 |
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"type": "text",
|
| 147 |
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"text": "",
|
| 148 |
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| 149 |
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| 154 |
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|
| 155 |
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},
|
| 156 |
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{
|
| 157 |
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"type": "text",
|
| 158 |
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"text": "2 Related Work ",
|
| 159 |
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"text_level": 1,
|
| 160 |
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| 161 |
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|
| 167 |
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},
|
| 168 |
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{
|
| 169 |
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"type": "text",
|
| 170 |
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"text": "2.1 Parameter-efficient Transfer Learning (PETL) ",
|
| 171 |
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"text_level": 1,
|
| 172 |
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"page_idx": 2
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| 179 |
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},
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| 180 |
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{
|
| 181 |
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"type": "text",
|
| 182 |
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"text": "PETL for NLP. In the past few years, large pre-trained language models have made huge success in NLP. Parameter-efficient transfer learning (PETL) is a research direction that reduces computational cost of adapting large pre-trained models to new tasks, by avoiding updates of the entire parameters. A popular line of work on PETL is to add a few trainable parameters and only tune them. For example, Adapters [48, 23] are small bottleneck modules that are inserted into transformer layers, and experiments have shown that training adapters with layer normalization layers is sufficient to achieve full fine-tuning performance. In a similar trend, LoRA [24] injects trainable rank decomposition matrices into a frozen pre-trained model. Instead of inserting new parameters into pre-trained models, prompt-based [31, 34] methods add trainable parameters to the input and keep the entire pre-trained model unchanged during training. The inserted prompts learn to make use of the knowledge of the pre-trained model to solve new tasks. Although the concept of adapter-based and prompt-based approaches is different, He et al. [18] unify the two lines of approaches (including LoRA) into adapter-based methods. In addition to approaches that introduce new parameters, there are also various methods [51, 58, 17] that select a sparse subset of parameters from the pre-trained model to update, without adding any new parameters. One of such representative methods is BitFit [58], which updates every bias term in the model. ",
|
| 183 |
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| 189 |
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"page_idx": 2
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| 190 |
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|
| 191 |
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|
| 192 |
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"type": "text",
|
| 193 |
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"text": "PETL for CV/VL. While most of the progress in PETL is made in NLP domain, researchers have also applied this technique to the CV [47, 46, 27, 59, 62, 28, 60, 21] and VL [52, 61, 62, 28, 60] domains. VL-Adapter [52] benchmarks adapter-based and prompt-based methods on multiple imagetext and video-text tasks, and shows adapters enable us to efficiently learn fusion information of vision and language. On the CV side, benefitting from the parameter efficiency of adapters and prompt-tuning, some works [62, 28, 60] apply these approaches to CLIP [43] to achieve strong few-shot performance in image classification tasks. Side-Tuning [59] uses an additive side network, which sums its representation with the backbone network in the last layer, to solve various tasks with ResNet [19] and BERT [8]. Although LST takes inspiration and has similarities to Side-Tuning, we argue that there are major differences in motivations, architecture designs, and applied tasks between the two methods. LST aims to reduce the memory requirement of current PETL methods, whereas Side-Tuning does not focus on memory reduction (sometimes their side network is even as big as the backbone network), but instead their motivation is to ease the forgetfulness in incremental learning. Our ladder side network is more robust than their design because the shortcuts fuse the intermediate information from the backbone network, and we also use layer dropping and network pruning techniques to make LST more efficient and stronger. Lastly, we further extend LST in VL architecture and demonstrate its usefulness on multiple VL tasks. ",
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"text": "Current PETL approaches explore how to achieve competitive results using as few parameters as possible. However, parameter efficiency does not necessarily mean memory efficiency. In this work, we propose LST that has these two benefits simultaneously. Concurrently, Liu et al. [37] also propose Y-tuning to address a similar issue; it exhausts all possible labels and feeds them into a model to select the best answer from the input. However, it is intractable oftentimes to list all answers in some tasks, for example, regression and open-ended generation tasks. On the other hand, LST is more flexible in applying to different architectures and tasks. We show that LST can outperform Y-tuning with fewer parameter updates in Table 2. ",
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"text": "2.2 Memory-Efficient Training ",
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"text": "Memory-efficient training aims to reduce the memory cost when training neural networks. Some approaches achieve this goal by cutting down the storage of intermediate activations, which dominate the training cost, to release a large amount of memory. The design of reversible neural networks [15, 29, 41] allows the model not to save intermediate activations because each layer’s activations can be reconstructed from the next layer’s. Gradient checkpointing [5] proposes to trade computation for memory by dropping some intermediate activations and recovering them from an extra forward pass. LST uses a different approach to cut the storage of activations; it keeps the backbone model frozen and constructs a side network for cheaper training. Since the backbone model is not updated, LST is not only memory-efficient but also parameter-efficient. Furthermore, memory saving from reversible neural networks and checkpointing is agnostic to memory saving from LST. Researchers can combine those methods with LST if they pursue a higher level of memory efficiency. ",
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"text": "",
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"text": "Another line of memory-efficient methods is network compression, which compresses a backbone network to a smaller one, and the generated network is cheaper for both training and inference. Two popular approaches for compression are network pruning and distillation. Network distillation [22, 30] constructs a student network and force it to have same output distribution of the teacher network over a chosen dataset. Network pruning [12, 13] makes models lighter by learning importance scores of parameters and trimming unimportant parameters or neurons. While PETL still uses those untrained parameters in the forward pass, network compression entirely discards them or sets them to zero. As a result, network compression can generate models for faster inference speed while PETL can achieve better performance by updating fewer parameters. ",
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"text": "In this paper, we explore using network pruning or network distillation [22] (used in Side-tuning) to extract a sub-network that contains critical information of the backbone model and use it for initializing our side network. For distillation, we did not follow the original Side-tuning to apply distillation with large-scale pre-training datasets (e.g., C4 [44]) because it makes distillation hard to conduct with limited resources and ultimately violates our goal of “efficient training.” Also, using an extra pre-training dataset during fine-tuning is unfair to other approaches. We use the standard distillation procedure with the T5 [44] pre-training objective to train the side network. That is, the student (side) network learns to predict the masked spans and match the output distribution of the teacher (backbone) network simultaneously. We show the comparison between distillation-based and pruning-based initializations in Figure 8. ",
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"text": "We primarily use the network pruning method proposed by Li et al. [33] to initialize the side network because of its efficiency, and we describe the approach in detail in Section 3.3. The standard procedure of network pruning is (1) learn [12, 13] or heuristically define [33] an \"importance measure\" to identify the importance of parameters, (2) prune $p \\%$ of parameters with lower importance scores, (3) repeat the first and second steps until reaching the target sparsity. The rewinding procedure enables pruning techniques to find a more sparse sub-network. In this paper, to keep the whole pruning process efficient, we either use weights magnitude [33] or Fisher Information [51, 35] as importance measures, and reach the target sparsity in one shot. As a PETL method, LST makes use of intermediate information from the backbone model as the inputs, and we empirically demonstrate those additional inputs significantly improve performance in Figure 8. ",
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"text": "3 Ladder Side-Tuning (LST) ",
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"text": "We introduce Ladder Side-Tuning (LST), a new PETL technique that can also reduce training memory requirements by substantial amounts than previous methods. In Section 3.1, we analyze the computational cost for fine-tuning with trainable modules in backbone models. Then we explain the architectural details (Section 3.2), structural weight initialization based on network pruning (Section 3.3), and dropping side network layers for more efficiency (Section 3.4). ",
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"text": "3.1 Dependency on Backpropagation through Large Backbone Model ",
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"text": "We consider a $N$ multilayer perceptron (MLP): $f _ { N } ( f _ { N - 1 } ( \\dots f _ { 2 } ( f _ { 1 } ( x ) ) . . . ) )$ , where the $i ^ { t h }$ layer $f _ { i } ( x ) = \\sigma _ { i } ( W _ { i } x + b _ { i } )$ consists of weight $W _ { i }$ , bias $b _ { i }$ , and nonlinear function $\\sigma _ { i }$ . We denote the output of $i ^ { t h }$ layer as $a _ { i + 1 }$ and the pre-activation as $z _ { i + 1 }$ , where $a _ { i + 1 } = \\sigma _ { i } ( z _ { i + 1 } ) = \\sigma _ { i } ( W _ { i } a _ { i } + b _ { i } )$ . In backpropagation with loss $L$ , the gradient with respect to $W _ { i }$ and $b _ { i }$ : ",
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"text": "$$\n\\frac { \\partial L } { d W _ { i } } = \\frac { \\partial L } { \\partial a _ { i + 1 } } \\frac { \\partial a _ { i + 1 } } { \\partial z _ { i + 1 } } \\frac { \\partial z _ { i + 1 } } { \\partial W _ { i } } = \\frac { \\partial L } { \\partial a _ { i + 1 } } \\sigma _ { i } ^ { \\prime } a _ { i } , \\qquad \\frac { \\partial L } { d b _ { i } } = \\frac { \\partial L } { \\partial a _ { i + 1 } } \\sigma _ { i } ^ { \\prime }\n$$",
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"text": "where σ′i is the derivative of σi. ∂L∂ai+1 , the gradient with respect to $a _ { i }$ , can be calculated with the gradients with respect to $a _ { i + 2 }$ , using the chain rule: ",
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"img_path": "images/351fd4e526725b6033d582d89d1ebf77f42aac951de886ad0788831da26f01fb.jpg",
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"image_caption": [
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"Figure 3: Illustration of Ladder Side-Tuning (LST) with transformers described in Section 3.2. (a) shows a high-level overview of LST, and (b) shows LST with an encoder-decoder architecture. "
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"text": "$$\n\\frac { \\partial L } { \\partial a _ { i + 1 } } = \\frac { \\partial L } { \\partial a _ { i + 2 } } \\frac { \\partial a _ { i + 2 } } { \\partial z _ { i + 2 } } \\frac { \\partial z _ { i + 2 } } { \\partial a _ { i + 1 } } = \\frac { \\partial L } { \\partial a _ { i + 2 } } \\sigma _ { i + 1 } ^ { \\prime } W _ { i + 1 }\n$$",
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"text": "As shown in Equations (1) and (2), during backpropagation, there are two terms dominating the memory footprint: 1) $\\{ a \\}$ corresponding to updated parameters $\\{ W \\}$ and 2) $\\{ \\sigma ^ { \\prime } \\}$ that must be cached for the chain rule. Note that we use $\\{ \\cdot \\}$ to denote a set of activations, parameters, or gradients. Existing PETL methods, such as Adapters [23], LoRA [24], Prompt-tuning [31], and BitFit [58, 3] could reduce the memory footprint by making $| a |$ smaller, as they have fewer $\\{ W \\}$ to update, but do not reduce $| \\sigma ^ { \\prime } |$ , where $| \\cdot |$ means the size of set $\\{ \\cdot \\}$ . Since most activation functions do not change dimensions (i.e., $| a | = | \\sigma ^ { \\prime } | )$ , the memory footprint for backpropagation $| a | + | \\sigma ^ { \\prime } |$ can be reduced by up to $50 \\%$ by the PETL methods when they reduce the entire memory footprint for $| a |$ . By making the updated parameter do not require backpropagation through the backbone network, our LST can achieve better memory efficiency beyond $50 \\%$ , and we explain it below. ",
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"text": "3.2 Ladder Side Network for Transformers ",
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"text": "Unlike existing transfer learning methods that insert additional parameters inside a transformer network, we propose training a ladder side network, a small and separate network that takes intermediate activations from the backbone transformer as input and makes predictions. As illustrated in Figure 3 (a), since the ladder side network parameters $\\phi$ are not used during the forward pass of the backbone transformer with parameters $\\theta$ , the update of the ladder side network does not require expensive backpropagation of the large backbone transformer. Note that our LST method is not limited to a specific architecture. We provide a simplified overview of LST with an encoder architecture in Figure 3 (a) and an illustration of LST with an encoder-decoder architecture in Figure 3 (b). ",
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"text": "Lightweight architecture. Our side network $g$ is a lightweight version of the backbone transformer $f$ , where all weights and hidden state dimensions of in $g$ are $\\textstyle { \\frac { 1 } { r } }$ times of the original weights and hidden states of $f$ , where $r$ is a reduction factor (e.g. $r = 2$ , 4, 8, 16). For example, if the backbone $f$ has a 768-dimensional hidden state, then the side network $g$ with $r = 1 6$ has a hidden state of 48 dimensions $\\left( = 7 6 8 / 1 6 \\right)$ . The side network $g$ reuses frozen word embeddings (‘Emb’ in Figure 3 (a)) and the language model head (‘LM head’ in Figure 3 (a)) of the backbone $f$ . Following the analysis in Section 3.1, we also examine the memory cost of LST. Recall that original memory footprint for backpropagation is $| a | + | \\sigma ^ { \\prime } |$ . Because we do not have to run a backward pass through the backbone network, we can only consider the gradients for the side network, whose memory footprint is $\\frac { | a | + | \\sigma ^ { \\prime } | } { r }$ . Therefore, LST has a better memory efficiency than other PETL methods (saving up to $50 \\%$ ) as long as $r$ is greater than 2 (we find 8 works well in most experiments). ",
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"image_caption": [
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"Figure 4: Illustration of (a) Structural Weight Initialization (Section 3.3) and (b) Layer Dropping (Section 3.4). In our experiments, we find that initialization of side network parameters from backbone network parameters improves performance, and dropping some shortcut connections improves efficiency without hurting performance. "
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"text": "Gated ladder connections. Although Zhang et al. [59] found that late fusion to combine the representations of the backbone and the side network works well with convolutional networks for CV tasks, in our experiments, we find that late fusion hurts the performance of the transformer architecture in NLP tasks (see Figure 8 in Section 5 for details). To address this, we use the shortcut connection (called ladder, due to the overall shape created from the multiple shortcut connections) from intermediate activations from the backbone $f$ to the side network $g$ and find it helpful. We learn linear projections to downsample $\\scriptstyle ( \\times { \\frac { 1 } { r } } )$ the intermediate activations (including word embeddings) of $f$ to low-dimensional attention blocks in $g$ . Then, we learn a linear projection to upsample $( \\times r )$ the side network output to the dimension of the original language model head. The linear projections are illustrated as green trapezoids in Figure 3 (a). The $i ^ { \\tilde { t } h }$ transformer layer of the side network $g$ combines the activation of the backbone $h _ { i } ^ { f }$ and the activation of the previous layer of the side network $h _ { i - 1 } ^ { g }$ with learned gating: $\\mu _ { i } * h _ { i } ^ { f } + ( 1 - \\mu _ { i } ) * h _ { i - 1 } ^ { g }$ , where $\\begin{array} { r } { \\mu _ { i } = \\tt s i g m o i d ( \\frac { \\alpha _ { i } } { T } ) } \\end{array}$ is a gate parameterized with a learnable zero-initialized scalar $\\alpha _ { i }$ and temperature $T$ $( = 0 . 1 )$ ). We have also tried to use Adapter blocks to build the side network and replace the gating mechanism with cross-attentions, but we find the current design works the best (see ??). ",
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"text": "3.3 Structural Weight Initialization for Ladder Side Network ",
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"text": "We find it helpful to initialize the weights of the side network $\\phi$ from the weight of the backbone network $\\theta$ based on structural pruning [33], as shown in Figure 4 (a). Concretely, given a weight matrix $W \\in \\mathbb { R } ^ { d _ { o u t } \\times d _ { i n } }$ of the backbone network that maps the $d _ { i n }$ -dim vectors to the $d _ { o u t }$ -dim space, and the importance matrix of the weight $I \\in \\mathbb { R } ^ { d _ { o u t } \\times d _ { i n } }$ , we first calculate the importance score of each row $\\begin{array} { r } { s _ { i } = \\sum _ { j } | I _ { i , j } | } \\end{array}$ , denoting the importance of each weight vector. Note that the importance matrix $I$ used in this work are either weight magnitude [33] $\\mathit { \\Delta } M = W$ ) or empirical Fisher Information [51] $\\begin{array} { r } { ( I = F _ { W } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } ( \\nabla _ { W } \\log p ( y _ { i } | x _ { i } ) ) ^ { 2 } ; ( x _ { i } , y _ { i } ) , . . . , ( x _ { N } , y _ { N } ) } \\end{array}$ are samples from data). Then, we choose the rows of remaining rows to obtain a new weig $W$ which matrix $\\frac { d _ { o u t } } { r }$ importance scores and prune the. The columns of the weights and $W ^ { P } \\in \\mathbb { R } ^ { \\frac { d _ { o u t } } { r } \\times d _ { i n } }$ the importance matrix in the next layer corresponding to the pruned feature map are also pruned. By iterating this process, we obtain the set of weight matrices whose rows and columns are pruned $\\frac { \\texttt { i } } { r }$ times from the backbone network and use them to initialize the side network. In our experiments shown in Figure 7, we find that using Fisher information as an importance score metric generally performs well, and therefore we use it in our structural weight initialization. ",
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"type": "text",
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"text": "3.4 Layer Dropping in the Ladder Side Network ",
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"text_level": 1,
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"text": "We explore to increase efficiency of LST even further by making side network more compact, by dropping its intermediate transformer layers, as illustrated in Figure 4 (b). Similar to LayerDrop [11], we drop layers in the side network, and this can linearly reduce the memory and parameter requirements of LST. For instance, a side network with $N$ layers will only have $2 ^ { n { \\dot { d } } }$ , $4 ^ { t h }$ , $\\mathbf { \\bar { \\boldsymbol { 6 } } } ^ { t h }$ . . . layers left, after we drop half of the layers. Refer to Section 4 for more details on applying layer dropping on an encoder-decoder architecture. In Figure 6, we show that layer dropping can greatly boost the model’s efficiency without sacrificing performance. ",
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"type": "text",
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"text": "4 Experiment Setup ",
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"text": "Datasets. We evaluate LST on NLP and VL tasks. For NLP tasks, we use the GLUE [55] benchmark, which consists of seven classification and one regression task. The benchmark evaluate models on multiple diverse tasks over linguistic acceptability (CoLA [56]), sentiment analysis (SST2 [49]), similarity and paraphrase (MRPC [9], QQP [26], STS-B [4]) and natural language inference (MNLI [57], QNLI [45], RTE [2]). For VL tasks, we experiment with visual question answering (VQA [16], GQA [25]), visual reasoning $\\mathrm { ( N L V R ^ { 2 } }$ [50]) and image captioning (MSCOCO [6]) tasks. ",
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"text": "Baselines. We compare LST against the full fine-tuning and several popular PETL approaches on both NLP and VL setups. In Full fine-tuning, all the parameters are updated for a downstream task. Full fine-tuning is not parameter-efficient nor memory-efficient, but it serves as the upper bound of the fine-tuning performance. To compare to other PETL methods, we reproduce Adapters, where we inject small trainable modules after every attention and feed-forward layer, and we solely train those modules and layer normalization layers while keeping the rest of the model frozen. We also reproduce LoRA, which inserts trainable low-rank matrices into the model to parameterize the weights’ changes. In BitFit, we only update the bias terms over the course of training. Lastly, we compare our method to Prompt-tuning, where trainable prompt vectors are prepended to the input. We initialize the prompt vectors with the embedding of the pre-trained model’s vocabularies. ",
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"text": "Training and Evaluation Setup. For NLP tasks, we use T5 [44], a pre-trained encoder-decoder language model as our backbone. We use T5-base in most experiments, except that we scale up LST on T5-large and T5-3B to demonstrate its memory efficiency. The training and evaluation process follows the setup used by Mahabadi et al. [40]. Since there is no local test set, we split 1k samples from the training set as the new validation set and use the original validation set as the test set. For datasets whose samples are less than 10k (RTE, MRPC, STS-B, CoLA), we split the validation set into two equal-sized subsets and treat them as a new validation and test set. For MNLI, we use the mismatched set as the validation set and matched set as the test set. We train every approach with 10 epochs on large datasets and 20 epochs on small ones (RTE, MRPC, STS-B, CoLA) for complete convergence. We search for learning rates over $\\{ 3 \\times 1 0 ^ { - 4 } , 1 \\times 1 0 ^ { - 3 } , 3 \\times 1 0 ^ { - 3 } \\}$ for LST and LoRA[24], and we use the optimal learning rates that are used by Mahabadi et al. [40] for other methods. The reduction factor used in LST is set to 8 if not additionally specified. T5-base has 12 layers each in encoder and decoder, while T5-large and T5-3B have 24 layers each. In our experiments, we do not drop layers in T5-base unless we specially mention it. For T5-large and T5-3B, we drop 24 layers (12 layers each in encoder and decoder) and 46 layers (23 each) of the side network to make the memory usage close to our baselines. The experiments on T5 take around 12 hours to train with one A6000 GPU (48GB). ",
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"type": "text",
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"text": "For VL tasks, we experiment with CLIP-T5 [52], which is a VL architecture combining CLIP [43] and T5 [44]. We always freeze the CLIP and only train the T5 for new tasks. The CLIP visual representation is concatenated with the text embedding, and the combined input is fed to T5 to make predictions. A visual projection layer is added between CLIP and T5 to let the visual representation have the same dimension as the text embedding. To avoid updating the visual projection layer by the gradients from the backbone model, we do not feed combined inputs to the backbone model, but only text inputs. The combined inputs are fed to the side network, so we can achieve efficient training by only computing the gradients from the side network. Because the backbone network only uses texts as the input, the information from the backbone network via shortcut connections is only summed to the text part of the side network’s combined inputs. We follow the multi-tasking setting for training and evaluation used in VL-Adapter [52]. We report the performance on Karpathy test/test-dev/test-P/Karpathy test split fo $\\mathrm { r \\nabla { V O A } / \\bar { G O A } / N L V R ^ { 2 } / M \\bar { S } C O C O }$ , and train models for 20 epochs. We search learning rates over $\\{ 3 \\times 1 0 ^ { - 4 } , 1 \\times 1 0 ^ { - 3 } , 3 \\times 1 0 ^ { - 3 } \\}$ for PETL methods, and use $1 \\times 1 0 ^ { - 4 }$ used by Sung et al. [52] for full fine-tuning. We set the reduction factor for the side network to 4. We train CLIP-T5 for 16 hours on one A6000 GPU. In ??, we comprehensively list hyper-parameters for NLP and VL experiments in ?? and ??, respectively. ",
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"type": "table",
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"img_path": "images/19ad73546fee5154d75d1f45a965156cfb0bbca353efae765248dd330bf7d425.jpg",
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"table_caption": [
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"Table 1: Comparison between multiple parameter-efficient training methods on GLUE benchmark. We use T5-base if we don’t additionally specify. We report accuracy for SST-2, MNLI, QNLI and RTE. For CoLA and STS-B, we use Matthew’s Correlation and Pearson-Spearman Correlation as the metrics, respectively. For MRPC and QQP, we report the average of F1 score and accuracy. Each number in the table is the average result over three seeds, and the subscripts are standard deviations. For the results with †, we report the best performance out of three seeds due to the instability of the method. We report the maximum memory usage training and evaluating on RTE for each method. "
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"table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Update Param. per Task (%)</td><td colspan=\"2\">Memory Usage (GB)</td><td rowspan=\"2\">CoLA</td><td rowspan=\"2\">SST-2</td><td rowspan=\"2\">MRPC</td><td rowspan=\"2\">QQP</td><td rowspan=\"2\">MNLI</td><td rowspan=\"2\">QNLI</td><td rowspan=\"2\">RTE</td><td rowspan=\"2\">STS-B</td><td rowspan=\"2\">Avg.</td></tr><tr><td>Train</td><td>Inference</td></tr><tr><td>Full fine-tuning</td><td>100</td><td>17.6</td><td>0.86</td><td>62.82.5</td><td>93.90.6</td><td>91.91.0</td><td>89.90.4</td><td>86.20.4</td><td>92.50.3</td><td>74.11.0</td><td>90.30.1</td><td>85.20.4</td></tr><tr><td>Adapters</td><td>1.63</td><td>13.0</td><td>0.87</td><td>64.41.5</td><td>94.20.5</td><td>88.90.2</td><td>88.90.1</td><td>86.40.2</td><td>93.10.2</td><td>75.10.7</td><td>91.10.2</td><td>85.30.2</td></tr><tr><td>LoRA</td><td>1.71</td><td>12.6</td><td>0.86</td><td>63.30.1</td><td>94.30.1</td><td>90.10.7</td><td>89.00.1</td><td>86.30.1</td><td>93.20.1</td><td>75.53.3</td><td>90.90.0</td><td>85.30.5</td></tr><tr><td>BitFit</td><td>0.13</td><td>10.7</td><td>0.86</td><td>61.81.5</td><td>94.30.1 90.316.3</td><td>91.00.2</td><td>88.70.0</td><td>85.60</td><td>93.10.1</td><td>67.60.6</td><td>90.80.2</td><td>84.10.1</td></tr><tr><td>Prompt-tuning</td><td>0.03</td><td>22.2</td><td>0.87</td><td>02.5</td><td></td><td>74.60.0</td><td>88.50.2</td><td>82.50.9</td><td>92.50.2</td><td>59.52.9</td><td>90.10.1</td><td>72.21.6</td></tr><tr><td>Ladder Side-Tuning</td><td>1.74</td><td>5.5</td><td>0.88</td><td>58.13.2</td><td>94.10.3</td><td>90.41.0</td><td>88.80.1</td><td>85.60.1</td><td>93.30.1</td><td>71.92.1</td><td>90.70.2</td><td>84.10.5</td></tr><tr><td>Ladder Side-Tuning (T5-large)</td><td>1.23</td><td>12.2</td><td>2.88</td><td>65.31.9</td><td>95.70.1</td><td>91.61.0</td><td>89.70.0</td><td>88.60.0</td><td>94.10.2</td><td>79.90.0</td><td>92.40.1</td><td>87.10.2</td></tr><tr><td>Ladder Side-Tuning (T5-3B)</td><td>0.08</td><td>22.4</td><td>11.01</td><td>66.41.7</td><td>96.50.1</td><td>92.90.8</td><td>89.70.1</td><td>90.70.1</td><td>95.10.2</td><td>80.11.0</td><td>93.00.3</td><td>88.10.4</td></tr></table>",
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"table_caption": [
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"Table 3: Comparison between multiple parameter-efficient training methods on VQA, GQA, $\\mathrm { \\ N L V R ^ { 2 } }$ , and MSCOCO. We use T5-base for all approaches. We report accuracy for VQA, GQA and NLVR while we use CIDEr to evaluate MSCOCO. Each number in the table is the average result over three seeds, and the subscripts are standard deviations. "
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"table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Update Param. (%)</td><td colspan=\"2\">Memory Usage (GB)</td><td rowspan=\"2\">VQA</td><td rowspan=\"2\">GQA</td><td rowspan=\"2\">NLVR²</td><td rowspan=\"2\">MSCOCO</td><td rowspan=\"2\">Avg.</td></tr><tr><td>Train</td><td>Inference</td></tr><tr><td>Full fine-tuning</td><td>100</td><td>36.2</td><td>0.86</td><td>67.10.1</td><td>56.30.3</td><td>74.30.4</td><td>112.20.3</td><td>77.50.3</td></tr><tr><td>Adapters</td><td>7.98</td><td>28.4</td><td>0.93</td><td>67.10.1</td><td>56.00.4</td><td>72.70.3</td><td>111.80.1</td><td>76.90.2</td></tr><tr><td>LoRA</td><td>7.54</td><td>27.9</td><td>0.86</td><td>63.70.2</td><td>53.30.1</td><td>70.00.3</td><td>110.30.4</td><td>74.30.1</td></tr><tr><td>BitFit</td><td>0.83</td><td>22.7</td><td>0.86</td><td>55.10.2</td><td>45.50.2</td><td>51.71.1</td><td>101.20.2</td><td>63.40.1</td></tr><tr><td>Prompt-tuning</td><td>1.26</td><td>38.7</td><td>0.87</td><td>47.40.7</td><td>40.60.4</td><td>51.00.4</td><td>96.10.9</td><td>58.80.6</td></tr><tr><td>Ladder Side-Tuning</td><td>7.46</td><td>15.3</td><td>0.93</td><td>66.50.1</td><td>55.90.1</td><td>71.60.3</td><td>113.50.3</td><td>76.90.1</td></tr></table>",
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"type": "text",
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"text": "5 Experimental Results ",
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"text": "In this section, we show experiments to justify our design of LST and demonstrate that LST performs the best among all approaches in the scenario with limited memory. As the result, LST is the most efficient tool to fine-tune large-scale pre-trained models for real-world applications. ",
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"text": "LST outperforms other methods under similar memory usage. Figure 1 and Table 1 show the results on GLUE of different approaches applying on T5-base. We drop 6 layers (3 layers each in side encoder and decoder) for LST to match the parameter usage of the Adapter and LoRA. Under the same parameter usage, LST can save $6 9 \\%$ of memory cost to fully fine-tune the model, while Adapter and LoRA only save $26 \\%$ of that, leading to LST having a $2 . 7 \\mathbf { x }$ more memory saving. Compared to BitFit, LST achieves the same average performance but costs 5GB less GPU memory. LST also surpasses Prompt-tuning in terms of both performance and memory cost. To further take advantage of the memory efficiency of LST, we also train T5-large and T5-3B with LST. We find that with a similar budget of memory usage in Adapter and LoRA, LST with T5-large can surpass the performance of other methods by a large margin. The result on T5-3B also outperforms the result on T5-large, demonstrating the scalability of our memory-efficiency method on large language models. Furthermore, even though LST increases the model size, its additional inference memory usage is negligible as LST uses almost the same inference memory (0.88 GB) as the full fine-tuning (0.86 GB). ",
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"table_caption": [
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"Table 2: LST vs. Y-tuning. "
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"table_body": "<table><tr><td>Method</td><td>Update Param. per Task (%)</td><td>Avg. GLUE</td></tr><tr><td>Y-tuning</td><td>7.7</td><td>76.9</td></tr><tr><td>LST</td><td>2.6</td><td>82.1</td></tr></table>",
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"text": "In Table 2, we also compare LST with a concurrent work, Y-tuning [37] on GLUE tasks (except for STS-B) with BARTlarge [32] encoder as backbone. Following their experimental setup, we use a different learning rate and report the best accuracy out of three seeds for each task. Overall, LST outperforms Y-tuning by a large margin with fewer updated parameters. ",
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"text": "LST is competitive on VL tasks. As we have mentioned beforehand, we also extend LST on a multi-modal architecture, CLIP-T5, on multiple VL tasks, and we demonstrate the outcome in Table 3. ",
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+
"image_caption": [
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| 670 |
+
"Figure 5: The accuracy-memory trade-off for Adapter, LoRA, and Ladder Side-Tuning over GLUE tasks. We vary the reduction factor in Ladder Side-Tuning, hidden dimension in Adapter, rank in LoRA to get the architectures with different training costs. "
|
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| 672 |
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"img_path": "images/b9d2e9a09010521117db6fbd7cac2863898ba90610ce55d306f27b439e14191a.jpg",
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"image_caption": [
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| 685 |
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"Figure 6: The accuracy-memory trade-off for Adapter, LoRA, BitFit, LST over GLUE tasks. we drop $N \\in \\{ 0 , 6 , 1 2 , 1 8 \\}$ layers in an interleaving manner for LST while we gradually freeze the first $N \\in \\{ 0 , 6 , 1 2 , 1 8 \\}$ layers in other methods (also remove inserted parameters in those layers). "
|
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],
|
| 687 |
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| 688 |
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{
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"type": "text",
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| 698 |
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"text": "With similar parameter usage, LST is the only method that can fit into a single 16GB GPU. Besides the efficiency, it is as competitive as full fine-tuning and Adapter, outperforming other PETL approaches. ",
|
| 699 |
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"type": "text",
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"text": "LST performs the best in low-memory regime. To have a better understand of the memory advantage of LST, we adjust the hyper-parameters in our method (reduction factor $\\in \\{ 3 2 , 1 6 , 8 , 4 \\} \\dot { ) }$ , Adapter (hidden dimension $\\in \\{ 6 , 1 2 , 2 4 , 4 8 \\} \\rangle$ and LoRA (rank $\\in \\{ 4 , 8 , 1 6 , 3 2 \\} )$ to create multiple architectures with different memory costs. Figure 5 shows the performance and memory efficiency trade-off for all methods. We find the memory saving is not obvious for Adapter and LoRA, because the gradients of the backbone model’s intermediate outputs are still computed (see Section 3.1 for details). Even though the Adapter and LoRA can get slightly better memory efficiency by reducing the hidden dimension and the rank, we find that the performance drops significantly. On the other hand, LST is quite robust across a wide range of side network sizes. ",
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| 710 |
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|
| 719 |
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"type": "text",
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| 720 |
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"text": "We also consider another way to compare LoRA, Adapter and BitFit to LST in different memory budgets. While we drop $N \\in \\{ 0 , 6 , 1 2 , 1 8 \\}$ layers in an interleaving manner to improve memory efficiency, we freeze the first $N$ layers and remove the corresponding inserted modules in other approaches. With this, other methods can achieve better memory efficiency because gradients do not propagate to those earlier frozen layers. We discuss the layer dropping and layer freezing with details in the following. In LST, we drop $\\begin{array} { l } { { \\frac { N } { 2 } } } \\end{array}$ layers in both side encoder and side decoder. However, in other PETL approaches, we start from freezing layers in the encoder and then turn to freeze layers in the decoder (e.g. freezing 18 layers means freezing all encoder layers and first 6 decoder layers). We display the comparison in Figure 6, showing that LST has a better performance and memory trade-off and outperforms other methods in the low-memory regime. We also find that layer dropping generally reduces the training cost without hurting performance. ",
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| 721 |
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"text": "The ablation of weight initialization on the side network. We compare the different initialization strategies for the side network and demonstrate the results in Figure 7. “Random” denotes we randomly initialize the network while we use network pruning to select initial weights for the side network based on two importance measures, “Weight Magnitude” and “Fisher Information.” In general, the initialization from the pruned network helps no matter the size of the side network, showing the effectiveness of our network pruning strategy. ",
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|
| 741 |
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"type": "text",
|
| 742 |
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"text": "Comparison of LST to network compression methods and Side-tuning. In Section 3.2, we mention that shortcut connections are added to every layer of the side network. We justify this design by comparing LST to two types of approaches: (1) network compression, which discards all shortcut connections and the entire backbone model; (2) Side-tuning, which only adds one shortcut connection to merge representations right before the output layer. Note that we do not drop any layer in the side network but only remove the shortcuts in this experiment. We also compare both distillation-based and pruning-based initialization methods as we describe in Section 2.2. We set the reduction factor to 8 for all approaches. Figure 8 shows the comparison and LST outperforms the other two types of methods significantly. We conclude that PETL methods are stronger than network compression as they use the information from the backbone model. This also suggests that network compression approaches need to train more parameters to achieve the same level of performance as PETL methods. We also demonstrate the usefulness of intermediate shortcuts since LST surpasses Side-tuning by a large amount. Furthermore, we find that using distillation-based initialization or network pruning-based initialization provides similar accuracy in all setups. Note that our network pruning-based initialization method is more efficient since it does not involve training. ",
|
| 743 |
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| 753 |
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"img_path": "images/f3a63c89ce9f810877de748d2de37879fd182f37f1d4ef37ccd69fc60f7984f5.jpg",
|
| 754 |
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"image_caption": [
|
| 755 |
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"Figure 7: The ablation study on different initialization strategies. Y-axis denotes the average score over GLUE tasks. "
|
| 756 |
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|
| 757 |
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|
| 758 |
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"type": "image",
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| 768 |
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"img_path": "images/9ac5e05b8678a0d0cb96185b3591e8643fedcdae8f1803e2eb71e3ff963839e1.jpg",
|
| 769 |
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"image_caption": [
|
| 770 |
+
"Figure 8: The comparison between network compression, Side-Tuning, and LST on GLUE tasks. "
|
| 771 |
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],
|
| 772 |
+
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|
| 773 |
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"text": "",
|
| 784 |
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"type": "text",
|
| 794 |
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"text": "6 Conclusion ",
|
| 795 |
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"text_level": 1,
|
| 796 |
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"bbox": [
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| 803 |
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|
| 804 |
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|
| 805 |
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"type": "text",
|
| 806 |
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"text": "We propose Ladder Side-Tuning (LST), a parameter- and memory-efficient method to adapt a large backbone network. LST does not require backpropagation through the backbone network, which allows for significantly lower memory requirement during training than recently proposed parameterefficient training techniques. We demonstrate that LST allows users to adapt a larger and more powerful backbone network to target tasks with a limited memory requirement, which cannot be achieved with recent parameter-efficient techniques. We also show that LST achieves a more efficient accuracy-memory trade-off than recent baselines, the impact of weight initialization of side networks, and the usefulness of intermediate shortcut connections. Finally, we show that the LST can be also extended beyond NLP tasks, with strong results on VL tasks. We hope that LST helps users with limited computational resources tune larger models in diverse domains. ",
|
| 807 |
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"type": "text",
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| 817 |
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"text": "Acknowledgments ",
|
| 818 |
+
"text_level": 1,
|
| 819 |
+
"bbox": [
|
| 820 |
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176,
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"type": "text",
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+
"text": "We thank the reviewers, Muqeeth Mohammed, Derek Tam, Prateek Yadav, and Gedas Bertasius for their helpful discussions. This work was supported by ARO Award W911NF2110220, ONR Grant N000141812871, and NSF-AI Engage Institute DRL-211263. The views, opinions, and/or findings contained in this article are those of the authors and not of the funding agency. ",
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| 830 |
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"text": "(a) If your work uses existing assets, did you cite the creators? [Yes] \n(b) Did you mention the license of the assets? [No] \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ",
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"text": "5. If you used crowdsourcing or conducted research with human subjects... ",
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| 1 |
+
# Degradation-Aware Unfolding Half-Shuffle Transformer for Spectral Compressive Imaging
|
| 2 |
+
|
| 3 |
+
Yuanhao Cai $^ { 1 , 2 , * }$ , Jing Lin $^ { 1 , 2 , * }$ , Haoqian Wang 1,2,†, Xin Yuan 3, Henghui Ding 4, Yulun Zhang 4, Radu Timofte 4,5, Luc Van Gool 4
|
| 4 |
+
|
| 5 |
+
1 Shenzhen International Graduate School, Tsinghua University, 2 Shenzhen Institute of Future Media Technology, 3 Westlake University, 4 ETH Zürich, 5 University of Würzburg
|
| 6 |
+
|
| 7 |
+
# Abstract
|
| 8 |
+
|
| 9 |
+
In coded aperture snapshot spectral compressive imaging (CASSI) systems, hyperspectral image (HSI) reconstruction methods are employed to recover the spatialspectral signal from a compressed measurement. Among these algorithms, deep unfolding methods demonstrate promising performance but suffer from two issues. Firstly, they do not estimate the degradation patterns and ill-posedness degree from CASSI to guide the iterative learning. Secondly, they are mainly CNN-based, showing limitations in capturing long-range dependencies. In this paper, we propose a principled Degradation-Aware Unfolding Framework (DAUF) that estimates parameters from the compressed image and physical mask, and then uses these parameters to control each iteration. Moreover, we customize a novel Half-Shuffle Transformer (HST) that simultaneously captures local contents and non-local dependencies. By plugging HST into DAUF, we establish the first Transformer-based deep unfolding method, Degradation-Aware Unfolding Half-Shuffle Transformer (DAUHST), for HSI reconstruction. Experiments show that DAUHST surpasses state-of-the-art methods while requiring cheaper computational and memory costs. Code and models are publicly available at https://github.com/caiyuanhao1998/MST
|
| 10 |
+
|
| 11 |
+
# 1 Introduction
|
| 12 |
+
|
| 13 |
+
Hyperspectral images (HSIs) have more spectral bands than normal RGB images to store more detailed information. Thus, HSIs are widely applied in image recognition [1, 2, 3], object detection [4, 5, 6], tracking [7, 8, 9], medical image processing [10, 11, 12], remote sensing [13, 14, 15, 16], etc. To obtain HSIs, traditional imaging systems use spectrometers to scan the scenes along the spectral or spatial dimensions, usually requiring a long time. These imaging systems fail to capture dynamic objects. Recently, snapshot compressive imaging (SCI) systems [17, 18, 19] have been developed to capture HSIs at video rate. Among these SCI systems, coded aperture snapshot spectral imaging (CASSI) [17, 20, 21] stands out for its impressive performance. CASSI uses a coded aperture and a disperser to modulate the HSI signal at different wavelengths, and then mixes all modulated signal to generate a 2D compressed measurement. Subsequently, HSI restoration methods are employed to solve the CASSI inverse problem, i.e., restore the HSIs from the measurement. These methods are divided into four categories.
|
| 14 |
+
|
| 15 |
+
|
| 16 |
+
Figure 1: PSNR-FLOPS comparisons of DAUHST and SOTA unfolding methods.
|
| 17 |
+
|
| 18 |
+
properties and can be interpreted. Yet, these methods need manual parameter tweaking, which slows down reconstruction. Also, they suffer from limited representation capacity and generalization ability.
|
| 19 |
+
|
| 20 |
+
(ii) Plug-and-play $( \mathrm { P n P } )$ algorithms [29, 30, 31] plug pre-trained denoising networks into traditional model-based methods to solve the HSI reconstruction problem. Nonetheless, the pre-trained networks in $\mathrm { P n P }$ methods are fixed without re-training, therefore limiting the performance.
|
| 21 |
+
|
| 22 |
+
(iii) End-to-end (E2E) algorithms employ a powerful model, usually a convolutional neural network (CNN) [12, 20, 32, 33], to learn the E2E mapping function from a measurement to the desired HSIs. E2E methods enjoy the power of deep learning. However, they learn a brute-force mapping from the compressed measurement to the underlying spectral images, thereby ignoring the working principles of CASSI systems. They come without theoretically proven properties, interpretability, and flexibility because the imaging models widely differ from each other for various hardware systems.
|
| 23 |
+
|
| 24 |
+
(iv) Deep unfolding methods [34, 35, 36, 37, 38, 39] adopt a multi-stage network to map the measurement into the HSI cube. Each stage usually includes two phases, i.e., linear projection followed by passing the signal through a single-stage network that learns the underlying denoiser prior. In deep unfolding methods, the network architecture is intuitively interpretable by explicitly characterizing the image priors and the system imaging model. Besides, these methods also enjoy the power of deep learning and thus have great potential. Yet, this potential has not been fully explored.
|
| 25 |
+
|
| 26 |
+
Existing deep unfolding algorithms suffer from two issues. (a) The iterative learning is highly related to the CASSI system. However, current unfolding methods do not estimate CASSI degradation patterns and ill-posedness degree to adjust the linear projection and denoising network in each iteration. (b) Existing deep unfolding methods are mainly CNN-based, therefore showing limitations in capturing non-local self-similarity and long-range dependencies, both critical for HSI reconstruction.
|
| 27 |
+
|
| 28 |
+
Recently, the emerging Transformer [40] has provided a solution to tackle the drawbacks of CNN. Due to its strong capability in modeling the interactions of non-local spatial regions, Transformer has been widely applied in image classification [41, 42, 43], object detection [44, 45, 46], semantic segmentation [47, 48, 49], human pose estimation [50, 51, 52], image restoration [53, 54, 55], etc. Yet, the use of Transformer is confronted with two main issues. (a) The computational complexity of global Transformer [42] is quadratic to the spatial dimensions. This cost is sometimes unaffordable. (b) The receptive fields of local Transformer [41] are limited within position-specific windows. Thus, some tokens with highly-related contents can not match each other when computing self-attention.
|
| 29 |
+
|
| 30 |
+
To address the above problems, in this paper, we firstly formulate a principled Degradation-Aware Unfolding Framework (DAUF) based on maximum a posteriori (MAP) theory for HSI reconstruction. Different from previous deep unfolding methods, our DAUF implicitly estimates informative parameters from the degraded compressed measurement and the physical mask used in the modulation. Then DAUF feeds the parameters, which capture key cues of CASSI degradation patterns and ill-posedness degree, into each iteration to adaptively scale the linear projection and provide the noise level information for the denoising network. Secondly, we design a novel Half-Shuffle Transformer (HST) as the denoiser prior in each iteration. Our HST can jointly extract local contextual information and model non-local dependencies, while requiring much cheaper computational costs than global Transformer. We achieve this by customizing a Half-Shuffle Multi-head Self-Attention (HS-MSA) mechanism that composes the basic unit of HST. More specifically, our HS-MSA has two branches, i.e., local branch and non-local branch. The local branch calculates the self-attention within the local window while the non-local branch shuffles the tokens and captures cross-window interactions. We plug HST into DAUF to establish an iterative architecture, Degradation-Aware Unfolding Half-Shuffle Transformer (DAUHST). With the proposed techniques, DAUHST models dramatically outperform state-of-the-art (SOTA) deep unfolding methods with the same number of stages by over 4 dB, as shown in Fig. 1.
|
| 31 |
+
|
| 32 |
+
In a nutshell, our contributions can be summarized as follows:
|
| 33 |
+
|
| 34 |
+
(i) We formulate a principled MAP-based unfolding framework DAUF for HSI reconstruction.
|
| 35 |
+
|
| 36 |
+
(ii) We propose a novel Transformer HST and plug it into DAUF to establish DAUHST. To the best of our knowledge, DAUHST is the first Transformer-based deep unfolding method for HSI restoration.
|
| 37 |
+
|
| 38 |
+
(iii) DAUHST outperforms SOTA methods by a large margin while requiring cheaper computational and memory costs. Besides, DAUHST yields more visually pleasant results in real HSI reconstruction.
|
| 39 |
+
|
| 40 |
+
|
| 41 |
+
Figure 2: The architecture of our DAUF with $K$ stages (iterations). $\varepsilon$ estimates informative parameters from the compressed measurement $\mathbf { y }$ and sensing matrix $\Phi$ . The estimated parameters $_ { \pmb { \alpha } }$ and $\beta$ are fed into each stage of subsequent iterative learning. $\mathcal { P }$ and $\mathcal { D }$ denote the linear projection and denoising network in each stage.
|
| 42 |
+
|
| 43 |
+
# 2 Proposed Method
|
| 44 |
+
|
| 45 |
+
# 2.1 Degradation Model of CASSI
|
| 46 |
+
|
| 47 |
+
In CASSI, we denote the vectorized measurement as $\mathbf { y } \in \mathbb { R } ^ { n }$ , where $\begin{array} { r } { n = H ( W + d ( N _ { \lambda } - 1 ) ) . ~ H , W , } \end{array}$ , $d$ , and $N _ { \lambda }$ denote the HSI’s height, width, shifting step in dispersion, and total number of wavelengths. Given the vectorized shifted HSI signal $\mathbf { x } \in \mathbb { R } ^ { n N _ { \lambda } ^ { \mathbf { * } } }$ and the sensing matrix $\Phi \in \mathbb { R } ^ { n \times n N _ { \lambda } }$ that is determined by the physical mask, the degradation model of CASSI can be formulated as
|
| 48 |
+
|
| 49 |
+
$$
|
| 50 |
+
\mathbf { y } = \Phi \mathbf { x } + \mathbf { n } ,
|
| 51 |
+
$$
|
| 52 |
+
|
| 53 |
+
where $\mathbf { n } \in \mathbb { R } ^ { n }$ represents the vectorized imaging noise on the measurement. As analyzed in [56, 57, 58], $\Phi$ is a fat, sparse, and highly structured matrix that is hard to handle. Please refer to the supplementary material for details about the mathematical model of CASSI. Then the task of HSI reconstruction is given y (captured by the camera) and $\Phi$ (calibrated based on pre-design), solving $\mathbf { x }$ .
|
| 54 |
+
|
| 55 |
+
# 2.2 Degradation-Aware Unfolding Framework
|
| 56 |
+
|
| 57 |
+
Previous unfolding frameworks [34, 35, 36, 37] do not estimate the CASSI degradation patterns to adjust the iterative learning. To alleviate this limitation, we formulate a principled Degradation-Aware Unfolding Framework (DAUF) as depicted in Fig. 2. DAUF starts from the MAP theory. In particular, the original HSI signal could be estimated by minimizing the following energy function as
|
| 58 |
+
|
| 59 |
+
$$
|
| 60 |
+
\hat { \mathbf { x } } = \arg \operatorname* { m i n } _ { \mathbf { x } } \frac { 1 } { 2 } | | \mathbf { y } - \boldsymbol { \Phi } \mathbf { x } | | ^ { 2 } + \tau R ( \mathbf { x } ) ,
|
| 61 |
+
$$
|
| 62 |
+
|
| 63 |
+
where $\frac { 1 } { 2 } | | \mathbf { y } - \Phi \mathbf { x } | | ^ { 2 }$ is the data fidelity term, $R ( \mathbf { x } )$ is the image prior term, and $\tau$ is a hyperparameter balancing the importance. By introducing an auxiliary variable $\mathbf { z }$ , Eq. (2) can be reformulated as
|
| 64 |
+
|
| 65 |
+
$$
|
| 66 |
+
\hat { \mathbf { x } } = \arg \operatorname* { m i n } _ { \mathbf { x } } ~ \frac { 1 } { 2 } | | \mathbf { y } - \boldsymbol { \Phi } \mathbf { x } | | ^ { 2 } + \tau R ( \mathbf { z } ) , \quad s . t . ~ \mathbf { z } = \mathbf { x } .
|
| 67 |
+
$$
|
| 68 |
+
|
| 69 |
+
This is a constrained optimization problem. To obtain an unfolding inference, we adopt half-quadratic splitting (HQS) algorithm for its simplicity and fast convergence. Then Eq. (3) is solved by minimizing
|
| 70 |
+
|
| 71 |
+
$$
|
| 72 |
+
\mathcal { L } _ { \mu } ( \mathbf { x } , \mathbf { z } ) = \frac { 1 } { 2 } | | \mathbf { y } - \boldsymbol { \Phi } \mathbf { x } | | ^ { 2 } + \tau R ( \mathbf { z } ) + \frac { \mu } { 2 } | | \mathbf { z } - \mathbf { x } | | ^ { 2 } ,
|
| 73 |
+
$$
|
| 74 |
+
|
| 75 |
+
where $\mu$ is a penalty parameter that forces $\mathbf { x }$ and $\mathbf { z }$ to approach the same fixed point. Subsequently, Eq. (4) can be solved by decoupling $\mathbf { x }$ and $\mathbf { z }$ into the following two iterative sub-problems as
|
| 76 |
+
|
| 77 |
+
$$
|
| 78 |
+
\mathbf { x } _ { k + 1 } = \arg \operatorname* { m i n } _ { \mathbf { x } } ~ | | \mathbf { y } - \Phi \mathbf { x } | | ^ { 2 } + \mu | | \mathbf { x } - \mathbf { z } _ { k } | | ^ { 2 } , ~ \mathbf { z } _ { k + 1 } = \arg \operatorname* { m i n } _ { \mathbf { z } } \frac { \mu } { 2 } | | \mathbf { z } - \mathbf { x } _ { k + 1 } | | ^ { 2 } + \tau R ( \mathbf { z } ) ,
|
| 79 |
+
$$
|
| 80 |
+
|
| 81 |
+
where $k = 0 , 1 , \ \dots , K - 1$ indexes the iteration. Note that the data fidelity term is associated with a quadratic regularized least-squares problem, i.e., $\mathbf { x } _ { k + 1 }$ in Eq. (5). It has a closed-form solution as
|
| 82 |
+
|
| 83 |
+
$$
|
| 84 |
+
\mathbf { x } _ { k + 1 } = ( \Phi ^ { \mathsf { T } } \Phi + \mu \mathbf { I } ) ^ { - 1 } ( \Phi ^ { \mathsf { T } } \mathbf { y } + \mu \mathbf { z } _ { k } ) ,
|
| 85 |
+
$$
|
| 86 |
+
|
| 87 |
+
where $\mathbf { I }$ is an identity matrix. Since $\Phi$ is a fat matrix, $( \Phi ^ { \mathsf { T } } \Phi + \mu \mathbf { I } )$ will be large and thus we simplify the computation of the inverse problem $( \Phi ^ { \mathsf { T } } \Phi + \mu \mathbf { I } ) ^ { - 1 }$ by the matrix inversion formula as
|
| 88 |
+
|
| 89 |
+
$$
|
| 90 |
+
( \Phi ^ { \mathsf { T } } \Phi + \mu \mathbf { I } ) ^ { - 1 } = \mu ^ { - 1 } \mathbf { I } - \mu ^ { - 1 } \Phi ^ { \mathsf { T } } ( \mathbf { I } + \Phi \mu ^ { - 1 } \Phi ^ { \mathsf { T } } ) ^ { - 1 } \Phi \mu ^ { - 1 } .
|
| 91 |
+
$$
|
| 92 |
+
|
| 93 |
+
By plugging Eq. (7) into Eq. (6), we can reformulate Eq. (6) as
|
| 94 |
+
|
| 95 |
+
$$
|
| 96 |
+
\mathbf { x } _ { k + 1 } = { \frac { \Phi ^ { \mathsf { T } } \mathbf { y } + \mu \mathbf { z } _ { k } } { \mu } } - { \frac { \Phi ^ { \mathsf { T } } ( \mathbf { I } + \Phi \mu ^ { - 1 } \Phi ^ { \mathsf { T } } ) ^ { - 1 } \Phi \Phi ^ { \mathsf { T } } \mathbf { y } } { \mu ^ { 2 } } } - { \frac { \Phi ^ { \mathsf { T } } ( \mathbf { I } + \Phi \mu ^ { - 1 } \Phi ^ { \mathsf { T } } ) ^ { - 1 } \Phi \mathbf { z } _ { k } } { \mu } } ~ .
|
| 97 |
+
$$
|
| 98 |
+
|
| 99 |
+
In CASSI systems, $\Phi \Phi ^ { \mathsf { T } }$ is a diagonal matrix which can be defined as $\Phi \Phi ^ { \mathsf { T } } \stackrel { \mathrm { d e f } } { = } \mathrm { d i a g } \{ \psi _ { 1 } , \hdots , \psi _ { n } \}$ . By plugging $\Phi \Phi ^ { \mathsf { T } }$ into $( \mathbf { I } + \pmb { \Phi } \pmb { \mu } ^ { - 1 } \pmb { \Phi } ^ { \mathsf { T } } ) ^ { - 1 }$ and $( \mathbf { I } + \pmb { \Phi } \pmb { \mu } ^ { - 1 } \pmb { \Phi } ^ { \mathsf { T } } ) ^ { - 1 } \pmb { \Phi } \pmb { \Phi } ^ { \mathsf { T } }$ , we obtain:
|
| 100 |
+
|
| 101 |
+
$$
|
| 102 |
+
\begin{array} { r l r } & { } & { ( { \bf I } + \Phi \mu ^ { - 1 } \Phi ^ { \mathsf { T } } ) ^ { - 1 } = \mathsf { d i a g } \Big \{ \displaystyle \frac { \mu } { \mu + \psi _ { 1 } } , \ldots , \displaystyle \frac { \mu } { \mu + \psi _ { n } } \Big \} , } \\ & { } & { ( { \bf I } + \Phi \mu ^ { - 1 } \Phi ^ { \mathsf { T } } ) ^ { - 1 } \Phi \Phi ^ { \mathsf { T } } = \mathsf { d i a g } \Big \{ \displaystyle \frac { \mu \psi _ { 1 } } { \mu + \psi _ { 1 } } , \ldots , \displaystyle \frac { \mu \psi _ { n } } { \mu + \psi _ { n } } \Big \} . } \end{array}
|
| 103 |
+
$$
|
| 104 |
+
|
| 105 |
+
Let $\mathbf { y } \ { \stackrel { \mathrm { d e f } } { = } } \ [ y _ { 1 } , \dots , y _ { n } ] ^ { \mathsf { T } }$ and $[ \Phi \mathbf { z } _ { k } ] _ { i }$ denotes the $i$ -th element of $\Phi \mathbf { z } _ { k }$ . We plug Eq. (9) into Eq. (8) as
|
| 106 |
+
|
| 107 |
+
$$
|
| 108 |
+
\begin{array} { l } { { \displaystyle { \bf x } _ { k + 1 } = \frac { { \bf \Phi } { \bf \Phi } ^ { \mathsf { T } } { \bf y } } { \mu } + { \bf z } _ { k } - \frac { 1 } { \mu } { \bf \Phi } ^ { \mathsf { T } } \Big [ \frac { y _ { 1 } \psi _ { 1 } + \mu [ { \bf \Phi } { \bf z } _ { k } ] _ { 1 } } { \mu + \psi _ { 1 } } , \ldots , \frac { y _ { n } \psi _ { n } + \mu [ { \bf \Phi } { \bf z } _ { k } ] _ { n } } { \mu + \psi _ { n } } \Big ] ^ { \mathsf { T } } } } \\ { { \displaystyle ~ = { \bf z } _ { k } + { \bf \Phi } ^ { \mathsf { T } } \Big [ \frac { y _ { 1 } - [ { \bf \Phi } { \bf z } _ { k } ] _ { 1 } } { \mu + \psi _ { 1 } } , \ldots , \frac { y _ { n } - [ { \bf \Phi } { \bf z } _ { k } ] _ { n } } { \mu + \psi _ { n } } \Big ] ^ { \mathsf { T } } } . } \end{array}
|
| 109 |
+
$$
|
| 110 |
+
|
| 111 |
+
Note that $\{ y _ { i } - [ \Phi \mathbf { z } _ { k } ] _ { i } \} _ { i = 1 } ^ { n }$ can be directly updated by $\mathbf { y } - \Phi \mathbf { z } _ { k }$ , and $\{ \psi _ { i } \} _ { i = 1 } ^ { n }$ is pre-calculated and stored in $\Phi \Phi ^ { \mathsf { T } }$ . Thus, by element-wise computation in Eq. (10), $\mathbf { x } _ { k + 1 }$ can be updated very efficiently. According to Eq. (5), the penalty parameter $\mu$ should be large enough so that $\mathbf { x }$ and $\mathbf { z }$ can approach approximately the same fixed point. This indicates that $\mu$ controls the convergence and output of each iteration. Thus, instead of manually tweaking $\mu$ , we set $\mu$ as a series of iteration-specific parameters to be automatically estimated from the CASSI system. We denote $\mu$ in the $k$ -th iteration as $\mu _ { k }$ .
|
| 112 |
+
|
| 113 |
+
Returning to Eq. (5), we also set $\tau$ as iteration-specific parameters and ${ \mathbf z } _ { k + 1 }$ can be reformulated as
|
| 114 |
+
|
| 115 |
+
$$
|
| 116 |
+
\mathbf { z } _ { k + 1 } = \arg \operatorname* { m i n } _ { \mathbf { z } } \ { \frac { 1 } { 2 ( { \sqrt { \tau _ { k + 1 } / \mu _ { k + 1 } } } ) ^ { 2 } } } \left| | \mathbf { z } - \mathbf { x } _ { k + 1 } | \right| ^ { 2 } + R ( \mathbf { z } ) .
|
| 117 |
+
$$
|
| 118 |
+
|
| 119 |
+
From the perspective of Bayesian probability, Eq. (11) is equivalent to denoising image $\mathbf { x } _ { k + 1 }$ with a Gaussian noise at level pτk+1/µk+1 [29]. To conveniently solve Eq. (11), we set $\begin{array} { r } { \frac { 1 } { ( \sqrt { \tau _ { k + 1 } / \mu _ { k + 1 } } ) ^ { 2 } } = } \end{array}$ $\mu _ { k + 1 } / \tau _ { k + 1 }$ as parameters to be estimated from CASSI. Let $\alpha _ { k } \ { \stackrel { \mathrm { d e f } } { = } } \ \mu _ { k }$ , ${ \pmb { \alpha } } \ { \stackrel { \mathrm { d e f } } { = } } \ [ \alpha _ { 1 } , . . . , \alpha _ { K } ]$ , βk def = $\mu _ { k } / \tau _ { k }$ , and $\beta \stackrel { \mathrm { d e f } } { = } [ \beta _ { 1 } , . . . , \beta _ { K } ]$ . Then we can formulate our DAUF as an iterative scheme:
|
| 120 |
+
|
| 121 |
+
$$
|
| 122 |
+
\begin{array} { r } { ( \alpha , \beta ) = \mathcal { E } ( \mathbf { y } , \Phi ) , \quad \mathbf { x } _ { k + 1 } = \mathcal { P } ( \mathbf { y } , \mathbf { z } _ { k } , \alpha _ { k + 1 } , \Phi ) , \quad \mathbf { z } _ { k + 1 } = \mathcal { D } ( \mathbf { x } _ { k + 1 } , \beta _ { k + 1 } ) , } \end{array}
|
| 123 |
+
$$
|
| 124 |
+
|
| 125 |
+
where $\mathcal { E }$ denotes the parameter estimator that takes the compressed measurement $\mathbf { y }$ and the sensing matrix $\Phi$ of the CASSI system as inputs, $\mathcal { P }$ equivalent to Eq. (10) denotes the linear projection, and $\mathcal { D }$ represents the Gaussian denoiser solving Eq. (11). $\mathbf { z } _ { 0 }$ is initialized by passing the shifted $\mathbf { y }$ concatenated with $\Phi$ through a $c o n v 1 \times 1$ (convolution with $1 \times 1$ kernel). Fig. 2 shows the architecture of $\mathcal { E }$ . It consists of a $c o n v 1 \times 1 .$ , a strided $c o n v 3 \times 3$ , a global average pooling, and three fully connected layers. Through $\mathcal { E }$ , DAUF captures critical cues from CASSI by learning the degradation patterns and ill-posedness degree caused by the mask-modulation and dispersionintegration. Parameters $_ { \pmb { \alpha } }$ and $\beta$ estimated by $\mathcal { E }$ direct the iterative learning by adaptively scaling the linear projection in Eq. (10) and providing noise level information for the denoiser prior in Eq. (11).
|
| 126 |
+
|
| 127 |
+
# 2.3 Half-Shuffle Transformer
|
| 128 |
+
|
| 129 |
+
When designing the denoiser prior, previous deep unfolding methods [34, 35, 36, 37] mainly adopt CNNs, showing limitations in capturing long-range dependencies. Directly applying local and global Transformers will encounter two problems, i.e., limited receptive fields and nontrivial computational costs. To address these challenges, we propose Half-Shuffle Transformer (HST) to play the role of $\mathcal { D }$ .
|
| 130 |
+
|
| 131 |
+
Network Architecture. As shown in Fig. 3 (a), HST adopts a three-level U-shaped structure built by the basic unit Half-Shuffle Attention Block (HSAB). Firstly, HST uses a $c o n v 3 \times 3$ to map reshaped $\mathbf { x } _ { k }$ concatenated with stretched $\beta _ { k }$ into feature $\mathbf { X } _ { 0 } \in \mathbb { R } ^ { H \times \hat { W } \times C }$ , where $\hat { W } = W + d ( N _ { \lambda } - 1 )$ Secondly, $\mathbf { X } _ { 0 }$ passes through the encoder, bottleneck, and decoder to be embedded into deep feature $\mathbf { X } _ { d } \in \mathbb { R } ^ { H \times \hat { W } \times C }$ . Each level of the encoder or decoder contains an HSAB and a resizing module.
|
| 132 |
+
|
| 133 |
+
|
| 134 |
+
Figure 3: Diagram of HST. (a) HST adopts a U-shaped structure. (b) HSAB consists of an FFN, an HS-MSA, and two layer normalization. (c) Components of FFN. (d) HS-MSA contains local branch and non-local branch.
|
| 135 |
+
|
| 136 |
+
In Fig. 3 (b), HSAB consists of two layer normalization (LN), an HS-MSA, and a Feed-Forward Network (FFN) that is detailed in Fig. 3 (c). The downsampling and upsampling modules are strided $c o n v 4 \times 4$ and deconv $2 \times 2$ . Finally, a conv $3 \times 3$ operates on $\mathbf { X } _ { d }$ to generate a residual image $\mathbf { R } \in \mathbb { R } ^ { H \times \hat { W } \times N _ { \lambda } }$ . The output denoised image $\mathbf { z } _ { k }$ is obtained by the sum of $\mathbf { x } _ { k }$ and reshaped $\mathbf { R }$ .
|
| 137 |
+
|
| 138 |
+
Half-Shuffle Multi-head Self-Attention. The most important element of HSAB is the proposed Half-Shuffle Multi-head Self-Attention (HS-MSA) module. Fig. 3 (d) depicts the HS-MSA used in the first level. The input tokens of HS-MSA are denoted as $\mathbf { X } _ { i n } \in \mathbb { R } ^ { H \times \hat { W } \times C }$ . Subsequently, ${ \bf X } _ { i n }$ is linearly projected into query $\mathbf { Q } \in \mathbb { R } ^ { H \times \hat { W } \times C }$ , key $\mathbf { K } \in \mathbb { R } ^ { H \times \hat { W } \times C }$ , and value $\mathbf { V } \in \mathbb { R } ^ { H \times \hat { W } \times C }$ as
|
| 139 |
+
|
| 140 |
+
$$
|
| 141 |
+
\mathbf { Q } = \mathbf { X } _ { i n } \mathbf { W ^ { Q } } , \mathbf { K } = \mathbf { X } _ { i n } \mathbf { W ^ { K } } , \mathbf { V } = \mathbf { X } _ { i n } \mathbf { W ^ { V } } ,
|
| 142 |
+
$$
|
| 143 |
+
|
| 144 |
+
where $\mathbf { W ^ { Q } } , \mathbf { W ^ { K } } , \mathbf { W ^ { V } } \in \mathbb { R } ^ { C \times C }$ are learnable parameters and biases are omitted for simplification. Our HS-MSA combines the advantages of global MSA [42] and local window-based MSA [41], i.e., HS-MSA can jointly capture local contextual information through the local branch and model long-range dependencies through the non-local branch, all while being computationally cheaper than global MSA. Specifically, $\mathbf { Q } , \mathbf { K } , \mathbf { V }$ are split into two equal parts along the channel dimension as
|
| 145 |
+
|
| 146 |
+
$$
|
| 147 |
+
\mathbf { Q } = [ \mathbf { Q } _ { l } , \mathbf { Q } _ { n l } ] , ~ \mathbf { K } = [ \mathbf { K } _ { l } , \mathbf { K } _ { n l } ] , ~ \mathbf { V } = [ \mathbf { V } _ { l } , \mathbf { V } _ { n l } ] ,
|
| 148 |
+
$$
|
| 149 |
+
|
| 150 |
+
where $\mathbf { Q } _ { l } , \mathbf { K } _ { l } , \mathbf { V } _ { l } \ \in \ \mathbb { R } ^ { H \times \hat { W } \times \frac { C } { 2 } }$ are fed into the local branch to capture local contents, while ${ \bf Q } _ { n l } , { \bf K } _ { n l } , { \bf V } _ { n l } \in \mathbb { R } ^ { H \times { \hat { W } } \times \frac { C } { 2 } }$ pass through the non-local branch to model non-local dependencies.
|
| 151 |
+
|
| 152 |
+
Local Branch. The local branch computes MSA within position-specific windows. As shown in the upper path of Fig. 3 (d), $\mathbf { Q } _ { l } , \mathbf { K } _ { l } , \mathbf { V } _ { l }$ are partitioned into non-overlapping windows of size $M \times M$ . Then they are reshaped into $\begin{array} { r } { \mathbb { R } ^ { \frac { H \hat { W } } { M ^ { 2 } } \times M ^ { 2 } \times \frac { C } { 2 } } } \end{array}$ . Subsequently, $\mathbf { Q } _ { l } , \mathbf { K } _ { l } , \mathbf { V } _ { l }$ are split along the channel wise into $h$ heads: $\mathbf { \bar { Q } } _ { l } = [ \mathbf { Q } _ { l } ^ { 1 } , \dots , \mathbf { Q } _ { l } ^ { h } \ ]$ ], $\mathbf { K } _ { l } = [ \dot { \mathbf { K } } _ { l } ^ { 1 } , \dot { \mathbf { \Omega } } . \dot { \mathbf { \Omega } } . \mathbf { K } _ { l } ^ { h } ]$ , and $\mathbf { V } _ { l } = \left[ \mathbf { V } _ { l } ^ { 1 } , \ldots , \mathbf { V } _ { l } ^ { h } \right]$ . The dimension of each head is $\begin{array} { r } { d _ { h } = \frac { C } { 2 h } } \end{array}$ l l l l. Note that Fig. 3 (d) depicts the situation with $h = 1$ l and some details are omitted for simplification. The local self-attention $\mathbf { A } _ { l } ^ { i }$ is calculated inside each head as
|
| 153 |
+
|
| 154 |
+
$$
|
| 155 |
+
\mathbf { A } _ { l } ^ { i } = \operatorname { s o f t m a x } ( \frac { \mathbf { Q } _ { l } ^ { i } \mathbf { \mathbf { K } } _ { l } ^ { i ^ { \top } } } { \sqrt { d _ { h } } } + \mathbf { P } _ { l } ^ { i } ) \mathbf { V } _ { l } ^ { i } , i = 1 , \ldots , h ,
|
| 156 |
+
$$
|
| 157 |
+
|
| 158 |
+
where $\mathbf { P } _ { l } ^ { i } \in \mathbb { R } ^ { M ^ { 2 } \times M ^ { 2 } }$ are learnable parameters embedding the position information.
|
| 159 |
+
|
| 160 |
+
Non-local Branch. The non-local branch computes cross-window interactions through shuffle operations inspired by ShuffleNet [59]. In particular, ${ \bf Q } _ { n l } , { \bf K } _ { n l } , { \bf V } _ { n l } \in \mathbb { R } ^ { H \times \hat { W } \times \frac { C } { 2 } }$ are firstly partitioned into non-overlapping windows with size $M \times M$ . Then their shapes are transposed from $\begin{array} { r } { \mathbb { R } ^ { \frac { H \hat { W } } { M ^ { 2 } } \times M ^ { 2 } \times \frac { C } { 2 } } } \end{array}$ to $\begin{array} { r } { \mathbb { R } ^ { M ^ { 2 } \times \frac { H \hat { W } } { M ^ { 2 } } \times \frac { C } { 2 } } } \end{array}$ to shuffle the positions of tokens and establish inter-window dependencies. $\mathbf { Q } _ { n l } , \mathbf { K } _ { n l } , \mathbf { V } _ { n l }$ are split into $h$ heads: $\mathbf { Q } _ { n l } = [ \mathbf { Q } _ { n l } ^ { 1 } , \ldots , \mathbf { Q } _ { n l } ^ { h } ]$ , ${ \bf K } _ { n l } = [ { \bf K } _ { n l } ^ { 1 } , \ldots , { \bf K } _ { n l } ^ { \hat { h } } ]$ , and ${ { \bf { V } } _ { n l } } = [ { \bf { V } } _ { n l } ^ { 1 } , . . . , { \bf { V } } _ { n l } ^ { h } ]$ . Then the non-local self-attention $\mathbf { A } _ { n l } ^ { i }$ is computed in each head as
|
| 161 |
+
|
| 162 |
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<table><tr><td>Algorithms</td><td>Params</td><td>GFLOPS</td><td>S1</td><td>S2</td><td>S3</td><td>S4</td><td>S5</td><td>S6</td><td>S7</td><td>S8</td><td>S9</td><td>S10</td><td>Avg</td></tr><tr><td>TwIST [60]</td><td>-</td><td>-</td><td>25.16 0.700</td><td>23.02 0.604</td><td>21.40 0.711</td><td>30.19 0.851</td><td>21.41 0.635</td><td>20.95 0.644</td><td>22.20 0.643</td><td>21.82 0.650</td><td>22.42 0.690</td><td>22.67 0.569</td><td>23.12 0.669</td></tr><tr><td>GAP-TV [26]</td><td>-</td><td>-</td><td>26.82 0.754</td><td>22.89 0.610</td><td>26.31 0.802</td><td>30.65 0.852</td><td>23.64 0.703</td><td>21.85 0.663</td><td>23.76 0.688</td><td>21.98 0.655</td><td>22.63 0.682</td><td>23.10 0.584</td><td>24.36 0.669</td></tr><tr><td>DeSCI [23]</td><td></td><td></td><td>27.13 0.748</td><td>23.04 0.620</td><td>26.62 0.818</td><td>34.96 0.897</td><td>23.94 0.706</td><td>22.38 0.683</td><td>24.45 0.743</td><td>22.03 0.673</td><td>24.56 0.732</td><td>23.59 0.587</td><td>25.27 0.721</td></tr><tr><td>λ-Net [33]</td><td>62.64M</td><td>117.98</td><td>30.10 0.849</td><td>28.49 0.805 31.09</td><td>27.73 0.870</td><td>37.01 0.934</td><td>26.19 0.817</td><td>28.64 0.853</td><td>26.47 0.806</td><td>26.09 0.831</td><td>27.50 0.826</td><td>27.13 0.816</td><td>28.53 0.841</td></tr><tr><td>HSSP [35]</td><td>-</td><td>-</td><td>31.48 0.858 31.72</td><td>0.842 31.13</td><td>28.96 0.823 29.99</td><td>34.56 0.902</td><td>28.53 0.808</td><td>30.83 0.877</td><td>28.71 0.824</td><td>30.09 0.881</td><td>30.43 0.868</td><td>28.78 0.842</td><td>30.35 0.852</td></tr><tr><td>DNU [34]</td><td>1.19M</td><td>163.48</td><td>0.863 32.68</td><td>0.846 27.26</td><td>0.845 31.30</td><td>35.34 0.908</td><td>29.03 0.833</td><td>30.87 0.887</td><td>28.99 0.839</td><td>30.13 0.885</td><td>31.03 0.876</td><td>29.14 0.849</td><td>30.74 0.863</td></tr><tr><td>DIP-HSI [30]</td><td>33.85M</td><td>64.42</td><td>0.890 32.03</td><td>0.833 31.00</td><td>0.914 32.25</td><td>40.54 0.962 39.19</td><td>29.79 0.900</td><td>30.39 0.877</td><td>28.18 0.913 30.32</td><td>29.44 0.874</td><td>34.51 0.927 30.01</td><td>28.51 0.851</td><td>31.26 0.894</td></tr><tr><td>TSA-Net [20]</td><td>44.25M</td><td>110.06</td><td>0.892 33.26</td><td>0.858 32.09</td><td>0.915</td><td>0.953</td><td>29.39 0.884</td><td>31.44 0.908</td><td>0.878</td><td>29.35 0.888</td><td>0.890</td><td>29.59 0.874</td><td>31.46 0.894</td></tr><tr><td>DGSMP [38]</td><td>3.76M</td><td>646.65</td><td>0.915</td><td>0.898 33.26</td><td>33.06 0.925 34.28</td><td>40.54 0.964</td><td>28.86 0.882</td><td>33.08 0.937</td><td>30.74 0.886</td><td>31.55 0.923</td><td>31.66 0.911</td><td>31.44 0.925</td><td>32.63 0.917</td></tr><tr><td>GAP-Net [36]</td><td>4.27M</td><td>78.58</td><td>33.74 0.911 34.12</td><td>0.900 33.62</td><td>0.929 35.04</td><td>41.03 0.967</td><td>31.44 0.919</td><td>32.40 0.925</td><td>32.27 0.902</td><td>30.46 0.905</td><td>33.51 0.915</td><td>30.24 0.895</td><td>33.26 0.917</td></tr><tr><td>ADMM-Net [37]</td><td>4.27M</td><td>78.58</td><td>0.918 35.14</td><td>0.902 35.67</td><td>0.931 36.03</td><td>41.15 0.966 42.30</td><td>31.82 0.922</td><td>32.54 0.924 34.46</td><td>32.42 0.896 33.67</td><td>30.74 0.907</td><td>33.75 0.915 34.89</td><td>30.68 0.895</td><td>33.58 0.918</td></tr><tr><td>HDNet [32]</td><td>2.37M</td><td>154.76</td><td>0.935 35.40</td><td>0.940 35.87</td><td>0.943 36.51</td><td>0.969 42.27</td><td>32.69 0.946</td><td>0.952</td><td>0.926</td><td>32.48 0.941</td><td>0.942 35.39</td><td>32.38 0.937</td><td>34.97 0.943</td></tr><tr><td>MST-L [61]</td><td>2.03M</td><td>28.15</td><td>0.941 35.80</td><td>0.944 36.23</td><td>0.953 37.34</td><td>0.973 42.63</td><td>32.77 0.947</td><td>34.80 0.955</td><td>33.66 0.925 34.35</td><td>32.67 0.948 33.71</td><td>0.949 36.67</td><td>32.50 0.941</td><td>35.18 0.948</td></tr><tr><td>MST++ [62]</td><td>1.33M</td><td>19.42</td><td>0.943 35.96</td><td>0.947 36.84</td><td>0.957 38.16</td><td>0.973 42.44</td><td>33.38 0.952</td><td>35.38 0.957 35.72</td><td>0.934 34.86</td><td>0.953 34.34</td><td>0.953 36.51</td><td>33.38 0.945 33.09</td><td>35.99 0.951</td></tr><tr><td>CST-L [62]</td><td>3.00M</td><td>40.01</td><td>0.949 36.79</td><td>0.955 37.89</td><td>0.962 40.61</td><td>0.975 46.94</td><td>33.25 0.955</td><td>0.963 35.30</td><td>0.944 36.58</td><td>0.961 33.96</td><td>0.957 39.47</td><td>0.945 32.80</td><td>36.12 0.957 37.58</td></tr><tr><td>BIRNAT [63]</td><td>4.40M</td><td>2122.66</td><td>0.951 35.93</td><td>0.957 36.70</td><td>0.971 37.96</td><td>0.985</td><td>35.42 0.964</td><td>0.959</td><td>0.955 34.78</td><td>0.956 33.65</td><td>0.970 37.42</td><td>0.938</td><td>0.960</td></tr><tr><td> DAUHST-2stg</td><td>1.40M</td><td>18.44</td><td>0.943 36.59</td><td>0.946 37.93</td><td>0.959 39.32</td><td>44.38 0.978</td><td>34.13 0.954</td><td>35.43 0.957</td><td>0.940</td><td>0.950</td><td>0.955 38.54</td><td>33.07 0.941</td><td>36.34 0.952</td></tr><tr><td>DAUHST-3stg</td><td>2.08M</td><td>27.17</td><td>0.949 36.92</td><td>0.958 38.52</td><td>0.964 40.51</td><td>44.77 0.980</td><td>34.82 0.961</td><td>36.19 0.963</td><td>36.02 0.950</td><td>34.28 0.956 34.74</td><td>0.963 38.71</td><td>33.67 0.947</td><td>37.21 0.959</td></tr><tr><td>DAUHST-5stg</td><td>3.44M</td><td>44.61</td><td>0.955</td><td>0.962</td><td>0.967</td><td>45.09 0.980</td><td>35.33 0.964</td><td>36.56 0.965</td><td>36.82 0.958</td><td>0.959</td><td>0.963</td><td>34.27 0.952</td><td>37.75 0.962</td></tr><tr><td>DAUHST-9stg</td><td>6.15M</td><td>79.50</td><td>37.25 0.958</td><td>39.02 0.967</td><td>41.05 0.971</td><td>46.15 0.983</td><td>35.80 0.969</td><td>37.08 0.970</td><td>37.57 0.963</td><td>35.10 0.966</td><td>40.02 0.970</td><td>34.59 0.956</td><td>38.36 0.967</td></tr></table>
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Table 1: Comparisons between DAUHST and SOTA methods on 10 simulation scenes $( \mathbf { S } 1 { \sim } \mathbf { S } 1 0 )$ . Params, FLOPS, PSNR (upper entry in each cell), and SSIM (lower entry in each cell) are reported.
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$$
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\mathbf { A } _ { n l } ^ { i } = \mathrm { s o f t m a x } ( \frac { \mathbf { Q } _ { n l } ^ { i } \mathbf { K } _ { n l } ^ { i } \mathsf { T } } { \sqrt { d _ { h } } } + \mathbf { P } _ { n l } ^ { i } ) \mathbf { V } _ { n l } ^ { i } , i = 1 , \ldots , h ,
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$$
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$\mathbf { A } _ { n l } ^ { i } \in \mathbb { R } ^ { M ^ { 2 } \times \frac { H \hat { W } } { M ^ { 2 } } \times d _ { h } }$ $\mathbf { P } _ { n l } ^ { i } \in \mathbb { R } ^ { \frac { H \hat { W } } { M ^ { 2 } } \times \frac { H \hat { W } } { M ^ { 2 } } }$ are learnable parameters representing thunshuffled by being transposed to shape $\mathbb { R } ^ { \frac { H \hat { W } } { M ^ { 2 } } \times M ^ { 2 } \times d _ { h } }$ edding. Subsequently,. Then the outputs of local branch in Eq. (15) and non-local branch in Eq. (16) are aggregated by a linear projection as
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$$
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\mathrm { H S - M S A } ( \mathbf { X } _ { i n } ) = \sum _ { i = 1 } ^ { h } \mathbf { A } _ { l } ^ { i } \mathbf { W } _ { l } ^ { i } + \sum _ { i = 1 } ^ { h } \mathbf { A } _ { n l } ^ { i } \mathbf { W } _ { n l } ^ { i } ,
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$$
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where $\mathbf { W } _ { l } ^ { i } , \mathbf { W } _ { n l } ^ { i } \in \mathbb { R } ^ { d _ { h } \times C }$ refer to learnable parameters. We reshape the result of Eq. (17) to obtain the output $\mathbf { X } _ { o u t } \in \mathbb { R } ^ { H \times \hat { W } \times C }$ . Instead of globally sampling all tokens, HS-MSA builds inter-window correlations by shuffle operations. The self-attention is calculated in the local window but with tokens from non-local regions. Therefore, HS-MSA is much computationally cheaper than global MSA.
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# 3 Experiment
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# 3.1 Experiment Setup
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Similar to [20, 32, 36, 38, 61], 28 wavelengths are selected from $4 5 0 \mathrm { n m }$ to $6 5 0 \mathrm { n m }$ and derived by spectral interpolation manipulation for the HSI data. Simulation and real experiments are conducted.
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Simulation Dataset. We adopt two datasets, i.e., CAVE [64] and KAIST [65] for simulation experiments. The CAVE dataset consists of 32 HSIs with spatial size $5 1 2 \times 5 1 2$ . The KAIST dataset contains 30 HSIs of spatial size $2 7 0 4 \times 3 3 7 6$ . Following the settings of [20, 32, 36, 38, 61], the CAVE dataset is adopted as the training set while 10 scenes from the KAIST dataset are selected for testing.
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Real Dataset. Five real HSIs collected by the CASSI system developed in [20] are used for testing.
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Figure 4: Simulation HSI reconstruction comparisons of Scene 2 with 4 (out of 28) spectral channels. The top-middle shows the spectral curves corresponding to the two green boxes of the RGB image. The top-right depicts the enlarged patches corresponding to the yellow boxes in the bottom HSIs. Zoom in for a better view.
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Implementation Details. We implement DAUHST by Pytorch. All DAUHST models are trained with Adam [66] optimizer ( $\beta _ { 1 } = 0 . 9$ and $\beta _ { 2 } = 0 . 9 9 9 )$ ) using Cosine Annealing scheme [67] for 300 epochs on an RTX 3090 GPU. The initial learning rate is $4 \times 1 0 ^ { - 4 }$ . Patches with spatial sizes $2 5 6 \times 2 5 6$ and $6 6 0 \times 6 6 0$ are randomly cropped from the 3D HSI cubes with 28 channels as training samples for the simulation and real experiments. The shifting step $d$ in the dispersion is set to 2. The batch size is 5. We set the basic channel $C = N _ { \lambda } = 2 8$ to store HSI information. The weights of $\mathcal { D }$ in different stages are unshared. Data augmentation includes random rotation and flipping. The training objective is to minimize the Root Mean Square Error (RMSE) between reconstructed and ground-truth HSIs.
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# 3.2 Quantitative Comparisons with State-of-the-Art Methods
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Tab. 1 compares the results of DAUHST and 16 SOTA methods including three model-based methods (TwIST [60], GAP-TV [26], and DeSCI [23]), one $\mathrm { P n P }$ method (DIP-HSI [30]), seven E2E methods $\lambda$ -Net [33], TSA-Net [20], HDNet [32], MST [61], ${ \mathrm { M S T } } { + + }$ [62], CST [68], and BIRNAT [63]), and five deep unfolding methods (HSSP [35], DNU [34], DGSMP [38], GAP-Net [36], and ADMMNet [37]) on 10 simulation scenes. All algorithms are tested with the same settings as [38, 61].
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(i) Our best model DAUHST-9stg (9-stage DAUHST) yields very impressive results, i.e., 38.36 dB in PSNR and 0.967 in SSIM. DAUHST-9stg significantly outperforms two recent SOTA methods BIRNAT [63] and MST-L [61] by 0.78 and 3.18 dB, suggesting the effectiveness of our method.
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(ii) Our DAUHST models dramatically surpass SOTA methods while requiring cheaper computational and memory costs. For instance, when compared with the only one Transformer-based E2E method MST, our DAUHST-2stg outperforms MST-L by $1 . 1 6 \mathrm { d B }$ but only costs $6 8 . 9 \%$ $\left( 1 . 4 0 / 2 . 0 3 \right)$ Params and $6 5 . 5 \%$ (18.44 / 28.15) FLOPS. When compared with CNN-based E2E methods, DAUHST-3stg surpasses HDNet, TSA-Net, and $\lambda$ -Net by 2.24, 5.75, and 8.68 dB while only requiring $8 7 . 8 \%$ , $4 . 7 \%$ , $3 . 3 \%$ Params and $1 7 . 6 \%$ , $2 4 . 7 \%$ , $2 3 . 0 \%$ FLOPS. When compared with RNN-based E2E method BIRNAT, our DAUHST-5stg is 0.17 dB higher but only costs $2 . 1 \%$ FLOPS and $7 8 . 2 \%$ Params. Fig. 1 plots the PSNR-FLOPS comparisons of DAUHST and SOTA unfolding methods. DAUHST outperforms other competitors with the same number of stages by very large margins, over 4 dB.
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# 3.3 Qualitative Comparisons with State-of-the-Art Methods
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Simulation HSI Reconstruction. Fig. 4 depicts the simulation HSI reconstruction comparisons between our DAUHST and other SOTA methods on Scene 2 with 4 (out of 28) spectral channels. The top-right part shows the zoomed-in patches of the yellow boxes in the entire HSIs (bottom). As can be observed that our DAUHST-9stg is more favorable to reconstruct visually pleasant HSIs with more detailed contents, cleaner textures, and fewer artifacts while preserving the spatial smoothness of homogeneous regions. In contrast, previous methods either yield over-smooth results compromising fine-grained structures, or introduce undesired chromatic artifacts and blotchy textures that are absent in the ground truth (GT). The top-middle part illustrates the density-wavelength spectral curves corresponding to the green boxes identified as $a$ and $^ b$ in the RGB image (top-left). The spectral curves of DAUHST-9stg achieve the highest correlation and coincidence with the reference curves, showing the advantage of our proposed DAUHST in spectral-dimension consistency reconstruction.
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Figure 5: Real HSI reconstruction results of DAUHST-3stg and 9 SOTA methods on Scene 1 with 4 (out of 28) spectra. Only our method can clearly reconstruct the picked flower at all wavelengths. Zoom in for a better view.
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Real HSI Reconstruction. We further evaluate the effectiveness of DAUHST in real HSI reconstruction. Following the same settings as [20, 38, 61] for a fair comparison, we re-train DAUHST-3stg with the real mask on the CAVE and KAIST datasets jointly. To simulate the real imaging situations, the training samples are also injected with 11-bit shot noise. Fig. 5 shows the visual comparisons between our DAUHST-3stg and nine SOTA methods. In the top three rows, only our DAUHST-3stg can reconstruct the flower patch corresponding to the yellow box at all wavelengths while other methods all fail to recover the entire patch. In the bottom row, DAUHST-3stg restores more HSI structural details and clearer contents with fewer artifacts. In contrast, other methods recover blurry images, generate incomplete responses, and are susceptible to the noise corruption. This evidence suggests that DAUHST is more robust to the noise distortion and more effective in real HSI reconstruction.
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# 3.4 Ablation Study
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Break-down Ablation. We adopt baseline-1 that is derived by removing HS-MSA and DAUF from DAUHST-3stg to conduct the break-down ablation. Our goal is to study the effect of each component towards higher performance. Baseline-1 is cascaded end to end by three single-stage networks. As shown in Tab. 2a, baseline-1 achieves 33.05 dB. When we respectively apply DAUF and HS-MSA, the model achieves 2.32 and $2 . 4 4 \ : \mathrm { d B }$ improvements. When we exploit DAUF and HS-MSA jointly, the model gains by 4.16 dB. These results demonstrate the effectiveness of our DAUF and HS-MSA.
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Self-Attention Mechanism. To compare HS-MSA with other MSAs, we adopt baseline-2 that is obtained by removing HS-MSA from DAUHST-1stg to conduct the ablation in Tab. 2b. We remove different position embedding schemes to avoid their impacts and only compare MSAs. For fairness, we keep the Params of MSAs the same by fixing the number of channels and heads. Baseline-2 yields 32.79 dB. We apply global MSA (G-MSA) [42], Swin MSA (SW-MSA) [41], Spectral-wise MSA (S-MSA) [61], and HS-MSA. Note that we downsample the input feature maps of G-MSA to avoid memory bottlenecks. As shown in Tab. 2b, HS-MSA yields the most significant improvement of 1.26 dB, which is 0.42, 0.30, and $0 . 2 3 \mathrm { d B }$ higher than G-MSA, SW-MSA, and S-MSA. This superiority is mainly derived from HS-MSA’s ability to jointly capture local contents and non-local dependencies.
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Unfolding Framework. We compare our DAUF with previous unfolding frameworks including DNU [34], ADMM-Net [37], and GAP-Net [36]. For a fair comparison, we replace each single-stage network of DNU, ADMM-Net, and GAP-Net by our HST. 3-stage architecture is adopted to conduct ablations. The results are shown in Tab. 2c. Our DAUF significantly outperforms DNU, ADMM, and GAP by 2.59, 1.69, and 1.63 dB while adding only 0.05M Params and 0.94G FLOPS. This is mainly because DAUF uses the parameters estimated from the compressed measurement and physical mask in the CASSI system to direct the iterative learning. These parameters capture critical information of CASSI degradation patterns and ill-posedness degree, providing key cues for HSI reconstruction.
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Figure 6: Visualization of $\mathbf { z } _ { k }$ and $\mathbf { x } _ { k }$ with 4 (out of 28) spectral channels on Scene 1 in different iterations. The bottom-left corner plots the curves of $_ { \pmb { \alpha } }$ and $\beta$ changing with the iteration. Please zoom in for a better view.
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<table><tr><td>Baseline-1</td><td>DAUF</td><td>HS-MSA</td><td>PSNR</td><td>SSIM</td><td>Params (M)</td><td>FLOPS (G)</td></tr><tr><td>√</td><td></td><td></td><td>33.05</td><td>0.912</td><td>1.06</td><td>17.62</td></tr><tr><td>√</td><td>√</td><td></td><td>35.37</td><td>0.938</td><td>1.11</td><td>18.55</td></tr><tr><td>√</td><td></td><td>√</td><td>35.49</td><td>0.941</td><td>2.03</td><td>26.23</td></tr><tr><td>√</td><td>√</td><td>√</td><td>37.21</td><td>0.959</td><td>2.08</td><td>27.17</td></tr></table>
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(b) Ablation of various self-attention mechanisms.
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<table><tr><td>Method</td><td>Baseline-2</td><td>G-MSA</td><td>SW-MSA</td><td>S-MSA</td><td>HS-MSA</td></tr><tr><td>PSNR</td><td>32.79</td><td>33.63</td><td>33.75</td><td>33.82</td><td>34.05</td></tr><tr><td>SSIM</td><td>0.904</td><td>0.920</td><td>0.924</td><td>0.926</td><td>0.930</td></tr><tr><td>Params (M)</td><td>0.40</td><td>0.48</td><td>0.48</td><td>0.48</td><td>0.48</td></tr><tr><td>FLOPS (G)</td><td>6.85</td><td>10.30</td><td>9.41</td><td>8.89</td><td>9.72</td></tr></table>
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(c) Ablation of different unfolding frameworks. (d) Ablation to study the effect of parameters $_ { \pmb { \alpha } }$ and $\beta$ . Table 2: Ablation studies on simulation datasets [64, 65]. PSNR, SSIM, Params, and FLOPS are reported.
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(a) Break-down ablation towards higher performance.
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<table><tr><td>Framework</td><td>DNU [34]</td><td>ADMM[37]</td><td>GAP [36]</td><td>DAUF</td></tr><tr><td>PSNR</td><td>34.62</td><td>35.52</td><td>35.58</td><td>37.21</td></tr><tr><td>SSIM</td><td>0.930</td><td>0.942</td><td>0.943</td><td>0.959</td></tr><tr><td>Params (M)</td><td>2.03</td><td>2.03</td><td>2.03</td><td>2.08</td></tr><tr><td>FLOPS (G)</td><td>26.23</td><td>26.23</td><td>26.23</td><td>27.17</td></tr></table>
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<table><tr><td>Baseline-3</td><td>α</td><td>β</td><td>PSNR</td><td>SSIM</td><td>Params (M)</td><td>FLOPS (G)</td></tr><tr><td>√</td><td></td><td></td><td>36.49</td><td>0.952</td><td>2.03</td><td>26.23</td></tr><tr><td>√</td><td>√</td><td></td><td>36.94</td><td>0.957</td><td>2.08</td><td>27.10</td></tr><tr><td>广</td><td></td><td>√</td><td>36.83</td><td>0.956</td><td>2.08</td><td>27.17</td></tr><tr><td></td><td>√</td><td>√</td><td>37.21</td><td>0.959</td><td>2.08</td><td>27.17</td></tr></table>
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To study the effect of the estimated parameters $_ { \pmb { \alpha } }$ and $\beta$ , we perform a break-down ablation of DAUF. We adopt DAUHST-3stg as baseline-3 but $_ { \pmb { \alpha } }$ is set as learnable parameters instead of being estimated by $\mathcal { E }$ in Eq. (12) and $\beta$ is not fed into $\mathcal { D }$ . The results are shown in Tab. 2d. Baseline-3 yields 36.49 dB. When $_ \alpha$ is set to be estimated by $\mathcal { E }$ , baseline-3 is improved by 0.45 dB. When $\beta$ is fed into $\mathcal { D }$ , baseline-3 gains by $0 . 3 4 \mathrm { d B }$ . When $_ { \pmb { \alpha } }$ and $\beta$ are exploited jointly in the iterative learning, baseline-3 achieves a significant improvement of 0.72 dB. These results verify that the estimated parameters $_ \alpha$ and $\beta$ are beneficial for the linear projection and denoising network of deep unfolding methods.
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To further analyze the roles of the estimated parameters, we visualize $\mathbf { z } _ { k }$ and $\mathbf { x } _ { k }$ of Eq. (12), and plot the curves of $_ \alpha$ and $\beta$ as they change with the iteration in Fig. 6. We observe: (i) $\mathbf { z } _ { 0 }$ and $\mathbf { x } _ { 1 }$ yield either blurry or noisy images. There is a significant gap between them. Since $\alpha _ { k } = \mu _ { k }$ in Eq. (5) penalizes the differences between $\mathbf { z }$ and x, $\alpha _ { 1 }$ is estimated to be a large value. From the linear projection of the second iteration $( \mathbf { z } _ { 1 } \mathbf { x } _ { 2 } ,$ ) on, the gap between $\mathbf { z }$ and $\mathbf { x }$ decreases substantially. Therefore, $\alpha _ { k }$ are estimated to be small values when $k \geq 2$ . This indicates that $_ { \pmb { \alpha } }$ can adaptively scale the linear projection $\mathcal { P }$ . (ii) The noise corruption is severe in the first iteration. Thus, $\beta _ { 1 } =$ $\mu _ { 1 } / \tau _ { 1 } = 1 / ( \sqrt { \tau _ { 1 } / \mu _ { 1 } } ) ^ { 2 }$ , which is inversely proportional to the noise level, is estimated to be a small value. With further iterations, the noise level decreases, and thus the estimated $\beta _ { k }$ increases. These results demonstrate that $\beta$ can provide the information about noise level for the denoising network $\mathcal { D }$ .
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# 4 Conclusion
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In this paper, we remedy two issues of previous deep unfolding methods, i.e., they do not estimate informative parameters from the CASSI system to direct the iterative learning and they are mainly CNN-based showing limitations in capturing long-range dependencies. To cope with these challenges, we firstly formulate a principled MAP-based unfolding framework DAUF that estimates parameters from the compressed measurement and physical mask. Then the parameters, which capture critical cues of CASSI degradation patterns and ill-posedness degree, are fed into each iteration to contextually scale the linear projection and provide noise level information for the denoising network. Secondly, we propose a novel Transformer HST that can jointly extract local contents and model non-local dependencies. By plugging HST into DAUF, we derive the first Transformer-based unfolding method DAUHST for HSI reconstruction. Comprehensive experiments show that our DAUHST outperforms SOTA methods by a large margin while requiring much cheaper memory and computational costs.
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# Limitation and Social Impact
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The main limitation of our work is that the performance improvement of our method comes with lowering down the inference speed and increasing the model complexity. Until now, spectral snapshot compressive imaging reconstruction techniques have no negative social impact yet. Our proposed DAUHST does not present any negative foreseeable societal consequence, either.
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# Acknowledgement
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This work is partially supported by the NSFC fund (61831014), the Shenzhen Science and Technology Project under Grant (JSGG20210802153150005, CJGJZD20200617102601004). Xin Yuan acknowledges the support of NSFC (62271414), Westlake Foundation (2021B1501-2) and the Research Center for Industries of the Future (RCIF) at Westlake University.
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# Checklist
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1. For all authors...
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(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]
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(b) Did you describe the limitations of your work? [Yes]
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(c) Did you discuss any potential negative societal impacts of your work? [Yes]
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(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
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2. If you are including theoretical results...
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(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes]
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3. If you ran experiments...
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(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]
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(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]
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(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]
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(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
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4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
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(a) If your work uses existing assets, did you cite the creators? [Yes]
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(b) Did you mention the license of the assets? [Yes]
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(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We provide our code and models in the supplementary material
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(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes]
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(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes]
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5. If you used crowdsourcing or conducted research with human subjects...
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(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
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(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
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(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
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|
| 1 |
+
[
|
| 2 |
+
{
|
| 3 |
+
"type": "text",
|
| 4 |
+
"text": "Degradation-Aware Unfolding Half-Shuffle Transformer for Spectral Compressive Imaging ",
|
| 5 |
+
"text_level": 1,
|
| 6 |
+
"bbox": [
|
| 7 |
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| 8 |
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| 9 |
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| 10 |
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| 11 |
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],
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| 12 |
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"page_idx": 0
|
| 13 |
+
},
|
| 14 |
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{
|
| 15 |
+
"type": "text",
|
| 16 |
+
"text": "Yuanhao Cai $^ { 1 , 2 , * }$ , Jing Lin $^ { 1 , 2 , * }$ , Haoqian Wang 1,2,†, Xin Yuan 3, Henghui Ding 4, Yulun Zhang 4, Radu Timofte 4,5, Luc Van Gool 4 ",
|
| 17 |
+
"bbox": [
|
| 18 |
+
267,
|
| 19 |
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|
| 20 |
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730,
|
| 21 |
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|
| 22 |
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],
|
| 23 |
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"page_idx": 0
|
| 24 |
+
},
|
| 25 |
+
{
|
| 26 |
+
"type": "text",
|
| 27 |
+
"text": "1 Shenzhen International Graduate School, Tsinghua University, 2 Shenzhen Institute of Future Media Technology, 3 Westlake University, 4 ETH Zürich, 5 University of Würzburg ",
|
| 28 |
+
"bbox": [
|
| 29 |
+
289,
|
| 30 |
+
256,
|
| 31 |
+
709,
|
| 32 |
+
297
|
| 33 |
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],
|
| 34 |
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"page_idx": 0
|
| 35 |
+
},
|
| 36 |
+
{
|
| 37 |
+
"type": "text",
|
| 38 |
+
"text": "Abstract ",
|
| 39 |
+
"text_level": 1,
|
| 40 |
+
"bbox": [
|
| 41 |
+
462,
|
| 42 |
+
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|
| 43 |
+
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|
| 44 |
+
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|
| 45 |
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],
|
| 46 |
+
"page_idx": 0
|
| 47 |
+
},
|
| 48 |
+
{
|
| 49 |
+
"type": "text",
|
| 50 |
+
"text": "In coded aperture snapshot spectral compressive imaging (CASSI) systems, hyperspectral image (HSI) reconstruction methods are employed to recover the spatialspectral signal from a compressed measurement. Among these algorithms, deep unfolding methods demonstrate promising performance but suffer from two issues. Firstly, they do not estimate the degradation patterns and ill-posedness degree from CASSI to guide the iterative learning. Secondly, they are mainly CNN-based, showing limitations in capturing long-range dependencies. In this paper, we propose a principled Degradation-Aware Unfolding Framework (DAUF) that estimates parameters from the compressed image and physical mask, and then uses these parameters to control each iteration. Moreover, we customize a novel Half-Shuffle Transformer (HST) that simultaneously captures local contents and non-local dependencies. By plugging HST into DAUF, we establish the first Transformer-based deep unfolding method, Degradation-Aware Unfolding Half-Shuffle Transformer (DAUHST), for HSI reconstruction. Experiments show that DAUHST surpasses state-of-the-art methods while requiring cheaper computational and memory costs. Code and models are publicly available at https://github.com/caiyuanhao1998/MST ",
|
| 51 |
+
"bbox": [
|
| 52 |
+
232,
|
| 53 |
+
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|
| 54 |
+
766,
|
| 55 |
+
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|
| 56 |
+
],
|
| 57 |
+
"page_idx": 0
|
| 58 |
+
},
|
| 59 |
+
{
|
| 60 |
+
"type": "text",
|
| 61 |
+
"text": "1 Introduction ",
|
| 62 |
+
"text_level": 1,
|
| 63 |
+
"bbox": [
|
| 64 |
+
174,
|
| 65 |
+
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|
| 66 |
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312,
|
| 67 |
+
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|
| 68 |
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],
|
| 69 |
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"page_idx": 0
|
| 70 |
+
},
|
| 71 |
+
{
|
| 72 |
+
"type": "text",
|
| 73 |
+
"text": "Hyperspectral images (HSIs) have more spectral bands than normal RGB images to store more detailed information. Thus, HSIs are widely applied in image recognition [1, 2, 3], object detection [4, 5, 6], tracking [7, 8, 9], medical image processing [10, 11, 12], remote sensing [13, 14, 15, 16], etc. To obtain HSIs, traditional imaging systems use spectrometers to scan the scenes along the spectral or spatial dimensions, usually requiring a long time. These imaging systems fail to capture dynamic objects. Recently, snapshot compressive imaging (SCI) systems [17, 18, 19] have been developed to capture HSIs at video rate. Among these SCI systems, coded aperture snapshot spectral imaging (CASSI) [17, 20, 21] stands out for its impressive performance. CASSI uses a coded aperture and a disperser to modulate the HSI signal at different wavelengths, and then mixes all modulated signal to generate a 2D compressed measurement. Subsequently, HSI restoration methods are employed to solve the CASSI inverse problem, i.e., restore the HSIs from the measurement. These methods are divided into four categories. ",
|
| 74 |
+
"bbox": [
|
| 75 |
+
174,
|
| 76 |
+
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|
| 77 |
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|
| 78 |
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|
| 79 |
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],
|
| 80 |
+
"page_idx": 0
|
| 81 |
+
},
|
| 82 |
+
{
|
| 83 |
+
"type": "image",
|
| 84 |
+
"img_path": "images/0e8d7777db4aa8e3bbce96989c4a0b0fcab024fda634a991ec43700bcdc750e1.jpg",
|
| 85 |
+
"image_caption": [
|
| 86 |
+
"Figure 1: PSNR-FLOPS comparisons of DAUHST and SOTA unfolding methods. "
|
| 87 |
+
],
|
| 88 |
+
"image_footnote": [],
|
| 89 |
+
"bbox": [
|
| 90 |
+
578,
|
| 91 |
+
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|
| 92 |
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|
| 93 |
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780
|
| 94 |
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],
|
| 95 |
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"page_idx": 0
|
| 96 |
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},
|
| 97 |
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{
|
| 98 |
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"type": "text",
|
| 99 |
+
"text": "",
|
| 100 |
+
"bbox": [
|
| 101 |
+
174,
|
| 102 |
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820,
|
| 103 |
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823,
|
| 104 |
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847
|
| 105 |
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],
|
| 106 |
+
"page_idx": 0
|
| 107 |
+
},
|
| 108 |
+
{
|
| 109 |
+
"type": "text",
|
| 110 |
+
"text": "properties and can be interpreted. Yet, these methods need manual parameter tweaking, which slows down reconstruction. Also, they suffer from limited representation capacity and generalization ability. ",
|
| 111 |
+
"bbox": [
|
| 112 |
+
173,
|
| 113 |
+
92,
|
| 114 |
+
825,
|
| 115 |
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119
|
| 116 |
+
],
|
| 117 |
+
"page_idx": 1
|
| 118 |
+
},
|
| 119 |
+
{
|
| 120 |
+
"type": "text",
|
| 121 |
+
"text": "(ii) Plug-and-play $( \\mathrm { P n P } )$ algorithms [29, 30, 31] plug pre-trained denoising networks into traditional model-based methods to solve the HSI reconstruction problem. Nonetheless, the pre-trained networks in $\\mathrm { P n P }$ methods are fixed without re-training, therefore limiting the performance. ",
|
| 122 |
+
"bbox": [
|
| 123 |
+
176,
|
| 124 |
+
126,
|
| 125 |
+
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|
| 126 |
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167
|
| 127 |
+
],
|
| 128 |
+
"page_idx": 1
|
| 129 |
+
},
|
| 130 |
+
{
|
| 131 |
+
"type": "text",
|
| 132 |
+
"text": "(iii) End-to-end (E2E) algorithms employ a powerful model, usually a convolutional neural network (CNN) [12, 20, 32, 33], to learn the E2E mapping function from a measurement to the desired HSIs. E2E methods enjoy the power of deep learning. However, they learn a brute-force mapping from the compressed measurement to the underlying spectral images, thereby ignoring the working principles of CASSI systems. They come without theoretically proven properties, interpretability, and flexibility because the imaging models widely differ from each other for various hardware systems. ",
|
| 133 |
+
"bbox": [
|
| 134 |
+
173,
|
| 135 |
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|
| 136 |
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|
| 137 |
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|
| 138 |
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],
|
| 139 |
+
"page_idx": 1
|
| 140 |
+
},
|
| 141 |
+
{
|
| 142 |
+
"type": "text",
|
| 143 |
+
"text": "(iv) Deep unfolding methods [34, 35, 36, 37, 38, 39] adopt a multi-stage network to map the measurement into the HSI cube. Each stage usually includes two phases, i.e., linear projection followed by passing the signal through a single-stage network that learns the underlying denoiser prior. In deep unfolding methods, the network architecture is intuitively interpretable by explicitly characterizing the image priors and the system imaging model. Besides, these methods also enjoy the power of deep learning and thus have great potential. Yet, this potential has not been fully explored. ",
|
| 144 |
+
"bbox": [
|
| 145 |
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174,
|
| 146 |
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263,
|
| 147 |
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|
| 148 |
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347
|
| 149 |
+
],
|
| 150 |
+
"page_idx": 1
|
| 151 |
+
},
|
| 152 |
+
{
|
| 153 |
+
"type": "text",
|
| 154 |
+
"text": "Existing deep unfolding algorithms suffer from two issues. (a) The iterative learning is highly related to the CASSI system. However, current unfolding methods do not estimate CASSI degradation patterns and ill-posedness degree to adjust the linear projection and denoising network in each iteration. (b) Existing deep unfolding methods are mainly CNN-based, therefore showing limitations in capturing non-local self-similarity and long-range dependencies, both critical for HSI reconstruction. ",
|
| 155 |
+
"bbox": [
|
| 156 |
+
174,
|
| 157 |
+
353,
|
| 158 |
+
826,
|
| 159 |
+
422
|
| 160 |
+
],
|
| 161 |
+
"page_idx": 1
|
| 162 |
+
},
|
| 163 |
+
{
|
| 164 |
+
"type": "text",
|
| 165 |
+
"text": "Recently, the emerging Transformer [40] has provided a solution to tackle the drawbacks of CNN. Due to its strong capability in modeling the interactions of non-local spatial regions, Transformer has been widely applied in image classification [41, 42, 43], object detection [44, 45, 46], semantic segmentation [47, 48, 49], human pose estimation [50, 51, 52], image restoration [53, 54, 55], etc. Yet, the use of Transformer is confronted with two main issues. (a) The computational complexity of global Transformer [42] is quadratic to the spatial dimensions. This cost is sometimes unaffordable. (b) The receptive fields of local Transformer [41] are limited within position-specific windows. Thus, some tokens with highly-related contents can not match each other when computing self-attention. ",
|
| 166 |
+
"bbox": [
|
| 167 |
+
174,
|
| 168 |
+
428,
|
| 169 |
+
826,
|
| 170 |
+
540
|
| 171 |
+
],
|
| 172 |
+
"page_idx": 1
|
| 173 |
+
},
|
| 174 |
+
{
|
| 175 |
+
"type": "text",
|
| 176 |
+
"text": "To address the above problems, in this paper, we firstly formulate a principled Degradation-Aware Unfolding Framework (DAUF) based on maximum a posteriori (MAP) theory for HSI reconstruction. Different from previous deep unfolding methods, our DAUF implicitly estimates informative parameters from the degraded compressed measurement and the physical mask used in the modulation. Then DAUF feeds the parameters, which capture key cues of CASSI degradation patterns and ill-posedness degree, into each iteration to adaptively scale the linear projection and provide the noise level information for the denoising network. Secondly, we design a novel Half-Shuffle Transformer (HST) as the denoiser prior in each iteration. Our HST can jointly extract local contextual information and model non-local dependencies, while requiring much cheaper computational costs than global Transformer. We achieve this by customizing a Half-Shuffle Multi-head Self-Attention (HS-MSA) mechanism that composes the basic unit of HST. More specifically, our HS-MSA has two branches, i.e., local branch and non-local branch. The local branch calculates the self-attention within the local window while the non-local branch shuffles the tokens and captures cross-window interactions. We plug HST into DAUF to establish an iterative architecture, Degradation-Aware Unfolding Half-Shuffle Transformer (DAUHST). With the proposed techniques, DAUHST models dramatically outperform state-of-the-art (SOTA) deep unfolding methods with the same number of stages by over 4 dB, as shown in Fig. 1. ",
|
| 177 |
+
"bbox": [
|
| 178 |
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|
| 179 |
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|
| 180 |
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|
| 181 |
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|
| 182 |
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],
|
| 183 |
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"page_idx": 1
|
| 184 |
+
},
|
| 185 |
+
{
|
| 186 |
+
"type": "text",
|
| 187 |
+
"text": "In a nutshell, our contributions can be summarized as follows: ",
|
| 188 |
+
"bbox": [
|
| 189 |
+
173,
|
| 190 |
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773,
|
| 191 |
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581,
|
| 192 |
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787
|
| 193 |
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],
|
| 194 |
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"page_idx": 1
|
| 195 |
+
},
|
| 196 |
+
{
|
| 197 |
+
"type": "text",
|
| 198 |
+
"text": "(i) We formulate a principled MAP-based unfolding framework DAUF for HSI reconstruction. ",
|
| 199 |
+
"bbox": [
|
| 200 |
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174,
|
| 201 |
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794,
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| 202 |
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792,
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| 203 |
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|
| 204 |
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],
|
| 205 |
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"page_idx": 1
|
| 206 |
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},
|
| 207 |
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{
|
| 208 |
+
"type": "text",
|
| 209 |
+
"text": "(ii) We propose a novel Transformer HST and plug it into DAUF to establish DAUHST. To the best of our knowledge, DAUHST is the first Transformer-based deep unfolding method for HSI restoration. ",
|
| 210 |
+
"bbox": [
|
| 211 |
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174,
|
| 212 |
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| 213 |
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| 214 |
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843
|
| 215 |
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],
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| 216 |
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"page_idx": 1
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| 217 |
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| 218 |
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{
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| 219 |
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"type": "text",
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| 220 |
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"text": "(iii) DAUHST outperforms SOTA methods by a large margin while requiring cheaper computational and memory costs. Besides, DAUHST yields more visually pleasant results in real HSI reconstruction. ",
|
| 221 |
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{
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"type": "image",
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"img_path": "images/010aaa963c82f61092af9261d001e1bef634947558e446cecf1638db47f737ca.jpg",
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"image_caption": [
|
| 233 |
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"Figure 2: The architecture of our DAUF with $K$ stages (iterations). $\\varepsilon$ estimates informative parameters from the compressed measurement $\\mathbf { y }$ and sensing matrix $\\Phi$ . The estimated parameters $_ { \\pmb { \\alpha } }$ and $\\beta$ are fed into each stage of subsequent iterative learning. $\\mathcal { P }$ and $\\mathcal { D }$ denote the linear projection and denoising network in each stage. "
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"type": "text",
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"text": "2 Proposed Method ",
|
| 247 |
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"text_level": 1,
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"type": "text",
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"text": "2.1 Degradation Model of CASSI ",
|
| 259 |
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"text_level": 1,
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{
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"type": "text",
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| 270 |
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"text": "In CASSI, we denote the vectorized measurement as $\\mathbf { y } \\in \\mathbb { R } ^ { n }$ , where $\\begin{array} { r } { n = H ( W + d ( N _ { \\lambda } - 1 ) ) . ~ H , W , } \\end{array}$ , $d$ , and $N _ { \\lambda }$ denote the HSI’s height, width, shifting step in dispersion, and total number of wavelengths. Given the vectorized shifted HSI signal $\\mathbf { x } \\in \\mathbb { R } ^ { n N _ { \\lambda } ^ { \\mathbf { * } } }$ and the sensing matrix $\\Phi \\in \\mathbb { R } ^ { n \\times n N _ { \\lambda } }$ that is determined by the physical mask, the degradation model of CASSI can be formulated as ",
|
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{
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| 280 |
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"type": "equation",
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| 281 |
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"img_path": "images/9c0a732ce79992bccc4de1d36881c34130eae830668cd98218a9ff09bb210a9d.jpg",
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"text": "$$\n\\mathbf { y } = \\Phi \\mathbf { x } + \\mathbf { n } ,\n$$",
|
| 283 |
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{
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"type": "text",
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"text": "where $\\mathbf { n } \\in \\mathbb { R } ^ { n }$ represents the vectorized imaging noise on the measurement. As analyzed in [56, 57, 58], $\\Phi$ is a fat, sparse, and highly structured matrix that is hard to handle. Please refer to the supplementary material for details about the mathematical model of CASSI. Then the task of HSI reconstruction is given y (captured by the camera) and $\\Phi$ (calibrated based on pre-design), solving $\\mathbf { x }$ . ",
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"type": "text",
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"text": "2.2 Degradation-Aware Unfolding Framework ",
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"text_level": 1,
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"type": "text",
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"text": "Previous unfolding frameworks [34, 35, 36, 37] do not estimate the CASSI degradation patterns to adjust the iterative learning. To alleviate this limitation, we formulate a principled Degradation-Aware Unfolding Framework (DAUF) as depicted in Fig. 2. DAUF starts from the MAP theory. In particular, the original HSI signal could be estimated by minimizing the following energy function as ",
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},
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{
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"type": "equation",
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"img_path": "images/db7ab95f8704204480454be7f3adeb1e86ac6e53aacf0737e1f7ab6011107593.jpg",
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"text": "$$\n\\hat { \\mathbf { x } } = \\arg \\operatorname* { m i n } _ { \\mathbf { x } } \\frac { 1 } { 2 } | | \\mathbf { y } - \\boldsymbol { \\Phi } \\mathbf { x } | | ^ { 2 } + \\tau R ( \\mathbf { x } ) ,\n$$",
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"text_format": "latex",
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"bbox": [
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{
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"type": "text",
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"text": "where $\\frac { 1 } { 2 } | | \\mathbf { y } - \\Phi \\mathbf { x } | | ^ { 2 }$ is the data fidelity term, $R ( \\mathbf { x } )$ is the image prior term, and $\\tau$ is a hyperparameter balancing the importance. By introducing an auxiliary variable $\\mathbf { z }$ , Eq. (2) can be reformulated as ",
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},
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{
|
| 351 |
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"type": "equation",
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| 352 |
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"img_path": "images/d083f9020113051552c909f1fbdeffbebd01e0beedf7d533344578ccb2a88341.jpg",
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"text": "$$\n\\hat { \\mathbf { x } } = \\arg \\operatorname* { m i n } _ { \\mathbf { x } } ~ \\frac { 1 } { 2 } | | \\mathbf { y } - \\boldsymbol { \\Phi } \\mathbf { x } | | ^ { 2 } + \\tau R ( \\mathbf { z } ) , \\quad s . t . ~ \\mathbf { z } = \\mathbf { x } .\n$$",
|
| 354 |
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"text_format": "latex",
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],
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{
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"type": "text",
|
| 365 |
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"text": "This is a constrained optimization problem. To obtain an unfolding inference, we adopt half-quadratic splitting (HQS) algorithm for its simplicity and fast convergence. Then Eq. (3) is solved by minimizing ",
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| 366 |
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},
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{
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"type": "equation",
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"img_path": "images/6d89bedd9c87cea3f8307f3b732e3090671646a610ed5d2cc4418190a084a4e9.jpg",
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"text": "$$\n\\mathcal { L } _ { \\mu } ( \\mathbf { x } , \\mathbf { z } ) = \\frac { 1 } { 2 } | | \\mathbf { y } - \\boldsymbol { \\Phi } \\mathbf { x } | | ^ { 2 } + \\tau R ( \\mathbf { z } ) + \\frac { \\mu } { 2 } | | \\mathbf { z } - \\mathbf { x } | | ^ { 2 } ,\n$$",
|
| 378 |
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"text_format": "latex",
|
| 379 |
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"bbox": [
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{
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| 388 |
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"type": "text",
|
| 389 |
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"text": "where $\\mu$ is a penalty parameter that forces $\\mathbf { x }$ and $\\mathbf { z }$ to approach the same fixed point. Subsequently, Eq. (4) can be solved by decoupling $\\mathbf { x }$ and $\\mathbf { z }$ into the following two iterative sub-problems as ",
|
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{
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"type": "equation",
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|
| 401 |
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"text": "$$\n\\mathbf { x } _ { k + 1 } = \\arg \\operatorname* { m i n } _ { \\mathbf { x } } ~ | | \\mathbf { y } - \\Phi \\mathbf { x } | | ^ { 2 } + \\mu | | \\mathbf { x } - \\mathbf { z } _ { k } | | ^ { 2 } , ~ \\mathbf { z } _ { k + 1 } = \\arg \\operatorname* { m i n } _ { \\mathbf { z } } \\frac { \\mu } { 2 } | | \\mathbf { z } - \\mathbf { x } _ { k + 1 } | | ^ { 2 } + \\tau R ( \\mathbf { z } ) ,\n$$",
|
| 402 |
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"text_format": "latex",
|
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],
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"page_idx": 2
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},
|
| 411 |
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{
|
| 412 |
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"type": "text",
|
| 413 |
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"text": "where $k = 0 , 1 , \\ \\dots , K - 1$ indexes the iteration. Note that the data fidelity term is associated with a quadratic regularized least-squares problem, i.e., $\\mathbf { x } _ { k + 1 }$ in Eq. (5). It has a closed-form solution as ",
|
| 414 |
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},
|
| 422 |
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{
|
| 423 |
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"type": "equation",
|
| 424 |
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"img_path": "images/2e86cad508e15c334ac8a33246ed8f8ecc135adc293f3b2778309eb8426e85a8.jpg",
|
| 425 |
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"text": "$$\n\\mathbf { x } _ { k + 1 } = ( \\Phi ^ { \\mathsf { T } } \\Phi + \\mu \\mathbf { I } ) ^ { - 1 } ( \\Phi ^ { \\mathsf { T } } \\mathbf { y } + \\mu \\mathbf { z } _ { k } ) ,\n$$",
|
| 426 |
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"text_format": "latex",
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],
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| 433 |
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"page_idx": 2
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| 434 |
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},
|
| 435 |
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{
|
| 436 |
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"type": "text",
|
| 437 |
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"text": "where $\\mathbf { I }$ is an identity matrix. Since $\\Phi$ is a fat matrix, $( \\Phi ^ { \\mathsf { T } } \\Phi + \\mu \\mathbf { I } )$ will be large and thus we simplify the computation of the inverse problem $( \\Phi ^ { \\mathsf { T } } \\Phi + \\mu \\mathbf { I } ) ^ { - 1 }$ by the matrix inversion formula as ",
|
| 438 |
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"bbox": [
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],
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|
| 445 |
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},
|
| 446 |
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{
|
| 447 |
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"type": "equation",
|
| 448 |
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"img_path": "images/4ab6b5ab94328196202ef67695ea33e5911b64c6ecea5d4b3cb8b4834a0466a8.jpg",
|
| 449 |
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"text": "$$\n( \\Phi ^ { \\mathsf { T } } \\Phi + \\mu \\mathbf { I } ) ^ { - 1 } = \\mu ^ { - 1 } \\mathbf { I } - \\mu ^ { - 1 } \\Phi ^ { \\mathsf { T } } ( \\mathbf { I } + \\Phi \\mu ^ { - 1 } \\Phi ^ { \\mathsf { T } } ) ^ { - 1 } \\Phi \\mu ^ { - 1 } .\n$$",
|
| 450 |
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"text_format": "latex",
|
| 451 |
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],
|
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"page_idx": 2
|
| 458 |
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},
|
| 459 |
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{
|
| 460 |
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"type": "text",
|
| 461 |
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"text": "By plugging Eq. (7) into Eq. (6), we can reformulate Eq. (6) as ",
|
| 462 |
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"bbox": [
|
| 463 |
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|
| 464 |
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|
| 465 |
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586,
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| 466 |
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| 467 |
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],
|
| 468 |
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"page_idx": 3
|
| 469 |
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},
|
| 470 |
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{
|
| 471 |
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"type": "equation",
|
| 472 |
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"img_path": "images/9016e78f3687ca890cc1b5ffa19162e9a3b2c4c4dcda5132f110314fe6787109.jpg",
|
| 473 |
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"text": "$$\n\\mathbf { x } _ { k + 1 } = { \\frac { \\Phi ^ { \\mathsf { T } } \\mathbf { y } + \\mu \\mathbf { z } _ { k } } { \\mu } } - { \\frac { \\Phi ^ { \\mathsf { T } } ( \\mathbf { I } + \\Phi \\mu ^ { - 1 } \\Phi ^ { \\mathsf { T } } ) ^ { - 1 } \\Phi \\Phi ^ { \\mathsf { T } } \\mathbf { y } } { \\mu ^ { 2 } } } - { \\frac { \\Phi ^ { \\mathsf { T } } ( \\mathbf { I } + \\Phi \\mu ^ { - 1 } \\Phi ^ { \\mathsf { T } } ) ^ { - 1 } \\Phi \\mathbf { z } _ { k } } { \\mu } } ~ .\n$$",
|
| 474 |
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"text_format": "latex",
|
| 475 |
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"bbox": [
|
| 476 |
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|
| 477 |
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| 478 |
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781,
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| 479 |
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147
|
| 480 |
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],
|
| 481 |
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"page_idx": 3
|
| 482 |
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},
|
| 483 |
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{
|
| 484 |
+
"type": "text",
|
| 485 |
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"text": "In CASSI systems, $\\Phi \\Phi ^ { \\mathsf { T } }$ is a diagonal matrix which can be defined as $\\Phi \\Phi ^ { \\mathsf { T } } \\stackrel { \\mathrm { d e f } } { = } \\mathrm { d i a g } \\{ \\psi _ { 1 } , \\hdots , \\psi _ { n } \\}$ . By plugging $\\Phi \\Phi ^ { \\mathsf { T } }$ into $( \\mathbf { I } + \\pmb { \\Phi } \\pmb { \\mu } ^ { - 1 } \\pmb { \\Phi } ^ { \\mathsf { T } } ) ^ { - 1 }$ and $( \\mathbf { I } + \\pmb { \\Phi } \\pmb { \\mu } ^ { - 1 } \\pmb { \\Phi } ^ { \\mathsf { T } } ) ^ { - 1 } \\pmb { \\Phi } \\pmb { \\Phi } ^ { \\mathsf { T } }$ , we obtain: ",
|
| 486 |
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"bbox": [
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| 487 |
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| 489 |
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| 491 |
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],
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|
| 493 |
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},
|
| 494 |
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{
|
| 495 |
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"type": "equation",
|
| 496 |
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"img_path": "images/8d2e5eb5c9b35ee3f161dde6a50bcc334efc81b6306767957d4dd23f2399e39f.jpg",
|
| 497 |
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"text": "$$\n\\begin{array} { r l r } & { } & { ( { \\bf I } + \\Phi \\mu ^ { - 1 } \\Phi ^ { \\mathsf { T } } ) ^ { - 1 } = \\mathsf { d i a g } \\Big \\{ \\displaystyle \\frac { \\mu } { \\mu + \\psi _ { 1 } } , \\ldots , \\displaystyle \\frac { \\mu } { \\mu + \\psi _ { n } } \\Big \\} , } \\\\ & { } & { ( { \\bf I } + \\Phi \\mu ^ { - 1 } \\Phi ^ { \\mathsf { T } } ) ^ { - 1 } \\Phi \\Phi ^ { \\mathsf { T } } = \\mathsf { d i a g } \\Big \\{ \\displaystyle \\frac { \\mu \\psi _ { 1 } } { \\mu + \\psi _ { 1 } } , \\ldots , \\displaystyle \\frac { \\mu \\psi _ { n } } { \\mu + \\psi _ { n } } \\Big \\} . } \\end{array}\n$$",
|
| 498 |
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"text_format": "latex",
|
| 499 |
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"bbox": [
|
| 500 |
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| 501 |
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| 502 |
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686,
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| 503 |
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|
| 504 |
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],
|
| 505 |
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"page_idx": 3
|
| 506 |
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},
|
| 507 |
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{
|
| 508 |
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"type": "text",
|
| 509 |
+
"text": "Let $\\mathbf { y } \\ { \\stackrel { \\mathrm { d e f } } { = } } \\ [ y _ { 1 } , \\dots , y _ { n } ] ^ { \\mathsf { T } }$ and $[ \\Phi \\mathbf { z } _ { k } ] _ { i }$ denotes the $i$ -th element of $\\Phi \\mathbf { z } _ { k }$ . We plug Eq. (9) into Eq. (8) as ",
|
| 510 |
+
"bbox": [
|
| 511 |
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| 512 |
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| 513 |
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| 515 |
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|
| 516 |
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|
| 517 |
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},
|
| 518 |
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{
|
| 519 |
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"type": "equation",
|
| 520 |
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"img_path": "images/9ddd8cb8701944767eae1b5b0a9f9b223fe4d214911f111e99759df10326653e.jpg",
|
| 521 |
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"text": "$$\n\\begin{array} { l } { { \\displaystyle { \\bf x } _ { k + 1 } = \\frac { { \\bf \\Phi } { \\bf \\Phi } ^ { \\mathsf { T } } { \\bf y } } { \\mu } + { \\bf z } _ { k } - \\frac { 1 } { \\mu } { \\bf \\Phi } ^ { \\mathsf { T } } \\Big [ \\frac { y _ { 1 } \\psi _ { 1 } + \\mu [ { \\bf \\Phi } { \\bf z } _ { k } ] _ { 1 } } { \\mu + \\psi _ { 1 } } , \\ldots , \\frac { y _ { n } \\psi _ { n } + \\mu [ { \\bf \\Phi } { \\bf z } _ { k } ] _ { n } } { \\mu + \\psi _ { n } } \\Big ] ^ { \\mathsf { T } } } } \\\\ { { \\displaystyle ~ = { \\bf z } _ { k } + { \\bf \\Phi } ^ { \\mathsf { T } } \\Big [ \\frac { y _ { 1 } - [ { \\bf \\Phi } { \\bf z } _ { k } ] _ { 1 } } { \\mu + \\psi _ { 1 } } , \\ldots , \\frac { y _ { n } - [ { \\bf \\Phi } { \\bf z } _ { k } ] _ { n } } { \\mu + \\psi _ { n } } \\Big ] ^ { \\mathsf { T } } } . } \\end{array}\n$$",
|
| 522 |
+
"text_format": "latex",
|
| 523 |
+
"bbox": [
|
| 524 |
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259,
|
| 525 |
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286,
|
| 526 |
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735,
|
| 527 |
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359
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| 528 |
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],
|
| 529 |
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"page_idx": 3
|
| 530 |
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},
|
| 531 |
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{
|
| 532 |
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"type": "text",
|
| 533 |
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"text": "Note that $\\{ y _ { i } - [ \\Phi \\mathbf { z } _ { k } ] _ { i } \\} _ { i = 1 } ^ { n }$ can be directly updated by $\\mathbf { y } - \\Phi \\mathbf { z } _ { k }$ , and $\\{ \\psi _ { i } \\} _ { i = 1 } ^ { n }$ is pre-calculated and stored in $\\Phi \\Phi ^ { \\mathsf { T } }$ . Thus, by element-wise computation in Eq. (10), $\\mathbf { x } _ { k + 1 }$ can be updated very efficiently. According to Eq. (5), the penalty parameter $\\mu$ should be large enough so that $\\mathbf { x }$ and $\\mathbf { z }$ can approach approximately the same fixed point. This indicates that $\\mu$ controls the convergence and output of each iteration. Thus, instead of manually tweaking $\\mu$ , we set $\\mu$ as a series of iteration-specific parameters to be automatically estimated from the CASSI system. We denote $\\mu$ in the $k$ -th iteration as $\\mu _ { k }$ . ",
|
| 534 |
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"bbox": [
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| 541 |
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| 542 |
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|
| 543 |
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"type": "text",
|
| 544 |
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"text": "Returning to Eq. (5), we also set $\\tau$ as iteration-specific parameters and ${ \\mathbf z } _ { k + 1 }$ can be reformulated as ",
|
| 545 |
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| 546 |
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| 554 |
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"type": "equation",
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| 555 |
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"img_path": "images/f040d42713645e0798aa14c8f69ae43158ca7dbd0cae819f33b3956a43c4e0d0.jpg",
|
| 556 |
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"text": "$$\n\\mathbf { z } _ { k + 1 } = \\arg \\operatorname* { m i n } _ { \\mathbf { z } } \\ { \\frac { 1 } { 2 ( { \\sqrt { \\tau _ { k + 1 } / \\mu _ { k + 1 } } } ) ^ { 2 } } } \\left| | \\mathbf { z } - \\mathbf { x } _ { k + 1 } | \\right| ^ { 2 } + R ( \\mathbf { z } ) .\n$$",
|
| 557 |
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"text_format": "latex",
|
| 558 |
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"bbox": [
|
| 559 |
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| 560 |
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| 561 |
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| 562 |
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],
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| 564 |
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"page_idx": 3
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| 565 |
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| 567 |
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"type": "text",
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| 568 |
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"text": "From the perspective of Bayesian probability, Eq. (11) is equivalent to denoising image $\\mathbf { x } _ { k + 1 }$ with a Gaussian noise at level pτk+1/µk+1 [29]. To conveniently solve Eq. (11), we set $\\begin{array} { r } { \\frac { 1 } { ( \\sqrt { \\tau _ { k + 1 } / \\mu _ { k + 1 } } ) ^ { 2 } } = } \\end{array}$ $\\mu _ { k + 1 } / \\tau _ { k + 1 }$ as parameters to be estimated from CASSI. Let $\\alpha _ { k } \\ { \\stackrel { \\mathrm { d e f } } { = } } \\ \\mu _ { k }$ , ${ \\pmb { \\alpha } } \\ { \\stackrel { \\mathrm { d e f } } { = } } \\ [ \\alpha _ { 1 } , . . . , \\alpha _ { K } ]$ , βk def = $\\mu _ { k } / \\tau _ { k }$ , and $\\beta \\stackrel { \\mathrm { d e f } } { = } [ \\beta _ { 1 } , . . . , \\beta _ { K } ]$ . Then we can formulate our DAUF as an iterative scheme: ",
|
| 569 |
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"bbox": [
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| 577 |
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|
| 578 |
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"type": "equation",
|
| 579 |
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"img_path": "images/49e37e08e3e925f4db3ad6dc6c29f3e3b954d365b87b03d0c0f9fc801872b4f2.jpg",
|
| 580 |
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"text": "$$\n\\begin{array} { r } { ( \\alpha , \\beta ) = \\mathcal { E } ( \\mathbf { y } , \\Phi ) , \\quad \\mathbf { x } _ { k + 1 } = \\mathcal { P } ( \\mathbf { y } , \\mathbf { z } _ { k } , \\alpha _ { k + 1 } , \\Phi ) , \\quad \\mathbf { z } _ { k + 1 } = \\mathcal { D } ( \\mathbf { x } _ { k + 1 } , \\beta _ { k + 1 } ) , } \\end{array}\n$$",
|
| 581 |
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"text_format": "latex",
|
| 582 |
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"bbox": [
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},
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{
|
| 591 |
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"type": "text",
|
| 592 |
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"text": "where $\\mathcal { E }$ denotes the parameter estimator that takes the compressed measurement $\\mathbf { y }$ and the sensing matrix $\\Phi$ of the CASSI system as inputs, $\\mathcal { P }$ equivalent to Eq. (10) denotes the linear projection, and $\\mathcal { D }$ represents the Gaussian denoiser solving Eq. (11). $\\mathbf { z } _ { 0 }$ is initialized by passing the shifted $\\mathbf { y }$ concatenated with $\\Phi$ through a $c o n v 1 \\times 1$ (convolution with $1 \\times 1$ kernel). Fig. 2 shows the architecture of $\\mathcal { E }$ . It consists of a $c o n v 1 \\times 1 .$ , a strided $c o n v 3 \\times 3$ , a global average pooling, and three fully connected layers. Through $\\mathcal { E }$ , DAUF captures critical cues from CASSI by learning the degradation patterns and ill-posedness degree caused by the mask-modulation and dispersionintegration. Parameters $_ { \\pmb { \\alpha } }$ and $\\beta$ estimated by $\\mathcal { E }$ direct the iterative learning by adaptively scaling the linear projection in Eq. (10) and providing noise level information for the denoiser prior in Eq. (11). ",
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| 593 |
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"type": "text",
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"text": "2.3 Half-Shuffle Transformer ",
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| 604 |
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"text_level": 1,
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"type": "text",
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"text": "When designing the denoiser prior, previous deep unfolding methods [34, 35, 36, 37] mainly adopt CNNs, showing limitations in capturing long-range dependencies. Directly applying local and global Transformers will encounter two problems, i.e., limited receptive fields and nontrivial computational costs. To address these challenges, we propose Half-Shuffle Transformer (HST) to play the role of $\\mathcal { D }$ . ",
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| 626 |
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"text": "Network Architecture. As shown in Fig. 3 (a), HST adopts a three-level U-shaped structure built by the basic unit Half-Shuffle Attention Block (HSAB). Firstly, HST uses a $c o n v 3 \\times 3$ to map reshaped $\\mathbf { x } _ { k }$ concatenated with stretched $\\beta _ { k }$ into feature $\\mathbf { X } _ { 0 } \\in \\mathbb { R } ^ { H \\times \\hat { W } \\times C }$ , where $\\hat { W } = W + d ( N _ { \\lambda } - 1 )$ Secondly, $\\mathbf { X } _ { 0 }$ passes through the encoder, bottleneck, and decoder to be embedded into deep feature $\\mathbf { X } _ { d } \\in \\mathbb { R } ^ { H \\times \\hat { W } \\times C }$ . Each level of the encoder or decoder contains an HSAB and a resizing module. ",
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| 627 |
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"type": "image",
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"img_path": "images/2c74375f4ce16b15ceb2b9b732e8cd8b2d12d682847831973af32f45e645128f.jpg",
|
| 638 |
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"image_caption": [
|
| 639 |
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"Figure 3: Diagram of HST. (a) HST adopts a U-shaped structure. (b) HSAB consists of an FFN, an HS-MSA, and two layer normalization. (c) Components of FFN. (d) HS-MSA contains local branch and non-local branch. "
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"image_footnote": [],
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| 651 |
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"type": "text",
|
| 652 |
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"text": "In Fig. 3 (b), HSAB consists of two layer normalization (LN), an HS-MSA, and a Feed-Forward Network (FFN) that is detailed in Fig. 3 (c). The downsampling and upsampling modules are strided $c o n v 4 \\times 4$ and deconv $2 \\times 2$ . Finally, a conv $3 \\times 3$ operates on $\\mathbf { X } _ { d }$ to generate a residual image $\\mathbf { R } \\in \\mathbb { R } ^ { H \\times \\hat { W } \\times N _ { \\lambda } }$ . The output denoised image $\\mathbf { z } _ { k }$ is obtained by the sum of $\\mathbf { x } _ { k }$ and reshaped $\\mathbf { R }$ . ",
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| 662 |
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"type": "text",
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| 663 |
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"text": "Half-Shuffle Multi-head Self-Attention. The most important element of HSAB is the proposed Half-Shuffle Multi-head Self-Attention (HS-MSA) module. Fig. 3 (d) depicts the HS-MSA used in the first level. The input tokens of HS-MSA are denoted as $\\mathbf { X } _ { i n } \\in \\mathbb { R } ^ { H \\times \\hat { W } \\times C }$ . Subsequently, ${ \\bf X } _ { i n }$ is linearly projected into query $\\mathbf { Q } \\in \\mathbb { R } ^ { H \\times \\hat { W } \\times C }$ , key $\\mathbf { K } \\in \\mathbb { R } ^ { H \\times \\hat { W } \\times C }$ , and value $\\mathbf { V } \\in \\mathbb { R } ^ { H \\times \\hat { W } \\times C }$ as ",
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"type": "equation",
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"img_path": "images/7a951e5070d4b8949357091b4ed1cbc94593fc06d834a83a8f4181add8524226.jpg",
|
| 675 |
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"text": "$$\n\\mathbf { Q } = \\mathbf { X } _ { i n } \\mathbf { W ^ { Q } } , \\mathbf { K } = \\mathbf { X } _ { i n } \\mathbf { W ^ { K } } , \\mathbf { V } = \\mathbf { X } _ { i n } \\mathbf { W ^ { V } } ,\n$$",
|
| 676 |
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|
| 686 |
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"type": "text",
|
| 687 |
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"text": "where $\\mathbf { W ^ { Q } } , \\mathbf { W ^ { K } } , \\mathbf { W ^ { V } } \\in \\mathbb { R } ^ { C \\times C }$ are learnable parameters and biases are omitted for simplification. Our HS-MSA combines the advantages of global MSA [42] and local window-based MSA [41], i.e., HS-MSA can jointly capture local contextual information through the local branch and model long-range dependencies through the non-local branch, all while being computationally cheaper than global MSA. Specifically, $\\mathbf { Q } , \\mathbf { K } , \\mathbf { V }$ are split into two equal parts along the channel dimension as ",
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|
| 699 |
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"text": "$$\n\\mathbf { Q } = [ \\mathbf { Q } _ { l } , \\mathbf { Q } _ { n l } ] , ~ \\mathbf { K } = [ \\mathbf { K } _ { l } , \\mathbf { K } _ { n l } ] , ~ \\mathbf { V } = [ \\mathbf { V } _ { l } , \\mathbf { V } _ { n l } ] ,\n$$",
|
| 700 |
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"bbox": [
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| 708 |
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|
| 709 |
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{
|
| 710 |
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"type": "text",
|
| 711 |
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"text": "where $\\mathbf { Q } _ { l } , \\mathbf { K } _ { l } , \\mathbf { V } _ { l } \\ \\in \\ \\mathbb { R } ^ { H \\times \\hat { W } \\times \\frac { C } { 2 } }$ are fed into the local branch to capture local contents, while ${ \\bf Q } _ { n l } , { \\bf K } _ { n l } , { \\bf V } _ { n l } \\in \\mathbb { R } ^ { H \\times { \\hat { W } } \\times \\frac { C } { 2 } }$ pass through the non-local branch to model non-local dependencies. ",
|
| 712 |
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{
|
| 721 |
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"type": "text",
|
| 722 |
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"text": "Local Branch. The local branch computes MSA within position-specific windows. As shown in the upper path of Fig. 3 (d), $\\mathbf { Q } _ { l } , \\mathbf { K } _ { l } , \\mathbf { V } _ { l }$ are partitioned into non-overlapping windows of size $M \\times M$ . Then they are reshaped into $\\begin{array} { r } { \\mathbb { R } ^ { \\frac { H \\hat { W } } { M ^ { 2 } } \\times M ^ { 2 } \\times \\frac { C } { 2 } } } \\end{array}$ . Subsequently, $\\mathbf { Q } _ { l } , \\mathbf { K } _ { l } , \\mathbf { V } _ { l }$ are split along the channel wise into $h$ heads: $\\mathbf { \\bar { Q } } _ { l } = [ \\mathbf { Q } _ { l } ^ { 1 } , \\dots , \\mathbf { Q } _ { l } ^ { h } \\ ]$ ], $\\mathbf { K } _ { l } = [ \\dot { \\mathbf { K } } _ { l } ^ { 1 } , \\dot { \\mathbf { \\Omega } } . \\dot { \\mathbf { \\Omega } } . \\mathbf { K } _ { l } ^ { h } ]$ , and $\\mathbf { V } _ { l } = \\left[ \\mathbf { V } _ { l } ^ { 1 } , \\ldots , \\mathbf { V } _ { l } ^ { h } \\right]$ . The dimension of each head is $\\begin{array} { r } { d _ { h } = \\frac { C } { 2 h } } \\end{array}$ l l l l. Note that Fig. 3 (d) depicts the situation with $h = 1$ l and some details are omitted for simplification. The local self-attention $\\mathbf { A } _ { l } ^ { i }$ is calculated inside each head as ",
|
| 723 |
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"bbox": [
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| 732 |
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"type": "equation",
|
| 733 |
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"img_path": "images/a7132c15393218cf97a52c969d54341d0a0f50868dab80121e82117d6d5cde6f.jpg",
|
| 734 |
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"text": "$$\n\\mathbf { A } _ { l } ^ { i } = \\operatorname { s o f t m a x } ( \\frac { \\mathbf { Q } _ { l } ^ { i } \\mathbf { \\mathbf { K } } _ { l } ^ { i ^ { \\top } } } { \\sqrt { d _ { h } } } + \\mathbf { P } _ { l } ^ { i } ) \\mathbf { V } _ { l } ^ { i } , i = 1 , \\ldots , h ,\n$$",
|
| 735 |
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"text_format": "latex",
|
| 736 |
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"bbox": [
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},
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| 744 |
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{
|
| 745 |
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"type": "text",
|
| 746 |
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"text": "where $\\mathbf { P } _ { l } ^ { i } \\in \\mathbb { R } ^ { M ^ { 2 } \\times M ^ { 2 } }$ are learnable parameters embedding the position information. ",
|
| 747 |
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"bbox": [
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},
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| 755 |
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{
|
| 756 |
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"type": "text",
|
| 757 |
+
"text": "Non-local Branch. The non-local branch computes cross-window interactions through shuffle operations inspired by ShuffleNet [59]. In particular, ${ \\bf Q } _ { n l } , { \\bf K } _ { n l } , { \\bf V } _ { n l } \\in \\mathbb { R } ^ { H \\times \\hat { W } \\times \\frac { C } { 2 } }$ are firstly partitioned into non-overlapping windows with size $M \\times M$ . Then their shapes are transposed from $\\begin{array} { r } { \\mathbb { R } ^ { \\frac { H \\hat { W } } { M ^ { 2 } } \\times M ^ { 2 } \\times \\frac { C } { 2 } } } \\end{array}$ to $\\begin{array} { r } { \\mathbb { R } ^ { M ^ { 2 } \\times \\frac { H \\hat { W } } { M ^ { 2 } } \\times \\frac { C } { 2 } } } \\end{array}$ to shuffle the positions of tokens and establish inter-window dependencies. $\\mathbf { Q } _ { n l } , \\mathbf { K } _ { n l } , \\mathbf { V } _ { n l }$ are split into $h$ heads: $\\mathbf { Q } _ { n l } = [ \\mathbf { Q } _ { n l } ^ { 1 } , \\ldots , \\mathbf { Q } _ { n l } ^ { h } ]$ , ${ \\bf K } _ { n l } = [ { \\bf K } _ { n l } ^ { 1 } , \\ldots , { \\bf K } _ { n l } ^ { \\hat { h } } ]$ , and ${ { \\bf { V } } _ { n l } } = [ { \\bf { V } } _ { n l } ^ { 1 } , . . . , { \\bf { V } } _ { n l } ^ { h } ]$ . Then the non-local self-attention $\\mathbf { A } _ { n l } ^ { i }$ is computed in each head as ",
|
| 758 |
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"bbox": [
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},
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| 766 |
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{
|
| 767 |
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"type": "table",
|
| 768 |
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"img_path": "images/a2969de7fa51efd8a9f1e460a2aa8012b04a5e4d86ac1014c627791aedc58cc9.jpg",
|
| 769 |
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"table_caption": [],
|
| 770 |
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"table_footnote": [
|
| 771 |
+
"Table 1: Comparisons between DAUHST and SOTA methods on 10 simulation scenes $( \\mathbf { S } 1 { \\sim } \\mathbf { S } 1 0 )$ . Params, FLOPS, PSNR (upper entry in each cell), and SSIM (lower entry in each cell) are reported. "
|
| 772 |
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],
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| 773 |
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"table_body": "<table><tr><td>Algorithms</td><td>Params</td><td>GFLOPS</td><td>S1</td><td>S2</td><td>S3</td><td>S4</td><td>S5</td><td>S6</td><td>S7</td><td>S8</td><td>S9</td><td>S10</td><td>Avg</td></tr><tr><td>TwIST [60]</td><td>-</td><td>-</td><td>25.16 0.700</td><td>23.02 0.604</td><td>21.40 0.711</td><td>30.19 0.851</td><td>21.41 0.635</td><td>20.95 0.644</td><td>22.20 0.643</td><td>21.82 0.650</td><td>22.42 0.690</td><td>22.67 0.569</td><td>23.12 0.669</td></tr><tr><td>GAP-TV [26]</td><td>-</td><td>-</td><td>26.82 0.754</td><td>22.89 0.610</td><td>26.31 0.802</td><td>30.65 0.852</td><td>23.64 0.703</td><td>21.85 0.663</td><td>23.76 0.688</td><td>21.98 0.655</td><td>22.63 0.682</td><td>23.10 0.584</td><td>24.36 0.669</td></tr><tr><td>DeSCI [23]</td><td></td><td></td><td>27.13 0.748</td><td>23.04 0.620</td><td>26.62 0.818</td><td>34.96 0.897</td><td>23.94 0.706</td><td>22.38 0.683</td><td>24.45 0.743</td><td>22.03 0.673</td><td>24.56 0.732</td><td>23.59 0.587</td><td>25.27 0.721</td></tr><tr><td>λ-Net [33]</td><td>62.64M</td><td>117.98</td><td>30.10 0.849</td><td>28.49 0.805 31.09</td><td>27.73 0.870</td><td>37.01 0.934</td><td>26.19 0.817</td><td>28.64 0.853</td><td>26.47 0.806</td><td>26.09 0.831</td><td>27.50 0.826</td><td>27.13 0.816</td><td>28.53 0.841</td></tr><tr><td>HSSP [35]</td><td>-</td><td>-</td><td>31.48 0.858 31.72</td><td>0.842 31.13</td><td>28.96 0.823 29.99</td><td>34.56 0.902</td><td>28.53 0.808</td><td>30.83 0.877</td><td>28.71 0.824</td><td>30.09 0.881</td><td>30.43 0.868</td><td>28.78 0.842</td><td>30.35 0.852</td></tr><tr><td>DNU [34]</td><td>1.19M</td><td>163.48</td><td>0.863 32.68</td><td>0.846 27.26</td><td>0.845 31.30</td><td>35.34 0.908</td><td>29.03 0.833</td><td>30.87 0.887</td><td>28.99 0.839</td><td>30.13 0.885</td><td>31.03 0.876</td><td>29.14 0.849</td><td>30.74 0.863</td></tr><tr><td>DIP-HSI [30]</td><td>33.85M</td><td>64.42</td><td>0.890 32.03</td><td>0.833 31.00</td><td>0.914 32.25</td><td>40.54 0.962 39.19</td><td>29.79 0.900</td><td>30.39 0.877</td><td>28.18 0.913 30.32</td><td>29.44 0.874</td><td>34.51 0.927 30.01</td><td>28.51 0.851</td><td>31.26 0.894</td></tr><tr><td>TSA-Net [20]</td><td>44.25M</td><td>110.06</td><td>0.892 33.26</td><td>0.858 32.09</td><td>0.915</td><td>0.953</td><td>29.39 0.884</td><td>31.44 0.908</td><td>0.878</td><td>29.35 0.888</td><td>0.890</td><td>29.59 0.874</td><td>31.46 0.894</td></tr><tr><td>DGSMP [38]</td><td>3.76M</td><td>646.65</td><td>0.915</td><td>0.898 33.26</td><td>33.06 0.925 34.28</td><td>40.54 0.964</td><td>28.86 0.882</td><td>33.08 0.937</td><td>30.74 0.886</td><td>31.55 0.923</td><td>31.66 0.911</td><td>31.44 0.925</td><td>32.63 0.917</td></tr><tr><td>GAP-Net [36]</td><td>4.27M</td><td>78.58</td><td>33.74 0.911 34.12</td><td>0.900 33.62</td><td>0.929 35.04</td><td>41.03 0.967</td><td>31.44 0.919</td><td>32.40 0.925</td><td>32.27 0.902</td><td>30.46 0.905</td><td>33.51 0.915</td><td>30.24 0.895</td><td>33.26 0.917</td></tr><tr><td>ADMM-Net [37]</td><td>4.27M</td><td>78.58</td><td>0.918 35.14</td><td>0.902 35.67</td><td>0.931 36.03</td><td>41.15 0.966 42.30</td><td>31.82 0.922</td><td>32.54 0.924 34.46</td><td>32.42 0.896 33.67</td><td>30.74 0.907</td><td>33.75 0.915 34.89</td><td>30.68 0.895</td><td>33.58 0.918</td></tr><tr><td>HDNet [32]</td><td>2.37M</td><td>154.76</td><td>0.935 35.40</td><td>0.940 35.87</td><td>0.943 36.51</td><td>0.969 42.27</td><td>32.69 0.946</td><td>0.952</td><td>0.926</td><td>32.48 0.941</td><td>0.942 35.39</td><td>32.38 0.937</td><td>34.97 0.943</td></tr><tr><td>MST-L [61]</td><td>2.03M</td><td>28.15</td><td>0.941 35.80</td><td>0.944 36.23</td><td>0.953 37.34</td><td>0.973 42.63</td><td>32.77 0.947</td><td>34.80 0.955</td><td>33.66 0.925 34.35</td><td>32.67 0.948 33.71</td><td>0.949 36.67</td><td>32.50 0.941</td><td>35.18 0.948</td></tr><tr><td>MST++ [62]</td><td>1.33M</td><td>19.42</td><td>0.943 35.96</td><td>0.947 36.84</td><td>0.957 38.16</td><td>0.973 42.44</td><td>33.38 0.952</td><td>35.38 0.957 35.72</td><td>0.934 34.86</td><td>0.953 34.34</td><td>0.953 36.51</td><td>33.38 0.945 33.09</td><td>35.99 0.951</td></tr><tr><td>CST-L [62]</td><td>3.00M</td><td>40.01</td><td>0.949 36.79</td><td>0.955 37.89</td><td>0.962 40.61</td><td>0.975 46.94</td><td>33.25 0.955</td><td>0.963 35.30</td><td>0.944 36.58</td><td>0.961 33.96</td><td>0.957 39.47</td><td>0.945 32.80</td><td>36.12 0.957 37.58</td></tr><tr><td>BIRNAT [63]</td><td>4.40M</td><td>2122.66</td><td>0.951 35.93</td><td>0.957 36.70</td><td>0.971 37.96</td><td>0.985</td><td>35.42 0.964</td><td>0.959</td><td>0.955 34.78</td><td>0.956 33.65</td><td>0.970 37.42</td><td>0.938</td><td>0.960</td></tr><tr><td> DAUHST-2stg</td><td>1.40M</td><td>18.44</td><td>0.943 36.59</td><td>0.946 37.93</td><td>0.959 39.32</td><td>44.38 0.978</td><td>34.13 0.954</td><td>35.43 0.957</td><td>0.940</td><td>0.950</td><td>0.955 38.54</td><td>33.07 0.941</td><td>36.34 0.952</td></tr><tr><td>DAUHST-3stg</td><td>2.08M</td><td>27.17</td><td>0.949 36.92</td><td>0.958 38.52</td><td>0.964 40.51</td><td>44.77 0.980</td><td>34.82 0.961</td><td>36.19 0.963</td><td>36.02 0.950</td><td>34.28 0.956 34.74</td><td>0.963 38.71</td><td>33.67 0.947</td><td>37.21 0.959</td></tr><tr><td>DAUHST-5stg</td><td>3.44M</td><td>44.61</td><td>0.955</td><td>0.962</td><td>0.967</td><td>45.09 0.980</td><td>35.33 0.964</td><td>36.56 0.965</td><td>36.82 0.958</td><td>0.959</td><td>0.963</td><td>34.27 0.952</td><td>37.75 0.962</td></tr><tr><td>DAUHST-9stg</td><td>6.15M</td><td>79.50</td><td>37.25 0.958</td><td>39.02 0.967</td><td>41.05 0.971</td><td>46.15 0.983</td><td>35.80 0.969</td><td>37.08 0.970</td><td>37.57 0.963</td><td>35.10 0.966</td><td>40.02 0.970</td><td>34.59 0.956</td><td>38.36 0.967</td></tr></table>",
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"text": "$$\n\\mathbf { A } _ { n l } ^ { i } = \\mathrm { s o f t m a x } ( \\frac { \\mathbf { Q } _ { n l } ^ { i } \\mathbf { K } _ { n l } ^ { i } \\mathsf { T } } { \\sqrt { d _ { h } } } + \\mathbf { P } _ { n l } ^ { i } ) \\mathbf { V } _ { n l } ^ { i } , i = 1 , \\ldots , h ,\n$$",
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"text": "$\\mathbf { A } _ { n l } ^ { i } \\in \\mathbb { R } ^ { M ^ { 2 } \\times \\frac { H \\hat { W } } { M ^ { 2 } } \\times d _ { h } }$ $\\mathbf { P } _ { n l } ^ { i } \\in \\mathbb { R } ^ { \\frac { H \\hat { W } } { M ^ { 2 } } \\times \\frac { H \\hat { W } } { M ^ { 2 } } }$ are learnable parameters representing thunshuffled by being transposed to shape $\\mathbb { R } ^ { \\frac { H \\hat { W } } { M ^ { 2 } } \\times M ^ { 2 } \\times d _ { h } }$ edding. Subsequently,. Then the outputs of local branch in Eq. (15) and non-local branch in Eq. (16) are aggregated by a linear projection as ",
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"text": "$$\n\\mathrm { H S - M S A } ( \\mathbf { X } _ { i n } ) = \\sum _ { i = 1 } ^ { h } \\mathbf { A } _ { l } ^ { i } \\mathbf { W } _ { l } ^ { i } + \\sum _ { i = 1 } ^ { h } \\mathbf { A } _ { n l } ^ { i } \\mathbf { W } _ { n l } ^ { i } ,\n$$",
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"text": "where $\\mathbf { W } _ { l } ^ { i } , \\mathbf { W } _ { n l } ^ { i } \\in \\mathbb { R } ^ { d _ { h } \\times C }$ refer to learnable parameters. We reshape the result of Eq. (17) to obtain the output $\\mathbf { X } _ { o u t } \\in \\mathbb { R } ^ { H \\times \\hat { W } \\times C }$ . Instead of globally sampling all tokens, HS-MSA builds inter-window correlations by shuffle operations. The self-attention is calculated in the local window but with tokens from non-local regions. Therefore, HS-MSA is much computationally cheaper than global MSA. ",
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"text": "3 Experiment ",
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"text": "3.1 Experiment Setup ",
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"text": "Similar to [20, 32, 36, 38, 61], 28 wavelengths are selected from $4 5 0 \\mathrm { n m }$ to $6 5 0 \\mathrm { n m }$ and derived by spectral interpolation manipulation for the HSI data. Simulation and real experiments are conducted. ",
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"text": "Simulation Dataset. We adopt two datasets, i.e., CAVE [64] and KAIST [65] for simulation experiments. The CAVE dataset consists of 32 HSIs with spatial size $5 1 2 \\times 5 1 2$ . The KAIST dataset contains 30 HSIs of spatial size $2 7 0 4 \\times 3 3 7 6$ . Following the settings of [20, 32, 36, 38, 61], the CAVE dataset is adopted as the training set while 10 scenes from the KAIST dataset are selected for testing. ",
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"text": "Real Dataset. Five real HSIs collected by the CASSI system developed in [20] are used for testing. ",
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"img_path": "images/032c04fcb32f373cc866844e0d42013cfb01152a4524a93edd34fefe5b42725b.jpg",
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"image_caption": [
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"Figure 4: Simulation HSI reconstruction comparisons of Scene 2 with 4 (out of 28) spectral channels. The top-middle shows the spectral curves corresponding to the two green boxes of the RGB image. The top-right depicts the enlarged patches corresponding to the yellow boxes in the bottom HSIs. Zoom in for a better view. "
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"text": "Implementation Details. We implement DAUHST by Pytorch. All DAUHST models are trained with Adam [66] optimizer ( $\\beta _ { 1 } = 0 . 9$ and $\\beta _ { 2 } = 0 . 9 9 9 )$ ) using Cosine Annealing scheme [67] for 300 epochs on an RTX 3090 GPU. The initial learning rate is $4 \\times 1 0 ^ { - 4 }$ . Patches with spatial sizes $2 5 6 \\times 2 5 6$ and $6 6 0 \\times 6 6 0$ are randomly cropped from the 3D HSI cubes with 28 channels as training samples for the simulation and real experiments. The shifting step $d$ in the dispersion is set to 2. The batch size is 5. We set the basic channel $C = N _ { \\lambda } = 2 8$ to store HSI information. The weights of $\\mathcal { D }$ in different stages are unshared. Data augmentation includes random rotation and flipping. The training objective is to minimize the Root Mean Square Error (RMSE) between reconstructed and ground-truth HSIs. ",
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"text": "3.2 Quantitative Comparisons with State-of-the-Art Methods ",
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"text": "Tab. 1 compares the results of DAUHST and 16 SOTA methods including three model-based methods (TwIST [60], GAP-TV [26], and DeSCI [23]), one $\\mathrm { P n P }$ method (DIP-HSI [30]), seven E2E methods $\\lambda$ -Net [33], TSA-Net [20], HDNet [32], MST [61], ${ \\mathrm { M S T } } { + + }$ [62], CST [68], and BIRNAT [63]), and five deep unfolding methods (HSSP [35], DNU [34], DGSMP [38], GAP-Net [36], and ADMMNet [37]) on 10 simulation scenes. All algorithms are tested with the same settings as [38, 61]. ",
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"text": "(i) Our best model DAUHST-9stg (9-stage DAUHST) yields very impressive results, i.e., 38.36 dB in PSNR and 0.967 in SSIM. DAUHST-9stg significantly outperforms two recent SOTA methods BIRNAT [63] and MST-L [61] by 0.78 and 3.18 dB, suggesting the effectiveness of our method. ",
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"text": "(ii) Our DAUHST models dramatically surpass SOTA methods while requiring cheaper computational and memory costs. For instance, when compared with the only one Transformer-based E2E method MST, our DAUHST-2stg outperforms MST-L by $1 . 1 6 \\mathrm { d B }$ but only costs $6 8 . 9 \\%$ $\\left( 1 . 4 0 / 2 . 0 3 \\right)$ Params and $6 5 . 5 \\%$ (18.44 / 28.15) FLOPS. When compared with CNN-based E2E methods, DAUHST-3stg surpasses HDNet, TSA-Net, and $\\lambda$ -Net by 2.24, 5.75, and 8.68 dB while only requiring $8 7 . 8 \\%$ , $4 . 7 \\%$ , $3 . 3 \\%$ Params and $1 7 . 6 \\%$ , $2 4 . 7 \\%$ , $2 3 . 0 \\%$ FLOPS. When compared with RNN-based E2E method BIRNAT, our DAUHST-5stg is 0.17 dB higher but only costs $2 . 1 \\%$ FLOPS and $7 8 . 2 \\%$ Params. Fig. 1 plots the PSNR-FLOPS comparisons of DAUHST and SOTA unfolding methods. DAUHST outperforms other competitors with the same number of stages by very large margins, over 4 dB. ",
|
| 961 |
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"bbox": [
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"page_idx": 6
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| 968 |
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},
|
| 969 |
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{
|
| 970 |
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"type": "text",
|
| 971 |
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"text": "3.3 Qualitative Comparisons with State-of-the-Art Methods ",
|
| 972 |
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"text_level": 1,
|
| 973 |
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"bbox": [
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849
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"page_idx": 6
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| 981 |
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{
|
| 982 |
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"type": "text",
|
| 983 |
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"text": "Simulation HSI Reconstruction. Fig. 4 depicts the simulation HSI reconstruction comparisons between our DAUHST and other SOTA methods on Scene 2 with 4 (out of 28) spectral channels. The top-right part shows the zoomed-in patches of the yellow boxes in the entire HSIs (bottom). As can be observed that our DAUHST-9stg is more favorable to reconstruct visually pleasant HSIs with more detailed contents, cleaner textures, and fewer artifacts while preserving the spatial smoothness of homogeneous regions. In contrast, previous methods either yield over-smooth results compromising fine-grained structures, or introduce undesired chromatic artifacts and blotchy textures that are absent in the ground truth (GT). The top-middle part illustrates the density-wavelength spectral curves corresponding to the green boxes identified as $a$ and $^ b$ in the RGB image (top-left). The spectral curves of DAUHST-9stg achieve the highest correlation and coincidence with the reference curves, showing the advantage of our proposed DAUHST in spectral-dimension consistency reconstruction. ",
|
| 984 |
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"bbox": [
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"page_idx": 6
|
| 991 |
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},
|
| 992 |
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{
|
| 993 |
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"type": "image",
|
| 994 |
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"img_path": "images/84e48303aca5150a64b1f22d168b984ead9519022c80f459a3abd3d8e5e67095.jpg",
|
| 995 |
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"image_caption": [
|
| 996 |
+
"Figure 5: Real HSI reconstruction results of DAUHST-3stg and 9 SOTA methods on Scene 1 with 4 (out of 28) spectra. Only our method can clearly reconstruct the picked flower at all wavelengths. Zoom in for a better view. "
|
| 997 |
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],
|
| 998 |
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"image_footnote": [],
|
| 999 |
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"bbox": [
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| 1000 |
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265
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"page_idx": 7
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{
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| 1008 |
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"type": "text",
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| 1009 |
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"text": "",
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| 1010 |
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"bbox": [
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"type": "text",
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| 1020 |
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"text": "Real HSI Reconstruction. We further evaluate the effectiveness of DAUHST in real HSI reconstruction. Following the same settings as [20, 38, 61] for a fair comparison, we re-train DAUHST-3stg with the real mask on the CAVE and KAIST datasets jointly. To simulate the real imaging situations, the training samples are also injected with 11-bit shot noise. Fig. 5 shows the visual comparisons between our DAUHST-3stg and nine SOTA methods. In the top three rows, only our DAUHST-3stg can reconstruct the flower patch corresponding to the yellow box at all wavelengths while other methods all fail to recover the entire patch. In the bottom row, DAUHST-3stg restores more HSI structural details and clearer contents with fewer artifacts. In contrast, other methods recover blurry images, generate incomplete responses, and are susceptible to the noise corruption. This evidence suggests that DAUHST is more robust to the noise distortion and more effective in real HSI reconstruction. ",
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| 1021 |
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"bbox": [
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| 1030 |
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"type": "text",
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| 1031 |
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"text": "3.4 Ablation Study ",
|
| 1032 |
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"text_level": 1,
|
| 1033 |
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"bbox": [
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{
|
| 1042 |
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"type": "text",
|
| 1043 |
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"text": "Break-down Ablation. We adopt baseline-1 that is derived by removing HS-MSA and DAUF from DAUHST-3stg to conduct the break-down ablation. Our goal is to study the effect of each component towards higher performance. Baseline-1 is cascaded end to end by three single-stage networks. As shown in Tab. 2a, baseline-1 achieves 33.05 dB. When we respectively apply DAUF and HS-MSA, the model achieves 2.32 and $2 . 4 4 \\ : \\mathrm { d B }$ improvements. When we exploit DAUF and HS-MSA jointly, the model gains by 4.16 dB. These results demonstrate the effectiveness of our DAUF and HS-MSA. ",
|
| 1044 |
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"bbox": [
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{
|
| 1053 |
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"type": "text",
|
| 1054 |
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"text": "Self-Attention Mechanism. To compare HS-MSA with other MSAs, we adopt baseline-2 that is obtained by removing HS-MSA from DAUHST-1stg to conduct the ablation in Tab. 2b. We remove different position embedding schemes to avoid their impacts and only compare MSAs. For fairness, we keep the Params of MSAs the same by fixing the number of channels and heads. Baseline-2 yields 32.79 dB. We apply global MSA (G-MSA) [42], Swin MSA (SW-MSA) [41], Spectral-wise MSA (S-MSA) [61], and HS-MSA. Note that we downsample the input feature maps of G-MSA to avoid memory bottlenecks. As shown in Tab. 2b, HS-MSA yields the most significant improvement of 1.26 dB, which is 0.42, 0.30, and $0 . 2 3 \\mathrm { d B }$ higher than G-MSA, SW-MSA, and S-MSA. This superiority is mainly derived from HS-MSA’s ability to jointly capture local contents and non-local dependencies. ",
|
| 1055 |
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"bbox": [
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794
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| 1061 |
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"page_idx": 7
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| 1062 |
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|
| 1063 |
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{
|
| 1064 |
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"type": "text",
|
| 1065 |
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"text": "Unfolding Framework. We compare our DAUF with previous unfolding frameworks including DNU [34], ADMM-Net [37], and GAP-Net [36]. For a fair comparison, we replace each single-stage network of DNU, ADMM-Net, and GAP-Net by our HST. 3-stage architecture is adopted to conduct ablations. The results are shown in Tab. 2c. Our DAUF significantly outperforms DNU, ADMM, and GAP by 2.59, 1.69, and 1.63 dB while adding only 0.05M Params and 0.94G FLOPS. This is mainly because DAUF uses the parameters estimated from the compressed measurement and physical mask in the CASSI system to direct the iterative learning. These parameters capture critical information of CASSI degradation patterns and ill-posedness degree, providing key cues for HSI reconstruction. ",
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| 1066 |
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"bbox": [
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"page_idx": 7
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},
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{
|
| 1075 |
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"type": "table",
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| 1076 |
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"img_path": "images/0c1037291ccb4a8ae631a3cf47a5f2bc2a8b15f074aba96a52ef3025e1e21d50.jpg",
|
| 1077 |
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"table_caption": [
|
| 1078 |
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"Figure 6: Visualization of $\\mathbf { z } _ { k }$ and $\\mathbf { x } _ { k }$ with 4 (out of 28) spectral channels on Scene 1 in different iterations. The bottom-left corner plots the curves of $_ { \\pmb { \\alpha } }$ and $\\beta$ changing with the iteration. Please zoom in for a better view. "
|
| 1079 |
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],
|
| 1080 |
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"table_footnote": [],
|
| 1081 |
+
"table_body": "<table><tr><td>Baseline-1</td><td>DAUF</td><td>HS-MSA</td><td>PSNR</td><td>SSIM</td><td>Params (M)</td><td>FLOPS (G)</td></tr><tr><td>√</td><td></td><td></td><td>33.05</td><td>0.912</td><td>1.06</td><td>17.62</td></tr><tr><td>√</td><td>√</td><td></td><td>35.37</td><td>0.938</td><td>1.11</td><td>18.55</td></tr><tr><td>√</td><td></td><td>√</td><td>35.49</td><td>0.941</td><td>2.03</td><td>26.23</td></tr><tr><td>√</td><td>√</td><td>√</td><td>37.21</td><td>0.959</td><td>2.08</td><td>27.17</td></tr></table>",
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| 1082 |
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"bbox": [
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| 1088 |
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"page_idx": 8
|
| 1089 |
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},
|
| 1090 |
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{
|
| 1091 |
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"type": "table",
|
| 1092 |
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"img_path": "images/2eb6cb2d9502b77f3602b9d61db031bb15f55c66793706670771f38d3d88fe03.jpg",
|
| 1093 |
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"table_caption": [
|
| 1094 |
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"(b) Ablation of various self-attention mechanisms. "
|
| 1095 |
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],
|
| 1096 |
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"table_footnote": [],
|
| 1097 |
+
"table_body": "<table><tr><td>Method</td><td>Baseline-2</td><td>G-MSA</td><td>SW-MSA</td><td>S-MSA</td><td>HS-MSA</td></tr><tr><td>PSNR</td><td>32.79</td><td>33.63</td><td>33.75</td><td>33.82</td><td>34.05</td></tr><tr><td>SSIM</td><td>0.904</td><td>0.920</td><td>0.924</td><td>0.926</td><td>0.930</td></tr><tr><td>Params (M)</td><td>0.40</td><td>0.48</td><td>0.48</td><td>0.48</td><td>0.48</td></tr><tr><td>FLOPS (G)</td><td>6.85</td><td>10.30</td><td>9.41</td><td>8.89</td><td>9.72</td></tr></table>",
|
| 1098 |
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"bbox": [
|
| 1099 |
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| 1100 |
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| 1101 |
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820,
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353
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| 1104 |
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"page_idx": 8
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| 1105 |
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},
|
| 1106 |
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{
|
| 1107 |
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"type": "image",
|
| 1108 |
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"img_path": "images/0b5abcf0cbe034f9cf6123f2672b68fcf17039bc71016172bb97f402b02d47ab.jpg",
|
| 1109 |
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"image_caption": [
|
| 1110 |
+
"(c) Ablation of different unfolding frameworks. (d) Ablation to study the effect of parameters $_ { \\pmb { \\alpha } }$ and $\\beta$ . Table 2: Ablation studies on simulation datasets [64, 65]. PSNR, SSIM, Params, and FLOPS are reported. "
|
| 1111 |
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],
|
| 1112 |
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"image_footnote": [],
|
| 1113 |
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"bbox": [
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| 1114 |
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173,
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| 1115 |
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92,
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| 1116 |
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818,
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| 1117 |
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260
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| 1118 |
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|
| 1119 |
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"page_idx": 8
|
| 1120 |
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},
|
| 1121 |
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{
|
| 1122 |
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"type": "table",
|
| 1123 |
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"img_path": "images/1cddc8d9a9078ea14eba14a957da63e0a4855c41d9aabd200863c8684c75ca90.jpg",
|
| 1124 |
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"table_caption": [
|
| 1125 |
+
"(a) Break-down ablation towards higher performance. "
|
| 1126 |
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],
|
| 1127 |
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"table_footnote": [],
|
| 1128 |
+
"table_body": "<table><tr><td>Framework</td><td>DNU [34]</td><td>ADMM[37]</td><td>GAP [36]</td><td>DAUF</td></tr><tr><td>PSNR</td><td>34.62</td><td>35.52</td><td>35.58</td><td>37.21</td></tr><tr><td>SSIM</td><td>0.930</td><td>0.942</td><td>0.943</td><td>0.959</td></tr><tr><td>Params (M)</td><td>2.03</td><td>2.03</td><td>2.03</td><td>2.08</td></tr><tr><td>FLOPS (G)</td><td>26.23</td><td>26.23</td><td>26.23</td><td>27.17</td></tr></table>",
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| 1129 |
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"bbox": [
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| 1132 |
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| 1133 |
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428
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| 1135 |
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| 1136 |
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| 1137 |
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{
|
| 1138 |
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"type": "table",
|
| 1139 |
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"img_path": "images/2fe83d23c5708bb3a89afdc0d5248124506b7152f809e457d3e3adc7de833dc4.jpg",
|
| 1140 |
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"table_caption": [],
|
| 1141 |
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"table_footnote": [],
|
| 1142 |
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"table_body": "<table><tr><td>Baseline-3</td><td>α</td><td>β</td><td>PSNR</td><td>SSIM</td><td>Params (M)</td><td>FLOPS (G)</td></tr><tr><td>√</td><td></td><td></td><td>36.49</td><td>0.952</td><td>2.03</td><td>26.23</td></tr><tr><td>√</td><td>√</td><td></td><td>36.94</td><td>0.957</td><td>2.08</td><td>27.10</td></tr><tr><td>广</td><td></td><td>√</td><td>36.83</td><td>0.956</td><td>2.08</td><td>27.17</td></tr><tr><td></td><td>√</td><td>√</td><td>37.21</td><td>0.959</td><td>2.08</td><td>27.17</td></tr></table>",
|
| 1143 |
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"bbox": [
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| 1145 |
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| 1147 |
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428
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|
| 1149 |
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"page_idx": 8
|
| 1150 |
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},
|
| 1151 |
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{
|
| 1152 |
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"type": "text",
|
| 1153 |
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"text": "To study the effect of the estimated parameters $_ { \\pmb { \\alpha } }$ and $\\beta$ , we perform a break-down ablation of DAUF. We adopt DAUHST-3stg as baseline-3 but $_ { \\pmb { \\alpha } }$ is set as learnable parameters instead of being estimated by $\\mathcal { E }$ in Eq. (12) and $\\beta$ is not fed into $\\mathcal { D }$ . The results are shown in Tab. 2d. Baseline-3 yields 36.49 dB. When $_ \\alpha$ is set to be estimated by $\\mathcal { E }$ , baseline-3 is improved by 0.45 dB. When $\\beta$ is fed into $\\mathcal { D }$ , baseline-3 gains by $0 . 3 4 \\mathrm { d B }$ . When $_ { \\pmb { \\alpha } }$ and $\\beta$ are exploited jointly in the iterative learning, baseline-3 achieves a significant improvement of 0.72 dB. These results verify that the estimated parameters $_ \\alpha$ and $\\beta$ are beneficial for the linear projection and denoising network of deep unfolding methods. ",
|
| 1154 |
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"bbox": [
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|
| 1160 |
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"page_idx": 8
|
| 1161 |
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},
|
| 1162 |
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{
|
| 1163 |
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"type": "text",
|
| 1164 |
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"text": "To further analyze the roles of the estimated parameters, we visualize $\\mathbf { z } _ { k }$ and $\\mathbf { x } _ { k }$ of Eq. (12), and plot the curves of $_ \\alpha$ and $\\beta$ as they change with the iteration in Fig. 6. We observe: (i) $\\mathbf { z } _ { 0 }$ and $\\mathbf { x } _ { 1 }$ yield either blurry or noisy images. There is a significant gap between them. Since $\\alpha _ { k } = \\mu _ { k }$ in Eq. (5) penalizes the differences between $\\mathbf { z }$ and x, $\\alpha _ { 1 }$ is estimated to be a large value. From the linear projection of the second iteration $( \\mathbf { z } _ { 1 } \\mathbf { x } _ { 2 } ,$ ) on, the gap between $\\mathbf { z }$ and $\\mathbf { x }$ decreases substantially. Therefore, $\\alpha _ { k }$ are estimated to be small values when $k \\geq 2$ . This indicates that $_ { \\pmb { \\alpha } }$ can adaptively scale the linear projection $\\mathcal { P }$ . (ii) The noise corruption is severe in the first iteration. Thus, $\\beta _ { 1 } =$ $\\mu _ { 1 } / \\tau _ { 1 } = 1 / ( \\sqrt { \\tau _ { 1 } / \\mu _ { 1 } } ) ^ { 2 }$ , which is inversely proportional to the noise level, is estimated to be a small value. With further iterations, the noise level decreases, and thus the estimated $\\beta _ { k }$ increases. These results demonstrate that $\\beta$ can provide the information about noise level for the denoising network $\\mathcal { D }$ . ",
|
| 1165 |
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"bbox": [
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| 1171 |
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"page_idx": 8
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| 1172 |
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},
|
| 1173 |
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{
|
| 1174 |
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"type": "text",
|
| 1175 |
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"text": "4 Conclusion ",
|
| 1176 |
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"text_level": 1,
|
| 1177 |
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"bbox": [
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750
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"page_idx": 8
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| 1185 |
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{
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| 1186 |
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"type": "text",
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| 1187 |
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"text": "In this paper, we remedy two issues of previous deep unfolding methods, i.e., they do not estimate informative parameters from the CASSI system to direct the iterative learning and they are mainly CNN-based showing limitations in capturing long-range dependencies. To cope with these challenges, we firstly formulate a principled MAP-based unfolding framework DAUF that estimates parameters from the compressed measurement and physical mask. Then the parameters, which capture critical cues of CASSI degradation patterns and ill-posedness degree, are fed into each iteration to contextually scale the linear projection and provide noise level information for the denoising network. Secondly, we propose a novel Transformer HST that can jointly extract local contents and model non-local dependencies. By plugging HST into DAUF, we derive the first Transformer-based unfolding method DAUHST for HSI reconstruction. Comprehensive experiments show that our DAUHST outperforms SOTA methods by a large margin while requiring much cheaper memory and computational costs. ",
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| 1188 |
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},
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| 1196 |
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{
|
| 1197 |
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"type": "text",
|
| 1198 |
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"text": "Limitation and Social Impact ",
|
| 1199 |
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"text_level": 1,
|
| 1200 |
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"page_idx": 9
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},
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{
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| 1209 |
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"type": "text",
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| 1210 |
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"text": "The main limitation of our work is that the performance improvement of our method comes with lowering down the inference speed and increasing the model complexity. Until now, spectral snapshot compressive imaging reconstruction techniques have no negative social impact yet. Our proposed DAUHST does not present any negative foreseeable societal consequence, either. ",
|
| 1211 |
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"bbox": [
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"text": "This work is partially supported by the NSFC fund (61831014), the Shenzhen Science and Technology Project under Grant (JSGG20210802153150005, CJGJZD20200617102601004). Xin Yuan acknowledges the support of NSFC (62271414), Westlake Foundation (2021B1501-2) and the Research Center for Industries of the Future (RCIF) at Westlake University. ",
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"text": "[1] M. Fauvel, Y. Tarabalka, J. A. Benediktsson, J. Chanussot, and J. C. Tilton, “Advances in spectral-spatial classification of hyperspectral images,” Proceedings of the IEEE, 2012. [2] E. Maggiori, G. Charpiat, Y. Tarabalka, and P. Alliez, “Recurrent neural networks to correct satellite image classification maps,” Transactions on Geoscience and Remote Sensing, 2017. [3] F. Zhang, B. Du, and L. Zhang, “Scene classification via a gradient boosting random convolutional network framework,” Transactions on Geoscience and Remote Sensing, 2015. [4] M. H. Kim, T. A. Harvey, D. S. Kittle, H. Rushmeier, R. O. P. J. Dorsey, and D. J. Brady, “3d imaging spectroscopy for measuring hyperspectral patterns on solid objects,” ACM Transactions on on Graphics, 2012. [5] Z. Pan, G. Healey, M. Prasad, and B. Tromberg, “Face recognition in hyperspectral images,” TPAMI, 2003. [6] H. V. Nguyen, A. Banerjee, and R. 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Schmid, “Segmenter: Transformer for semantic segmentation,” in ICCV, 2021. \n[50] Y. Li, S. Zhang, Z. Wang, S. Yang, W. Yang, S.-T. Xia, and E. Zhou, “Tokenpose: Learning keypoint tokens for human pose estimation,” in ICCV, 2021. \n[51] S. Yang, Z. Quan, M. Nie, and W. Yang, “Transpose: Keypoint localization via transformer,” in ICCV, 2021. \n[52] K. Li, S. Wang, X. Zhang, Y. Xu, W. Xu, and Z. Tu, “Pose recognition with cascade transformers,” in CVPR, 2021. \n[53] H. Chen, Y. Wang, T. Guo, C. Xu, Y. Deng, Z. Liu, S. Ma, C. Xu, C. Xu, and W. Gao, “Pre-trained image processing transformer,” in CVPR, 2021. \n[54] J. Liang, J. Cao, G. Sun, K. Zhang, L. Van Gool, and R. Timofte, “Swinir: Image restoration using swin transformer,” in ICCVW, 2021. \n[55] Z. Wang, X. Cun, J. Bao, W. Zhou, J. Liu, and H. Li, “Uformer: A general u-shaped transformer for image restoration,” in CVPR, 2022. \n[56] J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Transactions on Information Theory, 2007. \n[57] D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory, 2006. \n[58] S. Jalali and X. Yuan, “Snapshot compressed sensing: Performance bounds and algorithms,” Transactions on Information Theory, 2019. \n[59] X. Zhang, X. Zhou, M. Lin, and J. Sun, “Shufflenet: An extremely efficient convolutional neural network for mobile devices,” in CVPR, 2018. \n[60] J. Bioucas-Dias and M. Figueiredo., “A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration.,” TIP, 2007. \n[61] Y. Cai, J. Lin, X. Hu, H. Wang, X. Yuan, Y. Zhang, R. Timofte, and L. V. Gool, “Mask-guided spectral-wise transformer for efficient hyperspectral image reconstruction,” in CVPR, 2022. \n[62] Y. Cai, J. Lin, Z. Lin, H. Wang, Y. Zhang, H. Pfister, R. Timofte, and L. V. Gool, “Mst $^ { + + }$ : Multi-stage spectral-wise transformer for efficient spectral reconstruction,” in CVPRW, 2022. \n[63] Z. Cheng, B. Chen, R. Lu, Z. Wang, H. Zhang, Z. Meng, and X. Yuan, “Recurrent neural networks for snapshot compressive imaging,” TPAMI, 2022. \n[64] J.-I. Park, M.-H. Lee, M. D. Grossberg, and S. K. Nayar, “Multispectral imaging using multiplexed illumination,” in ICCV, 2007. \n[65] I. Choi, M. Kim, D. Gutierrez, D. Jeon, and G. Nam, “High-quality hyperspectral reconstruction using a spectral prior,” in Technical report, 2017. \n[66] D. P. Kingma and J. L. Ba, “Adam: A method for stochastic optimization,” in ICLR, 2015. \n[67] I. Loshchilov and F. Hutter, “Sgdr: Stochastic gradient descent with warm restarts,” in ICLR, 2017. \n[68] Y. Cai, J. Lin, X. Hu, H. Wang, X. Yuan, Y. Zhang, R. Timofte, and L. V. Gool, “Coarse-to-fine sparse transformer for hyperspectral image reconstruction,” in ECCV, 2022. ",
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